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  "schema": "course-result-source-use/v1",
  "course": "foundations-of-von-neumann-algebras",
  "utc": "2026-10-05T03:05:20.398082+00:00",
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    {
      "id": "OA-FND-HB-01",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "1. Zorn's lemma",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "anchor": "oa-fnd-hb-01",
      "proof_locus": {
        "line": 30,
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      "full_conditions_and_proof": "## 1. Zorn's lemma\n\nA *chain* in a partially ordered set is a totally ordered subset. An *upper bound* of a subset \\(S\\) is an element \\(u\\) with \\(s\\le u\\) for all \\(s\\in S\\). An element \\(m\\) is *maximal* if \\(m\\le x\\) implies \\(x=m\\).\n\n**Theorem 1.1** (Zorn's lemma). Let \\((P,\\le)\\) be a nonempty partially ordered set in which every chain has an upper bound. Then \\(P\\) has a maximal element.\n\n**Proof.** Suppose not. Then every chain \\(C\\) has an upper bound that is not in \\(C\\): if \\(u\\) is an upper bound and \\(u\\in C\\), then \\(u\\) is the largest element of \\(C\\). Since \\(u\\) is not maximal, some \\(u'>u\\) exists, and \\(u'\\) is an upper bound outside \\(C\\). By the axiom of choice there is a function \\(g\\) assigning to each chain \\(C\\) an upper bound \\(g(C)\\notin C\\).\n\nFix \\(p_0\\in P\\). Call a subset \\(T\\subseteq P\\) a *tower* if it satisfies three conditions:\n1. \\(p_0\\in T\\);\n2. \\(T\\) is well-ordered by \\(\\le\\): every nonempty subset of \\(T\\) has a least element, so in particular \\(T\\) is a chain;\n3. for every \\(t\\in T\\), \\(t=g(\\{s\\in T:s<t\\})\\) when \\(t\\neq p_0\\), and every element of \\(T\\) is \\(\\ge p_0\\).\n\n*Claim: of two towers, one is an initial segment of the other.* Let \\(T,T'\\) be towers. Let \\(I\\) be the set of \\(t\\in T\\cap T'\\) such that \\(\\{s\\in T:s<t\\}=\\{s\\in T':s<t\\}\\) and this set lies in \\(T\\cap T'\\). Then \\(I\\) is an initial segment of both towers. If \\(I\\neq T\\) and \\(I\\neq T'\\), let \\(t\\) be least in \\(T\\setminus I\\) and \\(t'\\) least in \\(T'\\setminus I\\). Then \\(\\{s\\in T:s<t\\}=I=\\{s\\in T':s<t'\\}\\). If \\(I=\\varnothing\\), then \\(t=t'=p_0\\), since \\(p_0\\) is the least element of every tower. Otherwise condition 3 gives \\(t=g(I)=t'\\). In both cases \\(t=t'\\in I\\), a contradiction. So \\(I=T\\) or \\(I=T'\\). This proves the claim.\n\nLet \\(U\\) be the union of all towers. By the claim, \\(U\\) is well-ordered and every tower is an initial segment of \\(U\\), so \\(U\\) is a tower. Then \\(U\\cup\\{g(U)\\}\\) is also a tower: \\(g(U)\\) is an upper bound of \\(U\\) not in \\(U\\), so it is the largest element, and condition 3 holds for it. Hence \\(g(U)\\in U\\), contradicting \\(g(U)\\notin U\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": []
    },
    {
      "id": "OA-FND-HB-02",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "2. The Hahn–Banach theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "anchor": "oa-fnd-hb-02",
      "proof_locus": {
        "line": 47,
        "through_line": 118
      },
      "full_conditions_and_proof": "## 2. The Hahn–Banach theorem\n\nA map \\(p:V\\to\\mathbb R\\) on a real vector space is *sublinear* if \\(p(x+y)\\le p(x)+p(y)\\) and \\(p(tx)=tp(x)\\) for all \\(x,y\\in V\\) and \\(t\\ge0\\). A *seminorm* on a vector space over \\(\\mathbb K\\) is a sublinear map with \\(p(\\lambda x)=|\\lambda|p(x)\\) for all scalars \\(\\lambda\\).\n\n**Theorem 2.1** (Hahn–Banach, real form). Let \\(V\\) be a real vector space, \\(p\\) a sublinear functional on \\(V\\), \\(M\\) a subspace, and \\(f:M\\to\\mathbb R\\) linear with \\(f\\le p\\) on \\(M\\). Then \\(f\\) extends to a linear \\(F:V\\to\\mathbb R\\) with \\(F\\le p\\) on \\(V\\).\n\n**Proof.** *One step.* Let \\(x_0\\notin M\\) and \\(M_1=M+\\mathbb Rx_0\\). A linear extension \\(f_1\\) to \\(M_1\\) is determined by \\(c=f_1(x_0)\\). The condition \\(f_1\\le p\\) on \\(M_1\\) says \\(f(m)+tc\\le p(m+tx_0)\\) for all \\(m\\in M\\) and \\(t\\in\\mathbb R\\). Dividing by \\(|t|\\) and using positive homogeneity, this is equivalent to\n\\[\n\\begin{gathered}\nf(m')-p(m'-x_0)\\ \\\\\n\\le\\ c\\ \\\\\n\\le\\ p(m+x_0)-f(m)\\\\\n\\text{for all }m,m'\\in M.\n\\end{gathered}\n\\tag{2.1}\n\\]\nSuch a \\(c\\) exists because, for all \\(m,m'\\in M\\),\n\\[\n\\begin{gathered}\nf(m')+f(m)\\\\\n=f(m'+m)\\\\\n\\le p(m'+m)\\\\\n\\le p(m'-x_0)+p(m+x_0).\n\\end{gathered}\n\\]\n*Zorn.* Order the pairs \\((D,g)\\), with \\(M\\subseteq D\\) a subspace and \\(g\\) a linear extension of \\(f\\) to \\(D\\) with \\(g\\le p\\), by extension. A chain has an upper bound: the union of the domains, with the common extension. By Zorn's lemma (Theorem 1.1) there is a maximal pair \\((D,g)\\). If \\(D\\neq V\\), the one-step extension contradicts maximality. So \\(D=V\\). \\(\\square\\)\n\n**Theorem 2.2** (Hahn–Banach, seminorm form). Let \\(V\\) be a vector space over \\(\\mathbb K\\), \\(p\\) a seminorm on \\(V\\), \\(M\\) a subspace, and \\(f:M\\to\\mathbb K\\) linear with \\(|f|\\le p\\) on \\(M\\). Then \\(f\\) extends to a linear \\(F:V\\to\\mathbb K\\) with \\(|F|\\le p\\) on \\(V\\).\n\n**Proof.** *Real scalars.* Theorem 2.1 gives \\(F\\le p\\). Then \\(-F(x)=F(-x)\\le p(-x)=p(x)\\), so \\(|F|\\le p\\).\n\n*Complex scalars.* Let \\(u=\\operatorname{Re}f\\), a real-linear functional on \\(M\\) with \\(u\\le|f|\\le p\\). By Theorem 2.1, applied to \\(V\\) as a real space, \\(u\\) extends to a real-linear \\(U\\le p\\). Put \\(F(x)=U(x)-iU(ix)\\).\n- \\(F\\) is real-linear and \\[\n\\begin{gathered}\nF(ix)\\\\\n=U(ix)-iU(-x)\\\\\n=U(ix)+iU(x)\\\\\n=iF(x),\n\\end{gathered}\n\\] so \\(F\\) is complex-linear.\n- \\(\\operatorname{Re}F=U\\). On \\(M\\), \\(f\\) and \\(F\\) have the same real part, hence are equal, since a complex-linear functional is determined by its real part: \\(f(x)=\\operatorname{Re}f(x)-i\\operatorname{Re}f(ix)\\).\n- Given \\(x\\), choose \\(\\theta\\) with \\(e^{-i\\theta}F(x)=|F(x)|\\). Then \\[\n\\begin{gathered}\n|F(x)|\\\\\n=F(e^{-i\\theta}x)\\\\\n=U(e^{-i\\theta}x)\\\\\n\\le p(e^{-i\\theta}x)\\\\\n=p(x).\n\\end{gathered}\n\\] \\(\\square\\)\n\n**Corollary 2.3** (normed spaces). Let \\(E\\) be a normed space.\n1. Every \\(\\varphi\\in M^*\\), for a subspace \\(M\\subseteq E\\), extends to some \\(\\Phi\\in E^*\\) with \\(\\|\\Phi\\|=\\|\\varphi\\|\\).\n2. For every \\(x\\in E\\) there is \\(\\varphi\\in E^*\\) with \\(\\|\\varphi\\|\\le1\\) and \\(\\varphi(x)=\\|x\\|\\); if \\(x\\ne0\\), then \\(\\|\\varphi\\|=1\\). Hence \\(\\|x\\|=\\max_{\\|\\varphi\\|\\le1}|\\varphi(x)|\\), and \\(E^*\\) separates the points of \\(E\\).\n3. Let \\(M\\) be a closed subspace and \\(x\\notin M\\), with \\(d=\\operatorname{dist}(x,M)>0\\). Then some \\(\\varphi\\in E^*\\) has \\(\\varphi|_M=0\\), \\(\\varphi(x)=d\\) and \\(\\|\\varphi\\|=1\\). Consequently a subspace is dense exactly when the only bounded functional vanishing on it is \\(0\\).\n4. The canonical map \\(j:E\\to E^{**}\\), \\(j(x)(\\varphi)=\\varphi(x)\\), is a linear isometry.\n\n**Proof.** 1. Apply Theorem 2.2 with \\(p(y)=\\|\\varphi\\|\\,\\|y\\|\\). The extension satisfies \\(|\\Phi(y)|\\le\\|\\varphi\\|\\|y\\|\\), and \\(\\|\\Phi\\|\\ge\\|\\varphi\\|\\) because \\(\\Phi\\) extends \\(\\varphi\\).\n\n2. For \\(x=0\\) take \\(\\varphi=0\\). Otherwise apply 1 to \\(t x\\mapsto t\\|x\\|\\) on \\(\\mathbb Kx\\), a functional of norm \\(1\\). The maximum in the norm formula is attained at this \\(\\varphi\\), and \\(|\\varphi(x)|\\le\\|\\varphi\\|\\|x\\|\\) gives the other inequality.\n\n3. On \\(M+\\mathbb Kx\\) put \\(\\psi(m+tx)=td\\). This is well defined because \\(x\\notin M\\). It satisfies \\[\n\\begin{gathered}\n|\\psi(m+tx)|\\\\\n=|t|d\\\\\n\\le|t|\\,\\|x+m/t\\|\\\\\n=\\|m+tx\\|\n\\end{gathered}\n\\] for \\(t\\ne0\\), so \\(\\|\\psi\\|\\le1\\). Choosing \\(m_k\\in M\\) with \\(\\|x-m_k\\|\\to d\\) gives \\(\\psi(x-m_k)=d\\), so \\(\\|\\psi\\|=1\\). Extend by 1. For the consequence: if \\(M\\) is a subspace that is not dense, apply this to its closure \\(\\overline M\\) and some \\(x\\notin\\overline M\\).\n\n4. \\(j\\) is linear, and \\(\\|j(x)\\|=\\sup_{\\|\\varphi\\|\\le1}|\\varphi(x)|=\\|x\\|\\) by 2. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": []
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    {
      "id": "OA-FND-HB-03",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "3. The Baire category theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
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      "anchor": "oa-fnd-hb-03",
      "proof_locus": {
        "line": 119,
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      },
      "full_conditions_and_proof": "## 3. The Baire category theorem\n\n**Theorem 3.1** (Baire). Let \\((X,d)\\) be a nonempty complete metric space, and \\((U_n)_{n\\ge1}\\) a sequence of dense open subsets. Then \\(\\bigcap_nU_n\\) is dense. Equivalently, \\(X\\) is not a countable union of closed sets with empty interior.\n\n**Proof.** Let \\(W\\) be a nonempty open set. Since \\(U_1\\) is dense and open, \\(W\\cap U_1\\) is nonempty and open. Choose \\(x_1\\) and \\(0<r_1<1\\) with \\(\\overline B(x_1,r_1)\\subseteq W\\cap U_1\\).\n\nInductively, \\(B(x_n,r_n)\\cap U_{n+1}\\) is nonempty and open. Choose \\(x_{n+1}\\) and \\(0<r_{n+1}<r_n/2\\) with \\(\\overline B(x_{n+1},r_{n+1})\\subseteq B(x_n,r_n)\\cap U_{n+1}\\).\n\nFor \\(m>n\\), \\(x_m\\in B(x_n,r_n)\\) and \\(r_n<2^{1-n}\\), so \\((x_n)\\) is Cauchy. Let \\(x\\) be its limit. For each \\(n\\), the closed ball \\(\\overline B(x_n,r_n)\\) contains \\(x_m\\) for all \\(m\\ge n\\), hence contains \\(x\\). So \\(x\\in W\\cap\\bigcap_nU_n\\).\n\nFor the second form: the complement of a closed set with empty interior is a dense open set. If \\(X=\\bigcup_nF_n\\) with each \\(F_n\\) closed with empty interior, then the dense open sets \\(X\\setminus F_n\\) have empty intersection. This contradicts the first form, since \\(X\\ne\\varnothing\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": []
    },
    {
      "id": "OA-FND-HB-04",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "4. Uniform boundedness",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "anchor": "oa-fnd-hb-04",
      "proof_locus": {
        "line": 131,
        "through_line": 151
      },
      "full_conditions_and_proof": "## 4. Uniform boundedness\n\n**Lemma 4.1.** If \\(E\\) is a normed space and \\(F\\) a Banach space, then \\(B(E,F)\\) is a Banach space. In particular \\(E^*\\) is a Banach space for every normed space \\(E\\).\n\n**Proof.** Let \\((T_n)\\) be Cauchy. For each \\(x\\), \\(\\|T_nx-T_mx\\|\\le\\|T_n-T_m\\|\\|x\\|\\), so \\((T_nx)\\) is Cauchy. Put \\(Tx=\\lim T_nx\\). Then \\(T\\) is linear. Given \\(\\varepsilon>0\\), choose \\(N\\) with \\(\\|T_n-T_m\\|\\le\\varepsilon\\) for \\(n,m\\ge N\\). Letting \\(m\\to\\infty\\) in \\(\\|T_nx-T_mx\\|\\le\\varepsilon\\|x\\|\\) gives \\(\\|(T_n-T)x\\|\\le\\varepsilon\\|x\\|\\) for \\(n\\ge N\\). So \\(T_n-T\\) is bounded, hence so is \\(T\\), and \\(\\|T_n-T\\|\\le\\varepsilon\\) for \\(n\\ge N\\). \\(\\square\\)\n\n**Theorem 4.2** (uniform boundedness principle). Let \\(E\\) be a Banach space, \\(F\\) a normed space, and \\(\\mathcal T\\subseteq B(E,F)\\) a family with \\(\\sup_{T\\in\\mathcal T}\\|Tx\\|<\\infty\\) for every \\(x\\in E\\). Then \\(\\sup_{T\\in\\mathcal T}\\|T\\|<\\infty\\).\n\n**Proof.** If \\(E=\\{0\\}\\) there is nothing to prove. Put \\(F_n=\\{x\\in E:\\|Tx\\|\\le n\\text{ for all }T\\in\\mathcal T\\}\\). Each \\(F_n\\) is closed, as an intersection of closed sets. By hypothesis \\(\\bigcup_nF_n=E\\). By Theorem 3.1 some \\(F_N\\) contains a closed ball \\(\\overline B(x_0,r)\\) with \\(r>0\\). For \\(\\|y\\|\\le1\\), both \\(x_0+ry\\) and \\(x_0\\) lie in \\(F_N\\), so \\(r\\|Ty\\|\\le\\|T(x_0+ry)\\|+\\|Tx_0\\|\\le2N\\). Hence \\(\\|T\\|\\le2N/r\\) for every \\(T\\in\\mathcal T\\). \\(\\square\\)\n\n**Corollary 4.3.**\n1. Let \\(E\\) be a Banach space. If \\(\\varphi_n\\in E^*\\) and \\(\\varphi_n(x)\\) converges for every \\(x\\in E\\), then \\(\\sup_n\\|\\varphi_n\\|<\\infty\\), and \\(\\varphi(x)=\\lim\\varphi_n(x)\\) defines \\(\\varphi\\in E^*\\).\n2. Let \\(E\\) be a normed space and \\(S\\subseteq E\\) with \\(\\sup_{x\\in S}|\\varphi(x)|<\\infty\\) for every \\(\\varphi\\in E^*\\). Then \\(S\\) is norm bounded.\n3. Let \\(E\\) be a Banach space and \\(F\\) a normed space. If \\(T_n\\in B(E,F)\\) and \\(T_nx\\) converges for every \\(x\\), then \\(\\sup_n\\|T_n\\|<\\infty\\), and \\(Tx=\\lim T_nx\\) defines \\(T\\in B(E,F)\\) with \\(\\|T\\|\\le\\liminf_n\\|T_n\\|\\). In particular, a strongly convergent sequence of operators on a Hilbert space is norm bounded.\n\n**Proof.** 1 and 3. Convergent sequences are bounded, so Theorem 4.2 applies. The limit is linear, and \\(\\|Tx\\|=\\lim\\|T_nx\\|\\le\\liminf_n\\|T_n\\|\\,\\|x\\|\\).\n\n2. The maps \\(j(x)\\in E^{**}=B(E^*,\\mathbb K)\\), \\(x\\in S\\), are pointwise bounded on the Banach space \\(E^*\\) (Lemma 4.1). By Theorem 4.2 they are bounded in norm, and \\(\\|j(x)\\|=\\|x\\|\\) by Corollary 2.3(4). \\(\\square\\)\n\n**Example 4.4** (completeness cannot be dropped). Let \\(c_{00}\\) be the space of finitely supported sequences, with the supremum norm, and \\(\\varphi_n(x)=nx_n\\). For each \\(x\\), \\(\\varphi_n(x)=0\\) once \\(n\\) exceeds the support of \\(x\\), so the family is pointwise bounded. But \\(\\|\\varphi_n\\|=n\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-HB-05",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "5. The open mapping and closed graph theorems",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
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      "anchor": "oa-fnd-hb-05",
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      "full_conditions_and_proof": "## 5. The open mapping and closed graph theorems\n\n**Theorem 5.1** (open mapping). Let \\(E,F\\) be Banach spaces and \\(T\\in B(E,F)\\) surjective. Then \\(T\\) maps open sets to open sets. More precisely, there is \\(\\delta>0\\) with \\(T(B_E(0,1))\\supseteq B_F(0,\\delta)\\).\n\n**Proof.** *Step 1: a ball in the closure.* Since \\(T\\) is onto, \\(F=\\bigcup_n\\overline{T(B_E(0,n))}\\). By Theorem 3.1 some \\(\\overline{T(B_E(0,n))}\\) contains a ball \\(B_F(y_0,\\varepsilon)\\).\n\nIf \\(a,b\\in\\overline{T(B_E(0,n))}\\), then \\(a-b\\in\\overline{T(B_E(0,2n))}\\): write \\(a=\\lim Tx_k\\) and \\(b=\\lim Tz_k\\) with \\(\\|x_k\\|,\\|z_k\\|<n\\), and note \\(\\|x_k-z_k\\|<2n\\). For \\(\\|y\\|<\\varepsilon\\), write \\(y=(y_0+y)-y_0\\) with both terms in \\(\\overline{T(B_E(0,n))}\\). So \\(B_F(0,\\varepsilon)\\subseteq\\overline{T(B_E(0,2n))}\\). By scaling, with \\(\\eta=\\varepsilon/2n\\),\n\\[\n\\begin{gathered}\nB_F(0,\\eta r)\\\\\n\\subseteq\\overline{T(B_E(0,r))}\\\\\n\\text{for every }r>0.\n\\end{gathered}\n\\tag{5.1}\n\\]\n\n*Step 2: removing the closure.* Let \\(\\|y\\|<\\eta/2\\). By (5.1) with \\(r=1/2\\), choose \\(x_1\\) with \\(\\|x_1\\|<1/2\\) and \\(\\|y-Tx_1\\|<\\eta/4\\). Inductively, given \\(x_1,\\dots,x_n\\) with \\(\\|y-T(x_1+\\dots+x_n)\\|<\\eta2^{-n-1}\\), apply (5.1) with \\(r=2^{-n-1}\\). This gives \\(x_{n+1}\\) with \\(\\|x_{n+1}\\|<2^{-n-1}\\) and \\(\\|y-T(x_1+\\dots+x_{n+1})\\|<\\eta2^{-n-2}\\).\n\nThe series \\(\\sum_nx_n\\) converges absolutely, so it converges in the Banach space \\(E\\). Its sum \\(x\\) satisfies \\(\\|x\\|\\le\\sum\\|x_n\\|<1\\), and \\(Tx=y\\) by continuity. So \\(B_F(0,\\eta/2)\\subseteq T(B_E(0,1))\\). Take \\(\\delta=\\eta/2\\).\n\n*Step 3: open sets.* Let \\(U\\subseteq E\\) be open and \\(y=Tx\\) with \\(x\\in U\\). Choose \\(r>0\\) with \\(x+B_E(0,r)\\subseteq U\\). Then \\(T(U)\\supseteq y+rT(B_E(0,1))\\supseteq B_F(y,r\\delta)\\). \\(\\square\\)\n\n**Corollary 5.2** (inverse mapping). A bijective \\(T\\in B(E,F)\\) between Banach spaces has a bounded inverse. If two complete norms on one vector space satisfy \\(\\|\\cdot\\|_1\\le C\\|\\cdot\\|_2\\), they are equivalent.\n\n**Proof.** By Theorem 5.1, \\(\\|Tx\\|<\\delta\\) implies \\(\\|x\\|<1\\), so \\(\\|T^{-1}y\\|\\le\\delta^{-1}\\|y\\|\\). For the norms, apply this to the identity map from the second space to the first. \\(\\square\\)\n\n**Theorem 5.3** (closed graph). Let \\(T:E\\to F\\) be linear between Banach spaces, with closed graph \\(G=\\{(x,Tx):x\\in E\\}\\subseteq E\\oplus F\\). Then \\(T\\) is bounded.\n\n**Proof.** Give \\(E\\oplus F\\) the norm \\(\\|x\\|+\\|y\\|\\), a complete norm. The closed subspace \\(G\\) is then a Banach space. The map \\(\\pi:G\\to E\\), \\((x,Tx)\\mapsto x\\), is bounded and bijective. By Corollary 5.2 its inverse is bounded: \\(\\|x\\|+\\|Tx\\|\\le C\\|x\\|\\). \\(\\square\\)\n\n*Remark 5.4.* The proofs of Theorems 3.1, 4.2, 5.1 and 5.3 use only real scalars. So they hold for real Banach spaces, and they apply to conjugate-linear maps between complex Banach spaces, which are real-linear. The lesson on C\\*-algebras uses the closed graph theorem in this form.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-HB-06",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "6. Topological vector spaces and separation of convex sets",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
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      "anchor": "oa-fnd-hb-06",
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        "line": 184,
        "through_line": 257
      },
      "full_conditions_and_proof": "## 6. Topological vector spaces and separation of convex sets\n\nA *topological vector space* is a vector space with a topology for which addition and scalar multiplication are continuous. It is *locally convex* if its topology is defined by a family \\(\\mathcal P\\) of seminorms: the sets \\(\\{x:p_i(x-x_0)<\\varepsilon,\\ i=1,\\dots,n\\}\\), for \\(p_1,\\dots,p_n\\in\\mathcal P\\) and \\(\\varepsilon>0\\), form a base of neighbourhoods of \\(x_0\\). These basic neighbourhoods of \\(0\\) are convex and *balanced*: \\(\\lambda U\\subseteq U\\) for \\(|\\lambda|\\le1\\). Normed spaces are locally convex. We do not assume a topological vector space to be Hausdorff unless we say so.\n\n**Lemma 6.1** (Minkowski functionals). Let \\(U\\) be a convex open neighbourhood of \\(0\\) in a topological vector space \\(X\\), and \\(p_U(x)=\\inf\\{t>0:x\\in tU\\}\\).\n1. \\(p_U\\) is finite and sublinear.\n2. \\(U=\\{x:p_U(x)<1\\}\\).\n3. \\[\n\\begin{gathered}\n|p_U(x)-p_U(y)|\\\\\n\\le\\max(p_U(x-y),p_U(y-x)),\n\\end{gathered}\n\\] and \\(p_U<\\varepsilon\\) on \\(\\varepsilon U\\). So \\(p_U\\) is continuous.\n\n**Proof.** 1. \\(U\\) is *absorbing*: for each \\(x\\), the map \\(t\\mapsto tx\\) is continuous with value \\(0\\) at \\(t=0\\), so \\(tx\\in U\\) for small \\(t>0\\). Hence \\(p_U(x)<\\infty\\). Homogeneity, \\(p_U(sx)=sp_U(x)\\) for \\(s\\ge0\\), is immediate. Subadditivity follows from convexity: if \\(x\\in aU\\) and \\(y\\in bU\\) with \\(a,b>0\\), then\n\\[\n\\begin{gathered}\nx+y\\\\\n=(a+b)\\Big(\\frac a{a+b}\\frac xa+\\frac b{a+b}\\frac yb\\Big)\\in(a+b)U.\n\\end{gathered}\n\\]\n\n2. If \\(x\\in U\\), then \\((1+\\varepsilon)x\\in U\\) for small \\(\\varepsilon>0\\), because \\(U\\) is open and \\(s\\mapsto sx\\) is continuous. Then \\(p_U(x)\\le(1+\\varepsilon)^{-1}<1\\). Conversely, \\(p_U(x)<1\\) gives \\(x\\in tU\\) for some \\(t<1\\), and \\(tU\\subseteq U\\) because \\(U\\) is convex and contains \\(0\\).\n\n3. Subadditivity gives \\(p_U(x)\\le p_U(y)+p_U(x-y)\\) and \\(p_U(y)\\le p_U(x)+p_U(y-x)\\). If \\(x\\in\\varepsilon U\\), then \\(p_U(x)\\le\\varepsilon\\cdot p_U(x/\\varepsilon)<\\varepsilon\\) by 2. \\(\\square\\)\n\n**Theorem 6.2** (separation from an open convex set). Let \\(X\\) be a topological vector space, and \\(A,B\\subseteq X\\) nonempty disjoint convex sets with \\(A\\) open. Then there are a continuous linear functional \\(\\varphi\\) on \\(X\\) and \\(t\\in\\mathbb R\\) with\n\\[\n\\operatorname{Re}\\varphi(a)<t\\le\\operatorname{Re}\\varphi(b)\\qquad(a\\in A,\\ b\\in B).\n\\]\n\n**Proof.** *Real scalars.* Fix \\(a_0\\in A\\) and \\(b_0\\in B\\), and put \\(x_0=b_0-a_0\\) and \\(U=A-B+x_0\\).\n- \\(U\\) is convex, and open (a union of translates of \\(A\\)). It contains \\(0\\).\n- \\(x_0\\notin U\\), because \\(A\\cap B=\\varnothing\\). So \\(p_U(x_0)\\ge1\\) by Lemma 6.1(2).\n\nDefine \\(f(sx_0)=s\\) on \\(\\mathbb Rx_0\\). Then \\(f\\le p_U\\) there: for \\(s\\ge0\\), \\(f(sx_0)=s\\le sp_U(x_0)\\); for \\(s<0\\), \\(f(sx_0)<0\\le p_U(sx_0)\\). By Theorem 2.1, \\(f\\) extends to a linear \\(F\\le p_U\\) on \\(X\\).\n\n\\(F\\) is continuous: on the neighbourhood \\(U\\cap(-U)\\) of \\(0\\) we have \\(F<1\\) and \\(-F(x)=F(-x)<1\\), so \\(|F|<1\\) there, and a linear functional bounded on a neighbourhood of \\(0\\) is continuous.\n\nFor \\(a\\in A\\) and \\(b\\in B\\), \\(a-b+x_0\\in U\\), so \\[\n\\begin{gathered}\nF(a)-F(b)+1\\\\\n=F(a-b+x_0)\\\\\n\\le p_U(a-b+x_0)<1.\n\\end{gathered}\n\\] Hence \\(F(a)<F(b)\\).\n\n\\(F\\) is not zero, so it is an open map \\(X\\to\\mathbb R\\): \\(F(x_1)=1\\) for some \\(x_1\\), and \\(F(x+sx_1)=F(x)+s\\). So \\(F(A)\\) is an open interval. Let \\(t=\\sup F(A)\\). Then \\(t\\le\\inf F(B)\\), and \\(F(a)<t\\) for every \\(a\\in A\\), since \\(F(A)\\) is open.\n\n*Complex scalars.* Apply the real case to \\(X\\) as a real space, obtaining \\(F\\). Put \\(\\varphi(x)=F(x)-iF(ix)\\). As in the proof of Theorem 2.2, \\(\\varphi\\) is complex-linear with \\(\\operatorname{Re}\\varphi=F\\), and it is continuous. \\(\\square\\)\n\n**Theorem 6.3** (strict separation). Let \\(X\\) be a locally convex space, \\(K\\subseteq X\\) compact and convex, and \\(C\\subseteq X\\) closed and convex, both nonempty, with \\(K\\cap C=\\varnothing\\). Then there are a continuous linear functional \\(\\varphi\\) and numbers \\(t_1<t_2\\) with \\(\\operatorname{Re}\\varphi<t_1\\) on \\(K\\) and \\(\\operatorname{Re}\\varphi>t_2\\) on \\(C\\).\n\n**Proof.** *A uniform neighbourhood.* For each \\(k\\in K\\), \\(X\\setminus C\\) is a neighbourhood of \\(k\\). Choose a convex balanced open neighbourhood \\(V_k\\) of \\(0\\) with \\(k+V_k+V_k\\subseteq X\\setminus C\\); a basic neighbourhood of radius \\(\\varepsilon/2\\) works when \\(\\{x:p_i(x)<\\varepsilon\\}\\) fits. Finitely many sets \\(k_j+V_{k_j}\\) cover \\(K\\). Let \\(V=\\bigcap_jV_{k_j}\\). Then \\(K+V\\subseteq\\bigcup_j(k_j+V_{k_j}+V_{k_j})\\) does not meet \\(C\\).\n\n*Separation.* \\(A=K+V\\) is convex, open and disjoint from \\(C\\). Theorem 6.2 gives \\(\\varphi\\) and \\(t\\) with \\(\\operatorname{Re}\\varphi<t\\) on \\(A\\) and \\(\\operatorname{Re}\\varphi\\ge t\\) on \\(C\\). On the compact set \\(K\\subseteq A\\), the continuous function \\(\\operatorname{Re}\\varphi\\) attains a maximum \\(s<t\\). Take \\(t_1\\) and \\(t_2\\) with \\(s<t_1<t_2<t\\). \\(\\square\\)\n\n**Corollary 6.4.** Let \\(X\\) be a locally convex space.\n1. If \\(X\\) is Hausdorff, its continuous linear functionals separate points.\n2. A point \\(x\\) outside a closed convex set \\(C\\) is strictly separated from \\(C\\) by a continuous linear functional.\n3. Every continuous linear functional on a subspace \\(M\\subseteq X\\) extends to a continuous linear functional on \\(X\\).\n\n**Proof.** 1 and 2 are Theorem 6.3 with \\(K=\\{x\\}\\), and with \\(C=\\{y\\}\\) for 1.\n\n3. If \\(\\varphi\\) is continuous on \\(M\\), then \\(\\{m\\in M:|\\varphi(m)|<1\\}\\) contains a basic neighbourhood \\(\\{m\\in M:p_i(m)<\\varepsilon,\\ i\\le n\\}\\). Hence \\(|\\varphi(m)|\\le\\varepsilon^{-1}\\max_ip_i(m)\\) on \\(M\\): if this failed at some \\(m\\), a scalar multiple \\(m'\\) of \\(m\\) would have \\(\\max_ip_i(m')<\\varepsilon\\) and \\(|\\varphi(m')|\\ge1\\). The function \\(q=\\varepsilon^{-1}\\max_ip_i\\) is a continuous seminorm on \\(X\\). Theorem 2.2 extends \\(\\varphi\\) to \\(\\Phi\\) with \\(|\\Phi|\\le q\\), and \\(\\Phi\\) is continuous. \\(\\square\\)\n\n**Lemma 6.5** (Closed kernels). A linear functional \\(f\\) on a topological vector space \\(X\\) is continuous if and only if its kernel is closed.\n\n**Proof.** If \\(f\\) is continuous, \\(\\ker f=f^{-1}(0)\\) is closed. Conversely, let \\(\\ker f\\) be closed and \\(f\\neq0\\).\n- Choose \\(x_0\\) with \\(f(x_0)=1\\). The set \\(f^{-1}(1)=x_0+\\ker f\\) is closed and does not contain \\(0\\), so its complement \\(U\\) is a neighbourhood of \\(0\\).\n- *A balanced neighbourhood inside \\(U\\).* By continuity of \\((\\lambda,x)\\mapsto\\lambda x\\) at \\((0,0)\\), there are \\(\\delta>0\\) and a neighbourhood \\(W_0\\) of \\(0\\) with \\(\\lambda W_0\\subseteq U\\) for \\(|\\lambda|\\le\\delta\\). The set \\(V=\\bigcup_{0<|\\lambda|\\le\\delta}\\lambda W_0\\) is a neighbourhood of \\(0\\), since it contains \\(\\delta W_0\\). It is balanced: \\(\\mu V\\subseteq V\\) for \\(|\\mu|\\le1\\), with \\(0\\cdot V=\\{0\\}\\subseteq V\\). And \\(V\\subseteq U\\).\n- *\\(|f|<1\\) on \\(V\\).* If \\(v\\in V\\) had \\(|f(v)|\\geq1\\), then \\(v/f(v)\\in V\\), because \\(V\\) is balanced, and \\(f(v/f(v))=1\\). This contradicts \\(V\\cap f^{-1}(1)=\\varnothing\\).\n- So \\(|f|<\\varepsilon\\) on the neighbourhood \\(\\varepsilon V\\), for every \\(\\varepsilon>0\\), and \\(f\\) is continuous at \\(0\\). A linear map that is continuous at \\(0\\) is continuous. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": []
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    {
      "id": "OA-FND-HB-07",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "7. Finite-dimensional spaces",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "anchor": "oa-fnd-hb-07",
      "proof_locus": {
        "line": 258,
        "through_line": 276
      },
      "full_conditions_and_proof": "## 7. Finite-dimensional spaces\n\n**Theorem 7.1.** Let \\(X\\) be a Hausdorff topological vector space of finite dimension \\(n\\). Then every linear bijection \\(f:\\mathbb K^n\\to X\\) is a homeomorphism, where \\(\\mathbb K^n\\) has its Euclidean topology. Consequently:\n1. all Hausdorff vector topologies on a finite-dimensional space coincide;\n2. every finite-dimensional subspace of a Hausdorff topological vector space is closed;\n3. every linear map from a finite-dimensional Hausdorff topological vector space to a topological vector space is continuous.\n\n**Proof.** \\(f\\) is continuous, being built from addition and scalar multiplication.\n\n*The inverse is continuous.* The unit sphere \\(S\\subseteq\\mathbb K^n\\) is compact, so \\(f(S)\\) is compact, hence closed in the Hausdorff space \\(X\\), and \\(0\\notin f(S)\\). The open set \\(X\\setminus f(S)\\) contains a balanced neighbourhood \\(W\\) of \\(0\\): continuity of \\((\\lambda,x)\\mapsto\\lambda x\\) at \\((0,0)\\) gives \\(\\delta>0\\) and a neighbourhood \\(W_0\\) with \\(\\lambda W_0\\subseteq X\\setminus f(S)\\) for \\(|\\lambda|\\le\\delta\\); take \\(W=\\bigcup_{|\\lambda|\\le\\delta}\\lambda W_0\\).\n\nThe set \\(f^{-1}(W)\\) is balanced and disjoint from \\(S\\). So it lies in the open unit ball: if \\(v\\in f^{-1}(W)\\) with \\(|v|\\ge1\\), then \\(v/|v|\\in f^{-1}(W)\\cap S\\). So \\(f^{-1}\\) maps the neighbourhood \\(W\\) into the unit ball. A linear map that is bounded on a neighbourhood of \\(0\\) into a normed space is continuous.\n\n1 follows at once. For 3, compose with \\(f\\): a linear map on \\(\\mathbb K^n\\) is continuous.\n\n2. Let \\(M\\) be a finite-dimensional subspace and \\(x\\in\\overline M\\). The space \\(N=M+\\mathbb Kx\\) is finite-dimensional. By the main statement its topology is Euclidean, in which subspaces are closed. Every neighbourhood of \\(x\\) in \\(N\\) meets \\(M\\), so \\(x\\) lies in the closure of \\(M\\) in \\(N\\), which is \\(M\\). \\(\\square\\)\n\nThe Hausdorff hypothesis is necessary: with the indiscrete topology, \\(\\{0\\}\\) is not closed.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": []
    },
    {
      "id": "OA-FND-HB-08",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "8. Cardinal arithmetic",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "anchor": "oa-fnd-hb-08",
      "proof_locus": {
        "line": 277,
        "through_line": 312
      },
      "full_conditions_and_proof": "## 8. Cardinal arithmetic\n\nSeveral lessons count orthonormal bases or projections. They use the following facts. Write \\(|X|\\leq|Y|\\) if there is an injection \\(X\\to Y\\), and \\(|X|=|Y|\\) if there is a bijection.\n\n**Theorem 8.1** (Cantor–Schröder–Bernstein). If \\(|X|\\leq|Y|\\) and \\(|Y|\\leq|X|\\), then \\(|X|=|Y|\\).\n\n**Proof.** Let \\(f:X\\to Y\\) and \\(g:Y\\to X\\) be injections. Put \\(C_0=X\\setminus g(Y)\\), \\(C_{n+1}=g(f(C_n))\\) and \\(C=\\bigcup_nC_n\\). Define \\(h(x)=f(x)\\) for \\(x\\in C\\), and \\(h(x)=g^{-1}(x)\\) for \\(x\\notin C\\); this makes sense because \\(x\\notin C_0\\) means \\(x\\in g(Y)\\).\n- *\\(h\\) is injective.* It is injective on \\(C\\) and on \\(X\\setminus C\\). If \\(f(x)=g^{-1}(x')\\) with \\(x\\in C_n\\) and \\(x'\\notin C\\), then \\(x'=g(f(x))\\in C_{n+1}\\subseteq C\\), a contradiction.\n- *\\(h\\) is surjective.* Let \\(y\\in Y\\). If \\(g(y)\\notin C\\), then \\(h(g(y))=y\\). If \\(g(y)\\in C\\), then \\(g(y)\\notin C_0\\), so \\(g(y)\\in C_{n+1}=g(f(C_n))\\) for some \\(n\\). Since \\(g\\) is injective, \\(y=f(x)\\) with \\(x\\in C_n\\), and \\(h(x)=y\\). \\(\\square\\)\n\n**Theorem 8.2** (Cantor). For every set \\(X\\), \\(|X|\\leq|\\mathcal P(X)|\\) and \\(|X|\\neq|\\mathcal P(X)|\\).\n\n**Proof.** \\(x\\mapsto\\{x\\}\\) is an injection. If \\(F:X\\to\\mathcal P(X)\\) is any map, the set \\(D=\\{x:x\\notin F(x)\\}\\) is not in its range: \\(D=F(x)\\) would give \\(x\\in D\\iff x\\notin D\\). \\(\\square\\)\n\n**Proposition 8.3** (Exponents). For sets \\(X,Y,Z\\), the map that sends \\(F:Y\\times Z\\to X\\) to \\(z\\mapsto F(\\cdot,z)\\) is a bijection from \\(X^{Y\\times Z}\\) onto \\((X^Y)^Z\\). The map \\((m,n)\\mapsto2^m(2n+1)-1\\) is a bijection \\(\\mathbb N\\times\\mathbb N\\to\\mathbb N\\). Consequently \\(|(\\{0,1\\}^{\\mathbb N})^{\\mathbb N}|=|\\{0,1\\}^{\\mathbb N}|\\), that is, \\((2^{\\aleph_0})^{\\aleph_0}=2^{\\aleph_0}\\).\n\n**Proof.** The inverse of the first map sends \\(G\\) to \\((y,z)\\mapsto G(z)(y)\\). Every positive integer is uniquely \\(2^m\\) times an odd number \\(2n+1\\). \\(\\square\\)\n\n**Theorem 8.4.** Let \\(X\\) be an infinite set. Then:\n1. \\(|X\\times\\mathbb N|=|X|\\);\n2. \\(|X\\times\\{0,1\\}|=|X|\\);\n3. if \\((A_i)_{i\\in I}\\) is a family of countable sets indexed by an infinite set \\(I\\), then \\(|\\bigcup_iA_i|\\leq|I|\\).\n\n**Proof.** *\\(X\\) has a countably infinite subset.* Injections from initial segments \\(\\{0,\\dots,n-1\\}\\) or from \\(\\mathbb N\\) into \\(X\\), ordered by extension, satisfy the hypothesis of Zorn's lemma: the union of a chain is an upper bound. A maximal one is defined on all of \\(\\mathbb N\\), because a finite one can be extended by a point outside its finite range.\n\n(1) Let \\(P\\) be the set of pairs \\((A,\\varphi)\\) with \\(A\\subseteq X\\) and \\(\\varphi:A\\times\\mathbb N\\to A\\) a bijection, ordered by \\((A,\\varphi)\\leq(A',\\varphi')\\) if \\(A\\subseteq A'\\) and \\(\\varphi'\\) extends \\(\\varphi\\).\n- \\(P\\) is nonempty: a countably infinite \\(A_0\\subseteq X\\) has \\(|A_0\\times\\mathbb N|=|A_0|\\) by Proposition 8.3.\n- The union of a chain is an upper bound.\n- Let \\((A,\\varphi)\\) be maximal (Zorn's lemma). If \\(X\\setminus A\\) were infinite, it would contain a countably infinite \\(B\\), and a bijection \\(B\\times\\mathbb N\\to B\\) would extend \\(\\varphi\\) to \\(A\\cup B\\). So \\(X\\setminus A\\) is finite.\n- An infinite set \\(A\\) absorbs a finite set \\(F\\) disjoint from it: choose distinct \\(c_0,c_1,\\ldots\\) in \\(A\\) and \\(F=\\{f_1,\\dots,f_k\\}\\); the map sending \\(f_j\\mapsto c_{j-1}\\), \\(c_n\\mapsto c_{n+k}\\), and fixing the rest of \\(A\\), is a bijection \\(A\\cup F\\to A\\).\n- \\(A\\) is infinite, since \\(X\\) is infinite and \\(X\\setminus A\\) is finite. So \\(|X|=|A|\\). A bijection \\(X\\to A\\) induces a bijection \\(X\\times\\mathbb N\\to A\\times\\mathbb N\\). Hence \\(|X\\times\\mathbb N|=|A\\times\\mathbb N|=|A|=|X|\\).\n\n(2) \\(X\\) injects into \\(X\\times\\{0,1\\}\\), which injects into \\(X\\times\\mathbb N\\). Apply (1) and Theorem 8.1.\n\n(3) By the axiom of choice, choose for each \\(i\\) a surjection \\(s_i:\\mathbb N\\to A_i\\) (for \\(A_i=\\varnothing\\), skip \\(i\\)). The map \\((i,n)\\mapsto s_i(n)\\) is a surjection from a subset of \\(I\\times\\mathbb N\\) onto \\(\\bigcup_iA_i\\). Choosing one preimage for each point gives an injection of \\(\\bigcup_iA_i\\) into \\(I\\times\\mathbb N\\), and \\(|I\\times\\mathbb N|=|I|\\) by (1). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": []
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    {
      "id": "OA-FND-HB-09",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "name": "Well-ordering and infinite products",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
      "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "anchor": "oa-fnd-hb-09",
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        "line": 313,
        "through_line": 391
      },
      "full_conditions_and_proof": "### Well-ordering and infinite products\n\n**Theorem 8.5.** Every set can be well-ordered. If \\(X\\) is infinite, then \\(|X\\times X|=|X|\\). Consequently, if \\(0<|Y|\\leq|X|\\), then \\(|X\\times Y|=|X|\\). In particular an uncountable set can be partitioned into uncountably many subsets, each in bijection with the whole set.\n\n*Proof.* A *well-order* is a linear order in which every nonempty subset has a least member. Order the well-orders on subsets of \\(X\\) by extension as an initial segment: an extension may append elements, but may not insert elements before an old element. The union of a chain is again a well-order. To see this, take a member \\(a\\) of a nonempty subset \\(S\\) of the union and a chain member containing \\(a\\). All predecessors of \\(a\\) in the union already lie in that member. The nonempty set of elements of \\(S\\) at or before \\(a\\) therefore has a least member there, which is least in all of \\(S\\). Zorn's lemma gives a maximal such well-order. If its domain omitted a point of \\(X\\), appending that point would extend it. Its domain is therefore \\(X\\).\n\nHere are the order-type facts needed to count the square. Two well-orders have at most one order isomorphism between initial segments: if two such maps first differ at \\(a\\), their common image of the predecessors of \\(a\\) determines the least unused image of \\(a\\), a contradiction. Take the union of all these initial-segment isomorphisms for two well-orders. They are compatible by uniqueness, and their union has initial-segment domains and ranges. If neither order were exhausted, mapping the least remaining point of one to the least remaining point of the other would extend the union. Thus one order is isomorphic to an initial segment of the other.\n\nA well-order cannot be isomorphic to a proper initial segment of itself. Indeed, suppose \\(f\\) is such an isomorphism and \\(a\\) is the first point with \\(f(a)\\ne a\\). The predecessors of \\(a\\) are fixed, so order preservation forces \\(f(a)\\geq a\\). Equality is excluded, and \\(f(a)>a\\) would omit \\(a\\) from the initial-segment range. If there is no such \\(a\\), the range is the whole order. These contradictions prove the assertion. Hence order types are linearly ordered by proper initial-segment inclusion.\n\nEvery nonempty set of order types has a least one. Pick a type \\(\\tau\\) in it. If there are types below \\(\\tau\\), identify them with proper initial segments of a representative of \\(\\tau\\). Their endpoint set has a least member, giving the least type below \\(\\tau\\). If there are none, \\(\\tau\\) itself is least. An *initial order type*, or cardinal, is the least type of a well-order on a set of its given cardinality. Such a least type exists, since the orders on that set form a set. Injections respect cardinal types: if \\(A\\) injects into \\(B\\) but its cardinal type were larger, the type of \\(B\\) would be an initial segment of that of \\(A\\), giving an injection in the other direction. Cantor–Schröder–Bernstein would make their cardinalities, and hence their least types, equal. Every proper initial segment of an initial type \\(\\kappa\\) therefore has cardinality less than \\(\\kappa\\): equality would give a smaller well-order of the same set. No infinite initial type has a last point, because an infinite set absorbs one extra point by the explicit shifting bijection in the proof of Theorem 8.4.\n\nSuppose now that an infinite cardinal fails the square identity, and take the least such \\(\\kappa\\). This choice is legitimate within a set: below any proposed counterexample, all smaller cardinal types are represented by well-orders on subsets of that counterexample. Represent \\(\\kappa\\) by its initial well-order. Order pairs \\((\\alpha,\\beta)\\) first by \\(\\max(\\alpha,\\beta)\\), then by \\(\\alpha\\), then by \\(\\beta\\). This is a well-order: a nonempty collection of pairs has a least maximum, a least first coordinate among pairs with that maximum, and then a least second coordinate.\n\nThe predecessors of a pair with maximum \\(\\gamma\\) are contained in the square of the initial segment through \\(\\gamma\\). That segment is proper, since \\(\\kappa\\) has no last point. Its cardinal \\(\\mu\\) is less than \\(\\kappa\\). If \\(\\mu\\) is finite, its square is finite; if \\(\\mu\\) is infinite, minimality of \\(\\kappa\\) gives \\(\\mu^2=\\mu\\). Thus every predecessor set in the pair order has cardinality less than \\(\\kappa\\).\n\nLet \\(\\tau\\) be the type of this pair order. If \\(\\tau>\\kappa\\), comparison of well-orders embeds \\(\\kappa\\) as a proper initial segment of it. The least point outside that segment then has \\(\\kappa\\) predecessors, a contradiction. Thus \\(\\tau\\leq\\kappa\\), giving \\(|\\kappa\\times\\kappa|\\leq\\kappa\\). The map \\(\\alpha\\mapsto(\\alpha,\\alpha)\\) gives the reverse inequality, so Cantor–Schröder–Bernstein proves the square identity. This contradicts the choice of \\(\\kappa\\), and proves the identity for every infinite set.\n\nFor nonempty \\(Y\\) with an injection into \\(X\\), choose \\(y_0\\in Y\\). The maps \\(x\\mapsto(x,y_0)\\) and the coordinate injection into \\(X\\times X\\) give\n\\[\n|X|\\leq|X\\times Y|\\leq|X\\times X|=|X|.\n\\]\nApply Cantor–Schröder–Bernstein again. Finally, choose a bijection \\(b:X\\times X\\to X\\). The sets \\(b(X\\times\\{y\\})\\), \\(y\\in X\\), partition \\(X\\) and each is in bijection with \\(X\\). If \\(X\\) is uncountable, their index set is uncountable. \\(\\square\\)\n\n\n## Exercises\n\n**Exercise 1** (medium; two complete norms). Let \\(\\|\\cdot\\|_1\\) and \\(\\|\\cdot\\|_2\\) be complete norms on a vector space \\(V\\) with \\(\\|v\\|_1\\le C\\|v\\|_2\\) for all \\(v\\). Show that the norms are equivalent. Show by example that completeness of both norms is needed.\n\n*Solution.* Equivalence is Corollary 5.2. For an example in which the weaker norm is incomplete, take \\(V=C[0,1]\\) with \\(\\|\\cdot\\|_1\\) the \\(L^1\\) norm and \\(\\|\\cdot\\|_2\\) the supremum norm. Then \\(\\|f\\|_1\\le\\|f\\|_2\\), and \\(\\|\\cdot\\|_2\\) is complete. The functions \\(f_n(t)=\\max(0,1-nt)\\) have \\(\\|f_n\\|_2=1\\) and \\(\\|f_n\\|_1=1/(2n)\\), so the norms are not equivalent. If \\(\\|\\cdot\\|_1\\) were complete on \\(C[0,1]\\), Corollary 5.2 would contradict these norm ratios, so it is incomplete. For the other completeness assumption, retain \\(V=C[0,1]\\) with complete norm \\(\\|f\\|_1=\\|f\\|_\\infty\\). The completeness follows because a uniformly Cauchy sequence has a uniform limit, and uniform limits of continuous functions are continuous. On the polynomial subspace define \\(u_0(p)=p'(1)\\). It has a linear extension \\(u\\) to all of \\(V\\): order algebraic extensions by inclusion, take unions along chains, and use Zorn; if a maximal domain omitted \\(f\\), extend by \\(u(d+\\lambda f)=u(d)\\), a contradiction. Define \\(\\|f\\|_2=\\|f\\|_\\infty+|u(f)|\\), a norm dominating \\(\\|f\\|_1\\). For \\(f_n(t)=t^n\\), \\(\\|f_n\\|_1=1\\) and \\(\\|f_n\\|_2=1+n\\), so the norms are not equivalent. If \\(\\|\\cdot\\|_2\\) were complete, Corollary 5.2 would again give equivalence. Thus this stronger norm is incomplete while the weaker one is complete. \\(\\square\\)\n\n**Exercise 2** (medium; Banach limits). Let \\(\\ell^\\infty_{\\mathbb R}\\) be the real space of bounded real sequences and \\(S\\) the shift, \\((Sx)_n=x_{n+1}\\). Show that there is a linear \\(L:\\ell^\\infty_{\\mathbb R}\\to\\mathbb R\\) with \\(\\liminf_nx_n\\le L(x)\\le\\limsup_nx_n\\) and \\(L(Sx)=L(x)\\) for all \\(x\\).\n\n*Solution.* Put \\(p(x)=\\limsup_n\\frac1n\\sum_{k=1}^nx_k\\).\n- *\\(p\\) is sublinear:* the averages are linear in \\(x\\), and \\(\\limsup\\) is subadditive and positively homogeneous.\n- Theorem 2.1 extends \\(0\\) on the subspace \\(\\{0\\}\\) to a linear \\(L\\le p\\).\n- *Bounds:* \\(p(x)\\le\\limsup_nx_n\\), because averages of a sequence eventually below \\(c+\\varepsilon\\) are eventually below \\(c+2\\varepsilon\\). Applying this to \\(-x\\) gives \\(L(x)=-L(-x)\\ge-p(-x)\\ge\\liminf_nx_n\\).\n- *Shift invariance:* \\(\\frac1n\\sum_{k\\le n}(Sx-x)_k=\\frac{x_{n+1}-x_1}n\\to0\\). So \\(p(Sx-x)=0\\) and \\(p(x-Sx)=0\\). Hence \\(L(Sx-x)\\le0\\) and \\(L(x-Sx)\\le0\\), that is, \\(L(Sx)=L(x)\\). \\(\\square\\)\n\n**Exercise 3** (easy; closed convex sets are intersections of half-spaces). Let \\(E\\) be a normed space and \\(C\\subseteq E\\) closed and convex. Show that \\(C\\) is the intersection of the sets \\(\\{x:\\operatorname{Re}\\varphi(x)\\le t\\}\\), over all \\(\\varphi\\in E^*\\) and \\(t\\in\\mathbb R\\) with \\(C\\subseteq\\{\\operatorname{Re}\\varphi\\le t\\}\\).\n\n*Solution.* The intersection contains \\(C\\). If \\(x\\notin C\\) and \\(C\\) is nonempty, Corollary 6.4(2) gives \\(\\varphi\\) and \\(t\\) with \\(\\operatorname{Re}\\varphi\\le t\\) on \\(C\\) and \\(\\operatorname{Re}\\varphi(x)>t\\). So \\(x\\) is not in the intersection. If \\(C=\\varnothing\\), use \\(\\varphi=0\\) and \\(t=-1\\). \\(\\square\\)\n\n**Exercise 4** (medium; weakly continuous implies bounded). Let \\(T:E\\to F\\) be linear between Banach spaces, with \\(\\psi\\circ T\\in E^*\\) for every \\(\\psi\\in F^*\\). Show that \\(T\\) is bounded.\n\n*Solution.* By Theorem 5.3 it suffices to show that the graph is closed. Let \\(x_n\\to x\\) and \\(Tx_n\\to y\\). For \\(\\psi\\in F^*\\), \\(\\psi(Tx_n)\\to\\psi(Tx)\\) because \\(\\psi\\circ T\\) is continuous, and \\(\\psi(Tx_n)\\to\\psi(y)\\). So \\(\\psi(Tx)=\\psi(y)\\) for all \\(\\psi\\), and \\(Tx=y\\) by Corollary 2.3(2). \\(\\square\\)\n\n**Exercise 5** (medium; quotients). Let \\(E\\) be a Banach space and \\(M\\) a closed subspace. Show that \\(\\|x+M\\|=\\operatorname{dist}(x,M)\\) is a complete norm on \\(E/M\\). Deduce that a bounded linear surjection \\(T:E\\to F\\) onto a normed space \\(F\\) that is open forces \\(F\\) to be complete.\n\n*Solution.*\n- *A norm:* translating a representative by a member of \\(M\\) does not change the infimum. Nonzero-scalar representatives are scalar multiples of the old representatives, giving homogeneity. Taking the two independent infima in \\[\n\\begin{gathered}\n\\|(x+m)+(y+n)\\|\\\\\n\\leq\\|x+m\\|+\\|y+n\\|\n\\end{gathered}\n\\] gives the triangle inequality. Finally, \\(\\|x+M\\|=0\\) means \\(x\\in\\overline M=M\\).\n- *Completeness:* a normed space in which every absolutely convergent series converges is complete. From a Cauchy sequence choose a subsequence with successive distances at most \\(2^{-k}\\); the series of its differences converges, hence the subsequence converges by telescoping and the original Cauchy sequence has the same limit. Let \\(\\sum\\|x_n+M\\|<\\infty\\). Choose representatives with \\(\\|x_n\\|\\le\\|x_n+M\\|+2^{-n}\\). Then \\(\\sum x_n\\) converges in \\(E\\) to some \\(x\\), and \\[\n\\begin{gathered}\n\\|\\sum_{n\\le N}x_n+M-(x+M)\\|\\\\\n\\le\\|\\sum_{n>N}x_n\\|\\to0.\n\\end{gathered}\n\\]\n- *The deduction:* if \\(T\\) is open, then \\(T(B_E(0,1))\\supseteq B_F(0,\\delta)\\), so the induced bijection \\(\\tilde T:E/\\ker T\\to F\\) has \\(\\|\\tilde T^{-1}y\\|\\le\\delta^{-1}\\|y\\|\\). Hence \\(\\tilde T\\) is an isomorphism of normed spaces, and \\(F\\) is complete because \\(E/\\ker T\\) is. \\(\\square\\)\n\n**Exercise 6** (easy; an indiscrete example). Give a vector space with a vector topology in which a one-dimensional subspace is not closed, and explain which step of Theorem 7.1 fails.\n\n*Solution.* Give \\(\\mathbb K^2\\) the indiscrete topology, whose only open sets are the empty set and the whole space. Addition and scalar multiplication into this space are continuous because every map into an indiscrete space is continuous. Every proper nonzero one-dimensional subspace is dense and is not closed: the only closed sets are the empty set and the whole space. For a linear bijection \\(f:\\mathbb K^2\\to X\\), the image \\(f(S)\\) of the Euclidean unit sphere is compact, but it is a nonempty proper subset and hence is not closed. The only neighbourhood of zero is \\(X\\), so none can avoid \\(f(S)\\). This is exactly the compact-implies-closed and avoiding-neighbourhood step of Theorem 7.1 that needs Hausdorffness. \\(\\square\\)\n\n## Where this leads\n\n- Weak and weak\\* topologies, Banach–Alaoglu, Krein–Milman and Eberlein–Šmulian: [Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md).\n- Hilbert spaces and compact operators: [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md).\n- Banach algebras: [Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md).\n\n## References\n\nThe results are classical: Hahn (1927) and Banach (1929) for the extension theorem, Baire (1899), Banach and Steinhaus (1927), Banach and Schauder for the open mapping theorem, and Zorn (1935) for the maximality principle. Readers who want a single textbook account can use the core course Functional Analysis (J. M. Erdman, *Functional Analysis and Operator Algebras: An Introduction*, CC BY-SA 4.0), which states these theorems with hints for their proofs.\n\n*Freely accessible reading:* [J. van Neerven, *Functional Analysis*, §§4.2 and 5.1–5.3](https://arxiv.org/pdf/2112.11166v7) gives a route through Hahn–Banach, separation, Baire and the basic Banach-space theorems. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WT-01",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "1. Weak topologies",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
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      "anchor": "oa-fnd-wt-01",
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      },
      "full_conditions_and_proof": "## 1. Weak topologies\n\nLet \\(X\\) be a vector space and \\(Y\\) a vector space of linear functionals on \\(X\\). The *weak topology* \\(\\sigma(X,Y)\\) is the locally convex topology given by the seminorms \\(x\\mapsto|y(x)|\\), \\(y\\in Y\\). It is the coarsest topology making every \\(y\\in Y\\) continuous. It is Hausdorff exactly when \\(Y\\) separates the points of \\(X\\).\n\nFor a normed space \\(E\\):\n- the *weak topology* on \\(E\\) is \\(\\sigma(E,E^*)\\);\n- the *weak\\* topology* on \\(E^*\\) is \\(\\sigma(E^*,j(E))\\), where \\(j(x)(\\varphi)=\\varphi(x)\\).\n\nA net \\(x_i\\to x\\) weakly when \\(\\varphi(x_i)\\to\\varphi(x)\\) for every \\(\\varphi\\in E^*\\), and \\(\\varphi_i\\to\\varphi\\) weak\\* when \\(\\varphi_i(x)\\to\\varphi(x)\\) for every \\(x\\in E\\).\n\n**Lemma 1.1.** Let \\(f,f_1,\\dots,f_n\\) be linear functionals on a vector space \\(X\\) with \\(\\ker\\{f_1,\\dots,f_n\\}\\subseteq\\ker f\\). Then \\(f\\) is a linear combination of \\(f_1,\\dots,f_n\\).\n\n**Proof.** The map \\(\\pi:X\\to\\mathbb K^n\\), \\(x\\mapsto(f_1(x),\\dots,f_n(x))\\), has kernel inside \\(\\ker f\\). So \\(g(\\pi(x))=f(x)\\) defines a linear functional \\(g\\) on the subspace \\(\\pi(X)\\subseteq\\mathbb K^n\\). Extend \\(g\\) linearly to \\(\\mathbb K^n\\); then \\(g(t)=\\sum_ic_it_i\\), and \\(f=\\sum_ic_if_i\\). \\(\\square\\)\n\n**Theorem 1.2.** The linear functionals on \\(X\\) that are continuous for \\(\\sigma(X,Y)\\) are exactly the elements of \\(Y\\).\n\n**Proof.** Elements of \\(Y\\) are continuous by definition. Let \\(f\\) be continuous. Then \\(\\{x:|f(x)|<1\\}\\) contains a basic neighbourhood \\(\\{x:|y_i(x)|<\\varepsilon,\\ i\\le n\\}\\) with \\(y_i\\in Y\\). If \\(y_i(x)=0\\) for all \\(i\\), then \\(tx\\) lies in this neighbourhood for every scalar \\(t\\), so \\(|f(tx)|<1\\) for all \\(t\\), and \\(f(x)=0\\). By Lemma 1.1, \\(f\\) is a linear combination of the \\(y_i\\), so \\(f\\in Y\\). \\(\\square\\)\n\n**Corollary 1.3.** The weak topology of a normed space has continuous dual \\(E^*\\). The weak\\* topology of \\(E^*\\) has continuous dual \\(j(E)\\). Both topologies are Hausdorff: \\(E^*\\) separates points of \\(E\\) by the previous lesson, Corollary 2.3, and \\(j(E)\\) separates points of \\(E^*\\) trivially.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-WT-02",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "2. Tychonoff's theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
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      "anchor": "oa-fnd-wt-02",
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        "line": 40,
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      },
      "full_conditions_and_proof": "## 2. Tychonoff's theorem\n\nA *filter* on a set \\(S\\) is a nonempty family \\(\\mathcal F\\) of subsets with three properties:\n- \\(\\varnothing\\notin\\mathcal F\\);\n- \\(A\\cap B\\in\\mathcal F\\) whenever \\(A,B\\in\\mathcal F\\);\n- \\(B\\in\\mathcal F\\) whenever \\(B\\supseteq A\\in\\mathcal F\\).\n\nAn *ultrafilter* is a filter not properly contained in another. A filter on a topological space *converges* to \\(x\\) if it contains every neighbourhood of \\(x\\).\n\n**Lemma 2.1.**\n1. Every filter is contained in an ultrafilter.\n2. A filter \\(\\mathcal U\\) is an ultrafilter iff for every \\(A\\subseteq S\\), either \\(A\\in\\mathcal U\\) or \\(S\\setminus A\\in\\mathcal U\\).\n3. If \\(f:S\\to T\\) and \\(\\mathcal U\\) is an ultrafilter on \\(S\\), then \\(f_*\\mathcal U=\\{B\\subseteq T:f^{-1}(B)\\in\\mathcal U\\}\\) is an ultrafilter on \\(T\\).\n\n**Proof.** 1. Filters containing the given one, ordered by inclusion, have unions of chains as upper bounds. Zorn's lemma (previous lesson, Theorem 1.1) gives a maximal one.\n\n2. Suppose \\(\\mathcal U\\) is an ultrafilter and \\(A\\notin\\mathcal U\\). If \\(A\\cap U\\neq\\varnothing\\) for all \\(U\\in\\mathcal U\\), then the sets containing some \\(A\\cap U\\) form a filter that contains \\(\\mathcal U\\) and \\(A\\), contradicting maximality. So \\(A\\cap U=\\varnothing\\) for some \\(U\\in\\mathcal U\\), and then \\(S\\setminus A\\supseteq U\\) lies in \\(\\mathcal U\\). Conversely, a filter with this property is maximal: a larger filter would contain some \\(A\\notin\\mathcal U\\) together with \\(S\\setminus A\\in\\mathcal U\\), hence \\(\\varnothing\\).\n\n3. \\(f_*\\mathcal U\\) is a filter, and 2 applies because \\(f^{-1}(T\\setminus B)=S\\setminus f^{-1}(B)\\). \\(\\square\\)\n\n**Lemma 2.2.** A topological space \\(K\\) is compact iff every ultrafilter on \\(K\\) converges.\n\n**Proof.** *Compact implies convergence.* Suppose an ultrafilter \\(\\mathcal U\\) converges to no point. Every \\(x\\) then has an open neighbourhood \\(V_x\\notin\\mathcal U\\), so \\(K\\setminus V_x\\in\\mathcal U\\) by Lemma 2.1(2). Finitely many \\(V_{x_1},\\dots,V_{x_n}\\) cover \\(K\\). Then \\(\\bigcap_j(K\\setminus V_{x_j})=\\varnothing\\) lies in \\(\\mathcal U\\), which is impossible.\n\n*Convergence implies compact.* Let \\(\\mathcal O\\) be an open cover with no finite subcover. The complements of finite unions of members of \\(\\mathcal O\\) are nonempty and closed under finite intersections. So they generate a filter, contained in an ultrafilter \\(\\mathcal U\\) (Lemma 2.1(1)). Let \\(\\mathcal U\\) converge to \\(x\\), and pick \\(O\\in\\mathcal O\\) with \\(x\\in O\\). Then \\(O\\in\\mathcal U\\) and \\(K\\setminus O\\in\\mathcal U\\), so \\(\\varnothing\\in\\mathcal U\\), a contradiction. \\(\\square\\)\n\n**Theorem 2.3** (Tychonoff). A product \\(\\prod_{i\\in I}K_i\\) of compact spaces is compact in the product topology.\n\n**Proof.** Let \\(\\mathcal U\\) be an ultrafilter on the product and \\(\\pi_i\\) the projections. By Lemma 2.1(3) each \\((\\pi_i)_*\\mathcal U\\) is an ultrafilter on \\(K_i\\). By Lemma 2.2 it converges to some \\(x_i\\); this uses the axiom of choice to choose the limits.\n\nThe point \\(x=(x_i)\\) is a limit of \\(\\mathcal U\\). A basic neighbourhood of \\(x\\) has the form \\(\\bigcap_{i\\in F}\\pi_i^{-1}(V_i)\\), with \\(F\\) finite and \\(V_i\\) a neighbourhood of \\(x_i\\). Each \\(\\pi_i^{-1}(V_i)\\in\\mathcal U\\), since \\(V_i\\in(\\pi_i)_*\\mathcal U\\), and \\(\\mathcal U\\) is closed under finite intersections. By Lemma 2.2 the product is compact. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-WT-03",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "3. The Banach–Alaoglu theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
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        "line": 72,
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      "full_conditions_and_proof": "## 3. The Banach–Alaoglu theorem\n\n**Theorem 3.1** (Banach–Alaoglu). For every normed space \\(E\\), the closed unit ball \\(B^*\\) of \\(E^*\\) is weak\\* compact.\n\n**Proof.** For \\(x\\in E\\) let \\(D_x=\\{\\lambda\\in\\mathbb K:|\\lambda|\\le\\|x\\|\\}\\), and \\(P=\\prod_{x\\in E}D_x\\), compact by Theorem 2.3. The map \\(\\Phi:B^*\\to P\\), \\(\\varphi\\mapsto(\\varphi(x))_x\\), is injective.\n\n\\(\\Phi\\) is a homeomorphism onto its image, with \\(B^*\\) carrying the weak\\* topology: both topologies are the topology of pointwise convergence.\n\nThe image is closed in \\(P\\). It is the set of points \\((\\lambda_x)\\) satisfying \\(\\lambda_{x+y}=\\lambda_x+\\lambda_y\\) and \\(\\lambda_{tx}=t\\lambda_x\\) for all \\(x,y,t\\), and each of these conditions defines a closed set: such a point is a linear functional bounded by \\(\\|x\\|\\) at \\(x\\), hence an element of \\(B^*\\).\n\nA closed subset of a compact space is compact. \\(\\square\\)\n\n**Proposition 3.2.** If \\(E\\) is separable, the weak\\* topology on \\(B^*\\) is metrizable.\n\n**Proof.** Let \\((x_n)\\) be dense in the unit ball of \\(E\\). Put \\(d(\\varphi,\\psi)=\\sum_n2^{-n}|\\varphi(x_n)-\\psi(x_n)|\\) on \\(B^*\\).\n- \\(d\\) is a metric: if \\(d(\\varphi,\\psi)=0\\), then \\(\\varphi=\\psi\\) on a dense subset of the ball, so \\(\\varphi=\\psi\\).\n- The identity map from \\((B^*,\\text{weak}^*)\\) to \\((B^*,d)\\) is continuous: each term is continuous, and the series converges uniformly because \\(|\\varphi(x_n)-\\psi(x_n)|\\le2\\).\n- A continuous bijection from a compact space onto a Hausdorff space is a homeomorphism. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WT-04",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "4. Mazur's theorem",
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      "anchor": "oa-fnd-wt-04",
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        "line": 91,
        "through_line": 104
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      "full_conditions_and_proof": "## 4. Mazur's theorem\n\n**Theorem 4.1** (Mazur). A convex subset \\(C\\) of a normed space has the same closure in the weak and in the norm topology. In particular, norm-closed convex sets are weakly closed. If \\(x_n\\to x\\) weakly, then some sequence of convex combinations of the \\(x_n\\) converges to \\(x\\) in norm.\n\n**Proof.** The weak topology is coarser, so the norm closure \\(\\overline C\\) lies in the weak closure. Conversely, if \\(x\\notin\\overline C\\), the previous lesson, Corollary 6.4(2), gives \\(\\varphi\\in E^*\\) and \\(t\\) with \\(\\operatorname{Re}\\varphi\\le t\\) on \\(\\overline C\\) and \\(\\operatorname{Re}\\varphi(x)>t\\). The weakly open set \\(\\{\\operatorname{Re}\\varphi>t\\}\\) contains \\(x\\) and misses \\(C\\), so \\(x\\) is not in the weak closure.\n\nFor the last statement, apply this to the convex hull of \\(\\{x_n:n\\ge1\\}\\). The point \\(x\\) lies in its weak closure, hence in its norm closure. \\(\\square\\)\n\nThe same argument works in every locally convex space.\n\n**Theorem 4.2** (Closures of convex sets). Let \\((X,\\tau)\\) be a locally convex space with continuous dual \\(X^*\\). A convex set \\(C\\subseteq X\\) has the same closure for \\(\\tau\\) and for \\(\\sigma(X,X^*)\\). Consequently, two locally convex topologies on \\(X\\) with the same continuous linear functionals have the same closed convex sets.\n\n**Proof.** Every \\(\\varphi\\in X^*\\) is \\(\\tau\\)-continuous, so \\(\\sigma(X,X^*)\\subseteq\\tau\\), and the \\(\\tau\\)-closure \\(\\overline C\\) lies in the \\(\\sigma(X,X^*)\\)-closure. Let \\(x\\notin\\overline C\\). The set \\(\\overline C\\) is closed and convex. The previous lesson, Corollary 6.4(2), gives \\(\\varphi\\in X^*\\) and \\(t\\) with \\(\\operatorname{Re}\\varphi\\le t\\) on \\(\\overline C\\) and \\(\\operatorname{Re}\\varphi(x)>t\\). The \\(\\sigma(X,X^*)\\)-open set \\(\\{\\operatorname{Re}\\varphi>t\\}\\) contains \\(x\\) and misses \\(C\\). For the last claim, both topologies give the closure of \\(C\\) that \\(\\sigma(X,X^*)\\) gives. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces"
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    },
    {
      "id": "OA-FND-WT-05",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "5. Annihilators, bipolars and Goldstine's theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
      "source_sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "anchor": "oa-fnd-wt-05",
      "proof_locus": {
        "line": 105,
        "through_line": 162
      },
      "full_conditions_and_proof": "## 5. Annihilators, bipolars and Goldstine's theorem\n\nFor \\(L\\subseteq E\\) let \\(L^\\perp=\\{\\varphi\\in E^*:\\varphi|_L=0\\}\\). For \\(N\\subseteq E^*\\) let \\(N_\\perp=\\{x\\in E:\\varphi(x)=0\\ \\forall\\varphi\\in N\\}\\).\n\n**Theorem 5.1.** Let \\(E\\) be a normed space.\n1. For a subspace \\(L\\subseteq E\\), the norm closure of \\(L\\) is \\((L^\\perp)_\\perp\\).\n2. For a subspace \\(N\\subseteq E^*\\), the weak\\* closure of \\(N\\) is \\((N_\\perp)^\\perp\\).\n\n**Proof.** 1. The set \\((L^\\perp)_\\perp\\) is norm closed and contains \\(L\\). If \\(x\\notin\\overline L\\), the previous lesson, Corollary 2.3(3), gives \\(\\varphi\\in L^\\perp\\) with \\(\\varphi(x)\\ne0\\), so \\(x\\notin(L^\\perp)_\\perp\\).\n\n2. The set \\((N_\\perp)^\\perp\\) is weak\\* closed and contains \\(N\\). Let \\(\\psi\\) lie outside the weak\\* closure \\(\\overline N\\), a weak\\* closed subspace. The weak\\* topology is locally convex. By the previous lesson, Corollary 6.4(2), there is a weak\\* continuous functional \\(F\\) with \\(\\operatorname{Re}F(\\psi)>t\\ge\\operatorname{Re}F(\\overline N)\\) for some \\(t\\).\n\nSince \\(\\overline N\\) is a subspace, \\(F\\) vanishes on it: if \\(F(\\varphi)\\neq0\\) for some \\(\\varphi\\in\\overline N\\), suitable multiples \\(s\\varphi\\) would make \\(\\operatorname{Re}F(s\\varphi)\\) as large as we like. So \\(t\\ge0\\) and \\(\\operatorname{Re}F(\\psi)>0\\).\n\nBy Corollary 1.3, \\(F=j(x)\\) for some \\(x\\in E\\). Then \\(x\\in N_\\perp\\) and \\(\\psi(x)\\ne0\\), so \\(\\psi\\notin(N_\\perp)^\\perp\\). \\(\\square\\)\n\n**Theorem 5.2** (Goldstine). The image \\(j(B)\\) of the closed unit ball \\(B\\) of \\(E\\) is weak\\* dense in the closed unit ball of \\(E^{**}\\).\n\n**Proof.** Let \\(K\\) be the weak\\* closure of \\(j(B)\\). It is convex and lies in the unit ball of \\(E^{**}\\), which is weak\\* closed.\n\nSuppose \\(\\Phi\\) is in the unit ball of \\(E^{**}\\) but not in \\(K\\). The space \\((E^{**},\\sigma(E^{**},E^*))\\) is locally convex with continuous dual \\(E^*\\) (Theorem 1.2). The previous lesson, Corollary 6.4(2), gives \\(\\varphi\\in E^*\\) and \\(t\\) with \\(\\operatorname{Re}\\Phi(\\varphi)>t\\ge\\operatorname{Re}j(x)(\\varphi)=\\operatorname{Re}\\varphi(x)\\) for all \\(x\\in B\\).\n\nTaking the supremum over \\(B\\), and using that \\(B\\) is balanced, gives \\(\\|\\varphi\\|\\le t\\). Then \\(\\operatorname{Re}\\Phi(\\varphi)>t\\ge\\|\\varphi\\|\\ge\\|\\Phi\\|\\,\\|\\varphi\\|\\), which is impossible. \\(\\square\\)\n\n**Proposition 5.3** (second adjoints). For \\(T\\in B(E,F)\\) let \\(T^*\\in B(F^*,E^*)\\), \\(T^*\\psi=\\psi\\circ T\\), and \\(T^{**}=(T^*)^*\\).\n1. \\(\\|T^*\\|=\\|T\\|\\) and \\(\\|T^{**}\\|=\\|T\\|\\).\n2. \\(T^{**}\\circ j_E=j_F\\circ T\\).\n3. \\((ST)^{**}=S^{**}T^{**}\\).\n4. \\(T^{**}\\) is weak\\*–weak\\* continuous.\n\n**Proof.** 1. \\(\\|T^*\\psi\\|=\\sup_{\\|x\\|\\le1}|\\psi(Tx)|\\le\\|\\psi\\|\\|T\\|\\). Conversely, for \\(\\|x\\|\\le1\\) choose \\(\\psi\\) of norm at most \\(1\\) with \\(\\psi(Tx)=\\|Tx\\|\\) (previous lesson, Corollary 2.3(2)); then \\(\\|Tx\\|\\le\\|T^*\\|\\). Apply the same to \\(T^*\\).\n\n2. \\[\n\\begin{gathered}\nT^{**}(j_Ex)(\\psi)\\\\\n=j_Ex(T^*\\psi)\\\\\n=\\psi(Tx)\\\\\n=j_F(Tx)(\\psi).\n\\end{gathered}\n\\]\n\n3. \\((ST)^*=T^*S^*\\), so \\((ST)^{**}=S^{**}T^{**}\\).\n\n4. If \\(\\Phi_i\\to\\Phi\\) weak\\*, then \\[\n\\begin{gathered}\nT^{**}\\Phi_i(\\psi)\\\\\n=\\Phi_i(T^*\\psi)\\to\\Phi(T^*\\psi)\\\\\n=T^{**}\\Phi(\\psi)\n\\end{gathered}\n\\] for every \\(\\psi\\in F^*\\). \\(\\square\\)\n\n**Proposition 5.4** (The dual of a quotient). Let \\(M\\) be a closed subspace of a normed space \\(E\\), with quotient map \\(q:E\\to E/M\\) and quotient norm \\(\\|x+M\\|=\\operatorname{dist}(x,M)\\). Then \\(g\\mapsto g\\circ q\\) is an isometric isomorphism of \\((E/M)^*\\) onto \\(M^\\perp\\).\n\n**Proof.**\n- \\(g\\circ q\\) is bounded and vanishes on \\(M\\).\n- *Isometry.* \\(q\\) maps the open unit ball of \\(E\\) onto the open unit ball of \\(E/M\\): if \\(\\|x+M\\|<1\\), some \\(m\\in M\\) has \\(\\|x-m\\|<1\\). Hence \\(\\|g\\circ q\\|=\\|g\\|\\).\n- *Onto \\(M^\\perp\\).* Let \\(\\varphi\\in M^\\perp\\). Then \\(g(x+M)=\\varphi(x)\\) is well defined and linear. For every \\(m\\in M\\), \\(|g(x+M)|=|\\varphi(x-m)|\\le\\|\\varphi\\|\\,\\|x-m\\|\\). Taking the infimum over \\(m\\) gives \\(|g(x+M)|\\le\\|\\varphi\\|\\,\\|x+M\\|\\), so \\(g\\in(E/M)^*\\) and \\(g\\circ q=\\varphi\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces"
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    },
    {
      "id": "OA-FND-WT-06",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "6. The Krein–Milman theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
      "source_sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "anchor": "oa-fnd-wt-06",
      "proof_locus": {
        "line": 163,
        "through_line": 194
      },
      "full_conditions_and_proof": "## 6. The Krein–Milman theorem\n\nA point \\(e\\) of a convex set \\(K\\) is *extreme* if \\(e=tx+(1-t)y\\) with \\(x,y\\in K\\) and \\(0<t<1\\) forces \\(x=y=e\\). A nonempty closed subset \\(F\\subseteq K\\) is a *face* (an *extreme subset*) if whenever \\(tx+(1-t)y\\in F\\) with \\(x,y\\in K\\) and \\(0<t<1\\), both \\(x,y\\in F\\).\n\n**Theorem 6.1** (Krein–Milman). Let \\(X\\) be a Hausdorff locally convex space and \\(K\\subseteq X\\) a nonempty compact convex set. Then \\(K\\) has an extreme point, and \\(K\\) is the closed convex hull of its extreme points.\n\n**Proof.** *Faces contain extreme points.* Faces of \\(K\\), ordered by reverse inclusion, satisfy the hypothesis of Zorn's lemma. The intersection of a chain is nonempty, by compactness and the finite intersection property, and it is closed and a face. So there is a minimal face \\(F\\).\n\nSuppose \\(F\\) has two points \\(x\\ne y\\). Choose a continuous \\(\\varphi\\) with \\(\\operatorname{Re}\\varphi(x)\\ne\\operatorname{Re}\\varphi(y)\\); this exists by the previous lesson, Corollary 6.4(1), replacing \\(\\varphi\\) by \\(i\\varphi\\) if needed. The set \\(F'=\\{z\\in F:\\operatorname{Re}\\varphi(z)=\\max_F\\operatorname{Re}\\varphi\\}\\) is then:\n- nonempty, because \\(F\\) is compact;\n- closed, and a face of \\(F\\), hence of \\(K\\): if \\(tx'+(1-t)y'\\in F'\\) with \\(x',y'\\in K\\), then \\(x',y'\\in F\\) because \\(F\\) is a face, and both attain the maximum, since a strict average of two values at most the maximum equals the maximum only if both do;\n- smaller than \\(F\\), since it cannot contain both \\(x\\) and \\(y\\).\n\nThis contradicts minimality. So \\(F=\\{e\\}\\), and \\(e\\) is an extreme point of \\(K\\), because \\(\\{e\\}\\) is a face.\n\n*The closed convex hull.* Let \\(C\\) be the closed convex hull of the extreme points; it lies in \\(K\\). If some \\(x\\in K\\setminus C\\) existed, the previous lesson, Theorem 6.3, would give a continuous \\(\\varphi\\) and \\(t\\) with \\(\\operatorname{Re}\\varphi<t\\) on \\(C\\) and \\(\\operatorname{Re}\\varphi(x)>t\\).\n\nThe set \\(F=\\{z\\in K:\\operatorname{Re}\\varphi(z)=\\max_K\\operatorname{Re}\\varphi\\}\\) is a face of \\(K\\) disjoint from \\(C\\), because its points have \\(\\operatorname{Re}\\varphi\\ge\\operatorname{Re}\\varphi(x)>t\\). By the first part, applied to the compact convex set \\(F\\), it contains an extreme point of \\(F\\), which is an extreme point of \\(K\\) because \\(F\\) is a face, and so lies in \\(C\\). This is a contradiction. \\(\\square\\)\n\n**Theorem 6.2** (Milman). Let \\(X\\) be a Hausdorff locally convex space, \\(Q\\subseteq X\\) compact, and suppose that the closed convex hull \\(K\\) of \\(Q\\) is compact. Then every extreme point of \\(K\\) lies in \\(Q\\).\n\n**Proof.** Let \\(e\\in K\\) be extreme, and suppose \\(e\\notin Q\\).\n\n*A neighbourhood that keeps \\(e\\) away from \\(Q\\).* For each \\(q\\in Q\\), \\(e-q\\neq0\\). Since \\(X\\) is Hausdorff and locally convex, there is a convex balanced open neighbourhood \\(U_q\\) of \\(0\\) with \\(e-q\\notin U_q+U_q\\). Finitely many sets \\(q_j+U_{q_j}\\) cover \\(Q\\). Put \\(W=\\bigcap_jU_{q_j}\\). Then \\(e\\notin Q+W\\): if \\(e=q+w\\) with \\(q\\in q_j+U_{q_j}\\) and \\(w\\in W\\), then \\(e-q_j\\in U_{q_j}+U_{q_j}\\).\n- Choose a convex balanced open neighbourhood \\(W'\\) of \\(0\\) with \\(W'+W'\\subseteq W\\), and let \\(V\\) be the closure of \\(W'\\). Then \\(V\\) is closed and convex, and \\(V\\subseteq W'+W'\\subseteq W\\), because every point of the closure of \\(W'\\) lies in \\(y+W'\\) for some \\(y\\in W'\\). So \\(e\\notin Q+V\\).\n\n*Splitting \\(K\\).* Finitely many sets \\(p_k+W'\\), \\(p_k\\in Q\\), cover \\(Q\\). Let \\(K_k\\) be the closed convex hull of \\(Q\\cap(p_k+V)\\).\n- \\(K_k\\subseteq p_k+V\\), because \\(p_k+V\\) is closed and convex. And \\(K_k\\subseteq K\\) is closed, hence compact.\n- The convex hull \\(K'\\) of \\(K_1\\cup\\dots\\cup K_n\\) is the image of the compact set \\(K_1\\times\\dots\\times K_n\\times\\Delta_n\\), where \\(\\Delta_n\\) is the simplex of weights, under the continuous map \\((x_1,\\dots,x_n,t)\\mapsto\\sum_kt_kx_k\\). So \\(K'\\) is compact, hence closed. It is convex and contains \\(Q\\), so it contains \\(K\\).\n\n*Conclusion.* So \\(e=\\sum_kt_kx_k\\) with \\(x_k\\in K_k\\) and weights \\(t_k\\). Since \\(e\\) is extreme in \\(K\\), \\(e=x_k\\) for every \\(k\\) with \\(t_k>0\\): if \\(0<t_1<1\\), write \\(e=t_1x_1+(1-t_1)y\\) with \\(y\\in K\\), so \\(x_1=y=e\\), and continue with \\(y\\). Hence \\(e\\in K_k\\subseteq p_k+V\\subseteq Q+V\\) for some \\(k\\), a contradiction. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces"
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    },
    {
      "id": "OA-FND-WT-07",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "name": "7. The Eberlein–Šmulian theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
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      "anchor": "oa-fnd-wt-07",
      "proof_locus": {
        "line": 195,
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      "full_conditions_and_proof": "## 7. The Eberlein–Šmulian theorem\n\n**Lemma 7.1.** Let \\(E\\) be a normed space and \\(M\\subseteq E^{**}\\) a finite-dimensional subspace. Then there are finitely many \\(\\varphi_1,\\dots,\\varphi_m\\in E^*\\) of norm \\(1\\) with \\(\\max\\{0,|\\Phi(\\varphi_1)|,\\dots,|\\Phi(\\varphi_m)|\\}\\ge\\frac12\\|\\Phi\\|\\) for every \\(\\Phi\\in M\\). The family may be empty when \\(M=\\{0\\}\\).\n\n**Proof.** If \\(M=\\{0\\}\\), the empty family works. Otherwise the unit sphere \\(S_M\\) of \\(M\\) is nonempty and compact (previous lesson, Theorem 7.1). For each \\(\\Phi\\in S_M\\), choose \\(\\varphi_\\Phi\\) of norm \\(1\\) with \\(|\\Phi(\\varphi_\\Phi)|>3/4\\). The open sets \\(\\{\\Psi\\in S_M:\\|\\Psi-\\Phi\\|<1/4\\}\\) cover \\(S_M\\), so finitely many centres \\(\\Phi_1,\\dots,\\Phi_m\\) suffice. For \\(\\Psi\\in S_M\\) with \\(\\|\\Psi-\\Phi_k\\|<1/4\\), \\(|\\Psi(\\varphi_{\\Phi_k})|\\ge|\\Phi_k(\\varphi_{\\Phi_k})|-\\|\\Psi-\\Phi_k\\|>1/2\\). Scale for general \\(\\Phi\\in M\\). \\(\\square\\)\n\n**Theorem 7.2** (Eberlein–Šmulian). For a subset \\(A\\) of a Banach space \\(E\\), the following are equivalent:\n1. \\(A\\) is relatively weakly compact;\n2. every sequence in \\(A\\) has a subsequence that converges weakly to a point of \\(E\\);\n3. every sequence in \\(A\\) has a weak cluster point in \\(E\\).\n\n**Proof.** If \\(E=\\{0\\}\\) or \\(A=\\varnothing\\), all three assertions are immediate. Suppose otherwise. *2 implies 3.* The limit of a weakly convergent subsequence is a weak cluster point.\n\n*1 implies 3.* A sequence in the compact set \\(\\overline A^{\\,w}\\), the weak closure of \\(A\\), has a cluster point there. The terms \\(\\{x_n:n\\ge m\\}\\) form a decreasing family of sets with the finite intersection property, so their closures meet.\n\n*3 implies 1.*\n\n**Step 1: \\(A\\) is bounded.** Otherwise there is \\(\\varphi\\in E^*\\) with \\(\\sup_{x\\in A}|\\varphi(x)|=\\infty\\), by the previous lesson, Corollary 4.3(2). So there are \\(x_n\\in A\\) with \\(|\\varphi(x_n)|\\ge n\\). A weak cluster point \\(x\\) of \\((x_n)\\) would make \\(\\varphi(x)\\) a cluster point of \\((\\varphi(x_n))\\), which has none.\n\n**Step 2: the weak\\* closure of \\(j(A)\\) lies in \\(j(E)\\).** The weak\\* closure \\(\\overline{j(A)}^{w*}\\) is weak\\* compact by Theorem 3.1, since \\(A\\) is bounded. Let \\(\\Phi\\) be in it. We build \\(x_n\\in A\\) and functionals of norm \\(1\\) inductively.\n- Let \\(\\varphi_1\\) be any functional of norm \\(1\\), and choose \\(x_1\\in A\\) with \\(|(\\Phi-jx_1)(\\varphi_1)|<1\\).\n- Given \\(x_1,\\dots,x_n\\), let \\(M_n=\\operatorname{span}\\{\\Phi,\\Phi-jx_1,\\dots,\\Phi-jx_n\\}\\). Lemma 7.1 gives finitely many functionals of norm \\(1\\) that norm \\(M_n\\) up to the factor \\(\\frac12\\). Append them to the list \\(\\varphi_1,\\varphi_2,\\dots\\).\n- Since \\(\\Phi\\) lies in the weak\\* closure of \\(j(A)\\), choose \\(x_{n+1}\\in A\\) with \\(|(\\Phi-jx_{n+1})(\\varphi_k)|<1/(n+1)\\) for all functionals \\(\\varphi_k\\) listed so far.\n\nLet \\(x\\) be a weak cluster point of \\((x_n)\\).\n\n**Step 3: \\(\\Phi=j(x)\\).**\n- *\\(\\Psi=\\Phi-jx\\) lies in \\(\\overline{\\bigcup_nM_n}\\).* The point \\(x\\) is in the weak closure of \\(\\{x_n\\}\\), hence in the norm-closed linear span of \\(\\{x_n\\}\\) by Theorem 4.1. So \\(jx\\) is a norm limit of combinations \\(\\sum c_kjx_k\\). Then \\(\\Psi\\) is a norm limit of \\(\\Phi-\\sum c_kjx_k\\). Each of these lies in \\(M_n\\) for large \\(n\\): write \\[\n\\begin{gathered}\n\\Phi-\\sum c_kjx_k\\\\\n=(1-\\sum c_k)\\Phi+\\sum c_k(\\Phi-jx_k).\n\\end{gathered}\n\\]\n- *\\(\\|\\Psi\\|\\le2\\sup_k|\\Psi(\\varphi_k)|\\).* For \\(G\\in M_n\\), \\(\\|G\\|\\le2\\max_k|G(\\varphi_k)|\\), with \\(k\\) running over the functionals chosen for \\(M_n\\). Approximating \\(\\Psi\\) by such \\(G\\) gives \\(\\|\\Psi\\|\\le2\\sup_k|\\Psi(\\varphi_k)|+3\\|\\Psi-G\\|\\), and the last term tends to \\(0\\).\n- *\\(\\Psi(\\varphi_k)=0\\) for every \\(k\\).* For \\(n\\) large, \\(|\\Phi(\\varphi_k)-\\varphi_k(x_n)|<1/n\\). Since \\(\\varphi_k(x)\\) is a cluster point of \\((\\varphi_k(x_n))\\), we get \\(\\Phi(\\varphi_k)=\\varphi_k(x)\\).\n\nHence \\(\\Psi=0\\), that is, \\(\\Phi=jx\\in j(E)\\).\n\n**Step 4: conclusion.** The map \\(j\\) is a homeomorphism from \\((E,\\text{weak})\\) onto \\((j(E),\\text{weak}^*)\\): both topologies are pointwise convergence on \\(E^*\\). By Step 2, \\(\\overline{j(A)}^{w*}=j(K)\\) for some \\(K\\subseteq E\\). This \\(K\\) is weakly compact and contains \\(A\\), so \\(A\\) is relatively weakly compact.\n\n*1 implies 2.* Let \\((x_n)\\) be a sequence in \\(A\\).\n- Let \\(E_0\\) be the norm-closed span of \\(\\{x_n\\}\\), a separable closed subspace. By Theorem 4.1 it is weakly closed. The weak closure \\(K\\) of \\(\\{x_n\\}\\) is weakly compact by 1 and lies in \\(E_0\\).\n- Let \\((y_m)\\) be dense in \\(E_0\\). Choose \\(\\psi_m\\in E^*\\) of norm \\(1\\) with \\(\\psi_m(y_m)=\\|y_m\\|\\). The family \\((\\psi_m)\\) separates the points of \\(E_0\\): if \\(z\\ne0\\) in \\(E_0\\), choose \\(y_m\\) with \\(\\|z-y_m\\|<\\|z\\|/3\\); then \\(|\\psi_m(z)|\\ge\\|y_m\\|-\\|z-y_m\\|>\\|z\\|/3\\).\n- By 1 implies 3 and Step 1, \\(A\\) is norm bounded. The weakly closed norm ball containing it also contains \\(K\\), so this series is uniformly convergent on \\(K\\times K\\). The topology on \\(K\\) induced by \\(d(u,v)=\\sum_m2^{-m}|\\psi_m(u-v)|\\) is Hausdorff and coarser than the weak topology. A compact topology admits no strictly coarser Hausdorff topology, so the weak topology on \\(K\\) is this metric topology.\n- In a compact metric space the closures of the sequence tails have a common point \\(x\\), by the finite intersection property. Every ball about \\(x\\) contains arbitrarily late terms. Choose successively increasing indices whose terms lie in the balls of radii \\(1/k\\); the subsequence converges in the metric. Thus \\((x_n)\\) has a subsequence converging weakly in \\(K\\subseteq E\\). \\(\\square\\)\n\n## Exercises\n\n**Exercise 1** (medium). Show that in an infinite-dimensional normed space the weak closure of the unit sphere \\(\\{\\|x\\|=1\\}\\) is the closed unit ball.\n\n*Solution.*\n- *The closure lies in the ball:* the closed unit ball is convex and norm closed, so weakly closed (Theorem 4.1).\n- *Every point with \\(\\|x_0\\|<1\\) is in the closure.* A basic weak neighbourhood \\(U=\\{x:|\\varphi_i(x-x_0)|<\\varepsilon,\\ i\\le n\\}\\) contains \\(x_0+\\ker\\{\\varphi_1,\\dots,\\varphi_n\\}\\). This kernel is a nonzero subspace, because a linear map from an infinite-dimensional space to \\(\\mathbb K^n\\) has nonzero kernel. Pick \\(y\\ne0\\) in it. The function \\(t\\mapsto\\|x_0+ty\\|\\) is continuous, equals \\(\\|x_0\\|<1\\) at \\(t=0\\), and tends to infinity. Choose \\(T>0\\) with \\(\\|x_0+Ty\\|>1\\). Bisect the interval \\([0,T]\\), retaining endpoint values that bracket one. Its nested endpoints converge to a common real number \\(t\\); continuity gives \\(\\|x_0+ty\\|=1\\). This produces a point of the sphere in \\(U\\).\n- *Every point with \\(\\|x_0\\|=1\\) is on the sphere itself.* \\(\\square\\)\n\n**Exercise 2** (medium). Let \\(E=c_0\\), the null sequences with the supremum norm. Show that \\(\\delta_n\\to0\\) weakly but not in norm, and find convex combinations of the \\(\\delta_n\\) that converge to \\(0\\) in norm.\n\n*Solution.*\n- *Weak convergence:* put \\(a_k=\\varphi(\\delta_k)\\) for \\(\\varphi\\in c_0^*\\). On a finite set \\(F\\), choose \\(\\theta_k=\\overline{a_k}/|a_k|\\) when \\(a_k\\ne0\\), and zero otherwise. Then \\(\\|\\sum_{k\\in F}\\theta_k\\delta_k\\|_\\infty\\leq1\\), so \\(\\sum_{k\\in F}|a_k|\\leq\\|\\varphi\\|\\). Hence \\(a\\in\\ell^1\\). Finite cutoffs approximate each \\(x\\in c_0\\) in supremum norm, giving \\(\\varphi(x)=\\sum_ka_kx_k\\). Conversely this series defines a bounded functional for every \\(a\\in\\ell^1\\), with norm at most \\(\\|a\\|_1\\); the same finite phase tests give the reverse inequality. Thus \\(c_0^*=\\ell^1\\) isometrically, and \\(\\varphi_a(\\delta_n)=a_n\\to0\\).\n- *No norm convergence:* \\(\\|\\delta_n\\|=1\\).\n- *Convex combinations:* \\(\\frac1N(\\delta_1+\\dots+\\delta_N)\\) has norm \\(1/N\\to0\\), as Theorem 4.1 predicts. \\(\\square\\)\n\n**Exercise 3** (medium). Show that the closed unit ball of \\(\\ell^1\\) has extreme points \\(\\{\\lambda\\delta_n:|\\lambda|=1\\}\\), but that the closed unit ball of \\(c_0\\) has none. Deduce that \\(c_0\\) is not isometrically isomorphic to the dual of any normed space.\n\n*Solution.*\n- *\\(\\ell^1\\):* if \\(\\|x\\|_1<1\\), choose \\(0<\\varepsilon<1-\\|x\\|_1\\); the distinct vectors \\(x\\pm\\varepsilon\\delta_1\\) belong to the ball and average to \\(x\\). If \\(\\|x\\|_1=1\\) and \\(x_i,x_j\\ne0\\) with \\(i\\ne j\\), put \\(u_i=x_i/|x_i|\\), \\(u_j=x_j/|x_j|\\), and choose \\(0<\\varepsilon<\\min(|x_i|,|x_j|)\\). The two vectors \\(x\\pm\\varepsilon(u_i\\delta_i-u_j\\delta_j)\\) both have norm one, since their two altered coordinate magnitudes add to \\(|x_i|+|x_j|\\). They are distinct and average to \\(x\\). The remaining points are \\(\\lambda\\delta_n\\), \\(|\\lambda|=1\\), and these are extreme: if \\(\\lambda\\delta_n=(x+y)/2\\) with \\(\\|x\\|_1,\\|y\\|_1\\leq1\\), multiply the nth coordinate by \\(\\bar\\lambda\\). The real parts of both resulting numbers are at most one and average to one, so both numbers equal one. Hence \\(x_n=y_n=\\lambda\\), and the norm bounds force every other coordinate of \\(x,y\\) to vanish.\n- *\\(c_0\\):* given \\(x\\) in the unit ball, \\(|x_k|<1/2\\) for some \\(k\\). Then \\(x\\pm\\frac12\\delta_k\\) are in the ball and average to \\(x\\).\n- *Deduction:* the unit ball of a dual space is weak\\* compact and convex (Theorem 3.1), so by Theorem 6.1 it has extreme points. A linear isometric isomorphism preserves convex combinations and therefore extreme points of balls. \\(\\square\\)\n\n**Exercise 4** (easy). Show that a reflexive Banach space (one with \\(j(E)=E^{**}\\)) has a weakly compact closed unit ball, and deduce that every bounded sequence in it has a weakly convergent subsequence.\n\n*Solution.*\n- *Weak compactness:* the ball of \\(E^{**}=j(E)\\) is weak\\* compact by Theorem 3.1, applied to \\(E^*\\). By Step 4 of the proof of Theorem 7.2, \\(j\\) is a homeomorphism from the weak topology to the weak\\* topology. So the unit ball of \\(E\\) is weakly compact.\n- *Sequences:* Theorem 7.2, 1 ⇒ 2, gives the weakly convergent subsequence. \\(\\square\\)\n\n## Where this leads\n\nThe lesson [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md) uses these results for operators on Hilbert space. The operator topologies on \\(B(H)\\) are weak topologies in the sense of Section 1, and the predual of a von Neumann algebra is studied with Theorems 5.1 and 7.2 in [Polar decomposition of functionals and weak compactness in preduals](polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md).\n\n## References\n\nThe results are classical: Tychonoff (1930, 1935), Banach (1932) and Alaoglu (1940), Mazur (1933), Goldstine (1938), Krein and Milman (1940), Eberlein (1947) and Šmulian (1940). The proof of the Eberlein–Šmulian theorem follows R. Whitley, \"An elementary proof of the Eberlein–Šmulian theorem\", *Mathematische Annalen* 172 (1967) 116–118, written here in our own words.\n\n*Freely accessible reading:* [H. Vogt, *An Eberlein–Šmulian type result for the weak* topology*, Theorems 3–4](https://user.math.uni-bremen.de/hvogt/papers/vo09a.pdf) gives a route through weak-star tail-convex-hull criteria; the extra hull hypothesis is essential. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n\nWhitley’s original proof is also freely readable in [Göttingen’s digitized volume, printed pp. 116–118](https://gdz.sub.uni-goettingen.de/id/PPN235181684_0172?tify=%7B%22pages%22:%5B126%5D%7D).\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-HS-01",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "1. Positive semidefinite forms",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
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      "full_conditions_and_proof": "## 1. Positive semidefinite forms\n\nA *sesquilinear form* on a complex vector space \\(V\\) is a map \\(B:V\\times V\\to\\mathbb C\\), linear in the first variable and conjugate-linear in the second. It is *Hermitian* if \\(B(\\eta,\\xi)=\\overline{B(\\xi,\\eta)}\\), and *positive semidefinite* if \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\).\n\n**Proposition 1.1.**\n1. (Polarization) \\(4B(\\xi,\\eta)=\\sum_{k=0}^3i^kB(\\xi+i^k\\eta,\\xi+i^k\\eta)\\).\n2. A form with \\(B(\\xi,\\xi)\\in\\mathbb R\\) for all \\(\\xi\\) is Hermitian.\n3. (Cauchy–Schwarz) If \\(B\\) is positive semidefinite, then \\(|B(\\xi,\\eta)|^2\\le B(\\xi,\\xi)B(\\eta,\\eta)\\). Consequently \\(\\{\\xi:B(\\xi,\\xi)=0\\}\\) is a subspace, and \\(\\xi\\mapsto B(\\xi,\\xi)^{1/2}\\) is a seminorm.\n\n**Proof.**\n1. Expand each term by sesquilinearity: the terms \\(B(\\xi,\\xi)\\) and \\(B(\\eta,\\eta)\\) cancel, and the cross terms give \\(4B(\\xi,\\eta)\\).\n2. Apply 1 to \\(B(\\eta,\\xi)\\) and to \\(\\overline{B(\\xi,\\eta)}\\), and compare the expressions term by term, using that the diagonal values are real.\n3. By 2, \\(B\\) is Hermitian. For \\(t\\in\\mathbb R\\) and \\(\\theta\\) with \\(e^{-i\\theta}B(\\xi,\\eta)=|B(\\xi,\\eta)|\\), the quantity\n\\[\n\\begin{gathered}\n0\\\\\n\\le B(\\xi+te^{i\\theta}\\eta,\\ \\xi+te^{i\\theta}\\eta)\\\\\n=B(\\xi,\\xi)+2t|B(\\xi,\\eta)|+t^2B(\\eta,\\eta)\n\\end{gathered}\n\\]\nis a nonnegative quadratic polynomial in \\(t\\). If \\(B(\\eta,\\eta)>0\\), its discriminant is \\(\\le0\\). If \\(B(\\eta,\\eta)=0\\), the linear polynomial \\(B(\\xi,\\xi)+2t|B(\\xi,\\eta)|\\) is nonnegative for all \\(t\\), so \\(|B(\\xi,\\eta)|=0\\).\n\nFor the consequence, Cauchy–Schwarz gives \\(B(\\xi+\\eta,\\xi+\\eta)\\le(B(\\xi,\\xi)^{1/2}+B(\\eta,\\eta)^{1/2})^2\\). \\(\\square\\)\n\nAn inner product is a positive definite Hermitian form. A *Hilbert space* is an inner product space that is complete for \\(\\|\\xi\\|=\\langle\\xi,\\xi\\rangle^{1/2}\\). The *parallelogram law* \\(\\|\\xi+\\eta\\|^2+\\|\\xi-\\eta\\|^2=2\\|\\xi\\|^2+2\\|\\eta\\|^2\\) follows by expanding.\n\n### Completing normed and inner product spaces\n\n**Lemma (Completion).** A normed space \\(E\\) embeds linearly and isometrically as a dense subspace of a Banach space \\(\\widehat E\\). If its norm comes from an inner product, that product extends to \\(\\widehat E\\), which is then a Hilbert space. Every bounded linear map \\(T:E\\to F\\), with \\(F\\) a Banach space, has a unique bounded linear extension to \\(\\widehat E\\), of the same norm. In particular the Hilbert construction applies to a positive semidefinite form after taking its quotient by the null subspace.\n\n*Proof.* Let \\(\\mathscr S\\) be the vector space of Cauchy sequences in \\(E\\), with termwise operations. Identify \\(a=(a_n)\\) and \\(b=(b_n)\\) when \\(\\|a_n-b_n\\|\\to0\\). The triangle inequality shows that this is an equivalence relation compatible with the vector operations. The reverse triangle inequality gives\n\n\\[\n\\begin{gathered}\n|\\|a_n\\|-\\|a_m\\||\\leq\\|a_n-a_m\\|.\n\\end{gathered}\n\\]\n\nSo these norms have a limit, unchanged on replacing \\(a\\) by an equivalent sequence. Set \\(\\widehat E=\\mathscr S/\\mathord\\sim\\) and \\(\\|[a]\\|=\\lim_n\\|a_n\\|\\). Homogeneity and the triangle inequality pass to this limit. Its value is zero exactly when \\(a\\) is equivalent to the zero sequence, so this is a norm. Constant sequences give the asserted linear isometry \\(E\\to\\widehat E\\).\n\nThe constants \\(a_n\\) converge to \\([a]\\): if \\(\\|a_n-a_m\\|<\\varepsilon\\) for all sufficiently large \\(n,m\\), taking the limit in \\(m\\) gives \\(\\|a_n-[a]\\|\\leq\\varepsilon\\). This proves density. If \\((z_k)\\) is Cauchy in \\(\\widehat E\\), choose \\(y_k\\in E\\) with \\(\\|y_k-z_k\\|<1/k\\). Then\n\n\\[\n\\|y_k-y_l\\|\\leq1/k+\\|z_k-z_l\\|+1/l.\n\\]\n\nSo \\((y_k)\\) is Cauchy in \\(E\\), and represents \\(z=[(y_k)]\\). The density argument gives \\(y_k\\to z\\); therefore \\(\\|z_k-z\\|\\leq1/k+\\|y_k-z\\|\\to0\\). This proves completeness.\n\nIf \\(E\\) has an inner product, its Cauchy sequences are bounded: a sufficiently late tail lies within distance \\(1\\) of one term, and the remaining initial block is finite. Cauchy–Schwarz therefore shows that the inner products have limits, using\n\n\\[\n\\begin{gathered}\n|\\langle a_n,b_n\\rangle-\\langle a_m,b_m\\rangle|\\\\\n\\leq\\|a_n-a_m\\|\\,\\|b_n\\|\\\\\n+\\|a_m\\|\\,\\|b_n-b_m\\|.\n\\end{gathered}\n\\]\n\nThe same estimate makes the limit independent of the representatives. Sesquilinearity and symmetry pass to the limit; its diagonal value is \\(\\lim_n\\|a_n\\|^2=\\|[a]\\|^2\\), so it is a positive definite inner product inducing the complete norm.\n\nFor the extension put \\(\\widehat T[a]=\\lim_nTa_n\\). The limit exists in \\(F\\) because \\(\\|Ta_n-Ta_m\\|\\leq\\|T\\|\\|a_n-a_m\\|\\). Equivalent sequences give the same limit by that bound. Linearity passes to limits, and \\(\\|\\widehat T[a]\\|\\leq\\|T\\|\\|[a]\\|\\). Restriction to the constant sequences gives the reverse norm inequality. Density gives uniqueness. Finally Proposition 1.1 shows that a null vector for a positive semidefinite form is orthogonal to every vector, so the form descends to a positive definite inner product on its quotient. The construction just proved then applies. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-HS-02",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "2. The projection theorem and the Riesz–Fréchet theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
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      "anchor": "oa-fnd-hs-02",
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        "line": 84,
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      },
      "full_conditions_and_proof": "## 2. The projection theorem and the Riesz–Fréchet theorem\n\n**Theorem 2.1.** Let \\(C\\) be a nonempty closed convex subset of a real or complex Hilbert space \\(H\\), and \\(\\xi\\in H\\). There is exactly one \\(c_0\\in C\\) with \\(\\|\\xi-c_0\\|=\\operatorname{dist}(\\xi,C)\\).\n\n**Proof.** Let \\(d=\\operatorname{dist}(\\xi,C)\\) and \\(c_n\\in C\\) with \\(\\|\\xi-c_n\\|\\to d\\). The parallelogram law applied to \\(\\xi-c_n\\) and \\(\\xi-c_m\\), together with \\(\\frac{c_n+c_m}2\\in C\\), gives\n\\[\n\\begin{gathered}\n\\|c_n-c_m\\|^2\\\\\n=2\\|\\xi-c_n\\|^2+2\\|\\xi-c_m\\|^2\\\\\n-4\\Big\\|\\xi-\\frac{c_n+c_m}2\\Big\\|^2\\\\\n\\le2\\|\\xi-c_n\\|^2+2\\|\\xi-c_m\\|^2-4d^2\\to0.\n\\end{gathered}\n\\]\nSo \\((c_n)\\) is Cauchy. Its limit \\(c_0\\in C\\) attains the distance. Two minimizers form a minimizing sequence when alternated, so they are equal. \\(\\square\\)\n\n**Theorem 2.2** (projection theorem). Let \\(L\\) be a closed subspace of a real or complex Hilbert space \\(H\\). Then:\n1. \\(H=L\\oplus L^\\perp\\);\n2. the map \\(P:\\xi\\mapsto c_0\\) of Theorem 2.1 is the orthogonal projection onto \\(L\\), a bounded self-adjoint idempotent with \\(\\|P\\|\\le1\\);\n3. for every subspace \\(Y\\), \\(\\overline Y=(Y^\\perp)^\\perp\\).\n\n**Proof.** Let \\(c_0\\) be the nearest point of \\(L\\) to \\(\\xi\\). For \\(\\eta\\in L\\) and scalars \\(t\\), \\(\\|\\xi-c_0-t\\eta\\|^2\\ge\\|\\xi-c_0\\|^2\\). Expanding gives \\(-2\\operatorname{Re}(\\bar t\\langle\\xi-c_0,\\eta\\rangle)+|t|^2\\|\\eta\\|^2\\ge0\\). Small \\(t\\) of suitable phase force \\(\\langle\\xi-c_0,\\eta\\rangle=0\\). So \\(\\xi=c_0+(\\xi-c_0)\\) with \\(\\xi-c_0\\in L^\\perp\\). The decomposition is unique because \\(L\\cap L^\\perp=\\{0\\}\\).\n\nThe projection is linear and idempotent, with \\(\\|P\\xi\\|^2+\\|(1-P)\\xi\\|^2=\\|\\xi\\|^2\\). It is self-adjoint because \\(\\langle P\\xi,\\eta\\rangle=\\langle P\\xi,P\\eta\\rangle=\\langle\\xi,P\\eta\\rangle\\).\n\nFor 3: \\((Y^\\perp)^\\perp\\) is closed and contains \\(Y\\). By 1, applied to \\(\\overline Y\\), we have \\(H=\\overline Y\\oplus\\overline Y^\\perp\\) and \\(Y^\\perp=\\overline Y^\\perp\\). An element of \\((Y^\\perp)^\\perp\\) has zero component in \\(\\overline Y^\\perp\\). \\(\\square\\)\n\n**Theorem 2.3** (Riesz–Fréchet). Every bounded linear functional \\(f\\) on \\(H\\) is \\(f(\\xi)=\\langle\\xi,\\eta\\rangle\\) for exactly one \\(\\eta\\in H\\), and \\(\\|f\\|=\\|\\eta\\|\\).\n\n**Proof.** If \\(f=0\\), take \\(\\eta=0\\). Otherwise \\(\\ker f\\) is a closed proper subspace. By Theorem 2.2 there is a unit vector \\(u\\perp\\ker f\\). For every \\(\\xi\\), the vector \\(f(\\xi)u-f(u)\\xi\\) lies in \\(\\ker f\\), so it is orthogonal to \\(u\\). This gives \\(f(\\xi)=f(u)\\langle\\xi,u\\rangle=\\langle\\xi,\\overline{f(u)}u\\rangle\\).\n\nUniqueness: \\(\\langle\\xi,\\eta-\\eta'\\rangle=0\\) for all \\(\\xi\\) forces \\(\\eta=\\eta'\\). The norm equality is Cauchy–Schwarz together with \\(\\xi=\\eta\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-HS-03",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "3. Sesquilinear forms and adjoints",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
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      "anchor": "oa-fnd-hs-03",
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        "line": 116,
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      "full_conditions_and_proof": "## 3. Sesquilinear forms and adjoints\n\n**Theorem 3.1.** Let \\(B\\) be a sesquilinear form on \\(H\\) with \\(|B(\\xi,\\eta)|\\le C\\|\\xi\\|\\|\\eta\\|\\). There is exactly one \\(t\\in B(H)\\) with \\(B(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\), and \\(\\|t\\|=\\sup\\{|B(\\xi,\\eta)|:\\|\\xi\\|,\\|\\eta\\|\\le1\\}\\le C\\). If \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\), then \\(\\langle t\\xi,\\xi\\rangle\\ge0\\).\n\n**Proof.** For fixed \\(\\xi\\), the map \\(\\eta\\mapsto\\overline{B(\\xi,\\eta)}\\) is a bounded linear functional. By Theorem 2.3 it is \\(\\langle\\eta,t\\xi\\rangle\\) for a unique vector \\(t\\xi\\). Uniqueness makes \\(t\\) linear. The norm formula follows from \\(\\|t\\xi\\|=\\sup_{\\|\\eta\\|\\le1}|\\langle t\\xi,\\eta\\rangle|\\). \\(\\square\\)\n\n**Corollary 3.2.** Every \\(T\\in B(H)\\) has a unique adjoint \\(T^*\\in B(H)\\) with \\(\\langle T\\xi,\\eta\\rangle=\\langle\\xi,T^*\\eta\\rangle\\). Moreover:\n- \\(T^{**}=T\\), \\(\\|T^*\\|=\\|T\\|\\) and \\(\\|T^*T\\|=\\|T\\|^2\\);\n- \\((ST)^*=T^*S^*\\), and \\(T\\mapsto T^*\\) is conjugate-linear;\n- (complex scalars) if \\(\\langle T\\xi,\\xi\\rangle=0\\) for all \\(\\xi\\), then \\(T=0\\);\n- (complex scalars) if \\(\\langle T\\xi,\\xi\\rangle\\in\\mathbb R\\) for all \\(\\xi\\), then \\(T=T^*\\);\n- for self-adjoint \\(T\\), \\(\\|T\\|=\\sup_{\\|\\xi\\|=1}|\\langle T\\xi,\\xi\\rangle|\\).\n\n**Proof.** Apply Theorem 3.1 to \\(B(\\xi,\\eta)=\\langle\\xi,T\\eta\\rangle\\) to obtain \\(T^*\\). The algebraic rules follow from uniqueness. \\(\\|T\\xi\\|^2=\\langle T^*T\\xi,\\xi\\rangle\\le\\|T^*T\\|\\|\\xi\\|^2\\) gives \\(\\|T\\|^2\\le\\|T^*T\\|\\le\\|T^*\\|\\|T\\|\\). So \\(\\|T\\|\\le\\|T^*\\|\\), and by symmetry the two norms are equal. The two complex-scalar statements follow from Proposition 1.1(1) and (2), applied to \\(B(\\xi,\\eta)=\\langle T\\xi,\\eta\\rangle\\).\n\n*The last statement.* Let \\(m=\\sup_{\\|\\xi\\|=1}|\\langle T\\xi,\\xi\\rangle|\\), so \\(|\\langle T\\xi,\\xi\\rangle|\\le m\\|\\xi\\|^2\\). For unit \\(\\xi,\\eta\\), expanding and using self-adjointness gives\n\\[\n\\begin{gathered}\n4\\operatorname{Re}\\langle T\\xi,\\eta\\rangle\\\\\n=\\langle T(\\xi+\\eta),\\xi+\\eta\\rangle\\\\\n-\\langle T(\\xi-\\eta),\\xi-\\eta\\rangle\\\\\n\\le m(\\|\\xi+\\eta\\|^2+\\|\\xi-\\eta\\|^2)\\\\\n=4m.\n\\end{gathered}\n\\]\nReplacing \\(\\eta\\) by \\(e^{i\\theta}\\eta\\) gives \\(|\\langle T\\xi,\\eta\\rangle|\\le m\\), so \\(\\|T\\|\\le m\\). The reverse inequality is Cauchy–Schwarz. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-HS-04",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "4. Orthonormal bases",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
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      "anchor": "oa-fnd-hs-04",
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        "through_line": 171
      },
      "full_conditions_and_proof": "## 4. Orthonormal bases\n\nAn *orthonormal family* \\((e_i)_{i\\in I}\\) satisfies \\(\\langle e_i,e_j\\rangle=\\delta_{ij}\\). It is an *orthonormal basis* if its closed linear span is \\(H\\).\n\n**Theorem 4.1.** Let \\((e_i)_{i\\in I}\\) be an orthonormal family in \\(H\\).\n1. (Bessel) \\(\\sum_i|\\langle\\xi,e_i\\rangle|^2\\le\\|\\xi\\|^2\\). In particular \\(\\langle\\xi,e_i\\rangle\\ne0\\) for at most countably many \\(i\\).\n2. For \\((c_i)\\in\\ell^2(I)\\), the sum \\(\\sum_ic_ie_i\\) converges, as the net of finite partial sums, and \\(\\|\\sum_ic_ie_i\\|^2=\\sum_i|c_i|^2\\).\n3. The following are equivalent:\n   - the family is a basis;\n   - it is maximal among orthonormal families;\n   - \\(\\xi=\\sum_i\\langle\\xi,e_i\\rangle e_i\\) for every \\(\\xi\\);\n   - Parseval's identity \\(\\langle\\xi,\\eta\\rangle=\\sum_i\\langle\\xi,e_i\\rangle\\langle e_i,\\eta\\rangle\\) holds for all \\(\\xi,\\eta\\).\n4. Every orthonormal family is contained in an orthonormal basis. A separable Hilbert space has a finite or countable orthonormal basis.\n5. Any two orthonormal bases of \\(H\\) have the same cardinality, the *Hilbert dimension* of \\(H\\).\n\n**Proof.**\n1. For finite \\(F\\subseteq I\\), \\(\\xi-\\sum_{i\\in F}\\langle\\xi,e_i\\rangle e_i\\) is orthogonal to each \\(e_i\\) with \\(i\\in F\\). So \\(\\|\\xi\\|^2\\ge\\sum_{i\\in F}|\\langle\\xi,e_i\\rangle|^2\\). The set \\(\\{i:|\\langle\\xi,e_i\\rangle|>1/n\\}\\) is finite for every \\(n\\).\n\n2. Only countably many \\(c_i\\) are nonzero. The partial sums form a Cauchy net, because \\(\\|\\sum_{i\\in F\\setminus G}c_ie_i\\|^2=\\sum_{F\\setminus G}|c_i|^2\\).\n\n3. *Basis implies expansion.* The vector \\(\\xi-\\sum_i\\langle\\xi,e_i\\rangle e_i\\) is orthogonal to every \\(e_j\\), hence to their closed span \\(H\\), so it is zero.\n- *Expansion implies Parseval:* take inner products of the expansions.\n- *Parseval implies maximality:* a unit vector \\(e\\) orthogonal to all \\(e_i\\) would have \\(\\|e\\|^2=\\sum|\\langle e,e_i\\rangle|^2=0\\).\n- *Maximality implies basis:* if the closed span \\(L\\) were not \\(H\\), Theorem 2.2 would give a unit vector in \\(L^\\perp\\), which could be added to the family.\n\n4. Orthonormal families containing the given one, ordered by inclusion, have unions of chains as upper bounds. Zorn's lemma (first lesson, Theorem 1.1) gives a maximal one, a basis by 3. For separable \\(H\\), apply the Gram–Schmidt process to a dense sequence.\n\n5. If one basis is finite, \\(H\\) is finite-dimensional, and both bases have \\(\\dim H\\) elements. Suppose both are infinite, \\((e_i)_{i\\in I}\\) and \\((f_j)_{j\\in J}\\). Each \\(e_i\\) has at most countably many \\(j\\) with \\(\\langle e_i,f_j\\rangle\\ne0\\), by 1. Every \\(j\\) occurs for some \\(i\\), since \\(f_j\\ne0\\) is the sum of its expansion in the \\(e_i\\). So \\(J\\) is a union of countable sets indexed by \\(I\\), and \\(|J|\\le|I|\\) (first lesson, Theorem 8.4(3)). Symmetrically \\(|I|\\le|J|\\), and \\(|I|=|J|\\) by the Cantor–Schröder–Bernstein theorem (first lesson, Theorem 8.1). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-HS-05",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "5. Compact operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "anchor": "oa-fnd-hs-05",
      "proof_locus": {
        "line": 172,
        "through_line": 175
      },
      "full_conditions_and_proof": "## 5. Compact operators\n\n\\(T\\in B(H)\\) is *compact* if the image of the unit ball is relatively compact. \\(K(H)\\) is the set of compact operators and \\(F(H)\\) the set of finite-rank operators.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-HS-06",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "6. The spectral theorem for compact self-adjoint operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "anchor": "oa-fnd-hs-06",
      "proof_locus": {
        "line": 226,
        "through_line": 255
      },
      "full_conditions_and_proof": "## 6. The spectral theorem for compact self-adjoint operators\n\n**Lemma 6.1.** Let \\(T\\) be compact and self-adjoint, \\(T\\ne0\\). Then \\(\\|T\\|\\) or \\(-\\|T\\|\\) is an eigenvalue of \\(T\\).\n\n**Proof.** By Corollary 3.2, there are unit vectors \\(\\xi_n\\) with \\(\\langle T\\xi_n,\\xi_n\\rangle\\to\\lambda\\), where \\(|\\lambda|=\\|T\\|\\). Then\n\\[\n\\begin{gathered}\n\\|T\\xi_n-\\lambda\\xi_n\\|^2\\\\\n=\\|T\\xi_n\\|^2-2\\lambda\\langle T\\xi_n,\\xi_n\\rangle+\\lambda^2\\\\\n\\le2\\lambda^2-2\\lambda\\langle T\\xi_n,\\xi_n\\rangle\\to0.\n\\end{gathered}\n\\]\nA subsequence has \\(T\\xi_n\\to\\eta\\), by compactness. Then \\(\\lambda\\xi_n\\to\\eta\\), so \\(\\|\\eta\\|=|\\lambda|\\ne0\\), and \\(T\\eta=\\lim\\lambda T\\xi_n=\\lambda\\eta\\). \\(\\square\\)\n\n**Theorem 6.2.** Let \\(T\\) be compact and self-adjoint, \\(T\\ne0\\). There are an orthonormal family \\((e_n)_{n\\in N}\\), with \\(N=\\{1,\\dots,r\\}\\) or \\(N=\\mathbb N\\), and real numbers \\(\\lambda_n\\ne0\\) with \\(|\\lambda_1|\\ge|\\lambda_2|\\ge\\dots\\), tending to \\(0\\) if \\(N=\\mathbb N\\), such that\n\\[\nT=\\sum_n\\lambda_n\\theta_{e_n,e_n},\n\\]\nwith convergence in norm. The nonzero eigenvalues of \\(T\\) are the \\(\\lambda_n\\). The eigenspace of an eigenvalue \\(\\mu\\ne0\\) is spanned by the \\(e_n\\) with \\(\\lambda_n=\\mu\\) and is finite-dimensional. \\(T\\ge0\\) iff all \\(\\lambda_n>0\\).\n\n**Proof.** *The eigenvectors.* Put \\(H_1=H\\). Lemma 6.1 gives a unit eigenvector \\(e_1\\) with eigenvalue \\(\\lambda_1\\), \\(|\\lambda_1|=\\|T\\|\\). The space \\(H_2=\\{e_1\\}^\\perp\\) is invariant under \\(T\\), because \\(T\\) is self-adjoint, and \\(T|_{H_2}\\) is compact and self-adjoint. Repeat. The process stops if \\(T|_{H_{r+1}}=0\\). Otherwise it produces \\((e_n,\\lambda_n)\\) for all \\(n\\), with \\(|\\lambda_n|\\) nonincreasing.\n\n*The eigenvalues tend to \\(0\\).* If not, \\(|\\lambda_n|\\ge\\delta>0\\) for all \\(n\\). Then \\(\\|Te_n-Te_m\\|^2=\\lambda_n^2+\\lambda_m^2\\ge2\\delta^2\\), so \\((Te_n)\\) has no convergent subsequence, contradicting compactness.\n\n*The expansion.* \\(T-\\sum_{n\\le k}\\lambda_n\\theta_{e_n,e_n}\\) vanishes on \\(\\operatorname{span}\\{e_1,\\dots,e_k\\}\\) and equals \\(T\\) on \\(H_{k+1}\\). So its norm is \\(\\|T|_{H_{k+1}}\\|=|\\lambda_{k+1}|\\to0\\).\n\n*Eigenvalues and eigenspaces.* If \\(T\\xi=\\mu\\xi\\) with \\(\\mu\\ne0\\), then \\(\\mu\\xi=\\sum\\lambda_n\\langle\\xi,e_n\\rangle e_n\\). So \\(\\xi\\) lies in the closed span of the \\(e_n\\), and \\(\\langle\\xi,e_n\\rangle=0\\) unless \\(\\lambda_n=\\mu\\). Only finitely many \\(\\lambda_n\\) equal \\(\\mu\\), because \\(\\lambda_n\\to0\\).\n\n*Positivity.* \\(\\langle T\\xi,\\xi\\rangle=\\sum\\lambda_n|\\langle\\xi,e_n\\rangle|^2\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-HS-07",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "7. The Riesz theory of compact operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "anchor": "oa-fnd-hs-07",
      "proof_locus": {
        "line": 256,
        "through_line": 291
      },
      "full_conditions_and_proof": "## 7. The Riesz theory of compact operators\n\n**Theorem 7.1.** Let \\(T\\in K(H)\\) and \\(\\lambda\\in\\mathbb C\\setminus\\{0\\}\\), and put \\(S=T-\\lambda\\).\n1. \\(\\ker S\\) is finite-dimensional.\n2. \\(S(H)\\) is closed.\n3. If \\(S\\) is injective, it is surjective, hence invertible in \\(B(H)\\).\n4. Consequently, every nonzero point of the spectrum \\(\\sigma(T)\\) is an eigenvalue of finite multiplicity.\n5. The nonzero eigenvalues of \\(T\\) have no accumulation point other than \\(0\\). Each nonzero point of \\(\\sigma(T)\\) is isolated.\n\n**Proof.** 1. On \\(\\ker S\\), \\(T=\\lambda\\). So the unit ball of \\(\\ker S\\) equals \\(\\lambda^{-1}T(\\text{ball of }\\ker S)\\), which is relatively compact. An infinite orthonormal sequence in \\(\\ker S\\) would have mutual distances \\(\\sqrt2\\). So \\(\\ker S\\) is finite-dimensional.\n\n2. Let \\(Sx_n\\to y\\). Write \\(x_n=u_n+v_n\\) with \\(u_n\\in\\ker S\\) and \\(v_n\\perp\\ker S\\), so \\(Sv_n=Sx_n\\).\n\n*\\((v_n)\\) is bounded.* Otherwise, along a subsequence with \\(\\|v_n\\|\\to\\infty\\), the unit vectors \\(w_n=v_n/\\|v_n\\|\\) have \\(Sw_n\\to0\\). A further subsequence has \\(Tw_n\\to z\\). Then \\(\\lambda w_n=Tw_n-Sw_n\\to z\\), so \\(w_n\\to z/\\lambda\\), a unit vector orthogonal to \\(\\ker S\\) with \\(S(z/\\lambda)=0\\), which is impossible.\n\n*Conclusion.* A subsequence of the bounded sequence has \\(Tv_n\\to z'\\). Then \\(\\lambda v_n=Tv_n-Sv_n\\to z'-y\\), so \\(v_n\\to v=(z'-y)/\\lambda\\), and \\(Sv=y\\).\n\n3. Suppose \\(S\\) is injective but \\(S(H)\\ne H\\). Each \\(S^k\\) is of the form \\((-\\lambda)^k+T_k\\) with \\(T_k\\) compact, so \\(S^k(H)\\) is closed by 2. The sequence \\(H\\supsetneq S(H)\\supsetneq S^2(H)\\supsetneq\\dots\\) is strictly decreasing: if \\(S^k(H)=S^{k+1}(H)\\), take \\(x\\notin S(H)\\). Then \\(S^kx=S^{k+1}y\\) for some \\(y\\), so \\(S^k(x-Sy)=0\\), and injectivity gives \\(x=Sy\\), a contradiction.\n\nChoose unit vectors \\(x_k\\in S^k(H)\\ominus S^{k+1}(H)\\). For \\(k<m\\),\n\\[\nTx_k-Tx_m=\\lambda x_k+\\big(Sx_k-Sx_m-\\lambda x_m\\big),\n\\]\nand the bracket lies in \\(S^{k+1}(H)\\), which is orthogonal to \\(x_k\\). So \\(\\|Tx_k-Tx_m\\|\\ge|\\lambda|\\), and \\((Tx_k)\\) has no convergent subsequence, contradicting compactness. Hence \\(S\\) is surjective, and its inverse is bounded by the inverse mapping theorem (first lesson, Corollary 5.2).\n\n4. If \\(\\lambda\\ne0\\) is not an eigenvalue, then \\(S\\) is injective, hence invertible by 3, so \\(\\lambda\\notin\\sigma(T)\\). The multiplicity is finite by 1.\n\n5. Suppose \\(\\lambda_n\\to\\lambda\\ne0\\) are distinct eigenvalues with eigenvectors \\(e_n\\). Eigenvectors for distinct eigenvalues are linearly independent. Let \\(M_n=\\operatorname{span}\\{e_1,\\dots,e_n\\}\\) and choose unit vectors \\(y_n\\in M_n\\ominus M_{n-1}\\). Then \\((T-\\lambda_n)M_n\\subseteq M_{n-1}\\). For \\(m<n\\),\n\\[\n\\begin{gathered}\n\\frac{Ty_n}{\\lambda_n}-\\frac{Ty_m}{\\lambda_m}\\\\\n=y_n-\\Big(\\frac{(\\lambda_n-T)y_n}{\\lambda_n}+\\frac{Ty_m}{\\lambda_m}\\Big),\n\\end{gathered}\n\\]\nand the bracket lies in \\(M_{n-1}\\perp y_n\\). So the left side has norm \\(\\ge1\\). Since \\(\\|y_n/\\lambda_n\\|\\) is bounded, this contradicts compactness. Hence the nonzero eigenvalues accumulate only at \\(0\\), and by 4 every nonzero point of \\(\\sigma(T)\\) is isolated. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-HS-08",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "8. Hilbert tensor products",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "anchor": "oa-fnd-hs-08",
      "proof_locus": {
        "line": 292,
        "through_line": 332
      },
      "full_conditions_and_proof": "## 8. Hilbert tensor products\n\nFor a set \\(S\\), \\(\\ell^2(S)\\) is the space of functions \\(c:S\\to\\mathbb C\\) with \\(\\sum_s|c_s|^2<\\infty\\), with \\(\\langle c,d\\rangle=\\sum_sc_s\\bar d_s\\). It is a Hilbert space. Completeness: a Cauchy sequence \\((c^{(n)})\\) converges at each \\(s\\) to some \\(c_s\\). For every finite \\(F\\subseteq S\\), \\[\n\\begin{gathered}\n\\sum_{s\\in F}|c_s-c^{(n)}_s|^2\\\\\n=\\lim_m\\sum_{s\\in F}|c^{(m)}_s-c^{(n)}_s|^2\\\\\n\\le\\sup_{m\\ge n}\\|c^{(m)}-c^{(n)}\\|^2\n\\end{gathered}\n\\]. So \\(c-c^{(n)}\\in\\ell^2(S)\\) and \\(\\|c-c^{(n)}\\|\\to0\\). The functions \\(\\delta_s\\) form an orthonormal basis.\n\n**Theorem 8.1** (Hilbert tensor products). Let \\(H\\) and \\(K\\) be Hilbert spaces.\n1. There are a Hilbert space \\(H\\otimes K\\) and a bilinear map \\((\\xi,\\eta)\\mapsto\\xi\\otimes\\eta\\) from \\(H\\times K\\) to \\(H\\otimes K\\) with\n\\[\n\\langle\\xi\\otimes\\eta,\\xi'\\otimes\\eta'\\rangle=\\langle\\xi,\\xi'\\rangle\\langle\\eta,\\eta'\\rangle ,\n\\]\nsuch that the elementary tensors \\(\\xi\\otimes\\eta\\) span a dense subspace.\n2. If \\((e_i)_{i\\in I}\\) and \\((f_j)_{j\\in J}\\) are orthonormal bases of \\(H\\) and \\(K\\), then \\((e_i\\otimes f_j)\\) is an orthonormal basis of \\(H\\otimes K\\).\n3. (*Universal property.*) Let \\(B:H\\times K\\to\\mathcal K\\) be a bilinear map into a Hilbert space with \\(\\langle B(\\xi,\\eta),B(\\xi',\\eta')\\rangle=\\langle\\xi,\\xi'\\rangle\\langle\\eta,\\eta'\\rangle\\). There is exactly one isometry \\(U:H\\otimes K\\to\\mathcal K\\) with \\(U(\\xi\\otimes\\eta)=B(\\xi,\\eta)\\). It is unitary if the vectors \\(B(\\xi,\\eta)\\) span a dense subspace. So \\(H\\otimes K\\) is unique up to a unitary that matches the elementary tensors.\n\n**Proof.** (1) and (2). Fix orthonormal bases as in (2), and put \\(H\\otimes K=\\ell^2(I\\times J)\\) and \\((\\xi\\otimes\\eta)(i,j)=\\langle\\xi,e_i\\rangle\\langle\\eta,f_j\\rangle\\).\n- By Parseval's identity (Theorem 4.1(3)), \\(\\sum_{i,j}|\\langle\\xi,e_i\\rangle|^2|\\langle\\eta,f_j\\rangle|^2=\\|\\xi\\|^2\\|\\eta\\|^2\\), so \\(\\xi\\otimes\\eta\\in\\ell^2(I\\times J)\\).\n- The map is bilinear, and by Parseval's identity\n\\[\n\\begin{gathered}\n\\langle\\xi\\otimes\\eta,\\xi'\\otimes\\eta'\\rangle\\\\\n=\\sum_i\\langle\\xi,e_i\\rangle\\langle e_i,\\xi'\\rangle\\sum_j\\langle\\eta,f_j\\rangle\\langle f_j,\\eta'\\rangle\\\\\n=\\langle\\xi,\\xi'\\rangle\\langle\\eta,\\eta'\\rangle .\n\\end{gathered}\n\\]\n- \\(e_i\\otimes f_j=\\delta_{(i,j)}\\). These form an orthonormal basis, so the elementary tensors span a dense subspace.\n\n(3) For finite sums,\n\\[\n\\begin{gathered}\n\\Big\\|\\sum_kc_kB(\\xi_k,\\eta_k)\\Big\\|^2\\\\\n=\\sum_{k,l}c_k\\bar c_l\\langle\\xi_k,\\xi_l\\rangle\\langle\\eta_k,\\eta_l\\rangle\\\\\n=\\Big\\|\\sum_kc_k\\,\\xi_k\\otimes\\eta_k\\Big\\|^2 .\n\\end{gathered}\n\\]\nSo \\(\\sum_kc_k\\,\\xi_k\\otimes\\eta_k\\mapsto\\sum_kc_kB(\\xi_k,\\eta_k)\\) is well defined, since a combination equal to \\(0\\) goes to a vector of norm \\(0\\), and it is isometric. It extends by continuity to \\(H\\otimes K\\). Uniqueness holds because the elementary tensors are total. An isometry has closed range, so if its range is dense, it is onto. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-HS-09",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "9. Multiplication operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
      "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "anchor": "oa-fnd-hs-09",
      "proof_locus": {
        "line": 333,
        "through_line": 404
      },
      "full_conditions_and_proof": "## 9. Multiplication operators\n\nLet \\((Z,\\Sigma,\\nu)\\) be a \\(\\sigma\\)-finite measure space. \\(L^\\infty(\\nu)\\) is the space of essentially bounded measurable functions modulo null functions, with the essential supremum norm \\(\\|f\\|_\\infty\\). For \\(f\\in L^\\infty(\\nu)\\), \\(m_f\\) is the operator \\(\\xi\\mapsto f\\xi\\) on \\(L^2(\\nu)\\). The [measure-tools lesson](measure-and-hilbert-space-tools.md#3-the-complete-spaces-of-integrable-functions), Theorems 3.1–3.2, proves Hölder, completeness of \\(L^2\\), and density of simple functions on arbitrary measure spaces. Its proof uses elementary integration and convergence, not the operator theorem here. Thus the use of Hilbert representation in the later Radon–Nikodym theorem creates no circular dependence.\n\n**Theorem 9.1.**\n1. \\(f\\mapsto m_f\\) is an isometric unital \\(*\\)-homomorphism from \\(L^\\infty(\\nu)\\) into \\(B(L^2(\\nu))\\).\n2. An operator \\(T\\in B(L^2(\\nu))\\) commutes with every \\(m_f\\) if and only if \\(T=m_g\\) for some \\(g\\in L^\\infty(\\nu)\\). So the algebra \\(\\{m_f:f\\in L^\\infty(\\nu)\\}\\) equals its own commutant: it is *maximal abelian*.\n\n**Proof.** (1) \\(\\|f\\xi\\|_2\\le\\|f\\|_\\infty\\|\\xi\\|_2\\), and \\(\\langle f\\xi,\\eta\\rangle=\\langle\\xi,\\bar f\\eta\\rangle\\). For \\(\\varepsilon>0\\), the set \\(\\{|f|>\\|f\\|_\\infty-\\varepsilon\\}\\) has positive measure. Since \\(\\nu\\) is \\(\\sigma\\)-finite, it contains a set \\(E\\) with \\(0<\\nu(E)<\\infty\\). Then \\(\\|f1_E\\|_2\\ge(\\|f\\|_\\infty-\\varepsilon)\\|1_E\\|_2\\). So \\(\\|m_f\\|=\\|f\\|_\\infty\\).\n\n(2) *Reduction to a finite measure.* Write \\(Z\\) as a disjoint union of sets \\(Z_n\\in\\Sigma\\) of finite measure. Let \\(w=\\sum_n2^{-n}(1+\\nu(Z_n))^{-1}1_{Z_n}\\). Then \\(w>0\\) everywhere, and \\(\\lambda=w\\,\\nu\\) is a finite measure with the same null sets as \\(\\nu\\).\n- So \\(L^\\infty(\\lambda)=L^\\infty(\\nu)\\).\n- \\(V\\xi=w^{-1/2}\\xi\\) is a unitary from \\(L^2(\\nu)\\) onto \\(L^2(\\lambda)\\), with inverse \\(\\xi\\mapsto w^{1/2}\\xi\\).\n- \\(Vm_fV^{-1}=m_f\\).\n\nSo we may assume \\(\\nu(Z)<\\infty\\). Then \\(1\\in L^2(\\nu)\\), and \\(L^\\infty(\\nu)\\subseteq L^2(\\nu)\\).\n\n*The function \\(g\\).* Let \\(T\\) commute with every \\(m_f\\), and put \\(g=T1\\in L^2(\\nu)\\). For \\(f\\in L^\\infty(\\nu)\\),\n\\[\nTf=Tm_f1=m_fT1=fg .\n\\]\n- *\\(g\\) is essentially bounded.* For \\(c>\\|T\\|\\) let \\(E=\\{|g|>c\\}\\). Then \\(c^2\\nu(E)\\le\\|1_Eg\\|_2^2=\\|T1_E\\|_2^2\\le\\|T\\|^2\\nu(E)\\), so \\(\\nu(E)=0\\). Hence \\(\\|g\\|_\\infty\\le\\|T\\|\\).\n- *\\(T=m_g\\).* The two operators agree on \\(L^\\infty(\\nu)\\), which contains the simple functions. These are dense in \\(L^2(\\nu)\\) [Theorem 3.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#3-the-complete-spaces-of-integrable-functions).\n\nThe converse holds because the algebra is commutative. \\(\\square\\)\n\n## Exercises\n\n**Exercise 1** (easy; diagonal operators). Let \\((e_n)\\) be an orthonormal basis of \\(H\\) and \\(T=\\sum_n\\mu_n\\theta_{e_n,e_n}\\) for a bounded sequence \\((\\mu_n)\\). Show that \\(T\\) is compact iff \\(\\mu_n\\to0\\).\n\n*Solution.* If \\(\\mu_n\\to0\\), the finite-rank truncations converge to \\(T\\) in norm, so \\(T\\) is compact by Theorem 5.1(1). If \\(|\\mu_{n_k}|\\ge\\delta>0\\) along a subsequence, then \\(\\|Te_{n_k}-Te_{n_l}\\|^2\\ge2\\delta^2\\), so \\(T\\) is not compact. \\(\\square\\)\n\n**Exercise 2** (medium; Hilbert–Schmidt operators). Show that an operator \\(T\\) with \\(\\sum_n\\|Te_n\\|^2<\\infty\\) for some orthonormal basis \\((e_n)\\) is compact.\n\n*Solution.* Let \\(P_k\\) be the projection onto \\(\\operatorname{span}\\{e_1,\\dots,e_k\\}\\). Then \\[\n\\begin{gathered}\n\\|T(1-P_k)\\xi\\|\\\\\n\\le\\sum_{n>k}|\\langle\\xi,e_n\\rangle|\\|Te_n\\|\\\\\n\\le\\|\\xi\\|(\\sum_{n>k}\\|Te_n\\|^2)^{1/2}\n\\end{gathered}\n\\] by Cauchy–Schwarz. So \\(TP_k\\to T\\) in norm, and each \\(TP_k\\) has finite rank. \\(\\square\\)\n\n**Exercise 3** (hard; the Volterra operator). On \\(L^2[0,1]\\), let \\((V\\xi)(s)=\\int_0^s\\xi(t)\\,dt\\). Show that \\(V\\) is compact and that \\(\\sigma(V)=\\{0\\}\\).\n\n*Solution.* First \\(V\\) is bounded: Cauchy–Schwarz gives \\(|V\\xi(s)|^2\\leq s\\|\\xi\\|_2^2\\), hence \\(\\|V\\xi\\|_2^2\\leq\\frac12\\|\\xi\\|_2^2\\). Also \\(|V\\xi(s)-V\\xi(t)|\\leq |s-t|^{1/2}\\|\\xi\\|_2\\), so \\(V\\xi\\) is continuous.\n- *Compactness:* with an orthonormal basis \\((e_n)\\), \\[\n\\begin{gathered}\n\\sum_n\\|Ve_n\\|^2\\\\\n=\\int_0^1\\sum_n|\\langle 1_{[0,s]},\\overline{e_n}\\rangle|^2ds\\\\\n=\\int_0^1\\|1_{[0,s]}\\|^2ds\\\\\n=\\frac12\n\\end{gathered}\n\\], by Parseval applied to the basis \\((\\overline{e_n})\\) and [monotone convergence](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions). By Exercise 2, \\(V\\) is compact.\n- *Spectrum:* by Theorem 7.1(4), a nonzero point of \\(\\sigma(V)\\) would be an eigenvalue. If \\(V\\xi=\\lambda\\xi\\), then \\(\\xi=\\lambda^{-1}V\\xi\\) is continuous, then \\(C^1\\) by the fundamental theorem of calculus proved in [Lemma 6.1 of the measure-tools lesson](measure-and-hilbert-space-tools.md#6-one-variable-integral-identities), with \\(\\xi=\\lambda\\xi'\\) and \\(\\xi(0)=0\\). The function \\(e^{-s/\\lambda}\\xi(s)\\) has zero derivative and is zero at \\(s=0\\), so \\(\\xi=0\\). Finally \\(0\\in\\sigma(V)\\), because \\(V\\) is not invertible in infinite dimensions: a compact invertible operator would make the identity compact. \\(\\square\\)\n\n**Exercise 4** (medium; closed range of \\(1-T\\)). Let \\(T\\) be compact. Show that \\(\\dim\\ker(1-T)=\\dim\\ker(1-T^*)\\).\n\n*Solution.*\n- Both kernels are finite-dimensional (Theorem 7.1(1), applied to \\(T\\) and to \\(T^*\\), which is compact by Theorem 5.1(2)).\n- Let \\(S=1-T\\). Since \\(S(H)\\) is closed, \\(\\ker S^*=S(H)^\\perp\\), and \\(H=S(H)\\oplus\\ker S^*\\).\n- Suppose \\(\\dim\\ker S<\\dim\\ker S^*\\). Choose an injective linear map \\(A\\) from \\(\\ker S\\) into \\(\\ker S^*\\) that is not onto, and extend it by \\(0\\) on \\((\\ker S)^\\perp\\). This gives a finite-rank operator. Then \\(S'=S+A=1-(T-A)\\) is injective with range \\(S(H)+A(\\ker S)\\ne H\\), contradicting Theorem 7.1(3).\n- Exchanging \\(T\\) and \\(T^*\\) gives the reverse inequality. \\(\\square\\)\n\n## Where this leads\n\nThe continuous functional calculus and the operator algebra \\(B(H)\\) are developed in [C\\*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md). The spectral theorem for bounded self-adjoint operators with projection-valued measures, and Calkin's theorem on the ideals of \\(B(H)\\), are proved in [The spectral theorem for bounded self-adjoint operators](the-spectral-theorem-for-bounded-self-adjoint-operators.md). The trace class and the operator topologies are in [Compact and trace-class operators, the predual of B(H), and the operator topologies](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md). Hilbert tensor products are used for \\(L^2\\) of a product of Radon measures in [Haar measure on locally compact groups](haar-measure.md), and multiplication operators are the starting point of [Abelian operator algebras](abelian-operator-algebras.md).\n\n## References\n\nHilbert (1904–1910), Riesz (1907, 1918) and Fréchet (1907). The Riesz theory of compact operators is from F. Riesz, \"Über lineare Funktionalgleichungen\", *Acta Mathematica* 41 (1918) 71–98. The proofs are written here in our own words.\n\n*Freely accessible reading:* [Jesse Peterson, *Notes on operator algebras*, §3.3](https://math.vanderbilt.edu/peters10/teaching/spring2015/OperatorAlgebras.pdf) gives a route through compact-operator spectral arguments. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-HS-10",
      "unit": "hilbert-spaces-and-compact-operators",
      "name": "Metric compactness tools",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "source": "src/hilbert-spaces-and-compact-operators.md",
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      "full_conditions_and_proof": "### Metric compactness tools\n\nA metric space is *totally bounded* if, for every \\(\\varepsilon>0\\), finitely many balls of radius \\(\\varepsilon\\) cover it.\n\n**Lemma 5.0.** A metric space is compact if and only if it is complete and totally bounded. A compact metric space is sequentially compact and separable.\n\n*Proof.* In a compact space, every sequence has a point whose every neighbourhood contains infinitely many sequence terms, counted with their indices: otherwise a finite subcover of neighbourhoods containing only finitely many terms gives a contradiction. Choosing increasing indices in the balls of radius \\(1/k\\) at that point gives a convergent subsequence. A Cauchy sequence with a convergent subsequence converges to the same limit, so the space is complete. Compactness also gives finite covers by balls of any prescribed radius.\n\nConversely, in a totally bounded space, repeatedly choose an infinite set of remaining sequence indices lying in one ball of radius \\(2^{-k}\\). Take these sets nested and choose increasing indices from them. The resulting subsequence is Cauchy, since its terms from stage \\(k\\) onward are at mutual distance at most \\(2^{1-k}\\). Completeness gives its limit.\n\nTo obtain compactness, consider an open cover. There is a number \\(\\delta>0\\) such that every ball of radius \\(\\delta\\) is contained in a member of the cover. If there were no such number, choose \\(x_n\\) whose ball of radius \\(1/n\\) is contained in no member. A convergent subsequence tends to some \\(x\\) in a member \\(U\\). Choose \\(r>0\\) with \\(B(x,2r)\\subseteq U\\). For large subsequence indices, \\(d(x_n,x)<r\\) and \\(1/n<r\\), so \\(B(x_n,1/n)\\subseteq U\\), a contradiction. A finite cover by \\(\\delta\\)-balls now gives a finite subcover of the original cover. Finally choose finite \\(1/n\\)-nets for all \\(n\\); their union is countable and dense. \\(\\square\\)\n\nIn a finite-dimensional Hilbert space, bounded sets are totally bounded: express them in an orthonormal basis, bound each coordinate by Cauchy–Schwarz, and use a finite grid in the resulting bounded coordinate box. A finite-dimensional subspace is closed, because the coordinates of a convergent sequence converge and its limit is their finite basis expansion. Its bounded closed subsets are therefore complete and compact by the lemma.\n\n### Compactness and finite-rank approximation\n\n**Theorem 5.1.**\n1. \\(K(H)\\) is a norm-closed two-sided ideal of \\(B(H)\\) that contains \\(F(H)\\).\n2. \\(T\\) is compact iff \\(T^*\\) is compact.\n3. \\(K(H)\\) is the norm closure of \\(F(H)\\).\n4. A compact operator maps weakly convergent sequences to norm convergent sequences.\n5. Let \\((P_i)\\) be a net in \\(B(H)\\) with \\(\\|P_i\\|\\le1\\) and \\(P_i\\xi\\to\\xi\\) for every \\(\\xi\\in H\\). Then \\(\\|T-P_iT\\|\\to0\\) for every compact \\(T\\). This applies to the projections \\(P_n\\) onto \\(\\operatorname{span}\\{e_1,\\dots,e_n\\}\\) for an orthonormal basis \\((e_k)\\) of a separable \\(H\\) (Theorem 4.1(3)).\n\n**Proof.** 5. The closure \\(K\\) of \\(T(\\text{ball})\\) is compact. Given \\(\\varepsilon>0\\), cover \\(K\\) by finitely many balls \\(B(y_k,\\varepsilon)\\), and choose \\(i_0\\) with \\(\\|(1-P_i)y_k\\|<\\varepsilon\\) for every \\(k\\) and every \\(i\\ge i_0\\). For \\(y\\in K\\) with \\(\\|y-y_k\\|<\\varepsilon\\),\n\\[\n\\begin{gathered}\n\\|(1-P_i)y\\|\\\\\n\\le\\|(1-P_i)y_k\\|+\\|1-P_i\\|\\,\\|y-y_k\\|<3\\varepsilon .\n\\end{gathered}\n\\]\nSo \\(\\|T-P_iT\\|=\\sup_{y\\in K}\\|(1-P_i)y\\|\\le3\\varepsilon\\) for \\(i\\ge i_0\\).\n\n1. A finite-rank bounded operator maps the ball into a bounded subset of a finite-dimensional space, which is relatively compact. Sums of compact operators are compact, since the image of the ball lies in the sum of two relatively compact sets. Products with bounded operators are compact, since bounded operators are continuous.\n\n*Closedness.* Let \\(T_n\\to T\\) in norm with \\(T_n\\) compact, and \\(\\varepsilon>0\\). Choose \\(n\\) with \\(\\|T-T_n\\|<\\varepsilon/3\\), and cover \\(T_n(\\text{ball})\\) by finitely many \\(\\varepsilon/3\\)-balls. The \\(\\varepsilon\\)-balls with the same centres cover \\(T(\\text{ball})\\). So \\(T(\\text{ball})\\) is totally bounded, hence relatively compact, since \\(H\\) is complete.\n\n2. If \\(T\\) is compact, so is \\(TT^*\\). For \\(\\|\\xi_n\\|\\le1\\), choose a subsequence along which \\(TT^*\\xi_n\\) converges. Then\n\\[\n\\begin{gathered}\n\\|T^*(\\xi_n-\\xi_m)\\|^2\\\\\n=\\langle TT^*(\\xi_n-\\xi_m),\\xi_n-\\xi_m\\rangle\\\\\n\\le2\\|TT^*(\\xi_n-\\xi_m)\\|\\to0,\n\\end{gathered}\n\\]\nso \\(T^*\\xi_n\\) converges along the subsequence. Apply this to \\(T^*\\) for the converse.\n\n3. Let \\(T\\) be compact. The closure \\(K\\) of \\(T(\\text{ball})\\) is a compact metric space, so it has a countable dense subset. The range of \\(T\\) is the union of the sets \\(nT(\\text{ball})\\), so its closure \\(L\\) is separable. Let \\((e_k)\\) be an orthonormal basis of \\(L\\), and \\(P_n\\) the projection onto \\(\\operatorname{span}\\{e_1,\\dots,e_n\\}\\). Then \\(P_ny\\to y\\) for \\(y\\in L\\) (Theorem 4.1(3)), and \\(P_n=0\\) on \\(L^\\perp\\). The argument of 5, applied with \\(K\\subseteq L\\), gives \\(\\|T-P_nT\\|\\to0\\). Each \\(P_nT\\) has finite rank.\n\n4. Let \\(\\xi_n\\to\\xi\\) weakly. Then \\((\\xi_n)\\) is bounded (first lesson, Corollary 4.3(2)), and \\(T\\xi_n\\to T\\xi\\) weakly. If \\(\\|T\\xi_n-T\\xi\\|\\not\\to0\\), a subsequence stays at distance \\(\\ge\\varepsilon\\). A further subsequence converges in norm, by compactness, and its limit must be the weak limit \\(T\\xi\\), a contradiction. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CT-01",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "1. The index of a cycle",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
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      "full_conditions_and_proof": "## 1. The index of a cycle\n\nFor a cycle \\(\\Gamma\\) and \\(z\\notin\\Gamma^*\\), the *index* is\n\\[\n\\operatorname{Ind}_\\Gamma(z)=\\frac1{2\\pi i}\\int_\\Gamma\\frac{d\\lambda}{\\lambda-z}.\n\\]\n\n**Lemma 1.1.**\n1. \\(\\operatorname{Ind}_\\Gamma\\) takes integer values on \\(\\mathbb C\\setminus\\Gamma^*\\), is constant on each connected component, and is \\(0\\) on the unbounded component.\n2. For the circle \\(\\partial D(c,r)\\), the index is \\(1\\) on \\(D(c,r)\\) and \\(0\\) outside \\(\\bar D(c,r)\\).\n3. For the positively oriented boundary \\(\\partial Q\\) of a closed square \\(Q\\), the index is \\(1\\) on the interior of \\(Q\\) and \\(0\\) outside \\(Q\\).\n\n**Proof.** (1) *Integer values.* Let \\(\\gamma:[a,b]\\to\\mathbb C\\) be a closed path and \\(z\\notin\\gamma^*\\). Put\n\\[\n\\begin{gathered}\nh(t)\\\\\n=\\int_a^t\\frac{\\gamma'(s)}{\\gamma(s)-z}\\,ds,\\\\\nF(t)\\\\\n=e^{-h(t)}(\\gamma(t)-z).\n\\end{gathered}\n\\]\nWherever \\(\\gamma\\) is differentiable, \\(F'=e^{-h}\\big(-h'(\\gamma-z)+\\gamma'\\big)=0\\). \\(F\\) is continuous, so it is constant. Since \\(\\gamma(b)=\\gamma(a)\\neq z\\), \\(F(b)=F(a)\\) gives \\(e^{-h(b)}=1\\). So \\(h(b)/(2\\pi i)\\) is an integer. The index of a cycle is a sum of such integers.\n\n*Constancy.* Let \\(d(z)=\\operatorname{dist}(z,\\Gamma^*)\\). For \\(z,w\\notin\\Gamma^*\\),\n\\[\n\\begin{gathered}\n|\\operatorname{Ind}_\\Gamma(z)-\\operatorname{Ind}_\\Gamma(w)|\\\\\n=\\frac1{2\\pi}\\Big|\\int_\\Gamma\\frac{(z-w)\\,d\\lambda}{(\\lambda-z)(\\lambda-w)}\\Big|\\\\\n\\leq\\frac{\\ell(\\Gamma)|z-w|}{2\\pi d(z)d(w)} .\n\\end{gathered}\n\\]\nSo the index is continuous. A continuous integer-valued function is constant on connected sets. Also \\(|\\operatorname{Ind}_\\Gamma(z)|\\leq\\ell(\\Gamma)/(2\\pi d(z))<1\\) for \\(d(z)\\) large, so the index vanishes far out, hence on the unbounded component.\n\n(2) At the centre, \\(\\frac1{2\\pi i}\\int_0^{2\\pi}\\frac{ire^{it}}{re^{it}}\\,dt=1\\). The open disc is connected, and the complement of the closed disc is connected and unbounded. Apply (1).\n\n(3) After a translation and a scaling, which do not change the index, \\(Q=[-1,1]^2\\) with centre \\(0\\). Multiplication by \\(i\\) maps each side of \\(\\partial Q\\) onto the next one and leaves \\(d\\lambda/\\lambda\\) unchanged. So the four sides contribute equally. On the right side, \\(\\lambda=1+it\\) with \\(t\\in[-1,1]\\), and\n\\[\n\\begin{gathered}\n\\int_{-1}^1\\frac{i\\,dt}{1+it}\\\\\n=\\int_{-1}^1\\frac{t+i}{1+t^2}\\,dt\\\\\n=i\\int_{-1}^1\\frac{dt}{1+t^2}\\\\\n=\\frac{i\\pi}2 .\n\\end{gathered}\n\\]\nSo the index at \\(0\\) is \\(4\\cdot\\frac{i\\pi}2/(2\\pi i)=1\\), and it is \\(1\\) on the connected interior.\n\nThe complement of \\(Q\\) is connected: the ray from \\(0\\) through a point outside \\(Q\\) leaves the convex set \\(Q\\) for good. It is unbounded, so it lies in the unbounded component, where the index is \\(0\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CT-02",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "2. Cauchy's theorem in convex sets",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
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      "full_conditions_and_proof": "## 2. Cauchy's theorem in convex sets\n\n**Theorem 2.1** (Goursat). Let \\(V\\) be open, \\(p\\in V\\), and \\(f:V\\to\\mathbb C\\) continuous on \\(V\\) and holomorphic on \\(V\\setminus\\{p\\}\\). Then \\(\\int_{\\partial\\Delta}f=0\\) for every closed triangle \\(\\Delta\\subseteq V\\).\n\n**Proof.** If the vertices are collinear, the three segments go back and forth along one segment, and the integrals cancel. Let \\(\\Delta\\) be nondegenerate, with diameter \\(d\\) and perimeter \\(\\ell\\).\n\n*Case 1: \\(p\\notin\\Delta\\).* Let \\(c=|\\int_{\\partial\\Delta}f|\\).\n- The midpoints of the sides cut \\(\\Delta\\) into four triangles. Oriented like \\(\\Delta\\), their boundary integrals add up to \\(\\int_{\\partial\\Delta}f\\), because the inner edges cancel. So one of them, \\(\\Delta_1\\), has \\(|\\int_{\\partial\\Delta_1}f|\\geq c/4\\).\n- Repeating this gives closed triangles \\(\\Delta\\supseteq\\Delta_1\\supseteq\\Delta_2\\supseteq\\cdots\\) with \\(|\\int_{\\partial\\Delta_n}f|\\geq c/4^n\\), diameter \\(d/2^n\\) and perimeter \\(\\ell/2^n\\).\n- They have a common point \\(z_0\\in\\Delta\\), and \\(z_0\\neq p\\). Write \\(R(z)\\) for \\(f(z)-f(z_0)-f'(z_0)(z-z_0)\\). Given \\(\\varepsilon>0\\), choose \\(\\delta>0\\) with\n\\[\n\\begin{gathered}\n|f(z)-f(z_0)-f'(z_0)(z-z_0)|\\\\\n\\leq\\varepsilon|z-z_0|\\\\\n(|z-z_0|<\\delta).\n\\end{gathered}\n\\]\n- The function \\(f(z_0)+f'(z_0)(z-z_0)\\) is a polynomial, so its integral over \\(\\partial\\Delta_n\\) is \\(0\\). Once \\(d/2^n<\\delta\\),\n\\[\n\\begin{gathered}\n\\frac c{4^n}\\\\\n\\leq\\Big|\\int_{\\partial\\Delta_n}R(z)\\,dz\\Big|\\\\\n\\leq\\frac\\ell{2^n}\\cdot\\varepsilon\\frac d{2^n} .\n\\end{gathered}\n\\]\n- So \\(c\\leq\\varepsilon\\ell d\\) for every \\(\\varepsilon>0\\), and \\(c=0\\).\n\n*Case 2: \\(p\\) is a vertex.* Let the vertices be \\(p,b,e\\). Choose \\(b'\\in[p,b]\\) and \\(e'\\in[p,e]\\) near \\(p\\), different from \\(p\\).\n- \\(\\int_{\\partial\\Delta}f\\) is the sum of the boundary integrals over the triangles \\((p,b',e')\\), \\((b',b,e)\\) and \\((b',e,e')\\).\n- The last two do not contain \\(p\\), so their integrals vanish by Case 1.\n- The first is at most \\(\\ell(\\partial(p,b',e'))\\max_\\Delta|f|\\) in absolute value. This tends to \\(0\\) as \\(b',e'\\to p\\).\n\n*Case 3: \\(p\\in\\Delta\\) is not a vertex.* Cut \\(\\Delta\\) into the three triangles with vertex \\(p\\) and the sides of \\(\\Delta\\) as opposite sides. Their boundary integrals add up to \\(\\int_{\\partial\\Delta}f\\), and each vanishes by Case 2. \\(\\square\\)\n\n**Theorem 2.2** (Cauchy's theorem in a convex set). Let \\(V\\) be convex and open, \\(p\\in V\\), and \\(f\\) continuous on \\(V\\) and holomorphic on \\(V\\setminus\\{p\\}\\). Fix \\(a\\in V\\) and put \\(F(z)=\\int_{[a,z]}f\\). Then \\(F\\) is holomorphic on \\(V\\) with \\(F'=f\\). Hence \\(\\int_\\gamma f=0\\) for every closed path \\(\\gamma\\) in \\(V\\).\n\n**Proof.** For \\(z,z+h\\in V\\), the triangle with vertices \\(a,z,z+h\\) lies in \\(V\\), by convexity. By Theorem 2.1, \\(F(z+h)-F(z)=\\int_{[z,z+h]}f\\). So\n\\[\n\\begin{gathered}\n\\Big|\\frac{F(z+h)-F(z)}h-f(z)\\Big|\\\\\n=\\Big|\\frac1h\\int_{[z,z+h]}(f(w)-f(z))\\,dw\\Big|\\\\\n\\leq\\max_{w\\in[z,z+h]}|f(w)-f(z)|\\to0 .\n\\end{gathered}\n\\]\nThe last claim follows from the Conventions on primitives. \\(\\square\\)\n\n**Theorem 2.3** (Cauchy's formula in a convex set). Let \\(V\\) be convex and open, \\(f\\) holomorphic on \\(V\\), \\(\\gamma\\) a closed path in \\(V\\), and \\(z\\in V\\setminus\\gamma^*\\). Then\n\\[\nf(z)\\operatorname{Ind}_\\gamma(z)=\\frac1{2\\pi i}\\int_\\gamma\\frac{f(w)}{w-z}\\,dw .\n\\]\n\n**Proof.** Put \\(g(w)=(f(w)-f(z))/(w-z)\\) for \\(w\\neq z\\), and \\(g(z)=f'(z)\\). Then \\(g\\) is continuous on \\(V\\) and holomorphic on \\(V\\setminus\\{z\\}\\). By Theorem 2.2, \\(\\int_\\gamma g=0\\), which is the formula. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CT-03",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "3. Power series and their consequences",
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      "anchor": "oa-fnd-ct-03",
      "proof_locus": {
        "line": 201,
        "through_line": 290
      },
      "full_conditions_and_proof": "## 3. Power series and their consequences\n\n**Lemma 3.1** (Power series). Let \\(c\\in\\mathbb C\\), \\(0<R\\leq\\infty\\), and let \\(a_n\\in\\mathbb C\\) satisfy \\(\\sum_n|a_n|\\rho^n<\\infty\\) for every \\(0<\\rho<R\\). Then:\n- \\(f(z)=\\sum_na_n(z-c)^n\\) is holomorphic on \\(D(c,R)\\), with \\(f'(z)=\\sum_{n\\geq1}na_n(z-c)^{n-1}\\);\n- the coefficients of \\(f'\\) satisfy the same hypothesis.\n\nConsequently \\(f\\) has derivatives of all orders, and \\(a_n=f^{(n)}(c)/n!\\).\n\n**Proof.** Take \\(c=0\\), and fix \\(0<\\rho<\\rho'<R\\).\n- *The derived coefficients.* \\(n|a_n|\\rho^{n-1}\\leq|a_n|\\rho'^n\\cdot\\frac n\\rho(\\rho/\\rho')^n\\), and \\(n(\\rho/\\rho')^n\\to0\\). So \\(\\sum_{n\\geq1}n|a_n|\\rho^{n-1}<\\infty\\). In the same way \\(\\sum_{n\\geq2}n^2|a_n|\\rho^{n-2}<\\infty\\).\n- *An estimate.* The constant term has zero difference quotient and the \\(n=1\\) term has zero derivative remainder. For \\(n\\geq2\\) and \\(|z|,|w|\\leq\\rho\\) with \\(z\\neq w\\),\n\\[\n\\begin{gathered}\n\\frac{z^n-w^n}{z-w}-nw^{n-1}\\\\\n=\\sum_{k=0}^{n-1}w^k\\big(z^{n-1-k}-w^{n-1-k}\\big).\n\\end{gathered}\n\\]\n  Since \\(|z^m-w^m|\\leq m\\rho^{m-1}|z-w|\\), its absolute value is at most \\(n^2\\rho^{n-2}|z-w|\\).\n- *The derivative.* So\n\\[\n\\begin{gathered}\n\\Big|\\frac{f(z)-f(w)}{z-w}-\\sum_{n\\geq1}na_nw^{n-1}\\Big|\\\\\n\\leq|z-w|\\sum_{n\\geq2}n^2|a_n|\\rho^{n-2},\n\\end{gathered}\n\\]\n  which tends to \\(0\\) as \\(z\\to w\\).\n- Repeating, \\(f\\) has derivatives of all orders. Evaluating the series of \\(f^{(n)}\\) at \\(0\\) gives \\(f^{(n)}(0)=n!a_n\\). \\(\\square\\)\n\n**Theorem 3.2** (Holomorphic functions are analytic). Let \\(f\\) be holomorphic on \\(U\\supseteq D(c,R)\\). Then\n\\[\nf(z)=\\sum_na_n(z-c)^n\\qquad(z\\in D(c,R)),\n\\]\nwith \\(\\sum_n|a_n|\\rho^n<\\infty\\) for every \\(0<\\rho<R\\). For every \\(0<r<R\\),\n\\[\n\\begin{gathered}\na_n\\\\\n=\\frac{f^{(n)}(c)}{n!}\\\\\n=\\frac1{2\\pi i}\\int_{\\partial D(c,r)}\\frac{f(w)}{(w-c)^{n+1}}\\,dw,\\\\\n|a_n|\\\\\n\\leq\\frac{M(r)}{r^n},\n\\end{gathered}\n\\]\nwhere \\(M(r)=\\max_{|w-c|=r}|f(w)|\\) (*Cauchy's estimates*). In particular \\(f'\\) is holomorphic, and \\(f\\) has derivatives of all orders.\n\n**Proof.** Fix \\(0<r<R\\) and \\(z\\) with \\(|z-c|<r\\). The disc \\(D(c,R)\\) is convex, and the index of \\(\\partial D(c,r)\\) at \\(z\\) is \\(1\\) (Lemma 1.1). By Theorem 2.3,\n\\[\nf(z)=\\frac1{2\\pi i}\\int_{\\partial D(c,r)}\\frac{f(w)}{w-z}\\,dw .\n\\]\nFor \\(|w-c|=r\\),\n\\[\n\\frac1{w-z}=\\sum_n\\frac{(z-c)^n}{(w-c)^{n+1}},\n\\]\nuniformly in \\(w\\), because \\(|z-c|/r<1\\). Integrating term by term gives \\(f(z)=\\sum_na_n(r)(z-c)^n\\), with \\(a_n(r)\\) the integral in the statement and \\(|a_n(r)|\\leq M(r)r^{-n}\\). This series converges absolutely for \\(|z-c|<r\\). By Lemma 3.1, \\(a_n(r)=f^{(n)}(c)/n!\\), which does not depend on \\(r\\). Since \\(r<R\\) is arbitrary, the expansion holds on \\(D(c,R)\\). \\(\\square\\)\n\n**Corollary 3.3** (Liouville). A bounded holomorphic function \\(f:\\mathbb C\\to\\mathbb C\\) is constant.\n\n**Proof.** If \\(|f|\\leq M\\), Cauchy's estimates at \\(c=0\\) give \\(|a_n|\\leq M/r^n\\) for every \\(r>0\\). So \\(a_n=0\\) for \\(n\\geq1\\). \\(\\square\\)\n\n**Theorem 3.4** (Morera). Let \\(f\\) be continuous on \\(U\\), with \\(\\int_{\\partial\\Delta}f=0\\) for every closed triangle \\(\\Delta\\subseteq U\\). Then \\(f\\) is holomorphic.\n\n**Proof.** On each disc \\(D(c,r)\\subseteq U\\), the function \\(F(z)=\\int_{[c,z]}f\\) satisfies \\(F'=f\\), by the proof of Theorem 2.2, which used only the vanishing of triangle integrals. So \\(F\\) is holomorphic, and by Theorem 3.2 so is \\(F'=f\\). \\(\\square\\)\n\n**Corollary 3.5** (Weierstrass). Let \\(f_n\\) be holomorphic on \\(U\\), and let \\(f_n\\to f\\) uniformly on every compact subset of \\(U\\). Then \\(f\\) is holomorphic.\n\n**Proof.** \\(f\\) is continuous. For a closed triangle \\(\\Delta\\subseteq U\\), \\(\\int_{\\partial\\Delta}f=\\lim_n\\int_{\\partial\\Delta}f_n=0\\), because the convergence is uniform on the compact set \\(\\partial\\Delta\\). Apply Morera's theorem. \\(\\square\\)\n\n**Theorem 3.6** (Maximum modulus). Let \\(f\\) be continuous on the closed unit disc \\(\\bar D=\\bar D(0,1)\\) and holomorphic on \\(D(0,1)\\). Then\n\\[\n\\max_{\\bar D}|f|=\\max_{|z|=1}|f(z)| .\n\\]\nIn particular, if \\(f=0\\) on the unit circle, then \\(f=0\\).\n\n**Proof.** Let \\(M=\\max_{\\bar D}|f|\\), attained at \\(c\\). Suppose \\(|c|<1\\), and put \\(r_0=1-|c|\\).\n- *Mean values.* For \\(0<\\rho<r_0\\), \\(f\\) is holomorphic on the disc \\(D(c,r_0)\\). Theorem 3.2 at the centre, with \\(a_0=f(c)\\), gives\n\\[\nf(c)=\\frac1{2\\pi}\\int_0^{2\\pi}f(c+\\rho e^{it})\\,dt .\n\\]\n- So \\(M=|f(c)|\\leq\\frac1{2\\pi}\\int_0^{2\\pi}|f(c+\\rho e^{it})|\\,dt\\leq M\\). The integrand is continuous and at most \\(M\\), so it equals \\(M\\) everywhere: \\(|f|=M\\) on the circle \\(|z-c|=\\rho\\).\n- Let \\(\\rho\\to r_0\\). By continuity on \\(\\bar D\\), \\(|f|=M\\) at every point with \\(|z-c|=r_0\\). One of these points lies on the unit circle: \\(c+r_0c/|c|\\) if \\(c\\neq0\\), and any point of the circle if \\(c=0\\). \\(\\square\\)\n\n**Theorem 3.7** (Identity theorem). Let \\(U\\) be connected and \\(f\\) holomorphic on \\(U\\). If the zeros of \\(f\\) have an accumulation point in \\(U\\), then \\(f=0\\). In particular, two holomorphic functions on \\(U\\) that agree on a nonempty open subset agree everywhere.\n\n**Proof.** Let \\(Z\\) be the set of \\(z\\in U\\) with \\(f^{(n)}(z)=0\\) for every \\(n\\geq0\\).\n- \\(Z\\) is closed in \\(U\\), because each \\(f^{(n)}\\) is continuous (Theorem 3.2).\n- \\(Z\\) is open: if \\(z\\in Z\\) and \\(D(z,r)\\subseteq U\\), the Taylor expansion of Theorem 3.2 shows \\(f=0\\) on \\(D(z,r)\\), so all derivatives vanish there.\n- Let \\(a\\in U\\) be an accumulation point of zeros, and suppose \\(a\\notin Z\\). Let \\(m\\) be the least \\(n\\) with \\(f^{(n)}(a)\\neq0\\). By Theorem 3.2, \\(f(z)=(z-a)^mg(z)\\) near \\(a\\), where \\(g(z)=\\sum_{n\\geq m}a_n(z-a)^{n-m}\\) is continuous with \\(g(a)=a_m\\neq0\\). So \\(f\\) has no zeros near \\(a\\) other than \\(a\\), a contradiction. Hence \\(a\\in Z\\).\n- So \\(Z\\) is nonempty, open and closed in the connected set \\(U\\), and \\(Z=U\\).\n\nFor the last claim, apply this to the difference of the two functions. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces"
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    },
    {
      "id": "OA-FND-CT-04",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "4. Cauchy's theorem for cycles",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
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      "anchor": "oa-fnd-ct-04",
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      "full_conditions_and_proof": "## 4. Cauchy's theorem for cycles\n\n**Lemma 4.1.** The image of a path has empty interior. So does \\(\\Gamma^*\\) for a cycle \\(\\Gamma\\). Hence \\(U\\setminus\\Gamma^*\\neq\\varnothing\\) for every nonempty open \\(U\\).\n\n**Proof.** *One path.* Let \\(\\gamma:[a,b]\\to\\mathbb C\\) be a path, and \\(L=\\sup|\\gamma'|\\) over its pieces. Then \\(|\\gamma(s)-\\gamma(t)|\\leq L|s-t|\\). If \\(a=b\\) or \\(L=0\\), the image is a single point and has empty interior. Otherwise \\(L(b-a)>0\\). Suppose \\(\\gamma^*\\) contains a closed square \\(Q\\) of side \\(\\sigma\\).\n- Cut \\([a,b]\\) into \\(N\\) equal intervals. Their images are \\(N\\) sets of diameter at most \\(\\varepsilon=L(b-a)/N\\), and they cover \\(Q\\).\n- Let \\(M=\\lfloor\\sigma/(2\\varepsilon)\\rfloor\\), and take \\(N\\) so large that \\(M\\geq1\\). The \\((M+1)^2\\) grid points of \\(Q\\) with spacing \\(\\sigma/M\\) are at mutual distance at least \\(\\sigma/M\\geq2\\varepsilon>\\varepsilon\\). So each of the \\(N\\) sets contains at most one of them, and \\(N\\geq(M+1)^2>(\\sigma N/(2L(b-a)))^2\\).\n- This fails for large \\(N\\).\n\n*A cycle.* Let \\(\\Gamma\\) consist of \\(\\gamma_1,\\dots,\\gamma_m\\), and suppose a nonempty open set \\(W\\) lies in \\(\\Gamma^*\\). Then \\(W\\setminus\\gamma_1^*\\) is open, because \\(\\gamma_1^*\\) is compact. It is nonempty, because \\(\\gamma_1^*\\) has empty interior. It lies in \\(\\gamma_2^*\\cup\\dots\\cup\\gamma_m^*\\). Repeating, we reach a nonempty open set inside a single \\(\\gamma_k^*\\), which is impossible. \\(\\square\\)\n\n**Theorem 4.2** (Cauchy's theorem for cycles). Let \\(\\Gamma\\) be a cycle in \\(U\\) with \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for every \\(\\alpha\\notin U\\), and let \\(f\\) be holomorphic on \\(U\\). Then\n\\[\n\\begin{gathered}\nf(z)\\operatorname{Ind}_\\Gamma(z)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{f(w)}{w-z}\\,dw\\\\\n(z\\in U\\setminus\\Gamma^*),\n\\\\\n\\text{and}\\\\\n\\int_\\Gamma f(w)\\,dw\\\\\n=0 .\n\\end{gathered}\n\\]\n\n*Reference:* The Dixon argument for a closed curve is presented in [MIT OCW Lecture 13, by Zuoqin Wang for Sigurdur Helgason’s course](https://ocw.mit.edu/courses/18-112-functions-of-a-complex-variable-fall-2008/8793d412fa0da2a4183539f5d8f7e3fd_lecture13.pdf). The proof below includes the finite-cycle formulation.\n\n**Proof.** *Step 1: a continuous difference quotient.* Define \\(\\varphi\\) on \\(U\\times U\\) by\n\\[\n\\begin{gathered}\n\\varphi(z,w)\\\\\n=\\frac{f(w)-f(z)}{w-z}\\\\\n(z\\neq w),\\\\\n\\varphi(z,z)\\\\\n=f'(z).\n\\end{gathered}\n\\]\n\\(\\varphi\\) is continuous at points with \\(z\\neq w\\). Let \\(a\\in U\\) and \\(D(a,r)\\subseteq U\\).\n- \\(f'\\) is holomorphic, hence continuous (Theorem 3.2), and \\(f\\) is a primitive of \\(f'\\).\n- So for \\(z,w\\in D(a,r)\\), \\(f(w)-f(z)=\\int_{[z,w]}f'\\), which gives\n\\[\n\\varphi(z,w)=\\int_0^1f'\\big(z+t(w-z)\\big)\\,dt .\n\\]\n  This also holds for \\(z=w\\).\n- As \\((z,w)\\to(a,a)\\), the integrand tends to \\(f'(a)\\) uniformly in \\(t\\). So \\(\\varphi\\) is continuous at \\((a,a)\\).\n\n*Step 2: a holomorphic function on \\(U\\).* Put\n\\[\ng(z)=\\frac1{2\\pi i}\\int_\\Gamma\\varphi(z,w)\\,dw\\qquad(z\\in U).\n\\]\n- \\(g\\) is continuous, because \\(\\varphi\\) is uniformly continuous on \\(K\\times\\Gamma^*\\) for each compact \\(K\\subseteq U\\).\n- Let \\(\\Delta\\subseteq U\\) be a closed triangle. After parametrizing \\(\\partial\\Delta\\) and the paths of \\(\\Gamma\\), the double integral is an iterated integral of a continuous function on finitely many rectangles. Exchanging the order of integration gives\n\\[\n\\begin{gathered}\n\\int_{\\partial\\Delta}g(z)\\,dz\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\Big(\\int_{\\partial\\Delta}\\varphi(z,w)\\,dz\\Big)dw .\n\\end{gathered}\n\\]\n- For fixed \\(w\\), \\(z\\mapsto\\varphi(z,w)\\) is continuous on \\(U\\) and holomorphic on \\(U\\setminus\\{w\\}\\). By Goursat's theorem 2.1, the inner integral is \\(0\\).\n- By Morera's theorem, \\(g\\) is holomorphic on \\(U\\).\n\n*Step 3: an entire function.* Let \\(V_0=\\{z\\notin\\Gamma^*:\\operatorname{Ind}_\\Gamma(z)=0\\}\\). It is open by Lemma 1.1. It contains \\(\\mathbb C\\setminus U\\) by hypothesis, so \\(U\\cup V_0=\\mathbb C\\). Put\n\\[\ng_1(z)=\\frac1{2\\pi i}\\int_\\Gamma\\frac{f(w)}{w-z}\\,dw\\qquad(z\\in V_0).\n\\]\n- \\(g_1\\) is holomorphic on \\(V_0\\), by the argument of Step 2 applied to the continuous function \\((z,w)\\mapsto f(w)/(w-z)\\) on \\(V_0\\times\\Gamma^*\\), which is holomorphic in \\(z\\).\n- For \\(z\\in U\\cap V_0\\), \\(g(z)=g_1(z)-f(z)\\operatorname{Ind}_\\Gamma(z)=g_1(z)\\).\n- So \\(G=g\\) on \\(U\\) and \\(G=g_1\\) on \\(V_0\\) is a well-defined holomorphic function on \\(\\mathbb C\\).\n\n*Step 4: \\(G=0\\).* Choose \\(R\\) with \\(\\Gamma^*\\subseteq D(0,R)\\). The set \\(\\{|z|>R\\}\\) is connected and unbounded, so it lies in the unbounded component of \\(\\mathbb C\\setminus\\Gamma^*\\), and hence in \\(V_0\\). There\n\\[\n\\begin{gathered}\n|G(z)|\\\\\n=|g_1(z)|\\\\\n\\leq\\frac{\\ell(\\Gamma)\\max_{\\Gamma^*}|f|}{2\\pi(|z|-R)}\\to0\\\\\n(|z|\\to\\infty).\n\\end{gathered}\n\\]\nSo \\(G\\) is bounded: it is continuous on a large closed disc and small outside it. By Liouville's theorem, \\(G\\) is constant, and the constant is \\(0\\).\n\n*Step 5: conclusion.* For \\(z\\in U\\setminus\\Gamma^*\\),\n\\[\n0=g(z)=\\frac1{2\\pi i}\\int_\\Gamma\\frac{f(w)}{w-z}\\,dw-f(z)\\operatorname{Ind}_\\Gamma(z).\n\\]\nThis is the formula. If \\(U=\\varnothing\\) there is nothing to prove. Otherwise choose \\(z\\in U\\setminus\\Gamma^*\\) (Lemma 4.1), and apply the formula to the holomorphic function \\(w\\mapsto(w-z)f(w)\\), which vanishes at \\(z\\):\n\\[\n\\frac1{2\\pi i}\\int_\\Gamma f(w)\\,dw=0\\cdot\\operatorname{Ind}_\\Gamma(z)=0 .\\qquad\\square\n\\]\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-CT-05",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "5. Cycles that surround a compact set",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
      "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "anchor": "oa-fnd-ct-05",
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        "line": 379,
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      },
      "full_conditions_and_proof": "## 5. Cycles that surround a compact set\n\nA cycle \\(\\Gamma\\) *surrounds* a compact set \\(K\\) in an open set \\(U\\supseteq K\\) if\n- \\(\\Gamma^*\\subseteq U\\setminus K\\),\n- \\(\\operatorname{Ind}_\\Gamma=1\\) on \\(K\\), and\n- \\(\\operatorname{Ind}_\\Gamma=0\\) on \\(\\mathbb C\\setminus U\\).\n\n**Theorem 5.1.** Let \\(K\\subseteq U\\) with \\(K\\) compact and \\(U\\) open. Some cycle made of finitely many oriented segments surrounds \\(K\\) in \\(U\\).\n\n**Proof.** If \\(K=\\varnothing\\), the empty cycle will do. Otherwise choose \\(\\delta>0\\) with \\(2\\delta<\\operatorname{dist}(K,\\mathbb C\\setminus U)\\); any \\(\\delta>0\\) if \\(U=\\mathbb C\\).\n\n*The squares.* Consider the grid of closed squares \\([j\\delta,(j+1)\\delta]\\times[k\\delta,(k+1)\\delta]\\), \\(j,k\\in\\mathbb Z\\). Let \\(Q_1,\\dots,Q_m\\) be those that meet \\(K\\); there are finitely many, since \\(K\\) is bounded. Each \\(Q_i\\) has diameter \\(\\delta\\sqrt2<2\\delta\\), so it lies in \\(U\\).\n\n*The edges.* List the oriented edges of the positively oriented boundaries \\(\\partial Q_1,\\dots,\\partial Q_m\\). An edge shared by two of the \\(Q_i\\) occurs twice, with opposite orientations. Remove all these pairs, and let \\(\\mathcal E\\) be the remaining oriented edges.\n- (a) For every continuous \\(F\\) on the union of the edges, \\(\\sum_{e\\in\\mathcal E}\\int_eF=\\sum_i\\int_{\\partial Q_i}F\\), because the removed pairs contribute opposite amounts.\n- (b) Every \\(e\\in\\mathcal E\\) lies in \\(U\\setminus K\\). Indeed, \\(e\\) is an edge of exactly one \\(Q_i\\), so \\(e\\subseteq U\\). The grid square on the other side of \\(e\\) is not among the \\(Q_i\\), so it does not meet \\(K\\), and it contains \\(e\\).\n- (c) At every grid vertex, as many edges of \\(\\mathcal E\\) end as start. This holds for the boundary of each square, hence for the full list, and removing a pair of opposite edges preserves it.\n\n*A cycle.* By (c), \\(\\mathcal E\\) splits into closed paths. Start with any edge and keep following unused edges. At a vertex other than the starting one, more edges have arrived than have left, so by (c) an unused edge leaves it. Since there are finitely many edges, the walk returns to the starting vertex and closes a path. Remove its edges; (c) still holds; repeat. Let \\(\\Gamma\\) be the resulting cycle. By (b), \\(\\Gamma^*\\subseteq U\\setminus K\\).\n\n*Index off \\(U\\).* Let \\(\\alpha\\notin U\\). Then \\(\\alpha\\) lies in no \\(Q_i\\), so \\(\\operatorname{Ind}_{\\partial Q_i}(\\alpha)=0\\) for every \\(i\\) (Lemma 1.1(3)). By (a), \\(\\operatorname{Ind}_\\Gamma(\\alpha)=\\sum_i\\operatorname{Ind}_{\\partial Q_i}(\\alpha)=0\\).\n\n*Index on \\(K\\).*\n- Let \\(w\\) be an interior point of some \\(Q_{i_0}\\). It lies in no other \\(Q_i\\), so by (a) and Lemma 1.1(3), \\(\\operatorname{Ind}_\\Gamma(w)=1\\).\n- Let \\(z\\in K\\). It lies in some grid square, which meets \\(K\\), so it is some \\(Q_{i_0}\\). By (b), \\(z\\notin\\Gamma^*\\). Interior points \\(w\\) of \\(Q_{i_0}\\) with \\(|w-z|<\\operatorname{dist}(z,\\Gamma^*)\\) exist, and the segment \\([z,w]\\) misses \\(\\Gamma^*\\). So \\(z\\) and \\(w\\) lie in one component of \\(\\mathbb C\\setminus\\Gamma^*\\), and \\(\\operatorname{Ind}_\\Gamma(z)=\\operatorname{Ind}_\\Gamma(w)=1\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-CT-06",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "6. Functions with values in a Banach space",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
      "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "anchor": "oa-fnd-ct-06",
      "proof_locus": {
        "line": 405,
        "through_line": 486
      },
      "full_conditions_and_proof": "## 6. Functions with values in a Banach space\n\n**Theorem 6.1.** Let \\(X\\) be a Banach space and \\(g:U\\to X\\) a map such that \\(\\varphi\\circ g\\) is holomorphic for every \\(\\varphi\\in X^*\\).\n1. \\(g\\) is continuous; it is even Lipschitz on a neighbourhood of each point.\n2. For every cycle \\(\\Gamma\\) in \\(U\\) with \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for all \\(\\alpha\\notin U\\),\n\\[\n\\begin{gathered}\n\\int_\\Gamma g(w)\\,dw\\\\\n=0,\\\\\ng(z)\\operatorname{Ind}_\\Gamma(z)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{g(w)}{w-z}\\,dw\\\\\n(z\\in U\\setminus\\Gamma^*).\n\\end{gathered}\n\\]\n3. On each disc \\(D(c,R)\\subseteq U\\), \\(g(z)=\\sum_na_n(z-c)^n\\) with \\(a_n\\in X\\) and \\(\\sum_n\\|a_n\\|\\rho^n<\\infty\\) for \\(\\rho<R\\). In particular the difference quotients \\((g(z+h)-g(z))/h\\) converge in norm as \\(h\\to0\\).\n4. (*Liouville.*) If \\(U=\\mathbb C\\) and \\(g\\) is bounded, then \\(g\\) is constant.\n\n**Proof.** (1) Fix \\(a\\in U\\) and \\(r>0\\) with \\(\\bar D(a,2r)\\subseteq U\\). By compactness, \\(D(a,2r+\\varepsilon)\\subseteq U\\) for some \\(\\varepsilon>0\\); this disc is convex. Let \\(C=\\partial D(a,2r)\\), \\(\\varphi\\in X^*\\), and \\(z\\neq z'\\) in \\(D(a,r)\\).\n- By Theorem 2.3 for \\(\\varphi\\circ g\\), with index \\(1\\) at \\(z\\) and \\(z'\\),\n\\[\n\\begin{gathered}\n\\varphi(g(z))-\\varphi(g(z'))\\\\\n=\\frac{z-z'}{2\\pi i}\\int_C\\frac{\\varphi(g(w))}{(w-z)(w-z')}\\,dw .\n\\end{gathered}\n\\]\n- On \\(C\\), \\(|w-z|\\geq r\\) and \\(|w-z'|\\geq r\\). With \\(M_\\varphi=\\max_C|\\varphi\\circ g|\\), which is finite because \\(\\varphi\\circ g\\) is continuous, we get\n\\[\n\\begin{gathered}\n\\frac{|\\varphi(g(z))-\\varphi(g(z'))|}{|z-z'|}\\\\\n\\leq\\frac{4\\pi r}{2\\pi}\\cdot\\frac{M_\\varphi}{r^2}\\\\\n=\\frac{2M_\\varphi}r .\n\\end{gathered}\n\\]\n- So the set of quotients \\((g(z)-g(z'))/(z-z')\\), for \\(z\\neq z'\\) in \\(D(a,r)\\), is bounded under every \\(\\varphi\\in X^*\\). By the first lesson, Corollary 4.3(2), it is norm bounded. So \\(g\\) is Lipschitz on \\(D(a,r)\\).\n\n(2) By (1) the integrals exist. For every \\(\\varphi\\in X^*\\), \\(\\varphi\\) passes through the integrals, and Theorem 4.2 for \\(\\varphi\\circ g\\) gives the two identities after \\(\\varphi\\) is applied. Bounded functionals separate the points of \\(X\\) (the first lesson, Corollary 2.3(2)).\n\n(3) Let \\(0<r<R\\) and \\(|z-c|<r\\). The cycle \\(\\partial D(c,r)\\) has index \\(0\\) off \\(\\bar D(c,r)\\subseteq U\\). By (2),\n\\[\ng(z)=\\frac1{2\\pi i}\\int_{\\partial D(c,r)}\\frac{g(w)}{w-z}\\,dw .\n\\]\nExpanding as in the proof of Theorem 3.2 gives \\(g(z)=\\sum_na_n(z-c)^n\\), with \\(\\|a_n\\|\\leq\\max_{|w-c|=r}\\|g(w)\\|/r^n\\). The \\(a_n\\) do not depend on \\(r\\), because \\(\\varphi(a_n)=(\\varphi\\circ g)^{(n)}(c)/n!\\) for every \\(\\varphi\\). The proof of Lemma 3.1 works verbatim with norms in place of absolute values, so the difference quotients converge in norm.\n\n(4) Each \\(\\varphi\\circ g\\) is bounded and holomorphic on \\(\\mathbb C\\), hence constant. So \\(\\varphi(g(z)-g(0))=0\\) for every \\(\\varphi\\), and \\(g(z)=g(0)\\). \\(\\square\\)\n\n## Exercises\n\n**Exercise 1** (medium; The fundamental theorem of algebra). Show that every nonconstant polynomial \\(p\\) has a complex root.\n\n*Solution.* Write \\(p(z)=c_dz^d+\\dots+c_0\\) with \\(d\\geq1\\) and \\(c_d\\neq0\\). Then \\(|p(z)|\\geq|c_d||z|^d/2\\) for large \\(|z|\\), so \\(|p(z)|\\to\\infty\\). If \\(p\\) had no root, \\(1/p\\) would be holomorphic on \\(\\mathbb C\\). It would be bounded: continuous on a large closed disc and small outside. By Liouville's theorem it would be constant, and then so would \\(p\\). \\(\\square\\)\n\n**Exercise 2** (easy; The condition on the index). Let \\(U=\\mathbb C\\setminus\\{0\\}\\), \\(f(z)=1/z\\), and let \\(\\Gamma\\) be the unit circle. Compute \\(\\int_\\Gamma f\\), and say which hypothesis of Theorem 4.2 fails.\n\n*Solution.* \\(\\int_\\Gamma dz/z=2\\pi i\\operatorname{Ind}_\\Gamma(0)=2\\pi i\\neq0\\). The hypothesis that fails is \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for \\(\\alpha\\notin U\\): the point \\(0\\) is not in \\(U\\), and \\(\\operatorname{Ind}_\\Gamma(0)=1\\). \\(\\square\\)\n\n**Exercise 3** (medium; Schwarz's lemma). Let \\(f\\) be holomorphic on \\(D(0,1)\\) with \\(|f|\\leq1\\) and \\(f(0)=0\\). Show that \\(|f(z)|\\leq|z|\\) for all \\(z\\), and \\(|f'(0)|\\leq1\\).\n\n*Solution.*\n- By Theorem 3.2, \\(f(z)=\\sum_{n\\geq1}a_nz^n\\) on \\(D(0,1)\\). So \\(g(z)=\\sum_{n\\geq0}a_{n+1}z^n\\) is holomorphic on \\(D(0,1)\\) (Lemma 3.1), with \\(f(z)=zg(z)\\) and \\(g(0)=f'(0)\\).\n- For \\(0<r<1\\), the function \\(z\\mapsto g(rz)\\) is holomorphic on \\(D(0,1/r)\\supseteq\\bar D(0,1)\\). By Theorem 3.6, \\(|g(rz)|\\leq\\max_{|w|=1}|g(rw)|\\leq1/r\\) for \\(|z|\\leq1\\).\n- Letting \\(r\\to1\\) gives \\(|g|\\leq1\\) on \\(D(0,1)\\). \\(\\square\\)\n\n**Exercise 4** (medium; Polynomials converging on the circle). Let \\(p_n\\) be polynomials that converge uniformly on the unit circle \\(\\mathbb T\\). Show that they converge uniformly on \\(\\bar D(0,1)\\), and that the limit \\(G\\) is continuous on \\(\\bar D(0,1)\\) and holomorphic on \\(D(0,1)\\). Show that \\(G=0\\) if \\(G=0\\) on \\(\\mathbb T\\).\n\n*Solution.*\n- By Theorem 3.6 applied to \\(p_n-p_m\\), \\(\\max_{\\bar D}|p_n-p_m|=\\max_{\\mathbb T}|p_n-p_m|\\). So \\((p_n)\\) is uniformly Cauchy on \\(\\bar D\\), and its uniform limit \\(G\\) is continuous there.\n- \\(G\\) is holomorphic on \\(D(0,1)\\) by Corollary 3.5.\n- If \\(G=0\\) on \\(\\mathbb T\\), then \\(G=0\\) by Theorem 3.6. \\(\\square\\)\n\n## Where this leads\n\n[Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md) defines \\(f(x)\\) for \\(f\\) holomorphic near the spectrum of \\(x\\) as an integral over a cycle that surrounds the spectrum. Theorem 5.1 provides the cycle, and Theorems 4.2 and 6.1 show that the result does not depend on it. Exercise 4 is the step used in Example 3.4 of the lesson on C\\*-algebras. Liouville's theorem is used in [Kaplansky's density theorem and its consequences](kaplansky-s-density-theorem-and-its-consequences.md).\n\n## References\n\n- É. Goursat, \"Sur la définition générale des fonctions analytiques, d'après Cauchy\", *Transactions of the American Mathematical Society* 1 (1900), 14–16.\n\nThe proofs are written here in our own words.\n\n*Freely accessible reading:* [Zuoqin Wang; course instructor Sigurdur Helgason, *MIT OCW 18.112 Lecture 13*, Lecture 13, printed pp. 1–4](https://ocw.mit.edu/courses/18-112-functions-of-a-complex-variable-fall-2008/8793d412fa0da2a4183539f5d8f7e3fd_lecture13.pdf) gives a route through Dixon’s closed-curve proof; the finite-cycle proof and local inputs are included here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n\nFor an alternative grid proof, see [I. Černý, expanded author presentation (2012), pp. 1–5](https://matematika.cuni.cz/dl/cerny/cauchy.pdf). Its rectangular formula is an input; the local integral results are proved in this lesson.\n",
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      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "name": "Elementary integration tools",
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      "full_conditions_and_proof": "### Elementary integration tools\n\nThe following tools apply to functions with values in any real or complex Banach space \\(X\\). They supply the integration facts used throughout this lesson and in the Banach-algebra lesson.\n\n**Lemma 0.1** (Intervals, rectangles and parameters).\n\n1. A continuous \\(F:[a,b]\\to X\\) has a Riemann integral. It is linear, commutes with bounded linear maps, satisfies \\(\\|\\int_a^b F\\|\\leq\\int_a^b\\|F\\|\\), and respects uniform limits.\n2. The function \\(P(t)=\\int_a^t F(s)\\,ds\\) has derivative \\(P'(t)=F(t)\\) in the interior, with one-sided derivatives at the endpoints. If \\(G\\) is continuously differentiable, then \\(\\int_a^bG'(t)\\,dt=G(b)-G(a)\\). The chain rule and substitution hold for continuously differentiable real paths; a holomorphic primitive with continuous derivative satisfies the same chain rule along a piecewise continuously differentiable complex path.\n3. For continuous \\(F:[a,b]\\times[c,d]\\to X\\), the two iterated Riemann integrals agree.\n4. Suppose \\(K(t,s)\\) and its partial derivative \\(\\partial_tK(t,s)\\) are continuous on a rectangle. Then \\(t\\mapsto\\int_c^dK(t,s)\\,ds\\) is differentiable, with derivative \\(\\int_c^d\\partial_tK(t,s)\\,ds\\). When both variables range over \\([a,b]\\), the moving-endpoint integral satisfies\n\\[\n\\begin{gathered}\n\\frac{d}{dt}\\int_a^tK(t,s)\\,ds\n\\\\\n=K(t,t)+\\int_a^t\\partial_tK(t,s)\\,ds .\n\\end{gathered}\n\\]\n\n**Proof.** *Compactness and uniform continuity.* A closed bounded interval or rectangle is compact. To see the finite-cover statement directly, if an open cover has no finite subcover, bisect each coordinate interval and choose a resulting closed box still having no finite subcover. Continue with nested boxes. Completeness of the real numbers gives a common point, and the diameters tend to zero. A member of the cover containing that point contains every sufficiently small box, a contradiction. A continuous map on such a box is bounded and uniformly continuous. For the latter claim, given \\(\\varepsilon>0\\), choose for each \\(x\\) a radius \\(r_x>0\\) such that \\(\\|F(y)-F(x)\\|<\\varepsilon/2\\) when \\(|y-x|<2r_x\\) within the box. Finitely many balls \\(B(x_j,r_{x_j})\\) cover it. If \\(|y-z|<\\min_jr_{x_j}\\), choose a ball containing \\(y\\); both \\(y,z\\) are within \\(2r_{x_j}\\) of its centre, giving \\(\\|F(y)-F(z)\\|<\\varepsilon\\).\n\n*Riemann sums.* Assume \\(a<b\\); a degenerate interval has integral zero. Write\n\\[\n\\begin{gathered}\nS(F;\\mathcal P)\\\\\n=\\sum_j(t_j-t_{j-1})F(\\xi_j),\n\\\\\nt_{j-1}\\\\\n\\leq\\xi_j\\\\\n\\leq t_j,\n\\end{gathered}\n\\]\nfor a tagged partition, and let its mesh be the largest \\(t_j-t_{j-1}\\). Put \\(\\omega(\\delta)=\\sup\\{\\|F(u)-F(v)\\|:|u-v|\\leq\\delta\\}\\), so \\(\\omega(\\delta)\\to0\\). If a partition of mesh at most \\(\\delta\\) is refined, expanding each original summand over its subintervals shows that its sum differs from any tagged refined sum by at most \\((b-a)\\omega(\\delta)\\). Two partitions of mesh at most \\(\\delta\\) have a common refinement, so their sums differ by at most \\(2(b-a)\\omega(\\delta)\\). Sums along uniform partitions therefore converge by completeness of \\(X\\); comparison with those sums shows that all tagged sums tend to the same limit as their meshes tend to zero. This defines \\(\\int_a^bF\\). Passing the finite-sum identities to the limit proves linearity and commutation with bounded maps. The finite-sum estimate\n\\(\\|S(F;\\mathcal P)\\|\\leq\\sum_j(t_j-t_{j-1})\\|F(\\xi_j)\\|\\)\ngives the stated norm bound. In particular, a uniform change of size at most \\(\\eta\\) changes the integral by at most \\((b-a)\\eta\\), which proves the uniform-limit assertion. Splitting a partition at an interior point proves additivity over adjacent intervals. Define reversed integrals by changing sign.\n\n*The fundamental theorem and the chain rule.* For interior \\(t\\) and nonzero small \\(h\\), the norm of\n\\((P(t+h)-P(t))/h-F(t)\\)\nis at most \\(\\sup_{s\\text{ between }t,t+h}\\|F(s)-F(t)\\|\\), which tends to zero. This also gives the endpoint derivatives. To prove the converse, first recall the scalar mean value theorem: a continuous real function on an interval attains its extrema; if its endpoint values agree and it is not constant, some extremum is interior and its derivative is zero, since the two one-sided difference quotients have opposite signs. This is Rolle's theorem. Subtracting the line joining the endpoints gives the mean value theorem. Thus a scalar function with zero derivative is constant.\n\nIf \\(G\\) has continuous derivative \\(F\\), the function \\(G-P\\) has zero derivative. Compose it with each bounded linear functional on \\(X\\); in the complex case apply the real mean value theorem to both real and imaginary parts. These scalar functions are constant, and [Hahn–Banach separation, Corollary 2.3 of the first lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-02), shows that \\(G-P\\) itself is constant. This gives the converse fundamental theorem. For a differentiable real function \\(\\phi\\), inserting\n\\(\\phi(t+h)-\\phi(t)=\\phi'(t)h+o(|h|)\\)\ninto the differentiability expansion of \\(G\\) proves \\((G\\circ\\phi)'=(G'\\circ\\phi)\\phi'\\). The same calculation with complex increments proves the chain rule for a holomorphic \\(G\\) along a differentiable complex path. Continuity of the displayed derivative permits the fundamental theorem on each path piece. Real substitution follows by applying this chain rule to the primitive \\(P(u)=\\int_{u_0}^uF(v)\\,dv\\). No monotonicity of \\(\\phi\\) is needed for the oriented identity.\n\n*Rectangles.* Joint uniform continuity makes \\(x\\mapsto\\int_c^dF(x,y)\\,dy\\) continuous, since the norm of a difference is at most \\((d-c)\\sup_y\\|F(x,y)-F(x',y)\\|\\); the analogous statement holds in the other variable. Approximate \\(F\\) by the function constant on each cell of a rectangular grid, using a value of \\(F\\) at a corner of that cell. Give boundary points a consistent half-open-cell convention, with the final cells closed at the outer boundary. For grid mesh \\(\\delta\\), the uniform error is at most the modulus of continuity at \\(\\sqrt2\\delta\\). A one-variable step function with \\(m\\) division points is Riemann integrable: in a partition of mesh \\(\\eta\\), intervals meeting those points have total length at most \\(2m\\eta\\); all other summands give exactly the appropriate constant times length. The error on the exceptional intervals tends to zero, since the finitely many values are bounded. Each iterated integral of the rectangular step function is consequently the same finite sum\n\\(\\sum_{i,j}(t_i-t_{i-1})(s_j-s_{j-1})F(t_{i-1},s_{j-1})\\).\nBoth iterated integrals of \\(F\\) differ from this sum by at most the area of the rectangle times that modulus, which tends to zero. This proves (3), including its Banach-valued version.\n\n*Parameters.* Fix an interior parameter \\(t\\). The fundamental theorem, applied with \\(s\\) fixed, gives\n\\[\n\\frac{K(t+h,s)-K(t,s)}{h}\n=\\frac1h\\int_t^{t+h}\\partial_tK(u,s)\\,du .\n\\]\nJoint uniform continuity of \\(\\partial_tK\\) makes the right side converge to \\(\\partial_tK(t,s)\\) uniformly in \\(s\\). The uniform-limit assertion in (1) permits integration in \\(s\\), proving the fixed-endpoint formula. For \\(J(t)=\\int_a^tK(t,s)\\,ds\\), subtract to obtain\n\\[\n\\begin{gathered}\nJ(t+h)-J(t)\n\\\\\n=\\int_a^t\\big(K(t+h,s)-K(t,s)\\big)\\,ds\n\\\\\n+\\int_t^{t+h}K(t+h,s)\\,ds .\n\\end{gathered}\n\\]\nAfter division by \\(h\\), the first term has the limit already proved and the second tends to \\(K(t,t)\\) by joint continuity. This proves (4). \\(\\square\\)\n\n\n",
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      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "1. Banach algebras and C\\*-algebras",
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      "full_conditions_and_proof": "## 1. Banach algebras and C\\*-algebras\n\n**Definition 1.1.**\n1. An *involution* on an algebra \\(A\\) is a map \\(x\\mapsto x^*\\) with \\((x^*)^*=x\\), \\((x+y)^*=x^*+y^*\\), \\((\\lambda x)^*=\\bar\\lambda x^*\\) and \\((xy)^*=y^*x^*\\).\n2. An *involutive Banach algebra* is a Banach algebra together with an isometric involution: \\(\\|x^*\\|=\\|x\\|\\).\n3. A *C\\*-algebra* is an involutive Banach algebra in which \\(\\|x^*x\\|=\\|x^*\\|\\|x\\|\\) for every \\(x\\). Because the involution is isometric, this says \\(\\|x^*x\\|=\\|x\\|^2\\) (the *C\\*-identity*).\n\n**Proposition 1.2.**\n1. Multiplication is jointly continuous: \\[\n\\begin{gathered}\n\\|x_1y_1-x_2y_2\\|\\\\\n\\leq\\|x_1\\|\\|y_1-y_2\\|+\\|x_1-x_2\\|\\|y_2\\|.\n\\end{gathered}\n\\]\n2. For a Banach space \\(E\\), \\(B(E)\\) with the operator norm is a Banach algebra.\n3. (*The C\\*-identity alone is enough.*) Let \\(A\\) be an algebra with an involution and a complete submultiplicative norm such that \\(\\|x^*x\\|=\\|x\\|^2\\) for all \\(x\\). Then \\(\\|x^*\\|=\\|x\\|\\), so \\(A\\) is a C\\*-algebra.\n4. For an LCH space \\(\\Omega\\), \\(C_0(\\Omega)\\) is a commutative C\\*-algebra under pointwise operations, with \\(x^*=\\bar x\\). It is unital exactly when \\(\\Omega\\) is compact; then it is written \\(C(\\Omega)\\). This includes \\(\\Omega=\\varnothing\\): \\(C_0(\\varnothing)=\\{0\\}\\) is unital, and \\(\\varnothing\\) is compact.\n5. For a Hilbert space \\(H\\), \\(B(H)\\) with the operator adjoint is a C\\*-algebra. It is commutative exactly when \\(\\dim H\\leq1\\). Every norm-closed \\(*\\)-subalgebra of \\(B(H)\\) is a C\\*-algebra.\n\n**Proof.** (1) Write \\(x_1y_1-x_2y_2=x_1(y_1-y_2)+(x_1-x_2)y_2\\).\n\n(2) \\(\\|ST\\xi\\|\\leq\\|S\\|\\|T\\xi\\|\\leq\\|S\\|\\|T\\|\\|\\xi\\|\\). Let \\((T_n)\\) be Cauchy in operator norm. For each \\(\\xi\\), \\((T_n\\xi)\\) is Cauchy; put \\(T\\xi=\\lim_nT_n\\xi\\). Then \\(T\\) is linear, and \\(\\|T\\xi-T_n\\xi\\|\\leq\\sup_{m\\geq n}\\|T_m-T_n\\|\\,\\|\\xi\\|\\). So \\(T\\) is bounded and \\(T_n\\to T\\).\n\n(3) \\(\\|x\\|^2=\\|x^*x\\|\\leq\\|x^*\\|\\|x\\|\\), so \\(\\|x\\|\\leq\\|x^*\\|\\) (also when \\(x=0\\)). Apply this to \\(x^*\\).\n\n(4) Pointwise products and conjugates of functions in \\(C_0(\\Omega)\\) stay in \\(C_0(\\Omega)\\). The supremum norm is submultiplicative, conjugation is isometric, and \\(\\|\\bar xx\\|_\\infty=\\sup|x|^2=\\|x\\|_\\infty^2\\). For completeness, let \\((x_n)\\) be uniformly Cauchy. It converges uniformly to a continuous \\(x\\). Given \\(\\varepsilon>0\\), take \\(n\\) with \\(\\|x-x_n\\|_\\infty<\\varepsilon/2\\). Then \\(\\{|x|\\geq\\varepsilon\\}\\) is a closed subset of the compact set \\(\\{|x_n|\\geq\\varepsilon/2\\}\\), so \\(x\\in C_0(\\Omega)\\). If \\(\\Omega\\) is compact, the constant \\(1\\) lies in \\(C_0(\\Omega)\\). Conversely, let \\(u\\) be an identity. For \\(p\\in\\Omega\\), Urysohn's lemma gives \\(x\\in C_c(\\Omega)\\) with \\(x(p)=1\\), and \\(ux=x\\) gives \\(u(p)=1\\). So \\(u\\equiv1\\). Since \\(u\\) vanishes at infinity, \\(\\Omega=\\{|u|\\geq1/2\\}\\) is compact.\n\n(5) For \\(\\xi\\in H\\), \\(\\|T\\xi\\|^2=\\langle T^*T\\xi,\\xi\\rangle\\leq\\|T^*T\\|\\|\\xi\\|^2\\). So \\(\\|T\\|^2\\leq\\|T^*T\\|\\leq\\|T^*\\|\\|T\\|\\), which gives \\(\\|T\\|\\leq\\|T^*\\|\\). Since \\(T^{**}=T\\), also \\(\\|T^*\\|\\leq\\|T\\|\\); hence \\(\\|T^*\\|=\\|T\\|\\) and \\(\\|T^*T\\|=\\|T\\|^2\\). If \\(\\dim H\\leq1\\), then \\(B(H)=\\mathbb C1\\). If \\(e_1,e_2\\) are orthonormal, put \\(E_{ij}\\xi=\\langle\\xi,e_j\\rangle e_i\\). Then \\(E_{12}E_{21}=E_{11}\\neq E_{22}=E_{21}E_{12}\\). A norm-closed \\(*\\)-subalgebra is complete and inherits the C\\*-identity. \\(\\square\\)\n\nSo C\\*-algebras can be described in two ways. Concretely, they are the norm-closed \\(*\\)-subalgebras of the algebras \\(B(H)\\); part (5) shows that these are C\\*-algebras. Abstractly, they are the algebras that satisfy the axioms of Definition 1.1. The two descriptions give the same objects: every C\\*-algebra has a faithful representation on a Hilbert space, that is, it is isometrically \\(*\\)-isomorphic to a norm-closed \\(*\\)-subalgebra of some \\(B(H)\\). This is the Gelfand–Naimark theorem. Its proof needs positive functionals and the representations they define, which are beyond this lesson.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "full_conditions_and_proof": "## 2. Invertible elements and the Neumann series\n\nLet \\(A\\) be a unital algebra. An element \\(y\\) is a *left inverse* of \\(x\\) if \\(yx=1\\), and a *right inverse* if \\(xy=1\\). If \\(x\\) has both, they coincide: \\(y=y(xz)=(yx)z=z\\). Then \\(x\\) is *invertible*, and this element is its inverse \\(x^{-1}\\). The invertible elements form a group \\(G(A)\\), with \\((xy)^{-1}=y^{-1}x^{-1}\\). If \\(\\pi:A\\to B\\) is a unital homomorphism, that is \\(\\pi(1)=1\\), then \\(\\pi(G(A))\\subseteq G(B)\\) and \\(\\pi(x^{-1})=\\pi(x)^{-1}\\).\n\n**Proposition 2.1** (Neumann series). Let \\(A\\) be a unital Banach algebra and \\(\\|1-x\\|<1\\). Then \\(x\\) is invertible, and\n\\[\nx^{-1}=\\sum_{n=0}^\\infty(1-x)^n,\n\\tag{2.1}\n\\]\nwhere \\((1-x)^0=1\\) and the series converges in norm. Moreover \\(\\|x^{-1}\\|\\leq\\|1\\|+\\|1-x\\|/(1-\\|1-x\\|)\\).\n\n**Proof.** Put \\(y=1-x\\). Since \\(\\|y^n\\|\\leq\\|y\\|^n\\) for \\(n\\geq1\\), the partial sums \\(s_N=\\sum_{n=0}^Ny^n\\) form a Cauchy sequence, with limit \\(s\\). Now \\((1-y)s_N=s_N(1-y)=1-y^{N+1}\\), which tends to \\(1\\). So \\(xs=sx=1\\). The norm bound is the sum of the norms of the terms. \\(\\square\\)\n\n**Proposition 2.2** (the invertible group is open, and inversion is continuous). Let \\(A\\) be a nontrivial unital Banach algebra, \\(x_0\\in G(A)\\), and \\(\\|x-x_0\\|<1/\\|x_0^{-1}\\|\\). Then \\(x\\) is invertible,\n\\[\nx^{-1}=\\sum_{n=0}^\\infty\\big[x_0^{-1}(x_0-x)\\big]^n\\,x_0^{-1},\n\\tag{2.2}\n\\]\n\\[\n\\begin{gathered}\n\\|x^{-1}-x_0^{-1}\\|\\\\\n\\leq\\frac{\\|x_0^{-1}\\|^2\\,\\|x-x_0\\|}{1-\\|x_0^{-1}\\|\\,\\|x-x_0\\|}.\n\\end{gathered}\n\\tag{2.3}\n\\]\nConsequently \\(G(A)\\) is open, and \\(x\\mapsto x^{-1}\\) is continuous on \\(G(A)\\). The distance from \\(x_0\\in G(A)\\) to the set of noninvertible elements is at least \\(1/\\|x_0^{-1}\\|\\). If \\(x_n\\in G(A)\\) converge to a noninvertible element, then \\(\\|x_n^{-1}\\|\\to\\infty\\).\n\n**Proof.** Put \\(y=x_0^{-1}(x_0-x)\\), so \\(\\|y\\|\\leq\\|x_0^{-1}\\|\\|x-x_0\\|<1\\) and \\(x=x_0(1-y)\\). By (2.1), \\(1-y\\) is invertible with inverse \\(\\sum_ny^n\\). So \\(x\\) is a product of invertible elements, and \\(x^{-1}=(1-y)^{-1}x_0^{-1}\\), which is (2.2). The difference \\(x^{-1}-x_0^{-1}=\\sum_{n\\geq1}y^nx_0^{-1}\\) has norm at most \\(\\|x_0^{-1}\\|\\,\\|y\\|/(1-\\|y\\|)\\). Since \\(t\\mapsto t/(1-t)\\) increases on \\([0,1)\\), this gives (2.3). The distance statement restates the first claim. For the last one, if \\(x_n\\to x\\) and \\(x\\notin G(A)\\), then \\(\\|x-x_n\\|\\geq1/\\|x_n^{-1}\\|\\). \\(\\square\\)\n\nIn the zero algebra every element is invertible, with inverse \\(0\\), which is why Proposition 2.2 assumes \\(A\\neq\\{0\\}\\). [Corollary 5.6(4)](#oa-fnd-bn-09) weakens the hypothesis of Proposition 2.1 to \\(r(1-x)<1\\).\n\n## 3. Identities and the unitization\n\nThis section shows that the norm of an identity can be taken to be \\(1\\), adjoins an identity to an arbitrary algebra, and puts a C\\*-norm on the result when the algebra is a C\\*-algebra without identity.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-03",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "The norm of the identity",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-03",
      "proof_locus": {
        "line": 131,
        "through_line": 149
      },
      "full_conditions_and_proof": "### The norm of the identity\n\n**Proposition 3.1.** Let \\(A\\) be a nontrivial unital Banach algebra.\n1. \\(\\|1\\|\\geq1\\).\n2. Put \\(\\|x\\|_\\ell=\\sup\\{\\|xy\\|:\\|y\\|\\leq1\\}\\), the operator norm of left multiplication \\(L_x:y\\mapsto xy\\). This is an algebra norm on \\(A\\), with\n\\[\n\\|x\\|/\\|1\\|\\leq\\|x\\|_\\ell\\leq\\|x\\|\\quad\\text{and}\\quad\\|1\\|_\\ell=1 .\n\\]\nSo \\((A,\\|\\cdot\\|_\\ell)\\) is a Banach algebra whose norm is equivalent to the given one. If \\(\\|1\\|=1\\), the two norms coincide.\n3. If \\(A\\) has an isometric involution, then \\(1^*=1\\), and \\(N(x)=\\max(\\|x\\|_\\ell,\\|x^*\\|_\\ell)\\) is an equivalent Banach algebra norm with \\(N(x^*)=N(x)\\) and \\(N(1)=1\\).\n4. If \\(A\\) is a C\\*-algebra, then \\(\\|1\\|=1\\).\n\n**Proof.** (1) \\(\\|1\\|=\\|1\\cdot1\\|\\leq\\|1\\|^2\\) and \\(\\|1\\|\\neq0\\).\n(2) \\(\\|xy\\|\\leq\\|x\\|\\|y\\|\\) gives \\(\\|x\\|_\\ell\\leq\\|x\\|\\). Testing \\(L_x\\) at \\(y=1/\\|1\\|\\) gives \\(\\|x\\|_\\ell\\geq\\|x\\|/\\|1\\|\\). The map \\(x\\mapsto L_x\\) is linear and \\(L_{xy}=L_xL_y\\), so \\(\\|\\cdot\\|_\\ell\\) is a submultiplicative seminorm; by the lower bound it is a norm. Equivalent norms have the same Cauchy sequences, so it is complete. Finally \\(L_1\\) is the identity operator of the nonzero space \\(A\\), of norm \\(1\\). If \\(\\|1\\|=1\\), the two bounds coincide.\n(3) \\(1^*\\) is an identity: \\(1^*x=(x^*1)^*=x\\) and \\(x1^*=(1x^*)^*=x\\). So \\(1^*=1\\). The function \\(N\\) is a norm. It is submultiplicative because \\(\\|(xy)^*\\|_\\ell=\\|y^*x^*\\|_\\ell\\leq\\|y^*\\|_\\ell\\|x^*\\|_\\ell\\). Clearly \\(N(x^*)=N(x)\\), and \\(N(1)=1\\) because \\(1^*=1\\). For equivalence, \\(\\|x\\|/\\|1\\|\\leq N(x)\\leq\\max(\\|x\\|,\\|x^*\\|)=\\|x\\|\\).\n(4) \\(\\|1\\|=\\|1^*1\\|=\\|1\\|^2\\) and \\(\\|1\\|\\neq0\\). \\(\\square\\)\n\nThe zero algebra shows that \"nontrivial\" cannot be dropped: there \\(1=0\\), and no norm gives \\(\\|1\\|=1\\). The left-regular norm of (2) controls \\(\\|1\\|\\) but says nothing about the involution. Taking the maximum with \\(x\\mapsto\\|x^*\\|_\\ell\\) in (3) restores isometry without losing \\(N(1)=1\\). It is common to assume \\(\\|1\\|=1\\) whenever an identity exists. By (2) and (3) this costs nothing up to an equivalent norm. We keep the general norm, because every estimate below survives it.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    },
    {
      "id": "OA-FND-BN-04",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Adjoining an identity",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-04",
      "proof_locus": {
        "line": 150,
        "through_line": 180
      },
      "full_conditions_and_proof": "### Adjoining an identity\n\n**Construction 3.2.** For any algebra \\(A\\), let \\(A_1=A\\oplus\\mathbb C\\) with the linear structure of the direct sum and the product\n\\[\n\\begin{gathered}\n(a,\\lambda)(b,\\mu)\\\\\n=(ab+\\lambda b+\\mu a,\\ \\lambda\\mu).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A\\) has an involution, put \\((a,\\lambda)^*=(a^*,\\bar\\lambda)\\). Write \\(j(a)=(a,0)\\), \\(q(a,\\lambda)=\\lambda\\), and \\(1=(0,1)\\). If \\(A\\) is normed, put \\(\\|(a,\\lambda)\\|_1=\\|a\\|+|\\lambda|\\).\n\n**Proposition 3.3.**\n1. \\(A_1\\) is a unital algebra with identity \\((0,1)\\). The map \\(j\\) is an injective homomorphism onto an ideal \\(j(A)\\) of codimension one, and \\(q\\) is a unital homomorphism onto \\(\\mathbb C\\) with kernel \\(j(A)\\). \\(A_1\\) is commutative exactly when \\(A\\) is. If \\(A\\) has an involution, so does \\(A_1\\), and \\(j\\) and \\(q\\) preserve it.\n2. If \\(A\\) is a normed algebra, so is \\((A_1,\\|\\cdot\\|_1)\\), with \\(\\|1\\|_1=1\\). The map \\(j\\) is isometric and \\(|q(u)|\\leq\\|u\\|_1\\). If \\(A\\) is a Banach algebra, so is \\(A_1\\), and \\(j(A)\\) is closed. If the involution of \\(A\\) is isometric, so is that of \\(A_1\\).\n3. If \\(A\\) has an identity \\(1_A\\), then \\((a,\\lambda)\\mapsto(a+\\lambda1_A,\\lambda)\\) is an algebra isomorphism of \\(A_1\\) onto the product algebra \\(A\\times\\mathbb C\\). In particular \\(j(1_A)\\) is an idempotent of \\(A_1\\) different from the new identity \\((0,1)\\). The construction adds a new identity even when \\(A\\) has one.\n4. If \\(A\\) is a C\\*-algebra with a nonzero projection \\(p=p^*=p^2\\) (for example \\(A=\\mathbb C\\)), then \\((A_1,\\|\\cdot\\|_1)\\) is not a C\\*-algebra.\n\n**Proof.** (1) Both ways of bracketing \\((a,\\lambda)(b,\\mu)(c,\\nu)\\) give the first coordinate \\(abc+\\lambda bc+\\mu ac+\\nu ab+\\lambda\\mu c+\\lambda\\nu b+\\mu\\nu a\\) and the second coordinate \\(\\lambda\\mu\\nu\\). Bilinearity is clear, and \\((0,1)\\) is an identity. The formulas for \\(j\\) and \\(q\\) are homomorphic, and \\(\\ker q=j(A)\\), which is an ideal because \\((a,\\lambda)(b,0)=(ab+\\lambda b,0)\\) and \\((b,0)(a,\\lambda)=(ba+\\lambda b,0)\\). For the involution, both \\(\\big((a,\\lambda)(b,\\mu)\\big)^*\\) and \\((b,\\mu)^*(a,\\lambda)^*\\) equal \\((b^*a^*+\\bar\\lambda b^*+\\bar\\mu a^*,\\bar\\lambda\\bar\\mu)\\).\n(2) \\[\n\\begin{gathered}\n\\|ab+\\lambda b+\\mu a\\|+|\\lambda\\mu|\\\\\n\\leq\\|a\\|\\|b\\|+|\\lambda|\\|b\\|+|\\mu|\\|a\\|+|\\lambda||\\mu|\\\\\n=\\|(a,\\lambda)\\|_1\\|(b,\\mu)\\|_1.\n\\end{gathered}\n\\] Completeness of a sum norm on \\(A\\times\\mathbb C\\) follows coordinatewise. The rest is immediate.\n(3) Call the map \\(\\Phi\\). Both \\(\\Phi\\big((a,\\lambda)(b,\\mu)\\big)\\) and \\(\\Phi(a,\\lambda)\\Phi(b,\\mu)\\) equal \\((ab+\\lambda b+\\mu a+\\lambda\\mu1_A,\\lambda\\mu)\\). Its inverse is \\((a,\\lambda)\\mapsto(a-\\lambda1_A,\\lambda)\\). Under \\(\\Phi\\), \\(j(1_A)\\) becomes \\((1_A,0)\\), which is not the identity \\((1_A,1)\\).\n(4) In a C\\*-algebra, \\(\\|p\\|=\\|p^*p\\|=\\|p\\|^2\\), so \\(\\|p\\|=1\\). The element \\(u=(-2p,1)\\) is self-adjoint, and \\(u^*u=u^2=(4p^2-4p,1)=(0,1)\\). So \\(\\|u^*u\\|_1=1\\), while \\(\\|u\\|_1^2=9\\). \\(\\square\\)\n\nThe construction works for every \\(A\\), whether or not \\(A\\) already has an identity. Proposition 3.4 below replaces the sum norm by a C\\*-norm when \\(A\\) is a nonunital C\\*-algebra, and Section 4 uses \\(A_1\\) to define the quasi-spectrum.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-05",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "A C\\*-norm on the unitization",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-05",
      "proof_locus": {
        "line": 181,
        "through_line": 213
      },
      "full_conditions_and_proof": "### A C\\*-norm on the unitization\n\nLet \\(A\\) be a C\\*-algebra and \\(A_1\\) its unitization (Construction 3.2). Since \\(j(A)\\) is an ideal, each \\(u\\in A_1\\) acts on \\(A\\) by left multiplication, \\(b\\mapsto ub\\). We identify \\(a\\in A\\) with \\(j(a)\\) and write \\(u=a+\\lambda\\). Put\n\\[\np(u)=\\sup\\{\\|ub\\|:\\ b\\in A,\\ \\|b\\|\\leq1\\}.\n\\]\n\n**Proposition 3.4.**\n1. \\(p\\) is a submultiplicative seminorm on \\(A_1\\), \\(p(u)\\leq\\|u\\|_1\\), and \\(p(a)=\\|a\\|\\) for \\(a\\in A\\).\n2. \\(p(u)^2\\leq p(u^*u)\\leq p(u^*)p(u)\\). Consequently \\(p(u^*)=p(u)\\) and \\(p(u^*u)=p(u)^2\\).\n3. If \\(A\\) is not unital, then \\(p(u)=0\\) only for \\(u=0\\). If \\(A\\) has an identity \\(1_A\\), then \\(p(u)=0\\) exactly for \\(u\\in\\mathbb C(1_A-1)\\).\n4. If \\(A\\) is not unital, then \\((A_1,p)\\) is a unital C\\*-algebra containing \\(A\\) isometrically as a closed ideal of codimension one. Moreover \\(|\\lambda|\\leq p(a+\\lambda)\\), so \\(p\\) equals the norm \\(N(a,\\lambda)=\\max\\big(p(a+\\lambda),|\\lambda|\\big)\\).\n\n**Proof.** (1) \\(\\|(a+\\lambda)b\\|\\leq(\\|a\\|+|\\lambda|)\\|b\\|\\) gives \\(p(u)\\leq\\|u\\|_1\\). Left multiplication is linear in \\(u\\), and the operator of \\(uv\\) is the product of the operators of \\(u\\) and \\(v\\); so \\(p\\) is a submultiplicative seminorm. For \\(a\\in A\\), \\(p(a)\\leq\\|a\\|\\). If \\(a\\neq0\\), test at \\(b=a^*/\\|a\\|\\), which has norm one: \\(\\|ab\\|=\\|aa^*\\|/\\|a\\|=\\|a^*\\|^2/\\|a\\|=\\|a\\|\\).\n(2) Let \\(b\\in A\\) with \\(\\|b\\|\\leq1\\). The elements \\(ub\\) and \\(b^*(u^*u)b\\) lie in \\(A\\), and \\((ub)^*(ub)=b^*(u^*u)b\\). The C\\*-identity in \\(A\\) gives\n\\[\n\\begin{gathered}\n\\|ub\\|^2\\\\\n=\\|b^*(u^*u)b\\|\\\\\n\\leq\\|b^*\\|\\,\\|(u^*u)b\\|\\\\\n\\leq p(u^*u).\n\\end{gathered}\n\\]\nTake the supremum over \\(b\\), and use submultiplicativity. If \\(p(u)=0\\), the same inequality for \\(u^*\\) gives \\(p(u^*)^2\\leq p(u)p(u^*)=0\\). Otherwise it gives \\(p(u)\\leq p(u^*)\\) and \\(p(u^*)\\leq p(u)\\). In both cases \\(p(u^*)=p(u)\\), and then \\(p(u)^2\\leq p(u^*u)\\leq p(u)^2\\).\n(3) Let \\(p(a+\\lambda)=0\\), that is, \\(ab+\\lambda b=0\\) for all \\(b\\in A\\). If \\(\\lambda=0\\), then \\(\\|a\\|=p(a)=0\\). If \\(\\lambda\\neq0\\), put \\(e=-a/\\lambda\\in A\\); then \\(eb=b\\) for every \\(b\\), so \\(e\\) is a left identity of \\(A\\). Its adjoint is a right identity: \\(be^*=(eb^*)^*=b\\). Hence \\(e=ee^*=e^*\\), so \\(e\\) is a two-sided identity of \\(A\\), and \\(u=\\lambda(1-e)=-\\lambda(1_A-1)\\). So \\(p\\) has zero kernel when \\(A\\) is not unital. If \\(A\\) has an identity \\(1_A\\), then \\((1_A-1)b=b-b=0\\) for all \\(b\\), and \\(1_A-1\\neq0\\) in \\(A_1\\) because its scalar part is \\(-1\\).\n(4) By (1)–(3), \\(p\\) is a submultiplicative norm with the C\\*-identity and an isometric involution, and it agrees with the norm of \\(A\\) on \\(A\\). Next we prove completeness. Look at the functional \\(q(a+\\lambda)=\\lambda\\). Its kernel \\(A\\) is complete for \\(p\\), hence closed in \\((A_1,p)\\). A nonzero linear functional with closed kernel is bounded. Indeed, choose \\(u_0\\) with \\(q(u_0)=1\\) and put \\(\\delta=\\inf\\{p(u_0-k):k\\in A\\}>0\\); if \\(q(u)\\neq0\\), then \\(u/q(u)-u_0\\in A\\), so \\(p(u)\\geq\\delta|q(u)|\\). So \\(|\\lambda|\\leq\\delta^{-1}p(a+\\lambda)\\). Also \\(\\|a\\|=p(a)\\leq p(a+\\lambda)+|\\lambda|p(1)\\), and \\(p(1)\\leq1\\). So \\(p\\) is equivalent to the complete norm \\(\\|\\cdot\\|_1\\), and \\((A_1,p)\\) is a C\\*-algebra. It is unital with \\(p(1)=1\\) by Proposition 3.1(4), since \\(A\\neq\\{0\\}\\). Finally, \\(q\\) is a character of this Banach algebra, and characters of Banach algebras have norm at most \\(1\\) ([Proposition 10.3(1)](#oa-fnd-bn-17)). So \\(|q(u)|\\leq p(u)\\), and hence \\(N=p\\). \\(\\square\\)\n\nParts (1) and (2) hold for every C\\*-algebra. Part (3) gives the exact kernel of \\(p\\): the left-regular seminorm is a norm if and only if \\(A\\) is not unital. The norm \\(N=\\max(p,|q|)\\) keeps the scalar coordinate, and we call \\(A_1\\) with this norm the *forced unitization* of \\(A\\). It is a C\\*-algebra for every C\\*-algebra \\(A\\): for nonunital \\(A\\), (4) shows that \\(N=p\\), and for unital \\(A\\) see Exercise 5. The proof of (4) uses the bound \\(|\\omega(x)|\\leq\\|x\\|\\) for characters from Section 10. That bound is proved from Sections 3 and 4 alone, without this proposition, so the argument is not circular.\n\n**Example 3.5.** Let \\(\\Omega\\) be LCH and not compact, and \\(A=C_0(\\Omega)\\), which is not unital by [Proposition 1.2(4)](#oa-fnd-bn-01). For \\(u=f+\\lambda\\), \\(p(u)=\\sup_{\\omega\\in\\Omega}|f(\\omega)+\\lambda|\\). Indeed, \\(\\|ub\\|_\\infty\\leq\\sup|f+\\lambda|\\) when \\(\\|b\\|_\\infty\\leq1\\). Conversely, for \\(\\omega_0\\in\\Omega\\), Urysohn's lemma gives \\(b\\in C_c(\\Omega)\\) with \\(0\\leq b\\leq1\\) and \\(b(\\omega_0)=1\\), and then \\(\\|ub\\|_\\infty\\geq|f(\\omega_0)+\\lambda|\\). Since \\(f\\) vanishes at infinity, \\(|\\lambda|\\leq p(u)\\), as (4) predicts. If \\(\\Omega\\) is compact, \\(A\\) is unital with \\(1_A\\equiv1\\), and \\(p(1_A-1)=\\sup|1-1|=0\\), as (3) predicts.\n\n## 4. The spectrum\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-06",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Spectrum and quasi-spectrum",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
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        "through_line": 253
      },
      "full_conditions_and_proof": "### Spectrum and quasi-spectrum\n\n**Definition 4.1.** Let \\(A\\) be unital and \\(x\\in A\\). The *spectrum* of \\(x\\) is\n\\[\n\\sigma_A(x)=\\{\\lambda\\in\\mathbb C :\\ \\lambda-x\\ \\text{is not invertible}\\},\n\\]\nand its complement \\(\\rho_A(x)\\) is the *resolvent set*. For an arbitrary algebra \\(A\\) and \\(x\\in A\\), the *quasi-spectrum* is \\(\\sigma'_A(x)=\\sigma_{A_1}(j(x))\\), computed in the unitization of [Construction 3.2](#oa-fnd-bn-04). The *spectral radius* is \\(r(x)=\\sup\\{|\\lambda|:\\lambda\\in\\sigma'_A(x)\\}\\in[0,\\infty]\\).\n\nThe quasi-spectrum matters mainly when \\(A\\) has no identity, but we define it for every \\(A\\). When \\(A\\) is unital, part (1) below shows that it is \\(\\sigma_A(x)\\cup\\{0\\}\\), which has the same radius as \\(\\sigma_A(x)\\) whenever \\(\\sigma_A(x)\\) is not empty.\n\n**Proposition 4.2.**\n1. \\(0\\in\\sigma'_A(x)\\). If \\(A\\) is unital, \\(\\sigma'_A(x)=\\sigma_A(x)\\cup\\{0\\}\\). So \\(r(x)=\\sup\\{|\\lambda|:\\lambda\\in\\sigma_A(x)\\}\\) when \\(A\\) is unital and \\(\\sigma_A(x)\\neq\\varnothing\\).\n2. Let \\(A\\) be unital, \\(x,y\\in A\\) and \\(\\lambda\\neq0\\). If \\(\\lambda-xy\\) has inverse \\(u\\), then \\(\\lambda-yx\\) has inverse \\(\\lambda^{-1}(1+yux)\\). Hence \\(\\sigma_A(xy)\\cup\\{0\\}=\\sigma_A(yx)\\cup\\{0\\}\\). For every algebra, \\(\\sigma'_A(xy)=\\sigma'_A(yx)\\) and \\(r(xy)=r(yx)\\).\n3. If \\(\\pi:A\\to B\\) is a unital homomorphism, then \\(\\sigma_B(\\pi(x))\\subseteq\\sigma_A(x)\\). If \\(B\\) is a subalgebra of the unital algebra \\(A\\) with \\(1_A\\in B\\), then \\(\\sigma_A(x)\\subseteq\\sigma_B(x)\\) for \\(x\\in B\\).\n4. If \\(x\\in G(A)\\), then \\(0\\notin\\sigma_A(x)\\) and \\(\\sigma_A(x^{-1})=\\{\\lambda^{-1}:\\lambda\\in\\sigma_A(x)\\}\\).\n5. \\(\\sigma_A(x-c)=\\sigma_A(x)-c\\) for \\(c\\in\\mathbb C\\), and \\(\\sigma_A(cx)=c\\,\\sigma_A(x)\\) for \\(c\\neq0\\).\n\n**Proof.** (1) The map \\(q:A_1\\to\\mathbb C\\) is a unital homomorphism with \\(q(j(x))=0\\). If \\(j(x)\\) had an inverse \\(w\\), then \\(1=q(j(x))q(w)=0\\). If \\(A\\) is unital, Proposition 3.3(3) identifies \\(A_1\\) with \\(A\\times\\mathbb C\\) and \\(j(x)\\) with \\((x,0)\\). An element \\((a,\\mu)\\) of \\(A\\times\\mathbb C\\) is invertible exactly when \\(a\\in G(A)\\) and \\(\\mu\\neq0\\). So \\(\\lambda-(x,0)=(\\lambda-x,\\lambda)\\) fails to be invertible exactly when \\(\\lambda\\in\\sigma_A(x)\\) or \\(\\lambda=0\\).\n\n(2) From \\((\\lambda-xy)u=1\\) we get \\(xyu=\\lambda u-1\\), so \\(yxyux=\\lambda yux-yx\\). Hence\n\\[\n\\begin{gathered}\n(\\lambda-yx)(1+yux)\\\\\n=\\lambda+\\lambda yux-yx-yxyux\\\\\n=\\lambda .\n\\end{gathered}\n\\]\nFrom \\(u(\\lambda-xy)=1\\) we get \\(uxy=\\lambda u-1\\), so \\(yuxyx=\\lambda yux-yx\\), and \\((1+yux)(\\lambda-yx)=\\lambda\\) in the same way. For an arbitrary algebra, apply this in \\(A_1\\) to \\(j(x)\\) and \\(j(y)\\); both quasi-spectra contain \\(0\\).\n\n(3) \\(\\pi(\\lambda-x)=\\lambda-\\pi(x)\\), and \\(\\pi\\) maps invertible elements to invertible elements ([Section 2](#oa-fnd-bn-02)). For the second claim, an inverse in \\(B\\) is an inverse in \\(A\\).\n\n(4) For \\(\\lambda\\neq0\\), \\(\\lambda-x=(-\\lambda x)(\\lambda^{-1}-x^{-1})\\), and \\(-\\lambda x\\) is invertible and commutes with \\(\\lambda^{-1}-x^{-1}\\). So \\(\\lambda-x\\) is invertible exactly when \\(\\lambda^{-1}-x^{-1}\\) is.\n\n(5) \\(\\lambda-(x-c)=(\\lambda+c)-x\\) and \\(\\lambda-cx=c(\\lambda/c-x)\\). \\(\\square\\)\n\n**Examples 4.3.**\n- *The \\(\\{0\\}\\) in (2) is needed.* On \\(\\ell^2(\\mathbb N)\\) let \\(Se_n=e_{n+1}\\). Then \\(S^*e_0=0\\) and \\(S^*e_{n+1}=e_n\\), so \\(S^*S=1\\), while \\(SS^*\\) is the orthogonal projection \\(P\\) onto the closed span of \\(e_1,e_2,\\dots\\). Since \\(0\\neq P\\neq1\\), \\(\\sigma(P)=\\{0,1\\}\\): for \\(\\lambda\\notin\\{0,1\\}\\) the inverse of \\(\\lambda-P\\) is \\(\\lambda^{-1}(1-P)+(\\lambda-1)^{-1}P\\), while \\(P\\) and \\(1-P\\) have nonzero kernels. So \\(\\sigma(SS^*)=\\{0,1\\}\\) and \\(\\sigma(S^*S)=\\{1\\}\\).\n- *Matrices.* In \\(M_n(\\mathbb C)\\), \\(xy\\) is invertible exactly when \\(yx\\) is, because \\(\\det(xy)=\\det(yx)\\). So there \\(\\sigma(xy)=\\sigma(yx)\\).\n- *The spectrum depends on the algebra.* Example 4.6 below gives an element whose spectrum is the unit circle in one algebra and the closed unit disc in a closed subalgebra.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BN-07",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "The resolvent; the spectrum is compact",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-07",
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      "full_conditions_and_proof": "### The resolvent; the spectrum is compact\n\nLet \\(A\\) be a unital Banach algebra and \\(x\\in A\\). Put \\(c_1=\\max(1,\\|1\\|)\\). If \\(A=\\{0\\}\\), every resolvent is \\(0\\); there \\(1/0\\) is read as \\(\\infty\\), here and in Section 8.\n\n**Proposition 4.4.**\n1. If \\(|\\lambda|>\\|x\\|\\), then \\(\\lambda\\in\\rho_A(x)\\),\n\\[\nR_x(\\lambda)=\\sum_{n=0}^\\infty\\frac{x^n}{\\lambda^{n+1}},\n\\tag{4.1}\n\\]\nand \\(\\|R_x(\\lambda)\\|\\leq c_1/(|\\lambda|-\\|x\\|)\\).\n2. If \\(\\lambda_0\\in\\rho_A(x)\\) and \\(|\\lambda-\\lambda_0|<1/\\|R_x(\\lambda_0)\\|\\), then \\(\\lambda\\in\\rho_A(x)\\) and\n\\[\n\\begin{gathered}\nR_x(\\lambda)\\\\\n=\\sum_{n=0}^\\infty(\\lambda_0-\\lambda)^n\\,R_x(\\lambda_0)^{n+1}.\n\\end{gathered}\n\\tag{4.2}\n\\]\nHence \\(\\rho_A(x)\\) is open, and \\(\\operatorname{dist}(\\lambda_0,\\sigma_A(x))\\geq1/\\|R_x(\\lambda_0)\\|\\).\n3. \\(\\sigma_A(x)\\) is compact and lies in the closed disc of radius \\(\\|x\\|\\). For every Banach algebra \\(A\\) and \\(x\\in A\\), \\(\\sigma'_A(x)\\) is compact and not empty, and \\(r(x)\\leq\\|x\\|\\).\n4. (*Resolvent identity.*) For \\(\\lambda,\\mu\\in\\rho_A(x)\\), \\(R_x(\\lambda)-R_x(\\mu)=(\\mu-\\lambda)R_x(\\lambda)R_x(\\mu)\\). All resolvents \\(R_x(\\lambda)\\) commute with each other and with every element that commutes with \\(x\\).\n5. \\(R_x\\) is continuous on \\(\\rho_A(x)\\). For each \\(\\varphi\\in A^*\\), \\(\\varphi\\circ R_x\\) is given near each \\(\\lambda_0\\in\\rho_A(x)\\) by a power series in \\(\\lambda-\\lambda_0\\); so it is holomorphic, with a continuous derivative.\n6. \\(\\|R_x(\\lambda)\\|\\to0\\) as \\(|\\lambda|\\to\\infty\\), and \\(\\|R_x(\\lambda)\\|\\to\\infty\\) as \\(\\lambda\\in\\rho_A(x)\\) approaches a point of \\(\\sigma_A(x)\\).\n\n**Proof.** (1) Since \\(\\|x/\\lambda\\|<1\\), (2.1) makes \\(1-x/\\lambda\\) invertible with inverse \\(\\sum_n(x/\\lambda)^n\\). So \\(\\lambda-x=\\lambda(1-x/\\lambda)\\) is invertible, with inverse (4.1). As \\(\\|x^0\\|=\\|1\\|\\leq c_1\\) and \\(\\|x^n\\|\\leq\\|x\\|^n\\), the norm is at most \\(\\sum_nc_1\\|x\\|^n/|\\lambda|^{n+1}=c_1/(|\\lambda|-\\|x\\|)\\).\n(2) Write \\(\\lambda-x=(\\lambda_0-x)\\big[1-(\\lambda_0-\\lambda)R_x(\\lambda_0)\\big]\\). Since \\(\\|(\\lambda_0-\\lambda)R_x(\\lambda_0)\\|<1\\), the bracket is invertible by (2.1), with inverse \\(\\sum_n(\\lambda_0-\\lambda)^nR_x(\\lambda_0)^n\\), which commutes with \\(R_x(\\lambda_0)\\). This gives (4.2). The whole disc of radius \\(1/\\|R_x(\\lambda_0)\\|\\) about \\(\\lambda_0\\) lies in \\(\\rho_A(x)\\).\n(3) By (1) and (2), \\(\\sigma_A(x)\\) is closed and bounded. For an arbitrary Banach algebra, apply this in \\(A_1\\), where \\(\\|j(x)\\|_1=\\|x\\|\\). The quasi-spectrum contains \\(0\\) by Proposition 4.2(1).\n(4) \\(\\lambda-x\\) and \\(\\mu-x\\) commute, so their inverses commute, and \\[\n\\begin{gathered}\nR_x(\\lambda)-R_x(\\mu)\\\\\n=R_x(\\lambda)\\big[(\\mu-x)-(\\lambda-x)\\big]R_x(\\mu).\n\\end{gathered}\n\\] If \\(y\\) commutes with \\(x\\), it commutes with \\(\\lambda-x\\) and hence with its inverse.\n(5) The series (4.2) converges uniformly on smaller discs about \\(\\lambda_0\\). Apply \\(\\varphi\\) term by term.\n(6) The first claim is (1). The second follows from (2): \\(\\|R_x(\\lambda)\\|\\geq1/\\operatorname{dist}(\\lambda,\\sigma_A(x))\\). \\(\\square\\)\n\n**Examples 4.5** (completeness is needed).\n- Give \\(\\mathbb C[z]\\) the norm \\(\\|p\\|=\\max_{|z|\\leq1}|p(z)|\\). It is a unital normed algebra, but not complete. A nonzero polynomial multiple of \\(z-\\lambda\\) has degree at least one, so \\(z-\\lambda\\) is never invertible: \\(\\sigma(z)=\\mathbb C\\), which is not bounded.\n- In the field \\(\\mathbb C(z)\\) of rational functions, every \\(z-\\lambda\\) is invertible, so \\(\\sigma(z)=\\varnothing\\). By the Gelfand–Mazur theorem ([Corollary 5.3](#oa-fnd-bn-08)), \\(\\mathbb C(z)\\) carries no algebra norm at all.\n\n### The spectrum depends on the algebra\n\n**Example 4.6.** Let \\(B=C(\\mathbb T)\\), and let \\(A\\) be the closure in \\(B\\) of the polynomials in \\(z\\) (the disc algebra, seen on the circle). Then \\(\\sigma_B(z)=\\mathbb T\\), while \\(\\sigma_A(z)\\) is the closed unit disc. In general, if \\(A\\) is a closed subalgebra of a unital Banach algebra \\(B\\) with \\(1_B\\in A\\), then \\(\\sigma_B(x)\\subseteq\\sigma_A(x)\\), and every boundary point of \\(\\sigma_A(x)\\) lies in \\(\\sigma_B(x)\\).\n\n**Proof.** In \\(B\\), \\(z-\\lambda\\) is invertible exactly when it has no zero on \\(\\mathbb T\\), that is, when \\(|\\lambda|\\neq1\\). For \\(A\\), write \\(c_n(g)=\\frac1{2\\pi}\\int_0^{2\\pi}g(e^{i\\theta})e^{-in\\theta}\\,d\\theta\\). Polynomials in \\(z\\) have \\(c_n=0\\) for \\(n<0\\), and \\(|c_n(g)-c_n(p)|\\leq\\|g-p\\|_\\infty\\), so every \\(g\\in A\\) has \\(c_n(g)=0\\) for \\(n<0\\). Suppose \\(|\\lambda|<1\\) and \\((z-\\lambda)g=1\\) with \\(g\\in A\\). Comparing coefficients, \\(c_{n-1}(g)-\\lambda c_n(g)\\) is \\(1\\) for \\(n=0\\) and \\(0\\) otherwise. If \\(\\lambda=0\\), this gives \\(c_{-1}(g)=1\\), which is impossible. If \\(\\lambda\\neq0\\), it gives \\(c_0(g)=-1/\\lambda\\) and \\(c_n(g)=c_{n-1}(g)/\\lambda\\) for \\(n\\geq1\\), so \\(|c_n(g)|=|\\lambda|^{-n-1}\\to\\infty\\), although \\(|c_n(g)|\\leq\\|g\\|_\\infty\\). So the open disc lies in \\(\\sigma_A(z)\\). Since \\(\\sigma_A(z)\\) is closed and \\(\\|z\\|=1\\), \\(\\sigma_A(z)\\) is the closed disc.\n\nFor the general claim, \\(\\sigma_B(x)\\subseteq\\sigma_A(x)\\) is Proposition 4.2(3). Let \\(\\lambda\\) be a boundary point of \\(\\sigma_A(x)\\), and suppose \\(\\lambda\\notin\\sigma_B(x)\\). Choose \\(\\lambda_n\\in\\rho_A(x)\\) with \\(\\lambda_n\\to\\lambda\\). In \\(B\\), \\((\\lambda_n-x)^{-1}\\to(\\lambda-x)^{-1}\\) by continuity of inversion ([Proposition 2.2](#oa-fnd-bn-02)). The left sides lie in the closed set \\(A\\), so \\((\\lambda-x)^{-1}\\in A\\), and \\(\\lambda\\in\\rho_A(x)\\). This contradicts \\(\\lambda\\in\\sigma_A(x)\\), which holds because \\(\\sigma_A(x)\\) is closed. \\(\\square\\)\n\nThis cannot happen for a C\\*-subalgebra of a C\\*-algebra: there the two spectra agree. This is proved in the [next lesson, on C\\*-algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-06).\n\n## 5. Nonempty spectrum and the spectral radius formula\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-08",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "The spectrum is not empty; the Gelfand–Mazur theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-08",
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        "line": 307,
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      "full_conditions_and_proof": "### The spectrum is not empty; the Gelfand–Mazur theorem\n\n**Lemma 5.1** (circle means). Let \\(0\\leq r_1<r_2\\leq\\infty\\), and let \\(g\\) be holomorphic with a continuous derivative on the annulus \\(\\{r_1<|\\lambda|<r_2\\}\\). Then\n\\[\nM(\\rho)=\\frac1{2\\pi}\\int_0^{2\\pi}g(\\rho e^{i\\theta})\\,d\\theta\n\\]\ndoes not depend on \\(\\rho\\in(r_1,r_2)\\). If \\(g\\) is holomorphic with a continuous derivative on the disc \\(\\{|\\lambda|<r_2\\}\\), then \\(M(\\rho)=g(0)\\) for \\(0<\\rho<r_2\\).\n\n**Proof.** The integrand has the \\(\\rho\\)-derivative \\(g'(\\rho e^{i\\theta})e^{i\\theta}\\), which is continuous in \\((\\rho,\\theta)\\), so we may differentiate under the integral sign. For \\(\\rho>0\\),\n\\[\n\\begin{gathered}\nM'(\\rho)\\\\\n=\\frac1{2\\pi}\\int_0^{2\\pi}g'(\\rho e^{i\\theta})e^{i\\theta}\\,d\\theta\\\\\n=\\frac1{2\\pi i\\rho}\\int_0^{2\\pi}\\frac{d}{d\\theta}\\,g(\\rho e^{i\\theta})\\,d\\theta\\\\\n=0,\n\\end{gathered}\n\\]\nbecause \\(\\theta\\mapsto g(\\rho e^{i\\theta})\\) has period \\(2\\pi\\). In the second case \\(M(\\rho)\\to g(0)\\) as \\(\\rho\\to0\\), by continuity of \\(g\\) at \\(0\\). \\(\\square\\)\n\nThis lemma is all the complex analysis that this section needs.\n\n**Theorem 5.2** (the spectrum is not empty). If \\(A\\) is a nontrivial unital Banach algebra, then \\(\\sigma_A(x)\\neq\\varnothing\\) for every \\(x\\in A\\).\n\n**Proof.** Suppose \\(\\sigma_A(x)=\\varnothing\\), and let \\(\\varphi\\in A^*\\). The function \\(g=\\varphi\\circ R_x\\) is holomorphic on \\(\\mathbb C\\) with a continuous derivative, by [Proposition 4.4(5)](#oa-fnd-bn-07). By Lemma 5.1 with \\(r_2=\\infty\\), and by the bound \\(\\|R_x(\\lambda)\\|\\leq c_1/(|\\lambda|-\\|x\\|)\\) of Proposition 4.4(1), for \\(\\rho>\\|x\\|\\),\n\\[\n|g(0)|=|M(\\rho)|\\leq\\max_{|\\lambda|=\\rho}|g(\\lambda)|\\leq\\frac{c_1\\|\\varphi\\|}{\\rho-\\|x\\|}.\n\\]\nLetting \\(\\rho\\to\\infty\\) gives \\(\\varphi(R_x(0))=0\\). Since this holds for every \\(\\varphi\\), the Hahn–Banach theorem gives \\(R_x(0)=-x^{-1}=0\\). Then \\(1=xx^{-1}=0\\), which contradicts \\(A\\neq\\{0\\}\\). \\(\\square\\)\n\nIn the zero algebra, \\(0-0=0=1\\) is invertible, so \\(\\sigma(0)=\\varnothing\\); the hypothesis \\(A\\neq\\{0\\}\\) is needed. The usual proof applies Liouville's theorem to \\(\\varphi\\circ R_x\\); Lemma 5.1 is the part of that argument that is needed here.\n\n**Corollary 5.3** (Gelfand–Mazur theorem, for normed algebras). Let \\(A\\) be a nontrivial unital normed algebra, not necessarily complete, in which every nonzero element is invertible. Then \\(A=\\mathbb C1\\), and \\(\\lambda\\mapsto\\lambda1\\) is the only unital algebra isomorphism of \\(\\mathbb C\\) onto \\(A\\).\n\n**Proof.** Let \\(\\hat A\\) be the completion of \\(A\\). To extend multiplication explicitly, represent \\(x,y\\in\\hat A\\) by Cauchy sequences \\(x_n,y_n\\in A\\), and define \\(xy\\) to be the class of \\(x_ny_n\\). These sequences are bounded, and\n\\[\n\\begin{aligned}\n\\|x_ny_n-x_my_m\\|\n&\\leq\\|x_n\\|\\|y_n-y_m\\|\\\\\n&\\quad+\\|x_n-x_m\\|\\|y_m\\|\\longrightarrow0 .\n\\end{aligned}\n\\]\nThe same estimate for two choices of representing sequences proves independence of those choices. The bound \\(\\|xy\\|\\leq\\|x\\|\\|y\\|\\), bilinearity, associativity and the identity laws pass to the limit from \\(A\\). Thus \\(\\hat A\\) is a Banach algebra with the same identity, still nonzero under the isometric embedding of \\(A\\).\n\nLet \\(x\\in A\\). By Theorem 5.2 there is \\(\\lambda\\in\\sigma_{\\hat A}(x)\\). Then \\(x-\\lambda\\) is not invertible in \\(\\hat A\\), so it is not invertible in \\(A\\), and therefore \\(x-\\lambda=0\\). A unital algebra homomorphism \\(\\psi:\\mathbb C\\to A\\) satisfies \\(\\psi(\\lambda)=\\lambda\\psi(1)=\\lambda1\\). \\(\\square\\)\n\nComplex scalars matter: over the real field the statement fails, since \\(\\mathbb C\\) and the quaternions, with their usual absolute values, are real normed division algebras.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BN-09",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "The spectral radius formula",
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      "full_conditions_and_proof": "### The spectral radius formula\n\n**Theorem 5.4** (spectral radius formula). For every element \\(x\\) of a Banach algebra \\(A\\),\n\\[\n\\begin{gathered}\nr(x)\\\\\n=\\lim_{n\\to\\infty}\\|x^n\\|^{1/n}\\\\\n=\\inf_{n\\geq1}\\|x^n\\|^{1/n}.\n\\end{gathered}\n\\tag{5.1}\n\\]\n\nStep 1 below gives the lower spectral-radius bound directly, before any limit of roots is known to exist. Remark 5.5 supplies a second proof using submultiplicativity.\n\n**Proof.** By Proposition 4.2(1) and [Proposition 3.3(2)](#oa-fnd-bn-04) we may compute in \\(A_1\\), where \\(\\|j(x)^n\\|_1=\\|x^n\\|\\). So let \\(A\\) be unital. If \\(A=\\{0\\}\\), both sides are \\(0\\), since \\(\\sigma'_A(x)=\\{0\\}\\). Let \\(A\\) be nontrivial.\n\n*Step 1: \\(r(x)\\leq\\|x^n\\|^{1/n}\\) for every \\(n\\geq1\\).* Let \\(\\lambda\\in\\sigma_A(x)\\). In \\(A\\),\n\\[\n\\begin{gathered}\nx^n-\\lambda^n\\\\\n=(x-\\lambda)p(x)\\\\\n=p(x)(x-\\lambda),\\\\\np(x)\\\\\n=\\sum_{k=0}^{n-1}\\lambda^{n-1-k}x^k .\n\\end{gathered}\n\\]\nIf \\(x^n-\\lambda^n\\) had an inverse \\(w\\), then \\(p(x)w\\) would be a right inverse and \\(wp(x)\\) a left inverse of \\(x-\\lambda\\), which is not invertible. So \\(\\lambda^n\\in\\sigma_A(x^n)\\), and \\(|\\lambda|^n\\leq\\|x^n\\|\\) because the spectrum of \\(x^n\\) lies in the disc of radius \\(\\|x^n\\|\\) ([Proposition 4.4(1)](#oa-fnd-bn-07)). Take the supremum over \\(\\lambda\\).\n\n*Step 2: \\(\\limsup_n\\|x^n\\|^{1/n}\\leq r(x)\\).* Fix \\(n\\geq0\\) and \\(\\varphi\\in A^*\\). Since \\(\\sigma_A(x)\\) lies in the closed disc of radius \\(r(x)\\), the function \\(g(\\lambda)=\\lambda^{n+1}\\varphi(R_x(\\lambda))\\) is holomorphic with a continuous derivative on \\(\\{|\\lambda|>r(x)\\}\\) (Proposition 4.4(5)). For \\(\\rho>\\|x\\|\\) the series (4.1) converges uniformly on \\(|\\lambda|=\\rho\\), and integrating term by term gives\n\\[\n\\begin{gathered}\n\\frac1{2\\pi}\\int_0^{2\\pi}g(\\rho e^{i\\theta})\\,d\\theta\\\\\n=\\sum_{k\\geq0}\\varphi(x^k)\\,\\frac1{2\\pi}\\int_0^{2\\pi}(\\rho e^{i\\theta})^{n-k}\\,d\\theta\\\\\n=\\varphi(x^n).\n\\end{gathered}\n\\]\nBy Lemma 5.1, the left side is the same for every \\(\\rho>r(x)\\). So \\(|\\varphi(x^n)|\\leq\\rho^{n+1}m(\\rho)\\|\\varphi\\|\\) for every \\(\\rho>r(x)\\), where \\(m(\\rho)=\\max_{|\\lambda|=\\rho}\\|R_x(\\lambda)\\|\\) is finite because \\(R_x\\) is continuous. By the Hahn–Banach theorem, \\(\\|x^n\\|\\leq\\rho^{n+1}m(\\rho)\\) for all \\(n\\). Hence \\(\\limsup_n\\|x^n\\|^{1/n}\\leq\\rho\\), and we let \\(\\rho\\) decrease to \\(r(x)\\).\n\nSteps 1 and 2 give \\[\n\\begin{gathered}\nr(x)\\\\\n\\leq\\inf_n\\|x^n\\|^{1/n}\\\\\n\\leq\\liminf_n\\|x^n\\|^{1/n}\\\\\n\\leq\\limsup_n\\|x^n\\|^{1/n}\\\\\n\\leq r(x).\n\\end{gathered}\n\\] \\(\\square\\)\n\n**Remark 5.5.** Submultiplicativity alone shows that the limit in (5.1) exists. Here is the elementary argument often called Fekete’s lemma in this multiplicative form. Put \\(s_n=\\|x^n\\|\\). If some \\(s_k=0\\), all powers from \\(k\\) onwards are zero and the limit is zero. Otherwise fix \\(k\\), write \\(n=qk+r\\) with \\(0\\leq r<k\\), and put \\(C_k=\\max(1,s_1,\\ldots,s_{k-1})\\), with \\(C_1=1\\). Submultiplicativity gives \\(s_n\\leq C_ks_k^q\\), including \\(r=0\\) without an identity factor. Hence \\(\\limsup_ns_n^{1/n}\\leq s_k^{1/k}\\). Taking the infimum over \\(k\\), and noting that every \\(s_n^{1/n}\\) is at least that infimum, proves convergence to \\(\\inf_{k\\geq1}s_k^{1/k}\\). The content of Theorem 5.4 is that this limit is the spectral radius.\n\n**Corollary 5.6.**\n1. \\(r(x^k)=r(x)^k\\) and \\(r(cx)=|c|\\,r(x)\\).\n2. \\(r(xy)=r(yx)\\) for all \\(x,y\\).\n3. If \\(xy=yx\\), then \\(r(xy)\\leq r(x)r(y)\\) and \\(r(x+y)\\leq r(x)+r(y)\\).\n4. If \\(A\\) is unital and \\(r(1-x)<1\\), then \\(x\\) is invertible, and the series (2.1) converges absolutely.\n\n**Proof.** (1) follows from (5.1). (2) is Proposition 4.2(2). (3) \\((xy)^n=x^ny^n\\) gives the first bound. For the second, fix \\(\\varepsilon>0\\). By (5.1) there is \\(C\\geq1\\) with \\(\\|x^k\\|\\leq C(r(x)+\\varepsilon)^k\\) and \\(\\|y^k\\|\\leq C(r(y)+\\varepsilon)^k\\) for all \\(k\\geq0\\). The binomial theorem holds for commuting elements, and gives \\(\\|(x+y)^n\\|\\leq C^2(r(x)+r(y)+2\\varepsilon)^n\\). (4) Choose \\(\\rho\\) with \\(r(1-x)<\\rho<1\\). By (5.1), \\(\\|(1-x)^n\\|\\leq\\rho^n\\) for large \\(n\\), so the series converges absolutely, and the proof of Proposition 2.1 applies. \\(\\square\\)\n\n**Examples 5.7.**\n- A nonzero nilpotent element \\(N\\) (\\(N^m=0\\)) has \\(r(N)=0<\\|N\\|\\).\n- *The Volterra operator.* On \\(C([0,1])\\) with the supremum norm, let \\((Vf)(t)=\\int_0^tf(s)\\,ds\\). Put \\(W_n(t)=\\int_0^t(t-s)^{n-1}f(s)/(n-1)!\\,ds\\). The fundamental theorem gives \\(W_1=Vf\\). For \\(n\\geq2\\), the moving-endpoint formula in [Lemma 0.1(4) of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) gives \\(W_n'=W_{n-1}\\) and \\(W_n(0)=0\\). Thus \\(W_n=VW_{n-1}\\) by the fundamental theorem, and induction gives \\(V^nf=W_n\\). So \\(\\|V^n\\|=1/n!\\), with equality at \\(f\\equiv1\\) and \\(t=1\\). Since at least half of the factors of \\(n!\\) are at least \\(n/2\\), \\((n!)^{1/n}\\geq(n/2)^{1/2}\\to\\infty\\). So \\(r(V)=0\\). The spectrum is not empty (Theorem 5.2) and lies in \\(\\{0\\}\\), so \\(\\sigma(V)=\\{0\\}\\), although \\(V\\neq0\\).\n- *Commutativity is needed in (3).* In \\(M_2(\\mathbb C)\\), \\(E_{12}\\) and \\(E_{21}\\) square to \\(0\\), so each has spectral radius \\(0\\). But \\(E_{12}+E_{21}\\) has eigenvalues \\(\\pm1\\), and \\(E_{12}E_{21}=E_{11}\\) has eigenvalues \\(0,1\\); both have spectral radius \\(1\\).\n- *Self-adjoint elements of a C\\*-algebra.* If \\(x=x^*\\), then \\(\\|x^2\\|=\\|x^*x\\|=\\|x\\|^2\\), so \\(\\|x^{2^k}\\|=\\|x\\|^{2^k}\\) for all \\(k\\), and (5.1) along the subsequence \\(2^k\\) gives \\(r(x)=\\|x\\|\\). The same holds for normal elements; this is proved in the [next lesson, on C\\*-algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-02).\n\n**Remark 5.8** (at most one C\\*-norm). A \\(*\\)-algebra carries at most one norm that makes it a C\\*-algebra. Indeed, let \\(N\\) be such a norm and \\(u\\) an element. Since \\(u^*u\\) is self-adjoint, the last example gives \\(N(u)^2=N(u^*u)=r(u^*u)\\). The spectral radius \\(r(u^*u)\\) is determined by the quasi-spectrum of \\(u^*u\\), which is defined algebraically and does not depend on the norm. In particular, for a nonunital C\\*-algebra \\(A\\), the C\\*-norm \\(p\\) on \\(A_1\\) of [Proposition 3.4](#oa-fnd-bn-05) is the only norm that makes \\(A_1\\) a C\\*-algebra.\n\n## 6. The holomorphic functional calculus\n\nFor a polynomial \\(p\\), the element \\(p(x)\\) makes sense in every unital algebra. This section defines \\(f(x)\\) for every function \\(f\\) that is holomorphic on a neighbourhood of \\(\\sigma_A(x)\\), by Cauchy's integral formula with the resolvent in place of \\((\\lambda-z)^{-1}\\). It then proves that \\(f\\mapsto f(x)\\) is a homomorphism, and that the spectrum of \\(f(x)\\) is \\(f(\\sigma_A(x))\\).\n\n### Paths, cycles and integrals\n\nA *path* is a piecewise continuously differentiable map \\(\\gamma:[a,b]\\to\\mathbb C\\). It is *closed* if \\(\\gamma(a)=\\gamma(b)\\). A *cycle* \\(\\Gamma\\) is a finite family of closed paths \\(\\gamma_1,\\dots,\\gamma_m\\), read as a formal sum. \\(\\Gamma^*\\) is the union of their images, a compact set, and \\(\\ell(\\Gamma)\\) is the total length. For a continuous map \\(F\\) from \\(\\Gamma^*\\) into a Banach space \\(X\\),\n\\[\n\\int_\\Gamma F(\\lambda)\\,d\\lambda=\\sum_{k=1}^m\\int_{a_k}^{b_k}F(\\gamma_k(t))\\,\\gamma_k'(t)\\,dt .\n\\]\nEach term is the Riemann integral of a piecewise continuous \\(X\\)-valued function on an interval. It exists for the same reason as for scalar functions: uniform continuity makes the Riemann sums a Cauchy net, and \\(X\\) is complete. The integral is linear in \\(F\\), satisfies \\(\\|\\int_\\Gamma F\\|\\leq\\ell(\\Gamma)\\sup_{\\Gamma^*}\\|F\\|\\), and commutes with bounded linear maps: \\(T\\int_\\Gamma F=\\int_\\Gamma T\\circ F\\). This applies to functionals, and to multiplication on either side by a fixed element of a Banach algebra. Uniform limits pass through the integral. For a continuous \\(F\\) on \\(\\Gamma_1^*\\times\\Gamma_2^*\\) the two iterated integrals agree: apply a functional, use the scalar statement for continuous functions on rectangles, and then the Hahn–Banach theorem. The *index* of a point \\(z\\notin\\Gamma^*\\) is\n\\[\n\\operatorname{Ind}_\\Gamma(z)=\\frac1{2\\pi i}\\int_\\Gamma\\frac{d\\lambda}{\\lambda-z}.\n\\]\nA cycle \\(\\Gamma\\) *surrounds* a compact set \\(K\\) in an open set \\(U\\supseteq K\\) if \\(\\Gamma^*\\subseteq U\\setminus K\\), \\(\\operatorname{Ind}_\\Gamma=1\\) on \\(K\\), and \\(\\operatorname{Ind}_\\Gamma=0\\) on \\(\\mathbb C\\setminus U\\). The empty cycle surrounds the empty set.\n\nWe use two facts proved in [Cauchy's theorem for cycles and its consequences](cauchy-s-theorem-for-cycles-and-its-consequences.md), cited as *the lesson on Cauchy's theorem*.\n\n**Theorem 6.1** (Cauchy's theorem). Let \\(U\\subseteq\\mathbb C\\) be open and let \\(\\Gamma\\) be a cycle in \\(U\\) with \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for every \\(\\alpha\\notin U\\). Let \\(g:U\\to X\\) be a map into a Banach space such that \\(\\varphi\\circ g\\) is holomorphic for every \\(\\varphi\\in X^*\\). Then \\(g\\) is continuous, \\(\\int_\\Gamma g(\\lambda)\\,d\\lambda=0\\), and\n\\[\n\\begin{gathered}\ng(z)\\operatorname{Ind}_\\Gamma(z)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{g(\\lambda)}{\\lambda-z}\\,d\\lambda\\\\\n(z\\in U\\setminus\\Gamma^*).\n\\end{gathered}\n\\]\n\nThis is Theorem 6.1(1)–(2) of the lesson on Cauchy's theorem. Its scalar case is Theorem 4.2 there. Applying a functional \\(\\varphi\\in X^*\\) reduces both formulas to the scalar case, and the Hahn–Banach theorem then gives them in \\(X\\).\n\nWe use the formula only at points of index \\(1\\) or \\(0\\). At a point \\(z\\) of index \\(0\\) it is the first conclusion, applied to \\(\\lambda\\mapsto g(\\lambda)/(\\lambda-z)\\) on \\(U\\setminus\\{z\\}\\). Applied on a disc, the formula gives the Taylor expansion of a holomorphic function at each point, in the usual way.\n\n**Theorem 6.2** (surrounding cycles). If \\(K\\subseteq U\\subseteq\\mathbb C\\) with \\(K\\) compact and \\(U\\) open, some cycle made of finitely many oriented segments surrounds \\(K\\) in \\(U\\).\n\nThis is Theorem 5.1 of the lesson on Cauchy's theorem.\n\n**Lemma 6.3** (index). Let \\(\\Gamma\\) be a cycle. On \\(\\mathbb C\\setminus\\Gamma^*\\), the function \\(\\operatorname{Ind}_\\Gamma\\) takes integer values, is constant on each connected component, and is \\(0\\) on the unbounded component. For the positively oriented circle \\(|\\lambda-c|=\\rho\\), the index is \\(1\\) on the open disc and \\(0\\) outside the closed disc.\n\n**Proof.** Let \\(\\gamma:[a,b]\\to\\mathbb C\\) be a closed path and \\(z\\notin\\gamma([a,b])\\). Put \\(h(t)=\\int_a^t\\gamma'(s)/(\\gamma(s)-z)\\,ds\\). The continuous function \\(F(t)=e^{-h(t)}(\\gamma(t)-z)\\) has derivative \\(e^{-h}\\big(-h'(\\gamma-z)+\\gamma'\\big)=0\\) wherever \\(\\gamma\\) is differentiable, so \\(F\\) is constant. Since \\(\\gamma(b)=\\gamma(a)\\neq z\\), \\(F(b)=F(a)\\) gives \\(e^{-h(b)}=1\\), so \\(h(b)/(2\\pi i)\\) is an integer. The index of a cycle is a sum of such integers. For \\(z,w\\notin\\Gamma^*\\), with \\(d(\\cdot)=\\operatorname{dist}(\\cdot,\\Gamma^*)\\),\n\\[\n|\\operatorname{Ind}_\\Gamma(z)-\\operatorname{Ind}_\\Gamma(w)|\\leq\\frac{\\ell(\\Gamma)\\,|z-w|}{2\\pi\\,d(z)\\,d(w)} .\n\\]\nSo \\(\\operatorname{Ind}_\\Gamma\\) is continuous and integer-valued, hence constant on components. Also \\(|\\operatorname{Ind}_\\Gamma(z)|\\leq\\ell(\\Gamma)/(2\\pi d(z))<1\\) when \\(d(z)\\) is large, so the index is \\(0\\) far out, hence on the whole unbounded component. For the circle, the index at the centre is \\(\\frac1{2\\pi i}\\int_0^{2\\pi}\\frac{i\\rho e^{it}}{\\rho e^{it}}\\,dt=1\\). \\(\\square\\)\n\nTwo sets built from a cycle \\(\\Gamma\\) that surrounds a compact set \\(K\\) in an open set \\(U\\) are used below:\n\\[\n\\begin{gathered}\nO_\\Gamma\\\\\n=\\{z\\notin\\Gamma^*:\\operatorname{Ind}_\\Gamma(z)=1\\},\\\\\nK_\\Gamma\\\\\n=\\Gamma^*\\cup\\{z\\notin\\Gamma^*:\\operatorname{Ind}_\\Gamma(z)\\neq0\\}.\n\\end{gathered}\n\\]\nBy Lemma 6.3, \\(O_\\Gamma\\) is open, and it contains \\(K\\). The set \\(K_\\Gamma\\) is closed, since its complement is the open set where the index is \\(0\\), and bounded, since the index is \\(0\\) on the unbounded component; so \\(K_\\Gamma\\) is compact. Both sets lie in \\(U\\), because \\(\\Gamma^*\\subseteq U\\) and the index is \\(0\\) off \\(U\\).\n\n",
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      "id": "OA-FND-BN-10",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Definition of the calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
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      "full_conditions_and_proof": "### Definition of the calculus\n\n**Definition 6.4** (holomorphic functional calculus). Let \\(A\\) be a unital Banach algebra, \\(x\\in A\\), \\(U\\subseteq\\mathbb C\\) open with \\(\\sigma_A(x)\\subseteq U\\), and \\(f\\in H(U)\\). Choose a cycle \\(\\Gamma\\) that surrounds \\(\\sigma_A(x)\\) in \\(U\\), which exists by Theorem 6.2, and put\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma f(\\lambda)\\,(\\lambda-x)^{-1}\\,d\\lambda\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma f(\\lambda)\\,R_x(\\lambda)\\,d\\lambda .\n\\end{gathered}\n\\tag{6.1}\n\\]\nThe integrand is continuous on \\(\\Gamma^*\\), because \\(\\Gamma^*\\subseteq\\rho_A(x)\\) and the resolvent is continuous there ([Proposition 4.4(5)](#oa-fnd-bn-07)).\n\n**Proposition 6.5.**\n1. Every cycle that surrounds \\(\\sigma_A(x)\\) in \\(U\\) gives the same element \\(f(x)\\).\n2. If \\(f\\in H(U)\\) and \\(g\\in H(V)\\) agree on an open set \\(W\\) with \\(\\sigma_A(x)\\subseteq W\\subseteq U\\cap V\\), then \\(f(x)=g(x)\\). So \\(f(x)\\) depends only on the germ of \\(f\\) at \\(\\sigma_A(x)\\), and (6.1) defines \\(f(x)\\) for every \\(f\\) holomorphic on some neighbourhood of \\(\\sigma_A(x)\\). These functions form an algebra, with the operations taken on the intersection of the domains.\n3. For \\(\\varphi\\in A^*\\), \\(\\varphi(f(x))=\\frac1{2\\pi i}\\int_\\Gamma f(\\lambda)\\,\\varphi(R_x(\\lambda))\\,d\\lambda\\).\n4. \\(f(x)\\) commutes with every element of \\(A\\) that commutes with \\(x\\).\n5. If \\(\\chi:A\\to\\mathbb C\\) is a unital homomorphism, then \\(\\chi(f(x))=f(\\chi(x))\\).\n\n**Proof.** (1) Let \\(\\Gamma\\) and \\(\\Gamma'\\) both surround \\(\\sigma_A(x)\\) in \\(U\\). Put \\(V=U\\setminus\\sigma_A(x)\\), an open set, and let \\(\\Gamma-\\Gamma'\\) be the cycle formed by the paths of \\(\\Gamma\\) and the reversed paths of \\(\\Gamma'\\). It lies in \\(V\\). For \\(\\alpha\\notin V\\), either \\(\\alpha\\notin U\\), where both indices are \\(0\\), or \\(\\alpha\\in\\sigma_A(x)\\), where both are \\(1\\); so \\(\\operatorname{Ind}_{\\Gamma-\\Gamma'}(\\alpha)=0\\). The map \\(\\lambda\\mapsto f(\\lambda)R_x(\\lambda)\\) from \\(V\\) to \\(A\\) becomes holomorphic after any functional is applied (Proposition 4.4(5)). By Cauchy's theorem 6.1, its integral over \\(\\Gamma-\\Gamma'\\) is \\(0\\).\n(2) A cycle that surrounds \\(\\sigma_A(x)\\) in \\(W\\) also surrounds it in \\(U\\) and in \\(V\\), because its index is \\(0\\) off \\(W\\). Compute both \\(f(x)\\) and \\(g(x)\\) with it.\n(3) and (4) Functionals and multiplications pass through the integral, and every \\(R_x(\\lambda)\\) commutes with the elements that commute with \\(x\\) (Proposition 4.4(4)).\n(5) \\(\\chi\\) is continuous, because a character of a Banach algebra has norm at most \\(1\\) ([Proposition 10.3(1)](#oa-fnd-bn-17); its proof does not use the calculus). Applying \\(\\chi\\) to \\((\\lambda-x)R_x(\\lambda)=1\\) gives \\(\\chi(R_x(\\lambda))=(\\lambda-\\chi(x))^{-1}\\). Also \\(\\chi(x)\\in\\sigma_A(x)\\): otherwise \\(\\chi(x)-x\\) would be invertible, and \\(1=\\chi\\big((\\chi(x)-x)(\\chi(x)-x)^{-1}\\big)=0\\). By Cauchy's theorem 6.1 for the scalar function \\(f\\) at the point \\(\\chi(x)\\),\n\\[\n\\begin{gathered}\n\\chi(f(x))\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{f(\\lambda)}{\\lambda-\\chi(x)}\\,d\\lambda\\\\\n=f(\\chi(x))\\operatorname{Ind}_\\Gamma(\\chi(x))\\\\\n=f(\\chi(x)).\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Remark 6.6** (one circle or one path). Let \\(C=\\partial D(c,r)\\) be positively oriented, with \\(\\sigma_A(x)\\subseteq D(c,r)\\) and \\(\\bar D(c,r)\\subseteq U\\). Then [Lemma 1.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-01) gives index \\(1\\) on the spectrum and \\(0\\) outside \\(U\\), so this one circle surrounds \\(\\sigma_A(x)\\) in \\(U\\). More generally a single closed path with these two index conditions is a surrounding cycle and may be used in (6.1). The precise index conditions are all the calculus needs.\n\nThe requirement that the region bounded by \\(C\\) lie in \\(U\\) cannot be dropped. Take \\(A=\\mathbb C\\), \\(x=0\\), \\(a\\neq0\\), \\(U=\\mathbb C\\setminus\\{a\\}\\) and \\(f(\\lambda)=1/(\\lambda-a)\\). The circle \\(|\\lambda|=|a|/2\\) gives \\(\\frac1{2\\pi i}\\oint f(\\lambda)\\lambda^{-1}\\,d\\lambda=-1/a\\), while the circle \\(|\\lambda|=2|a|\\) gives \\(-1/a+1/a=0\\). If the second circle were allowed, it would give \\(f(x)=0\\) and, for \\(g(\\lambda)=\\lambda-a\\), \\(g(x)=-a\\), but \\((fg)(x)=1\\); so the product rule of Theorem 6.7 below would fail.\n\nEven with this requirement, one curve reaches fewer functions than cycles do. If \\(\\sigma_A(x)\\) has two separated pieces, the function that is \\(0\\) near one piece and \\(1\\) near the other is holomorphic near \\(\\sigma_A(x)\\), but no single curve whose inside lies in its domain encloses both pieces. Example 6.12 uses exactly this function.\n\n*Nonunital algebras.* If \\(A\\) has no identity, apply the calculus in \\(A_1\\) to \\(j(x)\\), whose spectrum is \\(\\sigma'_A(x)\\). Since \\(q\\) is a unital homomorphism with \\(q(j(x))=0\\), part (5) gives \\(q(f(x))=f(0)\\). So \\(f(x)\\in j(A)\\) exactly when \\(f(0)=0\\). The [next lesson](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-08) proves the analogue for the continuous functional calculus of a normal element of a C\\*-algebra.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BN-11",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "The calculus is a unital homomorphism",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
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      },
      "full_conditions_and_proof": "### The calculus is a unital homomorphism\n\n**Theorem 6.7** (the calculus is a homomorphism). Let \\(A\\) be a unital Banach algebra, \\(x\\in A\\), and \\(U\\supseteq\\sigma_A(x)\\) open. The map \\(f\\mapsto f(x)\\) from \\(H(U)\\) to \\(A\\) is a unital algebra homomorphism: it is linear, \\((fg)(x)=f(x)g(x)\\), \\(1(x)=1\\), and \\(u(x)=x\\) for \\(u(\\lambda)=\\lambda\\). So \\(p(x)\\) has its usual meaning for every polynomial \\(p\\). The map is continuous: if \\(f_n\\to f\\) uniformly on compact subsets of \\(U\\), then \\(f_n(x)\\to f(x)\\). Any two elements \\(f(x),g(x)\\) commute.\n\nThe proof below uses cycles satisfying the exact index conditions. Remark 6.6 explains the conditions required for a single curve, and Example 6.12 shows why cycles are useful for disconnected spectra.\n\n**Proof.** Linearity is clear.\n\n*Products.* Choose a cycle \\(\\Gamma_2\\) that surrounds \\(\\sigma_A(x)\\) in \\(U\\). The set \\(K_2=K_{\\Gamma_2}\\), defined after Lemma 6.3, is compact and lies in \\(U\\); choose a cycle \\(\\Gamma_1\\) that surrounds \\(K_2\\) in \\(U\\). Since \\(\\sigma_A(x)\\subseteq K_2\\), \\(\\Gamma_1\\) also surrounds \\(\\sigma_A(x)\\) in \\(U\\). Moreover \\(\\Gamma_1^*\\cap\\Gamma_2^*=\\varnothing\\), \\(\\operatorname{Ind}_{\\Gamma_2}(\\lambda)=0\\) for \\(\\lambda\\in\\Gamma_1^*\\) (such \\(\\lambda\\) lie outside \\(K_2\\)), and \\(\\operatorname{Ind}_{\\Gamma_1}(\\mu)=1\\) for \\(\\mu\\in\\Gamma_2^*\\subseteq K_2\\). By the resolvent identity, for \\(\\lambda\\in\\Gamma_1^*\\) and \\(\\mu\\in\\Gamma_2^*\\),\n\\[\nR_x(\\lambda)R_x(\\mu)=\\frac{R_x(\\mu)-R_x(\\lambda)}{\\lambda-\\mu}.\n\\tag{6.2}\n\\]\nWith the rules for integrals stated at the start of this section,\n\\[\n\\begin{gathered}\nf(x)g(x)\\\\\n=\\frac1{(2\\pi i)^2}\\int_{\\Gamma_1}\\int_{\\Gamma_2}f(\\lambda)g(\\mu)R_x(\\lambda)R_x(\\mu)\\,d\\mu\\,d\\lambda\\\\\n=I_1-I_2,\n\\end{gathered}\n\\]\nwhere\n\\[\n\\begin{gathered}\nI_1\\\\\n=\\frac1{(2\\pi i)^2}\\int_{\\Gamma_2}g(\\mu)R_x(\\mu)\\Big(\\int_{\\Gamma_1}\\frac{f(\\lambda)}{\\lambda-\\mu}\\,d\\lambda\\Big)d\\mu,\\\\\nI_2\\\\\n=\\frac1{(2\\pi i)^2}\\int_{\\Gamma_1}f(\\lambda)R_x(\\lambda)\\Big(\\int_{\\Gamma_2}\\frac{g(\\mu)}{\\lambda-\\mu}\\,d\\mu\\Big)d\\lambda .\n\\end{gathered}\n\\]\nBy Cauchy's theorem 6.1 for \\(g\\), the inner integral of \\(I_2\\) is \\(-2\\pi i\\operatorname{Ind}_{\\Gamma_2}(\\lambda)g(\\lambda)=0\\), so \\(I_2=0\\). By Cauchy's theorem 6.1 for \\(f\\), the inner integral of \\(I_1\\) is \\(2\\pi i\\operatorname{Ind}_{\\Gamma_1}(\\mu)f(\\mu)=2\\pi i f(\\mu)\\). So \\(I_1=\\frac1{2\\pi i}\\int_{\\Gamma_2}f(\\mu)g(\\mu)R_x(\\mu)\\,d\\mu=(fg)(x)\\).\n\n*The functions \\(1\\) and \\(u\\).* Both are entire, so by Proposition 6.5(2) we may compute them with \\(U=\\mathbb C\\) and \\(\\Gamma\\) the positively oriented circle \\(|\\lambda|=\\rho\\), where \\(\\rho>\\|x\\|\\). By Lemma 6.3, and since \\(\\sigma_A(x)\\) lies in the disc of radius \\(\\|x\\|\\) ([Proposition 4.4(1)](#oa-fnd-bn-07)), \\(\\Gamma\\) surrounds \\(\\sigma_A(x)\\) in \\(\\mathbb C\\). By (4.1), \\(\\lambda^mR_x(\\lambda)=\\sum_nx^n\\lambda^{m-n-1}\\) uniformly on \\(\\Gamma^*\\), and \\(\\frac1{2\\pi i}\\oint\\lambda^{m-n-1}\\,d\\lambda\\) is \\(1\\) if \\(n=m\\) and \\(0\\) otherwise. So \\(1(x)=x^0=1\\) and \\(u(x)=x\\).\n\n*Continuity.* With one fixed \\(\\Gamma\\), \\[\n\\begin{gathered}\n\\|f_n(x)-f(x)\\|\\\\\n\\leq\\frac{\\ell(\\Gamma)}{2\\pi}\\sup_{\\Gamma^*}|f_n-f|\\,\\sup_{\\Gamma^*}\\|R_x\\|.\n\\end{gathered}\n\\]\n\n*Commutation.* \\(f(x)g(x)=(fg)(x)=(gf)(x)=g(x)f(x)\\). \\(\\square\\)\n\n**Proposition 6.8** (power series). Let \\(f(\\lambda)=\\sum_nc_n(\\lambda-c)^n\\) converge for \\(|\\lambda-c|<\\rho_0\\), and let \\(r(x-c)<\\rho_0\\). Then \\(\\sigma_A(x)\\) lies in the disc \\(|\\lambda-c|<\\rho_0\\), and \\(f(x)=\\sum_nc_n(x-c)^n\\), with the series converging in norm. In particular, the calculus of \\(\\lambda\\mapsto e^\\lambda\\) is \\(\\exp x=\\sum_nx^n/n!\\) ([Section 7](#oa-fnd-bn-14)).\n\n**Proof.** By parts (5) and (1) of [Proposition 4.2](#oa-fnd-bn-06), \\(\\sigma_A(x)=c+\\sigma_A(x-c)\\) lies in \\(|\\lambda-c|\\leq r(x-c)\\). Choose \\(\\rho\\) with \\(r(x-c)<\\rho<\\rho_0\\), and let \\(\\Gamma\\) be the positively oriented circle \\(|\\lambda-c|=\\rho\\). By Lemma 6.3 it surrounds \\(\\sigma_A(x)\\) in the disc \\(\\{|\\lambda-c|<\\rho_0\\}\\). The series of \\(f\\) converges uniformly on \\(\\Gamma^*\\). Integrating term by term, and using the theorem for the polynomials \\((\\lambda-c)^n\\), gives \\(f(x)=\\sum_nc_n(x-c)^n\\); the partial sums converge in norm. \\(\\square\\)\n\n**Example 6.9.** Let \\(x=\\alpha+N\\) with \\(N^2=0\\) and \\(N\\neq0\\), for instance a \\(2\\times2\\) Jordan block. Then \\(\\sigma_A(x)=\\{\\alpha\\}\\) (the spectrum of \\(N\\) is not empty and \\(r(N)=0\\)), and by the proposition \\(f(x)=f(\\alpha)+f'(\\alpha)N\\) for every \\(f\\) holomorphic near \\(\\alpha\\). So \\(f(x)\\) is not determined by the values of \\(f\\) on \\(\\sigma_A(x)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-BN-12",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Spectral mapping and composition",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-12",
      "proof_locus": {
        "line": 565,
        "through_line": 604
      },
      "full_conditions_and_proof": "### Spectral mapping and composition\n\n**Theorem 6.10** (invertibility, spectral mapping and composition). Let \\(A\\) be a unital Banach algebra, \\(x\\in A\\), \\(U\\supseteq\\sigma_A(x)\\) open, and \\(f\\in H(U)\\).\n1. \\(f(x)\\) is invertible if and only if \\(f\\) has no zero on \\(\\sigma_A(x)\\). Then \\(f(x)^{-1}=(1/f)(x)\\), where \\(1/f\\) is holomorphic on the open set \\(\\{\\lambda\\in U:f(\\lambda)\\neq0\\}\\supseteq\\sigma_A(x)\\).\n2. (*Spectral mapping theorem.*) \\(\\sigma_A(f(x))=f(\\sigma_A(x))\\).\n3. (*Composition.*) Let \\(V\\supseteq f(\\sigma_A(x))\\) be open and \\(g\\in H(V)\\). Then \\(g\\circ f\\) is holomorphic on \\(U\\cap f^{-1}(V)\\), which contains \\(\\sigma_A(x)\\), and \\((g\\circ f)(x)=g(f(x))\\).\n\n**Proof.** (1) If \\(f\\) has no zero on \\(\\sigma_A(x)\\), then \\(1/f\\) is holomorphic on \\(W=\\{\\lambda\\in U:f(\\lambda)\\neq0\\}\\supseteq\\sigma_A(x)\\), and \\(f\\cdot(1/f)=1\\) on \\(W\\). By Theorem 6.7 and Proposition 6.5(2), \\(f(x)(1/f)(x)=(1/f)(x)f(x)=1\\). Conversely, let \\(f(\\alpha)=0\\) with \\(\\alpha\\in\\sigma_A(x)\\). The Taylor expansion of \\(f\\) at \\(\\alpha\\) (from Cauchy's theorem 6.1) has no constant term. So \\(k(\\lambda)=f(\\lambda)/(\\lambda-\\alpha)\\) for \\(\\lambda\\neq\\alpha\\), with \\(k(\\alpha)=f'(\\alpha)\\), defines \\(k\\in H(U)\\), and \\(f(\\lambda)=(\\lambda-\\alpha)k(\\lambda)\\). By Theorem 6.7, \\(f(x)=(x-\\alpha)k(x)=k(x)(x-\\alpha)\\). If \\(f(x)\\) had an inverse \\(w\\), then \\(k(x)w\\) would be a right inverse and \\(wk(x)\\) a left inverse of the noninvertible element \\(x-\\alpha\\).\n(2) For \\(\\beta\\in\\mathbb C\\), \\((f-\\beta)(x)=f(x)-\\beta\\) by Theorem 6.7. By (1), \\(\\beta\\in\\sigma_A(f(x))\\) if and only if \\(f-\\beta\\) has a zero on \\(\\sigma_A(x)\\), that is, \\(\\beta\\in f(\\sigma_A(x))\\).\n(3) The set \\(U_0=U\\cap f^{-1}(V)\\) is open, contains \\(\\sigma_A(x)\\), and \\(g\\circ f\\in H(U_0)\\). By (2), \\(f(\\sigma_A(x))=\\sigma_A(f(x))\\). Choose a cycle \\(\\Gamma_V\\) that surrounds \\(f(\\sigma_A(x))\\) in \\(V\\). The set \\(O=O_{\\Gamma_V}\\), defined after Lemma 6.3, is open, lies in \\(V\\), and contains \\(f(\\sigma_A(x))\\). So \\(W=U\\cap f^{-1}(O)\\) is an open subset of \\(U_0\\) that contains \\(\\sigma_A(x)\\). Choose a cycle \\(\\Gamma_W\\) that surrounds \\(\\sigma_A(x)\\) in \\(W\\). For \\(\\zeta\\in\\Gamma_V^*\\) and \\(\\lambda\\in W\\) we have \\(f(\\lambda)\\in O\\), so \\(f(\\lambda)\\neq\\zeta\\). Thus \\(\\lambda\\mapsto(\\zeta-f(\\lambda))^{-1}\\) is holomorphic on \\(W\\), and by (1), applied in \\(W\\),\n\\[\n(\\zeta-f(x))^{-1}=\\frac1{2\\pi i}\\int_{\\Gamma_W}(\\zeta-f(\\lambda))^{-1}R_x(\\lambda)\\,d\\lambda .\n\\]\nSince \\(\\Gamma_V\\) surrounds \\(\\sigma_A(f(x))\\) in \\(V\\), inserting this and exchanging the order of integration (the integrand is continuous on \\(\\Gamma_V^*\\times\\Gamma_W^*\\)) gives\n\\[\n\\begin{gathered}\ng(f(x))\\\\\n=\\frac1{2\\pi i}\\int_{\\Gamma_V}g(\\zeta)(\\zeta-f(x))^{-1}\\,d\\zeta\\\\\n=\\frac1{2\\pi i}\\int_{\\Gamma_W}\\Big(\\frac1{2\\pi i}\\int_{\\Gamma_V}\\frac{g(\\zeta)}{\\zeta-f(\\lambda)}\\,d\\zeta\\Big)R_x(\\lambda)\\,d\\lambda .\n\\end{gathered}\n\\]\nFor \\(\\lambda\\in\\Gamma_W^*\\), \\(f(\\lambda)\\in O\\subseteq V\\setminus\\Gamma_V^*\\), so by Cauchy's theorem 6.1 the inner integral is \\(g(f(\\lambda))\\operatorname{Ind}_{\\Gamma_V}(f(\\lambda))=g(f(\\lambda))\\). Hence \\[\n\\begin{gathered}\ng(f(x))\\\\\n=\\frac1{2\\pi i}\\int_{\\Gamma_W}(g\\circ f)(\\lambda)R_x(\\lambda)\\,d\\lambda\\\\\n=(g\\circ f)(x),\n\\end{gathered}\n\\] by Proposition 6.5(2). \\(\\square\\)\n\n**Remark 6.11** (cycles are needed). The proof of (1) uses the function \\(1/f\\), which is holomorphic only where \\(f\\) has no zero. There may be no single closed curve that encloses \\(\\sigma_A(x)\\) and has its inside in that domain, although the conclusion is true. For example, in \\(A=C(\\mathbb T)\\) with \\(x(z)=z\\), \\(f(\\lambda)=\\lambda\\) and \\(h=1/f\\), every closed path \\(C\\) disjoint from \\(\\mathbb T\\) and having index \\(1\\) on \\(\\mathbb T\\) also has index \\(1\\) at \\(0\\), where \\(h(\\lambda)=1/\\lambda\\) is not defined. Indeed, the connected image of \\(C\\) lies entirely inside or entirely outside the unit circle. In the first case \\(\\mathbb T\\) lies in its unbounded complementary component, so its index would be zero. In the second case the closed unit disc is a connected set disjoint from \\(C\\), so the index is constant there and equals \\(1\\) at \\(0\\). The cycle made of the circles \\(|\\lambda|=2\\) and \\(|\\lambda|=1/2\\), the second one reversed, surrounds \\(\\mathbb T\\) in \\(\\mathbb C\\setminus\\{0\\}\\), and (1) gives \\(h(x)=x^{-1}\\).\n\n*Characters.* For commutative \\(A\\), (2) also follows from Proposition 6.5(5) and the description of the spectrum by characters in [Theorem 11.1](#oa-fnd-bn-18): \\[\n\\begin{gathered}\n\\sigma_A(f(x))\\\\\n=\\{\\chi(f(x))\\}\\\\\n=\\{f(\\chi(x))\\}\\\\\n=f(\\sigma_A(x)),\n\\end{gathered}\n\\] with \\(\\chi\\) running over the characters.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-22",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Idempotents from a disconnected spectrum",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-22",
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        "line": 605,
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      },
      "full_conditions_and_proof": "### Idempotents from a disconnected spectrum\n\n**Example 6.12.** Let \\(A\\) be a unital Banach algebra, and let \\(x\\in A\\) have spectrum \\(\\sigma_A(x)=K_0\\cup K_1\\), where \\(K_0\\) and \\(K_1\\) are disjoint, compact and nonempty. Choose disjoint open sets \\(U_0\\supseteq K_0\\) and \\(U_1\\supseteq K_1\\), and let \\(f\\) be \\(0\\) on \\(U_0\\) and \\(1\\) on \\(U_1\\). Then \\(f\\in H(U_0\\cup U_1)\\) and \\(f^2=f\\). The element \\(e=f(x)\\) satisfies \\(e^2=e\\) (Theorem 6.7), \\(ex=xe\\) (Proposition 6.5(4)), and \\(\\sigma_A(e)=f(\\sigma_A(x))=\\{0,1\\}\\) (Theorem 6.10); so \\(e\\neq0\\) and \\(e\\neq1\\). No single closed curve whose closed inside lies in \\(U_0\\cup U_1\\) encloses \\(\\sigma_A(x)\\): the closed region bounded by it is connected, so it lies in \\(U_0\\) or in \\(U_1\\). So \\(e\\) is out of reach of a calculus built on one closed curve. For the matrix \\(x=\\begin{pmatrix}1&1\\\\0&2\\end{pmatrix}\\), with \\(K_0=\\{1\\}\\) and \\(K_1=\\{2\\}\\), Exercise 3 at the end of the lesson gives \\(e=x-1=\\begin{pmatrix}0&1\\\\0&1\\end{pmatrix}\\), which is indeed idempotent.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-BN-14",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "7. Exponential, logarithm and the invertible group",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-14",
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        "line": 609,
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      "full_conditions_and_proof": "## 7. Exponential, logarithm and the invertible group\n\nLet \\(A\\) be a nontrivial unital Banach algebra. Let \\(D=\\mathbb C\\setminus(-\\infty,0]\\), and let \\(\\operatorname{Log}\\lambda=\\ln|\\lambda|+i\\arg\\lambda\\) with \\(\\arg\\lambda\\in(-\\pi,\\pi)\\) be the principal logarithm on \\(D\\).\n\n*Facts about \\(\\operatorname{Log}\\).* It is continuous on \\(D\\), \\(e^{\\operatorname{Log}\\lambda}=\\lambda\\), and \\(\\operatorname{Log}(e^w)=w\\) when \\(|\\operatorname{Im}w|<\\pi\\). It is holomorphic with derivative \\(1/\\lambda\\): if \\(\\lambda\\to\\lambda_0\\) in \\(D\\), then \\(w=\\operatorname{Log}\\lambda\\to w_0=\\operatorname{Log}\\lambda_0\\), and \\[\n\\begin{gathered}\n(\\operatorname{Log}\\lambda-\\operatorname{Log}\\lambda_0)/(\\lambda-\\lambda_0)\\\\\n=(w-w_0)/(e^w-e^{w_0})\\to e^{-w_0}\\\\\n=1/\\lambda_0.\n\\end{gathered}\n\\] On the disc \\(|\\lambda-1|<1\\), which lies in \\(D\\), the series \\(-\\sum_{n\\geq1}(1-\\lambda)^n/n\\) has the same derivative \\(\\sum_{n\\geq1}(1-\\lambda)^{n-1}=1/\\lambda\\) and the same value \\(0\\) at \\(\\lambda=1\\). So it equals \\(\\operatorname{Log}\\) there.\n\n**Proposition 7.1.**\n1. \\(\\exp x=\\sum_{n\\geq0}x^n/n!\\) converges absolutely, and it is the calculus of \\(\\lambda\\mapsto e^\\lambda\\). If \\(xy=yx\\), then \\(\\exp(x+y)=\\exp x\\exp y\\). Hence \\(\\exp(-x)=(\\exp x)^{-1}\\), \\(\\exp(A)\\subseteq G(A)\\), and \\(t\\mapsto\\exp(tx)\\) is a norm-continuous homomorphism from \\((\\mathbb R,+)\\) into \\(G(A)\\).\n2. Let \\(G_0(A)\\) be the connected component of \\(1\\) in \\(G(A)\\), the *principal component*. It is an open and closed normal subgroup of \\(G(A)\\), and it consists of the elements that can be joined to \\(1\\) by a continuous path in \\(G(A)\\). It contains \\(\\exp(A)\\). The quotient \\(G(A)/G_0(A)\\) is the *index group*.\n3. If \\(\\sigma_A(x)\\subseteq D\\), put \\(\\log x=\\operatorname{Log}(x)\\), the calculus of \\(\\operatorname{Log}\\). Then \\(\\exp(\\log x)=x\\). If \\(r(x-1)<1\\), then \\(\\log x=-\\sum_{n\\geq1}(1-x)^n/n\\).\n4. If \\(\\sigma_A(x)\\) lies in the strip \\(S=\\{|\\operatorname{Im}\\lambda|<\\pi\\}\\), then \\(\\sigma_A(\\exp x)\\subseteq D\\) and \\(\\log(\\exp x)=x\\).\n5. \\(G_0(A)\\) is the subgroup generated by \\(\\exp(A)\\). If \\(A\\) is commutative, \\(G_0(A)=\\exp(A)\\).\n\n**Proof.** (1) \\(\\|x^n/n!\\|\\leq\\|x\\|^n/n!\\) for \\(n\\geq1\\). [Proposition 6.8](#oa-fnd-bn-11), with \\(c=0\\) and \\(\\rho_0=\\infty\\), identifies \\(\\exp x\\) with the calculus of \\(e^\\lambda\\). For commuting \\(x\\) and \\(y\\), the Cauchy product of the two absolutely convergent series is allowed, and the binomial theorem turns it into \\(\\sum_n(x+y)^n/n!\\). Also \\(\\exp0=1\\). For continuity, \\[\n\\begin{gathered}\n\\|\\exp(tx)-\\exp(t_0x)\\|\\\\\n\\leq\\|\\exp(t_0x)\\|\\,\\|\\exp((t-t_0)x)-1\\|\\\\\n\\leq\\|\\exp(t_0x)\\|\\big(e^{|t-t_0|\\|x\\|}-1\\big).\n\\end{gathered}\n\\]\n(2) \\(G(A)\\) is open ([Proposition 2.2](#oa-fnd-bn-02)), so each of its points has a ball around it inside \\(G(A)\\), and balls are convex. Hence the path components of \\(G(A)\\) are open. Each is also closed in \\(G(A)\\), since its complement is a union of path components. So the path component of \\(1\\) is connected, open and closed in \\(G(A)\\); it is therefore the connected component \\(G_0(A)\\). If \\(\\alpha\\) and \\(\\beta\\) are paths in \\(G(A)\\) from \\(1\\) to \\(a\\) and to \\(b\\), then \\(t\\mapsto\\alpha(t)\\beta(t)\\) and \\(t\\mapsto\\alpha(t)^{-1}\\) are paths from \\(1\\) to \\(ab\\) and to \\(a^{-1}\\); the second is continuous because inversion is continuous (Proposition 2.2). For \\(c\\in G(A)\\), \\(t\\mapsto c\\alpha(t)c^{-1}\\) is a path from \\(1\\) to \\(cac^{-1}\\). So \\(G_0(A)\\) is a normal subgroup. The path \\(t\\mapsto\\exp(tx)\\), \\(0\\leq t\\leq1\\), joins \\(1\\) to \\(\\exp x\\).\n(3) By the composition rule, [Theorem 6.10(3)](#oa-fnd-bn-12), with \\(f=\\operatorname{Log}\\) and \\(g=\\exp\\), \\(\\exp(\\log x)=(\\exp\\circ\\operatorname{Log})(x)=x\\), because \\(\\exp\\circ\\operatorname{Log}\\) is the identity function on \\(D\\). If \\(r(x-1)<1\\), Proposition 6.8 with \\(c=1\\) and \\(\\rho_0=1\\) gives the series.\n(4) By the spectral mapping theorem, Theorem 6.10(2), \\(\\sigma_A(\\exp x)=\\exp(\\sigma_A(x))\\). For \\(\\lambda=a+ib\\) with \\(|b|<\\pi\\), \\(e^\\lambda=e^ae^{ib}\\) is not in \\((-\\infty,0]\\). So \\(\\sigma_A(\\exp x)\\subseteq D\\). By Theorem 6.10(3) with \\(f=\\exp\\) and \\(g=\\operatorname{Log}\\), \\(\\log(\\exp x)=(\\operatorname{Log}\\circ\\exp)(x)\\). On the open set \\(S\\supseteq\\sigma_A(x)\\), \\(\\operatorname{Log}\\circ\\exp\\) is the identity function, so \\((\\operatorname{Log}\\circ\\exp)(x)=x\\) by [Proposition 6.5(2)](#oa-fnd-bn-10) and Theorem 6.7.\n(5) Let \\(\\Gamma_e\\) be the subgroup generated by \\(\\exp(A)\\). By (1) and (2), \\(\\Gamma_e\\subseteq G_0(A)\\). If \\(\\|y-1\\|<1\\), then \\(r(y-1)<1\\), so \\(\\sigma_A(y)\\) lies in the disc \\(|\\lambda-1|<1\\), inside \\(D\\), and \\(y=\\exp(\\log y)\\in\\exp(A)\\) by (3). If \\(g\\in\\Gamma_e\\) and \\(\\|z-g\\|<1/\\|g^{-1}\\|\\), then \\(\\|g^{-1}z-1\\|<1\\), so \\(z=g(g^{-1}z)\\in\\Gamma_e\\). So \\(\\Gamma_e\\) is open. Its complement in \\(G_0(A)\\) is a union of cosets \\(g\\Gamma_e\\), each open, so \\(\\Gamma_e\\) is also closed in \\(G_0(A)\\). Since \\(G_0(A)\\) is connected, \\(\\Gamma_e=G_0(A)\\). If \\(A\\) is commutative, \\(\\exp(A)\\) is already a subgroup by (1), so \\(G_0(A)=\\exp(A)\\). \\(\\square\\)\n\nPart (5) shows what the logarithm of (3) is good for.\n\n**Example 7.2** (a nontrivial index group). Let \\(A=C(\\mathbb T)\\), the continuous functions on the unit circle, and let \\(z\\) be the identity function. It is invertible, with inverse \\(\\bar z\\). By (5), \\(G_0(A)=\\exp(A)\\), and \\(\\exp g=e^{g}\\) pointwise. Suppose \\(z=e^{g}\\) with \\(g\\in C(\\mathbb T)\\), and put \\(h(t)=g(e^{it})-it\\) for \\(t\\in[0,2\\pi]\\). Then \\(e^{h(t)}=1\\), so \\(h\\) takes values in \\(2\\pi i\\mathbb Z\\); being continuous, it is constant. But \\(h(2\\pi)=g(1)-2\\pi i\\neq g(1)=h(0)\\). So \\(z\\notin G_0(A)\\), and the index group of \\(C(\\mathbb T)\\) is not trivial.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-13",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "8. Perturbation of the spectrum",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
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      "anchor": "oa-fnd-bn-13",
      "proof_locus": {
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      "full_conditions_and_proof": "## 8. Perturbation of the spectrum\n\n**Theorem 8.1** (upper semicontinuity of the spectrum). Let \\(A\\) be a unital Banach algebra and \\(U\\subseteq\\mathbb C\\) open. The set \\(E_U=\\{x\\in A:\\sigma_A(x)\\subseteq U\\}\\) is open. More precisely, if \\(\\sigma_A(x_0)\\subseteq U\\), then \\(M=\\sup_{\\lambda\\notin U}\\|R_{x_0}(\\lambda)\\|\\) is finite, and \\(\\sigma_A(x)\\subseteq U\\) whenever \\(M\\|x-x_0\\|<1\\).\n\n**Proof.** If \\(A=\\{0\\}\\), every spectrum is empty and \\(E_U=A\\); all resolvents have norm zero, so take \\(M=0\\). If \\(U=\\mathbb C\\), take \\(M=0\\) (the supremum of an empty family of nonnegative numbers); there is nothing to prove. Otherwise \\(F=\\mathbb C\\setminus U\\) is closed and contained in \\(\\rho_A(x_0)\\). The continuous function \\(\\lambda\\mapsto\\|R_{x_0}(\\lambda)\\|\\) is bounded on the compact set \\(F\\cap\\{|\\lambda|\\leq\\|x_0\\|+1\\}\\), and by the bound of [Proposition 4.4(1)](#oa-fnd-bn-07) it is at most \\(c_1\\) when \\(|\\lambda|\\geq\\|x_0\\|+1\\). So \\(M<\\infty\\). If \\(M\\|x-x_0\\|<1\\) and \\(\\lambda\\in F\\), then \\[\n\\begin{gathered}\n\\|(\\lambda-x)-(\\lambda-x_0)\\|\\\\\n=\\|x-x_0\\|<1/\\|R_{x_0}(\\lambda)\\|,\n\\end{gathered}\n\\] so \\(\\lambda-x\\) is invertible by [Proposition 2.2](#oa-fnd-bn-02). \\(\\square\\)\n\n**Theorem 8.2** (continuity of the calculus). Let \\(f\\in H(U)\\). The map \\(x\\mapsto f(x)\\) is continuous on the open set \\(E_U\\), and it is locally Lipschitz: each \\(x_0\\in E_U\\) has \\(\\delta>0\\) and \\(L\\) with \\(\\|f(x)-f(x_0)\\|\\leq L\\|x-x_0\\|\\) for \\(\\|x-x_0\\|<\\delta\\). In particular, for a compact \\(K\\subseteq U\\), the map is continuous on \\(A_K=\\{x:\\sigma_A(x)\\subseteq K\\}\\), which is contained in \\(E_U\\).\n\nThe set \\(A_K\\) need not be open: in a nontrivial algebra, for \\(K=\\{0\\}\\), it contains \\(0\\) but no \\(\\varepsilon1\\) with \\(\\varepsilon\\neq0\\). So continuity on the open set \\(E_U\\) is the stronger statement.\n\n**Proof.** If \\(A=\\{0\\}\\), the calculus is identically zero and the estimate holds with \\(\\delta=L=1\\). Otherwise choose a cycle \\(\\Gamma\\) that surrounds \\(\\sigma_A(x_0)\\) in \\(U\\). The open set \\(O_\\Gamma\\), defined after Lemma 6.3, contains \\(\\sigma_A(x_0)\\) and lies in \\(U\\). By Theorem 8.1 there is \\(\\delta_1>0\\) with \\(\\sigma_A(x)\\subseteq O_\\Gamma\\) for \\(\\|x-x_0\\|<\\delta_1\\). For such \\(x\\), \\(\\Gamma\\) surrounds \\(\\sigma_A(x)\\) in \\(U\\), so \\(f(x)\\) and \\(f(x_0)\\) are integrals over the same \\(\\Gamma\\). Let \\(M_\\Gamma=\\max_{\\Gamma^*}\\|R_{x_0}\\|\\), and suppose also \\(\\|x-x_0\\|\\leq1/(2M_\\Gamma)\\). Then (2.3), applied to \\(\\lambda-x_0\\) and \\(\\lambda-x\\), gives \\(\\|R_x(\\lambda)\\|\\leq2M_\\Gamma\\) on \\(\\Gamma^*\\). The second resolvent identity\n\\[\n\\begin{gathered}\nR_x(\\lambda)-R_{x_0}(\\lambda)\\\\\n=R_x(\\lambda)\\,(x-x_0)\\,R_{x_0}(\\lambda)\n\\end{gathered}\n\\tag{8.1}\n\\]\nthen gives \\(\\|R_x(\\lambda)-R_{x_0}(\\lambda)\\|\\leq2M_\\Gamma^2\\|x-x_0\\|\\) on \\(\\Gamma^*\\). Hence \\(\\|f(x)-f(x_0)\\|\\leq\\frac{\\ell(\\Gamma)}{\\pi}\\big(\\sup_{\\Gamma^*}|f|\\big)M_\\Gamma^2\\,\\|x-x_0\\|\\). \\(\\square\\)\n\n**Example 8.3** (the spectrum is not continuous). On \\(\\ell^2(\\mathbb Z)\\) with orthonormal basis \\((e_n)\\), a bounded sequence \\(w=(w_n)\\) defines the weighted shift \\(W_we_n=w_ne_{n+1}\\). It maps the basis to orthogonal vectors, so \\(\\|W_w\\|=\\sup_n|w_n|\\). Also \\(W_w^ke_n=w_nw_{n+1}\\cdots w_{n+k-1}e_{n+k}\\), so \\(\\|W_w^k\\|\\) is the supremum of the products of \\(k\\) consecutive weights. For \\(t\\in\\mathbb C\\) let \\(W_t\\) have weights \\(w_0=t\\) and \\(w_n=1\\) for \\(n\\neq0\\). Then \\(\\|W_t-W_0\\|=|t|\\).\n- *\\(\\sigma(W_0)\\) is the closed unit disc.* \\(\\|W_0\\|=1\\). For \\(|\\lambda|<1\\), the vector \\(v=\\sum_{n\\leq0}\\lambda^{-n}e_n\\) lies in \\(\\ell^2(\\mathbb Z)\\), and \\(W_0v=\\sum_{n\\leq-1}\\lambda^{-n}e_{n+1}=\\lambda v\\). So every \\(\\lambda\\) with \\(|\\lambda|<1\\) is an eigenvalue, and the closed spectrum contains the closed disc.\n- *For \\(t\\neq0\\), \\(\\sigma(W_t)\\) lies in the unit circle.* Every product of \\(k\\) consecutive weights contains the weight \\(t\\) at most once, so \\(\\|W_t^k\\|\\leq\\max(1,|t|)\\) and \\(r(W_t)\\leq1\\) by the spectral radius formula (5.1). \\(W_t\\) is invertible, with \\(W_t^{-1}e_{n+1}=w_n^{-1}e_n\\), and the same count gives \\(\\|W_t^{-k}\\|\\leq\\max(1,1/|t|)\\), so \\(r(W_t^{-1})\\leq1\\). By [Proposition 4.2(4)](#oa-fnd-bn-06), \\(\\sigma(W_t)\\) lies in \\(\\{|\\lambda|\\geq1\\}\\), hence in the circle.\n\nSo \\(W_t\\to W_0\\) in norm while every \\(\\sigma(W_t)\\), \\(t\\neq0\\), stays in the circle and \\(\\sigma(W_0)\\) is the whole disc. A small perturbation can make the spectrum much smaller; by Theorem 8.1, it can never make it much larger. Theorem 8.1 cannot be improved to continuity.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BN-15",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "9. Modular ideals, maximal ideals and quotient algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
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      "full_conditions_and_proof": "## 9. Modular ideals, maximal ideals and quotient algebras\n\n**Definition 9.1.** Let \\(A\\) be an algebra.\n1. A left ideal \\(L\\) is *modular* if some \\(u\\in A\\) satisfies \\(x-xu\\in L\\) for all \\(x\\in A\\); such a \\(u\\) is a *right unit modulo \\(L\\)*. A two-sided ideal \\(I\\) is *modular* if \\(A/I\\) is unital, that is, if some \\(u\\) satisfies \\(x-xu\\in I\\) and \\(x-ux\\in I\\) for all \\(x\\). In a commutative algebra the two notions agree for ideals. Modular ideals are also called *regular*, and \\(u\\) is then called an *identity modulo* the ideal. Every left ideal of a unital algebra is modular, with \\(u=1\\).\n2. A proper left ideal is *maximal* if no proper left ideal strictly contains it. Maximal two-sided ideals are defined in the same way among two-sided ideals.\n\n**Lemma 9.2** (no norm is needed). Let \\(L\\) be a left ideal of an algebra \\(A\\), with a right unit \\(u\\) modulo \\(L\\).\n1. Every left ideal that contains \\(L\\) is modular, with the same \\(u\\).\n2. \\(L=A\\) if and only if \\(u\\in L\\).\n3. If \\(L\\) is proper, it lies in a maximal left ideal, and that ideal is modular.\n\nThe same holds for two-sided modular ideals, with maximality among two-sided ideals.\n\n**Proof.** (1) is clear. (2) If \\(u\\in L\\), then \\(x=(x-xu)+xu\\in L\\) for every \\(x\\), because \\(xu\\in L\\). (3) Let \\(\\mathcal F\\) be the set of left ideals \\(J\\supseteq L\\) with \\(u\\notin J\\). By (2), \\(L\\in\\mathcal F\\). The union of a chain in \\(\\mathcal F\\) is a left ideal that contains \\(L\\) and not \\(u\\). So Zorn's lemma gives a maximal element \\(M\\) of \\(\\mathcal F\\), and \\(M\\) is proper. If \\(J\\) is a left ideal with \\(M\\subsetneq J\\), then \\(J\\notin\\mathcal F\\), so \\(u\\in J\\), and \\(J=A\\) by (1) and (2). So \\(M\\) is a maximal left ideal, and it is modular by (1). For two-sided ideals, run the same argument with two-sided ideals and a two-sided unit modulo \\(I\\). \\(\\square\\)\n\n**Proposition 9.3** (modular ideals stay away from the unit). Let \\(A\\) be a Banach algebra, not necessarily commutative, and \\(L\\) a proper modular left ideal with right unit \\(u\\). Then \\(\\|u-\\ell\\|\\geq1\\) for every \\(\\ell\\in L\\). Consequently the closure of \\(L\\) is again a proper modular left ideal, and every maximal modular left ideal is closed. The same holds for two-sided ideals.\n\n**Proof.** Suppose \\(\\|u-\\ell\\|<1\\) for some \\(\\ell\\in L\\). Then \\(y=\\sum_{n\\geq1}(u-\\ell)^n\\) converges, and \\(y(u-\\ell)=y-(u-\\ell)\\). Rearranged, this says \\(u=y-yu+y\\ell+\\ell\\). Here \\(y-yu\\in L\\) because \\(u\\) is a right unit modulo \\(L\\), \\(y\\ell\\in L\\) because \\(L\\) is a left ideal, and \\(\\ell\\in L\\). So \\(u\\in L\\), and \\(L=A\\) by Lemma 9.2, a contradiction. The closure \\(\\bar L\\) is a left ideal because multiplication is continuous. The element \\(u\\) is a right unit modulo \\(\\bar L\\), and \\(\\operatorname{dist}(u,\\bar L)=\\operatorname{dist}(u,L)\\geq1\\); so \\(u\\notin\\bar L\\), and \\(\\bar L\\neq A\\). If \\(L\\) is maximal, then \\(L=\\bar L\\). A two-sided unit modulo an ideal is in particular a right unit, so the two-sided case follows. \\(\\square\\)\n\n**Proposition 9.4** (quotient algebras). Let \\(I\\) be a closed ideal of a Banach algebra \\(A\\). With the quotient norm \\(\\|x+I\\|=\\inf\\{\\|x+k\\|:k\\in I\\}\\), the algebra \\(A/I\\) is a Banach algebra. If \\(I\\) is proper and modular, with unit \\(u\\) modulo \\(I\\), then \\(u+I\\) is the identity of \\(A/I\\), and \\(\\|u+I\\|\\geq1\\).\n\n**Proof.** First this formula defines a norm. If \\(\\|x+I\\|=0\\), there are \\(k_n\\in I\\) with \\(x+k_n\\to0\\), so \\(-x\\in I\\) by closedness; the converse is immediate. For a nonzero scalar \\(\\lambda\\), representatives of \\(\\lambda x+I\\) are exactly \\(\\lambda(x+k)\\), \\(k\\in I\\), which proves homogeneity. The triangle inequality follows from \\(\\|(x+k)+(y+m)\\|\\leq\\|x+k\\|+\\|y+m\\|\\) by taking the two infima independently. Also the quotient map is contractive, since the infimum is at most \\(\\|x\\|\\). The quotient of a Banach space by a closed subspace is complete: if \\(\\sum_n\\|x_n+I\\|<\\infty\\), choose representatives with \\(\\|x_n\\|\\leq\\|x_n+I\\|+2^{-n}\\); then \\(\\sum_nx_n\\) converges in \\(A\\), and its class is the sum of the series in \\(A/I\\); and this property implies completeness. Indeed, from a Cauchy sequence \\(y_n\\) in any such normed space choose a subsequence \\(y_{n_k}\\) with \\(\\|y_{n_{k+1}}-y_{n_k}\\|\\leq2^{-k}\\). The series of these differences converges, so the subsequence converges by telescoping; the original Cauchy sequence has the same limit. For \\(x,y\\in A\\) and \\(k,m\\in I\\), \\((x+k)(y+m)\\in xy+I\\), so \\(\\|xy+I\\|\\leq\\|x+k\\|\\|y+m\\|\\); take the infimum over \\(k\\) and \\(m\\). The last claim is Proposition 9.3. \\(\\square\\)\n\n**Example 9.5** (completeness is needed). In the normed algebra \\(\\mathbb C[z]\\) of [Examples 4.5](#oa-fnd-bn-07) (norm \\(\\max_{|z|\\leq1}|p(z)|\\)), the ideal \\(I=(z-2)\\mathbb C[z]\\) is maximal, since \\(\\mathbb C[z]/I\\cong\\mathbb C\\) through \\(p\\mapsto p(2)\\), and it is modular because \\(\\mathbb C[z]\\) is unital. But it is dense. With \\(p_N=-\\frac12\\sum_{n=0}^N(z/2)^n\\) we get \\((z-2)p_N-1=-(z/2)^{N+1}\\), of norm \\(2^{-N-1}\\). So the distance from \\(1\\) to \\(I\\) is \\(0\\), and this maximal ideal is not closed. The character \\(p\\mapsto p(2)\\) is unbounded, since the polynomials \\(z^n\\) have norm \\(1\\) and value \\(2^n\\). Compare [Proposition 10.3(1)](#oa-fnd-bn-17).\n\n## 10. Characters\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BN-16",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Maximal ideals and characters of commutative algebras",
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      "full_conditions_and_proof": "### Maximal ideals and characters of commutative algebras\n\n**Definition 10.1.** A *character* of an algebra \\(A\\) is a nonzero algebra homomorphism \\(A\\to\\mathbb C\\), and \\(\\operatorname{Ch}(A)\\) is the set of characters. A nonzero homomorphism into \\(\\mathbb C\\) is automatically onto, because its image is a nonzero subspace of \\(\\mathbb C\\).\n\n**Proposition 10.2.**\n1. In a unital commutative algebra, every noninvertible element lies in a maximal ideal. No norm is needed.\n2. Let \\(A\\) be a commutative Banach algebra and \\(\\mathfrak m\\) a maximal modular ideal, with unit \\(u\\) modulo \\(\\mathfrak m\\). Then \\(\\mathfrak m\\) is closed, and \\(A/\\mathfrak m\\) is a field. The map \\(\\lambda\\mapsto\\lambda(u+\\mathfrak m)\\) is the only unital algebra isomorphism of \\(\\mathbb C\\) onto \\(A/\\mathfrak m\\). Let \\(\\omega_{\\mathfrak m}(x)\\) be the number \\(\\lambda\\) with \\(x+\\mathfrak m=\\lambda(u+\\mathfrak m)\\). Then \\(\\omega_{\\mathfrak m}\\) is a character with kernel \\(\\mathfrak m\\).\n3. For a commutative Banach algebra \\(A\\), \\(\\mathfrak m\\mapsto\\omega_{\\mathfrak m}\\) is a bijection from the set \\(\\mathcal M(A)\\) of maximal modular ideals onto \\(\\operatorname{Ch}(A)\\). Its inverse is \\(\\omega\\mapsto\\ker\\omega\\).\n4. In any algebra, two characters with the same kernel are equal, and the kernel of a character is a maximal modular ideal of codimension one.\n\n**Proof.** (1) If \\(x\\) is not invertible, then \\(Ax\\) is an ideal (as \\(A\\) is commutative) that does not contain \\(1\\); indeed \\(1=ax\\) would make \\(x\\) invertible. By [Lemma 9.2](#oa-fnd-bn-15), with \\(u=1\\), it lies in a maximal ideal, which contains \\(x=1x\\).\n(2) \\(\\mathfrak m\\) is closed by Proposition 9.3. So \\(A/\\mathfrak m\\) is a commutative Banach algebra (Proposition 9.4) with identity \\(u+\\mathfrak m\\neq0\\). Its ideals correspond to the ideals of \\(A\\) that contain \\(\\mathfrak m\\), so its only ideals are \\(\\{0\\}\\) and itself. By (1), every nonzero element of \\(A/\\mathfrak m\\) is invertible; otherwise it would lie in a maximal ideal of \\(A/\\mathfrak m\\), which can only be \\(\\{0\\}\\). So \\(A/\\mathfrak m\\) is a field, and by the Gelfand–Mazur theorem ([Corollary 5.3](#oa-fnd-bn-08)) every element is a multiple of \\(u+\\mathfrak m\\). A unital algebra homomorphism \\(\\psi:\\mathbb C\\to A/\\mathfrak m\\) satisfies \\(\\psi(\\lambda)=\\lambda\\psi(1)\\), so it equals \\(\\lambda\\mapsto\\lambda(u+\\mathfrak m)\\). Hence \\(\\omega_{\\mathfrak m}\\) is the quotient map followed by the inverse of this isomorphism. It is a homomorphism, it is nonzero because \\(\\omega_{\\mathfrak m}(u)=1\\), and its kernel is \\(\\mathfrak m\\).\n(4) Let \\(\\omega\\) be a character, and choose \\(u\\) with \\(\\omega(u)=1\\). Then \\(x-xu\\) and \\(x-ux\\) lie in \\(\\ker\\omega\\) for every \\(x\\), so \\(\\ker\\omega\\) is a modular ideal. It has codimension one, so it is maximal. Let \\(\\omega'\\) be a character with the same kernel. From \\(x-\\omega(x)u\\in\\ker\\omega'\\) we get \\(\\omega'(x)=\\omega(x)\\omega'(u)\\), and from \\(u-u^2\\in\\ker\\omega'\\) we get \\(\\omega'(u)=\\omega'(u)^2\\). Since \\(\\omega'\\neq0\\), \\(\\omega'(u)\\neq0\\); so \\(\\omega'(u)=1\\) and \\(\\omega'=\\omega\\).\n(3) By (2), \\(\\ker\\omega_{\\mathfrak m}=\\mathfrak m\\). By (4), for \\(\\omega\\in\\operatorname{Ch}(A)\\), \\(\\ker\\omega\\in\\mathcal M(A)\\), and \\(\\omega_{\\ker\\omega}=\\omega\\) because both have the same kernel. \\(\\square\\)\n\n*Automatic continuity.* For a commutative Banach algebra, the continuity of characters can be read off from (2): the kernel of a character is a maximal modular ideal, hence closed, and a linear functional with closed kernel is continuous. Proposition 10.3(1) below gives a direct proof with the bound \\(\\|\\omega\\|\\leq1\\), for noncommutative algebras as well. Example 9.5 shows that completeness cannot be dropped.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      ]
    },
    {
      "id": "OA-FND-BN-17",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Characters are contractive; the character space",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
      "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "anchor": "oa-fnd-bn-17",
      "proof_locus": {
        "line": 719,
        "through_line": 738
      },
      "full_conditions_and_proof": "### Characters are contractive; the character space\n\n**Proposition 10.3.** Let \\(A\\) be a Banach algebra, not necessarily commutative.\n1. For every character \\(\\omega\\) and every \\(x\\in A\\), \\(\\omega(x)\\in\\sigma'_A(x)\\), and \\(|\\omega(x)|\\leq r(x)\\leq\\|x\\|\\). So every character is continuous, with \\(\\|\\omega\\|\\leq1\\). If \\(A\\) is unital, \\(\\omega(1)=1\\).\n2. With the weak\\* topology, \\(\\operatorname{Ch}(A)\\cup\\{0\\}\\) is compact, and \\(\\operatorname{Ch}(A)\\) is locally compact Hausdorff. If \\(A\\) is unital, \\(\\operatorname{Ch}(A)\\) is compact.\n3. For \\(x\\in A\\), the function \\(\\hat x(\\omega)=\\omega(x)\\) lies in \\(C_0(\\operatorname{Ch}(A))\\).\n4. The characters of \\(A_1\\) are \\(q\\) and the maps \\(\\omega_1(a+\\lambda)=\\omega(a)+\\lambda\\) with \\(\\omega\\in\\operatorname{Ch}(A)\\). The map \\(\\omega\\mapsto\\omega_1\\) is a homeomorphism of \\(\\operatorname{Ch}(A)\\) onto \\(\\operatorname{Ch}(A_1)\\setminus\\{q\\}\\), which is open in the compact space \\(\\operatorname{Ch}(A_1)\\).\n\n**Proof.** (1) The map \\(\\omega_1\\) of (4) is a unital homomorphism \\(A_1\\to\\mathbb C\\) (a direct check with (3.1)), and \\(\\omega_1(\\omega(x)-j(x))=0\\). If \\(\\omega(x)-j(x)\\) had an inverse \\(w\\) in \\(A_1\\), applying \\(\\omega_1\\) would give \\(1=0\\). So \\(\\omega(x)\\in\\sigma'_A(x)\\), and \\(|\\omega(x)|\\leq r(x)\\leq\\|x\\|\\) because the quasi-spectrum lies in the disc of radius \\(\\|x\\|\\) ([Proposition 4.4(3)](#oa-fnd-bn-07)). If \\(A\\) is unital, then \\(\\omega(1)^2=\\omega(1)\\), and \\(\\omega(1)=0\\) would give \\(\\omega(x)=\\omega(x)\\omega(1)=0\\) for all \\(x\\).\n(2) By (1), \\(\\operatorname{Ch}(A)\\cup\\{0\\}\\) lies in the closed unit ball of \\(A^*\\), which is weak\\* compact by the Banach–Alaoglu theorem. It is weak\\* closed: if a net of characters, or zeros, converges pointwise to \\(\\omega_0\\), then \\(\\omega_0\\) is linear, and \\(\\omega_0(xy)=\\lim\\omega_i(x)\\omega_i(y)=\\omega_0(x)\\omega_0(y)\\). So \\(\\operatorname{Ch}(A)\\cup\\{0\\}\\) is compact and Hausdorff. The point \\(0\\) is closed, so \\(\\operatorname{Ch}(A)\\) is open in it, hence locally compact Hausdorff. If \\(A\\) is unital, \\(\\operatorname{Ch}(A)=\\{\\omega\\in\\operatorname{Ch}(A)\\cup\\{0\\}:\\omega(1)=1\\}\\) is closed, hence compact.\n(3) \\(\\hat x\\) is weak\\* continuous by definition. For \\(\\varepsilon>0\\), the set \\(\\{\\omega:|\\omega(x)|\\geq\\varepsilon\\}\\) equals \\(\\{\\varphi\\in\\operatorname{Ch}(A)\\cup\\{0\\}:|\\varphi(x)|\\geq\\varepsilon\\}\\), since it misses \\(0\\). It is closed in a compact space, hence compact.\n(4) Let \\(\\chi\\) be a character of \\(A_1\\); then \\(\\chi(1)=1\\) by (1). Its restriction to \\(j(A)\\) is a homomorphism. If the restriction is \\(0\\), then \\(\\chi=q\\). Otherwise it is a character \\(\\omega\\) of \\(A\\), and \\(\\chi=\\omega_1\\). Each \\(\\omega_1\\) is a character. The map \\(\\omega\\mapsto\\omega_1\\) is a bijection onto \\(\\operatorname{Ch}(A_1)\\setminus\\{q\\}\\), continuous in both directions, because \\(\\omega_1(a+\\lambda)=\\omega(a)+\\lambda\\) and \\(\\omega=\\omega_1\\circ j\\). The space \\(\\operatorname{Ch}(A_1)\\) is compact by (2), and \\(\\operatorname{Ch}(A_1)\\setminus\\{q\\}\\) is open in it. \\(\\square\\)\n\nIn (4), when \\(\\operatorname{Ch}(A)\\) is not compact, \\(\\operatorname{Ch}(A_1)\\) is its one-point compactification, with \\(q\\) as the point at infinity. Below we use only the description of the points.\n\n**Examples 10.4.**\n- For \\(n\\geq2\\), \\(M_n(\\mathbb C)\\) has no characters. A character vanishes on \\(E_{ij}\\) for \\(i\\neq j\\), since \\(E_{ij}^2=0\\). Then \\(\\omega(E_{ii})=\\omega(E_{ij}E_{ji})=0\\) for every \\(i\\), and \\(\\omega(1)=\\sum_i\\omega(E_{ii})=0\\), which contradicts (1).\n- A nonzero Banach space with the zero product is a commutative Banach algebra without characters, since \\(\\omega(x)^2=\\omega(x^2)=0\\).\n- For normed algebras that are not complete, (1) fails: see Example 9.5.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "haar-measure-on-locally-compact-groups",
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
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        "the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-BN-18",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "11. The Gelfand representation",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
      "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "anchor": "oa-fnd-bn-18",
      "proof_locus": {
        "line": 739,
        "through_line": 757
      },
      "full_conditions_and_proof": "## 11. The Gelfand representation\n\n**Theorem 11.1** (Gelfand representation). Let \\(A\\) be a commutative Banach algebra, and define \\(\\mathcal G:A\\to C_0(\\operatorname{Ch}(A))\\) by \\(\\mathcal G(x)=\\hat x\\).\n1. \\(\\mathcal G\\) is an algebra homomorphism, and \\(\\|\\hat x\\|_\\infty=r(x)\\leq\\|x\\|\\).\n2. If \\(A\\) is unital, then \\(\\operatorname{Ch}(A)\\) is compact and \\(\\sigma_A(x)=\\hat x(\\operatorname{Ch}(A))\\).\n3. In general, \\(\\sigma'_A(x)=\\hat x(\\operatorname{Ch}(A))\\cup\\{0\\}\\).\n\n\n**Proof.** (2) Compactness is Proposition 10.3(2). If \\(\\lambda\\in\\sigma_A(x)\\), then \\(\\lambda-x\\) lies in a maximal ideal \\(\\mathfrak m\\) by Proposition 10.2(1), which is modular because \\(A\\) is unital, and \\(\\omega_{\\mathfrak m}(\\lambda-x)=0\\); that is, \\(\\lambda=\\hat x(\\omega_{\\mathfrak m})\\). Conversely, let \\(\\omega\\in\\operatorname{Ch}(A)\\). Then \\(\\omega(1)=1\\) (Proposition 10.3(1)) and \\(\\omega(x-\\omega(x))=0\\). If \\(x-\\omega(x)\\) had an inverse \\(w\\), then \\(1=\\omega\\big((x-\\omega(x))w\\big)=0\\). So \\(\\hat x(\\omega)=\\omega(x)\\in\\sigma_A(x)\\).\n(3) \\(A_1\\) is a unital commutative Banach algebra, so by (2) \\(\\sigma'_A(x)=\\sigma_{A_1}(j(x))=\\{\\chi(j(x)):\\chi\\in\\operatorname{Ch}(A_1)\\}\\). By Proposition 10.3(4) this set is \\[\n\\begin{gathered}\n\\{\\omega(x):\\omega\\in\\operatorname{Ch}(A)\\}\\cup\\{q(j(x))\\}\\\\\n=\\hat x(\\operatorname{Ch}(A))\\cup\\{0\\}.\n\\end{gathered}\n\\]\n(1) The operations on \\(C_0(\\operatorname{Ch}(A))\\) are pointwise, and \\(\\widehat{xy}(\\omega)=\\omega(xy)=\\hat x(\\omega)\\hat y(\\omega)\\). By (3), \\(r(x)=\\max\\big(\\sup|\\hat x|,0\\big)=\\|\\hat x\\|_\\infty\\); this includes the case \\(\\operatorname{Ch}(A)=\\varnothing\\), where \\(C_0(\\varnothing)=\\{0\\}\\). \\(\\square\\)\n\nFor the zero algebra, which is unital, \\(\\operatorname{Ch}(A)=\\varnothing=\\sigma_A(0)\\), in agreement with (2).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "haar-measure-on-locally-compact-groups",
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
        "hilbert-spaces-and-compact-operators",
        "cauchy-s-theorem-for-cycles-and-its-consequences",
        "the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity",
        "measure-and-hilbert-space-tools"
      ]
    },
    {
      "id": "OA-FND-BN-19",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "The radical and semisimple algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
      "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "anchor": "oa-fnd-bn-19",
      "proof_locus": {
        "line": 758,
        "through_line": 782
      },
      "full_conditions_and_proof": "### The radical and semisimple algebras\n\n**Definition 11.2.** Let \\(A\\) be a commutative Banach algebra. The map \\(\\mathcal G\\) of Theorem 11.1 is the *Gelfand representation*; \\(\\operatorname{Ch}(A)\\) is the *spectrum* (or character space) of \\(A\\), and its members are the *characters*. The kernel of \\(\\mathcal G\\) is the *radical* \\(\\operatorname{rad}(A)\\). The algebra is *semisimple* if \\(\\operatorname{rad}(A)=\\{0\\}\\).\n\n**Proposition 11.3.** Let \\(A\\) be a commutative Banach algebra.\n1. \\(\\operatorname{rad}(A)=\\{x:r(x)=0\\}=\\bigcap_{\\mathfrak m\\in\\mathcal M(A)}\\mathfrak m\\), which is \\(A\\) when \\(\\mathcal M(A)\\) is empty. It is a closed ideal.\n2. If \\(A\\) is semisimple, then \\(\\mathcal G\\) is an injective homomorphism of \\(A\\) onto a subalgebra of \\(C_0(\\operatorname{Ch}(A))\\), with \\(\\|\\hat x\\|_\\infty\\leq\\|x\\|\\). This subalgebra separates the points of \\(\\operatorname{Ch}(A)\\) and vanishes at no point.\n\n**Proof.** (1) \\(\\|\\hat x\\|_\\infty=r(x)\\) by Theorem 11.1. Also \\(\\hat x=0\\) means \\(\\omega(x)=0\\) for every character, that is, \\(x\\in\\ker\\omega_{\\mathfrak m}=\\mathfrak m\\) for every \\(\\mathfrak m\\in\\mathcal M(A)\\) ([Proposition 10.2(3)](#oa-fnd-bn-16)). The radical is the kernel of the continuous homomorphism \\(\\mathcal G\\), so it is a closed ideal.\n(2) Injectivity is the definition. Distinct characters differ at some \\(x\\), and a character \\(\\omega\\neq0\\) has some \\(\\hat x(\\omega)\\neq0\\). \\(\\square\\)\n\nSo every commutative semisimple Banach algebra is isomorphic, as an algebra, to an algebra of continuous functions that vanish at infinity on a locally compact Hausdorff space. The isomorphism is contractive, but in general it is not isometric ([Proposition 13.2(5)](#oa-fnd-bn-21)).\n\n**Example 11.4** (the characters of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be LCH. Every character \\(\\omega\\) of \\(C_0(\\Omega)\\) is an evaluation \\(f\\mapsto f(p)\\), and \\(p\\mapsto\\text{(evaluation at }p)\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\). So \\(\\hat f\\) is \\(f\\) itself, and \\(\\mathcal G\\) is isometric.\n\n*Proof.* First we find a point \\(p\\) at which every \\(f\\in\\ker\\omega\\) vanishes. Suppose, to the contrary, that for every \\(p\\in\\Omega\\) some \\(f_p\\in\\ker\\omega\\) has \\(f_p(p)\\neq0\\). Choose \\(g\\) with \\(\\omega(g)=1\\). The set \\(C=\\{|g|\\geq1/2\\}\\) is compact, and it is not empty, since \\(\\|g\\|_\\infty\\geq|\\omega(g)|=1\\) by [Proposition 10.3(1)](#oa-fnd-bn-17). So finitely many sets \\(\\{f_{p_i}\\neq0\\}\\) cover it, and \\(h=\\sum_i\\bar f_{p_i}f_{p_i}\\) lies in the ideal \\(\\ker\\omega\\) and is positive on \\(C\\). Let \\(m=\\min_Ch>0\\) and \\(s=1/\\max(h,m)\\), a bounded continuous function. Then \\(gs\\in C_0(\\Omega)\\), so \\(ghs=(gs)h\\in\\ker\\omega\\). On \\(C\\), \\(hs=1\\), so \\(g-ghs=0\\); off \\(C\\), \\(0\\leq hs\\leq1\\) gives \\(|g-ghs|\\leq|g|<1/2\\). So \\(\\|g-ghs\\|_\\infty\\leq1/2\\), while \\(\\omega(g-ghs)=1\\). This contradicts the bound \\(|\\omega(x)|\\leq\\|x\\|\\) of Proposition 10.3(1). Hence some \\(p\\) has \\(f(p)=0\\) for all \\(f\\in\\ker\\omega\\).\n\nNext, \\(\\omega\\) is evaluation at this \\(p\\). Evaluation at \\(p\\) is nonzero (Urysohn), and its kernel contains \\(\\ker\\omega\\); both kernels have codimension one, so they are equal, and [Proposition 10.2(4)](#oa-fnd-bn-16) gives \\(\\omega=\\) evaluation at \\(p\\).\n\nFinally, the map is a homeomorphism. Distinct points give distinct evaluations (Urysohn). The map is weak\\* continuous, since \\(p\\mapsto f(p)\\) is continuous for each \\(f\\). Its inverse is continuous too: if evaluations at \\(p_i\\) converge to evaluation at \\(p\\) but \\(p_i\\) stays outside a neighbourhood \\(U\\) of \\(p\\) along a subnet, a Urysohn function \\(f\\) with \\(f(p)=1\\) and support in \\(U\\) gives \\(0=f(p_i)\\to1\\), a contradiction.\n\n**Examples 11.5.**\n- *Radicals.* A Banach space with the zero product is its own radical. The dual numbers \\(\\mathbb C[\\varepsilon]\\), with \\(\\varepsilon^2=0\\) and norm \\(|a|+|b|\\) for \\(a+b\\varepsilon\\), are unital with radical \\(\\mathbb C\\varepsilon\\).\n- *The Wiener algebra* ([Section 13](#oa-fnd-bn-21)) is semisimple, and its Gelfand representation is injective but not isometric.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "haar-measure-on-locally-compact-groups",
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        "the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity",
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    },
    {
      "id": "OA-FND-BN-20",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "12. Group algebras and transformation-group algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
      "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "anchor": "oa-fnd-bn-20",
      "proof_locus": {
        "line": 783,
        "through_line": 956
      },
      "full_conditions_and_proof": "## 12. Group algebras and transformation-group algebras\n\nThis section builds the convolution algebras of a locally compact group and of its actions on a locally compact space. It decides when they have an identity and when they are commutative. It uses Haar measure, from the lesson [Haar measure on locally compact groups](haar-measure.md).\n\n*Setting.* \\(G\\) is a locally compact Hausdorff group with identity \\(e\\) and a left Haar measure \\(\\mu\\), written \\(ds\\) ([existence of Haar measure](haar-measure.md#oa-fnd-hm-05)). Compact sets have finite measure, and \\(\\mu\\) is outer regular: the measure of a Borel set is the infimum of the measures of the open sets that contain it ([Radon measures](haar-measure.md#oa-fnd-hm-01)). Nonempty open sets have positive measure ([positivity of Haar measure](haar-measure.md#oa-fnd-hm-06)). The modular function \\(\\Delta:G\\to(0,\\infty)\\) is the continuous homomorphism with \\(\\int f(ts)\\,dt=\\Delta(s)^{-1}\\int f(t)\\,dt\\) ([the modular function](haar-measure.md#oa-fnd-hm-07)). The inversion formula \\(\\int f(t^{-1})\\Delta(t)^{-1}\\,dt=\\int f(t)\\,dt\\) ([inversion](haar-measure.md#oa-fnd-hm-08)), applied to \\(t\\mapsto k(st)\\) after left invariance, gives for every \\(s\\in G\\)\n\\[\n\\begin{gathered}\n\\int k(t)\\,dt\\\\\n=\\int k(sr^{-1})\\,\\Delta(r)^{-1}\\,dr .\n\\end{gathered}\n\\tag{12.1}\n\\]\n\\(\\Omega\\) is an LCH space with a continuous right action \\((\\omega,s)\\mapsto\\omega s\\) such that \\(\\omega e=\\omega\\) and \\(\\omega(st)=(\\omega s)t\\). Each map \\(\\omega\\mapsto\\omega s\\) is a homeomorphism, with inverse \\(\\omega\\mapsto\\omega s^{-1}\\). On \\(K=C_c(\\Omega\\times G)\\) put\n\\[\n\\begin{gathered}\n(x\\star y)(\\omega,s)\\\\\n=\\int_Gx(\\omega,t)\\,y(\\omega t,t^{-1}s)\\,dt,\\\\\nx^\\sharp(\\omega,s)\\\\\n=\\Delta(s)^{-1}\\,\\overline{x(\\omega s,s^{-1})},\\\\\n\\|x\\|_1\\\\\n=\\int_GX(s)\\,ds,\n\\end{gathered}\n\\tag{12.2}\n\\]\nwhere \\(X(s)=\\sup_{\\omega\\in\\Omega}|x(\\omega,s)|\\). When \\(\\Omega\\) is a single point, \\(K=C_c(G)\\) with convolution and the \\(L^1\\) norm.\n\nFor \\(x\\in K\\), let \\(K_x\\) and \\(L_x\\) be the projections of \\(\\operatorname{supp}x\\) to \\(\\Omega\\) and to \\(G\\); both are compact. By the lemma on continuous functions of two variables in the section on [iterated integrals on products](haar-measure.md#oa-fnd-hm-03), with the roles of the factors exchanged, \\(s\\mapsto x(\\cdot,s)\\) is continuous from \\(G\\) into \\(C_0(\\Omega)\\) with the supremum norm. So \\(X\\) is continuous and vanishes off \\(L_x\\), and \\(\\|x\\|_1<\\infty\\). If \\(\\|x\\|_1=0\\), then \\(X=0\\), because a nonzero continuous function \\(X\\geq0\\) with compact support has positive integral ([positivity of Haar measure](haar-measure.md#oa-fnd-hm-06)). So \\(\\|\\cdot\\|_1\\) is a norm.\n\n*Uniform continuity.* For \\(x\\in K\\), \\(\\sup_{\\omega,s}|x(\\omega,sr)-x(\\omega,s)|\\to0\\) as \\(r\\to e\\). This is the usual proof of the uniform continuity of functions in \\(C_c(G)\\) ([topological groups](haar-measure.md#oa-fnd-hm-04)), with the point of \\(\\Omega\\) carried along. Let \\(\\eta>0\\). For each \\((p,s)\\in\\operatorname{supp}x\\), continuity of \\((p',v)\\mapsto x(p',sv)\\) at \\((p,e)\\) gives an open \\(P\\ni p\\) and a symmetric open neighbourhood \\(W\\) of \\(e\\) with \\(|x(p',sv)-x(p,s)|<\\eta/2\\) for \\(p'\\in P\\) and \\(v\\in WW\\). Finitely many of the sets \\(P\\times sW\\) cover \\(\\operatorname{supp}x\\); call them \\(P_j\\times s_jW_j\\), and let \\(W\\) be the intersection of the \\(W_j\\). Let \\(r\\in W\\). If \\((p,s)\\in\\operatorname{supp}x\\), then \\((p,s)\\in P_j\\times s_jW_j\\) for some \\(j\\); writing \\(s=s_jv\\), both \\(x(p,s)\\) and \\(x(p,sr)=x(p,s_j(vr))\\) are within \\(\\eta/2\\) of \\(x(p_j,s_j)\\), because \\(v\\) and \\(vr\\) lie in \\(W_jW_j\\). If \\((p,sr)\\in\\operatorname{supp}x\\), the same argument applies to \\(sr\\) and \\(r^{-1}\\in W\\). Otherwise both values are \\(0\\). So \\(|x(p,sr)-x(p,s)|<\\eta\\) whenever \\(r\\in W\\).\n\n**Proposition 12.1** (the transformation-group algebra). (12.2) makes \\(K\\) a \\(*\\)-algebra with a submultiplicative norm and an isometric involution. For \\(x,y,z\\in K\\):\n\\[\n\\begin{gathered}\nx\\star y\\in K,\\\\\n(x\\star y)\\star z\\\\\n=x\\star(y\\star z),\\\\\nx^\\sharp\\in K,\\\\\nx^{\\sharp\\sharp}\\\\\n=x,\\\\\n(x\\star y)^\\sharp\\\\\n=y^\\sharp\\star x^\\sharp,\\\\\n\\|x\\star y\\|_1\\\\\n\\leq\\|x\\|_1\\|y\\|_1,\\\\\n\\|x^\\sharp\\|_1\\\\\n=\\|x\\|_1,\n\\end{gathered}\n\\]\nand \\(\\star\\) is bilinear and \\(\\sharp\\) conjugate-linear. So the completion \\(\\mathfrak A(\\Omega,G)\\) of \\(K\\) is an involutive Banach algebra. The isometric identification \\(\\mathfrak A(\\Omega,G)\\cong L^1(G,C_0(\\Omega))\\) and density of \\(K\\) are proved just after the algebra laws.\n\n**Proof.** *\\(x\\star y\\in K\\).* For fixed \\((\\omega,s)\\), the integrand \\(t\\mapsto x(\\omega,t)y(\\omega t,t^{-1}s)\\) is continuous and vanishes off \\(L_x\\). If \\((x\\star y)(\\omega,s)\\neq0\\), some \\(t\\) has \\((\\omega,t)\\in\\operatorname{supp}x\\) and \\(t^{-1}s\\in L_y\\), so \\(\\omega\\in K_x\\) and \\(s\\in L_xL_y\\). So \\(x\\star y\\) vanishes outside the compact set \\(K_x\\times L_xL_y\\). For continuity, let \\(F(\\omega,s,t)\\) be the integrand, a continuous function on \\(\\Omega\\times G\\times G\\). Fix \\((\\omega_0,s_0)\\) and \\(\\eta>0\\). Each \\(t\\in L_x\\) has neighbourhoods \\(N_t\\) of \\((\\omega_0,s_0)\\) and \\(O_t\\) of \\(t\\) with \\(|F(\\omega,s,t')-F(\\omega_0,s_0,t)|<\\eta/2\\) on \\(N_t\\times O_t\\). Finitely many \\(O_{t_i}\\) cover \\(L_x\\); let \\(N\\) be the intersection of the corresponding \\(N_{t_i}\\). For \\((\\omega,s)\\in N\\) and \\(t'\\in L_x\\), comparing both \\(F(\\omega,s,t')\\) and \\(F(\\omega_0,s_0,t')\\) with \\(F(\\omega_0,s_0,t_i)\\) gives \\(|F(\\omega,s,t')-F(\\omega_0,s_0,t')|<\\eta\\). So \\(|(x\\star y)(\\omega,s)-(x\\star y)(\\omega_0,s_0)|\\leq\\eta\\,\\mu(L_x)\\).\n\n*The norm.* \\(|(x\\star y)(\\omega,s)|\\leq\\int X(t)Y(t^{-1}s)\\,dt\\). The function \\((t,s)\\mapsto X(t)Y(t^{-1}s)\\) lies in \\(C_c(G\\times G)\\), so its two iterated integrals agree ([iterated integrals on products](haar-measure.md#oa-fnd-hm-03)), and left invariance gives \\[\n\\begin{gathered}\n\\|x\\star y\\|_1\\\\\n\\leq\\int X(t)\\big(\\int Y(t^{-1}s)\\,ds\\big)dt\\\\\n=\\|x\\|_1\\|y\\|_1.\n\\end{gathered}\n\\]\n\n*Associativity.* For fixed \\((\\omega,s)\\),\n\\[\n\\begin{gathered}\n((x\\star y)\\star z)(\\omega,s)\\\\\n=\\iint x(\\omega,r)\\,y(\\omega r,r^{-1}t)\\,z(\\omega t,t^{-1}s)\\,dr\\,dt,\\\\\n(x\\star(y\\star z))(\\omega,s)\\\\\n=\\int x(\\omega,r)\\int y(\\omega r,v)\\,z(\\omega rv,v^{-1}r^{-1}s)\\,dv\\,dr .\n\\end{gathered}\n\\]\nIn the inner integral on the right, substitute \\(v=r^{-1}t\\) (left invariance); it becomes \\(\\int y(\\omega r,r^{-1}t)z(\\omega t,t^{-1}s)\\,dt\\). The integrand \\((r,t)\\mapsto x(\\omega,r)y(\\omega r,r^{-1}t)z(\\omega t,t^{-1}s)\\) is continuous and vanishes off \\(L_x\\times L_xL_y\\), so, as in the norm estimate, the order of integration may be exchanged.\n\n*The involution.* \\(x^\\sharp\\) is continuous because \\(\\Delta\\) is. If \\(x^\\sharp(\\omega,s)\\neq0\\), then \\((\\omega s,s^{-1})\\in\\operatorname{supp}x\\), so \\(s\\in L_x^{-1}\\) and \\(\\omega=(\\omega s)s^{-1}\\) lies in the compact image of \\(K_x\\times L_x\\) under the action. So \\(x^\\sharp\\in K\\). Next,\n\\[\n\\begin{gathered}\nx^{\\sharp\\sharp}(\\omega,s)\\\\\n=\\Delta(s)^{-1}\\overline{x^\\sharp(\\omega s,s^{-1})}\\\\\n=\\Delta(s)^{-1}\\Delta(s^{-1})^{-1}x(\\omega,s)\\\\\n=x(\\omega,s).\n\\end{gathered}\n\\] Since \\(\\omega\\mapsto\\omega s\\) is a bijection of \\(\\Omega\\), \\(\\sup_\\omega|x^\\sharp(\\omega,s)|=\\Delta(s)^{-1}X(s^{-1})\\), and the inversion formula gives \\(\\|x^\\sharp\\|_1=\\int\\Delta(s)^{-1}X(s^{-1})\\,ds=\\|x\\|_1\\). Finally, using \\(\\Delta(t)^{-1}\\Delta(t^{-1}s)^{-1}=\\Delta(s)^{-1}\\),\n\\[\n\\begin{gathered}\n(y^\\sharp\\star x^\\sharp)(\\omega,s)\\\\\n=\\Delta(s)^{-1}\\,\\overline{\\int x(\\omega s,s^{-1}t)\\,y(\\omega t,t^{-1})\\,dt},\\\\\n(x\\star y)^\\sharp(\\omega,s)\\\\\n=\\Delta(s)^{-1}\\,\\overline{\\int x(\\omega s,r)\\,y(\\omega sr,r^{-1}s^{-1})\\,dr},\n\\end{gathered}\n\\]\nand the substitution \\(r=s^{-1}t\\) turns the second integral into the first.\n\n*Completion.* The product and the involution are bounded, so they extend to the completion, and all the identities persist by continuity. \\(\\square\\)\n\n*The Bochner-space identification.* Here \\(L^1(G,E)\\), for a Banach space \\(E\\), can be defined as the completion, in the norm \\(\\int\\|f(s)\\|\\,ds\\), of finite-valued simple functions supported on Borel sets of finite Haar measure, after identifying functions of norm zero. This definition makes no separability or sigma-compactness assumption on \\(G\\) or \\(E\\).\n\nFor \\(E=C_0(\\Omega)\\), the map \\(x\\mapsto[s\\mapsto x(\\cdot,s)]\\) from \\(K\\) is isometric and takes values in this completion. Indeed, the coefficient map is norm continuous and vanishes off the compact \\(L_x\\), as proved before Proposition 12.1. Its image is compact. Choose finitely many norm balls of radius \\(\\varepsilon\\) covering that image, and partition \\(L_x\\) into the Borel preimages of these balls, assigning each point to the first ball containing its value. Replacing the coefficient on each part by that ball’s centre gives a finite-valued simple function with error at most \\(\\varepsilon\\mu(L_x)\\) in \\(L^1\\). If \\(\\mu(L_x)=0\\), the coefficient map already has norm zero.\n\nThe image of \\(K\\) is dense. To approximate a simple function \\(\\sum_{j=1}^na_j1_{E_j}\\) with \\(a_j\\in C_0(\\Omega)\\) and \\(\\mu(E_j)<\\infty\\), use [Proposition 3.1(4) of the Haar lesson](haar-measure.md#oa-fnd-hm-02) to choose \\(h_j\\in C_c(G)\\) close to \\(1_{E_j}\\) in scalar \\(L^1\\). The error in replacing the simple function by \\(\\sum_jh_ja_j\\) is at most \\(\\sum_j\\|a_j\\|\\|1_{E_j}-h_j\\|_1\\), so it can be made arbitrarily small. Next choose \\(b_j\\in C_c(\\Omega)\\) uniformly close to \\(a_j\\), using [Proposition 2.1(3) of the Stone–Weierstrass lesson](stone-weierstrass-c0.md#oa-fnd-sw-01). The function \\(x(\\omega,s)=\\sum_jb_j(\\omega)h_j(s)\\) belongs to \\(K\\), and the further error is at most \\(\\sum_j\\|a_j-b_j\\|_\\infty\\|h_j\\|_1\\). This too can be made arbitrarily small. Since simple functions are dense by the defining completion, \\(K\\) is dense in \\(L^1(G,C_0(\\Omega))\\).\n\nAn isometry extends uniquely to an isometry of completions; its range is closed and dense, hence all of the target. Consequently \\(\\mathfrak A(\\Omega,G)\\cong L^1(G,C_0(\\Omega))\\) isometrically. The product and involution are the unique continuous extensions of (12.2), and the laws proved on \\(K\\) persist by continuity. The later lesson *Recovering covariance with nonunital coefficients* in *Crossed products and the flow of weights* develops representations of this algebra. Its representation results are not needed for this identification.\n\n\n**Remark 12.2** (the factor \\(\\Delta(s)^{-1}\\)). With the product (12.2), the involution is isometric only with the factor \\(\\Delta(s)^{-1}\\), for the modular function normalized by \\(\\int f(ts)\\,dt=\\Delta(s)^{-1}\\int f(t)\\,dt\\). With the factor \\(\\Delta(s)\\) instead, the inversion formula would give \\(\\int\\Delta(s)X(s^{-1})\\,ds=\\int\\Delta(s)^{-2}X(s)\\,ds\\), which differs from \\(\\|x\\|_1\\) on every non-unimodular group for suitable \\(x\\). The right action enters the product as \\(y(\\omega t,t^{-1}s)\\) and the involution as \\(x(\\omega s,s^{-1})\\); Proposition 12.1 checks that these choices are compatible.\n\n**Proposition 12.3** (the group algebra). \\(L^1(G)\\), with \\((xy)(t)=\\int x(s)y(s^{-1}t)\\,ds\\) and \\(x^*(t)=\\Delta(t)^{-1}\\overline{x(t^{-1})}\\), is an involutive Banach algebra. It is \\(\\mathfrak A(\\Omega,G)\\) for \\(\\Omega\\) a single point: \\(C_c(G)\\) is dense in \\(L^1(G)\\) ([density of compactly supported functions](haar-measure.md#oa-fnd-hm-02)), \\(L^1(G)\\) is complete, and the \\(L^1\\) convolution extends the product of \\(C_c(G)\\) continuously ([convolution](haar-measure.md#oa-fnd-hm-12)). The algebra laws on all of \\(L^1(G)\\) are also proved directly in the lesson on Haar measure, in the section on [convolution](haar-measure.md#oa-fnd-hm-12).\n\n**Theorem 12.4** (when the transformation-group algebra has an identity). Let \\(\\Omega\\) be nonempty. Then \\(\\mathfrak A(\\Omega,G)\\) has an identity exactly when \\(\\Omega\\) is compact and \\(G\\) is discrete. In that case the identity is \\(\\varepsilon=c^{-1}1_{\\Omega\\times\\{e\\}}\\), where \\(c=\\mu(\\{e\\})>0\\); moreover \\(\\|\\varepsilon\\|_1=1\\) and \\(\\varepsilon^\\sharp=\\varepsilon\\).\n\nThe hypothesis that \\(\\Omega\\) is nonempty is needed. If \\(\\Omega=\\varnothing\\), then \\(K=\\{0\\}\\) and \\(\\mathfrak A=\\{0\\}\\), and the equivalence fails whichever convention is used for the zero algebra. If \\(\\{0\\}\\) counts as unital, the left side holds for every \\(G\\); if it does not, the left side fails for every \\(G\\). The right side holds exactly when \\(G\\) is discrete.\n\n**Proof.** *Sufficiency.* In a discrete group, points are open and compact sets are finite. By left invariance \\(\\mu(\\{s\\})=\\mu(\\{e\\})=c\\) for all \\(s\\), and \\(c>0\\) because nonempty open sets have positive measure. Every subset of \\(G\\) is open, and the measure of an open set is the supremum of the measures of its compact subsets ([Radon measures](haar-measure.md#oa-fnd-hm-01)); this gives \\(\\mu(E)=c\\cdot\\#E\\), and \\(\\int f\\,d\\mu=c\\sum_sf(s)\\) for \\(f\\geq0\\) or integrable. The function \\(\\varepsilon\\) is continuous, since \\(\\Omega\\times\\{e\\}\\) is open and closed, and it has compact support because \\(\\Omega\\) is compact. For \\(x\\in K\\), \\((\\varepsilon\\star x)(\\omega,s)=c\\cdot c^{-1}x(\\omega e,s)=x(\\omega,s)\\), and \\((x\\star\\varepsilon)(\\omega,s)=c\\,x(\\omega,s)\\,c^{-1}\\), since only \\(t=s\\) contributes. By continuity \\(\\varepsilon\\) is an identity of \\(\\mathfrak A\\). Also \\(\\|\\varepsilon\\|_1=c\\cdot c^{-1}=1\\), and \\(\\varepsilon^\\sharp=\\varepsilon\\) because \\(\\Delta\\equiv1\\) on a discrete group ([discrete groups are unimodular](haar-measure.md#oa-fnd-hm-07)).\n\n*A right approximate identity.* Let \\(\\Omega\\) be nonempty, and fix \\(\\omega_0\\in\\Omega\\). Let \\(\\lambda=(V,L)\\) run over the pairs of an open neighbourhood \\(V\\) of \\(e\\) and a compact \\(L\\subseteq\\Omega\\) containing \\(\\omega_0\\), directed by \\((V,L)\\leq(V',L')\\) when \\(V'\\subseteq V\\) and \\(L'\\supseteq L\\). For each \\(V\\) choose \\(h_V\\in C_c(G)\\) with \\(h_V\\geq0\\), \\(\\operatorname{supp}h_V\\subseteq V\\) and \\(\\int h_V=1\\) ([approximate identities](haar-measure.md#oa-fnd-hm-13)), and for each \\(L\\) choose \\(\\varphi_L\\in C_c(\\Omega)\\) with \\(0\\leq\\varphi_L\\leq1\\) and \\(\\varphi_L=1\\) on \\(L\\) (Urysohn's lemma). Put \\(\\varepsilon_\\lambda(\\omega,s)=\\varphi_L(\\omega)h_V(s)\\). Then \\(\\varepsilon_\\lambda\\in K\\), and \\(\\|\\varepsilon_\\lambda\\|_1=\\int h_V=1\\) because \\(\\sup\\varphi_L=\\varphi_L(\\omega_0)=1\\). We claim that \\(a\\star\\varepsilon_\\lambda\\to a\\) for every \\(a\\in\\mathfrak A\\).\n\nFirst let \\(x\\in K\\). The image \\(C_x\\) of \\(\\operatorname{supp}x\\) under the action \\((\\omega,t)\\mapsto\\omega t\\) is compact. If \\(L\\supseteq C_x\\), then \\(\\varphi_L(\\omega t)=1\\) whenever \\(x(\\omega,t)\\neq0\\), so by (12.1)\n\\[\n\\begin{gathered}\n(x\\star\\varepsilon_\\lambda)(\\omega,s)\\\\\n=\\int x(\\omega,t)\\,h_V(t^{-1}s)\\,dt\\\\\n=\\int x(\\omega,sr^{-1})\\,h_V(r)\\,\\Delta(r)^{-1}\\,dr .\n\\end{gathered}\n\\]\nSince \\(\\int h_V=1\\),\n\\[\n\\begin{gathered}\n(x\\star\\varepsilon_\\lambda)(\\omega,s)-x(\\omega,s)\\\\\n=\\int\\big[x(\\omega,sr^{-1})\\Delta(r)^{-1}-x(\\omega,s)\\big]h_V(r)\\,dr .\n\\end{gathered}\n\\]\nFix a compact neighbourhood \\(N_0\\) of \\(e\\), and take \\(V\\subseteq N_0\\). For \\(r\\in V\\), the bracket vanishes unless \\(s\\in L_xN_0\\), and its absolute value is at most\n\\[\n\\begin{gathered}\n\\beta(V)\\\\\n=\\sup_{r\\in V}\\big(|\\Delta(r)^{-1}-1|\\,\\|x\\|_\\infty\\\\\n+\\sup_{\\omega,s}|x(\\omega,sr^{-1})-x(\\omega,s)|\\big).\n\\end{gathered}\n\\] So \\(\\|x\\star\\varepsilon_\\lambda-x\\|_1\\leq\\beta(V)\\,\\mu(L_xN_0)\\). By continuity of \\(\\Delta\\) and the uniform continuity above, \\(\\beta(V)\\to0\\) as \\(V\\) shrinks. For \\(a\\in\\mathfrak A\\) and \\(x\\in K\\), \\(\\|a\\star\\varepsilon_\\lambda-a\\|\\leq2\\|a-x\\|+\\|x\\star\\varepsilon_\\lambda-x\\|\\), and \\(K\\) is dense.\n\n*Necessity.* Suppose \\(\\mathfrak A\\) has an identity \\(1_{\\mathfrak A}\\). Then \\(\\varepsilon_\\lambda=1_{\\mathfrak A}\\star\\varepsilon_\\lambda\\to1_{\\mathfrak A}\\), so the net \\((\\varepsilon_\\lambda)\\) is Cauchy: there is \\(\\lambda_0=(V_0,L_0)\\) with \\(\\|\\varepsilon_\\lambda-\\varepsilon_{\\lambda'}\\|_1<1/2\\) for all \\(\\lambda,\\lambda'\\geq\\lambda_0\\).\n\n(i) *\\(G\\) is discrete.* Suppose not. Then \\(\\mu(\\{e\\})=0\\). Otherwise every point would have the same positive measure; a compact neighbourhood of \\(e\\) would be finite, because compact sets have finite measure; and \\(\\{e\\}\\), the interior of that neighbourhood minus finitely many other points, would be open. By outer regularity, \\(e\\) has open neighbourhoods of arbitrarily small measure. Let \\(m_0=\\max h_{V_0}>0\\), and choose an open neighbourhood \\(V'\\subseteq V_0\\) of \\(e\\) with \\(\\mu(V')<1/(2m_0)\\). Then \\((V',L_0)\\geq\\lambda_0\\). Both functions carry the factor \\(\\varphi_{L_0}\\), whose supremum is \\(1\\), so\n\\[\n\\begin{gathered}\n\\|\\varepsilon_{(V_0,L_0)}-\\varepsilon_{(V',L_0)}\\|_1\\\\\n=\\int|h_{V_0}-h_{V'}|\\\\\n\\geq\\int_{V'}(h_{V'}-h_{V_0})\\\\\n\\geq1-m_0\\,\\mu(V')>\\tfrac12,\n\\end{gathered}\n\\]\na contradiction.\n\n(ii) *\\(\\Omega\\) is compact.* Suppose not. The support of \\(\\varphi_{L_0}\\) is compact, so some \\(\\omega_1\\in\\Omega\\) lies outside it. Put \\(L'=L_0\\cup\\{\\omega_1\\}\\). Then \\((V_0,L')\\geq\\lambda_0\\), and\n\\[\n\\begin{gathered}\n\\|\\varepsilon_{(V_0,L_0)}-\\varepsilon_{(V_0,L')}\\|_1\\\\\n=\\int\\sup_\\omega|\\varphi_{L_0}(\\omega)-\\varphi_{L'}(\\omega)|\\,h_{V_0}(s)\\,ds\\\\\n=\\|\\varphi_{L_0}-\\varphi_{L'}\\|_\\infty\\\\\n\\geq|0-1|\\\\\n=1,\n\\end{gathered}\n\\]\na contradiction. \\(\\square\\)\n\n**Corollary 12.5** (the identity of \\(L^1(G)\\)). \\(L^1(G)\\) has an identity exactly when \\(G\\) is discrete. The identity is then \\(c^{-1}1_{\\{e\\}}\\) with \\(c=\\mu(\\{e\\})\\). This is Theorem 12.4 with \\(\\Omega\\) a point, together with Proposition 12.3. For counting measure (\\(c=1\\)), the lesson on Haar measure proves the same by a different argument, in the section on [approximate identities](haar-measure.md#oa-fnd-hm-13).\n\n**Theorem 12.6** (commutativity of \\(L^1(G)\\)). \\(L^1(G)\\) is commutative if and only if \\(G\\) is abelian.\n\n**Proof.** If \\(G\\) is abelian, then \\(\\Delta\\equiv1\\), since abelian groups are unimodular ([the modular function](haar-measure.md#oa-fnd-hm-07)). When \\(\\Delta\\equiv1\\), the convolution \\((xy)(t)\\) can also be written as \\(\\int x(ts^{-1})y(s)\\,ds\\) ([convolution](haar-measure.md#oa-fnd-hm-12)). This gives \\[\n\\begin{gathered}\n(xy)(t)\\\\\n=\\int x(ts^{-1})y(s)\\,ds\\\\\n=\\int y(s)x(s^{-1}t)\\,ds\\\\\n=(yx)(t),\n\\end{gathered}\n\\] since \\(ts^{-1}=s^{-1}t\\). Conversely, suppose \\(L^1(G)\\) is commutative, and let \\(s,t\\in G\\) with \\(st\\neq ts\\). Choose disjoint open sets \\(O_1\\ni st\\) and \\(O_2\\ni ts\\), and then, by continuity of multiplication, open sets \\(V\\ni s\\) and \\(W\\ni t\\) with \\(VW\\subseteq O_1\\) and \\(WV\\subseteq O_2\\). By Urysohn's lemma choose nonzero \\(f,g\\in C_c(G)\\) with \\(f,g\\geq0\\), \\(\\operatorname{supp}f\\subseteq V\\) and \\(\\operatorname{supp}g\\subseteq W\\). Then \\(fg\\) and \\(gf\\) (convolutions) are continuous ([convolution](haar-measure.md#oa-fnd-hm-12)); \\(fg\\) vanishes outside the compact set \\(\\operatorname{supp}f\\cdot\\operatorname{supp}g\\subseteq O_1\\), and \\(gf\\) vanishes outside \\(O_2\\). They agree almost everywhere, hence everywhere: a nonzero continuous function is nonzero on a nonempty open set, which has positive measure. So \\(fg=gf\\) vanishes outside \\(O_1\\) and outside \\(O_2\\), which are disjoint; hence \\(fg=0\\). But \\(\\int fg=\\int f\\int g>0\\): the integral of a convolution is the product of the integrals, and \\(\\int f\\) and \\(\\int g\\) are positive because Haar measure gives positive integrals to nonzero functions \\(f\\geq0\\) in \\(C_c(G)\\). This is a contradiction. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BN-21",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "13. The Wiener algebra and Wiener's lemma",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "anchor": "oa-fnd-bn-21",
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      "full_conditions_and_proof": "## 13. The Wiener algebra and Wiener's lemma\n\nThis section works out the Gelfand theory of the algebra of absolutely convergent Fourier series and deduces Wiener's lemma. By part (1) of Proposition 13.2 below, this algebra is the group algebra of \\(\\mathbb Z\\) with counting measure, that is, \\(\\ell^1(\\mathbb Z)\\) with convolution.\n\n*Setting.* A continuous function on \\([0,1]\\) with \\(f(0)=f(1)\\) is the same as a continuous function on the circle \\(\\mathbb R/\\mathbb Z\\), and we treat it so. Put \\(e_n(s)=e^{2\\pi ins}\\) and \\(\\hat f(n)=\\int_0^1f(s)e^{-2\\pi ins}\\,ds\\). Let \\(W\\) be the set of continuous \\(1\\)-periodic \\(f\\) with \\(\\sum_n|\\hat f(n)|<\\infty\\), and put \\(\\|f\\|_W=\\sum_n|\\hat f(n)|\\). In this section \\(\\hat f(n)\\) is a Fourier coefficient; the Gelfand transform is written out in words.\n\n**Lemma 13.1** (uniqueness of Fourier coefficients). A continuous \\(1\\)-periodic function \\(h\\) with \\(\\hat h(n)=0\\) for all \\(n\\) is zero.\n\n**Proof.** The Fejér kernel \\(F_N=\\sum_{|n|\\leq N}\\big(1-\\frac{|n|}{N+1}\\big)e_n\\) equals \\(\\frac1{N+1}\\big|\\sum_{k=0}^Ne_k\\big|^2\\geq0\\) and has integral \\(1\\) over \\([0,1]\\). Since \\(|\\sum_{k=0}^Ne_k(t)|=|\\sin((N+1)\\pi t)/\\sin(\\pi t)|\\), for \\(0<\\delta<1/2\\) and \\(\\delta\\leq t\\leq1-\\delta\\) we have \\(F_N(t)\\leq1/\\big((N+1)\\sin^2(\\pi\\delta)\\big)\\). The function \\(\\sigma_N(s)=\\int_0^1h(s-t)F_N(t)\\,dt\\) equals \\(\\sum_{|n|\\leq N}\\big(1-\\frac{|n|}{N+1}\\big)\\hat h(n)e_n(s)=0\\). On the other hand, reading \\(t\\) modulo \\(1\\),\n\\[\n\\begin{gathered}\n|\\sigma_N(s)-h(s)|\\\\\n\\leq\\int_0^1|h(s-t)-h(s)|F_N(t)\\,dt\\\\\n\\leq\\sup_{|t|\\leq\\delta}|h(s-t)-h(s)|+\\frac{2\\|h\\|_\\infty}{(N+1)\\sin^2(\\pi\\delta)} .\n\\end{gathered}\n\\]\nBy uniform continuity the first term is small for small \\(\\delta\\); then the second is small for large \\(N\\). So \\(h=\\lim_N\\sigma_N=0\\). \\(\\square\\)\n\n**Proposition 13.2.**\n1. For \\(f\\in W\\), \\(f(s)=\\sum_n\\hat f(n)e_n(s)\\), uniformly in \\(s\\), and \\(\\|f\\|_\\infty\\leq\\|f\\|_W\\). \\(W\\) is a commutative unital Banach algebra under pointwise multiplication, with \\(\\widehat{fg}(n)=\\sum_k\\hat f(k)\\hat g(n-k)\\). The map \\(f\\mapsto(\\hat f(n))_n\\) is an isometric algebra isomorphism of \\(W\\) onto \\(\\ell^1(\\mathbb Z)\\) with convolution.\n2. For each \\(t\\in\\mathbb R/\\mathbb Z\\), \\(\\omega_t(f)=f(t)\\) is a character of \\(W\\).\n3. \\(e_1\\) is invertible in \\(W\\), and \\(\\|e_1\\|_W=\\|e_1^{-1}\\|_W=1\\). For every character \\(\\omega\\), \\(|\\omega(e_1)|=1\\), so \\(\\omega(e_1)=e^{2\\pi it}\\) for exactly one \\(t=t_\\omega\\in[0,1)\\).\n4. \\(\\omega(f)=f(t_\\omega)\\) for every \\(f\\in W\\).\n5. \\(t\\mapsto\\omega_t\\) is a homeomorphism of the circle \\(\\mathbb R/\\mathbb Z\\) onto \\(\\operatorname{Ch}(W)\\). The Gelfand transform of \\(f\\) is \\(f\\) itself, seen on the circle. \\(W\\) is semisimple, and its Gelfand representation is not isometric.\n6. (*Wiener's lemma.*) If \\(f\\in W\\) has no zero, then \\(1/f\\in W\\): the Fourier coefficients of \\(1/f\\) are absolutely summable.\n\nPart (5) identifies the character space with the circle. This is different from the closed interval: removing an interior point disconnects the interval, whereas removing any point leaves the circle connected.\n\n**Proof.** (1) Since \\(\\sum_n|\\hat f(n)|<\\infty\\), the series \\(\\sum_n\\hat f(n)e_n\\) converges uniformly to a continuous periodic function \\(g\\), whose coefficients are \\(\\hat f(n)\\) (integrate term by term). By Lemma 13.1, \\(f=g\\), and \\(|f(s)|\\leq\\sum_n|\\hat f(n)|\\). For \\(f,g\\in W\\), the product of the two absolutely convergent series may be rearranged: \\(fg=\\sum_nc_ne_n\\) with \\(c_n=\\sum_k\\hat f(k)\\hat g(n-k)\\) and \\(\\sum_n|c_n|\\leq\\|f\\|_W\\|g\\|_W\\). This series converges uniformly, so \\(\\widehat{fg}(n)=c_n\\), \\(fg\\in W\\) and \\(\\|fg\\|_W\\leq\\|f\\|_W\\|g\\|_W\\). The map \\(f\\mapsto(\\hat f(n))\\) is linear, isometric and multiplicative into \\(\\ell^1(\\mathbb Z)\\) with convolution. It is onto, because for \\(c\\in\\ell^1(\\mathbb Z)\\) the function \\(\\sum_nc_ne_n\\) lies in \\(W\\) and has coefficients \\(c\\). The space \\(\\ell^1(\\mathbb Z)\\) is \\(L^1\\) for counting measure and is complete by [Theorem 3.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#3-the-complete-spaces-of-integrable-functions). Thus \\(W\\) is complete. The constant \\(1=e_0\\) is the identity, of norm \\(1\\).\n(2) Evaluation is linear and multiplicative, and \\(\\omega_t(1)=1\\).\n(3) \\(e_1e_{-1}=1\\), and both have norm \\(1\\). Characters have norm at most \\(1\\) ([Proposition 10.3(1)](#oa-fnd-bn-17)), so \\(|\\omega(e_1)|\\leq1\\) and \\(|\\omega(e_1)|^{-1}=|\\omega(e_{-1})|\\leq1\\).\n(4) \\(\\omega(e_n)=\\omega(e_1)^n=e^{2\\pi int_\\omega}\\) for all \\(n\\in\\mathbb Z\\), negative \\(n\\) through inverses. The partial sums of \\(\\sum_n\\hat f(n)e_n\\) converge to \\(f\\) in \\(W\\), because the tails of \\(\\sum_n|\\hat f(n)|\\) tend to \\(0\\). Since \\(\\omega\\) is continuous (Proposition 10.3(1)), \\(\\omega(f)=\\sum_n\\hat f(n)e^{2\\pi int_\\omega}=f(t_\\omega)\\) by (1).\n(5) By (3) and (4), \\(\\omega=\\omega_{t_\\omega}\\), so \\(t\\mapsto\\omega_t\\) is onto \\(\\operatorname{Ch}(W)\\). It is one-to-one because \\(e_1\\) separates the points of the circle, and it is weak\\* continuous because \\(t\\mapsto f(t)\\) is continuous for each \\(f\\). A continuous bijection from a compact space onto a Hausdorff space is a homeomorphism. The Gelfand transform of \\(f\\) sends \\(\\omega_t\\) to \\(f(t)\\), so it vanishes only when \\(f=0\\), and \\(W\\) is semisimple. Its Gelfand norm is \\(\\|f\\|_\\infty\\). For \\(f=e_0+e_1-e_2\\), multiplying by \\(e_{-1}\\) gives \\[\n\\begin{gathered}\n|f(s)|\\\\\n=|e^{-2\\pi is}+1-e^{2\\pi is}|\\\\\n=|1-2i\\sin(2\\pi s)|\\\\\n\\leq\\sqrt5,\n\\end{gathered}\n\\] while \\(\\|f\\|_W=3\\).\n(6) If \\(f\\) has no zero, then the Gelfand transform of \\(f\\) has no zero on \\(\\operatorname{Ch}(W)\\), by (5). Since the spectrum of \\(f\\) is the range of its Gelfand transform ([Theorem 11.1(2)](#oa-fnd-bn-18)), \\(0\\notin\\sigma_W(f)\\). So \\(f\\) has an inverse \\(g\\in W\\), and \\(fg=1\\) pointwise gives \\(g=1/f\\). \\(\\square\\)\n\n## Exercises\n\n**Exercise 1** (medium; the algebra \\(\\ell^1(\\mathbb Z_{\\geq0})\\)). Let \\(A\\) be the space of sequences \\((x_n)_{n\\geq0}\\) with \\(\\|x\\|=\\sum_n|x_n|<\\infty\\), with the product \\((xy)_n=\\sum_{k=0}^nx_ky_{n-k}\\). Let \\(\\delta_m\\) be the sequence with \\(1\\) in place \\(m\\) and \\(0\\) elsewhere. Show that \\(\\operatorname{Ch}(A)\\) is homeomorphic to the closed unit disc \\(\\bar{\\mathbb D}\\), with Gelfand transform \\(\\hat x(z)=\\sum_nx_nz^n\\); that \\(A\\) is semisimple; and that \\(\\sigma_A(\\delta_1)=\\bar{\\mathbb D}\\).\n\n*Solution.* \\(A\\) is a commutative unital Banach algebra, with identity \\(\\delta_0\\); submultiplicativity follows by rearranging absolutely convergent double series, as in [Proposition 13.2(1)](#oa-fnd-bn-21). Also \\(\\delta_1^n=\\delta_n\\), and \\(x=\\sum_nx_n\\delta_n\\) in norm. For a character \\(\\omega\\), \\(z=\\omega(\\delta_1)\\) satisfies \\(|z|\\leq\\|\\delta_1\\|=1\\), because characters have norm at most \\(1\\) ([Proposition 10.3(1)](#oa-fnd-bn-17)), and by continuity \\(\\omega(x)=\\sum_nx_nz^n\\). Conversely, for \\(|z|\\leq1\\), \\(x\\mapsto\\sum_nx_nz^n\\) is a character, by the Cauchy product formula for absolutely convergent series. So \\(\\omega\\mapsto\\omega(\\delta_1)\\) is a bijection of \\(\\operatorname{Ch}(A)\\) onto \\(\\bar{\\mathbb D}\\). It is weak\\* continuous, and \\(\\operatorname{Ch}(A)\\) is compact (Proposition 10.3(2)), so it is a homeomorphism. If \\(\\hat x=0\\) on \\(\\bar{\\mathbb D}\\), then \\(x_n=\\frac1{2\\pi}\\int_0^{2\\pi}\\hat x(e^{i\\theta})e^{-in\\theta}\\,d\\theta=0\\) for all \\(n\\), integrating the uniformly convergent series term by term; so \\(A\\) is semisimple. Finally \\(\\sigma_A(\\delta_1)=\\hat\\delta_1(\\operatorname{Ch}(A))=\\bar{\\mathbb D}\\) by [Theorem 11.1(2)](#oa-fnd-bn-18).\n\n**Exercise 2** (medium; spectra of commuting elements). Let \\(A\\) be a unital Banach algebra and \\(x,y\\in A\\) with \\(xy=yx\\). Show that \\(\\sigma_A(x+y)\\subseteq\\sigma_A(x)+\\sigma_A(y)\\) and \\(\\sigma_A(xy)\\subseteq\\sigma_A(x)\\sigma_A(y)\\), and that commutativity cannot be dropped.\n\n*Solution.* The subalgebra generated by \\(1,x,y\\) is commutative, and the union of a chain of commutative subalgebras is commutative. By Zorn's lemma there is a maximal commutative subalgebra \\(B\\) containing \\(1,x,y\\). Its closure is commutative, by continuity of multiplication, so \\(B\\) is closed. If \\(b\\in B\\) is invertible in \\(A\\), then \\(b^{-1}\\) commutes with every \\(c\\in B\\) (multiply \\(cb=bc\\) by \\(b^{-1}\\) on both sides), so the algebra generated by \\(B\\) and \\(b^{-1}\\) is commutative, and maximality gives \\(b^{-1}\\in B\\). Hence \\(\\sigma_B(b)=\\sigma_A(b)\\) for \\(b\\in B\\). Since \\(B\\) is a commutative unital Banach algebra, [Theorem 11.1(2)](#oa-fnd-bn-18) gives \\[\n\\begin{gathered}\n\\sigma_A(x+y)\\\\\n=\\sigma_B(x+y)\\\\\n=\\{\\omega(x)+\\omega(y):\\omega\\in\\operatorname{Ch}(B)\\}\\\\\n\\subseteq\\sigma_B(x)+\\sigma_B(y)\\\\\n=\\sigma_A(x)+\\sigma_A(y),\n\\end{gathered}\n\\] and the same for \\(xy\\). Without commutativity, \\(E_{12}+E_{21}\\) has spectrum \\(\\{1,-1\\}\\), while \\(\\sigma(E_{12})+\\sigma(E_{21})=\\{0\\}\\). Also \\(E_{12}E_{21}=E_{11}\\) has spectrum \\(\\{0,1\\}\\), whereas \\(\\sigma(E_{12})\\sigma(E_{21})=\\{0\\}\\).\n\n**Exercise 3** (medium; polynomial identities). Let \\(P(\\lambda)=(\\lambda-\\alpha_1)\\cdots(\\lambda-\\alpha_n)\\) with distinct \\(\\alpha_i\\), and let \\(x\\) be an element of a nontrivial unital Banach algebra with \\(P(x)=0\\). Show that \\(\\sigma_A(x)\\subseteq\\{\\alpha_1,\\dots,\\alpha_n\\}\\), and that \\(f(x)=Q(x)\\) for every \\(f\\) holomorphic near these points, where \\(Q\\) is the polynomial of degree less than \\(n\\) with \\(Q(\\alpha_i)=f(\\alpha_i)\\). Compute the idempotent \\(e\\) of [Example 6.12](#oa-fnd-bn-22).\n\n*Solution.* By the spectral mapping theorem ([Theorem 6.10(2)](#oa-fnd-bn-12)), \\(P(\\sigma_A(x))=\\sigma_A(P(x))=\\sigma_A(0)=\\{0\\}\\), so \\(\\sigma_A(x)\\subseteq\\{\\alpha_1,\\dots,\\alpha_n\\}\\). Let \\(U\\) be a union of small disjoint discs around the \\(\\alpha_i\\) on which \\(f\\) is holomorphic. The function \\(f-Q\\) vanishes at each \\(\\alpha_i\\), and each \\(\\alpha_i\\) is a simple zero of \\(P\\). So, as in the proof of Theorem 6.10(1), \\(k=(f-Q)/P\\) extends holomorphically across each \\(\\alpha_i\\), and \\(f-Q=Pk\\) on \\(U\\). By [Theorem 6.7](#oa-fnd-bn-11), \\(f(x)-Q(x)=P(x)k(x)=0\\). For the matrix of Example 6.12, \\((x-1)(x-2)=0\\); with \\(f(1)=0\\) and \\(f(2)=1\\) we get \\(Q(\\lambda)=\\lambda-1\\), so \\(e=x-1\\).\n\n**Exercise 4** (hard; the index group has no torsion). Let \\(A\\) be a commutative nontrivial unital Banach algebra. Show that if \\(x\\in G(A)\\) and \\(x^n\\in G_0(A)\\) for some \\(n\\geq1\\), then \\(x\\in G_0(A)\\).\n\n*Solution.* By [Proposition 7.1(5)](#oa-fnd-bn-14), \\(x^n=\\exp y\\) for some \\(y\\). Put \\(z=\\exp(y/n)\\in G_0(A)\\) and \\(w=xz^{-1}\\). Since \\(A\\) is commutative, \\(w^n=x^nz^{-n}=\\exp(y)\\exp(-y)=1\\). By the spectral mapping theorem ([Theorem 6.10(2)](#oa-fnd-bn-12)), \\(\\{\\lambda^n:\\lambda\\in\\sigma_A(w)\\}=\\sigma_A(w^n)=\\{1\\}\\), so \\(\\sigma_A(w)\\) lies in the finite set of \\(n\\)-th roots of unity. Choose \\(0<\\rho<1\\) so small that the open discs of radius \\(\\rho\\) around these roots are disjoint (for \\(n\\geq2\\), adjacent roots are \\(2\\sin(\\pi/n)\\) apart, so \\(\\rho=1/2\\) is too large once \\(n\\geq7\\)). Write each root as \\(\\zeta=e^{i\\theta}\\) with \\(\\theta\\) real. On the disc around \\(\\zeta\\), \\(|\\lambda/\\zeta-1|=|\\lambda-\\zeta|<\\rho<1\\), so \\(L(\\lambda)=\\operatorname{Log}(\\lambda/\\zeta)+i\\theta\\) is holomorphic there and \\(e^{L(\\lambda)}=(\\lambda/\\zeta)\\zeta=\\lambda\\). Together these functions define \\(L\\in H(U)\\) on the union \\(U\\) of the discs, with \\(\\exp\\circ L=\\) identity. By the composition rule, Theorem 6.10(3), \\(\\exp(L(w))=w\\), so \\(w\\in\\exp(A)=G_0(A)\\), and \\(x=wz\\in G_0(A)\\). The set \\(U\\) is not connected, so this uses the calculus over cycles.\n\n**Exercise 5** (easy; the unitization of a unital C\\*-algebra). Let \\(A\\) be a C\\*-algebra with identity \\(1_A\\), and let \\(N(a+\\lambda)=\\max\\big(p(a+\\lambda),|\\lambda|\\big)\\) on \\(A_1\\), as in [Proposition 3.4](#oa-fnd-bn-05). Show that \\(p(a+\\lambda)=\\|a+\\lambda1_A\\|\\), and that \\((a,\\lambda)\\mapsto(a+\\lambda1_A,\\lambda)\\) is an isometric \\(*\\)-isomorphism of \\((A_1,N)\\) onto \\(A\\oplus\\mathbb C\\) with the norm \\(\\max(\\|b\\|,|\\mu|)\\). Conclude that \\((A_1,N)\\) is a C\\*-algebra, without using the inequality of Proposition 3.4(2).\n\n*Solution.* For \\(b\\in A\\), \\((a+\\lambda)b=(a+\\lambda1_A)b\\), so \\(p(a+\\lambda)\\) is the norm of left multiplication by the element \\(a+\\lambda1_A\\) of \\(A\\), which is \\(\\|a+\\lambda1_A\\|\\) by Proposition 3.4(1). So \\(N(a+\\lambda)=\\max(\\|a+\\lambda1_A\\|,|\\lambda|)\\), which is the norm of the image \\((a+\\lambda1_A,\\lambda)\\). The map is an algebra isomorphism by [Proposition 3.3(3)](#oa-fnd-bn-04), and it preserves the involution because \\((a+\\lambda1_A)^*=a^*+\\bar\\lambda1_A\\). The algebra \\(A\\oplus\\mathbb C\\) with componentwise operations and the maximum norm is a C\\*-algebra, because the C\\*-identity holds in each component. So \\((A_1,N)\\) is a C\\*-algebra.\n\n## Where this leads\n\n- The next lesson, [on C\\*-algebras and their continuous functional calculus](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md), specializes this theory to C\\*-algebras. There the norm of every normal element equals its spectral radius (compare the last of Examples 5.7), and a commutative C\\*-algebra is isometrically \\(*\\)-isomorphic to \\(C_0\\) of its character space, so that its Gelfand representation is isometric and onto. The spectrum of an element of a C\\*-subalgebra is the same in both algebras (contrast Example 4.6). The characters of \\(C_0(\\Omega)\\), which are needed there, are identified with the points of \\(\\Omega\\) in Example 11.4.\n- Representations of \\(L^1(G)\\) and of \\(\\mathfrak A(\\Omega,G)\\), and the crossed products they lead to, are the subject of the course *Crossed products and the flow of weights*.\n- Not treated here: the calculus of several commuting elements and joint spectra; the Shilov boundary; Wiener–Tauberian theorems; the Jacobson radical of noncommutative algebras; the identification of the index group of \\(C(\\mathbb T)\\) with \\(\\mathbb Z\\); a description of the image of the Gelfand map.\n\n## References\n\n\n*Freely accessible reading:* [Vahid Shirbisheh, *Lectures on C*-algebras*, §2.5](https://arxiv.org/html/1211.3404v2) gives a route through holomorphic functional calculus; this lesson includes the general cycle-index conditions. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BN-23",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "name": "Absolutely convergent products",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
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      "proof_locus": {
        "line": 44,
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      },
      "full_conditions_and_proof": "### Absolutely convergent products\n\n**Lemma 0.1** (Products and regrouping). In a Banach algebra, the product of two absolutely convergent series is the sum of the products of their terms, and that double series may be regrouped in any countable partition of its index set. In particular, for series indexed by nonnegative integers,\n\\[\n\\left(\\sum_{j\\geq0}a_j\\right)\n\\left(\\sum_{k\\geq0}b_k\\right)\n=\\sum_{n\\geq0}\\sum_{j+k=n}a_jb_k .\n\\]\nIf \\(xy=yx\\) in a unital algebra, then\n\\((x+y)^n=\\sum_{j=0}^n\\binom njx^jy^{n-j}\\).\n\n**Proof.** The scalar estimate\n\\(\\sum_{j,k}\\|a_jb_k\\|\\leq(\\sum_j\\|a_j\\|)(\\sum_k\\|b_k\\|)\\)\nfollows first for finite rectangles and then by taking suprema of finite subsums. For finite subsets \\(J,K\\), the norm sum outside \\(J\\times K\\) is at most\n\\[\n\\left(\\sum_{j\\notin J}\\|a_j\\|\\right)\\sum_k\\|b_k\\|\n+\\sum_j\\|a_j\\|\\left(\\sum_{k\\notin K}\\|b_k\\|\\right),\n\\]\nwhich tends to zero as \\(J,K\\) exhaust the indices. Thus finite double sums form a Cauchy net and converge by completeness; sums over rectangles converge to the product by continuity of multiplication. For any countable grouping, each group itself has an absolutely convergent sum, and the sum of the norms of the grouped sums is at most the original norm sum. The tail estimate shows that the grouped series has the same limit: first keep a finite rectangle with small complementary norm sum, then keep all groups containing its finitely many terms. This proves the regrouping claim, including the groups \\(j+k=n\\). Finally, multiplication of the binomial identity for \\(n\\) by \\(x+y\\), using \\(xy=yx\\) and \\(\\binom nj+\\binom n{j-1}=\\binom{n+1}j\\), proves it for \\(n+1\\); the case \\(n=0\\) is the identity. \\(\\square\\)\n\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-01",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "1. Self-adjoint, normal and unitary elements",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "anchor": "oa-fnd-cf-01",
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      "full_conditions_and_proof": "### 1. Self-adjoint, normal and unitary elements\n\n**Definition 1.1.** In an involutive Banach algebra \\(A\\), call \\(x\\) *self-adjoint* (or *hermitian*) if \\(x^*=x\\), and *normal* if \\(x^*x=xx^*\\). If \\(A\\) is unital, \\(x\\) is *unitary* if \\(x^*x=xx^*=1\\). A *projection* is an element \\(p\\) with \\(p=p^*=p^2\\).\n\n**Proposition 1.2.**\n1. (*Cartesian decomposition.*) Every \\(x\\in A\\) can be written in exactly one way as \\(x=x_1+ix_2\\) with \\(x_1,x_2\\in A_h\\), namely \\(x_1=\\frac12(x+x^*)\\) and \\(x_2=\\frac1{2i}(x-x^*)\\). Moreover \\(\\|x_1\\|\\leq\\|x\\|\\) and \\(\\|x_2\\|\\leq\\|x\\|\\). The set \\(A_h\\) is a closed real subspace, and \\(A=A_h\\oplus iA_h\\) as real Banach spaces.\n2. \\(x\\) is normal exactly when \\(x_1\\) and \\(x_2\\) commute. Self-adjoint and unitary elements are normal.\n3. If \\(A\\) is unital, then \\(1\\in A_h\\), and \\(U(A)\\) is a closed subgroup of \\(G(A)\\) with \\(u^{-1}=u^*\\in U(A)\\). If \\(A\\) is a nontrivial unital C\\*-algebra, then \\(\\|u\\|=1\\) for every unitary \\(u\\).\n4. In a C\\*-algebra, every projection has norm \\(0\\) or \\(1\\).\n\n**Proof.** (1) If \\(x=x_1+ix_2\\) with \\(x_1,x_2\\) self-adjoint, then \\(x^*=x_1-ix_2\\), and solving gives the two formulas. Conversely, these formulas define self-adjoint elements with \\(x_1+ix_2=x\\). Since the involution is isometric, \\(\\|x_1\\|\\leq\\frac12(\\|x\\|+\\|x^*\\|)=\\|x\\|\\), and the same holds for \\(x_2\\). \\(A_h\\) is the set of fixed points of the continuous real-linear map \\(x\\mapsto x^*\\), so it is closed.\n(2) Expanding gives\n\\[\n\\begin{gathered}\nx^*x\\\\\n=x_1^2+x_2^2+i(x_1x_2-x_2x_1),\\\\\nxx^*\\\\\n=x_1^2+x_2^2-i(x_1x_2-x_2x_1).\n\\end{gathered}\n\\]\nSo \\(x^*x=xx^*\\) exactly when \\(x_1x_2=x_2x_1\\). A self-adjoint element commutes with its adjoint, and so does a unitary \\(u\\), since \\(u^*u=1=uu^*\\).\n(3) The adjoint \\(1^*\\) is again an identity, because \\(1^*x=(x^*1)^*=x\\) and \\(x1^*=(1x^*)^*=x\\); so \\(1^*=1\\). If \\(u,v\\) are unitary, then \\((uv)^*(uv)=v^*u^*uv=1\\) and \\((uv)(uv)^*=1\\); and \\(u^{-1}=u^*\\) is unitary. If unitaries \\(u_n\\) converge to \\(u\\), continuity of the product and of the involution gives \\(u^*u=uu^*=1\\). In a nontrivial unital C\\*-algebra, \\(\\|u\\|^2=\\|u^*u\\|=\\|1\\|=1\\).\n(4) \\(\\|p\\|=\\|p^*p\\|=\\|p\\|^2\\). \\(\\square\\)\n\nIn the zero algebra, \\(0\\) is a unitary and a projection.\n\n**Theorem 1.3** (The norm of a normal element is its spectral radius). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\), \\(\\|h^{2^k}\\|=\\|h\\|^{2^k}\\) for all \\(k\\geq0\\), and \\(r(h)=\\|h\\|\\).\n2. For every normal \\(x\\in A\\), \\(r(x)=\\|x\\|\\).\n3. For every \\(x\\in A\\), \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\).\n4. (*The norm is determined by the algebra.*) If an algebra with involution is a C\\*-algebra for two norms, the two norms are equal.\n\n**Proof.** (1) \\(\\|h^2\\|=\\|h^*h\\|=\\|h\\|^2\\), and \\(h^{2^k}\\) is again self-adjoint, so induction gives the first claim. The [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09) \\(r(h)=\\lim_n\\|h^n\\|^{1/n}\\), taken along \\(n=2^k\\), gives \\(r(h)=\\|h\\|\\).\n(2) The element \\(x^*x\\) is self-adjoint, so \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\) by (1). If \\(x\\) is normal, \\(x^*\\) and \\(x\\) commute, and the spectral radius is submultiplicative on commuting elements (a corollary of the [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09)); so \\(r(x^*x)\\leq r(x^*)r(x)\\). In \\(\\widetilde A\\), \\(\\lambda-x\\) has an inverse \\(w\\) exactly when \\(\\bar\\lambda-x^*\\) has the inverse \\(w^*\\). So \\(\\sigma'_A(x^*)=\\overline{\\sigma'_A(x)}\\) and \\(r(x^*)=r(x)\\). Hence \\(\\|x\\|^2\\leq r(x)^2\\leq\\|x\\|^2\\), where the last step uses \\(r(x)\\leq\\|x\\|\\) ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)).\n(3) This was shown in the proof of (2).\n(4) The quasi-spectrum of an element depends only on the algebra, because invertibility in \\(A_1\\) is an algebraic property. So the spectral radius \\(r\\) does not depend on the norm. By (3), both norms satisfy \\(\\|x\\|^2=r(x^*x)\\). \\(\\square\\)\n\n**Example 1.4** (Normality is needed). In \\(M_2(\\mathbb C)\\), \\(E_{12}^2=0\\), so \\(r(E_{12})=0\\), while \\(\\|E_{12}\\|=1\\).\n\n**Proposition 1.5** (Spectra of adjoints, unitaries and self-adjoint elements). Let \\(A\\) be a C\\*-algebra.\n1. For every \\(x\\in A\\), \\(\\sigma'_A(x^*)=\\{\\bar\\lambda:\\lambda\\in\\sigma'_A(x)\\}\\).\n2. If \\(A\\) is unital and \\(u\\in U(A)\\), then \\(\\sigma_A(u)\\subseteq\\mathbb T=\\{\\lambda:|\\lambda|=1\\}\\).\n3. For \\(h\\in A_h\\), \\(\\sigma'_A(h)\\subseteq[-\\|h\\|,\\|h\\|]\\), and \\(\\sigma'_A(h)\\) contains \\(\\|h\\|\\) or \\(-\\|h\\|\\). If \\(A\\) is unital and nontrivial, the same holds for \\(\\sigma_A(h)\\).\n\n**Proof.** (1) was shown in the proof of Theorem 1.3(2).\n(2) In the zero algebra \\(\\sigma_A(u)=\\varnothing\\). Otherwise \\(\\|u\\|=\\|u^{-1}\\|=1\\) by Proposition 1.2(3). The spectrum of an element lies in the closed disc whose radius is its norm ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)), and \\(\\sigma_A(u^{-1})=\\{\\lambda^{-1}:\\lambda\\in\\sigma_A(u)\\}\\) ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)). So \\(\\sigma_A(u)\\) and \\(\\sigma_A(u^{-1})\\) lie in the closed unit disc, and every \\(\\lambda\\in\\sigma_A(u)\\) has \\(|\\lambda|\\leq1\\) and \\(|\\lambda|^{-1}\\leq1\\).\n(3) Work in the nontrivial unital C\\*-algebra \\(\\widetilde A\\). Put \\(u=\\exp(ih)=\\sum_n(ih)^n/n!\\) ([the exponential](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14)). The involution is continuous and conjugate linear, so applying it to the partial sums gives \\(u^*=\\exp(-ih)\\). Since \\(ih\\) and \\(-ih\\) commute, \\(\\exp(ih)\\exp(-ih)=\\exp(-ih)\\exp(ih)=1\\). So \\(u\\) is unitary, and by the [spectral mapping theorem](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-12) and (2),\n\\[\n\\{e^{i\\lambda}:\\lambda\\in\\sigma'_A(h)\\}=\\sigma_{\\widetilde A}(u)\\subseteq\\mathbb T .\n\\]\nSince \\(|e^{i\\lambda}|=e^{-\\operatorname{Im}\\lambda}\\), every \\(\\lambda\\in\\sigma'_A(h)\\) is real. By Theorem 1.3, \\(r(h)=\\|h\\|\\). The quasi-spectrum is compact and not empty ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)), so the supremum \\(\\|h\\|\\) of \\(|\\lambda|\\) over it is attained. For nontrivial unital \\(A\\), \\(\\sigma_A(h)\\) is not empty ([the spectrum is not empty](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-08)) and \\(\\sigma'_A(h)=\\sigma_A(h)\\cup\\{0\\}\\). If \\(h\\neq0\\), the point of modulus \\(\\|h\\|\\) lies in \\(\\sigma_A(h)\\); if \\(h=0\\), then \\(\\sigma_A(h)=\\{0\\}\\). \\(\\square\\)\n\n**Exercise 1.6** (easy; Powers of normal elements). Show that \\(\\|x^n\\|=\\|x\\|^n\\) for normal \\(x\\) and all \\(n\\geq1\\). Show by examples that the equality can fail for nonnormal \\(x\\), and that it can hold for all \\(n\\) without \\(x\\) being normal.\n\n*Solution.* \\(x^n\\) is normal, so \\(\\|x^n\\|=r(x^n)=r(x)^n=\\|x\\|^n\\), by Theorem 1.3 and the identity \\(r(x^n)=r(x)^n\\), which follows from the [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09). For \\(E_{12}\\in M_2(\\mathbb C)\\), \\(\\|E_{12}^2\\|=0<1\\). The unilateral shift \\(S\\) on \\(\\ell^2(\\mathbb N)\\) is an isometry, so \\(\\|S^n\\|=1=\\|S\\|^n\\), but \\(S^*S=1\\neq SS^*\\).\n\n",
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      "id": "OA-FND-CF-02",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "1. Self-adjoint, normal and unitary elements",
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      "full_conditions_and_proof": "### 1. Self-adjoint, normal and unitary elements\n\n**Definition 1.1.** In an involutive Banach algebra \\(A\\), call \\(x\\) *self-adjoint* (or *hermitian*) if \\(x^*=x\\), and *normal* if \\(x^*x=xx^*\\). If \\(A\\) is unital, \\(x\\) is *unitary* if \\(x^*x=xx^*=1\\). A *projection* is an element \\(p\\) with \\(p=p^*=p^2\\).\n\n**Proposition 1.2.**\n1. (*Cartesian decomposition.*) Every \\(x\\in A\\) can be written in exactly one way as \\(x=x_1+ix_2\\) with \\(x_1,x_2\\in A_h\\), namely \\(x_1=\\frac12(x+x^*)\\) and \\(x_2=\\frac1{2i}(x-x^*)\\). Moreover \\(\\|x_1\\|\\leq\\|x\\|\\) and \\(\\|x_2\\|\\leq\\|x\\|\\). The set \\(A_h\\) is a closed real subspace, and \\(A=A_h\\oplus iA_h\\) as real Banach spaces.\n2. \\(x\\) is normal exactly when \\(x_1\\) and \\(x_2\\) commute. Self-adjoint and unitary elements are normal.\n3. If \\(A\\) is unital, then \\(1\\in A_h\\), and \\(U(A)\\) is a closed subgroup of \\(G(A)\\) with \\(u^{-1}=u^*\\in U(A)\\). If \\(A\\) is a nontrivial unital C\\*-algebra, then \\(\\|u\\|=1\\) for every unitary \\(u\\).\n4. In a C\\*-algebra, every projection has norm \\(0\\) or \\(1\\).\n\n**Proof.** (1) If \\(x=x_1+ix_2\\) with \\(x_1,x_2\\) self-adjoint, then \\(x^*=x_1-ix_2\\), and solving gives the two formulas. Conversely, these formulas define self-adjoint elements with \\(x_1+ix_2=x\\). Since the involution is isometric, \\(\\|x_1\\|\\leq\\frac12(\\|x\\|+\\|x^*\\|)=\\|x\\|\\), and the same holds for \\(x_2\\). \\(A_h\\) is the set of fixed points of the continuous real-linear map \\(x\\mapsto x^*\\), so it is closed.\n(2) Expanding gives\n\\[\n\\begin{gathered}\nx^*x\\\\\n=x_1^2+x_2^2+i(x_1x_2-x_2x_1),\\\\\nxx^*\\\\\n=x_1^2+x_2^2-i(x_1x_2-x_2x_1).\n\\end{gathered}\n\\]\nSo \\(x^*x=xx^*\\) exactly when \\(x_1x_2=x_2x_1\\). A self-adjoint element commutes with its adjoint, and so does a unitary \\(u\\), since \\(u^*u=1=uu^*\\).\n(3) The adjoint \\(1^*\\) is again an identity, because \\(1^*x=(x^*1)^*=x\\) and \\(x1^*=(1x^*)^*=x\\); so \\(1^*=1\\). If \\(u,v\\) are unitary, then \\((uv)^*(uv)=v^*u^*uv=1\\) and \\((uv)(uv)^*=1\\); and \\(u^{-1}=u^*\\) is unitary. If unitaries \\(u_n\\) converge to \\(u\\), continuity of the product and of the involution gives \\(u^*u=uu^*=1\\). In a nontrivial unital C\\*-algebra, \\(\\|u\\|^2=\\|u^*u\\|=\\|1\\|=1\\).\n(4) \\(\\|p\\|=\\|p^*p\\|=\\|p\\|^2\\). \\(\\square\\)\n\nIn the zero algebra, \\(0\\) is a unitary and a projection.\n\n**Theorem 1.3** (The norm of a normal element is its spectral radius). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\), \\(\\|h^{2^k}\\|=\\|h\\|^{2^k}\\) for all \\(k\\geq0\\), and \\(r(h)=\\|h\\|\\).\n2. For every normal \\(x\\in A\\), \\(r(x)=\\|x\\|\\).\n3. For every \\(x\\in A\\), \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\).\n4. (*The norm is determined by the algebra.*) If an algebra with involution is a C\\*-algebra for two norms, the two norms are equal.\n\n**Proof.** (1) \\(\\|h^2\\|=\\|h^*h\\|=\\|h\\|^2\\), and \\(h^{2^k}\\) is again self-adjoint, so induction gives the first claim. The [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09) \\(r(h)=\\lim_n\\|h^n\\|^{1/n}\\), taken along \\(n=2^k\\), gives \\(r(h)=\\|h\\|\\).\n(2) The element \\(x^*x\\) is self-adjoint, so \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\) by (1). If \\(x\\) is normal, \\(x^*\\) and \\(x\\) commute, and the spectral radius is submultiplicative on commuting elements (a corollary of the [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09)); so \\(r(x^*x)\\leq r(x^*)r(x)\\). In \\(\\widetilde A\\), \\(\\lambda-x\\) has an inverse \\(w\\) exactly when \\(\\bar\\lambda-x^*\\) has the inverse \\(w^*\\). So \\(\\sigma'_A(x^*)=\\overline{\\sigma'_A(x)}\\) and \\(r(x^*)=r(x)\\). Hence \\(\\|x\\|^2\\leq r(x)^2\\leq\\|x\\|^2\\), where the last step uses \\(r(x)\\leq\\|x\\|\\) ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)).\n(3) This was shown in the proof of (2).\n(4) The quasi-spectrum of an element depends only on the algebra, because invertibility in \\(A_1\\) is an algebraic property. So the spectral radius \\(r\\) does not depend on the norm. By (3), both norms satisfy \\(\\|x\\|^2=r(x^*x)\\). \\(\\square\\)\n\n**Example 1.4** (Normality is needed). In \\(M_2(\\mathbb C)\\), \\(E_{12}^2=0\\), so \\(r(E_{12})=0\\), while \\(\\|E_{12}\\|=1\\).\n\n**Proposition 1.5** (Spectra of adjoints, unitaries and self-adjoint elements). Let \\(A\\) be a C\\*-algebra.\n1. For every \\(x\\in A\\), \\(\\sigma'_A(x^*)=\\{\\bar\\lambda:\\lambda\\in\\sigma'_A(x)\\}\\).\n2. If \\(A\\) is unital and \\(u\\in U(A)\\), then \\(\\sigma_A(u)\\subseteq\\mathbb T=\\{\\lambda:|\\lambda|=1\\}\\).\n3. For \\(h\\in A_h\\), \\(\\sigma'_A(h)\\subseteq[-\\|h\\|,\\|h\\|]\\), and \\(\\sigma'_A(h)\\) contains \\(\\|h\\|\\) or \\(-\\|h\\|\\). If \\(A\\) is unital and nontrivial, the same holds for \\(\\sigma_A(h)\\).\n\n**Proof.** (1) was shown in the proof of Theorem 1.3(2).\n(2) In the zero algebra \\(\\sigma_A(u)=\\varnothing\\). Otherwise \\(\\|u\\|=\\|u^{-1}\\|=1\\) by Proposition 1.2(3). The spectrum of an element lies in the closed disc whose radius is its norm ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)), and \\(\\sigma_A(u^{-1})=\\{\\lambda^{-1}:\\lambda\\in\\sigma_A(u)\\}\\) ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)). So \\(\\sigma_A(u)\\) and \\(\\sigma_A(u^{-1})\\) lie in the closed unit disc, and every \\(\\lambda\\in\\sigma_A(u)\\) has \\(|\\lambda|\\leq1\\) and \\(|\\lambda|^{-1}\\leq1\\).\n(3) Work in the nontrivial unital C\\*-algebra \\(\\widetilde A\\). Put \\(u=\\exp(ih)=\\sum_n(ih)^n/n!\\) ([the exponential](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14)). The involution is continuous and conjugate linear, so applying it to the partial sums gives \\(u^*=\\exp(-ih)\\). Since \\(ih\\) and \\(-ih\\) commute, \\(\\exp(ih)\\exp(-ih)=\\exp(-ih)\\exp(ih)=1\\). So \\(u\\) is unitary, and by the [spectral mapping theorem](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-12) and (2),\n\\[\n\\{e^{i\\lambda}:\\lambda\\in\\sigma'_A(h)\\}=\\sigma_{\\widetilde A}(u)\\subseteq\\mathbb T .\n\\]\nSince \\(|e^{i\\lambda}|=e^{-\\operatorname{Im}\\lambda}\\), every \\(\\lambda\\in\\sigma'_A(h)\\) is real. By Theorem 1.3, \\(r(h)=\\|h\\|\\). The quasi-spectrum is compact and not empty ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)), so the supremum \\(\\|h\\|\\) of \\(|\\lambda|\\) over it is attained. For nontrivial unital \\(A\\), \\(\\sigma_A(h)\\) is not empty ([the spectrum is not empty](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-08)) and \\(\\sigma'_A(h)=\\sigma_A(h)\\cup\\{0\\}\\). If \\(h\\neq0\\), the point of modulus \\(\\|h\\|\\) lies in \\(\\sigma_A(h)\\); if \\(h=0\\), then \\(\\sigma_A(h)=\\{0\\}\\). \\(\\square\\)\n\n**Exercise 1.6** (easy; Powers of normal elements). Show that \\(\\|x^n\\|=\\|x\\|^n\\) for normal \\(x\\) and all \\(n\\geq1\\). Show by examples that the equality can fail for nonnormal \\(x\\), and that it can hold for all \\(n\\) without \\(x\\) being normal.\n\n*Solution.* \\(x^n\\) is normal, so \\(\\|x^n\\|=r(x^n)=r(x)^n=\\|x\\|^n\\), by Theorem 1.3 and the identity \\(r(x^n)=r(x)^n\\), which follows from the [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09). For \\(E_{12}\\in M_2(\\mathbb C)\\), \\(\\|E_{12}^2\\|=0<1\\). The unilateral shift \\(S\\) on \\(\\ell^2(\\mathbb N)\\) is an isometry, so \\(\\|S^n\\|=1=\\|S\\|^n\\), but \\(S^*S=1\\neq SS^*\\).\n\n",
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    {
      "id": "OA-FND-CF-03",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "1. Self-adjoint, normal and unitary elements",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
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      "full_conditions_and_proof": "### 1. Self-adjoint, normal and unitary elements\n\n**Definition 1.1.** In an involutive Banach algebra \\(A\\), call \\(x\\) *self-adjoint* (or *hermitian*) if \\(x^*=x\\), and *normal* if \\(x^*x=xx^*\\). If \\(A\\) is unital, \\(x\\) is *unitary* if \\(x^*x=xx^*=1\\). A *projection* is an element \\(p\\) with \\(p=p^*=p^2\\).\n\n**Proposition 1.2.**\n1. (*Cartesian decomposition.*) Every \\(x\\in A\\) can be written in exactly one way as \\(x=x_1+ix_2\\) with \\(x_1,x_2\\in A_h\\), namely \\(x_1=\\frac12(x+x^*)\\) and \\(x_2=\\frac1{2i}(x-x^*)\\). Moreover \\(\\|x_1\\|\\leq\\|x\\|\\) and \\(\\|x_2\\|\\leq\\|x\\|\\). The set \\(A_h\\) is a closed real subspace, and \\(A=A_h\\oplus iA_h\\) as real Banach spaces.\n2. \\(x\\) is normal exactly when \\(x_1\\) and \\(x_2\\) commute. Self-adjoint and unitary elements are normal.\n3. If \\(A\\) is unital, then \\(1\\in A_h\\), and \\(U(A)\\) is a closed subgroup of \\(G(A)\\) with \\(u^{-1}=u^*\\in U(A)\\). If \\(A\\) is a nontrivial unital C\\*-algebra, then \\(\\|u\\|=1\\) for every unitary \\(u\\).\n4. In a C\\*-algebra, every projection has norm \\(0\\) or \\(1\\).\n\n**Proof.** (1) If \\(x=x_1+ix_2\\) with \\(x_1,x_2\\) self-adjoint, then \\(x^*=x_1-ix_2\\), and solving gives the two formulas. Conversely, these formulas define self-adjoint elements with \\(x_1+ix_2=x\\). Since the involution is isometric, \\(\\|x_1\\|\\leq\\frac12(\\|x\\|+\\|x^*\\|)=\\|x\\|\\), and the same holds for \\(x_2\\). \\(A_h\\) is the set of fixed points of the continuous real-linear map \\(x\\mapsto x^*\\), so it is closed.\n(2) Expanding gives\n\\[\n\\begin{gathered}\nx^*x\\\\\n=x_1^2+x_2^2+i(x_1x_2-x_2x_1),\\\\\nxx^*\\\\\n=x_1^2+x_2^2-i(x_1x_2-x_2x_1).\n\\end{gathered}\n\\]\nSo \\(x^*x=xx^*\\) exactly when \\(x_1x_2=x_2x_1\\). A self-adjoint element commutes with its adjoint, and so does a unitary \\(u\\), since \\(u^*u=1=uu^*\\).\n(3) The adjoint \\(1^*\\) is again an identity, because \\(1^*x=(x^*1)^*=x\\) and \\(x1^*=(1x^*)^*=x\\); so \\(1^*=1\\). If \\(u,v\\) are unitary, then \\((uv)^*(uv)=v^*u^*uv=1\\) and \\((uv)(uv)^*=1\\); and \\(u^{-1}=u^*\\) is unitary. If unitaries \\(u_n\\) converge to \\(u\\), continuity of the product and of the involution gives \\(u^*u=uu^*=1\\). In a nontrivial unital C\\*-algebra, \\(\\|u\\|^2=\\|u^*u\\|=\\|1\\|=1\\).\n(4) \\(\\|p\\|=\\|p^*p\\|=\\|p\\|^2\\). \\(\\square\\)\n\nIn the zero algebra, \\(0\\) is a unitary and a projection.\n\n**Theorem 1.3** (The norm of a normal element is its spectral radius). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\), \\(\\|h^{2^k}\\|=\\|h\\|^{2^k}\\) for all \\(k\\geq0\\), and \\(r(h)=\\|h\\|\\).\n2. For every normal \\(x\\in A\\), \\(r(x)=\\|x\\|\\).\n3. For every \\(x\\in A\\), \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\).\n4. (*The norm is determined by the algebra.*) If an algebra with involution is a C\\*-algebra for two norms, the two norms are equal.\n\n**Proof.** (1) \\(\\|h^2\\|=\\|h^*h\\|=\\|h\\|^2\\), and \\(h^{2^k}\\) is again self-adjoint, so induction gives the first claim. The [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09) \\(r(h)=\\lim_n\\|h^n\\|^{1/n}\\), taken along \\(n=2^k\\), gives \\(r(h)=\\|h\\|\\).\n(2) The element \\(x^*x\\) is self-adjoint, so \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\) by (1). If \\(x\\) is normal, \\(x^*\\) and \\(x\\) commute, and the spectral radius is submultiplicative on commuting elements (a corollary of the [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09)); so \\(r(x^*x)\\leq r(x^*)r(x)\\). In \\(\\widetilde A\\), \\(\\lambda-x\\) has an inverse \\(w\\) exactly when \\(\\bar\\lambda-x^*\\) has the inverse \\(w^*\\). So \\(\\sigma'_A(x^*)=\\overline{\\sigma'_A(x)}\\) and \\(r(x^*)=r(x)\\). Hence \\(\\|x\\|^2\\leq r(x)^2\\leq\\|x\\|^2\\), where the last step uses \\(r(x)\\leq\\|x\\|\\) ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)).\n(3) This was shown in the proof of (2).\n(4) The quasi-spectrum of an element depends only on the algebra, because invertibility in \\(A_1\\) is an algebraic property. So the spectral radius \\(r\\) does not depend on the norm. By (3), both norms satisfy \\(\\|x\\|^2=r(x^*x)\\). \\(\\square\\)\n\n**Example 1.4** (Normality is needed). In \\(M_2(\\mathbb C)\\), \\(E_{12}^2=0\\), so \\(r(E_{12})=0\\), while \\(\\|E_{12}\\|=1\\).\n\n**Proposition 1.5** (Spectra of adjoints, unitaries and self-adjoint elements). Let \\(A\\) be a C\\*-algebra.\n1. For every \\(x\\in A\\), \\(\\sigma'_A(x^*)=\\{\\bar\\lambda:\\lambda\\in\\sigma'_A(x)\\}\\).\n2. If \\(A\\) is unital and \\(u\\in U(A)\\), then \\(\\sigma_A(u)\\subseteq\\mathbb T=\\{\\lambda:|\\lambda|=1\\}\\).\n3. For \\(h\\in A_h\\), \\(\\sigma'_A(h)\\subseteq[-\\|h\\|,\\|h\\|]\\), and \\(\\sigma'_A(h)\\) contains \\(\\|h\\|\\) or \\(-\\|h\\|\\). If \\(A\\) is unital and nontrivial, the same holds for \\(\\sigma_A(h)\\).\n\n**Proof.** (1) was shown in the proof of Theorem 1.3(2).\n(2) In the zero algebra \\(\\sigma_A(u)=\\varnothing\\). Otherwise \\(\\|u\\|=\\|u^{-1}\\|=1\\) by Proposition 1.2(3). The spectrum of an element lies in the closed disc whose radius is its norm ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)), and \\(\\sigma_A(u^{-1})=\\{\\lambda^{-1}:\\lambda\\in\\sigma_A(u)\\}\\) ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)). So \\(\\sigma_A(u)\\) and \\(\\sigma_A(u^{-1})\\) lie in the closed unit disc, and every \\(\\lambda\\in\\sigma_A(u)\\) has \\(|\\lambda|\\leq1\\) and \\(|\\lambda|^{-1}\\leq1\\).\n(3) Work in the nontrivial unital C\\*-algebra \\(\\widetilde A\\). Put \\(u=\\exp(ih)=\\sum_n(ih)^n/n!\\) ([the exponential](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14)). The involution is continuous and conjugate linear, so applying it to the partial sums gives \\(u^*=\\exp(-ih)\\). Since \\(ih\\) and \\(-ih\\) commute, \\(\\exp(ih)\\exp(-ih)=\\exp(-ih)\\exp(ih)=1\\). So \\(u\\) is unitary, and by the [spectral mapping theorem](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-12) and (2),\n\\[\n\\{e^{i\\lambda}:\\lambda\\in\\sigma'_A(h)\\}=\\sigma_{\\widetilde A}(u)\\subseteq\\mathbb T .\n\\]\nSince \\(|e^{i\\lambda}|=e^{-\\operatorname{Im}\\lambda}\\), every \\(\\lambda\\in\\sigma'_A(h)\\) is real. By Theorem 1.3, \\(r(h)=\\|h\\|\\). The quasi-spectrum is compact and not empty ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)), so the supremum \\(\\|h\\|\\) of \\(|\\lambda|\\) over it is attained. For nontrivial unital \\(A\\), \\(\\sigma_A(h)\\) is not empty ([the spectrum is not empty](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-08)) and \\(\\sigma'_A(h)=\\sigma_A(h)\\cup\\{0\\}\\). If \\(h\\neq0\\), the point of modulus \\(\\|h\\|\\) lies in \\(\\sigma_A(h)\\); if \\(h=0\\), then \\(\\sigma_A(h)=\\{0\\}\\). \\(\\square\\)\n\n**Exercise 1.6** (easy; Powers of normal elements). Show that \\(\\|x^n\\|=\\|x\\|^n\\) for normal \\(x\\) and all \\(n\\geq1\\). Show by examples that the equality can fail for nonnormal \\(x\\), and that it can hold for all \\(n\\) without \\(x\\) being normal.\n\n*Solution.* \\(x^n\\) is normal, so \\(\\|x^n\\|=r(x^n)=r(x)^n=\\|x\\|^n\\), by Theorem 1.3 and the identity \\(r(x^n)=r(x)^n\\), which follows from the [spectral radius formula](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-09). For \\(E_{12}\\in M_2(\\mathbb C)\\), \\(\\|E_{12}^2\\|=0<1\\). The unilateral shift \\(S\\) on \\(\\ell^2(\\mathbb N)\\) is an isometry, so \\(\\|S^n\\|=1=\\|S\\|^n\\), but \\(S^*S=1\\neq SS^*\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-04",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "2. Commutative C\\*-algebras",
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      "full_conditions_and_proof": "### 2. Commutative C\\*-algebras\n\n**Theorem 2.1** (Commutative Gelfand–Naimark theorem). For an abelian C\\*-algebra \\(A\\), put \\(\\Omega=\\operatorname{Ch}(A)\\).\n1. Every character of \\(A\\) is a \\(*\\)-homomorphism: \\(\\omega(x^*)=\\overline{\\omega(x)}\\).\n2. The Gelfand representation \\(\\mathcal G(x)=\\hat x\\) is an isometric \\(*\\)-isomorphism of \\(A\\) onto \\(C_0(\\Omega)\\). In particular \\(A\\) is semisimple.\n3. If \\(A\\) is unital, \\(\\Omega\\) is compact and \\(A\\cong C(\\Omega)\\).\n\n\n**Proof.** Every element of \\(A\\) is normal. The [Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18) satisfies \\(\\|\\hat x\\|_\\infty=r(x)\\), and \\(r(x)=\\|x\\|\\) by Theorem 1.3. Thus \\(\\mathcal G\\) is an isometric homomorphism, and in particular injective. Let \\(h\\in A_h\\) and \\(\\omega\\in\\Omega\\). Then \\(\\omega(h)\\in\\sigma'_A(h)\\), because characters take their values in the quasi-spectrum ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)), and \\(\\sigma'_A(h)\\) is real by Proposition 1.5(3). For \\(x=x_1+ix_2\\) as in Proposition 1.2, \\(\\omega(x^*)=\\omega(x_1)-i\\omega(x_2)=\\overline{\\omega(x)}\\). This is (1), and it says \\(\\widehat{x^*}=\\overline{\\hat x}\\). The image \\(\\mathcal G(A)\\) is complete, hence closed. It is a self-adjoint subalgebra of \\(C_0(\\Omega)\\). It separates the points of \\(\\Omega\\), since distinct characters differ at some \\(x\\), and it vanishes nowhere, since characters are not zero. By the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09), \\(\\mathcal G(A)=C_0(\\Omega)\\). Part (3) holds because the character space of a unital commutative Banach algebra is compact ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). \\(\\square\\)\n\nFor \\(A=\\{0\\}\\) the statement reads \\(\\{0\\}\\cong C_0(\\varnothing)\\).\n\n**Proposition 2.2** (The characters of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be an LCH space, and let \\(\\operatorname{ev}_p(f)=f(p)\\) for \\(p\\in\\Omega\\) and \\(f\\in C_0(\\Omega)\\).\n1. \\(p\\mapsto\\operatorname{ev}_p\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\) with the weak\\* topology. Under it, the Gelfand transform of \\(f\\) is \\(f\\) itself.\n2. Two LCH spaces \\(\\Omega\\) and \\(\\Omega'\\) are homeomorphic if and only if \\(C_0(\\Omega)\\) and \\(C_0(\\Omega')\\) are isomorphic as algebras; a \\(*\\)-isomorphism is not needed.\n3. Every abelian C\\*-algebra \\(A\\) is \\(*\\)-isomorphic to \\(C_0(\\Omega)\\) for an LCH space \\(\\Omega\\), unique up to homeomorphism; one choice is \\(\\Omega=\\operatorname{Ch}(A)\\).\n\n**Proof.** (1) This is proved in [the characters of \\(C_0(\\Omega)\\)](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-19), by a compactness argument that uses no measure theory. Remark 2.3 gives a second proof.\n(2) If \\(\\Phi:C_0(\\Omega)\\to C_0(\\Omega')\\) is an algebra isomorphism, then \\(\\chi\\mapsto\\chi\\circ\\Phi\\) is a bijection of \\(\\operatorname{Ch}(C_0(\\Omega'))\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\). It and its inverse \\(\\chi'\\mapsto\\chi'\\circ\\Phi^{-1}\\) are weak\\* continuous, because they are continuous in each evaluation. Composing with the homeomorphisms of (1) gives a homeomorphism \\(\\Omega'\\to\\Omega\\). The converse is clear: a homeomorphism \\(\\tau\\) gives \\(f\\mapsto f\\circ\\tau\\).\n(3) Existence is Theorem 2.1; uniqueness is (2). \\(\\square\\)\n\n**Remark 2.3** (A second proof of (1), through measure theory). Let \\(\\rho\\) be a character of \\(C_0(\\Omega)\\). We show in four steps that \\(\\rho=\\operatorname{ev}_p\\) for some \\(p\\in\\Omega\\).\n- *\\(\\rho\\) is positive.* For \\(f\\geq0\\), \\(\\rho(f)=\\rho(f^{1/2})^2\\geq0\\), because \\(\\rho(f^{1/2})\\) is real by Theorem 2.1(1). So by the [Riesz representation theorem](haar-measure.md#oa-fnd-hm-01) there is a positive Radon measure \\(\\mu\\) on \\(\\Omega\\) with \\(\\rho(x)=\\int x\\,d\\mu\\) for \\(x\\in C_c(\\Omega)\\).\n- *\\(\\mu(\\Omega)=1\\).* By the same theorem, \\(\\mu(\\Omega)\\) is the supremum of \\(\\rho(k)\\) over \\(k\\in C_c(\\Omega)\\) with \\(0\\leq k\\leq1\\), and each such \\(\\rho(k)\\) is at most \\(\\|\\rho\\|\\leq1\\), because characters are contractive ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). Conversely, pick \\(g\\) with \\(\\rho(g)\\neq0\\) and put \\(f=\\bar gg/\\|g\\|^2\\). Then \\(0\\leq f\\leq1\\) and \\(\\rho(f)=|\\rho(g)|^2/\\|g\\|^2=t>0\\), so \\(\\rho(f^{1/n})=t^{1/n}\\to1\\) while \\(0\\leq f^{1/n}\\leq1\\). Here \\(\\rho(f^{1/n})=\\rho(f)^{1/n}\\), because the root \\(f^{1/n}\\) is a uniform limit of polynomials in \\(f\\) without constant term (the compact form of the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09)), and \\(\\rho\\) is continuous and multiplicative. For \\(\\eta>0\\), \\(k=\\max(f^{1/n}-\\eta,0)\\) lies in \\(C_c(\\Omega)\\), has \\(0\\leq k\\leq1\\) and \\(\\|k-f^{1/n}\\|\\leq\\eta\\), so \\(\\mu(\\Omega)\\geq\\rho(k)\\geq t^{1/n}-\\eta\\).\n- *The formula holds on \\(C_0(\\Omega)\\).* Since \\(\\mu\\) is finite, both sides of \\(\\rho(x)=\\int x\\,d\\mu\\) are continuous for the supremum norm, and \\(C_c(\\Omega)\\) is dense in \\(C_0(\\Omega)\\) ([spaces of continuous functions](stone-weierstrass-c0.md#oa-fnd-sw-01)).\n- *\\(\\mu\\) is a point mass.* With \\(\\mu(\\Omega)=1\\), \\(\\int|x-\\rho(x)|^2\\,d\\mu=\\rho(\\bar xx)-|\\rho(x)|^2=0\\) for every \\(x\\in C_0(\\Omega)\\). Let \\(S\\) be the set of points every open neighbourhood of which has positive measure. Each point outside \\(S\\) has an open null neighbourhood, so \\(\\Omega\\setminus S\\) is open. Every compact subset of this open set has a finite cover by those null neighbourhoods and hence has measure zero. Radon inner regularity on the open set gives \\(\\mu(\\Omega\\setminus S)=0\\). Since \\(\\mu(\\Omega)=1\\), \\(S\\) is nonempty. If \\(p\\in S\\) and \\(x(p)\\ne\\rho(x)\\), continuity gives an open neighbourhood of \\(p\\) on which \\(|x-\\rho(x)|^2\\) is bounded below by a positive number; its positive measure contradicts the displayed integral identity. Thus \\(x(p)=\\rho(x)\\) for every \\(p\\in S\\) and every \\(x\\in C_0(\\Omega)\\). Two distinct points of \\(S\\) would be separated by a compactly supported continuous function, by the earlier Urysohn lemma, which is impossible. Hence \\(S=\\{p\\}\\), its complement is null, and \\(\\mu=\\delta_p\\). Therefore \\(\\rho=\\operatorname{ev}_p\\).\n\nThe full argument above proves positivity, total mass and the extension from compactly supported functions before identifying the point mass.\n\n**Exercise 2.4** (hard; The Stone–Čech compactification). Let \\(\\Gamma\\) be a completely regular Hausdorff space and \\(A=C_b(\\Gamma)\\) with the supremum norm. For \\(\\gamma\\in\\Gamma\\) let \\(\\omega_\\gamma(x)=x(\\gamma)\\), and let \\(\\iota:\\Gamma\\to\\operatorname{Ch}(A)\\), \\(\\iota(\\gamma)=\\omega_\\gamma\\).\n(a) \\(A\\) is a C\\*-algebra with identity.\n(b) \\(\\iota\\) is a homeomorphism of \\(\\Gamma\\) onto a dense subset of \\(\\operatorname{Ch}(A)\\).\n(c) \\(\\iota(\\Gamma)\\) is open in \\(\\operatorname{Ch}(A)\\) if and only if \\(\\Gamma\\) is locally compact.\n(d) Every continuous map \\(f\\) of \\(\\Gamma\\) into a compact Hausdorff space \\(K\\) is \\(g\\circ\\iota\\) for exactly one continuous \\(g:\\operatorname{Ch}(A)\\to K\\).\n\n*Solution.* (a) \\(C_b(\\Gamma)\\) is a Banach space ([spaces of continuous functions](stone-weierstrass-c0.md#oa-fnd-sw-01)), closed under pointwise products and conjugation, with \\(\\sup|\\bar xx|=(\\sup|x|)^2\\), and the constant \\(1\\) is its identity.\n(b) \\(\\omega_\\gamma\\) is a character. \\(\\iota\\) is injective because points of a completely regular Hausdorff space are separated by bounded continuous functions. It is weak\\* continuous because each \\(\\gamma\\mapsto x(\\gamma)\\) is continuous. It is open onto its image: for \\(\\gamma\\in U\\) open, complete regularity gives \\(x\\in A\\) with \\(0\\leq x\\leq1\\), \\(x(\\gamma)=1\\) and \\(x=0\\) off \\(U\\); then \\(W=\\{\\omega:\\operatorname{Re}\\omega(x)>\\frac12\\}\\) is weak\\* open and \\(\\iota^{-1}(W)\\subseteq U\\) contains \\(\\gamma\\). *Density.* If some \\(\\omega_0\\) had a neighbourhood missing \\(\\iota(\\Gamma)\\), [Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16) on the compact Hausdorff space \\(\\operatorname{Ch}(A)\\) would give a continuous \\(F\\) with \\(F(\\omega_0)=1\\) and \\(F=0\\) on \\(\\overline{\\iota(\\Gamma)}\\). By Theorem 2.1, \\(F=\\hat x\\) for some \\(x\\in A\\); then \\(x(\\gamma)=F(\\omega_\\gamma)=0\\) for all \\(\\gamma\\), so \\(x=0\\) and \\(F=0\\), a contradiction.\n(c) If \\(\\iota(\\Gamma)\\) is open in the compact Hausdorff space \\(\\operatorname{Ch}(A)\\), it is locally compact, since open subspaces of compact Hausdorff spaces are ([locally compact spaces](stone-weierstrass-c0.md#oa-fnd-sw-15)); so \\(\\Gamma\\cong\\iota(\\Gamma)\\) is locally compact. Conversely let \\(\\Gamma\\) be locally compact and \\(\\gamma_0\\in\\Gamma\\). Urysohn's lemma in its locally compact form gives \\(x\\in C_c(\\Gamma)\\) with \\(0\\leq x\\leq1\\), \\(x(\\gamma_0)=1\\) and support in a compact set \\(C\\). The set \\(V=\\{\\omega:\\operatorname{Re}\\omega(x)>\\frac12\\}\\) is open and contains \\(\\omega_{\\gamma_0}\\). If \\(\\omega\\in V\\), density gives a net \\(\\omega_{\\gamma_\\alpha}\\to\\omega\\), eventually in \\(V\\), so eventually \\(x(\\gamma_\\alpha)>\\frac12\\) and \\(\\gamma_\\alpha\\in C\\); hence \\(\\omega\\) lies in the compact, hence closed, set \\(\\iota(C)\\). So \\(V\\subseteq\\iota(C)\\subseteq\\iota(\\Gamma)\\), and \\(\\iota(\\Gamma)\\) is open.\n(d) For \\(\\omega\\in\\operatorname{Ch}(A)\\), \\(\\varphi\\mapsto\\omega(\\varphi\\circ f)\\) is a character of \\(C(K)\\) (it is \\(1\\) at \\(1\\)), hence evaluation at a unique point \\(g(\\omega)\\in K\\) (Proposition 2.2). For \\(\\varphi\\in C(K)\\), \\(\\varphi\\circ g(\\omega)=\\omega(\\varphi\\circ f)\\) is continuous in \\(\\omega\\), and by Urysohn's lemma the topology of \\(K\\) is the weak topology defined by \\(C(K)\\), so \\(g\\) is continuous. Also \\(\\omega_\\gamma(\\varphi\\circ f)=\\varphi(f(\\gamma))\\), so \\(g(\\omega_\\gamma)=f(\\gamma)\\). Two continuous maps into a Hausdorff space that agree on a dense set agree everywhere, which gives uniqueness.\n\n*The Hausdorff condition.* In this lesson \"completely regular\" includes the Hausdorff (\\(T_1\\)) axiom. Without that axiom, (b) fails: on a two-point space with the indiscrete topology, every continuous function is constant, and \\(\\iota\\) is not injective.\n\n**Exercise 2.5** (medium; Separability). Let \\(A\\) be an abelian C\\*-algebra with character space \\(\\Omega\\). Show that \\(A\\) has a countable dense subset exactly when \\(\\Omega\\) has a countable base.\n\n*Solution.* By Theorem 2.1, \\(A=C_0(\\Omega)\\). If \\(\\Omega\\) has a countable base, its one-point compactification \\(\\Omega_\\infty\\) is compact metrizable (the metrizability theorem in [Urysohn's lemma, complete regularity and metrizability](stone-weierstrass-c0.md#oa-fnd-sw-16)). So \\(C(\\Omega_\\infty)\\) is separable ([density results](stone-weierstrass-c0.md#oa-fnd-sw-17)), and so is its subspace \\(C_0(\\Omega)\\), the functions on \\(\\Omega_\\infty\\) that vanish at \\(\\infty\\) ([\\(C_0(X)\\) and the one-point compactification](stone-weierstrass-c0.md#oa-fnd-sw-03)). Conversely, let \\((x_n)\\) be dense in \\(C_0(\\Omega)\\), and put \\(V_n=\\{|x_n|>\\frac23\\}\\), an open set. Let \\(p\\in U\\) with \\(U\\) open. Urysohn's lemma gives \\(f\\in C_c(\\Omega)\\) with \\(0\\leq f\\leq1\\), \\(f(p)=1\\) and support in \\(U\\). Take \\(n\\) with \\(\\|x_n-f\\|<\\frac13\\). Then \\(p\\in V_n\\), and on \\(V_n\\), \\(|f|>\\frac13\\), so \\(V_n\\subseteq U\\). So \\((V_n)\\) is a countable base.\n\n",
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    {
      "id": "OA-FND-CF-05",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "2. Commutative C\\*-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 2. Commutative C\\*-algebras\n\n**Theorem 2.1** (Commutative Gelfand–Naimark theorem). For an abelian C\\*-algebra \\(A\\), put \\(\\Omega=\\operatorname{Ch}(A)\\).\n1. Every character of \\(A\\) is a \\(*\\)-homomorphism: \\(\\omega(x^*)=\\overline{\\omega(x)}\\).\n2. The Gelfand representation \\(\\mathcal G(x)=\\hat x\\) is an isometric \\(*\\)-isomorphism of \\(A\\) onto \\(C_0(\\Omega)\\). In particular \\(A\\) is semisimple.\n3. If \\(A\\) is unital, \\(\\Omega\\) is compact and \\(A\\cong C(\\Omega)\\).\n\n\n**Proof.** Every element of \\(A\\) is normal. The [Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18) satisfies \\(\\|\\hat x\\|_\\infty=r(x)\\), and \\(r(x)=\\|x\\|\\) by Theorem 1.3. Thus \\(\\mathcal G\\) is an isometric homomorphism, and in particular injective. Let \\(h\\in A_h\\) and \\(\\omega\\in\\Omega\\). Then \\(\\omega(h)\\in\\sigma'_A(h)\\), because characters take their values in the quasi-spectrum ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)), and \\(\\sigma'_A(h)\\) is real by Proposition 1.5(3). For \\(x=x_1+ix_2\\) as in Proposition 1.2, \\(\\omega(x^*)=\\omega(x_1)-i\\omega(x_2)=\\overline{\\omega(x)}\\). This is (1), and it says \\(\\widehat{x^*}=\\overline{\\hat x}\\). The image \\(\\mathcal G(A)\\) is complete, hence closed. It is a self-adjoint subalgebra of \\(C_0(\\Omega)\\). It separates the points of \\(\\Omega\\), since distinct characters differ at some \\(x\\), and it vanishes nowhere, since characters are not zero. By the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09), \\(\\mathcal G(A)=C_0(\\Omega)\\). Part (3) holds because the character space of a unital commutative Banach algebra is compact ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). \\(\\square\\)\n\nFor \\(A=\\{0\\}\\) the statement reads \\(\\{0\\}\\cong C_0(\\varnothing)\\).\n\n**Proposition 2.2** (The characters of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be an LCH space, and let \\(\\operatorname{ev}_p(f)=f(p)\\) for \\(p\\in\\Omega\\) and \\(f\\in C_0(\\Omega)\\).\n1. \\(p\\mapsto\\operatorname{ev}_p\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\) with the weak\\* topology. Under it, the Gelfand transform of \\(f\\) is \\(f\\) itself.\n2. Two LCH spaces \\(\\Omega\\) and \\(\\Omega'\\) are homeomorphic if and only if \\(C_0(\\Omega)\\) and \\(C_0(\\Omega')\\) are isomorphic as algebras; a \\(*\\)-isomorphism is not needed.\n3. Every abelian C\\*-algebra \\(A\\) is \\(*\\)-isomorphic to \\(C_0(\\Omega)\\) for an LCH space \\(\\Omega\\), unique up to homeomorphism; one choice is \\(\\Omega=\\operatorname{Ch}(A)\\).\n\n**Proof.** (1) This is proved in [the characters of \\(C_0(\\Omega)\\)](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-19), by a compactness argument that uses no measure theory. Remark 2.3 gives a second proof.\n(2) If \\(\\Phi:C_0(\\Omega)\\to C_0(\\Omega')\\) is an algebra isomorphism, then \\(\\chi\\mapsto\\chi\\circ\\Phi\\) is a bijection of \\(\\operatorname{Ch}(C_0(\\Omega'))\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\). It and its inverse \\(\\chi'\\mapsto\\chi'\\circ\\Phi^{-1}\\) are weak\\* continuous, because they are continuous in each evaluation. Composing with the homeomorphisms of (1) gives a homeomorphism \\(\\Omega'\\to\\Omega\\). The converse is clear: a homeomorphism \\(\\tau\\) gives \\(f\\mapsto f\\circ\\tau\\).\n(3) Existence is Theorem 2.1; uniqueness is (2). \\(\\square\\)\n\n**Remark 2.3** (A second proof of (1), through measure theory). Let \\(\\rho\\) be a character of \\(C_0(\\Omega)\\). We show in four steps that \\(\\rho=\\operatorname{ev}_p\\) for some \\(p\\in\\Omega\\).\n- *\\(\\rho\\) is positive.* For \\(f\\geq0\\), \\(\\rho(f)=\\rho(f^{1/2})^2\\geq0\\), because \\(\\rho(f^{1/2})\\) is real by Theorem 2.1(1). So by the [Riesz representation theorem](haar-measure.md#oa-fnd-hm-01) there is a positive Radon measure \\(\\mu\\) on \\(\\Omega\\) with \\(\\rho(x)=\\int x\\,d\\mu\\) for \\(x\\in C_c(\\Omega)\\).\n- *\\(\\mu(\\Omega)=1\\).* By the same theorem, \\(\\mu(\\Omega)\\) is the supremum of \\(\\rho(k)\\) over \\(k\\in C_c(\\Omega)\\) with \\(0\\leq k\\leq1\\), and each such \\(\\rho(k)\\) is at most \\(\\|\\rho\\|\\leq1\\), because characters are contractive ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). Conversely, pick \\(g\\) with \\(\\rho(g)\\neq0\\) and put \\(f=\\bar gg/\\|g\\|^2\\). Then \\(0\\leq f\\leq1\\) and \\(\\rho(f)=|\\rho(g)|^2/\\|g\\|^2=t>0\\), so \\(\\rho(f^{1/n})=t^{1/n}\\to1\\) while \\(0\\leq f^{1/n}\\leq1\\). Here \\(\\rho(f^{1/n})=\\rho(f)^{1/n}\\), because the root \\(f^{1/n}\\) is a uniform limit of polynomials in \\(f\\) without constant term (the compact form of the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09)), and \\(\\rho\\) is continuous and multiplicative. For \\(\\eta>0\\), \\(k=\\max(f^{1/n}-\\eta,0)\\) lies in \\(C_c(\\Omega)\\), has \\(0\\leq k\\leq1\\) and \\(\\|k-f^{1/n}\\|\\leq\\eta\\), so \\(\\mu(\\Omega)\\geq\\rho(k)\\geq t^{1/n}-\\eta\\).\n- *The formula holds on \\(C_0(\\Omega)\\).* Since \\(\\mu\\) is finite, both sides of \\(\\rho(x)=\\int x\\,d\\mu\\) are continuous for the supremum norm, and \\(C_c(\\Omega)\\) is dense in \\(C_0(\\Omega)\\) ([spaces of continuous functions](stone-weierstrass-c0.md#oa-fnd-sw-01)).\n- *\\(\\mu\\) is a point mass.* With \\(\\mu(\\Omega)=1\\), \\(\\int|x-\\rho(x)|^2\\,d\\mu=\\rho(\\bar xx)-|\\rho(x)|^2=0\\) for every \\(x\\in C_0(\\Omega)\\). Let \\(S\\) be the set of points every open neighbourhood of which has positive measure. Each point outside \\(S\\) has an open null neighbourhood, so \\(\\Omega\\setminus S\\) is open. Every compact subset of this open set has a finite cover by those null neighbourhoods and hence has measure zero. Radon inner regularity on the open set gives \\(\\mu(\\Omega\\setminus S)=0\\). Since \\(\\mu(\\Omega)=1\\), \\(S\\) is nonempty. If \\(p\\in S\\) and \\(x(p)\\ne\\rho(x)\\), continuity gives an open neighbourhood of \\(p\\) on which \\(|x-\\rho(x)|^2\\) is bounded below by a positive number; its positive measure contradicts the displayed integral identity. Thus \\(x(p)=\\rho(x)\\) for every \\(p\\in S\\) and every \\(x\\in C_0(\\Omega)\\). Two distinct points of \\(S\\) would be separated by a compactly supported continuous function, by the earlier Urysohn lemma, which is impossible. Hence \\(S=\\{p\\}\\), its complement is null, and \\(\\mu=\\delta_p\\). Therefore \\(\\rho=\\operatorname{ev}_p\\).\n\nThe full argument above proves positivity, total mass and the extension from compactly supported functions before identifying the point mass.\n\n**Exercise 2.4** (hard; The Stone–Čech compactification). Let \\(\\Gamma\\) be a completely regular Hausdorff space and \\(A=C_b(\\Gamma)\\) with the supremum norm. For \\(\\gamma\\in\\Gamma\\) let \\(\\omega_\\gamma(x)=x(\\gamma)\\), and let \\(\\iota:\\Gamma\\to\\operatorname{Ch}(A)\\), \\(\\iota(\\gamma)=\\omega_\\gamma\\).\n(a) \\(A\\) is a C\\*-algebra with identity.\n(b) \\(\\iota\\) is a homeomorphism of \\(\\Gamma\\) onto a dense subset of \\(\\operatorname{Ch}(A)\\).\n(c) \\(\\iota(\\Gamma)\\) is open in \\(\\operatorname{Ch}(A)\\) if and only if \\(\\Gamma\\) is locally compact.\n(d) Every continuous map \\(f\\) of \\(\\Gamma\\) into a compact Hausdorff space \\(K\\) is \\(g\\circ\\iota\\) for exactly one continuous \\(g:\\operatorname{Ch}(A)\\to K\\).\n\n*Solution.* (a) \\(C_b(\\Gamma)\\) is a Banach space ([spaces of continuous functions](stone-weierstrass-c0.md#oa-fnd-sw-01)), closed under pointwise products and conjugation, with \\(\\sup|\\bar xx|=(\\sup|x|)^2\\), and the constant \\(1\\) is its identity.\n(b) \\(\\omega_\\gamma\\) is a character. \\(\\iota\\) is injective because points of a completely regular Hausdorff space are separated by bounded continuous functions. It is weak\\* continuous because each \\(\\gamma\\mapsto x(\\gamma)\\) is continuous. It is open onto its image: for \\(\\gamma\\in U\\) open, complete regularity gives \\(x\\in A\\) with \\(0\\leq x\\leq1\\), \\(x(\\gamma)=1\\) and \\(x=0\\) off \\(U\\); then \\(W=\\{\\omega:\\operatorname{Re}\\omega(x)>\\frac12\\}\\) is weak\\* open and \\(\\iota^{-1}(W)\\subseteq U\\) contains \\(\\gamma\\). *Density.* If some \\(\\omega_0\\) had a neighbourhood missing \\(\\iota(\\Gamma)\\), [Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16) on the compact Hausdorff space \\(\\operatorname{Ch}(A)\\) would give a continuous \\(F\\) with \\(F(\\omega_0)=1\\) and \\(F=0\\) on \\(\\overline{\\iota(\\Gamma)}\\). By Theorem 2.1, \\(F=\\hat x\\) for some \\(x\\in A\\); then \\(x(\\gamma)=F(\\omega_\\gamma)=0\\) for all \\(\\gamma\\), so \\(x=0\\) and \\(F=0\\), a contradiction.\n(c) If \\(\\iota(\\Gamma)\\) is open in the compact Hausdorff space \\(\\operatorname{Ch}(A)\\), it is locally compact, since open subspaces of compact Hausdorff spaces are ([locally compact spaces](stone-weierstrass-c0.md#oa-fnd-sw-15)); so \\(\\Gamma\\cong\\iota(\\Gamma)\\) is locally compact. Conversely let \\(\\Gamma\\) be locally compact and \\(\\gamma_0\\in\\Gamma\\). Urysohn's lemma in its locally compact form gives \\(x\\in C_c(\\Gamma)\\) with \\(0\\leq x\\leq1\\), \\(x(\\gamma_0)=1\\) and support in a compact set \\(C\\). The set \\(V=\\{\\omega:\\operatorname{Re}\\omega(x)>\\frac12\\}\\) is open and contains \\(\\omega_{\\gamma_0}\\). If \\(\\omega\\in V\\), density gives a net \\(\\omega_{\\gamma_\\alpha}\\to\\omega\\), eventually in \\(V\\), so eventually \\(x(\\gamma_\\alpha)>\\frac12\\) and \\(\\gamma_\\alpha\\in C\\); hence \\(\\omega\\) lies in the compact, hence closed, set \\(\\iota(C)\\). So \\(V\\subseteq\\iota(C)\\subseteq\\iota(\\Gamma)\\), and \\(\\iota(\\Gamma)\\) is open.\n(d) For \\(\\omega\\in\\operatorname{Ch}(A)\\), \\(\\varphi\\mapsto\\omega(\\varphi\\circ f)\\) is a character of \\(C(K)\\) (it is \\(1\\) at \\(1\\)), hence evaluation at a unique point \\(g(\\omega)\\in K\\) (Proposition 2.2). For \\(\\varphi\\in C(K)\\), \\(\\varphi\\circ g(\\omega)=\\omega(\\varphi\\circ f)\\) is continuous in \\(\\omega\\), and by Urysohn's lemma the topology of \\(K\\) is the weak topology defined by \\(C(K)\\), so \\(g\\) is continuous. Also \\(\\omega_\\gamma(\\varphi\\circ f)=\\varphi(f(\\gamma))\\), so \\(g(\\omega_\\gamma)=f(\\gamma)\\). Two continuous maps into a Hausdorff space that agree on a dense set agree everywhere, which gives uniqueness.\n\n*The Hausdorff condition.* In this lesson \"completely regular\" includes the Hausdorff (\\(T_1\\)) axiom. Without that axiom, (b) fails: on a two-point space with the indiscrete topology, every continuous function is constant, and \\(\\iota\\) is not injective.\n\n**Exercise 2.5** (medium; Separability). Let \\(A\\) be an abelian C\\*-algebra with character space \\(\\Omega\\). Show that \\(A\\) has a countable dense subset exactly when \\(\\Omega\\) has a countable base.\n\n*Solution.* By Theorem 2.1, \\(A=C_0(\\Omega)\\). If \\(\\Omega\\) has a countable base, its one-point compactification \\(\\Omega_\\infty\\) is compact metrizable (the metrizability theorem in [Urysohn's lemma, complete regularity and metrizability](stone-weierstrass-c0.md#oa-fnd-sw-16)). So \\(C(\\Omega_\\infty)\\) is separable ([density results](stone-weierstrass-c0.md#oa-fnd-sw-17)), and so is its subspace \\(C_0(\\Omega)\\), the functions on \\(\\Omega_\\infty\\) that vanish at \\(\\infty\\) ([\\(C_0(X)\\) and the one-point compactification](stone-weierstrass-c0.md#oa-fnd-sw-03)). Conversely, let \\((x_n)\\) be dense in \\(C_0(\\Omega)\\), and put \\(V_n=\\{|x_n|>\\frac23\\}\\), an open set. Let \\(p\\in U\\) with \\(U\\) open. Urysohn's lemma gives \\(f\\in C_c(\\Omega)\\) with \\(0\\leq f\\leq1\\), \\(f(p)=1\\) and support in \\(U\\). Take \\(n\\) with \\(\\|x_n-f\\|<\\frac13\\). Then \\(p\\in V_n\\), and on \\(V_n\\), \\(|f|>\\frac13\\), so \\(V_n\\subseteq U\\). So \\((V_n)\\) is a countable base.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CF-06",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "3. C\\*-subalgebras and spectral permanence",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
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      "full_conditions_and_proof": "### 3. C\\*-subalgebras and spectral permanence\n\n**Definition 3.1.** In a C\\*-algebra \\(A\\), a *C\\*-subalgebra* is a norm-closed subalgebra \\(B\\) with \\(B^*=B\\); with the inherited structure, \\(B\\) is itself a C\\*-algebra. If \\(A\\) is unital, \\(B\\) is a *unital C\\*-subalgebra* when \\(1_A\\in B\\). Intersections of C\\*-subalgebras are C\\*-subalgebras. So every \\(E\\subseteq A\\) lies in a smallest C\\*-subalgebra \\(C^*(E)\\), the C\\*-subalgebra *generated* by \\(E\\); for unital \\(A\\), \\(C^*(1,E)\\) is the smallest unital one. \\(C^*(E)\\) is the closure of the set of noncommutative polynomials without constant term in the elements of \\(E\\cup E^*\\). If \\(x\\) is normal, \\(C^*(x)\\) and \\(C^*(1,x)\\) are commutative, because \\(x\\), \\(x^*\\) and \\(1\\) commute and closures of commutative algebras are commutative.\n\n**Theorem 3.2** (Spectral permanence). Let \\(A\\) be a C\\*-algebra, \\(B\\subseteq A\\) a C\\*-subalgebra, and \\(x\\in B\\).\n1. If \\(A\\) is unital and \\(1_A\\in B\\), then \\(\\sigma_B(x)=\\sigma_A(x)\\). Equivalently, an element of \\(B\\) that is invertible in \\(A\\) has its inverse in \\(B\\).\n2. \\(\\sigma'_B(x)=\\sigma'_A(x)\\).\n3. If \\(A\\) and \\(B\\) are both unital, possibly with different identities, then \\(\\sigma_B(x)\\cup\\{0\\}=\\sigma_A(x)\\cup\\{0\\}\\).\n\nExample 3.3 explains why spectra in algebras with different identities must be compared after adjoining zero, as in (3).\n\n**Proof.** (1) Always \\(\\sigma_A(x)\\subseteq\\sigma_B(x)\\), because an inverse in \\(B\\) is an inverse in \\(A\\). It is enough to show that \\(y\\in B\\cap G(A)\\) implies \\(y^{-1}\\in B\\); apply this to \\(y=\\lambda-x\\).\n*First, \\(y\\) self-adjoint.* Then \\(0\\notin\\sigma_A(y)\\), and \\(\\sigma_B(y)\\subseteq\\mathbb R\\) by Proposition 1.5(3), applied in the C\\*-algebra \\(B\\). So \\(y-i/n\\) is invertible in \\(B\\) for every \\(n\\geq1\\). In \\(A\\), \\(y-i/n\\to y\\in G(A)\\), and inversion is continuous on \\(G(A)\\) ([invertible elements and the Neumann series](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-02)). So \\((y-i/n)^{-1}\\to y^{-1}\\), and \\(y^{-1}\\in B\\) because \\(B\\) is closed.\n*General \\(y\\).* The elements \\(y^*y\\) and \\(yy^*\\) of \\(B\\) are self-adjoint and invertible in \\(A\\). By the first step their inverses lie in \\(B\\). Then \\((y^*y)^{-1}y^*\\) is a left inverse and \\(y^*(yy^*)^{-1}\\) a right inverse of \\(y\\) in \\(B\\), so \\(y\\) is invertible in \\(B\\).\n(2) Let \\(B'=B+\\mathbb C1\\subseteq\\widetilde A\\). It is a unital C\\*-subalgebra of \\(\\widetilde A\\): it is \\(*\\)-closed, and closed because \\(B\\) is closed and has codimension at most one in it. Since \\(q(1)=1\\) and \\(q(B)=0\\), \\(B\\cap\\mathbb C1=\\{0\\}\\). So \\(b+\\lambda1\\mapsto(b,\\lambda)\\) is an algebra isomorphism of \\(B'\\) onto the unitization \\(B_1\\) of \\(B\\), and spectra agree under algebra isomorphisms. By (1) in \\(\\widetilde A\\),\n\\[\n\\sigma'_B(x)=\\sigma_{B_1}(x)=\\sigma_{B'}(x)=\\sigma_{\\widetilde A}(x)=\\sigma'_A(x).\n\\]\n(3) combines (2) with the relation \\(\\sigma'=\\sigma\\cup\\{0\\}\\) for unital algebras ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)). \\(\\square\\)\n\n**Example 3.3** (Why an identity is adjoined to every algebra). Part (2) needs the quasi-spectrum of a unital algebra to be computed after adjoining a new identity, as in the Conventions. If instead \\(\\sigma'_B=\\sigma_B\\) for unital \\(B\\), then (2) fails. Take \\(A=c_0\\), the null sequences, which is not unital, \\(B=\\mathbb Ce_1\\), which is unital with identity \\(e_1\\), and \\(x=e_1\\). Then \\(\\sigma'_A(e_1)=\\{0,1\\}\\) but \\(\\sigma_B(e_1)=\\{1\\}\\).\n\n**Example 3.4** (Closure under the involution is needed). Let \\(A=C(\\mathbb T)\\), and let \\(B\\) be the closure in \\(A\\) of the polynomials in \\(z\\). It is a closed unital subalgebra, but not \\(*\\)-closed. By the maximum modulus principle (Theorem 3.6 and Exercise 4 of [Cauchy's theorem for cycles and its consequences](cauchy-s-theorem-for-cycles-and-its-consequences.md)), a sequence of polynomials that converges uniformly on \\(\\mathbb T\\) converges uniformly on the closed disc \\(\\bar{\\mathbb D}\\); so each \\(g\\in B\\) extends to a function \\(G\\) continuous on \\(\\bar{\\mathbb D}\\) and holomorphic inside. If \\(|\\lambda|<1\\) and \\((z-\\lambda)g=1\\) on \\(\\mathbb T\\), then \\((w-\\lambda)G(w)-1\\) is holomorphic in \\(\\mathbb D\\), continuous on \\(\\bar{\\mathbb D}\\) and zero on \\(\\mathbb T\\), hence zero; at \\(w=\\lambda\\) this says \\(-1=0\\). So \\(\\sigma_B(z)\\) contains the open disc, and \\(\\sigma_B(z)=\\bar{\\mathbb D}\\), while \\(\\sigma_A(z)=\\mathbb T\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CF-12",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "4. Homomorphisms: contractivity, isometry and automatic continuity",
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        "line": 159,
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      },
      "full_conditions_and_proof": "### 4. Homomorphisms: contractivity, isometry and automatic continuity\n\n**Definition 4.1.** A *\\(*\\)-homomorphism* between algebras with involution is an algebra homomorphism \\(\\pi\\) with \\(\\pi(x^*)=\\pi(x)^*\\).\n\n**Theorem 4.2** (\\(*\\)-homomorphisms are contractive). Let \\(A\\) be a Banach algebra with an involution that is not assumed to be continuous, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) a \\(*\\)-homomorphism. Then for every \\(x\\in A\\)\n\\[\n\\begin{gathered}\n\\sigma'_B(\\pi(x))\\\\\n\\subseteq\\sigma'_A(x),\\\\\n\\|\\pi(x)\\|^2\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\|\\\\\n\\leq\\|x^*\\|\\,\\|x\\| .\n\\end{gathered}\n\\tag{4.1}\n\\]\nConsequently:\n1. if \\(\\|x^*\\|\\leq C\\|x\\|\\) for all \\(x\\), then \\(\\|\\pi(x)\\|\\leq C^{1/2}\\|x\\|\\);\n2. if the involution is isometric, then \\(\\|\\pi(x)\\|\\leq\\|x\\|\\);\n3. in particular every \\(*\\)-homomorphism between C\\*-algebras is contractive, whatever is assumed about its continuity.\n\n**Proof.** The map \\(\\pi_1(a+\\lambda)=\\pi(a)+\\lambda\\) is a unital homomorphism from \\(A_1\\) to \\(\\widetilde B\\), and unital homomorphisms map invertible elements to invertible elements. So \\(\\lambda\\notin\\sigma'_A(x)\\) implies \\(\\lambda\\notin\\sigma'_B(\\pi(x))\\), and \\(r_B(\\pi(y))\\leq r_A(y)\\) for all \\(y\\). The element \\(\\pi(x)^*\\pi(x)=\\pi(x^*x)\\) is self-adjoint, so by Theorem 1.3(1)\n\\[\n\\begin{gathered}\n\\|\\pi(x)\\|^2\\\\\n=\\|\\pi(x^*x)\\|\\\\\n=r_B(\\pi(x^*x))\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\| ,\n\\end{gathered}\n\\]\nwhere the last step uses \\(r_A(y)\\leq\\|y\\|\\) ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)). \\(\\square\\)\n\n**Example 4.3** (An isometric involution is needed for contractivity). Let \\(t>1\\), \\(S=\\operatorname{diag}(1,t)\\in M_2(\\mathbb C)\\), and \\(\\|x\\|_S=\\|SxS^{-1}\\|\\) (operator norm). This is a complete algebra norm on \\(M_2(\\mathbb C)\\), and the usual adjoint is an involution for it, continuous but not isometric. The identity map \\(\\pi\\) from \\((M_2(\\mathbb C),\\|\\cdot\\|_S)\\) to the C\\*-algebra \\(M_2(\\mathbb C)\\) is a \\(*\\)-homomorphism. Since \\(SE_{12}S^{-1}=t^{-1}E_{12}\\) and \\(SE_{21}S^{-1}=tE_{21}\\), we get \\(\\|\\pi(E_{12})\\|=1>t^{-1}=\\|E_{12}\\|_S\\). The bound (4.1) is sharp here: \\(\\|E_{12}^*\\|_S\\|E_{12}\\|_S=t\\cdot t^{-1}=1\\).\n\n**Theorem 4.4** (Injective homomorphisms do not shrink normal elements). Let \\(A\\) be a C\\*-algebra, \\(B\\) a Banach algebra and \\(\\pi:A\\to B\\) an injective algebra homomorphism. Neither continuity nor compatibility with an involution is assumed.\n1. \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\in A\\).\n2. If \\(B\\) has an involution and \\(\\pi\\) is a \\(*\\)-homomorphism, then \\(\\|x\\|^2\\leq\\|\\pi(x)^*\\|\\,\\|\\pi(x)\\|\\) for all \\(x\\). If moreover the involution of \\(B\\) is isometric, then \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for all \\(x\\).\n3. If, in (2), \\(\\pi(A)\\) is closed, the last inequality also follows from Theorem 4.2 applied to \\(\\pi^{-1}:\\pi(A)\\to A\\).\n\nThe proof below uses the closure of \\(\\pi(C^*(h))\\), so it makes no continuity assumption on \\(\\pi\\).\n\n**Proof.** (1) Let \\(x\\) be normal. Put \\(A_0=C^*(x)\\), a commutative C\\*-algebra (Definition 3.1), and let \\(B_0\\) be the closure of \\(\\pi(A_0)\\) in \\(B\\), a commutative Banach algebra. Adjoin identities: \\(\\widetilde{A_0}\\) is a commutative unital C\\*-algebra (Conventions), \\(\\widetilde{B_0}=B_0\\oplus\\mathbb C\\) with the sum norm is a commutative unital Banach algebra ([adjoining an identity](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-04)), and \\(\\tilde\\pi(a+\\lambda)=\\pi(a)+\\lambda\\) is an injective unital homomorphism. The character spaces \\(X=\\operatorname{Ch}(\\widetilde{A_0})\\) and \\(Y=\\operatorname{Ch}(\\widetilde{B_0})\\) are compact, because the algebras are unital ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). The map \\(\\tau(\\chi)=\\chi\\circ\\tilde\\pi\\) from \\(Y\\) to \\(X\\) is weak\\* continuous, so \\(F=\\tau(Y)\\) is compact, hence closed.\n*Claim: \\(F=X\\).* Suppose \\(\\omega_0\\in X\\setminus F\\). The space \\(X\\) is compact Hausdorff, hence regular, so it has an open \\(U\\ni\\omega_0\\) whose closure misses \\(F\\). By Urysohn's lemma and Theorem 2.1, there are \\(a,b\\in\\widetilde{A_0}\\) with \\(\\hat a(\\omega_0)=1\\), \\(\\hat a=0\\) off \\(U\\), \\(\\hat b=1\\) on \\(F\\) and \\(\\hat b=0\\) on \\(\\overline U\\). Then \\(\\hat a\\hat b=0\\), so \\(ab=0\\), and \\(a\\neq0\\). For every \\(\\chi\\in Y\\), \\(\\chi(\\tilde\\pi(b))=\\hat b(\\tau\\chi)=1\\). So no character of \\(\\widetilde{B_0}\\) vanishes at \\(\\tilde\\pi(b)\\), and \\(\\tilde\\pi(b)\\) is invertible, since in a unital commutative Banach algebra an element at which no character vanishes is invertible ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). Hence \\(\\tilde\\pi(a)=\\tilde\\pi(ab)\\tilde\\pi(b)^{-1}=0\\), and \\(a=0\\) by injectivity, a contradiction.\nNow, in a commutative Banach algebra the spectral radius is the largest modulus of the Gelfand transform ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). Using this in both algebras, the fact that \\(\\tau\\) maps \\(Y\\) onto \\(X\\), and Theorem 1.3 in \\(\\widetilde{A_0}\\),\n\\[\n\\begin{gathered}\n\\|\\pi(x)\\|\\\\\n\\geq r_{\\widetilde{B_0}}(\\pi(x))\\\\\n=\\max_{\\chi\\in Y}|\\chi(\\tilde\\pi(x))|\\\\\n=\\max_{\\omega\\in X}|\\omega(x)|\\\\\n=r_{\\widetilde{A_0}}(x)\\\\\n=\\|x\\| .\n\\end{gathered}\n\\]\n(2) \\(x^*x\\) is normal, so by (1), \\[\n\\begin{gathered}\n\\|x\\|^2\\\\\n=\\|x^*x\\|\\\\\n\\leq\\|\\pi(x^*x)\\|\\\\\n=\\|\\pi(x)^*\\pi(x)\\|\\\\\n\\leq\\|\\pi(x)^*\\|\\|\\pi(x)\\|.\n\\end{gathered}\n\\]\n(3) \\(\\pi(A)\\) is then a closed \\(*\\)-subalgebra of \\(B\\), an involutive Banach algebra, and \\(\\pi^{-1}\\) is a \\(*\\)-homomorphism from it to the C\\*-algebra \\(A\\). By Theorem 4.2, \\(\\|x\\|=\\|\\pi^{-1}(\\pi(x))\\|\\leq\\|\\pi(x)\\|\\). \\(\\square\\)\n\n**Example 4.5** (Normality is needed in (1)). With \\(S=\\operatorname{diag}(1,t)\\), \\(t>1\\), the map \\(\\pi(x)=SxS^{-1}\\) is an algebra automorphism of the C\\*-algebra \\(M_2(\\mathbb C)\\), not \\(*\\)-preserving. It satisfies \\(\\|\\pi(E_{12})\\|=t^{-1}<1=\\|E_{12}\\|\\).\n\n**Corollary 4.6.** An injective \\(*\\)-homomorphism from a C\\*-algebra into a C\\*-algebra is isometric. Its range is closed.\n\n**Proof.** The map is contractive by Theorem 4.2 and does not decrease norms by Theorem 4.4(2). An isometric image of a complete space is complete, hence closed. \\(\\square\\)\n\n**Theorem 4.7** (Automatic continuity). Let \\(A\\) be a Banach algebra, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) an injective algebra homomorphism whose range is self-adjoint: \\(\\pi(A)^*=\\pi(A)\\). Then \\(\\pi\\) is continuous.\n\n**Proof.** For \\(x\\in A\\), \\(\\pi(x)^*\\) lies in \\(\\pi(A)\\), so \\(x^\\sharp=\\pi^{-1}(\\pi(x)^*)\\) is well defined. It is an involution on \\(A\\): it is conjugate linear, \\((xy)^\\sharp=\\pi^{-1}(\\pi(y)^*\\pi(x)^*)=y^\\sharp x^\\sharp\\), and \\(x^{\\sharp\\sharp}=x\\). By construction \\(\\pi\\) is a \\(*\\)-homomorphism for \\(\\sharp\\). By (4.1), which does not need a continuous involution,\n\\[\n\\|\\pi(x)\\|^2\\leq\\|x^\\sharp\\|\\,\\|x\\|\\qquad(x\\in A).\n\\tag{4.2}\n\\]\nWe show that \\(\\sharp\\) has a closed graph. Let \\(x_n\\to x\\) and \\(x_n^\\sharp\\to y\\). By (4.2),\n\\[\n\\begin{gathered}\n\\|\\pi(x)-\\pi(x_n)\\|^2\\\\\n\\leq\\|x^\\sharp-x_n^\\sharp\\|\\,\\|x-x_n\\|\\to0,\\\\\n\\|\\pi(y)-\\pi(x_n^\\sharp)\\|^2\\\\\n\\leq\\|y^\\sharp-x_n\\|\\,\\|y-x_n^\\sharp\\|\\to0 ,\n\\end{gathered}\n\\]\nbecause in each product the first factor stays bounded and the second tends to \\(0\\). Since \\(\\pi(x_n^\\sharp)=\\pi(x_n)^*\\to\\pi(x)^*\\), we get \\(\\pi(y)=\\pi(x)^*=\\pi(x^\\sharp)\\), and \\(y=x^\\sharp\\). By the closed graph theorem, applied to the real-linear map \\(\\sharp\\), there is \\(k\\) with \\(\\|x^\\sharp\\|\\leq k\\|x\\|\\). Then (4.2) gives \\(\\|\\pi(x)\\|\\leq k^{1/2}\\|x\\|\\). \\(\\square\\)\n\n**Corollary 4.8.** Let \\(\\pi\\) be an algebra isomorphism of a C\\*-algebra \\(A\\) onto a C\\*-algebra \\(B\\), not assumed to preserve the involution. Then \\(\\pi\\) and \\(\\pi^{-1}\\) are continuous: \\(c\\|x\\|\\leq\\|\\pi(x)\\|\\leq C\\|x\\|\\) with constants \\(c,C>0\\). Moreover \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\).\n\n**Proof.** The range \\(B\\) is self-adjoint, so \\(\\pi\\) is continuous by Theorem 4.7. The inverse is an algebra isomorphism of \\(B\\) onto \\(A\\), so it is continuous for the same reason. The last claim is Theorem 4.4(1). \\(\\square\\)\n\n**Example 4.9** (A self-adjoint range is needed). Let \\(E=\\ell^2(\\mathbb N)\\) with the zero product; it is a commutative Banach algebra. Choose a discontinuous linear functional \\(\\varphi\\) on \\(E\\). One exists by the axiom of choice: extend the linearly independent unit vectors \\(e_1,e_2,\\dots\\) to a Hamel basis, and put \\(\\varphi(e_n)=n\\) and \\(\\varphi=0\\) on the other basis vectors. Let \\(D(\\xi)\\) be the diagonal operator on \\(\\ell^2(\\mathbb N)\\) with entries \\(\\xi_n\\); it is bounded, and \\(\\xi\\mapsto D(\\xi)\\) is injective. Put \\(T(\\xi)=D(\\xi)+\\varphi(\\xi)E_{12}\\), where \\(E_{12}e_2=e_1\\), and on \\(\\ell^2\\oplus\\ell^2\\) put\n\\[\n\\pi(\\xi)=\\begin{pmatrix}0&T(\\xi)\\\\0&0\\end{pmatrix}.\n\\]\nThen \\(\\pi(\\xi)\\pi(\\eta)=0=\\pi(\\xi\\eta)\\), so \\(\\pi\\) is a homomorphism into \\(B(\\ell^2\\oplus\\ell^2)\\). It is injective, since the diagonal of \\(T(\\xi)\\) is \\(\\xi\\). It is not continuous, since the matrix entry \\(\\langle T(\\xi)e_2,e_1\\rangle=\\varphi(\\xi)\\) is not. Its range consists of nonzero nilpotents and zero, and is not self-adjoint.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-13",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "4. Homomorphisms: contractivity, isometry and automatic continuity",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
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      "full_conditions_and_proof": "### 4. Homomorphisms: contractivity, isometry and automatic continuity\n\n**Definition 4.1.** A *\\(*\\)-homomorphism* between algebras with involution is an algebra homomorphism \\(\\pi\\) with \\(\\pi(x^*)=\\pi(x)^*\\).\n\n**Theorem 4.2** (\\(*\\)-homomorphisms are contractive). Let \\(A\\) be a Banach algebra with an involution that is not assumed to be continuous, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) a \\(*\\)-homomorphism. Then for every \\(x\\in A\\)\n\\[\n\\begin{gathered}\n\\sigma'_B(\\pi(x))\\\\\n\\subseteq\\sigma'_A(x),\\\\\n\\|\\pi(x)\\|^2\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\|\\\\\n\\leq\\|x^*\\|\\,\\|x\\| .\n\\end{gathered}\n\\tag{4.1}\n\\]\nConsequently:\n1. if \\(\\|x^*\\|\\leq C\\|x\\|\\) for all \\(x\\), then \\(\\|\\pi(x)\\|\\leq C^{1/2}\\|x\\|\\);\n2. if the involution is isometric, then \\(\\|\\pi(x)\\|\\leq\\|x\\|\\);\n3. in particular every \\(*\\)-homomorphism between C\\*-algebras is contractive, whatever is assumed about its continuity.\n\n**Proof.** The map \\(\\pi_1(a+\\lambda)=\\pi(a)+\\lambda\\) is a unital homomorphism from \\(A_1\\) to \\(\\widetilde B\\), and unital homomorphisms map invertible elements to invertible elements. So \\(\\lambda\\notin\\sigma'_A(x)\\) implies \\(\\lambda\\notin\\sigma'_B(\\pi(x))\\), and \\(r_B(\\pi(y))\\leq r_A(y)\\) for all \\(y\\). The element \\(\\pi(x)^*\\pi(x)=\\pi(x^*x)\\) is self-adjoint, so by Theorem 1.3(1)\n\\[\n\\begin{gathered}\n\\|\\pi(x)\\|^2\\\\\n=\\|\\pi(x^*x)\\|\\\\\n=r_B(\\pi(x^*x))\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\| ,\n\\end{gathered}\n\\]\nwhere the last step uses \\(r_A(y)\\leq\\|y\\|\\) ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)). \\(\\square\\)\n\n**Example 4.3** (An isometric involution is needed for contractivity). Let \\(t>1\\), \\(S=\\operatorname{diag}(1,t)\\in M_2(\\mathbb C)\\), and \\(\\|x\\|_S=\\|SxS^{-1}\\|\\) (operator norm). This is a complete algebra norm on \\(M_2(\\mathbb C)\\), and the usual adjoint is an involution for it, continuous but not isometric. The identity map \\(\\pi\\) from \\((M_2(\\mathbb C),\\|\\cdot\\|_S)\\) to the C\\*-algebra \\(M_2(\\mathbb C)\\) is a \\(*\\)-homomorphism. Since \\(SE_{12}S^{-1}=t^{-1}E_{12}\\) and \\(SE_{21}S^{-1}=tE_{21}\\), we get \\(\\|\\pi(E_{12})\\|=1>t^{-1}=\\|E_{12}\\|_S\\). The bound (4.1) is sharp here: \\(\\|E_{12}^*\\|_S\\|E_{12}\\|_S=t\\cdot t^{-1}=1\\).\n\n**Theorem 4.4** (Injective homomorphisms do not shrink normal elements). Let \\(A\\) be a C\\*-algebra, \\(B\\) a Banach algebra and \\(\\pi:A\\to B\\) an injective algebra homomorphism. Neither continuity nor compatibility with an involution is assumed.\n1. \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\in A\\).\n2. If \\(B\\) has an involution and \\(\\pi\\) is a \\(*\\)-homomorphism, then \\(\\|x\\|^2\\leq\\|\\pi(x)^*\\|\\,\\|\\pi(x)\\|\\) for all \\(x\\). If moreover the involution of \\(B\\) is isometric, then \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for all \\(x\\).\n3. If, in (2), \\(\\pi(A)\\) is closed, the last inequality also follows from Theorem 4.2 applied to \\(\\pi^{-1}:\\pi(A)\\to A\\).\n\nThe proof below uses the closure of \\(\\pi(C^*(h))\\), so it makes no continuity assumption on \\(\\pi\\).\n\n**Proof.** (1) Let \\(x\\) be normal. Put \\(A_0=C^*(x)\\), a commutative C\\*-algebra (Definition 3.1), and let \\(B_0\\) be the closure of \\(\\pi(A_0)\\) in \\(B\\), a commutative Banach algebra. Adjoin identities: \\(\\widetilde{A_0}\\) is a commutative unital C\\*-algebra (Conventions), \\(\\widetilde{B_0}=B_0\\oplus\\mathbb C\\) with the sum norm is a commutative unital Banach algebra ([adjoining an identity](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-04)), and \\(\\tilde\\pi(a+\\lambda)=\\pi(a)+\\lambda\\) is an injective unital homomorphism. The character spaces \\(X=\\operatorname{Ch}(\\widetilde{A_0})\\) and \\(Y=\\operatorname{Ch}(\\widetilde{B_0})\\) are compact, because the algebras are unital ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). The map \\(\\tau(\\chi)=\\chi\\circ\\tilde\\pi\\) from \\(Y\\) to \\(X\\) is weak\\* continuous, so \\(F=\\tau(Y)\\) is compact, hence closed.\n*Claim: \\(F=X\\).* Suppose \\(\\omega_0\\in X\\setminus F\\). The space \\(X\\) is compact Hausdorff, hence regular, so it has an open \\(U\\ni\\omega_0\\) whose closure misses \\(F\\). By Urysohn's lemma and Theorem 2.1, there are \\(a,b\\in\\widetilde{A_0}\\) with \\(\\hat a(\\omega_0)=1\\), \\(\\hat a=0\\) off \\(U\\), \\(\\hat b=1\\) on \\(F\\) and \\(\\hat b=0\\) on \\(\\overline U\\). Then \\(\\hat a\\hat b=0\\), so \\(ab=0\\), and \\(a\\neq0\\). For every \\(\\chi\\in Y\\), \\(\\chi(\\tilde\\pi(b))=\\hat b(\\tau\\chi)=1\\). So no character of \\(\\widetilde{B_0}\\) vanishes at \\(\\tilde\\pi(b)\\), and \\(\\tilde\\pi(b)\\) is invertible, since in a unital commutative Banach algebra an element at which no character vanishes is invertible ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). Hence \\(\\tilde\\pi(a)=\\tilde\\pi(ab)\\tilde\\pi(b)^{-1}=0\\), and \\(a=0\\) by injectivity, a contradiction.\nNow, in a commutative Banach algebra the spectral radius is the largest modulus of the Gelfand transform ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). Using this in both algebras, the fact that \\(\\tau\\) maps \\(Y\\) onto \\(X\\), and Theorem 1.3 in \\(\\widetilde{A_0}\\),\n\\[\n\\begin{gathered}\n\\|\\pi(x)\\|\\\\\n\\geq r_{\\widetilde{B_0}}(\\pi(x))\\\\\n=\\max_{\\chi\\in Y}|\\chi(\\tilde\\pi(x))|\\\\\n=\\max_{\\omega\\in X}|\\omega(x)|\\\\\n=r_{\\widetilde{A_0}}(x)\\\\\n=\\|x\\| .\n\\end{gathered}\n\\]\n(2) \\(x^*x\\) is normal, so by (1), \\[\n\\begin{gathered}\n\\|x\\|^2\\\\\n=\\|x^*x\\|\\\\\n\\leq\\|\\pi(x^*x)\\|\\\\\n=\\|\\pi(x)^*\\pi(x)\\|\\\\\n\\leq\\|\\pi(x)^*\\|\\|\\pi(x)\\|.\n\\end{gathered}\n\\]\n(3) \\(\\pi(A)\\) is then a closed \\(*\\)-subalgebra of \\(B\\), an involutive Banach algebra, and \\(\\pi^{-1}\\) is a \\(*\\)-homomorphism from it to the C\\*-algebra \\(A\\). By Theorem 4.2, \\(\\|x\\|=\\|\\pi^{-1}(\\pi(x))\\|\\leq\\|\\pi(x)\\|\\). \\(\\square\\)\n\n**Example 4.5** (Normality is needed in (1)). With \\(S=\\operatorname{diag}(1,t)\\), \\(t>1\\), the map \\(\\pi(x)=SxS^{-1}\\) is an algebra automorphism of the C\\*-algebra \\(M_2(\\mathbb C)\\), not \\(*\\)-preserving. It satisfies \\(\\|\\pi(E_{12})\\|=t^{-1}<1=\\|E_{12}\\|\\).\n\n**Corollary 4.6.** An injective \\(*\\)-homomorphism from a C\\*-algebra into a C\\*-algebra is isometric. Its range is closed.\n\n**Proof.** The map is contractive by Theorem 4.2 and does not decrease norms by Theorem 4.4(2). An isometric image of a complete space is complete, hence closed. \\(\\square\\)\n\n**Theorem 4.7** (Automatic continuity). Let \\(A\\) be a Banach algebra, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) an injective algebra homomorphism whose range is self-adjoint: \\(\\pi(A)^*=\\pi(A)\\). Then \\(\\pi\\) is continuous.\n\n**Proof.** For \\(x\\in A\\), \\(\\pi(x)^*\\) lies in \\(\\pi(A)\\), so \\(x^\\sharp=\\pi^{-1}(\\pi(x)^*)\\) is well defined. It is an involution on \\(A\\): it is conjugate linear, \\((xy)^\\sharp=\\pi^{-1}(\\pi(y)^*\\pi(x)^*)=y^\\sharp x^\\sharp\\), and \\(x^{\\sharp\\sharp}=x\\). By construction \\(\\pi\\) is a \\(*\\)-homomorphism for \\(\\sharp\\). By (4.1), which does not need a continuous involution,\n\\[\n\\|\\pi(x)\\|^2\\leq\\|x^\\sharp\\|\\,\\|x\\|\\qquad(x\\in A).\n\\tag{4.2}\n\\]\nWe show that \\(\\sharp\\) has a closed graph. Let \\(x_n\\to x\\) and \\(x_n^\\sharp\\to y\\). By (4.2),\n\\[\n\\begin{gathered}\n\\|\\pi(x)-\\pi(x_n)\\|^2\\\\\n\\leq\\|x^\\sharp-x_n^\\sharp\\|\\,\\|x-x_n\\|\\to0,\\\\\n\\|\\pi(y)-\\pi(x_n^\\sharp)\\|^2\\\\\n\\leq\\|y^\\sharp-x_n\\|\\,\\|y-x_n^\\sharp\\|\\to0 ,\n\\end{gathered}\n\\]\nbecause in each product the first factor stays bounded and the second tends to \\(0\\). Since \\(\\pi(x_n^\\sharp)=\\pi(x_n)^*\\to\\pi(x)^*\\), we get \\(\\pi(y)=\\pi(x)^*=\\pi(x^\\sharp)\\), and \\(y=x^\\sharp\\). By the closed graph theorem, applied to the real-linear map \\(\\sharp\\), there is \\(k\\) with \\(\\|x^\\sharp\\|\\leq k\\|x\\|\\). Then (4.2) gives \\(\\|\\pi(x)\\|\\leq k^{1/2}\\|x\\|\\). \\(\\square\\)\n\n**Corollary 4.8.** Let \\(\\pi\\) be an algebra isomorphism of a C\\*-algebra \\(A\\) onto a C\\*-algebra \\(B\\), not assumed to preserve the involution. Then \\(\\pi\\) and \\(\\pi^{-1}\\) are continuous: \\(c\\|x\\|\\leq\\|\\pi(x)\\|\\leq C\\|x\\|\\) with constants \\(c,C>0\\). Moreover \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\).\n\n**Proof.** The range \\(B\\) is self-adjoint, so \\(\\pi\\) is continuous by Theorem 4.7. The inverse is an algebra isomorphism of \\(B\\) onto \\(A\\), so it is continuous for the same reason. The last claim is Theorem 4.4(1). \\(\\square\\)\n\n**Example 4.9** (A self-adjoint range is needed). Let \\(E=\\ell^2(\\mathbb N)\\) with the zero product; it is a commutative Banach algebra. Choose a discontinuous linear functional \\(\\varphi\\) on \\(E\\). One exists by the axiom of choice: extend the linearly independent unit vectors \\(e_1,e_2,\\dots\\) to a Hamel basis, and put \\(\\varphi(e_n)=n\\) and \\(\\varphi=0\\) on the other basis vectors. Let \\(D(\\xi)\\) be the diagonal operator on \\(\\ell^2(\\mathbb N)\\) with entries \\(\\xi_n\\); it is bounded, and \\(\\xi\\mapsto D(\\xi)\\) is injective. Put \\(T(\\xi)=D(\\xi)+\\varphi(\\xi)E_{12}\\), where \\(E_{12}e_2=e_1\\), and on \\(\\ell^2\\oplus\\ell^2\\) put\n\\[\n\\pi(\\xi)=\\begin{pmatrix}0&T(\\xi)\\\\0&0\\end{pmatrix}.\n\\]\nThen \\(\\pi(\\xi)\\pi(\\eta)=0=\\pi(\\xi\\eta)\\), so \\(\\pi\\) is a homomorphism into \\(B(\\ell^2\\oplus\\ell^2)\\). It is injective, since the diagonal of \\(T(\\xi)\\) is \\(\\xi\\). It is not continuous, since the matrix entry \\(\\langle T(\\xi)e_2,e_1\\rangle=\\varphi(\\xi)\\) is not. Its range consists of nonzero nilpotents and zero, and is not self-adjoint.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-14",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "4. Homomorphisms: contractivity, isometry and automatic continuity",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 4. Homomorphisms: contractivity, isometry and automatic continuity\n\n**Definition 4.1.** A *\\(*\\)-homomorphism* between algebras with involution is an algebra homomorphism \\(\\pi\\) with \\(\\pi(x^*)=\\pi(x)^*\\).\n\n**Theorem 4.2** (\\(*\\)-homomorphisms are contractive). Let \\(A\\) be a Banach algebra with an involution that is not assumed to be continuous, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) a \\(*\\)-homomorphism. Then for every \\(x\\in A\\)\n\\[\n\\begin{gathered}\n\\sigma'_B(\\pi(x))\\\\\n\\subseteq\\sigma'_A(x),\\\\\n\\|\\pi(x)\\|^2\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\|\\\\\n\\leq\\|x^*\\|\\,\\|x\\| .\n\\end{gathered}\n\\tag{4.1}\n\\]\nConsequently:\n1. if \\(\\|x^*\\|\\leq C\\|x\\|\\) for all \\(x\\), then \\(\\|\\pi(x)\\|\\leq C^{1/2}\\|x\\|\\);\n2. if the involution is isometric, then \\(\\|\\pi(x)\\|\\leq\\|x\\|\\);\n3. in particular every \\(*\\)-homomorphism between C\\*-algebras is contractive, whatever is assumed about its continuity.\n\n**Proof.** The map \\(\\pi_1(a+\\lambda)=\\pi(a)+\\lambda\\) is a unital homomorphism from \\(A_1\\) to \\(\\widetilde B\\), and unital homomorphisms map invertible elements to invertible elements. So \\(\\lambda\\notin\\sigma'_A(x)\\) implies \\(\\lambda\\notin\\sigma'_B(\\pi(x))\\), and \\(r_B(\\pi(y))\\leq r_A(y)\\) for all \\(y\\). The element \\(\\pi(x)^*\\pi(x)=\\pi(x^*x)\\) is self-adjoint, so by Theorem 1.3(1)\n\\[\n\\begin{gathered}\n\\|\\pi(x)\\|^2\\\\\n=\\|\\pi(x^*x)\\|\\\\\n=r_B(\\pi(x^*x))\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\| ,\n\\end{gathered}\n\\]\nwhere the last step uses \\(r_A(y)\\leq\\|y\\|\\) ([the spectrum is compact](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-07)). \\(\\square\\)\n\n**Example 4.3** (An isometric involution is needed for contractivity). Let \\(t>1\\), \\(S=\\operatorname{diag}(1,t)\\in M_2(\\mathbb C)\\), and \\(\\|x\\|_S=\\|SxS^{-1}\\|\\) (operator norm). This is a complete algebra norm on \\(M_2(\\mathbb C)\\), and the usual adjoint is an involution for it, continuous but not isometric. The identity map \\(\\pi\\) from \\((M_2(\\mathbb C),\\|\\cdot\\|_S)\\) to the C\\*-algebra \\(M_2(\\mathbb C)\\) is a \\(*\\)-homomorphism. Since \\(SE_{12}S^{-1}=t^{-1}E_{12}\\) and \\(SE_{21}S^{-1}=tE_{21}\\), we get \\(\\|\\pi(E_{12})\\|=1>t^{-1}=\\|E_{12}\\|_S\\). The bound (4.1) is sharp here: \\(\\|E_{12}^*\\|_S\\|E_{12}\\|_S=t\\cdot t^{-1}=1\\).\n\n**Theorem 4.4** (Injective homomorphisms do not shrink normal elements). Let \\(A\\) be a C\\*-algebra, \\(B\\) a Banach algebra and \\(\\pi:A\\to B\\) an injective algebra homomorphism. Neither continuity nor compatibility with an involution is assumed.\n1. \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\in A\\).\n2. If \\(B\\) has an involution and \\(\\pi\\) is a \\(*\\)-homomorphism, then \\(\\|x\\|^2\\leq\\|\\pi(x)^*\\|\\,\\|\\pi(x)\\|\\) for all \\(x\\). If moreover the involution of \\(B\\) is isometric, then \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for all \\(x\\).\n3. If, in (2), \\(\\pi(A)\\) is closed, the last inequality also follows from Theorem 4.2 applied to \\(\\pi^{-1}:\\pi(A)\\to A\\).\n\nThe proof below uses the closure of \\(\\pi(C^*(h))\\), so it makes no continuity assumption on \\(\\pi\\).\n\n**Proof.** (1) Let \\(x\\) be normal. Put \\(A_0=C^*(x)\\), a commutative C\\*-algebra (Definition 3.1), and let \\(B_0\\) be the closure of \\(\\pi(A_0)\\) in \\(B\\), a commutative Banach algebra. Adjoin identities: \\(\\widetilde{A_0}\\) is a commutative unital C\\*-algebra (Conventions), \\(\\widetilde{B_0}=B_0\\oplus\\mathbb C\\) with the sum norm is a commutative unital Banach algebra ([adjoining an identity](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-04)), and \\(\\tilde\\pi(a+\\lambda)=\\pi(a)+\\lambda\\) is an injective unital homomorphism. The character spaces \\(X=\\operatorname{Ch}(\\widetilde{A_0})\\) and \\(Y=\\operatorname{Ch}(\\widetilde{B_0})\\) are compact, because the algebras are unital ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). The map \\(\\tau(\\chi)=\\chi\\circ\\tilde\\pi\\) from \\(Y\\) to \\(X\\) is weak\\* continuous, so \\(F=\\tau(Y)\\) is compact, hence closed.\n*Claim: \\(F=X\\).* Suppose \\(\\omega_0\\in X\\setminus F\\). The space \\(X\\) is compact Hausdorff, hence regular, so it has an open \\(U\\ni\\omega_0\\) whose closure misses \\(F\\). By Urysohn's lemma and Theorem 2.1, there are \\(a,b\\in\\widetilde{A_0}\\) with \\(\\hat a(\\omega_0)=1\\), \\(\\hat a=0\\) off \\(U\\), \\(\\hat b=1\\) on \\(F\\) and \\(\\hat b=0\\) on \\(\\overline U\\). Then \\(\\hat a\\hat b=0\\), so \\(ab=0\\), and \\(a\\neq0\\). For every \\(\\chi\\in Y\\), \\(\\chi(\\tilde\\pi(b))=\\hat b(\\tau\\chi)=1\\). So no character of \\(\\widetilde{B_0}\\) vanishes at \\(\\tilde\\pi(b)\\), and \\(\\tilde\\pi(b)\\) is invertible, since in a unital commutative Banach algebra an element at which no character vanishes is invertible ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). Hence \\(\\tilde\\pi(a)=\\tilde\\pi(ab)\\tilde\\pi(b)^{-1}=0\\), and \\(a=0\\) by injectivity, a contradiction.\nNow, in a commutative Banach algebra the spectral radius is the largest modulus of the Gelfand transform ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). Using this in both algebras, the fact that \\(\\tau\\) maps \\(Y\\) onto \\(X\\), and Theorem 1.3 in \\(\\widetilde{A_0}\\),\n\\[\n\\begin{gathered}\n\\|\\pi(x)\\|\\\\\n\\geq r_{\\widetilde{B_0}}(\\pi(x))\\\\\n=\\max_{\\chi\\in Y}|\\chi(\\tilde\\pi(x))|\\\\\n=\\max_{\\omega\\in X}|\\omega(x)|\\\\\n=r_{\\widetilde{A_0}}(x)\\\\\n=\\|x\\| .\n\\end{gathered}\n\\]\n(2) \\(x^*x\\) is normal, so by (1), \\[\n\\begin{gathered}\n\\|x\\|^2\\\\\n=\\|x^*x\\|\\\\\n\\leq\\|\\pi(x^*x)\\|\\\\\n=\\|\\pi(x)^*\\pi(x)\\|\\\\\n\\leq\\|\\pi(x)^*\\|\\|\\pi(x)\\|.\n\\end{gathered}\n\\]\n(3) \\(\\pi(A)\\) is then a closed \\(*\\)-subalgebra of \\(B\\), an involutive Banach algebra, and \\(\\pi^{-1}\\) is a \\(*\\)-homomorphism from it to the C\\*-algebra \\(A\\). By Theorem 4.2, \\(\\|x\\|=\\|\\pi^{-1}(\\pi(x))\\|\\leq\\|\\pi(x)\\|\\). \\(\\square\\)\n\n**Example 4.5** (Normality is needed in (1)). With \\(S=\\operatorname{diag}(1,t)\\), \\(t>1\\), the map \\(\\pi(x)=SxS^{-1}\\) is an algebra automorphism of the C\\*-algebra \\(M_2(\\mathbb C)\\), not \\(*\\)-preserving. It satisfies \\(\\|\\pi(E_{12})\\|=t^{-1}<1=\\|E_{12}\\|\\).\n\n**Corollary 4.6.** An injective \\(*\\)-homomorphism from a C\\*-algebra into a C\\*-algebra is isometric. Its range is closed.\n\n**Proof.** The map is contractive by Theorem 4.2 and does not decrease norms by Theorem 4.4(2). An isometric image of a complete space is complete, hence closed. \\(\\square\\)\n\n**Theorem 4.7** (Automatic continuity). Let \\(A\\) be a Banach algebra, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) an injective algebra homomorphism whose range is self-adjoint: \\(\\pi(A)^*=\\pi(A)\\). Then \\(\\pi\\) is continuous.\n\n**Proof.** For \\(x\\in A\\), \\(\\pi(x)^*\\) lies in \\(\\pi(A)\\), so \\(x^\\sharp=\\pi^{-1}(\\pi(x)^*)\\) is well defined. It is an involution on \\(A\\): it is conjugate linear, \\((xy)^\\sharp=\\pi^{-1}(\\pi(y)^*\\pi(x)^*)=y^\\sharp x^\\sharp\\), and \\(x^{\\sharp\\sharp}=x\\). By construction \\(\\pi\\) is a \\(*\\)-homomorphism for \\(\\sharp\\). By (4.1), which does not need a continuous involution,\n\\[\n\\|\\pi(x)\\|^2\\leq\\|x^\\sharp\\|\\,\\|x\\|\\qquad(x\\in A).\n\\tag{4.2}\n\\]\nWe show that \\(\\sharp\\) has a closed graph. Let \\(x_n\\to x\\) and \\(x_n^\\sharp\\to y\\). By (4.2),\n\\[\n\\begin{gathered}\n\\|\\pi(x)-\\pi(x_n)\\|^2\\\\\n\\leq\\|x^\\sharp-x_n^\\sharp\\|\\,\\|x-x_n\\|\\to0,\\\\\n\\|\\pi(y)-\\pi(x_n^\\sharp)\\|^2\\\\\n\\leq\\|y^\\sharp-x_n\\|\\,\\|y-x_n^\\sharp\\|\\to0 ,\n\\end{gathered}\n\\]\nbecause in each product the first factor stays bounded and the second tends to \\(0\\). Since \\(\\pi(x_n^\\sharp)=\\pi(x_n)^*\\to\\pi(x)^*\\), we get \\(\\pi(y)=\\pi(x)^*=\\pi(x^\\sharp)\\), and \\(y=x^\\sharp\\). By the closed graph theorem, applied to the real-linear map \\(\\sharp\\), there is \\(k\\) with \\(\\|x^\\sharp\\|\\leq k\\|x\\|\\). Then (4.2) gives \\(\\|\\pi(x)\\|\\leq k^{1/2}\\|x\\|\\). \\(\\square\\)\n\n**Corollary 4.8.** Let \\(\\pi\\) be an algebra isomorphism of a C\\*-algebra \\(A\\) onto a C\\*-algebra \\(B\\), not assumed to preserve the involution. Then \\(\\pi\\) and \\(\\pi^{-1}\\) are continuous: \\(c\\|x\\|\\leq\\|\\pi(x)\\|\\leq C\\|x\\|\\) with constants \\(c,C>0\\). Moreover \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\).\n\n**Proof.** The range \\(B\\) is self-adjoint, so \\(\\pi\\) is continuous by Theorem 4.7. The inverse is an algebra isomorphism of \\(B\\) onto \\(A\\), so it is continuous for the same reason. The last claim is Theorem 4.4(1). \\(\\square\\)\n\n**Example 4.9** (A self-adjoint range is needed). Let \\(E=\\ell^2(\\mathbb N)\\) with the zero product; it is a commutative Banach algebra. Choose a discontinuous linear functional \\(\\varphi\\) on \\(E\\). One exists by the axiom of choice: extend the linearly independent unit vectors \\(e_1,e_2,\\dots\\) to a Hamel basis, and put \\(\\varphi(e_n)=n\\) and \\(\\varphi=0\\) on the other basis vectors. Let \\(D(\\xi)\\) be the diagonal operator on \\(\\ell^2(\\mathbb N)\\) with entries \\(\\xi_n\\); it is bounded, and \\(\\xi\\mapsto D(\\xi)\\) is injective. Put \\(T(\\xi)=D(\\xi)+\\varphi(\\xi)E_{12}\\), where \\(E_{12}e_2=e_1\\), and on \\(\\ell^2\\oplus\\ell^2\\) put\n\\[\n\\pi(\\xi)=\\begin{pmatrix}0&T(\\xi)\\\\0&0\\end{pmatrix}.\n\\]\nThen \\(\\pi(\\xi)\\pi(\\eta)=0=\\pi(\\xi\\eta)\\), so \\(\\pi\\) is a homomorphism into \\(B(\\ell^2\\oplus\\ell^2)\\). It is injective, since the diagonal of \\(T(\\xi)\\) is \\(\\xi\\). It is not continuous, since the matrix entry \\(\\langle T(\\xi)e_2,e_1\\rangle=\\varphi(\\xi)\\) is not. Its range consists of nonzero nilpotents and zero, and is not self-adjoint.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CF-07",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "5. The continuous functional calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 5. The continuous functional calculus\n\n**Theorem 5.1** (The continuous functional calculus). Let \\(A\\) be a unital C\\*-algebra, \\(x\\in A\\) normal, \\(S=\\sigma_A(x)\\), and \\(\\iota\\in C(S)\\) the function \\(\\iota(\\lambda)=\\lambda\\).\n1. There is exactly one unital \\(*\\)-homomorphism \\(\\Phi_x:C(S)\\to A\\) with \\(\\Phi_x(\\iota)=x\\). It is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(C^*(1,x)\\). We write \\(f(x)=\\Phi_x(f)\\).\n2. For \\(f,g\\in C(S)\\) and scalars \\(\\alpha,\\beta\\): \\((\\alpha f+\\beta g)(x)=\\alpha f(x)+\\beta g(x)\\), \\((fg)(x)=f(x)g(x)\\), \\(\\bar f(x)=f(x)^*\\), \\(1(x)=1\\), and \\(p(x)=\\sum c_{jk}x^j(x^*)^k\\) for \\(p(\\lambda)=\\sum c_{jk}\\lambda^j\\bar\\lambda^k\\).\n3. (*Spectral mapping.*) \\(\\sigma_A(f(x))=f(S)\\), and \\(f(x)\\) is normal.\n4. \\(\\|f(x)\\|=\\|f\\|_S\\).\n5. (*Composition.*) If \\(g\\in C(f(S))\\), then \\((g\\circ f)(x)=g(f(x))\\).\n6. \\(f(x)\\) commutes with every \\(y\\in A\\) that commutes with \\(x\\) and \\(x^*\\).\n7. (*Independence of the algebra.*) If \\(B\\) is a unital C\\*-subalgebra of \\(A\\) containing \\(x\\), the calculus of \\(x\\) in \\(B\\) is the same map.\n8. (*Characters.*) If \\(D\\) is a commutative unital C\\*-subalgebra containing \\(x\\) and \\(\\chi\\in\\operatorname{Ch}(D)\\), then \\(\\chi(f(x))=f(\\chi(x))\\).\n9. (*Real form.*) If \\(f\\) is real, \\(f(x)\\) is self-adjoint. For \\(x\\in A_h\\), \\(f\\mapsto f(x)\\) maps \\(C(S;\\mathbb R)\\) isometrically onto the self-adjoint part of \\(C^*(1,x)\\).\n10. (*Agreement with the holomorphic calculus.*) If \\(F\\) is holomorphic on an open set \\(U\\supseteq S\\), the element \\(F(x)\\) given by the [holomorphic functional calculus](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-10) equals \\(\\Phi_x(F|_S)\\).\n\n**Proof.** If \\(A=\\{0\\}\\), then \\(S=\\varnothing\\), \\(C(S)=\\{0\\}\\), and everything is trivial. Let \\(A\\) be nontrivial.\n\n(1) *Existence.* \\(B=C^*(1,x)\\) is a commutative unital C\\*-algebra (Definition 3.1). Put \\(\\Omega=\\operatorname{Ch}(B)\\), a compact space, and \\(\\psi(\\omega)=\\omega(x)\\). By Theorem 2.1, \\(\\mathcal G_B:B\\to C(\\Omega)\\) is an isometric \\(*\\)-isomorphism. The map \\(\\psi\\) is continuous, and \\(\\psi(\\Omega)=\\sigma_B(x)\\), because in a unital commutative Banach algebra the spectrum of an element is the range of its Gelfand transform ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). By spectral permanence (Theorem 3.2(1)), \\(\\sigma_B(x)=S\\). The map \\(\\psi\\) is injective: if \\(\\omega(x)=\\omega'(x)\\), then also \\(\\omega(x^*)=\\omega'(x^*)\\) by Theorem 2.1(1), and \\(\\omega(1)=\\omega'(1)=1\\); so \\(\\omega\\) and \\(\\omega'\\) agree on the polynomials in \\(x\\) and \\(x^*\\), which are dense in \\(B\\), and characters are continuous ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). A continuous bijection of a compact space onto a Hausdorff space is a homeomorphism. So \\(f\\mapsto f\\circ\\psi\\) is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(C(\\Omega)\\), and \\(\\Phi_x(f)=\\mathcal G_B^{-1}(f\\circ\\psi)\\) is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(B\\). It is unital, and \\(\\Phi_x(\\iota)=\\mathcal G_B^{-1}(\\hat x)=x\\).\n*Uniqueness.* A unital \\(*\\)-homomorphism \\(\\Phi':C(S)\\to A\\) with \\(\\Phi'(\\iota)=x\\) agrees with \\(\\Phi_x\\) on the polynomials in \\(\\iota\\) and \\(\\bar\\iota\\), which are dense in \\(C(S)\\) ([density results](stone-weierstrass-c0.md#oa-fnd-sw-17)). It is contractive by Theorem 4.2. So \\(\\Phi'=\\Phi_x\\).\n(2) holds because \\(\\Phi_x\\) is a unital \\(*\\)-homomorphism with \\(\\Phi_x(\\bar\\iota)=x^*\\).\n(3) \\(\\sigma_A(f(x))=\\sigma_B(f(x))\\) by Theorem 3.2(1), and this equals \\(\\sigma_{C(S)}(f)\\) because \\(\\Phi_x\\) is an algebra isomorphism onto \\(B\\). In \\(C(S)\\), \\(f-\\mu\\) is invertible exactly when it has no zero on \\(S\\), so \\(\\sigma_{C(S)}(f)=f(S)\\). Images of commuting elements commute, so \\(f(x)\\) is normal.\n(4) \\(\\Phi_x\\) is isometric.\n(5) By (3), \\(f(x)\\) is normal with spectrum \\(f(S)\\). The map \\(g\\mapsto(g\\circ f)(x)\\) is a unital \\(*\\)-homomorphism \\(C(f(S))\\to A\\) that sends \\(\\iota\\) to \\(f(x)\\). By the uniqueness in (1), applied to \\(f(x)\\), it is \\(g\\mapsto g(f(x))\\).\n(6) \\(y\\) commutes with every polynomial in \\(x\\) and \\(x^*\\), hence with their limits.\n(7) \\(\\sigma_B(x)=\\sigma_A(x)\\) by Theorem 3.2(1). The calculus of \\(x\\) in \\(B\\) is a unital \\(*\\)-homomorphism into \\(A\\) sending \\(\\iota\\) to \\(x\\); apply the uniqueness in (1).\n(8) \\(\\chi\\) restricts to a character of \\(C^*(1,x)\\subseteq D\\), since \\(\\chi(1)=1\\). So \\(\\chi\\circ\\Phi_x\\) is a character of \\(C(S)\\), hence evaluation at some \\(s\\in S\\) (Proposition 2.2(1)), and \\(s=\\chi(\\Phi_x(\\iota))=\\chi(x)\\).\n(9) If \\(f=\\bar f\\), then \\(f(x)^*=\\bar f(x)=f(x)\\). If \\(y\\in C^*(1,x)\\) is self-adjoint and \\(y=f(x)\\), then \\(\\bar f(x)=y^*=y=f(x)\\), and injectivity gives \\(f=\\bar f\\).\n(10) For \\(\\lambda\\notin S\\), both \\((\\lambda-x)^{-1}\\) and \\(\\Phi_x((\\lambda-\\iota)^{-1})\\) are inverses of \\(\\lambda-x\\), so they are equal. Choose a cycle \\(\\Gamma\\) that surrounds \\(S\\) in \\(U\\), as in the definition of the holomorphic calculus. The map \\(\\lambda\\mapsto F(\\lambda)(\\lambda-\\iota)^{-1}\\) from \\(\\Gamma^*\\) to \\(C(S)\\) is continuous. At each \\(s\\in S\\), \\(\\frac1{2\\pi i}\\) times its integral takes the value \\(\\frac1{2\\pi i}\\int_\\Gamma F(\\lambda)(\\lambda-s)^{-1}\\,d\\lambda=F(s)\\), by Cauchy's integral formula, since \\(\\operatorname{Ind}_\\Gamma(s)=1\\); here evaluation at \\(s\\) is a bounded functional, so it passes through the integral. The bounded linear map \\(\\Phi_x\\) also passes through the integral, so\n\\[\n\\begin{gathered}\n\\Phi_x(F|_S)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma F(\\lambda)\\,\\Phi_x\\big((\\lambda-\\iota)^{-1}\\big)\\,d\\lambda\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma F(\\lambda)(\\lambda-x)^{-1}\\,d\\lambda\\\\\n=F(x).\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Remark 5.2.** The uniqueness in (1) holds even among unital algebra homomorphisms of \\(C(S)\\) *into* \\(C^*(1,x)\\) that send \\(\\iota\\) to \\(x\\), with no continuity or \\(*\\) assumed. If \\(\\Phi'\\) is one, \\(\\sigma=\\Phi_x^{-1}\\circ\\Phi'\\) is a unital algebra endomorphism of \\(C(S)\\) fixing \\(\\iota\\). For \\(s\\in S\\), \\(\\operatorname{ev}_s\\circ\\sigma\\) is a character, so it is \\(\\operatorname{ev}_t\\) for some \\(t\\) (Proposition 2.2), and \\(t=\\operatorname{ev}_s(\\sigma(\\iota))=s\\). So \\(\\sigma\\) is the identity.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-CF-08",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "5. The continuous functional calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
      "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "anchor": "oa-fnd-cf-08",
      "proof_locus": {
        "line": 259,
        "through_line": 296
      },
      "full_conditions_and_proof": "### 5. The continuous functional calculus\n\n**Theorem 5.1** (The continuous functional calculus). Let \\(A\\) be a unital C\\*-algebra, \\(x\\in A\\) normal, \\(S=\\sigma_A(x)\\), and \\(\\iota\\in C(S)\\) the function \\(\\iota(\\lambda)=\\lambda\\).\n1. There is exactly one unital \\(*\\)-homomorphism \\(\\Phi_x:C(S)\\to A\\) with \\(\\Phi_x(\\iota)=x\\). It is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(C^*(1,x)\\). We write \\(f(x)=\\Phi_x(f)\\).\n2. For \\(f,g\\in C(S)\\) and scalars \\(\\alpha,\\beta\\): \\((\\alpha f+\\beta g)(x)=\\alpha f(x)+\\beta g(x)\\), \\((fg)(x)=f(x)g(x)\\), \\(\\bar f(x)=f(x)^*\\), \\(1(x)=1\\), and \\(p(x)=\\sum c_{jk}x^j(x^*)^k\\) for \\(p(\\lambda)=\\sum c_{jk}\\lambda^j\\bar\\lambda^k\\).\n3. (*Spectral mapping.*) \\(\\sigma_A(f(x))=f(S)\\), and \\(f(x)\\) is normal.\n4. \\(\\|f(x)\\|=\\|f\\|_S\\).\n5. (*Composition.*) If \\(g\\in C(f(S))\\), then \\((g\\circ f)(x)=g(f(x))\\).\n6. \\(f(x)\\) commutes with every \\(y\\in A\\) that commutes with \\(x\\) and \\(x^*\\).\n7. (*Independence of the algebra.*) If \\(B\\) is a unital C\\*-subalgebra of \\(A\\) containing \\(x\\), the calculus of \\(x\\) in \\(B\\) is the same map.\n8. (*Characters.*) If \\(D\\) is a commutative unital C\\*-subalgebra containing \\(x\\) and \\(\\chi\\in\\operatorname{Ch}(D)\\), then \\(\\chi(f(x))=f(\\chi(x))\\).\n9. (*Real form.*) If \\(f\\) is real, \\(f(x)\\) is self-adjoint. For \\(x\\in A_h\\), \\(f\\mapsto f(x)\\) maps \\(C(S;\\mathbb R)\\) isometrically onto the self-adjoint part of \\(C^*(1,x)\\).\n10. (*Agreement with the holomorphic calculus.*) If \\(F\\) is holomorphic on an open set \\(U\\supseteq S\\), the element \\(F(x)\\) given by the [holomorphic functional calculus](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-10) equals \\(\\Phi_x(F|_S)\\).\n\n**Proof.** If \\(A=\\{0\\}\\), then \\(S=\\varnothing\\), \\(C(S)=\\{0\\}\\), and everything is trivial. Let \\(A\\) be nontrivial.\n\n(1) *Existence.* \\(B=C^*(1,x)\\) is a commutative unital C\\*-algebra (Definition 3.1). Put \\(\\Omega=\\operatorname{Ch}(B)\\), a compact space, and \\(\\psi(\\omega)=\\omega(x)\\). By Theorem 2.1, \\(\\mathcal G_B:B\\to C(\\Omega)\\) is an isometric \\(*\\)-isomorphism. The map \\(\\psi\\) is continuous, and \\(\\psi(\\Omega)=\\sigma_B(x)\\), because in a unital commutative Banach algebra the spectrum of an element is the range of its Gelfand transform ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). By spectral permanence (Theorem 3.2(1)), \\(\\sigma_B(x)=S\\). The map \\(\\psi\\) is injective: if \\(\\omega(x)=\\omega'(x)\\), then also \\(\\omega(x^*)=\\omega'(x^*)\\) by Theorem 2.1(1), and \\(\\omega(1)=\\omega'(1)=1\\); so \\(\\omega\\) and \\(\\omega'\\) agree on the polynomials in \\(x\\) and \\(x^*\\), which are dense in \\(B\\), and characters are continuous ([characters are contractive](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-17)). A continuous bijection of a compact space onto a Hausdorff space is a homeomorphism. So \\(f\\mapsto f\\circ\\psi\\) is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(C(\\Omega)\\), and \\(\\Phi_x(f)=\\mathcal G_B^{-1}(f\\circ\\psi)\\) is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(B\\). It is unital, and \\(\\Phi_x(\\iota)=\\mathcal G_B^{-1}(\\hat x)=x\\).\n*Uniqueness.* A unital \\(*\\)-homomorphism \\(\\Phi':C(S)\\to A\\) with \\(\\Phi'(\\iota)=x\\) agrees with \\(\\Phi_x\\) on the polynomials in \\(\\iota\\) and \\(\\bar\\iota\\), which are dense in \\(C(S)\\) ([density results](stone-weierstrass-c0.md#oa-fnd-sw-17)). It is contractive by Theorem 4.2. So \\(\\Phi'=\\Phi_x\\).\n(2) holds because \\(\\Phi_x\\) is a unital \\(*\\)-homomorphism with \\(\\Phi_x(\\bar\\iota)=x^*\\).\n(3) \\(\\sigma_A(f(x))=\\sigma_B(f(x))\\) by Theorem 3.2(1), and this equals \\(\\sigma_{C(S)}(f)\\) because \\(\\Phi_x\\) is an algebra isomorphism onto \\(B\\). In \\(C(S)\\), \\(f-\\mu\\) is invertible exactly when it has no zero on \\(S\\), so \\(\\sigma_{C(S)}(f)=f(S)\\). Images of commuting elements commute, so \\(f(x)\\) is normal.\n(4) \\(\\Phi_x\\) is isometric.\n(5) By (3), \\(f(x)\\) is normal with spectrum \\(f(S)\\). The map \\(g\\mapsto(g\\circ f)(x)\\) is a unital \\(*\\)-homomorphism \\(C(f(S))\\to A\\) that sends \\(\\iota\\) to \\(f(x)\\). By the uniqueness in (1), applied to \\(f(x)\\), it is \\(g\\mapsto g(f(x))\\).\n(6) \\(y\\) commutes with every polynomial in \\(x\\) and \\(x^*\\), hence with their limits.\n(7) \\(\\sigma_B(x)=\\sigma_A(x)\\) by Theorem 3.2(1). The calculus of \\(x\\) in \\(B\\) is a unital \\(*\\)-homomorphism into \\(A\\) sending \\(\\iota\\) to \\(x\\); apply the uniqueness in (1).\n(8) \\(\\chi\\) restricts to a character of \\(C^*(1,x)\\subseteq D\\), since \\(\\chi(1)=1\\). So \\(\\chi\\circ\\Phi_x\\) is a character of \\(C(S)\\), hence evaluation at some \\(s\\in S\\) (Proposition 2.2(1)), and \\(s=\\chi(\\Phi_x(\\iota))=\\chi(x)\\).\n(9) If \\(f=\\bar f\\), then \\(f(x)^*=\\bar f(x)=f(x)\\). If \\(y\\in C^*(1,x)\\) is self-adjoint and \\(y=f(x)\\), then \\(\\bar f(x)=y^*=y=f(x)\\), and injectivity gives \\(f=\\bar f\\).\n(10) For \\(\\lambda\\notin S\\), both \\((\\lambda-x)^{-1}\\) and \\(\\Phi_x((\\lambda-\\iota)^{-1})\\) are inverses of \\(\\lambda-x\\), so they are equal. Choose a cycle \\(\\Gamma\\) that surrounds \\(S\\) in \\(U\\), as in the definition of the holomorphic calculus. The map \\(\\lambda\\mapsto F(\\lambda)(\\lambda-\\iota)^{-1}\\) from \\(\\Gamma^*\\) to \\(C(S)\\) is continuous. At each \\(s\\in S\\), \\(\\frac1{2\\pi i}\\) times its integral takes the value \\(\\frac1{2\\pi i}\\int_\\Gamma F(\\lambda)(\\lambda-s)^{-1}\\,d\\lambda=F(s)\\), by Cauchy's integral formula, since \\(\\operatorname{Ind}_\\Gamma(s)=1\\); here evaluation at \\(s\\) is a bounded functional, so it passes through the integral. The bounded linear map \\(\\Phi_x\\) also passes through the integral, so\n\\[\n\\begin{gathered}\n\\Phi_x(F|_S)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma F(\\lambda)\\,\\Phi_x\\big((\\lambda-\\iota)^{-1}\\big)\\,d\\lambda\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma F(\\lambda)(\\lambda-x)^{-1}\\,d\\lambda\\\\\n=F(x).\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Remark 5.2.** The uniqueness in (1) holds even among unital algebra homomorphisms of \\(C(S)\\) *into* \\(C^*(1,x)\\) that send \\(\\iota\\) to \\(x\\), with no continuity or \\(*\\) assumed. If \\(\\Phi'\\) is one, \\(\\sigma=\\Phi_x^{-1}\\circ\\Phi'\\) is a unital algebra endomorphism of \\(C(S)\\) fixing \\(\\iota\\). For \\(s\\in S\\), \\(\\operatorname{ev}_s\\circ\\sigma\\) is a character, so it is \\(\\operatorname{ev}_t\\) for some \\(t\\) (Proposition 2.2), and \\(t=\\operatorname{ev}_s(\\sigma(\\iota))=s\\). So \\(\\sigma\\) is the identity.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
        "the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity",
        "haar-measure-on-locally-compact-groups",
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
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      ]
    },
    {
      "id": "OA-FND-CF-11",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "6. Continuity of the functional calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
      "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "anchor": "oa-fnd-cf-11",
      "proof_locus": {
        "line": 347,
        "through_line": 374
      },
      "full_conditions_and_proof": "### 6. Continuity of the functional calculus\n\nFor a C\\*-algebra \\(A\\) and a compact \\(K\\subseteq\\mathbb C\\), let \\(A_K\\) be the set of normal \\(x\\in A\\) with \\(\\sigma'_A(x)\\subseteq K\\). For \\(f\\in C(K)\\) and \\(x\\in A_K\\), \\(f(x)\\) is the calculus of Theorem 5.3, computed in \\(\\widetilde A\\); it lies in \\(A\\) when \\(f(0)=0\\). If \\(A\\) is unital, the same statements hold, with the same proofs, for the unital calculus on the set of normal \\(x\\) with \\(\\sigma_A(x)\\subseteq K\\).\n\n**Theorem 6.1** (Continuity of the calculus).\n1. (*Uniform continuity.*) For \\(f\\in C(K)\\), the map \\(x\\mapsto f(x)\\) is uniformly continuous on \\(A_K\\).\n2. (*Joint continuity.*) For \\(f,g\\in C(K)\\) and \\(x,y\\in A_K\\), \\[\n\\begin{gathered}\n\\|f(x)-g(y)\\|\\\\\n\\leq\\|f-g\\|_K+\\|g(x)-g(y)\\|.\n\\end{gathered}\n\\]\n3. (*Spectra move little.*) If \\(x_0\\in A\\) is normal and \\(y\\in A\\) is arbitrary, then every point of \\(\\sigma'_A(y)\\) lies within \\(\\|y-x_0\\|\\) of \\(\\sigma'_A(x_0)\\).\n4. (*Open sets.*) Let \\(U\\subseteq\\mathbb C\\) be open and \\(f\\in C(U)\\). Then \\(x\\mapsto f(x)\\) is continuous on the set of normal \\(x\\) with \\(\\sigma'_A(x)\\subseteq U\\), which is relatively open in the set of normal elements.\n\n**Proof.** In the unital calculus on \\(A=\\{0\\}\\), all conclusions have the immediate zero interpretation. Otherwise all the spectra in use are nonempty. (1) If \\(K=\\varnothing\\), the domain \\(A_K\\) is empty, so the assertion is vacuous. Let \\(M=\\max_K|\\lambda|\\). If \\(M=0\\), every \\(x\\in A_K\\) has norm zero by Theorem 1.3; the domain contains at most the element zero and the assertion is immediate. Now suppose \\(M>0\\), and let \\(\\varepsilon>0\\). For \\(x\\in A_K\\), \\(\\|x\\|=r(x)\\leq M\\) by Theorem 1.3. Choose \\(p(\\lambda)=\\sum_{j,k\\leq n}c_{jk}\\lambda^j\\bar\\lambda^k\\) with \\(\\|f-p\\|_K<\\varepsilon/3\\) ([density results](stone-weierstrass-c0.md#oa-fnd-sw-17)). For \\(x\\in A_K\\), \\(p(x)=\\sum c_{jk}x^j(x^*)^k\\) and \\(\\|f(x)-p(x)\\|\\leq\\|f-p\\|_K<\\varepsilon/3\\), by Theorem 5.1(2) and (4) in \\(\\widetilde A\\). The constant term cancels. For \\(j+k\\geq1\\) and \\(\\|x\\|,\\|y\\|\\leq M\\), telescoping each factor gives \\(\\|x^j(x^*)^k-y^j(y^*)^k\\|\\leq(j+k)M^{j+k-1}\\|x-y\\|\\). Thus \\(\\|p(x)-p(y)\\|\\leq L\\|x-y\\|\\), where \\(L=\\sum_{j+k\\geq1}|c_{jk}|(j+k)M^{j+k-1}\\), and\n\\[\n\\begin{gathered}\n\\|f(x)-f(y)\\|<\\tfrac{2\\varepsilon}3+L\\|x-y\\|<\\varepsilon\\\\\n\\text{when }\\|x-y\\|<\\varepsilon/(3\\max(1,L)) .\n\\end{gathered}\n\\]\n(2) \\(\\|f(x)-g(x)\\|=\\|(f-g)(x)\\|\\leq\\|f-g\\|_K\\) by Theorem 5.1(4).\n(3) Let \\(d=\\|y-x_0\\|\\) and let \\(\\lambda\\) have distance \\(\\delta>d\\) from \\(\\sigma'(x_0)\\). In \\(\\widetilde A\\), \\(x_0-\\lambda\\) is normal and invertible, and its inverse is the calculus of \\(t\\mapsto(t-\\lambda)^{-1}\\), of norm \\(1/\\delta\\) by Theorem 5.1(4). Then \\(y-\\lambda=(x_0-\\lambda)\\big(1+(x_0-\\lambda)^{-1}(y-x_0)\\big)\\). The bracket is invertible by the [Neumann series](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-02), because \\(\\|(x_0-\\lambda)^{-1}(y-x_0)\\|\\leq d/\\delta<1\\). So \\(\\lambda\\notin\\sigma'(y)\\).\n(4) Let \\(x_0\\) be normal with \\(\\sigma'(x_0)\\subseteq U\\). The compact set \\(\\sigma'(x_0)\\) has a positive distance \\(3\\eta\\) from the closed set \\(\\mathbb C\\setminus U\\) (take \\(\\eta=1\\) if \\(U=\\mathbb C\\)). The set \\(K=\\{\\lambda:\\operatorname{dist}(\\lambda,\\sigma'(x_0))\\leq\\eta\\}\\) is compact and lies in \\(U\\). By (3), every normal \\(x\\) with \\(\\|x-x_0\\|<\\eta\\) lies in \\(A_K\\). Apply (1) to \\(f|_K\\). \\(\\square\\)\n\nThe compact set \\(K\\) cannot be dropped from (1): for \\(f(t)=t^2\\) and self-adjoint \\(h\\), \\(\\|(h+\\delta)^2-h^2\\|=\\|2\\delta h+\\delta^2\\|\\) is unbounded in \\(h\\). For the powers \\(t^\\alpha\\) with \\(0<\\alpha\\leq1\\), Corollary 9.2 gives a modulus of continuity on all positive elements at once. Part (4) is the continuous analogue of the [continuity of the holomorphic calculus](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-13).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
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    },
    {
      "id": "OA-FND-CF-09",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "7. Absolute value, Jordan decomposition and unitaries",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "anchor": "oa-fnd-cf-09",
      "proof_locus": {
        "line": 375,
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      },
      "full_conditions_and_proof": "### 7. Absolute value, Jordan decomposition and unitaries\n\n**Definition 7.1.** For \\(h\\in A_h\\) put \\(|h|=(h^2)^{1/2}\\), \\(h_+=\\frac12(|h|+h)\\) and \\(h_-=\\frac12(|h|-h)\\). By the composition rule (Theorem 5.1(5)), these are the calculi of \\(t\\mapsto|t|\\), \\(t\\mapsto\\max(t,0)\\) and \\(t\\mapsto\\max(-t,0)\\) at \\(h\\). They vanish at \\(0\\), so they lie in \\(C^*(h)\\subseteq A\\) (Theorem 5.3). They are the *absolute value*, the *positive part* and the *negative part* of \\(h\\), and \\(h=h_+-h_-\\) is its *Jordan decomposition*.\n\n**Proposition 7.2.** Let \\(h\\in A_h\\).\n1. \\(h=h_+-h_-\\), \\(|h|=h_++h_-\\), \\(h_+h_-=h_-h_+=0\\), and the quasi-spectra of \\(h_+\\), \\(h_-\\) and \\(|h|\\) lie in \\([0,\\infty)\\). Also \\(\\|h_\\pm\\|\\leq\\|h\\|=\\||h|\\|=\\max(\\|h_+\\|,\\|h_-\\|)\\).\n2. (*Uniqueness.*) If \\(h=a-b\\) with \\(a,b\\in A_h\\), \\(\\sigma'(a)\\cup\\sigma'(b)\\subseteq[0,\\infty)\\) and \\(ab=0\\), then \\(a=h_+\\) and \\(b=h_-\\).\n\n**Proof.** (1) These are identities between continuous functions on \\(\\sigma'(h)\\subseteq\\mathbb R\\) (Proposition 1.5): \\(t=t_+-t_-\\), \\(|t|=t_++t_-\\), \\(t_+t_-=0\\), \\(t_\\pm\\geq0\\), \\(t_\\pm\\leq|t|\\), and \\(\\max|t|=\\max(\\max t_+,\\max t_-)\\). Apply Theorem 5.1(2)–(4) in \\(\\widetilde A\\).\n(2) From \\(ab=0\\) we get \\(ba=(ab)^*=0\\), so \\(a\\) and \\(b\\) commute, and \\(D=C^*(a,b)\\) is commutative. By Theorem 2.1, identify \\(D\\) with \\(C_0(\\Omega)\\). The functions \\(\\hat a,\\hat b\\) are real, and their values lie in \\(\\sigma'_D(a)=\\sigma'_A(a)\\) and \\(\\sigma'_A(b)\\): the values of a Gelfand transform lie in the quasi-spectrum ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)), and quasi-spectra do not depend on the C\\*-subalgebra (Theorem 3.2(2)). So \\(\\hat a,\\hat b\\geq0\\), and \\(\\hat a\\hat b=0\\). Hence \\(\\max(\\hat h,0)=\\hat a\\) at every point. By Theorem 5.3(4), \\(\\widehat{h_+}=\\max(\\hat h,0)\\). Since \\(\\mathcal G\\) is injective, \\(h_+=a\\), and then \\(h_-=h_+-h=b\\). \\(\\square\\)\n\nPart (1) says in particular that every self-adjoint element is a difference of two elements with nonnegative quasi-spectrum, each of norm at most \\(\\|h\\|\\).\n\n**Proposition 7.3** (Unitaries span a unital C\\*-algebra). Let \\(A\\) be a C\\*-algebra with identity.\n1. If \\(h\\in A_h\\) and \\(\\|h\\|\\leq1\\), then \\(u=h+i(1-h^2)^{1/2}\\) is unitary and \\(h=\\frac12(u+u^*)\\).\n2. If \\(\\|x\\|\\leq1\\), then \\(x=\\frac12(u_1+u_2)+\\frac i2(u_3+u_4)\\) with unitaries \\(u_j\\). Every \\(x\\) is a combination \\(\\sum_{j=1}^4c_ju_j\\) of four unitaries with \\(\\sum_j|c_j|\\leq2\\|x\\|\\).\n3. If \\(A\\) is not unital, every element of \\(A\\) is such a combination of unitaries of \\(\\widetilde A\\).\n\n**Proof.** (1) \\(\\sigma(h)\\subseteq[-1,1]\\) by Proposition 1.5(3). Let \\(f(t)=t+i\\sqrt{1-t^2}\\) there. Then \\(|f|=1\\) and \\(\\operatorname{Re}f(t)=t\\), so Theorem 5.1(2) gives \\(u^*u=uu^*=|f|^2(h)=1\\) and \\(\\frac12(u+u^*)=(\\operatorname{Re}f)(h)=h\\).\n(2) Write \\(x=x_1+ix_2\\) with \\(\\|x_j\\|\\leq1\\) (Proposition 1.2(1)). By (1), \\(x_1=\\frac12(u_1+u_1^*)\\) and \\(x_2=\\frac12(u_3+u_3^*)\\); take \\(u_2=u_1^*\\) and \\(u_4=u_3^*\\). For general \\(x\\neq0\\), apply this to \\(x/\\|x\\|\\).\n(3) Apply (2) in \\(\\widetilde A\\). \\(\\square\\)\n\n**Exercise 7.4** (medium; Unitaries with a gap in the spectrum). In a C\\*-algebra \\(A\\) with identity, let \\(u\\in U(A)\\) with \\(\\sigma(u)\\neq\\mathbb T\\). Prove that \\(u=\\exp(ih)\\) for some \\(h\\in A_h\\).\n\n*Solution.* In the zero algebra take \\(h=0\\). Otherwise pick \\(\\theta_0\\) with \\(e^{i\\theta_0}\\notin\\sigma(u)\\). The map \\(\\theta\\mapsto e^{i\\theta}\\) is a homeomorphism of \\((\\theta_0,\\theta_0+2\\pi)\\) onto \\(\\mathbb T\\setminus\\{e^{i\\theta_0}\\}\\); let \\(g\\) be its inverse, a real continuous function on the compact set \\(\\sigma(u)\\subseteq\\mathbb T\\setminus\\{e^{i\\theta_0}\\}\\) (Proposition 1.5(2)). Put \\(h=g(u)\\), which is self-adjoint by Theorem 5.1(9). The power series \\(\\exp(ih)\\) is the holomorphic calculus of \\(e^{i\\lambda}\\) at \\(h\\) ([the exponential](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14)), which equals the continuous calculus by Theorem 5.1(10). By the composition rule (Theorem 5.1(5)), \\(\\exp(ih)=(e^{ig})(u)=\\iota(u)=u\\), since \\(e^{ig(\\lambda)}=\\lambda\\) on \\(\\sigma(u)\\).\n\n**Exercise 7.5** (medium; Not every unitary is an exponential). In \\(C(\\mathbb T)\\), the unitary \\(u(\\lambda)=\\lambda\\) is not \\(\\exp(ih)\\) for any \\(h\\in C(\\mathbb T)\\), self-adjoint or not.\n\n*Solution.* In \\(C(\\mathbb T)\\), \\(\\exp(ih)\\) is the function \\(e^{ih}\\), because evaluation at a point is a character and passes through the power series. So \\(u=\\exp(ih)\\) would give \\(\\lambda=e^{ih(\\lambda)}\\) with \\(h\\) continuous. This is the case \\(g=ih\\) of the [example on the index group of \\(C(\\mathbb T)\\)](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14), which shows that the identity function is not \\(e^g\\) for any \\(g\\in C(\\mathbb T)\\): the continuous function \\(t\\mapsto g(e^{it})-it\\) on \\([0,2\\pi]\\) takes values in \\(2\\pi i\\mathbb Z\\), so it is constant, but its values at \\(0\\) and \\(2\\pi\\) differ by \\(2\\pi i\\). So Exercise 7.4 cannot be extended to all unitaries.\n\n**Exercise 7.6** (medium; Self-adjointness through the norm). In a nontrivial C\\*-algebra \\(A\\) with identity, prove that \\(x\\in A\\) is self-adjoint exactly when \\(\\lim_{t\\to0}\\frac1t(\\|1+itx\\|-1)=0\\). (In the zero algebra \\(\\|1+itx\\|=0\\), so the quotient is \\(-1/t\\) and the statement fails.)\n\n*Solution.* If \\(x=h\\) is self-adjoint, then \\[\n\\begin{gathered}\n\\|1+ith\\|^2\\\\\n=\\|(1-ith)(1+ith)\\|\\\\\n=\\|1+t^2h^2\\|\\\\\n=1+t^2\\|h\\|^2\n\\end{gathered}\n\\] by Theorem 5.1(4), so \\(0\\leq\\|1+ith\\|-1\\leq\\frac12t^2\\|h\\|^2\\), and the quotient tends to \\(0\\). Conversely, write \\(x=h+ik\\) with \\(h,k\\in A_h\\) and \\(k\\neq0\\). For real \\(t\\), the self-adjoint part of \\(1+itx=(1-tk)+ith\\) is \\(1-tk\\), and a self-adjoint part has norm at most that of the element (Proposition 1.2(1)). By Proposition 1.5(3) there is \\(\\mu\\in\\sigma(k)\\) with \\(|\\mu|=\\|k\\|>0\\), and \\(\\|1-tk\\|\\geq|1-t\\mu|\\), because \\(1-t\\mu\\in\\sigma(1-tk)\\). For real \\(t\\) of sign opposite to \\(\\mu\\), \\(|1-t\\mu|=1+|t|\\|k\\|\\). Along such \\(t\\to0\\), \\(\\big|\\frac1t(\\|1+itx\\|-1)\\big|\\geq\\|k\\|\\), so the limit is not \\(0\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-10",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "7. Absolute value, Jordan decomposition and unitaries",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 7. Absolute value, Jordan decomposition and unitaries\n\n**Definition 7.1.** For \\(h\\in A_h\\) put \\(|h|=(h^2)^{1/2}\\), \\(h_+=\\frac12(|h|+h)\\) and \\(h_-=\\frac12(|h|-h)\\). By the composition rule (Theorem 5.1(5)), these are the calculi of \\(t\\mapsto|t|\\), \\(t\\mapsto\\max(t,0)\\) and \\(t\\mapsto\\max(-t,0)\\) at \\(h\\). They vanish at \\(0\\), so they lie in \\(C^*(h)\\subseteq A\\) (Theorem 5.3). They are the *absolute value*, the *positive part* and the *negative part* of \\(h\\), and \\(h=h_+-h_-\\) is its *Jordan decomposition*.\n\n**Proposition 7.2.** Let \\(h\\in A_h\\).\n1. \\(h=h_+-h_-\\), \\(|h|=h_++h_-\\), \\(h_+h_-=h_-h_+=0\\), and the quasi-spectra of \\(h_+\\), \\(h_-\\) and \\(|h|\\) lie in \\([0,\\infty)\\). Also \\(\\|h_\\pm\\|\\leq\\|h\\|=\\||h|\\|=\\max(\\|h_+\\|,\\|h_-\\|)\\).\n2. (*Uniqueness.*) If \\(h=a-b\\) with \\(a,b\\in A_h\\), \\(\\sigma'(a)\\cup\\sigma'(b)\\subseteq[0,\\infty)\\) and \\(ab=0\\), then \\(a=h_+\\) and \\(b=h_-\\).\n\n**Proof.** (1) These are identities between continuous functions on \\(\\sigma'(h)\\subseteq\\mathbb R\\) (Proposition 1.5): \\(t=t_+-t_-\\), \\(|t|=t_++t_-\\), \\(t_+t_-=0\\), \\(t_\\pm\\geq0\\), \\(t_\\pm\\leq|t|\\), and \\(\\max|t|=\\max(\\max t_+,\\max t_-)\\). Apply Theorem 5.1(2)–(4) in \\(\\widetilde A\\).\n(2) From \\(ab=0\\) we get \\(ba=(ab)^*=0\\), so \\(a\\) and \\(b\\) commute, and \\(D=C^*(a,b)\\) is commutative. By Theorem 2.1, identify \\(D\\) with \\(C_0(\\Omega)\\). The functions \\(\\hat a,\\hat b\\) are real, and their values lie in \\(\\sigma'_D(a)=\\sigma'_A(a)\\) and \\(\\sigma'_A(b)\\): the values of a Gelfand transform lie in the quasi-spectrum ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)), and quasi-spectra do not depend on the C\\*-subalgebra (Theorem 3.2(2)). So \\(\\hat a,\\hat b\\geq0\\), and \\(\\hat a\\hat b=0\\). Hence \\(\\max(\\hat h,0)=\\hat a\\) at every point. By Theorem 5.3(4), \\(\\widehat{h_+}=\\max(\\hat h,0)\\). Since \\(\\mathcal G\\) is injective, \\(h_+=a\\), and then \\(h_-=h_+-h=b\\). \\(\\square\\)\n\nPart (1) says in particular that every self-adjoint element is a difference of two elements with nonnegative quasi-spectrum, each of norm at most \\(\\|h\\|\\).\n\n**Proposition 7.3** (Unitaries span a unital C\\*-algebra). Let \\(A\\) be a C\\*-algebra with identity.\n1. If \\(h\\in A_h\\) and \\(\\|h\\|\\leq1\\), then \\(u=h+i(1-h^2)^{1/2}\\) is unitary and \\(h=\\frac12(u+u^*)\\).\n2. If \\(\\|x\\|\\leq1\\), then \\(x=\\frac12(u_1+u_2)+\\frac i2(u_3+u_4)\\) with unitaries \\(u_j\\). Every \\(x\\) is a combination \\(\\sum_{j=1}^4c_ju_j\\) of four unitaries with \\(\\sum_j|c_j|\\leq2\\|x\\|\\).\n3. If \\(A\\) is not unital, every element of \\(A\\) is such a combination of unitaries of \\(\\widetilde A\\).\n\n**Proof.** (1) \\(\\sigma(h)\\subseteq[-1,1]\\) by Proposition 1.5(3). Let \\(f(t)=t+i\\sqrt{1-t^2}\\) there. Then \\(|f|=1\\) and \\(\\operatorname{Re}f(t)=t\\), so Theorem 5.1(2) gives \\(u^*u=uu^*=|f|^2(h)=1\\) and \\(\\frac12(u+u^*)=(\\operatorname{Re}f)(h)=h\\).\n(2) Write \\(x=x_1+ix_2\\) with \\(\\|x_j\\|\\leq1\\) (Proposition 1.2(1)). By (1), \\(x_1=\\frac12(u_1+u_1^*)\\) and \\(x_2=\\frac12(u_3+u_3^*)\\); take \\(u_2=u_1^*\\) and \\(u_4=u_3^*\\). For general \\(x\\neq0\\), apply this to \\(x/\\|x\\|\\).\n(3) Apply (2) in \\(\\widetilde A\\). \\(\\square\\)\n\n**Exercise 7.4** (medium; Unitaries with a gap in the spectrum). In a C\\*-algebra \\(A\\) with identity, let \\(u\\in U(A)\\) with \\(\\sigma(u)\\neq\\mathbb T\\). Prove that \\(u=\\exp(ih)\\) for some \\(h\\in A_h\\).\n\n*Solution.* In the zero algebra take \\(h=0\\). Otherwise pick \\(\\theta_0\\) with \\(e^{i\\theta_0}\\notin\\sigma(u)\\). The map \\(\\theta\\mapsto e^{i\\theta}\\) is a homeomorphism of \\((\\theta_0,\\theta_0+2\\pi)\\) onto \\(\\mathbb T\\setminus\\{e^{i\\theta_0}\\}\\); let \\(g\\) be its inverse, a real continuous function on the compact set \\(\\sigma(u)\\subseteq\\mathbb T\\setminus\\{e^{i\\theta_0}\\}\\) (Proposition 1.5(2)). Put \\(h=g(u)\\), which is self-adjoint by Theorem 5.1(9). The power series \\(\\exp(ih)\\) is the holomorphic calculus of \\(e^{i\\lambda}\\) at \\(h\\) ([the exponential](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14)), which equals the continuous calculus by Theorem 5.1(10). By the composition rule (Theorem 5.1(5)), \\(\\exp(ih)=(e^{ig})(u)=\\iota(u)=u\\), since \\(e^{ig(\\lambda)}=\\lambda\\) on \\(\\sigma(u)\\).\n\n**Exercise 7.5** (medium; Not every unitary is an exponential). In \\(C(\\mathbb T)\\), the unitary \\(u(\\lambda)=\\lambda\\) is not \\(\\exp(ih)\\) for any \\(h\\in C(\\mathbb T)\\), self-adjoint or not.\n\n*Solution.* In \\(C(\\mathbb T)\\), \\(\\exp(ih)\\) is the function \\(e^{ih}\\), because evaluation at a point is a character and passes through the power series. So \\(u=\\exp(ih)\\) would give \\(\\lambda=e^{ih(\\lambda)}\\) with \\(h\\) continuous. This is the case \\(g=ih\\) of the [example on the index group of \\(C(\\mathbb T)\\)](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14), which shows that the identity function is not \\(e^g\\) for any \\(g\\in C(\\mathbb T)\\): the continuous function \\(t\\mapsto g(e^{it})-it\\) on \\([0,2\\pi]\\) takes values in \\(2\\pi i\\mathbb Z\\), so it is constant, but its values at \\(0\\) and \\(2\\pi\\) differ by \\(2\\pi i\\). So Exercise 7.4 cannot be extended to all unitaries.\n\n**Exercise 7.6** (medium; Self-adjointness through the norm). In a nontrivial C\\*-algebra \\(A\\) with identity, prove that \\(x\\in A\\) is self-adjoint exactly when \\(\\lim_{t\\to0}\\frac1t(\\|1+itx\\|-1)=0\\). (In the zero algebra \\(\\|1+itx\\|=0\\), so the quotient is \\(-1/t\\) and the statement fails.)\n\n*Solution.* If \\(x=h\\) is self-adjoint, then \\[\n\\begin{gathered}\n\\|1+ith\\|^2\\\\\n=\\|(1-ith)(1+ith)\\|\\\\\n=\\|1+t^2h^2\\|\\\\\n=1+t^2\\|h\\|^2\n\\end{gathered}\n\\] by Theorem 5.1(4), so \\(0\\leq\\|1+ith\\|-1\\leq\\frac12t^2\\|h\\|^2\\), and the quotient tends to \\(0\\). Conversely, write \\(x=h+ik\\) with \\(h,k\\in A_h\\) and \\(k\\neq0\\). For real \\(t\\), the self-adjoint part of \\(1+itx=(1-tk)+ith\\) is \\(1-tk\\), and a self-adjoint part has norm at most that of the element (Proposition 1.2(1)). By Proposition 1.5(3) there is \\(\\mu\\in\\sigma(k)\\) with \\(|\\mu|=\\|k\\|>0\\), and \\(\\|1-tk\\|\\geq|1-t\\mu|\\), because \\(1-t\\mu\\in\\sigma(1-tk)\\). For real \\(t\\) of sign opposite to \\(\\mu\\), \\(|1-t\\mu|=1+|t|\\|k\\|\\). Along such \\(t\\to0\\), \\(\\big|\\frac1t(\\|1+itx\\|-1)\\big|\\geq\\|k\\|\\), so the limit is not \\(0\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CF-15",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "8. The positive cone and the order",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 8. The positive cone and the order\n\n**Lemma 8.1.** Let \\(h\\in A_h\\) and \\(t\\geq\\|h\\|\\). Then \\(\\sigma'_A(h)\\subseteq[0,\\infty)\\) if and only if \\(\\|t-h\\|\\leq t\\), the norm taken in \\(\\widetilde A\\).\n\n**Proof.** \\(\\sigma'(h)\\subseteq[-\\|h\\|,\\|h\\|]\\subseteq[-t,t]\\) by Proposition 1.5(3). The element \\(t-h\\) of \\(\\widetilde A\\) is self-adjoint, and by Theorem 5.1(4), \\[\n\\begin{gathered}\n\\|t-h\\|\\\\\n=\\max_{\\lambda\\in\\sigma'(h)}|t-\\lambda|\\\\\n=\\max_{\\lambda\\in\\sigma'(h)}(t-\\lambda).\n\\end{gathered}\n\\] This is at most \\(t\\) exactly when every \\(\\lambda\\in\\sigma'(h)\\) is \\(\\geq0\\). \\(\\square\\)\n\n**Theorem 8.2** (The positive cone). For \\(h\\in A_h\\) the following are equivalent:\n- (i) \\(\\sigma'_A(h)\\subseteq[0,\\infty)\\);\n- (ii) \\(h=y^*y\\) for some \\(y\\in A\\);\n- (iii) \\(h=k^2\\) for some \\(k\\in A_h\\).\n\nThese elements form a closed convex cone \\(A_+\\subseteq A_h\\), and \\(A_+\\cap(-A_+)=\\{0\\}\\).\n\n**Proof.** *\\(A_+\\), defined by (i), is a closed convex cone.* It is closed under multiplication by \\(c\\geq0\\), since \\(\\sigma'(ch)=c\\sigma'(h)\\). Let \\(a,b\\) satisfy (i), and put \\(s=\\|a\\|\\), \\(t=\\|b\\|\\). By the lemma, \\[\n\\begin{gathered}\n\\|(s+t)-(a+b)\\|\\\\\n\\leq\\|s-a\\|+\\|t-b\\|\\\\\n\\leq s+t,\n\\end{gathered}\n\\] and \\(s+t\\geq\\|a+b\\|\\); so \\(a+b\\) satisfies (i). If \\(h_n\\to h\\) with \\(h_n\\) satisfying (i), fix \\(t\\geq\\sup_n\\|h_n\\|\\). Then \\(\\|t-h_n\\|\\leq t\\) for all \\(n\\), so \\(\\|t-h\\|\\leq t\\) and \\(t\\geq\\|h\\|\\), and \\(h\\) satisfies (i). If \\(h\\) and \\(-h\\) satisfy (i), then \\(\\sigma'(h)=\\{0\\}\\) and \\(\\|h\\|=r(h)=0\\) by Theorem 1.3.\n*(i) ⇒ (iii).* \\(k=h^{1/2}\\) lies in \\(A_h\\) by Theorem 5.3, since \\(\\sqrt t\\) vanishes at \\(0\\).\n*(iii) ⇒ (ii).* Take \\(y=k\\).\n*(iii) ⇒ (i).* \\(\\sigma'(k^2)=\\{\\lambda^2:\\lambda\\in\\sigma'(k)\\}\\subseteq[0,\\infty)\\), by spectral mapping (Theorem 5.1(3) in \\(\\widetilde A\\)) and because \\(\\sigma'(k)\\) is real (Proposition 1.5(3)).\n*(ii) ⇒ (i).* Let \\(h=y^*y\\). By Proposition 7.2, \\(h=u^2-v^2\\) with \\(u=h_+^{1/2}\\), \\(v=h_-^{1/2}\\) in \\(A_h\\) and \\(uv=vu=0\\). Put \\(w=yv\\). Then\n\\[\nw^*w=vy^*yv=v(u^2-v^2)v=-v^4 .\n\\]\nWrite \\(w=k_1+ik_2\\) with \\(k_1,k_2\\in A_h\\). Then \\(ww^*=-w^*w+2k_1^2+2k_2^2=v^4+2k_1^2+2k_2^2\\), which lies in the convex cone \\(A_+\\) by (iii) ⇒ (i). Since \\(\\sigma'(w^*w)=\\sigma'(ww^*)\\) ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)), also \\(\\sigma'(w^*w)\\subseteq[0,\\infty)\\). So \\(w^*w=-v^4\\) lies in \\(A_+\\cap(-A_+)=\\{0\\}\\). Hence \\(v^4=0\\), so \\(v=0\\) (because \\(\\|v\\|^4=\\|v^4\\|\\) by Theorem 1.3(1)), \\(h_-=0\\), and \\(h=h_+\\) satisfies (i). \\(\\square\\)\n\n**Definition 8.3.** An element of \\(A_+\\) is *positive*, written \\(h\\geq0\\). For \\(h,k\\in A_h\\), \\(h\\leq k\\) means \\(k-h\\in A_+\\); by the theorem this is a partial order on \\(A_h\\), compatible with sums and with multiplication by nonnegative scalars. The *absolute value* of \\(x\\in A\\) is \\(|x|=(x^*x)^{1/2}\\). For \\(h\\in A_h\\) it agrees with Definition 7.1, by the composition rule (Theorem 5.1(5)).\n\n**Remark 8.4** (Where positivity is computed). Condition (i) uses the quasi-spectrum, which is the same in every C\\*-subalgebra containing \\(h\\) and in \\(\\widetilde A\\) (Theorem 3.2(2)). So \\(A_+=A\\cap\\widetilde A_+\\), and \\(B_+=B\\cap A_+\\) for a C\\*-subalgebra \\(B\\).\n\nThe next proposition collects the rules for working with the order; they are used constantly below. Inequalities involving scalars are read in \\(\\widetilde A\\).\n\n**Proposition 8.5** (Working with the order). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\) and \\(t\\in\\mathbb R\\), \\(h\\leq t\\) in the forced unitization if and only if \\(\\sigma'(h)\\subseteq(-\\infty,t]\\). If \\(A\\) is unital, comparison with its own identity instead gives \\(h\\leq t1_A\\) if and only if \\(\\sigma_A(h)\\subseteq(-\\infty,t]\\). In particular \\(-\\|h\\|\\leq h\\leq\\|h\\|\\). If \\(b\\geq0\\) and \\(-b\\leq h\\leq b\\), then \\(\\|h\\|\\leq\\|b\\|\\).\n2. \\(x^*x\\leq\\|x\\|^2\\) for \\(x\\in\\widetilde A\\).\n3. If \\(h\\leq k\\), then \\(c^*hc\\leq c^*kc\\) for every \\(c\\in\\widetilde A\\).\n4. If \\(0\\leq a\\leq b\\), then \\(\\|a\\|\\leq\\|b\\|\\). For \\(a\\geq0\\): \\(a\\leq1\\) if and only if \\(\\|a\\|\\leq1\\).\n5. If \\(a\\geq0\\), \\(c\\in\\widetilde A\\) and \\(c^*ac=0\\), then \\(a^{1/2}c=0\\) and \\(ac=0\\).\n6. (*Support.*) If \\(0\\leq v\\leq a\\) and \\(ab=0\\) for some \\(b\\in\\widetilde A\\), then \\(vb=0\\).\n7. Let \\(A\\) be unital and nontrivial, \\(\\varepsilon>0\\), and \\(\\varepsilon1_A\\leq a\\leq b\\). Then \\(a,b\\) are invertible in \\(A\\) and \\(b^{-1}\\leq a^{-1}\\). In this item all scalar bounds and inverses use \\(A\\) and its own identity \\(1_A\\), not a second forced unitization.\n8. Every \\(h\\in A_h\\) is \\(h_+-h_-\\) with \\(h_\\pm\\in A_+\\) and \\(\\|h_\\pm\\|\\leq\\|h\\|\\). Every \\(x\\in A\\) is a combination of four positive elements of norm at most \\(\\|x\\|\\).\n9. If \\(a,b\\geq0\\) commute, then \\(ab\\geq0\\).\n10. (*Positive square roots.*) Every \\(a\\in A_+\\) has exactly one \\(b\\in A_+\\) with \\(b^2=a\\), namely \\(b=a^{1/2}\\). It lies in \\(C^*(a)\\) and commutes with every element that commutes with \\(a\\). No unit and no invertibility are needed; for \\(A=\\{0\\}\\), \\(b=0\\).\n11. (*Positivity in \\(B(H)\\).*) For \\(a\\in B(H)\\), \\(a\\geq0\\) if and only if \\(\\langle a\\xi,\\xi\\rangle\\geq0\\) for every \\(\\xi\\in H\\); in that case \\(a=a^*\\) automatically. The same holds in every C\\*-subalgebra of \\(B(H)\\).\n12. A \\(*\\)-homomorphism \\(\\pi:A\\to B\\) of C\\*-algebras maps \\(A_+\\) into \\(B_+\\). If \\(\\pi\\) is injective and \\(h\\in A_h\\) has \\(\\pi(h)\\geq0\\), then \\(h\\geq0\\).\n\n**Proof.** (1) \\(\\sigma_{\\widetilde A}(t-h)=t-\\sigma'(h)\\). For the comparison inside a unital \\(A\\), use \\(\\sigma_A(t1_A-h)=t-\\sigma_A(h)\\) and Theorem 8.2: adjoining \\(0\\) to a spectrum does not change whether it is contained in \\([0,\\infty)\\). The scalar norm bounds follow from Proposition 1.5(3). Finally, \\(h\\leq b\\leq\\|b\\|\\) and \\(-h\\leq\\|b\\|\\) give \\(\\sigma'(h)\\subseteq[-\\|b\\|,\\|b\\|]\\), so \\(\\|h\\|=r(h)\\leq\\|b\\|\\).\n(2) \\(\\sigma'(x^*x)\\subseteq[0,\\|x^*x\\|]=[0,\\|x\\|^2]\\) by Theorem 8.2 and Proposition 1.5(3); apply (1).\n(3) \\(k-h=y^*y\\), so \\(c^*(k-h)c=(yc)^*(yc)\\geq0\\).\n(4) \\(a\\leq b\\leq\\|b\\|\\) gives \\(\\sigma'(a)\\subseteq[0,\\|b\\|]\\), so \\(\\|a\\|=r(a)\\leq\\|b\\|\\). The second claim is (1) with \\(t=1\\).\n(5) \\(\\|a^{1/2}c\\|^2=\\|c^*ac\\|=0\\), and \\(ac=a^{1/2}(a^{1/2}c)\\).\n(6) By (3), \\(0\\leq b^*vb\\leq b^*ab=0\\), so \\(b^*vb=0\\), because \\(A_+\\cap(-A_+)=\\{0\\}\\) (Theorem 8.2). Then (5) gives \\(vb=0\\).\n(7) \\(\\sigma(a)\\subseteq[\\varepsilon,\\infty)\\) by (1), so \\(a\\) is invertible, and likewise \\(b\\). Put \\(c=a^{-1/2}ba^{-1/2}\\). By (3), \\(c\\geq a^{-1/2}aa^{-1/2}=1\\), so \\(\\sigma(c)\\subseteq[1,\\infty)\\). The spectrum of an inverse consists of the inverses of the points of the spectrum ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)), so \\(\\sigma(c^{-1})\\subseteq(0,1]\\). So \\(c^{-1}=a^{1/2}b^{-1}a^{1/2}\\leq1\\), and conjugating by \\(a^{-1/2}\\) gives \\(b^{-1}\\leq a^{-1}\\).\n(8) This is Proposition 7.2(1) together with the Cartesian decomposition (Proposition 1.2(1)).\n(9) \\(C^*(a,b)\\) is commutative. By Theorem 2.1 its elements are functions, and positivity there is pointwise nonnegativity; positivity in it agrees with positivity in \\(A\\) (Remark 8.4). The product of two nonnegative functions is nonnegative.\n(10) Existence is Theorem 8.2, (i) ⇒ (iii). Uniqueness: let \\(b\\geq0\\) with \\(b^2=a\\). Then \\(b\\) commutes with \\(a\\), and \\(D=C^*(b)\\) is commutative and contains \\(a\\) and \\(a^{1/2}\\in C^*(a)\\). In \\(D\\cong C_0(\\Omega)\\), \\(\\hat b\\geq0\\) and \\(\\hat b^2=\\hat a\\), so \\(\\hat b=\\sqrt{\\hat a}=\\widehat{a^{1/2}}\\) by Theorem 5.3(4). Hence \\(b=a^{1/2}\\). The commutation claim is Theorem 5.1(6), since \\(a=a^*\\).\n(11) If \\(a=y^*y\\), then \\(\\langle a\\xi,\\xi\\rangle=\\|y\\xi\\|^2\\geq0\\). Conversely, suppose \\(\\langle a\\xi,\\xi\\rangle\\geq0\\) for all \\(\\xi\\). For an inner product linear in the first variable, polarization reads \\(4\\langle T\\xi,\\eta\\rangle=\\sum_{k=0}^3i^k\\langle T(\\xi+i^k\\eta),\\xi+i^k\\eta\\rangle\\). Since \\(\\langle a\\zeta,\\zeta\\rangle=\\langle\\zeta,a\\zeta\\rangle=\\langle a^*\\zeta,\\zeta\\rangle\\) for every \\(\\zeta\\), polarization gives \\(a=a^*\\). Let \\(\\lambda<0\\). Then \\(\\|(a-\\lambda)\\xi\\|\\|\\xi\\|\\geq\\langle(a-\\lambda)\\xi,\\xi\\rangle\\geq|\\lambda|\\|\\xi\\|^2\\). So \\(a-\\lambda\\) is bounded below; it is injective with closed range, and the orthogonal complement of its range is the kernel of \\((a-\\lambda)^*=a-\\lambda\\), which is \\(\\{0\\}\\). So \\(a-\\lambda\\) is invertible. Hence \\(\\sigma(a)\\subseteq[0,\\infty)\\), since it is real (Proposition 1.5). For a C\\*-subalgebra, use Remark 8.4.\n(12) \\(\\pi(y^*y)=\\pi(y)^*\\pi(y)\\). If \\(\\pi\\) is injective, it is an isometric \\(*\\)-isomorphism onto the closed range \\(\\pi(A)\\) (Corollary 4.6), so \\(\\sigma'_A(h)=\\sigma'_{\\pi(A)}(\\pi(h))=\\sigma'_B(\\pi(h))\\) by Theorem 3.2(2). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-16",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "8. The positive cone and the order",
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      "full_conditions_and_proof": "### 8. The positive cone and the order\n\n**Lemma 8.1.** Let \\(h\\in A_h\\) and \\(t\\geq\\|h\\|\\). Then \\(\\sigma'_A(h)\\subseteq[0,\\infty)\\) if and only if \\(\\|t-h\\|\\leq t\\), the norm taken in \\(\\widetilde A\\).\n\n**Proof.** \\(\\sigma'(h)\\subseteq[-\\|h\\|,\\|h\\|]\\subseteq[-t,t]\\) by Proposition 1.5(3). The element \\(t-h\\) of \\(\\widetilde A\\) is self-adjoint, and by Theorem 5.1(4), \\[\n\\begin{gathered}\n\\|t-h\\|\\\\\n=\\max_{\\lambda\\in\\sigma'(h)}|t-\\lambda|\\\\\n=\\max_{\\lambda\\in\\sigma'(h)}(t-\\lambda).\n\\end{gathered}\n\\] This is at most \\(t\\) exactly when every \\(\\lambda\\in\\sigma'(h)\\) is \\(\\geq0\\). \\(\\square\\)\n\n**Theorem 8.2** (The positive cone). For \\(h\\in A_h\\) the following are equivalent:\n- (i) \\(\\sigma'_A(h)\\subseteq[0,\\infty)\\);\n- (ii) \\(h=y^*y\\) for some \\(y\\in A\\);\n- (iii) \\(h=k^2\\) for some \\(k\\in A_h\\).\n\nThese elements form a closed convex cone \\(A_+\\subseteq A_h\\), and \\(A_+\\cap(-A_+)=\\{0\\}\\).\n\n**Proof.** *\\(A_+\\), defined by (i), is a closed convex cone.* It is closed under multiplication by \\(c\\geq0\\), since \\(\\sigma'(ch)=c\\sigma'(h)\\). Let \\(a,b\\) satisfy (i), and put \\(s=\\|a\\|\\), \\(t=\\|b\\|\\). By the lemma, \\[\n\\begin{gathered}\n\\|(s+t)-(a+b)\\|\\\\\n\\leq\\|s-a\\|+\\|t-b\\|\\\\\n\\leq s+t,\n\\end{gathered}\n\\] and \\(s+t\\geq\\|a+b\\|\\); so \\(a+b\\) satisfies (i). If \\(h_n\\to h\\) with \\(h_n\\) satisfying (i), fix \\(t\\geq\\sup_n\\|h_n\\|\\). Then \\(\\|t-h_n\\|\\leq t\\) for all \\(n\\), so \\(\\|t-h\\|\\leq t\\) and \\(t\\geq\\|h\\|\\), and \\(h\\) satisfies (i). If \\(h\\) and \\(-h\\) satisfy (i), then \\(\\sigma'(h)=\\{0\\}\\) and \\(\\|h\\|=r(h)=0\\) by Theorem 1.3.\n*(i) ⇒ (iii).* \\(k=h^{1/2}\\) lies in \\(A_h\\) by Theorem 5.3, since \\(\\sqrt t\\) vanishes at \\(0\\).\n*(iii) ⇒ (ii).* Take \\(y=k\\).\n*(iii) ⇒ (i).* \\(\\sigma'(k^2)=\\{\\lambda^2:\\lambda\\in\\sigma'(k)\\}\\subseteq[0,\\infty)\\), by spectral mapping (Theorem 5.1(3) in \\(\\widetilde A\\)) and because \\(\\sigma'(k)\\) is real (Proposition 1.5(3)).\n*(ii) ⇒ (i).* Let \\(h=y^*y\\). By Proposition 7.2, \\(h=u^2-v^2\\) with \\(u=h_+^{1/2}\\), \\(v=h_-^{1/2}\\) in \\(A_h\\) and \\(uv=vu=0\\). Put \\(w=yv\\). Then\n\\[\nw^*w=vy^*yv=v(u^2-v^2)v=-v^4 .\n\\]\nWrite \\(w=k_1+ik_2\\) with \\(k_1,k_2\\in A_h\\). Then \\(ww^*=-w^*w+2k_1^2+2k_2^2=v^4+2k_1^2+2k_2^2\\), which lies in the convex cone \\(A_+\\) by (iii) ⇒ (i). Since \\(\\sigma'(w^*w)=\\sigma'(ww^*)\\) ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)), also \\(\\sigma'(w^*w)\\subseteq[0,\\infty)\\). So \\(w^*w=-v^4\\) lies in \\(A_+\\cap(-A_+)=\\{0\\}\\). Hence \\(v^4=0\\), so \\(v=0\\) (because \\(\\|v\\|^4=\\|v^4\\|\\) by Theorem 1.3(1)), \\(h_-=0\\), and \\(h=h_+\\) satisfies (i). \\(\\square\\)\n\n**Definition 8.3.** An element of \\(A_+\\) is *positive*, written \\(h\\geq0\\). For \\(h,k\\in A_h\\), \\(h\\leq k\\) means \\(k-h\\in A_+\\); by the theorem this is a partial order on \\(A_h\\), compatible with sums and with multiplication by nonnegative scalars. The *absolute value* of \\(x\\in A\\) is \\(|x|=(x^*x)^{1/2}\\). For \\(h\\in A_h\\) it agrees with Definition 7.1, by the composition rule (Theorem 5.1(5)).\n\n**Remark 8.4** (Where positivity is computed). Condition (i) uses the quasi-spectrum, which is the same in every C\\*-subalgebra containing \\(h\\) and in \\(\\widetilde A\\) (Theorem 3.2(2)). So \\(A_+=A\\cap\\widetilde A_+\\), and \\(B_+=B\\cap A_+\\) for a C\\*-subalgebra \\(B\\).\n\nThe next proposition collects the rules for working with the order; they are used constantly below. Inequalities involving scalars are read in \\(\\widetilde A\\).\n\n**Proposition 8.5** (Working with the order). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\) and \\(t\\in\\mathbb R\\), \\(h\\leq t\\) in the forced unitization if and only if \\(\\sigma'(h)\\subseteq(-\\infty,t]\\). If \\(A\\) is unital, comparison with its own identity instead gives \\(h\\leq t1_A\\) if and only if \\(\\sigma_A(h)\\subseteq(-\\infty,t]\\). In particular \\(-\\|h\\|\\leq h\\leq\\|h\\|\\). If \\(b\\geq0\\) and \\(-b\\leq h\\leq b\\), then \\(\\|h\\|\\leq\\|b\\|\\).\n2. \\(x^*x\\leq\\|x\\|^2\\) for \\(x\\in\\widetilde A\\).\n3. If \\(h\\leq k\\), then \\(c^*hc\\leq c^*kc\\) for every \\(c\\in\\widetilde A\\).\n4. If \\(0\\leq a\\leq b\\), then \\(\\|a\\|\\leq\\|b\\|\\). For \\(a\\geq0\\): \\(a\\leq1\\) if and only if \\(\\|a\\|\\leq1\\).\n5. If \\(a\\geq0\\), \\(c\\in\\widetilde A\\) and \\(c^*ac=0\\), then \\(a^{1/2}c=0\\) and \\(ac=0\\).\n6. (*Support.*) If \\(0\\leq v\\leq a\\) and \\(ab=0\\) for some \\(b\\in\\widetilde A\\), then \\(vb=0\\).\n7. Let \\(A\\) be unital and nontrivial, \\(\\varepsilon>0\\), and \\(\\varepsilon1_A\\leq a\\leq b\\). Then \\(a,b\\) are invertible in \\(A\\) and \\(b^{-1}\\leq a^{-1}\\). In this item all scalar bounds and inverses use \\(A\\) and its own identity \\(1_A\\), not a second forced unitization.\n8. Every \\(h\\in A_h\\) is \\(h_+-h_-\\) with \\(h_\\pm\\in A_+\\) and \\(\\|h_\\pm\\|\\leq\\|h\\|\\). Every \\(x\\in A\\) is a combination of four positive elements of norm at most \\(\\|x\\|\\).\n9. If \\(a,b\\geq0\\) commute, then \\(ab\\geq0\\).\n10. (*Positive square roots.*) Every \\(a\\in A_+\\) has exactly one \\(b\\in A_+\\) with \\(b^2=a\\), namely \\(b=a^{1/2}\\). It lies in \\(C^*(a)\\) and commutes with every element that commutes with \\(a\\). No unit and no invertibility are needed; for \\(A=\\{0\\}\\), \\(b=0\\).\n11. (*Positivity in \\(B(H)\\).*) For \\(a\\in B(H)\\), \\(a\\geq0\\) if and only if \\(\\langle a\\xi,\\xi\\rangle\\geq0\\) for every \\(\\xi\\in H\\); in that case \\(a=a^*\\) automatically. The same holds in every C\\*-subalgebra of \\(B(H)\\).\n12. A \\(*\\)-homomorphism \\(\\pi:A\\to B\\) of C\\*-algebras maps \\(A_+\\) into \\(B_+\\). If \\(\\pi\\) is injective and \\(h\\in A_h\\) has \\(\\pi(h)\\geq0\\), then \\(h\\geq0\\).\n\n**Proof.** (1) \\(\\sigma_{\\widetilde A}(t-h)=t-\\sigma'(h)\\). For the comparison inside a unital \\(A\\), use \\(\\sigma_A(t1_A-h)=t-\\sigma_A(h)\\) and Theorem 8.2: adjoining \\(0\\) to a spectrum does not change whether it is contained in \\([0,\\infty)\\). The scalar norm bounds follow from Proposition 1.5(3). Finally, \\(h\\leq b\\leq\\|b\\|\\) and \\(-h\\leq\\|b\\|\\) give \\(\\sigma'(h)\\subseteq[-\\|b\\|,\\|b\\|]\\), so \\(\\|h\\|=r(h)\\leq\\|b\\|\\).\n(2) \\(\\sigma'(x^*x)\\subseteq[0,\\|x^*x\\|]=[0,\\|x\\|^2]\\) by Theorem 8.2 and Proposition 1.5(3); apply (1).\n(3) \\(k-h=y^*y\\), so \\(c^*(k-h)c=(yc)^*(yc)\\geq0\\).\n(4) \\(a\\leq b\\leq\\|b\\|\\) gives \\(\\sigma'(a)\\subseteq[0,\\|b\\|]\\), so \\(\\|a\\|=r(a)\\leq\\|b\\|\\). The second claim is (1) with \\(t=1\\).\n(5) \\(\\|a^{1/2}c\\|^2=\\|c^*ac\\|=0\\), and \\(ac=a^{1/2}(a^{1/2}c)\\).\n(6) By (3), \\(0\\leq b^*vb\\leq b^*ab=0\\), so \\(b^*vb=0\\), because \\(A_+\\cap(-A_+)=\\{0\\}\\) (Theorem 8.2). Then (5) gives \\(vb=0\\).\n(7) \\(\\sigma(a)\\subseteq[\\varepsilon,\\infty)\\) by (1), so \\(a\\) is invertible, and likewise \\(b\\). Put \\(c=a^{-1/2}ba^{-1/2}\\). By (3), \\(c\\geq a^{-1/2}aa^{-1/2}=1\\), so \\(\\sigma(c)\\subseteq[1,\\infty)\\). The spectrum of an inverse consists of the inverses of the points of the spectrum ([spectrum and quasi-spectrum](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-06)), so \\(\\sigma(c^{-1})\\subseteq(0,1]\\). So \\(c^{-1}=a^{1/2}b^{-1}a^{1/2}\\leq1\\), and conjugating by \\(a^{-1/2}\\) gives \\(b^{-1}\\leq a^{-1}\\).\n(8) This is Proposition 7.2(1) together with the Cartesian decomposition (Proposition 1.2(1)).\n(9) \\(C^*(a,b)\\) is commutative. By Theorem 2.1 its elements are functions, and positivity there is pointwise nonnegativity; positivity in it agrees with positivity in \\(A\\) (Remark 8.4). The product of two nonnegative functions is nonnegative.\n(10) Existence is Theorem 8.2, (i) ⇒ (iii). Uniqueness: let \\(b\\geq0\\) with \\(b^2=a\\). Then \\(b\\) commutes with \\(a\\), and \\(D=C^*(b)\\) is commutative and contains \\(a\\) and \\(a^{1/2}\\in C^*(a)\\). In \\(D\\cong C_0(\\Omega)\\), \\(\\hat b\\geq0\\) and \\(\\hat b^2=\\hat a\\), so \\(\\hat b=\\sqrt{\\hat a}=\\widehat{a^{1/2}}\\) by Theorem 5.3(4). Hence \\(b=a^{1/2}\\). The commutation claim is Theorem 5.1(6), since \\(a=a^*\\).\n(11) If \\(a=y^*y\\), then \\(\\langle a\\xi,\\xi\\rangle=\\|y\\xi\\|^2\\geq0\\). Conversely, suppose \\(\\langle a\\xi,\\xi\\rangle\\geq0\\) for all \\(\\xi\\). For an inner product linear in the first variable, polarization reads \\(4\\langle T\\xi,\\eta\\rangle=\\sum_{k=0}^3i^k\\langle T(\\xi+i^k\\eta),\\xi+i^k\\eta\\rangle\\). Since \\(\\langle a\\zeta,\\zeta\\rangle=\\langle\\zeta,a\\zeta\\rangle=\\langle a^*\\zeta,\\zeta\\rangle\\) for every \\(\\zeta\\), polarization gives \\(a=a^*\\). Let \\(\\lambda<0\\). Then \\(\\|(a-\\lambda)\\xi\\|\\|\\xi\\|\\geq\\langle(a-\\lambda)\\xi,\\xi\\rangle\\geq|\\lambda|\\|\\xi\\|^2\\). So \\(a-\\lambda\\) is bounded below; it is injective with closed range, and the orthogonal complement of its range is the kernel of \\((a-\\lambda)^*=a-\\lambda\\), which is \\(\\{0\\}\\). So \\(a-\\lambda\\) is invertible. Hence \\(\\sigma(a)\\subseteq[0,\\infty)\\), since it is real (Proposition 1.5). For a C\\*-subalgebra, use Remark 8.4.\n(12) \\(\\pi(y^*y)=\\pi(y)^*\\pi(y)\\). If \\(\\pi\\) is injective, it is an isometric \\(*\\)-isomorphism onto the closed range \\(\\pi(A)\\) (Corollary 4.6), so \\(\\sigma'_A(h)=\\sigma'_{\\pi(A)}(\\pi(h))=\\sigma'_B(\\pi(h))\\) by Theorem 3.2(2). \\(\\square\\)\n\n",
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      "id": "OA-FND-CF-17",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "9. Operator monotone powers and a Hölder estimate",
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      "full_conditions_and_proof": "### 9. Operator monotone powers and a Hölder estimate\n\n**Theorem 9.1** (Löwner–Heinz inequality). Let \\(A\\) be a C\\*-algebra, \\(0\\leq b\\leq a\\) and \\(0<\\alpha\\leq1\\). Then \\(b^\\alpha\\leq a^\\alpha\\).\n\nHere \\(t^\\alpha\\) vanishes at \\(0\\), so \\(a^\\alpha,b^\\alpha\\in A\\). The statement also makes sense for \\(\\alpha=0\\) when \\(a\\) and \\(b\\) are invertible, with \\(a^0=b^0=1\\).\n\n**Proof.** We may work in \\(\\widetilde A\\) (Theorem 5.3(3) and Remark 8.4), so let \\(A\\) be unital and nontrivial.\n*Step 1: \\(b\\geq\\varepsilon\\) for some \\(\\varepsilon>0\\).* Then \\(a\\geq b\\geq\\varepsilon\\), and both are invertible (Proposition 8.5(7)). Let \\(E=\\{\\alpha\\in[0,1]:b^\\alpha\\leq a^\\alpha\\}\\), with \\(a^0=b^0=1\\). It contains \\(0\\) and \\(1\\). It is closed: on a compact interval \\([\\varepsilon,M]\\), \\(t^\\beta\\to t^\\alpha\\) uniformly as \\(\\beta\\to\\alpha\\), so \\(\\alpha\\mapsto a^\\alpha\\) and \\(\\alpha\\mapsto b^\\alpha\\) are norm continuous (Theorem 5.1(4)), and \\(A_+\\) is closed. Let \\(\\alpha,\\beta\\in E\\) and \\(\\gamma=\\frac12(\\alpha+\\beta)\\). From \\(b^\\alpha\\leq a^\\alpha\\) and Proposition 8.5(3), \\(a^{-\\alpha/2}b^\\alpha a^{-\\alpha/2}\\leq1\\), so by Proposition 8.5(4)\n\\[\n\\|b^{\\alpha/2}a^{-\\alpha/2}\\|^2=\\|a^{-\\alpha/2}b^\\alpha a^{-\\alpha/2}\\|\\leq1,\n\\]\nand likewise \\(\\|b^{\\beta/2}a^{-\\beta/2}\\|\\leq1\\). Hence\n\\[\nT=(b^{\\beta/2}a^{-\\beta/2})^*(b^{\\alpha/2}a^{-\\alpha/2})=a^{-\\beta/2}b^\\gamma a^{-\\alpha/2}\n\\]\nhas \\(\\|T\\|\\leq1\\). Put \\(S=a^{-\\gamma/2}b^\\gamma a^{-\\gamma/2}\\) and \\(c=a^{(\\alpha-\\beta)/4}\\). Then \\(T=cSc^{-1}\\), because \\(\\frac{\\alpha-\\beta}4-\\frac\\gamma2=-\\frac\\beta2\\) and \\(-\\frac\\gamma2-\\frac{\\alpha-\\beta}4=-\\frac\\alpha2\\). Similar elements have the same spectrum, so \\(r(S)=r(T)\\leq1\\). But \\(S=(b^{\\gamma/2}a^{-\\gamma/2})^*(b^{\\gamma/2}a^{-\\gamma/2})\\geq0\\), so \\(\\|S\\|=r(S)\\leq1\\), \\(S\\leq1\\), and conjugating by \\(a^{\\gamma/2}\\) gives \\(b^\\gamma\\leq a^\\gamma\\). So \\(E\\) is closed under midpoints. It contains every dyadic rational of \\([0,1]\\), and being closed, all of \\([0,1]\\).\n*Step 2: the general case.* For \\(\\varepsilon>0\\), \\(\\varepsilon\\leq b+\\varepsilon\\leq a+\\varepsilon\\), so \\((b+\\varepsilon)^\\alpha\\leq(a+\\varepsilon)^\\alpha\\) by Step 1. For \\(t\\geq0\\), \\(0\\leq(t+\\varepsilon)^\\alpha-t^\\alpha\\leq\\varepsilon^\\alpha\\) (see the proof of Corollary 9.2), so \\(\\|(a+\\varepsilon)^\\alpha-a^\\alpha\\|\\leq\\varepsilon^\\alpha\\), and the same holds for \\(b\\). Let \\(\\varepsilon\\to0\\) and use that \\(A_+\\) is closed. \\(\\square\\)\n\n**Corollary 9.2** (A Hölder estimate). For all \\(a,b\\in A_+\\) and \\(0<\\alpha\\leq1\\),\n\\[\n\\|a^\\alpha-b^\\alpha\\|\\leq\\|a-b\\|^\\alpha .\n\\tag{9.1}\n\\]\nIn particular \\(\\|a^{1/2}-b^{1/2}\\|\\leq\\|a-b\\|^{1/2}\\), and \\(a\\mapsto a^\\alpha\\) is uniformly continuous on the whole cone \\(A_+\\).\n\n**Proof.** First, \\((s+t)^\\alpha\\leq s^\\alpha+t^\\alpha\\) for \\(s,t\\geq0\\): if \\(s+t>0\\), then \\(u^\\alpha\\geq u\\) on \\([0,1]\\) gives \\(\\big(\\frac s{s+t}\\big)^\\alpha+\\big(\\frac t{s+t}\\big)^\\alpha\\geq1\\). Put \\(c=\\|a-b\\|\\). Then \\(a\\leq b+c\\) by Proposition 8.5(1), so by the theorem in \\(\\widetilde A\\), \\(a^\\alpha\\leq(b+c)^\\alpha\\leq b^\\alpha+c^\\alpha\\); the second inequality is the scalar one applied through the calculus of \\(b\\). So \\(a^\\alpha-b^\\alpha\\leq c^\\alpha\\), and by symmetry \\(b^\\alpha-a^\\alpha\\leq c^\\alpha\\). Proposition 8.5(1) gives (9.1). \\(\\square\\)\n\n**Example 9.3** (No exponent greater than 1 is allowed). In \\(M_2(\\mathbb C)\\) let \\(a=\\begin{pmatrix}2&1\\\\1&1\\end{pmatrix}\\) and \\(b=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\\). Then \\(a-b=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\geq0\\), so \\(0\\leq b\\leq a\\). But \\(a^2-b^2=\\begin{pmatrix}4&3\\\\3&2\\end{pmatrix}\\) has determinant \\(-1\\), so it is not positive.\n\nThe same obstruction occurs for every \\(\\alpha>1\\). Set\n\\[\n\\begin{gathered}\nt=(\\alpha^2+1)^{1/(\\alpha-1)}>1,\\qquad\nD=\\begin{pmatrix}t&0\\\\0&1\\end{pmatrix},\\\\\nJ=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\geq0 .\n\\end{gathered}\n\\]\nWe compute the first-order change of the power using only the two-dimensional spectral calculus. The eigenvalues of \\(D+\\varepsilon J\\) are\n\\[\n\\lambda_\\pm(\\varepsilon)\n=\\frac{t+1+2\\varepsilon\n\\ \\pm\\sqrt{(t-1)^2+4\\varepsilon^2}}2 .\n\\]\nAt zero they are \\(t,1\\), respectively, and both have derivative \\(1\\). Its upper spectral projection is\n\\[\nP_+(\\varepsilon)\n=\\frac{D+\\varepsilon J-\\lambda_-(\\varepsilon)1}\n{\\lambda_+(\\varepsilon)-\\lambda_-(\\varepsilon)} .\n\\]\nThis formula follows by applying the scalar function that equals \\(1\\) at \\(\\lambda_+\\) and \\(0\\) at \\(\\lambda_-\\); thus \\(P_-=1-P_+\\). At zero the diagonal entries of \\(P_+\\) have derivative \\(0\\), and both off-diagonal entries have derivative \\(1/(t-1)\\). Differentiating\n\\((D+\\varepsilon J)^\\alpha=\\lambda_+^\\alpha P_++\\lambda_-^\\alpha P_-\\)\ntherefore gives the norm limit\n\\[\n\\begin{gathered}\n\\frac{(D+\\varepsilon J)^\\alpha-D^\\alpha}{\\varepsilon}\n\\longrightarrow\nL=\\begin{pmatrix}\\alpha t^{\\alpha-1}&q\\\\q&\\alpha\\end{pmatrix},\\\\\nq=\\frac{t^\\alpha-1}{t-1}.\n\\end{gathered}\n\\]\nSince \\(t>1\\), \\(t^\\alpha-1\\geq t^{\\alpha-1}(t-1)\\), so \\(q\\geq t^{\\alpha-1}\\). Consequently\n\\[\n\\det L\n=\\alpha^2t^{\\alpha-1}-q^2\n\\leq t^{\\alpha-1}(\\alpha^2-t^{\\alpha-1})<0 .\n\\]\nThe two eigenvalues of the self-adjoint matrix \\(L\\) have opposite signs. Choose a unit eigenvector for its negative eigenvalue. The displayed norm limit shows that the quadratic form of \\((D+\\varepsilon J)^\\alpha-D^\\alpha\\) on that vector is negative for every sufficiently small positive \\(\\varepsilon\\). Yet \\(D\\leq D+\\varepsilon J\\), and both matrices are positive. Thus \\(s\\mapsto s^\\alpha\\) fails to be operator monotone on \\([0,\\infty)\\) for every \\(\\alpha>1\\).\n\nThe Löwner–Heinz theorem for \\(0<\\alpha\\leq1\\), Theorem 9.1 above, proves the complementary range in full. Theorem 10.2 shows that in a C*-algebra where \\(t^2\\) is monotone, all elements commute.\n\nThe next proposition gives a second two-by-two argument, using an affine interpolant on the spectrum, and records the complete range for real exponents.\n\n**Proposition 9.4** (The full range of monotone powers). On positive definite matrices of every size, \\(x\\mapsto x^\\alpha\\), for real \\(\\alpha\\), preserves order exactly when \\(0\\leq\\alpha\\leq1\\). For every \\(\\alpha>1\\) there are already positive definite two-by-two matrices \\(0<b\\leq a\\) for which \\(b^\\alpha\\not\\leq a^\\alpha\\). For \\(0<\\alpha\\leq1\\), the order-preserving statement holds on the entire positive cone of every C\\*-algebra, including nonunital algebras, by Theorem 9.1. At \\(\\alpha=0\\) on positive definite matrices the function is the constant identity.\n\n**Proof.** The affirmative cases are Theorem 9.1 and the constant function. If \\(\\alpha<0\\), the scalar inequality \\(1<2\\) gives \\(1^\\alpha>2^\\alpha\\), so even scalar monotonicity fails.\n\nFix \\(\\alpha>1\\), and choose\n\\[\n\\begin{gathered}\ns=(2\\alpha^2)^{-1/(\\alpha-1)},\\\\\nb=\\begin{pmatrix}s&0\\\\0&1\\end{pmatrix},\\qquad\nh=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix},\\\\\na_t=b+th .\n\\end{gathered}\n\\]\nHere \\(0<s<1\\), \\(h\\geq0\\), and \\(a_t\\geq b>0\\) for every \\(t>0\\). We calculate the first-order change of \\(a_t^\\alpha\\) directly, rather than assuming a criterion for matrix monotonicity.\n\nPut \\(d=1-s>0\\). The two distinct eigenvalues of \\(a_t\\) are\n\\[\n\\lambda_\\pm(t)=\\frac{1+s+2t\\pm\\sqrt{d^2+4t^2}}2 .\n\\]\nSince\n\\[\n0\\leq\\sqrt{d^2+4t^2}-d\n=\\frac{4t^2}{\\sqrt{d^2+4t^2}+d}\\leq\\frac{2t^2}{d},\n\\]\nwe have \\(\\lambda_-(t)=s+t+O(t^2)\\) and \\(\\lambda_+(t)=1+t+O(t^2)\\). For sufficiently small \\(t\\), all these numbers stay in a fixed compact subinterval of \\((0,\\infty)\\).\n\nFor \\(f(u)=u^\\alpha\\), set\n\\[\nc_t=\\frac{f(\\lambda_+(t))-f(\\lambda_-(t))}\n{\\lambda_+(t)-\\lambda_-(t)} .\n\\]\nThe affine polynomial \\(f(\\lambda_-)+c_t(u-\\lambda_-)\\) agrees with \\(f\\) on the two-point spectrum of \\(a_t\\). Functional calculus therefore gives\n\\[\nf(a_t)=f(\\lambda_-(t))I+\nc_t\\bigl(a_t-\\lambda_-(t)I\\bigr).\n\\]\nThis is precisely Theorem 5.1 applied to two functions with equal values on the spectrum. As \\(t\\to0\\),\n\\[\nc_t\\longrightarrow c=\\frac{1-s^\\alpha}{1-s}>1.\n\\]\nThe upper-left entry is \\(f(s)+t f'(s)+o(t)\\): indeed \\(s+t-\\lambda_-(t)=O(t^2)\\), \\(c_t\\) stays bounded, and \\(f(\\lambda_-(t))=f(s)+t f'(s)+o(t)\\). Using the equivalent formula\n\\(f(a_t)=f(\\lambda_+(t))I+c_t(a_t-\\lambda_+(t)I)\\)\ngives the lower-right entry \\(f(1)+t f'(1)+o(t)\\). The off-diagonal entries are \\(t c_t\\). Thus, entrywise and hence in matrix norm,\n\\[\n\\frac{a_t^\\alpha-b^\\alpha}{t}\n\\longrightarrow\nL=\\begin{pmatrix}\\alpha s^{\\alpha-1}&c\\\\c&\\alpha\\end{pmatrix}.\n\\]\nBut\n\\[\n\\det L=\\alpha^2s^{\\alpha-1}-c^2\n=\\frac12-c^2<0.\n\\]\nBy continuity of the determinant, the self-adjoint matrix\n\\((a_t^\\alpha-b^\\alpha)/t\\) has negative determinant for all sufficiently small positive \\(t\\). It cannot be positive: a positive two-by-two matrix has nonnegative eigenvalues and therefore nonnegative determinant, by Theorem 8.5(11) and the finite-dimensional spectral calculus. Consequently \\(a_t^\\alpha-b^\\alpha\\not\\geq0\\), although \\(a_t\\geq b>0\\). This supplies a counterexample for each \\(\\alpha>1\\), and completes the classification. \\(\\square\\)\n\nThe matrices of divided differences that appear here are studied more generally in [Hiai and Sano's freely accessible paper](https://arxiv.org/html/1007.2478v2). Its Proposition 3.1 treats power functions; the argument above proves the monotonicity range within this programme and does not require the paper's external prerequisites.\n\n*Proposition 9.4 written and self-checked by GPT-6 Astra (OpenAI), Ultra, in Codex.*\n\n",
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      "name": "10. Order properties that force commutativity",
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      "full_conditions_and_proof": "### 10. Order properties that force commutativity\n\nIn an abelian C\\*-algebra the self-adjoint part behaves like a space of real functions: it is a lattice, and the order is compatible with products and squares. This section shows that each of these properties, and several related ones, forces the algebra to be abelian. The proofs use only the functional calculus and the order.\n\n**Definition 10.1.** \\(A_h\\) is a *lattice* if every pair \\(h,k\\in A_h\\) has a least upper bound \\(h\\vee k\\) for \\(\\leq\\); then \\(h\\wedge k=-((-h)\\vee(-k))\\) is a greatest lower bound. \\(A_h\\) has the *Riesz decomposition property* (RDP) if \\(0\\leq x\\leq y_1+y_2\\) with \\(y_1,y_2\\geq0\\) implies \\(x=x_1+x_2\\) with \\(0\\leq x_j\\leq y_j\\). It has the *interpolation property* (RIP) if, whenever \\(u_1,u_2\\leq v_1,v_2\\), some \\(w\\) has \\(u_1,u_2\\leq w\\leq v_1,v_2\\).\n\n**Theorem 10.2** (Order and commutativity). For a C\\*-algebra \\(A\\), the following are equivalent:\n- (a) \\(A\\) is abelian;\n- (b) \\(A_h\\) is a lattice;\n- (c) \\(A_h\\) has the RDP;\n- (c′) \\(A_h\\) has the RIP;\n- (d) \\(0\\leq b\\leq a\\) implies \\(b^2\\leq a^2\\);\n- (e) \\(ab+ba\\geq0\\) for all \\(a,b\\in A_+\\);\n- (f) \\(azb=0\\) whenever \\(a,b\\in A_+\\), \\(ab=0\\) and \\(z\\in A\\).\n\nIf \\(A\\) is abelian, the real space \\(A^*_h\\) of bounded hermitian functionals, ordered by \\(\\varphi\\leq\\psi\\) when \\(\\psi-\\varphi\\) is positive, is a lattice.\n\nThe proof uses three lemmas.\n\n**Lemma 10.3** (Orthogonal sums). Let \\(p_1,\\dots,p_n\\) and \\(q_1,\\dots,q_n\\) be positive elements of \\(\\widetilde A\\) of norm at most \\(1\\), with \\(p_kp_l=0\\) and \\(q_kq_l=0\\) for \\(k\\neq l\\). Then \\(\\|\\sum_kp_kyq_k\\|\\leq\\|y\\|\\) for every \\(y\\in\\widetilde A\\).\n\n**Proof.** Put \\(T=\\sum_kp_kyq_k\\). Since \\(p_kp_l=0\\) for \\(k\\neq l\\), \\(T^*T=\\sum_kq_ky^*p_k^2yq_k\\). Since \\(\\|p_k\\|\\leq1\\), \\(p_k^2\\leq1\\) (Proposition 8.5(1)). Conjugation preserves the order (Proposition 8.5(3)), and \\(y^*y\\leq\\|y\\|^2\\) (Proposition 8.5(2)); so \\(y^*p_k^2y\\leq y^*y\\leq\\|y\\|^2\\), and \\(q_ky^*p_k^2yq_k\\leq\\|y\\|^2q_k^2\\). So \\(0\\leq T^*T\\leq\\|y\\|^2Q\\) with \\(Q=\\sum_kq_k^2\\). The \\(q_k^2\\) are positive and \\(q_k^2q_l^2=0\\) for \\(k\\neq l\\), so \\(Q^m=\\sum_kq_k^{2m}\\). By Theorem 1.3(1), \\(\\|Q\\|^m=\\|Q^m\\|\\leq n\\) for every power \\(m\\) of \\(2\\), and letting \\(m\\to\\infty\\) gives \\(\\|Q\\|\\leq1\\). By Proposition 8.5(4), \\(\\|T\\|^2=\\|T^*T\\|\\leq\\|y\\|^2\\). \\(\\square\\)\n\n**Lemma 10.4** ((f) implies (a)). Assume (f).\n1. If \\(a\\in A_+\\), \\(c\\in\\widetilde A\\) and \\(ca=0\\), then \\(azc=0\\) for all \\(z\\in A\\). If \\(ac=0\\), then \\(cza=0\\) for all \\(z\\in A\\).\n2. Every \\(x\\in A_+\\) commutes with every \\(y\\in A\\). Hence \\(A\\) is abelian.\n\n**Proof.** (1) Let \\(ca=0\\) and \\(z\\in A\\). Then \\(ca^{1/2}=0\\), since \\(\\|ca^{1/2}\\|^2=\\|cac^*\\|=0\\); so \\(a^{1/2}c^*=0\\). Put \\(Y=a^{1/2}zc\\in A\\). The positive elements \\(a^{1/2}\\) and \\(Y^*Y=c^*z^*azc\\) satisfy \\(a^{1/2}Y^*Y=0\\). By (f), applied to them and to \\(zc\\in A\\), \\(a^{1/2}(zc)Y^*Y=YY^*Y=0\\). Then \\((Y^*Y)^2=Y^*(YY^*Y)=0\\), so \\(Y^*Y=0\\) by Theorem 1.3(1), and \\(Y=0\\). Hence \\(azc=a^{1/2}Y=0\\). If \\(ac=0\\), apply this to \\(c^*\\) (note \\(c^*a=(ac)^*=0\\)) and take adjoints.\n(2) Let \\(x\\in A_+\\), \\(x\\neq0\\), \\(y\\in A\\) and \\(\\delta>0\\). Let \\(N\\) be an integer with \\(N\\delta\\geq\\|x\\|\\), and for \\(j=0,\\dots,N\\) let \\(\\phi_j(t)=\\max(0,1-|t-j\\delta|/\\delta)\\) on \\([0,N\\delta]\\). On \\([0,N\\delta]\\), \\(\\sum_j\\phi_j=1\\) and \\(\\sum_jj\\delta\\,\\phi_j(t)=t\\), because both sides are linear between consecutive nodes \\(j\\delta\\) and agree there. Also \\(0\\leq\\phi_j\\leq1\\), \\(\\phi_j\\phi_k=0\\) when \\(|j-k|\\geq2\\), and \\(\\phi_j(0)=0\\) for \\(j\\geq1\\). Put \\(e_j=\\phi_j(x)\\in\\widetilde A\\), so \\(e_j\\in A_+\\) for \\(j\\geq1\\), \\(\\sum_je_j=1\\) and \\(\\sum_jj\\delta e_j=x\\). For \\(|j-k|\\geq2\\), \\(e_jye_k=0\\): if \\(j,k\\geq1\\) this is (f); if \\(j=0\\) it is (1) with \\(a=e_k\\), \\(c=e_0\\) and \\(e_ke_0=0\\); if \\(k=0\\) it is (1) with \\(a=e_j\\), \\(c=e_0\\) and \\(e_0e_j=0\\). Therefore\n\\[\n\\begin{gathered}\nxy-yx\\\\\n=\\sum_{j,k}(j-k)\\delta\\,e_jye_k\\\\\n=\\delta\\sum_j\\big(e_{j+1}ye_j-e_jye_{j+1}\\big).\n\\end{gathered}\n\\]\nSplit each of the two sums into the terms with \\(j\\) even and those with \\(j\\) odd. In each of the four parts, the left factors are pairwise orthogonal and so are the right factors, since their indices differ by at least \\(2\\). Lemma 10.3 bounds each part by \\(\\|y\\|\\). So \\(\\|xy-yx\\|\\leq4\\delta\\|y\\|\\) for every \\(\\delta>0\\), and \\(xy=yx\\). Since \\(A\\) is spanned by \\(A_+\\) (Proposition 8.5(8)), \\(A\\) is abelian. \\(\\square\\)\n\n**Lemma 10.5** ((c′) implies (f)). Assume (c′). Let \\(p,q\\in A_+\\) with \\(pq=0\\), and \\(z\\in A\\). Then \\(p^2zq^2=0\\).\n\n**Proof.** Put \\(W=pzq+qz^*p\\in A_h\\). *Positivity test:* if \\(\\alpha,\\beta>0\\) and \\(\\alpha\\beta\\geq\\|z\\|^2\\), then \\(\\alpha p^2+\\beta q^2+W\\geq0\\). Indeed, with \\(v=\\alpha^{1/2}p+\\alpha^{-1/2}zq\\),\n\\[\n\\begin{gathered}\nv^*v\\\\\n=\\alpha p^2+W+\\alpha^{-1}qz^*zq\\\\\n\\leq\\alpha p^2+W+\\alpha^{-1}\\|z\\|^2q^2\\\\\n\\leq\\alpha p^2+W+\\beta q^2,\n\\end{gathered}\n\\]\nusing Proposition 8.5(2)–(3).\nFix \\(s>0\\) and choose \\(\\lambda>0\\) with \\(s(s+2\\lambda)\\geq\\|z\\|^2\\). Put \\(u=\\lambda(p^2-q^2)\\). Since \\(p^2q^2=q^2p^2=0\\), \\(u^2=(\\lambda(p^2+q^2))^2\\), so \\(|u|=\\lambda(p^2+q^2)\\) by the uniqueness of positive square roots (Proposition 8.5(10)), and \\(u_+=\\lambda p^2\\), \\(u_-=\\lambda q^2\\) by the uniqueness of the Jordan decomposition (Proposition 7.2(2)). Put \\(n=s(p^2+q^2)+W\\) and \\(m=|u|+n\\). By the positivity test,\n\\[\n\\begin{gathered}\nm-u\\\\\n=sp^2+(s+2\\lambda)q^2+W\\\\\n\\geq0,\\\\\nm+u\\\\\n=(s+2\\lambda)p^2+sq^2+W\\\\\n\\geq0 .\n\\end{gathered}\n\\]\nSo \\(u,-u\\leq m\\), and also \\(u,-u\\leq|u|\\). By (c′) there is \\(w\\) with \\(u,-u\\leq w\\leq m,|u|\\). Put \\(e=|u|-w\\geq0\\). From \\(w\\geq-u\\), \\(e\\leq|u|+u=2\\lambda p^2\\); from \\(w\\geq u\\), \\(e\\leq2\\lambda q^2\\). The first bound and \\(p^2q^2=0\\) give \\(eq^2=0\\) (Proposition 8.5(6)). The second, conjugated by \\(e\\), gives \\(0\\leq e^3\\leq2\\lambda eq^2e=0\\). So \\(e^3=0\\), \\(e=0\\), \\(w=|u|\\), and \\(n=m-w\\geq0\\).\nFor \\(n\\geq0\\) and \\(c,d\\in A\\), \\[\n\\begin{gathered}\n\\|cnd\\|\\\\\n\\leq\\|n^{1/2}c^*\\|\\|n^{1/2}d\\|\\\\\n=\\|cnc^*\\|^{1/2}\\|d^*nd\\|^{1/2}.\n\\end{gathered}\n\\] Here \\(pnp=sp^4\\), \\(qnq=sq^4\\) and \\(pnq=p^2zq^2\\), because \\(pq=qp=0\\). So \\(\\|p^2zq^2\\|\\leq s\\|p\\|^2\\|q\\|^2\\). As \\(s>0\\) was arbitrary, \\(p^2zq^2=0\\). \\(\\square\\)\n\n**Proof of Theorem 10.2.** (a) ⇒ (b): by Theorem 2.1, \\(A=C_0(\\Omega)\\), where positivity is pointwise nonnegativity, because the quasi-spectrum of \\(f\\) is \\(f(\\Omega)\\cup\\{0\\}\\) ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)). The pointwise maximum \\(\\max(h,k)\\) lies in \\(C_0(\\Omega;\\mathbb R)\\) and is the least upper bound.\n(b) ⇒ (c′): \\(w=u_1\\vee u_2\\) works.\n(c′) ⇒ (c): let \\(0\\leq x\\leq y_1+y_2\\) with \\(y_j\\geq0\\). Then \\(0,\\,x-y_2\\leq x,\\,y_1\\). An interpolant \\(w\\) gives \\(x_1=w\\) and \\(x_2=x-w\\), with \\(0\\leq x_1\\leq y_1\\) and \\(0\\leq x_2\\leq y_2\\).\n(c) ⇒ (c′): let \\(u_1,u_2\\leq v_1,v_2\\). Then \\(0\\leq v_1-u_1\\leq(v_1-u_2)+(v_2-u_1)\\), because the difference is \\(v_2-u_2\\geq0\\). The RDP gives \\(v_1-u_1=a_1+a_2\\) with \\(0\\leq a_1\\leq v_1-u_2\\) and \\(0\\leq a_2\\leq v_2-u_1\\). Then \\(w=u_1+a_2=v_1-a_1\\) satisfies \\(u_1\\leq w\\), \\(w\\leq v_1\\), \\(u_2=v_1-(v_1-u_2)\\leq w\\) and \\(w\\leq u_1+(v_2-u_1)=v_2\\).\n(c′) ⇒ (f): let \\(a,b\\in A_+\\) with \\(ab=0\\). Then \\(ba=0\\), \\(C^*(a,b)\\) is commutative, and there \\(\\widehat{a^{1/2}}\\widehat{b^{1/2}}=\\sqrt{\\hat a\\hat b}=0\\) (Theorem 5.3(4)); so \\(p=a^{1/2}\\) and \\(q=b^{1/2}\\) satisfy \\(pq=0\\). Lemma 10.5 gives \\(azb=p^2zq^2=0\\).\n(f) ⇒ (a): Lemma 10.4.\n(a) ⇒ (d), (e): in \\(C_0(\\Omega)\\), \\(0\\leq g\\leq f\\) gives \\(g^2\\leq f^2\\) pointwise, and \\(ab+ba=2ab\\geq0\\).\n(d) ⇒ (e): let \\(a,b\\geq0\\) and \\(t>0\\). Then \\(a\\leq a+tb\\), so \\(a^2\\leq(a+tb)^2=a^2+t(ab+ba)+t^2b^2\\), that is \\(ab+ba+tb^2\\geq0\\). Let \\(t\\to0\\).\n(e) ⇒ (f): let \\(a,b\\in A_+\\) with \\(ab=0\\), and \\(y\\in A_+\\). Then \\(w=ay+ya\\geq0\\) and \\(bwb=bayb+byab=0\\). By Proposition 8.5(5), \\(wb=0\\), that is \\(ayb=-yab=0\\). Since \\(A\\) is spanned by \\(A_+\\), \\(azb=0\\) for every \\(z\\in A\\).\n\n*The dual lattice.* A bounded linear functional \\(\\varphi\\) is *hermitian* if \\(\\varphi(x^*)=\\overline{\\varphi(x)}\\), and *positive* if \\(\\varphi(A_+)\\subseteq[0,\\infty)\\). Let \\(A\\) be abelian and \\(\\varphi,\\psi\\in A^*_h\\). For \\(x\\in A_+\\) put\n\\[\nS(x)=\\sup\\{\\varphi(y)+\\psi(x-y):\\ 0\\leq y\\leq x\\}.\n\\]\nSince \\(0\\leq y\\leq x\\) implies \\(\\|y\\|,\\|x-y\\|\\leq\\|x\\|\\) (Proposition 8.5(4)), \\(-\\|\\psi\\|\\|x\\|\\leq\\psi(x)\\leq S(x)\\leq(\\|\\varphi\\|+\\|\\psi\\|)\\|x\\|\\). \\(S\\) is positively homogeneous. It is additive on \\(A_+\\). If \\(0\\leq y_j\\leq x_j\\), then \\(y_1+y_2\\) is admissible for \\(x_1+x_2\\), so \\(S(x_1+x_2)\\geq S(x_1)+S(x_2)\\). Conversely, if \\(0\\leq y\\leq x_1+x_2\\), the RDP, which holds by (a) ⇒ (c), splits \\(y=y_1+y_2\\) with \\(0\\leq y_j\\leq x_j\\), and \\(\\varphi(y)+\\psi(x_1+x_2-y)\\) is the sum of the two admissible values for \\(x_1\\) and \\(x_2\\). So \\(S(u-v)=S(u)-S(v)\\) is a well-defined real-linear functional on \\(A_h=A_+-A_+\\), bounded by \\(2(\\|\\varphi\\|+\\|\\psi\\|)\\|h\\|\\) at \\(h=h_+-h_-\\). Its complexification \\(S(x_1+ix_2)=S(x_1)+iS(x_2)\\) is a bounded hermitian functional. Taking \\(y=x\\) and \\(y=0\\) shows \\(S\\geq\\varphi\\) and \\(S\\geq\\psi\\). If \\(\\theta\\in A^*_h\\) and \\(\\theta\\geq\\varphi,\\psi\\), then \\(\\varphi(y)+\\psi(x-y)\\leq\\theta(y)+\\theta(x-y)=\\theta(x)\\), so \\(S\\leq\\theta\\). Thus \\(S=\\varphi\\vee\\psi\\). \\(\\square\\)\n\n*The converse.* If \\(A^*_h\\) is a lattice, then \\(A\\) is abelian. By the Hahn–Banach theorem, the lattice property of \\(A^*_h\\) yields the interpolation property of \\(A_h\\) up to an arbitrarily small error, and the proof of Lemma 10.5 survives such an error.\n\n*Step 1: approximate interpolation.* Let \\(A^*_h\\) be a lattice. The steps (b) ⇒ (c′) ⇒ (c) of the proof of Theorem 10.2 use only the order, so they show that \\(A^*_h\\) has the RDP. Let \\(u_1,u_2\\leq v_1,v_2\\) in \\(A_h\\), and \\(\\varepsilon>0\\). We show that some \\(w\\in A_h\\) satisfies \\(u_i-\\varepsilon\\leq w\\leq v_j+\\varepsilon\\) for \\(i,j=1,2\\). In the real Banach space \\((A_h)^4\\), normed by the largest of the four norms, let \\(P=(A_+)^4\\), a convex cone, and let \\(K\\) be the set of the points \\((w-u_1,\\,w-u_2,\\,v_1-w,\\,v_2-w)\\) with \\(w\\in A_h\\). A self-adjoint element of norm less than \\(\\varepsilon\\) is \\(\\geq-\\varepsilon\\) (Proposition 8.5(1)). So if the point of \\(K\\) given by \\(w\\) is at distance less than \\(\\varepsilon\\) from some point of \\(P\\), then this \\(w\\) works. Suppose that no point of \\(K\\) is. Then the open convex set \\(U\\) of the points at distance less than \\(\\varepsilon\\) from some point of \\(K\\) does not meet \\(P\\). By the separation theorem for an open convex set (Theorem 6.2 of [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md), over the real numbers), there are bounded real-linear functionals \\(\\varphi_1,\\dots,\\varphi_4\\) on \\(A_h\\) and a real \\(\\gamma\\) with \\(\\sum_l\\varphi_l(x_l)<\\gamma\\leq\\sum_l\\varphi_l(y_l)\\) for all \\(x\\in U\\) and \\(y\\in P\\). Since \\(P\\) is a cone containing \\(0\\), each \\(\\varphi_l\\) is nonnegative on \\(A_+\\), and \\(\\gamma\\leq0\\). At the point of \\(K\\) given by \\(w\\), the sum is\n\\[\n\\begin{gathered}\n(\\varphi_1+\\varphi_2-\\varphi_3-\\varphi_4)(w)\\\\\n+\\varphi_3(v_1)+\\varphi_4(v_2)\\\\\n-\\varphi_1(u_1)-\\varphi_2(u_2),\n\\end{gathered}\n\\]\nand it is negative for every \\(w\\in A_h\\). A nonzero real-linear functional takes arbitrarily large values, so \\(\\varphi_1+\\varphi_2=\\varphi_3+\\varphi_4\\), and \\(\\varphi_3(v_1)+\\varphi_4(v_2)<\\varphi_1(u_1)+\\varphi_2(u_2)\\). Extended by \\(\\varphi_l(x+iy)=\\varphi_l(x)+i\\varphi_l(y)\\) for \\(x,y\\in A_h\\), the \\(\\varphi_l\\) are positive functionals. As \\(0\\leq\\varphi_1\\leq\\varphi_3+\\varphi_4\\), the RDP gives \\(\\varphi_1=\\psi_{13}+\\psi_{14}\\) with \\(0\\leq\\psi_{13}\\leq\\varphi_3\\) and \\(0\\leq\\psi_{14}\\leq\\varphi_4\\). Then \\(\\psi_{23}=\\varphi_3-\\psi_{13}\\) and \\(\\psi_{24}=\\varphi_4-\\psi_{14}\\) are positive, and their sum is \\(\\varphi_2\\). Since each \\(u_i\\) lies below \\(v_1\\) and \\(v_2\\),\n\\[\n\\begin{gathered}\n\\varphi_1(u_1)+\\varphi_2(u_2)\\\\\n=\\sum_{i=1,2}\\big(\\psi_{i3}(u_i)+\\psi_{i4}(u_i)\\big)\\\\\n\\leq\\sum_{i=1,2}\\big(\\psi_{i3}(v_1)+\\psi_{i4}(v_2)\\big)\\\\\n=\\varphi_3(v_1)+\\varphi_4(v_2),\n\\end{gathered}\n\\]\na contradiction.\n\n*Step 2: Lemma 10.5 with an error.* Let \\(p,q\\in A_+\\) with \\(pq=0\\), and \\(z\\in A\\). Take \\(s\\), \\(\\lambda\\), \\(u\\), \\(n\\) and \\(m\\) as in the proof of Lemma 10.5, so that \\(u,-u\\leq m,|u|\\), \\(pnp=sp^4\\), \\(qnq=sq^4\\) and \\(pnq=p^2zq^2\\). Let \\(\\varepsilon>0\\). Step 1 gives \\(w\\in A_h\\) with \\(u-\\varepsilon,\\,-u-\\varepsilon\\leq w\\leq m+\\varepsilon,\\,|u|+\\varepsilon\\). Put \\(e=|u|+\\varepsilon-w\\geq0\\). Then \\(n+e=m+\\varepsilon-w\\geq0\\). Also \\(e\\leq|u|+u+2\\varepsilon=2\\lambda p^2+2\\varepsilon\\) and \\(e\\leq|u|-u+2\\varepsilon=2\\lambda q^2+2\\varepsilon\\). Conjugating (Proposition 8.5(3)) and using \\(pq=qp=0\\) gives \\(qeq\\leq2\\varepsilon q^2\\) and \\(pep\\leq2\\varepsilon p^2\\). The norm inequality at the end of the proof of Lemma 10.5, applied to \\(n+e\\) and to \\(e\\), and Proposition 8.5(4) give\n\\[\n\\begin{gathered}\n\\|p^2zq^2\\|\\\\\n\\leq\\|p(n+e)q\\|+\\|peq\\|\\\\\n\\leq\\big(s\\|p\\|^4+2\\varepsilon\\|p\\|^2\\big)^{1/2}\\big(s\\|q\\|^4+2\\varepsilon\\|q\\|^2\\big)^{1/2}\\\\\n+2\\varepsilon\\|p\\|\\|q\\|.\n\\end{gathered}\n\\]\nLetting \\(\\varepsilon\\to0\\) and then \\(s\\to0\\) gives \\(p^2zq^2=0\\). As in the step (c′) ⇒ (f), this gives (f), and Lemma 10.4 shows that \\(A\\) is abelian. \\(\\square\\)\n\n**Example 10.6** (\\(2\\times2\\) matrices violate every condition of Theorem 10.2). In \\(M_2(\\mathbb C)\\):\n- (f) fails: \\(E_{11}E_{22}=0\\), but \\(E_{11}E_{12}E_{22}=E_{12}\\neq0\\).\n- (e) fails: for \\(p=E_{11}\\) and \\(q=\\frac12\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\), \\(pq+qp=\\frac12\\begin{pmatrix}2&1\\\\1&0\\end{pmatrix}\\), whose determinant is \\(-\\frac14\\).\n- (d) fails: see Example 9.3.\n- (c) fails: \\(q\\leq E_{11}+E_{22}=1\\), but if \\(q=q_1+q_2\\) with \\(0\\leq q_1\\leq E_{11}\\) and \\(0\\leq q_2\\leq E_{22}\\), then \\(q_1E_{22}=0\\) and \\(q_2E_{11}=0\\) by Proposition 8.5(6), so \\(q_1\\) and \\(q_2\\) are diagonal, and so is \\(q\\), which it is not.\n\n**Exercise 10.7** (medium; The positive part is not monotone). In \\(M_2(\\mathbb C)\\), find self-adjoint \\(h\\leq k\\) with \\(h_+\\not\\leq k_+\\). Explain why this cannot happen in an abelian C\\*-algebra.\n\n*Solution.* Take \\(h=\\begin{pmatrix}1&0\\\\0&-1\\end{pmatrix}\\) and \\(k=h+\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}=\\begin{pmatrix}2&1\\\\1&0\\end{pmatrix}\\), so \\(h\\leq k\\). Here \\(h_+=E_{11}\\). The eigenvalues of \\(k\\) are \\(1\\pm\\sqrt2\\), so \\(k_+=(1+\\sqrt2)P\\), where \\(P\\) is the projection onto the line spanned by \\(v=(1,\\sqrt2-1)\\). If \\(E_{11}\\leq k_+\\), then for a unit vector \\(w\\perp v\\), \\(|\\langle w,e_1\\rangle|^2=\\langle E_{11}w,w\\rangle\\leq\\langle k_+w,w\\rangle=0\\), so \\(e_1\\perp w\\) and \\(e_1\\) would be a multiple of \\(v\\), which it is not. In an abelian algebra, \\(h_+=\\max(h,0)\\) pointwise, which is monotone. More generally, if \\(A_h\\) is a lattice, then \\(h\\vee0=h_+\\). Indeed, put \\(c=h\\vee0\\). Since \\(h_+\\) is an upper bound of \\(h\\) and \\(0\\), we have \\(0\\leq c\\leq h_+\\) and \\(0\\leq c-h\\leq h_+-h=h_-\\). By Proposition 8.5(6), \\(ch_-=0\\) and \\((c-h)h_+=0\\), so \\(ch_+=hh_+=h_+^2\\). Then \\(e=h_+-c\\) satisfies \\(0\\leq e\\leq h_+\\) and \\(eh_+=0\\), so \\(0\\leq e^3\\leq eh_+e=0\\), and \\(e=0\\). In a lattice \\(h\\mapsto h\\vee0\\) is monotone, so the example is one more witness that \\(M_2(\\mathbb C)_h\\) is not a lattice.\n\n## B. Localize, form a quotient, and lift back\n\n**Running computation.** In \\(A=C_0((0,1])\\), let \\(h(t)=t\\) and \\(e_\\varepsilon(t)=t/(t+\\varepsilon)\\). Each \\(e_\\varepsilon\\) belongs to \\(A\\) and is a positive contraction. For \\(f\\in A\\) and \\(\\delta>0\\), choose \\(s>0\\) so that \\(|f(t)|<\\delta\\) for \\(0<t<s\\). On that interval \\(|(1-e_\\varepsilon)f|<\\delta\\); on \\([s,1]\\) it is at most \\(\\varepsilon\\|f\\|/(s+\\varepsilon)\\). Thus \\(e_\\varepsilon f\\to f\\) uniformly as \\(\\varepsilon\\downarrow0\\). The cutoff estimates below abstract this calculation. They also work for one-sided ideals without requiring those ideals to be closed.\n\nOnce local units exist, the quotient norm can be computed by removing the part supported in a closed two-sided ideal. This gives the C*-identity on the quotient rather than assuming it. The commutative hull-and-kernel theorem identifies the geometry of this operation. We then return to general quotients to lift norm bounds, positivity and order intervals. These three questions belong together: what is forgotten, what norm survives, and which constrained representatives can be recovered?\n\n",
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      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "11. Approximate identities",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 11. Approximate identities\n\nAdjoining an identity is not always enough: the unitization of an ideal is no longer an ideal, so statements about ideals cannot be reduced to the unital case. Approximate identities replace the identity in such cases.\n\n**Definition 11.1.** Let \\(A\\) be a Banach algebra and \\(J\\subseteq A\\). A net \\((u_i)\\) in \\(A\\) is a *right approximate identity for \\(J\\)* if \\(\\|xu_i-x\\|\\to0\\) for every \\(x\\in J\\), a *left* one if \\(\\|u_ix-x\\|\\to0\\), and an *approximate identity for \\(J\\)* if both hold; for \\(J=A\\) we say \"of \\(A\\)\". It is *bounded* if \\(\\sup_i\\|u_i\\|<\\infty\\). In a C\\*-algebra one often also asks that \\(0\\leq u_i\\leq u_j\\) for \\(i\\leq j\\) and \\(\\|u_i\\|\\leq1\\); we call such a net *increasing and contractive*.\n\nFor \\(h\\in A_+\\), the *closed right ideal generated by \\(h\\)* is the smallest closed right ideal of \\(A\\) containing \\(h\\), the closure of \\(hA+\\mathbb Ch\\); similarly on the left. For \\(\\varepsilon>0\\) and \\(t\\geq0\\) put \\(f_\\varepsilon(t)=t/(t+\\varepsilon)\\).\n\n**Lemma 11.2.** Let \\(A\\) be a C\\*-algebra and \\(\\varepsilon>0\\).\n1. If \\(0\\leq h\\leq k\\), then \\(f_\\varepsilon(h)\\leq f_\\varepsilon(k)\\).\n2. Let \\(h\\in A_+\\). Then \\(f_\\varepsilon(h)\\in A_+\\), \\(\\|f_\\varepsilon(h)\\|<1\\), and \\(f_\\varepsilon(h)\\) increases as \\(\\varepsilon\\) decreases. For every \\(x\\) in the closed right ideal generated by \\(h\\), \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\) as \\(\\varepsilon\\to0\\). For every \\(x\\) in the closed left ideal generated by \\(h\\), \\(\\|xf_\\varepsilon(h)-x\\|\\to0\\). For \\(x=hy+\\mu h\\) with \\(y\\in A\\), \\(\\|f_\\varepsilon(h)x-x\\|\\leq\\varepsilon(\\|y\\|+|\\mu|)\\).\n\n**Proof.** (1) \\(\\varepsilon\\leq h+\\varepsilon\\leq k+\\varepsilon\\), so \\((k+\\varepsilon)^{-1}\\leq(h+\\varepsilon)^{-1}\\) by Proposition 8.5(7). Since \\(f_\\varepsilon(t)=1-\\varepsilon(t+\\varepsilon)^{-1}\\),\n\\[\n\\begin{gathered}\nf_\\varepsilon(h)\\\\\n=1-\\varepsilon(h+\\varepsilon)^{-1}\\\\\n\\leq1-\\varepsilon(k+\\varepsilon)^{-1}\\\\\n=f_\\varepsilon(k).\n\\end{gathered}\n\\]\n(2) \\(f_\\varepsilon\\) vanishes at \\(0\\) and takes values in \\([0,1)\\), so \\(f_\\varepsilon(h)\\in A_+\\) and \\(\\|f_\\varepsilon(h)\\|=\\|h\\|/(\\|h\\|+\\varepsilon)<1\\) (Theorem 5.3 and Theorem 5.1(4)). For \\(\\varepsilon'<\\varepsilon\\), \\(f_{\\varepsilon'}\\geq f_\\varepsilon\\) on \\([0,\\infty)\\), so \\(f_{\\varepsilon'}(h)\\geq f_\\varepsilon(h)\\). Next, \\((1-f_\\varepsilon(t))t=\\varepsilon f_\\varepsilon(t)\\), so \\(\\|(1-f_\\varepsilon(h))h\\|\\leq\\varepsilon\\), which gives the bound for \\(x=hy+\\mu h\\). Let \\(\\mathfrak m\\) be the set of \\(x\\in\\widetilde A\\) with \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\). It is a right ideal of \\(\\widetilde A\\), it contains \\(h\\), and it is closed: if \\(x_n\\to x\\) with \\(x_n\\in\\mathfrak m\\), then \\(\\|f_\\varepsilon(h)x-x\\|\\leq2\\|x-x_n\\|+\\|f_\\varepsilon(h)x_n-x_n\\|\\), because \\(\\|f_\\varepsilon(h)\\|\\leq1\\). So every element of the closed right ideal of \\(A\\) generated by \\(h\\) lies in \\(\\mathfrak m\\). The left statement follows by taking adjoints. \\(\\square\\)\n\n**Corollary 11.3** (The closed one-sided ideal generated by a positive element). Let \\(h\\in A_+\\) and let \\(\\mathfrak m\\) be the closed right ideal generated by \\(h\\).\n1. \\(x\\in\\mathfrak m\\) if and only if \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\).\n2. If \\(xx^*\\leq\\lambda h\\) for some \\(\\lambda>0\\), then \\(x\\in\\mathfrak m\\), and \\(\\|x-f_\\varepsilon(h)x\\|\\leq\\frac12(\\lambda\\varepsilon)^{1/2}\\).\n3. \\(\\mathfrak m\\) is the closure of \\(hA\\), and it is also the closed right ideal generated by \\(h^\\alpha\\), for every \\(\\alpha>0\\).\n\nThe mirror statements hold for left ideals, with \\(x^*x\\leq\\lambda h\\) in (2).\n\n**Proof.** (1) One direction is the Lemma. Conversely, \\(f_\\varepsilon(h)x=h\\big((h+\\varepsilon)^{-1}x\\big)\\in hA\\), because \\(A\\) is an ideal of \\(\\widetilde A\\). So a limit of such elements lies in the closure of \\(hA\\), which is inside \\(\\mathfrak m\\).\n(2) By Proposition 8.5(3)–(4),\n\\[\n\\begin{gathered}\n\\|(1-f_\\varepsilon(h))x\\|^2\\\\\n=\\|(1-f_\\varepsilon(h))xx^*(1-f_\\varepsilon(h))\\|\\\\\n\\leq\\lambda\\|(1-f_\\varepsilon(h))h(1-f_\\varepsilon(h))\\|\\\\\n=\\lambda\\max_{t\\in\\sigma'(h)}\\frac{\\varepsilon^2t}{(t+\\varepsilon)^2}\\\\\n\\leq\\frac{\\lambda\\varepsilon}4 ,\n\\end{gathered}\n\\]\nsince \\(4\\varepsilon t\\leq(t+\\varepsilon)^2\\). Now apply (1).\n(3) \\(f_\\varepsilon(h)h\\in hA\\) and \\(\\|h-f_\\varepsilon(h)h\\|\\leq\\varepsilon\\), so \\(h\\) lies in the closure of \\(hA\\). That closure is a closed right ideal, so it is \\(\\mathfrak m\\). For \\(\\alpha>0\\), \\(\\|(1-f_\\varepsilon(h))h^\\alpha\\|=\\max_t\\varepsilon t^\\alpha/(t+\\varepsilon)\\), which is at most \\(\\delta^\\alpha\\) on \\(t\\leq\\delta\\) and at most \\(\\|h\\|^\\alpha\\varepsilon/\\delta\\) on \\(t\\geq\\delta\\); so it tends to \\(0\\), and \\(h^\\alpha\\in\\mathfrak m\\) by (1). Exchanging the roles of \\(h\\) and \\(h^\\alpha\\) (with the exponent \\(1/\\alpha\\)) gives \\(h\\) in the closed right ideal generated by \\(h^\\alpha\\). \\(\\square\\)\n\nSince \\(h^2\\leq\\|h\\|h\\), the condition \\(xx^*\\leq\\lambda h^2\\) implies the hypothesis of (2). The converse fails: \\(x=h^{1/2}\\) satisfies \\(xx^*=h\\), but \\(h\\leq\\lambda h^2\\) fails for every \\(\\lambda\\) when \\(\\sigma'(h)\\) accumulates at \\(0\\) without being \\(\\{0\\}\\).\n\nThe weaker domination in (2) is the exact form used in Proposition 17.2; the preceding counterexample distinguishes it from domination by the square.\n\n**Theorem 11.4** (Approximate identities of one-sided ideals). Let \\(A\\) be a C\\*-algebra, \\(S_0\\) its open unit ball, and \\(\\mathfrak m\\) a left ideal of \\(A\\), not necessarily closed, with \\(\\mathfrak m_+=\\mathfrak m\\cap A_+\\). Order \\(\\Lambda=\\mathfrak m_+\\cap S_0\\) by \\(\\leq\\).\n1. \\(\\Lambda\\) is upward directed.\n2. The net \\((u)_{u\\in\\Lambda}\\), which is increasing and contractive, satisfies \\(\\|x-xu\\|\\to0\\) for every \\(x\\) in the closure \\(\\overline{\\mathfrak m}\\); that is, it is a right approximate identity for \\(\\overline{\\mathfrak m}\\).\n3. If \\(\\mathfrak m\\) is a right ideal, the same set \\(\\Lambda\\) is upward directed and is a left approximate identity for \\(\\overline{\\mathfrak m}\\). If \\(\\mathfrak m\\) is a two-sided ideal, \\(\\Lambda\\) is an increasing contractive approximate identity for \\(\\overline{\\mathfrak m}\\).\n\n**Proof.** A left ideal of \\(A\\) is also a left ideal of \\(\\widetilde A\\), since \\(\\widetilde Ax=Ax+\\mathbb Cx\\).\n(1) Let \\(u_1,u_2\\in\\Lambda\\). Since \\(\\|u_j\\|<1\\), \\(1-u_j\\) is invertible in \\(\\widetilde A\\), and \\(h_j=(1-u_j)^{-1}u_j\\) lies in \\(\\mathfrak m\\). It is the calculus of \\(t/(1-t)\\) at \\(u_j\\), so \\(h_j\\geq0\\). From \\(1+h_j=(1-u_j)^{-1}\\) we get \\(u_j=(1+h_j)^{-1}h_j=f_1(h_j)\\). Put \\(h=h_1+h_2\\) and \\(u=f_1(h)=(1+h)^{-1}h\\). Then \\(u\\in\\mathfrak m\\), \\(u\\geq0\\), \\(\\|u\\|=\\|h\\|/(1+\\|h\\|)<1\\), and \\(u\\geq f_1(h_j)=u_j\\) by Lemma 11.2(1).\n(2) Let \\(x\\in\\mathfrak m\\) and \\(\\varepsilon>0\\). Then \\(k=x^*x\\in\\mathfrak m_+\\), and \\(u_\\varepsilon=f_\\varepsilon(k)=(k+\\varepsilon)^{-1}k\\in\\Lambda\\). Since \\((1-f_\\varepsilon(t))t=\\varepsilon f_\\varepsilon(t)\\),\n\\[\n\\begin{gathered}\n\\|x(1-u_\\varepsilon)^{1/2}\\|^2\\\\\n=\\|(1-u_\\varepsilon)^{1/2}k(1-u_\\varepsilon)^{1/2}\\|\\\\\n=\\|(1-u_\\varepsilon)k\\|\\\\\n\\leq\\varepsilon .\n\\end{gathered}\n\\tag{11.1}\n\\]\nLet \\(v\\in\\Lambda\\) with \\(v\\geq u_\\varepsilon\\). Since \\(0\\leq1-v\\leq1-u_\\varepsilon\\leq1\\),\n\\[\n\\begin{gathered}\n\\|x-xv\\|\\\\\n\\leq\\|x(1-v)^{1/2}\\|\\,\\|(1-v)^{1/2}\\|\\\\\n\\leq\\|x(1-v)x^*\\|^{1/2}\\\\\n\\leq\\|x(1-u_\\varepsilon)x^*\\|^{1/2}\\\\\n=\\|x(1-u_\\varepsilon)^{1/2}\\|\\\\\n\\leq\\varepsilon^{1/2},\n\\end{gathered}\n\\]\nby Proposition 8.5(3)–(4). So \\(xv\\to x\\) along \\(\\Lambda\\) for \\(x\\in\\mathfrak m\\). For \\(x\\in\\overline{\\mathfrak m}\\), use \\(\\|x-xv\\|\\leq2\\|x-y\\|+\\|y-yv\\|\\) with \\(y\\in\\mathfrak m\\) close to \\(x\\), since \\(\\|v\\|\\leq1\\).\n(3) For a right ideal \\(\\mathfrak m\\), \\(\\mathfrak m^*=\\{x^*:x\\in\\mathfrak m\\}\\) is a left ideal with the same positive part, because positive elements are self-adjoint. Apply (1)–(2) to \\(\\mathfrak m^*\\) and take adjoints. A two-sided ideal is both. \\(\\square\\)\n\n**Corollary 11.5** (Existence; the separable case).\n1. Every C\\*-algebra has an increasing contractive approximate identity, for instance \\(A_+\\cap S_0\\) itself.\n2. If \\(A\\) is separable, it has one that is an increasing sequence \\((u_n)\\).\n\n**Proof.** (1) is the theorem with \\(\\mathfrak m=A\\). (2) \\(A_+\\cap S_0\\) is a subset of a separable metric space, so it has a dense sequence \\((v_n)\\). Choose \\(u_1=v_1\\), and by (1) of the theorem choose inductively \\(u_{n+1}\\in A_+\\cap S_0\\) with \\(u_{n+1}\\geq u_n\\) and \\(u_{n+1}\\geq v_1,\\dots,v_{n+1}\\). Let \\(x\\in A\\) and \\(\\varepsilon>0\\), and let \\(u_\\varepsilon\\) be as in (11.1), so \\(\\|x(1-u_\\varepsilon)^{1/2}\\|\\leq\\varepsilon^{1/2}\\). Pick \\(n\\) with \\(\\|v_n-u_\\varepsilon\\|\\leq\\varepsilon/(1+\\|x\\|)^2\\). By the Hölder estimate (9.1), applied in \\(\\widetilde A\\) to \\(1-v_n\\) and \\(1-u_\\varepsilon\\),\n\\[\n\\begin{gathered}\n\\|x(1-v_n)^{1/2}\\|\\\\\n\\leq\\|x(1-u_\\varepsilon)^{1/2}\\|+\\|x\\|\\,\\|v_n-u_\\varepsilon\\|^{1/2}\\\\\n\\leq2\\varepsilon^{1/2}.\n\\end{gathered}\n\\]\nFor \\(k\\geq n\\), \\(u_k\\geq v_n\\), and the chain of inequalities in the theorem's proof gives \\(\\|x-xu_k\\|\\leq\\|x(1-v_n)^{1/2}\\|\\leq2\\varepsilon^{1/2}\\). So \\(xu_k\\to x\\) for every \\(x\\), and since each \\(u_k\\) is self-adjoint, \\(\\|x-u_kx\\|=\\|x^*-x^*u_k\\|\\to0\\) too. \\(\\square\\)\n\nA separable C\\*-algebra even has a sequential approximate identity whose members commute with each other (Proposition 13.3(7)).\n\n**Exercise 11.6** (medium; Closed one-sided ideals are hereditary). If \\(\\mathfrak m\\) is a closed ideal of \\(A\\), \\(0\\leq x\\leq y\\) and \\(y\\in\\mathfrak m\\), then \\(x\\in\\mathfrak m\\). In fact this holds for every closed left ideal \\(\\mathfrak m\\).\n\n*Solution.* \\((x^{1/2})^*x^{1/2}=x\\leq y\\). By Corollary 11.3(2), in its left form, \\(x^{1/2}\\) lies in the closed left ideal generated by \\(y\\), which is inside \\(\\mathfrak m\\). Then \\(x=x^{1/2}x^{1/2}\\in\\mathfrak m\\), since \\(\\mathfrak m\\) is a left ideal. A second solution uses the approximate identity \\((u_i)\\) of \\(\\mathfrak m\\) from Theorem 11.4: \\[\n\\begin{gathered}\n\\|x^{1/2}-x^{1/2}u_i\\|^2\\\\\n=\\|(1-u_i)x(1-u_i)\\|\\\\\n\\leq\\|(1-u_i)y(1-u_i)\\|\\\\\n\\leq\\|y-yu_i\\|\\to0,\n\\end{gathered}\n\\] and \\(x^{1/2}u_i\\in\\mathfrak m\\).\n\n**Example 11.7** (A nonclosed ideal need not be self-adjoint). In \\(A=C([-1,1])\\), put \\(f(t)=t+i|t|\\) and \\(I=fA\\). This is a two-sided algebraic ideal. For \\(t>0\\) the ratio \\(\\overline{f(t)}/f(t)\\) is \\(-i\\), and for \\(t<0\\) it is \\(i\\). A continuous function cannot have these two one-sided limits at \\(0\\), so \\(\\bar f\\notin I\\), although \\(f\\in I\\). Thus \\(I^*\\ne I\\).\n\nIts closure is precisely \\(J=\\{h\\in A:h(0)=0\\}\\). Certainly \\(I\\subseteq J\\). Conversely, for \\(h\\in J\\) and \\(\\varepsilon>0\\), set\n\\[\nh_\\varepsilon(t)=\\frac{h(t)|f(t)|^2}{|f(t)|^2+\\varepsilon}\n=f(t)\\frac{h(t)\\overline{f(t)}}{|f(t)|^2+\\varepsilon}\\in I.\n\\]\nGiven \\(\\delta>0\\), continuity of \\(h\\) makes \\(|h|<\\delta\\) on some interval about \\(0\\). There \\(|h-h_\\varepsilon|\\leq\\delta\\); on the compact complement, \\(|f|^2\\) has a positive minimum, so \\(h_\\varepsilon\\to h\\) uniformly. Hence \\(\\overline I=J\\). In particular \\(\\bar f\\in\\overline I\\setminus I\\), so \\(I\\) is not closed.\n\nTheorem 11.4 still supplies positive approximate identities inside \\(I\\) for its closure. It does not assert that \\(I\\) is self-adjoint. The closedness hypothesis in Theorem 15.1 is what permits a C*-algebra quotient. For related examples and the closed-ideal proof, see [Blackadar, *Operator Algebras*, II.5.2.1 and II.5.1.1, corrected author version](https://bruceblackadar.com/Mathematics/Cycr.pdf).\n\n",
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      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "11. Approximate identities",
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      "full_conditions_and_proof": "### 11. Approximate identities\n\nAdjoining an identity is not always enough: the unitization of an ideal is no longer an ideal, so statements about ideals cannot be reduced to the unital case. Approximate identities replace the identity in such cases.\n\n**Definition 11.1.** Let \\(A\\) be a Banach algebra and \\(J\\subseteq A\\). A net \\((u_i)\\) in \\(A\\) is a *right approximate identity for \\(J\\)* if \\(\\|xu_i-x\\|\\to0\\) for every \\(x\\in J\\), a *left* one if \\(\\|u_ix-x\\|\\to0\\), and an *approximate identity for \\(J\\)* if both hold; for \\(J=A\\) we say \"of \\(A\\)\". It is *bounded* if \\(\\sup_i\\|u_i\\|<\\infty\\). In a C\\*-algebra one often also asks that \\(0\\leq u_i\\leq u_j\\) for \\(i\\leq j\\) and \\(\\|u_i\\|\\leq1\\); we call such a net *increasing and contractive*.\n\nFor \\(h\\in A_+\\), the *closed right ideal generated by \\(h\\)* is the smallest closed right ideal of \\(A\\) containing \\(h\\), the closure of \\(hA+\\mathbb Ch\\); similarly on the left. For \\(\\varepsilon>0\\) and \\(t\\geq0\\) put \\(f_\\varepsilon(t)=t/(t+\\varepsilon)\\).\n\n**Lemma 11.2.** Let \\(A\\) be a C\\*-algebra and \\(\\varepsilon>0\\).\n1. If \\(0\\leq h\\leq k\\), then \\(f_\\varepsilon(h)\\leq f_\\varepsilon(k)\\).\n2. Let \\(h\\in A_+\\). Then \\(f_\\varepsilon(h)\\in A_+\\), \\(\\|f_\\varepsilon(h)\\|<1\\), and \\(f_\\varepsilon(h)\\) increases as \\(\\varepsilon\\) decreases. For every \\(x\\) in the closed right ideal generated by \\(h\\), \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\) as \\(\\varepsilon\\to0\\). For every \\(x\\) in the closed left ideal generated by \\(h\\), \\(\\|xf_\\varepsilon(h)-x\\|\\to0\\). For \\(x=hy+\\mu h\\) with \\(y\\in A\\), \\(\\|f_\\varepsilon(h)x-x\\|\\leq\\varepsilon(\\|y\\|+|\\mu|)\\).\n\n**Proof.** (1) \\(\\varepsilon\\leq h+\\varepsilon\\leq k+\\varepsilon\\), so \\((k+\\varepsilon)^{-1}\\leq(h+\\varepsilon)^{-1}\\) by Proposition 8.5(7). Since \\(f_\\varepsilon(t)=1-\\varepsilon(t+\\varepsilon)^{-1}\\),\n\\[\n\\begin{gathered}\nf_\\varepsilon(h)\\\\\n=1-\\varepsilon(h+\\varepsilon)^{-1}\\\\\n\\leq1-\\varepsilon(k+\\varepsilon)^{-1}\\\\\n=f_\\varepsilon(k).\n\\end{gathered}\n\\]\n(2) \\(f_\\varepsilon\\) vanishes at \\(0\\) and takes values in \\([0,1)\\), so \\(f_\\varepsilon(h)\\in A_+\\) and \\(\\|f_\\varepsilon(h)\\|=\\|h\\|/(\\|h\\|+\\varepsilon)<1\\) (Theorem 5.3 and Theorem 5.1(4)). For \\(\\varepsilon'<\\varepsilon\\), \\(f_{\\varepsilon'}\\geq f_\\varepsilon\\) on \\([0,\\infty)\\), so \\(f_{\\varepsilon'}(h)\\geq f_\\varepsilon(h)\\). Next, \\((1-f_\\varepsilon(t))t=\\varepsilon f_\\varepsilon(t)\\), so \\(\\|(1-f_\\varepsilon(h))h\\|\\leq\\varepsilon\\), which gives the bound for \\(x=hy+\\mu h\\). Let \\(\\mathfrak m\\) be the set of \\(x\\in\\widetilde A\\) with \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\). It is a right ideal of \\(\\widetilde A\\), it contains \\(h\\), and it is closed: if \\(x_n\\to x\\) with \\(x_n\\in\\mathfrak m\\), then \\(\\|f_\\varepsilon(h)x-x\\|\\leq2\\|x-x_n\\|+\\|f_\\varepsilon(h)x_n-x_n\\|\\), because \\(\\|f_\\varepsilon(h)\\|\\leq1\\). So every element of the closed right ideal of \\(A\\) generated by \\(h\\) lies in \\(\\mathfrak m\\). The left statement follows by taking adjoints. \\(\\square\\)\n\n**Corollary 11.3** (The closed one-sided ideal generated by a positive element). Let \\(h\\in A_+\\) and let \\(\\mathfrak m\\) be the closed right ideal generated by \\(h\\).\n1. \\(x\\in\\mathfrak m\\) if and only if \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\).\n2. If \\(xx^*\\leq\\lambda h\\) for some \\(\\lambda>0\\), then \\(x\\in\\mathfrak m\\), and \\(\\|x-f_\\varepsilon(h)x\\|\\leq\\frac12(\\lambda\\varepsilon)^{1/2}\\).\n3. \\(\\mathfrak m\\) is the closure of \\(hA\\), and it is also the closed right ideal generated by \\(h^\\alpha\\), for every \\(\\alpha>0\\).\n\nThe mirror statements hold for left ideals, with \\(x^*x\\leq\\lambda h\\) in (2).\n\n**Proof.** (1) One direction is the Lemma. Conversely, \\(f_\\varepsilon(h)x=h\\big((h+\\varepsilon)^{-1}x\\big)\\in hA\\), because \\(A\\) is an ideal of \\(\\widetilde A\\). So a limit of such elements lies in the closure of \\(hA\\), which is inside \\(\\mathfrak m\\).\n(2) By Proposition 8.5(3)–(4),\n\\[\n\\begin{gathered}\n\\|(1-f_\\varepsilon(h))x\\|^2\\\\\n=\\|(1-f_\\varepsilon(h))xx^*(1-f_\\varepsilon(h))\\|\\\\\n\\leq\\lambda\\|(1-f_\\varepsilon(h))h(1-f_\\varepsilon(h))\\|\\\\\n=\\lambda\\max_{t\\in\\sigma'(h)}\\frac{\\varepsilon^2t}{(t+\\varepsilon)^2}\\\\\n\\leq\\frac{\\lambda\\varepsilon}4 ,\n\\end{gathered}\n\\]\nsince \\(4\\varepsilon t\\leq(t+\\varepsilon)^2\\). Now apply (1).\n(3) \\(f_\\varepsilon(h)h\\in hA\\) and \\(\\|h-f_\\varepsilon(h)h\\|\\leq\\varepsilon\\), so \\(h\\) lies in the closure of \\(hA\\). That closure is a closed right ideal, so it is \\(\\mathfrak m\\). For \\(\\alpha>0\\), \\(\\|(1-f_\\varepsilon(h))h^\\alpha\\|=\\max_t\\varepsilon t^\\alpha/(t+\\varepsilon)\\), which is at most \\(\\delta^\\alpha\\) on \\(t\\leq\\delta\\) and at most \\(\\|h\\|^\\alpha\\varepsilon/\\delta\\) on \\(t\\geq\\delta\\); so it tends to \\(0\\), and \\(h^\\alpha\\in\\mathfrak m\\) by (1). Exchanging the roles of \\(h\\) and \\(h^\\alpha\\) (with the exponent \\(1/\\alpha\\)) gives \\(h\\) in the closed right ideal generated by \\(h^\\alpha\\). \\(\\square\\)\n\nSince \\(h^2\\leq\\|h\\|h\\), the condition \\(xx^*\\leq\\lambda h^2\\) implies the hypothesis of (2). The converse fails: \\(x=h^{1/2}\\) satisfies \\(xx^*=h\\), but \\(h\\leq\\lambda h^2\\) fails for every \\(\\lambda\\) when \\(\\sigma'(h)\\) accumulates at \\(0\\) without being \\(\\{0\\}\\).\n\nThe weaker domination in (2) is the exact form used in Proposition 17.2; the preceding counterexample distinguishes it from domination by the square.\n\n**Theorem 11.4** (Approximate identities of one-sided ideals). Let \\(A\\) be a C\\*-algebra, \\(S_0\\) its open unit ball, and \\(\\mathfrak m\\) a left ideal of \\(A\\), not necessarily closed, with \\(\\mathfrak m_+=\\mathfrak m\\cap A_+\\). Order \\(\\Lambda=\\mathfrak m_+\\cap S_0\\) by \\(\\leq\\).\n1. \\(\\Lambda\\) is upward directed.\n2. The net \\((u)_{u\\in\\Lambda}\\), which is increasing and contractive, satisfies \\(\\|x-xu\\|\\to0\\) for every \\(x\\) in the closure \\(\\overline{\\mathfrak m}\\); that is, it is a right approximate identity for \\(\\overline{\\mathfrak m}\\).\n3. If \\(\\mathfrak m\\) is a right ideal, the same set \\(\\Lambda\\) is upward directed and is a left approximate identity for \\(\\overline{\\mathfrak m}\\). If \\(\\mathfrak m\\) is a two-sided ideal, \\(\\Lambda\\) is an increasing contractive approximate identity for \\(\\overline{\\mathfrak m}\\).\n\n**Proof.** A left ideal of \\(A\\) is also a left ideal of \\(\\widetilde A\\), since \\(\\widetilde Ax=Ax+\\mathbb Cx\\).\n(1) Let \\(u_1,u_2\\in\\Lambda\\). Since \\(\\|u_j\\|<1\\), \\(1-u_j\\) is invertible in \\(\\widetilde A\\), and \\(h_j=(1-u_j)^{-1}u_j\\) lies in \\(\\mathfrak m\\). It is the calculus of \\(t/(1-t)\\) at \\(u_j\\), so \\(h_j\\geq0\\). From \\(1+h_j=(1-u_j)^{-1}\\) we get \\(u_j=(1+h_j)^{-1}h_j=f_1(h_j)\\). Put \\(h=h_1+h_2\\) and \\(u=f_1(h)=(1+h)^{-1}h\\). Then \\(u\\in\\mathfrak m\\), \\(u\\geq0\\), \\(\\|u\\|=\\|h\\|/(1+\\|h\\|)<1\\), and \\(u\\geq f_1(h_j)=u_j\\) by Lemma 11.2(1).\n(2) Let \\(x\\in\\mathfrak m\\) and \\(\\varepsilon>0\\). Then \\(k=x^*x\\in\\mathfrak m_+\\), and \\(u_\\varepsilon=f_\\varepsilon(k)=(k+\\varepsilon)^{-1}k\\in\\Lambda\\). Since \\((1-f_\\varepsilon(t))t=\\varepsilon f_\\varepsilon(t)\\),\n\\[\n\\begin{gathered}\n\\|x(1-u_\\varepsilon)^{1/2}\\|^2\\\\\n=\\|(1-u_\\varepsilon)^{1/2}k(1-u_\\varepsilon)^{1/2}\\|\\\\\n=\\|(1-u_\\varepsilon)k\\|\\\\\n\\leq\\varepsilon .\n\\end{gathered}\n\\tag{11.1}\n\\]\nLet \\(v\\in\\Lambda\\) with \\(v\\geq u_\\varepsilon\\). Since \\(0\\leq1-v\\leq1-u_\\varepsilon\\leq1\\),\n\\[\n\\begin{gathered}\n\\|x-xv\\|\\\\\n\\leq\\|x(1-v)^{1/2}\\|\\,\\|(1-v)^{1/2}\\|\\\\\n\\leq\\|x(1-v)x^*\\|^{1/2}\\\\\n\\leq\\|x(1-u_\\varepsilon)x^*\\|^{1/2}\\\\\n=\\|x(1-u_\\varepsilon)^{1/2}\\|\\\\\n\\leq\\varepsilon^{1/2},\n\\end{gathered}\n\\]\nby Proposition 8.5(3)–(4). So \\(xv\\to x\\) along \\(\\Lambda\\) for \\(x\\in\\mathfrak m\\). For \\(x\\in\\overline{\\mathfrak m}\\), use \\(\\|x-xv\\|\\leq2\\|x-y\\|+\\|y-yv\\|\\) with \\(y\\in\\mathfrak m\\) close to \\(x\\), since \\(\\|v\\|\\leq1\\).\n(3) For a right ideal \\(\\mathfrak m\\), \\(\\mathfrak m^*=\\{x^*:x\\in\\mathfrak m\\}\\) is a left ideal with the same positive part, because positive elements are self-adjoint. Apply (1)–(2) to \\(\\mathfrak m^*\\) and take adjoints. A two-sided ideal is both. \\(\\square\\)\n\n**Corollary 11.5** (Existence; the separable case).\n1. Every C\\*-algebra has an increasing contractive approximate identity, for instance \\(A_+\\cap S_0\\) itself.\n2. If \\(A\\) is separable, it has one that is an increasing sequence \\((u_n)\\).\n\n**Proof.** (1) is the theorem with \\(\\mathfrak m=A\\). (2) \\(A_+\\cap S_0\\) is a subset of a separable metric space, so it has a dense sequence \\((v_n)\\). Choose \\(u_1=v_1\\), and by (1) of the theorem choose inductively \\(u_{n+1}\\in A_+\\cap S_0\\) with \\(u_{n+1}\\geq u_n\\) and \\(u_{n+1}\\geq v_1,\\dots,v_{n+1}\\). Let \\(x\\in A\\) and \\(\\varepsilon>0\\), and let \\(u_\\varepsilon\\) be as in (11.1), so \\(\\|x(1-u_\\varepsilon)^{1/2}\\|\\leq\\varepsilon^{1/2}\\). Pick \\(n\\) with \\(\\|v_n-u_\\varepsilon\\|\\leq\\varepsilon/(1+\\|x\\|)^2\\). By the Hölder estimate (9.1), applied in \\(\\widetilde A\\) to \\(1-v_n\\) and \\(1-u_\\varepsilon\\),\n\\[\n\\begin{gathered}\n\\|x(1-v_n)^{1/2}\\|\\\\\n\\leq\\|x(1-u_\\varepsilon)^{1/2}\\|+\\|x\\|\\,\\|v_n-u_\\varepsilon\\|^{1/2}\\\\\n\\leq2\\varepsilon^{1/2}.\n\\end{gathered}\n\\]\nFor \\(k\\geq n\\), \\(u_k\\geq v_n\\), and the chain of inequalities in the theorem's proof gives \\(\\|x-xu_k\\|\\leq\\|x(1-v_n)^{1/2}\\|\\leq2\\varepsilon^{1/2}\\). So \\(xu_k\\to x\\) for every \\(x\\), and since each \\(u_k\\) is self-adjoint, \\(\\|x-u_kx\\|=\\|x^*-x^*u_k\\|\\to0\\) too. \\(\\square\\)\n\nA separable C\\*-algebra even has a sequential approximate identity whose members commute with each other (Proposition 13.3(7)).\n\n**Exercise 11.6** (medium; Closed one-sided ideals are hereditary). If \\(\\mathfrak m\\) is a closed ideal of \\(A\\), \\(0\\leq x\\leq y\\) and \\(y\\in\\mathfrak m\\), then \\(x\\in\\mathfrak m\\). In fact this holds for every closed left ideal \\(\\mathfrak m\\).\n\n*Solution.* \\((x^{1/2})^*x^{1/2}=x\\leq y\\). By Corollary 11.3(2), in its left form, \\(x^{1/2}\\) lies in the closed left ideal generated by \\(y\\), which is inside \\(\\mathfrak m\\). Then \\(x=x^{1/2}x^{1/2}\\in\\mathfrak m\\), since \\(\\mathfrak m\\) is a left ideal. A second solution uses the approximate identity \\((u_i)\\) of \\(\\mathfrak m\\) from Theorem 11.4: \\[\n\\begin{gathered}\n\\|x^{1/2}-x^{1/2}u_i\\|^2\\\\\n=\\|(1-u_i)x(1-u_i)\\|\\\\\n\\leq\\|(1-u_i)y(1-u_i)\\|\\\\\n\\leq\\|y-yu_i\\|\\to0,\n\\end{gathered}\n\\] and \\(x^{1/2}u_i\\in\\mathfrak m\\).\n\n**Example 11.7** (A nonclosed ideal need not be self-adjoint). In \\(A=C([-1,1])\\), put \\(f(t)=t+i|t|\\) and \\(I=fA\\). This is a two-sided algebraic ideal. For \\(t>0\\) the ratio \\(\\overline{f(t)}/f(t)\\) is \\(-i\\), and for \\(t<0\\) it is \\(i\\). A continuous function cannot have these two one-sided limits at \\(0\\), so \\(\\bar f\\notin I\\), although \\(f\\in I\\). Thus \\(I^*\\ne I\\).\n\nIts closure is precisely \\(J=\\{h\\in A:h(0)=0\\}\\). Certainly \\(I\\subseteq J\\). Conversely, for \\(h\\in J\\) and \\(\\varepsilon>0\\), set\n\\[\nh_\\varepsilon(t)=\\frac{h(t)|f(t)|^2}{|f(t)|^2+\\varepsilon}\n=f(t)\\frac{h(t)\\overline{f(t)}}{|f(t)|^2+\\varepsilon}\\in I.\n\\]\nGiven \\(\\delta>0\\), continuity of \\(h\\) makes \\(|h|<\\delta\\) on some interval about \\(0\\). There \\(|h-h_\\varepsilon|\\leq\\delta\\); on the compact complement, \\(|f|^2\\) has a positive minimum, so \\(h_\\varepsilon\\to h\\) uniformly. Hence \\(\\overline I=J\\). In particular \\(\\bar f\\in\\overline I\\setminus I\\), so \\(I\\) is not closed.\n\nTheorem 11.4 still supplies positive approximate identities inside \\(I\\) for its closure. It does not assert that \\(I\\) is self-adjoint. The closedness hypothesis in Theorem 15.1 is what permits a C*-algebra quotient. For related examples and the closed-ideal proof, see [Blackadar, *Operator Algebras*, II.5.2.1 and II.5.1.1, corrected author version](https://bruceblackadar.com/Mathematics/Cycr.pdf).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-23",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "12. Approximate identities of convolution algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 12. Approximate identities of convolution algebras\n\nTwo Banach \\(*\\)-algebras built from a locally compact group, the group algebra \\(L^1(G)\\) and the transformation-group algebra, need not have an identity; both are constructed in [group algebras and transformation-group algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-20). This section shows that both have approximate identities of norm at most one. We use the lesson on Haar measure: \\(G\\) is a locally compact group with left Haar measure \\(ds\\) and modular function \\(\\Delta\\) ([the modular function](haar-measure.md#oa-fnd-hm-07)). \\(L^1(G)\\), with convolution \\(f*g(x)=\\int f(y)g(y^{-1}x)\\,dy\\) and involution \\(f^*(x)=\\Delta(x)^{-1}\\overline{f(x^{-1})}\\), is a Banach \\(*\\)-algebra ([convolution and continuity of translation](haar-measure.md#oa-fnd-hm-12)). Translations are \\(L_yf(x)=f(y^{-1}x)\\) and \\(R_yf(x)=f(xy)\\).\n\n**Proposition 12.1** (Approximate identities of \\(L^1(G)\\)). For each neighbourhood \\(V\\) of \\(e\\), let \\(u_V\\in L^1(G)\\) satisfy \\(u_V\\geq0\\), \\(u_V=0\\) off \\(V\\) and \\(\\int u_V=1\\); continuity, compact support and symmetry are not required. Direct the neighbourhoods by reverse inclusion. Then \\((u_V)\\) is an approximate identity of \\(L^1(G)\\) with \\(\\|u_V\\|_1=1\\).\n\n**Proof.** *Left.* For \\(f\\in L^1(G)\\), \\(\\|u_V*f-f\\|_1\\leq\\sup_{y\\in V}\\|L_yf-f\\|_1\\). This is the computation in the proof of [compactly supported approximate identities](haar-measure.md#oa-fnd-hm-13). That proof uses of \\(\\psi_U\\) only that it is nonnegative with integral one and vanishes off \\(U\\); compact support serves only to make its nonzero set \\(\\sigma\\)-finite, which holds for every integrable function. The right side tends to \\(0\\) by the continuity of translation in \\(L^1(G)\\).\n*Right.* By the third formula for convolution in [convolution and continuity of translation](haar-measure.md#oa-fnd-hm-12), \\(f*u(x)=\\int f(xy^{-1})\\,u(y)\\,\\Delta(y)^{-1}\\,dy\\). Put \\(T_yf=\\Delta(y)^{-1}R_{y^{-1}}f\\). Right translation satisfies \\(\\|R_{y^{-1}}f\\|_1=\\Delta(y)\\|f\\|_1\\) ([the modular function](haar-measure.md#oa-fnd-hm-07)), so \\(T_y\\) is an isometry of \\(L^1(G)\\), and\n\\[\n\\begin{gathered}\n\\|T_yf-f\\|_1\\\\\n\\leq|\\Delta(y)^{-1}-1|\\,\\|R_{y^{-1}}f\\|_1+\\|R_{y^{-1}}f-f\\|_1\\to0\\\\\n(y\\to e)\n\\end{gathered}\n\\]\nby the continuity of \\(\\Delta\\) and of translation. Since \\(f*u_V-f=\\int u_V(y)(T_yf-f)\\,dy\\) almost everywhere, the same Tonelli argument gives \\(\\|f*u_V-f\\|_1\\leq\\sup_{y\\in V}\\|T_yf-f\\|_1\\to0\\). \\(\\square\\)\n\nThe lesson on Haar measure proves the right-hand statement for symmetric \\(\\psi_U\\); the modular factor in \\(T_y\\) removes the need for symmetry.\n\n**Proposition 12.2** (Approximate identities of the transformation-group algebra). Let \\(\\Omega\\) be an LCH space on which \\(G\\) acts continuously from the right, \\((\\omega,s)\\mapsto\\omega s\\). Let \\(A\\) be the completion of \\(C_c(\\Omega\\times G)\\) for \\(\\|x\\|=\\int_G\\sup_\\omega|x(\\omega,s)|\\,ds\\), with the product \\((xy)(\\omega,s)=\\int_Gx(\\omega,t)\\,y(\\omega t,t^{-1}s)\\,dt\\). For each compact \\(K\\subseteq\\Omega\\) let \\(f_K\\in C_c(\\Omega)\\) with \\(0\\leq f_K\\leq1\\) and \\(f_K=1\\) on \\(K\\). For each neighbourhood \\(V\\) of \\(e\\) inside a fixed compact neighbourhood \\(V_0\\), let \\(u_V\\) be continuous, \\(u_V\\geq0\\), \\(u_V=0\\) off \\(V\\), \\(\\int u_V=1\\). Put \\(u_{K,V}(\\omega,s)=f_K(\\omega)u_V(s)\\), and direct the pairs by \\(K\\) increasing and \\(V\\) decreasing. Then \\((u_{K,V})\\) is an approximate identity of \\(A\\) with \\(\\|u_{K,V}\\|\\leq1\\).\n\nRestricting to \\(V\\subseteq V_0\\) only makes \\(u_{K,V}\\) compactly supported; the directed set is unchanged far out, which is all that matters for convergence.\n\n**Proof.** \\(\\|u_{K,V}\\|=\\int u_V\\sup_\\omega f_K\\leq1\\). Because the net is bounded, it is enough to prove \\(u_{K,V}x\\to x\\) and \\(xu_{K,V}\\to x\\) for \\(x\\in C_c(\\Omega\\times G)\\) (a \\(3\\varepsilon\\) argument). Let \\(C\\) be the support of \\(x\\), with projections \\(C_\\Omega\\) and \\(C_G\\).\n*Uniform continuity.* As \\(t\\to e\\) and \\(y\\to e\\),\n\\[\n\\begin{gathered}\n\\sup_{\\omega,s}|x(\\omega t,t^{-1}s)-x(\\omega,s)|\\to0,\\\\\n\\sup_{\\omega,s}|x(\\omega,sy^{-1})-x(\\omega,s)|\\to0 .\n\\end{gathered}\n\\]\nIf the first fails, there are \\(\\varepsilon>0\\), a net \\(t_\\alpha\\to e\\) and points \\((\\omega_\\alpha,s_\\alpha)\\) with the difference at least \\(\\varepsilon\\). Then \\((\\omega_\\alpha,s_\\alpha)\\) or \\((\\omega_\\alpha t_\\alpha,t_\\alpha^{-1}s_\\alpha)\\) lies in \\(C\\) for every \\(\\alpha\\); passing to a subnet, the same alternative holds throughout and those points converge in the compact set \\(C\\). By continuity of the action and of the group operations, the other point converges to the same limit, and both values of \\(x\\) converge to its value there, a contradiction. The second statement is proved the same way.\n*Left.* For \\(t\\in V\\subseteq V_0\\), \\(x(\\omega t,t^{-1}s)\\neq0\\) forces \\(\\omega\\in C_\\Omega V_0^{-1}\\) and \\(s\\in V_0C_G\\), both compact. Once \\(K\\supseteq C_\\Omega V_0^{-1}\\cup C_\\Omega\\), \\(f_K=1\\) wherever either \\(x(\\omega,s)\\) or the integrand is nonzero, so\n\\[\n\\begin{gathered}\n(u_{K,V}x-x)(\\omega,s)\\\\\n=\\int u_V(t)\\big(x(\\omega t,t^{-1}s)-x(\\omega,s)\\big)\\,dt ,\n\\end{gathered}\n\\]\nwhich vanishes for \\(s\\notin V_0C_G\\) and is bounded by the first supremum, over \\(t\\in V\\). So \\[\n\\begin{gathered}\n\\|u_{K,V}x-x\\|\\\\\n\\leq|V_0C_G|\\sup_{t\\in V,\\omega,s}|x(\\omega t,t^{-1}s)-x(\\omega,s)|\\to0,\n\\end{gathered}\n\\] where \\(|\\cdot|\\) is Haar measure.\n*Right.* \\((xu_{K,V})(\\omega,s)=\\int x(\\omega,t)f_K(\\omega t)u_V(t^{-1}s)\\,dt\\). If \\(x(\\omega,t)\\neq0\\), then \\(\\omega t\\in C_\\Omega C_G\\), a compact set; once \\(K\\supseteq C_\\Omega C_G\\), the factor \\(f_K(\\omega t)\\) is \\(1\\). The change of variables behind the third formula for convolution ([convolution and continuity of translation](haar-measure.md#oa-fnd-hm-12)) gives\n\\[\n\\begin{gathered}\n(xu_{K,V}-x)(\\omega,s)\\\\\n=\\int u_V(y)\\big(\\Delta(y)^{-1}x(\\omega,sy^{-1})-x(\\omega,s)\\big)\\,dy ,\n\\end{gathered}\n\\]\nwhich vanishes for \\(s\\notin C_GV_0\\), and whose modulus is at most \\[\n\\begin{gathered}\n\\sup_{y\\in V}\\big(|\\Delta(y)^{-1}-1|\\,\\|x\\|_\\infty\\\\\n+\\sup_{\\omega,s}|x(\\omega,sy^{-1})-x(\\omega,s)|\\big).\n\\end{gathered}\n\\] This tends to \\(0\\), and so does \\(\\|xu_{K,V}-x\\|\\), which is at most \\(|C_GV_0|\\) times it. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CF-22",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "13. Strictly positive elements and countable approximate identities",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 13. Strictly positive elements and countable approximate identities\n\nA C\\*-algebra has a sequential approximate identity exactly when it has a strictly positive element (Proposition 13.3). We first collect the facts about positive linear functionals that are needed. A linear functional \\(\\varphi\\) on a C\\*-algebra \\(A\\) is *positive* if \\(\\varphi(A_+)\\subseteq[0,\\infty)\\), and a *state* is a positive linear functional of norm one.\n\n**Lemma 13.1** (Positive linear functionals). Let \\(A\\) be a C\\*-algebra and \\(\\varphi\\) a positive linear functional on \\(A\\).\n1. \\(\\varphi\\) is bounded.\n2. \\(\\varphi\\) is hermitian, and \\(|\\varphi(y^*x)|^2\\leq\\varphi(x^*x)\\varphi(y^*y)\\) for all \\(x,y\\in A\\).\n3. \\(|\\varphi(x)|^2\\leq\\|\\varphi\\|\\varphi(x^*x)\\) for all \\(x\\in A\\).\n4. The set \\(Q\\) of positive linear functionals of norm at most one is weak\\* compact.\n5. If \\(A\\) is unital and nontrivial, then \\(\\|\\varphi\\|=\\varphi(1)\\).\n6. (*States.*) If \\(A\\) is unital and nontrivial, \\(h\\in A_h\\) and \\(\\lambda\\in\\sigma(h)\\), some state \\(\\omega\\) has \\(\\omega(h)=\\lambda\\).\n7. (*States.*) If \\(A\\neq0\\) and \\(a\\in A_+\\), some state \\(\\omega\\) has \\(\\omega(a)=\\|a\\|\\).\n\n**Proof.** (1) If not, since every element is a combination of four positive elements of no larger norm (Proposition 8.5(8)), there are \\(a_n\\in A_+\\) with \\(\\|a_n\\|\\leq1\\) and \\(\\varphi(a_n)\\geq4^n\\). The series \\(a=\\sum2^{-n}a_n\\) converges, \\(a\\geq2^{-n}a_n\\) because the rest of the series is positive, and so \\(\\varphi(a)\\geq2^n\\) for every \\(n\\), which is absurd.\n(2) For \\(h\\in A_h\\), \\(\\varphi(h)=\\varphi(h_+)-\\varphi(h_-)\\) is real (Proposition 8.5(8)), so \\(\\varphi\\) is hermitian. The form \\((x,y)\\mapsto\\varphi(y^*x)\\) is sesquilinear and positive semidefinite, so the inequality is the Cauchy–Schwarz inequality, Proposition 1.1(3) of [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md).\n(3) For \\(u\\) in the approximate identity of Theorem 11.4, \\(|\\varphi(ux)|^2\\leq\\varphi(u^2)\\varphi(x^*x)\\leq\\|\\varphi\\|\\varphi(x^*x)\\), and \\(\\varphi(ux)\\to\\varphi(x)\\).\n(4) \\(Q\\) lies in the closed unit ball of the dual of \\(A\\), which is weak\\* compact by the Banach–Alaoglu theorem (Theorem 3.1 of [Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md)), and positivity is a weak\\*-closed condition.\n(5) Let \\(\\|x\\|\\leq1\\). By (2) with \\(y=1\\), \\(|\\varphi(x)|^2\\leq\\varphi(x^*x)\\varphi(1)\\). Since \\(x^*x\\leq\\|x\\|^2\\leq1\\) (Proposition 8.5(2)), \\(\\varphi(x^*x)\\leq\\varphi(1)\\). So \\(|\\varphi(x)|\\leq\\varphi(1)\\), and \\(\\|\\varphi\\|\\leq\\varphi(1)\\leq\\|\\varphi\\|\\,\\|1\\|=\\|\\varphi\\|\\).\n(6) *A state of \\(C^*(1,h)\\).* By Theorem 5.1, \\(f(h)\\mapsto f(\\lambda)\\) is a well-defined linear functional \\(\\omega_0\\) on \\(B=C^*(1,h)\\). It is positive, because the positive elements of \\(B\\) are the \\(f(h)\\) with \\(f\\geq0\\) (Theorem 5.1 and Remark 8.4). By (5), \\(\\|\\omega_0\\|=\\omega_0(1)=1\\).\n*Extension.* By the Hahn–Banach theorem (Corollary 2.3(1) of [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md)), \\(\\omega_0\\) extends to \\(\\omega\\in A^*\\) with \\(\\|\\omega\\|=1=\\omega(1)\\).\n*Such an \\(\\omega\\) is positive.* Let \\(a\\in A_+\\) with \\(\\|a\\|\\leq1\\), and write \\(\\omega(a)=\\alpha+i\\beta\\).\n- For real \\(t\\), \\[\n\\begin{gathered}\n\\|a+it\\|^2\\\\\n=\\|(a-it)(a+it)\\|\\\\\n=\\|a^2+t^2\\|\\\\\n\\leq1+t^2,\n\\end{gathered}\n\\] so \\(\\alpha^2+(\\beta+t)^2=|\\omega(a+it)|^2\\leq1+t^2\\). That is, \\(\\alpha^2+\\beta^2+2\\beta t\\leq1\\) for every \\(t\\), so \\(\\beta=0\\).\n- \\(\\sigma(1-a)\\subseteq[0,1]\\), so \\(\\|1-a\\|\\leq1\\) (Theorem 1.3). Hence \\(|1-\\omega(a)|\\leq1\\), and \\(\\omega(a)\\geq0\\).\n\nSo \\(\\omega\\) is a state, and \\(\\omega(h)=\\omega_0(h)=\\lambda\\).\n(7) In \\(\\widetilde A\\), \\(\\sigma'(a)\\subseteq[0,\\|a\\|]\\), and \\(\\|a\\|=r(a)\\) lies in \\(\\sigma'(a)\\) (Theorem 1.3). By (6) in \\(\\widetilde A\\), some state \\(\\tilde\\omega\\) of \\(\\widetilde A\\) has \\(\\tilde\\omega(a)=\\|a\\|\\). Its restriction \\(\\omega\\) to \\(A\\) is positive with \\(\\|\\omega\\|\\leq1\\). If \\(a\\neq0\\), then \\(\\omega(a)=\\|a\\|\\) forces \\(\\|\\omega\\|=1\\). If \\(a=0\\), take any nonzero \\(b\\in A_+\\), for instance \\(b=x^*x\\) with \\(x\\neq0\\), and a state \\(\\omega\\) with \\(\\omega(b)=\\|b\\|\\); then \\(\\omega(a)=0=\\|a\\|\\). \\(\\square\\)\n\nParts (6) and (7) say that states exist in abundance.\n\n**Definition 13.2.** An element \\(a\\in A_+\\) is *strictly positive* if \\(\\varphi(a)>0\\) for every nonzero positive linear functional \\(\\varphi\\).\n\n**Proposition 13.3** (Strictly positive elements). Let \\(A\\) be a C\\*-algebra, \\(a\\in A_+\\), and \\(f_t(\\lambda)=\\lambda/(\\lambda+t)\\).\n1. If \\(A\\) is unital and nontrivial, \\(a\\) is strictly positive if and only if it is invertible.\n2. Every separable C\\*-algebra has a strictly positive element.\n3. \\(a\\) is strictly positive if and only if \\(\\|f_t(a)x-x\\|\\to0\\) as \\(t\\to0\\), for every \\(x\\in A\\).\n4. Let \\(a\\) be strictly positive, and let \\(\\pi:A\\to B(H)\\) be a *nondegenerate representation*: a \\(*\\)-homomorphism such that \\(\\pi(A)H\\) spans a dense subspace. Then \\(\\pi(f_t(a))\\xi\\to\\xi\\) for every \\(\\xi\\in H\\).\n5. If \\(a\\) is strictly positive, \\((f_{1/n}(a))_{n\\geq1}\\) is an increasing, contractive, commuting approximate identity.\n6. (*Converse of (5).*) If some sequence \\((e_n)\\) in \\(A\\) satisfies \\(\\|e_nx-x\\|\\to0\\) for every \\(x\\in A\\), for instance a sequential approximate identity, then \\(a=\\sum_n2^{-n}(1+\\|e_n\\|^2)^{-1}e_ne_n^*\\) is strictly positive. If the \\(e_n\\) are positive contractions, \\(a=\\sum_n2^{-n}e_n\\) is strictly positive as well.\n7. Every separable C\\*-algebra has an increasing, contractive approximate identity made of a commuting sequence.\n\n**Proof.** (1) If \\(a\\) is invertible, then \\(a\\geq m\\) with \\(m=\\min\\sigma(a)>0\\), and \\(\\varphi(a)\\geq m\\varphi(1)=m\\|\\varphi\\|>0\\) for \\(\\varphi\\neq0\\). If \\(a\\) is not invertible, \\(0\\in\\sigma(a)\\), and some state \\(\\omega\\) has \\(\\omega(a)=0\\).\n(2) If \\(A=0\\), \\(a=0\\) will do. Otherwise let \\((v_n)\\) be dense in \\(A_+\\cap\\{\\|x\\|\\leq1\\}\\) and \\(a=\\sum_n2^{-n}v_n\\). Let \\(\\varphi\\neq0\\) be positive. Since \\(A_+\\) spans \\(A\\), \\(\\varphi(v)>0\\) for some positive \\(v\\) of norm at most one, and by Lemma 13.1(1) and density, \\(\\varphi(v_n)>0\\) for some \\(n\\). Then \\(\\varphi(a)\\geq2^{-n}\\varphi(v_n)>0\\).\n(3) *Only if.* Fix \\(x\\in A\\). For \\(t>0\\) put \\(y_t=x^*(1-f_t(a))^2x\\in A_+\\) and \\(g_t(\\varphi)=\\varphi(y_t)\\) on \\(Q\\). Each \\(g_t\\) is weak\\* continuous. Since \\(1-f_t(\\lambda)=t/(\\lambda+t)\\) increases with \\(t\\), \\(y_s\\leq y_t\\) for \\(s<t\\), so \\(g_t\\) decreases as \\(t\\downarrow0\\), to a limit \\(g\\geq0\\).\n*Claim: \\(g=0\\).* Suppose \\(g(\\varphi)=L>0\\) for some \\(\\varphi\\in Q\\); then \\(\\varphi\\neq0\\). For \\(t>0\\) define \\(\\rho_t(y)=\\varphi\\big(x^*(1-f_t(a))\\,y\\,(1-f_t(a))x\\big)\\) for \\(y\\in A\\). It is a positive functional with \\(\\|\\rho_t\\|\\leq\\|x\\|^2\\). First,\n\\[\n\\rho_t(a)\\leq\\|x\\|^2\\,\\|(1-f_t(a))^2a\\|\\leq\\|x\\|^2t/4 ,\n\\]\nsince \\(\\lambda t^2/(\\lambda+t)^2\\leq t/4\\). Second, with \\(w_t=x^*(1-f_t(a))x\\in A_h\\), \\(\\rho_t(xx^*)=\\varphi(w_t^2)\\geq\\varphi(w_t)^2/\\|\\varphi\\|\\) by Lemma 13.1(3), and \\(\\varphi(w_t)\\geq\\varphi(y_t)=g_t(\\varphi)\\geq L\\) because \\(1-f_t\\geq(1-f_t)^2\\). So \\(\\rho_t(xx^*)\\geq L^2/\\|\\varphi\\|\\). By Lemma 13.1(4) the net \\((\\rho_t)_{t\\downarrow0}\\) has a weak\\* cluster point \\(\\rho\\), a positive functional with \\(\\rho(a)=0\\) and \\(\\rho(xx^*)\\geq L^2/\\|\\varphi\\|>0\\). This contradicts strict positivity.\n*Dini.* For \\(\\eta>0\\) the open sets \\(\\{g_t<\\eta\\}\\) increase as \\(t\\downarrow0\\) and cover the compact set \\(Q\\), so one of them is all of \\(Q\\), and \\(g_t<\\eta\\) on \\(Q\\) for all smaller \\(t\\). By the existence of states, \\(\\|(1-f_t(a))x\\|^2=\\|y_t\\|=\\sup_{\\varphi\\in Q}\\varphi(y_t)\\to0\\) (for \\(A=0\\) there is nothing to prove).\n*If.* Let \\(\\varphi\\geq0\\) with \\(\\varphi(a)=0\\). Since \\(f_t(\\lambda)^2\\leq f_t(\\lambda)\\leq\\lambda/t\\), \\(0\\leq\\varphi(f_t(a)^2)\\leq\\varphi(a)/t=0\\). By Lemma 13.1(2), \\(|\\varphi(f_t(a)x)|^2\\leq\\varphi(f_t(a)^2)\\varphi(x^*x)=0\\). Since \\(\\varphi\\) is continuous (Lemma 13.1(1)), \\(\\varphi(x)=\\lim_t\\varphi(f_t(a)x)=0\\). So \\(\\varphi=0\\).\n(4) For \\(x\\in A\\) and \\(\\eta\\in H\\), \\[\n\\begin{gathered}\n\\|\\pi(f_t(a))\\pi(x)\\eta-\\pi(x)\\eta\\|\\\\\n\\leq\\|f_t(a)x-x\\|\\,\\|\\eta\\|\\to0\n\\end{gathered}\n\\] by (3) and Theorem 4.2. Such vectors span a dense subspace, and \\(\\|\\pi(f_t(a))\\|\\leq1\\), so the convergence extends to all \\(\\xi\\).\n(5) The \\(f_{1/n}(a)\\) are positive, of norm less than one, functions of \\(a\\) (so they commute), and increasing in \\(n\\) (Lemma 11.2(2)). By (3) they form a left approximate identity, and as they are self-adjoint, a right one too.\n(6) Let \\(\\varphi\\geq0\\) with \\(\\varphi(a)=0\\). Each term of the series defining \\(a\\) is positive, and \\(a\\) minus that term is positive, so \\(\\varphi(e_ne_n^*)=0\\) for every \\(n\\). By Lemma 13.1(2) with \\(y=e_n^*\\), \\(|\\varphi(e_nx)|^2\\leq\\varphi(x^*x)\\varphi(e_ne_n^*)=0\\) for all \\(x\\), and \\(\\varphi(x)=\\lim_n\\varphi(e_nx)=0\\) by Lemma 13.1(1). For positive contractions and \\(a=\\sum_n2^{-n}e_n\\), the same argument gives \\(\\varphi(e_n)=0\\), and \\(\\varphi(e_ne_n^*)=\\varphi(e_n^2)\\leq\\varphi(e_n)=0\\) because \\(e_n^2\\leq e_n\\).\n(7) combines (2) and (5). \\(\\square\\)\n\nSo a C\\*-algebra has a strictly positive element exactly when it has a sequential approximate identity; such algebras are called *\\(\\sigma\\)-unital*. Proposition 13.4 gives a C\\*-algebra with no strictly positive element. That algebra is commutative, so all its approximate identities commute: the converse of (5) fails for commuting approximate identities that are not sequences.\n\n**Proposition 13.4** (A C\\*-algebra without a strictly positive element). Let \\((\\Gamma,\\Sigma,\\mu)\\) be a \\(\\sigma\\)-finite measure space without atoms, with \\(\\mu(\\Gamma)=\\infty\\). Let \\(A\\subseteq L^\\infty(\\Gamma,\\mu)\\) be the C\\*-subalgebra generated by the projections \\(1_E\\) with \\(\\mu(E)<\\infty\\), that is, the closed linear span of these projections. Then:\n1. for \\(h\\in A\\) and \\(\\eta>0\\), \\(\\mu(\\{|h|>\\eta\\})<\\infty\\); in particular \\(A\\) is not unital;\n2. for every \\(h\\in A_+\\) there is a state \\(\\varphi\\) of \\(A\\) with \\(\\varphi(h)=0\\);\n3. hence \\(A\\) has no strictly positive element and no sequential approximate identity (Proposition 13.3(6)), although it has the approximate identity of Theorem 11.4.\n\n**Proof.** The span of the \\(1_E\\) is a \\(*\\)-algebra because \\(1_E1_F=1_{E\\cap F}\\), so its closure is a commutative C\\*-algebra.\n(1) Choose a simple \\(s=\\sum_kc_k1_{E_k}\\) with \\(\\mu(E_k)<\\infty\\) and \\(\\|h-s\\|_\\infty<\\eta/2\\). Up to a null set, \\(\\{|h|>\\eta\\}\\subseteq\\{|s|>\\eta/2\\}\\subseteq\\bigcup_kE_k\\). If \\(A\\) had an identity \\(e\\), then \\(e1_E=1_E\\) for every \\(E\\) of finite measure, so \\(e=1\\) almost everywhere on each such \\(E\\), hence almost everywhere on \\(\\Gamma\\) (\\(\\sigma\\)-finiteness), and \\(\\mu(\\{|e|>1/2\\})=\\infty\\), contradicting (1).\n(2) Let \\(h\\in A_+\\), so \\(h\\geq0\\) almost everywhere. By (1), \\(F_n=\\{h\\leq1/n\\}\\) has infinite measure. Choose inductively \\(E_n\\subseteq F_n\\setminus(E_1\\cup\\dots\\cup E_{n-1})\\) with \\(0<\\mu(E_n)<\\infty\\). This is possible because the set on the right has infinite measure, and a set of infinite measure in a \\(\\sigma\\)-finite space contains a subset of positive finite measure. Since there are no atoms, \\(E_n\\) contains a set \\(e_n\\) with \\(0<\\mu(e_n)<2^{-n}\\): a set of positive measure that is not an atom splits into two parts of positive measure, one of them of at most half the measure, and we repeat. The sets \\(e_n\\) are disjoint, and \\(e=\\bigcup_ne_n\\) has \\(\\mu(e)\\leq1\\), so \\(1_e\\in A\\). The functional \\(\\varphi_n(y)=\\mu(e_n)^{-1}\\int_{e_n}y\\,d\\mu\\) is a state of \\(A\\), with \\(\\varphi_n(1_e)=1\\) and \\(0\\leq\\varphi_n(h)\\leq1/n\\). By the Banach–Alaoglu theorem the sequence has a weak\\* cluster point \\(\\varphi\\). It is positive, \\(\\|\\varphi\\|\\leq1\\), and \\(\\varphi(1_e)=1\\), so \\(\\varphi\\) is a state. For every \\(N\\) and \\(\\varepsilon>0\\) there is \\(n\\geq N\\) with \\(|\\varphi_n(h)-\\varphi(h)|<\\varepsilon\\), so \\(0\\leq\\varphi(h)<1/N+\\varepsilon\\). Hence \\(\\varphi(h)=0\\).\n(3) follows from (2) and Proposition 13.3(6). \\(\\square\\)\n\n**Example 13.5** (\\(c_0\\)). In \\(c_0\\), the sequences \\(e_n=(1,\\dots,1,0,0,\\dots)\\) (\\(n\\) ones) form an increasing contractive sequential approximate identity, and \\(a=(1,\\frac12,\\frac13,\\dots)\\) is strictly positive. Indeed, a positive functional is \\(\\varphi(x)=\\sum_k\\varphi_kx_k\\) with \\(\\varphi_k=\\varphi(\\delta_k)\\geq0\\) and \\(\\sum_k\\varphi_k<\\infty\\) (by Lemma 13.1(1) and \\(\\|e_n\\|=1\\)), and \\(\\varphi(a)=\\sum_k\\varphi_k/k>0\\) unless all \\(\\varphi_k=0\\). Here \\(\\varphi(x)=\\lim_n\\varphi(e_nx)=\\sum_k\\varphi_kx_k\\) uses that \\((e_n)\\) is an approximate identity.\n\n**Exercise 13.6** (medium; \\(\\sigma\\)-unital commutative algebras). Let \\(X\\) be an LCH space. Show that \\(C_0(X)\\) has a strictly positive element if and only if \\(X\\) is \\(\\sigma\\)-compact. Deduce that \\(C_0(X)\\) has a sequential approximate identity exactly when \\(X\\) is \\(\\sigma\\)-compact.\n\n*Solution.* If \\(a\\in C_0(X)_+\\) is strictly positive, then \\(a(p)>0\\) for every \\(p\\), since evaluation at \\(p\\) is a nonzero positive functional. So \\(X=\\bigcup_n\\{a\\geq1/n\\}\\), a countable union of compact sets. Conversely, let \\(X=\\bigcup_nK_n\\) with \\(K_n\\) compact, and choose \\(f_n\\in C_c(X)\\) with \\(0\\leq f_n\\leq1\\) and \\(f_n=1\\) on \\(K_n\\) ([Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16)). Then \\(a=\\sum_n2^{-n}f_n\\) is positive at every point. For \\(g\\in C_0(X)\\) and \\(\\eta>0\\), \\(|g|<\\eta\\) off a compact set \\(C\\), and \\(a\\geq c>0\\) on \\(C\\). So \\(\\|g-f_\\varepsilon(a)g\\|\\leq\\max\\big(\\eta,\\|g\\|\\varepsilon/(c+\\varepsilon)\\big)\\), which is at most \\(\\eta\\) for small \\(\\varepsilon\\). By Proposition 13.3(3), \\(a\\) is strictly positive. The last claim is Proposition 13.3(5)–(6). So the algebra of Proposition 13.4 is \\(C_0\\) of a space that is not \\(\\sigma\\)-compact.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-24",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "13. Strictly positive elements and countable approximate identities",
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      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 13. Strictly positive elements and countable approximate identities\n\nA C\\*-algebra has a sequential approximate identity exactly when it has a strictly positive element (Proposition 13.3). We first collect the facts about positive linear functionals that are needed. A linear functional \\(\\varphi\\) on a C\\*-algebra \\(A\\) is *positive* if \\(\\varphi(A_+)\\subseteq[0,\\infty)\\), and a *state* is a positive linear functional of norm one.\n\n**Lemma 13.1** (Positive linear functionals). Let \\(A\\) be a C\\*-algebra and \\(\\varphi\\) a positive linear functional on \\(A\\).\n1. \\(\\varphi\\) is bounded.\n2. \\(\\varphi\\) is hermitian, and \\(|\\varphi(y^*x)|^2\\leq\\varphi(x^*x)\\varphi(y^*y)\\) for all \\(x,y\\in A\\).\n3. \\(|\\varphi(x)|^2\\leq\\|\\varphi\\|\\varphi(x^*x)\\) for all \\(x\\in A\\).\n4. The set \\(Q\\) of positive linear functionals of norm at most one is weak\\* compact.\n5. If \\(A\\) is unital and nontrivial, then \\(\\|\\varphi\\|=\\varphi(1)\\).\n6. (*States.*) If \\(A\\) is unital and nontrivial, \\(h\\in A_h\\) and \\(\\lambda\\in\\sigma(h)\\), some state \\(\\omega\\) has \\(\\omega(h)=\\lambda\\).\n7. (*States.*) If \\(A\\neq0\\) and \\(a\\in A_+\\), some state \\(\\omega\\) has \\(\\omega(a)=\\|a\\|\\).\n\n**Proof.** (1) If not, since every element is a combination of four positive elements of no larger norm (Proposition 8.5(8)), there are \\(a_n\\in A_+\\) with \\(\\|a_n\\|\\leq1\\) and \\(\\varphi(a_n)\\geq4^n\\). The series \\(a=\\sum2^{-n}a_n\\) converges, \\(a\\geq2^{-n}a_n\\) because the rest of the series is positive, and so \\(\\varphi(a)\\geq2^n\\) for every \\(n\\), which is absurd.\n(2) For \\(h\\in A_h\\), \\(\\varphi(h)=\\varphi(h_+)-\\varphi(h_-)\\) is real (Proposition 8.5(8)), so \\(\\varphi\\) is hermitian. The form \\((x,y)\\mapsto\\varphi(y^*x)\\) is sesquilinear and positive semidefinite, so the inequality is the Cauchy–Schwarz inequality, Proposition 1.1(3) of [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md).\n(3) For \\(u\\) in the approximate identity of Theorem 11.4, \\(|\\varphi(ux)|^2\\leq\\varphi(u^2)\\varphi(x^*x)\\leq\\|\\varphi\\|\\varphi(x^*x)\\), and \\(\\varphi(ux)\\to\\varphi(x)\\).\n(4) \\(Q\\) lies in the closed unit ball of the dual of \\(A\\), which is weak\\* compact by the Banach–Alaoglu theorem (Theorem 3.1 of [Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md)), and positivity is a weak\\*-closed condition.\n(5) Let \\(\\|x\\|\\leq1\\). By (2) with \\(y=1\\), \\(|\\varphi(x)|^2\\leq\\varphi(x^*x)\\varphi(1)\\). Since \\(x^*x\\leq\\|x\\|^2\\leq1\\) (Proposition 8.5(2)), \\(\\varphi(x^*x)\\leq\\varphi(1)\\). So \\(|\\varphi(x)|\\leq\\varphi(1)\\), and \\(\\|\\varphi\\|\\leq\\varphi(1)\\leq\\|\\varphi\\|\\,\\|1\\|=\\|\\varphi\\|\\).\n(6) *A state of \\(C^*(1,h)\\).* By Theorem 5.1, \\(f(h)\\mapsto f(\\lambda)\\) is a well-defined linear functional \\(\\omega_0\\) on \\(B=C^*(1,h)\\). It is positive, because the positive elements of \\(B\\) are the \\(f(h)\\) with \\(f\\geq0\\) (Theorem 5.1 and Remark 8.4). By (5), \\(\\|\\omega_0\\|=\\omega_0(1)=1\\).\n*Extension.* By the Hahn–Banach theorem (Corollary 2.3(1) of [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md)), \\(\\omega_0\\) extends to \\(\\omega\\in A^*\\) with \\(\\|\\omega\\|=1=\\omega(1)\\).\n*Such an \\(\\omega\\) is positive.* Let \\(a\\in A_+\\) with \\(\\|a\\|\\leq1\\), and write \\(\\omega(a)=\\alpha+i\\beta\\).\n- For real \\(t\\), \\[\n\\begin{gathered}\n\\|a+it\\|^2\\\\\n=\\|(a-it)(a+it)\\|\\\\\n=\\|a^2+t^2\\|\\\\\n\\leq1+t^2,\n\\end{gathered}\n\\] so \\(\\alpha^2+(\\beta+t)^2=|\\omega(a+it)|^2\\leq1+t^2\\). That is, \\(\\alpha^2+\\beta^2+2\\beta t\\leq1\\) for every \\(t\\), so \\(\\beta=0\\).\n- \\(\\sigma(1-a)\\subseteq[0,1]\\), so \\(\\|1-a\\|\\leq1\\) (Theorem 1.3). Hence \\(|1-\\omega(a)|\\leq1\\), and \\(\\omega(a)\\geq0\\).\n\nSo \\(\\omega\\) is a state, and \\(\\omega(h)=\\omega_0(h)=\\lambda\\).\n(7) In \\(\\widetilde A\\), \\(\\sigma'(a)\\subseteq[0,\\|a\\|]\\), and \\(\\|a\\|=r(a)\\) lies in \\(\\sigma'(a)\\) (Theorem 1.3). By (6) in \\(\\widetilde A\\), some state \\(\\tilde\\omega\\) of \\(\\widetilde A\\) has \\(\\tilde\\omega(a)=\\|a\\|\\). Its restriction \\(\\omega\\) to \\(A\\) is positive with \\(\\|\\omega\\|\\leq1\\). If \\(a\\neq0\\), then \\(\\omega(a)=\\|a\\|\\) forces \\(\\|\\omega\\|=1\\). If \\(a=0\\), take any nonzero \\(b\\in A_+\\), for instance \\(b=x^*x\\) with \\(x\\neq0\\), and a state \\(\\omega\\) with \\(\\omega(b)=\\|b\\|\\); then \\(\\omega(a)=0=\\|a\\|\\). \\(\\square\\)\n\nParts (6) and (7) say that states exist in abundance.\n\n**Definition 13.2.** An element \\(a\\in A_+\\) is *strictly positive* if \\(\\varphi(a)>0\\) for every nonzero positive linear functional \\(\\varphi\\).\n\n**Proposition 13.3** (Strictly positive elements). Let \\(A\\) be a C\\*-algebra, \\(a\\in A_+\\), and \\(f_t(\\lambda)=\\lambda/(\\lambda+t)\\).\n1. If \\(A\\) is unital and nontrivial, \\(a\\) is strictly positive if and only if it is invertible.\n2. Every separable C\\*-algebra has a strictly positive element.\n3. \\(a\\) is strictly positive if and only if \\(\\|f_t(a)x-x\\|\\to0\\) as \\(t\\to0\\), for every \\(x\\in A\\).\n4. Let \\(a\\) be strictly positive, and let \\(\\pi:A\\to B(H)\\) be a *nondegenerate representation*: a \\(*\\)-homomorphism such that \\(\\pi(A)H\\) spans a dense subspace. Then \\(\\pi(f_t(a))\\xi\\to\\xi\\) for every \\(\\xi\\in H\\).\n5. If \\(a\\) is strictly positive, \\((f_{1/n}(a))_{n\\geq1}\\) is an increasing, contractive, commuting approximate identity.\n6. (*Converse of (5).*) If some sequence \\((e_n)\\) in \\(A\\) satisfies \\(\\|e_nx-x\\|\\to0\\) for every \\(x\\in A\\), for instance a sequential approximate identity, then \\(a=\\sum_n2^{-n}(1+\\|e_n\\|^2)^{-1}e_ne_n^*\\) is strictly positive. If the \\(e_n\\) are positive contractions, \\(a=\\sum_n2^{-n}e_n\\) is strictly positive as well.\n7. Every separable C\\*-algebra has an increasing, contractive approximate identity made of a commuting sequence.\n\n**Proof.** (1) If \\(a\\) is invertible, then \\(a\\geq m\\) with \\(m=\\min\\sigma(a)>0\\), and \\(\\varphi(a)\\geq m\\varphi(1)=m\\|\\varphi\\|>0\\) for \\(\\varphi\\neq0\\). If \\(a\\) is not invertible, \\(0\\in\\sigma(a)\\), and some state \\(\\omega\\) has \\(\\omega(a)=0\\).\n(2) If \\(A=0\\), \\(a=0\\) will do. Otherwise let \\((v_n)\\) be dense in \\(A_+\\cap\\{\\|x\\|\\leq1\\}\\) and \\(a=\\sum_n2^{-n}v_n\\). Let \\(\\varphi\\neq0\\) be positive. Since \\(A_+\\) spans \\(A\\), \\(\\varphi(v)>0\\) for some positive \\(v\\) of norm at most one, and by Lemma 13.1(1) and density, \\(\\varphi(v_n)>0\\) for some \\(n\\). Then \\(\\varphi(a)\\geq2^{-n}\\varphi(v_n)>0\\).\n(3) *Only if.* Fix \\(x\\in A\\). For \\(t>0\\) put \\(y_t=x^*(1-f_t(a))^2x\\in A_+\\) and \\(g_t(\\varphi)=\\varphi(y_t)\\) on \\(Q\\). Each \\(g_t\\) is weak\\* continuous. Since \\(1-f_t(\\lambda)=t/(\\lambda+t)\\) increases with \\(t\\), \\(y_s\\leq y_t\\) for \\(s<t\\), so \\(g_t\\) decreases as \\(t\\downarrow0\\), to a limit \\(g\\geq0\\).\n*Claim: \\(g=0\\).* Suppose \\(g(\\varphi)=L>0\\) for some \\(\\varphi\\in Q\\); then \\(\\varphi\\neq0\\). For \\(t>0\\) define \\(\\rho_t(y)=\\varphi\\big(x^*(1-f_t(a))\\,y\\,(1-f_t(a))x\\big)\\) for \\(y\\in A\\). It is a positive functional with \\(\\|\\rho_t\\|\\leq\\|x\\|^2\\). First,\n\\[\n\\rho_t(a)\\leq\\|x\\|^2\\,\\|(1-f_t(a))^2a\\|\\leq\\|x\\|^2t/4 ,\n\\]\nsince \\(\\lambda t^2/(\\lambda+t)^2\\leq t/4\\). Second, with \\(w_t=x^*(1-f_t(a))x\\in A_h\\), \\(\\rho_t(xx^*)=\\varphi(w_t^2)\\geq\\varphi(w_t)^2/\\|\\varphi\\|\\) by Lemma 13.1(3), and \\(\\varphi(w_t)\\geq\\varphi(y_t)=g_t(\\varphi)\\geq L\\) because \\(1-f_t\\geq(1-f_t)^2\\). So \\(\\rho_t(xx^*)\\geq L^2/\\|\\varphi\\|\\). By Lemma 13.1(4) the net \\((\\rho_t)_{t\\downarrow0}\\) has a weak\\* cluster point \\(\\rho\\), a positive functional with \\(\\rho(a)=0\\) and \\(\\rho(xx^*)\\geq L^2/\\|\\varphi\\|>0\\). This contradicts strict positivity.\n*Dini.* For \\(\\eta>0\\) the open sets \\(\\{g_t<\\eta\\}\\) increase as \\(t\\downarrow0\\) and cover the compact set \\(Q\\), so one of them is all of \\(Q\\), and \\(g_t<\\eta\\) on \\(Q\\) for all smaller \\(t\\). By the existence of states, \\(\\|(1-f_t(a))x\\|^2=\\|y_t\\|=\\sup_{\\varphi\\in Q}\\varphi(y_t)\\to0\\) (for \\(A=0\\) there is nothing to prove).\n*If.* Let \\(\\varphi\\geq0\\) with \\(\\varphi(a)=0\\). Since \\(f_t(\\lambda)^2\\leq f_t(\\lambda)\\leq\\lambda/t\\), \\(0\\leq\\varphi(f_t(a)^2)\\leq\\varphi(a)/t=0\\). By Lemma 13.1(2), \\(|\\varphi(f_t(a)x)|^2\\leq\\varphi(f_t(a)^2)\\varphi(x^*x)=0\\). Since \\(\\varphi\\) is continuous (Lemma 13.1(1)), \\(\\varphi(x)=\\lim_t\\varphi(f_t(a)x)=0\\). So \\(\\varphi=0\\).\n(4) For \\(x\\in A\\) and \\(\\eta\\in H\\), \\[\n\\begin{gathered}\n\\|\\pi(f_t(a))\\pi(x)\\eta-\\pi(x)\\eta\\|\\\\\n\\leq\\|f_t(a)x-x\\|\\,\\|\\eta\\|\\to0\n\\end{gathered}\n\\] by (3) and Theorem 4.2. Such vectors span a dense subspace, and \\(\\|\\pi(f_t(a))\\|\\leq1\\), so the convergence extends to all \\(\\xi\\).\n(5) The \\(f_{1/n}(a)\\) are positive, of norm less than one, functions of \\(a\\) (so they commute), and increasing in \\(n\\) (Lemma 11.2(2)). By (3) they form a left approximate identity, and as they are self-adjoint, a right one too.\n(6) Let \\(\\varphi\\geq0\\) with \\(\\varphi(a)=0\\). Each term of the series defining \\(a\\) is positive, and \\(a\\) minus that term is positive, so \\(\\varphi(e_ne_n^*)=0\\) for every \\(n\\). By Lemma 13.1(2) with \\(y=e_n^*\\), \\(|\\varphi(e_nx)|^2\\leq\\varphi(x^*x)\\varphi(e_ne_n^*)=0\\) for all \\(x\\), and \\(\\varphi(x)=\\lim_n\\varphi(e_nx)=0\\) by Lemma 13.1(1). For positive contractions and \\(a=\\sum_n2^{-n}e_n\\), the same argument gives \\(\\varphi(e_n)=0\\), and \\(\\varphi(e_ne_n^*)=\\varphi(e_n^2)\\leq\\varphi(e_n)=0\\) because \\(e_n^2\\leq e_n\\).\n(7) combines (2) and (5). \\(\\square\\)\n\nSo a C\\*-algebra has a strictly positive element exactly when it has a sequential approximate identity; such algebras are called *\\(\\sigma\\)-unital*. Proposition 13.4 gives a C\\*-algebra with no strictly positive element. That algebra is commutative, so all its approximate identities commute: the converse of (5) fails for commuting approximate identities that are not sequences.\n\n**Proposition 13.4** (A C\\*-algebra without a strictly positive element). Let \\((\\Gamma,\\Sigma,\\mu)\\) be a \\(\\sigma\\)-finite measure space without atoms, with \\(\\mu(\\Gamma)=\\infty\\). Let \\(A\\subseteq L^\\infty(\\Gamma,\\mu)\\) be the C\\*-subalgebra generated by the projections \\(1_E\\) with \\(\\mu(E)<\\infty\\), that is, the closed linear span of these projections. Then:\n1. for \\(h\\in A\\) and \\(\\eta>0\\), \\(\\mu(\\{|h|>\\eta\\})<\\infty\\); in particular \\(A\\) is not unital;\n2. for every \\(h\\in A_+\\) there is a state \\(\\varphi\\) of \\(A\\) with \\(\\varphi(h)=0\\);\n3. hence \\(A\\) has no strictly positive element and no sequential approximate identity (Proposition 13.3(6)), although it has the approximate identity of Theorem 11.4.\n\n**Proof.** The span of the \\(1_E\\) is a \\(*\\)-algebra because \\(1_E1_F=1_{E\\cap F}\\), so its closure is a commutative C\\*-algebra.\n(1) Choose a simple \\(s=\\sum_kc_k1_{E_k}\\) with \\(\\mu(E_k)<\\infty\\) and \\(\\|h-s\\|_\\infty<\\eta/2\\). Up to a null set, \\(\\{|h|>\\eta\\}\\subseteq\\{|s|>\\eta/2\\}\\subseteq\\bigcup_kE_k\\). If \\(A\\) had an identity \\(e\\), then \\(e1_E=1_E\\) for every \\(E\\) of finite measure, so \\(e=1\\) almost everywhere on each such \\(E\\), hence almost everywhere on \\(\\Gamma\\) (\\(\\sigma\\)-finiteness), and \\(\\mu(\\{|e|>1/2\\})=\\infty\\), contradicting (1).\n(2) Let \\(h\\in A_+\\), so \\(h\\geq0\\) almost everywhere. By (1), \\(F_n=\\{h\\leq1/n\\}\\) has infinite measure. Choose inductively \\(E_n\\subseteq F_n\\setminus(E_1\\cup\\dots\\cup E_{n-1})\\) with \\(0<\\mu(E_n)<\\infty\\). This is possible because the set on the right has infinite measure, and a set of infinite measure in a \\(\\sigma\\)-finite space contains a subset of positive finite measure. Since there are no atoms, \\(E_n\\) contains a set \\(e_n\\) with \\(0<\\mu(e_n)<2^{-n}\\): a set of positive measure that is not an atom splits into two parts of positive measure, one of them of at most half the measure, and we repeat. The sets \\(e_n\\) are disjoint, and \\(e=\\bigcup_ne_n\\) has \\(\\mu(e)\\leq1\\), so \\(1_e\\in A\\). The functional \\(\\varphi_n(y)=\\mu(e_n)^{-1}\\int_{e_n}y\\,d\\mu\\) is a state of \\(A\\), with \\(\\varphi_n(1_e)=1\\) and \\(0\\leq\\varphi_n(h)\\leq1/n\\). By the Banach–Alaoglu theorem the sequence has a weak\\* cluster point \\(\\varphi\\). It is positive, \\(\\|\\varphi\\|\\leq1\\), and \\(\\varphi(1_e)=1\\), so \\(\\varphi\\) is a state. For every \\(N\\) and \\(\\varepsilon>0\\) there is \\(n\\geq N\\) with \\(|\\varphi_n(h)-\\varphi(h)|<\\varepsilon\\), so \\(0\\leq\\varphi(h)<1/N+\\varepsilon\\). Hence \\(\\varphi(h)=0\\).\n(3) follows from (2) and Proposition 13.3(6). \\(\\square\\)\n\n**Example 13.5** (\\(c_0\\)). In \\(c_0\\), the sequences \\(e_n=(1,\\dots,1,0,0,\\dots)\\) (\\(n\\) ones) form an increasing contractive sequential approximate identity, and \\(a=(1,\\frac12,\\frac13,\\dots)\\) is strictly positive. Indeed, a positive functional is \\(\\varphi(x)=\\sum_k\\varphi_kx_k\\) with \\(\\varphi_k=\\varphi(\\delta_k)\\geq0\\) and \\(\\sum_k\\varphi_k<\\infty\\) (by Lemma 13.1(1) and \\(\\|e_n\\|=1\\)), and \\(\\varphi(a)=\\sum_k\\varphi_k/k>0\\) unless all \\(\\varphi_k=0\\). Here \\(\\varphi(x)=\\lim_n\\varphi(e_nx)=\\sum_k\\varphi_kx_k\\) uses that \\((e_n)\\) is an approximate identity.\n\n**Exercise 13.6** (medium; \\(\\sigma\\)-unital commutative algebras). Let \\(X\\) be an LCH space. Show that \\(C_0(X)\\) has a strictly positive element if and only if \\(X\\) is \\(\\sigma\\)-compact. Deduce that \\(C_0(X)\\) has a sequential approximate identity exactly when \\(X\\) is \\(\\sigma\\)-compact.\n\n*Solution.* If \\(a\\in C_0(X)_+\\) is strictly positive, then \\(a(p)>0\\) for every \\(p\\), since evaluation at \\(p\\) is a nonzero positive functional. So \\(X=\\bigcup_n\\{a\\geq1/n\\}\\), a countable union of compact sets. Conversely, let \\(X=\\bigcup_nK_n\\) with \\(K_n\\) compact, and choose \\(f_n\\in C_c(X)\\) with \\(0\\leq f_n\\leq1\\) and \\(f_n=1\\) on \\(K_n\\) ([Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16)). Then \\(a=\\sum_n2^{-n}f_n\\) is positive at every point. For \\(g\\in C_0(X)\\) and \\(\\eta>0\\), \\(|g|<\\eta\\) off a compact set \\(C\\), and \\(a\\geq c>0\\) on \\(C\\). So \\(\\|g-f_\\varepsilon(a)g\\|\\leq\\max\\big(\\eta,\\|g\\|\\varepsilon/(c+\\varepsilon)\\big)\\), which is at most \\(\\eta\\) for small \\(\\varepsilon\\). By Proposition 13.3(3), \\(a\\) is strictly positive. The last claim is Proposition 13.3(5)–(6). So the algebra of Proposition 13.4 is \\(C_0\\) of a space that is not \\(\\sigma\\)-compact.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-21",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "14. The asymmetric Riesz decomposition",
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      "full_conditions_and_proof": "### 14. The asymmetric Riesz decomposition\n\nIn an abelian C\\*-algebra, \\(A_h\\) has the Riesz decomposition property, and by Theorem 10.2 no other C\\*-algebra does. The following weaker decomposition holds in every C\\*-algebra. We state it for families whose sums converge in norm; finite families are a special case.\n\n**Theorem 14.1** (Asymmetric Riesz decomposition). Let \\((x_i)_{i\\in I}\\) and \\((y_j)_{j\\in J}\\) be families in a C\\*-algebra \\(A\\) such that the sums \\(\\sum_ix_i^*x_i\\) and \\(\\sum_jy_j^*y_j\\) converge unconditionally in norm (the nets of finite partial sums converge) to the same element \\(a\\). Then there are \\(z_{ij}\\in A\\) with\n\\[\n\\begin{gathered}\nx_ix_i^*\\\\\n=\\sum_jz_{ij}^*z_{ij}\\\\\n(i\\in I),\\\\\ny_jy_j^*\\\\\n=\\sum_iz_{ij}z_{ij}^*\\\\\n(j\\in J),\n\\end{gathered}\n\\tag{14.1}\n\\]\nboth sums converging unconditionally in norm.\n\n**Proof.** Work in \\(\\widetilde A\\). For \\(t>0\\) put \\(u_t=f_t(a)=(a+t)^{-1}a\\) and \\(z_{ij,t}=y_j(a+t)^{-1}a^{1/2}x_i^*\\in A\\).\n*Each \\(z_{ij,t}\\) converges as \\(t\\to0\\).* Since \\(y_j^*y_j\\leq a\\), Corollary 11.3(2), in its left form, gives \\(\\|y_j-y_ju_t\\|\\to0\\). For \\(s,t>0\\) put \\(d=(a+s)^{-1}-(a+t)^{-1}\\), a self-adjoint function of \\(a\\) with \\(da=u_s-u_t\\). Using \\(x_i^*x_i\\leq a\\) and Proposition 8.5(3),\n\\[\n\\begin{gathered}\n(z_{ij,s}-z_{ij,t})(z_{ij,s}-z_{ij,t})^*\\\\\n=y_jda^{1/2}x_i^*x_ia^{1/2}dy_j^*\\\\\n\\leq y_jda^2dy_j^*\\\\\n=\\big(y_j(u_s-u_t)\\big)\\big(y_j(u_s-u_t)\\big)^* .\n\\end{gathered}\n\\]\nSo \\(\\|z_{ij,s}-z_{ij,t}\\|\\leq\\|y_ju_s-y_ju_t\\|\\to0\\) as \\(s,t\\to0\\). Let \\(z_{ij}=\\lim_{t\\to0}z_{ij,t}\\).\n*The second identity.* Fix \\(j\\), and for a finite \\(F\\subseteq I\\) put \\(R_F=a-\\sum_{i\\in F}x_i^*x_i\\), which is positive, as a norm limit of finite sums of positive elements, and \\(\\|R_F\\|\\to0\\) along \\(F\\). Since \\[\n\\begin{gathered}\n(y_ju_t)(y_ju_t)^*\\\\\n=y_j(a+t)^{-1}a^{1/2}\\,a\\,a^{1/2}(a+t)^{-1}y_j^*,\n\\end{gathered}\n\\]\n\\[\n\\begin{gathered}\n(y_ju_t)(y_ju_t)^*-\\sum_{i\\in F}z_{ij,t}z_{ij,t}^*\\\\\n=g_tR_Fg_t^*,\\\\\ng_t\\\\\n=y_j(a+t)^{-1}a^{1/2}.\n\\end{gathered}\n\\]\nHere \\[\n\\begin{gathered}\n\\|g_t\\|^2\\\\\n=\\|(a+t)^{-1}a^{1/2}y_j^*y_ja^{1/2}(a+t)^{-1}\\|\\\\\n\\leq\\|a^2(a+t)^{-2}\\|\\\\\n\\leq1,\n\\end{gathered}\n\\] so the right side has norm at most \\(\\|R_F\\|\\). Let \\(t\\to0\\): \\(\\|y_jy_j^*-\\sum_{i\\in F}z_{ij}z_{ij}^*\\|\\leq\\|R_F\\|\\), which tends to \\(0\\).\n*The first identity.* In the same way, with \\(R'_{F'}=a-\\sum_{j\\in F'}y_j^*y_j\\) and \\(g'_t=x_ia^{1/2}(a+t)^{-1}\\), we get \\((x_iu_t)(x_iu_t)^*-\\sum_{j\\in F'}z_{ij,t}^*z_{ij,t}=g'_tR'_{F'}g_t'^*\\), with \\(\\|g'_t\\|\\leq1\\) because \\(x_i^*x_i\\leq a\\). Since \\(x_iu_t\\to x_i\\), the limit gives \\(\\|x_ix_i^*-\\sum_{j\\in F'}z_{ij}^*z_{ij}\\|\\leq\\|R'_{F'}\\|\\). \\(\\square\\)\n\nFor operators on a Hilbert space there is also a direct proof in the style of the polar decomposition; Exercise 14.3 gives it for finite families.\n\n**Corollary 14.2.** If \\(\\sum_jy_j^*y_j\\leq\\sum_ix_i^*x_i\\), with both sums as in the theorem, there are \\(z_{ij}\\) with \\(y_jy_j^*=\\sum_iz_{ij}z_{ij}^*\\) and \\(x_ix_i^*\\geq\\sum_jz_{ij}^*z_{ij}\\).\n\n**Proof.** Add to the family \\((y_j)\\) one element \\(y_\\star=\\big(\\sum_ix_i^*x_i-\\sum_jy_j^*y_j\\big)^{1/2}\\), so that the two sums become equal. The theorem gives \\(z_{ij}\\) and \\(z_{i\\star}\\) with \\(x_ix_i^*=\\sum_jz_{ij}^*z_{ij}+z_{i\\star}^*z_{i\\star}\\geq\\sum_jz_{ij}^*z_{ij}\\). \\(\\square\\)\n\n**Exercise 14.3** (medium; Factorization of operators). (a) Let \\(x:H\\to K_1\\) and \\(y:H\\to K_2\\) be bounded operators between Hilbert spaces with \\(x^*x\\leq y^*y\\). Then \\(x=cy\\) for some \\(c:K_2\\to K_1\\) with \\(\\|c\\|\\leq1\\), and \\(c\\) can be taken to vanish on the orthogonal complement of the range of \\(y\\). (b) Prove Theorem 14.1 for finite families in \\(A=B(H)\\) from (a).\n\n*Solution.* (a) By Proposition 8.5(11) in \\(B(H)\\), \\(\\|x\\xi\\|^2=\\langle x^*x\\xi,\\xi\\rangle\\leq\\langle y^*y\\xi,\\xi\\rangle=\\|y\\xi\\|^2\\). So \\(c_0(y\\xi)=x\\xi\\) is well defined and linear on the range of \\(y\\), with norm at most one. Extend it by continuity to the closure of the range, and by \\(0\\) on the orthogonal complement.\n(b) Let \\(\\sum_{i\\leq m}x_i^*x_i=\\sum_{j\\leq n}y_j^*y_j=a\\) in \\(B(H)\\). The column operators \\(X\\xi=(x_i\\xi)_i\\) from \\(H\\) to \\(H^m\\) and \\(Y\\xi=(y_j\\xi)_j\\) from \\(H\\) to \\(H^n\\) satisfy \\(X^*X=Y^*Y=a=(a^{1/2})^2\\). By (a), \\(X=Ca^{1/2}\\), with \\(C\\) isometric on the closure of the range of \\(a^{1/2}\\), since \\(\\|X\\xi\\|=\\|a^{1/2}\\xi\\|\\), and zero on its complement. So \\(C^*C=P\\), the projection onto that closure, and \\(Pa^{1/2}=a^{1/2}\\). Writing \\(C\\) as a column \\((c_i)\\) gives \\(x_i=c_ia^{1/2}\\) and \\(\\sum_ic_i^*c_i=P\\). In the same way \\(y_j=d_ja^{1/2}\\) with \\(\\sum_jd_j^*d_j=P\\). Put \\(z_{ij}=d_ja^{1/2}c_i^*\\). Then\n\\[\n\\begin{gathered}\n\\sum_iz_{ij}z_{ij}^*\\\\\n=d_ja^{1/2}Pa^{1/2}d_j^*\\\\\n=d_jad_j^*\\\\\n=y_jy_j^*,\\\\\n\\sum_jz_{ij}^*z_{ij}\\\\\n=c_ia^{1/2}Pa^{1/2}c_i^*\\\\\n=x_ix_i^* .\n\\end{gathered}\n\\]\n\n## Results used from other lessons\n\n- *Haar integration and convolution.* [Theorem 8.3 and Theorems 9.2–11.1 of the Haar lesson](haar-measure.md#oa-fnd-hm-05) give existence, uniqueness, the modular function and inversion for every locally compact group. [Theorem 5.1 there](haar-measure.md#oa-fnd-hm-09) proves Tonelli and Fubini for Borel functions with sigma-finite support in a Radon product; [Theorem 14.2](haar-measure.md#oa-fnd-hm-12) proves convolution bounds and translation continuity, and [Theorem 15.1](haar-measure.md#oa-fnd-hm-13) proves compactly supported approximate identities. Section 12 extends the L1 approximate-identity argument to all nonnegative normalized functions supported in shrinking neighbourhoods, retaining the correct modular factor on the right.\n- *Compact extension.* [Proposition 16.1(2) of the Stone–Weierstrass lesson](stone-weierstrass-c0.md#oa-fnd-sw-18) proves norm-preserving extension from a closed subset of a compact Hausdorff space. It supplies the alternative commutative lifting argument in Section 17; the main lifting proof works for every C*-algebra.\n- *The Banach–Alaoglu theorem* (Theorem 3.1 of [Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md)). The closed unit ball of the dual of a normed space is compact for the weak\\* topology. It is used in Section 13.\n- *The closed graph theorem* (Theorem 5.3 and Remark 5.4 of [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md)). A linear map between Banach spaces whose graph is closed is bounded. It holds for real Banach spaces too, so it applies to conjugate-linear maps between complex Banach spaces. It is used in Theorem 4.7.\n- *Separation of convex sets* (Theorem 6.2 of the same lesson). It is used, over the real numbers, in the proof of Theorem 10.2.\n- *The maximum modulus principle* (Theorem 3.6, Corollary 3.5 and Exercise 4 of [Cauchy's theorem for cycles and its consequences](cauchy-s-theorem-for-cycles-and-its-consequences.md)). A function that is continuous on the closed unit disc and holomorphic in the open disc attains the maximum of its modulus on the unit circle; and a uniform limit of holomorphic functions on an open set is holomorphic. They are used in Example 3.4.\n- *Hilbert spaces* (Proposition 1.1 of [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md)). The Cauchy–Schwarz inequality for positive semidefinite sesquilinear forms, used in Lemma 13.1.\n\n## Where this leads\n\nThe next step is the theory of states and representations: the Gelfand–Naimark–Segal construction, and the theorem that every C\\*-algebra has a faithful representation on a Hilbert space . In von Neumann algebras the continuous functional calculus of normal elements extends to bounded Borel functions.\n\n## References\n\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [B. Blackadar, *Operator Algebras*, II.1.6, II.2.2–II.3.2 and II.4–II.5](https://bruceblackadar.com/Mathematics/Cycr.pdf) gives a route through automatic continuity, continuous calculus, positive cones, ideals and quotients; the asymmetric Riesz proof is given fully here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-25",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "15. Closed ideals and quotients",
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      "full_conditions_and_proof": "### 15. Closed ideals and quotients\n\n**Theorem 15.1** (Quotients of C\\*-algebras). Let \\(A\\) be a C\\*-algebra and \\(\\mathfrak m\\subseteq A\\) a closed ideal.\n1. \\(\\mathfrak m^*=\\mathfrak m\\), so \\(\\mathfrak m\\) is a C\\*-subalgebra.\n2. \\(A/\\mathfrak m\\), with the quotient norm and \\((x+\\mathfrak m)^*=x^*+\\mathfrak m\\), is a C\\*-algebra, and the quotient map is a \\(*\\)-homomorphism.\n3. (*Quotient norm.*) Let \\((e_i)\\) be any net of positive contractions in \\(\\mathfrak m\\) with \\(\\|y-ye_i\\|\\to0\\) for every \\(y\\in\\mathfrak m\\), for instance the net of Theorem 11.4. Then for every \\(x\\in A\\),\n\\[\n\\|x+\\mathfrak m\\|=\\lim_i\\|x-xe_i\\| .\n\\tag{15.1}\n\\]\n\n**Proof.** (1) By Theorem 11.4(3), \\(\\Lambda=\\mathfrak m_+\\cap S_0\\) is a two-sided approximate identity for \\(\\mathfrak m\\). For \\(x\\in\\mathfrak m\\) and \\(u\\in\\Lambda\\), \\(x^*u\\in\\mathfrak m\\) and \\(\\|x^*-x^*u\\|=\\|x-ux\\|\\to0\\). So \\(x^*\\in\\mathfrak m\\), as \\(\\mathfrak m\\) is closed.\n(3) Let \\(y\\in\\mathfrak m\\). In \\(\\widetilde A\\), \\(0\\leq1-e_i\\leq1\\), so \\(\\|1-e_i\\|\\leq1\\), and\n\\[\n\\begin{gathered}\n\\|x-xe_i\\|\\\\\n=\\|(x+y)(1-e_i)-(y-ye_i)\\|\\\\\n\\leq\\|x+y\\|+\\|y-ye_i\\| .\n\\end{gathered}\n\\]\nSo \\(\\limsup_i\\|x-xe_i\\|\\leq\\|x+y\\|\\) for every \\(y\\in\\mathfrak m\\), that is, \\(\\limsup_i\\|x-xe_i\\|\\leq\\|x+\\mathfrak m\\|\\). Since \\(xe_i\\in\\mathfrak m\\), \\(\\|x+\\mathfrak m\\|\\leq\\|x-xe_i\\|\\) for every \\(i\\). Hence the limit exists and equals \\(\\|x+\\mathfrak m\\|\\).\n(2) \\(A/\\mathfrak m\\) is a Banach algebra ([quotient algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-15)). By (1) the involution is well defined, and it is isometric: \\(\\|x^*+\\mathfrak m\\|=\\inf_{y\\in\\mathfrak m}\\|x^*+y^*\\|=\\|x+\\mathfrak m\\|\\). Write \\(\\dot x=x+\\mathfrak m\\), and let \\((e_i)\\) be as in (3). For \\(z\\in\\mathfrak m\\), \\(\\|(1-e_i)z(1-e_i)\\|\\leq\\|z-ze_i\\|\\to0\\). So by (15.1)\n\\[\n\\begin{gathered}\n\\|\\dot x\\|^2\\\\\n=\\lim_i\\|(x-xe_i)^*(x-xe_i)\\|\\\\\n=\\lim_i\\|(1-e_i)(x^*x+z)(1-e_i)\\|\\\\\n\\leq\\|x^*x+z\\| .\n\\end{gathered}\n\\]\nTaking the infimum over \\(z\\), \\(\\|\\dot x\\|^2\\leq\\|\\dot x^*\\dot x\\|\\leq\\|\\dot x^*\\|\\|\\dot x\\|=\\|\\dot x\\|^2\\). So the C\\*-identity holds. \\(\\square\\)\n\nThe proof of (3) uses only positivity, contractivity and the approximation property of \\((e_i)\\); whether the net increases plays no part.\n\n**Examples 15.2** (The two hypotheses of (1) are needed).\n- *A closed left ideal need not be self-adjoint.* In \\(M_2(\\mathbb C)\\), \\(L=\\{x:xE_{11}=x\\}\\), the matrices with zero second column, is a closed left ideal. It contains \\(E_{21}\\) but not \\(E_{21}^*=E_{12}\\).\n- *An ideal that is not closed need not be self-adjoint.* In \\(C([0,1])\\), let \\(h(t)=te^{i/t}\\) for \\(t>0\\) and \\(h(0)=0\\), and let \\(I=hC([0,1])\\), an ideal. If \\(\\bar h=hg\\) with \\(g\\) continuous, then \\(g(t)=e^{-2i/t}\\) for \\(t>0\\), which has no limit at \\(0\\). So \\(\\bar h\\notin I\\).\n\n**Example 15.3** (The quotient norm formula). In \\(A=C([0,1])\\), let \\(\\mathfrak m=\\{g:g(0)=0\\}\\) and \\(e_n(t)=\\min(1,nt)\\), a positive contractive (and increasing) approximate identity of \\(\\mathfrak m\\). For \\(f\\in A\\), \\[\n\\begin{gathered}\n\\|f-fe_n\\|\\\\\n=\\sup_{t\\leq1/n}|f(t)|(1-nt)\\to|f(0)|\\\\\n=\\|f+\\mathfrak m\\|,\n\\end{gathered}\n\\] as (15.1) says.\n\n**Corollary 15.4** (Kernels and ranges of \\(*\\)-homomorphisms). Let \\(\\pi:A\\to B\\) be a \\(*\\)-homomorphism of C\\*-algebras. Then \\(\\ker\\pi\\) is a closed ideal, the range \\(\\pi(A)\\) is closed, hence a C\\*-subalgebra of \\(B\\), and \\(\\tilde\\pi(x+\\ker\\pi)=\\pi(x)\\) is an isometric \\(*\\)-isomorphism of \\(A/\\ker\\pi\\) onto \\(\\pi(A)\\). In particular \\(\\|\\pi(x)\\|=\\|x+\\ker\\pi\\|\\).\n\n**Proof.** \\(\\pi\\) is continuous (Theorem 4.2), so \\(\\ker\\pi\\) is a closed ideal, and \\(A/\\ker\\pi\\) is a C\\*-algebra (Theorem 15.1). The induced map \\(\\tilde\\pi\\) is an injective \\(*\\)-homomorphism into \\(B\\), hence isometric with closed range (Corollary 4.6). \\(\\square\\)\n\n**Exercise 15.5** (easy; Ideals of ideals). Let \\(I\\) be a closed ideal of \\(A\\) and \\(J\\) a closed ideal of the C\\*-algebra \\(I\\). Show that \\(J\\) is an ideal of \\(A\\).\n\n*Solution.* Let \\(x\\in J\\), \\(a\\in A\\), and let \\((u)\\) be the approximate identity of Theorem 11.4 for the closed ideal \\(J\\) of the C\\*-algebra \\(I\\). Then \\(ux\\to x\\), and \\(au\\in I\\), since \\(u\\in J\\subseteq I\\) and \\(I\\) is an ideal of \\(A\\). So \\((au)x\\in IJ\\subseteq J\\), and \\(ax=\\lim(au)x\\in J\\), as \\(J\\) is closed. Similarly \\(xa=\\lim x(ua)\\in J\\).\n\n**Exercise 15.6** (easy; A C\\*-subalgebra plus a closed ideal). If \\(B\\) is a C\\*-subalgebra and \\(\\mathfrak m\\) a closed ideal of \\(A\\), then \\(B+\\mathfrak m\\) is a C\\*-subalgebra.\n\n*Solution.* \\(B+\\mathfrak m\\) is a \\(*\\)-subalgebra, because \\((b+x)(b'+x')=bb'+(bx'+xb'+xx')\\) and \\(\\mathfrak m\\) is a self-adjoint ideal (Theorem 15.1). Let \\(\\pi:A\\to A/\\mathfrak m\\). By Corollary 15.4, applied to \\(\\pi|_B\\), the set \\(\\pi(B)\\) is closed. So \\(B+\\mathfrak m=\\pi^{-1}(\\pi(B))\\) is closed.\n\n",
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      "id": "OA-FND-CF-26",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "15. Closed ideals and quotients",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
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      "full_conditions_and_proof": "### 15. Closed ideals and quotients\n\n**Theorem 15.1** (Quotients of C\\*-algebras). Let \\(A\\) be a C\\*-algebra and \\(\\mathfrak m\\subseteq A\\) a closed ideal.\n1. \\(\\mathfrak m^*=\\mathfrak m\\), so \\(\\mathfrak m\\) is a C\\*-subalgebra.\n2. \\(A/\\mathfrak m\\), with the quotient norm and \\((x+\\mathfrak m)^*=x^*+\\mathfrak m\\), is a C\\*-algebra, and the quotient map is a \\(*\\)-homomorphism.\n3. (*Quotient norm.*) Let \\((e_i)\\) be any net of positive contractions in \\(\\mathfrak m\\) with \\(\\|y-ye_i\\|\\to0\\) for every \\(y\\in\\mathfrak m\\), for instance the net of Theorem 11.4. Then for every \\(x\\in A\\),\n\\[\n\\|x+\\mathfrak m\\|=\\lim_i\\|x-xe_i\\| .\n\\tag{15.1}\n\\]\n\n**Proof.** (1) By Theorem 11.4(3), \\(\\Lambda=\\mathfrak m_+\\cap S_0\\) is a two-sided approximate identity for \\(\\mathfrak m\\). For \\(x\\in\\mathfrak m\\) and \\(u\\in\\Lambda\\), \\(x^*u\\in\\mathfrak m\\) and \\(\\|x^*-x^*u\\|=\\|x-ux\\|\\to0\\). So \\(x^*\\in\\mathfrak m\\), as \\(\\mathfrak m\\) is closed.\n(3) Let \\(y\\in\\mathfrak m\\). In \\(\\widetilde A\\), \\(0\\leq1-e_i\\leq1\\), so \\(\\|1-e_i\\|\\leq1\\), and\n\\[\n\\begin{gathered}\n\\|x-xe_i\\|\\\\\n=\\|(x+y)(1-e_i)-(y-ye_i)\\|\\\\\n\\leq\\|x+y\\|+\\|y-ye_i\\| .\n\\end{gathered}\n\\]\nSo \\(\\limsup_i\\|x-xe_i\\|\\leq\\|x+y\\|\\) for every \\(y\\in\\mathfrak m\\), that is, \\(\\limsup_i\\|x-xe_i\\|\\leq\\|x+\\mathfrak m\\|\\). Since \\(xe_i\\in\\mathfrak m\\), \\(\\|x+\\mathfrak m\\|\\leq\\|x-xe_i\\|\\) for every \\(i\\). Hence the limit exists and equals \\(\\|x+\\mathfrak m\\|\\).\n(2) \\(A/\\mathfrak m\\) is a Banach algebra ([quotient algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-15)). By (1) the involution is well defined, and it is isometric: \\(\\|x^*+\\mathfrak m\\|=\\inf_{y\\in\\mathfrak m}\\|x^*+y^*\\|=\\|x+\\mathfrak m\\|\\). Write \\(\\dot x=x+\\mathfrak m\\), and let \\((e_i)\\) be as in (3). For \\(z\\in\\mathfrak m\\), \\(\\|(1-e_i)z(1-e_i)\\|\\leq\\|z-ze_i\\|\\to0\\). So by (15.1)\n\\[\n\\begin{gathered}\n\\|\\dot x\\|^2\\\\\n=\\lim_i\\|(x-xe_i)^*(x-xe_i)\\|\\\\\n=\\lim_i\\|(1-e_i)(x^*x+z)(1-e_i)\\|\\\\\n\\leq\\|x^*x+z\\| .\n\\end{gathered}\n\\]\nTaking the infimum over \\(z\\), \\(\\|\\dot x\\|^2\\leq\\|\\dot x^*\\dot x\\|\\leq\\|\\dot x^*\\|\\|\\dot x\\|=\\|\\dot x\\|^2\\). So the C\\*-identity holds. \\(\\square\\)\n\nThe proof of (3) uses only positivity, contractivity and the approximation property of \\((e_i)\\); whether the net increases plays no part.\n\n**Examples 15.2** (The two hypotheses of (1) are needed).\n- *A closed left ideal need not be self-adjoint.* In \\(M_2(\\mathbb C)\\), \\(L=\\{x:xE_{11}=x\\}\\), the matrices with zero second column, is a closed left ideal. It contains \\(E_{21}\\) but not \\(E_{21}^*=E_{12}\\).\n- *An ideal that is not closed need not be self-adjoint.* In \\(C([0,1])\\), let \\(h(t)=te^{i/t}\\) for \\(t>0\\) and \\(h(0)=0\\), and let \\(I=hC([0,1])\\), an ideal. If \\(\\bar h=hg\\) with \\(g\\) continuous, then \\(g(t)=e^{-2i/t}\\) for \\(t>0\\), which has no limit at \\(0\\). So \\(\\bar h\\notin I\\).\n\n**Example 15.3** (The quotient norm formula). In \\(A=C([0,1])\\), let \\(\\mathfrak m=\\{g:g(0)=0\\}\\) and \\(e_n(t)=\\min(1,nt)\\), a positive contractive (and increasing) approximate identity of \\(\\mathfrak m\\). For \\(f\\in A\\), \\[\n\\begin{gathered}\n\\|f-fe_n\\|\\\\\n=\\sup_{t\\leq1/n}|f(t)|(1-nt)\\to|f(0)|\\\\\n=\\|f+\\mathfrak m\\|,\n\\end{gathered}\n\\] as (15.1) says.\n\n**Corollary 15.4** (Kernels and ranges of \\(*\\)-homomorphisms). Let \\(\\pi:A\\to B\\) be a \\(*\\)-homomorphism of C\\*-algebras. Then \\(\\ker\\pi\\) is a closed ideal, the range \\(\\pi(A)\\) is closed, hence a C\\*-subalgebra of \\(B\\), and \\(\\tilde\\pi(x+\\ker\\pi)=\\pi(x)\\) is an isometric \\(*\\)-isomorphism of \\(A/\\ker\\pi\\) onto \\(\\pi(A)\\). In particular \\(\\|\\pi(x)\\|=\\|x+\\ker\\pi\\|\\).\n\n**Proof.** \\(\\pi\\) is continuous (Theorem 4.2), so \\(\\ker\\pi\\) is a closed ideal, and \\(A/\\ker\\pi\\) is a C\\*-algebra (Theorem 15.1). The induced map \\(\\tilde\\pi\\) is an injective \\(*\\)-homomorphism into \\(B\\), hence isometric with closed range (Corollary 4.6). \\(\\square\\)\n\n**Exercise 15.5** (easy; Ideals of ideals). Let \\(I\\) be a closed ideal of \\(A\\) and \\(J\\) a closed ideal of the C\\*-algebra \\(I\\). Show that \\(J\\) is an ideal of \\(A\\).\n\n*Solution.* Let \\(x\\in J\\), \\(a\\in A\\), and let \\((u)\\) be the approximate identity of Theorem 11.4 for the closed ideal \\(J\\) of the C\\*-algebra \\(I\\). Then \\(ux\\to x\\), and \\(au\\in I\\), since \\(u\\in J\\subseteq I\\) and \\(I\\) is an ideal of \\(A\\). So \\((au)x\\in IJ\\subseteq J\\), and \\(ax=\\lim(au)x\\in J\\), as \\(J\\) is closed. Similarly \\(xa=\\lim x(ua)\\in J\\).\n\n**Exercise 15.6** (easy; A C\\*-subalgebra plus a closed ideal). If \\(B\\) is a C\\*-subalgebra and \\(\\mathfrak m\\) a closed ideal of \\(A\\), then \\(B+\\mathfrak m\\) is a C\\*-subalgebra.\n\n*Solution.* \\(B+\\mathfrak m\\) is a \\(*\\)-subalgebra, because \\((b+x)(b'+x')=bb'+(bx'+xb'+xx')\\) and \\(\\mathfrak m\\) is a self-adjoint ideal (Theorem 15.1). Let \\(\\pi:A\\to A/\\mathfrak m\\). By Corollary 15.4, applied to \\(\\pi|_B\\), the set \\(\\pi(B)\\) is closed. So \\(B+\\mathfrak m=\\pi^{-1}(\\pi(B))\\) is closed.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CF-27",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "16. Ideals of commutative C\\*-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
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      "full_conditions_and_proof": "### 16. Ideals of commutative C\\*-algebras\n\n**Proposition 16.1** (Hull and kernel). Let \\(A\\) be abelian, with character space \\(\\Omega\\). For a closed set \\(\\Gamma\\subseteq\\Omega\\) and a closed ideal \\(\\mathfrak m\\) put\n\\[\n\\begin{gathered}\n\\mathfrak m_\\Gamma\\\\\n=\\{x\\in A: \\ \\omega(x)=0\\ \\text{for all }\\omega\\in\\Gamma\\},\\\\\n\\Gamma_{\\mathfrak m}\\\\\n=\\{\\omega\\in\\Omega: \\ \\omega(x)=0\\ \\text{for all }x\\in\\mathfrak m\\}.\n\\end{gathered}\n\\]\n1. \\(\\Gamma\\mapsto\\mathfrak m_\\Gamma\\) and \\(\\mathfrak m\\mapsto\\Gamma_{\\mathfrak m}\\) are mutually inverse bijections, reversing inclusion, between the closed subsets of \\(\\Omega\\) and the closed ideals of \\(A\\).\n2. If \\(\\rho:A\\to A/\\mathfrak m\\) is the quotient map, then \\(\\omega'\\mapsto\\omega'\\circ\\rho\\) is a homeomorphism of \\(\\operatorname{Ch}(A/\\mathfrak m)\\) onto \\(\\Gamma_{\\mathfrak m}\\).\n3. Restriction \\(\\omega\\mapsto\\omega|_{\\mathfrak m}\\) is a homeomorphism of \\(\\Omega\\setminus\\Gamma_{\\mathfrak m}\\) onto \\(\\operatorname{Ch}(\\mathfrak m)\\).\n\nIdentifying \\(A\\) with \\(C_0(\\Omega)\\) (Theorem 2.1 and Proposition 2.2): \\(\\mathfrak m_\\Gamma\\) is the set of functions vanishing on \\(\\Gamma\\); restriction to \\(\\Gamma\\) identifies \\(A/\\mathfrak m_\\Gamma\\) with \\(C_0(\\Gamma)\\); and \\(\\mathfrak m_\\Gamma\\) is \\(C_0(\\Omega\\setminus\\Gamma)\\). The ideal \\(\\mathfrak m_\\Gamma\\) is called the *kernel* of \\(\\Gamma\\), and \\(\\Gamma_{\\mathfrak m}\\) the *hull* of \\(\\mathfrak m\\).\n\n**Proof.** \\(\\mathfrak m_\\Gamma\\) is an intersection of kernels of characters, so it is a closed ideal; \\(\\Gamma_{\\mathfrak m}\\) is weak\\* closed. Clearly \\(\\mathfrak m\\subseteq\\mathfrak m_{\\Gamma_{\\mathfrak m}}\\) and \\(\\Gamma\\subseteq\\Gamma_{\\mathfrak m_\\Gamma}\\).\n(1) *\\(\\mathfrak m_{\\Gamma_{\\mathfrak m}}\\subseteq\\mathfrak m\\).* Let \\(x_0\\notin\\mathfrak m\\). Then \\(\\rho(x_0)\\neq0\\) in the abelian C\\*-algebra \\(A/\\mathfrak m\\) (Theorem 15.1). By Theorem 2.1 its Gelfand transform is not zero, so some character \\(\\omega'\\) of \\(A/\\mathfrak m\\) has \\(\\omega'(\\rho(x_0))\\neq0\\). Then \\(\\omega=\\omega'\\circ\\rho\\) is a character of \\(A\\) that vanishes on \\(\\mathfrak m\\), so \\(\\omega\\in\\Gamma_{\\mathfrak m}\\), and \\(\\omega(x_0)\\neq0\\); so \\(x_0\\notin\\mathfrak m_{\\Gamma_{\\mathfrak m}}\\).\n*\\(\\Gamma_{\\mathfrak m_\\Gamma}\\subseteq\\Gamma\\).* Let \\(\\omega_0\\notin\\Gamma\\). By [Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16) in its locally compact form, there is \\(f\\in C_c(\\Omega)\\) with \\(f(\\omega_0)=1\\) and \\(f=0\\) on \\(\\Gamma\\). The element \\(x\\) with \\(\\hat x=f\\) lies in \\(\\mathfrak m_\\Gamma\\) and \\(\\omega_0(x)\\neq0\\), so \\(\\omega_0\\notin\\Gamma_{\\mathfrak m_\\Gamma}\\).\n(2) A character \\(\\omega\\) of \\(A\\) vanishing on \\(\\mathfrak m\\) factors as \\(\\omega=\\omega'\\circ\\rho\\), with \\(\\omega'(x+\\mathfrak m)=\\omega(x)\\) a character of \\(A/\\mathfrak m\\); conversely \\(\\omega'\\circ\\rho\\) is a character (nonzero because \\(\\rho\\) is onto) vanishing on \\(\\mathfrak m\\). So the map is a bijection onto \\(\\Gamma_{\\mathfrak m}\\). It is weak\\* continuous, and so is its inverse: if \\(\\omega'_\\alpha\\circ\\rho\\to\\omega'\\circ\\rho\\) pointwise on \\(A\\), then \\(\\omega'_\\alpha\\to\\omega'\\) pointwise on \\(A/\\mathfrak m\\).\n(3) For \\(\\omega\\in\\Omega\\), \\(\\omega|_{\\mathfrak m}\\) is a homomorphism, nonzero exactly when \\(\\omega\\notin\\Gamma_{\\mathfrak m}\\). The restriction map \\(I\\) is weak\\* continuous.\n*Injective.* For \\(\\omega_1\\neq\\omega_2\\) in \\(\\Omega\\setminus\\Gamma_{\\mathfrak m}\\), Urysohn's lemma gives \\(f\\in C_c(\\Omega)\\) with \\(f(\\omega_1)=1\\) and \\(f=0\\) on the closed set \\(\\Gamma_{\\mathfrak m}\\cup\\{\\omega_2\\}\\). By (1), the element with transform \\(f\\) lies in \\(\\mathfrak m\\), and it separates \\(I(\\omega_1)\\) from \\(I(\\omega_2)\\).\n*Surjective.* Let \\(\\chi\\in\\operatorname{Ch}(\\mathfrak m)\\) and \\(e\\in\\mathfrak m\\) with \\(\\chi(e)=1\\). Put \\(\\omega(x)=\\chi(ex)\\) for \\(x\\in A\\), which makes sense because \\(ex\\in\\mathfrak m\\). Since \\(A\\) is commutative, \\(\\omega(xy)=\\chi(exy)\\chi(e)=\\chi(ex\\cdot ey)=\\omega(x)\\omega(y)\\), and for \\(x\\in\\mathfrak m\\), \\(\\omega(x)=\\chi(e)\\chi(x)=\\chi(x)\\). So \\(\\omega\\) is a character with \\(I(\\omega)=\\chi\\).\n*Open.* Let \\(\\omega_0\\in W\\subseteq\\Omega\\setminus\\Gamma_{\\mathfrak m}\\) with \\(W\\) open. Choose \\(f\\in C_c(\\Omega)\\) with \\(f(\\omega_0)=1\\) and support in \\(W\\), and let \\(x\\in\\mathfrak m\\) have \\(\\hat x=f\\). The set \\(\\{\\chi\\in\\operatorname{Ch}(\\mathfrak m):\\chi(x)\\neq0\\}\\) is open, contains \\(I(\\omega_0)\\), and lies in \\(I(W)\\), because \\(\\chi=I(\\omega)\\) with \\(\\omega(x)=f(\\omega)\\neq0\\) forces \\(\\omega\\in W\\). So \\(I\\) is open, hence a homeomorphism.\n*The concrete description.* Under \\(A=C_0(\\Omega)\\), (1) says that \\(\\mathfrak m_\\Gamma\\) is the set of functions vanishing on \\(\\Gamma\\). The map \\(f\\mapsto f|_\\Gamma\\) is a \\(*\\)-homomorphism into \\(C_0(\\Gamma)\\) with kernel \\(\\mathfrak m_\\Gamma\\). Its range is closed (Corollary 15.4), separates the points of \\(\\Gamma\\) and vanishes nowhere on it (Urysohn), so it is \\(C_0(\\Gamma)\\) by the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09). Likewise \\(\\mathfrak m_\\Gamma\\), as a C\\*-algebra, has character space \\(\\Omega\\setminus\\Gamma\\) by (3), so it is \\(C_0(\\Omega\\setminus\\Gamma)\\) by Theorem 2.1. \\(\\square\\)\n\n**Example 16.2** (\\(C_0(\\mathbb R)\\)). Let \\(A=C_0(\\mathbb R)\\). By Proposition 2.2, the characters are the evaluations, so \\(\\sigma'(x)=x(\\mathbb R)\\cup\\{0\\}\\), and by Theorem 5.3(4), \\(f(x)=f\\circ x\\) for \\(f(0)=0\\). Positivity is pointwise. The function \\(a(t)=e^{-t^2}\\) is strictly positive: \\(f_\\varepsilon(a)=a/(a+\\varepsilon)\\) tends to \\(1\\) uniformly on compact sets, and for \\(g\\in C_0(\\mathbb R)\\) and \\(\\eta>0\\), \\(|g|<\\eta\\) off a compact set \\(C\\), so \\[\n\\begin{gathered}\n\\|g-f_\\varepsilon(a)g\\|\\\\\n\\leq\\max(\\eta,\\|g\\|\\max_C|1-f_\\varepsilon(a)|)\\to\\eta;\n\\end{gathered}\n\\] this is Proposition 13.3(3) seen directly. The closed ideals correspond to closed \\(F\\subseteq\\mathbb R\\) (Proposition 16.1): for \\(F=\\{0\\}\\), \\(\\mathfrak m_F=\\{g:g(0)=0\\}\\cong C_0(\\mathbb R\\setminus\\{0\\})\\), and \\(A/\\mathfrak m_F\\cong\\mathbb C\\) through \\(g\\mapsto g(0)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-30",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "16. Ideals of commutative C\\*-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
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      "full_conditions_and_proof": "### 16. Ideals of commutative C\\*-algebras\n\n**Proposition 16.1** (Hull and kernel). Let \\(A\\) be abelian, with character space \\(\\Omega\\). For a closed set \\(\\Gamma\\subseteq\\Omega\\) and a closed ideal \\(\\mathfrak m\\) put\n\\[\n\\begin{gathered}\n\\mathfrak m_\\Gamma\\\\\n=\\{x\\in A: \\ \\omega(x)=0\\ \\text{for all }\\omega\\in\\Gamma\\},\\\\\n\\Gamma_{\\mathfrak m}\\\\\n=\\{\\omega\\in\\Omega: \\ \\omega(x)=0\\ \\text{for all }x\\in\\mathfrak m\\}.\n\\end{gathered}\n\\]\n1. \\(\\Gamma\\mapsto\\mathfrak m_\\Gamma\\) and \\(\\mathfrak m\\mapsto\\Gamma_{\\mathfrak m}\\) are mutually inverse bijections, reversing inclusion, between the closed subsets of \\(\\Omega\\) and the closed ideals of \\(A\\).\n2. If \\(\\rho:A\\to A/\\mathfrak m\\) is the quotient map, then \\(\\omega'\\mapsto\\omega'\\circ\\rho\\) is a homeomorphism of \\(\\operatorname{Ch}(A/\\mathfrak m)\\) onto \\(\\Gamma_{\\mathfrak m}\\).\n3. Restriction \\(\\omega\\mapsto\\omega|_{\\mathfrak m}\\) is a homeomorphism of \\(\\Omega\\setminus\\Gamma_{\\mathfrak m}\\) onto \\(\\operatorname{Ch}(\\mathfrak m)\\).\n\nIdentifying \\(A\\) with \\(C_0(\\Omega)\\) (Theorem 2.1 and Proposition 2.2): \\(\\mathfrak m_\\Gamma\\) is the set of functions vanishing on \\(\\Gamma\\); restriction to \\(\\Gamma\\) identifies \\(A/\\mathfrak m_\\Gamma\\) with \\(C_0(\\Gamma)\\); and \\(\\mathfrak m_\\Gamma\\) is \\(C_0(\\Omega\\setminus\\Gamma)\\). The ideal \\(\\mathfrak m_\\Gamma\\) is called the *kernel* of \\(\\Gamma\\), and \\(\\Gamma_{\\mathfrak m}\\) the *hull* of \\(\\mathfrak m\\).\n\n**Proof.** \\(\\mathfrak m_\\Gamma\\) is an intersection of kernels of characters, so it is a closed ideal; \\(\\Gamma_{\\mathfrak m}\\) is weak\\* closed. Clearly \\(\\mathfrak m\\subseteq\\mathfrak m_{\\Gamma_{\\mathfrak m}}\\) and \\(\\Gamma\\subseteq\\Gamma_{\\mathfrak m_\\Gamma}\\).\n(1) *\\(\\mathfrak m_{\\Gamma_{\\mathfrak m}}\\subseteq\\mathfrak m\\).* Let \\(x_0\\notin\\mathfrak m\\). Then \\(\\rho(x_0)\\neq0\\) in the abelian C\\*-algebra \\(A/\\mathfrak m\\) (Theorem 15.1). By Theorem 2.1 its Gelfand transform is not zero, so some character \\(\\omega'\\) of \\(A/\\mathfrak m\\) has \\(\\omega'(\\rho(x_0))\\neq0\\). Then \\(\\omega=\\omega'\\circ\\rho\\) is a character of \\(A\\) that vanishes on \\(\\mathfrak m\\), so \\(\\omega\\in\\Gamma_{\\mathfrak m}\\), and \\(\\omega(x_0)\\neq0\\); so \\(x_0\\notin\\mathfrak m_{\\Gamma_{\\mathfrak m}}\\).\n*\\(\\Gamma_{\\mathfrak m_\\Gamma}\\subseteq\\Gamma\\).* Let \\(\\omega_0\\notin\\Gamma\\). By [Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16) in its locally compact form, there is \\(f\\in C_c(\\Omega)\\) with \\(f(\\omega_0)=1\\) and \\(f=0\\) on \\(\\Gamma\\). The element \\(x\\) with \\(\\hat x=f\\) lies in \\(\\mathfrak m_\\Gamma\\) and \\(\\omega_0(x)\\neq0\\), so \\(\\omega_0\\notin\\Gamma_{\\mathfrak m_\\Gamma}\\).\n(2) A character \\(\\omega\\) of \\(A\\) vanishing on \\(\\mathfrak m\\) factors as \\(\\omega=\\omega'\\circ\\rho\\), with \\(\\omega'(x+\\mathfrak m)=\\omega(x)\\) a character of \\(A/\\mathfrak m\\); conversely \\(\\omega'\\circ\\rho\\) is a character (nonzero because \\(\\rho\\) is onto) vanishing on \\(\\mathfrak m\\). So the map is a bijection onto \\(\\Gamma_{\\mathfrak m}\\). It is weak\\* continuous, and so is its inverse: if \\(\\omega'_\\alpha\\circ\\rho\\to\\omega'\\circ\\rho\\) pointwise on \\(A\\), then \\(\\omega'_\\alpha\\to\\omega'\\) pointwise on \\(A/\\mathfrak m\\).\n(3) For \\(\\omega\\in\\Omega\\), \\(\\omega|_{\\mathfrak m}\\) is a homomorphism, nonzero exactly when \\(\\omega\\notin\\Gamma_{\\mathfrak m}\\). The restriction map \\(I\\) is weak\\* continuous.\n*Injective.* For \\(\\omega_1\\neq\\omega_2\\) in \\(\\Omega\\setminus\\Gamma_{\\mathfrak m}\\), Urysohn's lemma gives \\(f\\in C_c(\\Omega)\\) with \\(f(\\omega_1)=1\\) and \\(f=0\\) on the closed set \\(\\Gamma_{\\mathfrak m}\\cup\\{\\omega_2\\}\\). By (1), the element with transform \\(f\\) lies in \\(\\mathfrak m\\), and it separates \\(I(\\omega_1)\\) from \\(I(\\omega_2)\\).\n*Surjective.* Let \\(\\chi\\in\\operatorname{Ch}(\\mathfrak m)\\) and \\(e\\in\\mathfrak m\\) with \\(\\chi(e)=1\\). Put \\(\\omega(x)=\\chi(ex)\\) for \\(x\\in A\\), which makes sense because \\(ex\\in\\mathfrak m\\). Since \\(A\\) is commutative, \\(\\omega(xy)=\\chi(exy)\\chi(e)=\\chi(ex\\cdot ey)=\\omega(x)\\omega(y)\\), and for \\(x\\in\\mathfrak m\\), \\(\\omega(x)=\\chi(e)\\chi(x)=\\chi(x)\\). So \\(\\omega\\) is a character with \\(I(\\omega)=\\chi\\).\n*Open.* Let \\(\\omega_0\\in W\\subseteq\\Omega\\setminus\\Gamma_{\\mathfrak m}\\) with \\(W\\) open. Choose \\(f\\in C_c(\\Omega)\\) with \\(f(\\omega_0)=1\\) and support in \\(W\\), and let \\(x\\in\\mathfrak m\\) have \\(\\hat x=f\\). The set \\(\\{\\chi\\in\\operatorname{Ch}(\\mathfrak m):\\chi(x)\\neq0\\}\\) is open, contains \\(I(\\omega_0)\\), and lies in \\(I(W)\\), because \\(\\chi=I(\\omega)\\) with \\(\\omega(x)=f(\\omega)\\neq0\\) forces \\(\\omega\\in W\\). So \\(I\\) is open, hence a homeomorphism.\n*The concrete description.* Under \\(A=C_0(\\Omega)\\), (1) says that \\(\\mathfrak m_\\Gamma\\) is the set of functions vanishing on \\(\\Gamma\\). The map \\(f\\mapsto f|_\\Gamma\\) is a \\(*\\)-homomorphism into \\(C_0(\\Gamma)\\) with kernel \\(\\mathfrak m_\\Gamma\\). Its range is closed (Corollary 15.4), separates the points of \\(\\Gamma\\) and vanishes nowhere on it (Urysohn), so it is \\(C_0(\\Gamma)\\) by the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09). Likewise \\(\\mathfrak m_\\Gamma\\), as a C\\*-algebra, has character space \\(\\Omega\\setminus\\Gamma\\) by (3), so it is \\(C_0(\\Omega\\setminus\\Gamma)\\) by Theorem 2.1. \\(\\square\\)\n\n**Example 16.2** (\\(C_0(\\mathbb R)\\)). Let \\(A=C_0(\\mathbb R)\\). By Proposition 2.2, the characters are the evaluations, so \\(\\sigma'(x)=x(\\mathbb R)\\cup\\{0\\}\\), and by Theorem 5.3(4), \\(f(x)=f\\circ x\\) for \\(f(0)=0\\). Positivity is pointwise. The function \\(a(t)=e^{-t^2}\\) is strictly positive: \\(f_\\varepsilon(a)=a/(a+\\varepsilon)\\) tends to \\(1\\) uniformly on compact sets, and for \\(g\\in C_0(\\mathbb R)\\) and \\(\\eta>0\\), \\(|g|<\\eta\\) off a compact set \\(C\\), so \\[\n\\begin{gathered}\n\\|g-f_\\varepsilon(a)g\\|\\\\\n\\leq\\max(\\eta,\\|g\\|\\max_C|1-f_\\varepsilon(a)|)\\to\\eta;\n\\end{gathered}\n\\] this is Proposition 13.3(3) seen directly. The closed ideals correspond to closed \\(F\\subseteq\\mathbb R\\) (Proposition 16.1): for \\(F=\\{0\\}\\), \\(\\mathfrak m_F=\\{g:g(0)=0\\}\\cong C_0(\\mathbb R\\setminus\\{0\\})\\), and \\(A/\\mathfrak m_F\\cong\\mathbb C\\) through \\(g\\mapsto g(0)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CF-28",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "17. Lifting through quotients",
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      "full_conditions_and_proof": "### 17. Lifting through quotients\n\n**Proposition 17.1** (Lifting self-adjoint, positive and arbitrary elements). Let \\(\\pi:A\\to B\\) be a surjective \\(*\\)-homomorphism of C\\*-algebras.\n1. Every \\(k\\in B_h\\) is \\(\\pi(h)\\) for some \\(h\\in A_h\\) with \\(\\|h\\|=\\|k\\|\\). If \\(k\\geq0\\), \\(h\\) can be chosen with \\(h\\geq0\\) and \\(\\|h\\|=\\|k\\|\\).\n2. Every \\(k\\in B\\) is \\(\\pi(x)\\) for some \\(x\\in A\\) with \\(\\|x\\|=\\|k\\|\\).\n3. \\(\\pi(A_+)=B_+\\).\n\n**Proof.** (1) Let \\(h'\\in A\\) with \\(\\pi(h')=k\\). Then \\(h''=\\frac12(h'+h'^*)\\) is self-adjoint with \\(\\pi(h'')=k\\). Put \\(c=\\|k\\|\\) and \\(\\tau(t)=\\max(-c,\\min(t,c))\\), a continuous function with \\(\\tau(0)=0\\) and \\(|\\tau|\\leq c\\). Let \\(h=\\tau(h'')\\in A_h\\). Then \\(\\|h\\|\\leq c\\), and by Corollary 5.4(3), \\(\\pi(h)=\\tau(k)=k\\), since \\(\\sigma'(k)\\subseteq[-c,c]\\), where \\(\\tau\\) is the identity. So \\(c=\\|\\pi(h)\\|\\leq\\|h\\|\\leq c\\). If \\(k\\geq0\\), use \\(\\tau_+(t)=\\min(\\max(t,0),c)\\) instead; then \\(h\\geq0\\).\n(2) Let \\(x'\\in A\\) with \\(\\pi(x')=k\\), and \\(c=\\|k\\|\\); if \\(c=0\\) take \\(x=0\\). Let \\(g(t)=\\min(1,ct^{-1/2})\\) for \\(t>0\\) and \\(g(0)=1\\), a continuous function on \\([0,\\infty)\\) that equals \\(1\\) on \\([0,c^2]\\). Put \\(x=x'g(x'^*x')\\), which lies in \\(A\\) because \\(A\\) is an ideal of \\(\\widetilde A\\). Then\n\\[\n\\begin{gathered}\n\\|x\\|^2\\\\\n=\\|g(x'^*x')\\,x'^*x'\\,g(x'^*x')\\|\\\\\n=\\max_{t\\in\\sigma'(x'^*x')}tg(t)^2\\\\\n\\leq c^2 ,\n\\end{gathered}\n\\]\nsince \\(tg(t)^2=\\min(t,c^2)\\). By Corollary 5.4(2), \\(\\pi(x)=k\\,g(k^*k)=k\\), because \\(\\sigma'(k^*k)\\subseteq[0,c^2]\\), where \\(g=1\\). So \\(\\|x\\|=c\\).\n(3) \\(\\pi(A_+)\\subseteq B_+\\) by Proposition 8.5(12), and (1) gives the rest. \\(\\square\\)\n\nA second proof of (1) uses Tietze's extension theorem. The C\\*-algebra \\(C^*(h'')\\) is abelian, and restricting \\(\\pi\\) to it reduces (1) to the abelian case. There, by Proposition 16.1, the quotient map is restriction \\(C_0(\\Omega)\\to C_0(\\Gamma)\\), and a bounded extension exists by [Tietze's theorem](stone-weierstrass-c0.md#oa-fnd-sw-18) on the one-point compactification, which is compact Hausdorff but need not be metrizable. Neither proof needs separability.\n\n**Proposition 17.2** (Lifting \\(\\{y^*y\\leq\\pi(a)\\}\\)). Let \\(\\pi:A\\to B\\) be a surjective \\(*\\)-homomorphism of C\\*-algebras and \\(a\\in A_+\\). Then\n\\[\n\\begin{gathered}\n\\pi\\big(\\{x\\in A: \\ x^*x\\leq a\\}\\big)\\\\\n=\\{y\\in B: \\ y^*y\\leq\\pi(a)\\}.\n\\end{gathered}\n\\]\n\n**Proof.** Put \\(b=\\pi(a)\\). If \\(x^*x\\leq a\\), then \\(\\pi(x)^*\\pi(x)=\\pi(x^*x)\\leq b\\) (Proposition 8.5(12)). Conversely let \\(y^*y\\leq b\\).\n*Step 1: a self-adjoint \\(h\\leq a\\) with \\(\\pi(h)=y^*y\\).* Choose \\(h_0\\in A_h\\) with \\(\\pi(h_0)=y^*y\\) (Proposition 17.1(1)). Then \\(\\pi(a-h_0)=b-y^*y\\geq0\\). By Corollary 5.4(3), \\(\\pi((a-h_0)_\\pm)=(b-y^*y)_\\pm\\), which is \\(b-y^*y\\) for the sign \\(+\\) and \\(0\\) for the sign \\(-\\). Put \\(h=a-(a-h_0)_+\\). Then \\(h\\leq a\\) and \\(\\pi(h)=b-(b-y^*y)=y^*y\\).\n*Step 2: a dominating element.* Choose \\(z\\in A\\) with \\(\\pi(z)=y\\), and put \\(k=z^*z-h\\in A_h\\). Then \\(\\pi(k)=0\\), so \\(\\pi(k_+)=\\pi(k)_+=0\\). Put \\(c=a+k_+\\geq0\\). Since \\(k\\leq k_+\\), \\(z^*z=h+k\\leq a+k_+=c\\), and \\(\\pi(c)=b\\).\n*Step 3: the approximation.* For \\(s>0\\) put \\(v_s=f_s(c)=(s+c)^{-1}c\\) and \\(x_s=z(s+c)^{-1}c^{1/2}a^{1/2}\\in A\\). With \\(d=(s+c)^{-1}-(t+c)^{-1}\\), which commutes with \\(c\\) and has \\(dc=v_s-v_t\\),\n\\[\n\\begin{gathered}\n(x_s-x_t)^*(x_s-x_t)\\\\\n=a^{1/2}c^{1/2}d\\,z^*z\\,d\\,c^{1/2}a^{1/2}\\\\\n\\leq a^{1/2}c^{1/2}dcdc^{1/2}a^{1/2}\\\\\n=\\big((v_s-v_t)a^{1/2}\\big)^*\\big((v_s-v_t)a^{1/2}\\big).\n\\end{gathered}\n\\]\nSo \\(\\|x_s-x_t\\|\\leq\\|a^{1/2}v_s-a^{1/2}v_t\\|\\). Since \\((a^{1/2})^*a^{1/2}=a\\leq c\\), Corollary 11.3(2), in its left form, gives \\(a^{1/2}v_s\\to a^{1/2}\\) as \\(s\\to0\\). So \\((x_s)\\) is Cauchy; let \\(x=\\lim_{s\\to0}x_s\\in A\\). In the same way, \\(x_s^*x_s\\leq a^{1/2}v_s^2a^{1/2}\\leq a\\), and so \\(x^*x\\leq a\\), because \\(A_+\\) is closed. Finally, by Corollary 5.4(2),\n\\[\n\\pi(x_s)=y(s+b)^{-1}b^{1/2}b^{1/2}=yf_s(b)\\longrightarrow y ,\n\\]\nby Corollary 11.3(2) again, since \\(y^*y\\leq b\\). So \\(\\pi(x)=y\\). \\(\\square\\)\n\n**Corollary 17.3** (Order intervals; sums of closed ideals).\n1. (*Order intervals.*) \\[\n\\begin{gathered}\n\\pi(\\{p\\in A:0\\leq p\\leq a\\})\\\\\n=\\{q\\in B:0\\leq q\\leq\\pi(a)\\}.\n\\end{gathered}\n\\]\n2. For closed ideals \\(\\mathfrak m\\) and \\(\\mathfrak n\\) of \\(A\\), \\(\\mathfrak m+\\mathfrak n\\) is a closed ideal and \\((\\mathfrak m+\\mathfrak n)_+=\\mathfrak m_++\\mathfrak n_+\\).\n\n**Proof.** (1) If \\(0\\leq q\\leq\\pi(a)\\), then \\(y=q^{1/2}\\) satisfies \\(y^*y=q\\leq\\pi(a)\\). The Proposition gives \\(x\\) with \\(x^*x\\leq a\\) and \\(\\pi(x)=q^{1/2}\\), and \\(p=x^*x\\) works. The other inclusion is Proposition 8.5(12).\n(2) *Closed.* Let \\(\\rho_{\\mathfrak m}:A\\to A/\\mathfrak m\\). By Corollary 15.4, applied to the restriction of \\(\\rho_{\\mathfrak m}\\) to the C\\*-algebra \\(\\mathfrak n\\), the set \\(\\rho_{\\mathfrak m}(\\mathfrak n)\\) is closed, and \\(\\mathfrak m+\\mathfrak n=\\rho_{\\mathfrak m}^{-1}(\\rho_{\\mathfrak m}(\\mathfrak n))\\) is closed. It is clearly an ideal.\n*Positive parts.* \\(\\mathfrak m_++\\mathfrak n_+\\subseteq(\\mathfrak m+\\mathfrak n)_+\\) is clear. Let \\(a\\in(\\mathfrak m+\\mathfrak n)_+\\), and let \\(\\rho:A\\to A/(\\mathfrak m\\cap\\mathfrak n)\\) be the quotient map. The sets \\(I=\\rho(\\mathfrak m)\\) and \\(J=\\rho(\\mathfrak n)\\) are closed ideals (Corollary 15.4, and \\(\\rho\\) is onto), and \\(I\\cap J=0\\): if \\(\\rho(u)=\\rho(v)\\) with \\(u\\in\\mathfrak m\\) and \\(v\\in\\mathfrak n\\), then \\(u-v\\in\\mathfrak m\\cap\\mathfrak n\\), so \\(u\\in\\mathfrak n\\), \\(u\\in\\mathfrak m\\cap\\mathfrak n\\) and \\(\\rho(u)=0\\). Write \\(\\rho(a)=i+j\\) with \\(i\\in I\\) and \\(j\\in J\\). Then \\(i-i^*=j^*-j\\in I\\cap J=0\\), so \\(i\\) and \\(j\\) are self-adjoint, and \\(ij\\) and \\(ji\\) lie in \\(I\\cap J\\), so they vanish. In the commutative C\\*-algebra \\(C^*(i,j)\\), the functions \\(\\hat i\\) and \\(\\hat j\\) have disjoint nonzero sets and their sum is \\(\\geq0\\); so \\(i,j\\geq0\\). Lift \\(i\\) to some \\(x_0\\in\\mathfrak m_+\\) (Proposition 17.1(3), for the surjection \\(\\mathfrak m\\to I\\)). Since \\(\\rho(x_0^{1/2})^*\\rho(x_0^{1/2})=i\\leq\\rho(a)\\), the Proposition gives \\(x\\in A\\) with \\(x^*x\\leq a\\) and \\(\\rho(x)=\\rho(x_0^{1/2})\\). Then \\(x-x_0^{1/2}\\in\\mathfrak m\\cap\\mathfrak n\\subseteq\\mathfrak m\\), so \\(x\\in\\mathfrak m\\) and \\(p=x^*x\\in\\mathfrak m_+\\), with \\(p\\leq a\\) and \\(\\rho(p)=i\\). The element \\(q=a-p\\geq0\\) has \\(\\rho(q)=j=\\rho(n_0)\\) for some \\(n_0\\in\\mathfrak n\\), so \\(q\\in n_0+\\mathfrak m\\cap\\mathfrak n\\subseteq\\mathfrak n\\). Thus \\(a=p+q\\) with \\(p\\in\\mathfrak m_+\\) and \\(q\\in\\mathfrak n_+\\). \\(\\square\\)\n\nThe point of the proof of (2) is to apply Proposition 17.2 to the quotient by \\(\\mathfrak m\\cap\\mathfrak n\\), so that the correction term falls into \\(\\mathfrak m\\).\n\n**Exercise 17.4** (medium; Lifting unitaries). Let \\(A\\) be unital, \\(\\mathfrak m\\) a closed ideal and \\(\\pi:A\\to A/\\mathfrak m\\). (a) If \\(\\dot u\\in U(A/\\mathfrak m)\\) and \\(\\sigma(\\dot u)\\neq\\mathbb T\\), then \\(\\dot u=\\pi(u)\\) for a unitary \\(u\\in A\\). (b) More generally, every unitary in the connected component \\(U_0(A/\\mathfrak m)\\) of \\(1\\) in \\(U(A/\\mathfrak m)\\) is the image of a unitary in \\(U_0(A)\\).\n\n*Solution.* If \\(\\mathfrak m=A\\), then \\(A/\\mathfrak m=0\\) and \\(u=1\\) works; so let \\(\\mathfrak m\\neq A\\).\n*A lemma: in a nontrivial unital C\\*-algebra \\(D\\), \\(U_0(D)\\) is the set \\(E\\) of finite products \\(\\exp(ih_1)\\cdots\\exp(ih_n)\\) with \\(h_k\\in D_h\\).* Each \\(\\exp(ih)\\) is unitary (proof of Proposition 1.5(3)), and \\(\\exp(ih)^{-1}=\\exp(-ih)\\), so \\(E\\) is a subgroup of \\(U(D)\\). It is path connected through \\(t\\mapsto\\prod_k\\exp(ith_k)\\), so \\(E\\subseteq U_0(D)\\). It is open in \\(U(D)\\): if \\(w\\in E\\), \\(v\\in U(D)\\) and \\(\\|v-w\\|<2\\), then \\(\\|w^{-1}v-1\\|=\\|v-w\\|<2\\), so every \\(\\lambda\\in\\sigma(w^{-1}v)\\) has \\(|\\lambda-1|<2\\) and \\(-1\\notin\\sigma(w^{-1}v)\\); Exercise 7.4 gives \\(w^{-1}v=\\exp(ih)\\), so \\(v\\in E\\). An open subgroup is also closed, since its complement is a union of open cosets. So \\(E\\) is open and closed in \\(U(D)\\) and contains \\(1\\), and \\(E\\supseteq U_0(D)\\). This is the unitary counterpart of the description of [the principal component](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14) of the invertible group.\n(b) Write \\(\\dot u=\\prod_k\\exp(i\\dot h_k)\\), lift each \\(\\dot h_k\\) to \\(h_k\\in A_h\\) (Proposition 17.1(1)), and put \\(u=\\prod_k\\exp(ih_k)\\in U_0(A)\\). The unital homomorphism \\(\\pi\\) is continuous, so it passes through the exponential series, and \\(\\pi(u)=\\dot u\\).\n(a) By Exercise 7.4, \\(\\dot u=\\exp(i\\dot h)\\), which lies in \\(U_0(A/\\mathfrak m)\\); apply (b).\n\n**Exercise 17.5** (medium; A unitary that does not lift). Let \\(\\bar{\\mathbb D}\\) be the closed unit disc, \\(A=C(\\bar{\\mathbb D})\\), and \\(\\mathfrak m\\) the closed ideal of functions vanishing on \\(\\mathbb T\\) (Proposition 16.1), so that \\(A/\\mathfrak m=C(\\mathbb T)\\) by restriction. The unitary \\(\\dot u(\\lambda)=\\lambda\\) of \\(C(\\mathbb T)\\) is not the image of a unitary of \\(A\\).\n\n*Solution.* Suppose \\(u\\in C(\\bar{\\mathbb D})\\), \\(|u|=1\\), and \\(u=\\dot u\\) on \\(\\mathbb T\\). By uniform continuity choose \\(N\\) with \\(|u(p)-u(p')|<1\\) whenever \\(|p-p'|\\leq1/N\\). For \\(w\\in\\bar{\\mathbb D}\\), the numbers \\(u(kw/N)/u((k-1)w/N)\\), \\(k=1,\\dots,N\\), lie in \\(\\{\\zeta\\in\\mathbb T:|\\zeta-1|<1\\}\\), where the principal argument \\(\\operatorname{Arg}\\) is continuous. Fix \\(\\theta_0\\) with \\(e^{i\\theta_0}=u(0)\\) and put\n\\[\nh(w)=\\theta_0+\\sum_{k=1}^N\\operatorname{Arg}\\frac{u(kw/N)}{u((k-1)w/N)} .\n\\]\nThen \\(h\\) is real and continuous, and the product telescopes: \\(e^{ih(w)}=u(w)\\). Restricted to \\(\\mathbb T\\), this gives \\(\\lambda=e^{ih(\\lambda)}\\), which Exercise 7.5 rules out. By Exercise 17.4, \\(\\dot u\\notin U_0(C(\\mathbb T))\\); here \\(\\sigma(\\dot u)=\\mathbb T\\).\n\n## C. Choose local units and divide a positive element\n\nA net of local units always exists, but a sequence requires additional structure. Strictly positive elements provide a single element whose spectral cutoffs approximate the whole algebra; the countability theorem states the exact equivalence. Convolution gives another instructive family: concentration of a kernel near the group identity replaces a pointwise unit, and the involution and Haar conventions must be checked.\n\nThe last question is an order decomposition. If a positive element is dominated by a sum, can it be divided among the summands? Commuting functions suggest a pointwise answer, while general C*-algebras require the asymmetric bounds proved below. The full theorem retains arbitrary families, unconditional norm convergence and the support conclusions for closed one-sided ideals.\n\n",
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      "id": "OA-FND-CF-29",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "17. Lifting through quotients",
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      "full_conditions_and_proof": "### 17. Lifting through quotients\n\n**Proposition 17.1** (Lifting self-adjoint, positive and arbitrary elements). Let \\(\\pi:A\\to B\\) be a surjective \\(*\\)-homomorphism of C\\*-algebras.\n1. Every \\(k\\in B_h\\) is \\(\\pi(h)\\) for some \\(h\\in A_h\\) with \\(\\|h\\|=\\|k\\|\\). If \\(k\\geq0\\), \\(h\\) can be chosen with \\(h\\geq0\\) and \\(\\|h\\|=\\|k\\|\\).\n2. Every \\(k\\in B\\) is \\(\\pi(x)\\) for some \\(x\\in A\\) with \\(\\|x\\|=\\|k\\|\\).\n3. \\(\\pi(A_+)=B_+\\).\n\n**Proof.** (1) Let \\(h'\\in A\\) with \\(\\pi(h')=k\\). Then \\(h''=\\frac12(h'+h'^*)\\) is self-adjoint with \\(\\pi(h'')=k\\). Put \\(c=\\|k\\|\\) and \\(\\tau(t)=\\max(-c,\\min(t,c))\\), a continuous function with \\(\\tau(0)=0\\) and \\(|\\tau|\\leq c\\). Let \\(h=\\tau(h'')\\in A_h\\). Then \\(\\|h\\|\\leq c\\), and by Corollary 5.4(3), \\(\\pi(h)=\\tau(k)=k\\), since \\(\\sigma'(k)\\subseteq[-c,c]\\), where \\(\\tau\\) is the identity. So \\(c=\\|\\pi(h)\\|\\leq\\|h\\|\\leq c\\). If \\(k\\geq0\\), use \\(\\tau_+(t)=\\min(\\max(t,0),c)\\) instead; then \\(h\\geq0\\).\n(2) Let \\(x'\\in A\\) with \\(\\pi(x')=k\\), and \\(c=\\|k\\|\\); if \\(c=0\\) take \\(x=0\\). Let \\(g(t)=\\min(1,ct^{-1/2})\\) for \\(t>0\\) and \\(g(0)=1\\), a continuous function on \\([0,\\infty)\\) that equals \\(1\\) on \\([0,c^2]\\). Put \\(x=x'g(x'^*x')\\), which lies in \\(A\\) because \\(A\\) is an ideal of \\(\\widetilde A\\). Then\n\\[\n\\begin{gathered}\n\\|x\\|^2\\\\\n=\\|g(x'^*x')\\,x'^*x'\\,g(x'^*x')\\|\\\\\n=\\max_{t\\in\\sigma'(x'^*x')}tg(t)^2\\\\\n\\leq c^2 ,\n\\end{gathered}\n\\]\nsince \\(tg(t)^2=\\min(t,c^2)\\). By Corollary 5.4(2), \\(\\pi(x)=k\\,g(k^*k)=k\\), because \\(\\sigma'(k^*k)\\subseteq[0,c^2]\\), where \\(g=1\\). So \\(\\|x\\|=c\\).\n(3) \\(\\pi(A_+)\\subseteq B_+\\) by Proposition 8.5(12), and (1) gives the rest. \\(\\square\\)\n\nA second proof of (1) uses Tietze's extension theorem. The C\\*-algebra \\(C^*(h'')\\) is abelian, and restricting \\(\\pi\\) to it reduces (1) to the abelian case. There, by Proposition 16.1, the quotient map is restriction \\(C_0(\\Omega)\\to C_0(\\Gamma)\\), and a bounded extension exists by [Tietze's theorem](stone-weierstrass-c0.md#oa-fnd-sw-18) on the one-point compactification, which is compact Hausdorff but need not be metrizable. Neither proof needs separability.\n\n**Proposition 17.2** (Lifting \\(\\{y^*y\\leq\\pi(a)\\}\\)). Let \\(\\pi:A\\to B\\) be a surjective \\(*\\)-homomorphism of C\\*-algebras and \\(a\\in A_+\\). Then\n\\[\n\\begin{gathered}\n\\pi\\big(\\{x\\in A: \\ x^*x\\leq a\\}\\big)\\\\\n=\\{y\\in B: \\ y^*y\\leq\\pi(a)\\}.\n\\end{gathered}\n\\]\n\n**Proof.** Put \\(b=\\pi(a)\\). If \\(x^*x\\leq a\\), then \\(\\pi(x)^*\\pi(x)=\\pi(x^*x)\\leq b\\) (Proposition 8.5(12)). Conversely let \\(y^*y\\leq b\\).\n*Step 1: a self-adjoint \\(h\\leq a\\) with \\(\\pi(h)=y^*y\\).* Choose \\(h_0\\in A_h\\) with \\(\\pi(h_0)=y^*y\\) (Proposition 17.1(1)). Then \\(\\pi(a-h_0)=b-y^*y\\geq0\\). By Corollary 5.4(3), \\(\\pi((a-h_0)_\\pm)=(b-y^*y)_\\pm\\), which is \\(b-y^*y\\) for the sign \\(+\\) and \\(0\\) for the sign \\(-\\). Put \\(h=a-(a-h_0)_+\\). Then \\(h\\leq a\\) and \\(\\pi(h)=b-(b-y^*y)=y^*y\\).\n*Step 2: a dominating element.* Choose \\(z\\in A\\) with \\(\\pi(z)=y\\), and put \\(k=z^*z-h\\in A_h\\). Then \\(\\pi(k)=0\\), so \\(\\pi(k_+)=\\pi(k)_+=0\\). Put \\(c=a+k_+\\geq0\\). Since \\(k\\leq k_+\\), \\(z^*z=h+k\\leq a+k_+=c\\), and \\(\\pi(c)=b\\).\n*Step 3: the approximation.* For \\(s>0\\) put \\(v_s=f_s(c)=(s+c)^{-1}c\\) and \\(x_s=z(s+c)^{-1}c^{1/2}a^{1/2}\\in A\\). With \\(d=(s+c)^{-1}-(t+c)^{-1}\\), which commutes with \\(c\\) and has \\(dc=v_s-v_t\\),\n\\[\n\\begin{gathered}\n(x_s-x_t)^*(x_s-x_t)\\\\\n=a^{1/2}c^{1/2}d\\,z^*z\\,d\\,c^{1/2}a^{1/2}\\\\\n\\leq a^{1/2}c^{1/2}dcdc^{1/2}a^{1/2}\\\\\n=\\big((v_s-v_t)a^{1/2}\\big)^*\\big((v_s-v_t)a^{1/2}\\big).\n\\end{gathered}\n\\]\nSo \\(\\|x_s-x_t\\|\\leq\\|a^{1/2}v_s-a^{1/2}v_t\\|\\). Since \\((a^{1/2})^*a^{1/2}=a\\leq c\\), Corollary 11.3(2), in its left form, gives \\(a^{1/2}v_s\\to a^{1/2}\\) as \\(s\\to0\\). So \\((x_s)\\) is Cauchy; let \\(x=\\lim_{s\\to0}x_s\\in A\\). In the same way, \\(x_s^*x_s\\leq a^{1/2}v_s^2a^{1/2}\\leq a\\), and so \\(x^*x\\leq a\\), because \\(A_+\\) is closed. Finally, by Corollary 5.4(2),\n\\[\n\\pi(x_s)=y(s+b)^{-1}b^{1/2}b^{1/2}=yf_s(b)\\longrightarrow y ,\n\\]\nby Corollary 11.3(2) again, since \\(y^*y\\leq b\\). So \\(\\pi(x)=y\\). \\(\\square\\)\n\n**Corollary 17.3** (Order intervals; sums of closed ideals).\n1. (*Order intervals.*) \\[\n\\begin{gathered}\n\\pi(\\{p\\in A:0\\leq p\\leq a\\})\\\\\n=\\{q\\in B:0\\leq q\\leq\\pi(a)\\}.\n\\end{gathered}\n\\]\n2. For closed ideals \\(\\mathfrak m\\) and \\(\\mathfrak n\\) of \\(A\\), \\(\\mathfrak m+\\mathfrak n\\) is a closed ideal and \\((\\mathfrak m+\\mathfrak n)_+=\\mathfrak m_++\\mathfrak n_+\\).\n\n**Proof.** (1) If \\(0\\leq q\\leq\\pi(a)\\), then \\(y=q^{1/2}\\) satisfies \\(y^*y=q\\leq\\pi(a)\\). The Proposition gives \\(x\\) with \\(x^*x\\leq a\\) and \\(\\pi(x)=q^{1/2}\\), and \\(p=x^*x\\) works. The other inclusion is Proposition 8.5(12).\n(2) *Closed.* Let \\(\\rho_{\\mathfrak m}:A\\to A/\\mathfrak m\\). By Corollary 15.4, applied to the restriction of \\(\\rho_{\\mathfrak m}\\) to the C\\*-algebra \\(\\mathfrak n\\), the set \\(\\rho_{\\mathfrak m}(\\mathfrak n)\\) is closed, and \\(\\mathfrak m+\\mathfrak n=\\rho_{\\mathfrak m}^{-1}(\\rho_{\\mathfrak m}(\\mathfrak n))\\) is closed. It is clearly an ideal.\n*Positive parts.* \\(\\mathfrak m_++\\mathfrak n_+\\subseteq(\\mathfrak m+\\mathfrak n)_+\\) is clear. Let \\(a\\in(\\mathfrak m+\\mathfrak n)_+\\), and let \\(\\rho:A\\to A/(\\mathfrak m\\cap\\mathfrak n)\\) be the quotient map. The sets \\(I=\\rho(\\mathfrak m)\\) and \\(J=\\rho(\\mathfrak n)\\) are closed ideals (Corollary 15.4, and \\(\\rho\\) is onto), and \\(I\\cap J=0\\): if \\(\\rho(u)=\\rho(v)\\) with \\(u\\in\\mathfrak m\\) and \\(v\\in\\mathfrak n\\), then \\(u-v\\in\\mathfrak m\\cap\\mathfrak n\\), so \\(u\\in\\mathfrak n\\), \\(u\\in\\mathfrak m\\cap\\mathfrak n\\) and \\(\\rho(u)=0\\). Write \\(\\rho(a)=i+j\\) with \\(i\\in I\\) and \\(j\\in J\\). Then \\(i-i^*=j^*-j\\in I\\cap J=0\\), so \\(i\\) and \\(j\\) are self-adjoint, and \\(ij\\) and \\(ji\\) lie in \\(I\\cap J\\), so they vanish. In the commutative C\\*-algebra \\(C^*(i,j)\\), the functions \\(\\hat i\\) and \\(\\hat j\\) have disjoint nonzero sets and their sum is \\(\\geq0\\); so \\(i,j\\geq0\\). Lift \\(i\\) to some \\(x_0\\in\\mathfrak m_+\\) (Proposition 17.1(3), for the surjection \\(\\mathfrak m\\to I\\)). Since \\(\\rho(x_0^{1/2})^*\\rho(x_0^{1/2})=i\\leq\\rho(a)\\), the Proposition gives \\(x\\in A\\) with \\(x^*x\\leq a\\) and \\(\\rho(x)=\\rho(x_0^{1/2})\\). Then \\(x-x_0^{1/2}\\in\\mathfrak m\\cap\\mathfrak n\\subseteq\\mathfrak m\\), so \\(x\\in\\mathfrak m\\) and \\(p=x^*x\\in\\mathfrak m_+\\), with \\(p\\leq a\\) and \\(\\rho(p)=i\\). The element \\(q=a-p\\geq0\\) has \\(\\rho(q)=j=\\rho(n_0)\\) for some \\(n_0\\in\\mathfrak n\\), so \\(q\\in n_0+\\mathfrak m\\cap\\mathfrak n\\subseteq\\mathfrak n\\). Thus \\(a=p+q\\) with \\(p\\in\\mathfrak m_+\\) and \\(q\\in\\mathfrak n_+\\). \\(\\square\\)\n\nThe point of the proof of (2) is to apply Proposition 17.2 to the quotient by \\(\\mathfrak m\\cap\\mathfrak n\\), so that the correction term falls into \\(\\mathfrak m\\).\n\n**Exercise 17.4** (medium; Lifting unitaries). Let \\(A\\) be unital, \\(\\mathfrak m\\) a closed ideal and \\(\\pi:A\\to A/\\mathfrak m\\). (a) If \\(\\dot u\\in U(A/\\mathfrak m)\\) and \\(\\sigma(\\dot u)\\neq\\mathbb T\\), then \\(\\dot u=\\pi(u)\\) for a unitary \\(u\\in A\\). (b) More generally, every unitary in the connected component \\(U_0(A/\\mathfrak m)\\) of \\(1\\) in \\(U(A/\\mathfrak m)\\) is the image of a unitary in \\(U_0(A)\\).\n\n*Solution.* If \\(\\mathfrak m=A\\), then \\(A/\\mathfrak m=0\\) and \\(u=1\\) works; so let \\(\\mathfrak m\\neq A\\).\n*A lemma: in a nontrivial unital C\\*-algebra \\(D\\), \\(U_0(D)\\) is the set \\(E\\) of finite products \\(\\exp(ih_1)\\cdots\\exp(ih_n)\\) with \\(h_k\\in D_h\\).* Each \\(\\exp(ih)\\) is unitary (proof of Proposition 1.5(3)), and \\(\\exp(ih)^{-1}=\\exp(-ih)\\), so \\(E\\) is a subgroup of \\(U(D)\\). It is path connected through \\(t\\mapsto\\prod_k\\exp(ith_k)\\), so \\(E\\subseteq U_0(D)\\). It is open in \\(U(D)\\): if \\(w\\in E\\), \\(v\\in U(D)\\) and \\(\\|v-w\\|<2\\), then \\(\\|w^{-1}v-1\\|=\\|v-w\\|<2\\), so every \\(\\lambda\\in\\sigma(w^{-1}v)\\) has \\(|\\lambda-1|<2\\) and \\(-1\\notin\\sigma(w^{-1}v)\\); Exercise 7.4 gives \\(w^{-1}v=\\exp(ih)\\), so \\(v\\in E\\). An open subgroup is also closed, since its complement is a union of open cosets. So \\(E\\) is open and closed in \\(U(D)\\) and contains \\(1\\), and \\(E\\supseteq U_0(D)\\). This is the unitary counterpart of the description of [the principal component](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14) of the invertible group.\n(b) Write \\(\\dot u=\\prod_k\\exp(i\\dot h_k)\\), lift each \\(\\dot h_k\\) to \\(h_k\\in A_h\\) (Proposition 17.1(1)), and put \\(u=\\prod_k\\exp(ih_k)\\in U_0(A)\\). The unital homomorphism \\(\\pi\\) is continuous, so it passes through the exponential series, and \\(\\pi(u)=\\dot u\\).\n(a) By Exercise 7.4, \\(\\dot u=\\exp(i\\dot h)\\), which lies in \\(U_0(A/\\mathfrak m)\\); apply (b).\n\n**Exercise 17.5** (medium; A unitary that does not lift). Let \\(\\bar{\\mathbb D}\\) be the closed unit disc, \\(A=C(\\bar{\\mathbb D})\\), and \\(\\mathfrak m\\) the closed ideal of functions vanishing on \\(\\mathbb T\\) (Proposition 16.1), so that \\(A/\\mathfrak m=C(\\mathbb T)\\) by restriction. The unitary \\(\\dot u(\\lambda)=\\lambda\\) of \\(C(\\mathbb T)\\) is not the image of a unitary of \\(A\\).\n\n*Solution.* Suppose \\(u\\in C(\\bar{\\mathbb D})\\), \\(|u|=1\\), and \\(u=\\dot u\\) on \\(\\mathbb T\\). By uniform continuity choose \\(N\\) with \\(|u(p)-u(p')|<1\\) whenever \\(|p-p'|\\leq1/N\\). For \\(w\\in\\bar{\\mathbb D}\\), the numbers \\(u(kw/N)/u((k-1)w/N)\\), \\(k=1,\\dots,N\\), lie in \\(\\{\\zeta\\in\\mathbb T:|\\zeta-1|<1\\}\\), where the principal argument \\(\\operatorname{Arg}\\) is continuous. Fix \\(\\theta_0\\) with \\(e^{i\\theta_0}=u(0)\\) and put\n\\[\nh(w)=\\theta_0+\\sum_{k=1}^N\\operatorname{Arg}\\frac{u(kw/N)}{u((k-1)w/N)} .\n\\]\nThen \\(h\\) is real and continuous, and the product telescopes: \\(e^{ih(w)}=u(w)\\). Restricted to \\(\\mathbb T\\), this gives \\(\\lambda=e^{ih(\\lambda)}\\), which Exercise 7.5 rules out. By Exercise 17.4, \\(\\dot u\\notin U_0(C(\\mathbb T))\\); here \\(\\sigma(\\dot u)=\\mathbb T\\).\n\n## C. Choose local units and divide a positive element\n\nA net of local units always exists, but a sequence requires additional structure. Strictly positive elements provide a single element whose spectral cutoffs approximate the whole algebra; the countability theorem states the exact equivalence. Convolution gives another instructive family: concentration of a kernel near the group identity replaces a pointwise unit, and the involution and Haar conventions must be checked.\n\nThe last question is an order decomposition. If a positive element is dominated by a sum, can it be divided among the summands? Commuting functions suggest a pointwise answer, while general C*-algebras require the asymmetric bounds proved below. The full theorem retains arbitrary families, unconditional norm convergence and the support conclusions for closed one-sided ideals.\n\n",
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      "id": "OA-FND-CF-31",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "Fuglede’s theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
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      "full_conditions_and_proof": "#### Fuglede’s theorem\n\nThe continuous-calculus proof in Theorem 5.1(6) assumes commutation with both \\(x\\) and \\(x^*\\). Commutation with \\(x\\) alone implies the second condition, as follows. This theorem is used later in the [joint functional calculus, Sections 3.2–3.3 of the Kaplansky lesson](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-03).\n\n**Theorem (Fuglede).** If \\(x\\) is normal in a C*-algebra \\(A\\) and \\(xy=yx\\), then \\(x^*y=yx^*\\).\n\n**Proof.** Work in the forced unitization \\(\\widetilde A\\). The [exponential laws of Proposition 7.1 of the Banach-algebra lesson](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14) give\n\\[\nF(z)=e^{zx^*}y e^{-zx^*}\\qquad(z\\in\\mathbb C).\n\\]\nBecause \\(y\\) commutes with \\(x\\), it commutes with the exponential series \\(e^{\\bar z x}\\). Inserting \\(y=e^{-\\bar z x}y e^{\\bar z x}\\), and using normality to combine commuting exponentials, gives\n\\[\nF(z)=U(z)yU(z)^{-1},\n\\qquad U(z)=e^{zx^*-\\bar z x}.\n\\]\nThe exponent \\(X=zx^*-\\bar z x\\) satisfies \\(X^*=-X\\). Applying the adjoint to the norm-convergent exponential series gives \\(U(z)^*=e^{-X}=U(z)^{-1}\\), so \\(U(z)\\) is unitary. The C*-identity gives \\(\\|U(z)\\|=\\|U(z)^*\\|=1\\). Submultiplicativity, applied to this conjugation and its inverse, therefore gives \\(\\|F(z)\\|=\\|y\\|\\).\n\nThe absolutely convergent product series is\n\\[\n\\begin{gathered}\nF(z)\\\\\n=\\sum_{m\\geq0}z^m c_m,\n\\\\\nc_m\\\\\n=\\sum_{j+k=m}\\frac{(-1)^k}{j!\\,k!}(x^*)^j y(x^*)^k .\n\\end{gathered}\n\\]\nFor every \\(R>0\\), \\(\\sum_mR^m\\|c_m\\|\\leq\\|y\\|e^{2R\\|x\\|}\\). Thus, for each bounded linear functional \\(\\varphi\\) on \\(\\widetilde A\\), the function \\(\\varphi(F(z))=\\sum_mz^m\\varphi(c_m)\\) is entire by [Lemma 3.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-03). It is bounded by \\(\\|\\varphi\\|\\|y\\|\\), hence constant by [Liouville’s theorem, Corollary 3.3 there](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-03). Its coefficient of \\(z\\) is zero: \\(\\varphi(c_1)=0\\), where \\(c_1=x^*y-yx^*\\). [Hahn–Banach separation](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-02) now gives \\(c_1=0\\). \\(\\square\\)\n\nThis is the exponential and Liouville proof of the operator version in the Kaplansky lesson, using bounded linear functionals in place of vector coefficients. It works directly in every C*-algebra and requires no representation theorem.\n\n**Theorem 5.3** (The calculus without an identity). Let \\(A\\) be a C\\*-algebra, \\(x\\in A\\) normal and \\(S'=\\sigma'_A(x)\\), a compact set containing \\(0\\). For \\(f\\in C(S')\\), let \\(f(x)\\in\\widetilde A\\) be given by Theorem 5.1 in \\(\\widetilde A\\). Identify \\(C_0(S'\\setminus\\{0\\})\\) with \\(\\{f\\in C(S'):f(0)=0\\}\\).\n1. \\(q(f(x))=f(0)\\). So \\(f(x)\\in A\\) if and only if \\(f(0)=0\\).\n2. \\(f\\mapsto f(x)\\) is an isometric \\(*\\)-isomorphism of \\(C_0(S'\\setminus\\{0\\})\\) onto \\(C^*(x)\\).\n3. If \\(A\\) is unital and \\(f(0)=0\\), then \\(f(x)\\) equals the unital calculus \\((f|_{\\sigma_A(x)})(x)\\) of Theorem 5.1 in \\(A\\). So the two calculi never disagree.\n4. If \\(D\\subseteq A\\) is a commutative C\\*-subalgebra containing \\(x\\), \\(\\chi\\in\\operatorname{Ch}(D)\\) and \\(f(0)=0\\), then \\(\\chi(f(x))=f(\\chi(x))\\).\n\n**Proof.** (1) \\(q\\) is a character of the commutative unital C\\*-subalgebra \\(C^*(1,x)\\) of \\(\\widetilde A\\), and \\(q(x)=0\\). Apply Theorem 5.1(8). The kernel of \\(q\\) is \\(A\\).\n(2) The map is an isometric \\(*\\)-homomorphism into \\(A\\) by (1) and Theorem 5.1. The polynomials \\(p(\\lambda,\\bar\\lambda)\\) without constant term form a self-adjoint subalgebra of \\(C(S')\\) that separates points (through \\(\\iota\\)) and whose common zero set is \\(\\{0\\}\\). By the compact form of the [Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09), they are dense in \\(\\{f:f(0)=0\\}\\). Their images, the polynomials in \\(x\\) and \\(x^*\\) without constant term, are dense in \\(C^*(x)\\). The image of an isometry defined on a complete space is closed, so it is exactly \\(C^*(x)\\).\n(3) By the Conventions, \\(\\widetilde A\\cong A\\oplus\\mathbb C\\) with \\(x\\mapsto(x,0)\\), and \\(S'=\\sigma_A(x)\\cup\\{0\\}\\). The map \\(f\\mapsto\\big((f|_{\\sigma_A(x)})(x),\\,f(0)\\big)\\) is a unital \\(*\\)-homomorphism \\(C(S')\\to A\\oplus\\mathbb C\\) that sends \\(\\iota\\) to \\((x,0)\\). By uniqueness it is the calculus of \\((x,0)\\). For \\(f(0)=0\\) its second coordinate is \\(0\\).\n(4) By (2), \\(\\chi\\circ(f\\mapsto f(x))\\) is a \\(*\\)-homomorphism from \\(C_0(S'\\setminus\\{0\\})\\) to \\(\\mathbb C\\). If it is zero, then \\(\\chi(x)=0\\), and \\(\\chi(f(x))=0=f(0)=f(\\chi(x))\\). Otherwise it is a character, hence evaluation at a point \\(s\\) (Proposition 2.2(1)), and \\(s=\\chi(x)\\). \\(\\square\\)\n\n**Corollary 5.4** (The calculus commutes with \\(*\\)-homomorphisms). Let \\(\\pi:A\\to B\\) be a \\(*\\)-homomorphism of C\\*-algebras, and \\(a\\in A\\) normal. Let \\(\\tilde\\pi:\\widetilde A\\to\\widetilde B\\), \\(\\tilde\\pi(x+\\lambda)=\\pi(x)+\\lambda\\), be its unital extension. In each statement, \\(f(\\pi(a))\\) is the calculus of the restriction of \\(f\\) to the smaller spectrum.\n1. \\(\\sigma'_B(\\pi(a))\\subseteq\\sigma'_A(a)\\).\n2. \\(\\tilde\\pi(f(a))=f(\\pi(a))\\) for every \\(f\\in C(\\sigma'_A(a))\\).\n3. \\(\\pi(f(a))=f(\\pi(a))\\) for every \\(f\\in C(\\sigma'_A(a))\\) with \\(f(0)=0\\).\n4. If \\(A\\) and \\(B\\) are unital and \\(\\pi(1)=1\\), then \\(\\sigma_B(\\pi(a))\\subseteq\\sigma_A(a)\\), and \\(\\pi(f(a))=f(\\pi(a))\\) for every \\(f\\in C(\\sigma_A(a))\\).\n\n**Proof.** (1) is part of (4.1). (4) The maps \\(f\\mapsto\\pi(f(a))\\) and \\(f\\mapsto f(\\pi(a))\\) are unital \\(*\\)-homomorphisms from \\(C(\\sigma_A(a))\\) into \\(B\\). The first is a composition. The second is restriction to \\(\\sigma_B(\\pi(a))\\), which lies in \\(\\sigma_A(a)\\) because unital homomorphisms map invertible elements to invertible elements, followed by the calculus of \\(\\pi(a)\\). They agree at \\(\\iota\\), hence on polynomials in \\(\\iota\\) and \\(\\bar\\iota\\), which are dense ([density results](stone-weierstrass-c0.md#oa-fnd-sw-17)), and both are contractive by Theorem 4.2. (2) is (4) for the unital \\(*\\)-homomorphism \\(\\tilde\\pi\\) of C\\*-algebras. (3) follows from (2) and Theorem 5.3(1): \\(f(a)\\in A\\), and \\(\\tilde\\pi=\\pi\\) on \\(A\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-GN-01",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "1. Representations",
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      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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      "anchor": "oa-fnd-gn-01",
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      "full_conditions_and_proof": "### 1. Representations\n\nIn this section \\(A\\) is a \\(*\\)-algebra. No norm on \\(A\\) is needed.\n\n**Definition 1.1.** A *representation* of \\(A\\) on a Hilbert space \\(H\\) is a \\(*\\)-homomorphism \\(\\pi:A\\to B(H)\\), that is, a linear map with \\(\\pi(xy)=\\pi(x)\\pi(y)\\) and \\(\\pi(x^*)=\\pi(x)^*\\). We write \\((\\pi,H)\\), or just \\(\\pi\\). A bounded operator \\(T:H_1\\to H_2\\) *intertwines* representations \\((\\pi_1,H_1)\\) and \\((\\pi_2,H_2)\\) if \\(T\\pi_1(x)=\\pi_2(x)T\\) for all \\(x\\in A\\). The two are *unitarily equivalent*, written \\(\\pi_1\\cong\\pi_2\\), if some unitary operator intertwines them. A representation is *faithful* if it is injective. The *zero representation* on \\(H\\) sends every element to \\(0\\).\n\nWhen \\(A\\) is an involutive Banach algebra, every representation satisfies \\(\\|\\pi(x)\\|\\le\\|x\\|\\), and no continuity has to be assumed ([\\(*\\)-homomorphisms are contractive](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-12)). The proof there uses that the involution is isometric.\n\n**Definition 1.2.** Let \\((\\pi,H)\\) be a representation.\n1. A closed subspace \\(M\\subseteq H\\) is *invariant* if \\(\\pi(x)M\\subseteq M\\) for all \\(x\\in A\\). Then \\(\\pi_M(x)=\\pi(x)|_M\\) defines a representation on \\(M\\), a *subrepresentation* of \\(\\pi\\).\n2. The *essential subspace* \\(E_\\pi\\) is the closed linear span of all vectors \\(\\pi(x)\\xi\\) with \\(x\\in A\\) and \\(\\xi\\in H\\). The *null space* is \\(N_\\pi=\\{\\xi\\in H:\\pi(x)\\xi=0\\text{ for all }x\\in A\\}\\). The representation is *nondegenerate* if \\(E_\\pi=H\\).\n3. A vector \\(\\xi\\in H\\) is *cyclic* if the subspace \\(\\pi(A)\\xi=\\{\\pi(x)\\xi:x\\in A\\}\\) is dense in \\(H\\). A representation with a cyclic vector is *cyclic*.\n4. Let \\((\\pi_i,H_i)_{i\\in I}\\) be representations with \\(\\sup_i\\|\\pi_i(x)\\|<\\infty\\) for every \\(x\\); for involutive Banach algebras this holds automatically. Their *direct sum* \\(\\bigoplus_i\\pi_i\\) acts on the Hilbert space direct sum \\(\\bigoplus_iH_i\\) by \\((\\xi_i)_i\\mapsto(\\pi_i(x)\\xi_i)_i\\). Each such operator has norm at most \\(\\sup_i\\|\\pi_i(x)\\|\\), and the rules of a representation hold coordinate by coordinate. A vector \\((\\xi_i)_i\\) is killed by every \\(\\bigoplus_i\\pi_i(x)\\) exactly when each \\(\\xi_i\\) is killed by every \\(\\pi_i(x)\\), so the null space of the direct sum is the direct sum of the null spaces, and by Proposition 1.4(2) below the direct sum is nondegenerate exactly when every summand is.\n\n**Lemma 1.3** (Invariant subspaces and the commutant). Let \\(\\mathcal S\\subseteq B(H)\\) be a self-adjoint set of operators, that is, \\(T^*\\in\\mathcal S\\) whenever \\(T\\in\\mathcal S\\). Let \\(M\\) be a closed subspace with projection \\(p\\). The following are equivalent:\n- (a) \\(TM\\subseteq M\\) for every \\(T\\in\\mathcal S\\);\n- (b) \\(TM^\\perp\\subseteq M^\\perp\\) for every \\(T\\in\\mathcal S\\);\n- (c) \\(p\\in\\mathcal S'\\).\n\nIn particular, if \\(M\\) is invariant for a representation \\(\\pi\\), so is \\(M^\\perp\\), and the unitary \\(M\\oplus M^\\perp\\to H\\), \\((\\xi,\\eta)\\mapsto\\xi+\\eta\\), intertwines \\(\\pi_M\\oplus\\pi_{M^\\perp}\\) with \\(\\pi\\).\n\n**Proof.** (a) ⇒ (b). Let \\(\\eta\\in M^\\perp\\), \\(\\xi\\in M\\) and \\(T\\in\\mathcal S\\). Since \\(T^*\\in\\mathcal S\\), we get \\(\\langle T\\eta,\\xi\\rangle=\\langle\\eta,T^*\\xi\\rangle=0\\). So \\(T\\eta\\in M^\\perp\\).\n(b) ⇒ (a). Apply the same argument to \\(M^\\perp\\) and use \\(M^{\\perp\\perp}=M\\).\n(a) and (b) ⇒ (c). Condition (a) says \\(pTp=Tp\\), and (b) says \\((1-p)T(1-p)=T(1-p)\\), which simplifies to \\(pTp=pT\\). Hence \\(Tp=pT\\).\n(c) ⇒ (a). If \\(\\xi\\in M\\), then \\(T\\xi=Tp\\xi=pT\\xi\\in M\\).\nThe last statement applies this to \\(\\mathcal S=\\pi(A)\\), which is self-adjoint because \\(\\pi(x)^*=\\pi(x^*)\\). \\(\\square\\)\n\n**Proposition 1.4** (The nondegenerate part). Let \\((\\pi,H)\\) be a representation.\n1. \\(N_\\pi=E_\\pi^\\perp\\). Both subspaces are invariant, \\(\\pi\\) is unitarily equivalent to \\(\\pi_{E_\\pi}\\oplus0\\), where \\(0\\) is the zero representation on \\(N_\\pi\\), and \\(\\pi_{E_\\pi}\\) is nondegenerate.\n2. \\(\\pi\\) is nondegenerate exactly when \\(N_\\pi=\\{0\\}\\), that is, when every nonzero \\(\\xi\\) has \\(\\pi(x)\\xi\\ne0\\) for some \\(x\\).\n3. If \\(\\pi\\) is nondegenerate, then every \\(\\xi\\in H\\) lies in the closure of \\(\\pi(A)\\xi\\).\n4. Suppose that \\((u_i)\\) is a left approximate identity of the involutive Banach algebra \\(A\\), bounded by \\(\\gamma\\). Then \\(\\pi(u_i)\\) converges strongly to the projection onto \\(E_\\pi\\). So nondegeneracy of \\(\\pi\\) amounts to \\(\\pi(u_i)\\xi\\to\\xi\\) for every \\(\\xi\\in H\\).\n\n**Proof.** (1) A vector \\(\\eta\\) is orthogonal to every \\(\\pi(x)\\xi\\) exactly when \\(\\langle\\pi(x^*)\\eta,\\xi\\rangle=0\\) for all \\(x\\) and \\(\\xi\\), that is, when \\(\\pi(x^*)\\eta=0\\) for all \\(x\\). Since \\(x^*\\) runs through all of \\(A\\), this says \\(\\eta\\in N_\\pi\\). The subspace \\(E_\\pi\\) is invariant because \\(\\pi(y)\\pi(x)\\xi=\\pi(yx)\\xi\\), and \\(N_\\pi\\) is invariant because \\(\\pi(A)\\) maps it to \\(0\\). Lemma 1.3 gives the decomposition. The null space of \\(\\pi_{E_\\pi}\\) is \\(N_\\pi\\cap E_\\pi=\\{0\\}\\), so \\(\\pi_{E_\\pi}\\) is nondegenerate by (2), applied to \\(\\pi_{E_\\pi}\\).\n(2) This is (1).\n(3) The closure \\(M\\) of \\(\\pi(A)\\xi\\) is invariant; let \\(p\\) be its projection. By Lemma 1.3, \\(p\\) commutes with \\(\\pi(A)\\), so \\(\\pi(x)(1-p)\\xi=(1-p)\\pi(x)\\xi=0\\) for every \\(x\\), because \\(\\pi(x)\\xi\\in M\\). Thus \\((1-p)\\xi\\in N_\\pi=\\{0\\}\\), and \\(\\xi=p\\xi\\in M\\).\n(4) Representations of involutive Banach algebras are contractive, so \\(\\|\\pi(u_i)\\|\\le\\gamma\\). For a vector \\(\\pi(x)\\xi\\) we have \\(\\|\\pi(u_i)\\pi(x)\\xi-\\pi(x)\\xi\\|\\le\\|u_ix-x\\|\\,\\|\\xi\\|\\to0\\). By linearity, \\(\\pi(u_i)\\eta\\to\\eta\\) for every \\(\\eta\\) in the linear span of such vectors. This span is dense in \\(E_\\pi\\), and the operators \\(\\pi(u_i)\\) are uniformly bounded, so the convergence extends to all \\(\\eta\\in E_\\pi\\): given \\(\\varepsilon>0\\) and \\(\\eta'\\) in the span with \\(\\|\\eta-\\eta'\\|<\\varepsilon\\), we get \\(\\|\\pi(u_i)\\eta-\\eta\\|\\le(\\gamma+1)\\varepsilon+\\|\\pi(u_i)\\eta'-\\eta'\\|\\). On \\(N_\\pi\\) all \\(\\pi(u_i)\\) vanish. \\(\\square\\)\n\n**Proposition 1.5** (Decomposition into cyclic representations). Let \\((\\pi,H)\\) be nondegenerate. There is a family \\((\\xi_j)_{j\\in J}\\) of nonzero vectors such that the closures \\(H_j\\) of \\(\\pi(A)\\xi_j\\) are pairwise orthogonal and \\(H=\\bigoplus_jH_j\\). Each \\(H_j\\) is invariant and \\(\\xi_j\\) is a cyclic vector for \\(\\pi_{H_j}\\), so \\(\\pi\\cong\\bigoplus_j\\pi_{H_j}\\) splits into cyclic summands. If \\(H\\) is separable, \\(J\\) is countable. An arbitrary representation is the direct sum of a nondegenerate one, which decomposes in this way, and a zero representation.\n\n**Proof.** Let \\(\\mathcal F\\) be the collection of all sets \\(F\\) of nonzero vectors such that the closures of \\(\\pi(A)\\xi\\) and \\(\\pi(A)\\eta\\) are orthogonal for any two distinct \\(\\xi,\\eta\\in F\\). Order \\(\\mathcal F\\) by inclusion. The union of a chain in \\(\\mathcal F\\) lies in \\(\\mathcal F\\), because the condition concerns two vectors at a time. By Zorn's lemma there is a maximal \\(F\\in\\mathcal F\\). For \\(\\xi\\in F\\) let \\(H_\\xi\\) be the closure of \\(\\pi(A)\\xi\\); it is invariant, and \\(\\xi\\in H_\\xi\\) by Proposition 1.4(3). Let \\(K\\) be the closed linear span of the spaces \\(H_\\xi\\), \\(\\xi\\in F\\). It is invariant, so \\(K^\\perp\\) is invariant (Lemma 1.3). Suppose \\(K^\\perp\\) contains some \\(\\eta\\ne0\\). The closure of \\(\\pi(A)\\eta\\) lies in \\(K^\\perp\\), so it is orthogonal to every \\(H_\\xi\\). Moreover \\(\\eta\\notin F\\), because every element of \\(F\\) lies in \\(K\\). So \\(F\\cup\\{\\eta\\}\\in\\mathcal F\\), contradicting maximality. Hence \\(K=H=\\bigoplus_{\\xi\\in F}H_\\xi\\). The null space of \\(\\pi_{H_\\xi}\\) lies in \\(N_\\pi=\\{0\\}\\), so \\(\\pi_{H_\\xi}\\) is nondegenerate and \\(\\xi\\) is cyclic for it. Lemma 1.3 gives the unitary equivalence. If \\(H\\) is separable, unit vectors chosen one in each \\(H_\\xi\\) form an orthonormal family, which is countable. The last sentence is Proposition 1.4(1). \\(\\square\\)\n\n**Examples 1.6.**\n1. For \\(H\\ne\\{0\\}\\), the identity representation of \\(B(H)\\) on \\(H\\) is nondegenerate, and every nonzero vector \\(\\xi\\) is cyclic, since \\(\\theta_{\\eta,\\xi}\\xi=\\|\\xi\\|^2\\eta\\) for every \\(\\eta\\).\n2. Let \\(X\\) be a locally compact Hausdorff space, and let \\(C_0(X)\\) act on \\(\\ell^2(X)\\) by \\((\\pi(f)\\xi)(x)=f(x)\\xi(x)\\). Each line \\(\\mathbb C\\delta_x\\) is invariant, and \\(\\pi\\) is the direct sum of the one-dimensional representations \\(f\\mapsto f(x)\\). It is nondegenerate, because by Urysohn's lemma each \\(x\\) has some \\(f\\) with \\(f(x)\\ne0\\), and it is faithful, because \\(\\pi(f)\\delta_x=f(x)\\delta_x\\). If \\(X\\) is uncountable, \\(\\pi\\) is not cyclic: a vector \\(\\xi\\in\\ell^2(X)\\) is supported on a countable set, and so is every \\(\\pi(f)\\xi\\).\n3. The representation \\(x\\mapsto x\\oplus0\\) of \\(M_n(\\mathbb C)\\) on \\(\\mathbb C^n\\oplus\\mathbb C\\) is degenerate, with null space \\(0\\oplus\\mathbb C\\). Here the image of the identity is the projection onto the essential subspace, as in Proposition 1.4(4).\n\n",
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      "id": "OA-FND-GN-02",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "2. Irreducible representations and Schur's lemma",
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      "full_conditions_and_proof": "### 2. Irreducible representations and Schur's lemma\n\n**Definition 2.1.** A representation \\((\\pi,H)\\) of a \\(*\\)-algebra is *irreducible* if it is not the zero representation and its only closed invariant subspaces are \\(\\{0\\}\\) and \\(H\\).\n\nAn irreducible representation \\(\\pi\\) is nondegenerate, because its essential subspace is invariant and not \\(\\{0\\}\\). By Proposition 1.4, every nonzero vector \\(\\xi\\) then has \\(\\pi(A)\\xi\\ne\\{0\\}\\), and the closure of \\(\\pi(A)\\xi\\) is a nonzero invariant subspace. So every nonzero vector is cyclic. The zero representation on a one-dimensional space also has no invariant subspaces other than \\(\\{0\\}\\) and \\(H\\); the definition excludes it on purpose.\n\n**Theorem 2.2** (Schur's lemma). Let \\(H\\ne\\{0\\}\\) and let \\(\\mathcal S\\subseteq B(H)\\) be a self-adjoint set of operators. The following are equivalent:\n- (i) the only closed subspaces \\(M\\) with \\(TM\\subseteq M\\) for all \\(T\\in\\mathcal S\\) are \\(\\{0\\}\\) and \\(H\\);\n- (ii) \\(\\mathcal S'=\\mathbb C1\\).\n\n**Proof.** (ii) ⇒ (i). By Lemma 1.3, the projection of an invariant subspace lies in \\(\\mathcal S'\\), so it is \\(0\\) or \\(1\\).\n\n(i) ⇒ (ii). The commutant \\(\\mathcal S'\\) is a norm-closed subalgebra of \\(B(H)\\) containing \\(1\\). It is closed under adjoints: if \\(T\\in\\mathcal S'\\) and \\(S\\in\\mathcal S\\), then \\(S^*\\in\\mathcal S\\), so \\(TS^*=S^*T\\), and taking adjoints gives \\(ST^*=T^*S\\). So \\(\\mathcal S'\\) is a unital C\\*-subalgebra of \\(B(H)\\). Every \\(T\\in\\mathcal S'\\) is \\(h+ik\\) with \\(h=\\frac12(T+T^*)\\) and \\(k=\\frac1{2i}(T-T^*)\\) self-adjoint elements of \\(\\mathcal S'\\). So it suffices to show that each self-adjoint \\(h\\in\\mathcal S'\\) is a scalar multiple of \\(1\\).\n\nThe spectrum \\(\\sigma(h)\\) is a nonempty compact subset of \\(\\mathbb R\\) ([the spectrum is not empty](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-08); [spectra of self-adjoint elements](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-01)). Suppose it contains two points \\(\\lambda<\\mu\\). Put \\(\\delta=\\frac12(\\mu-\\lambda)\\), \\(f(t)=\\max(0,1-|t-\\lambda|/\\delta)\\) and \\(g(t)=\\max(0,1-|t-\\mu|/\\delta)\\). These are continuous, \\(f(\\lambda)=g(\\mu)=1\\), and \\(fg=0\\), since \\(f(t)>0\\) forces \\(|t-\\lambda|<\\delta\\) and \\(g(t)>0\\) forces \\(|t-\\mu|<\\delta\\). The [continuous functional calculus](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-07) in the C\\*-algebra \\(\\mathcal S'\\) gives \\(f(h),g(h)\\in\\mathcal S'\\) with \\(f(h)g(h)=(fg)(h)=0\\), \\(\\|f(h)\\|=\\max_{\\sigma(h)}|f|\\ge1\\) and \\(\\|g(h)\\|\\ge1\\). Let \\(M\\) be the closure of the range of \\(g(h)\\). For \\(T\\in\\mathcal S\\) we have \\(Tg(h)\\xi=g(h)T\\xi\\), so \\(M\\) is invariant under \\(\\mathcal S\\). It is not \\(\\{0\\}\\), since \\(g(h)\\ne0\\). It is not \\(H\\): the operator \\(f(h)\\) vanishes on the range of \\(g(h)\\), hence on \\(M\\), but \\(f(h)\\ne0\\). This contradicts (i). Therefore \\(\\sigma(h)=\\{\\lambda\\}\\) for a single \\(\\lambda\\). Then \\(h-\\lambda1\\) is self-adjoint with spectrum \\(\\{0\\}\\), and since the norm of a self-adjoint element equals its spectral radius ([the norm of a normal element](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-01)), \\(h=\\lambda1\\). \\(\\square\\)\n\nThe proof uses only the continuous functional calculus, not the spectral theorem.\n\n**Corollary 2.3.** A representation \\(\\pi\\) of a \\(*\\)-algebra is irreducible if and only if \\(\\pi\\ne0\\) and \\(\\pi(A)'=\\mathbb C1\\). Unitarily equivalent representations are irreducible together.\n\n**Proof.** Apply Theorem 2.2 to the self-adjoint set \\(\\mathcal S=\\pi(A)\\); if \\(\\pi\\ne0\\), then \\(H\\ne\\{0\\}\\). A unitary \\(U\\) with \\(U\\pi_1(x)=\\pi_2(x)U\\) carries the commutant of \\(\\pi_1(A)\\) onto that of \\(\\pi_2(A)\\) by \\(T\\mapsto UTU^*\\). \\(\\square\\)\n\n**Examples 2.4.**\n1. For \\(H\\ne\\{0\\}\\), both \\(B(H)\\) and \\(K(H)\\) act irreducibly on \\(H\\). Indeed, let \\(M\\ne\\{0\\}\\) be invariant under all rank-one operators and pick \\(\\xi\\ne0\\) in \\(M\\). Then \\(\\theta_{\\eta,\\xi}\\xi=\\|\\xi\\|^2\\eta\\in M\\) for every \\(\\eta\\), so \\(M=H\\).\n2. An irreducible representation \\(\\pi\\) of a commutative \\(*\\)-algebra is one-dimensional. Indeed \\(\\pi(A)\\subseteq\\pi(A)'=\\mathbb C1\\), so \\(\\pi(x)=\\chi(x)1\\) for a nonzero \\(*\\)-homomorphism \\(\\chi:A\\to\\mathbb C\\). Then every closed subspace is invariant, which forces \\(\\dim H=1\\). Conversely, every nonzero \\(*\\)-homomorphism \\(\\chi:A\\to\\mathbb C\\) is an irreducible representation on \\(\\mathbb C\\). For \\(A=C_0(X)\\) these are the point evaluations \\(f\\mapsto f(x)\\) ([the characters of \\(C_0(X)\\)](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04)).\n3. The representation \\(x\\mapsto x\\oplus x\\) of \\(M_n(\\mathbb C)\\) on \\(\\mathbb C^n\\oplus\\mathbb C^n\\) is not irreducible: \\(\\mathbb C^n\\oplus0\\) is invariant. Its commutant consists of the block matrices \\(\\begin{pmatrix}a1&b1\\\\c1&d1\\end{pmatrix}\\) with scalars \\(a,b,c,d\\), because each block must commute with all of \\(M_n(\\mathbb C)\\), which acts irreducibly on \\(\\mathbb C^n\\) by (1). So the commutant is a copy of \\(M_2(\\mathbb C)\\).\n\n**Proposition 2.5** (Restriction to an ideal). Let \\(J\\) be a two-sided ideal of a \\(*\\)-algebra \\(A\\) with \\(J^*=J\\), and let \\((\\pi,H)\\) be an irreducible representation of \\(A\\). Then the restriction \\(\\pi|_J\\) vanishes identically or is irreducible. In particular, for every closed two-sided ideal \\(J\\) of a C\\*-algebra, \\(\\pi|_J\\) is zero or irreducible, since such ideals are self-adjoint ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)).\n\n**Proof.** Let \\(E\\) be the essential subspace of \\(\\pi|_J\\), the closed linear span of the vectors \\(\\pi(j)\\xi\\) with \\(j\\in J\\). It is invariant under \\(\\pi(A)\\), since \\(\\pi(a)\\pi(j)\\xi=\\pi(aj)\\xi\\) and \\(aj\\in J\\). So either \\(E=\\{0\\}\\), and then \\(\\pi|_J=0\\), or \\(E=H\\). Assume \\(E=H\\), and let \\(M\\) be a closed subspace invariant under \\(\\pi(J)\\), with projection \\(p\\). The set \\(\\pi(J)\\) is self-adjoint, so \\(p\\) commutes with \\(\\pi(J)\\) (Lemma 1.3). The vectors \\(\\pi(j)p\\xi=p\\pi(j)\\xi\\) therefore span the image under \\(p\\) of the span of \\(\\pi(J)H\\). That span is dense in \\(H\\), and \\(p\\) is continuous, so its image is dense in \\(pH=M\\). Hence the closed linear span of \\(\\pi(J)M\\) is \\(M\\). This closed span is invariant under \\(\\pi(A)\\), because \\(\\pi(a)\\pi(j)m=\\pi(aj)m\\). By irreducibility of \\(\\pi\\), \\(M\\) is \\(\\{0\\}\\) or \\(H\\). Since \\(\\pi|_J\\ne0\\), it is irreducible. \\(\\square\\)\n\nThe ideal need not be closed, and no norm is involved. For a C\\*-algebra even self-adjointness can be dropped (Exercise 12.4). A subalgebra that is not an ideal can act reducibly: the diagonal matrices in \\(M_2(\\mathbb C)\\) leave \\(\\mathbb Ce_1\\) invariant.\n\n",
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    {
      "id": "OA-FND-GN-03",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "3. Positive functionals",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
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      "full_conditions_and_proof": "### 3. Positive functionals\n\nIn this section \\(A\\) is a \\(*\\)-algebra.\n\n**Definition 3.1.** A linear functional \\(\\omega\\) on \\(A\\) is *positive* if \\(\\omega(x^*x)\\ge0\\) for every \\(x\\in A\\). The *adjoint* of a linear functional \\(f\\) is \\(f^*(x)=\\overline{f(x^*)}\\), and \\(f\\) is *hermitian* if \\(f^*=f\\). For positive \\(\\psi,\\varphi\\) we write \\(\\psi\\le\\varphi\\) if \\(\\varphi-\\psi\\) is positive. A positive \\(\\omega\\) is *faithful* if \\(\\omega(x^*x)=0\\) only for \\(x=0\\). If \\(A\\) is normed, a *state* is a bounded positive functional of norm one.\n\nFor a representation \\((\\pi,H)\\) and \\(\\xi,\\eta\\in H\\), the *coefficient functional* \\(\\omega_{\\xi,\\eta}(x)=\\langle\\pi(x)\\xi,\\eta\\rangle\\) satisfies \\(\\omega_{\\xi,\\eta}^*=\\omega_{\\eta,\\xi}\\), because \\(\\overline{\\langle\\pi(x^*)\\xi,\\eta\\rangle}=\\overline{\\langle\\xi,\\pi(x)\\eta\\rangle}=\\langle\\pi(x)\\eta,\\xi\\rangle\\). The *vector functional* \\(\\omega_\\xi=\\omega_{\\xi,\\xi}\\) is positive, since \\(\\omega_\\xi(x^*x)=\\|\\pi(x)\\xi\\|^2\\).\n\n**Proposition 3.2** (Cauchy–Schwarz). Let \\(\\omega\\) be a positive functional on \\(A\\). For all \\(x,y\\in A\\),\n\\[\n\\begin{gathered}\n\\omega(y^*x)\\\\\n=\\overline{\\omega(x^*y)},\\\\\n|\\omega(y^*x)|^2\\\\\n\\le\\omega(x^*x)\\,\\omega(y^*y).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A\\) has an identity, then \\(\\omega(x^*)=\\overline{\\omega(x)}\\) and \\(|\\omega(x)|^2\\le\\omega(1)\\,\\omega(x^*x)\\).\n\n**Proof.** Put \\([x,y]=\\omega(y^*x)\\). It is linear in \\(x\\), conjugate linear in \\(y\\), and \\(q(x)=[x,x]\\ge0\\). Expanding \\(q(x+i^ky)\\) for \\(k=0,1,2,3\\) and summing with weights \\(i^k\\) gives the polarization identity \\(4[x,y]=\\sum_{k=0}^3i^k\\,q(x+i^ky)\\). Exchanging \\(x\\) and \\(y\\), and using \\(q(y+i^kx)=q(x+i^{-k}y)\\) (multiply the vector by \\(i^{-k}\\), a scalar of modulus one), gives \\(4[y,x]=\\sum_ki^kq(x+i^{-k}y)=\\overline{4[x,y]}\\), because each \\(q\\) is real. This is the first identity. For the inequality, let \\(t\\in\\mathbb R\\) and \\(\\lambda=-t[x,y]\\). Then \\[\n\\begin{gathered}\n0\\\\\n\\le q(x+\\lambda y)\\\\\n=q(x)-2t|[x,y]|^2+t^2|[x,y]|^2q(y).\n\\end{gathered}\n\\] If \\(q(y)=0\\), letting \\(t\\to\\infty\\) forces \\([x,y]=0\\). If \\(q(y)>0\\), take \\(t=1/q(y)\\) to get \\(|[x,y]|^2\\le q(x)q(y)\\). In the unital case, \\(1^*\\) is also an identity, so \\(1^*=1\\), and we take \\(y=1\\). \\(\\square\\)\n\n**Lemma 3.3** (The left kernel). Let \\(\\omega\\) be positive and \\(N_\\omega=\\{x\\in A:\\omega(x^*x)=0\\}\\). Then \\(N_\\omega=\\{x\\in A:\\omega(y^*x)=0\\text{ for all }y\\in A\\}\\), and \\(N_\\omega\\) is a left ideal of \\(A\\). Likewise \\(\\{x:\\omega(xx^*)=0\\}=N_\\omega^*\\) is a right ideal.\n\n**Proof.** If \\(\\omega(x^*x)=0\\), then \\(\\omega(y^*x)=0\\) for all \\(y\\) by (3.1); for the converse take \\(y=x\\). The second description shows that \\(N_\\omega\\) is a linear subspace. If \\(x\\in N_\\omega\\) and \\(a\\in A\\), then \\(\\omega(y^*ax)=\\omega((a^*y)^*x)=0\\) for all \\(y\\), so \\(ax\\in N_\\omega\\). The involution turns left ideals into right ideals. \\(\\square\\)\n\nWe call \\(N_\\omega\\) the *left kernel* of \\(\\omega\\).\n\n**Examples 3.4.**\n1. Vector functionals are positive.\n2. Let \\(X\\) be a locally compact Hausdorff space and \\(\\mu\\) a finite positive Radon measure on \\(X\\). Then \\(\\omega_\\mu(f)=\\int f\\,d\\mu\\) is a positive functional on \\(C_0(X)\\), since \\(\\omega_\\mu(\\bar ff)=\\int|f|^2d\\mu\\). It is faithful exactly when every nonempty open set has positive measure. If some nonempty open \\(U\\) has \\(\\mu(U)=0\\), Urysohn's lemma gives \\(f\\ne0\\) vanishing outside \\(U\\), with \\(\\omega_\\mu(\\bar ff)=0\\). Conversely, if \\(f\\ne0\\), then \\(|f|^2>c>0\\) on some nonempty open set, which has positive measure.\n3. Every linear functional on \\(M_n(\\mathbb C)\\) has the form \\(\\omega_\\rho(x)=\\operatorname{Tr}(\\rho x)\\) for exactly one matrix \\(\\rho\\), because the pairing \\((\\rho,x)\\mapsto\\operatorname{Tr}(\\rho x)\\) is nondegenerate. It is positive if and only if \\(\\rho\\ge0\\). If \\(\\rho\\ge0\\), then \\(\\operatorname{Tr}(\\rho x^*x)=\\operatorname{Tr}(x\\rho x^*)\\ge0\\). Conversely, for \\(\\xi\\ne0\\) the operator \\(\\theta_{\\xi,\\xi}=\\|\\xi\\|^{-2}\\theta_{\\xi,\\xi}^*\\theta_{\\xi,\\xi}\\) is of the form \\(x^*x\\), and \\(\\omega_\\rho(\\theta_{\\xi,\\xi})=\\langle\\rho\\xi,\\xi\\rangle\\). A positive \\(\\omega_\\rho\\) is faithful exactly when \\(\\rho\\) is invertible: if \\(\\rho\\ge c1\\) with \\(c>0\\), then \\(\\operatorname{Tr}(x\\rho x^*)\\ge c\\operatorname{Tr}(xx^*)\\), and if \\(\\rho\\xi=0\\) with \\(\\xi\\ne0\\), then \\(\\omega_\\rho(\\theta_{\\xi,\\xi}^*\\theta_{\\xi,\\xi})=\\|\\xi\\|^2\\langle\\rho\\xi,\\xi\\rangle=0\\).\n4. (Zero products.) Let \\(E\\) be a Banach space with an isometric conjugate-linear map \\(x\\mapsto x^*\\) satisfying \\(x^{**}=x\\), and make it an involutive Banach algebra with the zero product \\(xy=0\\). Every \\(x^*x\\) is \\(0\\), so every linear functional is positive. On \\(E=\\mathbb C\\) with complex conjugation, \\(\\omega(x)=ix\\) is positive but not hermitian: \\(\\omega(1^*)=i\\) while \\(\\overline{\\omega(1)}=-i\\). On an infinite-dimensional \\(E\\), such as \\(\\ell^2(\\mathbb N)\\) with coordinatewise conjugation, any linear functional that is not continuous (Example 6.5) is positive. So without an identity or an approximate identity, positivity implies neither the hermitian property nor continuity.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-GN-04",
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      "name": "4. Continuity and norms of positive functionals",
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      "full_conditions_and_proof": "### 4. Continuity and norms of positive functionals\n\nThroughout this section, \\(A\\) denotes an involutive Banach algebra.\n\n#### Algebras with an identity\n\n**Lemma 4.1** (Square roots near the identity). Let \\(A\\) be unital and \\(a\\in A\\) with \\(r(1-a)<1\\). Put \\(c_0=1\\) and \\(c_n=\\binom{1/2}{n}=\\frac{(1/2)(1/2-1)\\cdots(1/2-n+1)}{n!}\\) for \\(n\\geq1\\); the coefficient identity needed below is proved in the final background list. The series \\(b=\\sum_{n\\ge0}c_n(a-1)^n\\) converges absolutely in \\(A\\), \\(b^2=a\\), and \\(b\\) commutes with every element that commutes with \\(a\\). If \\(a=a^*\\), then \\(b=b^*\\).\n\n**Proof.** We have \\(c_0=1\\) and \\(|c_{n+1}|=|c_n|\\,|\\tfrac12-n|/(n+1)\\le|c_n|\\), so \\(|c_n|\\le1\\). Choose \\(q\\) with \\(r(1-a)<q<1\\). By the spectral radius formula, \\(\\|(a-1)^n\\|\\le q^n\\) for all large \\(n\\), so \\(\\sum_n|c_n|\\|(a-1)^n\\|<\\infty\\). The finite Vandermonde identity proved in the background list gives \\(\\sum_{k=0}^nc_kc_{n-k}=1\\) for \\(n=0,1\\) and \\(\\sum_{k=0}^nc_kc_{n-k}=0\\) for \\(n\\ge2\\). [Lemma 0.1 of the Banach-algebra lesson](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-23) proves that the Cauchy product of absolutely convergent series can be regrouped, so \\[\n\\begin{gathered}\nb^2\\\\\n=\\sum_n\\big(\\sum_kc_kc_{n-k}\\big)(a-1)^n\\\\\n=1+(a-1)\\\\\n=a.\n\\end{gathered}\n\\] The partial sums commute with every element commuting with \\(a\\), and so does their limit. If \\(a=a^*\\), the partial sums are self-adjoint because the \\(c_n\\) are real, and the involution is continuous, so \\(b^*=b\\). \\(\\square\\)\n\n**Proposition 4.2** (Positive functionals on unital algebras). Let \\(A\\) be unital and \\(\\omega\\) positive.\n1. For \\(h\\in A_h\\), \\(\\omega(h)\\) is real and \\(-r(h)\\,\\omega(1)\\le\\omega(h)\\le r(h)\\,\\omega(1)\\).\n2. For \\(x\\in A\\), \\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\omega(1)\\,\\omega(x^*x)\\\\\n\\le\\omega(1)^2\\,r(x^*x)\\\\\n\\le\\omega(1)^2\\|x\\|^2.\n\\end{gathered}\n\\]\n3. \\(\\omega\\) is bounded, and \\(\\|\\omega\\|\\le\\omega(1)\\le\\|1\\|\\,\\|\\omega\\|\\). In particular \\(\\|\\omega\\|=\\omega(1)\\) when \\(\\|1\\|=1\\).\n\n**Proof.** (1) \\(\\omega(h)=\\omega(h^*)=\\overline{\\omega(h)}\\) by Proposition 3.2. Let \\(t>r(h)\\) and \\(a=1-t^{-1}h\\). Then \\(r(1-a)=t^{-1}r(h)<1\\), so \\(a=b^2\\) with \\(b=b^*\\) by Lemma 4.1, and \\(\\omega(1)-t^{-1}\\omega(h)=\\omega(b^*b)\\ge0\\). Letting \\(t\\) decrease to \\(r(h)\\) gives \\(\\omega(h)\\le r(h)\\omega(1)\\). Apply this to \\(-h\\), which has the same spectral radius.\n(2) The first inequality is Proposition 3.2. The second is (1) for \\(h=x^*x\\). The third holds because \\(r(x^*x)\\le\\|x^*x\\|\\le\\|x^*\\|\\|x\\|=\\|x\\|^2\\).\n(3) By (2), \\(|\\omega(x)|\\le\\omega(1)\\|x\\|\\). And \\(\\omega(1)\\le\\|\\omega\\|\\|1\\|\\). \\(\\square\\)\n\n**Example 4.3** (The norm of the identity matters). Let \\(A=\\mathbb C^2\\) with coordinatewise product and conjugation and the norm \\(\\|(a,b)\\|=|a|+|b|\\). It is an involutive Banach algebra whose identity \\((1,1)\\) has norm \\(2\\). The functional \\(\\omega(a,b)=a+b\\) is positive, since \\(\\omega(|a|^2,|b|^2)=|a|^2+|b|^2\\). But \\(\\|\\omega\\|=\\sup|a+b|/(|a|+|b|)=1\\), while \\(\\omega(1)=2\\). So \\(\\|\\omega\\|<\\omega(1)\\) can happen when \\(\\|1\\|>1\\), and the second inequality in Proposition 4.2(3) is attained here.\n\n#### Without a continuity assumption on the involution\n\n**Remark 4.4** (Two different continuity questions). Positivity forces a linear functional to be continuous on a unital Banach algebra even when the involution itself is not continuous. In this subsection alone, the involution is assumed only to satisfy the algebraic rules in the conventions. The norm is submultiplicative and complete, but no bound on \\(a^*\\) is assumed. The identity need not have norm one. Jordan Bell's freely readable [notes on positive linear functionals](https://jordanbell.info/LaTeX/mathematics/gelfandtransform/) discuss this distinction. We give the square-root and functional-continuity arguments below using the programme's earlier proofs.\n\nThe difficulty is concrete: a norm limit of self-adjoint elements need not be self-adjoint when the involution is discontinuous. Thus the last sentence of the proof of Lemma 4.1 cannot be used here. Spectra provide a way to identify the root instead.\n\n**Lemma 4.4a** (Self-adjoint roots by spectral separation). Let \\(A\\ne\\{0\\}\\) be a unital complex Banach algebra with an algebraic involution. If \\(a=a^*\\) and\n\\[\n\\sigma_A(a)\\cap(-\\infty,0]=\\varnothing,\n\\]\nthen there is a self-adjoint \\(b\\) with \\(b^2=a\\). It is the unique square root of \\(a\\) whose spectrum lies in the open right half-plane, and it commutes with every element commuting with \\(a\\).\n\n**Proof.** First \\(1^*=1\\): taking adjoints of the two identity laws shows that \\(1^*\\) is an identity. If \\(x\\) is invertible, then \\((x^{-1})^*\\) is the inverse of \\(x^*\\), by taking adjoints of both inverse identities. Therefore\n\\[\n\\sigma_A(x^*)=\\overline{\\sigma_A(x)}.\n\\tag{4.4a}\n\\]\nThese are algebraic statements, not continuity statements.\n\nOn \\(U=\\mathbb C\\setminus(-\\infty,0]\\), define\n\\[\n\\begin{gathered}\nu(z)=\\sqrt{\\frac{|z|+\\operatorname{Re}z}{2}},\\\\\nf(z)=u(z)+\\frac{i\\operatorname{Im}z}{2u(z)}.\n\\end{gathered}\n\\]\nHere \\(u(z)>0\\). Direct multiplication gives \\(f(z)^2=z\\), and \\(f\\) is continuous with positive real part. For nonzero \\(t\\) small enough that \\(z+t\\in U\\),\n\\[\n\\begin{aligned}\n\\frac{f(z+t)-f(z)}{t}\n&=\\frac{1}{f(z+t)+f(z)}\\\\\n&\\longrightarrow\\frac{1}{2f(z)}.\n\\end{aligned}\n\\]\nThus \\(f\\) is holomorphic on \\(U\\). Apply the programme's [holomorphic functional calculus and spectral mapping theorem](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-12). The element \\(b=f(a)\\) satisfies \\(b^2=a\\) and \\(\\sigma_A(b)=f(\\sigma_A(a))\\), which lies in the open right half-plane. By [Proposition 6.5(4) of the Banach-algebra lesson](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-10), \\(b\\) commutes with everything commuting with \\(a\\).\n\nWe check uniqueness without taking an adjoint through the calculus. Suppose \\(c^2=a\\) and \\(\\sigma_A(c)\\) also lies in that half-plane. Since \\(c\\) commutes with its square \\(a\\), it commutes with \\(b\\). Let \\(B\\) be the closed unital subalgebra generated by \\(b,c\\) and all the resolvents\n\\[\n\\begin{gathered}\n(\\lambda1-b)^{-1},\\quad(\\mu1-c)^{-1},\\\\\n\\lambda\\notin\\sigma_A(b),\\quad\\mu\\notin\\sigma_A(c).\n\\end{gathered}\n\\]\nAll these generators commute. Indeed, an element commuting with an invertible element also commutes with its inverse, by multiplying the commutation identity on both sides by that inverse. Thus \\(B\\) is a commutative Banach algebra. Its spectra of \\(b\\) and \\(c\\) are exactly their spectra in \\(A\\): invertibility in \\(B\\) implies invertibility in \\(A\\), and every required resolvent in the reverse direction was included among the generators.\n\nFor each character \\(\\chi\\) of \\(B\\), both \\(\\chi(b)\\) and \\(\\chi(c)\\) have positive real part. Hence \\(\\chi(b+c)\\ne0\\). The [character criterion for invertibility](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18) makes \\(b+c\\) invertible in \\(B\\), and therefore in \\(A\\). Consequently\n\\[\n(b-c)(b+c)=b^2-c^2=0\n\\quad\\Longrightarrow\\quad b=c.\n\\]\nNow \\((b^*)^2=a^*=a\\), and (4.4a) places \\(\\sigma_A(b^*)\\) in the same half-plane. Uniqueness gives \\(b^*=b\\). No continuity of the involution has been used. \\(\\square\\)\n\n**Theorem 4.4b** (Automatic continuity with an arbitrary involution). Let \\(A\\) be a unital complex Banach algebra with an algebraic involution, and let \\(\\omega:A\\to\\mathbb C\\) be complex linear with \\(\\omega(x^*x)\\ge0\\) for every \\(x\\in A\\). Then \\(\\omega\\) is continuous. More precisely, there is a finite constant \\(C\\), depending only on the normed algebra and its involution, such that\n\\[\n\\begin{gathered}\n|\\omega(a)|\\le C\\omega(1)\\|a\\|\\\\\n(a\\in A)\n\\end{gathered}\n\\tag{4.4b}\n\\]\nfor every such \\(\\omega\\). If \\(\\omega(1)=0\\), then \\(\\omega=0\\).\n\n**Proof.** The zero algebra is immediate; assume \\(A\\ne\\{0\\}\\). The algebraic proof of Proposition 3.2 gives the hermitian property and Cauchy–Schwarz without any topological assumption. In particular \\(\\omega(1)\\ge0\\) and\n\\[\n|\\omega(a)|^2\\le\\omega(1)\\omega(a^*a).\n\\]\nIf \\(\\omega(1)=0\\), this proves the assertion. Otherwise put \\(F=\\omega/\\omega(1)\\), so \\(F(1)=1\\).\n\nFor \\(h=h^*\\) and real \\(R>r(h)\\), the spectra of \\(R1+h\\) and \\(R1-h\\) lie in the open right half-plane. Lemma 4.4a writes each as a self-adjoint square, so positivity gives\n\\[\n-R\\le F(h)\\le R.\n\\]\nLetting \\(R\\downarrow r(h)\\) yields\n\\[\n\\begin{gathered}\nF(h)\\in\\mathbb R,\\\\\n|F(h)|\\le r(h)\\le\\|h\\|.\n\\end{gathered}\n\\tag{4.4c}\n\\]\n\nLet \\(H=\\{h\\in A:h=h^*\\}\\), a real linear subspace, and let \\(X=\\overline H\\), with closure in the given norm. Estimate (4.4c) extends \\(F|_H\\) uniquely to a bounded real-linear map \\(\\Phi:X\\to\\mathbb R\\), of norm at most one. Explicitly, for \\(h_n\\to x\\) in \\(H\\), the estimate on \\(h_n-h_m\\) makes \\(F(h_n)\\) Cauchy; its limit is independent of the approximating sequence and defines \\(\\Phi(x)\\). We have not yet proved that \\(\\Phi\\) agrees with \\(F\\) at points of \\(X\\setminus H\\).\n\nThe key fact is that \\(\\Phi\\) vanishes on \\(X\\cap iX\\). For \\(z\\) in this intersection choose \\(u_n,v_n\\in H\\) with \\(u_n\\to z\\) and \\(v_n\\to-iz\\). Continuity of multiplication gives\n\\[\nu_n^2+v_n^2\\longrightarrow z^2+(-iz)^2=0.\n\\]\nEach summand has nonnegative \\(F\\)-value. Cauchy–Schwarz and (4.4c), applied to the self-adjoint sum, give\n\\[\n\\begin{aligned}\n|F(u_n)|^2\n&\\le F(u_n^2)\\\\\n&\\le F(u_n^2+v_n^2)\\\\\n&\\le\\|u_n^2+v_n^2\\|\\longrightarrow0.\n\\end{aligned}\n\\]\nBy the definition of \\(\\Phi\\), this proves \\(\\Phi(z)=0\\).\n\nConsider the bounded real-linear map\n\\[\n\\begin{aligned}\nS:X\\oplus_1X&\\longrightarrow A_{\\mathbb R},\\\\\nS(u,v)&=u+iv,\\\\\n\\|(u,v)\\|_1&=\\|u\\|+\\|v\\|.\n\\end{aligned}\n\\]\nIts domain is Banach: a Cauchy sequence is Cauchy in both closed-subspace coordinates, and the two coordinate limits give convergence in the sum norm. Its codomain is the underlying real Banach space of \\(A\\). It is onto, since the algebraic decomposition\n\\[\na=\\frac{a+a^*}{2}\n+i\\frac{a-a^*}{2i}\n\\]\nhas both components in \\(H\\subseteq X\\). The programme's [open mapping theorem](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-05) supplies \\(\\delta>0\\) such that \\(S\\) maps its open unit ball onto a set containing the ball of radius \\(\\delta\\). Scaling by \\(2\\|a\\|/\\delta\\), for \\(a\\ne0\\), gives some decomposition \\(a=u+iv\\) with\n\\[\n\\begin{gathered}\n\\|u\\|+\\|v\\|\\le C\\|a\\|,\\\\\nC=2/\\delta.\n\\end{gathered}\n\\tag{4.4d}\n\\]\nFor \\(a=0\\), take both components zero. The constant depends on \\(S\\), not on \\(F\\).\n\nDefine \\(T(u,v)=\\Phi(u)+i\\Phi(v)\\). If \\(S(u,v)=0\\), then \\(u=-iv\\) and \\(v=iu\\), so both \\(u\\) and \\(v\\) lie in \\(X\\cap iX\\). Thus \\(T(u,v)=0\\). This proves that\n\\[\nL(a)=\\Phi(u)+i\\Phi(v),\\qquad a=u+iv,\n\\]\nis independent of the chosen decomposition. It is real-linear, and replacing \\((u,v)\\) by \\((-v,u)\\) shows \\(L(ia)=iL(a)\\); hence it is complex-linear. Formula (4.4d) implies \\(|L(a)|\\le C\\|a\\|\\). Finally choose the original algebraic decomposition in \\(H+iH\\). Since \\(\\Phi|_H=F|_H\\), complex linearity of \\(F\\) gives \\(L(a)=F(a)\\). Therefore \\(F\\), and then \\(\\omega\\), is continuous, with (4.4b). \\(\\square\\)\n\n![A bounded map on two closed real subspaces descends through their sum because it vanishes on the kernel.](assets/positive-functional-continuity.png)\n\n*The descent in Theorem 4.4b.* The arrows are real-linear during the construction. The condition \\(\\ker S\\subseteq\\ker T\\) defines \\(L\\) independently of a decomposition; the open mapping estimate makes it bounded. The proof then checks complex linearity and identifies \\(L\\) with \\(F\\). This diagram depicts maps, not an orthogonal decomposition of \\(A\\).\n\n**Corollary 4.4c** (Spectral estimates and familiar special cases). Under the assumptions of Theorem 4.4b,\n\\[\n\\begin{aligned}\n|\\omega(x)|^2\n&\\le\\omega(1)\\omega(x^*x)\\\\\n&\\le\\omega(1)^2 r(x^*x).\n\\end{aligned}\n\\]\nIf \\(x\\) is normal, then \\(|\\omega(x)|\\le\\omega(1)r(x)\\). If \\(\\|x^*\\|\\le\\beta\\|x\\|\\) for all \\(x\\), then \\(\\|\\omega\\|\\le\\sqrt\\beta\\,\\omega(1)\\). If \\(A\\) is commutative, then\n\\[\n\\|\\omega\\|\\le\\omega(1)\\le\\|1\\|\\,\\|\\omega\\|;\n\\]\nin particular \\(\\|\\omega\\|=\\omega(1)\\) when \\(\\|1\\|=1\\), even without a continuous involution.\n\n**Proof.** Apply (4.4c), scaled back from \\(F\\) to \\(\\omega\\), to the self-adjoint element \\(x^*x\\); combine it with algebraic Cauchy–Schwarz. If \\(x\\) commutes with \\(x^*\\), form the closed commutative algebra generated by both elements and their resolvents, exactly as in Lemma 4.4a. For every character \\(\\chi\\),\n\\[\n\\begin{aligned}\n|\\chi(x^*x)|&=|\\chi(x^*)\\chi(x)|\\\\\n&\\le r_A(x^*)r_A(x)=r_A(x)^2.\n\\end{aligned}\n\\]\nThe character criterion identifies the spectrum of \\(x^*x\\) in this commutative algebra; its spectrum in \\(A\\) is contained in that spectrum. Thus \\(r_A(x^*x)\\le r_A(x)^2\\), which proves the normal-element estimate. The bounded-involution estimate follows instead from \\(r(x^*x)\\le\\|x^*\\|\\|x\\|\\le\\beta\\|x\\|^2\\). In a commutative algebra every element is normal, so the normal-element estimate gives \\(\\|\\omega\\|\\le\\omega(1)\\); evaluation at the identity gives the other inequality. \\(\\square\\)\n\n**Checkpoint (with solution).** Why would either of the following shortcuts be invalid: declaring \\(H\\) closed because it consists of self-adjoint elements, or identifying \\(\\Phi(z)\\) with \\(F(z)\\) before constructing \\(L\\)?\n\n*Solution.* The first would use continuity of the involution, which is not assumed. The second would use continuity of \\(F\\), which is the conclusion. We only extend the already bounded restriction \\(F|_H\\). The vanishing argument on \\(X\\cap iX\\) and the open mapping theorem then prove the needed agreement on all of \\(A\\). Without an identity the theorem need not hold: in the infinite-dimensional zero-product example 3.4(4), every linear functional is positive, including discontinuous ones.\n\nFrom the next paragraph onward, “involutive Banach algebra” again has the isometric-involution meaning fixed in the conventions. The broader theorem does not silently change the hypotheses of the GNS estimates that follow.\n\n#### Compressed functionals\n\n**Lemma 4.5.** Let \\(\\omega\\) be a positive functional on \\(A\\); no identity and no continuity are assumed. For every \\(a\\in A\\) the functional \\(\\omega_a(x)=\\omega(axa^*)\\) is positive and bounded, with \\(\\|\\omega_a\\|\\le\\omega(aa^*)\\). Consequently\n\\[\n\\begin{gathered}\n\\omega(x^*a^*ax)\\\\\n\\le\\|a\\|^2\\,\\omega(x^*x)\\\\\n(a,x\\in A).\n\\end{gathered}\n\\tag{4.1}\n\\]\n\n**Proof.** Since \\(A\\) is an ideal in the unitization \\(A_1\\), the formula \\(z\\mapsto\\omega(aza^*)\\) defines a linear functional on \\(A_1\\). It is positive, because \\(\\omega(az^*za^*)=\\omega\\big((za^*)^*(za^*)\\big)\\ge0\\) with \\(za^*\\in A\\). Its value at \\(1\\) is \\(\\omega(aa^*)\\), and \\(\\|1\\|=1\\) in \\(A_1\\), so Proposition 4.2(3) gives \\(|\\omega(aza^*)|\\le\\omega(aa^*)\\|z\\|\\) for \\(z\\in A_1\\), in particular for \\(z\\in A\\). For (4.1), apply this with \\(x^*\\) in place of \\(a\\) and \\(z=a^*a\\): the number \\(\\omega(x^*a^*ax)\\) is the value of the positive functional \\(\\omega_{x^*}\\) at \\(a^*a\\), so it is at most \\(\\omega(x^*x)\\|a^*a\\|\\le\\omega(x^*x)\\|a\\|^2\\). \\(\\square\\)\n\n#### Algebras with an approximate identity\n\n**Proposition 4.6.** Let \\(A\\) have an approximate identity \\((u_i)\\) bounded by \\(\\gamma\\), and let \\(\\omega\\) be positive and continuous. Then \\(\\omega\\) is hermitian, and\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\gamma^2\\|\\omega\\|\\,\\omega(x^*x)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{4.2}\n\\]\n\n**Proof.** The involution is isometric, so \\(\\|u_i^*x-x\\|=\\|x^*u_i-x^*\\|\\to0\\) for every \\(x\\). By continuity and (3.1), \\(\\omega(x^*)=\\lim_i\\omega(x^*u_i)=\\lim_i\\overline{\\omega(u_i^*x)}=\\overline{\\omega(x)}\\). Also \\(|\\omega(u_i^*x)|^2\\le\\omega(u_i^*u_i)\\,\\omega(x^*x)\\le\\gamma^2\\|\\omega\\|\\,\\omega(x^*x)\\), and \\(\\omega(u_i^*x)\\to\\omega(x)\\). \\(\\square\\)\n\nSection 6 shows that the continuity assumption is automatic. Without an approximate identity both conclusions can fail (Example 3.4(4)).\n\n#### C\\*-algebras\n\n**Theorem 4.7.** Let \\(A\\) be a C\\*-algebra and \\(\\omega\\) a positive linear functional on \\(A\\).\n1. \\(\\omega\\) is bounded. More precisely, \\(M=\\sup\\{\\omega(a):a\\in A_+,\\ \\|a\\|\\le1\\}\\) is finite and \\(\\|\\omega\\|\\le2M\\).\n2. If \\((u_i)\\) is a net of positive contractions with \\(u_ix\\to x\\) for every \\(x\\), then \\(\\|\\omega\\|=\\lim_i\\omega(u_i)=M\\). If \\(A\\) is unital and nonzero, \\(\\|\\omega\\|=\\omega(1)\\).\n3. \\(\\omega\\) is hermitian, and \\(|\\omega(x)|^2\\le\\|\\omega\\|\\,\\omega(x^*x)\\).\n4. If \\(\\psi\\) is also positive, then \\(\\|\\omega+\\psi\\|=\\|\\omega\\|+\\|\\psi\\|\\).\n\n**Proof.** (1) Suppose \\(M=\\infty\\). Choose \\(a_n\\in A_+\\) with \\(\\|a_n\\|\\le1\\) and \\(\\omega(a_n)\\ge4^n\\). The series \\(a=\\sum_n2^{-n}a_n\\) converges, and \\(a\\) and each \\(a-2^{-n}a_n\\) lie in the closed convex cone \\(A_+\\) ([the positive cone](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-15)). Hence \\(\\omega(a)\\ge2^{-n}\\omega(a_n)\\ge2^n\\) for every \\(n\\), which is impossible. So \\(M<\\infty\\). A self-adjoint \\(h\\) with \\(\\|h\\|\\le1\\) is \\(h_+-h_-\\) with \\(h_\\pm\\in A_+\\) and \\(\\|h_\\pm\\|\\le1\\) (same link), so \\(\\omega(h)=\\omega(h_+)-\\omega(h_-)\\) is a difference of two numbers in \\([0,M]\\), and \\(|\\omega(h)|\\le M\\). An arbitrary \\(x\\) with \\(\\|x\\|\\le1\\) is \\(x_1+ix_2\\) with self-adjoint \\(x_j\\) of norm at most \\(1\\) ([self-adjoint, normal and unitary elements](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-01)). So \\(|\\omega(x)|\\le2M\\).\n(2) Clearly \\(\\omega(u_i)\\le M\\le\\|\\omega\\|\\). Let \\(\\|x\\|\\le1\\). By continuity, \\(\\omega(x)=\\lim_i\\omega(u_ix)\\). By (3.1), \\(|\\omega(u_ix)|^2\\le\\omega(u_i^2)\\,\\omega(x^*x)\\), and \\(\\omega(x^*x)\\le\\|\\omega\\|\\). Since \\(0\\le u_i\\le1\\), the functional calculus gives \\(u_i^2\\le u_i\\), so \\(\\omega(u_i^2)\\le\\omega(u_i)\\). Hence \\(|\\omega(x)|^2\\le\\|\\omega\\|\\liminf_i\\omega(u_i)\\), and taking the supremum over \\(x\\) gives \\(\\|\\omega\\|\\le\\liminf_i\\omega(u_i)\\) when \\(\\omega\\ne0\\). Together, \\(\\lim_i\\omega(u_i)=M=\\|\\omega\\|\\). Such nets exist, for instance an approximate identity of positive contractions. In the unital case take the constant net \\(u_i=1\\), which has norm one.\n(3) This is Proposition 4.6 with \\(\\gamma=1\\), applied to an approximate identity of positive contractions.\n(4) Apply (2) to \\(\\omega\\), \\(\\psi\\) and \\(\\omega+\\psi\\) with the same net. \\(\\square\\)\n\nThe argument of (1) also shows that every positive linear map between C\\*-algebras is bounded; see [Completely positive maps](completely-positive-maps.md#oa-fnd-cm-03).\n\n### Worked checkpoint: a state's seminorm remembers one column\n\n*Adapted from A. A. Westerbaan, The Category of Von Neumann Algebras (2019), supplied LaTeX source, under [CC BY 4.0](https://creativecommons.org/licenses/by/4.0/). This worked checkpoint, including its adaptation and added solution, retains that licence. The adaptation uses this course's inner product convention and supplies the calculations.*\n\nOn \\(M_2(\\mathbb C)\\), take the vector state \\(\\omega(a)=\\langle ae_1,e_1\\rangle\\), where \\(e_1=(1,0)\\). Its seminorm is \\(\\|a\\|_\\omega=\\omega(a^*a)^{1/2}=\\|ae_1\\|\\). Thus it remembers the first column and forgets the second. The triangle inequality follows from the vector norm; the same formula gives \\(|\\omega(b^*a)|\\le\\|a\\|_\\omega\\|b\\|_\\omega\\) by Hilbert-space Cauchy–Schwarz.\n\nPut\n\\[\np=\\frac12\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix},\\qquad\nq=\\begin{pmatrix}0&0\\\\0&1\\end{pmatrix}.\n\\]\nBoth are orthogonal projections, hence have operator norm one. Direct multiplication gives \\(p^2=p\\), \\(\\|p\\|_\\omega=1/\\sqrt2\\), \\(\\|q\\|_\\omega=0\\), and\n\\[\nqp=\\frac12\\begin{pmatrix}0&0\\\\1&1\\end{pmatrix},\\qquad\n\\|qp\\|_\\omega=\\frac12.\n\\]\nConsequently the estimate \\(\\|ab\\|_\\omega\\le\\|a\\|_\\omega\\|b\\|\\) fails with \\(a=q,b=p\\); so does the estimate with \\(\\|b\\|_\\omega\\) on the right. Also \\(\\|p^*p\\|_\\omega=1/\\sqrt2\\ne\\|p\\|_\\omega^2=1/2\\). The seminorm is not a C*-norm. Taking \\(a=e_{12}\\) gives \\(\\|a\\|_\\omega=0\\) but \\(\\|a^*\\|_\\omega=1\\), so the null space is not self-adjoint.\n\nThe correct estimate has the factors in the opposite roles: \\(\\|ab\\|_\\omega=\\|abe_1\\|\\le\\|a\\|\\|be_1\\|=\\|a\\|\\|b\\|_\\omega\\). It implies that the null space is stable under multiplication on the left. In this example the map \\(a+N_\\omega\\mapsto ae_1\\) is well defined and isometric. It is onto \\(\\mathbb C^2\\), because every vector is the first column of some matrix. Under this identification the GNS action is the usual matrix action and the cyclic vector is \\(e_1\\). This is the entire construction in a model we can compute. Lemma 4.5 supplies the corresponding multiplication estimate in the general involutive Banach algebra.\n\n",
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      "name": "5. The Gelfand–Naimark–Segal construction",
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      "full_conditions_and_proof": "### 5. The Gelfand–Naimark–Segal construction\n\nThroughout this section, \\(A\\) denotes an involutive Banach algebra. We fix a positive functional \\(\\omega\\) on \\(A\\), not assumed to be continuous.\n\n**Construction 5.1.** By Lemma 3.3, the left kernel \\(N_\\omega\\) is a left ideal, and \\(\\omega(y^*x)=0\\) whenever \\(x\\in N_\\omega\\) or \\(y\\in N_\\omega\\); for \\(y\\in N_\\omega\\) use the first identity in (3.1). So\n\\[ \\langle x+N_\\omega,\\,y+N_\\omega\\rangle=\\omega(y^*x) \\tag{5.1} \\]\nis a well-defined inner product on the quotient space \\(A/N_\\omega\\); it is positive definite by the definition of \\(N_\\omega\\). The [completion lemma in Section 1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#completing-normed-and-inner-product-spaces) constructs the Hilbert completion \\(H_\\omega\\). Let \\(\\Lambda_\\omega:A\\to H_\\omega\\), \\(\\Lambda_\\omega(x)=x+N_\\omega\\). Then \\(\\Lambda_\\omega\\) is linear with dense range, and \\(\\langle\\Lambda_\\omega(x),\\Lambda_\\omega(y)\\rangle=\\omega(y^*x)\\).\n\n**Lemma 5.2.** For each \\(a\\in A\\) there is exactly one \\(\\pi_\\omega(a)\\in B(H_\\omega)\\) with \\(\\pi_\\omega(a)\\Lambda_\\omega(x)=\\Lambda_\\omega(ax)\\) for all \\(x\\in A\\). It satisfies \\(\\|\\pi_\\omega(a)\\|\\le\\|a\\|\\), and \\(\\pi_\\omega\\) is a representation of \\(A\\) on \\(H_\\omega\\).\n\n**Proof.** By (4.1), \\(\\|\\Lambda_\\omega(ax)\\|^2=\\omega(x^*a^*ax)\\le\\|a\\|^2\\|\\Lambda_\\omega(x)\\|^2\\). So \\(\\Lambda_\\omega(x)\\mapsto\\Lambda_\\omega(ax)\\) is well defined on \\(\\Lambda_\\omega(A)\\) (if \\(\\Lambda_\\omega(x)=0\\), then \\(\\Lambda_\\omega(ax)=0\\)), linear, and bounded by \\(\\|a\\|\\). It extends uniquely to \\(H_\\omega\\). The identities \\(\\pi_\\omega(ab)=\\pi_\\omega(a)\\pi_\\omega(b)\\) and linearity in \\(a\\) hold on \\(\\Lambda_\\omega(A)\\), hence everywhere. Finally\n\\[\n\\begin{gathered}\n\\langle\\pi_\\omega(a)\\Lambda_\\omega(x),\\Lambda_\\omega(y)\\rangle\\\\\n=\\omega(y^*ax)\\\\\n=\\omega((a^*y)^*x)\\\\\n=\\langle\\Lambda_\\omega(x),\\pi_\\omega(a^*)\\Lambda_\\omega(y)\\rangle,\n\\end{gathered}\n\\]\nand density gives \\(\\pi_\\omega(a)^*=\\pi_\\omega(a^*)\\). \\(\\square\\)\n\n**Theorem 5.3** (When the construction has a cyclic vector). The following are equivalent:\n- (i) there are a representation \\((\\pi,H)\\) of \\(A\\) and a vector \\(\\xi\\in H\\) with \\(\\omega(x)=\\langle\\pi(x)\\xi,\\xi\\rangle\\) for all \\(x\\);\n- (ii) there is a constant \\(C\\) with \\(|\\omega(x)|^2\\le C\\,\\omega(x^*x)\\) for all \\(x\\);\n- (iii) there is a vector \\(\\xi_\\omega\\in H_\\omega\\) with \\(\\omega(x)=\\langle\\Lambda_\\omega(x),\\xi_\\omega\\rangle\\) for all \\(x\\).\n\nWhen they hold, \\(\\xi_\\omega\\) is unique, \\(\\pi_\\omega(x)\\xi_\\omega=\\Lambda_\\omega(x)\\) for all \\(x\\), \\(\\xi_\\omega\\) is a cyclic vector for \\(\\pi_\\omega\\), and \\(\\omega(x)=\\langle\\pi_\\omega(x)\\xi_\\omega,\\xi_\\omega\\rangle\\). The smallest possible \\(C\\) in (ii) is \\(\\|\\xi_\\omega\\|^2\\). Moreover \\(\\omega\\) is hermitian and bounded, with \\(\\|\\omega\\|\\le\\|\\xi_\\omega\\|^2\\).\n\nWe call \\(\\omega\\) *representable* when these conditions hold.\n\n**Proof.** (i) ⇒ (ii). \\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n=|\\langle\\pi(x)\\xi,\\xi\\rangle|^2\\\\\n\\le\\|\\pi(x)\\xi\\|^2\\|\\xi\\|^2\\\\\n=\\|\\xi\\|^2\\omega(x^*x).\n\\end{gathered}\n\\]\n(ii) ⇒ (iii). If \\(x\\in N_\\omega\\), then \\(|\\omega(x)|^2\\le C\\omega(x^*x)=0\\). So \\(\\ell(\\Lambda_\\omega(x))=\\omega(x)\\) is a well-defined linear functional on \\(\\Lambda_\\omega(A)\\), with \\(|\\ell(\\Lambda_\\omega(x))|\\le C^{1/2}\\|\\Lambda_\\omega(x)\\|\\). It extends to a bounded functional on \\(H_\\omega\\), which by the Riesz representation theorem is \\(\\zeta\\mapsto\\langle\\zeta,\\xi_\\omega\\rangle\\) for a unique \\(\\xi_\\omega\\), with \\(\\|\\xi_\\omega\\|\\le C^{1/2}\\).\n(iii) ⇒ (i). For \\(x,y\\in A\\),\n\\[\n\\begin{gathered}\n\\langle\\Lambda_\\omega(x),\\pi_\\omega(y)\\xi_\\omega\\rangle\\\\\n=\\langle\\Lambda_\\omega(y^*x),\\xi_\\omega\\rangle\\\\\n=\\omega(y^*x)\\\\\n=\\langle\\Lambda_\\omega(x),\\Lambda_\\omega(y)\\rangle,\n\\end{gathered}\n\\]\nand since \\(\\Lambda_\\omega(A)\\) is dense, \\(\\pi_\\omega(y)\\xi_\\omega=\\Lambda_\\omega(y)\\). Hence \\(\\omega(x)=\\langle\\Lambda_\\omega(x),\\xi_\\omega\\rangle=\\langle\\pi_\\omega(x)\\xi_\\omega,\\xi_\\omega\\rangle\\), which is (i).\nThe remaining claims. The vector \\(\\xi_\\omega\\) is determined by (iii) because \\(\\Lambda_\\omega(A)\\) is dense, and \\(\\pi_\\omega(A)\\xi_\\omega=\\Lambda_\\omega(A)\\) is dense. From (iii), \\(|\\omega(x)|^2\\le\\|\\xi_\\omega\\|^2\\|\\Lambda_\\omega(x)\\|^2=\\|\\xi_\\omega\\|^2\\omega(x^*x)\\), so \\(C=\\|\\xi_\\omega\\|^2\\) works; and if \\(C\\) works, the proof of (ii) ⇒ (iii) gives \\(\\|\\xi_\\omega\\|^2\\le C\\). Finally \\(\\omega(x^*)=\\langle\\pi_\\omega(x)^*\\xi_\\omega,\\xi_\\omega\\rangle=\\overline{\\omega(x)}\\), and \\(|\\omega(x)|\\le\\|\\pi_\\omega(x)\\|\\|\\xi_\\omega\\|^2\\le\\|x\\|\\|\\xi_\\omega\\|^2\\). \\(\\square\\)\n\n**Theorem 5.4** (GNS with an approximate identity). Let \\(A\\) have an approximate identity \\((u_i)\\) bounded by \\(\\gamma\\), and let \\(\\omega\\) be positive and continuous. Then \\(\\omega\\) is representable, \\(\\|\\omega\\|\\le\\|\\xi_\\omega\\|^2\\le\\gamma^2\\|\\omega\\|\\), and \\(\\xi_\\omega=\\lim_i\\Lambda_\\omega(u_i)\\). If \\(\\gamma=1\\), in particular if \\(A\\) is a C\\*-algebra, then \\(\\|\\xi_\\omega\\|^2=\\|\\omega\\|\\). If \\(A\\) is unital, every positive \\(\\omega\\) is representable, with \\(\\xi_\\omega=\\Lambda_\\omega(1)\\) and \\(\\|\\xi_\\omega\\|^2=\\omega(1)\\).\n\n**Proof.** (4.2) is condition (ii) of Theorem 5.3 with \\(C=\\gamma^2\\|\\omega\\|\\), and the bounds follow from that theorem. The representation \\(\\pi_\\omega\\) has the cyclic vector \\(\\xi_\\omega\\), so it is nondegenerate, and Proposition 1.4(4) gives \\(\\Lambda_\\omega(u_i)=\\pi_\\omega(u_i)\\xi_\\omega\\to\\xi_\\omega\\). A C\\*-algebra has an approximate identity of positive contractions, and its positive functionals are continuous (Theorem 4.7). In the unital case, \\(\\langle\\Lambda_\\omega(x),\\Lambda_\\omega(1)\\rangle=\\omega(1^*x)=\\omega(x)\\), which is (iii) with \\(\\xi_\\omega=\\Lambda_\\omega(1)\\), and \\(\\|\\Lambda_\\omega(1)\\|^2=\\omega(1)\\). \\(\\square\\)\n\n**Theorem 5.5** (Uniqueness). Let \\(\\omega\\) be representable, and let \\((\\pi,H)\\) be a representation with a cyclic vector \\(\\xi\\) such that \\(\\omega(x)=\\langle\\pi(x)\\xi,\\xi\\rangle\\) for all \\(x\\). There is exactly one unitary \\(U:H_\\omega\\to H\\) with \\(U\\Lambda_\\omega(x)=\\pi(x)\\xi\\) for all \\(x\\). It intertwines \\(\\pi_\\omega\\) and \\(\\pi\\), and \\(U\\xi_\\omega=\\xi\\).\n\n**Proof.** Since \\(\\|\\pi(x)\\xi\\|^2=\\omega(x^*x)=\\|\\Lambda_\\omega(x)\\|^2\\), the map \\(\\Lambda_\\omega(x)\\mapsto\\pi(x)\\xi\\) is a well-defined linear isometry of \\(\\Lambda_\\omega(A)\\) onto \\(\\pi(A)\\xi\\). Both subspaces are dense, so it extends to a unitary \\(U\\), and there is no other choice. On the dense subspace, \\[\n\\begin{gathered}\nU\\pi_\\omega(a)\\Lambda_\\omega(x)\\\\\n=U\\Lambda_\\omega(ax)\\\\\n=\\pi(a)\\pi(x)\\xi\\\\\n=\\pi(a)U\\Lambda_\\omega(x),\n\\end{gathered}\n\\] so \\(U\\pi_\\omega(a)=\\pi(a)U\\). Then \\(\\pi(x)U\\xi_\\omega=U\\pi_\\omega(x)\\xi_\\omega=U\\Lambda_\\omega(x)=\\pi(x)\\xi\\) for all \\(x\\). A representation with a cyclic vector is nondegenerate, so \\(U\\xi_\\omega=\\xi\\) by Proposition 1.4(2). \\(\\square\\)\n\n**Definition 5.6.** For a representable \\(\\omega\\), the triple \\((\\pi_\\omega,H_\\omega,\\xi_\\omega)\\) is the *GNS triple* of \\(\\omega\\), and \\(\\pi_\\omega\\) is its *GNS representation*; we also call it the cyclic representation defined by \\(\\omega\\).\n\nTheorems 5.3–5.5 give a bijection between representable positive functionals and unitary equivalence classes of representations with a distinguished cyclic vector: \\(\\omega\\) goes to its GNS triple, and a triple \\((\\pi,H,\\xi)\\) goes back to the vector functional \\(\\omega_\\xi\\).\n\n**Examples 5.7.**\n1. If \\((\\pi,H)\\) has a cyclic vector \\(\\xi\\), the GNS triple of \\(\\omega_\\xi\\) is \\((\\pi,H,\\xi)\\), up to the unitary of Theorem 5.5.\n2. Let \\(\\mu\\) be a finite positive Radon measure on a locally compact Hausdorff space \\(X\\) and \\(\\omega(f)=\\int f\\,d\\mu\\) on \\(C_0(X)\\). Then \\(\\|\\Lambda_\\omega(f)\\|^2=\\int|f|^2d\\mu\\), so \\(\\Lambda_\\omega(f)\\mapsto f\\) is an isometry into \\(L^2(\\mu)\\). Its range contains \\(C_c(X)\\), which is dense in \\(L^2(\\mu)\\) ([approximation by continuous functions](haar-measure.md#oa-fnd-hm-02)). So \\(H_\\omega=L^2(\\mu)\\), \\(\\pi_\\omega(f)\\) is multiplication by \\(f\\), and \\(\\xi_\\omega\\) is the constant function \\(1\\), which lies in \\(L^2(\\mu)\\) because \\(\\mu\\) is finite: indeed \\(\\langle f,1\\rangle=\\int f\\,d\\mu=\\omega(f)\\). The left kernel consists of the functions that vanish on the support of \\(\\mu\\). Indeed, a continuous function nonzero at a support point is bounded away from zero on a neighborhood of positive measure. Conversely, the complement of the support is the union of open null sets; each compact subset of that union has a finite null cover, so Radon inner regularity makes the union null. For a finitely supported \\(\\mu\\), \\(\\dim H_\\omega\\) is the number of points in the support.\n3. On \\(M_n(\\mathbb C)\\), let \\(\\tau=\\frac1n\\operatorname{Tr}\\). Then \\(H_\\tau=M_n(\\mathbb C)\\) with \\(\\langle x,y\\rangle=\\frac1n\\operatorname{Tr}(y^*x)\\), \\(\\pi_\\tau(a)\\) is left multiplication by \\(a\\), and \\(\\xi_\\tau=1\\). Right multiplications commute with \\(\\pi_\\tau\\), so for \\(n\\ge2\\) the commutant is not \\(\\mathbb C1\\) and \\(\\pi_\\tau\\) is reducible. For \\(\\omega(x)=\\langle xe_1,e_1\\rangle\\), the left kernel is \\(\\{x:xe_1=0\\}\\), the map \\(\\Lambda_\\omega(x)\\mapsto xe_1\\) identifies \\(H_\\omega\\) with \\(\\mathbb C^n\\), \\(\\pi_\\omega\\) becomes the identity representation, and \\(\\xi_\\omega=e_1\\). Exercise 12.1 treats all positive functionals on \\(M_n(\\mathbb C)\\).\n4. Continuity is not enough without an approximate identity. On the algebra \\(\\mathbb C\\) with the zero product (Example 3.4(4)), \\(\\omega(x)=x\\) is positive and continuous. Its left kernel is everything, so \\(H_\\omega=\\{0\\}\\), and \\(\\omega\\) is not representable. In fact every representation of this algebra is zero, since \\(\\|\\pi(x)\\|^2=\\|\\pi(x^*x)\\|=0\\).\n\n",
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    {
      "id": "OA-FND-GN-06",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "6. Automatic continuity and the factorization theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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      "full_conditions_and_proof": "### 6. Automatic continuity and the factorization theorem\n\nLet \\(B\\) be a Banach algebra. A *left Banach \\(B\\)-module* is a Banach space \\(X\\) with a bilinear map \\(B\\times X\\to X\\), \\((b,v)\\mapsto b\\cdot v\\), such that \\((bc)\\cdot v=b\\cdot(c\\cdot v)\\) and \\(\\|b\\cdot v\\|\\le\\|b\\|\\|v\\|\\). The main example is \\(X=B\\) with \\(b\\cdot v=bv\\).\n\n**Theorem 6.1** (Cohen–Hewitt factorization). Let \\(B\\) be a Banach algebra with a left approximate identity \\((e_i)\\) bounded by \\(\\gamma\\), and let \\(X\\) be a left Banach \\(B\\)-module. Let \\(v\\in X\\) satisfy \\(e_i\\cdot v\\to v\\). For every \\(\\delta>0\\) there are \\(b\\in B\\) and \\(w\\in X\\) with\n\\[ v=b\\cdot w,\\qquad\\|b\\|\\le\\gamma,\\qquad\\|w-v\\|\\le\\delta. \\]\n\n**Proof.** Let \\(B_1=B\\oplus\\mathbb C\\) be the unitization with the norm \\(\\|b+\\lambda\\|=\\|b\\|+|\\lambda|\\) ([adjoining an identity](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-04)). The formula \\((b+\\lambda)\\cdot v=b\\cdot v+\\lambda v\\) makes \\(X\\) a left Banach \\(B_1\\)-module on which \\(1\\) acts as the identity. Two facts about \\((e_i)\\) are used: \\(\\|e_iz-z\\|\\to0\\) for every \\(z\\in B\\), and \\(e_i\\cdot(z\\cdot v)=(e_iz)\\cdot v\\to z\\cdot v\\) for every \\(z\\in B\\).\n\nFix \\(c=\\frac1{4(\\gamma+1)}\\). We choose elements \\(e_{(1)},e_{(2)},\\dots\\) of the approximate identity inductively and put\n\\[\n\\begin{gathered}\na_n\\\\\n=(1-c)^n1+\\sum_{k=1}^nc(1-c)^{k-1}e_{(k)}\\in B_1,\\\\\na_0\\\\\n=1.\n\\end{gathered}\n\\]\nThen \\(a_n=a_{n-1}+c(1-c)^{n-1}(e_{(n)}-1)\\). The map \\(b+\\lambda\\mapsto\\lambda\\) is a unital homomorphism \\(B_1\\to\\mathbb C\\) that sends \\(a_n\\) to \\((1-c)^n\\). So if \\(a_n\\) is invertible, then \\(a_n^{-1}=\\beta_n+z_n\\) with \\(\\beta_n=(1-c)^{-n}\\) and \\(z_n\\in B\\).\n\nSuppose \\(a_{n-1}\\) is invertible with \\(\\|a_{n-1}^{-1}\\|\\le2^{n-1}\\), and put \\(v_{n-1}=a_{n-1}^{-1}\\cdot v\\); for \\(n=1\\) these are \\(a_0^{-1}=1\\) and \\(v_0=v\\). For any \\(e\\) in the approximate identity,\n\\[\n\\begin{gathered}\na_{n-1}+c(1-c)^{n-1}(e-1)\\\\\n=(1+w_e)\\,a_{n-1},\\\\\nw_e\\\\\n=c(1-c)^{n-1}(e-1)a_{n-1}^{-1},\n\\end{gathered}\n\\]\nand \\((e-1)a_{n-1}^{-1}=\\beta_{n-1}(e-1)+(ez_{n-1}-z_{n-1})\\). Since \\(\\beta_{n-1}c(1-c)^{n-1}=c\\) and \\(\\|e-1\\|\\le\\gamma+1\\),\n\\[\n\\begin{gathered}\n\\|w_e\\|\\\\\n\\le c(\\gamma+1)+c(1-c)^{n-1}\\|ez_{n-1}-z_{n-1}\\|\\\\\n=\\tfrac14+c(1-c)^{n-1}\\|ez_{n-1}-z_{n-1}\\|.\n\\end{gathered}\n\\]\nAlso \\(v_{n-1}=\\beta_{n-1}v+z_{n-1}\\cdot v\\), so \\(e_i\\cdot v_{n-1}\\to v_{n-1}\\). Choose \\(e_{(n)}=e\\) so far out in the net that\n\\[\n\\begin{gathered}\nc(1-c)^{n-1}\\|ez_{n-1}-z_{n-1}\\|\\\\\n\\le\\tfrac14\\\\\n\\text{and}\\\\\n2^nc(1-c)^{n-1}\\|v_{n-1}-e\\cdot v_{n-1}\\|\\\\\n\\le2^{-n}\\delta.\n\\end{gathered}\n\\]\nThen \\(\\|w_e\\|\\le\\frac12\\), so \\(1+w_e\\) is invertible with \\(\\|(1+w_e)^{-1}\\|\\le2\\) (Neumann series). Hence \\(a_n=(1+w_e)a_{n-1}\\) is invertible and \\(\\|a_n^{-1}\\|\\le2\\|a_{n-1}^{-1}\\|\\le2^n\\). Put \\(v_n=a_n^{-1}\\cdot v\\). Since \\(a_n^{-1}-a_{n-1}^{-1}=a_n^{-1}(a_{n-1}-a_n)a_{n-1}^{-1}\\) and \\(a_{n-1}-a_n=c(1-c)^{n-1}(1-e_{(n)})\\),\n\\[\n\\begin{gathered}\nv_n-v_{n-1}\\\\\n=a_n^{-1}\\cdot\\big(c(1-c)^{n-1}(v_{n-1}-e_{(n)}\\cdot v_{n-1})\\big),\\\\\n\\|v_n-v_{n-1}\\|\\\\\n\\le2^{-n}\\delta.\n\\end{gathered}\n\\]\nSo \\((v_n)\\) is a Cauchy sequence. Its limit \\(w\\) satisfies \\(\\|w-v\\|\\le\\sum_n2^{-n}\\delta=\\delta\\). The elements \\(b_n=\\sum_{k=1}^nc(1-c)^{k-1}e_{(k)}\\) of \\(B\\) converge absolutely to some \\(b\\in B\\) with \\(\\|b\\|\\le\\gamma c\\sum_{k\\ge1}(1-c)^{k-1}=\\gamma\\), and \\[\n\\begin{gathered}\n\\|a_n-b\\|\\\\\n\\le(1-c)^n+\\gamma\\sum_{k>n}c(1-c)^{k-1}\\\\\n=(1+\\gamma)(1-c)^n\\to0.\n\\end{gathered}\n\\] Finally \\(v=a_n\\cdot v_n\\) for every \\(n\\), and \\[\n\\begin{gathered}\n\\|a_n\\cdot v_n-b\\cdot w\\|\\\\\n\\le\\|a_n-b\\|\\|v_n\\|+\\|b\\|\\|v_n-w\\|\\to0.\n\\end{gathered}\n\\] Hence \\(v=b\\cdot w\\). \\(\\square\\)\n\n**Corollary 6.2.** Suppose the Banach algebra \\(A\\) has a bounded approximate identity.\n1. Every \\(x\\in A\\) is a product \\(x=yz\\) of two elements of \\(A\\).\n2. If \\(x_n\\to0\\) in \\(A\\), there are \\(a,b\\in A\\) and a sequence \\(y_n\\to0\\) in \\(A\\) with \\(x_n=ay_nb\\) for every \\(n\\).\n\n**Proof.** Let \\((u_i)\\) be the approximate identity, bounded by \\(\\gamma\\).\n(1) Apply Theorem 6.1 to \\(X=A\\) with left multiplication.\n(2) Let \\(c_0(A)\\) consist of the sequences in \\(A\\) that tend to \\(0\\), with the supremum norm. It is complete: a uniformly Cauchy sequence of such sequences has coordinatewise limits in the Banach space \\(A\\); the Cauchy estimate passes to the limit uniformly in the coordinate, and the limit has vanishing tails by uniform approximation by one of the original null sequences. Thus it is a Banach space. It is a left Banach \\(A\\)-module under \\(a\\cdot(z_n)=(az_n)\\). For \\((z_n)\\in c_0(A)\\) and \\(\\varepsilon>0\\), choose \\(N\\) with \\(\\|z_n\\|\\le\\varepsilon/(\\gamma+1)\\) for \\(n>N\\). Then \\(\\|u_iz_n-z_n\\|\\le\\varepsilon\\) for all \\(n>N\\) and all \\(i\\), and for large \\(i\\) also for the finitely many \\(n\\le N\\). So \\(u_i\\cdot(z_n)\\to(z_n)\\), and Theorem 6.1 gives \\(x_n=az_n\\) with \\((z_n)\\in c_0(A)\\). Next, \\(c_0(A)\\) is a left Banach module over the opposite algebra \\(A^{\\rm op}\\), the same space with the product \\(a\\circ b=ba\\), under \\(b\\cdot(z_n)=(z_nb)\\); and \\((u_i)\\) is a left approximate identity for \\(A^{\\rm op}\\), because \\(u_i\\circ z=zu_i\\to z\\). The same argument gives \\(z_n=y_nb\\) with \\(y_n\\to0\\). \\(\\square\\)\n\n**Theorem 6.3** (Varopoulos). If an involutive Banach algebra \\(A\\) has a bounded approximate identity, then every positive linear functional on \\(A\\) is continuous.\n\n\n**Proof.** Let \\(\\omega\\) be positive and \\(x_n\\to0\\). By Corollary 6.2, \\(x_n=ay_nb\\) with \\(y_n\\to0\\). By (3.1), applied to \\((a^*)^*(y_nb)\\),\n\\[ |\\omega(ay_nb)|^2\\le\\omega(aa^*)\\,\\omega(b^*y_n^*y_nb). \\]\nBy Lemma 4.5, the functional \\(z\\mapsto\\omega(b^*zb)\\) is bounded by \\(\\omega(b^*b)\\), so \\[\n\\begin{gathered}\n\\omega(b^*y_n^*y_nb)\\\\\n\\le\\omega(b^*b)\\|y_n^*y_n\\|\\\\\n\\le\\omega(b^*b)\\|y_n\\|^2\\to0.\n\\end{gathered}\n\\] Hence \\(\\omega(x_n)\\to0\\). A linear functional mapping every null sequence to a null sequence is bounded: otherwise choose \\(z_n\\) with \\(\\|z_n\\|\\leq1\\) and \\(|\\omega(z_n)|\\geq n^2\\); then \\(z_n/n\\to0\\) while \\(|\\omega(z_n/n)|\\geq n\\), a contradiction. \\(\\square\\)\n\n**Corollary 6.4.** Suppose the involutive Banach algebra \\(A\\) has an approximate identity bounded by \\(\\gamma\\). Then every positive linear functional \\(\\omega\\) on \\(A\\) is continuous, hermitian and representable, and \\(|\\omega(x)|^2\\le\\gamma^2\\|\\omega\\|\\,\\omega(x^*x)\\). In particular Theorems 5.4 and 5.5 apply to every positive functional on \\(A\\).\n\n**Proof.** Combine Theorem 6.3, Proposition 4.6 and Theorem 5.4. \\(\\square\\)\n\n**Example 6.5** (The approximate identity is needed). Let \\(E=\\ell^2(\\mathbb N)\\) with the zero product and coordinatewise complex conjugation, as in Example 3.4(4). The algebraic basis extension used here follows directly from [Zorn’s lemma](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-01): partially order linearly independent sets containing a given independent family by inclusion. A chain union is independent because every finite relation lies in one chain member. A maximal such set spans the space, since a vector outside its span could be adjoined while preserving independence. Extend the unit vectors \\(e_1,e_2,\\dots\\) to a basis of \\(E\\) as a vector space (a Hamel basis, which exists by Zorn's lemma), and let \\(f\\) be the linear functional with \\(f(e_n)=n\\) and \\(f=0\\) on the remaining basis vectors. Then \\(f\\) is not bounded, and it is positive because every \\(x^*x\\) is \\(0\\).\n\n## C. Complete the representation picture and split states\n\nThe universal seminorm asks how large an element can look in any Hilbert-space representation. Dividing out its zero space and completing gives the enveloping C*-algebra. The traditional term “A*-algebra” records precisely the case in which this division loses no element; its full definition and universal property remain below. Group algebras and transformation-group algebras are concrete reasons to retain this broader branch.\n\nTo split a positive functional below another one, look inside the commutant of its GNS representation. The Radon–Nikodym lemma translates domination into an operator between zero and one. With Schur's lemma, this gives the equivalence between purity and irreducibility. Weak compactness then provides enough pure states to detect both elements and their norms, strengthening the earlier faithful direct-sum construction.\n\n### The enveloping algebra and its concrete models\n\n#### A\\*-algebras and the enveloping C\\*-algebra\n\n**Definition 7.3.** An involutive Banach algebra is an *A\\*-algebra* if it has a faithful representation. For an involutive Banach algebra \\(A\\) and \\(x\\in A\\) put\n\\[ \\|x\\|_{\\rm u}=\\sup\\{\\|\\pi(x)\\|:\\pi\\text{ a representation of }A\\}. \\]\nThe possible numbers \\(\\|\\pi(x)\\|\\) form a subset of the set \\([0,\\|x\\|]\\), defined by the condition that a representation attains that number. The set is nonempty because it contains zero, so its supremum is well defined. No direct sum over a proper class of representations is formed.\n\n**Proposition 7.4** (The enveloping C\\*-algebra). Let \\(A\\) be an involutive Banach algebra.\n1. \\(\\|\\cdot\\|_{\\rm u}\\) is a seminorm with \\(\\|x\\|_{\\rm u}\\le\\|x\\|\\), \\(\\|xy\\|_{\\rm u}\\le\\|x\\|_{\\rm u}\\|y\\|_{\\rm u}\\), \\(\\|x^*\\|_{\\rm u}=\\|x\\|_{\\rm u}\\) and \\(\\|x^*x\\|_{\\rm u}=\\|x\\|_{\\rm u}^2\\).\n2. \\(I=\\{x:\\|x\\|_{\\rm u}=0\\}\\) is the intersection of the kernels of all representations. It is a closed self-adjoint two-sided ideal. The completion \\(C^*(A)\\) of \\(A/I\\) for the norm \\(\\|x+I\\|=\\|x\\|_{\\rm u}\\) is a C\\*-algebra, the *enveloping C\\*-algebra* of \\(A\\). The map \\(j:A\\to C^*(A)\\), \\(j(x)=x+I\\), is a contractive \\(*\\)-homomorphism with dense range. It is injective exactly when \\(A\\) is an A\\*-algebra.\n3. (Universal property.) For every representation \\(\\pi\\) of \\(A\\) there is exactly one representation \\(\\tilde\\pi\\) of \\(C^*(A)\\) with \\(\\pi=\\tilde\\pi\\circ j\\), and every representation \\(\\sigma\\) of \\(C^*(A)\\) is of this form, with \\(\\pi=\\sigma\\circ j\\). The representations \\(\\pi\\) and \\(\\tilde\\pi\\) have the same closed invariant subspaces, the same essential subspace and the same commutant, so one is irreducible, or nondegenerate, exactly when the other is.\n4. If \\(A\\) is a C\\*-algebra, then \\(\\|x\\|_{\\rm u}=\\|x\\|\\), and \\(j\\) is an isometric \\(*\\)-isomorphism of \\(A\\) onto \\(C^*(A)\\).\n\n**Proof.** (1) For each representation, \\(x\\mapsto\\|\\pi(x)\\|\\) is a seminorm with these properties; for the last one, \\(\\|\\pi(x^*x)\\|=\\|\\pi(x)^*\\pi(x)\\|=\\|\\pi(x)\\|^2\\) in the C\\*-algebra \\(B(H)\\). Suprema of such seminorms keep the properties, and \\(\\|\\pi(x)\\|\\le\\|x\\|\\).\n(2) By (1), \\(I\\) is a linear subspace; it is closed since \\(\\|x\\|_{\\rm u}\\le\\|x\\|\\), a two-sided ideal since \\(\\|xy\\|_{\\rm u}\\le\\|x\\|\\,\\|y\\|_{\\rm u}\\) and \\(\\|xy\\|_{\\rm u}\\le\\|x\\|_{\\rm u}\\|y\\|\\), and self-adjoint. On \\(A/I\\) the formula \\(\\|x+I\\|=\\|x\\|_{\\rm u}\\) is well defined, because \\(\\big|\\|x\\|_{\\rm u}-\\|y\\|_{\\rm u}\\big|\\le\\|x-y\\|_{\\rm u}\\), and it is a norm with the properties in (1). Product and involution are uniformly continuous on bounded sets, so they extend to the completion, where the identities of (1) persist. So \\(C^*(A)\\) is a C\\*-algebra. The map \\(j\\) is a \\(*\\)-homomorphism with \\(\\|j(x)\\|=\\|x\\|_{\\rm u}\\le\\|x\\|\\), and its range \\(A/I\\) is dense. If \\(A\\) has a faithful representation, \\(I=\\{0\\}\\). Conversely, let \\(I=\\{0\\}\\) and \\(x\\ne0\\). Some representation \\(\\pi\\) and vector \\(\\zeta\\) have \\(\\pi(x)\\zeta\\ne0\\). The functional \\(\\langle\\pi(\\cdot)\\zeta,\\zeta\\rangle\\) is representable (Theorem 5.3), and its GNS representation \\(\\pi_x\\) has \\(\\|\\pi_x(x)\\xi\\|^2=\\|\\pi(x)\\zeta\\|^2>0\\) for its cyclic vector \\(\\xi\\). These functionals are chosen from a set, the positive functionals on \\(A\\), so the direct sum \\(\\bigoplus_{x\\ne0}\\pi_x\\) is defined, and it is faithful.\n(3) Since \\(\\|\\pi(x)\\|\\le\\|x\\|_{\\rm u}=\\|j(x)\\|\\), the formula \\(\\tilde\\pi(j(x))=\\pi(x)\\) is well defined and contractive on \\(j(A)\\), and it extends by continuity to a \\(*\\)-homomorphism on \\(C^*(A)\\). It is unique because \\(j(A)\\) is dense. If \\(\\sigma\\) is a representation of \\(C^*(A)\\), then \\(\\sigma\\circ j\\) is a representation of \\(A\\), and \\(\\sigma\\) is its extension by uniqueness. The set \\(\\pi(A)=\\tilde\\pi(j(A))\\) is norm-dense in \\(\\tilde\\pi(C^*(A))\\), because \\(\\tilde\\pi\\) is contractive. A closed subspace invariant under a set of operators is invariant under its norm closure, an operator commuting with a set commutes with its norm closure, and a set and its norm closure span the same essential subspace. So \\(\\pi\\) and \\(\\tilde\\pi\\) share these data.\n(4) By Theorem 7.2, \\(A\\) has a faithful representation, which is isometric; so \\(\\|x\\|\\le\\|x\\|_{\\rm u}\\le\\|x\\|\\). Then \\(j\\) is isometric, its range is complete, hence closed, and dense, so \\(j\\) is onto. \\(\\square\\)\n\n**Examples 7.5.**\n1. For the algebra of Example 3.4(4) with the zero product, every representation vanishes, so \\(C^*(A)=\\{0\\}\\).\n2. Let \\(E\\) be such a zero-product algebra and \\(A=E\\oplus\\mathbb C\\) its unitization. Every representation \\(\\pi\\) vanishes on \\(E\\), because \\(\\|\\pi(e)\\|^2=\\|\\pi(e^*e)\\|=0\\). So \\(C^*(A)=\\mathbb C\\) and \\(j(e+\\lambda)=\\lambda\\): an involutive Banach algebra with an identity need not be an A\\*-algebra.\n\n",
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      "name": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
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      "full_conditions_and_proof": "### 7. The Gelfand–Naimark theorem and enveloping C\\*-algebras\n\n#### Faithful representations of C\\*-algebras\n\n**Lemma 7.1** (Enough positive functionals). Let \\(A\\) be a C\\*-algebra and \\(a\\in A_+\\), \\(a\\ne0\\). Some positive linear functional \\(f\\) on \\(A\\) has \\(\\|f\\|\\le1\\) and \\(f(a)>0\\).\n\n**Proof.** Inside the real Banach space \\(A_h\\), the positive cone \\(A_+\\) is closed and convex, it is stable under multiplication by nonnegative numbers, and \\(A_+\\cap(-A_+)=\\{0\\}\\) ([the positive cone](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-15)). So \\(-a\\notin A_+\\). By the separation theorem, applied in \\(A_h\\) to the compact convex set \\(\\{-a\\}\\) and the closed convex set \\(A_+\\), there are a continuous real-linear functional \\(g\\) on \\(A_h\\) and real numbers \\(t_1<t_2\\) with \\(g(-a)<t_1<t_2<g(y)\\) for all \\(y\\in A_+\\). Since \\(0\\in A_+\\), \\(t_2<0\\), so \\(g(a)>-t_1>0\\). If \\(g(y)<0\\) for some \\(y\\in A_+\\), then \\(g(sy)\\to-\\infty\\) as \\(s\\to\\infty\\), although \\(sy\\in A_+\\); so \\(g\\ge0\\) on \\(A_+\\). Put \\(F(h+ik)=g(h)+ig(k)\\) for \\(h,k\\in A_h\\). It is complex linear, since \\[\n\\begin{gathered}\nF(i(h+ik))\\\\\n=F(-k+ih)\\\\\n=-g(k)+ig(h)\\\\\n=iF(h+ik).\n\\end{gathered}\n\\] It is positive, because \\(F(x^*x)=g(x^*x)\\ge0\\), and \\(F(a)=g(a)>0\\). By Theorem 4.7, \\(F\\) is bounded, and \\(f=F/\\|F\\|\\) works. \\(\\square\\)\n\n**Proposition 7.1a** (A state that attains a positive norm). If \\(0\\ne a\\in A_+\\), there is a state \\(\\omega\\) on \\(A\\) with \\(\\omega(a)=\\|a\\|\\). More generally, a state on a unital C*-subalgebra with the same identity extends to a state on the containing unital C*-algebra.\n\n**Proof.** First let \\(B\\) be a unital C*-algebra. A bounded linear functional \\(F\\) with \\(F(1)=\\|F\\|=1\\) is positive. Indeed, for \\(h=h^*\\) and real \\(t\\), continuous functional calculus gives\n\\[\n|1+itF(h)|^2\\leq\\|1+ith\\|^2\\leq1+t^2\\|h\\|^2.\n\\]\nThe left side is \\(1-2t\\operatorname{Im}F(h)+t^2|F(h)|^2\\). Dividing by \\(t\\) and letting \\(t\\) tend to zero from each side shows that \\(F(h)\\) is real. For \\(0\\leq b\\leq1\\), \\(\\|1-b\\|\\leq1\\), so \\(|1-F(b)|\\leq1\\) and therefore \\(F(b)\\geq0\\). Scaling proves positivity on all of \\(B_+\\).\n\nA state on a unital subalgebra has norm one and takes the value one at the common identity (Theorem 4.7). Complex Hahn–Banach extends it with the same norm, and the preceding criterion proves that the extension is a state.\n\nNow work in the C*-unitization \\(\\widetilde A\\). By continuous functional calculus, \\(D=C^*(1,a)\\) is \\(C(\\sigma_{\\widetilde A}(a))\\), with \\(a\\) corresponding to the coordinate function. Since \\(a\\) is positive, \\(\\|a\\|\\) belongs to its spectrum. Evaluation there is a state on \\(D\\); extend it to a state \\(F\\) on \\(\\widetilde A\\). Its restriction \\(\\omega\\) to \\(A\\) is positive, has norm at most one, and satisfies \\(\\omega(a)=\\|a\\|\\). Testing on \\(a/\\|a\\|\\) gives \\(\\|\\omega\\|\\geq1\\), so \\(\\omega\\) is a state. \\(\\square\\)\n\nThis strengthens the separating-functional construction in Lemma 7.1. Applied to \\(a=x^*x\\), it produces a unit GNS vector \\(\\xi_\\omega\\) with \\(\\|\\pi_\\omega(x)\\xi_\\omega\\|=\\|x\\|\\). Thus these cyclic representations detect the norm itself. The extension and norming construction are also developed in [Blackadar, *Operator Algebras*, II.6.2.5 and II.6.3.1–3, corrected author version](https://bruceblackadar.com/Mathematics/Cycr.pdf).\n\n**Theorem 7.2** (Gelfand–Naimark). Every C\\*-algebra \\(A\\) has a faithful representation. Hence \\(A\\) is isometrically \\(*\\)-isomorphic to a norm-closed \\(*\\)-subalgebra of \\(B(H)\\) for some Hilbert space \\(H\\).\n\n\n**Proof.** For each \\(x\\ne0\\) in \\(A\\), \\(x^*x\\ne0\\) by the C\\*-identity, and \\(x^*x\\in A_+\\). Lemma 7.1 gives a positive \\(f_x\\) with \\(f_x(x^*x)>0\\). Let \\((\\pi_x,H_x,\\xi_x)\\) be its GNS triple (Theorem 5.4). Then \\(\\|\\pi_x(x)\\xi_x\\|^2=f_x(x^*x)>0\\), so \\(\\pi_x(x)\\ne0\\). The direct sum \\(\\pi=\\bigoplus_{x\\ne0}\\pi_x\\) exists because every \\(\\|\\pi_x(y)\\|\\le\\|y\\|\\), and it is faithful. An injective \\(*\\)-homomorphism between C\\*-algebras is isometric and has closed range ([\\(*\\)-homomorphisms are contractive](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-12)), so \\(\\pi(A)\\) is a norm-closed \\(*\\)-subalgebra of \\(B(H)\\) that is isometrically \\(*\\)-isomorphic to \\(A\\). \\(\\square\\)\n\nTheorem 8.5 below shows that the summands can be taken irreducible.\n\n## B. Understand what an action can separate\n\nA faithful action distinguishes algebra elements. An irreducible action has no proper closed invariant subspace. These are different requirements: the direct sum used in the faithful-representation theorem can contain many irreducible pieces. Schur's lemma characterizes irreducibility through the commutant, and the examples compare a point evaluation of a commutative algebra with the defining action of a full matrix algebra.\n\nFor a general involutive Banach algebra, an arbitrary positive functional may not yet have the continuity and representability needed to speak of its cyclic vector. Factorization resolves that problem when a bounded approximate identity exists. Keeping this step before the general pure-state theorem ensures that its hypotheses produce an actual GNS triple.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-GN-08",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "Group algebras and transformation-group algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
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      "anchor": "oa-fnd-gn-08",
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      "full_conditions_and_proof": "#### Group algebras and transformation-group algebras\n\n**Example 7.6** (Group algebras). Let \\(G\\) be a locally compact group with left Haar measure \\(ds\\) and modular function \\(\\Delta\\), and let \\(L^1(G)\\) be the involutive Banach algebra with convolution \\((f*g)(t)=\\int f(s)g(s^{-1}t)\\,ds\\) and involution \\(f^*(t)=\\Delta(t)^{-1}\\overline{f(t^{-1})}\\) ([group algebras and transformation-group algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-20)). It has an approximate identity of norm one ([approximate identities of convolution algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-23)), so by Corollary 6.4 all its positive functionals are continuous and representable. The fully proved [group-representation lesson](24-group-representation-completions.md#oa-flow.grp.integration), in *Crossed products and the flow of weights*, supplies the following for every locally compact group. Its exact Haar and operator inputs are the full programme providers listed at the end of this lesson, together with Lemma 0.1 above.\n- The left regular representation \\(\\lambda(f)\\xi=f*\\xi\\) on \\(L^2(G)\\) is faithful and nondegenerate. So \\(L^1(G)\\) is an A\\*-algebra.\n- For a strongly continuous unitary representation \\(U\\) of \\(G\\), the formula \\(\\langle\\pi_U(f)\\xi,\\eta\\rangle=\\int f(s)\\langle U(s)\\xi,\\eta\\rangle\\,ds\\) defines a nondegenerate representation \\(\\pi_U\\) of \\(L^1(G)\\). Every nondegenerate representation \\(\\pi\\) of \\(L^1(G)\\) equals \\(\\pi_U\\) for exactly one such \\(U\\), recovered from \\(U(s)\\pi(f)\\xi=\\pi(\\lambda_sf)\\xi\\) with \\((\\lambda_sf)(t)=f(s^{-1}t)\\). A bounded operator intertwines \\(U\\) and \\(V\\) exactly when it intertwines \\(\\pi_U\\) and \\(\\pi_V\\); so the correspondence preserves unitary equivalence and irreducibility.\n\nThe enveloping C\\*-algebra \\(C^*(G)=C^*(L^1(G))\\) is the *group C\\*-algebra*. By Proposition 1.4, a degenerate representation of \\(L^1(G)\\) is a nondegenerate one plus a zero summand, so \\(\\|f\\|_{\\rm u}=\\sup_U\\|\\pi_U(f)\\|\\) over strongly continuous unitary representations \\(U\\). The norm closure of \\(\\lambda(L^1(G))\\) is the *reduced group C\\*-algebra* \\(C^*_r(G)\\). By Proposition 7.4(3), \\(\\lambda\\) extends to a \\(*\\)-homomorphism of \\(C^*(G)\\) onto \\(C^*_r(G)\\): its range is closed ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)) and contains the dense set \\(\\lambda(L^1(G))\\). The [four-neighbor-tree calculation in that lesson](24-group-representation-completions.md#oa-flow.grp.models) proves noninjectivity for the free group on two generators: the sum of the four generator and inverse point masses has full norm \\(4\\) and reduced norm \\(2\\sqrt3\\).\n\n**Proposition 7.7** (Transformation-group algebras). Let a locally compact group \\(G\\) act continuously on the right of a locally compact Hausdorff space \\(\\Omega\\), \\((\\omega,s)\\mapsto\\omega s\\), and put \\((\\alpha_sf)(\\omega)=f(\\omega s)\\) for \\(f\\in C_0(\\Omega)\\). Let \\(\\mathfrak A(\\Omega,G)\\) be the involutive Banach algebra obtained by completing \\(C_c(\\Omega\\times G)\\) for \\(\\|x\\|_1=\\int_G\\sup_\\omega|x(\\omega,s)|\\,ds\\), with\n\\[\n\\begin{gathered}\n(x\\star y)(\\omega,s)\\\\\n=\\int_Gx(\\omega,t)\\,y(\\omega t,t^{-1}s)\\,dt,\\\\\nx^\\sharp(\\omega,s)\\\\\n=\\Delta(s)^{-1}\\overline{x(\\omega s,s^{-1})}\n\\end{gathered}\n\\]\n([group algebras and transformation-group algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-20)). A *covariant pair* \\((\\rho,U)\\) on \\(H\\) consists of a nondegenerate representation \\(\\rho\\) of \\(C_0(\\Omega)\\) and a strongly continuous unitary representation \\(U\\) of \\(G\\) on \\(H\\) with\n\\[\n\\begin{gathered}\nU(s)\\rho(f)U(s)^*\\\\\n=\\rho(\\alpha_sf)\\\\\n(f\\in C_0(\\Omega),\\ s\\in G).\n\\end{gathered}\n\\tag{7.1}\n\\]\nIts *integrated form* on \\(x\\in C_c(\\Omega\\times G)\\) is defined weakly by \\(\\langle\\pi(x)\\xi,\\eta\\rangle=\\int_G\\langle\\rho(x(\\cdot,s))U(s)\\xi,\\eta\\rangle\\,ds\\), where \\(x(\\cdot,s)\\in C_c(\\Omega)\\). Then, for \\(f\\in C_0(\\Omega)\\), \\(t\\in G\\) and \\(x\\in C_c(\\Omega\\times G)\\),\n\\[\n\\begin{gathered}\n\\rho(f)\\pi(x)\\\\\n=\\pi(L_fx),\\\\\nU(t)\\pi(x)\\\\\n=\\pi(M_tx),\\\\\n(L_fx)(\\omega,s)\\\\\n=f(\\omega)x(\\omega,s),\\\\\n(M_tx)(\\omega,s)\\\\\n=x(\\omega t,t^{-1}s).\n\\end{gathered}\n\\tag{7.2}\n\\]\n\nThe identities (7.2) follow directly from the integrated convention (7.1); the [three-moving-points model](25-cstar-covariant-representations.md#oa-flow.ccov.transformations) gives an explicit finite check of both coordinate shifts.\n\n**Proof.** Both \\(L_fx\\) and \\(M_tx\\) lie in \\(C_c(\\Omega\\times G)\\), since \\((\\omega,s)\\mapsto(\\omega t,t^{-1}s)\\) is a homeomorphism. For the first identity, \\[\n\\begin{gathered}\n\\langle\\rho(f)\\pi(x)\\xi,\\eta\\rangle\\\\\n=\\int\\langle\\rho(x(\\cdot,s))U(s)\\xi,\\rho(f)^*\\eta\\rangle\\,ds\\\\\n=\\int\\langle\\rho(fx(\\cdot,s))U(s)\\xi,\\eta\\rangle\\,ds.\n\\end{gathered}\n\\] For the second, (7.1) gives \\(U(t)\\rho(g)=\\rho(\\alpha_tg)U(t)\\), so\n\\[\n\\begin{gathered}\n\\langle U(t)\\pi(x)\\xi,\\eta\\rangle\\\\\n=\\int\\langle\\rho(\\alpha_t(x(\\cdot,s)))U(ts)\\xi,\\eta\\rangle\\,ds\\\\\n=\\int\\langle\\rho(\\alpha_t(x(\\cdot,t^{-1}r)))U(r)\\xi,\\eta\\rangle\\,dr\n\\end{gathered}\n\\]\nby the substitution \\(r=ts\\), which preserves left Haar measure. Finally \\(\\alpha_t(x(\\cdot,t^{-1}r))(\\omega)=x(\\omega t,t^{-1}r)=(M_tx)(\\omega,r)\\). \\(\\square\\)\n\nThe [coefficient-recovery, group-recovery and regular-faithfulness proofs](25-cstar-covariant-representations.md#oa-flow.ccov.coefficientrecovery), in *Crossed products and the flow of weights*, supply the remaining facts for general C\\*-dynamical systems. The point-norm continuity required there follows from the compactness argument after Lemma 0.1, and the Banach-space identification is proved in [the Banach-algebra lesson](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-20). The integrated form of a covariant pair extends to a nondegenerate representation of \\(\\mathfrak A(\\Omega,G)\\). Every nondegenerate representation of \\(\\mathfrak A(\\Omega,G)\\) is the integrated form of exactly one covariant pair, recovered by (7.2). And for a faithful nondegenerate representation \\(\\rho\\) of \\(C_0(\\Omega)\\) on \\(K\\), such as the multiplication representation of Example 1.6(2), the covariant pair on \\(L^2(G,K)\\) given by \\((\\tilde\\rho(f)\\zeta)(s)=\\rho(\\alpha_{s^{-1}}f)\\zeta(s)\\) and \\((V(t)\\zeta)(s)=\\zeta(t^{-1}s)\\) integrates to a faithful representation. So \\(\\mathfrak A(\\Omega,G)\\) is an A\\*-algebra. Its enveloping C\\*-algebra \\(C^*(\\Omega,G)\\) is the *transformation-group C\\*-algebra*, or covariance C\\*-algebra, of \\((\\Omega,G)\\). Section 9 works out the case \\(\\Omega=G=\\mathbb R\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-GN-09",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "Separable algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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      "anchor": "oa-fnd-gn-09",
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        "line": 761,
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      "full_conditions_and_proof": "#### Separable algebras\n\n**Proposition 7.8.**\n1. Every cyclic representation of a separable involutive Banach algebra acts on a separable Hilbert space.\n2. A separable C\\*-algebra \\(A\\ne0\\) has a faithful state. More generally, a separable A\\*-algebra has a faithful continuous positive functional.\n3. Every separable A\\*-algebra, and in particular every separable C\\*-algebra, can be represented faithfully on some separable Hilbert space.\n\n**Proof.** (1) If \\(\\xi\\) is cyclic, \\(H\\) is the closure of \\(\\pi(A)\\xi\\), the image of a separable space under the continuous map \\(x\\mapsto\\pi(x)\\xi\\).\n(2) The zero algebra has the faithful positive functional zero and the faithful representation on the zero Hilbert space; no state is asserted in that case. Hence assume the algebra is nonzero. Let \\(A\\) be a separable C\\*-algebra and \\(Q\\) the set of positive functionals of norm at most one. For \\(a\\in A_+\\) put \\(s(a)=\\sup_{\\varphi\\in Q}\\varphi(a)\\). Then \\(|s(a)-s(b)|\\le\\|a-b\\|\\), and \\(s(a)>0\\) when \\(a\\ne0\\), by Lemma 7.1. Let \\((x_n)_{n\\ge1}\\) be dense in \\(A\\), and choose \\(\\varphi_n\\in Q\\) with \\(\\varphi_n(x_n^*x_n)\\ge\\frac12s(x_n^*x_n)\\). The series \\(\\omega=\\sum_n2^{-n}\\varphi_n\\) converges in norm to a positive functional. Let \\(x\\ne0\\) and \\(a=x^*x\\), so \\(s(a)>0\\). Since \\(y\\mapsto y^*y\\) is continuous, some \\(n\\) has \\(\\|x_n^*x_n-a\\|<\\frac14s(a)\\). Then \\(s(x_n^*x_n)>\\frac34s(a)\\), so \\(\\varphi_n(x_n^*x_n)>\\frac38s(a)\\), and\n\\[\n\\begin{gathered}\n\\varphi_n(a)\\\\\n\\ge\\varphi_n(x_n^*x_n)-\\|x_n^*x_n-a\\|\\\\\n\\gt\\tfrac38s(a)-\\tfrac14s(a)>0.\n\\end{gathered}\n\\]\nHence \\(\\omega(x^*x)\\ge2^{-n}\\varphi_n(a)>0\\). So \\(\\omega\\) is faithful, and so is the state \\(\\omega/\\|\\omega\\|\\). If \\(A\\) is a separable A\\*-algebra, \\(C^*(A)\\) is separable, because \\(j(A)\\) is dense in it. A faithful state \\(\\varphi\\) of \\(C^*(A)\\) gives \\(\\omega=\\varphi\\circ j\\), which is continuous and positive, and faithful because \\(\\omega(x^*x)=\\varphi(j(x)^*j(x))=0\\) forces \\(j(x)=0\\), hence \\(x=0\\).\n(3) Let \\(\\omega\\) be as in (2). It is representable, since \\(\\omega(x)=\\langle\\pi_\\varphi(j(x))\\xi_\\varphi,\\xi_\\varphi\\rangle\\); for a C\\*-algebra take \\(j\\) to be the identity. If \\(\\pi_\\omega(x)=0\\), then \\(\\omega(x^*x)=\\|\\pi_\\omega(x)\\xi_\\omega\\|^2=0\\), so \\(x=0\\). And \\(H_\\omega\\) is separable by (1). \\(\\square\\)\n\n**Examples 7.9** (Separability is not necessary, but some countability is).\n1. \\(B(\\ell^2(\\mathbb N))\\) is not separable: the diagonal operators with entries in \\(\\{0,1\\}\\) form an uncountable family at pairwise norm distance one, whereas a countable dense set would give distinct approximating points at distance less than \\(1/3\\) for every member of that family. Nevertheless, for an orthonormal basis \\((e_n)_{n\\ge1}\\) of \\(\\ell^2(\\mathbb N)\\), \\(\\omega=\\sum_{n\\ge1}2^{-n}\\omega_{e_n}\\) is a faithful state, since \\(\\omega(x^*x)=\\sum_n2^{-n}\\|xe_n\\|^2\\), and the identity representation is faithful on a separable space.\n2. Let \\(\\Gamma\\) be an uncountable set and \\(A=c_0(\\Gamma)\\), the functions on \\(\\Gamma\\) that tend to zero at infinity. Suppose \\(\\omega\\) were a faithful positive functional. The numbers \\(\\omega(\\delta_\\gamma)\\), \\(\\gamma\\in\\Gamma\\), would all be positive, so some \\(m\\) would have \\(\\omega(\\delta_\\gamma)>1/m\\) for infinitely many \\(\\gamma\\). For a finite set \\(F\\) of such \\(\\gamma\\), the element \\(\\sum_{\\gamma\\in F}\\delta_\\gamma\\) has norm one, while \\(\\omega(\\sum_{\\gamma\\in F}\\delta_\\gamma)>|F|/m\\), which exceeds \\(\\|\\omega\\|\\) for large \\(|F|\\). And a faithful representation \\(\\pi\\) would give uncountably many nonzero, mutually orthogonal projections \\(\\pi(\\delta_\\gamma)\\), which a separable Hilbert space cannot carry.\n\n",
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      "id": "OA-FND-GN-10",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "8. Pure states and irreducible representations",
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      "full_conditions_and_proof": "### 8. Pure states and irreducible representations\n\n#### Functionals below a given one\n\n**Lemma 8.1** (Radon–Nikodym lemma for positive functionals). Let \\(\\varphi\\) be a representable positive functional on \\(A\\), where \\(A\\) is an involutive Banach algebra; write \\((\\pi,H,\\xi)\\) for its GNS triple.\n1. For \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\), the functional \\(\\varphi_T(x)=\\langle\\pi(x)T\\xi,\\xi\\rangle\\) is positive, \\(\\varphi_T\\le\\varphi\\), and \\(\\varphi_T(y^*x)=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\). The map \\(T\\mapsto\\varphi_T\\) is affine and injective.\n2. If \\(\\psi\\) is positive and \\(\\psi\\le\\varphi\\), there is exactly one \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\) and \\(\\psi(y^*x)=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\) for all \\(x,y\\in A\\).\n3. If every element of \\(A\\) is a product of two elements, then \\(\\psi=\\varphi_T\\) in (2), and \\(T\\mapsto\\varphi_T\\) is a bijection from \\(\\{T\\in\\pi(A)':0\\le T\\le1\\}\\) onto \\(\\{\\psi\\text{ positive}:\\psi\\le\\varphi\\}\\). This applies when \\(A\\) has a bounded approximate identity (Corollary 6.2), in particular to C\\*-algebras.\n\n**Proof.** (1) Since \\(T\\) commutes with \\(\\pi(A)\\), \\(\\varphi_T(y^*x)=\\langle\\pi(y)^*\\pi(x)T\\xi,\\xi\\rangle=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\). With \\(y=x\\) and \\(\\zeta=\\pi(x)\\xi\\), this is \\(\\langle T\\zeta,\\zeta\\rangle\\in[0,\\|\\zeta\\|^2]=[0,\\varphi(x^*x)]\\), because \\(0\\le T\\le1\\). So \\(\\varphi_T\\) and \\(\\varphi-\\varphi_T\\) are positive. If \\(\\varphi_S=\\varphi_T\\), then \\(\\langle(S-T)\\pi(x)\\xi,\\pi(y)\\xi\\rangle=0\\) for all \\(x,y\\), and since \\(\\pi(A)\\xi\\) is dense, \\(S=T\\).\n(2) For \\(z\\in A\\) with \\(\\pi(z)\\xi=0\\) we have \\(0\\le\\psi(z^*z)\\le\\varphi(z^*z)=\\|\\pi(z)\\xi\\|^2=0\\), and then \\(\\psi(y^*z)=0\\) and \\(\\psi(z^*y)=0\\) for all \\(y\\), by (3.1). So \\(\\beta(\\pi(x)\\xi,\\pi(y)\\xi)=\\psi(y^*x)\\) is a well-defined sesquilinear form on the dense subspace \\(\\pi(A)\\xi\\). By (3.1) again, \\[\n\\begin{gathered}\n|\\psi(y^*x)|^2\\\\\n\\le\\psi(x^*x)\\psi(y^*y)\\\\\n\\le\\|\\pi(x)\\xi\\|^2\\|\\pi(y)\\xi\\|^2.\n\\end{gathered}\n\\] So \\(\\beta\\) extends to a bounded sesquilinear form on \\(H\\) with \\(0\\le\\beta(\\zeta,\\zeta)\\le\\|\\zeta\\|^2\\). There is a unique \\(T\\in B(H)\\) with \\(\\beta(\\zeta,\\zeta')=\\langle T\\zeta,\\zeta'\\rangle\\), and \\(0\\le T\\le1\\). For \\(a,x,y\\in A\\),\n\\[\n\\begin{gathered}\n\\langle T\\pi(a)\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\\\\n=\\psi(y^*ax)\\\\\n=\\psi((a^*y)^*x)\\\\\n=\\langle T\\pi(x)\\xi,\\pi(a^*)\\pi(y)\\xi\\rangle\\\\\n=\\langle\\pi(a)T\\pi(x)\\xi,\\pi(y)\\xi\\rangle,\n\\end{gathered}\n\\]\nso \\(T\\pi(a)=\\pi(a)T\\). Uniqueness holds because \\(T\\) is determined on the dense subspace.\n(3) Write \\(x=yz\\). Then \\(x=(y^*)^*z\\), and by (2) and (1), \\(\\psi(x)=\\langle T\\pi(z)\\xi,\\pi(y^*)\\xi\\rangle=\\varphi_T(x)\\). \\(\\square\\)\n\n#### Pure states\n\n**Definition 8.2.** A positive functional \\(\\varphi\\) is *pure* if every positive \\(\\psi\\le\\varphi\\) is a scalar multiple of \\(\\varphi\\). If \\(\\varphi(x^*x)>0\\) for some \\(x\\), the multiple is \\(\\lambda\\varphi\\) with \\(0\\le\\lambda\\le1\\), since both \\(\\lambda\\varphi\\) and \\((1-\\lambda)\\varphi\\) are positive. This holds for every representable \\(\\varphi\\ne0\\): then \\(\\xi_\\varphi\\ne0\\), and \\(\\varphi(x^*x)=\\|\\pi_\\varphi(x)\\xi_\\varphi\\|^2\\) cannot vanish for all \\(x\\), because \\(\\pi_\\varphi(A)\\xi_\\varphi\\) is dense in \\(H_\\varphi\\). Without such an \\(x\\) the multiple need not lie in \\([0,1]\\): on \\(\\mathbb C\\) with the zero product (Example 3.4(4)), \\(\\varphi(x)=x\\) is pure, and \\(2\\varphi\\le\\varphi\\). A *pure state* is a state that is pure. \\(P(A)\\) denotes the set of pure states.\n\n**Theorem 8.3.** Suppose the involutive Banach algebra \\(A\\) has a bounded approximate identity (for instance, \\(A\\) is a C\\*-algebra), and let \\(\\varphi\\ne0\\) be a positive functional on \\(A\\). Then \\(\\varphi\\) is pure if and only if its GNS representation \\(\\pi_\\varphi\\) is irreducible.\n\n**Proof.** By Corollary 6.4, \\(\\varphi\\) is representable; let \\((\\pi,H,\\xi)\\) be its GNS triple. Since \\(\\varphi\\ne0\\), \\(\\xi\\ne0\\), and \\(\\pi\\ne0\\) because \\(\\pi(A)\\xi\\) is dense in \\(H\\ne\\{0\\}\\).\nSuppose \\(\\pi\\) is irreducible, and let \\(\\psi\\le\\varphi\\) be positive. By Lemma 8.1(3), \\(\\psi=\\varphi_T\\) with \\(T\\in\\pi(A)'=\\mathbb C1\\) (Corollary 2.3). So \\(T=\\lambda1\\) and \\(\\psi=\\lambda\\varphi\\).\nConversely, suppose \\(\\varphi\\) is pure, and let \\(p\\in\\pi(A)'\\) be a projection. By Lemma 8.1(1), \\(\\varphi_p\\le\\varphi\\), so \\(\\varphi_p=\\lambda\\varphi=\\varphi_{\\lambda1}\\), and injectivity gives \\(p=\\lambda1\\). Thus \\(p\\) is \\(0\\) or \\(1\\). By Lemma 1.3, the only closed invariant subspaces are \\(\\{0\\}\\) and \\(H\\), and \\(\\pi\\) is irreducible. \\(\\square\\)\n\n**Proposition 8.4** (Pure states as extreme points). Let \\(A\\) be a C\\*-algebra, and let \\(Q(A)\\) be the set of positive functionals of norm at most one, with the weak\\* topology.\n1. \\(Q(A)\\) is compact and convex.\n2. The extreme points of \\(Q(A)\\) are \\(0\\) and the pure states.\n3. A state is pure if and only if it is an extreme point of the set \\(S(A)\\) of all states.\n4. \\(Q(A)\\) is the weak\\*-closed convex hull of \\(\\{0\\}\\cup P(A)\\). If \\(A\\) is unital, \\(S(A)\\) is weak\\*-compact and is the weak\\*-closed convex hull of \\(P(A)\\).\n\n**Proof.** (1) By Theorem 4.7, \\(Q(A)\\) is the set of those \\(\\omega\\) in the closed unit ball of \\(A^*\\) with \\(\\omega(x^*x)\\ge0\\) for all \\(x\\). The ball is weak\\*-compact (Banach–Alaoglu), and each condition \\(\\omega(x^*x)\\ge0\\) is weak\\*-closed. Convexity is clear.\n(2) Let \\(\\varphi\\ne0\\) be an extreme point. If \\(\\|\\varphi\\|<1\\), then \\(\\varphi=\\|\\varphi\\|\\cdot\\frac{\\varphi}{\\|\\varphi\\|}+(1-\\|\\varphi\\|)\\cdot0\\) writes \\(\\varphi\\) as a proper convex combination of two points of \\(Q(A)\\) different from \\(\\varphi\\). So \\(\\|\\varphi\\|=1\\). Let \\(\\psi\\le\\varphi\\) be positive and \\(\\chi=\\varphi-\\psi\\). By Theorem 4.7(4), \\(\\|\\psi\\|+\\|\\chi\\|=1\\). If \\(\\psi=0\\) or \\(\\chi=0\\), then \\(\\psi=0\\cdot\\varphi\\) or \\(\\psi=1\\cdot\\varphi\\). Otherwise \\(\\varphi=\\|\\psi\\|\\frac{\\psi}{\\|\\psi\\|}+\\|\\chi\\|\\frac{\\chi}{\\|\\chi\\|}\\), and extremality gives \\(\\psi/\\|\\psi\\|=\\varphi\\). So \\(\\varphi\\) is a pure state.\nConversely, \\(0\\) is extreme: if \\(0=t\\psi+(1-t)\\chi\\) with \\(\\psi,\\chi\\in Q(A)\\) and \\(0<t<1\\), then \\(\\psi\\) and \\(\\chi\\) vanish on \\(A_+\\), which spans \\(A\\), so \\(\\psi=\\chi=0\\). Let \\(\\varphi\\) be a pure state and \\(\\varphi=t\\psi+(1-t)\\chi\\) with \\(\\psi,\\chi\\in Q(A)\\) and \\(0<t<1\\). Then \\(t\\psi\\le\\varphi\\), so \\(t\\psi=\\lambda\\varphi\\), and similarly \\((1-t)\\chi=\\mu\\varphi\\). By Theorem 4.7(4), \\(1=t\\|\\psi\\|+(1-t)\\|\\chi\\|\\), which forces \\(\\|\\psi\\|=\\|\\chi\\|=1\\). So \\(\\lambda=t\\) and \\(\\psi=\\varphi\\); likewise \\(\\chi=\\varphi\\).\n(3) If a state \\(\\varphi\\) is a proper convex combination \\(t\\psi+(1-t)\\chi\\) with \\(\\psi,\\chi\\in Q(A)\\), the norm identity above makes \\(\\psi\\) and \\(\\chi\\) states. So \\(\\varphi\\) is extreme in \\(S(A)\\) if and only if it is extreme in \\(Q(A)\\), and (2) applies.\n(4) The weak\\* topology on \\(A^*\\) is locally convex, and its continuous linear functionals are the evaluations at elements of \\(A\\), which separate points. So the Krein–Milman theorem applies to \\(Q(A)\\), and (2) identifies its extreme points. If \\(A\\) is unital, \\(S(A)=\\{\\omega\\in Q(A):\\omega(1)=1\\}\\) by Theorem 4.7(2); this set is weak\\*-closed in \\(Q(A)\\), hence compact, and its extreme points are the pure states by (3). \\(\\square\\)\n\nWithout an identity \\(S(A)\\) need not be weak\\*-closed: Example 10.5(1) gives states converging to \\(0\\).\n\n#### Enough irreducible representations\n\n**Theorem 8.5.**\n1. Let \\(A\\) be an A\\*-algebra. For every nonzero \\(x\\in A\\) there is an irreducible representation \\(\\pi\\) of \\(A\\) with \\(\\pi(x)\\ne0\\).\n2. Let \\(A\\ne0\\) be a C\\*-algebra. For every \\(x\\in A\\) there is a pure state \\(\\varphi\\) with \\(\\varphi(x^*x)=\\|x\\|^2\\), and then \\(\\pi_\\varphi\\) is irreducible with \\(\\|\\pi_\\varphi(x)\\|=\\|x\\|\\). The direct sum of the GNS representations of all pure states of \\(A\\) is faithful.\n\n**Proof.** First, for a positive operator \\(T\\), \\(\\|T\\|=\\sup_{\\|\\zeta\\|\\le1}\\langle T\\zeta,\\zeta\\rangle\\). Indeed, the form \\((\\zeta,\\zeta')\\mapsto\\langle T\\zeta,\\zeta'\\rangle\\) is positive semidefinite, so the Cauchy–Schwarz inequality gives \\(|\\langle T\\zeta,\\zeta'\\rangle|^2\\le\\langle T\\zeta,\\zeta\\rangle\\langle T\\zeta',\\zeta'\\rangle\\), and we take the supremum over \\(\\zeta,\\zeta'\\) in the unit ball.\n(1) Put \\(B=C^*(A)\\) and \\(y=j(x)\\), which is nonzero because \\(j\\) is injective (Proposition 7.4). Let \\(\\sigma\\) be a faithful representation of \\(B\\) (Theorem 7.2). Since \\(\\sigma(y)\\ne0\\), some unit vector \\(\\zeta\\) has \\(\\langle\\sigma(y)\\zeta,\\zeta\\rangle\\ne0\\), because an operator \\(S\\) with \\(\\langle S\\zeta,\\zeta\\rangle=0\\) for all \\(\\zeta\\) is zero. The functional \\(\\omega(b)=\\langle\\sigma(b)\\zeta,\\zeta\\rangle\\) lies in \\(Q(B)\\), and \\(\\omega(y)\\ne0\\). By Proposition 8.4(4), \\(Q(B)\\) is the weak\\*-closed convex hull of \\(\\{0\\}\\cup P(B)\\), and evaluation at \\(y\\) is weak\\*-continuous and linear. If every pure state vanished at \\(y\\), every element of \\(Q(B)\\) would too. So some pure state \\(\\varphi\\) of \\(B\\) has \\(\\varphi(y)\\ne0\\). By Theorem 8.3, \\(\\pi_\\varphi\\) is irreducible, and \\(\\langle\\pi_\\varphi(y)\\xi_\\varphi,\\xi_\\varphi\\rangle=\\varphi(y)\\ne0\\). By Proposition 7.4(3), \\(\\pi=\\pi_\\varphi\\circ j\\) is an irreducible representation of \\(A\\) with \\(\\pi(x)\\ne0\\).\n(2) Let \\(a=x^*x\\). Every \\(\\omega\\in Q(A)\\) has \\(\\omega(a)\\le\\|a\\|\\). Take a faithful, hence isometric, representation \\(\\sigma\\) of \\(A\\). By the remark at the start, \\(\\|a\\|=\\|\\sigma(a)\\|=\\sup_{\\|\\zeta\\|\\le1}\\langle\\sigma(a)\\zeta,\\zeta\\rangle\\), and each functional \\(b\\mapsto\\langle\\sigma(b)\\zeta,\\zeta\\rangle\\) with \\(\\|\\zeta\\|\\le1\\) lies in \\(Q(A)\\). So \\(\\sup_{\\omega\\in Q(A)}\\omega(a)=\\|a\\|\\). The function \\(\\omega\\mapsto\\omega(a)\\) is weak\\*-continuous on the compact set \\(Q(A)\\), so the set \\(F=\\{\\omega\\in Q(A):\\omega(a)=\\|a\\|\\}\\) is nonempty and compact. It is a face: if \\(t\\omega_1+(1-t)\\omega_2\\in F\\) with \\(\\omega_1,\\omega_2\\in Q(A)\\) and \\(0<t<1\\), then the numbers \\(\\omega_1(a),\\omega_2(a)\\le\\|a\\|\\) average to \\(\\|a\\|\\), so both functionals lie in \\(F\\). By the Krein–Milman theorem, \\(F\\) has an extreme point \\(\\varphi\\), which is then extreme in \\(Q(A)\\). If \\(x\\ne0\\), then \\(\\varphi(a)=\\|a\\|>0\\), so \\(\\varphi\\ne0\\), and \\(\\varphi\\) is a pure state by Proposition 8.4(2). If \\(x=0\\), any pure state will do, and one exists by the case of a nonzero element. By Theorem 8.3, \\(\\pi_\\varphi\\) is irreducible. Then \\(\\|\\pi_\\varphi(x)\\|\\ge\\|\\pi_\\varphi(x)\\xi_\\varphi\\|=\\varphi(x^*x)^{1/2}=\\|x\\|\\), and \\(\\|\\pi_\\varphi(x)\\|\\le\\|x\\|\\). The direct sum \\(\\pi=\\bigoplus_{\\varphi\\in P(A)}\\pi_\\varphi\\) therefore satisfies \\(\\|\\pi(x)\\|\\ge\\|x\\|\\) for every \\(x\\), so it is faithful. \\(\\square\\)\n\n**Examples 8.6.**\n1. The pure states of \\(C_0(X)\\) are the point evaluations. If \\(\\varphi\\) is a pure state, \\(\\pi_\\varphi\\) is irreducible, hence one-dimensional (Example 2.4(2)): \\(\\pi_\\varphi(f)=f(x)1\\) for some \\(x\\in X\\), and \\[\n\\begin{gathered}\n\\varphi(f)\\\\\n=\\langle\\pi_\\varphi(f)\\xi_\\varphi,\\xi_\\varphi\\rangle\\\\\n=f(x)\\|\\xi_\\varphi\\|^2\\\\\n=f(x).\n\\end{gathered}\n\\] Conversely, a point evaluation has a one-dimensional GNS space, so its GNS representation is irreducible and it is pure.\n2. The pure states of \\(M_n(\\mathbb C)\\) are the vector states \\(x\\mapsto\\langle x\\xi,\\xi\\rangle\\) with \\(\\|\\xi\\|=1\\); see Exercise 12.1. The tracial state \\(\\frac1n\\operatorname{Tr}=\\frac1n\\sum_k\\omega_{e_k}\\) is not pure when \\(n\\ge2\\).\n3. On a nonunital C\\*-algebra such as \\(C_0(\\mathbb R)\\), the zero functional lies in the weak\\*-closure of \\(P(A)\\): the point evaluations at \\(n\\) tend to \\(0\\) as \\(n\\to\\infty\\). This is why \\(0\\) appears in Proposition 8.4(4).\n\n## D. Apply the construction to compactness, symmetry and size\n\nThe construction is useful beyond its existence theorem. Compact operators turn irreducibility into rank-one structure; translations give a concrete model of that structure. A group action asks whether the scalar observation is invariant, and the induced unitaries turn invariance and ergodicity into statements about the GNS space. The dimension results finally quantify how the supply of vectors constrains the algebra. Each application keeps its own assumptions rather than importing the hypotheses of a simpler matrix example.\n\n",
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      "id": "OA-FND-GN-11",
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      "name": "9. Irreducible representations and compact operators",
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      "full_conditions_and_proof": "### 9. Irreducible representations and compact operators\n\nThe facts about compact operators that we use are listed under \"Results used from other lessons\".\n\n#### Irreducible algebras of compact operators\n\n**Lemma 9.1.** Let \\(J\\subseteq K(H)\\) be a C\\*-subalgebra.\n1. If \\(k\\in J\\) is self-adjoint and \\(\\lambda\\ne0\\) is a point of \\(\\sigma(k)\\), then \\(J\\) contains a nonzero projection \\(q\\) of finite rank with \\(kq=\\lambda q\\), whose range lies in the range of \\(k\\).\n2. If \\(J\\ne\\{0\\}\\), then \\(J\\) contains a nonzero projection of finite rank.\n\n**Proof.** (1) The point \\(\\lambda\\) is isolated in \\(\\sigma(k)\\), so the function \\(\\chi\\) equal to \\(1\\) at \\(\\lambda\\) and to \\(0\\) elsewhere on \\(\\sigma(k)\\cup\\{0\\}\\) is continuous there and vanishes at \\(0\\). The functional calculus without an identity ([the continuous functional calculus](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-07)) gives \\(q=\\chi(k)\\in J\\). Since \\(\\chi=\\bar\\chi=\\chi^2\\), \\(q\\) is a projection, and \\(\\|q\\|=1\\). Since \\(t\\chi(t)=\\lambda\\chi(t)\\), \\(kq=\\lambda q\\), so the range of \\(q\\) lies in the kernel of \\(k-\\lambda\\), which is finite-dimensional. Writing \\(\\chi(t)=t\\cdot g(t)\\) with \\(g(t)=\\chi(t)/t\\), which is continuous on \\(\\sigma(k)\\cup\\{0\\}\\) and zero at \\(0\\), gives \\(q=k\\,g(k)\\); so the range of \\(q\\) lies in the range of \\(k\\).\n(2) Take \\(y\\ne0\\) in \\(J\\) and \\(k=y^*y\\ne0\\). It is positive, so \\(\\|k\\|\\in\\sigma(k)\\) ([spectra of self-adjoint elements](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-01)), and (1) applies with \\(\\lambda=\\|k\\|\\). \\(\\square\\)\n\n**Theorem 9.2** (Irreducible algebras of compact operators). Let \\(J\\subseteq K(H)\\) be a C\\*-subalgebra with \\(J\\ne\\{0\\}\\) whose only invariant closed subspaces are \\(\\{0\\}\\) and \\(H\\). Then \\(J=K(H)\\).\n\n**Proof.** By Lemma 9.1, \\(J\\) contains nonzero projections of finite rank; choose one, \\(p\\), of smallest rank.\n*Step 1: \\(pJp=\\mathbb Cp\\).* Let \\(y\\in J\\) be self-adjoint and \\(k=pyp\\in J\\). If \\(\\sigma(k)=\\{0\\}\\), then \\(k=0\\), since its norm is its spectral radius. Otherwise let \\(\\lambda\\ne0\\) be in \\(\\sigma(k)\\), and let \\(q\\) be the projection of Lemma 9.1(1). Its range lies in the range of \\(k\\), which lies in \\(pH\\); so \\(q\\le p\\), and \\(q\\) has rank at most that of \\(p\\). By minimality they have the same rank, so \\(q=p\\). Hence \\(kp=\\lambda p\\), and since \\(k=kp\\) (it vanishes on \\((pH)^\\perp\\)), \\(k=\\lambda p\\). For general \\(y\\in J\\), write \\(y=y_1+iy_2\\) with self-adjoint \\(y_j\\in J\\).\n*Step 2: \\(p\\) has rank one.* Let \\(\\xi\\in pH\\) be a unit vector. The closure of \\(J\\xi\\) is invariant and contains \\(\\xi=p\\xi\\), so it is \\(H\\). If \\(p\\) had rank at least two, there would be a unit vector \\(\\eta\\in pH\\) orthogonal to \\(\\xi\\). For \\(y\\in J\\), Step 1 gives \\(pyp=\\lambda_yp\\), so \\(\\langle y\\xi,\\eta\\rangle=\\langle pyp\\xi,\\eta\\rangle=\\lambda_y\\langle\\xi,\\eta\\rangle=0\\). Then \\(\\eta\\) is orthogonal to the dense set \\(J\\xi\\), a contradiction. So \\(p=\\theta_{\\xi,\\xi}\\).\n*Step 3: \\(J\\) contains every rank-one operator.* For \\(y,z\\in J\\) and \\(v\\in H\\), \\(y\\theta_{\\xi,\\xi}z^*v=\\langle z^*v,\\xi\\rangle y\\xi=\\langle v,z\\xi\\rangle y\\xi\\), so \\(ypz^*=\\theta_{y\\xi,z\\xi}\\in J\\). Given \\(\\eta,\\zeta\\in H\\), choose \\(y_n,z_n\\in J\\) with \\(y_n\\xi\\to\\eta\\) and \\(z_n\\xi\\to\\zeta\\). Then \\(\\theta_{y_n\\xi,z_n\\xi}\\to\\theta_{\\eta,\\zeta}\\) in norm, because \\(\\|\\theta_{a,b}-\\theta_{a',b'}\\|\\le\\|a-a'\\|\\|b\\|+\\|a'\\|\\|b-b'\\|\\). As \\(J\\) is closed, \\(\\theta_{\\eta,\\zeta}\\in J\\).\nEvery finite-rank operator is a finite sum of rank-one operators, so \\(J\\) contains them all, and hence their norm closure \\(K(H)\\). \\(\\square\\)\n\n**Corollary 9.3.** Let \\(A\\) be a C\\*-algebra and \\(\\pi\\) an irreducible representation of \\(A\\) on \\(H\\). Then either \\(\\pi(A)\\supseteq K(H)\\) or \\(\\pi(A)\\cap K(H)=\\{0\\}\\).\n\n**Proof.** The range \\(\\pi(A)\\) is a C\\*-subalgebra of \\(B(H)\\) ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)), and it acts irreducibly on \\(H\\). Since \\(K(H)\\) is a closed two-sided ideal of \\(B(H)\\), \\(J=\\pi(A)\\cap K(H)\\) is a closed two-sided ideal of \\(\\pi(A)\\). By Proposition 2.5, applied to the identity representation of \\(\\pi(A)\\), either \\(J=\\{0\\}\\) or \\(J\\) acts irreducibly on \\(H\\). In the second case Theorem 9.2 gives \\(J=K(H)\\). \\(\\square\\)\n\n**Lemma 9.3a** (Closed ideals of \\(B(H)\\)). If \\(H\\) is separable and infinite-dimensional, the only norm-closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\).\n\n**Proof.** The Hilbert-space lesson proves that \\(K(H)\\) is a norm-closed two-sided ideal and the norm closure of the finite-rank operators. It is nonzero and proper: a rank-one operator is compact, while the unit vectors of an infinite orthonormal sequence show that the identity is not compact.\n\nLet \\(J\\) be a nonzero norm-closed two-sided ideal, and choose \\(T\\in J\\) and \\(\\eta\\in H\\) with \\(T\\eta\\ne0\\). For arbitrary \\(u,v\\in H\\), direct calculation gives\n\\[\n\\theta_{u,T\\eta}\\,T\\,\\theta_{\\eta,v}\n=\\|T\\eta\\|^2\\theta_{u,v}.\n\\]\nThus \\(J\\) contains every rank-one operator, every finite-rank operator and, by norm closure, \\(K(H)\\).\n\nSuppose \\(J\\) also contains a noncompact operator \\(T\\), and set \\(S=T^*T\\in J\\). Then \\(S\\) is not compact. Indeed, if it were compact, the compact self-adjoint spectral theorem in the Hilbert-space lesson would give finite-rank spectral projections \\(P_n\\) with \\(\\|(1-P_n)S(1-P_n)\\|\\to0\\). The identity\n\\(\\|T(1-P_n)\\|^2=\\|(1-P_n)S(1-P_n)\\|\\)\nwould make \\(TP_n\\to T\\) in norm, contradicting noncompactness.\n\nFor \\(\\varepsilon>0\\), let \\(F_\\varepsilon=(S-\\varepsilon1)_+\\), defined by the continuous functional calculus already proved in the C*-algebra lesson. Its scalar defining function vanishes at zero, so approximation by polynomials without constant term gives \\(F_\\varepsilon\\in J\\). Also \\(\\|S-F_\\varepsilon\\|\\leq\\varepsilon\\). Therefore for at least one \\(\\varepsilon>0\\) the space \\(K=\\overline{F_\\varepsilon H}\\) is infinite-dimensional: otherwise every \\(F_\\varepsilon\\) would have finite rank and \\(S\\) would be compact.\n\nThe scalar function \\((s-\\varepsilon)\\max(s-\\varepsilon,0)^2\\) is nonnegative on \\(\\sigma(S)\\). Positivity of the continuous calculus gives\n\\[\n\\langle S F_\\varepsilon\\xi,F_\\varepsilon\\xi\\rangle\n\\geq\\varepsilon\\|F_\\varepsilon\\xi\\|^2.\n\\]\nContinuity extends this inequality to every vector in \\(K\\). As \\(H\\) and \\(K\\) are both separable infinite-dimensional Hilbert spaces, matching their countable orthonormal bases gives an isometry \\(V:H\\to K\\subseteq H\\). Hence \\(R=V^*SV\\in J\\) and \\(R\\geq\\varepsilon1\\). Its spectrum lies in \\([\\varepsilon,\\|R\\|]\\), so the continuous calculus applied to \\(s\\mapsto1/s\\) gives an inverse in \\(B(H)\\). The ideal contains \\(R^{-1}R=1\\), and thus \\(J=B(H)\\). This proves the three possibilities. \\(\\square\\)\n\n**Examples 9.4.**\n1. Any C\\*-algebra \\(A\\) with \\(K(H)\\subseteq A\\subseteq B(H)\\), such as \\(K(H)+\\mathbb C1\\), acts irreducibly on \\(H\\) (Example 2.4(1)), with the first alternative. So does the extension of the representation of Theorem 9.5 below to the transformation-group C\\*-algebra (Remark 9.6).\n2. Let \\(H\\) be separable and infinite-dimensional. The only closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\) (Lemma 9.3a above). So the Calkin algebra \\(\\mathcal Q=B(H)/K(H)\\), a unital C\\*-algebra ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)), has no closed two-sided ideals except \\(\\{0\\}\\) and \\(\\mathcal Q\\), because a closed ideal of \\(\\mathcal Q\\) pulls back to a closed ideal of \\(B(H)\\) containing \\(K(H)\\). It is infinite-dimensional: splitting \\(H\\) into infinitely many orthogonal infinite-dimensional subspaces gives infinitely many orthogonal projections that are not compact; their images in \\(\\mathcal Q\\) are nonzero and orthogonal, hence linearly independent. By Theorem 8.5, \\(\\mathcal Q\\) has an irreducible representation \\(\\sigma\\) on some \\(H_\\sigma\\). Its kernel is a proper closed ideal, so \\(\\sigma\\) is faithful. If \\(\\sigma(\\mathcal Q)\\) met \\(K(H_\\sigma)\\), then \\(\\sigma(\\mathcal Q)\\cap K(H_\\sigma)\\) would be a nonzero closed ideal of \\(\\sigma(\\mathcal Q)\\cong\\mathcal Q\\), hence all of \\(\\sigma(\\mathcal Q)\\). Then \\(\\sigma(1)\\), which is the identity operator because \\(\\sigma\\) is nondegenerate, would be compact, and \\(H_\\sigma\\) and \\(\\mathcal Q\\) would be finite-dimensional. So \\(\\sigma\\), composed with the quotient map, is an irreducible representation of \\(B(H)\\) with the second alternative.\n\n",
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      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "Translations on the line",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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      "anchor": "oa-fnd-gn-12",
      "proof_locus": {
        "line": 916,
        "through_line": 968
      },
      "full_conditions_and_proof": "#### Translations on the line\n\nLet \\(\\mathbb R\\) act on itself by translation, \\(\\omega t=\\omega+t\\), with Lebesgue measure as Haar measure and \\(\\Delta=1\\). For the transformation-group algebra \\(\\mathfrak A=\\mathfrak A(\\mathbb R,\\mathbb R)\\) of Proposition 7.7, the operations on \\(C_c(\\mathbb R^2)\\) read\n\\[\n\\begin{gathered}\n(x\\star y)(s,t)\\\\\n=\\int x(s,r)\\,y(s+r,t-r)\\,dr,\\\\\nx^\\sharp(s,t)\\\\\n=\\overline{x(s+t,-t)},\\\\\n\\|x\\|_1\\\\\n=\\int\\sup_s|x(s,t)|\\,dt.\n\\end{gathered}\n\\]\nOn \\(L^2(\\mathbb R)\\), let \\(\\rho(f)\\) be multiplication by \\(f\\in C_0(\\mathbb R)\\) and \\((U(t)\\xi)(s)=\\xi(s+t)\\). Then \\(U(t)\\rho(f)U(t)^*\\) is multiplication by \\(s\\mapsto f(s+t)\\), which is (7.1). The integrated form is\n\\[\n\\begin{gathered}\n(\\pi(x)\\xi)(s)\\\\\n=\\int x(s,t)\\,\\xi(s+t)\\,dt\\\\\n(x\\in C_c(\\mathbb R^2),\\ \\xi\\in L^2(\\mathbb R)).\n\\end{gathered}\n\\tag{9.1}\n\\]\n\n**Theorem 9.5.**\n1. For \\(x\\in C_c(\\mathbb R^2)\\), \\(\\pi(x)\\) is the integral operator with kernel \\(k_x(s,r)=x(s,r-s)\\), that is, \\((\\pi(x)\\xi)(s)=\\int k_x(s,r)\\xi(r)\\,dr\\). It is a Hilbert–Schmidt operator: for every orthonormal basis \\((e_n)\\) of \\(L^2(\\mathbb R)\\), \\(\\sum_n\\|\\pi(x)e_n\\|^2=\\|k_x\\|_{L^2(\\mathbb R^2)}^2=\\|x\\|_{L^2(\\mathbb R^2)}^2\\). In particular \\(\\|\\pi(x)\\|\\le\\|x\\|_{L^2(\\mathbb R^2)}\\), and \\(\\pi(x)\\) is compact.\n2. The map \\(x\\mapsto k_x\\) is a linear bijection of \\(C_c(\\mathbb R^2)\\) onto itself. Moreover \\(\\pi(x\\star y)=\\pi(x)\\pi(y)\\), \\(\\pi(x^\\sharp)=\\pi(x)^*\\) and \\(\\|\\pi(x)\\|\\le\\|x\\|_1\\). So \\(\\pi\\) extends to a representation of \\(\\mathfrak A\\), and \\(\\pi(\\mathfrak A)\\) consists of compact operators.\n3. The norm closure of \\(\\pi(\\mathfrak A)\\) is \\(K(L^2(\\mathbb R))\\). In particular \\(\\pi\\) is irreducible.\n\n**Proof.** (1) Substituting \\(r=s+t\\) in (9.1) gives the kernel. The function \\(k_x\\) is continuous with compact support, because \\((s,t)\\mapsto(s,s+t)\\) is a homeomorphism of \\(\\mathbb R^2\\), and the same substitution gives \\(\\int\\!\\int|k_x(s,r)|^2\\,dr\\,ds=\\int\\!\\int|x(s,t)|^2\\,dt\\,ds\\). For each \\(s\\), \\((\\pi(x)e_n)(s)=\\langle k_x(s,\\cdot),\\bar e_n\\rangle\\), and \\((\\bar e_n)\\) is again an orthonormal basis. By Parseval's identity, \\(\\sum_n|(\\pi(x)e_n)(s)|^2=\\int|k_x(s,r)|^2dr\\), and integrating over \\(s\\) (the terms are nonnegative) gives the Hilbert–Schmidt identity. The Cauchy–Schwarz inequality gives \\(|(\\pi(x)\\xi)(s)|^2\\le\\int|k_x(s,r)|^2dr\\,\\|\\xi\\|^2\\), and integrating gives \\(\\|\\pi(x)\\|\\le\\|k_x\\|_2\\). For compactness, finite sums \\(\\sum_i\\varphi_i(s)\\psi_i(r)\\) with \\(\\varphi_i,\\psi_i\\in C_c(\\mathbb R)\\) are dense in \\(L^2(\\mathbb R^2)\\) ([\\(L^2\\) of a product](haar-measure.md#oa-fnd-hm-11)). The operator with kernel \\(\\varphi(s)\\psi(r)\\) is \\(\\theta_{\\varphi,\\bar\\psi}\\), of rank at most one. By the norm bound, \\(\\pi(x)\\) is a norm limit of finite-rank operators, so it is compact.\n(2) The inverse of \\(x\\mapsto k_x\\) is \\(k\\mapsto x\\) with \\(x(s,t)=k(s,s+t)\\). The kernel of \\(\\pi(x)\\pi(y)\\) is \\[\n\\begin{gathered}\n\\int k_x(s,u)k_y(u,r)\\,du\\\\\n=\\int x(s,u-s)\\,y(u,r-u)\\,du,\n\\end{gathered}\n\\] and substituting \\(u=s+t\\) in \\[\n\\begin{gathered}\nk_{x\\star y}(s,r)\\\\\n=(x\\star y)(s,r-s)\\\\\n=\\int x(s,t)\\,y(s+t,r-s-t)\\,dt\n\\end{gathered}\n\\] gives the same integral. The kernel of \\(\\pi(x)^*\\) is \\(\\overline{k_x(r,s)}=\\overline{x(r,s-r)}\\), and \\(k_{x^\\sharp}(s,r)=x^\\sharp(s,r-s)=\\overline{x(r,s-r)}\\). For the bound, put \\(X(t)=\\sup_s|x(s,t)|\\). By the Cauchy–Schwarz inequality with the weight \\(X\\),\n\\[\n\\begin{gathered}\n|(\\pi(x)\\xi)(s)|^2\\\\\n\\le\\Big(\\int X(t)|\\xi(s+t)|\\,dt\\Big)^2\\\\\n\\le\\|x\\|_1\\int X(t)|\\xi(s+t)|^2\\,dt,\n\\end{gathered}\n\\]\nand integrating over \\(s\\) gives \\(\\|\\pi(x)\\xi\\|^2\\le\\|x\\|_1^2\\|\\xi\\|^2\\). So \\(\\pi\\) is a \\(*\\)-homomorphism on the dense \\(*\\)-subalgebra \\(C_c(\\mathbb R^2)\\) of \\(\\mathfrak A\\), contractive for \\(\\|\\cdot\\|_1\\), and it extends by continuity to a representation of \\(\\mathfrak A\\). Its values are norm limits of compact operators, hence compact.\n(3) By (2), the closure of \\(\\pi(\\mathfrak A)\\) lies in \\(K(L^2(\\mathbb R))\\). Conversely, for \\(\\eta,\\zeta\\in C_c(\\mathbb R)\\), the function \\(x(s,t)=\\eta(s)\\overline{\\zeta(s+t)}\\) lies in \\(C_c(\\mathbb R^2)\\) and \\(k_x(s,r)=\\eta(s)\\overline{\\zeta(r)}\\), so \\(\\pi(x)=\\theta_{\\eta,\\zeta}\\). Since \\(C_c(\\mathbb R)\\) is dense in \\(L^2(\\mathbb R)\\) and \\(\\|\\theta_{a,b}-\\theta_{a',b'}\\|\\le\\|a-a'\\|\\|b\\|+\\|a'\\|\\|b-b'\\|\\), every rank-one operator, hence every finite-rank operator, lies in the closure of \\(\\pi(\\mathfrak A)\\), and so does their norm closure \\(K(L^2(\\mathbb R))\\). Finally, \\(K(L^2(\\mathbb R))\\) acts irreducibly (Example 2.4(1)), and \\(\\pi(\\mathfrak A)\\) has the same invariant subspaces as its norm closure. \\(\\square\\)\n\n**Remark 9.6.** The extension \\(\\tilde\\pi\\) of \\(\\pi\\) to the transformation-group C\\*-algebra \\(C^*(\\mathbb R,\\mathbb R)\\) has closed range, which lies between \\(\\pi(\\mathfrak A)\\) and its closure. So \\(\\tilde\\pi(C^*(\\mathbb R,\\mathbb R))=K(L^2(\\mathbb R))\\): the first alternative of Corollary 9.3, with equality.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-GN-13",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "10. Invariant states of a group action",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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      "anchor": "oa-fnd-gn-13",
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      "full_conditions_and_proof": "### 10. Invariant states of a group action\n\nIn this section \\(A\\) is a C\\*-algebra, \\(G\\) a topological group, and \\(\\alpha:G\\to\\operatorname{Aut}(A)\\) a homomorphism into the group of \\(*\\)-automorphisms of \\(A\\). No continuity of \\(\\alpha\\) is assumed. A state \\(\\varphi\\) is *\\(\\alpha\\)-invariant* if \\(\\varphi\\circ\\alpha_s=\\varphi\\) for all \\(s\\in G\\). Let \\(S^\\alpha\\) be the set of \\(\\alpha\\)-invariant states, and \\(S_\\alpha\\) the set of those \\(\\varphi\\in S^\\alpha\\) for which\n\\[\n\\begin{gathered}\ns\\mapsto\\varphi(y^*\\alpha_s(x))\\ \\\\\n\\text{is continuous on}\\ G\\ \\\\\n\\text{for all}\\ x,y\\in A.\n\\end{gathered}\n\\tag{10.1}\n\\]\nBoth sets are convex.\n\n**Theorem 10.1** (Implementing an invariant state). Let \\(\\varphi\\in S^\\alpha\\) have GNS triple \\((\\pi,H,\\xi)\\).\n1. For each \\(s\\in G\\) there is a unique unitary \\(U(s)\\) on \\(H\\) with \\(U(s)\\pi(x)\\xi=\\pi(\\alpha_s(x))\\xi\\) for all \\(x\\). The map \\(s\\mapsto U(s)\\) is a homomorphism into the unitary group, and\n\\[\n\\begin{gathered}\nU(s)\\xi\\\\\n=\\xi,\\\\\nU(s)\\pi(x)U(s)^*\\\\\n=\\pi(\\alpha_s(x))\\\\\n(s\\in G,\\ x\\in A).\n\\end{gathered}\n\\tag{10.2}\n\\]\nA unitary representation satisfying (10.2) is necessarily this \\(U\\).\n2. The following are equivalent: (a) \\(\\varphi\\in S_\\alpha\\); (b) \\(\\varphi(x^*\\alpha_s(x))\\to\\varphi(x^*x)\\) as \\(s\\to e\\), for every \\(x\\in A\\); (c) \\(U\\) is strongly continuous.\n\n**Proof.** (1) Since \\(\\alpha_s\\) is a \\(*\\)-automorphism and \\(\\varphi\\) is invariant, \\[\n\\begin{gathered}\n\\|\\pi(\\alpha_s(x))\\xi\\|^2\\\\\n=\\varphi(\\alpha_s(x^*x))\\\\\n=\\varphi(x^*x)\\\\\n=\\|\\pi(x)\\xi\\|^2.\n\\end{gathered}\n\\] So \\(\\pi(x)\\xi\\mapsto\\pi(\\alpha_s(x))\\xi\\) is a well-defined linear isometry of the dense subspace \\(\\pi(A)\\xi\\) onto itself (\\(\\alpha_s\\) is onto), and it extends to a unitary \\(U(s)\\). The relations \\(U(st)=U(s)U(t)\\) and \\(U(e)=1\\) follow from \\(\\alpha_{st}=\\alpha_s\\alpha_t\\) and \\(\\alpha_e=\\mathrm{id}\\). On vectors \\(\\pi(y)\\xi\\), \\[\n\\begin{gathered}\nU(s)\\pi(x)\\pi(y)\\xi\\\\\n=\\pi(\\alpha_s(xy))\\xi\\\\\n=\\pi(\\alpha_s(x))U(s)\\pi(y)\\xi,\n\\end{gathered}\n\\] so \\(U(s)\\pi(x)=\\pi(\\alpha_s(x))U(s)\\). For \\(U(s)\\xi=\\xi\\), let \\((u_i)\\) be an approximate identity of positive contractions. The net \\((\\alpha_s(u_i))\\) is again one, because \\(\\alpha_s(u_i)y=\\alpha_s(u_i\\alpha_s^{-1}(y))\\to y\\) and \\(\\alpha_s\\) is isometric. By Proposition 1.4(4), applied to the nondegenerate \\(\\pi\\), both \\(\\pi(u_i)\\xi\\to\\xi\\) and \\(U(s)\\pi(u_i)\\xi=\\pi(\\alpha_s(u_i))\\xi\\to\\xi\\); hence \\(U(s)\\xi=\\xi\\). If a unitary representation \\(V\\) satisfies (10.2), then \\(V(s)\\pi(x)\\xi=\\pi(\\alpha_s(x))V(s)\\xi=\\pi(\\alpha_s(x))\\xi\\), so \\(V=U\\).\n(2) (a) ⇒ (b): take \\(y=x\\) in (10.1) and note \\(\\alpha_e(x)=x\\). (b) ⇒ (c): by invariance,\n\\[\n\\begin{gathered}\n\\|U(s)\\pi(x)\\xi-\\pi(x)\\xi\\|^2\\\\\n=2\\varphi(x^*x)-2\\operatorname{Re}\\varphi(x^*\\alpha_s(x))\\to0\\\\\n(s\\to e).\n\\end{gathered}\n\\]\nFor an arbitrary \\(\\eta\\) and \\(\\eta'\\in\\pi(A)\\xi\\), \\(\\|U(s)\\eta-\\eta\\|\\le2\\|\\eta-\\eta'\\|+\\|U(s)\\eta'-\\eta'\\|\\), so \\(U(s)\\eta\\to\\eta\\) as \\(s\\to e\\). At another point \\(s_0\\), \\(\\|U(s)\\eta-U(s_0)\\eta\\|=\\|U(s_0^{-1}s)\\eta-\\eta\\|\\to0\\) as \\(s\\to s_0\\). (c) ⇒ (a): \\[\n\\begin{gathered}\n\\varphi(y^*\\alpha_s(x))\\\\\n=\\langle\\pi(\\alpha_s(x))\\xi,\\pi(y)\\xi\\rangle\\\\\n=\\langle U(s)\\pi(x)\\xi,\\pi(y)\\xi\\rangle\n\\end{gathered}\n\\] is continuous in \\(s\\). \\(\\square\\)\n\n**Example 10.2** (An invariant state without continuity). The numbers \\(1\\) and \\(\\sqrt2\\) are linearly independent over \\(\\mathbb Q\\). Extend them to a basis of \\(\\mathbb R\\) as a vector space over \\(\\mathbb Q\\), using the basis-extension proof in Example 6.5, and let \\(\\vartheta:\\mathbb R\\to\\mathbb R\\) be the \\(\\mathbb Q\\)-linear map with \\(\\vartheta(1)=0\\), \\(\\vartheta(\\sqrt2)=1\\) and \\(\\vartheta=0\\) on the other basis elements. Then \\(\\vartheta\\) is additive, so \\(\\alpha_t(f)(z)=f(e^{i\\vartheta(t)}z)\\) defines a homomorphism of the usual topological group \\(\\mathbb R\\) into \\(\\operatorname{Aut}(C(\\mathbb T))\\). The state \\(\\varphi(f)=\\frac1{2\\pi}\\int_0^{2\\pi}f(e^{iu})\\,du\\) is invariant. For the coordinate function \\(x(z)=z\\), \\(\\varphi(x^*\\alpha_t(x))=e^{i\\vartheta(t)}\\). Choose rational numbers \\(r_n\\to\\sqrt2\\) and put \\(t_n=\\sqrt2-r_n\\), so \\(t_n\\to0\\). Then \\(\\vartheta(t_n)=1\\), so \\(\\varphi(x^*\\alpha_{t_n}(x))=e^{i}\\) does not tend to \\(\\varphi(x^*x)=1\\). So \\(\\varphi\\in S^\\alpha\\setminus S_\\alpha\\): the unitaries of Theorem 10.1(1) exist, but they do not depend continuously on \\(t\\).\n\n**Theorem 10.3** (Ergodic states). Let \\(\\varphi\\in S_\\alpha\\) with GNS triple \\((\\pi,H,\\xi)\\), and let \\(U\\) be as in Theorem 10.1. The following are equivalent:\n- (i) the only closed subspaces of \\(H\\) invariant under both \\(\\pi(A)\\) and \\(U(G)\\) are \\(\\{0\\}\\) and \\(H\\);\n- (ii) \\(\\pi(A)'\\cap U(G)'=\\mathbb C1\\);\n- (iii) \\(\\varphi\\) is an extreme point of \\(S_\\alpha\\);\n- (iv) \\(\\varphi\\) is an extreme point of \\(S^\\alpha\\).\n\nA state with these properties is called *ergodic*.\n\n**Proof.** (i) ⇔ (ii). Apply Schur's lemma (Theorem 2.2) to the self-adjoint set \\(\\pi(A)\\cup U(G)\\); note \\(U(s)^*=U(s^{-1})\\).\n(ii) ⇒ (iv). Let \\(\\varphi=t\\varphi_1+(1-t)\\varphi_2\\) with \\(\\varphi_1,\\varphi_2\\in S^\\alpha\\) and \\(0<t<1\\). Then \\(t\\varphi_1\\le\\varphi\\), and Lemma 8.1 gives \\(T\\in\\pi(A)'\\), \\(0\\le T\\le1\\), with \\(t\\varphi_1(y^*x)=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\). By invariance of \\(\\varphi_1\\) and Theorem 10.1,\n\\[\n\\begin{gathered}\n\\langle TU(s)\\pi(x)\\xi,U(s)\\pi(y)\\xi\\rangle\\\\\n=t\\varphi_1(\\alpha_s(y)^*\\alpha_s(x))\\\\\n=t\\varphi_1(\\alpha_s(y^*x))\\\\\n=t\\varphi_1(y^*x)\\\\\n=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle.\n\\end{gathered}\n\\]\nSo \\(U(s)^*TU(s)=T\\) on a dense set, hence everywhere, and \\(T\\in U(G)'\\). By (ii), \\(T=\\lambda1\\). Then \\(t\\varphi_1(y^*x)=\\lambda\\varphi(y^*x)\\) for all \\(x,y\\), and since every element of a C\\*-algebra is a product (Corollary 6.2), \\(t\\varphi_1=\\lambda\\varphi\\). Comparing norms gives \\(\\lambda=t\\), so \\(\\varphi_1=\\varphi\\); likewise \\(\\varphi_2=\\varphi\\).\n(iv) ⇒ (iii). This holds because \\(S_\\alpha\\subseteq S^\\alpha\\).\n(iii) ⇒ (ii). Suppose (ii) fails. The set \\(\\pi(A)'\\cap U(G)'\\) is the commutant of the self-adjoint set \\(\\pi(A)\\cup U(G)\\), so by Theorem 2.2 there is a closed subspace other than \\(\\{0\\}\\) and \\(H\\) invariant under both, and by Lemma 1.3 its projection \\(p\\) lies in \\(\\pi(A)'\\cap U(G)'\\), with \\(p\\ne0,1\\). If \\(p\\xi=0\\), then \\(p\\pi(x)\\xi=\\pi(x)p\\xi=0\\) for all \\(x\\), so \\(p=0\\); hence \\(p\\xi\\ne0\\), and likewise \\((1-p)\\xi\\ne0\\). Put \\(c=\\|p\\xi\\|^2\\in(0,1)\\) and\n\\[\n\\begin{gathered}\n\\varphi_1(x)\\\\\n=c^{-1}\\langle\\pi(x)p\\xi,p\\xi\\rangle,\\\\\n\\varphi_2(x)\\\\\n=(1-c)^{-1}\\langle\\pi(x)(1-p)\\xi,(1-p)\\xi\\rangle.\n\\end{gathered}\n\\]\nSince \\(p\\) commutes with \\(\\pi(A)\\), the cross terms in \\(\\langle\\pi(x)\\xi,\\xi\\rangle\\) vanish, and \\(\\varphi=c\\varphi_1+(1-c)\\varphi_2\\). A vector functional \\(\\langle\\pi(\\cdot)\\eta,\\eta\\rangle\\) of the nondegenerate representation \\(\\pi\\) has norm \\(\\|\\eta\\|^2\\), by Theorem 4.7(2) and Proposition 1.4(4); so \\(\\varphi_1\\) and \\(\\varphi_2\\) are states. They are invariant: \\[\n\\begin{gathered}\n\\langle\\pi(\\alpha_s(x))p\\xi,p\\xi\\rangle\\\\\n=\\langle\\pi(x)pU(s)^*\\xi,pU(s)^*\\xi\\rangle\\\\\n=\\langle\\pi(x)p\\xi,p\\xi\\rangle,\n\\end{gathered}\n\\] because \\(p\\) commutes with \\(U(s)\\) and \\(U(s)^*\\xi=\\xi\\). They satisfy (10.1): \\(\\langle\\pi(\\alpha_s(x))p\\xi,\\pi(y)p\\xi\\rangle=\\langle U(s)\\pi(x)p\\xi,\\pi(y)p\\xi\\rangle\\) is continuous in \\(s\\). So \\(\\varphi_1,\\varphi_2\\in S_\\alpha\\). Finally \\(\\varphi_1\\ne\\varphi\\): otherwise \\[\n\\begin{gathered}\n\\langle p\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\\\\n=\\langle\\pi(y^*x)p\\xi,p\\xi\\rangle\\\\\n=c\\varphi(y^*x)\\\\\n=c\\langle\\pi(x)\\xi,\\pi(y)\\xi\\rangle\n\\end{gathered}\n\\] for all \\(x,y\\), giving \\(p=c1\\), which is not a projection. So \\(\\varphi\\) is not extreme in \\(S_\\alpha\\). \\(\\square\\)\n\nFor the trivial group, \\(S_\\alpha=S^\\alpha=S(A)\\), and Theorem 10.3 reduces to Theorem 8.3 together with Proposition 8.4(3).\n\n**Theorem 10.4** (Existence of ergodic states). Suppose that \\(\\alpha\\) is pointwise norm continuous: \\(\\|\\alpha_s(x)-x\\|\\to0\\) as \\(s\\to e\\), for every \\(x\\in A\\).\n1. \\(S_\\alpha=S^\\alpha\\).\n2. If \\(A\\) is unital, \\(S^\\alpha\\) is weak\\*-compact, so it has extreme points whenever it is not empty.\n3. Without an identity, \\(S^\\alpha\\) need not be compact, but it still has extreme points whenever it is not empty.\n\n**Proof.** (1) For a state \\(\\varphi\\), \\[\n\\begin{gathered}\n|\\varphi(y^*\\alpha_s(x))-\\varphi(y^*\\alpha_{s_0}(x))|\\\\\n\\le\\|y\\|\\|\\alpha_s(x)-\\alpha_{s_0}(x)\\|\\\\\n=\\|y\\|\\|\\alpha_{s_0^{-1}s}(x)-x\\|,\n\\end{gathered}\n\\] which tends to \\(0\\) as \\(s\\to s_0\\).\n(2) \\(S^\\alpha\\) is the intersection of the weak\\*-compact set \\(S(A)\\) (Proposition 8.4(4)) with the weak\\*-closed sets \\(\\{\\varphi:\\varphi(\\alpha_s(x))=\\varphi(x)\\}\\). It is convex, and the Krein–Milman theorem applies.\n(3) The set \\(Q^\\alpha\\) of \\(\\alpha\\)-invariant elements of \\(Q(A)\\) is weak\\*-compact and convex by the same argument. If \\(S^\\alpha\\ne\\emptyset\\), then \\(Q^\\alpha\\ne\\{0\\}\\), and by the Krein–Milman theorem \\(Q^\\alpha\\) has an extreme point \\(\\varphi\\ne0\\). As in the proof of Proposition 8.4(2), \\(\\|\\varphi\\|=1\\): otherwise \\(\\varphi\\) is a proper convex combination of \\(\\varphi/\\|\\varphi\\|\\in Q^\\alpha\\) and \\(0\\). So \\(\\varphi\\in S^\\alpha\\), and it is extreme there because \\(S^\\alpha\\subseteq Q^\\alpha\\). Example 10.5(1) shows that compactness can fail. \\(\\square\\)\n\n**Examples 10.5.**\n1. Let \\(A=C_0(\\mathbb R)\\) and let \\(G\\) be the trivial group. Then \\(S^\\alpha=S(A)\\) contains the point evaluations \\(\\delta_n\\), which converge weak\\* to \\(0\\notin S(A)\\), because every \\(f\\in C_0(\\mathbb R)\\) has \\(f(n)\\to0\\). So \\(S^\\alpha\\) is not compact; its extreme points are the point evaluations (Example 8.6(1)).\n2. Let \\(\\mathbb R\\) act on \\(C_0(\\mathbb R)\\) by translation, \\(\\alpha_t(f)(s)=f(s+t)\\). This action is pointwise norm continuous, by the point-norm continuity argument after Lemma 0.1. It has no invariant state. Indeed, let \\(\\varphi\\) be an invariant positive functional and \\(f\\in C_c(\\mathbb R)\\) with \\(0\\le f\\le1\\), supported in \\([-N,N]\\), with \\(N\\geq1\\). The translates \\(\\alpha_{(2N+1)k}(f)\\), \\(k=1,\\dots,m\\), have disjoint supports, so their sum has norm at most \\(1\\), and \\(m\\varphi(f)=\\varphi\\big(\\sum_k\\alpha_{(2N+1)k}(f)\\big)\\le\\|\\varphi\\|\\). So \\(\\varphi(f)=0\\). Such \\(f\\), and their positive multiples, are dense in the positive cone, so \\(\\varphi=0\\) on \\(A_+\\), and \\(\\varphi=0\\).\n3. Let the circle group \\(\\mathbb T\\) act on \\(C(\\mathbb T)\\) by rotation, \\(\\alpha_w(f)(z)=f(wz)\\); the action is pointwise norm continuous. An invariant state \\(\\varphi\\) satisfies \\(\\varphi(z^n)=\\varphi(\\alpha_w(z^n))=w^n\\varphi(z^n)\\) for all \\(w\\in\\mathbb T\\), so \\(\\varphi(z^n)=0\\) for \\(n\\ne0\\), and \\(\\varphi(1)=1\\). The trigonometric polynomials are dense in \\(C(\\mathbb T)\\) ([the complex Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09)), so \\(\\varphi(f)=\\frac1{2\\pi}\\int_0^{2\\pi}f(e^{iu})\\,du\\) is the only invariant state. It is extreme in the one-point set \\(S^\\alpha\\), so it is ergodic. By Theorem 10.3, only the scalars commute with all multiplications and all rotations on \\(L^2(\\mathbb T)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-GN-14",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "11. The dimension of a GNS space",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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        "line": 1096,
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      "full_conditions_and_proof": "### 11. The dimension of a GNS space\n\nLet \\(A\\) be a \\(*\\)-algebra and \\(\\omega\\) a positive functional. Construction 5.1 of \\(H_\\omega\\) and \\(\\Lambda_\\omega\\) uses only Lemma 3.3 and (3.1), so it works for any \\(*\\)-algebra. Put \\(d(\\omega)=\\dim H_\\omega\\), a nonnegative integer or \\(\\infty\\); every infinite dimension is recorded as \\(\\infty\\).\n\n**Proposition 11.1** (Gram determinants). For \\(x_1,\\dots,x_n\\in A\\), the vectors \\(\\Lambda_\\omega(x_1),\\dots,\\Lambda_\\omega(x_n)\\) are linearly independent exactly when the Gram determinant \\(\\det\\big[\\omega(x_j^*x_i)\\big]_{i,j=1}^n\\) is nonzero.\n\n**Proof.** Put \\(v_i=\\Lambda_\\omega(x_i)\\) and \\(G_{ij}=\\omega(x_j^*x_i)=\\langle v_i,v_j\\rangle\\). For \\(c\\in\\mathbb C^n\\) and each \\(j\\), \\(\\sum_ic_iG_{ij}=\\langle\\sum_ic_iv_i,v_j\\rangle\\). If \\(\\sum_ic_iv_i=0\\) with \\(c\\ne0\\), these numbers all vanish, so the rows of \\(G\\) are dependent and \\(\\det G=0\\). Conversely, if \\(\\det G=0\\), some \\(c\\ne0\\) has \\(\\sum_ic_iG_{ij}=0\\) for all \\(j\\). Then \\(v=\\sum_ic_iv_i\\) is orthogonal to every \\(v_j\\), hence to itself, so \\(v=0\\). \\(\\square\\)\n\n**Proposition 11.2** (Lower semicontinuity). Give the set of positive functionals on \\(A\\) the topology of pointwise convergence on \\(A\\); on bounded positive functionals on a normed algebra, this is the weak\\* topology. For every integer \\(m\\ge0\\), the set \\(\\{\\omega:d(\\omega)>m\\}\\) is open. So \\(d\\) is lower semicontinuous.\n\n**Proof.** If \\(d(\\omega)>m\\), the dense subspace \\(\\Lambda_\\omega(A)\\) of \\(H_\\omega\\) contains \\(m+1\\) linearly independent vectors: otherwise it would have dimension at most \\(m\\), hence be closed, and \\(H_\\omega\\) would have dimension at most \\(m\\). Conversely, \\(m+1\\) independent vectors in \\(H_\\omega\\) give \\(d(\\omega)>m\\). By Proposition 11.1,\n\\[\n\\begin{gathered}\n\\{\\omega:d(\\omega)>m\\}\\\\\n=\\bigcup_{x_0,\\dots,x_m\\in A}\\{\\omega:\\det[\\omega(x_j^*x_i)]\\ne0\\}.\n\\end{gathered}\n\\]\nEach set in the union is open, because the determinant is a polynomial in finitely many evaluations \\(\\omega\\mapsto\\omega(x_j^*x_i)\\). \\(\\square\\)\n\n**Example 11.3** (Why all infinite dimensions count as \\(\\infty\\)). Let \\(\\Gamma\\) be an uncountable set and \\(K=\\{0,1\\}^\\Gamma\\), a compact abelian group under coordinatewise addition modulo \\(2\\) with the product topology. Compactness is [Tychonoff’s theorem](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-02); the finite discrete factors are compact, the product is Hausdorff because distinct points differ in a coordinate, and continuity of each coordinate proves continuity of the group operations. Let \\(\\mu\\) be its normalized Haar measure ([existence of Haar measure](haar-measure.md#oa-fnd-hm-05)) and \\(\\omega(f)=\\int f\\,d\\mu\\) on \\(A=C(K)\\). For a finite \\(F\\subseteq\\Gamma\\), the function \\(w_F(\\varepsilon)=\\prod_{\\gamma\\in F}(-1)^{\\varepsilon_\\gamma}\\) is a continuous character of \\(K\\), and \\(w_Fw_{F'}=w_{F\\triangle F'}\\). For \\(F\\ne\\emptyset\\) pick \\(h\\in K\\) with \\(w_F(h)=-1\\); invariance of \\(\\mu\\) gives \\(\\int w_F\\,d\\mu=\\int w_F(\\varepsilon+h)\\,d\\mu(\\varepsilon)=-\\int w_F\\,d\\mu\\), so \\(\\int w_F\\,d\\mu=0\\). Hence the \\(w_F\\) are orthonormal in \\(H_\\omega=L^2(\\mu)\\) (Example 5.7(2)), and there are uncountably many of them.\n\nOn the other hand, for a finite \\(F\\subseteq\\Gamma\\) let \\(\\mu_F\\) be the average of the point masses at the \\(2^{|F|}\\) points \\(\\varepsilon\\) with \\(\\varepsilon_\\gamma=0\\) for \\(\\gamma\\notin F\\). Then \\(d(\\mu_F)=2^{|F|}\\). For \\(S\\subseteq F\\), \\(\\int w_S\\,d\\mu_F\\) is \\(1\\) if \\(S=\\emptyset\\) and \\(0\\) otherwise, which is also \\(\\int w_S\\,d\\mu\\). The linear combinations of the \\(w_S\\) form a unital self-adjoint subalgebra of \\(C(K)\\) that separates points, so they are dense ([the complex Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09)). Since \\(\\|\\mu_F\\|=\\|\\mu\\|=1\\), it follows that \\(\\int f\\,d\\mu_F\\to\\int f\\,d\\mu\\) for every \\(f\\in C(K)\\), along the finite subsets \\(F\\) directed by inclusion. So the functionals \\(\\mu_F\\), each with finite \\(d\\), converge to \\(\\omega\\), whose GNS space has an uncountable orthonormal family. The set of positive functionals whose GNS space has an uncountable orthonormal family is therefore not open: Proposition 11.2 would fail if uncountable dimensions were distinguished from countable ones.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-GN-15",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "12. Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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        "line": 1119,
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      "full_conditions_and_proof": "## 12. Exercises\n\n**Exercise 12.1** (medium; Positive functionals on matrices). Let \\(\\rho\\in M_n(\\mathbb C)\\) be positive with \\(\\operatorname{Tr}\\rho=1\\) and rank \\(r\\), and let \\(\\omega(x)=\\operatorname{Tr}(\\rho x)\\). (a) Show that \\(N_\\omega=\\{x:x\\rho=0\\}\\) and \\(d(\\omega)=nr\\). (b) Show that \\(\\pi_\\omega\\) is unitarily equivalent to the direct sum of \\(r\\) copies of the identity representation of \\(M_n(\\mathbb C)\\) on \\(\\mathbb C^n\\). (c) Deduce that \\(\\omega\\) is pure exactly when \\(r=1\\), that is, when \\(\\omega(x)=\\langle x\\xi,\\xi\\rangle\\) for a unit vector \\(\\xi\\).\n\n*Solution.* (a) \\[\n\\begin{gathered}\n\\omega(x^*x)\\\\\n=\\operatorname{Tr}(\\rho^{1/2}x^*x\\rho^{1/2})\\\\\n=\\operatorname{Tr}\\big((x\\rho^{1/2})^*(x\\rho^{1/2})\\big),\n\\end{gathered}\n\\] which vanishes exactly when \\(x\\rho^{1/2}=0\\). Since \\(\\rho\\) and \\(\\rho^{1/2}\\) have the same range \\(R\\), this means that \\(x=0\\) on \\(R\\), or \\(x\\rho=0\\). Give \\(M_n(\\mathbb C)\\) the inner product \\(\\langle y,z\\rangle=\\operatorname{Tr}(z^*y)\\). The map \\(V:\\Lambda_\\omega(x)\\mapsto x\\rho^{1/2}\\) preserves inner products, since \\(\\operatorname{Tr}(\\rho^{1/2}z^*x\\rho^{1/2})=\\operatorname{Tr}(z^*x\\rho)=\\omega(z^*x)\\). Its range is \\(W=\\{y\\in M_n(\\mathbb C):y=0\\text{ on }R^\\perp\\}\\): every such \\(y\\) equals \\(x\\rho^{1/2}\\) with \\(x=y\\,(\\rho^{1/2}|_R)^{-1}\\) on \\(R\\) and \\(x=0\\) on \\(R^\\perp\\). A matrix in \\(W\\) is determined by its restriction to the \\(r\\)-dimensional space \\(R\\), a linear map from \\(R\\) to \\(\\mathbb C^n\\), so \\(\\dim W=nr\\).\n(b) Let \\(f_1,\\dots,f_r\\) be an orthonormal basis of \\(R\\). The map \\(y\\mapsto(yf_1,\\dots,yf_r)\\) is a unitary from \\(W\\) onto \\((\\mathbb C^n)^r\\), since \\(\\operatorname{Tr}(z^*y)=\\sum_k\\langle yf_k,zf_k\\rangle\\) for \\(y,z\\in W\\) (compute the trace in an orthonormal basis that extends \\((f_k)\\)). Under \\(V\\), \\(\\pi_\\omega(a)\\) becomes left multiplication by \\(a\\) on \\(W\\), and then \\(a\\oplus\\dots\\oplus a\\) on \\((\\mathbb C^n)^r\\).\n(c) The commutant of \\(r\\) copies of the identity representation consists of the \\(r\\times r\\) block matrices whose blocks commute with \\(M_n(\\mathbb C)\\), hence are scalars (Example 2.4(1) and Corollary 2.3); it is a copy of \\(M_r(\\mathbb C)\\). It equals \\(\\mathbb C1\\) exactly when \\(r=1\\). By Theorem 8.3, \\(\\omega\\) is pure exactly when \\(r=1\\), and then \\(\\rho=\\theta_{\\xi,\\xi}\\) for a unit vector \\(\\xi\\), so \\(\\omega(x)=\\langle x\\xi,\\xi\\rangle\\).\n\n**Exercise 12.2** (medium; The trace on a finite group algebra). Let \\(G\\) be a finite group and \\(A=\\mathbb C[G]\\), the functions on \\(G\\) with \\((f*g)(t)=\\sum_sf(s)g(s^{-1}t)\\), \\(f^*(t)=\\overline{f(t^{-1})}\\) and \\(\\|f\\|=\\sum_t|f(t)|\\). Let \\(\\tau(f)=f(e)\\). Show that \\(\\tau\\) is a faithful positive functional, that its GNS representation is the left regular representation \\(\\lambda(f)\\xi=f*\\xi\\) on \\(\\ell^2(G)\\) with cyclic vector \\(\\delta_e\\), and that \\(\\pi_\\tau\\) is irreducible only for the trivial group.\n\n*Solution.* \\[\n\\begin{gathered}\n(f^**g)(e)\\\\\n=\\sum_sf^*(s)g(s^{-1})\\\\\n=\\sum_s\\overline{f(s^{-1})}g(s^{-1})\\\\\n=\\langle g,f\\rangle_{\\ell^2}.\n\\end{gathered}\n\\] So \\(\\tau(f^*f)=\\|f\\|_{\\ell^2}^2\\), and \\(\\tau\\) is positive and faithful. Then \\(N_\\tau=\\{0\\}\\), and \\(\\Lambda_\\tau\\) identifies \\(H_\\tau\\) with \\(\\ell^2(G)\\) (a finite-dimensional space needs no completion), since \\(\\langle\\Lambda_\\tau(g),\\Lambda_\\tau(f)\\rangle=\\tau(f^*g)=\\langle g,f\\rangle_{\\ell^2}\\). Under this identification, \\(\\pi_\\tau(f)g=f*g=\\lambda(f)g\\). The algebra has the identity \\(\\delta_e\\), so \\(\\xi_\\tau=\\Lambda_\\tau(\\delta_e)=\\delta_e\\) (Theorem 5.4). The right translations \\((R_t\\xi)(s)=\\xi(st)\\) commute with every \\(\\lambda(f)\\), since \\[\n\\begin{gathered}\n(f*R_t\\xi)(u)\\\\\n=\\sum_sf(s)\\xi(s^{-1}ut)\\\\\n=(R_t(f*\\xi))(u).\n\\end{gathered}\n\\] For \\(t\\ne e\\), \\(R_t\\) is not a scalar, since \\(R_t\\delta_t=\\delta_e\\). So \\(\\pi_\\tau\\) is irreducible only when \\(G=\\{e\\}\\).\n\n**Exercise 12.3** (medium; Ergodic states of a permutation action). Let a group \\(G\\), with the discrete topology, act on a finite set \\(X\\), and let it act on \\(A=C(X)\\) by \\(\\alpha_s(f)(x)=f(s^{-1}x)\\). (a) Show that the invariant states are the functionals \\(f\\mapsto\\sum_xp(x)f(x)\\) given by probability vectors \\(p\\) that are constant on each orbit. (b) Show that the ergodic states are the uniform distributions on single orbits. (c) For the uniform distribution on an orbit \\(O\\), describe \\((\\pi,H,\\xi,U)\\) and check condition (ii) of Theorem 10.3 directly.\n\n*Solution.* (a) The states of \\(C(X)\\) are the functionals \\(f\\mapsto\\sum_xp(x)f(x)\\) with \\(p\\ge0\\) and \\(\\sum_xp(x)=1\\): positivity at the functions \\(\\delta_x=\\delta_x^*\\delta_x\\) forces \\(p\\ge0\\), and the norm is \\(\\sum_xp(x)=\\varphi(1)\\) (Theorem 4.7(2)). Invariance reads \\(\\sum_xp(x)f(s^{-1}x)=\\sum_yp(sy)f(y)\\) for all \\(f\\), that is, \\(p(sy)=p(y)\\) for all \\(s\\) and \\(y\\).\n(b) Every invariant \\(p\\) is \\(\\sum_Op(O)\\,u_O\\), where \\(u_O\\) is the uniform distribution on the orbit \\(O\\) and \\(p(O)=\\sum_{x\\in O}p(x)\\). If \\(u_O=tp+(1-t)q\\) with invariant \\(p,q\\) and \\(0<t<1\\), then \\(p\\) and \\(q\\) vanish off \\(O\\) and are constant on \\(O\\), so \\(p=q=u_O\\); hence \\(u_O\\) is extreme. A \\(p\\) that gives positive weight to two orbits is a proper convex combination of distinct invariant states, so it is not extreme. Since \\(G\\) is discrete, (10.1) holds automatically, and by Theorem 10.3 the ergodic states are the \\(u_O\\).\n(c) The left kernel of \\(u_O\\) consists of the functions vanishing on \\(O\\). So \\(H=\\ell^2(O)\\), \\(\\pi(f)\\) is multiplication by \\(f|_O\\), \\(\\xi=|O|^{-1/2}1_O\\), and \\((U(s)g)(x)=g(s^{-1}x)\\). An operator commuting with all multiplications preserves each line \\(\\mathbb C\\delta_x\\), because it commutes with the projection \\(\\pi(\\delta_x)\\) onto it; so it is multiplication by some function \\(h\\). Commuting with every \\(U(s)\\) forces \\(h(s^{-1}x)=h(x)\\), and since \\(G\\) acts transitively on \\(O\\), \\(h\\) is constant.\n\n**Exercise 12.4** (easy; Ideals that are not closed). Let \\(A\\) be a C\\*-algebra, \\(\\pi\\) an irreducible representation of \\(A\\) on \\(H\\), and \\(J\\) a two-sided ideal of \\(A\\) that is not assumed to be closed or self-adjoint. Show that either \\(\\pi(J)=\\{0\\}\\), or the only closed subspaces invariant under \\(\\pi(J)\\) are \\(\\{0\\}\\) and \\(H\\).\n\n*Solution.* The closure \\(\\bar J\\) is a closed two-sided ideal, hence self-adjoint, and by Proposition 2.5, \\(\\pi|_{\\bar J}\\) is zero or irreducible. Since \\(\\pi\\) is continuous, \\(\\pi(J)\\) is norm-dense in \\(\\pi(\\bar J)\\). If \\(\\pi(J)=\\{0\\}\\) there is nothing to show. Otherwise \\(\\pi(\\bar J)\\ne\\{0\\}\\), so \\(\\pi|_{\\bar J}\\) is irreducible. A closed subspace invariant under \\(\\pi(J)\\) is invariant under the norm closure of \\(\\pi(J)\\), which contains \\(\\pi(\\bar J)\\), so it is \\(\\{0\\}\\) or \\(H\\).\n\n**Exercise 12.5** (medium; Extending to the unitization). Let \\(A\\) be a C\\*-algebra, \\(\\tilde A=A+\\mathbb C1\\) its C\\*-unitization (see the conventions of [C\\*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md)), and \\(\\omega\\) a positive functional on \\(A\\). Show that \\(\\tilde\\omega(x+\\lambda)=\\omega(x)+\\lambda\\|\\omega\\|\\) is positive on \\(\\tilde A\\), that \\(\\|\\tilde\\omega\\|=\\|\\omega\\|\\), and that it is the only positive extension of \\(\\omega\\) with this norm.\n\n*Solution.* Let \\((u_i)\\) be an approximate identity of positive contractions. Then \\((u_i^2)\\) is one as well, since \\(\\|u_i^2x-x\\|\\le2\\|u_ix-x\\|\\), so \\(\\omega(u_i^2)\\to\\|\\omega\\|\\) by Theorem 4.7(2). For \\(z=x+\\lambda\\in\\tilde A\\), the element \\(zu_i=xu_i+\\lambda u_i\\) lies in \\(A\\), so \\(\\omega(u_iz^*zu_i)\\ge0\\). Now \\[\n\\begin{gathered}\nu_iz^*zu_i\\\\\n=u_ix^*xu_i+\\lambda u_ix^*u_i+\\bar\\lambda u_ixu_i+|\\lambda|^2u_i^2,\n\\end{gathered}\n\\] and \\(u_iyu_i\\to y\\) for every \\(y\\in A\\). Since \\(\\omega\\) is continuous, \\(\\omega(u_iz^*zu_i)\\) tends to \\(\\omega(x^*x+\\lambda x^*+\\bar\\lambda x)+|\\lambda|^2\\|\\omega\\|=\\tilde\\omega(z^*z)\\). So \\(\\tilde\\omega\\) is positive. By Theorem 4.7(2) in the unital algebra \\(\\tilde A\\), \\(\\|\\tilde\\omega\\|=\\tilde\\omega(1)=\\|\\omega\\|\\). If \\(\\Omega\\) is another positive extension with \\(\\|\\Omega\\|=\\|\\omega\\|\\), then \\(\\Omega(1)=\\|\\Omega\\|=\\|\\omega\\|\\) by the same theorem, so \\(\\Omega=\\tilde\\omega\\).\n\n## Results used from other lessons\n\n- *Zorn's lemma* (Theorem 1.1 of [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md), cited below as *the Hahn–Banach lesson*). It gives the maximal family in Proposition 1.5 and the vector-space bases in Examples 6.5 and 10.2.\n- *Convexity in dual spaces.* (i) Separation: if a compact convex set and a closed convex set in a real locally convex space are disjoint, there are a continuous linear functional \\(g\\) and numbers \\(t_1<t_2\\) with \\(g<t_1\\) on the first set and \\(g>t_2\\) on the second (the Hahn–Banach lesson, Theorem 6.3). (ii) Banach–Alaoglu: the closed unit ball of the dual of a normed space is weak\\*-compact (Theorem 3.1 of [Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md), cited below as *the lesson on weak topologies*). (iii) The weak\\* topology is locally convex, and its continuous linear functionals are the evaluations at points of the normed space (the lesson on weak topologies, Theorem 1.2 and Corollary 1.3). (iv) Krein–Milman: a nonempty compact convex set in a Hausdorff locally convex space is the closed convex hull of its extreme points; in particular it has an extreme point (the lesson on weak topologies, Theorem 6.1).\n- *Hilbert spaces* ([Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md), cited below as *the Hilbert-space lesson*: Theorem 2.3, Corollary 3.2 and Theorem 3.1). (i) Every bounded linear functional on \\(H\\) is \\(\\zeta\\mapsto\\langle\\zeta,\\eta\\rangle\\) for a unique \\(\\eta\\in H\\). (ii) If \\(\\langle T\\zeta,\\zeta\\rangle=0\\) for all \\(\\zeta\\), then \\(T=0\\). (iii) For every bounded sesquilinear form \\(\\beta\\) on \\(H\\) there is a unique \\(T\\in B(H)\\) with \\(\\beta(\\zeta,\\zeta')=\\langle T\\zeta,\\zeta'\\rangle\\), and \\(\\|T\\|=\\sup\\{|\\beta(\\zeta,\\zeta')|:\\|\\zeta\\|,\\|\\zeta'\\|\\le1\\}\\).\n- *Compact operators* (the Hilbert-space lesson, Theorems 5.1 and 7.1). An operator \\(T\\in B(H)\\) is compact if it maps the unit ball to a set with compact closure. The compact operators form a norm-closed two-sided ideal \\(K(H)\\) of \\(B(H)\\), closed under adjoints, and \\(K(H)\\) is the norm closure of the set of finite-rank operators. If \\(T\\) is compact and \\(\\lambda\\ne0\\) lies in \\(\\sigma(T)\\), then \\(\\lambda\\) is an isolated point of \\(\\sigma(T)\\) and the kernel of \\(T-\\lambda\\) is finite-dimensional.\n- *Closed ideals of \\(B(H)\\)* (Lemma 9.3a above). If \\(H\\) is separable and infinite-dimensional, the only closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\). This is used only in Example 9.4(2).\n- *Measure theory and vector integration.* [Proposition 3.1(4) and Theorem 6.2 of the Haar lesson](haar-measure.md#oa-fnd-hm-11) prove \\(C_c\\) density in scalar \\(L^2\\) and density of finite sums of products in \\(L^2\\) of a Radon product. [Theorem 5.1](haar-measure.md#oa-fnd-hm-09) proves Radon-product Tonelli and Fubini on sigma-finite supports, including completed classes by Borel representatives. [Theorems 2.1–2.2 and 3.1–3.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md) prove scalar convergence, Minkowski and completeness. Lemma 0.1 above proves the Bochner integrals, vector Fubini and Hilbert-valued tensor identification needed by the group examples. Parseval is [Theorem 4.1(3) of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-04). The real-line measure is constructed by interval covers in [Section 1 of the measure-tools lesson](measure-and-hilbert-space-tools.md#1-from-an-outer-measure-to-a-measure).\n- *The binomial series.* Put \\(c_n=\\binom{1/2}{n}\\). Then \\(|c_n|\\le1\\), because \\(|c_{n+1}/c_n|=|n-\\frac12|/(n+1)<1\\). And \\(\\sum_{k=0}^nc_kc_{n-k}\\) is \\(1\\) for \\(n\\le1\\) and \\(0\\) for \\(n\\ge2\\). Indeed, Vandermonde's identity \\(\\sum_k\\binom ak\\binom b{n-k}=\\binom{a+b}n\\) holds for all positive integers \\(a,b\\) by counting subsets, both sides are polynomials in \\((a,b)\\). Fix a positive integer \\(b\\); their difference, as a polynomial in \\(a\\), has infinitely many integer roots and is zero. Each coefficient of that polynomial is itself a polynomial in \\(b\\), zero at every positive integer, hence zero. Thus the identity holds for all complex \\(a,b\\), in particular \\(a=b=\\frac12\\), where the right side is \\(\\binom1n\\). So \\(\\big(\\sum_nc_nz^n\\big)^2=1+z\\) as power series, which is what Lemma 4.1 uses.\n- *Group and transformation-group representations.* The full forward/reverse correspondence, intertwiner preservation and regular faithfulness are [Sections Integration–Regular of the scalar provider](24-group-representation-completions.md#oa-flow.grp.integration) and [Sections Integration–Regular of the coefficient provider](25-cstar-covariant-representations.md#oa-flow.ccov.integration). The full free-tree norm-gap proof is [the scalar provider’s Models section](24-group-representation-completions.md#oa-flow.grp.models). Their Haar existence, modular/inversion, translation and approximate-identity inputs are [Sections 8–11 and 14–15 of the Haar lesson](haar-measure.md#oa-fnd-hm-05). Banach unitization and spectral radius are [Sections 3–5 of the Banach-algebra lesson](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-04); positivity, C*-unitization, approximate identities and closed-range isometry are [the continuous-calculus lesson](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md). The faithful nondegenerate coefficient representation used by the regular model follows from Theorem 7.2 and Proposition 1.4 here. Thus these examples depend only on the already established part of this lesson; the later invariant-state theorem in the coefficient provider is not an input. The examples are unused by other course proofs.\n\n## Where this leads\n\n- A state can be written as an average of pure states in many ways. [Integral representations of states](integral-representations-of-states.md) develops the theory of such decompositions.\n- The direct sum of the GNS representations of all states leads to the universal enveloping von Neumann algebra; see [The universal enveloping von Neumann algebra of a C\\*-algebra, and W\\*-algebras](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md).\n- Stinespring's theorem in [Completely positive maps](completely-positive-maps.md) extends the GNS construction from states to completely positive maps.\n- Weights are positive functionals that may take infinite values. Their GNS construction, the C\\*-case of the cyclic vector, and the splitting of a hermitian functional into positive parts are treated in the lessons *Finite weight domains and GNS*, *Bounded functionals and cyclic vectors* and *States and the positive spanning family* of the course *Modular Theory & Weights*.\n\n## References\n\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n- **Westerbaan.** A. A. Westerbaan, *The Category of Von Neumann Algebras*, doctoral thesis, Radboud University, 2019. [Human-authored LaTeX source](https://github.com/westerbaan/theses/blob/bff9e58239a125af7d77a7ebacd686a21f761e42/cstar.tex), [thesis record](https://arxiv.org/abs/1804.02203). CC BY 4.0; the credited worked checkpoint is adapted from this source.\n\n*Freely accessible reading:* [Kristin Courtney; Elizabeth Gillaspy; Lara Ismert, *Notes on C*-algebras: Notes and Exercises for GOALS*, §7.8–§7.12](https://www.ipam.ucla.edu/wp-content/uploads/2024/07/Notes_and_Exercises_for_GOALS.pdf) gives a route through unital GNS and norming-state constructions; omitted verifications and the nonunital and Banach-* cases are proved here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-GN-16",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "Vector integration used by the group examples",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
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      "anchor": "oa-fnd-gn-16",
      "proof_locus": {
        "line": 653,
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      "full_conditions_and_proof": "#### Vector integration used by the group examples\n\nThe following elementary construction closes the vector-valued integration inputs of the two group-representation providers. Scalar completeness, Minkowski, monotone convergence and Tonelli are [Theorems 2.1–2.2 and 3.1–3.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md). Haar density and the Radon-product Fubini theorem are [Proposition 3.1(4) and Theorem 5.1 of the Haar lesson](haar-measure.md#oa-fnd-hm-02). The product theorem is used on the sigma-finite supports of the integrands; no global sigma-finiteness of the group is required.\n\n**Lemma 0.1** (Vector integration and Hilbert-valued sections). Let \\(E\\) be a Banach space and let \\(\\mu\\) be Haar measure on a locally compact Hausdorff group \\(G\\).\n\n1. Finite-valued simple \\(E\\)-valued functions supported on sets of finite measure have an integral satisfying \\(\\|\\int f\\,d\\mu\\|\\leq\\int\\|f\\|\\,d\\mu\\). Their completion in this last norm is \\(L^1(G,E)\\). Integration and every bounded linear map extend to the completion, and bounded linear maps commute with integration. The space \\(C_c(G,E)\\) is dense there.\n2. For an \\(E\\)-valued integrand on a Radon product that is a norm limit in \\(L^1\\) of simple functions, with sigma-finite support, the two iterated Bochner integrals exist almost everywhere and equal the product integral. In particular this holds for the compactly supported continuous integrands and their \\(L^1\\) limits used below.\n3. For a Hilbert space \\(K\\), finite-valued simple functions complete in the norm \\((\\int\\|f\\|^2)^{1/2}\\) to a Hilbert space \\(L^2(G,K)\\). The finite sums \\(\\sum_jh_j\\eta_j\\), with \\(h_j\\in C_c(G)\\) and \\(\\eta_j\\in K\\), are dense. The map \\(h\\otimes\\eta\\mapsto[s\\mapsto h(s)\\eta]\\) extends to a unitary \\(L^2(G)\\otimes K\\to L^2(G,K)\\).\n\n**Proof.** (1) Write a simple function using pairwise disjoint measurable sets, \\(f=\\sum_ja_j1_{D_j}\\), and set \\(\\int f=\\sum_j\\mu(D_j)a_j\\). Refining two finite partitions into their intersections proves that this definition is independent of the representation; changes on null sets have integral zero. The triangle inequality gives the claimed bound. Thus a Cauchy sequence in the integral norm has Cauchy integrals in the complete space \\(E\\), so the integral extends uniquely to the completion. If \\(T:E\\to F\\) is bounded and linear, the simple-function identity \\(T\\int f=\\int Tf\\) and the estimate \\(\\|Tf\\|_1\\leq\\|T\\|\\|f\\|_1\\) give the same statements after completion.\n\nFor clarity, the completion has the usual function interpretation. From a Cauchy sequence of simple functions choose a subsequence \\((f_n)\\) with \\(\\sum_n\\|f_{n+1}-f_n\\|_1<\\infty\\). Scalar Tonelli shows that \\(\\sum_n\\|f_{n+1}(s)-f_n(s)\\|<\\infty\\) almost everywhere. Consequently \\(f_n(s)\\) has a limit in \\(E\\) there, and the integrated tail bound proves convergence in \\(L^1\\). The pointwise limit has an almost-everywhere separable range and is measurable as a limit of simple functions. Conversely such an \\(L^1\\) limit represents its completion element uniquely up to a null set. For \\(f\\in C_c(G,E)\\), cover its compact norm image by finitely many balls of radius \\(\\varepsilon\\), and partition its compact support into the measurable preimages, choosing the first ball when they overlap. The resulting simple function has error at most \\(\\varepsilon\\mu(\\operatorname{supp}f)\\). Thus \\(f\\) belongs to the completion. Finally, a simple function \\(\\sum_ja_j1_{D_j}\\) is approximated by \\(\\sum_ja_jh_j\\), where scalar Haar density makes \\(h_j\\in C_c(G)\\) close to \\(1_{D_j}\\) in \\(L^1\\); the error is bounded by \\(\\sum_j\\|a_j\\|\\|h_j-1_{D_j}\\|_1\\). This proves density.\n\n(2) For a simple \\(E\\)-valued function the assertion follows term by term from scalar Radon-product Fubini. For a general integrand choose simple approximants \\(F_n\\) with \\(\\sum_n\\|F_{n+1}-F_n\\|_{L^1}<\\infty\\). All their supports can be kept inside one countable union of the original finite-measure pieces. Scalar Tonelli for \\(\\sum_n\\|F_{n+1}-F_n\\|\\) shows that, for almost every first coordinate, the second-coordinate functions are Cauchy in \\(L^1(E)\\); the same holds with coordinates interchanged. Their limits represent the sections of \\(F\\), because the summable pointwise tails have the same almost-everywhere limit. Moreover\n\\[\n\\begin{gathered}\n\\int\\left\\|\\int(F-F_n)(s,t)\\,d\\mu_2(t)\\right\\|d\\mu_1(s)\n\\\\\n\\leq\\|F-F_n\\|_{L^1(\\mu_1\\times\\mu_2)}\\longrightarrow0.\n\\end{gathered}\n\\]\nThe corresponding reversed estimate holds as well. Passing to the limit in the simple-function identities proves the assertion. A compactly supported continuous function on a product has a compact norm image and admits the finite-ball simple approximations from (1), now on its finite-measure compact support. For arbitrary locally compact groups, countably many compact supports lie in a sigma-compact open subgroup: include a relatively compact open identity neighborhood, the supports, their inverses and all finite products. Its Haar measure is sigma-finite, since each compact set has finite measure. This gives the claimed local reduction for every particular limit calculation.\n\n(3) On simple \\(K\\)-valued functions put \\(\\langle f,g\\rangle=\\int\\langle f(s),g(s)\\rangle\\,d\\mu(s)\\). Scalar Cauchy–Schwarz shows boundedness in the two \\(L^2\\) norms, so it extends to the completion, which is a Hilbert space. Its function interpretation follows as in (1): choose a subsequence with summable \\(L^2\\) differences. Minkowski for finite sums and monotone convergence imply\n\\[\n\\begin{gathered}\n\\left\\|\\sum_n\\|f_{n+1}(\\cdot)-f_n(\\cdot)\\|\\right\\|_{L^2}\n\\\\\n\\leq\\sum_n\\|f_{n+1}-f_n\\|_{L^2}<\\infty.\n\\end{gathered}\n\\]\nThe norm sum is therefore finite almost everywhere, giving a pointwise limit and the same \\(L^2\\) tail estimate. Scalar \\(C_c\\) density approximates each finite-measure indicator in \\(L^2\\), and\n\\(\\|\\sum_j\\eta_j(h_j-1_{D_j})\\|_{L^2}\\leq\\sum_j\\|\\eta_j\\|\\|h_j-1_{D_j}\\|_{L^2}\\).\nThis proves the asserted vector density. For finite tensor sums, direct integration gives\n\\[\n\\begin{gathered}\n\\left\\langle\\sum_i h_i\\eta_i,\\sum_j g_j\\zeta_j\\right\\rangle\n\\\\\n=\\sum_{i,j}\\langle h_i,g_j\\rangle_{L^2(G)}\\langle\\eta_i,\\zeta_j\\rangle_K.\n\\end{gathered}\n\\]\nThis is exactly the defining inner product of the Hilbert tensor product, whose construction is [Theorem 8.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-08). The resulting isometry extends to the completion and is onto by the density just proved. \\(\\square\\)\n\nWe will also need point-norm continuity of a continuous right action on \\(C_0(\\Omega)\\). Here is the compactness argument. For \\(a\\in C_c(\\Omega)\\) and a relatively compact identity neighborhood \\(V\\), all supports of \\(\\alpha_s(a)\\), \\(s\\in\\overline V\\), lie in the compact set \\((\\operatorname{supp}a)\\overline V^{-1}\\). On this compact set, continuity of \\((\\omega,s)\\mapsto a(\\omega s)-a(\\omega)\\), and a finite cover of its zero slice at \\(s=e\\), give a bound tending uniformly to zero as \\(s\\to e\\); off the set both functions vanish. Thus \\(\\|\\alpha_s(a)-a\\|_\\infty\\to0\\). Density of \\(C_c(\\Omega)\\) in \\(C_0(\\Omega)\\), proved in [Proposition 2.1(3) of the Stone–Weierstrass lesson](stone-weierstrass-c0.md#oa-fnd-sw-01), and the isometry of each \\(\\alpha_s\\), extend this to every \\(a\\in C_0(\\Omega)\\).\n\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-ST-01",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "1. Bounded Borel functions",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
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      "anchor": "oa-fnd-st-01",
      "proof_locus": {
        "line": 39,
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      },
      "full_conditions_and_proof": "## 1. Bounded Borel functions\n\n**Lemma 1.1.** Let \\(K\\) be a metric space. Let \\(\\mathcal M\\) be a set of bounded complex functions on \\(K\\) with two properties:\n- \\(\\mathcal M\\) contains every bounded continuous function;\n- \\(\\mathcal M\\) contains the limit of every boundedly convergent sequence in \\(\\mathcal M\\).\n\nThen \\(\\mathcal M\\) contains every bounded Borel function on \\(K\\).\n\n**Proof.** The intersection of all sets with the two properties again has them. Call it \\(\\mathcal B\\); then \\(\\mathcal B\\subseteq\\mathcal M\\). Any subset of \\(\\mathcal B\\) with the two properties equals \\(\\mathcal B\\).\n\n*(a) \\(\\mathcal B\\) is closed under sums, products, scalar multiples and complex conjugation.*\n- Fix a bounded continuous \\(f\\). The set of \\(g\\in\\mathcal B\\) with \\(f+g\\in\\mathcal B\\) and \\(fg\\in\\mathcal B\\) contains the bounded continuous functions. It is closed under bounded convergence: if \\(g_n\\to g\\) boundedly, then \\(f+g_n\\to f+g\\) and \\(fg_n\\to fg\\) boundedly. So this set is \\(\\mathcal B\\).\n- Now fix \\(g\\in\\mathcal B\\). The set of \\(f\\in\\mathcal B\\) with \\(f+g\\in\\mathcal B\\) and \\(fg\\in\\mathcal B\\) contains the bounded continuous functions, by the first step. It is closed under bounded convergence for the same reason. So it is \\(\\mathcal B\\).\n- In the same way, the set of \\(g\\in\\mathcal B\\) with \\(cg\\in\\mathcal B\\) for all \\(c\\in\\mathbb C\\) and \\(\\bar g\\in\\mathcal B\\) is \\(\\mathcal B\\).\n\n*(b) \\(\\Sigma=\\{E\\subseteq K:1_E\\in\\mathcal B\\}\\) is a \\(\\sigma\\)-algebra.*\n- \\(K\\in\\Sigma\\), because \\(1_K\\) is continuous.\n- \\(1_{K\\setminus E}=1-1_E\\) and \\(1_{E\\cap F}=1_E1_F\\), so \\(\\Sigma\\) is closed under complements and finite intersections, hence under finite unions.\n- If \\(E_n\\in\\Sigma\\), the indicators of \\(E_1\\cup\\dots\\cup E_N\\) converge boundedly to the indicator of \\(\\bigcup_nE_n\\).\n\n*(c) \\(\\Sigma\\) contains the open sets, hence all Borel sets.* Let \\(U\\neq K\\) be open. The functions \\(g_n(x)=\\min\\big(1,n\\operatorname{dist}(x,K\\setminus U)\\big)\\) are continuous with values in \\([0,1]\\). They converge to \\(1_U\\) at every point, because \\(\\operatorname{dist}(x,K\\setminus U)>0\\) for \\(x\\in U\\). So \\(U\\in\\Sigma\\), and \\(\\Sigma\\) contains the \\(\\sigma\\)-algebra generated by the open sets.\n\n*(d) Conclusion.* By (a)–(c), \\(\\mathcal B\\) contains the Borel simple functions, the finite combinations of indicators of Borel sets. Every bounded Borel function \\(f\\) is a uniform limit of such functions. For real \\(f\\) with \\(0\\leq f\\leq M\\), take\n\\[\n\\begin{gathered}\nf_n\\\\\n=\\sum_{0\\leq j\\leq nM}\\frac jn\\,1_{\\{j/n\\leq f<(j+1)/n\\}},\\\\\n0\\leq f-f_n<\\tfrac1n ,\n\\end{gathered}\n\\]\nand split a complex \\(f\\) into the positive and negative parts of its real and imaginary parts. A uniformly convergent sequence of bounded functions converges boundedly. So \\(f\\in\\mathcal B\\subseteq\\mathcal M\\). \\(\\square\\)\n\nConversely, every function in \\(\\mathcal B\\) is Borel, since the bounded Borel functions have the two properties by the preceding measurability argument. So \\(\\mathcal B=B_b(K)\\).\n\n**Corollary 1.2** (Measures are determined by continuous functions). Let \\(K\\) be a metric space, let \\(\\mu_1,\\dots,\\mu_m\\) be finite measures on the Borel sets of \\(K\\), and let \\(c_1,\\dots,c_m\\in\\mathbb C\\). Suppose that\n\\[\n\\sum_jc_j\\int f\\,d\\mu_j=0\n\\]\nfor every bounded continuous \\(f\\). Then the same holds for every bounded Borel \\(f\\). In particular, two finite Borel measures on \\(K\\) with the same integrals of bounded continuous functions are equal.\n\n**Proof.** Let \\(\\mathcal M\\) be the set of bounded Borel \\(f\\) with \\(\\sum_jc_j\\int f\\,d\\mu_j=0\\). It contains the bounded continuous functions. It is closed under bounded convergence, by dominated convergence for each \\(\\mu_j\\) and because limits of Borel functions are Borel. Lemma 1.1 applies. For the last statement, take \\(m=2\\), \\(c=(1,-1)\\) and \\(f=1_E\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-ST-02",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "2. The spectral measure of a vector",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
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        "line": 81,
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      "full_conditions_and_proof": "## 2. The spectral measure of a vector\n\nFix a self-adjoint \\(h\\in B(H)\\) with spectrum \\(S\\).\n\n**Proposition 2.1.** For every \\(\\xi\\in H\\) there is exactly one Radon measure \\(\\mu_\\xi\\) on \\(S\\) with\n\\[\n\\langle f(h)\\xi,\\xi\\rangle=\\int_Sf\\,d\\mu_\\xi\\qquad(f\\in C(S)).\n\\]\nIt is finite, with \\(\\mu_\\xi(S)=\\|\\xi\\|^2\\). For \\(c\\in\\mathbb C\\), \\(\\mu_{c\\xi}=|c|^2\\mu_\\xi\\).\n\n**Proof.** The functional \\(I_\\xi(f)=\\langle f(h)\\xi,\\xi\\rangle\\) on \\(C(S)\\) is linear.\n- *It is positive.* If \\(f\\geq0\\), put \\(g=\\sqrt f\\). Then \\(g\\) is real, so \\(g(h)\\) is self-adjoint, and \\(f(h)=g(h)^2\\). Hence \\(\\langle f(h)\\xi,\\xi\\rangle=\\|g(h)\\xi\\|^2\\geq0\\).\n- *The measure.* \\(S\\) is compact, so \\(C(S)=C_c(S)\\). The Riesz representation theorem ([Haar measure on locally compact groups](haar-measure.md), Theorem 2.2) gives exactly one Radon measure \\(\\mu_\\xi\\) on \\(S\\) that represents \\(I_\\xi\\).\n- *Its total mass.* \\(\\mu_\\xi(S)\\) is finite, since \\(S\\) is compact. It equals \\(I_\\xi(1)=\\langle\\xi,\\xi\\rangle\\), since \\(1(h)=1\\).\n- *Scaling.* \\(|c|^2\\mu_\\xi\\) is a Radon measure that represents \\(I_{c\\xi}=|c|^2I_\\xi\\). By uniqueness, it is \\(\\mu_{c\\xi}\\). \\(\\square\\)\n\nWe also regard \\(\\mu_\\xi\\) as a measure on the Borel sets of \\(\\mathbb R\\), by \\(\\mu_\\xi(\\Delta)=\\mu_\\xi(\\Delta\\cap S)\\). It is the *spectral measure* of \\(\\xi\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-ST-03",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "3. The Borel functional calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
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      "anchor": "oa-fnd-st-03",
      "proof_locus": {
        "line": 99,
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      "full_conditions_and_proof": "## 3. The Borel functional calculus\n\n**Theorem 3.1.** Let \\(h\\in B(H)\\) be self-adjoint with spectrum \\(S\\). For every \\(f\\in B_b(S)\\) there is exactly one operator \\(f(h)\\in B(H)\\) with\n\\[\n\\begin{gathered}\n\\langle f(h)\\xi,\\xi\\rangle\\\\\n=\\int_Sf\\,d\\mu_\\xi\\\\\n(\\xi\\in H).\n\\end{gathered}\n\\tag{3.1}\n\\]\nThe map \\(f\\mapsto f(h)\\) has the following properties.\n1. For continuous \\(f\\) it is the continuous functional calculus.\n2. It is a unital \\(*\\)-homomorphism: it is linear, \\((fg)(h)=f(h)g(h)\\), \\(\\bar f(h)=f(h)^*\\) and \\(1(h)=1\\).\n3. \\(\\|f(h)\\xi\\|^2=\\int_S|f|^2\\,d\\mu_\\xi\\). Hence \\(\\|f(h)\\|\\leq\\|f\\|_S\\). If \\(f\\geq0\\), then \\(f(h)\\geq0\\). If \\(f\\) is real, then \\(f(h)\\) is self-adjoint.\n4. (*Bounded convergence.*) If \\(f_n\\to f\\) boundedly on \\(S\\), then \\(f_n(h)\\xi\\to f(h)\\xi\\) for every \\(\\xi\\in H\\).\n5. \\(f(h)\\) commutes with every operator that commutes with \\(h\\).\n6. \\(\\sigma(f(h))\\) is contained in the closure of \\(f(S)\\).\n\nFor a bounded Borel function \\(f\\) on \\(\\mathbb R\\) we write \\(f(h)\\) for \\((f|_S)(h)\\).\n\n**Proof.** *Uniqueness.* Two operators \\(T,T'\\) with \\(\\langle T\\xi,\\xi\\rangle=\\langle T'\\xi,\\xi\\rangle\\) for every \\(\\xi\\) are equal (the Hilbert-space lesson, Corollary 3.2).\n\n*A sesquilinear form.* For \\(f\\in B_b(S)\\) and \\(\\xi,\\eta\\in H\\) put\n\\[\n\\beta_f(\\xi,\\eta)=\\frac14\\sum_{k=0}^3i^k\\int_Sf\\,d\\mu_{\\xi+i^k\\eta}.\n\\]\nFor continuous \\(f\\), \\(\\beta_f(\\xi,\\eta)=\\langle f(h)\\xi,\\eta\\rangle\\). This is the polarization identity (the Hilbert-space lesson, Proposition 1.1(1)) for the sesquilinear form \\((\\xi,\\eta)\\mapsto\\langle f(h)\\xi,\\eta\\rangle\\).\n\n*\\(\\beta_f\\) is sesquilinear for every \\(f\\in B_b(S)\\).*\n- Fix \\(\\xi,\\xi',\\eta\\). The number \\(\\beta_f(\\xi+\\xi',\\eta)-\\beta_f(\\xi,\\eta)-\\beta_f(\\xi',\\eta)\\) has the form \\(\\sum_jc_j\\int f\\,d\\mu_j\\) for finitely many finite measures \\(\\mu_j\\) on \\(S\\), which do not depend on \\(f\\). It vanishes for continuous \\(f\\), so it vanishes for every \\(f\\in B_b(S)\\), by Corollary 1.2.\n- The same argument gives \\(\\beta_f(c\\xi,\\eta)=c\\beta_f(\\xi,\\eta)\\).\n- It also gives \\(\\beta_{\\bar f}(\\eta,\\xi)=\\overline{\\beta_f(\\xi,\\eta)}\\). For continuous \\(f\\) this reads \\(\\langle f(h)^*\\eta,\\xi\\rangle=\\overline{\\langle f(h)\\xi,\\eta\\rangle}\\), and both sides are combinations of integrals of \\(\\bar f\\).\n- Conjugate-linearity in the second variable follows from the last two points.\n\n*The diagonal.* By Proposition 2.1, \\(\\mu_{(1+i^k)\\xi}=|1+i^k|^2\\mu_\\xi\\), and \\(\\sum_ki^k|1+i^k|^2=4+2i+0-2i=4\\). So \\(\\beta_f(\\xi,\\xi)=\\int_Sf\\,d\\mu_\\xi\\).\n\n*A bound.* \\(|\\int f\\,d\\mu_\\zeta|\\leq\\|f\\|_S\\|\\zeta\\|^2\\), and \\(\\sum_k\\|\\xi+i^k\\eta\\|^2=4(\\|\\xi\\|^2+\\|\\eta\\|^2)\\), because the cross terms cancel. So\n\\[\n|\\beta_f(\\xi,\\eta)|\\leq\\|f\\|_S\\big(\\|\\xi\\|^2+\\|\\eta\\|^2\\big).\n\\]\nFor \\(\\xi,\\eta\\neq0\\), replace \\(\\xi\\) by \\(t\\xi\\) and \\(\\eta\\) by \\(t^{-1}\\eta\\) with \\(t=(\\|\\eta\\|/\\|\\xi\\|)^{1/2}\\). This does not change \\(\\beta_f(\\xi,\\eta)\\), and gives \\(|\\beta_f(\\xi,\\eta)|\\leq2\\|f\\|_S\\|\\xi\\|\\|\\eta\\|\\).\n\n*Existence.* By the Hilbert-space lesson, Theorem 3.1, there is \\(f(h)\\in B(H)\\) with \\(\\langle f(h)\\xi,\\eta\\rangle=\\beta_f(\\xi,\\eta)\\). It satisfies (3.1). For continuous \\(f\\), the continuous calculus satisfies (3.1) by the definition of \\(\\mu_\\xi\\). By uniqueness, the two agree, which is (1).\n\n*Weak continuity.* Let \\(f_n\\to f\\) boundedly. Then \\(\\int f_n\\,d\\mu_\\zeta\\to\\int f\\,d\\mu_\\zeta\\) for every \\(\\zeta\\), so\n\\[\n\\begin{gathered}\n\\langle f_n(h)\\xi,\\eta\\rangle\\\\\n=\\beta_{f_n}(\\xi,\\eta)\\to\\beta_f(\\xi,\\eta)\\\\\n=\\langle f(h)\\xi,\\eta\\rangle .\n\\end{gathered}\n\\tag{3.2}\n\\]\n\n(2) Linearity, \\(1(h)=1\\) and \\(\\bar f(h)=f(h)^*\\) follow from (3.1) and uniqueness. For example,\n\\[\n\\begin{gathered}\n\\langle\\bar f(h)\\xi,\\xi\\rangle\\\\\n=\\int\\bar f\\,d\\mu_\\xi\\\\\n=\\overline{\\langle f(h)\\xi,\\xi\\rangle}\\\\\n=\\langle f(h)^*\\xi,\\xi\\rangle .\n\\end{gathered}\n\\]\n*Multiplicativity, first step.* Let \\(\\mathcal M_1\\) be the set of \\(f\\in B_b(S)\\) with \\((fg)(h)=f(h)g(h)\\) for every \\(g\\in C(S)\\).\n- It contains \\(C(S)\\), by the C\\*-algebra lesson, Theorem 5.1(2).\n- It is closed under bounded convergence. Let \\(f_n\\to f\\) boundedly with \\(f_n\\in\\mathcal M_1\\), and let \\(g\\in C(S)\\). Then \\(f_ng\\to fg\\) boundedly. By (3.2), applied to \\(f_ng\\) and to \\(f_n\\) at the vector \\(g(h)\\xi\\),\n\\[\n\\begin{gathered}\n\\langle(fg)(h)\\xi,\\eta\\rangle\\\\\n=\\lim_n\\langle(f_ng)(h)\\xi,\\eta\\rangle\\\\\n=\\lim_n\\langle f_n(h)g(h)\\xi,\\eta\\rangle\\\\\n=\\langle f(h)g(h)\\xi,\\eta\\rangle .\n\\end{gathered}\n\\]\nBy Lemma 1.1, \\(\\mathcal M_1=B_b(S)\\).\n\n*Multiplicativity, second step.* Let \\(\\mathcal M_2\\) be the set of \\(g\\in B_b(S)\\) with \\((fg)(h)=f(h)g(h)\\) for every \\(f\\in B_b(S)\\).\n- It contains \\(C(S)\\), by the first step.\n- It is closed under bounded convergence, because \\(\\langle f(h)g_n(h)\\xi,\\eta\\rangle=\\langle g_n(h)\\xi,f(h)^*\\eta\\rangle\\) and (3.2) applies to the \\(g_n\\).\n\nSo \\(\\mathcal M_2=B_b(S)\\).\n\n(3) By (2) and (3.1),\n\\[\n\\begin{gathered}\n\\|f(h)\\xi\\|^2\\\\\n=\\langle f(h)^*f(h)\\xi,\\xi\\rangle\\\\\n=\\langle(\\bar ff)(h)\\xi,\\xi\\rangle\\\\\n=\\int|f|^2\\,d\\mu_\\xi\\\\\n\\leq\\|f\\|_S^2\\,\\mu_\\xi(S)\\\\\n=\\|f\\|_S^2\\|\\xi\\|^2 .\n\\end{gathered}\n\\]\nIf \\(f\\geq0\\), then \\(\\langle f(h)\\xi,\\xi\\rangle=\\int f\\,d\\mu_\\xi\\geq0\\) for every \\(\\xi\\), so \\(f(h)\\geq0\\). If \\(f\\) is real, then \\(f(h)^*=\\bar f(h)=f(h)\\).\n\n(4) By (2) and (3), \\(\\|(f_n(h)-f(h))\\xi\\|^2=\\int|f_n-f|^2\\,d\\mu_\\xi\\). This tends to \\(0\\) by dominated convergence.\n\n(5) Let \\(Th=hT\\). Since \\(h=h^*\\), \\(T\\) commutes with every \\(f(h)\\), \\(f\\in C(S)\\) (the C\\*-algebra lesson, Theorem 5.1(6)). The set of \\(f\\in B_b(S)\\) with \\(Tf(h)=f(h)T\\) is closed under bounded convergence, by (4). By Lemma 1.1 it is \\(B_b(S)\\).\n\n(6) Let \\(\\lambda\\) lie outside the closure of \\(f(S)\\). Then \\(g=(f-\\lambda)^{-1}\\) is a bounded Borel function on \\(S\\). By (2), \\(g(h)\\) is an inverse of \\(f(h)-\\lambda\\). \\(\\square\\)\n\nIn (6) the inclusion can be strict (Exercise 2).\n\n",
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      "id": "OA-FND-ST-04",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "4. Projection-valued measures and the spectral theorem",
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      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
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      "full_conditions_and_proof": "## 4. Projection-valued measures and the spectral theorem\n\n**Definition 4.1.** Let \\(X\\) be a metric space; we use \\(X=\\mathbb R\\) and \\(X=\\mathbb C\\). A *projection-valued measure* on \\(X\\), acting on \\(H\\), is a map \\(E\\) from the Borel sets of \\(X\\) to the projections of \\(B(H)\\) with two properties:\n- \\(E(X)=1\\);\n- for every \\(\\xi\\in H\\), the function \\(\\mu^E_\\xi(\\Delta)=\\langle E(\\Delta)\\xi,\\xi\\rangle=\\|E(\\Delta)\\xi\\|^2\\) is a measure.\n\n\\(E\\) is *supported* by a Borel set \\(Y\\) if \\(E(Y)=1\\).\n\n**Lemma 4.2.** Let \\(E\\) be a projection-valued measure.\n1. \\(E(\\varnothing)=0\\). If \\(\\Delta\\cap\\Delta'=\\varnothing\\), then \\(E(\\Delta\\cup\\Delta')=E(\\Delta)+E(\\Delta')\\) and \\(E(\\Delta)E(\\Delta')=0\\).\n2. \\(E(\\Delta\\cap\\Delta')=E(\\Delta)E(\\Delta')=E(\\Delta')E(\\Delta)\\) for all Borel sets \\(\\Delta,\\Delta'\\).\n3. If \\(\\Delta\\) is the union of disjoint Borel sets \\(\\Delta_n\\), then \\(E(\\Delta)\\xi=\\sum_nE(\\Delta_n)\\xi\\) for every \\(\\xi\\in H\\).\n\n**Proof.** (1) \\(\\|E(\\varnothing)\\xi\\|^2=\\mu^E_\\xi(\\varnothing)=0\\) for every \\(\\xi\\).\n- For disjoint \\(\\Delta,\\Delta'\\), additivity of \\(\\mu^E_\\xi\\) gives \\(\\langle E(\\Delta\\cup\\Delta')\\xi,\\xi\\rangle=\\langle(E(\\Delta)+E(\\Delta'))\\xi,\\xi\\rangle\\) for every \\(\\xi\\). So \\(E(\\Delta\\cup\\Delta')=E(\\Delta)+E(\\Delta')\\) (the Hilbert-space lesson, Corollary 3.2).\n- If \\(P\\), \\(Q\\) and \\(P+Q\\) are projections, then \\((P+Q)^2=P+Q\\) gives \\(PQ+QP=0\\). Multiplying by \\(P\\) on the left and on the right gives \\(PQ+PQP=0=PQP+QP\\). So \\(PQ=QP\\), and then \\(2PQ=0\\).\n\n(2) The sets \\(\\Delta\\cap\\Delta'\\), \\(\\Delta\\setminus\\Delta'\\) and \\(\\Delta'\\setminus\\Delta\\) are disjoint. By (1),\n\\[\n\\begin{gathered}\nE(\\Delta)\\\\\n=E(\\Delta\\cap\\Delta')+E(\\Delta\\setminus\\Delta'),\\\\\nE(\\Delta')\\\\\n=E(\\Delta\\cap\\Delta')+E(\\Delta'\\setminus\\Delta),\n\\end{gathered}\n\\]\nand the products of the projections of two disjoint sets vanish. Multiplying out gives \\(E(\\Delta)E(\\Delta')=E(\\Delta\\cap\\Delta')\\), and likewise in the other order.\n\n(3) The vectors \\(E(\\Delta_n)\\xi\\) are mutually orthogonal, by (1). Also by (1),\n\\[\n\\begin{gathered}\n\\Big\\|E(\\Delta)\\xi-\\sum_{n\\leq N}E(\\Delta_n)\\xi\\Big\\|^2\\\\\n=\\Big\\|E\\Big(\\Delta\\setminus\\bigcup_{n\\leq N}\\Delta_n\\Big)\\xi\\Big\\|^2\\\\\n=\\mu^E_\\xi\\Big(\\Delta\\setminus\\bigcup_{n\\leq N}\\Delta_n\\Big).\n\\end{gathered}\n\\]\nThis tends to \\(0\\), because \\(\\mu^E_\\xi\\) is a finite measure. \\(\\square\\)\n\n**Proposition 4.3** (Integration against a projection-valued measure). Let \\(E\\) be a projection-valued measure on \\(X\\). There is exactly one linear map \\(f\\mapsto\\int f\\,dE\\) from \\(B_b(X)\\) to \\(B(H)\\) with\n\\[\n\\begin{gathered}\n\\int1_\\Delta\\,dE\\\\\n=E(\\Delta)\\\\\n\\text{for every Borel set }\\Delta,\\\\\n\\Big\\|\\int f\\,dE\\Big\\|\\\\\n\\leq\\|f\\|_X .\n\\end{gathered}\n\\]\nIt is a unital \\(*\\)-homomorphism, and \\(\\big\\langle\\big(\\int f\\,dE\\big)\\xi,\\xi\\big\\rangle=\\int f\\,d\\mu^E_\\xi\\).\n\n**Proof.** *Simple functions.* For a Borel simple function \\(f=\\sum_jc_j1_{\\Delta_j}\\) with disjoint \\(\\Delta_j\\), put \\(\\int f\\,dE=\\sum_jc_jE(\\Delta_j)\\).\n- This does not depend on the representation: two representations with disjoint sets have a common refinement, and Lemma 4.2(1) shows that refining does not change the sum.\n- The vectors \\(E(\\Delta_j)\\xi\\) are orthogonal, and \\(\\sum_j\\|E(\\Delta_j)\\xi\\|^2=\\|E(\\bigcup_j\\Delta_j)\\xi\\|^2\\leq\\|\\xi\\|^2\\). So\n\\[\n\\begin{gathered}\n\\Big\\|\\Big(\\int f\\,dE\\Big)\\xi\\Big\\|^2\\\\\n=\\sum_j|c_j|^2\\|E(\\Delta_j)\\xi\\|^2\\\\\n\\leq\\|f\\|_X^2\\|\\xi\\|^2 .\n\\end{gathered}\n\\]\n- On simple functions the map is linear and multiplicative (Lemma 4.2(2)), it preserves the involution (each \\(E(\\Delta)\\) is self-adjoint), it is unital, and \\(\\langle(\\int f\\,dE)\\xi,\\xi\\rangle=\\sum_jc_j\\mu^E_\\xi(\\Delta_j)=\\int f\\,d\\mu^E_\\xi\\).\n\n*Extension.* The simple functions are dense in \\(B_b(X)\\) for \\(\\|\\cdot\\|_X\\) (Lemma 1.1, step (d)). A linear contraction defined on them extends uniquely to a linear contraction on \\(B_b(X)\\). The identities above pass to uniform limits, since \\(\\mu^E_\\xi\\) is finite. Any linear contraction with \\(\\int1_\\Delta\\,dE=E(\\Delta)\\) agrees with this one on simple functions, hence everywhere. \\(\\square\\)\n\n**Theorem 4.4** (Spectral theorem). Let \\(h\\in B(H)\\) be self-adjoint with spectrum \\(S\\).\n1. \\(E_h(\\Delta)=1_\\Delta(h)\\), for Borel sets \\(\\Delta\\subseteq\\mathbb R\\), is a projection-valued measure on \\(\\mathbb R\\), supported by \\(S\\subseteq[-\\|h\\|,\\|h\\|]\\). For every bounded Borel function \\(f\\) on \\(\\mathbb R\\), \\(\\int f\\,dE_h=f(h)\\). In particular\n\\[\nh=\\int\\iota_S\\,dE_h ,\n\\]\nwhere \\(\\iota_S(\\lambda)=\\lambda\\) on \\(S\\) and \\(\\iota_S=0\\) off \\(S\\). This is usually written \\(h=\\int\\lambda\\,dE_h(\\lambda)\\).\n2. (*Uniqueness.*) Let \\(E\\) be a projection-valued measure on \\(\\mathbb R\\), supported by a compact set \\(L\\), with \\(h=\\int\\iota_L\\,dE\\). Then \\(E=E_h\\).\n3. For \\(\\xi\\in H\\), the spectral measure satisfies \\(\\mu_\\xi(\\Delta)=\\langle E_h(\\Delta)\\xi,\\xi\\rangle=\\|E_h(\\Delta)\\xi\\|^2\\), \\(\\mu_\\xi(\\mathbb R)=\\|\\xi\\|^2\\), and \\(\\langle f(h)\\xi,\\xi\\rangle=\\int f\\,d\\mu_\\xi\\).\n4. Each \\(E_h(\\Delta)\\) commutes with every operator that commutes with \\(h\\).\n\n**Proof.** (1) \\(1_\\Delta\\) is real and \\(1_\\Delta^2=1_\\Delta\\), so \\(E_h(\\Delta)\\) is a projection, by Theorem 3.1(2). Also:\n- \\(E_h(\\mathbb R)=1(h)=1\\);\n- \\(\\langle E_h(\\Delta)\\xi,\\xi\\rangle=\\mu_\\xi(\\Delta\\cap S)\\) is a measure in \\(\\Delta\\);\n- \\(E_h(S)=1\\);\n- \\(S\\subseteq[-\\|h\\|,\\|h\\|]\\) (Conventions).\n\nThe maps \\(f\\mapsto\\int f\\,dE_h\\) and \\(f\\mapsto f(h)\\) are linear contractions on the bounded Borel functions on \\(\\mathbb R\\) (Proposition 4.3 and Theorem 3.1(3)). They agree on indicators, hence on simple functions, hence everywhere. For \\(f=\\iota_S\\), \\(f|_S\\) is the identity function of \\(S\\), and the continuous calculus sends it to \\(h\\).\n\n(2) Let \\(L'=L\\cup S\\), a compact set.\n- \\(E(\\mathbb R\\setminus L)=0\\), so \\(\\mu^E_\\xi\\) is carried by \\(L\\).\n- \\(\\int\\cdot\\,dE\\) is a unital homomorphism, so \\(\\int(p\\circ\\iota_L)\\,dE=p(h)\\) for every polynomial \\(p\\). Since \\(p\\circ\\iota_L=p\\) on \\(L\\),\n\\[\n\\int_Lp\\,d\\mu^E_\\xi=\\langle p(h)\\xi,\\xi\\rangle=\\int_Sp\\,d\\mu_\\xi .\n\\]\n- So the finite measures \\(\\mu^E_\\xi\\) and \\(\\mu_\\xi\\), restricted to the Borel subsets of \\(L'\\), have the same integrals of polynomials. The polynomials are uniformly dense in \\(C(L')\\) ([the Stone–Weierstrass lesson](stone-weierstrass-c0.md#oa-fnd-sw-17)), so the two measures have the same integrals of continuous functions on \\(L'\\). By Corollary 1.2 they are equal.\n- Both vanish off \\(L'\\). So \\(\\langle E(\\Delta)\\xi,\\xi\\rangle=\\mu_\\xi(\\Delta)=\\langle E_h(\\Delta)\\xi,\\xi\\rangle\\) for every \\(\\xi\\), and \\(E(\\Delta)=E_h(\\Delta)\\).\n\n(3) restates the definitions with Proposition 2.1 and (3.1). (4) is Theorem 3.1(5). \\(\\square\\)\n\n**Remark 4.5** (Exponentials). For \\(t\\in\\mathbb R\\), \\(e^{ith}\\) denotes \\(f(h)\\) with \\(f(\\lambda)=e^{it\\lambda}\\). Since \\(f\\) is continuous, this is the continuous calculus. The partial sums of \\(\\sum_n(it\\lambda)^n/n!\\) converge to \\(f\\) uniformly on \\(S\\), and the calculus is isometric. So \\(e^{ith}\\) is the sum of the norm-convergent series \\(\\sum_n(ith)^n/n!\\). It is unitary, because \\(\\bar ff=1\\).\n\n",
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      "id": "OA-FND-ST-05",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "5. Spectral projections and polar decomposition",
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      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
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      "full_conditions_and_proof": "## 5. Spectral projections and polar decomposition\n\nThe projections \\(E_h(\\Delta)\\) are the *spectral projections* of \\(h\\).\n\n**Proposition 5.1.** Let \\(h\\in B(H)\\) be self-adjoint, with spectrum \\(S\\) and \\(E=E_h\\).\n1. For every \\(n\\geq1\\) there are disjoint Borel sets \\(\\Delta_1,\\dots,\\Delta_m\\) and real numbers \\(c_1,\\dots,c_m\\) with \\(\\|h-\\sum_jc_jE(\\Delta_j)\\|\\leq1/n\\). So \\(h\\) is a norm limit of real linear combinations of its spectral projections.\n2. If \\(h\\geq0\\), the \\(c_j\\) can be taken \\(\\geq0\\) and the \\(\\Delta_j\\) inside \\([1/n,\\infty)\\). So \\(h\\) is a norm limit of nonnegative combinations of spectral projections of Borel sets bounded away from \\(0\\).\n3. \\(\\ker h=E(\\{0\\})H\\).\n4. Let \\(h\\geq0\\). Then \\(E((0,\\infty))\\) is the projection onto the closure of \\(hH\\). As \\(\\varepsilon\\downarrow0\\), the operators \\(h(h+\\varepsilon)^{-1}\\) increase and converge strongly to \\(E((0,\\infty))\\).\n\n**Proof.** (1) Let \\(s_n(\\lambda)=\\lfloor n\\lambda\\rfloor/n\\). Then \\(0\\leq\\lambda-s_n(\\lambda)<1/n\\), and on the bounded set \\(S\\) the function \\(s_n\\) takes finitely many values. So\n\\[\ns_n(h)=\\sum_k\\frac kn\\,E\\big([k/n,(k+1)/n)\\big),\n\\]\na finite sum. By Theorem 3.1(3), \\(\\|h-s_n(h)\\|\\leq\\sup_S|\\lambda-s_n(\\lambda)|\\leq1/n\\).\n\n(2) If \\(h\\geq0\\), then \\(S\\subseteq[0,\\infty)\\) (the C\\*-algebra lesson, Theorem 8.2). The term with \\(k=0\\) vanishes, so only the sets \\([k/n,(k+1)/n)\\) with \\(k\\geq1\\) occur.\n\n(3) \\(hE(\\{0\\})=(\\iota1_{\\{0\\}})(h)=0\\), since \\(\\lambda1_{\\{0\\}}(\\lambda)=0\\). Conversely, let \\(h\\xi=0\\).\n- Then \\(\\int\\lambda^2\\,d\\mu_\\xi(\\lambda)=\\|h\\xi\\|^2=0\\), by Theorem 3.1(3).\n- So \\(\\mu_\\xi(S\\setminus\\{0\\})=0\\).\n- Hence \\(\\|E(\\{0\\})\\xi\\|^2=\\mu_\\xi(\\{0\\})=\\|\\xi\\|^2\\), and \\(E(\\{0\\})\\xi=\\xi\\).\n\n(4) A vector \\(\\eta\\) is orthogonal to \\(hH\\) exactly when \\(h^*\\eta=h\\eta=0\\).\n- By the Hilbert-space lesson, Theorem 2.2(3), the closure of \\(hH\\) is \\((\\ker h)^\\perp\\).\n- By (3), \\[\n\\begin{gathered}\n(\\ker h)^\\perp\\\\\n=(1-E(\\{0\\}))H\\\\\n=E(S\\setminus\\{0\\})H\\\\\n=E((0,\\infty))H\n\\end{gathered}\n\\], because \\(S\\subseteq[0,\\infty)\\).\n- Let \\(f_\\varepsilon(\\lambda)=\\lambda/(\\lambda+\\varepsilon)\\) on \\([0,\\infty)\\). By the continuous calculus, \\(f_\\varepsilon(h)=h(h+\\varepsilon)^{-1}\\).\n- As \\(\\varepsilon\\) decreases, \\(f_\\varepsilon\\) increases, so \\(f_\\varepsilon(h)\\) increases (Theorem 3.1(3)).\n- \\[\n\\begin{gathered}\n\\|(E((0,\\infty))-f_\\varepsilon(h))\\xi\\|^2\\\\\n=\\int(1_{(0,\\infty)}-f_\\varepsilon)^2\\,d\\mu_\\xi\n\\end{gathered}\n\\]. The integrand decreases to \\(0\\) at every point of \\([0,\\infty)\\) as \\(\\varepsilon\\downarrow0\\). So the integral decreases, and it tends to \\(0\\) along \\(\\varepsilon=1/n\\) by dominated convergence. \\(\\square\\)\n\nA *partial isometry* is an operator \\(u\\) that is isometric on a closed subspace \\(L\\), its *initial space*, and zero on \\(L^\\perp\\). Its range \\(uH=u(L)\\) is closed, the *final space*.\n\n**Proposition 5.2** (Polar decomposition). Let \\(x\\in B(H)\\) and \\(|x|=(x^*x)^{1/2}\\). There is exactly one operator \\(u\\) with \\(x=u|x|\\) and \\(\\ker u=\\ker x\\). It is a partial isometry with initial space \\((\\ker x)^\\perp\\), the closure of \\(|x|H\\), and final space the closure of \\(xH\\). Moreover \\(u^*u\\) and \\(uu^*\\) are the projections onto these two spaces, and \\(u^*x=|x|\\).\n\n**Proof.**\n- *An isometry.* \\(\\||x|\\xi\\|^2=\\langle x^*x\\xi,\\xi\\rangle=\\|x\\xi\\|^2\\). So \\(|x|\\xi\\mapsto x\\xi\\) is a well-defined isometry from \\(|x|H\\) onto \\(xH\\). It extends to an isometry from the closure \\(L\\) of \\(|x|H\\) onto a closed subspace, which contains \\(xH\\) and lies in its closure, so it is the closure of \\(xH\\).\n- *The kernel.* By the norm identity, \\(\\ker|x|=\\ker x\\). Since \\(|x|\\) is self-adjoint, \\(L^\\perp=\\ker|x|\\) (the Hilbert-space lesson, Theorem 2.2). So \\(L=(\\ker x)^\\perp\\).\n- *The operator.* Let \\(u\\) be the isometry on \\(L\\) and \\(0\\) on \\(L^\\perp\\). Then \\(u|x|=x\\) and \\(\\ker u=L^\\perp=\\ker x\\).\n- *Uniqueness.* If \\(x=v|x|\\) and \\(\\ker v=\\ker x\\), then \\(v=u\\) on \\(|x|H\\), hence on \\(L\\) by continuity, and \\(v=0=u\\) on \\(L^\\perp=\\ker x\\).\n- *The projections.* \\(\\langle u^*u\\xi,\\xi\\rangle=\\|u\\xi\\|^2=\\|P_L\\xi\\|^2=\\langle P_L\\xi,\\xi\\rangle\\), where \\(P_L\\) is the projection onto \\(L\\). So \\(u^*u=P_L\\) (the Hilbert-space lesson, Corollary 3.2). Then \\(uP_L=u\\), so \\(uu^*\\) is a self-adjoint idempotent, and its range is \\(uH\\), the final space. Finally \\(u^*x=u^*u|x|=P_L|x|=|x|\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hilbert-spaces-and-compact-operators",
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    },
    {
      "id": "OA-FND-ST-06",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "6. Monotone convergence of operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
      "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "anchor": "oa-fnd-st-06",
      "proof_locus": {
        "line": 351,
        "through_line": 381
      },
      "full_conditions_and_proof": "## 6. Monotone convergence of operators\n\n**Theorem 6.1** (Vigier). Let \\((a_i)_{i\\in I}\\) be an increasing net of self-adjoint operators on \\(H\\) with \\(C=\\sup_i\\|a_i\\|<\\infty\\). Then:\n- there is a self-adjoint \\(a\\in B(H)\\) with \\(a_i\\xi\\to a\\xi\\) for every \\(\\xi\\);\n- \\(a\\) is the least upper bound of the \\(a_i\\) among self-adjoint operators.\n\nThe same holds for decreasing nets, with the greatest lower bound.\n\n**Proof.** *The limit.*\n- For each \\(\\xi\\), the number \\(\\langle a_i\\xi,\\xi\\rangle\\) is real, increases with \\(i\\), and is at most \\(C\\|\\xi\\|^2\\). So it converges.\n- By polarization (the Hilbert-space lesson, Proposition 1.1(1)), \\(\\beta(\\xi,\\eta)=\\lim_i\\langle a_i\\xi,\\eta\\rangle\\) exists for all \\(\\xi,\\eta\\).\n- \\(\\beta\\) is sesquilinear, \\(|\\beta(\\xi,\\eta)|\\leq C\\|\\xi\\|\\|\\eta\\|\\), and \\(\\beta(\\xi,\\xi)\\) is real.\n- By the Hilbert-space lesson, Theorem 3.1, there is \\(a\\in B(H)\\) with \\(\\langle a\\xi,\\eta\\rangle=\\beta(\\xi,\\eta)\\). It is self-adjoint by Corollary 3.2 of that lesson.\n\n*The least upper bound.* \\(\\langle a\\xi,\\xi\\rangle=\\sup_i\\langle a_i\\xi,\\xi\\rangle\\), so \\(a\\geq a_i\\) for every \\(i\\). If \\(b\\) is self-adjoint and \\(b\\geq a_i\\) for every \\(i\\), then \\(\\langle b\\xi,\\xi\\rangle\\geq\\langle a\\xi,\\xi\\rangle\\), so \\(b\\geq a\\).\n\n*Strong convergence.* Put \\(b_i=a-a_i\\geq0\\).\n- \\(\\|a\\|\\leq C\\), by the Hilbert-space lesson, Corollary 3.2, so \\(\\|b_i\\|\\leq2C\\).\n- For a positive operator \\(b\\), \\(b^2\\leq\\|b\\|b\\), because \\(\\lambda^2\\leq\\|b\\|\\lambda\\) on \\(\\sigma(b)\\subseteq[0,\\|b\\|]\\) (Theorem 3.1(3)).\n- Hence\n\\[\n\\begin{gathered}\n\\|b_i\\xi\\|^2\\\\\n=\\langle b_i^2\\xi,\\xi\\rangle\\\\\n\\leq2C\\langle b_i\\xi,\\xi\\rangle\\\\\n=2C\\big(\\langle a\\xi,\\xi\\rangle-\\langle a_i\\xi,\\xi\\rangle\\big)\\to0 .\n\\end{gathered}\n\\]\n\nFor a decreasing net, apply this to \\((-a_i)\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "hilbert-spaces-and-compact-operators",
        "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
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    },
    {
      "id": "OA-FND-ST-07",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "7. Calkin's theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
      "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "anchor": "oa-fnd-st-07",
      "proof_locus": {
        "line": 382,
        "through_line": 409
      },
      "full_conditions_and_proof": "## 7. Calkin's theorem\n\n**Theorem 7.1** (Calkin). Let \\(H\\) be a separable infinite-dimensional Hilbert space. The closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\).\n\n**Proof.** \\(K(H)\\) is a closed ideal (the Hilbert-space lesson, Theorem 5.1). It is not \\(B(H)\\), since an orthonormal sequence has no convergent subsequence, so the identity is not compact. Let \\(J\\neq\\{0\\}\\) be a closed two-sided ideal.\n\n*\\(J\\) contains \\(K(H)\\).* Take \\(x\\in J\\) and \\(\\zeta\\in H\\) with \\(x\\zeta\\neq0\\). For \\(\\alpha,\\beta\\in H\\),\n\\[\n\\theta_{\\alpha,x\\zeta}\\,x\\,\\theta_{\\zeta,\\beta}=\\|x\\zeta\\|^2\\,\\theta_{\\alpha,\\beta}.\n\\]\nIndeed, both sides send \\(\\gamma\\) to \\(\\langle\\gamma,\\beta\\rangle\\|x\\zeta\\|^2\\alpha\\). So \\(J\\) contains every rank-one operator, hence every finite-rank operator. Since \\(J\\) is closed, \\(K(H)\\subseteq J\\) (the Hilbert-space lesson, Theorem 5.1(3)).\n\n*If \\(J\\neq K(H)\\), then \\(J=B(H)\\).* Take \\(x\\in J\\setminus K(H)\\).\n- *\\(a=x^*x\\in J\\) is positive and not compact.* Suppose \\(a\\) were compact. Let \\((\\xi_n)\\) be bounded, with \\(\\|\\xi_n\\|\\leq M\\). Some subsequence has \\(a\\xi_n\\) convergent, and along it\n\\[\n\\begin{gathered}\n\\|x(\\xi_n-\\xi_m)\\|^2\\\\\n=\\langle a(\\xi_n-\\xi_m),\\xi_n-\\xi_m\\rangle\\\\\n\\leq2M\\|a(\\xi_n-\\xi_m)\\|\\to0 .\n\\end{gathered}\n\\]\n  So \\(x\\) would be compact, a contradiction.\n- *A spectral projection of infinite rank.* For \\(\\varepsilon>0\\) let \\(E_\\varepsilon=1_{[\\varepsilon,\\infty)}(a)\\). The function \\(\\lambda1_{[0,\\varepsilon)}(\\lambda)\\) is at most \\(\\varepsilon\\) on \\([0,\\infty)\\), so \\(\\|a-aE_\\varepsilon\\|\\leq\\varepsilon\\) by Theorem 3.1. If every \\(E_\\varepsilon\\) had finite rank, then every \\(aE_\\varepsilon\\) would have finite rank, and \\(a\\) would be compact. So some \\(E=E_\\varepsilon\\) has infinite-dimensional range.\n- *\\(E\\in J\\).* Let \\(g(\\lambda)=\\lambda^{-1}1_{[\\varepsilon,\\infty)}(\\lambda)\\), a bounded Borel function on \\([0,\\infty)\\). Then \\(g(a)a=(g\\iota)(a)=E\\). So \\(E\\in J\\).\n- *The identity is in \\(J\\).* \\(EH\\) is a closed infinite-dimensional subspace of the separable space \\(H\\), so it is separable. So \\(EH\\) and \\(H\\) both have countably infinite orthonormal bases \\((f_n)\\) and \\((e_n)\\) (the Hilbert-space lesson, Theorem 4.1(4)). The operator \\(v\\) with \\(ve_n=f_n\\) is an isometry (Theorem 4.1(2) of that lesson). Since \\(Ev=v\\), we get \\(1=v^*v=v^*Ev\\in J\\). So \\(J=B(H)\\). \\(\\square\\)\n\nSeparability is used only to find the isometry \\(v\\). Exercise 5 shows that the conclusion fails without it.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-ST-08",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "name": "8. Normal operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
      "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "anchor": "oa-fnd-st-08",
      "proof_locus": {
        "line": 410,
        "through_line": 519
      },
      "full_conditions_and_proof": "## 8. Normal operators\n\n**Theorem 8.1.** Let \\(n\\in B(H)\\) be normal, with spectrum \\(S\\subseteq\\mathbb C\\). Sections 2–5 hold with \\(h\\) replaced by \\(n\\) and \\(\\mathbb R\\) by \\(\\mathbb C\\), with the following changes. These conditions give the normal version without invoking Fuglede’s theorem:\n- in Theorem 3.1(5) and Theorem 4.4(4), the operator must commute with \\(n\\) and \\(n^*\\);\n- in Proposition 5.1(1), the coefficients \\(c_j\\) are complex, and a grid of mesh \\(1/(2n)\\) gives the same \\(1/n\\) error bound;\n- the positive-coefficient and positive-support assertions, Proposition 5.1(2) and (4), still assume that the operator is positive; such an operator is self-adjoint. The kernel formula in (3) holds for every normal operator, using \\(|z|^2\\) in its proof.\n\nIn particular, \\(E_n(\\Delta)=1_\\Delta(n)\\) is the only projection-valued measure on \\(\\mathbb C\\) that is supported by a compact set \\(L\\) and has \\(n=\\int\\iota_L\\,dE_n\\).\n\n**Proof.** The proofs in Sections 2–5 use only the following facts.\n- \\(S\\) is a compact metric space.\n- \\(f\\mapsto f(n)\\) is a unital \\(*\\)-homomorphism from \\(C(S)\\) to \\(B(H)\\) that maps real functions to self-adjoint operators. This holds for normal \\(n\\) by the C\\*-algebra lesson, Theorem 5.1(1), (2) and (9).\n- Theorem 3.1(5) uses the C\\*-algebra lesson, Theorem 5.1(6). That theorem needs commutation with \\(n\\) and \\(n^*\\).\n- The uniqueness proof uses the polynomials \\(p(\\lambda,\\bar\\lambda)\\), which are dense in \\(C(L')\\) for compact \\(L'\\subseteq\\mathbb C\\) ([the Stone–Weierstrass lesson](stone-weierstrass-c0.md#oa-fnd-sw-17)), and \\(\\int p(\\iota_L,\\bar\\iota_L)\\,dE=p(n,n^*)\\).\n- In Proposition 5.1(1), use \\(s_k(z)=(\\lfloor k\\operatorname{Re}z\\rfloor+i\\lfloor k\\operatorname{Im}z\\rfloor)/k\\), with \\(|z-s_k(z)|<\\sqrt2/k\\). Taking \\(k=2n\\) gives the \\(1/n\\) estimate in Proposition 5.1(1). Assertions about a positive operator reduce to the self-adjoint case. For a general normal operator the kernel proof uses \\(\\int|z|^2\\,d\\mu_\\xi=\\|n\\xi\\|^2\\) and the projection of \\(\\{0\\}\\). \\(\\square\\)\n\n**Proposition 8.2.** Let \\(n\\in B(H)\\) be normal.\n1. Every \\(\\lambda\\in\\sigma(n)\\) is an approximate eigenvalue: there are unit vectors \\(\\xi_k\\) with \\(\\|(n-\\lambda)\\xi_k\\|\\to0\\).\n2. \\(\\|n\\|=\\sup_{\\|\\xi\\|=1}|\\langle n\\xi,\\xi\\rangle|\\).\n\n**Proof.** (1) \\(m=n-\\lambda\\) is normal, so \\(\\|m^*\\xi\\|^2=\\langle mm^*\\xi,\\xi\\rangle=\\langle m^*m\\xi,\\xi\\rangle=\\|m\\xi\\|^2\\) for every \\(\\xi\\). Suppose \\(\\|m\\xi\\|\\geq c\\|\\xi\\|\\) for some \\(c>0\\) and all \\(\\xi\\).\n- Then \\(m\\) is injective, and its range is closed: if \\(m\\xi_j\\) is Cauchy, so is \\(\\xi_j\\).\n- The orthogonal complement of the range is \\(\\ker m^*\\), which equals \\(\\ker m=\\{0\\}\\). So \\(m\\) is bijective, and its inverse is bounded by \\(1/c\\).\n- This contradicts \\(\\lambda\\in\\sigma(n)\\). So no such \\(c\\) exists, which gives the vectors \\(\\xi_k\\).\n\n(2) The supremum is at most \\(\\|n\\|\\), by Cauchy–Schwarz. For \\(\\xi_k\\) as in (1), \\[\n\\begin{gathered}\n|\\langle n\\xi_k,\\xi_k\\rangle-\\lambda|\\\\\n=|\\langle(n-\\lambda)\\xi_k,\\xi_k\\rangle|\\\\\n\\leq\\|(n-\\lambda)\\xi_k\\|\\to0\n\\end{gathered}\n\\]. So the supremum is at least \\(|\\lambda|\\) for every \\(\\lambda\\in\\sigma(n)\\). By the C\\*-algebra lesson, Theorem 1.3, some \\(\\lambda\\in\\sigma(n)\\) has \\(|\\lambda|=r(n)=\\|n\\|\\). \\(\\square\\)\n\n## Exercises\n\n**Exercise 1** (easy; Multiplication operators). Let \\(\\mu\\) be a finite Borel measure on a compact set \\(K\\subseteq\\mathbb R\\), and let \\(h\\) be multiplication by \\(\\lambda\\) on \\(L^2(K,\\mu)\\). Let \\(\\operatorname{supp}\\mu\\) be the set of points all of whose neighbourhoods have positive measure. Show:\n- \\(\\sigma(h)=\\operatorname{supp}\\mu\\);\n- \\(f(h)\\) is multiplication by \\(f\\), for every bounded Borel \\(f\\);\n- \\(E_h(\\Delta)\\) is multiplication by \\(1_\\Delta\\);\n- the spectral measure of the constant function \\(1\\) is \\(\\mu\\).\n\n*Solution.* If the measure is zero, the space and all its operators are zero, the spectrum and support are empty, and all assertions hold in that sense. Otherwise:\n- *The support.* \\(K\\setminus\\operatorname{supp}\\mu\\) is the union of the open sets of measure \\(0\\). Since \\(\\mathbb R\\) has a countable base of open intervals, it is a countable union of null sets, hence null.\n- *\\(c\\notin\\operatorname{supp}\\mu\\).* Then \\(\\delta=\\operatorname{dist}(c,\\operatorname{supp}\\mu)>0\\). Almost everywhere \\(|\\lambda-c|\\geq\\delta\\), so multiplication by \\((\\lambda-c)^{-1}\\) is a bounded inverse of \\(h-c\\).\n- *\\(c\\in\\operatorname{supp}\\mu\\).* Let \\(U_k=(c-1/k,c+1/k)\\). Then \\(\\mu(U_k)>0\\). The unit vectors \\(\\xi_k=\\mu(U_k)^{-1/2}1_{U_k}\\) have \\(\\|(h-c)\\xi_k\\|\\leq1/k\\). So \\(h-c\\) is not bounded below, and not invertible. This proves \\(\\sigma(h)=\\operatorname{supp}\\mu=:S\\).\n- *Continuous \\(f\\).* For \\(f\\in C(S)\\), multiplication by \\(f\\) makes sense on \\(L^2(\\mu)\\), because \\(\\mu(K\\setminus S)=0\\). The map \\(f\\mapsto M_f\\) is a unital \\(*\\)-homomorphism into \\(B(H)\\) that sends \\(\\iota\\) to \\(h\\). By the uniqueness in the C\\*-algebra lesson, Theorem 5.1(1), \\(f(h)=M_f\\).\n- *Spectral measures.* Hence \\(\\langle f(h)\\xi,\\xi\\rangle=\\int f|\\xi|^2\\,d\\mu\\) for \\(f\\in C(S)\\). By Corollary 1.2, \\(\\mu_\\xi=|\\xi|^2\\mu\\) on \\(S\\).\n- *Borel \\(f\\).* For bounded Borel \\(f\\), \\(\\langle M_f\\xi,\\xi\\rangle=\\int f|\\xi|^2\\,d\\mu=\\int f\\,d\\mu_\\xi\\). By uniqueness in Theorem 3.1, \\(f(h)=M_f\\).\n- The last two claims are the cases \\(f=1_\\Delta\\) and \\(\\xi=1\\). \\(\\square\\)\n\n**Exercise 2** (medium; A dense set of eigenvalues). Let \\((q_k)\\) enumerate \\(\\mathbb Q\\cap[0,1]\\) without repetitions, and let \\(h\\) be the operator on \\(\\ell^2(\\mathbb N)\\) with \\(he_k=q_ke_k\\). Show:\n- \\(\\sigma(h)=[0,1]\\);\n- \\(E_h(\\{q_k\\})\\) is the projection onto \\(\\mathbb Ce_k\\);\n- \\(E_h([0,1]\\setminus\\mathbb Q)=0\\).\n\nConclude that \\(f=1_{[0,1]\\setminus\\mathbb Q}\\) has \\(f(h)=0\\), although the closure of \\(f(\\sigma(h))\\) is \\(\\{0,1\\}\\).\n\n*Solution.*\n- *The spectrum.* For \\(c\\notin[0,1]\\), the diagonal operator with entries \\((q_k-c)^{-1}\\) is bounded by \\(\\operatorname{dist}(c,[0,1])^{-1}\\) and inverts \\(h-c\\). Each \\(q_k\\) is an eigenvalue. The spectrum is closed, so it contains the closure of \\(\\{q_k\\}\\), which is \\([0,1]\\).\n- *Spectral measures.* For a polynomial \\(p\\), \\(\\langle p(h)\\xi,\\xi\\rangle=\\sum_kp(q_k)|\\xi_k|^2\\). By uniform approximation, the same holds for continuous \\(p\\). By Corollary 1.2, \\(\\mu_\\xi=\\sum_k|\\xi_k|^2\\delta_{q_k}\\).\n- *The projections.* \\(\\langle E_h(\\{q_k\\})\\xi,\\xi\\rangle=|\\xi_k|^2=\\langle\\theta_{e_k,e_k}\\xi,\\xi\\rangle\\). And \\(\\langle E_h([0,1]\\setminus\\mathbb Q)\\xi,\\xi\\rangle=0\\) for every \\(\\xi\\). \\(\\square\\)\n\n**Exercise 3** (medium; Unitaries are exponentials). Let \\(u\\in B(H)\\) be unitary. Show that \\(u=e^{ih}\\) for a self-adjoint \\(h\\) with \\(\\|h\\|\\leq\\pi\\) that commutes with every operator commuting with \\(u\\) and \\(u^*\\). Deduce that the unitary group of \\(B(H)\\) is path-connected in the norm topology.\n\n*Solution.*\n- *The logarithm.* \\(\\sigma(u)\\) lies in the unit circle \\(\\mathbb T\\) (the C\\*-algebra lesson, Proposition 1.5). Let \\(\\theta(e^{it})=t\\) for \\(t\\in(-\\pi,\\pi]\\). This is a Borel function on \\(\\mathbb T\\), continuous off \\(-1\\). Put \\(h=\\theta(u)\\), the calculus of the normal operator \\(u\\) (Theorem 8.1). It is self-adjoint, because \\(\\theta\\) is real, and \\(\\sigma(h)\\subseteq[-\\pi,\\pi]\\) by Theorem 3.1(6). It commutes with every operator that commutes with \\(u\\) and \\(u^*\\).\n- *A composition rule.* For a polynomial \\(p\\), \\(p(h)=(p\\circ\\theta)(u)\\), by the homomorphism property. Let \\(g\\) be continuous on \\([-\\pi,\\pi]\\), and let \\(p_j\\to g\\) uniformly there. Then \\(p_j(h)\\to g(h)\\), by the continuous calculus of \\(h\\). Also \\((p_j\\circ\\theta)(u)\\to(g\\circ\\theta)(u)\\), by Theorem 3.1(3). So \\(g(h)=(g\\circ\\theta)(u)\\).\n- *The exponential.* With \\(g(t)=e^{it}\\), \\(g\\circ\\theta\\) is the identity function on \\(\\mathbb T\\). So \\(e^{ih}=u\\).\n- *A path.* \\(u_t=e^{ith}\\), \\(t\\in[0,1]\\), is a path of unitaries from \\(1\\) to \\(u\\). It is norm-continuous: \\[\n\\begin{gathered}\n\\|e^{ith}-e^{ish}\\|\\\\\n\\leq\\sup_{|\\lambda|\\leq\\pi}|e^{it\\lambda}-e^{is\\lambda}|\\\\\n\\leq\\pi|t-s|\n\\end{gathered}\n\\]. \\(\\square\\)\n\n**Exercise 4** (easy; The Calkin algebra). Let \\(H\\) be separable and infinite-dimensional. Show that \\(B(H)/K(H)\\) has no closed two-sided ideals other than \\(\\{0\\}\\) and itself.\n\n*Solution.* Let \\(q\\) be the quotient map and \\(I\\) a closed two-sided ideal of the quotient.\n- \\(q^{-1}(I)\\) is a closed two-sided ideal of \\(B(H)\\) that contains \\(K(H)\\). By Calkin's theorem it is \\(K(H)\\) or \\(B(H)\\).\n- Since \\(q\\) is surjective, \\(I=q(q^{-1}(I))\\), which is \\(\\{0\\}\\) or the whole quotient. \\(\\square\\)\n\n**Exercise 5** (hard; Without separability). Let \\(H\\) be a Hilbert space that is not separable. Let \\(J\\) be the norm closure of the set of operators whose range is separable. Show that \\(J\\) is a closed two-sided ideal with \\(K(H)\\subsetneq J\\subsetneq B(H)\\).\n\n*Solution.*\n- *An ideal.* If \\(x\\) has separable range, so do \\(yx\\) (a continuous image of a separable set) and \\(xy\\) (its range lies in that of \\(x\\)). If \\(x\\) and \\(x'\\) have separable ranges, so does \\(x+x'\\). So these operators form a two-sided ideal, and its closure \\(J\\) is a closed two-sided ideal.\n- *\\(K(H)\\subseteq J\\).* A compact operator has separable range (the proof of the Hilbert-space lesson, Theorem 5.1(3)).\n- *\\(K(H)\\neq J\\).* The projection onto the closed span of a countably infinite orthonormal family has separable range, and it is not compact.\n- *\\(J\\neq B(H)\\).* Suppose \\(\\|1-x\\|<1\\) for some \\(x\\) with separable range. Then \\(x\\) is invertible (Neumann series), so its range is \\(H\\), which is not separable. So \\(1\\notin J\\). \\(\\square\\)\n\n**Exercise 6** (easy; The support of a positive operator). Let \\(h\\geq0\\). Show that \\(E_h((0,\\infty))\\) is the smallest projection \\(p\\) with \\(ph=h\\).\n\n*Solution.*\n- On \\(S\\subseteq[0,\\infty)\\), \\(\\lambda1_{(0,\\infty)}(\\lambda)=\\lambda\\). So \\(E_h((0,\\infty))h=h\\).\n- If \\(ph=h\\), then \\(p\\) is the identity on \\(hH\\), hence on its closure, which is \\(E_h((0,\\infty))H\\) by Proposition 5.1(4).\n- So \\(pE_h((0,\\infty))=E_h((0,\\infty))\\), which says \\(E_h((0,\\infty))\\leq p\\). \\(\\square\\)\n\n## Where this leads\n\nEvery operator that commutes with \\(h\\) commutes with all spectral projections of \\(h\\). So a von Neumann algebra contains the spectral projections and the bounded Borel functions of each of its self-adjoint elements. This is how the projections enter [The double commutant theorem](the-double-commutant-theorem.md) and the lessons after it. The spectral measures \\(\\mu_\\xi\\) and the unitary groups \\(e^{ith}\\) are used in [Compact and trace-class operators, the predual of B(H), and the operator topologies](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md). Calkin's theorem is used in [Representations and positive functionals: the GNS construction and the Gelfand–Naimark theorem](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md).\n\n## References\n\n- The spectral theorem for bounded self-adjoint operators goes back to Hilbert's work on quadratic forms in infinitely many variables (1906). The proof given here, through the continuous functional calculus and the Riesz representation theorem, is standard.\n- J. W. Calkin, \"Two-sided ideals and congruences in the ring of bounded operators in Hilbert space\", *Annals of Mathematics* 42 (1941), 839–873.\n- The monotone convergence theorem for operators is due to J.-P. Vigier (thesis, Geneva, 1946).\n\nThe proofs are written here in our own words.\n\n*Freely accessible reading:* [Jesse Peterson, *Notes on operator algebras*, §3.7](https://math.vanderbilt.edu/peters10/teaching/spring2015/OperatorAlgebras.pdf) gives a route through spectral measures and bounded Borel calculus; the exact null-set and essential-range statements are proved here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
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      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Compact and trace-class operators, the predual of B(H), and the operator topologies",
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      "full_conditions_and_proof": "# Compact and trace-class operators, the predual of B(H), and the operator topologies\n\n*Originally written by Claude Opus 5.5 (Anthropic), September 2026, with a separate historical AI spot-check; revised and self-checked by that writing AI in October 2026. GPT-6.1 Sol (OpenAI), at the Ultra setting, read and self-checked the full lesson and all five solutions, completed the measure, Fourier and Schatten-extension proofs and corrected the topology hypothesis and examples, October 2026. Public domain (CC0).*\n\nLet \\(H\\) be a Hilbert space. The algebra \\(B(H)\\) of all bounded operators on \\(H\\) is the dual of a Banach space, and this one fact produces most of the topologies used in the theory of operator algebras. This lesson builds the three Banach spaces behind it: the compact operators \\(K(H)\\), the trace-class operators \\(\\mathcal L^1(H)\\), and \\(B(H)\\) itself. The dual of \\(K(H)\\) is \\(\\mathcal L^1(H)\\), and the dual of \\(\\mathcal L^1(H)\\) is \\(B(H)\\), just as the dual of the sequence space \\(c_0\\) is \\(\\ell^1\\) and the dual of \\(\\ell^1\\) is \\(\\ell^\\infty\\). On the way we develop the Schmidt decomposition of a compact operator, its singular values, the Hilbert–Schmidt operators and the trace.\n\nThe second half studies the topologies of \\(B(H)\\). Besides the norm topology there are six locally convex topologies: the weak, strong and strong\\(^*\\) topologies, which test an operator on finitely many vectors, and their \\(\\sigma\\)-versions, which test it on square-summable sequences of vectors and come from the duality with \\(\\mathcal L^1(H)\\). We compare them, find their continuous linear functionals, show that a convex set has the same closure in all topologies with the same continuous functionals, prove the Krein–Šmulian theorem on weak\\(^*\\) closed convex sets, and decide when these topologies are metrizable.\n\nSeveral applications follow. We classify the two-sided ideals of \\(B(H)\\) by spaces of sequences, following Calkin, and show that the Calkin algebra \\(B(H)/K(H)\\) cannot act nontrivially on a separable Hilbert space. We show that every state that vanishes on the compact operators is a limit of vector states. The last section compares two small ideals through their effect on self-adjoint operators. By the Weyl–von Neumann theorem, when \\(H\\) is separable, a Hilbert–Schmidt perturbation of arbitrarily small norm makes any self-adjoint operator diagonal. By the Kato–Rosenblum theorem, a trace-class perturbation never changes the absolutely continuous part, up to unitary equivalence. So the Hilbert–Schmidt condition in the first theorem cannot be replaced by the trace-class condition.\n\nWe assume basic Hilbert space theory, the spectral theorem for compact self-adjoint operators and the continuous functional calculus, as in the lesson [C\\*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md). Sections 7 and 11 also use the spectral theorem in its projection-valued form, and Section 11 uses some Fourier analysis on the real line. Everything used without proof is stated in the section *Results used from other lessons*. The topologies of Sections 8 and 9 are used throughout the next lesson, [The double commutant theorem](the-double-commutant-theorem.md).\n\nFreely readable background is [van Neerven] and [Blackadar]. The trace-class duality and the scattering arguments used in Section 11 are proved in full below.\n\n## Conventions\n\nHilbert spaces are complex. They may have any dimension; separability is assumed only where it is stated. Inner products are linear in the first variable and conjugate linear in the second.\n\n- \\(B(H)\\) is the algebra of bounded operators on \\(H\\), \\(K(H)\\) the compact operators, \\(F(H)\\) the operators of finite rank, and \\(1\\) the identity. \\(B(H)_1\\) is the closed unit ball, \\(B(H)_{\\mathrm{sa}}\\) the set of self-adjoint operators, and \\(B(H)_+\\) the set of positive operators. An operator \\(x\\) is *positive*, \\(x\\ge0\\), when \\(\\langle x\\xi,\\xi\\rangle\\ge0\\) for all \\(\\xi\\). A positive \\(x\\) has a unique positive square root \\(x^{1/2}\\), and \\(|x|=(x^*x)^{1/2}\\).\n- For \\(\\xi,\\eta\\in H\\) we write \\(\\theta_{\\xi,\\eta}\\) for the operator \\(\\zeta\\mapsto\\langle\\zeta,\\eta\\rangle\\xi\\), and \\(\\omega_{\\xi,\\eta}\\) for the functional \\(x\\mapsto\\langle x\\xi,\\eta\\rangle\\) on \\(B(H)\\); \\(\\omega_\\xi=\\omega_{\\xi,\\xi}\\).\n- A family \\((a_i)_{i\\in I}\\) in a normed space is *summable* with sum \\(a\\) when the finite partial sums converge to \\(a\\) along the directed set of finite subsets of \\(I\\). For numbers \\(a_i\\ge0\\) the sum is the supremum of the finite partial sums, a number in \\([0,\\infty]\\). An absolutely summable family in a Banach space may be summed in any order and grouped in any way. Indeed, if \\(\\sum_i\\|a_i\\|<\\infty\\), each set \\(\\{i:\\|a_i\\|\\geq1/n\\}\\) is finite, so the nonzero support is countable. For every \\(\\varepsilon>0\\), a finite partial norm sum lies within \\(\\varepsilon\\) of the supremum, and all disjoint finite tail sums then have norm at most \\(\\varepsilon\\). Completeness gives the finite-subset-net limit. The same tail estimate shows that every ordering or grouping has this limit. It applies to scalars in particular, and to square-summable norms when proving countable support in \\(\\ell^2\\).\n- For a set \\(I\\): \\(\\ell^\\infty(I)\\) is the space of bounded families \\(\\lambda=(\\lambda_i)_{i\\in I}\\) of scalars; \\(c_0(I)\\) consists of those with only finitely many \\(|\\lambda_i|\\ge\\varepsilon\\) for every \\(\\varepsilon>0\\); \\(\\ell^1(I)\\) and \\(\\ell^2(I)\\) are the summable and square-summable families; \\(c_{00}(I)\\) the finitely supported ones; \\(\\delta_i\\) is the family with \\(1\\) at \\(i\\) and \\(0\\) elsewhere. For \\(I=\\mathbb N\\) we write \\(\\ell^\\infty,c_0,\\ell^1,\\ell^2,c_{00}\\).\n- \\(\\ell^2(I;H)\\) is the Hilbert space of families \\((\\xi_i)_{i\\in I}\\) in \\(H\\) with \\(\\sum_i\\|\\xi_i\\|^2<\\infty\\), with inner product \\(\\sum_i\\langle\\xi_i,\\eta_i\\rangle\\). A sequence \\((\\xi_n)\\) with \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\) is called *square summable*.\n- The *conjugate space* \\(\\overline H\\) consists of the symbols \\(\\bar\\xi\\), \\(\\xi\\in H\\), with \\(\\bar\\xi+\\bar\\eta=\\overline{\\xi+\\eta}\\), \\(c\\,\\bar\\xi=\\overline{\\bar c\\,\\xi}\\) and \\(\\langle\\bar\\xi,\\bar\\eta\\rangle=\\langle\\eta,\\xi\\rangle\\). It is a Hilbert space, and \\(\\xi\\mapsto\\bar\\xi\\) is conjugate linear and isometric.\n\n",
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      "id": "OA-FND-LT-16",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Results used from other lessons",
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      "full_conditions_and_proof": "## Results used from other lessons\n\nThe facts below are proved in these earlier lessons: [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md) (*the Hahn–Banach lesson*), [Weak topologies: Tychonoff, Banach–Alaoglu, Mazur, bipolars, Krein–Milman and Eberlein–Šmulian](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md) (*the lesson on weak topologies*), [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md) (*the Hilbert-space lesson*) and [The spectral theorem for bounded self-adjoint operators](the-spectral-theorem-for-bounded-self-adjoint-operators.md) (*the spectral-theorem lesson*). The exact measure proofs are [Theorems 1.1, 2.1–2.2, 3.1–3.2 and 4.1 of the measure-tools lesson](measure-and-hilbert-space-tools.md), the [generating-family lemma and probability products](the-double-commutant-theorem.md#oa-fnd-bi-18), and the real-line tools proved immediately below. Fremlin’s paragraph numbers remain human bibliographic references, rather than proof providers.\n\n1. **Hilbert spaces** (the Hilbert-space lesson). (a) *Riesz representation* (Theorem 2.3). Every bounded linear functional \\(f\\) on \\(H\\) has the form \\(f(\\xi)=\\langle\\xi,\\eta\\rangle\\) for a unique \\(\\eta\\in H\\). (b) *Bessel and Parseval* (Theorem 4.1(1) and (3)). For an orthonormal family \\((\\varepsilon_i)\\), \\(\\sum_i|\\langle\\xi,\\varepsilon_i\\rangle|^2\\le\\|\\xi\\|^2\\). If the family is an orthonormal basis, then \\(\\xi=\\sum_i\\langle\\xi,\\varepsilon_i\\rangle\\varepsilon_i\\) and \\(\\langle\\xi,\\eta\\rangle=\\sum_i\\langle\\xi,\\varepsilon_i\\rangle\\langle\\varepsilon_i,\\eta\\rangle\\). (c) Every orthonormal family is contained in an orthonormal basis (Theorem 4.1(4)). (d) *Cauchy–Schwarz for forms* (Proposition 1.1(3)). If \\(B\\) is a sesquilinear form on a complex vector space with \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\), then \\(|B(\\xi,\\eta)|^2\\le B(\\xi,\\xi)B(\\eta,\\eta)\\).\n2. **Compact operators** (the Hilbert-space lesson). (a) \\(K(H)\\) is a norm-closed two-sided ideal of \\(B(H)\\), closed under adjoints, and it is the norm closure of \\(F(H)\\) (Theorem 5.1). (b) *Spectral theorem for compact self-adjoint operators* (Theorem 6.2). If \\(x\\in K(H)\\) is self-adjoint and \\(x\\ne0\\), there are an orthonormal family \\((e_n)_{n\\in N}\\), with \\(N=\\{1,\\dots,r\\}\\) or \\(N=\\mathbb N\\), and real numbers \\(\\lambda_n\\ne0\\) with \\(|\\lambda_1|\\ge|\\lambda_2|\\ge\\cdots\\), tending to \\(0\\) when \\(N=\\mathbb N\\), such that \\(x=\\sum_n\\lambda_n\\theta_{e_n,e_n}\\) with convergence in norm. The nonzero eigenvalues of \\(x\\) are the \\(\\lambda_n\\); the eigenspace of an eigenvalue \\(\\mu\\ne0\\) is spanned by the \\(e_n\\) with \\(\\lambda_n=\\mu\\) and is finite-dimensional. If \\(x\\ge0\\), all \\(\\lambda_n\\) are positive.\n3. **Functional calculus.** (a) *Continuous calculus.* For a self-adjoint operator the continuous functional calculus exists; positive operators have unique positive square roots, and \\(x^{1/2}\\) commutes with every operator that commutes with \\(x\\). See the lesson on C\\*-algebras named above. (b) *Spectral theorem* (the spectral-theorem lesson, Theorems 3.1 and 4.4 and Remark 4.5). For \\(h\\in B(H)_{\\mathrm{sa}}\\) there is a unique projection-valued measure \\(E\\) on the Borel sets of \\(\\mathbb R\\), supported in \\([-\\|h\\|,\\|h\\|]\\), with \\(h=\\int\\lambda\\,dE(\\lambda)\\). For bounded Borel functions \\(f\\), the operators \\(f(h)=\\int f\\,dE\\) form a \\(*\\)-homomorphism \\(f\\mapsto f(h)\\) that extends the continuous calculus, with \\(\\|f(h)\\|\\le\\sup|f|\\) and \\(f(h)\\ge0\\) when \\(f\\ge0\\); each \\(f(h)\\) commutes with every operator that commutes with \\(h\\). For \\(\\xi\\in H\\), \\(\\mu_\\xi(\\Delta)=\\langle E(\\Delta)\\xi,\\xi\\rangle=\\|E(\\Delta)\\xi\\|^2\\) is a finite positive Borel measure, the *spectral measure* of \\(\\xi\\), with \\(\\mu_\\xi(\\mathbb R)=\\|\\xi\\|^2\\) and \\(\\langle f(h)\\xi,\\xi\\rangle=\\int f\\,d\\mu_\\xi\\). We write \\(e^{ith}\\) for \\(f(h)\\) with \\(f(\\lambda)=e^{it\\lambda}\\); it is the sum of the exponential series.\n4. **Measure theory and Fourier analysis.** (a) A finite positive Borel measure on \\(\\mathbb R\\) that vanishes on Lebesgue null sets has an \\(L^1\\) density: [Theorem 4.1 of the measure-tools lesson](measure-and-hilbert-space-tools.md#4-densities-and-bounded-functionals). Lebesgue measure and its sigma-finiteness are constructed in Lemma 0.1(b) below. (b) \\(C_c^\\infty(I)\\) is dense in \\(L^2(I)\\) for every open interval \\(I\\), by Lemma 0.1(c). (c) With \\(\\widehat w(s)=\\int e^{-is\\lambda}w(\\lambda)\\,d\\lambda\\), Theorem 0.3 gives \\(\\int|\\widehat w|^2=2\\pi\\int|w|^2\\) and the unitary extension of \\((2\\pi)^{-1/2}\\widehat w\\). Lemma 0.2 proves the normalized Gaussian transform with this same sign and constant. (d) The full Dynkin-system/generating-family proof is the lemma preceding the probability product theorem linked above. It also determines finite signed measures of finite variation: if their difference is \\(\\nu\\), the finite measures \\((|\\nu|+\\nu)/2\\) and \\((|\\nu|-\\nu)/2\\) are positive, since \\(|\\nu(E)|\\leq|\\nu|(E)\\); agreement on the generating family makes these two positive measures agree there, hence everywhere. Countable additivity of variation is proved in [the complex-measure tools](abelian-operator-algebras.md#oa-fnd-ao-23). (e) Monotone and dominated convergence are [Theorems 2.1–2.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions). Lemma 0.1(a) proves Tonelli and Fubini for arbitrary sigma-finite measure spaces; it imposes no topological condition on the spaces.\n5. **Locally convex spaces.** (a) *Hahn–Banach separation* (the Hahn–Banach lesson, Theorems 6.2 and 6.3). If \\(A\\) and \\(B\\) are disjoint nonempty convex subsets of a topological vector space and \\(A\\) is open, there are a continuous linear functional \\(f\\) and \\(\\gamma\\in\\mathbb R\\) with \\(\\operatorname{Re}f(a)<\\gamma\\le\\operatorname{Re}f(b)\\) for \\(a\\in A\\), \\(b\\in B\\). If the space is locally convex, \\(A\\) is compact and \\(B\\) is closed, then even \\(\\operatorname{Re}f(a)<\\gamma_1<\\gamma_2<\\operatorname{Re}f(b)\\). (b) *Weak topologies of a dual pair* (the lesson on weak topologies, Theorem 1.2). If \\(Y\\) is a vector space of linear functionals on a vector space \\(X\\) that separates the points of \\(X\\), then the topology \\(\\sigma(X,Y)\\) of pointwise convergence on \\(Y\\) is locally convex, and its continuous linear functionals are exactly the elements of \\(Y\\). (c) *Closures of convex sets* (the lesson on weak topologies, Theorem 4.2). In a locally convex space \\(X\\) with continuous dual \\(X^*\\), a convex set has the same closure in the given topology and in \\(\\sigma(X,X^*)\\). Hence two locally convex topologies on \\(X\\) with the same continuous linear functionals have the same closed convex sets. (d) *Banach–Alaoglu* (the lesson on weak topologies, Theorem 3.1). The closed unit ball of the dual \\(X^*\\) of a normed space \\(X\\) is compact for \\(\\sigma(X^*,X)\\). (e) *Annihilators* (the lesson on weak topologies, Theorem 5.1(2) and Proposition 5.4). Let \\(X\\) be a Banach space. For a subspace \\(N\\subseteq X^*\\), the \\(\\sigma(X^*,X)\\)-closure of \\(N\\) is \\(({}^\\perp N)^\\perp\\), where \\({}^\\perp N=\\{x:f(x)=0\\ \\text{for all}\\ f\\in N\\}\\) and \\(({}^\\perp N)^\\perp=\\{f\\in X^*:f=0\\ \\text{on}\\ {}^\\perp N\\}\\). For a closed subspace \\(M\\subseteq X\\) with quotient map \\(q:X\\to X/M\\), the map \\(g\\mapsto g\\circ q\\) is an isometric isomorphism of \\((X/M)^*\\) onto \\(M^\\perp\\). (f) *Norming* (the Hahn–Banach lesson, Corollary 2.3(2)). For every \\(z\\) in a normed space \\(Z\\), \\(\\|z\\|=\\sup\\{|g(z)|:g\\in Z^*,\\ \\|g\\|\\le1\\}\\). (g) *Krein–Milman and Milman* (the lesson on weak topologies, Theorems 6.1 and 6.2). In a Hausdorff locally convex space, a nonempty compact convex set is the closed convex hull of its extreme points; and if \\(Q\\) is compact and the closed convex hull of \\(Q\\) is compact, then every extreme point of that hull lies in \\(Q\\). (h) *Closed kernels* (the Hahn–Banach lesson, Lemma 6.5). A linear functional on a topological vector space is continuous if its kernel is closed. (i) *Uniform boundedness* (the Hahn–Banach lesson, Theorem 4.2). A family of bounded operators from a Banach space to a normed space that is bounded at each point is bounded in norm.\n\n",
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      "id": "OA-FND-LT-02",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "1. Forms, rank-one operators and vector functionals",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-02",
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        "line": 169,
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      "full_conditions_and_proof": "## 1. Forms, rank-one operators and vector functionals\n\nBounded operators on a Hilbert space are the same thing as bounded sesquilinear forms. This section sets up that dictionary, and the simplest operators and functionals, which are built from two vectors.\n\n**Definition 1.1.** A *sesquilinear form* on \\(H\\) is a map \\(B:H\\times H\\to\\mathbb C\\) that is linear in the first variable and conjugate linear in the second. (The same words are used for forms on any complex vector space, and for maps with values in any complex vector space.) The form is *hermitian* if \\(B(\\eta,\\xi)=\\overline{B(\\xi,\\eta)}\\) for all \\(\\xi,\\eta\\), *positive* if \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\), and *bounded* if\n\\[\n\\|B\\|=\\sup\\{|B(\\xi,\\eta)|:\\ \\|\\xi\\|\\le1,\\ \\|\\eta\\|\\le1\\}\n\\]\nis finite. The bounded forms make up a normed space \\(\\operatorname{Sesq}(H)\\).\n\n**Lemma 1.2** (polarization). For every sesquilinear form \\(B\\) on \\(H\\), or on any complex vector space, and all vectors \\(\\xi,\\eta\\),\n\\[\n\\begin{gathered}\nB(\\xi,\\eta)\\\\\n=\\frac14\\sum_{k=0}^{3}i^k\\,B(\\xi+i^k\\eta,\\ \\xi+i^k\\eta).\n\\end{gathered}\n\\tag{1.1}\n\\]\nSo \\(B\\) is determined by its values \\(B(\\zeta,\\zeta)\\). The form \\(B\\) is hermitian exactly when every value \\(B(\\zeta,\\zeta)\\) is real; in particular every positive form is hermitian.\n\n**Proof.** Because \\(B\\) is conjugate linear in the second variable,\n\\[\n\\begin{gathered}\nB(\\xi+i^k\\eta,\\xi+i^k\\eta)\\\\\n=B(\\xi,\\xi)+\\overline{i^k}\\,B(\\xi,\\eta)+i^k B(\\eta,\\xi)+B(\\eta,\\eta).\n\\end{gathered}\n\\]\nMultiply by \\(i^k\\) and add over \\(k=0,1,2,3\\). The terms with \\(B(\\xi,\\xi)\\) and \\(B(\\eta,\\eta)\\) carry the factor \\(\\sum_ki^k=0\\), and the terms with \\(B(\\eta,\\xi)\\) carry \\(\\sum_ki^{2k}=0\\), while \\(i^k\\overline{i^k}=1\\). What remains is \\(4B(\\xi,\\eta)\\), which is (1.1).\n\nIf \\(B\\) is hermitian, then \\(B(\\zeta,\\zeta)=\\overline{B(\\zeta,\\zeta)}\\) is real. Conversely, suppose every \\(B(\\zeta,\\zeta)\\) is real. Apply (1.1) to \\(B(\\eta,\\xi)\\). Since \\(\\eta+i^k\\xi=i^k(\\xi+i^{-k}\\eta)\\) and \\(B(c\\zeta,c\\zeta)=|c|^2B(\\zeta,\\zeta)\\), we get\n\\[\n\\begin{gathered}\nB(\\eta,\\xi)\\\\\n=\\frac14\\sum_k i^k\\,B(\\xi+i^{-k}\\eta,\\xi+i^{-k}\\eta)\\\\\n=\\frac14\\sum_j i^{-j}\\,B(\\xi+i^{j}\\eta,\\xi+i^{j}\\eta),\n\\end{gathered}\n\\]\nwhich is the complex conjugate of the right side of (1.1), because the values \\(B(\\xi+i^j\\eta,\\xi+i^j\\eta)\\) are real. \\(\\square\\)\n\n**Theorem 1.3.** For every bounded sesquilinear form \\(B\\) on \\(H\\) there is a unique \\(t\\in B(H)\\) with \\(B(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\) for all \\(\\xi,\\eta\\). The map \\(t\\mapsto B_t\\), \\(B_t(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\), is a linear isometry of \\(B(H)\\) onto \\(\\operatorname{Sesq}(H)\\); in particular \\(\\operatorname{Sesq}(H)\\) is a Banach space. The operator \\(t\\) is self-adjoint if and only if \\(B_t\\) is hermitian, and positive if and only if \\(B_t\\) is positive.\n\n**Proof.** Fix \\(\\xi\\). The map \\(\\eta\\mapsto\\overline{B(\\xi,\\eta)}\\) is linear and bounded by \\(\\|B\\|\\,\\|\\xi\\|\\,\\|\\eta\\|\\). By the Riesz theorem there is a unique vector, which we call \\(t\\xi\\), with \\(\\overline{B(\\xi,\\eta)}=\\langle\\eta,t\\xi\\rangle\\), that is, \\(B(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\); and \\(\\|t\\xi\\|\\le\\|B\\|\\,\\|\\xi\\|\\). The uniqueness makes \\(\\xi\\mapsto t\\xi\\) linear, so \\(t\\in B(H)\\) and \\(\\|t\\|\\le\\|B\\|\\).\n\nFor every \\(t\\in B(H)\\) we have \\(\\|B_t\\|=\\sup|\\langle t\\xi,\\eta\\rangle|=\\|t\\|\\): take \\(\\eta=t\\xi/\\|t\\xi\\|\\) when \\(t\\xi\\ne0\\). So \\(t\\mapsto B_t\\) is a linear isometry, and the first paragraph shows that it is onto. A normed space isometric to a Banach space is complete. The form \\(B_t\\) is hermitian exactly when \\(\\langle t\\xi,\\eta\\rangle=\\overline{\\langle t\\eta,\\xi\\rangle}=\\langle\\xi,t\\eta\\rangle\\) for all \\(\\xi,\\eta\\), that is, when \\(t=t^*\\). Positivity of \\(B_t\\) is positivity of \\(t\\), by the definitions. \\(\\square\\)\n\nBy Lemma 1.2, an operator \\(t\\) is self-adjoint exactly when \\(\\langle t\\zeta,\\zeta\\rangle\\) is real for all \\(\\zeta\\), and every positive operator is self-adjoint.\n\n**Lemma 1.4** (rank-one operators and vector functionals). Let \\(\\xi,\\eta,\\zeta,\\upsilon\\in H\\) and \\(a,b\\in B(H)\\).\n\n(a) \\(\\|\\theta_{\\xi,\\eta}\\|=\\|\\xi\\|\\,\\|\\eta\\|\\) and \\(\\theta_{\\xi,\\eta}^*=\\theta_{\\eta,\\xi}\\).\n\n(b) \\(a\\,\\theta_{\\xi,\\eta}\\,b=\\theta_{a\\xi,\\,b^*\\eta}\\) and \\(\\theta_{\\xi,\\eta}\\theta_{\\zeta,\\upsilon}=\\langle\\zeta,\\eta\\rangle\\,\\theta_{\\xi,\\upsilon}\\).\n\n(c) For a unit vector \\(\\xi\\), \\(\\theta_{\\xi,\\xi}\\) is the orthogonal projection onto \\(\\mathbb C\\xi\\).\n\n(d) Every operator of rank one equals \\(\\theta_{\\xi,\\eta}\\) for some nonzero \\(\\xi,\\eta\\). If \\(x\\in F(H)\\) and \\(\\varepsilon_1,\\dots,\\varepsilon_r\\) is an orthonormal basis of \\(xH\\), then \\(x=\\sum_{j=1}^r\\theta_{\\varepsilon_j,\\,x^*\\varepsilon_j}\\). So \\(F(H)\\) is the linear span of the rank-one operators, and it is a two-sided ideal of \\(B(H)\\) closed under adjoints.\n\n(e) \\(\\omega_{\\xi,\\eta}(\\theta_{\\zeta,\\upsilon})=\\langle\\zeta,\\eta\\rangle\\langle\\xi,\\upsilon\\rangle\\), and \\(\\|\\omega_{\\xi,\\eta}\\|=\\|\\xi\\|\\,\\|\\eta\\|\\), both as a functional on \\(B(H)\\) and on \\(K(H)\\).\n\n**Proof.** (a) \\(\\|\\theta_{\\xi,\\eta}\\zeta\\|=|\\langle\\zeta,\\eta\\rangle|\\,\\|\\xi\\|\\le\\|\\zeta\\|\\,\\|\\eta\\|\\,\\|\\xi\\|\\), with equality for \\(\\zeta=\\eta\\). Also \\[\n\\begin{gathered}\n\\langle\\theta_{\\xi,\\eta}\\zeta,\\upsilon\\rangle\\\\\n=\\langle\\zeta,\\eta\\rangle\\langle\\xi,\\upsilon\\rangle\\\\\n=\\langle\\zeta,\\langle\\upsilon,\\xi\\rangle\\eta\\rangle\\\\\n=\\langle\\zeta,\\theta_{\\eta,\\xi}\\upsilon\\rangle.\n\\end{gathered}\n\\]\n\n(b) \\(a\\theta_{\\xi,\\eta}b\\zeta=\\langle b\\zeta,\\eta\\rangle a\\xi=\\langle\\zeta,b^*\\eta\\rangle a\\xi\\), and \\(\\theta_{\\xi,\\eta}\\theta_{\\zeta,\\upsilon}\\kappa=\\langle\\kappa,\\upsilon\\rangle\\langle\\zeta,\\eta\\rangle\\xi\\).\n\n(c) \\(\\theta_{\\xi,\\xi}\\zeta=\\langle\\zeta,\\xi\\rangle\\xi\\).\n\n(d) The range \\(xH\\) is finite-dimensional. For every \\(\\zeta\\), \\(x\\zeta=\\sum_j\\langle x\\zeta,\\varepsilon_j\\rangle\\varepsilon_j=\\sum_j\\langle\\zeta,x^*\\varepsilon_j\\rangle\\varepsilon_j\\). For \\(r=1\\) this is \\(\\theta_{\\varepsilon_1,x^*\\varepsilon_1}\\), and \\(x^*\\varepsilon_1\\ne0\\) because \\(x\\ne0\\). By (a) and (b), adjoints of rank-one operators and products of them with bounded operators are again of rank at most one.\n\n(e) \\(\\omega_{\\xi,\\eta}(\\theta_{\\zeta,\\upsilon})=\\langle\\theta_{\\zeta,\\upsilon}\\xi,\\eta\\rangle=\\langle\\xi,\\upsilon\\rangle\\langle\\zeta,\\eta\\rangle\\). Clearly \\(|\\omega_{\\xi,\\eta}(x)|\\le\\|x\\|\\,\\|\\xi\\|\\,\\|\\eta\\|\\). On the other hand \\(\\omega_{\\xi,\\eta}(\\theta_{\\eta,\\xi})=\\|\\xi\\|^2\\|\\eta\\|^2\\) and \\(\\|\\theta_{\\eta,\\xi}\\|=\\|\\xi\\|\\,\\|\\eta\\|\\). Since \\(\\theta_{\\eta,\\xi}\\) is compact, the norm is attained on \\(K(H)\\) as well. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-LT-03",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "2. Compact operators and singular values",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-03",
      "proof_locus": {
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      "full_conditions_and_proof": "## 2. Compact operators and singular values\n\nA compact operator is a norm limit of operators of finite rank. Each one has a canonical form, the Schmidt decomposition. Its coefficients, the singular values, measure how well the operator can be approximated by operators of small rank.\n\n**Lemma 2.1.** Let \\(x\\in K(H)\\) and let \\((\\xi_n)\\) be an orthonormal sequence. Then \\(\\|x\\xi_n\\|\\to0\\).\n\n**Proof.** Let \\(\\varepsilon>0\\), and choose \\(f\\in F(H)\\) with \\(\\|x-f\\|<\\varepsilon\\). By Lemma 1.4(d), \\(f=\\sum_{j\\le r}\\theta_{\\varepsilon_j,f^*\\varepsilon_j}\\), so \\(f\\xi_n=\\sum_{j\\le r}\\langle\\xi_n,f^*\\varepsilon_j\\rangle\\varepsilon_j\\). By Bessel's inequality \\(\\langle\\xi_n,f^*\\varepsilon_j\\rangle\\to0\\) for each \\(j\\), so \\(f\\xi_n\\to0\\). Hence \\(\\limsup_n\\|x\\xi_n\\|\\le\\varepsilon\\). \\(\\square\\)\n\n**Example 2.2** (diagonal operators). Let \\((\\varepsilon_i)_{i\\in I}\\) be an orthonormal basis of \\(H\\) and \\(\\lambda\\in\\ell^\\infty(I)\\). Put\n\\[\nd_\\lambda\\zeta=\\sum_i\\lambda_i\\langle\\zeta,\\varepsilon_i\\rangle\\varepsilon_i .\n\\]\nThe series converges because its coefficients are square summable, and \\(\\|d_\\lambda\\zeta\\|^2=\\sum_i|\\lambda_i|^2|\\langle\\zeta,\\varepsilon_i\\rangle|^2\\le\\|\\lambda\\|_\\infty^2\\|\\zeta\\|^2\\). Since \\(d_\\lambda\\varepsilon_i=\\lambda_i\\varepsilon_i\\), \\(\\|d_\\lambda\\|=\\|\\lambda\\|_\\infty\\). One checks \\(d_\\lambda d_\\mu=d_{\\lambda\\mu}\\) and \\(d_\\lambda^*=d_{\\bar\\lambda}\\). The operator \\(d_\\lambda\\) is compact if and only if \\(\\lambda\\in c_0(I)\\). Indeed, if \\(\\lambda\\in c_0(I)\\) and \\(F\\subseteq I\\) is a finite set outside of which \\(|\\lambda_i|<\\varepsilon\\), then \\(d_{\\lambda1_F}\\) has finite rank and \\(\\|d_\\lambda-d_{\\lambda1_F}\\|\\le\\varepsilon\\). If \\(\\lambda\\notin c_0(I)\\), there are \\(\\varepsilon>0\\) and distinct \\(i_1,i_2,\\dots\\) with \\(|\\lambda_{i_n}|\\ge\\varepsilon\\), so \\(\\|d_\\lambda\\varepsilon_{i_n}\\|\\ge\\varepsilon\\), and \\(d_\\lambda\\) is not compact by Lemma 2.1. In this way \\(\\ell^\\infty(I)\\) and \\(c_0(I)\\) sit inside \\(B(H)\\) and \\(K(H)\\) as algebras of diagonal operators.\n\n**Theorem 2.3** (Schmidt decomposition). Let \\(x\\in K(H)\\), \\(x\\ne0\\). There are orthonormal families \\((e_n)_{n\\in N}\\) and \\((f_n)_{n\\in N}\\), where \\(N=\\{1,\\dots,r\\}\\) or \\(N=\\mathbb N\\), and numbers \\(s_1\\ge s_2\\ge\\cdots>0\\), tending to \\(0\\) when \\(N=\\mathbb N\\), such that\n\\[\n\\begin{gathered}\nx\\\\\n=\\sum_{n\\in N}s_n\\,\\theta_{f_n,e_n},\\\\\n\\text{that is,}\\\\\nx\\zeta\\\\\n=\\sum_n s_n\\langle\\zeta,e_n\\rangle f_n,\n\\end{gathered}\n\\tag{2.1}\n\\]\nwith convergence in norm. For every decomposition of this form:\n\n(a) \\(\\|x\\|=s_1\\), the kernel of \\(x\\) is \\(\\{e_n:n\\in N\\}^\\perp\\), and \\(xe_n=s_nf_n\\), \\(x^*f_n=s_ne_n\\);\n\n(b) \\(x^*=\\sum_ns_n\\theta_{e_n,f_n}\\), \\(|x|=\\sum_ns_n\\theta_{e_n,e_n}\\) and \\(|x^*|=\\sum_ns_n\\theta_{f_n,f_n}\\);\n\n(c) the numbers \\(s_n\\), listed with repetitions, are the nonzero eigenvalues of \\(|x|\\) counted with multiplicity; so they do not depend on the decomposition;\n\n(d) the operator \\(v\\zeta=\\sum_n\\langle\\zeta,f_n\\rangle e_n\\) satisfies \\(\\|v\\|\\le1\\), \\(vx=|x|\\) and \\(v^*|x|=x\\);\n\n(e) if \\(x\\) is self-adjoint, a decomposition exists with \\(f_n=\\pm e_n\\) for every \\(n\\), and if \\(x\\ge0\\), one with \\(f_n=e_n\\);\n\n(f) if \\(x\\) is self-adjoint and \\(x=\\sum_n\\lambda_n\\theta_{e_n,e_n}\\) as in (e), then the spectrum of \\(x\\) contains every \\(\\lambda_n\\), and \\(\\sigma(x)\\setminus\\{0\\}=\\{\\lambda_n:n\\in N\\}\\); if \\(H\\) is infinite-dimensional, then \\(0\\in\\sigma(x)\\), so \\(\\sigma(x)=\\{\\lambda_n:n\\in N\\}\\cup\\{0\\}\\).\n\n**Proof.** *Existence.* The operator \\(x^*x\\) is compact and positive, since \\(\\langle x^*x\\zeta,\\zeta\\rangle=\\|x\\zeta\\|^2\\). By Background 2(b), \\(x^*x=\\sum_n\\mu_n\\theta_{e_n,e_n}\\) with an orthonormal family \\((e_n)_{n\\in N}\\) and numbers \\(\\mu_1\\ge\\mu_2\\ge\\cdots>0\\), tending to \\(0\\) if \\(N=\\mathbb N\\). If \\(x\\zeta=0\\), then \\(\\mu_n\\langle\\zeta,e_n\\rangle=\\langle\\zeta,x^*xe_n\\rangle=\\langle x^*x\\zeta,e_n\\rangle=0\\) for all \\(n\\). Conversely, if \\(\\zeta\\perp e_n\\) for all \\(n\\), then \\(x^*x\\zeta=0\\) and \\(\\|x\\zeta\\|^2=\\langle x^*x\\zeta,\\zeta\\rangle=0\\). So the kernel of \\(x\\) is \\(\\{e_n\\}^\\perp\\).\n\nPut \\(s_n=\\mu_n^{1/2}\\) and \\(f_n=s_n^{-1}xe_n\\). Then \\(\\langle f_n,f_m\\rangle=(s_ns_m)^{-1}\\langle x^*xe_n,e_m\\rangle=\\delta_{nm}\\), so \\((f_n)\\) is orthonormal. Every \\(\\zeta\\in H\\) is \\(\\zeta=\\sum_n\\langle\\zeta,e_n\\rangle e_n+\\zeta_0\\) with \\(\\zeta_0\\) in the kernel of \\(x\\), so \\(x\\zeta=\\sum_n\\langle\\zeta,e_n\\rangle xe_n=\\sum_ns_n\\langle\\zeta,e_n\\rangle f_n\\). When \\(N=\\mathbb N\\), the remainder \\(\\zeta\\mapsto\\sum_{n>M}s_n\\langle\\zeta,e_n\\rangle f_n\\) has norm at most \\(s_{M+1}\\), because \\[\n\\begin{gathered}\n\\|\\sum_{n>M}s_n\\langle\\zeta,e_n\\rangle f_n\\|^2\\\\\n=\\sum_{n>M}s_n^2|\\langle\\zeta,e_n\\rangle|^2\\\\\n\\le s_{M+1}^2\\|\\zeta\\|^2.\n\\end{gathered}\n\\] So (2.1) converges in norm.\n\n*Properties.* Now let (2.1) be any decomposition of the stated form.\n\n(a) \\(\\|x\\zeta\\|^2=\\sum_ns_n^2|\\langle\\zeta,e_n\\rangle|^2\\le s_1^2\\|\\zeta\\|^2\\), with equality for \\(\\zeta=e_1\\). The same formula shows that \\(x\\zeta=0\\) exactly when \\(\\zeta\\perp e_n\\) for all \\(n\\). Clearly \\(xe_n=s_nf_n\\), and \\(x^*f_n=s_ne_n\\) follows from (b).\n\n(b) Taking adjoints term by term, which is allowed since the series converges in norm, gives \\(x^*=\\sum s_n\\theta_{e_n,f_n}\\). By Lemma 1.4(b), \\(\\theta_{e_n,f_n}\\theta_{f_m,e_m}=\\delta_{nm}\\theta_{e_n,e_m}\\), so \\(x^*x=\\sum_ns_n^2\\theta_{e_n,e_n}\\). The operator \\(\\sum_ns_n\\theta_{e_n,e_n}\\) is positive and its square is \\(x^*x\\) by the same computation. By uniqueness of positive square roots it equals \\(|x|\\). In the same way \\(xx^*=\\sum s_n^2\\theta_{f_n,f_n}\\) and \\(|x^*|=\\sum s_n\\theta_{f_n,f_n}\\).\n\n(c) By (b), \\(|x|\\zeta=\\sum_ns_n\\langle\\zeta,e_n\\rangle e_n\\), so \\(|x|e_n=s_ne_n\\). If \\(|x|\\zeta=\\mu\\zeta\\) with \\(\\mu\\ne0\\), then \\(\\zeta=\\mu^{-1}|x|\\zeta\\) lies in the closed span of the \\(e_n\\), and \\((s_n-\\mu)\\langle\\zeta,e_n\\rangle=0\\) for every \\(n\\). So the eigenspace of \\(\\mu\\) is spanned by the \\(e_n\\) with \\(s_n=\\mu\\), and the nonzero eigenvalues of \\(|x|\\), counted with multiplicity, are the \\(s_n\\) listed with repetitions.\n\n(d) \\(\\|v\\zeta\\|^2=\\sum_n|\\langle\\zeta,f_n\\rangle|^2\\le\\|\\zeta\\|^2\\). For every \\(\\zeta\\), \\[\n\\begin{gathered}\nvx\\zeta\\\\\n=\\sum_ns_n\\langle\\zeta,e_n\\rangle vf_n\\\\\n=\\sum_ns_n\\langle\\zeta,e_n\\rangle e_n\\\\\n=|x|\\zeta.\n\\end{gathered}\n\\] The adjoint is \\(v^*\\zeta=\\sum_n\\langle\\zeta,e_n\\rangle f_n\\), so \\(v^*|x|\\zeta=\\sum_ns_n\\langle\\zeta,e_n\\rangle f_n=x\\zeta\\).\n\n(e) If \\(x\\) is self-adjoint, Background 2(b) gives \\(x=\\sum\\lambda_n\\theta_{e_n,e_n}\\) with \\(|\\lambda_n|\\) decreasing; put \\(s_n=|\\lambda_n|\\) and \\(f_n=(\\operatorname{sign}\\lambda_n)\\,e_n\\). If \\(x\\ge0\\), all \\(\\lambda_n>0\\).\n\n(f) Each \\(\\lambda_n\\) is an eigenvalue, since \\(xe_n=\\lambda_ne_n\\). Let \\(\\mu\\notin\\{\\lambda_n:n\\in N\\}\\cup\\{0\\}\\). This set is closed, because \\(\\lambda_n\\to0\\) when \\(N=\\mathbb N\\), so \\(d=\\inf_n|\\lambda_n-\\mu|>0\\). Let \\(p\\) be the projection onto the closed span of the \\(e_n\\); its complement \\(1-p\\) projects onto the kernel of \\(x\\). The operator \\(r\\zeta=\\sum_n(\\lambda_n-\\mu)^{-1}\\langle\\zeta,e_n\\rangle e_n-\\mu^{-1}(1-p)\\zeta\\) is bounded, with \\(\\|r\\|\\le\\max(d^{-1},|\\mu|^{-1})\\), and one checks on each \\(e_n\\) and on the kernel of \\(x\\) that \\(r(x-\\mu)=(x-\\mu)r=1\\). So \\(\\mu\\notin\\sigma(x)\\). If \\(H\\) is infinite-dimensional and \\(0\\notin\\sigma(x)\\), then \\(1=x^{-1}x\\) would be compact, which contradicts Lemma 2.1 applied to any orthonormal sequence. \\(\\square\\)\n\nPart (d) is the polar decomposition of a compact operator, written out explicitly. The general polar decomposition of bounded operators is not needed in this lesson.\n\n**Definition 2.4.** The *singular values* of \\(x\\in K(H)\\) are the numbers \\(s_n(x)=s_n\\), \\(n\\in N\\), of any decomposition (2.1), followed by \\(s_n(x)=0\\) for \\(n>r\\) when \\(N=\\{1,\\dots,r\\}\\). For \\(x=0\\) all \\(s_n(x)=0\\). We write \\(s(x)=(s_n(x))_{n\\ge1}\\); it is a decreasing sequence in \\(c_0\\).\n\n**Proposition 2.5** (approximation numbers). For \\(x\\in K(H)\\) and \\(n\\ge1\\),\n\\[\n\\begin{gathered}\ns_n(x)\\\\\n=\\min\\{\\|x-f\\|:\\\\\n\\ f\\in F(H),\\ \\operatorname{rank}f<n\\}.\n\\end{gathered}\n\\tag{2.2}\n\\]\nConsequently, for \\(x,y\\in K(H)\\), \\(a,b\\in B(H)\\) and \\(m,n\\ge1\\):\n\n(a) \\(s_n(axb)\\le\\|a\\|\\,s_n(x)\\,\\|b\\|\\);\n\n(b) \\(s_{m+n-1}(x+y)\\le s_m(x)+s_n(y)\\);\n\n(c) \\(s_n(x^*)=s_n(|x|)=s_n(x)\\);\n\n(d) \\(|s_n(x)-s_n(y)|\\le\\|x-y\\|\\).\n\n**Proof.** We may assume \\(x\\ne0\\) and use (2.1). The operator \\(x_{n-1}=\\sum_{k<n}s_k\\theta_{f_k,e_k}\\) has rank less than \\(n\\), and \\(x-x_{n-1}=\\sum_{k\\ge n}s_k\\theta_{f_k,e_k}\\) has norm \\(s_n(x)\\) by Theorem 2.3(a) (or is \\(0\\) if \\(n>r\\)). Conversely, let \\(f\\in F(H)\\) with \\(\\operatorname{rank}f<n\\). If \\(s_n(x)=0\\) there is nothing to prove. Otherwise \\(e_1,\\dots,e_n\\) exist, and \\(f\\) restricted to their \\(n\\)-dimensional span has a nonzero kernel, since its range has dimension less than \\(n\\). Take a unit vector \\(\\zeta=\\sum_{k\\le n}c_ke_k\\) with \\(f\\zeta=0\\). Then \\(\\|(x-f)\\zeta\\|^2=\\|x\\zeta\\|^2=\\sum_{k\\le n}s_k^2|c_k|^2\\ge s_n^2\\). This proves (2.2).\n\n(a) \\(axb\\) is compact; if \\(\\operatorname{rank}f<n\\) then \\(\\operatorname{rank}(afb)<n\\), and \\(\\|axb-afb\\|\\le\\|a\\|\\,\\|x-f\\|\\,\\|b\\|\\). Take the minimum over \\(f\\).\n\n(b) If \\(\\operatorname{rank}f<m\\) and \\(\\operatorname{rank}g<n\\), then \\(\\operatorname{rank}(f+g)\\le m+n-2<m+n-1\\) and \\(\\|x+y-f-g\\|\\le\\|x-f\\|+\\|y-g\\|\\).\n\n(c) By Theorem 2.3(b), \\(x^*\\) and \\(|x|\\) have decompositions (2.1) with the same numbers \\(s_n\\).\n\n(d) \\(\\|x-f\\|\\le\\|x-y\\|+\\|y-f\\|\\) for every \\(f\\), and symmetrically. \\(\\square\\)\n\n**Example 2.6.** For \\(\\lambda\\in c_0(I)\\), the diagonal operator \\(d_\\lambda\\) of Example 2.2 satisfies \\(|d_\\lambda|=d_{|\\lambda|}=\\sum_i|\\lambda_i|\\theta_{\\varepsilon_i,\\varepsilon_i}\\). By Theorem 2.3(c), \\(s(d_\\lambda)\\) lists the nonzero numbers \\(|\\lambda_i|\\) in decreasing order, with repetitions, followed by zeros. Only countably many \\(\\lambda_i\\) are nonzero because \\(\\lambda\\in c_0(I)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-04",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "3. Hilbert–Schmidt operators",
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      "full_conditions_and_proof": "## 3. Hilbert–Schmidt operators\n\nAn operator is Hilbert–Schmidt when the squares of its matrix entries are summable. These operators form a Hilbert space, and on a space of square-integrable functions they are exactly the integral operators with square-integrable kernels.\n\n**Lemma 3.1.** Let \\(x\\in B(H)\\), and let \\((\\varepsilon_i)_{i\\in I}\\) and \\((\\varepsilon'_j)_{j\\in J}\\) be orthonormal bases. Then, in \\([0,\\infty]\\),\n\\[\n\\begin{gathered}\n\\sum_i\\|x\\varepsilon_i\\|^2\\\\\n=\\sum_{i,j}|\\langle x\\varepsilon_i,\\varepsilon'_j\\rangle|^2\\\\\n=\\sum_j\\|x^*\\varepsilon'_j\\|^2 .\n\\end{gathered}\n\\tag{3.1}\n\\]\nIn particular \\(\\sum_i\\|x\\varepsilon_i\\|^2\\) does not depend on the basis, and it does not change when \\(x\\) is replaced by \\(x^*\\).\n\n**Proof.** By Parseval, \\(\\|x\\varepsilon_i\\|^2=\\sum_j|\\langle x\\varepsilon_i,\\varepsilon'_j\\rangle|^2\\), and since \\(\\langle x\\varepsilon_i,\\varepsilon'_j\\rangle=\\langle\\varepsilon_i,x^*\\varepsilon'_j\\rangle\\), also \\(\\sum_i|\\langle x\\varepsilon_i,\\varepsilon'_j\\rangle|^2=\\|x^*\\varepsilon'_j\\|^2\\). A double sum of nonnegative terms may be computed in either order. The right side of (3.1) involves only \\((\\varepsilon'_j)\\); applying (3.1) with a third basis in place of \\((\\varepsilon_i)\\) shows that the left side is the same for all bases. \\(\\square\\)\n\n**Definition 3.2.** The *Hilbert–Schmidt norm* of \\(x\\in B(H)\\) is \\(\\|x\\|_2=(\\sum_i\\|x\\varepsilon_i\\|^2)^{1/2}\\in[0,\\infty]\\) for any orthonormal basis \\((\\varepsilon_i)\\). The operators with \\(\\|x\\|_2<\\infty\\) are the *Hilbert–Schmidt operators*; they form the set \\(\\mathcal L^2(H)\\).\n\n**Theorem 3.3.**\n\n(a) \\(\\|x\\|\\le\\|x\\|_2=\\|x^*\\|_2\\) and \\(\\|axb\\|_2\\le\\|a\\|\\,\\|x\\|_2\\,\\|b\\|\\) for \\(a,b\\in B(H)\\). So \\(\\mathcal L^2(H)\\) is a two-sided ideal of \\(B(H)\\), closed under adjoints.\n\n(b) \\(\\mathcal L^2(H)\\subseteq K(H)\\). A compact operator \\(x\\) is Hilbert–Schmidt if and only if \\(\\sum_ns_n(x)^2<\\infty\\), and then \\(\\|x\\|_2^2=\\sum_ns_n(x)^2\\).\n\n(c) For \\(x,y\\in\\mathcal L^2(H)\\), the sum \\(\\langle x,y\\rangle_2=\\sum_i\\langle x\\varepsilon_i,y\\varepsilon_i\\rangle\\) converges absolutely and does not depend on the basis. With this inner product \\(\\mathcal L^2(H)\\) is a Hilbert space with norm \\(\\|\\cdot\\|_2\\), and \\(F(H)\\) is dense in it.\n\n(d) \\(\\|\\theta_{\\xi,\\eta}\\|_2=\\|\\xi\\|\\,\\|\\eta\\|\\), and \\(\\langle\\theta_{\\xi,\\eta},\\theta_{\\xi',\\eta'}\\rangle_2=\\langle\\xi,\\xi'\\rangle\\langle\\eta',\\eta\\rangle\\).\n\n**Proof.** (a) For a unit vector \\(\\zeta\\), choose an orthonormal basis containing \\(\\zeta\\); then \\(\\|x\\zeta\\|^2\\le\\sum_i\\|x\\varepsilon_i\\|^2\\). The equality \\(\\|x^*\\|_2=\\|x\\|_2\\) is Lemma 3.1. Since \\(\\|ax\\varepsilon_i\\|\\le\\|a\\|\\,\\|x\\varepsilon_i\\|\\), \\(\\|ax\\|_2\\le\\|a\\|\\,\\|x\\|_2\\); then \\(\\|xb\\|_2=\\|b^*x^*\\|_2\\le\\|b\\|\\,\\|x\\|_2\\). Minkowski's inequality in \\(\\ell^2(I)\\) gives \\(\\|x+y\\|_2\\le\\|x\\|_2+\\|y\\|_2\\).\n\n(b) Let \\(x\\in\\mathcal L^2(H)\\) and \\(\\varepsilon>0\\). Choose a finite \\(F\\subseteq I\\) with \\(\\sum_{i\\notin F}\\|x\\varepsilon_i\\|^2<\\varepsilon^2\\), and let \\(p_F\\) be the projection onto the span of \\(\\{\\varepsilon_i:i\\in F\\}\\). Then \\(xp_F\\) has finite rank and, by (a) and the choice of the basis, \\[\n\\begin{gathered}\n\\|x-xp_F\\|\\\\\n\\le\\|x-xp_F\\|_2\\\\\n=(\\sum_{i\\notin F}\\|x\\varepsilon_i\\|^2)^{1/2}<\\varepsilon.\n\\end{gathered}\n\\] So \\(x\\) is a norm limit of finite-rank operators, hence compact, and \\(F(H)\\) is dense in \\(\\mathcal L^2(H)\\) for \\(\\|\\cdot\\|_2\\). For a compact \\(x\\) with decomposition (2.1), choose an orthonormal basis consisting of the \\(e_n\\) and a basis of the kernel of \\(x\\). Then \\(\\|x\\|_2^2=\\sum_n\\|xe_n\\|^2=\\sum_ns_n^2\\).\n\n(c) \\(|\\langle x\\varepsilon_i,y\\varepsilon_i\\rangle|\\le\\frac12(\\|x\\varepsilon_i\\|^2+\\|y\\varepsilon_i\\|^2)\\) gives absolute convergence. The form \\(\\langle\\cdot,\\cdot\\rangle_2\\) is sesquilinear, and \\(\\langle x,x\\rangle_2=\\|x\\|_2^2\\). By the polarization identity (1.1) it is determined by \\(\\|\\cdot\\|_2\\), which does not depend on the basis; and \\(\\|x\\|_2=0\\) forces \\(x=0\\) by (a). For completeness, let \\((x_k)\\) be Cauchy for \\(\\|\\cdot\\|_2\\). By (a) it is Cauchy in norm, so \\(x_k\\to x\\) in norm for some \\(x\\in B(H)\\). For finite \\(F\\subseteq I\\),\n\\[\n\\begin{gathered}\n\\sum_{i\\in F}\\|(x-x_k)\\varepsilon_i\\|^2\\\\\n=\\lim_m\\sum_{i\\in F}\\|(x_m-x_k)\\varepsilon_i\\|^2\\\\\n\\le\\sup_{m\\ge k}\\|x_m-x_k\\|_2^2 .\n\\end{gathered}\n\\]\nTaking the supremum over \\(F\\) gives \\(\\|x-x_k\\|_2\\le\\sup_{m\\ge k}\\|x_m-x_k\\|_2\\to0\\). So \\(x\\in\\mathcal L^2(H)\\) and \\(x_k\\to x\\) in \\(\\|\\cdot\\|_2\\). Density of \\(F(H)\\) was shown in (b).\n\n(d) \\(\\theta_{\\xi,\\eta}\\varepsilon_i=\\langle\\varepsilon_i,\\eta\\rangle\\xi\\), so \\(\\|\\theta_{\\xi,\\eta}\\|_2^2=\\sum_i|\\langle\\varepsilon_i,\\eta\\rangle|^2\\|\\xi\\|^2=\\|\\eta\\|^2\\|\\xi\\|^2\\), and \\[\n\\begin{gathered}\n\\langle\\theta_{\\xi,\\eta},\\theta_{\\xi',\\eta'}\\rangle_2\\\\\n=\\sum_i\\langle\\varepsilon_i,\\eta\\rangle\\overline{\\langle\\varepsilon_i,\\eta'\\rangle}\\langle\\xi,\\xi'\\rangle\\\\\n=\\langle\\xi,\\xi'\\rangle\\langle\\eta',\\eta\\rangle\n\\end{gathered}\n\\] by Parseval. \\(\\square\\)\n\n**Example 3.4** (Hilbert–Schmidt operators as integral operators). Let \\((X,\\mu)\\) be a \\(\\sigma\\)-finite measure space and \\(H=L^2(X,\\mu)\\). For \\(k\\in L^2(X\\times X,\\mu\\otimes\\mu)\\) define\n\\[\n(x_kf)(s)=\\int_Xk(s,t)f(t)\\,d\\mu(t).\n\\]\nThen \\(x_k\\in\\mathcal L^2(H)\\), \\(\\|x_k\\|_2=\\|k\\|_{L^2}\\), and \\(k\\mapsto x_k\\) is a unitary operator of \\(L^2(X\\times X,\\mu\\otimes\\mu)\\) onto \\(\\mathcal L^2(H)\\). So the Hilbert–Schmidt operators on \\(L^2(X,\\mu)\\) are exactly the integral operators with square-integrable kernels.\n\n*Proof.* By Tonelli's theorem \\(\\int\\!\\!\\int|k(s,t)|^2d\\mu(t)\\,d\\mu(s)=\\|k\\|^2<\\infty\\), so \\(k(s,\\cdot)\\in L^2(X,\\mu)\\) for almost every \\(s\\). For such \\(s\\) the integral defining \\((x_kf)(s)\\) converges absolutely, and \\(|(x_kf)(s)|\\le\\|k(s,\\cdot)\\|\\,\\|f\\|\\) by Cauchy–Schwarz. If \\(A\\subseteq X\\) has finite measure, then \\[\n\\begin{gathered}\n\\int_A\\int|k(s,t)f(t)|\\,d\\mu(t)\\,d\\mu(s)\\\\\n\\le\\mu(A)^{1/2}\\|k\\|\\,\\|f\\|<\\infty,\n\\end{gathered}\n\\] so Fubini's theorem shows that \\(x_kf\\) is measurable on \\(A\\); as \\(X\\) is a countable union of such sets, \\(x_kf\\) is measurable. Then \\(\\|x_kf\\|^2\\le\\|k\\|^2\\|f\\|^2\\). So \\(x_k\\in B(H)\\), \\(\\|x_k\\|\\le\\|k\\|\\), and \\(k\\mapsto x_k\\) is linear.\n\nFor a product kernel \\(k(s,t)=\\varphi(s)\\overline{\\psi(t)}\\) with \\(\\varphi,\\psi\\in L^2(X,\\mu)\\) we get \\(x_k=\\theta_{\\varphi,\\psi}\\). By Theorem 3.3(d),\n\\[\n\\begin{gathered}\n\\langle\\theta_{\\varphi,\\psi},\\theta_{\\varphi',\\psi'}\\rangle_2\\\\\n=\\langle\\varphi,\\varphi'\\rangle\\langle\\psi',\\psi\\rangle\\\\\n=\\langle\\varphi\\otimes\\bar\\psi,\\ \\varphi'\\otimes\\bar\\psi'\\rangle_{L^2(X\\times X)} .\n\\end{gathered}\n\\]\nSo \\(k\\mapsto x_k\\) is isometric from the span \\(P\\) of the product kernels into \\(\\mathcal L^2(H)\\). The span \\(P\\) is dense in \\(L^2(X\\times X)\\). Indeed, if \\(g\\in L^2(X\\times X)\\) is orthogonal to \\(P\\), let \\(X=\\bigcup_nA_n\\) with \\(A_n\\) increasing of finite measure. The finite signed measures \\(E\\mapsto\\int_E\\operatorname{Re}g\\) and \\(E\\mapsto\\int_E\\operatorname{Im}g\\) on \\(A_n\\times A_n\\) vanish on the rectangles \\(A\\times B\\subseteq A_n\\times A_n\\), which form a family closed under intersections that generates the product \\(\\sigma\\)-algebra; by Dynkin's lemma they vanish identically, so \\(g=0\\) almost everywhere on each \\(A_n\\times A_n\\), and hence almost everywhere on \\(X\\times X\\).\n\nConsequently \\(k\\mapsto x_k\\) extends from \\(P\\) to an isometry \\(U\\) of \\(L^2(X\\times X)\\) into \\(\\mathcal L^2(H)\\). If \\(k_n\\in P\\) and \\(k_n\\to k\\) in \\(L^2\\), then \\(x_{k_n}\\to x_k\\) in operator norm (since \\(\\|x_k-x_{k_n}\\|\\le\\|k-k_n\\|\\)), and \\(x_{k_n}\\to Uk\\) in \\(\\|\\cdot\\|_2\\), hence in operator norm. So \\(Uk=x_k\\). The range of \\(U\\) is closed and contains every \\(\\theta_{\\varphi,\\psi}\\), hence \\(F(H)\\), which is dense in \\(\\mathcal L^2(H)\\) by Theorem 3.3(c). So \\(U\\) is onto. \\(\\square\\)\n\n**Example 3.5** (an operator that is Hilbert–Schmidt but not of trace class). On \\(H=L^2(0,1)\\) let \\((Vf)(s)=\\int_0^sf(t)\\,dt\\), the integral operator with kernel \\(k(s,t)=1\\) for \\(t<s\\) and \\(0\\) otherwise. By Example 3.4, \\(V\\in\\mathcal L^2(H)\\) and \\(\\|V\\|_2^2=\\int_0^1\\!\\int_0^1k^2=\\frac12\\). We compute its singular values. The adjoint is \\((V^*g)(t)=\\int_t^1g(s)\\,ds\\). Let \\(\\mu>0\\) and \\(f\\ne0\\) with \\(V^*Vf=\\mu f\\). The function \\(u=Vf\\) is continuous with \\(u(0)=0\\): Cauchy–Schwarz gives \\(|u(t)-u(s)|\\leq|t-s|^{1/2}\\|f\\|_2\\). The continuous-integrand fundamental theorem in [Lemma 0.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) justifies the derivatives below. Then \\(\\mu f=V^*u\\) is continuous, so \\(u\\) is continuously differentiable with \\(u'=f\\). Next \\(\\mu f=V^*u\\) is continuously differentiable with \\((\\mu f)'=-u\\) and \\(\\mu f(1)=0\\). Thus \\(\\mu u''=-u\\), \\(u(0)=0\\) and \\(u'(1)=0\\). Put \\(a=\\mu^{-1/2}\\). The general solution of \\(u''+a^2u=0\\) is \\(A\\cos(at)+B\\sin(at)\\). To verify completeness of this list, choose \\(A=u(0)\\), \\(B=u'(0)/a\\), and subtract that solution. The difference \\(w\\) has \\(w(0)=w'(0)=0\\); differentiating \\(|w'|^2+a^2|w|^2\\) gives zero, so \\(w=0\\). Thus the boundary conditions leave precisely the multiples of \\(\\sin(t/\\sqrt\\mu)\\) with \\(\\cos(1/\\sqrt\\mu)=0\\). So the nonzero eigenvalues of \\(V^*V\\) are \\(\\mu_n=((n-\\frac12)\\pi)^{-2}\\), \\(n\\ge1\\), each with a one-dimensional eigenspace spanned by \\(\\cos((n-\\frac12)\\pi t)\\). One checks directly that these functions are eigenvectors. By Theorem 2.3(c),\n\\[\ns_n(V)=\\frac{1}{(n-\\frac12)\\pi},\\qquad n\\ge1 .\n\\]\nTheorem 3.3(b) and the independently computed kernel norm now give \\(\\frac4{\\pi^2}\\sum_n(2n-1)^{-2}=\\sum_ns_n(V)^2=\\|V\\|_2^2=\\frac12\\). In particular this calculation proves \\(\\sum_n(2n-1)^{-2}=\\pi^2/8\\), rather than assuming that sum. But \\(\\sum_ns_n(V)=\\infty\\), so \\(V\\) is not of trace class (Theorem 4.3 below).\n\n",
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    {
      "id": "OA-FND-LT-05",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "4. Trace-class operators and the trace",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
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      "full_conditions_and_proof": "## 4. Trace-class operators and the trace\n\nAn operator is of trace class when its matrix entries along pairs of orthonormal families have uniformly bounded absolute sums. For a compact operator this is the summability of its singular values. The trace is defined on these operators and has the properties of the trace of a matrix.\n\n**Definition 4.1.** For \\(x\\in B(H)\\) let\n\\[\n\\begin{gathered}\n\\|x\\|_1\\\\\n=\\sup\\Big\\{\\sum_{j\\in J}|\\langle x\\xi_j,\\eta_j\\rangle|\\Big\\}\\in[0,\\infty],\n\\end{gathered}\n\\tag{4.1}\n\\]\nthe supremum over all finite sets \\(J\\) and all orthonormal families \\((\\xi_j)_{j\\in J}\\) and \\((\\eta_j)_{j\\in J}\\) in \\(H\\). The operators with \\(\\|x\\|_1<\\infty\\) are the *trace-class* (or *nuclear*) operators; they form the set \\(\\mathcal L^1(H)\\). Since the terms are nonnegative, the supremum does not change if infinite index sets \\(J\\) are allowed.\n\n**Lemma 4.2.**\n\n(a) \\(\\mathcal L^1(H)\\) is a linear subspace of \\(B(H)\\), \\(\\|\\cdot\\|_1\\) is a norm on it, \\(\\|x\\|\\le\\|x\\|_1\\), and \\(\\|x^*\\|_1=\\|x\\|_1\\).\n\n(b) \\(\\|\\theta_{\\xi,\\eta}\\|_1=\\|\\xi\\|\\,\\|\\eta\\|\\).\n\n(c) If \\((x_\\alpha)\\) is a net with \\(\\langle x_\\alpha\\xi,\\eta\\rangle\\to\\langle x\\xi,\\eta\\rangle\\) for all \\(\\xi,\\eta\\), then \\(\\|x\\|_1\\le\\liminf_\\alpha\\|x_\\alpha\\|_1\\).\n\n(d) \\((\\mathcal L^1(H),\\|\\cdot\\|_1)\\) is a Banach space.\n\n**Proof.** (a) The inequalities \\(\\|x+y\\|_1\\le\\|x\\|_1+\\|y\\|_1\\) and \\(\\|cx\\|_1=|c|\\,\\|x\\|_1\\) hold term by term. With one pair of unit vectors, \\(|\\langle x\\xi,\\eta\\rangle|\\le\\|x\\|_1\\); the supremum over \\(\\xi,\\eta\\) is \\(\\|x\\|\\). Since \\(|\\langle x^*\\xi_j,\\eta_j\\rangle|=|\\langle x\\eta_j,\\xi_j\\rangle|\\), passing to \\(x^*\\) exchanges the two families.\n\n(b) By Cauchy–Schwarz and Bessel's inequality,\n\\[\n\\begin{gathered}\n\\sum_j|\\langle\\theta_{\\xi,\\eta}\\xi_j,\\eta_j\\rangle|\\\\\n=\\sum_j|\\langle\\xi_j,\\eta\\rangle|\\,|\\langle\\xi,\\eta_j\\rangle|\\\\\n\\le\\Big(\\sum_j|\\langle\\xi_j,\\eta\\rangle|^2\\Big)^{1/2}\\Big(\\sum_j|\\langle\\eta_j,\\xi\\rangle|^2\\Big)^{1/2}\\\\\n\\le\\|\\eta\\|\\,\\|\\xi\\| .\n\\end{gathered}\n\\]\nIf \\(\\xi,\\eta\\ne0\\), the single pair \\(\\xi_1=\\eta/\\|\\eta\\|\\), \\(\\eta_1=\\xi/\\|\\xi\\|\\) gives equality.\n\n(c) For fixed finite families, \\[\n\\begin{gathered}\n\\sum_j|\\langle x\\xi_j,\\eta_j\\rangle|\\\\\n=\\lim_\\alpha\\sum_j|\\langle x_\\alpha\\xi_j,\\eta_j\\rangle|\\\\\n\\le\\liminf_\\alpha\\|x_\\alpha\\|_1.\n\\end{gathered}\n\\]\n\n(d) Let \\((x_k)\\) be Cauchy for \\(\\|\\cdot\\|_1\\). By (a) it is Cauchy in norm, so \\(x_k\\to x\\) in norm. For fixed \\(k\\), apply (c) to the sequence \\((x_m-x_k)_m\\), which converges in norm to \\(x-x_k\\): \\(\\|x-x_k\\|_1\\le\\sup_{m\\ge k}\\|x_m-x_k\\|_1\\), which tends to \\(0\\). So \\(x=(x-x_k)+x_k\\in\\mathcal L^1(H)\\) and \\(x_k\\to x\\) in \\(\\|\\cdot\\|_1\\). \\(\\square\\)\n\n**Theorem 4.3.**\n\n(a) \\(\\|x\\|_2^2\\le\\|x\\|\\,\\|x\\|_1\\) for every \\(x\\in B(H)\\). Hence \\(\\mathcal L^1(H)\\subseteq\\mathcal L^2(H)\\subseteq K(H)\\).\n\n(b) For \\(x\\in K(H)\\), \\(\\|x\\|_1=\\sum_ns_n(x)\\) in \\([0,\\infty]\\). So \\(x\\in\\mathcal L^1(H)\\) if and only if \\(x\\) is compact and \\(\\sum_ns_n(x)<\\infty\\).\n\n(c) If \\(x\\in\\mathcal L^1(H)\\), every decomposition (2.1) of \\(x\\) converges in \\(\\|\\cdot\\|_1\\).\n\n**Proof.** (a) We may assume \\(\\|x\\|_1<\\infty\\). Let \\(\\xi_1,\\dots,\\xi_m\\) be orthonormal, let \\(L\\) be their span and \\(p\\) the projection onto \\(L\\). The operator \\(px^*xp\\) is positive and maps \\(L\\) into \\(L\\), so \\(L\\) has an orthonormal basis \\(\\zeta_1,\\dots,\\zeta_m\\) with \\(px^*xp\\,\\zeta_k=\\sigma_k^2\\zeta_k\\), \\(\\sigma_k\\ge0\\). Then \\(\\langle x\\zeta_k,x\\zeta_l\\rangle=\\langle px^*xp\\,\\zeta_k,\\zeta_l\\rangle=\\sigma_k^2\\delta_{kl}\\): the vectors \\(x\\zeta_k\\) are orthogonal, with norms \\(\\sigma_k\\). Choose orthonormal \\(\\eta_1,\\dots,\\eta_m\\) with \\(x\\zeta_k=\\sigma_k\\eta_k\\): put \\(\\eta_k=\\sigma_k^{-1}x\\zeta_k\\) when \\(\\sigma_k>0\\), and complete by further orthonormal vectors, which is possible since \\(\\dim H\\ge m\\). Then\n\\[\n\\sum_k\\sigma_k=\\sum_k\\langle x\\zeta_k,\\eta_k\\rangle\\le\\|x\\|_1 .\n\\]\nApply Lemma 3.1 to \\(xp\\) and to the two orthonormal bases of \\(H\\) obtained by completing \\((\\xi_i)\\) and \\((\\zeta_k)\\) with one and the same orthonormal basis of \\(L^\\perp\\), on which \\(xp\\) vanishes. This gives \\(\\sum_i\\|x\\xi_i\\|^2=\\sum_k\\|x\\zeta_k\\|^2=\\sum_k\\sigma_k^2\\). Since \\(\\sigma_k\\le\\|x\\|\\),\n\\[\n\\sum_{i\\le m}\\|x\\xi_i\\|^2=\\sum_k\\sigma_k^2\\le\\|x\\|\\sum_k\\sigma_k\\le\\|x\\|\\,\\|x\\|_1 .\n\\]\nEvery finite subfamily of an orthonormal basis is such a family, so \\(\\|x\\|_2^2\\le\\|x\\|\\,\\|x\\|_1\\). The inclusion \\(\\mathcal L^2(H)\\subseteq K(H)\\) is Theorem 3.3(b).\n\n(b) Let \\(x\\in K(H)\\), \\(x\\ne0\\), with decomposition (2.1), and let \\((\\xi_j)\\), \\((\\eta_j)\\) be finite orthonormal families. Since \\(\\langle x\\xi_j,\\eta_j\\rangle=\\sum_ns_n\\langle\\xi_j,e_n\\rangle\\langle f_n,\\eta_j\\rangle\\),\n\\[\n\\begin{gathered}\n\\sum_j|\\langle x\\xi_j,\\eta_j\\rangle|\\\\\n\\le\\sum_ns_n\\sum_j|\\langle\\xi_j,e_n\\rangle|\\,|\\langle f_n,\\eta_j\\rangle|\\\\\n\\le\\sum_ns_n\\Big(\\sum_j|\\langle e_n,\\xi_j\\rangle|^2\\Big)^{1/2}\\\\\n{}\\cdot\\Big(\\sum_j|\\langle f_n,\\eta_j\\rangle|^2\\Big)^{1/2}\\\\\n\\le\\sum_ns_n\n\\end{gathered}\n\\]\nby Cauchy–Schwarz and Bessel's inequality. So \\(\\|x\\|_1\\le\\sum_ns_n\\). Conversely, the families \\(\\xi_j=e_j\\), \\(\\eta_j=f_j\\), \\(j\\le M\\), give \\(\\sum_{j\\le M}\\langle xe_j,f_j\\rangle=\\sum_{j\\le M}s_j\\), so \\(\\|x\\|_1\\ge\\sum_{j\\le M}s_j\\) for every \\(M\\). Together with (a), which shows that a trace-class operator is compact, this proves (b).\n\n(c) The remainder \\(\\sum_{n>M}s_n\\theta_{f_n,e_n}\\) is compact and has a decomposition of the form (2.1) with the numbers \\(s_{M+1},s_{M+2},\\dots\\). By (b) its trace norm is \\(\\sum_{n>M}s_n\\), which tends to \\(0\\). \\(\\square\\)\n\n**Corollary 4.4.**\n\n(a) If \\(x\\in\\mathcal L^1(H)\\) and \\(a,b\\in B(H)\\), then \\(axb\\in\\mathcal L^1(H)\\) and \\(\\|axb\\|_1\\le\\|a\\|\\,\\|x\\|_1\\,\\|b\\|\\). So \\(\\mathcal L^1(H)\\) is a two-sided ideal of \\(B(H)\\), closed under adjoints.\n\n(b) \\(x\\in\\mathcal L^1(H)\\) if and only if \\(|x|\\in\\mathcal L^1(H)\\), if and only if \\(x^*\\in\\mathcal L^1(H)\\), and \\(\\|x\\|_1=\\||x|\\|_1=\\|x^*\\|_1\\).\n\n(c) \\(F(H)\\subseteq\\mathcal L^1(H)\\), and \\(F(H)\\) is dense in \\(\\mathcal L^1(H)\\) for \\(\\|\\cdot\\|_1\\). For \\(x\\in\\mathcal L^1(H)\\), \\(\\|x\\|\\le\\|x\\|_2\\le\\|x\\|_1\\).\n\n(d) If \\(\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\), the series \\(\\sum_n\\theta_{\\xi_n,\\eta_n}\\) converges in \\(\\|\\cdot\\|_1\\), and its sum \\(t\\) satisfies \\(\\|t\\|_1\\le\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|\\).\n\n**Proof.** (a) follows from Proposition 2.5(a) and Theorem 4.3(b). (b) By Lemma 4.2(a), \\(x^*\\in\\mathcal L^1(H)\\) exactly when \\(x\\in\\mathcal L^1(H)\\), and \\(\\|x^*\\|_1=\\|x\\|_1\\). If \\(x\\in\\mathcal L^1(H)\\), then \\(x\\) is compact by Theorem 4.3(a). If \\(|x|\\in\\mathcal L^1(H)\\), then \\(x\\) is compact too: \\(\\|x\\zeta\\|^2=\\langle x^*x\\zeta,\\zeta\\rangle=\\||x|\\zeta\\|^2\\) for every \\(\\zeta\\), so \\(\\|x\\|_2=\\||x|\\|_2\\), which is finite by Theorem 4.3(a), and \\(\\mathcal L^2(H)\\subseteq K(H)\\) by Theorem 3.3(b). For compact \\(x\\), Proposition 2.5(c) and Theorem 4.3(b) give \\(\\|x\\|_1=\\||x|\\|_1\\) in \\([0,\\infty]\\). (c) A finite-rank operator has only finitely many nonzero singular values. Density follows from Theorem 4.3(c). For \\(x\\in\\mathcal L^1(H)\\), \\(\\|x\\|_2^2=\\sum s_n^2\\le(\\sum s_n)^2\\). (d) By Lemma 4.2(b) the series converges absolutely in the Banach space \\(\\mathcal L^1(H)\\). \\(\\square\\)\n\n**Example 4.5** (three different norms). Let \\(\\xi\\perp\\eta\\) be unit vectors and \\(x=\\theta_{\\xi,\\eta}+\\theta_{\\eta,\\xi}\\). Then \\(x(\\xi\\pm\\eta)=\\pm(\\xi\\pm\\eta)\\) and \\(x\\) vanishes on \\(\\{\\xi,\\eta\\}^\\perp\\). So \\(x\\) is self-adjoint with eigenvalues \\(\\pm1\\), and \\(s(x)=(1,1,0,0,\\dots)\\). Hence \\(\\|x\\|=1\\), \\(\\|x\\|_2=\\sqrt2\\), \\(\\|x\\|_1=2\\), and \\(\\operatorname{Tr}(x)=0\\) (Theorem 4.6). For the diagonal operators of Example 2.2, Example 2.6 shows that \\(d_\\lambda\\in\\mathcal L^2(H)\\) exactly when \\(\\lambda\\in\\ell^2(I)\\) and \\(d_\\lambda\\in\\mathcal L^1(H)\\) exactly when \\(\\lambda\\in\\ell^1(I)\\), with \\(\\|d_\\lambda\\|_2=\\|\\lambda\\|_2\\) and \\(\\|d_\\lambda\\|_1=\\|\\lambda\\|_1\\). The Volterra operator of Example 3.5 lies in \\(\\mathcal L^2(H)\\) but not in \\(\\mathcal L^1(H)\\).\n\n**Theorem 4.6** (the trace). Let \\(x\\in\\mathcal L^1(H)\\) and let \\((\\varepsilon_i)_{i\\in I}\\) be an orthonormal basis. Then \\(\\sum_i\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\) converges absolutely, \\(\\sum_i|\\langle x\\varepsilon_i,\\varepsilon_i\\rangle|\\le\\|x\\|_1\\), and the sum, denoted \\(\\operatorname{Tr}(x)\\), does not depend on the basis. More precisely, if \\(x=\\sum_n\\theta_{\\xi_n,\\eta_n}\\) with \\(\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\), as in (2.1), then\n\\[\n\\operatorname{Tr}(x)=\\sum_n\\langle\\xi_n,\\eta_n\\rangle .\n\\tag{4.2}\n\\]\nThe trace is a linear functional on \\(\\mathcal L^1(H)\\) with \\(|\\operatorname{Tr}(x)|\\le\\|x\\|_1\\), \\(\\operatorname{Tr}(x^*)=\\overline{\\operatorname{Tr}(x)}\\), \\(\\operatorname{Tr}(x)\\ge0\\) for \\(x\\ge0\\), and \\(\\operatorname{Tr}(\\theta_{\\xi,\\eta})=\\langle\\xi,\\eta\\rangle\\). For \\(a\\in B(H)\\),\n\\[\n\\begin{gathered}\n\\operatorname{Tr}(ax)\\\\\n=\\operatorname{Tr}(xa),\\\\\n|\\operatorname{Tr}(ax)|\\\\\n\\le\\|a\\|\\,\\|x\\|_1,\n\\end{gathered}\n\\tag{4.3}\n\\]\nand in particular\n\\[\n\\operatorname{Tr}(a\\,\\theta_{\\xi,\\eta})=\\langle a\\xi,\\eta\\rangle=\\omega_{\\xi,\\eta}(a).\n\\tag{4.4}\n\\]\nFinally \\(\\|x\\|_1=\\operatorname{Tr}(|x|)\\).\n\n**Proof.** Taking \\(\\xi_j=\\eta_j=\\varepsilon_j\\) over finite subsets in (4.1) gives absolute convergence and the bound. Let \\(x=\\sum_n\\theta_{\\xi_n,\\eta_n}\\) with \\(\\sum\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\); the series converges in \\(\\|\\cdot\\|_1\\) by Corollary 4.4(d), hence in norm. Then \\(\\langle x\\varepsilon_i,\\varepsilon_i\\rangle=\\sum_n\\langle\\varepsilon_i,\\eta_n\\rangle\\langle\\xi_n,\\varepsilon_i\\rangle\\). The double family is absolutely summable:\n\\[\n\\begin{gathered}\n\\sum_n\\sum_i|\\langle\\varepsilon_i,\\eta_n\\rangle|\\,|\\langle\\xi_n,\\varepsilon_i\\rangle|\\\\\n\\le\\sum_n\\|\\eta_n\\|\\,\\|\\xi_n\\|<\\infty,\n\\end{gathered}\n\\]\nby Cauchy–Schwarz and Parseval in \\(i\\). So we may sum over \\(i\\) first: \\[\n\\begin{gathered}\n\\sum_i\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\\\\n=\\sum_n\\sum_i\\langle\\xi_n,\\varepsilon_i\\rangle\\langle\\varepsilon_i,\\eta_n\\rangle\\\\\n=\\sum_n\\langle\\xi_n,\\eta_n\\rangle,\n\\end{gathered}\n\\] by Parseval. The right side does not involve the basis. Every trace-class operator has such a representation, namely (2.1) with \\(\\xi_n=s_nf_n\\), \\(\\eta_n=e_n\\).\n\nLinearity is clear for a fixed basis. \\(\\operatorname{Tr}(x^*)=\\sum_i\\langle x^*\\varepsilon_i,\\varepsilon_i\\rangle=\\sum_i\\overline{\\langle x\\varepsilon_i,\\varepsilon_i\\rangle}\\), positivity is termwise, and \\(\\operatorname{Tr}(\\theta_{\\xi,\\eta})=\\langle\\xi,\\eta\\rangle\\) is (4.2). For (4.3), write \\(x=\\sum_ns_n\\theta_{f_n,e_n}\\) as in (2.1). By Lemma 1.4(b), \\(ax=\\sum_ns_n\\theta_{af_n,e_n}\\) and \\(xa=\\sum_ns_n\\theta_{f_n,a^*e_n}\\), both with \\(\\sum_ns_n\\|a\\|<\\infty\\). By (4.2), \\[\n\\begin{gathered}\n\\operatorname{Tr}(ax)\\\\\n=\\sum_ns_n\\langle af_n,e_n\\rangle\\\\\n=\\sum_ns_n\\langle f_n,a^*e_n\\rangle\\\\\n=\\operatorname{Tr}(xa).\n\\end{gathered}\n\\] The bound follows from \\(|\\operatorname{Tr}(ax)|\\le\\|ax\\|_1\\) and Corollary 4.4(a). Formula (4.4) is (4.2) for \\(a\\theta_{\\xi,\\eta}=\\theta_{a\\xi,\\eta}\\). Finally \\(|x|=\\sum s_n\\theta_{e_n,e_n}\\) by Theorem 2.3(b), so \\(\\operatorname{Tr}(|x|)=\\sum s_n=\\|x\\|_1\\). \\(\\square\\)\n\n**Proposition 4.7** (testing on one basis). Let \\(h\\in B(H)_+\\). The sum \\(\\sum_i\\langle h\\varepsilon_i,\\varepsilon_i\\rangle\\in[0,\\infty]\\) is the same for all orthonormal bases. It is finite if and only if \\(h\\in\\mathcal L^1(H)\\), and then it equals \\(\\operatorname{Tr}(h)=\\|h\\|_1\\). Consequently, an operator \\(x\\) is of trace class if and only if \\(\\sum_i\\langle|x|\\varepsilon_i,\\varepsilon_i\\rangle<\\infty\\) for one orthonormal basis; and then \\(\\operatorname{Tr}(x)=\\sum_i\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\) for every orthonormal basis, with absolute convergence.\n\n**Proof.** \\(\\sum_i\\langle h\\varepsilon_i,\\varepsilon_i\\rangle=\\sum_i\\|h^{1/2}\\varepsilon_i\\|^2=\\|h^{1/2}\\|_2^2\\), which does not depend on the basis (Lemma 3.1). If it is finite, then \\(h^{1/2}\\in\\mathcal L^2(H)\\subseteq K(H)\\), so \\(h=(h^{1/2})^2\\) is compact. Write \\(h=\\sum_ns_n\\theta_{e_n,e_n}\\) as in Theorem 2.3(e) and complete \\((e_n)\\) to an orthonormal basis: \\(\\sum_ns_n=\\sum_n\\langle he_n,e_n\\rangle\\le\\|h^{1/2}\\|_2^2<\\infty\\). By Theorem 4.3(b), \\(h\\in\\mathcal L^1(H)\\), and by Theorem 4.6 the sum over any basis is \\(\\operatorname{Tr}(h)=\\operatorname{Tr}(|h|)=\\|h\\|_1\\). If \\(h\\in\\mathcal L^1(H)\\), the sum is \\(\\operatorname{Tr}(h)<\\infty\\). The last statement follows from Corollary 4.4(b) and Theorem 4.6. \\(\\square\\)\n\n**Proposition 4.8** (positive parts). Every \\(x\\in\\mathcal L^1(H)\\) can be written \\(x=h_1-h_2+i(h_3-h_4)\\) with positive trace-class operators \\(h_1,\\dots,h_4\\). A self-adjoint \\(x\\in\\mathcal L^1(H)\\) can be written \\(x=h_1-h_2\\) with \\(h_1,h_2\\ge0\\) of trace class and \\(\\|x\\|_1=\\operatorname{Tr}(h_1)+\\operatorname{Tr}(h_2)\\).\n\n**Proof.** The operators \\(\\operatorname{Re}x=\\frac12(x+x^*)\\) and \\(\\operatorname{Im}x=\\frac1{2i}(x-x^*)\\) are self-adjoint, lie in \\(\\mathcal L^1(H)\\) by Corollary 4.4(b), and \\(x=\\operatorname{Re}x+i\\operatorname{Im}x\\). For self-adjoint \\(x\\in\\mathcal L^1(H)\\), \\(x\\ne0\\), Theorem 2.3(e) gives \\(x=\\sum_n\\lambda_n\\theta_{e_n,e_n}\\) with real \\(\\lambda_n\\) and \\(\\sum|\\lambda_n|=\\|x\\|_1\\). Put \\(h_1=\\sum_{\\lambda_n>0}\\lambda_n\\theta_{e_n,e_n}\\) and \\(h_2=\\sum_{\\lambda_n<0}|\\lambda_n|\\theta_{e_n,e_n}\\). \\(\\square\\)\n\n**Proposition 4.9** (products of Hilbert–Schmidt operators).\n\n(a) An operator \\(x\\) is Hilbert–Schmidt if and only if \\(x^*x\\) is of trace class, and then \\(\\operatorname{Tr}(x^*x)=\\|x\\|_2^2\\).\n\n(b) If \\(x,y\\in\\mathcal L^2(H)\\), then \\(xy\\in\\mathcal L^1(H)\\), \\(\\|xy\\|_1\\le\\|x\\|_2\\,\\|y\\|_2\\), and \\(\\operatorname{Tr}(y^*x)=\\langle x,y\\rangle_2\\).\n\n(c) Every \\(t\\in\\mathcal L^1(H)\\) is a product \\(t=xy\\) of two Hilbert–Schmidt operators, with \\(\\|x\\|_2^2=\\|y\\|_2^2=\\|t\\|_1\\).\n\n**Proof.** (a) \\(\\sum_i\\langle x^*x\\varepsilon_i,\\varepsilon_i\\rangle=\\sum_i\\|x\\varepsilon_i\\|^2\\); apply Proposition 4.7 to \\(h=x^*x\\). (b) For finite orthonormal families, extend each to an orthonormal basis and use Cauchy–Schwarz:\n\\[\n\\begin{gathered}\n\\sum_j|\\langle xy\\xi_j,\\eta_j\\rangle|\\\\\n=\\sum_j|\\langle y\\xi_j,x^*\\eta_j\\rangle|\\\\\n\\le\\Big(\\sum_j\\|y\\xi_j\\|^2\\Big)^{1/2}\\Big(\\sum_j\\|x^*\\eta_j\\|^2\\Big)^{1/2}\\\\\n\\le\\|y\\|_2\\,\\|x^*\\|_2\\\\\n=\\|x\\|_2\\,\\|y\\|_2 .\n\\end{gathered}\n\\]\nThen \\[\n\\begin{gathered}\n\\operatorname{Tr}(y^*x)\\\\\n=\\sum_i\\langle y^*x\\varepsilon_i,\\varepsilon_i\\rangle\\\\\n=\\sum_i\\langle x\\varepsilon_i,y\\varepsilon_i\\rangle\\\\\n=\\langle x,y\\rangle_2.\n\\end{gathered}\n\\] (c) With (2.1) for \\(t\\ne0\\), put \\(x=\\sum_ns_n^{1/2}\\theta_{f_n,e_n}\\) and \\(y=\\sum_ns_n^{1/2}\\theta_{e_n,e_n}\\). Both are Hilbert–Schmidt with \\(\\|x\\|_2^2=\\|y\\|_2^2=\\sum s_n\\) (Theorem 3.3(b)), and \\(xy=\\sum_ns_n\\theta_{f_n,e_n}=t\\) because \\(\\theta_{f_n,e_n}\\theta_{e_m,e_m}=\\delta_{nm}\\theta_{f_n,e_n}\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-06",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "5. Duality between compact, trace-class and bounded operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-06",
      "proof_locus": {
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      "full_conditions_and_proof": "## 5. Duality between compact, trace-class and bounded operators\n\nThe model is the chain of sequence spaces \\(c_0\\), \\(\\ell^1\\), \\(\\ell^\\infty\\), each the dual of the one before. Replacing sequences by operators and sums by traces gives the chain \\(K(H)\\), \\(\\mathcal L^1(H)\\), \\(B(H)\\). \n\n**Example 5.1** (the commutative model). Let \\(I\\) be a set and \\(\\langle\\lambda,\\alpha\\rangle=\\sum_i\\lambda_i\\alpha_i\\) for \\(\\lambda\\in\\ell^\\infty(I)\\), \\(\\alpha\\in\\ell^1(I)\\).\n\n(a) \\(\\alpha\\mapsto\\langle\\cdot,\\alpha\\rangle\\) is an isometric isomorphism of \\(\\ell^1(I)\\) onto \\(c_0(I)^*\\).\n\n(b) \\(\\lambda\\mapsto\\langle\\lambda,\\cdot\\rangle\\) is an isometric isomorphism of \\(\\ell^\\infty(I)\\) onto \\(\\ell^1(I)^*\\).\n\n*Proof.* In both cases \\(|\\langle\\lambda,\\alpha\\rangle|\\le\\|\\lambda\\|_\\infty\\|\\alpha\\|_1\\). (a) Let \\(f\\in c_0(I)^*\\) and \\(\\alpha_i=f(\\delta_i)\\). For a finite \\(F\\subseteq I\\) choose \\(c_i\\) with \\(|c_i|=1\\) and \\(c_i\\alpha_i=|\\alpha_i|\\); then \\(\\sum_{i\\in F}|\\alpha_i|=f(\\sum_{i\\in F}c_i\\delta_i)\\le\\|f\\|\\). So \\(\\alpha\\in\\ell^1(I)\\) with \\(\\|\\alpha\\|_1\\le\\|f\\|\\). The finite-cutoff vectors are dense in \\(c_0(I)\\): retaining \\(\\{i:|\\lambda_i|\\geq\\varepsilon\\}\\) leaves supremum-norm error at most \\(\\varepsilon\\). Thus the functionals \\(f\\) and \\(\\langle\\cdot,\\alpha\\rangle\\), agreeing on \\(c_{00}(I)\\), are equal. The bound \\(\\|\\alpha\\|_1\\leq\\|f\\|\\) together with the first inequality proves the isometry. (b) Let \\(g\\in\\ell^1(I)^*\\) and \\(\\lambda_i=g(\\delta_i)\\). Then \\(|\\lambda_i|\\le\\|g\\|\\), and finite cutoffs are dense in \\(\\ell^1(I)\\): a finite partial absolute sum within \\(\\varepsilon\\) of \\(\\|\\alpha\\|_1\\) leaves tail norm at most \\(\\varepsilon\\). Hence \\(g\\) and \\(\\langle\\lambda,\\cdot\\rangle\\), agreeing on \\(c_{00}(I)\\), are equal. Testing on \\(\\delta_i\\) shows \\(\\|\\langle\\lambda,\\cdot\\rangle\\|\\ge\\|\\lambda\\|_\\infty\\). \\(\\square\\)\n\nSo \\(c_0(I)^{**}=\\ell^\\infty(I)\\). Through the diagonal operators of Example 2.2 this model sits inside the operator picture: for \\(\\lambda\\in\\ell^\\infty(I)\\) and \\(\\alpha\\in\\ell^1(I)\\), \\(d_\\lambda d_\\alpha=d_{\\lambda\\alpha}\\) is of trace class and \\(\\operatorname{Tr}(d_\\lambda d_\\alpha)=\\sum_i\\lambda_i\\alpha_i\\) by (4.2).\n\n**Theorem 5.2** (the dual of \\(K(H)\\)). For \\(t\\in\\mathcal L^1(H)\\) let \\(\\varphi_t(x)=\\operatorname{Tr}(xt)\\), \\(x\\in K(H)\\). Then \\(t\\mapsto\\varphi_t\\) is an isometric linear isomorphism of \\(\\mathcal L^1(H)\\) onto \\(K(H)^*\\). Its inverse sends \\(\\omega\\in K(H)^*\\) to the operator \\(t(\\omega)\\) determined by\n\\[\n\\begin{gathered}\n\\langle t(\\omega)\\xi,\\eta\\rangle\\\\\n=\\omega(\\theta_{\\xi,\\eta})\\\\\n(\\xi,\\eta\\in H).\n\\end{gathered}\n\\tag{5.1}\n\\]\nConsequently every \\(\\omega\\in K(H)^*\\) can be written\n\\[\n\\begin{gathered}\n\\omega\\\\\n=\\sum_ns_n\\,\\omega_{f_n,e_n}\\ \\ \\\\\n(\\text{norm convergent}),\\\\\n\\|\\omega\\|\\\\\n=\\sum_ns_n,\n\\end{gathered}\n\\tag{5.2}\n\\]\nwith orthonormal families \\((e_n)\\), \\((f_n)\\) and numbers \\(s_n>0\\) with \\(\\sum s_n<\\infty\\); here \\(t(\\omega)=\\sum_ns_n\\theta_{f_n,e_n}\\). Conversely, for orthonormal sequences \\((\\xi_n)\\), \\((\\eta_n)\\) and \\(\\alpha\\in\\ell^1\\), the series \\(\\omega=\\sum_n\\alpha_n\\omega_{\\xi_n,\\eta_n}\\) converges in \\(K(H)^*\\), \\(t(\\omega)=\\sum_n\\alpha_n\\theta_{\\xi_n,\\eta_n}\\), and \\(\\|\\omega\\|=\\|\\alpha\\|_1\\).\n\n**Proof.** (i) By (4.3), \\(|\\varphi_t(x)|\\le\\|x\\|\\,\\|t\\|_1\\), so \\(\\|\\varphi_t\\|\\le\\|t\\|_1\\).\n\n(ii) Let \\(\\omega\\in K(H)^*\\). The map \\((\\xi,\\eta)\\mapsto\\omega(\\theta_{\\xi,\\eta})\\) is sesquilinear and bounded by \\(\\|\\omega\\|\\,\\|\\xi\\|\\,\\|\\eta\\|\\) (Lemma 1.4(a)). Theorem 1.3 gives a unique \\(t(\\omega)\\in B(H)\\) with (5.1).\n\n(iii) \\(\\|t(\\omega)\\|_1\\le\\|\\omega\\|\\). Let \\((\\xi_j)\\), \\((\\eta_j)\\) be finite orthonormal families, and choose \\(c_j\\) with \\(|c_j|=1\\) and \\(c_j\\langle t(\\omega)\\xi_j,\\eta_j\\rangle=|\\langle t(\\omega)\\xi_j,\\eta_j\\rangle|\\). With \\(y=\\sum_jc_j\\theta_{\\xi_j,\\eta_j}\\),\n\\[\n\\sum_j|\\langle t(\\omega)\\xi_j,\\eta_j\\rangle|=\\sum_jc_j\\,\\omega(\\theta_{\\xi_j,\\eta_j})=\\omega(y).\n\\]\nNow \\(y\\) has finite rank and \\[\n\\begin{gathered}\n\\|y\\zeta\\|^2\\\\\n=\\|\\sum_jc_j\\langle\\zeta,\\eta_j\\rangle\\xi_j\\|^2\\\\\n=\\sum_j|\\langle\\zeta,\\eta_j\\rangle|^2\\\\\n\\le\\|\\zeta\\|^2,\n\\end{gathered}\n\\] so \\(\\|y\\|\\le1\\) and \\(\\omega(y)\\le\\|\\omega\\|\\). Hence \\(t(\\omega)\\in\\mathcal L^1(H)\\) and \\(\\|t(\\omega)\\|_1\\le\\|\\omega\\|\\).\n\n(iv) \\(t(\\varphi_t)=t\\): by Lemma 1.4(b) and (4.2), \\[\n\\begin{gathered}\n\\varphi_t(\\theta_{\\xi,\\eta})\\\\\n=\\operatorname{Tr}(\\theta_{\\xi,\\eta}t)\\\\\n=\\operatorname{Tr}(\\theta_{\\xi,t^*\\eta})\\\\\n=\\langle\\xi,t^*\\eta\\rangle\\\\\n=\\langle t\\xi,\\eta\\rangle.\n\\end{gathered}\n\\]\n\n(v) \\(\\varphi_{t(\\omega)}=\\omega\\): both are bounded functionals on \\(K(H)\\), and by (4.4) and (5.1) they agree on every \\(\\theta_{\\xi,\\eta}\\), hence on \\(F(H)\\) (Lemma 1.4(d)), which is dense in \\(K(H)\\).\n\nBy (iv) and (v), \\(t\\mapsto\\varphi_t\\) is bijective with inverse \\(\\omega\\mapsto t(\\omega)\\), and (i) and (iii) give \\(\\|t\\|_1\\le\\|\\varphi_t\\|\\le\\|t\\|_1\\). For (5.2), write \\(t(\\omega)\\) as in (2.1); since \\(xt(\\omega)=\\sum_ns_n\\theta_{xf_n,e_n}\\), formula (4.2) gives \\(\\omega(x)=\\operatorname{Tr}(xt(\\omega))=\\sum_ns_n\\langle xf_n,e_n\\rangle\\), and \\(\\|\\omega\\|=\\|t(\\omega)\\|_1=\\sum s_n\\). The series converges in norm because \\(\\|\\omega_{f_n,e_n}\\|=1\\). For the converse, \\(t=\\sum\\alpha_n\\theta_{\\xi_n,\\eta_n}\\) is of trace class by Corollary 4.4(d) and \\(\\varphi_t=\\sum\\alpha_n\\omega_{\\xi_n,\\eta_n}\\) by (4.2). Writing \\(\\alpha_n=|\\alpha_n|c_n\\) with \\(|c_n|=1\\), we have \\(t=\\sum|\\alpha_n|\\theta_{c_n\\xi_n,\\eta_n}\\), which after ordering the nonzero \\(|\\alpha_n|\\) by size is a decomposition of the form (2.1). So \\(\\|\\varphi_t\\|=\\|t\\|_1=\\|\\alpha\\|_1\\) by Theorem 4.3(b). \\(\\square\\)\n\nThe proof shows in particular that for every \\(\\omega\\in K(H)^*\\) the operator \\(t(\\omega)\\) is compact and \\(\\sum_i|\\langle t(\\omega)\\xi_i,\\xi_i\\rangle|\\le\\|\\omega\\|\\) for every orthonormal family \\((\\xi_i)\\). This estimate is the heart of the theorem: a bounded functional on \\(K(H)\\) cannot put too much weight on the diagonal of any basis.\n\n**Proposition 5.3** (the module structure). For \\(a\\in B(H)\\) and \\(\\omega\\in K(H)^*\\) define \\((a\\omega)(x)=\\omega(xa)\\) and \\((\\omega a)(x)=\\omega(ax)\\), \\(x\\in K(H)\\). Then \\(t(a\\omega)=a\\,t(\\omega)\\) and \\(t(\\omega a)=t(\\omega)\\,a\\).\n\n**Proof.** Let \\(t=t(\\omega)\\), so \\(\\omega=\\varphi_t\\). By (4.3), \\((a\\omega)(x)=\\operatorname{Tr}(xat)=\\varphi_{at}(x)\\) and \\((\\omega a)(x)=\\operatorname{Tr}(axt)=\\operatorname{Tr}(xta)=\\varphi_{ta}(x)\\). Apply Theorem 5.2. \\(\\square\\)\n\nSo the two-sided ideal property of \\(\\mathcal L^1(H)\\) (Corollary 4.4(a)) is the same thing as the natural \\(B(H)\\)-module structure of \\(K(H)^*\\).\n\n**Theorem 5.4** (the dual of \\(\\mathcal L^1(H)\\)). For \\(a\\in B(H)\\) let \\(\\psi_a(t)=\\operatorname{Tr}(at)\\), \\(t\\in\\mathcal L^1(H)\\). Then \\(a\\mapsto\\psi_a\\) is an isometric linear isomorphism of \\(B(H)\\) onto \\(\\mathcal L^1(H)^*\\). Consequently \\(B(H)\\) is isometrically isomorphic to \\(K(H)^{**}\\): the operator \\(a\\) corresponds to the functional \\(\\omega\\mapsto\\operatorname{Tr}(a\\,t(\\omega))\\) on \\(K(H)^*\\), whose value at \\(\\omega_{\\xi,\\eta}\\) is \\(\\langle a\\xi,\\eta\\rangle\\). Under this identification the canonical embedding of \\(K(H)\\) into \\(K(H)^{**}\\) is the inclusion \\(K(H)\\subseteq B(H)\\).\n\n**Proof.** By (4.3), \\(\\|\\psi_a\\|\\le\\|a\\|\\). By (4.4), \\(\\psi_a(\\theta_{\\xi,\\eta})=\\langle a\\xi,\\eta\\rangle\\), and \\(\\|\\theta_{\\xi,\\eta}\\|_1=\\|\\xi\\|\\,\\|\\eta\\|\\) by Lemma 4.2(b); so \\(\\|\\psi_a\\|\\ge\\sup\\{|\\langle a\\xi,\\eta\\rangle|:\\|\\xi\\|,\\|\\eta\\|\\le1\\}=\\|a\\|\\). To see that the map is onto, let \\(\\psi\\in\\mathcal L^1(H)^*\\). The form \\((\\xi,\\eta)\\mapsto\\psi(\\theta_{\\xi,\\eta})\\) is sesquilinear and bounded by \\(\\|\\psi\\|\\,\\|\\xi\\|\\,\\|\\eta\\|\\), so Theorem 1.3 gives \\(a\\in B(H)\\) with \\(\\langle a\\xi,\\eta\\rangle=\\psi(\\theta_{\\xi,\\eta})=\\psi_a(\\theta_{\\xi,\\eta})\\). Then \\(\\psi\\) and \\(\\psi_a\\) agree on \\(F(H)\\), which is dense in \\(\\mathcal L^1(H)\\) (Corollary 4.4(c)), so \\(\\psi=\\psi_a\\).\n\nBy Theorem 5.2, \\(\\omega\\mapsto t(\\omega)\\) is an isometric isomorphism \\(K(H)^*\\to\\mathcal L^1(H)\\); composing its adjoint with \\(a\\mapsto\\psi_a\\) gives the isometric isomorphism \\(B(H)\\to K(H)^{**}\\), \\(a\\mapsto(\\omega\\mapsto\\psi_a(t(\\omega)))\\). Since \\(t(\\omega_{\\xi,\\eta})=\\theta_{\\xi,\\eta}\\) by (5.1) and Lemma 1.4(e), the value at \\(\\omega_{\\xi,\\eta}\\) is \\(\\operatorname{Tr}(a\\theta_{\\xi,\\eta})=\\langle a\\xi,\\eta\\rangle\\). For \\(x\\in K(H)\\), the canonical image of \\(x\\) is \\(\\omega\\mapsto\\omega(x)=\\varphi_{t(\\omega)}(x)=\\operatorname{Tr}(x\\,t(\\omega))\\), which is the functional attached to \\(a=x\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-LT-07",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "6. The predual of B(H)",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-07",
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      "full_conditions_and_proof": "## 6. The predual of B(H)\n\n**Definition 6.1.** A linear functional \\(\\omega\\) on \\(B(H)\\) is *normal* if \\(\\omega(x)=\\operatorname{Tr}(xt)\\) for some \\(t\\in\\mathcal L^1(H)\\). By Theorem 5.4 the operator \\(t\\) is unique; we write \\(t=t_\\omega\\). The normal functionals form a subspace \\(B(H)_*\\) of \\(B(H)^*\\), with the norm of \\(B(H)^*\\); it is the *predual* of \\(B(H)\\).\n\n**Theorem 6.2.**\n\n(a) \\(t\\mapsto\\operatorname{Tr}(\\cdot\\,t)\\) is an isometric isomorphism of \\(\\mathcal L^1(H)\\) onto \\(B(H)_*\\). In particular \\(B(H)_*\\) is a norm-closed subspace of \\(B(H)^*\\).\n\n(b) Restriction to \\(K(H)\\) is an isometric isomorphism of \\(B(H)_*\\) onto \\(K(H)^*\\). So every bounded functional on \\(K(H)\\) has exactly one normal extension to \\(B(H)\\), and it has the same norm.\n\n(c) The map sending \\(a\\in B(H)\\) to the functional \\(\\omega\\mapsto\\omega(a)\\) is an isometric isomorphism of \\(B(H)\\) onto \\((B(H)_*)^*\\).\n\n(d) Every \\(\\omega\\in B(H)_*\\) can be written \\(\\omega=\\sum_ns_n\\omega_{f_n,e_n}\\) with orthonormal families \\((e_n)\\), \\((f_n)\\) and \\(\\sum s_n=\\|\\omega\\|\\), or equivalently \\(\\omega=\\sum_n\\omega_{\\xi_n,\\eta_n}\\) with \\(\\sum\\|\\xi_n\\|^2=\\sum\\|\\eta_n\\|^2=\\|\\omega\\|\\), the series converging in norm. Conversely, if \\(\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\), then \\(\\sum_n\\omega_{\\xi_n,\\eta_n}\\) converges in norm to an element of \\(B(H)_*\\) of norm at most \\(\\sum\\|\\xi_n\\|\\,\\|\\eta_n\\|\\). In particular the finite sums of vector functionals are norm dense in \\(B(H)_*\\).\n\n(e) For \\(\\omega\\in B(H)_*\\), \\(\\omega(x)=\\operatorname{Tr}(xt_\\omega)=\\operatorname{Tr}(t_\\omega x)\\) and \\(\\omega(1)=\\operatorname{Tr}(t_\\omega)\\).\n\n**Proof.** (a) For \\(t\\in\\mathcal L^1(H)\\), \\(\\|\\operatorname{Tr}(\\cdot\\,t)\\|_{B(H)^*}\\le\\|t\\|_1\\) by (4.3). Its restriction to \\(K(H)\\) is \\(\\varphi_t\\), of norm \\(\\|t\\|_1\\) by Theorem 5.2, so the norm is exactly \\(\\|t\\|_1\\). Injectivity follows, and \\(B(H)_*\\) is complete, hence closed. (b) follows from (a) and Theorem 5.2. (c) is Theorem 5.4 transported by (a). (d) With (2.1) for \\(t_\\omega\\), \\(xt_\\omega=\\sum_ns_n\\theta_{xf_n,e_n}\\) and (4.2) give \\(\\omega(x)=\\sum_ns_n\\langle xf_n,e_n\\rangle\\) for every \\(x\\in B(H)\\), and \\(\\sum s_n=\\|t_\\omega\\|_1=\\|\\omega\\|\\). For the second form put \\(\\xi_n=s_n^{1/2}f_n\\), \\(\\eta_n=s_n^{1/2}e_n\\). For the converse, \\(t=\\sum_n\\theta_{\\xi_n,\\eta_n}\\) is of trace class with \\(\\|t\\|_1\\le\\sum\\|\\xi_n\\|\\,\\|\\eta_n\\|\\) (Corollary 4.4(d)), and \\(\\operatorname{Tr}(xt)=\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\) by (4.2). The partial sums of the first form are finite sums of vector functionals. (e) is (4.3). \\(\\square\\)\n\n**Proposition 6.3** (positive and hermitian normal functionals). Let \\(\\omega\\in B(H)_*\\).\n\n(a) \\(\\omega\\) is positive, that is \\(\\omega(x^*x)\\ge0\\) for all \\(x\\), if and only if \\(t_\\omega\\ge0\\). Then \\(\\omega=\\sum_ns_n\\omega_{e_n}\\) for an orthonormal family \\((e_n)\\) and numbers \\(s_n>0\\) with \\(\\sum s_n=\\omega(1)=\\|\\omega\\|\\); equivalently \\(\\omega=\\sum_n\\omega_{\\zeta_n}\\) with \\(\\sum\\|\\zeta_n\\|^2=\\omega(1)\\).\n\n(b) \\(\\omega\\) is hermitian, that is \\(\\omega(x^*)=\\overline{\\omega(x)}\\) for all \\(x\\), if and only if \\(t_\\omega\\) is self-adjoint. A hermitian \\(\\omega\\) is a difference \\(\\omega=\\omega_1-\\omega_2\\) of positive normal functionals with \\(\\|\\omega\\|=\\|\\omega_1\\|+\\|\\omega_2\\|\\).\n\n(c) Every normal functional can be written \\(\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with positive normal functionals \\(\\omega_1,\\dots,\\omega_4\\).\n\n**Proof.** Write \\(t=t_\\omega\\). (a) If \\(\\omega\\ge0\\), then by (4.4) \\(\\langle t\\xi,\\xi\\rangle=\\operatorname{Tr}(t\\theta_{\\xi,\\xi})=\\omega(\\theta_{\\xi,\\xi})\\ge0\\), since \\(\\theta_{\\xi,\\xi}=\\|\\xi\\|^2p\\) with \\(p=p^*p\\) a projection (Lemma 1.4(c)). Conversely, if \\(t\\ge0\\), Theorem 2.3(e) gives \\(t=\\sum_ns_n\\theta_{e_n,e_n}\\), so \\(\\omega(x)=\\sum_ns_n\\langle xe_n,e_n\\rangle\\) and \\(\\omega(x^*x)=\\sum_ns_n\\|xe_n\\|^2\\ge0\\). Then \\(\\|\\omega\\|=\\|t\\|_1=\\operatorname{Tr}(t)=\\omega(1)\\), and \\(\\zeta_n=s_n^{1/2}e_n\\) gives the second form.\n\n(b) \\(\\omega(x^*)=\\operatorname{Tr}(x^*t)\\), while \\(\\overline{\\omega(x)}=\\operatorname{Tr}((xt)^*)=\\operatorname{Tr}(t^*x^*)=\\operatorname{Tr}(x^*t^*)\\) by Theorem 4.6. So \\(\\omega\\) is hermitian exactly when \\(\\operatorname{Tr}(y(t-t^*))=0\\) for all \\(y\\in B(H)\\). Taking \\(y=\\theta_{\\xi,\\eta}\\) and using (4.3) and (4.4), this says \\(\\langle(t-t^*)\\xi,\\eta\\rangle=0\\) for all \\(\\xi,\\eta\\), that is, \\(t=t^*\\). Then Proposition 4.8 writes \\(t=h_1-h_2\\) with \\(h_k\\ge0\\) and \\(\\|t\\|_1=\\operatorname{Tr}(h_1)+\\operatorname{Tr}(h_2)\\); put \\(\\omega_k=\\operatorname{Tr}(\\cdot\\,h_k)\\).\n\n(c) Proposition 4.8. \\(\\square\\)\n\n**Remark 6.4.** When \\(H\\) is infinite-dimensional, \\(B(H)_*\\) is a proper subspace of \\(B(H)^*\\). By Theorem 6.2(b), a normal functional that vanishes on \\(K(H)\\) is zero, while Proposition 10.5(b) below produces states of \\(B(H)\\) that vanish on \\(K(H)\\). Sections 8 and 9 show that the normal functionals are exactly the linear functionals that are continuous for the \\(\\sigma\\)-weak topology. The name *normal* refers to an order property, continuity along bounded increasing nets, which is treated in the lesson [The universal enveloping von Neumann algebra of a C\\*-algebra, and W\\*-algebras](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-LT-08",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "7. Ideals of B(H) and the Calkin algebra",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-08",
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      "full_conditions_and_proof": "## 7. Ideals of B(H) and the Calkin algebra\n\nIn this section *ideal* means two-sided ideal of \\(B(H)\\), not necessarily closed. The finite-rank operators form the smallest nonzero ideal. When \\(H\\) is separable and infinite-dimensional, the compact operators form the largest proper ideal, and every proper ideal is determined by a space of sequences, the singular values of its elements. \n\n**Proposition 7.1.** Every nonzero ideal \\(J\\) of \\(B(H)\\) contains \\(F(H)\\). So \\(F(H)\\) is the smallest nonzero ideal, and \\(K(H)\\) is the smallest nonzero closed ideal.\n\n**Proof.** Take \\(x\\in J\\), \\(x\\ne0\\), and \\(\\xi\\) with \\(x\\xi\\ne0\\). By Lemma 1.4(b), \\(x\\theta_{\\xi,\\xi}x^*=\\theta_{x\\xi,x\\xi}\\in J\\). For the unit vector \\(\\eta=x\\xi/\\|x\\xi\\|\\), the projection \\(\\theta_{\\eta,\\eta}\\) is a multiple of it, so it lies in \\(J\\). For arbitrary \\(\\zeta,\\upsilon\\in H\\), Lemma 1.4(b) gives \\(\\theta_{\\zeta,\\upsilon}=\\theta_{\\zeta,\\eta}\\,\\theta_{\\eta,\\eta}\\,\\theta_{\\eta,\\upsilon}\\in J\\). By Lemma 1.4(d), \\(F(H)\\subseteq J\\). A closed nonzero ideal therefore contains the closure of \\(F(H)\\), which is \\(K(H)\\). \\(\\square\\)\n\n**Lemma 7.2.** Let \\(h\\in B(H)_+\\) be noncompact. There are \\(\\varepsilon>0\\) and a projection \\(e\\) of infinite rank that commutes with \\(h\\) and satisfies \\(\\varepsilon e\\le he\\le h\\).\n\n**Proof.** For \\(\\varepsilon>0\\) let \\(e_\\varepsilon=1_{[\\varepsilon,\\infty)}(h)\\), a spectral projection of \\(h\\) (Background 3(b)); it commutes with \\(h\\). On \\([0,\\infty)\\) we have \\(\\lambda1_{[\\varepsilon,\\infty)}(\\lambda)\\ge\\varepsilon1_{[\\varepsilon,\\infty)}(\\lambda)\\) and \\(0\\le\\lambda-\\lambda1_{[\\varepsilon,\\infty)}(\\lambda)=\\lambda1_{[0,\\varepsilon)}(\\lambda)\\le\\varepsilon\\). By the functional calculus, \\(\\varepsilon e_\\varepsilon\\le he_\\varepsilon\\), \\(0\\le h-he_\\varepsilon\\), and \\(\\|h-he_\\varepsilon\\|\\le\\varepsilon\\). If every \\(e_\\varepsilon\\) had finite rank, then every \\(he_\\varepsilon\\) would have finite rank, and \\(h\\) would be a norm limit of finite-rank operators, hence compact. So some \\(e=e_\\varepsilon\\) has infinite rank. \\(\\square\\)\n\n**Proposition 7.3.** Let \\(H\\) be separable and infinite-dimensional, and let \\(a\\in B(H)\\) be noncompact. There are \\(b,c\\in B(H)\\) with \\(bac=1\\). Consequently every proper ideal of \\(B(H)\\) is contained in \\(K(H)\\); so \\(K(H)\\) is the largest proper ideal, and the closed ideals of \\(B(H)\\) are exactly \\(0\\), \\(K(H)\\) and \\(B(H)\\).\n\n**Proof.** First, \\(a^*a\\) is not compact. Otherwise, let \\((\\zeta_k)\\) be a sequence with \\(\\|\\zeta_k\\|\\le M\\), and pass to a subsequence along which \\(a^*a\\zeta_k\\) converges. Since\n\\[\n\\begin{gathered}\n\\|a(\\zeta_k-\\zeta_l)\\|^2\\\\\n=\\langle a^*a(\\zeta_k-\\zeta_l),\\zeta_k-\\zeta_l\\rangle\\\\\n\\le2M\\,\\|a^*a(\\zeta_k-\\zeta_l)\\|,\n\\end{gathered}\n\\]\n\\((a\\zeta_k)\\) is a Cauchy sequence along it, so \\(a\\) would be compact. By Lemma 7.2 there are \\(\\varepsilon>0\\) and an infinite-rank projection \\(e\\) with \\(\\varepsilon e\\le a^*a\\). For \\(\\xi\\in eH\\), \\(\\|a\\xi\\|^2=\\langle a^*a\\xi,\\xi\\rangle\\ge\\varepsilon\\langle e\\xi,\\xi\\rangle=\\varepsilon\\|\\xi\\|^2\\). The space \\(eH\\) is infinite-dimensional and separable, like \\(H\\), so there is an isometry \\(c\\) of \\(H\\) onto \\(eH\\): send an orthonormal basis of \\(H\\) onto one of \\(eH\\). Then \\(\\|ac\\xi\\|\\ge\\varepsilon^{1/2}\\|\\xi\\|\\) for all \\(\\xi\\). So \\(ac\\) is injective with closed range \\(L=acH\\). Define \\(b\\) as the inverse of \\(ac\\) on \\(L\\) and \\(0\\) on \\(L^\\perp\\); then \\(\\|b\\|\\le\\varepsilon^{-1/2}\\) and \\(bac=1\\). If an ideal \\(J\\) contains a noncompact \\(a\\), then \\(1=bac\\in J\\) and \\(J=B(H)\\). With Proposition 7.1 this gives the last statement. \\(\\square\\)\n\n**Example 7.4** (separability is needed). Let \\(H\\) be nonseparable, and let \\(J\\) be the set of operators whose range lies in a separable closed subspace. If \\(x,y\\in J\\), then \\((x+y)H\\) lies in the closed span of two separable subspaces, which is separable; \\(axH\\) lies in the closure of \\(a\\) applied to a separable subspace, which is separable; and \\(xaH\\subseteq xH\\). So \\(J\\) is an ideal. It does not contain \\(1\\), and it contains the projection onto any separable infinite-dimensional subspace, which is not compact. So \\(K(H)\\) is not the largest proper ideal, and Proposition 7.3 fails without separability.\n\nFrom now on in this section, \\(H\\) is separable and infinite-dimensional. For \\(\\lambda\\in\\ell^\\infty\\) and an orthonormal sequence \\(\\xi=(\\xi_n)\\) in \\(H\\) let\n\\[\nd^\\xi_\\lambda=\\sum_n\\lambda_n\\theta_{\\xi_n,\\xi_n},\n\\]\nthe series converging strongly. It is the diagonal operator of Example 2.2 on the closed span of the \\(\\xi_n\\), extended by \\(0\\); so \\(\\|d^\\xi_\\lambda\\|=\\|\\lambda\\|_\\infty\\), \\(d^\\xi_\\lambda d^\\xi_\\mu=d^\\xi_{\\lambda\\mu}\\), and \\(d^\\xi_\\lambda\\) is compact exactly when \\(\\lambda\\in c_0\\).\n\n**Definition 7.5.** For an ideal \\(J\\) of \\(B(H)\\) let \\(E(J)\\) be the set of \\(\\lambda\\in\\ell^\\infty\\) such that \\(d^\\xi_\\lambda\\in J\\) for some orthonormal sequence \\(\\xi\\). A linear subspace \\(E\\) of \\(c_0\\) is *solid* if \\(\\mu\\in E\\) whenever \\(\\lambda\\in E\\) and \\(|\\mu_n|\\le|\\lambda_n|\\) for all \\(n\\); solid subspaces are the ideals of \\(c_0\\) in the sense of ordered vector spaces. A solid subspace is *symmetric* if \\((\\lambda_{\\pi(n)})_n\\in E\\) for every \\(\\lambda\\in E\\) and every bijection \\(\\pi\\) of \\(\\mathbb N\\). For a symmetric solid subspace \\(E\\) let\n\\[\nJ(E)=\\{x\\in K(H):\\ s(x)\\in E\\}.\n\\]\n\n**Lemma 7.6** (rearrangements). Let \\(E\\ne0\\) be a symmetric solid subspace of \\(c_0\\).\n\n(a) \\(E\\) contains \\(c_{00}\\).\n\n(b) Let \\(\\lambda\\in E\\) and \\(\\mu\\in c_0\\), and suppose there is an injective map \\(\\sigma\\) from \\(S=\\{n:\\mu_n\\ne0\\}\\) into \\(\\mathbb N\\) with \\(|\\mu_n|\\le|\\lambda_{\\sigma(n)}|\\) for \\(n\\in S\\). Then \\(\\mu\\in E\\).\n\nIn particular, \\(E\\) contains every subsequence of each of its elements, every sequence obtained from one of its elements by inserting zeros or by prefixing finitely many terms, and the sequence \\((\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2,\\dots)\\) for each \\(\\lambda\\in E\\).\n\n**Proof.** (a) Some \\(\\lambda\\in E\\) has a coordinate \\(\\lambda_m\\ne0\\). Then \\(|\\delta_m|\\le|\\lambda/\\lambda_m|\\) coordinatewise, so \\(\\delta_m\\in E\\); by symmetry every \\(\\delta_n\\in E\\), and \\(E\\) is a subspace.\n\n(b) If \\(S\\) is finite, \\(\\mu\\in c_{00}\\subseteq E\\). Suppose \\(S\\) is infinite, and first assume that \\(\\mathbb N\\setminus S\\) and \\(\\mathbb N\\setminus\\sigma(S)\\) are both infinite. Then \\(\\sigma\\) extends to a bijection \\(\\pi\\) of \\(\\mathbb N\\) (map \\(\\mathbb N\\setminus S\\) bijectively onto \\(\\mathbb N\\setminus\\sigma(S)\\)). The sequence \\(\\rho=(\\lambda_{\\pi(n)})_n\\) lies in \\(E\\) by symmetry, and \\(|\\mu_n|\\le|\\rho_n|\\) for all \\(n\\), so \\(\\mu\\in E\\) by solidity. In general, list \\(S=\\{k_1<k_2<\\cdots\\}\\) and put \\(S_1=\\{k_1,k_3,\\dots\\}\\), \\(S_2=\\{k_2,k_4,\\dots\\}\\). For \\(i=1,2\\) the complement of \\(S_i\\) contains \\(S_{3-i}\\) and the complement of \\(\\sigma(S_i)\\) contains \\(\\sigma(S_{3-i})\\), so both are infinite, and the first case gives \\(\\mu1_{S_i}\\in E\\). Hence \\(\\mu=\\mu1_{S_1}+\\mu1_{S_2}\\in E\\).\n\nFor the examples: a subsequence \\((\\lambda_{n_k})_k\\) uses \\(\\sigma(k)=n_k\\); inserting zeros uses the inverse of the placement map; prefixing finitely many terms gives a sequence in \\(c_{00}\\) plus a sequence of the previous kind; and \\((\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2,\\dots)\\) is the sum of the two sequences that carry \\(\\lambda_n\\) at the places \\(2n-1\\) and \\(2n\\) respectively. \\(\\square\\)\n\n**Theorem 7.7** (Calkin). Let \\(H\\) be separable and infinite-dimensional. For every nonzero proper ideal \\(J\\) of \\(B(H)\\), \\(E(J)\\) is a nonzero symmetric solid subspace of \\(c_0\\); for every nonzero symmetric solid subspace \\(E\\) of \\(c_0\\), \\(J(E)\\) is a nonzero proper ideal; and the maps \\(J\\mapsto E(J)\\) and \\(E\\mapsto J(E)\\) are inverse to each other. Moreover, \\(\\lambda\\in E(J)\\) if and only if \\(d^\\xi_\\lambda\\in J\\) for every orthonormal sequence \\(\\xi\\), and \\(x\\in J\\) if and only if \\(x\\) is compact and \\(s(x)\\in E(J)\\).\n\n\nUnder this correspondence \\(c_{00}\\), \\(\\ell^1\\), \\(\\ell^2\\) and \\(c_0\\) correspond to \\(F(H)\\), \\(\\mathcal L^1(H)\\), \\(\\mathcal L^2(H)\\) and \\(K(H)\\), by Theorems 3.3(b) and 4.3(b).\n\n**Proof.** *Step 1: \\(E(J)\\) does not depend on the orthonormal sequence.* Let \\(d^\\xi_\\lambda\\in J\\) and let \\(\\eta\\) be another orthonormal sequence. The operator \\(v\\zeta=\\sum_n\\langle\\zeta,\\xi_n\\rangle\\eta_n\\) has norm at most \\(1\\), maps \\(\\xi_n\\) to \\(\\eta_n\\), and \\(v^*\\zeta=\\sum_n\\langle\\zeta,\\eta_n\\rangle\\xi_n\\). Then \\(vd^\\xi_\\lambda v^*\\zeta=\\sum_n\\lambda_n\\langle\\zeta,\\eta_n\\rangle\\eta_n=d^\\eta_\\lambda\\zeta\\), so \\(d^\\eta_\\lambda\\in J\\).\n\n*Step 2: \\(E(J)\\) is a nonzero symmetric solid subspace of \\(c_0\\).* By Proposition 7.3, \\(J\\subseteq K(H)\\), so \\(E(J)\\subseteq c_0\\). Using one orthonormal sequence for all elements, \\(d^\\xi_{\\lambda+\\mu}=d^\\xi_\\lambda+d^\\xi_\\mu\\) and \\(d^\\xi_{c\\lambda}=cd^\\xi_\\lambda\\) show that \\(E(J)\\) is a subspace. If \\(|\\mu|\\le|\\lambda|\\) coordinatewise, put \\(\\beta_n=\\mu_n/\\lambda_n\\) when \\(\\lambda_n\\ne0\\) and \\(\\beta_n=0\\) otherwise; then \\(\\beta\\in\\ell^\\infty\\) and \\(d^\\xi_\\mu=d^\\xi_\\beta d^\\xi_\\lambda\\in J\\). For a bijection \\(\\pi\\), \\(d^\\xi_{\\lambda\\circ\\pi}=d^\\eta_\\lambda\\) with the orthonormal sequence \\(\\eta_m=\\xi_{\\pi^{-1}(m)}\\), which lies in \\(J\\) by Step 1. Finally \\(F(H)\\subseteq J\\) by Proposition 7.1, so \\(\\delta_1\\in E(J)\\).\n\n*Step 3: \\(J(E)\\) is a nonzero proper ideal.* Let \\(x,y\\in J(E)\\). By Proposition 2.5(b) with \\(m=n\\), \\(s_{2n-1}(x+y)\\le s_n(x)+s_n(y)\\), and \\(s_{2n}(x+y)\\le s_{2n-1}(x+y)\\). So, coordinatewise,\n\\[\n\\begin{gathered}\ns(x+y)\\\\\n\\le D\\big(s(x)+s(y)\\big),\\\\\nD(\\alpha)\\\\\n=(\\alpha_1,\\alpha_1,\\alpha_2,\\alpha_2,\\dots).\n\\end{gathered}\n\\]\nBy Lemma 7.6, \\(D(s(x)+s(y))\\in E\\), and by solidity \\(s(x+y)\\in E\\). Also \\(s(cx)=|c|\\,s(x)\\), and \\(s(axb)\\le\\|a\\|\\,\\|b\\|\\,s(x)\\) by Proposition 2.5(a). So \\(J(E)\\) is an ideal. It contains \\(F(H)\\), because \\(c_{00}\\subseteq E\\), and it lies in \\(K(H)\\ne B(H)\\).\n\n*Step 4: \\(E(J(E))=E\\).* Let \\(\\lambda\\in c_0\\) and let \\(\\xi\\) be an orthonormal sequence. By Example 2.6, \\(s(d^\\xi_\\lambda)\\) lists the nonzero \\(|\\lambda_n|\\) in decreasing order. So there are injective maps \\(\\sigma\\) from the support of \\(s(d^\\xi_\\lambda)\\) into \\(\\mathbb N\\) with \\(s_k(d^\\xi_\\lambda)=|\\lambda_{\\sigma(k)}|\\), and \\(\\tau\\) from the support of \\(\\lambda\\) into \\(\\mathbb N\\) with \\(|\\lambda_n|=s_{\\tau(n)}(d^\\xi_\\lambda)\\). If \\(\\lambda\\in E\\), then \\(s(d^\\xi_\\lambda)\\in E\\) by Lemma 7.6(b) with \\(\\sigma\\), so \\(d^\\xi_\\lambda\\in J(E)\\) and \\(\\lambda\\in E(J(E))\\). If \\(\\lambda\\in E(J(E))\\), then \\(d^\\eta_\\lambda\\in J(E)\\) for some \\(\\eta\\), so \\(s(d^\\eta_\\lambda)\\in E\\), and Lemma 7.6(b) with \\(\\tau\\) gives \\(\\lambda\\in E\\).\n\n*Step 5: \\(J(E(J))=J\\).* Let \\(x\\in J\\). Then \\(x\\) is compact. Write \\(x\\) as in (2.1). If the family \\((e_n)\\) is finite, extend it to an orthonormal sequence, which is possible since \\(H\\) is infinite-dimensional. By Theorem 2.3(b),(d), \\(|x|=vx\\in J\\) and \\(|x|=d^e_{s(x)}\\). So \\(s(x)\\in E(J)\\) and \\(x\\in J(E(J))\\). Conversely, if \\(x\\in J(E(J))\\), then \\(s(x)\\in E(J)\\), so \\(|x|=d^e_{s(x)}\\in J\\) by Step 1, and \\(x=v^*|x|\\in J\\) by Theorem 2.3(d).\n\nThe last two statements of the theorem are Step 1 and Step 5. \\(\\square\\)\n\n**Example 7.8** (why the doubling operation matters). One may try to describe an ideal \\(J\\) by the set \\(\\Sigma(J)=\\{s(x):x\\in J\\}\\) of decreasing sequences alone. This set is hereditary (a decreasing sequence below a member is a member) and closed under addition. These two properties do not suffice. Let \\(\\Sigma\\) be the set of decreasing nonnegative sequences \\(\\alpha\\) with \\(\\alpha_n\\le C2^{-n}\\) for some \\(C\\). It is hereditary and closed under addition, but \\(\\{x\\in K(H):s(x)\\in\\Sigma\\}\\) is not closed under addition. Indeed, let \\(\\xi\\) and \\(\\eta\\) be orthonormal sequences with \\(\\xi_n\\perp\\eta_m\\) for all \\(n,m\\), and \\(\\lambda_n=2^{-n}\\). Then \\(s(d^\\xi_\\lambda)=s(d^\\eta_\\lambda)=\\lambda\\in\\Sigma\\), while \\(d^\\xi_\\lambda+d^\\eta_\\lambda\\) has each eigenvalue \\(2^{-n}\\) twice, so \\(s_{2n}(d^\\xi_\\lambda+d^\\eta_\\lambda)=2^{-n}\\), which is not \\(O(2^{-2n})\\). The missing condition is closure under \\(D\\), which Theorem 7.7 obtains from symmetry and solidity (Lemma 7.6).\n\n*Human reference:* [Blackadar]. The counterexample above directly establishes the need for doubling; Theorem 7.7 includes it through symmetry and solidity.\n\n**Proposition 7.9** (the Calkin algebra). Let \\(H\\) be infinite-dimensional, \\(Q(H)=B(H)/K(H)\\) the Calkin algebra, and \\(q:B(H)\\to Q(H)\\) the quotient map. \\(Q(H)\\) is a C\\*-algebra, as a quotient of a C\\*-algebra by a closed ideal (lesson on C\\*-algebras, Section 15).\n\n(a) Fix an orthonormal sequence \\(\\xi\\). The map \\(\\lambda\\mapsto q(d^\\xi_\\lambda)\\) induces an isometric \\(*\\)-isomorphism of \\(\\ell^\\infty/c_0\\) onto a C\\*-subalgebra of \\(Q(H)\\).\n\n(b) Every representation of \\(Q(H)\\) on a separable Hilbert space is zero. If \\(H\\) is separable, \\(Q(H)\\) is simple: its only closed ideals are \\(0\\) and \\(Q(H)\\).\n\n**Proof.** (a) \\(\\lambda\\mapsto d^\\xi_\\lambda\\) is a \\(*\\)-homomorphism \\(\\ell^\\infty\\to B(H)\\), so \\(\\lambda\\mapsto q(d^\\xi_\\lambda)\\) is a \\(*\\)-homomorphism with kernel \\(\\{\\lambda:d^\\xi_\\lambda\\in K(H)\\}=c_0\\). The induced map on \\(\\ell^\\infty/c_0\\) is an injective \\(*\\)-homomorphism between C\\*-algebras, hence isometric with closed range (lesson on C\\*-algebras, Section 4).\n\n(b) If \\(H\\) is separable, the closed ideals of \\(Q(H)\\) correspond to the closed ideals of \\(B(H)\\) that contain \\(K(H)\\), which are \\(K(H)\\) and \\(B(H)\\) by Proposition 7.3; so \\(Q(H)\\) is simple. Let \\(\\pi:Q(H)\\to B(K)\\) be a nonzero \\(*\\)-representation, where \\(K\\) is a separable Hilbert space.\n\n*Case 1: \\(H\\) separable.* The kernel of \\(\\pi\\) is a closed ideal, not all of \\(Q(H)\\), so \\(\\pi\\) is injective. Choose an uncountable family \\((A_\\iota)_{\\iota\\in\\Omega}\\) of infinite subsets of \\(\\mathbb N\\) with \\(A_\\iota\\cap A_\\kappa\\) finite for \\(\\iota\\ne\\kappa\\). (Number the finite words in the letters \\(0,1\\) by \\(\\mathbb N\\), and for each infinite \\(0\\)–\\(1\\) sequence \\(\\iota\\) let \\(A_\\iota\\) be the set of numbers of its initial segments.) The elements \\(P_\\iota=q(d^\\xi_{1_{A_\\iota}})\\) are projections, nonzero because \\(1_{A_\\iota}\\notin c_0\\), and \\(P_\\iota P_\\kappa=q(d^\\xi_{1_{A_\\iota\\cap A_\\kappa}})=0\\) for \\(\\iota\\ne\\kappa\\). So the \\(\\pi(P_\\iota)\\) are uncountably many nonzero mutually orthogonal projections on \\(K\\). Unit vectors in their ranges form an uncountable orthonormal family in the separable space \\(K\\), which is impossible.\n\n*Case 2: \\(H\\) nonseparable.* [Theorem 8.5 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-09) proves that every uncountable set can be partitioned into uncountably many subsets, each in bijection with the whole set. Apply it to an orthonormal basis of \\(H\\). Let \\(P_\\iota\\) be the projection onto the closed span of the \\(\\iota\\)-th subfamily, and \\(v_\\iota\\) an isometry of \\(H\\) onto \\(P_\\iota H\\); then \\(v_\\iota^*P_\\iota v_\\iota=1\\). The map \\(\\pi\\circ q\\) is a nonzero \\(*\\)-representation of \\(B(H)\\). If \\(\\pi(q(P_\\iota))=0\\), then \\(\\pi(q(1))=\\pi(q(v_\\iota^*P_\\iota v_\\iota))=0\\) and \\(\\pi\\circ q=0\\). So the \\(\\pi(q(P_\\iota))\\) are uncountably many nonzero mutually orthogonal projections on \\(K\\), which is again impossible. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-09",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "8. Seven topologies on B(H)",
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      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-09",
      "proof_locus": {
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      "full_conditions_and_proof": "## 8. Seven topologies on B(H)\n\nBesides the norm topology, \\(B(H)\\) carries six weaker locally convex topologies. Three of them test an operator on finitely many vectors at a time. The other three test it on square-summable sequences of vectors; they come from the duality \\(B(H)=(B(H)_*)^*\\) of Theorem 6.2(c).\n\n**Definition 8.1.** Each topology below is the locally convex topology on \\(B(H)\\) generated by the listed seminorms. Here \\(\\xi,\\eta\\in H\\), and \\(\\omega\\) runs through \\(B(H)_*\\), or through the positive elements \\(B(H)_*^+\\) of \\(B(H)_*\\) where stated.\n\n| topology | seminorms |\n|---|---|\n| weak | \\(x\\mapsto\\lvert\\langle x\\xi,\\eta\\rangle\\rvert\\) |\n| strong | \\(x\\mapsto\\lVert x\\xi\\rVert\\) |\n| strong\\(^*\\) | \\(x\\mapsto(\\lVert x\\xi\\rVert^2+\\lVert x^*\\xi\\rVert^2)^{1/2}\\) |\n| \\(\\sigma\\)-weak | \\(x\\mapsto\\lvert\\omega(x)\\rvert\\), \\(\\omega\\in B(H)_*\\) |\n| \\(\\sigma\\)-strong | \\(x\\mapsto p_\\omega(x)=\\omega(x^*x)^{1/2}\\), \\(\\omega\\in B(H)_*^+\\) |\n| \\(\\sigma\\)-strong\\(^*\\) | \\(x\\mapsto(p_\\omega(x)^2+p_\\omega(x^*)^2)^{1/2}\\), \\(\\omega\\in B(H)_*^+\\) |\n| norm | \\(x\\mapsto\\lVert x\\rVert\\) |\n\nThe norm topology is also called the uniform topology, the \\(\\sigma\\)-weak topology the ultraweak topology, and the \\(\\sigma\\)-strong topology the ultrastrong topology. By definition the \\(\\sigma\\)-weak topology is \\(\\sigma(B(H),B(H)_*)\\), the weak\\(^*\\) topology of \\(B(H)\\) as the dual of \\(B(H)_*\\), or equivalently of \\(\\mathcal L^1(H)\\). The weak topology is \\(\\sigma(B(H),F_*)\\), where \\(F_*\\) denotes the space of finite sums of vector functionals \\(\\omega_{\\xi,\\eta}\\).\n\n**Lemma 8.2** (equivalent seminorms).\n\n(a) For \\(\\omega\\in B(H)_*^+\\) write \\(\\omega=\\sum_n\\omega_{\\zeta_n}\\) with \\(\\sum\\|\\zeta_n\\|^2<\\infty\\) (Proposition 6.3(a)). Then \\(p_\\omega(x)=(\\sum_n\\|x\\zeta_n\\|^2)^{1/2}\\), and \\(p_\\omega\\) is a seminorm. Conversely, for every square-summable sequence \\((\\xi_n)\\), \\((\\sum_n\\|x\\xi_n\\|^2)^{1/2}=p_\\omega(x)\\) with \\(\\omega=\\sum_n\\omega_{\\xi_n}\\in B(H)_*^+\\). So the \\(\\sigma\\)-strong topology is generated by the seminorms \\(x\\mapsto(\\sum_n\\|x\\xi_n\\|^2)^{1/2}\\), and the \\(\\sigma\\)-strong\\(^*\\) topology by \\(x\\mapsto(\\sum_n\\|x\\xi_n\\|^2+\\|x^*\\xi_n\\|^2)^{1/2}\\), with \\((\\xi_n)\\) square summable.\n\n(b) The \\(\\sigma\\)-weak topology is generated by the seminorms \\(x\\mapsto|\\sum_n\\langle x\\xi_n,\\eta_n\\rangle|\\) with \\((\\xi_n)\\), \\((\\eta_n)\\) square summable.\n\n(c) In each of the strong, strong\\(^*\\), \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, finitely many of the defining seminorms are dominated by a single seminorm of the same kind. So every neighbourhood of \\(x_0\\) contains a set \\(\\{x:p(x-x_0)<\\varepsilon\\}\\) with one seminorm \\(p\\) of that kind; for the strong topology \\(p\\) can be taken as \\((\\sum_{j\\le m}\\|x\\xi_j\\|^2)^{1/2}\\) for finitely many vectors, and similarly for the strong\\(^*\\) topology.\n\n**Proof.** (a) \\[\n\\begin{gathered}\np_\\omega(x)^2\\\\\n=\\omega(x^*x)\\\\\n=\\sum_n\\langle x^*x\\zeta_n,\\zeta_n\\rangle\\\\\n=\\sum_n\\|x\\zeta_n\\|^2.\n\\end{gathered}\n\\] This is the square of the norm of the vector \\((x\\zeta_n)_n\\) of \\(\\ell^2(\\mathbb N;H)\\), which depends linearly on \\(x\\); so \\(p_\\omega\\) is a seminorm. The converse holds because \\(\\sum_n\\omega_{\\xi_n}\\) is normal and positive by Theorem 6.2(d) and Proposition 6.3(a). (b) is Theorem 6.2(d). (c) For vectors, concatenate the finite families or the square-summable sequences; for positive normal functionals, \\(p_{\\omega_1}^2+\\dots+p_{\\omega_k}^2=p_{\\omega_1+\\dots+\\omega_k}^2\\). \\(\\square\\)\n\n**Proposition 8.3** (comparison). In the table\n\n| | weak type | strong type | strong\\(^*\\) type |\n|---|---|---|---|\n| plain | weak | strong | strong\\(^*\\) |\n| \\(\\sigma\\) | \\(\\sigma\\)-weak | \\(\\sigma\\)-strong | \\(\\sigma\\)-strong\\(^*\\) |\n\neach topology is finer than the topologies to its left and than the topology above it, and the norm topology is finer than all six. So the weak topology is the coarsest and the \\(\\sigma\\)-strong\\(^*\\) topology the finest of the six.\n\n**Proof.** A topology given by seminorms is finer than another if each seminorm of the second is dominated by finitely many seminorms of the first. Now \\(|\\langle x\\xi,\\eta\\rangle|\\le\\|x\\xi\\|\\,\\|\\eta\\|\\) and \\[\n\\begin{gathered}\n|\\sum_n\\langle x\\xi_n,\\eta_n\\rangle|\\\\\n\\le(\\sum_n\\|x\\xi_n\\|^2)^{1/2}(\\sum_n\\|\\eta_n\\|^2)^{1/2};\n\\end{gathered}\n\\] a starred seminorm dominates the unstarred one; a plain seminorm is the \\(\\sigma\\)-seminorm of a sequence with one nonzero term; and \\(p_\\omega(x)\\le\\omega(1)^{1/2}\\|x\\|\\). \\(\\square\\)\n\n**Example 8.4** (the comparisons are strict). Let \\(H\\) be infinite-dimensional and \\((\\xi_n)\\) an orthonormal sequence.\n\n(a) *Norm and \\(\\sigma\\)-strong\\(^*\\).* The projections \\(e_n=\\theta_{\\xi_n,\\xi_n}\\) tend to \\(0\\) \\(\\sigma\\)-strongly\\(^*\\), although \\(\\|e_n\\|=1\\). Indeed \\(p_\\omega(e_n)^2=\\omega(e_n)\\), and \\(\\sum_n\\omega(e_n)\\le\\omega(1)\\) for \\(\\omega\\in B(H)_*^+\\) because \\(e_1+\\dots+e_m\\le1\\).\n\n(b) *Starred and unstarred.* \\(\\theta_{\\xi_1,\\xi_n}\\to0\\) \\(\\sigma\\)-strongly: for square-summable \\((\\zeta_k)\\), \\(\\sum_k\\|\\theta_{\\xi_1,\\xi_n}\\zeta_k\\|^2=\\sum_k|\\langle\\zeta_k,\\xi_n\\rangle|^2\\to0\\) by dominated convergence, since each term tends to \\(0\\) by Bessel's inequality and is at most \\(\\|\\zeta_k\\|^2\\). But \\(\\theta_{\\xi_1,\\xi_n}^*=\\theta_{\\xi_n,\\xi_1}\\) and \\(\\|\\theta_{\\xi_n,\\xi_1}\\xi_1\\|=1\\), so the sequence does not tend to \\(0\\) strongly\\(^*\\). So the strong\\(^*\\) topology is finer than the strong one without being equal to it, and the same holds for the \\(\\sigma\\)-strong\\(^*\\) and \\(\\sigma\\)-strong topologies.\n\n(c) *Strong type and weak type.* \\(\\theta_{\\xi_n,\\xi_1}\\to0\\) \\(\\sigma\\)-weakly: for \\(\\omega\\in B(H)_*\\), (4.4) gives \\(\\omega(\\theta_{\\xi_n,\\xi_1})=\\operatorname{Tr}(\\theta_{\\xi_n,\\xi_1}t_\\omega)=\\langle\\xi_n,t_\\omega^*\\xi_1\\rangle\\to0\\). But \\(\\|\\theta_{\\xi_n,\\xi_1}\\xi_1\\|=1\\), so there is no strong convergence to \\(0\\).\n\nExample 9.2 below shows that each \\(\\sigma\\)-topology is strictly finer than its plain version. If \\(H\\) is finite-dimensional, all seven topologies coincide: for an orthonormal basis \\(\\varepsilon_1,\\dots,\\varepsilon_d\\), \\(x=\\sum_{i,j}\\langle x\\varepsilon_j,\\varepsilon_i\\rangle\\theta_{\\varepsilon_i,\\varepsilon_j}\\), so \\(\\|x\\|\\le\\sum_{i,j}|\\langle x\\varepsilon_j,\\varepsilon_i\\rangle|\\), and the weak topology is already the norm topology.\n\n**Lemma 8.5** (bounded sets). On every bounded subset of \\(B(H)\\), the weak and \\(\\sigma\\)-weak topologies coincide, the strong and \\(\\sigma\\)-strong topologies coincide, and the strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topologies coincide.\n\n**Proof.** Let \\(\\mathcal B\\) be bounded, say \\(\\|x\\|\\le r\\) for \\(x\\in\\mathcal B\\). Topologies are determined by their convergent nets, and each \\(\\sigma\\)-topology is finer than its plain version. So it suffices to show: if a net \\((x_\\alpha)\\) in \\(\\mathcal B\\) converges to \\(x\\in\\mathcal B\\) in a plain topology, it converges in the \\(\\sigma\\)-version. *Weak.* Let \\(\\omega\\in B(H)_*\\) and \\(\\varepsilon>0\\). By Theorem 6.2(d) there is a finite sum \\(\\omega_0\\) of vector functionals with \\(\\|\\omega-\\omega_0\\|<\\varepsilon\\). Then \\(|\\omega(x_\\alpha-x)|\\le2r\\varepsilon+|\\omega_0(x_\\alpha-x)|\\), and the last term tends to \\(0\\). *Strong.* For square-summable \\((\\xi_n)\\) and every \\(N\\),\n\\[\n\\begin{gathered}\n\\sum_n\\|(x_\\alpha-x)\\xi_n\\|^2\\\\\n\\le\\sum_{n\\le N}\\|(x_\\alpha-x)\\xi_n\\|^2+4r^2\\sum_{n>N}\\|\\xi_n\\|^2 .\n\\end{gathered}\n\\]\nChoose \\(N\\) so that the tail is small; the finitely many remaining terms tend to \\(0\\). *Strong\\(^*\\).* Apply the same estimate to \\(x_\\alpha^*-x^*\\) as well. \\(\\square\\)\n\n**Proposition 8.6** (continuity of the operations).\n\n(a) For fixed \\(a,b\\in B(H)\\), the map \\(x\\mapsto axb\\) is continuous for all seven topologies. In particular multiplication is separately continuous in each of them.\n\n(b) The adjoint \\(x\\mapsto x^*\\) is continuous for the norm, weak, \\(\\sigma\\)-weak, strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topologies. If \\(H\\) is infinite-dimensional, it is not continuous for the strong or the \\(\\sigma\\)-strong topology, not even on the unit ball.\n\n(c) If \\(x_\\alpha\\to x\\) and \\(y_\\alpha\\to y\\) strongly (resp. \\(\\sigma\\)-strongly) and \\(\\sup_\\alpha\\|x_\\alpha\\|<\\infty\\), then \\(x_\\alpha y_\\alpha\\to xy\\) strongly (resp. \\(\\sigma\\)-strongly). If both nets are bounded and converge strongly\\(^*\\) (resp. \\(\\sigma\\)-strongly\\(^*\\)), then \\(x_\\alpha y_\\alpha\\to xy\\) strongly\\(^*\\) (resp. \\(\\sigma\\)-strongly\\(^*\\)).\n\n(d) If \\(H\\) is infinite-dimensional, multiplication is not jointly continuous for the weak or the \\(\\sigma\\)-weak topology, not even on the unit ball.\n\n**Proof.** (a) \\(\\|axb\\xi\\|\\le\\|a\\|\\,\\|x(b\\xi)\\|\\) and \\(\\sum_n\\|axb\\xi_n\\|^2\\le\\|a\\|^2\\sum_n\\|x(b\\xi_n)\\|^2\\), where \\((b\\xi_n)\\) is again square summable; \\(\\langle axb\\xi_n,\\eta_n\\rangle=\\langle x(b\\xi_n),a^*\\eta_n\\rangle\\); and \\((axb)^*=b^*x^*a^*\\) handles the starred seminorms. (b) \\(\\sum_n\\langle x^*\\xi_n,\\eta_n\\rangle=\\overline{\\sum_n\\langle x\\eta_n,\\xi_n\\rangle}\\), and the starred seminorms are unchanged by \\(x\\mapsto x^*\\). The counterexample is Example 8.4(b), which lies in the unit ball. (c) \\[\n\\begin{gathered}\n\\|(x_\\alpha y_\\alpha-xy)\\xi\\|\\\\\n\\le\\|x_\\alpha\\|\\,\\|(y_\\alpha-y)\\xi\\|+\\|(x_\\alpha-x)y\\xi\\|,\n\\end{gathered}\n\\] and the same estimate holds in \\(\\ell^2(\\mathbb N;H)\\) for square-summable \\((\\xi_n)\\), with \\((y\\xi_n)\\) square summable. For the adjoints use \\((x_\\alpha y_\\alpha)^*=y_\\alpha^*x_\\alpha^*\\) and the bound on \\(\\|y_\\alpha\\|\\). (d) Choose an orthonormal sequence in \\(H\\), let \\(L\\) be its closed span and \\(p_L\\) its projection. On \\(L\\cong\\ell^2\\) let \\(S\\delta_k=\\delta_{k+1}\\), and extend \\(S\\) by zero on \\(L^\\perp\\). Then \\(S^{*n}S^n=p_L\\ne0\\) for all \\(n\\geq1\\), while both powers have norm at most one. In the following estimate use the coordinates of the \\(L\\)-components of \\(\\xi,\\eta\\). But \\(S^n\\to0\\) and \\(S^{*n}\\to0\\) weakly: \\[\n\\begin{gathered}\n|\\langle S^n\\xi,\\eta\\rangle|\\\\\n=|\\sum_k\\xi_k\\overline{\\eta_{k+n}}|\\\\\n\\le\\|\\xi\\|(\\sum_{k>n}|\\eta_k|^2)^{1/2}\\to0,\n\\end{gathered}\n\\] and \\(\\langle S^{*n}\\xi,\\eta\\rangle=\\overline{\\langle S^n\\eta,\\xi\\rangle}\\). By Lemma 8.5 the convergence is also \\(\\sigma\\)-weak. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-10",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "9. Continuous functionals and closed convex sets",
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      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "full_conditions_and_proof": "## 9. Continuous functionals and closed convex sets\n\nThe weak, strong and strong\\(^*\\) topologies have the same continuous linear functionals, and so do the three \\(\\sigma\\)-topologies. By the Hahn–Banach theorem this forces them to have the same closed convex sets. The Krein–Šmulian theorem then reduces \\(\\sigma\\)-weak closedness of a convex set to closedness of its bounded parts.\n\n**Theorem 9.1** (continuous functionals). Let \\(M\\subseteq B(H)\\) be a linear subspace and \\(\\omega\\) a linear functional on \\(M\\). Continuity refers to the topologies restricted to \\(M\\).\n\n(i) The following are equivalent: (1) \\(\\omega\\) is weakly continuous; (2) \\(\\omega\\) is strongly continuous; (3) \\(\\omega\\) is strongly\\(^*\\) continuous; (4) there are finitely many vectors \\(\\xi_k,\\eta_k\\) with \\(\\omega(x)=\\sum_k\\langle x\\xi_k,\\eta_k\\rangle\\) for \\(x\\in M\\).\n\n(ii) The following are equivalent: (1) \\(\\omega\\) is \\(\\sigma\\)-weakly continuous; (2) \\(\\omega\\) is \\(\\sigma\\)-strongly continuous; (3) \\(\\omega\\) is \\(\\sigma\\)-strongly\\(^*\\) continuous; (4) there are square-summable sequences \\((\\xi_n)\\), \\((\\eta_n)\\) with \\(\\omega(x)=\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\) for \\(x\\in M\\); (5) \\(\\omega\\) is the restriction of a normal functional on \\(B(H)\\).\n\nIn particular, for \\(M=B(H)\\), the continuous linear functionals of the weak, strong and strong\\(^*\\) topologies are the elements of \\(F_*\\), and those of the three \\(\\sigma\\)-topologies are the normal functionals.\n\n**Proof.** (i) (1)\\(\\Rightarrow\\)(2)\\(\\Rightarrow\\)(3) because the topologies get finer (Proposition 8.3), and (4)\\(\\Rightarrow\\)(1) is clear. (3)\\(\\Rightarrow\\)(4): By continuity at \\(0\\) and Lemma 8.2(c), there are vectors \\(\\xi_1,\\dots,\\xi_m\\) and \\(\\delta>0\\) such that \\(|\\omega(x)|<1\\) for all \\(x\\in M\\) with \\(q(x)<\\delta\\), where \\(q(x)=(\\sum_j\\|x\\xi_j\\|^2+\\|x^*\\xi_j\\|^2)^{1/2}\\). By homogeneity \\(|\\omega(x)|\\le\\delta^{-1}q(x)\\) on \\(M\\): if \\(q(x)=0\\), then \\(q(sx)<\\delta\\) for all \\(s>0\\), so \\(|\\omega(x)|<1/s\\) for all \\(s\\), and \\(\\omega(x)=0\\). Let \\(\\mathcal K=H^m\\oplus\\overline H^m\\) and\n\\[\nJ(x)=(x\\xi_1,\\dots,x\\xi_m,\\ \\overline{x^*\\xi_1},\\dots,\\overline{x^*\\xi_m}).\n\\]\nThe map \\(J:M\\to\\mathcal K\\) is complex linear, because \\((cx)^*\\xi=\\bar cx^*\\xi\\) and \\(\\overline{\\bar cv}=c\\bar v\\) in \\(\\overline H\\); and \\(\\|J(x)\\|=q(x)\\). So \\(\\omega=\\tilde\\omega\\circ J\\) for a well-defined linear functional \\(\\tilde\\omega\\) on \\(J(M)\\) with \\(|\\tilde\\omega(v)|\\le\\delta^{-1}\\|v\\|\\). Extend \\(\\tilde\\omega\\) by continuity to the closure of \\(J(M)\\) and by \\(0\\) on its orthogonal complement. The Riesz theorem gives \\(\\eta_1,\\dots,\\eta_m,\\zeta_1,\\dots,\\zeta_m\\in H\\) with\n\\[\n\\begin{gathered}\n\\omega(x)\\\\\n=\\sum_j\\langle x\\xi_j,\\eta_j\\rangle+\\sum_j\\langle\\overline{x^*\\xi_j},\\overline{\\zeta_j}\\rangle\\\\\n=\\sum_j\\langle x\\xi_j,\\eta_j\\rangle+\\sum_j\\langle\\zeta_j,x^*\\xi_j\\rangle\\\\\n=\\sum_j\\langle x\\xi_j,\\eta_j\\rangle+\\sum_j\\langle x\\zeta_j,\\xi_j\\rangle .\n\\end{gathered}\n\\]\nThis is (4).\n\n(ii) (1)\\(\\Rightarrow\\)(2)\\(\\Rightarrow\\)(3) by Proposition 8.3. (3)\\(\\Rightarrow\\)(4): the same argument, with a square-summable sequence \\((\\xi_n)\\) in place of \\(\\xi_1,\\dots,\\xi_m\\) and \\(\\mathcal K=\\ell^2(\\mathbb N;H)\\oplus\\ell^2(\\mathbb N;\\overline H)\\), gives square-summable \\((\\eta_n)\\), \\((\\zeta_n)\\) with \\(\\omega(x)=\\sum_n\\langle x\\xi_n,\\eta_n\\rangle+\\sum_n\\langle x\\zeta_n,\\xi_n\\rangle\\) on \\(M\\); interleaving the two sums gives one pair of square-summable sequences. (4)\\(\\Rightarrow\\)(5): \\(\\sum_n\\omega_{\\xi_n,\\eta_n}\\) is normal by Theorem 6.2(d), since \\(\\sum\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\) by Cauchy–Schwarz. (5)\\(\\Rightarrow\\)(1) holds by the definition of the \\(\\sigma\\)-weak topology. \\(\\square\\)\n\n**Example 9.2** (the \\(\\sigma\\)-topologies are strictly finer). Let \\(H\\) be infinite-dimensional, \\((\\xi_n)\\) orthonormal, and \\(\\omega=\\sum_n2^{-n}\\omega_{\\xi_n}\\). Then \\(\\omega\\) is normal, so it is \\(\\sigma\\)-weakly continuous. It is not strongly\\(^*\\) continuous: otherwise, by Theorem 9.1(i), \\(\\omega=\\sum_{k\\le m}\\omega_{\\xi'_k,\\eta'_k}\\), and then \\(t_\\omega=\\sum_{k\\le m}\\theta_{\\xi'_k,\\eta'_k}\\) would have finite rank (Theorem 6.2(a)), whereas \\(t_\\omega=\\sum_n2^{-n}\\theta_{\\xi_n,\\xi_n}\\) has infinite rank. Since each \\(\\sigma\\)-topology is finer than its plain version (Proposition 8.3) and \\(\\omega\\) is continuous for the \\(\\sigma\\)-weak topology but not for the strong\\(^*\\) topology, no \\(\\sigma\\)-topology equals its plain version: the \\(\\sigma\\)-weak topology differs from the weak one, the \\(\\sigma\\)-strong from the strong, and the \\(\\sigma\\)-strong\\(^*\\) from the strong\\(^*\\).\n\n**Theorem 9.3** (Krein–Šmulian). Let \\(X\\) be a Banach space and \\(K\\subseteq X^*\\) a convex set. If \\(K\\cap rB_{X^*}\\) is weak\\(^*\\) closed for every \\(r>0\\), then \\(K\\) is weak\\(^*\\) closed. Here \\(B_{X^*}\\) is the closed unit ball of \\(X^*\\) and weak\\(^*\\) means \\(\\sigma(X^*,X)\\).\n\n\n**Proof.** If \\(K=\\varnothing\\), the conclusion is immediate. Otherwise write \\(B=B_{X^*}\\), and for \\(F\\subseteq X\\) let \\(F^\\circ=\\{\\varphi\\in X^*:|\\varphi(x)|\\le1\\ \\text{for all}\\ x\\in F\\}\\). Each \\(F^\\circ\\) is weak\\(^*\\) closed, and \\(F^\\circ\\cap G^\\circ=(F\\cup G)^\\circ\\).\n\n*Step 1: reduction.* Let \\(\\varphi_0\\in X^*\\setminus K\\); we must find a weak\\(^*\\) neighbourhood of \\(\\varphi_0\\) that misses \\(K\\). The set \\(K-\\varphi_0\\) is convex, and \\[\n\\begin{gathered}\n(K-\\varphi_0)\\cap rB\\\\\n=\\big(K\\cap(r+\\|\\varphi_0\\|)B\\cap(\\varphi_0+rB)\\big)-\\varphi_0\n\\end{gathered}\n\\] is weak\\(^*\\) closed, because \\[\n\\begin{gathered}\n\\varphi_0+rB\\\\\n=\\{\\varphi:|\\varphi(x)-\\varphi_0(x)|\\le r\\|x\\|\\ \\text{for all}\\ x\\}\n\\end{gathered}\n\\] is weak\\(^*\\) closed. So we may assume \\(\\varphi_0=0\\notin K\\). The hypothesis makes \\(K\\) norm closed: a norm-convergent sequence in \\(K\\) is bounded, so it lies in some \\(K\\cap rB\\), which is weak\\(^*\\) closed and hence norm closed. So some ball \\(\\rho B\\), \\(\\rho>0\\), misses \\(K\\), and after replacing \\(K\\) by \\(\\rho^{-1}K\\) we may assume \\(K\\cap B=\\emptyset\\).\n\n*Step 2: finite sets.* We choose finite sets \\(F_1,F_2,\\dots\\subseteq X\\), with \\(\\|x\\|\\le1/n\\) for \\(x\\in F_n\\), such that for every \\(n\\ge1\\)\n\\[\nK\\cap nB\\cap F_1^\\circ\\cap\\dots\\cap F_{n-1}^\\circ=\\emptyset .\n\\tag{9.1}\n\\]\nFor \\(n=1\\) this says \\(K\\cap B=\\emptyset\\). Suppose \\(F_1,\\dots,F_{n-1}\\) are chosen, and put \\(Q=F_1^\\circ\\cap\\dots\\cap F_{n-1}^\\circ\\). Suppose that \\(K\\cap(n+1)B\\cap Q\\cap F^\\circ\\ne\\emptyset\\) for every finite set \\(F\\) of vectors of norm at most \\(1/n\\). These sets are weak\\(^*\\) closed subsets of \\((n+1)B\\), which is weak\\(^*\\) compact by the Banach–Alaoglu theorem, and they have the finite intersection property. So they have a common point \\(\\varphi\\). Then \\(\\varphi\\in K\\cap(n+1)B\\cap Q\\), and \\(|\\varphi(x)|\\le1\\) whenever \\(\\|x\\|\\le1/n\\), so \\(\\|\\varphi\\|\\le n\\) and \\(\\varphi\\in K\\cap nB\\cap Q\\), contradicting (9.1) for \\(n\\). Hence some finite \\(F_n\\) of vectors of norm at most \\(1/n\\) gives \\(K\\cap(n+1)B\\cap Q\\cap F_n^\\circ=\\emptyset\\), which is (9.1) for \\(n+1\\).\n\n*Step 3: a null sequence.* List the elements of \\(F_1,F_2,\\dots\\), in this order, as a sequence \\((x_k)\\), adding zeros if there are only finitely many. Then \\(x_k\\to0\\). Let \\(\\varphi\\in K\\). Then \\(\\|\\varphi\\|>1\\); choose an integer \\(n\\ge2\\) with \\(\\|\\varphi\\|\\le n\\). By (9.1), \\(\\varphi\\notin F_1^\\circ\\cap\\dots\\cap F_{n-1}^\\circ\\), so \\(|\\varphi(x_k)|>1\\) for some \\(k\\).\n\n*Step 4: separation.* The map \\(T\\varphi=(\\varphi(x_k))_k\\) is linear from \\(X^*\\) into \\(c_0\\). The convex set \\(T(K)\\) misses the open unit ball \\(U\\) of \\(c_0\\), by Step 3. By Hahn–Banach separation and Example 5.1(a), there are \\(\\alpha\\in\\ell^1\\), \\(\\alpha\\ne0\\), and \\(\\gamma\\in\\mathbb R\\) with\n\\[\n\\begin{gathered}\n\\operatorname{Re}\\sum_k\\alpha_ku_k<\\gamma\\\\\n\\le\\operatorname{Re}\\sum_k\\alpha_k\\varphi(x_k)\\\\\n(u\\in U,\\ \\varphi\\in K).\n\\end{gathered}\n\\]\nThe supremum of the left side over \\(u\\in U\\) is \\(\\|\\alpha\\|_1\\), so \\(\\gamma\\ge\\|\\alpha\\|_1\\). The series \\(x=\\|\\alpha\\|_1^{-1}\\sum_k\\alpha_kx_k\\) converges absolutely in \\(X\\), and \\(\\operatorname{Re}\\varphi(x)\\ge1\\) for all \\(\\varphi\\in K\\). So the weak\\(^*\\) open set \\(\\{\\varphi:\\operatorname{Re}\\varphi(x)<1\\}\\) contains \\(0\\) and misses \\(K\\). \\(\\square\\)\n\n**Theorem 9.4** (preduals of subspaces). Let \\(M\\subseteq B(H)\\) be a \\(\\sigma\\)-weakly closed linear subspace. Let \\(M_*\\) be the space of \\(\\sigma\\)-weakly continuous linear functionals on \\(M\\) and \\(M_\\sim\\) the space of weakly continuous ones, both with the norm of \\(M^*\\), and let \\(M_\\perp=\\{\\omega\\in B(H)_*:\\omega|_M=0\\}\\).\n\n(a) Restriction \\(\\omega\\mapsto\\omega|_M\\) maps \\(B(H)_*\\) onto \\(M_*\\), and it induces an isometric isomorphism of \\(B(H)_*/M_\\perp\\) onto \\(M_*\\). In particular \\(M_*\\) is complete, and it is a norm-closed subspace of \\(M^*\\).\n\n(b) The map sending \\(x\\in M\\) to the functional \\(\\varphi\\mapsto\\varphi(x)\\) is an isometric isomorphism of \\(M\\) onto \\((M_*)^*\\). Under it, \\(\\sigma(M,M_*)\\) is the \\(\\sigma\\)-weak topology restricted to \\(M\\).\n\n(c) \\(M_\\sim\\) is norm dense in \\(M_*\\).\n\n**Proof.** (a) Restriction maps onto \\(M_*\\) by Theorem 9.1(ii), and its kernel is \\(M_\\perp\\). Let \\(\\omega\\in B(H)_*\\). By Background 5(f), the quotient norm \\(\\|\\omega+M_\\perp\\|\\) is the supremum of \\(|g(\\omega+M_\\perp)|\\) over the unit ball of \\((B(H)_*/M_\\perp)^*\\). By Background 5(e), with \\(X=B(H)_*\\) and \\(X^*=B(H)\\) (Theorem 6.2(c)), this dual is isometrically \\[\n\\begin{gathered}\n(M_\\perp)^\\perp\\\\\n=\\{x\\in B(H):\\omega'(x)=0\\ \\text{for all}\\ \\omega'\\in M_\\perp\\},\n\\end{gathered}\n\\] and \\((M_\\perp)^\\perp\\) is the \\(\\sigma\\)-weak closure of \\(M\\), which is \\(M\\). Hence\n\\[\n\\begin{gathered}\n\\|\\omega+M_\\perp\\|\\\\\n=\\sup\\{|\\omega(x)|:\\ x\\in M,\\ \\|x\\|\\le1\\}\\\\\n=\\|\\omega|_M\\|_{M^*}.\n\\end{gathered}\n\\]\nSo restriction induces an isometry of the Banach space \\(B(H)_*/M_\\perp\\) onto \\(M_*\\).\n\n(b) By (a) and Background 5(e), \\((M_*)^*\\cong(B(H)_*/M_\\perp)^*\\cong(M_\\perp)^\\perp=M\\) isometrically. Following an element \\(x\\in M\\) through these identifications, it acts on \\(\\omega|_M\\) by \\(\\omega|_M\\mapsto\\omega(x)\\), which is evaluation at \\(x\\). The topology \\(\\sigma(M,M_*)\\) is pointwise convergence on the restrictions of normal functionals, which is the restricted \\(\\sigma\\)-weak topology.\n\n(c) By Theorem 9.1(i), \\(M_\\sim\\) consists of the restrictions of elements of \\(F_*\\). Since \\(F_*\\) is norm dense in \\(B(H)_*\\) (Theorem 6.2(d)) and restriction is contractive and onto \\(M_*\\), the image of \\(F_*\\) is dense in \\(M_*\\). \\(\\square\\)\n\n**Theorem 9.5** (continuity on the unit ball). Let \\(M\\subseteq B(H)\\) be a \\(\\sigma\\)-weakly closed subspace, \\(M_1=M\\cap B(H)_1\\), and \\(\\omega\\) a linear functional on \\(M\\). The following are equivalent: (1) \\(\\omega\\) is \\(\\sigma\\)-weakly continuous on \\(M\\); (2) \\(\\omega\\) is weakly continuous on \\(M_1\\); (3) \\(\\omega\\) is strongly continuous on \\(M_1\\); (4) \\(\\omega\\) is strongly\\(^*\\) continuous on \\(M_1\\). No boundedness of \\(\\omega\\) is assumed.\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): on \\(M_1\\) the \\(\\sigma\\)-weak and weak topologies coincide (Lemma 8.5). (2)\\(\\Rightarrow\\)(3)\\(\\Rightarrow\\)(4): Proposition 8.3.\n\n(4)\\(\\Rightarrow\\)(2): The set \\(M_1\\) is \\(\\sigma\\)-weakly closed in \\(B(H)_1\\), which is \\(\\sigma\\)-weakly compact by the Banach–Alaoglu theorem (\\(B(H)=\\mathcal L^1(H)^*\\), Theorem 5.4). So \\(M_1\\) is \\(\\sigma\\)-weakly compact, hence weakly compact (the weak topology is coarser and Hausdorff), hence weakly closed, hence strongly\\(^*\\) closed. For \\(c\\in\\mathbb R\\), the set \\(\\{x\\in M_1:\\operatorname{Re}\\omega(x)\\le c\\}\\) is convex and closed in \\(M_1\\) for the strong\\(^*\\) topology, so it is strongly\\(^*\\) closed in \\(B(H)\\). The weak and strong\\(^*\\) topologies have the same continuous linear functionals (Theorem 9.1(i) with \\(M=B(H)\\)), so by Background 5(c) this convex set is weakly closed. The same holds for the sets where \\(\\operatorname{Re}\\omega\\ge c\\), \\(\\operatorname{Im}\\omega\\le c\\) and \\(\\operatorname{Im}\\omega\\ge c\\). A real function whose sublevel and superlevel sets are all closed is continuous; so \\(\\operatorname{Re}\\omega\\) and \\(\\operatorname{Im}\\omega\\) are weakly continuous on \\(M_1\\).\n\n(2)\\(\\Rightarrow\\)(1): For \\(r>0\\), \\(\\ker\\omega\\cap rM_1=r(\\ker\\omega\\cap M_1)\\) is weakly closed, being the zero set of a weakly continuous function on the weakly closed set \\(M_1\\), rescaled. It is therefore \\(\\sigma\\)-weakly closed. By Theorem 9.4(b), \\(M\\) is the dual of the Banach space \\(M_*\\), its weak\\(^*\\) topology is the restricted \\(\\sigma\\)-weak topology, and its closed unit ball is \\(M_1\\). The Krein–Šmulian theorem shows that the convex set \\(\\ker\\omega\\) is \\(\\sigma\\)-weakly closed in \\(M\\). A linear functional with closed kernel is continuous (Background 5(h)). \\(\\square\\)\n\n**Theorem 9.6** (closed convex sets). Let \\(C\\subseteq B(H)\\) be convex.\n\n(a) \\(C\\) has the same closure in the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, and the same closure in the weak, strong and strong\\(^*\\) topologies.\n\n(b) The following are equivalent: (1) \\(C\\) is \\(\\sigma\\)-weakly closed; (2) \\(C\\) is \\(\\sigma\\)-strongly closed; (3) \\(C\\) is \\(\\sigma\\)-strongly\\(^*\\) closed; (4) \\(C\\cap rB(H)_1\\) is weakly closed for every \\(r>0\\); (5) \\(C\\cap rB(H)_1\\) is strongly closed for every \\(r>0\\); (6) \\(C\\cap rB(H)_1\\) is strongly\\(^*\\) closed for every \\(r>0\\).\n\n(c) The same holds for convex subsets of a \\(\\sigma\\)-weakly closed subspace \\(M\\), and for convex subsets of \\(B(H)_{\\mathrm{sa}}\\), with the relative topologies.\n\n**Proof.** (a) By Theorem 9.1 with \\(M=B(H)\\), the three \\(\\sigma\\)-topologies have the same continuous linear functionals, namely the normal ones, and the three plain topologies all have \\(F_*\\). Apply Background 5(c).\n\n(b) (1)\\(\\Leftrightarrow\\)(2)\\(\\Leftrightarrow\\)(3) by (a). (1)\\(\\Rightarrow\\)(4): \\(rB(H)_1\\) is \\(\\sigma\\)-weakly closed, since \\(\\|x\\|=\\sup\\{|\\omega(x)|:\\omega\\in B(H)_*,\\ \\|\\omega\\|\\le1\\}\\) by Theorem 6.2(c). So \\(C\\cap rB(H)_1\\) is a \\(\\sigma\\)-weakly closed subset of the \\(\\sigma\\)-weakly compact set \\(rB(H)_1\\), hence \\(\\sigma\\)-weakly compact, hence weakly compact and weakly closed. (4)\\(\\Rightarrow\\)(5)\\(\\Rightarrow\\)(6): a set closed in a coarser topology is closed in a finer one. (6)\\(\\Rightarrow\\)(4): \\(C\\cap rB(H)_1\\) is convex and strongly\\(^*\\) closed, hence weakly closed by (a). (4)\\(\\Rightarrow\\)(1): each \\(C\\cap rB(H)_1\\) is weakly closed, hence \\(\\sigma\\)-weakly closed. Since \\(B(H)=\\mathcal L^1(H)^*\\) with the \\(\\sigma\\)-weak topology as weak\\(^*\\) topology, the Krein–Šmulian theorem shows that \\(C\\) is \\(\\sigma\\)-weakly closed.\n\n(c) Let \\(M\\) be a \\(\\sigma\\)-weakly closed subspace, or \\(M=B(H)_{\\mathrm{sa}}\\), which is a \\(\\sigma\\)-weakly closed real subspace because the adjoint is \\(\\sigma\\)-weakly continuous (Proposition 8.6(b)). For \\(C\\subseteq M\\) and each of the six topologies, the closure of \\(C\\) in \\(M\\) is its closure in \\(B(H)\\) intersected with \\(M\\); so the equalities of (a) hold for closures in \\(M\\). Since \\(M\\) is \\(\\sigma\\)-weakly closed, it is closed in all three \\(\\sigma\\)-topologies, so \\(C\\) is closed in \\(M\\) for one of them exactly when it is closed in \\(B(H)\\). Finally \\(M\\cap rB(H)_1\\) is \\(\\sigma\\)-weakly compact, hence weakly closed, so a subset of \\(C\\cap rB(H)_1\\) is closed in \\(M\\) for one of the plain topologies exactly when it is closed in \\(B(H)\\). So (b) holds for \\(C\\) with the relative topologies. \\(\\square\\)\n\n**Corollary 9.7.**\n\n(a) The unit ball \\(B(H)_1\\) is \\(\\sigma\\)-weakly compact and weakly compact, and so are \\(\\{x\\in B(H)_{\\mathrm{sa}}:\\|x\\|\\le1\\}\\) and \\(\\{x:0\\le x\\le1\\}\\).\n\n(b) A linear subspace of \\(B(H)\\) is \\(\\sigma\\)-weakly closed if and only if it is \\(\\sigma\\)-strongly\\(^*\\) closed, if and only if its intersection with the unit ball is weakly closed.\n\n(c) On \\(B(H)_{\\mathrm{sa}}\\) the \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies coincide. The real-valued real-linear functionals on \\(B(H)_{\\mathrm{sa}}\\) that are continuous for this topology, or for the \\(\\sigma\\)-weak topology, are exactly the restrictions of hermitian normal functionals.\n\n**Proof.** (a) Banach–Alaoglu, since \\(B(H)=\\mathcal L^1(H)^*\\). The two sets are \\(\\sigma\\)-weakly closed subsets of the unit ball: the conditions \\(\\langle x\\xi,\\xi\\rangle\\in\\mathbb R\\) for all \\(\\xi\\) (which means \\(x=x^*\\), by Lemma 1.2) and \\(0\\le\\langle x\\xi,\\xi\\rangle\\le\\|\\xi\\|^2\\) are \\(\\sigma\\)-weakly closed. Weak compactness follows since the weak topology is coarser.\n\n(b) Theorem 9.6(b) with \\(r\\) scaled out: for a subspace \\(C\\), \\(C\\cap rB(H)_1=r(C\\cap B(H)_1)\\).\n\n(c) For \\(h\\) self-adjoint, \\(p_\\omega(h^*)=p_\\omega(h)\\). Let \\(\\rho\\) be a real-linear functional on \\(B(H)_{\\mathrm{sa}}\\) that is \\(\\sigma\\)-strongly continuous. As in the proof of Theorem 9.1, \\(|\\rho(h)|\\le Cp(h)\\) for a single \\(\\sigma\\)-strong seminorm \\(p\\). Define \\(\\omega(x)=\\rho(\\operatorname{Re}x)+i\\rho(\\operatorname{Im}x)\\), with \\(\\operatorname{Re}x=\\frac12(x+x^*)\\) and \\(\\operatorname{Im}x=\\frac1{2i}(x-x^*)\\). Since \\(\\operatorname{Re}(ix)=-\\operatorname{Im}x\\) and \\(\\operatorname{Im}(ix)=\\operatorname{Re}x\\), we get \\(\\omega(ix)=i\\omega(x)\\), so \\(\\omega\\) is complex linear. Also \\(p(\\operatorname{Re}x)\\) and \\(p(\\operatorname{Im}x)\\) are at most \\(\\frac12(p(x)+p(x^*))\\), so \\(|\\omega(x)|\\le C(p(x)+p(x^*))\\) and \\(\\omega\\) is \\(\\sigma\\)-strongly\\(^*\\) continuous, hence normal by Theorem 9.1(ii). It is hermitian and restricts to \\(\\rho\\). Conversely, a hermitian normal functional is real on \\(B(H)_{\\mathrm{sa}}\\) and \\(\\sigma\\)-weakly continuous, hence also \\(\\sigma\\)-strongly continuous. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-11",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "10. Metrizability and vector states",
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      "full_conditions_and_proof": "## 10. Metrizability and vector states\n\n**Proposition 10.1** (metrizability on bounded sets).\n\n(a) If \\(H\\) is separable, each of the six topologies weak, strong, strong\\(^*\\), \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) is metrizable on every bounded subset of \\(B(H)\\). Explicitly, let \\((\\zeta_n)\\) be dense in the unit ball of \\(H\\) and put\n\\[\n\\begin{gathered}\nd_w(x,y)\\\\\n=\\sum_{n,m}2^{-n-m}|\\langle(x-y)\\zeta_n,\\zeta_m\\rangle|,\\\\\nd_s(x,y)\\\\\n=\\sum_n2^{-n}\\|(x-y)\\zeta_n\\|,\\\\\nd_{s^*}(x,y)\\\\\n=d_s(x,y)+d_s(x^*,y^*).\n\\end{gathered}\n\\]\nOn each ball \\(rB(H)_1\\), \\(d_w\\) induces the weak and \\(\\sigma\\)-weak topology, \\(d_s\\) the strong and \\(\\sigma\\)-strong topology, and \\(d_{s^*}\\) the strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topology.\n\n(b) If \\(H\\) is not separable, none of the six topologies is metrizable on \\(B(H)_1\\).\n\n**Proof.** (a) The series converge on bounded sets, and each \\(d\\) is a metric: if \\(d_w(x,y)=0\\), then \\(\\langle(x-y)\\zeta_n,\\zeta_m\\rangle=0\\) for all \\(n,m\\), and \\(x=y\\) by density. Let \\((x_\\alpha)\\) be a net in \\(rB(H)_1\\) and \\(x\\in rB(H)_1\\). If \\(x_\\alpha\\to x\\) weakly, each term of \\(d_w(x_\\alpha,x)\\) tends to \\(0\\), and the terms with \\(n+m>N\\) contribute at most \\(2r\\sum_{n+m>N}2^{-n-m}\\), uniformly in \\(\\alpha\\); so \\(d_w(x_\\alpha,x)\\to0\\). Conversely, if \\(d_w(x_\\alpha,x)\\to0\\), then \\(\\langle(x_\\alpha-x)\\zeta_n,\\zeta_m\\rangle\\to0\\) for all \\(n,m\\). For \\(\\xi,\\eta\\) in the unit ball choose \\(\\zeta_n,\\zeta_m\\) close to them; then \\[\n\\begin{gathered}\n|\\langle(x_\\alpha-x)\\xi,\\eta\\rangle|\\\\\n\\le|\\langle(x_\\alpha-x)\\zeta_n,\\zeta_m\\rangle|\\\\\n+2r(\\|\\xi-\\zeta_n\\|+\\|\\eta-\\zeta_m\\|),\n\\end{gathered}\n\\] so \\(x_\\alpha\\to x\\) weakly. The same argument works for \\(d_s\\) and \\(d_{s^*}\\), and Lemma 8.5 transfers the result to the \\(\\sigma\\)-topologies.\n\n(b) The finite-rank projections form a directed set under \\(p\\le p'\\). Along it, \\((1-p)\\xi=0\\) as soon as \\(\\xi\\in pH\\), so the net \\(1-p\\) in \\(B(H)_1\\) tends to \\(0\\) strongly\\(^*\\), hence in each of the six topologies (on \\(B(H)_1\\) the strong\\(^*\\) topology is the finest of them, by Lemma 8.5). If one of the six topologies were metrizable on \\(B(H)_1\\), some sequence \\(1-p_k\\) would tend to \\(0\\) in it, hence weakly. Then \\(\\|(1-p_k)\\xi\\|^2=\\langle(1-p_k)\\xi,\\xi\\rangle\\to0\\) for every \\(\\xi\\), so every \\(\\xi\\) lies in the closure of \\(\\bigcup_kp_kH\\), which is separable. This contradicts the nonseparability of \\(H\\). \\(\\square\\)\n\n**Proposition 10.2** (the six topologies are not metrizable on B(H)). Let \\(H\\) be infinite-dimensional, let \\((e_n)\\) be a sequence of nonzero mutually orthogonal projections, let \\(c_n>0\\), and let \\(A=\\{c_ne_n:n\\ge1\\}\\).\n\n(a) If \\(\\sum_nc_n^{-2}=\\infty\\), then \\(0\\) is an accumulation point of \\(A\\) for the \\(\\sigma\\)-strong\\(^*\\) topology, and hence for all six topologies.\n\n(b) If \\(\\sum_nc_n^{-1}=\\infty\\), then \\(0\\) is an accumulation point of \\(A\\) for the \\(\\sigma\\)-weak topology.\n\n(c) Let \\(e_n=\\theta_{\\xi_n,\\xi_n}\\) for an orthonormal sequence \\((\\xi_n)\\). If \\(\\sum_nc_n^{-2}<\\infty\\), then \\(0\\) is not in the strong closure of \\(A\\); if \\(\\sum_nc_n^{-1}<\\infty\\), then \\(0\\) is not in the weak closure of \\(A\\).\n\n(d) If \\(c_n\\to\\infty\\), no subsequence of \\((c_ne_n)\\) converges weakly.\n\n(e) None of the six topologies is metrizable on \\(B(H)\\); indeed none of them has a countable base of neighbourhoods at \\(0\\).\n\n**Proof.** (a) By Lemma 8.2(c), every \\(\\sigma\\)-strong\\(^*\\) neighbourhood of \\(0\\) contains a set \\(\\{x:p_\\omega(x)^2+p_\\omega(x^*)^2<\\varepsilon^2\\}\\) with \\(\\omega\\in B(H)_*^+\\). Since \\(e_n=e_n^*=e_n^*e_n\\), \\(p_\\omega(c_ne_n)^2+p_\\omega(c_ne_n^*)^2=2c_n^2\\omega(e_n)\\). Now \\(\\sum_n\\omega(e_n)\\le\\omega(1)<\\infty\\), because \\(e_1+\\dots+e_m\\le1\\). If \\(2c_n^2\\omega(e_n)\\ge\\varepsilon^2\\) held for all \\(n\\ge N\\), then \\(\\omega(e_n)\\ge\\frac{\\varepsilon^2}2c_n^{-2}\\) would not be summable. So infinitely many \\(c_ne_n\\) lie in the neighbourhood. The other five topologies are coarser.\n\n(b) A basic \\(\\sigma\\)-weak neighbourhood of \\(0\\) is \\(\\{x:|\\omega_j(x)|<\\varepsilon,\\ j\\le m\\}\\) with \\(\\omega_j\\in B(H)_*\\). By Proposition 6.3(c) each \\(\\omega_j\\) is a combination \\(\\sum_lc_{jl}\\omega_{jl}\\) of positive normal functionals; put \\(\\omega=\\sum_{j,l}|c_{jl}|\\omega_{jl}\\). Then \\(|\\omega_j(e_n)|\\le\\omega(e_n)\\) and \\(\\sum_n\\omega(e_n)<\\infty\\). As in (a), \\(c_n\\omega(e_n)<\\varepsilon\\) for infinitely many \\(n\\).\n\n(c) If \\(\\sum c_n^{-2}<\\infty\\), the vector \\(\\xi=\\sum_nc_n^{-1}\\xi_n\\) exists and \\(\\|c_ne_n\\xi\\|=c_nc_n^{-1}=1\\) for all \\(n\\); so the strong neighbourhood \\(\\{x:\\|x\\xi\\|<1\\}\\) of \\(0\\) misses \\(A\\). If \\(\\sum c_n^{-1}<\\infty\\), the vector \\(\\xi=\\sum_nc_n^{-1/2}\\xi_n\\) exists and \\(\\langle c_ne_n\\xi,\\xi\\rangle=1\\) for all \\(n\\).\n\n(d) A weakly convergent sequence \\((x_k)\\) is bounded. Indeed, for each \\(\\xi\\) the functionals \\(\\eta\\mapsto\\langle\\eta,x_k\\xi\\rangle\\) are bounded at each point, so \\(\\sup_k\\|x_k\\xi\\|<\\infty\\) by uniform boundedness; applying uniform boundedness again gives \\(\\sup_k\\|x_k\\|<\\infty\\). But \\(\\|c_ne_n\\|=c_n\\to\\infty\\).\n\n(e) Take \\(c_n=\\sqrt n\\). By (a), \\(0\\) lies in the closure of \\(A\\) for each of the six topologies. If one of them had a countable neighbourhood base at \\(0\\), some sequence \\((a_k)\\) in \\(A\\) would converge to \\(0\\) in it, hence weakly. A sequence in \\(A\\) that takes some value \\(c_me_m\\ne0\\) infinitely often cannot converge to \\(0\\), since the topology is Hausdorff. So \\((a_k)\\) has a subsequence of the form \\((c_{n_k}e_{n_k})\\) with \\(n_k\\) strictly increasing, which converges weakly to \\(0\\). This contradicts (d). \\(\\square\\)\n\nWith \\(c_n=n\\), part (b) still gives an accumulation point for the \\(\\sigma\\)-weak and weak topologies, but part (c) shows that \\(\\sqrt n\\) cannot be replaced by \\(n\\) for the strong-type topologies.\n\nThe comparison just proved explains the choice \\(\\sqrt n\\,e_n\\): replacing it by \\(ne_n\\) retains weak-type accumulation but loses strong-type accumulation.\n\n**Definition 10.3.** A *state* of \\(B(H)\\) is a positive linear functional \\(\\varphi\\), that is \\(\\varphi(x^*x)\\ge0\\) for all \\(x\\), with \\(\\varphi(1)=1\\). The states form a convex set \\(S(H)\\). A *pure state* is an extreme point of \\(S(H)\\). A *vector state* is a state \\(\\omega_\\xi\\) with \\(\\|\\xi\\|=1\\); let \\(V\\) be the set of vector states and \\(\\overline V\\) its closure in the weak\\(^*\\) topology \\(\\sigma(B(H)^*,B(H))\\). Let \\(S_0(H)\\) be the set of states that vanish on \\(K(H)\\).\n\n**Lemma 10.4.** (a) Every positive linear functional \\(\\varphi\\) on \\(B(H)\\) is bounded, with \\(\\|\\varphi\\|=\\varphi(1)\\), and hermitian: \\(\\varphi(x^*)=\\overline{\\varphi(x)}\\). (b) \\(S(H)\\) is weak\\(^*\\) compact, and \\(S_0(H)\\) is a weak\\(^*\\) closed convex subset of it.\n\n**Proof.** (a) The form \\((x,y)\\mapsto\\varphi(y^*x)\\) on \\(B(H)\\) is sesquilinear and positive, so it is hermitian by Lemma 1.2, which gives \\(\\varphi(x^*)=\\overline{\\varphi(x)}\\) (take \\(y=1\\)), and Background 1(d) gives \\(|\\varphi(y^*x)|^2\\le\\varphi(x^*x)\\varphi(y^*y)\\). Since \\(x^*x\\le\\|x\\|^21\\), \\(\\varphi(x^*x)\\le\\|x\\|^2\\varphi(1)\\). With \\(y=1\\): \\(|\\varphi(x)|^2\\le\\varphi(1)\\varphi(x^*x)\\le\\varphi(1)^2\\|x\\|^2\\). And \\(\\varphi(1)\\le\\|\\varphi\\|\\). (b) By (a), \\(S(H)\\) lies in the unit ball of \\(B(H)^*\\) and is defined by weak\\(^*\\) closed conditions; apply Banach–Alaoglu. \\(S_0(H)\\) is cut out by the further conditions \\(\\varphi(x)=0\\), \\(x\\in K(H)\\). \\(\\square\\)\n\n**Proposition 10.5** (vector states).\n\n(a) Every pure state of \\(B(H)\\) lies in \\(\\overline V\\).\n\n(b) Let \\(H\\) be infinite-dimensional. Then \\(S_0(H)\\cap\\overline V\\ne\\emptyset\\). If \\(f\\in\\overline V\\cap S_0(H)\\) and \\(L\\subseteq H\\) is a finite-dimensional subspace, then \\(f\\) lies in the weak\\(^*\\) closure of \\(\\{\\omega_\\eta:\\eta\\in L^\\perp,\\ \\|\\eta\\|=1\\}\\).\n\n(c) \\(\\overline V\\cap S_0(H)\\) is convex.\n\n(d) \\(S_0(H)\\subseteq\\overline V\\): every state that vanishes on the compact operators is a weak\\(^*\\) limit of vector states.\n\n**Proof.** (a) First, the weak\\(^*\\) closed convex hull \\(C\\) of \\(V\\) is \\(S(H)\\). Otherwise take \\(\\varphi\\in S(H)\\setminus C\\). The space \\(B(H)^*\\) with the weak\\(^*\\) topology is locally convex, and its continuous linear functionals are the evaluations at elements of \\(B(H)\\) (Background 5(b)). By Background 5(a), applied to the compact set \\(\\{\\varphi\\}\\) and the closed convex set \\(C\\), there are \\(x\\in B(H)\\) and \\(\\gamma\\in\\mathbb R\\) with \\(\\operatorname{Re}\\varphi(x)>\\gamma\\ge\\operatorname{Re}\\psi(x)\\) for all \\(\\psi\\in C\\). States are hermitian (Lemma 10.4), so with \\(h=\\frac12(x+x^*)\\) we have \\(\\operatorname{Re}\\psi(x)=\\psi(h)\\) for every state. In particular \\(\\langle h\\xi,\\xi\\rangle\\le\\gamma\\) for every unit vector \\(\\xi\\), that is \\(h\\le\\gamma1\\), and so \\(\\varphi(h)\\le\\gamma\\) by positivity. This contradicts \\(\\varphi(h)=\\operatorname{Re}\\varphi(x)>\\gamma\\). Now \\(\\overline V\\) is weak\\(^*\\) compact, as a closed subset of \\(S(H)\\), and its closed convex hull \\(S(H)\\) is compact. By Milman's theorem (Background 5(g)) every extreme point of \\(S(H)\\) lies in \\(\\overline V\\).\n\n(b) Let \\((\\xi_n)\\) be an orthonormal sequence. The states \\(\\omega_{\\xi_n}\\) have a weak\\(^*\\) cluster point \\(f\\in\\overline V\\), by compactness. For \\(x\\in K(H)\\), \\(\\omega_{\\xi_n}(x)=\\langle x\\xi_n,\\xi_n\\rangle\\to0\\) by Lemma 2.1, so \\(f(x)=0\\) and \\(f\\in S_0(H)\\). For the second statement let \\(f=\\lim_\\alpha\\omega_{\\xi_\\alpha}\\) with unit vectors \\(\\xi_\\alpha\\), and let \\(p\\) be the projection onto \\(L\\). Since \\(p\\) has finite rank, \\(\\|p\\xi_\\alpha\\|^2=\\omega_{\\xi_\\alpha}(p)\\to f(p)=0\\). So eventually \\((1-p)\\xi_\\alpha\\ne0\\), and \\(\\eta_\\alpha=(1-p)\\xi_\\alpha/\\|(1-p)\\xi_\\alpha\\|\\) is a unit vector in \\(L^\\perp\\) with \\(\\|\\xi_\\alpha-\\eta_\\alpha\\|\\to0\\). Since \\(|\\omega_\\xi(x)-\\omega_\\eta(x)|\\le\\|x\\|(\\|\\xi\\|+\\|\\eta\\|)\\|\\xi-\\eta\\|\\), \\(\\omega_{\\eta_\\alpha}\\to f\\).\n\n(c) Let \\(f,g\\in\\overline V\\cap S_0(H)\\) and \\(0<\\lambda<1\\). A basic weak\\(^*\\) neighbourhood of \\(\\lambda f+(1-\\lambda)g\\) is given by finitely many operators \\(x_1,\\dots,x_n\\) and \\(\\varepsilon>0\\); we may assume \\(x_1=1\\), since adding an operator only shrinks the neighbourhood. Choose a unit vector \\(\\xi\\) with \\(|f(x_j)-\\omega_\\xi(x_j)|<\\varepsilon\\) for all \\(j\\). Apply (b) to \\(g\\) and \\(L=\\operatorname{span}\\{x_j\\xi,x_j^*\\xi:j\\le n\\}\\): there is a unit vector \\(\\eta\\in L^\\perp\\) with \\(|g(x_j)-\\omega_\\eta(x_j)|<\\varepsilon\\) for all \\(j\\). Put \\(\\zeta=\\lambda^{1/2}\\xi+(1-\\lambda)^{1/2}\\eta\\). Since \\(\\eta\\perp x_1\\xi=\\xi\\), \\(\\|\\zeta\\|=1\\). Since \\(\\langle x_j\\xi,\\eta\\rangle=0\\) and \\(\\langle x_j\\eta,\\xi\\rangle=\\langle\\eta,x_j^*\\xi\\rangle=0\\), we get \\(\\omega_\\zeta(x_j)=\\lambda\\omega_\\xi(x_j)+(1-\\lambda)\\omega_\\eta(x_j)\\), which is within \\(\\varepsilon\\) of \\(\\lambda f(x_j)+(1-\\lambda)g(x_j)\\). So \\(\\lambda f+(1-\\lambda)g\\in\\overline V\\), and it lies in \\(S_0(H)\\) because \\(S_0(H)\\) is convex.\n\n(d) If \\(H\\) is finite-dimensional, \\(S_0(H)=\\emptyset\\). Otherwise \\(S_0(H)\\) is a nonempty weak\\(^*\\) compact convex set, by (b) and Lemma 10.4. It is a face of \\(S(H)\\): if \\(f\\in S_0(H)\\) and \\(f=\\lambda\\psi_1+(1-\\lambda)\\psi_2\\) with \\(\\psi_i\\in S(H)\\) and \\(0<\\lambda<1\\), then \\(\\psi_1(k)=\\psi_2(k)=0\\) for every positive compact \\(k\\), because \\(0=f(k)\\) is a combination of two nonnegative numbers with positive coefficients; every compact operator is a linear combination of positive compact operators (Theorem 2.3(e) applied to \\(\\operatorname{Re}k\\) and \\(\\operatorname{Im}k\\)); so \\(\\psi_1,\\psi_2\\in S_0(H)\\). Hence every extreme point of \\(S_0(H)\\) is an extreme point of \\(S(H)\\), a pure state, and lies in \\(\\overline V\\) by (a). So the extreme points of \\(S_0(H)\\) lie in \\(\\overline V\\cap S_0(H)\\), which is weak\\(^*\\) closed and, by (c), convex. By the Krein–Milman theorem, \\(S_0(H)\\subseteq\\overline V\\cap S_0(H)\\). \\(\\square\\)\n\nThe states in \\(S_0(H)\\) are the *singular* states: by Theorem 6.2(b) a normal state is determined by its restriction to \\(K(H)\\), so no singular state is normal. Part (d) shows that even so, each of them is a limit of vector states.\n\n## 11. Self-adjoint operators modulo small ideals\n\nIn this section \\(h\\) and \\(k\\) denote bounded self-adjoint operators. We use the spectral theorem of Background 3(b): \\(E\\) is the spectral measure of \\(h\\), and \\(\\mu_\\xi(\\Delta)=\\|E(\\Delta)\\xi\\|^2\\) is the spectral measure of a vector \\(\\xi\\). Lebesgue measure on \\(\\mathbb R\\) is written \\(d\\lambda\\), and \"null set\" means Lebesgue null set. \n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-12",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "The Weyl–von Neumann theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "full_conditions_and_proof": "### The Weyl–von Neumann theorem\n\n**Lemma 11.1.** Let \\(h\\in B(H)_{\\mathrm{sa}}\\), \\(\\xi\\in H\\) and \\(m\\ge1\\). There is a projection \\(p\\) of rank at most \\(m\\) with \\(p\\xi=\\xi\\) and \\(\\|(1-p)hp\\|_2\\le\\|h\\|\\,m^{-1/2}\\).\n\n**Proof.** If \\(h=0\\), let \\(p\\) be the projection onto \\(\\mathbb C\\xi\\). Otherwise put \\(r=\\|h\\|\\) and \\(\\delta=2r/m\\), and divide \\([-r,r]\\) into the intervals \\(\\Delta_j=[-r+(j-1)\\delta,-r+j\\delta)\\), \\(j<m\\), and \\(\\Delta_m=[r-\\delta,r]\\), with midpoints \\(c_j\\). Let \\(e_j=E(\\Delta_j)\\). These are mutually orthogonal projections that commute with \\(h\\), \\(\\sum_je_j=E([-r,r])=1\\), and \\(\\|(h-c_j)e_j\\|\\le\\delta/2\\) by the functional calculus. Let \\(\\xi_j=e_j\\xi\\), let \\(p_j\\) be the projection onto \\(\\mathbb C\\xi_j\\) (zero if \\(\\xi_j=0\\)), and \\(p=\\sum_jp_j\\). Since \\(p_j\\le e_j\\), the \\(p_j\\) are mutually orthogonal, \\(p\\) is a projection of rank at most \\(m\\), \\(p_j\\xi=p_je_j\\xi=\\xi_j\\), and \\(p\\xi=\\sum_j\\xi_j=\\xi\\). Also \\(e_jp=p_j=pe_j\\), so \\(p\\) commutes with every \\(e_j\\).\n\nPut \\(y_j=(1-p)hp_j\\). Since \\((1-p)p_j=0\\), \\(y_j=(1-p)(h-c_j)e_jp_j=e_j(1-p)(h-c_j)p_j\\). So \\(y_j\\) has rank at most one, maps into \\(e_jH\\), and \\(\\|y_j\\|\\le\\delta/2\\). For a rank-one operator the Hilbert–Schmidt norm equals the norm (Theorem 3.3(b)), so \\(\\|y_j\\|_2\\le\\delta/2\\). The ranges of the \\(y_j\\) are mutually orthogonal, so for every vector \\(\\zeta\\), \\(\\|(1-p)hp\\,\\zeta\\|^2=\\sum_j\\|y_j\\zeta\\|^2\\). Summing over an orthonormal basis,\n\\[\n\\begin{gathered}\n\\|(1-p)hp\\|_2^2\\\\\n=\\sum_j\\|y_j\\|_2^2\\\\\n\\le m\\,\\frac{\\delta^2}4\\\\\n=\\frac{r^2}m .\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Theorem 11.2** (Weyl–von Neumann). Let \\(H\\) be separable, \\(h\\in B(H)_{\\mathrm{sa}}\\) and \\(\\varepsilon>0\\). Then \\(h=k+a\\), where \\(a\\) is a self-adjoint Hilbert–Schmidt operator with \\(\\|a\\|_2<\\varepsilon\\) and \\(k\\) is a self-adjoint operator for which \\(H\\) has an orthonormal basis of eigenvectors.\n\n**Proof.** Let \\((\\zeta_n)_{n\\ge1}\\) be a dense sequence in \\(H\\). We construct mutually orthogonal finite-rank projections \\(q_1,q_2,\\dots\\) and self-adjoint Hilbert–Schmidt operators \\(a_1,a_2,\\dots\\) with \\(\\|a_n\\|_2\\le\\varepsilon2^{-n-1}\\), such that, with \\(Q_n=q_1+\\dots+q_n\\) and \\(Q_0=0\\):\n\n(i) \\(\\zeta_n\\in Q_nH\\);\n\n(ii) \\[\n\\begin{gathered}\nk_n:\\\\\n=h-(a_1+\\dots+a_n)\\\\\n=\\sum_{j\\le n}q_jhq_j+(1-Q_n)h(1-Q_n).\n\\end{gathered}\n\\]\n\nSuppose \\(q_1,\\dots,q_n\\) and \\(a_1,\\dots,a_n\\) are constructed; for \\(n=0\\) nothing is given and (ii) reads \\(k_0=h\\). Apply Lemma 11.1 in the Hilbert space \\((1-Q_n)H\\) to the self-adjoint operator \\((1-Q_n)h(1-Q_n)\\), of norm at most \\(\\|h\\|\\), to the vector \\((1-Q_n)\\zeta_{n+1}\\), and to an \\(m\\) so large that \\(\\|h\\|m^{-1/2}\\le\\varepsilon2^{-n-3}\\). This gives a finite-rank projection \\(q=q_{n+1}\\le1-Q_n\\) with \\((1-Q_n)\\zeta_{n+1}\\in qH\\) and, with \\(Q_{n+1}=Q_n+q\\), \\(\\|(1-Q_{n+1})hq\\|_2\\le\\varepsilon2^{-n-3}\\). Put\n\\[\na_{n+1}=(1-Q_{n+1})hq+qh(1-Q_{n+1}).\n\\]\nIt is self-adjoint, and \\(\\|a_{n+1}\\|_2\\le2\\|(1-Q_{n+1})hq\\|_2\\le\\varepsilon2^{-n-2}\\) by Theorem 3.3(a). Condition (i) holds because \\(\\zeta_{n+1}=Q_n\\zeta_{n+1}+(1-Q_n)\\zeta_{n+1}\\in Q_{n+1}H\\). For (ii) write \\(1-Q_n=q+s\\) with \\(s=1-Q_{n+1}\\). Then \\((1-Q_n)h(1-Q_n)=qhq+qhs+shq+shs\\), so \\(k_{n+1}=k_n-a_{n+1}=\\sum_{j\\le n+1}q_jhq_j+shs\\), which is (ii) for \\(n+1\\).\n\nSince \\(\\mathcal L^2(H)\\) is complete and \\(\\sum\\|a_n\\|_2\\le\\varepsilon/2\\), the series \\(a=\\sum_na_n\\) converges in \\(\\|\\cdot\\|_2\\), hence in norm; \\(a\\) is self-adjoint and \\(\\|a\\|_2\\le\\varepsilon/2<\\varepsilon\\). Put \\(k=h-a=\\lim_nk_n\\), a norm limit. For \\(n\\ge j\\), (ii) gives \\(q_jk_n=q_jhq_j=k_nq_j\\), so \\(q_jk=kq_j\\) for every \\(j\\). The projections \\(Q_n\\) increase, and their ranges contain all \\(\\zeta_n\\), which are dense; so \\(Q_n\\xi\\to\\xi\\) for every \\(\\xi\\), and \\(H\\) is the orthogonal sum of the finite-dimensional subspaces \\(q_jH\\). Each \\(q_jH\\) is invariant under the self-adjoint \\(k\\), so it has an orthonormal basis of eigenvectors of \\(k\\). Together they form an orthonormal basis of \\(H\\). \\(\\square\\)\n\n**Example 11.3** (separability cannot be dropped). Let \\(\\Gamma\\) be an uncountable set, \\(H=\\ell^2(\\Gamma;L^2(0,1))\\), and let \\(h\\) act on each coordinate as the operator \\(m\\) of multiplication by the variable, \\((m g)(t)=tg(t)\\). Then \\(h\\ne k+a\\) for every Hilbert–Schmidt operator \\(a\\) and every operator \\(k\\) for which \\(H\\) has an orthonormal basis of eigenvectors. Suppose otherwise. Since \\(a\\) is compact, the closures of the ranges of \\(a\\) and \\(a^*\\) are separable, and so is the closed span \\(L\\) of the vectors \\(h^j v\\), \\(j\\ge0\\), with \\(v\\) in these ranges. \\(L\\) is invariant under \\(h\\), hence reduces \\(h\\). It contains the ranges of \\(a\\) and \\(a^*\\), so \\(L^\\perp\\) lies in the kernels of \\(a^*\\) and \\(a\\): both \\(a\\) and \\(a^*\\) vanish on \\(L^\\perp\\). Every vector of \\(H\\) has countably many nonzero coordinates, so a countable dense subset of \\(L\\) is supported in a countable set \\(\\Gamma_0\\subseteq\\Gamma\\), and \\(L\\) lies in the coordinates \\(\\Gamma_0\\). For \\(\\gamma\\notin\\Gamma_0\\), the subspace \\(X_\\gamma\\) of vectors supported at \\(\\gamma\\) lies in \\(L^\\perp\\) and reduces \\(h\\), \\(a\\) (which is \\(0\\) there) and hence \\(k\\). The projection \\(P\\) onto \\(X_\\gamma\\) commutes with \\(k\\), so it maps each eigenvector of \\(k\\) to an eigenvector or to \\(0\\). These images span a dense subspace of \\(X_\\gamma\\), so \\(k\\) has an eigenvector in \\(X_\\gamma\\). But on \\(X_\\gamma\\), \\(k=h\\) acts as \\(m\\), which has no eigenvectors: \\((t-c)g(t)=0\\) almost everywhere forces \\(g=0\\). This contradiction shows that Theorem 11.2 needs separability, even for Hilbert–Schmidt perturbations of arbitrary size.\n\n**Remark 11.4.** The Hilbert–Schmidt norm enters only through Lemma 11.1: a block of rank at most \\(m\\) whose pieces have norm at most \\(\\|h\\|/m\\) has Hilbert–Schmidt norm at most \\(\\|h\\|m^{-1/2}\\), which tends to \\(0\\). The trace norm of the same block is only bounded by \\(\\|h\\|\\). Theorem 11.12 and Remark 11.17 show that this is not a weakness of the proof: the theorem is false for trace-class perturbations.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-LT-13",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Absolutely continuous and singular parts",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      },
      "full_conditions_and_proof": "### Absolutely continuous and singular parts\n\n**Definition 11.5.** For \\(h\\in B(H)_{\\mathrm{sa}}\\) let \\(H_{\\mathrm{ac}}(h)\\) be the set of vectors \\(\\xi\\) whose spectral measure \\(\\mu_\\xi\\) vanishes on null sets, and \\(H_{\\mathrm s}(h)\\) the set of vectors whose spectral measure is concentrated on a null Borel set. The operator \\(h\\) is *absolutely continuous* if \\(H_{\\mathrm{ac}}(h)=H\\).\n\n**Proposition 11.6.** \\(H_{\\mathrm{ac}}(h)\\) and \\(H_{\\mathrm s}(h)\\) are closed subspaces, \\(H_{\\mathrm s}(h)=H_{\\mathrm{ac}}(h)^\\perp\\), and both are invariant under every \\(f(h)\\), \\(f\\) a bounded Borel function; in particular they reduce \\(h\\). Every eigenvector of \\(h\\) lies in \\(H_{\\mathrm s}(h)\\). We write \\(P_{\\mathrm{ac}}(h)\\) for the projection onto \\(H_{\\mathrm{ac}}(h)\\), and \\(h_{\\mathrm{ac}}\\) and \\(h_{\\mathrm s}\\) for the restrictions of \\(h\\) to the two subspaces; the spectral measures of \\(h_{\\mathrm{ac}}\\) are the \\(\\mu_\\xi\\), \\(\\xi\\in H_{\\mathrm{ac}}(h)\\).\n\n**Proof.** If \\(N\\) is a null set and \\(\\mu_\\xi(N)=\\mu_\\eta(N)=0\\), then \\(\\|E(N)(\\xi+\\eta)\\|\\le\\|E(N)\\xi\\|+\\|E(N)\\eta\\|=0\\); scalar multiples are clear, and \\(\\|E(N)\\xi\\|=\\lim\\|E(N)\\xi_k\\|\\) when \\(\\xi_k\\to\\xi\\). So \\(H_{\\mathrm{ac}}(h)\\) is a closed subspace. For bounded Borel \\(f\\), \\(E(N)\\) commutes with \\(f(h)\\), so \\(\\mu_{f(h)\\xi}(N)=\\|f(h)E(N)\\xi\\|^2\\le(\\sup|f|)^2\\mu_\\xi(N)\\); hence \\(H_{\\mathrm{ac}}(h)\\) is invariant.\n\nIf \\(\\mu_\\eta\\) is concentrated on a null set \\(N\\), that is \\(E(\\mathbb R\\setminus N)\\eta=0\\), then for \\(\\xi\\in H_{\\mathrm{ac}}(h)\\), \\(\\langle\\eta,\\xi\\rangle=\\langle E(N)\\eta,\\xi\\rangle=\\langle\\eta,E(N)\\xi\\rangle=0\\). Conversely let \\(\\eta\\perp H_{\\mathrm{ac}}(h)\\). Let \\(\\alpha\\) be the supremum of \\(\\mu_\\eta(N)\\) over null Borel sets \\(N\\), and choose null sets \\(N_k\\) with \\(\\mu_\\eta(N_k)\\to\\alpha\\). Their union \\(N\\) is null and \\(\\mu_\\eta(N)=\\alpha\\), so \\(\\mu_\\eta(N'\\setminus N)=0\\) for every null \\(N'\\). The vector \\(\\xi=E(\\mathbb R\\setminus N)\\eta\\) has \\(\\mu_\\xi(\\Delta)=\\mu_\\eta(\\Delta\\setminus N)\\), which vanishes on null sets; so \\(\\xi\\in H_{\\mathrm{ac}}(h)\\), and \\(0=\\langle\\eta,\\xi\\rangle=\\|E(\\mathbb R\\setminus N)\\eta\\|^2\\). So \\(\\mu_\\eta\\) is concentrated on \\(N\\). This proves \\(H_{\\mathrm s}(h)=H_{\\mathrm{ac}}(h)^\\perp\\), which is invariant under the \\(f(h)\\) because the family \\(\\{f(h)\\}\\) is closed under adjoints. If \\(h\\xi=c\\xi\\), then \\(\\langle f(h)\\xi,\\xi\\rangle=f(c)\\|\\xi\\|^2\\) for continuous \\(f\\), so \\(\\mu_\\xi\\) is the point mass \\(\\|\\xi\\|^2\\delta_c\\) (two finite measures on a compact interval with the same integrals of continuous functions agree on open intervals by monotone approximation, hence everywhere by Dynkin's lemma). \\(\\square\\)\n\n**Lemma 11.7** (cyclic subspaces). For \\(\\xi\\in H\\) let \\(Z(\\xi)\\) be the closure of \\(\\{f(h)\\xi:f\\ \\text{bounded Borel}\\}\\). Then \\(Z(\\xi)\\) reduces \\(h\\), and there is a unitary \\(U:Z(\\xi)\\to L^2(\\mathbb R,\\mu_\\xi)\\) with \\(U(f(h)\\xi)=f\\). Under \\(U\\), each \\(g(h)\\) restricted to \\(Z(\\xi)\\) becomes multiplication by \\(g\\); in particular \\(h\\) becomes multiplication by the variable \\(\\lambda\\).\n\n**Proof.** \\(\\|f(h)\\xi\\|^2=\\langle|f|^2(h)\\xi,\\xi\\rangle=\\int|f|^2d\\mu_\\xi\\). So \\(f(h)\\xi\\mapsto f\\) is well defined and isometric from a dense subspace of \\(Z(\\xi)\\) onto the bounded Borel functions, which are dense in \\(L^2(\\mu_\\xi)\\); it extends to a unitary \\(U\\). Since \\(g(h)f(h)\\xi=(gf)(h)\\xi\\), \\(Z(\\xi)\\) is invariant under each \\(g(h)\\), hence reducing, and \\(Ug(h)f(h)\\xi=gf\\). \\(\\square\\)\n\n**Proposition 11.8** (the model). The operator \\(h_{\\mathrm{ac}}\\) is unitarily equivalent to a direct sum \\(\\bigoplus_{i\\in I}m_{S_i}\\), where each \\(S_i\\subseteq[-\\|h\\|,\\|h\\|]\\) is a Borel set and \\(m_S\\) is multiplication by \\(\\lambda\\) on \\(L^2(S,d\\lambda)\\). If \\(H\\) is separable, \\(I\\) is countable. Conversely, every such direct sum is absolutely continuous. So \\(h\\) is absolutely continuous exactly when it is unitarily equivalent to such a direct sum.\n\n**Proof.** The spectral measure of \\(m_S\\) is \\(\\Delta\\mapsto\\) multiplication by \\(1_\\Delta\\): this is a projection-valued measure whose integral of \\(\\lambda\\) is \\(m_S\\), and the spectral measure is unique. So the spectral measure of a vector \\(g=(g_i)\\) of the direct sum is \\(\\Delta\\mapsto\\sum_i\\int_{\\Delta\\cap S_i}|g_i|^2d\\lambda\\), which vanishes on null sets. Conversely, by Zorn's lemma choose a maximal family \\((\\xi_i)_{i\\in I}\\) of nonzero vectors of \\(H_{\\mathrm{ac}}(h)\\) whose cyclic subspaces \\(Z(\\xi_i)\\) are mutually orthogonal. If a nonzero \\(\\zeta\\in H_{\\mathrm{ac}}(h)\\) were orthogonal to all \\(Z(\\xi_i)\\), then \\(\\langle f(h)\\zeta,g(h)\\xi_i\\rangle=\\langle\\zeta,(\\bar fg)(h)\\xi_i\\rangle=0\\), so \\(Z(\\zeta)\\perp Z(\\xi_i)\\) for all \\(i\\), contradicting maximality. So \\(H_{\\mathrm{ac}}(h)=\\bigoplus_iZ(\\xi_i)\\), and \\(I\\) is countable if \\(H\\) is separable. By the Radon–Nikodym theorem \\(\\mu_{\\xi_i}=\\rho_i\\,d\\lambda\\) with a Borel function \\(\\rho_i\\ge0\\), which we may take to vanish outside \\([-\\|h\\|,\\|h\\|]\\). Let \\(S_i=\\{\\rho_i>0\\}\\). The map \\(g\\mapsto g\\rho_i^{1/2}\\) is a unitary of \\(L^2(\\mu_{\\xi_i})\\) onto \\(L^2(S_i,d\\lambda)\\) that commutes with multiplication by \\(\\lambda\\). Combined with Lemma 11.7, this gives the unitary equivalence. \\(\\square\\)\n\n**Proposition 11.9** (multiplication operators). Let \\((X,\\mu)\\) be a \\(\\sigma\\)-finite measure space, \\(f:X\\to\\mathbb R\\) a bounded measurable function, and \\(m_f\\) multiplication by \\(f\\) on \\(L^2(X,\\mu)\\).\n\n(a) \\(m_f\\) is absolutely continuous exactly when \\(\\mu(f^{-1}(N))=0\\) for every null Borel set \\(N\\subseteq\\mathbb R\\).\n\n(b) Let \\(X\\) be a bounded open interval with Lebesgue measure and \\(f\\) continuously differentiable and bounded. Then \\(m_f\\) is absolutely continuous exactly when \\(\\{t:f'(t)=0\\}\\) is a null set.\n\n**Proof.** (a) As in Proposition 11.8, the spectral measure of \\(m_f\\) is \\(\\Delta\\mapsto\\) multiplication by \\(1_{f^{-1}(\\Delta)}\\), so \\(\\mu_g(\\Delta)=\\int_{f^{-1}(\\Delta)}|g|^2d\\mu\\). If every \\(f^{-1}(N)\\) is \\(\\mu\\)-null, every \\(\\mu_g\\) vanishes on null sets. If \\(\\mu(f^{-1}(N))>0\\) for some null \\(N\\), choose \\(A\\subseteq f^{-1}(N)\\) with \\(0<\\mu(A)<\\infty\\); then \\(g=1_A\\) has \\(\\mu_g(N)=\\mu(A)>0\\).\n\n(b) Let \\(Z=\\{f'=0\\}\\). If \\(Z\\) is null: the open set \\(\\{f'\\ne0\\}\\) is a countable union of open intervals \\(J\\), on each of which \\(f\\) is strictly monotone with a continuously differentiable inverse \\(g_J\\) defined on the interval \\(f(J)\\). The inverse derivative is \\(g_J'(f(t))=1/f'(t)\\): this follows by taking difference quotients in \\(f(g_J(y))=y\\), and continuity follows from continuity and nonvanishing of \\(f'\\). On compact subintervals its derivative is bounded, so the mean value theorem makes \\(g_J\\) Lipschitz. Lemma 0.1(b) then shows that it maps null sets to null sets, and \\(f^{-1}(N)\\cap J=g_J(N\\cap f(J))\\) is null for every null \\(N\\). Hence \\(f^{-1}(N)\\subseteq Z\\cup\\bigcup_J(f^{-1}(N)\\cap J)\\) is null, and (a) applies. Conversely suppose \\(Z\\) has positive measure. Then \\(f(Z)\\) is null. Indeed, let \\([c,d]\\subseteq X\\) and \\(\\varepsilon>0\\); by uniform continuity of \\(f'\\) on \\([c,d]\\), divide \\([c,d]\\) into finitely many intervals on each of which \\(f'\\) varies by less than \\(\\varepsilon\\). On each such interval that meets \\(Z\\) we have \\(|f'|<\\varepsilon\\), so by the mean value theorem its image is an interval of length at most \\(\\varepsilon\\) times its length. Hence \\(f(Z\\cap[c,d])\\) is covered by intervals of total length at most \\(\\varepsilon(d-c)\\); letting \\(\\varepsilon\\to0\\) and exhausting \\(X\\) by countably many \\([c,d]\\), \\(f(Z)\\) is null. Choose a null Borel set \\(N\\supseteq f(Z)\\). Then \\(f^{-1}(N)\\supseteq Z\\) has positive measure, and \\(m_f\\) is not absolutely continuous by (a). \\(\\square\\)\n\nThe Cantor function used in the next example has an elementary construction. Start with \\(c_0(t)=t\\) on \\([0,1]\\), and define \\(c_{n+1}(t)=c_n(3t)/2\\) on \\([0,1/3]\\), \\(c_{n+1}(t)=1/2\\) on \\([1/3,2/3]\\), and \\(c_{n+1}(t)=1/2+c_n(3t-2)/2\\) on \\([2/3,1]\\). Induction gives continuity, monotonicity and endpoint values zero and one. Moreover \\(\\|c_{n+1}-c_n\\|_\\infty\\leq2^{-n}\\|c_1-c_0\\|_\\infty\\), so the sequence is uniformly Cauchy and has a continuous nondecreasing limit \\(c\\). On every middle-third interval removed at stage \\(j\\), all later functions have the same constant dyadic value, hence so does \\(c\\). The remaining Cantor set is covered at stage \\(n\\) by \\(2^n\\) intervals of length \\(3^{-n}\\), so it is null by Lemma 0.1(b). There are only countably many removed intervals, and their union has full measure in \\((0,1)\\).\n\nFor example, multiplication by \\(t\\) on \\(L^2(0,1)\\) is absolutely continuous. At the other extreme, let \\(c\\) be the Cantor function on \\((0,1)\\). It is continuous and nondecreasing, and it is constant on each of the countably many open intervals removed in the construction of the Cantor set, whose total length is \\(1\\). So for almost every \\(t\\), \\(c(t)\\) lies in the countable set \\(C\\) of values taken on these intervals. Hence \\(L^2(0,1)\\) is the orthogonal sum of the spaces \\(L^2(c^{-1}(v))\\), \\(v\\in C\\), on each of which \\(m_c\\) acts as the scalar \\(v\\): every spectral measure of \\(m_c\\) is a countable sum of point masses, and \\(m_c\\) has an orthonormal basis of eigenvectors, although \\(c\\) is continuous and nondecreasing.\n\n**Lemma 11.10** (square-integrable Fourier transforms). Let \\(\\mu\\) be a finite positive Borel measure on \\(\\mathbb R\\) and \\(\\hat\\mu(s)=\\int e^{is\\lambda}d\\mu(\\lambda)\\).\n\n(a) If \\(\\hat\\mu\\in L^2(\\mathbb R)\\), then \\(\\mu=g\\,d\\lambda\\) with \\(g\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\).\n\n(b) If \\(\\mu=g\\,d\\lambda\\) with \\(g\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\), then \\(\\hat\\mu\\in L^2(\\mathbb R)\\) and \\(\\int|\\hat\\mu|^2ds=2\\pi\\int g^2d\\lambda\\).\n\n**Proof.** (b) \\(\\hat\\mu(s)=\\hat g(-s)\\); apply Plancherel (Background 4(c)). (a) For \\(\\sigma>0\\) let \\(\\rho_\\sigma(\\lambda)=\\int\\gamma_\\sigma(\\lambda-\\nu)\\,d\\mu(\\nu)\\), with the normal density \\(\\gamma_\\sigma\\) of Background 4(c). Then \\(0\\le\\rho_\\sigma\\le(2\\pi\\sigma^2)^{-1/2}\\mu(\\mathbb R)\\) and \\(\\int\\rho_\\sigma=\\mu(\\mathbb R)\\), so \\(\\rho_\\sigma\\in L^1\\cap L^2\\). By Fubini's theorem and the formula for the transform of \\(\\gamma_\\sigma\\),\n\\[\n\\begin{gathered}\n\\hat\\rho_\\sigma(s)\\\\\n=\\int e^{-is\\nu}\\Big(\\int e^{-is(\\lambda-\\nu)}\\gamma_\\sigma(\\lambda-\\nu)\\,d\\lambda\\Big)d\\mu(\\nu)\\\\\n=e^{-\\sigma^2s^2/2}\\,\\hat\\mu(-s).\n\\end{gathered}\n\\]\nAs \\(\\sigma\\to0\\), these functions converge in \\(L^2(ds)\\) to \\(\\hat\\mu(-s)\\), by dominated convergence. Since \\(w\\mapsto(2\\pi)^{-1/2}\\hat w\\) extends to a unitary operator of \\(L^2(\\mathbb R)\\), the functions \\(\\rho_\\sigma\\) converge in \\(L^2(d\\lambda)\\) to some \\(g\\) as \\(\\sigma\\to0\\). Since \\(\\rho_\\sigma\\ge0\\), also \\(g\\ge0\\) almost everywhere (a sequence \\(\\rho_{\\sigma_k}\\) converges to \\(g\\) almost everywhere). For continuous \\(\\varphi\\) with compact support, on the one hand \\(\\int\\varphi\\rho_\\sigma\\,d\\lambda\\to\\int\\varphi g\\,d\\lambda\\). On the other hand \\(\\int\\varphi\\rho_\\sigma\\,d\\lambda=\\int(\\int\\varphi(\\lambda)\\gamma_\\sigma(\\lambda-\\nu)\\,d\\lambda)\\,d\\mu(\\nu)\\), and the inner integral is bounded by \\(\\sup|\\varphi|\\) and tends to \\(\\varphi(\\nu)\\) for every \\(\\nu\\), by continuity of \\(\\varphi\\); so \\(\\int\\varphi\\rho_\\sigma\\,d\\lambda\\to\\int\\varphi\\,d\\mu\\). Hence \\(\\int\\varphi\\,d\\mu=\\int\\varphi g\\,d\\lambda\\). Taking \\(0\\le\\varphi_j\\uparrow1_{(c,d)}\\) and using monotone convergence, \\(\\mu((c,d))=\\int_c^dg\\,d\\lambda\\) for every bounded interval. Letting \\((c,d)\\) increase to \\(\\mathbb R\\) gives \\(\\int g=\\mu(\\mathbb R)<\\infty\\), so \\(g\\in L^1\\), and \\(\\mu=g\\,d\\lambda\\) by Dynkin's lemma. \\(\\square\\)\n\n**Proposition 11.11** (a Fourier criterion). Let \\(u(t)=e^{ith}\\), so that \\(\\langle u(t)\\xi,\\xi\\rangle=\\hat\\mu_\\xi(t)\\). Let \\(D\\) be the set of \\(\\xi\\in H\\) with \\(\\int_{\\mathbb R}|\\langle u(t)\\xi,\\xi\\rangle|^2dt<\\infty\\). Then \\(D\\subseteq H_{\\mathrm{ac}}(h)\\), and the set of \\(\\xi\\in H_{\\mathrm{ac}}(h)\\) whose spectral measure has a bounded density is contained in \\(D\\) and dense in \\(H_{\\mathrm{ac}}(h)\\). Consequently \\(h\\) is absolutely continuous exactly when \\(D\\) is dense in \\(H\\).\n\n**Proof.** \\(\\langle u(t)\\xi,\\xi\\rangle=\\int e^{it\\lambda}d\\mu_\\xi(\\lambda)\\) by Background 3(b). Lemma 11.10(a) gives \\(D\\subseteq H_{\\mathrm{ac}}(h)\\). If \\(\\mu_\\xi\\) has a bounded density, that density lies in \\(L^1\\cap L^2\\), and \\(\\xi\\in D\\) by Lemma 11.10(b). Now let \\(\\xi\\in H_{\\mathrm{ac}}(h)\\) with density \\(\\rho\\), a Borel function, and put \\(\\xi_n=E(\\{\\rho\\le n\\})\\xi\\). Then \\(\\mu_{\\xi_n}(\\Delta)=\\mu_\\xi(\\Delta\\cap\\{\\rho\\le n\\})\\) has the density \\(\\rho1_{\\{\\rho\\le n\\}}\\le n\\), and \\(\\|\\xi-\\xi_n\\|^2=\\mu_\\xi(\\{\\rho>n\\})=\\int_{\\{\\rho>n\\}}\\rho\\,d\\lambda\\to0\\). This proves the density statement. If \\(D\\) is dense in \\(H\\), then so is \\(H_{\\mathrm{ac}}(h)\\supseteq D\\), which is closed; so \\(H_{\\mathrm{ac}}(h)=H\\). The converse follows from the density statement. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-LT-14",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Trace-class perturbations: the Kato–Rosenblum theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-14",
      "proof_locus": {
        "line": 1227,
        "through_line": 1428
      },
      "full_conditions_and_proof": "### Trace-class perturbations: the Kato–Rosenblum theorem\n\nFor \\(a,b\\in B(H)_{\\mathrm{sa}}\\) we write \\(W(t)=W_{a,b}(t)=e^{ita}e^{-itb}\\), a unitary operator. The map \\(t\\mapsto W(t)\\) is differentiable in norm with derivative \\(ie^{ita}(a-b)e^{-itb}\\), so for \\(t,t'\\in\\mathbb R\\) and \\(\\zeta\\in H\\)\n\\[\n\\begin{gathered}\nW(t)\\zeta-W(t')\\zeta\\\\\n=i\\int_{t'}^te^{isa}(a-b)e^{-isb}\\zeta\\,ds .\n\\end{gathered}\n\\tag{11.1}\n\\]\nSince \\(e^{ita}-e^{itb}=(W(t)-W(0))e^{itb}\\), (11.1) with \\(t'=0\\) gives \\(\\|e^{ita}-e^{itb}\\|\\le|t|\\,\\|a-b\\|\\).\n\n**Theorem 11.12** (Kato–Rosenblum). Let \\(h,k\\in B(H)_{\\mathrm{sa}}\\) with \\(h-k\\in\\mathcal L^1(H)\\). Then the strong limit\n\\[\nW_+=\\lim_{t\\to\\infty}e^{ith}e^{-itk}P_{\\mathrm{ac}}(k)\n\\]\nexists. It is a partial isometry with initial space \\(H_{\\mathrm{ac}}(k)\\) and final space \\(H_{\\mathrm{ac}}(h)\\), and \\(hW_+=W_+k\\). In particular \\(h_{\\mathrm{ac}}\\) and \\(k_{\\mathrm{ac}}\\) are unitarily equivalent. The same holds for \\(t\\to-\\infty\\).\n\nThe proof takes the rest of this section. Its argument uses the following elementary stages: an estimate for the rate of convergence is proved first for perturbations for which convergence is easy, in a form that does not involve the limit, and then carried over to all trace-class perturbations by approximation. The operator \\(W_+\\) is called a *wave operator*. Nothing in the proof uses separability of \\(H\\).\n\n**Lemma 11.13** (an \\(L^2\\) estimate). Let \\(k\\in B(H)_{\\mathrm{sa}}\\) and let \\(\\zeta\\in H_{\\mathrm{ac}}(k)\\) have spectral density \\(\\rho\\le M\\) almost everywhere. Then for every \\(\\eta\\in H\\),\n\\[\n\\int_{\\mathbb R}|\\langle e^{-isk}\\zeta,\\eta\\rangle|^2ds\\le2\\pi M\\,\\|\\eta\\|^2 .\n\\]\n\n**Proof.** Let \\(U:Z(\\zeta)\\to L^2(\\mu_\\zeta)\\) be the unitary of Lemma 11.7 for \\(k\\), let \\(\\eta_Z\\) be the projection of \\(\\eta\\) onto \\(Z(\\zeta)\\), and \\(\\gamma=U\\eta_Z\\). Since \\(e^{-isk}\\zeta\\in Z(\\zeta)\\) and \\(Ue^{-isk}\\zeta=e^{-is\\lambda}\\),\n\\[\n\\begin{gathered}\n\\langle e^{-isk}\\zeta,\\eta\\rangle\\\\\n=\\langle e^{-isk}\\zeta,\\eta_Z\\rangle\\\\\n=\\int e^{-is\\lambda}\\,\\overline{\\gamma(\\lambda)}\\,\\rho(\\lambda)\\,d\\lambda\\\\\n=\\hat w(s),\\\\\nw\\\\\n=\\bar\\gamma\\rho .\n\\end{gathered}\n\\]\nNow \\(\\int|w|\\le(\\int|\\gamma|^2\\rho)^{1/2}(\\int\\rho)^{1/2}=\\|\\eta_Z\\|\\,\\|\\zeta\\|\\) and \\[\n\\begin{gathered}\n\\int|w|^2\\\\\n=\\int|\\gamma|^2\\rho^2\\\\\n\\le M\\int|\\gamma|^2\\rho\\\\\n=M\\|\\eta_Z\\|^2\\\\\n\\le M\\|\\eta\\|^2.\n\\end{gathered}\n\\] By Plancherel, \\(\\int|\\hat w|^2ds=2\\pi\\int|w|^2\\le2\\pi M\\|\\eta\\|^2\\). \\(\\square\\)\n\n**Lemma 11.14** (Cook's criterion). Let \\(a,b\\in B(H)_{\\mathrm{sa}}\\), and let \\(D\\subseteq H_{\\mathrm{ac}}(b)\\) be a set whose linear span is dense in \\(H_{\\mathrm{ac}}(b)\\). If \\(\\int_0^\\infty\\|(a-b)e^{-isb}\\zeta\\|\\,ds<\\infty\\) for every \\(\\zeta\\in D\\), then \\(\\lim_{t\\to\\infty}W_{a,b}(t)\\zeta\\) exists for every \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\).\n\n\n**Proof.** For \\(\\zeta\\in D\\), (11.1) gives \\[\n\\begin{gathered}\n\\|W(t)\\zeta-W(t')\\zeta\\|\\\\\n\\le\\int_{t'}^t\\|(a-b)e^{-isb}\\zeta\\|\\,ds\\to0\n\\end{gathered}\n\\] as \\(t,t'\\to\\infty\\). So \\(W(t)\\zeta\\) converges for \\(\\zeta\\) in the span of \\(D\\). For \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\) and \\(\\varepsilon>0\\) choose \\(\\zeta'\\) in the span with \\(\\|\\zeta-\\zeta'\\|<\\varepsilon\\); since \\(W(t)\\) is unitary, \\(\\|W(t)\\zeta-W(t')\\zeta\\|\\le2\\varepsilon+\\|W(t)\\zeta'-W(t')\\zeta'\\|\\). \\(\\square\\)\n\n**Lemma 11.15** (properties of wave operators). Let \\(a,b\\in B(H)_{\\mathrm{sa}}\\), and suppose that \\(\\lim_{t\\to\\infty}W(t)\\zeta\\) exists for every \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\), where \\(W=W_{a,b}\\). Let \\(W_+\\) equal this limit on \\(H_{\\mathrm{ac}}(b)\\) and \\(0\\) on \\(H_{\\mathrm{ac}}(b)^\\perp\\).\n\n(a) \\(\\|W_+\\zeta\\|=\\|\\zeta\\|\\) for \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\), and \\(\\|W_+\\|\\le1\\).\n\n(b) \\(e^{isa}W_+=W_+e^{isb}\\) for all \\(s\\in\\mathbb R\\), and \\(aW_+=W_+b\\).\n\n(c) For \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\), the spectral measure of \\(W_+\\zeta\\) for \\(a\\) equals the spectral measure of \\(\\zeta\\) for \\(b\\). In particular \\(W_+H_{\\mathrm{ac}}(b)\\subseteq H_{\\mathrm{ac}}(a)\\).\n\n(d) Suppose \\(a-b=\\sum_{n=1}^Nc_n\\theta_{f_n,f_n}\\) with real \\(c_n\\) and vectors \\(f_n\\in H\\), and let \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\) have spectral density bounded by \\(M\\). Put \\(\\eta_n(t)=\\int_t^\\infty|\\langle e^{-isb}\\zeta,f_n\\rangle|^2ds\\), finite by Lemma 11.13. Then for every \\(t\\),\n\\[\n\\begin{gathered}\n\\|W_+\\zeta-W(t)\\zeta\\|^2\\\\\n\\le2(2\\pi M)^{1/2}\\\\\n{}\\cdot\\Big(\\sum_n|c_n|\\,\\|f_n\\|^2\\Big)^{1/2}\\\\\n{}\\cdot\\Big(\\sum_n|c_n|\\,\\eta_n(t)\\Big)^{1/2}.\n\\end{gathered}\n\\tag{11.2}\n\\]\n\n**Proof.** (a) \\(\\|W_+\\zeta\\|=\\lim\\|W(t)\\zeta\\|=\\|\\zeta\\|\\). The operator \\(W_+\\) is linear, and it is isometric on \\(H_{\\mathrm{ac}}(b)\\) and zero on its complement.\n\n(b) Let \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\). Then \\(e^{-isb}\\zeta\\in H_{\\mathrm{ac}}(b)\\) by Proposition 11.6, and \\(W(t+s)=e^{isa}W(t)e^{-isb}\\). Letting \\(t\\to\\infty\\) gives \\(W_+\\zeta=e^{isa}W_+e^{-isb}\\zeta\\); replacing \\(\\zeta\\) by \\(e^{isb}\\zeta\\) gives \\(W_+e^{isb}\\zeta=e^{isa}W_+\\zeta\\). On \\(H_{\\mathrm{ac}}(b)^\\perp\\), which is invariant under \\(e^{isb}\\), both sides vanish. For \\(\\zeta\\in H\\), \\(s^{-1}(e^{isa}-1)W_+\\zeta\\to iaW_+\\zeta\\) and \\(W_+s^{-1}(e^{isb}-1)\\zeta\\to iW_+b\\zeta\\) as \\(s\\to0\\), because \\(s^{-1}(e^{isx}-1)\\to ix\\) in norm for bounded \\(x\\). So \\(aW_+=W_+b\\).\n\n(c) By (b), \\(p(a)W_+=W_+p(b)\\) for every polynomial \\(p\\). Let \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\). By (a) and polarization, \\(W_+\\) preserves inner products on \\(H_{\\mathrm{ac}}(b)\\), and \\(p(b)\\zeta\\in H_{\\mathrm{ac}}(b)\\); so\n\\[\n\\begin{gathered}\n\\langle p(a)W_+\\zeta,W_+\\zeta\\rangle\\\\\n=\\langle W_+p(b)\\zeta,W_+\\zeta\\rangle\\\\\n=\\langle p(b)\\zeta,\\zeta\\rangle .\n\\end{gathered}\n\\]\nThus the two spectral measures give the same integral to every polynomial. Both live on \\([-R,R]\\) with \\(R=\\max(\\|a\\|,\\|b\\|)\\). By the Weierstrass approximation theorem (Proposition 15.1(2) of [The Stone–Weierstrass theorem for functions vanishing at infinity](stone-weierstrass-c0.md#oa-fnd-sw-17), specialized to the compact real interval (where the coordinate and its conjugate agree)) they give the same integral to every continuous function on \\([-R,R]\\); by monotone approximation of indicator functions of open intervals and Dynkin's lemma they are equal.\n\n(d) Since \\(\\|W_+\\zeta\\|=\\|\\zeta\\|=\\|W(t)\\zeta\\|\\),\n\\[\n\\begin{gathered}\n\\|W_+\\zeta-W(t)\\zeta\\|^2\\\\\n=2\\operatorname{Re}\\langle W_+\\zeta-W(t)\\zeta,W_+\\zeta\\rangle .\n\\end{gathered}\n\\]\nBy (11.1), with \\(V=a-b\\),\n\\[\n\\begin{gathered}\n\\langle W_+\\zeta-W(t)\\zeta,W_+\\zeta\\rangle\\\\\n=\\lim_{r\\to\\infty}\\,i\\int_t^r\\langle e^{isa}Ve^{-isb}\\zeta,W_+\\zeta\\rangle\\,ds .\n\\end{gathered}\n\\]\nBy (b), \\(e^{-isa}W_+\\zeta=W_+e^{-isb}\\zeta\\), so the integrand is\n\\[\n\\begin{gathered}\n\\langle Ve^{-isb}\\zeta,W_+e^{-isb}\\zeta\\rangle\\\\\n=\\sum_nc_n\\langle e^{-isb}\\zeta,f_n\\rangle\\,\\langle W_+^*f_n,e^{-isb}\\zeta\\rangle .\n\\end{gathered}\n\\]\nBy Lemma 11.13 applied to \\(b\\) and \\(\\eta=W_+^*f_n\\), with \\(\\|W_+^*f_n\\|\\le\\|f_n\\|\\), \\(\\int_{\\mathbb R}|\\langle e^{-isb}\\zeta,W_+^*f_n\\rangle|^2ds\\le2\\pi M\\|f_n\\|^2\\). Let \\(\\eta_n(t)=\\int_t^\\infty|\\langle e^{-isb}\\zeta,f_n\\rangle|^2ds\\), which is finite by Lemma 11.13. The Cauchy–Schwarz inequality, first in \\(s\\) and then in \\(n\\), gives for every \\(r>t\\)\n\\[\n\\begin{gathered}\n\\Big|\\int_t^r\\langle Ve^{-isb}\\zeta,W_+e^{-isb}\\zeta\\rangle\\,ds\\Big|\\\\\n\\le\\sum_n|c_n|\\,\\eta_n(t)^{1/2}\\\\\n{}\\cdot(2\\pi M)^{1/2}\\|f_n\\|\\\\\n\\le(2\\pi M)^{1/2}\\Big(\\sum_n|c_n|\\,\\|f_n\\|^2\\Big)^{1/2}\\\\\n{}\\cdot\\Big(\\sum_n|c_n|\\eta_n(t)\\Big)^{1/2}.\n\\end{gathered}\n\\]\nThe same bound holds for the limit \\(r\\to\\infty\\), and (11.2) follows. \\(\\square\\)\n\nThe point of (11.2) is that its right side involves neither \\(W_+\\) nor \\(a\\). This is what allows passage to a limit in the perturbation.\n\n**Lemma 11.16** (a model for the absolutely continuous part). Let \\(k\\in B(H)_{\\mathrm{sa}}\\). There are a Hilbert space \\(H'\\), an absolutely continuous \\(k'\\in B(H')_{\\mathrm{sa}}\\), a bounded open interval \\(I\\), a set \\(J\\), and a unitary \\(\\Phi\\) of \\(H_{\\mathrm{ac}}(k)\\oplus H'\\) onto the Hilbert space \\(L^2(I;\\ell^2(J))\\) of families \\((g_j)_{j\\in J}\\) in \\(L^2(I)\\) with \\(\\sum_j\\|g_j\\|^2<\\infty\\), such that \\(H_{\\mathrm{ac}}(k\\oplus k')=H_{\\mathrm{ac}}(k)\\oplus H'\\) and \\(\\Phi(k\\oplus k')\\Phi^{-1}\\) is multiplication by \\(\\lambda\\) in each entry. Let \\(D\\) be the set of \\(\\zeta\\in H_{\\mathrm{ac}}(k\\oplus k')\\) such that \\(\\Phi\\zeta\\) has finitely many nonzero entries, each in \\(C_c^\\infty(I)\\). Then \\(D\\) is a dense subspace of \\(H_{\\mathrm{ac}}(k\\oplus k')\\), and for \\(\\zeta,\\eta\\in D\\):\n\n(i) the spectral density of \\(\\zeta\\) for \\(k\\oplus k'\\) is the bounded function \\(\\lambda\\mapsto\\sum_j|(\\Phi\\zeta)_j(\\lambda)|^2\\);\n\n(ii) the function \\(s\\mapsto\\langle e^{-is(k\\oplus k')}\\zeta,\\eta\\rangle\\) is integrable on \\(\\mathbb R\\).\n\n**Proof.** By Proposition 11.8, there is a unitary \\(\\Phi_0\\) of \\(H_{\\mathrm{ac}}(k)\\) onto \\(\\bigoplus_{j\\in J}L^2(S_j)\\) carrying \\(k_{\\mathrm{ac}}\\) to \\(\\bigoplus_jm_{S_j}\\), with \\(S_j\\subseteq[-\\|k\\|,\\|k\\|]\\). Let \\(I=(-\\|k\\|-1,\\|k\\|+1)\\), \\(H'=\\bigoplus_jL^2(I\\setminus S_j)\\) and \\(k'=\\bigoplus_jm_{I\\setminus S_j}\\), which is absolutely continuous by Proposition 11.8. The spectral measure of \\(k\\oplus k'\\) is the direct sum of the spectral measures, so the spectral measure of \\(\\zeta\\oplus\\zeta'\\) is \\(\\mu_\\zeta+\\mu_{\\zeta'}\\), and \\(H_{\\mathrm{ac}}(k\\oplus k')=H_{\\mathrm{ac}}(k)\\oplus H'\\). Regard \\(L^2(S_j)\\) and \\(L^2(I\\setminus S_j)\\) as the functions in \\(L^2(I)\\) that vanish off \\(S_j\\), respectively on \\(S_j\\); then \\(L^2(I)=L^2(S_j)\\oplus L^2(I\\setminus S_j)\\), and adding the entries defines \\(\\Phi\\). Density of \\(D\\) follows from Background 4(b). Statement (i) follows as in the proof of Proposition 11.8. For (ii), \\(\\langle e^{-is(k\\oplus k')}\\zeta,\\eta\\rangle=\\int_Ie^{-is\\lambda}w(\\lambda)\\,d\\lambda\\) with \\(w=\\sum_j(\\Phi\\zeta)_j\\overline{(\\Phi\\eta)_j}\\in C_c^\\infty(I)\\). Integrating by parts twice, this is at most \\(\\min(\\|w\\|_1,s^{-2}\\|w''\\|_1)\\) in absolute value, which is integrable in \\(s\\). \\(\\square\\)\n\n**Proof of Theorem 11.12.** Let \\(V=h-k\\). It is a self-adjoint trace-class operator, so by Theorem 2.3(e) \\(V=\\sum_nc_n\\theta_{f_n,f_n}\\) with an orthonormal family \\((f_n)\\), real \\(c_n\\ne0\\), and \\(\\sum|c_n|=\\|V\\|_1\\) by Theorem 4.3(b); the index \\(n\\) runs through \\(\\{1,\\dots,r\\}\\) or \\(\\mathbb N\\) (if \\(V=0\\), there is nothing to prove).\n\n*Step 1: enlarging the space.* Take \\(H'\\), \\(k'\\), \\(\\Phi\\) and \\(D\\) as in Lemma 11.16, and put \\(\\hat H=H\\oplus H'\\), \\(\\hat k=k\\oplus k'\\) and \\(\\hat h=h\\oplus k'\\). Then \\(\\hat h-\\hat k=V\\oplus0=\\sum_nc_n\\theta_{\\hat f_n,\\hat f_n}\\) with \\(\\hat f_n=f_n\\oplus0\\), and \\(\\hat W(t)=e^{it\\hat h}e^{-it\\hat k}=W_{h,k}(t)\\oplus1\\). Since \\(H_{\\mathrm{ac}}(k)\\oplus0\\subseteq H_{\\mathrm{ac}}(\\hat k)\\), it suffices to prove that \\(\\hat W(t)\\zeta\\) converges for every \\(\\zeta\\in H_{\\mathrm{ac}}(\\hat k)\\); by density of \\(D\\) and \\(\\|\\hat W(t)\\|=1\\) (the approximation step at the end of the proof of Lemma 11.14), it suffices to do this for \\(\\zeta\\in D\\).\n\n*Step 2: approximating the perturbation.* Let \\(g_n=P_{\\mathrm{ac}}(\\hat k)\\hat f_n\\). For each integer \\(m\\ge1\\) and each \\(n\\le m\\) choose \\(g^{(m)}_n\\in D\\) with \\(\\|g^{(m)}_n-g_n\\|\\le1/m\\), put \\(f^{(m)}_n=\\hat f_n-g_n+g^{(m)}_n\\), and let\n\\[\n\\begin{gathered}\nV_m\\\\\n=\\sum_{n\\le m}c_n\\theta_{f^{(m)}_n,f^{(m)}_n},\\\\\nh_m\\\\\n=\\hat k+V_m,\\\\\nW_m(t)\\\\\n=e^{ith_m}e^{-it\\hat k}.\n\\end{gathered}\n\\]\nThen \\(\\|f^{(m)}_n\\|\\le1+1/m\\le2\\). Since \\(\\theta_{x,x}-\\theta_{y,y}=\\theta_{x-y,x}+\\theta_{y,x-y}\\) has norm at most \\(\\|x-y\\|(\\|x\\|+\\|y\\|)\\),\n\\[\n\\begin{gathered}\n\\|V_m-(\\hat h-\\hat k)\\|\\\\\n\\le\\sum_{n>m}|c_n|+\\sum_{n\\le m}|c_n|\\,\\frac1m\\Big(2+\\frac1m\\Big)\\to0 .\n\\end{gathered}\n\\]\nSo \\(\\|h_m-\\hat h\\|\\to0\\), and by the estimate after (11.1), \\(W_m(t)\\to\\hat W(t)\\) in norm for each fixed \\(t\\).\n\n*Step 3: the approximants have wave operators.* Let \\(\\zeta\\in D\\). Since \\(e^{-is\\hat k}\\zeta\\in H_{\\mathrm{ac}}(\\hat k)\\) and \\(f^{(m)}_n-g^{(m)}_n=\\hat f_n-g_n\\) is orthogonal to \\(H_{\\mathrm{ac}}(\\hat k)\\),\n\\[\n\\langle e^{-is\\hat k}\\zeta,f^{(m)}_n\\rangle=\\langle e^{-is\\hat k}\\zeta,g^{(m)}_n\\rangle ,\n\\]\nwhich is integrable in \\(s\\) by Lemma 11.16(ii). Hence \\(\\|V_me^{-is\\hat k}\\zeta\\|\\le\\sum_{n\\le m}|c_n|\\,\\|f^{(m)}_n\\|\\,|\\langle e^{-is\\hat k}\\zeta,g^{(m)}_n\\rangle|\\) is integrable, and by Cook's criterion the limit \\(W_{m,+}\\zeta=\\lim_tW_m(t)\\zeta\\) exists for every \\(\\zeta\\in H_{\\mathrm{ac}}(\\hat k)\\).\n\n*Step 4: a uniform estimate.* Fix \\(\\zeta\\in D\\), and let \\(M\\) bound its spectral density (Lemma 11.16(i)). Put \\(\\eta_n(t)=\\int_t^\\infty|\\langle e^{-is\\hat k}\\zeta,\\hat f_n\\rangle|^2ds\\) and \\(\\eta(t)=\\sum_n|c_n|\\eta_n(t)\\). By Lemma 11.13, \\(\\eta_n(t)\\le2\\pi M\\); each \\(\\eta_n(t)\\to0\\) as \\(t\\to\\infty\\); and by dominated convergence \\(\\eta(t)\\to0\\). As in Step 3, \\(\\langle e^{-is\\hat k}\\zeta,\\hat f_n\\rangle=\\langle e^{-is\\hat k}\\zeta,g_n\\rangle\\), so\n\\[\n\\begin{gathered}\n\\int_t^\\infty|\\langle e^{-is\\hat k}\\zeta,f^{(m)}_n\\rangle|^2ds\\\\\n=\\int_t^\\infty|\\langle e^{-is\\hat k}\\zeta,g_n+(g^{(m)}_n-g_n)\\rangle|^2ds\\\\\n\\le2\\eta_n(t)+2\\cdot2\\pi M\\,m^{-2},\n\\end{gathered}\n\\]\nusing Lemma 11.13 for the second part. Lemma 11.15(d) for the pair \\((h_m,\\hat k)\\), together with \\(\\sum_{n\\le m}|c_n|\\,\\|f^{(m)}_n\\|^2\\le4\\|V\\|_1\\), gives\n\\[\n\\begin{gathered}\n\\|W_{m,+}\\zeta-W_m(t)\\zeta\\|^2\\\\\n\\le\\delta_m(t)^2\\\\\n:=4(2\\pi M)^{1/2}\\|V\\|_1^{1/2}\\\\\n{}\\cdot\\big(2\\eta(t)+4\\pi M\\|V\\|_1m^{-2}\\big)^{1/2}.\n\\end{gathered}\n\\]\n\n*Step 5: passing to the limit.* For \\(t,t'\\in\\mathbb R\\), \\(\\|W_m(t)\\zeta-W_m(t')\\zeta\\|\\le\\delta_m(t)+\\delta_m(t')\\). Let \\(m\\to\\infty\\) with \\(t,t'\\) fixed. By Step 2,\n\\[\n\\begin{gathered}\n\\|\\hat W(t)\\zeta-\\hat W(t')\\zeta\\|\\\\\n\\le\\delta(t)+\\delta(t'),\\\\\n\\delta(t)^2\\\\\n=4(2\\pi M)^{1/2}\\|V\\|_1^{1/2}(2\\eta(t))^{1/2}.\n\\end{gathered}\n\\]\nSince \\(\\delta(t)\\to0\\) as \\(t\\to\\infty\\), \\(\\hat W(t)\\zeta\\) is a Cauchy net and converges. By Step 1, \\(W_+(h,k):=\\lim_te^{ith}e^{-itk}P_{\\mathrm{ac}}(k)\\) exists.\n\n*Step 6: the conclusion.* Since \\(k-h=-V\\) is also of trace class, \\(W_+(k,h)=\\lim_te^{itk}e^{-ith}P_{\\mathrm{ac}}(h)\\) exists as well. By Lemma 11.15, \\(W_+(h,k)\\) maps \\(H_{\\mathrm{ac}}(k)\\) isometrically into \\(H_{\\mathrm{ac}}(h)\\), and \\(W_+(k,h)\\) maps \\(H_{\\mathrm{ac}}(h)\\) isometrically into \\(H_{\\mathrm{ac}}(k)\\). For \\(\\zeta\\in H_{\\mathrm{ac}}(k)\\),\n\\[\n\\begin{gathered}\ne^{itk}e^{-ith}\\,W_+(h,k)\\zeta-\\zeta\\\\\n=e^{itk}e^{-ith}\\big(W_+(h,k)\\zeta-e^{ith}e^{-itk}\\zeta\\big)\\to0,\n\\end{gathered}\n\\]\nand \\(W_+(h,k)\\zeta\\in H_{\\mathrm{ac}}(h)\\); so \\(W_+(k,h)W_+(h,k)\\zeta=\\zeta\\). In the same way \\(W_+(h,k)W_+(k,h)\\eta=\\eta\\) for \\(\\eta\\in H_{\\mathrm{ac}}(h)\\). Hence \\(W_+(h,k)\\) maps \\(H_{\\mathrm{ac}}(k)\\) onto \\(H_{\\mathrm{ac}}(h)\\): it is a partial isometry with these initial and final spaces, and \\(hW_+=W_+k\\) by Lemma 11.15(b). Restricted to \\(H_{\\mathrm{ac}}(k)\\) it is a unitary onto \\(H_{\\mathrm{ac}}(h)\\) carrying \\(k_{\\mathrm{ac}}\\) to \\(h_{\\mathrm{ac}}\\). For \\(t\\to-\\infty\\), apply the result to \\(-h\\) and \\(-k\\): \\(e^{it(-h)}e^{-it(-k)}=e^{-ith}e^{itk}\\), and \\(H_{\\mathrm{ac}}(-h)=H_{\\mathrm{ac}}(h)\\) because reflection preserves null sets. \\(\\square\\)\n\n**Remark 11.17** (trace class cannot replace Hilbert–Schmidt). If \\(H\\) has an orthonormal basis of eigenvectors of a self-adjoint \\(k\\), then \\(H_{\\mathrm s}(k)\\) contains this basis (Proposition 11.6) and is closed, so \\(H_{\\mathrm{ac}}(k)=0\\). If moreover \\(h-k\\) is of trace class, Theorem 11.12 gives \\(H_{\\mathrm{ac}}(h)\\cong H_{\\mathrm{ac}}(k)=0\\). So a self-adjoint operator \\(h\\) with a nonzero absolutely continuous part is never a diagonal operator plus a trace-class operator. Self-adjointness of the two summands is no restriction here: if \\(h=d_\\lambda+x\\) as in Example 2.2, with \\(x\\) of trace class, then also \\(h=\\frac12(h+h^*)=d_{\\operatorname{Re}\\lambda}+\\frac12(x+x^*)\\), and \\(d_{\\operatorname{Re}\\lambda}\\) is self-adjoint with the basis as eigenvectors. For instance, multiplication by \\(t\\) on \\(L^2(0,1)\\) is absolutely continuous (Proposition 11.9); by Theorem 11.2 it is a diagonal operator plus a self-adjoint Hilbert–Schmidt operator of arbitrarily small Hilbert–Schmidt norm, but it is not a diagonal operator plus a trace-class operator of any size.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-LT-15",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "12. Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
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      "anchor": "oa-fnd-lt-15",
      "proof_locus": {
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      },
      "full_conditions_and_proof": "## 12. Exercises\n\n**Exercise 12.1** (medium; the adjoint on normal operators). Show that the adjoint is strongly continuous on the set of normal operators: if \\(x_\\alpha\\) and \\(x\\) are normal and \\(x_\\alpha\\to x\\) strongly, then \\(x_\\alpha^*\\to x^*\\) strongly. Compare with Proposition 8.6(b).\n\n*Solution.* For a normal operator \\(y\\), \\(\\|y^*\\xi\\|^2=\\langle yy^*\\xi,\\xi\\rangle=\\langle y^*y\\xi,\\xi\\rangle=\\|y\\xi\\|^2\\). Hence\n\\[\n\\begin{gathered}\n\\|(x_\\alpha^*-x^*)\\xi\\|^2\\\\\n=\\|x_\\alpha\\xi\\|^2-2\\operatorname{Re}\\langle x_\\alpha^*\\xi,x^*\\xi\\rangle+\\|x^*\\xi\\|^2\\\\\n=\\|x_\\alpha\\xi\\|^2-2\\operatorname{Re}\\langle\\xi,x_\\alpha x^*\\xi\\rangle+\\|x\\xi\\|^2 .\n\\end{gathered}\n\\]\nAs \\(x_\\alpha\\to x\\) strongly, \\(\\|x_\\alpha\\xi\\|\\to\\|x\\xi\\|\\) and \\(x_\\alpha x^*\\xi\\to xx^*\\xi\\). So the right side tends to \\(2\\|x\\xi\\|^2-2\\langle\\xi,xx^*\\xi\\rangle=2\\|x\\xi\\|^2-2\\|x^*\\xi\\|^2=0\\). The operators \\(\\theta_{\\xi_1,\\xi_n}\\) of Example 8.4(b) are not normal, so there is no contradiction.\n\n**Exercise 12.2** (medium; cyclicity of the trace for Hilbert–Schmidt operators). Let \\(x,y\\in\\mathcal L^2(H)\\). Show that \\(xy\\) and \\(yx\\) are of trace class and that \\(\\operatorname{Tr}(xy)=\\operatorname{Tr}(yx)\\), although neither \\(x\\) nor \\(y\\) need be of trace class.\n\n*Solution.* Both products are of trace class by Proposition 4.9(b). Fix an orthonormal basis \\((\\varepsilon_i)\\). By Parseval,\n\\[\n\\begin{gathered}\n\\operatorname{Tr}(xy)\\\\\n=\\sum_i\\langle y\\varepsilon_i,x^*\\varepsilon_i\\rangle\\\\\n=\\sum_i\\sum_j\\langle y\\varepsilon_i,\\varepsilon_j\\rangle\\langle x\\varepsilon_j,\\varepsilon_i\\rangle ,\n\\end{gathered}\n\\]\nand in the same way \\(\\operatorname{Tr}(yx)=\\sum_j\\sum_i\\langle x\\varepsilon_j,\\varepsilon_i\\rangle\\langle y\\varepsilon_i,\\varepsilon_j\\rangle\\). The double family is absolutely summable, since by Cauchy–Schwarz and (3.1) its absolute sum is at most \\[\n\\begin{gathered}\n(\\sum_{i,j}|\\langle y\\varepsilon_i,\\varepsilon_j\\rangle|^2)^{1/2}(\\sum_{i,j}|\\langle x\\varepsilon_j,\\varepsilon_i\\rangle|^2)^{1/2}\\\\\n=\\|y\\|_2\\|x\\|_2.\n\\end{gathered}\n\\] So the two iterated sums agree. The Volterra operator \\(V\\) of Example 3.5 is Hilbert–Schmidt but not of trace class, and \\(V^*V\\) is of trace class.\n\n**Exercise 12.3** (medium; the diagonal algebra and its predual). Let \\((\\varepsilon_i)_{i\\in I}\\) be an orthonormal basis and \\(\\mathcal D=\\{d_\\lambda:\\lambda\\in\\ell^\\infty(I)\\}\\) (Example 2.2). Show that \\(\\mathcal D\\) is \\(\\sigma\\)-weakly closed, that \\(\\omega\\mapsto(\\omega(d_{\\delta_i}))_i\\) is an isometric isomorphism of \\(\\mathcal D_*\\) onto \\(\\ell^1(I)\\), and that on the unit ball of \\(\\mathcal D\\) the \\(\\sigma\\)-weak topology is the topology of coordinatewise convergence of \\(\\lambda\\).\n\n*Solution.* An operator \\(x\\) lies in \\(\\mathcal D\\) exactly when \\(\\langle x\\varepsilon_i,\\varepsilon_j\\rangle=0\\) for all \\(i\\ne j\\): then \\(x\\varepsilon_i=\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\varepsilon_i\\) and \\(x=d_\\lambda\\) with \\(\\lambda_i=\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\), \\(|\\lambda_i|\\le\\|x\\|\\). So \\(\\mathcal D\\) is the intersection of the kernels of the \\(\\sigma\\)-weakly continuous functionals \\(\\omega_{\\varepsilon_i,\\varepsilon_j}\\), \\(i\\ne j\\). For \\(\\omega\\in B(H)_*\\) with \\(t=t_\\omega\\), computing the trace in the basis \\((\\varepsilon_i)\\) gives \\(\\omega(d_\\lambda)=\\operatorname{Tr}(d_\\lambda t)=\\sum_i\\lambda_i\\alpha_i\\) with \\(\\alpha_i=\\langle t\\varepsilon_i,\\varepsilon_i\\rangle=\\omega(d_{\\delta_i})\\), and \\(\\sum_i|\\alpha_i|\\le\\|t\\|_1\\) by Theorem 4.6. Since \\(\\|d_\\lambda\\|=\\|\\lambda\\|_\\infty\\), the norm of \\(\\omega|_{\\mathcal D}\\) is the norm of the functional \\(\\lambda\\mapsto\\sum_i\\lambda_i\\alpha_i\\) on \\(\\ell^\\infty(I)\\). This is \\(\\|\\alpha\\|_1\\): it is at most \\(\\|\\alpha\\|_1\\), and its restriction to \\(c_0(I)\\) already has norm \\(\\|\\alpha\\|_1\\) by Example 5.1(a). Every \\(\\alpha\\in\\ell^1(I)\\) arises, from \\(t=d_\\alpha\\). By Theorem 9.4(a) the elements of \\(\\mathcal D_*\\) are the restrictions \\(\\omega|_{\\mathcal D}\\), so \\(\\omega|_{\\mathcal D}\\mapsto\\alpha\\) is an isometric isomorphism onto \\(\\ell^1(I)\\). On the unit ball, \\(\\sigma\\)-weak convergence \\(d_{\\lambda^\\beta}\\to d_\\lambda\\) means \\(\\sum_i\\lambda^\\beta_i\\alpha_i\\to\\sum_i\\lambda_i\\alpha_i\\) for all \\(\\alpha\\in\\ell^1(I)\\); for a bounded net this holds as soon as it holds for \\(\\alpha\\in c_{00}(I)\\), which is dense, that is, for coordinatewise convergence.\n\n**Exercise 12.4** (medium; the shift). Let \\(S\\) be the unilateral shift on \\(\\ell^2\\). Show that \\(S\\) is a strong limit of unitary operators, so that the unitary group is not strongly closed, but that \\(\\|S-u\\|\\ge1\\) for every unitary \\(u\\).\n\n*Solution.* Let \\(u_n\\) be the unitary with \\(u_n\\delta_j=\\delta_{j+1}\\) for \\(j\\le n\\), \\(u_n\\delta_{n+1}=\\delta_1\\), and \\(u_n\\delta_j=\\delta_j\\) for \\(j>n+1\\). For a finitely supported \\(\\xi\\), \\(u_n\\xi=S\\xi\\) as soon as \\(n\\) exceeds its support. Since all these operators have norm one and finitely supported vectors are dense, \\(u_n\\to S\\) strongly. For a unitary \\(u\\), \\(\\|S-u\\|=\\|u^*S-1\\|\\). The operator \\(u^*S\\) is an isometry whose range \\(u^*(S\\ell^2)\\) is a proper subspace, so it is not invertible and \\(0\\) lies in its spectrum. Then \\(-1\\) lies in the spectrum of \\(u^*S-1\\), and the norm of an operator is at least its spectral radius (lesson [Banach algebras, spectrum, holomorphic functional calculus and Gelfand theory](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md)). So \\(\\|S-u\\|\\ge1\\).\n\n**Exercise 12.5** (easy; the trace is not weak\\(^*\\) continuous). Let \\((\\xi_n)\\) be an orthonormal sequence. Show that \\(\\theta_{\\xi_n,\\xi_n}\\to0\\) in the weak\\(^*\\) topology \\(\\sigma(\\mathcal L^1(H),K(H))\\) that comes from Theorem 5.2, while \\(\\operatorname{Tr}(\\theta_{\\xi_n,\\xi_n})=1\\). Conclude that the trace is not continuous for this topology on the unit ball of \\(\\mathcal L^1(H)\\), although this ball is compact for it.\n\n*Solution.* For \\(x\\in K(H)\\), \\(\\operatorname{Tr}(x\\theta_{\\xi_n,\\xi_n})=\\langle x\\xi_n,\\xi_n\\rangle\\to0\\) by (4.4) and Lemma 2.1, while \\(\\operatorname{Tr}(\\theta_{\\xi_n,\\xi_n})=\\|\\xi_n\\|^2=1\\) and \\(\\|\\theta_{\\xi_n,\\xi_n}\\|_1=1\\). So the trace is not continuous on the unit ball, which is compact by the Banach–Alaoglu theorem and Theorem 5.2. The trace is the pairing with \\(1\\in B(H)\\), and \\(1\\notin K(H)\\) when \\(H\\) is infinite-dimensional.\n\n## Where this leads\n\n- *Von Neumann algebras.* The next lesson, [The double commutant theorem](the-double-commutant-theorem.md), shows that a nondegenerate \\(*\\)-algebra of operators equals its bicommutant exactly when it is closed in at least one of the six topologies of Section 8, and that it is then closed in all of them. Theorem 9.4 then makes every von Neumann algebra the dual of its predual. The order-theoretic meaning of normal functionals is treated in [The universal enveloping von Neumann algebra of a C\\*-algebra, and W\\*-algebras](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md).\n- *Another construction of the predual.* The lesson *Concrete preduals from Hilbert tensors* of the course *Modular theory and weights* builds the predual of a von Neumann algebra as a quotient of the completed projective tensor product of \\(H\\) with its conjugate space, and proves the continuous-dual statements of Theorem 9.1(ii) in that setting. The lesson *Detecting normal weights by finite observations* in the same course uses Theorem 9.1 and Theorem 9.6. It also uses the description of the \\(\\sigma\\)-strongly and \\(\\sigma\\)-weakly continuous real-linear functionals on the self-adjoint part of a von Neumann algebra \\(M\\); Corollary 9.7(c) gives this description for \\(M=B(H)\\).\n- *Schatten classes.* For \\(1\\leq p<\\infty\\), the symmetric solid subspace \\(\\ell^p\\subset c_0\\) corresponds under Theorem 7.7 to the ideal of compact operators with \\(\\sum_ns_n(x)^p<\\infty\\). These ideals increase with \\(p\\): an \\(\\ell^p\\) sequence is bounded, and \\(\\sum|s_n|^q\\leq\\|s\\|_\\infty^{q-p}\\sum|s_n|^p\\) for \\(q>p\\). The cases \\(p=1,2\\) are the trace and Hilbert–Schmidt classes. Proposition 11.18 gives the full proof that Theorem 11.2 extends to every finite \\(p>1\\), while Remark 11.17 gives the obstruction at \\(p=1\\). \n\n## References\n\n- [van Neerven] J. van Neerven, *Functional Analysis*, [corrected author version, arXiv:2112.11166v7](https://arxiv.org/pdf/2112.11166v7).\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [Jesse Peterson, *Notes on operator algebras*, §§3.1–3.4](https://math.vanderbilt.edu/peters10/teaching/spring2015/OperatorAlgebras.pdf) gives a route through trace duality and operator topologies; the compact-operator duality exercise and Krein–Šmulian are proved here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-LT-17",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Measure and Fourier tools on the real line",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
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      "anchor": "oa-fnd-lt-17",
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      },
      "full_conditions_and_proof": "### Measure and Fourier tools on the real line\n\nThe perturbation argument needs the real-line Fourier transform with its exact constant. We prove that version here. The measure results used in the proof are [Carathéodory, monotone and dominated convergence, Hölder, completeness and Radon–Nikodym](measure-and-hilbert-space-tools.md), Theorems 1.1, 2.1–2.2, 3.1–3.2 and 4.1 of *Measure and Hilbert space tools for Haar integration*. The construction below also uses [the probability product theorem and its generating-family lemma](the-double-commutant-theorem.md#oa-fnd-bi-18). Those proofs use the measure-tools lesson, and do not use the operator-topology or Fourier results of this lesson.\n\n**Lemma 0.1 (sigma-finite products and real-line measure).**\n\n(a) If \\((X,\\Sigma,\\mu)\\) and \\((Y,\\mathcal T,\\nu)\\) are sigma-finite measure spaces, their product measure exists on \\(\\Sigma\\otimes\\mathcal T\\). For a nonnegative measurable \\(F\\), both iterated integrals are measurable and\n\\[\n\\begin{gathered}\n\\int F\\,d(\\mu\\otimes\\nu)\n\\\\\n=\\int_X\\!\\int_Y F(x,y)\\,d\\nu(y)\\,d\\mu(x)\n\\\\\n=\\int_Y\\!\\int_X F(x,y)\\,d\\mu(x)\\,d\\nu(y).\n\\end{gathered}\n\\]\nFor an integrable complex \\(F\\), the sections are integrable almost everywhere and the same identities hold. Classes in the completed product may be represented by functions measurable for the uncompleted product; iterated integrals are taken using such representatives.\n\n(b) Lebesgue measure can be constructed as normalized Haar measure on \\(\\mathbb R\\), completed on its null sets. It is sigma-finite, translation invariant, and assigns length \\(b-a\\) to \\((a,b)\\). A set is null exactly when, for every \\(\\varepsilon>0\\), it has a countable cover by open intervals of total length less than \\(\\varepsilon\\). Every null set is contained in a null Borel set. A locally Lipschitz map between real intervals maps null sets to null sets.\n\n(c) \\(C_c^\\infty(I)\\) is dense in \\(L^2(I)\\) for every open interval \\(I\\). Translations are continuous in \\(L^2(\\mathbb R)\\), and convolution with a nonnegative integral-one kernel \\(\\alpha_j\\) converges to the identity in \\(L^2\\) if\n\\[\n\\int_{|t|>\\delta}\\alpha_j(t)\\,dt\\longrightarrow0\n\\quad\\text{for every }\\delta>0.\n\\]\n\n*Proof.* (a) First suppose both measures are finite and nonzero. Normalize them to probability measures, apply the proved probability product theorem to these two factors, and multiply the resulting measure by \\(\\mu(X)\\nu(Y)\\). If either total mass is zero use the zero product measure. For finite measures, let \\(\\mathcal D\\) consist of the measurable sets \\(E\\) for which the sections are measurable, \\(x\\mapsto\\nu(E_x)\\) is measurable and\n\\(\\int\\nu(E_x)\\,d\\mu(x)=(\\mu\\otimes\\nu)(E)\\).\nRectangles belong to \\(\\mathcal D\\). It is a Dynkin system: it contains the whole product, nested differences follow by subtraction of finite section measures and finite integrals, and countable disjoint unions follow by monotone convergence. The generating-family proof linked above therefore gives every product-measurable set. Interchanging the factors proves the other section identity. Nonnegative simple approximation followed by monotone convergence proves the formula for every \\(F\\geq0\\). Apply it to \\(|F|\\) and then to the positive and negative real and imaginary parts for integrable \\(F\\).\n\nFor sigma-finite measures, partition \\(X\\) and \\(Y\\) into countably many disjoint measurable pieces \\(X_n,Y_m\\) of finite measure, by disjointifying finite-measure exhaustions. On each \\(X_n\\times Y_m\\) use the finite construction and sum the resulting measures. Rectangles have the required product values: the double sum of \\(\\mu(A\\cap X_n)\\nu(B\\cap Y_m)\\) is \\(\\mu(A)\\nu(B)\\), with zero times infinity interpreted as zero. Two candidate measures agree on each finite block by generating-family uniqueness, hence on the product. Apply the finite-block integral identity and monotone convergence to the countable blocks to obtain both iterated formulas. A completed-measurable function has a product-measurable representative: replace the sets in its countable simple approximations by measurable sets differing by subsets of measurable null sets. Their union is one measurable null set. Tonelli says its sections are null almost everywhere, so the integral identities are independent of the representative.\n\n(b) [Haar existence and uniqueness](haar-measure.md#oa-fnd-hm-05), Theorems 8.3 and 9.2, give a nonzero Radon measure on \\(\\mathbb R\\), finite on compact sets and positive on nonempty open sets. Every singleton has the same mass \\(c\\) by translation invariance. Placing arbitrarily many distinct points in \\([0,1]\\) gives \\(Nc\\leq\\mu([0,1])\\), so \\(c=0\\). Normalize \\(\\mu([0,1])\\) to one; this mass is finite and positive. Partitioning into equal intervals gives \\(\\mu([0,q])=q\\) for nonnegative rational \\(q\\). Increasing rational endpoints to \\(t\\geq0\\), countable additivity gives \\(\\mu([0,t])=t\\). Translation and the zero mass of endpoints give every interval length. The intervals \\([-n,n]\\) show sigma-finiteness.\n\nEvery open subset of the line is a countable disjoint union of open intervals: each component is an interval and contains a different rational point. Its measure is the sum of their lengths. Outer regularity of Haar measure now says that a null Borel set has open interval covers of arbitrarily small total length. The same statement holds for a subset of a null Borel set. Conversely, such covers force outer measure zero; Carathéodory's construction includes these subsets in the completion. Intersecting open covers with total lengths tending to zero supplies a null Borel superset. If \\(g\\) is Lipschitz with constant \\(L\\) on an interval, the image of each covering interval has diameter at most \\(L\\) times its length; enlarge the image intervals by summably small amounts if necessary. The covering criterion shows that \\(g\\) maps null sets to null sets. A locally Lipschitz map is Lipschitz on a countable cover by compact subintervals, which gives the local assertion.\n\nIntegrals of continuous functions on compact intervals agree with Riemann integrals: uniform step-function approximations have integral error at most interval length times their uniform error. Reflection preserves Lebesgue measure. For \\(a>0\\), the pushforward under \\(x\\mapsto ax\\) assigns \\((b,c)\\) mass \\((c-b)/a\\); generating-family uniqueness on bounded intervals therefore gives \\(\\int F(ax)\\,dx=a^{-1}\\int F(y)\\,dy\\) for nonnegative or integrable \\(F\\). Thus the elementary substitution and integration-by-parts formulas from [Lemma 0.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) apply to the continuous integrands below.\n\n(c) The [Haar density proof](haar-measure.md#oa-fnd-hm-02), Proposition 3.1(4), supplies \\(C_c(\\mathbb R)\\)-density in \\(L^2(\\mathbb R)\\). For a proper open interval \\(I\\), let \\(d(x)=\\operatorname{dist}(x,\\mathbb R\\setminus I)\\) and set \\[\n\\begin{gathered}\n\\chi_n(x)\\\\\n=\\min(1,\\max(0,nd(x)-1))\\\\\n{}\\cdot\\min(1,\\max(0,n-|x|));\n\\end{gathered}\n\\] for \\(I=\\mathbb R\\) omit the first factor. These continuous cutoffs have compact support inside \\(I\\), increase to one there, and lie between zero and one. Extend a function on \\(I\\) by zero, approximate it by \\(C_c(\\mathbb R)\\), and multiply the approximants by a fixed \\(\\chi_n\\). The error does not increase; dominated convergence then gives \\(C_c(I)\\)-density. For an explicit smooth kernel set\n\\[\n\\beta(t)=\n\\begin{cases}\n\\exp\\big(-1/(1-t^2)\\big),&|t|<1,\\\\\n0,&|t|\\geq1.\n\\end{cases}\n\\]\nInside \\((-1,1)\\), every derivative is the exponential times a rational function with a finite power of \\(1-t^2\\) in its denominator. These derivatives tend to zero at the endpoints: \\(u^Ne^{-u}\\to0\\) as \\(u\\to\\infty\\), since the exponential series bounds \\(e^u\\) below by \\(u^{N+1}/(N+1)!\\). Inductively the extensions of all derivatives are continuous with zero endpoint values, so \\(\\beta\\) is smooth on the line. Normalize its positive finite integral and rescale to obtain \\(\\beta_\\varepsilon\\), supported in \\([-\\varepsilon,\\varepsilon]\\) with integral one. For \\(f\\in C_c(I)\\), differentiation of \\(f*\\beta_\\varepsilon\\) falls on the kernel; compact parameter differentiation from the Cauchy lemma justifies every derivative. For small \\(\\varepsilon\\) the support remains inside \\(I\\), and uniform continuity gives uniform convergence to \\(f\\). A common finite-length compact support gives convergence in \\(L^2\\).\n\nTranslations \\(\\tau_tf(x)=f(x-t)\\) are isometries by (b). For \\(f\\in C_c(\\mathbb R)\\), uniform continuity and a common compact support give \\(\\|\\tau_tf-f\\|_2\\to0\\) as \\(t\\to0\\); density and the isometry extend this to every \\(f\\in L^2\\). Cauchy–Schwarz with the probability density \\(\\alpha_j\\), followed by Tonelli, gives\n\\[\n\\|f*\\alpha_j-f\\|_2^2\n\\leq\\int\\alpha_j(t)\\|\\tau_tf-f\\|_2^2\\,dt.\n\\]\nIt also shows that the convolution integral exists almost everywhere. Choose a neighbourhood of zero on which the squared norm in the integrand is small, and bound it on the complement by \\(4\\|f\\|_2^2\\). The concentration hypothesis finishes the proof. \\(\\square\\)\n\n**Lemma 0.2 (Gaussian transform).** For \\(\\sigma>0\\) the function\n\\[\n\\gamma_\\sigma(x)=\\frac{1}{\\sqrt{2\\pi}\\,\\sigma}\n\\exp\\left(-\\frac{x^2}{2\\sigma^2}\\right)\n\\]\nhas integral one and transform\n\\[\n\\int_{\\mathbb R}e^{-itx}\\gamma_\\sigma(x)\\,dx\n=e^{-\\sigma^2t^2/2}.\n\\]\nIts mass outside any fixed neighbourhood of zero tends to zero as \\(\\sigma\\downarrow0\\).\n\n*Proof.* The integral \\(J=\\int e^{-x^2}\\,dx\\) is finite, since outside \\([-1,1]\\) its integrand is bounded by \\(e^{-|x|}\\). Tonelli gives \\(J^2=\\int_{\\mathbb R^2}e^{-x^2-y^2}\\,dx\\,dy\\). The disc of radius \\(r\\) has area\n\\[\n\\int_{-r}^r2\\sqrt{r^2-x^2}\\,dx=\\pi r^2:\n\\]\nits sections give the first expression by Tonelli, and \\(x=r\\sin\\theta\\), \\(-\\pi/2\\leq\\theta\\leq\\pi/2\\), gives the second by the elementary substitution formula. Write \\(e^{-x^2-y^2}=\\int_0^1 1_{\\{u<e^{-x^2-y^2}\\}}\\,du\\) and use Tonelli again. Then\n\\[\nJ^2=\\int_0^1\\pi(-\\log u)\\,du=\\pi.\n\\]\nThe last integral follows by integrating on \\([\\delta,1]\\) and letting \\(\\delta\\downarrow0\\); \\(\\delta\\log\\delta\\to0\\) by the same exponential estimate. Since \\(J>0\\), \\(J=\\sqrt\\pi\\). Scaling gives the stated normalization.\n\nLet \\(\\phi(t)=\\int e^{-itx}\\gamma_\\sigma(x)\\,dx\\). Dominated convergence differentiates under the integral, since the difference quotients are bounded by \\(|x|\\gamma_\\sigma(x)\\), which is integrable. Since \\(\\gamma_\\sigma'(x)=-x\\gamma_\\sigma(x)/\\sigma^2\\), integration by parts first on \\([-R,R]\\) and then letting \\(R\\to\\infty\\) gives\n\\[\n\\phi'(t)=i\\sigma^2\\int e^{-itx}\\gamma_\\sigma'(x)\\,dx\n=-\\sigma^2t\\phi(t).\n\\]\nThe boundary terms tend to zero and both improper integrals converge absolutely. Differentiating \\(e^{\\sigma^2t^2/2}\\phi(t)\\) shows it is constant, equal to \\(\\phi(0)=1\\). Finally, the substitution \\(x=\\sigma v\\) identifies the tail mass with \\(\\int_{|v|>\\delta/\\sigma}\\gamma_1(v)\\,dv\\), which tends to zero by integrability. \\(\\square\\)\n\n**Theorem 0.3 (real-line Plancherel).** For \\(f\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\), put \\(\\widehat f(t)=\\int e^{-itx}f(x)\\,dx\\). Then\n\\[\n\\int|\\widehat f(t)|^2\\,dt=2\\pi\\int|f(x)|^2\\,dx.\n\\]\nThe operator \\(T:f\\mapsto(2\\pi)^{-1/2}\\widehat f\\) extends uniquely to a unitary operator on \\(L^2(\\mathbb R)\\).\n\n*Proof.* For \\(\\varepsilon>0\\), Fubini and Lemma 0.2 give\n\\[\n\\begin{gathered}\n\\int e^{-\\varepsilon t^2}|\\widehat f(t)|^2\\,dt\n\\\\\n=\\sqrt{\\frac\\pi\\varepsilon}\n\\int\\!\\int f(x)\\overline{f(y)}\\\\\n{}\\cdot e^{-(x-y)^2/(4\\varepsilon)}\\,dx\\,dy\\\\\n\\\\\n=2\\pi\\langle f,f*\\gamma_{\\sqrt{2\\varepsilon}}\\rangle.\n\\end{gathered}\n\\]\nThe interchange is justified before computing the inner integral: the integral of the absolute value of the three-variable integrand is\n\\(\\|f\\|_1^2\\int e^{-\\varepsilon t^2}\\,dt<\\infty\\).\nThe last convolution converges to \\(f\\) in \\(L^2\\) by Lemmas 0.1(c) and 0.2. On the left let \\(\\varepsilon=1/n\\); monotone convergence proves the norm identity. The domain contains \\(C_c^\\infty\\) and is dense in \\(L^2\\), so completeness extends \\(T\\) to an isometry with closed range. Polarization gives inner-product preservation.\n\nWe prove surjectivity as well. If \\(g\\in C_c^\\infty(\\mathbb R)\\), two integrations by parts give\n\\[\n|\\widehat g(t)|\\leq\n\\min\\big(\\|g\\|_1,\\,|t|^{-2}\\|g''\\|_1\\big)\n\\quad(t\\ne0).\n\\]\nThus \\(\\widehat g\\in L^1\\cap L^2\\). Fubini and the Gaussian formula show that\n\\[\n\\frac1{2\\pi}\\int e^{itx}e^{-\\varepsilon t^2}\n\\widehat g(t)\\,dt\n=(g*\\gamma_{\\sqrt{2\\varepsilon}})(x).\n\\]\nThe absolute-integrability justification is \\(\\|g\\|_1\\int e^{-\\varepsilon t^2}\\,dt<\\infty\\). The right side tends to \\(g(x)\\) for every \\(x\\), by bounded uniform continuity and Gaussian concentration. Dominated convergence on the left, using \\(\\widehat g\\in L^1\\), gives\n\\[\ng(x)=\\frac1{2\\pi}\\int e^{itx}\\widehat g(t)\\,dt.\n\\]\nConsequently \\(T^2g(x)=g(-x)\\). The range of \\(T\\) therefore contains every reflection of a function in \\(C_c^\\infty\\), a dense subspace. Its closed range is all of \\(L^2\\). \\(\\square\\)\n\nLemmas 0.1–0.3 give the measure, product-integration and Fourier arguments used below, including their hypotheses, sign and normalization. The exact earlier programme proofs used in Lemma 0.1 are identified there.\n\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Measure and Fourier tools on the real line",
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      "full_conditions_and_proof": "### Measure and Fourier tools on the real line\n\nThe perturbation argument needs the real-line Fourier transform with its exact constant. We prove that version here. The measure results used in the proof are [Carathéodory, monotone and dominated convergence, Hölder, completeness and Radon–Nikodym](measure-and-hilbert-space-tools.md), Theorems 1.1, 2.1–2.2, 3.1–3.2 and 4.1 of *Measure and Hilbert space tools for Haar integration*. The construction below also uses [the probability product theorem and its generating-family lemma](the-double-commutant-theorem.md#oa-fnd-bi-18). Those proofs use the measure-tools lesson, and do not use the operator-topology or Fourier results of this lesson.\n\n**Lemma 0.1 (sigma-finite products and real-line measure).**\n\n(a) If \\((X,\\Sigma,\\mu)\\) and \\((Y,\\mathcal T,\\nu)\\) are sigma-finite measure spaces, their product measure exists on \\(\\Sigma\\otimes\\mathcal T\\). For a nonnegative measurable \\(F\\), both iterated integrals are measurable and\n\\[\n\\begin{gathered}\n\\int F\\,d(\\mu\\otimes\\nu)\n\\\\\n=\\int_X\\!\\int_Y F(x,y)\\,d\\nu(y)\\,d\\mu(x)\n\\\\\n=\\int_Y\\!\\int_X F(x,y)\\,d\\mu(x)\\,d\\nu(y).\n\\end{gathered}\n\\]\nFor an integrable complex \\(F\\), the sections are integrable almost everywhere and the same identities hold. Classes in the completed product may be represented by functions measurable for the uncompleted product; iterated integrals are taken using such representatives.\n\n(b) Lebesgue measure can be constructed as normalized Haar measure on \\(\\mathbb R\\), completed on its null sets. It is sigma-finite, translation invariant, and assigns length \\(b-a\\) to \\((a,b)\\). A set is null exactly when, for every \\(\\varepsilon>0\\), it has a countable cover by open intervals of total length less than \\(\\varepsilon\\). Every null set is contained in a null Borel set. A locally Lipschitz map between real intervals maps null sets to null sets.\n\n(c) \\(C_c^\\infty(I)\\) is dense in \\(L^2(I)\\) for every open interval \\(I\\). Translations are continuous in \\(L^2(\\mathbb R)\\), and convolution with a nonnegative integral-one kernel \\(\\alpha_j\\) converges to the identity in \\(L^2\\) if\n\\[\n\\int_{|t|>\\delta}\\alpha_j(t)\\,dt\\longrightarrow0\n\\quad\\text{for every }\\delta>0.\n\\]\n\n*Proof.* (a) First suppose both measures are finite and nonzero. Normalize them to probability measures, apply the proved probability product theorem to these two factors, and multiply the resulting measure by \\(\\mu(X)\\nu(Y)\\). If either total mass is zero use the zero product measure. For finite measures, let \\(\\mathcal D\\) consist of the measurable sets \\(E\\) for which the sections are measurable, \\(x\\mapsto\\nu(E_x)\\) is measurable and\n\\(\\int\\nu(E_x)\\,d\\mu(x)=(\\mu\\otimes\\nu)(E)\\).\nRectangles belong to \\(\\mathcal D\\). It is a Dynkin system: it contains the whole product, nested differences follow by subtraction of finite section measures and finite integrals, and countable disjoint unions follow by monotone convergence. The generating-family proof linked above therefore gives every product-measurable set. Interchanging the factors proves the other section identity. Nonnegative simple approximation followed by monotone convergence proves the formula for every \\(F\\geq0\\). Apply it to \\(|F|\\) and then to the positive and negative real and imaginary parts for integrable \\(F\\).\n\nFor sigma-finite measures, partition \\(X\\) and \\(Y\\) into countably many disjoint measurable pieces \\(X_n,Y_m\\) of finite measure, by disjointifying finite-measure exhaustions. On each \\(X_n\\times Y_m\\) use the finite construction and sum the resulting measures. Rectangles have the required product values: the double sum of \\(\\mu(A\\cap X_n)\\nu(B\\cap Y_m)\\) is \\(\\mu(A)\\nu(B)\\), with zero times infinity interpreted as zero. Two candidate measures agree on each finite block by generating-family uniqueness, hence on the product. Apply the finite-block integral identity and monotone convergence to the countable blocks to obtain both iterated formulas. A completed-measurable function has a product-measurable representative: replace the sets in its countable simple approximations by measurable sets differing by subsets of measurable null sets. Their union is one measurable null set. Tonelli says its sections are null almost everywhere, so the integral identities are independent of the representative.\n\n(b) [Haar existence and uniqueness](haar-measure.md#oa-fnd-hm-05), Theorems 8.3 and 9.2, give a nonzero Radon measure on \\(\\mathbb R\\), finite on compact sets and positive on nonempty open sets. Every singleton has the same mass \\(c\\) by translation invariance. Placing arbitrarily many distinct points in \\([0,1]\\) gives \\(Nc\\leq\\mu([0,1])\\), so \\(c=0\\). Normalize \\(\\mu([0,1])\\) to one; this mass is finite and positive. Partitioning into equal intervals gives \\(\\mu([0,q])=q\\) for nonnegative rational \\(q\\). Increasing rational endpoints to \\(t\\geq0\\), countable additivity gives \\(\\mu([0,t])=t\\). Translation and the zero mass of endpoints give every interval length. The intervals \\([-n,n]\\) show sigma-finiteness.\n\nEvery open subset of the line is a countable disjoint union of open intervals: each component is an interval and contains a different rational point. Its measure is the sum of their lengths. Outer regularity of Haar measure now says that a null Borel set has open interval covers of arbitrarily small total length. The same statement holds for a subset of a null Borel set. Conversely, such covers force outer measure zero; Carathéodory's construction includes these subsets in the completion. Intersecting open covers with total lengths tending to zero supplies a null Borel superset. If \\(g\\) is Lipschitz with constant \\(L\\) on an interval, the image of each covering interval has diameter at most \\(L\\) times its length; enlarge the image intervals by summably small amounts if necessary. The covering criterion shows that \\(g\\) maps null sets to null sets. A locally Lipschitz map is Lipschitz on a countable cover by compact subintervals, which gives the local assertion.\n\nIntegrals of continuous functions on compact intervals agree with Riemann integrals: uniform step-function approximations have integral error at most interval length times their uniform error. Reflection preserves Lebesgue measure. For \\(a>0\\), the pushforward under \\(x\\mapsto ax\\) assigns \\((b,c)\\) mass \\((c-b)/a\\); generating-family uniqueness on bounded intervals therefore gives \\(\\int F(ax)\\,dx=a^{-1}\\int F(y)\\,dy\\) for nonnegative or integrable \\(F\\). Thus the elementary substitution and integration-by-parts formulas from [Lemma 0.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) apply to the continuous integrands below.\n\n(c) The [Haar density proof](haar-measure.md#oa-fnd-hm-02), Proposition 3.1(4), supplies \\(C_c(\\mathbb R)\\)-density in \\(L^2(\\mathbb R)\\). For a proper open interval \\(I\\), let \\(d(x)=\\operatorname{dist}(x,\\mathbb R\\setminus I)\\) and set \\[\n\\begin{gathered}\n\\chi_n(x)\\\\\n=\\min(1,\\max(0,nd(x)-1))\\\\\n{}\\cdot\\min(1,\\max(0,n-|x|));\n\\end{gathered}\n\\] for \\(I=\\mathbb R\\) omit the first factor. These continuous cutoffs have compact support inside \\(I\\), increase to one there, and lie between zero and one. Extend a function on \\(I\\) by zero, approximate it by \\(C_c(\\mathbb R)\\), and multiply the approximants by a fixed \\(\\chi_n\\). The error does not increase; dominated convergence then gives \\(C_c(I)\\)-density. For an explicit smooth kernel set\n\\[\n\\beta(t)=\n\\begin{cases}\n\\exp\\big(-1/(1-t^2)\\big),&|t|<1,\\\\\n0,&|t|\\geq1.\n\\end{cases}\n\\]\nInside \\((-1,1)\\), every derivative is the exponential times a rational function with a finite power of \\(1-t^2\\) in its denominator. These derivatives tend to zero at the endpoints: \\(u^Ne^{-u}\\to0\\) as \\(u\\to\\infty\\), since the exponential series bounds \\(e^u\\) below by \\(u^{N+1}/(N+1)!\\). Inductively the extensions of all derivatives are continuous with zero endpoint values, so \\(\\beta\\) is smooth on the line. Normalize its positive finite integral and rescale to obtain \\(\\beta_\\varepsilon\\), supported in \\([-\\varepsilon,\\varepsilon]\\) with integral one. For \\(f\\in C_c(I)\\), differentiation of \\(f*\\beta_\\varepsilon\\) falls on the kernel; compact parameter differentiation from the Cauchy lemma justifies every derivative. For small \\(\\varepsilon\\) the support remains inside \\(I\\), and uniform continuity gives uniform convergence to \\(f\\). A common finite-length compact support gives convergence in \\(L^2\\).\n\nTranslations \\(\\tau_tf(x)=f(x-t)\\) are isometries by (b). For \\(f\\in C_c(\\mathbb R)\\), uniform continuity and a common compact support give \\(\\|\\tau_tf-f\\|_2\\to0\\) as \\(t\\to0\\); density and the isometry extend this to every \\(f\\in L^2\\). Cauchy–Schwarz with the probability density \\(\\alpha_j\\), followed by Tonelli, gives\n\\[\n\\|f*\\alpha_j-f\\|_2^2\n\\leq\\int\\alpha_j(t)\\|\\tau_tf-f\\|_2^2\\,dt.\n\\]\nIt also shows that the convolution integral exists almost everywhere. Choose a neighbourhood of zero on which the squared norm in the integrand is small, and bound it on the complement by \\(4\\|f\\|_2^2\\). The concentration hypothesis finishes the proof. \\(\\square\\)\n\n**Lemma 0.2 (Gaussian transform).** For \\(\\sigma>0\\) the function\n\\[\n\\gamma_\\sigma(x)=\\frac{1}{\\sqrt{2\\pi}\\,\\sigma}\n\\exp\\left(-\\frac{x^2}{2\\sigma^2}\\right)\n\\]\nhas integral one and transform\n\\[\n\\int_{\\mathbb R}e^{-itx}\\gamma_\\sigma(x)\\,dx\n=e^{-\\sigma^2t^2/2}.\n\\]\nIts mass outside any fixed neighbourhood of zero tends to zero as \\(\\sigma\\downarrow0\\).\n\n*Proof.* The integral \\(J=\\int e^{-x^2}\\,dx\\) is finite, since outside \\([-1,1]\\) its integrand is bounded by \\(e^{-|x|}\\). Tonelli gives \\(J^2=\\int_{\\mathbb R^2}e^{-x^2-y^2}\\,dx\\,dy\\). The disc of radius \\(r\\) has area\n\\[\n\\int_{-r}^r2\\sqrt{r^2-x^2}\\,dx=\\pi r^2:\n\\]\nits sections give the first expression by Tonelli, and \\(x=r\\sin\\theta\\), \\(-\\pi/2\\leq\\theta\\leq\\pi/2\\), gives the second by the elementary substitution formula. Write \\(e^{-x^2-y^2}=\\int_0^1 1_{\\{u<e^{-x^2-y^2}\\}}\\,du\\) and use Tonelli again. Then\n\\[\nJ^2=\\int_0^1\\pi(-\\log u)\\,du=\\pi.\n\\]\nThe last integral follows by integrating on \\([\\delta,1]\\) and letting \\(\\delta\\downarrow0\\); \\(\\delta\\log\\delta\\to0\\) by the same exponential estimate. Since \\(J>0\\), \\(J=\\sqrt\\pi\\). Scaling gives the stated normalization.\n\nLet \\(\\phi(t)=\\int e^{-itx}\\gamma_\\sigma(x)\\,dx\\). Dominated convergence differentiates under the integral, since the difference quotients are bounded by \\(|x|\\gamma_\\sigma(x)\\), which is integrable. Since \\(\\gamma_\\sigma'(x)=-x\\gamma_\\sigma(x)/\\sigma^2\\), integration by parts first on \\([-R,R]\\) and then letting \\(R\\to\\infty\\) gives\n\\[\n\\phi'(t)=i\\sigma^2\\int e^{-itx}\\gamma_\\sigma'(x)\\,dx\n=-\\sigma^2t\\phi(t).\n\\]\nThe boundary terms tend to zero and both improper integrals converge absolutely. Differentiating \\(e^{\\sigma^2t^2/2}\\phi(t)\\) shows it is constant, equal to \\(\\phi(0)=1\\). Finally, the substitution \\(x=\\sigma v\\) identifies the tail mass with \\(\\int_{|v|>\\delta/\\sigma}\\gamma_1(v)\\,dv\\), which tends to zero by integrability. \\(\\square\\)\n\n**Theorem 0.3 (real-line Plancherel).** For \\(f\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\), put \\(\\widehat f(t)=\\int e^{-itx}f(x)\\,dx\\). Then\n\\[\n\\int|\\widehat f(t)|^2\\,dt=2\\pi\\int|f(x)|^2\\,dx.\n\\]\nThe operator \\(T:f\\mapsto(2\\pi)^{-1/2}\\widehat f\\) extends uniquely to a unitary operator on \\(L^2(\\mathbb R)\\).\n\n*Proof.* For \\(\\varepsilon>0\\), Fubini and Lemma 0.2 give\n\\[\n\\begin{gathered}\n\\int e^{-\\varepsilon t^2}|\\widehat f(t)|^2\\,dt\n\\\\\n=\\sqrt{\\frac\\pi\\varepsilon}\n\\int\\!\\int f(x)\\overline{f(y)}\\\\\n{}\\cdot e^{-(x-y)^2/(4\\varepsilon)}\\,dx\\,dy\\\\\n\\\\\n=2\\pi\\langle f,f*\\gamma_{\\sqrt{2\\varepsilon}}\\rangle.\n\\end{gathered}\n\\]\nThe interchange is justified before computing the inner integral: the integral of the absolute value of the three-variable integrand is\n\\(\\|f\\|_1^2\\int e^{-\\varepsilon t^2}\\,dt<\\infty\\).\nThe last convolution converges to \\(f\\) in \\(L^2\\) by Lemmas 0.1(c) and 0.2. On the left let \\(\\varepsilon=1/n\\); monotone convergence proves the norm identity. The domain contains \\(C_c^\\infty\\) and is dense in \\(L^2\\), so completeness extends \\(T\\) to an isometry with closed range. Polarization gives inner-product preservation.\n\nWe prove surjectivity as well. If \\(g\\in C_c^\\infty(\\mathbb R)\\), two integrations by parts give\n\\[\n|\\widehat g(t)|\\leq\n\\min\\big(\\|g\\|_1,\\,|t|^{-2}\\|g''\\|_1\\big)\n\\quad(t\\ne0).\n\\]\nThus \\(\\widehat g\\in L^1\\cap L^2\\). Fubini and the Gaussian formula show that\n\\[\n\\frac1{2\\pi}\\int e^{itx}e^{-\\varepsilon t^2}\n\\widehat g(t)\\,dt\n=(g*\\gamma_{\\sqrt{2\\varepsilon}})(x).\n\\]\nThe absolute-integrability justification is \\(\\|g\\|_1\\int e^{-\\varepsilon t^2}\\,dt<\\infty\\). The right side tends to \\(g(x)\\) for every \\(x\\), by bounded uniform continuity and Gaussian concentration. Dominated convergence on the left, using \\(\\widehat g\\in L^1\\), gives\n\\[\ng(x)=\\frac1{2\\pi}\\int e^{itx}\\widehat g(t)\\,dt.\n\\]\nConsequently \\(T^2g(x)=g(-x)\\). The range of \\(T\\) therefore contains every reflection of a function in \\(C_c^\\infty\\), a dense subspace. Its closed range is all of \\(L^2\\). \\(\\square\\)\n\nLemmas 0.1–0.3 give the measure, product-integration and Fourier arguments used below, including their hypotheses, sign and normalization. The exact earlier programme proofs used in Lemma 0.1 are identified there.\n\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-LT-19",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Measure and Fourier tools on the real line",
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      "full_conditions_and_proof": "### Measure and Fourier tools on the real line\n\nThe perturbation argument needs the real-line Fourier transform with its exact constant. We prove that version here. The measure results used in the proof are [Carathéodory, monotone and dominated convergence, Hölder, completeness and Radon–Nikodym](measure-and-hilbert-space-tools.md), Theorems 1.1, 2.1–2.2, 3.1–3.2 and 4.1 of *Measure and Hilbert space tools for Haar integration*. The construction below also uses [the probability product theorem and its generating-family lemma](the-double-commutant-theorem.md#oa-fnd-bi-18). Those proofs use the measure-tools lesson, and do not use the operator-topology or Fourier results of this lesson.\n\n**Lemma 0.1 (sigma-finite products and real-line measure).**\n\n(a) If \\((X,\\Sigma,\\mu)\\) and \\((Y,\\mathcal T,\\nu)\\) are sigma-finite measure spaces, their product measure exists on \\(\\Sigma\\otimes\\mathcal T\\). For a nonnegative measurable \\(F\\), both iterated integrals are measurable and\n\\[\n\\begin{gathered}\n\\int F\\,d(\\mu\\otimes\\nu)\n\\\\\n=\\int_X\\!\\int_Y F(x,y)\\,d\\nu(y)\\,d\\mu(x)\n\\\\\n=\\int_Y\\!\\int_X F(x,y)\\,d\\mu(x)\\,d\\nu(y).\n\\end{gathered}\n\\]\nFor an integrable complex \\(F\\), the sections are integrable almost everywhere and the same identities hold. Classes in the completed product may be represented by functions measurable for the uncompleted product; iterated integrals are taken using such representatives.\n\n(b) Lebesgue measure can be constructed as normalized Haar measure on \\(\\mathbb R\\), completed on its null sets. It is sigma-finite, translation invariant, and assigns length \\(b-a\\) to \\((a,b)\\). A set is null exactly when, for every \\(\\varepsilon>0\\), it has a countable cover by open intervals of total length less than \\(\\varepsilon\\). Every null set is contained in a null Borel set. A locally Lipschitz map between real intervals maps null sets to null sets.\n\n(c) \\(C_c^\\infty(I)\\) is dense in \\(L^2(I)\\) for every open interval \\(I\\). Translations are continuous in \\(L^2(\\mathbb R)\\), and convolution with a nonnegative integral-one kernel \\(\\alpha_j\\) converges to the identity in \\(L^2\\) if\n\\[\n\\int_{|t|>\\delta}\\alpha_j(t)\\,dt\\longrightarrow0\n\\quad\\text{for every }\\delta>0.\n\\]\n\n*Proof.* (a) First suppose both measures are finite and nonzero. Normalize them to probability measures, apply the proved probability product theorem to these two factors, and multiply the resulting measure by \\(\\mu(X)\\nu(Y)\\). If either total mass is zero use the zero product measure. For finite measures, let \\(\\mathcal D\\) consist of the measurable sets \\(E\\) for which the sections are measurable, \\(x\\mapsto\\nu(E_x)\\) is measurable and\n\\(\\int\\nu(E_x)\\,d\\mu(x)=(\\mu\\otimes\\nu)(E)\\).\nRectangles belong to \\(\\mathcal D\\). It is a Dynkin system: it contains the whole product, nested differences follow by subtraction of finite section measures and finite integrals, and countable disjoint unions follow by monotone convergence. The generating-family proof linked above therefore gives every product-measurable set. Interchanging the factors proves the other section identity. Nonnegative simple approximation followed by monotone convergence proves the formula for every \\(F\\geq0\\). Apply it to \\(|F|\\) and then to the positive and negative real and imaginary parts for integrable \\(F\\).\n\nFor sigma-finite measures, partition \\(X\\) and \\(Y\\) into countably many disjoint measurable pieces \\(X_n,Y_m\\) of finite measure, by disjointifying finite-measure exhaustions. On each \\(X_n\\times Y_m\\) use the finite construction and sum the resulting measures. Rectangles have the required product values: the double sum of \\(\\mu(A\\cap X_n)\\nu(B\\cap Y_m)\\) is \\(\\mu(A)\\nu(B)\\), with zero times infinity interpreted as zero. Two candidate measures agree on each finite block by generating-family uniqueness, hence on the product. Apply the finite-block integral identity and monotone convergence to the countable blocks to obtain both iterated formulas. A completed-measurable function has a product-measurable representative: replace the sets in its countable simple approximations by measurable sets differing by subsets of measurable null sets. Their union is one measurable null set. Tonelli says its sections are null almost everywhere, so the integral identities are independent of the representative.\n\n(b) [Haar existence and uniqueness](haar-measure.md#oa-fnd-hm-05), Theorems 8.3 and 9.2, give a nonzero Radon measure on \\(\\mathbb R\\), finite on compact sets and positive on nonempty open sets. Every singleton has the same mass \\(c\\) by translation invariance. Placing arbitrarily many distinct points in \\([0,1]\\) gives \\(Nc\\leq\\mu([0,1])\\), so \\(c=0\\). Normalize \\(\\mu([0,1])\\) to one; this mass is finite and positive. Partitioning into equal intervals gives \\(\\mu([0,q])=q\\) for nonnegative rational \\(q\\). Increasing rational endpoints to \\(t\\geq0\\), countable additivity gives \\(\\mu([0,t])=t\\). Translation and the zero mass of endpoints give every interval length. The intervals \\([-n,n]\\) show sigma-finiteness.\n\nEvery open subset of the line is a countable disjoint union of open intervals: each component is an interval and contains a different rational point. Its measure is the sum of their lengths. Outer regularity of Haar measure now says that a null Borel set has open interval covers of arbitrarily small total length. The same statement holds for a subset of a null Borel set. Conversely, such covers force outer measure zero; Carathéodory's construction includes these subsets in the completion. Intersecting open covers with total lengths tending to zero supplies a null Borel superset. If \\(g\\) is Lipschitz with constant \\(L\\) on an interval, the image of each covering interval has diameter at most \\(L\\) times its length; enlarge the image intervals by summably small amounts if necessary. The covering criterion shows that \\(g\\) maps null sets to null sets. A locally Lipschitz map is Lipschitz on a countable cover by compact subintervals, which gives the local assertion.\n\nIntegrals of continuous functions on compact intervals agree with Riemann integrals: uniform step-function approximations have integral error at most interval length times their uniform error. Reflection preserves Lebesgue measure. For \\(a>0\\), the pushforward under \\(x\\mapsto ax\\) assigns \\((b,c)\\) mass \\((c-b)/a\\); generating-family uniqueness on bounded intervals therefore gives \\(\\int F(ax)\\,dx=a^{-1}\\int F(y)\\,dy\\) for nonnegative or integrable \\(F\\). Thus the elementary substitution and integration-by-parts formulas from [Lemma 0.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) apply to the continuous integrands below.\n\n(c) The [Haar density proof](haar-measure.md#oa-fnd-hm-02), Proposition 3.1(4), supplies \\(C_c(\\mathbb R)\\)-density in \\(L^2(\\mathbb R)\\). For a proper open interval \\(I\\), let \\(d(x)=\\operatorname{dist}(x,\\mathbb R\\setminus I)\\) and set \\[\n\\begin{gathered}\n\\chi_n(x)\\\\\n=\\min(1,\\max(0,nd(x)-1))\\\\\n{}\\cdot\\min(1,\\max(0,n-|x|));\n\\end{gathered}\n\\] for \\(I=\\mathbb R\\) omit the first factor. These continuous cutoffs have compact support inside \\(I\\), increase to one there, and lie between zero and one. Extend a function on \\(I\\) by zero, approximate it by \\(C_c(\\mathbb R)\\), and multiply the approximants by a fixed \\(\\chi_n\\). The error does not increase; dominated convergence then gives \\(C_c(I)\\)-density. For an explicit smooth kernel set\n\\[\n\\beta(t)=\n\\begin{cases}\n\\exp\\big(-1/(1-t^2)\\big),&|t|<1,\\\\\n0,&|t|\\geq1.\n\\end{cases}\n\\]\nInside \\((-1,1)\\), every derivative is the exponential times a rational function with a finite power of \\(1-t^2\\) in its denominator. These derivatives tend to zero at the endpoints: \\(u^Ne^{-u}\\to0\\) as \\(u\\to\\infty\\), since the exponential series bounds \\(e^u\\) below by \\(u^{N+1}/(N+1)!\\). Inductively the extensions of all derivatives are continuous with zero endpoint values, so \\(\\beta\\) is smooth on the line. Normalize its positive finite integral and rescale to obtain \\(\\beta_\\varepsilon\\), supported in \\([-\\varepsilon,\\varepsilon]\\) with integral one. For \\(f\\in C_c(I)\\), differentiation of \\(f*\\beta_\\varepsilon\\) falls on the kernel; compact parameter differentiation from the Cauchy lemma justifies every derivative. For small \\(\\varepsilon\\) the support remains inside \\(I\\), and uniform continuity gives uniform convergence to \\(f\\). A common finite-length compact support gives convergence in \\(L^2\\).\n\nTranslations \\(\\tau_tf(x)=f(x-t)\\) are isometries by (b). For \\(f\\in C_c(\\mathbb R)\\), uniform continuity and a common compact support give \\(\\|\\tau_tf-f\\|_2\\to0\\) as \\(t\\to0\\); density and the isometry extend this to every \\(f\\in L^2\\). Cauchy–Schwarz with the probability density \\(\\alpha_j\\), followed by Tonelli, gives\n\\[\n\\|f*\\alpha_j-f\\|_2^2\n\\leq\\int\\alpha_j(t)\\|\\tau_tf-f\\|_2^2\\,dt.\n\\]\nIt also shows that the convolution integral exists almost everywhere. Choose a neighbourhood of zero on which the squared norm in the integrand is small, and bound it on the complement by \\(4\\|f\\|_2^2\\). The concentration hypothesis finishes the proof. \\(\\square\\)\n\n**Lemma 0.2 (Gaussian transform).** For \\(\\sigma>0\\) the function\n\\[\n\\gamma_\\sigma(x)=\\frac{1}{\\sqrt{2\\pi}\\,\\sigma}\n\\exp\\left(-\\frac{x^2}{2\\sigma^2}\\right)\n\\]\nhas integral one and transform\n\\[\n\\int_{\\mathbb R}e^{-itx}\\gamma_\\sigma(x)\\,dx\n=e^{-\\sigma^2t^2/2}.\n\\]\nIts mass outside any fixed neighbourhood of zero tends to zero as \\(\\sigma\\downarrow0\\).\n\n*Proof.* The integral \\(J=\\int e^{-x^2}\\,dx\\) is finite, since outside \\([-1,1]\\) its integrand is bounded by \\(e^{-|x|}\\). Tonelli gives \\(J^2=\\int_{\\mathbb R^2}e^{-x^2-y^2}\\,dx\\,dy\\). The disc of radius \\(r\\) has area\n\\[\n\\int_{-r}^r2\\sqrt{r^2-x^2}\\,dx=\\pi r^2:\n\\]\nits sections give the first expression by Tonelli, and \\(x=r\\sin\\theta\\), \\(-\\pi/2\\leq\\theta\\leq\\pi/2\\), gives the second by the elementary substitution formula. Write \\(e^{-x^2-y^2}=\\int_0^1 1_{\\{u<e^{-x^2-y^2}\\}}\\,du\\) and use Tonelli again. Then\n\\[\nJ^2=\\int_0^1\\pi(-\\log u)\\,du=\\pi.\n\\]\nThe last integral follows by integrating on \\([\\delta,1]\\) and letting \\(\\delta\\downarrow0\\); \\(\\delta\\log\\delta\\to0\\) by the same exponential estimate. Since \\(J>0\\), \\(J=\\sqrt\\pi\\). Scaling gives the stated normalization.\n\nLet \\(\\phi(t)=\\int e^{-itx}\\gamma_\\sigma(x)\\,dx\\). Dominated convergence differentiates under the integral, since the difference quotients are bounded by \\(|x|\\gamma_\\sigma(x)\\), which is integrable. Since \\(\\gamma_\\sigma'(x)=-x\\gamma_\\sigma(x)/\\sigma^2\\), integration by parts first on \\([-R,R]\\) and then letting \\(R\\to\\infty\\) gives\n\\[\n\\phi'(t)=i\\sigma^2\\int e^{-itx}\\gamma_\\sigma'(x)\\,dx\n=-\\sigma^2t\\phi(t).\n\\]\nThe boundary terms tend to zero and both improper integrals converge absolutely. Differentiating \\(e^{\\sigma^2t^2/2}\\phi(t)\\) shows it is constant, equal to \\(\\phi(0)=1\\). Finally, the substitution \\(x=\\sigma v\\) identifies the tail mass with \\(\\int_{|v|>\\delta/\\sigma}\\gamma_1(v)\\,dv\\), which tends to zero by integrability. \\(\\square\\)\n\n**Theorem 0.3 (real-line Plancherel).** For \\(f\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\), put \\(\\widehat f(t)=\\int e^{-itx}f(x)\\,dx\\). Then\n\\[\n\\int|\\widehat f(t)|^2\\,dt=2\\pi\\int|f(x)|^2\\,dx.\n\\]\nThe operator \\(T:f\\mapsto(2\\pi)^{-1/2}\\widehat f\\) extends uniquely to a unitary operator on \\(L^2(\\mathbb R)\\).\n\n*Proof.* For \\(\\varepsilon>0\\), Fubini and Lemma 0.2 give\n\\[\n\\begin{gathered}\n\\int e^{-\\varepsilon t^2}|\\widehat f(t)|^2\\,dt\n\\\\\n=\\sqrt{\\frac\\pi\\varepsilon}\n\\int\\!\\int f(x)\\overline{f(y)}\\\\\n{}\\cdot e^{-(x-y)^2/(4\\varepsilon)}\\,dx\\,dy\\\\\n\\\\\n=2\\pi\\langle f,f*\\gamma_{\\sqrt{2\\varepsilon}}\\rangle.\n\\end{gathered}\n\\]\nThe interchange is justified before computing the inner integral: the integral of the absolute value of the three-variable integrand is\n\\(\\|f\\|_1^2\\int e^{-\\varepsilon t^2}\\,dt<\\infty\\).\nThe last convolution converges to \\(f\\) in \\(L^2\\) by Lemmas 0.1(c) and 0.2. On the left let \\(\\varepsilon=1/n\\); monotone convergence proves the norm identity. The domain contains \\(C_c^\\infty\\) and is dense in \\(L^2\\), so completeness extends \\(T\\) to an isometry with closed range. Polarization gives inner-product preservation.\n\nWe prove surjectivity as well. If \\(g\\in C_c^\\infty(\\mathbb R)\\), two integrations by parts give\n\\[\n|\\widehat g(t)|\\leq\n\\min\\big(\\|g\\|_1,\\,|t|^{-2}\\|g''\\|_1\\big)\n\\quad(t\\ne0).\n\\]\nThus \\(\\widehat g\\in L^1\\cap L^2\\). Fubini and the Gaussian formula show that\n\\[\n\\frac1{2\\pi}\\int e^{itx}e^{-\\varepsilon t^2}\n\\widehat g(t)\\,dt\n=(g*\\gamma_{\\sqrt{2\\varepsilon}})(x).\n\\]\nThe absolute-integrability justification is \\(\\|g\\|_1\\int e^{-\\varepsilon t^2}\\,dt<\\infty\\). The right side tends to \\(g(x)\\) for every \\(x\\), by bounded uniform continuity and Gaussian concentration. Dominated convergence on the left, using \\(\\widehat g\\in L^1\\), gives\n\\[\ng(x)=\\frac1{2\\pi}\\int e^{itx}\\widehat g(t)\\,dt.\n\\]\nConsequently \\(T^2g(x)=g(-x)\\). The range of \\(T\\) therefore contains every reflection of a function in \\(C_c^\\infty\\), a dense subspace. Its closed range is all of \\(L^2\\). \\(\\square\\)\n\nLemmas 0.1–0.3 give the measure, product-integration and Fourier arguments used below, including their hypotheses, sign and normalization. The exact earlier programme proofs used in Lemma 0.1 are identified there.\n\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    },
    {
      "id": "OA-FND-LT-20",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "name": "Diagonalization with a Schatten perturbation",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
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      "full_conditions_and_proof": "### Diagonalization with a Schatten perturbation\n\nFor a compact operator set \\(N_p(x)=(\\sum_ns_n(x)^p)^{1/p}\\), allowing infinity. The next proof uses approximation numbers and scalar Minkowski; it does not assume completeness or a triangle inequality for the operator quantity \\(N_p\\).\n\n**Proposition 11.18 (Kuroda's extension).** If \\(1<p<\\infty\\), \\(H\\) is separable, \\(h\\in B(H)_{\\mathrm{sa}}\\), and \\(\\varepsilon>0\\), there are a self-adjoint compact operator \\(a\\) with \\(N_p(a)<\\varepsilon\\) and a self-adjoint diagonal operator \\(k\\) such that \\(h=k+a\\).\n\n*Proof.* We first justify the series estimate needed for the construction. For compact \\(x_1,\\ldots,x_N\\), put \\(\\lambda_j=2^{-j}\\) and\n\\(k_j(n)=\\lfloor\\lambda_j(n-1)\\rfloor+1\\).\nSince \\(\\sum_{j=1}^N\\lambda_j<1\\),\n\\(\\sum_j(k_j(n)-1)\\leq n-1\\).\nChoose operators of rank at most \\(k_j(n)-1\\) approximating \\(x_j\\) within \\(s_{k_j(n)}(x_j)+\\delta\\), using the approximation formula (2.2) in Proposition 2.5. Their sum has rank at most \\(n-1\\); applying that formula again and letting \\(\\delta\\downarrow0\\) gives\n\\[\ns_n\\left(\\sum_{j=1}^Nx_j\\right)\n\\leq\\sum_{j=1}^Ns_{k_j(n)}(x_j).\n\\]\nEach integer value of \\(k_j(n)\\) occurs exactly \\(2^j\\) times as \\(n\\) runs through the positive integers. Scalar Minkowski, [Theorem 3.1 of the measure-tools lesson](measure-and-hilbert-space-tools.md#3-the-complete-spaces-of-integrable-functions) for counting measure, therefore yields\n\\[\nN_p\\left(\\sum_{j=1}^Nx_j\\right)\n\\leq\\sum_{j=1}^N2^{j/p}N_p(x_j).\n\\]\nIf \\(\\sum_j2^{j/p}N_p(x_j)<\\infty\\), then \\(\\sum_j\\|x_j\\|<\\infty\\), since \\(\\|x_j\\|=s_1(x_j)\\leq N_p(x_j)\\). The operator-norm sum \\(x\\) is compact. Proposition 2.5(d) gives convergence of every singular value of the partial sums to that of \\(x\\). Taking any finite number of singular values, passing to the limit, and then taking their supremum proves\n\\[\nN_p(x)\\leq\\sum_j2^{j/p}N_p(x_j).\n\\]\n\nNow refine Lemma 11.1. With its notation, \\(y_j=y_jp_j\\) has norm at most \\(r/m\\), its initial space lies in \\(p_jH\\), and its range lies in \\(e_jH\\). The initial spaces and ranges are each mutually orthogonal. Hence, for \\(b=(1-p)hp=\\sum_jy_j\\),\n\\[\n\\begin{gathered}\n\\|b\\zeta\\|^2\\\\\n=\\sum_j\\|y_jp_j\\zeta\\|^2\n\\\\\n\\leq(r/m)^2\\sum_j\\|p_j\\zeta\\|^2\n\\\\\n\\leq(r/m)^2\\|\\zeta\\|^2.\n\\end{gathered}\n\\]\nThus \\(\\|b\\|\\leq r/m\\) and \\(\\operatorname{rank}b\\leq m\\). The self-adjoint off-diagonal correction \\(b+b^*\\) has rank at most \\(2m\\) and norm at most \\(2r/m\\), so\n\\[\nN_p(b+b^*)\\leq2^{1+1/p}r\\,m^{1/p-1}.\n\\]\nFor \\(p>1\\) this tends to zero as \\(m\\to\\infty\\).\n\nRepeat the projection construction of Theorem 11.2, choosing the integer \\(m\\) at its \\(j\\)-th step large enough that the resulting correction satisfies\n\\(N_p(a_j)\\leq\\varepsilon 2^{-j(1+1/p)-2}\\).\nThe compressed self-adjoint operator has norm at most \\(\\|h\\|\\), so the preceding bound permits this choice; a zero compression gives correction zero. The finite-rank projections still contain the successive members of a dense sequence and the same block identity (ii) holds. The series estimate gives an operator-norm sum \\(a=\\sum_ja_j\\), self-adjoint and compact, with\n\\[\nN_p(a)\\leq\\sum_j2^{j/p}N_p(a_j)\n\\leq\\frac\\varepsilon4\\sum_{j=1}^\\infty2^{-j}\n=\\frac\\varepsilon4<\\varepsilon.\n\\]\nFor \\(k=h-a\\), the block identity passes to the operator-norm limit. Each finite-dimensional block reduces \\(k\\), their orthogonal sum is \\(H\\), and the compact self-adjoint spectral theorem on each block supplies an orthonormal eigenbasis. Their union diagonalizes \\(k\\), exactly as in Theorem 11.2. \\(\\square\\)\n\nThe restriction \\(p>1\\) is essential: the exponent \\(1/p-1\\) becomes zero at \\(p=1\\), and Remark 11.17 gives a spectral obstruction to trace-class diagonalization. A human reference is S. T. Kuroda, *On a theorem of Weyl–von Neumann*, 1958.\n\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-01",
      "unit": "the-double-commutant-theorem",
      "name": "1. Operator topologies",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
      "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "anchor": "oa-fnd-bi-01",
      "proof_locus": {
        "line": 39,
        "through_line": 105
      },
      "full_conditions_and_proof": "## 1. Operator topologies\n\nSix topologies on \\(B(H)\\) are used throughout. This section compares them and shows that on linear subspaces they produce at most two different closures.\n\n**Definition 1.1.** Each of the six topologies below is the locally convex topology on \\(B(H)\\) given by a family of seminorms. In the table, \\((\\xi_n)\\) and \\((\\eta_n)\\) are sequences in \\(H\\) with \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\) and \\(\\sum_n\\|\\eta_n\\|^2<\\infty\\).\n\n| topology | seminorms |\n|---|---|\n| weak | \\(x\\mapsto\\lvert\\langle x\\xi,\\eta\\rangle\\rvert\\) |\n| strong | \\(x\\mapsto\\lVert x\\xi\\rVert\\) |\n| strong\\(^*\\) | \\(x\\mapsto(\\lVert x\\xi\\rVert^2+\\lVert x^*\\xi\\rVert^2)^{1/2}\\) |\n| \\(\\sigma\\)-weak | \\(x\\mapsto\\lvert\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\rvert\\) |\n| \\(\\sigma\\)-strong | \\(x\\mapsto(\\sum_n\\lVert x\\xi_n\\rVert^2)^{1/2}\\) |\n| \\(\\sigma\\)-strong\\(^*\\) | \\(x\\mapsto(\\sum_n\\lVert x\\xi_n\\rVert^2+\\lVert x^*\\xi_n\\rVert^2)^{1/2}\\) |\n\nThe \\(\\sigma\\)-weak topology is also called the *ultraweak* topology. With \\(\\omega=\\sum_n\\omega_{\\xi_n}\\), the \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) seminorms read \\(\\omega(x^*x)^{1/2}\\) and \\((\\omega(x^*x)+\\omega(xx^*))^{1/2}\\). By Theorem 10.1, every positive \\(\\sigma\\)-weakly continuous functional on \\(B(H)\\) has this form \\(\\omega\\). For a set \\(S\\subseteq B(H)\\) we write \\(\\overline S^{\\,w}\\), \\(\\overline S^{\\,s}\\), \\(\\overline S^{\\,s*}\\), \\(\\overline S^{\\,\\sigma w}\\), \\(\\overline S^{\\,\\sigma s}\\), \\(\\overline S^{\\,\\sigma s*}\\) for its closures.\n\n**Lemma 1.2** (elementary facts).\n\n(a) *One seminorm is enough.* In the \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, finitely many seminorms are dominated by a single seminorm of the same kind: concatenate the sequences. In the strong and strong\\(^*\\) topologies, the seminorms of finitely many vectors \\(\\xi_1,\\dots,\\xi_m\\) are dominated by \\((\\sum_{j=1}^m\\lVert x\\xi_j\\rVert^2)^{1/2}\\), respectively \\((\\sum_{j=1}^m\\lVert x\\xi_j\\rVert^2+\\lVert x^*\\xi_j\\rVert^2)^{1/2}\\), the \\(\\sigma\\)-type seminorm of a finite sequence. So in each of the four strong-type topologies, every neighbourhood of \\(a\\) contains a set \\(\\{x:p(x-a)<\\varepsilon\\}\\) for one such seminorm \\(p\\).\n\n(b) *Comparisons.* A sequence with one nonzero term shows that each \\(\\sigma\\)-topology is finer than its plain counterpart. The Cauchy–Schwarz inequality \\(\\lvert\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\rvert\\le(\\sum_n\\|x\\xi_n\\|^2)^{1/2}(\\sum_n\\|\\eta_n\\|^2)^{1/2}\\) shows that the \\(\\sigma\\)-strong topology is finer than the \\(\\sigma\\)-weak one; in the same way the strong topology is finer than the weak one. Each starred topology is finer than its unstarred version. So the weak topology is the coarsest of the six and the \\(\\sigma\\)-strong\\(^*\\) topology is the finest, and all six are coarser than the norm topology. A finer topology has smaller closures:\n\\[\n\\begin{gathered}\n\\overline S^{\\,\\sigma s*}\\subseteq\\overline S^{\\,\\sigma s}\\subseteq\\overline S^{\\,\\sigma w}\\subseteq\\overline S^{\\,w},\\\\\n\\overline S^{\\,\\sigma s*}\\subseteq\\overline S^{\\,s*}\\subseteq\\overline S^{\\,s}\\subseteq\\overline S^{\\,w},\\\\\n\\overline S^{\\,\\sigma s}\\subseteq\\overline S^{\\,s}.\n\\end{gathered}\n\\]\n\n(c) *Continuity of the operations.* Addition and scalar multiplication are continuous. For fixed \\(a,b\\in B(H)\\), the map \\(x\\mapsto axb\\) is continuous in all six topologies. Indeed \\(\\|axb\\xi_n\\|\\le\\|a\\|\\,\\|x(b\\xi_n)\\|\\), \\(\\langle axb\\xi_n,\\eta_n\\rangle=\\langle x(b\\xi_n),a^*\\eta_n\\rangle\\), and \\(\\|(axb)^*\\xi_n\\|\\le\\|b\\|\\,\\|x^*(a^*\\xi_n)\\|\\), where \\((b\\xi_n)\\) and \\((a^*\\eta_n)\\) are again square summable. The adjoint \\(x\\mapsto x^*\\) is continuous for the weak, \\(\\sigma\\)-weak, strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topologies, because it maps their seminorms to seminorms of the same kind. (When \\(\\dim H=\\infty\\) it is not \\(\\sigma\\)-strongly continuous. For an orthonormal sequence \\((\\xi_n)\\), \\(\\theta_{\\xi_1,\\xi_n}\\to0\\) \\(\\sigma\\)-strongly, while \\(\\|\\theta_{\\xi_1,\\xi_n}^*\\xi_1\\|=1\\). The same example shows that it is not strongly continuous. We do not need this.)\n\n(d) *Bounded nets.* Let \\(x_\\lambda\\to x\\) strongly, with \\(C=\\sup_\\lambda\\|x_\\lambda\\|+\\|x\\|<\\infty\\). Then \\(x_\\lambda\\to x\\) \\(\\sigma\\)-strongly. Given \\((\\xi_n)\\) and \\(\\varepsilon>0\\), choose \\(N\\) with \\(C^2\\sum_{n>N}\\|\\xi_n\\|^2<\\varepsilon^2/2\\); then \\(\\sum_n\\|(x_\\lambda-x)\\xi_n\\|^2<\\varepsilon^2\\) as soon as the finitely many terms with \\(n\\le N\\) add up to less than \\(\\varepsilon^2/2\\). If moreover \\(x_\\lambda^*\\to x^*\\) strongly, the same argument gives \\(\\sigma\\)-strong\\(^*\\) convergence. A bounded weakly convergent net converges \\(\\sigma\\)-weakly, by the same tail estimate.\n\n(e) *Increasing nets.* If \\(0\\le x_\\lambda\\) increases and \\(\\sup_\\lambda\\|x_\\lambda\\|<\\infty\\), then \\(x_\\lambda\\) converges strongly to its least upper bound, by Vigier's theorem (V). The operators are self-adjoint, so by (d) the convergence is also \\(\\sigma\\)-strong\\(^*\\) and \\(\\sigma\\)-weak.\n\n**Lemma 1.3** (closures of subspaces). Let \\(S\\subseteq B(H)\\) be a complex linear subspace. Then\n\n1. \\(\\overline S^{\\,w}=\\overline S^{\\,s}=\\overline S^{\\,s*}\\), and\n2. \\(\\overline S^{\\,\\sigma w}=\\overline S^{\\,\\sigma s}=\\overline S^{\\,\\sigma s*}\\).\n\n**Proof.** (2) By Lemma 1.2(b), \\(\\overline S^{\\,\\sigma s*}\\subseteq\\overline S^{\\,\\sigma s}\\subseteq\\overline S^{\\,\\sigma w}\\). For the reverse inclusion let \\(a\\in\\overline S^{\\,\\sigma w}\\), let \\((\\xi_n)\\) be square summable, and let \\(\\varepsilon>0\\). By Lemma 1.2(a), it is enough to find \\(x\\in S\\) with \\(\\sum_n\\|(a-x)\\xi_n\\|^2+\\|(a-x)^*\\xi_n\\|^2<\\varepsilon^2\\).\n\nLet \\(\\mathcal K=\\ell^2(\\mathbb N;H)\\oplus\\ell^2(\\mathbb N;H)\\). We regard it as a *real* Hilbert space, with inner product \\(\\operatorname{Re}\\langle\\cdot,\\cdot\\rangle\\). The map \\(\\Phi(x)=((x\\xi_n)_n,(x^*\\xi_n)_n)\\) from \\(B(H)\\) to \\(\\mathcal K\\) is real linear, and \\(\\|\\Phi(x)\\|^2\\le2\\|x\\|^2\\sum_n\\|\\xi_n\\|^2\\). It is not complex linear, because of the adjoint. \\(\\Phi(S)\\) is a real subspace of \\(\\mathcal K\\). By the projection theorem (H), its closure is the set of vectors that are real-orthogonal to \\(\\Phi(S)^\\perp\\), the real orthogonal complement.\n\nTake \\((\\eta,\\zeta)\\in\\Phi(S)^\\perp\\), with \\(\\eta=(\\eta_n)\\) and \\(\\zeta=(\\zeta_n)\\). For \\(x\\in S\\) put \\(A(x)=\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\) and \\(B(x)=\\sum_n\\langle x^*\\xi_n,\\zeta_n\\rangle\\). Orthogonality to \\(\\Phi(x)\\) says \\(\\operatorname{Re}A(x)+\\operatorname{Re}B(x)=0\\). The operator \\(ix\\) also lies in \\(S\\), and \\((ix)^*=-ix^*\\). So orthogonality to \\(\\Phi(ix)\\) says \\(\\operatorname{Re}(iA(x))+\\operatorname{Re}(-iB(x))=0\\), that is, \\(\\operatorname{Im}A(x)=\\operatorname{Im}B(x)\\). Together, \\(A(x)=-\\overline{B(x)}\\). Since \\(\\overline{B(x)}=\\sum_n\\langle\\zeta_n,x^*\\xi_n\\rangle=\\sum_n\\langle x\\zeta_n,\\xi_n\\rangle\\), the complex linear functional\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\sum_n\\langle x\\xi_n,\\eta_n\\rangle+\\sum_n\\langle x\\zeta_n,\\xi_n\\rangle\n\\end{gathered}\n\\tag{1.1}\n\\]\nvanishes on \\(S\\). It is \\(\\sigma\\)-weakly continuous, being the sum of two functionals of the form \\(\\sum_n\\langle x\\alpha_n,\\beta_n\\rangle\\) with square-summable sequences. Since \\(f\\) vanishes on \\(S\\) and \\(a\\) lies in the \\(\\sigma\\)-weak closure of \\(S\\), also \\(f(a)=0\\). Reading the computation backwards, \\(\\operatorname{Re}A(a)+\\operatorname{Re}B(a)=0\\): the vector \\(\\Phi(a)\\) is real-orthogonal to \\((\\eta,\\zeta)\\). As \\((\\eta,\\zeta)\\) was arbitrary, \\(\\Phi(a)\\) lies in the closure of \\(\\Phi(S)\\). Choose \\(x\\in S\\) with \\(\\|\\Phi(a)-\\Phi(x)\\|<\\varepsilon\\). Since \\(\\Phi(a)-\\Phi(x)=\\Phi(a-x)\\), this is the required estimate.\n\n(1) Run the same argument with finitely many vectors \\(\\xi_1,\\dots,\\xi_m\\) in place of a square-summable sequence, starting from \\(a\\) in the weak closure of \\(S\\). The functional \\(f\\) is then weakly continuous, so again \\(f(a)=0\\). \\(\\square\\)\n\n*Remark 1.4 (unused extension, not proved here).* The equalities of part (2) also hold for convex sets, by the separation theory of locally convex spaces. This stronger statement is background context and is never used in this lesson. The linear-subspace result needed below has the full real-Hilbert projection proof above. The two triples of closures can differ for a subspace: [Exercise 1.5](#oa-fnd-bi-01) gives a subspace that is \\(\\sigma\\)-weakly closed but not weakly closed. For \\(*\\)-subalgebras all six closures agree ([Theorem 4.4](#oa-fnd-bi-07)).\n\n**Exercise 1.5** (medium; the two triples of closures differ for subspaces). On \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_n)\\), let \\(\\omega(x)=\\sum_n2^{-n}\\langle x\\delta_n,\\delta_n\\rangle\\). Show that \\(\\ker\\omega\\) is \\(\\sigma\\)-weakly closed but not weakly closed. So Lemma 1.3 cannot be extended to all six topologies for subspaces, while [Theorem 4.4](#oa-fnd-bi-07) does this for \\(*\\)-subalgebras.\n\n*Solution.* \\(\\omega\\) is \\(\\sigma\\)-weakly continuous, so its kernel is \\(\\sigma\\)-weakly closed.\n\n*A linear functional with closed kernel is continuous.* Let \\(f\\neq0\\) be a linear functional on \\(B(H)\\) whose kernel is weakly closed, and pick \\(x_0\\) with \\(f(x_0)=1\\). There are weak seminorms \\(p_1,\\dots,p_k\\) and \\(\\delta>0\\) such that the set \\(x_0+U\\), where \\(U=\\{x:p_j(x)<\\delta\\text{ for all }j\\}\\), misses \\(\\ker f\\). The set \\(U\\) is balanced: \\(tU\\subseteq U\\) for \\(|t|\\le1\\). If some \\(u\\in U\\) had \\(|f(u)|\\ge1\\), then \\(u'=-u/f(u)\\in U\\) and \\(f(x_0+u')=0\\), which is impossible. So \\(|f|<1\\) on \\(U\\), and by homogeneity \\(|f(x)|\\le\\delta^{-1}\\max_jp_j(x)\\). So \\(f\\) is weakly continuous.\n\n*\\(\\omega\\) is not weakly continuous.* Otherwise there would be vectors \\(\\xi_j,\\eta_j\\) (\\(j=1,\\dots,k\\)) and \\(C>0\\) with \\(|\\omega(x)|\\le C\\max_j|\\langle x\\xi_j,\\eta_j\\rangle|\\). The span of \\(\\delta_1,\\dots,\\delta_{k+1}\\) has dimension \\(k+1\\), so it contains a nonzero \\(v\\) orthogonal to \\(\\xi_1,\\dots,\\xi_k\\). For \\(x=\\theta_{v,v}\\), \\(\\langle x\\xi_j,\\eta_j\\rangle=\\langle\\xi_j,v\\rangle\\langle v,\\eta_j\\rangle=0\\) for all \\(j\\), while \\(\\omega(x)=\\sum_n2^{-n}|\\langle v,\\delta_n\\rangle|^2>0\\). This is a contradiction. By the previous paragraph, \\(\\ker\\omega\\) is not weakly closed. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-02",
      "unit": "the-double-commutant-theorem",
      "name": "2. Commutants and von Neumann algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "anchor": "oa-fnd-bi-02",
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        "line": 106,
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      "full_conditions_and_proof": "## 2. Commutants and von Neumann algebras\n\nThe commutant of a set of operators is the algebraic side of the theorem. This section collects its basic properties and the definitions built on it.\n\nFor \\(S\\subseteq B(H)\\), the *commutant* is \\(S'=\\{y\\in B(H):ys=sy\\text{ for all }s\\in S\\}\\), and \\(S''=(S')'\\), \\(S'''=(S'')'\\), and so on.\n\n**Proposition 2.1.** Let \\(S,T\\subseteq B(H)\\).\n\n1. \\(S'\\) is a subalgebra of \\(B(H)\\) that contains \\(1\\). It is closed in the weak topology, hence in each of the six topologies of Section 1 and in norm.\n2. \\(S\\subseteq S''\\). If \\(S\\subseteq T\\), then \\(T'\\subseteq S'\\). Moreover \\(S'=S'''\\). So all commutants of odd order equal \\(S'\\), and all commutants of even order, from the second on, equal \\(S''\\).\n3. \\((S\\cup T)'=S'\\cap T'\\).\n4. If \\(S^*=S\\), then \\(S'\\) is a \\(*\\)-subalgebra with \\((S')''=S'\\). In particular \\(S''\\) is a \\(*\\)-subalgebra with \\((S'')''=S''\\) that contains \\(S\\). Every \\(*\\)-subalgebra \\(N\\) with \\(S\\subseteq N=N''\\) contains \\(S''\\).\n5. (*Invariant subspaces.*) Let \\(S^*=S\\), let \\(L\\subseteq H\\) be a closed subspace, and let \\(p\\) be its projection. Then \\(SL\\subseteq L\\) if and only if \\(p\\in S'\\). In that case \\(SL^\\perp\\subseteq L^\\perp\\) as well.\n\n**Proof.** (1) If \\(y_1,y_2\\in S'\\) and \\(s\\in S\\), then \\(y_1y_2s=y_1sy_2=sy_1y_2\\). Linear combinations are handled the same way, and \\(1\\in S'\\). Let \\(y_\\lambda\\in S'\\) converge weakly to \\(y\\). For \\(s\\in S\\) and \\(\\xi,\\eta\\in H\\),\n\\[\n\\begin{gathered}\n\\langle(ys-sy)\\xi,\\eta\\rangle\\\\\n=\\lim_\\lambda\\big(\\langle y_\\lambda(s\\xi),\\eta\\rangle-\\langle y_\\lambda\\xi,s^*\\eta\\rangle\\big)\\\\\n=\\lim_\\lambda\\langle(y_\\lambda s-sy_\\lambda)\\xi,\\eta\\rangle\\\\\n=0 .\n\\end{gathered}\n\\]\nSo \\(y\\in S'\\). The other five topologies and the norm topology are finer than the weak one ([Lemma 1.2](#oa-fnd-bi-01)(b)), so \\(S'\\) is closed in them too.\n\n(2) Each \\(s\\in S\\) commutes with every element of \\(S'\\), so \\(s\\in S''\\). The reversal of inclusions is immediate from the definition. Applying the first statement to \\(S'\\) gives \\(S'\\subseteq S'''\\). Applying the reversal to \\(S\\subseteq S''\\) gives \\(S'''\\subseteq S'\\).\n\n(3) This is the definition.\n\n(4) Let \\(y\\in S'\\) and \\(s\\in S\\). Then \\(s^*\\in S\\), so \\(ys^*=s^*y\\). Taking adjoints, \\(sy^*=y^*s\\). So \\(y^*\\in S'\\), and \\(S'\\) is a \\(*\\)-subalgebra by (1). Next, \\((S')''=(S'')'=S'''=S'\\) by (2). The set \\(S'\\) is self-adjoint, so the same statements hold for \\(S''=(S')'\\). Finally, if \\(S\\subseteq N=N''\\), then (2) gives \\(S''\\subseteq N''=N\\).\n\n(5) Suppose \\(SL\\subseteq L\\). For \\(\\eta\\in L^\\perp\\), \\(\\zeta\\in L\\) and \\(s\\in S\\) we have \\(\\langle s\\eta,\\zeta\\rangle=\\langle\\eta,s^*\\zeta\\rangle=0\\), because \\(s^*\\in S\\). So \\(SL^\\perp\\subseteq L^\\perp\\). Hence, for every \\(\\xi\\), \\(sp\\xi\\in L\\) and \\(s(1-p)\\xi\\in L^\\perp\\). The first gives \\(psp=sp\\), the second \\(ps(1-p)=0\\), that is \\(ps=psp\\). So \\(ps=sp\\). Conversely, if \\(p\\in S'\\), then \\(s(p\\xi)=p(s\\xi)\\in L\\). \\(\\square\\)\n\n*Remark 2.2* (non-self-adjoint sets). Part (4) needs \\(S^*=S\\). On \\(\\mathbb C^2\\) let \\(N\\) be the matrix unit with \\(Ne_2=e_1\\) and \\(Ne_1=0\\), and \\(S=\\{N\\}\\). A direct computation gives \\(S'=\\{\\alpha1+\\beta N\\}\\), and then \\(S''=\\{1,N\\}'=S'\\). This algebra contains \\(N\\) but not \\(N^*\\).\n\n**Definition 2.3.** A *\\(*\\)-subalgebra* \\(M\\) of \\(B(H)\\) is a subalgebra closed under adjoints. It is *nondegenerate* if \\([MH]=H\\). We call a \\(*\\)-subalgebra \\(M\\) with \\(M=M''\\) a *von Neumann algebra*, and write \\(\\{M,H\\}\\) when the Hilbert space matters. The *centre* of \\(M\\) is \\(Z(M)=M\\cap M'\\), and \\(M\\) is a *factor* if \\(Z(M)=\\mathbb C1\\). For a self-adjoint set \\(S\\subseteq B(H)\\) we say that \\(S''\\) is *generated* by \\(S\\). By [Proposition 2.1](#oa-fnd-bi-02)(4), \\(S''\\) is a von Neumann algebra, and every von Neumann algebra that contains \\(S\\) also contains \\(S''\\).\n\nTwo von Neumann algebras \\(\\{M_1,H_1\\}\\) and \\(\\{M_2,H_2\\}\\) are *spatially isomorphic* if there is a unitary \\(U:H_1\\to H_2\\) (an isometry of \\(H_1\\) onto \\(H_2\\)) with \\(UM_1U^*=M_2\\); the map \\(x\\mapsto UxU^*\\) is then a *spatial isomorphism*. They are *isomorphic* if there is a bijective linear map \\(\\pi:M_1\\to M_2\\) that is multiplicative and satisfies \\(\\pi(x^*)=\\pi(x)^*\\). A spatial isomorphism is an isomorphism, but not conversely ([Example 5.5](#oa-fnd-bi-15)).\n\n**Proposition 2.4.**\n\n1. For a \\(*\\)-subalgebra \\(M\\), \\([MH]^\\perp=\\{\\xi\\in H:x\\xi=0\\text{ for all }x\\in M\\}\\). So \\(M\\) is nondegenerate exactly when no nonzero vector is killed by all of \\(M\\). Every \\(*\\)-subalgebra that contains \\(1\\) is nondegenerate.\n2. A von Neumann algebra \\(M\\) contains \\(1\\), is nondegenerate, and is closed in norm and in all six topologies. So it is a nondegenerate norm-closed \\(*\\)-algebra. \\(M'\\) is a von Neumann algebra, \\(Z(M)\\) is a commutative von Neumann algebra, and \\(Z(M')=Z(M)\\). In particular \\(M\\) is a factor if and only if \\(M'\\) is.\n3. A spatial isomorphism carries commutants to commutants and centres to centres: \\(UM_1'U^*=M_2'\\) and \\(UZ(M_1)U^*=Z(M_2)\\).\n4. \\(B(H)'=\\mathbb C1\\). So \\(B(H)\\) and \\(\\mathbb C1\\) are factors, each the commutant of the other.\n\n**Proof.** (1) A vector \\(\\eta\\) is orthogonal to every \\(x\\xi\\) exactly when \\(x^*\\eta=0\\) for every \\(x\\in M\\). As \\(M^*=M\\), this says \\(x\\eta=0\\) for every \\(x\\in M\\). If \\(1\\in M\\), then \\(\\eta=1\\eta=0\\).\n\n(2) \\(1\\in M'\\) gives \\(1\\in M''=M\\), and (1) gives nondegeneracy. \\(M=(M')'\\) is a commutant, so it is closed by [Proposition 2.1](#oa-fnd-bi-02)(1). \\(M'\\) is the commutant of the self-adjoint set \\(M\\), so it is a von Neumann algebra by [Proposition 2.1](#oa-fnd-bi-02)(4). Also \\(Z(M)=M''\\cap M'=(M'\\cup M)'\\) by [Proposition 2.1](#oa-fnd-bi-02)(3); this is the commutant of a self-adjoint set, hence a von Neumann algebra. It is commutative, because elements of \\(M'\\) commute with elements of \\(M\\). Finally \\(Z(M')=M'\\cap M''=M'\\cap M=Z(M)\\).\n\n(3) For \\(y\\in B(H_1)\\), \\(y\\) commutes with every \\(x\\in M_1\\) if and only if \\(UyU^*\\) commutes with every \\(UxU^*\\in M_2\\). So \\(UM_1'U^*=M_2'\\), and intersecting with \\(UM_1U^*=M_2\\) gives the centres.\n\n(4) If \\(H=\\{0\\}\\) there is nothing to prove. Let \\(y\\in B(H)'\\). For \\(\\xi,\\eta\\in H\\), \\(y\\theta_{\\xi,\\eta}=\\theta_{y\\xi,\\eta}\\) and \\(\\theta_{\\xi,\\eta}y=\\theta_{\\xi,y^*\\eta}\\), so \\(\\theta_{y\\xi,\\eta}=\\theta_{\\xi,y^*\\eta}\\). Fix a unit vector \\(\\eta\\) and apply both operators to \\(\\eta\\): \\(y\\xi=\\langle\\eta,y^*\\eta\\rangle\\xi=\\langle y\\eta,\\eta\\rangle\\xi\\) for every \\(\\xi\\). So \\(y\\) is a scalar. Trivially \\((\\mathbb C1)'=B(H)\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-03",
      "unit": "the-double-commutant-theorem",
      "name": "2. Commutants and von Neumann algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 2. Commutants and von Neumann algebras\n\nThe commutant of a set of operators is the algebraic side of the theorem. This section collects its basic properties and the definitions built on it.\n\nFor \\(S\\subseteq B(H)\\), the *commutant* is \\(S'=\\{y\\in B(H):ys=sy\\text{ for all }s\\in S\\}\\), and \\(S''=(S')'\\), \\(S'''=(S'')'\\), and so on.\n\n**Proposition 2.1.** Let \\(S,T\\subseteq B(H)\\).\n\n1. \\(S'\\) is a subalgebra of \\(B(H)\\) that contains \\(1\\). It is closed in the weak topology, hence in each of the six topologies of Section 1 and in norm.\n2. \\(S\\subseteq S''\\). If \\(S\\subseteq T\\), then \\(T'\\subseteq S'\\). Moreover \\(S'=S'''\\). So all commutants of odd order equal \\(S'\\), and all commutants of even order, from the second on, equal \\(S''\\).\n3. \\((S\\cup T)'=S'\\cap T'\\).\n4. If \\(S^*=S\\), then \\(S'\\) is a \\(*\\)-subalgebra with \\((S')''=S'\\). In particular \\(S''\\) is a \\(*\\)-subalgebra with \\((S'')''=S''\\) that contains \\(S\\). Every \\(*\\)-subalgebra \\(N\\) with \\(S\\subseteq N=N''\\) contains \\(S''\\).\n5. (*Invariant subspaces.*) Let \\(S^*=S\\), let \\(L\\subseteq H\\) be a closed subspace, and let \\(p\\) be its projection. Then \\(SL\\subseteq L\\) if and only if \\(p\\in S'\\). In that case \\(SL^\\perp\\subseteq L^\\perp\\) as well.\n\n**Proof.** (1) If \\(y_1,y_2\\in S'\\) and \\(s\\in S\\), then \\(y_1y_2s=y_1sy_2=sy_1y_2\\). Linear combinations are handled the same way, and \\(1\\in S'\\). Let \\(y_\\lambda\\in S'\\) converge weakly to \\(y\\). For \\(s\\in S\\) and \\(\\xi,\\eta\\in H\\),\n\\[\n\\begin{gathered}\n\\langle(ys-sy)\\xi,\\eta\\rangle\\\\\n=\\lim_\\lambda\\big(\\langle y_\\lambda(s\\xi),\\eta\\rangle-\\langle y_\\lambda\\xi,s^*\\eta\\rangle\\big)\\\\\n=\\lim_\\lambda\\langle(y_\\lambda s-sy_\\lambda)\\xi,\\eta\\rangle\\\\\n=0 .\n\\end{gathered}\n\\]\nSo \\(y\\in S'\\). The other five topologies and the norm topology are finer than the weak one ([Lemma 1.2](#oa-fnd-bi-01)(b)), so \\(S'\\) is closed in them too.\n\n(2) Each \\(s\\in S\\) commutes with every element of \\(S'\\), so \\(s\\in S''\\). The reversal of inclusions is immediate from the definition. Applying the first statement to \\(S'\\) gives \\(S'\\subseteq S'''\\). Applying the reversal to \\(S\\subseteq S''\\) gives \\(S'''\\subseteq S'\\).\n\n(3) This is the definition.\n\n(4) Let \\(y\\in S'\\) and \\(s\\in S\\). Then \\(s^*\\in S\\), so \\(ys^*=s^*y\\). Taking adjoints, \\(sy^*=y^*s\\). So \\(y^*\\in S'\\), and \\(S'\\) is a \\(*\\)-subalgebra by (1). Next, \\((S')''=(S'')'=S'''=S'\\) by (2). The set \\(S'\\) is self-adjoint, so the same statements hold for \\(S''=(S')'\\). Finally, if \\(S\\subseteq N=N''\\), then (2) gives \\(S''\\subseteq N''=N\\).\n\n(5) Suppose \\(SL\\subseteq L\\). For \\(\\eta\\in L^\\perp\\), \\(\\zeta\\in L\\) and \\(s\\in S\\) we have \\(\\langle s\\eta,\\zeta\\rangle=\\langle\\eta,s^*\\zeta\\rangle=0\\), because \\(s^*\\in S\\). So \\(SL^\\perp\\subseteq L^\\perp\\). Hence, for every \\(\\xi\\), \\(sp\\xi\\in L\\) and \\(s(1-p)\\xi\\in L^\\perp\\). The first gives \\(psp=sp\\), the second \\(ps(1-p)=0\\), that is \\(ps=psp\\). So \\(ps=sp\\). Conversely, if \\(p\\in S'\\), then \\(s(p\\xi)=p(s\\xi)\\in L\\). \\(\\square\\)\n\n*Remark 2.2* (non-self-adjoint sets). Part (4) needs \\(S^*=S\\). On \\(\\mathbb C^2\\) let \\(N\\) be the matrix unit with \\(Ne_2=e_1\\) and \\(Ne_1=0\\), and \\(S=\\{N\\}\\). A direct computation gives \\(S'=\\{\\alpha1+\\beta N\\}\\), and then \\(S''=\\{1,N\\}'=S'\\). This algebra contains \\(N\\) but not \\(N^*\\).\n\n**Definition 2.3.** A *\\(*\\)-subalgebra* \\(M\\) of \\(B(H)\\) is a subalgebra closed under adjoints. It is *nondegenerate* if \\([MH]=H\\). We call a \\(*\\)-subalgebra \\(M\\) with \\(M=M''\\) a *von Neumann algebra*, and write \\(\\{M,H\\}\\) when the Hilbert space matters. The *centre* of \\(M\\) is \\(Z(M)=M\\cap M'\\), and \\(M\\) is a *factor* if \\(Z(M)=\\mathbb C1\\). For a self-adjoint set \\(S\\subseteq B(H)\\) we say that \\(S''\\) is *generated* by \\(S\\). By [Proposition 2.1](#oa-fnd-bi-02)(4), \\(S''\\) is a von Neumann algebra, and every von Neumann algebra that contains \\(S\\) also contains \\(S''\\).\n\nTwo von Neumann algebras \\(\\{M_1,H_1\\}\\) and \\(\\{M_2,H_2\\}\\) are *spatially isomorphic* if there is a unitary \\(U:H_1\\to H_2\\) (an isometry of \\(H_1\\) onto \\(H_2\\)) with \\(UM_1U^*=M_2\\); the map \\(x\\mapsto UxU^*\\) is then a *spatial isomorphism*. They are *isomorphic* if there is a bijective linear map \\(\\pi:M_1\\to M_2\\) that is multiplicative and satisfies \\(\\pi(x^*)=\\pi(x)^*\\). A spatial isomorphism is an isomorphism, but not conversely ([Example 5.5](#oa-fnd-bi-15)).\n\n**Proposition 2.4.**\n\n1. For a \\(*\\)-subalgebra \\(M\\), \\([MH]^\\perp=\\{\\xi\\in H:x\\xi=0\\text{ for all }x\\in M\\}\\). So \\(M\\) is nondegenerate exactly when no nonzero vector is killed by all of \\(M\\). Every \\(*\\)-subalgebra that contains \\(1\\) is nondegenerate.\n2. A von Neumann algebra \\(M\\) contains \\(1\\), is nondegenerate, and is closed in norm and in all six topologies. So it is a nondegenerate norm-closed \\(*\\)-algebra. \\(M'\\) is a von Neumann algebra, \\(Z(M)\\) is a commutative von Neumann algebra, and \\(Z(M')=Z(M)\\). In particular \\(M\\) is a factor if and only if \\(M'\\) is.\n3. A spatial isomorphism carries commutants to commutants and centres to centres: \\(UM_1'U^*=M_2'\\) and \\(UZ(M_1)U^*=Z(M_2)\\).\n4. \\(B(H)'=\\mathbb C1\\). So \\(B(H)\\) and \\(\\mathbb C1\\) are factors, each the commutant of the other.\n\n**Proof.** (1) A vector \\(\\eta\\) is orthogonal to every \\(x\\xi\\) exactly when \\(x^*\\eta=0\\) for every \\(x\\in M\\). As \\(M^*=M\\), this says \\(x\\eta=0\\) for every \\(x\\in M\\). If \\(1\\in M\\), then \\(\\eta=1\\eta=0\\).\n\n(2) \\(1\\in M'\\) gives \\(1\\in M''=M\\), and (1) gives nondegeneracy. \\(M=(M')'\\) is a commutant, so it is closed by [Proposition 2.1](#oa-fnd-bi-02)(1). \\(M'\\) is the commutant of the self-adjoint set \\(M\\), so it is a von Neumann algebra by [Proposition 2.1](#oa-fnd-bi-02)(4). Also \\(Z(M)=M''\\cap M'=(M'\\cup M)'\\) by [Proposition 2.1](#oa-fnd-bi-02)(3); this is the commutant of a self-adjoint set, hence a von Neumann algebra. It is commutative, because elements of \\(M'\\) commute with elements of \\(M\\). Finally \\(Z(M')=M'\\cap M''=M'\\cap M=Z(M)\\).\n\n(3) For \\(y\\in B(H_1)\\), \\(y\\) commutes with every \\(x\\in M_1\\) if and only if \\(UyU^*\\) commutes with every \\(UxU^*\\in M_2\\). So \\(UM_1'U^*=M_2'\\), and intersecting with \\(UM_1U^*=M_2\\) gives the centres.\n\n(4) If \\(H=\\{0\\}\\) there is nothing to prove. Let \\(y\\in B(H)'\\). For \\(\\xi,\\eta\\in H\\), \\(y\\theta_{\\xi,\\eta}=\\theta_{y\\xi,\\eta}\\) and \\(\\theta_{\\xi,\\eta}y=\\theta_{\\xi,y^*\\eta}\\), so \\(\\theta_{y\\xi,\\eta}=\\theta_{\\xi,y^*\\eta}\\). Fix a unit vector \\(\\eta\\) and apply both operators to \\(\\eta\\): \\(y\\xi=\\langle\\eta,y^*\\eta\\rangle\\xi=\\langle y\\eta,\\eta\\rangle\\xi\\) for every \\(\\xi\\). So \\(y\\) is a scalar. Trivially \\((\\mathbb C1)'=B(H)\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-05",
      "unit": "the-double-commutant-theorem",
      "name": "3. Amplification",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 3. Amplification\n\nTo pass from approximation at one vector to approximation at a square-summable sequence of vectors, we replace \\(H\\) by a direct sum of copies of \\(H\\). Operators on the larger space are described by matrices with entries in \\(B(H)\\).\n\nLet \\(I\\) be a nonempty set and \\(\\tilde H=\\ell^2(I;H)\\), the space of families \\((\\xi_i)_{i\\in I}\\) in \\(H\\) with \\(\\sum_i\\|\\xi_i\\|^2<\\infty\\). It is the direct sum of copies of \\(H\\) indexed by \\(I\\). Let \\(U_j:H\\to\\tilde H\\) put a vector in place \\(j\\) and zeros elsewhere. Then \\(U_i^*U_j=\\delta_{ij}1_H\\), \\(U_i^*(\\xi_k)_k=\\xi_i\\), and \\(\\sum_iU_iU_i^*=1\\), with unconditional convergence on each vector: \\(\\|\\tilde\\xi-\\sum_{i\\in F}U_iU_i^*\\tilde\\xi\\|^2=\\sum_{i\\notin F}\\|\\xi_i\\|^2\\to0\\). For \\(X\\in B(\\tilde H)\\) define its *matrix entries*\n\\[\nX_{ij}=U_i^*XU_j\\in B(H).\n\\tag{3.1}\n\\]\n\n**Lemma 3.1** (matrices).\n\n1. \\(X\\) is determined by its entries, and \\(\\|X_{ij}\\|\\le\\|X\\|\\).\n2. \\((\\alpha X+\\beta Y)_{ij}=\\alpha X_{ij}+\\beta Y_{ij}\\) and \\((X^*)_{ij}=(X_{ji})^*\\). For each \\(\\xi\\in H\\),\n\\[\n(XY)_{ij}\\xi=\\sum_kX_{ik}Y_{kj}\\xi ,\n\\tag{3.2}\n\\]\nwith unconditional convergence in norm.\n3. The operators \\(E_{ij}=U_iU_j^*\\) (the *matrix units*) have entries \\((E_{ij})_{kl}=\\delta_{ki}\\delta_{jl}1_H\\).\n\n**Proof.** (1) \\(\\langle XU_j\\xi,U_i\\eta\\rangle=\\langle X_{ij}\\xi,\\eta\\rangle\\), and the vectors \\(U_j\\xi\\) span a dense subspace (the finitely supported families). If all entries vanish, \\(\\langle X\\zeta,\\zeta'\\rangle=0\\) on this dense subspace, so \\(X=0\\). The norm bound holds because \\(U_i^*\\) and \\(U_j\\) have norm at most \\(1\\).\n\n(2) Linearity is clear, and \\((X^*)_{ij}=U_i^*X^*U_j=(U_j^*XU_i)^*\\). For the product, \\[\n\\begin{gathered}\n(XY)_{ij}\\xi\\\\\n=U_i^*X\\big(\\sum_kU_kU_k^*\\big)YU_j\\xi\\\\\n=\\sum_kU_i^*XU_k\\,U_k^*YU_j\\xi.\n\\end{gathered}\n\\] The sum converges unconditionally because \\(\\sum_kU_kU_k^*\\eta\\) does and \\(U_i^*X\\) is bounded.\n\n(3) \\(U_k^*U_iU_j^*U_l=\\delta_{ki}\\delta_{jl}1_H\\). \\(\\square\\)\n\nThe *amplification* \\(\\pi:B(H)\\to B(\\tilde H)\\) is\n\\[\n\\begin{gathered}\n\\pi(x)(\\xi_i)_i\\\\\n=(x\\xi_i)_i,\\\\\n\\text{that is,}\\\\\n\\pi(x)_{ij}\\\\\n=\\delta_{ij}x .\n\\end{gathered}\n\\tag{3.3}\n\\]\nIt is an injective unital \\(*\\)-homomorphism with \\(\\|\\pi(x)\\|=\\|x\\|\\), and \\(\\pi(x)U_j=U_jx\\), \\(U_i^*\\pi(x)=xU_i^*\\).\n\n**Proposition 3.2** (the range of the amplification). An operator \\(X\\in B(\\tilde H)\\) lies in \\(\\pi(B(H))\\) if and only if it commutes with every matrix unit \\(E_{ij}\\).\n\n**Proof.** \\(\\pi(x)E_{ij}=U_ixU_j^*=E_{ij}\\pi(x)\\). Conversely, let \\(X\\) commute with every \\(E_{ij}\\). Since \\(U_j=E_{jj}U_j\\),\n\\[\nX_{ij}=U_i^*XE_{jj}U_j=U_i^*E_{jj}XU_j=\\delta_{ij}X_{jj}.\n\\]\nSince \\(U_i=E_{ij}U_j\\) and \\(U_i^*E_{ij}=U_j^*\\), also \\(X_{ii}=U_i^*XE_{ij}U_j=U_i^*E_{ij}XU_j=X_{jj}\\). So all diagonal entries equal one operator \\(x\\), and the other entries vanish. By Lemma 3.1(1), \\(X=\\pi(x)\\). \\(\\square\\)\n\n**Proposition 3.3** (the commutant of an amplified set). For every subset \\(S\\subseteq B(H)\\),\n\\[\n\\begin{gathered}\n\\pi(S)'\\\\\n=\\{X\\in B(\\tilde H):\\\\\nX_{ij}\\in S'\\text{ for all }i,j\\in I\\}.\n\\end{gathered}\n\\tag{3.4}\n\\]\n\n**Proof.** For \\(s\\in S\\), \\((X\\pi(s))_{ij}=U_i^*XU_js=X_{ij}s\\) and \\((\\pi(s)X)_{ij}=sX_{ij}\\). By Lemma 3.1(1), \\(X\\) commutes with \\(\\pi(s)\\) exactly when \\(X_{ij}s=sX_{ij}\\) for all \\(i,j\\). \\(\\square\\)\n\n**Corollary 3.4.** \\(\\pi(S)''=\\pi(S'')\\) for every subset \\(S\\subseteq B(H)\\).\n\n**Proof.** The entries of each \\(E_{ij}\\) are \\(0\\) or \\(1_H\\), which lie in \\(S'\\). So \\(E_{ij}\\in\\pi(S)'\\) by (3.4). Every \\(Y\\in\\pi(S)''\\) therefore commutes with every \\(E_{ij}\\), and \\(Y=\\pi(y)\\) for some \\(y\\) by Proposition 3.2. For \\(y\\in B(H)\\) and \\(X\\in B(\\tilde H)\\), the entries of \\(\\pi(y)X\\) and \\(X\\pi(y)\\) are \\(yX_{ij}\\) and \\(X_{ij}y\\), by (3.2) and (3.3). So \\(\\pi(y)\\in\\pi(S)''\\) exactly when \\(y\\) commutes with every entry of every \\(X\\in\\pi(S)'\\). By (3.4) these entries lie in \\(S'\\), and every \\(z\\in S'\\) occurs, as a diagonal entry of \\(\\pi(z)\\in\\pi(S)'\\). Hence \\(\\pi(y)\\in\\pi(S)''\\) if and only if \\(y\\in S''\\). \\(\\square\\)\n\n**Lemma 3.5** (nondegeneracy survives amplification). If \\(M\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\), then \\(\\pi(M)\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(\\tilde H)\\).\n\n**Proof.** \\([\\pi(M)\\tilde H]\\supseteq[\\pi(M)U_jH]=U_j[MH]=U_jH\\) for every \\(j\\), and these subspaces together span a dense subspace of \\(\\tilde H\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-06",
      "unit": "the-double-commutant-theorem",
      "name": "4. The double commutant theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      },
      "full_conditions_and_proof": "## 4. The double commutant theorem\n\nThe proof rests on one observation: for a nondegenerate \\(*\\)-algebra \\(M\\), every vector \\(\\xi\\) lies in the closed span of its orbit \\(M\\xi\\). Amplification then turns approximation at one vector into approximation in the \\(\\sigma\\)-strong topology.\n\n**Lemma 4.1.** Let \\(S\\subseteq B(H)\\) be closed under products and adjoints, and suppose no nonzero vector is killed by every element of \\(S\\). Then \\(\\xi\\in[S\\xi]\\) for every \\(\\xi\\in H\\). By [Proposition 2.4](#oa-fnd-bi-03)(1), this applies to every nondegenerate \\(*\\)-subalgebra.\n\n**Proof.** Let \\(L=[S\\xi]\\) and let \\(p\\) be its projection. Since \\(s(t\\xi)=(st)\\xi\\), the subspace \\(L\\) is invariant under \\(S\\). As \\(S^*=S\\), [Proposition 2.1](#oa-fnd-bi-02)(5) gives \\(p\\in S'\\). For \\(s\\in S\\), \\(s\\xi\\in L\\), so \\(0=(1-p)s\\xi=s(1-p)\\xi\\). Thus every \\(s\\in S\\) kills \\((1-p)\\xi\\), and so \\((1-p)\\xi=0\\). \\(\\square\\)\n\nBoth hypotheses matter; [Example 4.2](#oa-fnd-bi-06) shows what fails without them.\n\n**Example 4.2** (Lemma 4.1 needs adjoints and nondegeneracy). On \\(\\mathbb C^2\\), let \\(S\\) be the set of matrices whose second column is zero. It is closed under products, and \\([S\\mathbb C^2]=\\mathbb C^2\\), since \\(S\\xi=\\mathbb C^2\\) whenever the first coordinate of \\(\\xi\\) is nonzero. For \\(\\xi=(0,1)\\), however, \\(S\\xi=\\{0\\}\\), so \\(\\xi\\notin[S\\xi]\\). Here \\(S\\) is not closed under adjoints, and some nonzero vectors are killed by all of \\(S\\): for non-self-adjoint algebras, the spanning condition and the kernel condition of [Proposition 2.4](#oa-fnd-bi-03)(1) differ. Adjoints are needed even when the kernel condition holds. The set \\(T\\) of matrices whose second row is zero is closed under products and kills no nonzero vector, but for \\(\\xi=(0,1)\\), \\([T\\xi]=\\mathbb C(1,0)\\) does not contain \\(\\xi\\). For the \\(*\\)-algebra \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\([M\\xi]=\\{0\\}\\) for every \\(\\xi\\), so nondegeneracy is needed too.\n\n**Lemma 4.3.** Let \\(M\\) be a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\). Then \\([M\\xi]=[M''\\xi]\\) for every \\(\\xi\\in H\\). Consequently, for \\(a\\in M''\\), \\(\\xi\\in H\\) and \\(\\varepsilon>0\\) there is \\(b\\in M\\) with \\(\\|(a-b)\\xi\\|<\\varepsilon\\).\n\n**Proof.** \\(M\\subseteq M''\\) gives \\(\\subseteq\\). The projection \\(p\\) onto \\([M\\xi]\\) lies in \\(M'\\), as in the previous proof. Each \\(a\\in M''\\) commutes with \\(p\\), so \\(a\\) maps \\([M\\xi]\\) into itself ([Proposition 2.1](#oa-fnd-bi-02)(5)). By Lemma 4.1, \\(\\xi\\in[M\\xi]\\), so \\(a\\xi\\in[M\\xi]\\). Hence \\(M''\\xi\\subseteq[M\\xi]\\). Since \\(M\\xi\\) is a linear subspace, \\([M\\xi]\\) is its norm closure, and some \\(b\\xi\\) with \\(b\\in M\\) lies within \\(\\varepsilon\\) of \\(a\\xi\\). \\(\\square\\)\n\nWe state the theorem for an arbitrary \\(*\\)-subalgebra, closed or not, degenerate or not.\n\n**Theorem 4.4** (the double commutant theorem). Let \\(M\\) be a \\(*\\)-subalgebra of \\(B(H)\\), and let \\(e\\) be the projection onto \\([MH]\\).\n\n1. \\(e\\in M'\\cap M''\\), and \\(x=ex=xe\\) for every \\(x\\in M\\).\n2. The closures of \\(M\\) in the weak, strong, strong\\(^*\\), \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies coincide. Call this common closure \\(\\overline M\\). It is a \\(*\\)-subalgebra. The projection \\(e\\) lies in \\(\\overline M\\), is its unit (\\(x=ex=xe\\) for \\(x\\in\\overline M\\)), and is the greatest projection in \\(\\overline M\\).\n3. \\(M''\\) consists of the operators \\(x+\\alpha1\\) with \\(x\\in\\overline M\\) and \\(\\alpha\\in\\mathbb C\\), and\n\\[\n\\begin{gathered}\nM''\\\\\n=\\overline M+\\mathbb C1,\\\\\n\\overline M\\\\\n=eM''\\\\\n=M''e .\n\\end{gathered}\n\\tag{4.1}\n\\]\n4. \\(\\overline M=M''\\) if and only if \\(M\\) is nondegenerate.\n5. \\(M\\) is a von Neumann algebra if and only if \\(M\\) is nondegenerate and closed in at least one of the six topologies. It is then closed in all of them.\n\n\n*Remark 4.5.* Suppose that \\(M\\) is closed in one of the six topologies. Since the \\(\\sigma\\)-strong\\(^*\\) topology is the finest of the six, closedness for it is the weakest hypothesis of this kind. Then \\(\\overline M=M\\), and parts (2)–(4) say: \\(e\\) is the greatest projection of \\(M\\) and its unit; \\(M''=M+\\mathbb C1\\); and \\(M=M''\\) when \\(M\\) is nondegenerate. Part (2) is von Neumann's density theorem: a nondegenerate \\(*\\)-algebra is dense in its bicommutant, even \\(\\sigma\\)-strongly\\(^*\\).\n\n**Proof.** *Step 1 (the nondegenerate case).* Assume \\(M\\) is nondegenerate. Let \\(a\\in M''\\), let \\((\\xi_n)_{n\\in\\mathbb N}\\) be square summable, and let \\(\\varepsilon>0\\). Use the amplification of [Section 3](#oa-fnd-bi-05) with \\(I=\\mathbb N\\), and put \\(\\tilde\\xi=(\\xi_n)\\in\\tilde H\\). By Lemma 3.5, \\(\\pi(M)\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(\\tilde H)\\), and \\(\\pi(a)\\in\\pi(M'')=\\pi(M)''\\) by Corollary 3.4. Lemma 4.3 on \\(\\tilde H\\) gives \\(b\\in M\\) with \\(\\|(\\pi(a)-\\pi(b))\\tilde\\xi\\|<\\varepsilon\\), that is,\n\\[\n\\Big(\\sum_n\\|(a-b)\\xi_n\\|^2\\Big)^{1/2}<\\varepsilon .\n\\tag{4.2}\n\\]\nSince \\((\\xi_n)\\) and \\(\\varepsilon\\) were arbitrary, [Lemma 1.2](#oa-fnd-bi-01)(a) shows that every \\(\\sigma\\)-strong neighbourhood of \\(a\\) meets \\(M\\), so \\(a\\in\\overline M^{\\,\\sigma s}\\). Thus \\(M''\\subseteq\\overline M^{\\,\\sigma s}\\). On the other hand \\(M''\\) is weakly closed, being a commutant, and it contains \\(M\\), so \\(\\overline M^{\\,w}\\subseteq M''\\). The comparisons of [Lemma 1.2](#oa-fnd-bi-01)(b) now give\n\\[\n\\begin{gathered}\nM''\\subseteq\\overline M^{\\,\\sigma s}\\subseteq\\overline M^{\\,\\sigma w}\\\\\n\\subseteq\\overline M^{\\,w}\\subseteq M'',\\\\\n\\overline M^{\\,\\sigma s}\\subseteq\\overline M^{\\,s}\\subseteq\\overline M^{\\,w}.\n\\end{gathered}\n\\]\nSo the weak, strong, \\(\\sigma\\)-weak and \\(\\sigma\\)-strong closures all equal \\(M''\\). Since \\(M\\) is a linear subspace, [Lemma 1.3](#oa-fnd-bi-01) adds \\(\\overline M^{\\,s*}=\\overline M^{\\,w}\\) and \\(\\overline M^{\\,\\sigma s*}=\\overline M^{\\,\\sigma w}\\). So all six closures equal \\(M''\\). Here \\(e=1\\).\n\n*Step 2 (part 1).* \\([MH]\\) is invariant under \\(M\\), since \\(x(y\\xi)=(xy)\\xi\\), and \\(M^*=M\\); so \\(e\\in M'\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). \\([MH]\\) is also invariant under \\(M'\\), since \\(y'(x\\xi)=x(y'\\xi)\\), and \\(M'\\) is self-adjoint; so \\(e\\in M''\\). For \\(x\\in M\\), \\(xH\\subseteq[MH]\\), so \\(ex=x\\). Applying this to \\(x^*\\in M\\) and taking adjoints, \\(xe=x\\).\n\n*Step 3 (reduction to \\(eH\\)).* Put \\(K=eH\\). By Step 2, each \\(x\\in M\\) maps \\(K\\) into \\(K\\) and vanishes on \\(K^\\perp\\). Let \\(x_K\\in B(K)\\) be the restriction of \\(x\\) to \\(K\\), and \\(M_K=\\{x_K:x\\in M\\}\\). Define \\(\\iota:B(K)\\to B(H)\\) by \\(\\iota(y)\\xi=y(e\\xi)\\), extension by zero. Then \\(\\iota\\) is an injective \\(*\\)-homomorphism onto \\(eB(H)e\\), and \\(\\iota(x_K)=x\\) for \\(x\\in M\\). So \\(M_K\\) is a \\(*\\)-subalgebra of \\(B(K)\\) with \\(\\iota(M_K)=M\\), and it is nondegenerate: \\([M_KK]=[MeH]=[MH]=K\\).\n\nFor each of the six topologies \\(\\mathcal T\\), the map \\(\\iota\\) is a homeomorphism from \\(B(K)\\) onto \\(eB(H)e\\), and \\(eB(H)e\\) is \\(\\mathcal T\\)-closed in \\(B(H)\\). Indeed \\(\\iota(y)\\xi=y(e\\xi)\\), \\(\\iota(y)^*\\xi=y^*(e\\xi)\\) and \\(\\langle\\iota(y)\\xi,\\eta\\rangle=\\langle y(e\\xi),e\\eta\\rangle\\). So every \\(\\mathcal T\\)-seminorm of \\(\\iota(y)\\), built from vectors \\(\\xi_n,\\eta_n\\) of \\(H\\), is the \\(\\mathcal T\\)-seminorm of \\(y\\) built from \\(e\\xi_n,e\\eta_n\\in K\\); and every \\(\\mathcal T\\)-seminorm on \\(B(K)\\) arises this way. Also \\(eB(H)e=\\{z:z-eze=0\\}\\) is closed, because \\(z\\mapsto z-eze\\) is continuous by [Lemma 1.2](#oa-fnd-bi-01)(c). Hence the \\(\\mathcal T\\)-closure of \\(M=\\iota(M_K)\\) in \\(B(H)\\) is \\(\\iota\\) of the \\(\\mathcal T\\)-closure of \\(M_K\\) in \\(B(K)\\). By Step 1 on \\(K\\), the latter is \\((M_K)''\\) for every \\(\\mathcal T\\). So all six closures of \\(M\\) are equal:\n\\[\n\\overline M=\\iota\\big((M_K)''\\big).\n\\]\nThis is a \\(*\\)-subalgebra, and it contains \\(\\iota(1_K)=e\\). Every \\(z\\in\\overline M\\) satisfies \\(z=eze\\), so \\(e\\) is the unit of \\(\\overline M\\). If \\(f\\in\\overline M\\) is a projection, then \\(f=efe\\), so \\(\\langle f\\xi,\\xi\\rangle=\\langle fe\\xi,e\\xi\\rangle\\le\\|e\\xi\\|^2=\\langle e\\xi,\\xi\\rangle\\), and \\(f\\le e\\). This proves (2).\n\n*Step 4 (part 3).* \\(M''\\) is closed and contains \\(M\\), so \\(\\overline M\\subseteq M''\\); and \\(1\\in M''\\). So \\(\\overline M+\\mathbb C1\\subseteq M''\\). Conversely, let \\(y\\in M''\\). By Step 2, \\(e\\in M'\\), so \\(y\\) commutes with \\(e\\) and maps \\(K\\) and \\(K^\\perp\\) into themselves.\n\n- *On \\(K^\\perp\\).* Let \\(d\\in B(H)\\) satisfy \\(d=(1-e)d(1-e)\\). For \\(x\\in M\\), \\(xd=x(1-e)d(1-e)=0\\) and \\(dx=d(1-e)x=0\\). So \\(d\\in M'\\), and \\(y\\) commutes with \\(d\\). Every operator on \\(K^\\perp\\) extends by zero to such a \\(d\\). So the restriction of \\(y\\) to \\(K^\\perp\\) commutes with all of \\(B(K^\\perp)\\), and by [Proposition 2.4](#oa-fnd-bi-03)(4) it is a scalar \\(\\alpha1_{K^\\perp}\\) with \\(\\alpha\\in\\mathbb C\\). (If \\(K^\\perp=\\{0\\}\\), take \\(\\alpha=0\\).)\n- *On \\(K\\).* Let \\(c\\in(M_K)'\\). For \\(x\\in M\\) and \\(\\xi\\in H\\), the vector \\(c(e\\xi)\\) lies in \\(K\\), and \\(x\\xi\\in K\\) equals \\(ex\\xi\\). So\n\\[\n\\begin{gathered}\nx\\,\\iota(c)\\xi\\\\\n=x_Kc(e\\xi)\\\\\n=c\\,x_K(e\\xi)\\\\\n=c(xe\\xi)\\\\\n=c(ex\\xi)\\\\\n=\\iota(c)x\\xi .\n\\end{gathered}\n\\]\nSo \\(\\iota(c)\\in M'\\), and \\(y\\) commutes with \\(\\iota(c)\\). Restricting to \\(K\\), \\(y_Kc=cy_K\\). So \\(y_K\\in(M_K)''\\).\n\nNow \\(y-\\alpha1\\) vanishes on \\(K^\\perp\\), and its restriction to \\(K\\) is \\(y_K-\\alpha1_K\\in(M_K)''\\). So \\(y-\\alpha1=\\iota(y_K-\\alpha1_K)\\in\\overline M\\). This proves \\(M''=\\overline M+\\mathbb C1\\). For \\(x\\in\\overline M\\), \\(e(x+\\alpha1)=x+\\alpha e\\in\\overline M\\); hence \\(eM''\\subseteq\\overline M=e\\overline M\\subseteq eM''\\). The same holds with \\(e\\) on the right.\n\n*Step 5 (parts 4 and 5).* If \\(M\\) is nondegenerate, \\(e=1\\) and \\(\\overline M=M''\\) by Step 1. If not, \\(e\\neq1\\). Then \\(1\\notin\\overline M\\), because \\(1\\cdot e\\neq1\\), while \\(1\\in M''\\). For (5): if \\(M=M''\\), then \\(M\\) contains \\(1\\), is nondegenerate, and is closed in all six topologies ([Proposition 2.4](#oa-fnd-bi-03)(2)). Conversely, if \\(M\\) is nondegenerate and closed in some \\(\\mathcal T\\), then \\(M=\\overline M^{\\,\\mathcal T}=\\overline M=M''\\). \\(\\square\\)\n\n**Example 4.6** (compact operators). Let \\(\\dim H=\\infty\\). Then \\(K(H)\\) is a norm-closed \\(*\\)-subalgebra that contains every \\(\\theta_{\\xi,\\xi}\\), so it is nondegenerate. The proof of [Proposition 2.4](#oa-fnd-bi-03)(4) uses only rank-one operators, so \\(K(H)'=\\mathbb C1\\) and \\(K(H)''=B(H)\\). By [Theorem 4.4](#oa-fnd-bi-07), \\(K(H)\\) is dense in \\(B(H)\\) for all six topologies; for instance, the finite-rank projections onto spans of finitely many vectors of an orthonormal basis increase strongly to \\(1\\). But \\(1\\notin K(H)\\), so \\(K(H)\\) is not a von Neumann algebra. Being norm closed is not enough.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-07",
      "unit": "the-double-commutant-theorem",
      "name": "4. The double commutant theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 4. The double commutant theorem\n\nThe proof rests on one observation: for a nondegenerate \\(*\\)-algebra \\(M\\), every vector \\(\\xi\\) lies in the closed span of its orbit \\(M\\xi\\). Amplification then turns approximation at one vector into approximation in the \\(\\sigma\\)-strong topology.\n\n**Lemma 4.1.** Let \\(S\\subseteq B(H)\\) be closed under products and adjoints, and suppose no nonzero vector is killed by every element of \\(S\\). Then \\(\\xi\\in[S\\xi]\\) for every \\(\\xi\\in H\\). By [Proposition 2.4](#oa-fnd-bi-03)(1), this applies to every nondegenerate \\(*\\)-subalgebra.\n\n**Proof.** Let \\(L=[S\\xi]\\) and let \\(p\\) be its projection. Since \\(s(t\\xi)=(st)\\xi\\), the subspace \\(L\\) is invariant under \\(S\\). As \\(S^*=S\\), [Proposition 2.1](#oa-fnd-bi-02)(5) gives \\(p\\in S'\\). For \\(s\\in S\\), \\(s\\xi\\in L\\), so \\(0=(1-p)s\\xi=s(1-p)\\xi\\). Thus every \\(s\\in S\\) kills \\((1-p)\\xi\\), and so \\((1-p)\\xi=0\\). \\(\\square\\)\n\nBoth hypotheses matter; [Example 4.2](#oa-fnd-bi-06) shows what fails without them.\n\n**Example 4.2** (Lemma 4.1 needs adjoints and nondegeneracy). On \\(\\mathbb C^2\\), let \\(S\\) be the set of matrices whose second column is zero. It is closed under products, and \\([S\\mathbb C^2]=\\mathbb C^2\\), since \\(S\\xi=\\mathbb C^2\\) whenever the first coordinate of \\(\\xi\\) is nonzero. For \\(\\xi=(0,1)\\), however, \\(S\\xi=\\{0\\}\\), so \\(\\xi\\notin[S\\xi]\\). Here \\(S\\) is not closed under adjoints, and some nonzero vectors are killed by all of \\(S\\): for non-self-adjoint algebras, the spanning condition and the kernel condition of [Proposition 2.4](#oa-fnd-bi-03)(1) differ. Adjoints are needed even when the kernel condition holds. The set \\(T\\) of matrices whose second row is zero is closed under products and kills no nonzero vector, but for \\(\\xi=(0,1)\\), \\([T\\xi]=\\mathbb C(1,0)\\) does not contain \\(\\xi\\). For the \\(*\\)-algebra \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\([M\\xi]=\\{0\\}\\) for every \\(\\xi\\), so nondegeneracy is needed too.\n\n**Lemma 4.3.** Let \\(M\\) be a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\). Then \\([M\\xi]=[M''\\xi]\\) for every \\(\\xi\\in H\\). Consequently, for \\(a\\in M''\\), \\(\\xi\\in H\\) and \\(\\varepsilon>0\\) there is \\(b\\in M\\) with \\(\\|(a-b)\\xi\\|<\\varepsilon\\).\n\n**Proof.** \\(M\\subseteq M''\\) gives \\(\\subseteq\\). The projection \\(p\\) onto \\([M\\xi]\\) lies in \\(M'\\), as in the previous proof. Each \\(a\\in M''\\) commutes with \\(p\\), so \\(a\\) maps \\([M\\xi]\\) into itself ([Proposition 2.1](#oa-fnd-bi-02)(5)). By Lemma 4.1, \\(\\xi\\in[M\\xi]\\), so \\(a\\xi\\in[M\\xi]\\). Hence \\(M''\\xi\\subseteq[M\\xi]\\). Since \\(M\\xi\\) is a linear subspace, \\([M\\xi]\\) is its norm closure, and some \\(b\\xi\\) with \\(b\\in M\\) lies within \\(\\varepsilon\\) of \\(a\\xi\\). \\(\\square\\)\n\nWe state the theorem for an arbitrary \\(*\\)-subalgebra, closed or not, degenerate or not.\n\n**Theorem 4.4** (the double commutant theorem). Let \\(M\\) be a \\(*\\)-subalgebra of \\(B(H)\\), and let \\(e\\) be the projection onto \\([MH]\\).\n\n1. \\(e\\in M'\\cap M''\\), and \\(x=ex=xe\\) for every \\(x\\in M\\).\n2. The closures of \\(M\\) in the weak, strong, strong\\(^*\\), \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies coincide. Call this common closure \\(\\overline M\\). It is a \\(*\\)-subalgebra. The projection \\(e\\) lies in \\(\\overline M\\), is its unit (\\(x=ex=xe\\) for \\(x\\in\\overline M\\)), and is the greatest projection in \\(\\overline M\\).\n3. \\(M''\\) consists of the operators \\(x+\\alpha1\\) with \\(x\\in\\overline M\\) and \\(\\alpha\\in\\mathbb C\\), and\n\\[\n\\begin{gathered}\nM''\\\\\n=\\overline M+\\mathbb C1,\\\\\n\\overline M\\\\\n=eM''\\\\\n=M''e .\n\\end{gathered}\n\\tag{4.1}\n\\]\n4. \\(\\overline M=M''\\) if and only if \\(M\\) is nondegenerate.\n5. \\(M\\) is a von Neumann algebra if and only if \\(M\\) is nondegenerate and closed in at least one of the six topologies. It is then closed in all of them.\n\n\n*Remark 4.5.* Suppose that \\(M\\) is closed in one of the six topologies. Since the \\(\\sigma\\)-strong\\(^*\\) topology is the finest of the six, closedness for it is the weakest hypothesis of this kind. Then \\(\\overline M=M\\), and parts (2)–(4) say: \\(e\\) is the greatest projection of \\(M\\) and its unit; \\(M''=M+\\mathbb C1\\); and \\(M=M''\\) when \\(M\\) is nondegenerate. Part (2) is von Neumann's density theorem: a nondegenerate \\(*\\)-algebra is dense in its bicommutant, even \\(\\sigma\\)-strongly\\(^*\\).\n\n**Proof.** *Step 1 (the nondegenerate case).* Assume \\(M\\) is nondegenerate. Let \\(a\\in M''\\), let \\((\\xi_n)_{n\\in\\mathbb N}\\) be square summable, and let \\(\\varepsilon>0\\). Use the amplification of [Section 3](#oa-fnd-bi-05) with \\(I=\\mathbb N\\), and put \\(\\tilde\\xi=(\\xi_n)\\in\\tilde H\\). By Lemma 3.5, \\(\\pi(M)\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(\\tilde H)\\), and \\(\\pi(a)\\in\\pi(M'')=\\pi(M)''\\) by Corollary 3.4. Lemma 4.3 on \\(\\tilde H\\) gives \\(b\\in M\\) with \\(\\|(\\pi(a)-\\pi(b))\\tilde\\xi\\|<\\varepsilon\\), that is,\n\\[\n\\Big(\\sum_n\\|(a-b)\\xi_n\\|^2\\Big)^{1/2}<\\varepsilon .\n\\tag{4.2}\n\\]\nSince \\((\\xi_n)\\) and \\(\\varepsilon\\) were arbitrary, [Lemma 1.2](#oa-fnd-bi-01)(a) shows that every \\(\\sigma\\)-strong neighbourhood of \\(a\\) meets \\(M\\), so \\(a\\in\\overline M^{\\,\\sigma s}\\). Thus \\(M''\\subseteq\\overline M^{\\,\\sigma s}\\). On the other hand \\(M''\\) is weakly closed, being a commutant, and it contains \\(M\\), so \\(\\overline M^{\\,w}\\subseteq M''\\). The comparisons of [Lemma 1.2](#oa-fnd-bi-01)(b) now give\n\\[\n\\begin{gathered}\nM''\\subseteq\\overline M^{\\,\\sigma s}\\subseteq\\overline M^{\\,\\sigma w}\\\\\n\\subseteq\\overline M^{\\,w}\\subseteq M'',\\\\\n\\overline M^{\\,\\sigma s}\\subseteq\\overline M^{\\,s}\\subseteq\\overline M^{\\,w}.\n\\end{gathered}\n\\]\nSo the weak, strong, \\(\\sigma\\)-weak and \\(\\sigma\\)-strong closures all equal \\(M''\\). Since \\(M\\) is a linear subspace, [Lemma 1.3](#oa-fnd-bi-01) adds \\(\\overline M^{\\,s*}=\\overline M^{\\,w}\\) and \\(\\overline M^{\\,\\sigma s*}=\\overline M^{\\,\\sigma w}\\). So all six closures equal \\(M''\\). Here \\(e=1\\).\n\n*Step 2 (part 1).* \\([MH]\\) is invariant under \\(M\\), since \\(x(y\\xi)=(xy)\\xi\\), and \\(M^*=M\\); so \\(e\\in M'\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). \\([MH]\\) is also invariant under \\(M'\\), since \\(y'(x\\xi)=x(y'\\xi)\\), and \\(M'\\) is self-adjoint; so \\(e\\in M''\\). For \\(x\\in M\\), \\(xH\\subseteq[MH]\\), so \\(ex=x\\). Applying this to \\(x^*\\in M\\) and taking adjoints, \\(xe=x\\).\n\n*Step 3 (reduction to \\(eH\\)).* Put \\(K=eH\\). By Step 2, each \\(x\\in M\\) maps \\(K\\) into \\(K\\) and vanishes on \\(K^\\perp\\). Let \\(x_K\\in B(K)\\) be the restriction of \\(x\\) to \\(K\\), and \\(M_K=\\{x_K:x\\in M\\}\\). Define \\(\\iota:B(K)\\to B(H)\\) by \\(\\iota(y)\\xi=y(e\\xi)\\), extension by zero. Then \\(\\iota\\) is an injective \\(*\\)-homomorphism onto \\(eB(H)e\\), and \\(\\iota(x_K)=x\\) for \\(x\\in M\\). So \\(M_K\\) is a \\(*\\)-subalgebra of \\(B(K)\\) with \\(\\iota(M_K)=M\\), and it is nondegenerate: \\([M_KK]=[MeH]=[MH]=K\\).\n\nFor each of the six topologies \\(\\mathcal T\\), the map \\(\\iota\\) is a homeomorphism from \\(B(K)\\) onto \\(eB(H)e\\), and \\(eB(H)e\\) is \\(\\mathcal T\\)-closed in \\(B(H)\\). Indeed \\(\\iota(y)\\xi=y(e\\xi)\\), \\(\\iota(y)^*\\xi=y^*(e\\xi)\\) and \\(\\langle\\iota(y)\\xi,\\eta\\rangle=\\langle y(e\\xi),e\\eta\\rangle\\). So every \\(\\mathcal T\\)-seminorm of \\(\\iota(y)\\), built from vectors \\(\\xi_n,\\eta_n\\) of \\(H\\), is the \\(\\mathcal T\\)-seminorm of \\(y\\) built from \\(e\\xi_n,e\\eta_n\\in K\\); and every \\(\\mathcal T\\)-seminorm on \\(B(K)\\) arises this way. Also \\(eB(H)e=\\{z:z-eze=0\\}\\) is closed, because \\(z\\mapsto z-eze\\) is continuous by [Lemma 1.2](#oa-fnd-bi-01)(c). Hence the \\(\\mathcal T\\)-closure of \\(M=\\iota(M_K)\\) in \\(B(H)\\) is \\(\\iota\\) of the \\(\\mathcal T\\)-closure of \\(M_K\\) in \\(B(K)\\). By Step 1 on \\(K\\), the latter is \\((M_K)''\\) for every \\(\\mathcal T\\). So all six closures of \\(M\\) are equal:\n\\[\n\\overline M=\\iota\\big((M_K)''\\big).\n\\]\nThis is a \\(*\\)-subalgebra, and it contains \\(\\iota(1_K)=e\\). Every \\(z\\in\\overline M\\) satisfies \\(z=eze\\), so \\(e\\) is the unit of \\(\\overline M\\). If \\(f\\in\\overline M\\) is a projection, then \\(f=efe\\), so \\(\\langle f\\xi,\\xi\\rangle=\\langle fe\\xi,e\\xi\\rangle\\le\\|e\\xi\\|^2=\\langle e\\xi,\\xi\\rangle\\), and \\(f\\le e\\). This proves (2).\n\n*Step 4 (part 3).* \\(M''\\) is closed and contains \\(M\\), so \\(\\overline M\\subseteq M''\\); and \\(1\\in M''\\). So \\(\\overline M+\\mathbb C1\\subseteq M''\\). Conversely, let \\(y\\in M''\\). By Step 2, \\(e\\in M'\\), so \\(y\\) commutes with \\(e\\) and maps \\(K\\) and \\(K^\\perp\\) into themselves.\n\n- *On \\(K^\\perp\\).* Let \\(d\\in B(H)\\) satisfy \\(d=(1-e)d(1-e)\\). For \\(x\\in M\\), \\(xd=x(1-e)d(1-e)=0\\) and \\(dx=d(1-e)x=0\\). So \\(d\\in M'\\), and \\(y\\) commutes with \\(d\\). Every operator on \\(K^\\perp\\) extends by zero to such a \\(d\\). So the restriction of \\(y\\) to \\(K^\\perp\\) commutes with all of \\(B(K^\\perp)\\), and by [Proposition 2.4](#oa-fnd-bi-03)(4) it is a scalar \\(\\alpha1_{K^\\perp}\\) with \\(\\alpha\\in\\mathbb C\\). (If \\(K^\\perp=\\{0\\}\\), take \\(\\alpha=0\\).)\n- *On \\(K\\).* Let \\(c\\in(M_K)'\\). For \\(x\\in M\\) and \\(\\xi\\in H\\), the vector \\(c(e\\xi)\\) lies in \\(K\\), and \\(x\\xi\\in K\\) equals \\(ex\\xi\\). So\n\\[\n\\begin{gathered}\nx\\,\\iota(c)\\xi\\\\\n=x_Kc(e\\xi)\\\\\n=c\\,x_K(e\\xi)\\\\\n=c(xe\\xi)\\\\\n=c(ex\\xi)\\\\\n=\\iota(c)x\\xi .\n\\end{gathered}\n\\]\nSo \\(\\iota(c)\\in M'\\), and \\(y\\) commutes with \\(\\iota(c)\\). Restricting to \\(K\\), \\(y_Kc=cy_K\\). So \\(y_K\\in(M_K)''\\).\n\nNow \\(y-\\alpha1\\) vanishes on \\(K^\\perp\\), and its restriction to \\(K\\) is \\(y_K-\\alpha1_K\\in(M_K)''\\). So \\(y-\\alpha1=\\iota(y_K-\\alpha1_K)\\in\\overline M\\). This proves \\(M''=\\overline M+\\mathbb C1\\). For \\(x\\in\\overline M\\), \\(e(x+\\alpha1)=x+\\alpha e\\in\\overline M\\); hence \\(eM''\\subseteq\\overline M=e\\overline M\\subseteq eM''\\). The same holds with \\(e\\) on the right.\n\n*Step 5 (parts 4 and 5).* If \\(M\\) is nondegenerate, \\(e=1\\) and \\(\\overline M=M''\\) by Step 1. If not, \\(e\\neq1\\). Then \\(1\\notin\\overline M\\), because \\(1\\cdot e\\neq1\\), while \\(1\\in M''\\). For (5): if \\(M=M''\\), then \\(M\\) contains \\(1\\), is nondegenerate, and is closed in all six topologies ([Proposition 2.4](#oa-fnd-bi-03)(2)). Conversely, if \\(M\\) is nondegenerate and closed in some \\(\\mathcal T\\), then \\(M=\\overline M^{\\,\\mathcal T}=\\overline M=M''\\). \\(\\square\\)\n\n**Example 4.6** (compact operators). Let \\(\\dim H=\\infty\\). Then \\(K(H)\\) is a norm-closed \\(*\\)-subalgebra that contains every \\(\\theta_{\\xi,\\xi}\\), so it is nondegenerate. The proof of [Proposition 2.4](#oa-fnd-bi-03)(4) uses only rank-one operators, so \\(K(H)'=\\mathbb C1\\) and \\(K(H)''=B(H)\\). By [Theorem 4.4](#oa-fnd-bi-07), \\(K(H)\\) is dense in \\(B(H)\\) for all six topologies; for instance, the finite-rank projections onto spans of finitely many vectors of an orthonormal basis increase strongly to \\(1\\). But \\(1\\notin K(H)\\), so \\(K(H)\\) is not a von Neumann algebra. Being norm closed is not enough.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "the-double-commutant-theorem",
      "name": "5. Direct sums, reduced and induced algebras",
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      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 5. Direct sums, reduced and induced algebras\n\nTwo constructions produce new von Neumann algebras from old ones: direct sums, and cutting down by a projection of the algebra or of its commutant. For both we compute the commutant and the centre.\n\nLet \\(\\{M_i,H_i\\}_{i\\in I}\\) be von Neumann algebras, \\(I\\) any set. Let \\(H=\\bigoplus_iH_i\\) be the Hilbert sum, whose vectors are the families \\(\\bigoplus_i\\xi_i\\) with \\(\\sum_i\\|\\xi_i\\|^2<\\infty\\), and let \\(P_i\\) be the projection onto \\(H_i\\). For a family \\(x_i\\in B(H_i)\\) with \\(\\sup_i\\|x_i\\|<\\infty\\), put \\((\\bigoplus_ix_i)(\\bigoplus_i\\xi_i)=\\bigoplus_ix_i\\xi_i\\). This is a bounded operator with \\(\\|\\bigoplus_ix_i\\|=\\sup_i\\|x_i\\|\\): the bound \\(\\sum_i\\|x_i\\xi_i\\|^2\\le\\sup_i\\|x_i\\|^2\\sum_i\\|\\xi_i\\|^2\\) gives \\(\\le\\), and testing on vectors in a single \\(H_i\\) gives \\(\\ge\\).\n\n**Definition 5.1.** The *direct sum* \\(\\sum^\\oplus_iM_i\\), also written \\(\\sum^\\oplus_i\\{M_i,H_i\\}\\), is the set of operators \\(\\bigoplus_ix_i\\) with \\(x_i\\in M_i\\) and \\(\\sup_i\\|x_i\\|<\\infty\\).\n\n**Proposition 5.2.**\n\n1. \\(T\\in B(H)\\) commutes with every \\(P_i\\) if and only if \\(T=\\bigoplus_iT_i\\) for a bounded family \\(T_i\\in B(H_i)\\).\n2. \\(\\big(\\sum^\\oplus_iM_i\\big)'=\\sum^\\oplus_iM_i'\\).\n3. \\(\\sum^\\oplus_iM_i\\) is itself a von Neumann algebra, and its centre is \\(\\sum^\\oplus_iZ(M_i)\\).\n4. If at least two of the spaces \\(H_i\\) are nonzero, \\(\\sum^\\oplus_iM_i\\) is not a factor. If exactly one, \\(H_{i_0}\\), is nonzero, then \\(\\sum^\\oplus_iM_i\\) is spatially isomorphic to \\(M_{i_0}\\).\n\n**Proof.** (1) If \\(T\\) commutes with every \\(P_i\\), then \\(TH_i\\subseteq H_i\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). Let \\(T_i\\) be the restriction of \\(T\\) to \\(H_i\\); then \\(\\|T_i\\|\\le\\|T\\|\\). Since \\(\\xi=\\sum_iP_i\\xi\\) and \\(T\\) is continuous, \\(T\\xi=\\sum_iTP_i\\xi=\\bigoplus_iT_i\\xi_i\\). The converse is clear.\n\n(2) Each \\(P_i\\) lies in \\(\\sum^\\oplus M_i\\): take \\(1\\) in place \\(i\\) and \\(0\\) elsewhere. So \\(T\\in(\\sum^\\oplus M_i)'\\) commutes with every \\(P_i\\), and \\(T=\\bigoplus T_i\\) by (1). For \\(x\\in M_i\\), the operator with \\(x\\) in place \\(i\\) and \\(0\\) elsewhere lies in \\(\\sum^\\oplus M_i\\). Commuting with it gives \\(T_ix=xT_i\\). So \\(T_i\\in M_i'\\). Conversely, \\(\\bigoplus y_i\\) with \\(y_i\\in M_i'\\) commutes with every \\(\\bigoplus x_i\\), place by place.\n\n(3) \\(\\sum^\\oplus M_i\\) is a \\(*\\)-subalgebra: sums, products and adjoints are taken place by place, and the norms stay bounded. Each \\(M_i'\\) is a von Neumann algebra ([Proposition 2.4](#oa-fnd-bi-03)(2)), so (2) applies to the family \\((M_i')\\). It gives \\((\\sum^\\oplus M_i)''=(\\sum^\\oplus M_i')'=\\sum^\\oplus M_i''=\\sum^\\oplus M_i\\). The centre is \\((\\sum^\\oplus M_i)\\cap(\\sum^\\oplus M_i')\\); comparing places, this is \\(\\sum^\\oplus(M_i\\cap M_i')\\).\n\n(4) If \\(H_i\\neq0\\neq H_j\\) with \\(i\\neq j\\), then \\(P_i\\) is central by (3), and \\(P_i\\) is neither \\(0\\) nor \\(1\\). If only \\(H_{i_0}\\) is nonzero, the identification of \\(H\\) with \\(H_{i_0}\\) is a unitary that carries \\(\\sum^\\oplus M_i\\) onto \\(M_{i_0}\\), since the other places act on zero spaces. \\(\\square\\)\n\n*Remark 5.3.* The proof of (2) uses only that each \\(M_i\\) contains \\(0\\) and \\(1_{H_i}\\). Without units the formula can fail: for \\(S_1=S_2=\\{0\\}\\) on \\(\\mathbb C\\oplus\\mathbb C\\), the commutant of the direct sum is all of \\(B(\\mathbb C^2)\\), not the diagonal matrices.\n\n**Example 5.4** (a degenerate algebra). Let \\(p\\) be a projection with \\(p\\neq0\\) and \\(p\\neq1\\), and \\(M=\\mathbb Cp\\). This is a closed \\(*\\)-subalgebra, \\([MH]=pH\\), and \\(e=p\\). An operator commutes with \\(p\\) exactly when it is block diagonal for \\(H=pH\\oplus(1-p)H\\), so \\(M'=B(pH)\\oplus B((1-p)H)\\). By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\{\\alpha p+\\beta(1-p)\\}=M+\\mathbb C1\\). So \\(M''\\neq M=\\overline M=eM''\\), as (4.1) and [Theorem 4.4](#oa-fnd-bi-07)(4) predict.\n\n**Example 5.5** (an isomorphism that is not spatial). On \\(\\mathbb C^4\\) let \\(M_1=\\{\\operatorname{diag}(a,b,b,b)\\}\\) and \\(M_2=\\{\\operatorname{diag}(a,a,b,b)\\}\\), with \\(a,b\\in\\mathbb C\\). By [Proposition 5.2](#oa-fnd-bi-04) and [Proposition 2.4](#oa-fnd-bi-03)(4), these are von Neumann algebras: \\(M_1=\\mathbb C1_{\\mathbb C}\\oplus\\mathbb C1_{\\mathbb C^3}\\) and \\(M_2=\\mathbb C1_{\\mathbb C^2}\\oplus\\mathbb C1_{\\mathbb C^2}\\). The map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism. It is not spatial. A unitary \\(U\\) with \\(UM_1U^*=M_2\\) would carry the rank-one projection \\(\\operatorname{diag}(1,0,0,0)\\in M_1\\) to a rank-one projection in \\(M_2\\), but the projections of \\(M_2\\) have rank \\(0\\), \\(2\\) or \\(4\\).\n\n**Example 5.6** (commutants of direct sums with a common summand). Suppose \\(H\\neq\\{0\\}\\), and let \\(M=\\{x\\oplus x:x\\in B(H)\\}\\) on \\(H\\oplus H\\). This is \\(\\pi(B(H))\\) for \\(I=\\{1,2\\}\\) in [Section 3](#oa-fnd-bi-05), so \\(M''=M\\) by Corollary 3.4, and by (3.4) \\(M'\\) consists of the \\(2\\times2\\) scalar matrices tensored with \\(1_H\\). An operator in both \\(M\\) and \\(M'\\) is \\(\\pi(x)\\) with \\(x\\) scalar. So \\(M\\) is a factor, while the direct sum \\(B(H)\\oplus B(H)\\) of [Definition 5.1](#oa-fnd-bi-04) is not: its central projection \\(1_H\\oplus0\\) is neither zero nor the identity, by \\(H\\neq\\{0\\}\\). The two algebras are different subalgebras of \\(B(H\\oplus H)\\) built from the same summands. If \\(H=\\{0\\}\\), both constructions give the same zero algebra; its centre is \\(\\mathbb C1_H=\\{0\\}\\), so it satisfies this lesson's centre-based factor convention. Thus the contrast between the two constructions requires a nonzero summand.\n\nNow fix a von Neumann algebra \\(\\{M,H\\}\\) and a projection \\(e\\in M\\), and put \\(K=eH\\). For \\(x\\in M\\), the operator \\(exe\\) maps \\(K\\) into \\(K\\); write \\(x_e\\) for its restriction to \\(K\\). Each \\(x'\\in M'\\) commutes with \\(e\\), so it maps \\(K\\) into \\(K\\); write \\(x'_e\\) for its restriction. Put \\(M_e=\\{x_e:x\\in M\\}\\) and \\(M'_e=\\{x'_e:x'\\in M'\\}\\). Both are \\(*\\)-subalgebras of \\(B(K)\\) containing \\(1_K\\). The map \\(x'\\mapsto x'_e\\) is a unital \\(*\\)-homomorphism from \\(M'\\) onto \\(M'_e\\), because \\(e\\) commutes with \\(M'\\). Let \\(z\\) be the projection onto \\([MK]=[MeH]\\).\n\n**Definition 5.7.** On \\(K\\), the algebra \\(M_e\\) is called *reduced* (from \\(M\\)) and \\(M'_e\\) is called *induced* (from \\(M'\\)). The map \\(x'\\mapsto x'_e\\) from \\(M'\\) onto \\(M'_e\\) is the *induction*. (Part (3) of Theorem 5.8 shows that both algebras are von Neumann algebras.)\n\n**Theorem 5.8.**\n\n1. \\((M'_e)'=M_e\\).\n2. \\((M_e)'=M'_e\\).\n3. \\(M_e\\) and \\(M'_e\\) are von Neumann algebras on \\(K\\).\n4. \\(z\\in Z(M)\\), \\(e\\le z\\), and \\(z\\) is the smallest projection of \\(Z(M)\\) that majorizes \\(e\\). The kernel of the induction is \\(M'(1-z)\\). So the induction is injective if and only if \\([MeH]=H\\).\n5. \\(Z(M_e)=\\{c_e:c\\in Z(M)\\}\\). In particular, if \\(M\\) is a factor, so are \\(M_e\\) and \\(M'_e\\).\n\nBy symmetry, the same statements hold for a projection \\(e'\\in M'\\), with the roles of \\(M\\) and \\(M'\\) exchanged: apply the theorem to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).\n\n**Proof.** *The easy inclusions.* Each \\(x'\\in M'\\) commutes with each \\(exe\\), so \\(x'_e\\) commutes with \\(x_e\\). Thus \\(M'_e\\subseteq(M_e)'\\) and \\(M_e\\subseteq(M'_e)'\\).\n\n(1) Let \\(T\\in(M'_e)'\\), and define \\(\\tilde T\\in B(H)\\) by \\(\\tilde T\\xi=T(e\\xi)\\). For \\(x'\\in M'\\) and \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\tilde Tx'\\xi\\\\\n=T(ex'\\xi)\\\\\n=T(x'e\\xi)\\\\\n=Tx'_e(e\\xi)\\\\\n=x'_eT(e\\xi)\\\\\n=x'\\tilde T\\xi .\n\\end{gathered}\n\\]\nSo \\(\\tilde T\\in M''=M\\). Since \\(\\tilde T=e\\tilde Te\\), we get \\(T=\\tilde T_e\\in M_e\\).\n\n(2) Let \\(u'\\) be a unitary in \\((M_e)'\\). On the linear span \\(L_0\\) of the vectors \\(x\\xi\\) with \\(x\\in M\\) and \\(\\xi\\in K\\), try to define \\(u'_0\\big(\\sum_kx_k\\xi_k\\big)=\\sum_kx_ku'\\xi_k\\). For finitely many \\(x_k\\in M\\) and \\(\\xi_k\\in K\\),\n\\[\n\\begin{gathered}\n\\Big\\|\\sum_kx_ku'\\xi_k\\Big\\|^2\\\\\n=\\sum_{j,k}\\langle(ex_j^*x_ke)_e\\,u'\\xi_k,u'\\xi_j\\rangle\\\\\n=\\sum_{j,k}\\langle u'(ex_j^*x_ke)_e\\xi_k,u'\\xi_j\\rangle\\\\\n=\\Big\\|\\sum_kx_k\\xi_k\\Big\\|^2 .\n\\end{gathered}\n\\]\nThe first equality uses \\(u'\\xi_k\\in K\\); the second uses that \\(u'\\) commutes with \\(M_e\\); the third uses \\(u'^*u'=1_K\\). So \\(u'_0\\) is well defined on \\(L_0\\) (apply the identity to a difference of two representations of the same vector) and isometric. It extends to an isometry of \\(L=[MK]\\) into itself. Its range contains \\(L_0\\), because \\(u'\\) maps \\(K\\) onto \\(K\\); being closed, the range is all of \\(L\\). Put \\(u'_0=0\\) on \\(L^\\perp\\). The subspaces \\(L\\) and \\(L^\\perp\\) are invariant under \\(M\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). For \\(y\\in M\\), on \\(L_0\\) we have \\(u'_0y\\sum_kx_k\\xi_k=\\sum_kyx_ku'\\xi_k=yu'_0\\sum_kx_k\\xi_k\\), and by continuity this holds on \\(L\\); on \\(L^\\perp\\) both \\(u'_0y\\) and \\(yu'_0\\) vanish. So \\(u'_0\\in M'\\). For \\(\\xi\\in K\\), \\(\\xi=e\\xi\\) with \\(e\\in M\\), so \\(u'_0\\xi=eu'\\xi=u'\\xi\\). Thus \\(u'=(u'_0)_e\\in M'_e\\).\n\n\\((M_e)'\\) is a unital norm-closed \\(*\\)-algebra ([Proposition 2.1](#oa-fnd-bi-02)). By (U), each of its elements has the form \\(\\sum_{k=1}^4\\lambda_ku_k\\) with scalars \\(\\lambda_k\\) and unitaries \\(u_k\\in(M_e)'\\). \\(M'_e\\) is a linear space that contains all these unitaries, so \\((M_e)'\\subseteq M'_e\\).\n\n(3) By (1) and (2), \\(M_e\\) and \\(M'_e\\) are commutants of self-adjoint subsets of \\(B(K)\\). So both are von Neumann algebras ([Proposition 2.1](#oa-fnd-bi-02)(4)).\n\n(4) \\([MK]\\) is invariant under \\(M\\). It is invariant under \\(M'\\) too: for \\(y'\\in M'\\), \\(x\\in M\\) and \\(\\xi\\in K\\), \\(y'x\\xi=xy'\\xi\\) and \\(y'\\xi\\in K\\). Both sets are self-adjoint, so \\(z\\in M''\\cap M'=Z(M)\\). Since \\(e\\in M\\), \\(K=eK\\subseteq[MK]\\), so \\(e\\le z\\). If \\(c\\in Z(M)\\) is a projection with \\(e\\le c\\), then \\(cxe\\xi=xce\\xi=xe\\xi\\), so \\([MK]\\subseteq cH\\) and \\(z\\le c\\). For the kernel: if \\(x'_e=0\\), that is \\(x'e=0\\), then \\(x'xe\\xi=xx'e\\xi=0\\) for \\(x\\in M\\); so \\(x'\\) vanishes on \\(zH\\), and \\(x'=x'(1-z)\\in M'(1-z)\\). Conversely \\(y'(1-z)e=y'(e-ze)=0\\), since \\(ze=e\\). The kernel \\(M'(1-z)\\) is zero exactly when \\(1-z=0\\).\n\n(5) For \\(c\\in Z(M)\\), \\(c_e\\in M_e\\) and \\(c_e=c|_K\\in M'_e=(M_e)'\\) by (2); so \\(c_e\\in Z(M_e)\\). Conversely, let \\(t\\in Z(M_e)=M_e\\cap M'_e\\). Choose \\(x'\\in M'\\) with \\(x'_e=t\\). Replacing \\(x'\\) by \\(x'z\\) changes nothing on \\(K\\), so we may assume \\(x'=x'z=zx'\\). Let \\(w'\\in M'\\). Then \\((x'w'-w'x')_e=tw'_e-w'_et=0\\), since \\(t\\in M_e=(M'_e)'\\) by (1). So \\(q=x'w'-w'x'\\) lies in the kernel \\(M'(1-z)\\) by (4), that is \\(q=q(1-z)\\). Also \\(q=zq\\), because \\(z\\) is central and \\(x'=zx'=x'z\\). Hence \\(q=zq(1-z)=qz(1-z)=0\\). So \\(x'\\) commutes with \\(M'\\), \\(x'\\in M''=M\\), and \\(x'\\in Z(M)\\) with \\(x'_e=t\\). If \\(M\\) is a factor, \\(Z(M_e)=\\mathbb C1_K\\). The centre of \\(M'_e\\) is \\(M'_e\\cap(M'_e)'=M'_e\\cap M_e=Z(M_e)\\) by (1), so \\(M'_e\\) is a factor as well. \\(\\square\\)\n\n*Remark 5.9.* An unused alternative proof of (2), with square roots of operator matrices in place of unitaries, is in the lesson Spatial tensor products of von Neumann algebras. The induction need not be injective ([Exercise 5.10](#oa-fnd-bi-08)); parts (4) and (5) describe its kernel and the centres.\n\n**Exercise 5.10** (easy; the induction need not be injective). Let \\(H=H_1\\oplus H_2\\) with \\(H_1,H_2\\neq\\{0\\}\\), \\(M=B(H_1)\\oplus B(H_2)\\), and \\(e=1\\oplus0\\). Compute \\(M'\\), \\(M_e\\), \\(M'_e\\), the central projection \\(z\\) of Theorem 5.8(4), and the kernel of the induction.\n\n*Solution.* By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M'=\\{\\alpha1\\oplus\\beta1\\}\\). Here \\(K=H_1\\), \\(M_e=B(H_1)\\) and \\(M'_e=\\mathbb C1_{H_1}\\). The subspace \\([MK]=H_1\\), so \\(z=e\\). The induction sends \\(\\alpha1\\oplus\\beta1\\) to \\(\\alpha1_{H_1}\\); its kernel is \\(\\{0\\oplus\\beta1\\}=M'(1-z)\\), which is nonzero. As [Theorem 5.8](#oa-fnd-bi-08)(2) requires, \\((M_e)'=B(H_1)'=\\mathbb C1_{H_1}=M'_e\\). \\(\\square\\)\n\n",
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      "unit": "the-double-commutant-theorem",
      "name": "5. Direct sums, reduced and induced algebras",
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      "full_conditions_and_proof": "## 5. Direct sums, reduced and induced algebras\n\nTwo constructions produce new von Neumann algebras from old ones: direct sums, and cutting down by a projection of the algebra or of its commutant. For both we compute the commutant and the centre.\n\nLet \\(\\{M_i,H_i\\}_{i\\in I}\\) be von Neumann algebras, \\(I\\) any set. Let \\(H=\\bigoplus_iH_i\\) be the Hilbert sum, whose vectors are the families \\(\\bigoplus_i\\xi_i\\) with \\(\\sum_i\\|\\xi_i\\|^2<\\infty\\), and let \\(P_i\\) be the projection onto \\(H_i\\). For a family \\(x_i\\in B(H_i)\\) with \\(\\sup_i\\|x_i\\|<\\infty\\), put \\((\\bigoplus_ix_i)(\\bigoplus_i\\xi_i)=\\bigoplus_ix_i\\xi_i\\). This is a bounded operator with \\(\\|\\bigoplus_ix_i\\|=\\sup_i\\|x_i\\|\\): the bound \\(\\sum_i\\|x_i\\xi_i\\|^2\\le\\sup_i\\|x_i\\|^2\\sum_i\\|\\xi_i\\|^2\\) gives \\(\\le\\), and testing on vectors in a single \\(H_i\\) gives \\(\\ge\\).\n\n**Definition 5.1.** The *direct sum* \\(\\sum^\\oplus_iM_i\\), also written \\(\\sum^\\oplus_i\\{M_i,H_i\\}\\), is the set of operators \\(\\bigoplus_ix_i\\) with \\(x_i\\in M_i\\) and \\(\\sup_i\\|x_i\\|<\\infty\\).\n\n**Proposition 5.2.**\n\n1. \\(T\\in B(H)\\) commutes with every \\(P_i\\) if and only if \\(T=\\bigoplus_iT_i\\) for a bounded family \\(T_i\\in B(H_i)\\).\n2. \\(\\big(\\sum^\\oplus_iM_i\\big)'=\\sum^\\oplus_iM_i'\\).\n3. \\(\\sum^\\oplus_iM_i\\) is itself a von Neumann algebra, and its centre is \\(\\sum^\\oplus_iZ(M_i)\\).\n4. If at least two of the spaces \\(H_i\\) are nonzero, \\(\\sum^\\oplus_iM_i\\) is not a factor. If exactly one, \\(H_{i_0}\\), is nonzero, then \\(\\sum^\\oplus_iM_i\\) is spatially isomorphic to \\(M_{i_0}\\).\n\n**Proof.** (1) If \\(T\\) commutes with every \\(P_i\\), then \\(TH_i\\subseteq H_i\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). Let \\(T_i\\) be the restriction of \\(T\\) to \\(H_i\\); then \\(\\|T_i\\|\\le\\|T\\|\\). Since \\(\\xi=\\sum_iP_i\\xi\\) and \\(T\\) is continuous, \\(T\\xi=\\sum_iTP_i\\xi=\\bigoplus_iT_i\\xi_i\\). The converse is clear.\n\n(2) Each \\(P_i\\) lies in \\(\\sum^\\oplus M_i\\): take \\(1\\) in place \\(i\\) and \\(0\\) elsewhere. So \\(T\\in(\\sum^\\oplus M_i)'\\) commutes with every \\(P_i\\), and \\(T=\\bigoplus T_i\\) by (1). For \\(x\\in M_i\\), the operator with \\(x\\) in place \\(i\\) and \\(0\\) elsewhere lies in \\(\\sum^\\oplus M_i\\). Commuting with it gives \\(T_ix=xT_i\\). So \\(T_i\\in M_i'\\). Conversely, \\(\\bigoplus y_i\\) with \\(y_i\\in M_i'\\) commutes with every \\(\\bigoplus x_i\\), place by place.\n\n(3) \\(\\sum^\\oplus M_i\\) is a \\(*\\)-subalgebra: sums, products and adjoints are taken place by place, and the norms stay bounded. Each \\(M_i'\\) is a von Neumann algebra ([Proposition 2.4](#oa-fnd-bi-03)(2)), so (2) applies to the family \\((M_i')\\). It gives \\((\\sum^\\oplus M_i)''=(\\sum^\\oplus M_i')'=\\sum^\\oplus M_i''=\\sum^\\oplus M_i\\). The centre is \\((\\sum^\\oplus M_i)\\cap(\\sum^\\oplus M_i')\\); comparing places, this is \\(\\sum^\\oplus(M_i\\cap M_i')\\).\n\n(4) If \\(H_i\\neq0\\neq H_j\\) with \\(i\\neq j\\), then \\(P_i\\) is central by (3), and \\(P_i\\) is neither \\(0\\) nor \\(1\\). If only \\(H_{i_0}\\) is nonzero, the identification of \\(H\\) with \\(H_{i_0}\\) is a unitary that carries \\(\\sum^\\oplus M_i\\) onto \\(M_{i_0}\\), since the other places act on zero spaces. \\(\\square\\)\n\n*Remark 5.3.* The proof of (2) uses only that each \\(M_i\\) contains \\(0\\) and \\(1_{H_i}\\). Without units the formula can fail: for \\(S_1=S_2=\\{0\\}\\) on \\(\\mathbb C\\oplus\\mathbb C\\), the commutant of the direct sum is all of \\(B(\\mathbb C^2)\\), not the diagonal matrices.\n\n**Example 5.4** (a degenerate algebra). Let \\(p\\) be a projection with \\(p\\neq0\\) and \\(p\\neq1\\), and \\(M=\\mathbb Cp\\). This is a closed \\(*\\)-subalgebra, \\([MH]=pH\\), and \\(e=p\\). An operator commutes with \\(p\\) exactly when it is block diagonal for \\(H=pH\\oplus(1-p)H\\), so \\(M'=B(pH)\\oplus B((1-p)H)\\). By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\{\\alpha p+\\beta(1-p)\\}=M+\\mathbb C1\\). So \\(M''\\neq M=\\overline M=eM''\\), as (4.1) and [Theorem 4.4](#oa-fnd-bi-07)(4) predict.\n\n**Example 5.5** (an isomorphism that is not spatial). On \\(\\mathbb C^4\\) let \\(M_1=\\{\\operatorname{diag}(a,b,b,b)\\}\\) and \\(M_2=\\{\\operatorname{diag}(a,a,b,b)\\}\\), with \\(a,b\\in\\mathbb C\\). By [Proposition 5.2](#oa-fnd-bi-04) and [Proposition 2.4](#oa-fnd-bi-03)(4), these are von Neumann algebras: \\(M_1=\\mathbb C1_{\\mathbb C}\\oplus\\mathbb C1_{\\mathbb C^3}\\) and \\(M_2=\\mathbb C1_{\\mathbb C^2}\\oplus\\mathbb C1_{\\mathbb C^2}\\). The map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism. It is not spatial. A unitary \\(U\\) with \\(UM_1U^*=M_2\\) would carry the rank-one projection \\(\\operatorname{diag}(1,0,0,0)\\in M_1\\) to a rank-one projection in \\(M_2\\), but the projections of \\(M_2\\) have rank \\(0\\), \\(2\\) or \\(4\\).\n\n**Example 5.6** (commutants of direct sums with a common summand). Suppose \\(H\\neq\\{0\\}\\), and let \\(M=\\{x\\oplus x:x\\in B(H)\\}\\) on \\(H\\oplus H\\). This is \\(\\pi(B(H))\\) for \\(I=\\{1,2\\}\\) in [Section 3](#oa-fnd-bi-05), so \\(M''=M\\) by Corollary 3.4, and by (3.4) \\(M'\\) consists of the \\(2\\times2\\) scalar matrices tensored with \\(1_H\\). An operator in both \\(M\\) and \\(M'\\) is \\(\\pi(x)\\) with \\(x\\) scalar. So \\(M\\) is a factor, while the direct sum \\(B(H)\\oplus B(H)\\) of [Definition 5.1](#oa-fnd-bi-04) is not: its central projection \\(1_H\\oplus0\\) is neither zero nor the identity, by \\(H\\neq\\{0\\}\\). The two algebras are different subalgebras of \\(B(H\\oplus H)\\) built from the same summands. If \\(H=\\{0\\}\\), both constructions give the same zero algebra; its centre is \\(\\mathbb C1_H=\\{0\\}\\), so it satisfies this lesson's centre-based factor convention. Thus the contrast between the two constructions requires a nonzero summand.\n\nNow fix a von Neumann algebra \\(\\{M,H\\}\\) and a projection \\(e\\in M\\), and put \\(K=eH\\). For \\(x\\in M\\), the operator \\(exe\\) maps \\(K\\) into \\(K\\); write \\(x_e\\) for its restriction to \\(K\\). Each \\(x'\\in M'\\) commutes with \\(e\\), so it maps \\(K\\) into \\(K\\); write \\(x'_e\\) for its restriction. Put \\(M_e=\\{x_e:x\\in M\\}\\) and \\(M'_e=\\{x'_e:x'\\in M'\\}\\). Both are \\(*\\)-subalgebras of \\(B(K)\\) containing \\(1_K\\). The map \\(x'\\mapsto x'_e\\) is a unital \\(*\\)-homomorphism from \\(M'\\) onto \\(M'_e\\), because \\(e\\) commutes with \\(M'\\). Let \\(z\\) be the projection onto \\([MK]=[MeH]\\).\n\n**Definition 5.7.** On \\(K\\), the algebra \\(M_e\\) is called *reduced* (from \\(M\\)) and \\(M'_e\\) is called *induced* (from \\(M'\\)). The map \\(x'\\mapsto x'_e\\) from \\(M'\\) onto \\(M'_e\\) is the *induction*. (Part (3) of Theorem 5.8 shows that both algebras are von Neumann algebras.)\n\n**Theorem 5.8.**\n\n1. \\((M'_e)'=M_e\\).\n2. \\((M_e)'=M'_e\\).\n3. \\(M_e\\) and \\(M'_e\\) are von Neumann algebras on \\(K\\).\n4. \\(z\\in Z(M)\\), \\(e\\le z\\), and \\(z\\) is the smallest projection of \\(Z(M)\\) that majorizes \\(e\\). The kernel of the induction is \\(M'(1-z)\\). So the induction is injective if and only if \\([MeH]=H\\).\n5. \\(Z(M_e)=\\{c_e:c\\in Z(M)\\}\\). In particular, if \\(M\\) is a factor, so are \\(M_e\\) and \\(M'_e\\).\n\nBy symmetry, the same statements hold for a projection \\(e'\\in M'\\), with the roles of \\(M\\) and \\(M'\\) exchanged: apply the theorem to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).\n\n**Proof.** *The easy inclusions.* Each \\(x'\\in M'\\) commutes with each \\(exe\\), so \\(x'_e\\) commutes with \\(x_e\\). Thus \\(M'_e\\subseteq(M_e)'\\) and \\(M_e\\subseteq(M'_e)'\\).\n\n(1) Let \\(T\\in(M'_e)'\\), and define \\(\\tilde T\\in B(H)\\) by \\(\\tilde T\\xi=T(e\\xi)\\). For \\(x'\\in M'\\) and \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\tilde Tx'\\xi\\\\\n=T(ex'\\xi)\\\\\n=T(x'e\\xi)\\\\\n=Tx'_e(e\\xi)\\\\\n=x'_eT(e\\xi)\\\\\n=x'\\tilde T\\xi .\n\\end{gathered}\n\\]\nSo \\(\\tilde T\\in M''=M\\). Since \\(\\tilde T=e\\tilde Te\\), we get \\(T=\\tilde T_e\\in M_e\\).\n\n(2) Let \\(u'\\) be a unitary in \\((M_e)'\\). On the linear span \\(L_0\\) of the vectors \\(x\\xi\\) with \\(x\\in M\\) and \\(\\xi\\in K\\), try to define \\(u'_0\\big(\\sum_kx_k\\xi_k\\big)=\\sum_kx_ku'\\xi_k\\). For finitely many \\(x_k\\in M\\) and \\(\\xi_k\\in K\\),\n\\[\n\\begin{gathered}\n\\Big\\|\\sum_kx_ku'\\xi_k\\Big\\|^2\\\\\n=\\sum_{j,k}\\langle(ex_j^*x_ke)_e\\,u'\\xi_k,u'\\xi_j\\rangle\\\\\n=\\sum_{j,k}\\langle u'(ex_j^*x_ke)_e\\xi_k,u'\\xi_j\\rangle\\\\\n=\\Big\\|\\sum_kx_k\\xi_k\\Big\\|^2 .\n\\end{gathered}\n\\]\nThe first equality uses \\(u'\\xi_k\\in K\\); the second uses that \\(u'\\) commutes with \\(M_e\\); the third uses \\(u'^*u'=1_K\\). So \\(u'_0\\) is well defined on \\(L_0\\) (apply the identity to a difference of two representations of the same vector) and isometric. It extends to an isometry of \\(L=[MK]\\) into itself. Its range contains \\(L_0\\), because \\(u'\\) maps \\(K\\) onto \\(K\\); being closed, the range is all of \\(L\\). Put \\(u'_0=0\\) on \\(L^\\perp\\). The subspaces \\(L\\) and \\(L^\\perp\\) are invariant under \\(M\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). For \\(y\\in M\\), on \\(L_0\\) we have \\(u'_0y\\sum_kx_k\\xi_k=\\sum_kyx_ku'\\xi_k=yu'_0\\sum_kx_k\\xi_k\\), and by continuity this holds on \\(L\\); on \\(L^\\perp\\) both \\(u'_0y\\) and \\(yu'_0\\) vanish. So \\(u'_0\\in M'\\). For \\(\\xi\\in K\\), \\(\\xi=e\\xi\\) with \\(e\\in M\\), so \\(u'_0\\xi=eu'\\xi=u'\\xi\\). Thus \\(u'=(u'_0)_e\\in M'_e\\).\n\n\\((M_e)'\\) is a unital norm-closed \\(*\\)-algebra ([Proposition 2.1](#oa-fnd-bi-02)). By (U), each of its elements has the form \\(\\sum_{k=1}^4\\lambda_ku_k\\) with scalars \\(\\lambda_k\\) and unitaries \\(u_k\\in(M_e)'\\). \\(M'_e\\) is a linear space that contains all these unitaries, so \\((M_e)'\\subseteq M'_e\\).\n\n(3) By (1) and (2), \\(M_e\\) and \\(M'_e\\) are commutants of self-adjoint subsets of \\(B(K)\\). So both are von Neumann algebras ([Proposition 2.1](#oa-fnd-bi-02)(4)).\n\n(4) \\([MK]\\) is invariant under \\(M\\). It is invariant under \\(M'\\) too: for \\(y'\\in M'\\), \\(x\\in M\\) and \\(\\xi\\in K\\), \\(y'x\\xi=xy'\\xi\\) and \\(y'\\xi\\in K\\). Both sets are self-adjoint, so \\(z\\in M''\\cap M'=Z(M)\\). Since \\(e\\in M\\), \\(K=eK\\subseteq[MK]\\), so \\(e\\le z\\). If \\(c\\in Z(M)\\) is a projection with \\(e\\le c\\), then \\(cxe\\xi=xce\\xi=xe\\xi\\), so \\([MK]\\subseteq cH\\) and \\(z\\le c\\). For the kernel: if \\(x'_e=0\\), that is \\(x'e=0\\), then \\(x'xe\\xi=xx'e\\xi=0\\) for \\(x\\in M\\); so \\(x'\\) vanishes on \\(zH\\), and \\(x'=x'(1-z)\\in M'(1-z)\\). Conversely \\(y'(1-z)e=y'(e-ze)=0\\), since \\(ze=e\\). The kernel \\(M'(1-z)\\) is zero exactly when \\(1-z=0\\).\n\n(5) For \\(c\\in Z(M)\\), \\(c_e\\in M_e\\) and \\(c_e=c|_K\\in M'_e=(M_e)'\\) by (2); so \\(c_e\\in Z(M_e)\\). Conversely, let \\(t\\in Z(M_e)=M_e\\cap M'_e\\). Choose \\(x'\\in M'\\) with \\(x'_e=t\\). Replacing \\(x'\\) by \\(x'z\\) changes nothing on \\(K\\), so we may assume \\(x'=x'z=zx'\\). Let \\(w'\\in M'\\). Then \\((x'w'-w'x')_e=tw'_e-w'_et=0\\), since \\(t\\in M_e=(M'_e)'\\) by (1). So \\(q=x'w'-w'x'\\) lies in the kernel \\(M'(1-z)\\) by (4), that is \\(q=q(1-z)\\). Also \\(q=zq\\), because \\(z\\) is central and \\(x'=zx'=x'z\\). Hence \\(q=zq(1-z)=qz(1-z)=0\\). So \\(x'\\) commutes with \\(M'\\), \\(x'\\in M''=M\\), and \\(x'\\in Z(M)\\) with \\(x'_e=t\\). If \\(M\\) is a factor, \\(Z(M_e)=\\mathbb C1_K\\). The centre of \\(M'_e\\) is \\(M'_e\\cap(M'_e)'=M'_e\\cap M_e=Z(M_e)\\) by (1), so \\(M'_e\\) is a factor as well. \\(\\square\\)\n\n*Remark 5.9.* An unused alternative proof of (2), with square roots of operator matrices in place of unitaries, is in the lesson Spatial tensor products of von Neumann algebras. The induction need not be injective ([Exercise 5.10](#oa-fnd-bi-08)); parts (4) and (5) describe its kernel and the centres.\n\n**Exercise 5.10** (easy; the induction need not be injective). Let \\(H=H_1\\oplus H_2\\) with \\(H_1,H_2\\neq\\{0\\}\\), \\(M=B(H_1)\\oplus B(H_2)\\), and \\(e=1\\oplus0\\). Compute \\(M'\\), \\(M_e\\), \\(M'_e\\), the central projection \\(z\\) of Theorem 5.8(4), and the kernel of the induction.\n\n*Solution.* By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M'=\\{\\alpha1\\oplus\\beta1\\}\\). Here \\(K=H_1\\), \\(M_e=B(H_1)\\) and \\(M'_e=\\mathbb C1_{H_1}\\). The subspace \\([MK]=H_1\\), so \\(z=e\\). The induction sends \\(\\alpha1\\oplus\\beta1\\) to \\(\\alpha1_{H_1}\\); its kernel is \\(\\{0\\oplus\\beta1\\}=M'(1-z)\\), which is nonzero. As [Theorem 5.8](#oa-fnd-bi-08)(2) requires, \\((M_e)'=B(H_1)'=\\mathbb C1_{H_1}=M'_e\\). \\(\\square\\)\n\n",
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      "name": "5. Direct sums, reduced and induced algebras",
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      "full_conditions_and_proof": "## 5. Direct sums, reduced and induced algebras\n\nTwo constructions produce new von Neumann algebras from old ones: direct sums, and cutting down by a projection of the algebra or of its commutant. For both we compute the commutant and the centre.\n\nLet \\(\\{M_i,H_i\\}_{i\\in I}\\) be von Neumann algebras, \\(I\\) any set. Let \\(H=\\bigoplus_iH_i\\) be the Hilbert sum, whose vectors are the families \\(\\bigoplus_i\\xi_i\\) with \\(\\sum_i\\|\\xi_i\\|^2<\\infty\\), and let \\(P_i\\) be the projection onto \\(H_i\\). For a family \\(x_i\\in B(H_i)\\) with \\(\\sup_i\\|x_i\\|<\\infty\\), put \\((\\bigoplus_ix_i)(\\bigoplus_i\\xi_i)=\\bigoplus_ix_i\\xi_i\\). This is a bounded operator with \\(\\|\\bigoplus_ix_i\\|=\\sup_i\\|x_i\\|\\): the bound \\(\\sum_i\\|x_i\\xi_i\\|^2\\le\\sup_i\\|x_i\\|^2\\sum_i\\|\\xi_i\\|^2\\) gives \\(\\le\\), and testing on vectors in a single \\(H_i\\) gives \\(\\ge\\).\n\n**Definition 5.1.** The *direct sum* \\(\\sum^\\oplus_iM_i\\), also written \\(\\sum^\\oplus_i\\{M_i,H_i\\}\\), is the set of operators \\(\\bigoplus_ix_i\\) with \\(x_i\\in M_i\\) and \\(\\sup_i\\|x_i\\|<\\infty\\).\n\n**Proposition 5.2.**\n\n1. \\(T\\in B(H)\\) commutes with every \\(P_i\\) if and only if \\(T=\\bigoplus_iT_i\\) for a bounded family \\(T_i\\in B(H_i)\\).\n2. \\(\\big(\\sum^\\oplus_iM_i\\big)'=\\sum^\\oplus_iM_i'\\).\n3. \\(\\sum^\\oplus_iM_i\\) is itself a von Neumann algebra, and its centre is \\(\\sum^\\oplus_iZ(M_i)\\).\n4. If at least two of the spaces \\(H_i\\) are nonzero, \\(\\sum^\\oplus_iM_i\\) is not a factor. If exactly one, \\(H_{i_0}\\), is nonzero, then \\(\\sum^\\oplus_iM_i\\) is spatially isomorphic to \\(M_{i_0}\\).\n\n**Proof.** (1) If \\(T\\) commutes with every \\(P_i\\), then \\(TH_i\\subseteq H_i\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). Let \\(T_i\\) be the restriction of \\(T\\) to \\(H_i\\); then \\(\\|T_i\\|\\le\\|T\\|\\). Since \\(\\xi=\\sum_iP_i\\xi\\) and \\(T\\) is continuous, \\(T\\xi=\\sum_iTP_i\\xi=\\bigoplus_iT_i\\xi_i\\). The converse is clear.\n\n(2) Each \\(P_i\\) lies in \\(\\sum^\\oplus M_i\\): take \\(1\\) in place \\(i\\) and \\(0\\) elsewhere. So \\(T\\in(\\sum^\\oplus M_i)'\\) commutes with every \\(P_i\\), and \\(T=\\bigoplus T_i\\) by (1). For \\(x\\in M_i\\), the operator with \\(x\\) in place \\(i\\) and \\(0\\) elsewhere lies in \\(\\sum^\\oplus M_i\\). Commuting with it gives \\(T_ix=xT_i\\). So \\(T_i\\in M_i'\\). Conversely, \\(\\bigoplus y_i\\) with \\(y_i\\in M_i'\\) commutes with every \\(\\bigoplus x_i\\), place by place.\n\n(3) \\(\\sum^\\oplus M_i\\) is a \\(*\\)-subalgebra: sums, products and adjoints are taken place by place, and the norms stay bounded. Each \\(M_i'\\) is a von Neumann algebra ([Proposition 2.4](#oa-fnd-bi-03)(2)), so (2) applies to the family \\((M_i')\\). It gives \\((\\sum^\\oplus M_i)''=(\\sum^\\oplus M_i')'=\\sum^\\oplus M_i''=\\sum^\\oplus M_i\\). The centre is \\((\\sum^\\oplus M_i)\\cap(\\sum^\\oplus M_i')\\); comparing places, this is \\(\\sum^\\oplus(M_i\\cap M_i')\\).\n\n(4) If \\(H_i\\neq0\\neq H_j\\) with \\(i\\neq j\\), then \\(P_i\\) is central by (3), and \\(P_i\\) is neither \\(0\\) nor \\(1\\). If only \\(H_{i_0}\\) is nonzero, the identification of \\(H\\) with \\(H_{i_0}\\) is a unitary that carries \\(\\sum^\\oplus M_i\\) onto \\(M_{i_0}\\), since the other places act on zero spaces. \\(\\square\\)\n\n*Remark 5.3.* The proof of (2) uses only that each \\(M_i\\) contains \\(0\\) and \\(1_{H_i}\\). Without units the formula can fail: for \\(S_1=S_2=\\{0\\}\\) on \\(\\mathbb C\\oplus\\mathbb C\\), the commutant of the direct sum is all of \\(B(\\mathbb C^2)\\), not the diagonal matrices.\n\n**Example 5.4** (a degenerate algebra). Let \\(p\\) be a projection with \\(p\\neq0\\) and \\(p\\neq1\\), and \\(M=\\mathbb Cp\\). This is a closed \\(*\\)-subalgebra, \\([MH]=pH\\), and \\(e=p\\). An operator commutes with \\(p\\) exactly when it is block diagonal for \\(H=pH\\oplus(1-p)H\\), so \\(M'=B(pH)\\oplus B((1-p)H)\\). By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\{\\alpha p+\\beta(1-p)\\}=M+\\mathbb C1\\). So \\(M''\\neq M=\\overline M=eM''\\), as (4.1) and [Theorem 4.4](#oa-fnd-bi-07)(4) predict.\n\n**Example 5.5** (an isomorphism that is not spatial). On \\(\\mathbb C^4\\) let \\(M_1=\\{\\operatorname{diag}(a,b,b,b)\\}\\) and \\(M_2=\\{\\operatorname{diag}(a,a,b,b)\\}\\), with \\(a,b\\in\\mathbb C\\). By [Proposition 5.2](#oa-fnd-bi-04) and [Proposition 2.4](#oa-fnd-bi-03)(4), these are von Neumann algebras: \\(M_1=\\mathbb C1_{\\mathbb C}\\oplus\\mathbb C1_{\\mathbb C^3}\\) and \\(M_2=\\mathbb C1_{\\mathbb C^2}\\oplus\\mathbb C1_{\\mathbb C^2}\\). The map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism. It is not spatial. A unitary \\(U\\) with \\(UM_1U^*=M_2\\) would carry the rank-one projection \\(\\operatorname{diag}(1,0,0,0)\\in M_1\\) to a rank-one projection in \\(M_2\\), but the projections of \\(M_2\\) have rank \\(0\\), \\(2\\) or \\(4\\).\n\n**Example 5.6** (commutants of direct sums with a common summand). Suppose \\(H\\neq\\{0\\}\\), and let \\(M=\\{x\\oplus x:x\\in B(H)\\}\\) on \\(H\\oplus H\\). This is \\(\\pi(B(H))\\) for \\(I=\\{1,2\\}\\) in [Section 3](#oa-fnd-bi-05), so \\(M''=M\\) by Corollary 3.4, and by (3.4) \\(M'\\) consists of the \\(2\\times2\\) scalar matrices tensored with \\(1_H\\). An operator in both \\(M\\) and \\(M'\\) is \\(\\pi(x)\\) with \\(x\\) scalar. So \\(M\\) is a factor, while the direct sum \\(B(H)\\oplus B(H)\\) of [Definition 5.1](#oa-fnd-bi-04) is not: its central projection \\(1_H\\oplus0\\) is neither zero nor the identity, by \\(H\\neq\\{0\\}\\). The two algebras are different subalgebras of \\(B(H\\oplus H)\\) built from the same summands. If \\(H=\\{0\\}\\), both constructions give the same zero algebra; its centre is \\(\\mathbb C1_H=\\{0\\}\\), so it satisfies this lesson's centre-based factor convention. Thus the contrast between the two constructions requires a nonzero summand.\n\nNow fix a von Neumann algebra \\(\\{M,H\\}\\) and a projection \\(e\\in M\\), and put \\(K=eH\\). For \\(x\\in M\\), the operator \\(exe\\) maps \\(K\\) into \\(K\\); write \\(x_e\\) for its restriction to \\(K\\). Each \\(x'\\in M'\\) commutes with \\(e\\), so it maps \\(K\\) into \\(K\\); write \\(x'_e\\) for its restriction. Put \\(M_e=\\{x_e:x\\in M\\}\\) and \\(M'_e=\\{x'_e:x'\\in M'\\}\\). Both are \\(*\\)-subalgebras of \\(B(K)\\) containing \\(1_K\\). The map \\(x'\\mapsto x'_e\\) is a unital \\(*\\)-homomorphism from \\(M'\\) onto \\(M'_e\\), because \\(e\\) commutes with \\(M'\\). Let \\(z\\) be the projection onto \\([MK]=[MeH]\\).\n\n**Definition 5.7.** On \\(K\\), the algebra \\(M_e\\) is called *reduced* (from \\(M\\)) and \\(M'_e\\) is called *induced* (from \\(M'\\)). The map \\(x'\\mapsto x'_e\\) from \\(M'\\) onto \\(M'_e\\) is the *induction*. (Part (3) of Theorem 5.8 shows that both algebras are von Neumann algebras.)\n\n**Theorem 5.8.**\n\n1. \\((M'_e)'=M_e\\).\n2. \\((M_e)'=M'_e\\).\n3. \\(M_e\\) and \\(M'_e\\) are von Neumann algebras on \\(K\\).\n4. \\(z\\in Z(M)\\), \\(e\\le z\\), and \\(z\\) is the smallest projection of \\(Z(M)\\) that majorizes \\(e\\). The kernel of the induction is \\(M'(1-z)\\). So the induction is injective if and only if \\([MeH]=H\\).\n5. \\(Z(M_e)=\\{c_e:c\\in Z(M)\\}\\). In particular, if \\(M\\) is a factor, so are \\(M_e\\) and \\(M'_e\\).\n\nBy symmetry, the same statements hold for a projection \\(e'\\in M'\\), with the roles of \\(M\\) and \\(M'\\) exchanged: apply the theorem to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).\n\n**Proof.** *The easy inclusions.* Each \\(x'\\in M'\\) commutes with each \\(exe\\), so \\(x'_e\\) commutes with \\(x_e\\). Thus \\(M'_e\\subseteq(M_e)'\\) and \\(M_e\\subseteq(M'_e)'\\).\n\n(1) Let \\(T\\in(M'_e)'\\), and define \\(\\tilde T\\in B(H)\\) by \\(\\tilde T\\xi=T(e\\xi)\\). For \\(x'\\in M'\\) and \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\tilde Tx'\\xi\\\\\n=T(ex'\\xi)\\\\\n=T(x'e\\xi)\\\\\n=Tx'_e(e\\xi)\\\\\n=x'_eT(e\\xi)\\\\\n=x'\\tilde T\\xi .\n\\end{gathered}\n\\]\nSo \\(\\tilde T\\in M''=M\\). Since \\(\\tilde T=e\\tilde Te\\), we get \\(T=\\tilde T_e\\in M_e\\).\n\n(2) Let \\(u'\\) be a unitary in \\((M_e)'\\). On the linear span \\(L_0\\) of the vectors \\(x\\xi\\) with \\(x\\in M\\) and \\(\\xi\\in K\\), try to define \\(u'_0\\big(\\sum_kx_k\\xi_k\\big)=\\sum_kx_ku'\\xi_k\\). For finitely many \\(x_k\\in M\\) and \\(\\xi_k\\in K\\),\n\\[\n\\begin{gathered}\n\\Big\\|\\sum_kx_ku'\\xi_k\\Big\\|^2\\\\\n=\\sum_{j,k}\\langle(ex_j^*x_ke)_e\\,u'\\xi_k,u'\\xi_j\\rangle\\\\\n=\\sum_{j,k}\\langle u'(ex_j^*x_ke)_e\\xi_k,u'\\xi_j\\rangle\\\\\n=\\Big\\|\\sum_kx_k\\xi_k\\Big\\|^2 .\n\\end{gathered}\n\\]\nThe first equality uses \\(u'\\xi_k\\in K\\); the second uses that \\(u'\\) commutes with \\(M_e\\); the third uses \\(u'^*u'=1_K\\). So \\(u'_0\\) is well defined on \\(L_0\\) (apply the identity to a difference of two representations of the same vector) and isometric. It extends to an isometry of \\(L=[MK]\\) into itself. Its range contains \\(L_0\\), because \\(u'\\) maps \\(K\\) onto \\(K\\); being closed, the range is all of \\(L\\). Put \\(u'_0=0\\) on \\(L^\\perp\\). The subspaces \\(L\\) and \\(L^\\perp\\) are invariant under \\(M\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). For \\(y\\in M\\), on \\(L_0\\) we have \\(u'_0y\\sum_kx_k\\xi_k=\\sum_kyx_ku'\\xi_k=yu'_0\\sum_kx_k\\xi_k\\), and by continuity this holds on \\(L\\); on \\(L^\\perp\\) both \\(u'_0y\\) and \\(yu'_0\\) vanish. So \\(u'_0\\in M'\\). For \\(\\xi\\in K\\), \\(\\xi=e\\xi\\) with \\(e\\in M\\), so \\(u'_0\\xi=eu'\\xi=u'\\xi\\). Thus \\(u'=(u'_0)_e\\in M'_e\\).\n\n\\((M_e)'\\) is a unital norm-closed \\(*\\)-algebra ([Proposition 2.1](#oa-fnd-bi-02)). By (U), each of its elements has the form \\(\\sum_{k=1}^4\\lambda_ku_k\\) with scalars \\(\\lambda_k\\) and unitaries \\(u_k\\in(M_e)'\\). \\(M'_e\\) is a linear space that contains all these unitaries, so \\((M_e)'\\subseteq M'_e\\).\n\n(3) By (1) and (2), \\(M_e\\) and \\(M'_e\\) are commutants of self-adjoint subsets of \\(B(K)\\). So both are von Neumann algebras ([Proposition 2.1](#oa-fnd-bi-02)(4)).\n\n(4) \\([MK]\\) is invariant under \\(M\\). It is invariant under \\(M'\\) too: for \\(y'\\in M'\\), \\(x\\in M\\) and \\(\\xi\\in K\\), \\(y'x\\xi=xy'\\xi\\) and \\(y'\\xi\\in K\\). Both sets are self-adjoint, so \\(z\\in M''\\cap M'=Z(M)\\). Since \\(e\\in M\\), \\(K=eK\\subseteq[MK]\\), so \\(e\\le z\\). If \\(c\\in Z(M)\\) is a projection with \\(e\\le c\\), then \\(cxe\\xi=xce\\xi=xe\\xi\\), so \\([MK]\\subseteq cH\\) and \\(z\\le c\\). For the kernel: if \\(x'_e=0\\), that is \\(x'e=0\\), then \\(x'xe\\xi=xx'e\\xi=0\\) for \\(x\\in M\\); so \\(x'\\) vanishes on \\(zH\\), and \\(x'=x'(1-z)\\in M'(1-z)\\). Conversely \\(y'(1-z)e=y'(e-ze)=0\\), since \\(ze=e\\). The kernel \\(M'(1-z)\\) is zero exactly when \\(1-z=0\\).\n\n(5) For \\(c\\in Z(M)\\), \\(c_e\\in M_e\\) and \\(c_e=c|_K\\in M'_e=(M_e)'\\) by (2); so \\(c_e\\in Z(M_e)\\). Conversely, let \\(t\\in Z(M_e)=M_e\\cap M'_e\\). Choose \\(x'\\in M'\\) with \\(x'_e=t\\). Replacing \\(x'\\) by \\(x'z\\) changes nothing on \\(K\\), so we may assume \\(x'=x'z=zx'\\). Let \\(w'\\in M'\\). Then \\((x'w'-w'x')_e=tw'_e-w'_et=0\\), since \\(t\\in M_e=(M'_e)'\\) by (1). So \\(q=x'w'-w'x'\\) lies in the kernel \\(M'(1-z)\\) by (4), that is \\(q=q(1-z)\\). Also \\(q=zq\\), because \\(z\\) is central and \\(x'=zx'=x'z\\). Hence \\(q=zq(1-z)=qz(1-z)=0\\). So \\(x'\\) commutes with \\(M'\\), \\(x'\\in M''=M\\), and \\(x'\\in Z(M)\\) with \\(x'_e=t\\). If \\(M\\) is a factor, \\(Z(M_e)=\\mathbb C1_K\\). The centre of \\(M'_e\\) is \\(M'_e\\cap(M'_e)'=M'_e\\cap M_e=Z(M_e)\\) by (1), so \\(M'_e\\) is a factor as well. \\(\\square\\)\n\n*Remark 5.9.* An unused alternative proof of (2), with square roots of operator matrices in place of unitaries, is in the lesson Spatial tensor products of von Neumann algebras. The induction need not be injective ([Exercise 5.10](#oa-fnd-bi-08)); parts (4) and (5) describe its kernel and the centres.\n\n**Exercise 5.10** (easy; the induction need not be injective). Let \\(H=H_1\\oplus H_2\\) with \\(H_1,H_2\\neq\\{0\\}\\), \\(M=B(H_1)\\oplus B(H_2)\\), and \\(e=1\\oplus0\\). Compute \\(M'\\), \\(M_e\\), \\(M'_e\\), the central projection \\(z\\) of Theorem 5.8(4), and the kernel of the induction.\n\n*Solution.* By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M'=\\{\\alpha1\\oplus\\beta1\\}\\). Here \\(K=H_1\\), \\(M_e=B(H_1)\\) and \\(M'_e=\\mathbb C1_{H_1}\\). The subspace \\([MK]=H_1\\), so \\(z=e\\). The induction sends \\(\\alpha1\\oplus\\beta1\\) to \\(\\alpha1_{H_1}\\); its kernel is \\(\\{0\\oplus\\beta1\\}=M'(1-z)\\), which is nonzero. As [Theorem 5.8](#oa-fnd-bi-08)(2) requires, \\((M_e)'=B(H_1)'=\\mathbb C1_{H_1}=M'_e\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-16",
      "unit": "the-double-commutant-theorem",
      "name": "6. Algebras of compact operators",
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      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 6. Algebras of compact operators\n\nBy Example 4.6, \\(K(H)\\) is dense in \\(B(H)\\) when \\(\\dim H=\\infty\\). This section looks at norm-closed \\(*\\)-subalgebras of \\(K(H)\\). The double commutant theorem still recovers such an algebra, provided one intersects with \\(K(H)\\) only at the end. The section ends with the simple C\\(^*\\)-algebras that have a nonzero minimal projection: each of them is isomorphic to \\(K(H)\\) for some Hilbert space \\(H\\).\n\n**Example 6.1** (commutants taken inside \\(K(H)\\)). Let \\(H=\\ell^2\\oplus\\ell^2\\) and \\(A=K(\\ell^2)\\oplus K(\\ell^2)=\\{a\\oplus b\\}\\). This is a norm-closed nondegenerate \\(*\\)-subalgebra contained in \\(K(H)\\). An operator that commutes with every \\(a\\oplus0\\) and every \\(0\\oplus b\\) has zero off-diagonal blocks, because \\(K(\\ell^2)\\) is nondegenerate, and its diagonal blocks lie in \\(K(\\ell^2)'=\\mathbb C1\\) (Example 4.6). So \\(A'=\\{\\alpha1\\oplus\\beta1\\}\\). ([Proposition 5.2](#oa-fnd-bi-04)(2) does not apply directly, since \\(1\\notin K(\\ell^2)\\); nondegeneracy replaces the projections \\(P_i\\) used there.) None of these operators is compact unless \\(\\alpha=\\beta=0\\), so \\(A'\\cap K(H)=\\{0\\}\\), and \\(\\{0\\}'\\cap K(H)=K(H)\\). But \\(K(H)\\) contains the rank-one operator \\(\\theta_{0\\oplus\\delta_1,\\,\\delta_1\\oplus0}\\), which maps the first summand into the second and so is not in \\(A\\). Hence \\(\\{A'\\cap K(H)\\}'\\cap K(H)\\neq A\\): commutants taken inside \\(K(H)\\) do not recover \\(A\\). When \\(\\dim H<\\infty\\) this identity does hold. Then \\(K(H)=B(H)\\), and \\(A\\) is finite dimensional, hence closed in every topology, so the identity reads \\(A''=A\\), which is [Theorem 4.4](#oa-fnd-bi-07)(5).\n\n**Exercise 6.2** (hard; compact operators in the bicommutant). Let \\(A\\subseteq K(H)\\) be a nondegenerate norm-closed \\(*\\)-subalgebra. Show that \\(A''\\cap K(H)=A\\).\n\nExample 6.1 proves directly that taking the first commutant inside \\(K(H)\\) can fail to recover the algebra.\n\nNondegeneracy is necessary here. For the degenerate algebra \\(A=\\mathbb CE_{11}\\) on \\(\\mathbb C^2\\), Example 5.4 gives \\(A''\\cap K(\\mathbb C^2)=A''=\\{\\operatorname{diag}(\\alpha,\\beta)\\}\\neq A\\).\n\n*Solution.* The inclusion \\(A\\subseteq A''\\cap K(H)\\) is clear.\n\n*Step 1 (spectral projections lie in \\(A\\)).* Let \\(a\\in A\\), \\(a\\ge0\\), and let \\(t>0\\) with \\(t\\) not in the spectrum \\(\\sigma(a)\\). By (K), \\(a=\\sum_n\\alpha_n\\theta_{\\varepsilon_n,\\varepsilon_n}\\) for an orthonormal family \\((\\varepsilon_n)\\) and numbers \\(\\alpha_n>0\\) (tending to \\(0\\) if there are infinitely many), and \\(\\sigma(a)\\subseteq\\{\\alpha_n\\}\\cup\\{0\\}\\). Only finitely many \\(\\alpha_n\\) exceed \\(t\\). The indicator function \\(g\\) of \\((t,\\infty)\\) is continuous on \\(\\sigma(a)\\), because the closed set \\(\\sigma(a)\\) misses a neighbourhood of \\(t\\), and \\(g(0)=0\\). So \\(g(a)\\in A\\) by (C). It is the finite-rank projection \\(p_t=\\sum_{\\alpha_n>t}\\theta_{\\varepsilon_n,\\varepsilon_n}\\), and \\(\\|a(1-p_t)\\|\\le t\\).\n\n*Step 2 (an increasing net of finite-rank projections in \\(A\\) with supremum \\(1\\)).* Let \\(\\mathcal P\\) be the set of finite-rank projections in \\(A\\). For \\(p,q\\in\\mathcal P\\), \\(p+q\\) is a positive finite-rank element of \\(A\\). If \\(p+q=0\\), then \\(p=q=0\\) and their join is already in \\(\\mathcal P\\). Otherwise choose \\(t>0\\) below the smallest nonzero eigenvalue of \\(p+q\\); Step 1 puts the projection onto the range of \\(p+q\\) in \\(A\\). That range is \\((\\ker p\\cap\\ker q)^\\perp\\), because \\(\\langle(p+q)\\xi,\\xi\\rangle=\\|p\\xi\\|^2+\\|q\\xi\\|^2\\); it contains \\(pH\\) and \\(qH\\). So \\(\\mathcal P\\) is upward directed. As a net indexed by itself, it increases, so it converges strongly to its supremum \\(P\\) by Vigier's theorem (V). \\(P\\) is the projection onto the closed span of the ranges of the elements of \\(\\mathcal P\\). For \\(a\\in A\\) and \\(t\\notin\\sigma(aa^*)\\), Step 1 gives \\(p_t\\in\\mathcal P\\) commuting with \\(aa^*\\) with \\(\\|(1-p_t)a\\|^2=\\|(1-p_t)aa^*(1-p_t)\\|\\le t\\). Since \\(\\sigma(aa^*)\\) is countable (K), such \\(t>0\\) can be taken as small as we like. So each \\(a\\xi\\) is a limit of vectors \\(p_ta\\xi\\in PH\\). Thus \\([AH]\\subseteq PH\\), and nondegeneracy gives \\(P=1\\).\n\n*Step 3 (\\(A''p\\subseteq A\\) for \\(p\\in\\mathcal P\\)).* For \\(p=0\\) the assertion is immediate. Assume \\(p\\ne0\\), and let \\(\\varepsilon_1,\\dots,\\varepsilon_n\\) be an orthonormal basis of \\(pH\\). The set \\(Ap=\\{b\\in A:bp=b\\}\\) is a norm-closed subspace of \\(A\\). The map \\(\\Phi(z)=(z\\varepsilon_1,\\dots,z\\varepsilon_n)\\) from \\(B(H)p\\) to \\(H^n\\) is linear and injective, and \\(\\|z\\|\\le\\|\\Phi(z)\\|\\le\\sqrt n\\,\\|z\\|\\) for \\(z\\in B(H)p\\), since \\(z\\zeta=\\sum_k\\langle\\zeta,\\varepsilon_k\\rangle z\\varepsilon_k\\). So \\(\\Phi(Ap)\\) is a complete, hence closed, subspace of \\(H^n\\), and therefore weakly closed. Let \\(y\\in A''\\). By [Theorem 4.4](#oa-fnd-bi-07), \\(y\\) is the weak limit of a net \\((a_\\lambda)\\) in \\(A\\). Then \\(a_\\lambda p\\in Ap\\), and \\(\\Phi(a_\\lambda p)\\to\\Phi(yp)\\) weakly in \\(H^n\\). So \\(\\Phi(yp)\\in\\Phi(Ap)\\), and \\(yp\\in Ap\\subseteq A\\).\n\n*Step 4 (conclusion).* Let \\(T\\in A''\\cap K(H)\\). Along the net \\(\\mathcal P\\) of Step 2, \\(\\|T-Tp\\|=\\|(1-p)T^*\\|\\to0\\). Indeed, \\(T^*\\) is compact, so it is a norm limit of finite-rank operators \\(F=\\sum_{k=1}^m\\theta_{\\alpha_k,\\beta_k}\\) (K); \\(\\|(1-p)F\\|\\le\\sum_k\\|(1-p)\\alpha_k\\|\\,\\|\\beta_k\\|\\to0\\), and \\(\\|(1-p)(T^*-F)\\|\\le\\|T^*-F\\|\\). Each \\(Tp\\) lies in \\(A\\) by Step 3, and \\(A\\) is norm closed. So \\(T\\in A\\). \\(\\square\\)\n\n**Exercise 6.3** (medium; minimal projections and simple C\\(^*\\)-algebras). Here a projection \\(e\\) in a C\\(^*\\)-algebra \\(A\\) is *minimal* if \\(eAe=\\mathbb Ce\\). A representation \\(\\{\\pi,H\\}\\) is *irreducible* if \\(H\\) and \\(\\{0\\}\\) are its only closed invariant subspaces. \\(A\\) is *simple* if its only closed two-sided ideals are \\(\\{0\\}\\) and \\(A\\). Let \\(e\\neq0\\) be a minimal projection of \\(A\\). Prove (a) and (b).\n\n(a) *If \\(\\pi\\) is irreducible and \\(\\pi(e)\\neq0\\), then \\(\\pi(e)\\) is a projection of rank one.* It is a projection because \\(\\pi\\) is a \\(*\\)-homomorphism. Take a unit vector \\(\\xi\\in\\pi(e)H\\). The subspace \\([\\pi(A)\\xi]\\) is closed, invariant and contains \\(\\xi=\\pi(e)\\xi\\), so it is \\(H\\). Let \\(\\eta\\in\\pi(e)H\\) and choose \\(a_k\\in A\\) with \\(\\pi(a_k)\\xi\\to\\eta\\). Write \\(ea_ke=\\lambda_ke\\). Then \\(\\eta=\\pi(e)\\eta=\\lim_k\\pi(e)\\pi(a_k)\\pi(e)\\xi=\\lim_k\\lambda_k\\xi\\). So \\(\\eta\\in\\mathbb C\\xi\\), and \\(\\pi(e)H=\\mathbb C\\xi\\). \\(\\square\\)\n\n(b) *If \\(A\\) is simple, then \\(A\\cong K(H)\\) for some Hilbert space \\(H\\).* Let \\(H=Ae=\\{x\\in A:xe=x\\}\\), a norm-closed subspace of \\(A\\). For \\(\\xi,\\eta\\in H\\), \\(\\eta^*\\xi=e\\eta^*\\xi e\\in eAe\\), so \\(\\eta^*\\xi=\\langle\\xi,\\eta\\rangle e\\) for a unique scalar \\(\\langle\\xi,\\eta\\rangle\\). This is linear in \\(\\xi\\) and conjugate linear in \\(\\eta\\). Since \\(\\xi^*\\xi\\ge0\\) and \\(e\\neq0\\), \\(\\langle\\xi,\\xi\\rangle\\ge0\\), and \\(\\|\\xi\\|_A^2=\\|\\xi^*\\xi\\|=\\langle\\xi,\\xi\\rangle\\|e\\|=\\langle\\xi,\\xi\\rangle\\). So the Hilbert norm equals the C\\(^*\\)-norm, and \\(H\\) is complete: a Hilbert space, nonzero because \\(e\\in H\\). Put \\(\\pi(x)\\xi=x\\xi\\). Then \\(\\|\\pi(x)\\|\\le\\|x\\|\\), \\(\\pi\\) is multiplicative, and \\(\\langle\\pi(x)\\xi,\\eta\\rangle e=\\eta^*x\\xi=(x^*\\eta)^*\\xi=\\langle\\xi,\\pi(x^*)\\eta\\rangle e\\). So \\(\\pi\\) is a representation.\n\nFor \\(\\zeta,\\eta\\in H\\), \\(\\pi(\\zeta\\eta^*)\\xi=\\zeta\\eta^*\\xi=\\langle\\xi,\\eta\\rangle\\zeta e=\\theta_{\\zeta,\\eta}\\xi\\). So \\(\\pi(A)\\) contains every rank-one operator, hence every finite-rank operator. In particular \\(\\pi(e)=\\theta_{e,e}\\) has rank one. The kernel of \\(\\pi\\) is a closed two-sided ideal that does not contain \\(e\\), so it is \\(\\{0\\}\\) by simplicity. The set \\(J=\\{x\\in A:\\pi(x)\\in K(H)\\}\\) is a closed two-sided ideal, since \\(\\pi\\) is continuous and \\(K(H)\\) is a closed ideal of \\(B(H)\\). It contains \\(e\\), so \\(J=A\\) and \\(\\pi(A)\\subseteq K(H)\\). An injective \\(*\\)-homomorphism between C\\(^*\\)-algebras is isometric (C), so \\(\\pi(A)\\) is norm closed. It contains the finite-rank operators, whose closure is \\(K(H)\\) (K). So \\(\\pi(A)=K(H)\\), and \\(\\pi:A\\to K(H)\\) is a \\(*\\)-isomorphism. \\(\\square\\)\n\n*Remarks.* (i) The definition \\(eAe=\\mathbb Ce\\) matters. A minimal projection in this sense majorizes no nonzero projection other than itself. The converse fails in C\\(^*\\)-algebras, and with that weaker definition (a) is false. The algebra \\(B=\\{f\\in C([0,1],M_2(\\mathbb C)):f(0),f(1)\\in\\mathbb C1\\}\\) has only the projections \\(0\\) and \\(1\\): the rank of a continuous projection-valued function is constant on \\([0,1]\\), and at \\(t=0\\) it is \\(0\\) or \\(2\\). Evaluation at \\(t=1/2\\) is an irreducible representation on \\(\\mathbb C^2\\), since every matrix is \\(f(1/2)\\) for some \\(f\\in B\\). But it sends the projection \\(1\\) to a projection of rank two. (ii) If \"simple\" is read as having no two-sided ideals other than \\(\\{0\\}\\) and \\(A\\), closed or not, then the ideal \\(AeA\\) of finite sums \\(\\sum x_iey_i\\) equals \\(A\\). Since \\(\\pi(xey)\\) has rank at most one, \\(\\pi(A)\\) then consists of finite-rank operators, so \\(K(H)\\) consists of finite-rank operators and \\(\\dim H<\\infty\\): \\(A\\cong M_n(\\mathbb C)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-09",
      "unit": "the-double-commutant-theorem",
      "name": "7. Polar decomposition inside a von Neumann algebra",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 7. Polar decomposition inside a von Neumann algebra\n\nThe polar decomposition of an element of a von Neumann algebra stays inside the algebra. The reason is uniqueness: conjugating by a unitary of the commutant gives another polar decomposition, which must be the same one. As a consequence, every two-sided ideal of a von Neumann algebra is self-adjoint.\n\n**Lemma 7.1** (uniqueness of the polar decomposition). Let \\(x=wk\\), where \\(k\\ge0\\) and \\(w\\) is a partial isometry with \\(\\ker w=\\ker k\\). Then \\(k=|x|\\), and \\(w\\) is the partial isometry \\(u\\) of (P).\n\n**Proof.** \\(w^*w\\) is the projection onto \\((\\ker w)^\\perp=(\\ker k)^\\perp\\), which is the closure of \\(kH\\) because \\(k\\) is self-adjoint. So \\(w^*wk=k\\), and \\(x^*x=kw^*wk=k^2\\). By uniqueness of positive square roots (C), \\(k=|x|\\). On \\(kH\\), \\(w(k\\xi)=x\\xi=u(|x|\\xi)=u(k\\xi)\\), so \\(w=u\\) on the closure of \\(kH\\). Both vanish on \\(\\ker k=\\ker|x|=\\ker x\\); here \\(\\ker|x|=\\ker x\\) because \\(\\||x|\\xi\\|^2=\\langle x^*x\\xi,\\xi\\rangle=\\|x\\xi\\|^2\\). So \\(w=u\\). \\(\\square\\)\n\n**Proposition 7.2.** Let \\(x\\) be an element of a von Neumann algebra \\(M\\), with polar decomposition \\(x=u|x|\\). Then \\(u\\in M\\) and \\(|x|\\in M\\).\n\n**Proof.** Let \\(v\\) be a unitary in \\(M'\\). Then \\(x=vxv^*=(vuv^*)(v|x|v^*)\\). Here \\(v|x|v^*\\ge0\\), and \\(vuv^*\\) is a partial isometry with \\[\n\\begin{gathered}\n\\ker(vuv^*)\\\\\n=v\\ker u\\\\\n=v\\ker x\\\\\n=v\\ker|x|\\\\\n=\\ker(v|x|v^*).\n\\end{gathered}\n\\] By Lemma 7.1, \\(vuv^*=u\\) and \\(v|x|v^*=|x|\\). So \\(u\\) and \\(|x|\\) commute with every unitary of \\(M'\\). Since \\(M'\\) is a von Neumann algebra ([Proposition 2.4](#oa-fnd-bi-03)(2)), it is spanned by its unitaries (U). Hence \\(u,|x|\\in M''=M\\). \\(\\square\\)\n\nThat \\(|x|\\in M\\) also follows from (C). Here is the full strong-limit argument for \\(u\\in M\\). Let \\(p\\) be the projection onto \\([|x|H]\\) and put \\(h_\\varepsilon=|x|(|x|+\\varepsilon1)^{-1}\\). The calculus gives \\(0\\le h_\\varepsilon\\le1\\), and \\(h_\\varepsilon\\) vanishes on \\(\\ker|x|\\). For \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\|(h_\\varepsilon-1)|x|\\xi\\|\\\\\n=\\|\\varepsilon(|x|+\\varepsilon1)^{-1}|x|\\xi\\|\\\\\n\\le\\varepsilon\\|\\xi\\|.\n\\end{gathered}\n\\]\nThus \\(h_\\varepsilon\\to p\\) on the dense subspace \\(|x|H\\) of \\(pH\\), and the uniform contraction bound extends this convergence to \\(pH\\). It is zero on \\((1-p)H\\), so the convergence is strong on \\(H\\). Consequently \\(x(|x|+\\varepsilon1)^{-1}=uh_\\varepsilon\\to up=u\\) strongly. Each approximant lies in \\(M\\) by (C), and \\(M\\) is strongly closed, so \\(u\\in M\\).\n\n**Corollary 7.3** (ideals and the polar decomposition). Fix a von Neumann algebra \\(M\\) and \\(x\\in M\\) with \\(x=u|x|\\). No closure assumption is made on the ideals below.\n\n1. If \\(l\\) is a left ideal of \\(M\\) (a linear subspace with \\(Ml\\subseteq l\\)) and \\(x\\in l\\), then \\(|x|=u^*x\\in l\\).\n2. If \\(r\\) is a right ideal (\\(rM\\subseteq r\\)) and \\(x\\in r\\), then \\(|x^*|=xu^*\\in r\\).\n3. Every two-sided ideal \\(m\\) of \\(M\\) is self-adjoint.\n\n**Proof.** (1) \\(u^*x=u^*u|x|=|x|\\), since \\(u^*u\\) is the projection onto the closure of \\(|x|H\\), and \\(u^*\\in M\\). (2) \\(xx^*=u|x|^2u^*=(u|x|u^*)^2\\) and \\(u|x|u^*\\ge0\\), so \\(|x^*|=u|x|u^*=xu^*\\), with \\(u^*\\in M\\). (3) For \\(x\\in m\\), (1) gives \\(|x|\\in m\\), and then \\(x^*=|x|u^*\\in m\\). \\(\\square\\)\n\nThe polar decomposition inside \\(M\\) is essential in (3). In C\\(^*\\)-algebras, two-sided ideals that are not closed need not be self-adjoint ([Example 7.4](#oa-fnd-bi-09)).\n\n**Example 7.4** (a two-sided ideal that is not self-adjoint, outside von Neumann algebras). In the C\\(^*\\)-algebra \\(C[0,1]\\), let \\(g(t)=te^{i/t}\\) for \\(t>0\\) and \\(g(0)=0\\); \\(g\\) is continuous. The ideal \\(gC[0,1]\\) is two-sided, but it does not contain \\(\\bar g\\). Indeed, \\(\\bar g=gk\\) would force \\(k(t)=e^{-2i/t}\\) for \\(t>0\\), which has no limit as \\(t\\downarrow0\\): it equals \\(1\\) at \\(t=1/(\\ell\\pi)\\) and \\(-i\\) at \\(t=1/(\\ell\\pi+\\pi/4)\\) for every integer \\(\\ell\\ge1\\). In the von Neumann algebra \\(L^\\infty[0,1]\\) of multiplication operators on \\(L^2[0,1]\\) (its full multiplication-commutant proof is [Theorem 9.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09)), the same ideal does contain \\(\\bar g\\), because \\(e^{-2i/t}\\) is bounded and measurable. This matches [Corollary 7.3](#oa-fnd-bi-09)(3).\n\n**Exercise 7.5** (hard; asymmetric Riesz decomposition). In a von Neumann algebra \\(M\\), suppose that \\(\\sum_{i\\in I}x_i^*x_i=\\sum_{j\\in J}y_j^*y_j\\), the sums converging \\(\\sigma\\)-strongly. Show that there are \\(z_{ij}\\in M\\) with\n\\[\n\\begin{gathered}\n\\sum_jz_{ij}^*z_{ij}\\\\\n=x_ix_i^*\\quad(i\\in I),\\\\\n\\sum_iz_{ij}z_{ij}^*\\\\\n=y_jy_j^*\\quad(j\\in J).\n\\end{gathered}\n\\]\n\n*Solution.* Let \\(a\\) be the common sum. For finite \\(G\\subseteq I\\), \\(\\langle\\sum_{i\\in G}x_i^*x_i\\xi,\\xi\\rangle\\le\\langle a\\xi,\\xi\\rangle\\), because the partial sums increase and converge weakly to \\(a\\). So \\(\\|x_i\\xi\\|\\le\\|a^{1/2}\\xi\\|\\) for each \\(i\\). (Only weak convergence of the sums was used. For increasing nets of positive operators, weak convergence to \\(a\\) already implies convergence in all six topologies by [Lemma 1.2](#oa-fnd-bi-01)(e).)\n\n*The factors \\(s_i\\).* Let \\(p\\) be the projection onto the closure of \\(a^{1/2}H\\), which is the closure of \\(aH\\) because \\(\\ker a^{1/2}=\\ker a\\). The rule \\(s_i(a^{1/2}\\xi)=x_i\\xi\\) is well defined and contractive on \\(a^{1/2}H\\), by the bound above. Extend it by continuity to \\(pH\\), and by \\(0\\) on \\((1-p)H\\). Then \\(x_i=s_ia^{1/2}\\) and \\(s_i=s_ip\\). To see \\(s_i\\in M\\), let \\(v\\) be a unitary in \\(M'\\). It commutes with \\(a^{1/2}\\) and \\(x_i\\), and it maps \\(a^{1/2}H\\), hence \\(pH\\) and \\((1-p)H\\), onto themselves. So \\[\n\\begin{gathered}\nvs_iv^*(a^{1/2}\\xi)\\\\\n=vs_ia^{1/2}v^*\\xi\\\\\n=vx_iv^*\\xi\\\\\n=x_i\\xi\\\\\n=s_i(a^{1/2}\\xi),\n\\end{gathered}\n\\] and both \\(vs_iv^*\\) and \\(s_i\\) vanish on \\((1-p)H\\). Thus \\(vs_iv^*=s_i\\) for every unitary \\(v\\in M'\\). Since \\(M'\\) is spanned by its unitaries (U), \\(s_i\\in M''=M\\).\n\n*The sum \\(\\sum_is_i^*s_i\\).* For finite \\(G\\), \\[\n\\begin{gathered}\n\\langle\\sum_{i\\in G}s_i^*s_ia^{1/2}\\xi,a^{1/2}\\xi\\rangle\\\\\n=\\sum_{i\\in G}\\|x_i\\xi\\|^2\\\\\n\\le\\|a^{1/2}\\xi\\|^2.\n\\end{gathered}\n\\] Since \\(a^{1/2}H\\) is dense in \\(pH\\) and the operators vanish on \\((1-p)H\\), \\(\\sum_{i\\in G}s_i^*s_i\\le p\\). The partial sums increase, so they converge strongly to some \\(c\\le p\\), by Vigier's theorem (V). On the dense subspace, \\[\n\\begin{gathered}\n\\langle ca^{1/2}\\xi,a^{1/2}\\xi\\rangle\\\\\n=\\sum_i\\|x_i\\xi\\|^2\\\\\n=\\langle a\\xi,\\xi\\rangle\\\\\n=\\|a^{1/2}\\xi\\|^2.\n\\end{gathered}\n\\] So \\((p-c)^{1/2}\\) vanishes on a dense subspace of \\(pH\\), and \\(c=p\\). In the same way, \\(y_j=t_ja^{1/2}\\) with \\(t_j\\in M\\) and \\(\\sum_jt_j^*t_j=p\\).\n\n*The decomposition.* Put \\(z_{ij}=t_ja^{1/2}s_i^*\\in M\\). Using \\(pa^{1/2}=a^{1/2}\\),\n\\[\n\\begin{gathered}\n\\sum_jz_{ij}^*z_{ij}\\\\\n=s_ia^{1/2}\\Big(\\sum_jt_j^*t_j\\Big)a^{1/2}s_i^*\\\\\n=s_ia^{1/2}pa^{1/2}s_i^*\\\\\n=s_ias_i^*\\\\\n=x_ix_i^*,\n\\end{gathered}\n\\]\n\\[\n\\begin{gathered}\n\\sum_iz_{ij}z_{ij}^*\\\\\n=t_ja^{1/2}\\Big(\\sum_is_i^*s_i\\Big)a^{1/2}t_j^*\\\\\n=t_jat_j^*\\\\\n=y_jy_j^* .\n\\end{gathered}\n\\]\nThe sums converge strongly: for \\(R\\in B(H)\\), \\(\\|R(c_\\lambda-c)R^*\\xi\\|\\le\\|R\\|\\,\\|(c_\\lambda-c)R^*\\xi\\|\\), and the partial sums \\(c_\\lambda\\) of \\(\\sum_jt_j^*t_j\\) and \\(\\sum_is_i^*s_i\\) converge strongly. Since they are increasing and bounded, the convergence is also \\(\\sigma\\)-strong\\(^*\\) ([Lemma 1.2](#oa-fnd-bi-01)(e)). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-10",
      "unit": "the-double-commutant-theorem",
      "name": "8. One-sided ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 8. One-sided ideals\n\nA one-sided ideal of a von Neumann algebra need not be closed. Its closure is the same in all six topologies, and it is cut out by a single projection. The tool is an increasing approximate unit that lies in the ideal itself.\n\n**Lemma 8.1** (approximate units for right ideals). Let \\(M\\) be a von Neumann algebra on \\(H\\) and \\(r\\subseteq M\\) a right ideal (a linear subspace with \\(rM\\subseteq r\\)); \\(r\\) need not be closed. Let \\(f\\) be the projection onto \\([rH]\\).\n\n1. \\(f\\in M\\), and \\(fa=a\\) for every \\(a\\in r\\).\n2. There is an increasing net \\((u_\\lambda)\\) in \\(r\\), with \\(0\\le u_\\lambda\\le f\\), such that \\(\\|(1-u_\\lambda)a\\|\\to0\\) for every \\(a\\in r\\).\n3. \\(u_\\lambda\\to f\\) strongly, \\(\\sigma\\)-strongly\\(^*\\) and \\(\\sigma\\)-weakly.\n\n**Proof.** (1) \\([rH]\\) is invariant under \\(M'\\), since \\(y'a\\xi=ay'\\xi\\); and \\(M'\\) is self-adjoint. So \\(f\\in M''=M\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). For \\(a\\in r\\), \\(aH\\subseteq[rH]\\), so \\(fa=a\\).\n\n(2) If \\(r=\\{0\\}\\), take the one-term net \\(u=0\\); then \\(f=0\\). Otherwise \\(r\\) is an infinite set, being a nonzero complex vector space. Let \\(\\Lambda\\) be the set of finite nonempty subsets of \\(r\\), directed by inclusion. For \\(\\lambda\\in\\Lambda\\) with \\(n\\) elements, put\n\\[\n\\begin{gathered}\nv_\\lambda\\\\\n=\\sum_{a\\in\\lambda}aa^*,\\\\\nu_\\lambda\\\\\n=nv_\\lambda(1+nv_\\lambda)^{-1}\\\\\n=1-(1+nv_\\lambda)^{-1}.\n\\end{gathered}\n\\tag{8.1}\n\\]\nHere \\(v_\\lambda\\ge0\\), and \\(1+nv_\\lambda\\ge1\\) is invertible in \\(M\\) by (C). Since \\(a\\in r\\) and \\(a^*\\in M\\), \\(aa^*\\in r\\); so \\(v_\\lambda\\in r\\), and then \\(u_\\lambda=v_\\lambda\\cdot n(1+nv_\\lambda)^{-1}\\in r\\). By (C), \\(0\\le u_\\lambda\\le1\\), because \\(t\\mapsto nt/(1+nt)\\) takes values in \\([0,1)\\) for \\(t\\ge0\\). By (1), \\(fu_\\lambda=u_\\lambda\\), and taking adjoints \\(u_\\lambda f=u_\\lambda\\). So \\(u_\\lambda=fu_\\lambda f\\le f\\).\n\n*The net increases.* Let \\(\\lambda\\subseteq\\mu\\), with \\(n\\) and \\(m\\ge n\\) elements. Then \\(v_\\lambda\\le v_\\mu\\), so \\(nv_\\lambda\\le nv_\\mu\\le mv_\\mu\\) and \\(1+nv_\\lambda\\le1+mv_\\mu\\). Inversion reverses order (C), so \\((1+mv_\\mu)^{-1}\\le(1+nv_\\lambda)^{-1}\\), which says \\(u_\\lambda\\le u_\\mu\\).\n\n*The estimate.* Let \\(a\\in\\lambda\\). Then \\(aa^*\\le v_\\lambda\\), and \\(1-u_\\lambda=(1+nv_\\lambda)^{-1}\\) is self-adjoint and commutes with \\(v_\\lambda\\). Conjugating the inequality,\n\\[\n\\begin{gathered}\n(1-u_\\lambda)aa^*(1-u_\\lambda)\\\\\n\\le(1-u_\\lambda)v_\\lambda(1-u_\\lambda)\\\\\n=v_\\lambda(1+nv_\\lambda)^{-2}\\\\\n\\le\\frac1{4n}\\,1 ,\n\\end{gathered}\n\\tag{8.2}\n\\]\nthe last step by (C), since \\(t/(1+nt)^2\\le1/(4n)\\) for \\(t\\ge0\\) (as \\((1+nt)^2\\ge4nt\\)). With \\(b=(1-u_\\lambda)a\\), (8.2) and the C\\(^*\\)-identity \\(\\|b\\|^2=\\|bb^*\\|\\) give \\(\\|(1-u_\\lambda)a\\|^2\\le1/(4n)\\). Given \\(a\\in r\\) and \\(N\\ge1\\), choose \\(\\lambda_0\\in\\Lambda\\) that contains \\(a\\) and has at least \\(N\\) elements. Every \\(\\lambda\\supseteq\\lambda_0\\) contains \\(a\\) and has at least \\(N\\) elements, so \\(\\|(1-u_\\lambda)a\\|\\le1/(2\\sqrt N)\\). Hence \\(\\|(1-u_\\lambda)a\\|\\to0\\).\n\n(3) The net increases and is bounded by \\(1\\), so by Vigier's theorem (V) it converges strongly to its least upper bound \\(u\\in M\\), and \\(u\\le f\\). For \\(a\\in r\\) and \\(\\xi\\in H\\), \\((1-u)a\\xi=\\lim_\\lambda(1-u_\\lambda)a\\xi=0\\) by (2). So \\(u\\xi=\\xi\\) for all \\(\\xi\\in[rH]=fH\\), that is \\(uf=f\\). From \\(0\\le u\\le f\\) we get \\(\\|u^{1/2}(1-f)\\xi\\|^2\\le\\langle f(1-f)\\xi,(1-f)\\xi\\rangle=0\\), so \\(u(1-f)=0\\) and \\(u=uf=f\\). The other two modes of convergence follow from [Lemma 1.2](#oa-fnd-bi-01)(e). \\(\\square\\)\n\n*Remark 8.2.* In any C\\(^*\\)-algebra, the positive part of the open unit ball of a left ideal is upward directed and is a right approximate identity for the norm closure of the ideal; see [approximate identities of one-sided ideals](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-20). The net (8.1) is a direct construction with exactly the properties needed here: it is increasing, it lies in the ideal, and it converges strongly to \\(f\\).\n\n**Theorem 8.3** (closures of one-sided ideals). Let \\(M\\) be a von Neumann algebra on \\(H\\).\n\n1. Let \\(l\\) be a left ideal of \\(M\\), not necessarily closed, and let \\(f\\) be the projection onto \\([l^*H]\\), where \\(l^*=\\{x^*:x\\in l\\}\\). Then the closure of \\(l\\) in each of the six topologies is \\(Mf=\\{y\\in M:y=yf\\}\\).\n2. Let \\(r\\) be a right ideal, and \\(f\\) the projection onto \\([rH]\\). Then the closure of \\(r\\) in each of the six topologies is \\(fM\\).\n3. A left ideal that is closed in one of the six topologies is \\(Me\\) for exactly one projection \\(e\\in M\\). A closed right ideal is \\(eM\\) for exactly one projection \\(e\\in M\\).\n4. Let \\(m\\) be a two-sided ideal and \\(\\overline m\\) its closure, which is the same in all six topologies. Let \\(e\\) be the projection onto \\([mH]\\). Then \\(\\overline m=Me=eM\\), and \\(e\\in Z(M)\\).\n5. If \\(M\\) is a factor, every two-sided ideal other than \\(\\{0\\}\\) is dense in \\(M\\) for all six topologies.\n\n**Proof.** (2) By [Lemma 8.1](#oa-fnd-bi-10), there is an increasing net \\(u_\\lambda\\in r\\) with \\(u_\\lambda\\to f\\) strongly, and \\(fa=a\\) for \\(a\\in r\\). So \\(r\\subseteq fM\\). The set \\(fM=\\{y\\in M:(1-f)y=0\\}\\) is closed in all six topologies, since \\(M\\) is closed ([Proposition 2.4](#oa-fnd-bi-03)(2)) and \\(y\\mapsto(1-f)y\\) is continuous ([Lemma 1.2](#oa-fnd-bi-01)(c)). So each closure of \\(r\\) lies in \\(fM\\). Conversely, let \\(y\\in M\\). Then \\(u_\\lambda y\\in r\\), \\(\\|u_\\lambda y\\|\\le\\|y\\|\\), \\(u_\\lambda y\\to fy\\) strongly, and \\((u_\\lambda y)^*=y^*u_\\lambda\\to y^*f\\) strongly. By [Lemma 1.2](#oa-fnd-bi-01)(d), \\(u_\\lambda y\\to fy\\) \\(\\sigma\\)-strongly\\(^*\\). This is the finest of the six topologies, so \\(fy\\) lies in every one of the six closures of \\(r\\).\n\n(1) \\(l^*\\) is a right ideal, and its net \\(u_\\lambda\\) from [Lemma 8.1](#oa-fnd-bi-10) consists of self-adjoint elements, which therefore lie in \\(l\\). For \\(x\\in l\\), \\(fx^*=x^*\\), so \\(x=xf\\); thus \\(l\\subseteq Mf\\), and \\(Mf\\) is closed as in (2). For \\(y\\in M\\), \\(yu_\\lambda\\in l\\), and \\(yu_\\lambda\\to yf\\) \\(\\sigma\\)-strongly\\(^*\\) as in (2). So every closure of \\(l\\) equals \\(Mf\\).\n\n(3) A closed left ideal is its own closure, so it equals \\(Mf\\) by (1). If \\(Me=Me'\\) for projections \\(e,e'\\in M\\), then \\(e\\in Me'\\) gives \\(e=ee'\\), and likewise \\(e'=e'e\\). Hence \\(e=e^*=(ee')^*=e'e=e'\\). Right ideals are treated in the same way.\n\n(4) By (1) and (2), \\(\\overline m=Mf=f'M\\), with \\(f\\) the projection onto \\([m^*H]\\) and \\(f'\\) the projection onto \\([mH]\\). From \\(f\\in f'M\\) we get \\(f'f=f\\), and from \\(f'\\in Mf\\) we get \\(f'f=f'\\). So \\(f=f'=e\\). For \\(x\\in M\\), \\(ex\\in eM=Me\\) gives \\(ex=exe\\), and \\(xe\\in Me=eM\\) gives \\(xe=exe\\). So \\(ex=xe\\), and \\(e\\in Z(M)\\).\n\n(5) In a factor, the central projection \\(e\\) of (4) is \\(0\\) or \\(1\\). It is not \\(0\\), since \\(m\\neq0\\) and \\(m\\subseteq Me\\). So \\(e=1\\) and \\(\\overline m=M\\). \\(\\square\\)\n\n*Remark 8.4.* The theorem identifies the closure of any one-sided ideal. It also shows that for one-sided ideals, being closed in one of the six topologies is the same as being closed in all of them. For general subspaces this fails ([Exercise 1.5](#oa-fnd-bi-01)). Norm-closed ideals need not be weakly closed: \\(K(H)\\) is a norm-closed two-sided ideal of the factor \\(B(H)\\), and by (5) it is dense in \\(B(H)\\) when \\(\\dim H=\\infty\\) ([Example 4.6](#oa-fnd-bi-07)).\n\n**Proposition 8.5** (monotone approximation in two-sided ideals). Take a two-sided ideal \\(m\\) in a von Neumann algebra \\(M\\), not assumed closed. Let \\(e\\) be the projection onto \\([mH]\\), so that \\(\\overline m=Me\\) by Theorem 8.3, and let \\((u_\\lambda)\\) be the net of Lemma 8.1 for the right ideal \\(m\\). Let \\(x\\in\\overline m\\) be positive, that is \\(x\\in M_+\\) with \\(x=xe\\). Put \\(x_\\lambda=x^{1/2}u_\\lambda x^{1/2}\\). Then \\(x_\\lambda\\in m\\), \\(0\\le x_\\lambda\\le x\\), the net \\((x_\\lambda)\\) increases, and \\(x_\\lambda\\to x\\) strongly, \\(\\sigma\\)-strongly\\(^*\\) and \\(\\sigma\\)-weakly.\n\n**Proof.** \\(x^{1/2}\\in M\\) by (C). Since \\(u_\\lambda\\in m\\) and \\(m\\) is two-sided, \\(x_\\lambda\\in m\\). Conjugation by \\(x^{1/2}\\) preserves order, so \\(x_\\lambda\\ge0\\) and the net increases. From \\(x(1-e)=0\\) we get \\(\\|x^{1/2}(1-e)\\xi\\|^2=\\langle x(1-e)\\xi,(1-e)\\xi\\rangle=0\\), so \\(x^{1/2}=x^{1/2}e\\) and, taking adjoints, \\(x^{1/2}=ex^{1/2}\\). Hence \\(x^{1/2}ex^{1/2}=x\\). As \\(u_\\lambda\\le e\\), \\(x_\\lambda\\le x^{1/2}ex^{1/2}=x\\). For \\(\\xi\\in H\\), \\(x_\\lambda\\xi=x^{1/2}u_\\lambda(x^{1/2}\\xi)\\to x^{1/2}ex^{1/2}\\xi=x\\xi\\). The net is self-adjoint and bounded by \\(\\|x\\|\\), so [Lemma 1.2](#oa-fnd-bi-01)(d) gives the other two modes. \\(\\square\\)\n\nThe bound \\(x_\\lambda\\le x\\) is automatic for an increasing net with strong limit \\(x\\), but the construction gives it directly. Two-sidedness cannot be dropped: [Example 8.6](#oa-fnd-bi-11) is a left ideal for which the conclusion fails.\n\n**Example 8.6** (monotone approximation fails for one-sided ideals). Let \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_k)\\), let \\(P_n\\) be the projection onto the span of \\(\\delta_1,\\dots,\\delta_n\\), and let \\(l=\\{x\\in B(H):x=xP_n\\text{ for some }n\\}\\). This is a left ideal. Its adjoint set \\(l^*\\) consists of the operators with range in some \\(P_nH\\), so \\([l^*H]=H\\), and by [Theorem 8.3](#oa-fnd-bi-11)(1) the closure of \\(l\\) is \\(B(H)\\). Let \\(v=\\sum_k2^{-k}\\delta_k\\) and let \\(q\\) be the projection onto \\(\\mathbb Cv\\). Suppose \\(y\\in l\\) and \\(0\\le y\\le q\\). Then \\((1-q)y(1-q)\\le0\\), so \\(y^{1/2}(1-q)=0\\), and \\(y=qyq=cq\\) for some \\(c\\ge0\\). Also \\(y=yP_n\\) for some \\(n\\). If \\(c>0\\), then \\(q=qP_n\\), hence \\(q=P_nq\\) and \\(v\\in P_nH\\), which is false. So \\(y=0\\). An increasing net in \\(l_+\\) that converged strongly to \\(q\\) would lie below \\(q\\), since \\(\\langle y_\\lambda\\xi,\\xi\\rangle\\le\\lim_\\mu\\langle y_\\mu\\xi,\\xi\\rangle=\\langle q\\xi,\\xi\\rangle\\). So it would be zero, and could not converge to \\(q\\). The positive element \\(q\\) of the closure of \\(l\\) is therefore not the limit of any increasing net in \\(l_+\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-11",
      "unit": "the-double-commutant-theorem",
      "name": "8. One-sided ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 8. One-sided ideals\n\nA one-sided ideal of a von Neumann algebra need not be closed. Its closure is the same in all six topologies, and it is cut out by a single projection. The tool is an increasing approximate unit that lies in the ideal itself.\n\n**Lemma 8.1** (approximate units for right ideals). Let \\(M\\) be a von Neumann algebra on \\(H\\) and \\(r\\subseteq M\\) a right ideal (a linear subspace with \\(rM\\subseteq r\\)); \\(r\\) need not be closed. Let \\(f\\) be the projection onto \\([rH]\\).\n\n1. \\(f\\in M\\), and \\(fa=a\\) for every \\(a\\in r\\).\n2. There is an increasing net \\((u_\\lambda)\\) in \\(r\\), with \\(0\\le u_\\lambda\\le f\\), such that \\(\\|(1-u_\\lambda)a\\|\\to0\\) for every \\(a\\in r\\).\n3. \\(u_\\lambda\\to f\\) strongly, \\(\\sigma\\)-strongly\\(^*\\) and \\(\\sigma\\)-weakly.\n\n**Proof.** (1) \\([rH]\\) is invariant under \\(M'\\), since \\(y'a\\xi=ay'\\xi\\); and \\(M'\\) is self-adjoint. So \\(f\\in M''=M\\) by [Proposition 2.1](#oa-fnd-bi-02)(5). For \\(a\\in r\\), \\(aH\\subseteq[rH]\\), so \\(fa=a\\).\n\n(2) If \\(r=\\{0\\}\\), take the one-term net \\(u=0\\); then \\(f=0\\). Otherwise \\(r\\) is an infinite set, being a nonzero complex vector space. Let \\(\\Lambda\\) be the set of finite nonempty subsets of \\(r\\), directed by inclusion. For \\(\\lambda\\in\\Lambda\\) with \\(n\\) elements, put\n\\[\n\\begin{gathered}\nv_\\lambda\\\\\n=\\sum_{a\\in\\lambda}aa^*,\\\\\nu_\\lambda\\\\\n=nv_\\lambda(1+nv_\\lambda)^{-1}\\\\\n=1-(1+nv_\\lambda)^{-1}.\n\\end{gathered}\n\\tag{8.1}\n\\]\nHere \\(v_\\lambda\\ge0\\), and \\(1+nv_\\lambda\\ge1\\) is invertible in \\(M\\) by (C). Since \\(a\\in r\\) and \\(a^*\\in M\\), \\(aa^*\\in r\\); so \\(v_\\lambda\\in r\\), and then \\(u_\\lambda=v_\\lambda\\cdot n(1+nv_\\lambda)^{-1}\\in r\\). By (C), \\(0\\le u_\\lambda\\le1\\), because \\(t\\mapsto nt/(1+nt)\\) takes values in \\([0,1)\\) for \\(t\\ge0\\). By (1), \\(fu_\\lambda=u_\\lambda\\), and taking adjoints \\(u_\\lambda f=u_\\lambda\\). So \\(u_\\lambda=fu_\\lambda f\\le f\\).\n\n*The net increases.* Let \\(\\lambda\\subseteq\\mu\\), with \\(n\\) and \\(m\\ge n\\) elements. Then \\(v_\\lambda\\le v_\\mu\\), so \\(nv_\\lambda\\le nv_\\mu\\le mv_\\mu\\) and \\(1+nv_\\lambda\\le1+mv_\\mu\\). Inversion reverses order (C), so \\((1+mv_\\mu)^{-1}\\le(1+nv_\\lambda)^{-1}\\), which says \\(u_\\lambda\\le u_\\mu\\).\n\n*The estimate.* Let \\(a\\in\\lambda\\). Then \\(aa^*\\le v_\\lambda\\), and \\(1-u_\\lambda=(1+nv_\\lambda)^{-1}\\) is self-adjoint and commutes with \\(v_\\lambda\\). Conjugating the inequality,\n\\[\n\\begin{gathered}\n(1-u_\\lambda)aa^*(1-u_\\lambda)\\\\\n\\le(1-u_\\lambda)v_\\lambda(1-u_\\lambda)\\\\\n=v_\\lambda(1+nv_\\lambda)^{-2}\\\\\n\\le\\frac1{4n}\\,1 ,\n\\end{gathered}\n\\tag{8.2}\n\\]\nthe last step by (C), since \\(t/(1+nt)^2\\le1/(4n)\\) for \\(t\\ge0\\) (as \\((1+nt)^2\\ge4nt\\)). With \\(b=(1-u_\\lambda)a\\), (8.2) and the C\\(^*\\)-identity \\(\\|b\\|^2=\\|bb^*\\|\\) give \\(\\|(1-u_\\lambda)a\\|^2\\le1/(4n)\\). Given \\(a\\in r\\) and \\(N\\ge1\\), choose \\(\\lambda_0\\in\\Lambda\\) that contains \\(a\\) and has at least \\(N\\) elements. Every \\(\\lambda\\supseteq\\lambda_0\\) contains \\(a\\) and has at least \\(N\\) elements, so \\(\\|(1-u_\\lambda)a\\|\\le1/(2\\sqrt N)\\). Hence \\(\\|(1-u_\\lambda)a\\|\\to0\\).\n\n(3) The net increases and is bounded by \\(1\\), so by Vigier's theorem (V) it converges strongly to its least upper bound \\(u\\in M\\), and \\(u\\le f\\). For \\(a\\in r\\) and \\(\\xi\\in H\\), \\((1-u)a\\xi=\\lim_\\lambda(1-u_\\lambda)a\\xi=0\\) by (2). So \\(u\\xi=\\xi\\) for all \\(\\xi\\in[rH]=fH\\), that is \\(uf=f\\). From \\(0\\le u\\le f\\) we get \\(\\|u^{1/2}(1-f)\\xi\\|^2\\le\\langle f(1-f)\\xi,(1-f)\\xi\\rangle=0\\), so \\(u(1-f)=0\\) and \\(u=uf=f\\). The other two modes of convergence follow from [Lemma 1.2](#oa-fnd-bi-01)(e). \\(\\square\\)\n\n*Remark 8.2.* In any C\\(^*\\)-algebra, the positive part of the open unit ball of a left ideal is upward directed and is a right approximate identity for the norm closure of the ideal; see [approximate identities of one-sided ideals](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-20). The net (8.1) is a direct construction with exactly the properties needed here: it is increasing, it lies in the ideal, and it converges strongly to \\(f\\).\n\n**Theorem 8.3** (closures of one-sided ideals). Let \\(M\\) be a von Neumann algebra on \\(H\\).\n\n1. Let \\(l\\) be a left ideal of \\(M\\), not necessarily closed, and let \\(f\\) be the projection onto \\([l^*H]\\), where \\(l^*=\\{x^*:x\\in l\\}\\). Then the closure of \\(l\\) in each of the six topologies is \\(Mf=\\{y\\in M:y=yf\\}\\).\n2. Let \\(r\\) be a right ideal, and \\(f\\) the projection onto \\([rH]\\). Then the closure of \\(r\\) in each of the six topologies is \\(fM\\).\n3. A left ideal that is closed in one of the six topologies is \\(Me\\) for exactly one projection \\(e\\in M\\). A closed right ideal is \\(eM\\) for exactly one projection \\(e\\in M\\).\n4. Let \\(m\\) be a two-sided ideal and \\(\\overline m\\) its closure, which is the same in all six topologies. Let \\(e\\) be the projection onto \\([mH]\\). Then \\(\\overline m=Me=eM\\), and \\(e\\in Z(M)\\).\n5. If \\(M\\) is a factor, every two-sided ideal other than \\(\\{0\\}\\) is dense in \\(M\\) for all six topologies.\n\n**Proof.** (2) By [Lemma 8.1](#oa-fnd-bi-10), there is an increasing net \\(u_\\lambda\\in r\\) with \\(u_\\lambda\\to f\\) strongly, and \\(fa=a\\) for \\(a\\in r\\). So \\(r\\subseteq fM\\). The set \\(fM=\\{y\\in M:(1-f)y=0\\}\\) is closed in all six topologies, since \\(M\\) is closed ([Proposition 2.4](#oa-fnd-bi-03)(2)) and \\(y\\mapsto(1-f)y\\) is continuous ([Lemma 1.2](#oa-fnd-bi-01)(c)). So each closure of \\(r\\) lies in \\(fM\\). Conversely, let \\(y\\in M\\). Then \\(u_\\lambda y\\in r\\), \\(\\|u_\\lambda y\\|\\le\\|y\\|\\), \\(u_\\lambda y\\to fy\\) strongly, and \\((u_\\lambda y)^*=y^*u_\\lambda\\to y^*f\\) strongly. By [Lemma 1.2](#oa-fnd-bi-01)(d), \\(u_\\lambda y\\to fy\\) \\(\\sigma\\)-strongly\\(^*\\). This is the finest of the six topologies, so \\(fy\\) lies in every one of the six closures of \\(r\\).\n\n(1) \\(l^*\\) is a right ideal, and its net \\(u_\\lambda\\) from [Lemma 8.1](#oa-fnd-bi-10) consists of self-adjoint elements, which therefore lie in \\(l\\). For \\(x\\in l\\), \\(fx^*=x^*\\), so \\(x=xf\\); thus \\(l\\subseteq Mf\\), and \\(Mf\\) is closed as in (2). For \\(y\\in M\\), \\(yu_\\lambda\\in l\\), and \\(yu_\\lambda\\to yf\\) \\(\\sigma\\)-strongly\\(^*\\) as in (2). So every closure of \\(l\\) equals \\(Mf\\).\n\n(3) A closed left ideal is its own closure, so it equals \\(Mf\\) by (1). If \\(Me=Me'\\) for projections \\(e,e'\\in M\\), then \\(e\\in Me'\\) gives \\(e=ee'\\), and likewise \\(e'=e'e\\). Hence \\(e=e^*=(ee')^*=e'e=e'\\). Right ideals are treated in the same way.\n\n(4) By (1) and (2), \\(\\overline m=Mf=f'M\\), with \\(f\\) the projection onto \\([m^*H]\\) and \\(f'\\) the projection onto \\([mH]\\). From \\(f\\in f'M\\) we get \\(f'f=f\\), and from \\(f'\\in Mf\\) we get \\(f'f=f'\\). So \\(f=f'=e\\). For \\(x\\in M\\), \\(ex\\in eM=Me\\) gives \\(ex=exe\\), and \\(xe\\in Me=eM\\) gives \\(xe=exe\\). So \\(ex=xe\\), and \\(e\\in Z(M)\\).\n\n(5) In a factor, the central projection \\(e\\) of (4) is \\(0\\) or \\(1\\). It is not \\(0\\), since \\(m\\neq0\\) and \\(m\\subseteq Me\\). So \\(e=1\\) and \\(\\overline m=M\\). \\(\\square\\)\n\n*Remark 8.4.* The theorem identifies the closure of any one-sided ideal. It also shows that for one-sided ideals, being closed in one of the six topologies is the same as being closed in all of them. For general subspaces this fails ([Exercise 1.5](#oa-fnd-bi-01)). Norm-closed ideals need not be weakly closed: \\(K(H)\\) is a norm-closed two-sided ideal of the factor \\(B(H)\\), and by (5) it is dense in \\(B(H)\\) when \\(\\dim H=\\infty\\) ([Example 4.6](#oa-fnd-bi-07)).\n\n**Proposition 8.5** (monotone approximation in two-sided ideals). Take a two-sided ideal \\(m\\) in a von Neumann algebra \\(M\\), not assumed closed. Let \\(e\\) be the projection onto \\([mH]\\), so that \\(\\overline m=Me\\) by Theorem 8.3, and let \\((u_\\lambda)\\) be the net of Lemma 8.1 for the right ideal \\(m\\). Let \\(x\\in\\overline m\\) be positive, that is \\(x\\in M_+\\) with \\(x=xe\\). Put \\(x_\\lambda=x^{1/2}u_\\lambda x^{1/2}\\). Then \\(x_\\lambda\\in m\\), \\(0\\le x_\\lambda\\le x\\), the net \\((x_\\lambda)\\) increases, and \\(x_\\lambda\\to x\\) strongly, \\(\\sigma\\)-strongly\\(^*\\) and \\(\\sigma\\)-weakly.\n\n**Proof.** \\(x^{1/2}\\in M\\) by (C). Since \\(u_\\lambda\\in m\\) and \\(m\\) is two-sided, \\(x_\\lambda\\in m\\). Conjugation by \\(x^{1/2}\\) preserves order, so \\(x_\\lambda\\ge0\\) and the net increases. From \\(x(1-e)=0\\) we get \\(\\|x^{1/2}(1-e)\\xi\\|^2=\\langle x(1-e)\\xi,(1-e)\\xi\\rangle=0\\), so \\(x^{1/2}=x^{1/2}e\\) and, taking adjoints, \\(x^{1/2}=ex^{1/2}\\). Hence \\(x^{1/2}ex^{1/2}=x\\). As \\(u_\\lambda\\le e\\), \\(x_\\lambda\\le x^{1/2}ex^{1/2}=x\\). For \\(\\xi\\in H\\), \\(x_\\lambda\\xi=x^{1/2}u_\\lambda(x^{1/2}\\xi)\\to x^{1/2}ex^{1/2}\\xi=x\\xi\\). The net is self-adjoint and bounded by \\(\\|x\\|\\), so [Lemma 1.2](#oa-fnd-bi-01)(d) gives the other two modes. \\(\\square\\)\n\nThe bound \\(x_\\lambda\\le x\\) is automatic for an increasing net with strong limit \\(x\\), but the construction gives it directly. Two-sidedness cannot be dropped: [Example 8.6](#oa-fnd-bi-11) is a left ideal for which the conclusion fails.\n\n**Example 8.6** (monotone approximation fails for one-sided ideals). Let \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_k)\\), let \\(P_n\\) be the projection onto the span of \\(\\delta_1,\\dots,\\delta_n\\), and let \\(l=\\{x\\in B(H):x=xP_n\\text{ for some }n\\}\\). This is a left ideal. Its adjoint set \\(l^*\\) consists of the operators with range in some \\(P_nH\\), so \\([l^*H]=H\\), and by [Theorem 8.3](#oa-fnd-bi-11)(1) the closure of \\(l\\) is \\(B(H)\\). Let \\(v=\\sum_k2^{-k}\\delta_k\\) and let \\(q\\) be the projection onto \\(\\mathbb Cv\\). Suppose \\(y\\in l\\) and \\(0\\le y\\le q\\). Then \\((1-q)y(1-q)\\le0\\), so \\(y^{1/2}(1-q)=0\\), and \\(y=qyq=cq\\) for some \\(c\\ge0\\). Also \\(y=yP_n\\) for some \\(n\\). If \\(c>0\\), then \\(q=qP_n\\), hence \\(q=P_nq\\) and \\(v\\in P_nH\\), which is false. So \\(y=0\\). An increasing net in \\(l_+\\) that converged strongly to \\(q\\) would lie below \\(q\\), since \\(\\langle y_\\lambda\\xi,\\xi\\rangle\\le\\lim_\\mu\\langle y_\\mu\\xi,\\xi\\rangle=\\langle q\\xi,\\xi\\rangle\\). So it would be zero, and could not converge to \\(q\\). The positive element \\(q\\) of the closure of \\(l\\) is therefore not the limit of any increasing net in \\(l_+\\).\n\n",
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    {
      "id": "OA-FND-BI-12",
      "unit": "the-double-commutant-theorem",
      "name": "9. Cyclic and separating vectors; σ-finite algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 9. Cyclic and separating vectors; σ-finite algebras\n\nA set of vectors can generate the whole space under an algebra, or it can detect every nonzero element of the algebra. Passing to the commutant exchanges the two properties. Countable versions of them characterize the \\(\\sigma\\)-finite von Neumann algebras, which are those with a faithful positive normal functional.\n\n**Definition 9.1.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\). The set \\(\\mathfrak A\\) is *separating* for \\(S\\) if \\(x\\in S\\) and \\(x\\mathfrak A=\\{0\\}\\) imply \\(x=0\\). It is *cyclic* for \\(S\\) if \\([S\\mathfrak A]=H\\). A vector \\(\\xi\\) is separating or cyclic if \\(\\{\\xi\\}\\) is.\n\n**Proposition 9.2.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\).\n\n1. If \\(\\mathfrak A\\) is cyclic for \\(S\\), then \\(\\mathfrak A\\) is separating for \\(S'\\).\n2. If \\(M\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\), closed or not, and \\(\\mathfrak A\\) is separating for \\(M'\\), then \\(\\mathfrak A\\) is cyclic for \\(M\\).\n3. For a von Neumann algebra \\(M\\): \\(\\mathfrak A\\) is cyclic for \\(M\\) if and only if it is separating for \\(M'\\), and \\(\\mathfrak A\\) is separating for \\(M\\) if and only if it is cyclic for \\(M'\\).\n4. Nondegeneracy cannot be dropped in (2). For \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\(M'=B(H)\\), and \\(\\mathfrak A=H\\) is separating for \\(M'\\) but not cyclic for \\(M\\).\n\n**Proof.** (1) Let \\(x'\\in S'\\) with \\(x'\\mathfrak A=\\{0\\}\\). For \\(s\\in S\\) and \\(\\xi\\in\\mathfrak A\\), \\(x's\\xi=sx'\\xi=0\\). So \\(x'\\) vanishes on the span of \\(S\\mathfrak A\\) and, being continuous, on \\([S\\mathfrak A]=H\\).\n\n(2) Let \\(p\\) be the projection onto \\([M\\mathfrak A]\\). This subspace is invariant under \\(M\\), and \\(M^*=M\\), so \\(p\\in M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). By [Lemma 4.1](#oa-fnd-bi-06), each \\(\\xi\\in\\mathfrak A\\) lies in \\([M\\xi]\\subseteq[M\\mathfrak A]\\). So \\((1-p)\\mathfrak A=\\{0\\}\\). Since \\(1-p\\in M'\\) and \\(\\mathfrak A\\) is separating for \\(M'\\), \\(1-p=0\\).\n\n(3) \\(M\\) is nondegenerate ([Proposition 2.4](#oa-fnd-bi-03)(2)), so (1) and (2) give the first equivalence. For the second, apply the first to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).\n\n(4) Here \\([M\\mathfrak A]=\\{0\\}\\). \\(\\square\\)\n\nA vector \\(\\xi\\) is separating for \\(M\\) exactly when the positive functional \\(\\omega_\\xi\\) is faithful on \\(M\\), since \\(\\omega_\\xi(x^*x)=\\|x\\xi\\|^2\\). This is the idea behind the implication (2)\\(\\Rightarrow\\)(4) of [Theorem 9.5](#oa-fnd-bi-13).\n\n**Exercise 9.3** (medium; separating and cyclic vectors for \\(M_n\\otimes1_k\\)). Let \\(n,k\\ge1\\), \\(I=\\{1,\\dots,k\\}\\), and \\(M=\\pi(B(\\mathbb C^n))\\) on \\(\\tilde H=\\ell^2(I;\\mathbb C^n)\\) as in [Section 3](#oa-fnd-bi-05); this is \\(M_n(\\mathbb C)\\otimes1_k\\). Write a vector as \\(\\xi=(\\xi_1,\\dots,\\xi_k)\\), or as the \\(n\\times k\\) matrix \\(X\\) with columns \\(\\xi_j\\). Show that \\(M''=M\\) and that \\(M'\\) consists of the scalar \\(k\\times k\\) matrices tensored with \\(1_{\\mathbb C^n}\\). Show that \\(\\xi\\) is separating for \\(M\\) if and only if \\(\\operatorname{rank}X=n\\), and cyclic for \\(M\\) if and only if \\(\\operatorname{rank}X=k\\). Deduce that \\(M\\) has a separating vector if and only if \\(k\\ge n\\), a cyclic vector if and only if \\(k\\le n\\), and a vector that is both if and only if \\(k=n\\).\n\n*Solution.* By Corollary 3.4 and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\pi(B(\\mathbb C^n)'')=M\\). By (3.4), \\(M'\\) consists of the \\(k\\times k\\) matrices whose entries lie in \\(B(\\mathbb C^n)'=\\mathbb C1\\). *Separating:* \\(\\pi(x)\\xi=(x\\xi_1,\\dots,x\\xi_k)=0\\) forces \\(x=0\\) exactly when the columns \\(\\xi_j\\) span \\(\\mathbb C^n\\), that is \\(\\operatorname{rank}X=n\\). *Cyclic:* \\([M\\xi]=\\{(x\\xi_1,\\dots,x\\xi_k):x\\in M_n(\\mathbb C)\\}\\), a subspace since \\(M\\xi\\) is finite dimensional. If the \\(\\xi_j\\) are linearly independent, extend them to a basis of \\(\\mathbb C^n\\); a linear map \\(x\\) can then take any prescribed values on them, so \\([M\\xi]=\\tilde H\\). If \\(\\sum_jc_j\\xi_j=0\\) with some \\(c_j\\neq0\\), every vector of \\([M\\xi]\\) satisfies the same relation, so \\([M\\xi]\\neq\\tilde H\\). Hence \\(\\xi\\) is cyclic exactly when \\(\\operatorname{rank}X=k\\). An \\(n\\times k\\) matrix of rank \\(n\\) exists exactly when \\(n\\le k\\), and one of rank \\(k\\) exactly when \\(k\\le n\\). As a check against [Proposition 9.2](#oa-fnd-bi-12)(3): \\(M'\\) acts by \\(X\\mapsto XB^{\\mathsf T}\\) for scalar \\(k\\times k\\) matrices \\(B\\), and its orbit of \\(X\\) is the set of matrices whose columns lie in the column space of \\(X\\); this is everything exactly when \\(\\operatorname{rank}X=n\\). \\(\\square\\)\n\n**Definition 9.4.** A von Neumann algebra \\(M\\) is *\\(\\sigma\\)-finite* (also called *countably decomposable*) if every family of mutually orthogonal nonzero projections in \\(M\\) is countable.\n\nA linear functional \\(\\varphi\\) on \\(M\\) is *positive* if \\(\\varphi(x^*x)\\ge0\\) for all \\(x\\), and a positive \\(\\varphi\\) is *faithful* if \\(\\varphi(x^*x)=0\\) implies \\(x=0\\). A *state* is a positive functional with \\(\\varphi(1)=1\\). A linear functional on \\(M\\) is *normal* if it is \\(\\sigma\\)-weakly continuous.\n\n**Theorem 9.5** (\\(\\sigma\\)-finite von Neumann algebras). For a von Neumann algebra \\(M\\) on \\(H\\), the following are equivalent.\n\n1. \\(M\\) is \\(\\sigma\\)-finite.\n2. \\(H\\) contains a countable set that is separating for \\(M\\).\n3. \\(H\\) contains a countable set that is cyclic for \\(M'\\).\n4. There is a faithful positive \\(\\sigma\\)-weakly continuous linear functional on \\(M\\).\n5. There is a faithful positive linear functional on \\(M\\), with no continuity assumed.\n\nIf \\(H\\neq\\{0\\}\\), the functional in (4) can be taken to be a state, that is, with \\(\\varphi(1)=1\\).\n\n**Proof.** (2)\\(\\Leftrightarrow\\)(3) is [Proposition 9.2](#oa-fnd-bi-12)(3).\n\n(1)\\(\\Rightarrow\\)(3). Let \\(\\mathcal F\\) be the collection of sets \\(F\\) of nonzero vectors such that the subspaces \\([M'\\xi]\\), \\(\\xi\\in F\\), are mutually orthogonal. The empty set belongs to \\(\\mathcal F\\), and the union of a chain in \\(\\mathcal F\\) is in \\(\\mathcal F\\), because any two of its vectors lie in one member of the chain. By Zorn's lemma there is a maximal \\(F\\in\\mathcal F\\). For \\(\\xi\\in F\\), the projection \\(p_\\xi\\) onto \\([M'\\xi]\\) lies in \\(M\\): the subspace is invariant under the self-adjoint algebra \\(M'\\), so \\(p_\\xi\\in M''=M\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). Also \\(p_\\xi\\neq0\\), because \\(\\xi\\in[M'\\xi]\\) (as \\(1\\in M'\\)). Distinct vectors of \\(F\\) give orthogonal, hence distinct, projections. By (1), \\(F\\) is countable. Suppose \\([M'F]\\neq H\\), and pick a nonzero \\(\\eta\\perp[M'F]\\). For each \\(\\xi\\in F\\), \\(\\eta\\) lies in \\([M'\\xi]^\\perp\\), which is invariant under \\(M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)); so \\([M'\\eta]\\perp[M'\\xi]\\). Also \\(\\eta\\notin F\\), since \\(\\eta\\perp\\xi\\) for every \\(\\xi\\in F\\) and \\(\\eta\\neq0\\). So \\(F\\cup\\{\\eta\\}\\in\\mathcal F\\), contradicting maximality. Hence \\(F\\) is a countable cyclic set for \\(M'\\).\n\n(2)\\(\\Rightarrow\\)(4). Let \\(\\mathfrak A\\) be a countable separating set. Removing \\(0\\) keeps it separating. If nothing is left, every \\(x\\in M\\) is \\(0\\); then \\(M=\\{0\\}\\), so \\(H=\\{0\\}\\) (as \\(1\\in M\\)) and \\(\\varphi=0\\) is faithful. Otherwise list the nonzero vectors as \\(\\xi_1,\\xi_2,\\dots\\) (finitely or infinitely many) and put\n\\[\n\\varphi(x)=\\sum_n2^{-n}\\|\\xi_n\\|^{-2}\\langle x\\xi_n,\\xi_n\\rangle .\n\\tag{9.1}\n\\]\nThis is \\(\\sum_n\\langle x\\zeta_n,\\zeta_n\\rangle\\) with \\(\\zeta_n=2^{-n/2}\\xi_n/\\|\\xi_n\\|\\) and \\(\\sum_n\\|\\zeta_n\\|^2\\le1\\). So \\(\\varphi\\) is positive and \\(\\sigma\\)-weakly continuous. The functional (9.1) is faithful: if \\(\\varphi(x^*x)=\\sum_n2^{-n}\\|x\\xi_n\\|^2/\\|\\xi_n\\|^2=0\\), then \\(x\\xi_n=0\\) for all \\(n\\), so \\(x=0\\). Dividing by \\(\\varphi(1)>0\\) gives a faithful normal state.\n\n(4)\\(\\Rightarrow\\)(5) is trivial.\n\n(5)\\(\\Rightarrow\\)(1). Let \\(\\varphi\\) be faithful and positive, and let \\((e_i)_{i\\in I}\\) be mutually orthogonal nonzero projections in \\(M\\). Faithfulness gives \\(\\varphi(e_i)=\\varphi(e_i^*e_i)>0\\). For a finite \\(G\\subseteq I\\), \\(\\sum_{i\\in G}e_i\\) is a projection, so \\(1-\\sum_{i\\in G}e_i\\ge0\\) and \\(\\sum_{i\\in G}\\varphi(e_i)\\le\\varphi(1)\\). Hence \\(I_n=\\{i:\\varphi(e_i)\\ge1/n\\}\\) has at most \\(n\\varphi(1)\\) elements, and \\(I=\\bigcup_nI_n\\) is countable. \\(\\square\\)\n\n*Remark 9.6.* A countable separating set cannot always be replaced by one vector, and \\(\\sigma\\)-finiteness of \\(M\\) does not pass to \\(M'\\) ([Example 9.7](#oa-fnd-bi-13)). Every von Neumann algebra on a separable Hilbert space is \\(\\sigma\\)-finite: a countable dense subset is separating, because a bounded operator that vanishes on a dense set is zero.\n\n**Example 9.7** (separating vectors and \\(\\sigma\\)-finiteness). \\(M=B(\\mathbb C^2)\\) is \\(\\sigma\\)-finite, and a basis is a separating set. No single vector separates: for \\(\\xi\\neq0\\), choose a nonzero \\(\\eta\\perp\\xi\\); then \\(\\theta_{\\eta,\\eta}\\xi=0\\). Every nonzero vector is cyclic for \\(M\\). Now let \\(\\Gamma\\) be uncountable and \\(M=\\mathbb C1\\) on \\(\\ell^2(\\Gamma)\\). It is \\(\\sigma\\)-finite, and every nonzero vector separates it. But \\(M'=B(\\ell^2(\\Gamma))\\) contains the uncountable family of mutually orthogonal rank-one projections onto the basis vectors, so \\(M'\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), \\(B(\\ell^2(\\Gamma))\\) has no faithful positive functional at all.\n\n**Exercise 9.8** (easy; no faithful functional on a large \\(B(H)\\)). For an uncountable set \\(\\Gamma\\), show that \\(B(\\ell^2(\\Gamma))\\) has no faithful positive linear functional, normal or not, while its commutant \\(\\mathbb C1\\) has a faithful normal state.\n\n*Solution.* The rank-one projections onto the basis vectors \\(\\delta_\\gamma\\) form an uncountable family of mutually orthogonal nonzero projections, so \\(B(\\ell^2(\\Gamma))\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), it has no faithful positive functional. On \\(\\mathbb C1\\), \\(\\omega_\\xi\\) for a unit vector \\(\\xi\\) is a normal state, and \\(\\omega_\\xi((\\lambda1)^*(\\lambda1))=|\\lambda|^2\\) vanishes only for \\(\\lambda=0\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-13",
      "unit": "the-double-commutant-theorem",
      "name": "9. Cyclic and separating vectors; σ-finite algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "anchor": "oa-fnd-bi-13",
      "proof_locus": {
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        "through_line": 670
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      "full_conditions_and_proof": "## 9. Cyclic and separating vectors; σ-finite algebras\n\nA set of vectors can generate the whole space under an algebra, or it can detect every nonzero element of the algebra. Passing to the commutant exchanges the two properties. Countable versions of them characterize the \\(\\sigma\\)-finite von Neumann algebras, which are those with a faithful positive normal functional.\n\n**Definition 9.1.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\). The set \\(\\mathfrak A\\) is *separating* for \\(S\\) if \\(x\\in S\\) and \\(x\\mathfrak A=\\{0\\}\\) imply \\(x=0\\). It is *cyclic* for \\(S\\) if \\([S\\mathfrak A]=H\\). A vector \\(\\xi\\) is separating or cyclic if \\(\\{\\xi\\}\\) is.\n\n**Proposition 9.2.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\).\n\n1. If \\(\\mathfrak A\\) is cyclic for \\(S\\), then \\(\\mathfrak A\\) is separating for \\(S'\\).\n2. If \\(M\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\), closed or not, and \\(\\mathfrak A\\) is separating for \\(M'\\), then \\(\\mathfrak A\\) is cyclic for \\(M\\).\n3. For a von Neumann algebra \\(M\\): \\(\\mathfrak A\\) is cyclic for \\(M\\) if and only if it is separating for \\(M'\\), and \\(\\mathfrak A\\) is separating for \\(M\\) if and only if it is cyclic for \\(M'\\).\n4. Nondegeneracy cannot be dropped in (2). For \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\(M'=B(H)\\), and \\(\\mathfrak A=H\\) is separating for \\(M'\\) but not cyclic for \\(M\\).\n\n**Proof.** (1) Let \\(x'\\in S'\\) with \\(x'\\mathfrak A=\\{0\\}\\). For \\(s\\in S\\) and \\(\\xi\\in\\mathfrak A\\), \\(x's\\xi=sx'\\xi=0\\). So \\(x'\\) vanishes on the span of \\(S\\mathfrak A\\) and, being continuous, on \\([S\\mathfrak A]=H\\).\n\n(2) Let \\(p\\) be the projection onto \\([M\\mathfrak A]\\). This subspace is invariant under \\(M\\), and \\(M^*=M\\), so \\(p\\in M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). By [Lemma 4.1](#oa-fnd-bi-06), each \\(\\xi\\in\\mathfrak A\\) lies in \\([M\\xi]\\subseteq[M\\mathfrak A]\\). So \\((1-p)\\mathfrak A=\\{0\\}\\). Since \\(1-p\\in M'\\) and \\(\\mathfrak A\\) is separating for \\(M'\\), \\(1-p=0\\).\n\n(3) \\(M\\) is nondegenerate ([Proposition 2.4](#oa-fnd-bi-03)(2)), so (1) and (2) give the first equivalence. For the second, apply the first to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).\n\n(4) Here \\([M\\mathfrak A]=\\{0\\}\\). \\(\\square\\)\n\nA vector \\(\\xi\\) is separating for \\(M\\) exactly when the positive functional \\(\\omega_\\xi\\) is faithful on \\(M\\), since \\(\\omega_\\xi(x^*x)=\\|x\\xi\\|^2\\). This is the idea behind the implication (2)\\(\\Rightarrow\\)(4) of [Theorem 9.5](#oa-fnd-bi-13).\n\n**Exercise 9.3** (medium; separating and cyclic vectors for \\(M_n\\otimes1_k\\)). Let \\(n,k\\ge1\\), \\(I=\\{1,\\dots,k\\}\\), and \\(M=\\pi(B(\\mathbb C^n))\\) on \\(\\tilde H=\\ell^2(I;\\mathbb C^n)\\) as in [Section 3](#oa-fnd-bi-05); this is \\(M_n(\\mathbb C)\\otimes1_k\\). Write a vector as \\(\\xi=(\\xi_1,\\dots,\\xi_k)\\), or as the \\(n\\times k\\) matrix \\(X\\) with columns \\(\\xi_j\\). Show that \\(M''=M\\) and that \\(M'\\) consists of the scalar \\(k\\times k\\) matrices tensored with \\(1_{\\mathbb C^n}\\). Show that \\(\\xi\\) is separating for \\(M\\) if and only if \\(\\operatorname{rank}X=n\\), and cyclic for \\(M\\) if and only if \\(\\operatorname{rank}X=k\\). Deduce that \\(M\\) has a separating vector if and only if \\(k\\ge n\\), a cyclic vector if and only if \\(k\\le n\\), and a vector that is both if and only if \\(k=n\\).\n\n*Solution.* By Corollary 3.4 and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\pi(B(\\mathbb C^n)'')=M\\). By (3.4), \\(M'\\) consists of the \\(k\\times k\\) matrices whose entries lie in \\(B(\\mathbb C^n)'=\\mathbb C1\\). *Separating:* \\(\\pi(x)\\xi=(x\\xi_1,\\dots,x\\xi_k)=0\\) forces \\(x=0\\) exactly when the columns \\(\\xi_j\\) span \\(\\mathbb C^n\\), that is \\(\\operatorname{rank}X=n\\). *Cyclic:* \\([M\\xi]=\\{(x\\xi_1,\\dots,x\\xi_k):x\\in M_n(\\mathbb C)\\}\\), a subspace since \\(M\\xi\\) is finite dimensional. If the \\(\\xi_j\\) are linearly independent, extend them to a basis of \\(\\mathbb C^n\\); a linear map \\(x\\) can then take any prescribed values on them, so \\([M\\xi]=\\tilde H\\). If \\(\\sum_jc_j\\xi_j=0\\) with some \\(c_j\\neq0\\), every vector of \\([M\\xi]\\) satisfies the same relation, so \\([M\\xi]\\neq\\tilde H\\). Hence \\(\\xi\\) is cyclic exactly when \\(\\operatorname{rank}X=k\\). An \\(n\\times k\\) matrix of rank \\(n\\) exists exactly when \\(n\\le k\\), and one of rank \\(k\\) exactly when \\(k\\le n\\). As a check against [Proposition 9.2](#oa-fnd-bi-12)(3): \\(M'\\) acts by \\(X\\mapsto XB^{\\mathsf T}\\) for scalar \\(k\\times k\\) matrices \\(B\\), and its orbit of \\(X\\) is the set of matrices whose columns lie in the column space of \\(X\\); this is everything exactly when \\(\\operatorname{rank}X=n\\). \\(\\square\\)\n\n**Definition 9.4.** A von Neumann algebra \\(M\\) is *\\(\\sigma\\)-finite* (also called *countably decomposable*) if every family of mutually orthogonal nonzero projections in \\(M\\) is countable.\n\nA linear functional \\(\\varphi\\) on \\(M\\) is *positive* if \\(\\varphi(x^*x)\\ge0\\) for all \\(x\\), and a positive \\(\\varphi\\) is *faithful* if \\(\\varphi(x^*x)=0\\) implies \\(x=0\\). A *state* is a positive functional with \\(\\varphi(1)=1\\). A linear functional on \\(M\\) is *normal* if it is \\(\\sigma\\)-weakly continuous.\n\n**Theorem 9.5** (\\(\\sigma\\)-finite von Neumann algebras). For a von Neumann algebra \\(M\\) on \\(H\\), the following are equivalent.\n\n1. \\(M\\) is \\(\\sigma\\)-finite.\n2. \\(H\\) contains a countable set that is separating for \\(M\\).\n3. \\(H\\) contains a countable set that is cyclic for \\(M'\\).\n4. There is a faithful positive \\(\\sigma\\)-weakly continuous linear functional on \\(M\\).\n5. There is a faithful positive linear functional on \\(M\\), with no continuity assumed.\n\nIf \\(H\\neq\\{0\\}\\), the functional in (4) can be taken to be a state, that is, with \\(\\varphi(1)=1\\).\n\n**Proof.** (2)\\(\\Leftrightarrow\\)(3) is [Proposition 9.2](#oa-fnd-bi-12)(3).\n\n(1)\\(\\Rightarrow\\)(3). Let \\(\\mathcal F\\) be the collection of sets \\(F\\) of nonzero vectors such that the subspaces \\([M'\\xi]\\), \\(\\xi\\in F\\), are mutually orthogonal. The empty set belongs to \\(\\mathcal F\\), and the union of a chain in \\(\\mathcal F\\) is in \\(\\mathcal F\\), because any two of its vectors lie in one member of the chain. By Zorn's lemma there is a maximal \\(F\\in\\mathcal F\\). For \\(\\xi\\in F\\), the projection \\(p_\\xi\\) onto \\([M'\\xi]\\) lies in \\(M\\): the subspace is invariant under the self-adjoint algebra \\(M'\\), so \\(p_\\xi\\in M''=M\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). Also \\(p_\\xi\\neq0\\), because \\(\\xi\\in[M'\\xi]\\) (as \\(1\\in M'\\)). Distinct vectors of \\(F\\) give orthogonal, hence distinct, projections. By (1), \\(F\\) is countable. Suppose \\([M'F]\\neq H\\), and pick a nonzero \\(\\eta\\perp[M'F]\\). For each \\(\\xi\\in F\\), \\(\\eta\\) lies in \\([M'\\xi]^\\perp\\), which is invariant under \\(M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)); so \\([M'\\eta]\\perp[M'\\xi]\\). Also \\(\\eta\\notin F\\), since \\(\\eta\\perp\\xi\\) for every \\(\\xi\\in F\\) and \\(\\eta\\neq0\\). So \\(F\\cup\\{\\eta\\}\\in\\mathcal F\\), contradicting maximality. Hence \\(F\\) is a countable cyclic set for \\(M'\\).\n\n(2)\\(\\Rightarrow\\)(4). Let \\(\\mathfrak A\\) be a countable separating set. Removing \\(0\\) keeps it separating. If nothing is left, every \\(x\\in M\\) is \\(0\\); then \\(M=\\{0\\}\\), so \\(H=\\{0\\}\\) (as \\(1\\in M\\)) and \\(\\varphi=0\\) is faithful. Otherwise list the nonzero vectors as \\(\\xi_1,\\xi_2,\\dots\\) (finitely or infinitely many) and put\n\\[\n\\varphi(x)=\\sum_n2^{-n}\\|\\xi_n\\|^{-2}\\langle x\\xi_n,\\xi_n\\rangle .\n\\tag{9.1}\n\\]\nThis is \\(\\sum_n\\langle x\\zeta_n,\\zeta_n\\rangle\\) with \\(\\zeta_n=2^{-n/2}\\xi_n/\\|\\xi_n\\|\\) and \\(\\sum_n\\|\\zeta_n\\|^2\\le1\\). So \\(\\varphi\\) is positive and \\(\\sigma\\)-weakly continuous. The functional (9.1) is faithful: if \\(\\varphi(x^*x)=\\sum_n2^{-n}\\|x\\xi_n\\|^2/\\|\\xi_n\\|^2=0\\), then \\(x\\xi_n=0\\) for all \\(n\\), so \\(x=0\\). Dividing by \\(\\varphi(1)>0\\) gives a faithful normal state.\n\n(4)\\(\\Rightarrow\\)(5) is trivial.\n\n(5)\\(\\Rightarrow\\)(1). Let \\(\\varphi\\) be faithful and positive, and let \\((e_i)_{i\\in I}\\) be mutually orthogonal nonzero projections in \\(M\\). Faithfulness gives \\(\\varphi(e_i)=\\varphi(e_i^*e_i)>0\\). For a finite \\(G\\subseteq I\\), \\(\\sum_{i\\in G}e_i\\) is a projection, so \\(1-\\sum_{i\\in G}e_i\\ge0\\) and \\(\\sum_{i\\in G}\\varphi(e_i)\\le\\varphi(1)\\). Hence \\(I_n=\\{i:\\varphi(e_i)\\ge1/n\\}\\) has at most \\(n\\varphi(1)\\) elements, and \\(I=\\bigcup_nI_n\\) is countable. \\(\\square\\)\n\n*Remark 9.6.* A countable separating set cannot always be replaced by one vector, and \\(\\sigma\\)-finiteness of \\(M\\) does not pass to \\(M'\\) ([Example 9.7](#oa-fnd-bi-13)). Every von Neumann algebra on a separable Hilbert space is \\(\\sigma\\)-finite: a countable dense subset is separating, because a bounded operator that vanishes on a dense set is zero.\n\n**Example 9.7** (separating vectors and \\(\\sigma\\)-finiteness). \\(M=B(\\mathbb C^2)\\) is \\(\\sigma\\)-finite, and a basis is a separating set. No single vector separates: for \\(\\xi\\neq0\\), choose a nonzero \\(\\eta\\perp\\xi\\); then \\(\\theta_{\\eta,\\eta}\\xi=0\\). Every nonzero vector is cyclic for \\(M\\). Now let \\(\\Gamma\\) be uncountable and \\(M=\\mathbb C1\\) on \\(\\ell^2(\\Gamma)\\). It is \\(\\sigma\\)-finite, and every nonzero vector separates it. But \\(M'=B(\\ell^2(\\Gamma))\\) contains the uncountable family of mutually orthogonal rank-one projections onto the basis vectors, so \\(M'\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), \\(B(\\ell^2(\\Gamma))\\) has no faithful positive functional at all.\n\n**Exercise 9.8** (easy; no faithful functional on a large \\(B(H)\\)). For an uncountable set \\(\\Gamma\\), show that \\(B(\\ell^2(\\Gamma))\\) has no faithful positive linear functional, normal or not, while its commutant \\(\\mathbb C1\\) has a faithful normal state.\n\n*Solution.* The rank-one projections onto the basis vectors \\(\\delta_\\gamma\\) form an uncountable family of mutually orthogonal nonzero projections, so \\(B(\\ell^2(\\Gamma))\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), it has no faithful positive functional. On \\(\\mathbb C1\\), \\(\\omega_\\xi\\) for a unit vector \\(\\xi\\) is a normal state, and \\(\\omega_\\xi((\\lambda1)^*(\\lambda1))=|\\lambda|^2\\) vanishes only for \\(\\lambda=0\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-17",
      "unit": "the-double-commutant-theorem",
      "name": "9. Cyclic and separating vectors; σ-finite algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "proof_locus": {
        "line": 607,
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      "full_conditions_and_proof": "## 9. Cyclic and separating vectors; σ-finite algebras\n\nA set of vectors can generate the whole space under an algebra, or it can detect every nonzero element of the algebra. Passing to the commutant exchanges the two properties. Countable versions of them characterize the \\(\\sigma\\)-finite von Neumann algebras, which are those with a faithful positive normal functional.\n\n**Definition 9.1.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\). The set \\(\\mathfrak A\\) is *separating* for \\(S\\) if \\(x\\in S\\) and \\(x\\mathfrak A=\\{0\\}\\) imply \\(x=0\\). It is *cyclic* for \\(S\\) if \\([S\\mathfrak A]=H\\). A vector \\(\\xi\\) is separating or cyclic if \\(\\{\\xi\\}\\) is.\n\n**Proposition 9.2.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\).\n\n1. If \\(\\mathfrak A\\) is cyclic for \\(S\\), then \\(\\mathfrak A\\) is separating for \\(S'\\).\n2. If \\(M\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\), closed or not, and \\(\\mathfrak A\\) is separating for \\(M'\\), then \\(\\mathfrak A\\) is cyclic for \\(M\\).\n3. For a von Neumann algebra \\(M\\): \\(\\mathfrak A\\) is cyclic for \\(M\\) if and only if it is separating for \\(M'\\), and \\(\\mathfrak A\\) is separating for \\(M\\) if and only if it is cyclic for \\(M'\\).\n4. Nondegeneracy cannot be dropped in (2). For \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\(M'=B(H)\\), and \\(\\mathfrak A=H\\) is separating for \\(M'\\) but not cyclic for \\(M\\).\n\n**Proof.** (1) Let \\(x'\\in S'\\) with \\(x'\\mathfrak A=\\{0\\}\\). For \\(s\\in S\\) and \\(\\xi\\in\\mathfrak A\\), \\(x's\\xi=sx'\\xi=0\\). So \\(x'\\) vanishes on the span of \\(S\\mathfrak A\\) and, being continuous, on \\([S\\mathfrak A]=H\\).\n\n(2) Let \\(p\\) be the projection onto \\([M\\mathfrak A]\\). This subspace is invariant under \\(M\\), and \\(M^*=M\\), so \\(p\\in M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). By [Lemma 4.1](#oa-fnd-bi-06), each \\(\\xi\\in\\mathfrak A\\) lies in \\([M\\xi]\\subseteq[M\\mathfrak A]\\). So \\((1-p)\\mathfrak A=\\{0\\}\\). Since \\(1-p\\in M'\\) and \\(\\mathfrak A\\) is separating for \\(M'\\), \\(1-p=0\\).\n\n(3) \\(M\\) is nondegenerate ([Proposition 2.4](#oa-fnd-bi-03)(2)), so (1) and (2) give the first equivalence. For the second, apply the first to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).\n\n(4) Here \\([M\\mathfrak A]=\\{0\\}\\). \\(\\square\\)\n\nA vector \\(\\xi\\) is separating for \\(M\\) exactly when the positive functional \\(\\omega_\\xi\\) is faithful on \\(M\\), since \\(\\omega_\\xi(x^*x)=\\|x\\xi\\|^2\\). This is the idea behind the implication (2)\\(\\Rightarrow\\)(4) of [Theorem 9.5](#oa-fnd-bi-13).\n\n**Exercise 9.3** (medium; separating and cyclic vectors for \\(M_n\\otimes1_k\\)). Let \\(n,k\\ge1\\), \\(I=\\{1,\\dots,k\\}\\), and \\(M=\\pi(B(\\mathbb C^n))\\) on \\(\\tilde H=\\ell^2(I;\\mathbb C^n)\\) as in [Section 3](#oa-fnd-bi-05); this is \\(M_n(\\mathbb C)\\otimes1_k\\). Write a vector as \\(\\xi=(\\xi_1,\\dots,\\xi_k)\\), or as the \\(n\\times k\\) matrix \\(X\\) with columns \\(\\xi_j\\). Show that \\(M''=M\\) and that \\(M'\\) consists of the scalar \\(k\\times k\\) matrices tensored with \\(1_{\\mathbb C^n}\\). Show that \\(\\xi\\) is separating for \\(M\\) if and only if \\(\\operatorname{rank}X=n\\), and cyclic for \\(M\\) if and only if \\(\\operatorname{rank}X=k\\). Deduce that \\(M\\) has a separating vector if and only if \\(k\\ge n\\), a cyclic vector if and only if \\(k\\le n\\), and a vector that is both if and only if \\(k=n\\).\n\n*Solution.* By Corollary 3.4 and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\pi(B(\\mathbb C^n)'')=M\\). By (3.4), \\(M'\\) consists of the \\(k\\times k\\) matrices whose entries lie in \\(B(\\mathbb C^n)'=\\mathbb C1\\). *Separating:* \\(\\pi(x)\\xi=(x\\xi_1,\\dots,x\\xi_k)=0\\) forces \\(x=0\\) exactly when the columns \\(\\xi_j\\) span \\(\\mathbb C^n\\), that is \\(\\operatorname{rank}X=n\\). *Cyclic:* \\([M\\xi]=\\{(x\\xi_1,\\dots,x\\xi_k):x\\in M_n(\\mathbb C)\\}\\), a subspace since \\(M\\xi\\) is finite dimensional. If the \\(\\xi_j\\) are linearly independent, extend them to a basis of \\(\\mathbb C^n\\); a linear map \\(x\\) can then take any prescribed values on them, so \\([M\\xi]=\\tilde H\\). If \\(\\sum_jc_j\\xi_j=0\\) with some \\(c_j\\neq0\\), every vector of \\([M\\xi]\\) satisfies the same relation, so \\([M\\xi]\\neq\\tilde H\\). Hence \\(\\xi\\) is cyclic exactly when \\(\\operatorname{rank}X=k\\). An \\(n\\times k\\) matrix of rank \\(n\\) exists exactly when \\(n\\le k\\), and one of rank \\(k\\) exactly when \\(k\\le n\\). As a check against [Proposition 9.2](#oa-fnd-bi-12)(3): \\(M'\\) acts by \\(X\\mapsto XB^{\\mathsf T}\\) for scalar \\(k\\times k\\) matrices \\(B\\), and its orbit of \\(X\\) is the set of matrices whose columns lie in the column space of \\(X\\); this is everything exactly when \\(\\operatorname{rank}X=n\\). \\(\\square\\)\n\n**Definition 9.4.** A von Neumann algebra \\(M\\) is *\\(\\sigma\\)-finite* (also called *countably decomposable*) if every family of mutually orthogonal nonzero projections in \\(M\\) is countable.\n\nA linear functional \\(\\varphi\\) on \\(M\\) is *positive* if \\(\\varphi(x^*x)\\ge0\\) for all \\(x\\), and a positive \\(\\varphi\\) is *faithful* if \\(\\varphi(x^*x)=0\\) implies \\(x=0\\). A *state* is a positive functional with \\(\\varphi(1)=1\\). A linear functional on \\(M\\) is *normal* if it is \\(\\sigma\\)-weakly continuous.\n\n**Theorem 9.5** (\\(\\sigma\\)-finite von Neumann algebras). For a von Neumann algebra \\(M\\) on \\(H\\), the following are equivalent.\n\n1. \\(M\\) is \\(\\sigma\\)-finite.\n2. \\(H\\) contains a countable set that is separating for \\(M\\).\n3. \\(H\\) contains a countable set that is cyclic for \\(M'\\).\n4. There is a faithful positive \\(\\sigma\\)-weakly continuous linear functional on \\(M\\).\n5. There is a faithful positive linear functional on \\(M\\), with no continuity assumed.\n\nIf \\(H\\neq\\{0\\}\\), the functional in (4) can be taken to be a state, that is, with \\(\\varphi(1)=1\\).\n\n**Proof.** (2)\\(\\Leftrightarrow\\)(3) is [Proposition 9.2](#oa-fnd-bi-12)(3).\n\n(1)\\(\\Rightarrow\\)(3). Let \\(\\mathcal F\\) be the collection of sets \\(F\\) of nonzero vectors such that the subspaces \\([M'\\xi]\\), \\(\\xi\\in F\\), are mutually orthogonal. The empty set belongs to \\(\\mathcal F\\), and the union of a chain in \\(\\mathcal F\\) is in \\(\\mathcal F\\), because any two of its vectors lie in one member of the chain. By Zorn's lemma there is a maximal \\(F\\in\\mathcal F\\). For \\(\\xi\\in F\\), the projection \\(p_\\xi\\) onto \\([M'\\xi]\\) lies in \\(M\\): the subspace is invariant under the self-adjoint algebra \\(M'\\), so \\(p_\\xi\\in M''=M\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)). Also \\(p_\\xi\\neq0\\), because \\(\\xi\\in[M'\\xi]\\) (as \\(1\\in M'\\)). Distinct vectors of \\(F\\) give orthogonal, hence distinct, projections. By (1), \\(F\\) is countable. Suppose \\([M'F]\\neq H\\), and pick a nonzero \\(\\eta\\perp[M'F]\\). For each \\(\\xi\\in F\\), \\(\\eta\\) lies in \\([M'\\xi]^\\perp\\), which is invariant under \\(M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(5)); so \\([M'\\eta]\\perp[M'\\xi]\\). Also \\(\\eta\\notin F\\), since \\(\\eta\\perp\\xi\\) for every \\(\\xi\\in F\\) and \\(\\eta\\neq0\\). So \\(F\\cup\\{\\eta\\}\\in\\mathcal F\\), contradicting maximality. Hence \\(F\\) is a countable cyclic set for \\(M'\\).\n\n(2)\\(\\Rightarrow\\)(4). Let \\(\\mathfrak A\\) be a countable separating set. Removing \\(0\\) keeps it separating. If nothing is left, every \\(x\\in M\\) is \\(0\\); then \\(M=\\{0\\}\\), so \\(H=\\{0\\}\\) (as \\(1\\in M\\)) and \\(\\varphi=0\\) is faithful. Otherwise list the nonzero vectors as \\(\\xi_1,\\xi_2,\\dots\\) (finitely or infinitely many) and put\n\\[\n\\varphi(x)=\\sum_n2^{-n}\\|\\xi_n\\|^{-2}\\langle x\\xi_n,\\xi_n\\rangle .\n\\tag{9.1}\n\\]\nThis is \\(\\sum_n\\langle x\\zeta_n,\\zeta_n\\rangle\\) with \\(\\zeta_n=2^{-n/2}\\xi_n/\\|\\xi_n\\|\\) and \\(\\sum_n\\|\\zeta_n\\|^2\\le1\\). So \\(\\varphi\\) is positive and \\(\\sigma\\)-weakly continuous. The functional (9.1) is faithful: if \\(\\varphi(x^*x)=\\sum_n2^{-n}\\|x\\xi_n\\|^2/\\|\\xi_n\\|^2=0\\), then \\(x\\xi_n=0\\) for all \\(n\\), so \\(x=0\\). Dividing by \\(\\varphi(1)>0\\) gives a faithful normal state.\n\n(4)\\(\\Rightarrow\\)(5) is trivial.\n\n(5)\\(\\Rightarrow\\)(1). Let \\(\\varphi\\) be faithful and positive, and let \\((e_i)_{i\\in I}\\) be mutually orthogonal nonzero projections in \\(M\\). Faithfulness gives \\(\\varphi(e_i)=\\varphi(e_i^*e_i)>0\\). For a finite \\(G\\subseteq I\\), \\(\\sum_{i\\in G}e_i\\) is a projection, so \\(1-\\sum_{i\\in G}e_i\\ge0\\) and \\(\\sum_{i\\in G}\\varphi(e_i)\\le\\varphi(1)\\). Hence \\(I_n=\\{i:\\varphi(e_i)\\ge1/n\\}\\) has at most \\(n\\varphi(1)\\) elements, and \\(I=\\bigcup_nI_n\\) is countable. \\(\\square\\)\n\n*Remark 9.6.* A countable separating set cannot always be replaced by one vector, and \\(\\sigma\\)-finiteness of \\(M\\) does not pass to \\(M'\\) ([Example 9.7](#oa-fnd-bi-13)). Every von Neumann algebra on a separable Hilbert space is \\(\\sigma\\)-finite: a countable dense subset is separating, because a bounded operator that vanishes on a dense set is zero.\n\n**Example 9.7** (separating vectors and \\(\\sigma\\)-finiteness). \\(M=B(\\mathbb C^2)\\) is \\(\\sigma\\)-finite, and a basis is a separating set. No single vector separates: for \\(\\xi\\neq0\\), choose a nonzero \\(\\eta\\perp\\xi\\); then \\(\\theta_{\\eta,\\eta}\\xi=0\\). Every nonzero vector is cyclic for \\(M\\). Now let \\(\\Gamma\\) be uncountable and \\(M=\\mathbb C1\\) on \\(\\ell^2(\\Gamma)\\). It is \\(\\sigma\\)-finite, and every nonzero vector separates it. But \\(M'=B(\\ell^2(\\Gamma))\\) contains the uncountable family of mutually orthogonal rank-one projections onto the basis vectors, so \\(M'\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), \\(B(\\ell^2(\\Gamma))\\) has no faithful positive functional at all.\n\n**Exercise 9.8** (easy; no faithful functional on a large \\(B(H)\\)). For an uncountable set \\(\\Gamma\\), show that \\(B(\\ell^2(\\Gamma))\\) has no faithful positive linear functional, normal or not, while its commutant \\(\\mathbb C1\\) has a faithful normal state.\n\n*Solution.* The rank-one projections onto the basis vectors \\(\\delta_\\gamma\\) form an uncountable family of mutually orthogonal nonzero projections, so \\(B(\\ell^2(\\Gamma))\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), it has no faithful positive functional. On \\(\\mathbb C1\\), \\(\\omega_\\xi\\) for a unit vector \\(\\xi\\) is a normal state, and \\(\\omega_\\xi((\\lambda1)^*(\\lambda1))=|\\lambda|^2\\) vanishes only for \\(\\lambda=0\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-BI-14",
      "unit": "the-double-commutant-theorem",
      "name": "10. Positive normal functionals as sums of vector functionals",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "## 10. Positive normal functionals as sums of vector functionals\n\nA vector functional \\(\\omega_\\xi\\) is positive and normal. Conversely, every positive normal functional is a sum of countably many vector functionals: on an amplification it becomes a single vector functional.\n\n**Theorem 10.1.** Let \\(M\\) be a \\(*\\)-subalgebra of \\(B(H)\\) that contains \\(1\\), closed or not. Let \\(\\varphi:M\\to\\mathbb C\\) be linear and positive, and continuous for the \\(\\sigma\\)-strong topology restricted to \\(M\\). Then there is a sequence \\((\\xi_n)\\) in \\(H\\) with \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\) and\n\\[\n\\begin{gathered}\n\\varphi(x)\\\\\n=\\sum_n\\langle x\\xi_n,\\xi_n\\rangle\\\\\n(x\\in M).\n\\end{gathered}\n\\tag{10.1}\n\\]\nIf \\(\\varphi\\) is continuous for the strong topology, finitely many vectors suffice. Every \\(\\sigma\\)-weakly continuous functional is \\(\\sigma\\)-strongly continuous ([Lemma 1.2](#oa-fnd-bi-01)(b)). So the theorem applies to every positive normal functional on a von Neumann algebra.\n\n**Proof.** *Step 1.* By continuity at \\(0\\) and [Lemma 1.2](#oa-fnd-bi-01)(a), there are a square-summable \\((\\zeta_n)\\) and \\(\\delta>0\\) such that \\(p(x)=(\\sum_n\\|x\\zeta_n\\|^2)^{1/2}<\\delta\\) implies \\(|\\varphi(x)|<1\\), for \\(x\\in M\\). By homogeneity, \\(|\\varphi(x)|\\le\\delta^{-1}p(x)\\). (If \\(p(x)=0\\), then \\(p(tx)=0\\) for all \\(t\\), so \\(|\\varphi(x)|<1/t\\) for all \\(t>0\\).) Replace \\(\\zeta_n\\) by \\(\\zeta_n/\\delta\\). With \\(\\tilde H=\\ell^2(\\mathbb N;H)\\), the amplification \\(\\pi\\) of [Section 3](#oa-fnd-bi-05), and \\(\\tilde\\zeta=(\\zeta_n)\\in\\tilde H\\), this reads \\(|\\varphi(x)|\\le\\|\\pi(x)\\tilde\\zeta\\|\\).\n\n*Step 2.* Let \\(L=[\\pi(M)\\tilde\\zeta]\\). The rule \\(\\pi(x)\\tilde\\zeta\\mapsto\\varphi(x)\\) is well defined and bounded by Step 1. It extends to a bounded functional on \\(L\\), and the Riesz theorem (H) gives \\(\\tilde\\eta\\in L\\) with \\(\\varphi(x)=\\langle\\pi(x)\\tilde\\zeta,\\tilde\\eta\\rangle\\) for \\(x\\in M\\).\n\n*Step 3.* A positive functional on a unital \\(*\\)-algebra is hermitian: \\(\\varphi(x^*)=\\overline{\\varphi(x)}\\). Indeed, for \\(h=h^*\\in M\\), the numbers \\(\\varphi(1)\\), \\(\\varphi(h^2)\\) and \\(\\varphi((1+h)^2)=\\varphi(1)+2\\varphi(h)+\\varphi(h^2)\\) are real, so \\(\\varphi(h)\\) is real. Then write \\(x=h+ik\\) with \\(h=(x+x^*)/2\\) and \\(k=(x-x^*)/(2i)\\) in \\(M\\). Hence \\(\\varphi(x)=\\overline{\\varphi(x^*)}=\\overline{\\langle\\pi(x^*)\\tilde\\zeta,\\tilde\\eta\\rangle}=\\langle\\pi(x)\\tilde\\eta,\\tilde\\zeta\\rangle\\). Adding the two expressions for \\(\\varphi(x)\\), with \\(\\tilde\\theta=\\tilde\\zeta+\\tilde\\eta\\) and \\(\\tilde\\kappa=\\tilde\\zeta-\\tilde\\eta\\),\n\\[\n\\begin{gathered}\n4\\varphi(x)\\\\\n=2\\langle\\pi(x)\\tilde\\zeta,\\tilde\\eta\\rangle\\\\\n+2\\langle\\pi(x)\\tilde\\eta,\\tilde\\zeta\\rangle\\\\\n=\\langle\\pi(x)\\tilde\\theta,\\tilde\\theta\\rangle\\\\\n-\\langle\\pi(x)\\tilde\\kappa,\\tilde\\kappa\\rangle .\n\\end{gathered}\n\\tag{10.2}\n\\]\nApplied to \\(x^*x\\), (10.2) gives \\(4\\varphi(x^*x)\\le\\langle\\pi(x^*x)\\tilde\\theta,\\tilde\\theta\\rangle=\\|\\pi(x)\\tilde\\theta\\|^2\\), because \\(\\langle\\pi(x^*x)\\tilde\\kappa,\\tilde\\kappa\\rangle=\\|\\pi(x)\\tilde\\kappa\\|^2\\ge0\\).\n\n*Step 4.* The form \\((x,y)\\mapsto\\varphi(y^*x)\\) on \\(M\\) is sesquilinear, hermitian by Step 3, and positive semidefinite. The Cauchy–Schwarz inequality for such forms (H) gives \\(|\\varphi(y^*x)|^2\\le\\varphi(x^*x)\\varphi(y^*y)\\). With Step 3, \\(|\\varphi(y^*x)|\\le\\frac14\\|\\pi(x)\\tilde\\theta\\|\\,\\|\\pi(y)\\tilde\\theta\\|\\). Let \\(L_\\theta=[\\pi(M)\\tilde\\theta]\\). The rule \\(B(\\pi(x)\\tilde\\theta,\\pi(y)\\tilde\\theta)=\\varphi(y^*x)\\) is therefore a well-defined sesquilinear form on the subspace \\(\\pi(M)\\tilde\\theta\\), bounded by \\(1/4\\) and positive. It extends to \\(L_\\theta\\). By (H) there is \\(h\\in B(L_\\theta)\\) with \\(0\\le h\\le\\tfrac14\\,1\\) and \\(\\varphi(y^*x)=\\langle h\\pi(x)\\tilde\\theta,\\pi(y)\\tilde\\theta\\rangle\\) for all \\(x,y\\in M\\).\n\n*Step 5.* \\(L_\\theta\\) is invariant under \\(\\pi(M)\\), and \\(h\\) commutes with the restrictions: for \\(x,y,z\\in M\\),\n\\[\n\\begin{gathered}\n\\langle h\\pi(z)\\pi(x)\\tilde\\theta,\\pi(y)\\tilde\\theta\\rangle\\\\\n=\\varphi(y^*zx)\\\\\n=\\varphi((z^*y)^*x)\\\\\n=\\langle h\\pi(x)\\tilde\\theta,\\pi(z^*y)\\tilde\\theta\\rangle\\\\\n=\\langle\\pi(z)h\\pi(x)\\tilde\\theta,\\pi(y)\\tilde\\theta\\rangle .\n\\end{gathered}\n\\]\nBy density, \\(h\\pi(z)=\\pi(z)h\\) on \\(L_\\theta\\). By (C), \\(h^{1/2}\\) commutes with \\(\\pi(z)\\) on \\(L_\\theta\\) as well. Since \\(1\\in M\\), \\(\\tilde\\theta\\in L_\\theta\\). Put \\(\\tilde\\xi=h^{1/2}\\tilde\\theta\\in L_\\theta\\subseteq\\tilde H\\) and write \\(\\tilde\\xi=(\\xi_n)\\). Then for \\(x\\in M\\),\n\\[\n\\begin{gathered}\n\\varphi(x)\\\\\n=\\varphi(1^*x)\\\\\n=\\langle h\\pi(x)\\tilde\\theta,\\tilde\\theta\\rangle\\\\\n=\\langle\\pi(x)h^{1/2}\\tilde\\theta,h^{1/2}\\tilde\\theta\\rangle\\\\\n=\\sum_n\\langle x\\xi_n,\\xi_n\\rangle ,\n\\end{gathered}\n\\]\nwhich is (10.1), with \\(\\sum_n\\|\\xi_n\\|^2=\\|\\tilde\\xi\\|^2<\\infty\\).\n\n*The finite case.* If \\(\\varphi\\) is strongly continuous, Step 1 produces finitely many vectors \\(\\zeta_1,\\dots,\\zeta_m\\). Steps 2–5 then run in \\(\\ell^2(\\{1,\\dots,m\\};H)\\), and the resulting \\(\\tilde\\xi\\) has \\(m\\) components. \\(\\square\\)\n\n## Where this leads\n\n- *Kaplansky's density theorem* sharpens Theorem 4.4: the approximating nets can be chosen with the same norm bound as the operator they approximate. The density results proved here give nets with no norm bound.\n- *Normal functionals and maps*, and W\\*-algebras, are treated in the lesson [The universal enveloping von Neumann algebra of a C\\*-algebra, and W\\*-algebras](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md).\n- *Comparison of projections and the type decomposition* are treated in [Projections and types of von Neumann algebras](projections-and-types-of-von-neumann-algebras.md). The projection \\(z\\) of Theorem 5.8(4) is the *central support* of \\(e\\), which is studied there.\n- *Tensor products* of von Neumann algebras and their commutants are treated in Spatial tensor products of von Neumann algebras.\n\n## References\n\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n- [van Neerven] J. van Neerven, *Functional Analysis*, [corrected author version, arXiv:2112.11166v7](https://arxiv.org/pdf/2112.11166v7).\n\n*Freely accessible reading:* [Jesse Peterson, *Notes on operator algebras*, §3.5](https://math.vanderbilt.edu/peters10/teaching/spring2015/OperatorAlgebras.pdf) gives a route through the finite-amplification proof of the bicommutant theorem. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-18",
      "unit": "the-double-commutant-theorem",
      "name": "Probability products for the large-Hilbert-space example",
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      "source": "src/the-double-commutant-theorem.md",
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      "full_conditions_and_proof": "### Probability products for the large-Hilbert-space example\n\nExercise 9.9 uses independent coordinates indexed by an arbitrary set. A finite product is insufficient when uncountably many coordinates must supply orthogonal vectors. Here we construct the required measure without assuming that the factor spaces are standard Borel, or even that they have topologies.\n\nThe measure tools are [Carathéodory's theorem](measure-and-hilbert-space-tools.md#1-from-an-outer-measure-to-a-measure) and [monotone convergence](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions), Theorems 1.1 and 2.1 of *Measure and Hilbert space tools for Haar integration*. In particular, monotone convergence gives \\(\\int\\sum_n f_n=\\sum_n\\int f_n\\) for nonnegative measurable functions. Choice is used to select points in nonempty factor spaces. Fremlin's [§254](https://www1.essex.ac.uk/maths/people/fremlin/cont25.htm) is a reference for the classical product theorem.\n\n**Lemma (Uniqueness from a generating family).** Let \\(\\mathcal P\\) be a family of subsets of \\(X\\) containing \\(X\\) and closed under finite intersections. Two finite measures on \\(\\sigma(\\mathcal P)\\) that agree on \\(\\mathcal P\\) agree everywhere.\n\n*Proof.* A *Dynkin system* contains \\(X\\), is closed under the difference \\(B\\setminus A\\) when \\(A\\subseteq B\\) are members, and is closed under countable disjoint unions. Let \\(\\mathcal D\\) be the smallest Dynkin system containing \\(\\mathcal P\\), obtained by intersecting all such systems.\n\nFix \\(P\\in\\mathcal P\\). The members \\(B\\in\\mathcal D\\) for which \\(P\\cap B\\in\\mathcal D\\) form a Dynkin system. Indeed, they contain \\(X\\); intersections with \\(P\\) preserve nested differences and disjoint unions. This system contains \\(\\mathcal P\\), so it contains \\(\\mathcal D\\). Now fix any \\(B\\in\\mathcal D\\). The members \\(A\\in\\mathcal D\\) with \\(A\\cap B\\in\\mathcal D\\) again form a Dynkin system, and the preceding conclusion says that they contain \\(\\mathcal P\\). Thus \\(\\mathcal D\\) is closed under finite intersections. Nested differences give complements, and finite intersections give finite unions. Disjointifying a countable union then shows that \\(\\mathcal D\\) is a sigma-algebra. Hence \\(\\mathcal D=\\sigma(\\mathcal P)\\).\n\nFor the two measures, the sets on which their values agree form a Dynkin system: the value at \\(X\\) agrees, subtraction is valid for nested sets because the measures are finite, and countable additivity handles disjoint unions. This system contains \\(\\mathcal P\\), hence \\(\\sigma(\\mathcal P)\\). \\(\\square\\)\n\n**Theorem (Arbitrary products of probability spaces).** Let \\((X_i,\\Sigma_i,\\mu_i)_{i\\in I}\\) be any family of probability spaces. On \\(X=\\prod_{i\\in I}X_i\\), let \\(\\Sigma\\) be the sigma-algebra generated by the coordinate maps. There is a unique probability measure \\(\\mu\\) on \\(\\Sigma\\) such that\n\\[\n\\begin{gathered}\n\\mu\\{x:x_i\\in E_i\\text{ for }i\\in F\\}\n=\\prod_{i\\in F}\\mu_i(E_i),\\\\\nF\\subseteq I\\text{ finite},\\qquad E_i\\in\\Sigma_i.\n\\end{gathered}\n\\]\nIt has a complete extension obtained by Carathéodory's construction. Each coordinate map has distribution \\(\\mu_i\\). No cardinality, separability or completeness condition is imposed on the factors.\n\n*Proof.* A *rectangle cylinder* is the empty set or a set\n\\[\nC=\\{x:x_i\\in E_i\\text{ for }i\\in F\\},\n\\]\nwhere \\(F\\subseteq I\\) is finite and \\(E_i\\in\\Sigma_i\\).\nGive it weight \\(w(C)=\\prod_{i\\in F}\\mu_i(E_i)\\), with \\(w(\\varnothing)=0\\) and the empty product equal to one. This is well defined. For a nonempty cylinder, its projection onto each constrained coordinate is exactly \\(E_i\\), since points can be chosen in all the other nonempty factors. Adding a redundant full coordinate changes no weight. An empty cylinder has an empty constrained factor, and therefore weight zero.\n\n**A cover cannot cost less than one.** Suppose rectangle cylinders \\(C_n\\) cover \\(X\\). We claim \\(\\sum_nw(C_n)\\ge1\\). Only countably many coordinates occur among their finite constraint sets. If none occur, each \\(C_n\\) is empty or \\(X\\), and the claim follows. Otherwise enumerate these coordinates as \\(j_1,j_2,\\ldots\\), stopping after the last coordinate when the set is finite. Write \\(E_{n,j}\\) for the constraint of \\(C_n\\) at \\(j\\), taking \\(E_{n,j}=X_j\\) at unconstrained coordinates.\n\nLet \\(F_n\\) be the finite constraint set chosen for \\(C_n\\). Assume for contradiction that \\(S_0=\\sum_nw(C_n)<1\\). After selecting \\(x_{j_1},\\ldots,x_{j_k}\\), put\n\\[\n\\begin{gathered}\nJ_k=\\{j_1,\\ldots,j_k\\},\\\\\na_{n,k}=\\prod_{j\\in F_n\\setminus J_k}\\mu_j(E_{n,j}),\\\\\nS_k=\\sum_n a_{n,k}\n\\prod_{\\ell\\le k}1_{E_{n,j_\\ell}}(x_{j_\\ell}).\n\\end{gathered}\n\\]\nThe product defining \\(a_{n,k}\\) is finite. Regard \\(S_{k+1}\\) as a nonnegative measurable function of the next coordinate. Monotone convergence, applied to the series, gives\n\\[\n\\int_{X_{j_{k+1}}}S_{k+1}(t)\\,d\\mu_{j_{k+1}}(t)=S_k.\n\\]\nIf \\(S_k<1\\), some \\(t\\) has \\(S_{k+1}(t)<1\\); otherwise its integral over this probability space would be at least one. Select such a \\(t\\) and continue.\n\nOnce all the constrained coordinates of a particular \\(C_n\\) have been selected, membership in \\(C_n\\) would make its summand equal to one. This contradicts \\(S_k<1\\). The selected coordinates therefore avoid every \\(C_n\\). Choose arbitrary values at all remaining coordinates. The resulting point is outside their union, contradicting the cover. This proves the claim for finite as well as countably infinite sets of relevant coordinates.\n\n**An outer measure with the correct cylinder values.** For every \\(A\\subseteq X\\), define\n\\[\nm^*(A)=\\inf\\sum_nw(C_n),\n\\]\nwhere the infimum runs over all countable covers of \\(A\\) by rectangle cylinders \\(C_n\\).\nThis is an outer measure. Monotonicity is immediate. The empty set has a zero-cost cover, and combining covers whose errors are bounded by \\(\\varepsilon2^{-n}\\) proves countable subadditivity. Every set has a cover of cost one, since \\(X\\) itself is a cylinder. The cover claim gives \\(m^*(X)=1\\).\n\nWe also have \\(m^*(C)=w(C)\\) for each cylinder \\(C\\). The upper bound comes from the one-member cover. It settles the case \\(w(C)=0\\). If \\(w(C)>0\\), replace the finitely many constrained factor spaces by \\(E_i\\) with their relative sigma-algebras and conditional probabilities\n\\[\n\\mu_i^C(B)=\\frac{\\mu_i(B)}{\\mu_i(E_i)}\n\\]\nfor relatively measurable \\(B\\subseteq E_i\\).\nTheir Cartesian product is \\(C\\). A cover of \\(C\\) by cylinders \\(D_n\\) induces a cylinder cover \\(D_n\\cap C\\) in these conditional spaces. Its \\(n\\)-th weight is\n\\[\n\\frac{w(D_n\\cap C)}{w(C)}\n\\le\\frac{w(D_n)}{w(C)}.\n\\]\nThe cover claim, now for the conditional probability spaces, yields \\(w(C)\\le\\sum_nw(D_n)\\). Taking the infimum proves the lower bound.\n\n**Coordinate sets are measurable.** Fix \\(i\\) and \\(E\\in\\Sigma_i\\), and put \\(P=\\{x:x_i\\in E\\}\\). Every cylinder \\(C\\) splits into the cylinders \\(C\\cap P\\) and \\(C\\setminus P\\), with\n\\[\nw(C\\cap P)+w(C\\setminus P)=w(C).\n\\]\nApply this splitting to any cylinder cover of \\(A\\). It gives\n\\[\nm^*(A\\cap P)+m^*(A\\setminus P)\n\\le\\sum_nw(C_n).\n\\]\nTake the infimum over covers. The reverse inequality is outer subadditivity. Thus \\(P\\) satisfies Carathéodory's condition. Theorem 1.1 of the measure-tools lesson supplies a complete measure on all sets satisfying that condition. Its domain contains \\(\\Sigma\\), so its restriction \\(\\mu\\) to \\(\\Sigma\\) is a probability measure with the asserted cylinder values. These values also give the coordinate distributions.\n\nRectangle cylinders form a family containing \\(X\\), closed under finite intersections, and generating \\(\\Sigma\\). The uniqueness lemma proves uniqueness on \\(\\Sigma\\). The theorem does not assert that \\(\\Sigma\\) itself is complete; the Carathéodory extension provides completeness on its larger domain. \\(\\square\\)\n\n**Example (independent signs).** Give each \\(X_i=\\{-1,1\\}\\) its probability assigning mass \\(1/2\\) to each point. The coordinate functions \\(r_i(x)=x_i\\) have \\(L^2\\) norm one and integral zero. For \\(i\\ne j\\), the four two-coordinate cylinders have mass \\(1/4\\), so \\(\\int r_ir_j\\,d\\mu=0\\). An uncountable index set therefore gives an uncountable orthonormal family in \\(L^2(\\mu)\\). Exercise 9.9 extends this computation to factors with unequal probabilities and proves the cyclic and separating assertions.\n\n**Exercise (medium): countable coordinate dependence.** Show that every set in \\(\\Sigma\\) belongs to the sigma-algebra generated by some countable set of coordinates. Deduce that every complex \\(\\Sigma\\)-measurable function depends on countably many coordinates.\n\n*Solution.* The sets with the stated property form a sigma-algebra: complements retain the same coordinates, and a countable union uses the union of countably many countable coordinate sets. This sigma-algebra contains every coordinate cylinder, so it contains \\(\\Sigma\\). For a measurable complex function, apply the set assertion to the inverse images of a countable base of \\(\\mathbb C\\), and take the union \\(J\\) of their coordinate sets. If two points have the same coordinates in \\(J\\), they lie in exactly the same inverse images of these basic open sets. Their function values must be equal, since the base separates distinct points. Thus the function factors through the projection to \\(X_J\\). This assertion concerns \\(\\Sigma\\); completion can add null subsets involving further coordinates.\n\n\n**Exercise 9.9** (medium; a cyclic and separating vector in a large Hilbert space). Let \\((\\Gamma_i,\\mu_i)_{i\\in I}\\) be an uncountable family of probability spaces, let \\((\\Gamma,\\mu)\\) be the product space, \\(H=L^2(\\Gamma,\\mu)\\), and let \\(M\\) be the von Neumann algebra generated by the multiplication operators \\(m_f\\), for bounded measurable \\(f\\). Show that the function equal to \\(1\\) everywhere is cyclic and separating for \\(M\\). Suppose moreover that uncountably many factors are *nontrivial*: for uncountably many \\(i\\) there is a measurable \\(B_i\\subseteq\\Gamma_i\\) with \\(0<\\mu_i(B_i)<1\\). Show that \\(H\\) is then not separable. Some hypothesis of this kind is needed: if every \\(\\Gamma_i\\) is a one-point space, then \\(\\Gamma\\) is one point and \\(H=\\mathbb C\\) is separable.\n\nThe one-point example in the question proves that an uncountable index set alone does not imply nonseparability.\n\n*Solution.* The [probability product theorem just proved](#oa-fnd-bi-18) supplies the product measure on the coordinate sigma-algebra, with no restrictions on the factors. *Cyclic:* \\([M1]\\) contains every bounded measurable \\(f=m_f1\\). These are dense in \\(L^2\\): for \\(g\\in L^2\\), \\(g1_{\\{|g|\\le k\\}}\\to g\\) in \\(L^2\\) by dominated convergence. *Separating:* the operators \\(m_f\\) commute with each other, so \\(\\{m_f\\}\\subseteq\\{m_f\\}'=\\{m_f\\}'''=M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(2)). Hence \\([M'1]\\supseteq[\\{m_f\\}1]=L^2\\), so \\(1\\) is cyclic for \\(M'\\) and separating for \\(M=M''\\) by [Proposition 9.2](#oa-fnd-bi-12)(1). (In fact \\(M=\\{m_f\\}\\), by [Theorem 9.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09), since \\(\\mu\\) is finite.) *Not separable, under the nontriviality hypothesis:* let \\(J\\) be the uncountable set of nontrivial indices. For \\(i\\in J\\) let \\(c_i=\\mu_i(B_i)\\), let \\(\\gamma_i:\\Gamma\\to\\Gamma_i\\) be the coordinate map, and\n\\[\ng_i=\\frac{1_{B_i}\\circ\\gamma_i-c_i}{\\sqrt{c_i(1-c_i)}} .\n\\]\nThen \\[\n\\begin{gathered}\n\\int|g_i|^2\\,d\\mu\\\\\n=\\big(c_i(1-c_i)^2+(1-c_i)c_i^2\\big)/\\big(c_i(1-c_i)\\big)\\\\\n=1\n\\end{gathered}\n\\] and \\(\\int g_i\\,d\\mu=0\\). For \\(i\\neq j\\) the coordinates are independent under the product measure, so \\(\\int g_i\\bar g_j\\,d\\mu=\\int g_i\\,d\\mu\\int\\bar g_j\\,d\\mu=0\\). So \\((g_i)_{i\\in J}\\) is an uncountable orthonormal family. Distinct members are at distance \\(\\sqrt2\\), so the open balls of radius \\(\\sqrt2/2\\) around them are disjoint, and no countable set can meet all of them. Hence \\(H\\) is not separable. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-BI-19",
      "unit": "the-double-commutant-theorem",
      "name": "Probability products for the large-Hilbert-space example",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "source": "src/the-double-commutant-theorem.md",
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      "anchor": "oa-fnd-bi-19",
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      "full_conditions_and_proof": "### Probability products for the large-Hilbert-space example\n\nExercise 9.9 uses independent coordinates indexed by an arbitrary set. A finite product is insufficient when uncountably many coordinates must supply orthogonal vectors. Here we construct the required measure without assuming that the factor spaces are standard Borel, or even that they have topologies.\n\nThe measure tools are [Carathéodory's theorem](measure-and-hilbert-space-tools.md#1-from-an-outer-measure-to-a-measure) and [monotone convergence](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions), Theorems 1.1 and 2.1 of *Measure and Hilbert space tools for Haar integration*. In particular, monotone convergence gives \\(\\int\\sum_n f_n=\\sum_n\\int f_n\\) for nonnegative measurable functions. Choice is used to select points in nonempty factor spaces. Fremlin's [§254](https://www1.essex.ac.uk/maths/people/fremlin/cont25.htm) is a reference for the classical product theorem.\n\n**Lemma (Uniqueness from a generating family).** Let \\(\\mathcal P\\) be a family of subsets of \\(X\\) containing \\(X\\) and closed under finite intersections. Two finite measures on \\(\\sigma(\\mathcal P)\\) that agree on \\(\\mathcal P\\) agree everywhere.\n\n*Proof.* A *Dynkin system* contains \\(X\\), is closed under the difference \\(B\\setminus A\\) when \\(A\\subseteq B\\) are members, and is closed under countable disjoint unions. Let \\(\\mathcal D\\) be the smallest Dynkin system containing \\(\\mathcal P\\), obtained by intersecting all such systems.\n\nFix \\(P\\in\\mathcal P\\). The members \\(B\\in\\mathcal D\\) for which \\(P\\cap B\\in\\mathcal D\\) form a Dynkin system. Indeed, they contain \\(X\\); intersections with \\(P\\) preserve nested differences and disjoint unions. This system contains \\(\\mathcal P\\), so it contains \\(\\mathcal D\\). Now fix any \\(B\\in\\mathcal D\\). The members \\(A\\in\\mathcal D\\) with \\(A\\cap B\\in\\mathcal D\\) again form a Dynkin system, and the preceding conclusion says that they contain \\(\\mathcal P\\). Thus \\(\\mathcal D\\) is closed under finite intersections. Nested differences give complements, and finite intersections give finite unions. Disjointifying a countable union then shows that \\(\\mathcal D\\) is a sigma-algebra. Hence \\(\\mathcal D=\\sigma(\\mathcal P)\\).\n\nFor the two measures, the sets on which their values agree form a Dynkin system: the value at \\(X\\) agrees, subtraction is valid for nested sets because the measures are finite, and countable additivity handles disjoint unions. This system contains \\(\\mathcal P\\), hence \\(\\sigma(\\mathcal P)\\). \\(\\square\\)\n\n**Theorem (Arbitrary products of probability spaces).** Let \\((X_i,\\Sigma_i,\\mu_i)_{i\\in I}\\) be any family of probability spaces. On \\(X=\\prod_{i\\in I}X_i\\), let \\(\\Sigma\\) be the sigma-algebra generated by the coordinate maps. There is a unique probability measure \\(\\mu\\) on \\(\\Sigma\\) such that\n\\[\n\\begin{gathered}\n\\mu\\{x:x_i\\in E_i\\text{ for }i\\in F\\}\n=\\prod_{i\\in F}\\mu_i(E_i),\\\\\nF\\subseteq I\\text{ finite},\\qquad E_i\\in\\Sigma_i.\n\\end{gathered}\n\\]\nIt has a complete extension obtained by Carathéodory's construction. Each coordinate map has distribution \\(\\mu_i\\). No cardinality, separability or completeness condition is imposed on the factors.\n\n*Proof.* A *rectangle cylinder* is the empty set or a set\n\\[\nC=\\{x:x_i\\in E_i\\text{ for }i\\in F\\},\n\\]\nwhere \\(F\\subseteq I\\) is finite and \\(E_i\\in\\Sigma_i\\).\nGive it weight \\(w(C)=\\prod_{i\\in F}\\mu_i(E_i)\\), with \\(w(\\varnothing)=0\\) and the empty product equal to one. This is well defined. For a nonempty cylinder, its projection onto each constrained coordinate is exactly \\(E_i\\), since points can be chosen in all the other nonempty factors. Adding a redundant full coordinate changes no weight. An empty cylinder has an empty constrained factor, and therefore weight zero.\n\n**A cover cannot cost less than one.** Suppose rectangle cylinders \\(C_n\\) cover \\(X\\). We claim \\(\\sum_nw(C_n)\\ge1\\). Only countably many coordinates occur among their finite constraint sets. If none occur, each \\(C_n\\) is empty or \\(X\\), and the claim follows. Otherwise enumerate these coordinates as \\(j_1,j_2,\\ldots\\), stopping after the last coordinate when the set is finite. Write \\(E_{n,j}\\) for the constraint of \\(C_n\\) at \\(j\\), taking \\(E_{n,j}=X_j\\) at unconstrained coordinates.\n\nLet \\(F_n\\) be the finite constraint set chosen for \\(C_n\\). Assume for contradiction that \\(S_0=\\sum_nw(C_n)<1\\). After selecting \\(x_{j_1},\\ldots,x_{j_k}\\), put\n\\[\n\\begin{gathered}\nJ_k=\\{j_1,\\ldots,j_k\\},\\\\\na_{n,k}=\\prod_{j\\in F_n\\setminus J_k}\\mu_j(E_{n,j}),\\\\\nS_k=\\sum_n a_{n,k}\n\\prod_{\\ell\\le k}1_{E_{n,j_\\ell}}(x_{j_\\ell}).\n\\end{gathered}\n\\]\nThe product defining \\(a_{n,k}\\) is finite. Regard \\(S_{k+1}\\) as a nonnegative measurable function of the next coordinate. Monotone convergence, applied to the series, gives\n\\[\n\\int_{X_{j_{k+1}}}S_{k+1}(t)\\,d\\mu_{j_{k+1}}(t)=S_k.\n\\]\nIf \\(S_k<1\\), some \\(t\\) has \\(S_{k+1}(t)<1\\); otherwise its integral over this probability space would be at least one. Select such a \\(t\\) and continue.\n\nOnce all the constrained coordinates of a particular \\(C_n\\) have been selected, membership in \\(C_n\\) would make its summand equal to one. This contradicts \\(S_k<1\\). The selected coordinates therefore avoid every \\(C_n\\). Choose arbitrary values at all remaining coordinates. The resulting point is outside their union, contradicting the cover. This proves the claim for finite as well as countably infinite sets of relevant coordinates.\n\n**An outer measure with the correct cylinder values.** For every \\(A\\subseteq X\\), define\n\\[\nm^*(A)=\\inf\\sum_nw(C_n),\n\\]\nwhere the infimum runs over all countable covers of \\(A\\) by rectangle cylinders \\(C_n\\).\nThis is an outer measure. Monotonicity is immediate. The empty set has a zero-cost cover, and combining covers whose errors are bounded by \\(\\varepsilon2^{-n}\\) proves countable subadditivity. Every set has a cover of cost one, since \\(X\\) itself is a cylinder. The cover claim gives \\(m^*(X)=1\\).\n\nWe also have \\(m^*(C)=w(C)\\) for each cylinder \\(C\\). The upper bound comes from the one-member cover. It settles the case \\(w(C)=0\\). If \\(w(C)>0\\), replace the finitely many constrained factor spaces by \\(E_i\\) with their relative sigma-algebras and conditional probabilities\n\\[\n\\mu_i^C(B)=\\frac{\\mu_i(B)}{\\mu_i(E_i)}\n\\]\nfor relatively measurable \\(B\\subseteq E_i\\).\nTheir Cartesian product is \\(C\\). A cover of \\(C\\) by cylinders \\(D_n\\) induces a cylinder cover \\(D_n\\cap C\\) in these conditional spaces. Its \\(n\\)-th weight is\n\\[\n\\frac{w(D_n\\cap C)}{w(C)}\n\\le\\frac{w(D_n)}{w(C)}.\n\\]\nThe cover claim, now for the conditional probability spaces, yields \\(w(C)\\le\\sum_nw(D_n)\\). Taking the infimum proves the lower bound.\n\n**Coordinate sets are measurable.** Fix \\(i\\) and \\(E\\in\\Sigma_i\\), and put \\(P=\\{x:x_i\\in E\\}\\). Every cylinder \\(C\\) splits into the cylinders \\(C\\cap P\\) and \\(C\\setminus P\\), with\n\\[\nw(C\\cap P)+w(C\\setminus P)=w(C).\n\\]\nApply this splitting to any cylinder cover of \\(A\\). It gives\n\\[\nm^*(A\\cap P)+m^*(A\\setminus P)\n\\le\\sum_nw(C_n).\n\\]\nTake the infimum over covers. The reverse inequality is outer subadditivity. Thus \\(P\\) satisfies Carathéodory's condition. Theorem 1.1 of the measure-tools lesson supplies a complete measure on all sets satisfying that condition. Its domain contains \\(\\Sigma\\), so its restriction \\(\\mu\\) to \\(\\Sigma\\) is a probability measure with the asserted cylinder values. These values also give the coordinate distributions.\n\nRectangle cylinders form a family containing \\(X\\), closed under finite intersections, and generating \\(\\Sigma\\). The uniqueness lemma proves uniqueness on \\(\\Sigma\\). The theorem does not assert that \\(\\Sigma\\) itself is complete; the Carathéodory extension provides completeness on its larger domain. \\(\\square\\)\n\n**Example (independent signs).** Give each \\(X_i=\\{-1,1\\}\\) its probability assigning mass \\(1/2\\) to each point. The coordinate functions \\(r_i(x)=x_i\\) have \\(L^2\\) norm one and integral zero. For \\(i\\ne j\\), the four two-coordinate cylinders have mass \\(1/4\\), so \\(\\int r_ir_j\\,d\\mu=0\\). An uncountable index set therefore gives an uncountable orthonormal family in \\(L^2(\\mu)\\). Exercise 9.9 extends this computation to factors with unequal probabilities and proves the cyclic and separating assertions.\n\n**Exercise (medium): countable coordinate dependence.** Show that every set in \\(\\Sigma\\) belongs to the sigma-algebra generated by some countable set of coordinates. Deduce that every complex \\(\\Sigma\\)-measurable function depends on countably many coordinates.\n\n*Solution.* The sets with the stated property form a sigma-algebra: complements retain the same coordinates, and a countable union uses the union of countably many countable coordinate sets. This sigma-algebra contains every coordinate cylinder, so it contains \\(\\Sigma\\). For a measurable complex function, apply the set assertion to the inverse images of a countable base of \\(\\mathbb C\\), and take the union \\(J\\) of their coordinate sets. If two points have the same coordinates in \\(J\\), they lie in exactly the same inverse images of these basic open sets. Their function values must be equal, since the base separates distinct points. Thus the function factors through the projection to \\(X_J\\). This assertion concerns \\(\\Sigma\\); completion can add null subsets involving further coordinates.\n\n\n**Exercise 9.9** (medium; a cyclic and separating vector in a large Hilbert space). Let \\((\\Gamma_i,\\mu_i)_{i\\in I}\\) be an uncountable family of probability spaces, let \\((\\Gamma,\\mu)\\) be the product space, \\(H=L^2(\\Gamma,\\mu)\\), and let \\(M\\) be the von Neumann algebra generated by the multiplication operators \\(m_f\\), for bounded measurable \\(f\\). Show that the function equal to \\(1\\) everywhere is cyclic and separating for \\(M\\). Suppose moreover that uncountably many factors are *nontrivial*: for uncountably many \\(i\\) there is a measurable \\(B_i\\subseteq\\Gamma_i\\) with \\(0<\\mu_i(B_i)<1\\). Show that \\(H\\) is then not separable. Some hypothesis of this kind is needed: if every \\(\\Gamma_i\\) is a one-point space, then \\(\\Gamma\\) is one point and \\(H=\\mathbb C\\) is separable.\n\nThe one-point example in the question proves that an uncountable index set alone does not imply nonseparability.\n\n*Solution.* The [probability product theorem just proved](#oa-fnd-bi-18) supplies the product measure on the coordinate sigma-algebra, with no restrictions on the factors. *Cyclic:* \\([M1]\\) contains every bounded measurable \\(f=m_f1\\). These are dense in \\(L^2\\): for \\(g\\in L^2\\), \\(g1_{\\{|g|\\le k\\}}\\to g\\) in \\(L^2\\) by dominated convergence. *Separating:* the operators \\(m_f\\) commute with each other, so \\(\\{m_f\\}\\subseteq\\{m_f\\}'=\\{m_f\\}'''=M'\\) ([Proposition 2.1](#oa-fnd-bi-02)(2)). Hence \\([M'1]\\supseteq[\\{m_f\\}1]=L^2\\), so \\(1\\) is cyclic for \\(M'\\) and separating for \\(M=M''\\) by [Proposition 9.2](#oa-fnd-bi-12)(1). (In fact \\(M=\\{m_f\\}\\), by [Theorem 9.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09), since \\(\\mu\\) is finite.) *Not separable, under the nontriviality hypothesis:* let \\(J\\) be the uncountable set of nontrivial indices. For \\(i\\in J\\) let \\(c_i=\\mu_i(B_i)\\), let \\(\\gamma_i:\\Gamma\\to\\Gamma_i\\) be the coordinate map, and\n\\[\ng_i=\\frac{1_{B_i}\\circ\\gamma_i-c_i}{\\sqrt{c_i(1-c_i)}} .\n\\]\nThen \\[\n\\begin{gathered}\n\\int|g_i|^2\\,d\\mu\\\\\n=\\big(c_i(1-c_i)^2+(1-c_i)c_i^2\\big)/\\big(c_i(1-c_i)\\big)\\\\\n=1\n\\end{gathered}\n\\] and \\(\\int g_i\\,d\\mu=0\\). For \\(i\\neq j\\) the coordinates are independent under the product measure, so \\(\\int g_i\\bar g_j\\,d\\mu=\\int g_i\\,d\\mu\\int\\bar g_j\\,d\\mu=0\\). So \\((g_i)_{i\\in J}\\) is an uncountable orthonormal family. Distinct members are at distance \\(\\sqrt2\\), so the open balls of radius \\(\\sqrt2/2\\) around them are disjoint, and no countable set can meet all of them. Hence \\(H\\) is not separable. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-KD-01",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "1. Bounded and monotone nets",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "anchor": "oa-fnd-kd-01",
      "proof_locus": {
        "line": 35,
        "through_line": 71
      },
      "full_conditions_and_proof": "## 1. Bounded and monotone nets\n\nMost arguments in this lesson pass to limits along nets. Two facts make this possible: products behave well along bounded nets, and bounded monotone nets always converge.\n\n**Lemma 1.1** (bounded nets). Let \\((x_i)\\) and \\((y_i)\\) be nets in \\(B(H)\\) over the same directed set, with \\(\\sup_i\\|x_i\\|<\\infty\\).\n\n1. If \\(x_i\\to x\\) and \\(y_i\\to y\\) strongly, then \\(x_iy_i\\to xy\\) strongly.\n2. If moreover \\(\\sup_i\\|y_i\\|<\\infty\\), and \\(x_i\\to x\\) and \\(y_i\\to y\\) strongly\\(^*\\), then \\(x_iy_i\\to xy\\) strongly\\(^*\\).\n3. A bounded net that converges strongly also converges \\(\\sigma\\)-strongly and \\(\\sigma\\)-weakly, and a bounded net that converges strongly\\(^*\\) also converges \\(\\sigma\\)-strongly\\(^*\\). In particular \\(\\varphi(x_i)\\to\\varphi(x)\\) for every normal functional \\(\\varphi\\) on a von Neumann algebra that contains the net.\n\n**Proof.** (1) For \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\|(x_iy_i-xy)\\xi\\|\\\\\n\\le\\|x_i\\|\\,\\|(y_i-y)\\xi\\|+\\|(x_i-x)y\\xi\\|,\n\\end{gathered}\n\\]\nand both terms tend to \\(0\\). (2) By (1), \\(x_iy_i\\to xy\\) strongly. The net \\((y_i^*)\\) is bounded, \\(y_i^*\\to y^*\\) and \\(x_i^*\\to x^*\\) strongly, so (1) gives \\((x_iy_i)^*=y_i^*x_i^*\\to y^*x^*\\) strongly. (3) This is [Lemma 1.2(d) of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-01), together with the fact that the \\(\\sigma\\)-strong topology is finer than the \\(\\sigma\\)-weak one. \\(\\square\\)\n\n**Lemma 1.2** (closed sets). For \\(r\\ge0\\), the ball \\(rS\\), the set \\(B(H)_h\\) and the cone \\(B(H)_+\\) are closed in each of the six topologies. So is the set \\(\\{x:x\\ge c\\}\\) for a fixed self-adjoint \\(c\\).\n\n**Proof.** The weak topology is the coarsest of the six, so it suffices to show weak closedness. Let \\(x_i\\to x\\) weakly. If \\(\\|x_i\\|\\le r\\), then \\(|\\langle x\\xi,\\eta\\rangle|=\\lim|\\langle x_i\\xi,\\eta\\rangle|\\le r\\|\\xi\\|\\|\\eta\\|\\), so \\(\\|x\\|\\le r\\). If \\(x_i=x_i^*\\), then \\(\\langle x\\xi,\\eta\\rangle=\\lim\\langle\\xi,x_i\\eta\\rangle=\\langle\\xi,x\\eta\\rangle\\). If \\(x_i\\ge c\\), then \\(\\langle(x-c)\\xi,\\xi\\rangle=\\lim\\langle(x_i-c)\\xi,\\xi\\rangle\\ge0\\). \\(\\square\\)\n\n**Theorem 1.3** (Vigier's theorem). Let \\((x_i)\\) be an increasing net of self-adjoint operators with \\(\\sup_i\\|x_i\\|<\\infty\\). Then \\((x_i)\\) converges strongly to a self-adjoint operator \\(x\\), and \\(x\\) is the least upper bound of the net: \\(x_i\\le x\\) for all \\(i\\), and \\(x\\le y\\) for every self-adjoint \\(y\\) with \\(x_i\\le y\\) for all \\(i\\). If all \\(x_i\\) lie in a von Neumann algebra \\(M\\), so does \\(x\\). Decreasing nets behave in the same way, with greatest lower bounds.\n\n**Proof.** Let \\(C=\\sup_i\\|x_i\\|\\). For each \\(\\xi\\), the numbers \\(\\langle x_i\\xi,\\xi\\rangle\\) increase and are bounded by \\(C\\|\\xi\\|^2\\), so they converge. By polarization, \\(B(\\xi,\\eta)=\\lim_i\\langle x_i\\xi,\\eta\\rangle\\) exists for all \\(\\xi,\\eta\\). It is a sesquilinear form with \\(|B(\\xi,\\eta)|\\le C\\|\\xi\\|\\|\\eta\\|\\) and \\(B(\\eta,\\xi)=\\overline{B(\\xi,\\eta)}\\), so \\(B(\\xi,\\eta)=\\langle x\\xi,\\eta\\rangle\\) for a self-adjoint \\(x\\) with \\(\\|x\\|\\le C\\). Since \\(\\langle x_i\\xi,\\xi\\rangle\\) increases to \\(\\langle x\\xi,\\xi\\rangle\\), we get \\(x_i\\le x\\) for all \\(i\\); and if \\(x_i\\le y\\) for all \\(i\\), then \\(\\langle x\\xi,\\xi\\rangle=\\lim\\langle x_i\\xi,\\xi\\rangle\\le\\langle y\\xi,\\xi\\rangle\\).\n\nFor strong convergence put \\(d_i=x-x_i\\). Then \\(d_i\\ge0\\) and \\(\\|d_i\\|\\le2C\\). For a positive operator \\(d\\), \\(d^2\\le\\|d\\|d\\), because \\(t^2\\le\\|d\\|t\\) on the interval \\([0,\\|d\\|]\\), which contains the spectrum of \\(d\\). Hence\n\\[\n\\|d_i\\xi\\|^2=\\langle d_i^2\\xi,\\xi\\rangle\\le2C\\langle d_i\\xi,\\xi\\rangle\\to0 .\n\\]\nA von Neumann algebra is strongly closed, so \\(x\\in M\\) when all \\(x_i\\) lie in \\(M\\). For a decreasing net apply this to \\((-x_i)\\). \\(\\square\\)\n\n**Corollary 1.4** (monotone nets of projections). An increasing net of projections \\((p_i)\\) converges strongly to \\(\\bigvee_ip_i\\), and a decreasing net of projections converges strongly to \\(\\bigwedge_ip_i\\). The limits lie in every von Neumann algebra that contains the net.\n\n**Proof.** Let \\((p_i)\\) increase, with strong limit \\(p\\) (Theorem 1.3). By Lemma 1.1, \\(p_i=p_i^2\\to p^2\\) strongly, so \\(p=p^2\\), and \\(p\\) is a projection. From \\(p_i\\le p\\) we get \\(p_iH\\subseteq pH\\) for all \\(i\\). Every vector \\(p\\xi=\\lim p_i\\xi\\) lies in the closed span of the ranges \\(p_iH\\). So \\(pH\\) is exactly that closed span. For a decreasing net, apply this to the increasing net \\((1-p_i)\\): the limit is \\(1-\\bigvee_i(1-p_i)=\\bigwedge_ip_i\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-KD-02",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "2. Adjoints along nets of normal operators",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
      "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "anchor": "oa-fnd-kd-02",
      "proof_locus": {
        "line": 72,
        "through_line": 112
      },
      "full_conditions_and_proof": "## 2. Adjoints along nets of normal operators\n\nThe adjoint is not strongly continuous on \\(B(H)\\) when \\(\\dim H=\\infty\\). It is strongly continuous on the set of normal operators. We prove a slightly stronger statement, in which the approximating operators need only be hyponormal.\n\n**Definition 2.1.** An operator \\(a\\) is *hyponormal* if \\(\\|a^*\\xi\\|\\le\\|a\\xi\\|\\) for every \\(\\xi\\in H\\). Normal operators are hyponormal, and so are isometries, since \\(\\|v^*\\xi\\|\\le\\|\\xi\\|=\\|v\\xi\\|\\). An operator \\(a\\) is normal exactly when \\(\\|a^*\\xi\\|=\\|a\\xi\\|\\) for all \\(\\xi\\): by polarization, this says \\(\\langle a^*a\\xi,\\xi\\rangle=\\langle aa^*\\xi,\\xi\\rangle\\) for all \\(\\xi\\).\n\n**Lemma 2.2.** For all \\(a,b\\in B(H)\\) and \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\|(a^*-b^*)\\xi\\|^2\\\\\n=\\|a^*\\xi\\|^2-\\|b^*\\xi\\|^2\\\\\n-2\\operatorname{Re}\\langle(a-b)b^*\\xi,\\xi\\rangle .\n\\end{gathered}\n\\tag{2.1}\n\\]\n\n**Proof.** Expanding the square, \\[\n\\begin{gathered}\n\\|(a^*-b^*)\\xi\\|^2\\\\\n=\\|a^*\\xi\\|^2+\\|b^*\\xi\\|^2-2\\operatorname{Re}\\langle a^*\\xi,b^*\\xi\\rangle.\n\\end{gathered}\n\\] Now \\(\\operatorname{Re}\\langle a^*\\xi,b^*\\xi\\rangle=\\operatorname{Re}\\langle\\xi,ab^*\\xi\\rangle=\\operatorname{Re}\\langle ab^*\\xi,\\xi\\rangle\\), and \\(\\langle ab^*\\xi,\\xi\\rangle=\\langle(a-b)b^*\\xi,\\xi\\rangle+\\|b^*\\xi\\|^2\\). Substituting gives (2.1). \\(\\square\\)\n\n**Theorem 2.3.** Let \\((a_i)\\) be a net in \\(B(H)\\) that converges strongly to \\(b\\).\n\n1. If every \\(a_i\\) is hyponormal, then \\(b\\) is hyponormal.\n2. If every \\(a_i\\) is hyponormal and \\(b\\) is normal, then \\(a_i^*\\to b^*\\) strongly. So \\(a_i\\to b\\) strongly\\(^*\\).\n3. If every \\(a_i\\) is normal, then \\(a_i^*\\to b^*\\) strongly if and only if \\(b\\) is normal.\n\n**Proof.** (1) The adjoint is weakly continuous, so \\(a_i^*\\to b^*\\) weakly. Then \\(\\|b^*\\xi\\|^2=\\lim_i\\langle a_i^*\\xi,b^*\\xi\\rangle\\le\\liminf_i\\|a_i^*\\xi\\|\\,\\|b^*\\xi\\|\\), hence \\(\\|b^*\\xi\\|\\le\\liminf_i\\|a_i^*\\xi\\|\\le\\liminf_i\\|a_i\\xi\\|=\\|b\\xi\\|\\).\n\n(2) In (2.1) with \\(a=a_i\\), the last term tends to \\(0\\), because \\((a_i-b)\\eta\\to0\\) for the fixed vector \\(\\eta=b^*\\xi\\). Hyponormality gives \\(\\|a_i^*\\xi\\|^2\\le\\|a_i\\xi\\|^2\\to\\|b\\xi\\|^2=\\|b^*\\xi\\|^2\\). So \\(\\limsup_i\\|(a_i^*-b^*)\\xi\\|^2\\le0\\).\n\n(3) If \\(b\\) is normal, use (2). Conversely, if \\(a_i^*\\to b^*\\) strongly, then \\(\\|b^*\\xi\\|=\\lim\\|a_i^*\\xi\\|=\\lim\\|a_i\\xi\\|=\\|b\\xi\\|\\) for every \\(\\xi\\), so \\(b\\) is normal. \\(\\square\\)\n\n**Corollary 2.4.** On the set of normal operators, the strong and the strong\\(^*\\) topologies coincide, and the adjoint is strongly continuous there. A net of isometries that converges strongly to a unitary converges strongly\\(^*\\).\n\n**Proof.** Both statements are Theorem 2.3(2). \\(\\square\\)\n\n**Example 2.5** (the limit must be normal). Let \\((\\xi_k)_{k\\ge1}\\) be an orthonormal sequence in \\(H\\). For \\(n\\ge1\\) let \\(u_n\\) be the unitary with \\(u_n\\xi_k=\\xi_{k+1}\\) for \\(k<n\\), \\(u_n\\xi_n=\\xi_1\\), and \\(u_n=1\\) on the orthogonal complement of \\(\\xi_1,\\dots,\\xi_n\\). Let \\(v\\) be the isometry with \\(v\\xi_k=\\xi_{k+1}\\) for all \\(k\\) and \\(v=1\\) on the orthogonal complement of all \\(\\xi_k\\). For fixed \\(k\\), \\(u_n\\xi_k=v\\xi_k\\) as soon as \\(n>k\\), and \\(u_n=v\\) on the complement. The net is bounded, so \\(u_n\\to v\\) strongly. The limit is not normal, since \\(\\|v^*\\xi_1\\|=0\\ne1=\\|v\\xi_1\\|\\). As Theorem 2.3(3) predicts, the adjoints do not converge strongly: \\(u_n^*\\xi_1=\\xi_n\\), and these vectors are at mutual distance \\(\\sqrt2\\). So a strong limit of unitaries need not be unitary, and a strong limit of normal operators need not be normal.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-KD-03",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "3. Commuting normal operators and their joint functional calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "anchor": "oa-fnd-kd-03",
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        "line": 113,
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      },
      "full_conditions_and_proof": "## 3. Commuting normal operators and their joint functional calculus\n\nThe continuity theorem of Section 5 is proved for several commuting normal operators at once. This section provides the functional calculus it needs. The first step is Fuglede's theorem, which shows that commuting normal operators also commute with each other's adjoints.\n\n**Theorem 3.1** (Fuglede's theorem). Let \\(a\\in B(H)\\) be normal and let \\(b\\in B(H)\\) commute with \\(a\\). Then \\(b\\) commutes with \\(a^*\\).\n\n\n\n**Proof.** We use the exponential \\(\\exp x=\\sum_kx^k/k!\\) of [the lesson on Banach algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14): \\(\\exp(x+y)=\\exp x\\exp y\\) when \\(xy=yx\\). The adjoint is continuous and conjugate linear, so applying it to the partial sums gives \\((\\exp x)^*=\\exp(x^*)\\). For \\(z\\in\\mathbb C\\) define\n\\[\nF(z)=\\exp(za^*)\\,b\\,\\exp(-za^*).\n\\]\nSince \\(b\\) commutes with \\(a\\), it commutes with every partial sum of \\(\\exp(\\bar za)\\), hence with \\(\\exp(\\bar za)\\), and \\(b=\\exp(-\\bar za)\\,b\\,\\exp(\\bar za)\\). The operators \\(a\\) and \\(a^*\\) commute, so\n\\[\n\\begin{gathered}\nF(z)\\\\\n=\\exp(za^*-\\bar za)\\,b\\,\\exp(\\bar za-za^*)\\\\\n=U(z)\\,b\\,U(z)^{-1},\\\\\nU(z)\\\\\n=\\exp(za^*-\\bar za).\n\\end{gathered}\n\\]\nThe operator \\(X=za^*-\\bar za\\) satisfies \\(X^*=-X\\). Hence \\(U(z)^*=\\exp(X^*)=\\exp(-X)=U(z)^{-1}\\), so \\(U(z)\\) is unitary and \\(\\|F(z)\\|=\\|b\\|\\) for every \\(z\\).\n\nMultiplying the two absolutely convergent series gives a norm-convergent power series \\(F(z)=\\sum_{m\\ge0}z^mc_m\\) for all \\(z\\), with \\(c_0=b\\) and \\(c_1=a^*b-ba^*\\). For fixed \\(\\xi,\\eta\\in H\\), the function \\(z\\mapsto\\langle F(z)\\xi,\\eta\\rangle=\\sum_mz^m\\langle c_m\\xi,\\eta\\rangle\\) is therefore holomorphic on \\(\\mathbb C\\), and it is bounded by \\(\\|b\\|\\|\\xi\\|\\|\\eta\\|\\). By Liouville's theorem it is constant, so its coefficient of \\(z\\) vanishes: \\(\\langle(a^*b-ba^*)\\xi,\\eta\\rangle=0\\). As \\(\\xi,\\eta\\) are arbitrary, \\(a^*b=ba^*\\). \\(\\square\\)\n\n**Definition 3.2.** A *commuting normal tuple* is an \\(n\\)-tuple \\(a=(a_1,\\dots,a_n)\\) of normal operators on \\(H\\) with \\(a_ja_k=a_ka_j\\) for all \\(j,k\\). By Fuglede's theorem, each \\(a_j\\) also commutes with each \\(a_k^*\\). So the C\\(^*\\)-algebra \\(C^*(1,a)\\) generated by \\(1,a_1,\\dots,a_n\\) is commutative: it is the norm closure of the polynomials in the pairwise commuting operators \\(a_j,a_j^*\\). Its characters form a compact space \\(\\Omega\\), and the *joint spectrum* of \\(a\\) is\n\\[\n\\sigma(a)=\\{(\\chi(a_1),\\dots,\\chi(a_n)):\\chi\\in\\Omega\\}\\subseteq\\mathbb C^n .\n\\]\n\n**Theorem 3.3** (the joint functional calculus). Let \\(a\\) be a commuting normal tuple on \\(H\\ne\\{0\\}\\), and let \\(\\iota_j\\) be the \\(j\\)-th coordinate function on \\(\\mathbb C^n\\).\n\n1. The map \\(\\Psi(\\chi)=(\\chi(a_1),\\dots,\\chi(a_n))\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\sigma(a)\\). So \\(\\sigma(a)\\) is a nonempty compact set. It lies in \\(\\sigma(a_1)\\times\\dots\\times\\sigma(a_n)\\), and its projection to the \\(j\\)-th coordinate is \\(\\sigma(a_j)\\).\n2. There is exactly one unital \\(*\\)-homomorphism \\(f\\mapsto f(a)\\) from \\(C(\\sigma(a))\\) into \\(B(H)\\) with \\(\\iota_j(a)=a_j\\) for all \\(j\\). It is isometric, \\(\\|f(a)\\|=\\max_{\\sigma(a)}|f|\\), and its range is \\(C^*(1,a)\\). Each \\(f(a)\\) is normal.\n3. \\(f(a)\\) commutes with every operator that commutes with \\(a_1,\\dots,a_n\\).\n4. For \\(n=1\\) this is the continuous functional calculus of a normal operator.\n\nFor a closed set \\(G\\supseteq\\sigma(a)\\) and a continuous \\(f\\) on \\(G\\) we write \\(f(a)\\) for \\((f|_{\\sigma(a)})(a)\\). If \\(f\\) is bounded, then \\(\\|f(a)\\|\\le\\sup_G|f|\\).\n\n**Proof.** (1) Characters of the commutative C\\(^*\\)-algebra \\(B=C^*(1,a)\\) are \\(*\\)-homomorphisms ([Theorem 2.1 of the lesson on C\\*-algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04)), so a character is determined by its values at \\(a_1,\\dots,a_n\\): it then knows its values at the \\(a_j^*\\), at all polynomials in them, and, being continuous, on \\(B\\). So \\(\\Psi\\) is injective. It is continuous for the weak\\(^*\\) topology, and \\(\\Omega\\) is compact, so \\(\\Psi\\) is a homeomorphism onto its image \\(\\sigma(a)\\). Since \\(H\\ne\\{0\\}\\), \\(B\\ne\\{0\\}\\) has characters, and \\(\\sigma(a)\\) is not empty. In a commutative unital Banach algebra the spectrum of an element is the set of values of the characters on it ([the Gelfand representation](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18)), and the spectrum of \\(a_j\\) in \\(B\\) is its spectrum in \\(B(H)\\) ([spectral permanence](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-06)). So \\(\\{\\chi(a_j):\\chi\\in\\Omega\\}=\\sigma(a_j)\\), which gives the last two claims.\n\n(2) By the commutative Gelfand–Naimark theorem, the Gelfand transform \\(\\mathcal G\\) is an isometric \\(*\\)-isomorphism of \\(B\\) onto \\(C(\\Omega)\\). Composition with \\(\\Psi\\) is an isometric \\(*\\)-isomorphism of \\(C(\\sigma(a))\\) onto \\(C(\\Omega)\\). So \\(f(a)=\\mathcal G^{-1}(f\\circ\\Psi)\\) defines an isometric \\(*\\)-isomorphism of \\(C(\\sigma(a))\\) onto \\(B\\), and \\(\\iota_j(a)=\\mathcal G^{-1}(\\chi\\mapsto\\chi(a_j))=a_j\\). Elements of \\(B\\) are normal, because \\(B\\) is commutative. For uniqueness, a second unital \\(*\\)-homomorphism with the same values at the \\(\\iota_j\\) agrees with the first on the polynomials in the \\(\\iota_j\\) and \\(\\bar\\iota_j\\). These are dense in \\(C(\\sigma(a))\\) by [the Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09), since they separate points and contain the constants. Both maps are contractive ([Theorem 4.2 of the lesson on C\\*-algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-12)), so they agree.\n\n(3) An operator that commutes with all \\(a_j\\) commutes with all \\(a_j^*\\) by Fuglede's theorem, hence with all of \\(B\\).\n\n(4) For \\(n=1\\) both calculi are unital \\(*\\)-homomorphisms that send \\(\\iota\\) to \\(a_1\\), so they agree by uniqueness.\n\nFor the last statement, \\(\\|f(a)\\|=\\max_{\\sigma(a)}|f|\\le\\sup_G|f|\\). \\(\\square\\)\n\n**Example 3.4.** (a) Let \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_k)\\), and let \\(a_j\\) be the diagonal operator \\(a_j\\delta_k=\\lambda^{(k)}_j\\delta_k\\) for bounded sequences \\((\\lambda^{(k)}_j)_k\\), \\(j=1,\\dots,n\\). Put \\(\\lambda^{(k)}=(\\lambda^{(k)}_1,\\dots,\\lambda^{(k)}_n)\\), and let \\(K\\) be the closure of \\(\\{\\lambda^{(k)}:k\\in\\mathbb N\\}\\). Then \\(\\sigma(a)=K\\). Indeed, \\(f\\mapsto\\) the diagonal operator with entries \\(f(\\lambda^{(k)})\\) is a unital \\(*\\)-homomorphism from \\(C(K)\\) into \\(B(H)\\). It is isometric, because the points \\(\\lambda^{(k)}\\) are dense in \\(K\\), and it sends \\(\\iota_j\\) to \\(a_j\\). Its range is therefore \\(C^*(1,a)\\), isomorphic to \\(C(K)\\), and the characters of \\(C(K)\\) are the evaluations at points of \\(K\\) ([the characters of \\(C_0(\\Omega)\\)](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04)). (b) For a projection \\(p\\ne0,1\\), the pair \\((p,1-p)\\) has joint spectrum \\(\\{(1,0),(0,1)\\}\\), a proper subset of \\(\\sigma(p)\\times\\sigma(1-p)=\\{0,1\\}^2\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-KD-04",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "4. Pointwise limits in the functional calculus",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "full_conditions_and_proof": "## 4. Pointwise limits in the functional calculus\n\nThe continuous functional calculus does not produce spectral projections. A von Neumann algebra is strongly closed, so we can instead pass to strong limits of continuous functions of an operator. This section shows that bounded pointwise convergence of the functions gives strong convergence of the operators.\n\n**Definition 4.1.** Let \\(K\\subseteq\\mathbb C\\) be compact. We write \\(\\mathcal B_1(K)\\) for the set of functions \\(\\varphi:K\\to\\mathbb C\\) for which there are continuous functions \\(\\varphi_k\\) on \\(K\\) with \\(\\sup_k\\max_K|\\varphi_k|<\\infty\\) and \\(\\varphi_k(\\lambda)\\to\\varphi(\\lambda)\\) for every \\(\\lambda\\in K\\). This set is a \\(*\\)-algebra under pointwise operations, and it contains \\(C(K)\\). Corollaries 4.3, 4.5 and 4.6 below show that it contains indicator functions of intervals, the argument function on the unit circle and binary digits.\n\n**Theorem 4.2** (bounded pointwise limits). Let \\(a\\) be a normal operator with spectrum \\(K\\). For \\(\\varphi\\in\\mathcal B_1(K)\\) and any sequence \\((\\varphi_k)\\) as in Definition 4.1, the operators \\(\\varphi_k(a)\\) converge strongly to an operator \\(\\varphi(a)\\) that depends only on \\(\\varphi\\). The map \\(\\varphi\\mapsto\\varphi(a)\\) is a \\(*\\)-homomorphism from \\(\\mathcal B_1(K)\\) into \\(B(H)\\) that extends the continuous functional calculus. It satisfies \\(\\|\\varphi(a)\\|\\le\\sup_K|\\varphi|\\), and \\(\\varphi\\ge0\\) implies \\(\\varphi(a)\\ge0\\). Every \\(\\varphi(a)\\) lies in every von Neumann algebra that contains \\(a\\), and commutes with every operator that commutes with \\(a\\).\n\n**Proof.** For \\(\\xi\\in H\\), the map \\(f\\mapsto\\langle f(a)\\xi,\\xi\\rangle\\) is a positive linear functional on \\(C(K)\\), because \\(f\\ge0\\) implies \\(f(a)\\ge0\\). By the [Riesz representation theorem](haar-measure.md#oa-fnd-hm-01) there is a finite positive Borel measure \\(\\mu_\\xi\\) on \\(K\\) with\n\\[\n\\begin{gathered}\n\\langle f(a)\\xi,\\xi\\rangle\\\\\n=\\int_Kf\\,d\\mu_\\xi\\\\\n(f\\in C(K)),\n\\end{gathered}\n\\tag{4.1}\n\\]\nand \\(\\mu_\\xi(K)=\\|\\xi\\|^2\\). Functions in \\(\\mathcal B_1(K)\\) are Borel, being pointwise limits of continuous functions.\n\nLet \\(\\varphi_k\\to\\varphi\\) as in Definition 4.1, with \\(|\\varphi_k|\\le c\\). For \\(k,l\\), by (4.1) applied to \\(|\\varphi_k-\\varphi_l|^2\\),\n\\[\n\\begin{gathered}\n\\|(\\varphi_k(a)-\\varphi_l(a))\\xi\\|^2\\\\\n=\\langle|\\varphi_k-\\varphi_l|^2(a)\\xi,\\xi\\rangle\\\\\n=\\int|\\varphi_k-\\varphi_l|^2\\,d\\mu_\\xi\\\\\n\\le2\\int|\\varphi_k-\\varphi|^2\\,d\\mu_\\xi+2\\int|\\varphi-\\varphi_l|^2\\,d\\mu_\\xi .\n\\end{gathered}\n\\]\nThe integrands tend to \\(0\\) pointwise and are bounded by \\(4c^2\\), which is integrable for the finite measure \\(\\mu_\\xi\\). By dominated convergence the right side tends to \\(0\\). So \\((\\varphi_k(a)\\xi)\\) is a Cauchy sequence. Call its limit \\(\\varphi(a)\\xi\\). The map \\(\\varphi(a)\\) is linear, and \\[\n\\begin{gathered}\n\\|\\varphi(a)\\xi\\|^2\\\\\n=\\lim\\int|\\varphi_k|^2\\,d\\mu_\\xi\\\\\n=\\int|\\varphi|^2\\,d\\mu_\\xi\\\\\n\\le(\\sup_K|\\varphi|)^2\\|\\xi\\|^2.\n\\end{gathered}\n\\] If \\(\\psi_k\\to\\varphi\\) is a second such sequence, then \\(\\|(\\varphi_k(a)-\\psi_k(a))\\xi\\|^2=\\int|\\varphi_k-\\psi_k|^2\\,d\\mu_\\xi\\to0\\), so the limit does not depend on the sequence. A constant sequence shows that the new map extends the continuous calculus.\n\nLinearity is clear. Since \\(\\bar\\varphi_k\\to\\bar\\varphi\\), we get \\[\n\\begin{gathered}\n\\langle\\bar\\varphi(a)\\xi,\\eta\\rangle\\\\\n=\\lim\\langle\\varphi_k(a)^*\\xi,\\eta\\rangle\\\\\n=\\lim\\langle\\xi,\\varphi_k(a)\\eta\\rangle\\\\\n=\\langle\\xi,\\varphi(a)\\eta\\rangle,\n\\end{gathered}\n\\] so \\(\\bar\\varphi(a)=\\varphi(a)^*\\). If \\(\\psi_k\\to\\psi\\) as well, then \\(\\varphi_k\\psi_k\\to\\varphi\\psi\\) with a uniform bound, and \\((\\varphi_k\\psi_k)(a)=\\varphi_k(a)\\psi_k(a)\\to\\varphi(a)\\psi(a)\\) strongly by Lemma 1.1. So \\((\\varphi\\psi)(a)=\\varphi(a)\\psi(a)\\). If \\(\\varphi\\ge0\\), then \\(\\langle\\varphi(a)\\xi,\\xi\\rangle=\\lim\\int\\varphi_k\\,d\\mu_\\xi=\\int\\varphi\\,d\\mu_\\xi\\ge0\\), by dominated convergence again.\n\nA von Neumann algebra \\(N\\) that contains \\(a\\) contains \\(1\\), \\(a^*\\) and all norm limits of polynomials in \\(a,a^*\\), so it contains every \\(\\varphi_k(a)\\); it is strongly closed, so it contains \\(\\varphi(a)\\). An operator that commutes with \\(a\\) commutes with \\(a^*\\) (Theorem 3.1), hence with every \\(\\varphi_k(a)\\), hence with the strong limit \\(\\varphi(a)\\). \\(\\square\\)\n\n**Corollary 4.3** (spectral projections). Let \\(x\\) be a self-adjoint element of a von Neumann algebra \\(N\\), let \\(c\\in\\mathbb R\\), and let \\(p=1_{(c,\\infty)}(x)\\).\n\n1. \\(p\\) is a projection in \\(N\\), and it commutes with every operator that commutes with \\(x\\).\n2. \\((x-c)p\\ge0\\) and \\((x-c)(1-p)\\le0\\). In particular \\(cp\\le xp\\) and \\(x(1-p)\\le c(1-p)\\).\n3. If \\(c\\ge0\\) and \\(x=exe\\) for a projection \\(e\\), then \\(p\\le e\\).\n4. If \\(c<\\max\\sigma(x)\\), then \\(p\\ne0\\). If \\(c>\\min\\sigma(x)\\), then \\(p\\ne1\\).\n\n**Proof.** The functions \\(g_k(t)=\\min(1,\\max(0,k(t-c)))\\) are continuous, take values in \\([0,1]\\) and increase to \\(1_{(c,\\infty)}\\), so \\(1_{(c,\\infty)}\\in\\mathcal B_1(\\sigma(x))\\). (1) It is real and equal to its square, so \\(p=p^*=p^2\\) by Theorem 4.2, which also gives the rest. (2) The functions \\((t-c)1_{(c,\\infty)}(t)\\) and \\((c-t)(1-1_{(c,\\infty)}(t))\\) are nonnegative and lie in \\(\\mathcal B_1(\\sigma(x))\\); apply positivity in Theorem 4.2. (3) If \\(c\\ge0\\), then \\(g_k(0)=0\\), so each \\(g_k(x)\\) is a norm limit of polynomials in \\(x\\) without constant term. Each such polynomial \\(q(x)\\) satisfies \\(q(x)=eq(x)e\\). So \\(g_k(x)=eg_k(x)e\\) and, in the limit, \\(p=epe\\), which means \\(p\\le e\\). (4) If \\(p=0\\), then (2) gives \\(x-c=(x-c)(1-p)\\le0\\), so \\(\\sigma(x)\\subseteq(-\\infty,c]\\). If \\(p=1\\), then \\(x-c=(x-c)p\\ge0\\), so \\(\\sigma(x)\\subseteq[c,\\infty)\\). \\(\\square\\)\n\n**Corollary 4.4.** A von Neumann algebra \\(N\\) on \\(H\\ne\\{0\\}\\) whose only projections are \\(0\\) and \\(1\\) is \\(\\mathbb C1\\).\n\n**Proof.** Let \\(x\\in N\\) be self-adjoint. If \\(\\sigma(x)\\) contained two points \\(s<t\\), then any \\(c\\) with \\(s<c<t\\) would give a projection \\(1_{(c,\\infty)}(x)\\in N\\) different from \\(0\\) and \\(1\\) (Corollary 4.3(4)). So \\(\\sigma(x)=\\{\\lambda\\}\\), and \\(\\|x-\\lambda\\|\\), which is the spectral radius of the self-adjoint operator \\(x-\\lambda\\) ([the norm of a normal element](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-01)), is \\(0\\). Every element of \\(N\\) is a combination of two self-adjoint elements of \\(N\\). \\(\\square\\)\n\n**Corollary 4.5** (unitaries are exponentials). Let \\(u\\) be a unitary in a von Neumann algebra \\(N\\).\n\n1. There is a self-adjoint \\(h\\in N\\) with \\(\\|h\\|\\le\\pi\\) and \\(u=\\exp(ih)\\).\n2. If \\(\\|u-1\\|<2\\), put \\(\\alpha=2\\arcsin(\\|u-1\\|/2)<\\pi\\). Then \\(h\\) can be chosen in \\(C^*(1,u)\\), with \\(\\|h\\|\\le\\alpha\\).\n\n**Proof.** The spectrum of \\(u\\) lies in the unit circle \\(\\mathbb T\\). Let \\(\\arg:\\mathbb T\\to(-\\pi,\\pi]\\) be the argument, \\(\\arg(e^{i\\theta})=\\theta\\) for \\(-\\pi<\\theta\\le\\pi\\). For \\(k\\ge1\\) let \\(\\varphi_k(e^{i\\theta})=\\theta\\) for \\(\\theta\\in[-\\pi+1/k,\\pi]\\), and let \\(\\varphi_k\\) be linear in \\(\\theta\\) on \\([-\\pi,-\\pi+1/k]\\), from the value \\(\\pi\\) at \\(\\theta=-\\pi\\) to the value \\(-\\pi+1/k\\). Each \\(\\varphi_k\\) is continuous on \\(\\mathbb T\\), because its values at \\(\\theta=\\pi\\) and \\(\\theta=-\\pi\\), which describe the same point \\(-1\\), are both \\(\\pi\\). The values lie in \\([-\\pi,\\pi]\\), and \\(\\varphi_k\\to\\arg\\) pointwise on \\(\\mathbb T\\). So \\(\\arg\\in\\mathcal B_1(\\sigma(u))\\).\n\n(1) Put \\(h=\\arg(u)\\). By Theorem 4.2, \\(h\\) is a self-adjoint element of \\(N\\) with \\(\\|h\\|\\le\\pi\\), and \\(h^m=(\\arg^m)(u)\\) for every \\(m\\). The partial sums of \\(\\exp(ih)\\) are therefore \\(\\big(\\sum_{m\\le M}(i\\arg)^m/m!\\big)(u)\\). These functions converge uniformly on \\(\\mathbb T\\) to \\(e^{i\\arg}\\), which is the identity function \\(\\lambda\\mapsto\\lambda\\). By the norm bound of Theorem 4.2, \\(\\exp(ih)=(e^{i\\arg})(u)=u\\).\n\n(2) For \\(\\lambda\\in\\sigma(u)\\), \\(\\lambda-1\\) lies in the spectrum of \\(u-1\\), so \\(|\\lambda-1|\\le\\|u-1\\|\\). Since \\(|e^{i\\theta}-1|=2|\\sin(\\theta/2)|\\), the spectrum lies in the arc \\(\\{e^{i\\theta}:|\\theta|\\le\\alpha\\}\\). There \\(\\arg\\) is continuous, so \\(h=\\arg(u)\\) is given by the continuous calculus, lies in \\(C^*(1,u)\\), and satisfies \\(\\|h\\|\\le\\alpha\\). \\(\\square\\)\n\n**Corollary 4.6** (dyadic expansion). Every \\(x\\) in a von Neumann algebra \\(N\\) with \\(0\\le x\\le1\\) is a norm-convergent sum \\(x=\\sum_{k\\ge1}2^{-k}p_k\\) of projections \\(p_k\\in N\\), each a spectral projection of \\(x\\).\n\n**Proof.** For \\(t\\in[0,1)\\) let \\(d_k(t)\\in\\{0,1\\}\\) be the \\(k\\)-th binary digit of \\(t\\), from the expansion \\(t=\\sum_k2^{-k}d_k(t)\\) that does not end in an infinite string of ones, and put \\(d_k(1)=1\\) for all \\(k\\). Then \\(d_k\\) is the indicator function of a finite union of intervals of the form \\([j2^{-k},(j+1)2^{-k})\\), together with the point \\(1\\). Such functions lie in \\(\\mathcal B_1([0,1])\\): the indicator function of \\([\\alpha,\\beta)\\) is the pointwise limit of continuous piecewise linear functions with values in \\([0,1]\\) that are \\(1\\) on \\([\\alpha,\\beta-1/m]\\) and \\(0\\) outside \\((\\alpha-1/m,\\beta)\\), and the indicator function of the point \\(1\\) is the pointwise limit of \\(\\max(0,1-m|t-1|)\\). Put \\(p_k=d_k(x)\\), a projection in \\(N\\) by Theorem 4.2. For every \\(t\\in[0,1]\\), \\(|t-\\sum_{k\\le m}2^{-k}d_k(t)|\\le2^{-m}\\). By the norm bound of Theorem 4.2, \\(\\|x-\\sum_{k\\le m}2^{-k}p_k\\|\\le2^{-m}\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-KD-05",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "5. Kaplansky's continuity theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "full_conditions_and_proof": "## 5. Kaplansky's continuity theorem\n\nWe now come to Kaplansky's continuity theorem. It says that the functional calculus of a continuous function of at most linear growth is strongly continuous on commuting normal tuples, even along unbounded nets.\n\nFor a closed set \\(G\\subseteq\\mathbb C^n\\), let \\(\\mathcal N_G\\) be the set of commuting normal \\(n\\)-tuples \\(a\\) on \\(H\\) with \\(\\sigma(a)\\subseteq G\\). A net \\((a^{(i)})\\) in \\(\\mathcal N_G\\) *converges strongly* to \\(a\\) if \\(a^{(i)}_j\\to a_j\\) strongly for each \\(j\\). For \\(n=1\\) and \\(G=\\mathbb R\\), \\(\\mathcal N_G\\) is the set of self-adjoint operators; for \\(G=\\mathbb T\\) it is the set of unitaries. For an \\(n\\)-tuple \\(a\\) of arbitrary operators put\n\\[\nR(a)=\\Big(1+\\sum_{k=1}^na_k^*a_k\\Big)^{-1} .\n\\]\n\n**Lemma 5.1** (resolvent estimates). Let \\(a\\) and \\(b\\) be \\(n\\)-tuples of operators.\n\n1. \\(0\\le R(a)\\le1\\), \\(\\|a_jR(a)\\|\\le\\frac12\\), \\(\\|R(a)a_j^*\\|\\le\\frac12\\) and \\(\\|a_jR(a)a_k^*\\|\\le1\\) for all \\(j,k\\).\n2. \\[\n\\begin{gathered}\nR(a)-R(b)\\\\\n=\\sum_kR(a)\\big[(b_k^*-a_k^*)b_k\\\\\n+a_k^*(b_k-a_k)\\big]R(b).\n\\end{gathered}\n\\]\n3. If \\((a^{(i)})\\) is a net of \\(n\\)-tuples with \\(a^{(i)}_k\\to b_k\\) and \\(a^{(i)*}_k\\to b_k^*\\) strongly for every \\(k\\), then \\(R(a^{(i)})\\to R(b)\\) and \\(a^{(i)}_jR(a^{(i)})\\to b_jR(b)\\) strongly for every \\(j\\).\n\n**Proof.** (1) Put \\(X=\\sum_ka_k^*a_k\\ge0\\) and \\(R=R(a)=(1+X)^{-1}\\). By the functional calculus of \\(X\\), \\(0\\le R\\le1\\) and \\(RXR\\le\\frac14\\), since \\(t/(1+t)^2\\le\\frac14\\) for \\(t\\ge0\\). As \\(a_j^*a_j\\le X\\), we get \\(\\|a_jR\\|^2=\\|Ra_j^*a_jR\\|\\le\\|RXR\\|\\le\\frac14\\). The adjoint gives \\(\\|Ra_j^*\\|\\le\\frac12\\). Also \\[\n\\begin{gathered}\n\\|a_jR^{1/2}\\|^2\\\\\n=\\|R^{1/2}a_j^*a_jR^{1/2}\\|\\\\\n\\le\\|R^{1/2}XR^{1/2}\\|\\\\\n\\le1,\n\\end{gathered}\n\\] since \\(t/(1+t)\\le1\\); so \\(\\|a_jRa_k^*\\|\\le\\|a_jR^{1/2}\\|\\,\\|R^{1/2}a_k^*\\|\\le1\\).\n\n(2) \\[\n\\begin{gathered}\nR(a)-R(b)\\\\\n=R(a)\\big[(1+X_b)-(1+X_a)\\big]R(b),\n\\end{gathered}\n\\] with \\(X_a=\\sum_ka_k^*a_k\\) and \\(X_b\\) likewise, and \\(b_k^*b_k-a_k^*a_k=(b_k^*-a_k^*)b_k+a_k^*(b_k-a_k)\\).\n\n(3) Write \\(a=a^{(i)}\\). By (1) and (2),\n\\[\n\\begin{gathered}\n\\|(R(a)-R(b))\\xi\\|\\\\\n\\le\\sum_k\\Big(\\|(b_k^*-a_k^*)b_kR(b)\\xi\\|\\\\\n+\\tfrac12\\|(b_k-a_k)R(b)\\xi\\|\\Big),\n\\end{gathered}\n\\]\nand each term tends to \\(0\\), because the vectors \\(b_kR(b)\\xi\\) and \\(R(b)\\xi\\) are fixed. Next, \\[\n\\begin{gathered}\na_jR(a)-b_jR(b)\\\\\n=(a_j-b_j)R(b)+a_j(R(a)-R(b)).\n\\end{gathered}\n\\] Inserting (2) and using \\(\\|a_jR(a)\\|\\le\\frac12\\) and \\(\\|a_jR(a)a_k^*\\|\\le1\\),\n\\[\n\\begin{gathered}\n\\|(a_jR(a)-b_jR(b))\\xi\\|\\\\\n\\le\\|(a_j-b_j)R(b)\\xi\\|\\\\\n+\\sum_k\\Big(\\tfrac12\\|(b_k^*-a_k^*)b_kR(b)\\xi\\|\\\\\n+\\|(b_k-a_k)R(b)\\xi\\|\\Big)\\to0 .\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Theorem 5.2** (Kaplansky's continuity theorem). Let \\(G\\subseteq\\mathbb C^n\\) be closed, and let \\(f:G\\to\\mathbb C\\) be continuous with\n\\[\n\\sup_{\\lambda\\in G}\\frac{|f(\\lambda)|}{1+|\\lambda|}<\\infty .\n\\tag{5.1}\n\\]\nIf a net \\((a^{(i)})\\) in \\(\\mathcal N_G\\) converges strongly to \\(a\\in\\mathcal N_G\\), then \\(f(a^{(i)})\\to f(a)\\) strongly and strongly\\(^*\\).\n\n\n**Proof.** *Step 1: a closed \\(*\\)-algebra of good functions.* Let \\(\\mathcal C\\) be the set of bounded continuous functions \\(g\\) on \\(G\\) such that \\(a\\mapsto g(a)\\) is strongly continuous on \\(\\mathcal N_G\\). It contains the constants. It is closed under sums. It is closed under products: for \\(g,k\\in\\mathcal C\\),\n\\[\n\\begin{gathered}\ng(a^{(i)})k(a^{(i)})-g(a)k(a)\\\\\n=g(a^{(i)})\\big(k(a^{(i)})-k(a)\\big)\\\\\n+\\big(g(a^{(i)})-g(a)\\big)k(a),\n\\end{gathered}\n\\]\nand \\(\\|g(a^{(i)})\\|\\le\\sup_G|g|\\). It is closed under complex conjugation: \\(\\bar g(a)=g(a)^*\\), and the operators \\(g(a^{(i)})\\) are normal and converge strongly to the normal operator \\(g(a)\\), so their adjoints converge strongly (Theorem 2.3(3)). It is closed under uniform limits: if \\(g_m\\to g\\) uniformly on \\(G\\) with \\(g_m\\in\\mathcal C\\), then\n\\[\n\\begin{gathered}\n\\|(g(a^{(i)})-g(a))\\xi\\|\\\\\n\\le2\\sup_G|g-g_m|\\,\\|\\xi\\|+\\|(g_m(a^{(i)})-g_m(a))\\xi\\| .\n\\end{gathered}\n\\]\nFinally, strong convergence of \\(g(a^{(i)})\\) to \\(g(a)\\) implies strong\\(^*\\) convergence, since all these operators are normal (Corollary 2.4).\n\n*Step 2: two resolvent functions.* Let \\(r(\\lambda)=(1+|\\lambda|^2)^{-1}\\) and \\(s_j(\\lambda)=\\lambda_jr(\\lambda)\\). Since the calculus is a \\(*\\)-homomorphism with \\(\\iota_k(a)=a_k\\), we have \\(r(a)=R(a)\\) and \\(s_j(a)=a_jR(a)\\) for \\(a\\in\\mathcal N_G\\). If \\(a^{(i)}\\to a\\) strongly in \\(\\mathcal N_G\\), then also \\(a^{(i)*}_k\\to a_k^*\\) strongly, by Corollary 2.4. Lemma 5.1(3) now shows \\(r,s_j\\in\\mathcal C\\).\n\n*Step 3: functions vanishing at infinity.* The closed set \\(G\\) is locally compact. The functions \\(r\\) and \\(s_j\\) vanish at infinity on \\(G\\). The \\(*\\)-algebra they generate without constant terms consists of functions in \\(C_0(G)\\). It vanishes nowhere, because \\(r>0\\). It separates points: if \\(r(\\lambda)=r(\\mu)\\) and \\(s_j(\\lambda)=s_j(\\mu)\\) for all \\(j\\), then \\(|\\lambda|=|\\mu|\\) and \\(\\lambda_j=\\mu_j\\). By [the Stone–Weierstrass theorem for \\(C_0(X)\\)](stone-weierstrass-c0.md#oa-fnd-sw-09), this \\(*\\)-algebra is dense in \\(C_0(G)\\). It lies in \\(\\mathcal C\\) by Steps 1 and 2, and \\(\\mathcal C\\) is closed, so \\(C_0(G)\\subseteq\\mathcal C\\).\n\n*Step 4: products with a coordinate.* Let \\(k\\in\\mathcal C\\). Then \\(a\\mapsto a_jk(a)\\) is strongly continuous on \\(\\mathcal N_G\\). Indeed, \\(k(a)\\) commutes with \\(a_j\\) (Theorem 3.3(3)), so\n\\[\n\\begin{gathered}\na^{(i)}_jk(a^{(i)})-a_jk(a)\\\\\n=k(a^{(i)})\\big(a^{(i)}_j-a_j\\big)+\\big(k(a^{(i)})-k(a)\\big)a_j ,\n\\end{gathered}\n\\]\nand \\(\\|k(a^{(i)})\\|\\le\\sup_G|k|\\).\n\n*Step 5: bounded functions.* Let \\(g\\) be bounded and continuous on \\(G\\). Since \\(r(\\lambda)(1+|\\lambda|^2)=1\\),\n\\[\ng=gr+\\sum_j\\iota_j\\cdot(\\bar\\iota_jgr).\n\\tag{5.2}\n\\]\nHere \\(gr\\in C_0(G)\\), and \\(\\bar\\iota_jgr\\in C_0(G)\\) because \\(|\\lambda_j|\\,|g(\\lambda)|\\,r(\\lambda)\\le\\sup|g|\\cdot|\\lambda|/(1+|\\lambda|^2)\\). Applying the calculus to (5.2), \\(g(a)=(gr)(a)+\\sum_ja_j\\,(\\bar\\iota_jgr)(a)\\). The first term is strongly continuous in \\(a\\) by Step 3, and each summand by Steps 3 and 4. So \\(g\\in\\mathcal C\\).\n\n*Step 6: linear growth.* Let \\(f\\) satisfy (5.1) with constant \\(C\\). The identity (5.2) holds with \\(f\\) in place of \\(g\\). Now \\(fr\\in C_0(G)\\), since \\(|f|r\\le C(1+|\\lambda|)/(1+|\\lambda|^2)\\). Each \\(\\bar\\iota_jfr\\) is bounded and continuous, since \\(|\\lambda|(1+|\\lambda|)\\le2(1+|\\lambda|^2)\\) gives \\(|\\lambda_j||f|r\\le2C\\); so it lies in \\(\\mathcal C\\) by Step 5. By Steps 3 and 4, \\(f(a)=(fr)(a)+\\sum_ja_j(\\bar\\iota_jfr)(a)\\) is strongly continuous in \\(a\\), and strong\\(^*\\) continuity follows as in Step 1. \\(\\square\\)\n\n**Corollary 5.3.** Let \\(G\\subseteq\\mathbb C^n\\) be closed and \\(f:G\\to\\mathbb C\\) continuous.\n\n1. If \\(G\\) is compact, the calculus of \\(f\\) is strongly continuous on \\(\\mathcal N_G\\).\n2. For every \\(c>0\\), the calculus of \\(f\\) is strongly continuous on the set of \\(a\\in\\mathcal N_G\\) with \\(\\|a_j\\|\\le c\\) for all \\(j\\).\n3. If a sequence \\((a^{(m)})\\) in \\(\\mathcal N_G\\) converges strongly to \\(a\\in\\mathcal N_G\\), then \\(f(a^{(m)})\\to f(a)\\) strongly.\n4. For self-adjoint operators (\\(n=1\\), \\(G=\\mathbb R\\)), every continuous \\(f\\) with \\(|f(t)|\\le C(1+|t|)\\) acts strongly continuously. Examples are \\(|t|\\), \\(\\max(t,0)\\), \\(e^{it}\\), \\(2t/(1+t^2)\\), and any continuous function on a compact interval, applied to self-adjoint operators with spectrum in that interval.\n\n**Proof.** (1) A continuous function on a compact set is bounded, so (5.1) holds. (2) For a normal operator, the norm is the largest modulus of a point of the spectrum ([the norm of a normal element](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-01)), and \\(\\sigma(a_j)\\) is the \\(j\\)-th projection of \\(\\sigma(a)\\) (Theorem 3.3(1)). So the set in question is \\(\\mathcal N_{G_c}\\) with the compact set \\(G_c=\\{\\lambda\\in G:|\\lambda_j|\\le c\\text{ for all }j\\}\\), and (1) applies to \\(f|_{G_c}\\). (3) A strongly convergent sequence is bounded in norm by the uniform boundedness principle, so (2) applies. (4) is Theorem 5.2 and (1). \\(\\square\\)\n\n**Example 5.4** (the Cayley transform). For a self-adjoint \\(h\\) the *Cayley transform* \\(\\kappa(h)=(h-i)(h+i)^{-1}\\) is the calculus of the bounded continuous function \\((t-i)/(t+i)\\), which has modulus \\(1\\). So \\(\\kappa(h)\\) is unitary, and \\(\\kappa\\) is strongly continuous on self-adjoint operators by Corollary 5.3(4). A direct estimate is also available. Since \\(\\kappa(h)=1-2i(h+i)^{-1}\\), the resolvent identity gives\n\\[\n\\begin{gathered}\n\\kappa(h)-\\kappa(k)\\\\\n=2i\\,(h+i)^{-1}(h-k)(k+i)^{-1},\\\\\n\\text{so}\\\\\n\\|(\\kappa(h)-\\kappa(k))\\xi\\|\\\\\n\\le2\\|(h-k)(k+i)^{-1}\\xi\\|,\n\\end{gathered}\n\\]\nbecause \\(\\|(h+i)^{-1}\\|\\le1\\). For fixed \\(k\\) and \\(\\xi\\) the right side tends to \\(0\\) as \\(h\\to k\\) strongly.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-KD-06",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "6. Linear growth is necessary",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "full_conditions_and_proof": "## 6. Linear growth is necessary\n\nCondition (5.1) cannot be weakened. In infinite dimensions, a continuous function of faster growth fails to act strongly continuously, although it still acts continuously on bounded sets and along sequences (Corollary 5.3).\n\n**Theorem 6.1.** Let \\(H\\) be infinite-dimensional, \\(G\\subseteq\\mathbb C^n\\) closed, and \\(f:G\\to\\mathbb C\\) continuous. Suppose that for some \\(g_0\\in G\\) the calculus of \\(f\\) is strongly continuous on \\(\\mathcal N_G\\) at the constant tuple \\(g_0\\cdot1=(g_{0,1}1,\\dots,g_{0,n}1)\\). Then \\(f\\) satisfies (5.1). If \\(\\dim H<\\infty\\), every continuous \\(f\\) acts strongly continuously on \\(\\mathcal N_G\\).\n\n**Proof.** Suppose (5.1) fails. As \\(f\\) is bounded on the compact sets \\(\\{\\lambda\\in G:|\\lambda|\\le c\\}\\), there are \\(t_k\\in G\\) with \\(|t_k|\\to\\infty\\) and \\(|f(t_k)|\\ge k(1+|t_k|)\\). Then \\(|f(t_k)-f(g_0)|\\to\\infty\\) and \\(|t_k-g_0|/|f(t_k)-f(g_0)|\\to0\\). Dropping finitely many \\(k\\), we may assume \\(|f(t_k)-f(g_0)|\\ge1\\), and we put \\(c_k=|f(t_k)-f(g_0)|^{-1}\\in(0,1]\\) and \\(s_k=(1-c_k^2)^{1/2}\\).\n\nFix a unit vector \\(\\xi\\). Let \\(\\Lambda\\) be the set of pairs \\(\\lambda=(F,k)\\) with \\(F\\subseteq H\\) a finite-dimensional subspace and \\(k\\ge1\\), directed by \\((F,k)\\le(F',k')\\) if \\(F\\subseteq F'\\) and \\(k\\le k'\\). For each \\(\\lambda=(F,k)\\) choose a unit vector \\(\\eta_\\lambda\\) orthogonal to \\(F\\) and to \\(\\xi\\), which exists because \\(\\dim H=\\infty\\). Put \\(v_\\lambda=c_k\\xi+s_k\\eta_\\lambda\\), a unit vector, \\(p_\\lambda=\\theta_{v_\\lambda,v_\\lambda}\\), and\n\\[\n\\begin{gathered}\na^{(\\lambda)}\\\\\n=g_0\\cdot1+(t_k-g_0)\\,p_\\lambda,\\\\\n\\text{that is,}\\\\\na^{(\\lambda)}_j\\\\\n=g_{0,j}1+(t_{k,j}-g_{0,j})p_\\lambda .\n\\end{gathered}\n\\]\nThese are commuting normal operators. The algebra \\(C^*(1,a^{(\\lambda)})\\) is spanned by \\(1\\) and \\(p_\\lambda\\), and its characters send \\(p_\\lambda\\) to \\(0\\) or \\(1\\), so \\(\\sigma(a^{(\\lambda)})=\\{g_0,t_k\\}\\subseteq G\\). The map \\(\\varphi\\mapsto\\varphi(g_0)(1-p_\\lambda)+\\varphi(t_k)p_\\lambda\\) is a unital \\(*\\)-homomorphism on \\(C(\\{g_0,t_k\\})\\) that sends each \\(\\iota_j\\) to \\(a^{(\\lambda)}_j\\), so by Theorem 3.3(2) it is the calculus: \\(f(a^{(\\lambda)})=f(g_0)(1-p_\\lambda)+f(t_k)p_\\lambda\\).\n\n*Strong convergence of the tuples.* Let \\(\\omega\\in H\\), and choose \\(\\lambda_0=(F_0,k_0)\\) with \\(\\omega\\in F_0\\). For \\(\\lambda=(F,k)\\ge\\lambda_0\\), \\(\\eta_\\lambda\\perp\\omega\\), so \\(\\langle\\omega,v_\\lambda\\rangle=c_k\\langle\\omega,\\xi\\rangle\\) and\n\\[\n\\|(a^{(\\lambda)}_j-g_{0,j})\\omega\\|\\le|t_k-g_0|\\,c_k\\,|\\langle\\omega,\\xi\\rangle|\\to0 .\n\\]\n*No convergence of the calculus.* On the other hand, \\[\n\\begin{gathered}\n\\|(f(a^{(\\lambda)})-f(g_0))\\xi\\|\\\\\n=|f(t_k)-f(g_0)|\\,|\\langle\\xi,v_\\lambda\\rangle|\\\\\n=|f(t_k)-f(g_0)|\\,c_k\\\\\n=1\n\\end{gathered}\n\\] for every \\(\\lambda\\). So \\(f(a^{(\\lambda)})\\) does not converge strongly to \\(f(g_0\\cdot1)=f(g_0)1\\).\n\nIf \\(\\dim H<\\infty\\), the strong topology is the norm topology. A net that converges in norm is eventually bounded, and Corollary 5.3(2) applies to its tail. \\(\\square\\)\n\n**Example 6.2** (squaring). On an infinite-dimensional \\(H\\), the map \\(h\\mapsto h^2\\) on self-adjoint operators is not strongly continuous at \\(0\\). In the construction above take \\(f(t)=t^2\\), \\(g_0=0\\), \\(t_k=k\\) and \\(c_k=k^{-2}\\). Then \\(h_\\lambda=kp_\\lambda\\to0\\) strongly, but \\(\\|h_\\lambda^2\\xi\\|=k^2\\,|\\langle\\xi,v_\\lambda\\rangle|=1\\). By Corollary 5.3(3), no sequence can show this.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-KD-07",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "7. Kaplansky's density theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "anchor": "oa-fnd-kd-07",
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      "full_conditions_and_proof": "## 7. Kaplansky's density theorem\n\n**Theorem 7.1** (Kaplansky's density theorem). Let \\(A\\) be a \\(*\\)-subalgebra of \\(B(H)\\), not necessarily norm closed or nondegenerate, and let \\(M\\) be its weak closure.\n\n1. \\(A\\cap S\\) is strongly\\(^*\\) dense in \\(M\\cap S\\).\n2. \\(A_h\\cap S\\) is strongly\\(^*\\) dense in \\(M_h\\cap S\\).\n3. \\(A_+\\cap S\\) is strongly\\(^*\\) dense in \\(M_+\\cap S\\).\n4. In each of (1)–(3), the closure of the smaller set in any of the six operator topologies is the larger set.\n\n\n**Proof.** *Step 1: we may assume that \\(A\\) is norm closed.* Let \\(\\bar A\\) be the norm closure of \\(A\\). It is a concrete C\\(^*\\)-algebra with the same weak closure \\(M\\). Every \\(c\\in\\bar A\\cap S\\) is a norm limit of elements of \\(A\\cap S\\): if \\(c_m\\in A\\) and \\(c_m\\to c\\) in norm, then \\(c_m/\\max(1,\\|c_m\\|)\\in A\\cap S\\) also converges to \\(c\\). For self-adjoint \\(c\\) use \\(\\frac12(c_m+c_m^*)\\). For positive \\(c\\), approximate \\(c^{1/2}\\in\\bar A_h\\cap S\\) by \\(d_m\\in A_h\\cap S\\); then \\(d_m^2\\in A_+\\cap S\\) and \\(d_m^2\\to c\\) in norm. Norm convergence implies strong\\(^*\\) convergence, so it suffices to prove (1)–(3) for \\(\\bar A\\). From now on \\(A\\) is a C\\(^*\\)-algebra.\n\n*Step 2: self-adjoint elements.* Let \\(x\\in M_h\\cap S\\). Put \\(f(t)=2t/(1+t^2)\\) on \\(\\mathbb R\\) and \\(g(s)=s/(1+\\sqrt{1-s^2})\\) on \\([-1,1]\\). Both are continuous, \\(f(0)=g(0)=0\\), \\(|f|\\le1\\), and \\(f(g(s))=s\\) for \\(s\\in[-1,1]\\): with \\(s=\\sin\\theta\\) and \\(|\\theta|\\le\\pi/2\\), \\(g(s)=\\tan(\\theta/2)\\) and \\(f(\\tan(\\theta/2))=\\sin\\theta\\). The weak closure \\(M\\) is a norm-closed \\(*\\)-algebra, so \\(y=g(x)\\) lies in \\(M\\) ([the calculus without an identity](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-08)), and \\(f(y)=x\\) by the composition rule. By the double commutant theorem, \\(M\\) is also the strong\\(^*\\) closure of \\(A\\). Choose a net \\((c_i)\\) in \\(A\\) with \\(c_i\\to y\\) strongly\\(^*\\), and put \\(b_i=\\frac12(c_i+c_i^*)\\in A_h\\). Then \\(b_i\\to y\\) strongly. By Corollary 5.3(4), \\(f(b_i)\\to f(y)=x\\) strongly, hence strongly\\(^*\\), since these operators are self-adjoint. Each \\(f(b_i)\\) lies in \\(A_h\\), because \\(f(0)=0\\), and has norm at most \\(\\max|f|=1\\).\n\n*Step 3: arbitrary elements.* Let \\(x\\in M\\cap S\\). On \\(H\\oplus H\\) consider the \\(*\\)-algebra \\(M_2(A)\\) of \\(2\\times2\\) matrices with entries in \\(A\\). A net of \\(2\\times2\\) matrices converges weakly exactly when its four entries do, so the weak closure of \\(M_2(A)\\) is \\(M_2(M)\\). The self-adjoint matrix \\(X=\\begin{pmatrix}0&x\\\\x^*&0\\end{pmatrix}\\) lies in \\(M_2(M)\\), and \\(\\|X\\|=\\|x\\|\\le1\\), since \\(X^2=\\begin{pmatrix}xx^*&0\\\\0&x^*x\\end{pmatrix}\\). By Step 2 there is a net of self-adjoint \\(Y_i=\\begin{pmatrix}b_i&a_i\\\\a_i^*&d_i\\end{pmatrix}\\in M_2(A)\\) with \\(\\|Y_i\\|\\le1\\) and \\(Y_i\\to X\\) strongly. The corner entries of a matrix are compressions of it, so \\(\\|a_i\\|\\le1\\), \\(a_i\\to x\\) strongly and \\(a_i^*\\to x^*\\) strongly. So \\(a_i\\to x\\) strongly\\(^*\\), with \\(a_i\\in A\\cap S\\).\n\n*Step 4: positive elements.* Let \\(x\\in M_+\\cap S\\). Then \\(x^{1/2}\\in M_h\\cap S\\), and Step 2 gives \\(b_i\\in A_h\\cap S\\) with \\(b_i\\to x^{1/2}\\) strongly\\(^*\\). By Lemma 1.1(2), \\(b_i^2\\to x\\) strongly\\(^*\\), and \\(b_i^2\\in A_+\\cap S\\).\n\n*Step 5: the six topologies.* A bounded net that converges strongly\\(^*\\) converges \\(\\sigma\\)-strongly\\(^*\\) (Lemma 1.1(3)), and the \\(\\sigma\\)-strong\\(^*\\) topology is the finest of the six. So each larger set lies in each closure of the smaller one. Conversely, \\(M\\cap S\\), \\(M_h\\cap S\\) and \\(M_+\\cap S\\) are weakly closed, by Lemma 1.2 and because \\(M\\) is weakly closed. So the closures are exactly the larger sets. \\(\\square\\)\n\n**Remark 7.2** (norm density fails). In general \\(A\\cap S\\) is not norm dense in \\(M\\cap S\\). For an infinite-dimensional \\(H\\), the compact operators \\(K(H)\\) form a nondegenerate C\\(^*\\)-algebra whose weak closure is \\(B(H)\\) ([Example 4.6 of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-07)). For every compact \\(k\\), \\(\\|1-k\\|\\ge1\\): otherwise \\(k=1-(1-k)\\) would be invertible by the Neumann series, and \\(1=kk^{-1}\\) would be compact, which is false in infinite dimensions. So \\(1\\) is not a norm limit of elements of \\(K(H)\\cap S\\).\n\n**Example 7.3** (the algebraic structure is needed). Kaplansky's theorem fails for self-adjoint subspaces that are not algebras. On \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_m)\\), let \\(\\omega(x)=\\sum_m2^{-m}\\langle x\\delta_m,\\delta_m\\rangle\\) and \\(V=\\ker\\omega\\). Then \\(V^*=V\\). By [Exercise 1.5 of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-01), \\(V\\) is not weakly closed. Its weak closure is a weakly closed subspace that strictly contains a subspace of codimension one, so it is \\(B(H)\\). But \\(V\\cap S\\) is weakly closed. Indeed, a bounded weakly convergent net converges \\(\\sigma\\)-weakly ([Lemma 1.2(d) of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-01)), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous, so \\(\\omega\\) vanishes at every weak limit of a net in \\(V\\cap S\\). So the weak closure of \\(V\\cap S\\) is \\(V\\cap S\\), which does not contain \\(1\\in B(H)\\cap S\\).\n\n**Theorem 7.2** (The bounded-net algebra). Let \\(A\\subseteq B(H)\\) be a norm-closed *-subalgebra, with no nondegeneracy or identity assumption. Let \\(D\\) consist of the strong limits of norm-bounded nets in \\(A\\). Then \\(D\\) is the strong closure of \\(A\\), and\n\\[\nD\\cap B(H)_1=\\overline{A\\cap B(H)_1}^{\\,s}.\n\\]\nThe same statement holds for self-adjoint and positive unit balls. In the unrestricted unit ball, approximation can also be chosen strongly* convergent.\n\nThis is the bounded-net route of [G. A. Elliott and C. J. K. Griffin, *On a question of Kaplansky concerning his density theorem* (2024)](https://arxiv.org/html/2410.03668v1). It explains how bounded-set functional calculus suffices for density. The argument below uses the complete trace-duality and Krein–Šmulian proofs in the operator-topologies lesson: [trace duality, Theorem 5.4](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md#oa-fnd-lt-06), and [Krein–Šmulian and its bounded-ball consequence, Theorems 9.3 and 9.6](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md#oa-fnd-lt-10), together with [the double commutant lesson, Theorem 4.4](the-double-commutant-theorem.md#oa-fnd-bi-06). Those results do not use Kaplansky density. The stronger continuity theorem on unbounded normal nets remains Theorem 5.2 above.\n\n**Proof.** Sums and scalar multiples of bounded approximating nets show that \\(D\\) is a complex vector space containing \\(A\\). If \\(a_i\\to x\\) and \\(b_j\\to y\\) strongly, with respective uniform bounds \\(C,E\\), the product net satisfies\n\\[\n\\begin{gathered}\n\\|(a_i b_j-xy)\\xi\\|\\\\\n\\le C\\|(b_j-y)\\xi\\|+\\|(a_i-x)y\\xi\\|\\\\\n\\longrightarrow0.\n\\end{gathered}\n\\]\nIts norm is at most \\(CE\\), so \\(xy\\in D\\).\n\nAdjoints require convexity. If \\(a_i\\to x\\) strongly and \\(\\|a_i\\|\\le C\\), then \\(a_i^*\\to x^*\\) weakly. The set \\(A\\cap CB(H)_1\\) is convex. Weak and strong closures of a convex set agree, by Hahn–Banach separation, so \\(x^*\\) is a strong limit of a net from this same bounded set. Thus \\(D=D^*\\). Similarly, if \\(x=x^*\\in D\\), the net \\((a_i+a_i^*)/2\\) converges weakly to \\(x\\) and lies in the convex set \\(A_h\\cap CB(H)_1\\). Hence \\(x\\) has a bounded strong approximation by self-adjoint elements of \\(A\\).\n\nFor such a net \\(h_i\\to h\\), all spectra lie in a fixed interval \\([-C,C]\\), enlarging \\(C\\) if necessary. Every continuous real function \\(f\\) on that interval with \\(f(0)=0\\) satisfies \\(f(h_i)\\to f(h)\\) strongly. Indeed, approximate \\(f\\) uniformly by real polynomials and subtract their value at zero to obtain polynomials without constant term. Their functional calculi lie in \\(A\\); the error in operator norm is uniform over the whole net. Each polynomial converges strongly by bounded-net multiplication (Lemma 1.1). The same uniform error then proves strong convergence of \\(f(h_i)\\).\n\nWhen \\(\\|h\\|\\le1\\), choose \\(f(t)=\\max(-1,\\min(t,1))\\). Then \\(f(h)=h\\) and the \\(f(h_i)\\) are self-adjoint contractions in \\(A\\). If \\(0\\le h\\le1\\), use \\(f(t)=\\max(0,\\min(t,1))\\) to obtain positive contractions. Both functions vanish at zero, which is necessary when \\(A\\) has no identity.\n\nFor \\(x\\in D\\), each entry of\n\\[\nX=\\begin{pmatrix}0&x\\\\x^*&0\\end{pmatrix}\\in B(H\\oplus H)\n\\]\nbelongs to \\(D\\). A finite product of the directed sets for its entries gives a uniformly bounded strong approximating net from \\(M_2(A)\\): the matrix norm is at most the sum of its four entry norms. Thus \\(X\\) belongs to the bounded-net algebra associated with \\(M_2(A)\\). If \\(\\|x\\|\\le1\\), then \\(\\|X\\|\\le1\\), since \\(X^2=\\operatorname{diag}(xx^*,x^*x)\\). The preceding self-adjoint argument on \\(H\\oplus H\\) gives self-adjoint contractions \\(X_i\\in M_2(A)\\) tending strongly to \\(X\\). Their upper-right entries \\(c_i\\) have norm at most one. Testing \\(X_i\\) on \\((0,\\xi)\\) gives \\(c_i\\xi\\to x\\xi\\), and testing on \\((\\xi,0)\\) gives \\(c_i^*\\xi\\to x^*\\xi\\). Consequently \\(c_i\\to x\\) strongly*. This proves that every contraction of \\(D\\) lies in the strong closure of the unit ball of \\(A\\). The reverse inclusion follows directly from the definition of \\(D\\).\n\nIt remains to prove that \\(D\\) is closed; a union of bounded closures need not be closed merely by its definition. The preceding equality shows that \\(D\\cap B(H)_1\\) is strongly closed. Scaling shows the same for \\(D\\cap rB(H)_1\\), for every \\(r>0\\). Since \\(D\\) is convex, Theorem 9.6 of the operator-topologies lesson, proved from trace duality and Krein–Šmulian, makes \\(D\\) ultraweakly closed. Since \\(D\\) is a *-subalgebra, Theorem 4.4 of the double commutant lesson identifies its ultraweak and strong closures, also in the degenerate case. Hence \\(D\\) is strongly closed. It contains \\(A\\) and is contained in its strong closure, so equality follows. This proof has not used Theorem 7.1 or the unbounded-net continuity theorem. \\(\\square\\)\n\nFor a *-algebra \\(A_0\\) that is not norm closed, apply the theorem to its norm closure \\(A\\). The unit ball of \\(A_0\\) is norm dense in that of \\(A\\): if \\(\\|a\\|\\le1\\) and \\(\\|b-a\\|<\\delta\\), with \\(b\\in A_0\\), then \\(b/(1+\\delta)\\) is a contraction and tends in norm to \\(a\\). Self-adjoint \\(a\\) can first be approximated by \\((b+b^*)/2\\). Thus the general contraction density follows without changing the strong closure. Positivity for a nonclosed *-algebra is handled by the square-root approximation in the proof of Theorem 7.1; functional calculus need not stay in \\(A_0\\).\n\n\n",
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      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "8. The unitary group and its closures",
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      "full_conditions_and_proof": "## 8. The unitary group and its closures\n\n**Theorem 8.1.** Let \\(U(H)\\) be the unitary group of \\(B(H)\\).\n\n1. \\(U(H)\\) is closed for the strong\\(^*\\) topology.\n2. On the set of isometries, the weak and the strong topologies coincide. On \\(U(H)\\), all six operator topologies coincide.\n3. The strong closure of \\(U(H)\\) is the set of all isometries.\n4. If \\(\\dim H=\\infty\\), the weak closure of \\(U(H)\\) is the unit ball \\(S\\). In particular, it contains all projections and all partial isometries. If \\(\\dim H<\\infty\\), \\(U(H)\\) is compact.\n5. \\(U(H)\\) is complete for the strong\\(^*\\) uniform structure: if \\((u_i)\\) is a net of unitaries such that \\((u_i\\xi)\\) and \\((u_i^*\\xi)\\) are Cauchy nets for every \\(\\xi\\), then it converges strongly\\(^*\\) to a unitary. If \\(\\dim H=\\infty\\), \\(U(H)\\) is not complete for the strong uniform structure.\n6. \\(U(H)\\) is relatively compact in the weak topology. If \\(\\dim H=\\infty\\), it is not weakly compact.\n7. The weak closure of \\(U(H)\\) is closed under products and adjoints.\n\n**Proof.** (1) Let \\(u_i\\to u\\) strongly\\(^*\\) with \\(u_i\\) unitary. Then \\(\\|u\\xi\\|=\\lim\\|u_i\\xi\\|=\\|\\xi\\|\\) and \\(\\|u^*\\xi\\|=\\lim\\|u_i^*\\xi\\|=\\|\\xi\\|\\). So \\(u\\) and \\(u^*\\) are isometries, and \\(u\\) is unitary.\n\n(2) Let \\(v_i\\to v\\) weakly, where all \\(v_i\\) and \\(v\\) are isometries. Then\n\\[\n\\begin{gathered}\n\\|(v_i-v)\\xi\\|^2\\\\\n=\\|v_i\\xi\\|^2+\\|v\\xi\\|^2-2\\operatorname{Re}\\langle v_i\\xi,v\\xi\\rangle\\\\\n=2\\|\\xi\\|^2-2\\operatorname{Re}\\langle v_i\\xi,v\\xi\\rangle\\to2\\|\\xi\\|^2-2\\|v\\xi\\|^2\\\\\n=0 .\n\\end{gathered}\n\\]\nOn \\(U(H)\\), strong convergence implies strong\\(^*\\) convergence (Corollary 2.4), and \\(U(H)\\) is bounded, so Lemma 1.1(3) adds the three \\(\\sigma\\)-topologies.\n\n(3) A strong limit of isometries is an isometry, since \\(\\|v\\xi\\|=\\lim\\|v_i\\xi\\|\\). Conversely, let \\(v\\) be an isometry and \\(F\\subseteq H\\) a finite-dimensional subspace. Put \\(F'=F+vF\\). The map \\(v|_F\\) is a linear isometry of \\(F\\) onto \\(vF\\), and \\(F'\\ominus F\\) and \\(F'\\ominus vF\\) have the same finite dimension \\(\\dim F'-\\dim F\\). Extending \\(v|_F\\) by a unitary map of \\(F'\\ominus F\\) onto \\(F'\\ominus vF\\) gives a unitary \\(W\\) of \\(F'\\). Let \\(u_F\\) be \\(W\\) on \\(F'\\) and \\(1\\) on \\(F'^\\perp\\). Then \\(u_F\\) is unitary and \\(u_F=v\\) on \\(F\\). Directed by inclusion, \\(u_F\\to v\\) strongly: for \\(\\xi\\in F_0\\) and \\(F\\supseteq F_0\\), \\(u_F\\xi=v\\xi\\).\n\n(4) Let \\(\\dim H=\\infty\\) and \\(x\\in S\\). Let \\(F\\) be a finite-dimensional subspace and \\(c=P_Fx|_F\\), a contraction on \\(F\\), where \\(P_F\\) is the projection onto \\(F\\). Since \\(F^\\perp\\) is infinite-dimensional, it contains a subspace \\(E\\) with an isometry \\(J\\) of \\(F\\) onto \\(E\\). Put \\(D=(1-c^*c)^{1/2}\\) and \\(D_*=(1-cc^*)^{1/2}\\), operators on \\(F\\), and define \\(u\\) by\n\\[\n\\begin{gathered}\nu(\\xi+J\\eta+\\zeta)\\\\\n=\\big(c\\xi+D_*\\eta\\big)+J\\big(D\\xi-c^*\\eta\\big)+\\zeta\\\\\n(\\xi,\\eta\\in F,\\ \\zeta\\perp F\\oplus E).\n\\end{gathered}\n\\]\nOn \\(F\\oplus E\\), identified with \\(F\\oplus F\\), \\(u\\) is the block matrix \\(W=\\begin{pmatrix}c&D_*\\\\D&-c^*\\end{pmatrix}\\). The relations \\(cD=D_*c\\) and \\(c^*D_*=Dc^*\\) hold, because \\(cq(c^*c)=q(cc^*)c\\) for every polynomial \\(q\\), and hence for every continuous function in place of \\(q\\). With them,\n\\[\n\\begin{gathered}\nW^*W\\\\\n=\\begin{pmatrix}c^*c+D^2&c^*D_*-Dc^*\\\\D_*c-cD&D_*^2+cc^*\\end{pmatrix}\\\\\n=1,\\\\\nWW^*\\\\\n=\\begin{pmatrix}cc^*+D_*^2&cD-D_*c\\\\Dc^*-c^*D_*&D^2+c^*c\\end{pmatrix}\\\\\n=1 .\n\\end{gathered}\n\\]\nSo \\(u\\) is unitary, and \\(P_FuP_F=c=P_FxP_F\\). Hence \\(\\langle u\\xi,\\eta\\rangle=\\langle x\\xi,\\eta\\rangle\\) for all \\(\\xi,\\eta\\in F\\). Every weak neighbourhood of \\(x\\) is determined by finitely many vectors, and \\(F\\) can be chosen to contain them, so \\(x\\) lies in the weak closure of \\(U(H)\\). Conversely, the weak closure lies in \\(S\\), which is weakly closed (Lemma 1.2). If \\(\\dim H<\\infty\\), \\(U(H)\\) is closed and bounded in the finite-dimensional space \\(B(H)\\).\n\n(5) If \\((u_i\\xi)\\) and \\((u_i^*\\xi)\\) are Cauchy for every \\(\\xi\\), let \\(x\\xi\\) and \\(y\\xi\\) be their limits. Then \\(x\\) and \\(y\\) are linear with norm at most \\(1\\), and \\(\\langle x\\xi,\\eta\\rangle=\\lim\\langle\\xi,u_i^*\\eta\\rangle=\\langle\\xi,y\\eta\\rangle\\), so \\(y=x^*\\). Thus \\(u_i\\to x\\) strongly\\(^*\\), and \\(x\\) is unitary by (1). If \\(\\dim H=\\infty\\), the sequence \\((u_n)\\) of Example 2.5 converges strongly in \\(B(H)\\), so it is Cauchy for the strong uniform structure; its only possible limit is the non-unitary isometry \\(v\\).\n\n(6) The map \\(x\\mapsto(\\langle x\\xi,\\eta\\rangle)_{\\xi,\\eta}\\), over pairs of unit vectors, identifies \\(S\\) with its image in the product of closed unit discs, and the weak topology with the product topology. The image is closed, because a pointwise limit of bounded sesquilinear forms of norm at most \\(1\\) is again such a form and so comes from an operator in \\(S\\). By Tychonoff's theorem, \\(S\\) is weakly compact, so \\(U(H)\\subseteq S\\) is relatively compact. If \\(\\dim H=\\infty\\), \\(U(H)\\) is not weakly closed by (4), so it is not weakly compact.\n\n(7) If \\(\\dim H=\\infty\\), the weak closure is \\(S\\) by (4). If \\(\\dim H<\\infty\\), it is \\(U(H)\\) itself. \\(\\square\\)\n\nThe next result describes which positive contractions are products of two projections, a question that arises when one tries to reach all of \\(S\\) from projections and unitaries by products.\n\n**Proposition 8.2** (products of two projections). Let \\(0\\le h\\le1\\), and let \\(R\\) be the closure of the range of \\(h-h^2\\). The following are equivalent.\n\n1. \\(h=efe\\) for two projections \\(e\\) and \\(f\\).\n2. There is a linear isometry of \\(R\\) into \\(\\ker h\\).\n\nCondition (2) holds, for example, if \\(H\\) is separable and \\(\\ker h\\) is infinite-dimensional, or if \\(\\ker h\\) contains a closed subspace isometric to \\(H\\).\n\n**Proof.** Let \\(E\\) be the closure of the range of \\(h\\); then \\(\\ker h=E^\\perp\\), and \\(R\\subseteq E\\). Write \\(T\\) for the restriction of \\(h\\) to \\(E\\). Since \\(h-h^2\\) vanishes on \\(\\ker h\\), \\(R\\) is also the closure of the range of \\(T-T^2\\).\n\n(2)\\(\\Rightarrow\\)(1). Let \\(V_0:R\\to\\ker h\\) be a linear isometry, and let \\(V:E\\to\\ker h\\) be \\(V_0\\) on \\(R\\) and \\(0\\) on \\(E\\ominus R\\). Then \\(V^*V=P_R\\), the projection of \\(E\\) onto \\(R\\), which commutes with \\(T\\), and \\((T-T^2)^{1/2}P_R=(T-T^2)^{1/2}\\). With respect to \\(H=E\\oplus\\ker h\\) define\n\\[\n\\begin{gathered}\nf\\\\\n=\\begin{pmatrix}T&(T-T^2)^{1/2}V^*\\\\V(T-T^2)^{1/2}&V(1-T)V^*\\end{pmatrix},\\\\\ne\\\\\n=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}.\n\\end{gathered}\n\\]\nClearly \\(f=f^*\\) and \\(efe=h\\). To see \\(f^2=f\\), write \\(B=(T-T^2)^{1/2}V^*\\) and \\(D=V(1-T)V^*\\). Then \\[\n\\begin{gathered}\nT^2+BB^*\\\\\n=T^2+(T-T^2)^{1/2}P_R(T-T^2)^{1/2}\\\\\n=T;\n\\end{gathered}\n\\] \\[\n\\begin{gathered}\nTB+BD\\\\\n=\\big(T+(1-T)\\big)(T-T^2)^{1/2}V^*\\\\\n=B,\n\\end{gathered}\n\\] using \\(V^*V=P_R\\); and \\[\n\\begin{gathered}\nB^*B+D^2\\\\\n=V\\big((T-T^2)+(1-T)^2\\big)V^*\\\\\n=V(1-T)V^*\\\\\n=D.\n\\end{gathered}\n\\]\n\n(1)\\(\\Rightarrow\\)(2). Let \\(h=efe\\), and put \\(E'=eH\\). The range of \\(h\\) lies in \\(E'\\), so \\(E\\subseteq E'\\) and \\(E'^\\perp\\subseteq\\ker h\\). Let \\(T'\\) be the restriction of \\(h\\) to \\(E'\\) and \\(B'=ef(1-e)\\), viewed as an operator from \\(E'^\\perp\\) to \\(E'\\). The corner of \\(f^2=f\\) on \\(E'\\) gives \\(T'^2+B'B'^*=T'\\), so \\(B'B'^*=T'-T'^2\\). Define \\(V\\) on the range of \\((T'-T'^2)^{1/2}\\) by \\(V\\big((T'-T'^2)^{1/2}\\xi\\big)=B'^*\\xi\\). It is well defined and isometric, since \\(\\|B'^*\\xi\\|^2=\\langle B'B'^*\\xi,\\xi\\rangle=\\|(T'-T'^2)^{1/2}\\xi\\|^2\\). It extends to an isometry from the closure of that range, which is \\(R\\), into \\(E'^\\perp\\subseteq\\ker h\\).\n\nFor the last statement: if \\(H\\) is separable, then \\(R\\) and \\(\\ker h\\) have orthonormal bases that are finite or countable, the second one infinite, and mapping the first basis into the second gives an isometry. If there is an isometry of \\(H\\) into \\(\\ker h\\), restrict it to \\(R\\). \\(\\square\\)\n\n**Example 8.3** (an infinite-dimensional kernel is not enough). Let \\(K\\) be a nonseparable Hilbert space, for instance \\(\\ell^2(\\Gamma)\\) for an uncountable set \\(\\Gamma\\), let \\(N\\) be a separable infinite-dimensional Hilbert space, and let \\(h=\\frac12\\cdot1_K\\oplus0_N\\) on \\(H=K\\oplus N\\). Then \\(0\\le h\\le1\\), and \\(\\ker h=N\\) is infinite-dimensional. But \\(h-h^2=\\frac14\\cdot1_K\\oplus0\\), so \\(R=K\\) is nonseparable. An isometric image of a nonseparable space is nonseparable, while every subset of the separable space \\(N\\) is separable. So there is no isometry of \\(R\\) into \\(\\ker h\\), and \\(h\\) is not a product \\(efe\\) of two projections.\n\nProposition 8.2 proves the exact dimension criterion; Example 8.3 proves directly that infinite-dimensional null space alone does not suffice on a nonseparable space.\n\nIn finite dimensions, Proposition 8.2 says that \\(h=efe\\) exactly when the rank of \\(h-h^2\\) is at most \\(\\dim\\ker h\\). For example, \\(\\operatorname{diag}(\\frac12,0)\\) on \\(\\mathbb C^2\\) equals \\(efe\\) with \\(e=\\operatorname{diag}(1,0)\\) and \\(f\\) the projection onto \\(\\mathbb C(1,1)\\), while \\(\\frac12\\) on \\(\\mathbb C\\) is not a product of two projections of \\(\\mathbb C\\).\n\n",
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      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "8. The unitary group and its closures",
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      "full_conditions_and_proof": "## 8. The unitary group and its closures\n\n**Theorem 8.1.** Let \\(U(H)\\) be the unitary group of \\(B(H)\\).\n\n1. \\(U(H)\\) is closed for the strong\\(^*\\) topology.\n2. On the set of isometries, the weak and the strong topologies coincide. On \\(U(H)\\), all six operator topologies coincide.\n3. The strong closure of \\(U(H)\\) is the set of all isometries.\n4. If \\(\\dim H=\\infty\\), the weak closure of \\(U(H)\\) is the unit ball \\(S\\). In particular, it contains all projections and all partial isometries. If \\(\\dim H<\\infty\\), \\(U(H)\\) is compact.\n5. \\(U(H)\\) is complete for the strong\\(^*\\) uniform structure: if \\((u_i)\\) is a net of unitaries such that \\((u_i\\xi)\\) and \\((u_i^*\\xi)\\) are Cauchy nets for every \\(\\xi\\), then it converges strongly\\(^*\\) to a unitary. If \\(\\dim H=\\infty\\), \\(U(H)\\) is not complete for the strong uniform structure.\n6. \\(U(H)\\) is relatively compact in the weak topology. If \\(\\dim H=\\infty\\), it is not weakly compact.\n7. The weak closure of \\(U(H)\\) is closed under products and adjoints.\n\n**Proof.** (1) Let \\(u_i\\to u\\) strongly\\(^*\\) with \\(u_i\\) unitary. Then \\(\\|u\\xi\\|=\\lim\\|u_i\\xi\\|=\\|\\xi\\|\\) and \\(\\|u^*\\xi\\|=\\lim\\|u_i^*\\xi\\|=\\|\\xi\\|\\). So \\(u\\) and \\(u^*\\) are isometries, and \\(u\\) is unitary.\n\n(2) Let \\(v_i\\to v\\) weakly, where all \\(v_i\\) and \\(v\\) are isometries. Then\n\\[\n\\begin{gathered}\n\\|(v_i-v)\\xi\\|^2\\\\\n=\\|v_i\\xi\\|^2+\\|v\\xi\\|^2-2\\operatorname{Re}\\langle v_i\\xi,v\\xi\\rangle\\\\\n=2\\|\\xi\\|^2-2\\operatorname{Re}\\langle v_i\\xi,v\\xi\\rangle\\to2\\|\\xi\\|^2-2\\|v\\xi\\|^2\\\\\n=0 .\n\\end{gathered}\n\\]\nOn \\(U(H)\\), strong convergence implies strong\\(^*\\) convergence (Corollary 2.4), and \\(U(H)\\) is bounded, so Lemma 1.1(3) adds the three \\(\\sigma\\)-topologies.\n\n(3) A strong limit of isometries is an isometry, since \\(\\|v\\xi\\|=\\lim\\|v_i\\xi\\|\\). Conversely, let \\(v\\) be an isometry and \\(F\\subseteq H\\) a finite-dimensional subspace. Put \\(F'=F+vF\\). The map \\(v|_F\\) is a linear isometry of \\(F\\) onto \\(vF\\), and \\(F'\\ominus F\\) and \\(F'\\ominus vF\\) have the same finite dimension \\(\\dim F'-\\dim F\\). Extending \\(v|_F\\) by a unitary map of \\(F'\\ominus F\\) onto \\(F'\\ominus vF\\) gives a unitary \\(W\\) of \\(F'\\). Let \\(u_F\\) be \\(W\\) on \\(F'\\) and \\(1\\) on \\(F'^\\perp\\). Then \\(u_F\\) is unitary and \\(u_F=v\\) on \\(F\\). Directed by inclusion, \\(u_F\\to v\\) strongly: for \\(\\xi\\in F_0\\) and \\(F\\supseteq F_0\\), \\(u_F\\xi=v\\xi\\).\n\n(4) Let \\(\\dim H=\\infty\\) and \\(x\\in S\\). Let \\(F\\) be a finite-dimensional subspace and \\(c=P_Fx|_F\\), a contraction on \\(F\\), where \\(P_F\\) is the projection onto \\(F\\). Since \\(F^\\perp\\) is infinite-dimensional, it contains a subspace \\(E\\) with an isometry \\(J\\) of \\(F\\) onto \\(E\\). Put \\(D=(1-c^*c)^{1/2}\\) and \\(D_*=(1-cc^*)^{1/2}\\), operators on \\(F\\), and define \\(u\\) by\n\\[\n\\begin{gathered}\nu(\\xi+J\\eta+\\zeta)\\\\\n=\\big(c\\xi+D_*\\eta\\big)+J\\big(D\\xi-c^*\\eta\\big)+\\zeta\\\\\n(\\xi,\\eta\\in F,\\ \\zeta\\perp F\\oplus E).\n\\end{gathered}\n\\]\nOn \\(F\\oplus E\\), identified with \\(F\\oplus F\\), \\(u\\) is the block matrix \\(W=\\begin{pmatrix}c&D_*\\\\D&-c^*\\end{pmatrix}\\). The relations \\(cD=D_*c\\) and \\(c^*D_*=Dc^*\\) hold, because \\(cq(c^*c)=q(cc^*)c\\) for every polynomial \\(q\\), and hence for every continuous function in place of \\(q\\). With them,\n\\[\n\\begin{gathered}\nW^*W\\\\\n=\\begin{pmatrix}c^*c+D^2&c^*D_*-Dc^*\\\\D_*c-cD&D_*^2+cc^*\\end{pmatrix}\\\\\n=1,\\\\\nWW^*\\\\\n=\\begin{pmatrix}cc^*+D_*^2&cD-D_*c\\\\Dc^*-c^*D_*&D^2+c^*c\\end{pmatrix}\\\\\n=1 .\n\\end{gathered}\n\\]\nSo \\(u\\) is unitary, and \\(P_FuP_F=c=P_FxP_F\\). Hence \\(\\langle u\\xi,\\eta\\rangle=\\langle x\\xi,\\eta\\rangle\\) for all \\(\\xi,\\eta\\in F\\). Every weak neighbourhood of \\(x\\) is determined by finitely many vectors, and \\(F\\) can be chosen to contain them, so \\(x\\) lies in the weak closure of \\(U(H)\\). Conversely, the weak closure lies in \\(S\\), which is weakly closed (Lemma 1.2). If \\(\\dim H<\\infty\\), \\(U(H)\\) is closed and bounded in the finite-dimensional space \\(B(H)\\).\n\n(5) If \\((u_i\\xi)\\) and \\((u_i^*\\xi)\\) are Cauchy for every \\(\\xi\\), let \\(x\\xi\\) and \\(y\\xi\\) be their limits. Then \\(x\\) and \\(y\\) are linear with norm at most \\(1\\), and \\(\\langle x\\xi,\\eta\\rangle=\\lim\\langle\\xi,u_i^*\\eta\\rangle=\\langle\\xi,y\\eta\\rangle\\), so \\(y=x^*\\). Thus \\(u_i\\to x\\) strongly\\(^*\\), and \\(x\\) is unitary by (1). If \\(\\dim H=\\infty\\), the sequence \\((u_n)\\) of Example 2.5 converges strongly in \\(B(H)\\), so it is Cauchy for the strong uniform structure; its only possible limit is the non-unitary isometry \\(v\\).\n\n(6) The map \\(x\\mapsto(\\langle x\\xi,\\eta\\rangle)_{\\xi,\\eta}\\), over pairs of unit vectors, identifies \\(S\\) with its image in the product of closed unit discs, and the weak topology with the product topology. The image is closed, because a pointwise limit of bounded sesquilinear forms of norm at most \\(1\\) is again such a form and so comes from an operator in \\(S\\). By Tychonoff's theorem, \\(S\\) is weakly compact, so \\(U(H)\\subseteq S\\) is relatively compact. If \\(\\dim H=\\infty\\), \\(U(H)\\) is not weakly closed by (4), so it is not weakly compact.\n\n(7) If \\(\\dim H=\\infty\\), the weak closure is \\(S\\) by (4). If \\(\\dim H<\\infty\\), it is \\(U(H)\\) itself. \\(\\square\\)\n\nThe next result describes which positive contractions are products of two projections, a question that arises when one tries to reach all of \\(S\\) from projections and unitaries by products.\n\n**Proposition 8.2** (products of two projections). Let \\(0\\le h\\le1\\), and let \\(R\\) be the closure of the range of \\(h-h^2\\). The following are equivalent.\n\n1. \\(h=efe\\) for two projections \\(e\\) and \\(f\\).\n2. There is a linear isometry of \\(R\\) into \\(\\ker h\\).\n\nCondition (2) holds, for example, if \\(H\\) is separable and \\(\\ker h\\) is infinite-dimensional, or if \\(\\ker h\\) contains a closed subspace isometric to \\(H\\).\n\n**Proof.** Let \\(E\\) be the closure of the range of \\(h\\); then \\(\\ker h=E^\\perp\\), and \\(R\\subseteq E\\). Write \\(T\\) for the restriction of \\(h\\) to \\(E\\). Since \\(h-h^2\\) vanishes on \\(\\ker h\\), \\(R\\) is also the closure of the range of \\(T-T^2\\).\n\n(2)\\(\\Rightarrow\\)(1). Let \\(V_0:R\\to\\ker h\\) be a linear isometry, and let \\(V:E\\to\\ker h\\) be \\(V_0\\) on \\(R\\) and \\(0\\) on \\(E\\ominus R\\). Then \\(V^*V=P_R\\), the projection of \\(E\\) onto \\(R\\), which commutes with \\(T\\), and \\((T-T^2)^{1/2}P_R=(T-T^2)^{1/2}\\). With respect to \\(H=E\\oplus\\ker h\\) define\n\\[\n\\begin{gathered}\nf\\\\\n=\\begin{pmatrix}T&(T-T^2)^{1/2}V^*\\\\V(T-T^2)^{1/2}&V(1-T)V^*\\end{pmatrix},\\\\\ne\\\\\n=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}.\n\\end{gathered}\n\\]\nClearly \\(f=f^*\\) and \\(efe=h\\). To see \\(f^2=f\\), write \\(B=(T-T^2)^{1/2}V^*\\) and \\(D=V(1-T)V^*\\). Then \\[\n\\begin{gathered}\nT^2+BB^*\\\\\n=T^2+(T-T^2)^{1/2}P_R(T-T^2)^{1/2}\\\\\n=T;\n\\end{gathered}\n\\] \\[\n\\begin{gathered}\nTB+BD\\\\\n=\\big(T+(1-T)\\big)(T-T^2)^{1/2}V^*\\\\\n=B,\n\\end{gathered}\n\\] using \\(V^*V=P_R\\); and \\[\n\\begin{gathered}\nB^*B+D^2\\\\\n=V\\big((T-T^2)+(1-T)^2\\big)V^*\\\\\n=V(1-T)V^*\\\\\n=D.\n\\end{gathered}\n\\]\n\n(1)\\(\\Rightarrow\\)(2). Let \\(h=efe\\), and put \\(E'=eH\\). The range of \\(h\\) lies in \\(E'\\), so \\(E\\subseteq E'\\) and \\(E'^\\perp\\subseteq\\ker h\\). Let \\(T'\\) be the restriction of \\(h\\) to \\(E'\\) and \\(B'=ef(1-e)\\), viewed as an operator from \\(E'^\\perp\\) to \\(E'\\). The corner of \\(f^2=f\\) on \\(E'\\) gives \\(T'^2+B'B'^*=T'\\), so \\(B'B'^*=T'-T'^2\\). Define \\(V\\) on the range of \\((T'-T'^2)^{1/2}\\) by \\(V\\big((T'-T'^2)^{1/2}\\xi\\big)=B'^*\\xi\\). It is well defined and isometric, since \\(\\|B'^*\\xi\\|^2=\\langle B'B'^*\\xi,\\xi\\rangle=\\|(T'-T'^2)^{1/2}\\xi\\|^2\\). It extends to an isometry from the closure of that range, which is \\(R\\), into \\(E'^\\perp\\subseteq\\ker h\\).\n\nFor the last statement: if \\(H\\) is separable, then \\(R\\) and \\(\\ker h\\) have orthonormal bases that are finite or countable, the second one infinite, and mapping the first basis into the second gives an isometry. If there is an isometry of \\(H\\) into \\(\\ker h\\), restrict it to \\(R\\). \\(\\square\\)\n\n**Example 8.3** (an infinite-dimensional kernel is not enough). Let \\(K\\) be a nonseparable Hilbert space, for instance \\(\\ell^2(\\Gamma)\\) for an uncountable set \\(\\Gamma\\), let \\(N\\) be a separable infinite-dimensional Hilbert space, and let \\(h=\\frac12\\cdot1_K\\oplus0_N\\) on \\(H=K\\oplus N\\). Then \\(0\\le h\\le1\\), and \\(\\ker h=N\\) is infinite-dimensional. But \\(h-h^2=\\frac14\\cdot1_K\\oplus0\\), so \\(R=K\\) is nonseparable. An isometric image of a nonseparable space is nonseparable, while every subset of the separable space \\(N\\) is separable. So there is no isometry of \\(R\\) into \\(\\ker h\\), and \\(h\\) is not a product \\(efe\\) of two projections.\n\nProposition 8.2 proves the exact dimension criterion; Example 8.3 proves directly that infinite-dimensional null space alone does not suffice on a nonseparable space.\n\nIn finite dimensions, Proposition 8.2 says that \\(h=efe\\) exactly when the rank of \\(h-h^2\\) is at most \\(\\dim\\ker h\\). For example, \\(\\operatorname{diag}(\\frac12,0)\\) on \\(\\mathbb C^2\\) equals \\(efe\\) with \\(e=\\operatorname{diag}(1,0)\\) and \\(f\\) the projection onto \\(\\mathbb C(1,1)\\), while \\(\\frac12\\) on \\(\\mathbb C\\) is not a product of two projections of \\(\\mathbb C\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "the-double-commutant-theorem",
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      ]
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    {
      "id": "OA-FND-KD-10",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "9. Density of unitary groups",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "anchor": "oa-fnd-kd-10",
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        "line": 571,
        "through_line": 601
      },
      "full_conditions_and_proof": "## 9. Density of unitary groups\n\n**Theorem 9.1** (density of unitary groups). Let \\(A\\) be a concrete C\\(^*\\)-algebra on \\(H\\) with \\(1\\in A\\), and let \\(M=A''\\), its weak closure. For \\(\\lambda>0\\) put\n\\[\n\\begin{gathered}\nU(A,\\lambda)\\\\\n=\\{u\\in U(A):\\|u-1\\|\\le\\lambda\\},\\\\\nU(M,\\lambda)\\\\\n=\\{u\\in U(M):\\|u-1\\|\\le\\lambda\\}.\n\\end{gathered}\n\\]\nThen \\(U(M,\\lambda)\\) is the strong\\(^*\\) closure of \\(U(A,\\lambda)\\). In particular, \\(U(A)\\) is strongly\\(^*\\) dense in \\(U(M)\\).\n\n\n**Proof.** *\\(U(M,\\lambda)\\) is strongly\\(^*\\) closed.* It is the intersection of \\(U(H)\\), which is strongly\\(^*\\) closed (Theorem 8.1(1)), with \\(M\\), which is strongly closed, and with the ball \\(\\{x:\\|x-1\\|\\le\\lambda\\}\\), which is strongly closed (Lemma 1.2). It contains \\(U(A,\\lambda)\\), hence its strong\\(^*\\) closure.\n\n*Density.* Let \\(u\\in U(M,\\lambda)\\). Put \\(\\alpha=2\\arcsin(\\min(\\lambda,2)/2)\\in(0,\\pi]\\). There is a self-adjoint \\(h\\in M\\) with \\(\\|h\\|\\le\\alpha\\) and \\(u=\\exp(ih)\\): if \\(\\lambda<2\\) use Corollary 4.5(2), noting \\(2\\arcsin(\\|u-1\\|/2)\\le\\alpha\\); if \\(\\lambda\\ge2\\), then \\(\\alpha=\\pi\\) and Corollary 4.5(1) applies. By Kaplansky's density theorem 7.1(2), applied to \\(h/\\alpha\\), there is a net \\((h_i)\\) in \\(A_h\\) with \\(\\|h_i\\|\\le\\alpha\\) and \\(h_i\\to h\\) strongly. The unitaries \\(u_i=\\exp(ih_i)\\) lie in \\(A\\), because \\(A\\) is a unital C\\(^*\\)-algebra. They satisfy\n\\[\n\\begin{gathered}\n\\|u_i-1\\|\\\\\n=\\max_{t\\in\\sigma(h_i)}|e^{it}-1|\\\\\n\\le\\max_{|t|\\le\\alpha}2|\\sin(t/2)|\\\\\n=2\\sin(\\alpha/2)\\\\\n=\\min(\\lambda,2)\\\\\n\\le\\lambda,\n\\end{gathered}\n\\]\nso \\(u_i\\in U(A,\\lambda)\\). The function \\(t\\mapsto e^{it}\\) is bounded and continuous, so \\(u_i\\to\\exp(ih)=u\\) strongly by Corollary 5.3(4), and strongly\\(^*\\) by Corollary 2.4. \\(\\square\\)\n\n**Remark 9.2** (strong density ignores topological obstructions). The proof approximates every unitary of \\(M\\) by exponentials \\(\\exp(ih)\\) with \\(h\\in A_h\\). In norm this is impossible in general. Let \\(A\\) be the algebra of multiplication operators by continuous functions on \\(L^2(\\mathbb T)\\), for the normalized arc-length measure, which is isometrically isomorphic to \\(C(\\mathbb T)\\). The unitary \\(u\\) of multiplication by \\(z\\) lies in \\(A\\). Every \\(\\exp(ih)\\) with \\(h\\in A_h\\) lies in the connected component of \\(1\\) in the invertible group of \\(A\\). This component is closed in the invertible group, and the invertible element \\(u\\) does not lie in it ([Example 7.2 of the lesson on Banach algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-14)). So \\(u\\) is not a norm limit of such exponentials. It is a strong limit of them. Let \\(h_k\\in A_h\\) be multiplication by the real continuous function \\(\\varphi_k\\) on \\(\\mathbb T\\) from the proof of Corollary 4.5. Then \\(\\exp(ih_k)\\) is multiplication by \\(e^{i\\varphi_k}\\), and \\(e^{i\\varphi_k(\\lambda)}\\to e^{i\\arg\\lambda}=\\lambda\\) at every point, so \\(\\exp(ih_k)\\to u\\) strongly by dominated convergence.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "the-double-commutant-theorem",
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    {
      "id": "OA-FND-KD-11",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "10. Approximation on large projections: the noncommutative Egoroff and Lusin theorems",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "anchor": "oa-fnd-kd-11",
      "proof_locus": {
        "line": 602,
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      },
      "full_conditions_and_proof": "## 10. Approximation on large projections: the noncommutative Egoroff and Lusin theorems\n\nEgoroff's theorem turns almost everywhere convergence into uniform convergence off a small set, and Lusin's theorem makes a measurable function continuous off a small set. In a von Neumann algebra, \"off a small set\" becomes \"on a projection \\(f\\le e\\) with \\(\\varphi(e-f)\\) small\", where \\(\\varphi\\) is a positive normal functional and \\(e\\) a given projection. Throughout this section \\(M\\) denotes a von Neumann algebra on \\(H\\), and in Corollary 10.4 and Theorem 10.6, \\(A\\) is a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), with \\(M=A''\\). For a degenerate \\(A\\) one restricts to \\([AH]\\), where \\(A\\) acts nondegenerately and its weak closure is its bicommutant ([the double commutant theorem, Step 3](the-double-commutant-theorem.md#oa-fnd-bi-07)).\n\n**Lemma 10.1** (cutting down a null net). Let \\(e\\in M\\) be a projection and \\((x_i)\\) a bounded net in \\(M\\) with \\(x_ie\\to0\\) strongly, and let \\(\\varepsilon>0\\). Then there are projections \\(e_i\\le e\\) in \\(M\\) with \\(e_i\\to e\\) strongly and \\(\\|x_ie_i\\|\\le\\varepsilon\\) for every \\(i\\).\n\n**Proof.** Let \\(c=\\sup_i\\|x_i\\|\\) and \\(y_i=ex_i^*x_ie\\in M_+\\). Then \\(y_i=ey_ie\\), and \\(\\|y_i\\xi\\|\\le c\\|x_ie\\xi\\|\\to0\\), so \\(y_i\\to0\\) strongly. Let \\(p_i=1_{(\\varepsilon^2,\\infty)}(y_i)\\). By Corollary 4.3, \\(p_i\\) is a projection in \\(M\\) that commutes with \\(y_i\\) and with \\(e\\), \\(p_i\\le e\\), and \\(\\varepsilon^2p_i\\le y_ip_i\\le y_i\\), the last because \\(y_i(1-p_i)=(1-p_i)y_i(1-p_i)\\ge0\\). So \\(\\varepsilon^2\\|p_i\\xi\\|^2\\le\\langle y_i\\xi,\\xi\\rangle\\to0\\), and \\(p_i\\to0\\) strongly. Put \\(e_i=e-p_i\\), a projection with \\(e_i\\to e\\) strongly. Since \\(e_i\\le1-p_i\\) and \\(y_i(1-p_i)\\le\\varepsilon^2(1-p_i)\\) (Corollary 4.3(2)),\n\\[\n\\begin{gathered}\n\\|x_ie_i\\|^2\\\\\n=\\|e_iy_ie_i\\|\\\\\n=\\|e_iy_i(1-p_i)e_i\\|\\\\\n\\le\\varepsilon^2\\|e_i(1-p_i)e_i\\|\\\\\n\\le\\varepsilon^2 .\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Theorem 10.2** (noncommutative Egoroff theorem). Let \\(B\\subseteq M\\) be a bounded set, \\(x\\) an element of its strong closure, \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection and \\(\\varepsilon>0\\). Then there are a projection \\(f\\le e\\) in \\(M\\) and a sequence \\((b_k)\\) in \\(B\\) with\n\\[\n\\begin{gathered}\n\\lim_{k\\to\\infty}\\|(b_k-x)f\\|\\\\\n=0\\\\\n\\text{and}\\\\\n\\varphi(e-f)<\\varepsilon .\n\\end{gathered}\n\\]\n\n**Proof.** Choose a net \\((b_i)_{i\\in I}\\) in \\(B\\) with \\(b_i\\to x\\) strongly, and put \\(d_i=b_i-x\\), a bounded net with \\(d_i\\to0\\) strongly. We choose indices \\(i_1\\le i_2\\le\\cdots\\) and projections \\(e=f_0\\ge f_1\\ge f_2\\ge\\cdots\\) in \\(M\\) with\n\\[\n\\begin{gathered}\n\\|d_{i_k}f_k\\|\\\\\n\\le2^{-k}\\\\\n\\text{and}\\\\\n\\varphi(f_{k-1}-f_k)<2^{-k}\\varepsilon .\n\\end{gathered}\n\\tag{10.1}\n\\]\nSuppose \\(i_1,\\dots,i_{k-1}\\) and \\(f_0,\\dots,f_{k-1}\\) are chosen (for \\(k=1\\), let \\(i_0\\) be any index). The net \\((d_if_{k-1})_{i\\ge i_{k-1}}\\) is bounded and tends to \\(0\\) strongly. Lemma 10.1, with the projection \\(f_{k-1}\\), gives projections \\(g_i\\le f_{k-1}\\) with \\(g_i\\to f_{k-1}\\) strongly and \\(\\|d_ig_i\\|=\\|d_if_{k-1}g_i\\|\\le2^{-k}\\). The bounded net \\((f_{k-1}-g_i)\\) tends to \\(0\\) strongly and \\(\\varphi\\) is normal, so \\(\\varphi(f_{k-1}-g_i)\\to0\\) (Lemma 1.1(3)). Choose \\(i_k\\ge i_{k-1}\\) with \\(\\varphi(f_{k-1}-g_{i_k})<2^{-k}\\varepsilon\\), and put \\(f_k=g_{i_k}\\).\n\nLet \\(f\\) be the strong limit of the decreasing sequence \\((f_k)\\), a projection in \\(M\\) with \\(f\\le e\\) (Corollary 1.4). By normality, \\(\\varphi(e-f)=\\sum_k\\varphi(f_{k-1}-f_k)<\\varepsilon\\). For each \\(k\\), \\(f=f_kf\\), so \\(\\|(b_{i_k}-x)f\\|=\\|d_{i_k}f_kf\\|\\le2^{-k}\\). The sequence \\(b_k=b_{i_k}\\) has the required property. \\(\\square\\)\n\nThe construction needs care at one point. At step \\(k\\) the projection \\(f_k\\) is chosen for the single index \\(i_k\\), and nothing is claimed about \\(\\|d_if_k\\|\\) for other indices \\(i\\ge i_k\\); in general that norm stays large. For example, on \\(\\ell^2(\\mathbb N)\\) the operators \\(d_m=\\theta_{\\delta_1,\\delta_m}\\) tend to \\(0\\) strongly, and \\(\\|d_mf\\|=\\|f\\delta_m\\|\\) for every projection \\(f\\). For \\(\\varphi=\\omega_{\\delta_1}\\) and \\(f=1-\\theta_{\\delta_2,\\delta_2}\\) we have \\(\\varphi(1-f)=0\\), yet \\(\\|d_mf\\|=1\\) for every \\(m\\ne2\\). The theorem only claims, and the proof only gives, convergence along a diagonal sequence.\n\nTo control the norms of the approximants, we need to complete a column to a self-adjoint operator without increasing the norm.\n\n**Lemma 10.3** (self-adjoint completion). Let \\(a\\in M\\) be self-adjoint, \\(e\\in M\\) a projection and \\(c=\\|ae\\|\\). There is a self-adjoint \\(b\\in M\\) with \\(be=ae\\) and \\(\\|b\\|=c\\).\n\nThe proof below gives the required norm-preserving completion explicitly.\n\n**Proof.** If \\(c=0\\), take \\(b=0\\). Otherwise put \\(P=eae\\) and \\(Q=(1-e)ae\\). Then \\[\n\\begin{gathered}\nP^2+Q^*Q\\\\\n=ea(e+1-e)ae\\\\\n=(ae)^*(ae)\\\\\n\\le c^2e.\n\\end{gathered}\n\\] So \\(c^2e-P^2\\ge Q^*Q\\ge0\\), and we let \\(D=(c^2e-P^2)^{1/2}\\in M\\). It satisfies \\(D=eDe\\), and it commutes with \\(P\\).\n\n*A contraction \\(K\\in M\\) with \\(Q=KD\\).* Define \\(K\\) on the range of \\(D\\) by \\(K(D\\xi)=Q\\xi\\). This is well defined and contractive, because \\(\\|Q\\xi\\|^2=\\langle Q^*Q\\xi,\\xi\\rangle\\le\\langle D^2\\xi,\\xi\\rangle=\\|D\\xi\\|^2\\). Extend \\(K\\) by continuity to the closure of the range of \\(D\\), and by \\(0\\) on its orthogonal complement \\(\\ker D\\). Then \\(Q=KD\\) and \\(\\|K\\|\\le1\\). For a unitary \\(v\\in M'\\), \\(v\\) commutes with \\(D\\) and \\(Q\\), so \\[\n\\begin{gathered}\nvKv^*(D\\xi)\\\\\n=vKDv^*\\xi\\\\\n=vQv^*\\xi\\\\\n=Q\\xi\\\\\n=K(D\\xi),\n\\end{gathered}\n\\] and \\(vKv^*\\) vanishes on \\(\\ker D\\), which \\(v\\) maps onto itself. So \\(vKv^*=K\\). The unitaries of \\(M'\\) span \\(M'\\) ([unitaries span a unital C\\*-algebra](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-10)), so \\(K\\in M''=M\\). Since \\(\\ker D\\supseteq(1-e)H\\) and the range of \\(K\\) lies in the closure of the range of \\(Q\\), \\(K=(1-e)Ke\\).\n\n*The completion.* Put \\(b=P+KD+DK^*-KPK^*\\). Then \\(b=b^*\\). Since \\(K^*e=0\\), \\(be=P+KD=eae+(1-e)ae=ae\\). For the norm, let \\(J=\\begin{pmatrix}P&D\\\\D&-P\\end{pmatrix}\\) on \\(eH\\oplus eH\\). As \\(P\\) and \\(D\\) commute, \\(J^2=(P^2+D^2)\\oplus(P^2+D^2)=c^2e\\oplus c^2e\\), so \\(\\|J\\|=c\\). Let \\(W:eH\\oplus eH\\to H\\), \\(W(\\xi,\\eta)=\\xi+K\\eta\\). Since \\(\\xi\\perp K\\eta\\), \\(\\|W(\\xi,\\eta)\\|^2=\\|\\xi\\|^2+\\|K\\eta\\|^2\\le\\|\\xi\\|^2+\\|\\eta\\|^2\\), so \\(\\|W\\|\\le1\\), and \\(W^*\\zeta=(e\\zeta,K^*\\zeta)\\). A direct computation gives \\[\n\\begin{gathered}\nWJW^*\\zeta\\\\\n=Pe\\zeta+DK^*\\zeta+KDe\\zeta-KPK^*\\zeta\\\\\n=b\\zeta.\n\\end{gathered}\n\\] Hence \\(\\|b\\|\\le\\|J\\|=c\\), and \\(\\|b\\|\\ge\\|be\\|=c\\). \\(\\square\\)\n\n**Corollary 10.4** (approximation with norm control). Let \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection, \\(\\varepsilon>0\\) and \\(\\delta>0\\).\n\n1. For every \\(x\\in M\\) there are \\(a\\in A\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(\\|(x-a)f\\|<\\delta\\), \\(\\|a\\|\\le\\|xe\\|\\) and \\(\\varphi(e-f)<\\varepsilon\\).\n2. If \\(x\\) is self-adjoint, \\(a\\) can be chosen self-adjoint, with the same three properties.\n3. If \\(1\\in A\\) and \\(x\\in U(M)\\), \\(a\\) can be chosen in \\(U(A)\\) with \\(\\|(x-a)f\\|<\\delta\\), \\(\\|a-1\\|\\le\\|x-1\\|\\) and \\(\\varphi(e-f)<\\varepsilon\\).\n\n**Proof.** In each case we apply Theorem 10.2 to a bounded set \\(B\\) and an element \\(y\\) of its strong closure with \\(ye=xe\\). Then \\((b_k-y)f=(b_k-y)ef=(b_k-x)f\\), and we take \\(a=b_k\\) for large \\(k\\).\n\n(1) Let \\(y=xe\\) and \\(B=A\\cap\\|xe\\|S\\). By Kaplansky's density theorem 7.1(1), \\(y\\) lies in the strong closure of \\(B\\).\n\n(2) Let \\(y\\) be the self-adjoint completion of Lemma 10.3, with \\(ye=xe\\) and \\(\\|y\\|=\\|xe\\|\\), and \\(B=A_h\\cap\\|xe\\|S\\). By Theorem 7.1(2), \\(y\\) lies in the strong closure of \\(B\\).\n\n(3) Let \\(y=x\\) and \\(B=U(A,\\|x-1\\|)\\). By Theorem 9.1, \\(x\\) lies in its strong closure. \\(\\square\\)\n\n**Lemma 10.5** (correcting a unitary). Let \\(w\\in U(M)\\) and let \\(e\\in M\\) be a projection with \\(\\|(1-w)e\\|\\le\\frac18\\). There is \\(v\\in U(M)\\) with \\(ve=we\\) and \\(\\|1-v\\|\\le6\\|(1-w)e\\|\\).\n\n**Proof.** If \\(e=0\\), take \\(v=1\\); if \\(e=1\\), take \\(v=w\\). Suppose otherwise. Put \\(\\delta=\\|(1-w)e\\|\\) and \\(f=wew^*\\), a projection in \\(M\\) with \\(fw=we\\). Since \\(e-f=(1-w)e+we(1-w^*)\\) and \\(\\|e(1-w^*)\\|=\\delta\\), we get \\(\\theta:=\\|e-f\\|\\le2\\delta\\le\\frac14\\).\n\n*A unitary map of \\((1-e)H\\) onto \\((1-f)H\\).* Let \\(T=(1-f)(1-e)\\in M\\). Writing \\(1-f=(1-e)-(f-e)\\),\n\\[\n\\begin{gathered}\nT^*T\\\\\n=(1-e)(1-f)(1-e)\\\\\n=(1-e)-(1-e)(f-e)(1-e)\\\\\n\\ge(1-\\theta)(1-e),\n\\end{gathered}\n\\]\nand in the same way \\(TT^*\\ge(1-\\theta)(1-f)\\). In the von Neumann algebra \\((1-e)M(1-e)\\) with unit \\(1-e\\), \\(|T|=(T^*T)^{1/2}\\) is invertible; let \\(|T|^{-1}\\) be its inverse there, extended by \\(0\\) on \\(eH\\). Put \\(U=T|T|^{-1}\\in M\\). Then \\(U^*U=|T|^{-1}T^*T|T|^{-1}=1-e\\), so \\(UU^*\\) is a projection, the projection onto the range of \\(U\\), which equals the range of \\(T\\). That range lies in \\((1-f)H\\), and it contains the range of \\(TT^*\\), which is \\((1-f)H\\) because \\(TT^*\\) is invertible on \\((1-f)H\\). So \\(UU^*=1-f\\).\n\n*\\(U\\) is close to \\(1-e\\).* On \\((1-e)H\\), \\(T=(1-e)-(f-e)(1-e)\\), so\n\\[\n\\begin{gathered}\nU-(1-e)\\\\\n=(1-e)\\big(|T|^{-1}-(1-e)\\big)\\\\\n-(f-e)(1-e)|T|^{-1}.\n\\end{gathered}\n\\]\nThe spectrum of \\(|T|^2\\) in \\((1-e)M(1-e)\\) lies in \\([1-\\theta,1+\\theta]\\), and for \\(|s|\\le\\theta\\le\\frac14\\) we have \\(|(1+s)^{-1/2}-1|\\le\\theta\\) and \\((1+s)^{-1/2}\\le(1-\\theta)^{-1/2}\\le\\frac2{\\sqrt3}\\). So \\(\\|U-(1-e)\\|\\le\\theta+\\frac2{\\sqrt3}\\theta\\le2.2\\,\\theta\\le4.4\\,\\delta\\).\n\n*The unitary \\(v\\).* Put \\(v=we+U\\). Then \\(ve=we\\), because \\(Ue=0\\). Next, \\(U^*we=|T|^{-1}(1-e)(1-f)fw=0\\) and \\(eU^*=0\\), so the cross terms in \\(v^*v\\) and \\(vv^*\\) vanish. Hence \\(v^*v=ew^*we+U^*U=e+(1-e)=1\\) and \\(vv^*=wew^*+UU^*=f+(1-f)=1\\). Finally \\(1-v=(1-w)e+\\big((1-e)-U\\big)\\), so \\(\\|1-v\\|\\le\\delta+4.4\\,\\delta<6\\delta\\). \\(\\square\\)\n\n**Theorem 10.6** (noncommutative Lusin theorem). Let \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection, \\(\\varepsilon>0\\) and \\(\\delta>0\\).\n\n1. For every \\(x\\in M\\) there are \\(a\\in A\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(xf=af\\), \\(\\varphi(e-f)<\\varepsilon\\) and \\(\\|a\\|\\le(1+\\delta)\\|xf\\|\\).\n2. If \\(x\\) is self-adjoint, \\(a\\) can be chosen self-adjoint, with the same properties.\n3. If \\(1\\in A\\) and \\(x\\in U(M)\\), there are \\(a\\in U(A)\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(xf=af\\), \\(\\varphi(e-f)<\\varepsilon\\) and \\(\\|a-1\\|\\le\\|x-1\\|+\\delta\\).\n\n**Proof.** (1) and (2). If \\(xe=0\\), take \\(a=0\\) and \\(f=e\\). Otherwise, replacing \\(x\\) by \\(x/\\|xe\\|\\), we may assume \\(\\|xe\\|=1\\). Choose \\(\\delta'\\in(0,\\frac14)\\) with \\((1+\\delta')/(1-2\\delta')\\le1+\\delta\\). Choose a unit vector \\(\\zeta\\in eH\\) with \\(\\|x\\zeta\\|^2\\ge1-\\delta'\\), and put \\(\\psi=\\varphi+\\omega_\\zeta\\in M_*^+\\) and \\(\\varepsilon'=\\min(\\varepsilon,\\delta'^2)\\).\n\nWe choose elements \\(a_1,a_2,\\dots\\) of \\(A\\), self-adjoint in case (2), and projections \\(e=f_0\\ge f_1\\ge f_2\\ge\\cdots\\) in \\(M\\). Suppose \\(a_1,\\dots,a_{k-1}\\) and \\(f_0,\\dots,f_{k-1}\\) are chosen, and let \\(x_{k-1}=x-\\sum_{j<k}a_j\\), which is self-adjoint in case (2). Corollary 10.4, part (1) or (2), applied to \\(x_{k-1}\\), the projection \\(f_{k-1}\\), the functional \\(\\psi\\) and the numbers \\(2^{-k}\\varepsilon'\\) and \\(2^{-k}\\delta'\\), gives \\(a_k\\) and \\(f_k\\le f_{k-1}\\) with\n\\[\n\\begin{gathered}\n\\|x_kf_k\\|<2^{-k}\\delta',\\\\\n\\|a_k\\|\\\\\n\\le\\|x_{k-1}f_{k-1}\\|,\\\\\n\\psi(f_{k-1}-f_k)<2^{-k}\\varepsilon',\n\\end{gathered}\n\\tag{10.2}\n\\]\nwhere \\(x_k=x_{k-1}-a_k\\), since \\((x_{k-1}-a_k)f_k=x_kf_k\\).\n\nBy (10.2), \\(\\|a_1\\|\\le\\|xe\\|=1\\) and \\(\\|a_k\\|<2^{-k+1}\\delta'\\) for \\(k\\ge2\\). So \\(a=\\sum_ka_k\\) converges in norm, \\(a\\in A\\), and \\(\\|a\\|\\le1+\\delta'\\); \\(a\\) is self-adjoint in case (2). Let \\(f\\) be the strong limit of the decreasing sequence \\((f_k)\\). Then \\(f\\le e\\) is a projection in \\(M\\), and by normality \\(\\psi(e-f)<\\varepsilon'\\), so \\(\\varphi(e-f)<\\varepsilon\\). Since \\(f=f_kf\\), \\(\\|(x-\\sum_{j\\le k}a_j)f\\|=\\|x_kf_kf\\|<2^{-k}\\delta'\\); letting \\(k\\to\\infty\\) gives \\(xf=af\\).\n\nIt remains to bound \\(\\|xf\\|\\) from below. We have \\(\\omega_\\zeta(e-f)=\\|(e-f)\\zeta\\|^2<\\delta'^2\\), and \\(\\|x(e-f)\\|=\\|xe(e-f)\\|\\le1\\). Since \\(\\zeta=e\\zeta\\),\n\\[\n\\begin{gathered}\n\\|xf\\|\\\\\n\\ge\\|xf\\zeta\\|\\\\\n=\\|x\\zeta-x(e-f)\\zeta\\|\\\\\n\\ge(1-\\delta')^{1/2}-\\delta'\\\\\n\\ge1-2\\delta' .\n\\end{gathered}\n\\]\nHence \\(\\|a\\|\\le1+\\delta'\\le\\frac{1+\\delta'}{1-2\\delta'}\\|xf\\|\\le(1+\\delta)\\|xf\\|\\).\n\n(3) We may assume \\(\\delta\\le1\\). Put \\(w_1=x\\) and \\(f_1=e\\). We choose \\(u_k\\in U(A)\\), \\(w_{k+1}\\in U(M)\\) and projections \\(f_{k+1}\\le f_k\\) in \\(M\\), for \\(k=1,2,\\dots\\), with\n\\[\n\\begin{gathered}\nxf_{k+1}\\\\\n=u_1u_2\\cdots u_k\\,w_{k+1}f_{k+1},\\\\\n\\|1-u_k\\|\\\\\n\\le\\|1-w_k\\|,\\\\\n\\|1-w_{k+1}\\|\\\\\n\\le2^{-k}\\delta,\\\\\n\\varphi(f_k-f_{k+1})<2^{-k}\\varepsilon .\n\\end{gathered}\n\\tag{10.3}\n\\]\nGiven \\(w_k\\) and \\(f_k\\) with \\(xf_k=u_1\\cdots u_{k-1}w_kf_k\\) (for \\(k=1\\) this reads \\(xe=w_1e\\)), Corollary 10.4(3) gives \\(u_k\\in U(A)\\) and \\(f_{k+1}\\le f_k\\) with \\(\\|(w_k-u_k)f_{k+1}\\|<2^{-k-3}\\delta\\), \\(\\|1-u_k\\|\\le\\|1-w_k\\|\\) and \\(\\varphi(f_k-f_{k+1})<2^{-k}\\varepsilon\\). The unitary \\(u_k^*w_k\\in U(M)\\) satisfies \\[\n\\begin{gathered}\n\\|(1-u_k^*w_k)f_{k+1}\\|\\\\\n=\\|(u_k-w_k)f_{k+1}\\|<2^{-k-3}\\delta\\\\\n\\le\\frac18.\n\\end{gathered}\n\\] Lemma 10.5 gives \\(w_{k+1}\\in U(M)\\) with \\(w_{k+1}f_{k+1}=u_k^*w_kf_{k+1}\\) and \\(\\|1-w_{k+1}\\|\\le6\\cdot2^{-k-3}\\delta<2^{-k}\\delta\\). Then\n\\[\n\\begin{gathered}\nxf_{k+1}\\\\\n=xf_kf_{k+1}\\\\\n=u_1\\cdots u_{k-1}w_kf_{k+1}\\\\\n=u_1\\cdots u_{k-1}u_k\\,(u_k^*w_kf_{k+1})\\\\\n=u_1\\cdots u_k\\,w_{k+1}f_{k+1},\n\\end{gathered}\n\\]\nwhich completes the step.\n\nBy (10.3), \\(\\|1-u_1\\|\\le\\|1-x\\|\\) and \\(\\|1-u_k\\|\\le\\|1-w_k\\|<2^{-k+1}\\delta\\) for \\(k\\ge2\\). Let \\(v_k=u_1\\cdots u_k\\). Then \\(\\|v_{k+1}-v_k\\|=\\|v_k(u_{k+1}-1)\\|=\\|u_{k+1}-1\\|\\), so \\((v_k)\\) converges in norm to a unitary \\(a\\in A\\). From \\[\n\\begin{gathered}\n1-v_k\\\\\n=(1-u_1)+u_1(1-u_2)\\\\\n+\\dots+u_1\\cdots u_{k-1}(1-u_k)\n\\end{gathered}\n\\] we get \\(\\|1-a\\|\\le\\sum_k\\|1-u_k\\|\\le\\|1-x\\|+\\delta\\). Let \\(f\\) be the strong limit of \\((f_k)\\), so \\(f\\le e\\) and \\(\\varphi(e-f)<\\varepsilon\\). Multiplying (10.3) by \\(f\\) on the right, \\(xf=v_kw_{k+1}f\\), so\n\\[\n\\begin{gathered}\n\\|xf-af\\|\\\\\n\\le\\|v_k(w_{k+1}-1)f\\|+\\|(v_k-a)f\\|\\\\\n\\le2^{-k}\\delta+\\|v_k-a\\|\\to0 ,\n\\end{gathered}\n\\]\nand \\(xf=af\\). \\(\\square\\)\n\n**Example 10.7** (the commutative case). Let \\(\\mu\\) be Lebesgue measure on \\([0,1]\\), \\(M\\) the algebra of multiplication operators \\(m_g\\) by bounded measurable \\(g\\) on \\(L^2[0,1]\\), \\(A\\) the multiplication operators by continuous functions, and \\(\\varphi(m_g)=\\int g\\,d\\mu\\), the vector functional of the constant function \\(1\\). The projections of \\(M\\) are the operators \\(m_{1_E}\\), and \\(\\varphi(1-m_{1_E})=\\mu([0,1]\\setminus E)\\). A bounded sequence \\(g_k\\to g\\) almost everywhere gives \\(m_{g_k}\\to m_g\\) strongly, by dominated convergence. Theorem 10.2 then yields a set \\(E\\) with small complement on which a sequence of the \\(g_k\\) converges uniformly, up to a null set: a form of Egoroff's theorem. Theorem 10.6(1) yields a continuous \\(a\\) that agrees with a bounded measurable \\(g\\) almost everywhere on a set \\(E\\) with small complement, with \\(\\max|a|\\le(1+\\delta)\\operatorname{ess\\,sup}_E|g|\\): this is Lusin's theorem with control of the norm. (That \\(M\\) is the weak closure of \\(A\\) is shown in Exercise 14.2.)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-KD-12",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "10. Approximation on large projections: the noncommutative Egoroff and Lusin theorems",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "full_conditions_and_proof": "## 10. Approximation on large projections: the noncommutative Egoroff and Lusin theorems\n\nEgoroff's theorem turns almost everywhere convergence into uniform convergence off a small set, and Lusin's theorem makes a measurable function continuous off a small set. In a von Neumann algebra, \"off a small set\" becomes \"on a projection \\(f\\le e\\) with \\(\\varphi(e-f)\\) small\", where \\(\\varphi\\) is a positive normal functional and \\(e\\) a given projection. Throughout this section \\(M\\) denotes a von Neumann algebra on \\(H\\), and in Corollary 10.4 and Theorem 10.6, \\(A\\) is a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), with \\(M=A''\\). For a degenerate \\(A\\) one restricts to \\([AH]\\), where \\(A\\) acts nondegenerately and its weak closure is its bicommutant ([the double commutant theorem, Step 3](the-double-commutant-theorem.md#oa-fnd-bi-07)).\n\n**Lemma 10.1** (cutting down a null net). Let \\(e\\in M\\) be a projection and \\((x_i)\\) a bounded net in \\(M\\) with \\(x_ie\\to0\\) strongly, and let \\(\\varepsilon>0\\). Then there are projections \\(e_i\\le e\\) in \\(M\\) with \\(e_i\\to e\\) strongly and \\(\\|x_ie_i\\|\\le\\varepsilon\\) for every \\(i\\).\n\n**Proof.** Let \\(c=\\sup_i\\|x_i\\|\\) and \\(y_i=ex_i^*x_ie\\in M_+\\). Then \\(y_i=ey_ie\\), and \\(\\|y_i\\xi\\|\\le c\\|x_ie\\xi\\|\\to0\\), so \\(y_i\\to0\\) strongly. Let \\(p_i=1_{(\\varepsilon^2,\\infty)}(y_i)\\). By Corollary 4.3, \\(p_i\\) is a projection in \\(M\\) that commutes with \\(y_i\\) and with \\(e\\), \\(p_i\\le e\\), and \\(\\varepsilon^2p_i\\le y_ip_i\\le y_i\\), the last because \\(y_i(1-p_i)=(1-p_i)y_i(1-p_i)\\ge0\\). So \\(\\varepsilon^2\\|p_i\\xi\\|^2\\le\\langle y_i\\xi,\\xi\\rangle\\to0\\), and \\(p_i\\to0\\) strongly. Put \\(e_i=e-p_i\\), a projection with \\(e_i\\to e\\) strongly. Since \\(e_i\\le1-p_i\\) and \\(y_i(1-p_i)\\le\\varepsilon^2(1-p_i)\\) (Corollary 4.3(2)),\n\\[\n\\begin{gathered}\n\\|x_ie_i\\|^2\\\\\n=\\|e_iy_ie_i\\|\\\\\n=\\|e_iy_i(1-p_i)e_i\\|\\\\\n\\le\\varepsilon^2\\|e_i(1-p_i)e_i\\|\\\\\n\\le\\varepsilon^2 .\\\\\n\\square\n\\end{gathered}\n\\]\n\n**Theorem 10.2** (noncommutative Egoroff theorem). Let \\(B\\subseteq M\\) be a bounded set, \\(x\\) an element of its strong closure, \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection and \\(\\varepsilon>0\\). Then there are a projection \\(f\\le e\\) in \\(M\\) and a sequence \\((b_k)\\) in \\(B\\) with\n\\[\n\\begin{gathered}\n\\lim_{k\\to\\infty}\\|(b_k-x)f\\|\\\\\n=0\\\\\n\\text{and}\\\\\n\\varphi(e-f)<\\varepsilon .\n\\end{gathered}\n\\]\n\n**Proof.** Choose a net \\((b_i)_{i\\in I}\\) in \\(B\\) with \\(b_i\\to x\\) strongly, and put \\(d_i=b_i-x\\), a bounded net with \\(d_i\\to0\\) strongly. We choose indices \\(i_1\\le i_2\\le\\cdots\\) and projections \\(e=f_0\\ge f_1\\ge f_2\\ge\\cdots\\) in \\(M\\) with\n\\[\n\\begin{gathered}\n\\|d_{i_k}f_k\\|\\\\\n\\le2^{-k}\\\\\n\\text{and}\\\\\n\\varphi(f_{k-1}-f_k)<2^{-k}\\varepsilon .\n\\end{gathered}\n\\tag{10.1}\n\\]\nSuppose \\(i_1,\\dots,i_{k-1}\\) and \\(f_0,\\dots,f_{k-1}\\) are chosen (for \\(k=1\\), let \\(i_0\\) be any index). The net \\((d_if_{k-1})_{i\\ge i_{k-1}}\\) is bounded and tends to \\(0\\) strongly. Lemma 10.1, with the projection \\(f_{k-1}\\), gives projections \\(g_i\\le f_{k-1}\\) with \\(g_i\\to f_{k-1}\\) strongly and \\(\\|d_ig_i\\|=\\|d_if_{k-1}g_i\\|\\le2^{-k}\\). The bounded net \\((f_{k-1}-g_i)\\) tends to \\(0\\) strongly and \\(\\varphi\\) is normal, so \\(\\varphi(f_{k-1}-g_i)\\to0\\) (Lemma 1.1(3)). Choose \\(i_k\\ge i_{k-1}\\) with \\(\\varphi(f_{k-1}-g_{i_k})<2^{-k}\\varepsilon\\), and put \\(f_k=g_{i_k}\\).\n\nLet \\(f\\) be the strong limit of the decreasing sequence \\((f_k)\\), a projection in \\(M\\) with \\(f\\le e\\) (Corollary 1.4). By normality, \\(\\varphi(e-f)=\\sum_k\\varphi(f_{k-1}-f_k)<\\varepsilon\\). For each \\(k\\), \\(f=f_kf\\), so \\(\\|(b_{i_k}-x)f\\|=\\|d_{i_k}f_kf\\|\\le2^{-k}\\). The sequence \\(b_k=b_{i_k}\\) has the required property. \\(\\square\\)\n\nThe construction needs care at one point. At step \\(k\\) the projection \\(f_k\\) is chosen for the single index \\(i_k\\), and nothing is claimed about \\(\\|d_if_k\\|\\) for other indices \\(i\\ge i_k\\); in general that norm stays large. For example, on \\(\\ell^2(\\mathbb N)\\) the operators \\(d_m=\\theta_{\\delta_1,\\delta_m}\\) tend to \\(0\\) strongly, and \\(\\|d_mf\\|=\\|f\\delta_m\\|\\) for every projection \\(f\\). For \\(\\varphi=\\omega_{\\delta_1}\\) and \\(f=1-\\theta_{\\delta_2,\\delta_2}\\) we have \\(\\varphi(1-f)=0\\), yet \\(\\|d_mf\\|=1\\) for every \\(m\\ne2\\). The theorem only claims, and the proof only gives, convergence along a diagonal sequence.\n\nTo control the norms of the approximants, we need to complete a column to a self-adjoint operator without increasing the norm.\n\n**Lemma 10.3** (self-adjoint completion). Let \\(a\\in M\\) be self-adjoint, \\(e\\in M\\) a projection and \\(c=\\|ae\\|\\). There is a self-adjoint \\(b\\in M\\) with \\(be=ae\\) and \\(\\|b\\|=c\\).\n\nThe proof below gives the required norm-preserving completion explicitly.\n\n**Proof.** If \\(c=0\\), take \\(b=0\\). Otherwise put \\(P=eae\\) and \\(Q=(1-e)ae\\). Then \\[\n\\begin{gathered}\nP^2+Q^*Q\\\\\n=ea(e+1-e)ae\\\\\n=(ae)^*(ae)\\\\\n\\le c^2e.\n\\end{gathered}\n\\] So \\(c^2e-P^2\\ge Q^*Q\\ge0\\), and we let \\(D=(c^2e-P^2)^{1/2}\\in M\\). It satisfies \\(D=eDe\\), and it commutes with \\(P\\).\n\n*A contraction \\(K\\in M\\) with \\(Q=KD\\).* Define \\(K\\) on the range of \\(D\\) by \\(K(D\\xi)=Q\\xi\\). This is well defined and contractive, because \\(\\|Q\\xi\\|^2=\\langle Q^*Q\\xi,\\xi\\rangle\\le\\langle D^2\\xi,\\xi\\rangle=\\|D\\xi\\|^2\\). Extend \\(K\\) by continuity to the closure of the range of \\(D\\), and by \\(0\\) on its orthogonal complement \\(\\ker D\\). Then \\(Q=KD\\) and \\(\\|K\\|\\le1\\). For a unitary \\(v\\in M'\\), \\(v\\) commutes with \\(D\\) and \\(Q\\), so \\[\n\\begin{gathered}\nvKv^*(D\\xi)\\\\\n=vKDv^*\\xi\\\\\n=vQv^*\\xi\\\\\n=Q\\xi\\\\\n=K(D\\xi),\n\\end{gathered}\n\\] and \\(vKv^*\\) vanishes on \\(\\ker D\\), which \\(v\\) maps onto itself. So \\(vKv^*=K\\). The unitaries of \\(M'\\) span \\(M'\\) ([unitaries span a unital C\\*-algebra](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-10)), so \\(K\\in M''=M\\). Since \\(\\ker D\\supseteq(1-e)H\\) and the range of \\(K\\) lies in the closure of the range of \\(Q\\), \\(K=(1-e)Ke\\).\n\n*The completion.* Put \\(b=P+KD+DK^*-KPK^*\\). Then \\(b=b^*\\). Since \\(K^*e=0\\), \\(be=P+KD=eae+(1-e)ae=ae\\). For the norm, let \\(J=\\begin{pmatrix}P&D\\\\D&-P\\end{pmatrix}\\) on \\(eH\\oplus eH\\). As \\(P\\) and \\(D\\) commute, \\(J^2=(P^2+D^2)\\oplus(P^2+D^2)=c^2e\\oplus c^2e\\), so \\(\\|J\\|=c\\). Let \\(W:eH\\oplus eH\\to H\\), \\(W(\\xi,\\eta)=\\xi+K\\eta\\). Since \\(\\xi\\perp K\\eta\\), \\(\\|W(\\xi,\\eta)\\|^2=\\|\\xi\\|^2+\\|K\\eta\\|^2\\le\\|\\xi\\|^2+\\|\\eta\\|^2\\), so \\(\\|W\\|\\le1\\), and \\(W^*\\zeta=(e\\zeta,K^*\\zeta)\\). A direct computation gives \\[\n\\begin{gathered}\nWJW^*\\zeta\\\\\n=Pe\\zeta+DK^*\\zeta+KDe\\zeta-KPK^*\\zeta\\\\\n=b\\zeta.\n\\end{gathered}\n\\] Hence \\(\\|b\\|\\le\\|J\\|=c\\), and \\(\\|b\\|\\ge\\|be\\|=c\\). \\(\\square\\)\n\n**Corollary 10.4** (approximation with norm control). Let \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection, \\(\\varepsilon>0\\) and \\(\\delta>0\\).\n\n1. For every \\(x\\in M\\) there are \\(a\\in A\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(\\|(x-a)f\\|<\\delta\\), \\(\\|a\\|\\le\\|xe\\|\\) and \\(\\varphi(e-f)<\\varepsilon\\).\n2. If \\(x\\) is self-adjoint, \\(a\\) can be chosen self-adjoint, with the same three properties.\n3. If \\(1\\in A\\) and \\(x\\in U(M)\\), \\(a\\) can be chosen in \\(U(A)\\) with \\(\\|(x-a)f\\|<\\delta\\), \\(\\|a-1\\|\\le\\|x-1\\|\\) and \\(\\varphi(e-f)<\\varepsilon\\).\n\n**Proof.** In each case we apply Theorem 10.2 to a bounded set \\(B\\) and an element \\(y\\) of its strong closure with \\(ye=xe\\). Then \\((b_k-y)f=(b_k-y)ef=(b_k-x)f\\), and we take \\(a=b_k\\) for large \\(k\\).\n\n(1) Let \\(y=xe\\) and \\(B=A\\cap\\|xe\\|S\\). By Kaplansky's density theorem 7.1(1), \\(y\\) lies in the strong closure of \\(B\\).\n\n(2) Let \\(y\\) be the self-adjoint completion of Lemma 10.3, with \\(ye=xe\\) and \\(\\|y\\|=\\|xe\\|\\), and \\(B=A_h\\cap\\|xe\\|S\\). By Theorem 7.1(2), \\(y\\) lies in the strong closure of \\(B\\).\n\n(3) Let \\(y=x\\) and \\(B=U(A,\\|x-1\\|)\\). By Theorem 9.1, \\(x\\) lies in its strong closure. \\(\\square\\)\n\n**Lemma 10.5** (correcting a unitary). Let \\(w\\in U(M)\\) and let \\(e\\in M\\) be a projection with \\(\\|(1-w)e\\|\\le\\frac18\\). There is \\(v\\in U(M)\\) with \\(ve=we\\) and \\(\\|1-v\\|\\le6\\|(1-w)e\\|\\).\n\n**Proof.** If \\(e=0\\), take \\(v=1\\); if \\(e=1\\), take \\(v=w\\). Suppose otherwise. Put \\(\\delta=\\|(1-w)e\\|\\) and \\(f=wew^*\\), a projection in \\(M\\) with \\(fw=we\\). Since \\(e-f=(1-w)e+we(1-w^*)\\) and \\(\\|e(1-w^*)\\|=\\delta\\), we get \\(\\theta:=\\|e-f\\|\\le2\\delta\\le\\frac14\\).\n\n*A unitary map of \\((1-e)H\\) onto \\((1-f)H\\).* Let \\(T=(1-f)(1-e)\\in M\\). Writing \\(1-f=(1-e)-(f-e)\\),\n\\[\n\\begin{gathered}\nT^*T\\\\\n=(1-e)(1-f)(1-e)\\\\\n=(1-e)-(1-e)(f-e)(1-e)\\\\\n\\ge(1-\\theta)(1-e),\n\\end{gathered}\n\\]\nand in the same way \\(TT^*\\ge(1-\\theta)(1-f)\\). In the von Neumann algebra \\((1-e)M(1-e)\\) with unit \\(1-e\\), \\(|T|=(T^*T)^{1/2}\\) is invertible; let \\(|T|^{-1}\\) be its inverse there, extended by \\(0\\) on \\(eH\\). Put \\(U=T|T|^{-1}\\in M\\). Then \\(U^*U=|T|^{-1}T^*T|T|^{-1}=1-e\\), so \\(UU^*\\) is a projection, the projection onto the range of \\(U\\), which equals the range of \\(T\\). That range lies in \\((1-f)H\\), and it contains the range of \\(TT^*\\), which is \\((1-f)H\\) because \\(TT^*\\) is invertible on \\((1-f)H\\). So \\(UU^*=1-f\\).\n\n*\\(U\\) is close to \\(1-e\\).* On \\((1-e)H\\), \\(T=(1-e)-(f-e)(1-e)\\), so\n\\[\n\\begin{gathered}\nU-(1-e)\\\\\n=(1-e)\\big(|T|^{-1}-(1-e)\\big)\\\\\n-(f-e)(1-e)|T|^{-1}.\n\\end{gathered}\n\\]\nThe spectrum of \\(|T|^2\\) in \\((1-e)M(1-e)\\) lies in \\([1-\\theta,1+\\theta]\\), and for \\(|s|\\le\\theta\\le\\frac14\\) we have \\(|(1+s)^{-1/2}-1|\\le\\theta\\) and \\((1+s)^{-1/2}\\le(1-\\theta)^{-1/2}\\le\\frac2{\\sqrt3}\\). So \\(\\|U-(1-e)\\|\\le\\theta+\\frac2{\\sqrt3}\\theta\\le2.2\\,\\theta\\le4.4\\,\\delta\\).\n\n*The unitary \\(v\\).* Put \\(v=we+U\\). Then \\(ve=we\\), because \\(Ue=0\\). Next, \\(U^*we=|T|^{-1}(1-e)(1-f)fw=0\\) and \\(eU^*=0\\), so the cross terms in \\(v^*v\\) and \\(vv^*\\) vanish. Hence \\(v^*v=ew^*we+U^*U=e+(1-e)=1\\) and \\(vv^*=wew^*+UU^*=f+(1-f)=1\\). Finally \\(1-v=(1-w)e+\\big((1-e)-U\\big)\\), so \\(\\|1-v\\|\\le\\delta+4.4\\,\\delta<6\\delta\\). \\(\\square\\)\n\n**Theorem 10.6** (noncommutative Lusin theorem). Let \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection, \\(\\varepsilon>0\\) and \\(\\delta>0\\).\n\n1. For every \\(x\\in M\\) there are \\(a\\in A\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(xf=af\\), \\(\\varphi(e-f)<\\varepsilon\\) and \\(\\|a\\|\\le(1+\\delta)\\|xf\\|\\).\n2. If \\(x\\) is self-adjoint, \\(a\\) can be chosen self-adjoint, with the same properties.\n3. If \\(1\\in A\\) and \\(x\\in U(M)\\), there are \\(a\\in U(A)\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(xf=af\\), \\(\\varphi(e-f)<\\varepsilon\\) and \\(\\|a-1\\|\\le\\|x-1\\|+\\delta\\).\n\n**Proof.** (1) and (2). If \\(xe=0\\), take \\(a=0\\) and \\(f=e\\). Otherwise, replacing \\(x\\) by \\(x/\\|xe\\|\\), we may assume \\(\\|xe\\|=1\\). Choose \\(\\delta'\\in(0,\\frac14)\\) with \\((1+\\delta')/(1-2\\delta')\\le1+\\delta\\). Choose a unit vector \\(\\zeta\\in eH\\) with \\(\\|x\\zeta\\|^2\\ge1-\\delta'\\), and put \\(\\psi=\\varphi+\\omega_\\zeta\\in M_*^+\\) and \\(\\varepsilon'=\\min(\\varepsilon,\\delta'^2)\\).\n\nWe choose elements \\(a_1,a_2,\\dots\\) of \\(A\\), self-adjoint in case (2), and projections \\(e=f_0\\ge f_1\\ge f_2\\ge\\cdots\\) in \\(M\\). Suppose \\(a_1,\\dots,a_{k-1}\\) and \\(f_0,\\dots,f_{k-1}\\) are chosen, and let \\(x_{k-1}=x-\\sum_{j<k}a_j\\), which is self-adjoint in case (2). Corollary 10.4, part (1) or (2), applied to \\(x_{k-1}\\), the projection \\(f_{k-1}\\), the functional \\(\\psi\\) and the numbers \\(2^{-k}\\varepsilon'\\) and \\(2^{-k}\\delta'\\), gives \\(a_k\\) and \\(f_k\\le f_{k-1}\\) with\n\\[\n\\begin{gathered}\n\\|x_kf_k\\|<2^{-k}\\delta',\\\\\n\\|a_k\\|\\\\\n\\le\\|x_{k-1}f_{k-1}\\|,\\\\\n\\psi(f_{k-1}-f_k)<2^{-k}\\varepsilon',\n\\end{gathered}\n\\tag{10.2}\n\\]\nwhere \\(x_k=x_{k-1}-a_k\\), since \\((x_{k-1}-a_k)f_k=x_kf_k\\).\n\nBy (10.2), \\(\\|a_1\\|\\le\\|xe\\|=1\\) and \\(\\|a_k\\|<2^{-k+1}\\delta'\\) for \\(k\\ge2\\). So \\(a=\\sum_ka_k\\) converges in norm, \\(a\\in A\\), and \\(\\|a\\|\\le1+\\delta'\\); \\(a\\) is self-adjoint in case (2). Let \\(f\\) be the strong limit of the decreasing sequence \\((f_k)\\). Then \\(f\\le e\\) is a projection in \\(M\\), and by normality \\(\\psi(e-f)<\\varepsilon'\\), so \\(\\varphi(e-f)<\\varepsilon\\). Since \\(f=f_kf\\), \\(\\|(x-\\sum_{j\\le k}a_j)f\\|=\\|x_kf_kf\\|<2^{-k}\\delta'\\); letting \\(k\\to\\infty\\) gives \\(xf=af\\).\n\nIt remains to bound \\(\\|xf\\|\\) from below. We have \\(\\omega_\\zeta(e-f)=\\|(e-f)\\zeta\\|^2<\\delta'^2\\), and \\(\\|x(e-f)\\|=\\|xe(e-f)\\|\\le1\\). Since \\(\\zeta=e\\zeta\\),\n\\[\n\\begin{gathered}\n\\|xf\\|\\\\\n\\ge\\|xf\\zeta\\|\\\\\n=\\|x\\zeta-x(e-f)\\zeta\\|\\\\\n\\ge(1-\\delta')^{1/2}-\\delta'\\\\\n\\ge1-2\\delta' .\n\\end{gathered}\n\\]\nHence \\(\\|a\\|\\le1+\\delta'\\le\\frac{1+\\delta'}{1-2\\delta'}\\|xf\\|\\le(1+\\delta)\\|xf\\|\\).\n\n(3) We may assume \\(\\delta\\le1\\). Put \\(w_1=x\\) and \\(f_1=e\\). We choose \\(u_k\\in U(A)\\), \\(w_{k+1}\\in U(M)\\) and projections \\(f_{k+1}\\le f_k\\) in \\(M\\), for \\(k=1,2,\\dots\\), with\n\\[\n\\begin{gathered}\nxf_{k+1}\\\\\n=u_1u_2\\cdots u_k\\,w_{k+1}f_{k+1},\\\\\n\\|1-u_k\\|\\\\\n\\le\\|1-w_k\\|,\\\\\n\\|1-w_{k+1}\\|\\\\\n\\le2^{-k}\\delta,\\\\\n\\varphi(f_k-f_{k+1})<2^{-k}\\varepsilon .\n\\end{gathered}\n\\tag{10.3}\n\\]\nGiven \\(w_k\\) and \\(f_k\\) with \\(xf_k=u_1\\cdots u_{k-1}w_kf_k\\) (for \\(k=1\\) this reads \\(xe=w_1e\\)), Corollary 10.4(3) gives \\(u_k\\in U(A)\\) and \\(f_{k+1}\\le f_k\\) with \\(\\|(w_k-u_k)f_{k+1}\\|<2^{-k-3}\\delta\\), \\(\\|1-u_k\\|\\le\\|1-w_k\\|\\) and \\(\\varphi(f_k-f_{k+1})<2^{-k}\\varepsilon\\). The unitary \\(u_k^*w_k\\in U(M)\\) satisfies \\[\n\\begin{gathered}\n\\|(1-u_k^*w_k)f_{k+1}\\|\\\\\n=\\|(u_k-w_k)f_{k+1}\\|<2^{-k-3}\\delta\\\\\n\\le\\frac18.\n\\end{gathered}\n\\] Lemma 10.5 gives \\(w_{k+1}\\in U(M)\\) with \\(w_{k+1}f_{k+1}=u_k^*w_kf_{k+1}\\) and \\(\\|1-w_{k+1}\\|\\le6\\cdot2^{-k-3}\\delta<2^{-k}\\delta\\). Then\n\\[\n\\begin{gathered}\nxf_{k+1}\\\\\n=xf_kf_{k+1}\\\\\n=u_1\\cdots u_{k-1}w_kf_{k+1}\\\\\n=u_1\\cdots u_{k-1}u_k\\,(u_k^*w_kf_{k+1})\\\\\n=u_1\\cdots u_k\\,w_{k+1}f_{k+1},\n\\end{gathered}\n\\]\nwhich completes the step.\n\nBy (10.3), \\(\\|1-u_1\\|\\le\\|1-x\\|\\) and \\(\\|1-u_k\\|\\le\\|1-w_k\\|<2^{-k+1}\\delta\\) for \\(k\\ge2\\). Let \\(v_k=u_1\\cdots u_k\\). Then \\(\\|v_{k+1}-v_k\\|=\\|v_k(u_{k+1}-1)\\|=\\|u_{k+1}-1\\|\\), so \\((v_k)\\) converges in norm to a unitary \\(a\\in A\\). From \\[\n\\begin{gathered}\n1-v_k\\\\\n=(1-u_1)+u_1(1-u_2)\\\\\n+\\dots+u_1\\cdots u_{k-1}(1-u_k)\n\\end{gathered}\n\\] we get \\(\\|1-a\\|\\le\\sum_k\\|1-u_k\\|\\le\\|1-x\\|+\\delta\\). Let \\(f\\) be the strong limit of \\((f_k)\\), so \\(f\\le e\\) and \\(\\varphi(e-f)<\\varepsilon\\). Multiplying (10.3) by \\(f\\) on the right, \\(xf=v_kw_{k+1}f\\), so\n\\[\n\\begin{gathered}\n\\|xf-af\\|\\\\\n\\le\\|v_k(w_{k+1}-1)f\\|+\\|(v_k-a)f\\|\\\\\n\\le2^{-k}\\delta+\\|v_k-a\\|\\to0 ,\n\\end{gathered}\n\\]\nand \\(xf=af\\). \\(\\square\\)\n\n**Example 10.7** (the commutative case). Let \\(\\mu\\) be Lebesgue measure on \\([0,1]\\), \\(M\\) the algebra of multiplication operators \\(m_g\\) by bounded measurable \\(g\\) on \\(L^2[0,1]\\), \\(A\\) the multiplication operators by continuous functions, and \\(\\varphi(m_g)=\\int g\\,d\\mu\\), the vector functional of the constant function \\(1\\). The projections of \\(M\\) are the operators \\(m_{1_E}\\), and \\(\\varphi(1-m_{1_E})=\\mu([0,1]\\setminus E)\\). A bounded sequence \\(g_k\\to g\\) almost everywhere gives \\(m_{g_k}\\to m_g\\) strongly, by dominated convergence. Theorem 10.2 then yields a set \\(E\\) with small complement on which a sequence of the \\(g_k\\) converges uniformly, up to a null set: a form of Egoroff's theorem. Theorem 10.6(1) yields a continuous \\(a\\) that agrees with a bounded measurable \\(g\\) almost everywhere on a set \\(E\\) with small complement, with \\(\\max|a|\\le(1+\\delta)\\operatorname{ess\\,sup}_E|g|\\): this is Lusin's theorem with control of the norm. (That \\(M\\) is the weak closure of \\(A\\) is shown in Exercise 14.2.)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-KD-13",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "11. Kadison's transitivity theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
      "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "anchor": "oa-fnd-kd-13",
      "proof_locus": {
        "line": 803,
        "through_line": 838
      },
      "full_conditions_and_proof": "## 11. Kadison's transitivity theorem\n\nA concrete C\\(^*\\)-algebra \\(A\\) on \\(H\\) is *irreducible* if \\(A\\ne\\{0\\}\\) and the only closed subspaces of \\(H\\) that are invariant under \\(A\\) are \\(\\{0\\}\\) and \\(H\\).\n\n**Theorem 11.1** (Kadison's transitivity theorem). Let \\(A\\) be an irreducible concrete C\\(^*\\)-algebra on \\(H\\), let \\(e\\) be a projection of finite rank on \\(H\\), and let \\(\\varepsilon>0\\).\n\n1. For every \\(b\\in B(H)\\) there is \\(a\\in A\\) with \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|be\\|\\). In particular \\(Ae=B(H)e\\).\n2. If \\(b\\) is self-adjoint, \\(a\\) can be chosen self-adjoint with \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|be\\|\\).\n3. If \\(b\\ge0\\), \\(a\\) can be chosen with \\(a\\ge0\\), \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|b\\|\\).\n4. For every unitary \\(u\\in B(H)\\) there is a unitary \\(v\\) in the C\\(^*\\)-algebra \\(A+\\mathbb C1\\) with \\(ve=ue\\) and \\(\\|v-1\\|\\le\\|u-1\\|+\\varepsilon\\).\n\n\n**Proof.** *The weak closure is \\(B(H)\\).* The closed subspace \\([AH]\\) is invariant and not \\(\\{0\\}\\), because \\(A\\ne\\{0\\}\\); so \\(A\\) is nondegenerate. A projection in \\(A'\\) is the projection onto a closed invariant subspace ([Proposition 2.1(5) of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-02)), so the only projections of the von Neumann algebra \\(A'\\) are \\(0\\) and \\(1\\). By Corollary 4.4, \\(A'=\\mathbb C1\\), so \\(A''=B(H)\\), and \\(B(H)\\) is the weak closure of \\(A\\).\n\n*A normal functional that detects \\(e\\).* Let \\(\\varepsilon_1,\\dots,\\varepsilon_r\\) be an orthonormal basis of \\(eH\\) and \\(\\varphi(y)=\\sum_{j=1}^r\\langle y\\varepsilon_j,\\varepsilon_j\\rangle\\), the trace of \\(eye\\). It is a positive normal functional on \\(B(H)\\). For a projection \\(f\\le e\\), \\(\\varphi(e-f)\\) is the rank of \\(e-f\\), an integer. So \\(\\varphi(e-f)<\\frac12\\) forces \\(f=e\\).\n\n(1) and (2). Apply Theorem 10.6(1), respectively (2), to \\(x=b\\), \\(M=B(H)\\), the projection \\(e\\) and the functional \\(\\varphi\\), with \\(\\frac12\\) in place of \\(\\varepsilon\\) and \\(\\varepsilon\\) in place of \\(\\delta\\). The projection \\(f\\) it produces is \\(e\\), so \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|be\\|\\).\n\n(3) Let \\(e'\\) be the projection onto the finite-dimensional space \\(eH+b^{1/2}eH\\). Choose \\(\\varepsilon'>0\\) with \\((1+\\varepsilon')^2\\le1+\\varepsilon\\). By (2) there is a self-adjoint \\(c\\in A\\) with \\(ce'=b^{1/2}e'\\) and \\(\\|c\\|\\le(1+\\varepsilon')\\|b^{1/2}\\|\\). Put \\(a=c^2\\ge0\\). Since \\(e\\le e'\\), \\(ce=b^{1/2}e\\), and since \\(b^{1/2}e=e'b^{1/2}e\\),\n\\[\nae=c\\,b^{1/2}e=c\\,e'b^{1/2}e=b^{1/2}e'b^{1/2}e=be .\n\\]\nAlso \\(\\|a\\|=\\|c\\|^2\\le(1+\\varepsilon')^2\\|b\\|\\le(1+\\varepsilon)\\|b\\|\\).\n\n(4) \\(A+\\mathbb C1\\) is a unital concrete C\\(^*\\)-algebra with weak closure \\(B(H)\\). Apply Theorem 10.6(3) to it, with \\(x=u\\), the same \\(\\varphi\\), \\(\\frac12\\) in place of \\(\\varepsilon\\) and \\(\\varepsilon\\) in place of \\(\\delta\\). \\(\\square\\)\n\n**Corollary 11.2.** Let \\(A\\) be an irreducible concrete C\\(^*\\)-algebra on \\(H\\).\n\n1. If \\(\\xi_1,\\dots,\\xi_n\\in H\\) are linearly independent and \\(\\eta_1,\\dots,\\eta_n\\in H\\) are arbitrary, there is \\(a\\in A\\) with \\(a\\xi_j=\\eta_j\\) for all \\(j\\).\n2. For every \\(\\xi\\ne0\\), \\(A\\xi=H\\). So \\(H\\) has no subspaces invariant under \\(A\\) except \\(\\{0\\}\\) and \\(H\\), closed or not: \\(A\\) acts *algebraically irreducibly*.\n3. Let \\(\\pi\\) be a representation of a C\\(^*\\)-algebra \\(B\\) on \\(H\\) such that \\(\\pi(B)\\) is irreducible. Then \\(\\pi(B)\\xi=H\\) for every nonzero \\(\\xi\\in H\\).\n\n**Proof.** (1) Let \\(e\\) be the projection onto the span of the \\(\\xi_j\\). By linear independence there is \\(b\\in B(H)\\) with \\(b\\xi_j=\\eta_j\\) for all \\(j\\) and \\(b=0\\) on \\((eH)^\\perp\\). Theorem 11.1(1) gives \\(a\\in A\\) with \\(ae=be\\). (2) is (1) with \\(n=1\\). (3) The image of a C\\(^*\\)-algebra under a representation is norm closed ([ranges of \\(*\\)-homomorphisms](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)), so it is a concrete C\\(^*\\)-algebra; apply (2) to it. \\(\\square\\)\n\n**Example 11.3** (norm closedness is needed). Let \\(s\\) be the unilateral shift on \\(\\ell^2(\\{0,1,2,\\dots\\})\\), \\(s\\delta_m=\\delta_{m+1}\\), and let \\(P\\) be the \\(*\\)-algebra of polynomials in \\(s\\) and \\(s^*\\). Then \\(1-ss^*=\\theta_{\\delta_0,\\delta_0}\\), and \\(s^i(1-ss^*)s^{*j}=\\theta_{\\delta_i,\\delta_j}\\), so \\(P\\) contains all matrix units. If a closed subspace \\(W\\ne\\{0\\}\\) is invariant under \\(P\\), pick \\(w\\in W\\) and \\(j\\) with \\(\\langle w,\\delta_j\\rangle\\ne0\\); then \\(\\theta_{\\delta_i,\\delta_j}w=\\langle w,\\delta_j\\rangle\\delta_i\\in W\\) for all \\(i\\), so \\(W\\) is the whole space. Thus \\(P\\) is irreducible in the topological sense. But \\(s\\) and \\(s^*\\) map finitely supported vectors to finitely supported vectors, so \\(P\\delta_0\\) consists of finitely supported vectors and is not the whole space. So the \\(*\\)-algebra \\(P\\) is not algebraically irreducible, while its norm closure is, by Corollary 11.2.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-KD-14",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "12. Monotone limits: the up-down and up-down-up theorems",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
      "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "anchor": "oa-fnd-kd-14",
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      "full_conditions_and_proof": "## 12. Monotone limits: the up-down and up-down-up theorems\n\nKaplansky's theorem approximates elements of \\(A''\\) by bounded nets from \\(A\\). This section approximates self-adjoint elements of \\(A''\\) in the order: by increasing and decreasing nets, applied in turn.\n\n**Definition 12.1.** Let \\(X\\) be a set of self-adjoint operators on \\(H\\). Then \\(X^{\\nearrow}\\) consists of all strong limits of increasing *sequences* in \\(X\\) that are bounded in norm, and \\(X^{\\searrow}\\) of all strong limits of such decreasing sequences. The sets \\(X^{\\uparrow}\\) and \\(X^{\\downarrow}\\) are defined in the same way with *nets* in place of sequences. By Vigier's theorem 1.3 such limits exist, and they are least upper, respectively greatest lower, bounds.\n\n**Lemma 12.2.** Let \\(X,Y\\) be sets of self-adjoint operators.\n\n1. \\(X\\subseteq X^{\\nearrow}\\subseteq X^{\\uparrow}\\) and \\(X\\subseteq X^{\\searrow}\\subseteq X^{\\downarrow}\\). If \\(X\\subseteq Y\\), then \\(X^{\\uparrow}\\subseteq Y^{\\uparrow}\\), and likewise for the other three operations.\n2. If \\(X\\) is convex, or closed under sums, or closed under multiplication by nonnegative numbers, then so are \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\).\n3. \\(-(X^{\\uparrow})=(-X)^{\\downarrow}\\) and \\(-(X^{\\nearrow})=(-X)^{\\searrow}\\). More generally, for the order-reversing map \\(t\\mapsto1-t\\): \\(1-X^{\\uparrow}=(1-X)^{\\downarrow}\\) and \\(1-X^{\\nearrow}=(1-X)^{\\searrow}\\), and with the roles of increasing and decreasing exchanged.\n4. Let \\(X\\subseteq B(H)_+\\cap S\\), and let \\(g:[0,1]\\to[0,1]\\) be continuous and operator monotone on \\([0,1]\\) (Definition 12.3) with \\(g(X)\\subseteq X\\). Then \\(g(Y)\\subseteq Y\\) for \\(Y\\) each of \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\).\n5. If \\(X\\subseteq M_+\\cap S\\) for a von Neumann algebra \\(M\\), then \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\subseteq M_+\\cap S\\).\n\n**Proof.** (1) Use constant sequences, and note that a sequence is a net. (2) If \\(x_i\\uparrow x\\) and \\(y_j\\uparrow y\\) are increasing nets in \\(X\\), then for \\(t\\in[0,1]\\) the net \\(tx_i+(1-t)y_j\\), indexed by pairs \\((i,j)\\) with the product order, is increasing in \\(X\\) and converges strongly to \\(tx+(1-t)y\\). Sums and nonnegative multiples are handled the same way. For sequences use the pairs \\((m,m)\\). (3) Negation and \\(t\\mapsto1-t\\) reverse the order and preserve strong limits. (4) If \\(x_i\\uparrow x\\) in \\(X\\), then \\(g(x_i)\\) is an increasing net in \\(X\\), and \\(g(x_i)\\to g(x)\\) strongly by Corollary 5.3(1). So \\(g(x)\\in X^{\\uparrow}\\). The other cases are the same. (5) \\(M\\) is strongly closed, and so are \\(B(H)_+\\) and \\(S\\) (Lemma 1.2). \\(\\square\\)\n\n**Definition 12.3.** A real continuous function \\(g\\) on an interval \\(J\\) is *operator monotone on \\(J\\)* if \\(g(x)\\le g(y)\\) whenever \\(x\\le y\\) are self-adjoint operators with spectra in \\(J\\).\n\nEvery operator monotone function is increasing, as one sees on multiples of \\(1\\). The converse fails: \\(t\\mapsto t^2\\) is not operator monotone on \\([0,\\infty)\\), since \\(x=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\\le y=\\begin{pmatrix}2&1\\\\1&1\\end{pmatrix}\\), while \\(y^2-x^2=\\begin{pmatrix}4&3\\\\3&2\\end{pmatrix}\\) has determinant \\(-1\\) and so is not positive. By the [Löwner–Heinz inequality](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-17), \\(t\\mapsto t^\\alpha\\) is operator monotone on \\([0,\\infty)\\) for \\(0\\le\\alpha\\le1\\) (for \\(\\alpha=0\\) the function is constant). We need two simpler families.\n\n**Lemma 12.4** (resolvent-type operator monotone functions). Let \\(c>0\\).\n\n1. \\(g_c(t)=t/(c+t)\\) is operator monotone on \\([0,\\infty)\\). It takes values in \\([0,1)\\), \\(g_c(0)=0\\), and \\(g_c\\) increases as \\(c\\) decreases.\n2. \\(\\phi_c(t)=t/(1+c(1-t))\\) is operator monotone on \\([0,1]\\). It maps \\([0,1]\\) onto \\([0,1]\\), \\(\\phi_c(0)=0\\), \\(\\phi_c(1)=1\\), \\(\\phi_c\\) decreases as \\(c\\) increases, and \\(\\phi_c(t)\\to0\\) as \\(c\\to\\infty\\) for \\(t<1\\).\n3. For \\(\\alpha>0\\), the function \\(t/(\\alpha+(1-\\alpha)t)\\) is operator monotone on \\([0,1]\\), and for \\(0<\\alpha\\le1\\) also on \\([0,\\infty)\\).\n\n**Proof.** Inversion reverses the order on invertible positive operators ([working with the order](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-16)). (1) \\(g_c(t)=1-c(c+t)^{-1}\\). If \\(0\\le x\\le y\\), then \\(c+x\\le c+y\\), so \\((c+y)^{-1}\\le(c+x)^{-1}\\) and \\(g_c(x)\\le g_c(y)\\). (2) For \\(t\\in[0,1]\\), \\(\\phi_c(t)=\\frac{1+c}{c}\\big(1+c(1-t)\\big)^{-1}-\\frac1c\\). If \\(0\\le x\\le y\\le1\\), then \\(1+c(1-x)\\ge1+c(1-y)\\ge1\\), so \\(\\big(1+c(1-x)\\big)^{-1}\\le\\big(1+c(1-y)\\big)^{-1}\\) and \\(\\phi_c(x)\\le\\phi_c(y)\\). The other claims are elementary. (3) For \\(0<\\alpha<1\\) the function is \\((1-\\alpha)^{-1}g_c\\) with \\(c=\\alpha/(1-\\alpha)\\); for \\(\\alpha=1\\) it is \\(t\\); for \\(\\alpha>1\\) it is \\(\\phi_c\\) with \\(c=\\alpha-1\\). \\(\\square\\)\n\n**Lemma 12.5** (a projection on countably many vectors). Let \\(A\\) be a concrete C\\(^*\\)-algebra on \\(H\\) with weak closure \\(M\\), let \\(p\\in M\\) be a projection, and let \\(\\xi_1,\\xi_2,\\dots\\in H\\). There are \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\) and a projection \\(q\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\) such that, for all \\(k\\),\n\\[\n\\begin{gathered}\ny(1-p)\\xi_k\\\\\n=0,\\\\\nyp\\xi_k\\\\\n=p\\xi_k,\\\\\nq(1-p)\\xi_k\\\\\n=0,\\\\\nqp\\xi_k\\\\\n=p\\xi_k .\n\\end{gathered}\n\\]\n\n**Proof.** Scaling the vectors changes nothing, so we assume \\(\\|\\xi_k\\|\\le1\\). Put \\(\\eta_k=(1-p)\\xi_k\\) and \\(\\zeta_k=p\\xi_k\\). By Kaplansky's density theorem 7.1(3), \\(p\\) is a strong limit of a net in \\(A_+\\cap S\\). Since \\(p\\eta_k=0\\) and \\(p\\zeta_k=\\zeta_k\\), for every \\(m\\) we can choose \\(x_m\\in A_+\\cap S\\) with\n\\[\n\\begin{gathered}\n\\|x_m\\eta_k\\|\\\\\n\\le4^{-m}\\\\\n\\text{and}\\\\\n\\|(1-x_m)\\zeta_k\\|\\\\\n\\le\\tfrac1m\\\\\n(k\\\\\n\\le m).\n\\end{gathered}\n\\]\nFor \\(n\\le m\\) put \\(X_{n,m}=\\sum_{j=n}^m2^jx_j\\) and \\(y_{n,m}=g_1(X_{n,m})=X_{n,m}(1+X_{n,m})^{-1}\\). Then \\(y_{n,m}\\in A_+\\cap S\\), since \\(g_1(0)=0\\) and \\(0\\le g_1\\le1\\).\n\n*Monotonicity.* For fixed \\(n\\), \\(X_{n,m}\\) increases with \\(m\\), so \\(y_{n,m}\\) increases with \\(m\\) by Lemma 12.4(1), to a limit \\(y_n\\in(A_+\\cap S)^{\\nearrow}\\). Since \\(X_{n+1,m}\\le X_{n,m}\\), \\(y_{n+1}\\le y_n\\), and \\(y_n\\) decreases to a limit \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\).\n\n*The vectors \\(\\eta_k\\).* For \\(k\\le n\\le m\\), using \\(g_1(t)\\le t\\),\n\\[\n\\begin{gathered}\n\\langle y_{n,m}\\eta_k,\\eta_k\\rangle\\\\\n\\le\\langle X_{n,m}\\eta_k,\\eta_k\\rangle\\\\\n\\le\\sum_{j\\ge n}2^j\\|x_j\\eta_k\\|\\\\\n\\le\\sum_{j\\ge n}2^{-j}\\\\\n=2^{1-n}.\n\\end{gathered}\n\\]\nSo \\(0\\le\\langle y\\eta_k,\\eta_k\\rangle\\le\\langle y_n\\eta_k,\\eta_k\\rangle\\le2^{1-n}\\) for all \\(n\\ge k\\). Hence \\(\\|y^{1/2}\\eta_k\\|=0\\) and \\(y\\eta_k=0\\).\n\n*The vectors \\(\\zeta_k\\).* Since \\(X_{n,m}\\ge2^mx_m\\), \\(y_{n,m}\\ge g_1(2^mx_m)\\), so \\(1-y_{n,m}\\le(1+2^mx_m)^{-1}\\). For \\(t\\in[0,1]\\) and \\(b>0\\), \\((1+bt)^{-1}\\le(1+b(1-t))/(1+b)\\), because \\((1+bt)(1+b-bt)-(1+b)=b^2t(1-t)\\ge0\\). With \\(b=2^m\\) this gives \\(1-y_{n,m}\\le2^{-m}+(1-x_m)\\). For \\(k\\le m\\),\n\\[\n\\begin{gathered}\n\\langle(1-y_{n,m})\\zeta_k,\\zeta_k\\rangle\\\\\n\\le2^{-m}+\\|(1-x_m)\\zeta_k\\|\\\\\n\\le2^{-m}+\\tfrac1m .\n\\end{gathered}\n\\]\nLetting \\(m\\to\\infty\\), \\(\\langle(1-y_n)\\zeta_k,\\zeta_k\\rangle\\le0\\). Since \\(1-y_n\\ge0\\), \\((1-y_n)\\zeta_k=0\\) for all \\(n\\), and in the limit \\(y\\zeta_k=\\zeta_k\\).\n\n*The projection \\(q\\).* Let \\(\\phi_n\\) be the functions of Lemma 12.4(2) with \\(c=n\\). They decrease to the indicator function \\(1_{\\{1\\}}\\) of the point \\(1\\) on \\([0,1]\\). So \\(\\phi_n(y)\\) decreases, and its strong limit \\(q\\) is \\(1_{\\{1\\}}(y)\\), a projection (Theorem 4.2). We show \\(q\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\). For fixed \\(n\\), \\(\\phi_n(y_{n,m})\\in A_+\\cap S\\), since \\(\\phi_n(0)=0\\). It increases with \\(m\\) (Lemma 12.4(2)) and converges strongly to \\(\\phi_n(y_n)\\) (Corollary 5.3(1)). So \\(z_n=\\phi_n(y_n)\\in(A_+\\cap S)^{\\nearrow}\\). Next, \\(z_{n+1}=\\phi_{n+1}(y_{n+1})\\le\\phi_n(y_{n+1})\\le\\phi_n(y_n)=z_n\\), using \\(\\phi_{n+1}\\le\\phi_n\\) and operator monotonicity. Let \\(z\\) be the limit of the decreasing sequence \\((z_n)\\); then \\(z\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\). On one hand, \\(q\\le\\phi_n(y)\\le\\phi_n(y_n)=z_n\\) for all \\(n\\), so \\(q\\le z\\). On the other hand, for \\(k\\ge n\\), \\(\\phi_n(y_k)\\ge\\phi_k(y_k)=z_k\\ge z\\); letting \\(k\\to\\infty\\), \\(\\phi_n(y)\\ge z\\) by Corollary 5.3(1), and letting \\(n\\to\\infty\\), \\(q\\ge z\\). So \\(q=z\\).\n\nFinally, \\[\n\\begin{gathered}\n\\|q\\eta_k\\|^2\\\\\n=\\langle q\\eta_k,\\eta_k\\rangle\\\\\n\\le\\langle\\phi_1(y)\\eta_k,\\eta_k\\rangle\\\\\n\\le\\langle y\\eta_k,\\eta_k\\rangle\\\\\n=0,\n\\end{gathered}\n\\] since \\(\\phi_1(t)=t/(2-t)\\le t\\) on \\([0,1]\\). And \\(y\\zeta_k=\\zeta_k\\) implies \\(\\psi(y)\\zeta_k=\\psi(1)\\zeta_k\\) for every continuous \\(\\psi\\) (first for polynomials, then by uniform approximation), so \\(\\phi_n(y)\\zeta_k=\\zeta_k\\) for all \\(n\\), and \\(q\\zeta_k=\\zeta_k\\). \\(\\square\\)\n\n**Theorem 12.6** (the up-down theorem). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and suppose that \\(M=A''\\) is \\(\\sigma\\)-finite. Then\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\\\\n\\text{and}\\\\\nM_h\\\\\n=\\big((A_h)^{\\nearrow}\\big)^{\\searrow}.\n\\end{gathered}\n\\]\n\n\n**Proof.** The inclusions \\(\\supseteq\\) hold by Lemma 12.2(5), and for \\(M_h\\) because \\(M\\) is strongly closed.\n\n*Projections.* By [the characterization of \\(\\sigma\\)-finite von Neumann algebras](the-double-commutant-theorem.md#oa-fnd-bi-13), \\(H\\) contains a countable set \\(\\{\\xi_k\\}\\) that is separating for \\(M\\). Let \\(p\\in M\\) be a projection. Lemma 12.5 gives \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\subseteq M\\) with \\((y-p)\\xi_k=y(1-p)\\xi_k+(yp\\xi_k-p\\xi_k)=0\\) for all \\(k\\). As \\(y-p\\in M\\) vanishes on a separating set, \\(y=p\\).\n\n*The unit.* In particular \\(1=\\lim y_n\\) for a decreasing sequence \\((y_n)\\) in \\((A_+\\cap S)^{\\nearrow}\\). Each \\(y_n\\) satisfies \\(1\\le y_n\\le1\\), so \\(y_n=1\\), and \\(1\\in(A_+\\cap S)^{\\nearrow}\\).\n\n*Positive contractions.* Let \\(x\\in M_+\\cap S\\). By Corollary 4.6, \\(x=\\sum_k2^{-k}p_k\\) with projections \\(p_k\\in M\\). By the first step, each \\(p_k\\) is the limit of a decreasing sequence \\((z_{k,n})_n\\) in \\((A_+\\cap S)^{\\nearrow}\\). Put\n\\[\nx_n=\\sum_{k=1}^n2^{-k}z_{k,n}+2^{-n}\\cdot1 .\n\\]\nThis is a convex combination of elements of the convex set \\((A_+\\cap S)^{\\nearrow}\\) (Lemma 12.2(2)), since the weights add up to \\(1\\) and \\(1\\in(A_+\\cap S)^{\\nearrow}\\). So \\(x_n\\in(A_+\\cap S)^{\\nearrow}\\). The sequence decreases:\n\\[\n\\begin{gathered}\nx_n-x_{n+1}\\\\\n=\\sum_{k=1}^n2^{-k}(z_{k,n}-z_{k,n+1})+2^{-n-1}(1-z_{n+1,n+1})\\\\\n\\ge0 .\n\\end{gathered}\n\\]\nIt lies above \\(x\\): \\(z_{k,n}\\ge p_k\\) and \\(\\sum_{k>n}2^{-k}p_k\\le2^{-n}\\), so \\(x_n\\ge\\sum_{k\\le n}2^{-k}p_k+2^{-n}\\ge x\\). For \\(n\\ge m\\), using \\(z_{k,n}\\le1\\) for \\(k>m\\),\n\\[\n\\begin{gathered}\nx_n-x\\\\\n\\le\\sum_{k=1}^m2^{-k}(z_{k,n}-p_k)+\\sum_{k=m+1}^n2^{-k}+2^{-n}\\\\\n=\\sum_{k=1}^m2^{-k}(z_{k,n}-p_k)+2^{-m}.\n\\end{gathered}\n\\]\nLet \\(x'\\) be the strong limit of \\((x_n)\\). Letting \\(n\\to\\infty\\) gives \\(x\\le x'\\le x+2^{-m}\\) for every \\(m\\). So \\(x'=x\\), and \\(x\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\).\n\n*Self-adjoint elements.* Let \\(y\\in M_h\\), \\(y\\ne0\\). Then \\(x=(y+\\|y\\|)/(2\\|y\\|)\\in M_+\\cap S\\), and \\(y=2\\|y\\|x-\\|y\\|\\cdot1\\). Here \\(2\\|y\\|x\\in\\big((A_h)^{\\nearrow}\\big)^{\\searrow}\\) by Lemma 12.2(2). Since \\(1\\in(A_h)^{\\nearrow}\\), \\(-\\|y\\|\\cdot1\\in(A_h)^{\\searrow}\\subseteq\\big((A_h)^{\\nearrow}\\big)^{\\searrow}\\) by Lemma 12.2(1) and (3). The set \\(\\big((A_h)^{\\nearrow}\\big)^{\\searrow}\\) is closed under sums by Lemma 12.2(2). \\(\\square\\)\n\nWithout \\(\\sigma\\)-finiteness the up-down theorem fails, even with nets in place of sequences (Section 13). The remedy is a third, increasing, limit. First we remove the need for a unit.\n\n**Lemma 12.7** (adding a unit). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and \\(\\tilde A=A+\\mathbb C1\\).\n\n1. For \\(\\varepsilon>0\\) and \\(x\\in(\\tilde A_+\\cap S)^{\\nearrow}\\), \\((1+\\varepsilon)^{-1}(x+\\varepsilon)\\in(A_+\\cap S)^{\\uparrow}\\).\n2. \\(\\big((\\tilde A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\subseteq\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\).\n\n**Proof.** (1) Let \\(x_n\\in\\tilde A_+\\cap S\\) increase to \\(x\\), and put \\(x_0=0\\) and \\(d_n=x_n-x_{n-1}\\ge0\\). Write \\(d_n=c_n+\\alpha_n\\) with \\(c_n\\in A_h\\) and \\(\\alpha_n\\in\\mathbb R\\). We may take \\(\\alpha_n\\ge0\\): if \\(1\\in A\\), take \\(\\alpha_n=0\\); if not, \\(y+\\alpha\\mapsto\\alpha\\) is a \\(*\\)-homomorphism of \\(\\tilde A\\) onto \\(\\mathbb C\\), which maps the positive element \\(d_n\\) to a nonnegative number. By [approximate identities](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-20), \\(A\\) has an increasing approximate identity \\((u_i)_{i\\in I}\\) in \\(A_+\\cap S\\). Its strong limit \\(e_0\\) (Theorem 1.3) satisfies \\(e_0z=\\lim u_iz=z\\) for \\(z\\in A\\), so \\(e_0=1\\) on \\([AH]=H\\): \\(u_i\\to1\\) strongly.\n\n*A scalar inequality.* Let \\(c\\) be self-adjoint and \\(\\alpha\\ge0\\) with \\(c+\\alpha\\ge0\\), and let \\(0<\\beta\\le\\varepsilon'\\). Then\n\\[\nc+(\\alpha+\\varepsilon')\\,g_\\beta(|c|)\\ge0 ,\n\\tag{12.1}\n\\]\nwith \\(g_\\beta\\) as in Lemma 12.4(1). By the functional calculus of \\(c\\) it suffices to check \\(t+(\\alpha+\\varepsilon')|t|/(\\beta+|t|)\\ge0\\) for \\(t\\in\\sigma(c)\\). For \\(t\\ge0\\) this is clear. For \\(t<0\\), \\(\\alpha\\ge|t|\\), and the left side is at least \\[\n\\begin{gathered}\n-|t|+(|t|+\\varepsilon')|t|/(\\beta+|t|)\\\\\n=|t|(\\varepsilon'-\\beta)/(\\beta+|t|)\\\\\n\\ge0.\n\\end{gathered}\n\\]\n\n*The approximating net.* Let \\(\\Lambda\\) be the set of triples \\(\\lambda=(m,\\beta,i)\\) with \\(m\\ge1\\), \\(0<\\beta\\le\\varepsilon2^{-m}\\) and \\(i\\in I\\), directed by \\(m\\le m'\\), \\(\\beta\\ge\\beta'\\), \\(i\\le i'\\). For \\(n\\le m\\) put\n\\[\n\\begin{gathered}\nw_{n,\\beta,i}\\\\\n=c_n+(\\alpha_n+\\varepsilon2^{-n})\\,g_\\beta(u_i+|c_n|)\\in A_h,\\\\\nv_\\lambda\\\\\n=(1+\\varepsilon)^{-1}\\sum_{n=1}^mw_{n,\\beta,i}.\n\\end{gathered}\n\\]\nEach \\(w_{n,\\beta,i}\\) lies in \\(A\\), because \\(g_\\beta(0)=0\\) and \\(u_i+|c_n|\\in A_+\\). Since \\(u_i+|c_n|\\ge|c_n|\\) and \\(g_\\beta\\) is operator monotone, (12.1) with \\(\\varepsilon'=\\varepsilon2^{-n}\\ge\\beta\\) gives \\(w_{n,\\beta,i}\\ge0\\). Since \\(g_\\beta\\le1\\), \\(w_{n,\\beta,i}\\le c_n+\\alpha_n+\\varepsilon2^{-n}=d_n+\\varepsilon2^{-n}\\). So\n\\[\n\\begin{gathered}\n0\\\\\n\\le v_\\lambda\\\\\n\\le(1+\\varepsilon)^{-1}(x_m+\\varepsilon)\\\\\n\\le(1+\\varepsilon)^{-1}(x+\\varepsilon)\\\\\n\\le1 ,\n\\end{gathered}\n\\]\nand \\(v_\\lambda\\in A_+\\cap S\\). The net \\((v_\\lambda)\\) increases: \\(w_{n,\\beta,i}\\) increases when \\(i\\) increases or \\(\\beta\\) decreases (Lemma 12.4(1)), and a larger \\(m\\) adds nonnegative terms. Let \\(v\\) be its strong limit. Then \\(v\\le(1+\\varepsilon)^{-1}(x+\\varepsilon)\\).\n\nFor the reverse inequality fix \\(N\\), then \\(m\\ge N\\) and \\(\\beta\\in(0,\\varepsilon2^{-m}]\\). The terms with \\(n>N\\) are positive, so \\(v\\ge v_{(m,\\beta,i)}\\ge(1+\\varepsilon)^{-1}\\sum_{n\\le N}w_{n,\\beta,i}\\) for every \\(i\\). As \\(i\\to\\infty\\), \\(u_i+|c_n|\\to1+|c_n|\\) strongly within a bounded set, so \\(g_\\beta(u_i+|c_n|)\\to g_\\beta(1+|c_n|)\\) strongly (Corollary 5.3(1)). Since \\(1-g_\\beta(t)=\\beta/(\\beta+t)\\le\\beta\\) for \\(t\\ge1\\), \\(g_\\beta(1+|c_n|)\\ge(1-\\beta)1\\). By Lemma 1.2 the inequality passes to the limit, and\n\\[\n\\begin{gathered}\nv\\\\\n\\ge(1+\\varepsilon)^{-1}\\sum_{n\\le N}\\big(c_n+(\\alpha_n+\\varepsilon2^{-n})(1-\\beta)\\big)\\\\\n=(1+\\varepsilon)^{-1}\\Big(x_N+\\varepsilon(1-2^{-N})\\\\\n-\\beta\\sum_{n\\le N}(\\alpha_n+\\varepsilon2^{-n})\\Big).\n\\end{gathered}\n\\]\nHere \\(\\beta\\) can be arbitrarily small, so \\(v\\ge(1+\\varepsilon)^{-1}(x_N+\\varepsilon(1-2^{-N}))\\). Letting \\(N\\to\\infty\\), \\(v\\ge(1+\\varepsilon)^{-1}(x+\\varepsilon)\\). So \\((1+\\varepsilon)^{-1}(x+\\varepsilon)=v\\in(A_+\\cap S)^{\\uparrow}\\).\n\n(2) Let \\(x_j\\) be a decreasing net in \\((\\tilde A_+\\cap S)^{\\nearrow}\\) with limit \\(x\\). By (1), \\(y_{j,k}=(1+\\frac1k)^{-1}(x_j+\\frac1k)\\in(A_+\\cap S)^{\\uparrow}\\). For \\(0\\le t\\le1\\), \\(\\frac{kt+1}{k+1}-\\frac{(k+1)t+1}{k+2}=\\frac{1-t}{(k+1)(k+2)}\\ge0\\), so \\(y_{j,k}\\) decreases in \\(k\\); it also decreases in \\(j\\). The net \\((y_{j,k})\\) has limit \\(x\\). So \\(x\\in\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\). \\(\\square\\)\n\n**Lemma 12.8** (suprema of projections). Let \\(Y\\subseteq B(H)_+\\cap S\\) be convex, with \\(0\\in Y\\) and \\(g_\\beta(Y)\\subseteq Y\\) for all \\(\\beta\\in(0,1]\\). If \\((p_i)_{i\\in I}\\) is any family of projections in \\(Y^{\\uparrow}\\), then \\(\\bigvee_ip_i\\in Y^{\\uparrow}\\).\n\n**Proof.** Put \\(p=\\bigvee_ip_i\\). For each \\(i\\), choose an increasing net \\((x_{i,j})_{j\\in J_i}\\) in \\(Y\\) with limit \\(p_i\\), and add to \\(J_i\\) a new least element \\(0_i\\) with \\(x_{i,0_i}=0\\); the net is still increasing, since \\(x_{i,j}\\ge0\\). Let \\(\\Gamma\\) be the set of functions \\(\\gamma\\) on \\(I\\) with \\(\\gamma(i)\\in J_i\\) for all \\(i\\) and \\(\\gamma(i)=0_i\\) for all but finitely many \\(i\\), ordered pointwise. It is directed. For \\(\\gamma\\in\\Gamma\\) and \\(k\\ge1\\) put\n\\[\n\\begin{gathered}\nX_\\gamma\\\\\n=\\sum_ix_{i,\\gamma(i)},\\\\\ny_{\\gamma,k}\\\\\n=g_{1/k}(X_\\gamma)\\\\\n=X_\\gamma\\big(\\tfrac1k+X_\\gamma\\big)^{-1}.\n\\end{gathered}\n\\]\nThe sum is finite. If \\(\\gamma\\) has \\(N\\ge1\\) entries that are not least elements, then \\(X_\\gamma/N\\in Y\\) by convexity, and \\(y_{\\gamma,k}=g_{1/(kN)}(X_\\gamma/N)\\in Y\\); if \\(N=0\\), \\(y_{\\gamma,k}=0\\in Y\\).\n\nThe net \\((y_{\\gamma,k})\\) increases: \\(X_\\gamma\\) increases with \\(\\gamma\\), \\(g_{1/k}\\) is operator monotone (Lemma 12.4(1)), and \\(g_{1/k}\\le g_{1/k'}\\) for \\(k\\le k'\\). Let \\(y\\in Y^{\\uparrow}\\) be its limit. Then \\(0\\le y\\le1\\). Each \\(x_{i,j}\\le p_i\\le p\\), so \\(x_{i,j}=px_{i,j}p\\), hence \\(X_\\gamma=pX_\\gamma p\\), \\(y_{\\gamma,k}=py_{\\gamma,k}p\\) and \\(y=pyp\\).\n\nFix \\(i\\) and \\(k\\). For \\(j\\in J_i\\), take \\(\\gamma\\) with \\(\\gamma(i)=j\\) and \\(\\gamma(i')=0_{i'}\\) otherwise; then \\(y\\ge y_{\\gamma,k}=g_{1/k}(x_{i,j})\\). Letting \\(j\\) run through \\(J_i\\), \\(g_{1/k}(x_{i,j})\\to g_{1/k}(p_i)=\\frac k{k+1}p_i\\) strongly (Corollary 5.3(1)). By Lemma 1.2, \\(y\\ge\\frac k{k+1}p_i\\), and letting \\(k\\to\\infty\\), \\(y\\ge p_i\\). Hence \\(p_i(1-y)p_i\\le p_i(1-p_i)p_i=0\\), while \\(1-y\\ge0\\) because \\(y\\le1\\). So \\((1-y)^{1/2}p_i=0\\) and \\(yp_i=p_i\\). So \\(y\\xi=\\xi\\) for every \\(\\xi\\) in the range of some \\(p_i\\), hence for every \\(\\xi\\in pH\\): \\(yp=p\\). With \\(y=pyp\\) this gives \\(y=pyp=p\\). \\(\\square\\)\n\n**Theorem 12.9** (the up-down-up theorem). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and \\(M=A''\\). Then\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\Big(\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\\\\n\\text{and}\\\\\nM_h\\\\\n=\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}.\n\\end{gathered}\n\\]\nIf \\(1\\in A\\), then moreover \\(M_+\\cap S=\\Big(\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\).\n\n\n**Proof.** The inclusions \\(\\supseteq\\) hold by Lemma 12.2(5).\n\n*Step 1: the case \\(1\\in A\\).* Put \\(Y=\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\) and \\(Y'=(A_+\\cap S)^{\\searrow}\\). Both are convex subsets of \\(B(H)_+\\cap S\\) that contain \\(0\\) (Lemma 12.2(1) and (2)). For \\(\\beta\\in(0,1]\\), \\(g_\\beta\\) maps \\([0,1]\\) into \\([0,1]\\) and \\(g_\\beta(0)=0\\), so \\(g_\\beta(A_+\\cap S)\\subseteq A_+\\cap S\\); by Lemma 12.2(4), \\(g_\\beta(Y)\\subseteq Y\\) and \\(g_\\beta(Y')\\subseteq Y'\\). So Lemma 12.8 applies to \\(Y\\) and to \\(Y'\\).\n\n*Step 2: infima of projections in \\(Y\\).* Let \\((q_i)\\) be projections in \\(Y\\). Since \\(1\\in A\\), the map \\(t\\mapsto1-t\\) maps \\(A_+\\cap S\\) onto itself, and Lemma 12.2(3) gives \\(1-q_i\\in1-Y=\\big((A_+\\cap S)^{\\searrow}\\big)^{\\uparrow}=Y'^{\\uparrow}\\). By Lemma 12.8 applied to \\(Y'\\), \\(\\bigvee_i(1-q_i)\\in Y'^{\\uparrow}\\). Hence \\(\\bigwedge_iq_i=1-\\bigvee_i(1-q_i)\\in1-Y'^{\\uparrow}=Y\\).\n\n*Step 3: projections.* Let \\(p\\in M\\) be a projection. For \\(\\xi\\in pH\\) and \\(\\eta\\in(1-p)H\\), Lemma 12.5 applied to \\(p\\) and the two vectors \\(\\xi,\\eta\\) gives a projection \\(q_{\\xi,\\eta}\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\subseteq Y\\) with \\(q_{\\xi,\\eta}\\xi=\\xi\\) and \\(q_{\\xi,\\eta}\\eta=0\\). Put \\(p_\\xi=\\bigwedge_{\\eta\\in(1-p)H}q_{\\xi,\\eta}\\), which lies in \\(Y\\) by Step 2. Then \\(p_\\xi\\xi=\\xi\\), and \\(p_\\xi\\eta=p_\\xi q_{\\xi,\\eta}\\eta=0\\) for every \\(\\eta\\in(1-p)H\\), so \\(p_\\xi\\le p\\). Hence \\(p=\\bigvee_{\\xi\\in pH}p_\\xi\\), and \\(p\\in Y^{\\uparrow}\\) by Lemma 12.8.\n\n*Step 4: positive contractions.* Let \\(x\\in M_+\\cap S\\), and write \\(x=\\sum_k2^{-k}p_k\\) with projections \\(p_k\\in M\\) (Corollary 4.6). By Step 3 each \\(p_k\\) is the limit of an increasing net \\((y_{k,j})_{j\\in J_k}\\) in \\(Y\\); add a least element with value \\(0\\) as in Lemma 12.8. Let \\(\\Gamma\\) be the set of functions \\(\\gamma\\) on \\(\\{1,2,\\dots\\}\\) with \\(\\gamma(k)\\in J_k\\), equal to the least element for all but finitely many \\(k\\), ordered pointwise, and put \\(x_\\gamma=\\sum_k2^{-k}y_{k,\\gamma(k)}\\). The sum is finite, and \\(x_\\gamma\\in Y\\), as a convex combination of elements of \\(Y\\) and \\(0\\). The net \\((x_\\gamma)\\) increases, and \\(x_\\gamma\\le x\\). For every \\(N\\), \\(\\lim_\\gamma x_\\gamma\\ge\\sum_{k\\le N}2^{-k}p_k\\ge x-2^{-N}\\). So \\(x_\\gamma\\to x\\), and \\(x\\in Y^{\\uparrow}=\\Big(\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\). This proves the last statement of the theorem, and the first one for unital \\(A\\).\n\n*Step 5: the general case.* Let \\(\\tilde A=A+\\mathbb C1\\). Since \\(A\\) is nondegenerate, \\(1\\in A''\\), so \\(\\tilde A''=A''=M\\). By Step 4 for \\(\\tilde A\\), and Lemma 12.7(2),\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\Big(\\big((\\tilde A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\\\\n\\subseteq\\Big(\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}.\n\\end{gathered}\n\\]\n\n*Step 6: self-adjoint elements.* The decomposition used for Theorem 12.6 writes \\(y\\in M_h\\) as \\(2\\|y\\|x-\\|y\\|\\cdot1\\) with \\(x\\in M_+\\cap S\\). An increasing approximate identity of \\(A\\) converges strongly to \\(1\\) (proof of Lemma 12.7), so \\(1\\in(A_h)^{\\uparrow}\\) and \\(-\\|y\\|\\cdot1\\in(A_h)^{\\downarrow}\\subseteq\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\). This set is closed under sums and contains \\(2\\|y\\|x\\) by Lemma 12.2. \\(\\square\\)\n\n**Corollary 12.10** (the monotone closure criterion). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\).\n\n1. \\(A\\) is a von Neumann algebra if and only if the strong limit of every norm-bounded increasing net in \\(A_h\\) lies in \\(A\\).\n2. If \\(A''\\) is \\(\\sigma\\)-finite, for instance if \\(H\\) is separable, then \\(A\\) is a von Neumann algebra if and only if the strong limit of every norm-bounded increasing sequence in \\(A_h\\) lies in \\(A\\).\n\n**Proof.** A von Neumann algebra is strongly closed, which gives \"only if\" in both parts. (1) Suppose \\((A_h)^{\\uparrow}\\subseteq A_h\\). Then \\((A_h)^{\\downarrow}=-(A_h)^{\\uparrow}\\subseteq A_h\\) by Lemma 12.2(3), since \\(-A_h=A_h\\). Applying Lemma 12.2(1) three times, \\(\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\subseteq A_h\\). By Theorem 12.9, \\((A'')_h\\subseteq A_h\\), and every element of \\(A''\\) is a combination of two self-adjoint elements of \\(A''\\). So \\(A''=A\\). (2) The same argument with sequences, using Theorem 12.6. A von Neumann algebra on a separable Hilbert space is \\(\\sigma\\)-finite ([Remark 9.6 of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-13)). \\(\\square\\)\n\n**Example 12.11** (sequences are not enough without \\(\\sigma\\)-finiteness). Let \\(H=\\ell^2([0,1])\\), for counting measure on \\([0,1]\\), with basis \\((\\delta_t)\\), and let \\(A\\) be the set of multiplication operators \\(m_g\\) by bounded Borel functions \\(g\\) on \\([0,1]\\). Since \\(\\langle m_g\\delta_t,\\delta_t\\rangle=g(t)\\), \\(\\|m_g\\|=\\sup|g|\\), and \\(g\\) is determined by \\(m_g\\). Uniform limits of Borel functions are Borel, so \\(A\\) is a C\\(^*\\)-algebra; it contains \\(1\\). If \\(g_k\\) is a bounded increasing sequence of real Borel functions with pointwise limit \\(g\\), then \\(g\\) is Borel, and \\(m_{g_k}\\to m_g\\) strongly by dominated convergence (for counting measure). So \\(A\\) contains the strong limits of its bounded increasing sequences of self-adjoint elements. But \\(A\\) is not a von Neumann algebra. There are subsets \\(T\\subseteq[0,1]\\) that are not Borel: [Corollary 2.8 and Lemma 1.3 of the Polish-space lesson](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-12) prove that this uncountable Polish space has \\(2^{\\aleph_0}\\) points but at most that many Borel sets, whereas Cantor’s diagonal theorem gives strictly more subsets. The cardinal identities and diagonal theorem have full proofs in [Theorem 8.2 and Proposition 8.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08). The operators \\(m_{1_{T\\cap F}}\\), for finite \\(F\\subseteq[0,1]\\), lie in \\(A\\) and increase strongly to \\(m_{1_T}\\), which lies in the weak closure \\(A''\\) but not in \\(A\\). Here \\(A''\\) is not \\(\\sigma\\)-finite: it contains the uncountably many orthogonal projections \\(m_{1_{\\{t\\}}}\\).\n\n**Example 12.12** (nondegeneracy is needed). Let \\(H\\ne\\{0\\}\\) and \\(A=\\{0\\}\\). Then \\(A'=B(H)\\) and \\(A''=\\mathbb C1\\), but every monotone limit of elements of \\(A\\) is \\(0\\). So \\(1\\in A''_+\\cap S\\) is not in any of the sets built from \\(A\\) in Theorems 12.6 and 12.9. The same happens for every \\(A\\) with \\([AH]\\ne H\\): all monotone limits \\(x\\) of elements of \\(A\\) satisfy \\(x=exe\\) for the projection \\(e\\) onto \\([AH]\\), while \\(1\\in A''\\). Restricting to \\([AH]\\) shows that both theorems hold for every concrete C\\(^*\\)-algebra if \\(A''\\) is replaced by the weak closure of \\(A\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-KD-15",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "12. Monotone limits: the up-down and up-down-up theorems",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "anchor": "oa-fnd-kd-15",
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      "full_conditions_and_proof": "## 12. Monotone limits: the up-down and up-down-up theorems\n\nKaplansky's theorem approximates elements of \\(A''\\) by bounded nets from \\(A\\). This section approximates self-adjoint elements of \\(A''\\) in the order: by increasing and decreasing nets, applied in turn.\n\n**Definition 12.1.** Let \\(X\\) be a set of self-adjoint operators on \\(H\\). Then \\(X^{\\nearrow}\\) consists of all strong limits of increasing *sequences* in \\(X\\) that are bounded in norm, and \\(X^{\\searrow}\\) of all strong limits of such decreasing sequences. The sets \\(X^{\\uparrow}\\) and \\(X^{\\downarrow}\\) are defined in the same way with *nets* in place of sequences. By Vigier's theorem 1.3 such limits exist, and they are least upper, respectively greatest lower, bounds.\n\n**Lemma 12.2.** Let \\(X,Y\\) be sets of self-adjoint operators.\n\n1. \\(X\\subseteq X^{\\nearrow}\\subseteq X^{\\uparrow}\\) and \\(X\\subseteq X^{\\searrow}\\subseteq X^{\\downarrow}\\). If \\(X\\subseteq Y\\), then \\(X^{\\uparrow}\\subseteq Y^{\\uparrow}\\), and likewise for the other three operations.\n2. If \\(X\\) is convex, or closed under sums, or closed under multiplication by nonnegative numbers, then so are \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\).\n3. \\(-(X^{\\uparrow})=(-X)^{\\downarrow}\\) and \\(-(X^{\\nearrow})=(-X)^{\\searrow}\\). More generally, for the order-reversing map \\(t\\mapsto1-t\\): \\(1-X^{\\uparrow}=(1-X)^{\\downarrow}\\) and \\(1-X^{\\nearrow}=(1-X)^{\\searrow}\\), and with the roles of increasing and decreasing exchanged.\n4. Let \\(X\\subseteq B(H)_+\\cap S\\), and let \\(g:[0,1]\\to[0,1]\\) be continuous and operator monotone on \\([0,1]\\) (Definition 12.3) with \\(g(X)\\subseteq X\\). Then \\(g(Y)\\subseteq Y\\) for \\(Y\\) each of \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\).\n5. If \\(X\\subseteq M_+\\cap S\\) for a von Neumann algebra \\(M\\), then \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\subseteq M_+\\cap S\\).\n\n**Proof.** (1) Use constant sequences, and note that a sequence is a net. (2) If \\(x_i\\uparrow x\\) and \\(y_j\\uparrow y\\) are increasing nets in \\(X\\), then for \\(t\\in[0,1]\\) the net \\(tx_i+(1-t)y_j\\), indexed by pairs \\((i,j)\\) with the product order, is increasing in \\(X\\) and converges strongly to \\(tx+(1-t)y\\). Sums and nonnegative multiples are handled the same way. For sequences use the pairs \\((m,m)\\). (3) Negation and \\(t\\mapsto1-t\\) reverse the order and preserve strong limits. (4) If \\(x_i\\uparrow x\\) in \\(X\\), then \\(g(x_i)\\) is an increasing net in \\(X\\), and \\(g(x_i)\\to g(x)\\) strongly by Corollary 5.3(1). So \\(g(x)\\in X^{\\uparrow}\\). The other cases are the same. (5) \\(M\\) is strongly closed, and so are \\(B(H)_+\\) and \\(S\\) (Lemma 1.2). \\(\\square\\)\n\n**Definition 12.3.** A real continuous function \\(g\\) on an interval \\(J\\) is *operator monotone on \\(J\\)* if \\(g(x)\\le g(y)\\) whenever \\(x\\le y\\) are self-adjoint operators with spectra in \\(J\\).\n\nEvery operator monotone function is increasing, as one sees on multiples of \\(1\\). The converse fails: \\(t\\mapsto t^2\\) is not operator monotone on \\([0,\\infty)\\), since \\(x=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\\le y=\\begin{pmatrix}2&1\\\\1&1\\end{pmatrix}\\), while \\(y^2-x^2=\\begin{pmatrix}4&3\\\\3&2\\end{pmatrix}\\) has determinant \\(-1\\) and so is not positive. By the [Löwner–Heinz inequality](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-17), \\(t\\mapsto t^\\alpha\\) is operator monotone on \\([0,\\infty)\\) for \\(0\\le\\alpha\\le1\\) (for \\(\\alpha=0\\) the function is constant). We need two simpler families.\n\n**Lemma 12.4** (resolvent-type operator monotone functions). Let \\(c>0\\).\n\n1. \\(g_c(t)=t/(c+t)\\) is operator monotone on \\([0,\\infty)\\). It takes values in \\([0,1)\\), \\(g_c(0)=0\\), and \\(g_c\\) increases as \\(c\\) decreases.\n2. \\(\\phi_c(t)=t/(1+c(1-t))\\) is operator monotone on \\([0,1]\\). It maps \\([0,1]\\) onto \\([0,1]\\), \\(\\phi_c(0)=0\\), \\(\\phi_c(1)=1\\), \\(\\phi_c\\) decreases as \\(c\\) increases, and \\(\\phi_c(t)\\to0\\) as \\(c\\to\\infty\\) for \\(t<1\\).\n3. For \\(\\alpha>0\\), the function \\(t/(\\alpha+(1-\\alpha)t)\\) is operator monotone on \\([0,1]\\), and for \\(0<\\alpha\\le1\\) also on \\([0,\\infty)\\).\n\n**Proof.** Inversion reverses the order on invertible positive operators ([working with the order](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-16)). (1) \\(g_c(t)=1-c(c+t)^{-1}\\). If \\(0\\le x\\le y\\), then \\(c+x\\le c+y\\), so \\((c+y)^{-1}\\le(c+x)^{-1}\\) and \\(g_c(x)\\le g_c(y)\\). (2) For \\(t\\in[0,1]\\), \\(\\phi_c(t)=\\frac{1+c}{c}\\big(1+c(1-t)\\big)^{-1}-\\frac1c\\). If \\(0\\le x\\le y\\le1\\), then \\(1+c(1-x)\\ge1+c(1-y)\\ge1\\), so \\(\\big(1+c(1-x)\\big)^{-1}\\le\\big(1+c(1-y)\\big)^{-1}\\) and \\(\\phi_c(x)\\le\\phi_c(y)\\). The other claims are elementary. (3) For \\(0<\\alpha<1\\) the function is \\((1-\\alpha)^{-1}g_c\\) with \\(c=\\alpha/(1-\\alpha)\\); for \\(\\alpha=1\\) it is \\(t\\); for \\(\\alpha>1\\) it is \\(\\phi_c\\) with \\(c=\\alpha-1\\). \\(\\square\\)\n\n**Lemma 12.5** (a projection on countably many vectors). Let \\(A\\) be a concrete C\\(^*\\)-algebra on \\(H\\) with weak closure \\(M\\), let \\(p\\in M\\) be a projection, and let \\(\\xi_1,\\xi_2,\\dots\\in H\\). There are \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\) and a projection \\(q\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\) such that, for all \\(k\\),\n\\[\n\\begin{gathered}\ny(1-p)\\xi_k\\\\\n=0,\\\\\nyp\\xi_k\\\\\n=p\\xi_k,\\\\\nq(1-p)\\xi_k\\\\\n=0,\\\\\nqp\\xi_k\\\\\n=p\\xi_k .\n\\end{gathered}\n\\]\n\n**Proof.** Scaling the vectors changes nothing, so we assume \\(\\|\\xi_k\\|\\le1\\). Put \\(\\eta_k=(1-p)\\xi_k\\) and \\(\\zeta_k=p\\xi_k\\). By Kaplansky's density theorem 7.1(3), \\(p\\) is a strong limit of a net in \\(A_+\\cap S\\). Since \\(p\\eta_k=0\\) and \\(p\\zeta_k=\\zeta_k\\), for every \\(m\\) we can choose \\(x_m\\in A_+\\cap S\\) with\n\\[\n\\begin{gathered}\n\\|x_m\\eta_k\\|\\\\\n\\le4^{-m}\\\\\n\\text{and}\\\\\n\\|(1-x_m)\\zeta_k\\|\\\\\n\\le\\tfrac1m\\\\\n(k\\\\\n\\le m).\n\\end{gathered}\n\\]\nFor \\(n\\le m\\) put \\(X_{n,m}=\\sum_{j=n}^m2^jx_j\\) and \\(y_{n,m}=g_1(X_{n,m})=X_{n,m}(1+X_{n,m})^{-1}\\). Then \\(y_{n,m}\\in A_+\\cap S\\), since \\(g_1(0)=0\\) and \\(0\\le g_1\\le1\\).\n\n*Monotonicity.* For fixed \\(n\\), \\(X_{n,m}\\) increases with \\(m\\), so \\(y_{n,m}\\) increases with \\(m\\) by Lemma 12.4(1), to a limit \\(y_n\\in(A_+\\cap S)^{\\nearrow}\\). Since \\(X_{n+1,m}\\le X_{n,m}\\), \\(y_{n+1}\\le y_n\\), and \\(y_n\\) decreases to a limit \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\).\n\n*The vectors \\(\\eta_k\\).* For \\(k\\le n\\le m\\), using \\(g_1(t)\\le t\\),\n\\[\n\\begin{gathered}\n\\langle y_{n,m}\\eta_k,\\eta_k\\rangle\\\\\n\\le\\langle X_{n,m}\\eta_k,\\eta_k\\rangle\\\\\n\\le\\sum_{j\\ge n}2^j\\|x_j\\eta_k\\|\\\\\n\\le\\sum_{j\\ge n}2^{-j}\\\\\n=2^{1-n}.\n\\end{gathered}\n\\]\nSo \\(0\\le\\langle y\\eta_k,\\eta_k\\rangle\\le\\langle y_n\\eta_k,\\eta_k\\rangle\\le2^{1-n}\\) for all \\(n\\ge k\\). Hence \\(\\|y^{1/2}\\eta_k\\|=0\\) and \\(y\\eta_k=0\\).\n\n*The vectors \\(\\zeta_k\\).* Since \\(X_{n,m}\\ge2^mx_m\\), \\(y_{n,m}\\ge g_1(2^mx_m)\\), so \\(1-y_{n,m}\\le(1+2^mx_m)^{-1}\\). For \\(t\\in[0,1]\\) and \\(b>0\\), \\((1+bt)^{-1}\\le(1+b(1-t))/(1+b)\\), because \\((1+bt)(1+b-bt)-(1+b)=b^2t(1-t)\\ge0\\). With \\(b=2^m\\) this gives \\(1-y_{n,m}\\le2^{-m}+(1-x_m)\\). For \\(k\\le m\\),\n\\[\n\\begin{gathered}\n\\langle(1-y_{n,m})\\zeta_k,\\zeta_k\\rangle\\\\\n\\le2^{-m}+\\|(1-x_m)\\zeta_k\\|\\\\\n\\le2^{-m}+\\tfrac1m .\n\\end{gathered}\n\\]\nLetting \\(m\\to\\infty\\), \\(\\langle(1-y_n)\\zeta_k,\\zeta_k\\rangle\\le0\\). Since \\(1-y_n\\ge0\\), \\((1-y_n)\\zeta_k=0\\) for all \\(n\\), and in the limit \\(y\\zeta_k=\\zeta_k\\).\n\n*The projection \\(q\\).* Let \\(\\phi_n\\) be the functions of Lemma 12.4(2) with \\(c=n\\). They decrease to the indicator function \\(1_{\\{1\\}}\\) of the point \\(1\\) on \\([0,1]\\). So \\(\\phi_n(y)\\) decreases, and its strong limit \\(q\\) is \\(1_{\\{1\\}}(y)\\), a projection (Theorem 4.2). We show \\(q\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\). For fixed \\(n\\), \\(\\phi_n(y_{n,m})\\in A_+\\cap S\\), since \\(\\phi_n(0)=0\\). It increases with \\(m\\) (Lemma 12.4(2)) and converges strongly to \\(\\phi_n(y_n)\\) (Corollary 5.3(1)). So \\(z_n=\\phi_n(y_n)\\in(A_+\\cap S)^{\\nearrow}\\). Next, \\(z_{n+1}=\\phi_{n+1}(y_{n+1})\\le\\phi_n(y_{n+1})\\le\\phi_n(y_n)=z_n\\), using \\(\\phi_{n+1}\\le\\phi_n\\) and operator monotonicity. Let \\(z\\) be the limit of the decreasing sequence \\((z_n)\\); then \\(z\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\). On one hand, \\(q\\le\\phi_n(y)\\le\\phi_n(y_n)=z_n\\) for all \\(n\\), so \\(q\\le z\\). On the other hand, for \\(k\\ge n\\), \\(\\phi_n(y_k)\\ge\\phi_k(y_k)=z_k\\ge z\\); letting \\(k\\to\\infty\\), \\(\\phi_n(y)\\ge z\\) by Corollary 5.3(1), and letting \\(n\\to\\infty\\), \\(q\\ge z\\). So \\(q=z\\).\n\nFinally, \\[\n\\begin{gathered}\n\\|q\\eta_k\\|^2\\\\\n=\\langle q\\eta_k,\\eta_k\\rangle\\\\\n\\le\\langle\\phi_1(y)\\eta_k,\\eta_k\\rangle\\\\\n\\le\\langle y\\eta_k,\\eta_k\\rangle\\\\\n=0,\n\\end{gathered}\n\\] since \\(\\phi_1(t)=t/(2-t)\\le t\\) on \\([0,1]\\). And \\(y\\zeta_k=\\zeta_k\\) implies \\(\\psi(y)\\zeta_k=\\psi(1)\\zeta_k\\) for every continuous \\(\\psi\\) (first for polynomials, then by uniform approximation), so \\(\\phi_n(y)\\zeta_k=\\zeta_k\\) for all \\(n\\), and \\(q\\zeta_k=\\zeta_k\\). \\(\\square\\)\n\n**Theorem 12.6** (the up-down theorem). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and suppose that \\(M=A''\\) is \\(\\sigma\\)-finite. Then\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\\\\n\\text{and}\\\\\nM_h\\\\\n=\\big((A_h)^{\\nearrow}\\big)^{\\searrow}.\n\\end{gathered}\n\\]\n\n\n**Proof.** The inclusions \\(\\supseteq\\) hold by Lemma 12.2(5), and for \\(M_h\\) because \\(M\\) is strongly closed.\n\n*Projections.* By [the characterization of \\(\\sigma\\)-finite von Neumann algebras](the-double-commutant-theorem.md#oa-fnd-bi-13), \\(H\\) contains a countable set \\(\\{\\xi_k\\}\\) that is separating for \\(M\\). Let \\(p\\in M\\) be a projection. Lemma 12.5 gives \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\subseteq M\\) with \\((y-p)\\xi_k=y(1-p)\\xi_k+(yp\\xi_k-p\\xi_k)=0\\) for all \\(k\\). As \\(y-p\\in M\\) vanishes on a separating set, \\(y=p\\).\n\n*The unit.* In particular \\(1=\\lim y_n\\) for a decreasing sequence \\((y_n)\\) in \\((A_+\\cap S)^{\\nearrow}\\). Each \\(y_n\\) satisfies \\(1\\le y_n\\le1\\), so \\(y_n=1\\), and \\(1\\in(A_+\\cap S)^{\\nearrow}\\).\n\n*Positive contractions.* Let \\(x\\in M_+\\cap S\\). By Corollary 4.6, \\(x=\\sum_k2^{-k}p_k\\) with projections \\(p_k\\in M\\). By the first step, each \\(p_k\\) is the limit of a decreasing sequence \\((z_{k,n})_n\\) in \\((A_+\\cap S)^{\\nearrow}\\). Put\n\\[\nx_n=\\sum_{k=1}^n2^{-k}z_{k,n}+2^{-n}\\cdot1 .\n\\]\nThis is a convex combination of elements of the convex set \\((A_+\\cap S)^{\\nearrow}\\) (Lemma 12.2(2)), since the weights add up to \\(1\\) and \\(1\\in(A_+\\cap S)^{\\nearrow}\\). So \\(x_n\\in(A_+\\cap S)^{\\nearrow}\\). The sequence decreases:\n\\[\n\\begin{gathered}\nx_n-x_{n+1}\\\\\n=\\sum_{k=1}^n2^{-k}(z_{k,n}-z_{k,n+1})+2^{-n-1}(1-z_{n+1,n+1})\\\\\n\\ge0 .\n\\end{gathered}\n\\]\nIt lies above \\(x\\): \\(z_{k,n}\\ge p_k\\) and \\(\\sum_{k>n}2^{-k}p_k\\le2^{-n}\\), so \\(x_n\\ge\\sum_{k\\le n}2^{-k}p_k+2^{-n}\\ge x\\). For \\(n\\ge m\\), using \\(z_{k,n}\\le1\\) for \\(k>m\\),\n\\[\n\\begin{gathered}\nx_n-x\\\\\n\\le\\sum_{k=1}^m2^{-k}(z_{k,n}-p_k)+\\sum_{k=m+1}^n2^{-k}+2^{-n}\\\\\n=\\sum_{k=1}^m2^{-k}(z_{k,n}-p_k)+2^{-m}.\n\\end{gathered}\n\\]\nLet \\(x'\\) be the strong limit of \\((x_n)\\). Letting \\(n\\to\\infty\\) gives \\(x\\le x'\\le x+2^{-m}\\) for every \\(m\\). So \\(x'=x\\), and \\(x\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\).\n\n*Self-adjoint elements.* Let \\(y\\in M_h\\), \\(y\\ne0\\). Then \\(x=(y+\\|y\\|)/(2\\|y\\|)\\in M_+\\cap S\\), and \\(y=2\\|y\\|x-\\|y\\|\\cdot1\\). Here \\(2\\|y\\|x\\in\\big((A_h)^{\\nearrow}\\big)^{\\searrow}\\) by Lemma 12.2(2). Since \\(1\\in(A_h)^{\\nearrow}\\), \\(-\\|y\\|\\cdot1\\in(A_h)^{\\searrow}\\subseteq\\big((A_h)^{\\nearrow}\\big)^{\\searrow}\\) by Lemma 12.2(1) and (3). The set \\(\\big((A_h)^{\\nearrow}\\big)^{\\searrow}\\) is closed under sums by Lemma 12.2(2). \\(\\square\\)\n\nWithout \\(\\sigma\\)-finiteness the up-down theorem fails, even with nets in place of sequences (Section 13). The remedy is a third, increasing, limit. First we remove the need for a unit.\n\n**Lemma 12.7** (adding a unit). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and \\(\\tilde A=A+\\mathbb C1\\).\n\n1. For \\(\\varepsilon>0\\) and \\(x\\in(\\tilde A_+\\cap S)^{\\nearrow}\\), \\((1+\\varepsilon)^{-1}(x+\\varepsilon)\\in(A_+\\cap S)^{\\uparrow}\\).\n2. \\(\\big((\\tilde A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\subseteq\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\).\n\n**Proof.** (1) Let \\(x_n\\in\\tilde A_+\\cap S\\) increase to \\(x\\), and put \\(x_0=0\\) and \\(d_n=x_n-x_{n-1}\\ge0\\). Write \\(d_n=c_n+\\alpha_n\\) with \\(c_n\\in A_h\\) and \\(\\alpha_n\\in\\mathbb R\\). We may take \\(\\alpha_n\\ge0\\): if \\(1\\in A\\), take \\(\\alpha_n=0\\); if not, \\(y+\\alpha\\mapsto\\alpha\\) is a \\(*\\)-homomorphism of \\(\\tilde A\\) onto \\(\\mathbb C\\), which maps the positive element \\(d_n\\) to a nonnegative number. By [approximate identities](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-20), \\(A\\) has an increasing approximate identity \\((u_i)_{i\\in I}\\) in \\(A_+\\cap S\\). Its strong limit \\(e_0\\) (Theorem 1.3) satisfies \\(e_0z=\\lim u_iz=z\\) for \\(z\\in A\\), so \\(e_0=1\\) on \\([AH]=H\\): \\(u_i\\to1\\) strongly.\n\n*A scalar inequality.* Let \\(c\\) be self-adjoint and \\(\\alpha\\ge0\\) with \\(c+\\alpha\\ge0\\), and let \\(0<\\beta\\le\\varepsilon'\\). Then\n\\[\nc+(\\alpha+\\varepsilon')\\,g_\\beta(|c|)\\ge0 ,\n\\tag{12.1}\n\\]\nwith \\(g_\\beta\\) as in Lemma 12.4(1). By the functional calculus of \\(c\\) it suffices to check \\(t+(\\alpha+\\varepsilon')|t|/(\\beta+|t|)\\ge0\\) for \\(t\\in\\sigma(c)\\). For \\(t\\ge0\\) this is clear. For \\(t<0\\), \\(\\alpha\\ge|t|\\), and the left side is at least \\[\n\\begin{gathered}\n-|t|+(|t|+\\varepsilon')|t|/(\\beta+|t|)\\\\\n=|t|(\\varepsilon'-\\beta)/(\\beta+|t|)\\\\\n\\ge0.\n\\end{gathered}\n\\]\n\n*The approximating net.* Let \\(\\Lambda\\) be the set of triples \\(\\lambda=(m,\\beta,i)\\) with \\(m\\ge1\\), \\(0<\\beta\\le\\varepsilon2^{-m}\\) and \\(i\\in I\\), directed by \\(m\\le m'\\), \\(\\beta\\ge\\beta'\\), \\(i\\le i'\\). For \\(n\\le m\\) put\n\\[\n\\begin{gathered}\nw_{n,\\beta,i}\\\\\n=c_n+(\\alpha_n+\\varepsilon2^{-n})\\,g_\\beta(u_i+|c_n|)\\in A_h,\\\\\nv_\\lambda\\\\\n=(1+\\varepsilon)^{-1}\\sum_{n=1}^mw_{n,\\beta,i}.\n\\end{gathered}\n\\]\nEach \\(w_{n,\\beta,i}\\) lies in \\(A\\), because \\(g_\\beta(0)=0\\) and \\(u_i+|c_n|\\in A_+\\). Since \\(u_i+|c_n|\\ge|c_n|\\) and \\(g_\\beta\\) is operator monotone, (12.1) with \\(\\varepsilon'=\\varepsilon2^{-n}\\ge\\beta\\) gives \\(w_{n,\\beta,i}\\ge0\\). Since \\(g_\\beta\\le1\\), \\(w_{n,\\beta,i}\\le c_n+\\alpha_n+\\varepsilon2^{-n}=d_n+\\varepsilon2^{-n}\\). So\n\\[\n\\begin{gathered}\n0\\\\\n\\le v_\\lambda\\\\\n\\le(1+\\varepsilon)^{-1}(x_m+\\varepsilon)\\\\\n\\le(1+\\varepsilon)^{-1}(x+\\varepsilon)\\\\\n\\le1 ,\n\\end{gathered}\n\\]\nand \\(v_\\lambda\\in A_+\\cap S\\). The net \\((v_\\lambda)\\) increases: \\(w_{n,\\beta,i}\\) increases when \\(i\\) increases or \\(\\beta\\) decreases (Lemma 12.4(1)), and a larger \\(m\\) adds nonnegative terms. Let \\(v\\) be its strong limit. Then \\(v\\le(1+\\varepsilon)^{-1}(x+\\varepsilon)\\).\n\nFor the reverse inequality fix \\(N\\), then \\(m\\ge N\\) and \\(\\beta\\in(0,\\varepsilon2^{-m}]\\). The terms with \\(n>N\\) are positive, so \\(v\\ge v_{(m,\\beta,i)}\\ge(1+\\varepsilon)^{-1}\\sum_{n\\le N}w_{n,\\beta,i}\\) for every \\(i\\). As \\(i\\to\\infty\\), \\(u_i+|c_n|\\to1+|c_n|\\) strongly within a bounded set, so \\(g_\\beta(u_i+|c_n|)\\to g_\\beta(1+|c_n|)\\) strongly (Corollary 5.3(1)). Since \\(1-g_\\beta(t)=\\beta/(\\beta+t)\\le\\beta\\) for \\(t\\ge1\\), \\(g_\\beta(1+|c_n|)\\ge(1-\\beta)1\\). By Lemma 1.2 the inequality passes to the limit, and\n\\[\n\\begin{gathered}\nv\\\\\n\\ge(1+\\varepsilon)^{-1}\\sum_{n\\le N}\\big(c_n+(\\alpha_n+\\varepsilon2^{-n})(1-\\beta)\\big)\\\\\n=(1+\\varepsilon)^{-1}\\Big(x_N+\\varepsilon(1-2^{-N})\\\\\n-\\beta\\sum_{n\\le N}(\\alpha_n+\\varepsilon2^{-n})\\Big).\n\\end{gathered}\n\\]\nHere \\(\\beta\\) can be arbitrarily small, so \\(v\\ge(1+\\varepsilon)^{-1}(x_N+\\varepsilon(1-2^{-N}))\\). Letting \\(N\\to\\infty\\), \\(v\\ge(1+\\varepsilon)^{-1}(x+\\varepsilon)\\). So \\((1+\\varepsilon)^{-1}(x+\\varepsilon)=v\\in(A_+\\cap S)^{\\uparrow}\\).\n\n(2) Let \\(x_j\\) be a decreasing net in \\((\\tilde A_+\\cap S)^{\\nearrow}\\) with limit \\(x\\). By (1), \\(y_{j,k}=(1+\\frac1k)^{-1}(x_j+\\frac1k)\\in(A_+\\cap S)^{\\uparrow}\\). For \\(0\\le t\\le1\\), \\(\\frac{kt+1}{k+1}-\\frac{(k+1)t+1}{k+2}=\\frac{1-t}{(k+1)(k+2)}\\ge0\\), so \\(y_{j,k}\\) decreases in \\(k\\); it also decreases in \\(j\\). The net \\((y_{j,k})\\) has limit \\(x\\). So \\(x\\in\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\). \\(\\square\\)\n\n**Lemma 12.8** (suprema of projections). Let \\(Y\\subseteq B(H)_+\\cap S\\) be convex, with \\(0\\in Y\\) and \\(g_\\beta(Y)\\subseteq Y\\) for all \\(\\beta\\in(0,1]\\). If \\((p_i)_{i\\in I}\\) is any family of projections in \\(Y^{\\uparrow}\\), then \\(\\bigvee_ip_i\\in Y^{\\uparrow}\\).\n\n**Proof.** Put \\(p=\\bigvee_ip_i\\). For each \\(i\\), choose an increasing net \\((x_{i,j})_{j\\in J_i}\\) in \\(Y\\) with limit \\(p_i\\), and add to \\(J_i\\) a new least element \\(0_i\\) with \\(x_{i,0_i}=0\\); the net is still increasing, since \\(x_{i,j}\\ge0\\). Let \\(\\Gamma\\) be the set of functions \\(\\gamma\\) on \\(I\\) with \\(\\gamma(i)\\in J_i\\) for all \\(i\\) and \\(\\gamma(i)=0_i\\) for all but finitely many \\(i\\), ordered pointwise. It is directed. For \\(\\gamma\\in\\Gamma\\) and \\(k\\ge1\\) put\n\\[\n\\begin{gathered}\nX_\\gamma\\\\\n=\\sum_ix_{i,\\gamma(i)},\\\\\ny_{\\gamma,k}\\\\\n=g_{1/k}(X_\\gamma)\\\\\n=X_\\gamma\\big(\\tfrac1k+X_\\gamma\\big)^{-1}.\n\\end{gathered}\n\\]\nThe sum is finite. If \\(\\gamma\\) has \\(N\\ge1\\) entries that are not least elements, then \\(X_\\gamma/N\\in Y\\) by convexity, and \\(y_{\\gamma,k}=g_{1/(kN)}(X_\\gamma/N)\\in Y\\); if \\(N=0\\), \\(y_{\\gamma,k}=0\\in Y\\).\n\nThe net \\((y_{\\gamma,k})\\) increases: \\(X_\\gamma\\) increases with \\(\\gamma\\), \\(g_{1/k}\\) is operator monotone (Lemma 12.4(1)), and \\(g_{1/k}\\le g_{1/k'}\\) for \\(k\\le k'\\). Let \\(y\\in Y^{\\uparrow}\\) be its limit. Then \\(0\\le y\\le1\\). Each \\(x_{i,j}\\le p_i\\le p\\), so \\(x_{i,j}=px_{i,j}p\\), hence \\(X_\\gamma=pX_\\gamma p\\), \\(y_{\\gamma,k}=py_{\\gamma,k}p\\) and \\(y=pyp\\).\n\nFix \\(i\\) and \\(k\\). For \\(j\\in J_i\\), take \\(\\gamma\\) with \\(\\gamma(i)=j\\) and \\(\\gamma(i')=0_{i'}\\) otherwise; then \\(y\\ge y_{\\gamma,k}=g_{1/k}(x_{i,j})\\). Letting \\(j\\) run through \\(J_i\\), \\(g_{1/k}(x_{i,j})\\to g_{1/k}(p_i)=\\frac k{k+1}p_i\\) strongly (Corollary 5.3(1)). By Lemma 1.2, \\(y\\ge\\frac k{k+1}p_i\\), and letting \\(k\\to\\infty\\), \\(y\\ge p_i\\). Hence \\(p_i(1-y)p_i\\le p_i(1-p_i)p_i=0\\), while \\(1-y\\ge0\\) because \\(y\\le1\\). So \\((1-y)^{1/2}p_i=0\\) and \\(yp_i=p_i\\). So \\(y\\xi=\\xi\\) for every \\(\\xi\\) in the range of some \\(p_i\\), hence for every \\(\\xi\\in pH\\): \\(yp=p\\). With \\(y=pyp\\) this gives \\(y=pyp=p\\). \\(\\square\\)\n\n**Theorem 12.9** (the up-down-up theorem). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and \\(M=A''\\). Then\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\Big(\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\\\\n\\text{and}\\\\\nM_h\\\\\n=\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}.\n\\end{gathered}\n\\]\nIf \\(1\\in A\\), then moreover \\(M_+\\cap S=\\Big(\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\).\n\n\n**Proof.** The inclusions \\(\\supseteq\\) hold by Lemma 12.2(5).\n\n*Step 1: the case \\(1\\in A\\).* Put \\(Y=\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\) and \\(Y'=(A_+\\cap S)^{\\searrow}\\). Both are convex subsets of \\(B(H)_+\\cap S\\) that contain \\(0\\) (Lemma 12.2(1) and (2)). For \\(\\beta\\in(0,1]\\), \\(g_\\beta\\) maps \\([0,1]\\) into \\([0,1]\\) and \\(g_\\beta(0)=0\\), so \\(g_\\beta(A_+\\cap S)\\subseteq A_+\\cap S\\); by Lemma 12.2(4), \\(g_\\beta(Y)\\subseteq Y\\) and \\(g_\\beta(Y')\\subseteq Y'\\). So Lemma 12.8 applies to \\(Y\\) and to \\(Y'\\).\n\n*Step 2: infima of projections in \\(Y\\).* Let \\((q_i)\\) be projections in \\(Y\\). Since \\(1\\in A\\), the map \\(t\\mapsto1-t\\) maps \\(A_+\\cap S\\) onto itself, and Lemma 12.2(3) gives \\(1-q_i\\in1-Y=\\big((A_+\\cap S)^{\\searrow}\\big)^{\\uparrow}=Y'^{\\uparrow}\\). By Lemma 12.8 applied to \\(Y'\\), \\(\\bigvee_i(1-q_i)\\in Y'^{\\uparrow}\\). Hence \\(\\bigwedge_iq_i=1-\\bigvee_i(1-q_i)\\in1-Y'^{\\uparrow}=Y\\).\n\n*Step 3: projections.* Let \\(p\\in M\\) be a projection. For \\(\\xi\\in pH\\) and \\(\\eta\\in(1-p)H\\), Lemma 12.5 applied to \\(p\\) and the two vectors \\(\\xi,\\eta\\) gives a projection \\(q_{\\xi,\\eta}\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\subseteq Y\\) with \\(q_{\\xi,\\eta}\\xi=\\xi\\) and \\(q_{\\xi,\\eta}\\eta=0\\). Put \\(p_\\xi=\\bigwedge_{\\eta\\in(1-p)H}q_{\\xi,\\eta}\\), which lies in \\(Y\\) by Step 2. Then \\(p_\\xi\\xi=\\xi\\), and \\(p_\\xi\\eta=p_\\xi q_{\\xi,\\eta}\\eta=0\\) for every \\(\\eta\\in(1-p)H\\), so \\(p_\\xi\\le p\\). Hence \\(p=\\bigvee_{\\xi\\in pH}p_\\xi\\), and \\(p\\in Y^{\\uparrow}\\) by Lemma 12.8.\n\n*Step 4: positive contractions.* Let \\(x\\in M_+\\cap S\\), and write \\(x=\\sum_k2^{-k}p_k\\) with projections \\(p_k\\in M\\) (Corollary 4.6). By Step 3 each \\(p_k\\) is the limit of an increasing net \\((y_{k,j})_{j\\in J_k}\\) in \\(Y\\); add a least element with value \\(0\\) as in Lemma 12.8. Let \\(\\Gamma\\) be the set of functions \\(\\gamma\\) on \\(\\{1,2,\\dots\\}\\) with \\(\\gamma(k)\\in J_k\\), equal to the least element for all but finitely many \\(k\\), ordered pointwise, and put \\(x_\\gamma=\\sum_k2^{-k}y_{k,\\gamma(k)}\\). The sum is finite, and \\(x_\\gamma\\in Y\\), as a convex combination of elements of \\(Y\\) and \\(0\\). The net \\((x_\\gamma)\\) increases, and \\(x_\\gamma\\le x\\). For every \\(N\\), \\(\\lim_\\gamma x_\\gamma\\ge\\sum_{k\\le N}2^{-k}p_k\\ge x-2^{-N}\\). So \\(x_\\gamma\\to x\\), and \\(x\\in Y^{\\uparrow}=\\Big(\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\). This proves the last statement of the theorem, and the first one for unital \\(A\\).\n\n*Step 5: the general case.* Let \\(\\tilde A=A+\\mathbb C1\\). Since \\(A\\) is nondegenerate, \\(1\\in A''\\), so \\(\\tilde A''=A''=M\\). By Step 4 for \\(\\tilde A\\), and Lemma 12.7(2),\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\Big(\\big((\\tilde A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\\\\n\\subseteq\\Big(\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}.\n\\end{gathered}\n\\]\n\n*Step 6: self-adjoint elements.* The decomposition used for Theorem 12.6 writes \\(y\\in M_h\\) as \\(2\\|y\\|x-\\|y\\|\\cdot1\\) with \\(x\\in M_+\\cap S\\). An increasing approximate identity of \\(A\\) converges strongly to \\(1\\) (proof of Lemma 12.7), so \\(1\\in(A_h)^{\\uparrow}\\) and \\(-\\|y\\|\\cdot1\\in(A_h)^{\\downarrow}\\subseteq\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\). This set is closed under sums and contains \\(2\\|y\\|x\\) by Lemma 12.2. \\(\\square\\)\n\n**Corollary 12.10** (the monotone closure criterion). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\).\n\n1. \\(A\\) is a von Neumann algebra if and only if the strong limit of every norm-bounded increasing net in \\(A_h\\) lies in \\(A\\).\n2. If \\(A''\\) is \\(\\sigma\\)-finite, for instance if \\(H\\) is separable, then \\(A\\) is a von Neumann algebra if and only if the strong limit of every norm-bounded increasing sequence in \\(A_h\\) lies in \\(A\\).\n\n**Proof.** A von Neumann algebra is strongly closed, which gives \"only if\" in both parts. (1) Suppose \\((A_h)^{\\uparrow}\\subseteq A_h\\). Then \\((A_h)^{\\downarrow}=-(A_h)^{\\uparrow}\\subseteq A_h\\) by Lemma 12.2(3), since \\(-A_h=A_h\\). Applying Lemma 12.2(1) three times, \\(\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\subseteq A_h\\). By Theorem 12.9, \\((A'')_h\\subseteq A_h\\), and every element of \\(A''\\) is a combination of two self-adjoint elements of \\(A''\\). So \\(A''=A\\). (2) The same argument with sequences, using Theorem 12.6. A von Neumann algebra on a separable Hilbert space is \\(\\sigma\\)-finite ([Remark 9.6 of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-13)). \\(\\square\\)\n\n**Example 12.11** (sequences are not enough without \\(\\sigma\\)-finiteness). Let \\(H=\\ell^2([0,1])\\), for counting measure on \\([0,1]\\), with basis \\((\\delta_t)\\), and let \\(A\\) be the set of multiplication operators \\(m_g\\) by bounded Borel functions \\(g\\) on \\([0,1]\\). Since \\(\\langle m_g\\delta_t,\\delta_t\\rangle=g(t)\\), \\(\\|m_g\\|=\\sup|g|\\), and \\(g\\) is determined by \\(m_g\\). Uniform limits of Borel functions are Borel, so \\(A\\) is a C\\(^*\\)-algebra; it contains \\(1\\). If \\(g_k\\) is a bounded increasing sequence of real Borel functions with pointwise limit \\(g\\), then \\(g\\) is Borel, and \\(m_{g_k}\\to m_g\\) strongly by dominated convergence (for counting measure). So \\(A\\) contains the strong limits of its bounded increasing sequences of self-adjoint elements. But \\(A\\) is not a von Neumann algebra. There are subsets \\(T\\subseteq[0,1]\\) that are not Borel: [Corollary 2.8 and Lemma 1.3 of the Polish-space lesson](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-12) prove that this uncountable Polish space has \\(2^{\\aleph_0}\\) points but at most that many Borel sets, whereas Cantor’s diagonal theorem gives strictly more subsets. The cardinal identities and diagonal theorem have full proofs in [Theorem 8.2 and Proposition 8.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08). The operators \\(m_{1_{T\\cap F}}\\), for finite \\(F\\subseteq[0,1]\\), lie in \\(A\\) and increase strongly to \\(m_{1_T}\\), which lies in the weak closure \\(A''\\) but not in \\(A\\). Here \\(A''\\) is not \\(\\sigma\\)-finite: it contains the uncountably many orthogonal projections \\(m_{1_{\\{t\\}}}\\).\n\n**Example 12.12** (nondegeneracy is needed). Let \\(H\\ne\\{0\\}\\) and \\(A=\\{0\\}\\). Then \\(A'=B(H)\\) and \\(A''=\\mathbb C1\\), but every monotone limit of elements of \\(A\\) is \\(0\\). So \\(1\\in A''_+\\cap S\\) is not in any of the sets built from \\(A\\) in Theorems 12.6 and 12.9. The same happens for every \\(A\\) with \\([AH]\\ne H\\): all monotone limits \\(x\\) of elements of \\(A\\) satisfy \\(x=exe\\) for the projection \\(e\\) onto \\([AH]\\), while \\(1\\in A''\\). Restricting to \\([AH]\\) shows that both theorems hold for every concrete C\\(^*\\)-algebra if \\(A''\\) is replaced by the weak closure of \\(A\\).\n\n",
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      "id": "OA-FND-KD-16",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "13. The last increasing limit cannot be dropped",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "full_conditions_and_proof": "## 13. The last increasing limit cannot be dropped\n\nTheorem 12.9 uses three monotone limits where Theorem 12.6 uses two. The following example shows that the third one is needed when the von Neumann algebra is not \\(\\sigma\\)-finite.\n\n**Example 13.1.** Let \\(H_d=\\ell^2([0,1])\\) for counting measure, with basis \\((\\delta_t)_{t\\in[0,1]}\\), and \\(H_c=L^2[0,1]\\) for Lebesgue measure. For \\(g\\in C[0,1]\\) let \\(m_g\\) denote multiplication by \\(g\\) on either space, and let\n\\[\n\\begin{gathered}\nA\\\\\n=\\{m_g\\oplus m_g:g\\in C[0,1]\\}\\\\\n\\subseteq B(H_d\\oplus H_c),\\\\\nM\\\\\n=A'',\\\\\nz\\\\\n=1\\oplus0 .\n\\end{gathered}\n\\]\nSince \\(\\|m_g\\oplus m_g\\|=\\max|g|\\), \\(A\\) is a unital concrete C\\(^*\\)-algebra. We show:\n\n(a) Every element of \\((A_h)^{\\uparrow}\\) is \\(m_f\\oplus m_f\\) for a unique bounded lower semicontinuous \\(f\\) on \\([0,1]\\), namely the pointwise supremum of the net. Conversely every bounded lower semicontinuous \\(f\\) arises.\n\n(b) \\(z\\in M\\).\n\n(c) If \\(a\\in(A_h)^{\\uparrow}\\) and \\(a\\ge z\\), then \\(a\\ge1\\). Consequently \\(z\\notin\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\), and in particular \\(z\\notin\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\).\n\n(d) \\(z\\in\\big((A_+\\cap S)^{\\searrow}\\big)^{\\uparrow}\\), and \\(M\\) is not \\(\\sigma\\)-finite.\n\nSo \\(z\\in M_+\\cap S\\) is reached by three monotone limits, as Theorem 12.9 asserts, but not by the first two.\n\n*Proof of (a).* Let \\((g_i)\\) be an increasing net of real continuous functions with \\(|g_i|\\le C\\), and \\(f=\\sup_ig_i\\) pointwise. Then \\(f\\) is bounded and lower semicontinuous. For each rational \\(r\\), the open set \\(\\{f>r\\}\\) is the union of the open sets \\(\\{g_i>r\\}\\). For every rational-interval basic open set contained in some \\(\\{g_i>r\\}\\), choose one such index. There are countably many choices, and they cover \\(\\{f>r\\}\\), since each point has such a basic neighborhood inside one member of the cover. Collecting these indices over all rational \\(r\\) gives a sequence of indices, and since the index set is directed we can choose an increasing sequence \\(i_1\\le i_2\\le\\cdots\\) that eventually lies above each of them. Then \\(g_{i_n}\\to f\\) pointwise: if \\(r<f(t)\\) is rational, some chosen index \\(j\\) has \\(g_j(t)>r\\), and \\(g_{i_n}(t)\\ge g_j(t)\\) for large \\(n\\). For \\(\\xi\\) in \\(H_d\\) or \\(H_c\\) and \\(i\\ge i_n\\),\n\\[\n\\begin{gathered}\n\\|(m_f-m_{g_i})\\xi\\|^2\\\\\n=\\int(f-g_i)^2|\\xi|^2\\\\\n\\le2C\\int(f-g_{i_n})|\\xi|^2 ,\n\\end{gathered}\n\\]\nthe integral taken for counting measure or Lebesgue measure. The right side tends to \\(0\\) as \\(n\\to\\infty\\), by dominated convergence. So \\(m_{g_i}\\oplus m_{g_i}\\to m_f\\oplus m_f\\) strongly. Uniqueness holds because \\(f(t)=\\langle a\\delta_t,\\delta_t\\rangle\\). Conversely, let \\(f\\) be bounded and lower semicontinuous, with \\(|f|\\le C\\). The continuous functions \\(g\\) with \\(-C\\le g\\le f\\) form an upward directed family, since \\(\\max(g_1,g_2)\\) is again one, and it is bounded in norm. Its pointwise supremum is \\(f\\): put \\(g_n(t)=\\inf_{s\\in[0,1]}(f(s)+n|s-t|)\\). The triangle inequality gives \\(|g_n(t)-g_n(t')|\\leq n|t-t'|\\), so \\(g_n\\) is continuous. Also \\(-C\\leq g_n\\leq f\\leq C\\), and \\(g_n\\) increases in \\(n\\). Given \\(t\\) and \\(\\varepsilon>0\\), lower semicontinuity gives \\(\\rho>0\\) with \\(f(s)>f(t)-\\varepsilon\\) for \\(|s-t|<\\rho\\). Outside that neighborhood, \\(f(s)+n|s-t|\\geq-C+n\\rho\\geq f(t)-\\varepsilon\\) once \\(n\\) is large. Hence \\(g_n(t)\\geq f(t)-\\varepsilon\\), proving \\(g_n(t)\\uparrow f(t)\\). Indexed by itself, this family is an increasing net, and the corresponding net in \\(A_h\\) converges strongly to \\(m_f\\oplus m_f\\).\n\n*Proof of (b).* It suffices to show that every \\(T\\in A'\\) commutes with \\(z\\), that is, has no off-diagonal parts. Let \\(Q\\) be the part of \\(T\\) that maps \\(H_d\\) to \\(H_c\\). Compressing \\(T(m_g\\oplus m_g)=(m_g\\oplus m_g)T\\) gives \\(Qm_g=m_gQ\\) for all \\(g\\). For \\(t\\in[0,1]\\), \\(m_g\\delta_t=g(t)\\delta_t\\), so the function \\(\\eta_t=Q\\delta_t\\in L^2[0,1]\\) satisfies \\((g-g(t))\\eta_t=0\\) almost everywhere. With \\(g(s)=s\\) this says \\((s-t)\\eta_t(s)=0\\) for almost every \\(s\\), so \\(\\eta_t=0\\) almost everywhere. Hence \\(Q=0\\) on the dense span of the \\(\\delta_t\\), and \\(Q=0\\). The part of \\(T\\) from \\(H_c\\) to \\(H_d\\) is the adjoint of the corresponding part \\(Q'\\) of \\(T^*\\in A'\\), and \\(Q'=0\\) by the same argument. So \\(z\\in A''=M\\).\n\n*Proof of (c).* By (a), \\(a=m_f\\oplus m_f\\) with \\(f\\) bounded. From \\(a\\ge z\\), \\(f(t)=\\langle a\\delta_t,\\delta_t\\rangle\\ge\\langle z\\delta_t,\\delta_t\\rangle=1\\) for every \\(t\\). So \\(f\\ge1\\) everywhere, and \\(a\\ge1\\) on both summands. If \\(z\\) were the limit of a decreasing net \\((a_j)\\) in \\((A_h)^{\\uparrow}\\), then \\(a_j\\ge z\\), so \\(a_j\\ge1\\) for all \\(j\\), and \\(z\\ge1\\), which is false because \\(H_c\\ne\\{0\\}\\).\n\n*Proof of (d).* For \\(t\\in[0,1]\\) let \\(h_{t,n}(s)=\\max(0,1-n|s-t|)\\). As \\(n\\to\\infty\\), \\(h_{t,n}\\) decreases to \\(1_{\\{t\\}}\\). By dominated convergence, \\(m_{h_{t,n}}\\to m_{1_{\\{t\\}}}\\) strongly on \\(H_d\\) and \\(m_{h_{t,n}}\\to0\\) strongly on \\(H_c\\), since \\(\\{t\\}\\) is a null set. For a finite set \\(F\\subseteq[0,1]\\), the sums \\(\\sum_{t\\in F}h_{t,n}\\) have disjoint supports and values in \\([0,1]\\) once \\(n\\) is large, and they decrease to \\(1_F\\). So \\(z_F=m_{1_F}\\oplus0\\in(A_+\\cap S)^{\\searrow}\\). As \\(F\\) increases, \\(z_F\\) increases strongly to \\(z\\). So \\(z\\in\\big((A_+\\cap S)^{\\searrow}\\big)^{\\uparrow}\\). The projections \\(z_{\\{t\\}}\\), \\(t\\in[0,1]\\), lie in \\(M\\) and are mutually orthogonal and nonzero, so \\(M\\) is not \\(\\sigma\\)-finite. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-KD-18",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "14. Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
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      "full_conditions_and_proof": "## 14. Exercises\n\n**Exercise 14.1** (hard; a second proof of the density theorem). For \\(x\\in B(H)\\) put \\(F(x)=2x(1+x^*x)^{-1}\\) and, for \\(y\\in S\\), \\(G(y)=y\\big(1+(1-y^*y)^{1/2}\\big)^{-1}\\).\n\n(a) Show that \\(\\|F(x)\\|\\le1\\), \\(F(x)^*=F(x^*)\\), \\(G(F(x))=x\\) for \\(x\\in S\\), and \\(F(G(y))=y\\) for \\(y\\in S\\).\n\n(b) Show that \\(F\\) is continuous from \\(B(H)\\) to \\(B(H)\\) for the strong\\(^*\\) topology, along arbitrary nets.\n\n(c) Deduce Theorem 7.1(1) for a unital concrete C\\(^*\\)-algebra \\(A\\).\n\n*Solution.* (a) We use \\(xq(x^*x)=q(xx^*)x\\) for continuous \\(q\\), which holds for polynomials and then by uniform approximation. With \\(t=x^*x\\), \\(F(x)^*F(x)=4t(1+t)^{-2}\\le1\\), since \\(4t\\le(1+t)^2\\). Next, \\[\n\\begin{gathered}\nF(x)^*\\\\\n=2(1+x^*x)^{-1}x^*\\\\\n=2x^*(1+xx^*)^{-1}\\\\\n=F(x^*).\n\\end{gathered}\n\\] For \\(x\\in S\\), \\(t\\in[0,1]\\), and \\(1-4t/(1+t)^2=\\big((1-t)/(1+t)\\big)^2\\), so\n\\[\n\\begin{gathered}\nG(F(x))\\\\\n=2x(1+t)^{-1}\\Big(1+\\tfrac{1-t}{1+t}\\Big)^{-1}\\\\\n=2x(1+t)^{-1}\\tfrac{1+t}2\\\\\n=x ,\n\\end{gathered}\n\\]\nwhere all functions are evaluated at \\(t=x^*x\\). For \\(y\\in S\\) put \\(\\sigma=(1-y^*y)^{1/2}\\) and \\(\\chi=(1+\\sigma)^{-1}\\), functions of \\(y^*y\\). Then \\[\n\\begin{gathered}\nG(y)^*G(y)\\\\\n=y^*y\\chi^2\\\\\n=(1-\\sigma^2)(1+\\sigma)^{-2}\\\\\n=(1-\\sigma)(1+\\sigma)^{-1},\n\\end{gathered}\n\\] so \\(1+G(y)^*G(y)=2(1+\\sigma)^{-1}\\), and \\(F(G(y))=2y\\chi\\cdot\\frac{1+\\sigma}2=y\\).\n\n(b) Using \\(x(1+x^*x)^{-1}=(1+xx^*)^{-1}x\\),\n\\[\n\\begin{gathered}\nx(1+x^*x)^{-1}-y(1+y^*y)^{-1}\\\\\n=(1+xx^*)^{-1}\\big[x(1+y^*y)-(1+xx^*)y\\big]\\\\\n(1+y^*y)^{-1}\\\\\n=(1+xx^*)^{-1}\\big[(x-y)+x(y^*-x^*)y\\big]\\\\\n(1+y^*y)^{-1}.\n\\end{gathered}\n\\]\nSince \\(\\|(1+xx^*)^{-1}\\|\\le1\\) and \\(\\|(1+xx^*)^{-1}x\\|\\le\\frac12\\),\n\\[\n\\begin{gathered}\n\\tfrac12\\|(F(x)-F(y))\\xi\\|\\\\\n\\le\\|(x-y)(1+y^*y)^{-1}\\xi\\|\\\\\n+\\tfrac12\\|(y^*-x^*)y(1+y^*y)^{-1}\\xi\\| .\n\\end{gathered}\n\\]\nFor fixed \\(y\\) and \\(\\xi\\), the right side tends to \\(0\\) when \\(x\\to y\\) strongly\\(^*\\). Applying this to \\(x^*,y^*\\) and using \\(F(x)^*=F(x^*)\\) gives strong convergence of the adjoints.\n\n(c) \\(A\\) is unital, so \\(F(A)\\subseteq A\\cap S\\) and \\(G(A\\cap S)\\subseteq A\\); with (a), \\(F(A)=A\\cap S\\), and likewise \\(F(M)=M\\cap S\\) for the weak closure \\(M=A''\\). By the double commutant theorem \\(A\\) is strongly\\(^*\\) dense in \\(M\\), and by (b) \\(F(A)=A\\cap S\\) is strongly\\(^*\\) dense in \\(F(M)=M\\cap S\\). \\(\\square\\)\n\n**Exercise 14.2** (medium; continuous functions on \\(L^2[0,1]\\)). Let \\(\\mu\\) be Lebesgue measure on \\([0,1]\\), that is, the Radon measure that represents the Riemann integral, let \\(A=\\{m_g:g\\in C[0,1]\\}\\) act on \\(L^2[0,1]\\) by multiplication, and let \\(L=\\{m_f:f\\in L^\\infty[0,1]\\}\\).\n\n(a) For open \\(U\\subseteq[0,1]\\) show \\(m_{1_U}\\in(A_+\\cap S)^{\\nearrow}\\), and for closed \\(F\\) show \\(m_{1_F}\\in(A_+\\cap S)^{\\searrow}\\).\n\n(b) For every Borel set \\(E\\) show \\(m_{1_E}\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\).\n\n(c) Deduce that \\(A''=L\\), and compare with Theorem 12.6.\n\n*Solution.* (a) If \\(U=[0,1]\\), use \\(g_k=1\\); if \\(U=\\varnothing\\), use \\(g_k=0\\). Otherwise \\(g_k(t)=\\min(1,k\\operatorname{dist}(t,[0,1]\\setminus U))\\) is continuous with values in \\([0,1]\\) and increases pointwise to \\(1_U\\). So \\(m_{g_k}\\to m_{1_U}\\) strongly, by dominated convergence: \\(\\|(m_{1_U}-m_{g_k})\\xi\\|^2=\\int(1_U-g_k)^2|\\xi|^2\\,d\\mu\\). For closed \\(F\\), \\(1-g_k\\) built from \\(U=[0,1]\\setminus F\\) decreases to \\(1_F\\).\n\n(b) Radon measures are outer regular ([Definition 2.1 of the lesson on Haar measure](haar-measure.md#oa-fnd-hm-01)). So there are open sets \\(U_k\\supseteq E\\) with \\(\\mu(U_k\\setminus E)<1/k\\), and replacing \\(U_k\\) by \\(U_1\\cap\\dots\\cap U_k\\) we may assume that they decrease. Then \\(1_{U_k}\\) decreases to \\(1_G\\) with \\(G=\\bigcap_kU_k\\supseteq E\\) and \\(\\mu(G\\setminus E)=0\\). So \\(m_{1_E}=m_{1_G}\\) is the strong limit of the decreasing sequence \\(m_{1_{U_k}}\\), each of which lies in \\((A_+\\cap S)^{\\nearrow}\\) by (a).\n\n(c) \\(L\\) is a von Neumann algebra that equals its own commutant (Theorem 9.1 of [Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md)). Since \\(A\\subseteq L\\), \\(A'\\supseteq L'=L\\) and \\(A''\\subseteq L'=L\\). Conversely, by (b) and Lemma 12.2(5), \\(A''\\) contains every \\(m_{1_E}\\), hence the norm closure of the simple functions, which is \\(L\\). This is the up-down theorem made concrete: \\(A''\\) is \\(\\sigma\\)-finite, because \\(m_f1=0\\) forces \\(f=0\\), so the function \\(1\\) separates \\(A''\\), and every projection of \\(A''\\) is reached by one increasing and one decreasing sequence. \\(\\square\\)\n\n**Exercise 14.3** (easy; closedness of some classes of operators). Show that the self-adjoint operators and the positive operators form strongly closed sets, and that the normal operators form a strongly\\(^*\\) closed set. Show that if \\(\\dim H=\\infty\\), the normal operators and the unitaries do not form strongly closed sets.\n\n*Solution.* The first two sets are weakly closed by Lemma 1.2. If \\(a_i\\to a\\) strongly\\(^*\\) with \\(a_i\\) normal, then \\(\\|a\\xi\\|=\\lim\\|a_i\\xi\\|=\\lim\\|a_i^*\\xi\\|=\\|a^*\\xi\\|\\) for every \\(\\xi\\), so \\(a\\) is normal; no boundedness is needed. In infinite dimensions, the unitaries \\(u_n\\) of Example 2.5 converge strongly to the isometry \\(v\\), which is not normal and not unitary. \\(\\square\\)\n\n**Exercise 14.4** (medium; finite-rank perturbations of the identity). Let \\(\\dim H=\\infty\\) and \\(A=K(H)+\\mathbb C1\\), the compact operators with the identity adjoined.\n\n(a) Show directly that every unitary \\(u\\in B(H)\\) is the strong limit of unitaries \\(u_F\\) such that \\(u_F-1\\) has finite rank.\n\n(b) Show that if \\(\\|u-1\\|\\le\\lambda\\), then \\(u\\) is a strong\\(^*\\) limit of unitaries \\(v\\in A\\) with \\(\\|v-1\\|\\le\\lambda\\).\n\n(c) Show that the construction of (a) does not control \\(\\|u_F-1\\|\\).\n\n*Solution.* (a) For a finite-dimensional subspace \\(F\\), let \\(F'=F+uF\\). The argument of Theorem 8.1(3) extends \\(u|_F\\) to a unitary \\(W\\) of \\(F'\\). Let \\(u_F=W\\) on \\(F'\\) and \\(1\\) on \\(F'^\\perp\\). Then \\(u_F-1\\) vanishes on \\(F'^\\perp\\) and has range in \\(F'\\), so it has finite rank, and \\(u_F\\xi=u\\xi\\) for \\(\\xi\\in F\\). Directed by inclusion, \\(u_F\\to u\\) strongly, and strongly\\(^*\\) by Corollary 2.4.\n\n(b) \\(A\\) is a unital concrete C\\(^*\\)-algebra, and its weak closure is \\(B(H)\\), because already \\(K(H)\\) is weakly dense ([Example 4.6 of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-07)). Theorem 9.1 gives a net in \\(U(A,\\lambda)\\) that converges strongly\\(^*\\) to \\(u\\).\n\n(c) Let \\(\\xi,\\eta\\) be orthonormal, \\(0<\\theta<\\pi/2\\), and let \\(u\\) be the rotation by \\(\\theta\\) in the plane spanned by \\(\\xi,\\eta\\), and \\(1\\) on its orthogonal complement; then \\(\\|u-1\\|=2\\sin(\\theta/2)\\) is small for small \\(\\theta\\). For \\(F=\\mathbb C\\xi\\), \\(F'\\) is that plane, and \\(W\\) must send \\(\\xi\\) to \\(u\\xi\\) and the unit vector \\(\\zeta\\in F'\\ominus F\\) to a unit vector of \\(F'\\ominus uF\\). Choosing \\(W\\zeta=-u\\zeta\\) is allowed, and then \\(\\|(u_F-1)\\zeta\\|=\\|u\\zeta+\\zeta\\|\\ge2-\\|u\\zeta-\\zeta\\|\\), which is close to \\(2\\). \\(\\square\\)\n\n**Exercise 14.5** (medium; Lusin's theorem with a norm bound). With the notation of Exercise 14.2, let \\(f\\) be a bounded real Borel function on \\([0,1]\\), and \\(\\varepsilon,\\delta>0\\). Show that there are a Borel set \\(E\\) with \\(\\mu([0,1]\\setminus E)<\\varepsilon\\) and a real \\(g\\in C[0,1]\\) with \\(g=f\\) almost everywhere on \\(E\\) and \\(\\max|g|\\le(1+\\delta)\\operatorname{ess\\,sup}_E|f|\\).\n\n*Solution.* By Exercise 14.2(c), \\(m_f\\in A''\\). Every projection of \\(A''=L\\) is \\(m_{1_E}\\) for a Borel set \\(E\\), since a real \\(h\\) with \\(h^2=h\\) almost everywhere is an indicator function almost everywhere. The functional \\(\\varphi(m_h)=\\int h\\,d\\mu=\\langle m_h1,1\\rangle\\) is positive and normal. Apply Theorem 10.6(2) with \\(x=m_f\\), \\(e=1\\) and this \\(\\varphi\\): there are a projection \\(m_{1_E}\\) with \\(\\mu([0,1]\\setminus E)=\\varphi(1-m_{1_E})<\\varepsilon\\) and a self-adjoint \\(m_g\\in A\\) with \\(m_fm_{1_E}=m_gm_{1_E}\\) and \\(\\|m_g\\|\\le(1+\\delta)\\|m_fm_{1_E}\\|\\). The first relation says \\(f=g\\) almost everywhere on \\(E\\). Since \\(\\mu\\) gives positive measure to every nonempty open set, \\(\\|m_g\\|=\\max|g|\\) for continuous \\(g\\), and \\(\\|m_{f1_E}\\|=\\operatorname{ess\\,sup}_E|f|\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-KD-17",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "name": "15. Historical notes",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
      "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "anchor": "oa-fnd-kd-17",
      "proof_locus": {
        "line": 1229,
        "through_line": 1246
      },
      "full_conditions_and_proof": "## 15. Historical notes\n\nThe \\(\\sigma\\)-weak topology comes from trace-class duality, proved in the operator-topologies lesson. Its relationship with the strong and weak operator topologies, together with the double commutant theorem, supplies the prerequisites for the density arguments here. Peterson’s freely readable notes give a comparison for these constructions; the exact programme proofs are linked where they are used.\n\nThe proofs distinguish ordinary strong convergence from convergence of adjoints and from bounded strong convergence. This distinction is used in unitary approximation, the noncommutative Egoroff and Lusin arguments, transitivity and the up-down theorems. The bounded-net alternative in Theorem 7.2 develops Elliott and Griffin’s argument with its complete prerequisites.\n\n## Where this leads\n\n- The lesson [The universal enveloping von Neumann algebra of a \\(C^*\\)-algebra, and \\(W^*\\)-algebras](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md) uses Kaplansky's density theorem, the spectral projections of Section 4 and the monotone closure criterion of Corollary 12.10.\n- [Projections and types of von Neumann algebras](projections-and-types-of-von-neumann-algebras.md) uses spectral projections inside a von Neumann algebra (Corollaries 4.3 and 4.4).\n- Spatial tensor products of von Neumann algebras uses the density theorem to pass from algebraic tensor products to their weak closures with norm control.\n\n## References\n\n- [van Neerven] J. van Neerven, *Functional Analysis*, [corrected author version, arXiv:2112.11166v7](https://arxiv.org/pdf/2112.11166v7).\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [G. A. Elliott; C. J. K. Griffin, *On a question of Kaplansky concerning his density theorem*, main argument and concluding remark](https://arxiv.org/html/2410.03668v1) gives a route through the bounded-net method developed with its full prerequisites in Theorem 7.2. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CM-01",
      "unit": "completely-positive-maps",
      "name": "2. Matrices over a C\\(^*\\)-algebra",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "anchor": "oa-fnd-cm-01",
      "proof_locus": {
        "line": 54,
        "through_line": 130
      },
      "full_conditions_and_proof": "## 2. Matrices over a C\\(^*\\)-algebra\n\nLet \\(A\\) be a C\\(^*\\)-algebra and \\(n\\ge1\\). \\(M_n(A)\\) is the set of \\(n\\times n\\) matrices \\(a=[a_{ij}]\\) with entries in \\(A\\). With entrywise linear operations, the product \\((ab)_{ij}=\\sum_ka_{ik}b_{kj}\\) and the adjoint \\((a^*)_{ij}=(a_{ji})^*\\), it is a \\(*\\)-algebra. For a representation \\(\\pi\\) of \\(A\\) on \\(H\\) put\n\\[\n\\begin{gathered}\n\\pi^{(n)}(a)=\\sum_{i,j}R_i\\,\\pi(a_{ij})\\,R_j^*, \\\\\n(\\pi^{(n)}(a)\\zeta)_i=\\sum_j\\pi(a_{ij})\\zeta_j.\n\\end{gathered}\n\\tag{2.1}\n\\]\n\n**Proposition 2.1** (The C\\(^*\\)-algebra \\(M_n(A)\\)).\n\n1. \\(\\pi^{(n)}\\) is a \\(*\\)-homomorphism of \\(M_n(A)\\) into \\(B(H^n)\\), with entries \\(R_i^*\\pi^{(n)}(a)R_j=\\pi(a_{ij})\\), and\n\\[\n\\begin{gathered}\n\\max_{i,j}\\|\\pi(a_{ij})\\| \\\\\n\\le\\|\\pi^{(n)}(a)\\| \\\\\n\\le\\sum_{i,j}\\|\\pi(a_{ij})\\|.\n\\end{gathered}\n\\tag{2.2}\n\\]\nIt is injective if and only if \\(\\pi\\) is. Its range is the set of operators on \\(H^n\\) whose matrix entries all lie in \\(\\pi(A)\\).\n2. \\(M_n(A)\\) has exactly one norm in which it is a C\\(^*\\)-algebra. For every faithful \\(\\pi\\) it is \\(\\|a\\|=\\|\\pi^{(n)}(a)\\|\\), and \\(\\max_{i,j}\\|a_{ij}\\|\\le\\|a\\|\\le\\sum_{i,j}\\|a_{ij}\\|\\).\n3. For every representation \\(\\pi\\), \\(\\pi^{(n)}\\) is a representation of the C\\(^*\\)-algebra \\(M_n(A)\\). For a \\(*\\)-homomorphism \\(\\sigma:A\\to B\\), the entrywise map \\(\\sigma^{(n)}:M_n(A)\\to M_n(B)\\) is a \\(*\\)-homomorphism, injective when \\(\\sigma\\) is.\n4. If \\(B\\subseteq A\\) is a C\\(^*\\)-subalgebra, then \\(M_n(B)\\) is a C\\(^*\\)-subalgebra of \\(M_n(A)\\), and \\(M_n(B)_+=M_n(B)\\cap M_n(A)_+\\).\n5. \\(X\\mapsto\\sum_{i,j}R_iX_{ij}R_j^*\\) is a \\(*\\)-isomorphism of \\(M_n(B(H))\\) onto \\(B(H^n)\\). So \\(X\\in M_n(B(H))\\) is positive exactly when \\(\\sum_{i,j}\\langle X_{ij}\\zeta_j,\\zeta_i\\rangle\\ge0\\) for every \\(\\zeta\\in H^n\\).\n6. Let \\(\\Omega\\) be a locally compact Hausdorff space. Identify \\(F\\in M_n(C_0(\\Omega))\\) with the function \\(\\omega\\mapsto F(\\omega)=[F_{ij}(\\omega)]\\in M_n(\\mathbb C)\\). Then \\(\\|F\\|=\\sup_\\omega\\|F(\\omega)\\|\\), and \\(F\\ge0\\) if and only if \\(F(\\omega)\\ge0\\) for every \\(\\omega\\).\n7. (*Scalar compressions.*) For \\(a\\in M_n(A)\\) and a scalar \\(n\\times m\\) matrix \\(\\alpha\\), let \\(\\alpha^*a\\alpha\\in M_m(A)\\) have entries \\((\\alpha^*a\\alpha)_{pq}=\\sum_{i,j}\\overline{\\alpha_{ip}}\\,a_{ij}\\,\\alpha_{jq}\\). If \\(a\\ge0\\), then \\(\\alpha^*a\\alpha\\ge0\\). In particular, square corners of positive matrices are positive, so are the diagonal entries, so is \\(a\\oplus0\\), and so is every relabelling \\([a_{\\tau(i)\\tau(j)}]\\) by a permutation \\(\\tau\\).\n\n**Proof.** (1) The product rule gives \\[\n\\begin{gathered}\n(\\pi^{(n)}(a)\\pi^{(n)}(b))_{ik}\\\\\n=\\sum_j\\pi(a_{ij})\\pi(b_{jk})\\\\\n=\\pi((ab)_{ik}).\n\\end{gathered}\n\\] Also \\(\\pi^{(n)}(a)^*=\\sum_{i,j}R_j\\pi(a_{ij})^*R_i^*=\\pi^{(n)}(a^*)\\). The entry formula follows from \\(R_i^*R_k=\\delta_{ik}1\\). It gives the left inequality in (2.2), and (2.1) gives the right one. An operator is determined by its entries, so \\(\\pi^{(n)}(a)=0\\) exactly when every \\(\\pi(a_{ij})=0\\). Every \\(T\\in B(H^n)\\) equals \\(\\sum R_iT_{ij}R_j^*\\), which describes the range.\n\n(2) Let \\(\\pi\\) be faithful; such a \\(\\pi\\) exists by (B9). By (1), \\(\\pi^{(n)}\\) is injective. The injective \\(*\\)-homomorphism \\(\\pi\\) is isometric (B3), so (2.2) becomes the stated inequalities for \\(\\|a\\|:=\\|\\pi^{(n)}(a)\\|\\). The range \\(D=\\pi^{(n)}(M_n(A))\\) is closed. Indeed, if \\(\\pi^{(n)}(a^{(k)})\\to T\\), then by (2.2) every entry sequence \\(\\pi(a^{(k)}_{ij})\\) is Cauchy. As \\(\\pi\\) is isometric, \\(\\pi(A)\\) is complete, so the sequence converges to some \\(\\pi(a_{ij})\\), and then \\(\\pi^{(n)}(a^{(k)})\\to\\pi^{(n)}(a)\\) by (2.2), so \\(T\\in D\\). Thus \\(D\\) is a C\\(^*\\)-subalgebra of \\(B(H^n)\\), and \\(\\|\\cdot\\|\\) makes \\(M_n(A)\\) a C\\(^*\\)-algebra. If \\(\\|\\cdot\\|'\\) is another such norm, the identity map from \\((M_n(A),\\|\\cdot\\|)\\) to \\((M_n(A),\\|\\cdot\\|')\\) is an injective \\(*\\)-homomorphism between C\\(^*\\)-algebras, hence isometric by (B3). So the norm is unique, and in particular it does not depend on \\(\\pi\\).\n\n(3) The first claim is (1) combined with (2). For \\(\\sigma\\), the algebraic identities are those of (1), and injectivity is checked entry by entry.\n\n(4) A faithful representation of \\(A\\) restricts to a faithful representation of \\(B\\). By (2), the norm of \\(M_n(B)\\) is the restriction of that of \\(M_n(A)\\), and \\(M_n(B)\\) is complete, hence closed. The claim about positive elements is (P1).\n\n(5) By (1) with \\(\\pi=\\mathrm{id}_{B(H)}\\), the map is an injective \\(*\\)-homomorphism onto \\(B(H^n)\\), isometric by (B3). For \\(T=\\sum R_iX_{ij}R_j^*\\) we have \\(\\langle T\\zeta,\\zeta\\rangle=\\sum_{i,j}\\langle X_{ij}\\zeta_j,\\zeta_i\\rangle\\); now use (P2).\n\n(6) On \\(\\ell^2(\\Omega)\\), with orthonormal basis \\((\\delta_\\omega)\\), the operators \\(\\pi(f)\\delta_\\omega=f(\\omega)\\delta_\\omega\\) form a faithful representation of \\(C_0(\\Omega)\\). After the unitary that regroups \\(\\ell^2(\\Omega)^n\\) as \\(\\bigoplus_\\omega\\mathbb C^n\\), \\(\\pi^{(n)}(F)\\) becomes the block-diagonal operator \\(\\bigoplus_\\omega F(\\omega)\\). Its norm is \\(\\sup_\\omega\\|F(\\omega)\\|\\), and it is positive exactly when every block is. Now apply (2) and (P1).\n\n(7) Take a faithful \\(\\pi\\) on \\(H\\) and let \\(\\alpha\\otimes1:H^m\\to H^n\\) be \\((\\alpha\\otimes1)\\zeta=(\\sum_q\\alpha_{iq}\\zeta_q)_i\\). Then \\(R_i^*(\\alpha\\otimes1)R_q=\\alpha_{iq}1\\), so the \\((p,q)\\) entry of \\((\\alpha\\otimes1)^*\\pi^{(n)}(a)(\\alpha\\otimes1)\\) is \\(\\sum_{i,j}\\overline{\\alpha_{ip}}\\alpha_{jq}\\pi(a_{ij})=\\pi((\\alpha^*a\\alpha)_{pq})\\). Hence \\(\\pi^{(m)}(\\alpha^*a\\alpha)=(\\alpha\\otimes1)^*\\pi^{(n)}(a)(\\alpha\\otimes1)\\), which is positive when \\(\\pi^{(n)}(a)\\) is. By (1) and (P1), \\(\\alpha^*a\\alpha\\ge0\\). For the special cases take for \\(\\alpha\\) a coordinate inclusion \\(\\mathbb C^m\\to\\mathbb C^n\\), the matrix \\([1_n\\ 0]\\), or a permutation matrix. \\(\\square\\)\n\n**Lemma 2.2** (The positive cone of \\(M_n(A)\\)).\n\n1. For \\(a_1,\\dots,a_n\\in A\\), the matrix \\([a_i^*a_j]\\) is positive. Each positive element of \\(M_n(A)\\) can be written as the sum of \\(n\\) such matrices.\n2. \\(a\\in M_n(A)\\) is positive exactly when \\(\\sum_{i,j}x_i^*a_{ij}x_j\\ge0\\) in \\(A\\) for all \\(x_1,\\dots,x_n\\in A\\).\n3. Let \\(\\pi\\) be a representation of \\(A\\) on \\(H\\). If \\(a\\ge0\\), then \\(\\sum_{i,j}\\langle\\pi(a_{ij})\\zeta_j,\\zeta_i\\rangle\\ge0\\) for all \\(\\zeta\\in H^n\\). If \\(\\pi\\) is faithful, the converse holds.\n4. (*Two-by-two test.*) Let \\(x,y\\in B(H)\\) and \\(t>0\\). The operator \\(\\begin{bmatrix}t1&x\\\\x^*&y\\end{bmatrix}\\) on \\(H\\oplus H\\) is positive if and only if \\(y\\ge t^{-1}x^*x\\).\n\n**Proof.** (1) Let \\(c\\in M_n(A)\\) have first row \\((a_1,\\dots,a_n)\\) and all other rows zero. Then \\((c^*c)_{ij}=\\sum_k(c_{ki})^*c_{kj}=a_i^*a_j\\), so \\([a_i^*a_j]=c^*c\\ge0\\). Conversely, let \\(a\\ge0\\) and write \\(a=b^*b\\) (B1). Then \\(a_{ij}=\\sum_kb_{ki}^*b_{kj}\\), so \\(a=\\sum_{k=1}^n[b_{ki}^*b_{kj}]_{i,j}\\), one summand for each row of \\(b\\).\n\n(2) If \\(a=\\sum_k[b_{ki}^*b_{kj}]_{i,j}\\), then \\(\\sum_{i,j}x_i^*a_{ij}x_j=\\sum_kw_k^*w_k\\) with \\(w_k=\\sum_jb_{kj}x_j\\), which is positive. Conversely, assume the condition. Let \\(f\\) be a positive functional on \\(A\\) with GNS triple \\((\\pi_f,H_f,\\xi_f)\\) (B8). For \\(x_1,\\dots,x_n\\in A\\) and \\(\\zeta=(\\pi_f(x_1)\\xi_f,\\dots,\\pi_f(x_n)\\xi_f)\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_f^{(n)}(a)\\zeta,\\zeta\\rangle \\\\\n=\\sum_{i,j}\\langle\\pi_f(x_i^*a_{ij}x_j)\\xi_f,\\xi_f\\rangle \\\\\n=f\\Bigl(\\sum_{i,j}x_i^*a_{ij}x_j\\Bigr)\\ge0.\n\\end{gathered}\n\\]\nThese \\(\\zeta\\) are dense in \\(H_f^n\\), because \\(\\pi_f(A)\\xi_f\\) is dense in \\(H_f\\) and the coordinates can be chosen independently. So \\(\\pi_f^{(n)}(a)\\ge0\\) by (P2). Let \\(\\pi\\) be the direct sum of the \\(\\pi_f\\) over all positive functionals \\(f\\). Up to regrouping the summands, \\(\\pi^{(n)}(a)\\) is the direct sum of the \\(\\pi_f^{(n)}(a)\\), so it is positive. And \\(\\pi\\) is faithful, because for each nonzero \\(b\\in A\\) some positive functional \\(f_b\\) has \\(\\pi_{f_b}(b)\\ne0\\) (B9). So \\(\\pi^{(n)}\\) is injective by Proposition 2.1(1), and \\(a\\ge0\\) by (P1).\n\n(3) \\(\\pi^{(n)}\\) is a \\(*\\)-homomorphism, so \\(a=c^*c\\) gives \\(\\pi^{(n)}(a)=\\pi^{(n)}(c)^*\\pi^{(n)}(c)\\ge0\\), and \\(\\langle\\pi^{(n)}(a)\\zeta,\\zeta\\rangle\\) is the given sum. For faithful \\(\\pi\\), the converse is (P2) followed by (P1).\n\n(4) Write \\(S\\) for the operator. For \\((\\xi,\\eta)\\in H\\oplus H\\),\n\\[\n\\langle S(\\xi,\\eta),(\\xi,\\eta)\\rangle=t\\|\\xi\\|^2+2\\,\\mathrm{Re}\\,\\langle x\\eta,\\xi\\rangle+\\langle y\\eta,\\eta\\rangle .\n\\]\nIf \\(S\\ge0\\), take \\(\\xi=-t^{-1}x\\eta\\); the right side becomes \\(\\langle y\\eta,\\eta\\rangle-t^{-1}\\|x\\eta\\|^2\\), so \\(y\\ge t^{-1}x^*x\\). Conversely, let \\(R:H\\oplus H\\to H\\) be \\(R(\\xi,\\eta)=t^{1/2}\\xi+t^{-1/2}x\\eta\\). Then \\(R^*R=\\begin{bmatrix}t1&x\\\\x^*&t^{-1}x^*x\\end{bmatrix}\\), and \\(S=R^*R+\\bigl(0\\oplus(y-t^{-1}x^*x)\\bigr)\\), a sum of two positive operators. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CM-02",
      "unit": "completely-positive-maps",
      "name": "2. Matrices over a C\\(^*\\)-algebra",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "full_conditions_and_proof": "## 2. Matrices over a C\\(^*\\)-algebra\n\nLet \\(A\\) be a C\\(^*\\)-algebra and \\(n\\ge1\\). \\(M_n(A)\\) is the set of \\(n\\times n\\) matrices \\(a=[a_{ij}]\\) with entries in \\(A\\). With entrywise linear operations, the product \\((ab)_{ij}=\\sum_ka_{ik}b_{kj}\\) and the adjoint \\((a^*)_{ij}=(a_{ji})^*\\), it is a \\(*\\)-algebra. For a representation \\(\\pi\\) of \\(A\\) on \\(H\\) put\n\\[\n\\begin{gathered}\n\\pi^{(n)}(a)=\\sum_{i,j}R_i\\,\\pi(a_{ij})\\,R_j^*, \\\\\n(\\pi^{(n)}(a)\\zeta)_i=\\sum_j\\pi(a_{ij})\\zeta_j.\n\\end{gathered}\n\\tag{2.1}\n\\]\n\n**Proposition 2.1** (The C\\(^*\\)-algebra \\(M_n(A)\\)).\n\n1. \\(\\pi^{(n)}\\) is a \\(*\\)-homomorphism of \\(M_n(A)\\) into \\(B(H^n)\\), with entries \\(R_i^*\\pi^{(n)}(a)R_j=\\pi(a_{ij})\\), and\n\\[\n\\begin{gathered}\n\\max_{i,j}\\|\\pi(a_{ij})\\| \\\\\n\\le\\|\\pi^{(n)}(a)\\| \\\\\n\\le\\sum_{i,j}\\|\\pi(a_{ij})\\|.\n\\end{gathered}\n\\tag{2.2}\n\\]\nIt is injective if and only if \\(\\pi\\) is. Its range is the set of operators on \\(H^n\\) whose matrix entries all lie in \\(\\pi(A)\\).\n2. \\(M_n(A)\\) has exactly one norm in which it is a C\\(^*\\)-algebra. For every faithful \\(\\pi\\) it is \\(\\|a\\|=\\|\\pi^{(n)}(a)\\|\\), and \\(\\max_{i,j}\\|a_{ij}\\|\\le\\|a\\|\\le\\sum_{i,j}\\|a_{ij}\\|\\).\n3. For every representation \\(\\pi\\), \\(\\pi^{(n)}\\) is a representation of the C\\(^*\\)-algebra \\(M_n(A)\\). For a \\(*\\)-homomorphism \\(\\sigma:A\\to B\\), the entrywise map \\(\\sigma^{(n)}:M_n(A)\\to M_n(B)\\) is a \\(*\\)-homomorphism, injective when \\(\\sigma\\) is.\n4. If \\(B\\subseteq A\\) is a C\\(^*\\)-subalgebra, then \\(M_n(B)\\) is a C\\(^*\\)-subalgebra of \\(M_n(A)\\), and \\(M_n(B)_+=M_n(B)\\cap M_n(A)_+\\).\n5. \\(X\\mapsto\\sum_{i,j}R_iX_{ij}R_j^*\\) is a \\(*\\)-isomorphism of \\(M_n(B(H))\\) onto \\(B(H^n)\\). So \\(X\\in M_n(B(H))\\) is positive exactly when \\(\\sum_{i,j}\\langle X_{ij}\\zeta_j,\\zeta_i\\rangle\\ge0\\) for every \\(\\zeta\\in H^n\\).\n6. Let \\(\\Omega\\) be a locally compact Hausdorff space. Identify \\(F\\in M_n(C_0(\\Omega))\\) with the function \\(\\omega\\mapsto F(\\omega)=[F_{ij}(\\omega)]\\in M_n(\\mathbb C)\\). Then \\(\\|F\\|=\\sup_\\omega\\|F(\\omega)\\|\\), and \\(F\\ge0\\) if and only if \\(F(\\omega)\\ge0\\) for every \\(\\omega\\).\n7. (*Scalar compressions.*) For \\(a\\in M_n(A)\\) and a scalar \\(n\\times m\\) matrix \\(\\alpha\\), let \\(\\alpha^*a\\alpha\\in M_m(A)\\) have entries \\((\\alpha^*a\\alpha)_{pq}=\\sum_{i,j}\\overline{\\alpha_{ip}}\\,a_{ij}\\,\\alpha_{jq}\\). If \\(a\\ge0\\), then \\(\\alpha^*a\\alpha\\ge0\\). In particular, square corners of positive matrices are positive, so are the diagonal entries, so is \\(a\\oplus0\\), and so is every relabelling \\([a_{\\tau(i)\\tau(j)}]\\) by a permutation \\(\\tau\\).\n\n**Proof.** (1) The product rule gives \\[\n\\begin{gathered}\n(\\pi^{(n)}(a)\\pi^{(n)}(b))_{ik}\\\\\n=\\sum_j\\pi(a_{ij})\\pi(b_{jk})\\\\\n=\\pi((ab)_{ik}).\n\\end{gathered}\n\\] Also \\(\\pi^{(n)}(a)^*=\\sum_{i,j}R_j\\pi(a_{ij})^*R_i^*=\\pi^{(n)}(a^*)\\). The entry formula follows from \\(R_i^*R_k=\\delta_{ik}1\\). It gives the left inequality in (2.2), and (2.1) gives the right one. An operator is determined by its entries, so \\(\\pi^{(n)}(a)=0\\) exactly when every \\(\\pi(a_{ij})=0\\). Every \\(T\\in B(H^n)\\) equals \\(\\sum R_iT_{ij}R_j^*\\), which describes the range.\n\n(2) Let \\(\\pi\\) be faithful; such a \\(\\pi\\) exists by (B9). By (1), \\(\\pi^{(n)}\\) is injective. The injective \\(*\\)-homomorphism \\(\\pi\\) is isometric (B3), so (2.2) becomes the stated inequalities for \\(\\|a\\|:=\\|\\pi^{(n)}(a)\\|\\). The range \\(D=\\pi^{(n)}(M_n(A))\\) is closed. Indeed, if \\(\\pi^{(n)}(a^{(k)})\\to T\\), then by (2.2) every entry sequence \\(\\pi(a^{(k)}_{ij})\\) is Cauchy. As \\(\\pi\\) is isometric, \\(\\pi(A)\\) is complete, so the sequence converges to some \\(\\pi(a_{ij})\\), and then \\(\\pi^{(n)}(a^{(k)})\\to\\pi^{(n)}(a)\\) by (2.2), so \\(T\\in D\\). Thus \\(D\\) is a C\\(^*\\)-subalgebra of \\(B(H^n)\\), and \\(\\|\\cdot\\|\\) makes \\(M_n(A)\\) a C\\(^*\\)-algebra. If \\(\\|\\cdot\\|'\\) is another such norm, the identity map from \\((M_n(A),\\|\\cdot\\|)\\) to \\((M_n(A),\\|\\cdot\\|')\\) is an injective \\(*\\)-homomorphism between C\\(^*\\)-algebras, hence isometric by (B3). So the norm is unique, and in particular it does not depend on \\(\\pi\\).\n\n(3) The first claim is (1) combined with (2). For \\(\\sigma\\), the algebraic identities are those of (1), and injectivity is checked entry by entry.\n\n(4) A faithful representation of \\(A\\) restricts to a faithful representation of \\(B\\). By (2), the norm of \\(M_n(B)\\) is the restriction of that of \\(M_n(A)\\), and \\(M_n(B)\\) is complete, hence closed. The claim about positive elements is (P1).\n\n(5) By (1) with \\(\\pi=\\mathrm{id}_{B(H)}\\), the map is an injective \\(*\\)-homomorphism onto \\(B(H^n)\\), isometric by (B3). For \\(T=\\sum R_iX_{ij}R_j^*\\) we have \\(\\langle T\\zeta,\\zeta\\rangle=\\sum_{i,j}\\langle X_{ij}\\zeta_j,\\zeta_i\\rangle\\); now use (P2).\n\n(6) On \\(\\ell^2(\\Omega)\\), with orthonormal basis \\((\\delta_\\omega)\\), the operators \\(\\pi(f)\\delta_\\omega=f(\\omega)\\delta_\\omega\\) form a faithful representation of \\(C_0(\\Omega)\\). After the unitary that regroups \\(\\ell^2(\\Omega)^n\\) as \\(\\bigoplus_\\omega\\mathbb C^n\\), \\(\\pi^{(n)}(F)\\) becomes the block-diagonal operator \\(\\bigoplus_\\omega F(\\omega)\\). Its norm is \\(\\sup_\\omega\\|F(\\omega)\\|\\), and it is positive exactly when every block is. Now apply (2) and (P1).\n\n(7) Take a faithful \\(\\pi\\) on \\(H\\) and let \\(\\alpha\\otimes1:H^m\\to H^n\\) be \\((\\alpha\\otimes1)\\zeta=(\\sum_q\\alpha_{iq}\\zeta_q)_i\\). Then \\(R_i^*(\\alpha\\otimes1)R_q=\\alpha_{iq}1\\), so the \\((p,q)\\) entry of \\((\\alpha\\otimes1)^*\\pi^{(n)}(a)(\\alpha\\otimes1)\\) is \\(\\sum_{i,j}\\overline{\\alpha_{ip}}\\alpha_{jq}\\pi(a_{ij})=\\pi((\\alpha^*a\\alpha)_{pq})\\). Hence \\(\\pi^{(m)}(\\alpha^*a\\alpha)=(\\alpha\\otimes1)^*\\pi^{(n)}(a)(\\alpha\\otimes1)\\), which is positive when \\(\\pi^{(n)}(a)\\) is. By (1) and (P1), \\(\\alpha^*a\\alpha\\ge0\\). For the special cases take for \\(\\alpha\\) a coordinate inclusion \\(\\mathbb C^m\\to\\mathbb C^n\\), the matrix \\([1_n\\ 0]\\), or a permutation matrix. \\(\\square\\)\n\n**Lemma 2.2** (The positive cone of \\(M_n(A)\\)).\n\n1. For \\(a_1,\\dots,a_n\\in A\\), the matrix \\([a_i^*a_j]\\) is positive. Each positive element of \\(M_n(A)\\) can be written as the sum of \\(n\\) such matrices.\n2. \\(a\\in M_n(A)\\) is positive exactly when \\(\\sum_{i,j}x_i^*a_{ij}x_j\\ge0\\) in \\(A\\) for all \\(x_1,\\dots,x_n\\in A\\).\n3. Let \\(\\pi\\) be a representation of \\(A\\) on \\(H\\). If \\(a\\ge0\\), then \\(\\sum_{i,j}\\langle\\pi(a_{ij})\\zeta_j,\\zeta_i\\rangle\\ge0\\) for all \\(\\zeta\\in H^n\\). If \\(\\pi\\) is faithful, the converse holds.\n4. (*Two-by-two test.*) Let \\(x,y\\in B(H)\\) and \\(t>0\\). The operator \\(\\begin{bmatrix}t1&x\\\\x^*&y\\end{bmatrix}\\) on \\(H\\oplus H\\) is positive if and only if \\(y\\ge t^{-1}x^*x\\).\n\n**Proof.** (1) Let \\(c\\in M_n(A)\\) have first row \\((a_1,\\dots,a_n)\\) and all other rows zero. Then \\((c^*c)_{ij}=\\sum_k(c_{ki})^*c_{kj}=a_i^*a_j\\), so \\([a_i^*a_j]=c^*c\\ge0\\). Conversely, let \\(a\\ge0\\) and write \\(a=b^*b\\) (B1). Then \\(a_{ij}=\\sum_kb_{ki}^*b_{kj}\\), so \\(a=\\sum_{k=1}^n[b_{ki}^*b_{kj}]_{i,j}\\), one summand for each row of \\(b\\).\n\n(2) If \\(a=\\sum_k[b_{ki}^*b_{kj}]_{i,j}\\), then \\(\\sum_{i,j}x_i^*a_{ij}x_j=\\sum_kw_k^*w_k\\) with \\(w_k=\\sum_jb_{kj}x_j\\), which is positive. Conversely, assume the condition. Let \\(f\\) be a positive functional on \\(A\\) with GNS triple \\((\\pi_f,H_f,\\xi_f)\\) (B8). For \\(x_1,\\dots,x_n\\in A\\) and \\(\\zeta=(\\pi_f(x_1)\\xi_f,\\dots,\\pi_f(x_n)\\xi_f)\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_f^{(n)}(a)\\zeta,\\zeta\\rangle \\\\\n=\\sum_{i,j}\\langle\\pi_f(x_i^*a_{ij}x_j)\\xi_f,\\xi_f\\rangle \\\\\n=f\\Bigl(\\sum_{i,j}x_i^*a_{ij}x_j\\Bigr)\\ge0.\n\\end{gathered}\n\\]\nThese \\(\\zeta\\) are dense in \\(H_f^n\\), because \\(\\pi_f(A)\\xi_f\\) is dense in \\(H_f\\) and the coordinates can be chosen independently. So \\(\\pi_f^{(n)}(a)\\ge0\\) by (P2). Let \\(\\pi\\) be the direct sum of the \\(\\pi_f\\) over all positive functionals \\(f\\). Up to regrouping the summands, \\(\\pi^{(n)}(a)\\) is the direct sum of the \\(\\pi_f^{(n)}(a)\\), so it is positive. And \\(\\pi\\) is faithful, because for each nonzero \\(b\\in A\\) some positive functional \\(f_b\\) has \\(\\pi_{f_b}(b)\\ne0\\) (B9). So \\(\\pi^{(n)}\\) is injective by Proposition 2.1(1), and \\(a\\ge0\\) by (P1).\n\n(3) \\(\\pi^{(n)}\\) is a \\(*\\)-homomorphism, so \\(a=c^*c\\) gives \\(\\pi^{(n)}(a)=\\pi^{(n)}(c)^*\\pi^{(n)}(c)\\ge0\\), and \\(\\langle\\pi^{(n)}(a)\\zeta,\\zeta\\rangle\\) is the given sum. For faithful \\(\\pi\\), the converse is (P2) followed by (P1).\n\n(4) Write \\(S\\) for the operator. For \\((\\xi,\\eta)\\in H\\oplus H\\),\n\\[\n\\langle S(\\xi,\\eta),(\\xi,\\eta)\\rangle=t\\|\\xi\\|^2+2\\,\\mathrm{Re}\\,\\langle x\\eta,\\xi\\rangle+\\langle y\\eta,\\eta\\rangle .\n\\]\nIf \\(S\\ge0\\), take \\(\\xi=-t^{-1}x\\eta\\); the right side becomes \\(\\langle y\\eta,\\eta\\rangle-t^{-1}\\|x\\eta\\|^2\\), so \\(y\\ge t^{-1}x^*x\\). Conversely, let \\(R:H\\oplus H\\to H\\) be \\(R(\\xi,\\eta)=t^{1/2}\\xi+t^{-1/2}x\\eta\\). Then \\(R^*R=\\begin{bmatrix}t1&x\\\\x^*&t^{-1}x^*x\\end{bmatrix}\\), and \\(S=R^*R+\\bigl(0\\oplus(y-t^{-1}x^*x)\\bigr)\\), a sum of two positive operators. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    },
    {
      "id": "OA-FND-CM-03",
      "unit": "completely-positive-maps",
      "name": "3. Completely positive maps",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "full_conditions_and_proof": "## 3. Completely positive maps\n\n**Definition 3.1.** Let \\(A,B\\) be C\\(^*\\)-algebras and \\(\\varphi:A\\to B\\) linear. Its \\(n\\)-th amplification \\(\\varphi^{(n)}:M_n(A)\\to M_n(B)\\) applies \\(\\varphi\\) to every entry. The map \\(\\varphi\\) is *\\(n\\)-positive* if \\(\\varphi^{(n)}\\) maps positive elements to positive elements, and *completely positive* (CP) if it is \\(n\\)-positive for every \\(n\\). *Positive* means 1-positive.\n\n**Proposition 3.2.**\n\n1. (*A criterion.*) \\(\\varphi\\) is \\(n\\)-positive exactly when\n\\[\n\\begin{gathered}\n\\sum_{i,j=1}^ny_i^*\\,\\varphi(x_i^*x_j)\\,y_j\\ge0, \\\\\n\\text{for all }x_1,\\dots,x_n\\in A, \\\\\ny_1,\\dots,y_n\\in B.\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(B\\subseteq B(H)\\), this holds if and only if \\(\\sum_{i,j}\\langle\\varphi(x_i^*x_j)\\zeta_j,\\zeta_i\\rangle\\ge0\\) for all \\(x\\in A^n\\) and \\(\\zeta\\in H^n\\).\n2. An \\(n\\)-positive map is \\(k\\)-positive for every \\(k\\le n\\). A positive map is hermitian: \\(\\varphi(x^*)=\\varphi(x)^*\\). If \\(\\psi:B\\to C\\) is \\(n\\)-positive too, so is \\(\\psi\\circ\\varphi\\), because \\((\\psi\\circ\\varphi)^{(n)}=\\psi^{(n)}\\circ\\varphi^{(n)}\\).\n3. (*Examples.*) Every \\(*\\)-homomorphism is CP. For \\(V\\in B(H,K)\\), the map \\(y\\mapsto V^*yV\\) from \\(B(K)\\) to \\(B(H)\\) is CP. The transpose \\(t(x)=x^{\\mathsf T}\\) on \\(M_d(\\mathbb C)\\), \\(d\\ge2\\), is positive but not 2-positive.\n4. (*Automatic boundedness.*) A positive map \\(\\varphi:A\\to B\\) is bounded, and\n\\[\n\\|\\varphi\\|\\le4\\sup\\{\\|\\varphi(a)\\|:\\ a\\in A_+,\\ \\|a\\|\\le1\\}.\n\\]\n\n**Proof.** (1) Let \\(\\varphi\\) be \\(n\\)-positive. The matrix \\([x_i^*x_j]\\) is positive by Lemma 2.2(1), so \\([\\varphi(x_i^*x_j)]\\ge0\\), and Lemma 2.2(2), applied in \\(M_n(B)\\), gives (3.1). Conversely, assume (3.1). By Lemma 2.2(2) in \\(M_n(B)\\), \\([\\varphi(x_i^*x_j)]\\ge0\\) for every \\(x\\in A^n\\). By Lemma 2.2(1), every positive element of \\(M_n(A)\\) is a finite sum of such matrices \\([x_i^*x_j]\\); \\(\\varphi^{(n)}\\) is additive, and \\(M_n(B)_+\\) is a convex cone. For the second form, \\([\\varphi(x_i^*x_j)]\\) is positive in \\(M_n(B)\\) exactly when it is positive in \\(M_n(B(H))\\) (Proposition 2.1(4)), that is, when the stated quadratic form is nonnegative (Proposition 2.1(5)).\n\n(2) For \\(X\\in M_k(A)_+\\), \\(X\\oplus0\\in M_n(A)_+\\) by Proposition 2.1(7), and \\(\\varphi^{(n)}(X\\oplus0)=\\varphi^{(k)}(X)\\oplus0\\). Its corner \\(\\varphi^{(k)}(X)\\) is positive by Proposition 2.1(7) again. If \\(h\\in A_h\\), then \\(\\varphi(h)=\\varphi(h_+)-\\varphi(h_-)\\) is self-adjoint; for \\(x=h+ik\\), \\(\\varphi(x^*)=\\varphi(h)-i\\varphi(k)=\\varphi(x)^*\\). The rule for compositions holds entry by entry.\n\n(3) A \\(*\\)-homomorphism \\(\\sigma\\) has \\(*\\)-homomorphic amplifications (Proposition 2.1(3)), and \\(\\sigma^{(n)}(c^*c)=\\sigma^{(n)}(c)^*\\sigma^{(n)}(c)\\). For \\(y\\mapsto V^*yV\\), the amplification is \\(Y\\mapsto(V\\otimes1)^*Y(V\\otimes1)\\) on \\(B(K^n)=M_n(B(K))\\) (Proposition 2.1(5)), where \\((V\\otimes1)(\\zeta_j)_j=(V\\zeta_j)_j\\); this preserves positivity. For the transpose: if \\(x=y^*y\\), then \\(x^{\\mathsf T}=y^{\\mathsf T}\\bar y=(\\bar y)^*\\bar y\\ge0\\), where \\(\\bar y\\) has the conjugate entries. Now let \\(E=[e_{ij}]_{i,j=1,2}\\in M_2(M_d(\\mathbb C))\\). It equals \\([e_{1i}^*e_{1j}]\\), so \\(E\\ge0\\) by Lemma 2.2(1). Its image is \\(t^{(2)}(E)=[e_{ji}]_{i,j}\\). For \\(\\zeta=(\\varepsilon_2,-\\varepsilon_1)\\in\\mathbb C^d\\oplus\\mathbb C^d\\),\n\\[\n\\begin{gathered}\n\\sum_{i,j=1}^2\\langle e_{ji}\\zeta_j,\\zeta_i\\rangle \\\\\n=\\langle e_{11}\\varepsilon_2,\\varepsilon_2\\rangle-\\langle e_{21}\\varepsilon_1,\\varepsilon_2\\rangle \\\\\n-\\langle e_{12}\\varepsilon_2,\\varepsilon_1\\rangle+\\langle e_{22}\\varepsilon_1,\\varepsilon_1\\rangle \\\\\n=0-1-1+0=-2.\n\\end{gathered}\n\\]\nSo \\(t^{(2)}(E)\\) is not positive, by Proposition 2.1(5).\n\n(4) Let \\(M\\) be the supremum on the right, and suppose \\(M=\\infty\\). Choose \\(a_k\\in A_+\\) with \\(\\|a_k\\|\\le1\\) and \\(\\|\\varphi(a_k)\\|\\ge4^k\\). The series \\(a=\\sum_k2^{-k}a_k\\) converges, and \\(a-2^{-k}a_k\\) is a limit of positive partial sums, so \\(a\\ge2^{-k}a_k\\ge0\\) (\\(A_+\\) is closed). Hence \\(\\varphi(a)\\ge2^{-k}\\varphi(a_k)\\ge0\\), and by (P2), \\(\\|\\varphi(a)\\|\\ge2^{-k}\\|\\varphi(a_k)\\|\\ge2^k\\) for every \\(k\\), which is absurd. So \\(M<\\infty\\). A contraction \\(x\\) is a combination \\(h_+-h_-+i(k_+-k_-)\\) of four positive contractions (Section 1), so \\(\\|\\varphi(x)\\|\\le4M\\). \\(\\square\\)\n\nThe argument for (4) is the usual proof that positive functionals are bounded, carried out for maps; it also gives the explicit bound. Proposition 7.1(5) extends (4) to maps into dual spaces.\n\n**Exercise 3.3** (medium; Choi's criterion). Let \\(\\varphi:M_k(\\mathbb C)\\to B\\) be linear. Show that the following are equivalent: (a) \\(\\varphi\\) is CP; (b) \\(\\varphi\\) is \\(k\\)-positive; (c) the matrix \\(C_\\varphi=[\\varphi(e_{ij})]_{i,j=1}^k\\in M_k(B)\\) is positive. Compute \\(C_t\\) for the transpose.\n\n*Solution.* (a)\\(\\Rightarrow\\)(b) is clear. (b)\\(\\Rightarrow\\)(c): \\([e_{ij}]=[e_{1i}^*e_{1j}]\\) is positive by Lemma 2.2(1), and \\(C_\\varphi=\\varphi^{(k)}([e_{ij}])\\). (c)\\(\\Rightarrow\\)(a): let \\(x_1,\\dots,x_m\\in M_k(\\mathbb C)\\) and \\(y_1,\\dots,y_m\\in B\\). Since \\(x_p^*x_q=\\sum_{i,j,l}\\overline{(x_p)_{li}}\\,(x_q)_{lj}\\,e_{ij}\\),\n\\[\n\\begin{gathered}\n\\sum_{p,q}y_p^*\\varphi(x_p^*x_q)y_q \\\\\n=\\sum_{l=1}^k\\sum_{i,j=1}^kz_{li}^*\\,\\varphi(e_{ij})\\,z_{lj}, \\\\\nz_{lj}=\\sum_q(x_q)_{lj}\\,y_q.\n\\end{gathered}\n\\]\nFor each \\(l\\), the inner sum is nonnegative by Lemma 2.2(2) applied to \\(C_\\varphi\\ge0\\) in \\(M_k(B)\\). So (3.1) holds for every \\(m\\), and \\(\\varphi\\) is CP. For the transpose on \\(M_2(\\mathbb C)\\), \\(C_t=[e_{ji}]_{i,j}\\), which is the operator on \\(\\mathbb C^2\\oplus\\mathbb C^2\\) that exchanges the two tensor factors of \\(\\mathbb C^2\\otimes\\mathbb C^2\\); it has eigenvalue \\(-1\\) on \\(\\varepsilon_1\\otimes\\varepsilon_2-\\varepsilon_2\\otimes\\varepsilon_1\\), so it is not positive. This is the computation of Proposition 3.2(3) again. The equivalence (a)\\(\\Leftrightarrow\\)(b) is also the case \"\\(\\Omega\\) a point\" of Theorem 5.4(1).\n\n",
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      "name": "4. The Kadison–Schwarz inequality",
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      "full_conditions_and_proof": "## 4. The Kadison–Schwarz inequality\n\n**Theorem 4.1.** Let \\(\\varphi:A\\to B\\) be 2-positive.\n\n1. (*The Kadison–Schwarz inequality.*) \\(\\varphi(a)^*\\varphi(a)\\le\\|\\varphi\\|\\,\\varphi(a^*a)\\) for all \\(a\\in A\\).\n2. \\[\n\\begin{gathered}\n\\|\\varphi\\|\\\\\n=\\sup\\{\\|\\varphi(b)\\|:\\ b\\in A_+,\\ \\|b\\|\\le1\\}\\\\\n=\\lim_i\\|\\varphi(u_i)\\|\n\\end{gathered}\n\\] for every approximate identity \\((u_i)\\). If \\(A\\) has a unit, \\(\\|\\varphi\\|=\\|\\varphi(1)\\|\\).\n3. (*Multiplicative domain.*) Let \\(\\|\\varphi\\|\\le1\\). If \\(\\varphi(a^*a)=\\varphi(a)^*\\varphi(a)\\), then \\(\\varphi(xa)=\\varphi(x)\\varphi(a)\\) for all \\(x\\in A\\). If \\(\\varphi(aa^*)=\\varphi(a)\\varphi(a)^*\\), then \\(\\varphi(ax)=\\varphi(a)\\varphi(x)\\) for all \\(x\\in A\\).\n4. Positivity alone does not give (1). For the transpose \\(t\\) on \\(M_2(\\mathbb C)\\), \\(\\|t\\|=1\\), but \\(t(e_{12})^*t(e_{12})=e_{11}\\) is not below \\(t(e_{12}^*e_{12})=e_{22}\\).\n\n\n\n**Proof.** Represent \\(B\\) faithfully on \\(H\\); by (P1) we may compute with operators. (1) Let \\(a\\in A\\) and \\(R_i=\\begin{bmatrix}u_i&a\\\\0&0\\end{bmatrix}\\in M_2(A)\\). Then \\(R_i^*R_i=\\begin{bmatrix}u_i^2&u_ia\\\\a^*u_i&a^*a\\end{bmatrix}\\ge0\\), so\n\\[\n\\begin{bmatrix}\\varphi(u_i^2)&\\varphi(u_ia)\\\\\\varphi(u_ia)^*&\\varphi(a^*a)\\end{bmatrix}\\ge0 ,\n\\]\nusing \\(\\varphi(a^*u_i)=\\varphi(u_ia)^*\\) (Proposition 3.2(2)). Since \\(0\\le\\varphi(u_i^2)\\le\\|\\varphi\\|1\\) by (P2), adding \\((\\|\\varphi\\|1-\\varphi(u_i^2))\\oplus0\\ge0\\) gives \\(\\begin{bmatrix}\\|\\varphi\\|1&\\varphi(u_ia)\\\\\\varphi(u_ia)^*&\\varphi(a^*a)\\end{bmatrix}\\ge0\\). If \\(\\varphi=0\\) there is nothing to prove. Otherwise the two-by-two test, Lemma 2.2(4), gives \\(\\varphi(u_ia)^*\\varphi(u_ia)\\le\\|\\varphi\\|\\,\\varphi(a^*a)\\). Now let \\(i\\to\\infty\\): \\(u_ia\\to a\\), \\(\\varphi\\) is bounded, and the positive cone is closed.\n\n(2) Let \\(s=\\sup\\{\\|\\varphi(b)\\|:b\\in A_+,\\|b\\|\\le1\\}\\). For \\(\\|a\\|\\le1\\), (1) and (P2) give \\[\n\\begin{gathered}\n\\|\\varphi(a)\\|^2\\\\\n=\\|\\varphi(a)^*\\varphi(a)\\|\\\\\n\\le\\|\\varphi\\|\\,\\|\\varphi(a^*a)\\|\\\\\n\\le\\|\\varphi\\|s,\n\\end{gathered}\n\\] so \\(\\|\\varphi\\|\\le s\\). For \\(b\\in A_+\\) with \\(\\|b\\|\\le1\\), in a faithful representation of \\(A\\) we have \\(u_ibu_i\\le u_i^2\\le u_i\\), so \\(\\|\\varphi(u_ibu_i)\\|\\le\\|\\varphi(u_i)\\|\\) by positivity and (P2). As \\(u_ibu_i\\to b\\), \\(\\|\\varphi(b)\\|\\le\\sup_i\\|\\varphi(u_i)\\|\\). Thus \\(\\|\\varphi\\|\\le s\\le\\sup_i\\|\\varphi(u_i)\\|\\le\\|\\varphi\\|\\). The norms \\(\\|\\varphi(u_i)\\|\\) increase, so the supremum is a limit. With a unit, \\(0\\le\\varphi(b)\\le\\varphi(1)\\) for \\(0\\le b\\le1\\), so \\(s=\\|\\varphi(1)\\|\\).\n\n(3) By (1) and \\(\\|\\varphi\\|\\le1\\), \\(\\varphi(y)^*\\varphi(y)\\le\\varphi(y^*y)\\) for every \\(y\\). Take \\(y=a+tw\\) with \\(w\\in A\\) and \\(t\\) real. Expanding and using \\(\\varphi(a^*a)=\\varphi(a)^*\\varphi(a)\\),\n\\[\n\\begin{gathered}\n0\\le\\varphi(y^*y)-\\varphi(y)^*\\varphi(y) \\\\\n=t\\,(S+S^*)+t^2E, \\\\\nS=\\varphi(a^*w)-\\varphi(a)^*\\varphi(w), \\\\\nE=\\varphi(w^*w)-\\varphi(w)^*\\varphi(w),\n\\end{gathered}\n\\]\nwhere \\(\\varphi(w^*a)-\\varphi(w)^*\\varphi(a)=S^*\\) because \\(\\varphi\\) is hermitian. Divide by \\(|t|\\) and let \\(t\\to0\\) from each side: \\(S+S^*\\ge0\\) and \\(-(S+S^*)\\ge0\\), so \\(S+S^*=0\\). Replacing \\(w\\) by \\(iw\\) replaces \\(S\\) by \\(iS\\), so \\(i(S-S^*)=0\\) as well. Hence \\(S=0\\): \\(\\varphi(a^*w)=\\varphi(a)^*\\varphi(w)\\) for all \\(w\\). Taking adjoints and writing \\(x=w^*\\) gives \\(\\varphi(xa)=\\varphi(x)\\varphi(a)\\). For the second statement apply the first to \\(a^*\\), for which \\(\\varphi(a)=\\varphi(a^*)^*\\) turns the hypothesis into \\(\\varphi((a^*)^*a^*)=\\varphi(a^*)^*\\varphi(a^*)\\); this gives \\(\\varphi(xa^*)=\\varphi(x)\\varphi(a^*)\\) for all \\(x\\), and taking adjoints gives \\(\\varphi(ax^*)=\\varphi(a)\\varphi(x^*)\\) for all \\(x\\), which is the claim.\n\n(4) \\(e_{12}^*e_{12}=e_{22}\\) and \\(t(e_{22})=e_{22}\\), while \\(t(e_{12})^*t(e_{12})=e_{21}^*e_{21}=e_{12}e_{21}=e_{11}\\), and \\(e_{22}-e_{11}\\) is not positive. The transpose is isometric, since \\(x^{\\mathsf T}\\) is the adjoint of the entrywise conjugate of \\(x\\), and entrywise conjugation is implemented by the conjugation of \\(\\mathbb C^2\\). \\(\\square\\)\n\n*Remarks.* The two-by-two argument needs no dilation. For completely positive maps, (1) also follows from Stinespring's theorem; see the remark after Theorem 6.1. The case of positive maps and normal elements, Kadison's inequality, is Corollary 5.5. Part (3) is used in Exercise 4.3.\n\n### Jordan homomorphisms\n\n**Exercise 4.2** (easy; Amplification detects multiplicativity). Call a linear map \\(\\pi:A\\to B\\) between C\\(^*\\)-algebras a *Jordan \\(*\\)-homomorphism* if \\(\\pi(x^*)=\\pi(x)^*\\) for all \\(x\\) and \\(\\pi(h^2)=\\pi(h)^2\\) for self-adjoint \\(h\\). Let \\(\\pi:A\\to B\\) be linear and \\(n\\ge2\\), and suppose that \\(\\pi^{(n)}=\\pi\\otimes\\mathrm{id}_{M_n}:M_n(A)\\to M_n(B)\\) is a Jordan \\(*\\)-homomorphism. Show that \\(\\pi\\) is a \\(*\\)-homomorphism; in particular, if \\(\\pi\\) is bijective, it is a \\(*\\)-isomorphism. Show that the statement fails for \\(n=1\\).\n\n*Solution.* First, \\(\\pi^{(n)}(Z^2)=\\pi^{(n)}(Z)^2\\) for every \\(Z\\in M_n(A)\\), not only for self-adjoint \\(Z\\). Indeed, write \\(Z=H+iK\\) with \\(H,K\\) self-adjoint. Then \\(Z^2=H^2-K^2+i\\bigl((H+K)^2-H^2-K^2\\bigr)\\), each square on the right is a square of a self-adjoint matrix, and the same identity holds for \\(\\pi^{(n)}(Z)=\\pi^{(n)}(H)+i\\pi^{(n)}(K)\\), whose parts \\(\\pi^{(n)}(H)\\), \\(\\pi^{(n)}(K)\\) are self-adjoint. Now let \\(x,y\\in A\\) and \\(Z=xe_{12}+ye_{21}\\), the matrix with \\(x\\) in place \\((1,2)\\), \\(y\\) in place \\((2,1)\\), and zeros elsewhere; this uses \\(n\\ge2\\). Then \\(Z^2=xy\\,e_{11}+yx\\,e_{22}\\) and \\(\\pi^{(n)}(Z)^2=\\pi(x)\\pi(y)e_{11}+\\pi(y)\\pi(x)e_{22}\\). Comparing the \\((1,1)\\) entries gives \\(\\pi(xy)=\\pi(x)\\pi(y)\\). Comparing the \\((1,1)\\) entries of \\(\\pi^{(n)}((xe_{11})^*)=\\pi^{(n)}(xe_{11})^*\\) gives \\(\\pi(x^*)=\\pi(x)^*\\). So \\(\\pi\\) is a \\(*\\)-homomorphism. For \\(n=1\\), the transpose \\(t\\) on \\(M_2(\\mathbb C)\\) is a bijective Jordan \\(*\\)-homomorphism, since \\((h^{\\mathsf T})^2=(h^2)^{\\mathsf T}\\) and \\((x^*)^{\\mathsf T}=(x^{\\mathsf T})^*\\), but \\(t(e_{12}e_{21})=t(e_{11})=e_{11}\\), while \\(t(e_{12})t(e_{21})=e_{21}e_{12}=e_{22}\\).\n\n*Remarks.* Only the corner \\(M_2\\) is used, so \\(n=2\\) is the whole content. For the transpose, \\(t^{(2)}\\) is not even positive (Proposition 3.2(3)), while a Jordan \\(*\\)-homomorphism is positive, as \\(\\pi(h^2)=\\pi(h)^2\\ge0\\). Exercise 4.3 gives an order-theoretic version.\n\n**Exercise 4.3** (medium; Jordan maps that are 2-positive). Let \\(\\pi:A\\to B\\) be a Jordan \\(*\\)-homomorphism (Exercise 4.2). Show that \\(\\pi\\) is positive and contractive on self-adjoint elements, and that \\(\\pi\\) is a \\(*\\)-homomorphism exactly when it is 2-positive.\n\n*Solution.* Every positive element is \\(h^2\\) with \\(h\\) self-adjoint (B1), and \\(\\pi(h^2)=\\pi(h)^2\\ge0\\); so \\(\\pi\\) is positive and hence bounded (Proposition 3.2(4)). For self-adjoint \\(h\\) and \\(m\\ge1\\), induction on \\(m\\) gives \\(\\pi(h^{2^m})=\\pi(h)^{2^m}\\), so \\(\\|\\pi(h)\\|^{2^m}=\\|\\pi(h^{2^m})\\|\\le\\|\\pi\\|\\,\\|h\\|^{2^m}\\), using \\(\\|y^2\\|=\\|y\\|^2\\) for self-adjoint \\(y\\). Taking \\(2^m\\)-th roots and letting \\(m\\to\\infty\\) gives \\(\\|\\pi(h)\\|\\le\\|h\\|\\). A \\(*\\)-homomorphism is CP (Proposition 3.2(3)). Conversely, let \\(\\pi\\) be 2-positive. By Theorem 4.1(2), \\(\\|\\pi\\|=\\lim_i\\|\\pi(u_i)\\|\\le1\\), because the \\(u_i\\) are self-adjoint contractions. By Theorem 4.1(1), \\(\\pi(x)^*\\pi(x)\\le\\pi(x^*x)\\) and \\(\\pi(x)\\pi(x)^*\\le\\pi(xx^*)\\) for all \\(x\\). For \\(x=h+ik\\) with \\(h,k\\) self-adjoint, \\(x^*x+xx^*=2(h^2+k^2)\\), and likewise \\[\n\\begin{gathered}\n\\pi(x)^*\\pi(x)+\\pi(x)\\pi(x)^*\\\\\n=2(\\pi(h)^2+\\pi(k)^2)\\\\\n=2\\pi(h^2+k^2).\n\\end{gathered}\n\\] So \\(\\pi(x^*x+xx^*)=\\pi(x)^*\\pi(x)+\\pi(x)\\pi(x)^*\\). The two nonnegative differences therefore add up to \\(0\\), so both vanish: \\(\\pi(x^*x)=\\pi(x)^*\\pi(x)\\) for all \\(x\\). By Theorem 4.1(3), \\(\\pi(yx)=\\pi(y)\\pi(x)\\) for all \\(x,y\\). Together with \\(\\pi(x^*)=\\pi(x)^*\\), \\(\\pi\\) is a \\(*\\)-homomorphism. The transpose on \\(M_2(\\mathbb C)\\) shows that 2-positivity cannot be dropped.\n\n## 5. When positivity implies complete positivity\n\nThe transpose shows that a positive map need not be 2-positive. This section shows that positivity is enough when the target or the domain is commutative, and that \\(k\\)-positivity is enough when the target has a separating family of representations of dimension at most \\(k\\), or when the domain is \\(C_0(\\Omega,M_k)\\).\n\n",
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      "unit": "completely-positive-maps",
      "name": "Commutative targets, and targets with small representations",
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      "full_conditions_and_proof": "### Commutative targets, and targets with small representations\n\n**Theorem 5.1.**\n\n1. If \\(B\\) is abelian, every positive linear map \\(\\varphi:A\\to B\\) is completely positive. In particular, every positive linear functional is completely positive.\n2. A \\(k\\)-positive map \\(\\psi:A\\to M_k(\\mathbb C)\\) is completely positive.\n3. Let \\(n\\ge1\\), and let \\(B\\) have a separating family of representations \\(\\sigma_\\alpha\\) on spaces of dimension at most \\(n\\) (for every \\(b\\ne0\\) some \\(\\sigma_\\alpha(b)\\ne0\\)). Then every \\(n\\)-positive map \\(\\varphi:A\\to B\\) is completely positive. Since the irreducible representations of \\(B\\) separate its points (B9), this applies whenever every irreducible representation of \\(B\\) has dimension at most \\(n\\).\n\n*Further reading:* [Blackadar, II.6.9.10]. The proof below needs only a separating family of representations of dimension at most \\(n\\).\n\n**Proof.** (1) By the commutative Gelfand–Naimark theorem (B5), the Gelfand transform identifies \\(B\\) with \\(C_0(\\Omega)\\) by an isometric \\(*\\)-isomorphism, where \\(\\Omega\\) is the space of characters of \\(B\\). So an element \\(b\\in B\\) is positive exactly when \\(\\chi(b)\\ge0\\) for every character \\(\\chi\\), because positivity in \\(C_0(\\Omega)\\) is pointwise (Proposition 2.1(6) with \\(n=1\\)) and the isomorphism preserves and reflects positivity (P1). A character is positive, since \\(\\chi(b^*b)=|\\chi(b)|^2\\), so \\(\\chi\\circ\\varphi\\) is a positive functional on \\(A\\). For \\(x\\in A^n\\), \\(y\\in B^n\\) and a character \\(\\chi\\), put \\(w=\\sum_j\\chi(y_j)x_j\\). Then\n\\[\n\\begin{gathered}\n\\chi\\Bigl(\\sum_{i,j}y_i^*\\varphi(x_i^*x_j)y_j\\Bigr) \\\\\n=\\sum_{i,j}\\overline{\\chi(y_i)}\\,\\chi(y_j)\\,(\\chi\\circ\\varphi)(x_i^*x_j) \\\\\n=(\\chi\\circ\\varphi)(w^*w)\\ge0.\n\\end{gathered}\n\\]\nSo (3.1) holds for every \\(n\\). For a functional, take \\(B=\\mathbb C\\).\n\n(2) By Proposition 3.2(1), \\(\\psi\\) is \\(m\\)-positive exactly when \\(\\sum_{p,q=1}^m\\langle\\psi(x_p^*x_q)\\zeta_q,\\zeta_p\\rangle\\ge0\\) for all \\(x\\in A^m\\) and \\(\\zeta\\in(\\mathbb C^k)^m\\). Write \\(\\zeta_q=\\sum_{s=1}^kc_{qs}\\varepsilon_s\\) and put \\(w_t=\\sum_qc_{qt}x_q\\) for \\(t=1,\\dots,k\\). Then \\(w_s^*w_t=\\sum_{p,q}\\overline{c_{ps}}\\,c_{qt}\\,x_p^*x_q\\), and\n\\[\n\\sum_{p,q=1}^m\\langle\\psi(x_p^*x_q)\\zeta_q,\\zeta_p\\rangle=\\sum_{s,t=1}^k\\langle\\psi(w_s^*w_t)\\varepsilon_t,\\varepsilon_s\\rangle ,\n\\]\nwhich is nonnegative by \\(k\\)-positivity. So \\(m\\) terms reduce to \\(k\\) terms.\n\n(3) Let \\(\\sigma_\\alpha\\) act on \\(K_\\alpha\\), with \\(k_\\alpha=\\dim K_\\alpha\\le n\\), and identify \\(B(K_\\alpha)\\) with \\(M_{k_\\alpha}(\\mathbb C)\\). The \\(*\\)-homomorphism \\(\\sigma_\\alpha\\) is completely positive, so \\(\\sigma_\\alpha\\circ\\varphi\\) is \\(n\\)-positive (Proposition 3.2(2)–(3)); hence it is \\(k_\\alpha\\)-positive, and completely positive by (2). The direct sum \\(\\sigma=\\bigoplus_\\alpha\\sigma_\\alpha\\) is faithful, because the family separates points. For \\(X\\in M_m(A)_+\\), after regrouping \\((\\bigoplus_\\alpha K_\\alpha)^m\\) as \\(\\bigoplus_\\alpha K_\\alpha^m\\), the operator \\(\\sigma^{(m)}(\\varphi^{(m)}(X))\\) is \\(\\bigoplus_\\alpha(\\sigma_\\alpha\\circ\\varphi)^{(m)}(X)\\), which is positive. Since \\(\\sigma^{(m)}\\) is injective (Proposition 2.1(1)) and an injective \\(*\\)-homomorphism reflects positivity (P1), \\(\\varphi^{(m)}(X)\\ge0\\). Finally, for every nonzero \\(b\\) there is an irreducible representation \\(\\sigma_b\\) with \\(\\sigma_b(b)\\ne0\\) (B9); these form a separating family. \\(\\square\\)\n\n*Remarks.* (i) Part (1) is also the case \\(n=1\\) of (3): by (B5) the characters of an abelian \\(B\\) are a separating family of representations on \\(\\mathbb C\\). (ii) For \\(n=2\\) the hypothesis in (3) cannot be weakened to positivity: \\(M_2(\\mathbb C)\\) has its identity representation, of dimension 2, and the transpose is a positive map into \\(M_2(\\mathbb C)\\) that is not CP (Proposition 3.2(3)).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CM-08",
      "unit": "completely-positive-maps",
      "name": "Domains of the form \\(C_0(\\Omega,M_k)\\)",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "anchor": "oa-fnd-cm-08",
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        "line": 284,
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      },
      "full_conditions_and_proof": "### Domains of the form \\(C_0(\\Omega,M_k)\\)\n\nThroughout this subsection, \\(\\Omega\\) is a locally compact Hausdorff space.\n\n**Lemma 5.2** (Partitions of unity). Let \\(C\\subseteq\\Omega\\) be compact and \\(U_1,\\dots,U_L\\) open sets covering \\(C\\). There are \\(g_1,\\dots,g_L\\in C_c(\\Omega)\\) with \\(g_l\\ge0\\), \\(g_l=0\\) off \\(U_l\\), \\(\\sum_lg_l\\le1\\) on \\(\\Omega\\), and \\(\\sum_lg_l=1\\) on \\(C\\).\n\n**Proof.** For each \\(\\omega\\in C\\) pick \\(l(\\omega)\\) with \\(\\omega\\in U_{l(\\omega)}\\), and an open \\(W_\\omega\\ni\\omega\\) whose closure is compact and lies in \\(U_{l(\\omega)}\\); such a set exists by [the existence of neighbourhoods with compact closure](stone-weierstrass-c0.md#oa-fnd-sw-15) (B12). Finitely many \\(W_\\omega\\) cover \\(C\\). Let \\(C_l\\) be the union of the closures of the chosen \\(W_\\omega\\) with \\(l(\\omega)=l\\); it is compact and lies in \\(U_l\\). The locally compact form of [Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16) (B12), applied to \\(C_l\\) and the closed set \\(\\Omega\\setminus U_l\\), gives \\(h_l\\in C_c(\\Omega)\\) with \\(0\\le h_l\\le1\\), \\(h_l=1\\) on \\(C_l\\) and \\(h_l=0\\) off \\(U_l\\). Put \\(g_1=h_1\\) and \\(g_l=(1-h_1)\\cdots(1-h_{l-1})h_l\\). Then \\(0\\le g_l\\le h_l\\), and by induction \\(\\sum_{l\\le L}g_l=1-\\prod_{l\\le L}(1-h_l)\\), which lies in \\([0,1]\\) and equals \\(1\\) on \\(C\\), since every point of \\(C\\) lies in some \\(C_l\\). \\(\\square\\)\n\n**Lemma 5.3** (Approximation). Let \\(F:\\Omega\\to M_N(\\mathbb C)\\) be continuous and vanish at infinity, and let \\(\\varepsilon>0\\). There are points \\(\\omega_1,\\dots,\\omega_L\\) and functions \\(g_1,\\dots,g_L\\) as in Lemma 5.2 with\n\\[\n\\sup_{\\omega\\in\\Omega}\\Bigl\\|F(\\omega)-\\sum_lg_l(\\omega)F(\\omega_l)\\Bigr\\|\\le2\\varepsilon .\n\\]\n\n**Proof.** The set \\(C=\\{\\omega:\\|F(\\omega)\\|\\ge\\varepsilon\\}\\) is compact; if it is empty, take \\(L=0\\). For \\(p\\in C\\), the set \\(U_p=\\{\\omega:\\|F(\\omega)-F(p)\\|<\\varepsilon\\}\\) is an open neighbourhood of \\(p\\). Choose \\(p_1,\\dots,p_L\\) with \\(C\\subseteq\\bigcup_lU_{p_l}\\), take \\(g_l\\) from Lemma 5.2, and put \\(\\omega_l=p_l\\). For every \\(\\omega\\),\n\\[\n\\begin{gathered}\nF(\\omega)-\\sum_lg_l(\\omega)F(\\omega_l) \\\\\n=\\Bigl(1-\\sum_lg_l(\\omega)\\Bigr)F(\\omega) \\\\\n+\\sum_lg_l(\\omega)\\bigl(F(\\omega)-F(\\omega_l)\\bigr).\n\\end{gathered}\n\\]\nThe first term vanishes on \\(C\\) and has norm less than \\(\\varepsilon\\) off \\(C\\). In the second, \\(g_l(\\omega)\\ne0\\) forces \\(\\omega\\in U_{p_l}\\), so its norm is at most \\(\\sum_lg_l(\\omega)\\varepsilon\\le\\varepsilon\\). \\(\\square\\)\n\n**Theorem 5.4.**\n\n1. Let \\(k\\ge1\\). Every \\(k\\)-positive map \\(\\varphi\\) from \\(A=M_k(C_0(\\Omega))=C_0(\\Omega,M_k)\\) into a C\\(^*\\)-algebra \\(B\\) is completely positive.\n2. In particular, every positive map from an abelian C\\(^*\\)-algebra into a C\\(^*\\)-algebra is completely positive (\\(k=1\\)), and every \\(k\\)-positive map from \\(M_k(\\mathbb C)\\) is completely positive (\\(\\Omega\\) a point).\n\nThe proof below uses finite partitions of unity and scalar matrix compressions; it makes no point-dependent choice outside exceptional null sets.\n\n**Proof.** (1) Let \\(m\\ge1\\). By Lemma 2.2(1) it suffices to show \\(\\varphi^{(m)}(X)\\ge0\\) for \\(X=[x_p^*x_q]_{p,q=1}^m\\) with \\(x_p\\in A\\). Identify \\(M_m(A)\\) with \\(C_0(\\Omega,M_{mk})\\); by Proposition 2.1(6), with \\(n=mk\\), its norm is the supremum norm. The function \\(\\omega\\mapsto X(\\omega)=[x_p(\\omega)^*x_q(\\omega)]\\) is continuous and vanishes at infinity. Fix \\(\\varepsilon>0\\), take \\(\\omega_l\\) and \\(g_l\\) from Lemma 5.3, and put \\(G=\\sum_lg_l\\,X(\\omega_l)\\in M_m(A)\\), where each scalar matrix \\(X(\\omega_l)\\) is multiplied by the function \\(g_l\\). Then \\(\\|X-G\\|\\le2\\varepsilon\\).\n\nWe show \\(\\varphi^{(m)}(G)\\ge0\\). Fix \\(l\\), and write \\(g=g_l\\) and \\(c_p=x_p(\\omega_l)\\in M_k(\\mathbb C)\\). Since \\(\\sum_re_{r1}e_{1r}=1\\), we have \\(c_p^*c_q=\\sum_{r=1}^k(e_{1r}c_p)^*(e_{1r}c_q)\\), so\n\\[\n\\begin{gathered}\ng\\,X(\\omega_l)=\\sum_{r=1}^k[w_{rp}^*w_{rq}]_{p,q}, \\\\\nw_{rp}=g^{1/2}e_{1r}c_p\\in A.\n\\end{gathered}\n\\]\nFor fixed \\(r\\), every \\(w_{rp}\\) is a combination of the \\(k\\) elements \\(v_s=g^{1/2}e_{1s}\\): \\(w_{rp}=\\sum_s\\beta_{ps}v_s\\), where \\(\\beta_{ps}\\) is the \\((r,s)\\) entry of \\(c_p\\). Hence \\([w_{rp}^*w_{rq}]_{p,q}=\\alpha^*[v_s^*v_t]_{s,t}\\,\\alpha\\) in the notation of Proposition 2.1(7), with \\(\\alpha\\in M_{k,m}(\\mathbb C)\\), \\(\\alpha_{tq}=\\beta_{qt}\\). Now \\(\\varphi^{(m)}(\\alpha^*T\\alpha)=\\alpha^*\\varphi^{(k)}(T)\\alpha\\) for every \\(T\\in M_k(A)\\), since both sides are linear in \\(T\\) and agree entry by entry. The matrix \\([v_s^*v_t]\\) is positive (Lemma 2.2(1)), \\(\\varphi^{(k)}\\) preserves positivity, and so does \\(\\alpha^*(\\cdot)\\alpha\\) (Proposition 2.1(7)). So each piece of \\(G\\) has positive image, and \\(\\varphi^{(m)}(G)\\ge0\\).\n\nFinally, \\(\\varphi\\) is bounded (Proposition 3.2(4)), so \\(\\|\\varphi^{(m)}(Y)\\|\\le\\|\\varphi\\|\\sum_{p,q}\\|Y_{pq}\\|\\le m^2\\|\\varphi\\|\\,\\|Y\\|\\) by Proposition 2.1(2). Letting \\(\\varepsilon\\to0\\), \\(\\varphi^{(m)}(X)\\) is a norm limit of positive elements, hence positive.\n\n(2) An abelian \\(A\\) is \\(C_0(\\Omega)\\) up to an isometric \\(*\\)-isomorphism \\(\\Gamma\\) (B5), which is CP with CP inverse (Proposition 3.2(3)). By (1) with \\(k=1\\), \\(\\varphi\\circ\\Gamma^{-1}\\) is CP, and so is \\(\\varphi=(\\varphi\\circ\\Gamma^{-1})\\circ\\Gamma\\) by Proposition 3.2(2). If \\(\\Omega\\) is a point, \\(C_0(\\Omega,M_k)=M_k(\\mathbb C)\\). \\(\\square\\)\n\nWith \\(\\Omega\\) a point, (1) contains Choi's theorem that \\(k\\)-positive maps on \\(M_k(\\mathbb C)\\) are completely positive; Exercise 3.3 gives a direct proof. Proposition 7.1(7) extends (1) to maps into dual spaces.\n\n**Corollary 5.5** (Kadison's inequality). If \\(\\varphi:A\\to B\\) is positive and \\(a\\in A\\) is normal, then \\(\\varphi(a)^*\\varphi(a)\\le\\|\\varphi\\|\\,\\varphi(a^*a)\\). In particular \\(\\varphi(h)^2\\le\\|\\varphi\\|\\,\\varphi(h^2)\\) for \\(h\\in A_h\\).\n\n**Proof.** The C\\(^*\\)-subalgebra \\(C\\) generated by \\(a\\) is abelian, because \\(a\\) is normal. By (P1), \\(\\varphi|_C\\) is positive, hence CP by Theorem 5.4(2). Theorem 4.1(1), applied to \\(\\varphi|_C\\), gives \\(\\varphi(a)^*\\varphi(a)\\le\\|\\varphi|_C\\|\\,\\varphi(a^*a)\\le\\|\\varphi\\|\\,\\varphi(a^*a)\\). \\(\\square\\)\n\n**Example 5.6** (How far positivity goes). The transpose \\(t\\) on \\(M_2(\\mathbb C)\\) is positive, isometric and unital. It is not 2-positive (Proposition 3.2(3)), and it breaks the Kadison–Schwarz inequality (Theorem 4.1(4)). It still satisfies Kadison's inequality for normal elements (Corollary 5.5), for instance \\(t(h)^2=(h^2)^{\\mathsf T}=t(h^2)\\) for self-adjoint \\(h\\). Its restriction to any abelian C\\(^*\\)-subalgebra, such as the diagonal matrices, is CP, by Theorem 5.4(2). And its composition with the identification \\(\\Phi_1\\) of Proposition 7.5 is \\(\\Phi_2\\), which is not 2-positive.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CM-05",
      "unit": "completely-positive-maps",
      "name": "6. Stinespring's dilation theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "full_conditions_and_proof": "## 6. Stinespring's dilation theorem\n\n**Theorem 6.1** (Stinespring). Fix a Hilbert space \\(H\\) and a C\\(^*\\)-algebra \\(A\\).\n\n1. If \\(\\pi\\) is a representation of \\(A\\) on \\(K\\) and \\(V\\in B(H,K)\\), then \\(\\varphi(a)=V^*\\pi(a)V\\) is completely positive, and \\(\\|\\varphi\\|\\le\\|V\\|^2\\).\n2. Let \\(\\varphi:A\\to B(H)\\) be completely positive, and put \\(N=\\varphi(A)'\\). This is a von Neumann algebra, because \\(\\varphi(A)\\) is self-adjoint: positive maps are hermitian (Proposition 3.2(2)). There are a Hilbert space \\(K\\), a representation \\(\\pi\\) of \\(A\\) on \\(K\\), an operator \\(V\\in B(H,K)\\) and a unital normal representation \\(\\rho\\) of \\(N\\) on \\(K\\) such that\n\\[\n\\begin{gathered}\n\\varphi(a)=V^*\\pi(a)V, \\\\\nK=\\overline{\\operatorname{span}}\\,\\pi(A)VH, \\\\\n\\rho(N)\\subseteq\\pi(A)', \\\\\n\\rho(x)V=Vx, \\\\\na\\in A,\\quad x\\in N.\n\\end{gathered}\n\\tag{6.1}\n\\]\n3. Let \\((u_i)\\) be any approximate identity of \\(A\\). The increasing net \\((\\varphi(u_i))\\) converges weakly to \\(a_0:=V^*V\\), which is its least upper bound, and\n\\[\n\\begin{gathered}\n\\|\\varphi\\|=\\|V\\|^2=\\|a_0\\| \\\\\n=\\lim_i\\|\\varphi(u_i)\\|.\n\\end{gathered}\n\\tag{6.2}\n\\]\nIf \\(A\\) has a unit, then \\(a_0=\\varphi(1)\\) and \\(\\|\\varphi\\|=\\|\\varphi(1)\\|\\).\n\n*Further reading:* [Blackadar, II.6.9.7]. The norm equality is proved in (3).\n\n**Proof.** (1) For \\(x\\in A^n\\) and \\(y\\in B(H)^n\\), put \\(W=\\sum_j\\pi(x_j)Vy_j\\in B(H,K)\\). Then \\(\\sum_{i,j}y_i^*V^*\\pi(x_i^*x_j)Vy_j=W^*W\\ge0\\), and the criterion of Proposition 3.2(1) applies. Also \\(\\|V^*\\pi(a)V\\|\\le\\|V\\|^2\\|a\\|\\), since \\(\\pi\\) is contractive.\n\n(2) *Step 1: the form.* The algebraic tensor product \\(A\\odot H\\) means the following concrete quotient. Take the complex vector space freely spanned by symbols \\((a,\\xi)\\in A\\times H\\), and divide by the subspace generated by the additivity and scalar-linearity relations in each entry. Write \\(a\\otimes\\xi\\) for the class of \\((a,\\xi)\\). On this quotient put\n\\[\n\\begin{gathered}\n\\Bigl\\langle\\sum_ix_i\\otimes\\xi_i,\\ \\sum_jy_j\\otimes\\eta_j\\Bigr\\rangle_\\varphi \\\\\n=\\sum_{i,j}\\langle\\varphi(y_j^*x_i)\\xi_i,\\eta_j\\rangle.\n\\end{gathered}\n\\tag{6.3}\n\\]\nThe expression is bilinear in \\(x,\\xi\\) and conjugate-bilinear in \\(y,\\eta\\). It therefore vanishes on every defining relation in either variable, so it descends to a well-defined sesquilinear form on the quotient. For \\(\\zeta=\\sum_{i=1}^nx_i\\otimes\\xi_i\\) and \\(\\hat\\xi=(\\xi_1,\\dots,\\xi_n)\\in H^n\\), exchanging the names of the indices gives\n\\[\n\\langle\\zeta,\\zeta\\rangle_\\varphi=\\bigl\\langle\\varphi^{(n)}\\bigl([x_i^*x_j]\\bigr)\\hat\\xi,\\hat\\xi\\bigr\\rangle\\ \\ge\\ 0 ,\n\\tag{6.4}\n\\]\nbecause \\([x_i^*x_j]\\) is positive (Lemma 2.2(1)), \\(\\varphi^{(n)}\\) preserves positivity, and a positive operator matrix has a nonnegative quadratic form (Proposition 2.1(5)). So the form is positive semidefinite. By [Cauchy–Schwarz for forms](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-01), \\(N_0=\\{\\zeta:\\langle\\zeta,\\zeta\\rangle_\\varphi=0\\}\\) is the subspace of vectors orthogonal to everything. The [Hilbert completion lemma](hilbert-spaces-and-compact-operators.md#completing-normed-and-inner-product-spaces) constructs the Hilbert space \\(K\\) from \\((A\\odot H)/N_0\\), and let \\(q:A\\odot H\\to K\\) be the quotient map.\n\n*Step 2: the two actions.* For \\(a\\in A\\) and \\(b\\in N\\) define linear maps of \\(A\\odot H\\) by \\(\\pi_0(a)(x\\otimes\\xi)=ax\\otimes\\xi\\) and \\(\\rho_0(b)(x\\otimes\\xi)=x\\otimes b\\xi\\). They commute. We claim that, for all \\(\\zeta,\\eta\\in A\\odot H\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_0(a)\\zeta,\\pi_0(a)\\zeta\\rangle_\\varphi \\\\\n\\le\\|a\\|^2\\langle\\zeta,\\zeta\\rangle_\\varphi, \\\\\n\\langle\\rho_0(b)\\zeta,\\rho_0(b)\\zeta\\rangle_\\varphi \\\\\n\\le\\|b\\|^2\\langle\\zeta,\\zeta\\rangle_\\varphi,\n\\end{gathered}\n\\tag{6.5}\n\\]\n\\[\n\\begin{gathered}\n\\langle\\pi_0(a)\\zeta,\\eta\\rangle_\\varphi \\\\\n=\\langle\\zeta,\\pi_0(a^*)\\eta\\rangle_\\varphi, \\\\\n\\langle\\rho_0(b)\\zeta,\\eta\\rangle_\\varphi \\\\\n=\\langle\\zeta,\\rho_0(b^*)\\eta\\rangle_\\varphi.\n\\end{gathered}\n\\tag{6.6}\n\\]\nLet \\(\\zeta=\\sum_{i=1}^nx_i\\otimes\\xi_i\\), \\(X=[x_i^*x_j]\\) and \\(Y=[x_i^*a^*ax_j]\\). In a faithful representation \\(\\sigma\\) of \\(A\\) on \\(L\\), for \\(\\vartheta\\in L^n\\),\n\\[\n\\begin{gathered}\n\\langle\\sigma^{(n)}(Y)\\vartheta,\\vartheta\\rangle \\\\\n=\\Bigl\\|\\sigma(a)\\sum_j\\sigma(x_j)\\vartheta_j\\Bigr\\|^2 \\\\\n\\le\\|a\\|^2\\Bigl\\|\\sum_j\\sigma(x_j)\\vartheta_j\\Bigr\\|^2 \\\\\n=\\|a\\|^2\\langle\\sigma^{(n)}(X)\\vartheta,\\vartheta\\rangle.\n\\end{gathered}\n\\]\nSo \\(Y\\le\\|a\\|^2X\\) by Lemma 2.2(3), since \\(\\sigma\\) is faithful, and \\(\\varphi^{(n)}(Y)\\le\\|a\\|^2\\varphi^{(n)}(X)\\). As in (6.4), \\(\\langle\\pi_0(a)\\zeta,\\pi_0(a)\\zeta\\rangle_\\varphi=\\langle\\varphi^{(n)}(Y)\\hat\\xi,\\hat\\xi\\rangle\\), which gives the first bound. For the second, let \\(T=\\varphi^{(n)}(X)\\ge0\\) and \\(b_n=\\operatorname{diag}(b,\\dots,b)\\) on \\(H^n\\). Since \\(b\\) commutes with \\(\\varphi(A)\\), \\(b_n\\) commutes with \\(T\\), hence with \\(T^{1/2}\\), a norm limit of polynomials in \\(T\\). As in (6.4),\n\\[\n\\begin{gathered}\n\\langle\\rho_0(b)\\zeta,\\rho_0(b)\\zeta\\rangle_\\varphi \\\\\n=\\langle Tb_n\\hat\\xi,b_n\\hat\\xi\\rangle \\\\\n=\\|b_nT^{1/2}\\hat\\xi\\|^2 \\\\\n\\le\\|b\\|^2\\langle T\\hat\\xi,\\hat\\xi\\rangle.\n\\end{gathered}\n\\]\nFor (6.6), let \\(\\eta=\\sum_jy_j\\otimes\\eta_j\\). Both sides of the first identity equal \\(\\sum_{i,j}\\langle\\varphi(y_j^*ax_i)\\xi_i,\\eta_j\\rangle\\), since \\((a^*y_j)^*=y_j^*a\\). Both sides of the second equal \\(\\sum_{i,j}\\langle b\\,\\varphi(y_j^*x_i)\\xi_i,\\eta_j\\rangle\\), because \\(b\\) commutes with \\(\\varphi(A)\\).\n\nBy (6.5), \\(\\pi_0(a)\\) and \\(\\rho_0(b)\\) map \\(N_0\\) into itself and induce operators of norm at most \\(\\|a\\|\\) and \\(\\|b\\|\\) on \\((A\\odot H)/N_0\\). Let \\(\\pi(a)\\) and \\(\\rho(b)\\) be their extensions to \\(K\\). The maps \\(a\\mapsto\\pi_0(a)\\) and \\(b\\mapsto\\rho_0(b)\\) are linear and multiplicative, so by (6.6) \\(\\pi\\) is a representation of \\(A\\) and \\(\\rho\\) is a representation of \\(N\\), with \\(\\rho(1)=1\\). The ranges commute, because \\(\\pi_0\\) and \\(\\rho_0\\) commute.\n\n*Step 3: the operator \\(V\\).* Fix an approximate identity \\((u_i)\\), and put \\(V_i\\xi=q(u_i\\otimes\\xi)\\). Then \\(\\|V_i\\xi\\|^2=\\langle\\varphi(u_i^2)\\xi,\\xi\\rangle\\le\\|\\varphi\\|\\,\\|\\xi\\|^2\\); \\(\\varphi\\) is bounded by Proposition 3.2(4). Let \\(i\\le j\\) and \\(c=u_j-u_i\\). In a faithful representation, \\(0\\le c\\le u_j\\le1\\), so \\(c^2=c^{1/2}cc^{1/2}\\le c\\). Hence\n\\[\n\\begin{gathered}\n\\|V_j\\xi-V_i\\xi\\|^2=\\langle\\varphi(c^2)\\xi,\\xi\\rangle \\\\\n\\le\\langle\\varphi(u_j)\\xi,\\xi\\rangle-\\langle\\varphi(u_i)\\xi,\\xi\\rangle.\n\\end{gathered}\n\\]\nThe numbers \\(\\langle\\varphi(u_i)\\xi,\\xi\\rangle\\) increase with \\(i\\) and are at most \\(\\|\\varphi\\|\\,\\|\\xi\\|^2\\), so they converge. Comparing \\(V_i\\xi\\) and \\(V_j\\xi\\) with \\(V_{i_0}\\xi\\) for \\(i,j\\ge i_0\\) shows that \\((V_i\\xi)\\) is a Cauchy net. A Cauchy net in a complete metric space converges: choose increasing indices \\(i_n\\) so that every tail after \\(i_n\\) has diameter at most \\(1/n\\), take the limit of the Cauchy sequence at those indices, and use the same tail bound for the whole net. So \\(V\\xi=\\lim_iV_i\\xi\\) defines \\(V\\in B(H,K)\\) with \\(\\|V\\|^2\\le\\|\\varphi\\|\\).\n\n*Step 4: the identities.* Let \\(x\\in A\\) and \\(\\xi,\\eta\\in H\\).\n- (a) \\(V^*q(x\\otimes\\xi)=\\varphi(x)\\xi\\). Indeed \\[\n\\begin{gathered}\n\\langle q(x\\otimes\\xi),V\\eta\\rangle\\\\\n=\\lim_i\\langle q(x\\otimes\\xi),q(u_i\\otimes\\eta)\\rangle\\\\\n=\\lim_i\\langle\\varphi(u_ix)\\xi,\\eta\\rangle\\\\\n=\\langle\\varphi(x)\\xi,\\eta\\rangle,\n\\end{gathered}\n\\] since \\(u_ix\\to x\\) and \\(\\varphi\\) is bounded.\n- (b) \\(\\pi(x)V\\xi=q(x\\otimes\\xi)\\). Indeed \\(\\pi(x)V_i\\xi=q(xu_i\\otimes\\xi)\\), and \\[\n\\begin{gathered}\n\\|q(xu_i\\otimes\\xi)-q(x\\otimes\\xi)\\|^2\\\\\n\\le\\|\\varphi\\|\\,\\|xu_i-x\\|^2\\|\\xi\\|^2\\to0.\n\\end{gathered}\n\\]\n- (c) By (a) and (b), \\(V^*\\pi(x)V\\xi=\\varphi(x)\\xi\\). So \\(\\varphi=V^*\\pi(\\cdot)V\\).\n- (d) By (b), \\(q(A\\odot H)\\) is the span of \\(\\pi(A)VH\\). So \\(K\\) is the closed span of \\(\\pi(A)VH\\); in particular \\(\\pi\\) is nondegenerate.\n- (e) For \\(b\\in N\\), \\[\n\\begin{gathered}\n\\pi(x)\\rho(b)V\\xi\\\\\n=\\rho(b)q(x\\otimes\\xi)\\\\\n=q(x\\otimes b\\xi)\\\\\n=\\pi(x)Vb\\xi.\n\\end{gathered}\n\\] So \\(w=\\rho(b)V\\xi-Vb\\xi\\) satisfies \\(\\pi(x)w=0\\) for all \\(x\\). Then \\(\\langle w,\\pi(x^*)\\kappa\\rangle=0\\) for all \\(x\\) and \\(\\kappa\\), and \\(w=0\\) by (d). Hence \\(\\rho(b)V=Vb\\).\n\n*Step 5: \\(\\rho\\) is normal.* For \\(\\zeta=\\sum_ix_i\\otimes\\xi_i\\) and \\(\\eta=\\sum_jy_j\\otimes\\eta_j\\),\n\\[\n\\langle\\rho(b)q(\\zeta),q(\\eta)\\rangle=\\sum_{i,j}\\langle b\\,\\xi_i,\\ \\varphi(x_i^*y_j)\\eta_j\\rangle ,\n\\]\na finite sum of vector functionals of \\(b\\), hence ultraweakly continuous on \\(N\\). For arbitrary \\(\\kappa,\\lambda\\in K\\), choose \\(\\kappa_m\\to\\kappa\\) and \\(\\lambda_m\\to\\lambda\\) in \\(q(A\\odot H)\\). Then\n\\[\n\\begin{gathered}\n|\\langle\\rho(b)\\kappa,\\lambda\\rangle-\\langle\\rho(b)\\kappa_m,\\lambda_m\\rangle|\\\\\n\\le\\|b\\|\\,(\\|\\kappa-\\kappa_m\\|\\,\\|\\lambda\\|+\\|\\kappa_m\\|\\,\\|\\lambda-\\lambda_m\\|).\n\\end{gathered}\n\\]\nSo \\(b\\mapsto\\langle\\rho(b)\\kappa,\\lambda\\rangle\\) is a norm limit in \\(N^*\\) of elements of \\(N_*\\), and it lies in \\(N_*\\) because \\(N_*\\) is norm-closed (B11). Finally, let \\(\\psi=\\sum_n\\langle\\,\\cdot\\,\\kappa_n,\\lambda_n\\rangle\\in B(K)_*\\). The series \\(\\psi\\circ\\rho=\\sum_n\\langle\\rho(\\cdot)\\kappa_n,\\lambda_n\\rangle\\) converges in the norm of \\(N^*\\), since the \\(n\\)-th term has norm at most \\(\\|\\kappa_n\\|\\,\\|\\lambda_n\\|\\) and \\(\\sum_n\\|\\kappa_n\\|\\,\\|\\lambda_n\\|<\\infty\\). So \\(\\psi\\circ\\rho\\in N_*\\), and \\(\\rho\\) is normal.\n\n(3) Let \\((u_i)\\) be any approximate identity, not necessarily the one fixed in Step 3. By (b), \\(q(u_i\\otimes\\xi)=\\pi(u_i)V\\xi\\), and \\(\\pi(u_i)\\to1\\) strongly because \\(\\pi\\) is nondegenerate by (d) (Section 1). So \\(V\\xi=\\lim_iq(u_i\\otimes\\xi)\\), and (a) gives \\[\n\\begin{gathered}\n\\langle V\\xi,V\\xi\\rangle\\\\\n=\\lim_i\\langle V^*q(u_i\\otimes\\xi),\\xi\\rangle\\\\\n=\\lim_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle\\\\\n=\\sup_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle.\n\\end{gathered}\n\\] So \\(\\langle a_0\\xi,\\xi\\rangle=\\sup_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle\\) for every \\(\\xi\\). By polarization \\(\\varphi(u_i)\\to a_0\\) weakly, and every self-adjoint upper bound \\(c\\) of the net satisfies \\(\\langle c\\xi,\\xi\\rangle\\ge\\langle a_0\\xi,\\xi\\rangle\\). By (P2), \\(\\|a_0\\|=\\sup_{\\|\\xi\\|=1}\\sup_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle=\\sup_i\\|\\varphi(u_i)\\|\\), and this supremum is a limit because the norms increase. Now \\(\\|\\varphi\\|\\le\\|V\\|^2\\) by (1), while \\(\\|V\\|^2=\\|V^*V\\|=\\|a_0\\|=\\sup_i\\|\\varphi(u_i)\\|\\le\\|\\varphi\\|\\). If \\(A\\) has a unit, then \\(\\|u_i-1\\|=\\|u_i1-1\\|\\to0\\), so \\(a_0=\\lim_i\\varphi(u_i)=\\varphi(1)\\). \\(\\square\\)\n\n*Remark* (A second proof of the Kadison–Schwarz inequality for completely positive maps). If \\(\\varphi=V^*\\pi(\\cdot)V\\) as in (6.1), then \\[\n\\begin{gathered}\n\\varphi(a)^*\\varphi(a)\\\\\n=V^*\\pi(a)^*VV^*\\pi(a)V\\\\\n\\le\\|V\\|^2V^*\\pi(a^*a)V,\n\\end{gathered}\n\\] because \\(VV^*\\le\\|V\\|^21\\). By (6.2) the right side is \\(\\|\\varphi\\|\\,\\varphi(a^*a)\\). This gives Theorem 4.1(1) for completely positive maps.\n\n**Theorem 6.2** (Uniqueness of the minimal dilation, and when \\(\\rho\\) is faithful). Let \\(\\varphi\\), \\(N\\), \\(K\\), \\(\\pi\\), \\(V\\), \\(\\rho\\) and \\(a_0\\) be as in Theorem 6.1.\n\n1. Let \\(\\pi'\\) be a representation of \\(A\\) on \\(K'\\) and \\(V'\\in B(H,K')\\) with \\(\\varphi=V'^*\\pi'(\\cdot)V'\\) and \\(K'=\\overline{\\operatorname{span}}\\,\\pi'(A)V'H\\). There is exactly one unitary \\(U:K\\to K'\\) with \\(U\\pi(a)=\\pi'(a)U\\) for all \\(a\\in A\\) and \\(UV=V'\\). If a map \\(\\rho':N\\to B(K')\\) satisfies \\(\\rho'(N)\\subseteq\\pi'(A)'\\) and \\(\\rho'(x)V'=V'x\\) for all \\(x\\), then \\(\\rho'(x)=U\\rho(x)U^*\\). In particular \\(\\rho\\) is determined by (6.1), and any such \\(\\rho'\\) is automatically a unital normal representation.\n2. \\(\\ker\\rho=\\{x\\in N:\\ a_0x=0\\}=N(1-z)\\), where \\(z\\) is the projection onto the closed span of \\(N'a_0H\\). The projection \\(z\\) lies in the centre \\(N\\cap N'\\). So \\(\\rho\\) is faithful if and only if \\(z=1\\). This holds when \\(a_0\\) has zero kernel, in particular when \\(a_0=1\\).\n3. There are completely positive maps for which \\(\\rho\\) is not faithful. So a triple as in (6.1) with a faithful \\(\\rho\\) need not exist, and the uniqueness in (1) must be stated without faithfulness.\n4. If \\(A\\) has a unit, then \\(V\\xi=q(1\\otimes\\xi)\\) and \\(V^*V=\\varphi(1)\\). \\(V\\) is an isometry if and only if \\(\\varphi(1)=1\\); for general \\(A\\), if and only if \\(a_0=1\\). Then \\(\\rho\\) is faithful, and \\(\\varphi(a)\\) is the compression of \\(\\pi(a)\\) to the subspace \\(VH\\), identified with \\(H\\).\n\nThe explicit example in (3) proves that faithfulness cannot be required for every minimal dilation; (2) gives the exact condition.\n\n**Proof.** (1) For finite sums,\n\\[\n\\begin{gathered}\n\\Bigl\\|\\sum_i\\pi'(x_i)V'\\xi_i\\Bigr\\|^2 \\\\\n=\\sum_{i,j}\\langle V'^*\\pi'(x_j^*x_i)V'\\xi_i,\\xi_j\\rangle \\\\\n=\\sum_{i,j}\\langle\\varphi(x_j^*x_i)\\xi_i,\\xi_j\\rangle \\\\\n=\\Bigl\\|\\sum_i\\pi(x_i)V\\xi_i\\Bigr\\|^2.\n\\end{gathered}\n\\]\nSo \\(U_0:\\sum_i\\pi(x_i)V\\xi_i\\mapsto\\sum_i\\pi'(x_i)V'\\xi_i\\) is well defined and isometric from a dense subspace of \\(K\\) onto a dense subspace of \\(K'\\). It extends to a unitary \\(U\\). The relation \\(U\\pi(a)=\\pi'(a)U\\) holds on the dense subspace, hence everywhere. Both representations are nondegenerate, so \\(\\pi(u_i)\\to1\\) and \\(\\pi'(u_i)\\to1\\) strongly (Section 1). Hence \\(UV\\xi=\\lim_iU\\pi(u_i)V\\xi=\\lim_i\\pi'(u_i)V'\\xi=V'\\xi\\). Any unitary with the two properties satisfies \\(U\\pi(a)V\\xi=\\pi'(a)V'\\xi\\), so it is unique. For \\(\\rho'\\): on the total set of vectors \\(\\pi'(a)V'\\xi\\), \\(\\rho'(x)\\pi'(a)V'\\xi=\\pi'(a)\\rho'(x)V'\\xi=\\pi'(a)V'x\\xi\\). The operator \\(U\\rho(x)U^*\\) has the same values there, because \\[\n\\begin{gathered}\nU\\rho(x)U^*\\pi'(a)V'\\xi\\\\\n=U\\rho(x)\\pi(a)V\\xi\\\\\n=U\\pi(a)Vx\\xi\\\\\n=\\pi'(a)V'x\\xi.\n\\end{gathered}\n\\] Two bounded operators that agree on a total set are equal.\n\n(2) If \\(\\rho(x)=0\\), then \\(Vx=\\rho(x)V=0\\) and \\(a_0x=V^*Vx=0\\). If \\(a_0x=0\\), then \\(\\|Vx\\xi\\|^2=\\langle a_0x\\xi,x\\xi\\rangle=0\\), so \\(Vx=0\\), and \\(\\rho(x)\\pi(a)V\\xi=\\pi(a)Vx\\xi=0\\) on a total set; so \\(\\rho(x)=0\\). Next, \\(a_0\\) is a weak limit of elements of \\(\\varphi(A)\\subseteq N'\\), and \\(N'\\) is weakly closed, so \\(a_0\\in N'\\). Let \\(x\\in N\\); it commutes with \\(a_0\\in N'\\). If \\(a_0x=0\\), then \\(xa_0=0\\), and for \\(y\\in N'\\) we get \\(xya_0=yxa_0=0\\); so \\(x\\) vanishes on \\(N'a_0H\\), and \\(xz=0\\). Conversely, if \\(xz=0\\), then \\(xa_0=xza_0=0\\), because \\(1\\in N'\\) gives \\(a_0H\\subseteq zH\\); and \\(a_0x=xa_0=0\\), since \\(x\\) and \\(a_0\\) commute. The subspace \\(zH\\) is invariant under \\(N'\\), and under \\(N\\) because \\(xya_0\\xi=ya_0x\\xi\\) for \\(x\\in N\\), \\(y\\in N'\\). Both sets are self-adjoint, so \\(z\\in N''\\cap N'=N\\cap N'\\). Hence \\(\\ker\\rho=\\{x\\in N:xz=0\\}=N(1-z)\\). If \\(\\ker a_0=\\{0\\}\\), then \\(a_0H\\) is dense, because \\((a_0H)^\\perp=\\ker a_0\\), and \\(z=1\\).\n\n(3) Let \\(A=\\mathbb C\\), \\(H=\\mathbb C^2\\) and \\(\\varphi(\\lambda)=\\lambda e_{11}\\). With \\(V(\\xi_1,\\xi_2)=\\xi_1\\in\\mathbb C\\) and \\(\\pi(\\lambda)=\\lambda\\) on \\(K=\\mathbb C\\), we have \\(\\varphi(\\lambda)=V^*\\pi(\\lambda)V\\), so \\(\\varphi\\) is CP by Theorem 6.1(1), and \\(K\\) is spanned by \\(\\pi(\\mathbb C)VH\\). Here \\(N=\\varphi(\\mathbb C)'\\) is the algebra of diagonal matrices, and \\(\\rho(\\operatorname{diag}(s,t))V=V\\operatorname{diag}(s,t)\\) forces \\(\\rho(\\operatorname{diag}(s,t))=s\\). So \\(\\rho(\\operatorname{diag}(0,1))=0\\). By (1), every triple satisfying (6.1) is unitarily equivalent to this one, so none of them has a faithful \\(\\rho\\). In the notation of (2), \\(a_0=e_{11}\\), \\(N'=N\\) and \\(z=e_{11}\\).\n\n(4) If \\(A\\) has a unit, then \\(\\|u_i-1\\|\\to0\\), and \\(\\|V_i\\xi-q(1\\otimes\\xi)\\|^2\\le\\|\\varphi\\|\\,\\|u_i-1\\|^2\\|\\xi\\|^2\\) (Step 3 of the proof of Theorem 6.1); so \\(V\\xi=q(1\\otimes\\xi)\\), and \\(a_0=\\varphi(1)\\) by Theorem 6.1(3). \\(V\\) is an isometry exactly when \\(V^*V=a_0=1\\). Then \\(\\rho\\) is faithful by (2), and \\(VV^*\\) is the projection onto \\(VH\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CM-06",
      "unit": "completely-positive-maps",
      "name": "6. Stinespring's dilation theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "anchor": "oa-fnd-cm-06",
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      "full_conditions_and_proof": "## 6. Stinespring's dilation theorem\n\n**Theorem 6.1** (Stinespring). Fix a Hilbert space \\(H\\) and a C\\(^*\\)-algebra \\(A\\).\n\n1. If \\(\\pi\\) is a representation of \\(A\\) on \\(K\\) and \\(V\\in B(H,K)\\), then \\(\\varphi(a)=V^*\\pi(a)V\\) is completely positive, and \\(\\|\\varphi\\|\\le\\|V\\|^2\\).\n2. Let \\(\\varphi:A\\to B(H)\\) be completely positive, and put \\(N=\\varphi(A)'\\). This is a von Neumann algebra, because \\(\\varphi(A)\\) is self-adjoint: positive maps are hermitian (Proposition 3.2(2)). There are a Hilbert space \\(K\\), a representation \\(\\pi\\) of \\(A\\) on \\(K\\), an operator \\(V\\in B(H,K)\\) and a unital normal representation \\(\\rho\\) of \\(N\\) on \\(K\\) such that\n\\[\n\\begin{gathered}\n\\varphi(a)=V^*\\pi(a)V, \\\\\nK=\\overline{\\operatorname{span}}\\,\\pi(A)VH, \\\\\n\\rho(N)\\subseteq\\pi(A)', \\\\\n\\rho(x)V=Vx, \\\\\na\\in A,\\quad x\\in N.\n\\end{gathered}\n\\tag{6.1}\n\\]\n3. Let \\((u_i)\\) be any approximate identity of \\(A\\). The increasing net \\((\\varphi(u_i))\\) converges weakly to \\(a_0:=V^*V\\), which is its least upper bound, and\n\\[\n\\begin{gathered}\n\\|\\varphi\\|=\\|V\\|^2=\\|a_0\\| \\\\\n=\\lim_i\\|\\varphi(u_i)\\|.\n\\end{gathered}\n\\tag{6.2}\n\\]\nIf \\(A\\) has a unit, then \\(a_0=\\varphi(1)\\) and \\(\\|\\varphi\\|=\\|\\varphi(1)\\|\\).\n\n*Further reading:* [Blackadar, II.6.9.7]. The norm equality is proved in (3).\n\n**Proof.** (1) For \\(x\\in A^n\\) and \\(y\\in B(H)^n\\), put \\(W=\\sum_j\\pi(x_j)Vy_j\\in B(H,K)\\). Then \\(\\sum_{i,j}y_i^*V^*\\pi(x_i^*x_j)Vy_j=W^*W\\ge0\\), and the criterion of Proposition 3.2(1) applies. Also \\(\\|V^*\\pi(a)V\\|\\le\\|V\\|^2\\|a\\|\\), since \\(\\pi\\) is contractive.\n\n(2) *Step 1: the form.* The algebraic tensor product \\(A\\odot H\\) means the following concrete quotient. Take the complex vector space freely spanned by symbols \\((a,\\xi)\\in A\\times H\\), and divide by the subspace generated by the additivity and scalar-linearity relations in each entry. Write \\(a\\otimes\\xi\\) for the class of \\((a,\\xi)\\). On this quotient put\n\\[\n\\begin{gathered}\n\\Bigl\\langle\\sum_ix_i\\otimes\\xi_i,\\ \\sum_jy_j\\otimes\\eta_j\\Bigr\\rangle_\\varphi \\\\\n=\\sum_{i,j}\\langle\\varphi(y_j^*x_i)\\xi_i,\\eta_j\\rangle.\n\\end{gathered}\n\\tag{6.3}\n\\]\nThe expression is bilinear in \\(x,\\xi\\) and conjugate-bilinear in \\(y,\\eta\\). It therefore vanishes on every defining relation in either variable, so it descends to a well-defined sesquilinear form on the quotient. For \\(\\zeta=\\sum_{i=1}^nx_i\\otimes\\xi_i\\) and \\(\\hat\\xi=(\\xi_1,\\dots,\\xi_n)\\in H^n\\), exchanging the names of the indices gives\n\\[\n\\langle\\zeta,\\zeta\\rangle_\\varphi=\\bigl\\langle\\varphi^{(n)}\\bigl([x_i^*x_j]\\bigr)\\hat\\xi,\\hat\\xi\\bigr\\rangle\\ \\ge\\ 0 ,\n\\tag{6.4}\n\\]\nbecause \\([x_i^*x_j]\\) is positive (Lemma 2.2(1)), \\(\\varphi^{(n)}\\) preserves positivity, and a positive operator matrix has a nonnegative quadratic form (Proposition 2.1(5)). So the form is positive semidefinite. By [Cauchy–Schwarz for forms](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-01), \\(N_0=\\{\\zeta:\\langle\\zeta,\\zeta\\rangle_\\varphi=0\\}\\) is the subspace of vectors orthogonal to everything. The [Hilbert completion lemma](hilbert-spaces-and-compact-operators.md#completing-normed-and-inner-product-spaces) constructs the Hilbert space \\(K\\) from \\((A\\odot H)/N_0\\), and let \\(q:A\\odot H\\to K\\) be the quotient map.\n\n*Step 2: the two actions.* For \\(a\\in A\\) and \\(b\\in N\\) define linear maps of \\(A\\odot H\\) by \\(\\pi_0(a)(x\\otimes\\xi)=ax\\otimes\\xi\\) and \\(\\rho_0(b)(x\\otimes\\xi)=x\\otimes b\\xi\\). They commute. We claim that, for all \\(\\zeta,\\eta\\in A\\odot H\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_0(a)\\zeta,\\pi_0(a)\\zeta\\rangle_\\varphi \\\\\n\\le\\|a\\|^2\\langle\\zeta,\\zeta\\rangle_\\varphi, \\\\\n\\langle\\rho_0(b)\\zeta,\\rho_0(b)\\zeta\\rangle_\\varphi \\\\\n\\le\\|b\\|^2\\langle\\zeta,\\zeta\\rangle_\\varphi,\n\\end{gathered}\n\\tag{6.5}\n\\]\n\\[\n\\begin{gathered}\n\\langle\\pi_0(a)\\zeta,\\eta\\rangle_\\varphi \\\\\n=\\langle\\zeta,\\pi_0(a^*)\\eta\\rangle_\\varphi, \\\\\n\\langle\\rho_0(b)\\zeta,\\eta\\rangle_\\varphi \\\\\n=\\langle\\zeta,\\rho_0(b^*)\\eta\\rangle_\\varphi.\n\\end{gathered}\n\\tag{6.6}\n\\]\nLet \\(\\zeta=\\sum_{i=1}^nx_i\\otimes\\xi_i\\), \\(X=[x_i^*x_j]\\) and \\(Y=[x_i^*a^*ax_j]\\). In a faithful representation \\(\\sigma\\) of \\(A\\) on \\(L\\), for \\(\\vartheta\\in L^n\\),\n\\[\n\\begin{gathered}\n\\langle\\sigma^{(n)}(Y)\\vartheta,\\vartheta\\rangle \\\\\n=\\Bigl\\|\\sigma(a)\\sum_j\\sigma(x_j)\\vartheta_j\\Bigr\\|^2 \\\\\n\\le\\|a\\|^2\\Bigl\\|\\sum_j\\sigma(x_j)\\vartheta_j\\Bigr\\|^2 \\\\\n=\\|a\\|^2\\langle\\sigma^{(n)}(X)\\vartheta,\\vartheta\\rangle.\n\\end{gathered}\n\\]\nSo \\(Y\\le\\|a\\|^2X\\) by Lemma 2.2(3), since \\(\\sigma\\) is faithful, and \\(\\varphi^{(n)}(Y)\\le\\|a\\|^2\\varphi^{(n)}(X)\\). As in (6.4), \\(\\langle\\pi_0(a)\\zeta,\\pi_0(a)\\zeta\\rangle_\\varphi=\\langle\\varphi^{(n)}(Y)\\hat\\xi,\\hat\\xi\\rangle\\), which gives the first bound. For the second, let \\(T=\\varphi^{(n)}(X)\\ge0\\) and \\(b_n=\\operatorname{diag}(b,\\dots,b)\\) on \\(H^n\\). Since \\(b\\) commutes with \\(\\varphi(A)\\), \\(b_n\\) commutes with \\(T\\), hence with \\(T^{1/2}\\), a norm limit of polynomials in \\(T\\). As in (6.4),\n\\[\n\\begin{gathered}\n\\langle\\rho_0(b)\\zeta,\\rho_0(b)\\zeta\\rangle_\\varphi \\\\\n=\\langle Tb_n\\hat\\xi,b_n\\hat\\xi\\rangle \\\\\n=\\|b_nT^{1/2}\\hat\\xi\\|^2 \\\\\n\\le\\|b\\|^2\\langle T\\hat\\xi,\\hat\\xi\\rangle.\n\\end{gathered}\n\\]\nFor (6.6), let \\(\\eta=\\sum_jy_j\\otimes\\eta_j\\). Both sides of the first identity equal \\(\\sum_{i,j}\\langle\\varphi(y_j^*ax_i)\\xi_i,\\eta_j\\rangle\\), since \\((a^*y_j)^*=y_j^*a\\). Both sides of the second equal \\(\\sum_{i,j}\\langle b\\,\\varphi(y_j^*x_i)\\xi_i,\\eta_j\\rangle\\), because \\(b\\) commutes with \\(\\varphi(A)\\).\n\nBy (6.5), \\(\\pi_0(a)\\) and \\(\\rho_0(b)\\) map \\(N_0\\) into itself and induce operators of norm at most \\(\\|a\\|\\) and \\(\\|b\\|\\) on \\((A\\odot H)/N_0\\). Let \\(\\pi(a)\\) and \\(\\rho(b)\\) be their extensions to \\(K\\). The maps \\(a\\mapsto\\pi_0(a)\\) and \\(b\\mapsto\\rho_0(b)\\) are linear and multiplicative, so by (6.6) \\(\\pi\\) is a representation of \\(A\\) and \\(\\rho\\) is a representation of \\(N\\), with \\(\\rho(1)=1\\). The ranges commute, because \\(\\pi_0\\) and \\(\\rho_0\\) commute.\n\n*Step 3: the operator \\(V\\).* Fix an approximate identity \\((u_i)\\), and put \\(V_i\\xi=q(u_i\\otimes\\xi)\\). Then \\(\\|V_i\\xi\\|^2=\\langle\\varphi(u_i^2)\\xi,\\xi\\rangle\\le\\|\\varphi\\|\\,\\|\\xi\\|^2\\); \\(\\varphi\\) is bounded by Proposition 3.2(4). Let \\(i\\le j\\) and \\(c=u_j-u_i\\). In a faithful representation, \\(0\\le c\\le u_j\\le1\\), so \\(c^2=c^{1/2}cc^{1/2}\\le c\\). Hence\n\\[\n\\begin{gathered}\n\\|V_j\\xi-V_i\\xi\\|^2=\\langle\\varphi(c^2)\\xi,\\xi\\rangle \\\\\n\\le\\langle\\varphi(u_j)\\xi,\\xi\\rangle-\\langle\\varphi(u_i)\\xi,\\xi\\rangle.\n\\end{gathered}\n\\]\nThe numbers \\(\\langle\\varphi(u_i)\\xi,\\xi\\rangle\\) increase with \\(i\\) and are at most \\(\\|\\varphi\\|\\,\\|\\xi\\|^2\\), so they converge. Comparing \\(V_i\\xi\\) and \\(V_j\\xi\\) with \\(V_{i_0}\\xi\\) for \\(i,j\\ge i_0\\) shows that \\((V_i\\xi)\\) is a Cauchy net. A Cauchy net in a complete metric space converges: choose increasing indices \\(i_n\\) so that every tail after \\(i_n\\) has diameter at most \\(1/n\\), take the limit of the Cauchy sequence at those indices, and use the same tail bound for the whole net. So \\(V\\xi=\\lim_iV_i\\xi\\) defines \\(V\\in B(H,K)\\) with \\(\\|V\\|^2\\le\\|\\varphi\\|\\).\n\n*Step 4: the identities.* Let \\(x\\in A\\) and \\(\\xi,\\eta\\in H\\).\n- (a) \\(V^*q(x\\otimes\\xi)=\\varphi(x)\\xi\\). Indeed \\[\n\\begin{gathered}\n\\langle q(x\\otimes\\xi),V\\eta\\rangle\\\\\n=\\lim_i\\langle q(x\\otimes\\xi),q(u_i\\otimes\\eta)\\rangle\\\\\n=\\lim_i\\langle\\varphi(u_ix)\\xi,\\eta\\rangle\\\\\n=\\langle\\varphi(x)\\xi,\\eta\\rangle,\n\\end{gathered}\n\\] since \\(u_ix\\to x\\) and \\(\\varphi\\) is bounded.\n- (b) \\(\\pi(x)V\\xi=q(x\\otimes\\xi)\\). Indeed \\(\\pi(x)V_i\\xi=q(xu_i\\otimes\\xi)\\), and \\[\n\\begin{gathered}\n\\|q(xu_i\\otimes\\xi)-q(x\\otimes\\xi)\\|^2\\\\\n\\le\\|\\varphi\\|\\,\\|xu_i-x\\|^2\\|\\xi\\|^2\\to0.\n\\end{gathered}\n\\]\n- (c) By (a) and (b), \\(V^*\\pi(x)V\\xi=\\varphi(x)\\xi\\). So \\(\\varphi=V^*\\pi(\\cdot)V\\).\n- (d) By (b), \\(q(A\\odot H)\\) is the span of \\(\\pi(A)VH\\). So \\(K\\) is the closed span of \\(\\pi(A)VH\\); in particular \\(\\pi\\) is nondegenerate.\n- (e) For \\(b\\in N\\), \\[\n\\begin{gathered}\n\\pi(x)\\rho(b)V\\xi\\\\\n=\\rho(b)q(x\\otimes\\xi)\\\\\n=q(x\\otimes b\\xi)\\\\\n=\\pi(x)Vb\\xi.\n\\end{gathered}\n\\] So \\(w=\\rho(b)V\\xi-Vb\\xi\\) satisfies \\(\\pi(x)w=0\\) for all \\(x\\). Then \\(\\langle w,\\pi(x^*)\\kappa\\rangle=0\\) for all \\(x\\) and \\(\\kappa\\), and \\(w=0\\) by (d). Hence \\(\\rho(b)V=Vb\\).\n\n*Step 5: \\(\\rho\\) is normal.* For \\(\\zeta=\\sum_ix_i\\otimes\\xi_i\\) and \\(\\eta=\\sum_jy_j\\otimes\\eta_j\\),\n\\[\n\\langle\\rho(b)q(\\zeta),q(\\eta)\\rangle=\\sum_{i,j}\\langle b\\,\\xi_i,\\ \\varphi(x_i^*y_j)\\eta_j\\rangle ,\n\\]\na finite sum of vector functionals of \\(b\\), hence ultraweakly continuous on \\(N\\). For arbitrary \\(\\kappa,\\lambda\\in K\\), choose \\(\\kappa_m\\to\\kappa\\) and \\(\\lambda_m\\to\\lambda\\) in \\(q(A\\odot H)\\). Then\n\\[\n\\begin{gathered}\n|\\langle\\rho(b)\\kappa,\\lambda\\rangle-\\langle\\rho(b)\\kappa_m,\\lambda_m\\rangle|\\\\\n\\le\\|b\\|\\,(\\|\\kappa-\\kappa_m\\|\\,\\|\\lambda\\|+\\|\\kappa_m\\|\\,\\|\\lambda-\\lambda_m\\|).\n\\end{gathered}\n\\]\nSo \\(b\\mapsto\\langle\\rho(b)\\kappa,\\lambda\\rangle\\) is a norm limit in \\(N^*\\) of elements of \\(N_*\\), and it lies in \\(N_*\\) because \\(N_*\\) is norm-closed (B11). Finally, let \\(\\psi=\\sum_n\\langle\\,\\cdot\\,\\kappa_n,\\lambda_n\\rangle\\in B(K)_*\\). The series \\(\\psi\\circ\\rho=\\sum_n\\langle\\rho(\\cdot)\\kappa_n,\\lambda_n\\rangle\\) converges in the norm of \\(N^*\\), since the \\(n\\)-th term has norm at most \\(\\|\\kappa_n\\|\\,\\|\\lambda_n\\|\\) and \\(\\sum_n\\|\\kappa_n\\|\\,\\|\\lambda_n\\|<\\infty\\). So \\(\\psi\\circ\\rho\\in N_*\\), and \\(\\rho\\) is normal.\n\n(3) Let \\((u_i)\\) be any approximate identity, not necessarily the one fixed in Step 3. By (b), \\(q(u_i\\otimes\\xi)=\\pi(u_i)V\\xi\\), and \\(\\pi(u_i)\\to1\\) strongly because \\(\\pi\\) is nondegenerate by (d) (Section 1). So \\(V\\xi=\\lim_iq(u_i\\otimes\\xi)\\), and (a) gives \\[\n\\begin{gathered}\n\\langle V\\xi,V\\xi\\rangle\\\\\n=\\lim_i\\langle V^*q(u_i\\otimes\\xi),\\xi\\rangle\\\\\n=\\lim_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle\\\\\n=\\sup_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle.\n\\end{gathered}\n\\] So \\(\\langle a_0\\xi,\\xi\\rangle=\\sup_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle\\) for every \\(\\xi\\). By polarization \\(\\varphi(u_i)\\to a_0\\) weakly, and every self-adjoint upper bound \\(c\\) of the net satisfies \\(\\langle c\\xi,\\xi\\rangle\\ge\\langle a_0\\xi,\\xi\\rangle\\). By (P2), \\(\\|a_0\\|=\\sup_{\\|\\xi\\|=1}\\sup_i\\langle\\varphi(u_i)\\xi,\\xi\\rangle=\\sup_i\\|\\varphi(u_i)\\|\\), and this supremum is a limit because the norms increase. Now \\(\\|\\varphi\\|\\le\\|V\\|^2\\) by (1), while \\(\\|V\\|^2=\\|V^*V\\|=\\|a_0\\|=\\sup_i\\|\\varphi(u_i)\\|\\le\\|\\varphi\\|\\). If \\(A\\) has a unit, then \\(\\|u_i-1\\|=\\|u_i1-1\\|\\to0\\), so \\(a_0=\\lim_i\\varphi(u_i)=\\varphi(1)\\). \\(\\square\\)\n\n*Remark* (A second proof of the Kadison–Schwarz inequality for completely positive maps). If \\(\\varphi=V^*\\pi(\\cdot)V\\) as in (6.1), then \\[\n\\begin{gathered}\n\\varphi(a)^*\\varphi(a)\\\\\n=V^*\\pi(a)^*VV^*\\pi(a)V\\\\\n\\le\\|V\\|^2V^*\\pi(a^*a)V,\n\\end{gathered}\n\\] because \\(VV^*\\le\\|V\\|^21\\). By (6.2) the right side is \\(\\|\\varphi\\|\\,\\varphi(a^*a)\\). This gives Theorem 4.1(1) for completely positive maps.\n\n**Theorem 6.2** (Uniqueness of the minimal dilation, and when \\(\\rho\\) is faithful). Let \\(\\varphi\\), \\(N\\), \\(K\\), \\(\\pi\\), \\(V\\), \\(\\rho\\) and \\(a_0\\) be as in Theorem 6.1.\n\n1. Let \\(\\pi'\\) be a representation of \\(A\\) on \\(K'\\) and \\(V'\\in B(H,K')\\) with \\(\\varphi=V'^*\\pi'(\\cdot)V'\\) and \\(K'=\\overline{\\operatorname{span}}\\,\\pi'(A)V'H\\). There is exactly one unitary \\(U:K\\to K'\\) with \\(U\\pi(a)=\\pi'(a)U\\) for all \\(a\\in A\\) and \\(UV=V'\\). If a map \\(\\rho':N\\to B(K')\\) satisfies \\(\\rho'(N)\\subseteq\\pi'(A)'\\) and \\(\\rho'(x)V'=V'x\\) for all \\(x\\), then \\(\\rho'(x)=U\\rho(x)U^*\\). In particular \\(\\rho\\) is determined by (6.1), and any such \\(\\rho'\\) is automatically a unital normal representation.\n2. \\(\\ker\\rho=\\{x\\in N:\\ a_0x=0\\}=N(1-z)\\), where \\(z\\) is the projection onto the closed span of \\(N'a_0H\\). The projection \\(z\\) lies in the centre \\(N\\cap N'\\). So \\(\\rho\\) is faithful if and only if \\(z=1\\). This holds when \\(a_0\\) has zero kernel, in particular when \\(a_0=1\\).\n3. There are completely positive maps for which \\(\\rho\\) is not faithful. So a triple as in (6.1) with a faithful \\(\\rho\\) need not exist, and the uniqueness in (1) must be stated without faithfulness.\n4. If \\(A\\) has a unit, then \\(V\\xi=q(1\\otimes\\xi)\\) and \\(V^*V=\\varphi(1)\\). \\(V\\) is an isometry if and only if \\(\\varphi(1)=1\\); for general \\(A\\), if and only if \\(a_0=1\\). Then \\(\\rho\\) is faithful, and \\(\\varphi(a)\\) is the compression of \\(\\pi(a)\\) to the subspace \\(VH\\), identified with \\(H\\).\n\nThe explicit example in (3) proves that faithfulness cannot be required for every minimal dilation; (2) gives the exact condition.\n\n**Proof.** (1) For finite sums,\n\\[\n\\begin{gathered}\n\\Bigl\\|\\sum_i\\pi'(x_i)V'\\xi_i\\Bigr\\|^2 \\\\\n=\\sum_{i,j}\\langle V'^*\\pi'(x_j^*x_i)V'\\xi_i,\\xi_j\\rangle \\\\\n=\\sum_{i,j}\\langle\\varphi(x_j^*x_i)\\xi_i,\\xi_j\\rangle \\\\\n=\\Bigl\\|\\sum_i\\pi(x_i)V\\xi_i\\Bigr\\|^2.\n\\end{gathered}\n\\]\nSo \\(U_0:\\sum_i\\pi(x_i)V\\xi_i\\mapsto\\sum_i\\pi'(x_i)V'\\xi_i\\) is well defined and isometric from a dense subspace of \\(K\\) onto a dense subspace of \\(K'\\). It extends to a unitary \\(U\\). The relation \\(U\\pi(a)=\\pi'(a)U\\) holds on the dense subspace, hence everywhere. Both representations are nondegenerate, so \\(\\pi(u_i)\\to1\\) and \\(\\pi'(u_i)\\to1\\) strongly (Section 1). Hence \\(UV\\xi=\\lim_iU\\pi(u_i)V\\xi=\\lim_i\\pi'(u_i)V'\\xi=V'\\xi\\). Any unitary with the two properties satisfies \\(U\\pi(a)V\\xi=\\pi'(a)V'\\xi\\), so it is unique. For \\(\\rho'\\): on the total set of vectors \\(\\pi'(a)V'\\xi\\), \\(\\rho'(x)\\pi'(a)V'\\xi=\\pi'(a)\\rho'(x)V'\\xi=\\pi'(a)V'x\\xi\\). The operator \\(U\\rho(x)U^*\\) has the same values there, because \\[\n\\begin{gathered}\nU\\rho(x)U^*\\pi'(a)V'\\xi\\\\\n=U\\rho(x)\\pi(a)V\\xi\\\\\n=U\\pi(a)Vx\\xi\\\\\n=\\pi'(a)V'x\\xi.\n\\end{gathered}\n\\] Two bounded operators that agree on a total set are equal.\n\n(2) If \\(\\rho(x)=0\\), then \\(Vx=\\rho(x)V=0\\) and \\(a_0x=V^*Vx=0\\). If \\(a_0x=0\\), then \\(\\|Vx\\xi\\|^2=\\langle a_0x\\xi,x\\xi\\rangle=0\\), so \\(Vx=0\\), and \\(\\rho(x)\\pi(a)V\\xi=\\pi(a)Vx\\xi=0\\) on a total set; so \\(\\rho(x)=0\\). Next, \\(a_0\\) is a weak limit of elements of \\(\\varphi(A)\\subseteq N'\\), and \\(N'\\) is weakly closed, so \\(a_0\\in N'\\). Let \\(x\\in N\\); it commutes with \\(a_0\\in N'\\). If \\(a_0x=0\\), then \\(xa_0=0\\), and for \\(y\\in N'\\) we get \\(xya_0=yxa_0=0\\); so \\(x\\) vanishes on \\(N'a_0H\\), and \\(xz=0\\). Conversely, if \\(xz=0\\), then \\(xa_0=xza_0=0\\), because \\(1\\in N'\\) gives \\(a_0H\\subseteq zH\\); and \\(a_0x=xa_0=0\\), since \\(x\\) and \\(a_0\\) commute. The subspace \\(zH\\) is invariant under \\(N'\\), and under \\(N\\) because \\(xya_0\\xi=ya_0x\\xi\\) for \\(x\\in N\\), \\(y\\in N'\\). Both sets are self-adjoint, so \\(z\\in N''\\cap N'=N\\cap N'\\). Hence \\(\\ker\\rho=\\{x\\in N:xz=0\\}=N(1-z)\\). If \\(\\ker a_0=\\{0\\}\\), then \\(a_0H\\) is dense, because \\((a_0H)^\\perp=\\ker a_0\\), and \\(z=1\\).\n\n(3) Let \\(A=\\mathbb C\\), \\(H=\\mathbb C^2\\) and \\(\\varphi(\\lambda)=\\lambda e_{11}\\). With \\(V(\\xi_1,\\xi_2)=\\xi_1\\in\\mathbb C\\) and \\(\\pi(\\lambda)=\\lambda\\) on \\(K=\\mathbb C\\), we have \\(\\varphi(\\lambda)=V^*\\pi(\\lambda)V\\), so \\(\\varphi\\) is CP by Theorem 6.1(1), and \\(K\\) is spanned by \\(\\pi(\\mathbb C)VH\\). Here \\(N=\\varphi(\\mathbb C)'\\) is the algebra of diagonal matrices, and \\(\\rho(\\operatorname{diag}(s,t))V=V\\operatorname{diag}(s,t)\\) forces \\(\\rho(\\operatorname{diag}(s,t))=s\\). So \\(\\rho(\\operatorname{diag}(0,1))=0\\). By (1), every triple satisfying (6.1) is unitarily equivalent to this one, so none of them has a faithful \\(\\rho\\). In the notation of (2), \\(a_0=e_{11}\\), \\(N'=N\\) and \\(z=e_{11}\\).\n\n(4) If \\(A\\) has a unit, then \\(\\|u_i-1\\|\\to0\\), and \\(\\|V_i\\xi-q(1\\otimes\\xi)\\|^2\\le\\|\\varphi\\|\\,\\|u_i-1\\|^2\\|\\xi\\|^2\\) (Step 3 of the proof of Theorem 6.1); so \\(V\\xi=q(1\\otimes\\xi)\\), and \\(a_0=\\varphi(1)\\) by Theorem 6.1(3). \\(V\\) is an isometry exactly when \\(V^*V=a_0=1\\). Then \\(\\rho\\) is faithful by (2), and \\(VV^*\\) is the projection onto \\(VH\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CM-12",
      "unit": "completely-positive-maps",
      "name": "Examples of dilations",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "full_conditions_and_proof": "### Examples of dilations\n\n**Example 6.3** (A state is its own dilation). Let \\(f\\) be a positive functional on \\(A\\), viewed as a CP map into \\(B(\\mathbb C)=\\mathbb C\\) (Theorem 5.1(1)). In the proof of Theorem 6.1, \\(A\\odot\\mathbb C=A\\) and the form (6.3) is \\(\\langle x,y\\rangle_f=f(y^*x)\\), so \\(K\\) is the GNS space \\(H_f\\), \\(\\pi=\\pi_f\\), and \\(V1=\\xi_f\\), the limit of the classes of \\(u_i\\). Here \\(N=\\mathbb C\\) and \\(\\rho(\\lambda)=\\lambda1_{H_f}\\). If \\(f=0\\), then \\(H_f=\\{0\\}\\), and \\(\\pi,V,\\rho\\) are the zero operators, as the same formulas require. The identity \\(\\|f\\|=\\|V\\|^2=\\lim_if(u_i)\\) of (6.2) is the familiar formula for the norm of a positive functional, and \\(a_0=\\|f\\|\\).\n\n**Example 6.4** (Compressions and minimality). Let \\(\\sigma\\) be a representation of \\(A\\) on \\(L\\), and \\(V\\in B(H,L)\\). Then \\(\\varphi=V^*\\sigma(\\cdot)V\\) is CP (Theorem 6.1(1)). The closed span \\(K_1\\) of \\(\\sigma(A)VH\\) reduces \\(\\sigma\\): multiplication and adjoints preserve this span. Let \\(P_1\\) be its orthogonal projection, \\(\\pi=\\sigma|_{K_1}\\) and \\(V_1=P_1V\\). For every \\(a\\in A\\), \\(\\sigma(a)VH\\subseteq K_1\\). Since \\(K_1\\) reduces \\(\\sigma\\), the vector \\(\\sigma(a)(1-P_1)V\\xi\\) also belongs to \\(K_1^\\perp\\), hence is zero. Thus \\(\\sigma(a)V=P_1\\sigma(a)P_1V\\), and\n\\[\nV_1^*\\pi(a)V_1=V^*\\sigma(a)V=\\varphi(a).\n\\]\nMoreover \\(\\pi(A)V_1H=\\sigma(A)VH\\), so this triple is minimal and Theorem 6.2(1) identifies it with the Stinespring triple. If \\(\\sigma\\) is nondegenerate, \\(\\sigma(u_i)V\\xi\\to V\\xi\\) shows \\(VH\\subseteq K_1\\), and \\(V_1=V\\). Without nondegeneracy, \\(VH\\) may leave \\(K_1\\): for \\(A=\\mathbb C\\), \\(L=\\mathbb C^2\\), \\(\\sigma(\\lambda)=\\lambda e_{11}\\) and \\(V=1\\), the map \\(\\varphi(\\lambda)=\\lambda e_{11}\\) of Theorem 6.2(3) appears, with \\(K_1=\\mathbb C\\varepsilon_1\\), and the minimal dilation replaces \\(V\\) by the compression \\(e_{11}V\\).\n\n**Example 6.5** (A faithful \\(\\rho\\) with \\(a_0\\ne1\\)). Let \\(A=c_0\\), the sequences tending to \\(0\\), acting diagonally on \\(H=\\ell^2\\), and \\(\\varphi(x)=\\operatorname{diag}(x_n/n)\\). With \\(\\pi(x)=\\operatorname{diag}(x_n)\\) and \\(V=\\operatorname{diag}(n^{-1/2})\\), \\(\\varphi=V^*\\pi(\\cdot)V\\), and \\(\\pi(A)VH\\) contains every finitely supported sequence, so the triple is minimal. Here \\(N=\\varphi(A)'\\) is the algebra of diagonal operators, \\(\\rho(b)=b\\), and \\(a_0=V^*V=\\operatorname{diag}(1/n)\\). It is not \\(1\\) and not invertible, but it has zero kernel, so \\(\\rho\\) is faithful, as Theorem 6.2(2) predicts, although the condition \\(a_0=1\\) of Theorem 6.2(4) fails. The norm formula gives \\(\\|\\varphi\\|=\\|a_0\\|=1\\), attained on the approximate identity \\(u_i=1_{\\{1,\\dots,i\\}}\\).\n\n**Example 6.6** (The diagonal compression). On \\(M_d(\\mathbb C)\\) let \\(E(x)=\\sum_ie_{ii}xe_{ii}\\), which keeps the diagonal of \\(x\\). With \\(\\pi(x)=x\\oplus\\cdots\\oplus x\\) on \\((\\mathbb C^d)^d\\) and \\(V\\xi=(e_{11}\\xi,\\dots,e_{dd}\\xi)\\), \\(V^*\\pi(x)V=\\sum_ie_{ii}xe_{ii}=E(x)\\). The triple is minimal: \\(\\pi(x)V\\varepsilon_i\\) has \\(x\\varepsilon_i\\) in the \\(i\\)-th place and \\(0\\) elsewhere, and \\(x\\varepsilon_i\\) runs through \\(\\mathbb C^d\\). Here \\(N=E(M_d)'\\) is the diagonal algebra, and \\(\\rho(b)=b_{11}1\\oplus\\cdots\\oplus b_{dd}1\\) satisfies \\(\\rho(b)V=Vb\\). Since \\(E(1)=1\\), \\(V\\) is an isometry and \\(\\rho\\) is faithful (Theorem 6.2(4)). \\(E\\) is a unital CP map that is not multiplicative, and \\(E(x)^*E(x)\\le E(x^*x)\\) is the Kadison–Schwarz inequality of Theorem 4.1(1) with \\(\\|E\\|=1\\).\n\n### Positive definite functions on groups\n\n**Exercise 6.7** (hard; Completely positive definite functions). Let \\(G\\) be a topological group, \\(H\\) a Hilbert space, and \\(x:G\\to B(H)\\) a function that is *completely positive definite*: for all \\(s_1,\\dots,s_n\\in G\\), the operator matrix \\([x(s_i^{-1}s_j)]_{i,j}\\) is positive on \\(H^n\\). Suppose \\(x\\) is weakly continuous at the identity \\(e\\). Show that there are a strongly continuous unitary representation \\(U\\) of \\(G\\) on a Hilbert space \\(K\\) and \\(T\\in B(H,K)\\) with \\(x(s)=T^*U(s)T\\) for all \\(s\\), and \\(K=\\overline{\\operatorname{span}}\\,U(G)TH\\). Show that such a pair is unique up to a unitary \\(W\\) with \\(WU(s)W^*=U'(s)\\) and \\(WT=T'\\), and that \\(x\\) is then strongly continuous on all of \\(G\\).\n\n*Solution.* Let \\(K_0\\) be the space of finitely supported functions \\(g:G\\to H\\), with\n\\[\n\\langle g,g'\\rangle_x=\\sum_{s,t\\in G}\\langle x(t^{-1}s)g(s),g'(t)\\rangle .\n\\]\nIf \\(g\\) is supported in \\(\\{s_1,\\dots,s_n\\}\\), put \\(\\zeta_j=g(s_j)\\). Then \\(\\langle g,g\\rangle_x=\\sum_{i,j}\\langle x(s_i^{-1}s_j)\\zeta_j,\\zeta_i\\rangle\\ge0\\), by the hypothesis and Proposition 2.1(5). So the form is positive semidefinite, hence hermitian. Apply the [null-space quotient and Hilbert completion lemma](hilbert-spaces-and-compact-operators.md#completing-normed-and-inner-product-spaces), to get \\(K\\) and the class map \\(g\\mapsto[g]\\). For \\(r\\in G\\) let \\((U_0(r)g)(s)=g(r^{-1}s)\\). Substituting \\(s=rs'\\), \\(t=rt'\\) gives \\(\\langle U_0(r)g,U_0(r)g'\\rangle_x=\\langle g,g'\\rangle_x\\), because \\((rt')^{-1}(rs')=t'^{-1}s'\\). Also \\(U_0(r)U_0(r')=U_0(rr')\\) and \\(U_0(e)=1\\). So each \\(U_0(r)\\) induces an isometry \\(U(r)\\) of \\(K\\) with inverse \\(U(r^{-1})\\), and \\(U\\) is a unitary representation. Let \\(\\delta_s\\xi\\) be the function with value \\(\\xi\\) at \\(s\\) and \\(0\\) elsewhere, and put \\(T\\xi=[\\delta_e\\xi]\\). Then \\(\\|T\\xi\\|^2=\\langle x(e)\\xi,\\xi\\rangle\\le\\|x(e)\\|\\,\\|\\xi\\|^2\\), \\(U(s)T\\xi=[\\delta_s\\xi]\\), and\n\\[\n\\langle T^*U(s)T\\xi,\\eta\\rangle=\\langle[\\delta_s\\xi],[\\delta_e\\eta]\\rangle=\\langle x(s)\\xi,\\eta\\rangle .\n\\]\nSo \\(x(s)=T^*U(s)T\\). The vectors \\(U(s)T\\xi=[\\delta_s\\xi]\\) span \\([K_0]\\), which is dense.\n\n*Continuity.* \\[\n\\begin{gathered}\n\\|U(r)[\\delta_t\\xi]-[\\delta_t\\xi]\\|^2\\\\\n=2\\langle x(e)\\xi,\\xi\\rangle-2\\,\\mathrm{Re}\\,\\langle x(t^{-1}rt)\\xi,\\xi\\rangle,\n\\end{gathered}\n\\] since \\(\\langle[\\delta_{rt}\\xi],[\\delta_t\\xi]\\rangle=\\langle x(t^{-1}rt)\\xi,\\xi\\rangle\\). As \\(r\\to e\\), \\(t^{-1}rt\\to e\\), so the right side tends to \\(0\\) by weak continuity at \\(e\\). The unitaries \\(U(r)\\) are uniformly bounded and the vectors \\([\\delta_t\\xi]\\) span a dense subspace, so \\(U(r)\\to1\\) strongly as \\(r\\to e\\). Then \\(U(r)\\kappa-U(r_0)\\kappa=U(r_0)(U(r_0^{-1}r)\\kappa-\\kappa)\\to0\\) as \\(r\\to r_0\\), so \\(U\\) is strongly continuous, and so is \\(x=T^*U(\\cdot)T\\).\n\n*Uniqueness.* If \\((U',K',T')\\) is another such pair, then \\[\n\\begin{gathered}\n\\|\\sum_lU'(s_l)T'\\xi_l\\|^2\\\\\n=\\sum_{l,m}\\langle x(s_m^{-1}s_l)\\xi_l,\\xi_m\\rangle\\\\\n=\\|\\sum_lU(s_l)T\\xi_l\\|^2.\n\\end{gathered}\n\\] As in the proof of Theorem 6.2(1), \\(W:\\sum U(s_l)T\\xi_l\\mapsto\\sum U'(s_l)T'\\xi_l\\) extends to a unitary with the stated properties, and \\(WT=T'\\) because \\(T\\xi=U(e)T\\xi\\).\n\n*Remarks.* The necessity is also true: if \\(x(s)=T^*U(s)T\\), then \\([x(s_i^{-1}s_j)]=[(U(s_i)T)^*(U(s_j)T)]\\ge0\\). When \\(G\\) is locally compact the result can also be derived from Stinespring's theorem through the group C\\(^*\\)-algebra, but the direct construction is shorter, and it is Stinespring's construction with \\(A\\odot H\\) replaced by functions on \\(G\\).\n\n## 7. Completely positive maps and dual spaces\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-CM-09",
      "unit": "completely-positive-maps",
      "name": "Matrix order on dual spaces, and transposes",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "anchor": "oa-fnd-cm-09",
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      "full_conditions_and_proof": "### Matrix order on dual spaces, and transposes\n\nLet \\(A\\) be a C\\(^*\\)-algebra. Identify \\(M_n(A^*)\\) with the dual of \\(M_n(A)\\) through\n\\[\n\\begin{gathered}\n\\langle a,f\\rangle=\\sum_{i,j}f_{ij}(a_{ij}), \\\\\na\\in M_n(A),\\quad f\\in M_n(A^*),\n\\end{gathered}\n\\tag{7.1}\n\\]\nand call \\(f\\) positive if it is a positive functional on \\(M_n(A)\\). For a linear subspace \\(E\\) of \\(A\\) or of \\(A^*\\), give \\(M_n(E)\\) the order inherited from \\(M_n(A)\\) or \\(M_n(A^*)\\). A linear map \\(\\varphi:E\\to F\\) between two such subspaces is \\(n\\)-positive if \\(\\varphi^{(n)}\\) maps positive elements of \\(M_n(E)\\) to positive elements of \\(M_n(F)\\), and completely positive if it is \\(n\\)-positive for all \\(n\\). For \\(E=A\\) and \\(F=B\\) this is Definition 3.1.\n\nFor a bounded map \\(\\varphi:A\\to B\\), write \\(\\varphi^\\sharp:B^*\\to A^*\\), \\((\\varphi^\\sharp g)(a)=g(\\varphi(a))\\), for its transpose (Banach-space adjoint).\n\n**Proposition 7.1.**\n\n1. (7.1) is a linear bijection of \\(M_n(A^*)\\) onto \\(M_n(A)^*\\), with \\(\\max_{i,j}\\|f_{ij}\\|\\le\\|f\\|\\le\\sum_{i,j}\\|f_{ij}\\|\\). The cone \\(M_n(A^*)_+\\) is weak\\(^*\\)-closed, hence norm-closed.\n2. \\(f\\in M_n(A^*)\\) is positive exactly when \\(\\sum_{i,j}f_{ij}(a_i^*a_j)\\ge0\\) for all \\(a_1,\\dots,a_n\\in A\\).\n3. If \\(f\\in M_n(A^*)_+\\) and \\(\\alpha\\in M_{n,m}(\\mathbb C)\\), then \\(\\alpha^*f\\alpha:=\\bigl[\\sum_{i,j}\\overline{\\alpha_{ip}}f_{ij}\\alpha_{jq}\\bigr]_{p,q}\\) lies in \\(M_m(A^*)_+\\).\n4. A composition of \\(n\\)-positive maps between such subspaces is \\(n\\)-positive; so a composition of CP maps is CP.\n5. A positive linear map from a C\\(^*\\)-algebra \\(A\\) into a subspace \\(F\\) of a C\\(^*\\)-algebra \\(B\\) or of a dual \\(B^*\\) is bounded.\n6. Let \\(\\varphi:A\\to B\\) be a bounded linear map of C\\(^*\\)-algebras. Under (7.1), \\((\\varphi^{(n)})^\\sharp=(\\varphi^\\sharp)^{(n)}\\). The map \\(\\varphi\\) is positive exactly when \\(\\varphi^\\sharp\\) is. Hence \\(\\varphi\\) and \\(\\varphi^\\sharp\\) are \\(n\\)-positive together, and CP together.\n7. Theorem 5.4(1) holds for maps into a subspace \\(F\\) of a dual \\(B^*\\): every \\(k\\)-positive map from \\(C_0(\\Omega,M_k)\\) into \\(F\\) is completely positive.\n\n**Proof.** (1) As a Banach space, \\(M_n(A)\\) is the direct sum of \\(n^2\\) copies of \\(A\\), with a norm equivalent to the largest entry norm (Proposition 2.1(2)). A functional on it is the sum of its restrictions to the copies, which is (7.1). Then \\(|\\langle a,f\\rangle|\\le\\sum\\|f_{ij}\\|\\,\\|a_{ij}\\|\\le(\\sum\\|f_{ij}\\|)\\|a\\|\\). The matrix with the single nonzero entry \\(x\\) at \\((i,j)\\) has norm \\(\\|x\\|\\) by Proposition 2.1(2), which gives \\(\\|f_{ij}\\|\\le\\|f\\|\\). The cone is the intersection of the weak\\(^*\\)-closed half-spaces \\(\\{f:\\langle a,f\\rangle\\ge0\\}\\), \\(a\\in M_n(A)_+\\).\n\n(2) This is Lemma 2.2(1): the matrices \\([a_i^*a_j]\\) generate \\(M_n(A)_+\\) as a convex cone.\n\n(3) For \\(a\\in A^m\\), put \\(b_i=\\sum_q\\alpha_{iq}a_q\\). Then \\(\\sum_{p,q}(\\alpha^*f\\alpha)_{pq}(a_p^*a_q)=\\sum_{i,j}f_{ij}(b_i^*b_j)\\ge0\\) by (2).\n\n(4) \\((\\psi\\circ\\varphi)^{(n)}=\\psi^{(n)}\\circ\\varphi^{(n)}\\).\n\n(5) For \\(F\\subseteq B\\) this is Proposition 3.2(4). Let \\(F\\subseteq B^*\\). If \\(0\\le f\\le g\\) in \\(B^*\\), then \\[\n\\begin{gathered}\n\\|f\\|\\\\\n\\le4\\sup\\{f(c):c\\in B_+,\\|c\\|\\le1\\}\\\\\n\\le4\\sup\\{g(c):c\\in B_+,\\|c\\|\\le1\\}\\\\\n\\le4\\|g\\|;\n\\end{gathered}\n\\] the first inequality is Proposition 3.2(4) for the positive map \\(f:B\\to\\mathbb C\\). Now repeat the proof of Proposition 3.2(4) with \\(\\|\\varphi(a_k)\\|\\ge8^k\\): from \\(\\varphi(a)\\ge2^{-k}\\varphi(a_k)\\ge0\\) we get \\(\\|\\varphi(a)\\|\\ge\\tfrac14\\,2^{-k}8^k=\\tfrac14 4^k\\) for every \\(k\\), which is absurd. The bound by four times the supremum over positive contractions follows as before.\n\n(6) For \\(x\\in M_n(A)\\) and \\(f\\in M_n(B^*)\\),\n\\[\n\\begin{gathered}\n\\langle\\varphi^{(n)}(x),f\\rangle\\\\\n=\\sum_{i,j}f_{ij}(\\varphi(x_{ij}))\\\\\n=\\sum_{i,j}(\\varphi^\\sharp f_{ij})(x_{ij})\\\\\n=\\langle x,(\\varphi^\\sharp)^{(n)}f\\rangle.\n\\end{gathered}\n\\]\nIf \\(\\varphi\\) is positive and \\(g\\in B^*_+\\), then \\((\\varphi^\\sharp g)(a)=g(\\varphi(a))\\ge0\\) for \\(a\\ge0\\). Conversely, let \\(\\varphi^\\sharp\\) be positive, let \\(a\\ge0\\), and write \\(\\varphi(a)=h+ik\\) with \\(h,k\\in B_h\\). For every positive \\(g\\), \\(g(\\varphi(a))=(\\varphi^\\sharp g)(a)\\ge0\\) is real, while \\(g(h)\\) and \\(g(k)\\) are real (B7); so \\(g(k)=0\\). Take \\(g=\\langle\\sigma(\\cdot)\\zeta,\\zeta\\rangle\\) for a faithful representation \\(\\sigma\\) of \\(B\\). Then \\(\\langle\\sigma(k)\\zeta,\\zeta\\rangle=0\\) for all \\(\\zeta\\), so \\(\\sigma(k)=0\\) and \\(k=0\\); and \\(\\langle\\sigma(h)\\zeta,\\zeta\\rangle\\ge0\\) for all \\(\\zeta\\), so \\(h\\ge0\\) by (P2) and (P1). Thus \\(\\varphi(a)\\ge0\\). Apply this to the bounded map \\(\\varphi^{(n)}\\) between the C\\(^*\\)-algebras \\(M_n(A)\\) and \\(M_n(B)\\): \\(\\varphi^{(n)}\\) is positive exactly when \\((\\varphi^{(n)})^\\sharp=(\\varphi^\\sharp)^{(n)}\\) is.\n\n(7) The proof of Theorem 5.4(1) uses three facts about the target: in each \\(M_m(F)\\) the positive cone is norm-closed, scalar compressions preserve positivity, and \\(\\varphi\\) is bounded. For a subspace of \\(B^*\\) these are (1), (3) and (5). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CM-10",
      "unit": "completely-positive-maps",
      "name": "The commutant of a GNS representation",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
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      "full_conditions_and_proof": "### The commutant of a GNS representation\n\nLet \\(\\omega\\) be a positive linear functional on \\(A\\), and let \\((\\pi,H,\\xi)\\) be its GNS representation (B8): \\(\\omega(a)=\\langle\\pi(a)\\xi,\\xi\\rangle\\), and \\(\\pi(A)\\xi\\) is dense in \\(H\\). Let \\(C_\\omega^+\\) be the set of \\(f\\in A^*_+\\) with \\(f\\le\\alpha\\omega\\) for some \\(\\alpha\\ge0\\), and \\(C_\\omega\\) its linear span. Define\n\\[\n\\begin{gathered}\n\\theta_\\omega(x)(a)=\\langle\\pi(a)x\\xi,\\xi\\rangle, \\\\\nx\\in\\pi(A)',\\quad a\\in A.\n\\end{gathered}\n\\tag{7.2}\n\\]\n\n**Lemma 7.2** (Few orthogonal projections force finite dimension). Let \\(M\\) be a von Neumann algebra in which every family of pairwise orthogonal nonzero projections has at most \\(N\\) members. Then \\(\\dim M\\le N^2\\).\n\n**Proof.** If \\(M=\\{0\\}\\) there is nothing to prove. Choose a family \\(q_1,\\dots,q_r\\) of pairwise orthogonal nonzero projections in \\(M\\) with \\(r\\) as large as possible; \\(r\\le N\\). Then \\(\\sum_iq_i=1\\), since otherwise \\(1-\\sum_iq_i\\) could be added. Each \\(q=q_i\\) is minimal: a projection \\(e\\in M\\) with \\(0\\ne e\\ne q\\) and \\(e\\le q\\) would split \\(q\\) into \\(e\\) and \\(q-e\\).\n\n*Claim: \\(qMq=\\mathbb Cq\\).* \\(qMq\\) is a C\\(^*\\)-algebra with unit \\(q\\). Let \\(y\\in qMq\\) be self-adjoint, and suppose its spectrum in \\(qMq\\) contains two points \\(\\lambda\\ne\\mu\\). Choose continuous \\(f,g\\ge0\\) on the spectrum with disjoint supports and \\(f(\\lambda)=g(\\mu)=1\\). Then \\(f(y),g(y)\\in qMq\\) are nonzero, positive, and \\(f(y)g(y)=0\\). Let \\(e_f\\) and \\(e_g\\) be the projections onto the closures of their ranges. They are orthogonal, because the range of \\(g(y)\\) lies in the kernel of \\(f(y)\\). They lie below \\(q\\), because \\(f(y)=qf(y)\\) and \\(g(y)=qg(y)\\). They lie in \\(M\\): if \\(x'\\in M'\\), then \\(x'\\) commutes with \\(f(y)\\), so it maps the range of \\(f(y)\\) into itself, and \\(x'e_f=e_fx'e_f\\); the same for \\(x'^*\\), and taking adjoints gives \\(e_fx'=e_fx'e_f\\). So \\(x'e_f=e_fx'\\) for every \\(x'\\in M'\\), and \\(e_f\\in M''=M\\). So \\(e_f\\) is a projection in \\(M\\) with \\(0\\ne e_f\\le q-e_g<q\\), which contradicts minimality. Hence the spectrum of \\(y\\) is a single point \\(\\lambda\\), and \\(y=\\lambda q\\), since the norm of the self-adjoint element \\(y-\\lambda q\\) equals its spectral radius (B4). Writing a general element as \\(h+ik\\) proves the claim.\n\n*Claim: \\(\\dim q_iMq_j\\le1\\).* Let \\(x,y\\in q_iMq_j\\) with \\(x\\ne0\\). By the first claim, \\(x^*x=cq_j\\) with \\(c=\\|x\\|^2>0\\), and \\(yx^*=\\lambda q_i\\) for some \\(\\lambda\\). Then \\(cy=yx^*x=\\lambda q_ix=\\lambda x\\), so \\(y\\in\\mathbb Cx\\).\n\nEvery \\(x\\in M\\) equals \\(\\sum_{i,j}q_ixq_j\\), so \\(\\dim M\\le r^2\\le N^2\\). \\(\\square\\)\n\n**Theorem 7.3.**\n\n1. \\(\\theta_\\omega\\) is an injective CP map of \\(\\pi(A)'\\) onto \\(C_\\omega\\). It maps \\(\\pi(A)'_+\\) onto \\(C_\\omega^+\\), \\(C_\\omega\\cap A^*_+=C_\\omega^+\\), and \\(\\theta_\\omega(1)=\\omega\\). The inverse \\(\\theta_\\omega^{-1}:C_\\omega\\to\\pi(A)'\\) is CP, with \\(C_\\omega\\subseteq A^*\\) ordered as in Proposition 7.1.\n2. \\(\\|\\theta_\\omega(x)\\|\\le\\|\\omega\\|\\,\\|x\\|\\). In particular \\(\\theta_\\omega\\) is contractive when \\(\\omega\\) is a state.\n3. The following are equivalent: (a) \\(\\theta_\\omega^{-1}\\) is bounded for the norm of \\(A^*\\); (b) \\(C_\\omega\\) is norm-closed in \\(A^*\\); (c) \\(\\pi(A)'\\) is finite-dimensional.\n4. Let \\(B\\) be a unital C\\(^*\\)-algebra and \\(\\varphi:B\\to A^*\\) positive with \\(\\varphi(1)\\le\\alpha\\omega\\). Then \\(\\varphi(B)\\subseteq C_\\omega\\), and \\(\\psi=\\theta_\\omega^{-1}\\circ\\varphi:B\\to\\pi(A)'\\) is positive, hence bounded. If \\(\\varphi\\) is \\(n\\)-positive (CP), so is \\(\\psi\\). If \\(\\varphi\\) is 2-positive, then \\(\\|\\psi\\|=\\|\\psi(1)\\|\\le\\alpha\\), and \\(\\psi\\) is unital when \\(\\varphi(1)=\\omega\\).\n\nPart (3) characterizes boundedness of \\(\\theta_\\omega^{-1}\\), and (4) needs only positivity and domination by a multiple of \\(\\omega\\); complete positivity and equality at the identity are not needed for its factorization.\n\n**Proof.** (1) *Positivity and range.* For \\(x\\in\\pi(A)'_+\\), its square root \\(x^{1/2}\\) lies in the C\\(^*\\)-algebra \\(\\pi(A)'\\) and commutes with \\(\\pi(A)\\). So\n\\[\n\\theta_\\omega(x)(a^*a)=\\langle\\pi(a)^*\\pi(a)x\\xi,\\xi\\rangle=\\|x^{1/2}\\pi(a)\\xi\\|^2,\n\\]\nwhich lies between \\(0\\) and \\(\\|x\\|\\,\\omega(a^*a)\\). Hence \\(0\\le\\theta_\\omega(x)\\le\\|x\\|\\omega\\), and \\(\\theta_\\omega(x)\\in C_\\omega^+\\). Every element of \\(\\pi(A)'\\) is a combination of four positive ones, so \\(\\theta_\\omega(\\pi(A)')\\subseteq C_\\omega\\).\n\n*Onto \\(C_\\omega^+\\).* Let \\(f\\in A^*_+\\) with \\(f\\le\\alpha\\omega\\). By the Cauchy–Schwarz inequality, \\[\n\\begin{gathered}\n|f(b^*a)|^2\\\\\n\\le f(a^*a)f(b^*b)\\\\\n\\le\\alpha^2\\|\\pi(a)\\xi\\|^2\\|\\pi(b)\\xi\\|^2.\n\\end{gathered}\n\\] So \\(B_f(\\pi(a)\\xi,\\pi(b)\\xi)=f(b^*a)\\) is a well-defined bounded positive sesquilinear form on the dense subspace \\(\\pi(A)\\xi\\). It extends to \\(H\\), and there is \\(h\\in B(H)\\) with \\(0\\le h\\le\\alpha1\\) and \\(f(b^*a)=\\langle h\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\) (B10). For \\(c\\in A\\), \\[\n\\begin{gathered}\n\\langle h\\pi(c)\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\\\\n=f((c^*b)^*a)\\\\\n=\\langle h\\pi(a)\\xi,\\pi(c^*b)\\xi\\rangle\\\\\n=\\langle\\pi(c)h\\pi(a)\\xi,\\pi(b)\\xi\\rangle,\n\\end{gathered}\n\\] so \\(h\\in\\pi(A)'\\). Since \\(\\pi\\) is nondegenerate, \\(\\pi(u_i)\\xi\\to\\xi\\), and \\[\n\\begin{gathered}\nf(a)\\\\\n=\\lim_if(u_ia)\\\\\n=\\lim_i\\langle h\\pi(a)\\xi,\\pi(u_i)\\xi\\rangle\\\\\n=\\langle\\pi(a)h\\xi,\\xi\\rangle\\\\\n=\\theta_\\omega(h)(a).\n\\end{gathered}\n\\] As \\(C_\\omega\\) is spanned by \\(C_\\omega^+\\), \\(\\theta_\\omega\\) maps onto \\(C_\\omega\\).\n\n*Injectivity and the cones.* \\(\\theta_\\omega(x)(b^*a)=\\langle x\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\). If \\(\\theta_\\omega(x)=0\\), these numbers vanish on a dense set, so \\(x=0\\). If \\(\\theta_\\omega(x)\\ge0\\), then \\(\\langle x\\pi(a)\\xi,\\pi(a)\\xi\\rangle\\ge0\\) for all \\(a\\), so \\(x\\ge0\\) by (P2), and \\(\\theta_\\omega(x)\\le\\|x\\|\\omega\\). This gives \\(C_\\omega\\cap A^*_+=C_\\omega^+\\). Clearly \\(\\theta_\\omega(1)=\\omega\\).\n\n*\\(\\theta_\\omega\\) is CP.* The matrices \\([x_p^*x_q]\\) generate the positive cone of \\(M_n(\\pi(A)')\\) (Lemma 2.2(1)), and \\(f\\in M_n(A^*)\\) is positive exactly when \\(\\sum_{p,q}f_{pq}(a_p^*a_q)\\ge0\\) for all \\(a_1,\\dots,a_n\\in A\\) (Proposition 7.1(2)). So it suffices to check, for \\(x_1,\\dots,x_n\\in\\pi(A)'\\) and \\(a_1,\\dots,a_n\\in A\\),\n\\[\n\\begin{gathered}\n\\sum_{p,q}\\theta_\\omega(x_p^*x_q)(a_p^*a_q) \\\\\n=\\sum_{p,q}\\langle\\pi(a_p)^*x_p^*x_q\\pi(a_q)\\xi,\\xi\\rangle \\\\\n=\\Bigl\\|\\sum_qx_q\\pi(a_q)\\xi\\Bigr\\|^2\\ge0.\n\\end{gathered}\n\\]\n\n*\\(\\theta_\\omega^{-1}\\) is CP.* Let \\(f\\in M_n(C_\\omega)\\) be positive in \\(M_n(A^*)\\), and \\(X=[\\theta_\\omega^{-1}(f_{pq})]\\). For \\(\\zeta=(\\pi(a_1)\\xi,\\dots,\\pi(a_n)\\xi)\\), since each \\(\\theta_\\omega^{-1}(f_{pq})\\) commutes with \\(\\pi(A)\\),\n\\[\n\\begin{gathered}\n\\sum_{p,q}\\langle\\theta_\\omega^{-1}(f_{pq})\\pi(a_q)\\xi,\\pi(a_p)\\xi\\rangle \\\\\n=\\sum_{p,q}f_{pq}(a_p^*a_q)\\ge0.\n\\end{gathered}\n\\]\nSuch \\(\\zeta\\) are dense in \\(H^n\\). Positivity in \\(M_n(B(H))\\) is tested by this quadratic form (Proposition 2.1(5)), so \\(X\\ge0\\) in \\(M_n(B(H))\\), and hence in the C\\(^*\\)-subalgebra \\(M_n(\\pi(A)')\\) (Proposition 2.1(4)).\n\n(2) \\(|\\theta_\\omega(x)(a)|\\le\\|a\\|\\,\\|x\\|\\,\\|\\xi\\|^2\\), and \\(\\|\\xi\\|^2=\\lim_i\\langle\\pi(u_i)\\xi,\\xi\\rangle=\\lim_i\\omega(u_i)\\le\\|\\omega\\|\\).\n\n(3) (a)\\(\\Leftrightarrow\\)(b): \\(\\theta_\\omega\\) is a bounded bijection of the Banach space \\(\\pi(A)'\\) onto \\(C_\\omega\\). If \\(C_\\omega\\) is closed, the open mapping theorem makes \\(\\theta_\\omega^{-1}\\) bounded. If \\(\\theta_\\omega^{-1}\\) is bounded, \\(C_\\omega\\) is isomorphic to a Banach space, hence complete, hence closed. (c)\\(\\Rightarrow\\)(a): \\(C_\\omega\\) is then finite-dimensional, so [Theorem 7.1(3) of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-07) makes its linear map \\(\\theta_\\omega^{-1}\\) bounded. (a)\\(\\Rightarrow\\)(c): let \\(\\|\\theta_\\omega^{-1}(f)\\|\\le c\\|f\\|\\) on \\(C_\\omega\\). For a projection \\(p\\in\\pi(A)'\\), \\(\\theta_\\omega(p)(a)=\\langle\\pi(a)p\\xi,p\\xi\\rangle\\), so \\(\\|\\theta_\\omega(p)\\|\\le\\|p\\xi\\|^2\\). If \\(p\\ne0\\), then \\(1=\\|p\\|\\le c\\|p\\xi\\|^2\\). For pairwise orthogonal nonzero projections \\(p_1,\\dots,p_m\\) in \\(\\pi(A)'\\) this gives \\(m\\le c\\sum_l\\|p_l\\xi\\|^2\\le c\\|\\xi\\|^2\\). By Lemma 7.2, \\(\\pi(A)'\\) is finite-dimensional.\n\n(4) For \\(b\\in B_+\\), \\(0\\le b\\le\\|b\\|1\\), so \\(0\\le\\varphi(b)\\le\\|b\\|\\alpha\\omega\\), and \\(\\varphi(b)\\in C_\\omega^+\\). \\(B\\) is spanned by \\(B_+\\), so \\(\\varphi(B)\\subseteq C_\\omega\\). By (1), \\(\\theta_\\omega^{-1}\\) maps \\(C_\\omega^+\\) into \\(\\pi(A)'_+\\), so \\(\\psi\\) is positive, and bounded by Proposition 3.2(4). If \\(\\varphi\\) is \\(n\\)-positive, then \\(\\psi\\) is \\(n\\)-positive by (1) and Proposition 7.1(4). If \\(\\varphi\\) is 2-positive, Theorem 4.1(2) gives \\(\\|\\psi\\|=\\|\\psi(1)\\|\\). Now \\(\\alpha\\omega-\\varphi(1)\\) lies in \\(C_\\omega\\cap A^*_+=C_\\omega^+\\), so \\(\\alpha1-\\psi(1)=\\theta_\\omega^{-1}(\\alpha\\omega-\\varphi(1))\\ge0\\), and \\(\\|\\psi(1)\\|\\le\\alpha\\) by (P2). If \\(\\varphi(1)=\\omega\\), then \\(\\psi(1)=\\theta_\\omega^{-1}(\\omega)=1\\). \\(\\square\\)\n\n**Example 7.4** (An unbounded inverse). Let \\(A=C[0,1]\\) and \\(\\omega(a)=\\int_0^1a(t)\\,dt\\). The GNS space is \\(L^2[0,1]\\), \\(\\pi(a)\\) is multiplication by \\(a\\), and \\(\\xi=1\\). Multiplication by the indicator of an interval commutes with \\(\\pi(A)\\). The indicators of \\(I_k=[2^{-k},2^{-k+1})\\), \\(k\\ge1\\), give infinitely many pairwise orthogonal nonzero projections in \\(\\pi(A)'\\), so \\(\\theta_\\omega^{-1}\\) is unbounded by Theorem 7.3(3). Concretely, \\(\\theta_\\omega\\) sends multiplication by \\(1_{I_k}\\), an operator of norm \\(1\\), to the functional \\(a\\mapsto\\int_{I_k}a\\,dt\\), of norm \\(2^{-k}\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-CM-11",
      "unit": "completely-positive-maps",
      "name": "Transposes and the dual of \\(M_d(\\mathbb C)\\)",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "source": "src/completely-positive-maps.md",
      "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "anchor": "oa-fnd-cm-11",
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        "line": 709,
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      },
      "full_conditions_and_proof": "### Transposes and the dual of \\(M_d(\\mathbb C)\\)\n\nLet \\(d\\ge2\\), \\(A=M_d(\\mathbb C)\\), and \\(\\tau=d^{-1}\\mathrm{Tr}\\). Define \\(\\Phi_1,\\Phi_2:A\\to A^*\\) by\n\\[\n\\begin{gathered}\n\\Phi_1(a)(x)=\\sum_{i,j}a_{ij}x_{ij} \\\\\n=\\mathrm{Tr}(a^{\\mathsf T}x), \\\\\n\\Phi_2(a)(x)=\\sum_{i,j}a_{ji}x_{ij} \\\\\n=\\mathrm{Tr}(ax).\n\\end{gathered}\n\\tag{7.3}\n\\]\n\n**Proposition 7.5.**\n\n1. \\(\\Phi_1\\) and \\(\\Phi_2\\) are positive linear bijections of \\(A\\) onto \\(A^*\\).\n2. \\(\\Phi_1\\) is completely positive, and so is \\(\\Phi_1^{-1}\\).\n3. \\(\\Phi_2\\) is not 2-positive.\n4. Under the identification (7.1) for the algebra \\(\\mathbb C\\), a functional \\(f\\) on \\(M_d(\\mathbb C)\\) corresponds to the matrix \\([f(e_{ij})]\\), because \\(f(x)=\\sum_{i,j}f(e_{ij})x_{ij}\\). Then \\(\\Phi_1(a)\\) corresponds to \\([a_{ij}]\\) and \\(\\Phi_2(a)\\) to \\([a_{ji}]\\). Under the trace pairing, \\(f\\) corresponds instead to the matrix \\(D_f\\) with \\(f(x)=\\mathrm{Tr}(D_fx)\\); then \\(\\Phi_2(a)\\) corresponds to \\(a\\) and \\(\\Phi_1(a)\\) to \\(a^{\\mathsf T}\\).\n\nThe explicit computations in (2)–(4) show how switching between the entrywise pairing (7.1) and the trace pairing introduces a transpose.\n\n**Proof.** For finite matrices, \\(\\mathrm{Tr}(cd)=\\sum_{i,j}c_{ij}d_{ji}=\\mathrm{Tr}(dc)\\); this proves every cyclic trace identity used below. (1) If \\(a,x\\ge0\\), then \\(\\mathrm{Tr}(ax)=\\mathrm{Tr}(a^{1/2}xa^{1/2})\\ge0\\), and \\(a^{\\mathsf T}\\ge0\\) (Proposition 3.2(3)), so both maps are positive. If \\(\\mathrm{Tr}(ax)=0\\) for all \\(x\\), then \\(a=0\\); so both maps are injective, and they are onto by dimension.\n\n(2) The GNS representation of \\(\\tau\\) acts on \\(H_\\tau=M_d(\\mathbb C)\\) with \\(\\langle x,y\\rangle=\\tau(y^*x)\\), by \\(\\pi(a)x=ax\\), with \\(\\xi=1\\). An operator \\(T\\) that commutes with every \\(\\pi(a)\\) satisfies \\(T(x)=T(x1)=xT(1)\\), so \\(\\pi(A)'=\\{R_c:c\\in A\\}\\), where \\(R_cx=xc\\). With respect to \\(\\langle\\cdot,\\cdot\\rangle\\), \\(R_c^*=R_{c^*}\\), since \\(\\tau(y^*xc)=\\tau((yc^*)^*x)\\). The map \\(j(a)=R_{a^{\\mathsf T}}\\) is a \\(*\\)-isomorphism of \\(A\\) onto \\(\\pi(A)'\\): it is linear and bijective, \\(R_{a^{\\mathsf T}}R_{b^{\\mathsf T}}x=xb^{\\mathsf T}a^{\\mathsf T}=R_{(ab)^{\\mathsf T}}x\\), and \\(R_{a^{\\mathsf T}}^*=R_{(a^*)^{\\mathsf T}}\\). Next, by (7.2), \\(\\theta_\\tau(R_c)(x)=\\langle\\pi(x)R_c1,1\\rangle=\\tau(xc)\\), so\n\\[\n\\begin{gathered}\n\\Phi_1(a)(x)=\\mathrm{Tr}(xa^{\\mathsf T}) \\\\\n=d\\,\\theta_\\tau(j(a))(x), \\\\\n\\text{that is, }\\Phi_1=d\\,\\theta_\\tau\\circ j.\n\\end{gathered}\n\\]\nEvery \\(f\\in A^*_+\\) has the form \\(\\mathrm{Tr}(D\\,\\cdot)\\) with \\(D=[f(e_{ji})]_{i,j}\\): the matrix-unit expansion proves the formula, and \\(v^*Dv=f(vv^*)\\ge0\\) for every column \\(v\\), so \\(D\\ge0\\). Consequently \\(f\\le d\\|D\\|\\tau\\); hence \\(C_\\tau=A^*\\). The maps \\(\\theta_\\tau\\) and \\(\\theta_\\tau^{-1}\\) are CP (Theorem 7.3(1)), the \\(*\\)-isomorphisms \\(j\\) and \\(j^{-1}\\) are CP (Proposition 3.2(3)), and compositions of CP maps are CP (Proposition 7.1(4)). So \\(\\Phi_1\\) and \\(\\Phi_1^{-1}=j^{-1}\\circ\\theta_\\tau^{-1}\\circ d^{-1}\\) are CP.\n\n(3) The matrix \\(X=[e_{pq}]_{p,q=1,2}\\in M_2(A)\\) is positive (Lemma 2.2(1)). Test \\(\\Phi_2^{(2)}(X)\\) with Proposition 7.1(2), using \\(a_1=e_{12}\\), \\(a_2=-e_{11}\\) and \\(\\mathrm{Tr}(e_{pq}y)=y_{qp}\\):\n\\[\n\\begin{gathered}\n\\sum_{p,q=1}^2\\Phi_2(e_{pq})(a_p^*a_q) \\\\\n=\\sum_{p,q}(a_p^*a_q)_{qp} \\\\\n=(e_{22})_{11}+(-e_{21})_{21} \\\\\n+(-e_{12})_{12}+(e_{11})_{22} \\\\\n=-2<0.\n\\end{gathered}\n\\]\n\n(4) This is bookkeeping: \\(\\Phi_1(a)(e_{ij})=a_{ij}\\) and \\(\\Phi_2(a)(e_{ij})=a_{ji}\\); and \\(\\Phi_2(a)=\\mathrm{Tr}(a\\,\\cdot)\\), \\(\\Phi_1(a)=\\mathrm{Tr}(a^{\\mathsf T}\\cdot)\\). \\(\\square\\)\n\nSo whether the natural map from \\(M_d(\\mathbb C)\\) to its dual is completely positive depends on how the dual is identified with \\(M_d(\\mathbb C)\\), and the transpose is what separates the two choices. The proof of (2) shows where the transpose comes from: the commutant of the trace representation acts by right multiplication, which reverses products.\n\n## Where this leads\n\nThe following four extension results are further directions, not proved here and unused in the proofs of this lesson.\n\n- Theorem 5.1(3) has a counterpart for domains: an \\(n\\)-positive map from a C\\(^*\\)-algebra whose irreducible representations all have dimension at most \\(n\\) is completely positive. Theorem 5.4 is the case of the domains \\(C_0(\\Omega,M_k)\\).\n- Conversely to Theorem 5.1(3), if every \\(n\\)-positive map from every C\\(^*\\)-algebra into \\(B\\) is completely positive, then every irreducible representation of \\(B\\) has dimension at most \\(n\\).\n- A unital linear map between unital C\\(^*\\)-algebras, with nonzero target, is positive if and only if it is contractive [Blackadar, II.6.9.4]. So every positive unital map has norm \\(1=\\|\\varphi(1)\\|\\), while Theorem 4.1(2) gives \\(\\|\\varphi\\|=\\|\\varphi(1)\\|\\) for 2-positive maps that need not be unital.\n- Completely positive maps into \\(B(H)\\) extend from a C\\(^*\\)-subalgebra, and even from an operator system, to the whole algebra with the same norm [Blackadar, II.6.9.12]. This extension theorem leads to the theory of injective C\\(^*\\)-algebras.\n- Tensor products of completely positive maps, and normal completely positive maps between von Neumann algebras, are the next steps. They are used for tensor products of C\\(^*\\)-algebras and for conditional expectations.\n\n## References\n\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [Kristin Courtney; Elizabeth Gillaspy; Lara Ismert, *Notes on C*-algebras: Notes and Exercises for GOALS*, §10](https://www.ipam.ucla.edu/wp-content/uploads/2024/07/Notes_and_Exercises_for_GOALS.pdf) gives a route through matrix positivity, Stinespring dilation and multiplicative domains; omitted extension and nonunital steps are supplied here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-WA-01",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "1. Functionals as vector coefficients",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
      "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "anchor": "oa-fnd-wa-01",
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      },
      "full_conditions_and_proof": "## 1. Functionals as vector coefficients\n\nEvery bounded functional on a \\(C^*\\)-algebra is a coefficient of a representation. This is the key step in Section 3, where the bidual is identified with the von Neumann algebra of a single representation.\n\nFor a possibly degenerate representation \\(\\pi\\) of \\(A\\) on \\(H\\) and \\(\\xi,\\eta\\in H\\) write\n\\[\n\\begin{gathered}\n\\omega_{\\pi;\\xi,\\eta}(a)\\\\\n=\\langle\\pi(a)\\xi,\\eta\\rangle\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{1.1}\n\\]\nA functional \\(f\\) is *hermitian* when \\(f(a^*)=\\overline{f(a)}\\).\n\n**Proposition 1.1.** Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. Every hermitian \\(f\\in A^*\\) is a difference \\(f=f_+-f_-\\) of positive functionals with \\(\\|f\\|=\\|f_+\\|+\\|f_-\\|\\).\n2. Every \\(f\\in A^*\\) has the form \\(f=\\omega_{\\pi_\\psi;\\xi_\\psi,\\eta}\\), where \\(\\psi\\in A^*_+\\), \\((\\pi_\\psi,H_\\psi,\\xi_\\psi)\\) is its cyclic representation and \\(\\eta\\in H_\\psi\\). One can arrange \\(\\|\\xi_\\psi\\|\\,\\|\\eta\\|\\le4\\|f\\|\\).\n\n**Proof.** (1) If \\(A=0\\) or \\(f=0\\), take both parts zero. For a hermitian \\(f\\), its restriction \\(g\\) to the real Banach space \\(E=A_h\\) has the same norm as \\(f\\). Indeed, for any \\(a\\) choose a scalar \\(\\lambda\\) of modulus one with \\(f(\\lambda a)=|f(a)|\\). Then \\(h=(\\lambda a+(\\lambda a)^*)/2\\) satisfies \\(\\|h\\|\\le\\|a\\|\\) and \\(g(h)=|f(a)|\\).\n\nFirst assume \\(A\\) nonzero and unital. Its state set \\(S\\) is weak-star compact: it is the intersection of the closed dual unit ball, the closed positivity conditions and \\(\\rho(1)=1\\). In \\(E^*\\) put\n\\[\n\\begin{gathered}\nK\\\\\n=\\{\\alpha\\rho-\\beta\\sigma:\n\\\\\n\\rho,\\sigma\\in S,\\quad \\alpha,\\beta\\ge0,\\quad \\\\\n\\alpha+\\beta\\le1\\}.\n\\end{gathered}\n\\]\nThis is compact, since it is the continuous image of a compact finite product, and it is convex: positive combinations of states can be normalized by their total coefficient whenever that coefficient is nonzero. It is symmetric and lies in the unit ball of \\(E^*\\). Conversely, the state norming result above gives \\(\\sup_{k\\in K}k(h)=\\|h\\|\\) for every \\(h\\in E\\). If a functional of norm at most one were outside \\(K\\), strict separation of a point from a compact convex set in the weak-star topology would produce \\(h\\in E\\) with \\(g(h)>\\sup_{k\\in K}k(h)=\\|h\\|\\), a contradiction. Here weak-star continuous real functionals are evaluations, by [Theorem 1.2 of weak topologies](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-01). Thus \\(K\\) is the whole real dual unit ball. Applied to \\(g/\\|f\\|\\), this gives positive \\(f_+,f_-\\) with \\(f=f_+-f_-\\) and \\(\\|f_+\\|+\\|f_-\\|\\le\\|f\\|\\); the triangle inequality forces equality.\n\nFor nonunital \\(A\\), extend \\(g\\) by real Hahn–Banach to the self-adjoint part of its \\(C^*\\)-unitization, with unchanged norm. Complexify by \\(F(h+ik)=G(h)+iG(k)\\). This is a hermitian complex-linear extension of \\(f\\), and the phase argument of the first paragraph proves \\(\\|F\\|=\\|G\\|=\\|f\\|\\). Apply the unital result and restrict its two positive parts to \\(A\\). Their norms sum to at most \\(\\|f\\|\\), and again the triangle inequality gives equality. This proves existence without presupposing a product on the bidual.\n\n(2) Write \\(f=h_1+ih_2\\) with the hermitian functionals \\(h_1(a)=\\tfrac12\\big(f(a)+\\overline{f(a^*)}\\big)\\) and \\(h_2(a)=\\tfrac1{2i}\\big(f(a)-\\overline{f(a^*)}\\big)\\); each has norm at most \\(\\|f\\|\\). Applying (1) to \\(h_1\\) and to \\(h_2\\), we can write \\(f=\\sum_{k=0}^3i^kf_k\\) with \\(f_k\\in A^*_+\\) and \\(\\sum_k\\|f_k\\|\\le2\\|f\\|\\). Put \\(\\psi=\\sum_kf_k\\) and \\((\\pi,H,\\xi)=(\\pi_\\psi,H_\\psi,\\xi_\\psi)\\). For each \\(k\\) and \\(a\\in A\\), the Cauchy–Schwarz estimate for \\(f_k\\) and the inequality \\(f_k\\le\\psi\\) give\n\\[\n\\begin{gathered}\n|f_k(a)|^2\\\\\n\\le\\|f_k\\|\\,f_k(a^*a)\\\\\n\\le\\|f_k\\|\\,\\psi(a^*a)\\\\\n=\\|f_k\\|\\,\\|\\pi(a)\\xi\\|^2 .\n\\end{gathered}\n\\]\nSo \\(\\pi(a)\\xi\\mapsto f_k(a)\\) is a well-defined linear functional on the dense subspace \\(\\pi(A)\\xi\\), of norm at most \\(\\|f_k\\|^{1/2}\\). It extends to \\(H\\), and the Riesz theorem gives \\(\\eta_k\\in H\\) with \\(f_k(a)=\\langle\\pi(a)\\xi,\\eta_k\\rangle\\) and \\(\\|\\eta_k\\|\\le\\|f_k\\|^{1/2}\\). Put \\(\\eta=\\sum_k\\overline{i^k}\\,\\eta_k\\). Then \\(f=\\omega_{\\pi;\\xi,\\eta}\\). Finally \\(\\|\\xi\\|^2=\\|\\psi\\|\\le2\\|f\\|\\) and \\[\n\\begin{gathered}\n\\|\\eta\\|\\\\\n\\le\\sum_k\\|f_k\\|^{1/2}\\\\\n\\le2\\big(\\sum_k\\|f_k\\|\\big)^{1/2}\\\\\n\\le2(2\\|f\\|)^{1/2},\n\\end{gathered}\n\\] so \\(\\|\\xi\\|\\,\\|\\eta\\|\\le4\\|f\\|\\). \\(\\square\\)\n\n**Remark 1.2.** If \\(\\psi\\ne0\\), then \\(\\pi_\\psi\\) is unitarily equivalent to \\(\\pi_{\\psi/\\|\\psi\\|}\\): the same space, with cyclic vector \\(\\|\\psi\\|^{-1/2}\\xi_\\psi\\). Hence every \\(f\\in A^*\\) is a vector coefficient of the universal representation of [Section 3](#oa-fnd-wa-03), and even of the direct sum of the cyclic representations of the states. The decomposition in (1) is unique; see [Proposition 3.4](#oa-fnd-wa-04).\n\n### Constructing the bidual before using it\n\nFor \\(A\\ne0\\), let \\(\\pi_S\\) be the direct sum of its state GNS representations, and \\(M=\\pi_S(A)''\\). Here the Hilbert direct sum consists of families \\((\\xi_\\rho)_{\\rho\\in S}\\) with\n\\[\n\\|\\xi\\|^2=\\sup_{F\\subseteq S,\\ F\\ {\\rm finite}}\\sum_{\\rho\\in F}\\|\\xi_\\rho\\|^2<\\infty.\n\\]\nIts inner product is the sum of component inner products, absolutely convergent by finite-sum Cauchy–Schwarz. Finite-support vectors are dense by the definition of that supremum. For completeness, a Cauchy sequence has a limit in each complete component; passing its Cauchy estimates to every finite component sum puts the resulting family in the direct sum and proves norm convergence to it. The componentwise operator \\(\\bigoplus_\\rho\\pi_\\rho(a)\\) has norm at most \\(\\|a\\|\\), first on finite-support vectors and then by density. Products and adjoints act componentwise, so it is a representation. Nondegeneracy follows from nondegeneracy on each summand and finite-support density. It is faithful: if \\(a\\ne0\\), the state order test supplies a state nonzero on \\(a^*a\\), whose cyclic representation does not kill \\(a\\). It is therefore isometric. Let\n\\[\nR:M_*\\longrightarrow A^*,\\qquad R(\\varphi)=\\varphi\\circ\\pi_S.\n\\]\nBy positive and ordinary Kaplansky density, the unit ball of \\(\\pi_S(A)\\) is strongly dense, hence sigma-weakly dense, in the unit ball of \\(M\\). It follows that\n\\[\n\\begin{gathered}\n\\|R\\varphi\\|\\\\\n=\\sup_{\\|a\\|\\le1}|\\varphi(\\pi_S(a))|\n\\\\\n=\\sup_{x\\in M,\\ \\|x\\|\\le1}|\\varphi(x)|\\\\\n=\\|\\varphi\\|.\n\\end{gathered}\n\\]\nThe map \\(R\\) is onto. By Proposition 1.1(2) every functional is a vector coefficient of one positive-functional GNS representation. After normalizing that functional, Remark 1.2 places that coefficient in a state summand of \\(\\pi_S\\); vector coefficients are normal on \\(M\\). Consequently \\(R\\) is a surjective isometry. The concrete predual proof gives \\(M=(M_*)^*\\). Taking adjoints,\n\\[\nU=R^*:A^{**}\\longrightarrow M\n\\]\nis a surjective isometry and a homeomorphism for the two weak-star topologies; its inverse is \\((R^{-1})^*\\). For \\(a\\in A\\), pairing with every \\(\\varphi\\in M_*\\) shows \\(U(j(a))=\\pi_S(a)\\). Transport the product and involution of \\(M\\) through \\(U\\). They make \\(A^{**}\\) a von Neumann algebra, extend those of \\(A\\), and have the stated separate weak-star continuity. This structure is unique with these properties, because \\(j(A)\\) is weak-star dense and two successive limits, one variable at a time, determine every product.\n\nThe predual identification is the original evaluation pairing: \\(f=R\\varphi\\) has extension \\(\\hat f(X)=X(f)=\\varphi(U(X))\\), with norm \\(\\|f\\|\\). If \\(f\\ge0\\), approximate any positive contraction of \\(M\\) sigma-weakly by positive contractions of \\(\\pi_S(A)\\); the corresponding \\(a\\)'s are positive because \\(\\pi_S\\) reflects order. Their nonnegative values show \\(\\varphi\\ge0\\). Thus positive functionals extend positively. The density of the unit ball of \\(j(A)\\) follows from Kaplansky density, or Goldstine. For \\(A=0\\) all these spaces and maps are zero and the construction has the same meaning. No normal-order converse, normal-isomorphism theorem or extension-to-bidual theorem was used in this construction.\n\n",
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      "id": "OA-FND-WA-02",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "2. Extending a representation to the bidual",
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      "full_conditions_and_proof": "## 2. Extending a representation to the bidual\n\nA representation of \\(A\\) extends in exactly one way to a normal representation of \\(A^{**}\\). We allow degenerate representations, since they occur naturally; for instance, the restriction of a representation to a \\(C^*\\)-subalgebra is often degenerate. The price is that \\(\\pi(A)''\\) must be replaced by the \\(\\sigma\\)-weak closure of \\(\\pi(A)\\).\n\n**Lemma 2.1.** Let \\(\\pi:A\\to B(H)\\) be a possibly degenerate representation. Let \\(q\\) be the projection onto the essential space \\(H_0\\), the closed span of \\(\\pi(A)H\\), and let \\(N(\\pi)\\) be the \\(\\sigma\\)-weak closure of \\(\\pi(A)\\).\n\n1. \\(q\\) commutes with \\(\\pi(A)\\), \\(\\pi(a)=q\\pi(a)q\\), and \\(\\pi(e_i)\\to q\\) strongly for every approximate unit \\((e_i)\\) of \\(A\\). \\(N(\\pi)\\) acts on \\(H_0\\) as the von Neumann algebra \\((\\pi(A)|_{H_0})''\\) and is zero on \\(H_0^\\perp\\). Moreover\n\\[\n\\pi(A)''=N(\\pi)+\\mathbb C(1-q),\n\\tag{2.1}\n\\]\nand the sum is direct when \\(q\\neq1\\). So \\(N(\\pi)=\\pi(A)''\\) exactly when \\(\\pi\\) is nondegenerate.\n2. There is exactly one linear map \\(\\bar\\pi:A^{**}\\to B(H)\\) that is continuous from \\(\\sigma(A^{**},A^*)\\) to the \\(\\sigma\\)-weak topology and satisfies \\(\\bar\\pi\\circ j=\\pi\\). It is a normal \\(*\\)-homomorphism with \\(\\bar\\pi(1)=q\\) and \\(\\bar\\pi(A^{**})=N(\\pi)\\).\n3. \\(\\bar\\pi\\) maps the closed unit ball of \\(A^{**}\\) onto the closed unit ball of \\(N(\\pi)\\).\n4. There is a unique central projection \\(z(\\pi)\\in A^{**}\\) with \\(\\ker\\bar\\pi=A^{**}(1-z(\\pi))\\). The restriction of \\(\\bar\\pi\\) to \\(A^{**}z(\\pi)\\) is a \\(*\\)-isomorphism onto \\(N(\\pi)\\). It is normal, and so is its inverse.\n\nFor nondegenerate \\(\\pi\\) we get \\(q=1\\) and \\(N(\\pi)=\\mathcal M(\\pi)\\): then \\(\\bar\\pi\\) maps \\(A^{**}\\) onto \\(\\mathcal M(\\pi)\\), and the unit ball onto the unit ball.\n\n**Proof.** (1) \\(H_0\\) is invariant under \\(\\pi(A)\\), and \\(\\pi(A)\\) is closed under adjoints, so \\(q\\in\\pi(A)'\\). If \\(\\eta\\perp H_0\\), then \\(\\langle\\pi(a)\\eta,\\zeta\\rangle=\\langle\\eta,\\pi(a^*)\\zeta\\rangle=0\\) for all \\(\\zeta\\), so \\(\\pi(a)\\) vanishes on \\(H_0^\\perp\\) and \\(\\pi(a)=q\\pi(a)q\\). For an approximate unit, \\(\\pi(e_i)\\pi(a)\\zeta=\\pi(e_ia)\\zeta\\to\\pi(a)\\zeta\\), and \\(\\|\\pi(e_i)\\|\\le1\\); so \\(\\pi(e_i)\\to1\\) strongly on \\(H_0\\), while \\(\\pi(e_i)=0\\) on \\(H_0^\\perp\\). Thus \\(\\pi(e_i)\\to q\\) strongly. A bounded net that converges strongly also converges \\(\\sigma\\)-weakly, so \\(q\\in N(\\pi)\\).\n\nLet \\(\\pi_0\\) be the restriction of \\(\\pi\\) to \\(H_0\\); it is nondegenerate. The \\(\\sigma\\)-weak closure of \\(\\pi_0(A)\\) contains \\(1_{H_0}\\), by the previous paragraph, and it lies in \\(\\pi_0(A)''\\). Conversely, by Kaplansky's density theorem every element of the unit ball of \\(\\pi_0(A)''\\) is a strong limit of a net in the unit ball of \\(\\pi_0(A)\\), and a bounded strong limit is a \\(\\sigma\\)-weak limit. So the \\(\\sigma\\)-weak closure of \\(\\pi_0(A)\\) is \\(\\pi_0(A)''\\), and \\(N(\\pi)=\\pi_0(A)''\\oplus0\\).\n\nEvery \\(T\\in\\pi(A)'\\) commutes with \\(q\\), so \\(T=T_0\\oplus T_1\\) with \\(T_0\\in\\pi_0(A)'\\), and \\(T_1\\in B(H_0^\\perp)\\) is arbitrary because \\(\\pi(A)\\) vanishes on \\(H_0^\\perp\\). Thus \\(\\pi(A)'=\\pi_0(A)'\\oplus B(H_0^\\perp)\\). Its commutant is \\(\\pi_0(A)''\\oplus\\mathbb C1_{H_0^\\perp}\\), which is (2.1).\n\n(2) Define \\(S:B(H)_*\\to A^*\\) by \\(S(\\rho)=\\rho\\circ\\pi\\); it is contractive. The adjoint \\(T=S^*:A^{**}\\to B(H)\\) is normal and satisfies \\(T(j(a))=\\pi(a)\\). Weak-star density makes it the unique normal linear extension. For fixed \\(a\\in A\\), approximate \\(Y\\in A^{**}\\) weak-star by \\(j(b_i)\\). Separate continuity gives\n\\[\nT(j(a)Y)=\\pi(a)T(Y).\n\\]\nNext approximate \\(X\\) by \\(j(a_i)\\), keeping \\(Y\\) fixed; separate continuity gives \\(T(XY)=T(X)T(Y)\\). The same density argument with the weak-star continuous involutions proves \\(T(X^*)=T(X)^*\\). An increasing approximate identity has \\(j(e_i)\\to1\\) weak-star, as seen in the faithful nondegenerate realization constructed above. Hence \\(T(1)=q\\) by (1). The map is therefore zero on \\(H_0^\\perp\\); on \\(H_0\\) its range lies in \\(\\pi_0(A)''\\), by density and closedness.\n\n(3)–(4) The kernel of \\(T\\) is a sigma-weakly closed self-adjoint two-sided ideal. The full earlier ideal theorem gives \\(\\ker T=A^{**}(1-z)\\) for a unique central projection \\(z\\). The restriction\n\\[\nV=T|_{A^{**}z}\n\\]\nis injective and therefore isometric. Its image of the unit ball is sigma-weakly compact, by Banach–Alaoglu and normality, hence closed. This image contains every contraction in \\(\\pi_0(A)\\): write it as \\(T(j(a)z)\\), and use the isometry of \\(V\\) to obtain \\(\\|j(a)z\\|=\\|\\pi_0(a)\\|\\le1\\), regardless of the norm of the chosen \\(a\\). Kaplansky density then puts every contraction of \\(\\pi_0(A)''\\) in the image. Thus \\(V\\) maps the unit ball onto the unit ball of \\(N(\\pi)=\\pi_0(A)''\\oplus0\\), and scaling proves \\(T(A^{**})=N(\\pi)\\). Since \\(T(X)=T(Xz)\\) and \\(\\|Xz\\|\\le\\|X\\|\\), this is also the image of the whole bidual unit ball.\n\nFor the normal inverse, let \\(V_*:N(\\pi)_*\\to(A^{**}z)_*\\) be the preadjoint. Surjectivity on unit balls gives \\(\\|V_*\\rho\\|=\\|\\rho\\|\\), so its range is norm closed. If \\(X\\in A^{**}z\\) annihilates that range, all \\(\\rho\\) vanish on \\(V(X)\\), hence \\(V(X)=0\\) and \\(X=0\\). Hahn–Banach therefore makes the range norm dense, so \\(V_*\\) is a surjective isometry. Its inverse is bounded, and \\(V^{-1}=(V_*^{-1})^*\\) is normal. This proves the inverse directly, without using automatic normality of arbitrary isomorphisms. All assertions include \\(H_0=0\\). \\(\\square\\)\n\n### Second adjoints of homomorphisms\n\nFor a \\(*\\)-homomorphism \\(\\theta:A\\to B\\), the Banach-space second adjoint is weak-star continuous and extends \\(j_B\\theta\\). Using weak-star density first in one factor and then in the other, exactly as in the preceding proof, gives\n\\[\n\\begin{gathered}\n\\theta^{**}(XY)\\\\\n=\\theta^{**}(X)\\theta^{**}(Y),\n\\\\\n\\theta^{**}(X^*)\\\\\n=\\theta^{**}(X)^*.\n\\end{gathered}\n\\]\nFor an inclusion \\(i:B\\hookrightarrow A\\), Hahn–Banach extends each \\(f\\in B^*\\) to \\(F\\in A^*\\) with equal norm. Hence \\(i^*\\) maps the dual unit ball onto the dual unit ball, and \\(i^{**}\\) is isometric. Its range is \\((\\ker i^*)^\\perp\\): for an element \\(X\\) of this annihilator, define \\(Y(f)=X(F)\\) using any extension \\(F\\); the definition is independent of the extension and bounded by \\(\\|X\\|\\|f\\|\\), and \\(i^{**}Y=X\\). This range is weak-star closed. Goldstine density in \\(B^{**}\\) shows that it is precisely the weak-star closure of \\(j_A(B)\\). This proves (28) without inferring global closedness merely from closed unit balls.\n\n**Example 2.2** (Degenerate representations). Suppose \\(q\\neq1\\). By (2.1), \\(1_H\\in\\pi(A)''\\) but \\(1_H\\notin\\bar\\pi(A^{**})\\subseteq qB(H)q\\). So \\(\\bar\\pi\\) is not onto \\(\\mathcal M(\\pi)\\), and the unit ball of \\(\\mathcal M(\\pi)\\) is not the image of the unit ball of \\(A^{**}\\). The smallest example is \\(A=\\mathbb C\\) with \\(\\pi(\\lambda)=\\lambda\\oplus0\\) on \\(\\mathbb C^2\\). Then \\(A^{**}=\\mathbb C\\), \\(N(\\pi)=\\mathbb C\\oplus0\\) and \\(\\mathcal M(\\pi)=\\mathbb C\\oplus\\mathbb C\\). This is why Lemma 2.1 is stated with \\(N(\\pi)\\) in place of \\(\\mathcal M(\\pi)\\).\n\n",
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      "id": "OA-FND-WA-03",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "3. The universal enveloping von Neumann algebra",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "full_conditions_and_proof": "## 3. The universal enveloping von Neumann algebra\n\nOne representation sees all of \\(A\\) at once: the direct sum of the cyclic representations of all positive functionals. Its von Neumann algebra maps onto the von Neumann algebra of every other representation, and it is isomorphic to the bidual.\n\n**Definition 3.1.** A representation \\(\\pi\\) of \\(A\\) is *universal* if for every representation \\(\\rho\\) of \\(A\\) there is a normal \\(*\\)-homomorphism \\(\\tilde\\rho\\) of \\(\\mathcal M(\\pi)\\) onto \\(\\mathcal M(\\rho)\\) with \\(\\tilde\\rho\\circ\\pi=\\rho\\). Then \\(\\mathcal M(\\pi)\\) is called a *universal enveloping von Neumann algebra* for \\(A\\).\n\n**Proposition 3.2** (Uniqueness). Let \\(\\pi_1,\\pi_2\\) be universal. There is exactly one normal \\(*\\)-homomorphism \\(\\theta:\\mathcal M(\\pi_1)\\to\\mathcal M(\\pi_2)\\) with \\(\\theta\\circ\\pi_1=\\pi_2\\). It is a \\(*\\)-isomorphism, and its inverse is the unique normal \\(\\theta'\\) with \\(\\theta'\\circ\\pi_2=\\pi_1\\).\n\n**Proof.** Universality supplies normal maps \\(\\theta\\) and \\(\\theta'\\). The composite \\(\\theta'\\theta\\) is normal and fixes \\(\\pi_1(A)\\) pointwise. By [Lemma 2.1](#oa-fnd-wa-02)(1), \\(\\pi_1(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\mathcal M(\\pi_1)\\), so \\(\\theta'\\theta=\\mathrm{id}\\). In the same way \\(\\theta\\theta'=\\mathrm{id}\\). Two normal maps that agree on \\(\\pi_1(A)\\) agree everywhere, which gives uniqueness. \\(\\square\\)\n\n**Theorem 3.3** (The universal representation). Put\n\\[\n\\begin{gathered}\n\\pi_u\\\\\n=\\bigoplus_{\\omega\\in A^*_+}\\pi_\\omega\\\\\n\\text{on}\\\\\nH_u\\\\\n=\\bigoplus_{\\omega\\in A^*_+}H_\\omega ,\n\\end{gathered}\n\\tag{3.1}\n\\]\nwhere the summand for \\(\\omega=0\\) is the zero space. Then \\(\\pi_u\\) is universal. The extension \\(\\bar\\pi_u:A^{**}\\to\\mathcal M(\\pi_u)\\) is an isometric \\(*\\)-isomorphism, and it is a homeomorphism for \\(\\sigma(A^{**},A^*)\\) and the \\(\\sigma\\)-weak topology. The same holds for the direct sum over the states only. We call \\(\\pi_u\\) the *universal representation* of \\(A\\).\n\n**Proof.** Each summand is nondegenerate, so \\(\\pi_u\\) is. Let \\(\\bar\\pi_u(X)=0\\) and let \\(f\\in A^*\\). By [Proposition 1.1](#oa-fnd-wa-01)(2), \\(f=\\omega_{\\pi_\\psi;\\xi,\\eta}\\) for some \\(\\psi\\in A^*_+\\) and \\(\\xi,\\eta\\in H_\\psi\\). Place \\(\\xi,\\eta\\) in the summand \\(H_\\psi\\) of \\(H_u\\). The functional \\(Y\\mapsto\\langle\\bar\\pi_u(Y)\\xi,\\eta\\rangle\\) is normal and extends \\(f\\), so it is \\(\\hat f\\). Hence \\(X(f)=\\hat f(X)=\\langle\\bar\\pi_u(X)\\xi,\\eta\\rangle=0\\). As \\(f\\) is arbitrary, \\(X=0\\). So \\(\\bar\\pi_u\\) is injective and \\(z(\\pi_u)=1\\). By [Lemma 2.1](#oa-fnd-wa-02)(4), \\(\\bar\\pi_u\\) is a \\(*\\)-isomorphism onto \\(\\mathcal M(\\pi_u)\\) whose inverse is normal, and an injective \\(*\\)-homomorphism is isometric. A normal map with a normal inverse is a homeomorphism for the two weak\\(^*\\) topologies.\n\nUniversality: for a representation \\(\\rho\\), \\(\\bar\\rho\\) maps \\(A^{**}\\) onto \\(\\mathcal M(\\rho)\\) (Lemma 2.1(2)), and \\(\\tilde\\rho=\\bar\\rho\\circ\\bar\\pi_u^{-1}\\) is normal, onto, and satisfies \\(\\tilde\\rho\\circ\\pi_u=\\bar\\rho\\circ j=\\rho\\). For the sum over the states, repeat the argument, using Remark 1.2. By the uniqueness in Lemma 2.1(2), the normal extension of this sum is the realization of \\(A^{**}\\) recalled in the background. \\(\\square\\)\n\nFrom now on \\(\\tilde A\\) denotes \\(A^{**}\\) with its von Neumann algebra structure, and we call it *the* universal enveloping von Neumann algebra of \\(A\\). Its predual is \\(A^*\\). By Proposition 3.2, any universal representation gives a canonically isomorphic algebra.\n\nA first use of \\(\\tilde A\\): the decomposition of Proposition 1.1(1) is unique, and it can be read off from one norming element.\n\n**Proposition 3.4** (The norm-additive decomposition is unique). Let \\(f\\in A^*\\) be hermitian. There is a self-adjoint \\(X\\) in the unit ball of \\(\\tilde A\\) with \\(\\hat f(X)=\\|f\\|\\). For any such \\(X\\) let \\(E_+\\) and \\(E_-\\) be the projections onto the kernels of \\(1-X\\) and \\(1+X\\), that is \\(E_\\pm=1-s(1\\mp X)\\). Then \\(E_+E_-=0\\), and every decomposition \\(f=f_+-f_-\\) with \\(f_\\pm\\in A^*_+\\) and \\(\\|f\\|=\\|f_+\\|+\\|f_-\\|\\) satisfies\n\\[\n\\begin{gathered}\nf_+(a)\\\\\n=\\hat f(aE_+),\\\\\nf_-(a)\\\\\n=-\\hat f(aE_-)\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{3.2}\n\\]\nIn particular the decomposition of [Proposition 1.1](#oa-fnd-wa-01)(1) is unique, and the supports of \\(\\hat f_+\\) and \\(\\hat f_-\\) are orthogonal, lying under \\(E_+\\) and \\(E_-\\).\n\n**Proof.** The unit ball of \\(\\tilde A\\) is \\(\\sigma(\\tilde A,A^*)\\)-compact by the Banach–Alaoglu theorem, and \\(\\hat f\\) is continuous and has norm \\(\\|f\\|\\). So some \\(Y\\) in the ball has \\(|\\hat f(Y)|=\\|f\\|\\); after multiplying \\(Y\\) by a scalar of modulus one, \\(\\hat f(Y)=\\|f\\|\\). The involution of \\(\\tilde A\\) is \\(\\sigma\\)-weakly continuous and \\(j(A)\\) is dense, so \\(\\hat f\\) is hermitian. Hence \\(X=(Y+Y^*)/2\\) also has \\(\\hat f(X)=\\|f\\|\\), and \\(\\|X\\|\\le1\\).\n\nTake a decomposition as stated. The normal extensions \\(\\hat f_\\pm\\) are positive, and a positive normal functional takes its norm at \\(1\\), so \\(\\|\\hat f_\\pm\\|=\\hat f_\\pm(1)=\\|f_\\pm\\|\\). Since \\(-1\\le X\\le1\\),\n\\[\n\\begin{gathered}\n\\|f\\|\\\\\n=\\hat f_+(X)-\\hat f_-(X)\\\\\n\\le\\hat f_+(1)+\\hat f_-(1)\\\\\n=\\|f\\| .\n\\end{gathered}\n\\]\nSo \\(\\hat f_+(1-X)=0\\) and \\(\\hat f_-(1+X)=0\\). Write \\(1-X=d^2\\) with \\(d\\ge0\\). For every \\(Y\\in\\tilde A\\), the Cauchy–Schwarz inequality for \\(\\hat f_+\\) gives\n\\[\n\\begin{gathered}\n|\\hat f_+(Y(1-X))|^2\\\\\n=|\\hat f_+((Yd)d)|^2\\\\\n\\le\\hat f_+(Y(1-X)Y^*)\\,\\hat f_+(1-X)\\\\\n=0 .\n\\end{gathered}\n\\]\nRealize \\(\\tilde A\\) as a von Neumann algebra on a Hilbert space (Theorem 3.3). By the background fact on supports, \\((1-X)(1-X+\\varepsilon)^{-1}\\) increases to \\(s(1-X)=1-E_+\\) strongly and \\(\\sigma\\)-weakly as \\(\\varepsilon\\downarrow0\\). The element \\(Y(1-X)(1-X+\\varepsilon)^{-1}\\) equals \\(Y'(1-X)\\) with \\(Y'=Y(1-X+\\varepsilon)^{-1}\\), so \\(\\hat f_+\\) vanishes on it. Letting \\(\\varepsilon\\downarrow0\\) gives \\(\\hat f_+(Y(1-E_+))=0\\), that is \\(\\hat f_+(Y)=\\hat f_+(YE_+)\\). Taking adjoints, \\(\\hat f_+(Y)=\\hat f_+(E_+Y)\\) as well. In the same way \\(\\hat f_-(Y)=\\hat f_-(YE_-)=\\hat f_-(E_-Y)\\).\n\nThe kernels of \\(1-X\\) and \\(1+X\\) are eigenspaces of the self-adjoint \\(X\\) for different eigenvalues, so \\(E_+E_-=0\\). Therefore\n\\[\n\\begin{gathered}\n\\hat f(YE_+)\\\\\n=\\hat f_+(YE_+)-\\hat f_-(E_-YE_+E_-)\\\\\n=\\hat f_+(Y),\n\\end{gathered}\n\\]\nand in the same way \\(-\\hat f(YE_-)=\\hat f_-(Y)\\). Restricting to \\(Y\\in A\\) gives (3.2). Its right-hand side depends only on \\(f\\) and \\(X\\), not on the decomposition. \\(\\square\\)\n\n**Remark 3.5** (The hermitian polar decomposition). Formula (3.2) is the hermitian case of the polar decomposition of normal functionals. Put \\(W=E_+-E_-\\), a self-adjoint partial isometry, and \\(|\\hat f|=\\hat f_++\\hat f_-\\), a positive normal functional of norm \\(\\|f\\|\\). The two support identities give \\[\n\\begin{gathered}\n|\\hat f|(YW)\\\\\n=\\hat f_+(YWE_+)+\\hat f_-(YWE_-)\\\\\n=\\hat f_+(Y)-\\hat f_-(Y),\n\\end{gathered}\n\\] so \\(\\hat f=W|\\hat f|\\) in the notation (0.1). The proof of Proposition 3.4 needs only the bidual, supports and one norming element, and it works for every \\(C^*\\)-algebra.\n\n",
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      "id": "OA-FND-WA-04",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "3. The universal enveloping von Neumann algebra",
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      "full_conditions_and_proof": "## 3. The universal enveloping von Neumann algebra\n\nOne representation sees all of \\(A\\) at once: the direct sum of the cyclic representations of all positive functionals. Its von Neumann algebra maps onto the von Neumann algebra of every other representation, and it is isomorphic to the bidual.\n\n**Definition 3.1.** A representation \\(\\pi\\) of \\(A\\) is *universal* if for every representation \\(\\rho\\) of \\(A\\) there is a normal \\(*\\)-homomorphism \\(\\tilde\\rho\\) of \\(\\mathcal M(\\pi)\\) onto \\(\\mathcal M(\\rho)\\) with \\(\\tilde\\rho\\circ\\pi=\\rho\\). Then \\(\\mathcal M(\\pi)\\) is called a *universal enveloping von Neumann algebra* for \\(A\\).\n\n**Proposition 3.2** (Uniqueness). Let \\(\\pi_1,\\pi_2\\) be universal. There is exactly one normal \\(*\\)-homomorphism \\(\\theta:\\mathcal M(\\pi_1)\\to\\mathcal M(\\pi_2)\\) with \\(\\theta\\circ\\pi_1=\\pi_2\\). It is a \\(*\\)-isomorphism, and its inverse is the unique normal \\(\\theta'\\) with \\(\\theta'\\circ\\pi_2=\\pi_1\\).\n\n**Proof.** Universality supplies normal maps \\(\\theta\\) and \\(\\theta'\\). The composite \\(\\theta'\\theta\\) is normal and fixes \\(\\pi_1(A)\\) pointwise. By [Lemma 2.1](#oa-fnd-wa-02)(1), \\(\\pi_1(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\mathcal M(\\pi_1)\\), so \\(\\theta'\\theta=\\mathrm{id}\\). In the same way \\(\\theta\\theta'=\\mathrm{id}\\). Two normal maps that agree on \\(\\pi_1(A)\\) agree everywhere, which gives uniqueness. \\(\\square\\)\n\n**Theorem 3.3** (The universal representation). Put\n\\[\n\\begin{gathered}\n\\pi_u\\\\\n=\\bigoplus_{\\omega\\in A^*_+}\\pi_\\omega\\\\\n\\text{on}\\\\\nH_u\\\\\n=\\bigoplus_{\\omega\\in A^*_+}H_\\omega ,\n\\end{gathered}\n\\tag{3.1}\n\\]\nwhere the summand for \\(\\omega=0\\) is the zero space. Then \\(\\pi_u\\) is universal. The extension \\(\\bar\\pi_u:A^{**}\\to\\mathcal M(\\pi_u)\\) is an isometric \\(*\\)-isomorphism, and it is a homeomorphism for \\(\\sigma(A^{**},A^*)\\) and the \\(\\sigma\\)-weak topology. The same holds for the direct sum over the states only. We call \\(\\pi_u\\) the *universal representation* of \\(A\\).\n\n**Proof.** Each summand is nondegenerate, so \\(\\pi_u\\) is. Let \\(\\bar\\pi_u(X)=0\\) and let \\(f\\in A^*\\). By [Proposition 1.1](#oa-fnd-wa-01)(2), \\(f=\\omega_{\\pi_\\psi;\\xi,\\eta}\\) for some \\(\\psi\\in A^*_+\\) and \\(\\xi,\\eta\\in H_\\psi\\). Place \\(\\xi,\\eta\\) in the summand \\(H_\\psi\\) of \\(H_u\\). The functional \\(Y\\mapsto\\langle\\bar\\pi_u(Y)\\xi,\\eta\\rangle\\) is normal and extends \\(f\\), so it is \\(\\hat f\\). Hence \\(X(f)=\\hat f(X)=\\langle\\bar\\pi_u(X)\\xi,\\eta\\rangle=0\\). As \\(f\\) is arbitrary, \\(X=0\\). So \\(\\bar\\pi_u\\) is injective and \\(z(\\pi_u)=1\\). By [Lemma 2.1](#oa-fnd-wa-02)(4), \\(\\bar\\pi_u\\) is a \\(*\\)-isomorphism onto \\(\\mathcal M(\\pi_u)\\) whose inverse is normal, and an injective \\(*\\)-homomorphism is isometric. A normal map with a normal inverse is a homeomorphism for the two weak\\(^*\\) topologies.\n\nUniversality: for a representation \\(\\rho\\), \\(\\bar\\rho\\) maps \\(A^{**}\\) onto \\(\\mathcal M(\\rho)\\) (Lemma 2.1(2)), and \\(\\tilde\\rho=\\bar\\rho\\circ\\bar\\pi_u^{-1}\\) is normal, onto, and satisfies \\(\\tilde\\rho\\circ\\pi_u=\\bar\\rho\\circ j=\\rho\\). For the sum over the states, repeat the argument, using Remark 1.2. By the uniqueness in Lemma 2.1(2), the normal extension of this sum is the realization of \\(A^{**}\\) recalled in the background. \\(\\square\\)\n\nFrom now on \\(\\tilde A\\) denotes \\(A^{**}\\) with its von Neumann algebra structure, and we call it *the* universal enveloping von Neumann algebra of \\(A\\). Its predual is \\(A^*\\). By Proposition 3.2, any universal representation gives a canonically isomorphic algebra.\n\nA first use of \\(\\tilde A\\): the decomposition of Proposition 1.1(1) is unique, and it can be read off from one norming element.\n\n**Proposition 3.4** (The norm-additive decomposition is unique). Let \\(f\\in A^*\\) be hermitian. There is a self-adjoint \\(X\\) in the unit ball of \\(\\tilde A\\) with \\(\\hat f(X)=\\|f\\|\\). For any such \\(X\\) let \\(E_+\\) and \\(E_-\\) be the projections onto the kernels of \\(1-X\\) and \\(1+X\\), that is \\(E_\\pm=1-s(1\\mp X)\\). Then \\(E_+E_-=0\\), and every decomposition \\(f=f_+-f_-\\) with \\(f_\\pm\\in A^*_+\\) and \\(\\|f\\|=\\|f_+\\|+\\|f_-\\|\\) satisfies\n\\[\n\\begin{gathered}\nf_+(a)\\\\\n=\\hat f(aE_+),\\\\\nf_-(a)\\\\\n=-\\hat f(aE_-)\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{3.2}\n\\]\nIn particular the decomposition of [Proposition 1.1](#oa-fnd-wa-01)(1) is unique, and the supports of \\(\\hat f_+\\) and \\(\\hat f_-\\) are orthogonal, lying under \\(E_+\\) and \\(E_-\\).\n\n**Proof.** The unit ball of \\(\\tilde A\\) is \\(\\sigma(\\tilde A,A^*)\\)-compact by the Banach–Alaoglu theorem, and \\(\\hat f\\) is continuous and has norm \\(\\|f\\|\\). So some \\(Y\\) in the ball has \\(|\\hat f(Y)|=\\|f\\|\\); after multiplying \\(Y\\) by a scalar of modulus one, \\(\\hat f(Y)=\\|f\\|\\). The involution of \\(\\tilde A\\) is \\(\\sigma\\)-weakly continuous and \\(j(A)\\) is dense, so \\(\\hat f\\) is hermitian. Hence \\(X=(Y+Y^*)/2\\) also has \\(\\hat f(X)=\\|f\\|\\), and \\(\\|X\\|\\le1\\).\n\nTake a decomposition as stated. The normal extensions \\(\\hat f_\\pm\\) are positive, and a positive normal functional takes its norm at \\(1\\), so \\(\\|\\hat f_\\pm\\|=\\hat f_\\pm(1)=\\|f_\\pm\\|\\). Since \\(-1\\le X\\le1\\),\n\\[\n\\begin{gathered}\n\\|f\\|\\\\\n=\\hat f_+(X)-\\hat f_-(X)\\\\\n\\le\\hat f_+(1)+\\hat f_-(1)\\\\\n=\\|f\\| .\n\\end{gathered}\n\\]\nSo \\(\\hat f_+(1-X)=0\\) and \\(\\hat f_-(1+X)=0\\). Write \\(1-X=d^2\\) with \\(d\\ge0\\). For every \\(Y\\in\\tilde A\\), the Cauchy–Schwarz inequality for \\(\\hat f_+\\) gives\n\\[\n\\begin{gathered}\n|\\hat f_+(Y(1-X))|^2\\\\\n=|\\hat f_+((Yd)d)|^2\\\\\n\\le\\hat f_+(Y(1-X)Y^*)\\,\\hat f_+(1-X)\\\\\n=0 .\n\\end{gathered}\n\\]\nRealize \\(\\tilde A\\) as a von Neumann algebra on a Hilbert space (Theorem 3.3). By the background fact on supports, \\((1-X)(1-X+\\varepsilon)^{-1}\\) increases to \\(s(1-X)=1-E_+\\) strongly and \\(\\sigma\\)-weakly as \\(\\varepsilon\\downarrow0\\). The element \\(Y(1-X)(1-X+\\varepsilon)^{-1}\\) equals \\(Y'(1-X)\\) with \\(Y'=Y(1-X+\\varepsilon)^{-1}\\), so \\(\\hat f_+\\) vanishes on it. Letting \\(\\varepsilon\\downarrow0\\) gives \\(\\hat f_+(Y(1-E_+))=0\\), that is \\(\\hat f_+(Y)=\\hat f_+(YE_+)\\). Taking adjoints, \\(\\hat f_+(Y)=\\hat f_+(E_+Y)\\) as well. In the same way \\(\\hat f_-(Y)=\\hat f_-(YE_-)=\\hat f_-(E_-Y)\\).\n\nThe kernels of \\(1-X\\) and \\(1+X\\) are eigenspaces of the self-adjoint \\(X\\) for different eigenvalues, so \\(E_+E_-=0\\). Therefore\n\\[\n\\begin{gathered}\n\\hat f(YE_+)\\\\\n=\\hat f_+(YE_+)-\\hat f_-(E_-YE_+E_-)\\\\\n=\\hat f_+(Y),\n\\end{gathered}\n\\]\nand in the same way \\(-\\hat f(YE_-)=\\hat f_-(Y)\\). Restricting to \\(Y\\in A\\) gives (3.2). Its right-hand side depends only on \\(f\\) and \\(X\\), not on the decomposition. \\(\\square\\)\n\n**Remark 3.5** (The hermitian polar decomposition). Formula (3.2) is the hermitian case of the polar decomposition of normal functionals. Put \\(W=E_+-E_-\\), a self-adjoint partial isometry, and \\(|\\hat f|=\\hat f_++\\hat f_-\\), a positive normal functional of norm \\(\\|f\\|\\). The two support identities give \\[\n\\begin{gathered}\n|\\hat f|(YW)\\\\\n=\\hat f_+(YWE_+)+\\hat f_-(YWE_-)\\\\\n=\\hat f_+(Y)-\\hat f_-(Y),\n\\end{gathered}\n\\] so \\(\\hat f=W|\\hat f|\\) in the notation (0.1). The proof of Proposition 3.4 needs only the bidual, supports and one norming element, and it works for every \\(C^*\\)-algebra.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-05",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
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      "full_conditions_and_proof": "## 4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals\n\nMultiplication by elements of \\(A\\) acts on \\(A^*\\) from both sides. The norm-closed subspaces of \\(A^*\\) that are invariant under these actions are exactly the polars of the weak\\(^*\\) closed one-sided ideals of \\(\\tilde A\\), and each of them is cut out by a projection of \\(\\tilde A\\). This correspondence attaches projections to representations in Section 5 and to functionals in Section 11.\n\n**Definition 4.1.** Let \\(A\\) be a \\(C^*\\)-algebra, acting on \\(A^*\\) by (0.1) through \\(j\\): \\((af)(x)=f(xa)\\) and \\((fa)(x)=f(ax)\\). A subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), *right invariant* if \\(Va\\subseteq V\\) for all \\(a\\in A\\), and *invariant* if it is both.\n\nTwo facts organise what follows.\n\n1. Let \\(M\\) be a von Neumann algebra, regarded as a \\(C^*\\)-algebra. Then \\(M_*\\) is a norm-closed invariant subspace of \\(M^*\\). It is invariant because multiplication by a fixed element is \\(\\sigma\\)-weakly continuous, and it is norm closed by the background fact on preduals.\n2. The algebra \\(\\tilde A=A^{**}\\) also acts on \\(A^*\\), since \\(A^*\\) is its predual. For a norm-closed subspace \\(V\\subseteq A^*\\), invariance under \\(A\\) and invariance under \\(\\tilde A\\) are the same thing. This is [Theorem 4.3](#oa-fnd-wa-07)(1) with \\(S=j(A)\\). So it makes no difference whether \"invariant\" refers to \\(A\\) or to \\(\\tilde A\\).\n\nThe correspondence rests on a description of the weak\\(^*\\) closed one-sided ideals of a von Neumann algebra.\n\n**Lemma 4.2.** Let \\(J\\) be a right ideal of a von Neumann algebra \\(M\\), closed in the \\(\\sigma\\)-weak topology. There is a unique projection \\(p\\in M\\) with \\(J=pM\\). It lies in \\(J\\) and is a left identity for \\(J\\). Symmetrically, a \\(\\sigma\\)-weakly closed left ideal is \\(Mp\\) for a unique projection \\(p\\), and a \\(\\sigma\\)-weakly closed two-sided ideal is \\(Mp\\) with \\(p\\) central.\n\n**Proof.** If \\(x\\in J\\) then \\(xx^*\\in J\\). Let \\(h\\in J\\) be positive. For \\(\\varepsilon>0\\) the element \\(h(h+\\varepsilon)^{-1}\\) lies in \\(J\\), since \\(J\\) is a right ideal. These elements increase to \\(s(h)\\) as \\(\\varepsilon\\downarrow0\\), strongly and \\(\\sigma\\)-weakly (background fact on supports). So \\(s(h)\\in J\\). If \\(h,k\\in J\\) are positive, then \\(h+k\\in J\\) and \\(\\ker(h+k)=\\ker h\\cap\\ker k\\), because \\(\\langle(h+k)\\zeta,\\zeta\\rangle=0\\) forces both terms to vanish; so \\(s(h+k)=s(h)\\vee s(k)\\). Hence the supports of the positive elements of \\(J\\) form an upward directed family of projections in \\(J\\). Its least upper bound \\(p\\) is its strong and \\(\\sigma\\)-weak limit (background fact on monotone nets), so \\(p\\in J\\). For \\(x\\in J\\) we have \\(s(xx^*)\\le p\\), so \\((1-p)xx^*=0\\) and \\(\\|(1-p)x\\|^2=\\|(1-p)xx^*(1-p)\\|=0\\). Thus \\(x=px\\), and \\(J\\subseteq pM\\). Conversely \\(pM\\subseteq J\\) because \\(p\\in J\\).\n\nIf also \\(J=p'M\\), then \\(p'\\in pM\\) gives \\(pp'=p'\\), and \\(p\\in p'M\\) gives \\(p'p=p\\). Hence \\(p=p^*=(p'p)^*=pp'=p'\\).\n\nA \\(\\sigma\\)-weakly closed left ideal \\(L\\) has \\(L^*\\) a \\(\\sigma\\)-weakly closed right ideal, since the involution is \\(\\sigma\\)-weakly continuous. So \\(L^*=pM\\) and \\(L=Mp\\). A \\(\\sigma\\)-weakly closed two-sided ideal \\(J\\) is then both \\(pM\\) and \\(Mp'\\) with \\(p,p'\\in J\\). From \\(p'\\in pM\\) and \\(p\\in Mp'\\) we get \\(pp'=p'\\) and \\(pp'=p\\), so \\(p=p'\\). Thus \\(J=pM=Mp\\) is self-adjoint, and \\(p\\) is central: for \\(x\\in M\\), \\(xp\\in pM\\) and \\(px\\in Mp\\) give \\(xp=pxp=px\\). (For self-adjoint ideals this is the background fact on weak\\(^*\\) closed ideals; the lemma shows that self-adjointness is automatic.) \\(\\square\\)\n\n**Theorem 4.3** (Invariant subspaces and weak\\(^*\\) closed ideals). Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(S\\subseteq M\\) have \\(\\sigma\\)-weakly dense linear span, and let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(sV\\subseteq V\\) (resp. \\(Vs\\subseteq V\\)) for every \\(s\\in S\\). Then \\(V\\) is left (resp. right) invariant under all of \\(M\\).\n2. Taking polars, \\(V\\mapsto V^\\circ\\), is a bijection from the norm-closed subspaces \\(V\\subseteq M_*\\) that are invariant on the left (resp. right) onto the right (resp. left) ideals \\(J\\) of \\(M\\) that are \\(\\sigma\\)-weakly closed; its inverse is \\(J\\mapsto J^\\circ\\).\n3. A norm-closed subspace \\(V\\) invariant on the left (resp. right) has the form \\(V=M_*e\\) (resp. \\(V=eM_*\\)) for a unique projection \\(e\\in M\\), with \\(V^\\circ=(1-e)M\\) (resp. \\(M(1-e)\\)). This \\(e\\) is called the *support* of \\(V\\). A norm-closed \\(V\\) invariant on the left is invariant on both sides exactly when \\(e\\) is central, and then \\(M_*e=eM_*\\).\n\nFor a \\(C^*\\)-algebra \\(A\\), apply this to \\(M=\\tilde A\\), \\(M_*=A^*\\) and \\(S=j(A)\\). Then the norm-closed subspaces of \\(A^*\\) that are invariant on the left (resp. right) under \\(A\\), as in [Definition 4.1](#oa-fnd-wa-05), match the right (resp. left) ideals of \\(\\tilde A\\) that are \\(\\sigma\\)-weakly closed, and each such subspace is \\(A^*e\\) (resp. \\(eA^*\\)) for a unique projection \\(e\\in\\tilde A\\).\n\n**Proof.** We need two duality facts. If \\(V\\subseteq M_*\\) is a norm-closed subspace, then \\(V^{\\circ\\circ}=V\\): if \\(\\varphi_0\\notin V\\), the Hahn–Banach theorem gives a bounded functional on \\(M_*\\) that vanishes on \\(V\\) and not at \\(\\varphi_0\\); by \\(M=(M_*)^*\\) it is evaluation at some \\(x\\in V^\\circ\\), so \\(\\varphi_0\\notin V^{\\circ\\circ}\\). If \\(J\\subseteq M\\) is a \\(\\sigma\\)-weakly closed subspace, then \\(J^{\\circ\\circ}=J\\) (background fact on bipolars).\n\n(1) Take the left case. Put \\(J=V^\\circ\\); it is a \\(\\sigma\\)-weakly closed subspace. For \\(x\\in J\\), \\(s\\in S\\) and \\(\\varphi\\in V\\) we get \\(\\varphi(xs)=(s\\varphi)(x)=0\\), so \\(xs\\in J\\). The set \\(T=\\{m\\in M:Jm\\subseteq J\\}\\) is a linear subspace containing \\(S\\). It is \\(\\sigma\\)-weakly closed, because for fixed \\(x\\in J\\) the map \\(m\\mapsto xm\\) is \\(\\sigma\\)-weakly continuous and \\(J\\) is closed. So \\(T=M\\), and \\(J\\) is a right ideal. Now \\(V=V^{\\circ\\circ}=J^\\circ\\). For \\(\\varphi\\in J^\\circ\\), \\(a\\in M\\) and \\(x\\in J\\), \\((a\\varphi)(x)=\\varphi(xa)=0\\), so \\(a\\varphi\\in J^\\circ\\). The right case is symmetric.\n\n(2) By (1) with \\(S=M\\), the polar of a norm-closed subspace invariant on the left is a right ideal, and it is \\(\\sigma\\)-weakly closed. Conversely, for a right ideal \\(J\\) closed in that topology, \\(J^\\circ\\) is norm closed, and invariant on the left by the last step of (1). The two bipolar identities show that the maps are mutually inverse.\n\n(3) Let \\(V\\) be norm closed and left invariant. By (2) and [Lemma 4.2](#oa-fnd-wa-06), \\(V^\\circ=pM\\) for a unique projection \\(p\\). Put \\(e=1-p\\). Then\n\\[\n\\begin{gathered}\nV\\\\\n=(pM)^\\circ\\\\\n=\\{\\varphi:\\varphi(px)=0\\ \\forall x\\}\\\\\n=\\{\\varphi:\\varphi p=0\\}\\\\\n=\\{\\varphi:\\varphi=\\varphi e\\}\\\\\n=M_*e .\n\\end{gathered}\n\\tag{4.1}\n\\]\nUniqueness of \\(e\\) follows from uniqueness of \\(p\\). If \\(V\\) is also right invariant, then \\(V^\\circ\\) is a \\(\\sigma\\)-weakly closed two-sided ideal. By Lemma 4.2 it is \\(Mp'\\) with \\(p'\\) central; comparing with \\(V^\\circ=pM\\) and using uniqueness gives \\(p=p'\\), so \\(e\\) is central. Conversely, if \\(e\\) is central then \\(\\varphi e=e\\varphi\\) for every \\(\\varphi\\), so \\(M_*e=eM_*\\) is invariant on both sides. The right-handed statements follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\), \\(\\varphi^*(x)=\\overline{\\varphi(x^*)}\\), which maps \\(M_*e\\) onto \\(eM_*\\): indeed \\((\\varphi e)^*=e\\varphi^*\\).\n\nFor the \\(C^*\\)-algebra statement, \\(j(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A\\), so (1) turns invariance under \\(A\\) into invariance under \\(\\tilde A\\), and (2)–(3) apply. \\(\\square\\)\n\n**Meets and joins.** For a family \\((e_i)_{i\\in I}\\) of projections of a von Neumann algebra \\(M\\subseteq B(H)\\), the projection onto \\(\\bigcap_ie_iH\\) lies in \\(M\\). Indeed every \\(T\\in M'\\) commutes with each \\(e_i\\), so \\(\\bigcap_ie_iH\\) is invariant under \\(M'\\), and \\(M'\\) is closed under adjoints; hence the projection commutes with \\(M'\\) and lies in \\(M''=M\\) by the bicommutant theorem. Write \\(\\bigwedge_ie_i\\) for it, and put \\(\\bigvee_ie_i=1-\\bigwedge_i(1-e_i)\\), the projection onto the closed span of \\(\\bigcup_ie_iH\\). They are the greatest projection below every \\(e_i\\) and the least projection above every \\(e_i\\), so they do not depend on the realisation of \\(M\\). In particular they make sense in \\(\\tilde A\\).\n\n**Proposition 4.4** (Meets and joins through invariant subspaces). Let \\(M\\) be a von Neumann algebra with projections \\((e_i)_{i\\in I}\\), and put \\(p=\\bigwedge_ie_i\\) and \\(q=\\bigvee_ie_i\\). Then\n\\[\n\\begin{gathered}\nM_*p\\\\\n=\\bigcap_iM_*e_i,\\\\\npM_*\\\\\n=\\bigcap_ie_iM_*,\\\\\nM_*q\\\\\n=\\overline{\\operatorname{span}}\\bigcup_iM_*e_i,\\\\\nqM_*\\\\\n=\\overline{\\operatorname{span}}\\bigcup_ie_iM_* .\n\\end{gathered}\n\\tag{4.2}\n\\]\n\n**Proof.** By (4.1), \\(M_*f=((1-f)M)^\\circ\\) and \\((M_*f)^\\circ=(1-f)M\\) for every projection \\(f\\). Polars turn unions into intersections: \\((\\bigcup_iV_i)^\\circ=\\bigcap_iV_i^\\circ\\), for subspaces of \\(M_*\\) and of \\(M\\) alike.\n\nFor the first identity, \\(\\bigcap_iM_*e_i=\\bigcap_i((1-e_i)M)^\\circ\\) is the polar of the \\(\\sigma\\)-weakly closed right ideal generated by the \\(1-e_i\\). That ideal is \\(fM\\) with \\(f=\\bigvee_i(1-e_i)=1-p\\). Indeed \\(fM\\) contains each \\(1-e_i=f(1-e_i)\\), and a \\(\\sigma\\)-weakly closed right ideal \\(gM\\) containing every \\(1-e_i\\) has \\(g\\ge1-e_i\\) for all \\(i\\), hence \\(g\\ge f\\) and \\(fM\\subseteq gM\\). So \\(\\bigcap_iM_*e_i=((1-p)M)^\\circ=M_*p\\).\n\nFor the third identity, let \\(W\\) be the closed span of \\(\\bigcup_iM_*e_i\\). Then \\(W^\\circ=\\bigcap_i(1-e_i)M\\). An \\(x\\in M\\) lies in every \\((1-e_i)M\\) exactly when \\(e_ix=0\\) for all \\(i\\), that is, when the range of \\(x\\) lies in \\(\\bigcap_i(1-e_i)H\\), that is, when \\(x=(1-q)x\\). So \\(W^\\circ=(1-q)M\\) and \\(W=W^{\\circ\\circ}=((1-q)M)^\\circ=M_*q\\). The other two identities follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\) of Theorem 4.3. \\(\\square\\)\n\n",
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      "name": "4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals",
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      "full_conditions_and_proof": "## 4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals\n\nMultiplication by elements of \\(A\\) acts on \\(A^*\\) from both sides. The norm-closed subspaces of \\(A^*\\) that are invariant under these actions are exactly the polars of the weak\\(^*\\) closed one-sided ideals of \\(\\tilde A\\), and each of them is cut out by a projection of \\(\\tilde A\\). This correspondence attaches projections to representations in Section 5 and to functionals in Section 11.\n\n**Definition 4.1.** Let \\(A\\) be a \\(C^*\\)-algebra, acting on \\(A^*\\) by (0.1) through \\(j\\): \\((af)(x)=f(xa)\\) and \\((fa)(x)=f(ax)\\). A subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), *right invariant* if \\(Va\\subseteq V\\) for all \\(a\\in A\\), and *invariant* if it is both.\n\nTwo facts organise what follows.\n\n1. Let \\(M\\) be a von Neumann algebra, regarded as a \\(C^*\\)-algebra. Then \\(M_*\\) is a norm-closed invariant subspace of \\(M^*\\). It is invariant because multiplication by a fixed element is \\(\\sigma\\)-weakly continuous, and it is norm closed by the background fact on preduals.\n2. The algebra \\(\\tilde A=A^{**}\\) also acts on \\(A^*\\), since \\(A^*\\) is its predual. For a norm-closed subspace \\(V\\subseteq A^*\\), invariance under \\(A\\) and invariance under \\(\\tilde A\\) are the same thing. This is [Theorem 4.3](#oa-fnd-wa-07)(1) with \\(S=j(A)\\). So it makes no difference whether \"invariant\" refers to \\(A\\) or to \\(\\tilde A\\).\n\nThe correspondence rests on a description of the weak\\(^*\\) closed one-sided ideals of a von Neumann algebra.\n\n**Lemma 4.2.** Let \\(J\\) be a right ideal of a von Neumann algebra \\(M\\), closed in the \\(\\sigma\\)-weak topology. There is a unique projection \\(p\\in M\\) with \\(J=pM\\). It lies in \\(J\\) and is a left identity for \\(J\\). Symmetrically, a \\(\\sigma\\)-weakly closed left ideal is \\(Mp\\) for a unique projection \\(p\\), and a \\(\\sigma\\)-weakly closed two-sided ideal is \\(Mp\\) with \\(p\\) central.\n\n**Proof.** If \\(x\\in J\\) then \\(xx^*\\in J\\). Let \\(h\\in J\\) be positive. For \\(\\varepsilon>0\\) the element \\(h(h+\\varepsilon)^{-1}\\) lies in \\(J\\), since \\(J\\) is a right ideal. These elements increase to \\(s(h)\\) as \\(\\varepsilon\\downarrow0\\), strongly and \\(\\sigma\\)-weakly (background fact on supports). So \\(s(h)\\in J\\). If \\(h,k\\in J\\) are positive, then \\(h+k\\in J\\) and \\(\\ker(h+k)=\\ker h\\cap\\ker k\\), because \\(\\langle(h+k)\\zeta,\\zeta\\rangle=0\\) forces both terms to vanish; so \\(s(h+k)=s(h)\\vee s(k)\\). Hence the supports of the positive elements of \\(J\\) form an upward directed family of projections in \\(J\\). Its least upper bound \\(p\\) is its strong and \\(\\sigma\\)-weak limit (background fact on monotone nets), so \\(p\\in J\\). For \\(x\\in J\\) we have \\(s(xx^*)\\le p\\), so \\((1-p)xx^*=0\\) and \\(\\|(1-p)x\\|^2=\\|(1-p)xx^*(1-p)\\|=0\\). Thus \\(x=px\\), and \\(J\\subseteq pM\\). Conversely \\(pM\\subseteq J\\) because \\(p\\in J\\).\n\nIf also \\(J=p'M\\), then \\(p'\\in pM\\) gives \\(pp'=p'\\), and \\(p\\in p'M\\) gives \\(p'p=p\\). Hence \\(p=p^*=(p'p)^*=pp'=p'\\).\n\nA \\(\\sigma\\)-weakly closed left ideal \\(L\\) has \\(L^*\\) a \\(\\sigma\\)-weakly closed right ideal, since the involution is \\(\\sigma\\)-weakly continuous. So \\(L^*=pM\\) and \\(L=Mp\\). A \\(\\sigma\\)-weakly closed two-sided ideal \\(J\\) is then both \\(pM\\) and \\(Mp'\\) with \\(p,p'\\in J\\). From \\(p'\\in pM\\) and \\(p\\in Mp'\\) we get \\(pp'=p'\\) and \\(pp'=p\\), so \\(p=p'\\). Thus \\(J=pM=Mp\\) is self-adjoint, and \\(p\\) is central: for \\(x\\in M\\), \\(xp\\in pM\\) and \\(px\\in Mp\\) give \\(xp=pxp=px\\). (For self-adjoint ideals this is the background fact on weak\\(^*\\) closed ideals; the lemma shows that self-adjointness is automatic.) \\(\\square\\)\n\n**Theorem 4.3** (Invariant subspaces and weak\\(^*\\) closed ideals). Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(S\\subseteq M\\) have \\(\\sigma\\)-weakly dense linear span, and let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(sV\\subseteq V\\) (resp. \\(Vs\\subseteq V\\)) for every \\(s\\in S\\). Then \\(V\\) is left (resp. right) invariant under all of \\(M\\).\n2. Taking polars, \\(V\\mapsto V^\\circ\\), is a bijection from the norm-closed subspaces \\(V\\subseteq M_*\\) that are invariant on the left (resp. right) onto the right (resp. left) ideals \\(J\\) of \\(M\\) that are \\(\\sigma\\)-weakly closed; its inverse is \\(J\\mapsto J^\\circ\\).\n3. A norm-closed subspace \\(V\\) invariant on the left (resp. right) has the form \\(V=M_*e\\) (resp. \\(V=eM_*\\)) for a unique projection \\(e\\in M\\), with \\(V^\\circ=(1-e)M\\) (resp. \\(M(1-e)\\)). This \\(e\\) is called the *support* of \\(V\\). A norm-closed \\(V\\) invariant on the left is invariant on both sides exactly when \\(e\\) is central, and then \\(M_*e=eM_*\\).\n\nFor a \\(C^*\\)-algebra \\(A\\), apply this to \\(M=\\tilde A\\), \\(M_*=A^*\\) and \\(S=j(A)\\). Then the norm-closed subspaces of \\(A^*\\) that are invariant on the left (resp. right) under \\(A\\), as in [Definition 4.1](#oa-fnd-wa-05), match the right (resp. left) ideals of \\(\\tilde A\\) that are \\(\\sigma\\)-weakly closed, and each such subspace is \\(A^*e\\) (resp. \\(eA^*\\)) for a unique projection \\(e\\in\\tilde A\\).\n\n**Proof.** We need two duality facts. If \\(V\\subseteq M_*\\) is a norm-closed subspace, then \\(V^{\\circ\\circ}=V\\): if \\(\\varphi_0\\notin V\\), the Hahn–Banach theorem gives a bounded functional on \\(M_*\\) that vanishes on \\(V\\) and not at \\(\\varphi_0\\); by \\(M=(M_*)^*\\) it is evaluation at some \\(x\\in V^\\circ\\), so \\(\\varphi_0\\notin V^{\\circ\\circ}\\). If \\(J\\subseteq M\\) is a \\(\\sigma\\)-weakly closed subspace, then \\(J^{\\circ\\circ}=J\\) (background fact on bipolars).\n\n(1) Take the left case. Put \\(J=V^\\circ\\); it is a \\(\\sigma\\)-weakly closed subspace. For \\(x\\in J\\), \\(s\\in S\\) and \\(\\varphi\\in V\\) we get \\(\\varphi(xs)=(s\\varphi)(x)=0\\), so \\(xs\\in J\\). The set \\(T=\\{m\\in M:Jm\\subseteq J\\}\\) is a linear subspace containing \\(S\\). It is \\(\\sigma\\)-weakly closed, because for fixed \\(x\\in J\\) the map \\(m\\mapsto xm\\) is \\(\\sigma\\)-weakly continuous and \\(J\\) is closed. So \\(T=M\\), and \\(J\\) is a right ideal. Now \\(V=V^{\\circ\\circ}=J^\\circ\\). For \\(\\varphi\\in J^\\circ\\), \\(a\\in M\\) and \\(x\\in J\\), \\((a\\varphi)(x)=\\varphi(xa)=0\\), so \\(a\\varphi\\in J^\\circ\\). The right case is symmetric.\n\n(2) By (1) with \\(S=M\\), the polar of a norm-closed subspace invariant on the left is a right ideal, and it is \\(\\sigma\\)-weakly closed. Conversely, for a right ideal \\(J\\) closed in that topology, \\(J^\\circ\\) is norm closed, and invariant on the left by the last step of (1). The two bipolar identities show that the maps are mutually inverse.\n\n(3) Let \\(V\\) be norm closed and left invariant. By (2) and [Lemma 4.2](#oa-fnd-wa-06), \\(V^\\circ=pM\\) for a unique projection \\(p\\). Put \\(e=1-p\\). Then\n\\[\n\\begin{gathered}\nV\\\\\n=(pM)^\\circ\\\\\n=\\{\\varphi:\\varphi(px)=0\\ \\forall x\\}\\\\\n=\\{\\varphi:\\varphi p=0\\}\\\\\n=\\{\\varphi:\\varphi=\\varphi e\\}\\\\\n=M_*e .\n\\end{gathered}\n\\tag{4.1}\n\\]\nUniqueness of \\(e\\) follows from uniqueness of \\(p\\). If \\(V\\) is also right invariant, then \\(V^\\circ\\) is a \\(\\sigma\\)-weakly closed two-sided ideal. By Lemma 4.2 it is \\(Mp'\\) with \\(p'\\) central; comparing with \\(V^\\circ=pM\\) and using uniqueness gives \\(p=p'\\), so \\(e\\) is central. Conversely, if \\(e\\) is central then \\(\\varphi e=e\\varphi\\) for every \\(\\varphi\\), so \\(M_*e=eM_*\\) is invariant on both sides. The right-handed statements follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\), \\(\\varphi^*(x)=\\overline{\\varphi(x^*)}\\), which maps \\(M_*e\\) onto \\(eM_*\\): indeed \\((\\varphi e)^*=e\\varphi^*\\).\n\nFor the \\(C^*\\)-algebra statement, \\(j(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A\\), so (1) turns invariance under \\(A\\) into invariance under \\(\\tilde A\\), and (2)–(3) apply. \\(\\square\\)\n\n**Meets and joins.** For a family \\((e_i)_{i\\in I}\\) of projections of a von Neumann algebra \\(M\\subseteq B(H)\\), the projection onto \\(\\bigcap_ie_iH\\) lies in \\(M\\). Indeed every \\(T\\in M'\\) commutes with each \\(e_i\\), so \\(\\bigcap_ie_iH\\) is invariant under \\(M'\\), and \\(M'\\) is closed under adjoints; hence the projection commutes with \\(M'\\) and lies in \\(M''=M\\) by the bicommutant theorem. Write \\(\\bigwedge_ie_i\\) for it, and put \\(\\bigvee_ie_i=1-\\bigwedge_i(1-e_i)\\), the projection onto the closed span of \\(\\bigcup_ie_iH\\). They are the greatest projection below every \\(e_i\\) and the least projection above every \\(e_i\\), so they do not depend on the realisation of \\(M\\). In particular they make sense in \\(\\tilde A\\).\n\n**Proposition 4.4** (Meets and joins through invariant subspaces). Let \\(M\\) be a von Neumann algebra with projections \\((e_i)_{i\\in I}\\), and put \\(p=\\bigwedge_ie_i\\) and \\(q=\\bigvee_ie_i\\). Then\n\\[\n\\begin{gathered}\nM_*p\\\\\n=\\bigcap_iM_*e_i,\\\\\npM_*\\\\\n=\\bigcap_ie_iM_*,\\\\\nM_*q\\\\\n=\\overline{\\operatorname{span}}\\bigcup_iM_*e_i,\\\\\nqM_*\\\\\n=\\overline{\\operatorname{span}}\\bigcup_ie_iM_* .\n\\end{gathered}\n\\tag{4.2}\n\\]\n\n**Proof.** By (4.1), \\(M_*f=((1-f)M)^\\circ\\) and \\((M_*f)^\\circ=(1-f)M\\) for every projection \\(f\\). Polars turn unions into intersections: \\((\\bigcup_iV_i)^\\circ=\\bigcap_iV_i^\\circ\\), for subspaces of \\(M_*\\) and of \\(M\\) alike.\n\nFor the first identity, \\(\\bigcap_iM_*e_i=\\bigcap_i((1-e_i)M)^\\circ\\) is the polar of the \\(\\sigma\\)-weakly closed right ideal generated by the \\(1-e_i\\). That ideal is \\(fM\\) with \\(f=\\bigvee_i(1-e_i)=1-p\\). Indeed \\(fM\\) contains each \\(1-e_i=f(1-e_i)\\), and a \\(\\sigma\\)-weakly closed right ideal \\(gM\\) containing every \\(1-e_i\\) has \\(g\\ge1-e_i\\) for all \\(i\\), hence \\(g\\ge f\\) and \\(fM\\subseteq gM\\). So \\(\\bigcap_iM_*e_i=((1-p)M)^\\circ=M_*p\\).\n\nFor the third identity, let \\(W\\) be the closed span of \\(\\bigcup_iM_*e_i\\). Then \\(W^\\circ=\\bigcap_i(1-e_i)M\\). An \\(x\\in M\\) lies in every \\((1-e_i)M\\) exactly when \\(e_ix=0\\) for all \\(i\\), that is, when the range of \\(x\\) lies in \\(\\bigcap_i(1-e_i)H\\), that is, when \\(x=(1-q)x\\). So \\(W^\\circ=(1-q)M\\) and \\(W=W^{\\circ\\circ}=((1-q)M)^\\circ=M_*q\\). The other two identities follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\) of Theorem 4.3. \\(\\square\\)\n\n",
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      "name": "4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
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      "full_conditions_and_proof": "## 4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals\n\nMultiplication by elements of \\(A\\) acts on \\(A^*\\) from both sides. The norm-closed subspaces of \\(A^*\\) that are invariant under these actions are exactly the polars of the weak\\(^*\\) closed one-sided ideals of \\(\\tilde A\\), and each of them is cut out by a projection of \\(\\tilde A\\). This correspondence attaches projections to representations in Section 5 and to functionals in Section 11.\n\n**Definition 4.1.** Let \\(A\\) be a \\(C^*\\)-algebra, acting on \\(A^*\\) by (0.1) through \\(j\\): \\((af)(x)=f(xa)\\) and \\((fa)(x)=f(ax)\\). A subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), *right invariant* if \\(Va\\subseteq V\\) for all \\(a\\in A\\), and *invariant* if it is both.\n\nTwo facts organise what follows.\n\n1. Let \\(M\\) be a von Neumann algebra, regarded as a \\(C^*\\)-algebra. Then \\(M_*\\) is a norm-closed invariant subspace of \\(M^*\\). It is invariant because multiplication by a fixed element is \\(\\sigma\\)-weakly continuous, and it is norm closed by the background fact on preduals.\n2. The algebra \\(\\tilde A=A^{**}\\) also acts on \\(A^*\\), since \\(A^*\\) is its predual. For a norm-closed subspace \\(V\\subseteq A^*\\), invariance under \\(A\\) and invariance under \\(\\tilde A\\) are the same thing. This is [Theorem 4.3](#oa-fnd-wa-07)(1) with \\(S=j(A)\\). So it makes no difference whether \"invariant\" refers to \\(A\\) or to \\(\\tilde A\\).\n\nThe correspondence rests on a description of the weak\\(^*\\) closed one-sided ideals of a von Neumann algebra.\n\n**Lemma 4.2.** Let \\(J\\) be a right ideal of a von Neumann algebra \\(M\\), closed in the \\(\\sigma\\)-weak topology. There is a unique projection \\(p\\in M\\) with \\(J=pM\\). It lies in \\(J\\) and is a left identity for \\(J\\). Symmetrically, a \\(\\sigma\\)-weakly closed left ideal is \\(Mp\\) for a unique projection \\(p\\), and a \\(\\sigma\\)-weakly closed two-sided ideal is \\(Mp\\) with \\(p\\) central.\n\n**Proof.** If \\(x\\in J\\) then \\(xx^*\\in J\\). Let \\(h\\in J\\) be positive. For \\(\\varepsilon>0\\) the element \\(h(h+\\varepsilon)^{-1}\\) lies in \\(J\\), since \\(J\\) is a right ideal. These elements increase to \\(s(h)\\) as \\(\\varepsilon\\downarrow0\\), strongly and \\(\\sigma\\)-weakly (background fact on supports). So \\(s(h)\\in J\\). If \\(h,k\\in J\\) are positive, then \\(h+k\\in J\\) and \\(\\ker(h+k)=\\ker h\\cap\\ker k\\), because \\(\\langle(h+k)\\zeta,\\zeta\\rangle=0\\) forces both terms to vanish; so \\(s(h+k)=s(h)\\vee s(k)\\). Hence the supports of the positive elements of \\(J\\) form an upward directed family of projections in \\(J\\). Its least upper bound \\(p\\) is its strong and \\(\\sigma\\)-weak limit (background fact on monotone nets), so \\(p\\in J\\). For \\(x\\in J\\) we have \\(s(xx^*)\\le p\\), so \\((1-p)xx^*=0\\) and \\(\\|(1-p)x\\|^2=\\|(1-p)xx^*(1-p)\\|=0\\). Thus \\(x=px\\), and \\(J\\subseteq pM\\). Conversely \\(pM\\subseteq J\\) because \\(p\\in J\\).\n\nIf also \\(J=p'M\\), then \\(p'\\in pM\\) gives \\(pp'=p'\\), and \\(p\\in p'M\\) gives \\(p'p=p\\). Hence \\(p=p^*=(p'p)^*=pp'=p'\\).\n\nA \\(\\sigma\\)-weakly closed left ideal \\(L\\) has \\(L^*\\) a \\(\\sigma\\)-weakly closed right ideal, since the involution is \\(\\sigma\\)-weakly continuous. So \\(L^*=pM\\) and \\(L=Mp\\). A \\(\\sigma\\)-weakly closed two-sided ideal \\(J\\) is then both \\(pM\\) and \\(Mp'\\) with \\(p,p'\\in J\\). From \\(p'\\in pM\\) and \\(p\\in Mp'\\) we get \\(pp'=p'\\) and \\(pp'=p\\), so \\(p=p'\\). Thus \\(J=pM=Mp\\) is self-adjoint, and \\(p\\) is central: for \\(x\\in M\\), \\(xp\\in pM\\) and \\(px\\in Mp\\) give \\(xp=pxp=px\\). (For self-adjoint ideals this is the background fact on weak\\(^*\\) closed ideals; the lemma shows that self-adjointness is automatic.) \\(\\square\\)\n\n**Theorem 4.3** (Invariant subspaces and weak\\(^*\\) closed ideals). Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(S\\subseteq M\\) have \\(\\sigma\\)-weakly dense linear span, and let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(sV\\subseteq V\\) (resp. \\(Vs\\subseteq V\\)) for every \\(s\\in S\\). Then \\(V\\) is left (resp. right) invariant under all of \\(M\\).\n2. Taking polars, \\(V\\mapsto V^\\circ\\), is a bijection from the norm-closed subspaces \\(V\\subseteq M_*\\) that are invariant on the left (resp. right) onto the right (resp. left) ideals \\(J\\) of \\(M\\) that are \\(\\sigma\\)-weakly closed; its inverse is \\(J\\mapsto J^\\circ\\).\n3. A norm-closed subspace \\(V\\) invariant on the left (resp. right) has the form \\(V=M_*e\\) (resp. \\(V=eM_*\\)) for a unique projection \\(e\\in M\\), with \\(V^\\circ=(1-e)M\\) (resp. \\(M(1-e)\\)). This \\(e\\) is called the *support* of \\(V\\). A norm-closed \\(V\\) invariant on the left is invariant on both sides exactly when \\(e\\) is central, and then \\(M_*e=eM_*\\).\n\nFor a \\(C^*\\)-algebra \\(A\\), apply this to \\(M=\\tilde A\\), \\(M_*=A^*\\) and \\(S=j(A)\\). Then the norm-closed subspaces of \\(A^*\\) that are invariant on the left (resp. right) under \\(A\\), as in [Definition 4.1](#oa-fnd-wa-05), match the right (resp. left) ideals of \\(\\tilde A\\) that are \\(\\sigma\\)-weakly closed, and each such subspace is \\(A^*e\\) (resp. \\(eA^*\\)) for a unique projection \\(e\\in\\tilde A\\).\n\n**Proof.** We need two duality facts. If \\(V\\subseteq M_*\\) is a norm-closed subspace, then \\(V^{\\circ\\circ}=V\\): if \\(\\varphi_0\\notin V\\), the Hahn–Banach theorem gives a bounded functional on \\(M_*\\) that vanishes on \\(V\\) and not at \\(\\varphi_0\\); by \\(M=(M_*)^*\\) it is evaluation at some \\(x\\in V^\\circ\\), so \\(\\varphi_0\\notin V^{\\circ\\circ}\\). If \\(J\\subseteq M\\) is a \\(\\sigma\\)-weakly closed subspace, then \\(J^{\\circ\\circ}=J\\) (background fact on bipolars).\n\n(1) Take the left case. Put \\(J=V^\\circ\\); it is a \\(\\sigma\\)-weakly closed subspace. For \\(x\\in J\\), \\(s\\in S\\) and \\(\\varphi\\in V\\) we get \\(\\varphi(xs)=(s\\varphi)(x)=0\\), so \\(xs\\in J\\). The set \\(T=\\{m\\in M:Jm\\subseteq J\\}\\) is a linear subspace containing \\(S\\). It is \\(\\sigma\\)-weakly closed, because for fixed \\(x\\in J\\) the map \\(m\\mapsto xm\\) is \\(\\sigma\\)-weakly continuous and \\(J\\) is closed. So \\(T=M\\), and \\(J\\) is a right ideal. Now \\(V=V^{\\circ\\circ}=J^\\circ\\). For \\(\\varphi\\in J^\\circ\\), \\(a\\in M\\) and \\(x\\in J\\), \\((a\\varphi)(x)=\\varphi(xa)=0\\), so \\(a\\varphi\\in J^\\circ\\). The right case is symmetric.\n\n(2) By (1) with \\(S=M\\), the polar of a norm-closed subspace invariant on the left is a right ideal, and it is \\(\\sigma\\)-weakly closed. Conversely, for a right ideal \\(J\\) closed in that topology, \\(J^\\circ\\) is norm closed, and invariant on the left by the last step of (1). The two bipolar identities show that the maps are mutually inverse.\n\n(3) Let \\(V\\) be norm closed and left invariant. By (2) and [Lemma 4.2](#oa-fnd-wa-06), \\(V^\\circ=pM\\) for a unique projection \\(p\\). Put \\(e=1-p\\). Then\n\\[\n\\begin{gathered}\nV\\\\\n=(pM)^\\circ\\\\\n=\\{\\varphi:\\varphi(px)=0\\ \\forall x\\}\\\\\n=\\{\\varphi:\\varphi p=0\\}\\\\\n=\\{\\varphi:\\varphi=\\varphi e\\}\\\\\n=M_*e .\n\\end{gathered}\n\\tag{4.1}\n\\]\nUniqueness of \\(e\\) follows from uniqueness of \\(p\\). If \\(V\\) is also right invariant, then \\(V^\\circ\\) is a \\(\\sigma\\)-weakly closed two-sided ideal. By Lemma 4.2 it is \\(Mp'\\) with \\(p'\\) central; comparing with \\(V^\\circ=pM\\) and using uniqueness gives \\(p=p'\\), so \\(e\\) is central. Conversely, if \\(e\\) is central then \\(\\varphi e=e\\varphi\\) for every \\(\\varphi\\), so \\(M_*e=eM_*\\) is invariant on both sides. The right-handed statements follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\), \\(\\varphi^*(x)=\\overline{\\varphi(x^*)}\\), which maps \\(M_*e\\) onto \\(eM_*\\): indeed \\((\\varphi e)^*=e\\varphi^*\\).\n\nFor the \\(C^*\\)-algebra statement, \\(j(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A\\), so (1) turns invariance under \\(A\\) into invariance under \\(\\tilde A\\), and (2)–(3) apply. \\(\\square\\)\n\n**Meets and joins.** For a family \\((e_i)_{i\\in I}\\) of projections of a von Neumann algebra \\(M\\subseteq B(H)\\), the projection onto \\(\\bigcap_ie_iH\\) lies in \\(M\\). Indeed every \\(T\\in M'\\) commutes with each \\(e_i\\), so \\(\\bigcap_ie_iH\\) is invariant under \\(M'\\), and \\(M'\\) is closed under adjoints; hence the projection commutes with \\(M'\\) and lies in \\(M''=M\\) by the bicommutant theorem. Write \\(\\bigwedge_ie_i\\) for it, and put \\(\\bigvee_ie_i=1-\\bigwedge_i(1-e_i)\\), the projection onto the closed span of \\(\\bigcup_ie_iH\\). They are the greatest projection below every \\(e_i\\) and the least projection above every \\(e_i\\), so they do not depend on the realisation of \\(M\\). In particular they make sense in \\(\\tilde A\\).\n\n**Proposition 4.4** (Meets and joins through invariant subspaces). Let \\(M\\) be a von Neumann algebra with projections \\((e_i)_{i\\in I}\\), and put \\(p=\\bigwedge_ie_i\\) and \\(q=\\bigvee_ie_i\\). Then\n\\[\n\\begin{gathered}\nM_*p\\\\\n=\\bigcap_iM_*e_i,\\\\\npM_*\\\\\n=\\bigcap_ie_iM_*,\\\\\nM_*q\\\\\n=\\overline{\\operatorname{span}}\\bigcup_iM_*e_i,\\\\\nqM_*\\\\\n=\\overline{\\operatorname{span}}\\bigcup_ie_iM_* .\n\\end{gathered}\n\\tag{4.2}\n\\]\n\n**Proof.** By (4.1), \\(M_*f=((1-f)M)^\\circ\\) and \\((M_*f)^\\circ=(1-f)M\\) for every projection \\(f\\). Polars turn unions into intersections: \\((\\bigcup_iV_i)^\\circ=\\bigcap_iV_i^\\circ\\), for subspaces of \\(M_*\\) and of \\(M\\) alike.\n\nFor the first identity, \\(\\bigcap_iM_*e_i=\\bigcap_i((1-e_i)M)^\\circ\\) is the polar of the \\(\\sigma\\)-weakly closed right ideal generated by the \\(1-e_i\\). That ideal is \\(fM\\) with \\(f=\\bigvee_i(1-e_i)=1-p\\). Indeed \\(fM\\) contains each \\(1-e_i=f(1-e_i)\\), and a \\(\\sigma\\)-weakly closed right ideal \\(gM\\) containing every \\(1-e_i\\) has \\(g\\ge1-e_i\\) for all \\(i\\), hence \\(g\\ge f\\) and \\(fM\\subseteq gM\\). So \\(\\bigcap_iM_*e_i=((1-p)M)^\\circ=M_*p\\).\n\nFor the third identity, let \\(W\\) be the closed span of \\(\\bigcup_iM_*e_i\\). Then \\(W^\\circ=\\bigcap_i(1-e_i)M\\). An \\(x\\in M\\) lies in every \\((1-e_i)M\\) exactly when \\(e_ix=0\\) for all \\(i\\), that is, when the range of \\(x\\) lies in \\(\\bigcap_i(1-e_i)H\\), that is, when \\(x=(1-q)x\\). So \\(W^\\circ=(1-q)M\\) and \\(W=W^{\\circ\\circ}=((1-q)M)^\\circ=M_*q\\). The other two identities follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\) of Theorem 4.3. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-08",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
      "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "anchor": "oa-fnd-wa-08",
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      "full_conditions_and_proof": "## 4. Invariant subspaces of the dual and weak\\(^*\\) closed ideals\n\nMultiplication by elements of \\(A\\) acts on \\(A^*\\) from both sides. The norm-closed subspaces of \\(A^*\\) that are invariant under these actions are exactly the polars of the weak\\(^*\\) closed one-sided ideals of \\(\\tilde A\\), and each of them is cut out by a projection of \\(\\tilde A\\). This correspondence attaches projections to representations in Section 5 and to functionals in Section 11.\n\n**Definition 4.1.** Let \\(A\\) be a \\(C^*\\)-algebra, acting on \\(A^*\\) by (0.1) through \\(j\\): \\((af)(x)=f(xa)\\) and \\((fa)(x)=f(ax)\\). A subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), *right invariant* if \\(Va\\subseteq V\\) for all \\(a\\in A\\), and *invariant* if it is both.\n\nTwo facts organise what follows.\n\n1. Let \\(M\\) be a von Neumann algebra, regarded as a \\(C^*\\)-algebra. Then \\(M_*\\) is a norm-closed invariant subspace of \\(M^*\\). It is invariant because multiplication by a fixed element is \\(\\sigma\\)-weakly continuous, and it is norm closed by the background fact on preduals.\n2. The algebra \\(\\tilde A=A^{**}\\) also acts on \\(A^*\\), since \\(A^*\\) is its predual. For a norm-closed subspace \\(V\\subseteq A^*\\), invariance under \\(A\\) and invariance under \\(\\tilde A\\) are the same thing. This is [Theorem 4.3](#oa-fnd-wa-07)(1) with \\(S=j(A)\\). So it makes no difference whether \"invariant\" refers to \\(A\\) or to \\(\\tilde A\\).\n\nThe correspondence rests on a description of the weak\\(^*\\) closed one-sided ideals of a von Neumann algebra.\n\n**Lemma 4.2.** Let \\(J\\) be a right ideal of a von Neumann algebra \\(M\\), closed in the \\(\\sigma\\)-weak topology. There is a unique projection \\(p\\in M\\) with \\(J=pM\\). It lies in \\(J\\) and is a left identity for \\(J\\). Symmetrically, a \\(\\sigma\\)-weakly closed left ideal is \\(Mp\\) for a unique projection \\(p\\), and a \\(\\sigma\\)-weakly closed two-sided ideal is \\(Mp\\) with \\(p\\) central.\n\n**Proof.** If \\(x\\in J\\) then \\(xx^*\\in J\\). Let \\(h\\in J\\) be positive. For \\(\\varepsilon>0\\) the element \\(h(h+\\varepsilon)^{-1}\\) lies in \\(J\\), since \\(J\\) is a right ideal. These elements increase to \\(s(h)\\) as \\(\\varepsilon\\downarrow0\\), strongly and \\(\\sigma\\)-weakly (background fact on supports). So \\(s(h)\\in J\\). If \\(h,k\\in J\\) are positive, then \\(h+k\\in J\\) and \\(\\ker(h+k)=\\ker h\\cap\\ker k\\), because \\(\\langle(h+k)\\zeta,\\zeta\\rangle=0\\) forces both terms to vanish; so \\(s(h+k)=s(h)\\vee s(k)\\). Hence the supports of the positive elements of \\(J\\) form an upward directed family of projections in \\(J\\). Its least upper bound \\(p\\) is its strong and \\(\\sigma\\)-weak limit (background fact on monotone nets), so \\(p\\in J\\). For \\(x\\in J\\) we have \\(s(xx^*)\\le p\\), so \\((1-p)xx^*=0\\) and \\(\\|(1-p)x\\|^2=\\|(1-p)xx^*(1-p)\\|=0\\). Thus \\(x=px\\), and \\(J\\subseteq pM\\). Conversely \\(pM\\subseteq J\\) because \\(p\\in J\\).\n\nIf also \\(J=p'M\\), then \\(p'\\in pM\\) gives \\(pp'=p'\\), and \\(p\\in p'M\\) gives \\(p'p=p\\). Hence \\(p=p^*=(p'p)^*=pp'=p'\\).\n\nA \\(\\sigma\\)-weakly closed left ideal \\(L\\) has \\(L^*\\) a \\(\\sigma\\)-weakly closed right ideal, since the involution is \\(\\sigma\\)-weakly continuous. So \\(L^*=pM\\) and \\(L=Mp\\). A \\(\\sigma\\)-weakly closed two-sided ideal \\(J\\) is then both \\(pM\\) and \\(Mp'\\) with \\(p,p'\\in J\\). From \\(p'\\in pM\\) and \\(p\\in Mp'\\) we get \\(pp'=p'\\) and \\(pp'=p\\), so \\(p=p'\\). Thus \\(J=pM=Mp\\) is self-adjoint, and \\(p\\) is central: for \\(x\\in M\\), \\(xp\\in pM\\) and \\(px\\in Mp\\) give \\(xp=pxp=px\\). (For self-adjoint ideals this is the background fact on weak\\(^*\\) closed ideals; the lemma shows that self-adjointness is automatic.) \\(\\square\\)\n\n**Theorem 4.3** (Invariant subspaces and weak\\(^*\\) closed ideals). Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(S\\subseteq M\\) have \\(\\sigma\\)-weakly dense linear span, and let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(sV\\subseteq V\\) (resp. \\(Vs\\subseteq V\\)) for every \\(s\\in S\\). Then \\(V\\) is left (resp. right) invariant under all of \\(M\\).\n2. Taking polars, \\(V\\mapsto V^\\circ\\), is a bijection from the norm-closed subspaces \\(V\\subseteq M_*\\) that are invariant on the left (resp. right) onto the right (resp. left) ideals \\(J\\) of \\(M\\) that are \\(\\sigma\\)-weakly closed; its inverse is \\(J\\mapsto J^\\circ\\).\n3. A norm-closed subspace \\(V\\) invariant on the left (resp. right) has the form \\(V=M_*e\\) (resp. \\(V=eM_*\\)) for a unique projection \\(e\\in M\\), with \\(V^\\circ=(1-e)M\\) (resp. \\(M(1-e)\\)). This \\(e\\) is called the *support* of \\(V\\). A norm-closed \\(V\\) invariant on the left is invariant on both sides exactly when \\(e\\) is central, and then \\(M_*e=eM_*\\).\n\nFor a \\(C^*\\)-algebra \\(A\\), apply this to \\(M=\\tilde A\\), \\(M_*=A^*\\) and \\(S=j(A)\\). Then the norm-closed subspaces of \\(A^*\\) that are invariant on the left (resp. right) under \\(A\\), as in [Definition 4.1](#oa-fnd-wa-05), match the right (resp. left) ideals of \\(\\tilde A\\) that are \\(\\sigma\\)-weakly closed, and each such subspace is \\(A^*e\\) (resp. \\(eA^*\\)) for a unique projection \\(e\\in\\tilde A\\).\n\n**Proof.** We need two duality facts. If \\(V\\subseteq M_*\\) is a norm-closed subspace, then \\(V^{\\circ\\circ}=V\\): if \\(\\varphi_0\\notin V\\), the Hahn–Banach theorem gives a bounded functional on \\(M_*\\) that vanishes on \\(V\\) and not at \\(\\varphi_0\\); by \\(M=(M_*)^*\\) it is evaluation at some \\(x\\in V^\\circ\\), so \\(\\varphi_0\\notin V^{\\circ\\circ}\\). If \\(J\\subseteq M\\) is a \\(\\sigma\\)-weakly closed subspace, then \\(J^{\\circ\\circ}=J\\) (background fact on bipolars).\n\n(1) Take the left case. Put \\(J=V^\\circ\\); it is a \\(\\sigma\\)-weakly closed subspace. For \\(x\\in J\\), \\(s\\in S\\) and \\(\\varphi\\in V\\) we get \\(\\varphi(xs)=(s\\varphi)(x)=0\\), so \\(xs\\in J\\). The set \\(T=\\{m\\in M:Jm\\subseteq J\\}\\) is a linear subspace containing \\(S\\). It is \\(\\sigma\\)-weakly closed, because for fixed \\(x\\in J\\) the map \\(m\\mapsto xm\\) is \\(\\sigma\\)-weakly continuous and \\(J\\) is closed. So \\(T=M\\), and \\(J\\) is a right ideal. Now \\(V=V^{\\circ\\circ}=J^\\circ\\). For \\(\\varphi\\in J^\\circ\\), \\(a\\in M\\) and \\(x\\in J\\), \\((a\\varphi)(x)=\\varphi(xa)=0\\), so \\(a\\varphi\\in J^\\circ\\). The right case is symmetric.\n\n(2) By (1) with \\(S=M\\), the polar of a norm-closed subspace invariant on the left is a right ideal, and it is \\(\\sigma\\)-weakly closed. Conversely, for a right ideal \\(J\\) closed in that topology, \\(J^\\circ\\) is norm closed, and invariant on the left by the last step of (1). The two bipolar identities show that the maps are mutually inverse.\n\n(3) Let \\(V\\) be norm closed and left invariant. By (2) and [Lemma 4.2](#oa-fnd-wa-06), \\(V^\\circ=pM\\) for a unique projection \\(p\\). Put \\(e=1-p\\). Then\n\\[\n\\begin{gathered}\nV\\\\\n=(pM)^\\circ\\\\\n=\\{\\varphi:\\varphi(px)=0\\ \\forall x\\}\\\\\n=\\{\\varphi:\\varphi p=0\\}\\\\\n=\\{\\varphi:\\varphi=\\varphi e\\}\\\\\n=M_*e .\n\\end{gathered}\n\\tag{4.1}\n\\]\nUniqueness of \\(e\\) follows from uniqueness of \\(p\\). If \\(V\\) is also right invariant, then \\(V^\\circ\\) is a \\(\\sigma\\)-weakly closed two-sided ideal. By Lemma 4.2 it is \\(Mp'\\) with \\(p'\\) central; comparing with \\(V^\\circ=pM\\) and using uniqueness gives \\(p=p'\\), so \\(e\\) is central. Conversely, if \\(e\\) is central then \\(\\varphi e=e\\varphi\\) for every \\(\\varphi\\), so \\(M_*e=eM_*\\) is invariant on both sides. The right-handed statements follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\), \\(\\varphi^*(x)=\\overline{\\varphi(x^*)}\\), which maps \\(M_*e\\) onto \\(eM_*\\): indeed \\((\\varphi e)^*=e\\varphi^*\\).\n\nFor the \\(C^*\\)-algebra statement, \\(j(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A\\), so (1) turns invariance under \\(A\\) into invariance under \\(\\tilde A\\), and (2)–(3) apply. \\(\\square\\)\n\n**Meets and joins.** For a family \\((e_i)_{i\\in I}\\) of projections of a von Neumann algebra \\(M\\subseteq B(H)\\), the projection onto \\(\\bigcap_ie_iH\\) lies in \\(M\\). Indeed every \\(T\\in M'\\) commutes with each \\(e_i\\), so \\(\\bigcap_ie_iH\\) is invariant under \\(M'\\), and \\(M'\\) is closed under adjoints; hence the projection commutes with \\(M'\\) and lies in \\(M''=M\\) by the bicommutant theorem. Write \\(\\bigwedge_ie_i\\) for it, and put \\(\\bigvee_ie_i=1-\\bigwedge_i(1-e_i)\\), the projection onto the closed span of \\(\\bigcup_ie_iH\\). They are the greatest projection below every \\(e_i\\) and the least projection above every \\(e_i\\), so they do not depend on the realisation of \\(M\\). In particular they make sense in \\(\\tilde A\\).\n\n**Proposition 4.4** (Meets and joins through invariant subspaces). Let \\(M\\) be a von Neumann algebra with projections \\((e_i)_{i\\in I}\\), and put \\(p=\\bigwedge_ie_i\\) and \\(q=\\bigvee_ie_i\\). Then\n\\[\n\\begin{gathered}\nM_*p\\\\\n=\\bigcap_iM_*e_i,\\\\\npM_*\\\\\n=\\bigcap_ie_iM_*,\\\\\nM_*q\\\\\n=\\overline{\\operatorname{span}}\\bigcup_iM_*e_i,\\\\\nqM_*\\\\\n=\\overline{\\operatorname{span}}\\bigcup_ie_iM_* .\n\\end{gathered}\n\\tag{4.2}\n\\]\n\n**Proof.** By (4.1), \\(M_*f=((1-f)M)^\\circ\\) and \\((M_*f)^\\circ=(1-f)M\\) for every projection \\(f\\). Polars turn unions into intersections: \\((\\bigcup_iV_i)^\\circ=\\bigcap_iV_i^\\circ\\), for subspaces of \\(M_*\\) and of \\(M\\) alike.\n\nFor the first identity, \\(\\bigcap_iM_*e_i=\\bigcap_i((1-e_i)M)^\\circ\\) is the polar of the \\(\\sigma\\)-weakly closed right ideal generated by the \\(1-e_i\\). That ideal is \\(fM\\) with \\(f=\\bigvee_i(1-e_i)=1-p\\). Indeed \\(fM\\) contains each \\(1-e_i=f(1-e_i)\\), and a \\(\\sigma\\)-weakly closed right ideal \\(gM\\) containing every \\(1-e_i\\) has \\(g\\ge1-e_i\\) for all \\(i\\), hence \\(g\\ge f\\) and \\(fM\\subseteq gM\\). So \\(\\bigcap_iM_*e_i=((1-p)M)^\\circ=M_*p\\).\n\nFor the third identity, let \\(W\\) be the closed span of \\(\\bigcup_iM_*e_i\\). Then \\(W^\\circ=\\bigcap_i(1-e_i)M\\). An \\(x\\in M\\) lies in every \\((1-e_i)M\\) exactly when \\(e_ix=0\\) for all \\(i\\), that is, when the range of \\(x\\) lies in \\(\\bigcap_i(1-e_i)H\\), that is, when \\(x=(1-q)x\\). So \\(W^\\circ=(1-q)M\\) and \\(W=W^{\\circ\\circ}=((1-q)M)^\\circ=M_*q\\). The other two identities follow by applying the involution \\(\\varphi\\mapsto\\varphi^*\\) of Theorem 4.3. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "name": "5. Central supports of representations and quasi-equivalence",
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      "full_conditions_and_proof": "## 5. Central supports of representations and quasi-equivalence\n\nEach representation of \\(A\\) determines a central projection of \\(\\tilde A\\), its support. Two representations generate isomorphic von Neumann algebras, by an isomorphism compatible with \\(A\\), exactly when their supports agree.\n\n**Definition 5.1.** Representations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *quasi-equivalent*, written \\(\\pi_1\\sim\\pi_2\\), if there is a \\(*\\)-isomorphism \\(\\theta\\) of \\(\\mathcal M(\\pi_1)\\) onto \\(\\mathcal M(\\pi_2)\\) with \\(\\theta\\circ\\pi_1=\\pi_2\\). For a representation \\(\\pi\\), the *support* of \\(\\pi\\) is the central projection \\(z(\\pi)\\in\\tilde A\\) of [Lemma 2.1](#oa-fnd-wa-02)(4), so that \\(\\ker\\bar\\pi=\\tilde A(1-z(\\pi))\\). The subspace *associated with* \\(\\pi\\) is\n\\[\nV(\\pi)=\\{\\psi\\circ\\pi:\\psi\\in\\mathcal M(\\pi)_*\\}\\subseteq A^* .\n\\]\n\nWe do not ask \\(\\theta\\) to be \\(\\sigma\\)-weakly bicontinuous: every \\(*\\)-isomorphism between von Neumann algebras is normal with normal inverse (background), so that requirement would not change the relation.\n\n**Proposition 5.2.**\n\n1. \\(\\psi\\mapsto\\psi\\circ\\pi\\) is an isometry of \\(\\mathcal M(\\pi)_*\\) onto \\(V(\\pi)\\).\n2. \\(V(\\pi)=A^*z(\\pi)\\): a functional \\(f\\) lies in \\(V(\\pi)\\) exactly when \\(\\hat f=\\hat f z(\\pi)\\). In particular \\(V(\\pi)\\) is a norm-closed invariant subspace, and its support in the sense of [Theorem 4.3](#oa-fnd-wa-07)(3) is \\(z(\\pi)\\).\n3. Every norm-closed invariant subspace \\(V\\subseteq A^*\\) equals \\(V(\\rho)\\) for some representation \\(\\rho\\).\n\n**Proof.** (1) Let \\(\\psi\\in\\mathcal M(\\pi)_*\\). The functional \\(\\psi\\circ\\bar\\pi\\) is normal on \\(\\tilde A\\) and restricts to \\(\\psi\\circ\\pi\\) on \\(A\\), so it is \\(\\widehat{\\psi\\circ\\pi}\\). The norm of a functional equals the norm of its normal extension, and \\(\\bar\\pi\\) maps the unit ball onto the unit ball (Lemma 2.1(3)). Hence\n\\[\n\\begin{gathered}\n\\|\\psi\\circ\\pi\\|\\\\\n=\\sup_{\\|X\\|\\le1}|\\psi(\\bar\\pi(X))|\\\\\n=\\sup_{\\|y\\|\\le1,\\,y\\in\\mathcal M(\\pi)}|\\psi(y)|\\\\\n=\\|\\psi\\| .\n\\end{gathered}\n\\]\n\n(2) If \\(f=\\psi\\circ\\pi\\), then \\(\\hat f=\\psi\\circ\\bar\\pi\\) vanishes on \\(\\ker\\bar\\pi=\\tilde A(1-z(\\pi))\\), which says \\(\\hat f(X(1-z(\\pi)))=0\\) for all \\(X\\), that is \\(\\hat f=\\hat fz(\\pi)\\). Conversely, if \\(\\hat f\\) vanishes on \\(\\ker\\bar\\pi\\), put \\(\\psi=\\hat f\\circ(\\bar\\pi|_{\\tilde Az(\\pi)})^{-1}\\). This is normal because the inverse is normal (Lemma 2.1(4)), and \\(\\psi\\circ\\pi=f\\) because \\(\\bar\\pi(j(a)z(\\pi))=\\pi(a)\\). The support statement follows from Theorem 4.3(3), \\(z(\\pi)\\) being central.\n\n(3) By Theorem 4.3(3), \\(V=A^*z\\) for a central projection \\(z\\in\\tilde A\\). Realise \\(\\tilde A\\) on \\(H_u\\) through \\(\\bar\\pi_u\\) ([Theorem 3.3](#oa-fnd-wa-03)), let \\(K=\\bar\\pi_u(z)H_u\\), and let \\(\\rho(a)=\\pi_u(a)|_K\\). This is a representation because \\(\\bar\\pi_u(z)\\) commutes with \\(\\pi_u(A)\\). It is nondegenerate: for an approximate unit, \\(\\rho(e_i)\\to1_K\\) strongly, since \\(\\pi_u(e_i)\\to1\\) strongly. The map \\(X\\mapsto\\bar\\pi_u(X)|_K\\) is normal and extends \\(\\rho\\), so it is \\(\\bar\\rho\\) (Lemma 2.1(2)). It vanishes exactly when \\(\\bar\\pi_u(Xz)=0\\), that is when \\(Xz=0\\). So \\(z(\\rho)=z\\) and \\(V(\\rho)=A^*z=V\\) by (2). If \\(z=0\\) this is the zero representation on the zero space. \\(\\square\\)\n\n**Theorem 5.3** (Quasi-equivalence and supports). For representations \\(\\pi_1,\\pi_2\\) of \\(A\\) the following are equivalent: (i) \\(\\pi_1\\sim\\pi_2\\); (ii) \\(V(\\pi_1)=V(\\pi_2)\\); (iii) \\(z(\\pi_1)=z(\\pi_2)\\). When they hold, the isomorphism \\(\\theta\\) in (i) is unique, and \\(\\theta\\) and \\(\\theta^{-1}\\) are normal.\n\n**Proof.** (ii)\\(\\Leftrightarrow\\)(iii): \\(V(\\pi_k)=A^*z(\\pi_k)\\), and the support of an invariant subspace is unique (Theorem 4.3(3)).\n\n(iii)\\(\\Rightarrow\\)(i): let \\(z=z(\\pi_1)=z(\\pi_2)\\). By Lemma 2.1(4), each \\(\\bar\\pi_k\\) restricts to a \\(*\\)-isomorphism of \\(\\tilde Az\\) onto \\(\\mathcal M(\\pi_k)\\), normal with normal inverse. Put \\(\\theta=\\bar\\pi_2\\circ(\\bar\\pi_1|_{\\tilde Az})^{-1}\\). Since \\(\\bar\\pi_k(j(a))=\\bar\\pi_k(j(a)z)\\), we get \\(\\theta(\\pi_1(a))=\\pi_2(a)\\).\n\n(i)\\(\\Rightarrow\\)(iii): let \\(\\theta\\) be a \\(*\\)-isomorphism with \\(\\theta\\circ\\pi_1=\\pi_2\\). A \\(*\\)-isomorphism between von Neumann algebras is normal, and so is its inverse: use [Corollary 11.4](#oa-fnd-wa-24), whose proof from predual uniqueness is independent of this quasi-equivalence theorem. Then \\(\\theta\\circ\\bar\\pi_1\\) is a normal \\(*\\)-homomorphism extending \\(\\pi_2\\), so it equals \\(\\bar\\pi_2\\) (Lemma 2.1(2)). As \\(\\theta\\) is injective, \\(\\ker\\bar\\pi_2=\\ker\\bar\\pi_1\\), and so \\(z(\\pi_1)=z(\\pi_2)\\). Uniqueness of \\(\\theta\\): it is normal and determined on the \\(\\sigma\\)-weakly dense set \\(\\pi_1(A)\\). \\(\\square\\)\n\n**Remark 5.4** (Degenerate representations). With \\(N(\\pi)\\) of [Lemma 2.1](#oa-fnd-wa-02) in place of \\(\\mathcal M(\\pi)\\), everything above holds for possibly degenerate representations, with the same proofs. With \\(\\mathcal M(\\pi)=\\pi(A)''\\) it fails. Take \\(A=\\mathbb C\\), \\(\\pi_1=\\mathrm{id}\\) on \\(\\mathbb C\\) and \\(\\pi_2(\\lambda)=\\lambda\\oplus0\\) on \\(\\mathbb C^2\\). Both have support \\(1\\) and \\(V(\\pi_k)=A^*\\), but \\(\\mathcal M(\\pi_1)=\\mathbb C\\) and \\(\\mathcal M(\\pi_2)\\cong\\mathbb C^2\\) are not isomorphic.\n\nTwo examples show the bidual concretely.\n\nFor completeness the sequence-space dualities here are elementary. If \\(f\\in c_0^*\\), put \\(c_n=f(e_n)\\). Choosing phases on any finite set gives \\(\\sum_{n\\in F}|c_n|\\le\\|f\\|\\), so \\(c\\in\\ell^1\\). Finite sequences are norm dense in \\(c_0\\), hence \\(f(x)=\\sum_nc_nx_n\\), and the same finite phase tests give \\(\\|f\\|=\\|c\\|_1\\). Conversely each \\(\\ell^1\\) sequence defines such a functional. Similarly every \\(g\\in(\\ell^1)^*\\) is \\(g(c)=\\sum_nb_nc_n\\) for \\(b_n=g(e_n)\\in\\ell^\\infty\\), by density of finite sequences, and \\(\\|g\\|=\\|b\\|_\\infty\\). Thus \\(c_0^{**}=\\ell^\\infty\\), with evaluation embedding the usual inclusion.\n\n**Example 5.5** (\\(c_0^{**}=\\ell^\\infty\\)). The identity representation of \\(c_0\\) on \\(\\ell^2\\) is nondegenerate, and \\(\\mathcal M(\\mathrm{id})=\\ell^\\infty\\) (diagonal operators). Its normal functionals are \\(\\ell^1\\) (background fact on preduals), and their restrictions to \\(c_0\\) are all of \\(c_0^*=\\ell^1\\). So \\(V(\\mathrm{id})=c_0^*\\), \\(z(\\mathrm{id})=1\\), and \\(\\overline{\\mathrm{id}}:\\tilde{c_0}\\to\\ell^\\infty\\) is an isomorphism ([Proposition 5.2](#oa-fnd-wa-09) and Lemma 2.1(4)). Here \\(j(c_0)\\) is \\(\\sigma\\)-weakly dense but far from closed: the unit \\(1\\in\\ell^\\infty\\) is a weak\\(^*\\) limit of the finite sequences \\(1_{\\{1,\\dots,n\\}}\\).\n\n**Example 5.6** (\\(\\mathcal K(H)^{**}=B(H)\\)). The identity representation of \\(\\mathcal K(H)\\) is nondegenerate and irreducible, and \\(\\mathcal K(H)''=B(H)\\). We show that every positive \\(f\\in\\mathcal K(H)^*\\) is normal on \\(B(H)\\). For a finite-rank projection \\(P\\), the compression of \\(f\\) to \\(P\\mathcal K(H)P=B(PH)\\) is \\(x\\mapsto\\operatorname{Tr}(\\rho_Px)\\) for a unique \\(\\rho_P\\ge0\\) on \\(PH\\), with \\(\\operatorname{Tr}\\rho_P=f(P)\\le\\|f\\|\\), and \\(\\rho_P=P\\rho_QP\\) for \\(P\\le Q\\). So there is a positive trace-class \\(\\rho\\) with \\(P\\rho P=\\rho_P\\) and \\(\\operatorname{Tr}\\rho\\le\\|f\\|\\): put \\(\\langle\\rho\\xi,\\eta\\rangle=\\langle\\rho_P\\xi,\\eta\\rangle\\) for any \\(P\\) whose range contains \\(\\xi\\) and \\(\\eta\\), which the compatibility makes independent of \\(P\\), and note \\(\\sum_k\\langle\\rho e_k,e_k\\rangle=\\sup_P\\operatorname{Tr}\\rho_P\\le\\|f\\|\\) for an orthonormal basis \\((e_k)\\). For compact \\(x\\), \\(PxP\\to x\\) in norm along the finite-rank projections, so \\(f(x)=\\lim f(PxP)=\\lim\\operatorname{Tr}(\\rho PxP)=\\operatorname{Tr}(\\rho x)\\), and \\(x\\mapsto\\operatorname{Tr}(\\rho x)\\) is a vector series, hence normal. As positive functionals span \\(\\mathcal K(H)^*\\) (see the proof of Proposition 1.1(2)), \\(V(\\mathrm{id})=\\mathcal K(H)^*\\), so \\(z(\\mathrm{id})=1\\) and \\(\\mathcal K(H)^{**}\\cong B(H)\\).\n\nA representation of an ideal extends to the whole algebra, and the bidual shows why.\n\n**Proposition 5.7** (Extending representations from an ideal). Let \\(J\\) be a closed two-sided ideal of \\(A\\) and \\(\\pi\\) a representation of \\(J\\) on \\(H\\). There is exactly one representation \\(\\pi^A\\) of \\(A\\) on \\(H\\) with \\(\\pi^A|_J=\\pi\\). It satisfies \\(\\pi^A(a)\\pi(x)\\zeta=\\pi(ax)\\zeta\\) for \\(a\\in A\\), \\(x\\in J\\), \\(\\zeta\\in H\\), and \\(\\pi^A(A)''=\\pi(J)''\\).\n\n**Proof.** Let \\(i:J\\to A\\) be the inclusion. By the background fact on second adjoints, \\(i^{**}:J^{**}\\to\\tilde A\\) is an isometric normal \\(*\\)-homomorphism, and its range is the \\(\\sigma\\)-weak closure of \\(j_A(J)\\). This closure is a two-sided ideal of \\(\\tilde A\\): it is stable under multiplication by \\(j_A(A)\\) because \\(J\\) is an ideal and multiplication is separately continuous, and then under multiplication by \\(\\tilde A\\) by density of \\(j_A(A)\\). By [Lemma 4.2](#oa-fnd-wa-06) it equals \\(\\tilde Az\\) for a central projection \\(z\\). Let \\(\\bar\\pi:J^{**}\\to\\mathcal M(\\pi)=\\pi(J)''\\) be the normal extension ([Lemma 2.1](#oa-fnd-wa-02)). Define\n\\[\n\\pi^A(a)=\\bar\\pi\\big((i^{**})^{-1}(j_A(a)z)\\big)\\qquad(a\\in A).\n\\]\nThe map \\(a\\mapsto j_A(a)z\\) is a \\(*\\)-homomorphism into \\(\\tilde Az\\), because \\(z\\) is central; \\((i^{**})^{-1}\\) is a \\(*\\)-isomorphism of \\(\\tilde Az\\) onto \\(J^{**}\\); and \\(\\bar\\pi\\) is a \\(*\\)-homomorphism. So \\(\\pi^A\\) is a \\(*\\)-homomorphism. For \\(x\\in J\\), \\(j_A(x)=i^{**}(j_J(x))\\) lies in \\(\\tilde Az\\), so \\(\\pi^A(x)=\\bar\\pi(j_J(x))=\\pi(x)\\). Next, \\(\\pi(J)\\subseteq\\pi^A(A)\\subseteq\\bar\\pi(J^{**})=\\pi(J)''\\), so \\(\\pi^A(A)''=\\pi(J)''\\); in particular \\(\\pi^A\\) is nondegenerate. For uniqueness, any extension \\(\\rho\\) satisfies \\(\\rho(a)\\pi(x)\\zeta=\\rho(ax)\\zeta=\\pi(ax)\\zeta\\), and the vectors \\(\\pi(x)\\zeta\\) span a dense subspace. \\(\\square\\)\n\n**Exercise 5.8.** (easy) Let \\(A\\ne0\\) and let \\(\\pi=0\\) on a Hilbert space \\(H\\ne0\\). Compute \\(N(\\pi)\\), \\(\\mathcal M(\\pi)\\), \\(\\bar\\pi\\), \\(z(\\pi)\\) and \\(V(\\pi)\\), and say which statements of Lemma 2.1 would fail with \\(\\mathcal M(\\pi)\\) in place of \\(N(\\pi)\\).\n\n**Solution.** \\(N(\\pi)=\\{0\\}\\), \\(\\bar\\pi=0\\), \\(z(\\pi)=0\\) and \\(V(\\pi)=\\{0\\}\\). But \\(\\mathcal M(\\pi)=\\{0\\}''=\\mathbb C1\\). So \\(\\bar\\pi\\) is not onto \\(\\mathcal M(\\pi)\\), and the unit ball of \\(\\mathcal M(\\pi)\\) is not the image of that of \\(\\tilde A\\). With \\(N(\\pi)\\) in place of \\(\\mathcal M(\\pi)\\), [Lemma 2.1](#oa-fnd-wa-02) holds, in agreement with (2.1): \\(\\pi(A)''=\\{0\\}+\\mathbb C1\\). \\(\\square\\)\n\n",
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      "name": "5. Central supports of representations and quasi-equivalence",
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      "full_conditions_and_proof": "## 5. Central supports of representations and quasi-equivalence\n\nEach representation of \\(A\\) determines a central projection of \\(\\tilde A\\), its support. Two representations generate isomorphic von Neumann algebras, by an isomorphism compatible with \\(A\\), exactly when their supports agree.\n\n**Definition 5.1.** Representations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *quasi-equivalent*, written \\(\\pi_1\\sim\\pi_2\\), if there is a \\(*\\)-isomorphism \\(\\theta\\) of \\(\\mathcal M(\\pi_1)\\) onto \\(\\mathcal M(\\pi_2)\\) with \\(\\theta\\circ\\pi_1=\\pi_2\\). For a representation \\(\\pi\\), the *support* of \\(\\pi\\) is the central projection \\(z(\\pi)\\in\\tilde A\\) of [Lemma 2.1](#oa-fnd-wa-02)(4), so that \\(\\ker\\bar\\pi=\\tilde A(1-z(\\pi))\\). The subspace *associated with* \\(\\pi\\) is\n\\[\nV(\\pi)=\\{\\psi\\circ\\pi:\\psi\\in\\mathcal M(\\pi)_*\\}\\subseteq A^* .\n\\]\n\nWe do not ask \\(\\theta\\) to be \\(\\sigma\\)-weakly bicontinuous: every \\(*\\)-isomorphism between von Neumann algebras is normal with normal inverse (background), so that requirement would not change the relation.\n\n**Proposition 5.2.**\n\n1. \\(\\psi\\mapsto\\psi\\circ\\pi\\) is an isometry of \\(\\mathcal M(\\pi)_*\\) onto \\(V(\\pi)\\).\n2. \\(V(\\pi)=A^*z(\\pi)\\): a functional \\(f\\) lies in \\(V(\\pi)\\) exactly when \\(\\hat f=\\hat f z(\\pi)\\). In particular \\(V(\\pi)\\) is a norm-closed invariant subspace, and its support in the sense of [Theorem 4.3](#oa-fnd-wa-07)(3) is \\(z(\\pi)\\).\n3. Every norm-closed invariant subspace \\(V\\subseteq A^*\\) equals \\(V(\\rho)\\) for some representation \\(\\rho\\).\n\n**Proof.** (1) Let \\(\\psi\\in\\mathcal M(\\pi)_*\\). The functional \\(\\psi\\circ\\bar\\pi\\) is normal on \\(\\tilde A\\) and restricts to \\(\\psi\\circ\\pi\\) on \\(A\\), so it is \\(\\widehat{\\psi\\circ\\pi}\\). The norm of a functional equals the norm of its normal extension, and \\(\\bar\\pi\\) maps the unit ball onto the unit ball (Lemma 2.1(3)). Hence\n\\[\n\\begin{gathered}\n\\|\\psi\\circ\\pi\\|\\\\\n=\\sup_{\\|X\\|\\le1}|\\psi(\\bar\\pi(X))|\\\\\n=\\sup_{\\|y\\|\\le1,\\,y\\in\\mathcal M(\\pi)}|\\psi(y)|\\\\\n=\\|\\psi\\| .\n\\end{gathered}\n\\]\n\n(2) If \\(f=\\psi\\circ\\pi\\), then \\(\\hat f=\\psi\\circ\\bar\\pi\\) vanishes on \\(\\ker\\bar\\pi=\\tilde A(1-z(\\pi))\\), which says \\(\\hat f(X(1-z(\\pi)))=0\\) for all \\(X\\), that is \\(\\hat f=\\hat fz(\\pi)\\). Conversely, if \\(\\hat f\\) vanishes on \\(\\ker\\bar\\pi\\), put \\(\\psi=\\hat f\\circ(\\bar\\pi|_{\\tilde Az(\\pi)})^{-1}\\). This is normal because the inverse is normal (Lemma 2.1(4)), and \\(\\psi\\circ\\pi=f\\) because \\(\\bar\\pi(j(a)z(\\pi))=\\pi(a)\\). The support statement follows from Theorem 4.3(3), \\(z(\\pi)\\) being central.\n\n(3) By Theorem 4.3(3), \\(V=A^*z\\) for a central projection \\(z\\in\\tilde A\\). Realise \\(\\tilde A\\) on \\(H_u\\) through \\(\\bar\\pi_u\\) ([Theorem 3.3](#oa-fnd-wa-03)), let \\(K=\\bar\\pi_u(z)H_u\\), and let \\(\\rho(a)=\\pi_u(a)|_K\\). This is a representation because \\(\\bar\\pi_u(z)\\) commutes with \\(\\pi_u(A)\\). It is nondegenerate: for an approximate unit, \\(\\rho(e_i)\\to1_K\\) strongly, since \\(\\pi_u(e_i)\\to1\\) strongly. The map \\(X\\mapsto\\bar\\pi_u(X)|_K\\) is normal and extends \\(\\rho\\), so it is \\(\\bar\\rho\\) (Lemma 2.1(2)). It vanishes exactly when \\(\\bar\\pi_u(Xz)=0\\), that is when \\(Xz=0\\). So \\(z(\\rho)=z\\) and \\(V(\\rho)=A^*z=V\\) by (2). If \\(z=0\\) this is the zero representation on the zero space. \\(\\square\\)\n\n**Theorem 5.3** (Quasi-equivalence and supports). For representations \\(\\pi_1,\\pi_2\\) of \\(A\\) the following are equivalent: (i) \\(\\pi_1\\sim\\pi_2\\); (ii) \\(V(\\pi_1)=V(\\pi_2)\\); (iii) \\(z(\\pi_1)=z(\\pi_2)\\). When they hold, the isomorphism \\(\\theta\\) in (i) is unique, and \\(\\theta\\) and \\(\\theta^{-1}\\) are normal.\n\n**Proof.** (ii)\\(\\Leftrightarrow\\)(iii): \\(V(\\pi_k)=A^*z(\\pi_k)\\), and the support of an invariant subspace is unique (Theorem 4.3(3)).\n\n(iii)\\(\\Rightarrow\\)(i): let \\(z=z(\\pi_1)=z(\\pi_2)\\). By Lemma 2.1(4), each \\(\\bar\\pi_k\\) restricts to a \\(*\\)-isomorphism of \\(\\tilde Az\\) onto \\(\\mathcal M(\\pi_k)\\), normal with normal inverse. Put \\(\\theta=\\bar\\pi_2\\circ(\\bar\\pi_1|_{\\tilde Az})^{-1}\\). Since \\(\\bar\\pi_k(j(a))=\\bar\\pi_k(j(a)z)\\), we get \\(\\theta(\\pi_1(a))=\\pi_2(a)\\).\n\n(i)\\(\\Rightarrow\\)(iii): let \\(\\theta\\) be a \\(*\\)-isomorphism with \\(\\theta\\circ\\pi_1=\\pi_2\\). A \\(*\\)-isomorphism between von Neumann algebras is normal, and so is its inverse: use [Corollary 11.4](#oa-fnd-wa-24), whose proof from predual uniqueness is independent of this quasi-equivalence theorem. Then \\(\\theta\\circ\\bar\\pi_1\\) is a normal \\(*\\)-homomorphism extending \\(\\pi_2\\), so it equals \\(\\bar\\pi_2\\) (Lemma 2.1(2)). As \\(\\theta\\) is injective, \\(\\ker\\bar\\pi_2=\\ker\\bar\\pi_1\\), and so \\(z(\\pi_1)=z(\\pi_2)\\). Uniqueness of \\(\\theta\\): it is normal and determined on the \\(\\sigma\\)-weakly dense set \\(\\pi_1(A)\\). \\(\\square\\)\n\n**Remark 5.4** (Degenerate representations). With \\(N(\\pi)\\) of [Lemma 2.1](#oa-fnd-wa-02) in place of \\(\\mathcal M(\\pi)\\), everything above holds for possibly degenerate representations, with the same proofs. With \\(\\mathcal M(\\pi)=\\pi(A)''\\) it fails. Take \\(A=\\mathbb C\\), \\(\\pi_1=\\mathrm{id}\\) on \\(\\mathbb C\\) and \\(\\pi_2(\\lambda)=\\lambda\\oplus0\\) on \\(\\mathbb C^2\\). Both have support \\(1\\) and \\(V(\\pi_k)=A^*\\), but \\(\\mathcal M(\\pi_1)=\\mathbb C\\) and \\(\\mathcal M(\\pi_2)\\cong\\mathbb C^2\\) are not isomorphic.\n\nTwo examples show the bidual concretely.\n\nFor completeness the sequence-space dualities here are elementary. If \\(f\\in c_0^*\\), put \\(c_n=f(e_n)\\). Choosing phases on any finite set gives \\(\\sum_{n\\in F}|c_n|\\le\\|f\\|\\), so \\(c\\in\\ell^1\\). Finite sequences are norm dense in \\(c_0\\), hence \\(f(x)=\\sum_nc_nx_n\\), and the same finite phase tests give \\(\\|f\\|=\\|c\\|_1\\). Conversely each \\(\\ell^1\\) sequence defines such a functional. Similarly every \\(g\\in(\\ell^1)^*\\) is \\(g(c)=\\sum_nb_nc_n\\) for \\(b_n=g(e_n)\\in\\ell^\\infty\\), by density of finite sequences, and \\(\\|g\\|=\\|b\\|_\\infty\\). Thus \\(c_0^{**}=\\ell^\\infty\\), with evaluation embedding the usual inclusion.\n\n**Example 5.5** (\\(c_0^{**}=\\ell^\\infty\\)). The identity representation of \\(c_0\\) on \\(\\ell^2\\) is nondegenerate, and \\(\\mathcal M(\\mathrm{id})=\\ell^\\infty\\) (diagonal operators). Its normal functionals are \\(\\ell^1\\) (background fact on preduals), and their restrictions to \\(c_0\\) are all of \\(c_0^*=\\ell^1\\). So \\(V(\\mathrm{id})=c_0^*\\), \\(z(\\mathrm{id})=1\\), and \\(\\overline{\\mathrm{id}}:\\tilde{c_0}\\to\\ell^\\infty\\) is an isomorphism ([Proposition 5.2](#oa-fnd-wa-09) and Lemma 2.1(4)). Here \\(j(c_0)\\) is \\(\\sigma\\)-weakly dense but far from closed: the unit \\(1\\in\\ell^\\infty\\) is a weak\\(^*\\) limit of the finite sequences \\(1_{\\{1,\\dots,n\\}}\\).\n\n**Example 5.6** (\\(\\mathcal K(H)^{**}=B(H)\\)). The identity representation of \\(\\mathcal K(H)\\) is nondegenerate and irreducible, and \\(\\mathcal K(H)''=B(H)\\). We show that every positive \\(f\\in\\mathcal K(H)^*\\) is normal on \\(B(H)\\). For a finite-rank projection \\(P\\), the compression of \\(f\\) to \\(P\\mathcal K(H)P=B(PH)\\) is \\(x\\mapsto\\operatorname{Tr}(\\rho_Px)\\) for a unique \\(\\rho_P\\ge0\\) on \\(PH\\), with \\(\\operatorname{Tr}\\rho_P=f(P)\\le\\|f\\|\\), and \\(\\rho_P=P\\rho_QP\\) for \\(P\\le Q\\). So there is a positive trace-class \\(\\rho\\) with \\(P\\rho P=\\rho_P\\) and \\(\\operatorname{Tr}\\rho\\le\\|f\\|\\): put \\(\\langle\\rho\\xi,\\eta\\rangle=\\langle\\rho_P\\xi,\\eta\\rangle\\) for any \\(P\\) whose range contains \\(\\xi\\) and \\(\\eta\\), which the compatibility makes independent of \\(P\\), and note \\(\\sum_k\\langle\\rho e_k,e_k\\rangle=\\sup_P\\operatorname{Tr}\\rho_P\\le\\|f\\|\\) for an orthonormal basis \\((e_k)\\). For compact \\(x\\), \\(PxP\\to x\\) in norm along the finite-rank projections, so \\(f(x)=\\lim f(PxP)=\\lim\\operatorname{Tr}(\\rho PxP)=\\operatorname{Tr}(\\rho x)\\), and \\(x\\mapsto\\operatorname{Tr}(\\rho x)\\) is a vector series, hence normal. As positive functionals span \\(\\mathcal K(H)^*\\) (see the proof of Proposition 1.1(2)), \\(V(\\mathrm{id})=\\mathcal K(H)^*\\), so \\(z(\\mathrm{id})=1\\) and \\(\\mathcal K(H)^{**}\\cong B(H)\\).\n\nA representation of an ideal extends to the whole algebra, and the bidual shows why.\n\n**Proposition 5.7** (Extending representations from an ideal). Let \\(J\\) be a closed two-sided ideal of \\(A\\) and \\(\\pi\\) a representation of \\(J\\) on \\(H\\). There is exactly one representation \\(\\pi^A\\) of \\(A\\) on \\(H\\) with \\(\\pi^A|_J=\\pi\\). It satisfies \\(\\pi^A(a)\\pi(x)\\zeta=\\pi(ax)\\zeta\\) for \\(a\\in A\\), \\(x\\in J\\), \\(\\zeta\\in H\\), and \\(\\pi^A(A)''=\\pi(J)''\\).\n\n**Proof.** Let \\(i:J\\to A\\) be the inclusion. By the background fact on second adjoints, \\(i^{**}:J^{**}\\to\\tilde A\\) is an isometric normal \\(*\\)-homomorphism, and its range is the \\(\\sigma\\)-weak closure of \\(j_A(J)\\). This closure is a two-sided ideal of \\(\\tilde A\\): it is stable under multiplication by \\(j_A(A)\\) because \\(J\\) is an ideal and multiplication is separately continuous, and then under multiplication by \\(\\tilde A\\) by density of \\(j_A(A)\\). By [Lemma 4.2](#oa-fnd-wa-06) it equals \\(\\tilde Az\\) for a central projection \\(z\\). Let \\(\\bar\\pi:J^{**}\\to\\mathcal M(\\pi)=\\pi(J)''\\) be the normal extension ([Lemma 2.1](#oa-fnd-wa-02)). Define\n\\[\n\\pi^A(a)=\\bar\\pi\\big((i^{**})^{-1}(j_A(a)z)\\big)\\qquad(a\\in A).\n\\]\nThe map \\(a\\mapsto j_A(a)z\\) is a \\(*\\)-homomorphism into \\(\\tilde Az\\), because \\(z\\) is central; \\((i^{**})^{-1}\\) is a \\(*\\)-isomorphism of \\(\\tilde Az\\) onto \\(J^{**}\\); and \\(\\bar\\pi\\) is a \\(*\\)-homomorphism. So \\(\\pi^A\\) is a \\(*\\)-homomorphism. For \\(x\\in J\\), \\(j_A(x)=i^{**}(j_J(x))\\) lies in \\(\\tilde Az\\), so \\(\\pi^A(x)=\\bar\\pi(j_J(x))=\\pi(x)\\). Next, \\(\\pi(J)\\subseteq\\pi^A(A)\\subseteq\\bar\\pi(J^{**})=\\pi(J)''\\), so \\(\\pi^A(A)''=\\pi(J)''\\); in particular \\(\\pi^A\\) is nondegenerate. For uniqueness, any extension \\(\\rho\\) satisfies \\(\\rho(a)\\pi(x)\\zeta=\\rho(ax)\\zeta=\\pi(ax)\\zeta\\), and the vectors \\(\\pi(x)\\zeta\\) span a dense subspace. \\(\\square\\)\n\n**Exercise 5.8.** (easy) Let \\(A\\ne0\\) and let \\(\\pi=0\\) on a Hilbert space \\(H\\ne0\\). Compute \\(N(\\pi)\\), \\(\\mathcal M(\\pi)\\), \\(\\bar\\pi\\), \\(z(\\pi)\\) and \\(V(\\pi)\\), and say which statements of Lemma 2.1 would fail with \\(\\mathcal M(\\pi)\\) in place of \\(N(\\pi)\\).\n\n**Solution.** \\(N(\\pi)=\\{0\\}\\), \\(\\bar\\pi=0\\), \\(z(\\pi)=0\\) and \\(V(\\pi)=\\{0\\}\\). But \\(\\mathcal M(\\pi)=\\{0\\}''=\\mathbb C1\\). So \\(\\bar\\pi\\) is not onto \\(\\mathcal M(\\pi)\\), and the unit ball of \\(\\mathcal M(\\pi)\\) is not the image of that of \\(\\tilde A\\). With \\(N(\\pi)\\) in place of \\(\\mathcal M(\\pi)\\), [Lemma 2.1](#oa-fnd-wa-02) holds, in agreement with (2.1): \\(\\pi(A)''=\\{0\\}+\\mathbb C1\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "6. Minimal projections and finite-dimensional von Neumann algebras",
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      "full_conditions_and_proof": "## 6. Minimal projections and finite-dimensional von Neumann algebras\n\nThis section collects facts about minimal projections and finite-dimensional von Neumann algebras. They are used in several exercises, starting in Section 7. A projection \\(f\\) in an algebra \\(N\\) is *minimal* if \\(fNf=\\mathbb Cf\\) and \\(f\\ne0\\).\n\n**Lemma 6.1** (Minimal projections). Let \\(N\\) be a \\(C^*\\)-algebra and \\(f,g\\) minimal projections.\n\n1. \\(Nf\\) is a Hilbert space for \\(\\langle x,y\\rangle f=y^*x\\), and its Hilbert norm is the norm of \\(N\\).\n2. \\(\\dim fNg\\le1\\).\n3. Left multiplication \\(\\lambda(a)x=ax\\) is an irreducible representation of \\(N\\) on \\(Nf\\).\n4. If \\(N\\) is a von Neumann algebra, every bounded linear functional on \\(Nf\\) is the restriction of a normal functional on \\(N\\).\n\n**Proof.** (1) For \\(x,y\\in Nf\\), \\(y^*x=fy^*xf\\in fNf=\\mathbb Cf\\), which defines \\(\\langle x,y\\rangle\\). It is a positive definite hermitian form, and \\(\\|x\\|^2=\\|x^*x\\|=\\langle x,x\\rangle\\). \\(Nf=\\{x:xf=x\\}\\) is norm closed, hence complete. (2) If \\(0\\ne x\\in fNg\\) and \\(y\\in fNg\\), then \\(x^*y=cg\\) and \\(xx^*=\\|x\\|^2f\\), so \\(\\|x\\|^2y=xx^*y=cxg=cx\\). (3) \\(\\langle ax,y\\rangle f=y^*ax=(a^*y)^*x\\), so \\(\\lambda\\) is a \\(*\\)-representation. If \\(0\\ne x\\in Nf\\), then \\(f=\\|x\\|^{-2}x^*x\\in Nx\\), so \\(Nf\\subseteq Nx=\\lambda(N)x\\): every nonzero vector is cyclic. (4) The Riesz theorem gives \\(y\\in Nf\\) with \\(F(x)=\\langle x,y\\rangle\\), that is \\(F(x)f=y^*x\\). Choose a normal state \\(\\varphi\\) with \\(\\varphi(f)=1\\) (a vector state at a unit vector in the range of \\(f\\)). Since \\(w=\\varphi(w)f\\) for \\(w\\in fNf\\), we get \\(F(x)=\\varphi(y^*x)\\), a normal functional of \\(x\\). \\(\\square\\)\n\n**Lemma 6.2** (Infinite dimension). A von Neumann algebra \\(N\\) of infinite dimension contains an infinite sequence \\(p_1,p_2,\\dots\\) of mutually orthogonal nonzero projections. Then \\(c\\mapsto\\sum_nc_np_n\\) (a strong sum) is an isometric \\(*\\)-homomorphism of \\(\\ell^\\infty\\) into \\(N\\). Consequently \\(N\\) is neither reflexive nor norm separable.\n\n**Proof.** Suppose \\(N\\) has no infinite orthogonal family of nonzero projections. Let \\(C\\) be a maximal abelian \\(*\\)-subalgebra of \\(N\\); it exists by Zorn's lemma, and \\(C=C'\\cap N\\) is a von Neumann algebra. Let \\(r\\in C\\) be a nonzero projection that is not minimal in \\(C\\), and let \\(h\\in rCr\\) be self-adjoint and not a multiple of \\(r\\). By the background fact on spectral projections, \\(h=rhr\\) is a norm limit of real combinations of projections \\(Pr\\), where \\(P\\) runs over spectral projections of \\(h\\); these lie in the abelian algebra \\(C\\), so \\(Pr=rP\\) is a projection under \\(r\\). If every such \\(Pr\\) were \\(0\\) or \\(r\\), \\(h\\) would be a multiple of \\(r\\). So some \\(r'=Pr\\) has \\(0\\ne r'\\ne r\\). If some nonzero \\(r\\) majorised no minimal projection, split it as \\(r=r_1+r_1'\\) with both parts nonzero; retain \\(r_1\\) and record \\(r_1'\\). Each retained part still majorises no minimal projection, so repeating records infinitely many mutually orthogonal nonzero parts. This contradicts the hypothesis. Thus every nonzero projection of \\(C\\) majorises a minimal one. A maximal orthogonal family of minimal projections of \\(C\\) is then finite, say \\(q_1,\\dots,q_m\\), with sum \\(1\\), and \\(C=\\operatorname{span}\\{q_i\\}\\), since \\(cq_i\\in q_iCq_i=\\mathbb Cq_i\\). Each \\(q_i\\) is minimal in \\(N\\): a self-adjoint \\(h\\in q_iNq_i\\) commutes with every \\(q_j\\), hence with \\(C\\), so \\(h\\in C\\cap q_iNq_i=\\mathbb Cq_i\\). By Lemma 6.1(2), each \\(q_iNq_j\\) has dimension at most one, so \\(\\dim N\\le m^2\\). This proves the first claim by contraposition.\n\nFor bounded \\(c\\), the partial sums of \\(\\sum_nc_np_n\\) converge strongly, because the \\(p_n\\) are orthogonal: \\(\\|\\sum_{n=k}^lc_np_n\\zeta\\|^2=\\sum_{n=k}^l|c_n|^2\\|p_n\\zeta\\|^2\\). The map is a \\(*\\)-homomorphism, and \\(\\|\\sum_nc_np_n\\|=\\sup_n|c_n|\\). Its range is therefore a norm-closed subspace isometric to \\(\\ell^\\infty\\). Its \\(0\\)–\\(1\\) sequences are uncountably many at mutual distance one, so \\(\\ell^\\infty\\) is not separable. It is not reflexive either: \\(c_0\\) is a norm-closed subspace, and the sequence-space dualities above identify its bidual with \\(\\ell^\\infty\\), where the evaluation image misses the constant sequence \\(1\\). To justify inheritance, a closed subspace \\(Y\\) of a reflexive space \\(E\\) has weakly compact unit ball: it is weakly closed by Hahn–Banach, hence a closed subset of the weakly compact ball of \\(E\\). Hahn–Banach makes the relative weak topology the weak topology of \\(Y\\). Its evaluation image is consequently weak-star compact and closed in \\(Y^{**}\\); Goldstine makes it dense in the bidual unit ball, so it is the whole ball and \\(Y\\) is reflexive. A subspace of a separable metric space is separable: take a countable base and choose one point from each nonempty basic set in the subspace. These facts prove both conclusions for \\(N\\). \\(\\square\\)\n\n**Lemma 6.3** (Finite dimension). A finite-dimensional von Neumann algebra \\(N\\) with centre \\(\\mathbb C1\\) is isomorphic to \\(M_n(\\mathbb C)\\). A finite-dimensional von Neumann algebra is the finite direct sum of the algebras \\(Nc\\), \\(c\\) running over its minimal central projections, and each \\(Nc\\) is a matrix algebra.\n\n**Proof.** Let the centre be trivial. A nonzero projection \\(f\\) exists, and if \\(fNf\\ne\\mathbb Cf\\), a spectral projection of a non-scalar self-adjoint element of \\(fNf\\) is a smaller nonzero projection with a smaller corner; by finite dimension we reach a minimal projection \\(f\\). Let \\(\\lambda\\) be the representation of Lemma 6.1(3) on \\(Nf\\). Its kernel is a two-sided ideal, \\(\\sigma\\)-weakly closed because \\(N\\) is finite-dimensional, hence \\(N(1-c)\\) with \\(c\\) central (Lemma 4.2); as \\(\\lambda(f)\\ne0\\), \\(c=1\\) and \\(\\lambda\\) is injective. If \\(T\\) commutes with \\(\\lambda(N)\\), each spectral projection of the self-adjoint parts of \\(T\\) has an invariant range, which by irreducibility is \\(0\\) or everything; so \\(T\\) is scalar. Thus \\(\\lambda(N)'=\\mathbb C1\\) and \\(\\lambda(N)=\\lambda(N)''=B(Nf)\\cong M_n(\\mathbb C)\\), \\(n=\\dim Nf\\). In general the centre is a finite-dimensional abelian von Neumann algebra, so, as in the proof of Lemma 6.2, it is spanned by finitely many minimal central projections \\(c_1,\\dots,c_r\\) with sum \\(1\\). Then \\(N=\\bigoplus_kNc_k\\), and \\(Nc_k\\) has trivial centre. \\(\\square\\)\n\n**Lemma 6.4** (Isomorphisms of \\(B(H)\\) are spatial). Every \\(*\\)-isomorphism \\(\\theta:B(H_1)\\to B(H_2)\\) is \\(x\\mapsto UxU^*\\) for a unitary \\(U:H_1\\to H_2\\).\n\n**Proof.** If \\(H_1=0\\), then \\(B(H_2)=0\\), so \\(H_2=0\\), and the unique map of zero Hilbert spaces is the required unitary. Otherwise \\(H_2\\ne0\\). Fix a unit vector \\(\\xi\\in H_1\\) and let \\(P\\) be the projection onto \\(\\mathbb C\\xi\\). Then \\(PB(H_1)P=\\mathbb CP\\), so \\(\\theta(P)\\) is a nonzero projection \\(Q\\) with \\(QB(H_2)Q=\\mathbb CQ\\), hence of rank one: \\(Q\\) projects onto \\(\\mathbb C\\eta\\), \\(\\|\\eta\\|=1\\). For \\(x\\in B(H_1)\\), \\(Q\\theta(x^*x)Q=\\theta(Px^*xP)=\\|x\\xi\\|^2Q\\), so \\(\\|\\theta(x)\\eta\\|=\\|x\\xi\\|\\). Hence \\(U(x\\xi)=\\theta(x)\\eta\\) is a well-defined isometry from \\(B(H_1)\\xi=H_1\\) onto \\(\\theta(B(H_1))\\eta=B(H_2)\\eta=H_2\\). Finally \\(Ux(y\\xi)=\\theta(xy)\\eta=\\theta(x)U(y\\xi)\\), so \\(UxU^*=\\theta(x)\\). \\(\\square\\)\n\n**Lemma 6.5** (Finite joins of minimal projections). Let \\(N\\) be a von Neumann algebra containing minimal projections \\(e_1,\\dots,e_n\\). Then \\(e=\\bigvee_ke_k\\) is a sum of \\(m\\le n\\) mutually orthogonal minimal projections, and \\(\\dim eNe\\le m^2\\).\n\n**Proof.** Consider two projections: \\(e_1\\) and a minimal \\(g\\). The range of \\(e_1\\vee g\\) is the closure of \\(e_1H+(1-e_1)gH\\), so \\(e_1\\vee g=e_1+r\\), where \\(r\\) is the range projection of \\(x=(1-e_1)g\\). By polar decomposition, \\(r=uu^*\\) and \\(u^*u\\) is the support of \\(x^*x=gx^*xg\\), which lies under \\(g\\); as \\(g\\) is minimal, \\(u^*u\\in\\{0,g\\}\\). If \\(u^*u=g\\), then \\(rNr=u(gNg)u^*=\\mathbb Cr\\), so \\(r\\) is minimal; otherwise \\(r=0\\). Induction on \\(n\\) gives the first claim, and Lemma 6.1(2) gives \\(\\dim eNe\\le m^2\\). \\(\\square\\)\n\n**Exercise 6.6** (hard) (Compact unitary groups). For a von Neumann algebra \\(M\\) with unitary group \\(\\mathcal U_M\\), prove that \\(\\mathcal U_M\\) is compact exactly when \\(M\\) is \\(*\\)-isomorphic to a direct sum of full matrix algebras \\(M_{n_i}(\\mathbb C)\\). Deduce compactness of \\(\\mathcal U_M\\) for every abelian \\(M\\) in which each nonzero projection majorises a minimal one.\n\nWe use the \\(\\sigma\\)-weak topology. On \\(\\mathcal U_M\\) it coincides with the weak, strong and strong\\(^*\\) topologies of any faithful normal representation: on bounded sets \\(\\sigma\\)-weak and weak agree, and for unitaries \\(\\|(u_\\alpha-u)\\zeta\\|^2=2\\|\\zeta\\|^2-2\\operatorname{Re}\\langle u_\\alpha\\zeta,u\\zeta\\rangle\\) and \\(\\|(u_\\alpha^*-u^*)\\zeta\\|^2=2\\|\\zeta\\|^2-2\\operatorname{Re}\\langle u_\\alpha u^*\\zeta,\\zeta\\rangle\\). A \\(*\\)-isomorphism between von Neumann algebras is \\(\\sigma\\)-weakly bicontinuous (background; see also [Corollary 11.4](#oa-fnd-wa-24)), so this topology does not depend on the representation. Multiplication is jointly strongly continuous on bounded sets and \\(u\\mapsto u^*\\) is strong\\(^*\\) continuous, so \\(\\mathcal U_M\\) is a Hausdorff topological group. \"Direct sum\" means an \\(\\ell^\\infty\\)-direct sum over an arbitrary index set \\(I\\), each \\(n_i\\) finite. (In the norm topology, \\(\\mathcal U_M\\) is compact exactly when \\(\\dim M<\\infty\\): by Lemma 6.2 an infinite-dimensional \\(M\\) contains unitaries \\(1-2p_n\\) at mutual distance \\(2\\).)\n\n**Solution.** For \\(M=0\\), the unitary group has one element and the direct sum is empty. Otherwise, (\\(\\Leftarrow\\)) let \\(M=\\bigoplus_iM_{n_i}(\\mathbb C)\\) act on \\(\\bigoplus_i\\mathbb C^{n_i}\\). A bounded net converges weakly exactly when every coordinate converges, because vectors supported in single summands span a dense subspace. So \\(\\mathcal U_M\\) is homeomorphic to the product \\(\\prod_iU(n_i)\\), which is compact by Tychonoff's theorem. An isomorphic algebra has a homeomorphic unitary group, because \\(*\\)-isomorphisms are \\(\\sigma\\)-weakly bicontinuous.\n\n(\\(\\Rightarrow\\)) Let \\(\\mathcal U_M\\) be compact, realise \\(M\\) on \\(H\\), and let \\(\\mu\\) be the Haar probability measure of \\(\\mathcal U_M\\) ([existence of Haar measure](haar-measure.md#oa-fnd-hm-05)). It is invariant under left translation, including in the form \\(\\int F(vu)\\,d\\mu(u)=\\int F\\,d\\mu\\) for continuous \\(F\\), and it gives positive mass to nonempty open sets ([uniqueness and positivity of Haar measure](haar-measure.md#oa-fnd-hm-06)).\n\n*Step 1: a nonzero finite-dimensional invariant subspace.* Fix a unit vector \\(\\xi\\) and the projection \\(P\\) onto \\(\\mathbb C\\xi\\). The function \\(u\\mapsto uPu^*=\\langle\\cdot,u\\xi\\rangle u\\xi\\) is norm continuous, since \\(\\|uPu^*-vPv^*\\|\\le2\\|u\\xi-v\\xi\\|\\). Define \\(T\\) by \\(\\langle T\\zeta,\\zeta'\\rangle=\\int\\langle uPu^*\\zeta,\\zeta'\\rangle\\,d\\mu(u)\\). This is a bounded positive operator. Left invariance gives \\(vTv^*=T\\) for \\(v\\in\\mathcal U_M\\), so \\(T\\) commutes with \\(\\mathcal U_M\\), hence with \\(M\\), since the unitaries of \\(M\\) span \\(M\\): \\(T\\in M'\\). Next, \\(\\langle T\\xi,\\xi\\rangle=\\int|\\langle u\\xi,\\xi\\rangle|^2d\\mu(u)>0\\), since the integrand is continuous, nonnegative, and equal to \\(1\\) at \\(u=1\\). Also \\(T\\) is compact. Given \\(\\varepsilon>0\\), cover \\(\\mathcal U_M\\) by finitely many open sets on which \\(uPu^*\\) varies by less than \\(\\varepsilon\\), refine them to a Borel partition \\(B_1,\\dots,B_m\\), and pick \\(u_k\\in B_k\\). Then \\(\\|T-\\sum_k\\mu(B_k)u_kPu_k^*\\|\\le\\varepsilon\\), so \\(T\\) is a norm limit of finite-rank operators. For a positive compact \\(T\\ne0\\), \\(\\lambda=\\|T\\|\\) is an eigenvalue: choose unit \\(\\zeta_n\\) with \\(\\langle T\\zeta_n,\\zeta_n\\rangle\\to\\lambda\\); then \\(\\|T\\zeta_n-\\lambda\\zeta_n\\|^2\\le2\\lambda^2-2\\lambda\\langle T\\zeta_n,\\zeta_n\\rangle\\to0\\), and a subsequence of \\(T\\zeta_n\\) converges, to some \\(\\zeta\\) with \\(\\|\\zeta\\|=\\lambda\\) and \\(T\\zeta=\\lambda\\zeta\\). The eigenspace \\(E\\) is finite-dimensional: an infinite orthonormal sequence \\(\\zeta_k\\) in it would have \\(\\|T\\zeta_k-T\\zeta_l\\|=\\lambda\\sqrt2\\), contradicting compactness. Since \\(T\\in M'\\), \\(E\\) is invariant under \\(M\\).\n\n*Step 2: \\(H\\) is a direct sum of finite-dimensional invariant subspaces.* Take a maximal family of mutually orthogonal nonzero finite-dimensional \\(M\\)-invariant subspaces, and let \\(K\\) be their closed sum. \\(K^\\perp\\) is invariant because \\(M\\) is closed under adjoints. If \\(K^\\perp\\ne0\\), repeat Step 1 with \\(\\xi\\in K^\\perp\\): the operator \\(T\\) then maps into \\(K^\\perp\\), since \\(\\langle uPu^*\\zeta,\\zeta'\\rangle=0\\) for \\(\\zeta'\\in K\\), and its eigenspace lies in \\(K^\\perp\\). This contradicts maximality. So \\(H=\\bigoplus_kH_k\\) with each \\(H_k\\) finite-dimensional and invariant.\n\n*Step 3: the structure of \\(M\\).* The restriction \\(\\rho_k(x)=x|_{H_k}\\) is a normal \\(*\\)-homomorphism. Its kernel is \\(M(1-z_k)\\) with \\(z_k\\) central (Lemma 4.2), and \\(Mz_k\\cong\\rho_k(M)\\) is finite-dimensional. By Lemma 6.3, \\(z_k\\) is a finite sum of minimal central projections \\(c\\) of \\(M\\) with \\(Mc\\) a matrix algebra; these are minimal central in \\(M\\) because \\(z_k\\) is central. Since \\(\\bigcap_k\\ker\\rho_k=0\\), we have \\(\\bigvee_kz_k=1\\). Distinct minimal central projections are orthogonal, so the family of all minimal central projections \\(c\\) that occur has sum \\(1\\). Then \\(x\\mapsto(xc)_c\\) is an isomorphism of \\(M\\) onto \\(\\bigoplus_cMc\\cong\\bigoplus_cM_{n_c}(\\mathbb C)\\): it is injective because \\(\\sum_cc=1\\), and onto because a bounded family \\((x_c)\\) has the strong sum \\(\\sum_cx_c\\in M\\).\n\n*Atomic abelian algebras.* Let \\(M\\) be abelian and atomic (every nonzero projection majorises a minimal one). A maximal orthogonal family \\((e_i)\\) of minimal projections has sum \\(1\\), and \\(Me_i=\\mathbb Ce_i\\). So \\(M\\cong\\ell^\\infty(I)=\\bigoplus_iM_1(\\mathbb C)\\), whose unitary group \\(\\mathbb T^I\\) is compact. \\(\\square\\)\n\n",
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      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "6. Minimal projections and finite-dimensional von Neumann algebras",
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      "full_conditions_and_proof": "## 6. Minimal projections and finite-dimensional von Neumann algebras\n\nThis section collects facts about minimal projections and finite-dimensional von Neumann algebras. They are used in several exercises, starting in Section 7. A projection \\(f\\) in an algebra \\(N\\) is *minimal* if \\(fNf=\\mathbb Cf\\) and \\(f\\ne0\\).\n\n**Lemma 6.1** (Minimal projections). Let \\(N\\) be a \\(C^*\\)-algebra and \\(f,g\\) minimal projections.\n\n1. \\(Nf\\) is a Hilbert space for \\(\\langle x,y\\rangle f=y^*x\\), and its Hilbert norm is the norm of \\(N\\).\n2. \\(\\dim fNg\\le1\\).\n3. Left multiplication \\(\\lambda(a)x=ax\\) is an irreducible representation of \\(N\\) on \\(Nf\\).\n4. If \\(N\\) is a von Neumann algebra, every bounded linear functional on \\(Nf\\) is the restriction of a normal functional on \\(N\\).\n\n**Proof.** (1) For \\(x,y\\in Nf\\), \\(y^*x=fy^*xf\\in fNf=\\mathbb Cf\\), which defines \\(\\langle x,y\\rangle\\). It is a positive definite hermitian form, and \\(\\|x\\|^2=\\|x^*x\\|=\\langle x,x\\rangle\\). \\(Nf=\\{x:xf=x\\}\\) is norm closed, hence complete. (2) If \\(0\\ne x\\in fNg\\) and \\(y\\in fNg\\), then \\(x^*y=cg\\) and \\(xx^*=\\|x\\|^2f\\), so \\(\\|x\\|^2y=xx^*y=cxg=cx\\). (3) \\(\\langle ax,y\\rangle f=y^*ax=(a^*y)^*x\\), so \\(\\lambda\\) is a \\(*\\)-representation. If \\(0\\ne x\\in Nf\\), then \\(f=\\|x\\|^{-2}x^*x\\in Nx\\), so \\(Nf\\subseteq Nx=\\lambda(N)x\\): every nonzero vector is cyclic. (4) The Riesz theorem gives \\(y\\in Nf\\) with \\(F(x)=\\langle x,y\\rangle\\), that is \\(F(x)f=y^*x\\). Choose a normal state \\(\\varphi\\) with \\(\\varphi(f)=1\\) (a vector state at a unit vector in the range of \\(f\\)). Since \\(w=\\varphi(w)f\\) for \\(w\\in fNf\\), we get \\(F(x)=\\varphi(y^*x)\\), a normal functional of \\(x\\). \\(\\square\\)\n\n**Lemma 6.2** (Infinite dimension). A von Neumann algebra \\(N\\) of infinite dimension contains an infinite sequence \\(p_1,p_2,\\dots\\) of mutually orthogonal nonzero projections. Then \\(c\\mapsto\\sum_nc_np_n\\) (a strong sum) is an isometric \\(*\\)-homomorphism of \\(\\ell^\\infty\\) into \\(N\\). Consequently \\(N\\) is neither reflexive nor norm separable.\n\n**Proof.** Suppose \\(N\\) has no infinite orthogonal family of nonzero projections. Let \\(C\\) be a maximal abelian \\(*\\)-subalgebra of \\(N\\); it exists by Zorn's lemma, and \\(C=C'\\cap N\\) is a von Neumann algebra. Let \\(r\\in C\\) be a nonzero projection that is not minimal in \\(C\\), and let \\(h\\in rCr\\) be self-adjoint and not a multiple of \\(r\\). By the background fact on spectral projections, \\(h=rhr\\) is a norm limit of real combinations of projections \\(Pr\\), where \\(P\\) runs over spectral projections of \\(h\\); these lie in the abelian algebra \\(C\\), so \\(Pr=rP\\) is a projection under \\(r\\). If every such \\(Pr\\) were \\(0\\) or \\(r\\), \\(h\\) would be a multiple of \\(r\\). So some \\(r'=Pr\\) has \\(0\\ne r'\\ne r\\). If some nonzero \\(r\\) majorised no minimal projection, split it as \\(r=r_1+r_1'\\) with both parts nonzero; retain \\(r_1\\) and record \\(r_1'\\). Each retained part still majorises no minimal projection, so repeating records infinitely many mutually orthogonal nonzero parts. This contradicts the hypothesis. Thus every nonzero projection of \\(C\\) majorises a minimal one. A maximal orthogonal family of minimal projections of \\(C\\) is then finite, say \\(q_1,\\dots,q_m\\), with sum \\(1\\), and \\(C=\\operatorname{span}\\{q_i\\}\\), since \\(cq_i\\in q_iCq_i=\\mathbb Cq_i\\). Each \\(q_i\\) is minimal in \\(N\\): a self-adjoint \\(h\\in q_iNq_i\\) commutes with every \\(q_j\\), hence with \\(C\\), so \\(h\\in C\\cap q_iNq_i=\\mathbb Cq_i\\). By Lemma 6.1(2), each \\(q_iNq_j\\) has dimension at most one, so \\(\\dim N\\le m^2\\). This proves the first claim by contraposition.\n\nFor bounded \\(c\\), the partial sums of \\(\\sum_nc_np_n\\) converge strongly, because the \\(p_n\\) are orthogonal: \\(\\|\\sum_{n=k}^lc_np_n\\zeta\\|^2=\\sum_{n=k}^l|c_n|^2\\|p_n\\zeta\\|^2\\). The map is a \\(*\\)-homomorphism, and \\(\\|\\sum_nc_np_n\\|=\\sup_n|c_n|\\). Its range is therefore a norm-closed subspace isometric to \\(\\ell^\\infty\\). Its \\(0\\)–\\(1\\) sequences are uncountably many at mutual distance one, so \\(\\ell^\\infty\\) is not separable. It is not reflexive either: \\(c_0\\) is a norm-closed subspace, and the sequence-space dualities above identify its bidual with \\(\\ell^\\infty\\), where the evaluation image misses the constant sequence \\(1\\). To justify inheritance, a closed subspace \\(Y\\) of a reflexive space \\(E\\) has weakly compact unit ball: it is weakly closed by Hahn–Banach, hence a closed subset of the weakly compact ball of \\(E\\). Hahn–Banach makes the relative weak topology the weak topology of \\(Y\\). Its evaluation image is consequently weak-star compact and closed in \\(Y^{**}\\); Goldstine makes it dense in the bidual unit ball, so it is the whole ball and \\(Y\\) is reflexive. A subspace of a separable metric space is separable: take a countable base and choose one point from each nonempty basic set in the subspace. These facts prove both conclusions for \\(N\\). \\(\\square\\)\n\n**Lemma 6.3** (Finite dimension). A finite-dimensional von Neumann algebra \\(N\\) with centre \\(\\mathbb C1\\) is isomorphic to \\(M_n(\\mathbb C)\\). A finite-dimensional von Neumann algebra is the finite direct sum of the algebras \\(Nc\\), \\(c\\) running over its minimal central projections, and each \\(Nc\\) is a matrix algebra.\n\n**Proof.** Let the centre be trivial. A nonzero projection \\(f\\) exists, and if \\(fNf\\ne\\mathbb Cf\\), a spectral projection of a non-scalar self-adjoint element of \\(fNf\\) is a smaller nonzero projection with a smaller corner; by finite dimension we reach a minimal projection \\(f\\). Let \\(\\lambda\\) be the representation of Lemma 6.1(3) on \\(Nf\\). Its kernel is a two-sided ideal, \\(\\sigma\\)-weakly closed because \\(N\\) is finite-dimensional, hence \\(N(1-c)\\) with \\(c\\) central (Lemma 4.2); as \\(\\lambda(f)\\ne0\\), \\(c=1\\) and \\(\\lambda\\) is injective. If \\(T\\) commutes with \\(\\lambda(N)\\), each spectral projection of the self-adjoint parts of \\(T\\) has an invariant range, which by irreducibility is \\(0\\) or everything; so \\(T\\) is scalar. Thus \\(\\lambda(N)'=\\mathbb C1\\) and \\(\\lambda(N)=\\lambda(N)''=B(Nf)\\cong M_n(\\mathbb C)\\), \\(n=\\dim Nf\\). In general the centre is a finite-dimensional abelian von Neumann algebra, so, as in the proof of Lemma 6.2, it is spanned by finitely many minimal central projections \\(c_1,\\dots,c_r\\) with sum \\(1\\). Then \\(N=\\bigoplus_kNc_k\\), and \\(Nc_k\\) has trivial centre. \\(\\square\\)\n\n**Lemma 6.4** (Isomorphisms of \\(B(H)\\) are spatial). Every \\(*\\)-isomorphism \\(\\theta:B(H_1)\\to B(H_2)\\) is \\(x\\mapsto UxU^*\\) for a unitary \\(U:H_1\\to H_2\\).\n\n**Proof.** If \\(H_1=0\\), then \\(B(H_2)=0\\), so \\(H_2=0\\), and the unique map of zero Hilbert spaces is the required unitary. Otherwise \\(H_2\\ne0\\). Fix a unit vector \\(\\xi\\in H_1\\) and let \\(P\\) be the projection onto \\(\\mathbb C\\xi\\). Then \\(PB(H_1)P=\\mathbb CP\\), so \\(\\theta(P)\\) is a nonzero projection \\(Q\\) with \\(QB(H_2)Q=\\mathbb CQ\\), hence of rank one: \\(Q\\) projects onto \\(\\mathbb C\\eta\\), \\(\\|\\eta\\|=1\\). For \\(x\\in B(H_1)\\), \\(Q\\theta(x^*x)Q=\\theta(Px^*xP)=\\|x\\xi\\|^2Q\\), so \\(\\|\\theta(x)\\eta\\|=\\|x\\xi\\|\\). Hence \\(U(x\\xi)=\\theta(x)\\eta\\) is a well-defined isometry from \\(B(H_1)\\xi=H_1\\) onto \\(\\theta(B(H_1))\\eta=B(H_2)\\eta=H_2\\). Finally \\(Ux(y\\xi)=\\theta(xy)\\eta=\\theta(x)U(y\\xi)\\), so \\(UxU^*=\\theta(x)\\). \\(\\square\\)\n\n**Lemma 6.5** (Finite joins of minimal projections). Let \\(N\\) be a von Neumann algebra containing minimal projections \\(e_1,\\dots,e_n\\). Then \\(e=\\bigvee_ke_k\\) is a sum of \\(m\\le n\\) mutually orthogonal minimal projections, and \\(\\dim eNe\\le m^2\\).\n\n**Proof.** Consider two projections: \\(e_1\\) and a minimal \\(g\\). The range of \\(e_1\\vee g\\) is the closure of \\(e_1H+(1-e_1)gH\\), so \\(e_1\\vee g=e_1+r\\), where \\(r\\) is the range projection of \\(x=(1-e_1)g\\). By polar decomposition, \\(r=uu^*\\) and \\(u^*u\\) is the support of \\(x^*x=gx^*xg\\), which lies under \\(g\\); as \\(g\\) is minimal, \\(u^*u\\in\\{0,g\\}\\). If \\(u^*u=g\\), then \\(rNr=u(gNg)u^*=\\mathbb Cr\\), so \\(r\\) is minimal; otherwise \\(r=0\\). Induction on \\(n\\) gives the first claim, and Lemma 6.1(2) gives \\(\\dim eNe\\le m^2\\). \\(\\square\\)\n\n**Exercise 6.6** (hard) (Compact unitary groups). For a von Neumann algebra \\(M\\) with unitary group \\(\\mathcal U_M\\), prove that \\(\\mathcal U_M\\) is compact exactly when \\(M\\) is \\(*\\)-isomorphic to a direct sum of full matrix algebras \\(M_{n_i}(\\mathbb C)\\). Deduce compactness of \\(\\mathcal U_M\\) for every abelian \\(M\\) in which each nonzero projection majorises a minimal one.\n\nWe use the \\(\\sigma\\)-weak topology. On \\(\\mathcal U_M\\) it coincides with the weak, strong and strong\\(^*\\) topologies of any faithful normal representation: on bounded sets \\(\\sigma\\)-weak and weak agree, and for unitaries \\(\\|(u_\\alpha-u)\\zeta\\|^2=2\\|\\zeta\\|^2-2\\operatorname{Re}\\langle u_\\alpha\\zeta,u\\zeta\\rangle\\) and \\(\\|(u_\\alpha^*-u^*)\\zeta\\|^2=2\\|\\zeta\\|^2-2\\operatorname{Re}\\langle u_\\alpha u^*\\zeta,\\zeta\\rangle\\). A \\(*\\)-isomorphism between von Neumann algebras is \\(\\sigma\\)-weakly bicontinuous (background; see also [Corollary 11.4](#oa-fnd-wa-24)), so this topology does not depend on the representation. Multiplication is jointly strongly continuous on bounded sets and \\(u\\mapsto u^*\\) is strong\\(^*\\) continuous, so \\(\\mathcal U_M\\) is a Hausdorff topological group. \"Direct sum\" means an \\(\\ell^\\infty\\)-direct sum over an arbitrary index set \\(I\\), each \\(n_i\\) finite. (In the norm topology, \\(\\mathcal U_M\\) is compact exactly when \\(\\dim M<\\infty\\): by Lemma 6.2 an infinite-dimensional \\(M\\) contains unitaries \\(1-2p_n\\) at mutual distance \\(2\\).)\n\n**Solution.** For \\(M=0\\), the unitary group has one element and the direct sum is empty. Otherwise, (\\(\\Leftarrow\\)) let \\(M=\\bigoplus_iM_{n_i}(\\mathbb C)\\) act on \\(\\bigoplus_i\\mathbb C^{n_i}\\). A bounded net converges weakly exactly when every coordinate converges, because vectors supported in single summands span a dense subspace. So \\(\\mathcal U_M\\) is homeomorphic to the product \\(\\prod_iU(n_i)\\), which is compact by Tychonoff's theorem. An isomorphic algebra has a homeomorphic unitary group, because \\(*\\)-isomorphisms are \\(\\sigma\\)-weakly bicontinuous.\n\n(\\(\\Rightarrow\\)) Let \\(\\mathcal U_M\\) be compact, realise \\(M\\) on \\(H\\), and let \\(\\mu\\) be the Haar probability measure of \\(\\mathcal U_M\\) ([existence of Haar measure](haar-measure.md#oa-fnd-hm-05)). It is invariant under left translation, including in the form \\(\\int F(vu)\\,d\\mu(u)=\\int F\\,d\\mu\\) for continuous \\(F\\), and it gives positive mass to nonempty open sets ([uniqueness and positivity of Haar measure](haar-measure.md#oa-fnd-hm-06)).\n\n*Step 1: a nonzero finite-dimensional invariant subspace.* Fix a unit vector \\(\\xi\\) and the projection \\(P\\) onto \\(\\mathbb C\\xi\\). The function \\(u\\mapsto uPu^*=\\langle\\cdot,u\\xi\\rangle u\\xi\\) is norm continuous, since \\(\\|uPu^*-vPv^*\\|\\le2\\|u\\xi-v\\xi\\|\\). Define \\(T\\) by \\(\\langle T\\zeta,\\zeta'\\rangle=\\int\\langle uPu^*\\zeta,\\zeta'\\rangle\\,d\\mu(u)\\). This is a bounded positive operator. Left invariance gives \\(vTv^*=T\\) for \\(v\\in\\mathcal U_M\\), so \\(T\\) commutes with \\(\\mathcal U_M\\), hence with \\(M\\), since the unitaries of \\(M\\) span \\(M\\): \\(T\\in M'\\). Next, \\(\\langle T\\xi,\\xi\\rangle=\\int|\\langle u\\xi,\\xi\\rangle|^2d\\mu(u)>0\\), since the integrand is continuous, nonnegative, and equal to \\(1\\) at \\(u=1\\). Also \\(T\\) is compact. Given \\(\\varepsilon>0\\), cover \\(\\mathcal U_M\\) by finitely many open sets on which \\(uPu^*\\) varies by less than \\(\\varepsilon\\), refine them to a Borel partition \\(B_1,\\dots,B_m\\), and pick \\(u_k\\in B_k\\). Then \\(\\|T-\\sum_k\\mu(B_k)u_kPu_k^*\\|\\le\\varepsilon\\), so \\(T\\) is a norm limit of finite-rank operators. For a positive compact \\(T\\ne0\\), \\(\\lambda=\\|T\\|\\) is an eigenvalue: choose unit \\(\\zeta_n\\) with \\(\\langle T\\zeta_n,\\zeta_n\\rangle\\to\\lambda\\); then \\(\\|T\\zeta_n-\\lambda\\zeta_n\\|^2\\le2\\lambda^2-2\\lambda\\langle T\\zeta_n,\\zeta_n\\rangle\\to0\\), and a subsequence of \\(T\\zeta_n\\) converges, to some \\(\\zeta\\) with \\(\\|\\zeta\\|=\\lambda\\) and \\(T\\zeta=\\lambda\\zeta\\). The eigenspace \\(E\\) is finite-dimensional: an infinite orthonormal sequence \\(\\zeta_k\\) in it would have \\(\\|T\\zeta_k-T\\zeta_l\\|=\\lambda\\sqrt2\\), contradicting compactness. Since \\(T\\in M'\\), \\(E\\) is invariant under \\(M\\).\n\n*Step 2: \\(H\\) is a direct sum of finite-dimensional invariant subspaces.* Take a maximal family of mutually orthogonal nonzero finite-dimensional \\(M\\)-invariant subspaces, and let \\(K\\) be their closed sum. \\(K^\\perp\\) is invariant because \\(M\\) is closed under adjoints. If \\(K^\\perp\\ne0\\), repeat Step 1 with \\(\\xi\\in K^\\perp\\): the operator \\(T\\) then maps into \\(K^\\perp\\), since \\(\\langle uPu^*\\zeta,\\zeta'\\rangle=0\\) for \\(\\zeta'\\in K\\), and its eigenspace lies in \\(K^\\perp\\). This contradicts maximality. So \\(H=\\bigoplus_kH_k\\) with each \\(H_k\\) finite-dimensional and invariant.\n\n*Step 3: the structure of \\(M\\).* The restriction \\(\\rho_k(x)=x|_{H_k}\\) is a normal \\(*\\)-homomorphism. Its kernel is \\(M(1-z_k)\\) with \\(z_k\\) central (Lemma 4.2), and \\(Mz_k\\cong\\rho_k(M)\\) is finite-dimensional. By Lemma 6.3, \\(z_k\\) is a finite sum of minimal central projections \\(c\\) of \\(M\\) with \\(Mc\\) a matrix algebra; these are minimal central in \\(M\\) because \\(z_k\\) is central. Since \\(\\bigcap_k\\ker\\rho_k=0\\), we have \\(\\bigvee_kz_k=1\\). Distinct minimal central projections are orthogonal, so the family of all minimal central projections \\(c\\) that occur has sum \\(1\\). Then \\(x\\mapsto(xc)_c\\) is an isomorphism of \\(M\\) onto \\(\\bigoplus_cMc\\cong\\bigoplus_cM_{n_c}(\\mathbb C)\\): it is injective because \\(\\sum_cc=1\\), and onto because a bounded family \\((x_c)\\) has the strong sum \\(\\sum_cx_c\\in M\\).\n\n*Atomic abelian algebras.* Let \\(M\\) be abelian and atomic (every nonzero projection majorises a minimal one). A maximal orthogonal family \\((e_i)\\) of minimal projections has sum \\(1\\), and \\(Me_i=\\mathbb Ce_i\\). So \\(M\\cong\\ell^\\infty(I)=\\bigoplus_iM_1(\\mathbb C)\\), whose unitary group \\(\\mathbb T^I\\) is compact. \\(\\square\\)\n\n",
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      "name": "7. Pure states and Hilbert space quotients",
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      "full_conditions_and_proof": "## 7. Pure states and Hilbert space quotients\n\nLet \\(\\omega\\) be a state. The quotient of \\(A\\) by the left kernel of \\(\\omega\\) carries the quotient norm and, separately, the inner product \\(\\omega(b^*a)\\). For a pure state the two norms agree, so the quotient is already a Hilbert space. The proof combines the normal extension of Section 2 with two lemmas. The first is the step in the proof of the open mapping theorem that turns density into surjectivity.\n\n**Lemma 7.1.** Let \\(T:X\\to Y\\) be a bounded linear map from a Banach space to a normed space. Write \\(B_X,B_Y\\) for the closed unit balls. If \\(T(B_X)\\) is norm dense in \\(B_Y\\), then every \\(y\\in Y\\) with \\(\\|y\\|<1\\) is \\(Tx\\) for some \\(x\\in X\\) with \\(\\|x\\|<1\\).\n\n**Proof.** If \\(y=0\\), take \\(x=0\\). Otherwise put \\(s=\\|y\\|>0\\) and choose \\(\\varepsilon>0\\) with \\(s+\\varepsilon<1\\). By scaling, \\(T(tB_X)\\) is dense in \\(tB_Y\\) for every \\(t>0\\). Choose \\(x_1\\in sB_X\\) with \\(\\|y-Tx_1\\|<\\varepsilon/2\\). Then choose \\(x_2\\in(\\varepsilon/2)B_X\\) with \\(\\|y-Tx_1-Tx_2\\|<\\varepsilon/4\\), and so on: \\(x_n\\in(\\varepsilon2^{1-n})B_X\\) with \\(\\|y-T(x_1+\\dots+x_n)\\|<\\varepsilon2^{-n}\\). The series \\(x=\\sum_nx_n\\) converges, \\(\\|x\\|\\le s+\\varepsilon<1\\), and \\(Tx=y\\). \\(\\square\\)\n\nA positive functional \\(\\varphi\\) is *pure* if every positive \\(\\psi\\le\\varphi\\) is \\(\\lambda\\varphi\\) for some \\(\\lambda\\in[0,1]\\).\n\n**Lemma 7.2** (Pure states and irreducibility). A state \\(\\omega\\) is pure exactly when its cyclic representation is irreducible, that is \\(\\pi_\\omega(A)'=\\mathbb C1\\); then \\(\\pi_\\omega(A)''=B(H_\\omega)\\).\n\n**Proof.** Let \\((\\pi,H,\\xi)=(\\pi_\\omega,H_\\omega,\\xi_\\omega)\\). Suppose \\(\\omega\\) is pure and \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\). Then \\(\\psi(a)=\\langle\\pi(a)T\\xi,\\xi\\rangle\\) is positive, and \\(\\psi(a^*a)=\\|T^{1/2}\\pi(a)\\xi\\|^2\\le\\omega(a^*a)\\). So \\(\\psi=\\lambda\\omega\\), and \\(\\langle T\\pi(a)\\xi,\\pi(b)\\xi\\rangle=\\psi(b^*a)=\\lambda\\langle\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\). By density \\(T=\\lambda1\\). The commutant is a von Neumann algebra spanned by its positive contractions, so \\(\\pi(A)'=\\mathbb C1\\), and \\(\\pi(A)''=B(H)\\). Conversely, let \\(\\pi(A)'=\\mathbb C1\\) and \\(\\psi\\le\\omega\\). The form \\((\\pi(a)\\xi,\\pi(b)\\xi)\\mapsto\\psi(b^*a)\\) is bounded by Cauchy–Schwarz, so \\(\\psi(b^*a)=\\langle T\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\) for some \\(0\\le T\\le1\\). Replacing \\(a\\) by \\(ca\\) shows \\(T\\in\\pi(A)'\\), so \\(T=\\lambda1\\) and \\(\\psi(b^*a)=\\lambda\\omega(b^*a)\\). Letting \\(b\\) run through an approximate unit gives \\(\\psi=\\lambda\\omega\\). \\(\\square\\)\n\n**Proposition 7.3** (Hilbert space quotients). Let \\(\\omega\\) be a pure state of \\(A\\) with left kernel \\(N_\\omega=\\{a:\\omega(a^*a)=0\\}\\). Then the quotient norm of \\(A/N_\\omega\\) is\n\\[\n\\begin{gathered}\n\\|a+N_\\omega\\|\\\\\n=\\omega(a^*a)^{1/2}\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{7.1}\n\\]\nConsequently the quotient norm comes from the inner product \\(\\langle a+N_\\omega,b+N_\\omega\\rangle=\\omega(b^*a)\\), so the complete space \\(A/N_\\omega\\) is a Hilbert space, and \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) is a unitary map onto \\(H_\\omega\\).\n\n**Proof.** Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\), with \\(\\|\\xi\\|=1\\). If \\(n\\in N_\\omega\\), then \\(\\|\\pi(n)\\xi\\|^2=\\omega(n^*n)=0\\). Hence \\(\\omega(a^*a)^{1/2}=\\|\\pi(a+n)\\xi\\|\\le\\|a+n\\|\\), which gives \\(\\ge\\) in (7.1).\n\nFor \\(\\le\\), let \\(R:A\\to H\\), \\(R(a)=\\pi(a)\\xi\\). By Lemma 7.2, \\(\\mathcal M(\\pi)=B(H)\\). By [Lemma 2.1](#oa-fnd-wa-02)(3), \\(\\bar\\pi\\) maps the unit ball of \\(\\tilde A\\) onto the unit ball of \\(B(H)\\). The unit ball of \\(j(A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(\\tilde A\\) (background fact on the bidual), and \\(\\bar\\pi\\) is continuous, so \\(\\pi(B_A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(B(H)\\). The map \\(y\\mapsto y\\xi\\) is continuous from the \\(\\sigma\\)-weak topology to the weak topology of \\(H\\). Every \\(\\eta\\) with \\(\\|\\eta\\|\\le1\\) is \\(y\\xi\\) for the rank-one \\(y=\\langle\\cdot,\\xi\\rangle\\eta\\), which has \\(\\|y\\|\\le1\\). So \\(R(B_A)\\) is weakly dense in \\(B_H\\). It is convex, so by Mazur's theorem its norm closure contains \\(B_H\\). Lemma 7.1 now says: if \\(\\|\\pi(a)\\xi\\|<1\\), there is \\(b\\in A\\) with \\(\\|b\\|<1\\) and \\(\\pi(b)\\xi=\\pi(a)\\xi\\). Then \\(a-b\\in N_\\omega\\), so \\(\\|a+N_\\omega\\|\\le\\|b\\|<1\\). By homogeneity \\(\\|a+N_\\omega\\|\\le\\omega(a^*a)^{1/2}\\).\n\nThe subspace \\(N_\\omega\\) is closed, so \\(A/N_\\omega\\) is a Banach space. By (7.1) its norm comes from the inner product \\(\\omega(b^*a)\\), which is well defined on the quotient. The map \\(a+N_\\omega\\mapsto\\pi(a)\\xi\\) is isometric by (7.1), and onto because its range contains the open unit ball of \\(H\\). \\(\\square\\)\n\n**Remark 7.4.**\n\n1. The proof shows more: every \\(\\eta\\in H_\\omega\\) with \\(\\|\\eta\\|<1\\) is \\(\\pi_\\omega(b)\\xi_\\omega\\) for some \\(b\\) with \\(\\|b\\|<1\\). This is Kadison's transitivity theorem for one vector.\n2. *Unused alternate argument, not proved here.* One can also argue by duality: show that the polar in \\(A^*\\) of the left kernel of \\(\\hat\\omega\\) is a Hilbert space, and then that it is \\(\\sigma(A^*,A)\\)-closed, being reflexive. That last step uses the Krein–Šmulian theorem: the unit ball of a reflexive subspace is weakly compact, hence weak\\(^*\\) compact, and a subspace with weak\\(^*\\) closed unit ball is weak\\(^*\\) closed. The proof above avoids the Krein–Šmulian theorem.\n3. Purity is needed. Let \\(A=C[0,1]\\) and \\(\\omega(f)=\\int_0^1f\\,dt\\). Then \\(N_\\omega=\\{0\\}\\), so \\(A/N_\\omega=C[0,1]\\) with the supremum norm, which is not \\(\\omega(|f|^2)^{1/2}\\) and is not a Hilbert space norm. [Exercise 7.6](#oa-fnd-wa-17) determines exactly when \\(A/N_\\omega\\) is reflexive.\n\n**Exercise 7.5** (medium) (Minimal left ideals). Let \\(\\mathfrak m\\ne\\{0\\}\\) be a closed left ideal of a \\(C^*\\)-algebra \\(A\\) that contains no smaller nonzero closed left ideal. Show: (a) \\(\\dim(\\mathfrak m\\cap\\mathfrak m^*)=1\\); (b) \\(\\mathfrak m=Ae\\) for a minimal projection \\(e\\in A\\); (c) \\(\\mathfrak m\\) is a Hilbert space on which left multiplication is an irreducible representation of \\(A\\).\n\n**Solution.** (a) \\(B=\\mathfrak m\\cap\\mathfrak m^*\\) is a \\(C^*\\)-subalgebra: for \\(x,y\\in B\\), \\(xy\\in\\mathfrak m\\) and \\((xy)^*=y^*x^*\\in\\mathfrak m\\). It is nonzero: for \\(0\\ne x\\in\\mathfrak m\\), \\(x^*x\\in B\\) and \\(x^*x\\ne0\\).\n\nFirst we show: *a \\(C^*\\)-algebra \\(B\\) of dimension at least \\(2\\) contains nonzero positive \\(h,k\\) with \\(hk=0\\).* If some self-adjoint \\(a\\in B\\) has at least two nonzero points \\(t_1\\ne t_2\\) in its spectrum, take continuous \\(g_1,g_2\\ge0\\) vanishing at \\(0\\), with disjoint supports and \\(g_l(t_l)\\ne0\\), and put \\(h=g_1(a)\\), \\(k=g_2(a)\\). Otherwise every self-adjoint element is a real multiple of a projection. Since \\(\\dim B\\ge2\\), there are linearly independent projections \\(p,q\\), and \\(p+q=\\lambda r\\), \\(p-q=\\mu s\\) with projections \\(r,s\\). Squaring gives \\(p+q+pq+qp=\\lambda(p+q)\\) and \\(p+q-pq-qp=\\mu(p-q)\\). Adding, \\((2-\\lambda-\\mu)p+(2-\\lambda+\\mu)q=0\\), so \\(\\lambda=2\\) and \\(\\mu=0\\), that is \\(p=q\\): a contradiction.\n\nNow suppose \\(\\dim B\\ge2\\) and take such \\(h,k\\in B\\). The closed left ideal \\(\\overline{Ah}\\) lies in \\(\\mathfrak m\\) and contains \\(h=\\lim_nh^{1/n}h\\), so by minimality \\(\\overline{Ah}=\\mathfrak m\\ni k\\). Every \\(y\\in Ah\\) satisfies \\(yk=0\\), and by continuity so does every \\(y\\in\\overline{Ah}\\). With \\(y=k\\) this gives \\(k^2=0\\), so \\(k=0\\): a contradiction. Hence \\(\\dim B=1\\).\n\n(b) \\(B=\\mathbb Ce\\) with \\(e\\) a nonzero projection: take \\(0\\ne h\\in B_+\\); then \\(h^2=\\|h\\|h\\), and \\(e=h/\\|h\\|\\). The left ideal \\(Ae=\\{y:ye=y\\}\\) is closed, nonzero and contained in \\(\\mathfrak m\\), so \\(Ae=\\mathfrak m\\). Also \\(eAe\\subseteq\\mathfrak m\\cap\\mathfrak m^*=\\mathbb Ce\\), so \\(e\\) is minimal.\n\n(c) This is Lemma 6.1(1),(3) with \\(N=A\\) and \\(f=e\\). The representation is nondegenerate, since \\(u_\\lambda x\\to x\\) for an approximate unit. \\(\\square\\)\n\n**Exercise 7.6** (hard) (Reflexive quotients). Let \\(\\omega\\) be a state of \\(A\\). Prove that the quotient \\(A/N_\\omega\\), with its quotient norm, is reflexive exactly when \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) for finitely many pure states \\(\\omega_k\\) and weights \\(\\lambda_k>0\\) with sum one.\n\n**Solution.** Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\), \\(M=\\mathcal M(\\pi)\\), and \\(e'\\in M\\) the projection onto \\(\\overline{M'\\xi}\\). Then \\(y\\xi=0\\) exactly when \\(ye'=0\\): if \\(y\\xi=0\\) then \\(yM'\\xi=M'y\\xi=0\\), and conversely \\(\\xi=e'\\xi\\). Let \\(G:A/N_\\omega\\to H\\), \\(G(a+N_\\omega)=\\pi(a)\\xi\\); it is injective and contractive.\n\n(\\(\\Rightarrow\\)) Let \\(X=A/N_\\omega\\) be reflexive. Its unit ball \\(B_X\\) is weakly compact, so \\(G(B_X)\\) is weakly compact, hence norm closed. \\(G(B_X)\\) contains \\(\\{\\pi(a)\\xi:\\|a\\|<1\\}\\), whose closure is \\(\\{y\\xi:y\\in M,\\|y\\|\\le1\\}\\). Indeed, this last set is convex and weakly compact. As in the proof of [Proposition 7.3](#oa-fnd-wa-11), \\(\\pi(B_A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(M\\) ([Lemma 2.1](#oa-fnd-wa-02)(3), with the density of the unit ball of \\(j(A)\\) in that of \\(\\tilde A\\)), so the convex set \\(\\{\\pi(a)\\xi:\\|a\\|\\le1\\}\\) is weakly dense in it, hence norm dense by Mazur's theorem. Scaling replaces \\(\\|a\\|\\le1\\) by \\(\\|a\\|<1\\). For the reverse inclusion of sets, do not assume a quotient norm is attained. If \\(x=a+N_\\omega\\in B_X\\), choose \\(n_k\\in N_\\omega\\) with \\(\\|a+n_k\\|\\le1+1/k\\). The contractions \\(y_k=(1+1/k)^{-1}\\pi(a+n_k)\\) have a sigma-weakly convergent subnet in the compact unit ball of \\(M\\), to a contraction \\(y\\). Their vectors converge weakly to \\(y\\xi\\), while \\(y_k\\xi=(1+1/k)^{-1}G(x)\\) converges in norm to \\(G(x)\\). Thus \\(G(x)=y\\xi\\), proving \\(G(B_X)\\subseteq\\{y\\xi:\\|y\\|\\le1\\}\\). So \\(G(B_X)=\\{y\\xi:\\|y\\|\\le1\\}\\). Hence \\(\\|x\\|=\\min\\{\\|y\\|:y\\in M,\\ y\\xi=G(x)\\}\\) for \\(x\\in X\\), and this minimum is \\(\\|ye'\\|\\) for any \\(y\\) with \\(y\\xi=G(x)\\), because the competitors form \\(y+M(1-e')\\) and \\(\\|y-w\\|\\ge\\|(y-w)e'\\|=\\|ye'\\|\\) for \\(w\\in M(1-e')\\). Thus \\(X\\cong Me'\\) isometrically, via \\(a+N_\\omega\\mapsto\\pi(a)e'\\). So \\(Me'\\) is reflexive, and so is its closed subspace \\(e'Me'\\). By Lemma 6.2, \\(e'Me'\\) is finite-dimensional.\n\nThe vector state \\(\\omega_\\xi(y)=\\langle y\\xi,\\xi\\rangle\\) satisfies \\(\\omega_\\xi(y)=\\omega_\\xi(e'ye')\\) and is faithful on \\(e'Me'\\): if \\(y\\in(e'Me')_+\\) and \\(\\langle y\\xi,\\xi\\rangle=0\\), then \\(y^{1/2}\\xi=0\\), so \\(y^{1/2}e'=0\\) and \\(y=0\\). By Lemma 6.3, \\(e'Me'\\cong\\bigoplus_kM_{n_k}(\\mathbb C)\\). A state of a matrix algebra is \\(\\operatorname{Tr}(\\rho\\,\\cdot)\\) with \\(\\rho\\ge0\\), and diagonalising \\(\\rho\\) writes it as a convex combination of states \\(w\\mapsto c\\), where \\(qwq=cq\\) for a rank-one projection \\(q\\). So \\(\\omega_\\xi|_{e'Me'}=\\sum_l\\lambda_l\\varphi_l\\) with \\(\\lambda_l>0\\), \\(\\sum_l\\lambda_l=1\\), and \\(q_lwq_l=\\varphi_l(w)q_l\\) for minimal projections \\(q_l\\) of \\(e'Me'\\). These are minimal in \\(M\\), since \\(q_lMq_l=q_le'Me'q_l\\). Hence \\(\\omega=\\sum_l\\lambda_l\\psi_l\\), where \\(\\psi_l(a)q_l=q_l\\pi(a)q_l\\).\n\nEach \\(\\psi_l\\) is a pure state. Pick a unit vector \\(\\zeta\\in q_lH\\); then \\(\\psi_l(a)=\\langle\\pi(a)\\zeta,\\zeta\\rangle\\), and \\(\\psi_l\\) is a state. Its cyclic representation is \\(\\pi\\) restricted to \\(K=\\overline{\\pi(A)\\zeta}=\\overline{M\\zeta}\\) (by Kaplansky's density theorem). If \\(T\\) commutes with \\(\\pi(A)|_K\\), it commutes with \\(M|_K\\), so \\(T\\zeta=Tq_l\\zeta=q_lT\\zeta\\in q_lK=\\overline{q_lMq_l\\zeta}=\\mathbb C\\zeta\\). Say \\(T\\zeta=c\\zeta\\); then \\(Ty\\zeta=cy\\zeta\\) for \\(y\\in M\\), so \\(T=c\\) on \\(K\\). The representation is irreducible, and \\(\\psi_l\\) is pure by Lemma 7.2.\n\n(\\(\\Leftarrow\\)) Let \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) with pure states \\(\\omega_k\\) and \\(\\lambda_k>0\\). Let \\(e_k=s(\\hat\\omega_k)\\) in \\(\\tilde A\\). It is a minimal projection. Indeed \\(\\hat\\omega_k=\\omega_{\\xi_k}\\circ\\bar\\pi_{\\omega_k}\\) vanishes on \\(\\tilde A(1-z(\\pi_{\\omega_k}))\\), so \\(e_k\\le z(\\pi_{\\omega_k})\\); the isomorphism \\(\\tilde Az(\\pi_{\\omega_k})\\cong B(H_{\\omega_k})\\) of Lemma 2.1(4) carries \\(e_k\\) to the support of the vector state of the cyclic vector, a rank-one projection, which is minimal. The support of \\(\\hat\\omega\\) is \\(e=\\bigvee_ke_k\\), since a projection is killed by \\(\\hat\\omega\\) exactly when it is killed by every \\(\\hat\\omega_k\\). By Lemma 6.5, \\(e=f_1+\\dots+f_m\\) with orthogonal minimal \\(f_i\\). Then \\(\\tilde Ae=\\bigoplus_i\\tilde Af_i\\) as Banach spaces, with equivalent norms (\\(\\|xf_i\\|\\le\\|xe\\|\\le\\sum_i\\|xf_i\\|\\)). Each \\(\\tilde Af_i\\) is a Hilbert space whose bounded functionals are normal (Lemma 6.1(1),(4)). So \\(\\tilde Ae\\) is reflexive, and its Banach weak topology is the restriction of \\(\\sigma(\\tilde A,A^*)\\).\n\nConsider \\(T:A\\to\\tilde Ae\\), \\(T(a)=j(a)e\\). Its kernel is \\(N_\\omega\\), because \\(\\hat\\omega(x^*x)=0\\) exactly when \\(xe=0\\) (the left kernel of a normal positive functional is \\(\\tilde A(1-s)\\), by the background fact on supports). The unit ball of \\(j(A)\\) is \\(\\sigma\\)-weakly dense in that of \\(\\tilde A\\), and \\(x\\mapsto xe\\) is \\(\\sigma\\)-weakly continuous, so \\(T(B_A)\\) is dense in the unit ball \\(\\{xe:\\|x\\|\\le1\\}\\) of \\(\\tilde Ae\\) for the weak topology, hence in norm by Mazur's theorem. Lemma 7.1 gives \\(\\|a+N_\\omega\\|=\\|j(a)e\\|\\), and \\(T\\) induces an isometry of \\(A/N_\\omega\\) onto \\(\\tilde Ae\\). So \\(A/N_\\omega\\) is reflexive. \\(\\square\\)\n\nFor a pure state this is again Proposition 7.3, since then \\(e\\) is minimal and \\(\\tilde Ae\\) is a Hilbert space.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "name": "7. Pure states and Hilbert space quotients",
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      "full_conditions_and_proof": "## 7. Pure states and Hilbert space quotients\n\nLet \\(\\omega\\) be a state. The quotient of \\(A\\) by the left kernel of \\(\\omega\\) carries the quotient norm and, separately, the inner product \\(\\omega(b^*a)\\). For a pure state the two norms agree, so the quotient is already a Hilbert space. The proof combines the normal extension of Section 2 with two lemmas. The first is the step in the proof of the open mapping theorem that turns density into surjectivity.\n\n**Lemma 7.1.** Let \\(T:X\\to Y\\) be a bounded linear map from a Banach space to a normed space. Write \\(B_X,B_Y\\) for the closed unit balls. If \\(T(B_X)\\) is norm dense in \\(B_Y\\), then every \\(y\\in Y\\) with \\(\\|y\\|<1\\) is \\(Tx\\) for some \\(x\\in X\\) with \\(\\|x\\|<1\\).\n\n**Proof.** If \\(y=0\\), take \\(x=0\\). Otherwise put \\(s=\\|y\\|>0\\) and choose \\(\\varepsilon>0\\) with \\(s+\\varepsilon<1\\). By scaling, \\(T(tB_X)\\) is dense in \\(tB_Y\\) for every \\(t>0\\). Choose \\(x_1\\in sB_X\\) with \\(\\|y-Tx_1\\|<\\varepsilon/2\\). Then choose \\(x_2\\in(\\varepsilon/2)B_X\\) with \\(\\|y-Tx_1-Tx_2\\|<\\varepsilon/4\\), and so on: \\(x_n\\in(\\varepsilon2^{1-n})B_X\\) with \\(\\|y-T(x_1+\\dots+x_n)\\|<\\varepsilon2^{-n}\\). The series \\(x=\\sum_nx_n\\) converges, \\(\\|x\\|\\le s+\\varepsilon<1\\), and \\(Tx=y\\). \\(\\square\\)\n\nA positive functional \\(\\varphi\\) is *pure* if every positive \\(\\psi\\le\\varphi\\) is \\(\\lambda\\varphi\\) for some \\(\\lambda\\in[0,1]\\).\n\n**Lemma 7.2** (Pure states and irreducibility). A state \\(\\omega\\) is pure exactly when its cyclic representation is irreducible, that is \\(\\pi_\\omega(A)'=\\mathbb C1\\); then \\(\\pi_\\omega(A)''=B(H_\\omega)\\).\n\n**Proof.** Let \\((\\pi,H,\\xi)=(\\pi_\\omega,H_\\omega,\\xi_\\omega)\\). Suppose \\(\\omega\\) is pure and \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\). Then \\(\\psi(a)=\\langle\\pi(a)T\\xi,\\xi\\rangle\\) is positive, and \\(\\psi(a^*a)=\\|T^{1/2}\\pi(a)\\xi\\|^2\\le\\omega(a^*a)\\). So \\(\\psi=\\lambda\\omega\\), and \\(\\langle T\\pi(a)\\xi,\\pi(b)\\xi\\rangle=\\psi(b^*a)=\\lambda\\langle\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\). By density \\(T=\\lambda1\\). The commutant is a von Neumann algebra spanned by its positive contractions, so \\(\\pi(A)'=\\mathbb C1\\), and \\(\\pi(A)''=B(H)\\). Conversely, let \\(\\pi(A)'=\\mathbb C1\\) and \\(\\psi\\le\\omega\\). The form \\((\\pi(a)\\xi,\\pi(b)\\xi)\\mapsto\\psi(b^*a)\\) is bounded by Cauchy–Schwarz, so \\(\\psi(b^*a)=\\langle T\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\) for some \\(0\\le T\\le1\\). Replacing \\(a\\) by \\(ca\\) shows \\(T\\in\\pi(A)'\\), so \\(T=\\lambda1\\) and \\(\\psi(b^*a)=\\lambda\\omega(b^*a)\\). Letting \\(b\\) run through an approximate unit gives \\(\\psi=\\lambda\\omega\\). \\(\\square\\)\n\n**Proposition 7.3** (Hilbert space quotients). Let \\(\\omega\\) be a pure state of \\(A\\) with left kernel \\(N_\\omega=\\{a:\\omega(a^*a)=0\\}\\). Then the quotient norm of \\(A/N_\\omega\\) is\n\\[\n\\begin{gathered}\n\\|a+N_\\omega\\|\\\\\n=\\omega(a^*a)^{1/2}\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{7.1}\n\\]\nConsequently the quotient norm comes from the inner product \\(\\langle a+N_\\omega,b+N_\\omega\\rangle=\\omega(b^*a)\\), so the complete space \\(A/N_\\omega\\) is a Hilbert space, and \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) is a unitary map onto \\(H_\\omega\\).\n\n**Proof.** Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\), with \\(\\|\\xi\\|=1\\). If \\(n\\in N_\\omega\\), then \\(\\|\\pi(n)\\xi\\|^2=\\omega(n^*n)=0\\). Hence \\(\\omega(a^*a)^{1/2}=\\|\\pi(a+n)\\xi\\|\\le\\|a+n\\|\\), which gives \\(\\ge\\) in (7.1).\n\nFor \\(\\le\\), let \\(R:A\\to H\\), \\(R(a)=\\pi(a)\\xi\\). By Lemma 7.2, \\(\\mathcal M(\\pi)=B(H)\\). By [Lemma 2.1](#oa-fnd-wa-02)(3), \\(\\bar\\pi\\) maps the unit ball of \\(\\tilde A\\) onto the unit ball of \\(B(H)\\). The unit ball of \\(j(A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(\\tilde A\\) (background fact on the bidual), and \\(\\bar\\pi\\) is continuous, so \\(\\pi(B_A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(B(H)\\). The map \\(y\\mapsto y\\xi\\) is continuous from the \\(\\sigma\\)-weak topology to the weak topology of \\(H\\). Every \\(\\eta\\) with \\(\\|\\eta\\|\\le1\\) is \\(y\\xi\\) for the rank-one \\(y=\\langle\\cdot,\\xi\\rangle\\eta\\), which has \\(\\|y\\|\\le1\\). So \\(R(B_A)\\) is weakly dense in \\(B_H\\). It is convex, so by Mazur's theorem its norm closure contains \\(B_H\\). Lemma 7.1 now says: if \\(\\|\\pi(a)\\xi\\|<1\\), there is \\(b\\in A\\) with \\(\\|b\\|<1\\) and \\(\\pi(b)\\xi=\\pi(a)\\xi\\). Then \\(a-b\\in N_\\omega\\), so \\(\\|a+N_\\omega\\|\\le\\|b\\|<1\\). By homogeneity \\(\\|a+N_\\omega\\|\\le\\omega(a^*a)^{1/2}\\).\n\nThe subspace \\(N_\\omega\\) is closed, so \\(A/N_\\omega\\) is a Banach space. By (7.1) its norm comes from the inner product \\(\\omega(b^*a)\\), which is well defined on the quotient. The map \\(a+N_\\omega\\mapsto\\pi(a)\\xi\\) is isometric by (7.1), and onto because its range contains the open unit ball of \\(H\\). \\(\\square\\)\n\n**Remark 7.4.**\n\n1. The proof shows more: every \\(\\eta\\in H_\\omega\\) with \\(\\|\\eta\\|<1\\) is \\(\\pi_\\omega(b)\\xi_\\omega\\) for some \\(b\\) with \\(\\|b\\|<1\\). This is Kadison's transitivity theorem for one vector.\n2. *Unused alternate argument, not proved here.* One can also argue by duality: show that the polar in \\(A^*\\) of the left kernel of \\(\\hat\\omega\\) is a Hilbert space, and then that it is \\(\\sigma(A^*,A)\\)-closed, being reflexive. That last step uses the Krein–Šmulian theorem: the unit ball of a reflexive subspace is weakly compact, hence weak\\(^*\\) compact, and a subspace with weak\\(^*\\) closed unit ball is weak\\(^*\\) closed. The proof above avoids the Krein–Šmulian theorem.\n3. Purity is needed. Let \\(A=C[0,1]\\) and \\(\\omega(f)=\\int_0^1f\\,dt\\). Then \\(N_\\omega=\\{0\\}\\), so \\(A/N_\\omega=C[0,1]\\) with the supremum norm, which is not \\(\\omega(|f|^2)^{1/2}\\) and is not a Hilbert space norm. [Exercise 7.6](#oa-fnd-wa-17) determines exactly when \\(A/N_\\omega\\) is reflexive.\n\n**Exercise 7.5** (medium) (Minimal left ideals). Let \\(\\mathfrak m\\ne\\{0\\}\\) be a closed left ideal of a \\(C^*\\)-algebra \\(A\\) that contains no smaller nonzero closed left ideal. Show: (a) \\(\\dim(\\mathfrak m\\cap\\mathfrak m^*)=1\\); (b) \\(\\mathfrak m=Ae\\) for a minimal projection \\(e\\in A\\); (c) \\(\\mathfrak m\\) is a Hilbert space on which left multiplication is an irreducible representation of \\(A\\).\n\n**Solution.** (a) \\(B=\\mathfrak m\\cap\\mathfrak m^*\\) is a \\(C^*\\)-subalgebra: for \\(x,y\\in B\\), \\(xy\\in\\mathfrak m\\) and \\((xy)^*=y^*x^*\\in\\mathfrak m\\). It is nonzero: for \\(0\\ne x\\in\\mathfrak m\\), \\(x^*x\\in B\\) and \\(x^*x\\ne0\\).\n\nFirst we show: *a \\(C^*\\)-algebra \\(B\\) of dimension at least \\(2\\) contains nonzero positive \\(h,k\\) with \\(hk=0\\).* If some self-adjoint \\(a\\in B\\) has at least two nonzero points \\(t_1\\ne t_2\\) in its spectrum, take continuous \\(g_1,g_2\\ge0\\) vanishing at \\(0\\), with disjoint supports and \\(g_l(t_l)\\ne0\\), and put \\(h=g_1(a)\\), \\(k=g_2(a)\\). Otherwise every self-adjoint element is a real multiple of a projection. Since \\(\\dim B\\ge2\\), there are linearly independent projections \\(p,q\\), and \\(p+q=\\lambda r\\), \\(p-q=\\mu s\\) with projections \\(r,s\\). Squaring gives \\(p+q+pq+qp=\\lambda(p+q)\\) and \\(p+q-pq-qp=\\mu(p-q)\\). Adding, \\((2-\\lambda-\\mu)p+(2-\\lambda+\\mu)q=0\\), so \\(\\lambda=2\\) and \\(\\mu=0\\), that is \\(p=q\\): a contradiction.\n\nNow suppose \\(\\dim B\\ge2\\) and take such \\(h,k\\in B\\). The closed left ideal \\(\\overline{Ah}\\) lies in \\(\\mathfrak m\\) and contains \\(h=\\lim_nh^{1/n}h\\), so by minimality \\(\\overline{Ah}=\\mathfrak m\\ni k\\). Every \\(y\\in Ah\\) satisfies \\(yk=0\\), and by continuity so does every \\(y\\in\\overline{Ah}\\). With \\(y=k\\) this gives \\(k^2=0\\), so \\(k=0\\): a contradiction. Hence \\(\\dim B=1\\).\n\n(b) \\(B=\\mathbb Ce\\) with \\(e\\) a nonzero projection: take \\(0\\ne h\\in B_+\\); then \\(h^2=\\|h\\|h\\), and \\(e=h/\\|h\\|\\). The left ideal \\(Ae=\\{y:ye=y\\}\\) is closed, nonzero and contained in \\(\\mathfrak m\\), so \\(Ae=\\mathfrak m\\). Also \\(eAe\\subseteq\\mathfrak m\\cap\\mathfrak m^*=\\mathbb Ce\\), so \\(e\\) is minimal.\n\n(c) This is Lemma 6.1(1),(3) with \\(N=A\\) and \\(f=e\\). The representation is nondegenerate, since \\(u_\\lambda x\\to x\\) for an approximate unit. \\(\\square\\)\n\n**Exercise 7.6** (hard) (Reflexive quotients). Let \\(\\omega\\) be a state of \\(A\\). Prove that the quotient \\(A/N_\\omega\\), with its quotient norm, is reflexive exactly when \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) for finitely many pure states \\(\\omega_k\\) and weights \\(\\lambda_k>0\\) with sum one.\n\n**Solution.** Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\), \\(M=\\mathcal M(\\pi)\\), and \\(e'\\in M\\) the projection onto \\(\\overline{M'\\xi}\\). Then \\(y\\xi=0\\) exactly when \\(ye'=0\\): if \\(y\\xi=0\\) then \\(yM'\\xi=M'y\\xi=0\\), and conversely \\(\\xi=e'\\xi\\). Let \\(G:A/N_\\omega\\to H\\), \\(G(a+N_\\omega)=\\pi(a)\\xi\\); it is injective and contractive.\n\n(\\(\\Rightarrow\\)) Let \\(X=A/N_\\omega\\) be reflexive. Its unit ball \\(B_X\\) is weakly compact, so \\(G(B_X)\\) is weakly compact, hence norm closed. \\(G(B_X)\\) contains \\(\\{\\pi(a)\\xi:\\|a\\|<1\\}\\), whose closure is \\(\\{y\\xi:y\\in M,\\|y\\|\\le1\\}\\). Indeed, this last set is convex and weakly compact. As in the proof of [Proposition 7.3](#oa-fnd-wa-11), \\(\\pi(B_A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(M\\) ([Lemma 2.1](#oa-fnd-wa-02)(3), with the density of the unit ball of \\(j(A)\\) in that of \\(\\tilde A\\)), so the convex set \\(\\{\\pi(a)\\xi:\\|a\\|\\le1\\}\\) is weakly dense in it, hence norm dense by Mazur's theorem. Scaling replaces \\(\\|a\\|\\le1\\) by \\(\\|a\\|<1\\). For the reverse inclusion of sets, do not assume a quotient norm is attained. If \\(x=a+N_\\omega\\in B_X\\), choose \\(n_k\\in N_\\omega\\) with \\(\\|a+n_k\\|\\le1+1/k\\). The contractions \\(y_k=(1+1/k)^{-1}\\pi(a+n_k)\\) have a sigma-weakly convergent subnet in the compact unit ball of \\(M\\), to a contraction \\(y\\). Their vectors converge weakly to \\(y\\xi\\), while \\(y_k\\xi=(1+1/k)^{-1}G(x)\\) converges in norm to \\(G(x)\\). Thus \\(G(x)=y\\xi\\), proving \\(G(B_X)\\subseteq\\{y\\xi:\\|y\\|\\le1\\}\\). So \\(G(B_X)=\\{y\\xi:\\|y\\|\\le1\\}\\). Hence \\(\\|x\\|=\\min\\{\\|y\\|:y\\in M,\\ y\\xi=G(x)\\}\\) for \\(x\\in X\\), and this minimum is \\(\\|ye'\\|\\) for any \\(y\\) with \\(y\\xi=G(x)\\), because the competitors form \\(y+M(1-e')\\) and \\(\\|y-w\\|\\ge\\|(y-w)e'\\|=\\|ye'\\|\\) for \\(w\\in M(1-e')\\). Thus \\(X\\cong Me'\\) isometrically, via \\(a+N_\\omega\\mapsto\\pi(a)e'\\). So \\(Me'\\) is reflexive, and so is its closed subspace \\(e'Me'\\). By Lemma 6.2, \\(e'Me'\\) is finite-dimensional.\n\nThe vector state \\(\\omega_\\xi(y)=\\langle y\\xi,\\xi\\rangle\\) satisfies \\(\\omega_\\xi(y)=\\omega_\\xi(e'ye')\\) and is faithful on \\(e'Me'\\): if \\(y\\in(e'Me')_+\\) and \\(\\langle y\\xi,\\xi\\rangle=0\\), then \\(y^{1/2}\\xi=0\\), so \\(y^{1/2}e'=0\\) and \\(y=0\\). By Lemma 6.3, \\(e'Me'\\cong\\bigoplus_kM_{n_k}(\\mathbb C)\\). A state of a matrix algebra is \\(\\operatorname{Tr}(\\rho\\,\\cdot)\\) with \\(\\rho\\ge0\\), and diagonalising \\(\\rho\\) writes it as a convex combination of states \\(w\\mapsto c\\), where \\(qwq=cq\\) for a rank-one projection \\(q\\). So \\(\\omega_\\xi|_{e'Me'}=\\sum_l\\lambda_l\\varphi_l\\) with \\(\\lambda_l>0\\), \\(\\sum_l\\lambda_l=1\\), and \\(q_lwq_l=\\varphi_l(w)q_l\\) for minimal projections \\(q_l\\) of \\(e'Me'\\). These are minimal in \\(M\\), since \\(q_lMq_l=q_le'Me'q_l\\). Hence \\(\\omega=\\sum_l\\lambda_l\\psi_l\\), where \\(\\psi_l(a)q_l=q_l\\pi(a)q_l\\).\n\nEach \\(\\psi_l\\) is a pure state. Pick a unit vector \\(\\zeta\\in q_lH\\); then \\(\\psi_l(a)=\\langle\\pi(a)\\zeta,\\zeta\\rangle\\), and \\(\\psi_l\\) is a state. Its cyclic representation is \\(\\pi\\) restricted to \\(K=\\overline{\\pi(A)\\zeta}=\\overline{M\\zeta}\\) (by Kaplansky's density theorem). If \\(T\\) commutes with \\(\\pi(A)|_K\\), it commutes with \\(M|_K\\), so \\(T\\zeta=Tq_l\\zeta=q_lT\\zeta\\in q_lK=\\overline{q_lMq_l\\zeta}=\\mathbb C\\zeta\\). Say \\(T\\zeta=c\\zeta\\); then \\(Ty\\zeta=cy\\zeta\\) for \\(y\\in M\\), so \\(T=c\\) on \\(K\\). The representation is irreducible, and \\(\\psi_l\\) is pure by Lemma 7.2.\n\n(\\(\\Leftarrow\\)) Let \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) with pure states \\(\\omega_k\\) and \\(\\lambda_k>0\\). Let \\(e_k=s(\\hat\\omega_k)\\) in \\(\\tilde A\\). It is a minimal projection. Indeed \\(\\hat\\omega_k=\\omega_{\\xi_k}\\circ\\bar\\pi_{\\omega_k}\\) vanishes on \\(\\tilde A(1-z(\\pi_{\\omega_k}))\\), so \\(e_k\\le z(\\pi_{\\omega_k})\\); the isomorphism \\(\\tilde Az(\\pi_{\\omega_k})\\cong B(H_{\\omega_k})\\) of Lemma 2.1(4) carries \\(e_k\\) to the support of the vector state of the cyclic vector, a rank-one projection, which is minimal. The support of \\(\\hat\\omega\\) is \\(e=\\bigvee_ke_k\\), since a projection is killed by \\(\\hat\\omega\\) exactly when it is killed by every \\(\\hat\\omega_k\\). By Lemma 6.5, \\(e=f_1+\\dots+f_m\\) with orthogonal minimal \\(f_i\\). Then \\(\\tilde Ae=\\bigoplus_i\\tilde Af_i\\) as Banach spaces, with equivalent norms (\\(\\|xf_i\\|\\le\\|xe\\|\\le\\sum_i\\|xf_i\\|\\)). Each \\(\\tilde Af_i\\) is a Hilbert space whose bounded functionals are normal (Lemma 6.1(1),(4)). So \\(\\tilde Ae\\) is reflexive, and its Banach weak topology is the restriction of \\(\\sigma(\\tilde A,A^*)\\).\n\nConsider \\(T:A\\to\\tilde Ae\\), \\(T(a)=j(a)e\\). Its kernel is \\(N_\\omega\\), because \\(\\hat\\omega(x^*x)=0\\) exactly when \\(xe=0\\) (the left kernel of a normal positive functional is \\(\\tilde A(1-s)\\), by the background fact on supports). The unit ball of \\(j(A)\\) is \\(\\sigma\\)-weakly dense in that of \\(\\tilde A\\), and \\(x\\mapsto xe\\) is \\(\\sigma\\)-weakly continuous, so \\(T(B_A)\\) is dense in the unit ball \\(\\{xe:\\|x\\|\\le1\\}\\) of \\(\\tilde Ae\\) for the weak topology, hence in norm by Mazur's theorem. Lemma 7.1 gives \\(\\|a+N_\\omega\\|=\\|j(a)e\\|\\), and \\(T\\) induces an isometry of \\(A/N_\\omega\\) onto \\(\\tilde Ae\\). So \\(A/N_\\omega\\) is reflexive. \\(\\square\\)\n\nFor a pure state this is again Proposition 7.3, since then \\(e\\) is minimal and \\(\\tilde Ae\\) is a Hilbert space.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-17",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "7. Pure states and Hilbert space quotients",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "full_conditions_and_proof": "## 7. Pure states and Hilbert space quotients\n\nLet \\(\\omega\\) be a state. The quotient of \\(A\\) by the left kernel of \\(\\omega\\) carries the quotient norm and, separately, the inner product \\(\\omega(b^*a)\\). For a pure state the two norms agree, so the quotient is already a Hilbert space. The proof combines the normal extension of Section 2 with two lemmas. The first is the step in the proof of the open mapping theorem that turns density into surjectivity.\n\n**Lemma 7.1.** Let \\(T:X\\to Y\\) be a bounded linear map from a Banach space to a normed space. Write \\(B_X,B_Y\\) for the closed unit balls. If \\(T(B_X)\\) is norm dense in \\(B_Y\\), then every \\(y\\in Y\\) with \\(\\|y\\|<1\\) is \\(Tx\\) for some \\(x\\in X\\) with \\(\\|x\\|<1\\).\n\n**Proof.** If \\(y=0\\), take \\(x=0\\). Otherwise put \\(s=\\|y\\|>0\\) and choose \\(\\varepsilon>0\\) with \\(s+\\varepsilon<1\\). By scaling, \\(T(tB_X)\\) is dense in \\(tB_Y\\) for every \\(t>0\\). Choose \\(x_1\\in sB_X\\) with \\(\\|y-Tx_1\\|<\\varepsilon/2\\). Then choose \\(x_2\\in(\\varepsilon/2)B_X\\) with \\(\\|y-Tx_1-Tx_2\\|<\\varepsilon/4\\), and so on: \\(x_n\\in(\\varepsilon2^{1-n})B_X\\) with \\(\\|y-T(x_1+\\dots+x_n)\\|<\\varepsilon2^{-n}\\). The series \\(x=\\sum_nx_n\\) converges, \\(\\|x\\|\\le s+\\varepsilon<1\\), and \\(Tx=y\\). \\(\\square\\)\n\nA positive functional \\(\\varphi\\) is *pure* if every positive \\(\\psi\\le\\varphi\\) is \\(\\lambda\\varphi\\) for some \\(\\lambda\\in[0,1]\\).\n\n**Lemma 7.2** (Pure states and irreducibility). A state \\(\\omega\\) is pure exactly when its cyclic representation is irreducible, that is \\(\\pi_\\omega(A)'=\\mathbb C1\\); then \\(\\pi_\\omega(A)''=B(H_\\omega)\\).\n\n**Proof.** Let \\((\\pi,H,\\xi)=(\\pi_\\omega,H_\\omega,\\xi_\\omega)\\). Suppose \\(\\omega\\) is pure and \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\). Then \\(\\psi(a)=\\langle\\pi(a)T\\xi,\\xi\\rangle\\) is positive, and \\(\\psi(a^*a)=\\|T^{1/2}\\pi(a)\\xi\\|^2\\le\\omega(a^*a)\\). So \\(\\psi=\\lambda\\omega\\), and \\(\\langle T\\pi(a)\\xi,\\pi(b)\\xi\\rangle=\\psi(b^*a)=\\lambda\\langle\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\). By density \\(T=\\lambda1\\). The commutant is a von Neumann algebra spanned by its positive contractions, so \\(\\pi(A)'=\\mathbb C1\\), and \\(\\pi(A)''=B(H)\\). Conversely, let \\(\\pi(A)'=\\mathbb C1\\) and \\(\\psi\\le\\omega\\). The form \\((\\pi(a)\\xi,\\pi(b)\\xi)\\mapsto\\psi(b^*a)\\) is bounded by Cauchy–Schwarz, so \\(\\psi(b^*a)=\\langle T\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\) for some \\(0\\le T\\le1\\). Replacing \\(a\\) by \\(ca\\) shows \\(T\\in\\pi(A)'\\), so \\(T=\\lambda1\\) and \\(\\psi(b^*a)=\\lambda\\omega(b^*a)\\). Letting \\(b\\) run through an approximate unit gives \\(\\psi=\\lambda\\omega\\). \\(\\square\\)\n\n**Proposition 7.3** (Hilbert space quotients). Let \\(\\omega\\) be a pure state of \\(A\\) with left kernel \\(N_\\omega=\\{a:\\omega(a^*a)=0\\}\\). Then the quotient norm of \\(A/N_\\omega\\) is\n\\[\n\\begin{gathered}\n\\|a+N_\\omega\\|\\\\\n=\\omega(a^*a)^{1/2}\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{7.1}\n\\]\nConsequently the quotient norm comes from the inner product \\(\\langle a+N_\\omega,b+N_\\omega\\rangle=\\omega(b^*a)\\), so the complete space \\(A/N_\\omega\\) is a Hilbert space, and \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) is a unitary map onto \\(H_\\omega\\).\n\n**Proof.** Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\), with \\(\\|\\xi\\|=1\\). If \\(n\\in N_\\omega\\), then \\(\\|\\pi(n)\\xi\\|^2=\\omega(n^*n)=0\\). Hence \\(\\omega(a^*a)^{1/2}=\\|\\pi(a+n)\\xi\\|\\le\\|a+n\\|\\), which gives \\(\\ge\\) in (7.1).\n\nFor \\(\\le\\), let \\(R:A\\to H\\), \\(R(a)=\\pi(a)\\xi\\). By Lemma 7.2, \\(\\mathcal M(\\pi)=B(H)\\). By [Lemma 2.1](#oa-fnd-wa-02)(3), \\(\\bar\\pi\\) maps the unit ball of \\(\\tilde A\\) onto the unit ball of \\(B(H)\\). The unit ball of \\(j(A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(\\tilde A\\) (background fact on the bidual), and \\(\\bar\\pi\\) is continuous, so \\(\\pi(B_A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(B(H)\\). The map \\(y\\mapsto y\\xi\\) is continuous from the \\(\\sigma\\)-weak topology to the weak topology of \\(H\\). Every \\(\\eta\\) with \\(\\|\\eta\\|\\le1\\) is \\(y\\xi\\) for the rank-one \\(y=\\langle\\cdot,\\xi\\rangle\\eta\\), which has \\(\\|y\\|\\le1\\). So \\(R(B_A)\\) is weakly dense in \\(B_H\\). It is convex, so by Mazur's theorem its norm closure contains \\(B_H\\). Lemma 7.1 now says: if \\(\\|\\pi(a)\\xi\\|<1\\), there is \\(b\\in A\\) with \\(\\|b\\|<1\\) and \\(\\pi(b)\\xi=\\pi(a)\\xi\\). Then \\(a-b\\in N_\\omega\\), so \\(\\|a+N_\\omega\\|\\le\\|b\\|<1\\). By homogeneity \\(\\|a+N_\\omega\\|\\le\\omega(a^*a)^{1/2}\\).\n\nThe subspace \\(N_\\omega\\) is closed, so \\(A/N_\\omega\\) is a Banach space. By (7.1) its norm comes from the inner product \\(\\omega(b^*a)\\), which is well defined on the quotient. The map \\(a+N_\\omega\\mapsto\\pi(a)\\xi\\) is isometric by (7.1), and onto because its range contains the open unit ball of \\(H\\). \\(\\square\\)\n\n**Remark 7.4.**\n\n1. The proof shows more: every \\(\\eta\\in H_\\omega\\) with \\(\\|\\eta\\|<1\\) is \\(\\pi_\\omega(b)\\xi_\\omega\\) for some \\(b\\) with \\(\\|b\\|<1\\). This is Kadison's transitivity theorem for one vector.\n2. *Unused alternate argument, not proved here.* One can also argue by duality: show that the polar in \\(A^*\\) of the left kernel of \\(\\hat\\omega\\) is a Hilbert space, and then that it is \\(\\sigma(A^*,A)\\)-closed, being reflexive. That last step uses the Krein–Šmulian theorem: the unit ball of a reflexive subspace is weakly compact, hence weak\\(^*\\) compact, and a subspace with weak\\(^*\\) closed unit ball is weak\\(^*\\) closed. The proof above avoids the Krein–Šmulian theorem.\n3. Purity is needed. Let \\(A=C[0,1]\\) and \\(\\omega(f)=\\int_0^1f\\,dt\\). Then \\(N_\\omega=\\{0\\}\\), so \\(A/N_\\omega=C[0,1]\\) with the supremum norm, which is not \\(\\omega(|f|^2)^{1/2}\\) and is not a Hilbert space norm. [Exercise 7.6](#oa-fnd-wa-17) determines exactly when \\(A/N_\\omega\\) is reflexive.\n\n**Exercise 7.5** (medium) (Minimal left ideals). Let \\(\\mathfrak m\\ne\\{0\\}\\) be a closed left ideal of a \\(C^*\\)-algebra \\(A\\) that contains no smaller nonzero closed left ideal. Show: (a) \\(\\dim(\\mathfrak m\\cap\\mathfrak m^*)=1\\); (b) \\(\\mathfrak m=Ae\\) for a minimal projection \\(e\\in A\\); (c) \\(\\mathfrak m\\) is a Hilbert space on which left multiplication is an irreducible representation of \\(A\\).\n\n**Solution.** (a) \\(B=\\mathfrak m\\cap\\mathfrak m^*\\) is a \\(C^*\\)-subalgebra: for \\(x,y\\in B\\), \\(xy\\in\\mathfrak m\\) and \\((xy)^*=y^*x^*\\in\\mathfrak m\\). It is nonzero: for \\(0\\ne x\\in\\mathfrak m\\), \\(x^*x\\in B\\) and \\(x^*x\\ne0\\).\n\nFirst we show: *a \\(C^*\\)-algebra \\(B\\) of dimension at least \\(2\\) contains nonzero positive \\(h,k\\) with \\(hk=0\\).* If some self-adjoint \\(a\\in B\\) has at least two nonzero points \\(t_1\\ne t_2\\) in its spectrum, take continuous \\(g_1,g_2\\ge0\\) vanishing at \\(0\\), with disjoint supports and \\(g_l(t_l)\\ne0\\), and put \\(h=g_1(a)\\), \\(k=g_2(a)\\). Otherwise every self-adjoint element is a real multiple of a projection. Since \\(\\dim B\\ge2\\), there are linearly independent projections \\(p,q\\), and \\(p+q=\\lambda r\\), \\(p-q=\\mu s\\) with projections \\(r,s\\). Squaring gives \\(p+q+pq+qp=\\lambda(p+q)\\) and \\(p+q-pq-qp=\\mu(p-q)\\). Adding, \\((2-\\lambda-\\mu)p+(2-\\lambda+\\mu)q=0\\), so \\(\\lambda=2\\) and \\(\\mu=0\\), that is \\(p=q\\): a contradiction.\n\nNow suppose \\(\\dim B\\ge2\\) and take such \\(h,k\\in B\\). The closed left ideal \\(\\overline{Ah}\\) lies in \\(\\mathfrak m\\) and contains \\(h=\\lim_nh^{1/n}h\\), so by minimality \\(\\overline{Ah}=\\mathfrak m\\ni k\\). Every \\(y\\in Ah\\) satisfies \\(yk=0\\), and by continuity so does every \\(y\\in\\overline{Ah}\\). With \\(y=k\\) this gives \\(k^2=0\\), so \\(k=0\\): a contradiction. Hence \\(\\dim B=1\\).\n\n(b) \\(B=\\mathbb Ce\\) with \\(e\\) a nonzero projection: take \\(0\\ne h\\in B_+\\); then \\(h^2=\\|h\\|h\\), and \\(e=h/\\|h\\|\\). The left ideal \\(Ae=\\{y:ye=y\\}\\) is closed, nonzero and contained in \\(\\mathfrak m\\), so \\(Ae=\\mathfrak m\\). Also \\(eAe\\subseteq\\mathfrak m\\cap\\mathfrak m^*=\\mathbb Ce\\), so \\(e\\) is minimal.\n\n(c) This is Lemma 6.1(1),(3) with \\(N=A\\) and \\(f=e\\). The representation is nondegenerate, since \\(u_\\lambda x\\to x\\) for an approximate unit. \\(\\square\\)\n\n**Exercise 7.6** (hard) (Reflexive quotients). Let \\(\\omega\\) be a state of \\(A\\). Prove that the quotient \\(A/N_\\omega\\), with its quotient norm, is reflexive exactly when \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) for finitely many pure states \\(\\omega_k\\) and weights \\(\\lambda_k>0\\) with sum one.\n\n**Solution.** Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\), \\(M=\\mathcal M(\\pi)\\), and \\(e'\\in M\\) the projection onto \\(\\overline{M'\\xi}\\). Then \\(y\\xi=0\\) exactly when \\(ye'=0\\): if \\(y\\xi=0\\) then \\(yM'\\xi=M'y\\xi=0\\), and conversely \\(\\xi=e'\\xi\\). Let \\(G:A/N_\\omega\\to H\\), \\(G(a+N_\\omega)=\\pi(a)\\xi\\); it is injective and contractive.\n\n(\\(\\Rightarrow\\)) Let \\(X=A/N_\\omega\\) be reflexive. Its unit ball \\(B_X\\) is weakly compact, so \\(G(B_X)\\) is weakly compact, hence norm closed. \\(G(B_X)\\) contains \\(\\{\\pi(a)\\xi:\\|a\\|<1\\}\\), whose closure is \\(\\{y\\xi:y\\in M,\\|y\\|\\le1\\}\\). Indeed, this last set is convex and weakly compact. As in the proof of [Proposition 7.3](#oa-fnd-wa-11), \\(\\pi(B_A)\\) is \\(\\sigma\\)-weakly dense in the unit ball of \\(M\\) ([Lemma 2.1](#oa-fnd-wa-02)(3), with the density of the unit ball of \\(j(A)\\) in that of \\(\\tilde A\\)), so the convex set \\(\\{\\pi(a)\\xi:\\|a\\|\\le1\\}\\) is weakly dense in it, hence norm dense by Mazur's theorem. Scaling replaces \\(\\|a\\|\\le1\\) by \\(\\|a\\|<1\\). For the reverse inclusion of sets, do not assume a quotient norm is attained. If \\(x=a+N_\\omega\\in B_X\\), choose \\(n_k\\in N_\\omega\\) with \\(\\|a+n_k\\|\\le1+1/k\\). The contractions \\(y_k=(1+1/k)^{-1}\\pi(a+n_k)\\) have a sigma-weakly convergent subnet in the compact unit ball of \\(M\\), to a contraction \\(y\\). Their vectors converge weakly to \\(y\\xi\\), while \\(y_k\\xi=(1+1/k)^{-1}G(x)\\) converges in norm to \\(G(x)\\). Thus \\(G(x)=y\\xi\\), proving \\(G(B_X)\\subseteq\\{y\\xi:\\|y\\|\\le1\\}\\). So \\(G(B_X)=\\{y\\xi:\\|y\\|\\le1\\}\\). Hence \\(\\|x\\|=\\min\\{\\|y\\|:y\\in M,\\ y\\xi=G(x)\\}\\) for \\(x\\in X\\), and this minimum is \\(\\|ye'\\|\\) for any \\(y\\) with \\(y\\xi=G(x)\\), because the competitors form \\(y+M(1-e')\\) and \\(\\|y-w\\|\\ge\\|(y-w)e'\\|=\\|ye'\\|\\) for \\(w\\in M(1-e')\\). Thus \\(X\\cong Me'\\) isometrically, via \\(a+N_\\omega\\mapsto\\pi(a)e'\\). So \\(Me'\\) is reflexive, and so is its closed subspace \\(e'Me'\\). By Lemma 6.2, \\(e'Me'\\) is finite-dimensional.\n\nThe vector state \\(\\omega_\\xi(y)=\\langle y\\xi,\\xi\\rangle\\) satisfies \\(\\omega_\\xi(y)=\\omega_\\xi(e'ye')\\) and is faithful on \\(e'Me'\\): if \\(y\\in(e'Me')_+\\) and \\(\\langle y\\xi,\\xi\\rangle=0\\), then \\(y^{1/2}\\xi=0\\), so \\(y^{1/2}e'=0\\) and \\(y=0\\). By Lemma 6.3, \\(e'Me'\\cong\\bigoplus_kM_{n_k}(\\mathbb C)\\). A state of a matrix algebra is \\(\\operatorname{Tr}(\\rho\\,\\cdot)\\) with \\(\\rho\\ge0\\), and diagonalising \\(\\rho\\) writes it as a convex combination of states \\(w\\mapsto c\\), where \\(qwq=cq\\) for a rank-one projection \\(q\\). So \\(\\omega_\\xi|_{e'Me'}=\\sum_l\\lambda_l\\varphi_l\\) with \\(\\lambda_l>0\\), \\(\\sum_l\\lambda_l=1\\), and \\(q_lwq_l=\\varphi_l(w)q_l\\) for minimal projections \\(q_l\\) of \\(e'Me'\\). These are minimal in \\(M\\), since \\(q_lMq_l=q_le'Me'q_l\\). Hence \\(\\omega=\\sum_l\\lambda_l\\psi_l\\), where \\(\\psi_l(a)q_l=q_l\\pi(a)q_l\\).\n\nEach \\(\\psi_l\\) is a pure state. Pick a unit vector \\(\\zeta\\in q_lH\\); then \\(\\psi_l(a)=\\langle\\pi(a)\\zeta,\\zeta\\rangle\\), and \\(\\psi_l\\) is a state. Its cyclic representation is \\(\\pi\\) restricted to \\(K=\\overline{\\pi(A)\\zeta}=\\overline{M\\zeta}\\) (by Kaplansky's density theorem). If \\(T\\) commutes with \\(\\pi(A)|_K\\), it commutes with \\(M|_K\\), so \\(T\\zeta=Tq_l\\zeta=q_lT\\zeta\\in q_lK=\\overline{q_lMq_l\\zeta}=\\mathbb C\\zeta\\). Say \\(T\\zeta=c\\zeta\\); then \\(Ty\\zeta=cy\\zeta\\) for \\(y\\in M\\), so \\(T=c\\) on \\(K\\). The representation is irreducible, and \\(\\psi_l\\) is pure by Lemma 7.2.\n\n(\\(\\Leftarrow\\)) Let \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) with pure states \\(\\omega_k\\) and \\(\\lambda_k>0\\). Let \\(e_k=s(\\hat\\omega_k)\\) in \\(\\tilde A\\). It is a minimal projection. Indeed \\(\\hat\\omega_k=\\omega_{\\xi_k}\\circ\\bar\\pi_{\\omega_k}\\) vanishes on \\(\\tilde A(1-z(\\pi_{\\omega_k}))\\), so \\(e_k\\le z(\\pi_{\\omega_k})\\); the isomorphism \\(\\tilde Az(\\pi_{\\omega_k})\\cong B(H_{\\omega_k})\\) of Lemma 2.1(4) carries \\(e_k\\) to the support of the vector state of the cyclic vector, a rank-one projection, which is minimal. The support of \\(\\hat\\omega\\) is \\(e=\\bigvee_ke_k\\), since a projection is killed by \\(\\hat\\omega\\) exactly when it is killed by every \\(\\hat\\omega_k\\). By Lemma 6.5, \\(e=f_1+\\dots+f_m\\) with orthogonal minimal \\(f_i\\). Then \\(\\tilde Ae=\\bigoplus_i\\tilde Af_i\\) as Banach spaces, with equivalent norms (\\(\\|xf_i\\|\\le\\|xe\\|\\le\\sum_i\\|xf_i\\|\\)). Each \\(\\tilde Af_i\\) is a Hilbert space whose bounded functionals are normal (Lemma 6.1(1),(4)). So \\(\\tilde Ae\\) is reflexive, and its Banach weak topology is the restriction of \\(\\sigma(\\tilde A,A^*)\\).\n\nConsider \\(T:A\\to\\tilde Ae\\), \\(T(a)=j(a)e\\). Its kernel is \\(N_\\omega\\), because \\(\\hat\\omega(x^*x)=0\\) exactly when \\(xe=0\\) (the left kernel of a normal positive functional is \\(\\tilde A(1-s)\\), by the background fact on supports). The unit ball of \\(j(A)\\) is \\(\\sigma\\)-weakly dense in that of \\(\\tilde A\\), and \\(x\\mapsto xe\\) is \\(\\sigma\\)-weakly continuous, so \\(T(B_A)\\) is dense in the unit ball \\(\\{xe:\\|x\\|\\le1\\}\\) of \\(\\tilde Ae\\) for the weak topology, hence in norm by Mazur's theorem. Lemma 7.1 gives \\(\\|a+N_\\omega\\|=\\|j(a)e\\|\\), and \\(T\\) induces an isometry of \\(A/N_\\omega\\) onto \\(\\tilde Ae\\). So \\(A/N_\\omega\\) is reflexive. \\(\\square\\)\n\nFor a pure state this is again Proposition 7.3, since then \\(e\\) is minimal and \\(\\tilde Ae\\) is a Hilbert space.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-20",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "8. Projections of norm one",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "anchor": "oa-fnd-wa-20",
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      "full_conditions_and_proof": "## 8. Projections of norm one\n\nA contractive linear projection of a \\(C^*\\)-algebra onto a \\(C^*\\)-subalgebra is automatically positive and a bimodule map. This is Tomiyama's theorem, the main tool for Section 9. We begin with a criterion for positivity of a functional.\n\n**Lemma 8.1** (Positivity from a norming net). Let \\(\\omega\\in A^*\\). Suppose there is a net \\((a_\\lambda)\\) in \\(A_+\\) with \\(\\|a_\\lambda\\|\\le1\\) and \\(\\omega(a_\\lambda)\\to\\|\\omega\\|\\). Then \\(\\omega\\) is positive. In particular this holds when \\(\\omega(a)=\\|\\omega\\|\\) for a single \\(a\\in A_+\\) with \\(\\|a\\|\\le1\\).\n\nThe net form matters when the subalgebra has no unit: a positive functional may have no single norming element there (Example 8.3). The proof below establishes the form needed for Corollary 8.2 directly. \n\n**Proof.** If \\(\\omega=0\\), it is positive. Otherwise normalize and assume \\(\\|\\omega\\|=1\\). First let \\(A\\) be unital. For \\(0\\le t\\le1\\) and real \\(\\theta\\), \\(|t+e^{i\\theta}(1-t)|\\le1\\), so the functional calculus gives \\(\\|a_\\lambda+e^{i\\theta}(1-a_\\lambda)\\|\\le1\\). Choose \\(\\theta\\) with \\(e^{i\\theta}\\omega(1-a_\\lambda)=|\\omega(1-a_\\lambda)|\\). Then \\(\\operatorname{Re}\\omega(a_\\lambda)+|\\omega(1-a_\\lambda)|\\le1\\), so \\(|\\omega(1-a_\\lambda)|\\le1-\\operatorname{Re}\\omega(a_\\lambda)\\to0\\), and \\(\\omega(1)=\\lim\\omega(a_\\lambda)=1\\). Now let \\(h=h^*\\) and write \\(\\omega(h)=\\alpha+i\\beta\\). For real \\(t\\), \\(\\|h+it1\\|^2=\\|h^2+t^21\\|=\\|h\\|^2+t^2\\), so \\(\\alpha^2+(\\beta+t)^2\\le\\|h\\|^2+t^2\\), that is \\(\\alpha^2+\\beta^2+2\\beta t\\le\\|h\\|^2\\) for all \\(t\\). Hence \\(\\beta=0\\): \\(\\omega\\) is real on self-adjoint elements. If \\(0\\le h\\le1\\), then \\(\\|1-h\\|\\le1\\), so \\(1-\\omega(h)=\\omega(1-h)\\le1\\), and \\(\\omega(h)\\ge0\\). Scaling gives positivity.\n\nIf \\(A\\) has no unit, embed it in its unitization \\(A^\\sim\\), whose positive cone restricts to that of \\(A\\) (background), and take a Hahn–Banach extension \\(\\omega^\\sim\\) with \\(\\|\\omega^\\sim\\|=\\|\\omega\\|\\). The same net shows that \\(\\omega^\\sim\\) is positive, hence so is \\(\\omega\\). \\(\\square\\)\n\n**Corollary 8.2** (Norm-preserving extensions are positive). Let \\(B\\) be a \\(C^*\\)-subalgebra of \\(A\\) and \\(\\varphi\\in B^*_+\\). Every \\(\\omega\\in A^*\\) with \\(\\omega|_B=\\varphi\\) and \\(\\|\\omega\\|=\\|\\varphi\\|\\) is positive.\n\n**Proof.** Let \\((e_i)\\) be an approximate unit of \\(B\\). For the cyclic representation of \\(\\varphi\\), \\(\\pi_\\varphi(e_i)\\xi_\\varphi\\to\\xi_\\varphi\\) and \\(\\|\\xi_\\varphi\\|^2=\\|\\varphi\\|\\), so \\(\\varphi(e_i)\\to\\|\\varphi\\|\\). Then \\(\\omega(e_i)=\\varphi(e_i)\\to\\|\\omega\\|\\), with \\(e_i\\in A_+\\) and \\(\\|e_i\\|\\le1\\). Apply Lemma 8.1. \\(\\square\\)\n\n**Example 8.3** (No single norming element). If \\(B\\) has a unit, the single element \\(a=1_B\\) norms \\(\\varphi\\) in Corollary 8.2. Without a unit there may be no such element: for \\(B=A=c_0\\) and \\(\\varphi(x)=\\sum_n2^{-n}x_n\\), no \\(a\\in c_0\\) with \\(0\\le a\\le1\\) has \\(\\varphi(a)=1\\). This is why Lemma 8.1 is stated for nets.\n\n**Definition 8.4.** Let \\(B\\) be a \\(C^*\\)-subalgebra of \\(A\\). A *projection of norm one* of \\(A\\) onto \\(B\\) is a linear map \\(E:A\\to B\\) with \\(E(b)=b\\) for \\(b\\in B\\) and \\(\\|E(x)\\|\\le\\|x\\|\\) for \\(x\\in A\\). (If \\(B\\ne0\\) then \\(\\|E\\|=1\\); if \\(B=0\\) then \\(E=0\\). Such a map is also called a contractive retraction.)\n\n**Theorem 8.5** (Tomiyama's theorem). Let \\(E\\) be a projection of norm one of \\(A\\) onto \\(B\\). Then\n\n1. \\(E\\) is positive, \\(E(x^*x)\\ge0\\), and hence \\(E(x^*)=E(x)^*\\);\n2. \\(E(bxc)=bE(x)c\\) for \\(b,c\\in B\\) and \\(x\\in A\\);\n3. \\(E(x)^*E(x)\\le E(x^*x)\\) for \\(x\\in A\\);\n4. if \\(A\\) has a unit and \\(B\\ne0\\), then \\(B\\) has a unit and \\(E(1_A)=1_B\\).\n\n\n**Proof.** If \\(B=0\\) all claims are trivial, so let \\(B\\ne0\\).\n\n*Step 0: passage to the biduals.* Let \\(i:B\\to A\\) be the inclusion. By the background facts on second adjoints, \\(i^{**}:\\tilde B\\to\\tilde A\\) is an injective normal \\(*\\)-homomorphism whose range \\(R\\) is \\(\\sigma\\)-weakly closed. So \\(R\\) is a von Neumann algebra inside \\(\\tilde A\\), and its unit \\(e\\) is a nonzero projection of \\(\\tilde A\\), possibly different from \\(1\\). Put \\(P=i^{**}\\circ E^{**}:\\tilde A\\to R\\). It is \\(\\sigma\\)-weakly continuous, \\(\\|P\\|\\le1\\) because second adjoints keep the norm, and \\(P(Y)=Y\\) for \\(Y\\in R\\), because \\(E\\circ i=\\mathrm{id}_B\\) gives \\(E^{**}\\circ i^{**}=\\mathrm{id}\\). Also \\(P(j_A(x))=j_A(E(x))\\) by naturality of second adjoints. We prove (1)–(3) for \\(P\\), with \\(R\\) in place of \\(B\\), and then restrict.\n\n*Step 1: \\(P(1)=e\\).* Put \\(a=P(1-e)=P(1)-e\\in R\\). For real \\(t\\) with \\(|t|\\ge1\\), \\(a+te=P((1-e)+te)\\), and \\(\\|(1-e)+te\\|\\le\\max(1,|t|)=|t|\\). So \\(\\|a+te\\|\\le|t|\\). Let \\(h=(a+a^*)/2\\). A self-adjoint part has no larger norm, so \\(\\|h+te\\|\\le|t|\\). In the unital algebra \\(R\\), \\(\\|h+te\\|=\\max\\{|\\alpha+t|:\\alpha\\in\\sigma_R(h)\\}\\). A spectral value \\(\\alpha>0\\) would give \\(1+\\alpha\\le1\\) at \\(t=1\\), and \\(\\alpha<0\\) would give \\(1+|\\alpha|\\le1\\) at \\(t=-1\\). So \\(\\sigma_R(h)=\\{0\\}\\) and \\(h=0\\). The same argument applied to \\(-ia=P(-i(1-e))\\), using \\(\\|-i(1-e)+te\\|=\\max(1,|t|)\\), kills the self-adjoint part of \\(-ia\\), which is \\((a-a^*)/(2i)\\). So \\(a=0\\).\n\n*Step 2: \\(P\\) is positive.* Let \\(\\psi\\) be a positive functional on the unital \\(C^*\\)-algebra \\(R\\), so \\(\\|\\psi\\|=\\psi(e)\\). Then \\(\\|\\psi\\circ P\\|\\le\\|\\psi\\|\\) and \\((\\psi\\circ P)(1)=\\psi(e)=\\|\\psi\\|\\). By [Lemma 8.1](#oa-fnd-wa-20), \\(\\psi\\circ P\\) is positive on \\(\\tilde A\\). So for \\(X\\ge0\\), \\(\\psi(P(X))\\ge0\\) for every positive \\(\\psi\\) on \\(R\\), and \\(P(X)\\ge0\\) because states detect positivity. A positive map preserves adjoints.\n\n*Step 3: projections of \\(R\\) pass through \\(P\\).* Let \\(f\\in R\\) be a projection and \\(f'=e-f\\), also a projection of \\(R\\). We use one norm fact: if \\(u^*v=0\\) and \\(uv^*=0\\), then \\(\\|u+v\\|=\\max(\\|u\\|,\\|v\\|)\\). Indeed \\(\\|u+v\\|^2=\\|u^*u+v^*v\\|\\), and \\(u^*u\\), \\(v^*v\\) are positive with product \\(u^*(uv^*)v=0\\), so the norm of their sum is the larger norm.\n\nFix \\(x\\) with \\(\\|x\\|\\le1\\) and put \\(y=P(fx(1-f))\\in R\\).\n\n(a) \\(fyf=0\\). If \\(f=0\\), this is immediate. For \\(f\\ne0\\) and \\(t>0\\), \\(P(fx(1-f)+tf)=y+tf\\), and\n\\[\n\\begin{gathered}\n\\|fx(1-f)+tf\\|^2\\\\\n=\\|(fx(1-f)+tf)(fx(1-f)+tf)^*\\|\\\\\n=\\|fx(1-f)x^*f+t^2f\\|\\\\\n\\le1+t^2 .\n\\end{gathered}\n\\]\nOn the other hand \\(\\|y+tf\\|\\ge\\|fyf+tf\\|\\ge\\|k+tf\\|\\), where \\(k\\) is the self-adjoint part of \\(fyf\\), and \\(\\|k+tf\\|\\ge t+\\alpha\\) for every \\(\\alpha\\in\\sigma_{fRf}(k)\\). If some \\(\\alpha>0\\), then \\((t+\\alpha)^2\\le1+t^2\\), that is \\(2\\alpha t+\\alpha^2\\le1\\), for all \\(t>0\\), which is false. So \\(\\sigma(k)\\subseteq(-\\infty,0]\\). Replacing \\(x\\) by \\(-x\\) gives \\(\\sigma(k)\\subseteq[0,\\infty)\\), so \\(k=0\\). Replacing \\(x\\) by \\(ix\\) kills the other self-adjoint part. So \\(fyf=0\\).\n\n(b) \\(f'yf'=0\\). If \\(f'=0\\), this is immediate. For \\(f'\\ne0\\) and \\(t>0\\), \\(P(fx(1-f)+tf')=y+tf'\\). Here use the other order in the \\(C^*\\)-identity:\n\\[\n\\begin{gathered}\n\\|fx(1-f)+tf'\\|^2\\\\\n=\\|(fx(1-f)+tf')^*(fx(1-f)+tf')\\|\\\\\n=\\|(1-f)x^*fx(1-f)+t^2f'\\|\\\\\n\\le1+t^2 ,\n\\end{gathered}\n\\]\nbecause \\(ff'=0\\) kills the cross terms. The argument of (a), with \\(f'\\) in place of \\(f\\), gives \\(f'yf'=0\\).\n\n(c) \\(f'yf=0\\). Since \\(y\\in R\\), \\(y=(f+f')y(f+f')=fyf'+f'yf\\) by (a) and (b). Suppose \\(f'yf\\ne0\\). For \\(t>0\\),\n\\[\nP\\big(fx(1-f)+tf'yf\\big)=fyf'+(1+t)f'yf .\n\\]\nThe two terms on the right satisfy the hypothesis of the norm fact (they meet \\(ff'=0\\) on both sides), so the right side has norm \\((1+t)\\|f'yf\\|\\) once \\(t\\) is large. The two terms on the left satisfy it too: \\((1-f)x^*f\\cdot f'yf=0\\) and \\(fx(1-f)\\cdot fy^*f'=0\\). So the left side has norm at most \\(\\max(1,t\\|f'yf\\|)=t\\|f'yf\\|\\) for large \\(t\\). Contractivity of \\(P\\) gives \\((1+t)\\|f'yf\\|\\le t\\|f'yf\\|\\), which is false. So \\(f'yf=0\\), and \\(y=fyf'\\).\n\n(d) Conclusion. For \\(0\\le w\\le1\\), \\(0\\le fwf\\le f\\), so \\(0\\le P(fwf)\\le P(f)=f\\) and hence \\(P(fwf)=fP(fwf)f\\). (If \\(0\\le v\\le f\\) for a projection \\(f\\), then \\((1-f)v(1-f)\\le0\\), so \\(v^{1/2}(1-f)=0\\) and \\(v=fvf\\).) By linearity \\(P(fwf)=fP(fwf)f\\) for all \\(w\\). Likewise \\(0\\le(1-f)w(1-f)\\le1-f\\) gives \\(0\\le P((1-f)w(1-f))\\le P(1-f)=f'\\), by Step 1, so \\(fP((1-f)w(1-f))=0\\). By (c) applied to \\(x^*\\) and taking adjoints, \\(P((1-f)xf)=f'y'^*f\\) with \\(y'=P(fx^*(1-f))\\), so \\(fP((1-f)xf)=0\\). Now split \\[\n\\begin{gathered}\nx\\\\\n=fxf+fx(1-f)\\\\\n+(1-f)xf+(1-f)x(1-f)\n\\end{gathered}\n\\] and multiply \\(P(x)\\) by \\(f\\) on the left:\n\\[\nfP(x)=P(fxf)+y=P(fx).\n\\]\nBy scaling, \\(P(fx)=fP(x)\\) for all \\(x\\), and taking adjoints, \\(P(xf)=P(x)f\\).\n\n*Step 4: bimodularity.* A self-adjoint \\(b\\in R\\) is a norm limit of real combinations \\(b_k\\) of projections of the von Neumann algebra \\(R\\) (background fact on spectral projections). Step 3 gives \\(P(b_kX)=b_kP(X)\\), and \\(\\|P(bX)-P(b_kX)\\|\\le\\|b-b_k\\|\\|X\\|\\). So \\(P(bX)=bP(X)\\), and the same on the right. Splitting \\(b\\) into self-adjoint parts gives \\(P(bXc)=bP(X)c\\) for all \\(b,c\\in R\\).\n\n*Step 5: the Schwarz inequality.* With \\(r=X-P(X)\\), bimodularity and \\(P(P(X))=P(X)\\) give\n\\[\n0\\le P(r^*r)=P(X^*X)-P(X)^*P(X).\n\\]\n\n*Step 6: back to \\(A\\).* The map \\(j_A\\) is an injective \\(*\\)-homomorphism, so it reflects positivity, and \\(j_A(B)\\subseteq R\\). Applying Steps 2, 4 and 5 to \\(X=j_A(x)\\), and using \\(P\\circ j_A=j_A\\circ E\\), gives (1)–(3). For (4), \\(j_A(1_A)=1\\), so \\(j_A(E(1_A))=P(1)=e\\), the unit of \\(R\\supseteq j_A(B)\\). Hence \\(E(1_A)b=b=bE(1_A)\\) for \\(b\\in B\\). \\(\\square\\)\n\n**Remark 8.6** (Complete positivity; unused extension, not proved here). Projections of norm one are even completely positive: every matrix amplification \\(E_n:M_n(A)\\to M_n(B)\\) is positive and contractive. This strengthens (1)–(3). A further programme treatment is *Contractive retractions and conditional expectations* in *Modular theory and weights*. This extension is not an input to any proof or solution here.\n\n**Exercise 8.7.** (easy) Show that there is no projection of norm one of \\(\\ell^\\infty\\) onto \\(c_0\\), and none onto the space \\(c\\) of convergent sequences.\n\n**Solution.** \\(\\ell^\\infty\\) has a unit, so by Tomiyama's theorem ([Theorem 8.5](#oa-fnd-wa-21)(4)) the range of such a projection has a unit; \\(c_0\\) has none. Suppose \\(E:\\ell^\\infty\\to c\\) is one. The coordinate sequences \\(e_n\\) lie in \\(c\\), so bimodularity gives \\[\n\\begin{gathered}\nE(x)_ne_n\\\\\n=e_nE(x)\\\\\n=E(e_nx)\\\\\n=E(x_ne_n)\\\\\n=x_ne_n.\n\\end{gathered}\n\\] Hence \\(E(x)=x\\) for every \\(x\\in\\ell^\\infty\\), which is impossible for a non-convergent \\(x\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-21",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "8. Projections of norm one",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "anchor": "oa-fnd-wa-21",
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      "full_conditions_and_proof": "## 8. Projections of norm one\n\nA contractive linear projection of a \\(C^*\\)-algebra onto a \\(C^*\\)-subalgebra is automatically positive and a bimodule map. This is Tomiyama's theorem, the main tool for Section 9. We begin with a criterion for positivity of a functional.\n\n**Lemma 8.1** (Positivity from a norming net). Let \\(\\omega\\in A^*\\). Suppose there is a net \\((a_\\lambda)\\) in \\(A_+\\) with \\(\\|a_\\lambda\\|\\le1\\) and \\(\\omega(a_\\lambda)\\to\\|\\omega\\|\\). Then \\(\\omega\\) is positive. In particular this holds when \\(\\omega(a)=\\|\\omega\\|\\) for a single \\(a\\in A_+\\) with \\(\\|a\\|\\le1\\).\n\nThe net form matters when the subalgebra has no unit: a positive functional may have no single norming element there (Example 8.3). The proof below establishes the form needed for Corollary 8.2 directly. \n\n**Proof.** If \\(\\omega=0\\), it is positive. Otherwise normalize and assume \\(\\|\\omega\\|=1\\). First let \\(A\\) be unital. For \\(0\\le t\\le1\\) and real \\(\\theta\\), \\(|t+e^{i\\theta}(1-t)|\\le1\\), so the functional calculus gives \\(\\|a_\\lambda+e^{i\\theta}(1-a_\\lambda)\\|\\le1\\). Choose \\(\\theta\\) with \\(e^{i\\theta}\\omega(1-a_\\lambda)=|\\omega(1-a_\\lambda)|\\). Then \\(\\operatorname{Re}\\omega(a_\\lambda)+|\\omega(1-a_\\lambda)|\\le1\\), so \\(|\\omega(1-a_\\lambda)|\\le1-\\operatorname{Re}\\omega(a_\\lambda)\\to0\\), and \\(\\omega(1)=\\lim\\omega(a_\\lambda)=1\\). Now let \\(h=h^*\\) and write \\(\\omega(h)=\\alpha+i\\beta\\). For real \\(t\\), \\(\\|h+it1\\|^2=\\|h^2+t^21\\|=\\|h\\|^2+t^2\\), so \\(\\alpha^2+(\\beta+t)^2\\le\\|h\\|^2+t^2\\), that is \\(\\alpha^2+\\beta^2+2\\beta t\\le\\|h\\|^2\\) for all \\(t\\). Hence \\(\\beta=0\\): \\(\\omega\\) is real on self-adjoint elements. If \\(0\\le h\\le1\\), then \\(\\|1-h\\|\\le1\\), so \\(1-\\omega(h)=\\omega(1-h)\\le1\\), and \\(\\omega(h)\\ge0\\). Scaling gives positivity.\n\nIf \\(A\\) has no unit, embed it in its unitization \\(A^\\sim\\), whose positive cone restricts to that of \\(A\\) (background), and take a Hahn–Banach extension \\(\\omega^\\sim\\) with \\(\\|\\omega^\\sim\\|=\\|\\omega\\|\\). The same net shows that \\(\\omega^\\sim\\) is positive, hence so is \\(\\omega\\). \\(\\square\\)\n\n**Corollary 8.2** (Norm-preserving extensions are positive). Let \\(B\\) be a \\(C^*\\)-subalgebra of \\(A\\) and \\(\\varphi\\in B^*_+\\). Every \\(\\omega\\in A^*\\) with \\(\\omega|_B=\\varphi\\) and \\(\\|\\omega\\|=\\|\\varphi\\|\\) is positive.\n\n**Proof.** Let \\((e_i)\\) be an approximate unit of \\(B\\). For the cyclic representation of \\(\\varphi\\), \\(\\pi_\\varphi(e_i)\\xi_\\varphi\\to\\xi_\\varphi\\) and \\(\\|\\xi_\\varphi\\|^2=\\|\\varphi\\|\\), so \\(\\varphi(e_i)\\to\\|\\varphi\\|\\). Then \\(\\omega(e_i)=\\varphi(e_i)\\to\\|\\omega\\|\\), with \\(e_i\\in A_+\\) and \\(\\|e_i\\|\\le1\\). Apply Lemma 8.1. \\(\\square\\)\n\n**Example 8.3** (No single norming element). If \\(B\\) has a unit, the single element \\(a=1_B\\) norms \\(\\varphi\\) in Corollary 8.2. Without a unit there may be no such element: for \\(B=A=c_0\\) and \\(\\varphi(x)=\\sum_n2^{-n}x_n\\), no \\(a\\in c_0\\) with \\(0\\le a\\le1\\) has \\(\\varphi(a)=1\\). This is why Lemma 8.1 is stated for nets.\n\n**Definition 8.4.** Let \\(B\\) be a \\(C^*\\)-subalgebra of \\(A\\). A *projection of norm one* of \\(A\\) onto \\(B\\) is a linear map \\(E:A\\to B\\) with \\(E(b)=b\\) for \\(b\\in B\\) and \\(\\|E(x)\\|\\le\\|x\\|\\) for \\(x\\in A\\). (If \\(B\\ne0\\) then \\(\\|E\\|=1\\); if \\(B=0\\) then \\(E=0\\). Such a map is also called a contractive retraction.)\n\n**Theorem 8.5** (Tomiyama's theorem). Let \\(E\\) be a projection of norm one of \\(A\\) onto \\(B\\). Then\n\n1. \\(E\\) is positive, \\(E(x^*x)\\ge0\\), and hence \\(E(x^*)=E(x)^*\\);\n2. \\(E(bxc)=bE(x)c\\) for \\(b,c\\in B\\) and \\(x\\in A\\);\n3. \\(E(x)^*E(x)\\le E(x^*x)\\) for \\(x\\in A\\);\n4. if \\(A\\) has a unit and \\(B\\ne0\\), then \\(B\\) has a unit and \\(E(1_A)=1_B\\).\n\n\n**Proof.** If \\(B=0\\) all claims are trivial, so let \\(B\\ne0\\).\n\n*Step 0: passage to the biduals.* Let \\(i:B\\to A\\) be the inclusion. By the background facts on second adjoints, \\(i^{**}:\\tilde B\\to\\tilde A\\) is an injective normal \\(*\\)-homomorphism whose range \\(R\\) is \\(\\sigma\\)-weakly closed. So \\(R\\) is a von Neumann algebra inside \\(\\tilde A\\), and its unit \\(e\\) is a nonzero projection of \\(\\tilde A\\), possibly different from \\(1\\). Put \\(P=i^{**}\\circ E^{**}:\\tilde A\\to R\\). It is \\(\\sigma\\)-weakly continuous, \\(\\|P\\|\\le1\\) because second adjoints keep the norm, and \\(P(Y)=Y\\) for \\(Y\\in R\\), because \\(E\\circ i=\\mathrm{id}_B\\) gives \\(E^{**}\\circ i^{**}=\\mathrm{id}\\). Also \\(P(j_A(x))=j_A(E(x))\\) by naturality of second adjoints. We prove (1)–(3) for \\(P\\), with \\(R\\) in place of \\(B\\), and then restrict.\n\n*Step 1: \\(P(1)=e\\).* Put \\(a=P(1-e)=P(1)-e\\in R\\). For real \\(t\\) with \\(|t|\\ge1\\), \\(a+te=P((1-e)+te)\\), and \\(\\|(1-e)+te\\|\\le\\max(1,|t|)=|t|\\). So \\(\\|a+te\\|\\le|t|\\). Let \\(h=(a+a^*)/2\\). A self-adjoint part has no larger norm, so \\(\\|h+te\\|\\le|t|\\). In the unital algebra \\(R\\), \\(\\|h+te\\|=\\max\\{|\\alpha+t|:\\alpha\\in\\sigma_R(h)\\}\\). A spectral value \\(\\alpha>0\\) would give \\(1+\\alpha\\le1\\) at \\(t=1\\), and \\(\\alpha<0\\) would give \\(1+|\\alpha|\\le1\\) at \\(t=-1\\). So \\(\\sigma_R(h)=\\{0\\}\\) and \\(h=0\\). The same argument applied to \\(-ia=P(-i(1-e))\\), using \\(\\|-i(1-e)+te\\|=\\max(1,|t|)\\), kills the self-adjoint part of \\(-ia\\), which is \\((a-a^*)/(2i)\\). So \\(a=0\\).\n\n*Step 2: \\(P\\) is positive.* Let \\(\\psi\\) be a positive functional on the unital \\(C^*\\)-algebra \\(R\\), so \\(\\|\\psi\\|=\\psi(e)\\). Then \\(\\|\\psi\\circ P\\|\\le\\|\\psi\\|\\) and \\((\\psi\\circ P)(1)=\\psi(e)=\\|\\psi\\|\\). By [Lemma 8.1](#oa-fnd-wa-20), \\(\\psi\\circ P\\) is positive on \\(\\tilde A\\). So for \\(X\\ge0\\), \\(\\psi(P(X))\\ge0\\) for every positive \\(\\psi\\) on \\(R\\), and \\(P(X)\\ge0\\) because states detect positivity. A positive map preserves adjoints.\n\n*Step 3: projections of \\(R\\) pass through \\(P\\).* Let \\(f\\in R\\) be a projection and \\(f'=e-f\\), also a projection of \\(R\\). We use one norm fact: if \\(u^*v=0\\) and \\(uv^*=0\\), then \\(\\|u+v\\|=\\max(\\|u\\|,\\|v\\|)\\). Indeed \\(\\|u+v\\|^2=\\|u^*u+v^*v\\|\\), and \\(u^*u\\), \\(v^*v\\) are positive with product \\(u^*(uv^*)v=0\\), so the norm of their sum is the larger norm.\n\nFix \\(x\\) with \\(\\|x\\|\\le1\\) and put \\(y=P(fx(1-f))\\in R\\).\n\n(a) \\(fyf=0\\). If \\(f=0\\), this is immediate. For \\(f\\ne0\\) and \\(t>0\\), \\(P(fx(1-f)+tf)=y+tf\\), and\n\\[\n\\begin{gathered}\n\\|fx(1-f)+tf\\|^2\\\\\n=\\|(fx(1-f)+tf)(fx(1-f)+tf)^*\\|\\\\\n=\\|fx(1-f)x^*f+t^2f\\|\\\\\n\\le1+t^2 .\n\\end{gathered}\n\\]\nOn the other hand \\(\\|y+tf\\|\\ge\\|fyf+tf\\|\\ge\\|k+tf\\|\\), where \\(k\\) is the self-adjoint part of \\(fyf\\), and \\(\\|k+tf\\|\\ge t+\\alpha\\) for every \\(\\alpha\\in\\sigma_{fRf}(k)\\). If some \\(\\alpha>0\\), then \\((t+\\alpha)^2\\le1+t^2\\), that is \\(2\\alpha t+\\alpha^2\\le1\\), for all \\(t>0\\), which is false. So \\(\\sigma(k)\\subseteq(-\\infty,0]\\). Replacing \\(x\\) by \\(-x\\) gives \\(\\sigma(k)\\subseteq[0,\\infty)\\), so \\(k=0\\). Replacing \\(x\\) by \\(ix\\) kills the other self-adjoint part. So \\(fyf=0\\).\n\n(b) \\(f'yf'=0\\). If \\(f'=0\\), this is immediate. For \\(f'\\ne0\\) and \\(t>0\\), \\(P(fx(1-f)+tf')=y+tf'\\). Here use the other order in the \\(C^*\\)-identity:\n\\[\n\\begin{gathered}\n\\|fx(1-f)+tf'\\|^2\\\\\n=\\|(fx(1-f)+tf')^*(fx(1-f)+tf')\\|\\\\\n=\\|(1-f)x^*fx(1-f)+t^2f'\\|\\\\\n\\le1+t^2 ,\n\\end{gathered}\n\\]\nbecause \\(ff'=0\\) kills the cross terms. The argument of (a), with \\(f'\\) in place of \\(f\\), gives \\(f'yf'=0\\).\n\n(c) \\(f'yf=0\\). Since \\(y\\in R\\), \\(y=(f+f')y(f+f')=fyf'+f'yf\\) by (a) and (b). Suppose \\(f'yf\\ne0\\). For \\(t>0\\),\n\\[\nP\\big(fx(1-f)+tf'yf\\big)=fyf'+(1+t)f'yf .\n\\]\nThe two terms on the right satisfy the hypothesis of the norm fact (they meet \\(ff'=0\\) on both sides), so the right side has norm \\((1+t)\\|f'yf\\|\\) once \\(t\\) is large. The two terms on the left satisfy it too: \\((1-f)x^*f\\cdot f'yf=0\\) and \\(fx(1-f)\\cdot fy^*f'=0\\). So the left side has norm at most \\(\\max(1,t\\|f'yf\\|)=t\\|f'yf\\|\\) for large \\(t\\). Contractivity of \\(P\\) gives \\((1+t)\\|f'yf\\|\\le t\\|f'yf\\|\\), which is false. So \\(f'yf=0\\), and \\(y=fyf'\\).\n\n(d) Conclusion. For \\(0\\le w\\le1\\), \\(0\\le fwf\\le f\\), so \\(0\\le P(fwf)\\le P(f)=f\\) and hence \\(P(fwf)=fP(fwf)f\\). (If \\(0\\le v\\le f\\) for a projection \\(f\\), then \\((1-f)v(1-f)\\le0\\), so \\(v^{1/2}(1-f)=0\\) and \\(v=fvf\\).) By linearity \\(P(fwf)=fP(fwf)f\\) for all \\(w\\). Likewise \\(0\\le(1-f)w(1-f)\\le1-f\\) gives \\(0\\le P((1-f)w(1-f))\\le P(1-f)=f'\\), by Step 1, so \\(fP((1-f)w(1-f))=0\\). By (c) applied to \\(x^*\\) and taking adjoints, \\(P((1-f)xf)=f'y'^*f\\) with \\(y'=P(fx^*(1-f))\\), so \\(fP((1-f)xf)=0\\). Now split \\[\n\\begin{gathered}\nx\\\\\n=fxf+fx(1-f)\\\\\n+(1-f)xf+(1-f)x(1-f)\n\\end{gathered}\n\\] and multiply \\(P(x)\\) by \\(f\\) on the left:\n\\[\nfP(x)=P(fxf)+y=P(fx).\n\\]\nBy scaling, \\(P(fx)=fP(x)\\) for all \\(x\\), and taking adjoints, \\(P(xf)=P(x)f\\).\n\n*Step 4: bimodularity.* A self-adjoint \\(b\\in R\\) is a norm limit of real combinations \\(b_k\\) of projections of the von Neumann algebra \\(R\\) (background fact on spectral projections). Step 3 gives \\(P(b_kX)=b_kP(X)\\), and \\(\\|P(bX)-P(b_kX)\\|\\le\\|b-b_k\\|\\|X\\|\\). So \\(P(bX)=bP(X)\\), and the same on the right. Splitting \\(b\\) into self-adjoint parts gives \\(P(bXc)=bP(X)c\\) for all \\(b,c\\in R\\).\n\n*Step 5: the Schwarz inequality.* With \\(r=X-P(X)\\), bimodularity and \\(P(P(X))=P(X)\\) give\n\\[\n0\\le P(r^*r)=P(X^*X)-P(X)^*P(X).\n\\]\n\n*Step 6: back to \\(A\\).* The map \\(j_A\\) is an injective \\(*\\)-homomorphism, so it reflects positivity, and \\(j_A(B)\\subseteq R\\). Applying Steps 2, 4 and 5 to \\(X=j_A(x)\\), and using \\(P\\circ j_A=j_A\\circ E\\), gives (1)–(3). For (4), \\(j_A(1_A)=1\\), so \\(j_A(E(1_A))=P(1)=e\\), the unit of \\(R\\supseteq j_A(B)\\). Hence \\(E(1_A)b=b=bE(1_A)\\) for \\(b\\in B\\). \\(\\square\\)\n\n**Remark 8.6** (Complete positivity; unused extension, not proved here). Projections of norm one are even completely positive: every matrix amplification \\(E_n:M_n(A)\\to M_n(B)\\) is positive and contractive. This strengthens (1)–(3). A further programme treatment is *Contractive retractions and conditional expectations* in *Modular theory and weights*. This extension is not an input to any proof or solution here.\n\n**Exercise 8.7.** (easy) Show that there is no projection of norm one of \\(\\ell^\\infty\\) onto \\(c_0\\), and none onto the space \\(c\\) of convergent sequences.\n\n**Solution.** \\(\\ell^\\infty\\) has a unit, so by Tomiyama's theorem ([Theorem 8.5](#oa-fnd-wa-21)(4)) the range of such a projection has a unit; \\(c_0\\) has none. Suppose \\(E:\\ell^\\infty\\to c\\) is one. The coordinate sequences \\(e_n\\) lie in \\(c\\), so bimodularity gives \\[\n\\begin{gathered}\nE(x)_ne_n\\\\\n=e_nE(x)\\\\\n=E(e_nx)\\\\\n=E(x_ne_n)\\\\\n=x_ne_n.\n\\end{gathered}\n\\] Hence \\(E(x)=x\\) for every \\(x\\in\\ell^\\infty\\), which is impossible for a non-convergent \\(x\\). \\(\\square\\)\n\n",
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      "id": "OA-FND-WA-22",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "9. \\(W^*\\)-algebras and dual \\(C^*\\)-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "anchor": "oa-fnd-wa-22",
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      "full_conditions_and_proof": "## 9. \\(W^*\\)-algebras and dual \\(C^*\\)-algebras\n\nA von Neumann algebra is defined through a Hilbert space. Sakai's theorem says that the Hilbert space can be forgotten: the \\(C^*\\)-algebras that are isomorphic to von Neumann algebras are exactly those that are dual Banach spaces.\n\n**Definition 9.1.** A \\(C^*\\)-algebra \\(A\\) is a *\\(W^*\\)-algebra* if it has a faithful representation \\(\\pi\\) with \\(\\pi(A)=\\pi(A)''\\). Equivalently, some von Neumann algebra is the image of \\(A\\) under a \\(*\\)-isomorphism. Every \\(W^*\\)-algebra has a unit. The zero algebra is a \\(W^*\\)-algebra, acting on the zero space.\n\nA *predual* of a \\(C^*\\)-algebra \\(A\\) is a Banach space \\(F\\) with an isometric linear bijection \\(\\theta:A\\to F^*\\). It gives an isometry \\(\\iota_F:F\\to A^*\\), \\(\\iota_F(f)(a)=\\theta(a)(f)\\); it is isometric because \\(\\sup_{\\|a\\|\\le1}|\\theta(a)(f)|=\\|f\\|\\) by the Hahn–Banach theorem. We write \\(\\sigma(A,F)\\) for the weak\\(^*\\) topology carried over by \\(\\theta\\); its continuous functionals are exactly \\(\\iota_F(F)\\).\n\n**Theorem 9.2** (Sakai's theorem). For a \\(C^*\\)-algebra \\(A\\) the following are equivalent:\n\n1. \\(A\\) is a \\(W^*\\)-algebra;\n2. \\(A\\) has a predual \\(F\\).\n\nWhen (2) holds, there is a faithful representation \\(\\pi\\) with \\(\\pi(A)\\) a von Neumann algebra that is a homeomorphism from \\(\\sigma(A,F)\\) onto the \\(\\sigma\\)-weak topology; and \\(\\psi\\mapsto\\psi\\circ\\pi\\) maps \\(\\pi(A)_*\\) isometrically onto \\(\\iota_F(F)\\).\n\n*Reference:* [Sakai 1956] gives the original characterization using normal-state representations. The complete proof below uses the norm-one projection and bidual route.\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): if \\(\\pi\\) maps \\(A\\) \\(*\\)-isomorphically onto a von Neumann algebra \\(M\\), take \\(F=M_*\\) and \\(\\theta=\\) evaluation composed with \\(\\pi\\) (background fact on preduals).\n\n(2)\\(\\Rightarrow\\)(1). Let \\(\\iota=\\iota_F\\) and define \\(\\varepsilon=\\theta^{-1}\\circ\\iota^*:\\tilde A\\to A\\). For \\(a\\in A\\) and \\(f\\in F\\), \\(\\iota^*(j_A(a))(f)=j_A(a)(\\iota f)=\\theta(a)(f)\\), so \\(\\varepsilon\\circ j_A=\\mathrm{id}_A\\). Hence \\(E=j_A\\circ\\varepsilon\\) is a projection of norm at most one of \\(\\tilde A\\) onto the \\(C^*\\)-subalgebra \\(j_A(A)\\). By Tomiyama's theorem ([Theorem 8.5](#oa-fnd-wa-21)) \\(E\\) is positive and \\(j_A(A)\\)-bimodular. Its kernel \\(J=\\ker\\iota^*\\) is the annihilator of \\(\\iota(F)\\) in \\(\\tilde A\\), so it is \\(\\sigma(\\tilde A,A^*)\\)-closed. For \\(X\\in J\\) and \\(a,b\\in A\\), \\(E(j_A(a)Xj_A(b))=j_A(a)E(X)j_A(b)=0\\). So \\(J\\) is stable under multiplication by \\(j_A(A)\\) on both sides, and by density of \\(j_A(A)\\) and separate continuity, under \\(\\tilde A\\): \\(J\\) is a \\(\\sigma\\)-weakly closed two-sided ideal. By Lemma 4.2, \\(J=\\tilde A(1-z)\\) with \\(z\\) central.\n\n\\(E\\) is multiplicative: for \\(X,Y\\in\\tilde A\\), \\(X-E(X)\\in J\\) because \\(E\\) is idempotent, so \\((X-E(X))Y\\in J\\) and \\(E(XY)=E(E(X)Y)=E(X)E(Y)\\). Being positive, \\(E\\) is a \\(*\\)-homomorphism. Since \\(E(X)=E(Xz)\\), the restriction \\(\\Phi=\\varepsilon|_{\\tilde Az}\\) is a \\(*\\)-isomorphism of \\(\\tilde Az\\) onto \\(A\\). Now \\(\\tilde Az\\) is a von Neumann algebra: in the realisation \\(\\bar\\pi_u\\) of [Theorem 3.3](#oa-fnd-wa-03) it is \\(\\bar\\pi_u(\\tilde A)\\bar\\pi_u(z)\\), acting on \\(\\bar\\pi_u(z)H_u\\) (background fact on corners). So \\(\\pi(a)=\\bar\\pi_u(\\Phi^{-1}(a))|_{\\bar\\pi_u(z)H_u}\\) defines a faithful representation whose image equals its bicommutant, and \\(A\\) is a \\(W^*\\)-algebra.\n\nTopologies. The predual of \\(\\tilde Az\\) is \\(A^*z\\), the restrictions of functionals (background fact on corners). Let \\(\\iota_z:F\\to A^*z\\), \\(\\iota_z(f)=\\iota(f)z\\). For \\(Y\\in\\tilde Az\\), \\(\\theta(\\Phi(Y))(f)=Y(\\iota(f))=Y(\\iota_z(f))\\), so \\(\\theta\\circ\\Phi=(\\iota_z)^*\\). The map \\(\\theta\\circ\\Phi\\) is an isometric bijection, since an injective \\(*\\)-homomorphism is isometric. If a bounded map \\(T\\) between Banach spaces has an isometric bijective adjoint, then \\(T\\) is an isometric bijection: \\(\\|Tx\\|=\\sup|T^*(y^*)(x)|\\) over the unit ball, which \\(T^*\\) maps onto the unit ball; so \\(T\\) is isometric and has closed range, and its range is dense because \\(T^*\\) is injective. Hence \\(\\iota_z\\) is an isometric bijection of \\(F\\) onto \\((\\tilde Az)_*\\), and \\(\\Phi\\) and \\(\\Phi^{-1}\\) are adjoints of isometric bijections, hence weak\\(^*\\) continuous. Composing with the homeomorphism \\(\\bar\\pi_u\\) (Theorem 3.3), \\(\\pi\\) is a homeomorphism for \\(\\sigma(A,F)\\) and the \\(\\sigma\\)-weak topology, and the last claim follows. \\(\\square\\)\n\n**Consequences.** A \\(C^*\\)-algebra with a predual has a unit, since \\(\\tilde Az\\) does. So \\(c_0\\), which has no unit, is not isometric to the dual of any Banach space. For the moment we call \\(F\\) *a* predual; [Corollary 11.3](#oa-fnd-wa-24) shows that \\(\\iota_F(F)\\) does not depend on \\(F\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-10",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "10. Normal and singular parts",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "anchor": "oa-fnd-wa-10",
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        "line": 707,
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      "full_conditions_and_proof": "## 10. Normal and singular parts\n\nEvery bounded functional on a von Neumann algebra splits, in exactly one way and with additive norms, into a normal part and a singular part. The splitting comes from one central projection of the universal enveloping algebra.\n\nLet \\(M\\subseteq B(H)\\) be a von Neumann algebra, viewed as a \\(C^*\\)-algebra, with universal enveloping algebra \\(\\tilde M=M^{**}\\). The identity map \\(\\pi_0:M\\to B(H)\\) is a representation, and \\(\\mathcal M(\\pi_0)=M''=M\\). Put \\(z_0=z(\\pi_0)\\), a central projection of \\(\\tilde M\\). By [Proposition 5.2](#oa-fnd-wa-09)(2),\n\\[\nM_*=V(\\pi_0)=M^*z_0 .\n\\tag{10.1}\n\\]\n\n**Definition 10.1.** The elements of \\(M_*\\) are the *normal* functionals, and \\(M_*\\) is the *predual* of \\(M\\). The elements of \\(M_*^{\\perp}:=M^*(1-z_0)\\) are the *singular* functionals. (The symbol \\(M_*^\\perp\\) is only a name here; it is not an annihilator.)\n\nFor a \\(W^*\\)-algebra \\(A\\) with predual \\(F\\), normal and singular functionals are defined in the same way through the representation \\(\\pi\\) of [Theorem 9.2](#oa-fnd-wa-22). Since \\(\\mathcal M(\\pi)=\\pi(A)\\), [Proposition 5.2](#oa-fnd-wa-09)(2) gives the normal ones as \\(\\iota_F(F)=V(\\pi)=A^*z(\\pi)\\), and the singular ones are \\(A^*(1-z(\\pi))\\).\n\n**Lemma 10.2** (Norms split along central projections). Let \\(N\\) be a von Neumann algebra, \\(z\\) a central projection and \\(\\varphi\\in N_*\\). Then \\(\\|\\varphi\\|=\\|\\varphi z\\|+\\|\\varphi(1-z)\\|\\).\n\n**Proof.** The inequality \\(\\le\\) is the triangle inequality. Given \\(\\varepsilon>0\\), choose \\(x,y\\) in the unit ball with \\((\\varphi z)(x)\\ge\\|\\varphi z\\|-\\varepsilon\\) and \\((\\varphi(1-z))(y)\\ge\\|\\varphi(1-z)\\|-\\varepsilon\\), after rotating phases. Put \\(w=zx+(1-z)y\\). Because \\(z\\) is central, \\(w^*w=zx^*xz+(1-z)y^*y(1-z)\\le1\\), so \\(\\|w\\|\\le1\\). And \\[\n\\begin{gathered}\n\\varphi(w)\\\\\n=\\varphi(zx)+\\varphi((1-z)y)\\\\\n\\ge\\|\\varphi z\\|+\\|\\varphi(1-z)\\|-2\\varepsilon.\n\\end{gathered}\n\\] \\(\\square\\)\n\n**Theorem 10.3** (The normal–singular splitting).\n\n1. Every norm-closed left or right invariant subspace \\(V\\subseteq M^*\\) splits as\n\\[\n\\begin{gathered}\nV\\\\\n=(V\\cap M_*)\\\\\n\\oplus_1(V\\cap M_*^\\perp),\\\\\nV\\cap M_*\\\\\n=Vz_0,\\\\\nV\\cap M_*^\\perp\\\\\n=V(1-z_0),\n\\end{gathered}\n\\tag{10.2}\n\\]\nwhere \\(\\oplus_1\\) means \\(\\|\\varphi+\\psi\\|=\\|\\varphi\\|+\\|\\psi\\|\\) for \\(\\varphi\\in V\\cap M_*\\) and \\(\\psi\\in V\\cap M_*^\\perp\\).\n2. Let \\(\\pi\\) be a representation of the \\(C^*\\)-algebra \\(M\\) on \\(H_\\pi\\), and put \\(z=\\bar\\pi(z_0)\\), a central projection of \\(\\mathcal M(\\pi)\\). Then \\(\\pi_n(x)=\\pi(x)z\\) is a normal representation of \\(M\\) on \\(zH_\\pi\\), and \\(\\pi_s(x)=\\pi(x)(1-z)\\) is a representation of \\(M\\) on \\((1-z)H_\\pi\\) whose coefficient functionals \\(\\omega_{\\pi_s;\\xi,\\eta}\\) are all singular.\n\n**Proof.** (1) Let \\(V\\) be left invariant; the right case is the same. By [Theorem 4.3](#oa-fnd-wa-07)(1), applied to \\(\\tilde M\\) and \\(S=M\\), \\(V\\) is invariant under \\(\\tilde M\\), so \\(z_0V\\subseteq V\\) and \\((1-z_0)V\\subseteq V\\). Since \\(z_0\\) is central, \\(z_0\\varphi=\\varphi z_0\\). Every \\(\\varphi\\in V\\) is \\(\\varphi z_0+\\varphi(1-z_0)\\). If \\(\\varphi\\in V\\) and \\(\\varphi=\\varphi z_0\\), then \\(\\varphi\\in Vz_0\\); this gives \\(V\\cap M^*z_0=Vz_0\\), and likewise for \\(1-z_0\\). The norm identity is Lemma 10.2, applied in \\(\\tilde M\\).\n\n(2) \\(\\bar\\pi\\) maps \\(\\tilde M\\) onto \\(\\mathcal M(\\pi)\\), so it maps the centre into the centre and \\(z\\) is a central projection; also \\(\\pi_n(x)=\\bar\\pi(xz_0)\\). For \\(\\xi,\\eta\\in zH_\\pi\\), the coefficient \\(x\\mapsto\\langle\\pi_n(x)\\xi,\\eta\\rangle\\) is \\(\\big(z_0\\,(\\omega_{\\xi,\\eta}\\circ\\bar\\pi)\\big)\\) restricted to \\(M\\), which lies in \\(M^*z_0=M_*\\). Every \\(\\sigma\\)-weakly continuous functional on \\(B(zH_\\pi)\\) is a norm-convergent sum of vector functionals (background fact on preduals), and \\(M_*\\) is norm closed. So \\(\\psi\\circ\\pi_n\\in M_*\\) for every normal \\(\\psi\\), and \\(\\pi_n\\) is normal (background fact on normal maps). It is nondegenerate on \\(zH_\\pi\\) because \\(\\pi_n(1)=z\\). The same computation with \\(1-z_0\\) shows that the coefficients of \\(\\pi_s\\) lie in \\(M^*(1-z_0)\\). \\(\\square\\)\n\n**Normal and singular parts.** By (1) with \\(V=M^*\\), every \\(\\varphi\\in M^*\\) is uniquely \\(\\varphi=\\varphi_n+\\varphi_s\\) with \\(\\varphi_n\\in M_*\\) and \\(\\varphi_s\\in M_*^\\perp\\), namely \\(\\varphi_n=\\varphi z_0\\), and \\(\\|\\varphi\\|=\\|\\varphi_n\\|+\\|\\varphi_s\\|\\). If \\(\\varphi\\) is positive, so are both parts, because \\((\\varphi z_0)(x^*x)=\\hat\\varphi(z_0x^*xz_0)\\ge0\\). Since \\(M^*\\) is spanned by its positive elements (see the proof of [Proposition 1.1](#oa-fnd-wa-01)(2)), so is \\(M_*^\\perp\\). By (2), every representation splits as \\(\\pi=\\pi_n\\oplus\\pi_s\\), a normal part and a singular part.\n\n**Definition 10.4.** A bounded linear map \\(T:M\\to N\\) between von Neumann algebras is *normal* if \\(\\psi\\circ T\\in M_*\\) for every \\(\\psi\\in N_*\\), and *singular* if \\(\\psi\\circ T\\in M_*^\\perp\\) for every \\(\\psi\\in N_*\\).\n\n**Proposition 10.5** (Splitting of maps). Every bounded linear \\(T:M\\to N\\) is uniquely \\(T=T_n+T_s\\) with \\(T_n\\) normal and \\(T_s\\) singular. If \\(T\\) is positive, so are \\(T_n\\) and \\(T_s\\).\n\n**Proof.** Let \\(T_*:N_*\\to M^*=\\tilde M_*\\), \\(\\psi\\mapsto\\psi\\circ T\\), and let \\(\\tilde T=(T_*)^*:\\tilde M\\to N\\), using \\(N=(N_*)^*\\). For \\(x\\in M\\) and \\(\\psi\\in N_*\\), \\(\\psi(\\tilde T(j(x)))=j(x)(\\psi\\circ T)=\\psi(T(x))\\), so \\(\\tilde T\\) extends \\(T\\). Put \\(T_n(x)=\\tilde T(xz_0)\\) and \\(T_s(x)=\\tilde T(x(1-z_0))\\). Then \\(\\psi\\circ T_n=z_0(\\psi\\circ T)\\in M_*\\) and \\(\\psi\\circ T_s=(1-z_0)(\\psi\\circ T)\\in M_*^\\perp\\). If \\(T=T_1+T_2\\) is another such splitting, then for each \\(\\psi\\), \\(\\psi\\circ T=\\psi\\circ T_1+\\psi\\circ T_2\\) is the unique splitting of \\(\\psi\\circ T\\), so \\(\\psi\\circ T_1=\\psi\\circ T_n\\); since \\(N_*\\) separates \\(N\\), \\(T_1=T_n\\). If \\(T\\ge0\\), then \\(\\psi\\circ T\\) is positive for \\(\\psi\\in N_*^+\\), hence so are its two parts, and positivity in \\(N\\) is detected by \\(N_*^+\\) (background). \\(\\square\\)\n\n**Exercise 10.6** (medium) (Nearby pure states). Let \\(\\varphi,\\psi\\) be pure states of \\(A\\) with \\(\\|\\varphi-\\psi\\|<2\\). Show that \\(\\pi_\\varphi\\) and \\(\\pi_\\psi\\) are unitarily equivalent.\n\n**Solution.** Write \\(z_\\varphi=z(\\pi_\\varphi)\\) and \\(z_\\psi=z(\\pi_\\psi)\\). By Lemma 7.2, \\(\\mathcal M(\\pi_\\varphi)=B(H_\\varphi)\\), and \\(\\tilde Az_\\varphi\\cong B(H_\\varphi)\\) ([Lemma 2.1](#oa-fnd-wa-02)(4)). The centre of \\(B(H_\\varphi)\\) is \\(\\mathbb C1\\), so \\(z_\\varphi\\) is a minimal central projection of \\(\\tilde A\\); likewise \\(z_\\psi\\). For two minimal central projections, \\(z_\\varphi z_\\psi\\) is a central projection under both, so either \\(z_\\varphi z_\\psi=0\\) or \\(z_\\varphi=z_\\psi\\).\n\nSuppose \\(z_\\varphi z_\\psi=0\\). Since \\(\\varphi\\in V(\\pi_\\varphi)\\) and \\(\\psi\\in V(\\pi_\\psi)\\), [Proposition 5.2](#oa-fnd-wa-09)(2) gives \\(\\hat\\varphi=\\hat\\varphi z_\\varphi\\) and \\(\\hat\\psi=\\hat\\psi z_\\psi\\), so \\(\\hat\\psi z_\\varphi=0\\). [Lemma 10.2](#oa-fnd-wa-10), with the central projection \\(z_\\varphi\\), gives\n\\[\n\\begin{gathered}\n\\|\\varphi-\\psi\\|\\\\\n=\\|(\\hat\\varphi-\\hat\\psi)z_\\varphi\\|+\\|(\\hat\\varphi-\\hat\\psi)(1-z_\\varphi)\\|\\\\\n=\\|\\hat\\varphi\\|+\\|\\hat\\psi\\|\\\\\n=2 .\n\\end{gathered}\n\\]\nSo \\(\\|\\varphi-\\psi\\|<2\\) forces \\(z_\\varphi=z_\\psi\\). By [Theorem 5.3](#oa-fnd-wa-09) there is a \\(*\\)-isomorphism \\(\\theta:B(H_\\varphi)\\to B(H_\\psi)\\) with \\(\\theta\\circ\\pi_\\varphi=\\pi_\\psi\\). By Lemma 6.4, \\(\\theta(x)=UxU^*\\) for a unitary \\(U\\), so \\(U\\pi_\\varphi(a)U^*=\\pi_\\psi(a)\\). \\(\\square\\)\n\nThe bound \\(2\\) cannot be improved: two characters \\(\\delta_s\\ne\\delta_t\\) of \\(C(X)\\) are pure states with \\(\\|\\delta_s-\\delta_t\\|=2\\) and inequivalent (one-dimensional, different) representations.\n\n",
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      "id": "OA-FND-WA-16",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "10. Normal and singular parts",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
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      "full_conditions_and_proof": "## 10. Normal and singular parts\n\nEvery bounded functional on a von Neumann algebra splits, in exactly one way and with additive norms, into a normal part and a singular part. The splitting comes from one central projection of the universal enveloping algebra.\n\nLet \\(M\\subseteq B(H)\\) be a von Neumann algebra, viewed as a \\(C^*\\)-algebra, with universal enveloping algebra \\(\\tilde M=M^{**}\\). The identity map \\(\\pi_0:M\\to B(H)\\) is a representation, and \\(\\mathcal M(\\pi_0)=M''=M\\). Put \\(z_0=z(\\pi_0)\\), a central projection of \\(\\tilde M\\). By [Proposition 5.2](#oa-fnd-wa-09)(2),\n\\[\nM_*=V(\\pi_0)=M^*z_0 .\n\\tag{10.1}\n\\]\n\n**Definition 10.1.** The elements of \\(M_*\\) are the *normal* functionals, and \\(M_*\\) is the *predual* of \\(M\\). The elements of \\(M_*^{\\perp}:=M^*(1-z_0)\\) are the *singular* functionals. (The symbol \\(M_*^\\perp\\) is only a name here; it is not an annihilator.)\n\nFor a \\(W^*\\)-algebra \\(A\\) with predual \\(F\\), normal and singular functionals are defined in the same way through the representation \\(\\pi\\) of [Theorem 9.2](#oa-fnd-wa-22). Since \\(\\mathcal M(\\pi)=\\pi(A)\\), [Proposition 5.2](#oa-fnd-wa-09)(2) gives the normal ones as \\(\\iota_F(F)=V(\\pi)=A^*z(\\pi)\\), and the singular ones are \\(A^*(1-z(\\pi))\\).\n\n**Lemma 10.2** (Norms split along central projections). Let \\(N\\) be a von Neumann algebra, \\(z\\) a central projection and \\(\\varphi\\in N_*\\). Then \\(\\|\\varphi\\|=\\|\\varphi z\\|+\\|\\varphi(1-z)\\|\\).\n\n**Proof.** The inequality \\(\\le\\) is the triangle inequality. Given \\(\\varepsilon>0\\), choose \\(x,y\\) in the unit ball with \\((\\varphi z)(x)\\ge\\|\\varphi z\\|-\\varepsilon\\) and \\((\\varphi(1-z))(y)\\ge\\|\\varphi(1-z)\\|-\\varepsilon\\), after rotating phases. Put \\(w=zx+(1-z)y\\). Because \\(z\\) is central, \\(w^*w=zx^*xz+(1-z)y^*y(1-z)\\le1\\), so \\(\\|w\\|\\le1\\). And \\[\n\\begin{gathered}\n\\varphi(w)\\\\\n=\\varphi(zx)+\\varphi((1-z)y)\\\\\n\\ge\\|\\varphi z\\|+\\|\\varphi(1-z)\\|-2\\varepsilon.\n\\end{gathered}\n\\] \\(\\square\\)\n\n**Theorem 10.3** (The normal–singular splitting).\n\n1. Every norm-closed left or right invariant subspace \\(V\\subseteq M^*\\) splits as\n\\[\n\\begin{gathered}\nV\\\\\n=(V\\cap M_*)\\\\\n\\oplus_1(V\\cap M_*^\\perp),\\\\\nV\\cap M_*\\\\\n=Vz_0,\\\\\nV\\cap M_*^\\perp\\\\\n=V(1-z_0),\n\\end{gathered}\n\\tag{10.2}\n\\]\nwhere \\(\\oplus_1\\) means \\(\\|\\varphi+\\psi\\|=\\|\\varphi\\|+\\|\\psi\\|\\) for \\(\\varphi\\in V\\cap M_*\\) and \\(\\psi\\in V\\cap M_*^\\perp\\).\n2. Let \\(\\pi\\) be a representation of the \\(C^*\\)-algebra \\(M\\) on \\(H_\\pi\\), and put \\(z=\\bar\\pi(z_0)\\), a central projection of \\(\\mathcal M(\\pi)\\). Then \\(\\pi_n(x)=\\pi(x)z\\) is a normal representation of \\(M\\) on \\(zH_\\pi\\), and \\(\\pi_s(x)=\\pi(x)(1-z)\\) is a representation of \\(M\\) on \\((1-z)H_\\pi\\) whose coefficient functionals \\(\\omega_{\\pi_s;\\xi,\\eta}\\) are all singular.\n\n**Proof.** (1) Let \\(V\\) be left invariant; the right case is the same. By [Theorem 4.3](#oa-fnd-wa-07)(1), applied to \\(\\tilde M\\) and \\(S=M\\), \\(V\\) is invariant under \\(\\tilde M\\), so \\(z_0V\\subseteq V\\) and \\((1-z_0)V\\subseteq V\\). Since \\(z_0\\) is central, \\(z_0\\varphi=\\varphi z_0\\). Every \\(\\varphi\\in V\\) is \\(\\varphi z_0+\\varphi(1-z_0)\\). If \\(\\varphi\\in V\\) and \\(\\varphi=\\varphi z_0\\), then \\(\\varphi\\in Vz_0\\); this gives \\(V\\cap M^*z_0=Vz_0\\), and likewise for \\(1-z_0\\). The norm identity is Lemma 10.2, applied in \\(\\tilde M\\).\n\n(2) \\(\\bar\\pi\\) maps \\(\\tilde M\\) onto \\(\\mathcal M(\\pi)\\), so it maps the centre into the centre and \\(z\\) is a central projection; also \\(\\pi_n(x)=\\bar\\pi(xz_0)\\). For \\(\\xi,\\eta\\in zH_\\pi\\), the coefficient \\(x\\mapsto\\langle\\pi_n(x)\\xi,\\eta\\rangle\\) is \\(\\big(z_0\\,(\\omega_{\\xi,\\eta}\\circ\\bar\\pi)\\big)\\) restricted to \\(M\\), which lies in \\(M^*z_0=M_*\\). Every \\(\\sigma\\)-weakly continuous functional on \\(B(zH_\\pi)\\) is a norm-convergent sum of vector functionals (background fact on preduals), and \\(M_*\\) is norm closed. So \\(\\psi\\circ\\pi_n\\in M_*\\) for every normal \\(\\psi\\), and \\(\\pi_n\\) is normal (background fact on normal maps). It is nondegenerate on \\(zH_\\pi\\) because \\(\\pi_n(1)=z\\). The same computation with \\(1-z_0\\) shows that the coefficients of \\(\\pi_s\\) lie in \\(M^*(1-z_0)\\). \\(\\square\\)\n\n**Normal and singular parts.** By (1) with \\(V=M^*\\), every \\(\\varphi\\in M^*\\) is uniquely \\(\\varphi=\\varphi_n+\\varphi_s\\) with \\(\\varphi_n\\in M_*\\) and \\(\\varphi_s\\in M_*^\\perp\\), namely \\(\\varphi_n=\\varphi z_0\\), and \\(\\|\\varphi\\|=\\|\\varphi_n\\|+\\|\\varphi_s\\|\\). If \\(\\varphi\\) is positive, so are both parts, because \\((\\varphi z_0)(x^*x)=\\hat\\varphi(z_0x^*xz_0)\\ge0\\). Since \\(M^*\\) is spanned by its positive elements (see the proof of [Proposition 1.1](#oa-fnd-wa-01)(2)), so is \\(M_*^\\perp\\). By (2), every representation splits as \\(\\pi=\\pi_n\\oplus\\pi_s\\), a normal part and a singular part.\n\n**Definition 10.4.** A bounded linear map \\(T:M\\to N\\) between von Neumann algebras is *normal* if \\(\\psi\\circ T\\in M_*\\) for every \\(\\psi\\in N_*\\), and *singular* if \\(\\psi\\circ T\\in M_*^\\perp\\) for every \\(\\psi\\in N_*\\).\n\n**Proposition 10.5** (Splitting of maps). Every bounded linear \\(T:M\\to N\\) is uniquely \\(T=T_n+T_s\\) with \\(T_n\\) normal and \\(T_s\\) singular. If \\(T\\) is positive, so are \\(T_n\\) and \\(T_s\\).\n\n**Proof.** Let \\(T_*:N_*\\to M^*=\\tilde M_*\\), \\(\\psi\\mapsto\\psi\\circ T\\), and let \\(\\tilde T=(T_*)^*:\\tilde M\\to N\\), using \\(N=(N_*)^*\\). For \\(x\\in M\\) and \\(\\psi\\in N_*\\), \\(\\psi(\\tilde T(j(x)))=j(x)(\\psi\\circ T)=\\psi(T(x))\\), so \\(\\tilde T\\) extends \\(T\\). Put \\(T_n(x)=\\tilde T(xz_0)\\) and \\(T_s(x)=\\tilde T(x(1-z_0))\\). Then \\(\\psi\\circ T_n=z_0(\\psi\\circ T)\\in M_*\\) and \\(\\psi\\circ T_s=(1-z_0)(\\psi\\circ T)\\in M_*^\\perp\\). If \\(T=T_1+T_2\\) is another such splitting, then for each \\(\\psi\\), \\(\\psi\\circ T=\\psi\\circ T_1+\\psi\\circ T_2\\) is the unique splitting of \\(\\psi\\circ T\\), so \\(\\psi\\circ T_1=\\psi\\circ T_n\\); since \\(N_*\\) separates \\(N\\), \\(T_1=T_n\\). If \\(T\\ge0\\), then \\(\\psi\\circ T\\) is positive for \\(\\psi\\in N_*^+\\), hence so are its two parts, and positivity in \\(N\\) is detected by \\(N_*^+\\) (background). \\(\\square\\)\n\n**Exercise 10.6** (medium) (Nearby pure states). Let \\(\\varphi,\\psi\\) be pure states of \\(A\\) with \\(\\|\\varphi-\\psi\\|<2\\). Show that \\(\\pi_\\varphi\\) and \\(\\pi_\\psi\\) are unitarily equivalent.\n\n**Solution.** Write \\(z_\\varphi=z(\\pi_\\varphi)\\) and \\(z_\\psi=z(\\pi_\\psi)\\). By Lemma 7.2, \\(\\mathcal M(\\pi_\\varphi)=B(H_\\varphi)\\), and \\(\\tilde Az_\\varphi\\cong B(H_\\varphi)\\) ([Lemma 2.1](#oa-fnd-wa-02)(4)). The centre of \\(B(H_\\varphi)\\) is \\(\\mathbb C1\\), so \\(z_\\varphi\\) is a minimal central projection of \\(\\tilde A\\); likewise \\(z_\\psi\\). For two minimal central projections, \\(z_\\varphi z_\\psi\\) is a central projection under both, so either \\(z_\\varphi z_\\psi=0\\) or \\(z_\\varphi=z_\\psi\\).\n\nSuppose \\(z_\\varphi z_\\psi=0\\). Since \\(\\varphi\\in V(\\pi_\\varphi)\\) and \\(\\psi\\in V(\\pi_\\psi)\\), [Proposition 5.2](#oa-fnd-wa-09)(2) gives \\(\\hat\\varphi=\\hat\\varphi z_\\varphi\\) and \\(\\hat\\psi=\\hat\\psi z_\\psi\\), so \\(\\hat\\psi z_\\varphi=0\\). [Lemma 10.2](#oa-fnd-wa-10), with the central projection \\(z_\\varphi\\), gives\n\\[\n\\begin{gathered}\n\\|\\varphi-\\psi\\|\\\\\n=\\|(\\hat\\varphi-\\hat\\psi)z_\\varphi\\|+\\|(\\hat\\varphi-\\hat\\psi)(1-z_\\varphi)\\|\\\\\n=\\|\\hat\\varphi\\|+\\|\\hat\\psi\\|\\\\\n=2 .\n\\end{gathered}\n\\]\nSo \\(\\|\\varphi-\\psi\\|<2\\) forces \\(z_\\varphi=z_\\psi\\). By [Theorem 5.3](#oa-fnd-wa-09) there is a \\(*\\)-isomorphism \\(\\theta:B(H_\\varphi)\\to B(H_\\psi)\\) with \\(\\theta\\circ\\pi_\\varphi=\\pi_\\psi\\). By Lemma 6.4, \\(\\theta(x)=UxU^*\\) for a unitary \\(U\\), so \\(U\\pi_\\varphi(a)U^*=\\pi_\\psi(a)\\). \\(\\square\\)\n\nThe bound \\(2\\) cannot be improved: two characters \\(\\delta_s\\ne\\delta_t\\) of \\(C(X)\\) are pure states with \\(\\|\\delta_s-\\delta_t\\|=2\\) and inequivalent (one-dimensional, different) representations.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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      "id": "OA-FND-WA-18",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "11. Singular functionals and the uniqueness of the predual",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "anchor": "oa-fnd-wa-18",
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      "full_conditions_and_proof": "## 11. Singular functionals and the uniqueness of the predual\n\nA normal positive functional has a support projection; a singular one has none. This difference characterizes singular functionals by their values on projections alone. It follows that the predual of a \\(W^*\\)-algebra is unique, and that normality does not depend on the representation.\n\n**Lemma 11.1** (Supports of normal positive functionals). Let \\(A\\) be a \\(W^*\\)-algebra with a predual \\(A_*\\), and \\(\\omega\\) a nonzero normal positive functional. There is a unique nonzero projection \\(e\\in A\\) such that \\(\\omega=e\\omega=\\omega e\\) (so \\(\\omega(x)=\\omega(exe)\\)) and \\(\\omega\\) is faithful on \\(eAe\\). It is the *support* \\(s(\\omega)\\). Also \\(1-s(\\omega)\\) is the largest projection on which \\(\\omega\\) vanishes, so this is the support recalled in the background.\n\n**Proof.** Realise \\(A\\) as a von Neumann algebra with \\(\\sigma\\)-weak topology \\(\\sigma(A,A_*)\\) ([Theorem 9.2](#oa-fnd-wa-22)). Let \\(W\\) be the closed span of \\(\\{\\omega a:a\\in A\\}\\), where \\((\\omega a)(x)=\\omega(ax)\\). It is a norm-closed right invariant subspace of \\(A_*\\), and \\(\\omega=\\omega1\\in W\\). By [Theorem 4.3](#oa-fnd-wa-07)(3), \\(W=eA_*\\) for a projection \\(e\\), and \\(W^\\circ=A(1-e)\\). From \\(\\omega\\in eA_*\\) we get \\(\\omega=e\\omega\\). Since \\(\\omega\\) is positive, \\(\\omega^*=\\omega\\), and \\((e\\omega)^*=\\omega^*e\\), so \\(\\omega=\\omega e\\) as well; hence \\(\\omega=e\\omega e\\).\n\nLet \\(x\\in A\\) with \\(\\omega(x^*x)=0\\). By Cauchy–Schwarz, \\(|(\\omega a)(x)|^2=|\\omega(ax)|^2\\le\\omega(aa^*)\\,\\omega(x^*x)=0\\) for all \\(a\\). So \\(x\\in W^\\circ\\) and \\(xe=0\\). If \\(x\\in eAe\\), then \\(x=xe=0\\): \\(\\omega\\) is faithful on \\(eAe\\). In particular \\(e\\ne0\\), since \\(\\omega\\ne0\\).\n\nUniqueness. Let \\(f\\) be another projection with \\(\\omega=f\\omega f\\), faithful on \\(fAf\\). Then \\(\\omega(1-f)=\\omega(f(1-f)f)=0\\), so \\(x=(1-f)e\\) has \\(\\omega(x^*x)=\\omega(e(1-f)e)=\\omega(1-f)=0\\). Hence \\(xe=0\\), that is \\(e\\le f\\). Also \\(\\omega(f-e)=\\omega(1-e)-\\omega(1-f)=0\\), and \\(f-e\\ge0\\) lies in \\(fAf\\), so \\(f=e\\). Finally, \\(\\omega(1-e)=0\\), and a projection \\(q\\) with \\(\\omega(q)=0\\) has \\(\\omega(q^*q)=0\\), so \\(qe=0\\) and \\(q\\le1-e\\). \\(\\square\\)\n\nA singular state has no support in this sense. If \\(\\omega\\) is a singular state and \\(e\\) is any nonzero projection, Theorem 11.2 below gives a nonzero \\(e_0\\le e\\) with \\(\\omega(e_0)=0\\), so \\(\\omega\\) is not faithful on \\(eAe\\). An example is a state of \\(\\ell^\\infty\\) that vanishes on \\(c_0\\) ([Example 11.8](#oa-fnd-wa-30)).\n\nThroughout the rest of this section, \\(A\\) is a \\(W^*\\)-algebra, realised as a von Neumann algebra by [Theorem 9.2](#oa-fnd-wa-22), with predual \\(A_*\\). Its universal enveloping algebra is \\(\\tilde A\\), and \\(z_0\\in\\tilde A\\) is the central projection with \\(A_*=A^*z_0\\), as in (10.1) of [Section 10](#oa-fnd-wa-10).\n\n**Theorem 11.2** (Singular functionals). For a positive \\(\\omega\\in A^*\\), the following are equivalent:\n\n1. \\(\\omega\\) is singular;\n2. every nonzero projection \\(e\\in A\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\).\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). Split \\(\\omega=\\omega_n+\\omega_s\\) into its normal and singular parts, which are positive (Section 10). If \\(\\omega_n\\ne0\\), let \\(e=s(\\omega_n)\\) ([Lemma 11.1](#oa-fnd-wa-23)). By (2), some nonzero \\(e_0\\le e\\) has \\(\\omega(e_0)=0\\). Then \\(0\\le\\omega_n(e_0)\\le\\omega(e_0)=0\\), which contradicts the faithfulness of \\(\\omega_n\\) on \\(eAe\\). So \\(\\omega_n=0\\).\n\n(1)\\(\\Rightarrow\\)(2). Let \\(\\omega\\) be singular and \\(e\\ne0\\) a projection. If \\(\\omega(e)=0\\), take \\(e_0=e\\). Otherwise choose a normal positive \\(\\omega_0\\) with \\(\\omega_0(e)>\\omega(e)\\): a large multiple of a vector state at a unit vector in the range of \\(e\\). Let \\(\\mathcal F\\) be the set of projections \\(p\\le e\\) with \\(\\omega_0(p)\\le\\omega(p)\\), ordered as projections. It contains \\(0\\). If \\((p_i)\\) is a chain in \\(\\mathcal F\\), its least upper bound \\(p\\le e\\) satisfies \\(\\omega(p)\\ge\\sup_i\\omega(p_i)\\ge\\sup_i\\omega_0(p_i)=\\omega_0(p)\\), using positivity of \\(\\omega\\) and normality of \\(\\omega_0\\). So \\(p\\in\\mathcal F\\), and Zorn's lemma gives a maximal \\(p\\in\\mathcal F\\). As \\(e\\notin\\mathcal F\\), \\(e_0=e-p\\ne0\\). For a nonzero projection \\(q\\le e_0\\), maximality excludes \\(p+q\\in\\mathcal F\\), so \\(\\omega(q)<\\omega_0(q)\\); hence \\(\\omega(q)\\le\\omega_0(q)\\) for all projections \\(q\\le e_0\\). A positive \\(y\\in e_0Ae_0\\) is a norm limit of nonnegative combinations of such projections (spectral projections of \\(y\\) for Borel sets away from \\(0\\); background). So \\(e_0\\omega e_0\\le e_0\\omega_0e_0\\) as functionals on \\(A\\), where \\((e_0\\varphi e_0)(y)=\\varphi(e_0ye_0)\\). These inequalities persist for the normal extensions to \\(\\tilde A\\), and multiplying by the central projection \\(1-z_0\\) preserves order. Now \\(e_0\\omega_0e_0\\) is normal, so \\((1-z_0)e_0\\omega_0e_0=0\\), while \\(\\omega=(1-z_0)\\omega\\) gives \\((1-z_0)e_0\\omega e_0=e_0\\omega e_0\\). Hence \\(0\\le e_0\\omega e_0\\le0\\), and \\(\\omega(e_0)=(e_0\\omega e_0)(1)=0\\). \\(\\square\\)\n\n**Corollary 11.3** (Uniqueness of the predual). If \\(F_1,F_2\\) are preduals of a \\(C^*\\)-algebra \\(A\\), then \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\) in \\(A^*\\); equivalently \\(\\sigma(A,F_1)=\\sigma(A,F_2)\\). We write \\(A_*\\) for this space.\n\n**Proof.** By [Theorem 9.2](#oa-fnd-wa-22), \\(F_k\\) gives a faithful representation \\(\\pi_k\\) with \\(\\mathcal M(\\pi_k)=\\pi_k(A)\\) and \\(\\iota_{F_k}(F_k)=V(\\pi_k)=A^*z_k\\), where \\(z_k=z(\\pi_k)\\) ([Proposition 5.2](#oa-fnd-wa-09)). The \\(F_k\\)-singular functionals are \\(A^*(1-z_k)\\). Condition (2) of Theorem 11.2 refers only to the algebra and its projections. So a positive functional is \\(F_1\\)-singular exactly when it is \\(F_2\\)-singular. Each \\(A^*(1-z_k)\\) is spanned by its positive elements (Section 10), so \\(A^*(1-z_1)=A^*(1-z_2)\\). Supports of invariant subspaces are unique (Theorem 4.3(3)), so \\(z_1=z_2\\) and \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\). \\(\\square\\)\n\n**Corollary 11.4** (Isomorphisms are normal). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) between von Neumann algebras is a homeomorphism for the \\(\\sigma\\)-weak topologies, and also for the \\(\\sigma\\)-strong and the \\(\\sigma\\)-strong\\(^*\\) topologies.\n\n**Proof.** Let \\(F=\\{\\psi\\circ\\theta:\\psi\\in N_*\\}\\subseteq M^*\\). Since \\(\\theta\\) is an isometric bijection and \\(N=(N_*)^*\\), the pairing \\(\\langle x,\\psi\\circ\\theta\\rangle=\\psi(\\theta(x))\\) makes \\(F\\) a predual of \\(M\\) with \\(\\iota_F(F)=F\\). By Corollary 11.3, \\(F=M_*\\). So \\(\\psi\\circ\\theta\\in M_*\\) for all \\(\\psi\\in N_*\\): \\(\\theta\\) is normal (background fact on normal maps). The same holds for \\(\\theta^{-1}\\). For \\(\\psi\\in N_*^+\\), \\(\\psi(\\theta(x)^*\\theta(x))=(\\psi\\circ\\theta)(x^*x)\\) with \\(\\psi\\circ\\theta\\in M_*^+\\), and the seminorms \\(x\\mapsto\\omega(x^*x)^{1/2}\\), \\(\\omega\\) positive normal, generate the \\(\\sigma\\)-strong topology (background); adding the same seminorms at \\(x^*\\) gives the \\(\\sigma\\)-strong\\(^*\\) topology. \\(\\square\\)\n\nThe positive-map criterion proved after Corollary 11.5 gives a second proof: a \\(*\\)-isomorphism is an order isomorphism, so it preserves suprema of bounded increasing nets, and a positive map with that property is normal. Either way, the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies of a \\(W^*\\)-algebra do not depend on how it is represented, and neither do its normal and singular functionals. This is why the Hilbert space need not be named.\n\n**Corollary 11.5** (Complete additivity). For \\(\\omega\\in A^*\\) on a \\(W^*\\)-algebra \\(A\\), the following are equivalent:\n\n1. \\(\\omega\\) is normal;\n2. \\(\\omega\\) is completely additive: for every family \\((e_i)_{i\\in I}\\) of mutually orthogonal projections, the finite partial sums of \\(\\sum_i\\omega(e_i)\\) converge to \\(\\omega(\\sum_ie_i)\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). The finite partial sums \\(e_J=\\sum_{i\\in J}e_i\\) increase to \\(\\sum_ie_i\\), strongly and \\(\\sigma\\)-weakly (background fact on monotone nets), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous.\n\n(2)\\(\\Rightarrow\\)(1). Write \\(\\omega=\\omega_n+\\omega_s\\) (Section 10). By (1)\\(\\Rightarrow\\)(2), \\(\\omega_n\\) is completely additive, so \\(\\omega_s=\\omega-\\omega_n\\) is too. It remains to show that a completely additive singular functional \\(\\omega\\) is zero. Write \\(\\omega=\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with singular positive \\(\\omega_k\\) (Section 10), and put \\([\\omega]=\\omega_1+\\omega_2+\\omega_3+\\omega_4\\), a singular positive functional. Let \\(e\\) be a projection. By Zorn's lemma choose a maximal family \\((e_i)\\) of mutually orthogonal nonzero projections under \\(e\\) with \\([\\omega](e_i)=0\\). If \\(e-\\sum_ie_i\\ne0\\), Theorem 11.2 applied to \\([\\omega]\\) gives a nonzero projection under it on which \\([\\omega]\\) vanishes, against maximality. So \\(e=\\sum_ie_i\\). Since \\(0\\le\\omega_k\\le[\\omega]\\), each \\(\\omega_k(e_i)=0\\), so \\(\\omega(e_i)=0\\), and complete additivity gives \\(\\omega(e)=0\\). Thus \\(\\omega\\) vanishes on every projection, and since every self-adjoint element is a norm limit of combinations of projections (background fact on spectral projections), \\(\\omega=0\\). \\(\\square\\)\n\n### Positive maps and increasing suprema\n\nFor a positive functional \\(\\omega\\) on a von Neumann algebra, order normality implies complete additivity: apply it to the finite sums of an orthogonal family of projections. Corollary 11.5 then makes \\(\\omega\\) sigma-weakly continuous. The converse follows from bounded monotone convergence and the definition of the topology.\n\nLet \\(T:M\\to N\\) be a bounded positive map. If it preserves suprema of bounded increasing self-adjoint nets, then each \\(\\psi\\in N_*^+\\) has order-normal pullback \\(\\psi\\circ T\\), hence a sigma-weakly continuous pullback by the preceding paragraph. Positive normal functionals span \\(N_*\\), so the preadjoint test makes \\(T\\) normal. Conversely, if \\(T\\) is normal and \\(x_i\\uparrow x\\) in \\(M\\) is norm bounded, then \\(T(x_i)\\) is increasing and bounded. Its supremum \\(y\\) is its sigma-weak limit by bounded monotone convergence. Normality gives the same limit \\(T(x)\\), so \\(y=T(x)\\). Positive maps are automatically bounded by [Proposition 3.2(4) of completely positive maps](completely-positive-maps.md#oa-fnd-cm-03), so this also applies without an added boundedness hypothesis.\n\nThe earlier use of automatic normality in Theorem 5.3 is a forward reference to Corollary 11.4. Its proof uses predual uniqueness, the singular-functional projection criterion, Sakai's theorem, and the bidual/ideal results, all of which were proved without Theorem 5.3. Lemma 2.1's normal inverse was proved directly by its preadjoint. There is therefore no dependency cycle.\n\n**Remark 11.6** (Abelian subalgebras suffice). If the restriction of \\(\\omega\\in A^*\\) to every abelian \\(W^*\\)-subalgebra \\(B\\) of \\(A\\) (a \\(\\sigma\\)-weakly closed abelian \\(*\\)-subalgebra containing \\(1\\)) is normal, then \\(\\omega\\) is normal. Indeed, given orthogonal projections \\((e_i)\\), let \\(B\\) be the von Neumann algebra generated by them and \\(1\\); it is abelian and contains \\(\\sum_ie_i\\). Its \\(\\sigma\\)-weak topology is the relative one, so normality of \\(\\omega|_B\\) and (1)\\(\\Rightarrow\\)(2) in \\(B\\) give complete additivity on \\((e_i)\\). By (2)\\(\\Rightarrow\\)(1), \\(\\omega\\) is normal.\n\n**Example 11.7** (Which topologies are intrinsic). The \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies are (Corollary 11.4). The weak, strong and strong\\(^*\\) operator topologies are not. Let \\(M=B(H)\\) with \\(\\dim H=\\infty\\), \\(\\pi_1\\) the identity representation and \\(\\pi_2(x)=x\\oplus x\\oplus\\cdots\\) on \\(H\\oplus H\\oplus\\cdots\\). Both are faithful normal representations with von Neumann algebra images. The weak operator topology of \\(\\pi_2(M)\\), pulled back to \\(M\\), is the \\(\\sigma\\)-weak topology, since the vector functionals of \\(\\pi_2\\) are exactly the square-summable series \\(\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\). It is strictly finer than the weak operator topology of \\(\\pi_1\\). Indeed, with an orthonormal sequence \\((e_n)\\), the functional \\(\\varphi(x)=\\sum_nn^{-2}\\langle xe_n,e_n\\rangle\\) is \\(\\sigma\\)-weakly continuous. It is not weakly continuous: a weakly continuous functional has the form \\(\\sum_{k=1}^m\\langle x\\xi_k,\\eta_k\\rangle\\) (background fact on preduals). Choose a unit vector \\(\\zeta\\in\\operatorname{span}\\{e_1,\\dots,e_{m+1}\\}\\) orthogonal to \\(\\xi_1,\\dots,\\xi_m\\), and let \\(x\\) be the projection onto \\(\\mathbb C\\zeta\\). Then the finite sum vanishes at \\(x\\), but \\(\\varphi(x)=\\sum_nn^{-2}|\\langle\\zeta,e_n\\rangle|^2>0\\).\n\n**Example 11.8** (Normal and singular parts on \\(\\ell^\\infty\\)). In \\(\\ell^\\infty\\) every nonzero projection majorises some coordinate projection \\(e_n\\), and the \\(e_n\\) are minimal. By Theorem 11.2, a positive functional is singular exactly when it vanishes on every \\(e_n\\), that is on \\(c_0\\). Hence for any \\(\\varphi\\in(\\ell^\\infty)^*\\), \\(\\varphi_n(x)=\\sum_k\\varphi(e_k)x_k\\) and \\(\\varphi_s=\\varphi-\\varphi_n\\). A state that is a limit along a free ultrafilter is singular, and it has no support in the sense of [Lemma 11.1](#oa-fnd-wa-23).\n\n**Exercise 11.9.** (easy) Let \\(\\mathcal V\\) be a free ultrafilter on \\(\\mathbb N\\), \\(\\chi(x)=\\lim_{n\\to\\mathcal V}x_n\\) on \\(\\ell^\\infty\\), and \\(\\psi=\\tfrac12(\\delta_1+\\chi)\\), where \\(\\delta_1(x)=x_1\\). Find \\(\\psi_n\\), \\(\\psi_s\\), the cyclic representation \\(\\pi_\\psi\\), and its normal and singular parts.\n\n**Solution.** By Example 11.8, \\(\\chi\\) is singular, and \\(\\psi_n=\\tfrac12\\delta_1\\), \\(\\psi_s=\\tfrac12\\chi\\). Both \\(\\delta_1\\) and \\(\\chi\\) are characters, and they are distinct, so the cyclic representation of \\(\\psi\\) is \\(\\pi_\\psi(x)=\\operatorname{diag}(x_1,\\chi(x))\\) on \\(\\mathbb C^2\\), with cyclic vector \\((2^{-1/2},2^{-1/2})\\). Its normal part ([Theorem 10.3](#oa-fnd-wa-10)(2)) is the first coordinate, \\(x\\mapsto x_1\\), and its singular part is the second, \\(x\\mapsto\\chi(x)\\). \\(\\square\\)\n\n**Exercise 11.10** (medium) (When every state is normal). Prove that all states of a von Neumann algebra \\(M\\) are normal exactly when \\(\\dim M<\\infty\\).\n\n**Solution.** If \\(\\dim M<\\infty\\), every linear functional is continuous for the Hausdorff vector topology \\(\\sigma(M,M_*)\\), hence normal. If \\(\\dim M=\\infty\\), Lemma 6.2 gives orthogonal nonzero projections \\(p_1,p_2,\\dots\\). Choose unit vectors \\(\\xi_n\\) in the range of \\(p_n\\) and a free ultrafilter \\(\\mathcal V\\) on \\(\\mathbb N\\), and put \\(\\varphi(x)=\\lim_{n\\to\\mathcal V}\\langle x\\xi_n,\\xi_n\\rangle\\). This is a state. Here \\(\\varphi(p_k)=\\lim_{n\\to\\mathcal V}\\delta_{kn}=0\\) for every \\(k\\), while \\(\\varphi(\\sum_kp_k)=1\\). The finite partial sums of \\(\\sum_kp_k\\) increase to it, so \\(\\varphi\\) does not preserve this supremum and is not normal (background fact on monotone nets). \\(\\square\\)\n\n",
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      "name": "11. Singular functionals and the uniqueness of the predual",
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      "full_conditions_and_proof": "## 11. Singular functionals and the uniqueness of the predual\n\nA normal positive functional has a support projection; a singular one has none. This difference characterizes singular functionals by their values on projections alone. It follows that the predual of a \\(W^*\\)-algebra is unique, and that normality does not depend on the representation.\n\n**Lemma 11.1** (Supports of normal positive functionals). Let \\(A\\) be a \\(W^*\\)-algebra with a predual \\(A_*\\), and \\(\\omega\\) a nonzero normal positive functional. There is a unique nonzero projection \\(e\\in A\\) such that \\(\\omega=e\\omega=\\omega e\\) (so \\(\\omega(x)=\\omega(exe)\\)) and \\(\\omega\\) is faithful on \\(eAe\\). It is the *support* \\(s(\\omega)\\). Also \\(1-s(\\omega)\\) is the largest projection on which \\(\\omega\\) vanishes, so this is the support recalled in the background.\n\n**Proof.** Realise \\(A\\) as a von Neumann algebra with \\(\\sigma\\)-weak topology \\(\\sigma(A,A_*)\\) ([Theorem 9.2](#oa-fnd-wa-22)). Let \\(W\\) be the closed span of \\(\\{\\omega a:a\\in A\\}\\), where \\((\\omega a)(x)=\\omega(ax)\\). It is a norm-closed right invariant subspace of \\(A_*\\), and \\(\\omega=\\omega1\\in W\\). By [Theorem 4.3](#oa-fnd-wa-07)(3), \\(W=eA_*\\) for a projection \\(e\\), and \\(W^\\circ=A(1-e)\\). From \\(\\omega\\in eA_*\\) we get \\(\\omega=e\\omega\\). Since \\(\\omega\\) is positive, \\(\\omega^*=\\omega\\), and \\((e\\omega)^*=\\omega^*e\\), so \\(\\omega=\\omega e\\) as well; hence \\(\\omega=e\\omega e\\).\n\nLet \\(x\\in A\\) with \\(\\omega(x^*x)=0\\). By Cauchy–Schwarz, \\(|(\\omega a)(x)|^2=|\\omega(ax)|^2\\le\\omega(aa^*)\\,\\omega(x^*x)=0\\) for all \\(a\\). So \\(x\\in W^\\circ\\) and \\(xe=0\\). If \\(x\\in eAe\\), then \\(x=xe=0\\): \\(\\omega\\) is faithful on \\(eAe\\). In particular \\(e\\ne0\\), since \\(\\omega\\ne0\\).\n\nUniqueness. Let \\(f\\) be another projection with \\(\\omega=f\\omega f\\), faithful on \\(fAf\\). Then \\(\\omega(1-f)=\\omega(f(1-f)f)=0\\), so \\(x=(1-f)e\\) has \\(\\omega(x^*x)=\\omega(e(1-f)e)=\\omega(1-f)=0\\). Hence \\(xe=0\\), that is \\(e\\le f\\). Also \\(\\omega(f-e)=\\omega(1-e)-\\omega(1-f)=0\\), and \\(f-e\\ge0\\) lies in \\(fAf\\), so \\(f=e\\). Finally, \\(\\omega(1-e)=0\\), and a projection \\(q\\) with \\(\\omega(q)=0\\) has \\(\\omega(q^*q)=0\\), so \\(qe=0\\) and \\(q\\le1-e\\). \\(\\square\\)\n\nA singular state has no support in this sense. If \\(\\omega\\) is a singular state and \\(e\\) is any nonzero projection, Theorem 11.2 below gives a nonzero \\(e_0\\le e\\) with \\(\\omega(e_0)=0\\), so \\(\\omega\\) is not faithful on \\(eAe\\). An example is a state of \\(\\ell^\\infty\\) that vanishes on \\(c_0\\) ([Example 11.8](#oa-fnd-wa-30)).\n\nThroughout the rest of this section, \\(A\\) is a \\(W^*\\)-algebra, realised as a von Neumann algebra by [Theorem 9.2](#oa-fnd-wa-22), with predual \\(A_*\\). Its universal enveloping algebra is \\(\\tilde A\\), and \\(z_0\\in\\tilde A\\) is the central projection with \\(A_*=A^*z_0\\), as in (10.1) of [Section 10](#oa-fnd-wa-10).\n\n**Theorem 11.2** (Singular functionals). For a positive \\(\\omega\\in A^*\\), the following are equivalent:\n\n1. \\(\\omega\\) is singular;\n2. every nonzero projection \\(e\\in A\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\).\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). Split \\(\\omega=\\omega_n+\\omega_s\\) into its normal and singular parts, which are positive (Section 10). If \\(\\omega_n\\ne0\\), let \\(e=s(\\omega_n)\\) ([Lemma 11.1](#oa-fnd-wa-23)). By (2), some nonzero \\(e_0\\le e\\) has \\(\\omega(e_0)=0\\). Then \\(0\\le\\omega_n(e_0)\\le\\omega(e_0)=0\\), which contradicts the faithfulness of \\(\\omega_n\\) on \\(eAe\\). So \\(\\omega_n=0\\).\n\n(1)\\(\\Rightarrow\\)(2). Let \\(\\omega\\) be singular and \\(e\\ne0\\) a projection. If \\(\\omega(e)=0\\), take \\(e_0=e\\). Otherwise choose a normal positive \\(\\omega_0\\) with \\(\\omega_0(e)>\\omega(e)\\): a large multiple of a vector state at a unit vector in the range of \\(e\\). Let \\(\\mathcal F\\) be the set of projections \\(p\\le e\\) with \\(\\omega_0(p)\\le\\omega(p)\\), ordered as projections. It contains \\(0\\). If \\((p_i)\\) is a chain in \\(\\mathcal F\\), its least upper bound \\(p\\le e\\) satisfies \\(\\omega(p)\\ge\\sup_i\\omega(p_i)\\ge\\sup_i\\omega_0(p_i)=\\omega_0(p)\\), using positivity of \\(\\omega\\) and normality of \\(\\omega_0\\). So \\(p\\in\\mathcal F\\), and Zorn's lemma gives a maximal \\(p\\in\\mathcal F\\). As \\(e\\notin\\mathcal F\\), \\(e_0=e-p\\ne0\\). For a nonzero projection \\(q\\le e_0\\), maximality excludes \\(p+q\\in\\mathcal F\\), so \\(\\omega(q)<\\omega_0(q)\\); hence \\(\\omega(q)\\le\\omega_0(q)\\) for all projections \\(q\\le e_0\\). A positive \\(y\\in e_0Ae_0\\) is a norm limit of nonnegative combinations of such projections (spectral projections of \\(y\\) for Borel sets away from \\(0\\); background). So \\(e_0\\omega e_0\\le e_0\\omega_0e_0\\) as functionals on \\(A\\), where \\((e_0\\varphi e_0)(y)=\\varphi(e_0ye_0)\\). These inequalities persist for the normal extensions to \\(\\tilde A\\), and multiplying by the central projection \\(1-z_0\\) preserves order. Now \\(e_0\\omega_0e_0\\) is normal, so \\((1-z_0)e_0\\omega_0e_0=0\\), while \\(\\omega=(1-z_0)\\omega\\) gives \\((1-z_0)e_0\\omega e_0=e_0\\omega e_0\\). Hence \\(0\\le e_0\\omega e_0\\le0\\), and \\(\\omega(e_0)=(e_0\\omega e_0)(1)=0\\). \\(\\square\\)\n\n**Corollary 11.3** (Uniqueness of the predual). If \\(F_1,F_2\\) are preduals of a \\(C^*\\)-algebra \\(A\\), then \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\) in \\(A^*\\); equivalently \\(\\sigma(A,F_1)=\\sigma(A,F_2)\\). We write \\(A_*\\) for this space.\n\n**Proof.** By [Theorem 9.2](#oa-fnd-wa-22), \\(F_k\\) gives a faithful representation \\(\\pi_k\\) with \\(\\mathcal M(\\pi_k)=\\pi_k(A)\\) and \\(\\iota_{F_k}(F_k)=V(\\pi_k)=A^*z_k\\), where \\(z_k=z(\\pi_k)\\) ([Proposition 5.2](#oa-fnd-wa-09)). The \\(F_k\\)-singular functionals are \\(A^*(1-z_k)\\). Condition (2) of Theorem 11.2 refers only to the algebra and its projections. So a positive functional is \\(F_1\\)-singular exactly when it is \\(F_2\\)-singular. Each \\(A^*(1-z_k)\\) is spanned by its positive elements (Section 10), so \\(A^*(1-z_1)=A^*(1-z_2)\\). Supports of invariant subspaces are unique (Theorem 4.3(3)), so \\(z_1=z_2\\) and \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\). \\(\\square\\)\n\n**Corollary 11.4** (Isomorphisms are normal). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) between von Neumann algebras is a homeomorphism for the \\(\\sigma\\)-weak topologies, and also for the \\(\\sigma\\)-strong and the \\(\\sigma\\)-strong\\(^*\\) topologies.\n\n**Proof.** Let \\(F=\\{\\psi\\circ\\theta:\\psi\\in N_*\\}\\subseteq M^*\\). Since \\(\\theta\\) is an isometric bijection and \\(N=(N_*)^*\\), the pairing \\(\\langle x,\\psi\\circ\\theta\\rangle=\\psi(\\theta(x))\\) makes \\(F\\) a predual of \\(M\\) with \\(\\iota_F(F)=F\\). By Corollary 11.3, \\(F=M_*\\). So \\(\\psi\\circ\\theta\\in M_*\\) for all \\(\\psi\\in N_*\\): \\(\\theta\\) is normal (background fact on normal maps). The same holds for \\(\\theta^{-1}\\). For \\(\\psi\\in N_*^+\\), \\(\\psi(\\theta(x)^*\\theta(x))=(\\psi\\circ\\theta)(x^*x)\\) with \\(\\psi\\circ\\theta\\in M_*^+\\), and the seminorms \\(x\\mapsto\\omega(x^*x)^{1/2}\\), \\(\\omega\\) positive normal, generate the \\(\\sigma\\)-strong topology (background); adding the same seminorms at \\(x^*\\) gives the \\(\\sigma\\)-strong\\(^*\\) topology. \\(\\square\\)\n\nThe positive-map criterion proved after Corollary 11.5 gives a second proof: a \\(*\\)-isomorphism is an order isomorphism, so it preserves suprema of bounded increasing nets, and a positive map with that property is normal. Either way, the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies of a \\(W^*\\)-algebra do not depend on how it is represented, and neither do its normal and singular functionals. This is why the Hilbert space need not be named.\n\n**Corollary 11.5** (Complete additivity). For \\(\\omega\\in A^*\\) on a \\(W^*\\)-algebra \\(A\\), the following are equivalent:\n\n1. \\(\\omega\\) is normal;\n2. \\(\\omega\\) is completely additive: for every family \\((e_i)_{i\\in I}\\) of mutually orthogonal projections, the finite partial sums of \\(\\sum_i\\omega(e_i)\\) converge to \\(\\omega(\\sum_ie_i)\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). The finite partial sums \\(e_J=\\sum_{i\\in J}e_i\\) increase to \\(\\sum_ie_i\\), strongly and \\(\\sigma\\)-weakly (background fact on monotone nets), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous.\n\n(2)\\(\\Rightarrow\\)(1). Write \\(\\omega=\\omega_n+\\omega_s\\) (Section 10). By (1)\\(\\Rightarrow\\)(2), \\(\\omega_n\\) is completely additive, so \\(\\omega_s=\\omega-\\omega_n\\) is too. It remains to show that a completely additive singular functional \\(\\omega\\) is zero. Write \\(\\omega=\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with singular positive \\(\\omega_k\\) (Section 10), and put \\([\\omega]=\\omega_1+\\omega_2+\\omega_3+\\omega_4\\), a singular positive functional. Let \\(e\\) be a projection. By Zorn's lemma choose a maximal family \\((e_i)\\) of mutually orthogonal nonzero projections under \\(e\\) with \\([\\omega](e_i)=0\\). If \\(e-\\sum_ie_i\\ne0\\), Theorem 11.2 applied to \\([\\omega]\\) gives a nonzero projection under it on which \\([\\omega]\\) vanishes, against maximality. So \\(e=\\sum_ie_i\\). Since \\(0\\le\\omega_k\\le[\\omega]\\), each \\(\\omega_k(e_i)=0\\), so \\(\\omega(e_i)=0\\), and complete additivity gives \\(\\omega(e)=0\\). Thus \\(\\omega\\) vanishes on every projection, and since every self-adjoint element is a norm limit of combinations of projections (background fact on spectral projections), \\(\\omega=0\\). \\(\\square\\)\n\n### Positive maps and increasing suprema\n\nFor a positive functional \\(\\omega\\) on a von Neumann algebra, order normality implies complete additivity: apply it to the finite sums of an orthogonal family of projections. Corollary 11.5 then makes \\(\\omega\\) sigma-weakly continuous. The converse follows from bounded monotone convergence and the definition of the topology.\n\nLet \\(T:M\\to N\\) be a bounded positive map. If it preserves suprema of bounded increasing self-adjoint nets, then each \\(\\psi\\in N_*^+\\) has order-normal pullback \\(\\psi\\circ T\\), hence a sigma-weakly continuous pullback by the preceding paragraph. Positive normal functionals span \\(N_*\\), so the preadjoint test makes \\(T\\) normal. Conversely, if \\(T\\) is normal and \\(x_i\\uparrow x\\) in \\(M\\) is norm bounded, then \\(T(x_i)\\) is increasing and bounded. Its supremum \\(y\\) is its sigma-weak limit by bounded monotone convergence. Normality gives the same limit \\(T(x)\\), so \\(y=T(x)\\). Positive maps are automatically bounded by [Proposition 3.2(4) of completely positive maps](completely-positive-maps.md#oa-fnd-cm-03), so this also applies without an added boundedness hypothesis.\n\nThe earlier use of automatic normality in Theorem 5.3 is a forward reference to Corollary 11.4. Its proof uses predual uniqueness, the singular-functional projection criterion, Sakai's theorem, and the bidual/ideal results, all of which were proved without Theorem 5.3. Lemma 2.1's normal inverse was proved directly by its preadjoint. There is therefore no dependency cycle.\n\n**Remark 11.6** (Abelian subalgebras suffice). If the restriction of \\(\\omega\\in A^*\\) to every abelian \\(W^*\\)-subalgebra \\(B\\) of \\(A\\) (a \\(\\sigma\\)-weakly closed abelian \\(*\\)-subalgebra containing \\(1\\)) is normal, then \\(\\omega\\) is normal. Indeed, given orthogonal projections \\((e_i)\\), let \\(B\\) be the von Neumann algebra generated by them and \\(1\\); it is abelian and contains \\(\\sum_ie_i\\). Its \\(\\sigma\\)-weak topology is the relative one, so normality of \\(\\omega|_B\\) and (1)\\(\\Rightarrow\\)(2) in \\(B\\) give complete additivity on \\((e_i)\\). By (2)\\(\\Rightarrow\\)(1), \\(\\omega\\) is normal.\n\n**Example 11.7** (Which topologies are intrinsic). The \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies are (Corollary 11.4). The weak, strong and strong\\(^*\\) operator topologies are not. Let \\(M=B(H)\\) with \\(\\dim H=\\infty\\), \\(\\pi_1\\) the identity representation and \\(\\pi_2(x)=x\\oplus x\\oplus\\cdots\\) on \\(H\\oplus H\\oplus\\cdots\\). Both are faithful normal representations with von Neumann algebra images. The weak operator topology of \\(\\pi_2(M)\\), pulled back to \\(M\\), is the \\(\\sigma\\)-weak topology, since the vector functionals of \\(\\pi_2\\) are exactly the square-summable series \\(\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\). It is strictly finer than the weak operator topology of \\(\\pi_1\\). Indeed, with an orthonormal sequence \\((e_n)\\), the functional \\(\\varphi(x)=\\sum_nn^{-2}\\langle xe_n,e_n\\rangle\\) is \\(\\sigma\\)-weakly continuous. It is not weakly continuous: a weakly continuous functional has the form \\(\\sum_{k=1}^m\\langle x\\xi_k,\\eta_k\\rangle\\) (background fact on preduals). Choose a unit vector \\(\\zeta\\in\\operatorname{span}\\{e_1,\\dots,e_{m+1}\\}\\) orthogonal to \\(\\xi_1,\\dots,\\xi_m\\), and let \\(x\\) be the projection onto \\(\\mathbb C\\zeta\\). Then the finite sum vanishes at \\(x\\), but \\(\\varphi(x)=\\sum_nn^{-2}|\\langle\\zeta,e_n\\rangle|^2>0\\).\n\n**Example 11.8** (Normal and singular parts on \\(\\ell^\\infty\\)). In \\(\\ell^\\infty\\) every nonzero projection majorises some coordinate projection \\(e_n\\), and the \\(e_n\\) are minimal. By Theorem 11.2, a positive functional is singular exactly when it vanishes on every \\(e_n\\), that is on \\(c_0\\). Hence for any \\(\\varphi\\in(\\ell^\\infty)^*\\), \\(\\varphi_n(x)=\\sum_k\\varphi(e_k)x_k\\) and \\(\\varphi_s=\\varphi-\\varphi_n\\). A state that is a limit along a free ultrafilter is singular, and it has no support in the sense of [Lemma 11.1](#oa-fnd-wa-23).\n\n**Exercise 11.9.** (easy) Let \\(\\mathcal V\\) be a free ultrafilter on \\(\\mathbb N\\), \\(\\chi(x)=\\lim_{n\\to\\mathcal V}x_n\\) on \\(\\ell^\\infty\\), and \\(\\psi=\\tfrac12(\\delta_1+\\chi)\\), where \\(\\delta_1(x)=x_1\\). Find \\(\\psi_n\\), \\(\\psi_s\\), the cyclic representation \\(\\pi_\\psi\\), and its normal and singular parts.\n\n**Solution.** By Example 11.8, \\(\\chi\\) is singular, and \\(\\psi_n=\\tfrac12\\delta_1\\), \\(\\psi_s=\\tfrac12\\chi\\). Both \\(\\delta_1\\) and \\(\\chi\\) are characters, and they are distinct, so the cyclic representation of \\(\\psi\\) is \\(\\pi_\\psi(x)=\\operatorname{diag}(x_1,\\chi(x))\\) on \\(\\mathbb C^2\\), with cyclic vector \\((2^{-1/2},2^{-1/2})\\). Its normal part ([Theorem 10.3](#oa-fnd-wa-10)(2)) is the first coordinate, \\(x\\mapsto x_1\\), and its singular part is the second, \\(x\\mapsto\\chi(x)\\). \\(\\square\\)\n\n**Exercise 11.10** (medium) (When every state is normal). Prove that all states of a von Neumann algebra \\(M\\) are normal exactly when \\(\\dim M<\\infty\\).\n\n**Solution.** If \\(\\dim M<\\infty\\), every linear functional is continuous for the Hausdorff vector topology \\(\\sigma(M,M_*)\\), hence normal. If \\(\\dim M=\\infty\\), Lemma 6.2 gives orthogonal nonzero projections \\(p_1,p_2,\\dots\\). Choose unit vectors \\(\\xi_n\\) in the range of \\(p_n\\) and a free ultrafilter \\(\\mathcal V\\) on \\(\\mathbb N\\), and put \\(\\varphi(x)=\\lim_{n\\to\\mathcal V}\\langle x\\xi_n,\\xi_n\\rangle\\). This is a state. Here \\(\\varphi(p_k)=\\lim_{n\\to\\mathcal V}\\delta_{kn}=0\\) for every \\(k\\), while \\(\\varphi(\\sum_kp_k)=1\\). The finite partial sums of \\(\\sum_kp_k\\) increase to it, so \\(\\varphi\\) does not preserve this supremum and is not normal (background fact on monotone nets). \\(\\square\\)\n\n",
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      "full_conditions_and_proof": "## 11. Singular functionals and the uniqueness of the predual\n\nA normal positive functional has a support projection; a singular one has none. This difference characterizes singular functionals by their values on projections alone. It follows that the predual of a \\(W^*\\)-algebra is unique, and that normality does not depend on the representation.\n\n**Lemma 11.1** (Supports of normal positive functionals). Let \\(A\\) be a \\(W^*\\)-algebra with a predual \\(A_*\\), and \\(\\omega\\) a nonzero normal positive functional. There is a unique nonzero projection \\(e\\in A\\) such that \\(\\omega=e\\omega=\\omega e\\) (so \\(\\omega(x)=\\omega(exe)\\)) and \\(\\omega\\) is faithful on \\(eAe\\). It is the *support* \\(s(\\omega)\\). Also \\(1-s(\\omega)\\) is the largest projection on which \\(\\omega\\) vanishes, so this is the support recalled in the background.\n\n**Proof.** Realise \\(A\\) as a von Neumann algebra with \\(\\sigma\\)-weak topology \\(\\sigma(A,A_*)\\) ([Theorem 9.2](#oa-fnd-wa-22)). Let \\(W\\) be the closed span of \\(\\{\\omega a:a\\in A\\}\\), where \\((\\omega a)(x)=\\omega(ax)\\). It is a norm-closed right invariant subspace of \\(A_*\\), and \\(\\omega=\\omega1\\in W\\). By [Theorem 4.3](#oa-fnd-wa-07)(3), \\(W=eA_*\\) for a projection \\(e\\), and \\(W^\\circ=A(1-e)\\). From \\(\\omega\\in eA_*\\) we get \\(\\omega=e\\omega\\). Since \\(\\omega\\) is positive, \\(\\omega^*=\\omega\\), and \\((e\\omega)^*=\\omega^*e\\), so \\(\\omega=\\omega e\\) as well; hence \\(\\omega=e\\omega e\\).\n\nLet \\(x\\in A\\) with \\(\\omega(x^*x)=0\\). By Cauchy–Schwarz, \\(|(\\omega a)(x)|^2=|\\omega(ax)|^2\\le\\omega(aa^*)\\,\\omega(x^*x)=0\\) for all \\(a\\). So \\(x\\in W^\\circ\\) and \\(xe=0\\). If \\(x\\in eAe\\), then \\(x=xe=0\\): \\(\\omega\\) is faithful on \\(eAe\\). In particular \\(e\\ne0\\), since \\(\\omega\\ne0\\).\n\nUniqueness. Let \\(f\\) be another projection with \\(\\omega=f\\omega f\\), faithful on \\(fAf\\). Then \\(\\omega(1-f)=\\omega(f(1-f)f)=0\\), so \\(x=(1-f)e\\) has \\(\\omega(x^*x)=\\omega(e(1-f)e)=\\omega(1-f)=0\\). Hence \\(xe=0\\), that is \\(e\\le f\\). Also \\(\\omega(f-e)=\\omega(1-e)-\\omega(1-f)=0\\), and \\(f-e\\ge0\\) lies in \\(fAf\\), so \\(f=e\\). Finally, \\(\\omega(1-e)=0\\), and a projection \\(q\\) with \\(\\omega(q)=0\\) has \\(\\omega(q^*q)=0\\), so \\(qe=0\\) and \\(q\\le1-e\\). \\(\\square\\)\n\nA singular state has no support in this sense. If \\(\\omega\\) is a singular state and \\(e\\) is any nonzero projection, Theorem 11.2 below gives a nonzero \\(e_0\\le e\\) with \\(\\omega(e_0)=0\\), so \\(\\omega\\) is not faithful on \\(eAe\\). An example is a state of \\(\\ell^\\infty\\) that vanishes on \\(c_0\\) ([Example 11.8](#oa-fnd-wa-30)).\n\nThroughout the rest of this section, \\(A\\) is a \\(W^*\\)-algebra, realised as a von Neumann algebra by [Theorem 9.2](#oa-fnd-wa-22), with predual \\(A_*\\). Its universal enveloping algebra is \\(\\tilde A\\), and \\(z_0\\in\\tilde A\\) is the central projection with \\(A_*=A^*z_0\\), as in (10.1) of [Section 10](#oa-fnd-wa-10).\n\n**Theorem 11.2** (Singular functionals). For a positive \\(\\omega\\in A^*\\), the following are equivalent:\n\n1. \\(\\omega\\) is singular;\n2. every nonzero projection \\(e\\in A\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\).\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). Split \\(\\omega=\\omega_n+\\omega_s\\) into its normal and singular parts, which are positive (Section 10). If \\(\\omega_n\\ne0\\), let \\(e=s(\\omega_n)\\) ([Lemma 11.1](#oa-fnd-wa-23)). By (2), some nonzero \\(e_0\\le e\\) has \\(\\omega(e_0)=0\\). Then \\(0\\le\\omega_n(e_0)\\le\\omega(e_0)=0\\), which contradicts the faithfulness of \\(\\omega_n\\) on \\(eAe\\). So \\(\\omega_n=0\\).\n\n(1)\\(\\Rightarrow\\)(2). Let \\(\\omega\\) be singular and \\(e\\ne0\\) a projection. If \\(\\omega(e)=0\\), take \\(e_0=e\\). Otherwise choose a normal positive \\(\\omega_0\\) with \\(\\omega_0(e)>\\omega(e)\\): a large multiple of a vector state at a unit vector in the range of \\(e\\). Let \\(\\mathcal F\\) be the set of projections \\(p\\le e\\) with \\(\\omega_0(p)\\le\\omega(p)\\), ordered as projections. It contains \\(0\\). If \\((p_i)\\) is a chain in \\(\\mathcal F\\), its least upper bound \\(p\\le e\\) satisfies \\(\\omega(p)\\ge\\sup_i\\omega(p_i)\\ge\\sup_i\\omega_0(p_i)=\\omega_0(p)\\), using positivity of \\(\\omega\\) and normality of \\(\\omega_0\\). So \\(p\\in\\mathcal F\\), and Zorn's lemma gives a maximal \\(p\\in\\mathcal F\\). As \\(e\\notin\\mathcal F\\), \\(e_0=e-p\\ne0\\). For a nonzero projection \\(q\\le e_0\\), maximality excludes \\(p+q\\in\\mathcal F\\), so \\(\\omega(q)<\\omega_0(q)\\); hence \\(\\omega(q)\\le\\omega_0(q)\\) for all projections \\(q\\le e_0\\). A positive \\(y\\in e_0Ae_0\\) is a norm limit of nonnegative combinations of such projections (spectral projections of \\(y\\) for Borel sets away from \\(0\\); background). So \\(e_0\\omega e_0\\le e_0\\omega_0e_0\\) as functionals on \\(A\\), where \\((e_0\\varphi e_0)(y)=\\varphi(e_0ye_0)\\). These inequalities persist for the normal extensions to \\(\\tilde A\\), and multiplying by the central projection \\(1-z_0\\) preserves order. Now \\(e_0\\omega_0e_0\\) is normal, so \\((1-z_0)e_0\\omega_0e_0=0\\), while \\(\\omega=(1-z_0)\\omega\\) gives \\((1-z_0)e_0\\omega e_0=e_0\\omega e_0\\). Hence \\(0\\le e_0\\omega e_0\\le0\\), and \\(\\omega(e_0)=(e_0\\omega e_0)(1)=0\\). \\(\\square\\)\n\n**Corollary 11.3** (Uniqueness of the predual). If \\(F_1,F_2\\) are preduals of a \\(C^*\\)-algebra \\(A\\), then \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\) in \\(A^*\\); equivalently \\(\\sigma(A,F_1)=\\sigma(A,F_2)\\). We write \\(A_*\\) for this space.\n\n**Proof.** By [Theorem 9.2](#oa-fnd-wa-22), \\(F_k\\) gives a faithful representation \\(\\pi_k\\) with \\(\\mathcal M(\\pi_k)=\\pi_k(A)\\) and \\(\\iota_{F_k}(F_k)=V(\\pi_k)=A^*z_k\\), where \\(z_k=z(\\pi_k)\\) ([Proposition 5.2](#oa-fnd-wa-09)). The \\(F_k\\)-singular functionals are \\(A^*(1-z_k)\\). Condition (2) of Theorem 11.2 refers only to the algebra and its projections. So a positive functional is \\(F_1\\)-singular exactly when it is \\(F_2\\)-singular. Each \\(A^*(1-z_k)\\) is spanned by its positive elements (Section 10), so \\(A^*(1-z_1)=A^*(1-z_2)\\). Supports of invariant subspaces are unique (Theorem 4.3(3)), so \\(z_1=z_2\\) and \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\). \\(\\square\\)\n\n**Corollary 11.4** (Isomorphisms are normal). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) between von Neumann algebras is a homeomorphism for the \\(\\sigma\\)-weak topologies, and also for the \\(\\sigma\\)-strong and the \\(\\sigma\\)-strong\\(^*\\) topologies.\n\n**Proof.** Let \\(F=\\{\\psi\\circ\\theta:\\psi\\in N_*\\}\\subseteq M^*\\). Since \\(\\theta\\) is an isometric bijection and \\(N=(N_*)^*\\), the pairing \\(\\langle x,\\psi\\circ\\theta\\rangle=\\psi(\\theta(x))\\) makes \\(F\\) a predual of \\(M\\) with \\(\\iota_F(F)=F\\). By Corollary 11.3, \\(F=M_*\\). So \\(\\psi\\circ\\theta\\in M_*\\) for all \\(\\psi\\in N_*\\): \\(\\theta\\) is normal (background fact on normal maps). The same holds for \\(\\theta^{-1}\\). For \\(\\psi\\in N_*^+\\), \\(\\psi(\\theta(x)^*\\theta(x))=(\\psi\\circ\\theta)(x^*x)\\) with \\(\\psi\\circ\\theta\\in M_*^+\\), and the seminorms \\(x\\mapsto\\omega(x^*x)^{1/2}\\), \\(\\omega\\) positive normal, generate the \\(\\sigma\\)-strong topology (background); adding the same seminorms at \\(x^*\\) gives the \\(\\sigma\\)-strong\\(^*\\) topology. \\(\\square\\)\n\nThe positive-map criterion proved after Corollary 11.5 gives a second proof: a \\(*\\)-isomorphism is an order isomorphism, so it preserves suprema of bounded increasing nets, and a positive map with that property is normal. Either way, the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies of a \\(W^*\\)-algebra do not depend on how it is represented, and neither do its normal and singular functionals. This is why the Hilbert space need not be named.\n\n**Corollary 11.5** (Complete additivity). For \\(\\omega\\in A^*\\) on a \\(W^*\\)-algebra \\(A\\), the following are equivalent:\n\n1. \\(\\omega\\) is normal;\n2. \\(\\omega\\) is completely additive: for every family \\((e_i)_{i\\in I}\\) of mutually orthogonal projections, the finite partial sums of \\(\\sum_i\\omega(e_i)\\) converge to \\(\\omega(\\sum_ie_i)\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). The finite partial sums \\(e_J=\\sum_{i\\in J}e_i\\) increase to \\(\\sum_ie_i\\), strongly and \\(\\sigma\\)-weakly (background fact on monotone nets), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous.\n\n(2)\\(\\Rightarrow\\)(1). Write \\(\\omega=\\omega_n+\\omega_s\\) (Section 10). By (1)\\(\\Rightarrow\\)(2), \\(\\omega_n\\) is completely additive, so \\(\\omega_s=\\omega-\\omega_n\\) is too. It remains to show that a completely additive singular functional \\(\\omega\\) is zero. Write \\(\\omega=\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with singular positive \\(\\omega_k\\) (Section 10), and put \\([\\omega]=\\omega_1+\\omega_2+\\omega_3+\\omega_4\\), a singular positive functional. Let \\(e\\) be a projection. By Zorn's lemma choose a maximal family \\((e_i)\\) of mutually orthogonal nonzero projections under \\(e\\) with \\([\\omega](e_i)=0\\). If \\(e-\\sum_ie_i\\ne0\\), Theorem 11.2 applied to \\([\\omega]\\) gives a nonzero projection under it on which \\([\\omega]\\) vanishes, against maximality. So \\(e=\\sum_ie_i\\). Since \\(0\\le\\omega_k\\le[\\omega]\\), each \\(\\omega_k(e_i)=0\\), so \\(\\omega(e_i)=0\\), and complete additivity gives \\(\\omega(e)=0\\). Thus \\(\\omega\\) vanishes on every projection, and since every self-adjoint element is a norm limit of combinations of projections (background fact on spectral projections), \\(\\omega=0\\). \\(\\square\\)\n\n### Positive maps and increasing suprema\n\nFor a positive functional \\(\\omega\\) on a von Neumann algebra, order normality implies complete additivity: apply it to the finite sums of an orthogonal family of projections. Corollary 11.5 then makes \\(\\omega\\) sigma-weakly continuous. The converse follows from bounded monotone convergence and the definition of the topology.\n\nLet \\(T:M\\to N\\) be a bounded positive map. If it preserves suprema of bounded increasing self-adjoint nets, then each \\(\\psi\\in N_*^+\\) has order-normal pullback \\(\\psi\\circ T\\), hence a sigma-weakly continuous pullback by the preceding paragraph. Positive normal functionals span \\(N_*\\), so the preadjoint test makes \\(T\\) normal. Conversely, if \\(T\\) is normal and \\(x_i\\uparrow x\\) in \\(M\\) is norm bounded, then \\(T(x_i)\\) is increasing and bounded. Its supremum \\(y\\) is its sigma-weak limit by bounded monotone convergence. Normality gives the same limit \\(T(x)\\), so \\(y=T(x)\\). Positive maps are automatically bounded by [Proposition 3.2(4) of completely positive maps](completely-positive-maps.md#oa-fnd-cm-03), so this also applies without an added boundedness hypothesis.\n\nThe earlier use of automatic normality in Theorem 5.3 is a forward reference to Corollary 11.4. Its proof uses predual uniqueness, the singular-functional projection criterion, Sakai's theorem, and the bidual/ideal results, all of which were proved without Theorem 5.3. Lemma 2.1's normal inverse was proved directly by its preadjoint. There is therefore no dependency cycle.\n\n**Remark 11.6** (Abelian subalgebras suffice). If the restriction of \\(\\omega\\in A^*\\) to every abelian \\(W^*\\)-subalgebra \\(B\\) of \\(A\\) (a \\(\\sigma\\)-weakly closed abelian \\(*\\)-subalgebra containing \\(1\\)) is normal, then \\(\\omega\\) is normal. Indeed, given orthogonal projections \\((e_i)\\), let \\(B\\) be the von Neumann algebra generated by them and \\(1\\); it is abelian and contains \\(\\sum_ie_i\\). Its \\(\\sigma\\)-weak topology is the relative one, so normality of \\(\\omega|_B\\) and (1)\\(\\Rightarrow\\)(2) in \\(B\\) give complete additivity on \\((e_i)\\). By (2)\\(\\Rightarrow\\)(1), \\(\\omega\\) is normal.\n\n**Example 11.7** (Which topologies are intrinsic). The \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies are (Corollary 11.4). The weak, strong and strong\\(^*\\) operator topologies are not. Let \\(M=B(H)\\) with \\(\\dim H=\\infty\\), \\(\\pi_1\\) the identity representation and \\(\\pi_2(x)=x\\oplus x\\oplus\\cdots\\) on \\(H\\oplus H\\oplus\\cdots\\). Both are faithful normal representations with von Neumann algebra images. The weak operator topology of \\(\\pi_2(M)\\), pulled back to \\(M\\), is the \\(\\sigma\\)-weak topology, since the vector functionals of \\(\\pi_2\\) are exactly the square-summable series \\(\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\). It is strictly finer than the weak operator topology of \\(\\pi_1\\). Indeed, with an orthonormal sequence \\((e_n)\\), the functional \\(\\varphi(x)=\\sum_nn^{-2}\\langle xe_n,e_n\\rangle\\) is \\(\\sigma\\)-weakly continuous. It is not weakly continuous: a weakly continuous functional has the form \\(\\sum_{k=1}^m\\langle x\\xi_k,\\eta_k\\rangle\\) (background fact on preduals). Choose a unit vector \\(\\zeta\\in\\operatorname{span}\\{e_1,\\dots,e_{m+1}\\}\\) orthogonal to \\(\\xi_1,\\dots,\\xi_m\\), and let \\(x\\) be the projection onto \\(\\mathbb C\\zeta\\). Then the finite sum vanishes at \\(x\\), but \\(\\varphi(x)=\\sum_nn^{-2}|\\langle\\zeta,e_n\\rangle|^2>0\\).\n\n**Example 11.8** (Normal and singular parts on \\(\\ell^\\infty\\)). In \\(\\ell^\\infty\\) every nonzero projection majorises some coordinate projection \\(e_n\\), and the \\(e_n\\) are minimal. By Theorem 11.2, a positive functional is singular exactly when it vanishes on every \\(e_n\\), that is on \\(c_0\\). Hence for any \\(\\varphi\\in(\\ell^\\infty)^*\\), \\(\\varphi_n(x)=\\sum_k\\varphi(e_k)x_k\\) and \\(\\varphi_s=\\varphi-\\varphi_n\\). A state that is a limit along a free ultrafilter is singular, and it has no support in the sense of [Lemma 11.1](#oa-fnd-wa-23).\n\n**Exercise 11.9.** (easy) Let \\(\\mathcal V\\) be a free ultrafilter on \\(\\mathbb N\\), \\(\\chi(x)=\\lim_{n\\to\\mathcal V}x_n\\) on \\(\\ell^\\infty\\), and \\(\\psi=\\tfrac12(\\delta_1+\\chi)\\), where \\(\\delta_1(x)=x_1\\). Find \\(\\psi_n\\), \\(\\psi_s\\), the cyclic representation \\(\\pi_\\psi\\), and its normal and singular parts.\n\n**Solution.** By Example 11.8, \\(\\chi\\) is singular, and \\(\\psi_n=\\tfrac12\\delta_1\\), \\(\\psi_s=\\tfrac12\\chi\\). Both \\(\\delta_1\\) and \\(\\chi\\) are characters, and they are distinct, so the cyclic representation of \\(\\psi\\) is \\(\\pi_\\psi(x)=\\operatorname{diag}(x_1,\\chi(x))\\) on \\(\\mathbb C^2\\), with cyclic vector \\((2^{-1/2},2^{-1/2})\\). Its normal part ([Theorem 10.3](#oa-fnd-wa-10)(2)) is the first coordinate, \\(x\\mapsto x_1\\), and its singular part is the second, \\(x\\mapsto\\chi(x)\\). \\(\\square\\)\n\n**Exercise 11.10** (medium) (When every state is normal). Prove that all states of a von Neumann algebra \\(M\\) are normal exactly when \\(\\dim M<\\infty\\).\n\n**Solution.** If \\(\\dim M<\\infty\\), every linear functional is continuous for the Hausdorff vector topology \\(\\sigma(M,M_*)\\), hence normal. If \\(\\dim M=\\infty\\), Lemma 6.2 gives orthogonal nonzero projections \\(p_1,p_2,\\dots\\). Choose unit vectors \\(\\xi_n\\) in the range of \\(p_n\\) and a free ultrafilter \\(\\mathcal V\\) on \\(\\mathbb N\\), and put \\(\\varphi(x)=\\lim_{n\\to\\mathcal V}\\langle x\\xi_n,\\xi_n\\rangle\\). This is a state. Here \\(\\varphi(p_k)=\\lim_{n\\to\\mathcal V}\\delta_{kn}=0\\) for every \\(k\\), while \\(\\varphi(\\sum_kp_k)=1\\). The finite partial sums of \\(\\sum_kp_k\\) increase to it, so \\(\\varphi\\) does not preserve this supremum and is not normal (background fact on monotone nets). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "11. Singular functionals and the uniqueness of the predual",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
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      "full_conditions_and_proof": "## 11. Singular functionals and the uniqueness of the predual\n\nA normal positive functional has a support projection; a singular one has none. This difference characterizes singular functionals by their values on projections alone. It follows that the predual of a \\(W^*\\)-algebra is unique, and that normality does not depend on the representation.\n\n**Lemma 11.1** (Supports of normal positive functionals). Let \\(A\\) be a \\(W^*\\)-algebra with a predual \\(A_*\\), and \\(\\omega\\) a nonzero normal positive functional. There is a unique nonzero projection \\(e\\in A\\) such that \\(\\omega=e\\omega=\\omega e\\) (so \\(\\omega(x)=\\omega(exe)\\)) and \\(\\omega\\) is faithful on \\(eAe\\). It is the *support* \\(s(\\omega)\\). Also \\(1-s(\\omega)\\) is the largest projection on which \\(\\omega\\) vanishes, so this is the support recalled in the background.\n\n**Proof.** Realise \\(A\\) as a von Neumann algebra with \\(\\sigma\\)-weak topology \\(\\sigma(A,A_*)\\) ([Theorem 9.2](#oa-fnd-wa-22)). Let \\(W\\) be the closed span of \\(\\{\\omega a:a\\in A\\}\\), where \\((\\omega a)(x)=\\omega(ax)\\). It is a norm-closed right invariant subspace of \\(A_*\\), and \\(\\omega=\\omega1\\in W\\). By [Theorem 4.3](#oa-fnd-wa-07)(3), \\(W=eA_*\\) for a projection \\(e\\), and \\(W^\\circ=A(1-e)\\). From \\(\\omega\\in eA_*\\) we get \\(\\omega=e\\omega\\). Since \\(\\omega\\) is positive, \\(\\omega^*=\\omega\\), and \\((e\\omega)^*=\\omega^*e\\), so \\(\\omega=\\omega e\\) as well; hence \\(\\omega=e\\omega e\\).\n\nLet \\(x\\in A\\) with \\(\\omega(x^*x)=0\\). By Cauchy–Schwarz, \\(|(\\omega a)(x)|^2=|\\omega(ax)|^2\\le\\omega(aa^*)\\,\\omega(x^*x)=0\\) for all \\(a\\). So \\(x\\in W^\\circ\\) and \\(xe=0\\). If \\(x\\in eAe\\), then \\(x=xe=0\\): \\(\\omega\\) is faithful on \\(eAe\\). In particular \\(e\\ne0\\), since \\(\\omega\\ne0\\).\n\nUniqueness. Let \\(f\\) be another projection with \\(\\omega=f\\omega f\\), faithful on \\(fAf\\). Then \\(\\omega(1-f)=\\omega(f(1-f)f)=0\\), so \\(x=(1-f)e\\) has \\(\\omega(x^*x)=\\omega(e(1-f)e)=\\omega(1-f)=0\\). Hence \\(xe=0\\), that is \\(e\\le f\\). Also \\(\\omega(f-e)=\\omega(1-e)-\\omega(1-f)=0\\), and \\(f-e\\ge0\\) lies in \\(fAf\\), so \\(f=e\\). Finally, \\(\\omega(1-e)=0\\), and a projection \\(q\\) with \\(\\omega(q)=0\\) has \\(\\omega(q^*q)=0\\), so \\(qe=0\\) and \\(q\\le1-e\\). \\(\\square\\)\n\nA singular state has no support in this sense. If \\(\\omega\\) is a singular state and \\(e\\) is any nonzero projection, Theorem 11.2 below gives a nonzero \\(e_0\\le e\\) with \\(\\omega(e_0)=0\\), so \\(\\omega\\) is not faithful on \\(eAe\\). An example is a state of \\(\\ell^\\infty\\) that vanishes on \\(c_0\\) ([Example 11.8](#oa-fnd-wa-30)).\n\nThroughout the rest of this section, \\(A\\) is a \\(W^*\\)-algebra, realised as a von Neumann algebra by [Theorem 9.2](#oa-fnd-wa-22), with predual \\(A_*\\). Its universal enveloping algebra is \\(\\tilde A\\), and \\(z_0\\in\\tilde A\\) is the central projection with \\(A_*=A^*z_0\\), as in (10.1) of [Section 10](#oa-fnd-wa-10).\n\n**Theorem 11.2** (Singular functionals). For a positive \\(\\omega\\in A^*\\), the following are equivalent:\n\n1. \\(\\omega\\) is singular;\n2. every nonzero projection \\(e\\in A\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\).\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). Split \\(\\omega=\\omega_n+\\omega_s\\) into its normal and singular parts, which are positive (Section 10). If \\(\\omega_n\\ne0\\), let \\(e=s(\\omega_n)\\) ([Lemma 11.1](#oa-fnd-wa-23)). By (2), some nonzero \\(e_0\\le e\\) has \\(\\omega(e_0)=0\\). Then \\(0\\le\\omega_n(e_0)\\le\\omega(e_0)=0\\), which contradicts the faithfulness of \\(\\omega_n\\) on \\(eAe\\). So \\(\\omega_n=0\\).\n\n(1)\\(\\Rightarrow\\)(2). Let \\(\\omega\\) be singular and \\(e\\ne0\\) a projection. If \\(\\omega(e)=0\\), take \\(e_0=e\\). Otherwise choose a normal positive \\(\\omega_0\\) with \\(\\omega_0(e)>\\omega(e)\\): a large multiple of a vector state at a unit vector in the range of \\(e\\). Let \\(\\mathcal F\\) be the set of projections \\(p\\le e\\) with \\(\\omega_0(p)\\le\\omega(p)\\), ordered as projections. It contains \\(0\\). If \\((p_i)\\) is a chain in \\(\\mathcal F\\), its least upper bound \\(p\\le e\\) satisfies \\(\\omega(p)\\ge\\sup_i\\omega(p_i)\\ge\\sup_i\\omega_0(p_i)=\\omega_0(p)\\), using positivity of \\(\\omega\\) and normality of \\(\\omega_0\\). So \\(p\\in\\mathcal F\\), and Zorn's lemma gives a maximal \\(p\\in\\mathcal F\\). As \\(e\\notin\\mathcal F\\), \\(e_0=e-p\\ne0\\). For a nonzero projection \\(q\\le e_0\\), maximality excludes \\(p+q\\in\\mathcal F\\), so \\(\\omega(q)<\\omega_0(q)\\); hence \\(\\omega(q)\\le\\omega_0(q)\\) for all projections \\(q\\le e_0\\). A positive \\(y\\in e_0Ae_0\\) is a norm limit of nonnegative combinations of such projections (spectral projections of \\(y\\) for Borel sets away from \\(0\\); background). So \\(e_0\\omega e_0\\le e_0\\omega_0e_0\\) as functionals on \\(A\\), where \\((e_0\\varphi e_0)(y)=\\varphi(e_0ye_0)\\). These inequalities persist for the normal extensions to \\(\\tilde A\\), and multiplying by the central projection \\(1-z_0\\) preserves order. Now \\(e_0\\omega_0e_0\\) is normal, so \\((1-z_0)e_0\\omega_0e_0=0\\), while \\(\\omega=(1-z_0)\\omega\\) gives \\((1-z_0)e_0\\omega e_0=e_0\\omega e_0\\). Hence \\(0\\le e_0\\omega e_0\\le0\\), and \\(\\omega(e_0)=(e_0\\omega e_0)(1)=0\\). \\(\\square\\)\n\n**Corollary 11.3** (Uniqueness of the predual). If \\(F_1,F_2\\) are preduals of a \\(C^*\\)-algebra \\(A\\), then \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\) in \\(A^*\\); equivalently \\(\\sigma(A,F_1)=\\sigma(A,F_2)\\). We write \\(A_*\\) for this space.\n\n**Proof.** By [Theorem 9.2](#oa-fnd-wa-22), \\(F_k\\) gives a faithful representation \\(\\pi_k\\) with \\(\\mathcal M(\\pi_k)=\\pi_k(A)\\) and \\(\\iota_{F_k}(F_k)=V(\\pi_k)=A^*z_k\\), where \\(z_k=z(\\pi_k)\\) ([Proposition 5.2](#oa-fnd-wa-09)). The \\(F_k\\)-singular functionals are \\(A^*(1-z_k)\\). Condition (2) of Theorem 11.2 refers only to the algebra and its projections. So a positive functional is \\(F_1\\)-singular exactly when it is \\(F_2\\)-singular. Each \\(A^*(1-z_k)\\) is spanned by its positive elements (Section 10), so \\(A^*(1-z_1)=A^*(1-z_2)\\). Supports of invariant subspaces are unique (Theorem 4.3(3)), so \\(z_1=z_2\\) and \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\). \\(\\square\\)\n\n**Corollary 11.4** (Isomorphisms are normal). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) between von Neumann algebras is a homeomorphism for the \\(\\sigma\\)-weak topologies, and also for the \\(\\sigma\\)-strong and the \\(\\sigma\\)-strong\\(^*\\) topologies.\n\n**Proof.** Let \\(F=\\{\\psi\\circ\\theta:\\psi\\in N_*\\}\\subseteq M^*\\). Since \\(\\theta\\) is an isometric bijection and \\(N=(N_*)^*\\), the pairing \\(\\langle x,\\psi\\circ\\theta\\rangle=\\psi(\\theta(x))\\) makes \\(F\\) a predual of \\(M\\) with \\(\\iota_F(F)=F\\). By Corollary 11.3, \\(F=M_*\\). So \\(\\psi\\circ\\theta\\in M_*\\) for all \\(\\psi\\in N_*\\): \\(\\theta\\) is normal (background fact on normal maps). The same holds for \\(\\theta^{-1}\\). For \\(\\psi\\in N_*^+\\), \\(\\psi(\\theta(x)^*\\theta(x))=(\\psi\\circ\\theta)(x^*x)\\) with \\(\\psi\\circ\\theta\\in M_*^+\\), and the seminorms \\(x\\mapsto\\omega(x^*x)^{1/2}\\), \\(\\omega\\) positive normal, generate the \\(\\sigma\\)-strong topology (background); adding the same seminorms at \\(x^*\\) gives the \\(\\sigma\\)-strong\\(^*\\) topology. \\(\\square\\)\n\nThe positive-map criterion proved after Corollary 11.5 gives a second proof: a \\(*\\)-isomorphism is an order isomorphism, so it preserves suprema of bounded increasing nets, and a positive map with that property is normal. Either way, the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies of a \\(W^*\\)-algebra do not depend on how it is represented, and neither do its normal and singular functionals. This is why the Hilbert space need not be named.\n\n**Corollary 11.5** (Complete additivity). For \\(\\omega\\in A^*\\) on a \\(W^*\\)-algebra \\(A\\), the following are equivalent:\n\n1. \\(\\omega\\) is normal;\n2. \\(\\omega\\) is completely additive: for every family \\((e_i)_{i\\in I}\\) of mutually orthogonal projections, the finite partial sums of \\(\\sum_i\\omega(e_i)\\) converge to \\(\\omega(\\sum_ie_i)\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). The finite partial sums \\(e_J=\\sum_{i\\in J}e_i\\) increase to \\(\\sum_ie_i\\), strongly and \\(\\sigma\\)-weakly (background fact on monotone nets), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous.\n\n(2)\\(\\Rightarrow\\)(1). Write \\(\\omega=\\omega_n+\\omega_s\\) (Section 10). By (1)\\(\\Rightarrow\\)(2), \\(\\omega_n\\) is completely additive, so \\(\\omega_s=\\omega-\\omega_n\\) is too. It remains to show that a completely additive singular functional \\(\\omega\\) is zero. Write \\(\\omega=\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with singular positive \\(\\omega_k\\) (Section 10), and put \\([\\omega]=\\omega_1+\\omega_2+\\omega_3+\\omega_4\\), a singular positive functional. Let \\(e\\) be a projection. By Zorn's lemma choose a maximal family \\((e_i)\\) of mutually orthogonal nonzero projections under \\(e\\) with \\([\\omega](e_i)=0\\). If \\(e-\\sum_ie_i\\ne0\\), Theorem 11.2 applied to \\([\\omega]\\) gives a nonzero projection under it on which \\([\\omega]\\) vanishes, against maximality. So \\(e=\\sum_ie_i\\). Since \\(0\\le\\omega_k\\le[\\omega]\\), each \\(\\omega_k(e_i)=0\\), so \\(\\omega(e_i)=0\\), and complete additivity gives \\(\\omega(e)=0\\). Thus \\(\\omega\\) vanishes on every projection, and since every self-adjoint element is a norm limit of combinations of projections (background fact on spectral projections), \\(\\omega=0\\). \\(\\square\\)\n\n### Positive maps and increasing suprema\n\nFor a positive functional \\(\\omega\\) on a von Neumann algebra, order normality implies complete additivity: apply it to the finite sums of an orthogonal family of projections. Corollary 11.5 then makes \\(\\omega\\) sigma-weakly continuous. The converse follows from bounded monotone convergence and the definition of the topology.\n\nLet \\(T:M\\to N\\) be a bounded positive map. If it preserves suprema of bounded increasing self-adjoint nets, then each \\(\\psi\\in N_*^+\\) has order-normal pullback \\(\\psi\\circ T\\), hence a sigma-weakly continuous pullback by the preceding paragraph. Positive normal functionals span \\(N_*\\), so the preadjoint test makes \\(T\\) normal. Conversely, if \\(T\\) is normal and \\(x_i\\uparrow x\\) in \\(M\\) is norm bounded, then \\(T(x_i)\\) is increasing and bounded. Its supremum \\(y\\) is its sigma-weak limit by bounded monotone convergence. Normality gives the same limit \\(T(x)\\), so \\(y=T(x)\\). Positive maps are automatically bounded by [Proposition 3.2(4) of completely positive maps](completely-positive-maps.md#oa-fnd-cm-03), so this also applies without an added boundedness hypothesis.\n\nThe earlier use of automatic normality in Theorem 5.3 is a forward reference to Corollary 11.4. Its proof uses predual uniqueness, the singular-functional projection criterion, Sakai's theorem, and the bidual/ideal results, all of which were proved without Theorem 5.3. Lemma 2.1's normal inverse was proved directly by its preadjoint. There is therefore no dependency cycle.\n\n**Remark 11.6** (Abelian subalgebras suffice). If the restriction of \\(\\omega\\in A^*\\) to every abelian \\(W^*\\)-subalgebra \\(B\\) of \\(A\\) (a \\(\\sigma\\)-weakly closed abelian \\(*\\)-subalgebra containing \\(1\\)) is normal, then \\(\\omega\\) is normal. Indeed, given orthogonal projections \\((e_i)\\), let \\(B\\) be the von Neumann algebra generated by them and \\(1\\); it is abelian and contains \\(\\sum_ie_i\\). Its \\(\\sigma\\)-weak topology is the relative one, so normality of \\(\\omega|_B\\) and (1)\\(\\Rightarrow\\)(2) in \\(B\\) give complete additivity on \\((e_i)\\). By (2)\\(\\Rightarrow\\)(1), \\(\\omega\\) is normal.\n\n**Example 11.7** (Which topologies are intrinsic). The \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies are (Corollary 11.4). The weak, strong and strong\\(^*\\) operator topologies are not. Let \\(M=B(H)\\) with \\(\\dim H=\\infty\\), \\(\\pi_1\\) the identity representation and \\(\\pi_2(x)=x\\oplus x\\oplus\\cdots\\) on \\(H\\oplus H\\oplus\\cdots\\). Both are faithful normal representations with von Neumann algebra images. The weak operator topology of \\(\\pi_2(M)\\), pulled back to \\(M\\), is the \\(\\sigma\\)-weak topology, since the vector functionals of \\(\\pi_2\\) are exactly the square-summable series \\(\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\). It is strictly finer than the weak operator topology of \\(\\pi_1\\). Indeed, with an orthonormal sequence \\((e_n)\\), the functional \\(\\varphi(x)=\\sum_nn^{-2}\\langle xe_n,e_n\\rangle\\) is \\(\\sigma\\)-weakly continuous. It is not weakly continuous: a weakly continuous functional has the form \\(\\sum_{k=1}^m\\langle x\\xi_k,\\eta_k\\rangle\\) (background fact on preduals). Choose a unit vector \\(\\zeta\\in\\operatorname{span}\\{e_1,\\dots,e_{m+1}\\}\\) orthogonal to \\(\\xi_1,\\dots,\\xi_m\\), and let \\(x\\) be the projection onto \\(\\mathbb C\\zeta\\). Then the finite sum vanishes at \\(x\\), but \\(\\varphi(x)=\\sum_nn^{-2}|\\langle\\zeta,e_n\\rangle|^2>0\\).\n\n**Example 11.8** (Normal and singular parts on \\(\\ell^\\infty\\)). In \\(\\ell^\\infty\\) every nonzero projection majorises some coordinate projection \\(e_n\\), and the \\(e_n\\) are minimal. By Theorem 11.2, a positive functional is singular exactly when it vanishes on every \\(e_n\\), that is on \\(c_0\\). Hence for any \\(\\varphi\\in(\\ell^\\infty)^*\\), \\(\\varphi_n(x)=\\sum_k\\varphi(e_k)x_k\\) and \\(\\varphi_s=\\varphi-\\varphi_n\\). A state that is a limit along a free ultrafilter is singular, and it has no support in the sense of [Lemma 11.1](#oa-fnd-wa-23).\n\n**Exercise 11.9.** (easy) Let \\(\\mathcal V\\) be a free ultrafilter on \\(\\mathbb N\\), \\(\\chi(x)=\\lim_{n\\to\\mathcal V}x_n\\) on \\(\\ell^\\infty\\), and \\(\\psi=\\tfrac12(\\delta_1+\\chi)\\), where \\(\\delta_1(x)=x_1\\). Find \\(\\psi_n\\), \\(\\psi_s\\), the cyclic representation \\(\\pi_\\psi\\), and its normal and singular parts.\n\n**Solution.** By Example 11.8, \\(\\chi\\) is singular, and \\(\\psi_n=\\tfrac12\\delta_1\\), \\(\\psi_s=\\tfrac12\\chi\\). Both \\(\\delta_1\\) and \\(\\chi\\) are characters, and they are distinct, so the cyclic representation of \\(\\psi\\) is \\(\\pi_\\psi(x)=\\operatorname{diag}(x_1,\\chi(x))\\) on \\(\\mathbb C^2\\), with cyclic vector \\((2^{-1/2},2^{-1/2})\\). Its normal part ([Theorem 10.3](#oa-fnd-wa-10)(2)) is the first coordinate, \\(x\\mapsto x_1\\), and its singular part is the second, \\(x\\mapsto\\chi(x)\\). \\(\\square\\)\n\n**Exercise 11.10** (medium) (When every state is normal). Prove that all states of a von Neumann algebra \\(M\\) are normal exactly when \\(\\dim M<\\infty\\).\n\n**Solution.** If \\(\\dim M<\\infty\\), every linear functional is continuous for the Hausdorff vector topology \\(\\sigma(M,M_*)\\), hence normal. If \\(\\dim M=\\infty\\), Lemma 6.2 gives orthogonal nonzero projections \\(p_1,p_2,\\dots\\). Choose unit vectors \\(\\xi_n\\) in the range of \\(p_n\\) and a free ultrafilter \\(\\mathcal V\\) on \\(\\mathbb N\\), and put \\(\\varphi(x)=\\lim_{n\\to\\mathcal V}\\langle x\\xi_n,\\xi_n\\rangle\\). This is a state. Here \\(\\varphi(p_k)=\\lim_{n\\to\\mathcal V}\\delta_{kn}=0\\) for every \\(k\\), while \\(\\varphi(\\sum_kp_k)=1\\). The finite partial sums of \\(\\sum_kp_k\\) increase to it, so \\(\\varphi\\) does not preserve this supremum and is not normal (background fact on monotone nets). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-30",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "11. Singular functionals and the uniqueness of the predual",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
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      "full_conditions_and_proof": "## 11. Singular functionals and the uniqueness of the predual\n\nA normal positive functional has a support projection; a singular one has none. This difference characterizes singular functionals by their values on projections alone. It follows that the predual of a \\(W^*\\)-algebra is unique, and that normality does not depend on the representation.\n\n**Lemma 11.1** (Supports of normal positive functionals). Let \\(A\\) be a \\(W^*\\)-algebra with a predual \\(A_*\\), and \\(\\omega\\) a nonzero normal positive functional. There is a unique nonzero projection \\(e\\in A\\) such that \\(\\omega=e\\omega=\\omega e\\) (so \\(\\omega(x)=\\omega(exe)\\)) and \\(\\omega\\) is faithful on \\(eAe\\). It is the *support* \\(s(\\omega)\\). Also \\(1-s(\\omega)\\) is the largest projection on which \\(\\omega\\) vanishes, so this is the support recalled in the background.\n\n**Proof.** Realise \\(A\\) as a von Neumann algebra with \\(\\sigma\\)-weak topology \\(\\sigma(A,A_*)\\) ([Theorem 9.2](#oa-fnd-wa-22)). Let \\(W\\) be the closed span of \\(\\{\\omega a:a\\in A\\}\\), where \\((\\omega a)(x)=\\omega(ax)\\). It is a norm-closed right invariant subspace of \\(A_*\\), and \\(\\omega=\\omega1\\in W\\). By [Theorem 4.3](#oa-fnd-wa-07)(3), \\(W=eA_*\\) for a projection \\(e\\), and \\(W^\\circ=A(1-e)\\). From \\(\\omega\\in eA_*\\) we get \\(\\omega=e\\omega\\). Since \\(\\omega\\) is positive, \\(\\omega^*=\\omega\\), and \\((e\\omega)^*=\\omega^*e\\), so \\(\\omega=\\omega e\\) as well; hence \\(\\omega=e\\omega e\\).\n\nLet \\(x\\in A\\) with \\(\\omega(x^*x)=0\\). By Cauchy–Schwarz, \\(|(\\omega a)(x)|^2=|\\omega(ax)|^2\\le\\omega(aa^*)\\,\\omega(x^*x)=0\\) for all \\(a\\). So \\(x\\in W^\\circ\\) and \\(xe=0\\). If \\(x\\in eAe\\), then \\(x=xe=0\\): \\(\\omega\\) is faithful on \\(eAe\\). In particular \\(e\\ne0\\), since \\(\\omega\\ne0\\).\n\nUniqueness. Let \\(f\\) be another projection with \\(\\omega=f\\omega f\\), faithful on \\(fAf\\). Then \\(\\omega(1-f)=\\omega(f(1-f)f)=0\\), so \\(x=(1-f)e\\) has \\(\\omega(x^*x)=\\omega(e(1-f)e)=\\omega(1-f)=0\\). Hence \\(xe=0\\), that is \\(e\\le f\\). Also \\(\\omega(f-e)=\\omega(1-e)-\\omega(1-f)=0\\), and \\(f-e\\ge0\\) lies in \\(fAf\\), so \\(f=e\\). Finally, \\(\\omega(1-e)=0\\), and a projection \\(q\\) with \\(\\omega(q)=0\\) has \\(\\omega(q^*q)=0\\), so \\(qe=0\\) and \\(q\\le1-e\\). \\(\\square\\)\n\nA singular state has no support in this sense. If \\(\\omega\\) is a singular state and \\(e\\) is any nonzero projection, Theorem 11.2 below gives a nonzero \\(e_0\\le e\\) with \\(\\omega(e_0)=0\\), so \\(\\omega\\) is not faithful on \\(eAe\\). An example is a state of \\(\\ell^\\infty\\) that vanishes on \\(c_0\\) ([Example 11.8](#oa-fnd-wa-30)).\n\nThroughout the rest of this section, \\(A\\) is a \\(W^*\\)-algebra, realised as a von Neumann algebra by [Theorem 9.2](#oa-fnd-wa-22), with predual \\(A_*\\). Its universal enveloping algebra is \\(\\tilde A\\), and \\(z_0\\in\\tilde A\\) is the central projection with \\(A_*=A^*z_0\\), as in (10.1) of [Section 10](#oa-fnd-wa-10).\n\n**Theorem 11.2** (Singular functionals). For a positive \\(\\omega\\in A^*\\), the following are equivalent:\n\n1. \\(\\omega\\) is singular;\n2. every nonzero projection \\(e\\in A\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\).\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). Split \\(\\omega=\\omega_n+\\omega_s\\) into its normal and singular parts, which are positive (Section 10). If \\(\\omega_n\\ne0\\), let \\(e=s(\\omega_n)\\) ([Lemma 11.1](#oa-fnd-wa-23)). By (2), some nonzero \\(e_0\\le e\\) has \\(\\omega(e_0)=0\\). Then \\(0\\le\\omega_n(e_0)\\le\\omega(e_0)=0\\), which contradicts the faithfulness of \\(\\omega_n\\) on \\(eAe\\). So \\(\\omega_n=0\\).\n\n(1)\\(\\Rightarrow\\)(2). Let \\(\\omega\\) be singular and \\(e\\ne0\\) a projection. If \\(\\omega(e)=0\\), take \\(e_0=e\\). Otherwise choose a normal positive \\(\\omega_0\\) with \\(\\omega_0(e)>\\omega(e)\\): a large multiple of a vector state at a unit vector in the range of \\(e\\). Let \\(\\mathcal F\\) be the set of projections \\(p\\le e\\) with \\(\\omega_0(p)\\le\\omega(p)\\), ordered as projections. It contains \\(0\\). If \\((p_i)\\) is a chain in \\(\\mathcal F\\), its least upper bound \\(p\\le e\\) satisfies \\(\\omega(p)\\ge\\sup_i\\omega(p_i)\\ge\\sup_i\\omega_0(p_i)=\\omega_0(p)\\), using positivity of \\(\\omega\\) and normality of \\(\\omega_0\\). So \\(p\\in\\mathcal F\\), and Zorn's lemma gives a maximal \\(p\\in\\mathcal F\\). As \\(e\\notin\\mathcal F\\), \\(e_0=e-p\\ne0\\). For a nonzero projection \\(q\\le e_0\\), maximality excludes \\(p+q\\in\\mathcal F\\), so \\(\\omega(q)<\\omega_0(q)\\); hence \\(\\omega(q)\\le\\omega_0(q)\\) for all projections \\(q\\le e_0\\). A positive \\(y\\in e_0Ae_0\\) is a norm limit of nonnegative combinations of such projections (spectral projections of \\(y\\) for Borel sets away from \\(0\\); background). So \\(e_0\\omega e_0\\le e_0\\omega_0e_0\\) as functionals on \\(A\\), where \\((e_0\\varphi e_0)(y)=\\varphi(e_0ye_0)\\). These inequalities persist for the normal extensions to \\(\\tilde A\\), and multiplying by the central projection \\(1-z_0\\) preserves order. Now \\(e_0\\omega_0e_0\\) is normal, so \\((1-z_0)e_0\\omega_0e_0=0\\), while \\(\\omega=(1-z_0)\\omega\\) gives \\((1-z_0)e_0\\omega e_0=e_0\\omega e_0\\). Hence \\(0\\le e_0\\omega e_0\\le0\\), and \\(\\omega(e_0)=(e_0\\omega e_0)(1)=0\\). \\(\\square\\)\n\n**Corollary 11.3** (Uniqueness of the predual). If \\(F_1,F_2\\) are preduals of a \\(C^*\\)-algebra \\(A\\), then \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\) in \\(A^*\\); equivalently \\(\\sigma(A,F_1)=\\sigma(A,F_2)\\). We write \\(A_*\\) for this space.\n\n**Proof.** By [Theorem 9.2](#oa-fnd-wa-22), \\(F_k\\) gives a faithful representation \\(\\pi_k\\) with \\(\\mathcal M(\\pi_k)=\\pi_k(A)\\) and \\(\\iota_{F_k}(F_k)=V(\\pi_k)=A^*z_k\\), where \\(z_k=z(\\pi_k)\\) ([Proposition 5.2](#oa-fnd-wa-09)). The \\(F_k\\)-singular functionals are \\(A^*(1-z_k)\\). Condition (2) of Theorem 11.2 refers only to the algebra and its projections. So a positive functional is \\(F_1\\)-singular exactly when it is \\(F_2\\)-singular. Each \\(A^*(1-z_k)\\) is spanned by its positive elements (Section 10), so \\(A^*(1-z_1)=A^*(1-z_2)\\). Supports of invariant subspaces are unique (Theorem 4.3(3)), so \\(z_1=z_2\\) and \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\). \\(\\square\\)\n\n**Corollary 11.4** (Isomorphisms are normal). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) between von Neumann algebras is a homeomorphism for the \\(\\sigma\\)-weak topologies, and also for the \\(\\sigma\\)-strong and the \\(\\sigma\\)-strong\\(^*\\) topologies.\n\n**Proof.** Let \\(F=\\{\\psi\\circ\\theta:\\psi\\in N_*\\}\\subseteq M^*\\). Since \\(\\theta\\) is an isometric bijection and \\(N=(N_*)^*\\), the pairing \\(\\langle x,\\psi\\circ\\theta\\rangle=\\psi(\\theta(x))\\) makes \\(F\\) a predual of \\(M\\) with \\(\\iota_F(F)=F\\). By Corollary 11.3, \\(F=M_*\\). So \\(\\psi\\circ\\theta\\in M_*\\) for all \\(\\psi\\in N_*\\): \\(\\theta\\) is normal (background fact on normal maps). The same holds for \\(\\theta^{-1}\\). For \\(\\psi\\in N_*^+\\), \\(\\psi(\\theta(x)^*\\theta(x))=(\\psi\\circ\\theta)(x^*x)\\) with \\(\\psi\\circ\\theta\\in M_*^+\\), and the seminorms \\(x\\mapsto\\omega(x^*x)^{1/2}\\), \\(\\omega\\) positive normal, generate the \\(\\sigma\\)-strong topology (background); adding the same seminorms at \\(x^*\\) gives the \\(\\sigma\\)-strong\\(^*\\) topology. \\(\\square\\)\n\nThe positive-map criterion proved after Corollary 11.5 gives a second proof: a \\(*\\)-isomorphism is an order isomorphism, so it preserves suprema of bounded increasing nets, and a positive map with that property is normal. Either way, the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies of a \\(W^*\\)-algebra do not depend on how it is represented, and neither do its normal and singular functionals. This is why the Hilbert space need not be named.\n\n**Corollary 11.5** (Complete additivity). For \\(\\omega\\in A^*\\) on a \\(W^*\\)-algebra \\(A\\), the following are equivalent:\n\n1. \\(\\omega\\) is normal;\n2. \\(\\omega\\) is completely additive: for every family \\((e_i)_{i\\in I}\\) of mutually orthogonal projections, the finite partial sums of \\(\\sum_i\\omega(e_i)\\) converge to \\(\\omega(\\sum_ie_i)\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). The finite partial sums \\(e_J=\\sum_{i\\in J}e_i\\) increase to \\(\\sum_ie_i\\), strongly and \\(\\sigma\\)-weakly (background fact on monotone nets), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous.\n\n(2)\\(\\Rightarrow\\)(1). Write \\(\\omega=\\omega_n+\\omega_s\\) (Section 10). By (1)\\(\\Rightarrow\\)(2), \\(\\omega_n\\) is completely additive, so \\(\\omega_s=\\omega-\\omega_n\\) is too. It remains to show that a completely additive singular functional \\(\\omega\\) is zero. Write \\(\\omega=\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with singular positive \\(\\omega_k\\) (Section 10), and put \\([\\omega]=\\omega_1+\\omega_2+\\omega_3+\\omega_4\\), a singular positive functional. Let \\(e\\) be a projection. By Zorn's lemma choose a maximal family \\((e_i)\\) of mutually orthogonal nonzero projections under \\(e\\) with \\([\\omega](e_i)=0\\). If \\(e-\\sum_ie_i\\ne0\\), Theorem 11.2 applied to \\([\\omega]\\) gives a nonzero projection under it on which \\([\\omega]\\) vanishes, against maximality. So \\(e=\\sum_ie_i\\). Since \\(0\\le\\omega_k\\le[\\omega]\\), each \\(\\omega_k(e_i)=0\\), so \\(\\omega(e_i)=0\\), and complete additivity gives \\(\\omega(e)=0\\). Thus \\(\\omega\\) vanishes on every projection, and since every self-adjoint element is a norm limit of combinations of projections (background fact on spectral projections), \\(\\omega=0\\). \\(\\square\\)\n\n### Positive maps and increasing suprema\n\nFor a positive functional \\(\\omega\\) on a von Neumann algebra, order normality implies complete additivity: apply it to the finite sums of an orthogonal family of projections. Corollary 11.5 then makes \\(\\omega\\) sigma-weakly continuous. The converse follows from bounded monotone convergence and the definition of the topology.\n\nLet \\(T:M\\to N\\) be a bounded positive map. If it preserves suprema of bounded increasing self-adjoint nets, then each \\(\\psi\\in N_*^+\\) has order-normal pullback \\(\\psi\\circ T\\), hence a sigma-weakly continuous pullback by the preceding paragraph. Positive normal functionals span \\(N_*\\), so the preadjoint test makes \\(T\\) normal. Conversely, if \\(T\\) is normal and \\(x_i\\uparrow x\\) in \\(M\\) is norm bounded, then \\(T(x_i)\\) is increasing and bounded. Its supremum \\(y\\) is its sigma-weak limit by bounded monotone convergence. Normality gives the same limit \\(T(x)\\), so \\(y=T(x)\\). Positive maps are automatically bounded by [Proposition 3.2(4) of completely positive maps](completely-positive-maps.md#oa-fnd-cm-03), so this also applies without an added boundedness hypothesis.\n\nThe earlier use of automatic normality in Theorem 5.3 is a forward reference to Corollary 11.4. Its proof uses predual uniqueness, the singular-functional projection criterion, Sakai's theorem, and the bidual/ideal results, all of which were proved without Theorem 5.3. Lemma 2.1's normal inverse was proved directly by its preadjoint. There is therefore no dependency cycle.\n\n**Remark 11.6** (Abelian subalgebras suffice). If the restriction of \\(\\omega\\in A^*\\) to every abelian \\(W^*\\)-subalgebra \\(B\\) of \\(A\\) (a \\(\\sigma\\)-weakly closed abelian \\(*\\)-subalgebra containing \\(1\\)) is normal, then \\(\\omega\\) is normal. Indeed, given orthogonal projections \\((e_i)\\), let \\(B\\) be the von Neumann algebra generated by them and \\(1\\); it is abelian and contains \\(\\sum_ie_i\\). Its \\(\\sigma\\)-weak topology is the relative one, so normality of \\(\\omega|_B\\) and (1)\\(\\Rightarrow\\)(2) in \\(B\\) give complete additivity on \\((e_i)\\). By (2)\\(\\Rightarrow\\)(1), \\(\\omega\\) is normal.\n\n**Example 11.7** (Which topologies are intrinsic). The \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies are (Corollary 11.4). The weak, strong and strong\\(^*\\) operator topologies are not. Let \\(M=B(H)\\) with \\(\\dim H=\\infty\\), \\(\\pi_1\\) the identity representation and \\(\\pi_2(x)=x\\oplus x\\oplus\\cdots\\) on \\(H\\oplus H\\oplus\\cdots\\). Both are faithful normal representations with von Neumann algebra images. The weak operator topology of \\(\\pi_2(M)\\), pulled back to \\(M\\), is the \\(\\sigma\\)-weak topology, since the vector functionals of \\(\\pi_2\\) are exactly the square-summable series \\(\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\). It is strictly finer than the weak operator topology of \\(\\pi_1\\). Indeed, with an orthonormal sequence \\((e_n)\\), the functional \\(\\varphi(x)=\\sum_nn^{-2}\\langle xe_n,e_n\\rangle\\) is \\(\\sigma\\)-weakly continuous. It is not weakly continuous: a weakly continuous functional has the form \\(\\sum_{k=1}^m\\langle x\\xi_k,\\eta_k\\rangle\\) (background fact on preduals). Choose a unit vector \\(\\zeta\\in\\operatorname{span}\\{e_1,\\dots,e_{m+1}\\}\\) orthogonal to \\(\\xi_1,\\dots,\\xi_m\\), and let \\(x\\) be the projection onto \\(\\mathbb C\\zeta\\). Then the finite sum vanishes at \\(x\\), but \\(\\varphi(x)=\\sum_nn^{-2}|\\langle\\zeta,e_n\\rangle|^2>0\\).\n\n**Example 11.8** (Normal and singular parts on \\(\\ell^\\infty\\)). In \\(\\ell^\\infty\\) every nonzero projection majorises some coordinate projection \\(e_n\\), and the \\(e_n\\) are minimal. By Theorem 11.2, a positive functional is singular exactly when it vanishes on every \\(e_n\\), that is on \\(c_0\\). Hence for any \\(\\varphi\\in(\\ell^\\infty)^*\\), \\(\\varphi_n(x)=\\sum_k\\varphi(e_k)x_k\\) and \\(\\varphi_s=\\varphi-\\varphi_n\\). A state that is a limit along a free ultrafilter is singular, and it has no support in the sense of [Lemma 11.1](#oa-fnd-wa-23).\n\n**Exercise 11.9.** (easy) Let \\(\\mathcal V\\) be a free ultrafilter on \\(\\mathbb N\\), \\(\\chi(x)=\\lim_{n\\to\\mathcal V}x_n\\) on \\(\\ell^\\infty\\), and \\(\\psi=\\tfrac12(\\delta_1+\\chi)\\), where \\(\\delta_1(x)=x_1\\). Find \\(\\psi_n\\), \\(\\psi_s\\), the cyclic representation \\(\\pi_\\psi\\), and its normal and singular parts.\n\n**Solution.** By Example 11.8, \\(\\chi\\) is singular, and \\(\\psi_n=\\tfrac12\\delta_1\\), \\(\\psi_s=\\tfrac12\\chi\\). Both \\(\\delta_1\\) and \\(\\chi\\) are characters, and they are distinct, so the cyclic representation of \\(\\psi\\) is \\(\\pi_\\psi(x)=\\operatorname{diag}(x_1,\\chi(x))\\) on \\(\\mathbb C^2\\), with cyclic vector \\((2^{-1/2},2^{-1/2})\\). Its normal part ([Theorem 10.3](#oa-fnd-wa-10)(2)) is the first coordinate, \\(x\\mapsto x_1\\), and its singular part is the second, \\(x\\mapsto\\chi(x)\\). \\(\\square\\)\n\n**Exercise 11.10** (medium) (When every state is normal). Prove that all states of a von Neumann algebra \\(M\\) are normal exactly when \\(\\dim M<\\infty\\).\n\n**Solution.** If \\(\\dim M<\\infty\\), every linear functional is continuous for the Hausdorff vector topology \\(\\sigma(M,M_*)\\), hence normal. If \\(\\dim M=\\infty\\), Lemma 6.2 gives orthogonal nonzero projections \\(p_1,p_2,\\dots\\). Choose unit vectors \\(\\xi_n\\) in the range of \\(p_n\\) and a free ultrafilter \\(\\mathcal V\\) on \\(\\mathbb N\\), and put \\(\\varphi(x)=\\lim_{n\\to\\mathcal V}\\langle x\\xi_n,\\xi_n\\rangle\\). This is a state. Here \\(\\varphi(p_k)=\\lim_{n\\to\\mathcal V}\\delta_{kn}=0\\) for every \\(k\\), while \\(\\varphi(\\sum_kp_k)=1\\). The finite partial sums of \\(\\sum_kp_k\\) increase to it, so \\(\\varphi\\) does not preserve this supremum and is not normal (background fact on monotone nets). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-19",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "12. Normal representations",
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      "full_conditions_and_proof": "## 12. Normal representations\n\nThe image of a normal representation of a \\(W^*\\)-algebra is again a von Neumann algebra, and the representation is an isomorphism on a central corner of the algebra.\n\n**Proposition 12.1.** Let \\(A\\) be a \\(W^*\\)-algebra and \\(\\pi\\) a normal representation of \\(A\\) on \\(H\\). Then \\(\\ker\\pi=A(1-z)\\) for a central projection \\(z\\in A\\); the image satisfies \\(\\pi(A)=\\pi(A)''\\); and \\(\\pi\\) restricts to a \\(*\\)-isomorphism of \\(Az\\) onto \\(\\pi(A)\\), which is a homeomorphism for the \\(\\sigma\\)-weak topologies. The cyclic representation of a normal positive functional is normal.\n\n**Proof.** The kernel is a \\(\\sigma\\)-weakly closed two-sided ideal, so \\(\\ker\\pi=A(1-z)\\) with \\(z\\) central (Lemma 4.2), and \\(\\pi|_{Az}\\) is injective, hence isometric, with image \\(\\pi(A)\\). The unit ball of \\(Az\\) is \\(\\sigma\\)-weakly compact (Banach–Alaoglu with \\(Az=(A_*z)^*\\)), so its image, which is the unit ball of \\(\\pi(A)\\), is \\(\\sigma\\)-weakly compact, hence \\(\\sigma\\)-weakly closed. \\(\\pi(A)\\) contains \\(1_H\\), because \\(\\pi\\) is nondegenerate and \\(A\\) has a unit. By Kaplansky's density theorem, each element of the unit ball of \\(\\pi(A)''\\) is a strong limit of a net in the unit ball of \\(\\pi(A)\\), hence a \\(\\sigma\\)-weak limit, hence in \\(\\pi(A)\\). So \\(\\pi(A)=\\pi(A)''\\). Then \\(\\pi|_{Az}\\) is a \\(*\\)-isomorphism between von Neumann algebras, a homeomorphism by Corollary 11.4.\n\nLet \\(\\omega\\) be normal and positive with cyclic representation \\((\\pi_\\omega,H_\\omega,\\xi_\\omega)\\). For \\(a,b,x\\in A\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_\\omega(x)\\pi_\\omega(a)\\xi_\\omega,\\pi_\\omega(b)\\xi_\\omega\\rangle\\\\\n=\\omega(b^*xa)\\\\\n=(a\\omega b^*)(x),\n\\end{gathered}\n\\]\nand \\(a\\omega b^*\\in A_*\\) because \\(A_*\\) is invariant (the first fact after Definition 4.1). The map \\((\\zeta,\\eta)\\mapsto\\omega_{\\pi_\\omega;\\zeta,\\eta}\\) is continuous into \\(A^*\\) with \\(\\|\\omega_{\\pi_\\omega;\\zeta,\\eta}\\|\\le\\|\\zeta\\|\\|\\eta\\|\\), \\(\\pi_\\omega(A)\\xi_\\omega\\) is dense, and \\(A_*\\) is norm closed; so every vector coefficient of \\(\\pi_\\omega\\) is normal. Every \\(\\sigma\\)-weakly continuous functional on \\(B(H_\\omega)\\) is a norm-convergent sum of vector functionals (background fact on preduals), so \\(\\psi\\circ\\pi_\\omega\\in A_*\\) for all of them, and \\(\\pi_\\omega\\) is normal. \\(\\square\\)\n\n**Exercise 12.2** (hard) (Fixed points of a unitary group). Let \\(G\\) be a group of unitaries on \\(H\\), \\(M=G''\\), and \\(e_0\\) the projection onto \\(H_0=\\{\\xi:u\\xi=\\xi\\ \\forall u\\in G\\}\\). Show: (a) \\(e_0\\) is a central projection of \\(M\\) with \\(Me_0=\\mathbb Ce_0\\), so \\(M_{e_0}\\cong\\mathbb C\\) when \\(e_0\\ne0\\); (b) a nonempty closed convex \\(G\\)-invariant set \\(\\mathfrak L\\subseteq H\\) meets \\(H_0\\); (c) if \\(\\mathcal K\\) is the weakly (equivalently strongly) closed convex hull of \\(G\\) and \\(\\pi\\) is a normal representation of \\(M\\), then \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\) for every vector \\(\\xi\\) of \\(\\pi\\); (d) \\(e_0\\in\\mathcal K\\).\n\n**Solution.** \\(\\operatorname{span}G\\) is a unital \\(*\\)-algebra (\\(u^*=u^{-1}\\in G\\)), so it is \\(\\sigma\\)-weakly dense in \\(M\\) (by the bicommutant theorem, with Kaplansky's density theorem as in the proof of [Lemma 2.1](#oa-fnd-wa-02)).\n\n(a) \\(H_0\\) is invariant under each \\(u\\) and \\(u^*\\), so \\(e_0\\in G'\\), and \\(ue_0=e_0\\). Also \\(e_0\\) is the meet of the kernel projections of the \\(u-1\\), which lie in \\(M\\), so \\(e_0\\in M\\). Being in \\(G'=M'\\), it is central in \\(M\\). From \\(ue_0=e_0\\) we get \\(xe_0\\in\\mathbb Ce_0\\) for \\(x\\in\\operatorname{span}G\\), and by density \\(Me_0\\subseteq\\mathbb Ce_0\\).\n\n(b) Let \\(\\xi_0\\) be the unique point of \\(\\mathfrak L\\) of least norm (its existence and uniqueness have the full proof in [Lemma 1.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-02)). For \\(u\\in G\\), \\(u\\xi_0\\in\\mathfrak L\\) has the same norm, so \\(u\\xi_0=\\xi_0\\).\n\n(c) Let \\(\\pi\\) act on \\(K\\). Every \\(k\\in\\mathcal K\\) satisfies \\(e_0k=e_0\\), since \\(e_0u=(u^*e_0)^*=e_0\\) for \\(u\\in G\\) and this passes to convex combinations and weak limits. The set \\(\\mathcal K\\) is bounded, convex and \\(\\sigma\\)-weakly compact, and \\(u\\mathcal K\\subseteq\\mathcal K\\). Since \\(\\pi\\) is normal, \\(\\mathfrak L=\\pi(\\mathcal K)\\xi\\) is weakly compact and convex, hence closed and convex, and \\(\\pi(u)\\mathfrak L\\subseteq\\mathfrak L\\). By (b) it contains a vector \\(\\eta=\\pi(k)\\xi\\) fixed by \\(\\pi(G)\\).\n\nWe show that the \\(\\pi(G)\\)-fixed vectors are exactly the range of \\(\\pi(e_0)\\). The range consists of fixed vectors because \\(\\pi(u)\\pi(e_0)=\\pi(e_0)\\). Conversely, let \\(\\zeta\\ne0\\) be fixed and put \\(\\zeta'=\\pi(1-e_0)\\zeta\\), which is fixed too, since \\(\\pi(e_0)\\) commutes with \\(\\pi(G)\\). If \\(\\zeta'\\ne0\\), then \\(\\pi(x)\\zeta'\\in\\mathbb C\\zeta'\\) for \\(x\\in\\operatorname{span}G\\), and by normality and density for all \\(x\\in M\\). So \\(\\pi(x)\\zeta'=\\chi(x)\\zeta'\\) for a normal character \\(\\chi\\) of \\(M\\). Its kernel is a \\(\\sigma\\)-weakly closed ideal of codimension one, so it is \\(M(1-c)\\) with \\(c\\) a central projection (Lemma 4.2) and \\(Mc=\\mathbb Cc\\); thus \\(xc=\\chi(x)c\\). For \\(u\\in G\\), \\(uc=\\chi(u)c=c\\), so the range of \\(c\\) consists of fixed vectors and \\(c\\le e_0\\). Then \\(\\chi(e_0)=1\\), because \\(\\chi(c)=\\chi(ce_0)=\\chi(c)\\chi(e_0)\\) and \\(\\chi(c)=1\\). But \\(\\chi(e_0)\\zeta'=\\pi(e_0)\\pi(1-e_0)\\zeta=0\\), a contradiction. So \\(\\zeta=\\pi(e_0)\\zeta\\).\n\nHence \\(\\eta=\\pi(e_0)\\eta=\\pi(e_0k)\\xi=\\pi(e_0)\\xi\\), and \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\).\n\n(d) For \\(\\xi_1,\\dots,\\xi_n\\in H\\) let \\(\\mathcal K(\\xi_1,\\dots,\\xi_n)=\\{x\\in\\mathcal K:x\\xi_i=e_0\\xi_i,\\ i\\le n\\}\\). Apply (c) to the \\(n\\)-fold amplification \\(\\pi_n(x)=x\\oplus\\dots\\oplus x\\) on \\(H^n\\), which is normal, and to \\(\\xi=\\xi_1\\oplus\\dots\\oplus\\xi_n\\): this set is nonempty. The sets are weakly closed in the weakly compact \\(\\mathcal K\\), and any finitely many of them contain a common one. So their intersection, which is \\(\\{e_0\\}\\cap\\mathcal K\\), is nonempty. \\(\\square\\)\n\n**Exercise 12.3** (hard) (Faithfulness seen on a separable subalgebra). Let \\(A\\) be a separable \\(C^*\\)-algebra acting nondegenerately on \\(H\\ne0\\), and \\(M=A''\\). Assume that every normal state of \\(M\\) whose restriction to \\(A\\) is faithful is itself faithful. Show:\n(a) \\(M\\) has a faithful normal state \\(\\varphi\\);\n(b) every nonzero projection \\(e\\in M\\) majorises a nonzero positive element of \\(A\\), that is \\(eMe\\cap A\\ne\\{0\\}\\);\n(c) no singular state of \\(M\\) is faithful on \\(A\\);\n(d) if a state \\(\\omega\\) of \\(M\\) is faithful on \\(A\\), its normal part \\(\\omega_n\\) is faithful on \\(M\\);\n(e) \\(A\\) has a faithful state \\(\\omega=\\sum_n\\lambda_n\\omega_n\\) with pure states \\(\\omega_n\\), \\(\\lambda_n\\ge0\\), \\(\\sum_n\\lambda_n=1\\);\n(f) if \\(\\varphi\\) is the normal part of a Hahn–Banach extension \\(\\bar\\omega\\) of this \\(\\omega\\) to \\(M\\), then the cyclic representation \\(\\pi_\\varphi\\) of \\(M\\) is faithful and normal and \\(\\pi_\\varphi(M)\\) is atomic;\n(g) \\(M\\) is generated by its minimal projections, and \\(A\\) contains all of them.\n\n**Solution.** (a) \\(A_+\\setminus\\{0\\}\\) is separable; take a dense sequence \\((a_n)\\) in it and unit vectors \\(\\xi_n\\) with \\(\\langle a_n\\xi_n,\\xi_n\\rangle\\ge\\|a_n\\|/2\\). Then \\(\\varphi=\\sum_n2^{-n}\\omega_{\\xi_n}\\) is a normal state of \\(M\\). If \\(0\\ne x\\in A_+\\), choose \\(n\\) with \\(\\|x-a_n\\|<\\|x\\|/4\\); then \\(\\|a_n\\|\\ge3\\|x\\|/4\\) and \\(\\omega_{\\xi_n}(x)\\ge\\omega_{\\xi_n}(a_n)-\\|x-a_n\\|\\ge\\|x\\|/8>0\\). So \\(\\varphi|_A\\) is faithful, and by hypothesis \\(\\varphi\\) is faithful.\n\n(b) Let \\(e\\ne0\\) and suppose \\(eMe\\cap A=\\{0\\}\\). If \\(e=1\\) this contradicts \\(A\\ne0\\). Otherwise \\(e^\\perp=1-e\\ne0\\), and \\(\\psi=\\varphi(e^\\perp\\cdot e^\\perp)/\\varphi(e^\\perp)\\) is a normal state. If \\(a\\in A_+\\) and \\(\\psi(a)=0\\), then \\(e^\\perp ae^\\perp=0\\) by faithfulness of \\(\\varphi\\), so \\(a^{1/2}e^\\perp=0\\) and \\(a=eae\\in eMe\\cap A\\), hence \\(a=0\\). So \\(\\psi|_A\\) is faithful, and by hypothesis \\(\\psi\\) is faithful; but \\(\\psi(e)=0\\).\n\n(c) Let \\(\\omega\\) be a singular state. By [Theorem 11.2](#oa-fnd-wa-24), \\(1\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\). By (b) there is \\(0\\ne a\\in A_+\\) with \\(a\\in e_0Me_0\\), so \\(a\\le\\|a\\|e_0\\) and \\(\\omega(a)=0\\).\n\n(d) Let \\(\\omega|_A\\) be faithful and \\(e=1-s(\\omega_n)\\) (with \\(e=1\\) if \\(\\omega_n=0\\)). If \\(e\\ne0\\), Theorem 11.2 applied to the singular positive \\(\\omega_s\\) gives a nonzero \\(e_0\\le e\\) with \\(\\omega_s(e_0)=0\\), and \\(\\omega_n(e_0)=0\\) as \\(e_0\\le1-s(\\omega_n)\\). By (b) some \\(0\\ne a\\in A_+\\) has \\(a\\le\\|a\\|e_0\\), and then \\(\\omega(a)=0\\), a contradiction. So \\(s(\\omega_n)=1\\).\n\n(e) The pure states of \\(A\\) are norming on \\(A_+\\): \\(\\sup\\{\\rho(x):\\rho\\text{ pure}\\}=\\|x\\|\\) for \\(x\\in A_+\\). This is the background fact on pure states, which rests on the Krein–Milman theorem: the positive functionals of norm at most one form the weak\\(^*\\) closed convex hull of \\(0\\) and the pure states, and the states norm \\(A_+\\). With \\((a_n)\\) as in (a), choose pure states \\(\\omega_n\\) with \\(\\omega_n(a_n)\\ge\\|a_n\\|/2\\), and put \\(\\omega=\\sum_n2^{-n}\\omega_n\\). The estimate of (a) shows that \\(\\omega\\) is faithful.\n\n(f) By [Corollary 8.2](#oa-fnd-wa-20), \\(\\bar\\omega\\) is positive, hence a state. By (d), \\(\\varphi=\\bar\\omega_n\\) is faithful on \\(M\\), so \\(\\pi_\\varphi\\) is faithful: \\(\\pi_\\varphi(x)=0\\) gives \\(\\varphi(x^*x)=0\\). It is normal by [Proposition 12.1](#oa-fnd-wa-26). For atomicity, write \\(\\rho=\\bigoplus_n\\pi_{\\omega_n}\\) on \\(K=\\bigoplus_nK_n\\), a direct sum of irreducible representations (Lemma 7.2), and \\(N=\\rho(A)''\\).\n\n*\\(N\\) is atomic.* Let \\(Q_n\\in N'\\) be the projection onto \\(K_n\\). The compression \\(x\\mapsto xQ_n|_{K_n}\\) is a normal \\(*\\)-homomorphism of \\(N\\) whose image is a von Neumann algebra (Proposition 12.1) containing \\(\\pi_{\\omega_n}(A)\\), hence equal to \\(B(K_n)\\). Its kernel is \\(N(1-z_n)\\) with \\(z_n\\) central, so \\(Nz_n\\cong B(K_n)\\) and \\(z_n\\) is a minimal central projection. If \\(xz_n=0\\) for all \\(n\\) then \\(xQ_n=0\\) for all \\(n\\) and \\(x=0\\); so \\(\\bigvee_nz_n=1\\). Distinct minimal central projections are orthogonal, so \\(N=\\bigoplus_zNz\\) over the distinct \\(z_n\\), each summand \\(\\cong B(K)\\), in which rank-one projections are minimal. Hence \\(N\\) is atomic. The same argument shows that for every projection \\(P\\in N'\\), the algebra \\(N_P=NP|_{PK}\\) is isomorphic to \\(Nc\\) with \\(c\\) central (the kernel of \\(x\\mapsto xP\\) is \\(N(1-c)\\)), which is a direct sum of some of the summands; so \\(N_P\\) is atomic.\n\n*\\(\\pi_\\varphi(M)\\) is some \\(N_P\\).* Since \\(\\varphi\\le\\bar\\omega\\), \\(\\varphi|_A\\le\\omega\\). The vector \\(\\xi_\\varphi\\) is cyclic for \\(\\pi_\\varphi(A)\\), because \\(\\pi_\\varphi(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\pi_\\varphi(M)\\); so \\(\\pi_\\varphi|_A\\) is the cyclic representation of \\(\\varphi|_A\\). By the Radon–Nikodym step in the proof of Lemma 7.2, \\(\\varphi(a)=\\langle\\pi_\\omega(a)T^{1/2}\\xi_\\omega,T^{1/2}\\xi_\\omega\\rangle\\) for some \\(0\\le T\\le1\\) in \\(\\pi_\\omega(A)'\\). So \\(\\pi_\\varphi|_A\\) is unitarily equivalent to \\(\\pi_\\omega\\) restricted to the invariant subspace \\(\\overline{\\pi_\\omega(A)T^{1/2}\\xi_\\omega}\\): two cyclic representations with the same vector state are unitarily equivalent, through \\(\\pi_1(a)\\zeta_1\\mapsto\\pi_2(a)\\zeta_2\\), which preserves inner products. In the same way \\(\\pi_\\omega\\) is \\(\\rho\\) restricted to the cyclic subspace of \\(\\bigoplus_n\\lambda_n^{1/2}\\xi_{\\omega_n}\\). Hence \\(\\pi_\\varphi|_A\\) is unitarily equivalent to \\(\\rho\\) restricted to \\(PK\\) for a projection \\(P\\in\\rho(A)'=N'\\). Therefore \\(\\pi_\\varphi(M)=\\pi_\\varphi(A)''\\cong(\\rho(A)|_{PK})''=N_P\\), which is atomic, and \\(M\\cong\\pi_\\varphi(M)\\).\n\n(g) Let \\((e_i)\\) be a maximal family of mutually orthogonal minimal projections of the atomic algebra \\(M\\); then \\(\\sum_ie_i=1\\), and every \\(x\\in M\\) is the \\(\\sigma\\)-weak limit of \\(e_Fxe_F\\), \\(e_F=\\sum_{i\\in F}e_i\\), \\(F\\) finite. By Lemma 6.1(2), \\(e_iMe_j\\) is \\(0\\) or \\(\\mathbb Cv_{ij}\\) with \\(v_{ij}^*v_{ij}=e_j\\), \\(v_{ij}v_{ij}^*=e_i\\). For \\(i\\ne j\\) with \\(v=v_{ij}\\ne0\\), \\(g=\\tfrac12(e_i+e_j+v+v^*)\\) is a projection, rank one in \\((e_i+e_j)M(e_i+e_j)\\cong M_2(\\mathbb C)\\), hence minimal in \\(M\\). So \\(v=e_i(2g-e_i-e_j)e_j\\) lies in the von Neumann algebra generated by the minimal projections, and so do all \\(e_ixe_j\\) and all \\(x\\). Finally, if \\(e\\) is minimal in \\(M\\), (b) gives \\(0\\ne a\\in A\\cap eMe=A\\cap\\mathbb Ce\\), so \\(e\\in A\\). \\(\\square\\)\n\n",
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      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "12. Normal representations",
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      "full_conditions_and_proof": "## 12. Normal representations\n\nThe image of a normal representation of a \\(W^*\\)-algebra is again a von Neumann algebra, and the representation is an isomorphism on a central corner of the algebra.\n\n**Proposition 12.1.** Let \\(A\\) be a \\(W^*\\)-algebra and \\(\\pi\\) a normal representation of \\(A\\) on \\(H\\). Then \\(\\ker\\pi=A(1-z)\\) for a central projection \\(z\\in A\\); the image satisfies \\(\\pi(A)=\\pi(A)''\\); and \\(\\pi\\) restricts to a \\(*\\)-isomorphism of \\(Az\\) onto \\(\\pi(A)\\), which is a homeomorphism for the \\(\\sigma\\)-weak topologies. The cyclic representation of a normal positive functional is normal.\n\n**Proof.** The kernel is a \\(\\sigma\\)-weakly closed two-sided ideal, so \\(\\ker\\pi=A(1-z)\\) with \\(z\\) central (Lemma 4.2), and \\(\\pi|_{Az}\\) is injective, hence isometric, with image \\(\\pi(A)\\). The unit ball of \\(Az\\) is \\(\\sigma\\)-weakly compact (Banach–Alaoglu with \\(Az=(A_*z)^*\\)), so its image, which is the unit ball of \\(\\pi(A)\\), is \\(\\sigma\\)-weakly compact, hence \\(\\sigma\\)-weakly closed. \\(\\pi(A)\\) contains \\(1_H\\), because \\(\\pi\\) is nondegenerate and \\(A\\) has a unit. By Kaplansky's density theorem, each element of the unit ball of \\(\\pi(A)''\\) is a strong limit of a net in the unit ball of \\(\\pi(A)\\), hence a \\(\\sigma\\)-weak limit, hence in \\(\\pi(A)\\). So \\(\\pi(A)=\\pi(A)''\\). Then \\(\\pi|_{Az}\\) is a \\(*\\)-isomorphism between von Neumann algebras, a homeomorphism by Corollary 11.4.\n\nLet \\(\\omega\\) be normal and positive with cyclic representation \\((\\pi_\\omega,H_\\omega,\\xi_\\omega)\\). For \\(a,b,x\\in A\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_\\omega(x)\\pi_\\omega(a)\\xi_\\omega,\\pi_\\omega(b)\\xi_\\omega\\rangle\\\\\n=\\omega(b^*xa)\\\\\n=(a\\omega b^*)(x),\n\\end{gathered}\n\\]\nand \\(a\\omega b^*\\in A_*\\) because \\(A_*\\) is invariant (the first fact after Definition 4.1). The map \\((\\zeta,\\eta)\\mapsto\\omega_{\\pi_\\omega;\\zeta,\\eta}\\) is continuous into \\(A^*\\) with \\(\\|\\omega_{\\pi_\\omega;\\zeta,\\eta}\\|\\le\\|\\zeta\\|\\|\\eta\\|\\), \\(\\pi_\\omega(A)\\xi_\\omega\\) is dense, and \\(A_*\\) is norm closed; so every vector coefficient of \\(\\pi_\\omega\\) is normal. Every \\(\\sigma\\)-weakly continuous functional on \\(B(H_\\omega)\\) is a norm-convergent sum of vector functionals (background fact on preduals), so \\(\\psi\\circ\\pi_\\omega\\in A_*\\) for all of them, and \\(\\pi_\\omega\\) is normal. \\(\\square\\)\n\n**Exercise 12.2** (hard) (Fixed points of a unitary group). Let \\(G\\) be a group of unitaries on \\(H\\), \\(M=G''\\), and \\(e_0\\) the projection onto \\(H_0=\\{\\xi:u\\xi=\\xi\\ \\forall u\\in G\\}\\). Show: (a) \\(e_0\\) is a central projection of \\(M\\) with \\(Me_0=\\mathbb Ce_0\\), so \\(M_{e_0}\\cong\\mathbb C\\) when \\(e_0\\ne0\\); (b) a nonempty closed convex \\(G\\)-invariant set \\(\\mathfrak L\\subseteq H\\) meets \\(H_0\\); (c) if \\(\\mathcal K\\) is the weakly (equivalently strongly) closed convex hull of \\(G\\) and \\(\\pi\\) is a normal representation of \\(M\\), then \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\) for every vector \\(\\xi\\) of \\(\\pi\\); (d) \\(e_0\\in\\mathcal K\\).\n\n**Solution.** \\(\\operatorname{span}G\\) is a unital \\(*\\)-algebra (\\(u^*=u^{-1}\\in G\\)), so it is \\(\\sigma\\)-weakly dense in \\(M\\) (by the bicommutant theorem, with Kaplansky's density theorem as in the proof of [Lemma 2.1](#oa-fnd-wa-02)).\n\n(a) \\(H_0\\) is invariant under each \\(u\\) and \\(u^*\\), so \\(e_0\\in G'\\), and \\(ue_0=e_0\\). Also \\(e_0\\) is the meet of the kernel projections of the \\(u-1\\), which lie in \\(M\\), so \\(e_0\\in M\\). Being in \\(G'=M'\\), it is central in \\(M\\). From \\(ue_0=e_0\\) we get \\(xe_0\\in\\mathbb Ce_0\\) for \\(x\\in\\operatorname{span}G\\), and by density \\(Me_0\\subseteq\\mathbb Ce_0\\).\n\n(b) Let \\(\\xi_0\\) be the unique point of \\(\\mathfrak L\\) of least norm (its existence and uniqueness have the full proof in [Lemma 1.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-02)). For \\(u\\in G\\), \\(u\\xi_0\\in\\mathfrak L\\) has the same norm, so \\(u\\xi_0=\\xi_0\\).\n\n(c) Let \\(\\pi\\) act on \\(K\\). Every \\(k\\in\\mathcal K\\) satisfies \\(e_0k=e_0\\), since \\(e_0u=(u^*e_0)^*=e_0\\) for \\(u\\in G\\) and this passes to convex combinations and weak limits. The set \\(\\mathcal K\\) is bounded, convex and \\(\\sigma\\)-weakly compact, and \\(u\\mathcal K\\subseteq\\mathcal K\\). Since \\(\\pi\\) is normal, \\(\\mathfrak L=\\pi(\\mathcal K)\\xi\\) is weakly compact and convex, hence closed and convex, and \\(\\pi(u)\\mathfrak L\\subseteq\\mathfrak L\\). By (b) it contains a vector \\(\\eta=\\pi(k)\\xi\\) fixed by \\(\\pi(G)\\).\n\nWe show that the \\(\\pi(G)\\)-fixed vectors are exactly the range of \\(\\pi(e_0)\\). The range consists of fixed vectors because \\(\\pi(u)\\pi(e_0)=\\pi(e_0)\\). Conversely, let \\(\\zeta\\ne0\\) be fixed and put \\(\\zeta'=\\pi(1-e_0)\\zeta\\), which is fixed too, since \\(\\pi(e_0)\\) commutes with \\(\\pi(G)\\). If \\(\\zeta'\\ne0\\), then \\(\\pi(x)\\zeta'\\in\\mathbb C\\zeta'\\) for \\(x\\in\\operatorname{span}G\\), and by normality and density for all \\(x\\in M\\). So \\(\\pi(x)\\zeta'=\\chi(x)\\zeta'\\) for a normal character \\(\\chi\\) of \\(M\\). Its kernel is a \\(\\sigma\\)-weakly closed ideal of codimension one, so it is \\(M(1-c)\\) with \\(c\\) a central projection (Lemma 4.2) and \\(Mc=\\mathbb Cc\\); thus \\(xc=\\chi(x)c\\). For \\(u\\in G\\), \\(uc=\\chi(u)c=c\\), so the range of \\(c\\) consists of fixed vectors and \\(c\\le e_0\\). Then \\(\\chi(e_0)=1\\), because \\(\\chi(c)=\\chi(ce_0)=\\chi(c)\\chi(e_0)\\) and \\(\\chi(c)=1\\). But \\(\\chi(e_0)\\zeta'=\\pi(e_0)\\pi(1-e_0)\\zeta=0\\), a contradiction. So \\(\\zeta=\\pi(e_0)\\zeta\\).\n\nHence \\(\\eta=\\pi(e_0)\\eta=\\pi(e_0k)\\xi=\\pi(e_0)\\xi\\), and \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\).\n\n(d) For \\(\\xi_1,\\dots,\\xi_n\\in H\\) let \\(\\mathcal K(\\xi_1,\\dots,\\xi_n)=\\{x\\in\\mathcal K:x\\xi_i=e_0\\xi_i,\\ i\\le n\\}\\). Apply (c) to the \\(n\\)-fold amplification \\(\\pi_n(x)=x\\oplus\\dots\\oplus x\\) on \\(H^n\\), which is normal, and to \\(\\xi=\\xi_1\\oplus\\dots\\oplus\\xi_n\\): this set is nonempty. The sets are weakly closed in the weakly compact \\(\\mathcal K\\), and any finitely many of them contain a common one. So their intersection, which is \\(\\{e_0\\}\\cap\\mathcal K\\), is nonempty. \\(\\square\\)\n\n**Exercise 12.3** (hard) (Faithfulness seen on a separable subalgebra). Let \\(A\\) be a separable \\(C^*\\)-algebra acting nondegenerately on \\(H\\ne0\\), and \\(M=A''\\). Assume that every normal state of \\(M\\) whose restriction to \\(A\\) is faithful is itself faithful. Show:\n(a) \\(M\\) has a faithful normal state \\(\\varphi\\);\n(b) every nonzero projection \\(e\\in M\\) majorises a nonzero positive element of \\(A\\), that is \\(eMe\\cap A\\ne\\{0\\}\\);\n(c) no singular state of \\(M\\) is faithful on \\(A\\);\n(d) if a state \\(\\omega\\) of \\(M\\) is faithful on \\(A\\), its normal part \\(\\omega_n\\) is faithful on \\(M\\);\n(e) \\(A\\) has a faithful state \\(\\omega=\\sum_n\\lambda_n\\omega_n\\) with pure states \\(\\omega_n\\), \\(\\lambda_n\\ge0\\), \\(\\sum_n\\lambda_n=1\\);\n(f) if \\(\\varphi\\) is the normal part of a Hahn–Banach extension \\(\\bar\\omega\\) of this \\(\\omega\\) to \\(M\\), then the cyclic representation \\(\\pi_\\varphi\\) of \\(M\\) is faithful and normal and \\(\\pi_\\varphi(M)\\) is atomic;\n(g) \\(M\\) is generated by its minimal projections, and \\(A\\) contains all of them.\n\n**Solution.** (a) \\(A_+\\setminus\\{0\\}\\) is separable; take a dense sequence \\((a_n)\\) in it and unit vectors \\(\\xi_n\\) with \\(\\langle a_n\\xi_n,\\xi_n\\rangle\\ge\\|a_n\\|/2\\). Then \\(\\varphi=\\sum_n2^{-n}\\omega_{\\xi_n}\\) is a normal state of \\(M\\). If \\(0\\ne x\\in A_+\\), choose \\(n\\) with \\(\\|x-a_n\\|<\\|x\\|/4\\); then \\(\\|a_n\\|\\ge3\\|x\\|/4\\) and \\(\\omega_{\\xi_n}(x)\\ge\\omega_{\\xi_n}(a_n)-\\|x-a_n\\|\\ge\\|x\\|/8>0\\). So \\(\\varphi|_A\\) is faithful, and by hypothesis \\(\\varphi\\) is faithful.\n\n(b) Let \\(e\\ne0\\) and suppose \\(eMe\\cap A=\\{0\\}\\). If \\(e=1\\) this contradicts \\(A\\ne0\\). Otherwise \\(e^\\perp=1-e\\ne0\\), and \\(\\psi=\\varphi(e^\\perp\\cdot e^\\perp)/\\varphi(e^\\perp)\\) is a normal state. If \\(a\\in A_+\\) and \\(\\psi(a)=0\\), then \\(e^\\perp ae^\\perp=0\\) by faithfulness of \\(\\varphi\\), so \\(a^{1/2}e^\\perp=0\\) and \\(a=eae\\in eMe\\cap A\\), hence \\(a=0\\). So \\(\\psi|_A\\) is faithful, and by hypothesis \\(\\psi\\) is faithful; but \\(\\psi(e)=0\\).\n\n(c) Let \\(\\omega\\) be a singular state. By [Theorem 11.2](#oa-fnd-wa-24), \\(1\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\). By (b) there is \\(0\\ne a\\in A_+\\) with \\(a\\in e_0Me_0\\), so \\(a\\le\\|a\\|e_0\\) and \\(\\omega(a)=0\\).\n\n(d) Let \\(\\omega|_A\\) be faithful and \\(e=1-s(\\omega_n)\\) (with \\(e=1\\) if \\(\\omega_n=0\\)). If \\(e\\ne0\\), Theorem 11.2 applied to the singular positive \\(\\omega_s\\) gives a nonzero \\(e_0\\le e\\) with \\(\\omega_s(e_0)=0\\), and \\(\\omega_n(e_0)=0\\) as \\(e_0\\le1-s(\\omega_n)\\). By (b) some \\(0\\ne a\\in A_+\\) has \\(a\\le\\|a\\|e_0\\), and then \\(\\omega(a)=0\\), a contradiction. So \\(s(\\omega_n)=1\\).\n\n(e) The pure states of \\(A\\) are norming on \\(A_+\\): \\(\\sup\\{\\rho(x):\\rho\\text{ pure}\\}=\\|x\\|\\) for \\(x\\in A_+\\). This is the background fact on pure states, which rests on the Krein–Milman theorem: the positive functionals of norm at most one form the weak\\(^*\\) closed convex hull of \\(0\\) and the pure states, and the states norm \\(A_+\\). With \\((a_n)\\) as in (a), choose pure states \\(\\omega_n\\) with \\(\\omega_n(a_n)\\ge\\|a_n\\|/2\\), and put \\(\\omega=\\sum_n2^{-n}\\omega_n\\). The estimate of (a) shows that \\(\\omega\\) is faithful.\n\n(f) By [Corollary 8.2](#oa-fnd-wa-20), \\(\\bar\\omega\\) is positive, hence a state. By (d), \\(\\varphi=\\bar\\omega_n\\) is faithful on \\(M\\), so \\(\\pi_\\varphi\\) is faithful: \\(\\pi_\\varphi(x)=0\\) gives \\(\\varphi(x^*x)=0\\). It is normal by [Proposition 12.1](#oa-fnd-wa-26). For atomicity, write \\(\\rho=\\bigoplus_n\\pi_{\\omega_n}\\) on \\(K=\\bigoplus_nK_n\\), a direct sum of irreducible representations (Lemma 7.2), and \\(N=\\rho(A)''\\).\n\n*\\(N\\) is atomic.* Let \\(Q_n\\in N'\\) be the projection onto \\(K_n\\). The compression \\(x\\mapsto xQ_n|_{K_n}\\) is a normal \\(*\\)-homomorphism of \\(N\\) whose image is a von Neumann algebra (Proposition 12.1) containing \\(\\pi_{\\omega_n}(A)\\), hence equal to \\(B(K_n)\\). Its kernel is \\(N(1-z_n)\\) with \\(z_n\\) central, so \\(Nz_n\\cong B(K_n)\\) and \\(z_n\\) is a minimal central projection. If \\(xz_n=0\\) for all \\(n\\) then \\(xQ_n=0\\) for all \\(n\\) and \\(x=0\\); so \\(\\bigvee_nz_n=1\\). Distinct minimal central projections are orthogonal, so \\(N=\\bigoplus_zNz\\) over the distinct \\(z_n\\), each summand \\(\\cong B(K)\\), in which rank-one projections are minimal. Hence \\(N\\) is atomic. The same argument shows that for every projection \\(P\\in N'\\), the algebra \\(N_P=NP|_{PK}\\) is isomorphic to \\(Nc\\) with \\(c\\) central (the kernel of \\(x\\mapsto xP\\) is \\(N(1-c)\\)), which is a direct sum of some of the summands; so \\(N_P\\) is atomic.\n\n*\\(\\pi_\\varphi(M)\\) is some \\(N_P\\).* Since \\(\\varphi\\le\\bar\\omega\\), \\(\\varphi|_A\\le\\omega\\). The vector \\(\\xi_\\varphi\\) is cyclic for \\(\\pi_\\varphi(A)\\), because \\(\\pi_\\varphi(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\pi_\\varphi(M)\\); so \\(\\pi_\\varphi|_A\\) is the cyclic representation of \\(\\varphi|_A\\). By the Radon–Nikodym step in the proof of Lemma 7.2, \\(\\varphi(a)=\\langle\\pi_\\omega(a)T^{1/2}\\xi_\\omega,T^{1/2}\\xi_\\omega\\rangle\\) for some \\(0\\le T\\le1\\) in \\(\\pi_\\omega(A)'\\). So \\(\\pi_\\varphi|_A\\) is unitarily equivalent to \\(\\pi_\\omega\\) restricted to the invariant subspace \\(\\overline{\\pi_\\omega(A)T^{1/2}\\xi_\\omega}\\): two cyclic representations with the same vector state are unitarily equivalent, through \\(\\pi_1(a)\\zeta_1\\mapsto\\pi_2(a)\\zeta_2\\), which preserves inner products. In the same way \\(\\pi_\\omega\\) is \\(\\rho\\) restricted to the cyclic subspace of \\(\\bigoplus_n\\lambda_n^{1/2}\\xi_{\\omega_n}\\). Hence \\(\\pi_\\varphi|_A\\) is unitarily equivalent to \\(\\rho\\) restricted to \\(PK\\) for a projection \\(P\\in\\rho(A)'=N'\\). Therefore \\(\\pi_\\varphi(M)=\\pi_\\varphi(A)''\\cong(\\rho(A)|_{PK})''=N_P\\), which is atomic, and \\(M\\cong\\pi_\\varphi(M)\\).\n\n(g) Let \\((e_i)\\) be a maximal family of mutually orthogonal minimal projections of the atomic algebra \\(M\\); then \\(\\sum_ie_i=1\\), and every \\(x\\in M\\) is the \\(\\sigma\\)-weak limit of \\(e_Fxe_F\\), \\(e_F=\\sum_{i\\in F}e_i\\), \\(F\\) finite. By Lemma 6.1(2), \\(e_iMe_j\\) is \\(0\\) or \\(\\mathbb Cv_{ij}\\) with \\(v_{ij}^*v_{ij}=e_j\\), \\(v_{ij}v_{ij}^*=e_i\\). For \\(i\\ne j\\) with \\(v=v_{ij}\\ne0\\), \\(g=\\tfrac12(e_i+e_j+v+v^*)\\) is a projection, rank one in \\((e_i+e_j)M(e_i+e_j)\\cong M_2(\\mathbb C)\\), hence minimal in \\(M\\). So \\(v=e_i(2g-e_i-e_j)e_j\\) lies in the von Neumann algebra generated by the minimal projections, and so do all \\(e_ixe_j\\) and all \\(x\\). Finally, if \\(e\\) is minimal in \\(M\\), (b) gives \\(0\\ne a\\in A\\cap eMe=A\\cap\\mathbb Ce\\), so \\(e\\in A\\). \\(\\square\\)\n\n",
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      "full_conditions_and_proof": "## 12. Normal representations\n\nThe image of a normal representation of a \\(W^*\\)-algebra is again a von Neumann algebra, and the representation is an isomorphism on a central corner of the algebra.\n\n**Proposition 12.1.** Let \\(A\\) be a \\(W^*\\)-algebra and \\(\\pi\\) a normal representation of \\(A\\) on \\(H\\). Then \\(\\ker\\pi=A(1-z)\\) for a central projection \\(z\\in A\\); the image satisfies \\(\\pi(A)=\\pi(A)''\\); and \\(\\pi\\) restricts to a \\(*\\)-isomorphism of \\(Az\\) onto \\(\\pi(A)\\), which is a homeomorphism for the \\(\\sigma\\)-weak topologies. The cyclic representation of a normal positive functional is normal.\n\n**Proof.** The kernel is a \\(\\sigma\\)-weakly closed two-sided ideal, so \\(\\ker\\pi=A(1-z)\\) with \\(z\\) central (Lemma 4.2), and \\(\\pi|_{Az}\\) is injective, hence isometric, with image \\(\\pi(A)\\). The unit ball of \\(Az\\) is \\(\\sigma\\)-weakly compact (Banach–Alaoglu with \\(Az=(A_*z)^*\\)), so its image, which is the unit ball of \\(\\pi(A)\\), is \\(\\sigma\\)-weakly compact, hence \\(\\sigma\\)-weakly closed. \\(\\pi(A)\\) contains \\(1_H\\), because \\(\\pi\\) is nondegenerate and \\(A\\) has a unit. By Kaplansky's density theorem, each element of the unit ball of \\(\\pi(A)''\\) is a strong limit of a net in the unit ball of \\(\\pi(A)\\), hence a \\(\\sigma\\)-weak limit, hence in \\(\\pi(A)\\). So \\(\\pi(A)=\\pi(A)''\\). Then \\(\\pi|_{Az}\\) is a \\(*\\)-isomorphism between von Neumann algebras, a homeomorphism by Corollary 11.4.\n\nLet \\(\\omega\\) be normal and positive with cyclic representation \\((\\pi_\\omega,H_\\omega,\\xi_\\omega)\\). For \\(a,b,x\\in A\\),\n\\[\n\\begin{gathered}\n\\langle\\pi_\\omega(x)\\pi_\\omega(a)\\xi_\\omega,\\pi_\\omega(b)\\xi_\\omega\\rangle\\\\\n=\\omega(b^*xa)\\\\\n=(a\\omega b^*)(x),\n\\end{gathered}\n\\]\nand \\(a\\omega b^*\\in A_*\\) because \\(A_*\\) is invariant (the first fact after Definition 4.1). The map \\((\\zeta,\\eta)\\mapsto\\omega_{\\pi_\\omega;\\zeta,\\eta}\\) is continuous into \\(A^*\\) with \\(\\|\\omega_{\\pi_\\omega;\\zeta,\\eta}\\|\\le\\|\\zeta\\|\\|\\eta\\|\\), \\(\\pi_\\omega(A)\\xi_\\omega\\) is dense, and \\(A_*\\) is norm closed; so every vector coefficient of \\(\\pi_\\omega\\) is normal. Every \\(\\sigma\\)-weakly continuous functional on \\(B(H_\\omega)\\) is a norm-convergent sum of vector functionals (background fact on preduals), so \\(\\psi\\circ\\pi_\\omega\\in A_*\\) for all of them, and \\(\\pi_\\omega\\) is normal. \\(\\square\\)\n\n**Exercise 12.2** (hard) (Fixed points of a unitary group). Let \\(G\\) be a group of unitaries on \\(H\\), \\(M=G''\\), and \\(e_0\\) the projection onto \\(H_0=\\{\\xi:u\\xi=\\xi\\ \\forall u\\in G\\}\\). Show: (a) \\(e_0\\) is a central projection of \\(M\\) with \\(Me_0=\\mathbb Ce_0\\), so \\(M_{e_0}\\cong\\mathbb C\\) when \\(e_0\\ne0\\); (b) a nonempty closed convex \\(G\\)-invariant set \\(\\mathfrak L\\subseteq H\\) meets \\(H_0\\); (c) if \\(\\mathcal K\\) is the weakly (equivalently strongly) closed convex hull of \\(G\\) and \\(\\pi\\) is a normal representation of \\(M\\), then \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\) for every vector \\(\\xi\\) of \\(\\pi\\); (d) \\(e_0\\in\\mathcal K\\).\n\n**Solution.** \\(\\operatorname{span}G\\) is a unital \\(*\\)-algebra (\\(u^*=u^{-1}\\in G\\)), so it is \\(\\sigma\\)-weakly dense in \\(M\\) (by the bicommutant theorem, with Kaplansky's density theorem as in the proof of [Lemma 2.1](#oa-fnd-wa-02)).\n\n(a) \\(H_0\\) is invariant under each \\(u\\) and \\(u^*\\), so \\(e_0\\in G'\\), and \\(ue_0=e_0\\). Also \\(e_0\\) is the meet of the kernel projections of the \\(u-1\\), which lie in \\(M\\), so \\(e_0\\in M\\). Being in \\(G'=M'\\), it is central in \\(M\\). From \\(ue_0=e_0\\) we get \\(xe_0\\in\\mathbb Ce_0\\) for \\(x\\in\\operatorname{span}G\\), and by density \\(Me_0\\subseteq\\mathbb Ce_0\\).\n\n(b) Let \\(\\xi_0\\) be the unique point of \\(\\mathfrak L\\) of least norm (its existence and uniqueness have the full proof in [Lemma 1.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-02)). For \\(u\\in G\\), \\(u\\xi_0\\in\\mathfrak L\\) has the same norm, so \\(u\\xi_0=\\xi_0\\).\n\n(c) Let \\(\\pi\\) act on \\(K\\). Every \\(k\\in\\mathcal K\\) satisfies \\(e_0k=e_0\\), since \\(e_0u=(u^*e_0)^*=e_0\\) for \\(u\\in G\\) and this passes to convex combinations and weak limits. The set \\(\\mathcal K\\) is bounded, convex and \\(\\sigma\\)-weakly compact, and \\(u\\mathcal K\\subseteq\\mathcal K\\). Since \\(\\pi\\) is normal, \\(\\mathfrak L=\\pi(\\mathcal K)\\xi\\) is weakly compact and convex, hence closed and convex, and \\(\\pi(u)\\mathfrak L\\subseteq\\mathfrak L\\). By (b) it contains a vector \\(\\eta=\\pi(k)\\xi\\) fixed by \\(\\pi(G)\\).\n\nWe show that the \\(\\pi(G)\\)-fixed vectors are exactly the range of \\(\\pi(e_0)\\). The range consists of fixed vectors because \\(\\pi(u)\\pi(e_0)=\\pi(e_0)\\). Conversely, let \\(\\zeta\\ne0\\) be fixed and put \\(\\zeta'=\\pi(1-e_0)\\zeta\\), which is fixed too, since \\(\\pi(e_0)\\) commutes with \\(\\pi(G)\\). If \\(\\zeta'\\ne0\\), then \\(\\pi(x)\\zeta'\\in\\mathbb C\\zeta'\\) for \\(x\\in\\operatorname{span}G\\), and by normality and density for all \\(x\\in M\\). So \\(\\pi(x)\\zeta'=\\chi(x)\\zeta'\\) for a normal character \\(\\chi\\) of \\(M\\). Its kernel is a \\(\\sigma\\)-weakly closed ideal of codimension one, so it is \\(M(1-c)\\) with \\(c\\) a central projection (Lemma 4.2) and \\(Mc=\\mathbb Cc\\); thus \\(xc=\\chi(x)c\\). For \\(u\\in G\\), \\(uc=\\chi(u)c=c\\), so the range of \\(c\\) consists of fixed vectors and \\(c\\le e_0\\). Then \\(\\chi(e_0)=1\\), because \\(\\chi(c)=\\chi(ce_0)=\\chi(c)\\chi(e_0)\\) and \\(\\chi(c)=1\\). But \\(\\chi(e_0)\\zeta'=\\pi(e_0)\\pi(1-e_0)\\zeta=0\\), a contradiction. So \\(\\zeta=\\pi(e_0)\\zeta\\).\n\nHence \\(\\eta=\\pi(e_0)\\eta=\\pi(e_0k)\\xi=\\pi(e_0)\\xi\\), and \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\).\n\n(d) For \\(\\xi_1,\\dots,\\xi_n\\in H\\) let \\(\\mathcal K(\\xi_1,\\dots,\\xi_n)=\\{x\\in\\mathcal K:x\\xi_i=e_0\\xi_i,\\ i\\le n\\}\\). Apply (c) to the \\(n\\)-fold amplification \\(\\pi_n(x)=x\\oplus\\dots\\oplus x\\) on \\(H^n\\), which is normal, and to \\(\\xi=\\xi_1\\oplus\\dots\\oplus\\xi_n\\): this set is nonempty. The sets are weakly closed in the weakly compact \\(\\mathcal K\\), and any finitely many of them contain a common one. So their intersection, which is \\(\\{e_0\\}\\cap\\mathcal K\\), is nonempty. \\(\\square\\)\n\n**Exercise 12.3** (hard) (Faithfulness seen on a separable subalgebra). Let \\(A\\) be a separable \\(C^*\\)-algebra acting nondegenerately on \\(H\\ne0\\), and \\(M=A''\\). Assume that every normal state of \\(M\\) whose restriction to \\(A\\) is faithful is itself faithful. Show:\n(a) \\(M\\) has a faithful normal state \\(\\varphi\\);\n(b) every nonzero projection \\(e\\in M\\) majorises a nonzero positive element of \\(A\\), that is \\(eMe\\cap A\\ne\\{0\\}\\);\n(c) no singular state of \\(M\\) is faithful on \\(A\\);\n(d) if a state \\(\\omega\\) of \\(M\\) is faithful on \\(A\\), its normal part \\(\\omega_n\\) is faithful on \\(M\\);\n(e) \\(A\\) has a faithful state \\(\\omega=\\sum_n\\lambda_n\\omega_n\\) with pure states \\(\\omega_n\\), \\(\\lambda_n\\ge0\\), \\(\\sum_n\\lambda_n=1\\);\n(f) if \\(\\varphi\\) is the normal part of a Hahn–Banach extension \\(\\bar\\omega\\) of this \\(\\omega\\) to \\(M\\), then the cyclic representation \\(\\pi_\\varphi\\) of \\(M\\) is faithful and normal and \\(\\pi_\\varphi(M)\\) is atomic;\n(g) \\(M\\) is generated by its minimal projections, and \\(A\\) contains all of them.\n\n**Solution.** (a) \\(A_+\\setminus\\{0\\}\\) is separable; take a dense sequence \\((a_n)\\) in it and unit vectors \\(\\xi_n\\) with \\(\\langle a_n\\xi_n,\\xi_n\\rangle\\ge\\|a_n\\|/2\\). Then \\(\\varphi=\\sum_n2^{-n}\\omega_{\\xi_n}\\) is a normal state of \\(M\\). If \\(0\\ne x\\in A_+\\), choose \\(n\\) with \\(\\|x-a_n\\|<\\|x\\|/4\\); then \\(\\|a_n\\|\\ge3\\|x\\|/4\\) and \\(\\omega_{\\xi_n}(x)\\ge\\omega_{\\xi_n}(a_n)-\\|x-a_n\\|\\ge\\|x\\|/8>0\\). So \\(\\varphi|_A\\) is faithful, and by hypothesis \\(\\varphi\\) is faithful.\n\n(b) Let \\(e\\ne0\\) and suppose \\(eMe\\cap A=\\{0\\}\\). If \\(e=1\\) this contradicts \\(A\\ne0\\). Otherwise \\(e^\\perp=1-e\\ne0\\), and \\(\\psi=\\varphi(e^\\perp\\cdot e^\\perp)/\\varphi(e^\\perp)\\) is a normal state. If \\(a\\in A_+\\) and \\(\\psi(a)=0\\), then \\(e^\\perp ae^\\perp=0\\) by faithfulness of \\(\\varphi\\), so \\(a^{1/2}e^\\perp=0\\) and \\(a=eae\\in eMe\\cap A\\), hence \\(a=0\\). So \\(\\psi|_A\\) is faithful, and by hypothesis \\(\\psi\\) is faithful; but \\(\\psi(e)=0\\).\n\n(c) Let \\(\\omega\\) be a singular state. By [Theorem 11.2](#oa-fnd-wa-24), \\(1\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\). By (b) there is \\(0\\ne a\\in A_+\\) with \\(a\\in e_0Me_0\\), so \\(a\\le\\|a\\|e_0\\) and \\(\\omega(a)=0\\).\n\n(d) Let \\(\\omega|_A\\) be faithful and \\(e=1-s(\\omega_n)\\) (with \\(e=1\\) if \\(\\omega_n=0\\)). If \\(e\\ne0\\), Theorem 11.2 applied to the singular positive \\(\\omega_s\\) gives a nonzero \\(e_0\\le e\\) with \\(\\omega_s(e_0)=0\\), and \\(\\omega_n(e_0)=0\\) as \\(e_0\\le1-s(\\omega_n)\\). By (b) some \\(0\\ne a\\in A_+\\) has \\(a\\le\\|a\\|e_0\\), and then \\(\\omega(a)=0\\), a contradiction. So \\(s(\\omega_n)=1\\).\n\n(e) The pure states of \\(A\\) are norming on \\(A_+\\): \\(\\sup\\{\\rho(x):\\rho\\text{ pure}\\}=\\|x\\|\\) for \\(x\\in A_+\\). This is the background fact on pure states, which rests on the Krein–Milman theorem: the positive functionals of norm at most one form the weak\\(^*\\) closed convex hull of \\(0\\) and the pure states, and the states norm \\(A_+\\). With \\((a_n)\\) as in (a), choose pure states \\(\\omega_n\\) with \\(\\omega_n(a_n)\\ge\\|a_n\\|/2\\), and put \\(\\omega=\\sum_n2^{-n}\\omega_n\\). The estimate of (a) shows that \\(\\omega\\) is faithful.\n\n(f) By [Corollary 8.2](#oa-fnd-wa-20), \\(\\bar\\omega\\) is positive, hence a state. By (d), \\(\\varphi=\\bar\\omega_n\\) is faithful on \\(M\\), so \\(\\pi_\\varphi\\) is faithful: \\(\\pi_\\varphi(x)=0\\) gives \\(\\varphi(x^*x)=0\\). It is normal by [Proposition 12.1](#oa-fnd-wa-26). For atomicity, write \\(\\rho=\\bigoplus_n\\pi_{\\omega_n}\\) on \\(K=\\bigoplus_nK_n\\), a direct sum of irreducible representations (Lemma 7.2), and \\(N=\\rho(A)''\\).\n\n*\\(N\\) is atomic.* Let \\(Q_n\\in N'\\) be the projection onto \\(K_n\\). The compression \\(x\\mapsto xQ_n|_{K_n}\\) is a normal \\(*\\)-homomorphism of \\(N\\) whose image is a von Neumann algebra (Proposition 12.1) containing \\(\\pi_{\\omega_n}(A)\\), hence equal to \\(B(K_n)\\). Its kernel is \\(N(1-z_n)\\) with \\(z_n\\) central, so \\(Nz_n\\cong B(K_n)\\) and \\(z_n\\) is a minimal central projection. If \\(xz_n=0\\) for all \\(n\\) then \\(xQ_n=0\\) for all \\(n\\) and \\(x=0\\); so \\(\\bigvee_nz_n=1\\). Distinct minimal central projections are orthogonal, so \\(N=\\bigoplus_zNz\\) over the distinct \\(z_n\\), each summand \\(\\cong B(K)\\), in which rank-one projections are minimal. Hence \\(N\\) is atomic. The same argument shows that for every projection \\(P\\in N'\\), the algebra \\(N_P=NP|_{PK}\\) is isomorphic to \\(Nc\\) with \\(c\\) central (the kernel of \\(x\\mapsto xP\\) is \\(N(1-c)\\)), which is a direct sum of some of the summands; so \\(N_P\\) is atomic.\n\n*\\(\\pi_\\varphi(M)\\) is some \\(N_P\\).* Since \\(\\varphi\\le\\bar\\omega\\), \\(\\varphi|_A\\le\\omega\\). The vector \\(\\xi_\\varphi\\) is cyclic for \\(\\pi_\\varphi(A)\\), because \\(\\pi_\\varphi(A)\\) is \\(\\sigma\\)-weakly dense in \\(\\pi_\\varphi(M)\\); so \\(\\pi_\\varphi|_A\\) is the cyclic representation of \\(\\varphi|_A\\). By the Radon–Nikodym step in the proof of Lemma 7.2, \\(\\varphi(a)=\\langle\\pi_\\omega(a)T^{1/2}\\xi_\\omega,T^{1/2}\\xi_\\omega\\rangle\\) for some \\(0\\le T\\le1\\) in \\(\\pi_\\omega(A)'\\). So \\(\\pi_\\varphi|_A\\) is unitarily equivalent to \\(\\pi_\\omega\\) restricted to the invariant subspace \\(\\overline{\\pi_\\omega(A)T^{1/2}\\xi_\\omega}\\): two cyclic representations with the same vector state are unitarily equivalent, through \\(\\pi_1(a)\\zeta_1\\mapsto\\pi_2(a)\\zeta_2\\), which preserves inner products. In the same way \\(\\pi_\\omega\\) is \\(\\rho\\) restricted to the cyclic subspace of \\(\\bigoplus_n\\lambda_n^{1/2}\\xi_{\\omega_n}\\). Hence \\(\\pi_\\varphi|_A\\) is unitarily equivalent to \\(\\rho\\) restricted to \\(PK\\) for a projection \\(P\\in\\rho(A)'=N'\\). Therefore \\(\\pi_\\varphi(M)=\\pi_\\varphi(A)''\\cong(\\rho(A)|_{PK})''=N_P\\), which is atomic, and \\(M\\cong\\pi_\\varphi(M)\\).\n\n(g) Let \\((e_i)\\) be a maximal family of mutually orthogonal minimal projections of the atomic algebra \\(M\\); then \\(\\sum_ie_i=1\\), and every \\(x\\in M\\) is the \\(\\sigma\\)-weak limit of \\(e_Fxe_F\\), \\(e_F=\\sum_{i\\in F}e_i\\), \\(F\\) finite. By Lemma 6.1(2), \\(e_iMe_j\\) is \\(0\\) or \\(\\mathbb Cv_{ij}\\) with \\(v_{ij}^*v_{ij}=e_j\\), \\(v_{ij}v_{ij}^*=e_i\\). For \\(i\\ne j\\) with \\(v=v_{ij}\\ne0\\), \\(g=\\tfrac12(e_i+e_j+v+v^*)\\) is a projection, rank one in \\((e_i+e_j)M(e_i+e_j)\\cong M_2(\\mathbb C)\\), hence minimal in \\(M\\). So \\(v=e_i(2g-e_i-e_j)e_j\\) lies in the von Neumann algebra generated by the minimal projections, and so do all \\(e_ixe_j\\) and all \\(x\\). Finally, if \\(e\\) is minimal in \\(M\\), (b) gives \\(0\\ne a\\in A\\cap eMe=A\\cap\\mathbb Ce\\), so \\(e\\in A\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-27",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "13. Monotone closed \\(C^*\\)-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
      "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "anchor": "oa-fnd-wa-27",
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        "through_line": 1012
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      "full_conditions_and_proof": "## 13. Monotone closed \\(C^*\\)-algebras\n\nIn a von Neumann algebra every bounded increasing net of self-adjoint elements has a least upper bound. We now show that this order property, together with enough normal states, characterizes \\(W^*\\)-algebras.\n\n**Definition 13.1.** We call a \\(C^*\\)-algebra \\(A\\) *monotone closed* when each increasing, norm-bounded net of self-adjoint elements has a least upper bound among the self-adjoint elements of \\(A\\). A positive functional \\(\\omega\\) on such an \\(A\\) is *normal* if \\(\\omega(\\sup_ix_i)=\\sup_i\\omega(x_i)\\) for each such net \\((x_i)\\). We say \\(A\\) has *sufficiently many* normal positive functionals if for every nonzero \\(x\\in A_+\\) some normal positive \\(\\omega\\) has \\(\\omega(x)>0\\).\n\n**Proposition 13.2.**\n\n1. A \\(W^*\\)-algebra is monotone closed, and for it the two meanings of *normal positive functional* agree.\n2. A monotone closed \\(C^*\\)-algebra has a unit.\n\n**Proof.** (1) The first claim is the background fact on monotone nets, after adding a multiple of \\(1\\) to make the net positive. For the second, a \\(\\sigma\\)-weakly continuous positive functional preserves these suprema (background). Conversely, a positive functional that preserves them is completely additive, since the finite sums of orthogonal projections form an increasing net with supremum the full sum, and so it is \\(\\sigma\\)-weakly continuous by [Corollary 11.5](#oa-fnd-wa-25). (The background fact on positive normal maps gives a second proof.)\n\n(2) Let \\((u_i)\\) be an increasing approximate unit (background), and \\(e\\) its least upper bound. If \\(A=0\\) there is nothing to prove. Let \\(\\pi\\) be a faithful nondegenerate representation, for instance \\(\\pi_u\\). Then \\(\\pi(u_i)\\to1\\) strongly, and \\(\\pi(u_i)\\le\\pi(e)\\), so \\(\\pi(e)\\ge1\\). Take a continuous \\(g\\) on \\(\\mathbb R\\) with \\(g(0)=0\\) and \\(g=1\\) on \\([1,\\infty)\\). Then \\(g(e)\\in A\\) by the functional calculus, and \\(\\pi(g(e))=g(\\pi(e))=1\\) because the spectrum of \\(\\pi(e)\\) lies in \\([1,\\infty)\\). Since \\(\\pi\\) is faithful, \\(g(e)\\) is a unit for \\(A\\). (Then \\(e\\le1\\) and \\(\\pi(e)\\ge1\\) give \\(e=1\\).) \\(\\square\\)\n\n**Remark 13.3.** For abelian algebras monotone closedness is a property of the spectrum; see \"Where this leads\" below. We do not need this: the only abelian fact used below is proved where it is used, in Exercise 13.10(b).\n\n**Lemma 13.4.** Let \\(A\\) be monotone closed and \\(\\omega\\in A^*_+\\). If \\(\\omega\\) lies in the norm closure of the set of normal positive functionals, then \\(\\omega\\) is normal. In particular this holds if \\(\\|\\omega-\\omega_n\\|\\to0\\) for a sequence of normal positive \\(\\omega_n\\).\n\n**Proof.** Let \\((x_i)\\) be increasing and norm bounded in \\(A_h\\), with least upper bound \\(x\\), and put \\(C=\\max(\\|x\\|,\\sup_i\\|x_i\\|)\\), which is finite. Positivity gives \\(\\omega(x)\\ge\\sup_i\\omega(x_i)\\). Let \\(\\varepsilon>0\\) and choose a normal positive \\(\\omega'\\) with \\(\\|\\omega-\\omega'\\|<\\varepsilon\\), then an index \\(i_0\\) with \\(\\omega'(x_i)>\\omega'(x)-\\varepsilon\\) for \\(i\\ge i_0\\). For such \\(i\\),\n\\[\n\\begin{gathered}\n\\omega(x_i)\\\\\n\\ge\\omega'(x_i)-\\varepsilon C\\\\\n\\ge\\omega'(x)-\\varepsilon-\\varepsilon C\\\\\n\\ge\\omega(x)-\\varepsilon(1+2C).\n\\end{gathered}\n\\]\nSo \\(\\sup_i\\omega(x_i)\\ge\\omega(x)\\). \\(\\square\\)\n\n**Remark 13.5.** The constant \\(C\\) is needed, because the bound \\(\\|x_i\\|\\le\\|x\\|\\) can fail for an increasing net of self-adjoint elements: in \\(\\mathbb C\\), \\(x_1=-1\\le x_2=0\\) has supremum \\(x=0\\), whereas \\(\\|x_1\\|=1\\). The proof above uses the bound that holds for the entire net. \n\n**Proposition 13.6.** Let \\(A\\) be monotone closed and \\(\\omega\\) a normal positive functional with cyclic representation \\((\\pi,H,\\xi)\\). Then \\(\\pi(A)=\\pi(A)''\\).\n\nThe representation need not be faithful: a vector state on the first summand of \\(B(H_1)\\oplus B(H_2)\\) kills the second summand. Steps 2 and 3 prove that every bounded increasing net in the image lifts to a bounded increasing net in the appropriate central corner. This lifting justifies the use of the full monotone closure criterion.\n\n**Proof.** *Step 1: increasing nets go to strong limits.* Let \\((x_i)\\) be increasing and norm bounded in \\(A_h\\), with least upper bound \\(x\\). For a unitary \\(u\\in A\\), conjugation \\(y\\mapsto u^*yu\\) is an order automorphism of \\(A_h\\), so \\(u^*xu\\) is the least upper bound of \\((u^*x_iu)\\), and normality gives \\(\\omega(u^*x_iu)\\to\\omega(u^*xu)\\). Put \\(d_i=\\pi(x)-\\pi(x_i)\\ge0\\). Then\n\\[\n\\|d_i^{1/2}\\pi(u)\\xi\\|^2=\\omega(u^*xu)-\\omega(u^*x_iu)\\to0 .\n\\]\n\\(A\\) has a unit (Proposition 13.2(2)), so its unitaries span \\(A\\), and the vectors \\(\\pi(u)\\xi\\) span a dense subspace. The \\(d_i^{1/2}\\) are uniformly bounded, so \\(d_i^{1/2}\\to0\\) strongly, and \\(d_i=d_i^{1/2}d_i^{1/2}\\to0\\) strongly. So \\(\\pi(x_i)\\to\\pi(x)\\) strongly.\n\n*Step 2: the kernel of \\(\\pi\\) is cut out by a central projection.* Let \\(J=\\ker\\pi\\). By Step 1, if an increasing, norm-bounded net of self-adjoint elements of \\(J\\) has least upper bound \\(x\\) in \\(A\\), then \\(\\pi(x)=0\\), so \\(x\\in J\\). Let \\((u_\\lambda)\\) be an increasing approximate unit of the \\(C^*\\)-algebra \\(J\\) (background), and \\(p\\in J\\) its least upper bound in \\(A\\); then \\(0\\le p\\le1\\). Every \\(a\\in J_+\\) with \\(\\|a\\|\\le1\\) satisfies \\(a\\le p\\): indeed \\(u_\\lambda au_\\lambda\\le u_\\lambda^2\\le u_\\lambda\\le p\\), and \\(u_\\lambda au_\\lambda\\to a\\) in norm. Taking \\(a=p^{1/2}\\in J\\) gives \\(p^{1/2}\\le p\\); with \\(p\\le p^{1/2}\\), which holds for \\(0\\le p\\le1\\), we get \\(p=p^{1/2}\\), so \\(p\\) is a projection. For \\(a\\in J_+\\) with \\(\\|a\\|\\le1\\), \\(0\\le a\\le p\\) now gives \\((1-p)a(1-p)=0\\), so \\(a^{1/2}(1-p)=0\\) and \\(a=ap=pa\\); by linearity \\(x=xp=px\\) for all \\(x\\in J\\). So \\(J=Ap=pA\\), and \\(p\\) is central: for \\(y\\in A\\), \\(yp\\in J\\) and \\(py\\in J\\) give \\(yp=pyp=py\\). Put \\(z=1-p\\).\n\n*Step 3: monotone nets in \\(\\pi(A)\\) lift.* \\(\\pi(A)=\\pi(Az)\\), and \\(\\pi\\) is injective on \\(Az\\), so it is an order isomorphism of \\((Az)_h\\) onto \\(\\pi(A)_h\\): an injective \\(*\\)-homomorphism preserves spectra. \\(Az\\) is monotone closed with the same least upper bounds as \\(A\\): if \\((c_i)\\) is bounded increasing in \\((Az)_h\\) with least upper bound \\(c\\) in \\(A\\), then \\(zcz\\) is an upper bound, so \\(c\\le zcz\\), that is \\((1-z)c\\le0\\); and \\[\n\\begin{gathered}\n(1-z)c\\\\\n=(1-z)c(1-z)\\\\\n\\ge(1-z)c_i(1-z)\\\\\n=0.\n\\end{gathered}\n\\] So \\(c=zc\\in Az\\). Now let \\((T_i)\\) be increasing and norm bounded in \\(\\pi(A)_h\\). Then \\(T_i=\\pi(y_i)\\) for a unique increasing, norm-bounded net \\((y_i)\\) in \\((Az)_h\\), with least upper bound \\(y\\in Az\\), and \\(T_i\\to\\pi(y)\\) strongly by Step 1.\n\n*Step 4.* \\(\\pi(A)\\) is a nondegenerate \\(C^*\\)-algebra of operators, since \\(\\pi(1)=1\\), and by Step 3 the strong limit of each increasing, norm-bounded net of its self-adjoint elements lies in it. The full [monotone closure criterion, Corollary 12.10 of the density lesson](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-14), now gives \\(\\pi(A)=\\pi(A)''\\). \\(\\square\\)\n\n**Theorem 13.7** (Kadison's characterization). A \\(C^*\\)-algebra is a \\(W^*\\)-algebra exactly when it is monotone closed and has sufficiently many normal positive functionals.\n\n\n**Proof.** If \\(A\\) is a \\(W^*\\)-algebra, it is monotone closed by Proposition 13.2(1), and the vector states of a faithful normal realisation are normal (Corollary 11.4) and separate the positive elements. Conversely, let \\(\\pi=\\bigoplus_\\omega\\pi_\\omega\\), over all normal positive \\(\\omega\\). It is nondegenerate, and faithful: if \\(\\pi(x)=0\\) then \\(\\omega(x^*x)=\\|\\pi_\\omega(x)\\xi_\\omega\\|^2=0\\) for every normal \\(\\omega\\), so \\(x^*x=0\\). If \\((x_i)\\) in \\(A_h\\) is increasing and norm bounded, with least upper bound \\(x\\), Step 1 of the proof of Proposition 13.6 gives \\(\\pi_\\omega(x_i)\\to\\pi_\\omega(x)\\) strongly on each summand; the net is uniformly bounded, so \\(\\pi(x_i)\\to\\pi(x)\\) strongly. Since \\(\\pi\\) is faithful, it is an order isomorphism onto its image, so every increasing norm-bounded net in \\(\\pi(A)_h\\) comes from one in \\(A_h\\), and its strong limit lies in \\(\\pi(A)\\). The same full [monotone closure criterion](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-14) gives \\(\\pi(A)=\\pi(A)''\\), so \\(A\\) is a \\(W^*\\)-algebra. \\(\\square\\)\n\n**Remark 13.8** (Unused examples, not constructed or proved here). The second hypothesis is needed: there are monotone closed \\(C^*\\)-algebras that are not \\(W^*\\)-algebras. Dixmier's classical examples are abelian: \\(C(\\Omega)\\) for a stonean space \\(\\Omega\\) that carries too few normal measures [Dixmier 1951]. We do not construct them here.\n\n**Exercise 13.9.** (easy) Show that a von Neumann algebra that is separable in the norm topology is finite-dimensional.\n\n**Solution.** By [Lemma 6.2](#oa-fnd-wa-13), an infinite-dimensional von Neumann algebra contains an isometric copy of \\(\\ell^\\infty\\), which is not separable; a subset of a separable metric space is separable. \\(\\square\\)\n\n**Exercise 13.10** (hard) (Ranges of projections of norm one). Let \\(A\\) be a \\(C^*\\)-algebra acting on \\(H\\), let \\(M\\supseteq A\\) be a von Neumann algebra acting on \\(H\\), and let \\(\\varepsilon:M\\to A\\) be a projection of norm one. (a) Prove that each increasing, norm-bounded net \\((h_i)\\) of positive elements of \\(A\\) has a least upper bound in \\(A\\), and that this bound need not be the strong limit. (b) Prove that a separable such \\(A\\) is finite-dimensional.\n\n**Solution.** (a) Let \\(h\\) be the strong limit of \\((h_i)\\), which is its least upper bound in \\(M\\) (background fact on monotone nets), and put \\(k=\\varepsilon(h)\\). By Tomiyama's theorem ([Theorem 8.5](#oa-fnd-wa-21)) \\(\\varepsilon\\) is positive, so \\(h_i=\\varepsilon(h_i)\\le\\varepsilon(h)=k\\). If \\(a\\in A_h\\) is an upper bound, then \\(h\\le a\\) in \\(M\\), so \\(k=\\varepsilon(h)\\le\\varepsilon(a)=a\\). So \\(k\\) is the least upper bound in \\(A\\).\n\nIt can differ from the strong limit. Let \\(I\\) be the set of all ultrafilters on \\(\\mathbb N\\), including the principal ones \\(i_n\\). For \\(a\\in\\ell^\\infty\\), define \\(\\chi_i(a)=\\lim_{n\\to i}a_n\\). This limit exists uniquely in the compact closed disk containing the values of \\(a\\), by the ultrafilter compactness criterion. Continuity of scalar addition, multiplication and conjugation makes each \\(\\chi_i\\) a unital \\(*\\)-homomorphism. On \\(H=\\ell^2(I)\\), set\n\\[\n\\pi(a)\\delta_i=\\chi_i(a)\\delta_i,\\qquad\nM=\\ell^\\infty(I).\n\\]\nPrincipal ultrafilters give \\(\\|\\pi(a)\\|=\\|a\\|_\\infty\\), so \\(A=\\pi(\\ell^\\infty)\\) is a unital norm-closed \\(C^*\\)-subalgebra of the diagonal von Neumann algebra \\(M\\). Define\n\\[\n\\varepsilon(g)=\\pi((g(i_n))_{n\\ge1})\\qquad(g\\in M).\n\\]\nIt is contractive, fixes \\(A\\), and fixes \\(1\\), so it is a projection of norm one. For finite \\(F\\subseteq\\mathbb N\\), \\(\\pi(1_F)\\) is the diagonal projection on the principal coordinates \\(i_n\\) with \\(n\\in F\\): a free ultrafilter contains no finite set. These projections increase strongly to the projection \\(h\\) on the closed span of all principal coordinates. A free ultrafilter exists by (30), so \\(h\\ne1\\). But their least upper bound in \\(A\\) is \\(1\\): any upper bound \\(\\pi(a)\\) has \\(a_n\\ge1\\) on every principal coordinate, whence \\(\\chi_i(a)\\ge1\\) on all coordinates. Thus the least upper bound in \\(A\\) differs from the strong limit. This construction requires no Stone–Čech compactification theorem.\n\n(b) We may assume \\(A\\ne0\\). By Tomiyama's theorem, \\(A\\) has the unit \\(1_A=\\varepsilon(1)\\). By (a), after adding multiples of \\(1_A\\), every increasing norm-bounded net in \\(A_h\\) has a least upper bound in \\(A\\), namely \\(\\varepsilon\\) of its strong limit.\n\nSuppose \\(\\dim A=\\infty\\). Let \\(C\\) be a maximal abelian \\(*\\)-subalgebra of \\(A\\); it is norm closed and contains \\(1_A\\). It is infinite-dimensional: otherwise, as in the proof of Lemma 6.2, \\(C\\) is spanned by minimal projections \\(q_1,\\dots,q_m\\) with sum \\(1_A\\), each minimal in \\(A\\) by maximality, and \\(\\dim A\\le m^2\\) by Lemma 6.1(2). Least upper bounds of nets from \\(C\\) stay in \\(C\\): if \\(c\\) is the least upper bound of an increasing norm-bounded net in \\(C_h\\), and \\(u\\in C\\) is unitary, then \\(ucu^*\\) is the least upper bound of the same net, so \\(ucu^*=c\\). The unitaries span \\(C\\), so \\(c\\) commutes with \\(C\\), and by maximality \\(c\\in C\\).\n\n*\\(C\\) has a projection other than \\(0\\) and \\(1_A\\).* Take \\(h\\in C_h\\), \\(0\\le h\\le1_A\\), not a multiple of \\(1_A\\), with spectral values \\(s_1<s_2\\), and \\(t\\in(s_1,s_2)\\). Put \\(k=(h-t)_+\\) and \\(k'=(t-h)_+\\), both nonzero, with \\(kk'=0\\). The elements \\(a_n=(k/\\|k\\|)^{1/n}\\) and \\(b_m=(k'/\\|k'\\|)^{1/m}\\) increase in \\(n\\) and \\(m\\), with least upper bounds \\(p,p'\\in C\\). Each \\(a_n=a_{2n}^2\\le p^2\\), since \\(a_{2n}\\le p\\) and they commute; so \\(p\\le p^2\\le p\\), and \\(p\\) is a projection; likewise \\(p'\\). The functions of \\(h\\) behind \\(a_n\\) and \\(b_m\\) have disjoint supports and values in \\([0,1]\\), so \\(a_n\\le1_A-b_m\\). Hence \\(p\\le1_A-b_m\\) for all \\(m\\), then \\(p'\\le1_A-p\\), and \\(pp'=0\\). As \\(p\\ge a_1\\ne0\\) and \\(p'\\ne0\\), \\(p\\notin\\{0,1_A\\}\\).\n\nThe corners \\(pC\\) and \\((1_A-p)C\\) are again unital and abelian, and closed under these least upper bounds: if an increasing net in \\((pC)_h\\) is bounded in norm by \\(K\\), its least upper bound \\(c\\in C\\) satisfies \\(-Kp\\le c\\le Kp\\), so \\((1_A-p)c=(1_A-p)c(1_A-p)=0\\) and \\(c\\in pC\\). One of the two corners is infinite-dimensional. Repeating the argument inside it produces nonzero, mutually orthogonal projections \\(q_1,q_2,\\ldots\\) in \\(C\\). For \\(S\\subseteq\\mathbb N\\), let \\(Q_S\\) be the strong sum of the \\(q_n\\), \\(n\\in S\\), and \\(P_S=\\varepsilon(Q_S)\\in A\\). If \\(n\\in S\\setminus T\\), bimodularity gives \\(q_nP_Sq_n=\\varepsilon(q_nQ_Sq_n)=q_n\\) and \\(q_nP_Tq_n=\\varepsilon(q_nQ_Tq_n)=0\\). So \\(\\|P_S-P_T\\|\\ge1\\) for \\(S\\ne T\\): uncountably many elements at mutual distance at least one. So \\(A\\) is not separable. \\(\\square\\)\n\n**Exercise 13.11** (medium) (Algebras that are not dual spaces). Show that neither \\(c_0\\) nor \\(C[0,1]\\) is isometrically isomorphic to the dual of a Banach space.\n\n**Solution.** By Sakai's theorem ([Theorem 9.2](#oa-fnd-wa-22)) such an algebra would be a \\(W^*\\)-algebra, hence unital and monotone closed ([Proposition 13.2](#oa-fnd-wa-27)(1)). \\(c_0\\) has no unit. In \\(C[0,1]\\), the increasing sequence \\(f_n(t)=\\min(1,\\max(0,n(t-\\tfrac12)))\\) has no least upper bound. An upper bound \\(g\\) satisfies \\(g\\ge1\\) on \\((\\tfrac12,1]\\), hence on \\([\\tfrac12,1]\\), and \\(g\\ge0\\). Then \\(g>\\tfrac12\\) on some interval \\([\\tfrac12-\\delta,\\tfrac12]\\); subtracting from \\(g\\) a continuous bump \\(b\\) with \\(0\\le b\\le\\tfrac12\\), supported in \\((\\tfrac12-\\delta,\\tfrac12)\\), gives a strictly smaller upper bound, because \\(f_n=0\\) on \\([0,\\tfrac12]\\). \\(\\square\\)\n\n## Where this leads\n\nThe following extensions are not proved here and are not used in the proofs or solutions above. The polar-functional lesson proves the first direction later in the course.\n\n- **Polar decomposition of functionals.** Remark 3.5 is the hermitian case of a general fact: every normal functional \\(\\varphi\\) on a von Neumann algebra can be written \\(\\varphi=u|\\varphi|\\) in the notation (0.1), with \\(|\\varphi|\\) a positive normal functional of the same norm and \\(u\\) a partial isometry.\n- **Abelian algebras.** An abelian \\(C^*\\)-algebra \\(C(\\Omega)\\) is monotone closed exactly when \\(\\Omega\\) is stonean, and its normal positive functionals then correspond to normal measures. It is a \\(W^*\\)-algebra exactly when, in addition, \\(\\Omega\\) carries enough normal measures, that is, when \\(\\Omega\\) is hyperstonean [Dixmier 1951].\n\n## References\n\n- [Kostecki] R. P. Kostecki, *W\\*-algebras and noncommutative integration*, lecture notes, 2013, arXiv:1307.4818, https://arxiv.org/abs/1307.4818v5.\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n- [Dixmier 1951] J. Dixmier, *Sur certains espaces considérés par M. H. Stone*, Summa Brasiliensis Mathematicae 2 (1951), 151–182, [freely readable scan](https://dmitripavlov.org/scans/dixmier.pdf).\n- [Sakai 1956] S. Sakai, A characterization of W\\*-algebras, *Pacific Journal of Mathematics* 6 (1956), 763–773. https://doi.org/10.2140/pjm.1956.6.763 Freely readable [original journal PDF](https://msp.org/pjm/1956/6-4/pjm-v6-n4-p11-p.pdf).\n\n*Freely accessible reading:* [Shoichiro Sakai, *A characterization of W*-algebras*, printed pp. 763–773](https://msp.org/pjm/1956/6-4/pjm-v6-n4-p11-p.pdf) gives a route through the original abstract characterization by normal-state representations; this lesson uses the complete Tomiyama/bidual route. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-WA-28",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "13. Monotone closed \\(C^*\\)-algebras",
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      "full_conditions_and_proof": "## 13. Monotone closed \\(C^*\\)-algebras\n\nIn a von Neumann algebra every bounded increasing net of self-adjoint elements has a least upper bound. We now show that this order property, together with enough normal states, characterizes \\(W^*\\)-algebras.\n\n**Definition 13.1.** We call a \\(C^*\\)-algebra \\(A\\) *monotone closed* when each increasing, norm-bounded net of self-adjoint elements has a least upper bound among the self-adjoint elements of \\(A\\). A positive functional \\(\\omega\\) on such an \\(A\\) is *normal* if \\(\\omega(\\sup_ix_i)=\\sup_i\\omega(x_i)\\) for each such net \\((x_i)\\). We say \\(A\\) has *sufficiently many* normal positive functionals if for every nonzero \\(x\\in A_+\\) some normal positive \\(\\omega\\) has \\(\\omega(x)>0\\).\n\n**Proposition 13.2.**\n\n1. A \\(W^*\\)-algebra is monotone closed, and for it the two meanings of *normal positive functional* agree.\n2. A monotone closed \\(C^*\\)-algebra has a unit.\n\n**Proof.** (1) The first claim is the background fact on monotone nets, after adding a multiple of \\(1\\) to make the net positive. For the second, a \\(\\sigma\\)-weakly continuous positive functional preserves these suprema (background). Conversely, a positive functional that preserves them is completely additive, since the finite sums of orthogonal projections form an increasing net with supremum the full sum, and so it is \\(\\sigma\\)-weakly continuous by [Corollary 11.5](#oa-fnd-wa-25). (The background fact on positive normal maps gives a second proof.)\n\n(2) Let \\((u_i)\\) be an increasing approximate unit (background), and \\(e\\) its least upper bound. If \\(A=0\\) there is nothing to prove. Let \\(\\pi\\) be a faithful nondegenerate representation, for instance \\(\\pi_u\\). Then \\(\\pi(u_i)\\to1\\) strongly, and \\(\\pi(u_i)\\le\\pi(e)\\), so \\(\\pi(e)\\ge1\\). Take a continuous \\(g\\) on \\(\\mathbb R\\) with \\(g(0)=0\\) and \\(g=1\\) on \\([1,\\infty)\\). Then \\(g(e)\\in A\\) by the functional calculus, and \\(\\pi(g(e))=g(\\pi(e))=1\\) because the spectrum of \\(\\pi(e)\\) lies in \\([1,\\infty)\\). Since \\(\\pi\\) is faithful, \\(g(e)\\) is a unit for \\(A\\). (Then \\(e\\le1\\) and \\(\\pi(e)\\ge1\\) give \\(e=1\\).) \\(\\square\\)\n\n**Remark 13.3.** For abelian algebras monotone closedness is a property of the spectrum; see \"Where this leads\" below. We do not need this: the only abelian fact used below is proved where it is used, in Exercise 13.10(b).\n\n**Lemma 13.4.** Let \\(A\\) be monotone closed and \\(\\omega\\in A^*_+\\). If \\(\\omega\\) lies in the norm closure of the set of normal positive functionals, then \\(\\omega\\) is normal. In particular this holds if \\(\\|\\omega-\\omega_n\\|\\to0\\) for a sequence of normal positive \\(\\omega_n\\).\n\n**Proof.** Let \\((x_i)\\) be increasing and norm bounded in \\(A_h\\), with least upper bound \\(x\\), and put \\(C=\\max(\\|x\\|,\\sup_i\\|x_i\\|)\\), which is finite. Positivity gives \\(\\omega(x)\\ge\\sup_i\\omega(x_i)\\). Let \\(\\varepsilon>0\\) and choose a normal positive \\(\\omega'\\) with \\(\\|\\omega-\\omega'\\|<\\varepsilon\\), then an index \\(i_0\\) with \\(\\omega'(x_i)>\\omega'(x)-\\varepsilon\\) for \\(i\\ge i_0\\). For such \\(i\\),\n\\[\n\\begin{gathered}\n\\omega(x_i)\\\\\n\\ge\\omega'(x_i)-\\varepsilon C\\\\\n\\ge\\omega'(x)-\\varepsilon-\\varepsilon C\\\\\n\\ge\\omega(x)-\\varepsilon(1+2C).\n\\end{gathered}\n\\]\nSo \\(\\sup_i\\omega(x_i)\\ge\\omega(x)\\). \\(\\square\\)\n\n**Remark 13.5.** The constant \\(C\\) is needed, because the bound \\(\\|x_i\\|\\le\\|x\\|\\) can fail for an increasing net of self-adjoint elements: in \\(\\mathbb C\\), \\(x_1=-1\\le x_2=0\\) has supremum \\(x=0\\), whereas \\(\\|x_1\\|=1\\). The proof above uses the bound that holds for the entire net. \n\n**Proposition 13.6.** Let \\(A\\) be monotone closed and \\(\\omega\\) a normal positive functional with cyclic representation \\((\\pi,H,\\xi)\\). Then \\(\\pi(A)=\\pi(A)''\\).\n\nThe representation need not be faithful: a vector state on the first summand of \\(B(H_1)\\oplus B(H_2)\\) kills the second summand. Steps 2 and 3 prove that every bounded increasing net in the image lifts to a bounded increasing net in the appropriate central corner. This lifting justifies the use of the full monotone closure criterion.\n\n**Proof.** *Step 1: increasing nets go to strong limits.* Let \\((x_i)\\) be increasing and norm bounded in \\(A_h\\), with least upper bound \\(x\\). For a unitary \\(u\\in A\\), conjugation \\(y\\mapsto u^*yu\\) is an order automorphism of \\(A_h\\), so \\(u^*xu\\) is the least upper bound of \\((u^*x_iu)\\), and normality gives \\(\\omega(u^*x_iu)\\to\\omega(u^*xu)\\). Put \\(d_i=\\pi(x)-\\pi(x_i)\\ge0\\). Then\n\\[\n\\|d_i^{1/2}\\pi(u)\\xi\\|^2=\\omega(u^*xu)-\\omega(u^*x_iu)\\to0 .\n\\]\n\\(A\\) has a unit (Proposition 13.2(2)), so its unitaries span \\(A\\), and the vectors \\(\\pi(u)\\xi\\) span a dense subspace. The \\(d_i^{1/2}\\) are uniformly bounded, so \\(d_i^{1/2}\\to0\\) strongly, and \\(d_i=d_i^{1/2}d_i^{1/2}\\to0\\) strongly. So \\(\\pi(x_i)\\to\\pi(x)\\) strongly.\n\n*Step 2: the kernel of \\(\\pi\\) is cut out by a central projection.* Let \\(J=\\ker\\pi\\). By Step 1, if an increasing, norm-bounded net of self-adjoint elements of \\(J\\) has least upper bound \\(x\\) in \\(A\\), then \\(\\pi(x)=0\\), so \\(x\\in J\\). Let \\((u_\\lambda)\\) be an increasing approximate unit of the \\(C^*\\)-algebra \\(J\\) (background), and \\(p\\in J\\) its least upper bound in \\(A\\); then \\(0\\le p\\le1\\). Every \\(a\\in J_+\\) with \\(\\|a\\|\\le1\\) satisfies \\(a\\le p\\): indeed \\(u_\\lambda au_\\lambda\\le u_\\lambda^2\\le u_\\lambda\\le p\\), and \\(u_\\lambda au_\\lambda\\to a\\) in norm. Taking \\(a=p^{1/2}\\in J\\) gives \\(p^{1/2}\\le p\\); with \\(p\\le p^{1/2}\\), which holds for \\(0\\le p\\le1\\), we get \\(p=p^{1/2}\\), so \\(p\\) is a projection. For \\(a\\in J_+\\) with \\(\\|a\\|\\le1\\), \\(0\\le a\\le p\\) now gives \\((1-p)a(1-p)=0\\), so \\(a^{1/2}(1-p)=0\\) and \\(a=ap=pa\\); by linearity \\(x=xp=px\\) for all \\(x\\in J\\). So \\(J=Ap=pA\\), and \\(p\\) is central: for \\(y\\in A\\), \\(yp\\in J\\) and \\(py\\in J\\) give \\(yp=pyp=py\\). Put \\(z=1-p\\).\n\n*Step 3: monotone nets in \\(\\pi(A)\\) lift.* \\(\\pi(A)=\\pi(Az)\\), and \\(\\pi\\) is injective on \\(Az\\), so it is an order isomorphism of \\((Az)_h\\) onto \\(\\pi(A)_h\\): an injective \\(*\\)-homomorphism preserves spectra. \\(Az\\) is monotone closed with the same least upper bounds as \\(A\\): if \\((c_i)\\) is bounded increasing in \\((Az)_h\\) with least upper bound \\(c\\) in \\(A\\), then \\(zcz\\) is an upper bound, so \\(c\\le zcz\\), that is \\((1-z)c\\le0\\); and \\[\n\\begin{gathered}\n(1-z)c\\\\\n=(1-z)c(1-z)\\\\\n\\ge(1-z)c_i(1-z)\\\\\n=0.\n\\end{gathered}\n\\] So \\(c=zc\\in Az\\). Now let \\((T_i)\\) be increasing and norm bounded in \\(\\pi(A)_h\\). Then \\(T_i=\\pi(y_i)\\) for a unique increasing, norm-bounded net \\((y_i)\\) in \\((Az)_h\\), with least upper bound \\(y\\in Az\\), and \\(T_i\\to\\pi(y)\\) strongly by Step 1.\n\n*Step 4.* \\(\\pi(A)\\) is a nondegenerate \\(C^*\\)-algebra of operators, since \\(\\pi(1)=1\\), and by Step 3 the strong limit of each increasing, norm-bounded net of its self-adjoint elements lies in it. The full [monotone closure criterion, Corollary 12.10 of the density lesson](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-14), now gives \\(\\pi(A)=\\pi(A)''\\). \\(\\square\\)\n\n**Theorem 13.7** (Kadison's characterization). A \\(C^*\\)-algebra is a \\(W^*\\)-algebra exactly when it is monotone closed and has sufficiently many normal positive functionals.\n\n\n**Proof.** If \\(A\\) is a \\(W^*\\)-algebra, it is monotone closed by Proposition 13.2(1), and the vector states of a faithful normal realisation are normal (Corollary 11.4) and separate the positive elements. Conversely, let \\(\\pi=\\bigoplus_\\omega\\pi_\\omega\\), over all normal positive \\(\\omega\\). It is nondegenerate, and faithful: if \\(\\pi(x)=0\\) then \\(\\omega(x^*x)=\\|\\pi_\\omega(x)\\xi_\\omega\\|^2=0\\) for every normal \\(\\omega\\), so \\(x^*x=0\\). If \\((x_i)\\) in \\(A_h\\) is increasing and norm bounded, with least upper bound \\(x\\), Step 1 of the proof of Proposition 13.6 gives \\(\\pi_\\omega(x_i)\\to\\pi_\\omega(x)\\) strongly on each summand; the net is uniformly bounded, so \\(\\pi(x_i)\\to\\pi(x)\\) strongly. Since \\(\\pi\\) is faithful, it is an order isomorphism onto its image, so every increasing norm-bounded net in \\(\\pi(A)_h\\) comes from one in \\(A_h\\), and its strong limit lies in \\(\\pi(A)\\). The same full [monotone closure criterion](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-14) gives \\(\\pi(A)=\\pi(A)''\\), so \\(A\\) is a \\(W^*\\)-algebra. \\(\\square\\)\n\n**Remark 13.8** (Unused examples, not constructed or proved here). The second hypothesis is needed: there are monotone closed \\(C^*\\)-algebras that are not \\(W^*\\)-algebras. Dixmier's classical examples are abelian: \\(C(\\Omega)\\) for a stonean space \\(\\Omega\\) that carries too few normal measures [Dixmier 1951]. We do not construct them here.\n\n**Exercise 13.9.** (easy) Show that a von Neumann algebra that is separable in the norm topology is finite-dimensional.\n\n**Solution.** By [Lemma 6.2](#oa-fnd-wa-13), an infinite-dimensional von Neumann algebra contains an isometric copy of \\(\\ell^\\infty\\), which is not separable; a subset of a separable metric space is separable. \\(\\square\\)\n\n**Exercise 13.10** (hard) (Ranges of projections of norm one). Let \\(A\\) be a \\(C^*\\)-algebra acting on \\(H\\), let \\(M\\supseteq A\\) be a von Neumann algebra acting on \\(H\\), and let \\(\\varepsilon:M\\to A\\) be a projection of norm one. (a) Prove that each increasing, norm-bounded net \\((h_i)\\) of positive elements of \\(A\\) has a least upper bound in \\(A\\), and that this bound need not be the strong limit. (b) Prove that a separable such \\(A\\) is finite-dimensional.\n\n**Solution.** (a) Let \\(h\\) be the strong limit of \\((h_i)\\), which is its least upper bound in \\(M\\) (background fact on monotone nets), and put \\(k=\\varepsilon(h)\\). By Tomiyama's theorem ([Theorem 8.5](#oa-fnd-wa-21)) \\(\\varepsilon\\) is positive, so \\(h_i=\\varepsilon(h_i)\\le\\varepsilon(h)=k\\). If \\(a\\in A_h\\) is an upper bound, then \\(h\\le a\\) in \\(M\\), so \\(k=\\varepsilon(h)\\le\\varepsilon(a)=a\\). So \\(k\\) is the least upper bound in \\(A\\).\n\nIt can differ from the strong limit. Let \\(I\\) be the set of all ultrafilters on \\(\\mathbb N\\), including the principal ones \\(i_n\\). For \\(a\\in\\ell^\\infty\\), define \\(\\chi_i(a)=\\lim_{n\\to i}a_n\\). This limit exists uniquely in the compact closed disk containing the values of \\(a\\), by the ultrafilter compactness criterion. Continuity of scalar addition, multiplication and conjugation makes each \\(\\chi_i\\) a unital \\(*\\)-homomorphism. On \\(H=\\ell^2(I)\\), set\n\\[\n\\pi(a)\\delta_i=\\chi_i(a)\\delta_i,\\qquad\nM=\\ell^\\infty(I).\n\\]\nPrincipal ultrafilters give \\(\\|\\pi(a)\\|=\\|a\\|_\\infty\\), so \\(A=\\pi(\\ell^\\infty)\\) is a unital norm-closed \\(C^*\\)-subalgebra of the diagonal von Neumann algebra \\(M\\). Define\n\\[\n\\varepsilon(g)=\\pi((g(i_n))_{n\\ge1})\\qquad(g\\in M).\n\\]\nIt is contractive, fixes \\(A\\), and fixes \\(1\\), so it is a projection of norm one. For finite \\(F\\subseteq\\mathbb N\\), \\(\\pi(1_F)\\) is the diagonal projection on the principal coordinates \\(i_n\\) with \\(n\\in F\\): a free ultrafilter contains no finite set. These projections increase strongly to the projection \\(h\\) on the closed span of all principal coordinates. A free ultrafilter exists by (30), so \\(h\\ne1\\). But their least upper bound in \\(A\\) is \\(1\\): any upper bound \\(\\pi(a)\\) has \\(a_n\\ge1\\) on every principal coordinate, whence \\(\\chi_i(a)\\ge1\\) on all coordinates. Thus the least upper bound in \\(A\\) differs from the strong limit. This construction requires no Stone–Čech compactification theorem.\n\n(b) We may assume \\(A\\ne0\\). By Tomiyama's theorem, \\(A\\) has the unit \\(1_A=\\varepsilon(1)\\). By (a), after adding multiples of \\(1_A\\), every increasing norm-bounded net in \\(A_h\\) has a least upper bound in \\(A\\), namely \\(\\varepsilon\\) of its strong limit.\n\nSuppose \\(\\dim A=\\infty\\). Let \\(C\\) be a maximal abelian \\(*\\)-subalgebra of \\(A\\); it is norm closed and contains \\(1_A\\). It is infinite-dimensional: otherwise, as in the proof of Lemma 6.2, \\(C\\) is spanned by minimal projections \\(q_1,\\dots,q_m\\) with sum \\(1_A\\), each minimal in \\(A\\) by maximality, and \\(\\dim A\\le m^2\\) by Lemma 6.1(2). Least upper bounds of nets from \\(C\\) stay in \\(C\\): if \\(c\\) is the least upper bound of an increasing norm-bounded net in \\(C_h\\), and \\(u\\in C\\) is unitary, then \\(ucu^*\\) is the least upper bound of the same net, so \\(ucu^*=c\\). The unitaries span \\(C\\), so \\(c\\) commutes with \\(C\\), and by maximality \\(c\\in C\\).\n\n*\\(C\\) has a projection other than \\(0\\) and \\(1_A\\).* Take \\(h\\in C_h\\), \\(0\\le h\\le1_A\\), not a multiple of \\(1_A\\), with spectral values \\(s_1<s_2\\), and \\(t\\in(s_1,s_2)\\). Put \\(k=(h-t)_+\\) and \\(k'=(t-h)_+\\), both nonzero, with \\(kk'=0\\). The elements \\(a_n=(k/\\|k\\|)^{1/n}\\) and \\(b_m=(k'/\\|k'\\|)^{1/m}\\) increase in \\(n\\) and \\(m\\), with least upper bounds \\(p,p'\\in C\\). Each \\(a_n=a_{2n}^2\\le p^2\\), since \\(a_{2n}\\le p\\) and they commute; so \\(p\\le p^2\\le p\\), and \\(p\\) is a projection; likewise \\(p'\\). The functions of \\(h\\) behind \\(a_n\\) and \\(b_m\\) have disjoint supports and values in \\([0,1]\\), so \\(a_n\\le1_A-b_m\\). Hence \\(p\\le1_A-b_m\\) for all \\(m\\), then \\(p'\\le1_A-p\\), and \\(pp'=0\\). As \\(p\\ge a_1\\ne0\\) and \\(p'\\ne0\\), \\(p\\notin\\{0,1_A\\}\\).\n\nThe corners \\(pC\\) and \\((1_A-p)C\\) are again unital and abelian, and closed under these least upper bounds: if an increasing net in \\((pC)_h\\) is bounded in norm by \\(K\\), its least upper bound \\(c\\in C\\) satisfies \\(-Kp\\le c\\le Kp\\), so \\((1_A-p)c=(1_A-p)c(1_A-p)=0\\) and \\(c\\in pC\\). One of the two corners is infinite-dimensional. Repeating the argument inside it produces nonzero, mutually orthogonal projections \\(q_1,q_2,\\ldots\\) in \\(C\\). For \\(S\\subseteq\\mathbb N\\), let \\(Q_S\\) be the strong sum of the \\(q_n\\), \\(n\\in S\\), and \\(P_S=\\varepsilon(Q_S)\\in A\\). If \\(n\\in S\\setminus T\\), bimodularity gives \\(q_nP_Sq_n=\\varepsilon(q_nQ_Sq_n)=q_n\\) and \\(q_nP_Tq_n=\\varepsilon(q_nQ_Tq_n)=0\\). So \\(\\|P_S-P_T\\|\\ge1\\) for \\(S\\ne T\\): uncountably many elements at mutual distance at least one. So \\(A\\) is not separable. \\(\\square\\)\n\n**Exercise 13.11** (medium) (Algebras that are not dual spaces). Show that neither \\(c_0\\) nor \\(C[0,1]\\) is isometrically isomorphic to the dual of a Banach space.\n\n**Solution.** By Sakai's theorem ([Theorem 9.2](#oa-fnd-wa-22)) such an algebra would be a \\(W^*\\)-algebra, hence unital and monotone closed ([Proposition 13.2](#oa-fnd-wa-27)(1)). \\(c_0\\) has no unit. In \\(C[0,1]\\), the increasing sequence \\(f_n(t)=\\min(1,\\max(0,n(t-\\tfrac12)))\\) has no least upper bound. An upper bound \\(g\\) satisfies \\(g\\ge1\\) on \\((\\tfrac12,1]\\), hence on \\([\\tfrac12,1]\\), and \\(g\\ge0\\). Then \\(g>\\tfrac12\\) on some interval \\([\\tfrac12-\\delta,\\tfrac12]\\); subtracting from \\(g\\) a continuous bump \\(b\\) with \\(0\\le b\\le\\tfrac12\\), supported in \\((\\tfrac12-\\delta,\\tfrac12)\\), gives a strictly smaller upper bound, because \\(f_n=0\\) on \\([0,\\tfrac12]\\). \\(\\square\\)\n\n## Where this leads\n\nThe following extensions are not proved here and are not used in the proofs or solutions above. The polar-functional lesson proves the first direction later in the course.\n\n- **Polar decomposition of functionals.** Remark 3.5 is the hermitian case of a general fact: every normal functional \\(\\varphi\\) on a von Neumann algebra can be written \\(\\varphi=u|\\varphi|\\) in the notation (0.1), with \\(|\\varphi|\\) a positive normal functional of the same norm and \\(u\\) a partial isometry.\n- **Abelian algebras.** An abelian \\(C^*\\)-algebra \\(C(\\Omega)\\) is monotone closed exactly when \\(\\Omega\\) is stonean, and its normal positive functionals then correspond to normal measures. It is a \\(W^*\\)-algebra exactly when, in addition, \\(\\Omega\\) carries enough normal measures, that is, when \\(\\Omega\\) is hyperstonean [Dixmier 1951].\n\n## References\n\n- [Kostecki] R. P. Kostecki, *W\\*-algebras and noncommutative integration*, lecture notes, 2013, arXiv:1307.4818, https://arxiv.org/abs/1307.4818v5.\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n- [Dixmier 1951] J. Dixmier, *Sur certains espaces considérés par M. H. Stone*, Summa Brasiliensis Mathematicae 2 (1951), 151–182, [freely readable scan](https://dmitripavlov.org/scans/dixmier.pdf).\n- [Sakai 1956] S. Sakai, A characterization of W\\*-algebras, *Pacific Journal of Mathematics* 6 (1956), 763–773. https://doi.org/10.2140/pjm.1956.6.763 Freely readable [original journal PDF](https://msp.org/pjm/1956/6-4/pjm-v6-n4-p11-p.pdf).\n\n*Freely accessible reading:* [Shoichiro Sakai, *A characterization of W*-algebras*, printed pp. 763–773](https://msp.org/pjm/1956/6-4/pjm-v6-n4-p11-p.pdf) gives a route through the original abstract characterization by normal-state representations; this lesson uses the complete Tomiyama/bidual route. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "name": "13. Monotone closed \\(C^*\\)-algebras",
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      "full_conditions_and_proof": "## 13. Monotone closed \\(C^*\\)-algebras\n\nIn a von Neumann algebra every bounded increasing net of self-adjoint elements has a least upper bound. We now show that this order property, together with enough normal states, characterizes \\(W^*\\)-algebras.\n\n**Definition 13.1.** We call a \\(C^*\\)-algebra \\(A\\) *monotone closed* when each increasing, norm-bounded net of self-adjoint elements has a least upper bound among the self-adjoint elements of \\(A\\). A positive functional \\(\\omega\\) on such an \\(A\\) is *normal* if \\(\\omega(\\sup_ix_i)=\\sup_i\\omega(x_i)\\) for each such net \\((x_i)\\). We say \\(A\\) has *sufficiently many* normal positive functionals if for every nonzero \\(x\\in A_+\\) some normal positive \\(\\omega\\) has \\(\\omega(x)>0\\).\n\n**Proposition 13.2.**\n\n1. A \\(W^*\\)-algebra is monotone closed, and for it the two meanings of *normal positive functional* agree.\n2. A monotone closed \\(C^*\\)-algebra has a unit.\n\n**Proof.** (1) The first claim is the background fact on monotone nets, after adding a multiple of \\(1\\) to make the net positive. For the second, a \\(\\sigma\\)-weakly continuous positive functional preserves these suprema (background). Conversely, a positive functional that preserves them is completely additive, since the finite sums of orthogonal projections form an increasing net with supremum the full sum, and so it is \\(\\sigma\\)-weakly continuous by [Corollary 11.5](#oa-fnd-wa-25). (The background fact on positive normal maps gives a second proof.)\n\n(2) Let \\((u_i)\\) be an increasing approximate unit (background), and \\(e\\) its least upper bound. If \\(A=0\\) there is nothing to prove. Let \\(\\pi\\) be a faithful nondegenerate representation, for instance \\(\\pi_u\\). Then \\(\\pi(u_i)\\to1\\) strongly, and \\(\\pi(u_i)\\le\\pi(e)\\), so \\(\\pi(e)\\ge1\\). Take a continuous \\(g\\) on \\(\\mathbb R\\) with \\(g(0)=0\\) and \\(g=1\\) on \\([1,\\infty)\\). Then \\(g(e)\\in A\\) by the functional calculus, and \\(\\pi(g(e))=g(\\pi(e))=1\\) because the spectrum of \\(\\pi(e)\\) lies in \\([1,\\infty)\\). Since \\(\\pi\\) is faithful, \\(g(e)\\) is a unit for \\(A\\). (Then \\(e\\le1\\) and \\(\\pi(e)\\ge1\\) give \\(e=1\\).) \\(\\square\\)\n\n**Remark 13.3.** For abelian algebras monotone closedness is a property of the spectrum; see \"Where this leads\" below. We do not need this: the only abelian fact used below is proved where it is used, in Exercise 13.10(b).\n\n**Lemma 13.4.** Let \\(A\\) be monotone closed and \\(\\omega\\in A^*_+\\). If \\(\\omega\\) lies in the norm closure of the set of normal positive functionals, then \\(\\omega\\) is normal. In particular this holds if \\(\\|\\omega-\\omega_n\\|\\to0\\) for a sequence of normal positive \\(\\omega_n\\).\n\n**Proof.** Let \\((x_i)\\) be increasing and norm bounded in \\(A_h\\), with least upper bound \\(x\\), and put \\(C=\\max(\\|x\\|,\\sup_i\\|x_i\\|)\\), which is finite. Positivity gives \\(\\omega(x)\\ge\\sup_i\\omega(x_i)\\). Let \\(\\varepsilon>0\\) and choose a normal positive \\(\\omega'\\) with \\(\\|\\omega-\\omega'\\|<\\varepsilon\\), then an index \\(i_0\\) with \\(\\omega'(x_i)>\\omega'(x)-\\varepsilon\\) for \\(i\\ge i_0\\). For such \\(i\\),\n\\[\n\\begin{gathered}\n\\omega(x_i)\\\\\n\\ge\\omega'(x_i)-\\varepsilon C\\\\\n\\ge\\omega'(x)-\\varepsilon-\\varepsilon C\\\\\n\\ge\\omega(x)-\\varepsilon(1+2C).\n\\end{gathered}\n\\]\nSo \\(\\sup_i\\omega(x_i)\\ge\\omega(x)\\). \\(\\square\\)\n\n**Remark 13.5.** The constant \\(C\\) is needed, because the bound \\(\\|x_i\\|\\le\\|x\\|\\) can fail for an increasing net of self-adjoint elements: in \\(\\mathbb C\\), \\(x_1=-1\\le x_2=0\\) has supremum \\(x=0\\), whereas \\(\\|x_1\\|=1\\). The proof above uses the bound that holds for the entire net. \n\n**Proposition 13.6.** Let \\(A\\) be monotone closed and \\(\\omega\\) a normal positive functional with cyclic representation \\((\\pi,H,\\xi)\\). Then \\(\\pi(A)=\\pi(A)''\\).\n\nThe representation need not be faithful: a vector state on the first summand of \\(B(H_1)\\oplus B(H_2)\\) kills the second summand. Steps 2 and 3 prove that every bounded increasing net in the image lifts to a bounded increasing net in the appropriate central corner. This lifting justifies the use of the full monotone closure criterion.\n\n**Proof.** *Step 1: increasing nets go to strong limits.* Let \\((x_i)\\) be increasing and norm bounded in \\(A_h\\), with least upper bound \\(x\\). For a unitary \\(u\\in A\\), conjugation \\(y\\mapsto u^*yu\\) is an order automorphism of \\(A_h\\), so \\(u^*xu\\) is the least upper bound of \\((u^*x_iu)\\), and normality gives \\(\\omega(u^*x_iu)\\to\\omega(u^*xu)\\). Put \\(d_i=\\pi(x)-\\pi(x_i)\\ge0\\). Then\n\\[\n\\|d_i^{1/2}\\pi(u)\\xi\\|^2=\\omega(u^*xu)-\\omega(u^*x_iu)\\to0 .\n\\]\n\\(A\\) has a unit (Proposition 13.2(2)), so its unitaries span \\(A\\), and the vectors \\(\\pi(u)\\xi\\) span a dense subspace. The \\(d_i^{1/2}\\) are uniformly bounded, so \\(d_i^{1/2}\\to0\\) strongly, and \\(d_i=d_i^{1/2}d_i^{1/2}\\to0\\) strongly. So \\(\\pi(x_i)\\to\\pi(x)\\) strongly.\n\n*Step 2: the kernel of \\(\\pi\\) is cut out by a central projection.* Let \\(J=\\ker\\pi\\). By Step 1, if an increasing, norm-bounded net of self-adjoint elements of \\(J\\) has least upper bound \\(x\\) in \\(A\\), then \\(\\pi(x)=0\\), so \\(x\\in J\\). Let \\((u_\\lambda)\\) be an increasing approximate unit of the \\(C^*\\)-algebra \\(J\\) (background), and \\(p\\in J\\) its least upper bound in \\(A\\); then \\(0\\le p\\le1\\). Every \\(a\\in J_+\\) with \\(\\|a\\|\\le1\\) satisfies \\(a\\le p\\): indeed \\(u_\\lambda au_\\lambda\\le u_\\lambda^2\\le u_\\lambda\\le p\\), and \\(u_\\lambda au_\\lambda\\to a\\) in norm. Taking \\(a=p^{1/2}\\in J\\) gives \\(p^{1/2}\\le p\\); with \\(p\\le p^{1/2}\\), which holds for \\(0\\le p\\le1\\), we get \\(p=p^{1/2}\\), so \\(p\\) is a projection. For \\(a\\in J_+\\) with \\(\\|a\\|\\le1\\), \\(0\\le a\\le p\\) now gives \\((1-p)a(1-p)=0\\), so \\(a^{1/2}(1-p)=0\\) and \\(a=ap=pa\\); by linearity \\(x=xp=px\\) for all \\(x\\in J\\). So \\(J=Ap=pA\\), and \\(p\\) is central: for \\(y\\in A\\), \\(yp\\in J\\) and \\(py\\in J\\) give \\(yp=pyp=py\\). Put \\(z=1-p\\).\n\n*Step 3: monotone nets in \\(\\pi(A)\\) lift.* \\(\\pi(A)=\\pi(Az)\\), and \\(\\pi\\) is injective on \\(Az\\), so it is an order isomorphism of \\((Az)_h\\) onto \\(\\pi(A)_h\\): an injective \\(*\\)-homomorphism preserves spectra. \\(Az\\) is monotone closed with the same least upper bounds as \\(A\\): if \\((c_i)\\) is bounded increasing in \\((Az)_h\\) with least upper bound \\(c\\) in \\(A\\), then \\(zcz\\) is an upper bound, so \\(c\\le zcz\\), that is \\((1-z)c\\le0\\); and \\[\n\\begin{gathered}\n(1-z)c\\\\\n=(1-z)c(1-z)\\\\\n\\ge(1-z)c_i(1-z)\\\\\n=0.\n\\end{gathered}\n\\] So \\(c=zc\\in Az\\). Now let \\((T_i)\\) be increasing and norm bounded in \\(\\pi(A)_h\\). Then \\(T_i=\\pi(y_i)\\) for a unique increasing, norm-bounded net \\((y_i)\\) in \\((Az)_h\\), with least upper bound \\(y\\in Az\\), and \\(T_i\\to\\pi(y)\\) strongly by Step 1.\n\n*Step 4.* \\(\\pi(A)\\) is a nondegenerate \\(C^*\\)-algebra of operators, since \\(\\pi(1)=1\\), and by Step 3 the strong limit of each increasing, norm-bounded net of its self-adjoint elements lies in it. The full [monotone closure criterion, Corollary 12.10 of the density lesson](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-14), now gives \\(\\pi(A)=\\pi(A)''\\). \\(\\square\\)\n\n**Theorem 13.7** (Kadison's characterization). A \\(C^*\\)-algebra is a \\(W^*\\)-algebra exactly when it is monotone closed and has sufficiently many normal positive functionals.\n\n\n**Proof.** If \\(A\\) is a \\(W^*\\)-algebra, it is monotone closed by Proposition 13.2(1), and the vector states of a faithful normal realisation are normal (Corollary 11.4) and separate the positive elements. Conversely, let \\(\\pi=\\bigoplus_\\omega\\pi_\\omega\\), over all normal positive \\(\\omega\\). It is nondegenerate, and faithful: if \\(\\pi(x)=0\\) then \\(\\omega(x^*x)=\\|\\pi_\\omega(x)\\xi_\\omega\\|^2=0\\) for every normal \\(\\omega\\), so \\(x^*x=0\\). If \\((x_i)\\) in \\(A_h\\) is increasing and norm bounded, with least upper bound \\(x\\), Step 1 of the proof of Proposition 13.6 gives \\(\\pi_\\omega(x_i)\\to\\pi_\\omega(x)\\) strongly on each summand; the net is uniformly bounded, so \\(\\pi(x_i)\\to\\pi(x)\\) strongly. Since \\(\\pi\\) is faithful, it is an order isomorphism onto its image, so every increasing norm-bounded net in \\(\\pi(A)_h\\) comes from one in \\(A_h\\), and its strong limit lies in \\(\\pi(A)\\). The same full [monotone closure criterion](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-14) gives \\(\\pi(A)=\\pi(A)''\\), so \\(A\\) is a \\(W^*\\)-algebra. \\(\\square\\)\n\n**Remark 13.8** (Unused examples, not constructed or proved here). The second hypothesis is needed: there are monotone closed \\(C^*\\)-algebras that are not \\(W^*\\)-algebras. Dixmier's classical examples are abelian: \\(C(\\Omega)\\) for a stonean space \\(\\Omega\\) that carries too few normal measures [Dixmier 1951]. We do not construct them here.\n\n**Exercise 13.9.** (easy) Show that a von Neumann algebra that is separable in the norm topology is finite-dimensional.\n\n**Solution.** By [Lemma 6.2](#oa-fnd-wa-13), an infinite-dimensional von Neumann algebra contains an isometric copy of \\(\\ell^\\infty\\), which is not separable; a subset of a separable metric space is separable. \\(\\square\\)\n\n**Exercise 13.10** (hard) (Ranges of projections of norm one). Let \\(A\\) be a \\(C^*\\)-algebra acting on \\(H\\), let \\(M\\supseteq A\\) be a von Neumann algebra acting on \\(H\\), and let \\(\\varepsilon:M\\to A\\) be a projection of norm one. (a) Prove that each increasing, norm-bounded net \\((h_i)\\) of positive elements of \\(A\\) has a least upper bound in \\(A\\), and that this bound need not be the strong limit. (b) Prove that a separable such \\(A\\) is finite-dimensional.\n\n**Solution.** (a) Let \\(h\\) be the strong limit of \\((h_i)\\), which is its least upper bound in \\(M\\) (background fact on monotone nets), and put \\(k=\\varepsilon(h)\\). By Tomiyama's theorem ([Theorem 8.5](#oa-fnd-wa-21)) \\(\\varepsilon\\) is positive, so \\(h_i=\\varepsilon(h_i)\\le\\varepsilon(h)=k\\). If \\(a\\in A_h\\) is an upper bound, then \\(h\\le a\\) in \\(M\\), so \\(k=\\varepsilon(h)\\le\\varepsilon(a)=a\\). So \\(k\\) is the least upper bound in \\(A\\).\n\nIt can differ from the strong limit. Let \\(I\\) be the set of all ultrafilters on \\(\\mathbb N\\), including the principal ones \\(i_n\\). For \\(a\\in\\ell^\\infty\\), define \\(\\chi_i(a)=\\lim_{n\\to i}a_n\\). This limit exists uniquely in the compact closed disk containing the values of \\(a\\), by the ultrafilter compactness criterion. Continuity of scalar addition, multiplication and conjugation makes each \\(\\chi_i\\) a unital \\(*\\)-homomorphism. On \\(H=\\ell^2(I)\\), set\n\\[\n\\pi(a)\\delta_i=\\chi_i(a)\\delta_i,\\qquad\nM=\\ell^\\infty(I).\n\\]\nPrincipal ultrafilters give \\(\\|\\pi(a)\\|=\\|a\\|_\\infty\\), so \\(A=\\pi(\\ell^\\infty)\\) is a unital norm-closed \\(C^*\\)-subalgebra of the diagonal von Neumann algebra \\(M\\). Define\n\\[\n\\varepsilon(g)=\\pi((g(i_n))_{n\\ge1})\\qquad(g\\in M).\n\\]\nIt is contractive, fixes \\(A\\), and fixes \\(1\\), so it is a projection of norm one. For finite \\(F\\subseteq\\mathbb N\\), \\(\\pi(1_F)\\) is the diagonal projection on the principal coordinates \\(i_n\\) with \\(n\\in F\\): a free ultrafilter contains no finite set. These projections increase strongly to the projection \\(h\\) on the closed span of all principal coordinates. A free ultrafilter exists by (30), so \\(h\\ne1\\). But their least upper bound in \\(A\\) is \\(1\\): any upper bound \\(\\pi(a)\\) has \\(a_n\\ge1\\) on every principal coordinate, whence \\(\\chi_i(a)\\ge1\\) on all coordinates. Thus the least upper bound in \\(A\\) differs from the strong limit. This construction requires no Stone–Čech compactification theorem.\n\n(b) We may assume \\(A\\ne0\\). By Tomiyama's theorem, \\(A\\) has the unit \\(1_A=\\varepsilon(1)\\). By (a), after adding multiples of \\(1_A\\), every increasing norm-bounded net in \\(A_h\\) has a least upper bound in \\(A\\), namely \\(\\varepsilon\\) of its strong limit.\n\nSuppose \\(\\dim A=\\infty\\). Let \\(C\\) be a maximal abelian \\(*\\)-subalgebra of \\(A\\); it is norm closed and contains \\(1_A\\). It is infinite-dimensional: otherwise, as in the proof of Lemma 6.2, \\(C\\) is spanned by minimal projections \\(q_1,\\dots,q_m\\) with sum \\(1_A\\), each minimal in \\(A\\) by maximality, and \\(\\dim A\\le m^2\\) by Lemma 6.1(2). Least upper bounds of nets from \\(C\\) stay in \\(C\\): if \\(c\\) is the least upper bound of an increasing norm-bounded net in \\(C_h\\), and \\(u\\in C\\) is unitary, then \\(ucu^*\\) is the least upper bound of the same net, so \\(ucu^*=c\\). The unitaries span \\(C\\), so \\(c\\) commutes with \\(C\\), and by maximality \\(c\\in C\\).\n\n*\\(C\\) has a projection other than \\(0\\) and \\(1_A\\).* Take \\(h\\in C_h\\), \\(0\\le h\\le1_A\\), not a multiple of \\(1_A\\), with spectral values \\(s_1<s_2\\), and \\(t\\in(s_1,s_2)\\). Put \\(k=(h-t)_+\\) and \\(k'=(t-h)_+\\), both nonzero, with \\(kk'=0\\). The elements \\(a_n=(k/\\|k\\|)^{1/n}\\) and \\(b_m=(k'/\\|k'\\|)^{1/m}\\) increase in \\(n\\) and \\(m\\), with least upper bounds \\(p,p'\\in C\\). Each \\(a_n=a_{2n}^2\\le p^2\\), since \\(a_{2n}\\le p\\) and they commute; so \\(p\\le p^2\\le p\\), and \\(p\\) is a projection; likewise \\(p'\\). The functions of \\(h\\) behind \\(a_n\\) and \\(b_m\\) have disjoint supports and values in \\([0,1]\\), so \\(a_n\\le1_A-b_m\\). Hence \\(p\\le1_A-b_m\\) for all \\(m\\), then \\(p'\\le1_A-p\\), and \\(pp'=0\\). As \\(p\\ge a_1\\ne0\\) and \\(p'\\ne0\\), \\(p\\notin\\{0,1_A\\}\\).\n\nThe corners \\(pC\\) and \\((1_A-p)C\\) are again unital and abelian, and closed under these least upper bounds: if an increasing net in \\((pC)_h\\) is bounded in norm by \\(K\\), its least upper bound \\(c\\in C\\) satisfies \\(-Kp\\le c\\le Kp\\), so \\((1_A-p)c=(1_A-p)c(1_A-p)=0\\) and \\(c\\in pC\\). One of the two corners is infinite-dimensional. Repeating the argument inside it produces nonzero, mutually orthogonal projections \\(q_1,q_2,\\ldots\\) in \\(C\\). For \\(S\\subseteq\\mathbb N\\), let \\(Q_S\\) be the strong sum of the \\(q_n\\), \\(n\\in S\\), and \\(P_S=\\varepsilon(Q_S)\\in A\\). If \\(n\\in S\\setminus T\\), bimodularity gives \\(q_nP_Sq_n=\\varepsilon(q_nQ_Sq_n)=q_n\\) and \\(q_nP_Tq_n=\\varepsilon(q_nQ_Tq_n)=0\\). So \\(\\|P_S-P_T\\|\\ge1\\) for \\(S\\ne T\\): uncountably many elements at mutual distance at least one. So \\(A\\) is not separable. \\(\\square\\)\n\n**Exercise 13.11** (medium) (Algebras that are not dual spaces). Show that neither \\(c_0\\) nor \\(C[0,1]\\) is isometrically isomorphic to the dual of a Banach space.\n\n**Solution.** By Sakai's theorem ([Theorem 9.2](#oa-fnd-wa-22)) such an algebra would be a \\(W^*\\)-algebra, hence unital and monotone closed ([Proposition 13.2](#oa-fnd-wa-27)(1)). \\(c_0\\) has no unit. In \\(C[0,1]\\), the increasing sequence \\(f_n(t)=\\min(1,\\max(0,n(t-\\tfrac12)))\\) has no least upper bound. An upper bound \\(g\\) satisfies \\(g\\ge1\\) on \\((\\tfrac12,1]\\), hence on \\([\\tfrac12,1]\\), and \\(g\\ge0\\). Then \\(g>\\tfrac12\\) on some interval \\([\\tfrac12-\\delta,\\tfrac12]\\); subtracting from \\(g\\) a continuous bump \\(b\\) with \\(0\\le b\\le\\tfrac12\\), supported in \\((\\tfrac12-\\delta,\\tfrac12)\\), gives a strictly smaller upper bound, because \\(f_n=0\\) on \\([0,\\tfrac12]\\). \\(\\square\\)\n\n## Where this leads\n\nThe following extensions are not proved here and are not used in the proofs or solutions above. The polar-functional lesson proves the first direction later in the course.\n\n- **Polar decomposition of functionals.** Remark 3.5 is the hermitian case of a general fact: every normal functional \\(\\varphi\\) on a von Neumann algebra can be written \\(\\varphi=u|\\varphi|\\) in the notation (0.1), with \\(|\\varphi|\\) a positive normal functional of the same norm and \\(u\\) a partial isometry.\n- **Abelian algebras.** An abelian \\(C^*\\)-algebra \\(C(\\Omega)\\) is monotone closed exactly when \\(\\Omega\\) is stonean, and its normal positive functionals then correspond to normal measures. It is a \\(W^*\\)-algebra exactly when, in addition, \\(\\Omega\\) carries enough normal measures, that is, when \\(\\Omega\\) is hyperstonean [Dixmier 1951].\n\n## References\n\n- [Kostecki] R. P. Kostecki, *W\\*-algebras and noncommutative integration*, lecture notes, 2013, arXiv:1307.4818, https://arxiv.org/abs/1307.4818v5.\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n- [Dixmier 1951] J. Dixmier, *Sur certains espaces considérés par M. H. Stone*, Summa Brasiliensis Mathematicae 2 (1951), 151–182, [freely readable scan](https://dmitripavlov.org/scans/dixmier.pdf).\n- [Sakai 1956] S. Sakai, A characterization of W\\*-algebras, *Pacific Journal of Mathematics* 6 (1956), 763–773. https://doi.org/10.2140/pjm.1956.6.763 Freely readable [original journal PDF](https://msp.org/pjm/1956/6-4/pjm-v6-n4-p11-p.pdf).\n\n*Freely accessible reading:* [Shoichiro Sakai, *A characterization of W*-algebras*, printed pp. 763–773](https://msp.org/pjm/1956/6-4/pjm-v6-n4-p11-p.pdf) gives a route through the original abstract characterization by normal-state representations; this lesson uses the complete Tomiyama/bidual route. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
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      "id": "OA-FND-PB-01",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "1. Trees in Polish spaces",
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      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "### 1. Trees in Polish spaces\n\nTwo tree constructions run through this lesson. A countably branching tree of closed sets shows that every nonempty Polish space is a continuous image of the Baire space. A binary tree of open balls places a copy of the Cantor space inside every uncountable Polish space.\n\n**Theorem 1.1** (Continuous images of the Baire space). Let \\(X\\) be a nonempty Polish space with a complete compatible metric \\(d\\). There are nonempty closed sets \\(F_s\\subseteq X\\), \\(s\\in\\mathbb N^{<\\mathbb N}\\), with\n\\[\n\\begin{gathered}\nF_\\varnothing=X,\\\\\nF_s=\\bigcup_{n\\in\\mathbb N}F_{sn},\\\\\n\\operatorname{diam}F_s\\leq2^{-|s|}\\ \\ (|s|\\geq1).\n\\end{gathered}\n\\tag{1}\n\\]\nFor every family as in (1), \\(\\{\\varphi(t)\\}=\\bigcap_kF_{t|k}\\) defines a continuous map \\(\\varphi:\\Lambda\\to X\\) with \\(\\varphi(\\Lambda_s)=F_s\\) for all \\(s\\). In particular \\(\\varphi\\) maps \\(\\Lambda\\) onto \\(X\\).\n\n**Proof.** *The sets.* Let \\(F\\) be a nonempty closed set and \\(r>0\\). \\(F\\) is separable, so it has a dense sequence \\((a_n)\\); if \\(F\\) is finite, repeat its points. The closed balls \\(\\{x:d(x,a_n)\\leq r/2\\}\\) meet \\(F\\) in nonempty closed sets of diameter at most \\(r\\), and these cover \\(F\\), since each point of \\(F\\) is within \\(r/2\\) of some \\(a_n\\). Start with \\(F_\\varnothing=X\\), and apply this to \\(F_s\\) with \\(r=2^{-|s|-1}\\) to obtain the sets \\(F_{sn}\\).\n\n*The map.* For \\(t\\in\\Lambda\\), the sets \\(F_{t|k}\\) are nonempty, closed and decreasing, and their diameters tend to \\(0\\). By Cantor's intersection theorem (B1) they have exactly one common point \\(\\varphi(t)\\). If \\(t|k=u|k\\), then \\(\\varphi(t)\\) and \\(\\varphi(u)\\) lie in \\(F_{t|k}\\), so \\(d(\\varphi(t),\\varphi(u))\\leq2^{-k}\\); thus \\(\\varphi\\) is continuous. Clearly \\(\\varphi(\\Lambda_s)\\subseteq F_s\\). Conversely, let \\(x\\in F_s\\) with \\(|s|=k\\). Put \\(t|k=s\\) and choose \\(t_{k+1},t_{k+2},\\dots\\) one at a time so that \\(x\\in F_{t|j}\\) for every \\(j\\); this is possible because \\(F_{t|j}=\\bigcup_nF_{(t|j)n}\\). Then \\(x\\in\\bigcap_jF_{t|j}\\), so \\(\\varphi(t)=x\\) with \\(t\\in\\Lambda_s\\). \\(\\square\\)\n\n**Remark 1.2.** Both nonemptiness conditions are needed. The empty metric space is complete and separable, hence Polish, and no map from \\(\\Lambda\\) into it exists. And if one set \\(F_s\\) of the tree were empty, the intersection along every branch through \\(s\\) would be empty. Repeating points, as in the proof, keeps every piece nonempty.\n\n**Lemma 1.3** (Perfect set lemma). Every uncountable Polish space \\(X\\) contains a compact subset homeomorphic to \\(\\mathcal C\\). Hence \\(|X|=2^{\\aleph_0}\\).\n\n**Proof.** Let \\(d\\) be a complete compatible metric and \\((W_j)\\) a countable base. Let \\(X'\\) be the set of points all of whose neighbourhoods are uncountable. A point outside \\(X'\\) has a countable neighbourhood, hence a countable basic neighbourhood \\(W_j\\). So \\(X\\setminus X'\\) lies in the union of the countable sets \\(W_j\\), and it is countable. Since \\(X\\) is uncountable, \\(X'\\neq\\varnothing\\). If an open set \\(V\\) meets \\(X'\\), then \\(V\\) is uncountable while \\(V\\setminus X'\\) is countable, so \\(V\\cap X'\\) is uncountable.\n\nChoose open sets \\(U_s\\), \\(s\\in\\{0,1\\}^{<\\mathbb N}\\), each meeting \\(X'\\), as follows. Put \\(U_\\varnothing=X\\). Given \\(U_s\\), pick two different points \\(a,b\\in U_s\\cap X'\\) and \\(r>0\\) with \\(3r<d(a,b)\\), \\(r\\leq2^{-|s|-2}\\), and the balls of radius \\(2r\\) about \\(a\\) and \\(b\\) inside \\(U_s\\). Let \\(U_{s0}\\) and \\(U_{s1}\\) be the open balls of radius \\(r\\) about \\(a\\) and \\(b\\). Their closures lie in \\(U_s\\) and are disjoint, their diameters are at most \\(2^{-|s|-1}\\), and they meet \\(X'\\) at \\(a\\) and \\(b\\).\n\nFor \\(c\\in\\mathcal C\\), the closed sets \\(\\overline{U_{c|k}}\\), \\(k\\geq1\\), are nonempty and decreasing, and their diameters tend to \\(0\\). So they have one common point \\(e(c)\\), by Cantor's intersection theorem (B1). If \\(c|k=c'|k\\), then \\(d(e(c),e(c'))\\leq2^{-k}\\), so \\(e\\) is continuous. If \\(c\\) and \\(c'\\) first differ at place \\(k+1\\), then \\(e(c)\\) and \\(e(c')\\) lie in the disjoint closures of \\(U_{(c|k)0}\\) and \\(U_{(c|k)1}\\), so \\(e\\) is injective. The space \\(\\mathcal C\\) is compact (B2), and a continuous injection of a compact space into a Hausdorff space is a homeomorphism onto its image (B3). So \\(e\\) is a homeomorphism onto the compact set \\(e(\\mathcal C)\\). Finally \\(|X|\\geq|\\mathcal C|=2^{\\aleph_0}\\), and \\(|X|\\leq2^{\\aleph_0}\\) because \\(X\\) is separable and metrizable (F2). \\(\\square\\)\n\n### 2. Souslin sets\n\nA continuous image of a Polish space need not be Polish, and it need not even be Borel. Such images are the Souslin sets. This section shows that they are closed under countable unions, countable intersections and countable products, that disjoint Souslin sets can be separated by Borel sets, and that every Borel set is a Souslin set but not conversely.\n\n",
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      "id": "OA-FND-PB-02",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Definition and closure properties",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "#### Definition and closure properties\n\n**Definition 2.1.** A *Souslin space*, also called an analytic space, is a metrizable space onto which some Polish space maps continuously. A *Souslin set* in a topological space \\(Z\\) is a subset of \\(Z\\) that, with the relative topology, is a Souslin space.\n\nSo the Souslin sets of a metrizable space are exactly its Polish images. The empty space is a Souslin space. A nonempty Souslin space is the image of \\(\\Lambda\\) under a continuous map: compose a continuous map of \\(\\Lambda\\) onto the Polish space ([Theorem 1.1](#oa-fnd-pb-01)) with the given map. Whether a set is a Souslin set depends only on the set with its own topology. Hence a Souslin set of a subspace \\(Y\\subseteq Z\\) is a Souslin set of \\(Z\\), and a subset of \\(Y\\) that is a Souslin set of \\(Z\\) is one of \\(Y\\).\n\n**Proposition 2.2.**\n\n1. A Souslin space is separable, and its Borel structure is countably generated and countably separated.\n2. If \\(X\\) is a Souslin space, \\(Y\\) is metrizable and \\(f:X\\to Y\\) is continuous, then the subset \\(f(X)\\) of \\(Y\\) is a Souslin set.\n3. A countable product of Souslin spaces is a Souslin space.\n4. Let \\(Z\\) be a Hausdorff space and \\(A_1,A_2,\\ldots\\) Polish images in \\(Z\\). Then \\(\\bigcup_nA_n\\) and \\(\\bigcap_nA_n\\) are Polish images in \\(Z\\). In particular, in a metrizable space countable unions and countable intersections of Souslin sets are Souslin sets.\n5. Open subsets and closed subsets of a Souslin space are Souslin sets.\n6. Let \\(f:X\\to Y\\) be a continuous map between metrizable spaces, and \\(A\\subseteq X\\), \\(A'\\subseteq Y\\) Souslin sets. Then \\(f(A)\\) and \\(A\\cap f^{-1}(A')\\) are Souslin sets.\n\n**Proof.** (1) A continuous image of a separable space is separable. A separable metrizable space has a countable base: the balls with rational radii about the points of a countable dense set. This base generates the Borel sets (F1), and it separates points, because for \\(x\\neq y\\) a small ball about \\(x\\) misses \\(y\\).\n\n(2) Compose \\(f\\) with a continuous surjection from a Polish space onto \\(X\\). The image \\(f(X)\\) is metrizable, being a subset of \\(Y\\).\n\n(3) If \\(g_n:P_n\\to X_n\\) are continuous surjections from Polish spaces, the product map \\(\\prod_ng_n:\\prod_nP_n\\to\\prod_nX_n\\) is a continuous surjection. \\(\\prod_nP_n\\) is Polish, since countable products of Polish spaces are Polish (F2), and a countable product of metrizable spaces is metrizable.\n\n(4) Let \\(g_n:P_n\\to Z\\) be continuous with \\(g_n(P_n)=A_n\\) and \\(P_n\\) Polish. For the union, let \\(P\\) be the disjoint sum of the \\(P_n\\), which is Polish (F2), and let \\(g=g_n\\) on \\(P_n\\). For the intersection, let\n\\[\n\\begin{gathered}\nP=\\{(p_n)\\in\\textstyle\\prod_nP_n:\\\\\n \\ g_1(p_1)=g_n(p_n)\\text{ for all }n\\}.\n\\end{gathered}\n\\]\nThe diagonal of \\(Z\\times Z\\) is closed because \\(Z\\) is Hausdorff, and \\(P\\) is the intersection of its preimages under the continuous maps \\((p_n)\\mapsto(g_1(p_1),g_n(p_n))\\). So \\(P\\) is closed in \\(\\prod_nP_n\\), hence Polish (F2). The map \\((p_n)\\mapsto g_1(p_1)\\) is continuous on \\(P\\) and its image is \\(\\bigcap_nA_n\\): for \\(x\\) in the intersection, choose \\(p_n\\in g_n^{-1}(x)\\). Every subset of a metrizable space is metrizable, which gives the last sentence.\n\n(5) Let \\(g:P\\to X\\) be a continuous surjection from a Polish space. For \\(U\\) open in \\(X\\), \\(g^{-1}(U)\\) is open in \\(P\\), hence Polish (F3), and \\(g\\) maps it onto \\(U\\). For \\(F\\) closed, \\(g^{-1}(F)\\) is closed, hence Polish (F2), and \\(g\\) maps it onto \\(F\\).\n\n(6) \\(f(A)\\) is a Souslin set by (2), applied to \\(f\\) restricted to \\(A\\). Next, \\(A\\times A'\\) is a Souslin space by (3), and \\(G=\\{(a,b)\\in A\\times A':f(a)=b\\}\\) is closed in it because \\(Y\\) is Hausdorff; so \\(G\\) is a Souslin set by (5). The first projection is continuous and maps \\(G\\) onto \\(A\\cap f^{-1}(A')\\), so (2) applies. \\(\\square\\)\n\nBorel subsets of Souslin spaces are Souslin sets as well; this is Corollary 2.6 below.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PB-03",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "The separation theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "#### The separation theorem\n\nTwo subsets \\(E,E'\\) of a Borel space are *Borel-separated* if some Borel set \\(B\\) has \\(E\\subseteq B\\) and \\(B\\cap E'=\\varnothing\\).\n\n**Lemma 2.3.** Let \\(E_n,E'_m\\) (\\(n,m\\in\\mathbb N\\)) be subsets of a Borel space, and for all \\(n\\) and \\(m\\) let \\(B_{n,m}\\) be a Borel set that contains \\(E_n\\) and misses \\(E'_m\\). Then \\(\\bigcup_nE_n\\) and \\(\\bigcup_mE'_m\\) are Borel-separated by\n\\[\nB=\\bigcup_n\\bigcap_mB_{n,m}.\n\\tag{2}\n\\]\n\n**Proof.** \\(E_n\\subseteq\\bigcap_mB_{n,m}\\subseteq B\\). If \\(x\\in B\\), then \\(x\\in B_{n,m}\\) for one \\(n\\) and every \\(m\\), so \\(x\\) lies in no \\(E'_m\\). \\(\\square\\)\n\nSo, by (2), if \\(\\bigcup_nE_n\\) and \\(\\bigcup_mE'_m\\) are not Borel-separated, then some pair \\(E_n,E'_m\\) is not Borel-separated.\n\n**Theorem 2.4** (Lusin's separation theorem). Let \\(X\\) be a Hausdorff space and \\((X_n)_{n\\in I}\\), with \\(I\\) countable, pairwise disjoint Polish images in \\(X\\). There are pairwise disjoint Borel sets \\(B_n\\) with \\(X_n\\subseteq B_n\\). In particular this holds for pairwise disjoint Souslin sets of a metrizable space.\n\n\n**Proof.** *Two sets.* Let \\(Y,Y'\\) be disjoint Polish images. If \\(Y=\\varnothing\\), take \\(B=\\varnothing\\); if \\(Y'=\\varnothing\\), take \\(B=X\\). Otherwise there are continuous maps \\(f,f':\\Lambda\\to X\\) with \\(f(\\Lambda)=Y\\) and \\(f'(\\Lambda)=Y'\\): compose continuous maps of \\(\\Lambda\\) onto the Polish spaces ([Theorem 1.1](#oa-fnd-pb-01)) with the given maps. Put \\(Y_s=f(\\Lambda_s)\\) and \\(Y'_s=f'(\\Lambda_s)\\); then \\(Y_s=\\bigcup_nY_{sn}\\) and \\(Y'_s=\\bigcup_nY'_{sn}\\).\n\nSuppose \\(Y\\) and \\(Y'\\) are not Borel-separated. By Lemma 2.3 there are \\(n_1,m_1\\) with \\(Y_{(n_1)}\\) and \\(Y'_{(m_1)}\\) not Borel-separated. Applying the lemma again, to \\(Y_{(n_1)}=\\bigcup_nY_{(n_1,n)}\\) and \\(Y'_{(m_1)}=\\bigcup_mY'_{(m_1,m)}\\), and so on, we get \\(t,t'\\in\\Lambda\\) such that \\(Y_{t|k}\\) and \\(Y'_{t'|k}\\) are not Borel-separated for any \\(k\\). The points \\(f(t)\\in Y\\) and \\(f'(t')\\in Y'\\) are different, because \\(Y\\) and \\(Y'\\) are disjoint. Since \\(X\\) is Hausdorff, they have disjoint open neighbourhoods \\(V\\) and \\(W\\). The open set \\(f^{-1}(V)\\) contains \\(t\\), hence contains \\(\\Lambda_{t|k}\\) for all large \\(k\\); likewise \\(f'^{-1}(W)\\supseteq\\Lambda_{t'|k}\\) for all large \\(k\\). For such \\(k\\), \\(Y_{t|k}\\subseteq V\\) and \\(Y'_{t'|k}\\subseteq W\\subseteq X\\setminus V\\). So the open set \\(V\\) separates them, which is a contradiction.\n\n*Countably many sets.* Number the index set \\(1,2,\\ldots\\). For \\(n\\neq m\\) choose a Borel set \\(B_{n,m}\\supseteq X_n\\) with \\(B_{n,m}\\cap X_m=\\varnothing\\). Put \\(C_n=\\bigcap_{m\\neq n}B_{n,m}\\) and \\(B_n=C_n\\setminus(C_1\\cup\\cdots\\cup C_{n-1})\\). These sets are Borel and pairwise disjoint. \\(X_n\\subseteq C_n\\), and for \\(j<n\\), \\(X_n\\cap C_j\\subseteq X_n\\cap B_{j,n}=\\varnothing\\). So \\(X_n\\subseteq B_n\\). \\(\\square\\)\n\n**Corollary 2.5** (Souslin's theorem). Let \\(X\\) be a Hausdorff space and \\(A\\subseteq X\\). If \\(A\\) and \\(X\\setminus A\\) are both Polish images, then \\(A\\) is Borel. In particular, in a metrizable space a Souslin set whose complement is a Souslin set is Borel.\n\n\n**Proof.** By Theorem 2.4, some Borel set contains \\(A\\) and misses \\(X\\setminus A\\). Such a set is \\(A\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-PB-12",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Borel sets and Souslin sets",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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        "line": 164,
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      },
      "full_conditions_and_proof": "#### Borel sets and Souslin sets\n\n**Corollary 2.6** (Borel subsets of Souslin spaces). Every Borel set in a Souslin space is itself a Souslin set. In particular, all Borel subsets of Polish spaces are Souslin sets.\n\n**Proof.** Let \\(g:P\\to X\\) be a continuous surjection from a Polish space, and \\(B\\subseteq X\\) Borel. The set \\(g^{-1}(B)\\) is Borel in \\(P\\). By the change-of-topology theorem (F4)(c), \\(P\\) has a Polish topology, finer than the given one and with the same Borel sets, in which \\(g^{-1}(B)\\) is closed. Then \\(g^{-1}(B)\\) is Polish in the finer topology (F2), \\(g\\) is still continuous on it, and \\(g\\) maps it onto \\(B\\) because \\(g\\) is onto. \\(B\\) is metrizable as a subset of \\(X\\). \\(\\square\\)\n\n**Remark 2.7.** So in a Polish space the Borel sets are exactly the Souslin sets whose complements are Souslin sets: one direction is Corollary 2.6, the other is Souslin's theorem. Example 2.9 gives a Souslin set that is not Borel, so Souslin's theorem has content.\n\n**Corollary 2.8** (Counting). A Polish space has at most \\(2^{\\aleph_0}\\) Souslin sets, and so at most \\(2^{\\aleph_0}\\) Borel sets. Consequently \\(\\mathcal C\\) has subsets that are not Borel.\n\n**Proof.** Let \\(X\\) be Polish. A nonempty Souslin set of \\(X\\) has the form \\(h(\\Lambda)\\) with \\(h:\\Lambda\\to X\\) continuous, as noted after Definition 2.1. The sequences that equal \\(1\\) from some place on form a countable dense set \\(D_0\\subseteq\\Lambda\\), and a continuous map into a Hausdorff space is determined by its values on a dense set. Since \\(|X|\\leq2^{\\aleph_0}\\) (F2), there are at most \\(|X|^{\\aleph_0}\\leq(2^{\\aleph_0})^{\\aleph_0}=2^{\\aleph_0}\\) such maps. By Corollary 2.6, Borel sets are Souslin sets. The space \\(\\mathcal C\\) has \\(2^{2^{\\aleph_0}}\\) subsets, and \\(2^{2^{\\aleph_0}}>2^{\\aleph_0}\\) by Cantor's theorem. \\(\\square\\)\n\n**Example 2.9** (A Souslin set that is not Borel). Let \\(Z=\\mathcal C\\times\\Lambda\\), a Polish space, with a countable base \\((V_n)_{n\\geq1}\\).\n\n(a) The set \\[\n\\begin{gathered}\nO\\\\\n=\\{(y,z)\\in\\mathcal C\\times Z:\\ y_n=1\\text{ and }z\\in V_n\\\\\n\\text{ for some }n\\}\n\\end{gathered}\n\\] is open, being the union of the open sets \\(\\{y:y_n=1\\}\\times V_n\\). For an open \\(W\\subseteq Z\\), let \\(y_n=1\\) exactly when \\(V_n\\subseteq W\\); then the section \\(O_y=\\{z:(y,z)\\in O\\}\\) equals \\(W\\). So \\(F=(\\mathcal C\\times Z)\\setminus O\\) is a closed set whose sections \\(F_y\\) run through all closed subsets of \\(Z\\).\n\n(b) Let \\[\n\\begin{gathered}\nU\\\\\n=\\{(y,x)\\in\\mathcal C\\times\\mathcal C:\\ (y,x,w)\\in F\\\\\n\\text{ for some }w\\in\\Lambda\\},\n\\end{gathered}\n\\] the projection of \\(F\\) along \\(\\Lambda\\). It is a Souslin set of \\(\\mathcal C\\times\\mathcal C\\), being the image of the Polish space \\(F\\) under a continuous map. Every Souslin set \\(A\\subseteq\\mathcal C\\) is a section \\(U_y\\). If \\(A=\\varnothing\\), take \\(y\\) with \\(F_y=\\varnothing\\). Otherwise \\(A=g(\\Lambda)\\) with \\(g\\) continuous, as noted after Definition 2.1; the set \\(K=\\{(g(w),w):w\\in\\Lambda\\}\\) is closed in \\(Z\\) because \\(\\mathcal C\\) is Hausdorff, its projection to \\(\\mathcal C\\) is \\(A\\), and we take \\(y\\) with \\(F_y=K\\).\n\n(c) Let \\(D=\\{x\\in\\mathcal C:(x,x)\\in U\\}\\). It is the projection to \\(\\mathcal C\\) of the closed set \\(\\{(x,w)\\in\\mathcal C\\times\\Lambda:(x,x,w)\\in F\\}\\), so \\(D\\) is a Souslin set. If \\(D\\) were Borel, \\(\\mathcal C\\setminus D\\) would be Borel, hence a Souslin set (Corollary 2.6), hence \\(\\mathcal C\\setminus D=U_{y_0}\\) for some \\(y_0\\). Then \\(y_0\\in D\\) exactly when \\((y_0,y_0)\\in U\\), that is, exactly when \\(y_0\\in U_{y_0}=\\mathcal C\\setminus D\\). This is absurd. So \\(D\\) is a Souslin set that is not Borel, and by Souslin's theorem its complement is not a Souslin set.\n\nThe argument in (c) is Cantor's diagonal argument, applied to the universal set \\(U\\). It shows that the image of a closed set under a continuous map need not be Borel: \\(D\\) is the projection of a closed subset of \\(\\mathcal C\\times\\Lambda\\). This is a strong form of the fact that Borel maps need not carry Borel sets to Borel sets. Examples 5.9 and 6.4 return to \\(D\\).\n\n### 3. Lusin spaces and one-to-one images\n\nA one-to-one continuous image of a Polish space in a Hausdorff space is always Borel. This is the Lusin–Souslin theorem, the main result of this section. The tool is a change of topology that makes a Polish space zero-dimensional without changing its Borel sets. A zero-dimensional Polish space is a closed subset of the Baire space, where tree arguments apply.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PB-04",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Lusin spaces and closed subsets of the Baire space",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "#### Lusin spaces and closed subsets of the Baire space\n\n**Lemma 3.1** (Zero-dimensional refinement). Let \\((X,\\tau)\\) be a Polish space and \\(B_1,B_2,\\ldots\\) Borel sets. There is a zero-dimensional Polish topology \\(\\tau'\\supseteq\\tau\\) with the same Borel sets as \\(\\tau\\), in which every \\(B_n\\) is open and closed.\n\n**Proof.** We use the change-of-topology theorem (F4). By (F4)(c), each \\(B_n\\) is open and closed for some admissible topology \\(\\tau_n\\). By (F4)(b), the topology \\(\\tau''\\) generated by \\(\\tau\\) and all the \\(\\tau_n\\) is admissible; if there are no sets \\(B_n\\), then \\(\\tau''=\\tau\\). Each \\(B_n\\) is open and closed in \\(\\tau''\\), since this property passes to finer topologies.\n\nLet \\((U_m)\\) be a countable base of \\(\\tau''\\). By (F4)(a), applied to the Polish space \\((X,\\tau'')\\), the topology \\(\\tau''_m\\) generated by \\(\\tau''\\) and \\(X\\setminus U_m\\) is admissible for \\(\\tau''\\). Let \\(\\tau'\\) be generated by all the \\(\\tau''_m\\). By (F4)(b) it is admissible for \\(\\tau''\\), hence for \\(\\tau\\), because \\(\\tau''\\) and \\(\\tau\\) have the same Borel sets. Each \\(\\tau''_m\\) has the countable base formed by the sets \\(U_j\\) and \\(U_j\\setminus U_m\\). So the finite intersections of sets \\(U_j\\) and \\(X\\setminus U_{m}\\) form a base of \\(\\tau'\\). All these sets are open and closed in \\(\\tau'\\). So \\(\\tau'\\) is zero-dimensional. \\(\\square\\)\n\nThis is the change-of-topology technique of [the lesson on the Effros Borel structure](effros-borel-structure.md#oa-fnd-ef-02), used once more to make a whole base open and closed.\n\n**Definition 3.2.** A metrizable space \\(X\\) is a *Lusin space* if some zero-dimensional Polish space maps continuously and bijectively onto \\(X\\). A *Lusin set* in a topological space \\(Z\\) is a subset of \\(Z\\) that, with the relative topology, is a Lusin space.\n\n**Proposition 3.3.**\n\n1. A metrizable space is a Lusin space if and only if some Polish space, zero-dimensional or not, maps continuously and bijectively onto it. So zero-dimensionality in the definition costs nothing.\n2. Polish spaces are Lusin spaces, and Lusin spaces are Souslin spaces.\n3. In a Hausdorff space, a countable intersection of Lusin sets is a Lusin set.\n4. A countable union of pairwise disjoint Lusin sets of a metrizable space is a Lusin set.\n5. Open subsets, closed subsets and \\(G_\\delta\\) subsets of a Lusin space are Lusin sets.\n6. A countable product of Lusin spaces is a Lusin space.\n\nExample 3.8 proves that the intersection assertion (3) needs the Hausdorff hypothesis on the ambient space.\n\n**Proof.** (1) If \\(f:P\\to X\\) is a continuous bijection from a Polish space, refine the topology of \\(P\\) by Lemma 3.1, with no sets \\(B_n\\). The new topology is zero-dimensional and Polish, and \\(f\\) stays continuous because the topology became finer.\n\n(2) The identity map of \\(X\\), from a zero-dimensional refinement to the given topology, is a continuous bijection. The second statement is clear.\n\n(3) Let \\(f_n:P_n\\to L_n\\) be continuous bijections from zero-dimensional Polish spaces onto Lusin sets \\(L_n\\) of a Hausdorff space \\(Z\\). As in the proof of Proposition 2.2(4), the set \\(P=\\{(p_n):f_1(p_1)=f_n(p_n)\\text{ for all }n\\}\\) is closed in \\(\\prod_nP_n\\), because the diagonal of \\(Z\\times Z\\) is closed. The map \\((p_n)\\mapsto f_1(p_1)\\) sends \\(P\\) continuously onto \\(\\bigcap_nL_n\\). It is injective: \\(f_1(p_1)=f_1(q_1)\\) gives \\(p_1=q_1\\), then \\(f_n(p_n)=f_n(q_n)\\), so \\(p_n=q_n\\) for all \\(n\\). A product of zero-dimensional spaces is zero-dimensional, since boxes built from finitely many open and closed sets are open and closed and form a base; a subspace of a zero-dimensional space is zero-dimensional. Finally \\(\\bigcap_nL_n\\subseteq L_1\\) is metrizable.\n\n(4) With \\(f_n\\) as before and the \\(L_n\\) pairwise disjoint, the disjoint sum of the \\(P_n\\) is zero-dimensional and Polish, and the map equal to \\(f_n\\) on \\(P_n\\) is a continuous bijection onto \\(\\bigcup_nL_n\\). This union is metrizable, being a subset of a metrizable space.\n\n(5) Let \\(f:P\\to L\\) be a continuous bijection from a zero-dimensional Polish space. For \\(U\\) open in \\(L\\), \\(f^{-1}(U)\\) is open in \\(P\\), hence Polish (F3) and zero-dimensional, and \\(f\\) maps it bijectively onto \\(U\\). Closed sets are handled the same way, using (F2). A \\(G_\\delta\\) subset of \\(L\\) is a countable intersection of open subsets, which are Lusin sets of the metrizable space \\(L\\); apply (3).\n\n(6) The product of continuous bijections \\(f_n:P_n\\to L_n\\) is a continuous bijection \\(\\prod_nP_n\\to\\prod_nL_n\\). The domain is zero-dimensional and Polish, and the target is metrizable. \\(\\square\\)\n\n*Another proof of (2).* The irrational numbers \\(J\\) in \\(I=[0,1]\\) form a Lusin set. Indeed, \\(J\\) is a \\(G_\\delta\\) subset of \\(I\\), hence Polish (F3). The sets \\(J\\cap(r,s)\\) with \\(r,s\\) rational are open and closed in \\(J\\), because their endpoints are not in \\(J\\), and they form a neighbourhood base; so \\(J\\) is zero-dimensional. Then \\(I\\) is the disjoint union of \\(J\\) and countably many points, each a Lusin set, so \\(I\\) is a Lusin space by (4). Next \\(I^{\\mathbb N}\\) is a Lusin space by (6). Finally a Polish space is homeomorphic to a \\(G_\\delta\\) subset of \\(I^{\\mathbb N}\\) (F3), and so it is a Lusin set by (5).\n\n**Theorem 3.4** (Closed subsets of the Baire space).\n\n1. Every zero-dimensional Polish space is homeomorphic to a closed subset of \\(\\Lambda\\).\n2. For every Lusin space \\(X\\), in particular for every Polish space, there are a closed set \\(T\\subseteq\\Lambda\\) and a continuous bijection of \\(T\\) onto \\(X\\).\n3. The closed set \\(T\\) in (2) cannot in general be a product \\(\\prod_kN_k\\) in which each \\(N_k\\) is \\(\\mathbb N\\) or a finite set \\(\\{1,\\dots,N_k\\}\\): the Polish space \\(X=[0,1]\\cup\\{2\\}\\subseteq\\mathbb R\\) is not the image of any such product under a continuous bijection.\n\nThe explicit isolated-point argument in (3) proves that the closed tree in (2) cannot always be replaced by a product.\n\n**Proof.** (1) Let \\(P\\) be a zero-dimensional Polish space with a complete compatible metric \\(d\\).\n\n*Step 1: partitions.* Let \\(Q\\subseteq P\\) be open and closed, and \\(r>0\\). Each \\(x\\in Q\\) has an open and closed neighbourhood inside \\(Q\\cap\\{y:d(x,y)<r/3\\}\\). These neighbourhoods cover \\(Q\\). Since \\(Q\\) is second countable, it is Lindelöf (B4), so countably many of them, \\(D_1,D_2,\\ldots\\), cover \\(Q\\). The sets \\(D_j\\setminus(D_1\\cup\\cdots\\cup D_{j-1})\\) are open and closed, pairwise disjoint, of diameter at most \\(r\\), and they cover \\(Q\\). If there are only finitely many, add empty sets.\n\n*Step 2: the tree.* By Step 1, choose open and closed sets \\(P_s\\), \\(s\\in\\mathbb N^{<\\mathbb N}\\), with \\(P_\\varnothing=P\\), each \\(P_s\\) the disjoint union of the sets \\(P_{sn}\\) (\\(n\\in\\mathbb N\\)), and \\(\\operatorname{diam}P_s\\leq2^{-|s|}\\) for \\(|s|\\geq1\\). By induction, for each \\(k\\) the sets \\(P_s\\) with \\(|s|=k\\) partition \\(P\\). Let \\(T=\\{t\\in\\Lambda:P_{t|k}\\neq\\varnothing\\text{ for all }k\\}\\). If \\(t\\notin T\\), some \\(P_{t|k}\\) is empty and the whole cylinder \\(\\Lambda_{t|k}\\) misses \\(T\\); so \\(T\\) is closed. For \\(t\\in T\\), the sets \\(P_{t|k}\\) are nonempty, closed and decreasing with diameters tending to \\(0\\), so they have one common point \\(\\psi(t)\\), by Cantor's intersection theorem (B1).\n\n*Step 3: \\(\\psi:T\\to P\\) is a homeomorphism.* If \\(t|k=u|k\\), then \\(d(\\psi(t),\\psi(u))\\leq2^{-k}\\); so \\(\\psi\\) is continuous. If \\(t\\) and \\(u\\) first differ at place \\(k+1\\), then \\(\\psi(t)\\) and \\(\\psi(u)\\) lie in the disjoint sets \\(P_{t|k+1}\\) and \\(P_{u|k+1}\\); so \\(\\psi\\) is injective. For \\(x\\in P\\), at each level exactly one \\(P_s\\) contains \\(x\\), and these \\(s\\) extend each other; they define \\(t\\in T\\) with \\(\\psi(t)=x\\). The same argument shows \\(\\psi(T_s)=P_s\\) for every \\(s\\), because a point of \\(P_s\\) is reached only along sequences that extend \\(s\\). The sets \\(T_s\\) form a base of \\(T\\) and the sets \\(P_s\\) are open, so \\(\\psi\\) is open.\n\n(2) By Proposition 3.3(1), some zero-dimensional Polish space \\(P\\) maps continuously and bijectively onto \\(X\\), say by \\(f\\). With \\(\\psi\\) as in (1), \\(f\\circ\\psi\\) is a continuous bijection of the closed set \\(T\\) onto \\(X\\).\n\n(3) Suppose \\(\\varphi:\\prod_kN_k\\to X\\) is a continuous bijection. \\(X\\) is uncountable, so infinitely many \\(N_k\\) have at least two elements; otherwise the product would be countable. Then every nonempty open subset of the product has at least two points, because a basic open set restricts only finitely many coordinates. But \\(2\\) is an isolated point of \\(X\\), so \\(\\varphi^{-1}(\\{2\\})\\) is open, and it has exactly one point because \\(\\varphi\\) is bijective. This is a contradiction. \\(\\square\\)\n\n**Example 3.5** (A tree, not a product). By Theorem 3.4(3), no product \\(\\prod_kN_k\\) maps continuously and bijectively onto \\(X=[0,1]\\cup\\{2\\}\\). A closed subset of \\(\\Lambda\\) does. Take a closed \\(T_1\\subseteq\\Lambda\\) and a continuous bijection \\(\\chi:T_1\\to[0,1]\\) (Theorem 3.4(2)), and let\n\\[\n\\begin{gathered}\nT=\\{(1,t_1,t_2,\\dots):t\\in T_1\\}\\\\\n\\cup\\{(2,1,1,1,\\dots)\\}.\n\\end{gathered}\n\\]\n\\(T\\) is closed, \\((2,1,1,\\dots)\\) is an isolated point of \\(T\\), and the map sending \\((1,t_1,t_2,\\dots)\\) to \\(\\chi(t)\\) and \\((2,1,1,\\dots)\\) to \\(2\\) is a continuous bijection of \\(T\\) onto \\(X\\). Below the node \\((2)\\) the tree has a single branch, while below \\((1)\\) it has uncountably many. A product cannot express branching that depends on the node. This is why the partitions in the proof of Theorem 3.4 produce a closed subset of \\(\\Lambda\\) and not a product: the number of nonempty pieces depends on the piece being divided, not only on the level.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "The Lusin–Souslin theorem",
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      },
      "full_conditions_and_proof": "#### The Lusin–Souslin theorem\n\n**Theorem 3.6** (Lusin–Souslin). Let \\(P\\) be a Polish space, \\(X\\) a Hausdorff space and \\(g:P\\to X\\) continuous and injective. Then \\(g(B)\\) is Borel in \\(X\\) for every Borel set \\(B\\subseteq P\\). In particular \\(g(P)\\) is Borel, and \\(g\\) is a Borel isomorphism of \\(P\\) onto \\(g(P)\\).\n\n**Proof.** *Reduction.* Let \\(B\\subseteq P\\) be Borel. Lemma 3.1 gives a zero-dimensional Polish topology on \\(P\\), finer than the given one and with the same Borel sets, in which \\(B\\) is open and closed. With the finer relative topology, \\(B\\) is a zero-dimensional Polish space, and \\(g\\) restricted to \\(B\\) is still continuous and injective. So it suffices to prove that \\(g(P)\\) is Borel when \\(P\\) is zero-dimensional. By Theorem 3.4(1), we may then assume that \\(P=T\\) is a closed subset of \\(\\Lambda\\).\n\n*Separation along the tree.* For \\(s\\in\\mathbb N^{<\\mathbb N}\\) put \\(Y_s=g(T_s)\\). Each \\(T_s\\) is closed in \\(\\Lambda\\), hence Polish, so each \\(Y_s\\) is a Polish image in \\(X\\). Because \\(g\\) is injective, \\(Y_s\\) is the disjoint union of the sets \\(Y_{sn}\\). We choose Borel sets \\(B_s\\supseteq Y_s\\) by induction on \\(|s|\\). Put \\(B_\\varnothing=X\\). Given \\(B_s\\), the separation theorem ([Theorem 2.4](#oa-fnd-pb-03)) gives pairwise disjoint Borel sets \\(E_{sn}\\supseteq Y_{sn}\\). Put\n\\[\nB_{sn}=E_{sn}\\cap B_s\\cap\\overline{Y_{sn}}.\n\\tag{3}\n\\]\nThen \\(Y_{sn}\\subseteq B_{sn}\\subseteq B_s\\), the sets \\(B_{sn}\\) (\\(n\\in\\mathbb N\\)) are pairwise disjoint, and \\(B_{sn}\\subseteq\\overline{Y_{sn}}\\). By induction on \\(k\\), the sets \\(B_s\\) with \\(|s|=k\\) are pairwise disjoint: two different sequences of length \\(k\\) either have the same parent, or have disjoint ancestors.\n\n*The Borel set.* Let \\(B^*=\\bigcap_k\\bigcup_{|s|=k}B_s\\), a Borel set. If \\(t\\in T\\), then \\(g(t)\\in Y_{t|k}\\subseteq B_{t|k}\\) for every \\(k\\), so \\(g(T)\\subseteq B^*\\). Conversely, let \\(x\\in B^*\\). For each \\(k\\), exactly one \\(s^{(k)}\\) of length \\(k\\) has \\(x\\in B_{s^{(k)}}\\), because these sets are pairwise disjoint. By (3), \\(B_{s^{(k+1)}}\\) lies inside the set indexed by the first \\(k\\) terms of \\(s^{(k+1)}\\), so uniqueness shows that \\(s^{(k+1)}\\) extends \\(s^{(k)}\\). So there is \\(t\\in\\Lambda\\) with \\(x\\in B_{t|k}\\subseteq\\overline{Y_{t|k}}\\) for every \\(k\\). Each \\(Y_{t|k}\\) is then nonempty, so each \\(T_{t|k}\\) is nonempty, and \\(t\\in T\\) because \\(T\\) is closed. Put \\(y=g(t)\\). If \\(x\\neq y\\), choose disjoint open sets \\(U\\ni y\\) and \\(W\\ni x\\). By continuity \\(T_{t|k}\\subseteq g^{-1}(U)\\) for large \\(k\\), so \\(Y_{t|k}\\subseteq U\\) and \\(\\overline{Y_{t|k}}\\subseteq\\overline U\\), which misses \\(W\\). This contradicts \\(x\\in\\overline{Y_{t|k}}\\cap W\\). Hence \\(x=g(t)\\in g(T)\\), and \\(B^*=g(T)\\).\n\nFinally, \\(g\\) is Borel because it is continuous, and it carries Borel sets of \\(P\\) to Borel sets of \\(X\\), which are Borel in \\(g(P)\\). \\(\\square\\)\n\n**Corollary 3.7.**\n\n1. Every Borel set in a Lusin space is itself a Lusin set.\n2. A Lusin set in a Hausdorff space, in particular in a metrizable space, is Borel.\n3. If \\(L\\) is a Lusin space and \\(h:L\\to X\\) is a continuous injection into a Hausdorff space, then \\(h(L)\\) is Borel and \\(h\\) is a Borel isomorphism of \\(L\\) onto \\(h(L)\\).\n4. A Lusin space, with its Borel sets, is a standard Borel space.\n\n**Proof.** Let \\(f:P\\to L\\) be a continuous bijection from a Polish space. (1) For Borel \\(B\\subseteq L\\), the set \\(f^{-1}(B)\\) is Borel in \\(P\\). Refine the topology of \\(P\\) by Lemma 3.1, so that it becomes zero-dimensional and \\(f^{-1}(B)\\) becomes open and closed. Then \\(f^{-1}(B)\\) is a zero-dimensional Polish space that \\(f\\) maps continuously and bijectively onto \\(B\\), and \\(B\\) is metrizable. (2) Apply Theorem 3.6 to \\(f\\) followed by the inclusion. (3) Apply Theorem 3.6 to \\(h\\circ f\\): for Borel \\(B\\subseteq L\\), \\(h(B)=(h\\circ f)(f^{-1}(B))\\). (4) By Theorem 3.6, \\(f\\) is a Borel isomorphism of \\(P\\) onto \\(L\\). \\(\\square\\)\n\n*Another proof of (1).* The Lusin sets of \\(L\\) whose complements are Lusin sets form a \\(\\sigma\\)-algebra that contains the open sets; so this \\(\\sigma\\)-algebra contains every Borel set. Complements are built into the definition. Finite intersections are Lusin sets by Proposition 3.3(3). A countable union \\(\\bigcup_nA_n\\) is the disjoint union of the Lusin sets \\(A_n\\cap(L\\setminus A_1)\\cap\\cdots\\cap(L\\setminus A_{n-1})\\), hence a Lusin set by Proposition 3.3(4), and its complement \\(\\bigcap_n(L\\setminus A_n)\\) is a Lusin set by Proposition 3.3(3). Open sets are Lusin sets, and so are their complements, which are closed, by Proposition 3.3(5).\n\n**Example 3.8** (Lusin sets in a space that is not Hausdorff). Let \\(A\\subseteq[0,1]\\) be a set that is not Borel. It exists by the counting argument ([Corollary 2.8](#oa-fnd-pb-12)), because \\([0,1]\\) is Borel isomorphic to \\(\\mathcal C\\) ([Theorem 5.2](#oa-fnd-pb-07)). Glue two copies of \\([0,1]\\) along \\(A\\): let \\(X\\) be the quotient of \\([0,1]\\times\\{1,2\\}\\) by the relation \\((a,1)\\sim(a,2)\\) for \\(a\\in A\\), with quotient map \\(q\\) and the quotient topology. Put \\(L_i=q([0,1]\\times\\{i\\})\\).\n\n- For each \\(i\\), the map \\(t\\mapsto q(t,i)\\) is a homeomorphism of \\([0,1]\\) onto \\(L_i\\) with its relative topology. It is continuous and injective. For \\(V\\subseteq[0,1]\\) open, the preimage of \\(q(V\\times\\{1,2\\})\\) is \\(V\\times\\{1,2\\}\\), so \\(q(V\\times\\{1,2\\})\\) is open in \\(X\\). Its intersection with \\(L_1\\) is \\(q(V\\times\\{1\\})\\), because \\(q(t,2)\\) lies in \\(L_1\\) only when \\(t\\in A\\), and then \\(q(t,2)=q(t,1)\\). So the map is open onto \\(L_1\\), and likewise onto \\(L_2\\).\n- Hence \\(L_1\\) and \\(L_2\\) are Lusin sets (Proposition 3.3(2)).\n- \\(L_1\\cap L_2=q(A\\times\\{1\\})\\) is homeomorphic to \\(A\\subseteq[0,1]\\). If \\(A\\) were a Lusin space, it would be Borel in \\([0,1]\\) by Corollary 3.7(2). So \\(L_1\\cap L_2\\) is not a Lusin set.\n- \\(X\\) is not Hausdorff. \\(A\\) is not closed, so some \\(a\\notin A\\) is a limit of points \\(a_n\\in A\\). Then \\(q(a,1)\\neq q(a,2)\\), but every neighbourhood of either point contains \\(q(a_n,1)=q(a_n,2)\\) for large \\(n\\).\n\nSo the intersection rule of Proposition 3.3(3) fails here. In its proof, the fibre product \\(P\\) is closed only because the diagonal of a Hausdorff space is closed.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PB-06",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "4. Borel maps between Souslin spaces",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "### 4. Borel maps between Souslin spaces\n\nA map is determined by its graph. When the target is countably separated, the graph of a Borel map is a Borel set. This turns questions about Borel maps into questions about Souslin sets, and gives the Borel form of the Lusin–Souslin theorem.\n\n**Proposition 4.1** (Graphs of Borel maps). Let \\((X,\\Sigma)\\) be a measurable space and \\((Y,\\mathcal B_Y)\\) a Borel space whose points are separated by a sequence \\((S_n)\\) of Borel sets. For every measurable \\(f:X\\to Y\\), with graph \\(\\operatorname{Gr}f=\\{(x,f(x)):x\\in X\\}\\),\n\\[\n\\begin{gathered}\n(X\\times Y)\\setminus\\operatorname{Gr}f\\\\\n=\\bigcup_n\\Big(\\big(f^{-1}(S_n)\\times(Y\\setminus S_n)\\big)\\\\\n\\cup\\big(f^{-1}(Y\\setminus S_n)\\times S_n\\big)\\Big),\n\\end{gathered}\n\\tag{4}\n\\]\nso \\(\\operatorname{Gr}f\\) belongs to \\(\\Sigma\\otimes\\mathcal B_Y\\). Consequently:\n\n1. If \\(X\\) and \\(Y\\) are topological spaces, \\(f\\) is Borel, and countably many Borel sets separate the points of \\(Y\\) (for example, \\(Y\\) separable metrizable), then \\(\\operatorname{Gr}f\\) is a Borel subset of \\(X\\times Y\\).\n2. If \\(Y\\) is countably separated, the points fixed by a Borel map \\(h:Y\\to Y\\) form a Borel set.\n3. Conversely, if the diagonal \\(\\Delta_Y=\\{(y,y):y\\in Y\\}\\) lies in \\(\\mathcal B_Y\\otimes\\mathcal B_Y\\), then \\(Y\\) is countably separated.\n\n**Proof.** (4) holds because \\(f(x)\\neq y\\) exactly when some \\(S_n\\) contains one of the two points and not the other.\n\n(1) The projections of \\(X\\times Y\\) are continuous, so a product of Borel sets is Borel in \\(X\\times Y\\), and \\(\\mathcal B(X)\\otimes\\mathcal B(Y)\\subseteq\\mathcal B(X\\times Y)\\).\n\n(2) The map \\(y\\mapsto(y,h(y))\\) is measurable into \\((Y\\times Y,\\mathcal B_Y\\otimes\\mathcal B_Y)\\), because its two coordinates are measurable and the coordinate maps generate the product \\(\\sigma\\)-algebra (F1). The fixed points form the preimage of \\(\\Delta_Y\\). This diagonal is the graph of the identity, so it lies in \\(\\mathcal B_Y\\otimes\\mathcal B_Y\\) by (4).\n\n(3) For any family \\(\\mathcal G\\) of sets, the sets that lie in the \\(\\sigma\\)-algebra generated by some countable subfamily of \\(\\mathcal G\\) form a \\(\\sigma\\)-algebra containing \\(\\mathcal G\\). Hence \\(\\Delta_Y\\) lies in the \\(\\sigma\\)-algebra generated by countably many products \\(A_n\\times B_n\\) of Borel sets. Let \\(x\\neq y\\). If no \\(A_n\\) contains exactly one of \\(x,y\\), then the points \\((x,x)\\) and \\((y,x)\\) lie in the same products \\(A_n\\times B_n\\). The sets that contain both points or neither form a \\(\\sigma\\)-algebra, so this \\(\\sigma\\)-algebra contains \\(\\Delta_Y\\). But \\(\\Delta_Y\\) contains \\((x,x)\\) and not \\((y,x)\\). So the countable family \\((A_n)\\) separates points. \\(\\square\\)\n\n**Remark 4.2.** For separable metrizable \\(X\\) and \\(Y\\) one can also argue with open sets. Every open subset of \\(X\\times Y\\) is then a countable union of open boxes, so \\(\\mathcal B(X\\times Y)=\\mathcal B(X)\\otimes\\mathcal B(Y)\\) (F1), and the measurability of a map into \\(X\\times Y\\) can be tested on preimages of open boxes. For instance, if \\(\\varphi\\) is a Borel map of a separable metric space \\(Z\\) into itself, then \\(z\\mapsto(z,\\varphi(z))\\) is Borel into \\(Z\\times Z\\), and the fixed points of \\(\\varphi\\) form the Borel set \\(\\{z:d(z,\\varphi(z))=0\\}\\). For nonseparable metrizable spaces the open boxes of a product can generate a strictly smaller \\(\\sigma\\)-algebra than the open sets (Exercise 4), and then this argument does not apply. We do not treat that case. All our applications concern separable spaces, where part (1) applies, and part (1) needs no condition on \\(X\\) at all.\n\n**Theorem 4.3** (Images and preimages under Borel maps). Let \\(X\\) be a Souslin space, \\(Y\\) a separable metrizable space and \\(f:X\\to Y\\) a Borel map.\n\n1. \\(f(A)\\) is a Souslin set in \\(Y\\) for every Souslin set \\(A\\subseteq X\\), in particular for every Borel set.\n2. \\(f^{-1}(A')\\) is a Souslin set in \\(X\\) for every Souslin set \\(A'\\subseteq Y\\).\n3. If \\(f\\) is injective, it is a Borel isomorphism of \\(X\\) onto \\(f(X)\\).\n4. If \\(X\\) is a Lusin space and \\(f\\) is injective, then \\(f(B)\\) is Borel in \\(Y\\) for every Borel \\(B\\subseteq X\\).\n5. Let \\(X'\\) be a standard Borel space and \\(f':X'\\to Y\\) an injective Borel map. Then \\(f'(X')\\) is Borel in \\(Y\\), and \\(f'\\) is a Borel isomorphism of \\(X'\\) onto \\(f'(X')\\). In particular, a Borel bijection between standard Borel spaces is a Borel isomorphism.\n\n**Proof.** Let \\(\\overline Y\\) be a Polish space that contains \\(Y\\) as a subspace, for instance the completion of \\(Y\\) for a compatible metric. (One can also embed \\(Y\\) into the cube \\([0,1]^{\\mathbb N}\\) by \\(y\\mapsto(\\min(1,d(a_n,y)))_n\\), with \\((a_n)\\) a dense sequence. This is the embedding used in [the lesson on the Effros Borel structure](effros-borel-structure.md#oa-fnd-ef-02) for Polish spaces, and that part of the argument does not use completeness.) The map \\(f\\) is Borel as a map into \\(\\overline Y\\), because the Borel sets of \\(Y\\) are the traces of those of \\(\\overline Y\\). Since \\(\\overline Y\\) is separable and metrizable, \\(\\operatorname{Gr}f\\) is Borel in \\(X\\times\\overline Y\\) by Proposition 4.1(1).\n\n(1) \\(X\\times\\overline Y\\) is a Souslin space, because countable products of Souslin spaces are Souslin spaces ([Proposition 2.2](#oa-fnd-pb-02)(3)). So its Borel subset \\(\\operatorname{Gr}f\\) is a Souslin set ([Corollary 2.6](#oa-fnd-pb-12)). The set \\(A\\times\\overline Y\\) is a Souslin set by Proposition 2.2(3), and so is its intersection with \\(\\operatorname{Gr}f\\), by Proposition 2.2(4). The projection of this intersection to \\(\\overline Y\\) is \\(f(A)\\), which is therefore a Souslin set by Proposition 2.2(2). It is the same space whether we view it in \\(\\overline Y\\) or in \\(Y\\).\n\n(2) \\(f^{-1}(A')\\) is the projection to \\(X\\) of \\(\\operatorname{Gr}f\\cap(X\\times A')\\), which is a Souslin set as in (1). So it is a Souslin set by Proposition 2.2(2).\n\n(3) Let \\(B\\subseteq X\\) be Borel. By (1), \\(f(B)\\) and \\(f(X\\setminus B)\\) are Souslin sets. They are disjoint because \\(f\\) is injective, and their union is \\(f(X)\\). By the separation theorem ([Theorem 2.4](#oa-fnd-pb-03)), some Borel set \\(E\\subseteq Y\\) contains \\(f(B)\\) and misses \\(f(X\\setminus B)\\). So \\(f(B)=f(X)\\cap E\\) is Borel in \\(f(X)\\), and \\(f^{-1}:f(X)\\to X\\) is Borel.\n\n(4) \\(\\overline Y\\) is a Lusin space, being Polish, and so is \\(X\\times\\overline Y\\), as a product of Lusin spaces ([Proposition 3.3](#oa-fnd-pb-04)(2) and (6)). By [Corollary 3.7](#oa-fnd-pb-05)(1), the Borel set \\(\\operatorname{Gr}f\\cap(B\\times\\overline Y)\\) is a Lusin set. The projection to \\(\\overline Y\\) is continuous, and it is injective on the graph because \\(f\\) is injective. By Corollary 3.7(3), its image \\(f(B)\\) is Borel in \\(\\overline Y\\), hence in \\(Y\\).\n\n(5) Choose a Borel isomorphism \\(j\\) of a Polish space \\(P\\) onto \\(X'\\). \\(P\\) is a Lusin space and a Souslin space (Proposition 3.3(2)). Apply (4) and (3) to the injective Borel map \\(f'\\circ j\\). For the last sentence, let \\(f':X'\\to Y'\\) be a Borel bijection between standard Borel spaces, and \\(k\\) a Borel isomorphism of \\(Y'\\) onto a Polish space \\(Q\\). Then \\(k\\circ f'\\) is a Borel isomorphism of \\(X'\\) onto \\(Q\\), and so \\(f'=k^{-1}\\circ(k\\circ f')\\) is a Borel isomorphism onto \\(Y'\\). \\(\\square\\)\n\n### 5. Standard Borel spaces\n\nWith the Lusin–Souslin theorem in hand, standard Borel spaces can be classified completely: they are determined by their cardinality. The same tools show that a countable separating family of Borel sets already generates the whole Borel structure of a Souslin–Borel space.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-PB-07",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "The isomorphism theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "anchor": "oa-fnd-pb-07",
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      "full_conditions_and_proof": "#### The isomorphism theorem\n\n**Lemma 5.1** (Borel Schröder–Bernstein). Let \\(X\\) and \\(Y\\) be Borel spaces, and let \\(f:X\\to Y\\) and \\(g:Y\\to X\\) be injective Borel maps that carry Borel sets to Borel sets. Then \\(X\\) and \\(Y\\) are isomorphic.\n\n**Proof.** Put \\(A_0=X\\setminus g(Y)\\), \\(A_{n+1}=g(f(A_n))\\) and \\(A=\\bigcup_nA_n\\). These sets are Borel, because \\(f\\) and \\(g\\) carry Borel sets to Borel sets. Define \\(h(x)=f(x)\\) for \\(x\\in A\\) and \\(h(x)=g^{-1}(x)\\) for \\(x\\notin A\\); this makes sense because \\(X\\setminus A\\subseteq X\\setminus A_0=g(Y)\\).\n\nThe map \\(h\\) is injective on \\(A\\) and on \\(X\\setminus A\\). If \\(f(a)=g^{-1}(x)\\) with \\(a\\in A_n\\) and \\(x\\notin A\\), then \\(x=g(f(a))\\in A_{n+1}\\), which is impossible; so \\(h\\) is injective. Let \\(y\\in Y\\). If \\(g(y)\\notin A\\), then \\(h(g(y))=y\\). If \\(g(y)\\in A_n\\), then \\(n\\geq1\\), because \\(A_0\\) misses \\(g(Y)\\); so \\(g(y)=g(f(a))\\) for some \\(a\\in A_{n-1}\\), and \\(y=f(a)=h(a)\\). So \\(h\\) is a bijection. For Borel \\(E\\subseteq Y\\), \\(h^{-1}(E)=(A\\cap f^{-1}(E))\\cup((X\\setminus A)\\cap g(E))\\) is Borel. For Borel \\(B\\subseteq X\\), \\(h(B)=f(A\\cap B)\\cup g^{-1}(B\\setminus A)\\) is Borel. \\(\\square\\)\n\n**Theorem 5.2** (Isomorphism theorem). Consider a standard Borel space \\(X\\).\n\n1. When \\(X\\) is countable, every subset of \\(X\\) is Borel.\n2. If \\(X\\) is uncountable, \\(X\\) is isomorphic to \\(\\mathcal C\\), and therefore to \\([0,1]\\), to \\(\\mathbb R\\), to \\(\\Lambda\\), and to every other uncountable standard Borel space.\n3. Two standard Borel spaces are isomorphic if and only if they have the same cardinality. The possible cardinalities are the finite ones, \\(\\aleph_0\\) and \\(2^{\\aleph_0}\\).\n\n**Proof.** We may assume that \\(X\\) is a Polish space with its Borel sets.\n\n(1) Points are closed, and every subset is a countable union of points.\n\n(2) Let \\((U_n)\\) be a countable base and \\(F(x)=(1_{U_n}(x))_n\\in\\mathcal C\\). The map \\(F\\) is Borel, since its coordinates are Borel and the coordinates generate the Borel sets of \\(\\mathcal C\\) (F1). It is injective, since the base separates points. By [Theorem 4.3](#oa-fnd-pb-06)(5), \\(F\\) is a Borel isomorphism of \\(X\\) onto the Borel set \\(F(X)\\); in particular it carries Borel sets to Borel sets. The perfect set lemma ([Lemma 1.3](#oa-fnd-pb-01)) gives a homeomorphism \\(e\\) of \\(\\mathcal C\\) onto a compact set \\(e(\\mathcal C)\\subseteq X\\). Since \\(e\\) is a homeomorphism onto \\(e(\\mathcal C)\\), it carries Borel sets of \\(\\mathcal C\\) to Borel subsets of \\(e(\\mathcal C)\\); these are Borel in \\(X\\), because the compact set \\(e(\\mathcal C)\\) is closed. And \\(e\\) is continuous. The Borel Schröder–Bernstein lemma gives \\(X\\cong\\mathcal C\\). The spaces \\([0,1]\\), \\(\\mathbb R\\) and \\(\\Lambda\\) are uncountable and Polish.\n\n(3) Any bijection between countable standard Borel spaces is an isomorphism by (1); uncountable ones are all isomorphic to \\(\\mathcal C\\) by (2); and \\(|\\mathcal C|=2^{\\aleph_0}\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-PB-08",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Souslin–Borel spaces and countable separating families",
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      "full_conditions_and_proof": "#### Souslin–Borel spaces and countable separating families\n\n**Definition 5.3.** A *Souslin–Borel space* is a countably separated Borel space that is the image of some standard Borel space under a Borel map.\n\n**Definition 5.4.** In a standard Borel space \\(X\\), call \\(A\\subseteq X\\) a *Souslin set* if \\(A=\\varnothing\\) or \\(A=f(Z)\\) for some standard Borel space \\(Z\\) and some Borel map \\(f:Z\\to X\\).\n\n**Lemma 5.5.**\n\n1. For a Polish space \\(X\\) and a set \\(A\\subseteq X\\), \\(A\\) is a Souslin set in the topological sense of Definition 2.1 exactly when it is one in the sense of Definition 5.4. So being a Souslin set of a standard Borel space does not depend on the Polish topology used to describe its Borel sets.\n2. With its relative Borel structure, a Souslin set \\(A\\) of a standard Borel space is a Souslin–Borel space.\n3. For a Souslin–Borel space \\(X\\) and a sequence \\((B_n)\\) of Borel sets that separates points, the formula \\(\\varphi(x)=(1_{B_n}(x))_n\\) defines an injective Borel map \\(\\varphi:X\\to\\mathcal C\\) whose image \\(\\varphi(X)\\) is a Souslin set in \\(\\mathcal C\\).\n\n**Proof.** (1) A topological Souslin set is \\(g(P)\\) with \\(g\\) continuous, hence Borel, and \\(P\\) Polish, hence standard. Conversely, if \\(f:Z\\to X\\) is Borel and \\(j\\) is a Borel isomorphism of a Polish space \\(P\\) onto \\(Z\\), then \\(f(Z)=(f\\circ j)(P)\\) is a Souslin set by [Theorem 4.3](#oa-fnd-pb-06)(1). The second sentence follows, because Definition 5.4 mentions no topology.\n\n(2) The traces on \\(A\\) of a countable separating family of \\(X\\) separate the points of \\(A\\). If \\(A=f(Z)\\), then \\(f\\) is Borel as a map into \\(A\\) with its relative structure.\n\n(3) \\(\\varphi\\) is injective because \\((B_n)\\) separates points. It is Borel because its coordinates \\(1_{B_n}\\) are Borel and the coordinates generate \\(\\mathcal B(\\mathcal C)\\) (F1). If \\(X=f(Z)\\) with \\(Z\\) standard and \\(f\\) Borel, then \\(\\varphi(X)=(\\varphi\\circ f)(Z)\\) is a Souslin set by (1). \\(\\square\\)\n\n**Theorem 5.6** (Blackwell). Let \\(X\\) be a Souslin–Borel space, \\((B_n)\\) a countable family of Borel sets, and \\(\\mathcal B_0\\) the \\(\\sigma\\)-algebra they generate. A Borel set \\(B\\subseteq X\\) belongs to \\(\\mathcal B_0\\) exactly when it is a union of atoms of \\(\\mathcal B_0\\), that is: whenever \\(x\\in B\\) and \\(y\\notin B\\), some \\(B_n\\) contains exactly one of \\(x\\) and \\(y\\). In particular:\n\n1. every sequence of Borel sets that separates points generates the Borel structure of \\(X\\);\n2. the map \\(\\varphi\\) of Lemma 5.5(3) is a Borel isomorphism of \\(X\\) onto the Souslin set \\(\\varphi(X)\\subseteq\\mathcal C\\).\n\nExample 5.7 proves that the countability hypothesis in (1) is essential.\n\n**Proof.** If no \\(B_n\\) separates \\(x\\) and \\(y\\), then no set of \\(\\mathcal B_0\\) does, because the sets that contain both points or neither form a \\(\\sigma\\)-algebra containing every \\(B_n\\). This proves \"only if\".\n\nConversely, let \\(B\\) be Borel and a union of atoms, and let \\(\\varphi=(1_{B_n})_n:X\\to\\mathcal C\\), a Borel map (F1). The hypothesis says exactly that \\(\\varphi(B)\\cap\\varphi(X\\setminus B)=\\varnothing\\). Both sets are Souslin sets in \\(\\mathcal C\\). Indeed, write \\(X=f(Z)\\) with \\(Z\\) standard and \\(f\\) Borel. Then \\(f^{-1}(B)\\) is a Borel subset of \\(Z\\), hence standard (F4), and \\(\\varphi\\circ f\\) maps it onto \\(\\varphi(B)\\) because \\(f\\) is onto; so Lemma 5.5(1) applies, and likewise to \\(X\\setminus B\\). By the separation theorem ([Theorem 2.4](#oa-fnd-pb-03)), there is a Borel set \\(D\\subseteq\\mathcal C\\) with \\(\\varphi(B)\\subseteq D\\) and \\(D\\cap\\varphi(X\\setminus B)=\\varnothing\\). Then \\(B=\\varphi^{-1}(D)\\), and \\(\\varphi^{-1}(D)\\in\\mathcal B_0\\) because \\(\\varphi\\) is \\(\\mathcal B_0\\)-measurable (F1).\n\n(1) If \\((B_n)\\) separates points, the atoms are single points, and every Borel set is a union of atoms.\n\n(2) \\(\\varphi\\) is injective and Borel. The sets \\(E\\subseteq X\\) for which \\(\\varphi(E)\\) is Borel in \\(\\varphi(X)\\) include each \\(B_n\\), since \\(\\varphi(B_n)=\\varphi(X)\\cap\\{c:c_n=1\\}\\). They are closed under complements and countable unions, since \\(\\varphi\\) is injective. So they include \\(\\mathcal B_0\\), which is the whole Borel structure by (1). \\(\\square\\)\n\n*Another proof of (1).* Let \\((B_n)\\) separate points, let \\(B\\) be Borel, and compare \\(\\varphi\\) with \\(\\psi=(1_B,1_{B_1},1_{B_2},\\dots)\\). By Lemma 5.5(3), \\(\\psi(X)\\) is a Souslin set in \\(\\mathcal C\\). The map \\(\\varphi\\circ\\psi^{-1}\\), which drops the first coordinate, is a one-to-one continuous map of \\(\\psi(X)\\) onto \\(\\varphi(X)\\). By [Theorem 4.3](#oa-fnd-pb-06)(3) it is a Borel isomorphism. So \\(1_B\\) is a Borel function of \\(\\varphi\\): the set \\(\\varphi(B)\\) is Borel in \\(\\varphi(X)\\), say \\(\\varphi(B)=\\varphi(X)\\cap D\\) with \\(D\\) Borel in \\(\\mathcal C\\), and \\(B=\\varphi^{-1}(D)\\) lies in the \\(\\sigma\\)-algebra generated by the \\(B_n\\) (F1).\n\n**Example 5.7** (Countability is needed in (1)). The singletons of \\([0,1]\\) are Borel sets, and they separate points. The countable sets and their complements form a \\(\\sigma\\)-algebra that contains every singleton, and each of its members lies in the \\(\\sigma\\)-algebra generated by the singletons; so it is the \\(\\sigma\\)-algebra they generate. It does not contain \\([0,\\tfrac12]\\), which is uncountable and has an uncountable complement.\n\n**Corollary 5.8.**\n\n1. Let \\((X,\\mathcal B)\\) be a Souslin–Borel space and \\(\\mathcal B'\\subseteq\\mathcal B\\) a \\(\\sigma\\)-algebra that is countably generated and separates points. Then \\(\\mathcal B'=\\mathcal B\\).\n2. Let \\(f\\) be a Borel bijection of a Souslin–Borel space onto a Borel space whose points are separated by countably many Borel sets. Then \\(f\\) is a Borel isomorphism.\n\n**Proof.** (1) Let \\(\\mathcal B'\\) be generated by \\((G_n)\\). Since \\(\\mathcal B'\\) separates points, so does \\((G_n)\\), because the sets that do not separate two given points form a \\(\\sigma\\)-algebra. Apply Theorem 5.6(1).\n\n(2) Let \\(f:X\\to Y\\) be the bijection and \\((S_n)\\) a sequence of Borel sets separating the points of \\(Y\\). As \\(f\\) is bijective, the sets \\(f^{-1}(S_n)\\) separate the points of \\(X\\), so they generate its Borel structure by Theorem 5.6(1). The \\(\\sigma\\)-algebra they generate is \\(\\{f^{-1}(E):E\\in\\sigma(S_n:n)\\}\\). So each Borel \\(B\\subseteq X\\) equals \\(f^{-1}(E)\\) for a Borel \\(E\\subseteq Y\\), and \\(f(B)=E\\) because \\(f\\) is bijective. \\(\\square\\)\n\n**Example 5.9** (A Souslin–Borel space that is not standard). The Souslin set \\(D\\subseteq\\mathcal C\\) of [Example 2.9](#oa-fnd-pb-12), with its relative Borel structure, is a Souslin–Borel space, by Lemma 5.5(1) and (2). It is not standard. If it were, the inclusion \\(D\\to\\mathcal C\\) would be an injective Borel map on a standard Borel space, and \\(D\\) would be Borel by Theorem 4.3(5).\n\n## B. Integrate over the encoded space\n\nAn image need not be Borel to be measurable for the completed measures under consideration. Keep the distinction between universal measurability and Borel measurability visible while reading the measure proofs. The sigma-finite standard-model theorem supplies the measure structure used in later operator-algebra decompositions.\n\n",
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      "id": "OA-FND-PB-09",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "6. Measures on Souslin sets",
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      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "### 6. Measures on Souslin sets\n\nA Souslin set need not be Borel, but it is always measurable. For every \\(\\sigma\\)-finite measure it is squeezed between two Borel sets of equal measure, and for every measure it is measurable in Carathéodory's sense. As a consequence, every \\(\\sigma\\)-finite measure on a Souslin–Borel space lives on a standard Borel set.\n\nLet \\((X,\\mathcal B)\\) be a Borel space and \\(\\mu\\) a measure on it, that is, a countably additive map \\(\\mathcal B\\to[0,\\infty]\\). Put\n\\[\n\\begin{gathered}\n\\mu^*(E)\\\\\n=\\inf\\{\\mu(B):\\\\\nB\\in\\mathcal B,\\ E\\subseteq B\\},\\\\\nE\\subseteq X.\n\\end{gathered}\n\\tag{5}\n\\]\nA set \\(E\\) is *\\(\\mu\\)-measurable* if there are Borel sets \\(B_1\\subseteq E\\subseteq B_2\\) with \\(\\mu(B_2\\setminus B_1)=0\\). These sets form a \\(\\sigma\\)-algebra, the completion of \\(\\mathcal B\\) for \\(\\mu\\): complements and countable unions of such sandwiches are again sandwiches, and the differences stay null. A set \\(E\\) is *\\(\\mu^*\\)-measurable*, in Carathéodory's sense, if \\(\\mu^*(W)=\\mu^*(W\\cap E)+\\mu^*(W\\setminus E)\\) for every \\(W\\subseteq X\\). The \\(\\mu^*\\)-measurable sets form a \\(\\sigma\\)-algebra by Carathéodory's theorem, because \\(\\mu^*\\) is an outer measure: it is monotone, \\(\\mu^*(\\varnothing)=0\\), and it is countably subadditive by Lemma 6.1(1).\n\n**Lemma 6.1** (Outer measure).\n\n1. The function \\(\\mu^*\\) of (5) is countably subadditive, and every \\(E\\) has a Borel \\(H\\supseteq E\\) with \\(\\mu(H)=\\mu^*(E)\\).\n2. If \\(E_1\\subseteq E_2\\subseteq\\cdots\\), then \\(\\mu^*(\\bigcup_kE_k)=\\lim_k\\mu^*(E_k)\\).\n3. Borel sets are \\(\\mu^*\\)-measurable.\n4. If \\(\\mu\\) is \\(\\sigma\\)-finite, some finite measure \\(\\nu\\) has the same null sets, and a set is \\(\\mu\\)-measurable exactly when it is \\(\\nu\\)-measurable.\n\n**Proof.** (1) If \\(\\mu^*(E)=\\infty\\), take \\(H=X\\). Otherwise choose Borel \\(H_j\\supseteq E\\) with \\(\\mu(H_j)<\\mu^*(E)+1/j\\) and let \\(H=\\bigcap_jH_j\\). For subadditivity, the union of such sets for the \\(E_k\\) covers \\(\\bigcup_kE_k\\).\n\n(2) Choose \\(H_k\\) as in (1) for \\(E_k\\) and put \\(H'_k=\\bigcap_{j\\geq k}H_j\\). Then \\(E_k\\subseteq H'_k\\subseteq H_k\\), so \\(\\mu(H'_k)=\\mu^*(E_k)\\), and the \\(H'_k\\) increase. Hence \\[\n\\begin{gathered}\n\\mu^*(\\bigcup_kE_k)\\\\\n\\leq\\mu(\\bigcup_kH'_k)\\\\\n=\\lim_k\\mu^*(E_k)\\\\\n\\leq\\mu^*(\\bigcup_kE_k).\n\\end{gathered}\n\\]\n\n(3) Let \\(K\\) be Borel and \\(W\\subseteq X\\) with \\(\\mu^*(W)<\\infty\\); otherwise there is nothing to prove. With \\(H\\) as in (1) for \\(W\\), \\[\n\\begin{gathered}\n\\mu^*(W\\cap K)+\\mu^*(W\\setminus K)\\\\\n\\leq\\mu(H\\cap K)+\\mu(H\\setminus K)\\\\\n=\\mu^*(W).\n\\end{gathered}\n\\] The reverse inequality is subadditivity.\n\n(4) Write \\(X\\) as a disjoint union of Borel sets \\(X_n\\) with \\(\\mu(X_n)<\\infty\\), and put \\(\\nu(B)=\\sum_n2^{-n}\\mu(B\\cap X_n)/(1+\\mu(X_n))\\). Then \\(\\nu(X)\\leq1\\), and \\(\\nu(B)=0\\) exactly when \\(\\mu(B\\cap X_n)=0\\) for every \\(n\\), that is, when \\(\\mu(B)=0\\). The completion depends only on the null sets. \\(\\square\\)\n\n**Theorem 6.2** (Souslin sets are measurable). Let \\(X\\) be a Hausdorff space, \\(\\mu\\) a measure on \\(\\mathcal B(X)\\), and \\(A\\) a Polish image in \\(X\\), for instance a Souslin set of a metrizable \\(X\\).\n\n1. If \\(\\mu\\) is finite, then \\(\\mu^*(A)=\\sup\\{\\mu(K):K\\subseteq A\\text{ compact}\\}\\), and \\(A\\) is \\(\\mu\\)-measurable.\n2. If \\(\\mu\\) is \\(\\sigma\\)-finite, \\(A\\) is \\(\\mu\\)-measurable.\n3. For every measure \\(\\mu\\), \\(A\\) is \\(\\mu^*\\)-measurable.\n\nIn particular, in a standard Borel space every Souslin set is \\(\\mu\\)-measurable for every \\(\\sigma\\)-finite \\(\\mu\\). By [Lemma 5.5](#oa-fnd-pb-08)(1), it does not matter which Polish topology is used to describe the Borel sets.\n\n\n\n**Proof.** We may assume \\(A\\neq\\varnothing\\), so \\(A=g(\\Lambda)\\) with \\(g:\\Lambda\\to X\\) continuous, as noted after [Definition 2.1](#oa-fnd-pb-02). For a finite sequence \\(s=(n_1,\\dots,n_k)\\) let\n\\[\nL_s=\\{t\\in\\Lambda: \\ t_i\\leq n_i\\text{ for }i\\leq k\\},\n\\tag{6}\n\\]\na closed set. \\(L_\\varnothing=\\Lambda\\), and \\(L_{sm}\\) increases with \\(m\\) and has union \\(L_s\\).\n\n(1) Fix \\(\\varepsilon>0\\). The sets \\(g(L_{sm})\\) increase in \\(m\\) with union \\(g(L_s)\\). So by Lemma 6.1(2) we can choose \\(n_1,n_2,\\ldots\\) one at a time with \\(\\mu^*(g(L_{(n_1,\\dots,n_k)}))>\\mu^*(A)-\\varepsilon\\) for every \\(k\\). Write \\(L_k=L_{(n_1,\\dots,n_k)}\\), with \\(L_s\\) as in (6), and \\(K=\\bigcap_kL_k=\\prod_i\\{1,\\dots,n_i\\}\\), a compact set by (B2). Then \\(C=g(K)\\) is a compact subset of \\(A\\).\n\n*Claim: \\(\\bigcap_k\\overline{g(L_k)}\\subseteq C\\).* Let \\(y\\notin C\\). Because \\(C\\) is compact and \\(X\\) is Hausdorff, there are disjoint open sets \\(U\\ni y\\) and \\(W\\supseteq C\\); then \\(\\overline U\\cap W=\\varnothing\\). Let \\(F=g^{-1}(\\overline U)\\). It is a closed subset of \\(\\Lambda\\), and it is disjoint from \\(K\\), because \\(g(K)=C\\subseteq W\\).\n\nSuppose \\(F\\cap L_k\\neq\\varnothing\\) for every \\(k\\). Call a finite sequence \\(s=(s_1,\\dots,s_j)\\) *good* if \\(s_i\\leq n_i\\) for \\(i\\leq j\\) and \\(F\\cap L_k\\cap\\Lambda_s\\neq\\varnothing\\) for all \\(k\\geq j\\). The empty sequence is good, by our assumption. Let \\(s\\) be good with \\(|s|=j\\). For each \\(k\\geq j+1\\), every point of \\(L_k\\cap\\Lambda_s\\) has \\((j+1)\\)-st entry at most \\(n_{j+1}\\), so it lies in some \\(\\Lambda_{sm}\\) with \\(m\\leq n_{j+1}\\). Hence some such \\(m\\) has \\(F\\cap L_k\\cap\\Lambda_{sm}\\neq\\varnothing\\). There are only finitely many such \\(m\\), so one \\(m\\) serves infinitely many \\(k\\), and then all \\(k\\geq j+1\\), because the \\(L_k\\) decrease. So every good sequence has a good one-step extension, and we obtain \\(t\\in K\\) with every \\(t|j\\) good. Each cylinder \\(\\Lambda_{t|j}\\) meets \\(F\\). These cylinders form a neighbourhood base at \\(t\\), and \\(F\\) is closed, so \\(t\\in F\\cap K\\), which is impossible. Hence \\(F\\cap L_k=\\varnothing\\) for some \\(k\\). Then \\(U\\) is a neighbourhood of \\(y\\) that misses \\(g(L_k)\\), so \\(y\\notin\\overline{g(L_k)}\\). This proves the claim.\n\nSince \\(\\mu\\) is finite and the closed sets \\(\\overline{g(L_k)}\\) decrease,\n\\[\n\\begin{gathered}\n\\mu(C)\\\\\n\\geq\\mu\\Big(\\bigcap_k\\overline{g(L_k)}\\Big)\\\\\n=\\lim_k\\mu\\big(\\overline{g(L_k)}\\big)\\\\\n\\geq\\lim_k\\mu^*\\big(g(L_k)\\big)\\\\\n\\geq\\mu^*(A)-\\varepsilon .\n\\end{gathered}\n\\]\nHere the first inequality is the claim, the equality is continuity from above of the finite measure \\(\\mu\\), and the next inequality holds because \\(\\overline{g(L_k)}\\) is a Borel set containing \\(g(L_k)\\). So the supremum equals \\(\\mu^*(A)\\). Choose compact \\(C_j\\subseteq A\\) with \\(\\mu(C_j)>\\mu^*(A)-1/j\\), let \\(K'=\\bigcup_jC_j\\), a Borel set because compact subsets of a Hausdorff space are closed, and let \\(H\\supseteq A\\) be Borel with \\(\\mu(H)=\\mu^*(A)\\) (Lemma 6.1(1)). Then \\(K'\\subseteq A\\subseteq H\\) and \\(\\mu(H\\setminus K')\\leq\\mu^*(A)-\\mu(C_j)<1/j\\) for every \\(j\\), so \\(\\mu(H\\setminus K')=0\\).\n\n(2) Apply (1) to the finite measure \\(\\nu\\) of Lemma 6.1(4), which has the same measurable sets as \\(\\mu\\).\n\n(3) Let \\(W\\subseteq X\\) with \\(\\mu^*(W)<\\infty\\), and \\(H\\supseteq W\\) Borel with \\(\\mu(H)=\\mu^*(W)\\). Part (1), for the finite measure \\(\\mu_H(B)=\\mu(B\\cap H)\\), gives Borel sets \\(K'\\subseteq A\\subseteq L'\\) with \\(\\mu(H\\cap(L'\\setminus K'))=0\\). Then \\[\n\\begin{gathered}\n\\mu^*(W\\cap A)\\\\\n\\leq\\mu^*(W\\cap K')+\\mu(H\\cap(L'\\setminus K'))\\\\\n=\\mu^*(W\\cap K'),\n\\end{gathered}\n\\] and \\(W\\setminus A\\subseteq W\\setminus K'\\). By Lemma 6.1(3), \\[\n\\begin{gathered}\n\\mu^*(W\\cap A)+\\mu^*(W\\setminus A)\\\\\n\\leq\\mu^*(W\\cap K')+\\mu^*(W\\setminus K')\\\\\n=\\mu^*(W).\n\\end{gathered}\n\\] Subadditivity gives the reverse inequality. \\(\\square\\)\n\n**Corollary 6.3** (Standard measures). Let \\(X\\) be a Souslin–Borel space and \\(\\mu\\) a \\(\\sigma\\)-finite measure on it. Then \\(\\mu\\) is standard: some Borel set \\(N\\subseteq X\\) with \\(\\mu(N)=0\\) has \\(X\\setminus N\\) standard. One can take \\(X\\setminus N\\) isomorphic to a \\(\\sigma\\)-compact subset of \\(\\mathcal C\\).\n\n**Proof.** By [Theorem 5.6](#oa-fnd-pb-08)(2), we may assume that \\(X\\) is a Souslin set in \\(\\mathcal C\\) with its relative Borel structure. Put \\(\\nu(E)=\\mu(E\\cap X)\\) for Borel \\(E\\subseteq\\mathcal C\\). This is a \\(\\sigma\\)-finite measure on \\(\\mathcal C\\): if \\(X=\\bigcup_nX_n\\) with \\(\\mu(X_n)<\\infty\\) and \\(X_n=X\\cap E_n\\) with \\(E_n\\) Borel in \\(\\mathcal C\\), then \\(\\nu(E_n)<\\infty\\) and \\(\\nu(\\mathcal C\\setminus\\bigcup_nE_n)=0\\). Apply Theorem 6.2(1) to a finite measure with the same null sets as \\(\\nu\\) (Lemma 6.1(4)). It gives a \\(\\sigma\\)-compact set \\(X_0\\subseteq X\\) and a Borel set \\(X_1\\supseteq X\\) with \\(\\nu(X_1\\setminus X_0)=0\\). Let \\(N=X\\setminus X_0=X\\cap(\\mathcal C\\setminus X_0)\\), a Borel subset of \\(X\\). Then \\(\\mu(N)=\\nu(\\mathcal C\\setminus X_0)=\\nu(X_1\\setminus X_0)=0\\), because \\(\\nu(\\mathcal C\\setminus X_1)=\\mu(X\\setminus X_1)=0\\). Finally \\(X\\setminus N=X_0\\) is a Borel subset of \\(\\mathcal C\\), hence standard (F4). \\(\\square\\)\n\n**Example 6.4** (\\(\\sigma\\)-finiteness is needed). Let \\(D\\subseteq\\mathcal C\\) be the Souslin set of [Example 2.9](#oa-fnd-pb-12). The counting measure on \\(D\\), which gives each subset its number of points, is a measure that is not standard: its only null set is \\(\\varnothing\\), and \\(D\\) is not standard (Example 5.9). \\(D\\) is uncountable, since countable sets are Borel, so this measure is not \\(\\sigma\\)-finite. Hence Corollary 6.3 needs \\(\\sigma\\)-finiteness. On the other hand, every \\(\\sigma\\)-finite Borel measure on \\(\\mathcal C\\) still measures \\(D\\), by Theorem 6.2.\n\n## C. Choose a representative with the right measurability\n\nA section is a right inverse of a surjection. Its existence as a set map says nothing about its regularity. The proofs below construct representatives through shrinking choices and state exactly when the section is Borel and when it is measurable for all the specified measures. Closed equivalence classes and cosets are two separate applications of this mechanism.\n\n### 7. Choosing points in a Borel way\n\nGiven an equivalence relation, or a map onto a set, we often need to choose one point from each class or fibre and to make the choice measurable. This section makes a Borel choice when the classes are closed and saturations of open sets are Borel. It applies this to coset spaces of Polish groups. For an arbitrary Borel surjection it makes a choice that is measurable for every \\(\\sigma\\)-finite measure.\n\n",
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      "id": "OA-FND-PB-10",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Borel transversals and coset spaces",
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      "full_conditions_and_proof": "#### Borel transversals and coset spaces\n\n**Theorem 7.1** (Borel transversals). Let \\(X\\) be a separable metrizable space with a compatible metric \\(d\\), and \\(R\\) an equivalence relation on \\(X\\) whose classes are complete for \\(d\\); for example, \\(X\\) Polish, \\(d\\) complete and every class closed. For \\(E\\subseteq X\\) let \\(R(E)\\) be the set of points equivalent to a point of \\(E\\), and let \\([x]\\) be the class of \\(x\\). Assume that \\(R(U)\\) is Borel for every open \\(U\\subseteq X\\). Then:\n\n1. There is a Borel map \\(\\eta:X\\to X\\) with \\(\\eta(x)\\in[x]\\) for all \\(x\\), and \\(\\eta(x)=\\eta(y)\\) whenever \\([x]=[y]\\).\n2. The set \\(S=\\{x:\\eta(x)=x\\}\\) is Borel and meets every class exactly once.\n\nIf \\(R(F)\\) is Borel for every closed \\(F\\subseteq X\\), then \\(R(U)\\) is Borel for every open \\(U\\), and the same conclusions hold.\n\nExample 7.2 proves that replacing the open sets by their closures can lose a class. The closed-saturation hypothesis is therefore reduced to the open-saturation case before using the construction.\n\n**Proof.** *A system of open sets.* Choose open sets \\(X(s)\\), for finite sequences \\(s\\) of length at least \\(1\\), with\n\\[\n\\begin{gathered}\nX=\\bigcup_nX(n),\\\\\nX(s)=\\bigcup_nX(sn),\\\\\n\\overline{X(sn)}\\subseteq X(s),\\\\\n\\operatorname{diam}X(s)\\leq2^{-|s|}.\n\\end{gathered}\n\\tag{7}\n\\]\nThis is possible. An open set \\(V\\) is a union of open balls \\(\\{y:d(x,y)<r_x\\}\\), \\(x\\in V\\), with \\(r_x\\leq2^{-k-2}\\) and the ball of radius \\(2r_x\\) inside \\(V\\). These balls have closures inside \\(V\\) and diameters at most \\(2^{-k-1}\\), and countably many of them cover \\(V\\), because \\(V\\) is second countable and hence Lindelöf (B4). Index them by \\(\\mathbb N\\), adding empty sets if needed. By induction, for each \\(k\\) the sets \\(X(s)\\) with \\(|s|=k\\) cover \\(X\\).\n\n*The lexicographic choice.* Order \\(\\mathbb N^k\\) lexicographically: \\(s\\) comes before \\(u\\) if \\(s_i<u_i\\) at the first place where they differ. This is a well-ordering: in any nonempty subset choose the least first coordinate, then the least second coordinate among tuples with that first coordinate, and continue through the finitely many coordinates. For a class \\(H\\) and \\(k\\geq1\\), let \\(\\kappa_k(H)\\) be the first \\(s\\in\\mathbb N^k\\) with \\(H\\cap X(s)\\neq\\varnothing\\).\n\n(a) *\\(\\kappa_{k+1}(H)\\) extends \\(\\kappa_k(H)\\).* Write \\(\\kappa_{k+1}(H)=(u,m)\\) with \\(u\\in\\mathbb N^k\\). \\(H\\) meets \\(X(u,m)\\subseteq X(u)\\), so \\(\\kappa_k(H)\\) is \\(u\\) or comes before \\(u\\). If some \\(v\\in\\mathbb N^k\\) before \\(u\\) had \\(H\\cap X(v)\\neq\\varnothing\\), then \\(H\\) would meet some \\(X(v,j)\\), and \\((v,j)\\) comes before \\((u,m)\\), against the choice of \\((u,m)\\). So \\(\\kappa_k(H)=u\\).\n\n(b) For \\(s\\in\\mathbb N^k\\),\n\\[\n\\begin{gathered}\n\\{x:\\kappa_k([x])=s\\}\\\\\n=R(X(s))\\setminus\\\\\n\\bigcup\\{R(X(v)):v\\in\\mathbb N^k\\text{ comes before }s\\},\n\\end{gathered}\n\\]\na Borel set, since \\([x]\\) meets \\(X(v)\\) exactly when \\(x\\in R(X(v))\\), and the union is countable.\n\n(c) The sets \\(H\\cap X(\\kappa_k(H))\\) are nonempty and decrease in \\(k\\), by (a). The closure in \\(H\\) of \\(H\\cap X(\\kappa_{k+1}(H))\\) lies in \\(H\\cap\\overline{X(\\kappa_{k+1}(H))}\\subseteq H\\cap X(\\kappa_k(H))\\). These closures are nonempty, closed in \\(H\\) and decreasing, with diameters at most \\(2^{-k-1}\\). As \\((H,d)\\) is complete, they have exactly one common point \\(p(H)\\), by Cantor's intersection theorem (B1), and \\(p(H)\\) is also the only point of \\(\\bigcap_k(H\\cap X(\\kappa_k(H)))\\).\n\n*The selector.* Put \\(\\eta(x)=p([x])\\). Then \\(\\eta(x)\\in[x]\\), and \\(\\eta\\) is constant on classes. For each finite \\(u\\) with \\(X(u)\\neq\\varnothing\\) fix a point \\(c(u)\\in X(u)\\), and let \\(\\eta_k(x)=c(\\kappa_k([x]))\\). Each \\(\\eta_k\\) is Borel, being constant on each of the countably many Borel sets of (b). Both \\(\\eta(x)\\) and \\(\\eta_k(x)\\) lie in \\(X(\\kappa_k([x]))\\), so \\(d(\\eta(x),\\eta_k(x))\\leq2^{-k}\\). A pointwise limit of Borel maps into a metric space is Borel: for closed \\(F\\), \\[\n\\begin{gathered}\n\\eta^{-1}(F)\\\\\n=\\bigcap_j\\bigcup_n\\bigcap_{k\\geq n}\\eta_k^{-1}(\\{y:d(y,F)<1/j\\}).\n\\end{gathered}\n\\] So \\(\\eta\\) is Borel.\n\n(2) \\(S\\) consists of the points fixed by the Borel map \\(\\eta\\). The separable metrizable space \\(X\\) is countably separated, so \\(S\\) is Borel by [Proposition 4.1](#oa-fnd-pb-06)(2). A point \\(x\\) lies in \\(S\\) exactly when it is the point \\(p([x])\\) of its class.\n\nFor the last sentence: an open set \\(U\\) is the union of the closed sets \\(\\{x:d(x,X\\setminus U)\\geq1/n\\}\\), and \\(R\\) commutes with unions. \\(\\square\\)\n\n*Remarks.* (a) The proof uses only the saturations of the countably many sets \\(X(s)\\). (b) The sets \\(X(s)\\) must be open. If one runs the construction with their closures instead, step (a) can fail: a class may meet \\(\\overline{X(u)}\\) only in a boundary point that lies in no \\(\\overline{X(uj)}\\), and then the choice at the next level moves to another branch. Example 7.2 gives a class that the resulting set misses entirely. This is why the version with closed sets is proved by showing that the saturations of open sets are then Borel, as in the last paragraph of the proof.\n\n**Example 7.2** (Closures instead of open sets). Let \\(X=[-1,1]\\) with \\(d(x,y)=|x-y|\\), and let \\(x\\mathrel Ry\\) mean \\(|x|=|y|\\). The classes \\(\\{x,-x\\}\\) are closed, and \\(R(E)=E\\cup(-E)\\) is open for open \\(E\\) and closed for closed \\(E\\). So both forms of the hypothesis of Theorem 7.1 hold, and the theorem gives a Borel transversal, for instance \\([0,1]\\). Choose a system (7) with\n\\[\n\\begin{gathered}\nX(1)=(0,\\tfrac14),\\\\\nX(2)=(-\\tfrac38,-\\tfrac18),\\\\\nX(1,j)=(2^{-j-3},\\tfrac14-2^{-j-3}),\\\\\nX(2,j)=(-\\tfrac38+2^{-j-3},-\\tfrac18-2^{-j-3})\\\\\n(j\\geq1),\n\\end{gathered}\n\\]\nand the remaining sets chosen in any way that satisfies (7). These sets do satisfy (7): the \\(X(1,j)\\) increase to \\(X(1)\\), the \\(X(2,j)\\) increase to \\(X(2)\\), their closures lie inside, and all their diameters are at most \\(\\tfrac14\\). Now run the construction with closures. At level \\(k\\) it keeps, from each class, the points that lie in \\(\\overline{X(s)}\\) for the first \\(s\\in\\mathbb N^k\\) whose closure meets the class. Take the class \\(H=\\{\\tfrac14,-\\tfrac14\\}\\).\n\n- Level 1: \\(\\overline{X(1)}=[0,\\tfrac14]\\) meets \\(H\\), so the level-1 set keeps only the point \\(\\tfrac14\\) of \\(H\\).\n- Level 2: every \\(\\overline{X(1,j)}\\) lies in \\((0,\\tfrac14)\\) and misses \\(H\\), while \\(\\overline{X(2,1)}=[-\\tfrac5{16},-\\tfrac3{16}]\\) contains \\(-\\tfrac14\\). So the level-2 set keeps only the point \\(-\\tfrac14\\) of \\(H\\).\n\nThe final set is the intersection of the sets of all levels, so it misses \\(H\\) entirely, and it is not a transversal. Writing \\(S_k\\) for the set kept at level \\(k\\), what fails is the inclusion of the closure of \\(H\\cap S_{k+1}\\) in \\(H\\cap S_k\\). With the open sets \\(X(s)\\), step (a) of the proof of Theorem 7.1 rules this out.\n\n**Example 7.3** (Both hypotheses are needed).\n\n(a) *Closed classes.* On \\(X=\\mathbb R\\) let \\(x\\mathrel Ry\\) mean \\(x-y\\in\\mathbb Q\\). The classes \\(x+\\mathbb Q\\) are countable and dense, so not closed, and \\(R(U)=U+\\mathbb Q\\) is open for open \\(U\\). No Lebesgue measurable set \\(S\\) meets every class exactly once. If one did, \\(\\mathbb R\\) would be the disjoint union of the translates \\(S+q\\), \\(q\\in\\mathbb Q\\). If \\(\\lambda(S)=0\\), then \\(\\lambda(\\mathbb R)=0\\). If \\(\\lambda(S)>0\\), some \\(S_0=S\\cap[n,n+1]\\) has \\(\\lambda(S_0)>0\\), and the disjoint translates \\(S_0+q\\), \\(q\\in\\mathbb Q\\cap[0,1]\\), all lie in \\([n,n+2]\\) and all have measure \\(\\lambda(S_0)\\); infinitely many of them give infinite measure inside \\([n,n+2]\\). Both cases are absurd. In particular no Borel transversal exists.\n\n(b) *Borel saturations.* Let \\(A\\subseteq\\mathcal C\\) be a set that is not Borel ([Corollary 2.8](#oa-fnd-pb-12)). On \\(X=\\mathcal C\\times\\{0,1\\}\\) let \\((c,i)\\mathrel R(c',i')\\) mean that \\(c=c'\\) and either \\(i=i'\\) or \\(c\\in A\\). The classes have one or two points, so they are closed. The saturation of the open set \\(\\mathcal C\\times\\{1\\}\\) is \\((\\mathcal C\\times\\{1\\})\\cup(A\\times\\{0\\})\\), which is not Borel, since its preimage under \\(c\\mapsto(c,0)\\) is \\(A\\). Suppose \\(S\\) were a Borel transversal. For \\(c\\notin A\\), both one-point classes \\(\\{(c,0)\\}\\) and \\(\\{(c,1)\\}\\) must meet \\(S\\); for \\(c\\in A\\), exactly one of \\((c,0),(c,1)\\) lies in \\(S\\). So \\(\\{c:(c,0)\\in S\\text{ and }(c,1)\\in S\\}=\\mathcal C\\setminus A\\), which would be Borel. This is a contradiction.\n\nSo countable classes do not suffice, and neither do finite classes with a saturation that is not Borel.\n\n**Proposition 7.4** (Borel sections for coset spaces). Let \\(G\\) be a Polish group, that is, a topological group whose topology is Polish, and \\(H\\) a closed subgroup. Let \\(q:G\\to G/H\\) be the map onto the space of left cosets, and give \\(G/H\\) the quotient topology and its Borel sets.\n\n1. \\(q\\) is continuous and open, and \\(G/H\\) is Hausdorff.\n2. There is a Borel set \\(S\\subseteq G\\) that meets every left coset exactly once and contains \\(1\\).\n3. The map \\(\\sigma:G/H\\to G\\) that sends each coset to its point in \\(S\\) is Borel, satisfies \\(q\\circ\\sigma=\\mathrm{id}\\) and \\(\\sigma(H)=1\\), and is a Borel isomorphism of \\(G/H\\) onto \\(S\\). In particular \\(G/H\\) is a standard Borel space.\n\nThe same holds for right cosets.\n\n**Proof.** (1) \\(q\\) is continuous by definition. For open \\(V\\), \\(q^{-1}(q(V))=VH\\) is the union of the open sets \\(Vh\\), so \\(q(V)\\) is open. Let \\(gH\\neq g'H\\). The set \\(gHg'^{-1}\\) is closed, because \\(H\\) is closed and translations are homeomorphisms. It does not contain \\(1\\), since \\(1=gkg'^{-1}\\) with \\(k\\in H\\) would give \\(g'H=gH\\). By (B5), applied to the open neighbourhood \\(G\\setminus gHg'^{-1}\\) of \\(1\\), there is an open neighbourhood \\(V=V^{-1}\\) of \\(1\\) with \\(V^2\\cap gHg'^{-1}=\\varnothing\\). If \\(vgh=v'g'h'\\) with \\(v,v'\\in V\\) and \\(h,h'\\in H\\), then \\(v'^{-1}v=g'h'h^{-1}g^{-1}\\in V^2\\cap g'Hg^{-1}\\). This set is empty, because \\(g'Hg^{-1}=(gHg'^{-1})^{-1}\\) and \\(V^2\\) is symmetric. So \\(q(Vg)\\) and \\(q(Vg')\\) are disjoint open neighbourhoods of \\(gH\\) and \\(g'H\\).\n\n(2) Let \\(x\\mathrel Ry\\) mean \\(x^{-1}y\\in H\\). The classes are the cosets \\(xH\\), which are closed, and \\(R(U)=UH\\) is open for open \\(U\\). Theorem 7.1 gives a Borel transversal \\(S_0\\). Then \\(S=(S_0\\setminus H)\\cup\\{1\\}\\) is Borel and still a transversal, because \\(H\\) is the coset of \\(1\\).\n\n(3) \\(q\\) restricted to \\(S\\) is a continuous bijection onto \\(G/H\\). \\(S\\) is a Borel subset of the Polish space \\(G\\), so by [Lemma 3.1](#oa-fnd-pb-04) it carries a Polish topology, finer than its relative topology, with the same Borel sets; \\(q\\) is still continuous and injective on it. By the Lusin–Souslin theorem ([Theorem 3.6](#oa-fnd-pb-05)), \\(q\\) carries Borel subsets of \\(S\\) to Borel subsets of the Hausdorff space \\(G/H\\). So \\(\\sigma=(q|_S)^{-1}\\) is Borel. It carries Borel sets of \\(G/H\\) to Borel subsets of \\(S\\), because \\(q\\) is continuous. \\(S\\) is standard, being a Borel subset of a Polish space (F4), hence so is \\(G/H\\). For right cosets, compose with the homeomorphism \\(g\\mapsto g^{-1}\\), which carries \\(gH\\) to \\(Hg^{-1}\\). \\(\\square\\)\n\n**Example 7.5** (Borel is the best one can ask). For \\(G=\\mathbb R\\) and \\(H=\\mathbb Z\\), \\(S=[0,1)\\) is a Borel transversal, and \\(\\sigma(x+\\mathbb Z)\\) is the fractional part of \\(x\\). No continuous section exists. Here \\(G/H\\) is a circle, and a continuous injective map \\(\\sigma\\) from a circle into \\(\\mathbb R\\) is impossible: it takes its largest and smallest values at two different points, each of the two arcs joining these points is mapped onto the whole interval between the two values, and so the values strictly between them are taken twice.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    },
    {
      "id": "OA-FND-PB-11",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Measurable sections",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "anchor": "oa-fnd-pb-11",
      "proof_locus": {
        "line": 634,
        "through_line": 684
      },
      "full_conditions_and_proof": "#### Measurable sections\n\n**Lemma 7.6** (Lexicographic order on \\(\\Lambda\\)). For \\(t\\neq u\\) in \\(\\Lambda\\) write \\(t\\prec u\\) if \\(t_i<u_i\\) at the first index \\(i\\) with \\(t_i\\neq u_i\\), and \\(t\\preceq u\\) if \\(t\\prec u\\) or \\(t=u\\).\n\n1. \\(\\prec\\) is a strict total order on \\(\\Lambda\\).\n2. For every \\(u\\in\\Lambda\\), the set \\(\\{t:t\\prec u\\}\\) is open.\n3. Every nonempty closed set \\(F\\subseteq\\Lambda\\) has a least element.\n4. For \\(s=(s_1,\\dots,s_k)\\) with \\(k\\geq1\\), put \\(a_s=(s_1,\\dots,s_k,1,1,\\dots)\\) and \\(b_s=(s_1,\\dots,s_{k-1},s_k+1,1,1,\\dots)\\). Then\n\\[\n\\Lambda_s=\\{t\\in\\Lambda: \\ a_s\\preceq t\\prec b_s\\}.\n\\tag{8}\n\\]\n\n**Proof.** (1) Two different sequences have a first place of difference, so any two are comparable. If \\(t\\prec u\\prec v\\) with first differences at \\(i\\) (for \\(t,u\\)) and \\(j\\) (for \\(u,v\\)), then \\(t\\) and \\(v\\) agree before \\(\\min(i,j)\\), and at \\(\\min(i,j)\\) the entry of \\(t\\) is smaller; so \\(t\\prec v\\).\n\n(2) If \\(t\\prec u\\) with first difference at \\(i\\), then every \\(t'\\in\\Lambda_{t|i}\\) satisfies \\(t'\\prec u\\). So \\(\\{t:t\\prec u\\}\\) is a union of cylinders.\n\n(3) Let \\(u_1\\) be the least first entry of elements of \\(F\\), and, recursively, \\(u_{k+1}\\) the least \\((k+1)\\)-st entry among the elements of \\(F\\) that begin with \\((u_1,\\dots,u_k)\\); such elements exist by the previous step. Every cylinder \\(\\Lambda_{u|k}\\) meets \\(F\\) and \\(F\\) is closed, so \\(u=(u_k)\\in F\\). If \\(t\\in F\\), \\(t\\neq u\\), first differs from \\(u\\) at \\(i\\), then \\(t\\) begins with \\(u|i-1\\), so \\(t_i\\geq u_i\\) by the choice of \\(u_i\\), and \\(t_i\\neq u_i\\); hence \\(u\\prec t\\).\n\n(4) If \\(t\\in\\Lambda_s\\), then \\(t\\) agrees with \\(a_s\\) up to place \\(k\\) and has entries at least \\(1\\) afterwards, so \\(a_s\\preceq t\\); and \\(t\\) first differs from \\(b_s\\) at place \\(k\\), with \\(t_k=s_k<s_k+1\\), so \\(t\\prec b_s\\). Conversely, let \\(a_s\\preceq t\\prec b_s\\), and suppose \\(t|k\\neq s\\), with first difference at \\(i\\leq k\\). Since \\(a_s\\preceq t\\), \\(t_i>s_i\\). If \\(i<k\\), then \\(t\\) and \\(b_s\\) also first differ at \\(i\\), with \\(t_i>s_i\\), so \\(b_s\\prec t\\). If \\(i=k\\), then \\(t_k\\geq s_k+1\\): if \\(t_k>s_k+1\\), then \\(b_s\\prec t\\); if \\(t_k=s_k+1\\), then \\(t\\) agrees with \\(b_s\\) up to \\(k\\) and every later entry of \\(b_s\\) is \\(1\\), at most the entry of \\(t\\), so \\(b_s\\preceq t\\). Each case contradicts \\(t\\prec b_s\\). \\(\\square\\)\n\nIn (3) the entries must be minimized one after another, among the elements that begin with the entries already chosen. Minimizing each entry over all of \\(F\\) separately can leave \\(F\\): for the closed set \\(F=\\{(1,2,2,2,\\dots),(2,1,1,1,\\dots)\\}\\) it gives \\((1,1,1,\\dots)\\notin F\\).\n\n**Theorem 7.7** (Measurable sections; Jankov–von Neumann). Let \\(X\\) be a Souslin space, \\(Y\\) a separable metrizable space and \\(f:X\\to Y\\) a Borel map. Then \\(f(X)\\) is a Souslin set, and there is a map \\(\\varphi:f(X)\\to X\\) with \\(f(\\varphi(y))=y\\) for every \\(y\\in f(X)\\), such that for every Borel set \\(B\\subseteq X\\) the set \\(\\varphi^{-1}(B)\\) belongs to the \\(\\sigma\\)-algebra generated by the Souslin subsets of \\(f(X)\\). Consequently \\(\\varphi^{-1}(B)\\) is \\(\\mu^*\\)-measurable for every measure \\(\\mu\\) on \\(\\mathcal B(Y)\\), and \\(\\mu\\)-measurable for every \\(\\sigma\\)-finite \\(\\mu\\). One map \\(\\varphi\\) serves all measures at once.\n\nIn particular, if \\(X\\) and \\(Y\\) are Souslin spaces, \\(f\\) maps \\(X\\) onto \\(Y\\), and \\(\\mu\\) is a \\(\\sigma\\)-finite measure on \\(Y\\), then there is a \\(\\mu\\)-measurable map \\(\\varphi:Y\\to X\\) with \\(f\\circ\\varphi=\\mathrm{id}_Y\\).\n\nThe two-point example after Lemma 7.6 proves that independent coordinate minima can leave a closed fibre; the proof below minimizes within the previously chosen prefix.\n\n**Proof.** If \\(X=\\varnothing\\) there is nothing to prove. Let \\(\\overline Y\\) be a Polish space containing \\(Y\\), as in the proof of [Theorem 4.3](#oa-fnd-pb-06). As in the proof of part (1) there, \\(\\operatorname{Gr}f\\) is a nonempty Souslin set in \\(X\\times\\overline Y\\). So there is a continuous map \\(g\\) of \\(\\Lambda\\) onto \\(\\operatorname{Gr}f\\), as noted after [Definition 2.1](#oa-fnd-pb-02). Let \\(h=\\pi_{\\overline Y}\\circ g\\), with \\(\\pi_{\\overline Y}\\) the projection. Then \\(h\\) is continuous and \\(h(\\Lambda)=f(X)\\), a Souslin set. For \\(y\\in f(X)\\) the set \\(h^{-1}(y)\\) is nonempty and closed; let \\(\\psi(y)\\) be its least element for \\(\\prec\\) (Lemma 7.6(3)). Put \\(\\varphi=\\pi_X\\circ g\\circ\\psi\\). The point \\(g(\\psi(y))\\) lies on the graph and its second coordinate is \\(h(\\psi(y))=y\\); so it equals \\((\\varphi(y),y)\\), and \\(f(\\varphi(y))=y\\).\n\nLet \\(\\mathcal A\\) be the \\(\\sigma\\)-algebra on \\(f(X)\\) generated by its Souslin subsets. For \\(u\\in\\Lambda\\),\n\\[\n\\psi^{-1}(\\{t:t\\prec u\\})=h(\\{t:t\\prec u\\}),\n\\tag{9}\n\\]\nbecause \\(\\psi(y)\\prec u\\) holds exactly when some \\(t\\in h^{-1}(y)\\) has \\(t\\prec u\\), \\(\\psi(y)\\) being the least element of \\(h^{-1}(y)\\). The set \\(\\{t:t\\prec u\\}\\) is open by Lemma 7.6(2), hence Polish (F3), so the right side of (9) is a Souslin set and lies in \\(\\mathcal A\\). By (8), \\[\n\\begin{gathered}\n\\psi^{-1}(\\Lambda_s)\\\\\n=\\psi^{-1}(\\{t:t\\prec b_s\\})\\setminus\\psi^{-1}(\\{t:t\\prec a_s\\})\n\\end{gathered}\n\\] lies in \\(\\mathcal A\\) for every \\(s\\). The sets \\(E\\subseteq\\Lambda\\) with \\(\\psi^{-1}(E)\\in\\mathcal A\\) form a \\(\\sigma\\)-algebra containing all cylinders. The cylinders form a countable base, which generates the Borel sets of \\(\\Lambda\\) (F1), so this \\(\\sigma\\)-algebra contains every Borel set of \\(\\Lambda\\). For Borel \\(B\\subseteq X\\), the set \\((\\pi_X\\circ g)^{-1}(B)\\) is Borel in \\(\\Lambda\\), and \\(\\varphi^{-1}(B)=\\psi^{-1}((\\pi_X\\circ g)^{-1}(B))\\) lies in \\(\\mathcal A\\).\n\nEvery Souslin subset of \\(f(X)\\) is a Souslin set in \\(Y\\). So it is \\(\\mu^*\\)-measurable by [Theorem 6.2](#oa-fnd-pb-09)(3), and \\(\\mu\\)-measurable for \\(\\sigma\\)-finite \\(\\mu\\) by Theorem 6.2(2). The \\(\\mu^*\\)-measurable sets form a \\(\\sigma\\)-algebra on \\(Y\\), and so do the \\(\\mu\\)-measurable sets. The set \\(f(X)\\) is itself one of these Souslin subsets, so the \\(\\mu^*\\)-measurable subsets of \\(f(X)\\) form a \\(\\sigma\\)-algebra on \\(f(X)\\): the complement in \\(f(X)\\) of such a set \\(E\\) is \\(f(X)\\cap(Y\\setminus E)\\). The same holds for the \\(\\mu\\)-measurable subsets, and both \\(\\sigma\\)-algebras contain \\(\\mathcal A\\). \\(\\square\\)\n\n**Remark 7.8** (Borel sections). We do not claim Borel sections in this generality. They exist in one case: if \\(X\\) is Polish, the fibres of \\(f\\) are closed, and \\(f^{-1}(f(U))\\) is Borel for every open \\(U\\subseteq X\\), then Theorem 7.1 gives a Borel set \\(S\\subseteq X\\) meeting each fibre exactly once, and \\((f|_S)^{-1}\\) is Borel on \\(f(X)\\) by Theorem 4.3(5).\n\n## D. Recover topology on a group quotient\n\nFor a Polish group and a closed subgroup, the quotient must first be shown Polish in its quotient topology. The metric construction and the open-homomorphism theorem provide that topology; the Borel sections from the preceding route then apply to the resulting standard Borel space. A measurable representative does not replace either topological theorem.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      ]
    },
    {
      "id": "OA-FND-PB-13",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "8. Topological quotients and open homomorphisms",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
      "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "anchor": "oa-fnd-pb-13",
      "proof_locus": {
        "line": 685,
        "through_line": 692
      },
      "full_conditions_and_proof": "### 8. Topological quotients and open homomorphisms\n\nA Borel section answers a measurable question. For a closed subgroup we can say more: the quotient topology itself is Polish. A surjective continuous homomorphism between Polish groups also carries open sets to open sets. These two assertions have different proofs. Category gives the open mapping theorem; completeness of an open image gives the quotient theorem.\n\nWe use the Baire category theorem from [Hahn–Banach, Baire and the basic theorems on Banach spaces](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-03). A set is *nowhere dense* if its closure has empty interior, and *meagre* if it is a countable union of nowhere dense sets. It is *comeagre in an open set* if its complement there is meagre. A set has the *Baire property* if its symmetric difference with an open set is meagre. Translations, inversion and other homeomorphisms preserve these notions.\n\n Invariant metrization is the Birkhoff–Kakutani theorem; the game criterion for completeness is due to Choquet. We prove the forms needed here.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    },
    {
      "id": "OA-FND-PB-14",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Category and the open mapping theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "anchor": "oa-fnd-pb-14",
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      "full_conditions_and_proof": "#### Category and the open mapping theorem\n\n**Lemma 8.1** (The Baire property of Polish images). A continuous image of a Polish space in a Polish space has the Baire property.\n\n**Proof.** Write the image as \\(A=f(P)\\), and fix complete compatible metrics on \\(P\\) and on the target \\(Y\\). The empty image causes no difficulty. By Theorem 1.1, there is a countably branching family of nonempty closed sets \\(F_s\\) in \\(P\\), with \\(F_s=\\bigcup_jF_{sj}\\), \\(F_\\varnothing=P\\), and diameters tending to zero along branches. Set \\(A_s=f(F_s)\\). Thus \\(A_s=\\bigcup_jA_{sj}\\).\n\nFirst observe that if a subset \\(E\\) is nonmeagre in an open set \\(W\\), some nonempty open \\(V\\subseteq W\\) has \\(E\\) nonmeagre in every nonempty open subset of \\(V\\). Indeed, let \\(O\\) be the union of the open subsets of \\(W\\) in which \\(E\\) is meagre. Second countability makes \\(E\\cap O\\) meagre: a countable subfamily of these open sets covers \\(O\\). The relatively closed set \\(W\\setminus O\\) cannot be nowhere dense in \\(W\\), for otherwise \\(E\\cap W\\) would be meagre. Its relative interior contains the required \\(V\\).\n\nSuppose \\(A_s\\) is nonmeagre in every nonempty open subset of a nonempty open set \\(V_s\\). In any nonempty open \\(W\\subseteq V_s\\), some \\(A_{sj}\\cap W\\) is nonmeagre, since their union is \\(A_s\\cap W\\). The observation gives a nonempty open \\(V\\subseteq W\\) where \\(A_{sj}\\) is nonmeagre in every nonempty open subset. Shrink \\(V\\) so that its closure lies in \\(W\\) and its diameter is at most \\(2^{-|s|-1}\\). There is a maximal pairwise disjoint family of such \\(V\\)'s, each labelled with its chosen \\(j\\). The family is countable, because disjoint nonempty open sets contain distinct members of a countable base. Its union is dense in \\(V_s\\): an open part of its complement would allow one more member. This constructs children of each \\(V_s\\). Distinct children may have the same source label; the tree of open sets retains the complete source label along each branch.\n\nStarting in any open region \\(V_\\varnothing\\) where \\(A\\) is everywhere nonmeagre, iterate this construction. At each level the union of the open sets is dense and open in \\(V_\\varnothing\\). Their intersection is comeagre there by Baire. A point \\(y\\) in this intersection determines a unique nested branch of open sets and source labels \\(s_n\\). Since \\(A_{s_n}\\) is nonmeagre in every nonempty open subset of \\(V_{s_n}\\), it is dense in \\(V_{s_n}\\). Choose \\(a_n\\in F_{s_n}\\) with \\(d_Y(f(a_n),y)<1/n\\). The sets \\(F_{s_n}\\) are nested, closed, and have diameters tending to zero. Hence \\(a_n\\to a\\in\\bigcap_nF_{s_n}\\), by completeness. Continuity gives \\(f(a)=y\\). Thus \\(A\\) contains a comeagre subset of \\(V_\\varnothing\\).\n\nWe have proved that every nonmeagre Polish image contains a comeagre subset of some nonempty open set. Let \\(U\\) be the union of all open sets in which \\(A\\) is comeagre. Second countability shows that \\(U\\setminus A\\) is meagre. The remainder \\(A\\setminus U\\) is again a Polish image: it is the image under \\(f\\) of the closed Polish subspace \\(f^{-1}(Y\\setminus U)\\). If that remainder were nonmeagre, the assertion just proved would make it comeagre in a nonempty open \\(V\\). Then \\(A\\) would be comeagre in \\(V\\), so \\(V\\subseteq U\\), whereas \\(A\\setminus U\\) is empty there. This contradicts Baire. Consequently \\(A\\setminus U\\) is meagre, and \\(A\\mathbin\\triangle U\\) is meagre. \\(\\square\\)\n\n**Theorem 8.2** (Pettis). Let \\(G\\) be a Polish group and \\(A\\subseteq G\\) have the Baire property and be nonmeagre. Both \\(A^{-1}A\\) and \\(AA^{-1}\\) contain an open neighbourhood of the identity.\n\n**Proof.** There is a nonempty open \\(O\\) in which \\(A\\) is comeagre. Fix \\(x\\in O\\). Continuity of \\(g\\mapsto xg\\) gives a neighbourhood \\(V\\) of the identity such that \\(xV\\subseteq O\\). For \\(g\\in V\\), the set \\(O\\cap Og^{-1}\\) is nonempty and open: it contains \\(x\\). The sets \\(A\\) and \\(Ag^{-1}\\) are comeagre there. Their intersection is nonempty by Baire. A point \\(a\\in A\\cap Ag^{-1}\\) satisfies \\(a\\in A\\) and \\(ag\\in A\\), whence \\(g=a^{-1}(ag)\\in A^{-1}A\\). Apply this result to \\(A^{-1}\\) to obtain a neighbourhood in \\(AA^{-1}\\). \\(\\square\\)\n\n**Theorem 8.3** (Open mapping for Polish groups). A continuous surjective homomorphism \\(\\phi:G\\to L\\) between Polish groups is open. It therefore induces a topological group isomorphism \\(G/\\ker\\phi\\cong L\\).\n\n**Proof.** Let \\(U\\) be a neighbourhood of the identity in \\(G\\). Choose an open symmetric neighbourhood \\(V\\) with \\(V^{-1}V\\subseteq U\\). Separability gives a countable set \\(D\\) with \\(G=DV\\): for each \\(g\\), the open set \\(gV^{-1}\\) meets a countable dense set. Surjectivity gives \\(L=\\bigcup_{d\\in D}\\phi(d)\\phi(V)\\). The open subspace \\(V\\) is Polish by (F3), so its continuous image \\(\\phi(V)\\) has the Baire property by Lemma 8.1. It cannot be meagre, since otherwise the nonempty Baire space \\(L\\) would be meagre in itself. Pettis gives an identity neighbourhood in\n\\[\n\\phi(V)^{-1}\\phi(V)=\\phi(V^{-1}V)\\subseteq\\phi(U).\n\\]\nIf \\(W\\subseteq G\\) is open and \\(g\\in W\\), choose such a \\(U\\) with \\(gU\\subseteq W\\). The set \\(\\phi(g)\\phi(U)\\) contains a neighbourhood of \\(\\phi(g)\\) lying in \\(\\phi(W)\\). Hence \\(\\phi(W)\\) is open. The kernel is closed. The induced bijection from the quotient is continuous by the definition of the quotient topology, and open by what we have just proved. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-PB-15",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "A quotient metric need not be complete",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "anchor": "oa-fnd-pb-15",
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        "line": 719,
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      "full_conditions_and_proof": "#### A quotient metric need not be complete\n\n**Lemma 8.4** (Birkhoff–Kakutani invariant metrization). Every first countable Hausdorff topological group has a compatible right invariant metric. The metric is not asserted to be complete.\n\n**Proof.** Choose symmetric identity neighbourhoods \\(V_n\\), \\(n\\geq0\\), with \\(V_0=G\\), \\(V_{n+1}^3\\subseteq V_n\\), and with \\((V_n)\\) a neighbourhood base. Continuity of multiplication and inversion permits this recursive choice, while intersection with the \\(n\\)-th member of an initial countable base ensures the base condition. Put\n\\[\n\\delta(x,y)=\\inf\\{2^{-n}:xy^{-1}\\in V_n\\},\n\\]\n\\[\nd(x,y)=\\inf_{\\substack{m\\geq1,\\ x_0=x\\\\x_m=y}}\n\\sum_{i=1}^m\\delta(x_{i-1},x_i).\n\\]\nThe infimum defining \\(\\delta\\) is positive and attained when \\(x\\ne y\\), since \\(\\bigcap_nV_n=\\{1\\}\\); for \\(x=y\\) it is zero.\n\nWe need a finite product observation. If \\(g_i\\in V_{n_i}\\) and \\(\\sum_i2^{-n_i}<2^{-n}\\), then \\(g_1\\cdots g_m\\in V_n\\). Prove this by induction on \\(m\\), for all \\(n\\) at once. Let a middle term be one whose interval in the cumulative sum contains half the total. Each side has total at most half the original total, so by induction its product is in \\(V_{n+1}\\). The middle term itself has weight less than \\(2^{-n}\\), so it too is in \\(V_{n+1}\\). The product is in \\(V_{n+1}^3\\subseteq V_n\\). Empty sides mean the identity.\n\nApply the observation to the increments \\(x_{i-1}x_i^{-1}\\), whose product telescopes to \\(xy^{-1}\\). It gives\n\\[\n\\tfrac12\\delta(x,y)\\leq d(x,y)\\leq\\delta(x,y).\n\\]\nFor the lower bound, if \\(\\delta(x,y)=2^{-k}\\) and a chain had total less than \\(2^{-k-1}\\), the observation would put \\(xy^{-1}\\) in \\(V_{k+1}\\), a contradiction. The upper bound uses the one-edge chain. Chain concatenation proves the triangle inequality. Symmetry and right invariance hold for \\(\\delta\\), and then for \\(d\\) by reversing or right translating chains. The comparison shows that \\(d\\) separates points and induces the given topology, because the \\(V_n\\)'s are a neighbourhood base. \\(\\square\\)\n\n**Lemma 8.5** (Metrizing cosets). If \\(H\\) is a closed subgroup of a first countable Hausdorff group \\(G\\), the space of left cosets \\(G/H\\) has a compatible metric\n\\[\n\\begin{aligned}\nD(gH,kH)&=\\inf_{h,l\\in H}d(gh,kl)\\\\\n&=\\inf_{h\\in H}d(gh,k),\n\\end{aligned}\n\\]\nwhere \\(d\\) is a compatible right invariant metric on \\(G\\). The quotient map is open. If \\(G\\) is separable, so is \\(G/H\\).\n\n**Proof.** The second formula follows by right translating both entries by \\(l^{-1}\\). It also shows positivity for distinct cosets: distance zero would put \\(k\\) in the closed set \\(gH\\). For the triangle inequality choose almost minimizing pairs from the first and second cosets and from the second and third. Right translate the first pair by an element of \\(H\\) so that the two middle representatives agree; this leaves its distance unchanged. Apply the triangle inequality for \\(d\\), then take infima. Symmetry is immediate. Moreover\n\\[\nB_D(kH,r)=q(B_d(k,r)).\n\\]\nIndeed, a coset is within \\(r\\) of \\(kH\\) precisely when one of its representatives is within \\(r\\) of \\(k\\). The map \\(q\\) is continuous for \\(D\\), since \\(D(q(g),q(k))\\leq d(g,k)\\). It is open for the quotient topology because \\(q^{-1}(q(W))=WH\\) is open whenever \\(W\\) is open. The ball identity proves that the metric and quotient topologies coincide. The continuous image of a countable dense set is dense. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-PB-16",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "name": "Completeness of an open image",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "source": "src/polish-spaces-and-standard-borel-spaces.md",
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      "full_conditions_and_proof": "#### Completeness of an open image\n\nFor completeness we prove the topological tool used next. A *strong Choquet play* in a space \\(Y\\) consists of points and nonempty open sets\n\\[\n\\begin{gathered}\nx_n\\in U_n,\\qquad x_n\\in B_n\\subseteq U_n,\\\\\nU_{n+1}\\subseteq B_n.\n\\end{gathered}\n\\]\nThe first player chooses \\((x_n,U_n)\\); the second chooses \\(B_n\\). A strategy for the second player is *winning* if every play following it has \\(\\bigcap_nB_n\\ne\\varnothing\\).\n\n#### Completing a metric space\n\n**Lemma (Metric completion).** Every metric space \\((Y,r)\\) embeds isometrically as a dense subspace of a complete metric space \\((Z,\\bar r)\\). If \\(Y\\) is separable, so is \\(Z\\). No compatibility with a group operation is asserted.\n\n*Proof.* Let \\(\\mathscr S\\) be the set of Cauchy sequences in \\(Y\\). For \\(a=(a_n)\\) and \\(b=(b_n)\\), the inequality\n\\[\n|r(a_n,b_n)-r(a_m,b_m)|\n\\leq r(a_n,a_m)+r(b_n,b_m)\n\\]\nshows that \\(r(a_n,b_n)\\) is a Cauchy sequence of real numbers. Its limit \\(R(a,b)\\) exists. Symmetry and the triangle inequality pass to this limit. Thus the relation \\(a\\sim b\\) defined by \\(R(a,b)=0\\) is an equivalence relation. The triangle inequality also shows that replacing either sequence by an equivalent one preserves \\(R\\). The set \\(Z=\\mathscr S/\\mathord\\sim\\), with \\(\\bar r([a],[b])=R(a,b)\\), is therefore a metric space. Constant sequences embed \\(Y\\) isometrically.\n\nFor \\(a\\in\\mathscr S\\), let \\(y_n\\in Z\\) denote the constant sequence at \\(a_n\\). Given \\(\\varepsilon>0\\), the Cauchy condition gives \\(r(a_n,a_m)<\\varepsilon\\) for all sufficiently large \\(n,m\\). Taking the limit in \\(m\\) gives \\(\\bar r(y_n,[a])\\leq\\varepsilon\\) for all sufficiently large \\(n\\). Consequently \\(y_n\\to[a]\\), and the embedded copy of \\(Y\\) is dense.\n\nTo prove completeness, let \\((z_k)\\) be a Cauchy sequence in \\(Z\\). By density choose \\(y_k\\in Y\\) with \\(\\bar r(y_k,z_k)<1/k\\). Then\n\\[\nr(y_k,y_l)\n\\leq 1/k+\\bar r(z_k,z_l)+1/l,\n\\]\nso \\((y_k)\\) is Cauchy in \\(Y\\) and defines \\(z=[(y_k)]\\in Z\\). The preceding paragraph gives \\(y_k\\to z\\); the bound \\(\\bar r(z_k,z)\\leq1/k+\\bar r(y_k,z)\\) gives \\(z_k\\to z\\). Hence \\(Z\\) is complete. Finally a countable dense subset of \\(Y\\) remains dense in \\(Z\\): approximate first by a point of \\(Y\\), then by a point of that subset, using the triangle inequality. \\(\\square\\)\n\n**Lemma 8.6** (Completeness criterion). A second countable metrizable space with a winning strong Choquet strategy is Polish.\n\n**Proof.** We give the construction, including its covering step. Fix a compatible metric \\(r\\), bounded by \\(1\\), and let \\(Z\\) be its completion. The metric-completion lemma just proved constructs \\(Z\\), proves its completeness, and shows that \\(Y\\) embeds densely and isometrically. A countable dense subset of \\(Y\\) remains dense in \\(Z\\), so \\(Z\\) is Polish.\n\n*Locally finite refinements.* Every countable open cover \\((O_i)\\) of a metric space has a countable open refinement \\((T_j)\\) that covers the space, is locally finite, and has each \\(\\overline{T_j}\\) contained in one \\(O_i\\). Here is a direct proof. Put \\(A_n=\\bigcup_{i\\leq n}O_i\\) and\n\\[\n\\begin{aligned}\nF_n&=\\{y:r(y,Y\\setminus A_n)\\geq1/n\\},\\\\\nL_n&=\\operatorname{int}F_{n+1}\\setminus F_{n-1},\n\\end{aligned}\n\\]\nwith \\(F_0=\\varnothing\\) and distance to the empty set interpreted as infinity. The \\(F_n\\)'s increase, \\(F_n\\subseteq\\operatorname{int}F_{n+1}\\), and their interiors cover \\(Y\\). Hence the \\(L_n\\)'s cover \\(Y\\) and are locally finite. Write \\(t_i(y)=\\min(1,r(y,Y\\setminus O_i))\\), taking \\(t_i=1\\) if \\(O_i=Y\\). Refine \\(L_n\\) by the finitely many open sets\n\\[\n\\begin{aligned}\nT_{n,i}&=L_n\\cap\\\\\n&\\quad\\left\\{y:t_i(y)>\n\\frac{\\max_{k\\leq n+1}t_k(y)}{2(n+1)}\\right\\},\n\\end{aligned}\n\\]\nwhere \\(i\\leq n+1\\). They cover \\(L_n\\). Their closures lie in \\(F_{n+1}\\subseteq A_{n+1}\\); on this closed set the displayed maximum is positive. The non-strict inequality on the closure therefore forces \\(t_i>0\\), so \\(\\overline{T_{n,i}}\\subseteq O_i\\). Local finiteness follows from that of the \\(L_n\\)'s. This proves the covering assertion, also for an open subspace with its relative metric.\n\n*Extending a locally finite family.* If \\((T_i)\\) is a locally finite family of open sets in an open subspace \\(V\\subseteq Y\\), its members have open extensions in \\(Z\\) that are locally finite on some open neighbourhood of \\(V\\) in \\(Z\\). To see this, the canonical extension of \\(T_i\\), considered as an open set of \\(Y\\), is\n\\[\n\\widehat T_i=Z\\setminus\\overline{Y\\setminus T_i}^{\\,Z}.\n\\]\nIts trace on \\(Y\\) is \\(T_i\\). At each \\(y\\in V\\), choose a ball in \\(Y\\) contained in \\(V\\) and meeting only finitely many \\(T_i\\)'s. A smaller corresponding ball in \\(Z\\) meets only these canonical extensions: any nonempty open intersection with another extension would meet the dense subset \\(Y\\), contradicting the choice of ball. The union of these smaller balls is the required open neighbourhood. Extensions may be intersected with any prescribed open sets containing their traces. They may also be kept within distance \\(\\varepsilon\\) of their traces, by intersection with \\(\\bigcup_{t\\in T_i}B_Z(t,\\varepsilon)\\). Thus a family of sufficiently small trace diameters has extensions of arbitrarily small prescribed diameters.\n\n*A tree of plays.* Start with the empty play. For every \\(y\\in Y\\), choose a first-player neighbourhood \\(U_y\\ni y\\) of diameter less than \\(1/4\\), and let \\(B_y\\ni y\\) be the response of the strategy. Choose a countable subcover from these responses. Refine it by the locally finite family just constructed, with each trace \\(T_s\\) having closure inside its assigned response \\(B_s\\). Attach the corresponding first move and response to \\(s\\).\n\nRecursively, at a node \\(s\\), for each \\(y\\in T_s\\) choose a next move \\((y,U_y)\\) with \\(U_y\\subseteq T_s\\) and diameter less than \\(2^{-|s|-3}\\). Its strategy response contains \\(y\\). A countable subcover of these responses, refined as above, gives children whose traces cover \\(T_s\\); each child retains its assigned move, response and the previous history. Every descending branch is consequently a legal play of the strategy.\n\nUse the extension step at each node. Extend its children inside the node's current open extension, with diameters at most \\(2^{-|s|-1}\\). Shrink the parent's extension to the open neighbourhood of its trace on which the children are locally finite, and restrict the child extensions to that neighbourhood. This shrinking does not change any trace on \\(Y\\). At the root the same step starts in \\(Z\\). Each node's final extension is fixed when its children are chosen; later shrinkings affect only descendants. Denote these final open extensions by \\(W_s\\). Children lie inside their final parent; the family of children is locally finite there; and \\(\\operatorname{diam}W_s\\leq2^{-|s|}\\). At each depth the traces cover \\(Y\\). Put\n\\[\nO_n=\\bigcup_{|s|=n}W_s.\n\\]\nThese are open sets in \\(Z\\) containing \\(Y\\).\n\nLet \\(z\\in\\bigcap_nO_n\\). At each depth some node contains \\(z\\). There are only finitely many such first-level nodes, and only finitely many such children of any node containing \\(z\\), by local finiteness. Children have parents containing \\(z\\). The elementary finite-branching tree argument therefore gives an infinite branch \\(s_1,s_2,\\ldots\\) with \\(z\\in W_{s_n}\\): from a node having descendants at arbitrarily large depths, choose a child with the same property. Its associated play is won, so some \\(y\\in Y\\) belongs to every response \\(B_{s_n}\\). Choose \\(t_n\\in T_{s_n}\\subseteq B_{s_n}\\). The diameter bounds on \\(W_{s_n}\\) and on the move containing \\(B_{s_n}\\) give \\(r_Z(z,t_n)\\to0\\) and \\(r(y,t_n)\\to0\\). Hence \\(z=y\\in Y\\). We have proved \\(Y=\\bigcap_nO_n\\), a \\(G_\\delta\\) in the Polish space \\(Z\\). By (F3), \\(Y\\) is Polish. \\(\\square\\)\n\n**Theorem 8.7** (Open images). Let \\(P\\) be Polish, \\(Y\\) metrizable, and \\(f:P\\to Y\\) continuous, open and surjective. Then \\(Y\\) is Polish.\n\n**Proof.** The images of a countable base of \\(P\\) form a countable base of \\(Y\\). Fix a complete compatible metric on \\(P\\). The second player wins the strong Choquet game on \\(Y\\) by lifting moves. For the first move \\((y_1,U_1)\\), choose \\(a_1\\in f^{-1}(y_1)\\), and a nonempty open \\(V_1\\ni a_1\\) of diameter at most \\(1/2\\), whose closure lies in \\(f^{-1}(U_1)\\). Respond with \\(B_1=f(V_1)\\), which is open and contains \\(y_1\\).\n\nFor a later move \\((y_n,U_n)\\), with \\(U_n\\subseteq f(V_{n-1})\\), choose \\(a_n\\in V_{n-1}\\cap f^{-1}(y_n)\\). Choose \\(V_n\\ni a_n\\) of diameter at most \\(2^{-n}\\), with \\(\\overline{V_n}\\subseteq V_{n-1}\\cap f^{-1}(U_n)\\), and respond with \\(B_n=f(V_n)\\). The closed sets \\(\\overline{V_n}\\) are nonempty, nested, and have diameters tending to zero. Completeness gives a common point \\(a\\). Since \\(\\overline{V_{n+1}}\\subseteq V_n\\), this point lies in every \\(V_n\\), so \\(f(a)\\in\\bigcap_nB_n\\). Lemma 8.6 now proves that \\(Y\\) is Polish. \\(\\square\\)\n\n**Theorem 8.8** (Polish coset spaces). If \\(G\\) is a Polish group and \\(H\\subseteq G\\) is a closed subgroup, \\(G/H\\), with the quotient topology, is Polish. The same holds for \\(H\\backslash G\\). No normality is required. When \\(H\\) is normal, the quotient is a Polish topological group.\n\n**Proof.** Lemmas 8.4 and 8.5 give a compatible metric on \\(G/H\\) and show that \\(q:G\\to G/H\\) is continuous, open and surjective. Theorem 8.7 gives completeness for a compatible metric on the quotient. Inversion identifies the left and right coset spaces homeomorphically. If \\(H\\) is normal, multiplication and inversion descend continuously: \\(q\\times q\\) is an open quotient map, and the group operations commute with the quotient maps. \\(\\square\\)\n\nThe invariant metric in Lemma 8.4 was used only for metrization. Its completeness was never assumed. The complete metric on \\(G\\) used in Theorem 8.7 can be entirely different.\n\n**Example 8.9** (A quotient and its section). The quotient \\(\\mathbb R/\\mathbb Z\\) is homeomorphic to the circle through \\(x+\\mathbb Z\\mapsto e^{2\\pi ix}\\). The displayed map is well defined, continuous and bijective: two real numbers have the same exponential exactly when their difference is an integer. The quotient is compact since it is the image of \\([0,1]\\), and the circle is Hausdorff. The compact-to-Hausdorff argument (B3) therefore proves the homeomorphism. It is Polish by Theorem 8.8. The Borel section from Example 7.5 takes values in \\([0,1)\\), but is not continuous. Polish topology on the quotient does not make every Borel choice continuous.\n\n**Example 8.10** (Why both Polish hypotheses matter for open mapping). Give \\(\\mathbb R\\) its discrete topology. The identity homomorphism to \\(\\mathbb R\\) with its usual topology is continuous and onto, but is not open, since the image of a singleton is not open. The discrete source is complete but is not separable. Separability cannot simply be omitted from Theorem 8.3.\n\n## Exercises\n\n**Exercise 1** (medium; Borel graphs). Let \\(X\\) and \\(Y\\) be standard Borel spaces and \\(f:X\\to Y\\) a map. Show that \\(f\\) is Borel if and only if its graph is a Borel subset of \\(X\\times Y\\), with the product Borel structure.\n\n*Solution.* If \\(f\\) is Borel, the graph lies in the product \\(\\sigma\\)-algebra by Proposition 4.1, since \\(Y\\) is countably separated. Conversely, let the graph \\(G\\) be Borel. Choose Polish topologies on \\(X\\) and \\(Y\\) that give their Borel sets; then the product \\(\\sigma\\)-algebra is the Borel structure of the Polish space \\(X\\times Y\\) (F1). For Borel \\(E\\subseteq Y\\),\n\\[\n\\begin{gathered}\nf^{-1}(E)\\\\\n=\\pi_X\\big(G\\cap(X\\times E)\\big),\\\\\nX\\setminus f^{-1}(E)\\\\\n=\\pi_X\\big(G\\cap(X\\times(Y\\setminus E))\\big).\n\\end{gathered}\n\\]\nThe sets inside the brackets are Borel in \\(X\\times Y\\), hence Souslin sets (Corollary 2.6), and the projection \\(\\pi_X\\) is continuous. So \\(f^{-1}(E)\\) and its complement are Souslin sets of \\(X\\), and \\(f^{-1}(E)\\) is Borel by Souslin's theorem (Corollary 2.5).\n\n**Exercise 2** (medium; Quotients by Borel surjections). Let \\(Z\\) be a standard Borel space, \\(Y\\) a Borel space whose points are separated by countably many Borel sets, and \\(f:Z\\to Y\\) a Borel surjection. Show that \\(E\\subseteq Y\\) is Borel if and only if \\(f^{-1}(E)\\) is Borel in \\(Z\\). Show by an example that the hypothesis \"countably separated\" cannot be dropped.\n\n*Solution.* \"Only if\" is clear. \\(Y\\) is a Souslin–Borel space by definition; let \\(\\varphi:Y\\to\\mathcal C\\) be the injective Borel map of Lemma 5.5(3). Suppose \\(f^{-1}(E)\\) is Borel. Then \\(\\varphi(E)=(\\varphi\\circ f)(f^{-1}(E))\\) and \\(\\varphi(Y\\setminus E)=(\\varphi\\circ f)(Z\\setminus f^{-1}(E))\\) are Souslin sets in \\(\\mathcal C\\), by Lemma 5.5(1), because Borel subsets of \\(Z\\) are standard (F4). They are disjoint, since \\(\\varphi\\) is injective and \\(f\\) is onto. By the separation theorem (Theorem 2.4), some Borel \\(D\\subseteq\\mathcal C\\) contains \\(\\varphi(E)\\) and misses \\(\\varphi(Y\\setminus E)\\); then \\(E=\\varphi^{-1}(D)\\) is Borel.\n\nFor the example, let \\(Y=[0,1]\\) with the \\(\\sigma\\)-algebra of countable and co-countable sets, and let \\(f\\) be the identity map from \\([0,1]\\) with its Borel sets. \\(f\\) is Borel and onto, and \\(f^{-1}([0,\\tfrac12])\\) is Borel, but \\([0,\\tfrac12]\\) is not in the \\(\\sigma\\)-algebra of \\(Y\\). This \\(Y\\) is not countably separated. Given countably many countable or co-countable sets, replace each co-countable one by its complement; a set and its complement separate the same pairs. The uncountably many points outside the union of the resulting countable sets are separated by none of them.\n\n**Exercise 3** (medium; Orbits of a compact group). Let a compact metrizable group \\(K\\) act continuously on a Polish space \\(X\\), that is, \\((k,x)\\mapsto k\\cdot x\\) is a continuous action. Show that some Borel set meets every orbit exactly once, and that the orbit space \\(X/K\\), with the quotient topology and its Borel sets, is a standard Borel space.\n\n*Solution.* Orbits \\(K\\cdot x\\) are compact, hence closed, since \\(X\\) is Hausdorff. For open \\(U\\), the saturation \\(K\\cdot U=\\bigcup_kk\\cdot U\\) is open, because each \\(x\\mapsto k\\cdot x\\) is a homeomorphism. So Theorem 7.1 gives a Borel transversal \\(S\\).\n\nNext, \\(X/K\\) is Hausdorff. First, \\(K\\cdot F\\) is closed for closed \\(F\\): if \\(k_n\\cdot z_n\\to w\\) with \\(z_n\\in F\\), pass to a subsequence with \\(k_n\\to k\\), which is possible because \\(K\\) is compact and metrizable; then \\(z_n=k_n^{-1}\\cdot(k_n\\cdot z_n)\\to k^{-1}\\cdot w\\) by continuity, so \\(k^{-1}\\cdot w\\in F\\) and \\(w\\in K\\cdot F\\). Now let \\(K\\cdot x\\neq K\\cdot y\\). These disjoint compact sets have disjoint open neighbourhoods \\(U\\) and \\(W\\). The sets \\(U'=X\\setminus K\\cdot(X\\setminus U)\\) and \\(W'=X\\setminus K\\cdot(X\\setminus W)\\) are open and saturated, \\(K\\cdot x\\subseteq U'\\subseteq U\\) and \\(K\\cdot y\\subseteq W'\\subseteq W\\). Their images in \\(X/K\\) are open, since a saturated open set is the full preimage of its image, and they are disjoint.\n\nFinally, the quotient map is continuous and injective on \\(S\\), and \\(S\\) is a Borel subset of a Polish space. As in the proof of Proposition 7.4(3), it is a Borel isomorphism of \\(S\\) onto \\(X/K\\), so \\(X/K\\) is standard.\n\n**Exercise 4** (medium; Product \\(\\sigma\\)-algebras). Let \\(D\\) be a set of cardinality greater than \\(2^{\\aleph_0}\\), with the discrete topology. Show that the diagonal of \\(D\\times D\\) is Borel in the discrete space \\(D\\times D\\) but does not belong to \\(\\mathcal B(D)\\otimes\\mathcal B(D)\\). Conclude that the open boxes of \\(D\\times D\\) generate a strictly smaller \\(\\sigma\\)-algebra than its open sets.\n\n*Solution.* \\(D\\times D\\) is discrete, so every subset is open. If the diagonal belonged to \\(\\mathcal B(D)\\otimes\\mathcal B(D)\\), then \\(D\\) would be countably separated by Proposition 4.1(3). A countable separating family \\((S_n)\\) gives an injective map \\(x\\mapsto(1_{S_n}(x))_n\\) of \\(D\\) into \\(\\mathcal C\\), so \\(|D|\\leq2^{\\aleph_0}\\), which is false. The open boxes \\(U\\times V\\) generate \\(\\mathcal B(D)\\otimes\\mathcal B(D)\\), which misses the diagonal. So testing a map into \\(D\\times D\\) on preimages of open boxes proves measurability for \\(\\mathcal B(D)\\otimes\\mathcal B(D)\\) only, and not for \\(\\mathcal B(D\\times D)\\) (compare Remark 4.2).\n\n**Exercise 5** (medium; An explicit isomorphism). Build a Borel isomorphism of \\(\\mathcal C\\) onto \\([0,1]\\) by hand, using binary expansions and moving a countable set. Show that no homeomorphism exists.\n\n*Solution.* Let \\(\\beta(c)=\\sum_kc_k2^{-k}\\). It is continuous and onto. Two different sequences have the same image exactly when one has the form \\(w0111\\cdots\\) and the other \\(w1000\\cdots\\) for a finite word \\(w\\); the common image is a dyadic rational in \\((0,1)\\). Indeed, if \\(c\\) and \\(c'\\) first differ at place \\(k\\), with \\(c_k=0\\) and \\(c'_k=1\\), then \\[\n\\begin{gathered}\n\\beta(c')-\\beta(c)\\\\\n=2^{-k}+\\sum_{j>k}(c'_j-c_j)2^{-j}\\\\\n\\geq2^{-k}-\\sum_{j>k}2^{-j}\\\\\n=0,\n\\end{gathered}\n\\] with equality exactly when \\(c_j=1\\) and \\(c'_j=0\\) for all \\(j>k\\).\n\nLet \\(E\\) be the countable set of sequences that end in \\(1\\)'s but are not all \\(1\\)'s. Then \\(\\beta\\) is a bijection of \\(\\mathcal C\\setminus E\\) onto \\([0,1]\\): a dyadic rational in \\((0,1)\\) keeps its preimage ending in \\(0\\)'s, the points \\(0\\) and \\(1\\) have the single preimages \\(000\\cdots\\) and \\(111\\cdots\\), and every other point has one preimage, which is not in \\(E\\). The inverse can be given explicitly. For \\(0\\leq t<1\\), put\n\\[\nb_n(t)=\\lfloor2^nt\\rfloor-2\\lfloor2^{n-1}t\\rfloor,\n\\qquad n\\geq1,\n\\]\nand put \\(b_n(1)=1\\). Each digit lies in \\(\\{0,1\\}\\). For \\(t<1\\), the first \\(n\\) binary terms sum to \\(2^{-n}\\lfloor2^nt\\rfloor\\), which tends to \\(t\\). Dyadic points receive an expansion ending in zeros; a nondyadic point cannot have an expansion ending in ones, since such an expansion sums to a dyadic number. Thus \\(b(t)\\in\\mathcal C\\setminus E\\), and the uniqueness already proved gives \\(b=\\beta^{-1}\\) on that set. The floor function is Borel because each integer level set is a half-open interval. Hence each digit is Borel, and (F1) shows that \\(b\\) is Borel. Therefore \\(\\beta\\) is a Borel isomorphism of \\(\\mathcal C\\setminus E\\) onto \\([0,1]\\).\n\nNext let \\(e_n\\in\\mathcal C\\) have a single \\(1\\), at place \\(n\\); these points are not in \\(E\\). Enumerate \\(E=\\{f_1,f_2,\\dots\\}\\) and define \\(\\theta:\\mathcal C\\to\\mathcal C\\setminus E\\) by \\(\\theta(f_n)=e_{2n-1}\\), \\(\\theta(e_n)=e_{2n}\\), and \\(\\theta(c)=c\\) otherwise. \\(\\theta\\) is a bijection that moves only a countable set, all of whose subsets are Borel, so \\(\\theta\\) and \\(\\theta^{-1}\\) are Borel. Then \\(\\beta\\circ\\theta\\) is a Borel isomorphism of \\(\\mathcal C\\) onto \\([0,1]\\). No homeomorphism exists, since \\([0,1]\\) is connected and \\(\\mathcal C\\) is not.\n\n**Exercise 6** (easy; First application). Let \\(\\phi:G\\to L\\) be a continuous bijective homomorphism of Polish groups. Prove that its inverse is continuous. Apply this to show that two Polish group topologies on the same group coincide whenever one is finer than the other.\n\n*Solution.* By Theorem 8.3, \\(\\phi\\) is open. Therefore \\((\\phi^{-1})^{-1}(U)=\\phi(U)\\) is open for every open \\(U\\subseteq G\\). For the second assertion use the identity map from the finer topology to the coarser one. It is a continuous bijective homomorphism between Polish groups, so its inverse is continuous as well.\n\n**Exercise 7** (easy; Closedness and quotient topology). Show that \\(\\mathbb R/\\mathbb Q\\), with the quotient topology, is not Hausdorff. In fact show that its only open sets are the empty set and the entire quotient. Explain why a standard Borel structure on a set, even if one is chosen separately, would not repair this topology.\n\n*Solution.* A nonempty open saturated subset \\(W\\subseteq\\mathbb R\\) contains an interval \\(I\\). Since \\(I+\\mathbb Q=\\mathbb R\\), saturation forces \\(W=\\mathbb R\\). Thus the quotient has the indiscrete topology and more than one point; it is not Hausdorff and cannot be Polish. A separately assigned sigma-algebra does not change its open sets. The closed-subgroup hypothesis in Theorem 8.8 is a topological requirement.\n\n**Exercise 8** (medium; Compact orbits, a stronger conclusion). In Exercise 3, prove that the quotient \\(X/K\\) is Polish, rather than merely standard Borel.\n\n*Solution.* Start with any bounded complete compatible metric \\(r\\) on \\(X\\), and set \\(r_K(x,y)=\\max_{k\\in K}r(kx,ky)\\). Compactness and continuity make the maximum finite. This is a \\(K\\)-invariant metric and dominates \\(r\\). It has the same topology as \\(r\\): if \\(x_n\\to x\\), continuity of the action is uniform on the compact set \\(K\\times\\{x\\}\\), so \\(\\max_k r(kx_n,kx)\\to0\\). For clarity, if this failed choose \\(k_n\\in K\\) with a fixed positive lower bound; a convergent subsequence of \\(k_n\\) and joint continuity give a contradiction. An \\(r_K\\)-Cauchy sequence converges for \\(r\\), and hence for \\(r_K\\), so \\(r_K\\) is complete. Define \\(D(Kx,Ky)=\\min_{k\\in K}r_K(x,ky)\\). The minimum exists; it is positive for distinct compact orbits, and invariance proves the triangle inequality by aligning middle representatives. Its open balls are images of \\(r_K\\)-balls, so it induces the quotient topology. The quotient map is continuous, open and onto. Theorem 8.7 proves that \\(X/K\\) is Polish. This supplies the stronger topological conclusion while retaining the Borel transversal of Exercise 3.\n\n## Where this leads\n\n- *Decomposition theory.* By the isomorphism theorem (Theorem 5.2), an uncountable standard Borel space can always be replaced by \\([0,1]\\) or \\(\\mathcal C\\). Direct integrals and disintegrations of measures are set up over standard Borel spaces for this reason. The measurable sections of Theorem 7.7 are used there to choose measurable families, for instance of unitaries.\n- *The Borel space of von Neumann algebras.* The von Neumann algebras on a separable Hilbert space form a standard Borel space ([The Effros Borel structure](effros-borel-structure.md)). In the study of this space, Theorem 4.3(5) shows that a unitary orbit is Borel, once the orbit is written as a one-to-one Borel image of a standard Borel space. Proposition 7.4 and the separation theorem enter the same study.\n- *Extensions of Polish groups.* If \\(E\\) is a Polish group and \\(A\\) a closed subgroup, Proposition 7.4 gives a Borel section \\(\\sigma:E/A\\to E\\) with \\(\\sigma(A)=1\\). Such normalized sections are used in the study of crossed products and the flow of weights. Theorems 8.3 and 8.8 prove that \\(E/A\\) is Polish and that continuous surjective homomorphisms of Polish groups are open.\n- *Unused extension, not proved here: countable-to-one maps.* Theorem 7.1 also gives Borel right inverses of continuous countable-to-one maps \\(f\\) from a Polish space into a Polish space, once one knows Lusin's theorem that such maps carry Borel sets to Borel sets, which is not proved here. The classes are the fibres of \\(f\\), and the saturation \\(f^{-1}(f(U))\\) of an open set \\(U\\) is Borel.\n- *Closed sets.* The nonempty closed subsets of a Polish space form a standard Borel space for the Effros Borel structure; see [the lesson on the Effros Borel structure](effros-borel-structure.md#oa-fnd-ef-05).\n- *Not treated here:* the Borel hierarchy, co-analytic and projective sets, the second separation theorem and finer uniformization theorems, Borel sections of arbitrary Borel surjections, and nonseparable spaces.\n\n## References\n\nOpen ones first.\n\n\n*Freely accessible reading:* [Anush Tserunyan, *Introduction to descriptive set theory*, §§1–6, 12A–12C and 13A–13C](https://www.math.mcgill.ca/atserunyan/Teaching_notes/dst_lectures.pdf) gives a route through Polish models, analytic separation and Borel inversion; the exercises and the group-quotient proof are supplied here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n\nFor countable generators and the precise measurability of prefix selection, see [Fremlin, Chapter 42, 423J, 423N–423P and 424B–424G](https://www1.essex.ac.uk/maths/people/fremlin/chap42.pdf).\n",
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      "full_conditions_and_proof": "## Prerequisites\n\nThe following facts are proved in [the lesson on the Effros Borel structure](effros-borel-structure.md#oa-fnd-ef-01).\n\n- **(F1)** *Generators.* A map into a Borel space is measurable when its compositions with a generating family of maps are measurable. A countable base of a topological space generates its Borel sets. In a countable product of second countable spaces, the coordinate maps generate the Borel sets.\n- **(F2)** *A stock of Polish spaces.* Closed subsets and countable products of Polish spaces are Polish, and \\(\\mathbb N^{\\mathbb N}\\) is Polish. A separable metrizable space has at most \\(2^{\\aleph_0}\\) points. A countable disjoint sum of Polish spaces is Polish. The prerequisite lesson treats two pieces, and its metric (distance \\(1\\) between pieces, \\(\\min(d,1)\\) inside a piece) works for countably many pieces: a Cauchy sequence eventually stays in one piece, and the union of countable dense subsets of the pieces is countable and dense.\n- **(F3)** *Polish subspaces*, proved in [the prerequisite lesson](effros-borel-structure.md#oa-fnd-ef-02). Every \\(G_\\delta\\) subset of a Polish space, in particular every open subset, is Polish. A Polish space is homeomorphic to a \\(G_\\delta\\) subset of \\([0,1]^{\\mathbb N}\\).\n- **(F4)** *Change of topology*, also proved in [the prerequisite lesson](effros-borel-structure.md#oa-fnd-ef-02). Call a topology \\(\\tau'\\) *admissible* for a Polish space \\((X,\\tau)\\) if \\(\\tau'\\) is Polish, \\(\\tau'\\supseteq\\tau\\), and \\(\\tau'\\) has the same Borel sets as \\(\\tau\\).\n  - (a) For \\(\\tau\\)-open \\(U\\), the topology generated by \\(\\tau\\) and \\(X\\setminus U\\) is admissible.\n  - (b) The topology generated by countably many admissible topologies is admissible. A countable base of it consists of finite intersections of members of countable bases of the given topologies.\n  - (c) Every Borel set is open and closed for some admissible topology.\n\n  Consequently each Borel set in a standard Borel space is itself standard.\n\nThe measure prerequisites are proved in [Measure and Hilbert space tools for Haar integration](measure-and-hilbert-space-tools.md).\n\n- *Carathéodory's theorem*, [Theorem 1.1](measure-and-hilbert-space-tools.md#1-from-an-outer-measure-to-a-measure). Let \\(\\mu^*\\) be an outer measure on \\(X\\). The sets \\(E\\) with \\(\\mu^*(W)=\\mu^*(W\\cap E)+\\mu^*(W\\setminus E)\\) for every \\(W\\subseteq X\\) form a sigma-algebra. The restriction of \\(\\mu^*\\) to it is a complete measure. No finiteness or countability hypothesis is needed.\n- *Lebesgue measure*, constructed in the example following that theorem. There is a complete measure \\(\\lambda\\) on the Lebesgue measurable subsets of \\(\\mathbb R\\), containing the Borel sets, with \\(\\lambda([a,b])=b-a\\) for \\(a\\leq b\\). Translating interval covers proves that translation preserves both measurable sets and their measures.\n\n",
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      "full_conditions_and_proof": "## Prerequisites\n\nThe following facts are proved in [the lesson on the Effros Borel structure](effros-borel-structure.md#oa-fnd-ef-01).\n\n- **(F1)** *Generators.* A map into a Borel space is measurable when its compositions with a generating family of maps are measurable. A countable base of a topological space generates its Borel sets. In a countable product of second countable spaces, the coordinate maps generate the Borel sets.\n- **(F2)** *A stock of Polish spaces.* Closed subsets and countable products of Polish spaces are Polish, and \\(\\mathbb N^{\\mathbb N}\\) is Polish. A separable metrizable space has at most \\(2^{\\aleph_0}\\) points. A countable disjoint sum of Polish spaces is Polish. The prerequisite lesson treats two pieces, and its metric (distance \\(1\\) between pieces, \\(\\min(d,1)\\) inside a piece) works for countably many pieces: a Cauchy sequence eventually stays in one piece, and the union of countable dense subsets of the pieces is countable and dense.\n- **(F3)** *Polish subspaces*, proved in [the prerequisite lesson](effros-borel-structure.md#oa-fnd-ef-02). Every \\(G_\\delta\\) subset of a Polish space, in particular every open subset, is Polish. A Polish space is homeomorphic to a \\(G_\\delta\\) subset of \\([0,1]^{\\mathbb N}\\).\n- **(F4)** *Change of topology*, also proved in [the prerequisite lesson](effros-borel-structure.md#oa-fnd-ef-02). Call a topology \\(\\tau'\\) *admissible* for a Polish space \\((X,\\tau)\\) if \\(\\tau'\\) is Polish, \\(\\tau'\\supseteq\\tau\\), and \\(\\tau'\\) has the same Borel sets as \\(\\tau\\).\n  - (a) For \\(\\tau\\)-open \\(U\\), the topology generated by \\(\\tau\\) and \\(X\\setminus U\\) is admissible.\n  - (b) The topology generated by countably many admissible topologies is admissible. A countable base of it consists of finite intersections of members of countable bases of the given topologies.\n  - (c) Every Borel set is open and closed for some admissible topology.\n\n  Consequently each Borel set in a standard Borel space is itself standard.\n\nThe measure prerequisites are proved in [Measure and Hilbert space tools for Haar integration](measure-and-hilbert-space-tools.md).\n\n- *Carathéodory's theorem*, [Theorem 1.1](measure-and-hilbert-space-tools.md#1-from-an-outer-measure-to-a-measure). Let \\(\\mu^*\\) be an outer measure on \\(X\\). The sets \\(E\\) with \\(\\mu^*(W)=\\mu^*(W\\cap E)+\\mu^*(W\\setminus E)\\) for every \\(W\\subseteq X\\) form a sigma-algebra. The restriction of \\(\\mu^*\\) to it is a complete measure. No finiteness or countability hypothesis is needed.\n- *Lebesgue measure*, constructed in the example following that theorem. There is a complete measure \\(\\lambda\\) on the Lebesgue measurable subsets of \\(\\mathbb R\\), containing the Borel sets, with \\(\\lambda([a,b])=b-a\\) for \\(a\\leq b\\). Translating interval covers proves that translation preserves both measurable sets and their measures.\n\n",
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      "full_conditions_and_proof": "### Cardinality and elementary topology\n\nChoice is understood throughout, as in [Section 1 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-01). Its [Theorem 8.2 and Proposition 8.3](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08) give the cardinal facts used here: \\(|X|<|\\mathcal P(X)|=2^{|X|}\\), \\((\\kappa^\\lambda)^\\mu=\\kappa^{\\lambda\\cdot\\mu}\\), and \\((2^{\\aleph_0})^{\\aleph_0}=2^{\\aleph_0}\\). The pairing of countable indices is independent of whether they start at zero or one.\n\nTranslations and inversion in a topological group are homeomorphisms, by [Proposition 7.1(1) of the Haar lesson](haar-measure.md#oa-fnd-hm-04). Two disjoint compact sets in a Hausdorff space have disjoint open neighbourhoods; its [Theorem 2.4 proof, part (3)](haar-measure.md#oa-fnd-hm-01), gives the finite-cover argument. We will also use the following metric fact.\n\n**Lemma (Sequential compactness).** A compact metrizable space is sequentially compact.\n\n*Proof.* Consider a sequence in a compact metric space. Some point \\(x\\) has every neighbourhood containing infinitely many sequence terms, counted with their indices. Otherwise choose at each point an open neighbourhood containing only finitely many terms; a finite subcover would contain only finitely many terms in total, a contradiction. Recursively choose increasing indices \\(n_k\\) with \\(d(x_{n_k},x)<1/k\\). Then \\(x_{n_k}\\to x\\). \\(\\square\\)\n\n\n## Elementary facts\n\nThese five facts are used throughout. Each has a short proof.\n\n- **(B1)** (Cantor's intersection theorem) In a complete metric space, a decreasing sequence of nonempty closed sets \\(F_k\\) whose diameters tend to \\(0\\) has exactly one common point. Pick \\(x_k\\in F_k\\); for \\(j\\geq k\\), \\(d(x_j,x_k)\\leq\\operatorname{diam}F_k\\), so the sequence is Cauchy. Its limit lies in every closed \\(F_k\\), and two common points are at distance at most \\(\\operatorname{diam}F_k\\) for every \\(k\\).\n- **(B2)** A countable product \\(\\prod_iF_i\\) of finite discrete sets is compact. Suppose an open cover has no finite subcover of \\(\\prod_iF_i\\). Call a finite sequence \\(s\\) with \\(s_i\\in F_i\\) *bad* if the cylinder of points beginning with \\(s\\) has no finite subcover. The empty sequence is bad, and a bad \\(s\\) has a bad one-step extension, since the finitely many extensions cover its cylinder. So some point \\(t\\) has every \\(t|j\\) bad. But \\(t\\) lies in a member of the cover, which contains the cylinder of some \\(t|j\\), a contradiction. In particular \\(\\mathcal C\\) is compact.\n- **(B3)** A continuous injection \\(f\\) of a compact space \\(K\\) into a Hausdorff space is a homeomorphism onto \\(f(K)\\): closed subsets of \\(K\\) are compact, so their images are compact, hence closed.\n- **(B4)** A second countable space is Lindelöf. Given an open cover, choose for each basic open set that lies in some member of the cover one such member. These countably many members cover the space, because each point lies in a basic open set that lies inside some member of the cover.\n- **(B5)** In a topological group, every neighbourhood \\(W\\) of \\(1\\) contains \\(V^2\\) for some open neighbourhood \\(V\\) of \\(1\\) with \\(V=V^{-1}\\). Continuity of multiplication at \\((1,1)\\) gives open neighbourhoods \\(V_1,V_2\\) of \\(1\\) with \\(V_1V_2\\subseteq W\\); take \\(V=V_0\\cap V_0^{-1}\\) with \\(V_0=V_1\\cap V_2\\), which is open because inversion is a homeomorphism.\n\n## A. Encode an image and recover its parameter\n\nTrees encode successive approximations, and the Baire space supplies a common domain for those encodings. Separation of disjoint Souslin sets is the step that makes injective images behave better than arbitrary images. Lusin–Souslin then turns injectivity into measurable inversion. The standard Borel classification records what survives when only measurable sets, rather than a topology, are retained.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-01",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "1. Unital functionals and the numerical range",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "full_conditions_and_proof": "### 1. Unital functionals and the numerical range\n\nIn Sections 1–3, \\(A\\) is a unital Banach algebra, so \\(\\|1\\|=1\\). The numerical-range arguments in Sections 1–3 are proved below.\n\n**Definition 1.1.** A functional \\(\\omega\\in A^*\\) is *unital* if \\(\\omega(1)=1=\\|\\omega\\|\\). Let \\(V(A)\\) be the set of unital functionals. The *numerical range* of \\(a\\in A\\) is\n\\[\nW(a)=W_A(a)=\\{\\omega(a):\\ \\omega\\in V(A)\\},\n\\]\nand the *numerical radius* of \\(a\\) is \\(v(a)=\\max\\{|z|:z\\in W(a)\\}\\).\n\nFor example, let \\(A=B(H)\\) for a Hilbert space \\(H\\neq0\\), and let \\(\\xi\\in H\\) be a unit vector. The functional \\(x\\mapsto\\langle x\\xi,\\xi\\rangle\\) takes the value \\(\\|\\xi\\|^2=1\\) at \\(1\\), and \\(|\\langle x\\xi,\\xi\\rangle|\\leq\\|x\\|\\). So it is unital, and \\(W(x)\\) contains every value of the quadratic form \\(\\xi\\mapsto\\langle x\\xi,\\xi\\rangle\\) on the unit sphere. For a general C\\*-algebra, Corollary 3.6 shows that the unital functionals are exactly the states.\n\n**Proposition 1.2.** Let \\(a,b\\in A\\).\n1. \\(V(A)=\\{\\omega\\in A^*:\\ \\|\\omega\\|\\leq1,\\ \\omega(1)=1\\}\\). This set is nonempty, convex and weak\\*-compact.\n2. \\(W(a)\\) is a nonempty compact convex subset of the closed disc of radius \\(\\|a\\|\\).\n3. \\(W(\\alpha+\\beta a)=\\alpha+\\beta W(a)\\) for \\(\\alpha,\\beta\\in\\mathbb C\\), and \\(W(a+b)\\subseteq W(a)+W(b)\\).\n\n**Proof.** (1) If \\(\\omega(1)=1\\), then \\(1=|\\omega(1)|\\leq\\|\\omega\\|\\,\\|1\\|=\\|\\omega\\|\\). So \\(\\|\\omega\\|\\leq1\\) and \\(\\omega(1)=1\\) together give \\(\\|\\omega\\|=1\\). The set is convex, and it is weak\\*-closed in the closed unit ball of \\(A^*\\), which is weak\\*-compact by (H3). By (H2) some \\(\\omega\\) has \\(\\|\\omega\\|\\leq1\\) and \\(\\omega(1)=\\|1\\|=1\\), so the set is not empty.\n\n(2) The map \\(\\omega\\mapsto\\omega(a)\\) is affine and weak\\*-continuous. It carries the nonempty compact convex set \\(V(A)\\) onto a nonempty compact convex set, and \\(|\\omega(a)|\\leq\\|\\omega\\|\\,\\|a\\|=\\|a\\|\\).\n\n(3) For \\(\\omega\\in V(A)\\), \\(\\omega(\\alpha+\\beta a)=\\alpha+\\beta\\omega(a)\\) and \\(\\omega(a+b)=\\omega(a)+\\omega(b)\\). \\(\\square\\)\n\nThe next theorem shows that \\(W(a)\\) depends only on the norm on the space spanned by \\(1\\) and \\(a\\).\n\n**Theorem 1.3.** Let \\(a\\in A\\) and \\(\\alpha\\in\\mathbb C\\). The following are equivalent.\n1. \\(\\alpha\\in W(a)\\).\n2. The formula \\(f(\\lambda+\\mu a)=\\lambda+\\mu\\alpha\\), for \\(\\lambda,\\mu\\in\\mathbb C\\), defines a linear functional \\(f\\) on \\(\\operatorname{span}\\{1,a\\}\\) with \\(|f(y)|\\leq\\|y\\|\\).\n3. \\(|\\lambda+\\mu\\alpha|\\leq\\|\\lambda+\\mu a\\|\\) for all \\(\\lambda,\\mu\\in\\mathbb C\\).\n4. \\(|\\lambda+\\alpha|\\leq\\|\\lambda+a\\|\\) for all \\(\\lambda\\in\\mathbb C\\).\n\n**Proof.** (1) ⇒ (4). If \\(\\alpha=\\omega(a)\\) with \\(\\omega\\in V(A)\\), then \\(|\\lambda+\\alpha|=|\\omega(\\lambda+a)|\\leq\\|\\lambda+a\\|\\).\n\n(4) ⇒ (3). For \\(\\mu=0\\) the claim reads \\(|\\lambda|\\leq\\|\\lambda1\\|\\), which is an equality. For \\(\\mu\\neq0\\), apply (4) to \\(\\lambda/\\mu\\) and multiply by \\(|\\mu|\\).\n\n(3) ⇒ (2). If \\(\\lambda+\\mu a=\\lambda'+\\mu'a\\), then (3) for the pair \\((\\lambda-\\lambda',\\mu-\\mu')\\) gives \\(|(\\lambda-\\lambda')+(\\mu-\\mu')\\alpha|\\leq0\\), so \\(\\lambda+\\mu\\alpha=\\lambda'+\\mu'\\alpha\\). Hence \\(f\\) is well defined, also when \\(1\\) and \\(a\\) are linearly dependent. It is linear, and (3) is the bound \\(|f(y)|\\leq\\|y\\|\\).\n\n(2) ⇒ (1). By (H2), \\(f\\) extends to some \\(\\omega\\in A^*\\) with \\(\\|\\omega\\|\\leq1\\). Then \\(\\omega(1)=f(1)=1\\), so \\(\\omega\\in V(A)\\) by Proposition 1.2(1), and \\(\\omega(a)=f(a)=\\alpha\\). \\(\\square\\)\n\n**Corollary 1.4.** Let \\(a\\in A\\).\n1. If \\(B\\subseteq A\\) is a closed subalgebra that contains \\(1\\) and \\(a\\), with the norm of \\(A\\), then \\(W_B(a)=W_A(a)\\).\n2. \\(\\sigma(a)\\subseteq W(a)\\). Hence \\(W(a)\\) contains the convex hull of \\(\\sigma(a)\\), and \\(r(a)\\leq v(a)\\leq\\|a\\|\\).\n\n**Proof.** (1) Condition (4) of Theorem 1.3 involves only the norms of the elements \\(\\lambda+a\\), which lie in \\(B\\).\n\n(2) Let \\(\\alpha\\in\\sigma(a)\\) and \\(\\lambda\\in\\mathbb C\\). By (B1), \\(\\lambda+\\alpha\\in\\sigma(\\lambda+a)\\), so \\(|\\lambda+\\alpha|\\leq r(\\lambda+a)\\leq\\|\\lambda+a\\|\\). Theorem 1.3 gives \\(\\alpha\\in W(a)\\). The rest follows from the convexity of \\(W(a)\\) and from Proposition 1.2(2). \\(\\square\\)\n\nPart (1) is in contrast with the spectrum, which can shrink when the algebra grows; see the example of the disc algebra in [the lesson on Banach algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md).\n\n**Proposition 1.4a** (Stability under norm perturbation). For \\(a,b\\in A\\), the Hausdorff distance between their numerical ranges is at most \\(\\|a-b\\|\\). In particular, both the numerical radius and the largest real part of the numerical range change by at most \\(\\|a-b\\|\\).\n\nHere the Hausdorff distance of nonempty compact sets \\(K,L\\subseteq\\mathbb C\\) is the larger of \\(\\sup_{z\\in K}\\inf_{w\\in L}|z-w|\\) and \\(\\sup_{w\\in L}\\inf_{z\\in K}|z-w|\\). The numerical-range estimate is Lemma 2.1 of [H. Blazhko, D. Homza, F. L. Schwenninger, J. de Vries and M. Wojtylak, *The algebraic numerical range as a spectral set in Banach algebras* (2025)](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/algebraic-numerical-range-as-a-spectral-set-in-banach-algebras/96536D155B032F6C67B750582F5DBF07#sec2).\n\n**Proof.** Put \\(\\delta=\\|a-b\\|\\). If \\(z\\in W(a)\\), choose \\(\\omega\\in V(A)\\) with \\(z=\\omega(a)\\). The point \\(w=\\omega(b)\\) belongs to \\(W(b)\\), and\n\\[\n|z-w|=|\\omega(a-b)|\\le\\delta.\n\\]\nThus every point of \\(W(a)\\) is within \\(\\delta\\) of \\(W(b)\\). Interchanging \\(a\\) and \\(b\\) gives the other half of the Hausdorff estimate. Since \\(|z|\\le |w|+\\delta\\), taking maxima gives \\(v(a)\\le v(b)+\\delta\\); interchanging the elements yields the absolute-difference bound. Likewise \\(\\operatorname{Re}z\\le\\operatorname{Re}w+\\delta\\), which proves the claim about largest real parts. The same argument applies to \\(e^{i\\theta}a,e^{i\\theta}b\\) for every real \\(\\theta\\), with the same constant. It needs only the norm-one functionals and their common domain, so it also holds in the normed-space formulation of Remark 1.5. \\(\\square\\)\n\n**Remark 1.5** (only the norm and the unit matter). Apart from the spectrum in Corollary 1.4(2), nothing in this section used the product of \\(A\\). For a complex normed space \\(E\\) and a vector \\(e\\in E\\) with \\(\\|e\\|=1\\), define \\(V(E,e)\\) and \\(W(x)\\), \\(x\\in E\\), as above with \\(e\\) in place of \\(1\\). Proposition 1.2, Theorem 1.3 and Corollary 1.4(1) hold with the same proofs, and so do parts (1) and (2) of Theorem 2.3 below. The product enters through the spectrum and through the exponential in Theorem 2.3(3).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-02",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "2. The numerical range from the norm",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "full_conditions_and_proof": "### 2. The numerical range from the norm\n\nFor \\(a\\in A\\) let \\(m(a)=\\max\\operatorname{Re}W(a)\\), the largest real part of a point of \\(W(a)\\); it exists because \\(W(a)\\) is compact. For real \\(\\theta\\), \\(W(e^{i\\theta}a)=e^{i\\theta}W(a)\\) by Proposition 1.2(3), so\n\\[\nm(e^{i\\theta}a)=\\max_{z\\in W(a)}\\operatorname{Re}\\big(e^{i\\theta}z\\big).\n\\]\nBy the next lemma these numbers determine \\(W(a)\\).\n\n**Lemma 2.1** (support functions). Let \\(K\\subseteq\\mathbb C\\) be nonempty, compact and convex, and put \\(h_K(\\theta)=\\max_{z\\in K}\\operatorname{Re}(e^{i\\theta}z)\\) for \\(\\theta\\in\\mathbb R\\). Then\n\\[\n\\begin{gathered}\nK\\\\\n=\\{z\\in\\mathbb C :\\ \\operatorname{Re}(e^{i\\theta}z)\\leq h_K(\\theta)\\text{ for all }\\theta\\in\\mathbb R\\}.\n\\end{gathered}\n\\]\nHence two nonempty compact convex sets with the same function \\(h\\) are equal. If \\(h_K(\\theta)=R\\) for every \\(\\theta\\), then \\(K\\) is the closed disc \\(\\{|z|\\leq R\\}\\).\n\n**Proof.** Every point of \\(K\\) lies in the set on the right. Let \\(p\\notin K\\), and let \\(q\\in K\\) be a point nearest to \\(p\\), which exists because \\(K\\) is compact. For \\(z\\in K\\) and \\(0<s\\leq1\\) the point \\(q+s(z-q)\\) lies in \\(K\\), so\n\\[\n\\begin{gathered}\n|p-q|^2\\\\\n\\leq|p-q-s(z-q)|^2\\\\\n=|p-q|^2-2s\\operatorname{Re}\\big(\\overline{(p-q)}(z-q)\\big)\\\\\n+s^2|z-q|^2 .\n\\end{gathered}\n\\]\nDividing by \\(s\\) and letting \\(s\\to0\\) gives \\(\\operatorname{Re}\\big(\\overline{(p-q)}(z-q)\\big)\\leq0\\). Choose \\(\\theta\\) with \\(e^{i\\theta}=\\overline{(p-q)}/|p-q|\\). Then \\(\\operatorname{Re}(e^{i\\theta}z)\\leq\\operatorname{Re}(e^{i\\theta}q)\\) for all \\(z\\in K\\), so \\(h_K(\\theta)=\\operatorname{Re}(e^{i\\theta}q)\\), while \\(\\operatorname{Re}(e^{i\\theta}p)=\\operatorname{Re}(e^{i\\theta}q)+|p-q|>h_K(\\theta)\\). So \\(p\\) is not in the set on the right. If \\(h_K\\equiv R\\), the width \\(h_K(\\theta)+h_K(\\theta+\\pi)=2R\\) is nonnegative, so \\(R\\geq0\\). The closed disc of radius \\(R\\) is compact and convex, and its function \\(h\\) is constant, equal to \\(R\\); this gives the last claim. \\(\\square\\)\n\n**Lemma 2.2** (subadditive functions near zero). Let \\(F:[0,\\infty)\\to\\mathbb R\\) satisfy \\(F(0)=0\\), \\(F(s+t)\\leq F(s)+F(t)\\) for \\(s,t\\geq0\\), and \\(F(t)\\to0\\) as \\(t\\to0^+\\). Suppose that \\(\\alpha=\\sup_{t>0}F(t)/t\\) is finite. Then \\(F(t)/t\\to\\alpha\\) as \\(t\\to0^+\\).\n\n**Proof.** Clearly \\(\\limsup_{t\\to0^+}F(t)/t\\leq\\alpha\\). Fix \\(t>0\\). For \\(0<s<t\\) write \\(t=ns+\\delta\\) with an integer \\(n\\geq1\\) and \\(0\\leq\\delta<s\\). Subadditivity gives \\(F(t)\\leq nF(s)+F(\\delta)\\), so\n\\[\n\\frac{F(t)}{t}\\leq\\frac{ns}{t}\\cdot\\frac{F(s)}{s}+\\frac{F(\\delta)}{t}.\n\\tag{2.1}\n\\]\nAs \\(s\\to0^+\\), \\(ns/t\\to1\\) because \\(t-s<ns\\leq t\\), and \\(F(\\delta)\\to0\\) because \\(0\\leq\\delta<s\\). Choose \\(s_k\\to0^+\\) with \\(F(s_k)/s_k\\to L=\\liminf_{s\\to0^+}F(s)/s\\). If \\(L\\) were \\(-\\infty\\), the right side of (2.1) would tend to \\(-\\infty\\) along \\(s_k\\), which is impossible; so \\(L\\) is finite, and (2.1) gives \\(F(t)/t\\leq L\\). Taking the supremum over \\(t\\) gives \\(\\alpha\\leq L\\). \\(\\square\\)\n\n**Theorem 2.3.** Let \\(a\\in A\\).\n1. The function \\(g(u)=\\|u+a\\|-u\\), \\(u\\in\\mathbb R\\), is nonincreasing, and\n\\[\n\\begin{gathered}\nm(a)\\\\\n=\\inf_{u\\in\\mathbb R}g(u)\\\\\n=\\lim_{u\\to+\\infty}\\big(\\|u+a\\|-u\\big)\\\\\n=\\inf_{t>0}\\frac{\\|1+ta\\|-1}{t}\\\\\n=\\lim_{t\\to0^+}\\frac{\\|1+ta\\|-1}{t}.\n\\end{gathered}\n\\tag{2.2}\n\\]\n2. \\(\\operatorname{Re}W(a)=[-m(-a),\\,m(a)]\\).\n3. We have\n\\[\n\\begin{gathered}\nm(a)\\\\\n=\\lim_{t\\to0^+}\\frac{\\|e^{ta}\\|-1}{t}\\\\\n=\\lim_{t\\to0^+}\\frac{\\log\\|e^{ta}\\|}{t}\\\\\n=\\sup_{t>0}\\frac{\\log\\|e^{ta}\\|}{t}.\n\\end{gathered}\n\\tag{2.3}\n\\]\nIn particular \\(\\|e^{ta}\\|\\leq e^{t\\,m(a)}\\) for all \\(t\\geq0\\), and \\(m(a)\\) is the smallest real number \\(w\\) with \\(\\|e^{ta}\\|\\leq e^{tw}\\) for all \\(t\\geq0\\).\n\n**Proof.** (1) For \\(u<u'\\), \\(\\|u'+a\\|\\leq\\|u+a\\|+(u'-u)\\), so \\(g(u')\\leq g(u)\\). For \\(\\alpha\\in W(a)\\) and real \\(u\\), Theorem 1.3(4) gives \\(u+\\operatorname{Re}\\alpha\\leq|u+\\alpha|\\leq\\|u+a\\|\\), that is, \\(\\operatorname{Re}\\alpha\\leq g(u)\\). Hence \\(m(a)\\leq c\\), where \\(c=\\inf_ug(u)\\); in particular \\(c\\) is finite.\n\nFor the reverse inequality we find \\(\\omega\\in V(A)\\) with \\(\\operatorname{Re}\\omega(a)=c\\). Regard \\(A\\) as a real vector space. If \\(a=s_01\\) with \\(s_0\\) real, then \\(W(a)=\\{s_0\\}\\) and \\(g(u)=|u+s_0|-u=s_0\\) for \\(u\\geq-s_0\\), so \\(c=s_0=m(a)\\). Otherwise \\(1\\) and \\(a\\) are linearly independent over \\(\\mathbb R\\), and \\(\\psi_0(s+ra)=s+rc\\), for \\(s,r\\in\\mathbb R\\), defines a real-linear functional on their real span. We check that \\(\\psi_0(y)\\leq\\|y\\|\\) there. For \\(r=0\\) this reads \\(s\\leq|s|\\). For \\(r>0\\), since \\(c\\leq g(s/r)\\),\n\\[\n\\begin{gathered}\ns+rc\\\\\n\\leq r\\big(s/r+g(s/r)\\big)\\\\\n=r\\,\\|s/r+a\\|\\\\\n=\\|s+ra\\|.\n\\end{gathered}\n\\]\nFor \\(r<0\\) put \\(q=-r>0\\) and \\(w=s/q\\). For every real \\(u\\),\n\\[\n\\begin{gathered}\ng(u)-\\big(w-\\|w-a\\|\\big)\\\\\n=\\|u+a\\|+\\|w-a\\|-(u+w)\\\\\n\\geq\\|(u+w)1\\|-(u+w)\\\\\n\\geq0,\n\\end{gathered}\n\\]\nso \\(c\\geq w-\\|w-a\\|\\), and \\(s+rc=q(w-c)\\leq q\\|w-a\\|=\\|s+ra\\|\\). By (H1), \\(\\psi_0\\) extends to a real-linear \\(\\psi\\) on \\(A\\) with \\(\\psi\\leq\\|\\cdot\\|\\); applied to \\(-x\\), this gives \\(|\\psi(x)|\\leq\\|x\\|\\). By (H2), \\(\\omega(x)=\\psi(x)-i\\psi(ix)\\) is complex-linear, \\(\\operatorname{Re}\\omega=\\psi\\) and \\(\\|\\omega\\|\\leq1\\). Now \\(\\operatorname{Re}\\omega(1)=\\psi(1)=1\\) and \\(|\\omega(1)|\\leq1\\), so \\(\\omega(1)=1\\). Thus \\(\\omega\\in V(A)\\) and \\(\\operatorname{Re}\\omega(a)=\\psi(a)=c\\), and \\(m(a)=c\\).\n\nFinally, \\((\\|1+ta\\|-1)/t=g(1/t)\\) for \\(t>0\\). Since \\(g\\) is nonincreasing, its infimum over \\(u>0\\) equals its infimum over \\(\\mathbb R\\) and its limit as \\(u\\to+\\infty\\), which is the limit of \\(g(1/t)\\) as \\(t\\to0^+\\). This proves (2.2).\n\n(2) \\(\\operatorname{Re}W(a)\\) is the image of the compact convex set \\(W(a)\\) under the real-linear map \\(\\operatorname{Re}\\), so it is a compact interval. Its maximum is \\(m(a)\\). Its minimum is \\(-\\max\\operatorname{Re}(-W(a))=-m(-a)\\), since \\(W(-a)=-W(a)\\).\n\n(3) *The first limit.* For \\(t\\geq0\\),\n\\[\n\\begin{gathered}\n\\Big|\\,\\|e^{ta}\\|-\\|1+ta\\|\\,\\Big|\\\\\n\\leq\\Big\\|\\sum_{n\\geq2}\\frac{t^na^n}{n!}\\Big\\|\\\\\n\\leq\\sum_{n\\geq2}\\frac{(t\\|a\\|)^n}{n!}\\\\\n\\leq t^2\\|a\\|^2e^{t\\|a\\|}.\n\\end{gathered}\n\\]\nDivide by \\(t\\) and use (2.2).\n\n*The second limit.* Put \\(h(t)=\\|e^{ta}\\|\\). It is positive, because \\(e^{ta}\\) is invertible (B3), and \\(h(t)\\to\\|1\\|=1\\) as \\(t\\to0\\). The inequalities \\(1-1/y\\leq\\log y\\leq y-1\\) for \\(y>0\\) give\n\\[\n\\frac{h(t)-1}{t\\,h(t)}\\leq\\frac{\\log h(t)}{t}\\leq\\frac{h(t)-1}{t},\n\\]\nand both bounds tend to \\(m(a)\\) by the first limit.\n\n*The supremum.* The function \\(F(t)=\\log h(t)\\) has \\(F(0)=0\\) and \\(F(t)\\to0\\) as \\(t\\to0^+\\). It is subadditive, because \\(e^{(s+t)a}=e^{sa}e^{ta}\\) by (B3), so \\(h(s+t)\\leq h(s)h(t)\\). And \\(F(t)\\leq t\\|a\\|\\), since \\(h(t)\\leq e^{t\\|a\\|}\\). By Lemma 2.2, \\(\\sup_{t>0}F(t)/t=\\lim_{t\\to0^+}F(t)/t=m(a)\\). So \\(F(t)\\leq t\\,m(a)\\) for \\(t>0\\). If \\(h(t)\\leq e^{tw}\\) for all \\(t>0\\), then \\(F(t)/t\\leq w\\) for all \\(t>0\\), and \\(m(a)\\leq w\\). \\(\\square\\)\n\nSo \\(m(a)\\) is the right derivative at \\(0\\) of \\(t\\mapsto\\|1+ta\\|\\), and also the exact exponential growth rate of \\(\\|e^{ta}\\|\\) for \\(t\\geq0\\).\n\n**Corollary 2.4** (dissipative elements). \\(\\operatorname{Re}W(a)\\subseteq(-\\infty,0]\\) if and only if \\(\\|e^{ta}\\|\\leq1\\) for all \\(t\\geq0\\).\n\n**Proof.** Both conditions say that \\(m(a)\\leq0\\), by Theorem 2.3(3). \\(\\square\\)\n\nFor bounded operators, the dissipativity condition is expressed directly by the numerical-range inequality proved here.\n\n**Example 2.5** (a nilpotent matrix and two norms). Let \\(N=\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\in M_2(\\mathbb C)\\). Then \\(N^2=0\\), \\(\\sigma(N)=\\{0\\}\\) and \\(r(N)=0\\). We compute \\(W(N)\\) for two algebra norms on \\(M_2(\\mathbb C)\\) with \\(\\|1\\|=1\\), using Theorem 2.3(1) and Lemma 2.1. For \\(z\\in\\mathbb C\\), \\(1+zN=\\begin{pmatrix}1&z\\\\0&1\\end{pmatrix}\\).\n\n(a) *The operator norm for the Euclidean norm on \\(\\mathbb C^2\\).* Then \\(M_2(\\mathbb C)=B(\\mathbb C^2)\\) is a C\\*-algebra, and by (C1), \\(\\|1+zN\\|^2=\\|(1+zN)^*(1+zN)\\|\\) is the largest eigenvalue of the self-adjoint matrix \\(\\begin{pmatrix}1&z\\\\\\bar z&1+|z|^2\\end{pmatrix}\\). This matrix has trace \\(2+|z|^2\\) and determinant \\(1\\), so its eigenvalues are \\(\\big(2+|z|^2\\pm|z|\\sqrt{|z|^2+4}\\big)/2\\), and the larger one equals \\(\\big((|z|+\\sqrt{|z|^2+4})/2\\big)^2\\). For \\(t>0\\) and real \\(\\theta\\),\n\\[\n\\begin{gathered}\n\\frac{\\|1+te^{i\\theta}N\\|-1}{t}\\\\\n=\\frac{t+\\sqrt{t^2+4}-2}{2t}\\\\\n=\\frac12+\\frac{t}{2\\big(\\sqrt{t^2+4}+2\\big)}\\longrightarrow\\frac12 .\n\\end{gathered}\n\\]\nSo \\(m(e^{i\\theta}N)=1/2\\) for every \\(\\theta\\), and \\(W(N)=\\{|z|\\leq1/2\\}\\).\n\n(b) *The operator norm for the maximum norm on \\(\\mathbb C^2\\)*, which is the largest row sum of absolute values. Here \\(\\|1+zN\\|=1+|z|\\), so \\(m(e^{i\\theta}N)=1\\) for every \\(\\theta\\), and \\(W(N)=\\{|z|\\leq1\\}\\).\n\nIn both cases \\(\\|N\\|=1\\). In (a), \\(r(N)=0<v(N)=\\tfrac12<\\|N\\|\\); in (b), \\(v(N)=\\|N\\|\\). So the numerical range depends on the norm, and not only on the algebra.\n\n",
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      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "3. Hermitian elements",
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      "full_conditions_and_proof": "### 3. Hermitian elements\n\n**Definition 3.1.** An element \\(h\\in A\\) is *hermitian* if \\(W(h)\\subseteq\\mathbb R\\).\n\nBy Proposition 1.2(3), the hermitian elements form a real vector space that contains \\(\\mathbb R1\\). It is closed, since \\(\\omega(h_n)\\to\\omega(h)\\) for every \\(\\omega\\in V(A)\\) when \\(h_n\\to h\\). By Corollary 1.4(2), a hermitian element has real spectrum.\n\n**Theorem 3.2.** For \\(h\\in A\\) the following are equivalent.\n1. \\(h\\) is hermitian.\n2. \\(\\|e^{ith}\\|=1\\) for every \\(t\\in\\mathbb R\\).\n3. \\(\\|e^{ith}\\|\\leq1\\) for every \\(t\\in\\mathbb R\\).\n4. \\(\\|1+ith\\|=1+o(t)\\) as \\(t\\to0\\) through real values; that is, \\((\\|1+ith\\|-1)/t\\to0\\).\n\n**Proof.** Since \\(W(\\pm ih)=\\pm iW(h)\\), we have \\(m(ih)=-\\min\\operatorname{Im}W(h)\\) and \\(m(-ih)=\\max\\operatorname{Im}W(h)\\). So \\(h\\) is hermitian exactly when \\(m(ih)\\leq0\\) and \\(m(-ih)\\leq0\\), and then both numbers are \\(0\\).\n\n(1) ⇔ (3). By Theorem 2.3(3), \\(m(ih)\\leq0\\) exactly when \\(\\|e^{tih}\\|\\leq1\\) for all \\(t\\geq0\\), and \\(m(-ih)\\leq0\\) exactly when \\(\\|e^{-tih}\\|\\leq1\\) for all \\(t\\geq0\\).\n\n(2) ⇔ (3). If (3) holds, then \\(1=\\|1\\|=\\|e^{ith}e^{-ith}\\|\\leq\\|e^{ith}\\|\\,\\|e^{-ith}\\|\\leq1\\), so both norms equal \\(1\\).\n\n(1) ⇔ (4). By Theorem 2.3(1), \\(m(\\pm ih)=\\lim_{t\\to0^+}(\\|1\\pm ith\\|-1)/t\\). If \\(h\\) is hermitian, both limits are \\(0\\), which is (4). Conversely, (4) gives \\(m(ih)=m(-ih)=0\\). \\(\\square\\)\n\nThe proof of Sinclair's theorem below uses the power series of the arcsine. We derive the facts we need.\n\n**Lemma 3.3** (the arcsine series). Let \\(c_n=\\binom{2n}{n}4^{-n}/(2n+1)\\) for \\(n\\geq0\\), and \\(S(z)=\\sum_{n\\geq0}c_nz^{2n+1}\\) for \\(|z|<1\\).\n1. \\(c_n>0\\), the series converges for \\(|z|<1\\), and \\(\\sum_nc_n\\leq\\pi/2\\).\n2. Let \\(0<c'<\\pi/2\\), choose \\(\\delta>0\\) with \\(\\sin^2c'+\\sinh^2\\delta<1\\), and let \\(R=\\{x+iy:\\ |x|<c',\\ |y|<\\delta\\}\\). Then \\(|\\sin z|<1\\) and \\(S(\\sin z)=z\\) for every \\(z\\in R\\).\n\n**Proof.** Since \\(\\binom{2n}{n}\\leq4^n\\), \\(0<c_n\\leq1\\), so the series converges for \\(|z|<1\\). Let \\(P(w)=\\sum_n\\binom{2n}{n}w^n\\) for real \\(|w|<1/4\\). From \\((n+1)\\binom{2n+2}{n+1}=2(2n+1)\\binom{2n}{n}\\) we get\n\\[\n\\begin{gathered}\nP'(w)\\\\\n=\\sum_{n\\geq0}(n+1)\\binom{2n+2}{n+1}w^n\\\\\n=4wP'(w)+2P(w),\n\\end{gathered}\n\\]\nso \\[\n\\begin{gathered}\n\\big((1-4w)^{1/2}P(w)\\big)'\\\\\n=(1-4w)^{-1/2}\\big((1-4w)P'(w)-2P(w)\\big)\\\\\n=0.\n\\end{gathered}\n\\] As \\(P(0)=1\\), \\(P(w)=(1-4w)^{-1/2}\\). Differentiating \\(S\\) term by term, \\[\n\\begin{gathered}\nS'(y)\\\\\n=\\sum_n\\binom{2n}{n}4^{-n}y^{2n}\\\\\n=P(y^2/4)\\\\\n=(1-y^2)^{-1/2}\n\\end{gathered}\n\\] for \\(-1<y<1\\). For \\(|x|<\\pi/2\\), \\(\\cos x>0\\), so\n\\[\n\\frac{d}{dx}S(\\sin x)=\\frac{\\cos x}{\\sqrt{1-\\sin^2x}}=1,\n\\]\nand \\(S(\\sin0)=0\\). Hence \\(S(\\sin x)=x\\) for \\(|x|<\\pi/2\\).\n\nFor \\(z=x+iy\\in R\\), \\[\n\\begin{gathered}\n|\\sin z|^2\\\\\n=\\sin^2x+\\sinh^2y<\\sin^2c'+\\sinh^2\\delta<1.\n\\end{gathered}\n\\] So \\(z\\mapsto S(\\sin z)-z\\) is holomorphic on the connected open set \\(R\\) and vanishes on the real segment \\((-c',c')\\); by the identity theorem it vanishes on \\(R\\).\n\nFinally, for \\(0<x<\\pi/2\\) and every \\(N\\), \\(\\sum_{n\\leq N}c_n\\sin^{2n+1}x\\leq S(\\sin x)=x<\\pi/2\\). Letting \\(x\\) increase to \\(\\pi/2\\) gives \\(\\sum_{n\\leq N}c_n\\leq\\pi/2\\) for every \\(N\\). \\(\\square\\)\n\n**Theorem 3.4** (Sinclair). If \\(h\\in A\\) is hermitian, then \\(\\|h\\|=r(h)\\), and\n\\[\nW(h)=[\\min\\sigma(h),\\ \\max\\sigma(h)],\n\\]\nthe convex hull of \\(\\sigma(h)\\).\n\n\n**Proof.** *Step 1: if \\(h\\) is hermitian and \\(r(h)<\\pi/2\\), then \\(\\|h\\|\\leq\\pi/2\\).* The spectrum of \\(h\\) is real, so \\(\\sigma(h)\\subseteq[-r(h),r(h)]\\). Choose \\(c'\\) with \\(r(h)<c'<\\pi/2\\), and let \\(\\delta\\) and the rectangle \\(R\\) be as in Lemma 3.3; then \\(R\\) is an open set containing \\(\\sigma(h)\\). Let \\(s=\\sin h\\) be the functional calculus of \\(\\sin\\) at \\(h\\). Since \\(\\sin z=(e^{iz}-e^{-iz})/(2i)\\), (B2) and (B3) give \\(s=(e^{ih}-e^{-ih})/(2i)\\), and Theorem 3.2 gives\n\\[\n\\|s\\|\\leq\\tfrac12\\big(\\|e^{ih}\\|+\\|e^{-ih}\\|\\big)=1 .\n\\]\nBy spectral mapping (B2), \\(\\sigma(s)=\\sin(\\sigma(h))\\subseteq\\sin([-r(h),r(h)])\\), which lies in \\((-1,1)\\); so \\(r(s)<1\\). By the power series rule in (B2), the functional calculus of \\(S\\) at \\(s\\) is \\(S(s)=\\sum_nc_ns^{2n+1}\\), with convergence in norm. By the composition rule in (B2), \\(S(s)=S(\\sin h)=(S\\circ\\sin)(h)\\). By Lemma 3.3, \\(S\\circ\\sin\\) is the identity function on the open set \\(R\\supseteq\\sigma(h)\\), so \\((S\\circ\\sin)(h)=h\\) by (B2). Therefore\n\\[\n\\begin{gathered}\n\\|h\\|\\\\\n=\\Big\\|\\sum_nc_ns^{2n+1}\\Big\\|\\\\\n\\leq\\sum_nc_n\\|s\\|^{2n+1}\\\\\n\\leq\\sum_nc_n\\\\\n\\leq\\frac\\pi2 .\n\\end{gathered}\n\\]\n\n*Step 2: \\(\\|h\\|=r(h)\\).* Let \\(h\\) be hermitian. For real \\(t>0\\) with \\(t\\,r(h)<\\pi/2\\), the element \\(th\\) is hermitian and \\(r(th)=t\\,r(h)\\), so Step 1 gives \\(t\\|h\\|\\leq\\pi/2\\). If \\(r(h)=0\\), this holds for every \\(t>0\\), and \\(h=0\\). If \\(r(h)>0\\), let \\(t\\) increase to \\(\\pi/(2r(h))\\); this gives \\(\\|h\\|\\leq r(h)\\). The reverse inequality holds for every element (B1).\n\n*Step 3: the numerical range.* The spectrum of \\(h\\) lies in \\([-\\|h\\|,\\|h\\|]\\). For \\(u\\geq\\|h\\|\\), the element \\(u+h\\) is hermitian and \\(\\sigma(u+h)=u+\\sigma(h)\\subseteq[0,\\infty)\\), so by Step 2, \\(\\|u+h\\|=r(u+h)=u+\\max\\sigma(h)\\). Thus \\(g(u)=\\|u+h\\|-u=\\max\\sigma(h)\\) for \\(u\\geq\\|h\\|\\), and Theorem 2.3(1) gives \\(m(h)=\\max\\sigma(h)\\). Since \\(W(h)\\) is real, \\(\\max W(h)=m(h)=\\max\\sigma(h)\\). The same argument for the hermitian element \\(-h\\) gives \\[\n\\begin{gathered}\n\\min W(h)\\\\\n=-\\max W(-h)\\\\\n=-\\max\\sigma(-h)\\\\\n=\\min\\sigma(h).\n\\end{gathered}\n\\] As \\(W(h)\\) is a compact convex subset of \\(\\mathbb R\\), it is the interval between these two numbers. \\(\\square\\)\n\nThe same renorming trick that makes an operator group isometric gives a version for bounded groups.\n\n**Corollary 3.5** (bounded groups). Let \\(a\\in A\\), and suppose \\(M=\\sup_{t\\in\\mathbb R}\\|e^{ita}\\|\\) is finite. Then \\(\\|a\\|\\leq M\\,r(a)\\). The case \\(M=1\\) is the first statement of Theorem 3.4.\n\n**Proof.** For \\(x\\in A\\) put \\(|x|=\\sup_{t\\in\\mathbb R}\\|e^{ita}x\\|\\). Then \\(\\|x\\|\\leq|x|\\leq M\\|x\\|\\) (take \\(t=0\\), and use \\(\\|e^{ita}x\\|\\leq\\|e^{ita}\\|\\|x\\|\\)). So \\(E=(A,|\\cdot|)\\) is a Banach space, and the algebra \\(B(E)\\) of bounded operators on \\(E\\), with the operator norm \\(|\\cdot|_{\\mathrm{op}}\\), is a unital Banach algebra whose identity has norm \\(1\\). For \\(b\\in A\\) let \\(L_b\\) be left multiplication, \\(L_bx=bx\\). Then \\(|L_bx|\\leq M\\|bx\\|\\leq M\\|b\\|\\,|x|\\), so \\(|L_b|_{\\mathrm{op}}\\leq M\\|b\\|\\), and \\(b\\mapsto L_b\\) is a continuous unital homomorphism \\(A\\to B(E)\\). Applying it to the exponential series gives \\(e^{isL_a}=L_{e^{isa}}\\). For \\(x\\in A\\) and real \\(s\\), by (B3),\n\\[\n\\begin{gathered}\n|e^{isL_a}x|\\\\\n=|e^{isa}x|\\\\\n=\\sup_t\\|e^{ita}e^{isa}x\\|\\\\\n=\\sup_t\\|e^{i(t+s)a}x\\|\\\\\n=|x| .\n\\end{gathered}\n\\]\nSo every \\(e^{isL_a}\\) is an isometry of \\(E\\), and \\(|e^{isL_a}|_{\\mathrm{op}}=1\\). By Theorem 3.2, applied in \\(B(E)\\), \\(L_a\\) is hermitian there, and Theorem 3.4 gives \\(|L_a|_{\\mathrm{op}}=r_{B(E)}(L_a)\\). By (B1),\n\\[\n\\begin{gathered}\nr_{B(E)}(L_a)\\\\\n=\\lim_n|L_{a^n}|_{\\mathrm{op}}^{1/n}\\\\\n\\leq\\lim_n\\big(M\\|a^n\\|\\big)^{1/n}\\\\\n=r(a).\n\\end{gathered}\n\\]\nFinally \\(\\|a\\|=\\|a1\\|\\leq|L_a1|\\leq|L_a|_{\\mathrm{op}}\\,|1|\\leq r(a)\\,M\\), because \\(|1|=\\sup_t\\|e^{ita}\\|=M\\). \\(\\square\\)\n\n**Corollary 3.6** (C\\*-algebras). Let \\(A\\) be a unital C\\*-algebra with \\(1\\neq0\\).\n1. The hermitian elements of \\(A\\) are exactly its self-adjoint elements.\n2. The unital functionals are exactly the states. So \\(W(a)=\\{\\varphi(a):\\ \\varphi\\text{ a state}\\}\\).\n3. For \\(h=h^*\\), \\(W(h)=[\\min\\sigma(h),\\max\\sigma(h)]\\), and \\(v(h)=\\|h\\|\\).\n\n**Proof.** *Self-adjoint elements are hermitian.* Let \\(h=h^*\\) and \\(t\\in\\mathbb R\\). The involution is isometric and conjugate-linear, so applying it to the partial sums of the exponential series gives \\((e^{ith})^*=e^{-ith}\\). By (B3) and (C1), \\(\\|e^{ith}\\|^2=\\|e^{-ith}e^{ith}\\|=\\|1\\|=1\\). So \\(h\\) is hermitian by Theorem 3.2.\n\n(3) follows from this and Theorem 3.4. Also \\[\n\\begin{gathered}\nv(h)\\\\\n=\\max(|\\min\\sigma(h)|,|\\max\\sigma(h)|)\\\\\n=r(h)\\\\\n=\\|h\\|\n\\end{gathered}\n\\] by (C1).\n\n(1) It remains to show that a hermitian \\(x\\) is self-adjoint. Write \\(x=h+ik\\) with \\(h=(x+x^*)/2\\) and \\(k=(x-x^*)/(2i)\\), both self-adjoint. Then \\(ik=x-h\\) is hermitian, as a difference of hermitian elements. So \\(W(ik)=iW(k)\\) is contained in \\(\\mathbb R\\cap i\\mathbb R=\\{0\\}\\), and \\(W(k)=\\{0\\}\\). By (3), \\(\\sigma(k)=\\{0\\}\\), and \\(\\|k\\|=r(k)=0\\) by (C1). So \\(x=h\\).\n\n(2) Let \\(\\omega\\in V(A)\\) and \\(a\\in A_+\\). Then \\(\\omega(a)\\in W(a)=[\\min\\sigma(a),\\max\\sigma(a)]\\subseteq[0,\\infty)\\) by (3). So \\(\\omega\\) is positive and of norm one: a state. Conversely, let \\(\\varphi\\) be a state. For \\(x,y\\in A\\) the form \\(\\langle x,y\\rangle_\\varphi=\\varphi(y^*x)\\) is sesquilinear, and \\(\\langle x,x\\rangle_\\varphi\\geq0\\) by (C2). The polarization and quadratic-minimization proof in [Proposition 3.2 of the GNS lesson](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-03) shows that this form is hermitian and satisfies Cauchy–Schwarz, so\n\\[\n\\begin{gathered}\n|\\varphi(x)|^2\\\\\n=|\\langle x,1\\rangle_\\varphi|^2\\\\\n\\leq\\varphi(x^*x)\\,\\varphi(1)\\\\\n\\leq\\|x\\|^2\\varphi(1)^2,\n\\end{gathered}\n\\]\nusing \\(\\varphi(\\|x\\|^2-x^*x)\\geq0\\) from (C2). Hence \\(1=\\|\\varphi\\|\\leq\\varphi(1)\\leq\\|\\varphi\\|\\,\\|1\\|=1\\), so \\(\\varphi(1)=1\\) and \\(\\varphi\\in V(A)\\). \\(\\square\\)\n\nIn a Banach algebra with an isometric involution, self-adjoint elements need not be hermitian, even when their spectrum is real and their norm equals their spectral radius.\n\n**Example 3.7** (group algebras). Let \\(G\\) be a discrete group with identity \\(e\\), and \\(A=\\ell^1(G)\\) as in (B4). Write \\(a\\in A\\) as \\(a=a(e)\\delta_e+b\\) with \\(b(e)=0\\). Then\n\\[\nW(a)=\\{z\\in\\mathbb C :\\ |z-a(e)|\\leq\\|b\\|_1\\}.\n\\]\nIndeed, for \\(z\\in\\mathbb C\\), \\(\\delta_e\\) and \\(zb\\) have disjoint supports, so \\(\\|\\delta_e+zb\\|_1=1+|z|\\,\\|b\\|_1\\). For \\(t>0\\) and real \\(\\theta\\) this gives \\((\\|1+te^{i\\theta}b\\|-1)/t=\\|b\\|_1\\). By Theorem 2.3(1), \\(m(e^{i\\theta}b)=\\|b\\|_1\\) for all \\(\\theta\\), and Lemma 2.1 gives \\(W(b)=\\{|z|\\leq\\|b\\|_1\\}\\). Then \\(W(a)=a(e)+W(b)\\) by Proposition 1.2(3).\n\nSo the hermitian elements of \\(\\ell^1(G)\\) are exactly the real multiples of \\(\\delta_e\\). The self-adjoint elements, those with \\(a(g^{-1})=\\overline{a(g)}\\), form a much larger set when \\(G\\neq\\{e\\}\\). For \\(G=\\mathbb Z\\) and \\(a=\\delta_1+\\delta_{-1}\\): \\(a\\) is self-adjoint, \\(\\sigma(a)=[-2,2]\\) by (B4), and \\(\\|a\\|_1=2=r(a)\\); but \\(W(a)\\) is the closed disc of radius \\(2\\). Exercise 2 gives a self-adjoint element whose spectrum is not even real.\n\n",
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      "name": "4. Positive matrices and the Perron–Frobenius eigenvalue",
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      "full_conditions_and_proof": "### 4. Positive matrices and the Perron–Frobenius eigenvalue\n\n The notation for vectors and matrices is fixed in the Conventions.\n\n**Lemma 4.1.** Let \\(T\\in M_n(\\mathbb C)\\) and \\(x,y,u\\in\\mathbb C^n\\).\n1. If \\(T\\geq0\\), then \\(|Tx|\\leq T|x|\\), and \\(0\\leq x\\leq y\\) implies \\(0\\leq Tx\\leq Ty\\).\n2. If \\(T\\gg0\\), \\(x\\geq0\\) and \\(x\\neq0\\), then \\(Tx\\gg0\\).\n3. If \\(|x|\\leq y\\), then \\(\\|x\\|\\leq\\|y\\|\\). If \\(T\\geq0\\), then \\(\\|T\\|=\\|T\\mathbf 1\\|\\).\n4. If \\(u\\gg0\\), \\(x\\geq0\\) and \\(x\\neq0\\), then \\(u^{\\top}x>0\\).\n\n**Proof.** (1) \\(|(Tx)_i|=|\\sum_jt_{ij}x_j|\\leq\\sum_jt_{ij}|x_j|\\); the second claim is clear. (2) Each \\((Tx)_i=\\sum_jt_{ij}x_j\\) has a positive term. (3) The first claim is clear; for \\(T\\geq0\\), \\((T\\mathbf 1)_i=\\sum_jt_{ij}\\) is the \\(i\\)-th row sum. (4) \\(u^{\\top}x=\\sum_iu_ix_i\\) has a positive term and no negative one. \\(\\square\\)\n\n**Theorem 4.2** (the spectral radius is an eigenvalue). Let \\(T\\geq0\\) and \\(\\rho=r(T)\\). Then \\(\\rho\\) is an eigenvalue of \\(T\\) with an eigenvector \\(x\\geq0\\), \\(x\\neq0\\).\n\n**Proof.** For \\(|\\lambda|>\\rho\\) the series \\(R(\\lambda)=\\sum_{k\\geq0}T^k/\\lambda^{k+1}\\) converges absolutely, since \\(\\|T^k\\|\\leq(\\rho+\\varepsilon)^k\\) for large \\(k\\) when \\(\\rho+\\varepsilon<|\\lambda|\\) (B1). It is the inverse of \\(\\lambda-T\\), because \\[\n\\begin{gathered}\n(\\lambda-T)\\sum_{k\\leq K}T^k/\\lambda^{k+1}\\\\\n=1-T^{K+1}/\\lambda^{K+1}\\to1.\n\\end{gathered}\n\\] For real \\(t>\\rho\\) every term is \\(\\geq0\\), so \\(R(t)\\geq0\\). By Lemma 4.1(1), \\(|R(\\lambda)x|\\leq R(|\\lambda|)|x|\\) termwise, so \\(\\|R(\\lambda)\\|\\leq\\|R(|\\lambda|)\\|\\) by Lemma 4.1(3).\n\nChoose \\(\\mu\\in\\sigma(T)\\) with \\(|\\mu|=\\rho\\), and \\(\\zeta\\) with \\(|\\zeta|=1\\) and \\(\\mu=\\rho\\zeta\\). For \\(t>\\rho\\), put \\(\\lambda=t\\zeta\\). Then \\(|\\lambda|=t\\) and \\(|\\lambda-\\mu|=t-\\rho\\), so by (B1)\n\\[\n\\|R(t)\\|\\geq\\|R(\\lambda)\\|\\geq\\frac1{\\operatorname{dist}(\\lambda,\\sigma(T))}\\geq\\frac1{t-\\rho}.\n\\]\nSince \\(R(t)\\geq0\\), \\(\\|R(t)\\|=\\|R(t)\\mathbf 1\\|\\) by Lemma 4.1(3). Put \\(y_t=R(t)\\mathbf 1\\geq0\\) and \\(x_t=y_t/\\|y_t\\|\\). Then \\(x_t\\geq0\\), \\(\\|x_t\\|=1\\), and \\((t-T)x_t=\\mathbf 1/\\|y_t\\|\\), whose norm is at most \\(t-\\rho\\). Choose \\(t_k\\downarrow\\rho\\) such that \\(x_{t_k}\\) converges, to \\(x\\) say. Then \\(x\\geq0\\), \\(\\|x\\|=1\\), and \\((\\rho-T)x=\\lim_k(t_k-T)x_{t_k}=0\\). \\(\\square\\)\n\n**Definition 4.3.** A matrix \\(T\\geq0\\) is *primitive* if \\(T^m\\gg0\\) for some \\(m\\geq1\\). Every \\(T\\gg0\\) is primitive. For a primitive \\(T\\), \\(\\rho=r(T)\\) is the *Perron–Frobenius eigenvalue* of \\(T\\), and a vector \\(v\\gg0\\) with \\(Tv=\\rho v\\) is a *Perron–Frobenius eigenvector*. By the next theorem such \\(v\\) exists and is unique up to a positive factor.\n\n**Theorem 4.4** (Perron–Frobenius, primitive matrices). Let \\(T\\geq0\\) be primitive and \\(\\rho=r(T)\\).\n1. \\(\\rho>0\\), and there is \\(v\\gg0\\) with \\(Tv=\\rho v\\).\n2. Every \\(y\\in\\mathbb C^n\\) with \\(Ty=\\rho y\\) is a multiple of \\(v\\).\n3. Every eigenvalue \\(\\lambda\\neq\\rho\\) of \\(T\\) has \\(|\\lambda|<\\rho\\).\n4. There is \\(u\\gg0\\) with \\(u^{\\top}T=\\rho u^{\\top}\\). The space \\(\\mathbb C^n\\) is the direct sum of \\(\\mathbb Cv\\) and \\(\\ker u^{\\top}\\), both invariant under \\(T\\), and \\(\\rho\\) is a simple root of the characteristic polynomial of \\(T\\).\n5. If \\(x\\geq0\\), \\(x\\neq0\\), is an eigenvector of \\(T\\) for any eigenvalue, then \\(x\\) is a positive multiple of \\(v\\), and \\(Tx=\\rho x\\).\n6. (Max–min formula)\n\\[\n\\begin{gathered}\n\\rho\\\\\n=\\max_{x\\geq0,\\ x\\neq0}\\ \\min_{i:\\,x_i>0}\\frac{(Tx)_i}{x_i}\\\\\n=\\min_{x\\gg0}\\ \\max_i\\frac{(Tx)_i}{x_i}.\n\\end{gathered}\n\\]\n7. \\((T/\\rho)^k\\to P=vu^{\\top}/(u^{\\top}v)\\) as \\(k\\to\\infty\\), where \\(P\\) is the projection onto \\(\\mathbb Cv\\) along \\(\\ker u^{\\top}\\). Consequently \\(T^kw/\\|T^kw\\|\\to v/\\|v\\|\\) for every \\(w\\geq0\\), \\(w\\neq0\\).\n\nThe strict-positivity case and the max–min formula are both proved below.\n\n**Proof.** *Step A: parts (1)–(3) when \\(T\\gg0\\).*\n\n(1) Theorem 4.2 gives \\(x\\geq0\\), \\(x\\neq0\\), with \\(Tx=\\rho x\\). By Lemma 4.1(2), \\(\\rho x=Tx\\gg0\\). So \\(\\rho>0\\) and \\(x\\gg0\\); put \\(v=x\\).\n\n(2) Since \\(T\\) and \\(\\rho\\) are real, the real and imaginary parts of \\(y\\) satisfy the same equation; so we may assume \\(y\\) real and \\(y\\neq0\\). Replacing \\(y\\) by \\(-y\\) if necessary, \\(y\\) has a positive entry. Let \\(c=\\min\\{v_i/y_i:\\ y_i>0\\}>0\\), attained at the index \\(j\\), and put \\(z=v-cy\\). Then \\(z\\geq0\\): for \\(y_i>0\\) by the choice of \\(c\\), and for \\(y_i\\leq0\\) because \\(v_i>0\\). Also \\(z_j=0\\) and \\(Tz=\\rho z\\). If \\(z\\neq0\\), then \\(\\rho z=Tz\\gg0\\) by Lemma 4.1(2), which contradicts \\(z_j=0\\). So \\(z=0\\) and \\(y=v/c\\).\n\n(3) Let \\(Tw=\\lambda w\\) with \\(w\\neq0\\) and \\(|\\lambda|=\\rho\\). Then \\(\\rho|w|=|Tw|\\leq T|w|\\) by Lemma 4.1(1). Put \\(d=T|w|-\\rho|w|\\geq0\\), and suppose \\(d\\neq0\\). Then \\(Td\\gg0\\), that is, \\(Tq\\gg\\rho q\\) for \\(q=T|w|\\gg0\\). Let \\(\\varepsilon=\\min_i\\big((Tq)_i-\\rho q_i\\big)/q_i>0\\). Then \\(Tq\\geq(\\rho+\\varepsilon)q\\), and by Lemma 4.1(1) and induction \\(T^kq\\geq(\\rho+\\varepsilon)^kq\\) for all \\(k\\). By Lemma 4.1(3), \\(\\|T^k\\|\\,\\|q\\|\\geq\\|T^kq\\|\\geq(\\rho+\\varepsilon)^k\\|q\\|\\), so \\(\\|T^k\\|^{1/k}\\geq\\rho+\\varepsilon\\) for all \\(k\\), which contradicts \\(\\|T^k\\|^{1/k}\\to\\rho\\) (B1). Hence \\(T|w|=\\rho|w|=|Tw|\\). For each \\(i\\), \\(|\\sum_jt_{ij}w_j|=\\sum_jt_{ij}|w_j|\\) with all \\(t_{ij}>0\\); by the case of equality in the triangle inequality, all nonzero \\(w_j\\) have the same argument, so \\(w=e^{i\\vartheta}|w|\\) for some real \\(\\vartheta\\). Then \\(\\lambda w=Tw=e^{i\\vartheta}T|w|=e^{i\\vartheta}\\rho|w|=\\rho w\\), so \\(\\lambda=\\rho\\).\n\n*Step B: parts (1)–(3) when \\(T^m\\gg0\\).* Put \\(T_1=T^m\\). By (B1), \\(r(T_1)=\\rho^m\\), and Step A applies to \\(T_1\\): \\(\\rho^m>0\\), and there is \\(v\\gg0\\) with \\(T_1v=\\rho^mv\\) that spans the solutions of \\(T_1y=\\rho^my\\). By Theorem 4.2 there is \\(x\\geq0\\), \\(x\\neq0\\), with \\(Tx=\\rho x\\). Then \\(T_1x=\\rho^mx\\), so \\(x\\) is a multiple of \\(v\\), and a positive one because \\(x\\geq0\\). So \\(Tv=\\rho v\\) and \\(\\rho>0\\); this is (1). If \\(Ty=\\rho y\\), then \\(T_1y=\\rho^my\\) and \\(y\\in\\mathbb Cv\\); this is (2). If \\(Tw=\\lambda w\\) with \\(w\\neq0\\) and \\(|\\lambda|=\\rho\\), then \\(T_1w=\\lambda^mw\\) with \\(|\\lambda^m|=\\rho^m\\). By Step A(3) for \\(T_1\\), \\(\\lambda^m=\\rho^m\\); then \\(w\\in\\mathbb Cv\\), and \\(\\lambda w=Tw=\\rho w\\) gives \\(\\lambda=\\rho\\). This is (3).\n\n*Step C: parts (4)–(7).* The matrix \\(T^{\\top}\\geq0\\) is primitive, since \\((T^{\\top})^m=(T^m)^{\\top}\\gg0\\), and it has the same characteristic polynomial as \\(T\\), hence \\(r(T^{\\top})=\\rho\\). Part (1) for \\(T^{\\top}\\) gives \\(u\\gg0\\) with \\(T^{\\top}u=\\rho u\\), that is, \\(u^{\\top}T=\\rho u^{\\top}\\).\n\n(4) Put \\(K=\\ker u^{\\top}\\), a subspace of dimension \\(n-1\\). Since \\(u^{\\top}v>0\\) by Lemma 4.1(4), \\(v\\notin K\\), and \\(\\mathbb C^n=\\mathbb Cv\\oplus K\\). Both summands are invariant: \\(Tv=\\rho v\\), and for \\(x\\in K\\), \\(u^{\\top}Tx=\\rho u^{\\top}x=0\\). If \\(\\rho\\) were an eigenvalue of \\(T|_K\\), a corresponding eigenvector would lie in \\(\\mathbb Cv\\cap K=\\{0\\}\\) by (2), which is absurd. In a basis adapted to the decomposition, \\(T\\) is block diagonal, so \\(\\det(\\lambda-T)=(\\lambda-\\rho)\\det(\\lambda-T|_K)\\) with \\(\\det(\\rho-T|_K)\\neq0\\). So \\(\\rho\\) is a simple root.\n\n(5) If \\(Tx=\\kappa x\\) with \\(x\\geq0\\), \\(x\\neq0\\), then \\(\\rho u^{\\top}x=u^{\\top}Tx=\\kappa u^{\\top}x\\), and \\(u^{\\top}x>0\\) by Lemma 4.1(4). So \\(\\kappa=\\rho\\), and \\(x\\in\\mathbb Cv\\) by (2); as \\(x\\geq0\\), it is a positive multiple of \\(v\\).\n\n(6) For \\(x\\geq0\\), \\(x\\neq0\\), let \\(\\underline r(x)=\\min_{i:\\,x_i>0}(Tx)_i/x_i\\). Then \\(\\underline r(x)\\,x\\leq Tx\\), since the coordinates with \\(x_i=0\\) give \\(0\\leq(Tx)_i\\). Applying \\(u^{\\top}\\) gives \\(\\underline r(x)\\,u^{\\top}x\\leq\\rho u^{\\top}x\\), so \\(\\underline r(x)\\leq\\rho\\), with equality at \\(x=v\\). For \\(x\\gg0\\) let \\(\\overline r(x)=\\max_i(Tx)_i/x_i\\). Then \\(Tx\\leq\\overline r(x)\\,x\\), so \\(\\rho u^{\\top}x\\leq\\overline r(x)\\,u^{\\top}x\\) and \\(\\rho\\leq\\overline r(x)\\), with equality at \\(x=v\\).\n\n(7) If \\(K=\\{0\\}\\), then \\(P=1\\) and \\(T/\\rho=1\\), so the assertion is immediate. Otherwise every eigenvalue of \\(T|_K\\) is an eigenvalue of \\(T\\) other than \\(\\rho\\), so it has modulus less than \\(\\rho\\) by (3). Hence the spectral radius of \\(T|_K/\\rho\\) is less than \\(1\\), and \\((T|_K/\\rho)^k\\to0\\) by (B1). Every \\(x\\in\\mathbb C^n\\) is \\(Px+(x-Px)\\) with \\(Px=(u^{\\top}x/u^{\\top}v)\\,v\\) and \\(x-Px\\in K\\), so \\((T/\\rho)^kx=Px+(T|_K/\\rho)^k(x-Px)\\to Px\\). For \\(w\\geq0\\), \\(w\\neq0\\), we have \\(u^{\\top}w>0\\), so \\(T^kw/\\rho^k\\) tends to the nonzero vector \\(Pw\\), a positive multiple of \\(v\\); normalizing gives the last claim. \\(\\square\\)\n\nA matrix \\(T\\geq0\\) is *irreducible* if for all \\(i\\neq j\\) some power \\(T^k\\), \\(k\\geq1\\), has \\((T^k)_{ij}>0\\); for \\(n=1\\) every matrix counts as irreducible. Primitive matrices are irreducible, but not conversely.\n\n**Proposition 4.5** (irreducible matrices). Let \\(T\\geq0\\) be irreducible and \\(\\rho=r(T)\\).\n1. \\(1+T\\) is primitive, and \\(r(1+T)=1+\\rho\\).\n2. There is \\(v\\gg0\\) with \\(Tv=\\rho v\\). Parts (2), (4), (5) and (6) of Theorem 4.4 hold for \\(T\\), and \\(\\rho>0\\) when \\(n\\geq2\\).\n3. Parts (3) and (7) of Theorem 4.4 can fail: for \\(T=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\\), which is irreducible, \\(\\rho=1\\), \\(-1\\) is an eigenvalue, and \\(T^k\\) alternates between \\(1\\) and \\(T\\).\n\n\n**Proof.** (1) Let \\(n\\geq2\\) and \\(i\\neq j\\), and choose \\(k\\) with \\((T^k)_{ij}>0\\). Expanding the matrix product, \\((T^k)_{ij}\\) is a sum of products \\(t_{i_0i_1}t_{i_1i_2}\\cdots t_{i_{k-1}i_k}\\) over chains with \\(i_0=i\\) and \\(i_k=j\\), all terms \\(\\geq0\\); so some chain has all its factors positive. Removing the loops of this chain gives a chain from \\(i\\) to \\(j\\) with distinct indices, hence of length \\(l\\leq n-1\\), and \\((T^l)_{ij}>0\\). Now \\((1+T)^{n-1}=\\sum_{l=0}^{n-1}\\binom{n-1}{l}T^l\\) has all entries positive: the off-diagonal ones by what we just showed, the diagonal ones because of the term \\(l=0\\). For \\(n=1\\), \\(1+T\\gg0\\) directly. By (B1), \\(\\sigma(1+T)=1+\\sigma(T)\\), and \\(|1+\\lambda|\\leq1+|\\lambda|\\leq1+\\rho\\) for \\(\\lambda\\in\\sigma(T)\\), with equality at \\(\\lambda=\\rho\\in\\sigma(T)\\) (Theorem 4.2). So \\(r(1+T)=1+\\rho\\).\n\n(2) Theorem 4.4 applies to \\(1+T\\). The equations \\((1+T)y=(1+\\rho)y\\) and \\(Ty=\\rho y\\) are the same; \\(\\det\\big(\\lambda-(1+T)\\big)=\\det\\big((\\lambda-1)-T\\big)\\), so multiplicities of roots correspond; \\(1+T\\) and \\(T\\) have the same eigenvectors; and \\(\\big((1+T)x\\big)_i/x_i=1+(Tx)_i/x_i\\). This transfers (1), (2), (4), (5) and (6). If \\(n\\geq2\\) and \\(\\rho=0\\), then \\(Tv=0\\) with \\(v\\gg0\\) forces \\(T=0\\), which is not irreducible.\n\n(3) The eigenvalues of this \\(T\\) are \\(\\pm1\\), and \\(T^2=1\\). \\(\\square\\)\n\n**Example 4.6** (the hypotheses matter). For the \\(2\\times2\\) identity matrix, \\(\\rho=1\\) has a two-dimensional eigenspace, and \\((1,0)\\) is a nonnegative eigenvector that is not \\(\\gg0\\). For \\(\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\), \\(\\rho=1\\) is a double root, and the only eigenvectors are the multiples of \\((1,0)\\). For \\(\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\), \\(\\rho=0\\). None of these three matrices is irreducible. On the other hand, \\(T=\\begin{pmatrix}1&2\\\\3&4\\end{pmatrix}\\gg0\\) has eigenvalues \\((5\\pm\\sqrt{33})/2\\); here \\(\\rho=(5+\\sqrt{33})/2\\), the other eigenvalue has modulus less than \\(1\\), and \\(v=(2,\\rho-1)\\gg0\\) is a Perron–Frobenius eigenvector, since its first coordinate gives \\(2+2(\\rho-1)=2\\rho\\) and its second \\(6+4(\\rho-1)=\\rho(\\rho-1)\\), which is the equation \\(\\rho^2=5\\rho+2\\).\n\n## B. Code obstructions and construct Borel choices\n\nBegin here after the Borel-model lesson if the problem is measurable selection. The background proofs establish the order-type and boundedness tools before they are used in tree coding. The representation of co-Souslin sets by well-orders leads to reduction and separation; these in turn yield the countable-to-one and closed-set selector theorems. Each implication is proved below, including the auxiliary ordinal statements.\n\n### Background proofs for the selection route\n\n*From [Polish spaces and standard Borel spaces](polish-spaces-and-standard-borel-spaces.md) and [The Effros Borel structure](effros-borel-structure.md).* Let \\(X\\) and \\(Y\\) be standard Borel spaces.\n\n- **(D1)** Borel sets are Souslin sets. Countable unions and countable intersections of Souslin sets are Souslin sets. If \\(A\\subseteq X\\) and \\(B\\subseteq Y\\) are Souslin, so is \\(A\\times B\\subseteq X\\times Y\\). If \\(f:X\\to Y\\) is Borel, then \\(f(A)\\) is Souslin for every Souslin \\(A\\subseteq X\\), and \\(f^{-1}(B)\\) is Souslin for every Souslin \\(B\\subseteq Y\\). See [closure properties](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-02), [Borel sets and Souslin sets](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-12) and [Borel maps between Souslin spaces](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-06).\n- **(D2)** (Souslin’s theorem) A Souslin set whose complement is a Souslin set is Borel ([the separation theorem](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-03)).\n- **(D3)** A nonempty Souslin subset of a Polish space \\(Z\\) has the form \\(g(\\Lambda)\\) for a continuous \\(g:\\Lambda\\to Z\\) ([closure properties](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-02)).\n- **(D4)** There is a Souslin set \\(U\\subseteq\\mathcal C\\times\\mathcal C\\) such that every Souslin subset of \\(\\mathcal C\\) equals \\(U_y=\\{x:(y,x)\\in U\\}\\) for some \\(y\\in\\mathcal C\\); and there is a Souslin set \\(D\\subseteq\\mathcal C\\) that is not Borel, so that \\(\\mathcal C\\setminus D\\) is not a Souslin set ([a Souslin set that is not Borel](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-12)).\n- **(D5)** (Change of topology) If \\((X,\\tau)\\) is Polish and \\(B_1,B_2,\\ldots\\) are Borel, there is a Polish topology \\(\\tau'\\supseteq\\tau\\) with the same Borel sets in which every \\(B_n\\) is open and closed ([zero-dimensional refinement](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-04)). Borel subsets of standard Borel spaces, with the relative Borel structure, are again standard ([Borel subsets of standard spaces](effros-borel-structure.md#oa-fnd-ef-02)). Open subsets of Polish spaces are Polish ([Polish subspaces](effros-borel-structure.md#oa-fnd-ef-02)).\n- **(D6)** Every Borel subset of a Polish space is the image of a closed subset of \\(\\Lambda\\) under a continuous bijection ([closed subsets of the Baire space](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-04) and [Lusin sets](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-05)).\n- **(D7)** If a Borel map on a standard Borel space is injective and takes values in a separable metrizable space, then its image is Borel and the map is a Borel isomorphism onto the image. The graph of a Borel map \\(f:X\\to Y\\) is a Borel subset of \\(X\\times Y\\) ([Borel maps between Souslin spaces](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-06)).\n- **(D8)** For a Polish space \\(X\\), the *Effros Borel structure* on the set \\(\\mathcal C_0(X)\\) of nonempty closed subsets is the \\(\\sigma\\)-algebra generated by the sets \\(\\{F:F\\cap U\\neq\\varnothing\\}\\), \\(U\\subseteq X\\) open. It is standard ([spaces of closed sets](effros-borel-structure.md#oa-fnd-ef-05)).\n- **(D9)** (Cantor's intersection theorem) In a complete metric space, a decreasing sequence of nonempty closed sets with diameters tending to \\(0\\) has exactly one common point ([Polish spaces and standard Borel spaces](polish-spaces-and-standard-borel-spaces.md), elementary fact B1).\n\n*Well-orders.*\n\n- **(S1)** If \\((W,<)\\) is well-ordered and \\(f:W\\to W\\) is strictly increasing, then \\(f(w)\\geq w\\) for all \\(w\\) (Theorem 0.1 below). So no well-ordered set is isomorphic to a proper initial segment \\(\\{w:w<q\\}\\) of itself (Theorem 0.1).\n- **(S2)** Every well-ordered set is isomorphic to exactly one ordinal, its *order type*. Ordinals are linearly ordered, every nonempty set of ordinals has a least element, and each ordinal is the set of smaller ordinals, so that \\(\\alpha<\\beta\\) makes \\(\\alpha\\) a proper initial segment of \\(\\beta\\) (Theorem 0.1). For two well-ordered sets, exactly one of the following holds: they are isomorphic, the first is isomorphic to a proper initial segment of the second, or the second is isomorphic to a proper initial segment of the first (Theorem 0.1).\n\nWe use elementary finite-dimensional linear algebra over \\(\\mathbb C\\). The remaining topological and scalar facts are justified immediately below. The background statements are proved below.\n\n#### Well-orders and their order types\n\nThe following proof supplies (S1)–(S2). We use the usual axioms of set theory, including Separation and Replacement, to form subsets and images of sets; no collection of all ordinals is treated as a set. An *initial segment* of a well-order is a downward closed subset, possibly the whole order. A proper initial segment is exactly the set of predecessors of its least omitted point.\n\n**Theorem 0.1** (Well-orders and ordinals). If a strictly increasing map \\(f:W\\to W\\) acts on a well-ordered set, then \\(f(w)\\geq w\\) for every \\(w\\). No well-order is isomorphic to a proper initial segment of itself. Every well-order is isomorphic to exactly one ordinal. Ordinals are linearly ordered by membership; every nonempty set of ordinals has a least member, and an ordinal is exactly the set of its predecessors. Consequently two well-orders are either isomorphic or exactly one is isomorphic to a proper initial segment of the other. A countable well-order has a countable ordinal as its order type.\n\n*Proof.* **Induction and increasing maps.** A property holds throughout a well-order if, whenever it holds at all predecessors of a point, it holds at that point: a nonempty set of failures would have a least member. If \\(w\\) is the least point with \\(f(w)<w\\), put \\(v=f(w)\\). Then \\(v<w\\), so \\(f(v)\\geq v=f(w)\\), whereas strict increase gives \\(f(v)<f(w)\\), a contradiction. Thus \\(f(w)\\geq w\\). An isomorphism of \\(W\\) onto the predecessors of \\(q\\in W\\) would give \\(f(q)<q\\), which is impossible.\n\n**Comparison of well-orders.** Consider all order isomorphisms between initial segments of two well-orders \\(W,V\\). Any two agree on their common domain. Otherwise take the first point where they disagree. Their values at all preceding points agree, and each value at the new point must be the least element of \\(V\\) outside that common predecessor image. The values therefore agree, a contradiction. These maps are subsets of the set \\(W\\times V\\). Their union is an order isomorphism between initial segments: domains are nested initial segments, compatibility makes the union a function, and the ranges are initial segments too. If both domain and range were proper, map the least unused point of \\(W\\) to the least unused point of \\(V\\). This would give another initial-segment isomorphism strictly extending the union, a contradiction. Thus one order is exhausted. The alternatives cannot overlap, since composing competing comparisons would make a well-order isomorphic to a proper initial segment of itself.\n\n**Ordinals.** A set \\(\\alpha\\) is an ordinal if it is transitive (\\(x\\in\\alpha\\) implies \\(x\\subseteq\\alpha\\)) and membership is a strict well-order on \\(\\alpha\\). If \\(x\\in\\alpha\\), its predecessors in \\(\\alpha\\) are precisely the members of \\(x\\): transitivity of \\(\\alpha\\) puts those members in \\(\\alpha\\). Also \\(x\\) is transitive, because \\(z\\in y\\in x\\), with all three in \\(\\alpha\\), implies \\(z\\in x\\) by transitivity of the membership order. Thus \\(x\\) is itself an ordinal. In particular the proper initial segments of an ordinal are its members. No ordinal belongs to itself: if \\(\\alpha\\in\\alpha\\), membership restricted to \\(\\alpha\\) would have the forbidden loop \\(\\alpha\\in\\alpha\\).\n\nAn isomorphism between initial segments of ordinals is the identity on its domain. Induct on \\(x\\) in that domain. The predecessors of its image are the images of the predecessors of \\(x\\), so\n\\[\nf(x)=\\{f(y):y\\in x\\}=\\{y:y\\in x\\}=x.\n\\]\nThe preceding comparison theorem now implies that for any two ordinals \\(\\alpha,\\beta\\), exactly one of \\(\\alpha=\\beta\\), \\(\\alpha\\in\\beta\\), \\(\\beta\\in\\alpha\\) holds. Moreover \\(\\beta+1:=\\beta\\cup\\{\\beta\\}\\) is an ordinal: it is transitive, its old elements retain their membership order, and \\(\\beta\\) is the new last element. Given a nonempty set \\(S\\) of ordinals, choose \\(\\beta\\in S\\). The nonempty subset \\(S\\cap(\\beta+1)\\) has a least member \\(\\gamma\\) in this well-order. Every member of \\(S\\) outside \\(\\beta+1\\) is larger than \\(\\beta\\), so \\(\\gamma\\) is least in all of \\(S\\). Since all members of an ordinal are smaller ordinals and comparison identifies every smaller ordinal with a proper initial segment, an ordinal is exactly the set of smaller ordinals.\n\n**Constructing the order type.** We justify the recursion\n\\[\nF(w)=\\{F(u):u<w\\}\n\\]\non a well-order \\(W\\). Call a function on an initial segment *compatible* if it satisfies this formula at every point of its domain. Two compatible functions agree on the intersection of their domains: the first disagreement would have exactly the same predecessor values on both sides. By Separation form the initial segment \\(D\\subseteq W\\) consisting of points contained in some compatible domain. At each point of \\(D\\) its compatible value exists and is unique. Replacement therefore supplies the function \\(F_D\\) with those values. It obeys the recursion, because any compatible domain containing a point contains all its predecessors. If \\(D\\ne W\\), let \\(w\\) be its least omitted point. Then \\(D=\\{u:u<w\\}\\), and extending \\(F_D\\) by the value \\(\\{F_D(u):u<w\\}\\) at \\(w\\) gives a compatible function containing \\(w\\), a contradiction. Hence \\(D=W\\), and the recursion defines a unique function on all of \\(W\\).\n\nInductively \\(F(w)\\) is an ordinal and \\(u\\mapsto F(u)\\), for \\(u<w\\), is an isomorphism onto \\(F(w)\\). Indeed, predecessor values are ordinals, and \\(u<v<w\\) implies \\(F(u)\\in F(v)\\) by the recursion. They are distinct, since an ordinal cannot belong to itself. A membership in the reverse direction would give \\(F(v)\\in F(u)\\in F(v)\\), and transitivity of the ordinal \\(F(v)\\) would force \\(F(v)\\in F(v)\\), a contradiction. Thus membership agrees exactly with the predecessor order. The set of predecessor values is transitive, since every member of \\(F(u)\\) is an earlier value, and it is well-ordered by pulling nonempty subsets back to the predecessor order. This proves the induction assertion. The same argument applied to all values shows that\n\\[\n\\alpha=\\{F(w):w\\in W\\}\n\\]\nis an ordinal and \\(F:W\\to\\alpha\\) is an order isomorphism. Two ordinals isomorphic to \\(W\\) are isomorphic to each other; the identity result above forces equality. Finally this isomorphism is a bijection, so if \\(W\\) is countable, so is \\(\\alpha\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-05",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "5. Souslin and co-Souslin sets",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "anchor": "oa-fnd-dt-05",
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        "line": 555,
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      "full_conditions_and_proof": "### 5. Souslin and co-Souslin sets\n\nFrom now on \\(X\\) and \\(Y\\) are standard Borel spaces, unless something else is said.\n\n**Definition 5.1.** A subset of \\(X\\) is *co-Souslin* if its complement is a Souslin set.\n\nWhen a Polish topology generating the Borel sets is fixed, the Souslin sets are the analytic sets, and the co-Souslin sets are the sets denoted by \\(\\Pi^1_1\\).\n\n**Proposition 5.2.**\n1. Borel sets are both Souslin and co-Souslin. A set that is both Souslin and co-Souslin is Borel.\n2. Countable unions and countable intersections of co-Souslin sets are co-Souslin.\n3. If \\(f:X\\to Y\\) is Borel and \\(C\\subseteq Y\\) is co-Souslin, then \\(f^{-1}(C)\\) is co-Souslin.\n4. If \\(S\\subseteq X\\times Y\\) is Souslin, its projection \\(\\{x:(x,y)\\in S\\text{ for some }y\\}\\) is Souslin. If \\(C\\subseteq X\\times Y\\) is co-Souslin, then \\(\\{x:(x,y)\\in C\\text{ for all }y\\}\\) is co-Souslin.\n5. Some co-Souslin set in \\(\\mathcal C\\) is not a Souslin set, namely \\(\\mathcal C\\setminus D\\) for the set \\(D\\) of (D4).\n\n**Proof.** (1) is (D1) and (D2). (2) and (3) follow from (D1) by taking complements: the complement of a countable intersection is a countable union, and \\(X\\setminus f^{-1}(C)=f^{-1}(Y\\setminus C)\\). (4) Fix Polish topologies generating the Borel sets. The projection is continuous, hence Borel, and (D1) applies. The second set is the complement of the projection of the Souslin set \\((X\\times Y)\\setminus C\\). (5) is (D4). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-DT-06",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "6. Trees on a countable set",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "full_conditions_and_proof": "### 6. Trees on a countable set\n\nIn Sections 6 and 7, \\(Q\\) and \\(P\\) are countable nonempty sets with the discrete topology. \\(Q^{<\\mathbb N}\\) consists of the finite sequences \\(s=(s_1,\\dots,s_k)\\) with entries in \\(Q\\), including the empty sequence \\(\\varnothing\\); \\(|s|=k\\) is the length, and \\(s{}^\\frown a=(s_1,\\dots,s_k,a)\\). We write \\(s\\sqsubseteq t\\) if \\(s\\) is an initial segment of \\(t\\), that is, \\(|s|\\leq|t|\\) and \\(t_i=s_i\\) for \\(i\\leq|s|\\), and \\(s\\sqsubset t\\) if moreover \\(s\\neq t\\). For \\(x\\in Q^{\\mathbb N}\\) and \\(k\\geq0\\), \\(x|k=(x_1,\\dots,x_k)\\). The space \\(Q^{\\mathbb N}\\) has the product topology. It is Polish, and the cylinders \\(O_s=\\{x\\in Q^{\\mathbb N}:\\ x|\\,|s|=s\\}\\) are open and closed and form a base; for each \\(x\\), the cylinders \\(O_{x|k}\\), \\(k\\geq0\\), form a neighbourhood base at \\(x\\). For \\(Q=\\mathbb N\\) this is the Baire space \\(\\Lambda\\), and \\(O_s=\\Lambda_s\\).\n\nSubsets of \\(Q^{<\\mathbb N}\\) are points of the Polish space \\(2^{Q^{<\\mathbb N}}\\) (Conventions). The next lemma shows that this Borel structure is the Effros structure (D8), so no choice is involved.\n\n**Lemma 6.1.** Let \\(Z\\) be a countable set with the discrete topology. On the set of nonempty subsets of \\(Z\\), which are the nonempty closed subsets, the Effros Borel structure equals the Borel structure of \\(2^Z\\setminus\\{\\varnothing\\}\\).\n\n**Proof.** Every subset of \\(Z\\) is open. The Effros structure is generated by the sets \\(\\{S:S\\cap U\\neq\\varnothing\\}=\\bigcup_{z\\in U}\\{S:z\\in S\\}\\), and conversely \\(\\{S:z\\in S\\}=\\{S:S\\cap\\{z\\}\\neq\\varnothing\\}\\). So both \\(\\sigma\\)-algebras are generated by the sets \\(\\{S:z\\in S\\}\\), \\(z\\in Z\\). \\(\\square\\)\n\n**Definition 6.2.** A *tree* on \\(Q\\) is a set \\(T\\subseteq Q^{<\\mathbb N}\\) such that \\(s\\in T\\) whenever \\(s\\sqsubseteq t\\in T\\). Its *body* is\n\\[\n[T]=\\{x\\in Q^{\\mathbb N}:\\ x|k\\in T\\text{ for all }k\\geq0\\}.\n\\]\nThe tree \\(T\\) is *well-founded* if \\([T]=\\varnothing\\) and *ill-founded* otherwise, and *pruned* if every \\(s\\in T\\) has an extension \\(s{}^\\frown a\\in T\\). We write \\(\\mathrm{Tree}(Q)\\) and \\(\\mathrm{WF}(Q)\\) for the sets of trees and of well-founded trees on \\(Q\\), and \\(\\mathrm{PT}(Q)\\) for the set of nonempty pruned trees.\n\nA tree is well-founded exactly when it contains no infinite chain \\(s^1\\sqsubset s^2\\sqsubset\\cdots\\). Indeed, the lengths along such a chain tend to infinity, and there is a unique \\(x\\in Q^{\\mathbb N}\\) with \\(x|\\,|s^k|=s^k\\) for all \\(k\\). Each \\(x|j\\) is an initial segment of some \\(s^k\\), so it lies in \\(T\\), and \\(x\\in[T]\\). Conversely, \\(x\\in[T]\\) gives the chain \\(x|1\\sqsubset x|2\\sqsubset\\cdots\\). The empty set is a well-founded tree, and every nonempty tree contains \\(\\varnothing\\).\n\n**Proposition 6.3.**\n1. \\(\\mathrm{Tree}(Q)\\) is closed in \\(2^{Q^{<\\mathbb N}}\\), and \\([T]\\) is closed in \\(Q^{\\mathbb N}\\) for every tree \\(T\\).\n2. For a closed set \\(F\\subseteq Q^{\\mathbb N}\\), the set \\(T_F=\\{x|k:\\ x\\in F,\\ k\\geq0\\}\\) is a pruned tree with \\([T_F]=F\\). The map \\(F\\mapsto T_F\\) is a Borel isomorphism of the Effros space \\(\\mathcal C_0(Q^{\\mathbb N})\\) onto \\(\\mathrm{PT}(Q)\\), which is a \\(G_\\delta\\) subset of \\(2^{Q^{<\\mathbb N}}\\). The inverse map is \\(T\\mapsto[T]\\).\n3. The set \\(\\mathrm{WF}(Q)\\) is co-Souslin in \\(2^{Q^{<\\mathbb N}}\\).\n\n**Proof.** (1) A set \\(T\\) is a tree when, for every pair \\(s\\sqsubseteq t\\), either \\(t\\notin T\\) or \\(s\\in T\\). For one pair this condition defines an open and closed set, and there are countably many pairs. If \\(x\\notin[T]\\), then \\(x|k\\notin T\\) for some \\(k\\), and the whole open set \\(O_{x|k}\\) misses \\([T]\\).\n\n(2) The initial segments of \\(x|k\\) are the \\(x|j\\), \\(j\\leq k\\), so \\(T_F\\) is a tree; it is pruned, since \\(x|k\\sqsubset x|(k+1)\\). Clearly \\(F\\subseteq[T_F]\\). If \\(y\\in[T_F]\\), then for each \\(k\\) some \\(x^k\\in F\\) has \\(x^k|k=y|k\\); so \\(x^k\\to y\\), and \\(y\\in F\\) because \\(F\\) is closed.\n\nFor \\(s\\in Q^{<\\mathbb N}\\), \\(\\{F:s\\in T_F\\}=\\{F:F\\cap O_s\\neq\\varnothing\\}\\), a generator of the Effros structure (D8). The coordinate maps generate the Borel sets of \\(2^{Q^{<\\mathbb N}}\\), so \\(F\\mapsto T_F\\) is Borel. It is injective because \\(F=[T_F]\\), and its values are nonempty pruned trees. Conversely, let \\(T\\in\\mathrm{PT}(Q)\\) and \\(s\\in T\\). Fix an enumeration of \\(Q\\), and extend \\(s\\) one step at a time, each time by the first \\(a\\) in the enumeration that keeps the sequence in \\(T\\). This defines \\(x\\in[T]\\) with \\(x|\\,|s|=s\\). So \\([T]\\) is nonempty and closed, and \\(T=T_{[T]}\\): every \\(s\\in T\\) is an initial segment of a point of \\([T]\\), and \\(T_{[T]}\\subseteq T\\) by the definition of \\([T]\\). So \\(F\\mapsto T_F\\) maps onto \\(\\mathrm{PT}(Q)\\), with inverse \\(T\\mapsto[T]\\). The inverse is Borel: an open set \\(U\\subseteq Q^{\\mathbb N}\\) is a countable union of cylinders \\(O_s\\), and for \\(T\\in\\mathrm{PT}(Q)\\), \\([T]\\cap O_s\\neq\\varnothing\\) exactly when \\(s\\in T\\); so \\(\\{T:[T]\\cap U\\neq\\varnothing\\}\\) is a countable union of Borel sets. Finally,\n\\[\n\\begin{gathered}\n\\mathrm{PT}(Q)\\\\\n=\\mathrm{Tree}(Q)\\cap\\{T :\\varnothing\\in T\\}\\\\\n\\cap\\bigcap_{s\\in Q^{<\\mathbb N}}\\Big(\\{T:s\\notin T\\}\\\\\n\\cup\\bigcup_{a\\in Q}\\{T:s{}^\\frown a\\in T\\}\\Big),\n\\end{gathered}\n\\]\nthe intersection of a closed set with countably many open sets.\n\n(3) The set \\(\\{(T,x)\\in2^{Q^{<\\mathbb N}}\\times Q^{\\mathbb N}:\\ x|k\\in T\\text{ for all }k\\}\\) is closed, because each condition \\(x|k\\in T\\) involves one coordinate of \\(T\\) and finitely many coordinates of \\(x\\). Its projection to \\(2^{Q^{<\\mathbb N}}\\) is the set of subsets \\(T\\) with a branch, which is Souslin by Proposition 5.2(4). The complement of this projection is co-Souslin, and its intersection with the closed set \\(\\mathrm{Tree}(Q)\\) is \\(\\mathrm{WF}(Q)\\). \\(\\square\\)\n\nWe now order finite sequences of natural numbers so that well-founded trees become well-ordered.\n\n**Definition 6.4** (the Kleene–Brouwer order). For distinct \\(s,t\\in\\mathbb N^{<\\mathbb N}\\), put \\(s<_{\\mathrm{KB}}t\\) if either \\(t\\sqsubset s\\), or there is an index \\(i\\leq\\min(|s|,|t|)\\) with \\(s_j=t_j\\) for \\(j<i\\) and \\(s_i<t_i\\). Write \\(s\\leq_{\\mathrm{KB}}t\\) if \\(s<_{\\mathrm{KB}}t\\) or \\(s=t\\).\n\nSo a proper extension of \\(t\\) comes before \\(t\\), and sequences that branch apart are compared at the first place where they differ.\n\n**Lemma 6.5.** \\(<_{\\mathrm{KB}}\\) is a strict linear order on \\(\\mathbb N^{<\\mathbb N}\\). The empty sequence is its largest element, and \\(s{}^\\frown a<_{\\mathrm{KB}}s\\) for all \\(s\\) and \\(a\\).\n\n**Proof.** Two distinct sequences are comparable: either one is a proper initial segment of the other, or they first differ at some index \\(i\\leq\\min(|s|,|t|)\\). The two alternatives of the definition exclude each other and cannot hold in both directions, so \\(<_{\\mathrm{KB}}\\) is asymmetric. For transitivity let \\(s<_{\\mathrm{KB}}t<_{\\mathrm{KB}}u\\). There are four cases.\n- If \\(t\\sqsubset s\\) and \\(u\\sqsubset t\\), then \\(u\\sqsubset s\\), so \\(s<_{\\mathrm{KB}}u\\).\n- If \\(t\\sqsubset s\\), and \\(t,u\\) first differ at \\(i\\) with \\(t_i<u_i\\), then \\(i\\leq|t|\\), so \\(s\\) agrees with \\(t\\) up to \\(i\\), and \\(s,u\\) first differ at \\(i\\) with \\(s_i=t_i<u_i\\).\n- If \\(s,t\\) first differ at \\(i\\) with \\(s_i<t_i\\), and \\(u\\sqsubset t\\): when \\(|u|\\geq i\\), \\(s\\) and \\(u\\) first differ at \\(i\\) with \\(s_i<t_i=u_i\\); when \\(|u|<i\\), \\(u\\) is a proper initial segment of \\(s\\), because \\(s_j=t_j=u_j\\) for \\(j\\leq|u|\\) and \\(|s|\\geq i>|u|\\).\n- If \\(s,t\\) first differ at \\(i\\) with \\(s_i<t_i\\), and \\(t,u\\) first differ at \\(i'\\) with \\(t_{i'}<u_{i'}\\), let \\(j=\\min(i,i')\\). Then \\(s\\) and \\(u\\) agree before \\(j\\), and \\(s_j<u_j\\): if \\(i<i'\\), \\(s_i<t_i=u_i\\); if \\(i'<i\\), \\(s_{i'}=t_{i'}<u_{i'}\\); if \\(i=i'\\), \\(s_i<t_i<u_i\\).\n\nIn each case \\(s<_{\\mathrm{KB}}u\\). Every nonempty sequence properly extends \\(\\varnothing\\), so \\(\\varnothing\\) is the largest element. The last claim holds because \\(s\\sqsubset s{}^\\frown a\\). \\(\\square\\)\n\n**Theorem 6.6.** A tree \\(T\\) on \\(\\mathbb N\\) is well-founded if and only if \\(<_{\\mathrm{KB}}\\) restricted to \\(T\\) is a well-order.\n\n**Proof.** If \\(x\\in[T]\\), then \\(x|0>_{\\mathrm{KB}}x|1>_{\\mathrm{KB}}x|2>_{\\mathrm{KB}}\\cdots\\) by Lemma 6.5, an infinite strictly decreasing sequence in \\(T\\); so \\(<_{\\mathrm{KB}}\\) is not a well-order on \\(T\\).\n\nIf a linear order on the countable set of nodes were not a well-order, a nonempty subset without a least node would give an infinite decreasing sequence: fix an enumeration, start with one node of the subset, and repeatedly take the first smaller node in that enumeration. Thus absence of decreasing sequences suffices. Conversely, let \\(T\\) be well-founded, and suppose \\(s^0>_{\\mathrm{KB}}s^1>_{\\mathrm{KB}}s^2>_{\\mathrm{KB}}\\cdots\\) in \\(T\\). We build \\(x\\in[T]\\), which is a contradiction. We show by induction on \\(m\\geq0\\) that there are \\(x_1,\\dots,x_m\\in\\mathbb N\\) and an index \\(k_m\\) such that \\(s^k\\) extends \\(x|m=(x_1,\\dots,x_m)\\) for all \\(k\\geq k_m\\). For \\(m=0\\) take \\(k_0=0\\). Given \\(m\\), put \\(w=x|m\\). The terms of the sequence are distinct, so at most one \\(k\\geq k_m\\) has \\(s^k=w\\); hence there is \\(k'_m\\geq k_m\\) such that \\(s^k\\) properly extends \\(w\\) for all \\(k\\geq k'_m\\), and the entry \\(s^k_{m+1}\\) is defined. For \\(k\\geq k'_m\\), \\(s^{k+1}<_{\\mathrm{KB}}s^k\\) and both properly extend \\(w\\). Either \\(s^{k+1}\\) extends \\(s^k\\), and then \\(s^{k+1}_{m+1}=s^k_{m+1}\\); or they first differ at an index \\(i>m\\) with \\(s^{k+1}_i<s^k_i\\), and then \\(s^{k+1}_{m+1}\\leq s^k_{m+1}\\). So the entries \\(s^k_{m+1}\\), \\(k\\geq k'_m\\), form a nonincreasing sequence in \\(\\mathbb N\\), which is eventually constant, say equal to \\(x_{m+1}\\) for \\(k\\geq k_{m+1}\\). This completes the induction. Each \\(x|m\\) is an initial segment of some \\(s^k\\in T\\), so \\(x|m\\in T\\) for all \\(m\\), and \\(x\\in[T]\\). \\(\\square\\)\n\nFor example, the whole set \\(\\mathbb N^{<\\mathbb N}\\) is an ill-founded tree, and \\((1)>_{\\mathrm{KB}}(1,1)>_{\\mathrm{KB}}(1,1,1)>_{\\mathrm{KB}}\\cdots\\). The well-founded tree \\(\\{\\varnothing,(1),(2),(3),\\dots\\}\\) is ordered as \\((1)<_{\\mathrm{KB}}(2)<_{\\mathrm{KB}}(3)<_{\\mathrm{KB}}\\cdots<_{\\mathrm{KB}}\\varnothing\\), a well-order of type \\(\\omega+1\\).\n\nFinally we look at trees on a product. A sequence \\((q_1,p_1),\\dots,(q_k,p_k)\\) in \\(Q\\times P\\) is the same as a pair \\((s,t)\\in Q^{<\\mathbb N}\\times P^{<\\mathbb N}\\) with \\(|s|=|t|\\), and \\((Q\\times P)^{\\mathbb N}\\) is homeomorphic to \\(Q^{\\mathbb N}\\times P^{\\mathbb N}\\). So a tree on \\(Q\\times P\\) is a set \\(T\\) of such pairs, closed under taking initial segments of both entries at once, and \\([T]\\subseteq Q^{\\mathbb N}\\times P^{\\mathbb N}\\). For \\(x\\in Q^{\\mathbb N}\\), the *section* of \\(T\\) at \\(x\\) is\n\\[\nT(x)=\\{t\\in P^{<\\mathbb N}:\\ (x|\\,|t|,\\,t)\\in T\\},\n\\]\na tree on \\(P\\).\n\n**Lemma 6.7** (sections). Let \\(T\\) be a tree on \\(Q\\times P\\).\n1. \\([T(x)]=\\{y\\in P^{\\mathbb N}:(x,y)\\in[T]\\}\\) for every \\(x\\in Q^{\\mathbb N}\\).\n2. The map \\((T,x)\\mapsto T(x)\\), from \\(\\mathrm{Tree}(Q\\times P)\\times Q^{\\mathbb N}\\) to \\(\\mathrm{Tree}(P)\\), is continuous.\n3. Let \\(C\\subseteq Q^{\\mathbb N}\\times P^{\\mathbb N}\\) be closed and \\(T_C\\) its tree (Proposition 6.3(2)). Then \\(x\\mapsto T_C(x)\\) is continuous, and \\(x\\) lies outside the projection \\(\\{x:(x,y)\\in C\\text{ for some }y\\}\\) if and only if \\(T_C(x)\\in\\mathrm{WF}(P)\\).\n\n**Proof.** (1) \\(y\\in[T(x)]\\) means \\(y|k\\in T(x)\\), that is, \\((x|k,y|k)\\in T\\), for all \\(k\\); this says \\((x,y)\\in[T]\\).\n\n(2) For fixed \\(t\\) with \\(|t|=k\\), the condition \\(t\\in T(x)\\) holds exactly when \\((x,T)\\) lies in \\(\\bigcup_{|s|=k}\\big(O_s\\times\\{T:(s,t)\\in T\\}\\big)\\). This set is open, and so is its complement \\(\\bigcup_{|s|=k}\\big(O_s\\times\\{T:(s,t)\\notin T\\}\\big)\\), because the cylinders \\(O_s\\) with \\(|s|=k\\) partition \\(Q^{\\mathbb N}\\). So each coordinate of \\(T(x)\\) is a continuous function of \\((T,x)\\).\n\n(3) By Proposition 6.3(2), \\([T_C]=C\\), so by (1), \\([T_C(x)]\\) is the set of \\(y\\) with \\((x,y)\\in C\\). It is empty exactly when \\(x\\) is outside the projection of \\(C\\). Continuity is (2) with \\(T=T_C\\) fixed. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-DT-07",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "7. Countable orders and well-orders",
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      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "anchor": "oa-fnd-dt-07",
      "proof_locus": {
        "line": 653,
        "through_line": 693
      },
      "full_conditions_and_proof": "### 7. Countable orders and well-orders\n\nRelations on \\(Q\\) are subsets \\(R\\subseteq Q\\times Q\\), that is, points of \\(2^{Q\\times Q}\\). The *domain* of \\(R\\) is \\(D(R)=\\{q:(q,q)\\in R\\}\\). We write \\(q\\leq_Rq'\\) for \\((q,q')\\in R\\), and \\(q<_Rq'\\) if moreover \\(q\\neq q'\\).\n\n**Definition 7.1.** A relation \\(R\\) is an *order* if\n1. \\(q\\leq_Rq'\\) implies \\(q,q'\\in D(R)\\);\n2. \\(q\\leq_Rq'\\) and \\(q'\\leq_Rq\\) imply \\(q=q'\\);\n3. \\(q\\leq_Rq'\\leq_Rq''\\) implies \\(q\\leq_Rq''\\).\n\nIt is a *linear order* if moreover any \\(q,q'\\in D(R)\\) satisfy \\(q\\leq_Rq'\\) or \\(q'\\leq_Rq\\), and a *well-order* if moreover every nonempty subset of \\(D(R)\\) has an \\(R\\)-least element. We write \\(\\mathrm{PO}(Q)\\supseteq\\mathrm{LO}(Q)\\supseteq\\mathrm{WO}(Q)\\) for these sets of relations.\n\nSo an order is a reflexive partial order on its domain that relates nothing outside the domain. Since \\(Q\\) is countable, a linear order \\(R\\) is a well-order exactly when there is no sequence \\((z_k)_{k\\geq1}\\) in \\(D(R)\\) with \\(z_{k+1}<_Rz_k\\) for all \\(k\\). Indeed, such a sequence has no least element. Conversely, if a nonempty \\(S\\subseteq D(R)\\) has no least element, fix an enumeration of \\(Q\\), pick \\(z_1\\in S\\), and let \\(z_{k+1}\\) be the first element of \\(S\\) in the enumeration with \\(z_{k+1}<_Rz_k\\).\n\n**Proposition 7.2.**\n1. \\(\\mathrm{PO}(Q)\\) and \\(\\mathrm{LO}(Q)\\) are closed subsets of \\(2^{Q\\times Q}\\). So they are compact metrizable spaces and standard Borel spaces.\n2. \\(\\mathrm{WO}(Q)\\) is co-Souslin, in \\(\\mathrm{LO}(Q)\\) and in \\(2^{Q\\times Q}\\).\n3. For \\(R\\in\\mathrm{LO}(Q)\\), let \\(T_R\\) consist of the finite sequences \\((z_1,\\dots,z_k)\\) with entries in \\(D(R)\\) and \\(z_{i+1}<_Rz_i\\) for \\(i<k\\). Then \\(T_R\\) is a tree, \\(R\\mapsto T_R\\) is continuous from \\(\\mathrm{LO}(Q)\\) to \\(2^{Q^{<\\mathbb N}}\\), and \\(R\\in\\mathrm{WO}(Q)\\) if and only if \\(T_R\\in\\mathrm{WF}(Q)\\).\n\n**Proof.** (1) For fixed elements, each condition in Definition 7.1 and the linearity condition involve finitely many coordinates of \\(R\\), so they define open and closed sets; \\(\\mathrm{PO}(Q)\\) and \\(\\mathrm{LO}(Q)\\) are countable intersections of such sets. Closed subsets of \\(2^{Q\\times Q}\\) are compact and metrizable, hence Polish.\n\n(2) The set of pairs \\((R,z)\\in\\mathrm{LO}(Q)\\times Q^{\\mathbb N}\\) with \\(z_{k+1}<_Rz_k\\) for all \\(k\\) is closed, since each condition involves finitely many coordinates of \\((R,z)\\). By the remark before the proposition, its projection to \\(\\mathrm{LO}(Q)\\) is \\(\\mathrm{LO}(Q)\\setminus\\mathrm{WO}(Q)\\), which is therefore Souslin by Proposition 5.2(4). So \\(\\mathrm{WO}(Q)\\) is co-Souslin in \\(\\mathrm{LO}(Q)\\), and in \\(2^{Q\\times Q}\\) because \\(\\mathrm{LO}(Q)\\) is closed there.\n\n(3) Initial segments of strictly decreasing sequences are strictly decreasing, so \\(T_R\\) is a tree. For fixed \\(s=(z_1,\\dots,z_k)\\), the condition \\(s\\in T_R\\) involves the coordinates \\((z_i,z_i)\\) and \\((z_{i+1},z_i)\\) of \\(R\\) and the conditions \\(z_{i+1}\\neq z_i\\); so it defines an open and closed set, and \\(R\\mapsto T_R\\) is continuous. The body \\([T_R]\\) is the set of infinite strictly decreasing sequences, which is empty exactly when \\(R\\) is a well-order. \\(\\square\\)\n\nStrict decrease matters in (3). If the tree were built from sequences with \\(z_{i+1}\\leq_Rz_i\\), every constant sequence \\((z,z,\\dots,z)\\) with \\(z\\in D(R)\\) would belong to it, and for a nonempty domain the tree would never be well-founded.\n\nTo compare well-orders we use embeddings.\n\n**Definition 7.3.** For \\(R,S\\in\\mathrm{LO}(\\mathbb N)\\), write \\(R\\preceq S\\) if some injective map \\(\\varphi:D(R)\\to D(S)\\) satisfies \\(m\\leq_Rn\\iff\\varphi(m)\\leq_S\\varphi(n)\\) for \\(m,n\\in D(R)\\). Write \\(R\\prec S\\) if such a \\(\\varphi\\) can be chosen with values in \\(\\{n\\in D(S):\\ n<_Sq\\}\\) for some \\(q\\in D(S)\\).\n\n**Lemma 7.4.**\n1. The sets \\(\\{(R,S):R\\preceq S\\}\\) and \\(\\{(R,S):R\\prec S\\}\\) are Souslin subsets of \\(\\mathrm{LO}(\\mathbb N)\\times\\mathrm{LO}(\\mathbb N)\\).\n2. If \\(R\\preceq S\\) and \\(S\\in\\mathrm{WO}(\\mathbb N)\\), then \\(R\\in\\mathrm{WO}(\\mathbb N)\\).\n3. Let \\(R,S\\in\\mathrm{WO}(\\mathbb N)\\) have order types \\(|R|\\) and \\(|S|\\) (S2). Then \\(R\\preceq S\\) if and only if \\(|R|\\leq|S|\\), and \\(R\\prec S\\) if and only if \\(|R|<|S|\\).\n\n**Proof.** (1) Extending \\(\\varphi\\) arbitrarily outside \\(D(R)\\), \\(R\\preceq S\\) holds exactly when some \\(\\varphi\\in\\Lambda\\) satisfies, for all \\(m,n\\in\\mathbb N\\) with \\(m,n\\in D(R)\\): \\(\\varphi(m)\\neq\\varphi(n)\\) if \\(m\\neq n\\), and \\((m,n)\\in R\\iff(\\varphi(m),\\varphi(n))\\in S\\). For fixed \\(m,n\\), the set of \\((\\varphi,S)\\) with \\((\\varphi(m),\\varphi(n))\\in S\\) is the union over \\(p,p'\\) of the open sets \\(\\{\\varphi:\\varphi(m)=p,\\ \\varphi(n)=p'\\}\\times\\{S:(p,p')\\in S\\}\\), and its complement is the analogous union; so it is open and closed. Hence the set of \\((\\varphi,R,S)\\) satisfying all conditions is closed in \\(\\Lambda\\times\\mathrm{LO}(\\mathbb N)^2\\), and its projection is Souslin by Proposition 5.2(4). For \\(\\prec\\), add a variable \\(q\\in\\mathbb N\\) and the conditions \\(q\\in D(S)\\) and \\(\\varphi(m)<_Sq\\) for \\(m\\in D(R)\\), which are of the same kind.\n\n(2) \\(\\varphi\\) carries a strictly \\(R\\)-decreasing sequence to a strictly \\(S\\)-decreasing one, because it is injective and preserves the order in both directions.\n\n(3) Let \\(S_{<q}\\) denote the restriction of \\(S\\) to \\(\\{n\\in D(S):n<_Sq\\}\\), a proper initial segment. If \\(|R|\\leq|S|\\), then by (S2) \\(R\\) is isomorphic to \\(S\\) or to some \\(S_{<q}\\); in either case \\(R\\preceq S\\), and in the second case \\(R\\prec S\\). Suppose now \\(R\\preceq S\\) through \\(\\varphi\\), but \\(|S|<|R|\\). By (S2), \\(S\\) is isomorphic to some \\(R_{<q}\\) through \\(\\psi\\). Then \\(\\psi\\circ\\varphi\\) is a strictly increasing map of \\(D(R)\\) into \\(\\{n:n<_Rq\\}\\), so \\(\\psi(\\varphi(q))<_Rq\\), which contradicts (S1). Hence \\(R\\preceq S\\) implies \\(|R|\\leq|S|\\). If \\(R\\prec S\\) through a map into \\(S_{<q}\\), then \\(R\\preceq S_{<q}\\), so \\(|R|\\leq|S_{<q}|\\); and \\(|S_{<q}|<|S|\\), since \\(S_{<q}\\preceq S\\) and \\(S_{<q}\\) is not isomorphic to \\(S\\) by (S1). If \\(|R|<|S|\\), then \\(R\\) is isomorphic to a proper initial segment of \\(S\\) by (S2), so \\(R\\prec S\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-08",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "8. Every co-Souslin set comes from the well-orders",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
      "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "anchor": "oa-fnd-dt-08",
      "proof_locus": {
        "line": 694,
        "through_line": 731
      },
      "full_conditions_and_proof": "### 8. Every co-Souslin set comes from the well-orders\n\nWe code trees on \\(\\mathbb N\\) by linear orders on \\(\\mathbb N\\). Fix a bijection \\(e:\\mathbb N\\to\\mathbb N^{<\\mathbb N}\\): list words in increasing order of the weight \\(|s|+\\sum_js_j\\), and lexicographically within each weight. Every weight class is finite, every word occurs, and there are infinitely many words, so this produces exactly such a bijection.\n\n**Lemma 8.1.** For \\(T\\subseteq\\mathbb N^{<\\mathbb N}\\) let\n\\[\n\\begin{gathered}\nK(T)\\\\\n=\\{(m,n)\\in\\mathbb N\\times\\mathbb N :\\\\\n\\ e(m)\\in T,\\ e(n)\\in T,\\\\\n\\ e(m)\\leq_{\\mathrm{KB}}e(n)\\}.\n\\end{gathered}\n\\]\nThen \\(K\\) is a continuous map of \\(2^{\\mathbb N^{<\\mathbb N}}\\) into \\(\\mathrm{LO}(\\mathbb N)\\), the domain of \\(K(T)\\) is \\(e^{-1}(T)\\), and \\(e\\) is an order isomorphism of \\(\\big(e^{-1}(T),\\leq_{K(T)}\\big)\\) onto \\((T,\\leq_{\\mathrm{KB}})\\). In particular, when \\(T\\) is a tree, \\(K(T)\\) is a well-order exactly when \\(T\\) is well-founded.\n\n**Proof.** The coordinate \\((m,n)\\) of \\(K(T)\\) is \\(0\\) when \\(e(n)<_{\\mathrm{KB}}e(m)\\), and otherwise equals \\(1_T(e(m))\\,1_T(e(n))\\); so it is continuous in \\(T\\). The order properties follow from Lemma 6.5, and the last claim from Theorem 6.6. \\(\\square\\)\n\n**Theorem 8.2** (universality of the well-orders). Let \\(X\\) be a standard Borel space and \\(A\\subseteq X\\) a co-Souslin set.\n1. There is a Borel map \\(F:X\\to\\mathrm{LO}(\\mathbb N)\\) with \\(A=F^{-1}(\\mathrm{WO}(\\mathbb N))\\).\n2. If \\(X=\\Lambda\\), the map \\(F\\) can be chosen continuous.\n\n**Proof.** (1) Fix a Polish topology on \\(X\\) that generates its Borel sets, and a compatible metric. If \\(A=X\\), take \\(F\\) constant, equal to the empty relation, which is a well-order. Otherwise \\(X\\setminus A\\) is a nonempty Souslin set, so \\(X\\setminus A=g(\\Lambda)\\) for a continuous \\(g:\\Lambda\\to X\\) by (D3). For \\(x\\in X\\) put\n\\[\n\\tau(x)=\\{s\\in\\mathbb N^{<\\mathbb N}:\\ x\\in\\overline{g(\\Lambda_s)}\\}.\n\\]\nIf \\(s\\sqsubseteq t\\), then \\(\\Lambda_t\\subseteq\\Lambda_s\\), so \\(\\tau(x)\\) is a tree. The map \\(\\tau:X\\to2^{\\mathbb N^{<\\mathbb N}}\\) is Borel, since \\(\\{x:s\\in\\tau(x)\\}=\\overline{g(\\Lambda_s)}\\) is closed for each \\(s\\).\n\n*Claim: \\(\\tau(x)\\) is ill-founded if and only if \\(x\\in g(\\Lambda)\\).* If \\(x=g(z)\\), then \\(x\\in g(\\Lambda_{z|k})\\) for all \\(k\\), so \\(z\\in[\\tau(x)]\\). Conversely, let \\(z\\in[\\tau(x)]\\), so that \\(x\\in\\overline{g(\\Lambda_{z|k})}\\) for all \\(k\\), and suppose \\(x\\neq g(z)\\). Choose disjoint open sets \\(U\\ni g(z)\\) and \\(V\\ni x\\) with \\(\\overline U\\cap V=\\varnothing\\), for instance balls of radius one third of the distance. By continuity, \\(\\Lambda_{z|k}\\subseteq g^{-1}(U)\\) for large \\(k\\), so \\(\\overline{g(\\Lambda_{z|k})}\\subseteq\\overline U\\), which misses \\(V\\ni x\\). This is a contradiction, so \\(x=g(z)\\).\n\nPut \\(F=K\\circ\\tau\\), a Borel map into \\(\\mathrm{LO}(\\mathbb N)\\) by Lemma 8.1. By the claim and Lemma 8.1, \\(F(x)\\in\\mathrm{WO}(\\mathbb N)\\) exactly when \\(\\tau(x)\\) is well-founded, that is, when \\(x\\in A\\).\n\n(2) If \\(A=\\Lambda\\), take \\(F\\) constant. Otherwise \\(\\Lambda\\setminus A=g(\\Lambda)\\) for a continuous \\(g:\\Lambda\\to\\Lambda\\). The set \\(C=\\{(g(y),y):y\\in\\Lambda\\}\\) is closed in \\(\\Lambda\\times\\Lambda\\), and its projection to the first factor is \\(\\Lambda\\setminus A\\). By Lemma 6.7(3), \\(x\\mapsto T_C(x)\\) is continuous, and \\(x\\in A\\) exactly when \\(T_C(x)\\) is well-founded. So \\(F(x)=K(T_C(x))\\) is continuous and \\(A=F^{-1}(\\mathrm{WO}(\\mathbb N))\\) by Lemma 8.1. \\(\\square\\)\n\n**Corollary 8.3.** \\(\\mathrm{WO}(\\mathbb N)\\) and \\(\\mathrm{WF}(\\mathbb N)\\) are co-Souslin but not Souslin. In particular they are not Borel.\n\n\n**Proof.** They are co-Souslin by Propositions 7.2(2) and 6.3(3). If \\(\\mathrm{WO}(\\mathbb N)\\) were Souslin, then by Theorem 8.2 and (D1) every co-Souslin subset of \\(\\mathcal C\\) would be Souslin, against Proposition 5.2(5). By Proposition 7.2(3), \\(\\mathrm{WO}(\\mathbb N)\\) is the preimage of \\(\\mathrm{WF}(\\mathbb N)\\) under the continuous map \\(R\\mapsto T_R\\) on \\(\\mathrm{LO}(\\mathbb N)\\); so if \\(\\mathrm{WF}(\\mathbb N)\\) were Souslin, \\(\\mathrm{WO}(\\mathbb N)\\) would be too. Borel sets are Souslin (D1). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-DT-09",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "9. Reduction and the second separation theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
      "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "anchor": "oa-fnd-dt-09",
      "proof_locus": {
        "line": 732,
        "through_line": 757
      },
      "full_conditions_and_proof": "### 9. Reduction and the second separation theorem\n\n**Theorem 9.1** (reduction). Let \\(X\\) be a standard Borel space and \\(A_n\\), \\(n\\in I\\), countably many co-Souslin subsets of \\(X\\). There are pairwise disjoint co-Souslin sets \\(C_n\\subseteq A_n\\) with \\(\\bigcup_nC_n=\\bigcup_nA_n\\).\n\nMarker’s freely readable notes give the two-set rank argument; the proof here establishes the stated countable reduction.\n\n**Proof.** Number \\(I\\) by an initial segment of \\(\\mathbb N\\). By Theorem 8.2, choose Borel maps \\(F_n:X\\to\\mathrm{LO}(\\mathbb N)\\) with \\(A_n=F_n^{-1}(\\mathrm{WO}(\\mathbb N))\\). Let \\(C_n\\) be the set of \\(x\\in A_n\\) such that\n\\[\n\\begin{gathered}\nF_m(x)\\not\\preceq F_n(x)\\ \\text{ for all }m<n,\\\\\nF_m(x)\\not\\prec F_n(x)\\ \\text{ for all }m>n .\n\\end{gathered}\n\\tag{9.1}\n\\]\nEach set \\(\\{x:F_m(x)\\preceq F_n(x)\\}\\) is the preimage of a Souslin set (Lemma 7.4(1)) under the Borel map \\(x\\mapsto(F_m(x),F_n(x))\\), hence Souslin by (D1); the same holds for \\(\\prec\\). So \\(C_n\\) is the intersection of \\(A_n\\) with countably many co-Souslin sets, and it is co-Souslin (Proposition 5.2(2)).\n\nFix \\(x\\), and let \\(J=\\{n:x\\in A_n\\}\\). For \\(n\\in J\\) let \\(\\alpha_n\\) be the order type of the well-order \\(F_n(x)\\). Let \\(n\\in J\\). If \\(m\\notin J\\), then \\(F_m(x)\\) is not a well-order, so neither \\(F_m(x)\\preceq F_n(x)\\) nor \\(F_m(x)\\prec F_n(x)\\) holds, by Lemma 7.4(2). If \\(m\\in J\\), then \\(F_m(x)\\preceq F_n(x)\\) means \\(\\alpha_m\\leq\\alpha_n\\), and \\(F_m(x)\\prec F_n(x)\\) means \\(\\alpha_m<\\alpha_n\\), by Lemma 7.4(3). So (9.1) says: \\(\\alpha_m>\\alpha_n\\) for \\(m\\in J\\) with \\(m<n\\), and \\(\\alpha_m\\geq\\alpha_n\\) for \\(m\\in J\\) with \\(m>n\\). In other words, \\(x\\in C_n\\) exactly when \\(\\alpha_n\\) is the least of the ordinals \\(\\alpha_m\\), \\(m\\in J\\), and \\(n\\) is the first index where this least value occurs. A nonempty set of ordinals has a least element (S2), so every \\(x\\in\\bigcup_nA_n\\) lies in exactly one \\(C_n\\). \\(\\square\\)\n\n**Corollary 9.2** (second separation theorem). Let \\(X\\) be a standard Borel space.\n1. If \\(A_n\\), \\(n\\in I\\), are countably many Souslin subsets of \\(X\\), there are pairwise disjoint co-Souslin sets \\(C_n\\) with \\(A_n\\setminus\\bigcup_{m\\neq n}A_m\\subseteq C_n\\).\n2. In particular, for Souslin sets \\(A\\) and \\(B\\) there are disjoint co-Souslin sets \\(C\\) and \\(D\\) with \\(A\\setminus B\\subseteq C\\) and \\(B\\setminus A\\subseteq D\\).\n\n**Proof.** (1) The sets \\(E_n=X\\setminus\\bigcup_{m\\neq n}A_m\\) are co-Souslin by (D1). Theorem 9.1 gives pairwise disjoint co-Souslin sets \\(C_n\\subseteq E_n\\) with \\(\\bigcup_nC_n=\\bigcup_nE_n\\). Let \\(x\\in A_n\\setminus\\bigcup_{m\\neq n}A_m\\). Then \\(x\\in E_n\\), and \\(x\\notin E_k\\) for \\(k\\neq n\\), because \\(x\\in A_n\\) and \\(n\\neq k\\). So \\(x\\) lies in some \\(C_k\\subseteq E_k\\), and necessarily \\(k=n\\). (2) is (1) for two sets. \\(\\square\\)\n\nLusin’s separation theorem separates *disjoint* Souslin sets by Borel sets ([the separation theorem](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-03)). In Corollary 9.2 the sets \\(C\\) and \\(D\\) cannot always be chosen Borel; Exercise 5 gives an example.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-10",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "10. Sets of uniqueness and countable-to-one maps",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "anchor": "oa-fnd-dt-10",
      "proof_locus": {
        "line": 758,
        "through_line": 823
      },
      "full_conditions_and_proof": "### 10. Sets of uniqueness and countable-to-one maps\n\n**Theorem 10.1** (sets of uniqueness). Let \\(X,Y\\) be standard Borel spaces and \\(G\\subseteq X\\times Y\\) a Borel set. The set of \\(y\\in Y\\) for which exactly one \\(x\\in X\\) has \\((x,y)\\in G\\) is co-Souslin. In particular, for a Borel map \\(f:X\\to Y\\), the set\n\\[\nZ_f=\\{y\\in Y :\\ f^{-1}(y)\\text{ has exactly one point}\\}\n\\]\nis co-Souslin.\n\n\n**Proof.** The second statement is the first for the graph of \\(f\\), which is Borel by (D7).\n\n*Reduction.* Fix Polish topologies on \\(X\\) and \\(Y\\) that generate their Borel sets, and a compatible metric on \\(Y\\). By (D6) there are a closed set \\(F\\subseteq\\Lambda\\) and a continuous bijection \\(k:F\\to G\\). Put \\(p=\\pi_Y\\circ k:F\\to Y\\), a continuous map. For each \\(y\\), \\(k\\) maps \\(p^{-1}(y)\\) bijectively onto \\(\\{(x,y)\\in G\\}\\). So the set in the theorem is \\(Z_p\\), and it suffices to show that \\(Z_p\\) is co-Souslin.\n\n*The sets.* For \\(s\\in\\mathbb N^{<\\mathbb N}\\) put \\(A_s=p(F\\cap\\Lambda_s)\\), a Souslin set (a continuous image of a closed subset of \\(\\Lambda\\), or empty), and for \\(|s|=k\\geq1\\) let \\(B_s=\\bigcup\\{A_t:\\ |t|=k,\\ t\\neq s\\}\\), a Souslin set. Fix \\(k\\geq1\\). By Corollary 9.2(1), applied to the countable family \\((A_s)_{|s|=k}\\), there are pairwise disjoint co-Souslin sets \\(C_s\\), \\(|s|=k\\), with \\(A_s\\setminus B_s\\subseteq C_s\\). For \\(|s|\\geq1\\) put\n\\[\n\\begin{gathered}\nC'_s\\\\\n=C_s\\cap\\bigcap_{1\\leq j<|s|}C_{s|j},\\\\\nC^*_s\\\\\n=C'_s\\cap\\overline{A_s}\\cap(Y\\setminus B_s).\n\\end{gathered}\n\\]\nThese sets are co-Souslin, and for each \\(k\\) the sets \\(C^*_s\\), \\(|s|=k\\), are pairwise disjoint.\n\n*Claim: \\(A_s\\setminus B_s\\subseteq C^*_s\\).* Let \\(y\\in A_s\\setminus B_s\\) with \\(|s|=k\\). Then \\(y\\in C_s\\), \\(y\\in\\overline{A_s}\\) and \\(y\\notin B_s\\). Let \\(1\\leq j<k\\). Then \\(y\\in A_s\\subseteq A_{s|j}\\). If \\(y\\in A_u\\) for some \\(u\\neq s|j\\) with \\(|u|=j\\), then \\(y=p(z)\\) for some \\(z\\in F\\cap\\Lambda_u\\), and \\(z|k\\neq s\\), so \\(y\\in A_{z|k}\\subseteq B_s\\), which is false. So \\(y\\in A_{s|j}\\setminus B_{s|j}\\subseteq C_{s|j}\\). Hence \\(y\\in C^*_s\\).\n\nLet \\(Z'=\\bigcap_{k\\geq1}\\bigcup_{|s|=k}C^*_s\\), a co-Souslin set by Proposition 5.2(2). We show \\(Z_p=Z'\\).\n\n*\\(Z_p\\subseteq Z'\\).* Let \\(p^{-1}(y)=\\{z\\}\\) and \\(k\\geq1\\). Then \\(y\\in A_{z|k}\\). If \\(y\\in A_t\\) with \\(|t|=k\\) and \\(t\\neq z|k\\), then \\(y\\) would have a preimage in \\(\\Lambda_t\\), different from \\(z\\). So \\(y\\in A_{z|k}\\setminus B_{z|k}\\subseteq C^*_{z|k}\\) by the claim.\n\n*\\(Z'\\subseteq Z_p\\).* Let \\(y\\in Z'\\). For each \\(k\\geq1\\) there is exactly one \\(s^k\\) with \\(|s^k|=k\\) and \\(y\\in C^*_{s^k}\\), by disjointness. For \\(k\\geq1\\), \\(y\\in C^*_{s^{k+1}}\\subseteq C_{s^{k+1}|k}\\) and \\(y\\in C^*_{s^k}\\subseteq C_{s^k}\\); as the sets \\(C_t\\), \\(|t|=k\\), are disjoint, \\(s^{k+1}|k=s^k\\). So there is \\(z\\in\\Lambda\\) with \\(z|k=s^k\\) for all \\(k\\). Now \\(y\\in\\overline{A_{z|k}}\\), so \\(A_{z|k}\\neq\\varnothing\\) and \\(F\\cap\\Lambda_{z|k}\\neq\\varnothing\\) for all \\(k\\). The cylinders \\(\\Lambda_{z|k}\\) form a neighbourhood base at \\(z\\) and \\(F\\) is closed, so \\(z\\in F\\). If \\(y\\neq p(z)\\), choose open \\(U\\ni p(z)\\) and \\(V\\ni y\\) with \\(\\overline U\\cap V=\\varnothing\\). By continuity \\(p(F\\cap\\Lambda_{z|k})\\subseteq U\\) for large \\(k\\), so \\(\\overline{A_{z|k}}\\subseteq\\overline U\\) misses \\(y\\); this is false. So \\(y=p(z)\\). If also \\(y=p(z')\\) with \\(z'\\in F\\), \\(z'\\neq z\\), choose \\(k\\) with \\(z'|k\\neq z|k\\). Then \\(y\\in A_{z'|k}\\subseteq B_{z|k}\\), while \\(y\\in C^*_{z|k}\\subseteq Y\\setminus B_{z|k}\\). This is impossible, so \\(p^{-1}(y)=\\{z\\}\\) and \\(y\\in Z_p\\). \\(\\square\\)\n\nThe set \\(\\{y:f^{-1}(y)\\text{ has at least two points}\\}\\) is the projection of the Borel set \\(\\{(x,x')\\in X\\times X:x\\neq x',\\ f(x)=f(x')\\}\\) under \\((x,x')\\mapsto f(x)\\), so it is Souslin, and \"at most one preimage\" is a co-Souslin condition for simple reasons. The content of Theorem 10.1 is that \"at least one preimage\", which by itself is only a Souslin condition, becomes co-Souslin in combination with \"at most one\".\n\n**Example 10.2** (the theorem is sharp). Let \\(D\\subseteq\\mathcal C\\) be the Souslin set of (D4) that is not Borel, and write \\(D=g(\\Lambda)\\) with \\(g\\) continuous (D3).\n\n(a) *\\(Z_f\\) need not be Borel.* Let \\(X\\) be the disjoint union of \\(\\Lambda\\) and \\(\\mathcal C\\), a Polish space, and let \\(f:X\\to\\mathcal C\\) be \\(g\\) on \\(\\Lambda\\) and the identity on \\(\\mathcal C\\). Then \\(f^{-1}(y)\\) consists of \\(y\\) and the points of \\(g^{-1}(y)\\), so it is a singleton exactly when \\(y\\notin D\\). Hence \\(Z_f=\\mathcal C\\setminus D\\), which is co-Souslin and not Borel.\n\n(b) *The domain must be standard.* Give \\(D\\) its relative Borel structure and let \\(f:D\\to\\mathcal C\\) be the inclusion. Every fibre has at most one point, and \\(Z_f=D\\). This set is not co-Souslin: otherwise it would be Borel by (D2).\n\n**Lemma 10.3** (isolated points). Let \\(L\\) be a nonempty countable closed subset of a Polish space. Then every nonempty \\(S\\subseteq L\\) contains a point \\(q\\) with an open set \\(V\\) such that \\(V\\cap S=\\{q\\}\\).\n\n**Proof.** First, a nonempty countable complete metric space \\((M,d)\\) has an isolated point. If \\(M\\) is finite, every point is isolated. Otherwise enumerate \\(M=\\{l_1,l_2,\\dots\\}\\), and suppose no point is isolated. Then every open ball in \\(M\\) is infinite, since a ball with finitely many points would contain an isolated point. Choose recursively points \\(c_j\\in M\\) and radii \\(0<r_j<2^{-j}\\) such that the closed ball \\(\\overline B_j=\\{l:d(l,c_j)\\leq r_j\\}\\) lies in the open ball \\(\\{l:d(l,c_{j-1})<r_{j-1}\\}\\) (in \\(M\\) itself for \\(j=1\\)) and does not contain \\(l_j\\). This is possible: the open ball is infinite, so it contains some \\(c_j\\neq l_j\\), and a small radius does the rest. The closed balls \\(\\overline B_j\\) decrease and their diameters tend to \\(0\\), so by (D9) they have a common point. It differs from every \\(l_j\\), which is absurd.\n\nNow let \\(\\emptyset\\neq S\\subseteq L\\). The closure \\(\\overline S\\subseteq L\\) is nonempty, countable and closed, hence complete for a complete compatible metric. So it has an isolated point \\(q\\): some open \\(V\\) has \\(V\\cap\\overline S=\\{q\\}\\). Since \\(q\\in\\overline S\\), \\(V\\) meets \\(S\\), and \\(V\\cap S\\subseteq\\{q\\}\\); so \\(V\\cap S=\\{q\\}\\). \\(\\square\\)\n\n**Theorem 10.4** (countable-to-one maps). Let \\(X,Y\\) be standard Borel spaces and \\(f:X\\to Y\\) a Borel map such that every fibre \\(f^{-1}(y)\\) is countable.\n1. \\(f(B)\\) is Borel for every Borel set \\(B\\subseteq X\\); in particular \\(f(X)\\) is Borel.\n2. There is a Borel map \\(g:f(X)\\to X\\) with \\(f(g(y))=y\\) for all \\(y\\in f(X)\\).\n\n\n**Proof.** *Reduction.* Fix Polish topologies on \\(X\\) and \\(Y\\) that generate their Borel sets, and a countable base \\((V_m)\\) of \\(Y\\). By (D5) we may refine the topology of \\(X\\), without changing its Borel sets, so that every \\(f^{-1}(V_m)\\) is open. Then \\(f\\) is continuous, and each fibre \\(f^{-1}(y)\\) is closed and countable. Let \\((U_n)\\) be a countable base of \\(X\\).\n\n(1) Let \\(B\\subseteq X\\) be Borel, and for each \\(n\\) let \\(B_n\\) be the set of \\(y\\) such that \\(f^{-1}(y)\\cap B\\cap U_n\\) has exactly one point. The set \\(B\\cap U_n\\) is a standard Borel space (D5), and Theorem 10.1, applied to the restriction of \\(f\\) to it, shows that \\(B_n\\) is co-Souslin. Clearly \\(B_n\\subseteq f(B)\\). Conversely, let \\(y\\in f(B)\\). Lemma 10.3, applied to \\(L=f^{-1}(y)\\) and \\(S=f^{-1}(y)\\cap B\\), gives a point \\(q\\) and an open set \\(V\\) with \\(V\\cap S=\\{q\\}\\). A basic set \\(U_n\\) with \\(q\\in U_n\\subseteq V\\) then has \\(f^{-1}(y)\\cap B\\cap U_n=\\{q\\}\\), so \\(y\\in B_n\\). Hence \\(f(B)=\\bigcup_nB_n\\) is co-Souslin (Proposition 5.2(2)). It is also Souslin (D1), so it is Borel (D2).\n\n(2) For each \\(n\\), the set \\(M_n\\) of \\(y\\) with at least two preimages in \\(U_n\\) is the image of the Borel set \\[\n\\begin{gathered}\nP_n\\\\\n=\\{(x,x')\\in U_n\\times U_n:\\ x\\neq x',\\ f(x)=f(x')\\}\n\\end{gathered}\n\\] under the map \\((x,x')\\mapsto f(x)\\). The set \\(P_n\\) is a standard Borel space (D5), and on \\(P_n\\) this map is Borel with countable fibres, since the fibre over \\(y\\) lies in \\(f^{-1}(y)\\times f^{-1}(y)\\), a countable set by the explicit pairing in [Proposition 8.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08). By (1), applied to this map on \\(P_n\\), \\(M_n\\) is Borel. So is \\(f(U_n)\\), by (1). Hence \\(B'_n=f(U_n)\\setminus M_n\\), the set of \\(y\\) with exactly one preimage in \\(U_n\\), is Borel. Every \\(y\\in f(X)\\) lies in some \\(B'_n\\): apply Lemma 10.3 to \\(S=L=f^{-1}(y)\\) and choose a basic set as in (1). So \\(\\bigcup_nB'_n=f(X)\\). Put \\(E_n=B'_n\\setminus(B'_1\\cup\\dots\\cup B'_{n-1})\\), a Borel partition of \\(f(X)\\). On the Borel set \\(X_n=U_n\\cap f^{-1}(E_n)\\), \\(f\\) is injective with image \\(E_n\\). By (D7), \\(f|_{X_n}\\) is a Borel isomorphism of \\(X_n\\) onto \\(E_n\\). Define \\(g=(f|_{X_n})^{-1}\\) on \\(E_n\\). Then \\(g\\) is Borel on each set of a countable Borel partition of \\(f(X)\\), hence Borel, and \\(f(g(y))=y\\). \\(\\square\\)\n\n**Example 10.5** (the hypotheses are needed). (a) *Countable fibres.* The set \\(D\\) of (D4) is not Borel, but it is \\(g(\\Lambda)\\) for a continuous \\(g\\) on the standard Borel space \\(\\Lambda\\). By Theorem 10.4, some fibres of \\(g\\) are uncountable. (b) *A standard domain.* The inclusion of \\(D\\) into \\(\\mathcal C\\), as in Example 10.2(b), is injective, and its image \\(D\\) is not Borel.\n\nPart (2) can also be derived from the Borel transversal theorem of [Polish spaces and standard Borel spaces](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-10): after the reduction the fibres of \\(f\\) are closed, and \\(f^{-1}(f(U))\\) is Borel for every open \\(U\\) by part (1).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-11",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "11. Borel selectors for closed sets",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      "full_conditions_and_proof": "### 11. Borel selectors for closed sets\n\nGiven a nonempty closed subset of a Polish space, we want to choose one of its points, or a dense sequence of its points, in a Borel way. We prove this for families of closed sets indexed by an arbitrary measurable space. The Effros space of all nonempty closed sets is one such family.\n\n**Theorem 11.1** (Kuratowski–Ryll-Nardzewski). Let \\((\\Omega,\\mathcal S)\\) be a measurable space, \\(X\\) a Polish space, and \\(\\Phi\\) a map that assigns to each \\(\\omega\\in\\Omega\\) a nonempty closed set \\(\\Phi(\\omega)\\subseteq X\\), such that\n\\[\n\\begin{gathered}\n\\{\\omega:\\ \\Phi(\\omega)\\cap U\\neq\\varnothing\\}\\in\\mathcal S\\\\\n\\text{for every open }U\\\\\n\\subseteq X.\n\\end{gathered}\n\\tag{11.1}\n\\]\n1. For every open \\(U\\subseteq X\\) there is an \\(\\mathcal S\\)-measurable \\(f:\\Omega\\to X\\) with \\(f(\\omega)\\in\\Phi(\\omega)\\) for all \\(\\omega\\), and \\(f(\\omega)\\in U\\) whenever \\(\\Phi(\\omega)\\cap U\\neq\\varnothing\\).\n2. There are \\(\\mathcal S\\)-measurable maps \\(f_1,f_2,\\ldots:\\Omega\\to X\\) such that, for every \\(\\omega\\), all \\(f_n(\\omega)\\) lie in \\(\\Phi(\\omega)\\) and \\(\\{f_n(\\omega):n\\geq1\\}\\) is dense in \\(\\Phi(\\omega)\\).\n3. Conversely, if maps as in (2) exist, then \\(\\Phi\\) satisfies (11.1).\n\n**Proof.** If \\(\\Omega=\\varnothing\\), the unique empty maps prove all assertions. Otherwise the nonempty values of \\(\\Phi\\) force \\(X\\ne\\varnothing\\). *Step 1: one selector.* Let \\(d\\) be a complete compatible metric on \\(X\\) and \\((x_i)_{i\\geq1}\\) a dense sequence. For \\(x\\in X\\) and \\(r>0\\), \\[\n\\begin{gathered}\n\\{\\omega:d(x,\\Phi(\\omega))<r\\}\\\\\n=\\{\\omega:\\Phi(\\omega)\\cap B(x,r)\\neq\\varnothing\\}\\in\\mathcal S,\n\\end{gathered}\n\\] where \\(B(x,r)\\) is the open ball. For each \\(\\omega\\) we choose indices \\(\\iota_k(\\omega)\\), \\(k\\geq0\\), and put \\(p_k(\\omega)=x_{\\iota_k(\\omega)}\\), so that\n\\[\n\\begin{gathered}\nd\\big(p_k(\\omega),\\Phi(\\omega)\\big)<2^{-k},\\\\\nd\\big(p_{k+1}(\\omega),p_k(\\omega)\\big)<2^{-k}.\n\\end{gathered}\n\\tag{11.2}\n\\]\nLet \\(\\iota_0(\\omega)\\) be the least \\(i\\) with \\(d(x_i,\\Phi(\\omega))<1\\). Given \\(\\iota_k(\\omega)\\), let \\(\\iota_{k+1}(\\omega)\\) be the least \\(i\\) with \\(d(x_i,\\Phi(\\omega))<2^{-k-1}\\) and \\(d(x_i,p_k(\\omega))<2^{-k}\\). Such \\(i\\) exist. For \\(k=0\\) this follows from density. For the recursion, pick \\(z\\in\\Phi(\\omega)\\) with \\(d(p_k(\\omega),z)<2^{-k}\\), and then \\(x_i\\) with \\(d(x_i,z)<\\min\\big(2^{-k-1},\\,2^{-k}-d(p_k(\\omega),z)\\big)\\).\n\nEach \\(\\iota_k\\) is \\(\\mathcal S\\)-measurable, by induction: \\(\\{\\iota_{k+1}=i\\}\\) is the union over \\(j\\) of the sets \\(\\{\\iota_k=j\\}\\cap H_{k,i,j}\\setminus\\bigcup_{i'<i}H_{k,i',j}\\), where \\(H_{k,i,j}=\\{\\omega:d(x_i,\\Phi(\\omega))<2^{-k-1}\\}\\) if \\(d(x_i,x_j)<2^{-k}\\), and \\(H_{k,i,j}=\\varnothing\\) otherwise. So each \\(p_k\\) is measurable, taking countably many values on measurable sets. By (11.2), \\((p_k(\\omega))_k\\) is a Cauchy sequence, with \\(d(p_k(\\omega),p_l(\\omega))<2^{1-k}\\) for \\(l>k\\). Its limit \\(p(\\omega)\\) satisfies \\(d(p(\\omega),\\Phi(\\omega))=\\lim_kd(p_k(\\omega),\\Phi(\\omega))=0\\), because \\(y\\mapsto d(y,\\Phi(\\omega))\\) is \\(1\\)-Lipschitz; so \\(p(\\omega)\\in\\Phi(\\omega)\\), as \\(\\Phi(\\omega)\\) is closed. Since \\(d(p_k,p)\\leq2^{1-k}\\), for every closed \\(E\\subseteq X\\)\n\\[\np^{-1}(E)=\\bigcap_kp_k^{-1}\\big(\\{y:\\ d(y,E)\\leq2^{1-k}\\}\\big),\n\\]\nwhich lies in \\(\\mathcal S\\). Closed sets generate the Borel sets, so \\(p\\) is measurable.\n\n*Step 2: selectors into an open set.* Let \\(U\\subseteq X\\) be open and nonempty, and \\(\\Omega_U=\\{\\omega:\\Phi(\\omega)\\cap U\\neq\\varnothing\\}\\in\\mathcal S\\). By (D5), \\(U\\) is Polish; fix a complete metric on \\(U\\) compatible with its topology. For \\(\\omega\\in\\Omega_U\\), \\(\\Phi(\\omega)\\cap U\\) is a nonempty closed subset of \\(U\\). For \\(V\\) open in \\(U\\), hence open in \\(X\\), \\[\n\\begin{gathered}\n\\{\\omega\\in\\Omega_U :\\Phi(\\omega)\\cap U\\cap V\\neq\\varnothing\\}\\\\\n=\\{\\omega:\\Phi(\\omega)\\cap V\\neq\\varnothing\\}\\in\\mathcal S.\n\\end{gathered}\n\\] So Step 1, applied on the measurable space \\(\\Omega_U\\) with the relative \\(\\sigma\\)-algebra, to \\(U\\) and to \\(\\omega\\mapsto\\Phi(\\omega)\\cap U\\), gives a measurable \\(p_U :\\Omega_U\\to U\\) with \\(p_U(\\omega)\\in\\Phi(\\omega)\\cap U\\). Put \\(f=p_U\\) on \\(\\Omega_U\\) and \\(f=p\\) on \\(\\Omega\\setminus\\Omega_U\\), with \\(p\\) from Step 1. Then \\(f\\) is measurable, \\(f(\\omega)\\in\\Phi(\\omega)\\) always, and \\(f(\\omega)\\in U\\) when \\(\\omega\\in\\Omega_U\\). For \\(U=\\varnothing\\) take \\(f=p\\). This proves (1).\n\n*Step 3: a dense sequence.* Let \\((U_n)\\) be a countable base of \\(X\\), and let \\(f_n\\) be the map of (1) for \\(U_n\\). A nonempty relatively open subset of \\(\\Phi(\\omega)\\) contains a nonempty set \\(\\Phi(\\omega)\\cap U_n\\), and then \\(f_n(\\omega)\\) lies in it. This proves (2).\n\n*Step 4: the converse.* If the \\(f_n\\) are as in (2) and \\(U\\) is open, then by density \\(\\{\\omega:\\Phi(\\omega)\\cap U\\neq\\varnothing\\}=\\bigcup_nf_n^{-1}(U)\\), which lies in \\(\\mathcal S\\). \\(\\square\\)\n\n**Corollary 11.2** (Borel selectors on the Effros space). Let \\(X\\) be a Polish space and \\(\\mathcal C_0(X)\\) the standard Borel space of its nonempty closed subsets (D8).\n1. For every open \\(U\\subseteq X\\) there is a Borel map \\(f:\\mathcal C_0(X)\\to X\\) with \\(f(F)\\in F\\) for all \\(F\\), and \\(f(F)\\in U\\) whenever \\(F\\cap U\\neq\\varnothing\\).\n2. There are Borel maps \\(f_n:\\mathcal C_0(X)\\to X\\) such that \\(\\{f_n(F):n\\geq1\\}\\) is a dense subset of \\(F\\) for every \\(F\\).\n\n**Proof.** Apply Theorem 11.1 to \\(\\Omega=\\mathcal C_0(X)\\) with the Effros structure and \\(\\Phi(F)=F\\). Condition (11.1) holds by the definition of the Effros structure. \\(\\square\\)\n\nIn (1) the selector must be allowed to leave \\(U\\) when \\(F\\) misses \\(U\\), since then no point of \\(F\\) lies in \\(U\\); what (1) guarantees is that it always stays in \\(F\\). A choice function that took a fixed value outside \\(F\\) for those \\(F\\) would not serve in (2), where every \\(f_n(F)\\) must lie in \\(F\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-DT-12",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "name": "Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
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      },
      "full_conditions_and_proof": "## Exercises\n\n**Exercise 1** (medium; commutative case). Let \\(K\\) be a nonempty compact Hausdorff space and \\(A=C(K)\\) with the supremum norm. Show that for every \\(f\\in A\\), \\(W(f)\\) is the closed convex hull of \\(f(K)=\\sigma(f)\\).\n\n*Solution.* Let \\(R=\\|f\\|\\). For \\(|w|\\leq R\\) and \\(0<t\\leq1/R\\) (any \\(t>0\\) if \\(R=0\\)),\n\\[\n1+t\\operatorname{Re}w\\leq|1+tw|\\leq1+t\\operatorname{Re}w+\\tfrac12t^2|w|^2 ;\n\\]\nthe first inequality is \\(\\operatorname{Re}z\\leq|z|\\), and the second follows by squaring, since \\[\n\\begin{gathered}\n(1+t\\operatorname{Re}w+\\tfrac12t^2|w|^2)^2-|1+tw|^2\\\\\n=(t\\operatorname{Re}w+\\tfrac12t^2|w|^2)^2\\\\\n\\geq0\n\\end{gathered}\n\\] and the base is nonnegative. Apply this to \\(w=e^{i\\theta}f(k)\\) and take the maximum over \\(k\\in K\\):\n\\[\n\\begin{gathered}\n\\Big|\\frac{\\|1+te^{i\\theta}f\\|-1}{t}-\\max_{k\\in K}\\operatorname{Re}\\big(e^{i\\theta}f(k)\\big)\\Big|\\\\\n\\leq\\tfrac12tR^2 .\n\\end{gathered}\n\\]\nBy Theorem 2.3(1), \\(m(e^{i\\theta}f)=\\max_k\\operatorname{Re}(e^{i\\theta}f(k))\\), which is the function \\(h\\) of Lemma 2.1 for the closed convex hull of \\(f(K)\\): the maximum of a real-linear function over the closed convex hull of a compact set equals its maximum over the set. Both \\(W(f)\\) and this hull are nonempty, compact and convex, so they are equal by Lemma 2.1. Finally, \\(\\sigma(f)=f(K)\\), because \\(\\lambda-f\\) is invertible in \\(C(K)\\) exactly when it has no zero.\n\n**Exercise 2** (easy; a self-adjoint element with nonreal spectrum). On \\(A=\\mathbb C^2\\), with coordinatewise operations and the norm \\(\\|(z,w)\\|=\\max(|z|,|w|)\\), put \\((z,w)^*=(\\bar w,\\bar z)\\). Show that this is an isometric involution, that \\(A\\) is not a C\\*-algebra, and that \\(a=(i,-i)\\) is self-adjoint and not hermitian. Compute \\(\\sigma(a)\\) and \\(W(a)\\).\n\n*Solution.* The map is conjugate-linear, \\((x^*)^*=x\\), and \\((xy)^*=x^*y^*=y^*x^*\\) because the product is commutative; it preserves the maximum norm. For \\(x=(1,0)\\), \\(x^*x=(0,1)(1,0)=0\\), while \\(\\|x\\|^2=1\\); so the C\\*-identity fails. Next, \\(a^*=(\\overline{-i},\\overline{i})=(i,-i)=a\\). The algebra is \\(C(\\{1,2\\})\\), so by Exercise 1, \\(\\sigma(a)=\\{i,-i\\}\\) and \\(W(a)\\) is the segment from \\(-i\\) to \\(i\\). This is not contained in \\(\\mathbb R\\), so \\(a\\) is not hermitian. Note that the spectrum of \\(a\\) is not real, which cannot happen for a hermitian element (Corollary 1.4(2)).\n\n**Exercise 3** (easy; stochastic matrices). Let \\(T\\gg0\\) be an \\(n\\times n\\) matrix whose rows sum to \\(1\\). Show that \\(r(T)=1\\), and that \\(T^k\\) converges to the matrix all of whose rows equal the vector \\(u\\gg0\\) determined by \\(u^{\\top}T=u^{\\top}\\) and \\(\\sum_iu_i=1\\).\n\n*Solution.* \\(T\\mathbf 1=\\mathbf 1\\), so \\(\\mathbf 1\\geq0\\) is an eigenvector for the eigenvalue \\(1\\). By Theorem 4.4(5), \\(1=r(T)\\), and \\(v=\\mathbf 1\\) is a Perron–Frobenius eigenvector. Let \\(u\\gg0\\) be as in Theorem 4.4(4), normalized by \\(u^{\\top}\\mathbf 1=1\\); by Theorem 4.4(2) applied to \\(T^{\\top}\\), it is the only such vector. By Theorem 4.4(7), \\(T^k\\to\\mathbf 1u^{\\top}/(u^{\\top}\\mathbf 1)=\\mathbf 1u^{\\top}\\), the matrix whose rows all equal \\(u^{\\top}\\).\n\n**Exercise 4** (medium; operations on relations). Let \\(Q\\) be countable. For relations \\(R,S\\) on \\(Q\\) put \\[\n\\begin{gathered}\nR\\ast S\\\\\n=\\{(q,q',q''):(q,q')\\in R,\\ (q',q'')\\in S\\}\\\\\n\\subseteq Q^3\n\\end{gathered}\n\\] and \\[\n\\begin{gathered}\nR\\circ S\\\\\n=\\{(q,q''):(q,q',q'')\\in R\\ast S\\text{ for some }q'\\}.\n\\end{gathered}\n\\] Show that \\((R,S)\\mapsto R\\ast S\\) is continuous, that \\((R,S)\\mapsto R\\circ S\\) is Borel, and that it is not continuous when \\(Q\\) is infinite. Use this to show once more that \\(\\mathrm{PO}(Q)\\) is a Borel set, from the description: \\(R\\in\\mathrm{PO}(Q)\\) if and only if \\(R\\cap R^{-1}\\) is the diagonal of \\(D(R)\\), both projections of \\(R\\) equal \\(D(R)\\), and \\(R\\circ R\\subseteq R\\).\n\n*Solution.* The coordinate \\((q,q',q'')\\) of \\(R\\ast S\\) is \\(1_R(q,q')1_S(q',q'')\\), a continuous function of \\((R,S)\\). The set of \\((R,S)\\) with \\((q,q'')\\in R\\circ S\\) is the union over \\(q'\\) of the open and closed sets \\(\\{(q,q')\\in R\\}\\times\\{(q',q'')\\in S\\}\\); so it is open, its complement is closed, and each coordinate of \\(R\\circ S\\) is a Borel function of \\((R,S)\\). Hence the map is Borel. If \\(Q\\) is infinite, pick distinct \\(a,b,c_1,c_2,\\ldots\\in Q\\), and put \\(R_n=\\{(a,c_n)\\}\\) and \\(S=\\{(c_n,b):n\\geq1\\}\\). Every coordinate of \\(R_n\\) is eventually \\(0\\), so \\(R_n\\to\\varnothing\\). But \\(R_n\\circ S=\\{(a,b)\\}\\) for all \\(n\\), while \\(\\varnothing\\circ S=\\varnothing\\). For the last claim: \\(R\\mapsto R^{-1}\\), \\(R\\mapsto D(R)\\) and \\(R\\mapsto\\{(q,q):q\\in D(R)\\}\\) are continuous; the projections \\(\\{q:\\exists q'\\,(q,q')\\in R\\}\\) and \\(\\{q':\\exists q\\,(q,q')\\in R\\}\\) are Borel functions of \\(R\\), for the same reason as \\(R\\circ S\\); and \\(R\\mapsto(R\\circ R,R)\\) is Borel, while the inclusion relation \\(\\{(P',P''):P'\\subseteq P''\\}\\) is closed. Each of the three conditions defines the preimage of a closed set under a Borel map, and the set of pairs of equal points is closed. So \\(\\mathrm{PO}(Q)\\) is Borel. Proposition 7.2(1) shows more: it is closed.\n\n**Exercise 5** (hard; the second separation theorem cannot use Borel sets). Using the universal Souslin set \\(U\\subseteq\\mathcal C\\times\\mathcal C\\) of (D4), find disjoint co-Souslin sets in \\(\\mathcal C\\times\\mathcal C\\) that no Borel set separates. Conclude that in Corollary 9.2(2) the sets \\(C\\) and \\(D\\) cannot always be chosen Borel.\n\n*Solution.* Let \\(V=(\\mathcal C\\times\\mathcal C)\\setminus U\\). Every co-Souslin subset of \\(\\mathcal C\\) is a section \\(V_y=\\{x:(y,x)\\in V\\}\\). Let \\(h:\\mathcal C\\times\\mathcal C\\to\\mathcal C\\) be the homeomorphism that interleaves coordinates, \\(h(u,u')=(u_1,u'_1,u_2,u'_2,\\dots)\\), and write \\(h^{-1}(w)=(\\pi(w),\\pi'(w))\\). The sets\n\\[\n\\begin{gathered}\nA\\\\\n=\\{(w,x):\\ (\\pi(w),x)\\in V\\},\\\\\nB\\\\\n=\\{(w,x):\\ (\\pi'(w),x)\\in V\\}\n\\end{gathered}\n\\]\nare co-Souslin in \\(\\mathcal C\\times\\mathcal C\\), as preimages of \\(V\\) under continuous maps (Proposition 5.2(3)). By Theorem 9.1 there are disjoint co-Souslin sets \\(A'\\subseteq A\\) and \\(B'\\subseteq B\\) with \\(A'\\cup B'=A\\cup B\\). Suppose a Borel set \\(E\\) had \\(A'\\subseteq E\\) and \\(E\\cap B'=\\varnothing\\). Let \\(G\\subseteq\\mathcal C\\) be any Borel set. Both \\(G\\) and \\(\\mathcal C\\setminus G\\) are co-Souslin, so \\(G=V_u\\) and \\(\\mathcal C\\setminus G=V_{u'}\\) for some \\(u,u'\\). For \\(w=h(u,u')\\) the sections at \\(w\\) are \\(A_w=G\\) and \\(B_w=\\mathcal C\\setminus G\\). So \\(A'_w\\subseteq G\\) and \\(B'_w\\subseteq\\mathcal C\\setminus G\\) are disjoint with union \\(\\mathcal C\\), which forces \\(A'_w=G\\) and \\(B'_w=\\mathcal C\\setminus G\\). Then \\(E_w\\) contains \\(G\\) and misses \\(\\mathcal C\\setminus G\\), so \\(E_w=G\\). Thus every Borel subset of \\(\\mathcal C\\) is a section of the one Borel set \\(E\\). But \\(G_0=\\{x:(x,x)\\notin E\\}\\) is Borel, so \\(G_0=E_{w_0}\\) for some \\(w_0\\), and then \\(w_0\\in G_0\\) if and only if \\((w_0,w_0)\\notin E\\), if and only if \\(w_0\\notin E_{w_0}=G_0\\). This contradiction shows that no Borel set separates \\(A'\\) from \\(B'\\).\n\nNow put \\(A''=(\\mathcal C\\times\\mathcal C)\\setminus B'\\) and \\(B''=(\\mathcal C\\times\\mathcal C)\\setminus A'\\), two Souslin sets. Since \\(A'\\) and \\(B'\\) are disjoint, \\(A''\\setminus B''=A'\\) and \\(B''\\setminus A''=B'\\). Disjoint Borel sets \\(C\\supseteq A'\\) and \\(D\\supseteq B'\\) would give a Borel set \\(C\\) that separates \\(A'\\) from \\(B'\\), which is impossible.\n\n## References\n\nOpen ones first.\n\n\n*Freely accessible reading:* [Hanna Blazhko; Daniil Homza; Felix L. Schwenninger; Jens de Vries; Michał Wojtylak, *The algebraic numerical range as a spectral set in Banach algebras*, §2, Lemma 2.1](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/algebraic-numerical-range-as-a-spectral-set-in-banach-algebras/96536D155B032F6C67B750582F5DBF07) gives a route through algebraic numerical ranges and the perturbation estimate of Proposition 1.4a; the classical facts referenced there are proved here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n\nFor tree ranks and two-set coanalytic reduction, see [D. Marker, §5, pp. 43–48](https://homepages.math.uic.edu/~marker/math512/dst.pdf). Sections 6–9 here include the complete countable reduction and second separation arguments.\n",
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      "id": "OA-FND-AO-01",
      "unit": "abelian-operator-algebras",
      "name": "1. Four examples",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "full_conditions_and_proof": "### 1. Four examples\n\nThe following examples show the phenomena that the lesson explains. When an example uses a later result, the result is named.\n\n**Example 1.1** (Diagonal operators). Let \\(I\\) be a set, \\(H=\\ell^2(I)\\) with its standard basis \\((\\delta_i)_{i\\in I}\\), and for a bounded function \\(f\\) on \\(I\\) let \\(M_f\\) be the diagonal operator \\(M_f\\delta_i=f(i)\\delta_i\\). The algebra \\(\\mathcal A=\\{M_f:f\\in\\ell^\\infty(I)\\}\\) is abelian, and it is its own commutant. Indeed, if \\(T\\) commutes with each rank-one projection \\(M_{1_{\\{i\\}}}\\), then \\(T\\) maps \\(\\mathbb C\\delta_i\\) into itself, so \\(T\\delta_i=t_i\\delta_i\\) with \\(|t_i|\\le\\|T\\|\\), and \\(T=M_t\\). So \\(\\mathcal A\\) is a maximal abelian von Neumann algebra. In \\(\\ell^\\infty(I)\\) a bounded increasing net of real functions \\(f_\\alpha\\) has its pointwise supremum \\(f\\) as least upper bound, and \\(M_{f_\\alpha}\\to M_f\\) strongly. Indeed, for a fixed vector, choose a finite set of coordinates with arbitrarily small squared-norm tail. Pointwise convergence handles the finite set, and the uniform bound on \\(f_\\alpha-f\\) controls the tail.\n\nIf \\(I=\\{i_1,i_2,\\dots\\}\\) is countable (listed without repetitions, with a finite sum when \\(I\\) is finite), the vector \\(\\xi=\\sum_k2^{-k}\\delta_{i_k}\\) is cyclic for \\(\\mathcal A\\), because \\(M_{1_{\\{i_k\\}}}\\xi=2^{-k}\\delta_{i_k}\\). For \\(I=\\varnothing\\), \\(H=\\{0\\}\\) and the zero vector is cyclic. If \\(I\\) is uncountable, no vector is cyclic: a vector \\(\\xi\\in\\ell^2(I)\\) has only countably many nonzero coordinates (the sets where \\(|\\xi_i|\\geq1/n\\) are finite), and \\(\\mathcal A\\xi\\) lies in the closed subspace spanned by the corresponding basis vectors. The spectrum of \\(\\ell^\\infty(I)\\) is the Stone–Čech compactification \\(\\beta I\\) of the discrete space \\(I\\).\n\n**Example 1.2** (Multiplication operators on the unit interval). Let \\(m\\) be Lebesgue measure on \\([0,1]\\) and \\(\\mathcal A=\\{M_f:f\\in L^\\infty[0,1]\\}\\) on \\(L^2[0,1]\\), where \\(M_f\\xi=f\\xi\\). This is a maximal abelian von Neumann algebra, and the constant function \\(1\\) is a cyclic vector (Theorem 3.1). The multiplications by continuous functions form a C\\*-subalgebra that is not a von Neumann algebra. For instance, \\(g_n(t)=\\min\\{1,\\max\\{0,n(t-\\frac12)\\}\\}\\) increases to the indicator function of \\((\\frac12,1]\\), and \\(M_{g_n}\\) converges strongly to multiplication by that indicator. In \\(C[0,1]\\) itself the sequence \\((g_n)\\) has no least upper bound. An upper bound \\(g\\) satisfies \\(g\\ge1\\) on \\((\\frac12,1]\\), so \\(g(\\frac12)\\ge1\\) by continuity, and \\(g\\ge0\\). Hence \\(g>\\frac12\\) on some interval \\([\\frac12-\\delta,\\frac12]\\). Subtracting a continuous bump \\(b\\) with \\(0\\le b\\le\\frac12\\), supported in \\((\\frac12-\\delta,\\frac12)\\) and not identically zero, gives a smaller upper bound, because every \\(g_n\\) vanishes on \\([0,\\frac12]\\). So \\(C[0,1]\\) lacks least upper bounds that \\(L^\\infty[0,1]\\) has. The reason is topological: \\([0,1]\\) is connected, while spectra of von Neumann algebras are extremally disconnected (Sections 4 and 7).\n\n**Example 1.3** (Multiplicity two). On \\(L^2[0,1]\\oplus L^2[0,1]\\) let \\(\\mathcal A_2=\\{M_f\\oplus M_f:f\\in L^\\infty[0,1]\\}\\). This abelian von Neumann algebra is isomorphic to the algebra of Example 1.2 by \\(M_f\\mapsto M_f\\oplus M_f\\). But its commutant contains the operators \\(\\begin{pmatrix}\\alpha&\\beta\\\\\\gamma&\\delta\\end{pmatrix}\\) with scalar entries, which do not commute with each other. So \\(\\mathcal A_2\\) is not maximal abelian, and by Corollary 3.3 it has no cyclic vector. The two algebras are isomorphic but not spatially isomorphic, since their commutants differ (see Exercise 12.4).\n\n**Example 1.4** (An \\(L^\\infty\\) space that is not a von Neumann algebra on its \\(L^2\\) space). Let \\(X\\) be an uncountable set, \\(\\Sigma\\) the \\(\\sigma\\)-algebra of sets that are countable or have countable complement, and \\(\\nu(E)\\) the number of points of \\(E\\) if \\(E\\) is countable and \\(\\nu(E)=\\infty\\) otherwise. This is a measure, and only the empty set has measure zero.\n\n*Measurable functions are constant off a countable set.* Let \\(f:X\\to\\mathbb R\\) be \\(\\Sigma\\)-measurable. If \\(\\{f>r\\}\\) had countable complement for every rational \\(r\\), then \\(\\bigcap_r\\{f>r\\}=\\varnothing\\) would have countable complement, which is absurd. If \\(\\{f>r\\}\\) were countable for every rational \\(r\\), then \\(X=\\bigcup_r\\{f>r\\}\\) would be countable. The sets \\(\\{f>r\\}\\) decrease in \\(r\\), so there is a number \\(c\\) with \\(\\{f>r\\}\\) co-countable for rational \\(r<c\\) and countable for rational \\(r>c\\). Then \\(\\{f\\ne c\\}\\) is contained in \\(\\bigcup_{r<c}\\{f\\le r\\}\\cup\\bigcup_{r>c}\\{f>r\\}\\), a countable set. The same holds for complex functions, by taking real and imaginary parts.\n\n*The spaces.* If \\(\\int|f|^2d\\nu<\\infty\\), the constant \\(c\\) must be \\(0\\), since the co-countable set \\(\\{f=c\\}\\) has infinite measure. Conversely, every function with countable support is measurable. So \\(L^2(X,\\Sigma,\\nu)=\\ell^2(X)\\), with the same norm. Since only \\(\\varnothing\\) is null, \\(L^\\infty(X,\\Sigma,\\nu)\\) consists of the bounded functions that are constant off a countable set, acting on \\(\\ell^2(X)\\) by diagonal operators.\n\n*The algebra is too small.* Every singleton is measurable, so an operator that commutes with \\(L^\\infty(X,\\Sigma,\\nu)\\) commutes with the projections onto the basis vectors, and it is diagonal, as in Example 1.1. So the commutant of the multiplication algebra is the algebra of all diagonal operators \\(M_t\\), \\(t\\in\\ell^\\infty(X)\\), and its bicommutant is again this algebra. [Proposition 8.5 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08) supplies a partition of \\(X\\) into two uncountable sets. The multiplication operator by the indicator of one of them lies in the bicommutant but not in the multiplication algebra. So the multiplication algebra of \\(L^\\infty(X,\\Sigma,\\nu)\\) is not a von Neumann algebra.\n\nWhether \\(L^\\infty\\) acts on \\(L^2\\) as a von Neumann algebra is therefore a property of the measure. It holds for finite measures (Remark 3.2), for sigma-finite measures and for Radon measures on locally compact spaces with the local conventions (Proposition 3.2a below). Proposition 3.2b proves the finite-piece gluing criterion used here; Example 3.2c shows why a local-null convention matters.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AO-02",
      "unit": "abelian-operator-algebras",
      "name": "2. Tools from topology, measure theory and operator theory",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "anchor": "oa-fnd-ao-02",
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        "line": 45,
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      "full_conditions_and_proof": "### 2. Tools from topology, measure theory and operator theory\n\nThis section fixes conventions and supplies the measure, category and order tools used throughout.\n\n*Spaces.* Compact and locally compact spaces are Hausdorff. For a subset \\(A\\) of a topological space, \\(\\overline A\\) is its closure and \\(A^\\circ\\) its interior. A set is *clopen* if it is closed and open. A set \\(R\\) is *rare* (or nowhere dense) if \\(\\overline R\\) has empty interior. A set is *meager* (or of the first category) if it is a countable union of rare sets. A space is a *Baire space* if every meager subset has empty interior. For a compact space \\(\\Omega\\), \\(C(\\Omega)\\) is the C\\*-algebra of continuous complex functions and \\(C_{\\mathbb R}(\\Omega)\\) its real part. For a locally compact space \\(\\Gamma\\), \\(C_0(\\Gamma)\\) and \\(C_c(\\Gamma)\\) are the continuous functions that vanish at infinity and those with compact support. For any set \\(X\\), \\(\\ell^\\infty(X)\\) denotes the C\\*-algebra formed by the bounded complex functions on \\(X\\). Real functions are ordered pointwise.\n\n*Regularizations.* For a bounded real function \\(f\\) on a topological space \\(X\\) put\n\\[\n\\begin{gathered}\nf_*(x)=\\sup_{U\\ni x}\\,\\inf_{y\\in U}f(y),\\\\\nf^*(x)=\\inf_{U\\ni x}\\,\\sup_{y\\in U}f(y),\n\\end{gathered}\n\\tag{2.1}\n\\]\nwhere \\(U\\) runs over the open neighbourhoods of \\(x\\). We call \\(f_*\\) the *lower* and \\(f^*\\) the *upper regularization* of \\(f\\). A real function \\(g\\) is *lower semicontinuous* (lsc) if every set \\(\\{g>t\\}\\) is open, and *upper semicontinuous* (usc) if \\(-g\\) is lsc.\n\n*Measures.* For a compact space \\(\\Omega\\), a *Radon measure* on \\(\\Omega\\) is a finite positive Borel measure that is outer regular and inner regular on open sets, as in [Haar measure on locally compact groups](haar-measure.md). We identify it with the positive linear functional \\(x\\mapsto\\mu(x)=\\int x\\,d\\mu\\) on \\(C(\\Omega)\\), which determines it (Riesz representation theorem, see Background). A finite Radon measure is inner regular on all Borel sets: \\(\\mu(E)=\\sup\\{\\mu(K):K\\subseteq E\\text{ compact}\\}\\). A set is *\\(\\mu\\)-null* if it lies in a Borel set of measure zero, and *\\(\\mu\\)-measurable* if it differs from a Borel set by a \\(\\mu\\)-null set. The *support* of \\(\\mu\\) is the complement of the union of all open \\(\\mu\\)-null sets. For Radon measures on locally compact spaces that need not be finite, \\(L^\\infty(\\Gamma,\\mu)\\) is formed with locally null sets, as in [Vector-valued functions, tensor products with \\(L^p\\), and preduals](vector-valued-integration-and-preduals.md); for finite measures on compact spaces this is the usual space.\n\n*Operators.* Hilbert spaces are complex, and inner products are linear in the first variable. For a von Neumann algebra \\(M\\), \\(M_h\\) is the set of its self-adjoint elements and \\(M_+\\) the set of its positive elements. A von Neumann algebra \\(M\\) is *generated* by a subset \\(S\\) if \\(M=(S\\cup S^*)''\\), the smallest von Neumann algebra containing \\(S\\). A \\(*\\)-isomorphism is a bijective \\(*\\)-homomorphism. For a vector \\(\\xi\\), \\(\\omega_\\xi(x)=\\langle x\\xi,\\xi\\rangle\\).\n\n#### Regularity after completing the measure\n\n**Completed regularity.** The inner and outer regularity of a finite Radon measure also hold for its completed measurable sets. Indeed, write \\(E\\triangle B\\subseteq Z\\), where \\(B,Z\\) are Borel and \\(\\mu(Z)=0\\). Given \\(\\varepsilon>0\\), choose a compact \\(K_0\\subseteq B\\) with \\(\\mu(B\\setminus K_0)<\\varepsilon/2\\), and an open \\(U\\supseteq Z\\) with \\(\\mu(U)<\\varepsilon/2\\). Then \\(K=K_0\\setminus U\\) is compact, \\(K\\subseteq E\\), and \\(\\mu(E\\setminus K)<\\varepsilon\\). Choose also an open \\(V\\supseteq B\\) with \\(\\mu(V\\setminus B)<\\varepsilon/2\\). The open set \\(V\\cup U\\) contains \\(E\\) and satisfies \\(\\mu((V\\cup U)\\setminus E)<\\varepsilon\\). Thus every completed measurable set admits the same compact and open approximations as a Borel set. This argument applies to finite Radon measures on locally compact spaces as well.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AO-03",
      "unit": "abelian-operator-algebras",
      "name": "3. Measures and cyclic representations",
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      "full_conditions_and_proof": "### 3. Measures and cyclic representations\n\nIn this section \\(\\Omega\\) is locally compact and \\(\\mu\\) is a finite positive Radon measure on it. For \\(f\\in L^\\infty(\\Omega,\\mu)\\) let \\(M_f\\) be the operator \\(\\xi\\mapsto f\\xi\\) on \\(L^2(\\Omega,\\mu)\\). The *multiplication representation* of \\(C_0(\\Omega)\\) is \\(\\pi_\\mu(x)=M_x\\). The Riesz representation theorem identifies finite positive Radon measures on \\(\\Omega\\) with positive linear functionals on \\(C_0(\\Omega)\\), and complex Radon measures with bounded linear functionals (see Background). A representation \\(\\pi\\) of a C\\*-algebra on \\(H\\) has a *cyclic vector* \\(\\xi\\) if \\(\\pi(A)\\xi\\) is dense in \\(H\\).\n\n**Theorem 3.1** (Cyclic representations of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be locally compact, with a finite positive Radon measure \\(\\mu\\).\n\n1. The constant function \\(1\\) is a cyclic vector for \\(\\pi_\\mu\\), and \\(\\langle\\pi_\\mu(x)1,1\\rangle=\\int x\\,d\\mu\\) for \\(x\\in C_0(\\Omega)\\).\n2. Let \\(\\pi\\) be a representation of \\(C_0(\\Omega)\\) on \\(H\\) with a cyclic vector \\(\\xi\\) such that \\(\\langle\\pi(x)\\xi,\\xi\\rangle=\\int x\\,d\\mu\\) for all \\(x\\). Then there is exactly one unitary \\(U:L^2(\\Omega,\\mu)\\to H\\) with \\(U\\pi_\\mu(x)=\\pi(x)U\\) for all \\(x\\) and \\(U1=\\xi\\).\n3. \\[\n\\begin{gathered}\n\\pi_\\mu(C_0(\\Omega))'\\\\\n=\\{M_f:f\\in L^\\infty(\\Omega,\\mu)\\}\\\\\n=\\pi_\\mu(C_0(\\Omega))''.\n\\end{gathered}\n\\] So the von Neumann algebra generated by \\(\\pi_\\mu(C_0(\\Omega))\\) is the algebra of multiplications by \\(L^\\infty(\\Omega,\\mu)\\), it is maximal abelian, and \\(f\\mapsto M_f\\) is an isometric \\(*\\)-isomorphism of \\(L^\\infty(\\Omega,\\mu)\\) onto it.\n\nPart (2) identifies \\(\\pi_\\mu\\) with the GNS representation of the positive functional \\(x\\mapsto\\int x\\,d\\mu\\). Its cyclic vector has squared norm \\(\\mu(\\Omega)\\); the functional is a state when this mass is one. The nonunital construction and its uniqueness are proved in [the GNS lesson, Theorems 5.4–5.5](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-05).\n\n**Proof.** (1) Since \\(\\mu\\) is finite, \\(1\\in L^2(\\Omega,\\mu)\\) and \\(\\pi_\\mu(C_0(\\Omega))1=C_0(\\Omega)\\), which contains \\(C_c(\\Omega)\\) and is therefore dense in \\(L^2(\\Omega,\\mu)\\) by the density theorem for Radon measures in [Haar measure on locally compact groups](haar-measure.md). The formula is the definition of \\(\\pi_\\mu\\).\n\n(2) For \\(x\\in C_0(\\Omega)\\), \\(\\|\\pi(x)\\xi\\|^2=\\langle\\pi(\\bar xx)\\xi,\\xi\\rangle=\\int|x|^2d\\mu=\\|x\\|_{L^2}^2\\). So \\(x\\mapsto\\pi(x)\\xi\\), defined on the dense subspace \\(C_0(\\Omega)\\) of \\(L^2(\\Omega,\\mu)\\), is well defined and isometric, and its range \\(\\pi(C_0(\\Omega))\\xi\\) is dense in \\(H\\). It extends to a unitary \\(U\\). For \\(x,y\\in C_0(\\Omega)\\), \\(U\\pi_\\mu(x)y=\\pi(xy)\\xi=\\pi(x)Uy\\), and by density \\(U\\pi_\\mu(x)=\\pi(x)U\\). For every \\(x\\), \\(\\langle U1,\\pi(x)\\xi\\rangle=\\langle U1,Ux\\rangle=\\int\\bar x\\,d\\mu=\\langle\\xi,\\pi(x)\\xi\\rangle\\), since \\(\\langle\\pi(\\bar x)\\xi,\\xi\\rangle=\\int\\bar x\\,d\\mu\\). The vectors \\(\\pi(x)\\xi\\) are dense, so \\(U1=\\xi\\). Any unitary with the two properties maps \\(x=\\pi_\\mu(x)1\\) to \\(\\pi(x)\\xi\\), so it equals \\(U\\).\n\n(3) Let \\(T\\) commute with every \\(M_x\\), \\(x\\in C_0(\\Omega)\\), and put \\(g=T1\\in L^2(\\Omega,\\mu)\\). For \\(x\\in C_0(\\Omega)\\), \\(Tx=TM_x1=M_xT1=xg\\). Let \\(h\\) be a bounded completed measurable function, and choose \\(x_n\\in C_0(\\Omega)\\) with \\(x_n\\to h\\) in \\(L^2(\\Omega,\\mu)\\). Then \\(Tx_n\\to Th\\) in \\(L^2\\), and \\(x_ng\\to hg\\) in \\(L^1\\) by the Cauchy–Schwarz inequality. Choose a subsequence whose squared \\(L^2\\) errors in the first convergence and \\(L^1\\) errors in the second have finite sums. Monotone convergence shows that both error series are finite almost everywhere, so this subsequence converges almost everywhere in both cases and gives \\(Th=hg\\). Let \\(c>\\|T\\|\\) and \\(E=\\{|g|>c\\}\\). Then\n\\[\nc^2\\mu(E)\\le\\int_E|g|^2\\,d\\mu=\\|T1_E\\|^2\\le\\|T\\|^2\\mu(E),\n\\]\nso \\(\\mu(E)=0\\). Hence \\(g\\in L^\\infty(\\Omega,\\mu)\\) with \\(\\|g\\|_\\infty\\le\\|T\\|\\). The operators \\(T\\) and \\(M_g\\) agree on the bounded functions, which are dense in \\(L^2\\), so \\(T=M_g\\).\n\nWrite \\(\\mathcal A=\\{M_f:f\\in L^\\infty\\}\\). We have shown \\(\\pi_\\mu(C_0(\\Omega))'\\subseteq\\mathcal A\\), and the converse inclusion holds because multiplication operators commute. \\(\\mathcal A\\) is abelian, so \\(\\mathcal A\\subseteq\\mathcal A'\\). Since \\(\\pi_\\mu(C_0(\\Omega))\\subseteq\\mathcal A\\), taking commutants gives \\(\\mathcal A'\\subseteq\\pi_\\mu(C_0(\\Omega))'=\\mathcal A\\). So \\(\\mathcal A'=\\mathcal A\\), and \\(\\pi_\\mu(C_0(\\Omega))''=\\mathcal A'=\\mathcal A\\). Clearly \\(\\|M_f\\|\\le\\|f\\|_\\infty\\). If \\(c<\\|f\\|_\\infty\\), the set \\(F=\\{|f|>c\\}\\) has positive measure and \\(\\|M_f1_F\\|\\ge c\\|1_F\\|\\), so \\(\\|M_f\\|=\\|f\\|_\\infty\\). The algebraic properties hold pointwise. \\(\\square\\)\n\n**Remark 3.2** (Finite measure spaces). The proof of (3) uses only one property of \\(C_0(\\Omega)\\): it consists of bounded functions and is dense in \\(L^2\\). So for every finite measure space \\((X,\\Sigma,\\nu)\\) and every set \\(S\\) of bounded measurable functions whose span is dense in \\(L^2(X,\\nu)\\), the commutant of \\(\\{M_s:s\\in S\\}\\) is \\(\\{M_f:f\\in L^\\infty(X,\\nu)\\}\\). Taking \\(S=L^\\infty\\), the algebra \\(L^\\infty(X,\\nu)\\) is maximal abelian on \\(L^2(X,\\nu)\\) for every finite measure. Proposition 3.2a extends this argument to sigma-finite measures and to arbitrary Radon measures on locally compact spaces. Example 1.4 shows that it fails for some measure spaces.\n\n#### Multiplication for sigma-finite and Radon measures\n\n**Proposition 3.2a** (Multiplication beyond finite measures). Multiplication gives an isometric unital \\(*\\)-isomorphism of \\(L^\\infty\\) onto a maximal abelian von Neumann algebra on \\(L^2\\) in either of the following settings: an arbitrary sigma-finite measure space, or a positive Radon measure on a locally compact Hausdorff space, with measurable functions and equality understood locally in the second setting.\n\n*Proof.* For a sigma-finite measure space, partition the space into countably many measurable sets \\(E_i\\) of finite measure. The projections \\(P_i=M_{1_{E_i}}\\) are pairwise orthogonal and their sum is \\(1\\) strongly, because the squared norm of the omitted tail is the tail of the convergent series \\(\\sum_i\\int_{E_i}|\\xi|^2\\). If \\(T\\) commutes with every bounded multiplication, it commutes with these projections. On \\(P_iL^2=L^2(E_i)\\), Remark 3.2 gives \\(T|_{P_iL^2}=M_{g_i}\\) with \\(\\|g_i\\|_\\infty\\leq\\|T\\|\\). Choose measurable representatives bounded everywhere by \\(\\|T\\|\\), and glue them on the \\(E_i\\). The resulting bounded measurable \\(g\\) satisfies \\(T=M_g\\), first on finite sums of the subspaces and then on all of \\(L^2\\) by continuity. The multiplication algebra is therefore its own commutant. Its operator norm equals the essential supremum norm: a level set of positive measure meets some \\(E_i\\) in a set of positive finite measure, and the indicator of that intersection tests the required lower bound.\n\nFor a Radon measure, [Lemma 2.1 of the vector-valued-functions lesson](vector-valued-integration-and-preduals.md#oa-fnd-vv-03) proves a decomposition into pairwise disjoint compact pieces \\(K_i\\), with locally null complement \\(N\\), such that every compact set meets only countably many pieces. Its gluing assertion says that scalar functions measurable on the pieces are locally measurable after being set to zero on \\(N\\); Lemma 1.2(4) identifies measurability on each compact piece with completed Borel measurability. These conclusions use only Radon regularity and Zorn's lemma, and their full proofs are given there. [Lemma 2.3(2)–(3) of the same lesson](vector-valued-integration-and-preduals.md#oa-fnd-vv-03) proves\n\\[\n\\|\\xi\\|_2^2=\\sum_i\\int_{K_i}|\\xi|^2\\,d\\mu.\n\\]\nThus \\(L^2\\) is the Hilbert sum of the finite-measure spaces \\(L^2(K_i)\\). The identification is onto: a square-summable family of piecewise \\(L^2\\) classes has only countably many nonzero components; choose completed measurable representatives on those pieces, put zero on all others and on \\(N\\), and apply the scalar gluing assertion. The displayed integral identity gives its squared norm and its prescribed components. In the compact-set approximation used by that gluing proof, a test set of measure zero needs no pieces: take the approximating compact subset to be empty, with error zero. For a test set of positive measure, the list of meeting pieces is nonempty, and the finite-tail approximation uses an integer \\(J\\geq1\\). In particular every vector has only countably many nonzero components, and the finite partial sums of \\(P_i=M_{1_{K_i}}\\) converge strongly to \\(1\\).\n\nAgain an operator commuting with every multiplication restricts on each piece to \\(M_{g_i}\\), by Remark 3.2. Truncate representatives on each piece so that \\(|g_i|\\leq\\|T\\|\\) everywhere. The gluing assertion gives a locally measurable bounded function \\(g\\), zero on \\(N\\), and the Hilbert-sum identity gives \\(T=M_g\\). Conversely all scalar multiplications commute, so the multiplication algebra equals its commutant and is a von Neumann algebra. If \\(0<c<\\|g\\|_\\infty\\), the level set \\(\\{|g|>c\\}\\) is not locally null. It therefore meets a compact set in a completed measurable set \\(F\\) with \\(0<\\mu(F)<\\infty\\); testing on \\(1_F\\) gives \\(\\|M_g\\|\\geq c\\). Together with the upper bound this proves isometry and faithfulness. If the norm is zero, the assertion is immediate. No lifting or general decomposable-operator theorem is used. \\(\\square\\)\n\nThe local integral used here is the supremum of integrals over compact subsets, as defined and justified in [Definitions 2.2–2.4 of the same provider](vector-valued-integration-and-preduals.md#oa-fnd-vv-03). It ignores locally null sets even when their outer-regular Borel measure is not zero. For a finite measure on a compact space it is the usual integral.\n\n**Proposition 3.2b** (Finite-piece gluing beyond sigma-finiteness). Let \\((X,\\Sigma,\\mu)\\) admit a partition \\((E_i)_{i\\in I}\\) into measurable sets of finite measure such that a set \\(S\\subseteq X\\) belongs to \\(\\Sigma\\) exactly when each \\(S\\cap E_i\\) is measurable, and then\n\\[\n\\mu(S)=\\sum_{i\\in I}\\mu(S\\cap E_i).\n\\]\nThe sum means the supremum of the finite partial sums. Such a measure space is called *strictly localizable*. No countability of \\(I\\) is required. Multiplication is an isometric unital *-isomorphism of \\(L^\\infty(X,\\mu)\\) onto a maximal abelian von Neumann algebra on \\(L^2(X,\\mu)\\).\n\n**Proof.** Integrals of nonnegative simple functions split over the pieces by the measure identity. Increasing simple approximations and monotone convergence give the same identity for every nonnegative measurable function. In particular,\n\\[\n\\begin{gathered}\nL^2(X,\\mu)=\\bigoplus_{i\\in I}L^2(E_i,\\mu),\\\\\n\\|\\xi\\|_2^2=\\sum_i\\|\\xi|_{E_i}\\|_2^2.\n\\end{gathered}\n\\]\nEach square-summable family has only countably many nonzero components: for every positive integer \\(n\\), only finitely many components have squared norm at least \\(1/n\\). Representatives on those components glue measurably by the partition hypothesis. Thus the displayed identification is onto, and the finite sums of \\(P_i=M_{1_{E_i}}\\) tend strongly to the identity.\n\nSuppose \\(T\\) commutes with every multiplication. It commutes with every \\(P_i\\), so its restriction \\(T_i\\) acts on \\(L^2(E_i)\\) and commutes with all bounded multipliers there; a function on \\(E_i\\), extended by zero, is measurable on \\(X\\). The finite-measure argument of Remark 3.2 gives \\(T_i=M_{g_i}\\) with \\(\\|g_i\\|_\\infty\\leq\\|T\\|\\). Choose measurable representatives bounded everywhere by \\(\\|T\\|\\), putting zero on their measurable exceptional null sets. Piecewise gluing yields a measurable bounded \\(g\\) on \\(X\\). On finite-component vectors \\(T=M_g\\), and density gives this equality on all of \\(L^2\\). Hence the multiplication algebra is its own commutant, and consequently is weakly closed.\n\nFor \\(0<c<\\|g\\|_\\infty\\), the measurable set \\(S=\\{|g|>c\\}\\) has positive measure. The measure identity gives an \\(i\\) with \\(0<\\mu(S\\cap E_i)<\\infty\\). Testing \\(M_g\\) on \\(1_{S\\cap E_i}\\) gives \\(\\|M_g\\|\\geq c\\). The reverse inequality follows from \\(\\int|g\\xi|^2\\leq\\|g\\|_\\infty^2\\int|\\xi|^2\\), proving isometry. Algebra, adjoint and identity are preserved pointwise. \\(\\square\\)\n\nThis isolates the gluing mechanism used in Proposition 3.2a. In its Radon case the exact compact-piece provider and the local integral remain essential: one cannot silently replace local equality by equality for an arbitrary outer-regular Borel measure. For the distinction between localizability, strict localizability and gluing, see [Bouafia and De Pauw, *Localizable locally determined measurable spaces with negligibles*, §§4.1–4.7 and 6.1–6.4](https://arxiv.org/html/2105.11331v1). Their terminology for measurable spaces with negligible sets is broader; the finite-piece operator proof above concerns the stated measure-space hypothesis.\n\n**Example 3.2c** (A measure whose multiplication representation loses the identity). Let \\(X\\) be uncountable, let \\(\\Sigma\\) consist of the countable and co-countable subsets, and put \\(\\mu(S)=0\\) for countable \\(S\\), \\(\\mu(S)=\\infty\\) otherwise. This is countably additive: a disjoint sequence contains at most one co-countable member; if it contains none, its union is countable. Every integrable squared modulus has null positive level sets, since \\(\\mu(\\{|\\xi|>1/n\\})\\leq n^2\\int|\\xi|^2<\\infty\\), and every finite-measure set here is null. Thus \\(L^2(X,\\mu)=\\{0\\}\\), whereas the constant function \\(1\\) has \\(L^\\infty\\)-norm one. Multiplication is not faithful. The measure is not semi-finite: its positive-measure sets contain no subsets of positive finite measure. Finite-piece gluing in Proposition 3.2b prevents this loss, and Proposition 3.2a already specifies conventions that prevent it in the Radon model.\n\n**Corollary 3.3** (Abelian algebras with a cyclic vector). Suppose that the abelian von Neumann algebra \\(M\\subseteq B(H)\\) has a cyclic vector \\(\\xi\\). Let \\(K\\) be the spectrum of \\(M\\), write \\(x\\mapsto\\hat x\\) for the Gelfand isomorphism of \\(M\\) onto \\(C(K)\\), and let \\(\\mu\\) be the Radon measure on \\(K\\) with \\(\\int\\hat x\\,d\\mu=\\langle x\\xi,\\xi\\rangle\\). Then:\n\n1. there is a unitary \\(U:L^2(K,\\mu)\\to H\\) with \\(U1=\\xi\\) and \\(UM_{\\hat x}U^*=x\\) for \\(x\\in M\\);\n2. \\(M\\) is maximal abelian, \\(M'=M\\);\n3. every bounded Borel function on \\(K\\) agrees \\(\\mu\\)-almost everywhere with a continuous function.\n\n**Proof.** Since \\(M\\) is a unital abelian C\\*-algebra, \\(K\\) is compact and \\(M\\cong C(K)\\). The map \\(\\pi(\\hat x)=x\\) is a representation of \\(C(K)\\) on \\(H\\) with cyclic vector \\(\\xi\\), and \\(\\langle\\pi(\\hat x)\\xi,\\xi\\rangle=\\int\\hat x\\,d\\mu\\). Theorem 3.1(2) gives (1). By Theorem 3.1(3), \\(M'=U\\pi_\\mu(C(K))'U^*\\) is abelian, so \\(M'\\subseteq M''=M\\). Since \\(M\\) is abelian, \\(M\\subseteq M'\\). Hence \\(M=M'\\). Finally \\[\n\\begin{gathered}\nU\\pi_\\mu(C(K))U^*\\\\\n=M\\\\\n=M'\\\\\n=U\\{M_f:f\\in L^\\infty\\}U^*,\n\\end{gathered}\n\\] so every \\(M_f\\) is some \\(M_{\\hat x}\\), which is (3). \\(\\square\\)\n\nPart (3) is a first sign of the special topology of \\(K\\): on \\([0,1]\\) with Lebesgue measure, the indicator of \\([0,\\frac12]\\) agrees almost everywhere with no continuous function. Section 5 explains (3) through normal measures.\n\n**Corollary 3.4** (Equivalent measures give equivalent representations). Let \\(\\Omega\\) be locally compact, and let \\(\\mu\\) and \\(\\nu\\) be finite positive Radon measures on it. The representations \\(\\pi_\\mu\\) and \\(\\pi_\\nu\\) are unitarily equivalent if and only if \\(\\mu\\) and \\(\\nu\\) have the same null sets.\n\n**Proof.** Suppose they have the same null sets. By the Radon–Nikodym theorem (see Background) \\(\\nu=h\\mu\\) with \\(h\\ge0\\) integrable. The set \\(\\{h=0\\}\\) is \\(\\nu\\)-null, hence \\(\\mu\\)-null. The map \\(V\\xi=h^{1/2}\\xi\\) from \\(L^2(\\Omega,\\nu)\\) to \\(L^2(\\Omega,\\mu)\\) is isometric, since \\(\\int|\\xi|^2d\\nu=\\int|\\xi|^2h\\,d\\mu\\). It is onto, since \\(\\eta=V(h^{-1/2}\\eta)\\) for \\(\\eta\\in L^2(\\Omega,\\mu)\\). It commutes with every multiplication operator, so \\(V\\pi_\\nu(x)=\\pi_\\mu(x)V\\).\n\nConversely, let \\(U:L^2(\\Omega,\\mu)\\to L^2(\\Omega,\\nu)\\) be unitary with \\(U\\pi_\\mu(x)=\\pi_\\nu(x)U\\), and put \\(g=U^*1\\in L^2(\\Omega,\\mu)\\). For \\(x\\in C_0(\\Omega)\\),\n\\[\n\\begin{gathered}\n\\int x\\,d\\nu\\\\\n=\\langle\\pi_\\nu(x)1,1\\rangle\\\\\n=\\langle U\\pi_\\mu(x)U^*1,1\\rangle\\\\\n=\\langle\\pi_\\mu(x)g,g\\rangle\\\\\n=\\int x|g|^2d\\mu .\n\\end{gathered}\n\\]\nThe finite measure \\(\\lambda=|g|^2\\mu\\) is again a Radon measure. Indeed, for \\(\\varepsilon>0\\) there is \\(\\delta>0\\) with \\(\\lambda(E)<\\varepsilon\\) whenever \\(\\mu(E)<\\delta\\), because \\(|g|^2\\) is integrable. So the outer regularity and the inner regularity of \\(\\mu\\) pass to \\(\\lambda\\). By the uniqueness in the Riesz representation theorem, \\(\\nu=\\lambda\\), so every \\(\\mu\\)-null set is \\(\\nu\\)-null. By symmetry the converse holds. \\(\\square\\)\n\n**Example 3.5.** On \\([0,1]\\) let \\(m\\) be Lebesgue measure and \\(\\delta_{1/2}\\) the point mass at \\(\\frac12\\). The measures \\(m+\\delta_{1/2}\\) and \\(2m+3\\delta_{1/2}\\) have the same null sets, so their multiplication representations are equivalent. The measures \\(m\\) and \\(m+\\delta_{1/2}\\) do not, and indeed \\(\\pi_{m+\\delta_{1/2}}(C[0,1])''\\) contains a rank-one projection, multiplication by \\(1_{\\{1/2\\}}\\), while \\(\\pi_m(C[0,1])''=L^\\infty[0,1]\\) contains none.\n\n",
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      "full_conditions_and_proof": "### 3. Measures and cyclic representations\n\nIn this section \\(\\Omega\\) is locally compact and \\(\\mu\\) is a finite positive Radon measure on it. For \\(f\\in L^\\infty(\\Omega,\\mu)\\) let \\(M_f\\) be the operator \\(\\xi\\mapsto f\\xi\\) on \\(L^2(\\Omega,\\mu)\\). The *multiplication representation* of \\(C_0(\\Omega)\\) is \\(\\pi_\\mu(x)=M_x\\). The Riesz representation theorem identifies finite positive Radon measures on \\(\\Omega\\) with positive linear functionals on \\(C_0(\\Omega)\\), and complex Radon measures with bounded linear functionals (see Background). A representation \\(\\pi\\) of a C\\*-algebra on \\(H\\) has a *cyclic vector* \\(\\xi\\) if \\(\\pi(A)\\xi\\) is dense in \\(H\\).\n\n**Theorem 3.1** (Cyclic representations of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be locally compact, with a finite positive Radon measure \\(\\mu\\).\n\n1. The constant function \\(1\\) is a cyclic vector for \\(\\pi_\\mu\\), and \\(\\langle\\pi_\\mu(x)1,1\\rangle=\\int x\\,d\\mu\\) for \\(x\\in C_0(\\Omega)\\).\n2. Let \\(\\pi\\) be a representation of \\(C_0(\\Omega)\\) on \\(H\\) with a cyclic vector \\(\\xi\\) such that \\(\\langle\\pi(x)\\xi,\\xi\\rangle=\\int x\\,d\\mu\\) for all \\(x\\). Then there is exactly one unitary \\(U:L^2(\\Omega,\\mu)\\to H\\) with \\(U\\pi_\\mu(x)=\\pi(x)U\\) for all \\(x\\) and \\(U1=\\xi\\).\n3. \\[\n\\begin{gathered}\n\\pi_\\mu(C_0(\\Omega))'\\\\\n=\\{M_f:f\\in L^\\infty(\\Omega,\\mu)\\}\\\\\n=\\pi_\\mu(C_0(\\Omega))''.\n\\end{gathered}\n\\] So the von Neumann algebra generated by \\(\\pi_\\mu(C_0(\\Omega))\\) is the algebra of multiplications by \\(L^\\infty(\\Omega,\\mu)\\), it is maximal abelian, and \\(f\\mapsto M_f\\) is an isometric \\(*\\)-isomorphism of \\(L^\\infty(\\Omega,\\mu)\\) onto it.\n\nPart (2) identifies \\(\\pi_\\mu\\) with the GNS representation of the positive functional \\(x\\mapsto\\int x\\,d\\mu\\). Its cyclic vector has squared norm \\(\\mu(\\Omega)\\); the functional is a state when this mass is one. The nonunital construction and its uniqueness are proved in [the GNS lesson, Theorems 5.4–5.5](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-05).\n\n**Proof.** (1) Since \\(\\mu\\) is finite, \\(1\\in L^2(\\Omega,\\mu)\\) and \\(\\pi_\\mu(C_0(\\Omega))1=C_0(\\Omega)\\), which contains \\(C_c(\\Omega)\\) and is therefore dense in \\(L^2(\\Omega,\\mu)\\) by the density theorem for Radon measures in [Haar measure on locally compact groups](haar-measure.md). The formula is the definition of \\(\\pi_\\mu\\).\n\n(2) For \\(x\\in C_0(\\Omega)\\), \\(\\|\\pi(x)\\xi\\|^2=\\langle\\pi(\\bar xx)\\xi,\\xi\\rangle=\\int|x|^2d\\mu=\\|x\\|_{L^2}^2\\). So \\(x\\mapsto\\pi(x)\\xi\\), defined on the dense subspace \\(C_0(\\Omega)\\) of \\(L^2(\\Omega,\\mu)\\), is well defined and isometric, and its range \\(\\pi(C_0(\\Omega))\\xi\\) is dense in \\(H\\). It extends to a unitary \\(U\\). For \\(x,y\\in C_0(\\Omega)\\), \\(U\\pi_\\mu(x)y=\\pi(xy)\\xi=\\pi(x)Uy\\), and by density \\(U\\pi_\\mu(x)=\\pi(x)U\\). For every \\(x\\), \\(\\langle U1,\\pi(x)\\xi\\rangle=\\langle U1,Ux\\rangle=\\int\\bar x\\,d\\mu=\\langle\\xi,\\pi(x)\\xi\\rangle\\), since \\(\\langle\\pi(\\bar x)\\xi,\\xi\\rangle=\\int\\bar x\\,d\\mu\\). The vectors \\(\\pi(x)\\xi\\) are dense, so \\(U1=\\xi\\). Any unitary with the two properties maps \\(x=\\pi_\\mu(x)1\\) to \\(\\pi(x)\\xi\\), so it equals \\(U\\).\n\n(3) Let \\(T\\) commute with every \\(M_x\\), \\(x\\in C_0(\\Omega)\\), and put \\(g=T1\\in L^2(\\Omega,\\mu)\\). For \\(x\\in C_0(\\Omega)\\), \\(Tx=TM_x1=M_xT1=xg\\). Let \\(h\\) be a bounded completed measurable function, and choose \\(x_n\\in C_0(\\Omega)\\) with \\(x_n\\to h\\) in \\(L^2(\\Omega,\\mu)\\). Then \\(Tx_n\\to Th\\) in \\(L^2\\), and \\(x_ng\\to hg\\) in \\(L^1\\) by the Cauchy–Schwarz inequality. Choose a subsequence whose squared \\(L^2\\) errors in the first convergence and \\(L^1\\) errors in the second have finite sums. Monotone convergence shows that both error series are finite almost everywhere, so this subsequence converges almost everywhere in both cases and gives \\(Th=hg\\). Let \\(c>\\|T\\|\\) and \\(E=\\{|g|>c\\}\\). Then\n\\[\nc^2\\mu(E)\\le\\int_E|g|^2\\,d\\mu=\\|T1_E\\|^2\\le\\|T\\|^2\\mu(E),\n\\]\nso \\(\\mu(E)=0\\). Hence \\(g\\in L^\\infty(\\Omega,\\mu)\\) with \\(\\|g\\|_\\infty\\le\\|T\\|\\). The operators \\(T\\) and \\(M_g\\) agree on the bounded functions, which are dense in \\(L^2\\), so \\(T=M_g\\).\n\nWrite \\(\\mathcal A=\\{M_f:f\\in L^\\infty\\}\\). We have shown \\(\\pi_\\mu(C_0(\\Omega))'\\subseteq\\mathcal A\\), and the converse inclusion holds because multiplication operators commute. \\(\\mathcal A\\) is abelian, so \\(\\mathcal A\\subseteq\\mathcal A'\\). Since \\(\\pi_\\mu(C_0(\\Omega))\\subseteq\\mathcal A\\), taking commutants gives \\(\\mathcal A'\\subseteq\\pi_\\mu(C_0(\\Omega))'=\\mathcal A\\). So \\(\\mathcal A'=\\mathcal A\\), and \\(\\pi_\\mu(C_0(\\Omega))''=\\mathcal A'=\\mathcal A\\). Clearly \\(\\|M_f\\|\\le\\|f\\|_\\infty\\). If \\(c<\\|f\\|_\\infty\\), the set \\(F=\\{|f|>c\\}\\) has positive measure and \\(\\|M_f1_F\\|\\ge c\\|1_F\\|\\), so \\(\\|M_f\\|=\\|f\\|_\\infty\\). The algebraic properties hold pointwise. \\(\\square\\)\n\n**Remark 3.2** (Finite measure spaces). The proof of (3) uses only one property of \\(C_0(\\Omega)\\): it consists of bounded functions and is dense in \\(L^2\\). So for every finite measure space \\((X,\\Sigma,\\nu)\\) and every set \\(S\\) of bounded measurable functions whose span is dense in \\(L^2(X,\\nu)\\), the commutant of \\(\\{M_s:s\\in S\\}\\) is \\(\\{M_f:f\\in L^\\infty(X,\\nu)\\}\\). Taking \\(S=L^\\infty\\), the algebra \\(L^\\infty(X,\\nu)\\) is maximal abelian on \\(L^2(X,\\nu)\\) for every finite measure. Proposition 3.2a extends this argument to sigma-finite measures and to arbitrary Radon measures on locally compact spaces. Example 1.4 shows that it fails for some measure spaces.\n\n#### Multiplication for sigma-finite and Radon measures\n\n**Proposition 3.2a** (Multiplication beyond finite measures). Multiplication gives an isometric unital \\(*\\)-isomorphism of \\(L^\\infty\\) onto a maximal abelian von Neumann algebra on \\(L^2\\) in either of the following settings: an arbitrary sigma-finite measure space, or a positive Radon measure on a locally compact Hausdorff space, with measurable functions and equality understood locally in the second setting.\n\n*Proof.* For a sigma-finite measure space, partition the space into countably many measurable sets \\(E_i\\) of finite measure. The projections \\(P_i=M_{1_{E_i}}\\) are pairwise orthogonal and their sum is \\(1\\) strongly, because the squared norm of the omitted tail is the tail of the convergent series \\(\\sum_i\\int_{E_i}|\\xi|^2\\). If \\(T\\) commutes with every bounded multiplication, it commutes with these projections. On \\(P_iL^2=L^2(E_i)\\), Remark 3.2 gives \\(T|_{P_iL^2}=M_{g_i}\\) with \\(\\|g_i\\|_\\infty\\leq\\|T\\|\\). Choose measurable representatives bounded everywhere by \\(\\|T\\|\\), and glue them on the \\(E_i\\). The resulting bounded measurable \\(g\\) satisfies \\(T=M_g\\), first on finite sums of the subspaces and then on all of \\(L^2\\) by continuity. The multiplication algebra is therefore its own commutant. Its operator norm equals the essential supremum norm: a level set of positive measure meets some \\(E_i\\) in a set of positive finite measure, and the indicator of that intersection tests the required lower bound.\n\nFor a Radon measure, [Lemma 2.1 of the vector-valued-functions lesson](vector-valued-integration-and-preduals.md#oa-fnd-vv-03) proves a decomposition into pairwise disjoint compact pieces \\(K_i\\), with locally null complement \\(N\\), such that every compact set meets only countably many pieces. Its gluing assertion says that scalar functions measurable on the pieces are locally measurable after being set to zero on \\(N\\); Lemma 1.2(4) identifies measurability on each compact piece with completed Borel measurability. These conclusions use only Radon regularity and Zorn's lemma, and their full proofs are given there. [Lemma 2.3(2)–(3) of the same lesson](vector-valued-integration-and-preduals.md#oa-fnd-vv-03) proves\n\\[\n\\|\\xi\\|_2^2=\\sum_i\\int_{K_i}|\\xi|^2\\,d\\mu.\n\\]\nThus \\(L^2\\) is the Hilbert sum of the finite-measure spaces \\(L^2(K_i)\\). The identification is onto: a square-summable family of piecewise \\(L^2\\) classes has only countably many nonzero components; choose completed measurable representatives on those pieces, put zero on all others and on \\(N\\), and apply the scalar gluing assertion. The displayed integral identity gives its squared norm and its prescribed components. In the compact-set approximation used by that gluing proof, a test set of measure zero needs no pieces: take the approximating compact subset to be empty, with error zero. For a test set of positive measure, the list of meeting pieces is nonempty, and the finite-tail approximation uses an integer \\(J\\geq1\\). In particular every vector has only countably many nonzero components, and the finite partial sums of \\(P_i=M_{1_{K_i}}\\) converge strongly to \\(1\\).\n\nAgain an operator commuting with every multiplication restricts on each piece to \\(M_{g_i}\\), by Remark 3.2. Truncate representatives on each piece so that \\(|g_i|\\leq\\|T\\|\\) everywhere. The gluing assertion gives a locally measurable bounded function \\(g\\), zero on \\(N\\), and the Hilbert-sum identity gives \\(T=M_g\\). Conversely all scalar multiplications commute, so the multiplication algebra equals its commutant and is a von Neumann algebra. If \\(0<c<\\|g\\|_\\infty\\), the level set \\(\\{|g|>c\\}\\) is not locally null. It therefore meets a compact set in a completed measurable set \\(F\\) with \\(0<\\mu(F)<\\infty\\); testing on \\(1_F\\) gives \\(\\|M_g\\|\\geq c\\). Together with the upper bound this proves isometry and faithfulness. If the norm is zero, the assertion is immediate. No lifting or general decomposable-operator theorem is used. \\(\\square\\)\n\nThe local integral used here is the supremum of integrals over compact subsets, as defined and justified in [Definitions 2.2–2.4 of the same provider](vector-valued-integration-and-preduals.md#oa-fnd-vv-03). It ignores locally null sets even when their outer-regular Borel measure is not zero. For a finite measure on a compact space it is the usual integral.\n\n**Proposition 3.2b** (Finite-piece gluing beyond sigma-finiteness). Let \\((X,\\Sigma,\\mu)\\) admit a partition \\((E_i)_{i\\in I}\\) into measurable sets of finite measure such that a set \\(S\\subseteq X\\) belongs to \\(\\Sigma\\) exactly when each \\(S\\cap E_i\\) is measurable, and then\n\\[\n\\mu(S)=\\sum_{i\\in I}\\mu(S\\cap E_i).\n\\]\nThe sum means the supremum of the finite partial sums. Such a measure space is called *strictly localizable*. No countability of \\(I\\) is required. Multiplication is an isometric unital *-isomorphism of \\(L^\\infty(X,\\mu)\\) onto a maximal abelian von Neumann algebra on \\(L^2(X,\\mu)\\).\n\n**Proof.** Integrals of nonnegative simple functions split over the pieces by the measure identity. Increasing simple approximations and monotone convergence give the same identity for every nonnegative measurable function. In particular,\n\\[\n\\begin{gathered}\nL^2(X,\\mu)=\\bigoplus_{i\\in I}L^2(E_i,\\mu),\\\\\n\\|\\xi\\|_2^2=\\sum_i\\|\\xi|_{E_i}\\|_2^2.\n\\end{gathered}\n\\]\nEach square-summable family has only countably many nonzero components: for every positive integer \\(n\\), only finitely many components have squared norm at least \\(1/n\\). Representatives on those components glue measurably by the partition hypothesis. Thus the displayed identification is onto, and the finite sums of \\(P_i=M_{1_{E_i}}\\) tend strongly to the identity.\n\nSuppose \\(T\\) commutes with every multiplication. It commutes with every \\(P_i\\), so its restriction \\(T_i\\) acts on \\(L^2(E_i)\\) and commutes with all bounded multipliers there; a function on \\(E_i\\), extended by zero, is measurable on \\(X\\). The finite-measure argument of Remark 3.2 gives \\(T_i=M_{g_i}\\) with \\(\\|g_i\\|_\\infty\\leq\\|T\\|\\). Choose measurable representatives bounded everywhere by \\(\\|T\\|\\), putting zero on their measurable exceptional null sets. Piecewise gluing yields a measurable bounded \\(g\\) on \\(X\\). On finite-component vectors \\(T=M_g\\), and density gives this equality on all of \\(L^2\\). Hence the multiplication algebra is its own commutant, and consequently is weakly closed.\n\nFor \\(0<c<\\|g\\|_\\infty\\), the measurable set \\(S=\\{|g|>c\\}\\) has positive measure. The measure identity gives an \\(i\\) with \\(0<\\mu(S\\cap E_i)<\\infty\\). Testing \\(M_g\\) on \\(1_{S\\cap E_i}\\) gives \\(\\|M_g\\|\\geq c\\). The reverse inequality follows from \\(\\int|g\\xi|^2\\leq\\|g\\|_\\infty^2\\int|\\xi|^2\\), proving isometry. Algebra, adjoint and identity are preserved pointwise. \\(\\square\\)\n\nThis isolates the gluing mechanism used in Proposition 3.2a. In its Radon case the exact compact-piece provider and the local integral remain essential: one cannot silently replace local equality by equality for an arbitrary outer-regular Borel measure. For the distinction between localizability, strict localizability and gluing, see [Bouafia and De Pauw, *Localizable locally determined measurable spaces with negligibles*, §§4.1–4.7 and 6.1–6.4](https://arxiv.org/html/2105.11331v1). Their terminology for measurable spaces with negligible sets is broader; the finite-piece operator proof above concerns the stated measure-space hypothesis.\n\n**Example 3.2c** (A measure whose multiplication representation loses the identity). Let \\(X\\) be uncountable, let \\(\\Sigma\\) consist of the countable and co-countable subsets, and put \\(\\mu(S)=0\\) for countable \\(S\\), \\(\\mu(S)=\\infty\\) otherwise. This is countably additive: a disjoint sequence contains at most one co-countable member; if it contains none, its union is countable. Every integrable squared modulus has null positive level sets, since \\(\\mu(\\{|\\xi|>1/n\\})\\leq n^2\\int|\\xi|^2<\\infty\\), and every finite-measure set here is null. Thus \\(L^2(X,\\mu)=\\{0\\}\\), whereas the constant function \\(1\\) has \\(L^\\infty\\)-norm one. Multiplication is not faithful. The measure is not semi-finite: its positive-measure sets contain no subsets of positive finite measure. Finite-piece gluing in Proposition 3.2b prevents this loss, and Proposition 3.2a already specifies conventions that prevent it in the Radon model.\n\n**Corollary 3.3** (Abelian algebras with a cyclic vector). Suppose that the abelian von Neumann algebra \\(M\\subseteq B(H)\\) has a cyclic vector \\(\\xi\\). Let \\(K\\) be the spectrum of \\(M\\), write \\(x\\mapsto\\hat x\\) for the Gelfand isomorphism of \\(M\\) onto \\(C(K)\\), and let \\(\\mu\\) be the Radon measure on \\(K\\) with \\(\\int\\hat x\\,d\\mu=\\langle x\\xi,\\xi\\rangle\\). Then:\n\n1. there is a unitary \\(U:L^2(K,\\mu)\\to H\\) with \\(U1=\\xi\\) and \\(UM_{\\hat x}U^*=x\\) for \\(x\\in M\\);\n2. \\(M\\) is maximal abelian, \\(M'=M\\);\n3. every bounded Borel function on \\(K\\) agrees \\(\\mu\\)-almost everywhere with a continuous function.\n\n**Proof.** Since \\(M\\) is a unital abelian C\\*-algebra, \\(K\\) is compact and \\(M\\cong C(K)\\). The map \\(\\pi(\\hat x)=x\\) is a representation of \\(C(K)\\) on \\(H\\) with cyclic vector \\(\\xi\\), and \\(\\langle\\pi(\\hat x)\\xi,\\xi\\rangle=\\int\\hat x\\,d\\mu\\). Theorem 3.1(2) gives (1). By Theorem 3.1(3), \\(M'=U\\pi_\\mu(C(K))'U^*\\) is abelian, so \\(M'\\subseteq M''=M\\). Since \\(M\\) is abelian, \\(M\\subseteq M'\\). Hence \\(M=M'\\). Finally \\[\n\\begin{gathered}\nU\\pi_\\mu(C(K))U^*\\\\\n=M\\\\\n=M'\\\\\n=U\\{M_f:f\\in L^\\infty\\}U^*,\n\\end{gathered}\n\\] so every \\(M_f\\) is some \\(M_{\\hat x}\\), which is (3). \\(\\square\\)\n\nPart (3) is a first sign of the special topology of \\(K\\): on \\([0,1]\\) with Lebesgue measure, the indicator of \\([0,\\frac12]\\) agrees almost everywhere with no continuous function. Section 5 explains (3) through normal measures.\n\n**Corollary 3.4** (Equivalent measures give equivalent representations). Let \\(\\Omega\\) be locally compact, and let \\(\\mu\\) and \\(\\nu\\) be finite positive Radon measures on it. The representations \\(\\pi_\\mu\\) and \\(\\pi_\\nu\\) are unitarily equivalent if and only if \\(\\mu\\) and \\(\\nu\\) have the same null sets.\n\n**Proof.** Suppose they have the same null sets. By the Radon–Nikodym theorem (see Background) \\(\\nu=h\\mu\\) with \\(h\\ge0\\) integrable. The set \\(\\{h=0\\}\\) is \\(\\nu\\)-null, hence \\(\\mu\\)-null. The map \\(V\\xi=h^{1/2}\\xi\\) from \\(L^2(\\Omega,\\nu)\\) to \\(L^2(\\Omega,\\mu)\\) is isometric, since \\(\\int|\\xi|^2d\\nu=\\int|\\xi|^2h\\,d\\mu\\). It is onto, since \\(\\eta=V(h^{-1/2}\\eta)\\) for \\(\\eta\\in L^2(\\Omega,\\mu)\\). It commutes with every multiplication operator, so \\(V\\pi_\\nu(x)=\\pi_\\mu(x)V\\).\n\nConversely, let \\(U:L^2(\\Omega,\\mu)\\to L^2(\\Omega,\\nu)\\) be unitary with \\(U\\pi_\\mu(x)=\\pi_\\nu(x)U\\), and put \\(g=U^*1\\in L^2(\\Omega,\\mu)\\). For \\(x\\in C_0(\\Omega)\\),\n\\[\n\\begin{gathered}\n\\int x\\,d\\nu\\\\\n=\\langle\\pi_\\nu(x)1,1\\rangle\\\\\n=\\langle U\\pi_\\mu(x)U^*1,1\\rangle\\\\\n=\\langle\\pi_\\mu(x)g,g\\rangle\\\\\n=\\int x|g|^2d\\mu .\n\\end{gathered}\n\\]\nThe finite measure \\(\\lambda=|g|^2\\mu\\) is again a Radon measure. Indeed, for \\(\\varepsilon>0\\) there is \\(\\delta>0\\) with \\(\\lambda(E)<\\varepsilon\\) whenever \\(\\mu(E)<\\delta\\), because \\(|g|^2\\) is integrable. So the outer regularity and the inner regularity of \\(\\mu\\) pass to \\(\\lambda\\). By the uniqueness in the Riesz representation theorem, \\(\\nu=\\lambda\\), so every \\(\\mu\\)-null set is \\(\\nu\\)-null. By symmetry the converse holds. \\(\\square\\)\n\n**Example 3.5.** On \\([0,1]\\) let \\(m\\) be Lebesgue measure and \\(\\delta_{1/2}\\) the point mass at \\(\\frac12\\). The measures \\(m+\\delta_{1/2}\\) and \\(2m+3\\delta_{1/2}\\) have the same null sets, so their multiplication representations are equivalent. The measures \\(m\\) and \\(m+\\delta_{1/2}\\) do not, and indeed \\(\\pi_{m+\\delta_{1/2}}(C[0,1])''\\) contains a rank-one projection, multiplication by \\(1_{\\{1/2\\}}\\), while \\(\\pi_m(C[0,1])''=L^\\infty[0,1]\\) contains none.\n\n",
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      "unit": "abelian-operator-algebras",
      "name": "4. Stonean spaces",
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      "full_conditions_and_proof": "### 4. Stonean spaces\n\nAbelian von Neumann algebras have many projections, and the projections of \\(C(\\Omega)\\) are the indicators of clopen sets. So the spectrum of such an algebra has many clopen sets. This section studies the compact spaces that have enough of them to make \\(C_{\\mathbb R}(\\Omega)\\) complete as an ordered set. A reference for this section and the next two is [Kostecki, Sections 5.6 and 5.8].\n\n**Definition 4.1.** A Hausdorff space \\(X\\) is called *extremally disconnected* when each open set \\(U\\subseteq X\\) has an open closure \\(\\overline U\\). A compact extremally disconnected space is called *stonean*.\n\n**Lemma 4.2.** Let \\(X\\) be a Hausdorff space.\n\n1. \\(X\\) is extremally disconnected if and only if any two disjoint open subsets have disjoint closures. In that case the interior of every closed set is clopen.\n2. An open subspace of an extremally disconnected space is extremally disconnected. A clopen subset of a stonean space is stonean.\n3. In a stonean space the clopen sets form a base of the topology. So every open set is the union of the clopen sets it contains, and these form an increasing net under inclusion.\n\n**Proof.** (1) Let \\(X\\) be extremally disconnected and let \\(U,V\\) be disjoint open sets. Then \\(V\\cap\\overline U=\\varnothing\\), because \\(V\\) is open. Since \\(\\overline U\\) is open, \\(X\\setminus\\overline U\\) is closed, and it contains \\(V\\), hence \\(\\overline V\\). Conversely, assume the condition, and let \\(U\\) be open. Then \\(V=X\\setminus\\overline U\\) is open and disjoint from \\(U\\), so \\(\\overline U\\cap\\overline V=\\varnothing\\). Since \\(\\overline U\\cup V=X\\), this gives \\(\\overline U=X\\setminus\\overline V\\), which is open. For a closed set \\(F\\), \\(F^\\circ=X\\setminus\\overline{X\\setminus F}\\), the complement of the closure of an open set; so \\(F^\\circ\\) is clopen.\n\n(2) If \\(W\\) is open in an open subspace \\(U\\), then \\(W\\) is open in \\(X\\), and its closure in \\(U\\) is \\(\\overline W\\cap U\\), which is open. A clopen subset of a compact space is compact.\n\n(3) Let \\(x\\in U\\) with \\(U\\) open. A compact space is regular, so there is an open \\(V\\) with \\(x\\in V\\subseteq\\overline V\\subseteq U\\), and \\(\\overline V\\) is clopen. Finite unions of clopen sets are clopen. \\(\\square\\)\n\n**Theorem 4.3** (Order completeness and stonean spaces). For a compact space \\(\\Omega\\) the following are equivalent.\n\n1. \\(\\Omega\\) is stonean.\n2. Every nonempty subset of \\(C_{\\mathbb R}(\\Omega)\\) that is bounded above has a supremum in \\(C_{\\mathbb R}(\\Omega)\\), that is, a least upper bound for the pointwise order.\n3. Every bounded lsc function on \\(\\Omega\\) agrees with a continuous function outside a meager set.\n\nWhen they hold, the following is true as well. For every bounded lsc function \\(g\\), the continuous function in (3) is unique, it is the upper regularization \\(g^*\\), and \\(g^*\\ge g\\). For every nonempty set \\(S\\subseteq C_{\\mathbb R}(\\Omega)\\) that is bounded above, the supremum of \\(S\\) in \\(C_{\\mathbb R}(\\Omega)\\) is \\(g^*\\), where \\(g=\\sup_{s\\in S}s\\) is the pointwise supremum; so it agrees with the pointwise supremum outside a meager set.\n\n\n**Proof.** *Uniqueness in (3).* Two continuous functions that agree outside a meager set agree on a dense set, by Lemma 2.1(2), hence everywhere.\n\n*Identification of the continuous function.* Let \\(g\\) be bounded and lsc, and let \\(f\\) be continuous with \\(f=g\\) outside a meager set \\(M\\). The function \\(g-f\\) is lsc, so \\(\\{g>f\\}\\) is open. It lies in \\(M\\), so it is empty by Lemma 2.1(2), and \\(f\\ge g\\). Since \\(f\\) is usc, Lemma 2.2(1) gives \\(f\\ge g^*\\). Conversely, fix \\(\\omega\\) and \\(\\varepsilon>0\\), and let \\(U\\) be an open neighbourhood of \\(\\omega\\) on which \\(f>f(\\omega)-\\varepsilon\\). The set \\(U\\setminus M\\) is not empty, again by Lemma 2.1(2), and \\(g=f\\) there. So \\(\\sup_Ug\\ge f(\\omega)-\\varepsilon\\), and the same holds for every smaller neighbourhood. Hence \\(g^*(\\omega)\\ge f(\\omega)-\\varepsilon\\). So \\(f=g^*\\).\n\n(1)\\(\\Rightarrow\\)(3). Let \\(g\\) be bounded and lsc. After an affine change we may assume \\(0\\le g\\le1\\). For real \\(\\lambda\\) let \\(F(\\lambda)=\\{g\\le\\lambda\\}\\), a closed set, and \\(C(\\lambda)=F(\\lambda)^\\circ\\), which is clopen by Lemma 4.2(1). Both increase with \\(\\lambda\\), \\(F(1)=C(1)=\\Omega\\), and \\(C(\\lambda)\\subseteq F(\\lambda)\\). For \\(n\\ge1\\) define\n\\[\nf_n=\\sum_{k=1}^{2^n}\\frac k{2^n}\\big(1_{C(k2^{-n})}-1_{C((k-1)2^{-n})}\\big).\n\\]\nIt is continuous, because the sets \\(C(\\lambda)\\) are clopen. The sets \\(C(0)\\) and \\(C(k2^{-n})\\setminus C((k-1)2^{-n})\\), \\(1\\le k\\le2^n\\), partition \\(\\Omega\\); \\(f_n\\) is \\(0\\) on the first and \\(k2^{-n}\\) on the others. A point of \\(C(k2^{-n})\\setminus C((k-1)2^{-n})\\) lies in exactly one of the two sets \\(C((2k-1)2^{-n-1})\\setminus C((2k-2)2^{-n-1})\\) and \\(C(2k\\,2^{-n-1})\\setminus C((2k-1)2^{-n-1})\\), so \\(f_{n+1}\\) takes the value \\(k2^{-n}-2^{-n-1}\\) or \\(k2^{-n}\\) there. Hence \\(\\|f_n-f_{n+1}\\|\\le2^{-n-1}\\), and the sequence \\((f_n)\\) is uniformly Cauchy; its uniform limit \\(f\\) is continuous.\n\nLet \\(M\\) be the union of the sets \\(F(\\lambda)\\setminus C(\\lambda)\\) over the dyadic rationals \\(\\lambda\\in[0,1]\\). Each of them is rare by Lemma 2.1(1), so \\(M\\) is meager. Let \\(\\omega\\notin M\\). If \\(\\omega\\in C(0)\\), then \\(g(\\omega)=0=f_n(\\omega)\\). Otherwise \\(\\omega\\in C(k2^{-n})\\setminus C((k-1)2^{-n})\\) for some \\(k\\ge1\\). Then \\(g(\\omega)\\le k2^{-n}\\), and \\(\\omega\\notin F((k-1)2^{-n})\\), because on the dyadic levels \\(F\\) and \\(C\\) agree outside \\(M\\); so \\(g(\\omega)>(k-1)2^{-n}\\). Hence \\(|g(\\omega)-f_n(\\omega)|\\le2^{-n}\\) for every \\(n\\), and \\(g(\\omega)=f(\\omega)\\). So \\(g=f\\) outside the meager set \\(M\\).\n\n(3)\\(\\Rightarrow\\)(2). Let \\(S\\) be nonempty and bounded above by \\(b\\), fix \\(s_0\\in S\\), and let \\(g=\\sup_{s\\in S}s\\). Then \\(g\\) is lsc and \\(s_0\\le g\\le b\\). Let \\(f\\) be the continuous function given by (3). As shown above, \\(f\\ge g\\), so \\(f\\) is an upper bound of \\(S\\). If \\(h\\in C_{\\mathbb R}(\\Omega)\\) is an upper bound of \\(S\\), then \\(h\\ge g=f\\) outside a meager set, that is, on a dense set, so \\(h\\ge f\\). Hence \\(f\\) is the least upper bound, and \\(f=g^*\\).\n\n(2)\\(\\Rightarrow\\)(1). Let \\(G\\) be open, and let \\(S\\) be the set of \\(x\\in C_{\\mathbb R}(\\Omega)\\) with \\(0\\le x\\le1_G\\). It is bounded above by \\(1\\); let \\(f\\) be its least upper bound. By Urysohn's lemma, for every \\(\\omega\\in G\\) some \\(x\\in S\\) has \\(x(\\omega)=1\\). Hence \\(0\\le f\\le1\\), \\(f=1\\) on \\(G\\), and \\(f=1\\) on \\(\\overline G\\) by continuity. Suppose \\(f(\\omega_0)>0\\) for some \\(\\omega_0\\notin\\overline G\\). By Urysohn's lemma there is \\(h\\in C_{\\mathbb R}(\\Omega)\\) with \\(0\\le h\\le1\\), \\(h=1\\) on \\(\\overline G\\) and \\(h(\\omega_0)=0\\). Then \\(fh\\) is an upper bound of \\(S\\): on \\(\\overline G\\) it equals \\(f\\), and outside \\(G\\) every \\(x\\in S\\) vanishes. But \\(fh\\le f\\) and \\(fh(\\omega_0)=0<f(\\omega_0)\\), which contradicts the choice of \\(f\\). So \\(f=1_{\\overline G}\\). Since \\(f\\) is continuous, \\(\\overline G\\) is open. \\(\\square\\)\n\n**Theorem 4.4** (Extending continuous functions). Let \\(\\Omega\\) be a stonean space and \\(D\\subseteq\\Omega\\) a subset that is dense or open. Every bounded continuous function on \\(D\\) extends to a continuous function on \\(\\Omega\\). If \\(D\\) is dense, the extension is unique, and \\(x\\mapsto x|_D\\) is an isometric \\(*\\)-isomorphism of \\(C(\\Omega)\\) onto \\(C_b(D)\\). So \\(\\Omega\\) is the Stone–Čech compactification of every dense subset.\n\n**Proof.** First let \\(D\\) be dense and \\(f:D\\to[0,1]\\) continuous; the general case follows by taking real and imaginary parts and an affine change. For a rational \\(r\\) let \\(O_r\\) be the union of all open \\(V\\subseteq\\Omega\\) with \\(V\\cap D\\subseteq\\{f<r\\}\\), and let \\(C_r=\\overline{O_r}\\), a clopen set.\n\n*Step 1.* \\(O_r\\cap D=\\{d\\in D:f(d)<r\\}\\). One inclusion is the definition. If \\(f(d)<r\\), continuity of \\(f\\) at \\(d\\) gives an open \\(V\\ni d\\) with \\(f<r\\) on \\(V\\cap D\\), so \\(d\\in O_r\\).\n\n*Step 2.* If \\(r<s\\), then \\(O_r\\subseteq O_s\\) and \\(C_r\\subseteq C_s\\). If \\(r\\le0\\), then \\(O_r\\cap D=\\varnothing\\), so the open set \\(O_r\\) is empty because \\(D\\) is dense, and \\(C_r=\\varnothing\\). If \\(r>1\\), then \\(O_r=\\Omega\\).\n\n*Step 3.* Define \\(F(\\omega)=\\inf\\{r\\in\\mathbb Q:\\omega\\in C_r\\}\\). By Step 2, \\(0\\le F\\le1\\). For real \\(t\\),\n\\[\n\\{F<t\\}=\\bigcup_{r<t}C_r,\\qquad\\{F>t\\}=\\bigcup_{s>t}(\\Omega\\setminus C_s),\n\\]\nwith \\(r,s\\) rational. The first identity is the definition of an infimum. For the second, if \\(F(\\omega)>t\\), choose a rational \\(s\\) with \\(t<s<F(\\omega)\\); then \\(\\omega\\notin C_s\\). If \\(\\omega\\notin C_s\\) with \\(s>t\\), then \\(\\omega\\notin C_r\\) for every \\(r\\le s\\), and \\(F(\\omega)\\ge s>t\\). Both unions consist of clopen sets, so \\(F\\) is continuous.\n\n*Step 4.* \\(F=f\\) on \\(D\\). If \\(d\\in D\\) and \\(f(d)<r\\), then \\(d\\in O_r\\subseteq C_r\\) by Step 1, so \\(F(d)\\le f(d)\\). Let \\(r<f(d)\\), choose \\(s\\) with \\(r<s<f(d)\\), and an open \\(V\\ni d\\) with \\(f>s\\) on \\(V\\cap D\\). If \\(d\\in C_r=\\overline{O_r}\\), then \\(V\\cap O_r\\) is a nonempty open set, and it contains a point \\(d'\\in D\\). But then \\(f(d')<r\\) by Step 1 and \\(f(d')>s>r\\), which is impossible. So \\(d\\notin C_r\\) whenever \\(r<f(d)\\), and \\(F(d)\\ge f(d)\\).\n\nNow let \\(D\\) be open. Its closure \\(\\overline D\\) is clopen, hence stonean, and \\(D\\) is dense in it. Extend \\(f\\) to \\(\\overline D\\) as above and by \\(0\\) on \\(\\Omega\\setminus\\overline D\\); the result is continuous because \\(\\overline D\\) is clopen.\n\nFor dense \\(D\\), two continuous extensions agree on \\(D\\), hence everywhere. Restriction to \\(D\\) is an injective \\(*\\)-homomorphism of \\(C(\\Omega)\\) into \\(C_b(D)\\). It preserves the supremum norm, since \\(D\\) is dense, and it is onto by what we proved. The spectrum of \\(C_b(D)\\) is the Stone–Čech compactification of \\(D\\), with \\(D\\) embedded by point evaluations; [Exercise 2.4 of the C\\*-algebra lesson](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04) proves the construction and its universal compact-target extension property. A subspace of a compact Hausdorff space is completely regular: an open neighbourhood in the subspace is the intersection with an ambient open set, and an ambient Urysohn function separating the point from the complement restricts to the required bounded function. Thus this provider applies to \\(D\\). Under the isomorphism the evaluation at \\(d\\in D\\) corresponds to the evaluation at \\(d\\), so \\(\\Omega\\) is \\(\\beta D\\). \\(\\square\\)\n\n**Example 4.5.** (a) For every set \\(I\\) the Stone–Čech compactification \\(\\beta I\\) of the discrete space \\(I\\) is stonean. Indeed \\(C(\\beta I)\\cong\\ell^\\infty(I)\\), and every nonempty set of real functions on \\(I\\) that is bounded above has a pointwise supremum, which is its least upper bound in \\(\\ell^\\infty(I)\\); Theorem 4.3 applies.\n\n(b) \\([0,1]\\) is not stonean: the closure of \\([0,\\frac12)\\) is not open. In fact no infinite compact metrizable space is stonean (Exercise 12.1), and so the Cantor set, although it has a base of clopen sets, is not stonean.\n\n(c) If \\(\\Omega\\) is stonean and \\(\\omega\\) is not an isolated point, then \\(\\Omega\\setminus\\{\\omega\\}\\) is open and dense, and by Theorem 4.4 it has \\(\\Omega\\) as its Stone–Čech compactification.\n\n",
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      "unit": "abelian-operator-algebras",
      "name": "4. Stonean spaces",
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      "full_conditions_and_proof": "### 4. Stonean spaces\n\nAbelian von Neumann algebras have many projections, and the projections of \\(C(\\Omega)\\) are the indicators of clopen sets. So the spectrum of such an algebra has many clopen sets. This section studies the compact spaces that have enough of them to make \\(C_{\\mathbb R}(\\Omega)\\) complete as an ordered set. A reference for this section and the next two is [Kostecki, Sections 5.6 and 5.8].\n\n**Definition 4.1.** A Hausdorff space \\(X\\) is called *extremally disconnected* when each open set \\(U\\subseteq X\\) has an open closure \\(\\overline U\\). A compact extremally disconnected space is called *stonean*.\n\n**Lemma 4.2.** Let \\(X\\) be a Hausdorff space.\n\n1. \\(X\\) is extremally disconnected if and only if any two disjoint open subsets have disjoint closures. In that case the interior of every closed set is clopen.\n2. An open subspace of an extremally disconnected space is extremally disconnected. A clopen subset of a stonean space is stonean.\n3. In a stonean space the clopen sets form a base of the topology. So every open set is the union of the clopen sets it contains, and these form an increasing net under inclusion.\n\n**Proof.** (1) Let \\(X\\) be extremally disconnected and let \\(U,V\\) be disjoint open sets. Then \\(V\\cap\\overline U=\\varnothing\\), because \\(V\\) is open. Since \\(\\overline U\\) is open, \\(X\\setminus\\overline U\\) is closed, and it contains \\(V\\), hence \\(\\overline V\\). Conversely, assume the condition, and let \\(U\\) be open. Then \\(V=X\\setminus\\overline U\\) is open and disjoint from \\(U\\), so \\(\\overline U\\cap\\overline V=\\varnothing\\). Since \\(\\overline U\\cup V=X\\), this gives \\(\\overline U=X\\setminus\\overline V\\), which is open. For a closed set \\(F\\), \\(F^\\circ=X\\setminus\\overline{X\\setminus F}\\), the complement of the closure of an open set; so \\(F^\\circ\\) is clopen.\n\n(2) If \\(W\\) is open in an open subspace \\(U\\), then \\(W\\) is open in \\(X\\), and its closure in \\(U\\) is \\(\\overline W\\cap U\\), which is open. A clopen subset of a compact space is compact.\n\n(3) Let \\(x\\in U\\) with \\(U\\) open. A compact space is regular, so there is an open \\(V\\) with \\(x\\in V\\subseteq\\overline V\\subseteq U\\), and \\(\\overline V\\) is clopen. Finite unions of clopen sets are clopen. \\(\\square\\)\n\n**Theorem 4.3** (Order completeness and stonean spaces). For a compact space \\(\\Omega\\) the following are equivalent.\n\n1. \\(\\Omega\\) is stonean.\n2. Every nonempty subset of \\(C_{\\mathbb R}(\\Omega)\\) that is bounded above has a supremum in \\(C_{\\mathbb R}(\\Omega)\\), that is, a least upper bound for the pointwise order.\n3. Every bounded lsc function on \\(\\Omega\\) agrees with a continuous function outside a meager set.\n\nWhen they hold, the following is true as well. For every bounded lsc function \\(g\\), the continuous function in (3) is unique, it is the upper regularization \\(g^*\\), and \\(g^*\\ge g\\). For every nonempty set \\(S\\subseteq C_{\\mathbb R}(\\Omega)\\) that is bounded above, the supremum of \\(S\\) in \\(C_{\\mathbb R}(\\Omega)\\) is \\(g^*\\), where \\(g=\\sup_{s\\in S}s\\) is the pointwise supremum; so it agrees with the pointwise supremum outside a meager set.\n\n\n**Proof.** *Uniqueness in (3).* Two continuous functions that agree outside a meager set agree on a dense set, by Lemma 2.1(2), hence everywhere.\n\n*Identification of the continuous function.* Let \\(g\\) be bounded and lsc, and let \\(f\\) be continuous with \\(f=g\\) outside a meager set \\(M\\). The function \\(g-f\\) is lsc, so \\(\\{g>f\\}\\) is open. It lies in \\(M\\), so it is empty by Lemma 2.1(2), and \\(f\\ge g\\). Since \\(f\\) is usc, Lemma 2.2(1) gives \\(f\\ge g^*\\). Conversely, fix \\(\\omega\\) and \\(\\varepsilon>0\\), and let \\(U\\) be an open neighbourhood of \\(\\omega\\) on which \\(f>f(\\omega)-\\varepsilon\\). The set \\(U\\setminus M\\) is not empty, again by Lemma 2.1(2), and \\(g=f\\) there. So \\(\\sup_Ug\\ge f(\\omega)-\\varepsilon\\), and the same holds for every smaller neighbourhood. Hence \\(g^*(\\omega)\\ge f(\\omega)-\\varepsilon\\). So \\(f=g^*\\).\n\n(1)\\(\\Rightarrow\\)(3). Let \\(g\\) be bounded and lsc. After an affine change we may assume \\(0\\le g\\le1\\). For real \\(\\lambda\\) let \\(F(\\lambda)=\\{g\\le\\lambda\\}\\), a closed set, and \\(C(\\lambda)=F(\\lambda)^\\circ\\), which is clopen by Lemma 4.2(1). Both increase with \\(\\lambda\\), \\(F(1)=C(1)=\\Omega\\), and \\(C(\\lambda)\\subseteq F(\\lambda)\\). For \\(n\\ge1\\) define\n\\[\nf_n=\\sum_{k=1}^{2^n}\\frac k{2^n}\\big(1_{C(k2^{-n})}-1_{C((k-1)2^{-n})}\\big).\n\\]\nIt is continuous, because the sets \\(C(\\lambda)\\) are clopen. The sets \\(C(0)\\) and \\(C(k2^{-n})\\setminus C((k-1)2^{-n})\\), \\(1\\le k\\le2^n\\), partition \\(\\Omega\\); \\(f_n\\) is \\(0\\) on the first and \\(k2^{-n}\\) on the others. A point of \\(C(k2^{-n})\\setminus C((k-1)2^{-n})\\) lies in exactly one of the two sets \\(C((2k-1)2^{-n-1})\\setminus C((2k-2)2^{-n-1})\\) and \\(C(2k\\,2^{-n-1})\\setminus C((2k-1)2^{-n-1})\\), so \\(f_{n+1}\\) takes the value \\(k2^{-n}-2^{-n-1}\\) or \\(k2^{-n}\\) there. Hence \\(\\|f_n-f_{n+1}\\|\\le2^{-n-1}\\), and the sequence \\((f_n)\\) is uniformly Cauchy; its uniform limit \\(f\\) is continuous.\n\nLet \\(M\\) be the union of the sets \\(F(\\lambda)\\setminus C(\\lambda)\\) over the dyadic rationals \\(\\lambda\\in[0,1]\\). Each of them is rare by Lemma 2.1(1), so \\(M\\) is meager. Let \\(\\omega\\notin M\\). If \\(\\omega\\in C(0)\\), then \\(g(\\omega)=0=f_n(\\omega)\\). Otherwise \\(\\omega\\in C(k2^{-n})\\setminus C((k-1)2^{-n})\\) for some \\(k\\ge1\\). Then \\(g(\\omega)\\le k2^{-n}\\), and \\(\\omega\\notin F((k-1)2^{-n})\\), because on the dyadic levels \\(F\\) and \\(C\\) agree outside \\(M\\); so \\(g(\\omega)>(k-1)2^{-n}\\). Hence \\(|g(\\omega)-f_n(\\omega)|\\le2^{-n}\\) for every \\(n\\), and \\(g(\\omega)=f(\\omega)\\). So \\(g=f\\) outside the meager set \\(M\\).\n\n(3)\\(\\Rightarrow\\)(2). Let \\(S\\) be nonempty and bounded above by \\(b\\), fix \\(s_0\\in S\\), and let \\(g=\\sup_{s\\in S}s\\). Then \\(g\\) is lsc and \\(s_0\\le g\\le b\\). Let \\(f\\) be the continuous function given by (3). As shown above, \\(f\\ge g\\), so \\(f\\) is an upper bound of \\(S\\). If \\(h\\in C_{\\mathbb R}(\\Omega)\\) is an upper bound of \\(S\\), then \\(h\\ge g=f\\) outside a meager set, that is, on a dense set, so \\(h\\ge f\\). Hence \\(f\\) is the least upper bound, and \\(f=g^*\\).\n\n(2)\\(\\Rightarrow\\)(1). Let \\(G\\) be open, and let \\(S\\) be the set of \\(x\\in C_{\\mathbb R}(\\Omega)\\) with \\(0\\le x\\le1_G\\). It is bounded above by \\(1\\); let \\(f\\) be its least upper bound. By Urysohn's lemma, for every \\(\\omega\\in G\\) some \\(x\\in S\\) has \\(x(\\omega)=1\\). Hence \\(0\\le f\\le1\\), \\(f=1\\) on \\(G\\), and \\(f=1\\) on \\(\\overline G\\) by continuity. Suppose \\(f(\\omega_0)>0\\) for some \\(\\omega_0\\notin\\overline G\\). By Urysohn's lemma there is \\(h\\in C_{\\mathbb R}(\\Omega)\\) with \\(0\\le h\\le1\\), \\(h=1\\) on \\(\\overline G\\) and \\(h(\\omega_0)=0\\). Then \\(fh\\) is an upper bound of \\(S\\): on \\(\\overline G\\) it equals \\(f\\), and outside \\(G\\) every \\(x\\in S\\) vanishes. But \\(fh\\le f\\) and \\(fh(\\omega_0)=0<f(\\omega_0)\\), which contradicts the choice of \\(f\\). So \\(f=1_{\\overline G}\\). Since \\(f\\) is continuous, \\(\\overline G\\) is open. \\(\\square\\)\n\n**Theorem 4.4** (Extending continuous functions). Let \\(\\Omega\\) be a stonean space and \\(D\\subseteq\\Omega\\) a subset that is dense or open. Every bounded continuous function on \\(D\\) extends to a continuous function on \\(\\Omega\\). If \\(D\\) is dense, the extension is unique, and \\(x\\mapsto x|_D\\) is an isometric \\(*\\)-isomorphism of \\(C(\\Omega)\\) onto \\(C_b(D)\\). So \\(\\Omega\\) is the Stone–Čech compactification of every dense subset.\n\n**Proof.** First let \\(D\\) be dense and \\(f:D\\to[0,1]\\) continuous; the general case follows by taking real and imaginary parts and an affine change. For a rational \\(r\\) let \\(O_r\\) be the union of all open \\(V\\subseteq\\Omega\\) with \\(V\\cap D\\subseteq\\{f<r\\}\\), and let \\(C_r=\\overline{O_r}\\), a clopen set.\n\n*Step 1.* \\(O_r\\cap D=\\{d\\in D:f(d)<r\\}\\). One inclusion is the definition. If \\(f(d)<r\\), continuity of \\(f\\) at \\(d\\) gives an open \\(V\\ni d\\) with \\(f<r\\) on \\(V\\cap D\\), so \\(d\\in O_r\\).\n\n*Step 2.* If \\(r<s\\), then \\(O_r\\subseteq O_s\\) and \\(C_r\\subseteq C_s\\). If \\(r\\le0\\), then \\(O_r\\cap D=\\varnothing\\), so the open set \\(O_r\\) is empty because \\(D\\) is dense, and \\(C_r=\\varnothing\\). If \\(r>1\\), then \\(O_r=\\Omega\\).\n\n*Step 3.* Define \\(F(\\omega)=\\inf\\{r\\in\\mathbb Q:\\omega\\in C_r\\}\\). By Step 2, \\(0\\le F\\le1\\). For real \\(t\\),\n\\[\n\\{F<t\\}=\\bigcup_{r<t}C_r,\\qquad\\{F>t\\}=\\bigcup_{s>t}(\\Omega\\setminus C_s),\n\\]\nwith \\(r,s\\) rational. The first identity is the definition of an infimum. For the second, if \\(F(\\omega)>t\\), choose a rational \\(s\\) with \\(t<s<F(\\omega)\\); then \\(\\omega\\notin C_s\\). If \\(\\omega\\notin C_s\\) with \\(s>t\\), then \\(\\omega\\notin C_r\\) for every \\(r\\le s\\), and \\(F(\\omega)\\ge s>t\\). Both unions consist of clopen sets, so \\(F\\) is continuous.\n\n*Step 4.* \\(F=f\\) on \\(D\\). If \\(d\\in D\\) and \\(f(d)<r\\), then \\(d\\in O_r\\subseteq C_r\\) by Step 1, so \\(F(d)\\le f(d)\\). Let \\(r<f(d)\\), choose \\(s\\) with \\(r<s<f(d)\\), and an open \\(V\\ni d\\) with \\(f>s\\) on \\(V\\cap D\\). If \\(d\\in C_r=\\overline{O_r}\\), then \\(V\\cap O_r\\) is a nonempty open set, and it contains a point \\(d'\\in D\\). But then \\(f(d')<r\\) by Step 1 and \\(f(d')>s>r\\), which is impossible. So \\(d\\notin C_r\\) whenever \\(r<f(d)\\), and \\(F(d)\\ge f(d)\\).\n\nNow let \\(D\\) be open. Its closure \\(\\overline D\\) is clopen, hence stonean, and \\(D\\) is dense in it. Extend \\(f\\) to \\(\\overline D\\) as above and by \\(0\\) on \\(\\Omega\\setminus\\overline D\\); the result is continuous because \\(\\overline D\\) is clopen.\n\nFor dense \\(D\\), two continuous extensions agree on \\(D\\), hence everywhere. Restriction to \\(D\\) is an injective \\(*\\)-homomorphism of \\(C(\\Omega)\\) into \\(C_b(D)\\). It preserves the supremum norm, since \\(D\\) is dense, and it is onto by what we proved. The spectrum of \\(C_b(D)\\) is the Stone–Čech compactification of \\(D\\), with \\(D\\) embedded by point evaluations; [Exercise 2.4 of the C\\*-algebra lesson](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04) proves the construction and its universal compact-target extension property. A subspace of a compact Hausdorff space is completely regular: an open neighbourhood in the subspace is the intersection with an ambient open set, and an ambient Urysohn function separating the point from the complement restricts to the required bounded function. Thus this provider applies to \\(D\\). Under the isomorphism the evaluation at \\(d\\in D\\) corresponds to the evaluation at \\(d\\), so \\(\\Omega\\) is \\(\\beta D\\). \\(\\square\\)\n\n**Example 4.5.** (a) For every set \\(I\\) the Stone–Čech compactification \\(\\beta I\\) of the discrete space \\(I\\) is stonean. Indeed \\(C(\\beta I)\\cong\\ell^\\infty(I)\\), and every nonempty set of real functions on \\(I\\) that is bounded above has a pointwise supremum, which is its least upper bound in \\(\\ell^\\infty(I)\\); Theorem 4.3 applies.\n\n(b) \\([0,1]\\) is not stonean: the closure of \\([0,\\frac12)\\) is not open. In fact no infinite compact metrizable space is stonean (Exercise 12.1), and so the Cantor set, although it has a base of clopen sets, is not stonean.\n\n(c) If \\(\\Omega\\) is stonean and \\(\\omega\\) is not an isolated point, then \\(\\Omega\\setminus\\{\\omega\\}\\) is open and dense, and by Theorem 4.4 it has \\(\\Omega\\) as its Stone–Čech compactification.\n\n",
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      "name": "5. Normal measures",
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      "full_conditions_and_proof": "### 5. Normal measures\n\n**Definition 5.1.** Let \\(\\Omega\\) be stonean. A Radon measure \\(\\mu\\) on \\(\\Omega\\) is *normal* if \\(\\mu(f)=\\sup_i\\mu(f_i)\\) for every increasing net \\((f_i)\\) in \\(C_{\\mathbb R}(\\Omega)\\) with \\(\\sup_i\\|f_i\\|<\\infty\\), where \\(f\\) is its least upper bound in \\(C_{\\mathbb R}(\\Omega)\\). A complex Radon measure is called *normal* when it is a finite sum of complex multiples of positive normal measures.\n\nThe least upper bound \\(f\\) exists by Theorem 4.3. Normality asks that the integral see \\(f\\), and not only the pointwise supremum, which may be smaller on a meager set.\n\n**Theorem 5.2** (Normal measures kill rare sets). Let \\(\\Omega\\) be stonean. For a Radon measure \\(\\mu\\) on \\(\\Omega\\) the following are equivalent.\n\n1. \\(\\mu\\) is normal.\n2. Every rare set is \\(\\mu\\)-null.\n3. Every meager set is \\(\\mu\\)-null.\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). A rare set lies in its closure, which is a closed rare set, so let \\(R\\) be closed and rare. The open set \\(\\Omega\\setminus R\\) is dense. By Lemma 4.2(3) it is the union of the increasing net \\((C_i)\\) of clopen sets contained in it. The net \\((1_{C_i})\\) in \\(C_{\\mathbb R}(\\Omega)\\) has least upper bound \\(1\\): an upper bound is at least \\(1\\) on the dense set \\(\\Omega\\setminus R\\), hence everywhere. By normality and Lemma 2.3(1), \\(\\mu(\\Omega)=\\sup_i\\mu(C_i)=\\mu(\\Omega\\setminus R)\\). So \\(\\mu(R)=0\\).\n\n(2)\\(\\Rightarrow\\)(3). A meager set lies in a countable union of closed rare sets.\n\n(3)\\(\\Rightarrow\\)(1). Let \\((f_i)\\) be as in Definition 5.1 with least upper bound \\(f\\), and let \\(g=\\sup_if_i\\) pointwise. By Theorem 4.3, \\(f=g\\) outside a meager set, which is null. By Lemma 2.3(2), \\(\\mu(f)=\\int g\\,d\\mu=\\sup_i\\mu(f_i)\\). \\(\\square\\)\n\n**Lemma 5.3** (Open kernels of measurable sets). Let \\(\\Omega\\) be stonean, \\(\\mu\\) a normal measure on \\(\\Omega\\), and \\(E\\) a \\(\\mu\\)-measurable set. There is an open set \\(G\\subseteq E\\) with \\(\\mu(E\\setminus G)=0\\).\n\n**Proof.** By inner regularity there are compact sets \\(K_n\\subseteq E\\) with \\(\\mu(E\\setminus K_n)<1/n\\). The interior \\(K_n^\\circ\\) is clopen (Lemma 4.2(1)), and \\(K_n\\setminus K_n^\\circ\\) is rare (Lemma 2.1(1)), hence null (Theorem 5.2). The open set \\(G=\\bigcup_nK_n^\\circ\\) lies in \\(E\\), and \\(\\mu(E\\setminus G)\\le\\mu(E\\setminus K_n)+\\mu(K_n\\setminus K_n^\\circ)<1/n\\) for every \\(n\\). \\(\\square\\)\n\n**Corollary 5.4.** Let \\(\\Omega\\) be stonean and \\(\\mu\\) a normal measure on \\(\\Omega\\).\n\n1. The closure of a \\(\\mu\\)-null set is \\(\\mu\\)-null.\n2. If \\(E\\) is \\(\\mu\\)-measurable, the sets \\(E\\), \\(\\overline E\\), \\(E^\\circ\\), \\((\\overline E)^\\circ\\) and \\(\\overline{E^\\circ}\\) differ from each other by null sets. The last two are clopen. So every measurable set agrees up to a null set with a clopen set.\n3. The support of \\(\\mu\\) is clopen.\n4. Let \\(S\\) be the support of \\(\\mu\\). A \\(\\mu\\)-measurable set \\(E\\) is null if and only if \\(E\\cap S\\) is rare. In particular, two normal measures with the same support have the same null sets, and if \\(S=\\Omega\\), the measurable null sets are exactly the measurable rare sets.\n\n**Proof.** (1) Let \\(N\\) be null. It is measurable, so Lemma 5.3 applied to \\(\\Omega\\setminus N\\) gives an open \\(G\\subseteq\\Omega\\setminus N\\) with \\(\\mu(\\Omega\\setminus G)=0\\). The closed set \\(\\Omega\\setminus G\\) contains \\(N\\), hence \\(\\overline N\\).\n\n(2) Lemma 5.3 gives an open \\(G\\subseteq E\\) with \\(E\\setminus G\\) null, and \\(G\\subseteq E^\\circ\\); so \\(E\\setminus E^\\circ\\) is null. Applied to \\(\\Omega\\setminus E\\), it gives an open \\(G'\\subseteq\\Omega\\setminus E\\) with \\((\\Omega\\setminus E)\\setminus G'\\) null. Then \\(\\overline E\\subseteq\\Omega\\setminus G'\\), so \\(\\overline E\\setminus E\\) is null. The other two sets lie between \\(E^\\circ\\) and \\(\\overline E\\). They are clopen by Lemma 4.2(1) and Definition 4.1.\n\n(3) Let \\(V\\) be the union of all open null sets. By Lemma 2.3(1), applied to the net of finite unions, \\(V\\) is null. By (1) so is \\(\\overline V\\), which is open because \\(\\Omega\\) is stonean. So \\(\\overline V\\subseteq V\\), and \\(V\\) is clopen. The support \\(\\Omega\\setminus V\\) is clopen.\n\n(4) \\(\\mu(E)=\\mu(E\\cap S)\\). If \\(E\\cap S\\) is null, so is its closure by (1), and the interior of that closure is an open null set inside \\(S\\). An open subset of the support with measure zero is empty, since it lies in \\(V\\). So \\(E\\cap S\\) is rare. The converse is Theorem 5.2. \\(\\square\\)\n\n**Theorem 5.5** (Measurable functions are almost continuous). Let \\(\\Omega\\) be stonean, \\(\\mu\\) a normal measure on \\(\\Omega\\), and \\(f\\) a bounded real \\(\\mu\\)-measurable function. Then \\(f=f_*=f^*\\) almost everywhere, the functions \\((f_*)^*\\) and \\((f^*)_*\\) are continuous, and both agree with \\(f\\) almost everywhere. Consequently, every element of \\(L^\\infty(\\Omega,\\mu)\\) contains a continuous function. Let \\(S\\) be the support of \\(\\mu\\). The map that sends \\(x\\in C(S)\\) to the class of its extension by \\(0\\) is an isometric \\(*\\)-isomorphism\n\\[\nC(S)\\longrightarrow L^\\infty(\\Omega,\\mu).\n\\tag{5.1}\n\\]\nIf \\(S=\\Omega\\), the map \\(x\\mapsto[x]\\) from \\(C(\\Omega)\\) onto \\(L^\\infty(\\Omega,\\mu)\\) is an isometric \\(*\\)-isomorphism.\n\n**Proof.** *Step 1.* Choose simple functions \\(s_n\\) with \\(|f-s_n|\\le1/n\\) everywhere, each of the form \\(s_n=\\sum_kc_k1_{E_k}\\) for a finite partition of \\(\\Omega\\) into measurable sets \\(E_k\\). Lemma 5.3 gives open sets \\(G_k\\subseteq E_k\\) with \\(\\mu(E_k\\setminus G_k)=0\\). Their union \\(W_n\\) is open, \\(\\mu(\\Omega\\setminus W_n)=0\\), and \\(s_n\\) is constant on each \\(G_k\\). Put \\(W=\\bigcap_nW_n\\); then \\(\\mu(\\Omega\\setminus W)=0\\). Let \\(\\omega\\in W\\) and \\(n\\ge1\\). The point \\(\\omega\\) lies in one of the sets \\(G_k\\) that make up \\(W_n\\), and for \\(\\omega'\\in G_k\\),\n\\[\n\\begin{gathered}\n|f(\\omega')-f(\\omega)|\\\\\n\\le|f(\\omega')-s_n(\\omega')|+|s_n(\\omega)-f(\\omega)|\\le2/n .\n\\end{gathered}\n\\]\nSo \\(f\\) is continuous at \\(\\omega\\), and \\(f_*(\\omega)=f(\\omega)=f^*(\\omega)\\) by Lemma 2.2(1).\n\n*Step 2.* The function \\(f_*\\) is bounded and lsc. By Theorem 4.3, \\((f_*)^*\\) is continuous and agrees with \\(f_*\\) outside a meager set, which is null (Theorem 5.2). Applying this to \\(-f\\), and using \\((-f)^*=-f_*\\) and \\((-f)_*=-f^*\\), shows that \\((f^*)_*\\) is continuous and agrees with \\(f^*\\) almost everywhere. With Step 1, all five functions agree almost everywhere.\n\n*Step 3.* For continuous \\(x\\), the set \\(\\{|x|>c\\}\\) is open, so it is null exactly when it misses \\(S\\). Hence \\(\\|[x]\\|_\\infty=\\sup_S|x|\\). Since \\(S\\) is clopen, every continuous function on \\(S\\) is the restriction of a continuous function on \\(\\Omega\\) (extend by \\(0\\)). So (5.1) is well defined and isometric, and by Step 2 it is onto. \\(\\square\\)\n\n**Theorem 5.6** (Normal and singular parts). Let \\(\\Omega\\) be stonean. Every Radon measure \\(\\mu\\) on \\(\\Omega\\) can be written in exactly one way as \\(\\mu=\\mu_n+\\mu_s\\), where \\(\\mu_n\\) is a normal measure and \\(\\mu_s\\) is a Radon measure concentrated on a meager set, that is, \\(\\mu_s(\\Omega\\setminus M)=0\\) for some meager Borel set \\(M\\).\n\n**Proof.** *Existence.* Let \\(\\alpha\\) be the supremum of \\(\\mu(R)\\) over closed rare sets \\(R\\). Finite unions of closed rare sets are closed and rare, so there are closed rare sets \\(R_1\\subseteq R_2\\subseteq\\cdots\\) with \\(\\mu(R_n)\\to\\alpha\\). Their union \\(M\\) is a meager Borel set with \\(\\mu(M)=\\alpha\\). Put \\(\\mu_s(E)=\\mu(E\\cap M)\\) and \\(\\mu_n(E)=\\mu(E\\setminus M)\\). Both are finite Borel measures below \\(\\mu\\), and they inherit outer regularity and inner regularity on open sets from \\(\\mu\\), since for instance \\(\\mu_s(U\\setminus E)\\le\\mu(U\\setminus E)\\). Let \\(R\\) be closed and rare. Each \\(R\\cup R_n\\) is closed and rare, so \\(\\mu(R\\cup R_n)\\le\\alpha\\), and letting \\(n\\to\\infty\\) gives \\(\\mu(R\\cup M)\\le\\alpha=\\mu(M)\\). Hence \\(\\mu_n(R)=\\mu(R\\setminus M)=0\\). By Theorem 5.2, \\(\\mu_n\\) is normal.\n\n*Uniqueness.* Let \\(\\mu=\\nu_n+\\nu_s\\) be another such decomposition, with \\(\\nu_s\\) concentrated on the meager set \\(M'\\). The set \\(M\\cup M'\\) is meager, so \\(\\mu_n\\) and \\(\\nu_n\\) vanish on it. For a Borel set \\(E\\), \\[\n\\begin{gathered}\n\\mu_s(E)\\\\\n=\\mu_s(E\\cap(M\\cup M'))\\\\\n=\\mu(E\\cap(M\\cup M')),\n\\end{gathered}\n\\] and in the same way \\(\\nu_s(E)=\\mu(E\\cap(M\\cup M'))\\). So \\(\\mu_s=\\nu_s\\) and \\(\\mu_n=\\nu_n\\). \\(\\square\\)\n\n**Corollary 5.7.** If a stonean space carries no normal measure other than \\(0\\), every Radon measure on it is concentrated on a meager set.\n\n**Proof.** In the decomposition \\(\\mu=\\mu_n+\\mu_s\\) of Theorem 5.6 the normal part \\(\\mu_n\\) is \\(0\\), so \\(\\mu=\\mu_s\\). \\(\\square\\)\n\n**Example 5.8** (The space \\(\\beta I\\)). Let \\(I\\) be a set and \\(\\Omega=\\beta I\\), which is stonean by Example 4.5. The points of \\(I\\) are isolated in \\(\\beta I\\). Indeed, the indicator \\(1_{\\{i\\}}\\in\\ell^\\infty(I)=C(\\beta I)\\) is the indicator of a clopen set, and a character \\(\\omega\\) of \\(\\ell^\\infty(I)\\) with \\(\\omega(1_{\\{i\\}})=1\\) satisfies \\(\\omega(f)=\\omega(f1_{\\{i\\}})=f(i)\\) for all \\(f\\), so this clopen set is \\(\\{i\\}\\). So a set that contains a point of \\(I\\) is not rare. A set \\(R\\subseteq\\beta I\\setminus I\\) is rare: its closure lies in the closed set \\(\\beta I\\setminus I\\), which has empty interior because \\(I\\) is dense. So the rare sets are exactly the subsets of \\(\\beta I\\setminus I\\), and by Theorem 5.2 the normal measures are the Radon measures \\(\\mu\\) with \\(\\mu(\\beta I\\setminus I)=0\\), that is, \\(\\mu=\\sum_{i\\in I}c_i\\delta_i\\) with \\(c_i\\ge0\\) and \\(\\sum_ic_i<\\infty\\). The decomposition of Theorem 5.6 is \\(\\mu_n=\\mu(\\,\\cdot\\cap I)\\) and \\(\\mu_s=\\mu(\\,\\cdot\\cap(\\beta I\\setminus I))\\). For instance, a point mass at a point of \\(\\beta I\\setminus I\\) is purely singular.\n\n",
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      "unit": "abelian-operator-algebras",
      "name": "5. Normal measures",
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      "full_conditions_and_proof": "### 5. Normal measures\n\n**Definition 5.1.** Let \\(\\Omega\\) be stonean. A Radon measure \\(\\mu\\) on \\(\\Omega\\) is *normal* if \\(\\mu(f)=\\sup_i\\mu(f_i)\\) for every increasing net \\((f_i)\\) in \\(C_{\\mathbb R}(\\Omega)\\) with \\(\\sup_i\\|f_i\\|<\\infty\\), where \\(f\\) is its least upper bound in \\(C_{\\mathbb R}(\\Omega)\\). A complex Radon measure is called *normal* when it is a finite sum of complex multiples of positive normal measures.\n\nThe least upper bound \\(f\\) exists by Theorem 4.3. Normality asks that the integral see \\(f\\), and not only the pointwise supremum, which may be smaller on a meager set.\n\n**Theorem 5.2** (Normal measures kill rare sets). Let \\(\\Omega\\) be stonean. For a Radon measure \\(\\mu\\) on \\(\\Omega\\) the following are equivalent.\n\n1. \\(\\mu\\) is normal.\n2. Every rare set is \\(\\mu\\)-null.\n3. Every meager set is \\(\\mu\\)-null.\n\n**Proof.** (1)\\(\\Rightarrow\\)(2). A rare set lies in its closure, which is a closed rare set, so let \\(R\\) be closed and rare. The open set \\(\\Omega\\setminus R\\) is dense. By Lemma 4.2(3) it is the union of the increasing net \\((C_i)\\) of clopen sets contained in it. The net \\((1_{C_i})\\) in \\(C_{\\mathbb R}(\\Omega)\\) has least upper bound \\(1\\): an upper bound is at least \\(1\\) on the dense set \\(\\Omega\\setminus R\\), hence everywhere. By normality and Lemma 2.3(1), \\(\\mu(\\Omega)=\\sup_i\\mu(C_i)=\\mu(\\Omega\\setminus R)\\). So \\(\\mu(R)=0\\).\n\n(2)\\(\\Rightarrow\\)(3). A meager set lies in a countable union of closed rare sets.\n\n(3)\\(\\Rightarrow\\)(1). Let \\((f_i)\\) be as in Definition 5.1 with least upper bound \\(f\\), and let \\(g=\\sup_if_i\\) pointwise. By Theorem 4.3, \\(f=g\\) outside a meager set, which is null. By Lemma 2.3(2), \\(\\mu(f)=\\int g\\,d\\mu=\\sup_i\\mu(f_i)\\). \\(\\square\\)\n\n**Lemma 5.3** (Open kernels of measurable sets). Let \\(\\Omega\\) be stonean, \\(\\mu\\) a normal measure on \\(\\Omega\\), and \\(E\\) a \\(\\mu\\)-measurable set. There is an open set \\(G\\subseteq E\\) with \\(\\mu(E\\setminus G)=0\\).\n\n**Proof.** By inner regularity there are compact sets \\(K_n\\subseteq E\\) with \\(\\mu(E\\setminus K_n)<1/n\\). The interior \\(K_n^\\circ\\) is clopen (Lemma 4.2(1)), and \\(K_n\\setminus K_n^\\circ\\) is rare (Lemma 2.1(1)), hence null (Theorem 5.2). The open set \\(G=\\bigcup_nK_n^\\circ\\) lies in \\(E\\), and \\(\\mu(E\\setminus G)\\le\\mu(E\\setminus K_n)+\\mu(K_n\\setminus K_n^\\circ)<1/n\\) for every \\(n\\). \\(\\square\\)\n\n**Corollary 5.4.** Let \\(\\Omega\\) be stonean and \\(\\mu\\) a normal measure on \\(\\Omega\\).\n\n1. The closure of a \\(\\mu\\)-null set is \\(\\mu\\)-null.\n2. If \\(E\\) is \\(\\mu\\)-measurable, the sets \\(E\\), \\(\\overline E\\), \\(E^\\circ\\), \\((\\overline E)^\\circ\\) and \\(\\overline{E^\\circ}\\) differ from each other by null sets. The last two are clopen. So every measurable set agrees up to a null set with a clopen set.\n3. The support of \\(\\mu\\) is clopen.\n4. Let \\(S\\) be the support of \\(\\mu\\). A \\(\\mu\\)-measurable set \\(E\\) is null if and only if \\(E\\cap S\\) is rare. In particular, two normal measures with the same support have the same null sets, and if \\(S=\\Omega\\), the measurable null sets are exactly the measurable rare sets.\n\n**Proof.** (1) Let \\(N\\) be null. It is measurable, so Lemma 5.3 applied to \\(\\Omega\\setminus N\\) gives an open \\(G\\subseteq\\Omega\\setminus N\\) with \\(\\mu(\\Omega\\setminus G)=0\\). The closed set \\(\\Omega\\setminus G\\) contains \\(N\\), hence \\(\\overline N\\).\n\n(2) Lemma 5.3 gives an open \\(G\\subseteq E\\) with \\(E\\setminus G\\) null, and \\(G\\subseteq E^\\circ\\); so \\(E\\setminus E^\\circ\\) is null. Applied to \\(\\Omega\\setminus E\\), it gives an open \\(G'\\subseteq\\Omega\\setminus E\\) with \\((\\Omega\\setminus E)\\setminus G'\\) null. Then \\(\\overline E\\subseteq\\Omega\\setminus G'\\), so \\(\\overline E\\setminus E\\) is null. The other two sets lie between \\(E^\\circ\\) and \\(\\overline E\\). They are clopen by Lemma 4.2(1) and Definition 4.1.\n\n(3) Let \\(V\\) be the union of all open null sets. By Lemma 2.3(1), applied to the net of finite unions, \\(V\\) is null. By (1) so is \\(\\overline V\\), which is open because \\(\\Omega\\) is stonean. So \\(\\overline V\\subseteq V\\), and \\(V\\) is clopen. The support \\(\\Omega\\setminus V\\) is clopen.\n\n(4) \\(\\mu(E)=\\mu(E\\cap S)\\). If \\(E\\cap S\\) is null, so is its closure by (1), and the interior of that closure is an open null set inside \\(S\\). An open subset of the support with measure zero is empty, since it lies in \\(V\\). So \\(E\\cap S\\) is rare. The converse is Theorem 5.2. \\(\\square\\)\n\n**Theorem 5.5** (Measurable functions are almost continuous). Let \\(\\Omega\\) be stonean, \\(\\mu\\) a normal measure on \\(\\Omega\\), and \\(f\\) a bounded real \\(\\mu\\)-measurable function. Then \\(f=f_*=f^*\\) almost everywhere, the functions \\((f_*)^*\\) and \\((f^*)_*\\) are continuous, and both agree with \\(f\\) almost everywhere. Consequently, every element of \\(L^\\infty(\\Omega,\\mu)\\) contains a continuous function. Let \\(S\\) be the support of \\(\\mu\\). The map that sends \\(x\\in C(S)\\) to the class of its extension by \\(0\\) is an isometric \\(*\\)-isomorphism\n\\[\nC(S)\\longrightarrow L^\\infty(\\Omega,\\mu).\n\\tag{5.1}\n\\]\nIf \\(S=\\Omega\\), the map \\(x\\mapsto[x]\\) from \\(C(\\Omega)\\) onto \\(L^\\infty(\\Omega,\\mu)\\) is an isometric \\(*\\)-isomorphism.\n\n**Proof.** *Step 1.* Choose simple functions \\(s_n\\) with \\(|f-s_n|\\le1/n\\) everywhere, each of the form \\(s_n=\\sum_kc_k1_{E_k}\\) for a finite partition of \\(\\Omega\\) into measurable sets \\(E_k\\). Lemma 5.3 gives open sets \\(G_k\\subseteq E_k\\) with \\(\\mu(E_k\\setminus G_k)=0\\). Their union \\(W_n\\) is open, \\(\\mu(\\Omega\\setminus W_n)=0\\), and \\(s_n\\) is constant on each \\(G_k\\). Put \\(W=\\bigcap_nW_n\\); then \\(\\mu(\\Omega\\setminus W)=0\\). Let \\(\\omega\\in W\\) and \\(n\\ge1\\). The point \\(\\omega\\) lies in one of the sets \\(G_k\\) that make up \\(W_n\\), and for \\(\\omega'\\in G_k\\),\n\\[\n\\begin{gathered}\n|f(\\omega')-f(\\omega)|\\\\\n\\le|f(\\omega')-s_n(\\omega')|+|s_n(\\omega)-f(\\omega)|\\le2/n .\n\\end{gathered}\n\\]\nSo \\(f\\) is continuous at \\(\\omega\\), and \\(f_*(\\omega)=f(\\omega)=f^*(\\omega)\\) by Lemma 2.2(1).\n\n*Step 2.* The function \\(f_*\\) is bounded and lsc. By Theorem 4.3, \\((f_*)^*\\) is continuous and agrees with \\(f_*\\) outside a meager set, which is null (Theorem 5.2). Applying this to \\(-f\\), and using \\((-f)^*=-f_*\\) and \\((-f)_*=-f^*\\), shows that \\((f^*)_*\\) is continuous and agrees with \\(f^*\\) almost everywhere. With Step 1, all five functions agree almost everywhere.\n\n*Step 3.* For continuous \\(x\\), the set \\(\\{|x|>c\\}\\) is open, so it is null exactly when it misses \\(S\\). Hence \\(\\|[x]\\|_\\infty=\\sup_S|x|\\). Since \\(S\\) is clopen, every continuous function on \\(S\\) is the restriction of a continuous function on \\(\\Omega\\) (extend by \\(0\\)). So (5.1) is well defined and isometric, and by Step 2 it is onto. \\(\\square\\)\n\n**Theorem 5.6** (Normal and singular parts). Let \\(\\Omega\\) be stonean. Every Radon measure \\(\\mu\\) on \\(\\Omega\\) can be written in exactly one way as \\(\\mu=\\mu_n+\\mu_s\\), where \\(\\mu_n\\) is a normal measure and \\(\\mu_s\\) is a Radon measure concentrated on a meager set, that is, \\(\\mu_s(\\Omega\\setminus M)=0\\) for some meager Borel set \\(M\\).\n\n**Proof.** *Existence.* Let \\(\\alpha\\) be the supremum of \\(\\mu(R)\\) over closed rare sets \\(R\\). Finite unions of closed rare sets are closed and rare, so there are closed rare sets \\(R_1\\subseteq R_2\\subseteq\\cdots\\) with \\(\\mu(R_n)\\to\\alpha\\). Their union \\(M\\) is a meager Borel set with \\(\\mu(M)=\\alpha\\). Put \\(\\mu_s(E)=\\mu(E\\cap M)\\) and \\(\\mu_n(E)=\\mu(E\\setminus M)\\). Both are finite Borel measures below \\(\\mu\\), and they inherit outer regularity and inner regularity on open sets from \\(\\mu\\), since for instance \\(\\mu_s(U\\setminus E)\\le\\mu(U\\setminus E)\\). Let \\(R\\) be closed and rare. Each \\(R\\cup R_n\\) is closed and rare, so \\(\\mu(R\\cup R_n)\\le\\alpha\\), and letting \\(n\\to\\infty\\) gives \\(\\mu(R\\cup M)\\le\\alpha=\\mu(M)\\). Hence \\(\\mu_n(R)=\\mu(R\\setminus M)=0\\). By Theorem 5.2, \\(\\mu_n\\) is normal.\n\n*Uniqueness.* Let \\(\\mu=\\nu_n+\\nu_s\\) be another such decomposition, with \\(\\nu_s\\) concentrated on the meager set \\(M'\\). The set \\(M\\cup M'\\) is meager, so \\(\\mu_n\\) and \\(\\nu_n\\) vanish on it. For a Borel set \\(E\\), \\[\n\\begin{gathered}\n\\mu_s(E)\\\\\n=\\mu_s(E\\cap(M\\cup M'))\\\\\n=\\mu(E\\cap(M\\cup M')),\n\\end{gathered}\n\\] and in the same way \\(\\nu_s(E)=\\mu(E\\cap(M\\cup M'))\\). So \\(\\mu_s=\\nu_s\\) and \\(\\mu_n=\\nu_n\\). \\(\\square\\)\n\n**Corollary 5.7.** If a stonean space carries no normal measure other than \\(0\\), every Radon measure on it is concentrated on a meager set.\n\n**Proof.** In the decomposition \\(\\mu=\\mu_n+\\mu_s\\) of Theorem 5.6 the normal part \\(\\mu_n\\) is \\(0\\), so \\(\\mu=\\mu_s\\). \\(\\square\\)\n\n**Example 5.8** (The space \\(\\beta I\\)). Let \\(I\\) be a set and \\(\\Omega=\\beta I\\), which is stonean by Example 4.5. The points of \\(I\\) are isolated in \\(\\beta I\\). Indeed, the indicator \\(1_{\\{i\\}}\\in\\ell^\\infty(I)=C(\\beta I)\\) is the indicator of a clopen set, and a character \\(\\omega\\) of \\(\\ell^\\infty(I)\\) with \\(\\omega(1_{\\{i\\}})=1\\) satisfies \\(\\omega(f)=\\omega(f1_{\\{i\\}})=f(i)\\) for all \\(f\\), so this clopen set is \\(\\{i\\}\\). So a set that contains a point of \\(I\\) is not rare. A set \\(R\\subseteq\\beta I\\setminus I\\) is rare: its closure lies in the closed set \\(\\beta I\\setminus I\\), which has empty interior because \\(I\\) is dense. So the rare sets are exactly the subsets of \\(\\beta I\\setminus I\\), and by Theorem 5.2 the normal measures are the Radon measures \\(\\mu\\) with \\(\\mu(\\beta I\\setminus I)=0\\), that is, \\(\\mu=\\sum_{i\\in I}c_i\\delta_i\\) with \\(c_i\\ge0\\) and \\(\\sum_ic_i<\\infty\\). The decomposition of Theorem 5.6 is \\(\\mu_n=\\mu(\\,\\cdot\\cap I)\\) and \\(\\mu_s=\\mu(\\,\\cdot\\cap(\\beta I\\setminus I))\\). For instance, a point mass at a point of \\(\\beta I\\setminus I\\) is purely singular.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-AO-09",
      "unit": "abelian-operator-algebras",
      "name": "6. Hyperstonean spaces and the decomposition of a stonean space",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "anchor": "oa-fnd-ao-09",
      "proof_locus": {
        "line": 518,
        "through_line": 545
      },
      "full_conditions_and_proof": "### 6. Hyperstonean spaces and the decomposition of a stonean space\n\n**Definition 6.1.** Let \\(\\Omega\\) be stonean. A family \\(\\mathfrak F\\) of normal measures is *sufficient* if for every nonzero \\(x\\in C(\\Omega)\\) with \\(x\\ge0\\) some \\(\\mu\\in\\mathfrak F\\) has \\(\\mu(x)>0\\). The space \\(\\Omega\\) is *hyperstonean* if the family of all normal measures on \\(\\Omega\\) is sufficient.\n\n**Lemma 6.2.** A family \\(\\mathfrak F\\) of normal measures is sufficient exactly when the supports of its members have a dense union.\n\n**Proof.** Suppose these supports have a dense union, and let \\(x\\ge0\\) be nonzero. Then the open set \\(\\{x>0\\}\\) meets the support of some \\(\\mu\\in\\mathfrak F\\). A nonempty open subset of the support has positive measure, so \\(\\mu(x)>0\\). If the union is not dense, there is a nonempty clopen set \\(C\\) disjoint from all supports (Lemma 4.2(3)), and \\(\\mu(1_C)=0\\) for every \\(\\mu\\in\\mathfrak F\\). \\(\\square\\)\n\n**Proposition 6.3** (Rare sets in a hyperstonean space). Let \\(\\Omega\\) be hyperstonean and \\(\\mathfrak F\\) a sufficient family of normal measures.\n\n1. A set \\(A\\subseteq\\Omega\\) is rare exactly when it is \\(\\mu\\)-null for all \\(\\mu\\in\\mathfrak F\\). In particular every meager set is rare.\n2. If \\(E\\) is \\(\\mu\\)-measurable for every \\(\\mu\\in\\mathfrak F\\), then \\((\\overline E)^\\circ=\\overline{E^\\circ}\\), and the sets \\(E\\), \\(\\overline E\\), \\(E^\\circ\\) and \\((\\overline E)^\\circ\\) differ from each other by rare sets.\n3. If \\(f\\) is bounded, real and \\(\\mu\\)-measurable for every \\(\\mu\\in\\mathfrak F\\), then \\((f_*)^*=(f^*)_*\\). This function is continuous, and it agrees with \\(f\\), \\(f_*\\) and \\(f^*\\) outside a rare set.\n\n**Proof.** (1) A rare set is null for every normal measure (Theorem 5.2). Conversely, let \\(A\\) be null for every \\(\\mu\\in\\mathfrak F\\). By Corollary 5.4(1), \\(\\overline A\\) is null for every \\(\\mu\\in\\mathfrak F\\), and so is the open set \\((\\overline A)^\\circ\\). If it were not empty, it would contain a nonempty clopen set \\(C\\), and \\(1_C\\) would contradict sufficiency. So \\(A\\) is rare. A meager set is null for every normal measure, hence rare.\n\n(2) Both \\((\\overline E)^\\circ\\) and \\(\\overline{E^\\circ}\\) are clopen, and each differs from \\(E\\) by a set that is null for every \\(\\mu\\in\\mathfrak F\\) (Corollary 5.4(2)). So their symmetric difference is a clopen set that is null for every \\(\\mu\\in\\mathfrak F\\); by (1) it is rare and open, hence empty. The differences in the second statement are null for every \\(\\mu\\in\\mathfrak F\\), hence rare by (1).\n\n(3) Both functions are continuous (Theorem 5.5) and agree \\(\\mu\\)-almost everywhere for every \\(\\mu\\in\\mathfrak F\\). The set where they differ is open and null for every \\(\\mu\\in\\mathfrak F\\), hence empty by (1). The exceptional sets in Theorem 5.5 are null for every \\(\\mu\\in\\mathfrak F\\), hence rare. \\(\\square\\)\n\n## C. Recognize von Neumann algebras and classify their models\n\nWe can now combine two independent tests: order completeness gives a stonean spectrum, while enough normal observations give a hyperstonean spectrum. The equivalence theorem constructs a von Neumann representation from these observations and recovers them from vector functionals in the opposite direction. It also explains why an abstract isomorphism need not be implemented by a unitary in a particular representation.\n\n**Checkpoint: multiplicity versus the algebra.** Multiplication by \\(f\\) on \\(L^2[0,1]\\) and by \\(f\\oplus f\\) on its doubled Hilbert space obey the same algebraic rules. In the doubled space the operator swapping the summands commutes with every multiplication operator. The commutant has therefore changed. The spatial-isomorphism theorem needs its cyclicity and maximality hypotheses; Example 7.5 records the failures when they are removed. Atomic and diffuse decomposition concerns the algebra's projections, whereas this multiplicity concerns its action.\n\nThe countable-generation results then compress many commuting observables into one self-adjoint generator. The interval model for the diffuse case retains its separate sigma-finiteness and nonzero hypotheses.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-AO-10",
      "unit": "abelian-operator-algebras",
      "name": "The three clopen regions of a stonean space",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "anchor": "oa-fnd-ao-10",
      "proof_locus": {
        "line": 733,
        "through_line": 756
      },
      "full_conditions_and_proof": "### The three clopen regions of a stonean space\n\n**Theorem 6.4** (Decomposition of a stonean space). Let \\(\\Omega\\) be stonean. There is exactly one partition of \\(\\Omega\\) into three clopen sets \\(\\Omega_1\\), \\(\\Omega_2\\), \\(\\Omega_3\\) with the following properties.\n\n1. \\(\\Omega_1\\) is hyperstonean.\n2. \\(\\Omega_2\\) contains a dense meager subset.\n3. Every meager subset of \\(\\Omega_3\\) is rare, and every Radon measure on \\(\\Omega_3\\) is concentrated on a closed rare subset of \\(\\Omega_3\\).\n\nMoreover, \\(\\Omega_1\\) is the closure of the union of all supports of normal measures on \\(\\Omega\\), and \\(\\Omega_2\\cup\\Omega_3\\) carries no normal measure other than \\(0\\). Any of the three parts may be empty; Section 9 gives stonean spaces with \\(\\Omega=\\Omega_2\\) and with \\(\\Omega=\\Omega_3\\).\n\n**Proof.** We use two facts about a clopen set \\(C\\). A subset of \\(C\\) is rare in \\(C\\) exactly when it is rare in \\(\\Omega\\), because closures in \\(C\\) are closures in \\(\\Omega\\) and \\(C\\) is open. Consequently the normal measures on \\(C\\) are the normal measures on \\(\\Omega\\) that vanish outside \\(C\\), and the restriction \\(\\mu(\\,\\cdot\\cap C)\\) of a normal measure to \\(C\\) is normal.\n\n*The first part.* Let \\(F\\) be the union of all supports of normal measures on \\(\\Omega\\). The supports are clopen (Corollary 5.4(3)), so \\(F\\) is open, and \\(\\Omega_1=\\overline F\\) is clopen. The normal measures on \\(\\Omega_1\\) have supports whose union \\(F\\) is dense in \\(\\Omega_1\\), so \\(\\Omega_1\\) is hyperstonean by Lemma 6.2.\n\n*The second part.* Call a nonempty open set \\(U\\) *thin* if it contains a meager subset that is dense in \\(U\\). By Zorn's lemma there is a maximal family \\((G_j)_{j\\in J}\\) of pairwise disjoint thin sets. Let \\(G=\\bigcup_jG_j\\) and \\(\\Omega_2=\\overline G\\), a clopen set. For each \\(j\\) choose a meager \\(M_j\\subseteq G_j\\) that is dense in \\(G_j\\). By Lemma 2.1(3) the union \\(\\bigcup_jM_j\\) is meager, and it is dense in \\(\\Omega_2\\).\n\n*The first two parts are disjoint.* Let \\(\\mu\\) be normal and suppose \\(W=G_j\\cap\\operatorname{supp}\\mu\\) is not empty. Then \\(M_j\\cap W\\) is dense in the open set \\(W\\) and meager, hence null, and by Corollary 5.4(1) its closure is null. This closure contains \\(W\\), a nonempty open subset of the support, which is impossible. So \\(F\\cap G=\\varnothing\\). Since \\(F\\) is open, \\(F\\cap\\overline G=\\varnothing\\), and since \\(\\overline G\\) is open, \\(\\overline F\\cap\\overline G=\\varnothing\\).\n\n*The third part.* Put \\(\\Omega_3=\\Omega\\setminus(\\Omega_1\\cup\\Omega_2)\\), a clopen set. Let \\(M\\subseteq\\Omega_3\\) be meager, and suppose that \\(\\overline M\\) has a nonempty interior \\(U\\). Then \\(U\\subseteq\\Omega_3\\), and \\(M\\cap U\\) is a meager subset dense in \\(U\\), so \\(U\\) is thin and disjoint from every \\(G_j\\). This contradicts maximality, so \\(M\\) is rare. A normal measure is supported in \\(\\Omega_1\\), so \\(\\Omega_2\\cup\\Omega_3\\) carries no normal measure other than \\(0\\). Let \\(\\mu\\) be a Radon measure on \\(\\Omega_3\\), extended by \\(0\\) to \\(\\Omega\\). In the decomposition of Theorem 5.6 the normal part is supported in \\(\\Omega_1\\cap\\Omega_3=\\varnothing\\), so \\(\\mu\\) is concentrated on a meager set \\(M\\), which we may take inside \\(\\Omega_3\\). Then \\(M\\) is rare, and \\(\\mu\\) is concentrated on the closed rare set \\(\\overline M\\).\n\n*Uniqueness.* Let \\(\\Omega_1',\\Omega_2',\\Omega_3'\\) be another clopen partition with properties 1–3. A normal measure on \\(\\Omega_1'\\) is a normal measure on \\(\\Omega\\), so its support lies in \\(F\\); as \\(\\Omega_1'\\) is hyperstonean, these supports are dense in \\(\\Omega_1'\\), and \\(\\Omega_1'\\subseteq\\overline F=\\Omega_1\\). If the open set \\(W=\\Omega_2'\\setminus\\Omega_2\\) were not empty, it would be thin, because the dense meager subset of \\(\\Omega_2'\\) meets it in a dense meager subset; and it would be disjoint from \\(G\\), against maximality. So \\(\\Omega_2'\\subseteq\\Omega_2\\). If \\(W=\\Omega_1\\cap\\Omega_3'\\) were not empty, some normal measure would give \\(W\\) positive measure, since \\(\\Omega_1\\) is hyperstonean; its restriction to \\(W\\) would be a nonzero normal measure concentrated on the rare set of property 3, which is impossible. If \\(W=\\Omega_2\\cap\\Omega_3'\\) were not empty, the dense meager subset of \\(\\Omega_2\\) would meet \\(W\\) in a meager set dense in \\(W\\); by property 3 this set is rare, and a rare set is dense in no nonempty open set. So \\(\\Omega_3'\\subseteq\\Omega_3\\). Since both triples are partitions of \\(\\Omega\\), the three inclusions are equalities. \\(\\square\\)\n\nIn particular, three conditions on a stonean space are equivalent: it has no nonempty clopen subset that is hyperstonean; \\(\\Omega_1=\\varnothing\\); it carries no normal measure other than \\(0\\). Indeed, \\(\\Omega_1\\) is itself a clopen hyperstonean set; a nonempty clopen hyperstonean set carries a nonzero normal measure, which is also normal on the whole space; and the support of a nonzero normal measure is a nonempty clopen set that is hyperstonean by Lemma 6.2. When the three conditions hold, Corollary 5.7 shows that every Radon measure on the space is concentrated on a meager set.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-AO-11",
      "unit": "abelian-operator-algebras",
      "name": "Atomic and diffuse regions",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "anchor": "oa-fnd-ao-11",
      "proof_locus": {
        "line": 620,
        "through_line": 637
      },
      "full_conditions_and_proof": "### Atomic and diffuse regions\n\n**Proposition 6.5** (Atomic and diffuse parts). Let \\(\\Omega\\) be stonean, let \\(I\\) be the set of its isolated points, and put \\(\\Omega_d=\\overline I\\) and \\(\\Omega_c=\\Omega\\setminus\\Omega_d\\).\n\n1. \\(\\Omega_d\\) and \\(\\Omega_c\\) are clopen. \\(I\\) is an open discrete subset of \\(\\Omega_d\\) that is dense in \\(\\Omega_d\\), and \\(\\Omega_c\\) has no isolated points. The space \\(\\Omega_d\\) is the Stone–Čech compactification of the discrete space \\(I\\), so \\(C(\\Omega_d)\\cong\\ell^\\infty(I)\\).\n2. If \\(\\Omega=A\\cup B\\) is a partition into clopen sets such that some open discrete subset of \\(A\\) is dense in \\(A\\) and \\(B\\) has no isolated points, then \\(A=\\Omega_d\\) and \\(B=\\Omega_c\\).\n3. If \\(\\Omega\\) is hyperstonean, so are \\(\\Omega_d\\) and \\(\\Omega_c\\).\n\nThe clopen hypothesis in (2) is essential, as the complete counterexample in Example 6.6 proves.\n\n**Proof.** (1) \\(I\\) is open, as a union of open points, so \\(\\Omega_d=\\overline I\\) is clopen, and so is \\(\\Omega_c\\). An isolated point of the open set \\(\\Omega_c\\) would be isolated in \\(\\Omega\\) and so lie in \\(I\\subseteq\\Omega_d\\). The space \\(\\Omega_d\\) is stonean and \\(I\\) is dense in it, so Theorem 4.4 identifies \\(\\Omega_d\\) with \\(\\beta I\\) and \\(C(\\Omega_d)\\) with \\(C_b(I)=\\ell^\\infty(I)\\).\n\n(2) Let \\(D\\) be an open discrete subset of \\(A\\) that is dense in \\(A\\). Each point of \\(D\\) is open in \\(D\\), \\(D\\) is open in \\(A\\), and \\(A\\) is open in \\(\\Omega\\); so \\(D\\subseteq I\\). As \\(A\\) is closed, \\(A=\\overline D\\subseteq\\overline I=\\Omega_d\\). A point of \\(I\\) lying in the open set \\(B\\) would be isolated in \\(B\\), so \\(I\\subseteq A\\), and \\(\\Omega_d\\subseteq A\\) because \\(A\\) is closed.\n\n(3) If \\(\\mathfrak F\\) is a sufficient family of normal measures on \\(\\Omega\\) and \\(C\\) is clopen, the restrictions of its members to \\(C\\) form a sufficient family on \\(C\\), by Lemma 6.2. \\(\\square\\)\n\n**Example 6.6** (The parts must be clopen). In \\(\\beta\\mathbb N\\) the isolated points are the points of \\(\\mathbb N\\), so \\(\\Omega_d=\\beta\\mathbb N\\) and \\(\\Omega_c=\\varnothing\\). The partition into \\(A=\\mathbb N\\) and \\(B=\\beta\\mathbb N\\setminus\\mathbb N\\) also has the two properties of Proposition 6.5(2), except that the parts are not clopen. Clearly \\(\\mathbb N\\) is open, discrete and dense in itself. To see that \\(B\\) has no isolated points, let \\(p\\in B\\) and let \\(C\\) be a clopen neighbourhood of \\(p\\) in \\(\\beta\\mathbb N\\). The set \\(C\\cap\\mathbb N\\) is infinite: otherwise \\(C=\\overline{C\\cap\\mathbb N}\\), which holds because \\(\\mathbb N\\) is dense and \\(C\\) is open, would be a finite subset of \\(\\mathbb N\\) and could not contain \\(p\\). Split \\(C\\cap\\mathbb N\\) into two disjoint infinite sets \\(N_1,N_2\\). They are open, so their closures are disjoint (Lemma 4.2(1)) and lie in \\(C\\). The closure of an infinite subset of \\(\\mathbb N\\) is compact and so is not a subset of the discrete set \\(\\mathbb N\\); hence each closure contains a point of \\(B\\). So \\(C\\cap B\\) has at least two points, and \\(p\\) is not isolated in \\(B\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AO-12",
      "unit": "abelian-operator-algebras",
      "name": "7. Spectra of abelian von Neumann algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "proof_locus": {
        "line": 546,
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      },
      "full_conditions_and_proof": "### 7. Spectra of abelian von Neumann algebras\n\nWe can now say which compact spaces are spectra of abelian von Neumann algebras, and describe the algebras as \\(L^\\infty\\) spaces. For a family \\((B_i)_{i\\in I}\\) of C\\*-algebras, \\(\\prod_iB_i\\) denotes the C\\*-algebra of bounded families \\((b_i)\\) with the supremum norm.\n\n**Theorem 7.1** (Abelian von Neumann algebras and hyperstonean spaces). Let \\(A\\) be an abelian C\\*-algebra and \\(\\Omega\\) its spectrum. The following are equivalent.\n\n1. \\(\\Omega\\) is hyperstonean.\n2. \\(A\\) is \\(*\\)-isomorphic to some von Neumann algebra: it has a faithful representation \\(\\pi\\) such that \\(\\pi(A)\\) is a von Neumann algebra.\n3. \\(A\\cong L^\\infty(\\Gamma,\\mu)\\) for a locally compact space \\(\\Gamma\\) and a positive Radon measure \\(\\mu\\) on \\(\\Gamma\\).\n4. \\(A\\cong L^\\infty(\\Gamma,\\mu)\\), where \\(\\Gamma\\) is the union of pairwise disjoint compact open sets \\(\\Gamma_i\\), \\(i\\in I\\), each of them hyperstonean, and \\(\\mu\\) is a Radon measure on \\(\\Gamma\\) whose restriction \\(\\mu_i\\) to each \\(\\Gamma_i\\) is a normal measure with support \\(\\Gamma_i\\).\n\nIn (4) one can take \\(\\Gamma\\) to be an open dense subset of \\(\\Omega\\), and then \\(A\\cong C(\\Omega)\\cong\\prod_iC(\\Gamma_i)\\cong\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\).\n\n*Reference:* [Dixmier, “Sur certains espaces considérés par M. H. Stone” (1951), §5 Theorem 1 and §6 Theorem 2](https://dmitripavlov.org/scans/dixmier.pdf).\n\n**Proof.** If \\(A=0\\), its spectrum is empty, which is hyperstonean; take \\(\\Gamma=\\varnothing\\) and the zero Hilbert space to obtain all four conditions. Assume henceforth that \\(A\\ne0\\). (2)\\(\\Rightarrow\\)(1). We may assume that \\(A=M\\) acts on a Hilbert space \\(H\\) as a von Neumann algebra. The Gelfand isomorphism is an isomorphism of ordered spaces from \\(M_h\\) onto \\(C_{\\mathbb R}(\\Omega)\\), because the positive elements of \\(C(\\Omega)\\) are the nonnegative functions. Let \\(S\\subseteq M_h\\) be nonempty and bounded above by \\(b\\), and fix \\(s_0\\in S\\). The maxima of the finite subsets of \\(S\\) that contain \\(s_0\\) form an increasing net with the same upper bounds as \\(S\\), and it is bounded in norm, since it lies between \\(s_0\\) and \\(b\\). By Lemma 2.4 its strong limit is its least upper bound in \\(M_h\\). So \\(C_{\\mathbb R}(\\Omega)\\) is conditionally complete, and \\(\\Omega\\) is stonean by Theorem 4.3. For \\(\\xi\\in H\\) let \\(\\mu_\\xi\\) be the Radon measure on \\(\\Omega\\) given by the positive functional \\(\\omega_\\xi\\). If \\((x_i)\\) is a bounded increasing net in \\(M_h\\) with least upper bound \\(x\\), then \\(x_i\\to x\\) strongly by Lemma 2.4, so \\(\\omega_\\xi(x_i)\\to\\omega_\\xi(x)\\). So \\(\\mu_\\xi\\) is normal. If \\(x\\in M_+\\) is not zero, then \\(\\omega_\\xi(x)>0\\) for some \\(\\xi\\). So the measures \\(\\mu_\\xi\\) form a sufficient family, and \\(\\Omega\\) is hyperstonean.\n\n(1)\\(\\Rightarrow\\)(4). The space \\(\\Omega\\) is compact, so \\(A\\) is unital and \\(A\\cong C(\\Omega)\\). By Zorn's lemma choose a maximal family \\((\\mu_i)_{i\\in I}\\) of nonzero normal measures with pairwise disjoint supports \\(\\Gamma_i\\). These supports are clopen (Corollary 5.4(3)), so \\(\\Gamma=\\bigcup_i\\Gamma_i\\) is open. It is dense. Otherwise the clopen set \\(W=\\Omega\\setminus\\overline\\Gamma\\) is not empty, some normal measure \\(\\mu\\) has \\(\\mu(W)>0\\) because \\(\\Omega\\) is hyperstonean, and the restriction of \\(\\mu\\) to \\(W\\) is a nonzero normal measure whose support lies in \\(W\\); adding it to the family contradicts maximality.\n\nRestriction \\(x\\mapsto(x|_{\\Gamma_i})_i\\) is a \\(*\\)-homomorphism from \\(C(\\Omega)\\) to \\(\\prod_iC(\\Gamma_i)\\). It is isometric because \\(\\Gamma\\) is dense. It is onto: a bounded family \\((x_i)\\) defines a bounded continuous function on the open set \\(\\Gamma\\), since the \\(\\Gamma_i\\) are open, and Theorem 4.4 extends it to \\(\\Omega\\). Each \\(\\Gamma_i\\) is clopen, hence stonean, and \\(\\mu_i\\) is a normal measure on it with full support; so \\(\\Gamma_i\\) is hyperstonean by Lemma 6.2. By Theorem 5.5, \\(C(\\Gamma_i)\\cong L^\\infty(\\Gamma_i,\\mu_i)\\). The subspace \\(\\Gamma\\) is locally compact and is the topological disjoint union of the compact open sets \\(\\Gamma_i\\). For \\(x\\in C_c(\\Gamma)\\) the compact support of \\(x\\) meets only finitely many \\(\\Gamma_i\\), so \\(\\mu(x)=\\sum_i\\int_{\\Gamma_i}x\\,d\\mu_i\\) is a finite sum. It is a positive linear functional on \\(C_c(\\Gamma)\\), hence a Radon measure on \\(\\Gamma\\), and its restriction to \\(\\Gamma_i\\) is \\(\\mu_i\\). A set \\(N\\subseteq\\Gamma\\) is locally \\(\\mu\\)-null if and only if every \\(N\\cap\\Gamma_i\\) is \\(\\mu_i\\)-null, and a function on \\(\\Gamma\\) is \\(\\mu\\)-measurable if and only if each restriction to \\(\\Gamma_i\\) is \\(\\mu_i\\)-measurable, because every compact subset of \\(\\Gamma\\) lies in finitely many \\(\\Gamma_i\\). So restriction is an isometric \\(*\\)-isomorphism of \\(L^\\infty(\\Gamma,\\mu)\\) onto \\(\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\). Composing, \\[\n\\begin{gathered}\nA\\\\\n\\cong C(\\Omega)\\\\\n\\cong\\prod_iC(\\Gamma_i)\\\\\n\\cong\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\\\\n\\cong L^\\infty(\\Gamma,\\mu).\n\\end{gathered}\n\\]\n\n(4)\\(\\Rightarrow\\)(3) is clear.\n\n(3)\\(\\Rightarrow\\)(2). The representation of \\(L^\\infty(\\Gamma,\\mu)\\) on \\(L^2(\\Gamma,\\mu)\\) by multiplication operators is faithful, and its image is a maximal abelian von Neumann algebra, for every Radon measure; both facts are proved in Proposition 3.2a. \\(\\square\\)\n\n**Remark 7.2** (Examples). The spectrum of \\(\\ell^\\infty(I)\\) is \\(\\beta I\\), and its normal measures were found in Example 5.8. The spectrum of \\(L^\\infty[0,1]\\) is a hyperstonean space without isolated points, since \\(L^\\infty[0,1]\\) has no minimal projections (Exercise 12.3). Every abelian von Neumann algebra of infinite dimension has an infinite hyperstonean space as its spectrum, and such a space is never metrizable (Exercise 12.1). The algebra \\(C[0,1]\\) is isomorphic to no von Neumann algebra, since \\([0,1]\\) is not stonean. Section 9 gives an abelian C\\*-algebra whose spectrum is stonean but carries no normal measure at all; it is monotone complete but isomorphic to no von Neumann algebra.\n\n**Proposition 7.3** (Isomorphisms preserve monotone limits). Let \\(\\theta:M\\to N\\) be a \\(*\\)-isomorphism between von Neumann algebras on Hilbert spaces \\(H\\) and \\(K\\). Then \\(\\theta\\) maps \\(M_+\\) onto \\(N_+\\), and for every bounded increasing net \\((x_i)\\) in \\(M_h\\) with strong limit \\(x\\), the net \\((\\theta(x_i))\\) converges strongly to \\(\\theta(x)\\). In particular \\(\\omega_\\eta\\circ\\theta\\) preserves the limits of such nets, for every \\(\\eta\\in K\\).\n\n**Proof.** The positive elements of a C\\*-algebra are the elements \\(y^*y\\), so \\(\\theta(M_+)\\subseteq N_+\\) and \\(\\theta^{-1}(N_+)\\subseteq M_+\\). Hence \\(\\theta\\) is an isomorphism of ordered spaces from \\(M_h\\) onto \\(N_h\\). By Lemma 2.4, the net \\((x_i)\\) has \\(x\\) as least upper bound in \\(M_h\\), and \\((\\theta(x_i))\\) has its strong limit \\(y\\) as least upper bound in \\(N_h\\). Since \\(\\theta(x)\\) is an upper bound of \\((\\theta(x_i))\\), \\(y\\le\\theta(x)\\). Since \\(\\theta^{-1}(y)\\) is an upper bound of \\((x_i)\\), \\(x\\le\\theta^{-1}(y)\\), that is, \\(\\theta(x)\\le y\\). \\(\\square\\)\n\nSo \\(*\\)-isomorphisms of von Neumann algebras need no continuity assumption: they automatically respect monotone limits.\n\n**Theorem 7.4** (Cyclic vectors and spatial isomorphisms).\n\n1. Let \\(M\\subseteq B(H)\\) and \\(N\\subseteq B(K)\\) be abelian von Neumann algebras with cyclic vectors \\(\\xi\\) and \\(\\eta\\). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) is spatial: there is a unitary \\(U:H\\to K\\) with \\(\\theta(x)=UxU^*\\) for all \\(x\\in M\\).\n2. An abelian von Neumann algebra has a cyclic vector exactly when it is both maximal abelian and \\(\\sigma\\)-finite.\n3. Every \\(*\\)-isomorphism between maximal abelian von Neumann algebras is spatial.\n\n**Proof.** If one algebra or Hilbert space is zero, the isomorphism forces both algebras to be zero; their units are the identity operators, so both Hilbert spaces are zero and the unique empty unitary applies. The zero vector is cyclic and the zero algebra is maximal abelian and sigma-finite. We may therefore assume the spaces in use are nonzero. (1) Let \\(\\Omega\\) be the spectrum of \\(M\\), which is stonean by Theorem 7.1, and \\(x\\mapsto\\hat x\\) the Gelfand isomorphism of \\(M\\) onto \\(C(\\Omega)\\). Let \\(\\mu\\) and \\(\\nu\\) be the Radon measures on \\(\\Omega\\) with \\(\\int\\hat x\\,d\\mu=\\langle x\\xi,\\xi\\rangle\\) and \\(\\int\\hat x\\,d\\nu=\\langle\\theta(x)\\eta,\\eta\\rangle\\); the second functional is positive by Proposition 7.3. Both measures are normal, by Lemma 2.4 and Proposition 7.3, because the supremum in \\(C_{\\mathbb R}(\\Omega)\\) of an increasing net that is bounded in norm corresponds to its least upper bound in \\(M_h\\). Both have support \\(\\Omega\\). Indeed, \\(\\xi\\) is cyclic for \\(M\\), hence for the larger algebra \\(M'\\), so \\(\\xi\\) separates \\(M\\) (see [The double commutant theorem](the-double-commutant-theorem.md), Proposition 9.2). Thus \\(\\langle x\\xi,\\xi\\rangle=\\|x^{1/2}\\xi\\|^2>0\\) for nonzero \\(x\\in M_+\\). In the same way \\(\\eta\\) separates \\(N\\), and \\(\\theta\\) is injective. If the support of a positive functional on \\(C(\\Omega)\\) were not all of \\(\\Omega\\), Urysohn's lemma would give a nonzero \\(x\\ge0\\) that vanishes on the support, with integral \\(0\\). By Corollary 5.4(4), \\(\\mu\\) and \\(\\nu\\) have the same null sets. By Corollary 3.4 there is a unitary \\(V:L^2(\\Omega,\\mu)\\to L^2(\\Omega,\\nu)\\) with \\(V\\pi_\\mu(f)=\\pi_\\nu(f)V\\). The representations \\(\\hat x\\mapsto x\\) on \\(H\\) and \\(\\hat x\\mapsto\\theta(x)\\) on \\(K\\) have the cyclic vectors \\(\\xi\\) and \\(\\eta\\), so Theorem 3.1(2) gives unitaries \\(U_1:L^2(\\Omega,\\mu)\\to H\\) and \\(U_2:L^2(\\Omega,\\nu)\\to K\\) with \\(U_1\\pi_\\mu(\\hat x)U_1^*=x\\) and \\(U_2\\pi_\\nu(\\hat x)U_2^*=\\theta(x)\\). Then \\(U=U_2VU_1^*\\) satisfies \\[\n\\begin{gathered}\nUxU^*\\\\\n=U_2V\\pi_\\mu(\\hat x)V^*U_2^*\\\\\n=U_2\\pi_\\nu(\\hat x)U_2^*\\\\\n=\\theta(x).\n\\end{gathered}\n\\]\n\n(2) Let \\(M\\) have a cyclic vector \\(\\xi\\). It is maximal abelian by Corollary 3.3. If \\((p_j)\\) are pairwise orthogonal nonzero projections in \\(M\\), then \\(\\langle p_j\\xi,\\xi\\rangle>0\\) because \\(\\xi\\) separates \\(M\\), and every finite sum of these numbers is at most \\(\\|\\xi\\|^2\\). So the family is countable, and \\(M\\) is \\(\\sigma\\)-finite.\n\nConversely, let \\(M\\) be maximal abelian and \\(\\sigma\\)-finite. By Zorn's lemma choose a maximal family of unit vectors \\(\\xi_k\\) such that the subspaces \\([M\\xi_k]\\) are pairwise orthogonal. The projection \\(p_k\\) onto \\([M\\xi_k]\\) lies in \\(M'\\), because the subspace is invariant under the self-adjoint algebra \\(M\\), and \\(M'=M\\). The \\(p_k\\) are nonzero and pairwise orthogonal, so there are countably many; list them without repetitions as \\(p_1,p_2,\\dots\\), using a finite list and finite sum when the family is finite. Their sum is \\(1\\). Indeed, let \\(\\zeta\\) be orthogonal to every \\([M\\xi_k]\\). Each \\(p_k\\) commutes with \\(M\\), so \\(p_kx\\zeta=xp_k\\zeta=0\\) for \\(x\\in M\\), and \\([M\\zeta]\\) is orthogonal to all \\([M\\xi_k]\\). By maximality \\(\\zeta=0\\). Put \\(\\xi=\\sum_k2^{-k}\\xi_k\\). Then \\(p_k\\xi=2^{-k}\\xi_k\\) and \\(p_k\\in M\\), so \\([M\\xi]\\) contains every \\([M\\xi_k]\\). Hence \\(\\xi\\) is cyclic.\n\n(3) Let \\(\\theta:M\\to N\\) be a \\(*\\)-isomorphism, where \\(M\\subseteq B(H)\\) and \\(N\\subseteq B(K)\\) are maximal abelian. As in the proof of (2), without the countability, choose unit vectors \\(\\xi_k\\), \\(k\\in\\Lambda\\), such that the projections \\(e_k\\) onto \\([M\\xi_k]\\) are pairwise orthogonal, lie in \\(M\\), and have sum \\(1\\). Put \\(f_k=\\theta(e_k)\\), pairwise orthogonal projections in \\(N\\). By Proposition 7.3, applied to the finite partial sums, \\(\\sum_kf_k=\\theta(1)=1\\). Since \\(e_k\\in M=M'\\) and \\(f_k\\in N=N'\\), the algebras \\(M_k=\\{x|_{e_kH}:x\\in M\\}\\) and \\(N_k=\\{y|_{f_kK}:y\\in N\\}\\) are von Neumann algebras on \\(e_kH\\) and \\(f_kK\\), and their commutants are \\(\\{x'|_{e_kH}:x'\\in M'\\}=M_k\\) and \\(N_k\\) (see [The double commutant theorem](the-double-commutant-theorem.md), Theorem 5.8). So both are maximal abelian. The kernel of \\(x\\mapsto x|_{e_kH}\\) is \\(M(1-e_k)\\), which \\(\\theta\\) maps onto \\(N(1-f_k)\\), the kernel of \\(y\\mapsto y|_{f_kK}\\). So \\(\\theta\\) induces a \\(*\\)-isomorphism \\(\\theta_k:M_k\\to N_k\\). The algebra \\(M_k\\) has the cyclic vector \\(\\xi_k\\), so it is \\(\\sigma\\)-finite by (2); \\(\\sigma\\)-finiteness is defined through orthogonal projections, so \\(N_k\\) is \\(\\sigma\\)-finite too, and by (2) it has a cyclic vector. By (1) there is a unitary \\(U_k:e_kH\\to f_kK\\) with \\(\\theta_k(z)=U_kzU_k^*\\). The unitary \\(U=\\bigoplus_kU_k:H\\to K\\) satisfies, for \\(x\\in M\\),\n\\[\n\\begin{gathered}\n\\theta(x)\\\\\n=\\sum_k\\theta(x)f_k\\\\\n=\\sum_k\\theta(xe_k)\\\\\n=\\sum_kU_k(x|_{e_kH})U_k^*\\\\\n=UxU^*,\n\\end{gathered}\n\\]\nwith sums converging strongly. \\(\\square\\)\n\n**Example 7.5.** (a) The algebra \\(\\ell^\\infty(I)\\) on \\(\\ell^2(I)\\), for uncountable \\(I\\), is maximal abelian but not \\(\\sigma\\)-finite, and it has no cyclic vector (Example 1.1). (b) The algebra \\(\\mathcal A_2\\) of Example 1.3 is \\(\\sigma\\)-finite but not maximal abelian, and the isomorphism \\(M_f\\mapsto M_f\\oplus M_f\\) from the algebra of Example 1.2 onto it is not spatial. So maximality cannot be dropped in Theorem 7.4(3). (c) On \\(\\mathbb C^4\\) the map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism of abelian von Neumann algebras that is not spatial, since it sends a projection of rank one to a projection of rank two.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "abelian-operator-algebras",
      "name": "7. Spectra of abelian von Neumann algebras",
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      "source": "src/abelian-operator-algebras.md",
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      "full_conditions_and_proof": "### 7. Spectra of abelian von Neumann algebras\n\nWe can now say which compact spaces are spectra of abelian von Neumann algebras, and describe the algebras as \\(L^\\infty\\) spaces. For a family \\((B_i)_{i\\in I}\\) of C\\*-algebras, \\(\\prod_iB_i\\) denotes the C\\*-algebra of bounded families \\((b_i)\\) with the supremum norm.\n\n**Theorem 7.1** (Abelian von Neumann algebras and hyperstonean spaces). Let \\(A\\) be an abelian C\\*-algebra and \\(\\Omega\\) its spectrum. The following are equivalent.\n\n1. \\(\\Omega\\) is hyperstonean.\n2. \\(A\\) is \\(*\\)-isomorphic to some von Neumann algebra: it has a faithful representation \\(\\pi\\) such that \\(\\pi(A)\\) is a von Neumann algebra.\n3. \\(A\\cong L^\\infty(\\Gamma,\\mu)\\) for a locally compact space \\(\\Gamma\\) and a positive Radon measure \\(\\mu\\) on \\(\\Gamma\\).\n4. \\(A\\cong L^\\infty(\\Gamma,\\mu)\\), where \\(\\Gamma\\) is the union of pairwise disjoint compact open sets \\(\\Gamma_i\\), \\(i\\in I\\), each of them hyperstonean, and \\(\\mu\\) is a Radon measure on \\(\\Gamma\\) whose restriction \\(\\mu_i\\) to each \\(\\Gamma_i\\) is a normal measure with support \\(\\Gamma_i\\).\n\nIn (4) one can take \\(\\Gamma\\) to be an open dense subset of \\(\\Omega\\), and then \\(A\\cong C(\\Omega)\\cong\\prod_iC(\\Gamma_i)\\cong\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\).\n\n*Reference:* [Dixmier, “Sur certains espaces considérés par M. H. Stone” (1951), §5 Theorem 1 and §6 Theorem 2](https://dmitripavlov.org/scans/dixmier.pdf).\n\n**Proof.** If \\(A=0\\), its spectrum is empty, which is hyperstonean; take \\(\\Gamma=\\varnothing\\) and the zero Hilbert space to obtain all four conditions. Assume henceforth that \\(A\\ne0\\). (2)\\(\\Rightarrow\\)(1). We may assume that \\(A=M\\) acts on a Hilbert space \\(H\\) as a von Neumann algebra. The Gelfand isomorphism is an isomorphism of ordered spaces from \\(M_h\\) onto \\(C_{\\mathbb R}(\\Omega)\\), because the positive elements of \\(C(\\Omega)\\) are the nonnegative functions. Let \\(S\\subseteq M_h\\) be nonempty and bounded above by \\(b\\), and fix \\(s_0\\in S\\). The maxima of the finite subsets of \\(S\\) that contain \\(s_0\\) form an increasing net with the same upper bounds as \\(S\\), and it is bounded in norm, since it lies between \\(s_0\\) and \\(b\\). By Lemma 2.4 its strong limit is its least upper bound in \\(M_h\\). So \\(C_{\\mathbb R}(\\Omega)\\) is conditionally complete, and \\(\\Omega\\) is stonean by Theorem 4.3. For \\(\\xi\\in H\\) let \\(\\mu_\\xi\\) be the Radon measure on \\(\\Omega\\) given by the positive functional \\(\\omega_\\xi\\). If \\((x_i)\\) is a bounded increasing net in \\(M_h\\) with least upper bound \\(x\\), then \\(x_i\\to x\\) strongly by Lemma 2.4, so \\(\\omega_\\xi(x_i)\\to\\omega_\\xi(x)\\). So \\(\\mu_\\xi\\) is normal. If \\(x\\in M_+\\) is not zero, then \\(\\omega_\\xi(x)>0\\) for some \\(\\xi\\). So the measures \\(\\mu_\\xi\\) form a sufficient family, and \\(\\Omega\\) is hyperstonean.\n\n(1)\\(\\Rightarrow\\)(4). The space \\(\\Omega\\) is compact, so \\(A\\) is unital and \\(A\\cong C(\\Omega)\\). By Zorn's lemma choose a maximal family \\((\\mu_i)_{i\\in I}\\) of nonzero normal measures with pairwise disjoint supports \\(\\Gamma_i\\). These supports are clopen (Corollary 5.4(3)), so \\(\\Gamma=\\bigcup_i\\Gamma_i\\) is open. It is dense. Otherwise the clopen set \\(W=\\Omega\\setminus\\overline\\Gamma\\) is not empty, some normal measure \\(\\mu\\) has \\(\\mu(W)>0\\) because \\(\\Omega\\) is hyperstonean, and the restriction of \\(\\mu\\) to \\(W\\) is a nonzero normal measure whose support lies in \\(W\\); adding it to the family contradicts maximality.\n\nRestriction \\(x\\mapsto(x|_{\\Gamma_i})_i\\) is a \\(*\\)-homomorphism from \\(C(\\Omega)\\) to \\(\\prod_iC(\\Gamma_i)\\). It is isometric because \\(\\Gamma\\) is dense. It is onto: a bounded family \\((x_i)\\) defines a bounded continuous function on the open set \\(\\Gamma\\), since the \\(\\Gamma_i\\) are open, and Theorem 4.4 extends it to \\(\\Omega\\). Each \\(\\Gamma_i\\) is clopen, hence stonean, and \\(\\mu_i\\) is a normal measure on it with full support; so \\(\\Gamma_i\\) is hyperstonean by Lemma 6.2. By Theorem 5.5, \\(C(\\Gamma_i)\\cong L^\\infty(\\Gamma_i,\\mu_i)\\). The subspace \\(\\Gamma\\) is locally compact and is the topological disjoint union of the compact open sets \\(\\Gamma_i\\). For \\(x\\in C_c(\\Gamma)\\) the compact support of \\(x\\) meets only finitely many \\(\\Gamma_i\\), so \\(\\mu(x)=\\sum_i\\int_{\\Gamma_i}x\\,d\\mu_i\\) is a finite sum. It is a positive linear functional on \\(C_c(\\Gamma)\\), hence a Radon measure on \\(\\Gamma\\), and its restriction to \\(\\Gamma_i\\) is \\(\\mu_i\\). A set \\(N\\subseteq\\Gamma\\) is locally \\(\\mu\\)-null if and only if every \\(N\\cap\\Gamma_i\\) is \\(\\mu_i\\)-null, and a function on \\(\\Gamma\\) is \\(\\mu\\)-measurable if and only if each restriction to \\(\\Gamma_i\\) is \\(\\mu_i\\)-measurable, because every compact subset of \\(\\Gamma\\) lies in finitely many \\(\\Gamma_i\\). So restriction is an isometric \\(*\\)-isomorphism of \\(L^\\infty(\\Gamma,\\mu)\\) onto \\(\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\). Composing, \\[\n\\begin{gathered}\nA\\\\\n\\cong C(\\Omega)\\\\\n\\cong\\prod_iC(\\Gamma_i)\\\\\n\\cong\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\\\\n\\cong L^\\infty(\\Gamma,\\mu).\n\\end{gathered}\n\\]\n\n(4)\\(\\Rightarrow\\)(3) is clear.\n\n(3)\\(\\Rightarrow\\)(2). The representation of \\(L^\\infty(\\Gamma,\\mu)\\) on \\(L^2(\\Gamma,\\mu)\\) by multiplication operators is faithful, and its image is a maximal abelian von Neumann algebra, for every Radon measure; both facts are proved in Proposition 3.2a. \\(\\square\\)\n\n**Remark 7.2** (Examples). The spectrum of \\(\\ell^\\infty(I)\\) is \\(\\beta I\\), and its normal measures were found in Example 5.8. The spectrum of \\(L^\\infty[0,1]\\) is a hyperstonean space without isolated points, since \\(L^\\infty[0,1]\\) has no minimal projections (Exercise 12.3). Every abelian von Neumann algebra of infinite dimension has an infinite hyperstonean space as its spectrum, and such a space is never metrizable (Exercise 12.1). The algebra \\(C[0,1]\\) is isomorphic to no von Neumann algebra, since \\([0,1]\\) is not stonean. Section 9 gives an abelian C\\*-algebra whose spectrum is stonean but carries no normal measure at all; it is monotone complete but isomorphic to no von Neumann algebra.\n\n**Proposition 7.3** (Isomorphisms preserve monotone limits). Let \\(\\theta:M\\to N\\) be a \\(*\\)-isomorphism between von Neumann algebras on Hilbert spaces \\(H\\) and \\(K\\). Then \\(\\theta\\) maps \\(M_+\\) onto \\(N_+\\), and for every bounded increasing net \\((x_i)\\) in \\(M_h\\) with strong limit \\(x\\), the net \\((\\theta(x_i))\\) converges strongly to \\(\\theta(x)\\). In particular \\(\\omega_\\eta\\circ\\theta\\) preserves the limits of such nets, for every \\(\\eta\\in K\\).\n\n**Proof.** The positive elements of a C\\*-algebra are the elements \\(y^*y\\), so \\(\\theta(M_+)\\subseteq N_+\\) and \\(\\theta^{-1}(N_+)\\subseteq M_+\\). Hence \\(\\theta\\) is an isomorphism of ordered spaces from \\(M_h\\) onto \\(N_h\\). By Lemma 2.4, the net \\((x_i)\\) has \\(x\\) as least upper bound in \\(M_h\\), and \\((\\theta(x_i))\\) has its strong limit \\(y\\) as least upper bound in \\(N_h\\). Since \\(\\theta(x)\\) is an upper bound of \\((\\theta(x_i))\\), \\(y\\le\\theta(x)\\). Since \\(\\theta^{-1}(y)\\) is an upper bound of \\((x_i)\\), \\(x\\le\\theta^{-1}(y)\\), that is, \\(\\theta(x)\\le y\\). \\(\\square\\)\n\nSo \\(*\\)-isomorphisms of von Neumann algebras need no continuity assumption: they automatically respect monotone limits.\n\n**Theorem 7.4** (Cyclic vectors and spatial isomorphisms).\n\n1. Let \\(M\\subseteq B(H)\\) and \\(N\\subseteq B(K)\\) be abelian von Neumann algebras with cyclic vectors \\(\\xi\\) and \\(\\eta\\). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) is spatial: there is a unitary \\(U:H\\to K\\) with \\(\\theta(x)=UxU^*\\) for all \\(x\\in M\\).\n2. An abelian von Neumann algebra has a cyclic vector exactly when it is both maximal abelian and \\(\\sigma\\)-finite.\n3. Every \\(*\\)-isomorphism between maximal abelian von Neumann algebras is spatial.\n\n**Proof.** If one algebra or Hilbert space is zero, the isomorphism forces both algebras to be zero; their units are the identity operators, so both Hilbert spaces are zero and the unique empty unitary applies. The zero vector is cyclic and the zero algebra is maximal abelian and sigma-finite. We may therefore assume the spaces in use are nonzero. (1) Let \\(\\Omega\\) be the spectrum of \\(M\\), which is stonean by Theorem 7.1, and \\(x\\mapsto\\hat x\\) the Gelfand isomorphism of \\(M\\) onto \\(C(\\Omega)\\). Let \\(\\mu\\) and \\(\\nu\\) be the Radon measures on \\(\\Omega\\) with \\(\\int\\hat x\\,d\\mu=\\langle x\\xi,\\xi\\rangle\\) and \\(\\int\\hat x\\,d\\nu=\\langle\\theta(x)\\eta,\\eta\\rangle\\); the second functional is positive by Proposition 7.3. Both measures are normal, by Lemma 2.4 and Proposition 7.3, because the supremum in \\(C_{\\mathbb R}(\\Omega)\\) of an increasing net that is bounded in norm corresponds to its least upper bound in \\(M_h\\). Both have support \\(\\Omega\\). Indeed, \\(\\xi\\) is cyclic for \\(M\\), hence for the larger algebra \\(M'\\), so \\(\\xi\\) separates \\(M\\) (see [The double commutant theorem](the-double-commutant-theorem.md), Proposition 9.2). Thus \\(\\langle x\\xi,\\xi\\rangle=\\|x^{1/2}\\xi\\|^2>0\\) for nonzero \\(x\\in M_+\\). In the same way \\(\\eta\\) separates \\(N\\), and \\(\\theta\\) is injective. If the support of a positive functional on \\(C(\\Omega)\\) were not all of \\(\\Omega\\), Urysohn's lemma would give a nonzero \\(x\\ge0\\) that vanishes on the support, with integral \\(0\\). By Corollary 5.4(4), \\(\\mu\\) and \\(\\nu\\) have the same null sets. By Corollary 3.4 there is a unitary \\(V:L^2(\\Omega,\\mu)\\to L^2(\\Omega,\\nu)\\) with \\(V\\pi_\\mu(f)=\\pi_\\nu(f)V\\). The representations \\(\\hat x\\mapsto x\\) on \\(H\\) and \\(\\hat x\\mapsto\\theta(x)\\) on \\(K\\) have the cyclic vectors \\(\\xi\\) and \\(\\eta\\), so Theorem 3.1(2) gives unitaries \\(U_1:L^2(\\Omega,\\mu)\\to H\\) and \\(U_2:L^2(\\Omega,\\nu)\\to K\\) with \\(U_1\\pi_\\mu(\\hat x)U_1^*=x\\) and \\(U_2\\pi_\\nu(\\hat x)U_2^*=\\theta(x)\\). Then \\(U=U_2VU_1^*\\) satisfies \\[\n\\begin{gathered}\nUxU^*\\\\\n=U_2V\\pi_\\mu(\\hat x)V^*U_2^*\\\\\n=U_2\\pi_\\nu(\\hat x)U_2^*\\\\\n=\\theta(x).\n\\end{gathered}\n\\]\n\n(2) Let \\(M\\) have a cyclic vector \\(\\xi\\). It is maximal abelian by Corollary 3.3. If \\((p_j)\\) are pairwise orthogonal nonzero projections in \\(M\\), then \\(\\langle p_j\\xi,\\xi\\rangle>0\\) because \\(\\xi\\) separates \\(M\\), and every finite sum of these numbers is at most \\(\\|\\xi\\|^2\\). So the family is countable, and \\(M\\) is \\(\\sigma\\)-finite.\n\nConversely, let \\(M\\) be maximal abelian and \\(\\sigma\\)-finite. By Zorn's lemma choose a maximal family of unit vectors \\(\\xi_k\\) such that the subspaces \\([M\\xi_k]\\) are pairwise orthogonal. The projection \\(p_k\\) onto \\([M\\xi_k]\\) lies in \\(M'\\), because the subspace is invariant under the self-adjoint algebra \\(M\\), and \\(M'=M\\). The \\(p_k\\) are nonzero and pairwise orthogonal, so there are countably many; list them without repetitions as \\(p_1,p_2,\\dots\\), using a finite list and finite sum when the family is finite. Their sum is \\(1\\). Indeed, let \\(\\zeta\\) be orthogonal to every \\([M\\xi_k]\\). Each \\(p_k\\) commutes with \\(M\\), so \\(p_kx\\zeta=xp_k\\zeta=0\\) for \\(x\\in M\\), and \\([M\\zeta]\\) is orthogonal to all \\([M\\xi_k]\\). By maximality \\(\\zeta=0\\). Put \\(\\xi=\\sum_k2^{-k}\\xi_k\\). Then \\(p_k\\xi=2^{-k}\\xi_k\\) and \\(p_k\\in M\\), so \\([M\\xi]\\) contains every \\([M\\xi_k]\\). Hence \\(\\xi\\) is cyclic.\n\n(3) Let \\(\\theta:M\\to N\\) be a \\(*\\)-isomorphism, where \\(M\\subseteq B(H)\\) and \\(N\\subseteq B(K)\\) are maximal abelian. As in the proof of (2), without the countability, choose unit vectors \\(\\xi_k\\), \\(k\\in\\Lambda\\), such that the projections \\(e_k\\) onto \\([M\\xi_k]\\) are pairwise orthogonal, lie in \\(M\\), and have sum \\(1\\). Put \\(f_k=\\theta(e_k)\\), pairwise orthogonal projections in \\(N\\). By Proposition 7.3, applied to the finite partial sums, \\(\\sum_kf_k=\\theta(1)=1\\). Since \\(e_k\\in M=M'\\) and \\(f_k\\in N=N'\\), the algebras \\(M_k=\\{x|_{e_kH}:x\\in M\\}\\) and \\(N_k=\\{y|_{f_kK}:y\\in N\\}\\) are von Neumann algebras on \\(e_kH\\) and \\(f_kK\\), and their commutants are \\(\\{x'|_{e_kH}:x'\\in M'\\}=M_k\\) and \\(N_k\\) (see [The double commutant theorem](the-double-commutant-theorem.md), Theorem 5.8). So both are maximal abelian. The kernel of \\(x\\mapsto x|_{e_kH}\\) is \\(M(1-e_k)\\), which \\(\\theta\\) maps onto \\(N(1-f_k)\\), the kernel of \\(y\\mapsto y|_{f_kK}\\). So \\(\\theta\\) induces a \\(*\\)-isomorphism \\(\\theta_k:M_k\\to N_k\\). The algebra \\(M_k\\) has the cyclic vector \\(\\xi_k\\), so it is \\(\\sigma\\)-finite by (2); \\(\\sigma\\)-finiteness is defined through orthogonal projections, so \\(N_k\\) is \\(\\sigma\\)-finite too, and by (2) it has a cyclic vector. By (1) there is a unitary \\(U_k:e_kH\\to f_kK\\) with \\(\\theta_k(z)=U_kzU_k^*\\). The unitary \\(U=\\bigoplus_kU_k:H\\to K\\) satisfies, for \\(x\\in M\\),\n\\[\n\\begin{gathered}\n\\theta(x)\\\\\n=\\sum_k\\theta(x)f_k\\\\\n=\\sum_k\\theta(xe_k)\\\\\n=\\sum_kU_k(x|_{e_kH})U_k^*\\\\\n=UxU^*,\n\\end{gathered}\n\\]\nwith sums converging strongly. \\(\\square\\)\n\n**Example 7.5.** (a) The algebra \\(\\ell^\\infty(I)\\) on \\(\\ell^2(I)\\), for uncountable \\(I\\), is maximal abelian but not \\(\\sigma\\)-finite, and it has no cyclic vector (Example 1.1). (b) The algebra \\(\\mathcal A_2\\) of Example 1.3 is \\(\\sigma\\)-finite but not maximal abelian, and the isomorphism \\(M_f\\mapsto M_f\\oplus M_f\\) from the algebra of Example 1.2 onto it is not spatial. So maximality cannot be dropped in Theorem 7.4(3). (c) On \\(\\mathbb C^4\\) the map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism of abelian von Neumann algebras that is not spatial, since it sends a projection of rank one to a projection of rank two.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "abelian-operator-algebras",
      "name": "8. Countably generated abelian von Neumann algebras",
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      "full_conditions_and_proof": "### 8. Countably generated abelian von Neumann algebras\n\nThis section shows that an abelian von Neumann algebra generated by countably many operators is generated by one self-adjoint operator, and that the nonzero \\(\\sigma\\)-finite ones without minimal projections are all isomorphic to \\(L^\\infty[0,1]\\). Separability of the Hilbert space is one way to ensure countable generation, but it is not needed.\n\n**Lemma 8.1** (Singly generated algebras of functions). Let \\(\\Gamma\\) be a compact space.\n\n1. \\(C(\\Gamma)\\) is generated as a C\\*-algebra by a single self-adjoint element if and only if \\(\\Gamma\\) is homeomorphic to a compact subset of \\(\\mathbb R\\).\n2. Suppose a sequence \\((E_n)\\) of clopen sets separates the points of \\(\\Gamma\\): for \\(\\gamma\\ne\\gamma'\\) some \\(E_n\\) contains exactly one of them. Then \\(a=\\sum_n3^{-n}(2\\cdot1_{E_n}-1)\\) generates \\(C(\\Gamma)\\) as a C\\*-algebra.\n\n**Proof.** If \\(\\Gamma=\\varnothing\\), then \\(C(\\Gamma)=0\\), generated by its zero self-adjoint element, and \\(\\Gamma\\) is the empty compact subset of \\(\\mathbb R\\); (2) also gives the zero generator. Assume now that \\(\\Gamma\\ne\\varnothing\\). (1) Let \\(a\\) generate \\(C(\\Gamma)\\). The C\\*-algebra generated by \\(a\\) consists of uniform limits of polynomials in \\(a\\) without constant term, so its members are constant on the level sets of \\(a\\). It is all of \\(C(\\Gamma)\\), which separates the points of \\(\\Gamma\\) by Urysohn's lemma. So the real function \\(a\\) is injective, and a continuous injection of a compact space into \\(\\mathbb R\\) is a homeomorphism onto its image. Conversely, let \\(h:\\Gamma\\to\\mathbb R\\) be a homeomorphism onto its image, and put \\(a=h-\\min h+1\\). The C\\*-algebra generated by \\(a\\) separates points and vanishes nowhere, so it is \\(C(\\Gamma)\\) by the Stone–Weierstrass theorem.\n\n(2) The series converges uniformly, so \\(a\\) is continuous and real. Let \\(\\gamma\\ne\\gamma'\\), and let \\(n\\) be the first index for which \\(E_n\\) contains exactly one of them. The terms before \\(n\\) agree at \\(\\gamma\\) and \\(\\gamma'\\), so\n\\[\n\\begin{gathered}\n|a(\\gamma)-a(\\gamma')|\\\\\n\\ge2\\cdot3^{-n}-\\sum_{k>n}2\\cdot3^{-k}\\\\\n=3^{-n}>0 .\n\\end{gathered}\n\\]\nSo \\(a\\) is injective, and \\(|a|\\ge\\frac13-\\sum_{k\\ge2}3^{-k}=\\frac16\\) everywhere. The proof of (1) applies to \\(a\\) directly. \\(\\square\\)\n\n**Theorem 8.2** (Countable generation). For an abelian von Neumann algebra \\(M\\) the following are equivalent.\n\n1. \\(M\\) is generated by countably many elements.\n2. \\(M\\) is generated by countably many projections.\n3. \\(M\\) is generated by one self-adjoint element.\n\nOn a separable Hilbert space, every abelian von Neumann algebra has these properties.\n\n**Proof.** For \\(M=0\\), all three assertions hold with the zero generator and zero projection. Assume \\(M\\ne0\\); finite generating lists can be extended to sequences by appending zeros. (3)\\(\\Rightarrow\\)(1) is clear.\n\n(1)\\(\\Rightarrow\\)(2). Replacing each generator \\(x\\) by \\(\\frac12(x+x^*)\\) and \\(\\frac1{2i}(x-x^*)\\), we may assume the generators \\(h_1,h_2,\\dots\\) are self-adjoint. By Theorem 7.1 the spectrum \\(\\Omega\\) of \\(M\\) is stonean; write \\(\\hat x\\) for the Gelfand transform. For self-adjoint \\(h\\in M\\) and rational \\(\\lambda\\), let \\(e_h(\\lambda)\\in M\\) be the projection whose transform is the indicator of the clopen set \\(C_\\lambda=\\overline{\\{\\hat h<\\lambda\\}}\\). Then \\(\\{\\hat h<\\lambda\\}\\subseteq C_\\lambda\\subseteq\\{\\hat h\\le\\lambda\\}\\). Given \\(\\varepsilon>0\\), choose rationals \\(\\lambda_0<\\lambda_1<\\dots<\\lambda_N\\) with \\(\\lambda_0<\\min\\hat h\\), \\(\\lambda_N>\\max\\hat h\\) and \\(\\lambda_k-\\lambda_{k-1}<\\varepsilon\\). Then \\(C_{\\lambda_0}=\\varnothing\\) and \\(C_{\\lambda_N}=\\Omega\\). On \\(C_{\\lambda_k}\\setminus C_{\\lambda_{k-1}}\\) we have \\(\\lambda_{k-1}\\le\\hat h\\le\\lambda_k\\). Hence\n\\[\n\\Big\\|h-\\sum_{k=1}^N\\lambda_k\\big(e_h(\\lambda_k)-e_h(\\lambda_{k-1})\\big)\\Big\\|\\le\\varepsilon .\n\\]\nSo each \\(h_n\\) is a norm limit of linear combinations of the countably many projections \\(e_{h_n}(\\lambda)\\), and these projections generate \\(M\\).\n\n(2)\\(\\Rightarrow\\)(3). Let the projections \\(P_1,P_2,\\dots\\) generate \\(M\\), and put \\(a=\\sum_k3^{-k}(2P_k-1)\\in M_h\\). Let \\(C\\) be the C\\*-algebra generated by \\(1\\) and the \\(P_k\\), and \\(\\Gamma\\) its spectrum. The transforms of the \\(P_k\\) are indicators of clopen sets \\(E_k\\), and they separate the points of \\(\\Gamma\\), since a character of \\(C\\) is determined by its values on generators. By Lemma 8.1(2) the transform of \\(a\\) generates \\(C(\\Gamma)\\), so every \\(P_k\\) lies in the C\\*-algebra generated by \\(a\\). Hence \\(M\\) is generated by \\(a\\).\n\n*Separable Hilbert spaces.* Let \\((\\zeta_k)\\) be dense in \\(H\\). On the unit ball of \\(B(H)\\), the strong topology is the topology of the map \\(b\\mapsto(b\\zeta_k)_k\\) into \\(H^{\\mathbb N}\\), because on bounded sets strong convergence follows from convergence on a dense set. The space \\(H^{\\mathbb N}\\) is separable and metrizable, and so is every subspace of it. So the unit ball of \\(M\\) has a countable strongly dense subset. The von Neumann algebra that this subset generates contains the unit ball of \\(M\\), hence equals \\(M\\). \\(\\square\\)\n\n**Lemma 8.3** (Diffuse measures on the unit interval). Let \\(\\nu\\) be a Radon probability measure on \\([0,1]\\) with no atoms, let \\(m\\) be Lebesgue measure, and put \\(F(t)=\\nu([0,t])\\). Then \\(V\\varphi=\\varphi\\circ F\\) defines a unitary \\(V:L^2([0,1],m)\\to L^2([0,1],\\nu)\\), and\n\\[\n\\begin{gathered}\nV\\{M_\\varphi:\\varphi\\in L^\\infty(m)\\}V^*\\\\\n=\\{M_f:f\\in L^\\infty(\\nu)\\},\\\\\nVM_\\varphi V^*\\\\\n=M_{\\varphi\\circ F}.\n\\end{gathered}\n\\tag{8.1}\n\\]\nIn particular \\(L^\\infty([0,1],\\nu)\\) is isomorphic to \\(L^\\infty[0,1]\\).\n\n**Proof.** Since \\(\\nu\\) has no atoms, \\(F\\) is continuous and nondecreasing, with \\(F(0)=0\\) and \\(F(1)=1\\).\n\n*The image of \\(\\nu\\) under \\(F\\) is \\(m\\).* For \\(s\\in[0,1]\\) let \\(t_s\\) be the largest \\(t\\) with \\(F(t)\\le s\\). Then \\(\\{F\\le s\\}=[0,t_s]\\), and \\(F(t_s)=s\\): this is clear if \\(t_s=1\\), and otherwise \\(F>s\\) to the right of \\(t_s\\), so \\(F(t_s)\\ge s\\) by continuity. Hence \\(\\nu(\\{F\\le s\\})=s=m([0,s])\\). The intervals \\([0,s]\\) are closed under intersections and generate the Borel sets, so \\(\\nu(F^{-1}(B))=m(B)\\) for every Borel set \\(B\\), by the [finite-measure uniqueness lemma in the double-commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-18). Consequently \\(\\int\\varphi\\circ F\\,d\\nu=\\int\\varphi\\,dm\\) for bounded or nonnegative Borel \\(\\varphi\\), and \\(V\\) is a well-defined isometry.\n\n*\\(V\\) is onto.* Let \\(q(s)\\) be the least \\(t\\) with \\(F(t)\\ge s\\). It is nondecreasing, hence Borel. For every \\(t\\), \\(q(F(t))\\le t\\) and \\(F(q(F(t)))=F(t)\\). If \\(q(F(t))<t\\), then \\(F\\) is constant on the nondegenerate interval \\([q(F(t)),t]\\), and \\(t\\) is not the left end of the maximal interval of constancy that contains it. The maximal nondegenerate intervals of constancy are disjoint, so there are countably many, and each has \\(\\nu\\)-measure \\(0\\), because \\(\\nu\\) has no atoms. So \\(q\\circ F=\\mathrm{id}\\) \\(\\nu\\)-almost everywhere. For a polynomial \\(p\\), the function \\(p\\circ q\\) is bounded and Borel, and \\(V(p\\circ q)=p\\) \\(\\nu\\)-almost everywhere. So the range of \\(V\\) contains the polynomials. These are dense in \\(C[0,1]\\) and hence in \\(L^2([0,1],\\nu)\\) (Lemma 2.3(3)). The range of an isometry is closed, so \\(V\\) is onto.\n\n*The algebras.* Clearly \\(VM_\\varphi=M_{\\varphi\\circ F}V\\) for bounded Borel \\(\\varphi\\). So \\(V\\{M_\\varphi\\}V^*\\) is a maximal abelian algebra (Theorem 3.1(3)) contained in the abelian algebra \\(\\{M_f:f\\in L^\\infty(\\nu)\\}\\). If \\(\\mathcal A\\subseteq\\mathcal B\\) with \\(\\mathcal A\\) maximal abelian and \\(\\mathcal B\\) abelian, then \\(\\mathcal B\\subseteq\\mathcal A'=\\mathcal A\\). This gives (8.1). \\(\\square\\)\n\n**Theorem 8.4** (Diffuse countably generated algebras). For an abelian von Neumann algebra \\(M\\) the following are equivalent.\n\n1. \\(M\\) is isomorphic to \\(L^\\infty[0,1]\\), with Lebesgue measure.\n2. \\(M\\ne0\\), and \\(M\\) is countably generated, \\(\\sigma\\)-finite, and without minimal projections.\n\nIn particular, if a nonzero separable Hilbert space carries an abelian von Neumann algebra without minimal projections, that algebra is isomorphic to \\(L^\\infty[0,1]\\). The condition \\(M\\ne0\\) excludes the zero algebra on the zero space, which has the other three properties.\n\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). *Step 1: a cyclic vector.* Let \\(M\\) act on \\(H\\). Choose a maximal family of unit vectors \\(\\xi_k\\) such that the subspaces \\([M'\\xi_k]\\) are pairwise orthogonal. Their projections \\(p_k\\) lie in \\(M''=M\\) and are pairwise orthogonal, so they are countably many, listed without repetitions and with finite sums for a finite family, and their sum is \\(1\\) by maximality. Put \\(\\xi=\\sum_k2^{-k}\\xi_k\\). Since \\(M\\) is abelian, \\(p_k\\in M'\\), and \\(p_k\\xi=2^{-k}\\xi_k\\); so \\([M'\\xi]\\) contains every \\([M'\\xi_k]\\), and \\(\\xi\\) is cyclic for \\(M'\\). Hence \\(\\xi\\) separates \\(M\\). Let \\(e\\) be the projection onto \\([M\\xi]\\); it lies in \\(M'\\). The map \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-isomorphism of \\(M\\) onto the von Neumann algebra \\(M_e=\\{x|_{eH}:x\\in M\\}\\) (see [The double commutant theorem](the-double-commutant-theorem.md), Theorem 5.8); it is injective because \\(xe\\xi=x\\xi=0\\) forces \\(x=0\\). The vector \\(\\xi\\) is cyclic for \\(M_e\\). The algebra \\(M_e\\) again has the properties in (2). Being \\(\\sigma\\)-finite and having minimal projections are properties defined through projections, so they pass through \\(*\\)-isomorphisms. By Theorem 8.2, \\(M\\) is generated by one self-adjoint element \\(a\\), and \\(a|_{eH}\\) generates \\(M_e\\). Indeed, the restriction map is strongly continuous and maps the \\(*\\)-algebra generated by \\(1\\) and \\(a\\) onto the one generated by \\(1\\) and \\(a|_{eH}\\); since \\(M\\) is the strong closure of the former, \\(M_e\\) lies in the von Neumann algebra generated by \\(a|_{eH}\\), and the converse inclusion is clear. So we may assume that \\(M\\) has a cyclic vector \\(\\xi\\) with \\(\\|\\xi\\|=1\\).\n\n*Step 2: a multiplication algebra.* By Theorem 8.2, \\(M\\) is generated by a self-adjoint \\(a\\). \\(M\\) is not \\(\\mathbb C1\\), since \\(1\\) would then be a minimal projection. After an affine change we may assume that the spectrum \\(\\sigma(a)\\) lies in \\([0,1]\\). The continuous functional calculus identifies \\(C(\\sigma(a))\\) with the C\\*-algebra generated by \\(1\\) and \\(a\\). Let \\(\\nu\\) be the Radon probability measure on \\(\\sigma(a)\\), extended by \\(0\\) to \\([0,1]\\), with \\(\\int g\\,d\\nu=\\langle g(a)\\xi,\\xi\\rangle\\). The vector \\(\\xi\\) is cyclic for the C\\*-algebra generated by \\(1\\) and \\(a\\), because its strong closure is \\(M\\). Theorem 3.1(2) gives a unitary \\(U:L^2([0,1],\\nu)\\to H\\) with \\(UM_gU^*=g(a)\\) for continuous \\(g\\). By Theorem 3.1(3), \\(U\\{M_f:f\\in L^\\infty(\\nu)\\}U^*\\) is the von Neumann algebra generated by \\(a\\), which is \\(M\\).\n\n*Step 3: no atoms.* If \\(\\nu(\\{t\\})>0\\), then \\(M_{1_{\\{t\\}}}\\) is a nonzero projection, and \\(M_fM_{1_{\\{t\\}}}=f(t)M_{1_{\\{t\\}}}\\) for every \\(f\\). So it is a minimal projection, and so is its image in \\(M\\), which is excluded.\n\n*Step 4.* By Lemma 8.3, \\(M\\cong L^\\infty([0,1],\\nu)\\cong L^\\infty[0,1]\\).\n\n(1)\\(\\Rightarrow\\)(2). The algebra \\(L^\\infty[0,1]\\) on \\(L^2[0,1]\\) is \\(\\sigma\\)-finite, because \\(\\sum_jm(E_j)\\le1\\) for disjoint sets. It has no minimal projection: if \\(m(E)>0\\), the continuous function \\(t\\mapsto m(E\\cap[0,t])\\) takes the value \\(\\frac12m(E)\\), so \\(E\\) splits into two sets of positive measure. Both properties are defined through projections, so they pass to every algebra \\(M\\) isomorphic to \\(L^\\infty[0,1]\\). Let \\(\\theta:L^\\infty[0,1]\\to M\\) be a \\(*\\)-isomorphism and \\(a=\\theta(\\iota)\\), where \\(\\iota(t)=t\\). Let \\(N\\subseteq M\\) be the von Neumann algebra generated by \\(a\\), and \\(B=\\theta^{-1}(N)\\). Then \\(B\\) is a C\\*-subalgebra of \\(L^\\infty[0,1]\\) that contains \\(1\\) and \\(\\iota\\), hence all continuous functions. Whenever an increasing net in \\(B_h\\) is bounded in norm, its least upper bound lies in \\(B\\): the image net in \\(N\\) converges strongly to an element of \\(N\\), and by Proposition 7.3 its preimage is the least upper bound in \\(L^\\infty[0,1]\\). Let \\(\\mathcal S\\) be the family of Borel sets \\(E\\) with \\(1_E\\in B\\). For open \\(U\\ne[0,1]\\) the functions \\(g_n(t)=\\min\\{1,n\\operatorname{dist}(t,[0,1]\\setminus U)\\}\\) are continuous and increase to \\(1_U\\), and \\(M_{g_n}\\to M_{1_U}\\) strongly by dominated convergence, so \\(U\\in\\mathcal S\\); and \\([0,1]\\in\\mathcal S\\) because \\(1\\in B\\). The family \\(\\mathcal S\\) is closed under complements (\\(1-1_E\\)), finite intersections (products) and increasing countable unions (least upper bounds), so it is a \\(\\sigma\\)-algebra containing the open sets, and it contains every Borel set. Then \\(B\\) contains all simple Borel functions and, being norm closed, all of \\(L^\\infty[0,1]\\). So \\(N=M\\), and \\(M\\) is generated by \\(a\\).\n\nThe last statement follows from Theorem 8.2 and the fact that von Neumann algebras acting on separable Hilbert spaces are \\(\\sigma\\)-finite: an uncountable family of pairwise orthogonal nonzero projections would give an uncountable orthonormal family of vectors. \\(\\square\\)\n\n**Example 8.5** (Each hypothesis is needed).\n\n(a) *\\(\\sigma\\)-finiteness.* Let \\(M=\\prod_{t\\in[0,1]}L^\\infty[0,1]\\), acting on the Hilbert sum \\(\\bigoplus_{t\\in[0,1]}L^2[0,1]\\) by \\((f_t)_t\\mapsto\\bigoplus_tM_{f_t}\\). It is a von Neumann algebra, because each summand is maximal abelian and the commutant of a direct sum is the direct sum of the commutants ([The double commutant theorem](the-double-commutant-theorem.md), Proposition 5.2). It has no minimal projection: a nonzero projection of \\(M\\) has a coordinate \\(1_E\\) with \\(m(E)>0\\), and \\(E\\) splits into two sets of positive measure, as in the proof of Theorem 8.4. It is not \\(\\sigma\\)-finite: the projections onto the summands are uncountably many. It is countably generated. Let \\(a\\) act as the scalar \\(t\\) on the summand with index \\(t\\), and \\(b\\) as \\(M_\\iota\\) on every summand. An operator commuting with \\(a\\) maps each eigenspace \\(\\ker(a-t)\\), which is the summand with index \\(t\\), into itself, so the projections onto the summands lie in the von Neumann algebra generated by \\(a\\). An operator commuting with \\(a\\) and \\(b\\) is therefore a bounded family of operators that commute with \\(M_\\iota\\), hence of multiplication operators (Theorem 3.1(3)). So the commutant of \\(\\{a,b\\}\\) is \\(M\\), and \\(\\{a,b\\}''=M'=M\\). So \\(M\\) is not isomorphic to \\(L^\\infty[0,1]\\).\n\n(b) *Countable generation.* Let \\(J\\) be uncountable, \\(P\\) the product of the fair coin measures on \\(\\{0,1\\}^J\\), defined on the product \\(\\sigma\\)-algebra, and \\(M=L^\\infty(P)\\) on \\(L^2(P)\\). By Remark 3.2 it is maximal abelian, and \\(1\\) is a cyclic vector because \\(L^\\infty(P)\\) is dense in \\(L^2(P)\\); so it is \\(\\sigma\\)-finite (Theorem 7.4(2)). It has no minimal projection. Every measurable set \\(E\\) depends only on the coordinates in some countable set \\(J_E\\), because the sets with this property form a \\(\\sigma\\)-algebra containing the cylinder sets. For \\(j\\notin J_E\\), independence of the coordinates gives \\(P(E\\cap\\{\\gamma_j=0\\})=\\frac12P(E)\\), which splits \\(E\\). Finally, \\(M\\) is not countably generated. Otherwise \\(M\\) is generated by one self-adjoint \\(a\\) (Theorem 8.2), and \\(L^2(P)=[M1]\\) is the closure of the set of vectors \\(p(a)1\\), with \\(p\\) a polynomial whose coefficients have rational real and imaginary parts; so \\(L^2(P)\\) would be separable. But it is not ([The double commutant theorem](the-double-commutant-theorem.md), Exercise 9.9).\n\n(c) *No minimal projections.* \\(\\ell^\\infty(\\mathbb N)\\) is countably generated and \\(\\sigma\\)-finite, but each point gives a minimal projection.\n\n## D. Find the limits of the measure picture\n\nAn algebra can have every bounded increasing supremum and still fail to be a von Neumann algebra. To see the obstruction rather than just its name, partition a stonean spectrum by where normal measures can live, then build the other regions using Borel functions modulo meager functions. Here the negligible sets come from category, not from a fixed measure. The quotient remains a C*-algebra and is order complete, but its normal observations can all vanish.\n\nThis construction also makes the three-region decomposition useful: it predicts precisely which part an example realizes. The final extension theorem asks a different question—when continuous functions defined on a closed subspace can be extended by a positive norm-one operator—and identifies stonean spaces through that property.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AO-15",
      "unit": "abelian-operator-algebras",
      "name": "8. Countably generated abelian von Neumann algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "full_conditions_and_proof": "### 8. Countably generated abelian von Neumann algebras\n\nThis section shows that an abelian von Neumann algebra generated by countably many operators is generated by one self-adjoint operator, and that the nonzero \\(\\sigma\\)-finite ones without minimal projections are all isomorphic to \\(L^\\infty[0,1]\\). Separability of the Hilbert space is one way to ensure countable generation, but it is not needed.\n\n**Lemma 8.1** (Singly generated algebras of functions). Let \\(\\Gamma\\) be a compact space.\n\n1. \\(C(\\Gamma)\\) is generated as a C\\*-algebra by a single self-adjoint element if and only if \\(\\Gamma\\) is homeomorphic to a compact subset of \\(\\mathbb R\\).\n2. Suppose a sequence \\((E_n)\\) of clopen sets separates the points of \\(\\Gamma\\): for \\(\\gamma\\ne\\gamma'\\) some \\(E_n\\) contains exactly one of them. Then \\(a=\\sum_n3^{-n}(2\\cdot1_{E_n}-1)\\) generates \\(C(\\Gamma)\\) as a C\\*-algebra.\n\n**Proof.** If \\(\\Gamma=\\varnothing\\), then \\(C(\\Gamma)=0\\), generated by its zero self-adjoint element, and \\(\\Gamma\\) is the empty compact subset of \\(\\mathbb R\\); (2) also gives the zero generator. Assume now that \\(\\Gamma\\ne\\varnothing\\). (1) Let \\(a\\) generate \\(C(\\Gamma)\\). The C\\*-algebra generated by \\(a\\) consists of uniform limits of polynomials in \\(a\\) without constant term, so its members are constant on the level sets of \\(a\\). It is all of \\(C(\\Gamma)\\), which separates the points of \\(\\Gamma\\) by Urysohn's lemma. So the real function \\(a\\) is injective, and a continuous injection of a compact space into \\(\\mathbb R\\) is a homeomorphism onto its image. Conversely, let \\(h:\\Gamma\\to\\mathbb R\\) be a homeomorphism onto its image, and put \\(a=h-\\min h+1\\). The C\\*-algebra generated by \\(a\\) separates points and vanishes nowhere, so it is \\(C(\\Gamma)\\) by the Stone–Weierstrass theorem.\n\n(2) The series converges uniformly, so \\(a\\) is continuous and real. Let \\(\\gamma\\ne\\gamma'\\), and let \\(n\\) be the first index for which \\(E_n\\) contains exactly one of them. The terms before \\(n\\) agree at \\(\\gamma\\) and \\(\\gamma'\\), so\n\\[\n\\begin{gathered}\n|a(\\gamma)-a(\\gamma')|\\\\\n\\ge2\\cdot3^{-n}-\\sum_{k>n}2\\cdot3^{-k}\\\\\n=3^{-n}>0 .\n\\end{gathered}\n\\]\nSo \\(a\\) is injective, and \\(|a|\\ge\\frac13-\\sum_{k\\ge2}3^{-k}=\\frac16\\) everywhere. The proof of (1) applies to \\(a\\) directly. \\(\\square\\)\n\n**Theorem 8.2** (Countable generation). For an abelian von Neumann algebra \\(M\\) the following are equivalent.\n\n1. \\(M\\) is generated by countably many elements.\n2. \\(M\\) is generated by countably many projections.\n3. \\(M\\) is generated by one self-adjoint element.\n\nOn a separable Hilbert space, every abelian von Neumann algebra has these properties.\n\n**Proof.** For \\(M=0\\), all three assertions hold with the zero generator and zero projection. Assume \\(M\\ne0\\); finite generating lists can be extended to sequences by appending zeros. (3)\\(\\Rightarrow\\)(1) is clear.\n\n(1)\\(\\Rightarrow\\)(2). Replacing each generator \\(x\\) by \\(\\frac12(x+x^*)\\) and \\(\\frac1{2i}(x-x^*)\\), we may assume the generators \\(h_1,h_2,\\dots\\) are self-adjoint. By Theorem 7.1 the spectrum \\(\\Omega\\) of \\(M\\) is stonean; write \\(\\hat x\\) for the Gelfand transform. For self-adjoint \\(h\\in M\\) and rational \\(\\lambda\\), let \\(e_h(\\lambda)\\in M\\) be the projection whose transform is the indicator of the clopen set \\(C_\\lambda=\\overline{\\{\\hat h<\\lambda\\}}\\). Then \\(\\{\\hat h<\\lambda\\}\\subseteq C_\\lambda\\subseteq\\{\\hat h\\le\\lambda\\}\\). Given \\(\\varepsilon>0\\), choose rationals \\(\\lambda_0<\\lambda_1<\\dots<\\lambda_N\\) with \\(\\lambda_0<\\min\\hat h\\), \\(\\lambda_N>\\max\\hat h\\) and \\(\\lambda_k-\\lambda_{k-1}<\\varepsilon\\). Then \\(C_{\\lambda_0}=\\varnothing\\) and \\(C_{\\lambda_N}=\\Omega\\). On \\(C_{\\lambda_k}\\setminus C_{\\lambda_{k-1}}\\) we have \\(\\lambda_{k-1}\\le\\hat h\\le\\lambda_k\\). Hence\n\\[\n\\Big\\|h-\\sum_{k=1}^N\\lambda_k\\big(e_h(\\lambda_k)-e_h(\\lambda_{k-1})\\big)\\Big\\|\\le\\varepsilon .\n\\]\nSo each \\(h_n\\) is a norm limit of linear combinations of the countably many projections \\(e_{h_n}(\\lambda)\\), and these projections generate \\(M\\).\n\n(2)\\(\\Rightarrow\\)(3). Let the projections \\(P_1,P_2,\\dots\\) generate \\(M\\), and put \\(a=\\sum_k3^{-k}(2P_k-1)\\in M_h\\). Let \\(C\\) be the C\\*-algebra generated by \\(1\\) and the \\(P_k\\), and \\(\\Gamma\\) its spectrum. The transforms of the \\(P_k\\) are indicators of clopen sets \\(E_k\\), and they separate the points of \\(\\Gamma\\), since a character of \\(C\\) is determined by its values on generators. By Lemma 8.1(2) the transform of \\(a\\) generates \\(C(\\Gamma)\\), so every \\(P_k\\) lies in the C\\*-algebra generated by \\(a\\). Hence \\(M\\) is generated by \\(a\\).\n\n*Separable Hilbert spaces.* Let \\((\\zeta_k)\\) be dense in \\(H\\). On the unit ball of \\(B(H)\\), the strong topology is the topology of the map \\(b\\mapsto(b\\zeta_k)_k\\) into \\(H^{\\mathbb N}\\), because on bounded sets strong convergence follows from convergence on a dense set. The space \\(H^{\\mathbb N}\\) is separable and metrizable, and so is every subspace of it. So the unit ball of \\(M\\) has a countable strongly dense subset. The von Neumann algebra that this subset generates contains the unit ball of \\(M\\), hence equals \\(M\\). \\(\\square\\)\n\n**Lemma 8.3** (Diffuse measures on the unit interval). Let \\(\\nu\\) be a Radon probability measure on \\([0,1]\\) with no atoms, let \\(m\\) be Lebesgue measure, and put \\(F(t)=\\nu([0,t])\\). Then \\(V\\varphi=\\varphi\\circ F\\) defines a unitary \\(V:L^2([0,1],m)\\to L^2([0,1],\\nu)\\), and\n\\[\n\\begin{gathered}\nV\\{M_\\varphi:\\varphi\\in L^\\infty(m)\\}V^*\\\\\n=\\{M_f:f\\in L^\\infty(\\nu)\\},\\\\\nVM_\\varphi V^*\\\\\n=M_{\\varphi\\circ F}.\n\\end{gathered}\n\\tag{8.1}\n\\]\nIn particular \\(L^\\infty([0,1],\\nu)\\) is isomorphic to \\(L^\\infty[0,1]\\).\n\n**Proof.** Since \\(\\nu\\) has no atoms, \\(F\\) is continuous and nondecreasing, with \\(F(0)=0\\) and \\(F(1)=1\\).\n\n*The image of \\(\\nu\\) under \\(F\\) is \\(m\\).* For \\(s\\in[0,1]\\) let \\(t_s\\) be the largest \\(t\\) with \\(F(t)\\le s\\). Then \\(\\{F\\le s\\}=[0,t_s]\\), and \\(F(t_s)=s\\): this is clear if \\(t_s=1\\), and otherwise \\(F>s\\) to the right of \\(t_s\\), so \\(F(t_s)\\ge s\\) by continuity. Hence \\(\\nu(\\{F\\le s\\})=s=m([0,s])\\). The intervals \\([0,s]\\) are closed under intersections and generate the Borel sets, so \\(\\nu(F^{-1}(B))=m(B)\\) for every Borel set \\(B\\), by the [finite-measure uniqueness lemma in the double-commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-18). Consequently \\(\\int\\varphi\\circ F\\,d\\nu=\\int\\varphi\\,dm\\) for bounded or nonnegative Borel \\(\\varphi\\), and \\(V\\) is a well-defined isometry.\n\n*\\(V\\) is onto.* Let \\(q(s)\\) be the least \\(t\\) with \\(F(t)\\ge s\\). It is nondecreasing, hence Borel. For every \\(t\\), \\(q(F(t))\\le t\\) and \\(F(q(F(t)))=F(t)\\). If \\(q(F(t))<t\\), then \\(F\\) is constant on the nondegenerate interval \\([q(F(t)),t]\\), and \\(t\\) is not the left end of the maximal interval of constancy that contains it. The maximal nondegenerate intervals of constancy are disjoint, so there are countably many, and each has \\(\\nu\\)-measure \\(0\\), because \\(\\nu\\) has no atoms. So \\(q\\circ F=\\mathrm{id}\\) \\(\\nu\\)-almost everywhere. For a polynomial \\(p\\), the function \\(p\\circ q\\) is bounded and Borel, and \\(V(p\\circ q)=p\\) \\(\\nu\\)-almost everywhere. So the range of \\(V\\) contains the polynomials. These are dense in \\(C[0,1]\\) and hence in \\(L^2([0,1],\\nu)\\) (Lemma 2.3(3)). The range of an isometry is closed, so \\(V\\) is onto.\n\n*The algebras.* Clearly \\(VM_\\varphi=M_{\\varphi\\circ F}V\\) for bounded Borel \\(\\varphi\\). So \\(V\\{M_\\varphi\\}V^*\\) is a maximal abelian algebra (Theorem 3.1(3)) contained in the abelian algebra \\(\\{M_f:f\\in L^\\infty(\\nu)\\}\\). If \\(\\mathcal A\\subseteq\\mathcal B\\) with \\(\\mathcal A\\) maximal abelian and \\(\\mathcal B\\) abelian, then \\(\\mathcal B\\subseteq\\mathcal A'=\\mathcal A\\). This gives (8.1). \\(\\square\\)\n\n**Theorem 8.4** (Diffuse countably generated algebras). For an abelian von Neumann algebra \\(M\\) the following are equivalent.\n\n1. \\(M\\) is isomorphic to \\(L^\\infty[0,1]\\), with Lebesgue measure.\n2. \\(M\\ne0\\), and \\(M\\) is countably generated, \\(\\sigma\\)-finite, and without minimal projections.\n\nIn particular, if a nonzero separable Hilbert space carries an abelian von Neumann algebra without minimal projections, that algebra is isomorphic to \\(L^\\infty[0,1]\\). The condition \\(M\\ne0\\) excludes the zero algebra on the zero space, which has the other three properties.\n\n\n**Proof.** (2)\\(\\Rightarrow\\)(1). *Step 1: a cyclic vector.* Let \\(M\\) act on \\(H\\). Choose a maximal family of unit vectors \\(\\xi_k\\) such that the subspaces \\([M'\\xi_k]\\) are pairwise orthogonal. Their projections \\(p_k\\) lie in \\(M''=M\\) and are pairwise orthogonal, so they are countably many, listed without repetitions and with finite sums for a finite family, and their sum is \\(1\\) by maximality. Put \\(\\xi=\\sum_k2^{-k}\\xi_k\\). Since \\(M\\) is abelian, \\(p_k\\in M'\\), and \\(p_k\\xi=2^{-k}\\xi_k\\); so \\([M'\\xi]\\) contains every \\([M'\\xi_k]\\), and \\(\\xi\\) is cyclic for \\(M'\\). Hence \\(\\xi\\) separates \\(M\\). Let \\(e\\) be the projection onto \\([M\\xi]\\); it lies in \\(M'\\). The map \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-isomorphism of \\(M\\) onto the von Neumann algebra \\(M_e=\\{x|_{eH}:x\\in M\\}\\) (see [The double commutant theorem](the-double-commutant-theorem.md), Theorem 5.8); it is injective because \\(xe\\xi=x\\xi=0\\) forces \\(x=0\\). The vector \\(\\xi\\) is cyclic for \\(M_e\\). The algebra \\(M_e\\) again has the properties in (2). Being \\(\\sigma\\)-finite and having minimal projections are properties defined through projections, so they pass through \\(*\\)-isomorphisms. By Theorem 8.2, \\(M\\) is generated by one self-adjoint element \\(a\\), and \\(a|_{eH}\\) generates \\(M_e\\). Indeed, the restriction map is strongly continuous and maps the \\(*\\)-algebra generated by \\(1\\) and \\(a\\) onto the one generated by \\(1\\) and \\(a|_{eH}\\); since \\(M\\) is the strong closure of the former, \\(M_e\\) lies in the von Neumann algebra generated by \\(a|_{eH}\\), and the converse inclusion is clear. So we may assume that \\(M\\) has a cyclic vector \\(\\xi\\) with \\(\\|\\xi\\|=1\\).\n\n*Step 2: a multiplication algebra.* By Theorem 8.2, \\(M\\) is generated by a self-adjoint \\(a\\). \\(M\\) is not \\(\\mathbb C1\\), since \\(1\\) would then be a minimal projection. After an affine change we may assume that the spectrum \\(\\sigma(a)\\) lies in \\([0,1]\\). The continuous functional calculus identifies \\(C(\\sigma(a))\\) with the C\\*-algebra generated by \\(1\\) and \\(a\\). Let \\(\\nu\\) be the Radon probability measure on \\(\\sigma(a)\\), extended by \\(0\\) to \\([0,1]\\), with \\(\\int g\\,d\\nu=\\langle g(a)\\xi,\\xi\\rangle\\). The vector \\(\\xi\\) is cyclic for the C\\*-algebra generated by \\(1\\) and \\(a\\), because its strong closure is \\(M\\). Theorem 3.1(2) gives a unitary \\(U:L^2([0,1],\\nu)\\to H\\) with \\(UM_gU^*=g(a)\\) for continuous \\(g\\). By Theorem 3.1(3), \\(U\\{M_f:f\\in L^\\infty(\\nu)\\}U^*\\) is the von Neumann algebra generated by \\(a\\), which is \\(M\\).\n\n*Step 3: no atoms.* If \\(\\nu(\\{t\\})>0\\), then \\(M_{1_{\\{t\\}}}\\) is a nonzero projection, and \\(M_fM_{1_{\\{t\\}}}=f(t)M_{1_{\\{t\\}}}\\) for every \\(f\\). So it is a minimal projection, and so is its image in \\(M\\), which is excluded.\n\n*Step 4.* By Lemma 8.3, \\(M\\cong L^\\infty([0,1],\\nu)\\cong L^\\infty[0,1]\\).\n\n(1)\\(\\Rightarrow\\)(2). The algebra \\(L^\\infty[0,1]\\) on \\(L^2[0,1]\\) is \\(\\sigma\\)-finite, because \\(\\sum_jm(E_j)\\le1\\) for disjoint sets. It has no minimal projection: if \\(m(E)>0\\), the continuous function \\(t\\mapsto m(E\\cap[0,t])\\) takes the value \\(\\frac12m(E)\\), so \\(E\\) splits into two sets of positive measure. Both properties are defined through projections, so they pass to every algebra \\(M\\) isomorphic to \\(L^\\infty[0,1]\\). Let \\(\\theta:L^\\infty[0,1]\\to M\\) be a \\(*\\)-isomorphism and \\(a=\\theta(\\iota)\\), where \\(\\iota(t)=t\\). Let \\(N\\subseteq M\\) be the von Neumann algebra generated by \\(a\\), and \\(B=\\theta^{-1}(N)\\). Then \\(B\\) is a C\\*-subalgebra of \\(L^\\infty[0,1]\\) that contains \\(1\\) and \\(\\iota\\), hence all continuous functions. Whenever an increasing net in \\(B_h\\) is bounded in norm, its least upper bound lies in \\(B\\): the image net in \\(N\\) converges strongly to an element of \\(N\\), and by Proposition 7.3 its preimage is the least upper bound in \\(L^\\infty[0,1]\\). Let \\(\\mathcal S\\) be the family of Borel sets \\(E\\) with \\(1_E\\in B\\). For open \\(U\\ne[0,1]\\) the functions \\(g_n(t)=\\min\\{1,n\\operatorname{dist}(t,[0,1]\\setminus U)\\}\\) are continuous and increase to \\(1_U\\), and \\(M_{g_n}\\to M_{1_U}\\) strongly by dominated convergence, so \\(U\\in\\mathcal S\\); and \\([0,1]\\in\\mathcal S\\) because \\(1\\in B\\). The family \\(\\mathcal S\\) is closed under complements (\\(1-1_E\\)), finite intersections (products) and increasing countable unions (least upper bounds), so it is a \\(\\sigma\\)-algebra containing the open sets, and it contains every Borel set. Then \\(B\\) contains all simple Borel functions and, being norm closed, all of \\(L^\\infty[0,1]\\). So \\(N=M\\), and \\(M\\) is generated by \\(a\\).\n\nThe last statement follows from Theorem 8.2 and the fact that von Neumann algebras acting on separable Hilbert spaces are \\(\\sigma\\)-finite: an uncountable family of pairwise orthogonal nonzero projections would give an uncountable orthonormal family of vectors. \\(\\square\\)\n\n**Example 8.5** (Each hypothesis is needed).\n\n(a) *\\(\\sigma\\)-finiteness.* Let \\(M=\\prod_{t\\in[0,1]}L^\\infty[0,1]\\), acting on the Hilbert sum \\(\\bigoplus_{t\\in[0,1]}L^2[0,1]\\) by \\((f_t)_t\\mapsto\\bigoplus_tM_{f_t}\\). It is a von Neumann algebra, because each summand is maximal abelian and the commutant of a direct sum is the direct sum of the commutants ([The double commutant theorem](the-double-commutant-theorem.md), Proposition 5.2). It has no minimal projection: a nonzero projection of \\(M\\) has a coordinate \\(1_E\\) with \\(m(E)>0\\), and \\(E\\) splits into two sets of positive measure, as in the proof of Theorem 8.4. It is not \\(\\sigma\\)-finite: the projections onto the summands are uncountably many. It is countably generated. Let \\(a\\) act as the scalar \\(t\\) on the summand with index \\(t\\), and \\(b\\) as \\(M_\\iota\\) on every summand. An operator commuting with \\(a\\) maps each eigenspace \\(\\ker(a-t)\\), which is the summand with index \\(t\\), into itself, so the projections onto the summands lie in the von Neumann algebra generated by \\(a\\). An operator commuting with \\(a\\) and \\(b\\) is therefore a bounded family of operators that commute with \\(M_\\iota\\), hence of multiplication operators (Theorem 3.1(3)). So the commutant of \\(\\{a,b\\}\\) is \\(M\\), and \\(\\{a,b\\}''=M'=M\\). So \\(M\\) is not isomorphic to \\(L^\\infty[0,1]\\).\n\n(b) *Countable generation.* Let \\(J\\) be uncountable, \\(P\\) the product of the fair coin measures on \\(\\{0,1\\}^J\\), defined on the product \\(\\sigma\\)-algebra, and \\(M=L^\\infty(P)\\) on \\(L^2(P)\\). By Remark 3.2 it is maximal abelian, and \\(1\\) is a cyclic vector because \\(L^\\infty(P)\\) is dense in \\(L^2(P)\\); so it is \\(\\sigma\\)-finite (Theorem 7.4(2)). It has no minimal projection. Every measurable set \\(E\\) depends only on the coordinates in some countable set \\(J_E\\), because the sets with this property form a \\(\\sigma\\)-algebra containing the cylinder sets. For \\(j\\notin J_E\\), independence of the coordinates gives \\(P(E\\cap\\{\\gamma_j=0\\})=\\frac12P(E)\\), which splits \\(E\\). Finally, \\(M\\) is not countably generated. Otherwise \\(M\\) is generated by one self-adjoint \\(a\\) (Theorem 8.2), and \\(L^2(P)=[M1]\\) is the closure of the set of vectors \\(p(a)1\\), with \\(p\\) a polynomial whose coefficients have rational real and imaginary parts; so \\(L^2(P)\\) would be separable. But it is not ([The double commutant theorem](the-double-commutant-theorem.md), Exercise 9.9).\n\n(c) *No minimal projections.* \\(\\ell^\\infty(\\mathbb N)\\) is countably generated and \\(\\sigma\\)-finite, but each point gives a minimal projection.\n\n## D. Find the limits of the measure picture\n\nAn algebra can have every bounded increasing supremum and still fail to be a von Neumann algebra. To see the obstruction rather than just its name, partition a stonean spectrum by where normal measures can live, then build the other regions using Borel functions modulo meager functions. Here the negligible sets come from category, not from a fixed measure. The quotient remains a C*-algebra and is order complete, but its normal observations can all vanish.\n\nThis construction also makes the three-region decomposition useful: it predicts precisely which part an example realizes. The final extension theorem asks a different question—when continuous functions defined on a closed subspace can be extended by a positive norm-one operator—and identifies stonean spaces through that property.\n\n",
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      "id": "OA-FND-AO-16",
      "unit": "abelian-operator-algebras",
      "name": "9. Borel functions modulo meager functions",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "anchor": "oa-fnd-ao-16",
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      "full_conditions_and_proof": "### 9. Borel functions modulo meager functions\n\nThis section builds stonean spaces that carry no normal measure: one kind for the part \\(\\Omega_2\\) and one for the part \\(\\Omega_3\\) of Theorem 6.4. The construction goes back to [Dixmier 1951].\n\nLet \\(\\Gamma\\) be a topological space. No separation axiom is needed for the construction, for Lemmas 9.1 and 9.2 or for Theorem 9.3; Theorems 9.5 and 9.6 say which ones they use. Let \\(\\mathcal B(\\Gamma)\\) be the C\\*-algebra of bounded complex Borel functions on \\(\\Gamma\\), with pointwise operations and the supremum norm. Let \\(\\mathcal B_0(\\Gamma)\\) be the set of \\(f\\in\\mathcal B(\\Gamma)\\) for which \\(\\{f\\ne0\\}\\) is meager. It is a two-sided ideal, closed under conjugation, and norm closed: a uniform limit of functions \\(f_n\\in\\mathcal B_0(\\Gamma)\\) vanishes outside the meager set \\(\\bigcup_n\\{f_n\\ne0\\}\\). So\n\\[\n\\mathcal D(\\Gamma)=\\mathcal B(\\Gamma)/\\mathcal B_0(\\Gamma)\n\\]\nis an abelian C\\*-algebra with unit. We write \\(\\tilde f\\) for the class of \\(f\\), and \\(\\Omega_\\Gamma\\) for the spectrum of \\(\\mathcal D(\\Gamma)\\).\n\n*Order.* For real \\(f\\in\\mathcal B(\\Gamma)\\), \\(\\tilde f\\ge0\\) if and only if \\(\\{f<0\\}\\) is meager. Indeed, if \\(\\{f<0\\}\\) is meager, then the negative part of \\(f\\) lies in \\(\\mathcal B_0(\\Gamma)\\), and \\(\\tilde f\\) is the square of the class of \\(\\sqrt{\\max(f,0)}\\). Conversely, if \\(\\tilde f\\ge0\\), then \\(\\tilde f=\\tilde g^*\\tilde g\\) for some \\(g\\), so \\(f-|g|^2\\in\\mathcal B_0(\\Gamma)\\), and \\(f\\ge0\\) outside a meager set. The self-adjoint elements are the classes of real functions, and the class of \\(\\max(f,g)\\) is the supremum of \\(\\tilde f\\) and \\(\\tilde g\\) in the order.\n\n**Lemma 9.1** (The Baire property). Every bounded real Borel function on \\(\\Gamma\\) agrees outside a meager set with a bounded lsc function, and also with a bounded usc function. In particular, for every Borel set \\(E\\) there is an open set \\(U\\) such that the symmetric difference \\(E\\triangle U\\) is meager.\n\n**Proof.** Let \\(\\mathcal K\\) be the set of bounded real functions on \\(\\Gamma\\) that agree outside a meager set with a bounded lsc function.\n\n*Step 1.* \\(\\mathcal K\\) contains the bounded lsc functions, trivially, and the bounded usc functions: a bounded usc \\(f\\) agrees with the lsc function \\(f_*\\) outside a meager set, by Lemma 2.2(2).\n\n*Step 2.* \\(\\mathcal K\\) is a real vector space, closed under \\(\\max\\) and \\(\\min\\), and contains the constants. Sums, positive multiples, maxima and minima of lsc functions are lsc, and a countable union of meager sets is meager. If \\(f\\) agrees with an lsc \\(g\\) outside a meager set, then \\(-f\\) agrees with the usc function \\(-g\\), hence with a bounded lsc function, outside a meager set.\n\n*Step 3.* \\(\\mathcal K\\) is closed under pointwise limits of uniformly bounded sequences. Let \\(f_k\\in\\mathcal K\\), \\(|f_k|\\le c\\), and \\(f_k\\to f\\) pointwise. For \\(n<m\\) put \\(h_{n,m}=\\max(f_{n+1},\\dots,f_m)\\in\\mathcal K\\), and choose an lsc \\(g_{n,m}\\) with values in \\([-c,c]\\) that agrees with \\(h_{n,m}\\) outside a meager set \\(M_{n,m}\\); truncating an lsc function at \\(\\pm c\\) keeps it lsc. The function \\(g_n=\\sup_mg_{n,m}\\) is lsc, and outside the meager set \\(M_n=\\bigcup_mM_{n,m}\\) it equals \\(H_n=\\sup_{k>n}f_k\\). By Lemma 2.2(2), \\(g_n\\) agrees with its upper regularization \\(u_n=(g_n)^*\\) outside a meager set \\(M_n'\\). Put \\(v_n=\\min(u_1,\\dots,u_n)\\), a usc function. Outside the meager set \\(M^\\sharp=\\bigcup_n(M_n\\cup M_n')\\) we have \\(v_n=\\min(H_1,\\dots,H_n)=H_n\\), because \\(H_n\\) decreases in \\(n\\). So \\(v=\\inf_nv_n\\) is usc, and outside \\(M^\\sharp\\) it equals \\(\\lim_nH_n=\\limsup_kf_k=f\\). By Step 1, \\(v\\in\\mathcal K\\), hence \\(f\\in\\mathcal K\\).\n\n*Step 4.* The sets \\(E\\) with \\(1_E\\in\\mathcal K\\) contain the open sets (whose indicators are lsc). They are closed under complements, by Step 2, and under countable unions, since \\(1_{E_1\\cup\\dots\\cup E_n}\\) is a maximum and increases to \\(1_{\\bigcup_nE_n}\\) (Steps 2 and 3). So they include all Borel sets. Hence \\(\\mathcal K\\) contains the simple Borel functions, and by Step 3 their uniform limits, which are all bounded real Borel functions. The statement about usc functions follows by applying this to \\(-f\\). Finally, if \\(1_E\\) agrees with an lsc \\(g\\) outside a meager set \\(M\\), then \\(U=\\{g>\\frac12\\}\\) is open and \\(E\\triangle U\\subseteq M\\). \\(\\square\\)\n\n**Lemma 9.2** (Clopen subsets of \\(\\Omega_\\Gamma\\)). For a Borel set \\(E\\subseteq\\Gamma\\), the class \\(\\tilde1_E\\) is a projection; let \\(\\widehat E\\subseteq\\Omega_\\Gamma\\) be the clopen set on which its Gelfand transform equals \\(1\\).\n\n1. \\(\\widehat E=\\varnothing\\) if and only if \\(E\\) is meager, and \\(\\widehat E\\subseteq\\widehat F\\) if and only if \\(E\\setminus F\\) is meager. Moreover \\(\\widehat{E\\cap F}=\\widehat E\\cap\\widehat F\\) and \\(\\widehat{\\Gamma\\setminus E}=\\Omega_\\Gamma\\setminus\\widehat E\\).\n2. Every clopen subset of \\(\\Omega_\\Gamma\\) is \\(\\widehat U\\) for some open set \\(U\\subseteq\\Gamma\\).\n3. If \\(\\Gamma\\) is a Baire space, then \\(\\widehat U\\ne\\varnothing\\) for every nonempty open set \\(U\\).\n\n**Proof.** (1) \\(\\tilde1_E=0\\) means \\(1_E\\in\\mathcal B_0(\\Gamma)\\). By the description of the order, \\(\\tilde1_E\\le\\tilde1_F\\) means \\(1_E\\le1_F\\) outside a meager set. The other two statements follow from \\(1_{E\\cap F}=1_E1_F\\) and \\(1_{\\Gamma\\setminus E}=1-1_E\\). (2) A clopen set is the set where the transform of a projection \\(\\tilde p\\) equals \\(1\\), with \\(p\\) real. Since \\(\\tilde p^2=\\tilde p\\), the set \\(\\{p^2\\ne p\\}\\) is meager, so \\(p=1_E\\) outside a meager set, with \\(E=\\{p=1\\}\\). By Lemma 9.1 there is an open \\(U\\) with \\(E\\triangle U\\) meager, and then \\(\\widehat E=\\widehat U\\). (3) In a Baire space a nonempty open set is not meager. \\(\\square\\)\n\n**Theorem 9.3** (Every stonean space is a space of this kind).\n\n1. For every topological space \\(\\Gamma\\), the space \\(\\Omega_\\Gamma\\) is stonean. More precisely, let \\((\\tilde f_i)\\) be an increasing net in \\(\\mathcal D(\\Gamma)_h\\) that is bounded in norm, and let the \\(f_i\\) be lsc representatives with a common bound. Then the supremum of \\((\\tilde f_i)\\) in \\(\\mathcal D(\\Gamma)_h\\) is the class of the pointwise supremum \\(\\sup_if_i\\).\n2. If \\(\\Omega\\) is stonean, then \\(x\\mapsto\\tilde x\\) is a \\(*\\)-isomorphism of \\(C(\\Omega)\\) onto \\(\\mathcal D(\\Omega)\\). So every stonean space \\(\\Omega\\) is homeomorphic to \\(\\Omega_\\Omega\\).\n\n**Proof.** (1) Take an increasing net \\((\\tilde f_i)\\) in \\(\\mathcal D(\\Gamma)_h\\) that is bounded in norm. By Lemma 9.1 and truncation we may choose lsc representatives \\(f_i\\) with \\(|f_i|\\le c\\). Put \\(f=\\sup_if_i\\), a bounded lsc function. Since \\(f\\ge f_i\\), \\(\\tilde f\\) is an upper bound. Let \\(\\tilde g\\) be another upper bound, with \\(g\\) a bounded usc representative (Lemma 9.1). For each \\(i\\), \\(f_i\\le g\\) outside a meager set. The set \\(A=\\{f>g\\}\\) is open, since \\(f-g\\) is lsc. By Zorn's lemma choose a maximal family \\((G_j)\\) of pairwise disjoint nonempty open sets such that each \\(G_j\\cap A\\) is meager, and put \\(G=\\bigcup_jG_j\\).\n\nWe claim that \\(G\\) is dense. If not, \\(V=\\Gamma\\setminus\\overline G\\) is a nonempty open set. If \\(V\\cap A=\\varnothing\\), then \\(V\\) could be added to the family. Otherwise pick \\(\\gamma\\in V\\cap A\\). Then \\(f(\\gamma)>g(\\gamma)\\), so \\(f_i(\\gamma)>g(\\gamma)\\) for some \\(i\\). The set \\(W=V\\cap\\{f_i>g\\}\\) is open, because \\(f_i-g\\) is lsc, and it contains \\(\\gamma\\). It is meager, since \\(f_i\\le g\\) outside a meager set. So \\(W\\cap A\\) is meager, and \\(W\\) could be added to the family. Both cases contradict maximality.\n\nSo \\(\\Gamma\\setminus G\\) is closed with empty interior, that is, rare. By Lemma 2.1(3), \\(\\bigcup_j(G_j\\cap A)\\) is meager. Hence \\(A=\\bigcup_j(G_j\\cap A)\\cup(A\\setminus G)\\) is meager, which means \\(\\tilde f\\le\\tilde g\\). So \\(\\tilde f\\) is the least upper bound. For a nonempty set \\(S\\subseteq\\mathcal D(\\Gamma)_h\\) that is bounded above, the classes of maxima of finitely many representatives form a bounded increasing net with the same upper bounds, as in the proof of Theorem 7.1. So \\(\\mathcal D(\\Gamma)_h\\) is conditionally complete. Through the Gelfand isomorphism, which preserves order, so is \\(C_{\\mathbb R}(\\Omega_\\Gamma)\\), and \\(\\Omega_\\Gamma\\) is stonean by Theorem 4.3.\n\n(2) The map is a \\(*\\)-homomorphism. It is injective: if \\(x\\) is continuous and \\(\\{x\\ne0\\}\\) is meager, this open set is empty by Lemma 2.1(2). It is onto: a bounded real Borel function agrees outside a meager set with an lsc function (Lemma 9.1), which agrees outside a meager set with a continuous function (Theorem 4.3). Isomorphic commutative C\\*-algebras have homeomorphic spectra. \\(\\square\\)\n\n**Remark 9.4.** If \\(\\Gamma\\) is a Baire space, the proof of Theorem 9.3(1) shows more: the set \\(A\\) is empty. Indeed, if \\(\\gamma\\in A\\), then the nonempty open set \\(\\{f_i>g\\}\\) is meager for a suitable \\(i\\), which is impossible in a Baire space. So then \\(f\\le g\\) everywhere.\n\n**Theorem 9.5** (A stonean space with a dense meager subset). Let \\(\\Gamma\\) be a regular Baire space that contains a dense meager subset; here *regular* means that a point and a closed set not containing it have disjoint open neighbourhoods. Then \\(\\Omega_\\Gamma\\) contains a dense meager subset. So \\(\\Omega_\\Gamma\\) is its own second part in Theorem 6.4, and it carries no normal measure other than \\(0\\). If moreover \\(\\Gamma\\) is Hausdorff, has no isolated points and has a countable family of nonempty open sets such that every nonempty open set contains one of them, then \\(\\Omega_\\Gamma\\) has no isolated points and has a countable dense subset. All of this applies to \\(\\Gamma=[0,1]\\).\n\n**Proof.** Let \\(A=\\bigcup_nA_n\\) be dense, with each \\(A_n\\) rare. Replacing \\(A_n\\) by its closure, we may assume that each \\(A_n\\) is closed. For each \\(n\\) let \\(B_n\\) be the intersection of the sets \\(\\widehat U\\) over all open \\(U\\supseteq A_n\\). It is closed.\n\n*Each \\(B_n\\) is rare.* Suppose \\(B_n\\) contains a nonempty open set. Then it contains a nonempty clopen set, of the form \\(\\widehat V\\) with \\(V\\) open and not meager (Lemma 9.2). For every open \\(U\\supseteq A_n\\), \\(\\widehat V\\subseteq\\widehat U\\), so \\(V\\setminus U\\) is meager. The open set \\(V\\setminus A_n\\) is not meager, because \\(V\\) is not and \\(A_n\\) is rare; so it contains a point \\(\\gamma\\). As \\(\\Gamma\\) is regular and \\(A_n\\) is closed, there are disjoint open sets \\(P\\ni\\gamma\\) and \\(U\\supseteq A_n\\). The set \\(V\\cap P\\) is open and nonempty, hence not meager, because \\(\\Gamma\\) is a Baire space. But \\(V\\cap P\\subseteq V\\setminus U\\), which is meager. This is a contradiction.\n\n*The union \\(B=\\bigcup_nB_n\\) is dense.* Suppose a nonempty clopen set \\(\\widehat V\\), with \\(V\\) open and not meager, misses every \\(B_n\\). The open sets \\(U\\supseteq A_n\\) are closed under finite intersections, and so are the sets \\(\\widehat U\\) (Lemma 9.2(1)). The compact set \\(\\widehat V\\) misses their intersection \\(B_n\\), so it misses one of them: there is an open \\(U_n\\supseteq A_n\\) with \\(\\widehat V\\cap\\widehat{U_n}=\\varnothing\\), that is, \\(V\\cap U_n\\) is meager. The open set \\(U=\\bigcup_nU_n\\) contains \\(A\\), so it is dense, and \\(V\\cap U=\\bigcup_n(V\\cap U_n)\\) is meager. But \\(V\\cap U\\) is a nonempty open set, because \\(V\\) is open and nonempty and \\(U\\) is dense. This contradicts the Baire property of \\(\\Gamma\\).\n\nSo \\(B\\) is a dense meager subset of \\(\\Omega_\\Gamma\\). A normal measure vanishes on \\(B\\) (Theorem 5.2), hence on \\(\\overline B=\\Omega_\\Gamma\\) (Corollary 5.4(1)).\n\n*Separability.* Let \\((V_n)\\) be the countable family, and choose \\(\\omega_n\\in\\widehat{V_n}\\), which is not empty by Lemma 9.2(3). A nonempty clopen subset of \\(\\Omega_\\Gamma\\) is \\(\\widehat U\\) with \\(U\\) open and not meager, and \\(U\\) contains some \\(V_n\\); so \\(\\omega_n\\in\\widehat{V_n}\\subseteq\\widehat U\\). So \\(\\{\\omega_n\\}\\) is dense. If \\(\\{\\omega\\}\\) were open, it would be clopen, say \\(\\{\\omega\\}=\\widehat U\\) with \\(U\\) nonempty and open. Since \\(\\Gamma\\) is Hausdorff without isolated points, \\(U\\) contains two disjoint nonempty open sets \\(U_1,U_2\\), and \\(\\widehat{U_1}\\), \\(\\widehat{U_2}\\) would be disjoint nonempty subsets of \\(\\{\\omega\\}\\).\n\nFor \\(\\Gamma=[0,1]\\): it is a compact metric space, hence regular, Hausdorff and a Baire space; the rational points form a dense meager set; the open intervals with rational end points form the countable family; and there are no isolated points. \\(\\square\\)\n\nSo \\(C(\\Omega_{[0,1]})\\cong\\mathcal D([0,1])\\) is an abelian C\\*-algebra in which every nonempty bounded set of self-adjoint elements has a least upper bound, but which is isomorphic to no von Neumann algebra.\n\n**Theorem 9.6** (Stonean spaces on which every measure lives on a rare set). Let \\(\\Gamma\\) be a topological space with a base \\(\\mathfrak B\\) of nonempty open sets such that\n\n\\((\\ast)\\) for every decreasing sequence \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\), the intersection \\(\\bigcap_nB_n\\) contains a member of \\(\\mathfrak B\\).\n\n1. If \\(G_1,G_2,\\dots\\) are dense open subsets of \\(\\Gamma\\), the interior of \\(\\bigcap_nG_n\\) is dense. So \\(\\Gamma\\) is a Baire space, and every meager subset of \\(\\Gamma\\) is rare.\n2. The sets \\(\\widehat B\\), \\(B\\in\\mathfrak B\\), are nonempty clopen sets, and every nonempty open subset of \\(\\Omega_\\Gamma\\) contains one of them. Every nonempty clopen subset of \\(\\widehat B\\) contains \\(\\widehat C\\) for some \\(C\\in\\mathfrak B\\) with \\(C\\subseteq B\\). For every decreasing sequence \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\) there is \\(C\\in\\mathfrak B\\) with \\(\\widehat C\\subseteq\\bigcap_n\\widehat{B_n}\\).\n3. Every meager subset of \\(\\Omega_\\Gamma\\) is rare.\n4. If \\(\\Gamma\\) is Hausdorff and has no isolated points, every nonempty clopen subset of \\(\\Omega_\\Gamma\\) is uncountable, and every Radon measure on \\(\\Omega_\\Gamma\\) has a rare support.\n\nIf \\(\\Gamma\\) is a nonempty Hausdorff space without isolated points, \\(\\Omega_\\Gamma\\) is a nonempty stonean space that is its own third part in Theorem 6.4.\n\nExample 9.8 proves that for the space of Example 9.7 no base of \\(\\Omega_\\Gamma\\) satisfies \\((\\ast)\\), so (2) is stated for sequences that decrease in \\(\\Gamma\\).\n\n**Proof.** (1) Let \\(V\\) be a nonempty open set. Choose \\(B_1\\in\\mathfrak B\\) with \\(B_1\\subseteq V\\cap G_1\\), then \\(B_2\\in\\mathfrak B\\) with \\(B_2\\subseteq B_1\\cap G_2\\), and so on; each of these open sets is nonempty because the \\(G_n\\) are dense. By \\((\\ast)\\), \\(\\bigcap_nB_n\\) contains some \\(B\\in\\mathfrak B\\), and \\(B\\) is an open subset of \\(V\\cap\\bigcap_nG_n\\). So the interior of \\(\\bigcap_nG_n\\) meets \\(V\\). If \\(M=\\bigcup_nR_n\\) is meager, the sets \\(G_n=\\Gamma\\setminus\\overline{R_n}\\) are dense and open, and \\(M\\) lies in the complement of the interior of \\(\\bigcap_nG_n\\), a closed set with empty interior. So \\(M\\) is rare, and in particular has empty interior.\n\n(2) By (1) and Lemma 9.2(3), \\(\\widehat B\\ne\\varnothing\\). A nonempty open subset of \\(\\Omega_\\Gamma\\) contains a nonempty clopen set, which is \\(\\widehat E\\) for an open \\(E\\) that is not meager (Lemma 9.2); \\(E\\) contains some \\(B\\in\\mathfrak B\\), and \\(\\widehat B\\subseteq\\widehat E\\). If a nonempty clopen set \\(\\widehat E\\), with \\(E\\) open, lies in \\(\\widehat B\\), then \\(E\\setminus B\\) is meager. So \\(E\\cap B\\) is not meager, hence not empty, and it contains some \\(C\\in\\mathfrak B\\); then \\(C\\subseteq B\\) and \\(\\widehat C\\subseteq\\widehat E\\). Finally, if \\(B_1\\supseteq B_2\\supseteq\\cdots\\), then \\((\\ast)\\) gives \\(C\\in\\mathfrak B\\) with \\(C\\subseteq B_n\\), hence \\(\\widehat C\\subseteq\\widehat{B_n}\\), for all \\(n\\).\n\n(3) Let \\(R_1,R_2,\\dots\\) be rare subsets of \\(\\Omega_\\Gamma\\) and \\(W\\) a nonempty open set. The open set \\(W\\setminus\\overline{R_1}\\) is not empty, so by (2) it contains \\(\\widehat{B_1}\\) for some \\(B_1\\in\\mathfrak B\\). The open set \\(\\widehat{B_1}\\setminus\\overline{R_2}\\) is not empty and contains a nonempty clopen set; by (2) this contains \\(\\widehat{B_2}\\) with \\(B_2\\in\\mathfrak B\\) and \\(B_2\\subseteq B_1\\). Continuing, we get \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\) with \\(\\widehat{B_n}\\cap R_n=\\varnothing\\). By (2) some nonempty \\(\\widehat C\\) lies in every \\(\\widehat{B_n}\\). It is an open subset of \\(W\\) that misses \\(\\bigcup_nR_n\\). So \\(\\bigcup_nR_n\\) is dense in no nonempty open set, that is, it is rare.\n\n(4) Let \\(W\\) be a nonempty clopen subset of \\(\\Omega_\\Gamma\\). It contains some \\(\\widehat B\\). The open set \\(B\\) contains two disjoint nonempty open sets, since \\(\\Gamma\\) is Hausdorff and has no isolated points, and they give two disjoint nonempty clopen subsets of \\(\\widehat B\\) (Lemma 9.2). So no clopen set is a single point; since the clopen sets form a base, \\(W\\) has no isolated points. A compact space without isolated points is uncountable: if it were countable, it would be a countable union of rare singletons, against Lemma 2.1(2).\n\nNow let \\(\\mu\\) be a Radon measure on \\(\\Omega_\\Gamma\\). Only countably many points have positive measure, so every nonempty clopen set contains a point \\(\\omega\\) with \\(\\mu(\\{\\omega\\})=0\\). By outer regularity and Lemma 4.2(3), such a point has clopen neighbourhoods of arbitrarily small measure. Start from a nonempty clopen \\(W\\), and choose \\(B_1\\in\\mathfrak B\\) with \\(\\widehat{B_1}\\subseteq W\\). Given \\(B_n\\), choose \\(\\omega\\in\\widehat{B_n}\\) with \\(\\mu(\\{\\omega\\})=0\\), a clopen neighbourhood \\(Q\\subseteq\\widehat{B_n}\\) of \\(\\omega\\) with \\(\\mu(Q)<1/(n+1)\\), and, by (2), \\(B_{n+1}\\in\\mathfrak B\\) with \\(B_{n+1}\\subseteq B_n\\) and \\(\\widehat{B_{n+1}}\\subseteq Q\\). By (2) again some nonempty \\(\\widehat C\\) lies in all \\(\\widehat{B_n}\\), so \\(\\mu(\\widehat C)=0\\). Thus every nonempty clopen set contains a nonempty open null set. If the support \\(S\\) of \\(\\mu\\) had interior points, its interior would contain a nonempty clopen set, and hence a nonempty open null set, which is impossible inside the support. So \\(S\\) is rare.\n\nThe final statement: \\(\\Omega_\\Gamma\\) is not empty, because \\(\\Gamma\\) is a nonempty Baire space and so is not meager in itself (Lemma 9.2(1)). By (3), no meager set is dense in a nonempty open set, so the second part is empty. The support of a nonzero normal measure is a nonempty clopen set (Corollary 5.4(3)), which is not rare; by (4) there is no such measure, so the first part is empty. \\(\\square\\)\n\n**Example 9.7** (A space satisfying \\((\\ast)\\)). Let \\(I\\) be an uncountable set and \\(\\Gamma=\\{0,1\\}^I\\). For a countable set \\(J\\subseteq I\\) and \\(\\alpha\\in\\{0,1\\}^J\\) put \\(U(J,\\alpha)=\\{\\gamma\\in\\Gamma:\\gamma_j=\\alpha_j\\text{ for all }j\\in J\\}\\). The intersection of two such sets is empty or again of this form, so they form a base \\(\\mathfrak B\\) of a topology. Two different points differ at some coordinate \\(i\\), and the sets \\(U(\\{i\\},\\cdot)\\) through them are disjoint; so \\(\\Gamma\\) is Hausdorff. Each \\(U(J,\\alpha)\\) is also closed, since its complement is the union of the sets \\(U(\\{j\\},1-\\alpha_j)\\), \\(j\\in J\\). So \\(\\Gamma\\) has a base of clopen sets, and it is completely regular, because the indicators of these sets are continuous. No point is isolated, because a basic set fixes only countably many of the uncountably many coordinates. If \\(U(J_1,\\alpha_1)\\supseteq U(J_2,\\alpha_2)\\supseteq\\cdots\\), then \\(J_1\\subseteq J_2\\subseteq\\cdots\\) and each \\(\\alpha_{n+1}\\) extends \\(\\alpha_n\\), so the intersection is \\(U(\\bigcup_nJ_n,\\bigcup_n\\alpha_n)\\), which lies in \\(\\mathfrak B\\). So Theorem 9.6 applies: \\(\\Omega_\\Gamma\\) is a stonean space in which every meager set is rare and every Radon measure has a rare support.\n\n**Example 9.8** (No base of \\(\\Omega_\\Gamma\\) satisfies \\((\\ast)\\)). Keep \\(\\Gamma\\) of Example 9.7, and choose distinct indices \\(j_1,j_2,\\dots\\) in \\(I\\). Let \\(V_k\\) be the set of \\(\\gamma\\) with \\(\\gamma_{j_1}=\\dots=\\gamma_{j_{k-1}}=0\\) and \\(\\gamma_{j_k}=1\\); these are disjoint basic sets. Put \\(R_n=\\bigcup_{k\\ge n}V_k\\). It is the intersection of \\(U(\\{j_1,\\dots,j_{n-1}\\},0)\\) with the complement of the basic set \\(Z=U(\\{j_k:k\\ge1\\},0)\\), so it is clopen. The sets \\(R_n\\) decrease, they are nonempty, and \\(\\bigcap_nR_n=\\varnothing\\), because a point of \\(R_n\\) lies in exactly one \\(V_k\\), and then \\(k\\ge n\\).\n\nIn \\(\\Omega_\\Gamma\\) the clopen sets \\(\\widehat{R_n}\\) decrease and are nonempty, by Lemma 9.2(3), since \\(\\Gamma\\) is a Baire space by Theorem 9.6(1). By compactness they have a common point \\(\\omega\\). Their intersection has empty interior. Otherwise it contains a nonempty clopen set \\(\\widehat E\\), with \\(E\\) not meager, and \\(E\\setminus R_n\\) is meager for every \\(n\\); then \\(E\\subseteq\\bigcup_n(E\\setminus R_n)\\cup\\bigcap_nR_n\\) is meager. Now let \\(\\mathfrak G\\) be any base of \\(\\Omega_\\Gamma\\). Choose \\(W_1\\in\\mathfrak G\\) with \\(\\omega\\in W_1\\subseteq\\widehat{R_1}\\), and inductively \\(W_{n+1}\\in\\mathfrak G\\) with \\(\\omega\\in W_{n+1}\\subseteq W_n\\cap\\widehat{R_{n+1}}\\). Then \\((W_n)\\) decreases in \\(\\mathfrak G\\), and \\(\\bigcap_nW_n\\subseteq\\bigcap_n\\widehat{R_n}\\) has empty interior, so it contains no nonempty member of \\(\\mathfrak G\\). This is why Theorem 9.6(2) speaks of the sets \\(\\widehat B\\), which form a base for the nonempty open sets in the weaker sense that every nonempty open set contains one of them, and of sequences that decrease in \\(\\Gamma\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AO-17",
      "unit": "abelian-operator-algebras",
      "name": "9. Borel functions modulo meager functions",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "anchor": "oa-fnd-ao-17",
      "proof_locus": {
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      "full_conditions_and_proof": "### 9. Borel functions modulo meager functions\n\nThis section builds stonean spaces that carry no normal measure: one kind for the part \\(\\Omega_2\\) and one for the part \\(\\Omega_3\\) of Theorem 6.4. The construction goes back to [Dixmier 1951].\n\nLet \\(\\Gamma\\) be a topological space. No separation axiom is needed for the construction, for Lemmas 9.1 and 9.2 or for Theorem 9.3; Theorems 9.5 and 9.6 say which ones they use. Let \\(\\mathcal B(\\Gamma)\\) be the C\\*-algebra of bounded complex Borel functions on \\(\\Gamma\\), with pointwise operations and the supremum norm. Let \\(\\mathcal B_0(\\Gamma)\\) be the set of \\(f\\in\\mathcal B(\\Gamma)\\) for which \\(\\{f\\ne0\\}\\) is meager. It is a two-sided ideal, closed under conjugation, and norm closed: a uniform limit of functions \\(f_n\\in\\mathcal B_0(\\Gamma)\\) vanishes outside the meager set \\(\\bigcup_n\\{f_n\\ne0\\}\\). So\n\\[\n\\mathcal D(\\Gamma)=\\mathcal B(\\Gamma)/\\mathcal B_0(\\Gamma)\n\\]\nis an abelian C\\*-algebra with unit. We write \\(\\tilde f\\) for the class of \\(f\\), and \\(\\Omega_\\Gamma\\) for the spectrum of \\(\\mathcal D(\\Gamma)\\).\n\n*Order.* For real \\(f\\in\\mathcal B(\\Gamma)\\), \\(\\tilde f\\ge0\\) if and only if \\(\\{f<0\\}\\) is meager. Indeed, if \\(\\{f<0\\}\\) is meager, then the negative part of \\(f\\) lies in \\(\\mathcal B_0(\\Gamma)\\), and \\(\\tilde f\\) is the square of the class of \\(\\sqrt{\\max(f,0)}\\). Conversely, if \\(\\tilde f\\ge0\\), then \\(\\tilde f=\\tilde g^*\\tilde g\\) for some \\(g\\), so \\(f-|g|^2\\in\\mathcal B_0(\\Gamma)\\), and \\(f\\ge0\\) outside a meager set. The self-adjoint elements are the classes of real functions, and the class of \\(\\max(f,g)\\) is the supremum of \\(\\tilde f\\) and \\(\\tilde g\\) in the order.\n\n**Lemma 9.1** (The Baire property). Every bounded real Borel function on \\(\\Gamma\\) agrees outside a meager set with a bounded lsc function, and also with a bounded usc function. In particular, for every Borel set \\(E\\) there is an open set \\(U\\) such that the symmetric difference \\(E\\triangle U\\) is meager.\n\n**Proof.** Let \\(\\mathcal K\\) be the set of bounded real functions on \\(\\Gamma\\) that agree outside a meager set with a bounded lsc function.\n\n*Step 1.* \\(\\mathcal K\\) contains the bounded lsc functions, trivially, and the bounded usc functions: a bounded usc \\(f\\) agrees with the lsc function \\(f_*\\) outside a meager set, by Lemma 2.2(2).\n\n*Step 2.* \\(\\mathcal K\\) is a real vector space, closed under \\(\\max\\) and \\(\\min\\), and contains the constants. Sums, positive multiples, maxima and minima of lsc functions are lsc, and a countable union of meager sets is meager. If \\(f\\) agrees with an lsc \\(g\\) outside a meager set, then \\(-f\\) agrees with the usc function \\(-g\\), hence with a bounded lsc function, outside a meager set.\n\n*Step 3.* \\(\\mathcal K\\) is closed under pointwise limits of uniformly bounded sequences. Let \\(f_k\\in\\mathcal K\\), \\(|f_k|\\le c\\), and \\(f_k\\to f\\) pointwise. For \\(n<m\\) put \\(h_{n,m}=\\max(f_{n+1},\\dots,f_m)\\in\\mathcal K\\), and choose an lsc \\(g_{n,m}\\) with values in \\([-c,c]\\) that agrees with \\(h_{n,m}\\) outside a meager set \\(M_{n,m}\\); truncating an lsc function at \\(\\pm c\\) keeps it lsc. The function \\(g_n=\\sup_mg_{n,m}\\) is lsc, and outside the meager set \\(M_n=\\bigcup_mM_{n,m}\\) it equals \\(H_n=\\sup_{k>n}f_k\\). By Lemma 2.2(2), \\(g_n\\) agrees with its upper regularization \\(u_n=(g_n)^*\\) outside a meager set \\(M_n'\\). Put \\(v_n=\\min(u_1,\\dots,u_n)\\), a usc function. Outside the meager set \\(M^\\sharp=\\bigcup_n(M_n\\cup M_n')\\) we have \\(v_n=\\min(H_1,\\dots,H_n)=H_n\\), because \\(H_n\\) decreases in \\(n\\). So \\(v=\\inf_nv_n\\) is usc, and outside \\(M^\\sharp\\) it equals \\(\\lim_nH_n=\\limsup_kf_k=f\\). By Step 1, \\(v\\in\\mathcal K\\), hence \\(f\\in\\mathcal K\\).\n\n*Step 4.* The sets \\(E\\) with \\(1_E\\in\\mathcal K\\) contain the open sets (whose indicators are lsc). They are closed under complements, by Step 2, and under countable unions, since \\(1_{E_1\\cup\\dots\\cup E_n}\\) is a maximum and increases to \\(1_{\\bigcup_nE_n}\\) (Steps 2 and 3). So they include all Borel sets. Hence \\(\\mathcal K\\) contains the simple Borel functions, and by Step 3 their uniform limits, which are all bounded real Borel functions. The statement about usc functions follows by applying this to \\(-f\\). Finally, if \\(1_E\\) agrees with an lsc \\(g\\) outside a meager set \\(M\\), then \\(U=\\{g>\\frac12\\}\\) is open and \\(E\\triangle U\\subseteq M\\). \\(\\square\\)\n\n**Lemma 9.2** (Clopen subsets of \\(\\Omega_\\Gamma\\)). For a Borel set \\(E\\subseteq\\Gamma\\), the class \\(\\tilde1_E\\) is a projection; let \\(\\widehat E\\subseteq\\Omega_\\Gamma\\) be the clopen set on which its Gelfand transform equals \\(1\\).\n\n1. \\(\\widehat E=\\varnothing\\) if and only if \\(E\\) is meager, and \\(\\widehat E\\subseteq\\widehat F\\) if and only if \\(E\\setminus F\\) is meager. Moreover \\(\\widehat{E\\cap F}=\\widehat E\\cap\\widehat F\\) and \\(\\widehat{\\Gamma\\setminus E}=\\Omega_\\Gamma\\setminus\\widehat E\\).\n2. Every clopen subset of \\(\\Omega_\\Gamma\\) is \\(\\widehat U\\) for some open set \\(U\\subseteq\\Gamma\\).\n3. If \\(\\Gamma\\) is a Baire space, then \\(\\widehat U\\ne\\varnothing\\) for every nonempty open set \\(U\\).\n\n**Proof.** (1) \\(\\tilde1_E=0\\) means \\(1_E\\in\\mathcal B_0(\\Gamma)\\). By the description of the order, \\(\\tilde1_E\\le\\tilde1_F\\) means \\(1_E\\le1_F\\) outside a meager set. The other two statements follow from \\(1_{E\\cap F}=1_E1_F\\) and \\(1_{\\Gamma\\setminus E}=1-1_E\\). (2) A clopen set is the set where the transform of a projection \\(\\tilde p\\) equals \\(1\\), with \\(p\\) real. Since \\(\\tilde p^2=\\tilde p\\), the set \\(\\{p^2\\ne p\\}\\) is meager, so \\(p=1_E\\) outside a meager set, with \\(E=\\{p=1\\}\\). By Lemma 9.1 there is an open \\(U\\) with \\(E\\triangle U\\) meager, and then \\(\\widehat E=\\widehat U\\). (3) In a Baire space a nonempty open set is not meager. \\(\\square\\)\n\n**Theorem 9.3** (Every stonean space is a space of this kind).\n\n1. For every topological space \\(\\Gamma\\), the space \\(\\Omega_\\Gamma\\) is stonean. More precisely, let \\((\\tilde f_i)\\) be an increasing net in \\(\\mathcal D(\\Gamma)_h\\) that is bounded in norm, and let the \\(f_i\\) be lsc representatives with a common bound. Then the supremum of \\((\\tilde f_i)\\) in \\(\\mathcal D(\\Gamma)_h\\) is the class of the pointwise supremum \\(\\sup_if_i\\).\n2. If \\(\\Omega\\) is stonean, then \\(x\\mapsto\\tilde x\\) is a \\(*\\)-isomorphism of \\(C(\\Omega)\\) onto \\(\\mathcal D(\\Omega)\\). So every stonean space \\(\\Omega\\) is homeomorphic to \\(\\Omega_\\Omega\\).\n\n**Proof.** (1) Take an increasing net \\((\\tilde f_i)\\) in \\(\\mathcal D(\\Gamma)_h\\) that is bounded in norm. By Lemma 9.1 and truncation we may choose lsc representatives \\(f_i\\) with \\(|f_i|\\le c\\). Put \\(f=\\sup_if_i\\), a bounded lsc function. Since \\(f\\ge f_i\\), \\(\\tilde f\\) is an upper bound. Let \\(\\tilde g\\) be another upper bound, with \\(g\\) a bounded usc representative (Lemma 9.1). For each \\(i\\), \\(f_i\\le g\\) outside a meager set. The set \\(A=\\{f>g\\}\\) is open, since \\(f-g\\) is lsc. By Zorn's lemma choose a maximal family \\((G_j)\\) of pairwise disjoint nonempty open sets such that each \\(G_j\\cap A\\) is meager, and put \\(G=\\bigcup_jG_j\\).\n\nWe claim that \\(G\\) is dense. If not, \\(V=\\Gamma\\setminus\\overline G\\) is a nonempty open set. If \\(V\\cap A=\\varnothing\\), then \\(V\\) could be added to the family. Otherwise pick \\(\\gamma\\in V\\cap A\\). Then \\(f(\\gamma)>g(\\gamma)\\), so \\(f_i(\\gamma)>g(\\gamma)\\) for some \\(i\\). The set \\(W=V\\cap\\{f_i>g\\}\\) is open, because \\(f_i-g\\) is lsc, and it contains \\(\\gamma\\). It is meager, since \\(f_i\\le g\\) outside a meager set. So \\(W\\cap A\\) is meager, and \\(W\\) could be added to the family. Both cases contradict maximality.\n\nSo \\(\\Gamma\\setminus G\\) is closed with empty interior, that is, rare. By Lemma 2.1(3), \\(\\bigcup_j(G_j\\cap A)\\) is meager. Hence \\(A=\\bigcup_j(G_j\\cap A)\\cup(A\\setminus G)\\) is meager, which means \\(\\tilde f\\le\\tilde g\\). So \\(\\tilde f\\) is the least upper bound. For a nonempty set \\(S\\subseteq\\mathcal D(\\Gamma)_h\\) that is bounded above, the classes of maxima of finitely many representatives form a bounded increasing net with the same upper bounds, as in the proof of Theorem 7.1. So \\(\\mathcal D(\\Gamma)_h\\) is conditionally complete. Through the Gelfand isomorphism, which preserves order, so is \\(C_{\\mathbb R}(\\Omega_\\Gamma)\\), and \\(\\Omega_\\Gamma\\) is stonean by Theorem 4.3.\n\n(2) The map is a \\(*\\)-homomorphism. It is injective: if \\(x\\) is continuous and \\(\\{x\\ne0\\}\\) is meager, this open set is empty by Lemma 2.1(2). It is onto: a bounded real Borel function agrees outside a meager set with an lsc function (Lemma 9.1), which agrees outside a meager set with a continuous function (Theorem 4.3). Isomorphic commutative C\\*-algebras have homeomorphic spectra. \\(\\square\\)\n\n**Remark 9.4.** If \\(\\Gamma\\) is a Baire space, the proof of Theorem 9.3(1) shows more: the set \\(A\\) is empty. Indeed, if \\(\\gamma\\in A\\), then the nonempty open set \\(\\{f_i>g\\}\\) is meager for a suitable \\(i\\), which is impossible in a Baire space. So then \\(f\\le g\\) everywhere.\n\n**Theorem 9.5** (A stonean space with a dense meager subset). Let \\(\\Gamma\\) be a regular Baire space that contains a dense meager subset; here *regular* means that a point and a closed set not containing it have disjoint open neighbourhoods. Then \\(\\Omega_\\Gamma\\) contains a dense meager subset. So \\(\\Omega_\\Gamma\\) is its own second part in Theorem 6.4, and it carries no normal measure other than \\(0\\). If moreover \\(\\Gamma\\) is Hausdorff, has no isolated points and has a countable family of nonempty open sets such that every nonempty open set contains one of them, then \\(\\Omega_\\Gamma\\) has no isolated points and has a countable dense subset. All of this applies to \\(\\Gamma=[0,1]\\).\n\n**Proof.** Let \\(A=\\bigcup_nA_n\\) be dense, with each \\(A_n\\) rare. Replacing \\(A_n\\) by its closure, we may assume that each \\(A_n\\) is closed. For each \\(n\\) let \\(B_n\\) be the intersection of the sets \\(\\widehat U\\) over all open \\(U\\supseteq A_n\\). It is closed.\n\n*Each \\(B_n\\) is rare.* Suppose \\(B_n\\) contains a nonempty open set. Then it contains a nonempty clopen set, of the form \\(\\widehat V\\) with \\(V\\) open and not meager (Lemma 9.2). For every open \\(U\\supseteq A_n\\), \\(\\widehat V\\subseteq\\widehat U\\), so \\(V\\setminus U\\) is meager. The open set \\(V\\setminus A_n\\) is not meager, because \\(V\\) is not and \\(A_n\\) is rare; so it contains a point \\(\\gamma\\). As \\(\\Gamma\\) is regular and \\(A_n\\) is closed, there are disjoint open sets \\(P\\ni\\gamma\\) and \\(U\\supseteq A_n\\). The set \\(V\\cap P\\) is open and nonempty, hence not meager, because \\(\\Gamma\\) is a Baire space. But \\(V\\cap P\\subseteq V\\setminus U\\), which is meager. This is a contradiction.\n\n*The union \\(B=\\bigcup_nB_n\\) is dense.* Suppose a nonempty clopen set \\(\\widehat V\\), with \\(V\\) open and not meager, misses every \\(B_n\\). The open sets \\(U\\supseteq A_n\\) are closed under finite intersections, and so are the sets \\(\\widehat U\\) (Lemma 9.2(1)). The compact set \\(\\widehat V\\) misses their intersection \\(B_n\\), so it misses one of them: there is an open \\(U_n\\supseteq A_n\\) with \\(\\widehat V\\cap\\widehat{U_n}=\\varnothing\\), that is, \\(V\\cap U_n\\) is meager. The open set \\(U=\\bigcup_nU_n\\) contains \\(A\\), so it is dense, and \\(V\\cap U=\\bigcup_n(V\\cap U_n)\\) is meager. But \\(V\\cap U\\) is a nonempty open set, because \\(V\\) is open and nonempty and \\(U\\) is dense. This contradicts the Baire property of \\(\\Gamma\\).\n\nSo \\(B\\) is a dense meager subset of \\(\\Omega_\\Gamma\\). A normal measure vanishes on \\(B\\) (Theorem 5.2), hence on \\(\\overline B=\\Omega_\\Gamma\\) (Corollary 5.4(1)).\n\n*Separability.* Let \\((V_n)\\) be the countable family, and choose \\(\\omega_n\\in\\widehat{V_n}\\), which is not empty by Lemma 9.2(3). A nonempty clopen subset of \\(\\Omega_\\Gamma\\) is \\(\\widehat U\\) with \\(U\\) open and not meager, and \\(U\\) contains some \\(V_n\\); so \\(\\omega_n\\in\\widehat{V_n}\\subseteq\\widehat U\\). So \\(\\{\\omega_n\\}\\) is dense. If \\(\\{\\omega\\}\\) were open, it would be clopen, say \\(\\{\\omega\\}=\\widehat U\\) with \\(U\\) nonempty and open. Since \\(\\Gamma\\) is Hausdorff without isolated points, \\(U\\) contains two disjoint nonempty open sets \\(U_1,U_2\\), and \\(\\widehat{U_1}\\), \\(\\widehat{U_2}\\) would be disjoint nonempty subsets of \\(\\{\\omega\\}\\).\n\nFor \\(\\Gamma=[0,1]\\): it is a compact metric space, hence regular, Hausdorff and a Baire space; the rational points form a dense meager set; the open intervals with rational end points form the countable family; and there are no isolated points. \\(\\square\\)\n\nSo \\(C(\\Omega_{[0,1]})\\cong\\mathcal D([0,1])\\) is an abelian C\\*-algebra in which every nonempty bounded set of self-adjoint elements has a least upper bound, but which is isomorphic to no von Neumann algebra.\n\n**Theorem 9.6** (Stonean spaces on which every measure lives on a rare set). Let \\(\\Gamma\\) be a topological space with a base \\(\\mathfrak B\\) of nonempty open sets such that\n\n\\((\\ast)\\) for every decreasing sequence \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\), the intersection \\(\\bigcap_nB_n\\) contains a member of \\(\\mathfrak B\\).\n\n1. If \\(G_1,G_2,\\dots\\) are dense open subsets of \\(\\Gamma\\), the interior of \\(\\bigcap_nG_n\\) is dense. So \\(\\Gamma\\) is a Baire space, and every meager subset of \\(\\Gamma\\) is rare.\n2. The sets \\(\\widehat B\\), \\(B\\in\\mathfrak B\\), are nonempty clopen sets, and every nonempty open subset of \\(\\Omega_\\Gamma\\) contains one of them. Every nonempty clopen subset of \\(\\widehat B\\) contains \\(\\widehat C\\) for some \\(C\\in\\mathfrak B\\) with \\(C\\subseteq B\\). For every decreasing sequence \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\) there is \\(C\\in\\mathfrak B\\) with \\(\\widehat C\\subseteq\\bigcap_n\\widehat{B_n}\\).\n3. Every meager subset of \\(\\Omega_\\Gamma\\) is rare.\n4. If \\(\\Gamma\\) is Hausdorff and has no isolated points, every nonempty clopen subset of \\(\\Omega_\\Gamma\\) is uncountable, and every Radon measure on \\(\\Omega_\\Gamma\\) has a rare support.\n\nIf \\(\\Gamma\\) is a nonempty Hausdorff space without isolated points, \\(\\Omega_\\Gamma\\) is a nonempty stonean space that is its own third part in Theorem 6.4.\n\nExample 9.8 proves that for the space of Example 9.7 no base of \\(\\Omega_\\Gamma\\) satisfies \\((\\ast)\\), so (2) is stated for sequences that decrease in \\(\\Gamma\\).\n\n**Proof.** (1) Let \\(V\\) be a nonempty open set. Choose \\(B_1\\in\\mathfrak B\\) with \\(B_1\\subseteq V\\cap G_1\\), then \\(B_2\\in\\mathfrak B\\) with \\(B_2\\subseteq B_1\\cap G_2\\), and so on; each of these open sets is nonempty because the \\(G_n\\) are dense. By \\((\\ast)\\), \\(\\bigcap_nB_n\\) contains some \\(B\\in\\mathfrak B\\), and \\(B\\) is an open subset of \\(V\\cap\\bigcap_nG_n\\). So the interior of \\(\\bigcap_nG_n\\) meets \\(V\\). If \\(M=\\bigcup_nR_n\\) is meager, the sets \\(G_n=\\Gamma\\setminus\\overline{R_n}\\) are dense and open, and \\(M\\) lies in the complement of the interior of \\(\\bigcap_nG_n\\), a closed set with empty interior. So \\(M\\) is rare, and in particular has empty interior.\n\n(2) By (1) and Lemma 9.2(3), \\(\\widehat B\\ne\\varnothing\\). A nonempty open subset of \\(\\Omega_\\Gamma\\) contains a nonempty clopen set, which is \\(\\widehat E\\) for an open \\(E\\) that is not meager (Lemma 9.2); \\(E\\) contains some \\(B\\in\\mathfrak B\\), and \\(\\widehat B\\subseteq\\widehat E\\). If a nonempty clopen set \\(\\widehat E\\), with \\(E\\) open, lies in \\(\\widehat B\\), then \\(E\\setminus B\\) is meager. So \\(E\\cap B\\) is not meager, hence not empty, and it contains some \\(C\\in\\mathfrak B\\); then \\(C\\subseteq B\\) and \\(\\widehat C\\subseteq\\widehat E\\). Finally, if \\(B_1\\supseteq B_2\\supseteq\\cdots\\), then \\((\\ast)\\) gives \\(C\\in\\mathfrak B\\) with \\(C\\subseteq B_n\\), hence \\(\\widehat C\\subseteq\\widehat{B_n}\\), for all \\(n\\).\n\n(3) Let \\(R_1,R_2,\\dots\\) be rare subsets of \\(\\Omega_\\Gamma\\) and \\(W\\) a nonempty open set. The open set \\(W\\setminus\\overline{R_1}\\) is not empty, so by (2) it contains \\(\\widehat{B_1}\\) for some \\(B_1\\in\\mathfrak B\\). The open set \\(\\widehat{B_1}\\setminus\\overline{R_2}\\) is not empty and contains a nonempty clopen set; by (2) this contains \\(\\widehat{B_2}\\) with \\(B_2\\in\\mathfrak B\\) and \\(B_2\\subseteq B_1\\). Continuing, we get \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\) with \\(\\widehat{B_n}\\cap R_n=\\varnothing\\). By (2) some nonempty \\(\\widehat C\\) lies in every \\(\\widehat{B_n}\\). It is an open subset of \\(W\\) that misses \\(\\bigcup_nR_n\\). So \\(\\bigcup_nR_n\\) is dense in no nonempty open set, that is, it is rare.\n\n(4) Let \\(W\\) be a nonempty clopen subset of \\(\\Omega_\\Gamma\\). It contains some \\(\\widehat B\\). The open set \\(B\\) contains two disjoint nonempty open sets, since \\(\\Gamma\\) is Hausdorff and has no isolated points, and they give two disjoint nonempty clopen subsets of \\(\\widehat B\\) (Lemma 9.2). So no clopen set is a single point; since the clopen sets form a base, \\(W\\) has no isolated points. A compact space without isolated points is uncountable: if it were countable, it would be a countable union of rare singletons, against Lemma 2.1(2).\n\nNow let \\(\\mu\\) be a Radon measure on \\(\\Omega_\\Gamma\\). Only countably many points have positive measure, so every nonempty clopen set contains a point \\(\\omega\\) with \\(\\mu(\\{\\omega\\})=0\\). By outer regularity and Lemma 4.2(3), such a point has clopen neighbourhoods of arbitrarily small measure. Start from a nonempty clopen \\(W\\), and choose \\(B_1\\in\\mathfrak B\\) with \\(\\widehat{B_1}\\subseteq W\\). Given \\(B_n\\), choose \\(\\omega\\in\\widehat{B_n}\\) with \\(\\mu(\\{\\omega\\})=0\\), a clopen neighbourhood \\(Q\\subseteq\\widehat{B_n}\\) of \\(\\omega\\) with \\(\\mu(Q)<1/(n+1)\\), and, by (2), \\(B_{n+1}\\in\\mathfrak B\\) with \\(B_{n+1}\\subseteq B_n\\) and \\(\\widehat{B_{n+1}}\\subseteq Q\\). By (2) again some nonempty \\(\\widehat C\\) lies in all \\(\\widehat{B_n}\\), so \\(\\mu(\\widehat C)=0\\). Thus every nonempty clopen set contains a nonempty open null set. If the support \\(S\\) of \\(\\mu\\) had interior points, its interior would contain a nonempty clopen set, and hence a nonempty open null set, which is impossible inside the support. So \\(S\\) is rare.\n\nThe final statement: \\(\\Omega_\\Gamma\\) is not empty, because \\(\\Gamma\\) is a nonempty Baire space and so is not meager in itself (Lemma 9.2(1)). By (3), no meager set is dense in a nonempty open set, so the second part is empty. The support of a nonzero normal measure is a nonempty clopen set (Corollary 5.4(3)), which is not rare; by (4) there is no such measure, so the first part is empty. \\(\\square\\)\n\n**Example 9.7** (A space satisfying \\((\\ast)\\)). Let \\(I\\) be an uncountable set and \\(\\Gamma=\\{0,1\\}^I\\). For a countable set \\(J\\subseteq I\\) and \\(\\alpha\\in\\{0,1\\}^J\\) put \\(U(J,\\alpha)=\\{\\gamma\\in\\Gamma:\\gamma_j=\\alpha_j\\text{ for all }j\\in J\\}\\). The intersection of two such sets is empty or again of this form, so they form a base \\(\\mathfrak B\\) of a topology. Two different points differ at some coordinate \\(i\\), and the sets \\(U(\\{i\\},\\cdot)\\) through them are disjoint; so \\(\\Gamma\\) is Hausdorff. Each \\(U(J,\\alpha)\\) is also closed, since its complement is the union of the sets \\(U(\\{j\\},1-\\alpha_j)\\), \\(j\\in J\\). So \\(\\Gamma\\) has a base of clopen sets, and it is completely regular, because the indicators of these sets are continuous. No point is isolated, because a basic set fixes only countably many of the uncountably many coordinates. If \\(U(J_1,\\alpha_1)\\supseteq U(J_2,\\alpha_2)\\supseteq\\cdots\\), then \\(J_1\\subseteq J_2\\subseteq\\cdots\\) and each \\(\\alpha_{n+1}\\) extends \\(\\alpha_n\\), so the intersection is \\(U(\\bigcup_nJ_n,\\bigcup_n\\alpha_n)\\), which lies in \\(\\mathfrak B\\). So Theorem 9.6 applies: \\(\\Omega_\\Gamma\\) is a stonean space in which every meager set is rare and every Radon measure has a rare support.\n\n**Example 9.8** (No base of \\(\\Omega_\\Gamma\\) satisfies \\((\\ast)\\)). Keep \\(\\Gamma\\) of Example 9.7, and choose distinct indices \\(j_1,j_2,\\dots\\) in \\(I\\). Let \\(V_k\\) be the set of \\(\\gamma\\) with \\(\\gamma_{j_1}=\\dots=\\gamma_{j_{k-1}}=0\\) and \\(\\gamma_{j_k}=1\\); these are disjoint basic sets. Put \\(R_n=\\bigcup_{k\\ge n}V_k\\). It is the intersection of \\(U(\\{j_1,\\dots,j_{n-1}\\},0)\\) with the complement of the basic set \\(Z=U(\\{j_k:k\\ge1\\},0)\\), so it is clopen. The sets \\(R_n\\) decrease, they are nonempty, and \\(\\bigcap_nR_n=\\varnothing\\), because a point of \\(R_n\\) lies in exactly one \\(V_k\\), and then \\(k\\ge n\\).\n\nIn \\(\\Omega_\\Gamma\\) the clopen sets \\(\\widehat{R_n}\\) decrease and are nonempty, by Lemma 9.2(3), since \\(\\Gamma\\) is a Baire space by Theorem 9.6(1). By compactness they have a common point \\(\\omega\\). Their intersection has empty interior. Otherwise it contains a nonempty clopen set \\(\\widehat E\\), with \\(E\\) not meager, and \\(E\\setminus R_n\\) is meager for every \\(n\\); then \\(E\\subseteq\\bigcup_n(E\\setminus R_n)\\cup\\bigcap_nR_n\\) is meager. Now let \\(\\mathfrak G\\) be any base of \\(\\Omega_\\Gamma\\). Choose \\(W_1\\in\\mathfrak G\\) with \\(\\omega\\in W_1\\subseteq\\widehat{R_1}\\), and inductively \\(W_{n+1}\\in\\mathfrak G\\) with \\(\\omega\\in W_{n+1}\\subseteq W_n\\cap\\widehat{R_{n+1}}\\). Then \\((W_n)\\) decreases in \\(\\mathfrak G\\), and \\(\\bigcap_nW_n\\subseteq\\bigcap_n\\widehat{R_n}\\) has empty interior, so it contains no nonempty member of \\(\\mathfrak G\\). This is why Theorem 9.6(2) speaks of the sets \\(\\widehat B\\), which form a base for the nonempty open sets in the weaker sense that every nonempty open set contains one of them, and of sequences that decrease in \\(\\Gamma\\).\n\n",
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      "name": "10. The dual of \\(L^\\infty\\)",
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      "full_conditions_and_proof": "### 10. The dual of \\(L^\\infty\\)\n\nIntegrable functions give bounded linear functionals on \\(L^\\infty\\). When \\(L^\\infty\\) is infinite-dimensional they are not all of its dual: the dual consists of finitely additive measures. We prove this in a setting that covers \\(L^\\infty\\) of a Radon measure, the algebras \\(\\ell^\\infty(X)\\), and the algebras \\(\\mathcal D(\\Gamma)\\) of Section 9.\n\nLet \\(X\\) be a set, \\(\\Sigma\\) a \\(\\sigma\\)-algebra of subsets of \\(X\\), and \\(\\mathcal I\\subseteq\\Sigma\\) a \\(\\sigma\\)-ideal: countable unions of members of \\(\\mathcal I\\) lie in \\(\\mathcal I\\), and so do the members of \\(\\Sigma\\) that are contained in a member of \\(\\mathcal I\\). Let \\(L^\\infty(\\Sigma,\\mathcal I)\\) be the space of bounded \\(\\Sigma\\)-measurable complex functions, modulo those that vanish outside a member of \\(\\mathcal I\\), with the norm\n\\[\n\\|f\\|_\\infty=\\min\\{c\\ge0:\\{|f|>c\\}\\in\\mathcal I\\};\n\\]\nthe minimum is attained because \\(\\mathcal I\\) is closed under countable unions. It is an abelian C\\*-algebra. Three examples:\n\n- for a locally compact \\(\\Gamma\\) with a Radon measure \\(\\mu\\), \\(\\Sigma\\) the \\(\\mu\\)-measurable sets and \\(\\mathcal I\\) the locally null ones, this is \\(L^\\infty(\\Gamma,\\mu)\\);\n- for \\(\\Sigma\\) the set of all subsets of \\(X\\) and \\(\\mathcal I=\\{\\varnothing\\}\\), it is \\(\\ell^\\infty(X)\\);\n- for \\(\\Sigma\\) the Borel sets of a topological space \\(\\Gamma\\) and \\(\\mathcal I\\) the meager Borel sets, it is \\(\\mathcal D(\\Gamma)\\).\n\nLet \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) be the set of functions \\(\\nu:\\Sigma\\to\\mathbb C\\) that are finitely additive, bounded (\\(\\sup_{E\\in\\Sigma}|\\nu(E)|<\\infty\\)), and vanish on \\(\\mathcal I\\). The *variation* of \\(\\nu\\) on \\(E\\in\\Sigma\\) is \\(|\\nu|(E)=\\sup\\sum_k|\\nu(E_k)|\\), over the finite partitions of \\(E\\) into sets \\(E_k\\in\\Sigma\\).\n\n**Lemma 10.1.** Let \\(\\nu\\in\\operatorname{ba}(\\Sigma,\\mathcal I)\\). Then \\(|\\nu|(X)\\le4\\sup_E|\\nu(E)|\\). If \\(E_1,\\dots,E_n\\in\\Sigma\\) are disjoint, then \\(\\sum_k|\\nu|(E_k)=|\\nu|(\\bigcup_kE_k)\\). So \\(|\\nu|\\) is a bounded, positive, finitely additive function on \\(\\Sigma\\), and \\(\\|\\nu\\|=|\\nu|(X)\\) is a norm on \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\).\n\n**Proof.** Let \\((E_k)\\) be a finite partition of \\(X\\). Let \\(P\\) be the union of the \\(E_k\\) with \\(\\operatorname{Re}\\nu(E_k)\\ge0\\) and \\(Q\\) the union of the others. Then \\[\n\\begin{gathered}\n\\sum_k|\\operatorname{Re}\\nu(E_k)|\\\\\n=\\operatorname{Re}\\nu(P)-\\operatorname{Re}\\nu(Q)\\\\\n\\le2\\sup|\\nu|,\n\\end{gathered}\n\\] and the same holds for the imaginary parts. Since \\(|z|\\le|\\operatorname{Re}z|+|\\operatorname{Im}z|\\), the first claim follows. For the second, partitions of the sets \\(E_k\\) together form a partition of their union, which gives \\(\\le\\). Conversely, if \\((A_l)\\) is a finite partition of the union, then \\[\n\\begin{gathered}\n\\sum_l|\\nu(A_l)|\\\\\n\\le\\sum_l\\sum_k|\\nu(A_l\\cap E_k)|\\\\\n\\le\\sum_k|\\nu|(E_k).\n\\end{gathered}\n\\] Finally, \\(|\\nu+\\nu'|(X)\\le|\\nu|(X)+|\\nu'|(X)\\), and \\(|\\nu(E)|\\le|\\nu|(X)\\) for every \\(E\\), so \\(|\\nu|(X)=0\\) only for \\(\\nu=0\\). \\(\\square\\)\n\n**Theorem 10.2** (The dual of \\(L^\\infty\\)). Let \\(\\nu\\in\\operatorname{ba}(\\Sigma,\\mathcal I)\\). For a simple function \\(s=\\sum_kc_k1_{E_k}\\), with \\((E_k)\\) a finite partition of \\(X\\) into sets of \\(\\Sigma\\), put \\(\\int s\\,d\\nu=\\sum_kc_k\\nu(E_k)\\). This extends to a bounded linear functional \\(\\varphi_\\nu\\) on \\(L^\\infty(\\Sigma,\\mathcal I)\\). The map \\(\\nu\\mapsto\\varphi_\\nu\\) is a linear bijection of \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) onto the dual of \\(L^\\infty(\\Sigma,\\mathcal I)\\), and it is isometric for the norm \\(\\|\\nu\\|=|\\nu|(X)\\); in particular \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) is a Banach space. The functional \\(\\varphi_\\nu\\) is positive if and only if \\(\\nu\\ge0\\).\n\n**Proof.** The value \\(\\sum_kc_k\\nu(E_k)\\) does not depend on the partition, since two partitions have a common refinement and \\(\\nu\\) is additive. It is linear in \\(s\\). If \\(E_k\\in\\mathcal I\\), then \\(\\nu(E_k)=0\\); if not, then \\(|c_k|\\le\\|s\\|_\\infty\\), because \\(E_k\\subseteq\\{|s|>\\|s\\|_\\infty\\}\\in\\mathcal I\\) would hold otherwise. So\n\\[\n\\Big|\\int s\\,d\\nu\\Big|\\le\\sum_{E_k\\notin\\mathcal I}|c_k||\\nu(E_k)|\\le\\|s\\|_\\infty\\,|\\nu|(X).\n\\]\nEvery bounded measurable function is a uniform limit of simple functions, so \\(\\int\\cdot\\,d\\nu\\) extends by continuity to a functional \\(\\varphi_\\nu\\) on \\(L^\\infty(\\Sigma,\\mathcal I)\\) with \\(\\|\\varphi_\\nu\\|\\le|\\nu|(X)\\); it vanishes on functions that vanish outside a member of \\(\\mathcal I\\), by the estimate. Conversely, take any \\(\\varphi\\) in the dual of \\(L^\\infty(\\Sigma,\\mathcal I)\\), and put \\(\\nu(E)=\\varphi(1_E)\\). Then \\(\\nu\\) is finitely additive, \\(|\\nu(E)|\\le\\|\\varphi\\|\\), and \\(\\nu\\) vanishes on \\(\\mathcal I\\). For a finite partition \\((E_k)\\) choose numbers \\(c_k\\) of modulus one with \\(c_k\\nu(E_k)=|\\nu(E_k)|\\). Then\n\\[\n\\sum_k|\\nu(E_k)|=\\varphi\\Big(\\sum_kc_k1_{E_k}\\Big)\\le\\|\\varphi\\| ,\n\\]\nso \\(|\\nu|(X)\\le\\|\\varphi\\|\\). The functionals \\(\\varphi\\) and \\(\\varphi_\\nu\\) agree on simple functions, hence everywhere. Different \\(\\nu\\) give different functionals, since \\(\\varphi_\\nu(1_E)=\\nu(E)\\). So the map is bijective and isometric. If \\(\\nu\\ge0\\), then \\(\\int s\\,d\\nu\\ge0\\) for simple \\(s\\ge0\\), and by approximation \\(\\varphi_\\nu\\ge0\\); conversely \\(\\nu(E)=\\varphi_\\nu(1_E)\\). \\(\\square\\)\n\nFor a finite measure \\(\\mu\\) with \\(\\mathcal I\\) its null sets, the countably additive members of \\(\\operatorname{ba}\\) are exactly the measures \\(h\\,d\\mu\\) with \\(h\\in L^1(\\mu)\\), by the [complex Radon–Nikodym theorem of Section 2](#oa-fnd-ao-23); so \\(L^1(\\mu)\\) is the part of the dual of \\(L^\\infty(\\mu)\\) that consists of countably additive measures. For \\(\\ell^\\infty(\\mathbb N)\\), the countably additive members are the functions \\(E\\mapsto\\sum_{n\\in E}c_n\\) with \\(c\\in\\ell^1(\\mathbb N)\\). The dual of \\(\\ell^\\infty(\\mathbb N)\\) is much larger (Exercise 12.2).\n\n**Theorem 10.3** (Phillips' lemma). Let \\(\\Gamma\\) be a set and \\((\\nu_n)\\) a sequence in \\(\\operatorname{ba}(2^\\Gamma,\\{\\varnothing\\})=\\ell^\\infty(\\Gamma)^*\\) with \\(\\sup_n|\\nu_n|(\\Gamma)<\\infty\\). If \\(\\nu_n(E)\\to0\\) for every \\(E\\subseteq\\Gamma\\), then\n\\[\n\\lim_{n\\to\\infty}\\sum_{\\gamma\\in\\Gamma}|\\nu_n(\\{\\gamma\\})|=0 .\n\\]\n\n\n**Proof.** Put \\(C=\\sup_n|\\nu_n|(\\Gamma)\\) and \\(\\sigma_n=\\sum_\\gamma|\\nu_n(\\{\\gamma\\})|\\). By Lemma 10.1, applied to finite sets of points, \\(\\sigma_n\\le C\\). Since \\((\\nu_n)\\) is bounded and \\(\\nu_n(E)\\to0\\) for every \\(E\\), also \\(\\nu_n(f)\\to0\\) for every \\(f\\in\\ell^\\infty(\\Gamma)\\): approximate \\(f\\) uniformly by simple functions. Suppose the conclusion fails. Passing to a subsequence, we may assume \\(\\sigma_n>5\\delta\\) for all \\(n\\), for some \\(\\delta>0\\).\n\n*Step 1: a sliding hump.* We choose indices \\(n_1<n_2<\\cdots\\) and pairwise disjoint finite sets \\(F_1,F_2,\\dots\\subseteq\\Gamma\\) with\n\\[\n\\sum_{\\gamma\\notin F_k}|\\nu_{n_k}(\\{\\gamma\\})|<\\delta\\qquad(k\\ge1).\n\\tag{10.1}\n\\]\nTake \\(n_1=1\\) and a finite \\(F_1\\) that carries all but \\(\\delta\\) of the convergent sum \\(\\sigma_1\\). Given \\(n_1,\\dots,n_k\\) and \\(F_1,\\dots,F_k\\), let \\(P=F_1\\cup\\dots\\cup F_k\\), a finite set. Since \\(\\nu_n(\\{\\gamma\\})\\to0\\) for each \\(\\gamma\\in P\\), there is \\(n_{k+1}>n_k\\) with \\(\\sum_{\\gamma\\in P}|\\nu_{n_{k+1}}(\\{\\gamma\\})|<\\delta/2\\). Then choose a finite \\(F_{k+1}\\subseteq\\Gamma\\setminus P\\) with \\(\\sum_{\\gamma\\notin P\\cup F_{k+1}}|\\nu_{n_{k+1}}(\\{\\gamma\\})|<\\delta/2\\). This gives (10.1) for \\(k+1\\).\n\n*Step 2: thinning out.* Write \\(\\lambda_k=\\nu_{n_k}\\). We find an infinite set \\(K\\subseteq\\mathbb N\\) such that\n\\[\n\\begin{gathered}\n|\\lambda_k|\\Big(\\\\\n\\bigcup\\{F_j:j\\in K,\\ j>k\\}\\Big)<\\delta\\\\\n(k\\in K).\n\\end{gathered}\n\\tag{10.2}\n\\]\nPut \\(K_0=\\mathbb N\\) and \\(k_1=\\min K_0\\). Choose an integer \\(N>C/\\delta\\), and split \\(K_0\\setminus\\{k_1\\}\\) into \\(N\\) disjoint infinite sets. The corresponding unions of the \\(F_j\\) are disjoint, so by Lemma 10.1 their \\(|\\lambda_{k_1}|\\)-variations add up to at most \\(C\\), and one of them, indexed by an infinite set \\(K_1\\), has variation less than \\(\\delta\\). Put \\(k_2=\\min K_1\\), split \\(K_1\\setminus\\{k_2\\}\\) in the same way with respect to \\(\\lambda_{k_2}\\), and obtain \\(K_2\\subseteq K_1\\); and so on. The set \\(K=\\{k_1,k_2,\\dots\\}\\) is infinite, and the elements of \\(K\\) after \\(k_i\\) lie in \\(K_i\\). So (10.2) holds for \\(k=k_i\\).\n\n*Step 3: a test function.* For \\(k\\in K\\) and \\(\\gamma\\in F_k\\) let \\(f(\\gamma)\\) be the number of modulus one with \\(f(\\gamma)\\lambda_k(\\{\\gamma\\})=|\\lambda_k(\\{\\gamma\\})|\\), and let \\(f=0\\) elsewhere. Then \\(f\\in\\ell^\\infty(\\Gamma)\\), \\(\\|f\\|\\le1\\). Fix \\(k\\in K\\), let \\(P_k\\) be the union of the \\(F_j\\) with \\(j\\in K\\), \\(j<k\\), and \\(Q_k\\) the union of those with \\(j>k\\). Then\n\\[\n\\begin{gathered}\n\\lambda_k(f)\\\\\n=\\sum_{\\gamma\\in F_k}|\\lambda_k(\\{\\gamma\\})|\\\\\n+\\sum_{\\gamma\\in P_k}f(\\gamma)\\lambda_k(\\{\\gamma\\})\\\\\n+\\lambda_k(f1_{Q_k}).\n\\end{gathered}\n\\]\nThe first term exceeds \\(5\\delta-\\delta=4\\delta\\), by (10.1). The second is at most \\(\\delta\\) in modulus, again by (10.1), because the finite set \\(P_k\\) misses \\(F_k\\). The third is at most \\(|\\lambda_k|(Q_k)\\|f\\|<\\delta\\), by the estimate in the proof of Theorem 10.2 and (10.2). So \\(|\\lambda_k(f)|>2\\delta\\) for every \\(k\\in K\\). But \\(\\lambda_k(f)=\\nu_{n_k}(f)\\to0\\). This contradiction proves the theorem. \\(\\square\\)\n\n**Corollary 10.4** (Weak and norm convergence in \\(\\ell^1\\)). Let \\(\\Gamma\\) be a set. Every weakly convergent sequence in \\(\\ell^1(\\Gamma)\\) converges in norm. More generally, every weakly Cauchy sequence in \\(\\ell^1(\\Gamma)\\) converges in norm.\n\n\n**Proof.** Let \\(g_n\\to0\\) weakly in \\(\\ell^1(\\Gamma)\\), whose dual is \\(\\ell^\\infty(\\Gamma)\\). By the [uniform boundedness principle, Theorem 4.1 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-04) the sequence is bounded. Put \\(\\nu_n(E)=\\sum_{\\gamma\\in E}g_n(\\gamma)\\). Then \\(\\nu_n\\in\\ell^\\infty(\\Gamma)^*\\), \\(|\\nu_n|(\\Gamma)\\le\\|g_n\\|_1\\), and \\(\\nu_n(E)=\\langle g_n,1_E\\rangle\\to0\\) for every \\(E\\). By Theorem 10.3, \\(\\|g_n\\|_1=\\sum_\\gamma|\\nu_n(\\{\\gamma\\})|\\to0\\). If \\((f_n)\\) converges weakly to \\(f\\), apply this to \\(g_n=f_n-f\\). If \\((f_n)\\) is only weakly Cauchy and not norm Cauchy, there are \\(\\varepsilon>0\\) and indices \\(p_k<q_k\\), both tending to infinity, with \\(\\|f_{p_k}-f_{q_k}\\|_1\\ge\\varepsilon\\). The sequence \\(f_{p_k}-f_{q_k}\\) converges weakly to \\(0\\), hence in norm, which is a contradiction. So \\((f_n)\\) is norm Cauchy and converges, since \\(\\ell^1(\\Gamma)\\) is complete. \\(\\square\\)\n\n**Example 10.5** (The hypotheses matter). The unit vectors \\(e_n\\in\\ell^1(\\mathbb N)\\) have norm one. They converge to \\(0\\) against every element of \\(c_0\\), the predual of \\(\\ell^1\\), but not weakly: pairing with the constant function \\(1\\in\\ell^\\infty\\) gives \\(1\\). Correspondingly, the point evaluations \\(\\nu_n=\\delta_n\\in\\ell^\\infty(\\mathbb N)^*\\) satisfy \\(\\nu_n(E)\\to0\\) for every finite set \\(E\\), but not for \\(E=\\mathbb N\\), and \\(\\sum_k|\\nu_n(\\{k\\})|=1\\) for all \\(n\\). So in Theorem 10.3 the hypothesis is needed for all subsets \\(E\\), not only for the finite ones.\n\n## B. Which measures see the order on the spectrum?\n\nThe Gelfand representation replaces an abelian unital C*-algebra by \\(C(\\Omega)\\). A bounded pointwise supremum of continuous functions need not be continuous, so the supremum *in this ordered algebra* cannot simply be assumed to be the pointwise one. The stonean criterion below resolves that question through the closure of open sets. Normality then asks whether integration preserves those algebraic suprema.\n\nFor a finite spectrum the distinction disappears: all functions are continuous, all sets are clopen, and every finite measure preserves increasing bounded nets. On an infinite spectrum, the distinction is the main issue. A sufficient family of normal measures must detect every nonzero positive function; one measure need not suffice. The support criterion expresses detection geometrically and is the bridge to the operator representation.\n\n",
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      "unit": "abelian-operator-algebras",
      "name": "10. The dual of \\(L^\\infty\\)",
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      "full_conditions_and_proof": "### 10. The dual of \\(L^\\infty\\)\n\nIntegrable functions give bounded linear functionals on \\(L^\\infty\\). When \\(L^\\infty\\) is infinite-dimensional they are not all of its dual: the dual consists of finitely additive measures. We prove this in a setting that covers \\(L^\\infty\\) of a Radon measure, the algebras \\(\\ell^\\infty(X)\\), and the algebras \\(\\mathcal D(\\Gamma)\\) of Section 9.\n\nLet \\(X\\) be a set, \\(\\Sigma\\) a \\(\\sigma\\)-algebra of subsets of \\(X\\), and \\(\\mathcal I\\subseteq\\Sigma\\) a \\(\\sigma\\)-ideal: countable unions of members of \\(\\mathcal I\\) lie in \\(\\mathcal I\\), and so do the members of \\(\\Sigma\\) that are contained in a member of \\(\\mathcal I\\). Let \\(L^\\infty(\\Sigma,\\mathcal I)\\) be the space of bounded \\(\\Sigma\\)-measurable complex functions, modulo those that vanish outside a member of \\(\\mathcal I\\), with the norm\n\\[\n\\|f\\|_\\infty=\\min\\{c\\ge0:\\{|f|>c\\}\\in\\mathcal I\\};\n\\]\nthe minimum is attained because \\(\\mathcal I\\) is closed under countable unions. It is an abelian C\\*-algebra. Three examples:\n\n- for a locally compact \\(\\Gamma\\) with a Radon measure \\(\\mu\\), \\(\\Sigma\\) the \\(\\mu\\)-measurable sets and \\(\\mathcal I\\) the locally null ones, this is \\(L^\\infty(\\Gamma,\\mu)\\);\n- for \\(\\Sigma\\) the set of all subsets of \\(X\\) and \\(\\mathcal I=\\{\\varnothing\\}\\), it is \\(\\ell^\\infty(X)\\);\n- for \\(\\Sigma\\) the Borel sets of a topological space \\(\\Gamma\\) and \\(\\mathcal I\\) the meager Borel sets, it is \\(\\mathcal D(\\Gamma)\\).\n\nLet \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) be the set of functions \\(\\nu:\\Sigma\\to\\mathbb C\\) that are finitely additive, bounded (\\(\\sup_{E\\in\\Sigma}|\\nu(E)|<\\infty\\)), and vanish on \\(\\mathcal I\\). The *variation* of \\(\\nu\\) on \\(E\\in\\Sigma\\) is \\(|\\nu|(E)=\\sup\\sum_k|\\nu(E_k)|\\), over the finite partitions of \\(E\\) into sets \\(E_k\\in\\Sigma\\).\n\n**Lemma 10.1.** Let \\(\\nu\\in\\operatorname{ba}(\\Sigma,\\mathcal I)\\). Then \\(|\\nu|(X)\\le4\\sup_E|\\nu(E)|\\). If \\(E_1,\\dots,E_n\\in\\Sigma\\) are disjoint, then \\(\\sum_k|\\nu|(E_k)=|\\nu|(\\bigcup_kE_k)\\). So \\(|\\nu|\\) is a bounded, positive, finitely additive function on \\(\\Sigma\\), and \\(\\|\\nu\\|=|\\nu|(X)\\) is a norm on \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\).\n\n**Proof.** Let \\((E_k)\\) be a finite partition of \\(X\\). Let \\(P\\) be the union of the \\(E_k\\) with \\(\\operatorname{Re}\\nu(E_k)\\ge0\\) and \\(Q\\) the union of the others. Then \\[\n\\begin{gathered}\n\\sum_k|\\operatorname{Re}\\nu(E_k)|\\\\\n=\\operatorname{Re}\\nu(P)-\\operatorname{Re}\\nu(Q)\\\\\n\\le2\\sup|\\nu|,\n\\end{gathered}\n\\] and the same holds for the imaginary parts. Since \\(|z|\\le|\\operatorname{Re}z|+|\\operatorname{Im}z|\\), the first claim follows. For the second, partitions of the sets \\(E_k\\) together form a partition of their union, which gives \\(\\le\\). Conversely, if \\((A_l)\\) is a finite partition of the union, then \\[\n\\begin{gathered}\n\\sum_l|\\nu(A_l)|\\\\\n\\le\\sum_l\\sum_k|\\nu(A_l\\cap E_k)|\\\\\n\\le\\sum_k|\\nu|(E_k).\n\\end{gathered}\n\\] Finally, \\(|\\nu+\\nu'|(X)\\le|\\nu|(X)+|\\nu'|(X)\\), and \\(|\\nu(E)|\\le|\\nu|(X)\\) for every \\(E\\), so \\(|\\nu|(X)=0\\) only for \\(\\nu=0\\). \\(\\square\\)\n\n**Theorem 10.2** (The dual of \\(L^\\infty\\)). Let \\(\\nu\\in\\operatorname{ba}(\\Sigma,\\mathcal I)\\). For a simple function \\(s=\\sum_kc_k1_{E_k}\\), with \\((E_k)\\) a finite partition of \\(X\\) into sets of \\(\\Sigma\\), put \\(\\int s\\,d\\nu=\\sum_kc_k\\nu(E_k)\\). This extends to a bounded linear functional \\(\\varphi_\\nu\\) on \\(L^\\infty(\\Sigma,\\mathcal I)\\). The map \\(\\nu\\mapsto\\varphi_\\nu\\) is a linear bijection of \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) onto the dual of \\(L^\\infty(\\Sigma,\\mathcal I)\\), and it is isometric for the norm \\(\\|\\nu\\|=|\\nu|(X)\\); in particular \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) is a Banach space. The functional \\(\\varphi_\\nu\\) is positive if and only if \\(\\nu\\ge0\\).\n\n**Proof.** The value \\(\\sum_kc_k\\nu(E_k)\\) does not depend on the partition, since two partitions have a common refinement and \\(\\nu\\) is additive. It is linear in \\(s\\). If \\(E_k\\in\\mathcal I\\), then \\(\\nu(E_k)=0\\); if not, then \\(|c_k|\\le\\|s\\|_\\infty\\), because \\(E_k\\subseteq\\{|s|>\\|s\\|_\\infty\\}\\in\\mathcal I\\) would hold otherwise. So\n\\[\n\\Big|\\int s\\,d\\nu\\Big|\\le\\sum_{E_k\\notin\\mathcal I}|c_k||\\nu(E_k)|\\le\\|s\\|_\\infty\\,|\\nu|(X).\n\\]\nEvery bounded measurable function is a uniform limit of simple functions, so \\(\\int\\cdot\\,d\\nu\\) extends by continuity to a functional \\(\\varphi_\\nu\\) on \\(L^\\infty(\\Sigma,\\mathcal I)\\) with \\(\\|\\varphi_\\nu\\|\\le|\\nu|(X)\\); it vanishes on functions that vanish outside a member of \\(\\mathcal I\\), by the estimate. Conversely, take any \\(\\varphi\\) in the dual of \\(L^\\infty(\\Sigma,\\mathcal I)\\), and put \\(\\nu(E)=\\varphi(1_E)\\). Then \\(\\nu\\) is finitely additive, \\(|\\nu(E)|\\le\\|\\varphi\\|\\), and \\(\\nu\\) vanishes on \\(\\mathcal I\\). For a finite partition \\((E_k)\\) choose numbers \\(c_k\\) of modulus one with \\(c_k\\nu(E_k)=|\\nu(E_k)|\\). Then\n\\[\n\\sum_k|\\nu(E_k)|=\\varphi\\Big(\\sum_kc_k1_{E_k}\\Big)\\le\\|\\varphi\\| ,\n\\]\nso \\(|\\nu|(X)\\le\\|\\varphi\\|\\). The functionals \\(\\varphi\\) and \\(\\varphi_\\nu\\) agree on simple functions, hence everywhere. Different \\(\\nu\\) give different functionals, since \\(\\varphi_\\nu(1_E)=\\nu(E)\\). So the map is bijective and isometric. If \\(\\nu\\ge0\\), then \\(\\int s\\,d\\nu\\ge0\\) for simple \\(s\\ge0\\), and by approximation \\(\\varphi_\\nu\\ge0\\); conversely \\(\\nu(E)=\\varphi_\\nu(1_E)\\). \\(\\square\\)\n\nFor a finite measure \\(\\mu\\) with \\(\\mathcal I\\) its null sets, the countably additive members of \\(\\operatorname{ba}\\) are exactly the measures \\(h\\,d\\mu\\) with \\(h\\in L^1(\\mu)\\), by the [complex Radon–Nikodym theorem of Section 2](#oa-fnd-ao-23); so \\(L^1(\\mu)\\) is the part of the dual of \\(L^\\infty(\\mu)\\) that consists of countably additive measures. For \\(\\ell^\\infty(\\mathbb N)\\), the countably additive members are the functions \\(E\\mapsto\\sum_{n\\in E}c_n\\) with \\(c\\in\\ell^1(\\mathbb N)\\). The dual of \\(\\ell^\\infty(\\mathbb N)\\) is much larger (Exercise 12.2).\n\n**Theorem 10.3** (Phillips' lemma). Let \\(\\Gamma\\) be a set and \\((\\nu_n)\\) a sequence in \\(\\operatorname{ba}(2^\\Gamma,\\{\\varnothing\\})=\\ell^\\infty(\\Gamma)^*\\) with \\(\\sup_n|\\nu_n|(\\Gamma)<\\infty\\). If \\(\\nu_n(E)\\to0\\) for every \\(E\\subseteq\\Gamma\\), then\n\\[\n\\lim_{n\\to\\infty}\\sum_{\\gamma\\in\\Gamma}|\\nu_n(\\{\\gamma\\})|=0 .\n\\]\n\n\n**Proof.** Put \\(C=\\sup_n|\\nu_n|(\\Gamma)\\) and \\(\\sigma_n=\\sum_\\gamma|\\nu_n(\\{\\gamma\\})|\\). By Lemma 10.1, applied to finite sets of points, \\(\\sigma_n\\le C\\). Since \\((\\nu_n)\\) is bounded and \\(\\nu_n(E)\\to0\\) for every \\(E\\), also \\(\\nu_n(f)\\to0\\) for every \\(f\\in\\ell^\\infty(\\Gamma)\\): approximate \\(f\\) uniformly by simple functions. Suppose the conclusion fails. Passing to a subsequence, we may assume \\(\\sigma_n>5\\delta\\) for all \\(n\\), for some \\(\\delta>0\\).\n\n*Step 1: a sliding hump.* We choose indices \\(n_1<n_2<\\cdots\\) and pairwise disjoint finite sets \\(F_1,F_2,\\dots\\subseteq\\Gamma\\) with\n\\[\n\\sum_{\\gamma\\notin F_k}|\\nu_{n_k}(\\{\\gamma\\})|<\\delta\\qquad(k\\ge1).\n\\tag{10.1}\n\\]\nTake \\(n_1=1\\) and a finite \\(F_1\\) that carries all but \\(\\delta\\) of the convergent sum \\(\\sigma_1\\). Given \\(n_1,\\dots,n_k\\) and \\(F_1,\\dots,F_k\\), let \\(P=F_1\\cup\\dots\\cup F_k\\), a finite set. Since \\(\\nu_n(\\{\\gamma\\})\\to0\\) for each \\(\\gamma\\in P\\), there is \\(n_{k+1}>n_k\\) with \\(\\sum_{\\gamma\\in P}|\\nu_{n_{k+1}}(\\{\\gamma\\})|<\\delta/2\\). Then choose a finite \\(F_{k+1}\\subseteq\\Gamma\\setminus P\\) with \\(\\sum_{\\gamma\\notin P\\cup F_{k+1}}|\\nu_{n_{k+1}}(\\{\\gamma\\})|<\\delta/2\\). This gives (10.1) for \\(k+1\\).\n\n*Step 2: thinning out.* Write \\(\\lambda_k=\\nu_{n_k}\\). We find an infinite set \\(K\\subseteq\\mathbb N\\) such that\n\\[\n\\begin{gathered}\n|\\lambda_k|\\Big(\\\\\n\\bigcup\\{F_j:j\\in K,\\ j>k\\}\\Big)<\\delta\\\\\n(k\\in K).\n\\end{gathered}\n\\tag{10.2}\n\\]\nPut \\(K_0=\\mathbb N\\) and \\(k_1=\\min K_0\\). Choose an integer \\(N>C/\\delta\\), and split \\(K_0\\setminus\\{k_1\\}\\) into \\(N\\) disjoint infinite sets. The corresponding unions of the \\(F_j\\) are disjoint, so by Lemma 10.1 their \\(|\\lambda_{k_1}|\\)-variations add up to at most \\(C\\), and one of them, indexed by an infinite set \\(K_1\\), has variation less than \\(\\delta\\). Put \\(k_2=\\min K_1\\), split \\(K_1\\setminus\\{k_2\\}\\) in the same way with respect to \\(\\lambda_{k_2}\\), and obtain \\(K_2\\subseteq K_1\\); and so on. The set \\(K=\\{k_1,k_2,\\dots\\}\\) is infinite, and the elements of \\(K\\) after \\(k_i\\) lie in \\(K_i\\). So (10.2) holds for \\(k=k_i\\).\n\n*Step 3: a test function.* For \\(k\\in K\\) and \\(\\gamma\\in F_k\\) let \\(f(\\gamma)\\) be the number of modulus one with \\(f(\\gamma)\\lambda_k(\\{\\gamma\\})=|\\lambda_k(\\{\\gamma\\})|\\), and let \\(f=0\\) elsewhere. Then \\(f\\in\\ell^\\infty(\\Gamma)\\), \\(\\|f\\|\\le1\\). Fix \\(k\\in K\\), let \\(P_k\\) be the union of the \\(F_j\\) with \\(j\\in K\\), \\(j<k\\), and \\(Q_k\\) the union of those with \\(j>k\\). Then\n\\[\n\\begin{gathered}\n\\lambda_k(f)\\\\\n=\\sum_{\\gamma\\in F_k}|\\lambda_k(\\{\\gamma\\})|\\\\\n+\\sum_{\\gamma\\in P_k}f(\\gamma)\\lambda_k(\\{\\gamma\\})\\\\\n+\\lambda_k(f1_{Q_k}).\n\\end{gathered}\n\\]\nThe first term exceeds \\(5\\delta-\\delta=4\\delta\\), by (10.1). The second is at most \\(\\delta\\) in modulus, again by (10.1), because the finite set \\(P_k\\) misses \\(F_k\\). The third is at most \\(|\\lambda_k|(Q_k)\\|f\\|<\\delta\\), by the estimate in the proof of Theorem 10.2 and (10.2). So \\(|\\lambda_k(f)|>2\\delta\\) for every \\(k\\in K\\). But \\(\\lambda_k(f)=\\nu_{n_k}(f)\\to0\\). This contradiction proves the theorem. \\(\\square\\)\n\n**Corollary 10.4** (Weak and norm convergence in \\(\\ell^1\\)). Let \\(\\Gamma\\) be a set. Every weakly convergent sequence in \\(\\ell^1(\\Gamma)\\) converges in norm. More generally, every weakly Cauchy sequence in \\(\\ell^1(\\Gamma)\\) converges in norm.\n\n\n**Proof.** Let \\(g_n\\to0\\) weakly in \\(\\ell^1(\\Gamma)\\), whose dual is \\(\\ell^\\infty(\\Gamma)\\). By the [uniform boundedness principle, Theorem 4.1 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-04) the sequence is bounded. Put \\(\\nu_n(E)=\\sum_{\\gamma\\in E}g_n(\\gamma)\\). Then \\(\\nu_n\\in\\ell^\\infty(\\Gamma)^*\\), \\(|\\nu_n|(\\Gamma)\\le\\|g_n\\|_1\\), and \\(\\nu_n(E)=\\langle g_n,1_E\\rangle\\to0\\) for every \\(E\\). By Theorem 10.3, \\(\\|g_n\\|_1=\\sum_\\gamma|\\nu_n(\\{\\gamma\\})|\\to0\\). If \\((f_n)\\) converges weakly to \\(f\\), apply this to \\(g_n=f_n-f\\). If \\((f_n)\\) is only weakly Cauchy and not norm Cauchy, there are \\(\\varepsilon>0\\) and indices \\(p_k<q_k\\), both tending to infinity, with \\(\\|f_{p_k}-f_{q_k}\\|_1\\ge\\varepsilon\\). The sequence \\(f_{p_k}-f_{q_k}\\) converges weakly to \\(0\\), hence in norm, which is a contradiction. So \\((f_n)\\) is norm Cauchy and converges, since \\(\\ell^1(\\Gamma)\\) is complete. \\(\\square\\)\n\n**Example 10.5** (The hypotheses matter). The unit vectors \\(e_n\\in\\ell^1(\\mathbb N)\\) have norm one. They converge to \\(0\\) against every element of \\(c_0\\), the predual of \\(\\ell^1\\), but not weakly: pairing with the constant function \\(1\\in\\ell^\\infty\\) gives \\(1\\). Correspondingly, the point evaluations \\(\\nu_n=\\delta_n\\in\\ell^\\infty(\\mathbb N)^*\\) satisfy \\(\\nu_n(E)\\to0\\) for every finite set \\(E\\), but not for \\(E=\\mathbb N\\), and \\(\\sum_k|\\nu_n(\\{k\\})|=1\\) for all \\(n\\). So in Theorem 10.3 the hypothesis is needed for all subsets \\(E\\), not only for the finite ones.\n\n## B. Which measures see the order on the spectrum?\n\nThe Gelfand representation replaces an abelian unital C*-algebra by \\(C(\\Omega)\\). A bounded pointwise supremum of continuous functions need not be continuous, so the supremum *in this ordered algebra* cannot simply be assumed to be the pointwise one. The stonean criterion below resolves that question through the closure of open sets. Normality then asks whether integration preserves those algebraic suprema.\n\nFor a finite spectrum the distinction disappears: all functions are continuous, all sets are clopen, and every finite measure preserves increasing bounded nets. On an infinite spectrum, the distinction is the main issue. A sufficient family of normal measures must detect every nonzero positive function; one measure need not suffice. The support criterion expresses detection geometrically and is the bridge to the operator representation.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AO-20",
      "unit": "abelian-operator-algebras",
      "name": "11. Stonean spaces and injectivity",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "full_conditions_and_proof": "### 11. Stonean spaces and injectivity\n\nA Banach space \\(Y\\) is *injective* if for every Banach space \\(E\\), every subspace \\(F\\subseteq E\\) and every bounded linear map \\(T:F\\to Y\\), there is a linear \\(\\tilde T:E\\to Y\\) that extends \\(T\\) with \\(\\|\\tilde T\\|=\\|T\\|\\). The scalars are injective by the Hahn–Banach theorem, and so is \\(\\ell^\\infty(X)\\), by extending coordinate by coordinate. We show that \\(C(\\Omega)\\) is injective when \\(\\Omega\\) is stonean. The key is a positive projection of \\(\\ell^\\infty(\\Omega)\\) onto \\(C(\\Omega)\\), obtained from a Hahn–Banach argument with values in the ordered space \\(C_{\\mathbb R}(\\Omega)\\).\n\n**Lemma 11.1** (Extending into \\(C_{\\mathbb R}(\\Omega)\\)). Let \\(\\Omega\\) be stonean, \\(V\\) a real vector space, \\(W\\subseteq V\\) a subspace, and \\(p:V\\to C_{\\mathbb R}(\\Omega)\\) sublinear, that is, \\(p(v+w)\\le p(v)+p(w)\\) and \\(p(tv)=tp(v)\\) for \\(t\\ge0\\). If \\(T_0:W\\to C_{\\mathbb R}(\\Omega)\\) is linear and \\(T_0\\le p\\) on \\(W\\), then \\(T_0\\) has a linear extension \\(T:V\\to C_{\\mathbb R}(\\Omega)\\) with \\(T\\le p\\) on \\(V\\).\n\n**Proof.** *One step.* Let \\(v_0\\notin W\\). For \\(w,w'\\in W\\),\n\\[\n\\begin{gathered}\nT_0(w)+T_0(w')\\\\\n=T_0(w+w')\\\\\n\\le p(w+w')\\\\\n\\le p(w-v_0)+p(w'+v_0),\n\\end{gathered}\n\\]\nso every element of \\(L=\\{T_0(w)-p(w-v_0):w\\in W\\}\\) lies below every element of \\(U=\\{p(w'+v_0)-T_0(w'):w'\\in W\\}\\). By Theorem 4.3, \\(L\\) has a least upper bound \\(a\\), and \\(a\\le u\\) for every \\(u\\in U\\). Define \\(T_1(w+tv_0)=T_0(w)+ta\\) on \\(W+\\mathbb Rv_0\\). For \\(t>0\\), \\(a\\le p(w/t+v_0)-T_0(w/t)\\) gives \\(T_1(w+tv_0)\\le p(w+tv_0)\\). For \\(t=-s<0\\), \\(a\\ge T_0(w/s)-p(w/s-v_0)\\) gives \\(T_1(w-sv_0)\\le p(w-sv_0)\\). So \\(T_1\\le p\\).\n\n*Zorn.* Order the pairs \\((W',T')\\), with \\(W\\subseteq W'\\) a subspace and \\(T'\\) a linear extension of \\(T_0\\) with \\(T'\\le p\\), by extension. A chain has an upper bound, its union. A maximal pair has \\(W'=V\\), by the first step. \\(\\square\\)\n\n**Theorem 11.2** (A positive projection onto \\(C(\\Omega)\\), and injectivity). Let \\(\\Omega\\) be stonean. For a bounded real function \\(f\\) on \\(\\Omega\\) let \\(m(f)\\) be the supremum in \\(C_{\\mathbb R}(\\Omega)\\) of \\(\\{g\\in C_{\\mathbb R}(\\Omega):g\\le f\\}\\), and \\(M(f)\\) the infimum there of \\(\\{g\\in C_{\\mathbb R}(\\Omega):g\\ge f\\}\\); both exist by Theorem 4.3, and \\(M(f)=-m(-f)\\).\n\n1. \\(m(f)\\le M(f)\\). The map \\(m\\) is superadditive and \\(M\\) is subadditive; both are positively homogeneous and equal to the identity on \\(C_{\\mathbb R}(\\Omega)\\). Moreover \\(m(f)=(f_*)^*\\), which agrees with \\(f_*\\) outside a meager set.\n2. There is a linear map \\(\\varepsilon:\\ell^\\infty(\\Omega)\\to C(\\Omega)\\) such that \\(\\varepsilon(g)=g\\) for \\(g\\in C(\\Omega)\\), \\(m(f)\\le\\varepsilon(f)\\le M(f)\\) for real \\(f\\), \\(\\varepsilon(f)\\ge0\\) for \\(f\\ge0\\), \\(\\|\\varepsilon\\|\\leq1\\), with equality when \\(\\Omega\\ne\\varnothing\\), and \\(\\varepsilon(gf)=g\\,\\varepsilon(f)\\) for \\(g\\in C(\\Omega)\\) and \\(f\\in\\ell^\\infty(\\Omega)\\).\n3. \\(C(\\Omega)\\) is injective: for every complex Banach space \\(E\\), every subspace \\(F\\subseteq E\\) and every bounded linear \\(T:F\\to C(\\Omega)\\) there is a linear \\(\\tilde T:E\\to C(\\Omega)\\) extending \\(T\\) with \\(\\|\\tilde T\\|=\\|T\\|\\).\n\nThe real and complex cases are both proved below. No converse is used in the subsequent arguments.\n\n**Proof.** (1) If \\(g\\le f\\le h\\) with \\(g,h\\) continuous, then \\(g\\le h\\); so \\(m(f)\\le h\\) for every such \\(h\\), and \\(m(f)\\le M(f)\\). For continuous \\(g_1\\le f_1\\) and \\(g_2\\le f_2\\), \\(g_1+g_2\\le m(f_1+f_2)\\). Taking the least upper bound over \\(g_1\\) gives \\(m(f_1)\\le m(f_1+f_2)-g_2\\), and then over \\(g_2\\) gives \\(m(f_1)+m(f_2)\\le m(f_1+f_2)\\). Homogeneity and \\(m(g)=g\\) for continuous \\(g\\) are clear, and the statements on \\(M\\) follow from \\(M(f)=-m(-f)\\). A continuous \\(g\\) satisfies \\(g\\le f\\) exactly when \\(g\\le f_*\\) (Lemma 2.2(1)). The pointwise supremum of these \\(g\\) is \\(f_*\\). Indeed, the constant \\(-\\|f\\|\\) is one of them, and given \\(\\omega\\) and \\(t\\) with \\(-\\|f\\|<t<f_*(\\omega)\\), Urysohn's lemma gives \\(u\\in C(\\Omega)\\) with \\(0\\le u\\le1\\), \\(u(\\omega)=1\\) and \\(u=0\\) outside the open set \\(\\{f_*>t\\}\\), and \\(g=-\\|f\\|+(t+\\|f\\|)u\\) satisfies \\(g\\le f_*\\) and \\(g(\\omega)=t\\). By Theorem 4.3, \\(m(f)=(f_*)^*\\), and Lemma 2.2(2) gives the last claim.\n\n(2) Apply Lemma 11.1 to the real bounded functions \\(V\\), the subspace \\(W=C_{\\mathbb R}(\\Omega)\\), \\(p=M\\) and \\(T_0=\\mathrm{id}\\); \\(M\\) is sublinear by (1), and \\(T_0\\le M\\) on \\(W\\). We get a linear \\(\\varepsilon_{\\mathbb R}\\le M\\) that is the identity on \\(C_{\\mathbb R}(\\Omega)\\). Then \\(\\varepsilon_{\\mathbb R}(f)=-\\varepsilon_{\\mathbb R}(-f)\\ge-M(-f)=m(f)\\). If \\(f\\ge0\\), then \\(0\\) is a continuous minorant, so \\(\\varepsilon_{\\mathbb R}(f)\\ge m(f)\\ge0\\). Put \\(\\varepsilon(f_1+if_2)=\\varepsilon_{\\mathbb R}(f_1)+i\\varepsilon_{\\mathbb R}(f_2)\\) for real \\(f_1,f_2\\); this is complex linear.\n\n*Norm.* If \\(\\Omega=\\varnothing\\), both function spaces and the projection are zero, and \\(\\|\\varepsilon\\|=0\\). Otherwise, for \\(\\omega\\in\\Omega\\), \\(\\varphi(f)=\\varepsilon(f)(\\omega)\\) is a positive linear functional on \\(\\ell^\\infty(\\Omega)\\) with \\(\\varphi(1)=1\\). It is real on real functions, so \\((f,h)\\mapsto\\varphi(\\bar hf)\\) is a positive hermitian form, and the [positive-form Cauchy–Schwarz proof](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-01) gives \\(|\\varphi(f)|^2\\le\\varphi(|f|^2)\\varphi(1)\\le\\|f\\|^2\\). So \\(\\|\\varepsilon(f)\\|\\le\\|f\\|\\), and \\(\\|\\varepsilon\\|=1\\) since \\(\\varepsilon(1)=1\\).\n\n*Module property.* Let \\(C\\) be clopen and \\(0\\le f\\le1\\). Then \\(0\\le\\varepsilon(1_Cf)\\le\\varepsilon(1_C)=1_C\\), so \\(\\varepsilon(1_Cf)\\) vanishes outside \\(C\\), that is, \\(\\varepsilon(1_Cf)=1_C\\varepsilon(1_Cf)\\). In the same way \\(\\varepsilon((1-1_C)f)=(1-1_C)\\varepsilon((1-1_C)f)\\). Adding, \\(\\varepsilon(f)=1_C\\varepsilon(1_Cf)+(1-1_C)\\varepsilon((1-1_C)f)\\), and multiplying by \\(1_C\\) gives \\(1_C\\varepsilon(f)=\\varepsilon(1_Cf)\\). Every \\(f\\in\\ell^\\infty(\\Omega)\\) is a linear combination of four functions with values in \\([0,1]\\), so \\(\\varepsilon(gf)=g\\varepsilon(f)\\) whenever \\(g\\) is a linear combination of indicators of clopen sets. These combinations form a unital self-adjoint subalgebra of \\(C(\\Omega)\\) that separates points (Lemma 4.2(3)), hence a dense one (Stone–Weierstrass). By continuity the identity holds for all \\(g\\in C(\\Omega)\\).\n\n(3) For each \\(\\omega\\in\\Omega\\), \\(x\\mapsto T(x)(\\omega)\\) is a linear functional on \\(F\\) of norm at most \\(\\|T\\|\\). By [the complex Hahn–Banach extension theorem, Corollary 2.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-02) it has an extension to \\(E\\) of the same norm; call its value at \\(x\\in E\\) \\(T'(x)(\\omega)\\). This defines a linear map \\(T':E\\to\\ell^\\infty(\\Omega)\\) that extends \\(T\\), with \\(\\|T'\\|\\le\\|T\\|\\). Put \\(\\tilde T=\\varepsilon\\circ T'\\). It takes values in \\(C(\\Omega)\\), it extends \\(T\\) because \\(\\varepsilon\\) is the identity on \\(C(\\Omega)\\), and \\(\\|\\tilde T\\|\\le\\|\\varepsilon\\|\\|T'\\|\\le\\|T\\|\\). The reverse inequality holds for every extension. \\(\\square\\)\n\n**Example 11.3** (The map \\(m\\) is not additive). Let \\(\\Omega=\\Omega_{[0,1]}\\), the stonean space of Theorem 9.5. It has a countable dense subset \\(D\\) and no isolated points. Each point of \\(D\\) is a rare set, so \\(D\\) is meager and \\(\\Omega\\setminus D\\) is dense (Lemma 2.1(2)). A continuous \\(g\\le1_D\\) satisfies \\(g\\le0\\) on the dense set \\(\\Omega\\setminus D\\), hence everywhere; so \\(m(1_D)=0\\). In the same way \\(m(1_{\\Omega\\setminus D})=0\\), because \\(D\\) is dense. But \\(m(1_D+1_{\\Omega\\setminus D})=m(1)=1\\). So \\(m\\) is not additive, and the linear projection \\(\\varepsilon\\) of Theorem 11.2 cannot be taken to be \\(m\\). For this \\(\\varepsilon\\), \\(\\varepsilon(1_D)+\\varepsilon(1_{\\Omega\\setminus D})=1\\), so at least one of the two values differs from the value of \\(m\\).\n\nExample 11.3 proves that \\(m\\) need not be linear; the linear projection \\(\\varepsilon\\) therefore comes from Lemma 11.1.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AO-21",
      "unit": "abelian-operator-algebras",
      "name": "12. Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
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      "full_conditions_and_proof": "## 12. Exercises\n\n**Exercise 12.1** (medium; No convergent sequences). Let \\(X\\) be an extremally disconnected Hausdorff space.\n\n(a) Show that every convergent sequence in \\(X\\) is eventually constant.\n\n(b) Deduce that an infinite stonean space is not metrizable, and that an abelian von Neumann algebra of infinite dimension is never norm separable.\n\n*Solution.* (a) Let \\(x_n\\to x\\), and suppose the sequence is not eventually constant. Infinitely many terms differ from \\(x\\), and a value \\(y\\ne x\\) occurs only finitely often, since a constant subsequence equal to \\(y\\) cannot converge to \\(x\\) in a Hausdorff space. So, passing to a subsequence, we may assume that the \\(x_n\\) are distinct and different from \\(x\\). Put \\(n_1=1\\) and choose disjoint open sets \\(U_1\\ni x_{n_1}\\) and \\(O_1\\ni x\\). Since \\(x_n\\to x\\), some \\(n_2>n_1\\) has \\(x_{n_2}\\in O_1\\). Separate \\(x_{n_2}\\) from \\(x\\) by disjoint open sets and intersect them with \\(O_1\\); this gives disjoint open sets \\(U_2\\ni x_{n_2}\\) and \\(O_2\\ni x\\) inside \\(O_1\\). Continuing, we get open sets \\(U_k\\ni x_{n_k}\\) and \\(O_k\\ni x\\) with \\(U_{k+1}\\cup O_{k+1}\\subseteq O_k\\) and \\(U_k\\cap O_k=\\varnothing\\). The \\(U_k\\) are pairwise disjoint: for \\(j<k\\), \\(U_k\\subseteq O_j\\), which misses \\(U_j\\). Put \\(U=\\bigcup_kU_{2k}\\) and \\(V=\\bigcup_kU_{2k+1}\\). These are disjoint open sets, and \\(x\\) lies in the closure of both, because \\(x_{n_{2k}}\\to x\\) and \\(x_{n_{2k+1}}\\to x\\). This contradicts Lemma 4.2(1).\n\n(b) An infinite compact metric space contains a sequence of distinct points, and by compactness this sequence has a convergent subsequence, which is not eventually constant. So an infinite stonean space is not metrizable, by (a). Take an abelian von Neumann algebra \\(M\\) of infinite dimension, and let \\(\\Omega\\) be its spectrum. If \\(\\Omega\\) were finite, \\(M\\cong C(\\Omega)\\) would have dimension equal to the number of points. So \\(\\Omega\\) is infinite, and it is stonean (Theorem 7.1). If \\(C(\\Omega)\\) had a dense sequence \\((x_n)\\) in its unit ball, then \\(d(\\omega,\\omega')=\\sum_n2^{-n}|x_n(\\omega)-x_n(\\omega')|\\) would be a metric, because the \\(x_n\\) separate points, and it would define the topology of \\(\\Omega\\), since the identity from the compact space \\(\\Omega\\) to the metric space \\((\\Omega,d)\\) is continuous and bijective. This is impossible, so \\(M\\) is not separable.\n\n**Exercise 12.2** (easy; States of \\(\\ell^\\infty\\) that vanish on \\(c_0\\)). Let \\(\\varphi\\) be a state of \\(\\ell^\\infty(\\mathbb N)\\) with \\(\\varphi(x)=0\\) for all \\(x\\in c_0\\), and let \\(\\mu\\) be the Radon measure on \\(\\beta\\mathbb N\\) with \\(\\varphi(x)=\\int\\hat x\\,d\\mu\\).\n\n(a) Show that \\(\\mu(\\mathbb N)=0\\). Conclude that \\(\\mu\\) is purely singular in the sense of Theorem 5.6, and that \\(\\varphi\\) is not of the form \\(x\\mapsto\\sum_nc_nx_n\\) with \\(c\\in\\ell^1\\).\n\n(b) Show that such states exist.\n\n*Solution.* (a) By Example 5.8, the transform of \\(1_{\\{n\\}}\\) is the indicator of the isolated point \\(n\\). Since \\(1_{\\{n\\}}\\in c_0\\), \\(\\mu(\\{n\\})=\\varphi(1_{\\{n\\}})=0\\), and \\(\\mu(\\mathbb N)=\\sum_n\\mu(\\{n\\})=0\\). So \\(\\mu\\) is concentrated on \\(\\beta\\mathbb N\\setminus\\mathbb N\\), which is rare (Example 5.8), and its normal part is \\(0\\). If \\(\\varphi(x)=\\sum_nc_nx_n\\), then \\(c_n=\\varphi(1_{\\{n\\}})=0\\) for all \\(n\\), so \\(\\varphi=0\\), which contradicts \\(\\varphi(1)=1\\).\n\n(b) The space \\(\\beta\\mathbb N\\) is compact and \\(\\mathbb N\\) is an infinite discrete subset, so \\(\\beta\\mathbb N\\setminus\\mathbb N\\) is not empty; let \\(p\\) be one of its points. Put \\(\\varphi(x)=\\hat x(p)\\), a character and hence a state. If \\(x\\) has finite support, \\(\\hat x\\) vanishes outside finitely many isolated points of \\(\\mathbb N\\), so \\(\\hat x(p)=0\\). Every \\(x\\in c_0\\) is a uniform limit of such functions, so \\(\\varphi(x)=0\\).\n\n**Exercise 12.3** (medium; Minimal projections and isolated points). Let \\(M\\) be a commutative von Neumann algebra, with spectrum \\(\\Omega\\). For a projection \\(e\\in M\\) let \\(C_e\\) be the clopen set on which its transform is \\(1\\).\n\n(a) Show that \\(e\\ne0\\) is a minimal projection if and only if \\(C_e\\) is a single point, which is then isolated.\n\n(b) Show that \\(M\\) is *atomic*, that is, every nonzero projection majorizes a minimal projection, if and only if \\(\\Omega=\\Omega_d\\) in Proposition 6.5, and that then \\(M\\cong\\ell^\\infty(I)\\), where \\(I\\) is the set of minimal projections.\n\n(c) Show that \\(M\\cong\\ell^\\infty(I)\\oplus M_c\\), where the abelian von Neumann algebra \\(M_c\\) has no minimal projections; \\(I\\) may be empty and \\(M_c\\) may be \\(0\\).\n\n(d) Show that the spectrum of \\(L^\\infty[0,1]\\) has no isolated points.\n\n*Solution.* (a) The projections of \\(C(\\Omega)\\) are the indicators of clopen sets, and \\(e\\le f\\) exactly when \\(C_e\\subseteq C_f\\). If \\(C_e=\\{\\omega\\}\\), a nonzero projection \\(f\\le e\\) has \\(\\varnothing\\ne C_f\\subseteq\\{\\omega\\}\\), so \\(f=e\\). If \\(C_e\\) has two points, Lemma 4.2(3) gives a clopen \\(C\\subseteq C_e\\) that contains one of them and not the other, and \\(1_C\\) is a nonzero projection strictly below \\(e\\). A clopen singleton is an isolated point.\n\n(b) By (a), minimal projections correspond to isolated points. Every nonzero projection majorizes a minimal one exactly when every nonempty clopen set contains an isolated point, that is, when the isolated points are dense (Lemma 4.2(3)), that is, when \\(\\Omega_d=\\Omega\\). Then Proposition 6.5(1) gives \\(M\\cong C(\\Omega_d)\\cong\\ell^\\infty(I)\\).\n\n(c) Since \\(\\Omega_d\\) and \\(\\Omega_c\\) are clopen, \\(C(\\Omega)=C(\\Omega_d)\\oplus C(\\Omega_c)\\). The first summand is \\(\\ell^\\infty(I)\\), and the second has no minimal projections by (a), since \\(\\Omega_c\\) has no isolated points. Both summands are von Neumann algebras: they are \\(Mz\\) and \\(M(1-z)\\) for the projection \\(z\\) with \\(C_z=\\Omega_d\\).\n\n(d) \\(L^\\infty[0,1]\\) has no minimal projection (proof of Theorem 8.4), so by (a) its spectrum has no isolated points.\n\n**Exercise 12.4** (easy; Multiplicity). Let \\(M_1=\\{M_f:f\\in L^\\infty[0,1]\\}\\) on \\(L^2[0,1]\\), and let \\(M_2=\\mathcal A_2\\) be the algebra of Example 1.3 on \\(L^2[0,1]\\oplus L^2[0,1]\\).\n\n(a) Show that \\(M_2'\\) consists of the operator matrices \\(\\begin{pmatrix}M_{f_{11}}&M_{f_{12}}\\\\M_{f_{21}}&M_{f_{22}}\\end{pmatrix}\\) with \\(f_{ij}\\in L^\\infty[0,1]\\).\n\n(b) Show that \\(M_1\\) and \\(M_2\\) are isomorphic but not spatially isomorphic, and name the hypothesis of Theorem 7.4(3) that fails.\n\n*Solution.* (a) An operator on \\(L^2\\oplus L^2\\) is a matrix \\((T_{ij})\\) of operators on \\(L^2[0,1]\\). It commutes with \\(M_f\\oplus M_f\\) exactly when each \\(T_{ij}\\) commutes with \\(M_f\\). For all \\(f\\in L^\\infty\\), this means \\(T_{ij}\\in M_1'=M_1\\) (Theorem 3.1(3)). (b) \\(M_f\\mapsto M_f\\oplus M_f\\) is a \\(*\\)-isomorphism onto \\(M_2\\). A spatial isomorphism \\(x\\mapsto UxU^*\\) carries commutants onto commutants. \\(M_1'=M_1\\) is abelian, while \\(M_2'\\) is not, by (a). So no unitary implements an isomorphism between them. The hypothesis that fails is maximality: \\(M_2\\) is not maximal abelian.\n\n**Exercise 12.5** (easy; A sliding hump without Phillips' lemma). Let \\((g_n)\\) be a sequence in \\(\\ell^1(\\mathbb N)\\) that converges weakly to \\(0\\), and suppose the \\(g_n\\) have pairwise disjoint finite supports \\(F_n\\). Show directly that \\(\\|g_n\\|_1\\to0\\).\n\n*Solution.* Otherwise there are \\(\\varepsilon>0\\) and infinitely many \\(n\\), forming a set \\(N\\), with \\(\\|g_n\\|_1\\ge\\varepsilon\\). Define \\(x\\in\\ell^\\infty\\) by letting \\(x(k)\\) be the number of modulus one with \\(x(k)g_n(k)=|g_n(k)|\\) for \\(k\\in F_n\\) and \\(n\\in N\\), and \\(x(k)=0\\) elsewhere. The supports are disjoint, so \\(x\\) is well defined and \\(\\|x\\|\\le1\\). For \\(n\\in N\\), \\(\\langle g_n,x\\rangle=\\sum_{k\\in F_n}|g_n(k)|=\\|g_n\\|_1\\ge\\varepsilon\\), which contradicts weak convergence to \\(0\\). Steps 1 and 2 of the proof of Theorem 10.3 reduce the general case to almost disjoint supports of this kind.\n\n",
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      "id": "OA-FND-AO-22",
      "unit": "abelian-operator-algebras",
      "name": "Background proofs",
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      "full_conditions_and_proof": "## Background proofs\n\nEach background statement below has a full programme proof. Historical books remain further reading; they do not carry a missing prerequisite. The scalar multiplication extension is proved in Proposition 3.2a, and the elementary summable-family tools are proved after this list.\n\n- **Commutative C\\*-algebras.** The Gelfand transform maps every abelian C\\*-algebra \\(A\\) isometrically and \\(*\\)-isomorphically onto \\(C_0(\\Omega)\\), where \\(\\Omega\\) is its space of characters; \\(\\Omega\\) is compact if \\(A\\) has a unit, and isomorphic algebras have homeomorphic spectra. A self-adjoint element is positive exactly when its transform is nonnegative, and the positive elements of a C\\*-algebra are the elements \\(y^*y\\). The quotient of a C\\*-algebra by a closed two-sided ideal is a C\\*-algebra. For a self-adjoint \\(a\\) in a unital C\\*-algebra, \\(g\\mapsto g(a)\\) is an isometric \\(*\\)-isomorphism of \\(C(\\sigma(a))\\) onto the C\\*-algebra generated by \\(1\\) and \\(a\\). All this is proved in [C\\*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md).\n- **Topology.** Urysohn's lemma: in a compact space, or for a compact and a disjoint closed set in a locally compact space, there are continuous functions with values in \\([0,1]\\) that are \\(1\\) on the first set and \\(0\\) on the second; in particular compact spaces are regular. The Stone–Weierstrass theorem: a self-adjoint subalgebra of \\(C_0(X)\\) that separates points and vanishes nowhere is dense; in particular polynomials are dense in \\(C[0,1]\\). Both are proved in [The Stone–Weierstrass theorem for functions vanishing at infinity](stone-weierstrass-c0.md). The Stone–Čech compactification of a completely regular space \\(D\\) is the spectrum of \\(C_b(D)\\), with \\(D\\) embedded by point evaluations; [Exercise 2.4 of the C\\*-algebra lesson](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04) proves the full construction, topological embedding, density, and unique extension of every continuous map to a compact Hausdorff target. This last property defines the Stone–Čech compactification.\n- **Riesz representation theorem.** For a locally compact Hausdorff space \\(X\\), a positive linear functional on \\(C_c(X)\\) is integration against a unique Radon measure. The measure is finite on compact sets, outer regular on Borel sets and inner regular on open sets. It is also inner regular on every Borel set of finite measure, hence on all Borel sets when the measure is finite. These are [Theorem 2.2 and Proposition 2.3 of the Haar lesson](haar-measure.md#oa-fnd-hm-01). Its Theorem 2.4 proves that every bounded functional on \\(C_0(X)\\) is integration against a unique finite complex Radon measure \\(\\nu\\), with norm \\(|\\nu|(X)\\), and that the measure is positive exactly when the functional is. The [density of \\(C_c(X)\\) in \\(L^p\\), \\(1\\leq p<\\infty\\)](haar-measure.md#oa-fnd-hm-02), is Proposition 3.1(4); it needs no global sigma-finiteness.\n- **Measure theory.** [Theorems 2.1–2.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions) prove monotone and dominated convergence and Fatou on arbitrary measure spaces. Small-set control of integrable densities and the complex Radon–Nikodym theorem, with a sigma-finite reference measure and finite total variation, are proved in [Section 2 above](#oa-fnd-ao-23). For positive sigma-finite measures the density theorem is [Theorem 4.1 of the measure-tools lesson](measure-and-hilbert-space-tools.md#4-densities-and-bounded-functionals). The [finite-measure uniqueness lemma and arbitrary probability product theorem](the-double-commutant-theorem.md#oa-fnd-bi-18) are proved in Section 9 of the double-commutant lesson, on the coordinate sigma-algebra and without topological assumptions on the factors.\n- **Radon measures that are not \\(\\sigma\\)-finite.** Let \\(\\Gamma\\) be locally compact and \\(\\mu\\) a positive Radon measure on it. The space \\(L^\\infty(\\Gamma,\\mu)\\) is formed from \\(\\mu\\)-measurable functions modulo locally null sets; its multiplication representation on \\(L^2(\\Gamma,\\mu)\\) is faithful, and its image is a maximal abelian von Neumann algebra. Proposition 3.2a proves this directly, using the full Radon decomposition and local integration proofs in Sections 1–2 of [Vector-valued functions, tensor products with \\(L^p\\), and preduals](vector-valued-integration-and-preduals.md#oa-fnd-vv-03).\n- **Von Neumann algebras.** The double commutant theorem: a \\(*\\)-algebra of operators that contains \\(1\\) is strongly dense in its bicommutant, and \\(S''\\) is the smallest von Neumann algebra containing a self-adjoint set \\(S\\). A closed subspace is invariant under a self-adjoint set \\(S\\) exactly when its projection lies in \\(S'\\). The commutant of a direct sum of von Neumann algebras is the direct sum of the commutants. For a projection \\(e\\) in the commutant of a von Neumann algebra \\(M\\), the operators \\(x|_{eH}\\), \\(x\\in M\\), form a von Neumann algebra on \\(eH\\), with commutant \\(\\{ex'e|_{eH}:x'\\in M'\\}\\). A vector is cyclic for \\(M\\) exactly when it is separating for \\(M'\\). The Hilbert space \\(L^2\\) of a product of uncountably many nontrivial probability spaces is not separable. All of these are proved in [The double commutant theorem](the-double-commutant-theorem.md) (Propositions 2.1, 5.2 and 9.2, Theorems 4.4 and 5.8, and Exercise 9.9).\n- **Functional analysis.** The complex Hahn–Banach theorem extends every bounded functional from a subspace of a normed space with the same norm: [Theorem 2.2 and Corollary 2.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-02) give the full proof. Its [Theorem 4.1](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-04) proves that a pointwise bounded family of bounded operators on a Banach space is bounded in operator norm. Its [Theorem 1.1](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-01) proves Zorn's lemma from the axiom of choice. The [Hilbert-space lesson, Sections 1–3](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-01), supplies full polarization, positive-form Cauchy–Schwarz and bounded-form representation proofs used in Lemma 2.4 and Theorem 11.2.\n\n### The elementary duality of summable families\n\nFor any set \\(J\\), define \\(\\sum_{j\\in J}|a_j|\\) as the supremum of its finite partial sums. A family with finite sum has countable support: for each positive integer \\(n\\), only finitely many coordinates have \\(|a_j|\\geq1/n\\), and their union contains every nonzero coordinate. Finite truncations are consequently dense in \\(\\ell^1(J)\\).\n\nThe space \\(\\ell^1(J)\\) is complete. For a norm-Cauchy sequence \\(a^{(n)}\\), each coordinate has a limit \\(a_j\\). If \\(\\|a^{(n)}-a^{(m)}\\|_1\\leq\\varepsilon\\) for \\(m,n\\geq N\\), then for every finite \\(F\\subseteq J\\), taking the coordinate limits gives \\(\\sum_{j\\in F}|a^{(n)}_j-a_j|\\leq\\varepsilon\\) for \\(n\\geq N\\). Taking the supremum over \\(F\\) shows that \\(a-a^{(N)}\\in\\ell^1(J)\\), so \\(a\\in\\ell^1(J)\\), and that \\(a^{(n)}\\to a\\) in norm.\n\nEvery \\(b\\in\\ell^\\infty(J)\\) defines a bounded linear functional \\(a\\mapsto\\sum_ja_jb_j\\) on \\(\\ell^1(J)\\), of norm \\(\\|b\\|_\\infty\\), by absolute convergence and testing individual coordinates. Conversely, if \\(\\Lambda\\in\\ell^1(J)^*\\), put \\(b_j=\\Lambda(\\delta_j)\\). Then \\(|b_j|\\leq\\|\\Lambda\\|\\), and linearity on finite truncations followed by norm continuity gives \\(\\Lambda(a)=\\sum_ja_jb_j\\). Thus \\(\\ell^1(J)^*=\\ell^\\infty(J)\\) isometrically, including the empty index set. This is the full duality and completeness argument used in Corollary 10.4.\n\n## References\n\n- [Kostecki] R. P. Kostecki, *W\\*-algebras and noncommutative integration*, survey, version 5, 2014, arXiv:1307.4818. https://arxiv.org/abs/1307.4818v5\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n- [Dixmier 1951] J. Dixmier, *Sur certains espaces considérés par M. H. Stone*, Summa Brasiliensis Mathematicae 2 (1951), 151–182, [freely readable scan](https://dmitripavlov.org/scans/dixmier.pdf).\n- [van Neerven] J. van Neerven, *Functional Analysis*, [corrected author version, arXiv:2112.11166v7](https://arxiv.org/pdf/2112.11166v7).\n\n*Freely accessible reading:* [Jesse Peterson, *Notes on operator algebras*, §3.8](https://math.vanderbilt.edu/peters10/teaching/spring2015/OperatorAlgebras.pdf) gives a route through cyclic abelian models; the arbitrary decomposition, local-null convention and finite-piece gluing are fully proved here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AO-23",
      "unit": "abelian-operator-algebras",
      "name": "Densities of complex measures",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "source": "src/abelian-operator-algebras.md",
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      "full_conditions_and_proof": "#### Densities of complex measures\n\nThe [measure tools for Haar integration](measure-and-hilbert-space-tools.md#4-densities-and-bounded-functionals), Theorem 4.1, prove Radon–Nikodym for positive sigma-finite measures. We need its complex version as well. The finite complex Radon measures used here have finite total variation, by [Theorem 2.4 of the Haar lesson](haar-measure.md#oa-fnd-hm-01). The following argument works on any sigma-algebra and requires no topology.\n\n**Lemma (Small sets have small integrals).** If \\(g\\geq0\\) is integrable, then for every \\(\\varepsilon>0\\) there is \\(\\delta>0\\) such that \\(\\int_Eg\\,d\\mu<\\varepsilon\\) whenever \\(\\mu(E)<\\delta\\).\n\n*Proof.* The integrable functions \\(g1_{\\{g>n\\}}\\) tend to zero almost everywhere and are bounded by \\(g\\). Dominated convergence, [Theorem 2.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions), gives an integer \\(n\\geq1\\) for which their integral is less than \\(\\varepsilon/2\\). Take \\(\\delta=\\varepsilon/(2n)\\). Splitting \\(E\\) at the level \\(n\\) gives \\(\\int_Eg\\leq n\\mu(E)+\\int_{\\{g>n\\}}g<\\varepsilon\\). \\(\\square\\)\n\nIn particular an integrable density against a finite Radon measure on a compact space again gives a finite Radon measure. Approximate a Borel set from inside by a compact set and from outside by an open set in the original measure; the lemma makes the corresponding errors small for the density measure.\n\n**Theorem (Complex Radon–Nikodym).** Let \\(\\mu\\) be a positive sigma-finite measure, and let \\(\\nu\\) be a countably additive complex measure on the same sigma-algebra with finite total variation. Suppose \\(\\nu(E)=0\\) whenever \\(\\mu(E)=0\\). There is a unique \\(h\\in L^1(\\mu)\\), up to almost-everywhere equality, with\n\\[\n\\nu(E)=\\int_Eh\\,d\\mu\n\\]\nfor every measurable \\(E\\). Its variation satisfies \\(|\\nu|(E)=\\int_E|h|\\,d\\mu\\).\n\n*Proof.* Define \\(|\\nu|(E)\\) as the supremum of \\(\\sum_j|\\nu(E_j)|\\) over finite measurable partitions of \\(E\\). It is a finite positive measure. To see countable additivity, let \\(E=\\bigcup_nE_n\\) be disjoint. Restrict a finite partition of \\(E\\) to each \\(E_n\\); countable additivity of \\(\\nu\\) and the triangle inequality give \\(|\\nu|(E)\\leq\\sum_n|\\nu|(E_n)\\). Conversely, combine partitions of the first \\(N\\) sets whose costs are arbitrarily close to their variations, and add the remaining part of \\(E\\) as one piece. This gives \\(|\\nu|(E)\\geq\\sum_{n\\leq N}|\\nu|(E_n)\\). Let \\(N\\) increase.\n\nIf \\(\\mu(E)=0\\), every measurable subset of \\(E\\) is also \\(\\mu\\)-null and therefore \\(\\nu\\)-null. The partition definition gives \\(|\\nu|(E)=0\\). The four set functions\n\\[\n\\begin{gathered}\n\\alpha_\\pm=\\tfrac12\\bigl(|\\nu|\\pm\\operatorname{Re}\\nu\\bigr),\\\\\n\\beta_\\pm=\\tfrac12\\bigl(|\\nu|\\pm\\operatorname{Im}\\nu\\bigr)\n\\end{gathered}\n\\]\nare finite positive measures, because \\(|\\operatorname{Re}\\nu(E)|\\) and \\(|\\operatorname{Im}\\nu(E)|\\) are at most \\(|\\nu|(E)\\). They are absolutely continuous with respect to \\(\\mu\\). Apply the positive Radon–Nikodym theorem to obtain their nonnegative densities \\(a_\\pm,b_\\pm\\). Each is integrable, since its integral over the whole space is the finite mass of its measure. Then\n\\[\nh=a_+-a_-+i(b_+-b_-)\n\\]\nis integrable and gives the asserted formula.\n\nIf an integrable \\(k\\) has zero integral on every measurable set, its real part cannot be positive on a set of positive measure: test the sets \\(\\{\\operatorname{Re}k>1/n\\}\\). These sets have finite measure by integrability, and their integrals would be strictly positive. Apply the same argument to the negative real part and both signs of the imaginary part. Thus \\(k=0\\) almost everywhere, proving uniqueness.\n\nFor every finite partition of \\(E\\), \\(\\sum_j|\\int_{E_j}h|\\leq\\int_E|h|\\), so \\(|\\nu|(E)\\leq\\int_E|h|\\). For the reverse inequality let \\(u=1_E\\overline h/|h|\\), with value zero where \\(h=0\\). Approximate this bounded measurable function uniformly by simple functions of supremum norm at most one. For such a simple function supported on \\(E\\), the partition definition bounds the absolute value of its \\(\\nu\\)-integral by \\(|\\nu|(E)\\). Uniform convergence is valid both against the finite variation measure and against \\(|h|\\mu\\). Hence\n\\[\n\\int_E|h|\\,d\\mu=\\int u\\,d\\nu\\leq|\\nu|(E).\n\\]\nThe integral on the left is real and nonnegative. This proves the variation identity. \\(\\square\\)\n\n**Example.** For Lebesgue measure \\(\\lambda\\) on \\([0,1]\\), put \\(\\nu(E)=i\\lambda(E\\cap[0,1/2])-i\\lambda(E\\cap(1/2,1])\\). Its density is \\(i\\) on the first half and \\(-i\\) on the second. Thus \\(|\\nu|([0,1])=1\\), although \\(\\nu([0,1])=0\\). Cancellation affects the complex mass but not its variation.\n\n**Exercise (medium): equivalent weighted representations.** Let \\(\\mu\\) be a finite measure and \\(h>0\\) almost everywhere with \\(\\int h\\,d\\mu<\\infty\\). Put \\(\\nu=h\\mu\\). Prove that \\(V:L^2(\\nu)\\to L^2(\\mu)\\), \\(Vf=h^{1/2}f\\), is unitary and intertwines multiplication by every bounded measurable function.\n\n*Solution.* The identity \\(\\|Vf\\|_{L^2(\\mu)}^2=\\int|f|^2h\\,d\\mu=\\|f\\|_{L^2(\\nu)}^2\\) gives isometry and well-definedness. The null sets of the two measures agree: if \\(\\int_Eh=0\\), each \\(E\\cap\\{h\\geq1/n\\}\\) is \\(\\mu\\)-null, and their union differs from \\(E\\) only by the null set where \\(h=0\\). The inverse is \\(g\\mapsto h^{-1/2}g\\), with arbitrary value on that null set; its norm identity is the reverse one. Finally \\(h^{1/2}(ag)=a(h^{1/2}g)\\). This is the mechanism used in Corollary 3.4.\n\n#### Category and operator order\n\n\n**Lemma 2.1** (Rare and meager sets). Let \\(X\\) be a topological space.\n\n1. A set is rare exactly when its closure is rare. Subsets of rare sets and finite unions of rare sets are rare. For an open set \\(G\\) the set \\(\\overline G\\setminus G\\) is rare, and for a closed set \\(F\\) the set \\(F\\setminus F^\\circ\\) is rare.\n2. (*Baire category theorem for compact spaces.*) Every compact space is a Baire space. Equivalently, a countable intersection of dense open subsets is dense, and the complement of a meager set is dense.\n3. Let \\((G_j)_{j\\in J}\\) be pairwise disjoint open sets and \\(R_j\\subseteq G_j\\). If every \\(R_j\\) is rare, then \\(\\bigcup_jR_j\\) is rare. If every \\(R_j\\) is meager, then \\(\\bigcup_jR_j\\) is meager.\n\n**Proof.** (1) The first two statements follow from the definition. If \\(R\\) and \\(S\\) are rare, the complements of \\(\\overline R\\) and \\(\\overline S\\) are dense open sets. A dense open set meets every nonempty open set \\(U\\) in a nonempty open set, and this set meets the second dense open set. So the intersection of the two complements is dense, and \\(\\overline R\\cup\\overline S\\) has empty interior. If an open set \\(V\\) lies in \\(\\overline G\\setminus G\\), then \\(V\\subseteq\\overline G\\) and \\(V\\cap G=\\varnothing\\), which forces \\(V=\\varnothing\\). If an open set \\(V\\) lies in \\(F\\setminus F^\\circ\\), then \\(V\\subseteq F\\), so \\(V\\subseteq F^\\circ\\), and again \\(V=\\varnothing\\).\n\n(2) Let \\(D_1,D_2,\\dots\\) be dense open sets and \\(V\\) a nonempty open set. A compact space is regular: a point and a closed set not containing it have disjoint neighbourhoods (by Urysohn's lemma, see Background). So there is a nonempty open \\(V_1\\) with \\(\\overline{V_1}\\subseteq V\\cap D_1\\), then a nonempty open \\(V_2\\) with \\(\\overline{V_2}\\subseteq V_1\\cap D_2\\), and so on. The compact sets \\(\\overline{V_n}\\) decrease and are nonempty, so they have a common point, which lies in \\(V\\cap\\bigcap_nD_n\\). If \\(M=\\bigcup_nR_n\\) is meager, the sets \\(D_n=X\\setminus\\overline{R_n}\\) are dense and open, and \\(\\bigcap_nD_n\\) is disjoint from \\(M\\); so \\(X\\setminus M\\) is dense and \\(M\\) has empty interior. The complete-metric Baire theorem, although not needed for this compact-space argument, has its full nested-ball proof in [Theorem 3.1 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-03).\n\n(3) Put \\(R=\\bigcup_jR_j\\) and \\(G=\\bigcup_jG_j\\). Suppose a nonempty open \\(U\\) lies in \\(\\overline R\\). Since \\(R\\subseteq G\\), \\(U\\) meets \\(G\\), so \\(W=U\\cap G_j\\) is nonempty for some \\(j\\). Let \\(x\\in W\\) and let \\(V\\subseteq W\\) be an open neighbourhood of \\(x\\). Then \\(V\\) meets \\(R\\), and \\(V\\cap R=V\\cap R_j\\) because the sets \\(G_k\\) are disjoint. So \\(x\\in\\overline{R_j}\\). Hence \\(W\\subseteq\\overline{R_j}\\), which is impossible for a rare set \\(R_j\\). For the meager case write \\(R_j=\\bigcup_nR_{j,n}\\) with rare sets \\(R_{j,n}\\subseteq G_j\\). By the rare case each \\(\\bigcup_jR_{j,n}\\) is rare, and \\(\\bigcup_jR_j=\\bigcup_n\\bigcup_jR_{j,n}\\). \\(\\square\\)\n\n**Lemma 2.2** (Regularizations). Let \\(f\\) be a bounded real function on a topological space \\(X\\).\n\n1. \\(f_*\\le f\\le f^*\\). The function \\(f_*\\) is lsc, and it is the largest lsc function below \\(f\\); likewise \\(f^*\\) is the smallest usc function above \\(f\\). If \\(f\\) is continuous at \\(x\\), then \\(f_*(x)=f(x)=f^*(x)\\).\n2. If \\(f\\) is lsc, then \\(\\{f^*\\ne f\\}\\) is meager. If \\(f\\) is usc, then \\(\\{f_*\\ne f\\}\\) is meager.\n\nNo separation axiom is needed.\n\n**Proof.** (1) The inequalities hold because every neighbourhood of \\(x\\) contains \\(x\\). If \\(f_*(x)>t\\), there is an open \\(U\\ni x\\) with \\(\\inf_Uf>t\\), and then \\(f_*(y)\\ge\\inf_Uf>t\\) for every \\(y\\in U\\). So \\(\\{f_*>t\\}\\) is open. If \\(g\\le f\\) is lsc and \\(t<g(x)\\), the open set \\(U=\\{g>t\\}\\) contains \\(x\\) and \\(\\inf_Uf\\ge t\\), so \\(f_*(x)\\ge t\\); hence \\(g\\le f_*\\). The statements on \\(f^*\\) follow by applying these to \\(-f\\). Continuity at \\(x\\) means that \\(\\inf_Uf\\) and \\(\\sup_Uf\\) tend to \\(f(x)\\) as \\(U\\) shrinks.\n\n(2) Let \\(f\\) be lsc. The function \\(f^*-f\\) is usc, so \\(A_n=\\{f^*-f\\ge1/n\\}\\) is closed. Suppose a nonempty open \\(U\\) lies in \\(A_n\\). For \\(y\\in U\\), \\(U\\) is a neighbourhood of \\(y\\), so \\(f^*(y)\\le\\sup_Uf\\); hence \\(\\sup_Uf^*=\\sup_Uf\\). On \\(U\\) we have \\(f\\le f^*-1/n\\), so \\(\\sup_Uf\\le\\sup_Uf-1/n\\), which is absurd. So every \\(A_n\\) is rare, and \\(\\{f^*\\ne f\\}=\\bigcup_nA_n\\) is meager. The usc case follows by applying this to \\(-f\\), since \\((-f)^*=-f_*\\). \\(\\square\\)\n\n**Lemma 2.3** (Radon measures and increasing nets). Let \\(\\mu\\) be a Radon measure on a compact space \\(\\Omega\\).\n\n1. If \\((U_i)\\) is an increasing net of open sets, then \\(\\mu(\\bigcup_iU_i)=\\sup_i\\mu(U_i)\\).\n2. Let \\((f_i)\\) be an increasing net of real continuous functions with \\(\\sup_i\\|f_i\\|<\\infty\\), and let \\(f\\) be its pointwise supremum. Then \\(f\\) is lsc and \\(\\int f\\,d\\mu=\\sup_i\\mu(f_i)\\).\n3. \\(C(\\Omega)\\) is dense in \\(L^2(\\Omega,\\mu)\\).\n\n**Proof.** (1) Put \\(U=\\bigcup_iU_i\\). By inner regularity on open sets, \\(\\mu(U)\\) is the supremum of \\(\\mu(K)\\) over compact \\(K\\subseteq U\\). Each such \\(K\\) is covered by finitely many \\(U_i\\), hence by a single one, since the net increases.\n\n(2) A supremum of continuous functions is lsc. Adding a constant, we may assume \\(0\\le f_i\\le c\\) for all \\(i\\). Fix \\(\\varepsilon>0\\) and an integer \\(N\\geq1\\) with \\(N\\ge c/\\varepsilon\\). For a function \\(g\\) with values in \\([0,c]\\) put \\(s(g)=\\varepsilon\\sum_{k=1}^N1_{\\{g>k\\varepsilon\\}}\\). If \\(j\\varepsilon<g(x)\\le(j+1)\\varepsilon\\), exactly the terms with \\(k\\le j\\) are present, so \\(s(g)(x)=j\\varepsilon\\). Hence \\(g-\\varepsilon\\le s(g)\\le g\\). The sets \\(\\{f>k\\varepsilon\\}=\\bigcup_i\\{f_i>k\\varepsilon\\}\\) are unions of increasing nets of open sets. By (1), and since the net is directed and there are only \\(N\\) values of \\(k\\), there is an index \\(i\\) with \\(\\mu(\\{f_i>k\\varepsilon\\})\\ge\\mu(\\{f>k\\varepsilon\\})-\\varepsilon/N\\) for all \\(k\\le N\\). Then\n\\[\n\\begin{gathered}\n\\mu(f_i)\\\\\n\\ge\\int s(f_i)\\,d\\mu\\\\\n\\ge\\int s(f)\\,d\\mu-\\varepsilon^2\\\\\n\\ge\\int f\\,d\\mu-\\varepsilon\\mu(\\Omega)-\\varepsilon^2 .\n\\end{gathered}\n\\]\nSince \\(f_i\\le f\\) for all \\(i\\), this proves (2).\n\n(3) This is the density of \\(C_c\\) in \\(L^p\\) for Radon measures, proved in [Haar measure on locally compact groups](haar-measure.md). \\(\\square\\)\n\n**Lemma 2.4** (Increasing nets of operators). Let \\((x_i)\\) be an increasing net of self-adjoint operators on a Hilbert space \\(H\\) with \\(\\sup_i\\|x_i\\|<\\infty\\). Then \\((x_i)\\) converges strongly to a self-adjoint operator \\(x\\), and among the self-adjoint operators on \\(H\\) the net has \\(x\\) as its least upper bound. If all \\(x_i\\) lie in a von Neumann algebra \\(M\\), then \\(x\\in M\\), and \\(x\\) is also its least upper bound in \\(M_h\\).\n\n**Proof.** For each \\(\\xi\\), the numbers \\(\\langle x_i\\xi,\\xi\\rangle\\) increase and are bounded, so they converge. By polarization, \\(\\langle x_i\\xi,\\eta\\rangle\\) converges for all \\(\\xi,\\eta\\), and the limit is a bounded hermitian sesquilinear form. [Theorem 3.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-03) gives a self-adjoint \\(x\\) with \\(\\langle x_i\\xi,\\eta\\rangle\\to\\langle x\\xi,\\eta\\rangle\\), and \\(x\\ge x_i\\) for every \\(i\\). For a positive operator \\(T\\), [Proposition 1.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-01), applied to the positive form \\((\\eta,\\zeta)\\mapsto\\langle T\\eta,\\zeta\\rangle\\) gives \\[\n\\begin{gathered}\n\\|T\\xi\\|^2\\\\\n=\\langle T\\xi,T\\xi\\rangle\\\\\n\\le\\langle T\\xi,\\xi\\rangle^{1/2}\\langle T^2\\xi,T\\xi\\rangle^{1/2}\\\\\n\\le\\langle T\\xi,\\xi\\rangle^{1/2}\\|T\\|^{1/2}\\|T\\xi\\|,\n\\end{gathered}\n\\] and hence \\(\\|T\\xi\\|^2\\le\\|T\\|\\,\\langle T\\xi,\\xi\\rangle\\). With \\(T=x-x_i\\), whose norm stays bounded, this shows \\(\\|(x-x_i)\\xi\\|\\to0\\). If \\(y\\) is self-adjoint and \\(y\\ge x_i\\) for all \\(i\\), then \\(\\langle y\\xi,\\xi\\rangle\\ge\\lim_i\\langle x_i\\xi,\\xi\\rangle=\\langle x\\xi,\\xi\\rangle\\), so \\(y\\ge x\\). A von Neumann algebra is strongly closed, so \\(x\\in M\\) when all \\(x_i\\) are. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "3. The projection lattice and equivalence of projections",
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      "full_conditions_and_proof": "## 3. The projection lattice and equivalence of projections\n\n**Proposition 3.1** (The projection lattice). Let \\(\\{e_i\\}_{i\\in I}\\) be any family in \\(\\mathcal P(M)\\). The projection \\(\\bigwedge_ie_i\\) onto \\(\\bigcap_ie_iH\\) and the projection \\(\\bigvee_ie_i\\) onto \\([\\bigcup_ie_iH]\\) lie in \\(M\\). They are the greatest lower bound and the least upper bound of the family, both in \\(\\mathcal P(M)\\) and in \\(\\mathcal P(B(H))\\). So \\(\\mathcal P(M)\\) is a complete lattice whose meets and joins are those of \\(B(H)\\).\n\n**Proof.** Put \\(L=\\bigcap_ie_iH\\). Take \\(x'\\in M'\\). It commutes with every \\(e_i\\), so it maps each \\(e_iH\\) into itself, and therefore maps \\(L\\) into \\(L\\). So does \\(x'^*\\), since \\(M'\\) is closed under adjoints. A closed subspace that is invariant under an operator and under its adjoint reduces the operator. Hence the projection onto \\(L\\) commutes with every \\(x'\\in M'\\) and lies in \\(M''=M\\). A projection \\(g\\) lies below every \\(e_i\\) exactly when \\(gH\\subseteq L\\), so this projection is the greatest lower bound in \\(\\mathcal P(B(H))\\), and a fortiori in \\(\\mathcal P(M)\\). Apply this to the family \\(1-e_i\\). The projection onto \\(\\bigcap_i(e_iH)^\\perp=(\\bigcup_ie_iH)^\\perp\\) lies in \\(M\\), and so does its complement \\(\\bigvee_ie_i\\). Taking complements reverses the order, so \\(\\bigvee_ie_i\\) is the least upper bound. \\(\\square\\)\n\n\\(\\mathcal P(M)\\) is the *projection lattice* of \\(M\\). For mutually orthogonal \\(e_i\\), \\(\\bigvee_ie_i=\\sum_ie_i\\), the strong sum. De Morgan's laws \\((\\bigvee_ie_i)^\\perp=\\bigwedge_ie_i^\\perp\\) hold.\n\n**Definition 3.2** (Equivalence of projections). Let \\(e,f\\in\\mathcal P(M)\\).\n\n1. \\(e\\) and \\(f\\) are *equivalent*, \\(e\\sim f\\), if some \\(u\\in M\\) satisfies \\(u^*u=e\\) and \\(uu^*=f\\). Then \\(u\\) is a partial isometry with *initial projection* \\(e\\) and *final projection* \\(f\\), and we say that \\(u\\) implements \\(e\\sim f\\).\n2. \\(e\\precsim f\\), also written \\(f\\succsim e\\), if \\(e\\sim f_1\\) for some \\(f_1\\in\\mathcal P(M)\\) with \\(f_1\\le f\\).\n3. \\(e\\prec f\\) if \\(e\\precsim f\\) and \\(e\\) is not equivalent to \\(f\\).\n\nThese relations are also applied to closed subspaces whose projections lie in \\(M\\).\n\n**Lemma 3.3.**\n\n1. If \\(u\\in M\\) and \\(u^*u=e\\) is a projection, then \\(u=ue\\), \\(uu^*\\) is a projection, and \\(u\\) maps \\(eH\\) isometrically onto \\(uu^*H\\).\n2. \\(\\sim\\) is an equivalence relation, and \\(\\precsim\\) is reflexive and transitive.\n3. (*Additivity*) Let \\(\\{e_i\\}_{i\\in I}\\) and \\(\\{f_i\\}_{i\\in I}\\) be families of mutually orthogonal projections. If \\(e_i\\sim f_i\\) for every \\(i\\), then \\(\\sum_ie_i\\sim\\sum_if_i\\). The same holds with \\(\\precsim\\) in place of \\(\\sim\\).\n4. (*Central cuts*) If \\(e\\sim f\\) (or \\(e\\precsim f\\)) and \\(z\\) is a central projection, then \\(ze\\sim zf\\) (or \\(ze\\precsim zf\\)).\n\n**Proof.** (1) \\((u-ue)^*(u-ue)=e-e-e+e=0\\), so \\(u=ue\\). Then \\((uu^*)^2=ueu^*=uu^*\\), and \\(\\|u\\xi\\|^2=\\langle u^*u\\xi,\\xi\\rangle=\\|e\\xi\\|^2\\). So \\(u\\) is isometric on \\(eH\\) and zero on \\(e^\\perp H\\). Its range \\(u(eH)\\) is closed, and \\(uu^*\\) is the projection onto it.\n\n(2) \\(e\\) implements \\(e\\sim e\\). If \\(u\\) implements \\(e\\sim f\\), then \\(u^*\\) implements \\(f\\sim e\\). If \\(u\\) implements \\(e\\sim f\\) and \\(v\\) implements \\(f\\sim g\\), then \\(fu=u\\) because \\(u=uu^*u\\). Hence \\((vu)^*(vu)=u^*fu=e\\) and \\((vu)(vu)^*=vfv^*=g\\). For \\(\\precsim\\), let \\(e\\sim f_1\\le f\\), and let \\(v\\) implement \\(f\\sim g_1\\le g\\). The operator \\(vf_1\\) has initial projection \\(f_1v^*vf_1=f_1\\) and final projection \\(vf_1v^*\\le vfv^*=g_1\\le g\\). So \\(e\\sim f_1\\sim vf_1v^*\\le g\\).\n\n(3) Let \\(u_i\\) implement \\(e_i\\sim f_i\\). For \\(\\xi\\in H\\), the vectors \\(u_i\\xi=u_ie_i\\xi\\) lie in the mutually orthogonal subspaces \\(f_iH\\), and \\(\\|u_i\\xi\\|=\\|e_i\\xi\\|\\). As \\(\\sum_i\\|e_i\\xi\\|^2=\\|(\\sum_ie_i)\\xi\\|^2\\le\\|\\xi\\|^2\\), the sum \\(u\\xi=\\sum_iu_i\\xi\\) converges and \\(\\|u\\xi\\|=\\|(\\sum_ie_i)\\xi\\|\\). So \\(u\\) is a partial isometry with initial projection \\(\\sum_ie_i\\). Its range is \\([\\bigcup_if_iH]\\), so \\(uu^*=\\sum_if_i\\). Being a strong limit of finite sums, \\(u\\) lies in \\(M\\). For \\(\\precsim\\), apply this to \\(e_i\\sim f_i'\\le f_i\\) and note \\(\\sum_if_i'\\le\\sum_if_i\\).\n\n(4) If \\(u\\) implements \\(e\\sim f\\), then \\(zu\\) implements \\(ze\\sim zf\\), because \\(z\\) is central. For \\(\\precsim\\), cut the subprojection by \\(z\\) as well. \\(\\square\\)\n\n**Definition 3.4** (Central support). For \\(e\\in\\mathcal P(M)\\), the *central support* \\(c(e)\\) is the least central projection that majorizes \\(e\\). It exists: \\(Z=(M\\cup M')'\\) is a von Neumann algebra whose projections are the central projections, so by [Proposition 3.1](#oa-fnd-ty-01), applied to \\(Z\\), the meet of all central projections that majorize \\(e\\) is again central. It majorizes \\(e\\), and it is the least such projection.\n\n**Proposition 3.5** (Central supports).\n\n1. \\(c(e)\\) is the projection onto \\([MeH]\\). For a central projection \\(z\\), \\(ze=0\\) if and only if \\(zc(e)=0\\). Moreover \\(c(ze)=zc(e)\\).\n2. If \\(e\\sim f\\) then \\(c(e)=c(f)\\). If \\(e\\precsim f\\) then \\(c(e)\\le c(f)\\).\n3. (*Induction*) The map \\(x'\\mapsto x'e|_{eH}\\) is a \\(*\\)-homomorphism of \\(M'\\) onto the induced algebra \\(M'e\\), with kernel \\(M'(1-c(e))\\). Likewise, for \\(e'\\in\\mathcal P(M')\\), the map \\(x\\mapsto xe'|_{e'H}\\) is a \\(*\\)-homomorphism of \\(M\\) onto \\(Me'\\) with kernel \\(M(1-c(e'))\\); here \\(c(e')\\) is the central support of \\(e'\\) in \\(M'\\), the projection onto \\([M'e'H]\\).\n4. (*Centre of a reduced algebra*) The centre of \\(eMe\\) (on \\(eH\\)) is \\(Ze=\\{ae:a\\in Z\\}\\), and \\(a\\mapsto ae|_{eH}\\) is an isomorphism of \\(Zc(e)\\) onto it. In particular every projection in the centre of \\(eMe\\) is \\(ze\\) for a central projection \\(z\\).\n\n**Proof.** (1) The subspace \\([MeH]\\) is invariant under \\(M\\). It is also invariant under \\(M'\\), since \\(x'ye=yex'\\) for \\(x'\\in M'\\) and \\(y\\in M\\). Both algebras are closed under adjoints, so the projection \\(q\\) onto \\([MeH]\\) commutes with \\(M\\) and with \\(M'\\). Hence \\(q\\in M''\\cap M'=Z\\), and \\(q\\ge e\\) because \\(1\\in M\\). If \\(z\\) is central and \\(z\\ge e\\), then \\(zye=yze=ye\\) for \\(y\\in M\\), so \\([MeH]\\subseteq zH\\) and \\(q\\le z\\). So \\(q=c(e)\\). If \\(z\\) is central and \\(ze=0\\), then \\(zye=yze=0\\), so \\(z\\) vanishes on \\([MeH]\\) and \\(zc(e)=0\\). The converse follows from \\(e=c(e)e\\). Finally, a central projection \\(w\\) satisfies \\(w(ze)=0\\) iff \\((wz)c(e)=0\\), iff \\(w\\le1-zc(e)\\). The largest such \\(w\\) is \\(1-c(ze)\\) by what was just shown, so \\(c(ze)=zc(e)\\).\n\n(2) Let \\(u\\) implement \\(e\\sim f\\), and let \\(z\\) be central. From \\(u=ue\\), \\(e=u^*u\\), \\(u=fu\\) and \\(f=uu^*\\) we get \\(ze=0\\iff zu=0\\iff zf=0\\). By (1), \\(zc(e)=0\\iff zc(f)=0\\). Taking \\(z=1-c(e)\\) and \\(z=1-c(f)\\) gives \\(c(f)\\le c(e)\\le c(f)\\). The second claim follows, since \\(f_1\\le f\\) implies \\(c(f_1)\\le c(f)\\).\n\n(3) The map is a \\(*\\)-homomorphism because \\(e\\) commutes with \\(M'\\), and its image is \\(M'e\\) by definition. Further, \\(x'e=0\\) iff \\(x'\\) vanishes on \\(eH\\), iff \\(x'\\) vanishes on \\([MeH]\\) (as \\(x'ye=yx'e\\)), iff \\(x'c(e)=0\\). The second statement is the first one with the roles of \\(M\\) and \\(M'\\) exchanged.\n\n(4) By Fact 2.1, \\(eMe\\) and \\(M'e\\) are each other's commutants on \\(eH\\), so they have the same centre. By (3), the induction restricts to an isomorphism of \\(M'c(e)\\) onto \\(M'e\\), and an isomorphism maps centre onto centre. The centre of \\(M'c(e)\\) is \\(Zc(e)\\): if \\(y'c(e)\\) (\\(y'\\in M'\\)) commutes with \\(M'c(e)\\), it also commutes with \\(M'(1-c(e))\\) (both products vanish), hence with all of \\(M'\\), so it lies in \\(M'\\cap M''=Z\\). Therefore the centre of \\(M'e\\) is \\(\\{ac(e)e:a\\in Z\\}=Ze\\). On \\(Zc(e)\\) the map \\(a\\mapsto ae|_{eH}\\) is the restriction of the induction, which is injective there. An isomorphism maps projections to projections. \\(\\square\\)\n\n",
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      "id": "OA-FND-TY-02",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "3. The projection lattice and equivalence of projections",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
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      "full_conditions_and_proof": "## 3. The projection lattice and equivalence of projections\n\n**Proposition 3.1** (The projection lattice). Let \\(\\{e_i\\}_{i\\in I}\\) be any family in \\(\\mathcal P(M)\\). The projection \\(\\bigwedge_ie_i\\) onto \\(\\bigcap_ie_iH\\) and the projection \\(\\bigvee_ie_i\\) onto \\([\\bigcup_ie_iH]\\) lie in \\(M\\). They are the greatest lower bound and the least upper bound of the family, both in \\(\\mathcal P(M)\\) and in \\(\\mathcal P(B(H))\\). So \\(\\mathcal P(M)\\) is a complete lattice whose meets and joins are those of \\(B(H)\\).\n\n**Proof.** Put \\(L=\\bigcap_ie_iH\\). Take \\(x'\\in M'\\). It commutes with every \\(e_i\\), so it maps each \\(e_iH\\) into itself, and therefore maps \\(L\\) into \\(L\\). So does \\(x'^*\\), since \\(M'\\) is closed under adjoints. A closed subspace that is invariant under an operator and under its adjoint reduces the operator. Hence the projection onto \\(L\\) commutes with every \\(x'\\in M'\\) and lies in \\(M''=M\\). A projection \\(g\\) lies below every \\(e_i\\) exactly when \\(gH\\subseteq L\\), so this projection is the greatest lower bound in \\(\\mathcal P(B(H))\\), and a fortiori in \\(\\mathcal P(M)\\). Apply this to the family \\(1-e_i\\). The projection onto \\(\\bigcap_i(e_iH)^\\perp=(\\bigcup_ie_iH)^\\perp\\) lies in \\(M\\), and so does its complement \\(\\bigvee_ie_i\\). Taking complements reverses the order, so \\(\\bigvee_ie_i\\) is the least upper bound. \\(\\square\\)\n\n\\(\\mathcal P(M)\\) is the *projection lattice* of \\(M\\). For mutually orthogonal \\(e_i\\), \\(\\bigvee_ie_i=\\sum_ie_i\\), the strong sum. De Morgan's laws \\((\\bigvee_ie_i)^\\perp=\\bigwedge_ie_i^\\perp\\) hold.\n\n**Definition 3.2** (Equivalence of projections). Let \\(e,f\\in\\mathcal P(M)\\).\n\n1. \\(e\\) and \\(f\\) are *equivalent*, \\(e\\sim f\\), if some \\(u\\in M\\) satisfies \\(u^*u=e\\) and \\(uu^*=f\\). Then \\(u\\) is a partial isometry with *initial projection* \\(e\\) and *final projection* \\(f\\), and we say that \\(u\\) implements \\(e\\sim f\\).\n2. \\(e\\precsim f\\), also written \\(f\\succsim e\\), if \\(e\\sim f_1\\) for some \\(f_1\\in\\mathcal P(M)\\) with \\(f_1\\le f\\).\n3. \\(e\\prec f\\) if \\(e\\precsim f\\) and \\(e\\) is not equivalent to \\(f\\).\n\nThese relations are also applied to closed subspaces whose projections lie in \\(M\\).\n\n**Lemma 3.3.**\n\n1. If \\(u\\in M\\) and \\(u^*u=e\\) is a projection, then \\(u=ue\\), \\(uu^*\\) is a projection, and \\(u\\) maps \\(eH\\) isometrically onto \\(uu^*H\\).\n2. \\(\\sim\\) is an equivalence relation, and \\(\\precsim\\) is reflexive and transitive.\n3. (*Additivity*) Let \\(\\{e_i\\}_{i\\in I}\\) and \\(\\{f_i\\}_{i\\in I}\\) be families of mutually orthogonal projections. If \\(e_i\\sim f_i\\) for every \\(i\\), then \\(\\sum_ie_i\\sim\\sum_if_i\\). The same holds with \\(\\precsim\\) in place of \\(\\sim\\).\n4. (*Central cuts*) If \\(e\\sim f\\) (or \\(e\\precsim f\\)) and \\(z\\) is a central projection, then \\(ze\\sim zf\\) (or \\(ze\\precsim zf\\)).\n\n**Proof.** (1) \\((u-ue)^*(u-ue)=e-e-e+e=0\\), so \\(u=ue\\). Then \\((uu^*)^2=ueu^*=uu^*\\), and \\(\\|u\\xi\\|^2=\\langle u^*u\\xi,\\xi\\rangle=\\|e\\xi\\|^2\\). So \\(u\\) is isometric on \\(eH\\) and zero on \\(e^\\perp H\\). Its range \\(u(eH)\\) is closed, and \\(uu^*\\) is the projection onto it.\n\n(2) \\(e\\) implements \\(e\\sim e\\). If \\(u\\) implements \\(e\\sim f\\), then \\(u^*\\) implements \\(f\\sim e\\). If \\(u\\) implements \\(e\\sim f\\) and \\(v\\) implements \\(f\\sim g\\), then \\(fu=u\\) because \\(u=uu^*u\\). Hence \\((vu)^*(vu)=u^*fu=e\\) and \\((vu)(vu)^*=vfv^*=g\\). For \\(\\precsim\\), let \\(e\\sim f_1\\le f\\), and let \\(v\\) implement \\(f\\sim g_1\\le g\\). The operator \\(vf_1\\) has initial projection \\(f_1v^*vf_1=f_1\\) and final projection \\(vf_1v^*\\le vfv^*=g_1\\le g\\). So \\(e\\sim f_1\\sim vf_1v^*\\le g\\).\n\n(3) Let \\(u_i\\) implement \\(e_i\\sim f_i\\). For \\(\\xi\\in H\\), the vectors \\(u_i\\xi=u_ie_i\\xi\\) lie in the mutually orthogonal subspaces \\(f_iH\\), and \\(\\|u_i\\xi\\|=\\|e_i\\xi\\|\\). As \\(\\sum_i\\|e_i\\xi\\|^2=\\|(\\sum_ie_i)\\xi\\|^2\\le\\|\\xi\\|^2\\), the sum \\(u\\xi=\\sum_iu_i\\xi\\) converges and \\(\\|u\\xi\\|=\\|(\\sum_ie_i)\\xi\\|\\). So \\(u\\) is a partial isometry with initial projection \\(\\sum_ie_i\\). Its range is \\([\\bigcup_if_iH]\\), so \\(uu^*=\\sum_if_i\\). Being a strong limit of finite sums, \\(u\\) lies in \\(M\\). For \\(\\precsim\\), apply this to \\(e_i\\sim f_i'\\le f_i\\) and note \\(\\sum_if_i'\\le\\sum_if_i\\).\n\n(4) If \\(u\\) implements \\(e\\sim f\\), then \\(zu\\) implements \\(ze\\sim zf\\), because \\(z\\) is central. For \\(\\precsim\\), cut the subprojection by \\(z\\) as well. \\(\\square\\)\n\n**Definition 3.4** (Central support). For \\(e\\in\\mathcal P(M)\\), the *central support* \\(c(e)\\) is the least central projection that majorizes \\(e\\). It exists: \\(Z=(M\\cup M')'\\) is a von Neumann algebra whose projections are the central projections, so by [Proposition 3.1](#oa-fnd-ty-01), applied to \\(Z\\), the meet of all central projections that majorize \\(e\\) is again central. It majorizes \\(e\\), and it is the least such projection.\n\n**Proposition 3.5** (Central supports).\n\n1. \\(c(e)\\) is the projection onto \\([MeH]\\). For a central projection \\(z\\), \\(ze=0\\) if and only if \\(zc(e)=0\\). Moreover \\(c(ze)=zc(e)\\).\n2. If \\(e\\sim f\\) then \\(c(e)=c(f)\\). If \\(e\\precsim f\\) then \\(c(e)\\le c(f)\\).\n3. (*Induction*) The map \\(x'\\mapsto x'e|_{eH}\\) is a \\(*\\)-homomorphism of \\(M'\\) onto the induced algebra \\(M'e\\), with kernel \\(M'(1-c(e))\\). Likewise, for \\(e'\\in\\mathcal P(M')\\), the map \\(x\\mapsto xe'|_{e'H}\\) is a \\(*\\)-homomorphism of \\(M\\) onto \\(Me'\\) with kernel \\(M(1-c(e'))\\); here \\(c(e')\\) is the central support of \\(e'\\) in \\(M'\\), the projection onto \\([M'e'H]\\).\n4. (*Centre of a reduced algebra*) The centre of \\(eMe\\) (on \\(eH\\)) is \\(Ze=\\{ae:a\\in Z\\}\\), and \\(a\\mapsto ae|_{eH}\\) is an isomorphism of \\(Zc(e)\\) onto it. In particular every projection in the centre of \\(eMe\\) is \\(ze\\) for a central projection \\(z\\).\n\n**Proof.** (1) The subspace \\([MeH]\\) is invariant under \\(M\\). It is also invariant under \\(M'\\), since \\(x'ye=yex'\\) for \\(x'\\in M'\\) and \\(y\\in M\\). Both algebras are closed under adjoints, so the projection \\(q\\) onto \\([MeH]\\) commutes with \\(M\\) and with \\(M'\\). Hence \\(q\\in M''\\cap M'=Z\\), and \\(q\\ge e\\) because \\(1\\in M\\). If \\(z\\) is central and \\(z\\ge e\\), then \\(zye=yze=ye\\) for \\(y\\in M\\), so \\([MeH]\\subseteq zH\\) and \\(q\\le z\\). So \\(q=c(e)\\). If \\(z\\) is central and \\(ze=0\\), then \\(zye=yze=0\\), so \\(z\\) vanishes on \\([MeH]\\) and \\(zc(e)=0\\). The converse follows from \\(e=c(e)e\\). Finally, a central projection \\(w\\) satisfies \\(w(ze)=0\\) iff \\((wz)c(e)=0\\), iff \\(w\\le1-zc(e)\\). The largest such \\(w\\) is \\(1-c(ze)\\) by what was just shown, so \\(c(ze)=zc(e)\\).\n\n(2) Let \\(u\\) implement \\(e\\sim f\\), and let \\(z\\) be central. From \\(u=ue\\), \\(e=u^*u\\), \\(u=fu\\) and \\(f=uu^*\\) we get \\(ze=0\\iff zu=0\\iff zf=0\\). By (1), \\(zc(e)=0\\iff zc(f)=0\\). Taking \\(z=1-c(e)\\) and \\(z=1-c(f)\\) gives \\(c(f)\\le c(e)\\le c(f)\\). The second claim follows, since \\(f_1\\le f\\) implies \\(c(f_1)\\le c(f)\\).\n\n(3) The map is a \\(*\\)-homomorphism because \\(e\\) commutes with \\(M'\\), and its image is \\(M'e\\) by definition. Further, \\(x'e=0\\) iff \\(x'\\) vanishes on \\(eH\\), iff \\(x'\\) vanishes on \\([MeH]\\) (as \\(x'ye=yx'e\\)), iff \\(x'c(e)=0\\). The second statement is the first one with the roles of \\(M\\) and \\(M'\\) exchanged.\n\n(4) By Fact 2.1, \\(eMe\\) and \\(M'e\\) are each other's commutants on \\(eH\\), so they have the same centre. By (3), the induction restricts to an isomorphism of \\(M'c(e)\\) onto \\(M'e\\), and an isomorphism maps centre onto centre. The centre of \\(M'c(e)\\) is \\(Zc(e)\\): if \\(y'c(e)\\) (\\(y'\\in M'\\)) commutes with \\(M'c(e)\\), it also commutes with \\(M'(1-c(e))\\) (both products vanish), hence with all of \\(M'\\), so it lies in \\(M'\\cap M''=Z\\). Therefore the centre of \\(M'e\\) is \\(\\{ac(e)e:a\\in Z\\}=Ze\\). On \\(Zc(e)\\) the map \\(a\\mapsto ae|_{eH}\\) is the restriction of the induction, which is injective there. An isomorphism maps projections to projections. \\(\\square\\)\n\n",
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    {
      "id": "OA-FND-TY-04",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "4. Supports, cyclic projections and the parallelogram law",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
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      "full_conditions_and_proof": "## 4. Supports, cyclic projections and the parallelogram law\n\nEvery operator in \\(M\\) has a left and a right support in \\(M\\), and the two supports are equivalent. This is the basic way to produce equivalent projections.\n\n**Definition 4.1** (Left and right supports). For \\(x\\in M\\), the *left support* \\(\\mathrm l(x)\\) is the least projection \\(p\\in M\\) with \\(px=x\\), and the *right support* \\(\\mathrm r(x)\\) is the least projection \\(q\\in M\\) with \\(xq=x\\).\n\n**Lemma 4.2.** \\(\\mathrm l(x)\\) is the projection onto \\([xH]\\), and \\(\\mathrm r(x)=\\mathrm l(x^*)\\) is the projection onto \\([x^*H]=(\\ker x)^\\perp\\). Both lie in \\(M\\).\n\n**Proof.** For a projection \\(p\\), \\(px=x\\) iff \\(xH\\subseteq pH\\), iff \\(p\\) majorizes the projection onto \\([xH]\\). That projection lies in \\(M\\): it is \\(uu^*\\) for the partial isometry \\(u\\) of the polar decomposition (Fact 2.2). Next, \\(xq=x\\) iff \\(qx^*=x^*\\), iff \\(q\\ge\\mathrm l(x^*)\\). Finally \\([x^*H]^\\perp=\\ker x\\). \\(\\square\\)\n\n**Proposition 4.3.** \\(\\mathrm l(x)\\sim\\mathrm r(x)\\) for every \\(x\\in M\\).\n\n**Proof.** In the polar decomposition \\(x=u|x|\\) of Fact 2.2, \\(u^*u=\\mathrm r(x)\\) and \\(uu^*=\\mathrm l(x)\\). \\(\\square\\)\n\n**Proposition 4.4** (The parallelogram law). For \\(e,f\\in\\mathcal P(M)\\), \\((e\\vee f)-e\\sim f-(e\\wedge f)\\).\n\n**Proof.** Put \\(x=e^\\perp f\\in M\\). A vector \\(\\xi\\) lies in \\(\\ker x\\) iff \\(f\\xi\\in eH\\). Writing \\(\\xi=f^\\perp\\xi+f\\xi\\), this gives \\(\\ker x=f^\\perp H\\oplus(eH\\cap fH)\\), with projection \\(f^\\perp+e\\wedge f\\). Hence \\(\\mathrm r(x)=f-e\\wedge f\\). In the same way, \\(\\xi\\in\\ker x^*=\\ker(fe^\\perp)\\) iff \\(e^\\perp\\xi\\in f^\\perp H\\). Writing \\(\\xi=e\\xi+e^\\perp\\xi\\), we get \\(\\ker x^*=eH\\oplus(e^\\perp H\\cap f^\\perp H)\\), with projection \\(e+e^\\perp\\wedge f^\\perp=e+(e\\vee f)^\\perp\\). Hence \\(\\mathrm l(x)=(e\\vee f)-e\\). Now apply Proposition 4.3. \\(\\square\\)\n\nVector functionals also have supports in \\(M\\): they are the cyclic projections.\n\n**Definition 4.5** (Cyclic projections). For \\(\\xi\\in H\\), \\(p_\\xi\\) is the projection onto \\([M'\\xi]\\) and \\(p'_\\xi\\) the projection onto \\([M\\xi]\\). These are the *cyclic projections* of \\(M\\) and of \\(M'\\) defined by \\(\\xi\\).\n\n**Lemma 4.6.** \\(p_\\xi\\in M\\), and \\(p_\\xi\\) is the least projection \\(p\\in M\\) with \\(p\\xi=\\xi\\). Equivalently, \\(p_\\xi\\) is the support of the restriction of \\(\\omega_\\xi\\) to \\(M\\). The same holds for \\(p'_\\xi\\) with \\(M'\\) in place of \\(M\\).\n\n**Proof.** \\([M'\\xi]\\) is invariant under \\(M'\\), which is closed under adjoints, so \\(p_\\xi\\) commutes with \\(M'\\) and lies in \\(M\\). As \\(1\\in M'\\), \\(p_\\xi\\xi=\\xi\\). If \\(p\\in\\mathcal P(M)\\) and \\(p\\xi=\\xi\\), then \\(px'\\xi=x'p\\xi=x'\\xi\\) for \\(x'\\in M'\\), so \\(p\\ge p_\\xi\\). Finally \\(\\omega_\\xi(1-p)=\\|(1-p)\\xi\\|^2\\), which vanishes iff \\(p\\xi=\\xi\\). \\(\\square\\)\n\n**Exercise 4.7.** (medium) Let \\(\\mathfrak m\\) be a left ideal of \\(M\\), not necessarily closed, and \\(\\mathcal P=\\mathcal P(M)\\).\n\n- (a) \\(\\mathfrak m=\\{xh:\\ x\\in M,\\ h\\in\\mathfrak m\\cap M_+\\}\\).\n- (b) The linear span of \\(\\{xe:\\ x\\in M,\\ e\\in\\mathcal P\\cap\\mathfrak m\\}\\) is norm dense in \\(\\mathfrak m\\). So a norm-closed left ideal is determined by its projections.\n\n*Solution.* (a) If \\(y\\in\\mathfrak m\\) has polar decomposition \\(y=u|y|\\) (Fact 2.2), then \\(|y|=u^*y\\in\\mathfrak m\\cap M_+\\) and \\(y=u|y|\\). Conversely \\(xh\\in\\mathfrak m\\) for \\(h\\in\\mathfrak m\\).\n\n(b) Each \\(xe\\) with \\(e\\in\\mathcal P\\cap\\mathfrak m\\) lies in \\(\\mathfrak m\\). Let \\(y=u|y|\\in\\mathfrak m\\), put \\(h=|y|\\in\\mathfrak m\\cap M_+\\), and let \\(\\varepsilon>0\\). The spectral projection \\(p=\\chi_{(\\varepsilon,\\infty)}(h)\\) equals \\(k(h)h\\) with \\(k(t)=t^{-1}\\chi_{(\\varepsilon,\\infty)}(t)\\), a bounded Borel function (Fact 2.8), so \\(p\\in\\mathcal P\\cap\\mathfrak m\\), and \\(\\|h-hp\\|=\\|h\\chi_{[0,\\varepsilon]}(h)\\|\\le\\varepsilon\\). Then \\(yp=u(hp)\\), so \\(\\|y-yp\\|\\le\\|h-hp\\|\\le\\varepsilon\\), and \\(yp\\) has the form \\(xe\\). If \\(\\mathfrak m\\) is norm closed, it is the closed linear span of these products, which depends only on \\(\\mathfrak m\\cap\\mathcal P\\). \\(\\square\\)\n\n",
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      "id": "OA-FND-TY-07",
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      "name": "4. Supports, cyclic projections and the parallelogram law",
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      "full_conditions_and_proof": "## 4. Supports, cyclic projections and the parallelogram law\n\nEvery operator in \\(M\\) has a left and a right support in \\(M\\), and the two supports are equivalent. This is the basic way to produce equivalent projections.\n\n**Definition 4.1** (Left and right supports). For \\(x\\in M\\), the *left support* \\(\\mathrm l(x)\\) is the least projection \\(p\\in M\\) with \\(px=x\\), and the *right support* \\(\\mathrm r(x)\\) is the least projection \\(q\\in M\\) with \\(xq=x\\).\n\n**Lemma 4.2.** \\(\\mathrm l(x)\\) is the projection onto \\([xH]\\), and \\(\\mathrm r(x)=\\mathrm l(x^*)\\) is the projection onto \\([x^*H]=(\\ker x)^\\perp\\). Both lie in \\(M\\).\n\n**Proof.** For a projection \\(p\\), \\(px=x\\) iff \\(xH\\subseteq pH\\), iff \\(p\\) majorizes the projection onto \\([xH]\\). That projection lies in \\(M\\): it is \\(uu^*\\) for the partial isometry \\(u\\) of the polar decomposition (Fact 2.2). Next, \\(xq=x\\) iff \\(qx^*=x^*\\), iff \\(q\\ge\\mathrm l(x^*)\\). Finally \\([x^*H]^\\perp=\\ker x\\). \\(\\square\\)\n\n**Proposition 4.3.** \\(\\mathrm l(x)\\sim\\mathrm r(x)\\) for every \\(x\\in M\\).\n\n**Proof.** In the polar decomposition \\(x=u|x|\\) of Fact 2.2, \\(u^*u=\\mathrm r(x)\\) and \\(uu^*=\\mathrm l(x)\\). \\(\\square\\)\n\n**Proposition 4.4** (The parallelogram law). For \\(e,f\\in\\mathcal P(M)\\), \\((e\\vee f)-e\\sim f-(e\\wedge f)\\).\n\n**Proof.** Put \\(x=e^\\perp f\\in M\\). A vector \\(\\xi\\) lies in \\(\\ker x\\) iff \\(f\\xi\\in eH\\). Writing \\(\\xi=f^\\perp\\xi+f\\xi\\), this gives \\(\\ker x=f^\\perp H\\oplus(eH\\cap fH)\\), with projection \\(f^\\perp+e\\wedge f\\). Hence \\(\\mathrm r(x)=f-e\\wedge f\\). In the same way, \\(\\xi\\in\\ker x^*=\\ker(fe^\\perp)\\) iff \\(e^\\perp\\xi\\in f^\\perp H\\). Writing \\(\\xi=e\\xi+e^\\perp\\xi\\), we get \\(\\ker x^*=eH\\oplus(e^\\perp H\\cap f^\\perp H)\\), with projection \\(e+e^\\perp\\wedge f^\\perp=e+(e\\vee f)^\\perp\\). Hence \\(\\mathrm l(x)=(e\\vee f)-e\\). Now apply Proposition 4.3. \\(\\square\\)\n\nVector functionals also have supports in \\(M\\): they are the cyclic projections.\n\n**Definition 4.5** (Cyclic projections). For \\(\\xi\\in H\\), \\(p_\\xi\\) is the projection onto \\([M'\\xi]\\) and \\(p'_\\xi\\) the projection onto \\([M\\xi]\\). These are the *cyclic projections* of \\(M\\) and of \\(M'\\) defined by \\(\\xi\\).\n\n**Lemma 4.6.** \\(p_\\xi\\in M\\), and \\(p_\\xi\\) is the least projection \\(p\\in M\\) with \\(p\\xi=\\xi\\). Equivalently, \\(p_\\xi\\) is the support of the restriction of \\(\\omega_\\xi\\) to \\(M\\). The same holds for \\(p'_\\xi\\) with \\(M'\\) in place of \\(M\\).\n\n**Proof.** \\([M'\\xi]\\) is invariant under \\(M'\\), which is closed under adjoints, so \\(p_\\xi\\) commutes with \\(M'\\) and lies in \\(M\\). As \\(1\\in M'\\), \\(p_\\xi\\xi=\\xi\\). If \\(p\\in\\mathcal P(M)\\) and \\(p\\xi=\\xi\\), then \\(px'\\xi=x'p\\xi=x'\\xi\\) for \\(x'\\in M'\\), so \\(p\\ge p_\\xi\\). Finally \\(\\omega_\\xi(1-p)=\\|(1-p)\\xi\\|^2\\), which vanishes iff \\(p\\xi=\\xi\\). \\(\\square\\)\n\n**Exercise 4.7.** (medium) Let \\(\\mathfrak m\\) be a left ideal of \\(M\\), not necessarily closed, and \\(\\mathcal P=\\mathcal P(M)\\).\n\n- (a) \\(\\mathfrak m=\\{xh:\\ x\\in M,\\ h\\in\\mathfrak m\\cap M_+\\}\\).\n- (b) The linear span of \\(\\{xe:\\ x\\in M,\\ e\\in\\mathcal P\\cap\\mathfrak m\\}\\) is norm dense in \\(\\mathfrak m\\). So a norm-closed left ideal is determined by its projections.\n\n*Solution.* (a) If \\(y\\in\\mathfrak m\\) has polar decomposition \\(y=u|y|\\) (Fact 2.2), then \\(|y|=u^*y\\in\\mathfrak m\\cap M_+\\) and \\(y=u|y|\\). Conversely \\(xh\\in\\mathfrak m\\) for \\(h\\in\\mathfrak m\\).\n\n(b) Each \\(xe\\) with \\(e\\in\\mathcal P\\cap\\mathfrak m\\) lies in \\(\\mathfrak m\\). Let \\(y=u|y|\\in\\mathfrak m\\), put \\(h=|y|\\in\\mathfrak m\\cap M_+\\), and let \\(\\varepsilon>0\\). The spectral projection \\(p=\\chi_{(\\varepsilon,\\infty)}(h)\\) equals \\(k(h)h\\) with \\(k(t)=t^{-1}\\chi_{(\\varepsilon,\\infty)}(t)\\), a bounded Borel function (Fact 2.8), so \\(p\\in\\mathcal P\\cap\\mathfrak m\\), and \\(\\|h-hp\\|=\\|h\\chi_{[0,\\varepsilon]}(h)\\|\\le\\varepsilon\\). Then \\(yp=u(hp)\\), so \\(\\|y-yp\\|\\le\\|h-hp\\|\\le\\varepsilon\\), and \\(yp\\) has the form \\(xe\\). If \\(\\mathfrak m\\) is norm closed, it is the closed linear span of these products, which depends only on \\(\\mathfrak m\\cap\\mathcal P\\). \\(\\square\\)\n\n",
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      "id": "OA-FND-TY-03",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "5. The comparison theorem",
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      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 5. The comparison theorem\n\nTwo ingredients lead to the comparison theorem: a Schröder–Bernstein theorem for projections, and a test for when two projections have equivalent nonzero subprojections.\n\n**Proposition 5.1** (Schröder–Bernstein for projections). If \\(e\\precsim f\\) and \\(f\\precsim e\\), then \\(e\\sim f\\).\n\nThe proof makes one \"Hilbert hotel\" shift inside \\(e\\).\n\n**Proof.** *Step 1.* Let \\(e_2\\le e_1\\le e\\) be projections with \\(e\\sim e_2\\). We show \\(e\\sim e_1\\). Let \\(w\\) implement \\(e\\sim e_2\\). Then \\(w=we\\) and \\(ew=w\\), so the powers \\(w^n\\) have initial projection \\(e\\), and their final projections \\(w^nw^{*n}\\) decrease, since \\[\n\\begin{gathered}\nw^{n+1}w^{*(n+1)}\\\\\n=w^n(ww^*)w^{*n}\\\\\n\\le w^new^{*n}\\\\\n=w^nw^{*n}.\n\\end{gathered}\n\\] Put \\(g=e-e_1\\) and \\(g_n=w^ngw^{*n}\\) for \\(n\\ge0\\). As \\(g\\le e-e_2=e-ww^*\\), we get \\(g_n\\le w^nw^{*n}-w^{n+1}w^{*(n+1)}\\). The right-hand sides are differences of a decreasing sequence, so the \\(g_n\\) are mutually orthogonal. Put \\(G=\\sum_ng_n\\le e\\) and \\(t=wG+(e-G)\\). Since \\(wg_nw^*=g_{n+1}\\), the operator \\(wG\\) maps \\(GH\\) isometrically onto \\(\\sum_{n\\ge1}g_nH\\subseteq GH\\), while \\(e-G\\) is the identity on \\((e-G)H\\), which is orthogonal to \\(GH\\). The cross terms in \\(t^*t\\) and \\(tt^*\\) vanish, and\n\\[\n\\begin{gathered}\nt^*t\\\\\n=G+(e-G)\\\\\n=e,\\\\\ntt^*\\\\\n=\\sum_{n\\ge1}g_n+(e-G)\\\\\n=e-g_0\\\\\n=e_1 .\n\\end{gathered}\n\\]\nSo \\(e\\sim e_1\\).\n\n*Step 2.* Let \\(u\\) implement \\(e\\sim f_1\\le f\\) and \\(v\\) implement \\(f\\sim e_1\\le e\\). Put \\(e_2=vf_1v^*\\). Then \\(e_2\\le vfv^*=e_1\\), and \\(vf_1\\) implements \\(f_1\\sim e_2\\), so \\(e\\sim e_2\\). Step 1 gives \\(e\\sim e_1\\), and \\(e_1\\sim f\\). \\(\\square\\)\n\n**Definition 5.2.** Projections \\(e,f\\in\\mathcal P(M)\\) are *centrally orthogonal* if \\(c(e)c(f)=0\\).\n\n**Lemma 5.3** (Central orthogonality). For \\(e,f\\in\\mathcal P(M)\\) the following are equivalent.\n\n1. \\(c(e)c(f)\\ne0\\).\n2. \\(eMf\\ne\\{0\\}\\).\n3. There are nonzero projections \\(e_1\\le e\\) and \\(f_1\\le f\\) in \\(M\\) with \\(e_1\\sim f_1\\).\n\n**Proof.** (3)⇒(2). If \\(u\\) implements \\(e_1\\sim f_1\\), then \\(u=f_1ue_1\\in fMe\\), so \\(0\\ne u^*\\in eMf\\).\n\n(2)⇒(1). If \\(c(e)c(f)=0\\), then \\(eyf=ec(e)\\,y\\,c(f)f=ey\\,c(e)c(f)\\,f=0\\) for all \\(y\\in M\\).\n\n(1)⇒(2). Suppose \\(eMf=\\{0\\}\\). Then \\(fMe=(eMf)^*=\\{0\\}\\), so \\(f\\) vanishes on \\([MeH]=c(e)H\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)). Thus \\(fc(e)=0\\), so \\(f\\le1-c(e)\\) and \\(c(f)\\le1-c(e)\\), which contradicts (1).\n\n(2)⇒(3). Take \\(x=eyf\\ne0\\). Then \\(ex=x\\) and \\(xf=x\\), so \\(\\mathrm l(x)\\le e\\) and \\(\\mathrm r(x)\\le f\\). These supports are nonzero because \\(x\\ne0\\), and they are equivalent by Proposition 4.3. \\(\\square\\)\n\n**Corollary 5.4.** If \\(e\\ne0\\) and \\(c(e)\\le c(f)\\), there are nonzero \\(e_1\\le e\\) and \\(f_1\\le f\\) with \\(e_1\\sim f_1\\). Indeed \\(c(e)c(f)=c(e)\\ne0\\).\n\n**Theorem 5.5** (The comparison theorem). For \\(e,f\\in\\mathcal P(M)\\) there is a central projection \\(z\\) with\n\\[\nze\\precsim zf\\qquad\\text{and}\\qquad(1-z)f\\precsim(1-z)e .\n\\]\nIf \\(M\\) is a factor, exactly one of \\(e\\prec f\\), \\(e\\sim f\\), \\(f\\prec e\\) holds.\n\n\n\n**Proof.** Call a family \\(\\{(e_i,f_i)\\}_{i\\in I}\\) *admissible* if the \\(e_i\\) are mutually orthogonal nonzero subprojections of \\(e\\), the \\(f_i\\) are mutually orthogonal subprojections of \\(f\\), and \\(e_i\\sim f_i\\) for each \\(i\\). The union of a chain of admissible families is admissible, so Zorn's lemma gives a maximal one. Put \\(e_0=\\sum_ie_i\\) and \\(f_0=\\sum_if_i\\). Then \\(e_0\\sim f_0\\) by additivity (Lemma 3.3(3)). If \\(e-e_0\\) and \\(f-f_0\\) were not centrally orthogonal, Lemma 5.3 would give nonzero \\(e'\\le e-e_0\\) and \\(f'\\le f-f_0\\) with \\(e'\\sim f'\\). Adding the pair \\((e',f')\\) would contradict maximality. So \\(c(e-e_0)\\,c(f-f_0)=0\\).\n\nPut \\(z=1-c(e-e_0)\\). Then \\(z(e-e_0)=0\\), and \\[\n\\begin{gathered}\n(1-z)(f-f_0)\\\\\n=c(e-e_0)c(f-f_0)(f-f_0)\\\\\n=0.\n\\end{gathered}\n\\] Cutting by a central projection preserves equivalence ([Lemma 3.3(4)](#oa-fnd-ty-02)), so\n\\[\n\\begin{gathered}\nze\\\\\n=ze_0\\sim zf_0\\\\\n\\le zf,\\\\\n(1-z)f\\\\\n=(1-z)f_0\\sim(1-z)e_0\\\\\n\\le(1-z)e .\n\\end{gathered}\n\\]\n\nLet \\(M\\) be a factor. Then \\(z\\in\\{0,1\\}\\), so \\(e\\precsim f\\) or \\(f\\precsim e\\). If both hold, \\(e\\sim f\\) by Proposition 5.1. If only \\(e\\precsim f\\) holds, then \\(e\\) is not equivalent to \\(f\\) (else \\(f\\precsim e\\)), so \\(e\\prec f\\). Symmetrically, if only \\(f\\precsim e\\) holds, then \\(f\\prec e\\). The three cases exclude each other: \\(e\\prec f\\) rules out \\(e\\sim f\\) by definition, and \\(e\\prec f\\) together with \\(f\\prec e\\) would give \\(e\\sim f\\) by Proposition 5.1. \\(\\square\\)\n\n**Exercise 5.6.** (hard) Let \\(\\alpha\\) be an automorphism of \\(M\\). Suppose that some \\(e\\in\\mathcal P(M)\\) with \\(c(e)=1\\) and some \\(u\\in M\\) satisfy \\(\\alpha(x)=uxu^*\\) for all \\(x\\in eMe\\). Show that \\(\\alpha\\) is inner.\n\n*Solution.* *Normalizing \\(u\\).* Put \\(v=ue\\). Then \\(vv^*=ueu^*=\\alpha(e)\\) is a projection, so \\(v\\) is a partial isometry ([Lemma 3.3(1)](#oa-fnd-ty-02) applied to \\(v^*\\)), and \\(q=v^*v=eu^*ue\\) is a projection below \\(e\\). For \\(y\\in eMe\\), \\(\\alpha(y)=vyv^*\\). Since \\(vq=v\\), the element \\(e-q\\in eMe\\) satisfies \\(\\alpha(e-q)=vv^*-vqv^*=0\\); as \\(\\alpha\\) is injective, \\(q=e\\). So \\(v^*v=e\\), \\(vv^*=\\alpha(e)\\), and \\(\\alpha(y)v=vyv^*v=vy\\) for \\(y\\in eMe\\).\n\n*Building the unitary.* Since \\(c(e)=1\\), a maximal family of mutually orthogonal projections each subequivalent to \\(e\\) has sum \\(1\\) (otherwise Corollary 5.4, applied to the remainder and \\(e\\), would enlarge it). So there are partial isometries \\(v_i\\) with \\(v_i^*v_i\\le e\\), mutually orthogonal final projections \\(v_iv_i^*\\), and \\(\\sum_iv_iv_i^*=1\\). Put \\(t_i=\\alpha(v_i)vv_i^*\\). Using \\(\\alpha(y)=vyv^*\\) for \\(y=v_i^*v_i\\in eMe\\) and \\(v_i=v_ie\\), one finds that \\(t_i\\) has initial projection \\(v_iv_i^*\\) and final projection \\(\\alpha(v_iv_i^*)\\). The initial projections are mutually orthogonal with sum \\(1\\). So are the final ones: \\(\\alpha\\) preserves orthogonality and the order of projections, hence least upper bounds (Proposition 3.1), so \\(\\sum_i\\alpha(v_iv_i^*)=\\alpha(1)=1\\). By the proof of [Lemma 3.3(3)](#oa-fnd-ty-02), \\(w=\\sum_it_i\\) converges strongly to a unitary in \\(M\\).\n\n*It implements \\(\\alpha\\).* As \\(v_i^*v_j=0\\) for \\(i\\ne j\\), \\[\n\\begin{gathered}\nwv_j\\\\\n=t_jv_j\\\\\n=\\alpha(v_j)vv_j^*v_j\\\\\n=\\alpha(v_j)\\alpha(v_j^*v_j)v\\\\\n=\\alpha(v_j)v.\n\\end{gathered}\n\\] For \\(x\\in M\\), \\(xv_j=\\sum_iv_iv_i^*xv_j\\) strongly, and \\(v_i^*xv_j\\in eMe\\), so\n\\[\n\\begin{gathered}\nwxv_j\\\\\n=\\sum_i\\alpha(v_i)v(v_i^*xv_j)\\\\\n=\\sum_i\\alpha(v_i)\\alpha(v_i^*xv_j)v\\\\\n=\\sum_i\\alpha(v_iv_i^*)\\,\\alpha(xv_j)v\\\\\n=\\alpha(xv_j)v\\\\\n=\\alpha(x)wv_j .\n\\end{gathered}\n\\]\nThe ranges of the \\(v_j\\) span a dense subspace of \\(H\\), so \\(wx=\\alpha(x)w\\), that is, \\(\\alpha(x)=wxw^*\\). No normality of \\(\\alpha\\) was used. \\(\\square\\)\n\n",
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      "id": "OA-FND-TY-05",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "5. The comparison theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 5. The comparison theorem\n\nTwo ingredients lead to the comparison theorem: a Schröder–Bernstein theorem for projections, and a test for when two projections have equivalent nonzero subprojections.\n\n**Proposition 5.1** (Schröder–Bernstein for projections). If \\(e\\precsim f\\) and \\(f\\precsim e\\), then \\(e\\sim f\\).\n\nThe proof makes one \"Hilbert hotel\" shift inside \\(e\\).\n\n**Proof.** *Step 1.* Let \\(e_2\\le e_1\\le e\\) be projections with \\(e\\sim e_2\\). We show \\(e\\sim e_1\\). Let \\(w\\) implement \\(e\\sim e_2\\). Then \\(w=we\\) and \\(ew=w\\), so the powers \\(w^n\\) have initial projection \\(e\\), and their final projections \\(w^nw^{*n}\\) decrease, since \\[\n\\begin{gathered}\nw^{n+1}w^{*(n+1)}\\\\\n=w^n(ww^*)w^{*n}\\\\\n\\le w^new^{*n}\\\\\n=w^nw^{*n}.\n\\end{gathered}\n\\] Put \\(g=e-e_1\\) and \\(g_n=w^ngw^{*n}\\) for \\(n\\ge0\\). As \\(g\\le e-e_2=e-ww^*\\), we get \\(g_n\\le w^nw^{*n}-w^{n+1}w^{*(n+1)}\\). The right-hand sides are differences of a decreasing sequence, so the \\(g_n\\) are mutually orthogonal. Put \\(G=\\sum_ng_n\\le e\\) and \\(t=wG+(e-G)\\). Since \\(wg_nw^*=g_{n+1}\\), the operator \\(wG\\) maps \\(GH\\) isometrically onto \\(\\sum_{n\\ge1}g_nH\\subseteq GH\\), while \\(e-G\\) is the identity on \\((e-G)H\\), which is orthogonal to \\(GH\\). The cross terms in \\(t^*t\\) and \\(tt^*\\) vanish, and\n\\[\n\\begin{gathered}\nt^*t\\\\\n=G+(e-G)\\\\\n=e,\\\\\ntt^*\\\\\n=\\sum_{n\\ge1}g_n+(e-G)\\\\\n=e-g_0\\\\\n=e_1 .\n\\end{gathered}\n\\]\nSo \\(e\\sim e_1\\).\n\n*Step 2.* Let \\(u\\) implement \\(e\\sim f_1\\le f\\) and \\(v\\) implement \\(f\\sim e_1\\le e\\). Put \\(e_2=vf_1v^*\\). Then \\(e_2\\le vfv^*=e_1\\), and \\(vf_1\\) implements \\(f_1\\sim e_2\\), so \\(e\\sim e_2\\). Step 1 gives \\(e\\sim e_1\\), and \\(e_1\\sim f\\). \\(\\square\\)\n\n**Definition 5.2.** Projections \\(e,f\\in\\mathcal P(M)\\) are *centrally orthogonal* if \\(c(e)c(f)=0\\).\n\n**Lemma 5.3** (Central orthogonality). For \\(e,f\\in\\mathcal P(M)\\) the following are equivalent.\n\n1. \\(c(e)c(f)\\ne0\\).\n2. \\(eMf\\ne\\{0\\}\\).\n3. There are nonzero projections \\(e_1\\le e\\) and \\(f_1\\le f\\) in \\(M\\) with \\(e_1\\sim f_1\\).\n\n**Proof.** (3)⇒(2). If \\(u\\) implements \\(e_1\\sim f_1\\), then \\(u=f_1ue_1\\in fMe\\), so \\(0\\ne u^*\\in eMf\\).\n\n(2)⇒(1). If \\(c(e)c(f)=0\\), then \\(eyf=ec(e)\\,y\\,c(f)f=ey\\,c(e)c(f)\\,f=0\\) for all \\(y\\in M\\).\n\n(1)⇒(2). Suppose \\(eMf=\\{0\\}\\). Then \\(fMe=(eMf)^*=\\{0\\}\\), so \\(f\\) vanishes on \\([MeH]=c(e)H\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)). Thus \\(fc(e)=0\\), so \\(f\\le1-c(e)\\) and \\(c(f)\\le1-c(e)\\), which contradicts (1).\n\n(2)⇒(3). Take \\(x=eyf\\ne0\\). Then \\(ex=x\\) and \\(xf=x\\), so \\(\\mathrm l(x)\\le e\\) and \\(\\mathrm r(x)\\le f\\). These supports are nonzero because \\(x\\ne0\\), and they are equivalent by Proposition 4.3. \\(\\square\\)\n\n**Corollary 5.4.** If \\(e\\ne0\\) and \\(c(e)\\le c(f)\\), there are nonzero \\(e_1\\le e\\) and \\(f_1\\le f\\) with \\(e_1\\sim f_1\\). Indeed \\(c(e)c(f)=c(e)\\ne0\\).\n\n**Theorem 5.5** (The comparison theorem). For \\(e,f\\in\\mathcal P(M)\\) there is a central projection \\(z\\) with\n\\[\nze\\precsim zf\\qquad\\text{and}\\qquad(1-z)f\\precsim(1-z)e .\n\\]\nIf \\(M\\) is a factor, exactly one of \\(e\\prec f\\), \\(e\\sim f\\), \\(f\\prec e\\) holds.\n\n\n\n**Proof.** Call a family \\(\\{(e_i,f_i)\\}_{i\\in I}\\) *admissible* if the \\(e_i\\) are mutually orthogonal nonzero subprojections of \\(e\\), the \\(f_i\\) are mutually orthogonal subprojections of \\(f\\), and \\(e_i\\sim f_i\\) for each \\(i\\). The union of a chain of admissible families is admissible, so Zorn's lemma gives a maximal one. Put \\(e_0=\\sum_ie_i\\) and \\(f_0=\\sum_if_i\\). Then \\(e_0\\sim f_0\\) by additivity (Lemma 3.3(3)). If \\(e-e_0\\) and \\(f-f_0\\) were not centrally orthogonal, Lemma 5.3 would give nonzero \\(e'\\le e-e_0\\) and \\(f'\\le f-f_0\\) with \\(e'\\sim f'\\). Adding the pair \\((e',f')\\) would contradict maximality. So \\(c(e-e_0)\\,c(f-f_0)=0\\).\n\nPut \\(z=1-c(e-e_0)\\). Then \\(z(e-e_0)=0\\), and \\[\n\\begin{gathered}\n(1-z)(f-f_0)\\\\\n=c(e-e_0)c(f-f_0)(f-f_0)\\\\\n=0.\n\\end{gathered}\n\\] Cutting by a central projection preserves equivalence ([Lemma 3.3(4)](#oa-fnd-ty-02)), so\n\\[\n\\begin{gathered}\nze\\\\\n=ze_0\\sim zf_0\\\\\n\\le zf,\\\\\n(1-z)f\\\\\n=(1-z)f_0\\sim(1-z)e_0\\\\\n\\le(1-z)e .\n\\end{gathered}\n\\]\n\nLet \\(M\\) be a factor. Then \\(z\\in\\{0,1\\}\\), so \\(e\\precsim f\\) or \\(f\\precsim e\\). If both hold, \\(e\\sim f\\) by Proposition 5.1. If only \\(e\\precsim f\\) holds, then \\(e\\) is not equivalent to \\(f\\) (else \\(f\\precsim e\\)), so \\(e\\prec f\\). Symmetrically, if only \\(f\\precsim e\\) holds, then \\(f\\prec e\\). The three cases exclude each other: \\(e\\prec f\\) rules out \\(e\\sim f\\) by definition, and \\(e\\prec f\\) together with \\(f\\prec e\\) would give \\(e\\sim f\\) by Proposition 5.1. \\(\\square\\)\n\n**Exercise 5.6.** (hard) Let \\(\\alpha\\) be an automorphism of \\(M\\). Suppose that some \\(e\\in\\mathcal P(M)\\) with \\(c(e)=1\\) and some \\(u\\in M\\) satisfy \\(\\alpha(x)=uxu^*\\) for all \\(x\\in eMe\\). Show that \\(\\alpha\\) is inner.\n\n*Solution.* *Normalizing \\(u\\).* Put \\(v=ue\\). Then \\(vv^*=ueu^*=\\alpha(e)\\) is a projection, so \\(v\\) is a partial isometry ([Lemma 3.3(1)](#oa-fnd-ty-02) applied to \\(v^*\\)), and \\(q=v^*v=eu^*ue\\) is a projection below \\(e\\). For \\(y\\in eMe\\), \\(\\alpha(y)=vyv^*\\). Since \\(vq=v\\), the element \\(e-q\\in eMe\\) satisfies \\(\\alpha(e-q)=vv^*-vqv^*=0\\); as \\(\\alpha\\) is injective, \\(q=e\\). So \\(v^*v=e\\), \\(vv^*=\\alpha(e)\\), and \\(\\alpha(y)v=vyv^*v=vy\\) for \\(y\\in eMe\\).\n\n*Building the unitary.* Since \\(c(e)=1\\), a maximal family of mutually orthogonal projections each subequivalent to \\(e\\) has sum \\(1\\) (otherwise Corollary 5.4, applied to the remainder and \\(e\\), would enlarge it). So there are partial isometries \\(v_i\\) with \\(v_i^*v_i\\le e\\), mutually orthogonal final projections \\(v_iv_i^*\\), and \\(\\sum_iv_iv_i^*=1\\). Put \\(t_i=\\alpha(v_i)vv_i^*\\). Using \\(\\alpha(y)=vyv^*\\) for \\(y=v_i^*v_i\\in eMe\\) and \\(v_i=v_ie\\), one finds that \\(t_i\\) has initial projection \\(v_iv_i^*\\) and final projection \\(\\alpha(v_iv_i^*)\\). The initial projections are mutually orthogonal with sum \\(1\\). So are the final ones: \\(\\alpha\\) preserves orthogonality and the order of projections, hence least upper bounds (Proposition 3.1), so \\(\\sum_i\\alpha(v_iv_i^*)=\\alpha(1)=1\\). By the proof of [Lemma 3.3(3)](#oa-fnd-ty-02), \\(w=\\sum_it_i\\) converges strongly to a unitary in \\(M\\).\n\n*It implements \\(\\alpha\\).* As \\(v_i^*v_j=0\\) for \\(i\\ne j\\), \\[\n\\begin{gathered}\nwv_j\\\\\n=t_jv_j\\\\\n=\\alpha(v_j)vv_j^*v_j\\\\\n=\\alpha(v_j)\\alpha(v_j^*v_j)v\\\\\n=\\alpha(v_j)v.\n\\end{gathered}\n\\] For \\(x\\in M\\), \\(xv_j=\\sum_iv_iv_i^*xv_j\\) strongly, and \\(v_i^*xv_j\\in eMe\\), so\n\\[\n\\begin{gathered}\nwxv_j\\\\\n=\\sum_i\\alpha(v_i)v(v_i^*xv_j)\\\\\n=\\sum_i\\alpha(v_i)\\alpha(v_i^*xv_j)v\\\\\n=\\sum_i\\alpha(v_iv_i^*)\\,\\alpha(xv_j)v\\\\\n=\\alpha(xv_j)v\\\\\n=\\alpha(x)wv_j .\n\\end{gathered}\n\\]\nThe ranges of the \\(v_j\\) span a dense subspace of \\(H\\), so \\(wx=\\alpha(x)w\\), that is, \\(\\alpha(x)=wxw^*\\). No normality of \\(\\alpha\\) was used. \\(\\square\\)\n\n",
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      "id": "OA-FND-TY-06",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "5. The comparison theorem",
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      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 5. The comparison theorem\n\nTwo ingredients lead to the comparison theorem: a Schröder–Bernstein theorem for projections, and a test for when two projections have equivalent nonzero subprojections.\n\n**Proposition 5.1** (Schröder–Bernstein for projections). If \\(e\\precsim f\\) and \\(f\\precsim e\\), then \\(e\\sim f\\).\n\nThe proof makes one \"Hilbert hotel\" shift inside \\(e\\).\n\n**Proof.** *Step 1.* Let \\(e_2\\le e_1\\le e\\) be projections with \\(e\\sim e_2\\). We show \\(e\\sim e_1\\). Let \\(w\\) implement \\(e\\sim e_2\\). Then \\(w=we\\) and \\(ew=w\\), so the powers \\(w^n\\) have initial projection \\(e\\), and their final projections \\(w^nw^{*n}\\) decrease, since \\[\n\\begin{gathered}\nw^{n+1}w^{*(n+1)}\\\\\n=w^n(ww^*)w^{*n}\\\\\n\\le w^new^{*n}\\\\\n=w^nw^{*n}.\n\\end{gathered}\n\\] Put \\(g=e-e_1\\) and \\(g_n=w^ngw^{*n}\\) for \\(n\\ge0\\). As \\(g\\le e-e_2=e-ww^*\\), we get \\(g_n\\le w^nw^{*n}-w^{n+1}w^{*(n+1)}\\). The right-hand sides are differences of a decreasing sequence, so the \\(g_n\\) are mutually orthogonal. Put \\(G=\\sum_ng_n\\le e\\) and \\(t=wG+(e-G)\\). Since \\(wg_nw^*=g_{n+1}\\), the operator \\(wG\\) maps \\(GH\\) isometrically onto \\(\\sum_{n\\ge1}g_nH\\subseteq GH\\), while \\(e-G\\) is the identity on \\((e-G)H\\), which is orthogonal to \\(GH\\). The cross terms in \\(t^*t\\) and \\(tt^*\\) vanish, and\n\\[\n\\begin{gathered}\nt^*t\\\\\n=G+(e-G)\\\\\n=e,\\\\\ntt^*\\\\\n=\\sum_{n\\ge1}g_n+(e-G)\\\\\n=e-g_0\\\\\n=e_1 .\n\\end{gathered}\n\\]\nSo \\(e\\sim e_1\\).\n\n*Step 2.* Let \\(u\\) implement \\(e\\sim f_1\\le f\\) and \\(v\\) implement \\(f\\sim e_1\\le e\\). Put \\(e_2=vf_1v^*\\). Then \\(e_2\\le vfv^*=e_1\\), and \\(vf_1\\) implements \\(f_1\\sim e_2\\), so \\(e\\sim e_2\\). Step 1 gives \\(e\\sim e_1\\), and \\(e_1\\sim f\\). \\(\\square\\)\n\n**Definition 5.2.** Projections \\(e,f\\in\\mathcal P(M)\\) are *centrally orthogonal* if \\(c(e)c(f)=0\\).\n\n**Lemma 5.3** (Central orthogonality). For \\(e,f\\in\\mathcal P(M)\\) the following are equivalent.\n\n1. \\(c(e)c(f)\\ne0\\).\n2. \\(eMf\\ne\\{0\\}\\).\n3. There are nonzero projections \\(e_1\\le e\\) and \\(f_1\\le f\\) in \\(M\\) with \\(e_1\\sim f_1\\).\n\n**Proof.** (3)⇒(2). If \\(u\\) implements \\(e_1\\sim f_1\\), then \\(u=f_1ue_1\\in fMe\\), so \\(0\\ne u^*\\in eMf\\).\n\n(2)⇒(1). If \\(c(e)c(f)=0\\), then \\(eyf=ec(e)\\,y\\,c(f)f=ey\\,c(e)c(f)\\,f=0\\) for all \\(y\\in M\\).\n\n(1)⇒(2). Suppose \\(eMf=\\{0\\}\\). Then \\(fMe=(eMf)^*=\\{0\\}\\), so \\(f\\) vanishes on \\([MeH]=c(e)H\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)). Thus \\(fc(e)=0\\), so \\(f\\le1-c(e)\\) and \\(c(f)\\le1-c(e)\\), which contradicts (1).\n\n(2)⇒(3). Take \\(x=eyf\\ne0\\). Then \\(ex=x\\) and \\(xf=x\\), so \\(\\mathrm l(x)\\le e\\) and \\(\\mathrm r(x)\\le f\\). These supports are nonzero because \\(x\\ne0\\), and they are equivalent by Proposition 4.3. \\(\\square\\)\n\n**Corollary 5.4.** If \\(e\\ne0\\) and \\(c(e)\\le c(f)\\), there are nonzero \\(e_1\\le e\\) and \\(f_1\\le f\\) with \\(e_1\\sim f_1\\). Indeed \\(c(e)c(f)=c(e)\\ne0\\).\n\n**Theorem 5.5** (The comparison theorem). For \\(e,f\\in\\mathcal P(M)\\) there is a central projection \\(z\\) with\n\\[\nze\\precsim zf\\qquad\\text{and}\\qquad(1-z)f\\precsim(1-z)e .\n\\]\nIf \\(M\\) is a factor, exactly one of \\(e\\prec f\\), \\(e\\sim f\\), \\(f\\prec e\\) holds.\n\n\n\n**Proof.** Call a family \\(\\{(e_i,f_i)\\}_{i\\in I}\\) *admissible* if the \\(e_i\\) are mutually orthogonal nonzero subprojections of \\(e\\), the \\(f_i\\) are mutually orthogonal subprojections of \\(f\\), and \\(e_i\\sim f_i\\) for each \\(i\\). The union of a chain of admissible families is admissible, so Zorn's lemma gives a maximal one. Put \\(e_0=\\sum_ie_i\\) and \\(f_0=\\sum_if_i\\). Then \\(e_0\\sim f_0\\) by additivity (Lemma 3.3(3)). If \\(e-e_0\\) and \\(f-f_0\\) were not centrally orthogonal, Lemma 5.3 would give nonzero \\(e'\\le e-e_0\\) and \\(f'\\le f-f_0\\) with \\(e'\\sim f'\\). Adding the pair \\((e',f')\\) would contradict maximality. So \\(c(e-e_0)\\,c(f-f_0)=0\\).\n\nPut \\(z=1-c(e-e_0)\\). Then \\(z(e-e_0)=0\\), and \\[\n\\begin{gathered}\n(1-z)(f-f_0)\\\\\n=c(e-e_0)c(f-f_0)(f-f_0)\\\\\n=0.\n\\end{gathered}\n\\] Cutting by a central projection preserves equivalence ([Lemma 3.3(4)](#oa-fnd-ty-02)), so\n\\[\n\\begin{gathered}\nze\\\\\n=ze_0\\sim zf_0\\\\\n\\le zf,\\\\\n(1-z)f\\\\\n=(1-z)f_0\\sim(1-z)e_0\\\\\n\\le(1-z)e .\n\\end{gathered}\n\\]\n\nLet \\(M\\) be a factor. Then \\(z\\in\\{0,1\\}\\), so \\(e\\precsim f\\) or \\(f\\precsim e\\). If both hold, \\(e\\sim f\\) by Proposition 5.1. If only \\(e\\precsim f\\) holds, then \\(e\\) is not equivalent to \\(f\\) (else \\(f\\precsim e\\)), so \\(e\\prec f\\). Symmetrically, if only \\(f\\precsim e\\) holds, then \\(f\\prec e\\). The three cases exclude each other: \\(e\\prec f\\) rules out \\(e\\sim f\\) by definition, and \\(e\\prec f\\) together with \\(f\\prec e\\) would give \\(e\\sim f\\) by Proposition 5.1. \\(\\square\\)\n\n**Exercise 5.6.** (hard) Let \\(\\alpha\\) be an automorphism of \\(M\\). Suppose that some \\(e\\in\\mathcal P(M)\\) with \\(c(e)=1\\) and some \\(u\\in M\\) satisfy \\(\\alpha(x)=uxu^*\\) for all \\(x\\in eMe\\). Show that \\(\\alpha\\) is inner.\n\n*Solution.* *Normalizing \\(u\\).* Put \\(v=ue\\). Then \\(vv^*=ueu^*=\\alpha(e)\\) is a projection, so \\(v\\) is a partial isometry ([Lemma 3.3(1)](#oa-fnd-ty-02) applied to \\(v^*\\)), and \\(q=v^*v=eu^*ue\\) is a projection below \\(e\\). For \\(y\\in eMe\\), \\(\\alpha(y)=vyv^*\\). Since \\(vq=v\\), the element \\(e-q\\in eMe\\) satisfies \\(\\alpha(e-q)=vv^*-vqv^*=0\\); as \\(\\alpha\\) is injective, \\(q=e\\). So \\(v^*v=e\\), \\(vv^*=\\alpha(e)\\), and \\(\\alpha(y)v=vyv^*v=vy\\) for \\(y\\in eMe\\).\n\n*Building the unitary.* Since \\(c(e)=1\\), a maximal family of mutually orthogonal projections each subequivalent to \\(e\\) has sum \\(1\\) (otherwise Corollary 5.4, applied to the remainder and \\(e\\), would enlarge it). So there are partial isometries \\(v_i\\) with \\(v_i^*v_i\\le e\\), mutually orthogonal final projections \\(v_iv_i^*\\), and \\(\\sum_iv_iv_i^*=1\\). Put \\(t_i=\\alpha(v_i)vv_i^*\\). Using \\(\\alpha(y)=vyv^*\\) for \\(y=v_i^*v_i\\in eMe\\) and \\(v_i=v_ie\\), one finds that \\(t_i\\) has initial projection \\(v_iv_i^*\\) and final projection \\(\\alpha(v_iv_i^*)\\). The initial projections are mutually orthogonal with sum \\(1\\). So are the final ones: \\(\\alpha\\) preserves orthogonality and the order of projections, hence least upper bounds (Proposition 3.1), so \\(\\sum_i\\alpha(v_iv_i^*)=\\alpha(1)=1\\). By the proof of [Lemma 3.3(3)](#oa-fnd-ty-02), \\(w=\\sum_it_i\\) converges strongly to a unitary in \\(M\\).\n\n*It implements \\(\\alpha\\).* As \\(v_i^*v_j=0\\) for \\(i\\ne j\\), \\[\n\\begin{gathered}\nwv_j\\\\\n=t_jv_j\\\\\n=\\alpha(v_j)vv_j^*v_j\\\\\n=\\alpha(v_j)\\alpha(v_j^*v_j)v\\\\\n=\\alpha(v_j)v.\n\\end{gathered}\n\\] For \\(x\\in M\\), \\(xv_j=\\sum_iv_iv_i^*xv_j\\) strongly, and \\(v_i^*xv_j\\in eMe\\), so\n\\[\n\\begin{gathered}\nwxv_j\\\\\n=\\sum_i\\alpha(v_i)v(v_i^*xv_j)\\\\\n=\\sum_i\\alpha(v_i)\\alpha(v_i^*xv_j)v\\\\\n=\\sum_i\\alpha(v_iv_i^*)\\,\\alpha(xv_j)v\\\\\n=\\alpha(xv_j)v\\\\\n=\\alpha(x)wv_j .\n\\end{gathered}\n\\]\nThe ranges of the \\(v_j\\) span a dense subspace of \\(H\\), so \\(wx=\\alpha(x)w\\), that is, \\(\\alpha(x)=wxw^*\\). No normality of \\(\\alpha\\) was used. \\(\\square\\)\n\n",
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      "id": "OA-FND-TY-09",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "6. Finite, infinite and abelian projections",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 6. Finite, infinite and abelian projections\n\nFiniteness is the analogue of finite dimension: a finite projection is not equivalent to a proper subprojection of itself.\n\n**Definition 6.1.** A projection \\(e\\in\\mathcal P(M)\\) is\n\n- *finite* if \\(e\\sim f\\le e\\) implies \\(f=e\\), and *infinite* otherwise;\n- *purely infinite* if \\(0\\) is its only finite subprojection in \\(M\\);\n- *properly infinite* if each nonzero central cut \\(ze\\) (\\(z\\) a central projection) is infinite;\n- *abelian* if \\(eMe\\) is commutative.\n\nThe same four words are applied to \\(M\\) itself when the projection \\(1\\) has the property. By these definitions, \\(0\\) is finite, abelian, properly infinite and purely infinite.\n\n**Lemma 6.2.**\n\n1. A subprojection of a finite (or abelian) projection is finite (or abelian). A projection equivalent to a finite (or abelian) projection is finite (or abelian). A projection that is subequivalent to a finite projection is finite.\n2. Abelian projections are finite. Minimal projections, meaning nonzero projections whose only subprojections are \\(0\\) and themselves, are abelian.\n3. \\(e\\) is finite (or abelian) in \\(M\\) if and only if \\(eMe\\) is a finite (or commutative) algebra. A nonzero central cut \\(ze\\) of a properly infinite \\(e\\) is properly infinite, and a central cut of \\(e\\) that is finite is \\(0\\). A purely infinite projection is properly infinite.\n4. If \\(e\\) is abelian, then \\(eMe=Ze\\), and every subprojection \\(f\\) of \\(e\\) satisfies \\(f=c(f)e\\).\n\n**Proof.** (1) Let \\(e\\) be finite, \\(f\\le e\\), and \\(f\\sim f_1\\le f\\). Additivity gives \\(e=(e-f)+f\\sim(e-f)+f_1\\le e\\), so \\((e-f)+f_1=e\\) and \\(f_1=f\\). Next let \\(v\\) implement \\(e\\sim g\\), and let \\(g\\sim g_1\\le g\\). The operator \\(v^*g_1\\) has initial projection \\(g_1vv^*g_1=g_1\\) and final projection \\(v^*g_1v\\le v^*gv=e\\). So \\(e\\sim g\\sim g_1\\sim v^*g_1v\\le e\\). As \\(e\\) is finite, \\(v^*g_1v=e\\), and then \\(g_1=vv^*g_1vv^*=vev^*=g\\). So \\(g\\) is finite. If \\(e\\sim f_1\\le f\\) with \\(f\\) finite, then \\(f_1\\) is finite, and so is \\(e\\). For abelian projections: if \\(f\\le e\\), the elements of \\(fMf\\) lie in \\(eMe\\), so they commute; and if \\(u\\) implements \\(e\\sim g\\), then \\(y\\mapsto uyu^*\\) is an isomorphism of \\(eMe\\) onto \\(gMg\\), with inverse \\(y\\mapsto u^*yu\\).\n\n(2) Let \\(e\\) be abelian and let \\(u\\) implement \\(e\\sim f\\le e\\). Then \\(u=fue\\in eMe\\), so \\(u^*u=uu^*\\), that is, \\(e=f\\). Let \\(e\\) be minimal. The algebra \\(eMe\\) on \\(eH\\) is a von Neumann algebra (Fact 2.1) whose only projections are \\(0\\) and \\(e\\). The spectral projections of a self-adjoint element of \\(eMe\\) lie in \\(eMe\\) (Fact 2.8), so each self-adjoint element is a real multiple of \\(e\\). Hence \\(eMe=\\mathbb Ce\\).\n\n(3) A partial isometry \\(u\\) with \\(u^*u=e\\) and \\(uu^*\\le e\\) satisfies \\(u=uu^*u\\in eMe\\). So finiteness of \\(e\\) means the same in \\(M\\) and in \\(eMe\\), and so does commutativity. If \\(e\\) is properly infinite and \\(z,z'\\) are central with \\(z'(ze)\\ne0\\), then \\(z'ze=(z'z)e\\) is infinite. A finite nonzero central cut \\(ze\\) would contradict the definition. If \\(e\\) is purely infinite and \\(ze\\ne0\\), then \\(ze\\) is a nonzero subprojection of \\(e\\), hence infinite.\n\n(4) \\(eMe\\) is a commutative von Neumann algebra on \\(eH\\), so it is its own centre, which is \\(Ze\\) by [Proposition 3.5(4)](#oa-fnd-ty-02). A subprojection \\(f\\) of \\(e\\) lies in \\(eMe\\), so \\(f=ze\\) for a central projection \\(z\\), again by [Proposition 3.5(4)](#oa-fnd-ty-02). Then \\(c(f)=zc(e)\\) by [Proposition 3.5(1)](#oa-fnd-ty-02), and \\(c(f)e=zc(e)e=ze=f\\). \\(\\square\\)\n\n**Lemma 6.3** (Centrally orthogonal sums). The sum of a family of mutually centrally orthogonal abelian (or finite) projections is abelian (or finite).\n\n**Proof.** Let \\(e=\\sum_ie_i\\) with \\(c(e_i)c(e_j)=0\\) for \\(i\\ne j\\), and write \\(c_i=c(e_i)\\). Then \\(c_ie=e_i\\), because \\(c_ie_j=c_ic_je_j=0\\) for \\(j\\ne i\\). For \\(x\\in M\\) and \\(i\\ne j\\), \\(e_ixe_j=e_ic_i\\,x\\,c_je_j=e_ixe_jc_ic_j=0\\). Hence \\(exe=\\sum_ie_ixe_i\\), a strong sum. If every \\(e_i\\) is abelian, then for \\(x,y\\in M\\), \\((exe)(eye)=\\sum_i(e_ixe_i)(e_iye_i)\\), and the terms commute inside \\(e_iMe_i\\); so \\(eMe\\) is commutative. If every \\(e_i\\) is finite and \\(e\\sim f\\le e\\), then \\(c_ie\\sim c_if\\le c_ie\\) by [Lemma 3.3(4)](#oa-fnd-ty-02), that is, \\(e_i\\sim c_if\\le e_i\\). So \\(c_if=e_i\\). As \\(f\\le e\\le\\sum_ic_i\\), we get \\(f=\\sum_ic_if=\\sum_ie_i=e\\). \\(\\square\\)\n\nFor a central decomposition \\(1=\\sum_i z_i\\), the map \\(x\\mapsto(xz_i)_i\\) identifies \\(M\\) with the algebra of bounded families \\(x_i\\in Mz_i\\), with norm \\(\\sup_i\\|x_i\\|\\). Indeed, the finite sums of such a family converge strongly: for each \\(\\xi\\), their orthogonal components have total squared norm at most \\((\\sup_i\\|x_i\\|)^2\\|\\xi\\|^2\\). Their bounded strong limit belongs to \\(M\\), has the prescribed central components, and has that supremum norm. Thus algebra direct sums \\(\\bigoplus_i Mz_i\\) below mean this bounded product, acting on the Hilbert direct sum \\(\\bigoplus_i z_iH\\).\n\n",
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      "id": "OA-FND-TY-10",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "7. The type decomposition",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 7. The type decomposition\n\n**Definition 7.1** (Types). \\(M\\) is\n\n- *of type I* if below each nonzero central projection there is a nonzero abelian projection;\n- *of type II* if \\(0\\) is its only abelian projection, while below each nonzero central projection there is a nonzero finite projection;\n- *of type III* if \\(M\\) is purely infinite: \\(0\\) is its only finite projection;\n- *of type II\\(_1\\)* if it is of type II and finite, and *of type II\\(_\\infty\\)* if it is of type II and \\(0\\) is its only finite central projection;\n- *semifinite* if it has no nonzero central summand of type III.\n\nFor a central projection \\(z\\), the projections of the central summand \\(Mz\\) (on \\(zH\\)) are the projections of \\(M\\) below \\(z\\), its central projections are the central projections of \\(M\\) below \\(z\\), and equivalence among them is the same as in \\(M\\). So each type passes to central summands. Each type is also invariant under isomorphism.\n\n**Theorem 7.2** (The type decomposition). There are unique mutually orthogonal central projections \\(z_{\\rm I}\\), \\(z_{{\\rm II}_1}\\), \\(z_{{\\rm II}_\\infty}\\), \\(z_{\\rm III}\\) with sum \\(1\\) such that \\(Mz_{\\rm I}\\), \\(Mz_{{\\rm II}_1}\\), \\(Mz_{{\\rm II}_\\infty}\\) and \\(Mz_{\\rm III}\\) are of types I, II\\(_1\\), II\\(_\\infty\\) and III. Every projection \\(e\\in M\\) can be written in exactly one way as \\(e=e_1+e_2\\) with \\(e_1\\) and \\(e_2\\) centrally orthogonal, \\(e_1\\) finite and \\(e_2\\) properly infinite.\n\n**Proof.** *Existence.* Zorn's lemma gives a maximal family \\(\\{a_k\\}\\) of mutually centrally orthogonal nonzero abelian projections. Their sum \\(a\\) is abelian (Lemma 6.3); put \\(z_{\\rm I}=c(a)\\). If \\(z\\le z_{\\rm I}\\) is central and nonzero, then \\(za\\ne0\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)), and \\(za\\) is abelian. So \\(Mz_{\\rm I}\\) is of type I. The summand \\(M(1-z_{\\rm I})\\) has no nonzero abelian projection \\(q\\): such a \\(q\\) would have \\(c(q)\\le1-z_{\\rm I}\\), so it would be centrally orthogonal to every \\(a_k\\), against maximality.\n\nNext take a maximal family \\(\\{b_k\\}\\) of mutually centrally orthogonal nonzero finite projections below \\(1-z_{\\rm I}\\). Their sum \\(b\\) is finite (Lemma 6.3); put \\(z_{\\rm II}=c(b)\\le1-z_{\\rm I}\\). The summand \\(Mz_{\\rm II}\\) has no nonzero abelian projection, and each nonzero central \\(z\\le z_{\\rm II}\\) majorizes the nonzero finite projection \\(zb\\). So \\(Mz_{\\rm II}\\) is of type II. Put \\(z_{\\rm III}=1-z_{\\rm I}-z_{\\rm II}\\). A nonzero finite projection below \\(z_{\\rm III}\\) would be centrally orthogonal to all \\(b_k\\), against maximality. So \\(Mz_{\\rm III}\\) is of type III.\n\nFinally take a maximal family \\(\\{c_k\\}\\) of mutually orthogonal nonzero finite central projections below \\(z_{\\rm II}\\). Orthogonal central projections are centrally orthogonal, so \\(z_{{\\rm II}_1}=\\sum_kc_k\\) is finite (Lemma 6.3), and \\(Mz_{{\\rm II}_1}\\) is of type II\\(_1\\). By maximality, \\(Mz_{{\\rm II}_\\infty}\\), with \\(z_{{\\rm II}_\\infty}=z_{\\rm II}-z_{{\\rm II}_1}\\), has no nonzero finite central projection, so it is of type II\\(_\\infty\\).\n\n*Uniqueness.* A nonzero algebra of type I has a nonzero abelian projection, and one of type II or III has none. A nonzero algebra of type I or II has a nonzero finite projection, and one of type III has none. Let \\(z'_{\\rm I},z'_{{\\rm II}_1},z'_{{\\rm II}_\\infty},z'_{\\rm III}\\) be a second family as in the theorem. The algebra \\(Mz'_{\\rm I}(1-z_{\\rm I})\\) is a central summand both of \\(Mz'_{\\rm I}\\) and of \\(M(z_{\\rm II}+z_{\\rm III})\\), so it is \\(0\\). Hence \\(z'_{\\rm I}\\le z_{\\rm I}\\), and by symmetry \\(z'_{\\rm I}=z_{\\rm I}\\). Next, \\(Mz'_{\\rm III}z_{\\rm II}\\) is a summand of an algebra of type III and of one of type II, so it is \\(0\\); and \\(z'_{\\rm III}z_{\\rm I}=z'_{\\rm III}z'_{\\rm I}=0\\). So \\(z'_{\\rm III}\\le z_{\\rm III}\\), and by symmetry they are equal. Then \\(z'_{{\\rm II}_1}+z'_{{\\rm II}_\\infty}=z_{{\\rm II}_1}+z_{{\\rm II}_\\infty}\\). The central projection \\(z'_{{\\rm II}_1}z_{{\\rm II}_\\infty}\\) lies below the finite projection \\(z'_{{\\rm II}_1}\\), so it is finite, and it lies in a summand of type II\\(_\\infty\\); so it is \\(0\\). Hence \\(z'_{{\\rm II}_1}\\le z_{{\\rm II}_1}\\). By symmetry they are equal, and then \\(z'_{{\\rm II}_\\infty}=z_{{\\rm II}_\\infty}\\).\n\n*Projections.* Let \\(e\\in\\mathcal P(M)\\). Take a maximal family \\(\\{c_k\\}\\) of mutually orthogonal central projections below \\(c(e)\\) such that each \\(c_ke\\) is nonzero and finite. Put \\(c=\\sum_kc_k\\), \\(e_1=ce\\) and \\(e_2=(1-c)e\\). The projection \\(e_1=\\sum_kc_ke\\) is finite by Lemma 6.3. It is centrally orthogonal to \\(e_2\\), since \\(c(e_1)\\le c\\) and \\(c(e_2)\\le1-c\\). If \\(z\\) is central and \\(ze_2\\) is finite and nonzero, then \\(z'=z(1-c)c(e)\\) is a central projection below \\(c(e)\\), orthogonal to \\(c\\), and \\(z'e=ze_2\\) is finite and nonzero, against maximality. So \\(e_2\\) is properly infinite. For uniqueness, let \\(e=e_1'+e_2'\\) be a second such decomposition and put \\(a=c(e_1')\\). Then \\(e_1'=ae\\), since \\(ae=e_1'+ae_2'\\) and \\(ae_2'=ac(e_2')e_2'=0\\); and \\(e_2'=(1-a)e\\). The projection \\(a(1-c)e\\) equals \\((1-c)e_1'\\le e_1'\\), so it is finite; it also equals \\(ae_2\\), a central cut of the properly infinite \\(e_2\\); so it is \\(0\\) (Lemma 6.2(3)). Likewise \\(c(1-a)e=(1-a)e_1\\) is finite and is a central cut of \\(e_2'\\), so it is \\(0\\). Hence \\(ae=ace=ce\\), that is, \\(e_1'=e_1\\). \\(\\square\\)\n\n**Corollary 7.3.** A nonzero factor is of exactly one of the types I, II\\(_1\\), II\\(_\\infty\\), III.\n\n**Proof.** Its only central projections are \\(0\\) and \\(1\\), so exactly one of the four projections of Theorem 7.2 equals \\(1\\). By the facts listed at the start of the uniqueness step in that proof, a nonzero algebra cannot be of two of these types. \\(\\square\\)\n\n**Lemma 7.4** (Good projections).\n\n1. In a type I algebra, every nonzero projection majorizes a nonzero abelian projection, and some abelian projection has central support \\(1\\).\n2. In a semifinite algebra, below each nonzero central projection there is a nonzero finite projection, and some finite projection has central support \\(1\\).\n\n**Proof.** (1) Let \\(q\\ne0\\). The central projection \\(c(q)\\) majorizes a nonzero abelian projection \\(e_1\\), and \\(c(e_1)\\le c(q)\\). Corollary 5.4 gives nonzero \\(e_2\\le e_1\\) and \\(q_1\\le q\\) with \\(e_2\\sim q_1\\), and \\(q_1\\) is abelian by Lemma 6.2(1). For the second claim, take a maximal family \\(\\{e_k\\}\\) of mutually centrally orthogonal nonzero abelian projections. Their sum \\(e\\) is abelian (Lemma 6.3), and \\(c(e)=\\bigvee_kc(e_k)\\), because a central projection majorizes \\(e\\) iff it majorizes every \\(e_k\\). If \\(c(e)\\ne1\\), the central projection \\(1-c(e)\\) majorizes a nonzero abelian projection, which is centrally orthogonal to every \\(e_k\\), against maximality. (2) If \\(M\\) is semifinite, then \\(z_{\\rm III}=0\\) in Theorem 7.2. A nonzero central \\(z\\) has \\(zz_{\\rm I}\\ne0\\) or \\(zz_{\\rm II}\\ne0\\). In the first case it majorizes a nonzero abelian projection, which is finite (Lemma 6.2(2)); in the second case it majorizes a nonzero finite projection by the definition of type II. The rest is the argument of (1), with Lemma 6.3 for finite projections. \\(\\square\\)\n\n**Example 7.5** (\\(B(H)\\)). Assume \\(H\\ne0\\); the zero algebra has only its zero projection and is covered separately by the conventions. Here \\(M'=\\mathbb C1\\). Two projections are equivalent iff their ranges have the same Hilbert dimension, since a partial isometry is a unitary between its initial and final spaces. A projection is finite iff it has finite rank: an infinite-dimensional \\(eH\\) is unitarily equivalent to a proper subspace of itself (shift a countable part of an orthonormal basis), while in finite dimension \\(e\\sim f\\le e\\) forces \\(f=e\\). The abelian projections are those of rank at most \\(1\\), and \\(B(H)\\) is of type I\\(_{\\dim H}\\) in the sense of Definition 10.1.\n\n**Example 7.6** (Atomic algebras). For \\(H\\ne0\\), \\(B(H)\\) is a type I factor: its commutant is \\(\\mathbb C1\\), and a rank-one projection is minimal, hence abelian (Lemma 6.2(2)). More generally, call \\(M\\) *atomic* if every nonzero projection majorizes a minimal projection. An atomic algebra is of type I, because minimal projections are abelian. Equivalently, \\(1\\) is a sum of minimal projections. Indeed, if \\(M\\) is atomic, a maximal orthogonal family of minimal projections has sum \\(1\\). Conversely, let \\(1=\\sum_ip_i\\) with minimal \\(p_i\\), and let \\(e\\ne0\\). Then \\(ep_i\\ne0\\) for some \\(i\\), and Lemma 5.3 gives nonzero \\(e_1\\le e\\) and \\(f_1\\le p_i\\) with \\(e_1\\sim f_1\\). Then \\(f_1=p_i\\). If \\(u\\) implements \\(p_i\\sim e_1\\) and \\(g\\le e_1\\), then \\(u^*gu\\le p_i\\) is \\(0\\) or \\(p_i\\), so \\(g=u(u^*gu)u^*\\) is \\(0\\) or \\(e_1\\): the projection \\(e_1\\) is minimal.\n\n**Example 7.7** (Commutative algebras). In a commutative \\(A\\), \\(e\\sim f\\) iff \\(e=f\\) (since \\(u^*u=uu^*\\)), every projection is abelian and finite, \\(c(e)=e\\), and \\(A\\) is of type I\\(_1\\) in the sense of Definition 10.1.\n\n",
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      "id": "OA-FND-TY-11",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "8. Matrix units and tensor splitting",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
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      "full_conditions_and_proof": "## 8. Matrix units and tensor splitting\n\nIf the unit of \\(M\\) is a sum of mutually equivalent projections, then \\(M\\) is a matrix algebra over one of its corners. This section makes that precise. We first describe operators on \\(K\\otimes\\ell^2(I)\\) by matrices.\n\nLet \\(N\\) be a von Neumann algebra on \\(K\\) and \\(I\\) a nonempty set. On \\(K\\otimes\\ell^2(I)\\) put \\(V_i\\zeta=\\zeta\\otimes\\delta_i\\). The *entries* of \\(T\\in B(K\\otimes\\ell^2(I))\\) are \\(T_{ij}=V_i^*TV_j\\in B(K)\\). An operator is determined by its entries, \\((T^*)_{ij}=(T_{ji})^*\\), and \\((ST)_{ij}=\\sum_kS_{ik}T_{kj}\\), the sum converging strongly. Put\n\\[\n\\begin{gathered}\n\\mathbb M_I(N)\\\\\n=\\{T\\in B(K\\otimes\\ell^2(I)):\\\\\n\\ T_{ij}\\in N\\ \\text{for all }i,j\\},\\\\\nN\\otimes1\\\\\n=\\{x\\otimes1:\\ x\\in N\\}.\n\\end{gathered}\n\\tag{8.1}\n\\]\nThe operator \\(x\\otimes1\\) has entries \\(\\delta_{ij}x\\), and \\(E_{ij}\\) are the matrix units of \\(B(\\ell^2(I))\\). We write \\(\\mathbb M_n(N)\\) when \\(I=\\{1,\\ldots,n\\}\\), and \\(\\mathbb M_\\alpha(N)\\) when only the cardinal \\(\\alpha=|I|\\) matters; \\(\\ell^2(\\alpha)\\) is \\(\\ell^2(I)\\) for a set of that cardinality.\n\n**Lemma 8.2** (Matrix algebras).\n\n1. \\((N\\otimes1)'=\\mathbb M_I(N')\\).\n2. \\(\\mathbb M_I(N)'=N'\\otimes1\\).\n3. \\(\\mathbb M_I(N)\\) and \\(N\\otimes1\\) are von Neumann algebras, and \\(N\\otimes1\\) together with \\(1\\otimes B(\\ell^2(I))\\) generates \\(\\mathbb M_I(N)\\) as a von Neumann algebra. We also write \\(N\\bar\\otimes B(\\ell^2(I))\\) for it.\n4. The centre of \\(\\mathbb M_I(N)\\) is \\((N\\cap N')\\otimes1\\).\n\n**Proof.** (1) \\((T(x\\otimes1))_{ij}=T_{ij}x\\) and \\(((x\\otimes1)T)_{ij}=xT_{ij}\\). So \\(T\\) commutes with \\(N\\otimes1\\) iff all its entries lie in \\(N'\\).\n\n(2) The operators \\(1\\otimes E_{kl}\\) lie in \\(\\mathbb M_I(N)\\), since their entries are \\(0\\) or \\(1\\). Suppose \\(S\\) commutes with all of them. Comparing entries, \\((S(1\\otimes E_{kl}))_{ij}=\\delta_{lj}S_{ik}\\) and \\(((1\\otimes E_{kl})S)_{ij}=\\delta_{ik}S_{lj}\\). Taking \\(j=l\\) gives \\(S_{ik}=\\delta_{ik}S_{ll}\\) for all \\(i,k,l\\). So \\(S=s\\otimes1\\) for a single \\(s\\in B(K)\\). If \\(S\\) also commutes with every \\(x\\otimes1\\), \\(x\\in N\\), then \\(s\\in N'\\). Conversely, \\(s\\otimes1\\) with \\(s\\in N'\\) commutes with every \\(T\\in\\mathbb M_I(N)\\), entry by entry.\n\n(3) By (2), and then (1) for \\(N'\\): \\(\\mathbb M_I(N)''=(N'\\otimes1)'=\\mathbb M_I(N'')=\\mathbb M_I(N)\\). By (1), and then (2) for \\(N'\\): \\((N\\otimes1)''=\\mathbb M_I(N')'=N''\\otimes1=N\\otimes1\\). The von Neumann algebra \\(R\\) generated by \\(N\\otimes1\\) and \\(1\\otimes B(\\ell^2(I))\\) lies in \\(\\mathbb M_I(N)\\). By the computation in (2), the commutant of \\(1\\otimes B(\\ell^2(I))\\) is \\(B(K)\\otimes1\\), so \\(R'=\\mathbb M_I(N')\\cap(B(K)\\otimes1)=N'\\otimes1\\), and \\(R=R''=\\mathbb M_I(N)\\).\n\n(4) The centre is \\(\\mathbb M_I(N)\\cap(N'\\otimes1)=\\{s\\otimes1:\\ s\\in N\\cap N'\\}\\). \\(\\square\\)\n\n**Definition 8.3** (Matrix units). A *matrix unit* in \\(M\\) indexed by \\(I\\) is a family \\(\\{w_{ij}\\}_{i,j\\in I}\\subseteq M\\) with \\(w_{ij}^*=w_{ji}\\), \\(w_{ij}w_{kl}=\\delta_{jk}w_{il}\\), and \\(\\sum_iw_{ii}=1\\).\n\n**Proposition 8.4** (Splitting off a matrix algebra). Let \\(\\{e_i\\}_{i\\in I}\\) be mutually orthogonal, mutually equivalent projections with \\(\\sum_ie_i=1\\). Fix \\(i_0\\in I\\), put \\(e=e_{i_0}\\), and choose \\(u_i\\) implementing \\(e\\sim e_i\\), with \\(u_{i_0}=e\\). Then \\(w_{ij}=u_iu_j^*\\) is a matrix unit, and \\(W:eH\\otimes\\ell^2(I)\\to H\\), \\(W(\\zeta\\otimes\\delta_i)=u_i\\zeta\\), is a unitary with\n\\[\n\\begin{gathered}\nW^*MW\\\\\n=\\mathbb M_I(eMe)\\\\\n=eMe\\bar\\otimes B(\\ell^2(I)),\\\\\nW^*M'W\\\\\n=M'e\\otimes1 .\n\\end{gathered}\n\\]\nSo \\(\\{M,H\\}\\cong\\{eMe,eH\\}\\otimes\\{B(\\ell^2(I)),\\ell^2(I)\\}\\).\n\n**Proof.** We have \\(u_j^*u_k=u_j^*e_je_ku_k=0\\) for \\(j\\ne k\\), \\(u_j^*u_j=e\\), and \\(u_ie=u_i\\). Hence \\(w_{ij}^*=w_{ji}\\), \\(w_{ij}w_{kl}=u_iu_j^*u_ku_l^*=\\delta_{jk}u_ieu_l^*=\\delta_{jk}w_{il}\\), and \\(w_{ii}=e_i\\). For \\(\\zeta,\\zeta'\\in eH\\), \\(\\langle u_i\\zeta,u_k\\zeta'\\rangle=\\langle u_k^*u_i\\zeta,\\zeta'\\rangle=\\delta_{ik}\\langle\\zeta,\\zeta'\\rangle\\), so \\(W\\) is isometric. Its range contains every \\(u_i(eH)=e_iH\\), so \\(W\\) is onto. Since \\(W^*\\eta=\\sum_iu_i^*\\eta\\otimes\\delta_i\\), the entries of \\(W^*xW\\) are \\(u_i^*xu_j|_{eH}\\). For \\(x\\in M\\) they lie in \\(eMe\\), so \\(W^*MW\\subseteq\\mathbb M_I(eMe)\\). For \\(x'\\in M'\\) they are \\(x'u_i^*u_j|_{eH}=\\delta_{ij}x'e|_{eH}\\), so \\(W^*M'W=M'e\\otimes1\\). By Fact 2.1, \\(M'e=(eMe)'\\) on \\(eH\\), and by Lemma 8.2(2), \\(W^*M'W=(eMe)'\\otimes1=\\mathbb M_I(eMe)'\\). Hence \\[\n\\begin{gathered}\nW^*MW\\\\\n=(W^*M'W)'\\\\\n=\\mathbb M_I(eMe)''\\\\\n=\\mathbb M_I(eMe),\n\\end{gathered}\n\\] where the last step is Lemma 8.2(3). \\(\\square\\)\n\n**Lemma 8.5** (Equivalent projections have isomorphic corners). If \\(u\\) implements \\(e\\sim f\\), then \\(x\\mapsto uxu^*\\) is an isomorphism of \\(eMe\\) onto \\(fMf\\), implemented by the unitary \\(u|_{eH}:eH\\to fH\\). So \\(\\{eMe,eH\\}\\cong\\{fMf,fH\\}\\).\n\n**Proof.** \\(u(eMe)u^*\\subseteq fMf\\) and \\(u^*(fMf)u\\subseteq eMe\\), and the two maps are inverse to each other. For \\(\\zeta\\in eH\\) and \\(x\\in eMe\\), \\((uxu^*)(u\\zeta)=ux\\zeta\\). \\(\\square\\)\n\n",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "9. Comparing and counting abelian projections",
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      "full_conditions_and_proof": "## 9. Comparing and counting abelian projections\n\nAn abelian projection is subequivalent to every projection whose central support is at least as large. So two abelian projections with the same central support are equivalent, and the number of such projections needed to fill a central projection is well defined. Matrix algebras over commutative algebras supply the examples and the counting tool.\n\n**Proposition 9.1** (Matrices over a commutative algebra). Let \\(A\\) be a commutative von Neumann algebra acting on \\(K\\).\n\n1. For every nonzero Hilbert space \\(L\\), \\(A\\bar\\otimes B(L)\\) is of type I. For a rank-one projection \\(q\\) on \\(L\\), \\(1\\otimes q\\) is abelian with central support \\(1\\).\n2. For finite \\(n\\), the matrix algebra \\(\\mathbb M_n(A)\\) of (8.1) is finite and of type I. The linear map \\(\\Phi(x)=\\frac1n\\sum_ix_{ii}\\), with values in \\(A\\) (identified with the diagonal copy \\(A\\otimes1\\)), satisfies\n\\[\n\\begin{gathered}\n\\Phi(a)\\\\\n=a,\\\\\n\\Phi(axb)\\\\\n=a\\Phi(x)b\\ \\ (a,b\\in A),\\\\\n\\Phi(x^*x)\\\\\n=\\Phi(xx^*)\\\\\n\\ge0,\n\\end{gathered}\n\\]\nand \\(\\Phi(x^*x)=0\\) only if \\(x=0\\).\n\n**Proof.** (1) Choose an orthonormal basis of \\(L\\) that contains a unit vector of \\(qL\\); then \\(L=\\ell^2(I)\\) and \\(q=E_{i_0i_0}\\). For \\(T\\in\\mathbb M_I(A)\\), \\((1\\otimes q)T(1\\otimes q)=T_{i_0i_0}\\otimes q\\), so \\((1\\otimes q)\\mathbb M_I(A)(1\\otimes q)=A\\otimes q\\) is commutative. The centre is \\(A\\otimes1\\) by Lemma 8.2(4). If \\((a\\otimes1)(1\\otimes q)=a\\otimes q=0\\), then \\(a=0\\); so no nonzero central projection is orthogonal to \\(1\\otimes q\\), and \\(c(1\\otimes q)=1\\). Every nonzero central projection \\(z\\) therefore majorizes the nonzero abelian projection \\(z(1\\otimes q)\\).\n\n(2) \\(\\mathbb M_n(A)\\) is of type I by (1). The first two properties hold because \\(a\\in A\\) acts as \\(\\mathrm{diag}(a,\\ldots,a)\\) and \\(A\\) is commutative, so \\((axb)_{ii}=ax_{ii}b\\). Next, \\(n\\Phi(x^*x)=\\sum_{i,k}x_{ki}^*x_{ki}\\) and \\(n\\Phi(xx^*)=\\sum_{i,k}x_{ik}x_{ik}^*\\). These agree, since every entry is a normal operator (\\(A\\) is commutative), and they are positive. If \\(\\Phi(x^*x)=0\\), each \\(x_{ki}^*x_{ki}\\) vanishes, so \\(x=0\\). Finally, if \\(v^*v=1\\) and \\(vv^*=f\\), then \\(\\Phi(1-f)=\\Phi(v^*v)-\\Phi(vv^*)=0\\) and \\(1-f=(1-f)^*(1-f)\\), so \\(f=1\\): the algebra is finite. \\(\\square\\)\n\n**Definition 9.2** (Algebra-valued traces). Let \\(A\\) be a \\(*\\)-subalgebra of \\(M\\). An *\\(A\\)-valued trace* on \\(M\\) is a linear map \\(\\Phi:M\\to A\\) with \\(\\Phi(a)=a\\), \\(\\Phi(axb)=a\\Phi(x)b\\) for \\(a,b\\in A\\), and \\(\\Phi(x^*x)=\\Phi(xx^*)\\ge0\\) for \\(x\\in M\\). It is *faithful* if \\(\\Phi(x^*x)=0\\) only for \\(x=0\\). When \\(A\\) is the centre of \\(M\\), \\(\\Phi\\) is a *centre-valued trace*.\n\nAn \\(A\\)-valued trace is positive, since every positive element of \\(M\\) has the form \\(x^*x\\); hence it is monotone. It is constant on equivalence classes: if \\(u\\) implements \\(e\\sim f\\), then \\(\\Phi(e)=\\Phi(u^*u)=\\Phi(uu^*)=\\Phi(f)\\). Centre-valued traces on finite algebras are the subject of the lesson Traces on von Neumann algebras.\n\n**Lemma 9.3** (Comparing abelian projections). Let \\(e\\) be abelian and \\(f\\in\\mathcal P(M)\\) with \\(c(e)\\le c(f)\\) (equivalently, \\(e\\le c(f)\\)). Then \\(e\\precsim f\\). If \\(f\\) is also abelian and \\(c(e)=c(f)\\), then \\(e\\sim f\\).\n\nThe direction of subequivalence matters: Remark 9.4 gives the rank-one matrix test. \n\n**Proof.** First, if \\(f\\precsim e\\) and \\(c(e)\\le c(f)\\), then \\(f\\sim e\\). Indeed \\(f\\sim e_1\\le e\\), where \\(e_1=c(e_1)e\\) by Lemma 6.2(4) and \\(c(e_1)=c(f)\\ge c(e)\\) by [Proposition 3.5(2)](#oa-fnd-ty-02); so \\(e_1\\ge c(e)e=e\\), and \\(e_1=e\\). Now take \\(z\\) from the comparison theorem (Theorem 5.5) with \\(ze\\precsim zf\\) and \\((1-z)f\\precsim(1-z)e\\). The projection \\((1-z)e\\) is abelian (Lemma 6.2(1)), and by Proposition 3.5(1), \\[\n\\begin{gathered}\nc((1-z)e)\\\\\n=(1-z)c(e)\\\\\n\\le(1-z)c(f)\\\\\n=c((1-z)f).\n\\end{gathered}\n\\] By the first step, \\((1-z)f\\sim(1-z)e\\). Adding (Lemma 3.3(3)), \\(e=ze+(1-z)e\\precsim zf+(1-z)f=f\\). If \\(f\\) is abelian with \\(c(f)=c(e)\\), exchange the roles and use Proposition 5.1. \\(\\square\\)\n\n**Remark 9.4.** The relation in Lemma 9.3 cannot be reversed. In \\(M=M_2(\\mathbb C)\\), the projection \\(e=E_{11}\\) is abelian and \\(c(e)=1=c(1)\\), but \\(1\\precsim E_{11}\\) fails, since \\(1\\) is not equivalent to any projection of rank one.\n\n**Lemma 9.5** (Counting abelian projections). Let \\(c\\ne0\\) be a central projection. Let \\(\\{e_i\\}_{i\\in I}\\) be mutually orthogonal abelian projections with \\(c(e_i)=c\\) and \\(\\sum_ie_i=c\\). Let \\(\\{f_j\\}_{j\\in J}\\) be mutually orthogonal abelian projections with \\(c(f_j)=c\\) and \\(\\sum_jf_j\\le c\\). Then \\(|J|\\le|I|\\). Consequently, if also \\(\\sum_jf_j=c\\), then \\(|I|=|J|\\).\n\n**Proof.** Replacing \\(M\\) by \\(Mc\\), we may assume \\(c=1\\). By Lemma 9.3, all the \\(e_i\\) and \\(f_j\\) are equivalent to one another.\n\n*\\(I\\) finite, \\(|I|=n\\).* By Proposition 8.4, \\(W^*MW=\\mathbb M_n(A)\\) with \\(A=e_{i_0}Me_{i_0}\\) commutative, and \\(W^*e_iW=1\\otimes E_{ii}\\). Carry the trace \\(\\Phi\\) of Proposition 9.1(2) over to \\(M\\). Then \\(\\Phi(e_i)=\\frac1n1\\), and \\(\\Phi(f_j)=\\Phi(e_{i_0})=\\frac1n1\\) because \\(f_j\\sim e_{i_0}\\). For a finite set \\(F\\subseteq J\\), \\(\\sum_{j\\in F}f_j\\le1\\), and \\(\\Phi\\) is monotone, so \\(\\frac{|F|}n1\\le1\\) and \\(|F|\\le n\\). Hence \\(|J|\\le n\\).\n\n*\\(I\\) infinite.* Fix \\(i_0\\), a unit vector \\(\\zeta\\in e_{i_0}H\\), and partial isometries \\(v_i\\) implementing \\(e_{i_0}\\sim e_i\\). Put \\(\\varphi_i=\\omega_{v_i\\zeta}\\) and \\(z_0=c(p_\\zeta)\\), which is nonzero because \\(p_\\zeta\\zeta=\\zeta\\ne0\\). Since \\(p_\\zeta\\le e_{i_0}\\) ([Lemma 4.6](#oa-fnd-ty-07)) and \\(e_{i_0}\\) is abelian, \\(p_\\zeta=z_0e_{i_0}\\) (Lemma 6.2(4)). Suppose \\(\\varphi_i(f_j)=\\|f_jv_i\\zeta\\|^2=0\\). Then the operator \\(f_jv_i\\in M\\) vanishes on \\([M'\\zeta]=p_\\zeta H\\), so \\(f_jv_iz_0e_{i_0}=0\\). As \\(v_i=v_ie_{i_0}\\), this gives \\(z_0f_jv_i=0\\), and so \\(z_0f_je_i=z_0f_jv_iv_i^*=0\\). Now fix \\(j\\). Since \\(c(f_j)=1\\), we have \\(z_0f_j\\ne0\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)), while \\(\\sum_iz_0f_je_i=z_0f_j\\). So \\(z_0f_je_i\\ne0\\), and hence \\(\\varphi_i(f_j)>0\\), for some \\(i\\). Thus \\(J=\\bigcup_iJ_i\\) with \\(J_i=\\{j:\\varphi_i(f_j)>0\\}\\). Each \\(J_i\\) is countable, because \\(\\sum_{j\\in F}\\varphi_i(f_j)=\\varphi_i(\\sum_{j\\in F}f_j)\\le1\\) for every finite \\(F\\subseteq J\\). Therefore \\(|J|\\le|I|\\cdot\\aleph_0=|I|\\) (Fact 2.9).\n\nThe last statement follows by symmetry and the Cantor–Schröder–Bernstein theorem for cardinals. \\(\\square\\)\n\n",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "9. Comparing and counting abelian projections",
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      "full_conditions_and_proof": "## 9. Comparing and counting abelian projections\n\nAn abelian projection is subequivalent to every projection whose central support is at least as large. So two abelian projections with the same central support are equivalent, and the number of such projections needed to fill a central projection is well defined. Matrix algebras over commutative algebras supply the examples and the counting tool.\n\n**Proposition 9.1** (Matrices over a commutative algebra). Let \\(A\\) be a commutative von Neumann algebra acting on \\(K\\).\n\n1. For every nonzero Hilbert space \\(L\\), \\(A\\bar\\otimes B(L)\\) is of type I. For a rank-one projection \\(q\\) on \\(L\\), \\(1\\otimes q\\) is abelian with central support \\(1\\).\n2. For finite \\(n\\), the matrix algebra \\(\\mathbb M_n(A)\\) of (8.1) is finite and of type I. The linear map \\(\\Phi(x)=\\frac1n\\sum_ix_{ii}\\), with values in \\(A\\) (identified with the diagonal copy \\(A\\otimes1\\)), satisfies\n\\[\n\\begin{gathered}\n\\Phi(a)\\\\\n=a,\\\\\n\\Phi(axb)\\\\\n=a\\Phi(x)b\\ \\ (a,b\\in A),\\\\\n\\Phi(x^*x)\\\\\n=\\Phi(xx^*)\\\\\n\\ge0,\n\\end{gathered}\n\\]\nand \\(\\Phi(x^*x)=0\\) only if \\(x=0\\).\n\n**Proof.** (1) Choose an orthonormal basis of \\(L\\) that contains a unit vector of \\(qL\\); then \\(L=\\ell^2(I)\\) and \\(q=E_{i_0i_0}\\). For \\(T\\in\\mathbb M_I(A)\\), \\((1\\otimes q)T(1\\otimes q)=T_{i_0i_0}\\otimes q\\), so \\((1\\otimes q)\\mathbb M_I(A)(1\\otimes q)=A\\otimes q\\) is commutative. The centre is \\(A\\otimes1\\) by Lemma 8.2(4). If \\((a\\otimes1)(1\\otimes q)=a\\otimes q=0\\), then \\(a=0\\); so no nonzero central projection is orthogonal to \\(1\\otimes q\\), and \\(c(1\\otimes q)=1\\). Every nonzero central projection \\(z\\) therefore majorizes the nonzero abelian projection \\(z(1\\otimes q)\\).\n\n(2) \\(\\mathbb M_n(A)\\) is of type I by (1). The first two properties hold because \\(a\\in A\\) acts as \\(\\mathrm{diag}(a,\\ldots,a)\\) and \\(A\\) is commutative, so \\((axb)_{ii}=ax_{ii}b\\). Next, \\(n\\Phi(x^*x)=\\sum_{i,k}x_{ki}^*x_{ki}\\) and \\(n\\Phi(xx^*)=\\sum_{i,k}x_{ik}x_{ik}^*\\). These agree, since every entry is a normal operator (\\(A\\) is commutative), and they are positive. If \\(\\Phi(x^*x)=0\\), each \\(x_{ki}^*x_{ki}\\) vanishes, so \\(x=0\\). Finally, if \\(v^*v=1\\) and \\(vv^*=f\\), then \\(\\Phi(1-f)=\\Phi(v^*v)-\\Phi(vv^*)=0\\) and \\(1-f=(1-f)^*(1-f)\\), so \\(f=1\\): the algebra is finite. \\(\\square\\)\n\n**Definition 9.2** (Algebra-valued traces). Let \\(A\\) be a \\(*\\)-subalgebra of \\(M\\). An *\\(A\\)-valued trace* on \\(M\\) is a linear map \\(\\Phi:M\\to A\\) with \\(\\Phi(a)=a\\), \\(\\Phi(axb)=a\\Phi(x)b\\) for \\(a,b\\in A\\), and \\(\\Phi(x^*x)=\\Phi(xx^*)\\ge0\\) for \\(x\\in M\\). It is *faithful* if \\(\\Phi(x^*x)=0\\) only for \\(x=0\\). When \\(A\\) is the centre of \\(M\\), \\(\\Phi\\) is a *centre-valued trace*.\n\nAn \\(A\\)-valued trace is positive, since every positive element of \\(M\\) has the form \\(x^*x\\); hence it is monotone. It is constant on equivalence classes: if \\(u\\) implements \\(e\\sim f\\), then \\(\\Phi(e)=\\Phi(u^*u)=\\Phi(uu^*)=\\Phi(f)\\). Centre-valued traces on finite algebras are the subject of the lesson Traces on von Neumann algebras.\n\n**Lemma 9.3** (Comparing abelian projections). Let \\(e\\) be abelian and \\(f\\in\\mathcal P(M)\\) with \\(c(e)\\le c(f)\\) (equivalently, \\(e\\le c(f)\\)). Then \\(e\\precsim f\\). If \\(f\\) is also abelian and \\(c(e)=c(f)\\), then \\(e\\sim f\\).\n\nThe direction of subequivalence matters: Remark 9.4 gives the rank-one matrix test. \n\n**Proof.** First, if \\(f\\precsim e\\) and \\(c(e)\\le c(f)\\), then \\(f\\sim e\\). Indeed \\(f\\sim e_1\\le e\\), where \\(e_1=c(e_1)e\\) by Lemma 6.2(4) and \\(c(e_1)=c(f)\\ge c(e)\\) by [Proposition 3.5(2)](#oa-fnd-ty-02); so \\(e_1\\ge c(e)e=e\\), and \\(e_1=e\\). Now take \\(z\\) from the comparison theorem (Theorem 5.5) with \\(ze\\precsim zf\\) and \\((1-z)f\\precsim(1-z)e\\). The projection \\((1-z)e\\) is abelian (Lemma 6.2(1)), and by Proposition 3.5(1), \\[\n\\begin{gathered}\nc((1-z)e)\\\\\n=(1-z)c(e)\\\\\n\\le(1-z)c(f)\\\\\n=c((1-z)f).\n\\end{gathered}\n\\] By the first step, \\((1-z)f\\sim(1-z)e\\). Adding (Lemma 3.3(3)), \\(e=ze+(1-z)e\\precsim zf+(1-z)f=f\\). If \\(f\\) is abelian with \\(c(f)=c(e)\\), exchange the roles and use Proposition 5.1. \\(\\square\\)\n\n**Remark 9.4.** The relation in Lemma 9.3 cannot be reversed. In \\(M=M_2(\\mathbb C)\\), the projection \\(e=E_{11}\\) is abelian and \\(c(e)=1=c(1)\\), but \\(1\\precsim E_{11}\\) fails, since \\(1\\) is not equivalent to any projection of rank one.\n\n**Lemma 9.5** (Counting abelian projections). Let \\(c\\ne0\\) be a central projection. Let \\(\\{e_i\\}_{i\\in I}\\) be mutually orthogonal abelian projections with \\(c(e_i)=c\\) and \\(\\sum_ie_i=c\\). Let \\(\\{f_j\\}_{j\\in J}\\) be mutually orthogonal abelian projections with \\(c(f_j)=c\\) and \\(\\sum_jf_j\\le c\\). Then \\(|J|\\le|I|\\). Consequently, if also \\(\\sum_jf_j=c\\), then \\(|I|=|J|\\).\n\n**Proof.** Replacing \\(M\\) by \\(Mc\\), we may assume \\(c=1\\). By Lemma 9.3, all the \\(e_i\\) and \\(f_j\\) are equivalent to one another.\n\n*\\(I\\) finite, \\(|I|=n\\).* By Proposition 8.4, \\(W^*MW=\\mathbb M_n(A)\\) with \\(A=e_{i_0}Me_{i_0}\\) commutative, and \\(W^*e_iW=1\\otimes E_{ii}\\). Carry the trace \\(\\Phi\\) of Proposition 9.1(2) over to \\(M\\). Then \\(\\Phi(e_i)=\\frac1n1\\), and \\(\\Phi(f_j)=\\Phi(e_{i_0})=\\frac1n1\\) because \\(f_j\\sim e_{i_0}\\). For a finite set \\(F\\subseteq J\\), \\(\\sum_{j\\in F}f_j\\le1\\), and \\(\\Phi\\) is monotone, so \\(\\frac{|F|}n1\\le1\\) and \\(|F|\\le n\\). Hence \\(|J|\\le n\\).\n\n*\\(I\\) infinite.* Fix \\(i_0\\), a unit vector \\(\\zeta\\in e_{i_0}H\\), and partial isometries \\(v_i\\) implementing \\(e_{i_0}\\sim e_i\\). Put \\(\\varphi_i=\\omega_{v_i\\zeta}\\) and \\(z_0=c(p_\\zeta)\\), which is nonzero because \\(p_\\zeta\\zeta=\\zeta\\ne0\\). Since \\(p_\\zeta\\le e_{i_0}\\) ([Lemma 4.6](#oa-fnd-ty-07)) and \\(e_{i_0}\\) is abelian, \\(p_\\zeta=z_0e_{i_0}\\) (Lemma 6.2(4)). Suppose \\(\\varphi_i(f_j)=\\|f_jv_i\\zeta\\|^2=0\\). Then the operator \\(f_jv_i\\in M\\) vanishes on \\([M'\\zeta]=p_\\zeta H\\), so \\(f_jv_iz_0e_{i_0}=0\\). As \\(v_i=v_ie_{i_0}\\), this gives \\(z_0f_jv_i=0\\), and so \\(z_0f_je_i=z_0f_jv_iv_i^*=0\\). Now fix \\(j\\). Since \\(c(f_j)=1\\), we have \\(z_0f_j\\ne0\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)), while \\(\\sum_iz_0f_je_i=z_0f_j\\). So \\(z_0f_je_i\\ne0\\), and hence \\(\\varphi_i(f_j)>0\\), for some \\(i\\). Thus \\(J=\\bigcup_iJ_i\\) with \\(J_i=\\{j:\\varphi_i(f_j)>0\\}\\). Each \\(J_i\\) is countable, because \\(\\sum_{j\\in F}\\varphi_i(f_j)=\\varphi_i(\\sum_{j\\in F}f_j)\\le1\\) for every finite \\(F\\subseteq J\\). Therefore \\(|J|\\le|I|\\cdot\\aleph_0=|I|\\) (Fact 2.9).\n\nThe last statement follows by symmetry and the Cantor–Schröder–Bernstein theorem for cardinals. \\(\\square\\)\n\n",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "10. Homogeneous algebras and the structure of type I algebras",
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      "full_conditions_and_proof": "## 10. Homogeneous algebras and the structure of type I algebras\n\n**Definition 10.1** (Homogeneous algebras). Let \\(\\alpha\\) be a nonzero cardinal. A central projection \\(z\\) is *\\(\\alpha\\)-homogeneous* if \\(z=\\sum_{i\\in I}e_i\\) for mutually orthogonal abelian projections \\(e_i\\) with \\(c(e_i)=z\\) and \\(|I|=\\alpha\\). A type I algebra is *of type I\\(_\\alpha\\)* if \\(1\\) is \\(\\alpha\\)-homogeneous.\n\n**Lemma 10.2.**\n\n1. A central projection below an \\(\\alpha\\)-homogeneous one is \\(\\alpha\\)-homogeneous. The sum of mutually orthogonal \\(\\alpha\\)-homogeneous central projections is \\(\\alpha\\)-homogeneous.\n2. A nonzero central projection is \\(\\alpha\\)-homogeneous for at most one \\(\\alpha\\).\n3. If \\(M\\) is of type I, then below each nonzero central projection there is a nonzero central projection that is \\(\\alpha\\)-homogeneous for some \\(\\alpha\\).\n4. If \\(M\\cong A\\bar\\otimes B(L)\\) with \\(A\\) abelian and \\(\\dim L=\\alpha\\), then \\(1\\) is \\(\\alpha\\)-homogeneous in \\(M\\).\n\n**Proof.** (1) Let \\(z=\\sum_ie_i\\) as in the definition and let \\(z_1\\le z\\) be central. Then \\(z_1e_i\\) is abelian, \\(c(z_1e_i)=z_1c(e_i)=z_1\\), and \\(\\sum_iz_1e_i=z_1\\). For a sum \\(\\sum_kz_k\\), index all the families by one set \\(I\\) with \\(|I|=\\alpha\\), and put \\(e_i=\\sum_ke_i^{(k)}\\). The summands have mutually orthogonal central supports \\(z_k\\), so \\(e_i\\) is abelian (Lemma 6.3), \\(c(e_i)=\\sum_kz_k\\), and \\(\\sum_ie_i=\\sum_kz_k\\).\n\n(2) This is Lemma 9.5 in \\(Mz\\).\n\n(3) Let \\(c\\ne0\\) be central. It majorizes a nonzero abelian projection \\(e\\); put \\(c_1=c(e)\\le c\\). Take a maximal family \\(\\{e_i\\}_{i\\in I}\\) of mutually orthogonal abelian projections with \\(c(e_i)=c_1\\) (it may start with \\(e\\)), and put \\(r=c_1-\\sum_ie_i\\). Suppose \\(c(r)=c_1\\). Take a maximal family of mutually centrally orthogonal nonzero abelian projections below \\(r\\), and let \\(p\\) be its sum, which is abelian (Lemma 6.3). If \\(d=c(r)-c(p)\\) were nonzero, then \\(dr\\ne0\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)), and by [Lemma 7.4(1)](#oa-fnd-ty-10), \\(dr\\) would majorize a nonzero abelian projection centrally orthogonal to that family, against maximality. So \\(c(p)=c(r)=c_1\\). Then \\(p\\) is an abelian projection with central support \\(c_1\\), orthogonal to every \\(e_i\\), against the maximality of \\(\\{e_i\\}\\). Hence \\(z=c_1-c(r)\\ne0\\). As \\(zr=0\\), \\(z=zc_1=\\sum_ize_i\\), where each \\(ze_i\\) is abelian with \\(c(ze_i)=z\\). So \\(z\\le c\\) is \\(|I|\\)-homogeneous.\n\n(4) By Proposition 9.1(1), the projections \\(1\\otimes E_{ii}\\) for an orthonormal basis of \\(L\\) are abelian, have central support \\(1\\), are mutually orthogonal and add up to \\(1\\). An isomorphism preserves all of this, including the sum (it preserves the order of projections, hence least upper bounds). \\(\\square\\)\n\n**Theorem 10.3** (Structure of type I algebras). Let \\(M\\) be of type I. For each nonzero cardinal \\(\\alpha\\) there is a largest \\(\\alpha\\)-homogeneous central projection \\(z_\\alpha\\). The \\(z_\\alpha\\) are mutually orthogonal, \\(\\sum_\\alpha z_\\alpha=1\\), and \\(z_\\alpha=0\\) unless \\(\\alpha\\le\\dim H\\). For each \\(\\alpha\\) with \\(z_\\alpha\\ne0\\),\n\\[\n\\{Mz_\\alpha,z_\\alpha H\\}\\cong\\{A_\\alpha\\bar\\otimes B(\\ell^2(\\alpha)),\\,K_\\alpha\\otimes\\ell^2(\\alpha)\\},\n\\]\nwhere \\(A_\\alpha\\) is commutative and isomorphic to \\(Zz_\\alpha\\). Hence \\(M\\cong\\bigoplus_\\alpha A_\\alpha\\bar\\otimes B(\\ell^2(\\alpha))\\). The family \\(\\{z_\\alpha\\}\\) is unique: if \\(\\{c_\\alpha\\}\\) are mutually orthogonal central projections with \\(\\sum_\\alpha c_\\alpha=1\\) and \\(Mc_\\alpha\\cong B_\\alpha\\bar\\otimes B(\\ell^2(\\alpha))\\) with \\(B_\\alpha\\) abelian, then \\(c_\\alpha=z_\\alpha\\) for every \\(\\alpha\\). Finally, \\(M\\) is finite if and only if \\(z_\\alpha=0\\) for every infinite \\(\\alpha\\).\n\nThe sum and uniqueness are proved explicitly below. \n\n**Proof.** *Set of indices.* Every nonzero \\(\\alpha\\)-homogeneous projection is a sum of \\(\\alpha\\) nonzero orthogonal projections. Choosing a unit vector in each range gives an orthonormal set, so \\(\\alpha\\le\\dim H\\) by the Hilbert basis theorem. Consequently all nonzero terms in the sum are indexed by the set of nonzero cardinals at most \\(\\dim H\\); for larger cardinals the largest homogeneous projection is zero. All sums below are over this set.\n\n*Largest homogeneous projections.* Take a maximal family of mutually orthogonal nonzero \\(\\alpha\\)-homogeneous central projections, and let \\(z_\\alpha\\) be its sum. It is \\(\\alpha\\)-homogeneous by Lemma 10.2(1). If \\(z\\) is \\(\\alpha\\)-homogeneous and \\(z(1-z_\\alpha)\\ne0\\), then \\(z(1-z_\\alpha)\\) is \\(\\alpha\\)-homogeneous and orthogonal to the family, against maximality. So \\(z\\le z_\\alpha\\).\n\n*Orthogonality and sum.* For \\(\\alpha\\ne\\beta\\), \\(z_\\alpha z_\\beta\\) is both \\(\\alpha\\)- and \\(\\beta\\)-homogeneous (Lemma 10.2(1)), so it is \\(0\\) (Lemma 10.2(2)). If \\(1-\\sum_\\alpha z_\\alpha\\ne0\\), Lemma 10.2(3) gives a nonzero \\(\\gamma\\)-homogeneous \\(z\\le1-\\sum_\\alpha z_\\alpha\\). But \\(z\\le z_\\gamma\\), so \\(z=zz_\\gamma=0\\), a contradiction. If \\(z_\\alpha\\ne0\\), it is a sum of \\(\\alpha\\) mutually orthogonal nonzero projections, so \\(\\alpha\\le\\dim H\\).\n\n*Structure.* Write \\(z_\\alpha=\\sum_{i\\in I}e_i\\) as in the definition. The \\(e_i\\) are mutually equivalent by Lemma 9.3, so Proposition 8.4 in \\(Mz_\\alpha\\) gives \\(\\{Mz_\\alpha,z_\\alpha H\\}\\cong\\{A_\\alpha\\bar\\otimes B(\\ell^2(I)),K_\\alpha\\otimes\\ell^2(I)\\}\\) with \\(A_\\alpha=e_{i_0}Me_{i_0}\\) on \\(K_\\alpha=e_{i_0}H\\). This algebra is commutative, and by Lemma 6.2(4) and [Proposition 3.5(4)](#oa-fnd-ty-02) it equals \\(Ze_{i_0}\\cong Zc(e_{i_0})=Zz_\\alpha\\).\n\n*Uniqueness.* By Lemma 10.2(4), carried through the isomorphism \\(Mc_\\alpha\\cong B_\\alpha\\bar\\otimes B(\\ell^2(\\alpha))\\), the projection \\(c_\\alpha\\) is \\(\\alpha\\)-homogeneous, so \\(c_\\alpha\\le z_\\alpha\\). Then \\(z_\\alpha=\\sum_\\beta z_\\alpha c_\\beta=z_\\alpha c_\\alpha\\le c_\\alpha\\), because \\(z_\\alpha c_\\beta\\le z_\\alpha z_\\beta=0\\) for \\(\\beta\\ne\\alpha\\).\n\n*Finiteness.* Let \\(\\alpha\\) be infinite with \\(z_\\alpha\\ne0\\). Choose \\(i_0\\in I\\) and a bijection \\(\\sigma:I\\to I\\setminus\\{i_0\\}\\). Additivity gives \\(z_\\alpha=\\sum_ie_i\\sim\\sum_ie_{\\sigma(i)}=z_\\alpha-e_{i_0}<z_\\alpha\\), so \\(z_\\alpha\\) is infinite, and \\(M\\) is not finite. Conversely, if \\(z_\\alpha=0\\) for every infinite \\(\\alpha\\), each \\(Mz_n\\cong\\mathbb M_n(A_n)\\) is finite (Proposition 9.1(2)), and \\(1=\\sum_nz_n\\) is a sum of centrally orthogonal finite projections, which is finite (Lemma 6.3). \\(\\square\\)\n\n**Corollary 10.4** (Type I factors). Every type I factor is isomorphic to \\(B(L)\\) for some Hilbert space \\(L\\), and \\(B(L_1)\\cong B(L_2)\\) if and only if \\(\\dim L_1=\\dim L_2\\).\n\n**Proof.** The central projections are \\(0\\) and \\(1\\), so \\(1=z_\\alpha\\) for exactly one \\(\\alpha\\). In a factor, a nonzero abelian \\(e\\) satisfies \\(eMe=Ze=\\mathbb Ce\\), so \\(e\\) is minimal. With Lemma 6.2(2), the abelian projections are exactly the minimal ones (and \\(0\\)). So \\(A_\\alpha=e_{i_0}Me_{i_0}=\\mathbb C\\) and \\(M\\cong B(\\ell^2(\\alpha))\\). If \\(B(L_1)\\cong B(L_2)\\), the isomorphism carries the rank-one projections of an orthonormal basis of \\(L_1\\), which are mutually orthogonal abelian projections with central support \\(1\\) and sum \\(1\\), to a family of the same kind in \\(B(L_2)\\) of cardinality \\(\\dim L_1\\). Lemma 9.5 compares it with the family given by a basis of \\(L_2\\), so \\(\\dim L_1=\\dim L_2\\). The converse is clear. \\(\\square\\)\n\n**Remark 10.5.** By Corollary 10.4, a type I\\(_n\\) factor with \\(n\\) finite is a copy of the matrix algebra \\(M_n(\\mathbb C)\\).\n\n**Example 10.6** (A type decomposition). Let \\(M=\\mathbb C\\oplus M_2(\\mathbb C)\\oplus B(\\ell^2(\\mathbb N))\\oplus\\mathbb M_3(L^\\infty[0,1])\\) on \\(\\mathbb C\\oplus\\mathbb C^2\\oplus\\ell^2(\\mathbb N)\\oplus(L^2[0,1]\\otimes\\mathbb C^3)\\). It is of type I, with \\(z_1=1\\oplus0\\oplus0\\oplus0\\), \\(z_2=0\\oplus1\\oplus0\\oplus0\\), \\(z_3=0\\oplus0\\oplus0\\oplus1\\) and \\(z_{\\aleph_0}=0\\oplus0\\oplus1\\oplus0\\) in Theorem 10.3. It is not finite, and \\(z_{\\aleph_0}\\) is the properly infinite part of \\(1\\). For \\(e=0\\oplus E_{11}\\oplus p\\oplus0\\) with \\(p\\) of infinite rank, the decomposition of Theorem 7.2 is \\(e_1=0\\oplus E_{11}\\oplus0\\oplus0\\) (finite) and \\(e_2=0\\oplus0\\oplus p\\oplus0\\) (properly infinite). If \\(p\\) has finite rank, \\(e\\) is finite.\n\n**Exercise 10.7.** (medium) Let \\(M\\) be of type I. Show that an automorphism \\(\\alpha\\) of \\(M\\) that fixes every element of the centre is inner.\n\n*Solution.* Let \\(e\\) be abelian with \\(c(e)=1\\) ([Lemma 7.4(1)](#oa-fnd-ty-10)). The projection \\(\\alpha(e)\\) is abelian, since \\(\\alpha\\) maps \\(eMe\\) onto \\(\\alpha(e)M\\alpha(e)\\). It has central support \\(\\alpha(c(e))=1\\), because \\(\\alpha\\) maps the central projections above \\(e\\) onto the central projections above \\(\\alpha(e)\\), preserving order, and fixes the centre. By Lemma 9.3, some \\(v\\) implements \\(e\\sim\\alpha(e)\\). For \\(y\\in eMe=Ze\\) (Lemma 6.2(4)), write \\(y=ae\\) with \\(a\\in Z\\); then \\(\\alpha(y)=a\\alpha(e)=avv^*=v(ae)v^*=vyv^*\\). So \\(\\alpha\\) agrees with \\(x\\mapsto vxv^*\\) on \\(eMe\\), and Exercise 5.6 shows that \\(\\alpha\\) is inner. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-TY-15",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "11. Type I algebras and their commutants",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 11. Type I algebras and their commutants\n\nA von Neumann algebra with a commutative commutant is of type I, and the commutant of a type I algebra is again of type I. With both algebras in view, a type I algebra has a spatial normal form.\n\n**Lemma 11.1** (Cyclic commutative algebras are maximal abelian). Let \\(A\\) be a commutative von Neumann algebra on \\(K\\), and \\(\\xi\\in K\\) a cyclic vector for it. Then \\(A\\) is maximal abelian: \\(A'=A\\).\n\n**Proof.** We know \\(A\\subseteq A'\\), and \\(A'\\) is spanned by its positive contractions. So let \\(S\\in A'\\) with \\(0\\le S\\le1\\); we show \\(S\\in A\\). Choose \\(a_n\\in A\\) with \\(a_n\\xi\\to S\\xi\\). For \\(b\\in A\\),\n\\[\n\\begin{gathered}\n\\langle a_n^*\\xi,b\\xi\\rangle\\\\\n=\\langle\\xi,a_nb\\xi\\rangle\\\\\n=\\langle\\xi,ba_n\\xi\\rangle\\to\\langle\\xi,bS\\xi\\rangle\\\\\n=\\langle\\xi,Sb\\xi\\rangle\\\\\n=\\langle S\\xi,b\\xi\\rangle ,\n\\end{gathered}\n\\]\nand \\(\\|a_n^*\\xi\\|=\\|a_n\\xi\\|\\) because \\(a_n\\) is normal. So \\(a_n^*\\xi\\to S\\xi\\) weakly, with \\(\\|a_n^*\\xi\\|\\to\\|S\\xi\\|\\), hence in norm. The self-adjoint elements \\(h_n=\\frac12(a_n+a_n^*)\\in A\\) satisfy \\(h_n\\xi\\to S\\xi\\). Let \\(g(t)=\\min(\\max(t,0),1)\\). Then \\(|g(s)-g(t)|\\le|s-t|\\) and \\(g(S)=S\\). The operators \\(h_n\\) and \\(S\\) commute, since \\(S\\in A'\\), so they lie in a commutative C\\(^*\\)-algebra \\(C(\\Omega)\\), where \\[\n\\begin{gathered}\n(g(h_n)-g(S))^*(g(h_n)-g(S))\\\\\n\\le(h_n-S)^*(h_n-S)\n\\end{gathered}\n\\] holds pointwise. Therefore \\(\\|(g(h_n)-S)\\xi\\|\\le\\|(h_n-S)\\xi\\|\\to0\\). The operators \\(b_n=g(h_n)\\) lie in \\(A\\), \\(\\|b_n\\|\\le1\\), and \\(b_n\\xi\\to S\\xi\\). For \\(c\\in A\\), \\(b_nc\\xi=cb_n\\xi\\to cS\\xi=Sc\\xi\\). So \\(b_n\\to S\\) strongly on the dense set \\(A\\xi\\), and, being bounded, strongly on \\(K\\). As \\(A\\) is strongly closed, \\(S\\in A\\). \\(\\square\\)\n\n**Proposition 11.2** (Type I through the commutant). \\(M\\) is of type I exactly when some faithful normal representation \\(\\pi\\) of \\(M\\) has a commutative commutant \\(\\pi(M)'\\).\n\n**Proof.** \"⇐\". By Fact 2.6, \\(\\pi(M)\\) is a von Neumann algebra isomorphic to \\(M\\), and types are invariant under isomorphism. So we may assume that \\(M\\) acts on \\(H\\) with \\(M'\\) commutative. Then \\(M'\\subseteq M''=M\\), so \\(M'=Z\\). For \\(\\xi\\in H\\), \\(p_\\xi\\) is the projection onto \\([Z\\xi]\\). The induced algebra \\(M'p_\\xi=Zp_\\xi\\) on \\(p_\\xi H\\) is commutative and has the cyclic vector \\(\\xi\\), so it is maximal abelian by Lemma 11.1. By Fact 2.1 its commutant on \\(p_\\xi H\\) is \\(p_\\xi Mp_\\xi\\). Hence \\(p_\\xi Mp_\\xi=Zp_\\xi\\) is commutative: \\(p_\\xi\\) is abelian. If \\(p\\in\\mathcal P(M)\\) is nonzero, pick a nonzero \\(\\xi\\in pH\\); then \\(0\\ne p_\\xi\\le p\\) ([Lemma 4.6](#oa-fnd-ty-07)). So every nonzero projection majorizes a nonzero abelian one, and \\(M\\) is of type I.\n\n\"⇒\". By Corollary 11.3 below, whose proof uses only \"⇐\", the commutant \\(M'\\) is of type I. So \\(M'\\) has an abelian projection \\(e'\\) with central support \\(1\\) in \\(M'\\) ([Lemma 7.4(1)](#oa-fnd-ty-10)). By [Proposition 3.5(3)](#oa-fnd-ty-02), \\(x\\mapsto xe'|_{e'H}\\) is a \\(*\\)-homomorphism of \\(M\\) onto the induced algebra \\(Me'\\), and it is injective, because its kernel is \\(M(1-c(e'))=\\{0\\}\\). It is normal, being a restriction. By Fact 2.1, the commutant of \\(Me'\\) on \\(e'H\\) is \\(e'M'e'\\), which is commutative. \\(\\square\\)\n\n**Corollary 11.3.** If \\(M\\) is of type I, so is \\(M'\\).\n\n**Proof.** Let \\(e\\in M\\) be abelian with \\(c(e)=1\\) ([Lemma 7.4(1)](#oa-fnd-ty-10)). By [Proposition 3.5(3)](#oa-fnd-ty-02), the induction \\(x'\\mapsto x'e|_{eH}\\) is an isomorphism of \\(M'\\) onto \\(M'e\\), a von Neumann algebra on \\(eH\\) whose commutant \\(eMe\\) is commutative (Fact 2.1). By \"⇐\" of Proposition 11.2, applied to the identity representation of \\(M'e\\), the algebra \\(M'e\\) is of type I, and so is \\(M'\\cong M'e\\). \\(\\square\\)\n\n**Theorem 11.4** (The spatial form of a type I algebra). Let \\(M\\) be of type I on \\(H\\). By Corollary 11.3, \\(M'\\) is of type I as well, and its centre is \\(Z\\), so Theorem 10.3 applies to both algebras. Let \\(z_\\alpha\\) be the largest \\(\\alpha\\)-homogeneous central projection of \\(M\\) and \\(z'_\\beta\\) the largest \\(\\beta\\)-homogeneous central projection of \\(M'\\). Put \\(z_{\\alpha,\\beta}=z_\\alpha z'_\\beta\\). These central projections are mutually orthogonal with sum \\(1\\). For each \\((\\alpha,\\beta)\\) with \\(z_{\\alpha,\\beta}\\ne0\\) there are a Hilbert space \\(K_{\\alpha,\\beta}\\), a maximal abelian algebra \\(A_{\\alpha,\\beta}\\) on it, and a unitary \\(W:K_{\\alpha,\\beta}\\otimes\\ell^2(\\alpha)\\otimes\\ell^2(\\beta)\\to z_{\\alpha,\\beta}H\\) such that \\(W^*(Mz_{\\alpha,\\beta})W=\\mathbb M_\\alpha(A_{\\alpha,\\beta})\\otimes1_{\\ell^2(\\beta)}\\), and \\(W^*(M'z_{\\alpha,\\beta})W\\) is the algebra of operators whose matrix entries with respect to \\(\\ell^2(\\beta)\\) lie in \\(A_{\\alpha,\\beta}\\otimes1_{\\ell^2(\\alpha)}\\). That is,\n\\[\n\\begin{gathered}\n\\{M,H\\}\\\\\n\\cong\\bigoplus_{\\alpha,\\beta}\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\\\\n\\otimes\\{B(\\ell^2(\\alpha)),\\ell^2(\\alpha)\\}\\\\\n\\otimes\\{\\mathbb C,\\ell^2(\\beta)\\},\n\\end{gathered}\n\\]\n\\[\n\\begin{gathered}\n\\{M',H\\}\\\\\n\\cong\\bigoplus_{\\alpha,\\beta}\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\\\\n\\otimes\\{\\mathbb C,\\ell^2(\\alpha)\\}\\\\\n\\otimes\\{B(\\ell^2(\\beta)),\\ell^2(\\beta)\\},\n\\end{gathered}\n\\]\nwith the same unitary. The decomposition is unique: the projections \\(z_{\\alpha,\\beta}\\) are determined by \\(M\\), and each \\(\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\) is determined up to spatial isomorphism.\n\n**Proof.** The \\(z_{\\alpha,\\beta}\\) are central and mutually orthogonal, and \\(\\sum_{\\alpha,\\beta}z_{\\alpha,\\beta}=(\\sum_\\alpha z_\\alpha)(\\sum_\\beta z'_\\beta)=1\\). Fix \\((\\alpha,\\beta)\\) with \\(z=z_{\\alpha,\\beta}\\ne0\\). Cutting the homogeneous decompositions by \\(z\\) ([Lemma 10.2(1)](#oa-fnd-ty-14)), \\(z=\\sum_{i\\in I}e_i\\) with abelian \\(e_i\\in M\\), \\(c(e_i)=z\\), \\(|I|=\\alpha\\), and \\(z=\\sum_{j\\in J}e'_j\\) with \\(e'_j\\in M'\\) abelian in \\(M'\\), of central support \\(z\\), \\(|J|=\\beta\\). Fix \\(i_0\\) and \\(j_0\\), put \\(e=e_{i_0}\\) and \\(e'=e'_{j_0}\\), and choose \\(u_i\\in M\\) implementing \\(e\\sim e_i\\) and \\(v_j\\in M'\\) implementing \\(e'\\sim e'_j\\) (Lemma 9.3 in \\(M\\) and in \\(M'\\)). Let \\(K=ee'H\\); the projections \\(e\\) and \\(e'\\) commute.\n\n*The algebra \\(A\\).* Let \\(A=\\{ye'|_K:\\ y\\in eMe\\}\\). This is the induced algebra of the reduced algebra \\(eMe\\) (on \\(eH\\)) by the projection \\(e'|_{eH}\\), which lies in its commutant \\(M'e\\). By Fact 2.1, used twice, \\(A\\) is a von Neumann algebra acting on \\(K\\), with commutant \\(\\{y'e|_K:\\ y'\\in e'M'e'\\}\\). Both are images of commutative algebras, \\(eMe\\) and \\(e'M'e'\\), under \\(*\\)-homomorphisms, so both are commutative. Hence \\(A\\subseteq A'\\subseteq A''=A\\): the algebra \\(A\\) is maximal abelian.\n\n*The unitary.* Put \\(W(\\zeta\\otimes\\delta_i\\otimes\\delta_j)=u_iv_j\\zeta\\) for \\(\\zeta\\in K\\). The \\(u\\)'s commute with the \\(v\\)'s, \\(u_k^*u_i=\\delta_{ik}e\\), and \\(v_l^*v_j=\\delta_{jl}e'\\). So\n\\[\n\\langle u_iv_j\\zeta,u_kv_l\\zeta'\\rangle=\\langle u_k^*u_iv_l^*v_j\\zeta,\\zeta'\\rangle=\\delta_{ik}\\delta_{jl}\\langle\\zeta,\\zeta'\\rangle ,\n\\]\nand \\(W\\) is isometric. The partial isometry \\(u_iv_j\\) has initial projection \\(ee'\\) and final projection \\(e_ie'_j\\), so \\(u_iv_jK=e_ie'_jH\\). As \\(\\sum_{i,j}e_ie'_j=z\\), \\(W\\) is a unitary onto \\(zH\\).\n\n*The two algebras.* Let \\(x\\in M\\) and \\(\\zeta\\in K\\). Then \\(xu_kv_l\\zeta=\\sum_{i,j}e_ie'_jxu_kv_l\\zeta\\). Here \\(e'_jxu_kv_l\\zeta=xu_ke'_jv_l\\zeta=\\delta_{jl}xu_kv_l\\zeta\\), and \\(e_ixu_kv_l\\zeta=u_iv_l(u_i^*xu_k)\\zeta\\) with \\(u_i^*xu_k\\in eMe\\). Hence\n\\[\n\\begin{gathered}\nW^*xW(\\zeta\\otimes\\delta_k\\otimes\\delta_l)\\\\\n=\\sum_i(u_i^*xu_k)e'\\zeta\\otimes\\delta_i\\otimes\\delta_l ,\n\\end{gathered}\n\\]\nso \\(W^*(Mz)W\\) lies in \\(R_1=\\mathbb M_I(A)\\otimes1_{\\ell^2(J)}\\). In the same way, for \\(y'\\in M'\\), \\(W^*y'W(\\zeta\\otimes\\delta_k\\otimes\\delta_l)=\\sum_j(v_j^*y'v_l)e\\zeta\\otimes\\delta_k\\otimes\\delta_j\\), so \\(W^*(M'z)W\\) lies in the algebra \\(R_2\\) of operators whose \\(\\ell^2(J)\\)-entries lie in \\(A'\\otimes1_{\\ell^2(I)}=A\\otimes1_{\\ell^2(I)}\\). By [Lemma 8.2(2)](#oa-fnd-ty-11), \\(\\mathbb M_I(A)'=A'\\otimes1_{\\ell^2(I)}\\), and then [Lemma 8.2(1)](#oa-fnd-ty-11) for the amplification by \\(\\ell^2(J)\\) gives \\(R_1'=R_2\\). Since \\(Mz\\) and \\(M'z\\) are each other's commutants on \\(zH\\),\n\\[\n\\begin{gathered}\nW^*(Mz)W\\\\\n=\\bigl(W^*(M'z)W\\bigr)'\\\\\n\\supseteq R_2'\\\\\n=R_1''\\\\\n=R_1\\\\\n\\supseteq W^*(Mz)W .\n\\end{gathered}\n\\]\nSo \\(W^*(Mz)W=R_1\\) and \\(W^*(M'z)W=R_1'=R_2\\).\n\n*Uniqueness.* The \\(z_\\alpha\\) are determined by \\(M\\) and the \\(z'_\\beta\\) by \\(M'\\) (Theorem 10.3 for each), so the \\(z_{\\alpha,\\beta}\\) are determined by \\(M\\). Let \\(\\{c_{\\alpha,\\beta}\\}\\) be the central projections of the summands in a second decomposition of this form. Then \\(Mc_\\alpha\\), with \\(c_\\alpha=\\sum_\\beta c_{\\alpha,\\beta}\\), is isomorphic to an abelian algebra tensor \\(B(\\ell^2(\\alpha))\\), so \\(c_\\alpha=z_\\alpha\\) by the uniqueness in Theorem 10.3. In the same way \\(\\sum_\\alpha c_{\\alpha,\\beta}=z'_\\beta\\) (Theorem 10.3 in \\(M'\\)), and therefore \\(c_{\\alpha,\\beta}=z_\\alpha z'_\\beta=z_{\\alpha,\\beta}\\). Next, let \\(e_1,e_2\\in M\\) be abelian with central support \\(z\\), and \\(e'_1,e'_2\\in M'\\) abelian in \\(M'\\) with central support \\(z\\). Choose \\(u\\in M\\) implementing \\(e_1\\sim e_2\\) and \\(v\\in M'\\) implementing \\(e'_1\\sim e'_2\\) (Lemma 9.3). The unitary \\(uv:e_1e'_1H\\to e_2e'_2H\\) carries \\(ye'_1\\) to \\((uyu^*)e'_2\\) for \\(y\\in e_1Me_1\\). So the algebra \\(A\\) does not depend on the choices, up to spatial isomorphism. Finally, in a model \\(\\mathbb M_\\alpha(B)\\otimes1\\) on \\(L\\otimes\\ell^2(\\alpha)\\otimes\\ell^2(\\beta)\\) with \\(\\{B,L\\}\\) maximal abelian and the commutant as described, the projection \\(1\\otimes E_{11}\\otimes1\\) is abelian with central support \\(1\\), and so is \\(1\\otimes1\\otimes E_{11}\\) in the commutant (Proposition 9.1(1) for each). The algebra built from this pair is \\(\\{B,L\\}\\) itself. So any such model has \\(\\{B,L\\}\\) spatially isomorphic to \\(\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\). \\(\\square\\)\n\n**Corollary 11.5** (Commutative von Neumann algebras and multiplicity). A commutative von Neumann algebra \\(\\{A,H\\}\\) is spatially isomorphic to \\(\\bigoplus_\\beta\\{A_\\beta\\otimes1,K_\\beta\\otimes\\ell^2(\\beta)\\}\\) with each \\(\\{A_\\beta,K_\\beta\\}\\) maximal abelian. The central projections of the summands are unique, and each \\(\\{A_\\beta,K_\\beta\\}\\) is unique up to spatial isomorphism.\n\n**Proof.** \\(A\\) is of type I, and \\(1\\) is abelian with central support \\(1\\), so \\(1\\) is \\(1\\)-homogeneous: \\(z_1=1\\) and \\(z_{1,\\beta}=z'_\\beta\\). Apply Theorem 11.4 with \\(\\alpha=1\\), where \\(B(\\ell^2(1))=\\mathbb C\\). The cardinal \\(\\beta\\) is the *multiplicity* of the summand: the homogeneity of \\(A'\\) there. \\(\\square\\)\n\n**Example 11.6** (Multiplicity \\(n\\)). For \\(A=L^\\infty[0,1]\\otimes1\\) on \\(L^2[0,1]\\otimes\\mathbb C^n\\), [Lemma 8.2(1)](#oa-fnd-ty-11) gives \\(A'=\\mathbb M_n(L^\\infty[0,1])\\), because \\(L^\\infty[0,1]\\) is maximal abelian on \\(L^2[0,1]\\) (Lemma 11.1, with the cyclic vector \\(1\\)). This algebra is of type I\\(_n\\) (Proposition 9.1(1) and Lemma 10.2(4)). So in Corollary 11.5 only \\(\\beta=n\\) occurs: the multiplicity is \\(n\\).\n\n",
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      "id": "OA-FND-TY-16",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "11. Type I algebras and their commutants",
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      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 11. Type I algebras and their commutants\n\nA von Neumann algebra with a commutative commutant is of type I, and the commutant of a type I algebra is again of type I. With both algebras in view, a type I algebra has a spatial normal form.\n\n**Lemma 11.1** (Cyclic commutative algebras are maximal abelian). Let \\(A\\) be a commutative von Neumann algebra on \\(K\\), and \\(\\xi\\in K\\) a cyclic vector for it. Then \\(A\\) is maximal abelian: \\(A'=A\\).\n\n**Proof.** We know \\(A\\subseteq A'\\), and \\(A'\\) is spanned by its positive contractions. So let \\(S\\in A'\\) with \\(0\\le S\\le1\\); we show \\(S\\in A\\). Choose \\(a_n\\in A\\) with \\(a_n\\xi\\to S\\xi\\). For \\(b\\in A\\),\n\\[\n\\begin{gathered}\n\\langle a_n^*\\xi,b\\xi\\rangle\\\\\n=\\langle\\xi,a_nb\\xi\\rangle\\\\\n=\\langle\\xi,ba_n\\xi\\rangle\\to\\langle\\xi,bS\\xi\\rangle\\\\\n=\\langle\\xi,Sb\\xi\\rangle\\\\\n=\\langle S\\xi,b\\xi\\rangle ,\n\\end{gathered}\n\\]\nand \\(\\|a_n^*\\xi\\|=\\|a_n\\xi\\|\\) because \\(a_n\\) is normal. So \\(a_n^*\\xi\\to S\\xi\\) weakly, with \\(\\|a_n^*\\xi\\|\\to\\|S\\xi\\|\\), hence in norm. The self-adjoint elements \\(h_n=\\frac12(a_n+a_n^*)\\in A\\) satisfy \\(h_n\\xi\\to S\\xi\\). Let \\(g(t)=\\min(\\max(t,0),1)\\). Then \\(|g(s)-g(t)|\\le|s-t|\\) and \\(g(S)=S\\). The operators \\(h_n\\) and \\(S\\) commute, since \\(S\\in A'\\), so they lie in a commutative C\\(^*\\)-algebra \\(C(\\Omega)\\), where \\[\n\\begin{gathered}\n(g(h_n)-g(S))^*(g(h_n)-g(S))\\\\\n\\le(h_n-S)^*(h_n-S)\n\\end{gathered}\n\\] holds pointwise. Therefore \\(\\|(g(h_n)-S)\\xi\\|\\le\\|(h_n-S)\\xi\\|\\to0\\). The operators \\(b_n=g(h_n)\\) lie in \\(A\\), \\(\\|b_n\\|\\le1\\), and \\(b_n\\xi\\to S\\xi\\). For \\(c\\in A\\), \\(b_nc\\xi=cb_n\\xi\\to cS\\xi=Sc\\xi\\). So \\(b_n\\to S\\) strongly on the dense set \\(A\\xi\\), and, being bounded, strongly on \\(K\\). As \\(A\\) is strongly closed, \\(S\\in A\\). \\(\\square\\)\n\n**Proposition 11.2** (Type I through the commutant). \\(M\\) is of type I exactly when some faithful normal representation \\(\\pi\\) of \\(M\\) has a commutative commutant \\(\\pi(M)'\\).\n\n**Proof.** \"⇐\". By Fact 2.6, \\(\\pi(M)\\) is a von Neumann algebra isomorphic to \\(M\\), and types are invariant under isomorphism. So we may assume that \\(M\\) acts on \\(H\\) with \\(M'\\) commutative. Then \\(M'\\subseteq M''=M\\), so \\(M'=Z\\). For \\(\\xi\\in H\\), \\(p_\\xi\\) is the projection onto \\([Z\\xi]\\). The induced algebra \\(M'p_\\xi=Zp_\\xi\\) on \\(p_\\xi H\\) is commutative and has the cyclic vector \\(\\xi\\), so it is maximal abelian by Lemma 11.1. By Fact 2.1 its commutant on \\(p_\\xi H\\) is \\(p_\\xi Mp_\\xi\\). Hence \\(p_\\xi Mp_\\xi=Zp_\\xi\\) is commutative: \\(p_\\xi\\) is abelian. If \\(p\\in\\mathcal P(M)\\) is nonzero, pick a nonzero \\(\\xi\\in pH\\); then \\(0\\ne p_\\xi\\le p\\) ([Lemma 4.6](#oa-fnd-ty-07)). So every nonzero projection majorizes a nonzero abelian one, and \\(M\\) is of type I.\n\n\"⇒\". By Corollary 11.3 below, whose proof uses only \"⇐\", the commutant \\(M'\\) is of type I. So \\(M'\\) has an abelian projection \\(e'\\) with central support \\(1\\) in \\(M'\\) ([Lemma 7.4(1)](#oa-fnd-ty-10)). By [Proposition 3.5(3)](#oa-fnd-ty-02), \\(x\\mapsto xe'|_{e'H}\\) is a \\(*\\)-homomorphism of \\(M\\) onto the induced algebra \\(Me'\\), and it is injective, because its kernel is \\(M(1-c(e'))=\\{0\\}\\). It is normal, being a restriction. By Fact 2.1, the commutant of \\(Me'\\) on \\(e'H\\) is \\(e'M'e'\\), which is commutative. \\(\\square\\)\n\n**Corollary 11.3.** If \\(M\\) is of type I, so is \\(M'\\).\n\n**Proof.** Let \\(e\\in M\\) be abelian with \\(c(e)=1\\) ([Lemma 7.4(1)](#oa-fnd-ty-10)). By [Proposition 3.5(3)](#oa-fnd-ty-02), the induction \\(x'\\mapsto x'e|_{eH}\\) is an isomorphism of \\(M'\\) onto \\(M'e\\), a von Neumann algebra on \\(eH\\) whose commutant \\(eMe\\) is commutative (Fact 2.1). By \"⇐\" of Proposition 11.2, applied to the identity representation of \\(M'e\\), the algebra \\(M'e\\) is of type I, and so is \\(M'\\cong M'e\\). \\(\\square\\)\n\n**Theorem 11.4** (The spatial form of a type I algebra). Let \\(M\\) be of type I on \\(H\\). By Corollary 11.3, \\(M'\\) is of type I as well, and its centre is \\(Z\\), so Theorem 10.3 applies to both algebras. Let \\(z_\\alpha\\) be the largest \\(\\alpha\\)-homogeneous central projection of \\(M\\) and \\(z'_\\beta\\) the largest \\(\\beta\\)-homogeneous central projection of \\(M'\\). Put \\(z_{\\alpha,\\beta}=z_\\alpha z'_\\beta\\). These central projections are mutually orthogonal with sum \\(1\\). For each \\((\\alpha,\\beta)\\) with \\(z_{\\alpha,\\beta}\\ne0\\) there are a Hilbert space \\(K_{\\alpha,\\beta}\\), a maximal abelian algebra \\(A_{\\alpha,\\beta}\\) on it, and a unitary \\(W:K_{\\alpha,\\beta}\\otimes\\ell^2(\\alpha)\\otimes\\ell^2(\\beta)\\to z_{\\alpha,\\beta}H\\) such that \\(W^*(Mz_{\\alpha,\\beta})W=\\mathbb M_\\alpha(A_{\\alpha,\\beta})\\otimes1_{\\ell^2(\\beta)}\\), and \\(W^*(M'z_{\\alpha,\\beta})W\\) is the algebra of operators whose matrix entries with respect to \\(\\ell^2(\\beta)\\) lie in \\(A_{\\alpha,\\beta}\\otimes1_{\\ell^2(\\alpha)}\\). That is,\n\\[\n\\begin{gathered}\n\\{M,H\\}\\\\\n\\cong\\bigoplus_{\\alpha,\\beta}\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\\\\n\\otimes\\{B(\\ell^2(\\alpha)),\\ell^2(\\alpha)\\}\\\\\n\\otimes\\{\\mathbb C,\\ell^2(\\beta)\\},\n\\end{gathered}\n\\]\n\\[\n\\begin{gathered}\n\\{M',H\\}\\\\\n\\cong\\bigoplus_{\\alpha,\\beta}\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\\\\n\\otimes\\{\\mathbb C,\\ell^2(\\alpha)\\}\\\\\n\\otimes\\{B(\\ell^2(\\beta)),\\ell^2(\\beta)\\},\n\\end{gathered}\n\\]\nwith the same unitary. The decomposition is unique: the projections \\(z_{\\alpha,\\beta}\\) are determined by \\(M\\), and each \\(\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\) is determined up to spatial isomorphism.\n\n**Proof.** The \\(z_{\\alpha,\\beta}\\) are central and mutually orthogonal, and \\(\\sum_{\\alpha,\\beta}z_{\\alpha,\\beta}=(\\sum_\\alpha z_\\alpha)(\\sum_\\beta z'_\\beta)=1\\). Fix \\((\\alpha,\\beta)\\) with \\(z=z_{\\alpha,\\beta}\\ne0\\). Cutting the homogeneous decompositions by \\(z\\) ([Lemma 10.2(1)](#oa-fnd-ty-14)), \\(z=\\sum_{i\\in I}e_i\\) with abelian \\(e_i\\in M\\), \\(c(e_i)=z\\), \\(|I|=\\alpha\\), and \\(z=\\sum_{j\\in J}e'_j\\) with \\(e'_j\\in M'\\) abelian in \\(M'\\), of central support \\(z\\), \\(|J|=\\beta\\). Fix \\(i_0\\) and \\(j_0\\), put \\(e=e_{i_0}\\) and \\(e'=e'_{j_0}\\), and choose \\(u_i\\in M\\) implementing \\(e\\sim e_i\\) and \\(v_j\\in M'\\) implementing \\(e'\\sim e'_j\\) (Lemma 9.3 in \\(M\\) and in \\(M'\\)). Let \\(K=ee'H\\); the projections \\(e\\) and \\(e'\\) commute.\n\n*The algebra \\(A\\).* Let \\(A=\\{ye'|_K:\\ y\\in eMe\\}\\). This is the induced algebra of the reduced algebra \\(eMe\\) (on \\(eH\\)) by the projection \\(e'|_{eH}\\), which lies in its commutant \\(M'e\\). By Fact 2.1, used twice, \\(A\\) is a von Neumann algebra acting on \\(K\\), with commutant \\(\\{y'e|_K:\\ y'\\in e'M'e'\\}\\). Both are images of commutative algebras, \\(eMe\\) and \\(e'M'e'\\), under \\(*\\)-homomorphisms, so both are commutative. Hence \\(A\\subseteq A'\\subseteq A''=A\\): the algebra \\(A\\) is maximal abelian.\n\n*The unitary.* Put \\(W(\\zeta\\otimes\\delta_i\\otimes\\delta_j)=u_iv_j\\zeta\\) for \\(\\zeta\\in K\\). The \\(u\\)'s commute with the \\(v\\)'s, \\(u_k^*u_i=\\delta_{ik}e\\), and \\(v_l^*v_j=\\delta_{jl}e'\\). So\n\\[\n\\langle u_iv_j\\zeta,u_kv_l\\zeta'\\rangle=\\langle u_k^*u_iv_l^*v_j\\zeta,\\zeta'\\rangle=\\delta_{ik}\\delta_{jl}\\langle\\zeta,\\zeta'\\rangle ,\n\\]\nand \\(W\\) is isometric. The partial isometry \\(u_iv_j\\) has initial projection \\(ee'\\) and final projection \\(e_ie'_j\\), so \\(u_iv_jK=e_ie'_jH\\). As \\(\\sum_{i,j}e_ie'_j=z\\), \\(W\\) is a unitary onto \\(zH\\).\n\n*The two algebras.* Let \\(x\\in M\\) and \\(\\zeta\\in K\\). Then \\(xu_kv_l\\zeta=\\sum_{i,j}e_ie'_jxu_kv_l\\zeta\\). Here \\(e'_jxu_kv_l\\zeta=xu_ke'_jv_l\\zeta=\\delta_{jl}xu_kv_l\\zeta\\), and \\(e_ixu_kv_l\\zeta=u_iv_l(u_i^*xu_k)\\zeta\\) with \\(u_i^*xu_k\\in eMe\\). Hence\n\\[\n\\begin{gathered}\nW^*xW(\\zeta\\otimes\\delta_k\\otimes\\delta_l)\\\\\n=\\sum_i(u_i^*xu_k)e'\\zeta\\otimes\\delta_i\\otimes\\delta_l ,\n\\end{gathered}\n\\]\nso \\(W^*(Mz)W\\) lies in \\(R_1=\\mathbb M_I(A)\\otimes1_{\\ell^2(J)}\\). In the same way, for \\(y'\\in M'\\), \\(W^*y'W(\\zeta\\otimes\\delta_k\\otimes\\delta_l)=\\sum_j(v_j^*y'v_l)e\\zeta\\otimes\\delta_k\\otimes\\delta_j\\), so \\(W^*(M'z)W\\) lies in the algebra \\(R_2\\) of operators whose \\(\\ell^2(J)\\)-entries lie in \\(A'\\otimes1_{\\ell^2(I)}=A\\otimes1_{\\ell^2(I)}\\). By [Lemma 8.2(2)](#oa-fnd-ty-11), \\(\\mathbb M_I(A)'=A'\\otimes1_{\\ell^2(I)}\\), and then [Lemma 8.2(1)](#oa-fnd-ty-11) for the amplification by \\(\\ell^2(J)\\) gives \\(R_1'=R_2\\). Since \\(Mz\\) and \\(M'z\\) are each other's commutants on \\(zH\\),\n\\[\n\\begin{gathered}\nW^*(Mz)W\\\\\n=\\bigl(W^*(M'z)W\\bigr)'\\\\\n\\supseteq R_2'\\\\\n=R_1''\\\\\n=R_1\\\\\n\\supseteq W^*(Mz)W .\n\\end{gathered}\n\\]\nSo \\(W^*(Mz)W=R_1\\) and \\(W^*(M'z)W=R_1'=R_2\\).\n\n*Uniqueness.* The \\(z_\\alpha\\) are determined by \\(M\\) and the \\(z'_\\beta\\) by \\(M'\\) (Theorem 10.3 for each), so the \\(z_{\\alpha,\\beta}\\) are determined by \\(M\\). Let \\(\\{c_{\\alpha,\\beta}\\}\\) be the central projections of the summands in a second decomposition of this form. Then \\(Mc_\\alpha\\), with \\(c_\\alpha=\\sum_\\beta c_{\\alpha,\\beta}\\), is isomorphic to an abelian algebra tensor \\(B(\\ell^2(\\alpha))\\), so \\(c_\\alpha=z_\\alpha\\) by the uniqueness in Theorem 10.3. In the same way \\(\\sum_\\alpha c_{\\alpha,\\beta}=z'_\\beta\\) (Theorem 10.3 in \\(M'\\)), and therefore \\(c_{\\alpha,\\beta}=z_\\alpha z'_\\beta=z_{\\alpha,\\beta}\\). Next, let \\(e_1,e_2\\in M\\) be abelian with central support \\(z\\), and \\(e'_1,e'_2\\in M'\\) abelian in \\(M'\\) with central support \\(z\\). Choose \\(u\\in M\\) implementing \\(e_1\\sim e_2\\) and \\(v\\in M'\\) implementing \\(e'_1\\sim e'_2\\) (Lemma 9.3). The unitary \\(uv:e_1e'_1H\\to e_2e'_2H\\) carries \\(ye'_1\\) to \\((uyu^*)e'_2\\) for \\(y\\in e_1Me_1\\). So the algebra \\(A\\) does not depend on the choices, up to spatial isomorphism. Finally, in a model \\(\\mathbb M_\\alpha(B)\\otimes1\\) on \\(L\\otimes\\ell^2(\\alpha)\\otimes\\ell^2(\\beta)\\) with \\(\\{B,L\\}\\) maximal abelian and the commutant as described, the projection \\(1\\otimes E_{11}\\otimes1\\) is abelian with central support \\(1\\), and so is \\(1\\otimes1\\otimes E_{11}\\) in the commutant (Proposition 9.1(1) for each). The algebra built from this pair is \\(\\{B,L\\}\\) itself. So any such model has \\(\\{B,L\\}\\) spatially isomorphic to \\(\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\). \\(\\square\\)\n\n**Corollary 11.5** (Commutative von Neumann algebras and multiplicity). A commutative von Neumann algebra \\(\\{A,H\\}\\) is spatially isomorphic to \\(\\bigoplus_\\beta\\{A_\\beta\\otimes1,K_\\beta\\otimes\\ell^2(\\beta)\\}\\) with each \\(\\{A_\\beta,K_\\beta\\}\\) maximal abelian. The central projections of the summands are unique, and each \\(\\{A_\\beta,K_\\beta\\}\\) is unique up to spatial isomorphism.\n\n**Proof.** \\(A\\) is of type I, and \\(1\\) is abelian with central support \\(1\\), so \\(1\\) is \\(1\\)-homogeneous: \\(z_1=1\\) and \\(z_{1,\\beta}=z'_\\beta\\). Apply Theorem 11.4 with \\(\\alpha=1\\), where \\(B(\\ell^2(1))=\\mathbb C\\). The cardinal \\(\\beta\\) is the *multiplicity* of the summand: the homogeneity of \\(A'\\) there. \\(\\square\\)\n\n**Example 11.6** (Multiplicity \\(n\\)). For \\(A=L^\\infty[0,1]\\otimes1\\) on \\(L^2[0,1]\\otimes\\mathbb C^n\\), [Lemma 8.2(1)](#oa-fnd-ty-11) gives \\(A'=\\mathbb M_n(L^\\infty[0,1])\\), because \\(L^\\infty[0,1]\\) is maximal abelian on \\(L^2[0,1]\\) (Lemma 11.1, with the cyclic vector \\(1\\)). This algebra is of type I\\(_n\\) (Proposition 9.1(1) and Lemma 10.2(4)). So in Corollary 11.5 only \\(\\beta=n\\) occurs: the multiplicity is \\(n\\).\n\n",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "12. The algebra generated by two projections",
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      "full_conditions_and_proof": "## 12. The algebra generated by two projections\n\nLet \\(e\\) and \\(f\\) be projections on \\(H\\), and \\(M=\\{e,f\\}''\\) the von Neumann algebra they generate. This section describes \\(M\\) completely. It is the direct sum of a commutative algebra of dimension at most \\(4\\) and an algebra of type I\\(_2\\). On the second summand the pair has a normal form given by two commuting operators, which play the parts of a cosine and a sine.\n\n**Lemma 12.1** (The diagonal part). The four projections\n\\[\n\\begin{gathered}\np_{11}\\\\\n=e\\wedge f,\\\\\np_{10}\\\\\n=e\\wedge f^\\perp,\\\\\np_{01}\\\\\n=e^\\perp\\wedge f,\\\\\np_{00}\\\\\n=e^\\perp\\wedge f^\\perp\n\\end{gathered}\n\\]\nlie in \\(M\\), are mutually orthogonal, and are central in \\(M\\). Let \\(z=1-(p_{11}+p_{10}+p_{01}+p_{00})\\), \\(H_0=zH\\), \\(e_0=ez\\) and \\(f_0=fz\\).\n\n1. \\(e=p_{11}+p_{10}+e_0\\) and \\(f=p_{11}+p_{01}+f_0\\). In particular \\(e\\) commutes with \\(e\\wedge f+e^\\perp\\wedge f\\), and \\(f\\) commutes with \\(e\\wedge f+e\\wedge f^\\perp\\).\n2. (*Generic position*) On \\(H_0\\), with complements taken in \\(H_0\\): \\[\n\\begin{gathered}\ne_0\\wedge f_0\\\\\n=e_0\\wedge f_0^\\perp\\\\\n=e_0^\\perp\\wedge f_0\\\\\n=e_0^\\perp\\wedge f_0^\\perp\\\\\n=0.\n\\end{gathered}\n\\] Hence all four joins \\(e_0\\vee f_0\\), \\(e_0^\\perp\\vee f_0\\), \\(e_0\\vee f_0^\\perp\\), \\(e_0^\\perp\\vee f_0^\\perp\\) equal \\(z\\).\n3. \\(M(1-z)\\), on \\((1-z)H\\), is the linear span of the nonzero \\(p_{ab}\\); it is commutative and has dimension at most \\(4\\). And \\(Mz\\), on \\(H_0\\), equals \\(\\{e_0,f_0\\}''\\).\n\n**Proof.** The \\(p_{ab}\\) lie in \\(M\\) by Proposition 3.1. Each lies below or is orthogonal to \\(e\\), and likewise for \\(f\\), so it commutes with \\(e\\) and \\(f\\), hence with \\(M=\\{e,f\\}''\\); so it is central. Two different \\(p_{ab}\\) differ in at least one index and are then orthogonal.\n\n(1) \\(ep_{11}=p_{11}\\), \\(ep_{10}=p_{10}\\) and \\(ep_{01}=ep_{00}=0\\), so \\[\n\\begin{gathered}\ne\\\\\n=e(p_{11}+p_{10}+p_{01}+p_{00})+ez\\\\\n=p_{11}+p_{10}+e_0.\n\\end{gathered}\n\\] The formula for \\(f\\) is the same. So \\(e\\) commutes with \\(p_{11}+p_{01}\\), and \\(f\\) with \\(p_{11}+p_{10}\\).\n\n(2) A vector of \\(e_0H_0\\cap f_0H_0\\) lies in \\(eH\\cap fH=p_{11}H\\), which is orthogonal to \\(H_0\\); so it is \\(0\\). The other three cases are the same, with \\(p_{10}\\), \\(p_{01}\\) and \\(p_{00}\\). The joins follow by De Morgan's laws in \\(H_0\\).\n\n(3) The span \\(S\\) of the nonzero \\(p_{ab}\\) is a commutative \\(*\\)-algebra on \\((1-z)H\\) spanned by orthogonal projections with sum \\(1-z\\); its commutant consists of the operators that commute with each \\(p_{ab}\\), and \\(S\\) equals its own bicommutant. \\(S\\) contains \\(e(1-z)=p_{11}+p_{10}\\) and \\(f(1-z)=p_{11}+p_{01}\\), and each \\(p_{ab}\\) is a product of \\(e(1-z)\\) or its complement with \\(f(1-z)\\) or its complement; so \\(S=\\{e(1-z),f(1-z)\\}''\\). Now \\(Mz\\supseteq\\{e_0,f_0\\}''\\), because \\(Mz\\) acts on \\(H_0\\) as a von Neumann algebra (Fact 2.1) and contains \\(e_0\\) and \\(f_0\\). Conversely, if \\(y\\in B(H_0)\\) commutes with \\(e_0\\) and \\(f_0\\), then \\(y\\oplus0\\) on \\(H=H_0\\oplus(1-z)H\\) commutes with \\(e\\) and \\(f\\), hence with \\(M\\), so \\(y\\) commutes with \\(Mz\\). Thus \\(Mz\\subseteq\\{e_0,f_0\\}''\\). The same argument gives \\(M(1-z)=\\{e(1-z),f(1-z)\\}''=S\\). \\(\\square\\)\n\n**Proposition 12.2** (A pair in generic position). Suppose \\(e\\wedge f=e\\wedge f^\\perp=e^\\perp\\wedge f=e^\\perp\\wedge f^\\perp=0\\) (so \\(z=1\\) above) and \\(H\\ne0\\). Let \\(M=\\{e,f\\}''\\).\n\n1. \\(e\\sim f\\sim e^\\perp\\sim f^\\perp\\) in \\(M\\).\n2. The operator \\(a=e^\\perp fe\\) is injective on \\(eH\\) and has dense range in \\(e^\\perp H\\). In its polar decomposition \\(a=u|a|\\), \\(u\\in M\\), \\(u^*u=e\\) and \\(uu^*=e^\\perp\\).\n3. The map \\(W:eH\\otimes\\mathbb C^2\\to H\\), \\(W(\\zeta_1\\otimes\\delta_1+\\zeta_2\\otimes\\delta_2)=\\zeta_1+u\\zeta_2\\), is unitary, and\n\\[\n\\begin{gathered}\nW^*eW\\\\\n=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\\\\\nW^*fW\\\\\n=\\begin{pmatrix}C^2&CS\\\\CS&S^2\\end{pmatrix},\n\\end{gathered}\n\\tag{12.3}\n\\]\nwhere \\(C=(efe|_{eH})^{1/2}\\) and \\(S=(1-C^2)^{1/2}\\) are commuting, injective, positive contractions on \\(eH\\) with \\(C^2+S^2=1\\).\n4. \\(W^*MW=\\mathbb M_2(A)\\) with \\(A=\\{C^2\\}''\\) on \\(eH\\), a commutative von Neumann algebra. So \\(M\\) is of type I\\(_2\\).\n5. \\(W^*|e-f|W=S\\otimes1\\) and \\(W^*|e-f^\\perp|W=C\\otimes1\\).\n\n\n**Proof.** (1) Apply the parallelogram law (Proposition 4.4) to the pair \\((f^\\perp,e)\\): \\((f^\\perp\\vee e)-f^\\perp\\sim e-e\\wedge f^\\perp\\). Here \\(f^\\perp\\vee e=(f\\wedge e^\\perp)^\\perp=1\\) and \\(e\\wedge f^\\perp=0\\), so \\(f\\sim e\\). The pair \\((e,f)\\) gives \\(e^\\perp=(e\\vee f)-e\\sim f-e\\wedge f=f\\). The pair \\((e^\\perp,f^\\perp)\\) gives \\(e=(e^\\perp\\vee f^\\perp)-e^\\perp\\sim f^\\perp-e^\\perp\\wedge f^\\perp=f^\\perp\\).\n\n(2) Let \\(\\zeta\\in eH\\) with \\(e^\\perp f\\zeta=0\\). Then \\(f\\zeta\\in eH\\), so \\(\\zeta-f\\zeta\\in eH\\cap f^\\perp H=\\{0\\}\\), and \\(\\zeta=f\\zeta\\in eH\\cap fH=\\{0\\}\\). The same argument, with \\(e\\) and \\(e^\\perp\\) exchanged, shows that \\(a^*=efe^\\perp\\) is injective on \\(e^\\perp H\\). As \\(a\\) vanishes on \\(e^\\perp H\\) and maps into \\(e^\\perp H\\), we get \\(\\mathrm r(a)=e\\) and \\(\\mathrm l(a)=e^\\perp\\), and Fact 2.2 gives \\(u\\).\n\n(3) \\(u\\) maps \\(eH\\) onto \\(e^\\perp H\\), so \\(W\\) is unitary. The entries of \\(W^*XW\\) are \\(eXe\\), \\(eXu\\), \\(u^*Xe\\) and \\(u^*Xu\\), restricted to \\(eH\\). For \\(X=e\\) this gives the first matrix. For \\(X=f\\), the \\((2,1)\\) entry is \\(u^*fe=u^*e^\\perp fe=u^*u|a|=|a|\\), using \\(u^*=u^*e^\\perp\\); the \\((1,2)\\) entry is its adjoint \\(|a|\\); and the diagonal entries are \\(F_{11}=efe\\) and \\(F_{22}=u^*fu\\). Comparing entries in \\(f^2=f\\) gives\n\\[\n\\begin{gathered}\nF_{11}^2+|a|^2\\\\\n=F_{11},\\\\\nF_{11}|a|+|a|F_{22}\\\\\n=|a|,\\\\\n|a|^2+F_{22}^2\\\\\n=F_{22}.\n\\end{gathered}\n\\]\nThe first identity shows that \\(|a|^2\\) commutes with \\(F_{11}\\), so its square root \\(|a|\\) does too; the third gives the same for \\(F_{22}\\). The second then reads \\(|a|(F_{11}+F_{22}-1)=0\\). Since \\(|a|\\) is injective on \\(eH\\) (\\(\\ker|a|=\\ker a\\)), \\(F_{11}+F_{22}=1\\). So \\(F_{11}=C^2\\), \\(F_{22}=1-C^2=S^2\\), and \\(|a|=(F_{11}(1-F_{11}))^{1/2}=CS\\). \\(C\\) and \\(S\\) are injective because \\(CS=|a|\\) is.\n\n(4) The entries of \\(W^*eW\\) and \\(W^*fW\\) lie in \\(A\\), since \\(C\\), \\(S\\) and \\(CS\\) are continuous functions of \\(C^2\\). By [Lemma 8.2(3)](#oa-fnd-ty-11), \\(\\mathbb M_2(A)\\) is a von Neumann algebra; it contains \\(W^*eW\\) and \\(W^*fW\\), and so it contains \\(W^*MW=\\{W^*eW,W^*fW\\}''\\). Conversely, \\(W^*MW\\) contains \\(W^*uW=1\\otimes E_{21}\\), hence all the \\(1\\otimes E_{ij}\\). Its corner \\((1\\otimes E_{11})W^*MW(1\\otimes E_{11})\\) acts on \\(eH\\otimes\\delta_1\\) as a von Neumann algebra (Fact 2.1), and it contains \\(C^2\\otimes E_{11}\\), hence \\(A\\otimes E_{11}\\). So \\(a\\otimes E_{ij}=(1\\otimes E_{i1})(a\\otimes E_{11})(1\\otimes E_{1j})\\) lies in \\(W^*MW\\) for \\(a\\in A\\), and \\(W^*MW=\\mathbb M_2(A)\\). By Proposition 9.1(1), \\(1\\otimes E_{11}\\) and \\(1\\otimes E_{22}\\) are orthogonal abelian projections with central support \\(1\\) and sum \\(1\\), so \\(M\\) is of type I\\(_2\\).\n\n(5) \\(W^*(e-f)W=\\begin{pmatrix}S^2&-CS\\\\-CS&-S^2\\end{pmatrix}\\), whose square is \\(S^2\\otimes1\\), because \\(C\\) and \\(S\\) commute and \\(C^2+S^2=1\\). Likewise \\[\n\\begin{gathered}\nW^*(e-f^\\perp)W\\\\\n=W^*(e+f-1)W\\\\\n=\\begin{pmatrix}C^2&CS\\\\CS&-C^2\\end{pmatrix},\n\\end{gathered}\n\\] whose square is \\(C^2\\otimes1\\). Take positive square roots. \\(\\square\\)\n\n**Theorem 12.4** (The algebra of two projections). Let \\(e\\) and \\(f\\) be projections on \\(H\\), and \\(M=\\{e,f\\}''\\).\n\n1. \\(M\\) is of type I.\n2. There is exactly one central projection \\(z\\) of \\(M\\) such that \\(Mz\\) is of type I\\(_2\\) and \\(M(1-z)\\) is commutative. Moreover \\(\\dim M(1-z)\\le4\\).\n\n**Proof.** Take \\(z\\) from Lemma 12.1. By its part (3), \\(M(1-z)\\) is commutative of dimension at most \\(4\\). If \\(z\\ne0\\), the pair \\(e_0,f_0\\) is in generic position on \\(H_0\\) and \\(Mz=\\{e_0,f_0\\}''\\), so \\(Mz\\) is of type I\\(_2\\) by Proposition 12.2. (If \\(z=0\\), then \\(Mz=\\{0\\}\\), which is of every type.) So \\(M\\) is the direct sum of a commutative algebra and an algebra of type I\\(_2\\), hence of type I. For uniqueness, let \\(z'\\) be a second such projection. The algebra \\(Mz'(1-z)\\) is a central summand of the type I\\(_2\\) algebra \\(Mz'\\) and of the commutative algebra \\(M(1-z)\\). A nonzero algebra of type I\\(_2\\) is not commutative: cutting \\(1=e_1+e_2\\) (abelian, central support \\(1\\), and \\(e_1\\sim e_2\\) by Lemma 9.3) by a nonzero central projection gives two orthogonal, nonzero, equivalent projections, while in a commutative algebra equivalent projections are equal. So \\(z'(1-z)=0\\), \\(z'\\le z\\), and by symmetry \\(z'=z\\). \\(\\square\\)\n\n**Definition 12.5** (Sine and cosine). For projections \\(e\\) and \\(f\\), we call \\(\\sin(e,f)=|e-f|\\) the *sine* of the pair and \\(\\cos(e,f)=|e-f^\\perp|\\) its *cosine*.\n\n**Remark 12.6.** The identity \\((e-f)^2+(e+f-1)^2=1\\), which holds for any two projections, gives \\(\\sin(e,f)^2+\\cos(e,f)^2=1\\). Moreover \\((e-f)^2\\) commutes with \\(e\\) and with \\(f\\) (both products equal \\(e-efe\\) for \\(e\\), and similarly for \\(f\\)), so \\(\\sin(e,f)\\) and \\(\\cos(e,f)\\) are central in \\(\\{e,f\\}''\\). On the generic part they are \\(S\\otimes1\\) and \\(C\\otimes1\\), by Proposition 12.2(5). On \\((1-z)H\\) they are \\(p_{10}+p_{01}\\) and \\(p_{11}+p_{00}\\). If \\(H=\\mathbb C^2\\) and \\(eH\\), \\(fH\\) are lines at an angle \\(\\theta\\) with \\(0<\\theta<\\pi/2\\), the pair is in generic position, \\(C=\\cos\\theta\\) and \\(S=\\sin\\theta\\). So \\(C\\) and \\(S\\) generalize \\(\\cos\\theta\\) and \\(\\sin\\theta\\) for a pair of lines.\n\n**Example 12.7** (Two lines in \\(\\mathbb C^2\\)). Let \\(eH=\\mathbb C\\delta_1\\) and \\(fH=\\mathbb C(\\cos\\theta\\,\\delta_1+\\sin\\theta\\,\\delta_2)\\) with \\(0<\\theta<\\pi/2\\). The four corner projections vanish, so \\(z=1\\) and \\(M=M_2(\\mathbb C)\\), of type I\\(_2\\). Here \\(efe=\\cos^2\\theta\\,e\\), so in (12.3) \\(C=\\cos\\theta\\) and \\(S=\\sin\\theta\\) (scalars on the line \\(eH\\)), \\(|e-f|=\\sin\\theta\\cdot1\\) and \\(|e-f^\\perp|=\\cos\\theta\\cdot1\\). For \\(\\theta=0\\) or \\(\\theta=\\pi/2\\) the projections commute, \\(z=0\\), and \\(M\\) is commutative.\n\n",
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      "id": "OA-FND-TY-17",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "13. Orthogonal families of equivalent projections, and halving",
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      "full_conditions_and_proof": "## 13. Orthogonal families of equivalent projections, and halving\n\nThis section shows when projections can be halved. In an algebra without a type I part every projection is the sum of two equivalent halves, and a properly infinite projection is the sum of two, or of countably many, copies of itself. The tool is a maximal family of mutually orthogonal, mutually equivalent projections.\n\n**Proposition 13.1** (Orthogonal families of equivalent projections). Let \\(\\{e_i\\}_{i\\in I}\\), with \\(I\\ne\\emptyset\\), be mutually orthogonal, mutually equivalent, nonzero projections. There are a nonzero central projection \\(z\\) and mutually orthogonal, mutually equivalent projections \\(\\{f_j\\}_{j\\in J}\\), with \\(J\\supseteq I\\), such that \\(f_i=ze_i\\) for \\(i\\in I\\) and\n\\[\nf_0:=z-\\sum_{j\\in J}f_j\\prec f_j\\qquad\\text{for every }j\\in J .\n\\]\nIf \\(J\\) is infinite, the \\(f_j\\) can be replaced by mutually orthogonal, mutually equivalent projections with \\(f_i\\sim ze_i\\) for \\(i\\in I\\) and \\(f_0=0\\). The \\(e_i\\) must be nonzero: for a family of zero projections the conclusion fails ([Example 16.1](#oa-fnd-ty-22)).\n\nNonzero projections are essential for the strict comparison conclusion, as Example 16.1 verifies. \n\n**Proof.** Fix \\(i_1\\in I\\). By Zorn's lemma the family extends to a maximal family \\(\\{e_j\\}_{j\\in J}\\) of mutually orthogonal projections equivalent to \\(e_{i_1}\\). Put \\(e_0=1-\\sum_je_j\\). The comparison theorem (Theorem 5.5), applied to \\(e_0\\) and \\(e_{i_1}\\), gives a central \\(z\\) with \\(ze_0\\precsim ze_{i_1}\\) and \\((1-z)e_{i_1}\\precsim(1-z)e_0\\). Central cuts preserve equivalence ([Lemma 3.3(4)](#oa-fnd-ty-02)), so the same holds with any \\(e_j\\) in place of \\(e_{i_1}\\). If \\(z\\) were \\(0\\), then \\(e_{i_1}\\precsim e_0\\): some nonzero \\(g\\le e_0\\) would satisfy \\(g\\sim e_{i_1}\\), and adding \\(g\\) would contradict maximality. So \\(z\\ne0\\). Put \\(f_j=ze_j\\) for \\(j\\in J\\). Then \\(f_0=ze_0=z-\\sum_jf_j\\), and \\(f_0\\precsim f_j\\). If also \\(f_j\\precsim f_0\\), then together with \\((1-z)e_j\\precsim(1-z)e_0\\), additivity gives \\(e_j\\precsim e_0\\), which contradicts maximality as before. So \\(f_0\\prec f_j\\).\n\nLet \\(J\\) be infinite. Fix \\(j_0\\in J\\) and a bijection \\(\\sigma:J\\to J\\setminus\\{j_0\\}\\). Then \\(\\sum_jf_j\\sim\\sum_jf_{\\sigma(j)}=\\sum_{j\\ne j_0}f_j\\), and \\(f_0\\precsim f_{j_0}\\). Additivity gives\n\\[\n\\begin{gathered}\nz\\\\\n=f_0+\\sum_{j\\in J}f_j\\ \\precsim\\ f_{j_0}+\\sum_{j\\ne j_0}f_j\\\\\n=\\sum_{j\\in J}f_j\\\\\n\\le z ,\n\\end{gathered}\n\\tag{13.2}\n\\]\nso \\(z\\sim\\sum_jf_j\\) by Proposition 5.1. If \\(w\\) implements \\(\\sum_jf_j\\sim z\\), the projections \\(wf_jw^*\\) are mutually orthogonal, each is equivalent to \\(f_j\\) (through \\(wf_j\\)), and they add up to \\(z\\). \\(\\square\\)\n\n**Proposition 13.3** (Algebras without a type I part). The following are equivalent.\n\n1. \\(M\\) has no nonzero central summand of type I, that is, \\(z_{\\rm I}=0\\) in Theorem 7.2.\n2. \\(M\\) has no nonzero abelian projection.\n3. Every projection of \\(M\\) is the sum of two orthogonal, equivalent projections.\n\n**Proof.** (1)⇔(2). In the proof of Theorem 7.2, \\(z_{\\rm I}=c(a)\\) for the sum \\(a\\) of a maximal family of mutually centrally orthogonal nonzero abelian projections, and \\(z_{\\rm I}\\) is unique. If \\(M\\) has a nonzero abelian projection, that family is not empty, so \\(z_{\\rm I}\\ne0\\). Conversely, if \\(z_{\\rm I}\\ne0\\), the type I summand \\(Mz_{\\rm I}\\) has a nonzero abelian projection.\n\n(3)⇒(2). Let \\(e\\) be abelian, and \\(e=e_1+e_2\\) with \\(e_1\\perp e_2\\) and \\(e_1\\sim e_2\\). By Lemma 6.2(4), \\(e_k=c(e_k)e\\), and \\(c(e_1)=c(e_2)\\) by [Proposition 3.5(2)](#oa-fnd-ty-02). So \\(e_1=e_2\\), and orthogonality forces \\(e_1=e_2=0\\), hence \\(e=0\\).\n\n(2)⇒(3). First, every nonzero projection \\(e\\) majorizes two orthogonal, equivalent, nonzero projections. The algebra \\(eMe\\) is not commutative, and it is the norm-closed span of its projections (Fact 2.8), so some projection \\(p\\in eMe\\) fails to commute with some element of \\(eMe\\). Then \\(pM(e-p)\\ne\\{0\\}\\): otherwise, for \\(x\\in eMe\\), \\(px(e-p)=0\\) and \\((e-p)xp=(px^*(e-p))^*=0\\), so \\(px=pxp=xp\\). Lemma 5.3 gives nonzero \\(e_1\\le p\\) and \\(e_2\\le e-p\\) with \\(e_1\\sim e_2\\). Next, let \\(e\\) be any projection, and take a maximal family of pairs \\((a_k,b_k)\\) of nonzero projections below \\(e\\) such that all the \\(a_k\\) and \\(b_k\\) are mutually orthogonal and \\(a_k\\sim b_k\\). If \\(r=e-\\sum_k(a_k+b_k)\\) were nonzero, the first step inside \\(r\\) would enlarge the family. So \\(e=\\sum_ka_k+\\sum_kb_k\\), and \\(\\sum_ka_k\\sim\\sum_kb_k\\) by additivity. \\(\\square\\)\n\n**Proposition 13.4** (Properly infinite algebras halve). If \\(M\\) is properly infinite, there is \\(e\\in\\mathcal P(M)\\) with \\(e\\sim1-e\\sim1\\).\n\n**Proof.** *Step 1.* Every nonzero central \\(z\\) majorizes a nonzero central \\(z_1\\) that contains a projection \\(a\\) with \\(a\\sim z_1-a\\sim z_1\\). Indeed, \\(z\\) is infinite, because \\(M\\) is properly infinite, so some \\(u\\) has \\(u^*u=z\\) and \\(uu^*<z\\). The powers \\(u^n\\) have initial projection \\(z\\) and decreasing final projections, so the projections \\[\n\\begin{gathered}\np_n\\\\\n=u^{n-1}(z-uu^*)u^{*(n-1)}\\\\\n=u^{n-1}u^{*(n-1)}-u^nu^{*n},\n\\end{gathered}\n\\] \\(n\\ge1\\) (with \\(u^0=z\\), the unit of \\(Mz\\)), are mutually orthogonal; they are nonzero and equivalent to \\(p_1\\) through \\(u^{n-1}p_1\\). Proposition 13.1 in \\(Mz\\) gives a nonzero central \\(z_1\\le z\\) and an infinite family \\(\\{q_j\\}_{j\\in J}\\) of mutually orthogonal, mutually equivalent projections with \\(\\sum_jq_j=z_1\\). Split \\(J=J_1\\sqcup J_2\\) with \\(|J_1|=|J_2|=|J|\\) (Fact 2.9), and put \\(a=\\sum_{j\\in J_1}q_j\\). Additivity gives \\(a\\sim\\sum_{j\\in J_2}q_j=z_1-a\\) and \\(a\\sim\\sum_{j\\in J}q_j=z_1\\).\n\n*Step 2.* Take a maximal family \\(\\{e_k\\}\\) of nonzero projections with mutually orthogonal central supports such that \\(e_k\\sim c(e_k)-e_k\\sim c(e_k)\\) for each \\(k\\). If \\(c=\\sum_kc(e_k)\\) were not \\(1\\), Step 1 with \\(z=1-c\\) would give a projection \\(a\\) with \\(c(a)=c(z_1)=z_1\\le1-c\\) and \\(a\\sim z_1-a\\sim z_1\\), enlarging the family. So \\(\\sum_kc(e_k)=1\\). Put \\(e=\\sum_ke_k\\). By additivity, \\(1=\\sum_kc(e_k)\\sim\\sum_ke_k=e\\) and \\(e\\sim\\sum_k(c(e_k)-e_k)=1-e\\). \\(\\square\\)\n\n**Corollary 13.5** (Division by \\(\\aleph_0\\)). If \\(e\\) is properly infinite, then \\(e=\\sum_{n\\ge1}e_n\\) with mutually orthogonal \\(e_n\\sim e\\).\n\n**Proof.** The algebra \\(eMe\\) is properly infinite: its central projections are the \\(ze\\) ([Proposition 3.5(4)](#oa-fnd-ty-02)), and each nonzero \\(ze\\) is infinite. Proposition 13.4 in \\(eMe\\) gives \\(e=a_1+b_1\\) with \\(a_1\\sim b_1\\sim e\\). The projection \\(b_1\\) is properly infinite as well: its central cuts \\(zb_1\\sim ze\\) are infinite when nonzero. So \\(b_1=a_2+b_2\\) with \\(a_2\\sim b_2\\sim b_1\\), and so on. The \\(a_n\\) are mutually orthogonal and equivalent to \\(e\\), and \\(q=\\sum_na_n\\) satisfies \\(e\\sim a_1\\le q\\le e\\), so \\(q\\sim e\\) (Proposition 5.1). If \\(w\\) implements \\(q\\sim e\\), the projections \\(e_n=wa_nw^*\\) are mutually orthogonal (conjugation by \\(w\\) is an isomorphism of \\(qMq\\) onto \\(eMe\\)), \\(e_n\\sim a_n\\sim e\\), and \\(\\sum_ne_n=wqw^*=e\\). \\(\\square\\)\n\n",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "14. Finite projections: the modular law and unitary equivalence",
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      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 14. Finite projections: the modular law and unitary equivalence\n\nFinite projections behave like finite-dimensional subspaces: they satisfy the modular law, and equivalent finite projections are unitarily equivalent.\n\n**Theorem 14.1** (Finite projections form a modular lattice). If \\(e\\) and \\(f\\) are finite, then \\(e\\vee f\\) and \\(e\\wedge f\\) are finite, so the finite projections form a sublattice of \\(\\mathcal P(M)\\). This sublattice is modular. More generally, the modular law\n\\[\n(e\\vee f)\\wedge g=e\\vee(f\\wedge g)\\qquad(e\\le g)\n\\]\nholds for all \\(e,f,g\\in\\mathcal P(M)\\) with \\(e\\le g\\) as soon as \\(g\\) is finite.\n\n**Proof.** The projection \\(e\\wedge f\\le e\\) is finite (Lemma 6.2(1)). By the parallelogram law (Proposition 4.4), \\((e\\vee f)-e\\sim f-e\\wedge f\\le f\\), so \\(g'=(e\\vee f)-e\\) is finite, and \\(e\\vee f=e+g'\\) with \\(e\\perp g'\\). It remains to show: *if \\(a\\) and \\(b\\) are orthogonal finite projections, then \\(q=a+b\\) is finite.*\n\nWrite \\(q=q_1+q_2\\) as in Theorem 7.2, with \\(q_1\\) finite, \\(q_2\\) properly infinite and the two centrally orthogonal, and put \\(c=c(q_2)\\), so that \\(q_2=cq\\). Suppose \\(c\\ne0\\). The projections \\(ca\\) and \\(cb\\) are finite and orthogonal, and \\(cq=ca+cb\\) is properly infinite and nonzero. We work in \\(N=cqMcq\\), whose unit \\(cq\\) is properly infinite, and write \\(a,b,q\\) for \\(ca,cb,cq\\). By Proposition 13.4 in \\(N\\), \\(q=r+s\\) with \\(r\\sim s\\sim q\\). The comparison theorem (Theorem 5.5) in \\(N\\) gives a central projection \\(w\\) of \\(N\\) with \\(w(a\\wedge r)\\precsim w(b\\wedge s)\\) and \\((q-w)(b\\wedge s)\\precsim(q-w)(a\\wedge r)\\).\n\nOn \\(w\\): \\(wr=w(a\\wedge r)+w(r-a\\wedge r)\\), and \\(r-a\\wedge r\\sim(a\\vee r)-a\\) by Proposition 4.4. The projections \\(b\\wedge s\\) and \\((a\\vee r)-a\\) are orthogonal, because \\(b\\wedge s\\) is orthogonal to \\(a\\) and to \\(r\\), hence to \\(a\\vee r\\); and both lie below \\(b=q-a\\). Hence\n\\[\nwr\\ \\precsim\\ w(b\\wedge s)+w((a\\vee r)-a)\\ \\le\\ wb ,\n\\]\nso \\(wr\\) is finite. As \\(wr\\sim wq=w\\), the central projection \\(w\\) of \\(N\\) is finite, and since the unit of \\(N\\) is properly infinite, \\(w=0\\).\n\nHence \\(b\\wedge s\\precsim a\\wedge r\\). The same computation with the roles exchanged gives\n\\[\n\\begin{gathered}\ns\\\\\n=b\\wedge s+(s-b\\wedge s)\\ \\\\\n\\sim\\ b\\wedge s+((b\\vee s)-b)\\ \\\\\n\\precsim\\ a\\wedge r+((b\\vee s)-b)\\ \\\\\n\\le\\ a ,\n\\end{gathered}\n\\]\nsince \\(a\\wedge r\\) is orthogonal to \\(b\\) and to \\(s\\), and both summands lie below \\(a=q-b\\). So \\(s\\precsim a\\) is finite, while \\(s\\sim q\\) is properly infinite and nonzero. This contradiction shows \\(c=0\\), so \\(q\\) is finite.\n\n*The modular law.* Let \\(e\\le g\\) with \\(g\\) finite, and put \\(h=(e\\vee f)\\wedge g\\) and \\(k=e\\vee(f\\wedge g)\\). Then \\(k\\le h\\), since \\(e\\) and \\(f\\wedge g\\) lie below both \\(e\\vee f\\) and \\(g\\). Next, \\(h\\vee f=e\\vee f=k\\vee f\\), because \\(e\\le h\\le e\\vee f\\) and \\(e\\le k\\le e\\vee f\\). And \\(h\\wedge f=f\\wedge g=k\\wedge f\\), because \\(h\\wedge f=(e\\vee f)\\wedge g\\wedge f=g\\wedge f\\) and \\(f\\wedge g\\le k\\wedge f\\le h\\wedge f\\). By Proposition 4.4,\n\\[\n\\begin{gathered}\nh-f\\wedge g\\\\\n=h-h\\wedge f\\ \\sim\\ (h\\vee f)-f\\\\\n=(e\\vee f)-f ,\n\\end{gathered}\n\\]\nand in the same way \\(k-f\\wedge g\\sim(e\\vee f)-f\\). Adding \\(f\\wedge g\\) gives \\(h\\sim k\\). As \\(k\\le h\\le g\\) and \\(g\\) is finite, \\(h\\) is finite, and so \\(h=k\\). \\(\\square\\)\n\n**Proposition 14.2** (Unitary equivalence of finite projections). If \\(e\\sim f\\) and \\(e\\) is finite, then \\(1-e\\sim1-f\\), and \\(ueu^*=f\\) for some unitary \\(u\\in M\\).\n\n**Proof.** \\(f\\) is finite (Lemma 6.2(1)), and so is \\(g=e\\vee f\\) (Theorem 14.1). In the finite algebra \\(N=gMg\\) apply the comparison theorem (Theorem 5.5) to \\(g-e\\) and \\(g-f\\): a central projection \\(w\\) of \\(N\\) has \\(w(g-e)\\precsim w(g-f)\\) and \\((g-w)(g-f)\\precsim(g-w)(g-e)\\). Choose \\(h\\le w(g-f)\\) with \\(w(g-e)\\sim h\\). Then \\(w=we+w(g-e)\\sim wf+h\\le w\\), because \\(we\\sim wf\\) and \\(wf\\perp h\\). As \\(w\\) is finite, \\(wf+h=w\\), so \\(h=w(g-f)\\) and \\(w(g-e)\\sim w(g-f)\\). In the same way \\((g-w)(g-e)\\sim(g-w)(g-f)\\). Adding, \\(g-e\\sim g-f\\); adding \\(1-g\\), \\(1-e\\sim1-f\\). Let \\(v\\) implement \\(e\\sim f\\) and \\(v'\\) implement \\(1-e\\sim1-f\\). Then \\(v=fve\\) and \\(v'=(1-f)v'(1-e)\\), so the cross terms vanish, and \\(u=v+v'\\) satisfies \\(u^*u=e+(1-e)=1\\), \\(uu^*=f+(1-f)=1\\) and \\(ueu^*=vv^*=f\\). \\(\\square\\)\n\n**Exercise 14.3.** (easy) Show that \\(M\\) is finite if and only if, for all \\(e,f\\in\\mathcal P(M)\\), \\(e\\precsim f\\) implies \\(1-f\\precsim1-e\\).\n\n*Solution.* Let \\(M\\) be finite and \\(e\\sim e_1\\le f\\). All projections are finite (Lemma 6.2(1)), so Proposition 14.2 gives \\(1-e\\sim1-e_1\\ge1-f\\); hence \\(1-f\\precsim1-e\\). Conversely, if \\(1\\sim f\\), then \\(1\\precsim f\\), so \\(1-f\\precsim1-1=0\\) and \\(f=1\\). \\(\\square\\)\n\n**Exercise 14.4.** (hard) Projections \\(e,f\\) are *unitarily equivalent* if \\(ueu^*=f\\) for a unitary \\(u\\in M\\); then \\(e\\sim f\\). Show that if \\(e\\sim f\\), there are orthogonal decompositions \\(e=\\sum_ie_i\\) and \\(f=\\sum_if_i\\) such that \\(e_i\\) and \\(f_i\\) are unitarily equivalent for each \\(i\\).\n\n*Solution.* Unitary equivalence of \\(e_i\\) and \\(f_i\\) holds exactly when \\(e_i\\sim f_i\\) and \\(1-e_i\\sim1-f_i\\): one direction is clear, and the other is the last step of the proof of Proposition 14.2. Let \\(v\\) implement \\(e\\sim f\\). Write \\(e=e_1+e_2\\) as in Theorem 7.2, with \\(e_1\\) finite and \\(e_2\\) properly infinite, centrally orthogonal, and put \\(c=c(e_1)\\); then \\(e_1=ce\\) and \\(e_2=(1-c)e\\), and \\(f=cf+(1-c)f\\) with \\(cf\\sim ce\\) and \\((1-c)f\\sim(1-c)e\\).\n\n- The finite pair \\(ce\\), \\(cf\\) is unitarily equivalent by Proposition 14.2.\n- For the properly infinite pair, put \\(f_2=(1-c)f\\). The algebra \\(e_2Me_2\\) is properly infinite, as in the proof of Corollary 13.5, so Proposition 13.4 in \\(e_2Me_2\\) gives \\(e_2=a_1+a_2\\) with \\(a_1\\sim a_2\\sim e_2\\). The partial isometry \\(w=(1-c)v\\) implements \\(e_2\\sim f_2\\); put \\(b_k=wa_kw^*\\), so \\(f_2=b_1+b_2\\) and \\(a_k\\sim b_k\\). If \\(t\\) implements \\(a_2\\sim e_2\\), then \\((1-e_2)+t\\) is a partial isometry with initial projection \\((1-e_2)+a_2=1-a_1\\) and final projection \\(1\\) (the cross terms vanish). So \\(1-a_1\\sim1\\), and in the same way \\(1-b_1=(1-f_2)+b_2\\sim1\\). Hence \\(1-a_1\\sim1-b_1\\), and \\(a_1\\), \\(b_1\\) are unitarily equivalent. By symmetry, so are \\(a_2\\) and \\(b_2\\).\n\nSo \\(e=ce+a_1+a_2\\) and \\(f=cf+b_1+b_2\\) is a decomposition of the required kind, with at most three nonzero pieces. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-TY-19",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "15. Countable sums and properly infinite algebras",
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      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 15. Countable sums and properly infinite algebras\n\nA properly infinite projection absorbs countably many projections that are subequivalent to it. Under a countability hypothesis this gives the normal form \\(M\\cong N\\bar\\otimes B(\\ell^2(\\mathbb N))\\) for properly infinite algebras.\n\n**Definition 15.1** (Locally \\(\\sigma\\)-finite projections). A projection \\(f\\) is *locally \\(\\sigma\\)-finite* if every central projection \\(z\\) with \\(zf\\ne0\\) majorizes a central projection \\(z'\\) such that \\(z'f\\) is nonzero and \\(\\sigma\\)-finite. \\(M\\) is *locally \\(\\sigma\\)-finite* if \\(1\\) is. Every \\(\\sigma\\)-finite projection is locally \\(\\sigma\\)-finite.\n\nBy Zorn's lemma, \\(M\\) is locally \\(\\sigma\\)-finite exactly when \\(1\\) is a sum of mutually orthogonal central projections \\(z_k\\) with every \\(Mz_k\\) \\(\\sigma\\)-finite. Indeed, if \\(M\\) is locally \\(\\sigma\\)-finite, a maximal family of mutually orthogonal nonzero central projections \\(z_k\\) with \\(Mz_k\\) \\(\\sigma\\)-finite has sum \\(1\\), since otherwise \\(1-\\sum_kz_k\\) would majorize a further member. Conversely, if \\(1=\\sum_kz_k\\) with every \\(Mz_k\\) \\(\\sigma\\)-finite, and \\(z\\) is a nonzero central projection, then \\(zz_k\\ne0\\) for some \\(k\\), and \\(Mzz_k\\) is \\(\\sigma\\)-finite, since orthogonal families in \\(Mzz_k\\) are orthogonal families in \\(Mz_k\\). Such algebras are also called *locally countably decomposable*.\n\n**Proposition 15.2** (Properly infinite projections absorb countable sums). Let \\(e\\) be a properly infinite projection.\n\n1. If \\(f_1,f_2,\\ldots\\) are mutually orthogonal projections with \\(f_n\\precsim e\\), then \\(\\sum_nf_n\\precsim e\\).\n2. If \\(f\\) is \\(\\sigma\\)-finite and \\(c(f)\\le c(e)\\), then \\(f\\precsim e\\).\n3. If \\(f\\) is locally \\(\\sigma\\)-finite and \\(c(f)\\le c(e)\\), then \\(f\\precsim e\\).\n4. If \\(M\\) is locally \\(\\sigma\\)-finite and \\(c(e)=1\\), then \\(e\\sim1\\). In particular, in a \\(\\sigma\\)-finite factor any two infinite projections are equivalent.\n\nThe countability conditions cannot be dropped ([Example 16.4](#oa-fnd-ty-22)).\n\n**Proof.** (1) By Corollary 13.5, \\(e=\\sum_ne_n\\) with mutually orthogonal \\(e_n\\sim e\\). Then \\(f_n\\precsim e\\sim e_n\\), and additivity gives \\(\\sum_nf_n\\precsim\\sum_ne_n=e\\).\n\n(2) Take a maximal family \\(\\{f_k\\}\\) of mutually orthogonal nonzero subprojections of \\(f\\) with \\(f_k\\precsim e\\). It is countable, because \\(f\\) is \\(\\sigma\\)-finite. If \\(r=f-\\sum_kf_k\\) were nonzero, then \\(c(r)\\le c(f)\\le c(e)\\), and Corollary 5.4 would give a nonzero \\(r_1\\le r\\) with \\(r_1\\precsim e\\), against maximality. So \\(f=\\sum_kf_k\\), and (1) applies (a finite family is padded with zeros).\n\n(3) Take a maximal family \\(\\{z_k\\}\\) of mutually orthogonal central projections below \\(c(f)\\) such that each \\(z_kf\\) is nonzero and \\(\\sigma\\)-finite. If \\(z=c(f)-\\sum_kz_k\\) were nonzero, then \\(zf\\ne0\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)), and the hypothesis would give a further member of the family. So \\(f=\\sum_kz_kf\\). Each \\(z_ke\\) is nonzero, because \\(0\\ne z_k\\le c(e)\\), and it is properly infinite (Lemma 6.2(3)); moreover \\(c(z_kf)=z_kc(f)\\le z_kc(e)=c(z_ke)\\). By (2), \\(z_kf\\precsim z_ke\\), and additivity over the orthogonal central pieces gives \\(f\\precsim\\sum_kz_ke\\le e\\).\n\n(4) The projection \\(1\\) is locally \\(\\sigma\\)-finite and \\(c(1)=1=c(e)\\), so \\(1\\precsim e\\) by (3), and \\(e\\sim1\\) by Proposition 5.1. In a factor, infinite projections are properly infinite, nonzero projections have central support \\(1\\), and in a \\(\\sigma\\)-finite factor every projection is \\(\\sigma\\)-finite; so two infinite projections are subequivalent to each other, hence equivalent. \\(\\square\\)\n\n**Proposition 15.3** (Properly infinite semifinite algebras). Let \\(M\\) be properly infinite and semifinite, and let \\(f\\) be a finite projection with \\(c(f)=1\\) ([Lemma 7.4(2)](#oa-fnd-ty-10)).\n\n1. There are mutually orthogonal central projections \\(z_\\alpha\\), indexed by the infinite cardinals \\(\\alpha\\le|M|\\), some possibly \\(0\\), with \\(\\sum_\\alpha z_\\alpha=1\\) and \\[\n\\begin{gathered}\n\\{Mz_\\alpha,z_\\alpha H\\}\\\\\n\\cong\\{N_\\alpha\\bar\\otimes B(\\ell^2(\\alpha)),\\,fz_\\alpha H\\otimes\\ell^2(\\alpha)\\},\n\\end{gathered}\n\\] where \\(N_\\alpha=fz_\\alpha Mfz_\\alpha\\) is finite. The \\(N_\\alpha\\) are not unique ([Example 16.5](#oa-fnd-ty-22)).\n2. If \\(M\\) is locally \\(\\sigma\\)-finite, then \\(M\\cong fMf\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n3. More generally, if \\(M\\) is properly infinite and locally \\(\\sigma\\)-finite (not necessarily semifinite) and \\(p\\) is any projection with \\(c(p)=1\\), then \\(1=\\sum_{n\\in\\mathbb N}p_n\\) with mutually orthogonal \\(p_n\\sim p\\), and \\(M\\cong pMp\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n\n*Reference:* The decomposition of \\(1\\) in part (3) is [Blackadar, III.1.3.6].\n\nThe further trace theory studies uniqueness of the family \\(\\{z_\\alpha\\}\\) in (1); see Traces on von Neumann algebras. That extension is not proved or used here. The present proof establishes existence; when the countability hypothesis in (2) holds, it constructs a single amplification by \\(\\ell^2(\\mathbb N)\\).\n\n**Proof.** (1) Take a maximal family \\(\\{c_k\\}\\) of mutually orthogonal nonzero central projections such that each \\(c_k=\\sum_{j\\in J_k}f_{k,j}\\) for mutually orthogonal projections \\(f_{k,j}\\sim fc_k\\). Suppose \\(c=1-\\sum_kc_k\\ne0\\); then \\(fc\\ne0\\) since \\(c(f)=1\\). Apply Proposition 13.1 in \\(Mc\\) to the one-element family \\(\\{fc\\}\\). It gives a nonzero central \\(c''\\le c\\) and mutually orthogonal, mutually equivalent projections \\(\\{g_j\\}_{j\\in J}\\), one of which is \\(fc''\\), with \\(g_0=c''-\\sum_jg_j\\prec g_j\\). If \\(J\\) were finite, \\(c''=g_0+\\sum_jg_j\\) would be a finite sum of finite projections (each \\(g_j\\sim fc''\\) is finite, and \\(g_0\\precsim g_j\\)), hence finite by Theorem 14.1. But \\(c''\\ne0\\) is infinite, because \\(M\\) is properly infinite. So \\(J\\) is infinite, and Proposition 13.1 lets us take \\(g_0=0\\) with every \\(g_j\\sim fc''\\). Then \\(c''\\) can be added to the family, a contradiction. Hence \\(\\sum_kc_k=1\\), and by the same finiteness argument every \\(J_k\\) is infinite. For an infinite cardinal \\(\\alpha\\) put \\(z_\\alpha=\\sum\\{c_k:\\ |J_k|=\\alpha\\}\\). Fix a set \\(S_\\alpha\\) with \\(|S_\\alpha|=\\alpha\\) and bijections \\(\\beta_k:S_\\alpha\\to J_k\\) for the \\(k\\) with \\(|J_k|=\\alpha\\), and put \\(f_{\\alpha,s}=\\sum_kf_{k,\\beta_k(s)}\\). These are mutually orthogonal, \\(f_{\\alpha,s}\\sim\\sum_kfc_k=fz_\\alpha\\) (additivity over orthogonal central pieces), and \\(\\sum_sf_{\\alpha,s}=z_\\alpha\\). By Proposition 8.4 in \\(Mz_\\alpha\\) and [Lemma 8.5](#oa-fnd-ty-11), \\[\n\\begin{gathered}\n\\{Mz_\\alpha,z_\\alpha H\\}\\\\\n\\cong\\{fz_\\alpha Mfz_\\alpha\\bar\\otimes B(\\ell^2(S_\\alpha)),fz_\\alpha H\\otimes\\ell^2(S_\\alpha)\\}.\n\\end{gathered}\n\\] The algebra \\(fz_\\alpha Mfz_\\alpha\\) is finite, because \\(fz_\\alpha\\) is (Lemma 6.2(1) and (3)). If \\(z_\\alpha\\ne0\\), then \\(fz_\\alpha\\ne0\\) because \\(c(f)=1\\), so the \\(f_{\\alpha,s}\\) are nonzero and mutually orthogonal; they are \\(\\alpha\\) distinct elements of \\(M\\), and \\(\\alpha\\le|M|\\).\n\n(3) Take a maximal family \\(\\{c_k\\}\\) of mutually orthogonal nonzero central projections such that each \\(c_k=\\sum_{n\\in\\mathbb N}p_{k,n}\\) for mutually orthogonal projections \\(p_{k,n}\\sim c_kp\\). Suppose \\(c=1-\\sum_kc_k\\ne0\\), and choose a nonzero central \\(c'\\le c\\) with \\(Mc'\\) \\(\\sigma\\)-finite. In \\(N=Mc'\\), whose unit \\(c'\\) is properly infinite, the projection \\(q=c'p\\) is nonzero. By Corollary 13.5, \\(c'=\\sum_ng_n\\) with \\(g_n\\sim c'\\ge q\\), so each \\(g_n\\) contains a copy of \\(q\\); thus \\(N\\) contains an infinite sequence of mutually orthogonal projections equivalent to \\(q\\). Extend it to a maximal family \\(\\{q_i\\}_{i\\in I}\\) of mutually orthogonal projections equivalent to \\(q\\). The set \\(I\\) is infinite, and countable because \\(N\\) is \\(\\sigma\\)-finite; put \\(r=\\sum_iq_i\\). The comparison theorem (Theorem 5.5) in \\(N\\) gives a central \\(w\\le c'\\) with \\(w(c'-r)\\precsim wq\\) and \\((c'-w)q\\precsim(c'-w)(c'-r)\\). If \\(w=0\\), then \\(q\\precsim c'-r\\), and a copy of \\(q\\) orthogonal to all the \\(q_i\\) would contradict maximality. So \\(w\\ne0\\). Fix \\(i_0\\in I\\) and a bijection \\(\\sigma:I\\to I\\setminus\\{i_0\\}\\). Since \\(w(c'-r)\\precsim wq\\sim wq_{i_0}\\) and \\(wr=\\sum_iwq_i\\sim\\sum_iwq_{\\sigma(i)}\\), additivity gives\n\\[\n\\begin{gathered}\nw\\\\\n=w(c'-r)+wr\\ \\precsim\\ wq_{i_0}+\\sum_{i\\ne i_0}wq_i\\\\\n=wr\\\\\n\\le w ,\n\\end{gathered}\n\\]\nso \\(w\\sim wr\\) (Proposition 5.1). If \\(t\\) implements \\(wr\\sim w\\), the projections \\(t(wq_i)t^*\\) are mutually orthogonal, equivalent to \\(wq=wp\\), and add up to \\(w\\). After relabeling \\(I\\) as \\(\\mathbb N\\), \\(w\\le c\\) can be added to \\(\\{c_k\\}\\), a contradiction. So \\(\\sum_kc_k=1\\). The projections \\(p_n=\\sum_kp_{k,n}\\) are mutually orthogonal, \\(p_n\\sim\\sum_kc_kp=p\\), and \\(\\sum_np_n=\\sum_kc_k=1\\). Proposition 8.4 and [Lemma 8.5](#oa-fnd-ty-11) give \\(M\\cong pMp\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n\n(2) This is (3) with \\(p=f\\). \\(\\square\\)\n\n**Exercise 15.4.** (hard) Let \\(M\\) on \\(H\\) and \\(N\\) on \\(K\\) be von Neumann algebras, with \\(H\\) and \\(K\\) separable, and let \\(\\pi:M\\to N\\) be an isomorphism. Show that \\(x\\otimes1\\mapsto\\pi(x)\\otimes1\\) is a spatial isomorphism of \\(\\{M\\otimes1,H\\otimes\\ell^2(\\mathbb N)\\}\\) onto \\(\\{N\\otimes1,K\\otimes\\ell^2(\\mathbb N)\\}\\). Show that separability can be replaced by \\(\\sigma\\)-finiteness of \\(M'\\) and \\(N'\\), and that it cannot simply be dropped.\n\n*Solution.* On \\(L=(H\\otimes\\ell^2)\\oplus(K\\otimes\\ell^2)\\) put \\(\\sigma(x)=(x\\otimes1)\\oplus(\\pi(x)\\otimes1)\\). By Fact 2.6, \\(\\pi\\) is \\(\\sigma\\)-weakly continuous, so \\(\\sigma\\) is a faithful normal representation, and \\(P=\\sigma(M)\\) is a von Neumann algebra. The projections \\(p=1\\oplus0\\) and \\(q=0\\oplus1\\) lie in \\(P'\\).\n\n- *Central supports.* The centre of \\(P'\\) is that of \\(P\\), namely \\(\\sigma(Z)\\). If \\(c_0\\in Z\\) is a projection with \\(\\sigma(c_0)p=0\\), then \\(c_0\\otimes1=0\\) and \\(c_0=0\\). So the central projection \\(1-c(p)\\), which is orthogonal to \\(p\\), is \\(0\\), and \\(c(p)=1\\). As \\(\\pi\\) is injective, \\(c(q)=1\\) in the same way.\n- *Proper infiniteness.* By Fact 2.1 for \\(P'\\) and \\(p\\), the reduced algebra \\(pP'p\\) on \\(H\\otimes\\ell^2\\) is the commutant of \\(Pp=M\\otimes1\\), which is \\(\\mathbb M_{\\mathbb N}(M')\\) by [Lemma 8.2(1)](#oa-fnd-ty-11). Its central projections are the \\(c_0\\otimes1\\) with \\(c_0\\in Z\\), and each nonzero one is infinite: \\(c_0\\otimes s\\), with \\(s\\) the unilateral shift, implements \\(c_0\\otimes1\\sim c_0\\otimes(1-E_{00})\\). So \\(p\\) is properly infinite (Lemma 6.2(3)), and so is \\(q\\).\n- *\\(\\sigma\\)-finiteness.* If \\(M'\\) is \\(\\sigma\\)-finite, then by Facts 2.3 and 2.4 there is a countable set \\(\\{\\kappa_m\\}\\) that is separating for \\(M'\\), hence cyclic for \\(M\\). The countable set \\(\\{\\kappa_m\\otimes\\delta_n\\}\\) is cyclic for \\(M\\otimes1\\), hence separating for its commutant \\(pP'p\\), so \\(p\\) is \\(\\sigma\\)-finite (Fact 2.4). When \\(H\\) is separable, a countable dense subset of \\(H\\) is cyclic for \\(M\\). The same holds for \\(q\\).\n\nProposition 15.2(2) gives \\(q\\precsim p\\) and \\(p\\precsim q\\), and Proposition 5.1 gives \\(p\\sim q\\) in \\(P'\\). If \\(v\\) implements \\(p\\sim q\\), then \\(V=v|_{pL}:H\\otimes\\ell^2\\to K\\otimes\\ell^2\\) is unitary, and \\(V(x\\otimes1)=(\\pi(x)\\otimes1)V\\) because \\(v\\) commutes with \\(\\sigma(x)\\).\n\nCountability cannot be dropped. Let \\(M=\\mathbb C1\\) on \\(H=\\mathbb C\\), \\(N=\\mathbb C1\\) on \\(K\\) with \\(\\dim K=\\aleph_1\\), and \\(\\pi(\\lambda1)=\\lambda1\\). A unitary \\(H\\otimes\\ell^2\\to K\\otimes\\ell^2\\) would require \\(\\aleph_0=\\aleph_1\\). Here \\(N'=B(K)\\) is not \\(\\sigma\\)-finite. \\(\\square\\)\n\n**Exercise 15.5.** (hard) Let \\(\\mathcal J\\) be a two-sided ideal of \\(M\\), not necessarily closed, and \\(\\mathcal P=\\mathcal P(M)\\).\n\n- (a) If \\(e,f\\in\\mathcal P\\), \\(e\\precsim f\\) and \\(f\\in\\mathcal J\\), then \\(e\\in\\mathcal J\\).\n- (b) \\(\\mathcal P\\cap\\mathcal J\\) is a sublattice of \\(\\mathcal P\\).\n- (c) If \\(\\mathcal P_0\\) is a nonempty sublattice of \\(\\mathcal P\\) such that \\(e\\precsim f\\in\\mathcal P_0\\) implies \\(e\\in\\mathcal P_0\\), then \\(\\mathcal J_0=\\{x\\in M:\\mathrm l(x)\\in\\mathcal P_0\\}\\) is a two-sided ideal and \\(\\mathcal J_0\\cap\\mathcal P=\\mathcal P_0\\).\n- (d) For every two-sided ideal \\(\\mathcal J\\), the ideal \\(\\mathcal J_0=\\{x:\\mathrm l(x)\\in\\mathcal P\\cap\\mathcal J\\}\\) satisfies \\(\\mathcal J_0\\subseteq\\mathcal J\\subseteq\\overline{\\mathcal J_0}\\) (norm closure).\n- (e) The norm-closed two-sided ideals of a factor are totally ordered by inclusion.\n- (f) A factor that is finite, or \\(\\sigma\\)-finite and of type III, has no two-sided ideals other than \\(\\{0\\}\\) and itself; in particular it is simple.\n- (g) Conversely, a factor that is infinite and semifinite, or of type III and not \\(\\sigma\\)-finite, has a proper nonzero norm-closed two-sided ideal. So a factor is simple exactly when it is finite, or \\(\\sigma\\)-finite of type III.\n\n*Solution.* (a) Let \\(u\\) implement \\(e\\sim f_1\\le f\\). Then \\(f_1=f_1f\\in\\mathcal J\\) and \\(e=u^*u=u^*f_1u\\in\\mathcal J\\).\n\n(b) If \\(e,f\\in\\mathcal P\\cap\\mathcal J\\), then \\(e\\wedge f=(e\\wedge f)e\\in\\mathcal J\\). By Proposition 4.4, \\((e\\vee f)-e\\sim f-e\\wedge f\\le f\\), so \\((e\\vee f)-e\\in\\mathcal J\\) by (a), and \\(e\\vee f=e+((e\\vee f)-e)\\in\\mathcal J\\).\n\n(c) Choose \\(f\\in\\mathcal P_0\\), using the nonempty hypothesis. Then \\(0\\precsim f\\) gives \\(0\\in\\mathcal P_0\\). An empty lattice would give an empty \\(\\mathcal J_0\\), which is not an ideal. For \\(x,y\\in M\\) and \\(\\lambda\\ne0\\): \\(\\mathrm l(x+y)\\le\\mathrm l(x)\\vee\\mathrm l(y)\\); \\(\\mathrm l(\\lambda x)=\\mathrm l(x)\\); \\(\\mathrm l(xy)\\le\\mathrm l(x)\\); and \\(\\mathrm l(yx)=\\mathrm l(y\\,\\mathrm l(x))\\sim\\mathrm r(y\\,\\mathrm l(x))\\le\\mathrm l(x)\\) by Proposition 4.3, since \\([yxH]=[y\\,\\mathrm l(x)H]\\). As \\(\\mathcal P_0\\) is a lattice and closed under subequivalence (in particular under \\(\\le\\)), \\(\\mathcal J_0\\) is closed under sums, scalar multiples and products on either side. For \\(e\\in\\mathcal P\\), \\(\\mathrm l(e)=e\\), so \\(\\mathcal J_0\\cap\\mathcal P=\\mathcal P_0\\).\n\n(d) By (a) and (b), \\(\\mathcal P_0=\\mathcal P\\cap\\mathcal J\\) satisfies the hypothesis of (c), so \\(\\mathcal J_0\\) is an ideal. If \\(x\\in\\mathcal J_0\\), then \\(x=\\mathrm l(x)x\\in\\mathcal J\\). Now let \\(x\\in\\mathcal J\\) and \\(\\varepsilon>0\\). The spectral projection \\(p=\\chi_{(\\varepsilon,\\infty)}(xx^*)\\) equals \\(k(xx^*)\\,xx^*\\) with \\(k(t)=t^{-1}\\chi_{(\\varepsilon,\\infty)}(t)\\) bounded (Fact 2.8), so \\(p\\in\\mathcal J\\cap\\mathcal P\\). The element \\(y=px\\) has \\(\\mathrm l(y)\\le p\\), so \\(\\mathrm l(y)=\\mathrm l(y)p\\in\\mathcal J\\) and \\(y\\in\\mathcal J_0\\). Finally \\(\\|x-y\\|^2=\\|(1-p)xx^*(1-p)\\|\\le\\varepsilon\\). So \\(\\mathcal J\\subseteq\\overline{\\mathcal J_0}\\).\n\n(e) By (d), a closed ideal \\(\\mathcal J\\) equals \\(\\overline{\\mathcal J_0}\\), and \\(\\mathcal J_0\\) depends only on \\(\\mathcal P\\cap\\mathcal J\\). So \\(\\mathcal P\\cap\\mathcal J\\subseteq\\mathcal P\\cap\\mathcal K\\) implies \\(\\mathcal J\\subseteq\\mathcal K\\) for closed ideals \\(\\mathcal J,\\mathcal K\\). Suppose \\(\\mathcal P\\cap\\mathcal J\\not\\subseteq\\mathcal K\\), and pick \\(e\\in\\mathcal P\\cap\\mathcal J\\) outside \\(\\mathcal K\\). For every \\(f\\in\\mathcal P\\cap\\mathcal K\\), \\(e\\precsim f\\) is impossible by (a), so \\(f\\prec e\\) by Theorem 5.5 (\\(M\\) is a factor), and \\(f\\in\\mathcal J\\) by (a). Hence \\(\\mathcal P\\cap\\mathcal K\\subseteq\\mathcal P\\cap\\mathcal J\\), and \\(\\mathcal K\\subseteq\\mathcal J\\).\n\n(f) Let \\(\\mathcal J\\ne\\{0\\}\\) and \\(0\\ne x\\in\\mathcal J\\). As in (d), for small \\(\\varepsilon\\) the projection \\(p=\\chi_{(\\varepsilon,\\infty)}(xx^*)\\) is a nonzero element of \\(\\mathcal J\\). If \\(M\\) is \\(\\sigma\\)-finite of type III, \\(p\\) is infinite, so \\(p\\sim1\\) by Proposition 15.2(4), and \\(1\\in\\mathcal J\\) by (a). If \\(M\\) is finite, apply Proposition 13.1 to the family \\(\\{p\\}\\). As \\(M\\) is a factor, \\(z=1\\), and we get mutually orthogonal projections \\(\\{f_j\\}_{j\\in J}\\) equivalent to \\(p\\) with \\(f_0=1-\\sum_jf_j\\prec f_j\\). The set \\(J\\) is finite: otherwise \\(\\sum_jf_j\\sim\\sum_{j\\ne j_0}f_j<\\sum_jf_j\\) would be an infinite projection in a finite algebra. So \\(1=f_0+\\sum_jf_j\\) is a finite sum of projections subequivalent to \\(p\\), all in \\(\\mathcal J\\) by (a), and \\(1\\in\\mathcal J\\).\n\n(g) Let \\(\\mathcal P_0\\) be the set of finite projections (if \\(M\\) is infinite and semifinite) or of \\(\\sigma\\)-finite projections (if \\(M\\) is of type III and not \\(\\sigma\\)-finite). In both cases \\(\\mathcal P_0\\) is a sublattice closed under subequivalence. For finite projections this is Theorem 14.1 with Lemma 6.2(1). For \\(\\sigma\\)-finite ones: a projection subequivalent to a \\(\\sigma\\)-finite one is \\(\\sigma\\)-finite; and \\(e\\vee f=e+((e\\vee f)-e)\\) with \\((e\\vee f)-e\\precsim f\\) (Proposition 4.4), where a sum of two orthogonal \\(\\sigma\\)-finite projections \\(e,h\\) is \\(\\sigma\\)-finite, because faithful normal positive functionals \\(\\varphi_1\\) on \\(eMe\\) and \\(\\varphi_2\\) on \\(hMh\\) (Fact 2.4) give the faithful normal functional \\(x\\mapsto\\varphi_1(exe)+\\varphi_2(hxh)\\) on \\((e+h)M(e+h)\\). For faithfulness, if \\(x\\ge0\\) has value zero, faithfulness on the two corners gives \\(exe=hxh=0\\). Hence \\(x^{1/2}e=x^{1/2}h=0\\), so \\(x^{1/2}(e+h)=0\\) and \\(x=0\\). Normality follows from the normality of the two corner functionals and fixed compression. By (c), \\(\\mathcal J_0=\\{x:\\mathrm l(x)\\in\\mathcal P_0\\}\\) is an ideal. It is nonzero: a semifinite algebra has nonzero finite projections, and every cyclic projection \\(p_\\xi\\) with \\(\\xi\\ne0\\) is \\(\\sigma\\)-finite, because \\(\\omega_\\xi\\) is faithful on \\(p_\\xi Mp_\\xi\\) (Fact 2.4). Its norm closure is a closed ideal, and it is proper: if \\(\\|1-x\\|<1\\) for some \\(x\\in\\mathcal J_0\\), then \\(x\\) is invertible, \\(\\mathrm l(x)=1\\), and \\(1\\) would be finite (respectively \\(\\sigma\\)-finite). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "15. Countable sums and properly infinite algebras",
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      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 15. Countable sums and properly infinite algebras\n\nA properly infinite projection absorbs countably many projections that are subequivalent to it. Under a countability hypothesis this gives the normal form \\(M\\cong N\\bar\\otimes B(\\ell^2(\\mathbb N))\\) for properly infinite algebras.\n\n**Definition 15.1** (Locally \\(\\sigma\\)-finite projections). A projection \\(f\\) is *locally \\(\\sigma\\)-finite* if every central projection \\(z\\) with \\(zf\\ne0\\) majorizes a central projection \\(z'\\) such that \\(z'f\\) is nonzero and \\(\\sigma\\)-finite. \\(M\\) is *locally \\(\\sigma\\)-finite* if \\(1\\) is. Every \\(\\sigma\\)-finite projection is locally \\(\\sigma\\)-finite.\n\nBy Zorn's lemma, \\(M\\) is locally \\(\\sigma\\)-finite exactly when \\(1\\) is a sum of mutually orthogonal central projections \\(z_k\\) with every \\(Mz_k\\) \\(\\sigma\\)-finite. Indeed, if \\(M\\) is locally \\(\\sigma\\)-finite, a maximal family of mutually orthogonal nonzero central projections \\(z_k\\) with \\(Mz_k\\) \\(\\sigma\\)-finite has sum \\(1\\), since otherwise \\(1-\\sum_kz_k\\) would majorize a further member. Conversely, if \\(1=\\sum_kz_k\\) with every \\(Mz_k\\) \\(\\sigma\\)-finite, and \\(z\\) is a nonzero central projection, then \\(zz_k\\ne0\\) for some \\(k\\), and \\(Mzz_k\\) is \\(\\sigma\\)-finite, since orthogonal families in \\(Mzz_k\\) are orthogonal families in \\(Mz_k\\). Such algebras are also called *locally countably decomposable*.\n\n**Proposition 15.2** (Properly infinite projections absorb countable sums). Let \\(e\\) be a properly infinite projection.\n\n1. If \\(f_1,f_2,\\ldots\\) are mutually orthogonal projections with \\(f_n\\precsim e\\), then \\(\\sum_nf_n\\precsim e\\).\n2. If \\(f\\) is \\(\\sigma\\)-finite and \\(c(f)\\le c(e)\\), then \\(f\\precsim e\\).\n3. If \\(f\\) is locally \\(\\sigma\\)-finite and \\(c(f)\\le c(e)\\), then \\(f\\precsim e\\).\n4. If \\(M\\) is locally \\(\\sigma\\)-finite and \\(c(e)=1\\), then \\(e\\sim1\\). In particular, in a \\(\\sigma\\)-finite factor any two infinite projections are equivalent.\n\nThe countability conditions cannot be dropped ([Example 16.4](#oa-fnd-ty-22)).\n\n**Proof.** (1) By Corollary 13.5, \\(e=\\sum_ne_n\\) with mutually orthogonal \\(e_n\\sim e\\). Then \\(f_n\\precsim e\\sim e_n\\), and additivity gives \\(\\sum_nf_n\\precsim\\sum_ne_n=e\\).\n\n(2) Take a maximal family \\(\\{f_k\\}\\) of mutually orthogonal nonzero subprojections of \\(f\\) with \\(f_k\\precsim e\\). It is countable, because \\(f\\) is \\(\\sigma\\)-finite. If \\(r=f-\\sum_kf_k\\) were nonzero, then \\(c(r)\\le c(f)\\le c(e)\\), and Corollary 5.4 would give a nonzero \\(r_1\\le r\\) with \\(r_1\\precsim e\\), against maximality. So \\(f=\\sum_kf_k\\), and (1) applies (a finite family is padded with zeros).\n\n(3) Take a maximal family \\(\\{z_k\\}\\) of mutually orthogonal central projections below \\(c(f)\\) such that each \\(z_kf\\) is nonzero and \\(\\sigma\\)-finite. If \\(z=c(f)-\\sum_kz_k\\) were nonzero, then \\(zf\\ne0\\) ([Proposition 3.5(1)](#oa-fnd-ty-02)), and the hypothesis would give a further member of the family. So \\(f=\\sum_kz_kf\\). Each \\(z_ke\\) is nonzero, because \\(0\\ne z_k\\le c(e)\\), and it is properly infinite (Lemma 6.2(3)); moreover \\(c(z_kf)=z_kc(f)\\le z_kc(e)=c(z_ke)\\). By (2), \\(z_kf\\precsim z_ke\\), and additivity over the orthogonal central pieces gives \\(f\\precsim\\sum_kz_ke\\le e\\).\n\n(4) The projection \\(1\\) is locally \\(\\sigma\\)-finite and \\(c(1)=1=c(e)\\), so \\(1\\precsim e\\) by (3), and \\(e\\sim1\\) by Proposition 5.1. In a factor, infinite projections are properly infinite, nonzero projections have central support \\(1\\), and in a \\(\\sigma\\)-finite factor every projection is \\(\\sigma\\)-finite; so two infinite projections are subequivalent to each other, hence equivalent. \\(\\square\\)\n\n**Proposition 15.3** (Properly infinite semifinite algebras). Let \\(M\\) be properly infinite and semifinite, and let \\(f\\) be a finite projection with \\(c(f)=1\\) ([Lemma 7.4(2)](#oa-fnd-ty-10)).\n\n1. There are mutually orthogonal central projections \\(z_\\alpha\\), indexed by the infinite cardinals \\(\\alpha\\le|M|\\), some possibly \\(0\\), with \\(\\sum_\\alpha z_\\alpha=1\\) and \\[\n\\begin{gathered}\n\\{Mz_\\alpha,z_\\alpha H\\}\\\\\n\\cong\\{N_\\alpha\\bar\\otimes B(\\ell^2(\\alpha)),\\,fz_\\alpha H\\otimes\\ell^2(\\alpha)\\},\n\\end{gathered}\n\\] where \\(N_\\alpha=fz_\\alpha Mfz_\\alpha\\) is finite. The \\(N_\\alpha\\) are not unique ([Example 16.5](#oa-fnd-ty-22)).\n2. If \\(M\\) is locally \\(\\sigma\\)-finite, then \\(M\\cong fMf\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n3. More generally, if \\(M\\) is properly infinite and locally \\(\\sigma\\)-finite (not necessarily semifinite) and \\(p\\) is any projection with \\(c(p)=1\\), then \\(1=\\sum_{n\\in\\mathbb N}p_n\\) with mutually orthogonal \\(p_n\\sim p\\), and \\(M\\cong pMp\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n\n*Reference:* The decomposition of \\(1\\) in part (3) is [Blackadar, III.1.3.6].\n\nThe further trace theory studies uniqueness of the family \\(\\{z_\\alpha\\}\\) in (1); see Traces on von Neumann algebras. That extension is not proved or used here. The present proof establishes existence; when the countability hypothesis in (2) holds, it constructs a single amplification by \\(\\ell^2(\\mathbb N)\\).\n\n**Proof.** (1) Take a maximal family \\(\\{c_k\\}\\) of mutually orthogonal nonzero central projections such that each \\(c_k=\\sum_{j\\in J_k}f_{k,j}\\) for mutually orthogonal projections \\(f_{k,j}\\sim fc_k\\). Suppose \\(c=1-\\sum_kc_k\\ne0\\); then \\(fc\\ne0\\) since \\(c(f)=1\\). Apply Proposition 13.1 in \\(Mc\\) to the one-element family \\(\\{fc\\}\\). It gives a nonzero central \\(c''\\le c\\) and mutually orthogonal, mutually equivalent projections \\(\\{g_j\\}_{j\\in J}\\), one of which is \\(fc''\\), with \\(g_0=c''-\\sum_jg_j\\prec g_j\\). If \\(J\\) were finite, \\(c''=g_0+\\sum_jg_j\\) would be a finite sum of finite projections (each \\(g_j\\sim fc''\\) is finite, and \\(g_0\\precsim g_j\\)), hence finite by Theorem 14.1. But \\(c''\\ne0\\) is infinite, because \\(M\\) is properly infinite. So \\(J\\) is infinite, and Proposition 13.1 lets us take \\(g_0=0\\) with every \\(g_j\\sim fc''\\). Then \\(c''\\) can be added to the family, a contradiction. Hence \\(\\sum_kc_k=1\\), and by the same finiteness argument every \\(J_k\\) is infinite. For an infinite cardinal \\(\\alpha\\) put \\(z_\\alpha=\\sum\\{c_k:\\ |J_k|=\\alpha\\}\\). Fix a set \\(S_\\alpha\\) with \\(|S_\\alpha|=\\alpha\\) and bijections \\(\\beta_k:S_\\alpha\\to J_k\\) for the \\(k\\) with \\(|J_k|=\\alpha\\), and put \\(f_{\\alpha,s}=\\sum_kf_{k,\\beta_k(s)}\\). These are mutually orthogonal, \\(f_{\\alpha,s}\\sim\\sum_kfc_k=fz_\\alpha\\) (additivity over orthogonal central pieces), and \\(\\sum_sf_{\\alpha,s}=z_\\alpha\\). By Proposition 8.4 in \\(Mz_\\alpha\\) and [Lemma 8.5](#oa-fnd-ty-11), \\[\n\\begin{gathered}\n\\{Mz_\\alpha,z_\\alpha H\\}\\\\\n\\cong\\{fz_\\alpha Mfz_\\alpha\\bar\\otimes B(\\ell^2(S_\\alpha)),fz_\\alpha H\\otimes\\ell^2(S_\\alpha)\\}.\n\\end{gathered}\n\\] The algebra \\(fz_\\alpha Mfz_\\alpha\\) is finite, because \\(fz_\\alpha\\) is (Lemma 6.2(1) and (3)). If \\(z_\\alpha\\ne0\\), then \\(fz_\\alpha\\ne0\\) because \\(c(f)=1\\), so the \\(f_{\\alpha,s}\\) are nonzero and mutually orthogonal; they are \\(\\alpha\\) distinct elements of \\(M\\), and \\(\\alpha\\le|M|\\).\n\n(3) Take a maximal family \\(\\{c_k\\}\\) of mutually orthogonal nonzero central projections such that each \\(c_k=\\sum_{n\\in\\mathbb N}p_{k,n}\\) for mutually orthogonal projections \\(p_{k,n}\\sim c_kp\\). Suppose \\(c=1-\\sum_kc_k\\ne0\\), and choose a nonzero central \\(c'\\le c\\) with \\(Mc'\\) \\(\\sigma\\)-finite. In \\(N=Mc'\\), whose unit \\(c'\\) is properly infinite, the projection \\(q=c'p\\) is nonzero. By Corollary 13.5, \\(c'=\\sum_ng_n\\) with \\(g_n\\sim c'\\ge q\\), so each \\(g_n\\) contains a copy of \\(q\\); thus \\(N\\) contains an infinite sequence of mutually orthogonal projections equivalent to \\(q\\). Extend it to a maximal family \\(\\{q_i\\}_{i\\in I}\\) of mutually orthogonal projections equivalent to \\(q\\). The set \\(I\\) is infinite, and countable because \\(N\\) is \\(\\sigma\\)-finite; put \\(r=\\sum_iq_i\\). The comparison theorem (Theorem 5.5) in \\(N\\) gives a central \\(w\\le c'\\) with \\(w(c'-r)\\precsim wq\\) and \\((c'-w)q\\precsim(c'-w)(c'-r)\\). If \\(w=0\\), then \\(q\\precsim c'-r\\), and a copy of \\(q\\) orthogonal to all the \\(q_i\\) would contradict maximality. So \\(w\\ne0\\). Fix \\(i_0\\in I\\) and a bijection \\(\\sigma:I\\to I\\setminus\\{i_0\\}\\). Since \\(w(c'-r)\\precsim wq\\sim wq_{i_0}\\) and \\(wr=\\sum_iwq_i\\sim\\sum_iwq_{\\sigma(i)}\\), additivity gives\n\\[\n\\begin{gathered}\nw\\\\\n=w(c'-r)+wr\\ \\precsim\\ wq_{i_0}+\\sum_{i\\ne i_0}wq_i\\\\\n=wr\\\\\n\\le w ,\n\\end{gathered}\n\\]\nso \\(w\\sim wr\\) (Proposition 5.1). If \\(t\\) implements \\(wr\\sim w\\), the projections \\(t(wq_i)t^*\\) are mutually orthogonal, equivalent to \\(wq=wp\\), and add up to \\(w\\). After relabeling \\(I\\) as \\(\\mathbb N\\), \\(w\\le c\\) can be added to \\(\\{c_k\\}\\), a contradiction. So \\(\\sum_kc_k=1\\). The projections \\(p_n=\\sum_kp_{k,n}\\) are mutually orthogonal, \\(p_n\\sim\\sum_kc_kp=p\\), and \\(\\sum_np_n=\\sum_kc_k=1\\). Proposition 8.4 and [Lemma 8.5](#oa-fnd-ty-11) give \\(M\\cong pMp\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n\n(2) This is (3) with \\(p=f\\). \\(\\square\\)\n\n**Exercise 15.4.** (hard) Let \\(M\\) on \\(H\\) and \\(N\\) on \\(K\\) be von Neumann algebras, with \\(H\\) and \\(K\\) separable, and let \\(\\pi:M\\to N\\) be an isomorphism. Show that \\(x\\otimes1\\mapsto\\pi(x)\\otimes1\\) is a spatial isomorphism of \\(\\{M\\otimes1,H\\otimes\\ell^2(\\mathbb N)\\}\\) onto \\(\\{N\\otimes1,K\\otimes\\ell^2(\\mathbb N)\\}\\). Show that separability can be replaced by \\(\\sigma\\)-finiteness of \\(M'\\) and \\(N'\\), and that it cannot simply be dropped.\n\n*Solution.* On \\(L=(H\\otimes\\ell^2)\\oplus(K\\otimes\\ell^2)\\) put \\(\\sigma(x)=(x\\otimes1)\\oplus(\\pi(x)\\otimes1)\\). By Fact 2.6, \\(\\pi\\) is \\(\\sigma\\)-weakly continuous, so \\(\\sigma\\) is a faithful normal representation, and \\(P=\\sigma(M)\\) is a von Neumann algebra. The projections \\(p=1\\oplus0\\) and \\(q=0\\oplus1\\) lie in \\(P'\\).\n\n- *Central supports.* The centre of \\(P'\\) is that of \\(P\\), namely \\(\\sigma(Z)\\). If \\(c_0\\in Z\\) is a projection with \\(\\sigma(c_0)p=0\\), then \\(c_0\\otimes1=0\\) and \\(c_0=0\\). So the central projection \\(1-c(p)\\), which is orthogonal to \\(p\\), is \\(0\\), and \\(c(p)=1\\). As \\(\\pi\\) is injective, \\(c(q)=1\\) in the same way.\n- *Proper infiniteness.* By Fact 2.1 for \\(P'\\) and \\(p\\), the reduced algebra \\(pP'p\\) on \\(H\\otimes\\ell^2\\) is the commutant of \\(Pp=M\\otimes1\\), which is \\(\\mathbb M_{\\mathbb N}(M')\\) by [Lemma 8.2(1)](#oa-fnd-ty-11). Its central projections are the \\(c_0\\otimes1\\) with \\(c_0\\in Z\\), and each nonzero one is infinite: \\(c_0\\otimes s\\), with \\(s\\) the unilateral shift, implements \\(c_0\\otimes1\\sim c_0\\otimes(1-E_{00})\\). So \\(p\\) is properly infinite (Lemma 6.2(3)), and so is \\(q\\).\n- *\\(\\sigma\\)-finiteness.* If \\(M'\\) is \\(\\sigma\\)-finite, then by Facts 2.3 and 2.4 there is a countable set \\(\\{\\kappa_m\\}\\) that is separating for \\(M'\\), hence cyclic for \\(M\\). The countable set \\(\\{\\kappa_m\\otimes\\delta_n\\}\\) is cyclic for \\(M\\otimes1\\), hence separating for its commutant \\(pP'p\\), so \\(p\\) is \\(\\sigma\\)-finite (Fact 2.4). When \\(H\\) is separable, a countable dense subset of \\(H\\) is cyclic for \\(M\\). The same holds for \\(q\\).\n\nProposition 15.2(2) gives \\(q\\precsim p\\) and \\(p\\precsim q\\), and Proposition 5.1 gives \\(p\\sim q\\) in \\(P'\\). If \\(v\\) implements \\(p\\sim q\\), then \\(V=v|_{pL}:H\\otimes\\ell^2\\to K\\otimes\\ell^2\\) is unitary, and \\(V(x\\otimes1)=(\\pi(x)\\otimes1)V\\) because \\(v\\) commutes with \\(\\sigma(x)\\).\n\nCountability cannot be dropped. Let \\(M=\\mathbb C1\\) on \\(H=\\mathbb C\\), \\(N=\\mathbb C1\\) on \\(K\\) with \\(\\dim K=\\aleph_1\\), and \\(\\pi(\\lambda1)=\\lambda1\\). A unitary \\(H\\otimes\\ell^2\\to K\\otimes\\ell^2\\) would require \\(\\aleph_0=\\aleph_1\\). Here \\(N'=B(K)\\) is not \\(\\sigma\\)-finite. \\(\\square\\)\n\n**Exercise 15.5.** (hard) Let \\(\\mathcal J\\) be a two-sided ideal of \\(M\\), not necessarily closed, and \\(\\mathcal P=\\mathcal P(M)\\).\n\n- (a) If \\(e,f\\in\\mathcal P\\), \\(e\\precsim f\\) and \\(f\\in\\mathcal J\\), then \\(e\\in\\mathcal J\\).\n- (b) \\(\\mathcal P\\cap\\mathcal J\\) is a sublattice of \\(\\mathcal P\\).\n- (c) If \\(\\mathcal P_0\\) is a nonempty sublattice of \\(\\mathcal P\\) such that \\(e\\precsim f\\in\\mathcal P_0\\) implies \\(e\\in\\mathcal P_0\\), then \\(\\mathcal J_0=\\{x\\in M:\\mathrm l(x)\\in\\mathcal P_0\\}\\) is a two-sided ideal and \\(\\mathcal J_0\\cap\\mathcal P=\\mathcal P_0\\).\n- (d) For every two-sided ideal \\(\\mathcal J\\), the ideal \\(\\mathcal J_0=\\{x:\\mathrm l(x)\\in\\mathcal P\\cap\\mathcal J\\}\\) satisfies \\(\\mathcal J_0\\subseteq\\mathcal J\\subseteq\\overline{\\mathcal J_0}\\) (norm closure).\n- (e) The norm-closed two-sided ideals of a factor are totally ordered by inclusion.\n- (f) A factor that is finite, or \\(\\sigma\\)-finite and of type III, has no two-sided ideals other than \\(\\{0\\}\\) and itself; in particular it is simple.\n- (g) Conversely, a factor that is infinite and semifinite, or of type III and not \\(\\sigma\\)-finite, has a proper nonzero norm-closed two-sided ideal. So a factor is simple exactly when it is finite, or \\(\\sigma\\)-finite of type III.\n\n*Solution.* (a) Let \\(u\\) implement \\(e\\sim f_1\\le f\\). Then \\(f_1=f_1f\\in\\mathcal J\\) and \\(e=u^*u=u^*f_1u\\in\\mathcal J\\).\n\n(b) If \\(e,f\\in\\mathcal P\\cap\\mathcal J\\), then \\(e\\wedge f=(e\\wedge f)e\\in\\mathcal J\\). By Proposition 4.4, \\((e\\vee f)-e\\sim f-e\\wedge f\\le f\\), so \\((e\\vee f)-e\\in\\mathcal J\\) by (a), and \\(e\\vee f=e+((e\\vee f)-e)\\in\\mathcal J\\).\n\n(c) Choose \\(f\\in\\mathcal P_0\\), using the nonempty hypothesis. Then \\(0\\precsim f\\) gives \\(0\\in\\mathcal P_0\\). An empty lattice would give an empty \\(\\mathcal J_0\\), which is not an ideal. For \\(x,y\\in M\\) and \\(\\lambda\\ne0\\): \\(\\mathrm l(x+y)\\le\\mathrm l(x)\\vee\\mathrm l(y)\\); \\(\\mathrm l(\\lambda x)=\\mathrm l(x)\\); \\(\\mathrm l(xy)\\le\\mathrm l(x)\\); and \\(\\mathrm l(yx)=\\mathrm l(y\\,\\mathrm l(x))\\sim\\mathrm r(y\\,\\mathrm l(x))\\le\\mathrm l(x)\\) by Proposition 4.3, since \\([yxH]=[y\\,\\mathrm l(x)H]\\). As \\(\\mathcal P_0\\) is a lattice and closed under subequivalence (in particular under \\(\\le\\)), \\(\\mathcal J_0\\) is closed under sums, scalar multiples and products on either side. For \\(e\\in\\mathcal P\\), \\(\\mathrm l(e)=e\\), so \\(\\mathcal J_0\\cap\\mathcal P=\\mathcal P_0\\).\n\n(d) By (a) and (b), \\(\\mathcal P_0=\\mathcal P\\cap\\mathcal J\\) satisfies the hypothesis of (c), so \\(\\mathcal J_0\\) is an ideal. If \\(x\\in\\mathcal J_0\\), then \\(x=\\mathrm l(x)x\\in\\mathcal J\\). Now let \\(x\\in\\mathcal J\\) and \\(\\varepsilon>0\\). The spectral projection \\(p=\\chi_{(\\varepsilon,\\infty)}(xx^*)\\) equals \\(k(xx^*)\\,xx^*\\) with \\(k(t)=t^{-1}\\chi_{(\\varepsilon,\\infty)}(t)\\) bounded (Fact 2.8), so \\(p\\in\\mathcal J\\cap\\mathcal P\\). The element \\(y=px\\) has \\(\\mathrm l(y)\\le p\\), so \\(\\mathrm l(y)=\\mathrm l(y)p\\in\\mathcal J\\) and \\(y\\in\\mathcal J_0\\). Finally \\(\\|x-y\\|^2=\\|(1-p)xx^*(1-p)\\|\\le\\varepsilon\\). So \\(\\mathcal J\\subseteq\\overline{\\mathcal J_0}\\).\n\n(e) By (d), a closed ideal \\(\\mathcal J\\) equals \\(\\overline{\\mathcal J_0}\\), and \\(\\mathcal J_0\\) depends only on \\(\\mathcal P\\cap\\mathcal J\\). So \\(\\mathcal P\\cap\\mathcal J\\subseteq\\mathcal P\\cap\\mathcal K\\) implies \\(\\mathcal J\\subseteq\\mathcal K\\) for closed ideals \\(\\mathcal J,\\mathcal K\\). Suppose \\(\\mathcal P\\cap\\mathcal J\\not\\subseteq\\mathcal K\\), and pick \\(e\\in\\mathcal P\\cap\\mathcal J\\) outside \\(\\mathcal K\\). For every \\(f\\in\\mathcal P\\cap\\mathcal K\\), \\(e\\precsim f\\) is impossible by (a), so \\(f\\prec e\\) by Theorem 5.5 (\\(M\\) is a factor), and \\(f\\in\\mathcal J\\) by (a). Hence \\(\\mathcal P\\cap\\mathcal K\\subseteq\\mathcal P\\cap\\mathcal J\\), and \\(\\mathcal K\\subseteq\\mathcal J\\).\n\n(f) Let \\(\\mathcal J\\ne\\{0\\}\\) and \\(0\\ne x\\in\\mathcal J\\). As in (d), for small \\(\\varepsilon\\) the projection \\(p=\\chi_{(\\varepsilon,\\infty)}(xx^*)\\) is a nonzero element of \\(\\mathcal J\\). If \\(M\\) is \\(\\sigma\\)-finite of type III, \\(p\\) is infinite, so \\(p\\sim1\\) by Proposition 15.2(4), and \\(1\\in\\mathcal J\\) by (a). If \\(M\\) is finite, apply Proposition 13.1 to the family \\(\\{p\\}\\). As \\(M\\) is a factor, \\(z=1\\), and we get mutually orthogonal projections \\(\\{f_j\\}_{j\\in J}\\) equivalent to \\(p\\) with \\(f_0=1-\\sum_jf_j\\prec f_j\\). The set \\(J\\) is finite: otherwise \\(\\sum_jf_j\\sim\\sum_{j\\ne j_0}f_j<\\sum_jf_j\\) would be an infinite projection in a finite algebra. So \\(1=f_0+\\sum_jf_j\\) is a finite sum of projections subequivalent to \\(p\\), all in \\(\\mathcal J\\) by (a), and \\(1\\in\\mathcal J\\).\n\n(g) Let \\(\\mathcal P_0\\) be the set of finite projections (if \\(M\\) is infinite and semifinite) or of \\(\\sigma\\)-finite projections (if \\(M\\) is of type III and not \\(\\sigma\\)-finite). In both cases \\(\\mathcal P_0\\) is a sublattice closed under subequivalence. For finite projections this is Theorem 14.1 with Lemma 6.2(1). For \\(\\sigma\\)-finite ones: a projection subequivalent to a \\(\\sigma\\)-finite one is \\(\\sigma\\)-finite; and \\(e\\vee f=e+((e\\vee f)-e)\\) with \\((e\\vee f)-e\\precsim f\\) (Proposition 4.4), where a sum of two orthogonal \\(\\sigma\\)-finite projections \\(e,h\\) is \\(\\sigma\\)-finite, because faithful normal positive functionals \\(\\varphi_1\\) on \\(eMe\\) and \\(\\varphi_2\\) on \\(hMh\\) (Fact 2.4) give the faithful normal functional \\(x\\mapsto\\varphi_1(exe)+\\varphi_2(hxh)\\) on \\((e+h)M(e+h)\\). For faithfulness, if \\(x\\ge0\\) has value zero, faithfulness on the two corners gives \\(exe=hxh=0\\). Hence \\(x^{1/2}e=x^{1/2}h=0\\), so \\(x^{1/2}(e+h)=0\\) and \\(x=0\\). Normality follows from the normality of the two corner functionals and fixed compression. By (c), \\(\\mathcal J_0=\\{x:\\mathrm l(x)\\in\\mathcal P_0\\}\\) is an ideal. It is nonzero: a semifinite algebra has nonzero finite projections, and every cyclic projection \\(p_\\xi\\) with \\(\\xi\\ne0\\) is \\(\\sigma\\)-finite, because \\(\\omega_\\xi\\) is faithful on \\(p_\\xi Mp_\\xi\\) (Fact 2.4). Its norm closure is a closed ideal, and it is proper: if \\(\\|1-x\\|<1\\) for some \\(x\\in\\mathcal J_0\\), then \\(x\\) is invertible, \\(\\mathrm l(x)=1\\), and \\(1\\) would be finite (respectively \\(\\sigma\\)-finite). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-TY-22",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "16. Counterexamples",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "source": "src/projections-and-types-of-von-neumann-algebras.md",
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      "full_conditions_and_proof": "## 16. Counterexamples\n\nThe examples below show that several hypotheses in Sections 13–15 cannot be dropped, and that the corners in Proposition 15.3(1) are not unique.\n\n**Example 16.1** (A family of zero projections). If \\(I\\ne\\emptyset\\) and all \\(e_i=0\\), then \\(f_i\\sim ze_i=0\\), so every \\(f_j=0\\), and \\(f_0=z\\prec0\\) is impossible (\\(z\\precsim0\\) forces \\(z=0\\sim0\\)). This is why Proposition 13.1 asks for nonzero \\(e_i\\).\n\n**Example 16.2** (Modularity needs finiteness). Let \\(K=\\ell^2(\\mathbb N)\\) and \\(T\\delta_n=\\delta_n/(n+1)\\). Then \\(T\\) is injective with dense range, and \\(w=\\sum_n\\delta_n/(n+1)\\) is not in the range of \\(T\\) (its preimage would be \\(\\sum_n\\delta_n\\notin K\\)). On \\(H=K\\oplus K\\), let \\(e\\) be the projection onto \\(K\\oplus0\\), \\(f\\) the projection onto the graph \\(\\{(\\xi,T\\xi)\\}\\), and \\(g\\) the projection onto \\((K\\oplus0)+\\mathbb C(0,w)\\); these subspaces are closed, and \\(e\\le g\\). If \\((\\xi,T\\xi)=(x,\\lambda w)\\), then \\(T\\xi=\\lambda w\\) forces \\(\\lambda=0\\) and \\(\\xi=0\\); so \\(f\\wedge g=0\\) and \\(e\\vee(f\\wedge g)=e\\). But \\((K\\oplus0)+\\text{graph}=K\\oplus T(K)\\) is dense in \\(H\\), so \\(e\\vee f=1\\) and \\((e\\vee f)\\wedge g=g\\ne e\\). So the modular law fails in \\(B(H)\\), with \\(g\\) infinite. Exercise 16.6 extends this to every algebra that is not finite.\n\n**Example 16.3** (The shift). On \\(\\ell^2(\\mathbb N)\\) the unilateral shift \\(s\\) satisfies \\(s^*s=1\\) and \\(ss^*=1-p_0\\), where \\(p_0\\) is the projection onto \\(\\mathbb C\\delta_0\\). So \\(1\\sim1-p_0\\), and \\(1\\) is infinite. The projections \\(1\\) and \\(1-p_0\\) are equivalent but not unitarily equivalent, since their complements \\(0\\) and \\(p_0\\) are not equivalent. So Proposition 14.2 needs finiteness.\n\n**Example 16.4** (Countability in Propositions 15.2 and 15.3). Let \\(H\\) have Hilbert dimension \\(\\aleph_1\\), \\(M=B(H)\\), and \\(e\\) the projection onto a separable infinite-dimensional subspace. Then \\(e\\) is properly infinite (\\(M\\) is a factor and \\(e\\) is infinite), and \\(c(e)=1=c(1)\\). But \\(1\\not\\precsim e\\), since an isometry of \\(H\\) into \\(eH\\) would give \\(\\dim H\\le\\aleph_0\\). So the conclusions of Proposition 15.2(2)–(3) fail for \\(f=1\\), which is neither \\(\\sigma\\)-finite nor locally \\(\\sigma\\)-finite. Also \\(M\\) is properly infinite and semifinite, and a rank-one \\(f\\) is finite with \\(c(f)=1\\), but \\(M\\not\\cong fMf\\bar\\otimes B(\\ell^2(\\mathbb N))=B(\\ell^2(\\mathbb N))\\) by Corollary 10.4. So Proposition 15.3(2) needs its countability hypothesis. In Proposition 15.3(1), \\(z_{\\aleph_1}=1\\).\n\n**Example 16.5** (Non-uniqueness in Proposition 15.3). \\(B(\\ell^2(\\mathbb N))\\cong\\mathbb C\\bar\\otimes B(\\ell^2(\\mathbb N))\\cong M_2(\\mathbb C)\\bar\\otimes B(\\ell^2(\\mathbb N))\\), because \\(\\mathbb C^2\\otimes\\ell^2(\\mathbb N)\\cong\\ell^2(\\mathbb N)\\). Both \\(\\mathbb C\\) and \\(M_2(\\mathbb C)\\) are finite; they are the corners \\(fMf\\) for an \\(f\\) of rank one and of rank two.\n\n**Exercise 16.6.** (hard) Show that the projection lattice \\(\\mathcal P(M)\\) is modular if and only if \\(M\\) is finite.\n\n*Solution.* If \\(M\\) is finite, every projection is finite, and Theorem 14.1 applies. Suppose \\(M\\) is not finite. By Theorem 7.2 the properly infinite part \\(z\\) of the projection \\(1\\) is a nonzero central projection, and \\(Mz\\) is properly infinite. By Corollary 13.5, \\(z=\\sum_{n\\in\\mathbb N}q_n\\) with mutually orthogonal \\(q_n\\sim z\\), and Proposition 8.4 gives a unitary \\(W\\) with \\(W^*(Mz)W=\\mathbb M_{\\mathbb N}(N)\\), \\(N=q_1Mq_1\\) on \\(L=q_1H\\). So \\(Mz\\) contains \\(W(1\\otimes B(\\ell^2(\\mathbb N)))W^*\\). For projections \\(a,b\\) on \\(\\ell^2(\\mathbb N)\\), the range of \\(1\\otimes a\\) is \\(L\\otimes a\\ell^2\\), and \\((L\\otimes A)\\cap(L\\otimes B)=L\\otimes(A\\cap B)\\) for closed subspaces \\(A,B\\) (decompose along an orthonormal basis of \\(L\\)). So \\(a\\mapsto W(1\\otimes a)W^*\\) preserves meets, and by De Morgan's laws joins, which in \\(\\mathcal P(M)\\) are those of \\(B(H)\\) (Proposition 3.1). Identify \\(\\ell^2(\\mathbb N)\\) with \\(K\\oplus K\\) as in [Example 16.2](#oa-fnd-ty-22), and take the projections \\(e\\le g\\) and \\(f\\) found there. Their images violate the modular law in \\(\\mathcal P(M)\\). \\(\\square\\)\n\n## 17. Comparison with the commutant: the transfer theorem\n\nThis section compares the cyclic projections of \\(M\\) (Definition 4.5) with those of \\(M'\\). The main result, the transfer theorem, says that \\(p_\\xi\\succsim p_\\eta\\) in \\(M\\) exactly when \\(p'_\\xi\\succsim p'_\\eta\\) in \\(M'\\). Its key step rotates a vector functional into a positive one. This is the vector case of the polar decomposition of normal functionals. Two elementary lemmas prepare it.\n\n**Lemma 17.1** (A positivity test). Let \\(A\\) be a unital C\\(^*\\)-algebra and \\(\\omega\\) a bounded linear functional on \\(A\\). If \\(\\omega(h)=\\|\\omega\\|\\) for some \\(h\\in A\\) with \\(0\\le h\\le1\\), then \\(\\omega\\) is positive.\n\n**Proof.** Put \\(R=\\|\\omega\\|\\); we may assume \\(R>0\\). For \\(-1\\le s\\le1\\), the self-adjoint element \\(h+s(1-h)\\) has its spectrum in \\([-1,1]\\): each point \\(\\lambda+s(1-\\lambda)\\) with \\(\\lambda\\in[0,1]\\) is a convex combination of \\(1\\) and \\(s\\). So \\(|R+s\\,\\omega(1-h)|=|\\omega(h+s(1-h))|\\le R\\). Writing \\(a=\\omega(1-h)\\), this says \\(2sR\\operatorname{Re}a+s^2|a|^2\\le0\\) for all \\(s\\in[-1,1]\\). Small \\(s\\) of both signs force \\(\\operatorname{Re}a=0\\), and then \\(a=0\\). So \\(\\omega(1)=R\\).\n\nNow let \\(x=x^*\\), \\(\\|x\\|\\le1\\), and \\(\\omega(x)=\\alpha+i\\beta\\). For real \\(t\\), \\(\\|1+itx\\|^2=\\|1+t^2x^2\\|\\le1+t^2\\), so \\((R-t\\beta)^2+t^2\\alpha^2\\le R^2(1+t^2)\\). This reads \\(-2tR\\beta+t^2(\\alpha^2+\\beta^2)\\le R^2t^2\\). Divide by \\(|t|\\) and let \\(t\\to0\\) from each side: \\(\\beta=0\\). So \\(\\omega\\) is real on self-adjoint elements. If \\(0\\le x\\le1\\), then \\(\\|1-x\\|\\le1\\), so \\(|R-\\omega(x)|\\le R\\) with \\(\\omega(x)\\) real, and hence \\(\\omega(x)\\ge0\\). \\(\\square\\)\n\n**Lemma 17.2** (Dropping a projection). Let \\(\\varphi\\) be a weakly continuous linear functional on \\(M\\), let \\(e\\in\\mathcal P(M)\\), and put \\(\\varphi_e(x)=\\varphi(xe)\\). If \\(\\|\\varphi_e\\|=\\|\\varphi\\|\\), then \\(\\varphi_e=\\varphi\\).\n\n**Proof.** We may assume \\(\\|\\varphi\\|=1\\). Put \\(f=1-e\\), and suppose that \\(\\varphi(yf)\\ne0\\) for some \\(y\\in M\\). Rescaling \\(y\\) gives \\(b\\) in the unit ball with \\(\\varphi(bf)=\\delta>0\\). The functional \\(\\varphi_e\\) is weakly continuous and the unit ball of \\(M\\) is weakly compact, so \\(|\\varphi_e|\\) attains its norm there. After a rotation, \\(\\varphi(ae)=1\\) for some \\(a\\) in the unit ball. For \\(t>0\\), \\((ae+tbf)(ae+tbf)^*=aea^*+t^2bfb^*\\), because \\(ef=0\\). So \\(\\|ae+tbf\\|^2\\le1+t^2\\). But \\(\\varphi(ae+tbf)=1+t\\delta\\). Hence \\((1+t\\delta)^2\\le1+t^2\\), that is, \\(2\\delta\\le t(1-\\delta^2)\\) for every \\(t>0\\). This is impossible for \\(\\delta>0\\). So \\(\\varphi(yf)=0\\) for all \\(y\\), which says \\(\\varphi=\\varphi_e\\). \\(\\square\\)\n\n**Lemma 17.3** (Rotating a vector functional). Let \\(\\xi,\\eta\\in H\\) and \\(\\varphi(x)=\\langle x\\eta,\\xi\\rangle\\) for \\(x\\in M\\). There is a partial isometry \\(u\\in M\\) such that\n\\[\n\\begin{gathered}\n\\varphi(xuu^*)\\\\\n=\\varphi(x)\\\\\n(x\\in M)\\\\\n\\text{and}\\\\\nx\\mapsto\\varphi(xu^*)\\ \\text{is positive on }M.\n\\end{gathered}\n\\tag{17.4}\n\\]\nIf \\(\\eta\\in[M\\xi]\\), then \\(uu^*\\eta=\\eta\\), and the vector \\(\\eta_2=u^*\\eta\\) lies in \\([M'\\xi]\\cap[M\\xi]\\).\n\n**Proof.** If \\(\\varphi=0\\), take \\(u=0\\). Then \\(\\eta\\perp x^*\\xi\\) for all \\(x\\in M\\), so an \\(\\eta\\in[M\\xi]\\) must be \\(0\\), and the last statement holds trivially. Otherwise let \\(R=\\|\\varphi\\|>0\\). The functional \\(\\varphi\\) is weakly continuous and the unit ball of \\(M\\) is weakly compact, so \\(\\varphi(a)=R\\) for some \\(a\\) in the unit ball. Let \\(a^*=u|a^*|\\) be the polar decomposition (Fact 2.2). Then \\(a=|a^*|u^*\\) with \\(u\\in M\\) and \\(0\\le|a^*|\\le1\\). The functional \\(\\omega(x)=\\varphi(xu^*)\\) has norm at most \\(R\\), and \\(\\omega(|a^*|)=\\varphi(a)=R\\). By Lemma 17.1, \\(\\omega\\) is positive. Next put \\(e=uu^*\\). Then \\(ae=|a^*|u^*uu^*=|a^*|u^*=a\\), so \\(\\varphi_e(a)=R\\) and \\(\\|\\varphi_e\\|=\\|\\varphi\\|\\). Lemma 17.2 gives \\(\\varphi(xuu^*)=\\varphi(x)\\). This proves (17.4).\n\nNow let \\(\\eta\\in[M\\xi]\\). The first property in (17.4) says \\(\\langle uu^*\\eta-\\eta,x^*\\xi\\rangle=0\\) for all \\(x\\in M\\). The vector \\(uu^*\\eta-\\eta\\) lies in \\([M\\xi]\\), which is invariant under \\(M\\), and it is orthogonal to \\([M\\xi]\\). So it is zero. Put \\(\\eta_2=u^*\\eta\\in[M\\xi]\\). The functional \\(\\psi(x)=\\langle x\\eta_2,\\xi\\rangle=\\varphi(xu^*)\\) is positive, hence hermitian: \\(\\psi(x^*)=\\overline{\\psi(x)}\\). As \\(p_\\xi\\xi=\\xi\\), we have \\(\\psi(x)=\\langle p_\\xi x\\eta_2,\\xi\\rangle=\\psi(p_\\xi x)\\), and therefore\n\\[\n\\psi(x)=\\overline{\\psi(x^*)}=\\overline{\\psi(p_\\xi x^*)}=\\psi(xp_\\xi).\n\\]\nSo \\(\\langle x(\\eta_2-p_\\xi\\eta_2),\\xi\\rangle=0\\) for every \\(x\\in M\\). The vector \\(\\eta_2-p_\\xi\\eta_2\\) lies in \\([M\\xi]\\) and is orthogonal to it, so it vanishes. Thus \\(\\eta_2=p_\\xi\\eta_2\\in[M'\\xi]\\). \\(\\square\\)\n\n**Theorem 17.5** (The transfer theorem). For \\(\\xi,\\eta\\in H\\),\n\\[\np_\\xi\\succsim p_\\eta\\ \\text{ in } M\\quad\\Longleftrightarrow\\quad p'_\\xi\\succsim p'_\\eta\\ \\text{ in } M'.\n\\]\n\n**Proof.** *Step 1 (moving a vector inside its orbit).* If \\(v\\in M'\\) is a partial isometry with \\(v^*v\\eta=\\eta\\), then \\(p_{v\\eta}=p_\\eta\\). Indeed \\([M'v\\eta]\\subseteq[M'\\eta]\\) because \\(v\\in M'\\), and \\(\\eta=v^*(v\\eta)\\) gives the reverse inclusion.\n\n*Step 2 (the core).* If \\(\\eta\\in[M\\xi]\\), then \\(p_\\eta\\precsim p_\\xi\\). Take \\(u\\) and \\(\\eta_2=u^*\\eta\\) from Lemma 17.3. The projection \\(uu^*\\in M\\) fixes \\(\\eta\\) and commutes with \\(M'\\), so it fixes \\([M'\\eta]\\) pointwise, and \\(p_\\eta\\le uu^*\\). The operator \\(w=u^*p_\\eta\\) therefore has initial projection \\(p_\\eta uu^*p_\\eta=p_\\eta\\). Its final projection is the projection onto \\(u^*[M'\\eta]\\). Since \\(u^*\\) is isometric on \\(uu^*H\\supseteq[M'\\eta]\\), this subspace is closed and equals \\([u^*M'\\eta]=[M'u^*\\eta]=[M'\\eta_2]\\). So \\(p_\\eta\\sim p_{\\eta_2}\\). Finally \\(\\eta_2\\in[M'\\xi]\\) gives \\(p_{\\eta_2}\\le p_\\xi\\) (Lemma 4.6).\n\n*Step 3.* Suppose \\(p'_\\xi\\succsim p'_\\eta\\): some \\(v\\in M'\\) has \\(v^*v=p'_\\eta\\) and \\(vv^*\\le p'_\\xi\\). Put \\(\\eta_1=v\\eta\\). Then \\(\\eta_1\\in vv^*H\\subseteq[M\\xi]\\), and \\(v^*v\\eta=p'_\\eta\\eta=\\eta\\), so \\(p_{\\eta_1}=p_\\eta\\) by Step 1. Step 2 gives \\(p_\\eta=p_{\\eta_1}\\precsim p_\\xi\\). This proves \"⇐\". The implication \"⇒\" is the same statement for the von Neumann algebra \\(M'\\), whose commutant is \\(M\\), with the roles of \\(p\\) and \\(p'\\) exchanged. \\(\\square\\)\n\n**Example 17.6** (\\(B(H)\\) and the scalars). For \\(M=B(H)\\) and \\(\\xi\\ne0\\), \\(p_\\xi\\) is the projection onto \\(\\mathbb C\\xi\\) and \\(p'_\\xi=1\\); Theorem 17.5 reads: \\(p_\\xi\\succsim p_\\eta\\) iff \\(\\xi\\ne0\\) or \\(\\eta=0\\), iff \\(p'_\\xi\\succsim p'_\\eta\\). The algebra \\(M=\\mathbb C1\\) on \\(H\\ne0\\) is this example with \\(M\\) and \\(M'\\) exchanged: for \\(\\xi\\ne0\\), \\(p_\\xi=1\\) and \\(p'_\\xi\\) is the projection onto \\(\\mathbb C\\xi\\). Theorem 17.5 is symmetric under this exchange.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "projections-and-types-of-von-neumann-algebras",
      "name": "18. Vector states and spatial isomorphisms",
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      "full_conditions_and_proof": "## 18. Vector states and spatial isomorphisms\n\nThe transfer theorem answers three natural questions: when one cyclic representation of a C\\(^*\\)-algebra is contained in another; which normal functionals are vector functionals; and, for algebras with cyclic and separating vectors, whether an isomorphism is implemented by a unitary.\n\n**Proposition 18.1** (Subrepresentations and cyclic projections). Let \\(A\\) be a C\\(^*\\)-algebra, \\(\\{\\pi,H\\}\\) a nondegenerate representation of \\(A\\), \\(\\xi,\\eta\\in H\\), \\(\\varphi=\\omega_\\xi\\circ\\pi\\), \\(\\psi=\\omega_\\eta\\circ\\pi\\), and \\(N=\\pi(A)''\\). The following are equivalent.\n\n1. The cyclic representation \\(\\pi_\\varphi\\) of \\(\\varphi\\) *embeds* in \\(\\pi_\\psi\\): some isometry \\(H_\\varphi\\to H_\\psi\\) intertwines \\(\\pi_\\varphi\\) with \\(\\pi_\\psi\\). (Its range is then an invariant subspace, so this says that \\(\\pi_\\varphi\\) is carried by a unitary onto a subrepresentation of \\(\\pi_\\psi\\).)\n2. \\(p'_\\xi\\precsim p'_\\eta\\) in \\(N'=\\pi(A)'\\).\n3. \\(p_\\xi\\precsim p_\\eta\\) in \\(N\\).\n\nIn the universal representation (Fact 2.7), \\(p_\\xi=s(\\varphi)\\) and \\(p_\\eta=s(\\psi)\\). So \\(\\pi_\\varphi\\) embeds in \\(\\pi_\\psi\\) exactly when \\(s(\\varphi)\\precsim s(\\psi)\\) in \\(\\tilde A\\).\n\n**Proof.** Since \\(\\pi\\) is nondegenerate, \\(\\pi(A)\\) is strongly dense in \\(N\\) by the double commutant theorem (Fact 2.7). So every vector \\(a\\xi\\) with \\(a\\in N\\) is a limit of vectors \\(\\pi(b)\\xi\\) with \\(b\\in A\\), and \\([N\\xi]=[\\pi(A)\\xi]\\). Thus \\(p'_\\xi\\) is the projection onto \\([\\pi(A)\\xi]\\). The map \\(\\pi_\\varphi(a)\\xi_\\varphi\\mapsto\\pi(a)\\xi\\) is isometric, since both sides have squared norm \\(\\varphi(a^*a)\\). It carries a dense subspace of \\(H_\\varphi\\) onto a dense subspace of \\(p'_\\xi H\\), so it extends to a unitary \\(U_\\xi:H_\\varphi\\to p'_\\xi H\\) that intertwines \\(\\pi_\\varphi\\) with the restriction of \\(\\pi\\) to \\(p'_\\xi H\\). The same holds for \\(\\eta\\) and \\(\\psi\\).\n\n(2)⇒(1). Let \\(v\\in\\pi(A)'\\) have \\(v^*v=p'_\\xi\\) and \\(vv^*\\le p'_\\eta\\). The range \\(L=vv^*H\\) is invariant under \\(\\pi(A)\\), and \\(v\\) restricts to a unitary \\(p'_\\xi H\\to L\\) that intertwines the restrictions of \\(\\pi\\). Composing with \\(U_\\xi\\) and \\(U_\\eta^{-1}\\) shows that \\(\\pi_\\varphi\\) is equivalent to the restriction of \\(\\pi_\\psi\\) to the invariant subspace \\(U_\\eta^*L\\).\n\n(1)⇒(2). Let \\(T:H_\\varphi\\to L_0\\) be a unitary onto a closed \\(\\pi_\\psi\\)-invariant subspace, intertwining \\(\\pi_\\varphi\\) with the restriction of \\(\\pi_\\psi\\). Define \\(v=U_\\eta TU_\\xi^*\\) on \\(p'_\\xi H\\) and \\(v=0\\) on its orthogonal complement. On \\(p'_\\xi H\\), \\(v\\) intertwines. On \\((p'_\\xi H)^\\perp\\), both \\(v\\pi(a)\\) and \\(\\pi(a)v\\) vanish, because \\(\\pi(a)\\) leaves that space invariant. So \\(v\\in\\pi(A)'\\), \\(v^*v=p'_\\xi\\) and \\(vv^*\\le p'_\\eta\\).\n\n(2)⇔(3) is Theorem 17.5 in \\(N\\), whose commutant is \\(\\pi(A)'\\).\n\nIn the state-indexed universal representation, normalize each nonzero functional and rescale its cyclic vector as in Fact 2.7; use the zero vector for a zero functional. This gives \\(\\xi,\\eta\\in H_u\\) with \\(\\varphi=\\omega_\\xi\\circ\\pi_u\\) and \\(\\psi=\\omega_\\eta\\circ\\pi_u\\). The restriction of \\(\\omega_\\xi\\) to \\(\\tilde A\\) is \\(\\sigma\\)-weakly continuous and extends \\(\\varphi\\), so it is \\(\\tilde\\varphi\\), and its support is \\(p_\\xi\\) ([Lemma 4.6](#oa-fnd-ty-07)). The same holds for \\(\\psi\\). \\(\\square\\)\n\n**Proposition 18.2** (Which normal functionals are vector functionals). Let \\(\\xi_0\\in H\\) and let \\(\\varphi\\) be a positive normal functional on \\(M\\). Then \\(\\varphi=\\omega_\\xi|_M\\) for some \\(\\xi\\in[M\\xi_0]\\) if and only if \\(s(\\varphi)\\precsim p_{\\xi_0}\\). In particular, if \\(M\\) has a separating vector \\(\\xi_0\\), then \\(p_{\\xi_0}=1\\), and every positive normal functional on \\(M\\) is a vector functional \\(\\omega_\\xi\\) with \\(\\xi\\in[M\\xi_0]\\).\n\n**Proof.** If \\(\\varphi=\\omega_\\xi\\) with \\(\\xi\\in[M\\xi_0]\\), then \\(s(\\varphi)=p_\\xi\\) ([Lemma 4.6](#oa-fnd-ty-07)), and \\(p_\\xi\\precsim p_{\\xi_0}\\) by Step 2 of the proof of Theorem 17.5.\n\nConversely, suppose \\(s(\\varphi)\\precsim p_{\\xi_0}\\). By Fact 2.5, \\(\\varphi=\\sum_n\\omega_{\\xi_n}\\) with \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\). On \\(K=H\\otimes\\ell^2(\\mathbb N)\\), with orthonormal basis \\((\\delta_n)\\) of \\(\\ell^2(\\mathbb N)\\), let \\(N=M\\otimes1\\). By [Lemma 8.2](#oa-fnd-ty-11), \\(N\\) is a von Neumann algebra, its commutant \\(N'\\) consists of the operators whose matrix entries lie in \\(M'\\), and \\(x\\mapsto x\\otimes1\\) is an isomorphism of \\(M\\) onto \\(N\\). Put \\(\\zeta=\\sum_n\\xi_n\\otimes\\delta_n\\) and \\(\\zeta_0=\\xi_0\\otimes\\delta_1\\). Then \\(\\omega_\\zeta(x\\otimes1)=\\varphi(x)\\). A projection \\(p\\otimes1\\) of \\(N\\) fixes \\(\\zeta\\) iff \\(\\varphi(1-p)=0\\), so the cyclic projection of \\(N\\) at \\(\\zeta\\) is \\(p^N_\\zeta=s(\\varphi)\\otimes1\\). Next, \\([N'\\zeta_0]=[M'\\xi_0]\\otimes\\ell^2(\\mathbb N)\\): the inclusion \"⊇\" holds because \\(N'\\) contains the operators \\(x'\\otimes1\\) and the matrix units \\(1\\otimes E_{n1}\\), and \"⊆\" holds because every \\(T\\in N'\\) maps \\(\\zeta_0\\) to \\(\\sum_nT_{n1}\\xi_0\\otimes\\delta_n\\) with \\(T_{n1}\\in M'\\). So \\(p^N_{\\zeta_0}=p_{\\xi_0}\\otimes1\\). The isomorphism \\(x\\mapsto x\\otimes1\\) turns \\(s(\\varphi)\\precsim p_{\\xi_0}\\) into \\(p^N_\\zeta\\precsim p^N_{\\zeta_0}\\). By Theorem 17.5 in \\(N\\), \\(p'^N_\\zeta\\precsim p'^N_{\\zeta_0}\\) in \\(N'\\): some \\(v\\in N'\\) has \\(v^*v=p'^N_\\zeta\\) and \\(vv^*\\le p'^N_{\\zeta_0}\\). The range of \\(p'^N_{\\zeta_0}\\) is \\([N\\zeta_0]=[M\\xi_0]\\otimes\\delta_1\\). Hence \\(v\\zeta=\\xi\\otimes\\delta_1\\) with \\(\\xi\\in[M\\xi_0]\\), and for \\(x\\in M\\)\n\\[\n\\begin{gathered}\n\\omega_\\xi(x)\\\\\n=\\langle(x\\otimes1)v\\zeta,v\\zeta\\rangle\\\\\n=\\langle(x\\otimes1)v^*v\\zeta,\\zeta\\rangle\\\\\n=\\omega_\\zeta(x\\otimes1)\\\\\n=\\varphi(x).\n\\end{gathered}\n\\]\nIf \\(\\xi_0\\) is separating for \\(M\\), it is cyclic for \\(M'\\) (Fact 2.3), so \\(p_{\\xi_0}=1\\) and the condition holds for every \\(\\varphi\\). \\(\\square\\)\n\n**Example 18.3** (A normal state that is not a vector state). Let \\(M=M_2(\\mathbb C)\\) on \\(\\mathbb C^2\\) and \\(\\varphi=\\frac12\\mathrm{tr}\\). Its support is \\(1\\). Each vector functional \\(\\omega_\\xi\\) has support \\(p_\\xi\\), the rank-one projection onto \\(\\mathbb C\\xi\\) (here \\(M'=\\mathbb C1\\)). So \\(\\varphi\\) is not a vector functional, in line with Proposition 18.2: \\(s(\\varphi)=1\\not\\precsim p_{\\xi_0}\\) for every \\(\\xi_0\\). Accordingly \\(M\\) has no separating vector. On \\(\\mathbb C^2\\otimes\\mathbb C^2\\), with \\(M\\) acting as \\(M\\otimes1\\), the vector \\((\\delta_1\\otimes\\delta_1+\\delta_2\\otimes\\delta_2)/\\sqrt2\\) is separating and gives \\(\\varphi\\).\n\n**Lemma 18.4** (Interpolating two vectors). Let \\(\\xi_1,\\xi_2\\in H\\) with \\(p'_{\\xi_1}\\sim p'_{\\xi_2}\\) in \\(M'\\); by Theorem 17.5 and Proposition 5.1 this is the same as \\(p_{\\xi_1}\\sim p_{\\xi_2}\\) in \\(M\\). Then some \\(\\xi_0\\in H\\) has \\(p'_{\\xi_0}=p'_{\\xi_1}\\) and \\(p_{\\xi_0}=p_{\\xi_2}\\).\n\n**Proof.** Let \\(u\\in M'\\) implement \\(p'_{\\xi_2}\\sim p'_{\\xi_1}\\), and put \\(\\xi_0=u\\xi_2\\). Since \\(u\\) commutes with \\(M\\) and maps \\([M\\xi_2]\\) isometrically onto \\(p'_{\\xi_1}H\\), we get \\([M\\xi_0]=u[M\\xi_2]=p'_{\\xi_1}H\\). Since \\(u^*u\\xi_2=\\xi_2\\), Step 1 of the proof of Theorem 17.5 gives \\(p_{\\xi_0}=p_{\\xi_2}\\). \\(\\square\\)\n\n**Proposition 18.5** (A cyclic and separating vector). If some vector is cyclic for \\(M\\) and some vector is separating for \\(M\\), then a single vector is both cyclic and separating.\n\n**Proof.** Let \\(\\xi_1\\) be cyclic and \\(\\xi_2\\) separating for \\(M\\). Then \\(p'_{\\xi_1}=1\\), and \\(p_{\\xi_2}=1\\) by Fact 2.3. As \\(p_{\\xi_1}\\le1=p_{\\xi_2}\\), Theorem 17.5 gives \\(1=p'_{\\xi_1}\\precsim p'_{\\xi_2}\\), and Proposition 5.1 gives \\(p'_{\\xi_2}\\sim1=p'_{\\xi_1}\\). Lemma 18.4 yields \\(\\xi_0\\) with \\(p'_{\\xi_0}=1\\) and \\(p_{\\xi_0}=1\\). So \\(\\xi_0\\) is cyclic for \\(M\\) and for \\(M'\\), hence also separating for \\(M\\) (Fact 2.3). \\(\\square\\)\n\n**Proposition 18.6** (Isomorphisms are spatial). Let \\(\\{M_1,H_1\\}\\) and \\(\\{M_2,H_2\\}\\) be von Neumann algebras with cyclic and separating vectors \\(\\xi_1\\) and \\(\\xi_2\\). Every isomorphism \\(\\pi:M_1\\to M_2\\) is spatial.\n\n**Proof.** By Fact 2.6, \\(\\pi\\) is \\(\\sigma\\)-weakly continuous, so \\(\\psi=\\omega_{\\xi_2}\\circ\\pi\\) is a positive normal functional on \\(M_1\\). It is faithful: \\(\\psi(x^*x)=\\|\\pi(x)\\xi_2\\|^2=0\\) forces \\(\\pi(x)=0\\) and \\(x=0\\). By Proposition 18.2, applied to \\(M_1\\) and its separating vector \\(\\xi_1\\), \\(\\psi=\\omega_\\zeta\\) for some \\(\\zeta\\in H_1\\). Faithfulness makes \\(\\zeta\\) separating for \\(M_1\\), so \\(p_\\zeta=1=p_{\\xi_1}\\) (Fact 2.3), and Theorem 17.5 with Proposition 5.1 gives \\(p'_\\zeta\\sim p'_{\\xi_1}=1\\). Lemma 18.4 gives \\(\\zeta_0=u\\zeta\\), with \\(u\\in M_1'\\) isometric on \\([M_1\\zeta]\\), such that \\(p'_{\\zeta_0}=1\\). Then \\(\\omega_{\\zeta_0}=\\omega_\\zeta=\\psi\\) on \\(M_1\\), because \\(\\langle xu\\zeta,u\\zeta\\rangle=\\langle u^*u\\,x\\zeta,\\zeta\\rangle=\\langle x\\zeta,\\zeta\\rangle\\).\n\nDefine \\(U\\) on \\(M_2\\xi_2=\\pi(M_1)\\xi_2\\) by \\(U\\pi(x)\\xi_2=x\\zeta_0\\). It is well defined and isometric, since \\(\\|x\\zeta_0\\|^2=\\psi(x^*x)=\\|\\pi(x)\\xi_2\\|^2\\). Its domain is dense in \\(H_2\\) and its range is dense in \\(H_1\\), so \\(U\\) extends to a unitary \\(H_2\\to H_1\\). For \\(x,y\\in M_1\\), \\(U\\pi(y)\\pi(x)\\xi_2=yx\\zeta_0=yU\\pi(x)\\xi_2\\). So \\(U\\pi(y)=yU\\), that is, \\(\\pi(y)=U^*yU\\). \\(\\square\\)\n\n## Where this leads\n\nThe following topics extend this lesson. Their further assertions and constructions are not proved here and are not inputs to any proof or solution above.\n\n- *Traces.* Traces and dimension functions are built on the comparison theory of this lesson. The lesson Traces on von Neumann algebras constructs the centre-valued trace of a finite algebra, of which the map \\(\\Phi\\) of Proposition 9.1(2) is the simplest case. It also proves that the family \\(\\{z_\\alpha\\}\\) of Proposition 15.3(1) is unique.\n- *Types II and III.* This lesson gives examples of type I only. Factors of types II and III exist, but their construction needs other tools, such as group von Neumann algebras, infinite tensor products or crossed products.\n- *Multiplicity.* The spatial form of type I algebras (Theorem 11.4) and the normal form of properly infinite algebras (Proposition 15.3) are the starting points of the multiplicity theory of normal representations.\n\n## References\n\n- [Kostecki] R. P. Kostecki, *W\\(^*\\)-algebras and noncommutative integration*, [arXiv:1307.4818](https://arxiv.org/abs/1307.4818v5), 2013.\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [B. Blackadar, *Operator Algebras*, III.1.1–III.1.4](https://bruceblackadar.com/Mathematics/Cycr.pdf) gives a route through projection comparison and type decomposition; all cardinal and boundary cases stated here retain their complete proofs. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
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      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "1. Cutting a functional by a projection",
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      "full_conditions_and_proof": "## 1. Cutting a functional by a projection\n\nCutting a functional by a projection \\(e\\) means passing from \\(\\varphi\\) to \\(e\\varphi\\), that is, to \\(x\\mapsto\\varphi(xe)\\). The first result says how much norm can survive the two cuts \\(e\\varphi\\) and \\((1-e)\\varphi\\). Its main use is the special case: if the cut by \\(e\\) keeps all of the norm, it keeps all of the functional.\n\n**Lemma 1.1.** Let \\(B\\) be a \\(C^*\\)-algebra, \\(\\varphi\\in B^*\\) and \\(e\\in B\\) a projection. Put \\(f=1-e\\), computed in the unitization if \\(B\\) has no unit, so that \\(xf=x-xe\\in B\\) for \\(x\\in B\\). Then\n\\[\n\\begin{gathered}\n\\|e\\varphi\\|^2+\\|f\\varphi\\|^2\\\\\n\\le\\|\\varphi\\|^2\\\\\n\\text{and}\\\\\n\\|\\varphi e\\|^2+\\|\\varphi f\\|^2\\\\\n\\le\\|\\varphi\\|^2 .\n\\end{gathered}\n\\tag{1.1}\n\\]\nIn particular, if \\(\\|e\\varphi\\|=\\|\\varphi\\|\\) then \\(e\\varphi=\\varphi\\), and if \\(\\|\\varphi e\\|=\\|\\varphi\\|\\) then \\(\\varphi e=\\varphi\\).\n\n**Proof.** Put \\(\\alpha=\\|e\\varphi\\|\\) and \\(\\beta=\\|f\\varphi\\|\\), and let \\(\\varepsilon>0\\). Choose \\(a,b\\) in the unit ball of \\(B\\) with \\(\\varphi(ae)\\ge\\alpha-\\varepsilon\\) and \\(\\varphi(bf)\\ge\\beta-\\varepsilon\\); multiplying \\(a\\) and \\(b\\) by scalars of modulus one makes these numbers real. For \\(s,t\\ge0\\) with \\(s^2+t^2=1\\), put \\(y=s\\,ae+t\\,bf\\in B\\). Since \\(ef=0\\), the cross terms of \\(yy^*\\) vanish:\n\\[\n\\begin{gathered}\nyy^*\\\\\n=s^2\\,aea^*+t^2\\,bfb^* ,\\\\\n\\text{so}\\\\\n\\|y\\|^2\\\\\n=\\|yy^*\\|\\\\\n\\le s^2+t^2\\\\\n=1 .\n\\end{gathered}\n\\]\nHence \\(\\|\\varphi\\|\\ge\\operatorname{Re}\\varphi(y)\\ge s(\\alpha-\\varepsilon)+t(\\beta-\\varepsilon)\\). If \\(\\alpha^2+\\beta^2>0\\), take \\((s,t)=(\\alpha,\\beta)/(\\alpha^2+\\beta^2)^{1/2}\\) and let \\(\\varepsilon\\to0\\): this gives \\(\\|\\varphi\\|\\ge(\\alpha^2+\\beta^2)^{1/2}\\). The second inequality follows by applying the first to \\(\\varphi^*\\), since \\((\\varphi e)^*=e\\varphi^*\\) and adjoints keep norms. If \\(\\|e\\varphi\\|=\\|\\varphi\\|\\), then \\(\\|f\\varphi\\|=0\\), so \\(\\varphi=e\\varphi+f\\varphi=e\\varphi\\). \\(\\square\\)\n\n**Example 1.2** (The two extremes). Let \\(B=B(H)\\), \\(\\xi,\\eta\\in H\\) and \\(\\varphi=\\omega_{\\xi,\\eta}\\). Then \\(\\|\\omega_{\\xi,\\eta}\\|=\\|\\xi\\|\\|\\eta\\|\\): the inequality \\(\\le\\) is Cauchy–Schwarz, and for nonzero \\(\\xi,\\eta\\), \\(x=\\theta_{\\eta,\\xi}/(\\|\\xi\\|\\|\\eta\\|)\\) attains it. If either vector is zero, the functional and both sides are zero. Since \\(e\\omega_{\\xi,\\eta}=\\omega_{e\\xi,\\eta}\\),\n\\[\n\\begin{gathered}\n\\|e\\varphi\\|^2+\\|(1-e)\\varphi\\|^2\\\\\n=\\big(\\|e\\xi\\|^2+\\|(1-e)\\xi\\|^2\\big)\\|\\eta\\|^2\\\\\n=\\|\\varphi\\|^2 .\n\\end{gathered}\n\\]\nSo (1.1) is an equality for vector functionals, and it cannot be improved. At the other extreme, for a central projection \\(e\\) of a von Neumann algebra and a normal \\(\\varphi\\), the norm is additive: \\(\\|\\varphi\\|=\\|e\\varphi\\|+\\|(1-e)\\varphi\\|\\) ([Lemma 10.2](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md#oa-fnd-wa-10)). Every functional lies between the two: \\[\n\\begin{gathered}\n(\\|e\\varphi\\|^2+\\|(1-e)\\varphi\\|^2)^{1/2}\\\\\n\\le\\|\\varphi\\|\\\\\n\\le\\|e\\varphi\\|+\\|(1-e)\\varphi\\|.\n\\end{gathered}\n\\]\n\nA normal functional determines two projections. Let \\(\\varphi\\in M_*\\). The set \\(R_\\varphi=\\{y\\in M:\\varphi(yx)=0\\text{ for all }x\\in M\\}\\) is a \\(\\sigma\\)-weakly closed right ideal, because \\(y\\mapsto\\varphi(yx)\\) is \\(\\sigma\\)-weakly continuous for each \\(x\\). By background fact 19, \\(R_\\varphi=pM\\) for a unique projection \\(p\\). Likewise \\[\n\\begin{gathered}\nL_\\varphi\\\\\n=\\{y\\in M:\\varphi(xy)=0\\text{ for all }x\\in M\\}\\\\\n=Mq\n\\end{gathered}\n\\] for a unique projection \\(q\\).\n\n**Definition 1.3.** The *right support* of \\(\\varphi\\in M_*\\) is \\(s_r(\\varphi)=1-p\\) and the *left support* is \\(s_l(\\varphi)=1-q\\), with \\(p,q\\) as above.\n\n**Lemma 1.4.** Let \\(\\varphi\\in M_*\\).\n\n1. For a projection \\(e\\), \\(\\varphi=\\varphi e\\) exactly when \\(e\\ge s_r(\\varphi)\\), and \\(\\varphi=e\\varphi\\) exactly when \\(e\\ge s_l(\\varphi)\\). So \\(s_r(\\varphi)\\) is the least projection \\(e\\) with \\(\\varphi(ex)=\\varphi(x)\\) for all \\(x\\), and \\(s_l(\\varphi)\\) is the least projection \\(f\\) with \\(\\varphi(xf)=\\varphi(x)\\) for all \\(x\\).\n2. \\(\\varphi(x)=\\varphi(s_r(\\varphi)\\,x\\,s_l(\\varphi))\\) for all \\(x\\in M\\).\n3. \\(s_l(\\varphi^*)=s_r(\\varphi)\\) and \\(s_r(\\varphi^*)=s_l(\\varphi)\\).\n4. If \\(\\omega\\) is positive, \\(s_l(\\omega)=s_r(\\omega)=s(\\omega)\\). If \\(\\varphi\\) is hermitian, \\(s_l(\\varphi)=s_r(\\varphi)\\).\n\n**Proof.** (1) \\(\\varphi=\\varphi e\\) means \\(\\varphi((1-e)x)=0\\) for all \\(x\\), that is \\(1-e\\in R_\\varphi=pM\\). This holds exactly when \\(p(1-e)=1-e\\), that is \\(1-e\\le p\\), that is \\(e\\ge1-p\\). The left case is the same with \\(L_\\varphi=Mq\\).\n\n(2) With \\(e=s_r(\\varphi)\\) and \\(f=s_l(\\varphi)\\): \\(\\varphi(x)=\\varphi(ex)=\\varphi(exf)\\) by (1), applied twice.\n\n(3) \\((\\varphi e)^*=e\\varphi^*\\), so \\(\\varphi=\\varphi e\\) exactly when \\(\\varphi^*=e\\varphi^*\\). Now use (1) for \\(\\varphi\\) and \\(\\varphi^*\\).\n\n(4) From \\(\\omega=\\omega s(\\omega)=s(\\omega)\\omega\\) and (1), both supports lie under \\(s(\\omega)\\). If \\(\\omega=\\omega e\\), then \\(\\omega(1-e)=\\omega(e(1-e))=0\\), so \\(1-e\\le1-s(\\omega)\\), that is \\(e\\ge s(\\omega)\\); the left support is handled in the same way. For hermitian \\(\\varphi\\), (3) gives \\(s_l(\\varphi)=s_r(\\varphi^*)=s_r(\\varphi)\\). \\(\\square\\)\n\nFor the vector functional \\(\\omega_{\\xi,\\eta}\\) on \\(B(H)\\), with \\(\\xi,\\eta\\ne0\\): \\(\\omega_{\\xi,\\eta}(ex)=\\langle x\\xi,e\\eta\\rangle\\) and \\(\\omega_{\\xi,\\eta}(xf)=\\langle xf\\xi,\\eta\\rangle\\). So \\(s_r(\\omega_{\\xi,\\eta})\\) is the projection onto \\(\\mathbb C\\eta\\) and \\(s_l(\\omega_{\\xi,\\eta})\\) the projection onto \\(\\mathbb C\\xi\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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        "the-double-commutant-theorem",
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    {
      "id": "OA-FND-PD-02",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "2. The polar decomposition",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
      "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "anchor": "oa-fnd-pd-02",
      "proof_locus": {
        "line": 168,
        "through_line": 369
      },
      "full_conditions_and_proof": "## 2. The polar decomposition\n\nWe first record a comparison principle for positive functionals. It gives uniqueness in the polar decomposition, and in Section 3 it characterizes the absolute value.\n\n**Lemma 2.1** (Comparison of positive functionals). Let \\(A\\) be a \\(C^*\\)-algebra and \\(\\omega,\\omega_1\\in A^*_+\\) with \\(\\|\\omega\\|=\\|\\omega_1\\|\\) and\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\|\\omega_1\\|\\,\\omega_1(x^*x)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{2.1}\n\\]\nThen \\(\\omega=\\omega_1\\).\n\n**Proof.** Put \\(c=\\|\\omega_1\\|\\); if \\(c=0\\) both functionals vanish. Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega_1\\), so \\(\\omega_1(x^*x)=\\|\\pi(x)\\xi\\|^2\\) and \\(\\|\\xi\\|^2=c\\). By (2.1), \\(\\pi(x)\\xi\\mapsto\\omega(x)\\) is a well-defined linear functional on the dense subspace \\(\\pi(A)\\xi\\), of norm at most \\(c^{1/2}\\). The Riesz theorem gives \\(\\eta\\in H\\) with \\(\\|\\eta\\|\\le c^{1/2}\\) and \\(\\omega(x)=\\langle\\pi(x)\\xi,\\eta\\rangle\\) for \\(x\\in A\\). For an approximate unit \\((u_\\lambda)\\), \\(\\omega(u_\\lambda)\\to\\|\\omega\\|=c\\) and \\(\\pi(u_\\lambda)\\xi\\to\\xi\\) (background fact 12). Hence \\(\\langle\\xi,\\eta\\rangle=c\\), and\n\\[\n\\begin{gathered}\n\\|\\xi-\\eta\\|^2\\\\\n=\\|\\xi\\|^2-2\\operatorname{Re}\\langle\\xi,\\eta\\rangle+\\|\\eta\\|^2\\\\\n\\le c-2c+c\\\\\n=0 .\n\\end{gathered}\n\\]\nSo \\(\\eta=\\xi\\) and \\(\\omega(x)=\\langle\\pi(x)\\xi,\\xi\\rangle=\\omega_1(x)\\). \\(\\square\\)\n\n**Theorem 2.2** (Polar decomposition). Let \\(M\\) be a von Neumann algebra and \\(\\varphi\\in M_*\\). There is exactly one pair \\((v,\\omega)\\) of a partial isometry \\(v\\in M\\) and a positive normal functional \\(\\omega\\) with\n\\[\n\\varphi=v\\omega\\qquad\\text{and}\\qquad v^*v=s(\\omega).\n\\tag{2.2}\n\\]\nFor this pair:\n\n1. \\(\\omega=v^*\\varphi\\), and \\(\\|\\omega\\|=\\|\\varphi\\|=\\varphi(v^*)\\);\n2. \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\,\\omega(xx^*)\\) for all \\(x\\in M\\);\n3. \\(v^*v=s_r(\\varphi)\\) and \\(vv^*=s_l(\\varphi)\\);\n4. \\(\\varphi^*=v^*\\psi\\) with \\(\\psi=v\\omega v^*\\), that is \\(\\psi(x)=\\omega(v^*xv)\\), and \\((v^*,\\psi)\\) is the pair (2.2) for \\(\\varphi^*\\). Moreover \\(\\varphi=\\psi v\\).\n\n\n**Proof.** *Existence.* For \\(\\varphi=0\\), take \\(\\omega=0\\) and \\(v=0\\); any pair with the stated support condition must also have both entries zero, since \\(\\|\\omega\\|=\\|\\varphi\\|\\) by the argument below and \\(v^*v=0\\). We may therefore assume \\(\\varphi\\ne0\\). The unit ball \\(M_1\\) is \\(\\sigma\\)-weakly compact and \\(\\varphi\\) is \\(\\sigma\\)-weakly continuous, so \\(|\\varphi|\\) attains its supremum on \\(M_1\\); after a rotation there is \\(a\\in M_1\\) with \\(\\varphi(a)=\\|\\varphi\\|\\). Let \\(a^*=u|a^*|\\) be the polar decomposition of \\(a^*\\) (background fact 17), where \\(|a^*|=(aa^*)^{1/2}\\) and \\(u\\in M\\) is a partial isometry with \\(u^*u=s(|a^*|)\\), the range projection of \\(|a^*|\\). Then \\(a=|a^*|u^*\\) and \\(0\\le|a^*|\\le1\\).\n\nPut \\(\\omega_0=u^*\\varphi\\), that is \\(\\omega_0(x)=\\varphi(xu^*)\\). Then \\(\\|\\omega_0\\|\\le\\|\\varphi\\|\\) and \\(\\omega_0(|a^*|)=\\varphi(|a^*|u^*)=\\varphi(a)=\\|\\varphi\\|\\ge\\|\\omega_0\\|\\). So \\(\\omega_0\\) attains its norm at the positive contraction \\(|a^*|\\), and it is positive by background fact 21, with \\(\\|\\omega_0\\|=\\|\\varphi\\|\\).\n\nNext let \\(p=uu^*\\). Since \\(u^*p=u^*\\), we have \\(ap=|a^*|u^*p=a\\), so \\((p\\varphi)(a)=\\varphi(ap)=\\|\\varphi\\|\\). Hence \\(\\|p\\varphi\\|=\\|\\varphi\\|\\), and Lemma 1.1 gives \\(p\\varphi=\\varphi\\). Therefore \\(u\\omega_0=u(u^*\\varphi)=p\\varphi=\\varphi\\).\n\nFinally \\((u^*u)\\omega_0=u^*(uu^*\\varphi)=u^*\\varphi=\\omega_0\\), that is \\(\\omega_0(xu^*u)=\\omega_0(x)\\) for all \\(x\\). At \\(x=1\\) this gives \\(\\omega_0(1-u^*u)=0\\), so \\(s(\\omega_0)\\le u^*u\\). Put \\(v=us(\\omega_0)\\). It is a partial isometry with \\(v^*v=s(\\omega_0)u^*us(\\omega_0)=s(\\omega_0)\\), and \\(v\\omega_0=u(s(\\omega_0)\\omega_0)=u\\omega_0=\\varphi\\). So \\((v,\\omega_0)\\) satisfies (2.2).\n\n*Properties (1) and (2) for any pair (2.2).* Let \\((v,\\omega)\\) satisfy (2.2) and put \\(s=s(\\omega)\\). Then \\(v^*\\varphi=(v^*v)\\omega=s\\omega=\\omega\\). Also \\(\\varphi(v^*)=\\omega(v^*v)=\\omega(s)=\\omega(1)=\\|\\omega\\|\\), and \\(\\|\\varphi\\|\\le\\|v\\|\\|\\omega\\|\\le\\|\\omega\\|=\\varphi(v^*)\\le\\|\\varphi\\|\\). So \\(\\|\\omega\\|=\\|\\varphi\\|=\\varphi(v^*)\\). By the Cauchy–Schwarz inequality,\n\\[\n\\begin{gathered}\n|\\varphi(x)|^2\\\\\n=|\\omega(xv)|^2\\\\\n\\le\\omega(xx^*)\\,\\omega(v^*v)\\\\\n=\\|\\varphi\\|\\,\\omega(xx^*) .\n\\end{gathered}\n\\tag{2.3}\n\\]\n\n*Uniqueness.* Let \\((v,\\omega)\\) and \\((v_1,\\omega_1)\\) both satisfy (2.2). By (1) and (2) for the second pair, \\(\\|\\omega_1\\|=\\|\\varphi\\|\\) and \\(|\\varphi(y)|^2\\le\\|\\varphi\\|\\,\\omega_1(yy^*)\\). Since \\(\\omega(x)=(v^*\\varphi)(x)=\\varphi(xv^*)\\),\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n=|\\varphi(xv^*)|^2\\\\\n\\le\\|\\varphi\\|\\,\\omega_1(xv^*vx^*)\\\\\n\\le\\|\\varphi\\|\\,\\omega_1(xx^*) .\n\\end{gathered}\n\\]\nReplacing \\(x\\) by \\(x^*\\) and using that \\(\\omega\\) is hermitian, \\(|\\omega(x)|^2\\le\\|\\omega_1\\|\\,\\omega_1(x^*x)\\). Lemma 2.1 gives \\(\\omega=\\omega_1\\). Then \\((v-v_1)\\omega=0\\), so \\[\n\\begin{gathered}\n\\omega((v-v_1)^*(v-v_1))\\\\\n=((v-v_1)\\omega)((v-v_1)^*)\\\\\n=0.\n\\end{gathered}\n\\] By background fact 18, \\((v-v_1)s(\\omega)=0\\). As \\(v=vs(\\omega)\\) and \\(v_1=v_1s(\\omega)\\), we get \\(v=v_1\\).\n\n*(3).* Let \\(s=s(\\omega)=v^*v\\). Since \\(\\omega(y)=\\omega(sys)\\) and \\(vs=v\\), we get \\(\\varphi(sx)=\\omega(sxv)=\\omega(sxvs)=\\omega(xv)=\\varphi(x)\\), so \\(\\varphi=\\varphi s\\) and \\(s_r(\\varphi)\\le s\\) (Lemma 1.4). Conversely, if \\(\\varphi=\\varphi e\\), then \\[\n\\begin{gathered}\n\\omega(x)\\\\\n=\\varphi(xv^*)\\\\\n=\\varphi(exv^*)\\\\\n=\\omega(exv^*v)\\\\\n=\\omega(exs)\\\\\n=\\omega(ex),\n\\end{gathered}\n\\] so \\(\\omega=\\omega e\\) and \\(e\\ge s\\) by Lemma 1.4(4). Hence \\(s_r(\\varphi)=v^*v\\). Now put \\(q=vv^*\\). Since \\(qv=v\\), \\(\\varphi(xq)=\\omega(xqv)=\\omega(xv)=\\varphi(x)\\), so \\(\\varphi=q\\varphi\\) and \\(s_l(\\varphi)\\le q\\). Conversely, let \\(\\varphi=f\\varphi\\) for a projection \\(f\\). Then \\(\\omega(x(1-f)v)=\\varphi(x(1-f))=0\\) for all \\(x\\). Taking \\(x=v^*(1-f)\\) gives \\(\\omega\\big(((1-f)v)^*(1-f)v\\big)=0\\), so \\((1-f)vs=0\\) by background fact 18. Thus \\((1-f)v=0\\), \\(fv=v\\), and \\(f\\ge vv^*\\). Hence \\(s_l(\\varphi)=vv^*\\).\n\n*(4).* \\(\\psi=v\\omega v^*\\) is positive and normal. Since \\(\\omega\\) is hermitian and \\(\\omega(y)=\\omega(ys)\\),\n\\[\n\\begin{gathered}\n\\varphi^*(x)\\\\\n=\\overline{\\omega(x^*v)}\\\\\n=\\omega(v^*x)\\\\\n=\\omega(v^*xs)\\\\\n=\\omega(v^*xv^*v)\\\\\n=\\psi(xv^*)\\\\\n=(v^*\\psi)(x).\n\\end{gathered}\n\\]\nThe support of \\(\\psi\\) is \\(vv^*=(v^*)^*v^*\\): first \\(\\psi(1-vv^*)=\\omega(v^*v-v^*vv^*v)=0\\); and if \\(g\\) is a projection with \\(\\psi(1-g)=0\\), then \\(\\omega(((1-g)v)^*(1-g)v)=0\\), so \\((1-g)v=(1-g)vs=0\\) and \\(g\\ge vv^*\\). So \\((v^*,\\psi)\\) satisfies (2.2) for \\(\\varphi^*\\). Finally \\[\n\\begin{gathered}\n(\\psi v)(x)\\\\\n=\\psi(vx)\\\\\n=\\omega(v^*vxv)\\\\\n=\\omega(sxv)\\\\\n=\\omega(xv)\\\\\n=\\varphi(x),\n\\end{gathered}\n\\] using \\(\\omega=\\omega s\\). \\(\\square\\)\n\n**Definition 2.3.** The functional \\(\\omega\\) of Theorem 2.2 is the *absolute value* \\(|\\varphi|\\) of \\(\\varphi\\), and \\(\\varphi=v|\\varphi|\\) is the *polar decomposition* of \\(\\varphi\\). By Theorem 2.2,\n\\[\n\\begin{gathered}\n\\||\\varphi|\\|\\\\\n=\\||\\varphi^*|\\|\\\\\n=\\|\\varphi\\|,\\\\\n|\\varphi|\\\\\n=v^*\\varphi,\\\\\n|\\varphi^*|\\\\\n=v|\\varphi|v^*,\\\\\n\\varphi\\\\\n=v|\\varphi|\\\\\n=|\\varphi^*|v .\n\\end{gathered}\n\\tag{2.4}\n\\]\nAlso \\(v^*|\\varphi^*|v=|\\varphi|\\), since \\(\\omega(v^*vxv^*v)=\\omega(x)\\). If \\(\\varphi\\) is positive, then \\(\\varphi=s(\\varphi)\\varphi\\) is its polar decomposition, so \\(|\\varphi|=\\varphi\\) and \\(v=s(\\varphi)\\).\n\nThree examples show what the absolute value is in familiar cases.\n\n**Example 2.4** (Matrices). Let \\(M=M_n(\\mathbb C)\\). Every functional is \\(\\varphi(x)=\\operatorname{Tr}(\\rho x)\\) for a unique matrix \\(\\rho\\). Let \\(\\rho=w|\\rho|\\) be the polar decomposition of the matrix \\(\\rho\\). Then \\(|\\varphi|=\\operatorname{Tr}(|\\rho|\\,\\cdot\\,)\\) and \\(v=w\\). Indeed, \\((w|\\varphi|)(x)=\\operatorname{Tr}(|\\rho|xw)=\\operatorname{Tr}(w|\\rho|x)=\\varphi(x)\\). And the support of \\(\\operatorname{Tr}(\\sigma\\,\\cdot\\,)\\), for \\(\\sigma\\ge0\\), is the range projection of \\(\\sigma\\): \\(\\operatorname{Tr}(\\sigma(1-p))=\\operatorname{Tr}((1-p)\\sigma(1-p))\\) vanishes exactly when \\(\\sigma^{1/2}(1-p)=0\\), that is when \\(p\\) majorizes the range projection of \\(\\sigma\\). For \\(\\sigma=|\\rho|\\) this range projection is \\(w^*w\\). So (2.2) holds. By Theorem 2.2(3), \\(s_l(\\varphi)=ww^*\\) is the range projection of \\(\\rho\\), and \\(s_r(\\varphi)=w^*w\\) is the projection onto \\((\\ker\\rho)^\\perp\\).\n\n**Example 2.5** (Vector functionals). Let \\(M=B(H)\\) and \\(\\xi,\\eta\\ne0\\). Then\n\\[\n\\begin{gathered}\n|\\omega_{\\xi,\\eta}|\\\\\n=\\frac{\\|\\xi\\|}{\\|\\eta\\|}\\,\\omega_\\eta,\\\\\nv\\\\\n=\\frac{\\theta_{\\xi,\\eta}}{\\|\\xi\\|\\|\\eta\\|} .\n\\end{gathered}\n\\tag{2.5}\n\\]\nIndeed, \\(v^*v=\\theta_{\\eta,\\eta}/\\|\\eta\\|^2\\) is the projection onto \\(\\mathbb C\\eta\\), which is the support of \\(\\omega_\\eta\\); and \\(v\\eta=(\\|\\eta\\|/\\|\\xi\\|)\\xi\\), so \\[\n\\begin{gathered}\n(\\|\\xi\\|/\\|\\eta\\|)\\,\\omega_\\eta(xv)\\\\\n=(\\|\\xi\\|/\\|\\eta\\|)\\langle xv\\eta,\\eta\\rangle\\\\\n=\\langle x\\xi,\\eta\\rangle.\n\\end{gathered}\n\\] So the absolute value of \\(\\omega_{\\xi,\\eta}\\) lives on \\(\\eta\\), and that of \\(\\omega_{\\xi,\\eta}^*=\\omega_{\\eta,\\xi}\\) lives on \\(\\xi\\).\n\n**Example 2.6** (Commutative algebras). Let \\((\\Gamma,\\mu)\\) be a \\(\\sigma\\)-finite measure space and \\(M=L^\\infty(\\Gamma,\\mu)\\) acting on \\(L^2(\\Gamma,\\mu)\\) (background fact 28). The normal functionals are exactly \\(\\varphi_h(x)=\\int xh\\,d\\mu\\) with \\(h\\in L^1(\\Gamma,\\mu)\\), and \\(\\|\\varphi_h\\|=\\|h\\|_1\\). Indeed, \\(\\varphi_h=\\omega_{f,g}\\) for \\(f=|h|^{1/2}\\operatorname{sgn}h\\) and \\(g=|h|^{1/2}\\), so \\(\\varphi_h\\) is normal. Conversely, a \\(\\sigma\\)-weakly continuous functional on \\(M\\) extends to a \\(\\sigma\\)-weakly continuous functional on \\(B(L^2)\\) (background fact 3, since the \\(\\sigma\\)-weak topology of \\(M\\) is the relative one, and the \\(\\sigma\\)-weak topology is locally convex). That extension is \\(\\sum_n\\omega_{\\xi_n,\\eta_n}\\) with \\(\\sum_n\\|\\xi_n\\|\\|\\eta_n\\|<\\infty\\) (background fact 27), so on \\(M\\) it is \\(\\varphi_h\\) with \\(h=\\sum_n\\xi_n\\bar\\eta_n\\in L^1\\). Finally \\(\\|\\varphi_h\\|\\le\\|h\\|_1\\), and \\(x=\\overline{\\operatorname{sgn}h}\\) gives equality. Now put \\(v=\\operatorname{sgn}h\\), with \\(\\operatorname{sgn}h=h/|h|\\) where \\(h\\ne0\\) and \\(0\\) elsewhere. Then \\(v\\varphi_{|h|}=\\varphi_h\\), and \\(v^*v=1_{\\{h\\ne0\\}}\\) is the support of \\(\\varphi_{|h|}\\). So\n\\[\n|\\varphi_h|=\\varphi_{|h|} ,\n\\]\nthe familiar total variation. The same argument works for counting measure on any set \\(\\Gamma\\), even when \\(\\Gamma\\) is uncountable: \\(\\ell^\\infty(\\Gamma)\\), acting diagonally on \\(\\ell^2(\\Gamma)\\), is a von Neumann algebra, because an operator that commutes with every coordinate projection is diagonal, so \\(\\ell^\\infty(\\Gamma)'=\\ell^\\infty(\\Gamma)\\). Thus \\(\\ell^\\infty(\\Gamma)_*=\\ell^1(\\Gamma)\\), and the absolute value is taken coordinatewise.\n\nFor a \\(C^*\\)-algebra the partial isometry lives in the bidual.\n\n**Theorem 2.7** (Polar decomposition on a \\(C^*\\)-algebra). Let \\(A\\) be a \\(C^*\\)-algebra and \\(f\\in A^*\\). There is exactly one pair of a partial isometry \\(v\\in\\tilde A\\) and a positive \\(\\omega\\in A^*\\) with \\(f=v\\omega\\) and \\(v^*v=s(\\omega)\\), the support of \\(\\omega\\) in \\(\\tilde A\\). We write \\(|f|=\\omega\\). Then \\(\\||f|\\|=\\|f\\|\\) and \\(|f(x)|^2\\le\\|f\\|\\,|f|(xx^*)\\) for all \\(x\\in\\tilde A\\), in particular for all \\(x\\in A\\).\n\n**Proof.** Apply Theorem 2.2 to the von Neumann algebra \\(\\tilde A\\), whose predual is \\(A^*\\). \\(\\square\\)\n\nExamples in Section 3 show that \\(v\\) need not lie in \\(A\\), even when \\(f\\) is positive.\n\nThe hermitian case of the polar decomposition is the norm-additive splitting of a hermitian functional into positive parts. The lesson on the universal enveloping algebra proves existence before the bidual construction and proves uniqueness directly ([Proposition 3.4](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md#oa-fnd-wa-04)); here both drop out of Theorem 2.2.\n\n**Corollary 2.8** (Hermitian functionals). Let \\(\\varphi\\in M_*\\) be hermitian, with polar decomposition \\(\\varphi=v|\\varphi|\\), and put \\(s=s(|\\varphi|)\\).\n\n1. \\(v=v^*\\), \\(|\\varphi^*|=|\\varphi|\\), and \\(v|\\varphi|=|\\varphi|v\\).\n2. \\(e=\\tfrac12(s+v)\\) and \\(f=\\tfrac12(s-v)\\) are orthogonal projections with \\(e+f=s\\) and \\(e-f=v\\). The functionals \\(\\varphi_+=e|\\varphi|=\\tfrac12(|\\varphi|+\\varphi)\\) and \\(\\varphi_-=f|\\varphi|=\\tfrac12(|\\varphi|-\\varphi)\\) are positive and normal, \\(s(\\varphi_+)=e\\), \\(s(\\varphi_-)=f\\), \\(\\varphi=\\varphi_+-\\varphi_-\\) and \\(\\|\\varphi\\|=\\|\\varphi_+\\|+\\|\\varphi_-\\|\\).\n3. If \\(\\varphi=\\psi_1-\\psi_2\\) with \\(\\psi_1,\\psi_2\\in M_*^+\\) and \\(\\|\\varphi\\|=\\|\\psi_1\\|+\\|\\psi_2\\|\\), then \\(\\psi_1=\\varphi_+\\) and \\(\\psi_2=\\varphi_-\\).\n4. For \\(\\psi_1,\\psi_2\\in M_*^+\\): \\(\\|\\psi_1-\\psi_2\\|=\\|\\psi_1\\|+\\|\\psi_2\\|\\) exactly when \\(s(\\psi_1)s(\\psi_2)=0\\).\n\n**Proof.** (1) Since \\(\\varphi^*=\\varphi\\), Theorem 2.2(4) and uniqueness give \\(v^*=v\\) and \\(v|\\varphi|v^*=|\\varphi|\\). Then, by the module rules, \\(|\\varphi|v=(v|\\varphi|v^*)v=v|\\varphi|(v^*v)=v(|\\varphi|s)=v|\\varphi|\\).\n\n(2) \\(v\\) is a self-adjoint partial isometry with \\(v^2=v^*v=s\\) and \\(vs=sv=v\\). So \\(e^2=\\tfrac14(s+2v+s)=e\\), \\(f^2=f\\), \\(ef=\\tfrac14(s-v^2)=0\\), \\(e+f=s\\) and \\(e-f=v\\). By (1), \\(e\\) commutes with \\(|\\varphi|\\): \\(e|\\varphi|=|\\varphi|e\\). Hence \\[\n\\begin{gathered}\n(e|\\varphi|)(x)\\\\\n=|\\varphi|(xe)\\\\\n=|\\varphi|(xe\\cdot e)\\\\\n=(|\\varphi|e)(xe)\\\\\n=|\\varphi|(exe),\n\\end{gathered}\n\\] which is positive in \\(x\\); likewise for \\(f\\). Next \\(\\varphi_+-\\varphi_-=(e-f)|\\varphi|=v|\\varphi|=\\varphi\\) and \\(\\varphi_++\\varphi_-=s|\\varphi|=|\\varphi|\\), which give the formulas \\(\\tfrac12(|\\varphi|\\pm\\varphi)\\) and \\[\n\\begin{gathered}\n\\|\\varphi_+\\|+\\|\\varphi_-\\|\\\\\n=|\\varphi|(e)+|\\varphi|(f)\\\\\n=|\\varphi|(s)\\\\\n=\\|\\varphi\\|.\n\\end{gathered}\n\\] Since \\(\\varphi_+(1-e)=|\\varphi|(e(1-e)e)=0\\), \\(s(\\varphi_+)\\le e\\); and if \\(g\\le e\\) is a projection with \\(\\varphi_+(e-g)=0\\), then \\(|\\varphi|(e-g)=0\\) with \\(e-g\\le s\\), so \\(e=g\\) by faithfulness of \\(|\\varphi|\\) on \\(sMs\\). Thus \\(s(\\varphi_+)=e\\), and similarly \\(s(\\varphi_-)=f\\).\n\n(3) By Theorem 2.2(1), \\(\\|\\varphi\\|=\\varphi(v^*)=\\varphi(e-f)\\), so\n\\[\n\\begin{gathered}\n\\psi_1(1)+\\psi_2(1)\\\\\n=\\psi_1(e)-\\psi_1(f)-\\psi_2(e)+\\psi_2(f).\n\\end{gathered}\n\\]\nHere \\(\\psi_1(e)\\le\\psi_1(1)\\), \\(\\psi_2(f)\\le\\psi_2(1)\\) and \\(\\psi_1(f),\\psi_2(e)\\ge0\\). So all four inequalities are equalities: \\(\\psi_1(1-e)=0\\) and \\(\\psi_2(1-f)=0\\), that is \\(s(\\psi_1)\\le e\\) and \\(s(\\psi_2)\\le f\\). Then \\(e\\psi_1=\\psi_1\\), because \\(\\psi_1(xe)=\\psi_1(xes(\\psi_1))=\\psi_1(xs(\\psi_1))=\\psi_1(x)\\); and \\(e\\psi_2=0\\), because \\(\\psi_2(xe)=\\psi_2(xes(\\psi_2))=0\\) as \\(es(\\psi_2)=efs(\\psi_2)=0\\). Therefore \\(\\psi_1=e(\\psi_1-\\psi_2)=e\\varphi=ev|\\varphi|=e|\\varphi|=\\varphi_+\\), since \\(ev=e(e-f)=e\\). And \\(\\psi_2=\\psi_1-\\varphi=\\varphi_-\\).\n\n(4) If \\(s(\\psi_1)s(\\psi_2)=0\\), then \\(x=s(\\psi_1)-s(\\psi_2)\\) has norm at most one and \\((\\psi_1-\\psi_2)(x)=\\psi_1(1)+\\psi_2(1)\\), so \\(\\|\\psi_1-\\psi_2\\|\\ge\\|\\psi_1\\|+\\|\\psi_2\\|\\); the other inequality is the triangle inequality. Conversely, if the norms add, the proof of (3), applied to \\(\\varphi=\\psi_1-\\psi_2\\), gives \\(s(\\psi_1)\\le e\\) and \\(s(\\psi_2)\\le f\\), which are orthogonal. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-03",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "2. The polar decomposition",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 2. The polar decomposition\n\nWe first record a comparison principle for positive functionals. It gives uniqueness in the polar decomposition, and in Section 3 it characterizes the absolute value.\n\n**Lemma 2.1** (Comparison of positive functionals). Let \\(A\\) be a \\(C^*\\)-algebra and \\(\\omega,\\omega_1\\in A^*_+\\) with \\(\\|\\omega\\|=\\|\\omega_1\\|\\) and\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\|\\omega_1\\|\\,\\omega_1(x^*x)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{2.1}\n\\]\nThen \\(\\omega=\\omega_1\\).\n\n**Proof.** Put \\(c=\\|\\omega_1\\|\\); if \\(c=0\\) both functionals vanish. Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega_1\\), so \\(\\omega_1(x^*x)=\\|\\pi(x)\\xi\\|^2\\) and \\(\\|\\xi\\|^2=c\\). By (2.1), \\(\\pi(x)\\xi\\mapsto\\omega(x)\\) is a well-defined linear functional on the dense subspace \\(\\pi(A)\\xi\\), of norm at most \\(c^{1/2}\\). The Riesz theorem gives \\(\\eta\\in H\\) with \\(\\|\\eta\\|\\le c^{1/2}\\) and \\(\\omega(x)=\\langle\\pi(x)\\xi,\\eta\\rangle\\) for \\(x\\in A\\). For an approximate unit \\((u_\\lambda)\\), \\(\\omega(u_\\lambda)\\to\\|\\omega\\|=c\\) and \\(\\pi(u_\\lambda)\\xi\\to\\xi\\) (background fact 12). Hence \\(\\langle\\xi,\\eta\\rangle=c\\), and\n\\[\n\\begin{gathered}\n\\|\\xi-\\eta\\|^2\\\\\n=\\|\\xi\\|^2-2\\operatorname{Re}\\langle\\xi,\\eta\\rangle+\\|\\eta\\|^2\\\\\n\\le c-2c+c\\\\\n=0 .\n\\end{gathered}\n\\]\nSo \\(\\eta=\\xi\\) and \\(\\omega(x)=\\langle\\pi(x)\\xi,\\xi\\rangle=\\omega_1(x)\\). \\(\\square\\)\n\n**Theorem 2.2** (Polar decomposition). Let \\(M\\) be a von Neumann algebra and \\(\\varphi\\in M_*\\). There is exactly one pair \\((v,\\omega)\\) of a partial isometry \\(v\\in M\\) and a positive normal functional \\(\\omega\\) with\n\\[\n\\varphi=v\\omega\\qquad\\text{and}\\qquad v^*v=s(\\omega).\n\\tag{2.2}\n\\]\nFor this pair:\n\n1. \\(\\omega=v^*\\varphi\\), and \\(\\|\\omega\\|=\\|\\varphi\\|=\\varphi(v^*)\\);\n2. \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\,\\omega(xx^*)\\) for all \\(x\\in M\\);\n3. \\(v^*v=s_r(\\varphi)\\) and \\(vv^*=s_l(\\varphi)\\);\n4. \\(\\varphi^*=v^*\\psi\\) with \\(\\psi=v\\omega v^*\\), that is \\(\\psi(x)=\\omega(v^*xv)\\), and \\((v^*,\\psi)\\) is the pair (2.2) for \\(\\varphi^*\\). Moreover \\(\\varphi=\\psi v\\).\n\n\n**Proof.** *Existence.* For \\(\\varphi=0\\), take \\(\\omega=0\\) and \\(v=0\\); any pair with the stated support condition must also have both entries zero, since \\(\\|\\omega\\|=\\|\\varphi\\|\\) by the argument below and \\(v^*v=0\\). We may therefore assume \\(\\varphi\\ne0\\). The unit ball \\(M_1\\) is \\(\\sigma\\)-weakly compact and \\(\\varphi\\) is \\(\\sigma\\)-weakly continuous, so \\(|\\varphi|\\) attains its supremum on \\(M_1\\); after a rotation there is \\(a\\in M_1\\) with \\(\\varphi(a)=\\|\\varphi\\|\\). Let \\(a^*=u|a^*|\\) be the polar decomposition of \\(a^*\\) (background fact 17), where \\(|a^*|=(aa^*)^{1/2}\\) and \\(u\\in M\\) is a partial isometry with \\(u^*u=s(|a^*|)\\), the range projection of \\(|a^*|\\). Then \\(a=|a^*|u^*\\) and \\(0\\le|a^*|\\le1\\).\n\nPut \\(\\omega_0=u^*\\varphi\\), that is \\(\\omega_0(x)=\\varphi(xu^*)\\). Then \\(\\|\\omega_0\\|\\le\\|\\varphi\\|\\) and \\(\\omega_0(|a^*|)=\\varphi(|a^*|u^*)=\\varphi(a)=\\|\\varphi\\|\\ge\\|\\omega_0\\|\\). So \\(\\omega_0\\) attains its norm at the positive contraction \\(|a^*|\\), and it is positive by background fact 21, with \\(\\|\\omega_0\\|=\\|\\varphi\\|\\).\n\nNext let \\(p=uu^*\\). Since \\(u^*p=u^*\\), we have \\(ap=|a^*|u^*p=a\\), so \\((p\\varphi)(a)=\\varphi(ap)=\\|\\varphi\\|\\). Hence \\(\\|p\\varphi\\|=\\|\\varphi\\|\\), and Lemma 1.1 gives \\(p\\varphi=\\varphi\\). Therefore \\(u\\omega_0=u(u^*\\varphi)=p\\varphi=\\varphi\\).\n\nFinally \\((u^*u)\\omega_0=u^*(uu^*\\varphi)=u^*\\varphi=\\omega_0\\), that is \\(\\omega_0(xu^*u)=\\omega_0(x)\\) for all \\(x\\). At \\(x=1\\) this gives \\(\\omega_0(1-u^*u)=0\\), so \\(s(\\omega_0)\\le u^*u\\). Put \\(v=us(\\omega_0)\\). It is a partial isometry with \\(v^*v=s(\\omega_0)u^*us(\\omega_0)=s(\\omega_0)\\), and \\(v\\omega_0=u(s(\\omega_0)\\omega_0)=u\\omega_0=\\varphi\\). So \\((v,\\omega_0)\\) satisfies (2.2).\n\n*Properties (1) and (2) for any pair (2.2).* Let \\((v,\\omega)\\) satisfy (2.2) and put \\(s=s(\\omega)\\). Then \\(v^*\\varphi=(v^*v)\\omega=s\\omega=\\omega\\). Also \\(\\varphi(v^*)=\\omega(v^*v)=\\omega(s)=\\omega(1)=\\|\\omega\\|\\), and \\(\\|\\varphi\\|\\le\\|v\\|\\|\\omega\\|\\le\\|\\omega\\|=\\varphi(v^*)\\le\\|\\varphi\\|\\). So \\(\\|\\omega\\|=\\|\\varphi\\|=\\varphi(v^*)\\). By the Cauchy–Schwarz inequality,\n\\[\n\\begin{gathered}\n|\\varphi(x)|^2\\\\\n=|\\omega(xv)|^2\\\\\n\\le\\omega(xx^*)\\,\\omega(v^*v)\\\\\n=\\|\\varphi\\|\\,\\omega(xx^*) .\n\\end{gathered}\n\\tag{2.3}\n\\]\n\n*Uniqueness.* Let \\((v,\\omega)\\) and \\((v_1,\\omega_1)\\) both satisfy (2.2). By (1) and (2) for the second pair, \\(\\|\\omega_1\\|=\\|\\varphi\\|\\) and \\(|\\varphi(y)|^2\\le\\|\\varphi\\|\\,\\omega_1(yy^*)\\). Since \\(\\omega(x)=(v^*\\varphi)(x)=\\varphi(xv^*)\\),\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n=|\\varphi(xv^*)|^2\\\\\n\\le\\|\\varphi\\|\\,\\omega_1(xv^*vx^*)\\\\\n\\le\\|\\varphi\\|\\,\\omega_1(xx^*) .\n\\end{gathered}\n\\]\nReplacing \\(x\\) by \\(x^*\\) and using that \\(\\omega\\) is hermitian, \\(|\\omega(x)|^2\\le\\|\\omega_1\\|\\,\\omega_1(x^*x)\\). Lemma 2.1 gives \\(\\omega=\\omega_1\\). Then \\((v-v_1)\\omega=0\\), so \\[\n\\begin{gathered}\n\\omega((v-v_1)^*(v-v_1))\\\\\n=((v-v_1)\\omega)((v-v_1)^*)\\\\\n=0.\n\\end{gathered}\n\\] By background fact 18, \\((v-v_1)s(\\omega)=0\\). As \\(v=vs(\\omega)\\) and \\(v_1=v_1s(\\omega)\\), we get \\(v=v_1\\).\n\n*(3).* Let \\(s=s(\\omega)=v^*v\\). Since \\(\\omega(y)=\\omega(sys)\\) and \\(vs=v\\), we get \\(\\varphi(sx)=\\omega(sxv)=\\omega(sxvs)=\\omega(xv)=\\varphi(x)\\), so \\(\\varphi=\\varphi s\\) and \\(s_r(\\varphi)\\le s\\) (Lemma 1.4). Conversely, if \\(\\varphi=\\varphi e\\), then \\[\n\\begin{gathered}\n\\omega(x)\\\\\n=\\varphi(xv^*)\\\\\n=\\varphi(exv^*)\\\\\n=\\omega(exv^*v)\\\\\n=\\omega(exs)\\\\\n=\\omega(ex),\n\\end{gathered}\n\\] so \\(\\omega=\\omega e\\) and \\(e\\ge s\\) by Lemma 1.4(4). Hence \\(s_r(\\varphi)=v^*v\\). Now put \\(q=vv^*\\). Since \\(qv=v\\), \\(\\varphi(xq)=\\omega(xqv)=\\omega(xv)=\\varphi(x)\\), so \\(\\varphi=q\\varphi\\) and \\(s_l(\\varphi)\\le q\\). Conversely, let \\(\\varphi=f\\varphi\\) for a projection \\(f\\). Then \\(\\omega(x(1-f)v)=\\varphi(x(1-f))=0\\) for all \\(x\\). Taking \\(x=v^*(1-f)\\) gives \\(\\omega\\big(((1-f)v)^*(1-f)v\\big)=0\\), so \\((1-f)vs=0\\) by background fact 18. Thus \\((1-f)v=0\\), \\(fv=v\\), and \\(f\\ge vv^*\\). Hence \\(s_l(\\varphi)=vv^*\\).\n\n*(4).* \\(\\psi=v\\omega v^*\\) is positive and normal. Since \\(\\omega\\) is hermitian and \\(\\omega(y)=\\omega(ys)\\),\n\\[\n\\begin{gathered}\n\\varphi^*(x)\\\\\n=\\overline{\\omega(x^*v)}\\\\\n=\\omega(v^*x)\\\\\n=\\omega(v^*xs)\\\\\n=\\omega(v^*xv^*v)\\\\\n=\\psi(xv^*)\\\\\n=(v^*\\psi)(x).\n\\end{gathered}\n\\]\nThe support of \\(\\psi\\) is \\(vv^*=(v^*)^*v^*\\): first \\(\\psi(1-vv^*)=\\omega(v^*v-v^*vv^*v)=0\\); and if \\(g\\) is a projection with \\(\\psi(1-g)=0\\), then \\(\\omega(((1-g)v)^*(1-g)v)=0\\), so \\((1-g)v=(1-g)vs=0\\) and \\(g\\ge vv^*\\). So \\((v^*,\\psi)\\) satisfies (2.2) for \\(\\varphi^*\\). Finally \\[\n\\begin{gathered}\n(\\psi v)(x)\\\\\n=\\psi(vx)\\\\\n=\\omega(v^*vxv)\\\\\n=\\omega(sxv)\\\\\n=\\omega(xv)\\\\\n=\\varphi(x),\n\\end{gathered}\n\\] using \\(\\omega=\\omega s\\). \\(\\square\\)\n\n**Definition 2.3.** The functional \\(\\omega\\) of Theorem 2.2 is the *absolute value* \\(|\\varphi|\\) of \\(\\varphi\\), and \\(\\varphi=v|\\varphi|\\) is the *polar decomposition* of \\(\\varphi\\). By Theorem 2.2,\n\\[\n\\begin{gathered}\n\\||\\varphi|\\|\\\\\n=\\||\\varphi^*|\\|\\\\\n=\\|\\varphi\\|,\\\\\n|\\varphi|\\\\\n=v^*\\varphi,\\\\\n|\\varphi^*|\\\\\n=v|\\varphi|v^*,\\\\\n\\varphi\\\\\n=v|\\varphi|\\\\\n=|\\varphi^*|v .\n\\end{gathered}\n\\tag{2.4}\n\\]\nAlso \\(v^*|\\varphi^*|v=|\\varphi|\\), since \\(\\omega(v^*vxv^*v)=\\omega(x)\\). If \\(\\varphi\\) is positive, then \\(\\varphi=s(\\varphi)\\varphi\\) is its polar decomposition, so \\(|\\varphi|=\\varphi\\) and \\(v=s(\\varphi)\\).\n\nThree examples show what the absolute value is in familiar cases.\n\n**Example 2.4** (Matrices). Let \\(M=M_n(\\mathbb C)\\). Every functional is \\(\\varphi(x)=\\operatorname{Tr}(\\rho x)\\) for a unique matrix \\(\\rho\\). Let \\(\\rho=w|\\rho|\\) be the polar decomposition of the matrix \\(\\rho\\). Then \\(|\\varphi|=\\operatorname{Tr}(|\\rho|\\,\\cdot\\,)\\) and \\(v=w\\). Indeed, \\((w|\\varphi|)(x)=\\operatorname{Tr}(|\\rho|xw)=\\operatorname{Tr}(w|\\rho|x)=\\varphi(x)\\). And the support of \\(\\operatorname{Tr}(\\sigma\\,\\cdot\\,)\\), for \\(\\sigma\\ge0\\), is the range projection of \\(\\sigma\\): \\(\\operatorname{Tr}(\\sigma(1-p))=\\operatorname{Tr}((1-p)\\sigma(1-p))\\) vanishes exactly when \\(\\sigma^{1/2}(1-p)=0\\), that is when \\(p\\) majorizes the range projection of \\(\\sigma\\). For \\(\\sigma=|\\rho|\\) this range projection is \\(w^*w\\). So (2.2) holds. By Theorem 2.2(3), \\(s_l(\\varphi)=ww^*\\) is the range projection of \\(\\rho\\), and \\(s_r(\\varphi)=w^*w\\) is the projection onto \\((\\ker\\rho)^\\perp\\).\n\n**Example 2.5** (Vector functionals). Let \\(M=B(H)\\) and \\(\\xi,\\eta\\ne0\\). Then\n\\[\n\\begin{gathered}\n|\\omega_{\\xi,\\eta}|\\\\\n=\\frac{\\|\\xi\\|}{\\|\\eta\\|}\\,\\omega_\\eta,\\\\\nv\\\\\n=\\frac{\\theta_{\\xi,\\eta}}{\\|\\xi\\|\\|\\eta\\|} .\n\\end{gathered}\n\\tag{2.5}\n\\]\nIndeed, \\(v^*v=\\theta_{\\eta,\\eta}/\\|\\eta\\|^2\\) is the projection onto \\(\\mathbb C\\eta\\), which is the support of \\(\\omega_\\eta\\); and \\(v\\eta=(\\|\\eta\\|/\\|\\xi\\|)\\xi\\), so \\[\n\\begin{gathered}\n(\\|\\xi\\|/\\|\\eta\\|)\\,\\omega_\\eta(xv)\\\\\n=(\\|\\xi\\|/\\|\\eta\\|)\\langle xv\\eta,\\eta\\rangle\\\\\n=\\langle x\\xi,\\eta\\rangle.\n\\end{gathered}\n\\] So the absolute value of \\(\\omega_{\\xi,\\eta}\\) lives on \\(\\eta\\), and that of \\(\\omega_{\\xi,\\eta}^*=\\omega_{\\eta,\\xi}\\) lives on \\(\\xi\\).\n\n**Example 2.6** (Commutative algebras). Let \\((\\Gamma,\\mu)\\) be a \\(\\sigma\\)-finite measure space and \\(M=L^\\infty(\\Gamma,\\mu)\\) acting on \\(L^2(\\Gamma,\\mu)\\) (background fact 28). The normal functionals are exactly \\(\\varphi_h(x)=\\int xh\\,d\\mu\\) with \\(h\\in L^1(\\Gamma,\\mu)\\), and \\(\\|\\varphi_h\\|=\\|h\\|_1\\). Indeed, \\(\\varphi_h=\\omega_{f,g}\\) for \\(f=|h|^{1/2}\\operatorname{sgn}h\\) and \\(g=|h|^{1/2}\\), so \\(\\varphi_h\\) is normal. Conversely, a \\(\\sigma\\)-weakly continuous functional on \\(M\\) extends to a \\(\\sigma\\)-weakly continuous functional on \\(B(L^2)\\) (background fact 3, since the \\(\\sigma\\)-weak topology of \\(M\\) is the relative one, and the \\(\\sigma\\)-weak topology is locally convex). That extension is \\(\\sum_n\\omega_{\\xi_n,\\eta_n}\\) with \\(\\sum_n\\|\\xi_n\\|\\|\\eta_n\\|<\\infty\\) (background fact 27), so on \\(M\\) it is \\(\\varphi_h\\) with \\(h=\\sum_n\\xi_n\\bar\\eta_n\\in L^1\\). Finally \\(\\|\\varphi_h\\|\\le\\|h\\|_1\\), and \\(x=\\overline{\\operatorname{sgn}h}\\) gives equality. Now put \\(v=\\operatorname{sgn}h\\), with \\(\\operatorname{sgn}h=h/|h|\\) where \\(h\\ne0\\) and \\(0\\) elsewhere. Then \\(v\\varphi_{|h|}=\\varphi_h\\), and \\(v^*v=1_{\\{h\\ne0\\}}\\) is the support of \\(\\varphi_{|h|}\\). So\n\\[\n|\\varphi_h|=\\varphi_{|h|} ,\n\\]\nthe familiar total variation. The same argument works for counting measure on any set \\(\\Gamma\\), even when \\(\\Gamma\\) is uncountable: \\(\\ell^\\infty(\\Gamma)\\), acting diagonally on \\(\\ell^2(\\Gamma)\\), is a von Neumann algebra, because an operator that commutes with every coordinate projection is diagonal, so \\(\\ell^\\infty(\\Gamma)'=\\ell^\\infty(\\Gamma)\\). Thus \\(\\ell^\\infty(\\Gamma)_*=\\ell^1(\\Gamma)\\), and the absolute value is taken coordinatewise.\n\nFor a \\(C^*\\)-algebra the partial isometry lives in the bidual.\n\n**Theorem 2.7** (Polar decomposition on a \\(C^*\\)-algebra). Let \\(A\\) be a \\(C^*\\)-algebra and \\(f\\in A^*\\). There is exactly one pair of a partial isometry \\(v\\in\\tilde A\\) and a positive \\(\\omega\\in A^*\\) with \\(f=v\\omega\\) and \\(v^*v=s(\\omega)\\), the support of \\(\\omega\\) in \\(\\tilde A\\). We write \\(|f|=\\omega\\). Then \\(\\||f|\\|=\\|f\\|\\) and \\(|f(x)|^2\\le\\|f\\|\\,|f|(xx^*)\\) for all \\(x\\in\\tilde A\\), in particular for all \\(x\\in A\\).\n\n**Proof.** Apply Theorem 2.2 to the von Neumann algebra \\(\\tilde A\\), whose predual is \\(A^*\\). \\(\\square\\)\n\nExamples in Section 3 show that \\(v\\) need not lie in \\(A\\), even when \\(f\\) is positive.\n\nThe hermitian case of the polar decomposition is the norm-additive splitting of a hermitian functional into positive parts. The lesson on the universal enveloping algebra proves existence before the bidual construction and proves uniqueness directly ([Proposition 3.4](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md#oa-fnd-wa-04)); here both drop out of Theorem 2.2.\n\n**Corollary 2.8** (Hermitian functionals). Let \\(\\varphi\\in M_*\\) be hermitian, with polar decomposition \\(\\varphi=v|\\varphi|\\), and put \\(s=s(|\\varphi|)\\).\n\n1. \\(v=v^*\\), \\(|\\varphi^*|=|\\varphi|\\), and \\(v|\\varphi|=|\\varphi|v\\).\n2. \\(e=\\tfrac12(s+v)\\) and \\(f=\\tfrac12(s-v)\\) are orthogonal projections with \\(e+f=s\\) and \\(e-f=v\\). The functionals \\(\\varphi_+=e|\\varphi|=\\tfrac12(|\\varphi|+\\varphi)\\) and \\(\\varphi_-=f|\\varphi|=\\tfrac12(|\\varphi|-\\varphi)\\) are positive and normal, \\(s(\\varphi_+)=e\\), \\(s(\\varphi_-)=f\\), \\(\\varphi=\\varphi_+-\\varphi_-\\) and \\(\\|\\varphi\\|=\\|\\varphi_+\\|+\\|\\varphi_-\\|\\).\n3. If \\(\\varphi=\\psi_1-\\psi_2\\) with \\(\\psi_1,\\psi_2\\in M_*^+\\) and \\(\\|\\varphi\\|=\\|\\psi_1\\|+\\|\\psi_2\\|\\), then \\(\\psi_1=\\varphi_+\\) and \\(\\psi_2=\\varphi_-\\).\n4. For \\(\\psi_1,\\psi_2\\in M_*^+\\): \\(\\|\\psi_1-\\psi_2\\|=\\|\\psi_1\\|+\\|\\psi_2\\|\\) exactly when \\(s(\\psi_1)s(\\psi_2)=0\\).\n\n**Proof.** (1) Since \\(\\varphi^*=\\varphi\\), Theorem 2.2(4) and uniqueness give \\(v^*=v\\) and \\(v|\\varphi|v^*=|\\varphi|\\). Then, by the module rules, \\(|\\varphi|v=(v|\\varphi|v^*)v=v|\\varphi|(v^*v)=v(|\\varphi|s)=v|\\varphi|\\).\n\n(2) \\(v\\) is a self-adjoint partial isometry with \\(v^2=v^*v=s\\) and \\(vs=sv=v\\). So \\(e^2=\\tfrac14(s+2v+s)=e\\), \\(f^2=f\\), \\(ef=\\tfrac14(s-v^2)=0\\), \\(e+f=s\\) and \\(e-f=v\\). By (1), \\(e\\) commutes with \\(|\\varphi|\\): \\(e|\\varphi|=|\\varphi|e\\). Hence \\[\n\\begin{gathered}\n(e|\\varphi|)(x)\\\\\n=|\\varphi|(xe)\\\\\n=|\\varphi|(xe\\cdot e)\\\\\n=(|\\varphi|e)(xe)\\\\\n=|\\varphi|(exe),\n\\end{gathered}\n\\] which is positive in \\(x\\); likewise for \\(f\\). Next \\(\\varphi_+-\\varphi_-=(e-f)|\\varphi|=v|\\varphi|=\\varphi\\) and \\(\\varphi_++\\varphi_-=s|\\varphi|=|\\varphi|\\), which give the formulas \\(\\tfrac12(|\\varphi|\\pm\\varphi)\\) and \\[\n\\begin{gathered}\n\\|\\varphi_+\\|+\\|\\varphi_-\\|\\\\\n=|\\varphi|(e)+|\\varphi|(f)\\\\\n=|\\varphi|(s)\\\\\n=\\|\\varphi\\|.\n\\end{gathered}\n\\] Since \\(\\varphi_+(1-e)=|\\varphi|(e(1-e)e)=0\\), \\(s(\\varphi_+)\\le e\\); and if \\(g\\le e\\) is a projection with \\(\\varphi_+(e-g)=0\\), then \\(|\\varphi|(e-g)=0\\) with \\(e-g\\le s\\), so \\(e=g\\) by faithfulness of \\(|\\varphi|\\) on \\(sMs\\). Thus \\(s(\\varphi_+)=e\\), and similarly \\(s(\\varphi_-)=f\\).\n\n(3) By Theorem 2.2(1), \\(\\|\\varphi\\|=\\varphi(v^*)=\\varphi(e-f)\\), so\n\\[\n\\begin{gathered}\n\\psi_1(1)+\\psi_2(1)\\\\\n=\\psi_1(e)-\\psi_1(f)-\\psi_2(e)+\\psi_2(f).\n\\end{gathered}\n\\]\nHere \\(\\psi_1(e)\\le\\psi_1(1)\\), \\(\\psi_2(f)\\le\\psi_2(1)\\) and \\(\\psi_1(f),\\psi_2(e)\\ge0\\). So all four inequalities are equalities: \\(\\psi_1(1-e)=0\\) and \\(\\psi_2(1-f)=0\\), that is \\(s(\\psi_1)\\le e\\) and \\(s(\\psi_2)\\le f\\). Then \\(e\\psi_1=\\psi_1\\), because \\(\\psi_1(xe)=\\psi_1(xes(\\psi_1))=\\psi_1(xs(\\psi_1))=\\psi_1(x)\\); and \\(e\\psi_2=0\\), because \\(\\psi_2(xe)=\\psi_2(xes(\\psi_2))=0\\) as \\(es(\\psi_2)=efs(\\psi_2)=0\\). Therefore \\(\\psi_1=e(\\psi_1-\\psi_2)=e\\varphi=ev|\\varphi|=e|\\varphi|=\\varphi_+\\), since \\(ev=e(e-f)=e\\). And \\(\\psi_2=\\psi_1-\\varphi=\\varphi_-\\).\n\n(4) If \\(s(\\psi_1)s(\\psi_2)=0\\), then \\(x=s(\\psi_1)-s(\\psi_2)\\) has norm at most one and \\((\\psi_1-\\psi_2)(x)=\\psi_1(1)+\\psi_2(1)\\), so \\(\\|\\psi_1-\\psi_2\\|\\ge\\|\\psi_1\\|+\\|\\psi_2\\|\\); the other inequality is the triangle inequality. Conversely, if the norms add, the proof of (3), applied to \\(\\varphi=\\psi_1-\\psi_2\\), gives \\(s(\\psi_1)\\le e\\) and \\(s(\\psi_2)\\le f\\), which are orthogonal. \\(\\square\\)\n\n",
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      "id": "OA-FND-PD-04",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "3. How the absolute value is determined",
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      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 3. How the absolute value is determined\n\nThe absolute value was defined through the partial isometry \\(v\\). It is also the only positive functional of the right size that dominates \\(\\varphi\\) in the sense of (2.3). This description does not mention \\(v\\), does not require normality, and works for every \\(C^*\\)-algebra.\n\n**Theorem 3.1** (Characterization of the absolute value). Let \\(A\\) be a \\(C^*\\)-algebra and \\(f\\in A^*\\). Then \\(|f|\\) is the only positive functional \\(\\omega\\) on \\(A\\) with \\(\\|\\omega\\|\\le\\|f\\|\\) and\n\\[\n\\begin{gathered}\n|f(x)|^2\\\\\n\\le\\|f\\|\\,\\omega(xx^*)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A=M\\) is a von Neumann algebra and \\(f\\in M_*\\), then the \\(|f|\\) of Theorem 2.2 is the only positive functional on \\(M\\), normal or not, with these two properties. In both cases such an \\(\\omega\\) has \\(\\|\\omega\\|=\\|f\\|\\).\n\n**Proof.** *The von Neumann case.* By Theorem 2.2, \\(|f|\\) has the two properties. Let \\(\\omega\\) be any positive functional on \\(M\\) with the two properties, and let \\(f=v|f|\\). If \\(f=0\\) then \\(\\omega=0\\). Otherwise, (3.1) at \\(x=v^*\\) gives \\(\\|f\\|^2=f(v^*)^2\\le\\|f\\|\\,\\omega(v^*v)\\le\\|f\\|\\,\\|\\omega\\|\\), so \\(\\|\\omega\\|=\\|f\\|\\). Since \\(|f|(x)=f(xv^*)\\),\n\\[\n\\begin{gathered}\n||f|(x)|^2\\\\\n=|f(xv^*)|^2\\\\\n\\le\\|f\\|\\,\\omega(xv^*vx^*)\\\\\n\\le\\|f\\|\\,\\omega(xx^*).\n\\end{gathered}\n\\]\nReplacing \\(x\\) by \\(x^*\\), \\(||f|(x)|^2\\le\\|\\omega\\|\\,\\omega(x^*x)\\). Lemma 2.1, applied in the \\(C^*\\)-algebra \\(M\\), gives \\(|f|=\\omega\\). No normality of \\(\\omega\\) was used.\n\n*The general case.* By Theorem 2.7, \\(|f|\\) has the two properties. Let \\(\\omega\\in A^*_+\\) have them, and let \\((\\pi,H,\\xi)\\) be its cyclic representation. By (3.1) with \\(x=y^*\\), \\(|f(y^*)|\\le\\|f\\|^{1/2}\\|\\pi(y)\\xi\\|\\). So \\(\\pi(y)\\xi\\mapsto\\overline{f(y^*)}\\) is a bounded linear functional on \\(\\pi(A)\\xi\\), and the Riesz theorem gives \\(\\eta\\in H\\) with \\(\\|\\eta\\|\\le\\|f\\|^{1/2}\\) and \\(f(y^*)=\\overline{\\langle\\pi(y)\\xi,\\eta\\rangle}=\\langle\\pi(y^*)\\eta,\\xi\\rangle\\). Thus \\(f(x)=\\langle\\pi(x)\\eta,\\xi\\rangle\\) for \\(x\\in A\\). Let \\(\\bar\\pi\\) be the normal extension of \\(\\pi\\) to \\(\\tilde A\\) (background fact 14). The normal functionals \\(X\\mapsto\\langle\\bar\\pi(X)\\eta,\\xi\\rangle\\) and \\(X\\mapsto\\langle\\bar\\pi(X)\\xi,\\xi\\rangle\\) agree with \\(f\\) and \\(\\omega\\) on the dense subalgebra \\(A\\), so they are the normal extensions. Hence, for \\(X\\in\\tilde A\\),\n\\[\n\\begin{gathered}\n|f(X)|^2\\\\\n=|\\langle\\eta,\\bar\\pi(X^*)\\xi\\rangle|^2\\\\\n\\le\\|f\\|\\,\\|\\bar\\pi(X^*)\\xi\\|^2\\\\\n=\\|f\\|\\,\\omega(XX^*) .\n\\end{gathered}\n\\]\nSo the normal extension of \\(\\omega\\) has the two properties in \\(\\tilde A\\), and the von Neumann case gives \\(\\omega=|f|\\). \\(\\square\\)\n\n**Corollary 3.2.** Let \\(M\\) be a von Neumann algebra and \\(\\varphi\\in M_*\\). The absolute value of \\(\\varphi\\) as a functional on the \\(C^*\\)-algebra \\(M\\) (Theorem 2.7, through \\(M^{**}\\)) is the absolute value of Theorem 2.2. In particular it is normal.\n\n**Proof.** Both are positive functionals on \\(M\\) of norm \\(\\|\\varphi\\|\\) that satisfy (3.1) on \\(M\\). By the general case of Theorem 3.1 they coincide. \\(\\square\\)\n\n**Example 3.3** (Where the partial isometry lives). Let \\(A=C[0,1]\\) and \\(\\lambda(x)=\\int_0^1x\\,dt\\).\n\n1. For \\(s\\ne t\\) in \\([0,1]\\), \\(|\\delta_s-\\delta_t|=\\delta_s+\\delta_t\\). Indeed, \\(\\omega=\\delta_s+\\delta_t\\) is positive with \\(\\|\\omega\\|=2=\\|\\delta_s-\\delta_t\\|\\) (test on a continuous \\(x\\) with \\(|x|\\le1\\), \\(x(s)=1\\), \\(x(t)=-1\\)), and \\[\n\\begin{gathered}\n|x(s)-x(t)|^2\\\\\n\\le2(|x(s)|^2+|x(t)|^2)\\\\\n=2\\,\\omega(xx^*).\n\\end{gathered}\n\\] Theorem 3.1 applies.\n2. Let \\(\\sigma=1_{[0,1/2)}-1_{[1/2,1]}\\) and \\(f(x)=\\int_0^1x\\sigma\\,dt\\). Then \\(\\|f\\|=1\\): the bound \\(\\le\\) is clear, and continuous functions \\(x_n\\) with \\(|x_n|\\le1\\) that equal \\(\\sigma\\) outside an interval of length \\(1/n\\) give \\(f(x_n)\\ge1-2/n\\). Also \\(\\|\\lambda\\|=1\\) and \\(|f(x)|^2\\le\\big(\\int|x|\\big)^2\\le\\int|x|^2=\\lambda(xx^*)\\). By Theorem 3.1, \\(|f|=\\lambda\\). Now suppose the partial isometry \\(v\\) of \\(f\\) lay in \\(A\\), say \\(v=a\\). Then \\(f(x)=\\lambda(xa)=\\int_0^1xa\\,dt\\) for all \\(x\\in A\\). Take \\(x\\ge0\\) continuous, supported in \\([\\tfrac12-\\delta,\\tfrac12]\\), with \\(\\int x=1\\). Then \\(1=f(x)=\\int xa\\), and as \\(\\delta\\to0\\) this tends to \\(a(\\tfrac12)\\), by continuity of \\(a\\). So \\(a(\\tfrac12)=1\\). The same test with support in \\([\\tfrac12,\\tfrac12+\\delta]\\) gives \\(a(\\tfrac12)=-1\\), a contradiction. So \\(v\\notin A\\).\n3. Even for positive functionals, \\(v=s(f)\\) need not lie in \\(A\\). Let \\(A=K(H)\\), whose bidual is \\(B(H)\\) (background fact 14), let \\((e_n)\\) be an infinite orthonormal sequence, and let \\(f=\\sum_n2^{-n}\\omega_{e_n}\\). Its normal extension to \\(B(H)\\) is given by the same formula. Its support is the projection \\(P\\) onto the closed span of the \\(e_n\\): \\(f(1-P)=0\\), and a projection \\(q\\le P\\) with \\(f(q)=0\\) has \\(qe_n=0\\) for all \\(n\\), so \\(q=0\\). So \\(v=P\\) has infinite rank and is not compact.\n\nThe next result replaces the triangle inequality, which fails for absolute values (Example 3.5).\n\n**Proposition 3.4** (A weak triangle inequality). Let \\(A\\) be a \\(C^*\\)-algebra and \\((f_k)\\) a finite or infinite sequence in \\(A^*\\) with \\(\\sum_k\\|f_k\\|<\\infty\\). Put \\(f=\\sum_kf_k\\). Then for every \\(x\\in\\tilde A\\),\n\\[\n\\begin{gathered}\n||f|(x)|^2\\\\\n\\le\\Big(\\sum_k\\|f_k\\|\\Big)\\Big(\\sum_k|f_k|(xx^*)\\Big).\n\\end{gathered}\n\\tag{3.2}\n\\]\n\n**Proof.** Let \\(f=w|f|\\) and \\(f_k=u_k|f_k|\\) be polar decompositions in \\(\\tilde A\\). The series \\(f=\\sum_kf_k\\) converges in norm, so \\(|f|=w^*f=\\sum_kw^*u_k|f_k|\\), that is \\(|f|(x)=\\sum_k|f_k|(xw^*u_k)\\). By the Cauchy–Schwarz inequality for \\(|f_k|\\),\n\\[\n\\begin{gathered}\n||f_k|(xw^*u_k)|^2\\\\\n\\le|f_k|(xx^*)\\,|f_k|(u_k^*ww^*u_k)\\\\\n\\le|f_k|(xx^*)\\,\\|f_k\\| .\n\\end{gathered}\n\\]\nSo \\(||f|(x)|\\le\\sum_k\\|f_k\\|^{1/2}|f_k|(xx^*)^{1/2}\\), and the Cauchy–Schwarz inequality for sequences gives (3.2). \\(\\square\\)\n\nFor commutative algebras the absolute value is the total variation of a measure (Example 2.6), and then \\(|f+g|\\le|f|+|g|\\). This fails in general.\n\n**Example 3.5** (No triangle inequality). In \\(M_2(\\mathbb C)\\) let \\(\\varphi(x)=\\operatorname{Tr}(E_{11}x)\\) and \\(\\psi(x)=\\operatorname{Tr}(E_{12}x)\\), with matrix units \\(E_{ij}\\). By Example 2.4, \\(|\\varphi|=\\operatorname{Tr}(E_{11}\\,\\cdot\\,)\\) and \\(|\\psi|=\\operatorname{Tr}(E_{22}\\,\\cdot\\,)\\), because \\((E_{12}^*E_{12})^{1/2}=E_{22}\\). For \\(\\rho=E_{11}+E_{12}\\) we have \\(\\rho^*\\rho=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\), so \\(|\\rho|=2^{-1/2}\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\) and \\(|\\varphi+\\psi|=\\operatorname{Tr}(|\\rho|\\,\\cdot\\,)\\). Let \\(P\\) be the projection onto \\(2^{-1/2}(1,1)\\). Then \\(|\\varphi+\\psi|(P)=\\sqrt2\\), while \\((|\\varphi|+|\\psi|)(P)=\\operatorname{Tr}(P)=1\\). So \\(|\\varphi+\\psi|\\le|\\varphi|+|\\psi|\\) fails. Inequality (3.2) holds here with equality at \\(x=P\\): \\(2\\le(1+1)\\cdot1\\).\n\n",
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      "id": "OA-FND-PD-05",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "3. How the absolute value is determined",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 3. How the absolute value is determined\n\nThe absolute value was defined through the partial isometry \\(v\\). It is also the only positive functional of the right size that dominates \\(\\varphi\\) in the sense of (2.3). This description does not mention \\(v\\), does not require normality, and works for every \\(C^*\\)-algebra.\n\n**Theorem 3.1** (Characterization of the absolute value). Let \\(A\\) be a \\(C^*\\)-algebra and \\(f\\in A^*\\). Then \\(|f|\\) is the only positive functional \\(\\omega\\) on \\(A\\) with \\(\\|\\omega\\|\\le\\|f\\|\\) and\n\\[\n\\begin{gathered}\n|f(x)|^2\\\\\n\\le\\|f\\|\\,\\omega(xx^*)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A=M\\) is a von Neumann algebra and \\(f\\in M_*\\), then the \\(|f|\\) of Theorem 2.2 is the only positive functional on \\(M\\), normal or not, with these two properties. In both cases such an \\(\\omega\\) has \\(\\|\\omega\\|=\\|f\\|\\).\n\n**Proof.** *The von Neumann case.* By Theorem 2.2, \\(|f|\\) has the two properties. Let \\(\\omega\\) be any positive functional on \\(M\\) with the two properties, and let \\(f=v|f|\\). If \\(f=0\\) then \\(\\omega=0\\). Otherwise, (3.1) at \\(x=v^*\\) gives \\(\\|f\\|^2=f(v^*)^2\\le\\|f\\|\\,\\omega(v^*v)\\le\\|f\\|\\,\\|\\omega\\|\\), so \\(\\|\\omega\\|=\\|f\\|\\). Since \\(|f|(x)=f(xv^*)\\),\n\\[\n\\begin{gathered}\n||f|(x)|^2\\\\\n=|f(xv^*)|^2\\\\\n\\le\\|f\\|\\,\\omega(xv^*vx^*)\\\\\n\\le\\|f\\|\\,\\omega(xx^*).\n\\end{gathered}\n\\]\nReplacing \\(x\\) by \\(x^*\\), \\(||f|(x)|^2\\le\\|\\omega\\|\\,\\omega(x^*x)\\). Lemma 2.1, applied in the \\(C^*\\)-algebra \\(M\\), gives \\(|f|=\\omega\\). No normality of \\(\\omega\\) was used.\n\n*The general case.* By Theorem 2.7, \\(|f|\\) has the two properties. Let \\(\\omega\\in A^*_+\\) have them, and let \\((\\pi,H,\\xi)\\) be its cyclic representation. By (3.1) with \\(x=y^*\\), \\(|f(y^*)|\\le\\|f\\|^{1/2}\\|\\pi(y)\\xi\\|\\). So \\(\\pi(y)\\xi\\mapsto\\overline{f(y^*)}\\) is a bounded linear functional on \\(\\pi(A)\\xi\\), and the Riesz theorem gives \\(\\eta\\in H\\) with \\(\\|\\eta\\|\\le\\|f\\|^{1/2}\\) and \\(f(y^*)=\\overline{\\langle\\pi(y)\\xi,\\eta\\rangle}=\\langle\\pi(y^*)\\eta,\\xi\\rangle\\). Thus \\(f(x)=\\langle\\pi(x)\\eta,\\xi\\rangle\\) for \\(x\\in A\\). Let \\(\\bar\\pi\\) be the normal extension of \\(\\pi\\) to \\(\\tilde A\\) (background fact 14). The normal functionals \\(X\\mapsto\\langle\\bar\\pi(X)\\eta,\\xi\\rangle\\) and \\(X\\mapsto\\langle\\bar\\pi(X)\\xi,\\xi\\rangle\\) agree with \\(f\\) and \\(\\omega\\) on the dense subalgebra \\(A\\), so they are the normal extensions. Hence, for \\(X\\in\\tilde A\\),\n\\[\n\\begin{gathered}\n|f(X)|^2\\\\\n=|\\langle\\eta,\\bar\\pi(X^*)\\xi\\rangle|^2\\\\\n\\le\\|f\\|\\,\\|\\bar\\pi(X^*)\\xi\\|^2\\\\\n=\\|f\\|\\,\\omega(XX^*) .\n\\end{gathered}\n\\]\nSo the normal extension of \\(\\omega\\) has the two properties in \\(\\tilde A\\), and the von Neumann case gives \\(\\omega=|f|\\). \\(\\square\\)\n\n**Corollary 3.2.** Let \\(M\\) be a von Neumann algebra and \\(\\varphi\\in M_*\\). The absolute value of \\(\\varphi\\) as a functional on the \\(C^*\\)-algebra \\(M\\) (Theorem 2.7, through \\(M^{**}\\)) is the absolute value of Theorem 2.2. In particular it is normal.\n\n**Proof.** Both are positive functionals on \\(M\\) of norm \\(\\|\\varphi\\|\\) that satisfy (3.1) on \\(M\\). By the general case of Theorem 3.1 they coincide. \\(\\square\\)\n\n**Example 3.3** (Where the partial isometry lives). Let \\(A=C[0,1]\\) and \\(\\lambda(x)=\\int_0^1x\\,dt\\).\n\n1. For \\(s\\ne t\\) in \\([0,1]\\), \\(|\\delta_s-\\delta_t|=\\delta_s+\\delta_t\\). Indeed, \\(\\omega=\\delta_s+\\delta_t\\) is positive with \\(\\|\\omega\\|=2=\\|\\delta_s-\\delta_t\\|\\) (test on a continuous \\(x\\) with \\(|x|\\le1\\), \\(x(s)=1\\), \\(x(t)=-1\\)), and \\[\n\\begin{gathered}\n|x(s)-x(t)|^2\\\\\n\\le2(|x(s)|^2+|x(t)|^2)\\\\\n=2\\,\\omega(xx^*).\n\\end{gathered}\n\\] Theorem 3.1 applies.\n2. Let \\(\\sigma=1_{[0,1/2)}-1_{[1/2,1]}\\) and \\(f(x)=\\int_0^1x\\sigma\\,dt\\). Then \\(\\|f\\|=1\\): the bound \\(\\le\\) is clear, and continuous functions \\(x_n\\) with \\(|x_n|\\le1\\) that equal \\(\\sigma\\) outside an interval of length \\(1/n\\) give \\(f(x_n)\\ge1-2/n\\). Also \\(\\|\\lambda\\|=1\\) and \\(|f(x)|^2\\le\\big(\\int|x|\\big)^2\\le\\int|x|^2=\\lambda(xx^*)\\). By Theorem 3.1, \\(|f|=\\lambda\\). Now suppose the partial isometry \\(v\\) of \\(f\\) lay in \\(A\\), say \\(v=a\\). Then \\(f(x)=\\lambda(xa)=\\int_0^1xa\\,dt\\) for all \\(x\\in A\\). Take \\(x\\ge0\\) continuous, supported in \\([\\tfrac12-\\delta,\\tfrac12]\\), with \\(\\int x=1\\). Then \\(1=f(x)=\\int xa\\), and as \\(\\delta\\to0\\) this tends to \\(a(\\tfrac12)\\), by continuity of \\(a\\). So \\(a(\\tfrac12)=1\\). The same test with support in \\([\\tfrac12,\\tfrac12+\\delta]\\) gives \\(a(\\tfrac12)=-1\\), a contradiction. So \\(v\\notin A\\).\n3. Even for positive functionals, \\(v=s(f)\\) need not lie in \\(A\\). Let \\(A=K(H)\\), whose bidual is \\(B(H)\\) (background fact 14), let \\((e_n)\\) be an infinite orthonormal sequence, and let \\(f=\\sum_n2^{-n}\\omega_{e_n}\\). Its normal extension to \\(B(H)\\) is given by the same formula. Its support is the projection \\(P\\) onto the closed span of the \\(e_n\\): \\(f(1-P)=0\\), and a projection \\(q\\le P\\) with \\(f(q)=0\\) has \\(qe_n=0\\) for all \\(n\\), so \\(q=0\\). So \\(v=P\\) has infinite rank and is not compact.\n\nThe next result replaces the triangle inequality, which fails for absolute values (Example 3.5).\n\n**Proposition 3.4** (A weak triangle inequality). Let \\(A\\) be a \\(C^*\\)-algebra and \\((f_k)\\) a finite or infinite sequence in \\(A^*\\) with \\(\\sum_k\\|f_k\\|<\\infty\\). Put \\(f=\\sum_kf_k\\). Then for every \\(x\\in\\tilde A\\),\n\\[\n\\begin{gathered}\n||f|(x)|^2\\\\\n\\le\\Big(\\sum_k\\|f_k\\|\\Big)\\Big(\\sum_k|f_k|(xx^*)\\Big).\n\\end{gathered}\n\\tag{3.2}\n\\]\n\n**Proof.** Let \\(f=w|f|\\) and \\(f_k=u_k|f_k|\\) be polar decompositions in \\(\\tilde A\\). The series \\(f=\\sum_kf_k\\) converges in norm, so \\(|f|=w^*f=\\sum_kw^*u_k|f_k|\\), that is \\(|f|(x)=\\sum_k|f_k|(xw^*u_k)\\). By the Cauchy–Schwarz inequality for \\(|f_k|\\),\n\\[\n\\begin{gathered}\n||f_k|(xw^*u_k)|^2\\\\\n\\le|f_k|(xx^*)\\,|f_k|(u_k^*ww^*u_k)\\\\\n\\le|f_k|(xx^*)\\,\\|f_k\\| .\n\\end{gathered}\n\\]\nSo \\(||f|(x)|\\le\\sum_k\\|f_k\\|^{1/2}|f_k|(xx^*)^{1/2}\\), and the Cauchy–Schwarz inequality for sequences gives (3.2). \\(\\square\\)\n\nFor commutative algebras the absolute value is the total variation of a measure (Example 2.6), and then \\(|f+g|\\le|f|+|g|\\). This fails in general.\n\n**Example 3.5** (No triangle inequality). In \\(M_2(\\mathbb C)\\) let \\(\\varphi(x)=\\operatorname{Tr}(E_{11}x)\\) and \\(\\psi(x)=\\operatorname{Tr}(E_{12}x)\\), with matrix units \\(E_{ij}\\). By Example 2.4, \\(|\\varphi|=\\operatorname{Tr}(E_{11}\\,\\cdot\\,)\\) and \\(|\\psi|=\\operatorname{Tr}(E_{22}\\,\\cdot\\,)\\), because \\((E_{12}^*E_{12})^{1/2}=E_{22}\\). For \\(\\rho=E_{11}+E_{12}\\) we have \\(\\rho^*\\rho=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\), so \\(|\\rho|=2^{-1/2}\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\) and \\(|\\varphi+\\psi|=\\operatorname{Tr}(|\\rho|\\,\\cdot\\,)\\). Let \\(P\\) be the projection onto \\(2^{-1/2}(1,1)\\). Then \\(|\\varphi+\\psi|(P)=\\sqrt2\\), while \\((|\\varphi|+|\\psi|)(P)=\\operatorname{Tr}(P)=1\\). So \\(|\\varphi+\\psi|\\le|\\varphi|+|\\psi|\\) fails. Inequality (3.2) holds here with equality at \\(x=P\\): \\(2\\le(1+1)\\cdot1\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-06",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "4. Domination by a positive functional",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 4. Domination by a positive functional\n\nFor a positive functional \\(\\varphi\\) and an element \\(a\\), the functional \\(a\\varphi\\) need not be positive. But when \\(a\\varphi\\) is hermitian it is squeezed between \\(\\pm r(a)\\varphi\\). The proof squares \\(a\\) repeatedly, which is why the spectral radius, not the norm, appears.\n\n**Proposition 4.1.** Let \\(B\\) be a \\(C^*\\)-algebra, \\(\\varphi\\in B^*_+\\) and \\(a\\in B\\), and suppose that \\(a\\varphi\\) is hermitian, that is \\(\\varphi(ya)=\\varphi(a^*y)\\) for all \\(y\\in B\\). Then\n\\[\n\\begin{gathered}\n|\\varphi(ha)|\\\\\n\\le r(a)\\,\\varphi(h)\\\\\n(h\\in B_+),\n\\end{gathered}\n\\tag{4.1}\n\\]\nthat is, \\(-r(a)\\varphi\\le a\\varphi\\le r(a)\\varphi\\).\n\n**Proof.** *Step 1.* By induction on \\(m\\), \\(\\varphi(ya^m)=\\varphi((a^*)^my)\\) for all \\(y\\): \\[\n\\begin{gathered}\n\\varphi(ya^{m+1})\\\\\n=\\varphi((ya^m)a)\\\\\n=\\varphi(a^*ya^m)\\\\\n=\\varphi((a^*)^{m}a^*y).\n\\end{gathered}\n\\] Hence for \\(m\\ge1\\) and \\(c=a^{m}\\), the functional \\(c\\varphi\\) is hermitian, and \\(\\varphi(yc^2)=\\varphi((ya^m)a^m)=\\varphi((a^m)^*ya^m)\\), so \\(c^2\\varphi\\) is positive.\n\n*Step 2.* Let \\(h\\in B_+\\) and let \\(c\\) be an element with \\(c\\varphi\\) hermitian. By the Cauchy–Schwarz inequality and \\(\\varphi(c^*(hc))=\\varphi((hc)c)\\),\n\\[\n\\begin{gathered}\n|\\varphi(hc)|^2\\\\\n=|\\varphi(h^{1/2}\\cdot h^{1/2}c)|^2\\\\\n\\le\\varphi(h)\\,\\varphi(c^*hc)\\\\\n=\\varphi(h)\\,\\varphi(hc^2),\n\\end{gathered}\n\\]\nand \\(\\varphi(hc^2)=\\varphi(c^*hc)\\ge0\\).\n\n*Step 3.* Apply Step 2 with \\(c=a,a^2,a^4,\\dots\\). By induction,\n\\[\n\\begin{gathered}\n|\\varphi(ha)|\\\\\n\\le\\varphi(h)^{1-2^{-n}}\\varphi(ha^{2^n})^{2^{-n}}\\\\\n\\le\\varphi(h)^{1-2^{-n}}\\big(\\|\\varphi\\|\\|h\\|\\big)^{2^{-n}}\\|a^{2^n}\\|^{2^{-n}} .\n\\end{gathered}\n\\]\nIf \\(\\varphi(h)=0\\), Step 2 already gives \\(\\varphi(ha)=0\\). Otherwise let \\(n\\to\\infty\\): the first factor tends to \\(\\varphi(h)\\), the second to \\(1\\), and the third to \\(r(a)\\) by the spectral radius formula. \\(\\square\\)\n\n**Corollary 4.2.** Let \\(M\\) be a von Neumann algebra, \\(\\varphi\\in M_*^+\\) and \\(a\\in M\\). Then\n\\[\n|a\\varphi|\\le\\|as(\\varphi)\\|\\,\\varphi\\le\\|a\\|\\,\\varphi .\n\\tag{4.2}\n\\]\nThe same holds for \\(\\varphi\\in A^*_+\\) and \\(a\\in\\tilde A\\), for any \\(C^*\\)-algebra \\(A\\).\n\n**Proof.** Put \\(s=s(\\varphi)\\). Since \\(\\varphi=s\\varphi\\), \\(a\\varphi=(as)\\varphi\\). Let \\(a\\varphi=u|a\\varphi|\\). Then \\(|a\\varphi|=u^*a\\varphi=(u^*as)\\varphi\\) is positive, hence hermitian, so Proposition 4.1 applies to \\(c=u^*as\\): for \\(h\\ge0\\), \\[\n\\begin{gathered}\n|a\\varphi|(h)\\\\\n=\\varphi(hu^*as)\\\\\n\\le r(u^*as)\\varphi(h)\\\\\n\\le\\|as\\|\\varphi(h).\n\\end{gathered}\n\\] For a \\(C^*\\)-algebra, work in \\(\\tilde A\\). \\(\\square\\)\n\n**Example 4.3.** Let \\(M=B(H)\\), \\(\\xi\\) a unit vector and \\(\\varphi=\\omega_\\xi\\).\n\n1. Left multiplication: \\(a\\omega_\\xi=\\omega_{a\\xi,\\xi}\\), so \\(|a\\omega_\\xi|=\\|a\\xi\\|\\,\\omega_\\xi\\) by (2.5). Since \\(s(\\omega_\\xi)=\\theta_{\\xi,\\xi}\\), \\(\\|as(\\omega_\\xi)\\|=\\|a\\xi\\|\\): the first inequality in (4.2) is an equality.\n2. Right multiplication has no such bound. \\(\\omega_\\xi a=\\omega_{\\xi,a^*\\xi}\\), so \\(|\\omega_\\xi a|=\\|a^*\\xi\\|^{-1}\\omega_{a^*\\xi}\\) when \\(a^*\\xi\\ne0\\). If \\(a^*\\xi\\notin\\mathbb C\\xi\\), the support of \\(|\\omega_\\xi a|\\) is not under \\(s(\\omega_\\xi)\\), so \\(|\\omega_\\xi a|\\le C\\omega_\\xi\\) fails for every \\(C\\).\n3. The spectral radius matters. In \\(M_2(\\mathbb C)\\) let \\(a=E_{12}\\), which has \\(r(a)=0\\), and \\(\\varphi=\\operatorname{Tr}(\\rho\\,\\cdot\\,)\\) with \\(\\rho=\\begin{pmatrix}p&q\\\\\\bar q&t\\end{pmatrix}\\ge0\\). Since \\((a\\varphi)(x)=\\operatorname{Tr}(a\\rho x)\\), \\(a\\varphi\\) is hermitian exactly when \\(a\\rho=\\rho a^*\\), that is \\(\\begin{pmatrix}\\bar q&t\\\\0&0\\end{pmatrix}=\\begin{pmatrix}q&0\\\\t&0\\end{pmatrix}\\). This forces \\(t=0\\), hence \\(q=0\\) by positivity, and then \\(a\\rho=0\\). So \\(a\\varphi=0\\), as (4.1) predicts.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-07",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "5. Continuity of the absolute value",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 5. Continuity of the absolute value\n\n**Theorem 5.1.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(\\varphi,\\psi\\in A^*\\) (for instance \\(A=M\\) and \\(\\varphi,\\psi\\in M_*\\)). Then\n\\[\n\\begin{gathered}\n\\||\\varphi|-|\\psi|\\|\\\\\n\\le\\|\\varphi-\\psi\\|+2\\big(\\|\\varphi\\|\\,\\|\\varphi-\\psi\\|\\big)^{1/2}.\n\\end{gathered}\n\\tag{5.1}\n\\]\nThe same bound holds for \\(\\||\\varphi^*|-|\\psi^*|\\|\\). So \\(\\varphi\\mapsto|\\varphi|\\) is norm continuous, uniformly on bounded sets.\n\n**Proof.** Work in \\(\\tilde A\\) (or in \\(M\\)), with \\(\\varphi=v|\\varphi|\\) and \\(\\psi=w|\\psi|\\). Since \\(|\\psi|=w^*\\psi\\), \\(|\\psi|(x)=\\psi(xw^*)\\). Also \\(\\varphi(xw^*)=|\\varphi|(xw^*v)\\). So\n\\[\n\\begin{gathered}\n|\\varphi|(x)-|\\psi|(x)\\\\\n=|\\varphi|\\big(x(1-w^*v)\\big)+(\\varphi-\\psi)(xw^*).\n\\end{gathered}\n\\]\nThe second term has modulus at most \\(\\|\\varphi-\\psi\\|\\|x\\|\\). For the first, the Cauchy–Schwarz inequality gives\n\\[\n\\begin{gathered}\n\\big||\\varphi|(x(1-w^*v))\\big|^2\\\\\n\\le|\\varphi|(xx^*)\\;|\\varphi|\\big((1-v^*w)(1-w^*v)\\big).\n\\end{gathered}\n\\]\nHere \\(|\\varphi|(xx^*)\\le\\|\\varphi\\|\\|x\\|^2\\). Expanding, and using \\(|\\varphi|(v^*ww^*v)\\le|\\varphi|(v^*v)=\\|\\varphi\\|\\) and \\(|\\varphi|(w^*v)=\\varphi(w^*)\\),\n\\[\n\\begin{gathered}\n|\\varphi|\\big((1-v^*w)(1-w^*v)\\big)\\\\\n\\le2\\|\\varphi\\|-2\\operatorname{Re}\\varphi(w^*).\n\\end{gathered}\n\\]\nSince \\(\\psi(w^*)=|\\psi|(w^*w)=\\|\\psi\\|\\), \\(\\operatorname{Re}\\varphi(w^*)\\ge\\|\\psi\\|-\\|\\varphi-\\psi\\|\\). So the last quantity is at most \\(2(\\|\\varphi\\|-\\|\\psi\\|+\\|\\varphi-\\psi\\|)\\le4\\|\\varphi-\\psi\\|\\). Altogether \\[\n\\begin{gathered}\n||\\varphi|(x)-|\\psi|(x)|\\\\\n\\le\\big(\\|\\varphi-\\psi\\|+2(\\|\\varphi\\|\\|\\varphi-\\psi\\|)^{1/2}\\big)\\|x\\|,\n\\end{gathered}\n\\] which is (5.1). Apply this to \\(\\varphi^*,\\psi^*\\) for the second claim. \\(\\square\\)\n\n**Remark 5.2.** By symmetry \\(\\|\\varphi\\|\\) can be replaced by \\(\\min(\\|\\varphi\\|,\\|\\psi\\|)\\) in (5.1). In the commutative case of Example 2.6, the pointwise inequality \\(||h|-|k||\\le|h-k|\\) gives the Lipschitz bound \\(\\||\\varphi_h|-|\\varphi_k|\\|\\le\\|\\varphi_h-\\varphi_k\\|\\).\n\nWeak\\(^*\\) convergence alone does not move the absolute values along (Example 5.4), but convergence of the norms is enough.\n\n**Theorem 5.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\((f_i)\\) a net in \\(A^*\\) with \\(f_i\\to f\\) in \\(\\sigma(A^*,A)\\) and \\(\\|f_i\\|\\to\\|f\\|\\). Then \\(|f_i|\\to|f|\\) in \\(\\sigma(A^*,A)\\). Likewise, if \\((\\varphi_i)\\) is a net in the predual of a von Neumann algebra \\(M\\) with \\(\\varphi_i\\to\\varphi\\) weakly and \\(\\|\\varphi_i\\|\\to\\|\\varphi\\|\\), then \\(|\\varphi_i|\\to|\\varphi|\\) weakly.\n\n**Proof.** Eventually \\(\\||f_i|\\|=\\|f_i\\|\\le\\|f\\|+1\\), so the tail of the net \\((|f_i|)\\) lies in a weak\\(^*\\) compact ball. It is therefore enough to show that every weak\\(^*\\) cluster point \\(\\omega\\) of \\((|f_i|)\\) equals \\(|f|\\). Let a subnet \\(|f_j|\\) converge to \\(\\omega\\). Then \\(\\omega\\) is positive, and \\(\\|\\omega\\|\\le\\liminf_j\\||f_j|\\|=\\|f\\|\\), because the norm is weak\\(^*\\) lower semicontinuous. By Theorem 2.7, \\(|f_j(x)|^2\\le\\|f_j\\|\\,|f_j|(xx^*)\\) for \\(x\\in A\\), and in the limit \\(|f(x)|^2\\le\\|f\\|\\,\\omega(xx^*)\\). Theorem 3.1 gives \\(\\omega=|f|\\).\n\nFor the predual, apply the same argument in \\(M^*\\) with the topology \\(\\sigma(M^*,M)\\), whose restriction to \\(M_*\\) is the weak topology. A cluster point \\(\\omega\\in M^*\\) of \\((|\\varphi_i|)\\) is positive, has \\(\\|\\omega\\|\\le\\|\\varphi\\|\\) and satisfies (3.1) on \\(M\\), so \\(\\omega=|\\varphi|\\) by the von Neumann case of Theorem 3.1, which allows \\(\\omega\\) to be singular. \\(\\square\\)\n\n**Example 5.4.** On \\(A=C[0,1]\\), \\(f_n=\\delta_{1/n}-\\delta_0\\) tends to \\(0\\) weak\\(^*\\). By Example 3.3(1), \\(|f_n|=\\delta_{1/n}+\\delta_0\\), which tends to \\(2\\delta_0\\ne|0|\\). Here \\(\\|f_n\\|=2\\) does not tend to \\(\\|0\\|\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-08",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "5. Continuity of the absolute value",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
      "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "anchor": "oa-fnd-pd-08",
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        "line": 515,
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      "full_conditions_and_proof": "## 5. Continuity of the absolute value\n\n**Theorem 5.1.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(\\varphi,\\psi\\in A^*\\) (for instance \\(A=M\\) and \\(\\varphi,\\psi\\in M_*\\)). Then\n\\[\n\\begin{gathered}\n\\||\\varphi|-|\\psi|\\|\\\\\n\\le\\|\\varphi-\\psi\\|+2\\big(\\|\\varphi\\|\\,\\|\\varphi-\\psi\\|\\big)^{1/2}.\n\\end{gathered}\n\\tag{5.1}\n\\]\nThe same bound holds for \\(\\||\\varphi^*|-|\\psi^*|\\|\\). So \\(\\varphi\\mapsto|\\varphi|\\) is norm continuous, uniformly on bounded sets.\n\n**Proof.** Work in \\(\\tilde A\\) (or in \\(M\\)), with \\(\\varphi=v|\\varphi|\\) and \\(\\psi=w|\\psi|\\). Since \\(|\\psi|=w^*\\psi\\), \\(|\\psi|(x)=\\psi(xw^*)\\). Also \\(\\varphi(xw^*)=|\\varphi|(xw^*v)\\). So\n\\[\n\\begin{gathered}\n|\\varphi|(x)-|\\psi|(x)\\\\\n=|\\varphi|\\big(x(1-w^*v)\\big)+(\\varphi-\\psi)(xw^*).\n\\end{gathered}\n\\]\nThe second term has modulus at most \\(\\|\\varphi-\\psi\\|\\|x\\|\\). For the first, the Cauchy–Schwarz inequality gives\n\\[\n\\begin{gathered}\n\\big||\\varphi|(x(1-w^*v))\\big|^2\\\\\n\\le|\\varphi|(xx^*)\\;|\\varphi|\\big((1-v^*w)(1-w^*v)\\big).\n\\end{gathered}\n\\]\nHere \\(|\\varphi|(xx^*)\\le\\|\\varphi\\|\\|x\\|^2\\). Expanding, and using \\(|\\varphi|(v^*ww^*v)\\le|\\varphi|(v^*v)=\\|\\varphi\\|\\) and \\(|\\varphi|(w^*v)=\\varphi(w^*)\\),\n\\[\n\\begin{gathered}\n|\\varphi|\\big((1-v^*w)(1-w^*v)\\big)\\\\\n\\le2\\|\\varphi\\|-2\\operatorname{Re}\\varphi(w^*).\n\\end{gathered}\n\\]\nSince \\(\\psi(w^*)=|\\psi|(w^*w)=\\|\\psi\\|\\), \\(\\operatorname{Re}\\varphi(w^*)\\ge\\|\\psi\\|-\\|\\varphi-\\psi\\|\\). So the last quantity is at most \\(2(\\|\\varphi\\|-\\|\\psi\\|+\\|\\varphi-\\psi\\|)\\le4\\|\\varphi-\\psi\\|\\). Altogether \\[\n\\begin{gathered}\n||\\varphi|(x)-|\\psi|(x)|\\\\\n\\le\\big(\\|\\varphi-\\psi\\|+2(\\|\\varphi\\|\\|\\varphi-\\psi\\|)^{1/2}\\big)\\|x\\|,\n\\end{gathered}\n\\] which is (5.1). Apply this to \\(\\varphi^*,\\psi^*\\) for the second claim. \\(\\square\\)\n\n**Remark 5.2.** By symmetry \\(\\|\\varphi\\|\\) can be replaced by \\(\\min(\\|\\varphi\\|,\\|\\psi\\|)\\) in (5.1). In the commutative case of Example 2.6, the pointwise inequality \\(||h|-|k||\\le|h-k|\\) gives the Lipschitz bound \\(\\||\\varphi_h|-|\\varphi_k|\\|\\le\\|\\varphi_h-\\varphi_k\\|\\).\n\nWeak\\(^*\\) convergence alone does not move the absolute values along (Example 5.4), but convergence of the norms is enough.\n\n**Theorem 5.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\((f_i)\\) a net in \\(A^*\\) with \\(f_i\\to f\\) in \\(\\sigma(A^*,A)\\) and \\(\\|f_i\\|\\to\\|f\\|\\). Then \\(|f_i|\\to|f|\\) in \\(\\sigma(A^*,A)\\). Likewise, if \\((\\varphi_i)\\) is a net in the predual of a von Neumann algebra \\(M\\) with \\(\\varphi_i\\to\\varphi\\) weakly and \\(\\|\\varphi_i\\|\\to\\|\\varphi\\|\\), then \\(|\\varphi_i|\\to|\\varphi|\\) weakly.\n\n**Proof.** Eventually \\(\\||f_i|\\|=\\|f_i\\|\\le\\|f\\|+1\\), so the tail of the net \\((|f_i|)\\) lies in a weak\\(^*\\) compact ball. It is therefore enough to show that every weak\\(^*\\) cluster point \\(\\omega\\) of \\((|f_i|)\\) equals \\(|f|\\). Let a subnet \\(|f_j|\\) converge to \\(\\omega\\). Then \\(\\omega\\) is positive, and \\(\\|\\omega\\|\\le\\liminf_j\\||f_j|\\|=\\|f\\|\\), because the norm is weak\\(^*\\) lower semicontinuous. By Theorem 2.7, \\(|f_j(x)|^2\\le\\|f_j\\|\\,|f_j|(xx^*)\\) for \\(x\\in A\\), and in the limit \\(|f(x)|^2\\le\\|f\\|\\,\\omega(xx^*)\\). Theorem 3.1 gives \\(\\omega=|f|\\).\n\nFor the predual, apply the same argument in \\(M^*\\) with the topology \\(\\sigma(M^*,M)\\), whose restriction to \\(M_*\\) is the weak topology. A cluster point \\(\\omega\\in M^*\\) of \\((|\\varphi_i|)\\) is positive, has \\(\\|\\omega\\|\\le\\|\\varphi\\|\\) and satisfies (3.1) on \\(M\\), so \\(\\omega=|\\varphi|\\) by the von Neumann case of Theorem 3.1, which allows \\(\\omega\\) to be singular. \\(\\square\\)\n\n**Example 5.4.** On \\(A=C[0,1]\\), \\(f_n=\\delta_{1/n}-\\delta_0\\) tends to \\(0\\) weak\\(^*\\). By Example 3.3(1), \\(|f_n|=\\delta_{1/n}+\\delta_0\\), which tends to \\(2\\delta_0\\ne|0|\\). Here \\(\\|f_n\\|=2\\) does not tend to \\(\\|0\\|\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "the-double-commutant-theorem",
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    {
      "id": "OA-FND-PD-09",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "6. Invariant subspaces, hereditary cones and one-sided ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
      "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "anchor": "oa-fnd-pd-09",
      "proof_locus": {
        "line": 567,
        "through_line": 672
      },
      "full_conditions_and_proof": "## 6. Invariant subspaces, hereditary cones and one-sided ideals\n\nA subset \\(C\\subseteq M_*^+\\) is a *hereditary cone* if it is a convex cone (closed under sums and multiplication by nonnegative scalars) and \\(0\\le\\psi\\le\\omega\\in C\\) implies \\(\\psi\\in C\\). The positive part of a closed invariant subspace is such a cone, and the cone determines the subspace.\n\n**Theorem 6.1.** Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(MV\\subseteq V\\), and \\(V=M_*e\\) as in background fact 20. Then \\[\n\\begin{gathered}\nV_+\\\\\n=V\\cap M_*^+\\\\\n=\\{\\omega\\in M_*^+:\\omega(1-e)=0\\}\n\\end{gathered}\n\\] is a norm-closed hereditary cone, and \\(V=MV_+=\\{a\\omega:a\\in M,\\ \\omega\\in V_+\\}\\).\n2. Let \\(C\\subseteq M_*^+\\) be a norm-closed hereditary cone. Then \\(V=MC\\) is a norm-closed subspace with \\(MV\\subseteq V\\) and \\(V\\cap M_*^+=C\\).\n3. Hence \\(V\\mapsto V\\cap M_*^+\\) is a bijection from the norm-closed left invariant subspaces of \\(M_*\\) onto the norm-closed hereditary cones in \\(M_*^+\\), with inverse \\(C\\mapsto MC\\). Both correspond to projections \\(e\\): \\(V=M_*e\\) and \\(C=\\{\\omega\\ge0:\\omega(1-e)=0\\}\\).\n\nThe right-handed statements (subspaces with \\(VM\\subseteq V\\), and \\(CM\\)) follow by applying \\(\\varphi\\mapsto\\varphi^*\\).\n\n**Proof.** (1) For \\(\\omega\\ge0\\): if \\(\\omega=\\omega e\\), then \\(\\omega(1-e)=\\omega(e(1-e))=0\\); conversely, if \\(\\omega(1-e)=0\\) then \\(s(\\omega)\\le e\\) and \\(\\omega(ex)=\\omega(s(\\omega)ex)=\\omega(s(\\omega)x)=\\omega(x)\\). This gives the formula for \\(V_+\\), which is visibly a norm-closed hereditary cone. If \\(\\varphi\\in V\\) has polar decomposition \\(\\varphi=v|\\varphi|\\), then \\(|\\varphi|=v^*\\varphi\\in V\\), so \\(|\\varphi|\\in V_+\\) and \\(\\varphi\\in MV_+\\). The inclusion \\(MV_+\\subseteq V\\) is invariance.\n\n(2) *Step 1: \\(MC\\cap M_*^+=C\\).* Clearly \\(C\\subseteq MC\\). If \\(\\varphi=a\\omega\\ge0\\) with \\(\\omega\\in C\\), then \\(\\varphi=|a\\omega|\\le\\|a\\|\\omega\\) by Corollary 4.2, so \\(\\varphi\\in C\\).\n\n*Step 2: \\(MC\\) is norm closed.* Let \\(a_n\\omega_n\\to\\varphi\\) in norm, with \\(\\omega_n\\in C\\). If \\(a_n\\omega_n=u_n|a_n\\omega_n|\\), then \\(|a_n\\omega_n|=(u_n^*a_n)\\omega_n\\in MC\\cap M_*^+=C\\). By Theorem 5.1, \\(|a_n\\omega_n|\\to|\\varphi|\\), so \\(|\\varphi|\\in C\\) and \\(\\varphi=u|\\varphi|\\in MC\\).\n\n*Step 3: for \\(\\chi\\in M_*^+\\), the norm closure of \\(M\\chi\\) is \\(M_*s(\\chi)=\\{\\psi:\\psi=\\psi s(\\chi)\\}\\).* The set \\(M_*s(\\chi)\\) is norm closed and contains \\(M\\chi\\), since \\((a\\chi)(s(\\chi)x)=\\chi(s(\\chi)xa)=\\chi(xa)\\). If some \\(\\psi\\in M_*s(\\chi)\\) were not in the closure of the subspace \\(M\\chi\\), the Hahn–Banach theorem would give \\(y\\in M=(M_*)^*\\) with \\(\\chi(ya)=0\\) for all \\(a\\) and \\(\\psi(y)\\ne0\\). With \\(a=y^*\\), \\(\\chi(yy^*)=0\\), so \\(y^*s(\\chi)=0\\) (background fact 18) and \\(s(\\chi)y=0\\). Then \\(\\psi(y)=\\psi(s(\\chi)y)=0\\), a contradiction.\n\n*Step 4: \\(MC\\) is a subspace.* It is closed under scalar multiples. Let \\(a\\omega,b\\psi\\in MC\\) and \\(\\chi=\\omega+\\psi\\in C\\). Since \\(\\omega\\le\\chi\\), \\(\\omega(1-s(\\chi))=0\\) and \\(s(\\omega)\\le s(\\chi)\\); so \\(a\\omega\\in M_*s(\\omega)\\subseteq M_*s(\\chi)\\), and likewise \\(b\\psi\\in M_*s(\\chi)\\). By Step 3, \\(a\\omega+b\\psi\\) lies in the closure of \\(M\\chi\\subseteq MC\\), which is contained in \\(MC\\) by Step 2.\n\nInvariance \\(M(MC)\\subseteq MC\\) is clear.\n\n(3) By (1), \\(V=M(V\\cap M_*^+)\\); by (2), \\((MC)\\cap M_*^+=C\\). \\(\\square\\)\n\n**Example 6.2** (Convexity is needed). In \\(M=\\mathbb C^2\\), \\(M_*^+\\) is the quadrant \\([0,\\infty)^2\\). The union of the two axes is a closed cone, and it is hereditary, but it is not convex. It is not the positive part of any subspace, because such a positive part is convex.\n\nFor a \\(C^*\\)-algebra \\(A\\), a subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), where \\((af)(x)=f(xa)\\).\n\n**Corollary 6.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(V\\subseteq A^*\\) a norm-closed left invariant subspace, and put \\(V_+=V\\cap A^*_+\\). Then \\(V\\) is invariant under \\(\\tilde A\\), \\(V_+\\) is a norm-closed hereditary cone, \\(V=\\tilde AV_+\\), and \\(V\\) is the norm closure of the set \\(AV_+=\\{af:a\\in A,\\ f\\in V_+\\}\\). In particular a nonzero \\(V\\) contains a nonzero positive functional, and \\(V\\) is determined by \\(V_+\\).\n\n**Proof.** By background fact 20 with the total set \\(A\\subseteq\\tilde A\\), \\(V\\) is invariant under \\(\\tilde A\\), and Theorem 6.1 applies to \\(\\tilde A\\) and \\(V\\subseteq\\tilde A_*=A^*\\). Let \\(f\\in V\\) with \\(f=u|f|\\); then \\(|f|=u^*f\\in V_+\\). The norm closure of \\(A|f|\\) is a norm-closed subspace invariant under \\(A\\), hence under \\(\\tilde A\\); so it contains \\(u|f|=f\\). Thus \\(V\\subseteq\\overline{AV_+}\\subseteq V\\). If \\(f\\ne0\\), then \\(|f|\\in V_+\\) is nonzero. \\(\\square\\)\n\nTo pass from norm-closed to weak\\(^*\\) closed subspaces we need a closedness test on the unit ball.\n\n**Lemma 6.4** (Krein–Šmulian, subspace case). Let \\(X\\) be a Banach space and \\(V\\subseteq X^*\\) a linear subspace whose intersection with the closed unit ball \\(B\\) of \\(X^*\\) is weak\\(^*\\) closed. Then \\(V\\) is weak\\(^*\\) closed.\n\n\n**Proof.** Let \\(\\varphi_0\\in X^*\\setminus V\\). We find \\(x\\in X\\) that is annihilated by \\(V\\) but not by \\(\\varphi_0\\); by background fact 7 this proves the lemma.\n\n*Step 1.* For \\(r>0\\), \\(V\\cap rB=r(V\\cap B)\\) is weak\\(^*\\) closed. Hence \\(V\\) is norm closed: a norm-convergent sequence in \\(V\\) is bounded, so it lies in some \\(V\\cap rB\\), and norm limits are weak\\(^*\\) limits. So \\(d=\\operatorname{dist}(\\varphi_0,V)>0\\); replacing \\(\\varphi_0\\) by a multiple, we may assume \\(d>1\\). Put \\(C=\\varphi_0+V\\), a convex set with \\(C\\cap B=\\emptyset\\). For \\(r>0\\), \\(C\\cap rB=rB\\cap\\big(\\varphi_0+V\\cap(r+\\|\\varphi_0\\|)B\\big)\\) is weak\\(^*\\) closed.\n\n*Step 2.* For a finite set \\(F\\subseteq X\\) write \\(F^\\circ=\\{\\varphi:|\\varphi(x)|\\le1\\ \\forall x\\in F\\}\\), a weak\\(^*\\) closed set; \\(\\emptyset^\\circ=X^*\\). We choose finite sets \\(F_1,F_2,\\dots\\) with \\(F_n\\subseteq n^{-1}B_X\\) (where \\(B_X\\) is the unit ball of \\(X\\)) such that, with \\(G_n=F_1\\cup\\dots\\cup F_n\\) and \\(G_0=\\emptyset\\),\n\\[\n\\begin{gathered}\nC\\cap(n+1)B\\cap G_n^\\circ\\\\\n=\\emptyset\\\\\n(n\\\\\n\\ge0).\n\\end{gathered}\n\\tag{6.1}\n\\]\nFor \\(n=0\\) this is \\(C\\cap B=\\emptyset\\). Suppose (6.1) holds for \\(n-1\\) and fails for every choice of \\(F_n\\). Then the weak\\(^*\\) closed sets \\(C\\cap(n+1)B\\cap G_{n-1}^\\circ\\cap F^\\circ\\), for finite \\(F\\subseteq n^{-1}B_X\\), are nonempty. Any finitely many of them contain a common one (take the union of the \\(F\\)'s), and they lie in the weak\\(^*\\) compact set \\((n+1)B\\). So they have a common point \\(\\varphi\\). Then \\(|\\varphi(x)|\\le1\\) for all \\(x\\in n^{-1}B_X\\), that is \\(\\|\\varphi\\|\\le n\\), and \\(\\varphi\\in C\\cap nB\\cap G_{n-1}^\\circ\\), which contradicts (6.1) for \\(n-1\\).\n\n*Step 3.* List the elements of \\(F_1,F_2,\\dots\\) in order as a sequence \\((x_k)\\); then \\(x_k\\to0\\), since each \\(F_n\\) is finite and lies in \\(n^{-1}B_X\\) (pad with zeros if the list is finite). Every \\(\\varphi\\in C\\) lies in some \\((n+1)B\\), so by (6.1) it is not in \\(G_n^\\circ\\): \\(\\sup_k|\\varphi(x_k)|>1\\).\n\n*Step 4.* Let \\(T:X^*\\to c_0\\), \\(T\\varphi=(\\varphi(x_k))_k\\). Then \\(T(C)\\) is convex and misses the open unit ball \\(U\\) of \\(c_0\\). By background fact 3 there are a nonzero bounded functional \\(\\Lambda\\) on \\(c_0\\) and a real \\(\\gamma\\) with \\(\\operatorname{Re}\\Lambda(u)<\\gamma\\le\\operatorname{Re}\\Lambda(y)\\) for \\(u\\in U\\), \\(y\\in T(C)\\). Every bounded functional on \\(c_0\\) is \\(\\Lambda(y)=\\sum_k\\lambda_ky_k\\) with \\(\\sum_k|\\lambda_k|=\\|\\Lambda\\|\\): put \\(\\lambda_k=\\Lambda(\\delta_k)\\); testing \\(\\Lambda\\) on \\(\\sum_{k\\le n}\\overline{\\operatorname{sgn}\\lambda_k}\\,\\delta_k\\) gives \\(\\sum_{k\\le n}|\\lambda_k|\\le\\|\\Lambda\\|\\), and \\(y=\\lim_n\\sum_{k\\le n}y_k\\delta_k\\) in \\(c_0\\). Normalize \\(\\|\\Lambda\\|=1\\); then \\(\\sup_U\\operatorname{Re}\\Lambda=1\\), so \\(\\gamma\\ge1\\). Put \\(x=\\sum_k\\lambda_kx_k\\), an absolutely convergent series in \\(X\\). Then \\(\\operatorname{Re}\\varphi(x)=\\operatorname{Re}\\Lambda(T\\varphi)\\ge1\\) for all \\(\\varphi\\in C\\).\n\n*Step 5.* For \\(\\psi\\in V\\) and real \\(t\\), \\(\\varphi_0+t\\psi\\in C\\), so \\(\\operatorname{Re}\\varphi_0(x)+t\\operatorname{Re}\\psi(x)\\ge1\\) for all \\(t\\); hence \\(\\operatorname{Re}\\psi(x)=0\\), and with \\(i\\psi\\) also \\(\\operatorname{Im}\\psi(x)=0\\). So \\(V\\) annihilates \\(x\\), while \\(\\operatorname{Re}\\varphi_0(x)\\ge1\\). \\(\\square\\)\n\n**Theorem 6.5.** Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. If \\(V\\subseteq A^*\\) is a weak\\(^*\\) closed left invariant subspace, then \\(V\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone.\n2. If \\(C\\subseteq A^*_+\\) is a weak\\(^*\\) closed hereditary cone, then \\(\\tilde AC\\) is a weak\\(^*\\) closed left invariant subspace with \\(\\tilde AC\\cap A^*_+=C\\).\n3. The map \\(\\mathfrak r\\mapsto\\mathfrak r^\\perp\\cap A^*_+\\), where \\(\\mathfrak r^\\perp=\\{f\\in A^*:f(\\mathfrak r)=0\\}\\), is a bijection from the closed right ideals of \\(A\\) onto the weak\\(^*\\) closed hereditary cones in \\(A^*_+\\). Its inverse is \\(C\\mapsto\\{x\\in A:\\omega(xx^*)=0\\ \\forall\\omega\\in C\\}\\), and \\(\\mathfrak r^\\perp=\\tilde A(\\mathfrak r^\\perp\\cap A^*_+)\\).\n\n**Proof.** (1) The positive cone \\(A^*_+\\) is weak\\(^*\\) closed, and Corollary 6.3 applies to \\(V\\), which is norm closed.\n\n(2) \\(C\\) is norm closed, so by Corollary 6.3 (through Theorem 6.1 for \\(\\tilde A\\)) \\(V=\\tilde AC\\) is a left invariant subspace, closed in norm, with \\(V\\cap A^*_+=C\\), and \\(V=A^*e\\) for a projection \\(e\\in\\tilde A\\) with \\(s(\\rho)\\le e\\) for every \\(\\rho\\in C\\). By Lemma 6.4 it suffices to show that \\(V\\cap B\\) is weak\\(^*\\) closed, \\(B\\) being the unit ball of \\(A^*\\). Let \\((f_i)\\) be a net in \\(V\\cap B\\) with weak\\(^*\\) limit \\(f\\). The functionals \\(|f_i|\\) lie in \\(V\\cap A^*_+=C\\) and in \\(B\\). Passing to a subnet, \\(|f_i|\\to\\rho\\) weak\\(^*\\), and \\(\\rho\\in C\\) since \\(C\\) is weak\\(^*\\) closed. From \\(|f_i(x)|^2\\le|f_i|(xx^*)\\) we get \\(|f(x)|^2\\le\\rho(xx^*)\\) for \\(x\\in A\\). The argument in the general case of Theorem 3.1 extends this inequality to \\(\\tilde A\\): \\(|f(X)|^2\\le\\rho(XX^*)\\) for all \\(X\\in\\tilde A\\). Let \\(f=w|f|\\). Then \\[\n\\begin{gathered}\n||f|(X)|^2\\\\\n=|f(Xw^*)|^2\\\\\n\\le\\rho(Xw^*wX^*)\\\\\n\\le\\rho(XX^*).\n\\end{gathered}\n\\] At \\(X=1-s(\\rho)\\) this gives \\(|f|(1-s(\\rho))=0\\), so \\(s(|f|)\\le s(\\rho)\\le e\\). Hence \\(|f|=|f|e\\in V\\), so \\(|f|\\in C\\) and \\(f=w|f|\\in V\\).\n\n(3) Let \\(\\mathfrak r\\) be a closed right ideal. Then \\(\\mathfrak r^\\perp\\) is weak\\(^*\\) closed and left invariant, since \\((af)(x)=f(xa)\\) and \\(xa\\in\\mathfrak r\\) for \\(x\\in\\mathfrak r\\). So \\(C_{\\mathfrak r}=\\mathfrak r^\\perp\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone, and \\(\\mathfrak r^\\perp=\\tilde AC_{\\mathfrak r}\\) by Corollary 6.3. We recover \\(\\mathfrak r\\). If \\(x\\in\\mathfrak r\\) then \\(xx^*\\in\\mathfrak r\\), so \\(\\omega(xx^*)=0\\) for \\(\\omega\\in C_{\\mathfrak r}\\). Conversely, if \\(\\omega(xx^*)=0\\) for all \\(\\omega\\in C_{\\mathfrak r}\\), then for \\(a\\in A\\), \\(|(a\\omega)(x)|^2=|\\omega(xa)|^2\\le\\omega(xx^*)\\omega(a^*a)=0\\). So \\(x\\) is annihilated by \\(AC_{\\mathfrak r}\\), hence by its norm closure \\(\\mathfrak r^\\perp\\) (Corollary 6.3), and \\(x\\in\\mathfrak r\\) by background fact 7.\n\nConversely, let \\(C\\) be a weak\\(^*\\) closed hereditary cone and \\(V=\\tilde AC\\), weak\\(^*\\) closed by (2). Then \\(\\mathfrak r=V_\\perp\\) is a closed right ideal: \\(f(xa)=(af)(x)=0\\) for \\(x\\in\\mathfrak r\\), \\(a\\in A\\), \\(f\\in V\\). By background fact 7, \\(\\mathfrak r^\\perp=V\\), so \\(\\mathfrak r^\\perp\\cap A^*_+=C\\), and the formula for \\(\\mathfrak r\\) follows from the first part. \\(\\square\\)\n\nApplying the involution, closed left ideals \\(\\mathfrak m\\) correspond in the same way to weak\\(^*\\) closed hereditary cones, through \\(\\mathfrak m\\mapsto\\{\\omega\\ge0:\\omega(\\mathfrak m)=0\\}\\) and \\(C\\mapsto\\{x:\\omega(x^*x)=0\\ \\forall\\omega\\in C\\}\\). Extreme points turn this into a statement about pure states. For a state \\(\\omega\\) let \\(N_\\omega=\\{x\\in A:\\omega(x^*x)=0\\}\\), its left kernel.\n\n**Corollary 6.6** (Left ideals and pure states). Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. Every closed left ideal \\(\\mathfrak m\\) is the intersection of the left kernels \\(N_\\omega\\) of the pure states \\(\\omega\\) with \\(\\mathfrak m\\subseteq N_\\omega\\). (For \\(\\mathfrak m=A\\) the family is empty and the intersection is \\(A\\).)\n2. For a pure state \\(\\omega\\), \\(N_\\omega\\) is a maximal closed left ideal. Every proper closed left ideal is contained in some \\(N_\\omega\\), and every maximal closed left ideal is some \\(N_\\omega\\).\n\nThe same holds for closed right ideals with right kernels \\(\\{x:\\omega(xx^*)=0\\}\\).\n\n**Proof.** (1) Let \\(C=\\{\\omega\\in A^*_+:\\omega(\\mathfrak m)=0\\}\\), so that \\(\\mathfrak m=\\{x:\\omega(x^*x)=0\\ \\forall\\omega\\in C\\}\\) by the left-ideal form of Theorem 6.5(3). The set \\(K=\\{\\omega\\in C:\\|\\omega\\|\\le1\\}\\) is convex and weak\\(^*\\) compact. Let \\(\\omega\\) be a nonzero extreme point of \\(K\\). Then \\(\\|\\omega\\|=1\\), for otherwise \\(\\omega=\\|\\omega\\|(\\omega/\\|\\omega\\|)+(1-\\|\\omega\\|)\\cdot0\\). If \\(0\\le\\psi\\le\\omega\\) and \\(\\psi\\ne0,\\omega\\), then \\(\\psi\\) and \\(\\omega-\\psi\\) lie in \\(C\\) (heredity), and \\[\n\\begin{gathered}\n\\omega\\\\\n=\\|\\psi\\|(\\psi/\\|\\psi\\|)+\\|\\omega-\\psi\\|\\big((\\omega-\\psi)/\\|\\omega-\\psi\\|\\big)\n\\end{gathered}\n\\] with \\(\\|\\psi\\|+\\|\\omega-\\psi\\|=1\\) (background fact 12); extremality forces \\(\\psi=\\|\\psi\\|\\omega\\). So \\(\\omega\\) is a pure state. By the Krein–Milman theorem, \\(K\\) is the weak\\(^*\\) closed convex hull of \\(0\\) and the pure states in \\(C\\). For \\(x\\in A\\), \\(\\omega\\mapsto\\omega(x^*x)\\) is affine and weak\\(^*\\) continuous; so if it vanishes at every pure state in \\(C\\), it vanishes on \\(K\\), hence on \\(C\\), and \\(x\\in\\mathfrak m\\). Finally a pure state \\(\\omega\\) lies in \\(C\\) exactly when \\(\\mathfrak m\\subseteq N_\\omega\\). If \\(\\omega(\\mathfrak m)=0\\), then \\(\\omega(y^*y)=0\\) for \\(y\\in\\mathfrak m\\), since \\(y^*y\\in\\mathfrak m\\). Conversely, if \\(\\omega(y^*y)=0\\) for \\(y\\in\\mathfrak m\\), then \\(|\\omega(u_\\lambda y)|^2\\le\\omega(u_\\lambda^2)\\omega(y^*y)=0\\) and \\(\\omega(y)=\\lim_\\lambda\\omega(u_\\lambda y)=0\\).\n\n(2) \\(N_\\omega\\ne A\\), since \\(\\omega\\ne0\\). By background fact 13, \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) identifies \\(A/N_\\omega\\) isometrically with \\(H_\\omega\\), and it carries left multiplication by \\(b\\) to \\(\\pi_\\omega(b)\\). If \\(\\mathfrak m\\supsetneq N_\\omega\\) is a closed left ideal, its image is a closed, nonzero, \\(\\pi_\\omega(A)\\)-invariant subspace of \\(H_\\omega\\) (the image of a closed subspace containing the kernel of a quotient map is closed). As \\(\\pi_\\omega\\) is irreducible, the image is \\(H_\\omega\\), and \\(\\mathfrak m=A\\). So \\(N_\\omega\\) is maximal. If \\(\\mathfrak m\\) is proper, the family in (1) is not empty, so \\(\\mathfrak m\\subseteq N_\\omega\\) for some pure \\(\\omega\\); if \\(\\mathfrak m\\) is maximal, \\(\\mathfrak m=N_\\omega\\). The right-handed version follows by taking adjoints. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "6. Invariant subspaces, hereditary cones and one-sided ideals",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 6. Invariant subspaces, hereditary cones and one-sided ideals\n\nA subset \\(C\\subseteq M_*^+\\) is a *hereditary cone* if it is a convex cone (closed under sums and multiplication by nonnegative scalars) and \\(0\\le\\psi\\le\\omega\\in C\\) implies \\(\\psi\\in C\\). The positive part of a closed invariant subspace is such a cone, and the cone determines the subspace.\n\n**Theorem 6.1.** Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(MV\\subseteq V\\), and \\(V=M_*e\\) as in background fact 20. Then \\[\n\\begin{gathered}\nV_+\\\\\n=V\\cap M_*^+\\\\\n=\\{\\omega\\in M_*^+:\\omega(1-e)=0\\}\n\\end{gathered}\n\\] is a norm-closed hereditary cone, and \\(V=MV_+=\\{a\\omega:a\\in M,\\ \\omega\\in V_+\\}\\).\n2. Let \\(C\\subseteq M_*^+\\) be a norm-closed hereditary cone. Then \\(V=MC\\) is a norm-closed subspace with \\(MV\\subseteq V\\) and \\(V\\cap M_*^+=C\\).\n3. Hence \\(V\\mapsto V\\cap M_*^+\\) is a bijection from the norm-closed left invariant subspaces of \\(M_*\\) onto the norm-closed hereditary cones in \\(M_*^+\\), with inverse \\(C\\mapsto MC\\). Both correspond to projections \\(e\\): \\(V=M_*e\\) and \\(C=\\{\\omega\\ge0:\\omega(1-e)=0\\}\\).\n\nThe right-handed statements (subspaces with \\(VM\\subseteq V\\), and \\(CM\\)) follow by applying \\(\\varphi\\mapsto\\varphi^*\\).\n\n**Proof.** (1) For \\(\\omega\\ge0\\): if \\(\\omega=\\omega e\\), then \\(\\omega(1-e)=\\omega(e(1-e))=0\\); conversely, if \\(\\omega(1-e)=0\\) then \\(s(\\omega)\\le e\\) and \\(\\omega(ex)=\\omega(s(\\omega)ex)=\\omega(s(\\omega)x)=\\omega(x)\\). This gives the formula for \\(V_+\\), which is visibly a norm-closed hereditary cone. If \\(\\varphi\\in V\\) has polar decomposition \\(\\varphi=v|\\varphi|\\), then \\(|\\varphi|=v^*\\varphi\\in V\\), so \\(|\\varphi|\\in V_+\\) and \\(\\varphi\\in MV_+\\). The inclusion \\(MV_+\\subseteq V\\) is invariance.\n\n(2) *Step 1: \\(MC\\cap M_*^+=C\\).* Clearly \\(C\\subseteq MC\\). If \\(\\varphi=a\\omega\\ge0\\) with \\(\\omega\\in C\\), then \\(\\varphi=|a\\omega|\\le\\|a\\|\\omega\\) by Corollary 4.2, so \\(\\varphi\\in C\\).\n\n*Step 2: \\(MC\\) is norm closed.* Let \\(a_n\\omega_n\\to\\varphi\\) in norm, with \\(\\omega_n\\in C\\). If \\(a_n\\omega_n=u_n|a_n\\omega_n|\\), then \\(|a_n\\omega_n|=(u_n^*a_n)\\omega_n\\in MC\\cap M_*^+=C\\). By Theorem 5.1, \\(|a_n\\omega_n|\\to|\\varphi|\\), so \\(|\\varphi|\\in C\\) and \\(\\varphi=u|\\varphi|\\in MC\\).\n\n*Step 3: for \\(\\chi\\in M_*^+\\), the norm closure of \\(M\\chi\\) is \\(M_*s(\\chi)=\\{\\psi:\\psi=\\psi s(\\chi)\\}\\).* The set \\(M_*s(\\chi)\\) is norm closed and contains \\(M\\chi\\), since \\((a\\chi)(s(\\chi)x)=\\chi(s(\\chi)xa)=\\chi(xa)\\). If some \\(\\psi\\in M_*s(\\chi)\\) were not in the closure of the subspace \\(M\\chi\\), the Hahn–Banach theorem would give \\(y\\in M=(M_*)^*\\) with \\(\\chi(ya)=0\\) for all \\(a\\) and \\(\\psi(y)\\ne0\\). With \\(a=y^*\\), \\(\\chi(yy^*)=0\\), so \\(y^*s(\\chi)=0\\) (background fact 18) and \\(s(\\chi)y=0\\). Then \\(\\psi(y)=\\psi(s(\\chi)y)=0\\), a contradiction.\n\n*Step 4: \\(MC\\) is a subspace.* It is closed under scalar multiples. Let \\(a\\omega,b\\psi\\in MC\\) and \\(\\chi=\\omega+\\psi\\in C\\). Since \\(\\omega\\le\\chi\\), \\(\\omega(1-s(\\chi))=0\\) and \\(s(\\omega)\\le s(\\chi)\\); so \\(a\\omega\\in M_*s(\\omega)\\subseteq M_*s(\\chi)\\), and likewise \\(b\\psi\\in M_*s(\\chi)\\). By Step 3, \\(a\\omega+b\\psi\\) lies in the closure of \\(M\\chi\\subseteq MC\\), which is contained in \\(MC\\) by Step 2.\n\nInvariance \\(M(MC)\\subseteq MC\\) is clear.\n\n(3) By (1), \\(V=M(V\\cap M_*^+)\\); by (2), \\((MC)\\cap M_*^+=C\\). \\(\\square\\)\n\n**Example 6.2** (Convexity is needed). In \\(M=\\mathbb C^2\\), \\(M_*^+\\) is the quadrant \\([0,\\infty)^2\\). The union of the two axes is a closed cone, and it is hereditary, but it is not convex. It is not the positive part of any subspace, because such a positive part is convex.\n\nFor a \\(C^*\\)-algebra \\(A\\), a subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), where \\((af)(x)=f(xa)\\).\n\n**Corollary 6.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(V\\subseteq A^*\\) a norm-closed left invariant subspace, and put \\(V_+=V\\cap A^*_+\\). Then \\(V\\) is invariant under \\(\\tilde A\\), \\(V_+\\) is a norm-closed hereditary cone, \\(V=\\tilde AV_+\\), and \\(V\\) is the norm closure of the set \\(AV_+=\\{af:a\\in A,\\ f\\in V_+\\}\\). In particular a nonzero \\(V\\) contains a nonzero positive functional, and \\(V\\) is determined by \\(V_+\\).\n\n**Proof.** By background fact 20 with the total set \\(A\\subseteq\\tilde A\\), \\(V\\) is invariant under \\(\\tilde A\\), and Theorem 6.1 applies to \\(\\tilde A\\) and \\(V\\subseteq\\tilde A_*=A^*\\). Let \\(f\\in V\\) with \\(f=u|f|\\); then \\(|f|=u^*f\\in V_+\\). The norm closure of \\(A|f|\\) is a norm-closed subspace invariant under \\(A\\), hence under \\(\\tilde A\\); so it contains \\(u|f|=f\\). Thus \\(V\\subseteq\\overline{AV_+}\\subseteq V\\). If \\(f\\ne0\\), then \\(|f|\\in V_+\\) is nonzero. \\(\\square\\)\n\nTo pass from norm-closed to weak\\(^*\\) closed subspaces we need a closedness test on the unit ball.\n\n**Lemma 6.4** (Krein–Šmulian, subspace case). Let \\(X\\) be a Banach space and \\(V\\subseteq X^*\\) a linear subspace whose intersection with the closed unit ball \\(B\\) of \\(X^*\\) is weak\\(^*\\) closed. Then \\(V\\) is weak\\(^*\\) closed.\n\n\n**Proof.** Let \\(\\varphi_0\\in X^*\\setminus V\\). We find \\(x\\in X\\) that is annihilated by \\(V\\) but not by \\(\\varphi_0\\); by background fact 7 this proves the lemma.\n\n*Step 1.* For \\(r>0\\), \\(V\\cap rB=r(V\\cap B)\\) is weak\\(^*\\) closed. Hence \\(V\\) is norm closed: a norm-convergent sequence in \\(V\\) is bounded, so it lies in some \\(V\\cap rB\\), and norm limits are weak\\(^*\\) limits. So \\(d=\\operatorname{dist}(\\varphi_0,V)>0\\); replacing \\(\\varphi_0\\) by a multiple, we may assume \\(d>1\\). Put \\(C=\\varphi_0+V\\), a convex set with \\(C\\cap B=\\emptyset\\). For \\(r>0\\), \\(C\\cap rB=rB\\cap\\big(\\varphi_0+V\\cap(r+\\|\\varphi_0\\|)B\\big)\\) is weak\\(^*\\) closed.\n\n*Step 2.* For a finite set \\(F\\subseteq X\\) write \\(F^\\circ=\\{\\varphi:|\\varphi(x)|\\le1\\ \\forall x\\in F\\}\\), a weak\\(^*\\) closed set; \\(\\emptyset^\\circ=X^*\\). We choose finite sets \\(F_1,F_2,\\dots\\) with \\(F_n\\subseteq n^{-1}B_X\\) (where \\(B_X\\) is the unit ball of \\(X\\)) such that, with \\(G_n=F_1\\cup\\dots\\cup F_n\\) and \\(G_0=\\emptyset\\),\n\\[\n\\begin{gathered}\nC\\cap(n+1)B\\cap G_n^\\circ\\\\\n=\\emptyset\\\\\n(n\\\\\n\\ge0).\n\\end{gathered}\n\\tag{6.1}\n\\]\nFor \\(n=0\\) this is \\(C\\cap B=\\emptyset\\). Suppose (6.1) holds for \\(n-1\\) and fails for every choice of \\(F_n\\). Then the weak\\(^*\\) closed sets \\(C\\cap(n+1)B\\cap G_{n-1}^\\circ\\cap F^\\circ\\), for finite \\(F\\subseteq n^{-1}B_X\\), are nonempty. Any finitely many of them contain a common one (take the union of the \\(F\\)'s), and they lie in the weak\\(^*\\) compact set \\((n+1)B\\). So they have a common point \\(\\varphi\\). Then \\(|\\varphi(x)|\\le1\\) for all \\(x\\in n^{-1}B_X\\), that is \\(\\|\\varphi\\|\\le n\\), and \\(\\varphi\\in C\\cap nB\\cap G_{n-1}^\\circ\\), which contradicts (6.1) for \\(n-1\\).\n\n*Step 3.* List the elements of \\(F_1,F_2,\\dots\\) in order as a sequence \\((x_k)\\); then \\(x_k\\to0\\), since each \\(F_n\\) is finite and lies in \\(n^{-1}B_X\\) (pad with zeros if the list is finite). Every \\(\\varphi\\in C\\) lies in some \\((n+1)B\\), so by (6.1) it is not in \\(G_n^\\circ\\): \\(\\sup_k|\\varphi(x_k)|>1\\).\n\n*Step 4.* Let \\(T:X^*\\to c_0\\), \\(T\\varphi=(\\varphi(x_k))_k\\). Then \\(T(C)\\) is convex and misses the open unit ball \\(U\\) of \\(c_0\\). By background fact 3 there are a nonzero bounded functional \\(\\Lambda\\) on \\(c_0\\) and a real \\(\\gamma\\) with \\(\\operatorname{Re}\\Lambda(u)<\\gamma\\le\\operatorname{Re}\\Lambda(y)\\) for \\(u\\in U\\), \\(y\\in T(C)\\). Every bounded functional on \\(c_0\\) is \\(\\Lambda(y)=\\sum_k\\lambda_ky_k\\) with \\(\\sum_k|\\lambda_k|=\\|\\Lambda\\|\\): put \\(\\lambda_k=\\Lambda(\\delta_k)\\); testing \\(\\Lambda\\) on \\(\\sum_{k\\le n}\\overline{\\operatorname{sgn}\\lambda_k}\\,\\delta_k\\) gives \\(\\sum_{k\\le n}|\\lambda_k|\\le\\|\\Lambda\\|\\), and \\(y=\\lim_n\\sum_{k\\le n}y_k\\delta_k\\) in \\(c_0\\). Normalize \\(\\|\\Lambda\\|=1\\); then \\(\\sup_U\\operatorname{Re}\\Lambda=1\\), so \\(\\gamma\\ge1\\). Put \\(x=\\sum_k\\lambda_kx_k\\), an absolutely convergent series in \\(X\\). Then \\(\\operatorname{Re}\\varphi(x)=\\operatorname{Re}\\Lambda(T\\varphi)\\ge1\\) for all \\(\\varphi\\in C\\).\n\n*Step 5.* For \\(\\psi\\in V\\) and real \\(t\\), \\(\\varphi_0+t\\psi\\in C\\), so \\(\\operatorname{Re}\\varphi_0(x)+t\\operatorname{Re}\\psi(x)\\ge1\\) for all \\(t\\); hence \\(\\operatorname{Re}\\psi(x)=0\\), and with \\(i\\psi\\) also \\(\\operatorname{Im}\\psi(x)=0\\). So \\(V\\) annihilates \\(x\\), while \\(\\operatorname{Re}\\varphi_0(x)\\ge1\\). \\(\\square\\)\n\n**Theorem 6.5.** Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. If \\(V\\subseteq A^*\\) is a weak\\(^*\\) closed left invariant subspace, then \\(V\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone.\n2. If \\(C\\subseteq A^*_+\\) is a weak\\(^*\\) closed hereditary cone, then \\(\\tilde AC\\) is a weak\\(^*\\) closed left invariant subspace with \\(\\tilde AC\\cap A^*_+=C\\).\n3. The map \\(\\mathfrak r\\mapsto\\mathfrak r^\\perp\\cap A^*_+\\), where \\(\\mathfrak r^\\perp=\\{f\\in A^*:f(\\mathfrak r)=0\\}\\), is a bijection from the closed right ideals of \\(A\\) onto the weak\\(^*\\) closed hereditary cones in \\(A^*_+\\). Its inverse is \\(C\\mapsto\\{x\\in A:\\omega(xx^*)=0\\ \\forall\\omega\\in C\\}\\), and \\(\\mathfrak r^\\perp=\\tilde A(\\mathfrak r^\\perp\\cap A^*_+)\\).\n\n**Proof.** (1) The positive cone \\(A^*_+\\) is weak\\(^*\\) closed, and Corollary 6.3 applies to \\(V\\), which is norm closed.\n\n(2) \\(C\\) is norm closed, so by Corollary 6.3 (through Theorem 6.1 for \\(\\tilde A\\)) \\(V=\\tilde AC\\) is a left invariant subspace, closed in norm, with \\(V\\cap A^*_+=C\\), and \\(V=A^*e\\) for a projection \\(e\\in\\tilde A\\) with \\(s(\\rho)\\le e\\) for every \\(\\rho\\in C\\). By Lemma 6.4 it suffices to show that \\(V\\cap B\\) is weak\\(^*\\) closed, \\(B\\) being the unit ball of \\(A^*\\). Let \\((f_i)\\) be a net in \\(V\\cap B\\) with weak\\(^*\\) limit \\(f\\). The functionals \\(|f_i|\\) lie in \\(V\\cap A^*_+=C\\) and in \\(B\\). Passing to a subnet, \\(|f_i|\\to\\rho\\) weak\\(^*\\), and \\(\\rho\\in C\\) since \\(C\\) is weak\\(^*\\) closed. From \\(|f_i(x)|^2\\le|f_i|(xx^*)\\) we get \\(|f(x)|^2\\le\\rho(xx^*)\\) for \\(x\\in A\\). The argument in the general case of Theorem 3.1 extends this inequality to \\(\\tilde A\\): \\(|f(X)|^2\\le\\rho(XX^*)\\) for all \\(X\\in\\tilde A\\). Let \\(f=w|f|\\). Then \\[\n\\begin{gathered}\n||f|(X)|^2\\\\\n=|f(Xw^*)|^2\\\\\n\\le\\rho(Xw^*wX^*)\\\\\n\\le\\rho(XX^*).\n\\end{gathered}\n\\] At \\(X=1-s(\\rho)\\) this gives \\(|f|(1-s(\\rho))=0\\), so \\(s(|f|)\\le s(\\rho)\\le e\\). Hence \\(|f|=|f|e\\in V\\), so \\(|f|\\in C\\) and \\(f=w|f|\\in V\\).\n\n(3) Let \\(\\mathfrak r\\) be a closed right ideal. Then \\(\\mathfrak r^\\perp\\) is weak\\(^*\\) closed and left invariant, since \\((af)(x)=f(xa)\\) and \\(xa\\in\\mathfrak r\\) for \\(x\\in\\mathfrak r\\). So \\(C_{\\mathfrak r}=\\mathfrak r^\\perp\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone, and \\(\\mathfrak r^\\perp=\\tilde AC_{\\mathfrak r}\\) by Corollary 6.3. We recover \\(\\mathfrak r\\). If \\(x\\in\\mathfrak r\\) then \\(xx^*\\in\\mathfrak r\\), so \\(\\omega(xx^*)=0\\) for \\(\\omega\\in C_{\\mathfrak r}\\). Conversely, if \\(\\omega(xx^*)=0\\) for all \\(\\omega\\in C_{\\mathfrak r}\\), then for \\(a\\in A\\), \\(|(a\\omega)(x)|^2=|\\omega(xa)|^2\\le\\omega(xx^*)\\omega(a^*a)=0\\). So \\(x\\) is annihilated by \\(AC_{\\mathfrak r}\\), hence by its norm closure \\(\\mathfrak r^\\perp\\) (Corollary 6.3), and \\(x\\in\\mathfrak r\\) by background fact 7.\n\nConversely, let \\(C\\) be a weak\\(^*\\) closed hereditary cone and \\(V=\\tilde AC\\), weak\\(^*\\) closed by (2). Then \\(\\mathfrak r=V_\\perp\\) is a closed right ideal: \\(f(xa)=(af)(x)=0\\) for \\(x\\in\\mathfrak r\\), \\(a\\in A\\), \\(f\\in V\\). By background fact 7, \\(\\mathfrak r^\\perp=V\\), so \\(\\mathfrak r^\\perp\\cap A^*_+=C\\), and the formula for \\(\\mathfrak r\\) follows from the first part. \\(\\square\\)\n\nApplying the involution, closed left ideals \\(\\mathfrak m\\) correspond in the same way to weak\\(^*\\) closed hereditary cones, through \\(\\mathfrak m\\mapsto\\{\\omega\\ge0:\\omega(\\mathfrak m)=0\\}\\) and \\(C\\mapsto\\{x:\\omega(x^*x)=0\\ \\forall\\omega\\in C\\}\\). Extreme points turn this into a statement about pure states. For a state \\(\\omega\\) let \\(N_\\omega=\\{x\\in A:\\omega(x^*x)=0\\}\\), its left kernel.\n\n**Corollary 6.6** (Left ideals and pure states). Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. Every closed left ideal \\(\\mathfrak m\\) is the intersection of the left kernels \\(N_\\omega\\) of the pure states \\(\\omega\\) with \\(\\mathfrak m\\subseteq N_\\omega\\). (For \\(\\mathfrak m=A\\) the family is empty and the intersection is \\(A\\).)\n2. For a pure state \\(\\omega\\), \\(N_\\omega\\) is a maximal closed left ideal. Every proper closed left ideal is contained in some \\(N_\\omega\\), and every maximal closed left ideal is some \\(N_\\omega\\).\n\nThe same holds for closed right ideals with right kernels \\(\\{x:\\omega(xx^*)=0\\}\\).\n\n**Proof.** (1) Let \\(C=\\{\\omega\\in A^*_+:\\omega(\\mathfrak m)=0\\}\\), so that \\(\\mathfrak m=\\{x:\\omega(x^*x)=0\\ \\forall\\omega\\in C\\}\\) by the left-ideal form of Theorem 6.5(3). The set \\(K=\\{\\omega\\in C:\\|\\omega\\|\\le1\\}\\) is convex and weak\\(^*\\) compact. Let \\(\\omega\\) be a nonzero extreme point of \\(K\\). Then \\(\\|\\omega\\|=1\\), for otherwise \\(\\omega=\\|\\omega\\|(\\omega/\\|\\omega\\|)+(1-\\|\\omega\\|)\\cdot0\\). If \\(0\\le\\psi\\le\\omega\\) and \\(\\psi\\ne0,\\omega\\), then \\(\\psi\\) and \\(\\omega-\\psi\\) lie in \\(C\\) (heredity), and \\[\n\\begin{gathered}\n\\omega\\\\\n=\\|\\psi\\|(\\psi/\\|\\psi\\|)+\\|\\omega-\\psi\\|\\big((\\omega-\\psi)/\\|\\omega-\\psi\\|\\big)\n\\end{gathered}\n\\] with \\(\\|\\psi\\|+\\|\\omega-\\psi\\|=1\\) (background fact 12); extremality forces \\(\\psi=\\|\\psi\\|\\omega\\). So \\(\\omega\\) is a pure state. By the Krein–Milman theorem, \\(K\\) is the weak\\(^*\\) closed convex hull of \\(0\\) and the pure states in \\(C\\). For \\(x\\in A\\), \\(\\omega\\mapsto\\omega(x^*x)\\) is affine and weak\\(^*\\) continuous; so if it vanishes at every pure state in \\(C\\), it vanishes on \\(K\\), hence on \\(C\\), and \\(x\\in\\mathfrak m\\). Finally a pure state \\(\\omega\\) lies in \\(C\\) exactly when \\(\\mathfrak m\\subseteq N_\\omega\\). If \\(\\omega(\\mathfrak m)=0\\), then \\(\\omega(y^*y)=0\\) for \\(y\\in\\mathfrak m\\), since \\(y^*y\\in\\mathfrak m\\). Conversely, if \\(\\omega(y^*y)=0\\) for \\(y\\in\\mathfrak m\\), then \\(|\\omega(u_\\lambda y)|^2\\le\\omega(u_\\lambda^2)\\omega(y^*y)=0\\) and \\(\\omega(y)=\\lim_\\lambda\\omega(u_\\lambda y)=0\\).\n\n(2) \\(N_\\omega\\ne A\\), since \\(\\omega\\ne0\\). By background fact 13, \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) identifies \\(A/N_\\omega\\) isometrically with \\(H_\\omega\\), and it carries left multiplication by \\(b\\) to \\(\\pi_\\omega(b)\\). If \\(\\mathfrak m\\supsetneq N_\\omega\\) is a closed left ideal, its image is a closed, nonzero, \\(\\pi_\\omega(A)\\)-invariant subspace of \\(H_\\omega\\) (the image of a closed subspace containing the kernel of a quotient map is closed). As \\(\\pi_\\omega\\) is irreducible, the image is \\(H_\\omega\\), and \\(\\mathfrak m=A\\). So \\(N_\\omega\\) is maximal. If \\(\\mathfrak m\\) is proper, the family in (1) is not empty, so \\(\\mathfrak m\\subseteq N_\\omega\\) for some pure \\(\\omega\\); if \\(\\mathfrak m\\) is maximal, \\(\\mathfrak m=N_\\omega\\). The right-handed version follows by taking adjoints. \\(\\square\\)\n\n",
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      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "6. Invariant subspaces, hereditary cones and one-sided ideals",
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      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 6. Invariant subspaces, hereditary cones and one-sided ideals\n\nA subset \\(C\\subseteq M_*^+\\) is a *hereditary cone* if it is a convex cone (closed under sums and multiplication by nonnegative scalars) and \\(0\\le\\psi\\le\\omega\\in C\\) implies \\(\\psi\\in C\\). The positive part of a closed invariant subspace is such a cone, and the cone determines the subspace.\n\n**Theorem 6.1.** Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(MV\\subseteq V\\), and \\(V=M_*e\\) as in background fact 20. Then \\[\n\\begin{gathered}\nV_+\\\\\n=V\\cap M_*^+\\\\\n=\\{\\omega\\in M_*^+:\\omega(1-e)=0\\}\n\\end{gathered}\n\\] is a norm-closed hereditary cone, and \\(V=MV_+=\\{a\\omega:a\\in M,\\ \\omega\\in V_+\\}\\).\n2. Let \\(C\\subseteq M_*^+\\) be a norm-closed hereditary cone. Then \\(V=MC\\) is a norm-closed subspace with \\(MV\\subseteq V\\) and \\(V\\cap M_*^+=C\\).\n3. Hence \\(V\\mapsto V\\cap M_*^+\\) is a bijection from the norm-closed left invariant subspaces of \\(M_*\\) onto the norm-closed hereditary cones in \\(M_*^+\\), with inverse \\(C\\mapsto MC\\). Both correspond to projections \\(e\\): \\(V=M_*e\\) and \\(C=\\{\\omega\\ge0:\\omega(1-e)=0\\}\\).\n\nThe right-handed statements (subspaces with \\(VM\\subseteq V\\), and \\(CM\\)) follow by applying \\(\\varphi\\mapsto\\varphi^*\\).\n\n**Proof.** (1) For \\(\\omega\\ge0\\): if \\(\\omega=\\omega e\\), then \\(\\omega(1-e)=\\omega(e(1-e))=0\\); conversely, if \\(\\omega(1-e)=0\\) then \\(s(\\omega)\\le e\\) and \\(\\omega(ex)=\\omega(s(\\omega)ex)=\\omega(s(\\omega)x)=\\omega(x)\\). This gives the formula for \\(V_+\\), which is visibly a norm-closed hereditary cone. If \\(\\varphi\\in V\\) has polar decomposition \\(\\varphi=v|\\varphi|\\), then \\(|\\varphi|=v^*\\varphi\\in V\\), so \\(|\\varphi|\\in V_+\\) and \\(\\varphi\\in MV_+\\). The inclusion \\(MV_+\\subseteq V\\) is invariance.\n\n(2) *Step 1: \\(MC\\cap M_*^+=C\\).* Clearly \\(C\\subseteq MC\\). If \\(\\varphi=a\\omega\\ge0\\) with \\(\\omega\\in C\\), then \\(\\varphi=|a\\omega|\\le\\|a\\|\\omega\\) by Corollary 4.2, so \\(\\varphi\\in C\\).\n\n*Step 2: \\(MC\\) is norm closed.* Let \\(a_n\\omega_n\\to\\varphi\\) in norm, with \\(\\omega_n\\in C\\). If \\(a_n\\omega_n=u_n|a_n\\omega_n|\\), then \\(|a_n\\omega_n|=(u_n^*a_n)\\omega_n\\in MC\\cap M_*^+=C\\). By Theorem 5.1, \\(|a_n\\omega_n|\\to|\\varphi|\\), so \\(|\\varphi|\\in C\\) and \\(\\varphi=u|\\varphi|\\in MC\\).\n\n*Step 3: for \\(\\chi\\in M_*^+\\), the norm closure of \\(M\\chi\\) is \\(M_*s(\\chi)=\\{\\psi:\\psi=\\psi s(\\chi)\\}\\).* The set \\(M_*s(\\chi)\\) is norm closed and contains \\(M\\chi\\), since \\((a\\chi)(s(\\chi)x)=\\chi(s(\\chi)xa)=\\chi(xa)\\). If some \\(\\psi\\in M_*s(\\chi)\\) were not in the closure of the subspace \\(M\\chi\\), the Hahn–Banach theorem would give \\(y\\in M=(M_*)^*\\) with \\(\\chi(ya)=0\\) for all \\(a\\) and \\(\\psi(y)\\ne0\\). With \\(a=y^*\\), \\(\\chi(yy^*)=0\\), so \\(y^*s(\\chi)=0\\) (background fact 18) and \\(s(\\chi)y=0\\). Then \\(\\psi(y)=\\psi(s(\\chi)y)=0\\), a contradiction.\n\n*Step 4: \\(MC\\) is a subspace.* It is closed under scalar multiples. Let \\(a\\omega,b\\psi\\in MC\\) and \\(\\chi=\\omega+\\psi\\in C\\). Since \\(\\omega\\le\\chi\\), \\(\\omega(1-s(\\chi))=0\\) and \\(s(\\omega)\\le s(\\chi)\\); so \\(a\\omega\\in M_*s(\\omega)\\subseteq M_*s(\\chi)\\), and likewise \\(b\\psi\\in M_*s(\\chi)\\). By Step 3, \\(a\\omega+b\\psi\\) lies in the closure of \\(M\\chi\\subseteq MC\\), which is contained in \\(MC\\) by Step 2.\n\nInvariance \\(M(MC)\\subseteq MC\\) is clear.\n\n(3) By (1), \\(V=M(V\\cap M_*^+)\\); by (2), \\((MC)\\cap M_*^+=C\\). \\(\\square\\)\n\n**Example 6.2** (Convexity is needed). In \\(M=\\mathbb C^2\\), \\(M_*^+\\) is the quadrant \\([0,\\infty)^2\\). The union of the two axes is a closed cone, and it is hereditary, but it is not convex. It is not the positive part of any subspace, because such a positive part is convex.\n\nFor a \\(C^*\\)-algebra \\(A\\), a subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), where \\((af)(x)=f(xa)\\).\n\n**Corollary 6.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(V\\subseteq A^*\\) a norm-closed left invariant subspace, and put \\(V_+=V\\cap A^*_+\\). Then \\(V\\) is invariant under \\(\\tilde A\\), \\(V_+\\) is a norm-closed hereditary cone, \\(V=\\tilde AV_+\\), and \\(V\\) is the norm closure of the set \\(AV_+=\\{af:a\\in A,\\ f\\in V_+\\}\\). In particular a nonzero \\(V\\) contains a nonzero positive functional, and \\(V\\) is determined by \\(V_+\\).\n\n**Proof.** By background fact 20 with the total set \\(A\\subseteq\\tilde A\\), \\(V\\) is invariant under \\(\\tilde A\\), and Theorem 6.1 applies to \\(\\tilde A\\) and \\(V\\subseteq\\tilde A_*=A^*\\). Let \\(f\\in V\\) with \\(f=u|f|\\); then \\(|f|=u^*f\\in V_+\\). The norm closure of \\(A|f|\\) is a norm-closed subspace invariant under \\(A\\), hence under \\(\\tilde A\\); so it contains \\(u|f|=f\\). Thus \\(V\\subseteq\\overline{AV_+}\\subseteq V\\). If \\(f\\ne0\\), then \\(|f|\\in V_+\\) is nonzero. \\(\\square\\)\n\nTo pass from norm-closed to weak\\(^*\\) closed subspaces we need a closedness test on the unit ball.\n\n**Lemma 6.4** (Krein–Šmulian, subspace case). Let \\(X\\) be a Banach space and \\(V\\subseteq X^*\\) a linear subspace whose intersection with the closed unit ball \\(B\\) of \\(X^*\\) is weak\\(^*\\) closed. Then \\(V\\) is weak\\(^*\\) closed.\n\n\n**Proof.** Let \\(\\varphi_0\\in X^*\\setminus V\\). We find \\(x\\in X\\) that is annihilated by \\(V\\) but not by \\(\\varphi_0\\); by background fact 7 this proves the lemma.\n\n*Step 1.* For \\(r>0\\), \\(V\\cap rB=r(V\\cap B)\\) is weak\\(^*\\) closed. Hence \\(V\\) is norm closed: a norm-convergent sequence in \\(V\\) is bounded, so it lies in some \\(V\\cap rB\\), and norm limits are weak\\(^*\\) limits. So \\(d=\\operatorname{dist}(\\varphi_0,V)>0\\); replacing \\(\\varphi_0\\) by a multiple, we may assume \\(d>1\\). Put \\(C=\\varphi_0+V\\), a convex set with \\(C\\cap B=\\emptyset\\). For \\(r>0\\), \\(C\\cap rB=rB\\cap\\big(\\varphi_0+V\\cap(r+\\|\\varphi_0\\|)B\\big)\\) is weak\\(^*\\) closed.\n\n*Step 2.* For a finite set \\(F\\subseteq X\\) write \\(F^\\circ=\\{\\varphi:|\\varphi(x)|\\le1\\ \\forall x\\in F\\}\\), a weak\\(^*\\) closed set; \\(\\emptyset^\\circ=X^*\\). We choose finite sets \\(F_1,F_2,\\dots\\) with \\(F_n\\subseteq n^{-1}B_X\\) (where \\(B_X\\) is the unit ball of \\(X\\)) such that, with \\(G_n=F_1\\cup\\dots\\cup F_n\\) and \\(G_0=\\emptyset\\),\n\\[\n\\begin{gathered}\nC\\cap(n+1)B\\cap G_n^\\circ\\\\\n=\\emptyset\\\\\n(n\\\\\n\\ge0).\n\\end{gathered}\n\\tag{6.1}\n\\]\nFor \\(n=0\\) this is \\(C\\cap B=\\emptyset\\). Suppose (6.1) holds for \\(n-1\\) and fails for every choice of \\(F_n\\). Then the weak\\(^*\\) closed sets \\(C\\cap(n+1)B\\cap G_{n-1}^\\circ\\cap F^\\circ\\), for finite \\(F\\subseteq n^{-1}B_X\\), are nonempty. Any finitely many of them contain a common one (take the union of the \\(F\\)'s), and they lie in the weak\\(^*\\) compact set \\((n+1)B\\). So they have a common point \\(\\varphi\\). Then \\(|\\varphi(x)|\\le1\\) for all \\(x\\in n^{-1}B_X\\), that is \\(\\|\\varphi\\|\\le n\\), and \\(\\varphi\\in C\\cap nB\\cap G_{n-1}^\\circ\\), which contradicts (6.1) for \\(n-1\\).\n\n*Step 3.* List the elements of \\(F_1,F_2,\\dots\\) in order as a sequence \\((x_k)\\); then \\(x_k\\to0\\), since each \\(F_n\\) is finite and lies in \\(n^{-1}B_X\\) (pad with zeros if the list is finite). Every \\(\\varphi\\in C\\) lies in some \\((n+1)B\\), so by (6.1) it is not in \\(G_n^\\circ\\): \\(\\sup_k|\\varphi(x_k)|>1\\).\n\n*Step 4.* Let \\(T:X^*\\to c_0\\), \\(T\\varphi=(\\varphi(x_k))_k\\). Then \\(T(C)\\) is convex and misses the open unit ball \\(U\\) of \\(c_0\\). By background fact 3 there are a nonzero bounded functional \\(\\Lambda\\) on \\(c_0\\) and a real \\(\\gamma\\) with \\(\\operatorname{Re}\\Lambda(u)<\\gamma\\le\\operatorname{Re}\\Lambda(y)\\) for \\(u\\in U\\), \\(y\\in T(C)\\). Every bounded functional on \\(c_0\\) is \\(\\Lambda(y)=\\sum_k\\lambda_ky_k\\) with \\(\\sum_k|\\lambda_k|=\\|\\Lambda\\|\\): put \\(\\lambda_k=\\Lambda(\\delta_k)\\); testing \\(\\Lambda\\) on \\(\\sum_{k\\le n}\\overline{\\operatorname{sgn}\\lambda_k}\\,\\delta_k\\) gives \\(\\sum_{k\\le n}|\\lambda_k|\\le\\|\\Lambda\\|\\), and \\(y=\\lim_n\\sum_{k\\le n}y_k\\delta_k\\) in \\(c_0\\). Normalize \\(\\|\\Lambda\\|=1\\); then \\(\\sup_U\\operatorname{Re}\\Lambda=1\\), so \\(\\gamma\\ge1\\). Put \\(x=\\sum_k\\lambda_kx_k\\), an absolutely convergent series in \\(X\\). Then \\(\\operatorname{Re}\\varphi(x)=\\operatorname{Re}\\Lambda(T\\varphi)\\ge1\\) for all \\(\\varphi\\in C\\).\n\n*Step 5.* For \\(\\psi\\in V\\) and real \\(t\\), \\(\\varphi_0+t\\psi\\in C\\), so \\(\\operatorname{Re}\\varphi_0(x)+t\\operatorname{Re}\\psi(x)\\ge1\\) for all \\(t\\); hence \\(\\operatorname{Re}\\psi(x)=0\\), and with \\(i\\psi\\) also \\(\\operatorname{Im}\\psi(x)=0\\). So \\(V\\) annihilates \\(x\\), while \\(\\operatorname{Re}\\varphi_0(x)\\ge1\\). \\(\\square\\)\n\n**Theorem 6.5.** Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. If \\(V\\subseteq A^*\\) is a weak\\(^*\\) closed left invariant subspace, then \\(V\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone.\n2. If \\(C\\subseteq A^*_+\\) is a weak\\(^*\\) closed hereditary cone, then \\(\\tilde AC\\) is a weak\\(^*\\) closed left invariant subspace with \\(\\tilde AC\\cap A^*_+=C\\).\n3. The map \\(\\mathfrak r\\mapsto\\mathfrak r^\\perp\\cap A^*_+\\), where \\(\\mathfrak r^\\perp=\\{f\\in A^*:f(\\mathfrak r)=0\\}\\), is a bijection from the closed right ideals of \\(A\\) onto the weak\\(^*\\) closed hereditary cones in \\(A^*_+\\). Its inverse is \\(C\\mapsto\\{x\\in A:\\omega(xx^*)=0\\ \\forall\\omega\\in C\\}\\), and \\(\\mathfrak r^\\perp=\\tilde A(\\mathfrak r^\\perp\\cap A^*_+)\\).\n\n**Proof.** (1) The positive cone \\(A^*_+\\) is weak\\(^*\\) closed, and Corollary 6.3 applies to \\(V\\), which is norm closed.\n\n(2) \\(C\\) is norm closed, so by Corollary 6.3 (through Theorem 6.1 for \\(\\tilde A\\)) \\(V=\\tilde AC\\) is a left invariant subspace, closed in norm, with \\(V\\cap A^*_+=C\\), and \\(V=A^*e\\) for a projection \\(e\\in\\tilde A\\) with \\(s(\\rho)\\le e\\) for every \\(\\rho\\in C\\). By Lemma 6.4 it suffices to show that \\(V\\cap B\\) is weak\\(^*\\) closed, \\(B\\) being the unit ball of \\(A^*\\). Let \\((f_i)\\) be a net in \\(V\\cap B\\) with weak\\(^*\\) limit \\(f\\). The functionals \\(|f_i|\\) lie in \\(V\\cap A^*_+=C\\) and in \\(B\\). Passing to a subnet, \\(|f_i|\\to\\rho\\) weak\\(^*\\), and \\(\\rho\\in C\\) since \\(C\\) is weak\\(^*\\) closed. From \\(|f_i(x)|^2\\le|f_i|(xx^*)\\) we get \\(|f(x)|^2\\le\\rho(xx^*)\\) for \\(x\\in A\\). The argument in the general case of Theorem 3.1 extends this inequality to \\(\\tilde A\\): \\(|f(X)|^2\\le\\rho(XX^*)\\) for all \\(X\\in\\tilde A\\). Let \\(f=w|f|\\). Then \\[\n\\begin{gathered}\n||f|(X)|^2\\\\\n=|f(Xw^*)|^2\\\\\n\\le\\rho(Xw^*wX^*)\\\\\n\\le\\rho(XX^*).\n\\end{gathered}\n\\] At \\(X=1-s(\\rho)\\) this gives \\(|f|(1-s(\\rho))=0\\), so \\(s(|f|)\\le s(\\rho)\\le e\\). Hence \\(|f|=|f|e\\in V\\), so \\(|f|\\in C\\) and \\(f=w|f|\\in V\\).\n\n(3) Let \\(\\mathfrak r\\) be a closed right ideal. Then \\(\\mathfrak r^\\perp\\) is weak\\(^*\\) closed and left invariant, since \\((af)(x)=f(xa)\\) and \\(xa\\in\\mathfrak r\\) for \\(x\\in\\mathfrak r\\). So \\(C_{\\mathfrak r}=\\mathfrak r^\\perp\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone, and \\(\\mathfrak r^\\perp=\\tilde AC_{\\mathfrak r}\\) by Corollary 6.3. We recover \\(\\mathfrak r\\). If \\(x\\in\\mathfrak r\\) then \\(xx^*\\in\\mathfrak r\\), so \\(\\omega(xx^*)=0\\) for \\(\\omega\\in C_{\\mathfrak r}\\). Conversely, if \\(\\omega(xx^*)=0\\) for all \\(\\omega\\in C_{\\mathfrak r}\\), then for \\(a\\in A\\), \\(|(a\\omega)(x)|^2=|\\omega(xa)|^2\\le\\omega(xx^*)\\omega(a^*a)=0\\). So \\(x\\) is annihilated by \\(AC_{\\mathfrak r}\\), hence by its norm closure \\(\\mathfrak r^\\perp\\) (Corollary 6.3), and \\(x\\in\\mathfrak r\\) by background fact 7.\n\nConversely, let \\(C\\) be a weak\\(^*\\) closed hereditary cone and \\(V=\\tilde AC\\), weak\\(^*\\) closed by (2). Then \\(\\mathfrak r=V_\\perp\\) is a closed right ideal: \\(f(xa)=(af)(x)=0\\) for \\(x\\in\\mathfrak r\\), \\(a\\in A\\), \\(f\\in V\\). By background fact 7, \\(\\mathfrak r^\\perp=V\\), so \\(\\mathfrak r^\\perp\\cap A^*_+=C\\), and the formula for \\(\\mathfrak r\\) follows from the first part. \\(\\square\\)\n\nApplying the involution, closed left ideals \\(\\mathfrak m\\) correspond in the same way to weak\\(^*\\) closed hereditary cones, through \\(\\mathfrak m\\mapsto\\{\\omega\\ge0:\\omega(\\mathfrak m)=0\\}\\) and \\(C\\mapsto\\{x:\\omega(x^*x)=0\\ \\forall\\omega\\in C\\}\\). Extreme points turn this into a statement about pure states. For a state \\(\\omega\\) let \\(N_\\omega=\\{x\\in A:\\omega(x^*x)=0\\}\\), its left kernel.\n\n**Corollary 6.6** (Left ideals and pure states). Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. Every closed left ideal \\(\\mathfrak m\\) is the intersection of the left kernels \\(N_\\omega\\) of the pure states \\(\\omega\\) with \\(\\mathfrak m\\subseteq N_\\omega\\). (For \\(\\mathfrak m=A\\) the family is empty and the intersection is \\(A\\).)\n2. For a pure state \\(\\omega\\), \\(N_\\omega\\) is a maximal closed left ideal. Every proper closed left ideal is contained in some \\(N_\\omega\\), and every maximal closed left ideal is some \\(N_\\omega\\).\n\nThe same holds for closed right ideals with right kernels \\(\\{x:\\omega(xx^*)=0\\}\\).\n\n**Proof.** (1) Let \\(C=\\{\\omega\\in A^*_+:\\omega(\\mathfrak m)=0\\}\\), so that \\(\\mathfrak m=\\{x:\\omega(x^*x)=0\\ \\forall\\omega\\in C\\}\\) by the left-ideal form of Theorem 6.5(3). The set \\(K=\\{\\omega\\in C:\\|\\omega\\|\\le1\\}\\) is convex and weak\\(^*\\) compact. Let \\(\\omega\\) be a nonzero extreme point of \\(K\\). Then \\(\\|\\omega\\|=1\\), for otherwise \\(\\omega=\\|\\omega\\|(\\omega/\\|\\omega\\|)+(1-\\|\\omega\\|)\\cdot0\\). If \\(0\\le\\psi\\le\\omega\\) and \\(\\psi\\ne0,\\omega\\), then \\(\\psi\\) and \\(\\omega-\\psi\\) lie in \\(C\\) (heredity), and \\[\n\\begin{gathered}\n\\omega\\\\\n=\\|\\psi\\|(\\psi/\\|\\psi\\|)+\\|\\omega-\\psi\\|\\big((\\omega-\\psi)/\\|\\omega-\\psi\\|\\big)\n\\end{gathered}\n\\] with \\(\\|\\psi\\|+\\|\\omega-\\psi\\|=1\\) (background fact 12); extremality forces \\(\\psi=\\|\\psi\\|\\omega\\). So \\(\\omega\\) is a pure state. By the Krein–Milman theorem, \\(K\\) is the weak\\(^*\\) closed convex hull of \\(0\\) and the pure states in \\(C\\). For \\(x\\in A\\), \\(\\omega\\mapsto\\omega(x^*x)\\) is affine and weak\\(^*\\) continuous; so if it vanishes at every pure state in \\(C\\), it vanishes on \\(K\\), hence on \\(C\\), and \\(x\\in\\mathfrak m\\). Finally a pure state \\(\\omega\\) lies in \\(C\\) exactly when \\(\\mathfrak m\\subseteq N_\\omega\\). If \\(\\omega(\\mathfrak m)=0\\), then \\(\\omega(y^*y)=0\\) for \\(y\\in\\mathfrak m\\), since \\(y^*y\\in\\mathfrak m\\). Conversely, if \\(\\omega(y^*y)=0\\) for \\(y\\in\\mathfrak m\\), then \\(|\\omega(u_\\lambda y)|^2\\le\\omega(u_\\lambda^2)\\omega(y^*y)=0\\) and \\(\\omega(y)=\\lim_\\lambda\\omega(u_\\lambda y)=0\\).\n\n(2) \\(N_\\omega\\ne A\\), since \\(\\omega\\ne0\\). By background fact 13, \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) identifies \\(A/N_\\omega\\) isometrically with \\(H_\\omega\\), and it carries left multiplication by \\(b\\) to \\(\\pi_\\omega(b)\\). If \\(\\mathfrak m\\supsetneq N_\\omega\\) is a closed left ideal, its image is a closed, nonzero, \\(\\pi_\\omega(A)\\)-invariant subspace of \\(H_\\omega\\) (the image of a closed subspace containing the kernel of a quotient map is closed). As \\(\\pi_\\omega\\) is irreducible, the image is \\(H_\\omega\\), and \\(\\mathfrak m=A\\). So \\(N_\\omega\\) is maximal. If \\(\\mathfrak m\\) is proper, the family in (1) is not empty, so \\(\\mathfrak m\\subseteq N_\\omega\\) for some pure \\(\\omega\\); if \\(\\mathfrak m\\) is maximal, \\(\\mathfrak m=N_\\omega\\). The right-handed version follows by taking adjoints. \\(\\square\\)\n\n",
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      "id": "OA-FND-PD-12",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "7. Phillips's lemma and Schur's theorem",
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      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 7. Phillips's lemma and Schur's theorem\n\nFor a set \\(\\Gamma\\), a bounded functional \\(\\mu\\) on \\(\\ell^\\infty(\\Gamma)\\) is a finitely additive set function \\(E\\mapsto\\mu(E)=\\mu(1_E)\\). The next lemma says that if such functionals tend to zero on every set, their parts on points tend to zero in \\(\\ell^1\\)-norm. It is the tool behind Section 8.\n\n**Lemma 7.1** (Phillips's lemma). Let \\((\\mu_n)\\) be a bounded sequence in \\(\\ell^\\infty(\\Gamma)^*\\) with \\(\\mu_n(E)\\to0\\) for every \\(E\\subseteq\\Gamma\\). Then\n\\[\n\\sum_{\\gamma\\in\\Gamma}|\\mu_n(\\{\\gamma\\})|\\to0 .\n\\]\n\n\n**Proof.** For \\(\\mu\\in\\ell^\\infty(\\Gamma)^*\\) and \\(E\\subseteq\\Gamma\\) put \\(|\\mu|(E)=\\sup\\{|\\mu(x)|:\\|x\\|\\le1,\\ x=0\\text{ off }E\\}\\). Three facts:\n\n- (a) If \\(E_1,\\dots,E_m\\) are disjoint, \\(\\sum_j|\\mu|(E_j)\\le\\|\\mu\\|\\). Choose \\(x_j\\) supported in \\(E_j\\) with \\(\\|x_j\\|\\le1\\) and \\(\\mu(x_j)\\ge|\\mu|(E_j)-\\varepsilon\\); then \\(x=\\sum_jx_j\\) has \\(\\|x\\|\\le1\\).\n- (b) \\(\\sum_{\\gamma\\in E}|\\mu(\\{\\gamma\\})|\\le|\\mu|(E)\\), by testing on \\(\\sum_{\\gamma\\in F}\\overline{\\operatorname{sgn}\\mu(\\{\\gamma\\})}\\,1_{\\{\\gamma\\}}\\) for finite \\(F\\subseteq E\\). In particular \\(a_\\mu(\\gamma)=\\mu(\\{\\gamma\\})\\) defines \\(a_\\mu\\in\\ell^1(\\Gamma)\\).\n- (c) For finite \\(F\\) and any \\(x\\), \\(\\mu(x1_F)=\\sum_{\\gamma\\in F}x(\\gamma)\\mu(\\{\\gamma\\})\\), by finite additivity.\n\nLet \\(c=\\sup_n\\|\\mu_n\\|\\) and \\(a_n=a_{\\mu_n}\\). Suppose the lemma fails. Passing to a subsequence, which keeps the hypotheses, we may assume \\(\\|a_n\\|_1\\ge5\\varepsilon\\) for all \\(n\\), for some \\(\\varepsilon>0\\). By hypothesis \\(a_n(\\gamma)\\to0\\) for each \\(\\gamma\\), so \\(\\sum_{\\gamma\\in D}|a_n(\\gamma)|\\to0\\) for every finite \\(D\\).\n\n*Humps.* Choose indices \\(m_1<m_2<\\cdots\\) and disjoint finite sets \\(E_1,E_2,\\dots\\) as follows. Given \\(E_1,\\dots,E_{k-1}\\), with union \\(D\\), choose \\(m_k>m_{k-1}\\) with \\(\\sum_{\\gamma\\in D}|a_{m_k}(\\gamma)|<\\varepsilon/2\\); then \\(\\sum_{\\gamma\\notin D}|a_{m_k}(\\gamma)|>4\\varepsilon\\), so there is a finite \\(E_k\\) disjoint from \\(D\\) with \\(\\sum_{\\gamma\\in E_k}|a_{m_k}(\\gamma)|\\ge4\\varepsilon\\).\n\n*Thinning.* Fix an integer \\(p>c/\\varepsilon\\). If \\(L\\) is an infinite set of indices and \\(\\mu\\) a functional, split \\(L\\) into \\(p\\) disjoint infinite sets \\(L_1,\\dots,L_p\\). The sets \\(\\bigcup_{l\\in L_r}E_l\\) are disjoint, so by (a) some \\(r\\) has \\(|\\mu|(\\bigcup_{l\\in L_r}E_l)\\le c/p<\\varepsilon\\). Starting from \\(L_0=\\mathbb N\\), define \\(k_j=\\min L_{j-1}\\) and apply this to \\(\\mu=\\mu_{m_{k_j}}\\) and \\(L=L_{j-1}\\setminus\\{k_j\\}\\), to get an infinite \\(L_j\\subseteq L_{j-1}\\setminus\\{k_j\\}\\) with \\(|\\mu_{m_{k_j}}|\\big(\\bigcup_{l\\in L_j}E_l\\big)<\\varepsilon\\). Then \\(k_1<k_2<\\cdots\\), and \\(k_i\\in L_j\\) for \\(i>j\\).\n\n*Test function.* Let \\(\\nu_j=\\mu_{m_{k_j}}\\) and define \\(x\\in\\ell^\\infty(\\Gamma)\\) by \\(x(\\gamma)=\\overline{\\operatorname{sgn}a_{m_{k_i}}(\\gamma)}\\) for \\(\\gamma\\in E_{k_i}\\), \\(i\\ge1\\), and \\(x=0\\) elsewhere; \\(\\|x\\|\\le1\\). Fix \\(j\\) and split \\(x=x1_{P}+x1_{E_{k_j}}+x1_{T}\\), where \\(P=\\bigcup_{i<j}E_{k_i}\\) and \\(T=\\bigcup_{i>j}E_{k_i}\\). By (c) and the choice of \\(m_{k_j}\\) (the set \\(P\\) lies in the \\(D\\) used at that step), \\(|\\nu_j(x1_P)|<\\varepsilon/2\\). By (c), \\(\\nu_j(x1_{E_{k_j}})=\\sum_{\\gamma\\in E_{k_j}}|a_{m_{k_j}}(\\gamma)|\\ge4\\varepsilon\\). And \\(T\\subseteq\\bigcup_{l\\in L_j}E_l\\), so \\(|\\nu_j(x1_T)|\\le|\\nu_j|(T)<\\varepsilon\\). Hence \\(|\\nu_j(x)|>2\\varepsilon\\) for every \\(j\\).\n\n*Contradiction.* The function \\(x\\) is a uniform limit of finite linear combinations of indicator functions (cut the unit disc into finitely many small pieces). Each \\(\\nu_j(1_E)\\to0\\) and \\(\\sup_j\\|\\nu_j\\|\\le c\\), so \\(\\nu_j(x)\\to0\\). \\(\\square\\)\n\n**Corollary 7.2** (Schur's theorem). In \\(\\ell^1(\\Gamma)\\) every weakly convergent sequence converges in norm.\n\n\n**Proof.** Let \\(g_n\\to g\\) weakly; replacing \\(g_n\\) by \\(g_n-g\\), let \\(g=0\\). The sequence is bounded (background fact 2). Put \\(\\mu_n(x)=\\sum_\\gamma x(\\gamma)g_n(\\gamma)\\) on \\(\\ell^\\infty(\\Gamma)\\). For \\(E\\subseteq\\Gamma\\), \\(g\\mapsto\\sum_{\\gamma\\in E}g(\\gamma)\\) is a bounded functional on \\(\\ell^1(\\Gamma)\\), so \\(\\mu_n(E)\\to0\\). Lemma 7.1 gives \\(\\|g_n\\|_1=\\sum_\\gamma|\\mu_n(\\{\\gamma\\})|\\to0\\). \\(\\square\\)\n\nSo in \\(\\ell^1\\) weak and norm convergence of sequences agree. Section 12 shows how much of this survives in \\(B(H)_*\\) and other atomic preduals.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-13",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "8. Normal and singular parts along weak\\(^*\\) convergent sequences",
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      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 8. Normal and singular parts along weak\\(^*\\) convergent sequences\n\nFor the rest of the lesson we attach to each functional one positive functional that controls it.\n\n**Lemma 8.1.** For \\(\\varphi\\in M^*\\) put \\(\\varphi^\\sharp=|\\varphi|+|\\varphi^*|\\), the absolute values being taken in the \\(C^*\\)-algebra \\(M\\) (Theorem 2.7).\n\n1. \\(\\varphi^\\sharp\\ge0\\), \\(\\|\\varphi^\\sharp\\|=2\\|\\varphi\\|\\), and \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(xx^*)\\), \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(x^*x)\\).\n2. If \\(\\varphi\\) is normal, so is \\(\\varphi^\\sharp\\), and \\(\\varphi(x)=\\varphi(gxg)\\) with \\(g=s(\\varphi^\\sharp)\\). If \\(\\varphi\\) is singular, so is \\(\\varphi^\\sharp\\).\n3. If \\(p\\) is a projection with \\(\\varphi^\\sharp(p)=0\\), then \\(\\varphi(p)=0\\).\n\n**Proof.** (1) follows from (2.4) and Theorem 2.7, the second inequality by applying the first to \\(\\varphi^*\\) and \\(x^*\\). (2) If \\(\\varphi\\) is normal, \\(|\\varphi|\\) and \\(|\\varphi^*|\\) are normal by Corollary 3.2. Their supports are \\(s_r(\\varphi)\\) and \\(s_l(\\varphi)\\) (Theorem 2.2), both under \\(g\\), so \\[\n\\begin{gathered}\n\\varphi(gxg)\\\\\n=\\varphi(s_r(\\varphi)\\,gxg\\,s_l(\\varphi))\\\\\n=\\varphi(s_r(\\varphi)\\,x\\,s_l(\\varphi))\\\\\n=\\varphi(x)\n\\end{gathered}\n\\] by Lemma 1.4(2). If \\(\\varphi\\) is singular, then \\(\\varphi\\in M^*(1-z_0)\\) (background fact 22), and so is \\(\\varphi^*\\) since \\(z_0\\) is central. The space \\(M^*(1-z_0)\\) is invariant under \\(M^{**}\\), so \\(|\\varphi|=v^*\\varphi\\) and \\(|\\varphi^*|\\) lie in it. (3) By (1), \\(|\\varphi(p)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(p)=0\\). \\(\\square\\)\n\n**Theorem 8.2.** Let \\((\\varphi_k)\\) be a sequence in \\(M^*\\) that converges to \\(\\varphi\\) in \\(\\sigma(M^*,M)\\). Then \\(\\varphi_k^{\\rm n}\\to\\varphi^{\\rm n}\\) and \\(\\varphi_k^{\\rm s}\\to\\varphi^{\\rm s}\\) in \\(\\sigma(M^*,M)\\).\n\n**Proof.** The splitting is linear, so we may assume \\(\\varphi=0\\). By background fact 2, \\(c=\\sup_k\\|\\varphi_k\\|<\\infty\\), and \\(\\|\\varphi_k^{\\rm n}\\|,\\|\\varphi_k^{\\rm s}\\|\\le c\\). Since \\(\\varphi_k^{\\rm s}=\\varphi_k-\\varphi_k^{\\rm n}\\), it suffices to show \\(\\varphi_k^{\\rm n}\\to0\\). The sequence is bounded and the span of the projections is norm dense in \\(M\\) (background fact 16), so it suffices to show \\(\\varphi_k^{\\rm n}(p)\\to0\\) for each projection \\(p\\ne0\\).\n\nLet \\(\\omega=\\sum_k2^{-k}(\\varphi_k^{\\rm s})^\\sharp\\), a norm-convergent series of singular positive functionals (Lemma 8.1), so \\(\\omega\\) is singular and positive. By Zorn's lemma choose a maximal family \\((p_i)_{i\\in I}\\) of mutually orthogonal nonzero projections under \\(p\\) with \\(\\omega(p_i)=0\\). Then \\(\\sum_ip_i=p\\): otherwise \\(p-\\sum_ip_i\\) majorizes a nonzero projection on which \\(\\omega\\) vanishes (background fact 23), against maximality. For each \\(k\\) and \\(i\\), \\((\\varphi_k^{\\rm s})^\\sharp(p_i)\\le2^k\\omega(p_i)=0\\), so \\(\\varphi_k^{\\rm s}(p_i)=0\\) (Lemma 8.1(3)) and \\(\\varphi_k(p_i)=\\varphi_k^{\\rm n}(p_i)\\).\n\nThe map \\(\\rho:\\ell^\\infty(I)\\to M\\), \\(\\rho(x)=\\sum_ix(i)p_i\\) (a strong sum), is a contractive \\(*\\)-homomorphism. Put \\(\\mu_k=\\varphi_k\\circ\\rho\\). Then \\(\\|\\mu_k\\|\\le c\\), and \\(\\mu_k(E)=\\varphi_k(\\sum_{i\\in E}p_i)\\to0\\) for every \\(E\\subseteq I\\). By Phillips's lemma,\n\\[\n\\sum_i|\\varphi_k^{\\rm n}(p_i)|=\\sum_i|\\mu_k(\\{i\\})|\\to0 .\n\\]\nAs \\(\\varphi_k^{\\rm n}\\) is normal, it is completely additive, so \\(|\\varphi_k^{\\rm n}(p)|=|\\sum_i\\varphi_k^{\\rm n}(p_i)|\\le\\sum_i|\\varphi_k^{\\rm n}(p_i)|\\to0\\). \\(\\square\\)\n\n**Corollary 8.3** (Weak sequential completeness).\n\n1. The predual \\(M_*\\) of a von Neumann algebra is weakly sequentially complete: every weakly Cauchy sequence in \\(M_*\\) converges weakly to an element of \\(M_*\\).\n2. The dual \\(A^*\\) of a \\(C^*\\)-algebra is weakly sequentially complete.\n\n**Proof.** (1) If \\((\\varphi_k)\\) is weakly Cauchy, \\(\\varphi_k(x)\\) converges for every \\(x\\in M=(M_*)^*\\). By uniform boundedness the limit \\(\\varphi(x)\\) defines \\(\\varphi\\in M^*\\), and \\(\\varphi_k\\to\\varphi\\) in \\(\\sigma(M^*,M)\\). By Theorem 8.2, \\(\\varphi^{\\rm s}=\\lim\\varphi_k^{\\rm s}=0\\), so \\(\\varphi\\in M_*\\). (2) \\(A^*=\\tilde A_*\\) and \\((\\tilde A_*)^*=\\tilde A\\). \\(\\square\\)\n\n**Example 8.4** (Sequences cannot be replaced by nets). Let \\(M\\) be infinite-dimensional. It has a non-normal state (background fact 23), whose singular part, normalized, is a singular state \\(\\chi\\). The normal states are weak\\(^*\\) dense in the state space. Indeed, otherwise the separation theorem in \\((M^*,\\sigma(M^*,M))\\), whose dual is \\(M\\), gives a self-adjoint \\(k\\in M\\) and a state \\(\\chi'\\) with \\(\\chi'(k)>\\sup\\{\\omega(k):\\omega\\text{ a normal state}\\}\\). The right side is at least \\(\\sup_{\\|\\zeta\\|=1}\\langle k\\zeta,\\zeta\\rangle=\\max\\sigma(k)\\), computed in a faithful normal representation; and \\(\\chi'(k)\\le\\max\\sigma(k)\\) because \\(k\\le\\max\\sigma(k)1\\). This is a contradiction. So there is a net of normal states \\(\\omega_i\\to\\chi\\). Their normal parts are \\(\\omega_i\\), which tend to \\(\\chi\\), not to \\(\\chi^{\\rm n}=0\\). In particular the net \\((\\omega_i)\\) is weakly Cauchy in \\(M_*\\) without a weak limit in \\(M_*\\).\n\nThe singular functionals form a norm-closed subspace. Its weak\\(^*\\) closure can be large, but countable subsets cannot escape from it.\n\n**Proposition 8.5.** The \\(\\sigma(M^*,M)\\)-closure of any countable set of singular functionals consists of singular functionals.\n\n**Proof.** If the countable set contains zero, handle that singular point separately; a finite union with the closed singleton \\(\\{0\\}\\) has the corresponding union of closures. An empty set needs no argument. Enumerate the remaining nonzero elements as \\(\\{\\varphi_k\\}\\subseteq M_*^\\perp\\), repeating elements if the nonempty set is finite, and put \\(\\omega=\\sum_k2^{-k}\\varphi_k^\\sharp/\\|\\varphi_k\\|\\), a singular positive functional. If \\(\\omega(q)=0\\) for a projection \\(q\\), then \\(\\varphi_k(q)=0\\) for all \\(k\\) (Lemma 8.1(3)). Let \\(\\varphi\\) lie in the weak\\(^*\\) closure of \\(\\{\\varphi_k\\}\\), and let \\(p\\) be a projection. The functional \\(\\omega+(\\varphi^{\\rm s})^\\sharp\\) is singular and positive. As in the proof of Theorem 8.2, take a maximal orthogonal family \\((p_i)\\) of nonzero projections under \\(p\\) on which it vanishes; then \\(\\sum_ip_i=p\\). For finite \\(F\\) put \\(p_F=\\sum_{i\\in F}p_i\\). Then \\(\\omega(p_F)=0\\), so \\(\\varphi_k(p_F)=0\\) for all \\(k\\); since \\(\\psi\\mapsto\\psi(p_F)\\) is weak\\(^*\\) continuous and vanishes on the set \\(\\{\\varphi_k\\}\\), \\(\\varphi(p_F)=0\\). Also \\(\\varphi^{\\rm s}(p_F)=0\\). So \\(\\varphi^{\\rm n}(p_F)=0\\), and normality gives \\(\\varphi^{\\rm n}(p)=\\lim_F\\varphi^{\\rm n}(p_F)=0\\). Thus \\(\\varphi^{\\rm n}\\) vanishes on all projections, so \\(\\varphi^{\\rm n}=0\\) and \\(\\varphi\\) is singular. \\(\\square\\)\n\n**Example 8.6** (Countability is needed). Let \\(M=L^\\infty[0,1]\\) with Lebesgue measure. We show that the singular functionals are weak\\(^*\\) dense in \\(M^*\\). By background fact 7 it suffices to show that no \\(x\\ne0\\) in \\(M\\) is annihilated by all singular functionals.\n\nFirst, for every measurable \\(E\\) of positive measure there is a singular state \\(\\psi\\) of \\(M\\) with \\(\\psi(1_E)=1\\). The algebra \\(N=1_EM\\cong L^\\infty(E)\\) is infinite-dimensional: \\(t\\mapsto\\mu(E\\cap[0,t])\\) is continuous because its increments have absolute value at most the length of the interval. Bisect its positive total mass repeatedly by the intermediate value property, obtaining infinitely many disjoint subsets of positive measure and hence independent indicator functions; so it has a singular state \\(\\psi_1\\) (as in Example 8.4). Put \\(\\psi(y)=\\psi_1(y1_E)\\). This is a state of \\(M\\) with \\(\\psi(1_E)=1\\). It is singular by background fact 23: if \\(q\\ne0\\) is a projection of \\(M\\) and \\(q1_E=0\\), then \\(\\psi(q)=0\\); otherwise \\(q1_E\\) is a nonzero projection of \\(N\\), which majorizes a nonzero projection \\(q_0\\) of \\(N\\) with \\(\\psi_1(q_0)=0\\), and \\(q_0\\le q\\), \\(\\psi(q_0)=0\\).\n\nNow let \\(x\\ne0\\). Choose \\(\\varepsilon>0\\) such that \\(\\{|x|\\ge\\varepsilon\\}\\) has positive measure. Cover the bounded annulus \\(\\{\\varepsilon\\le|z|\\le\\|x\\|_\\infty\\}\\) by finitely many disks of radius \\(\\varepsilon/4\\), with centres \\(c\\) in the annulus. At least one preimage of a disk meets that set in positive measure. For its nonzero centre \\(c\\), \\(E=\\{t:|x(t)-c|\\le|c|/2\\}\\) therefore has positive measure. With \\(\\psi\\) as above, \\(\\psi(x)=\\psi(x1_E)\\) (Cauchy–Schwarz, as \\(\\psi(1-1_E)=0\\)), and \\(|\\psi(x1_E)-c|=|\\psi((x-c)1_E)|\\le|c|/2\\). So \\(\\psi(x)\\ne0\\). Hence Lebesgue measure, a normal state, is a weak\\(^*\\) limit of a net of singular functionals, and Proposition 8.5 fails for uncountable sets.\n\nIn atomic algebras, such as \\(\\ell^\\infty\\) and \\(B(H)\\), the singular functionals do form a weak\\(^*\\) closed subspace.\n\n**Proposition 8.7.** Let \\(M\\) be atomic, that is, every nonzero projection majorizes a minimal projection, and let \\(I_0\\) be the norm-closed two-sided ideal generated by the minimal projections. Then \\(\\varphi\\in M^*\\) is singular exactly when \\(\\varphi(I_0)=0\\). So the singular functionals form a weak\\(^*\\) closed subspace, and Proposition 8.5 holds for every set of singular functionals. For \\(M=\\ell^\\infty\\), \\(I_0=c_0\\); for \\(M=B(H)\\), \\(I_0\\) is the algebra of compact operators.\n\n**Proof.** Let \\(\\psi\\in M^*\\) be singular and positive, and \\(f\\) a minimal projection. The only nonzero projection under \\(f\\) is \\(f\\), so \\(\\psi(f)=0\\) by background fact 23. Let \\(x\\in M\\). Since \\(fx^*xf\\in fMf=\\mathbb Cf\\), \\(fx^*xf=\\mu f\\) with \\(\\mu=\\|xf\\|^2\\). If \\(\\mu>0\\), \\(w=\\mu^{-1/2}xf\\) is a partial isometry with \\(w^*w=f\\), and \\(f'=ww^*\\) is minimal, since \\(f'Mf'=wfMfw^*=\\mathbb Cf'\\); so \\(\\psi(xfx^*)=\\mu\\psi(f')=0\\). This also holds if \\(\\mu=0\\). For \\(y\\in M\\), \\((xfy)(xfy)^*\\le\\|y\\|^2xfx^*\\), and the Cauchy–Schwarz inequality gives \\(|\\psi(xfy)|^2\\le\\psi(1)\\,\\psi((xfy)(xfy)^*)=0\\). So \\(\\psi\\) vanishes on \\(I_0\\). Singular functionals are linear combinations of positive singular ones (background fact 22), so they all vanish on \\(I_0\\).\n\nConversely, let \\(\\varphi(I_0)=0\\). Then \\(\\varphi^{\\rm s}(I_0)=0\\) by the first part, so \\(\\varphi^{\\rm n}(I_0)=0\\). Write \\(1=\\sum_jf_j\\) with mutually orthogonal minimal projections (background fact 26). The finite partial sums \\(e_F=\\sum_{j\\in F}f_j\\) lie in \\(I_0\\) and increase to \\(1\\), so for \\(x\\in M\\), \\(e_Fxe_F\\in I_0\\) and \\(e_Fxe_F\\to x\\) \\(\\sigma\\)-weakly (background facts 15 and 16). Hence \\(\\varphi^{\\rm n}(x)=\\lim_F\\varphi^{\\rm n}(e_Fxe_F)=0\\), and \\(\\varphi\\) is singular. So the singular functionals are the annihilator of \\(I_0\\), which is weak\\(^*\\) closed.\n\nIn \\(\\ell^\\infty\\) the minimal projections are the coordinate projections, and they generate \\(c_0\\). In \\(B(H)\\) they are the rank-one projections, and they generate the closure of the finite-rank operators, which is the algebra of compact operators. \\(\\square\\)\n\n",
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      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "8. Normal and singular parts along weak\\(^*\\) convergent sequences",
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      "full_conditions_and_proof": "## 8. Normal and singular parts along weak\\(^*\\) convergent sequences\n\nFor the rest of the lesson we attach to each functional one positive functional that controls it.\n\n**Lemma 8.1.** For \\(\\varphi\\in M^*\\) put \\(\\varphi^\\sharp=|\\varphi|+|\\varphi^*|\\), the absolute values being taken in the \\(C^*\\)-algebra \\(M\\) (Theorem 2.7).\n\n1. \\(\\varphi^\\sharp\\ge0\\), \\(\\|\\varphi^\\sharp\\|=2\\|\\varphi\\|\\), and \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(xx^*)\\), \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(x^*x)\\).\n2. If \\(\\varphi\\) is normal, so is \\(\\varphi^\\sharp\\), and \\(\\varphi(x)=\\varphi(gxg)\\) with \\(g=s(\\varphi^\\sharp)\\). If \\(\\varphi\\) is singular, so is \\(\\varphi^\\sharp\\).\n3. If \\(p\\) is a projection with \\(\\varphi^\\sharp(p)=0\\), then \\(\\varphi(p)=0\\).\n\n**Proof.** (1) follows from (2.4) and Theorem 2.7, the second inequality by applying the first to \\(\\varphi^*\\) and \\(x^*\\). (2) If \\(\\varphi\\) is normal, \\(|\\varphi|\\) and \\(|\\varphi^*|\\) are normal by Corollary 3.2. Their supports are \\(s_r(\\varphi)\\) and \\(s_l(\\varphi)\\) (Theorem 2.2), both under \\(g\\), so \\[\n\\begin{gathered}\n\\varphi(gxg)\\\\\n=\\varphi(s_r(\\varphi)\\,gxg\\,s_l(\\varphi))\\\\\n=\\varphi(s_r(\\varphi)\\,x\\,s_l(\\varphi))\\\\\n=\\varphi(x)\n\\end{gathered}\n\\] by Lemma 1.4(2). If \\(\\varphi\\) is singular, then \\(\\varphi\\in M^*(1-z_0)\\) (background fact 22), and so is \\(\\varphi^*\\) since \\(z_0\\) is central. The space \\(M^*(1-z_0)\\) is invariant under \\(M^{**}\\), so \\(|\\varphi|=v^*\\varphi\\) and \\(|\\varphi^*|\\) lie in it. (3) By (1), \\(|\\varphi(p)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(p)=0\\). \\(\\square\\)\n\n**Theorem 8.2.** Let \\((\\varphi_k)\\) be a sequence in \\(M^*\\) that converges to \\(\\varphi\\) in \\(\\sigma(M^*,M)\\). Then \\(\\varphi_k^{\\rm n}\\to\\varphi^{\\rm n}\\) and \\(\\varphi_k^{\\rm s}\\to\\varphi^{\\rm s}\\) in \\(\\sigma(M^*,M)\\).\n\n**Proof.** The splitting is linear, so we may assume \\(\\varphi=0\\). By background fact 2, \\(c=\\sup_k\\|\\varphi_k\\|<\\infty\\), and \\(\\|\\varphi_k^{\\rm n}\\|,\\|\\varphi_k^{\\rm s}\\|\\le c\\). Since \\(\\varphi_k^{\\rm s}=\\varphi_k-\\varphi_k^{\\rm n}\\), it suffices to show \\(\\varphi_k^{\\rm n}\\to0\\). The sequence is bounded and the span of the projections is norm dense in \\(M\\) (background fact 16), so it suffices to show \\(\\varphi_k^{\\rm n}(p)\\to0\\) for each projection \\(p\\ne0\\).\n\nLet \\(\\omega=\\sum_k2^{-k}(\\varphi_k^{\\rm s})^\\sharp\\), a norm-convergent series of singular positive functionals (Lemma 8.1), so \\(\\omega\\) is singular and positive. By Zorn's lemma choose a maximal family \\((p_i)_{i\\in I}\\) of mutually orthogonal nonzero projections under \\(p\\) with \\(\\omega(p_i)=0\\). Then \\(\\sum_ip_i=p\\): otherwise \\(p-\\sum_ip_i\\) majorizes a nonzero projection on which \\(\\omega\\) vanishes (background fact 23), against maximality. For each \\(k\\) and \\(i\\), \\((\\varphi_k^{\\rm s})^\\sharp(p_i)\\le2^k\\omega(p_i)=0\\), so \\(\\varphi_k^{\\rm s}(p_i)=0\\) (Lemma 8.1(3)) and \\(\\varphi_k(p_i)=\\varphi_k^{\\rm n}(p_i)\\).\n\nThe map \\(\\rho:\\ell^\\infty(I)\\to M\\), \\(\\rho(x)=\\sum_ix(i)p_i\\) (a strong sum), is a contractive \\(*\\)-homomorphism. Put \\(\\mu_k=\\varphi_k\\circ\\rho\\). Then \\(\\|\\mu_k\\|\\le c\\), and \\(\\mu_k(E)=\\varphi_k(\\sum_{i\\in E}p_i)\\to0\\) for every \\(E\\subseteq I\\). By Phillips's lemma,\n\\[\n\\sum_i|\\varphi_k^{\\rm n}(p_i)|=\\sum_i|\\mu_k(\\{i\\})|\\to0 .\n\\]\nAs \\(\\varphi_k^{\\rm n}\\) is normal, it is completely additive, so \\(|\\varphi_k^{\\rm n}(p)|=|\\sum_i\\varphi_k^{\\rm n}(p_i)|\\le\\sum_i|\\varphi_k^{\\rm n}(p_i)|\\to0\\). \\(\\square\\)\n\n**Corollary 8.3** (Weak sequential completeness).\n\n1. The predual \\(M_*\\) of a von Neumann algebra is weakly sequentially complete: every weakly Cauchy sequence in \\(M_*\\) converges weakly to an element of \\(M_*\\).\n2. The dual \\(A^*\\) of a \\(C^*\\)-algebra is weakly sequentially complete.\n\n**Proof.** (1) If \\((\\varphi_k)\\) is weakly Cauchy, \\(\\varphi_k(x)\\) converges for every \\(x\\in M=(M_*)^*\\). By uniform boundedness the limit \\(\\varphi(x)\\) defines \\(\\varphi\\in M^*\\), and \\(\\varphi_k\\to\\varphi\\) in \\(\\sigma(M^*,M)\\). By Theorem 8.2, \\(\\varphi^{\\rm s}=\\lim\\varphi_k^{\\rm s}=0\\), so \\(\\varphi\\in M_*\\). (2) \\(A^*=\\tilde A_*\\) and \\((\\tilde A_*)^*=\\tilde A\\). \\(\\square\\)\n\n**Example 8.4** (Sequences cannot be replaced by nets). Let \\(M\\) be infinite-dimensional. It has a non-normal state (background fact 23), whose singular part, normalized, is a singular state \\(\\chi\\). The normal states are weak\\(^*\\) dense in the state space. Indeed, otherwise the separation theorem in \\((M^*,\\sigma(M^*,M))\\), whose dual is \\(M\\), gives a self-adjoint \\(k\\in M\\) and a state \\(\\chi'\\) with \\(\\chi'(k)>\\sup\\{\\omega(k):\\omega\\text{ a normal state}\\}\\). The right side is at least \\(\\sup_{\\|\\zeta\\|=1}\\langle k\\zeta,\\zeta\\rangle=\\max\\sigma(k)\\), computed in a faithful normal representation; and \\(\\chi'(k)\\le\\max\\sigma(k)\\) because \\(k\\le\\max\\sigma(k)1\\). This is a contradiction. So there is a net of normal states \\(\\omega_i\\to\\chi\\). Their normal parts are \\(\\omega_i\\), which tend to \\(\\chi\\), not to \\(\\chi^{\\rm n}=0\\). In particular the net \\((\\omega_i)\\) is weakly Cauchy in \\(M_*\\) without a weak limit in \\(M_*\\).\n\nThe singular functionals form a norm-closed subspace. Its weak\\(^*\\) closure can be large, but countable subsets cannot escape from it.\n\n**Proposition 8.5.** The \\(\\sigma(M^*,M)\\)-closure of any countable set of singular functionals consists of singular functionals.\n\n**Proof.** If the countable set contains zero, handle that singular point separately; a finite union with the closed singleton \\(\\{0\\}\\) has the corresponding union of closures. An empty set needs no argument. Enumerate the remaining nonzero elements as \\(\\{\\varphi_k\\}\\subseteq M_*^\\perp\\), repeating elements if the nonempty set is finite, and put \\(\\omega=\\sum_k2^{-k}\\varphi_k^\\sharp/\\|\\varphi_k\\|\\), a singular positive functional. If \\(\\omega(q)=0\\) for a projection \\(q\\), then \\(\\varphi_k(q)=0\\) for all \\(k\\) (Lemma 8.1(3)). Let \\(\\varphi\\) lie in the weak\\(^*\\) closure of \\(\\{\\varphi_k\\}\\), and let \\(p\\) be a projection. The functional \\(\\omega+(\\varphi^{\\rm s})^\\sharp\\) is singular and positive. As in the proof of Theorem 8.2, take a maximal orthogonal family \\((p_i)\\) of nonzero projections under \\(p\\) on which it vanishes; then \\(\\sum_ip_i=p\\). For finite \\(F\\) put \\(p_F=\\sum_{i\\in F}p_i\\). Then \\(\\omega(p_F)=0\\), so \\(\\varphi_k(p_F)=0\\) for all \\(k\\); since \\(\\psi\\mapsto\\psi(p_F)\\) is weak\\(^*\\) continuous and vanishes on the set \\(\\{\\varphi_k\\}\\), \\(\\varphi(p_F)=0\\). Also \\(\\varphi^{\\rm s}(p_F)=0\\). So \\(\\varphi^{\\rm n}(p_F)=0\\), and normality gives \\(\\varphi^{\\rm n}(p)=\\lim_F\\varphi^{\\rm n}(p_F)=0\\). Thus \\(\\varphi^{\\rm n}\\) vanishes on all projections, so \\(\\varphi^{\\rm n}=0\\) and \\(\\varphi\\) is singular. \\(\\square\\)\n\n**Example 8.6** (Countability is needed). Let \\(M=L^\\infty[0,1]\\) with Lebesgue measure. We show that the singular functionals are weak\\(^*\\) dense in \\(M^*\\). By background fact 7 it suffices to show that no \\(x\\ne0\\) in \\(M\\) is annihilated by all singular functionals.\n\nFirst, for every measurable \\(E\\) of positive measure there is a singular state \\(\\psi\\) of \\(M\\) with \\(\\psi(1_E)=1\\). The algebra \\(N=1_EM\\cong L^\\infty(E)\\) is infinite-dimensional: \\(t\\mapsto\\mu(E\\cap[0,t])\\) is continuous because its increments have absolute value at most the length of the interval. Bisect its positive total mass repeatedly by the intermediate value property, obtaining infinitely many disjoint subsets of positive measure and hence independent indicator functions; so it has a singular state \\(\\psi_1\\) (as in Example 8.4). Put \\(\\psi(y)=\\psi_1(y1_E)\\). This is a state of \\(M\\) with \\(\\psi(1_E)=1\\). It is singular by background fact 23: if \\(q\\ne0\\) is a projection of \\(M\\) and \\(q1_E=0\\), then \\(\\psi(q)=0\\); otherwise \\(q1_E\\) is a nonzero projection of \\(N\\), which majorizes a nonzero projection \\(q_0\\) of \\(N\\) with \\(\\psi_1(q_0)=0\\), and \\(q_0\\le q\\), \\(\\psi(q_0)=0\\).\n\nNow let \\(x\\ne0\\). Choose \\(\\varepsilon>0\\) such that \\(\\{|x|\\ge\\varepsilon\\}\\) has positive measure. Cover the bounded annulus \\(\\{\\varepsilon\\le|z|\\le\\|x\\|_\\infty\\}\\) by finitely many disks of radius \\(\\varepsilon/4\\), with centres \\(c\\) in the annulus. At least one preimage of a disk meets that set in positive measure. For its nonzero centre \\(c\\), \\(E=\\{t:|x(t)-c|\\le|c|/2\\}\\) therefore has positive measure. With \\(\\psi\\) as above, \\(\\psi(x)=\\psi(x1_E)\\) (Cauchy–Schwarz, as \\(\\psi(1-1_E)=0\\)), and \\(|\\psi(x1_E)-c|=|\\psi((x-c)1_E)|\\le|c|/2\\). So \\(\\psi(x)\\ne0\\). Hence Lebesgue measure, a normal state, is a weak\\(^*\\) limit of a net of singular functionals, and Proposition 8.5 fails for uncountable sets.\n\nIn atomic algebras, such as \\(\\ell^\\infty\\) and \\(B(H)\\), the singular functionals do form a weak\\(^*\\) closed subspace.\n\n**Proposition 8.7.** Let \\(M\\) be atomic, that is, every nonzero projection majorizes a minimal projection, and let \\(I_0\\) be the norm-closed two-sided ideal generated by the minimal projections. Then \\(\\varphi\\in M^*\\) is singular exactly when \\(\\varphi(I_0)=0\\). So the singular functionals form a weak\\(^*\\) closed subspace, and Proposition 8.5 holds for every set of singular functionals. For \\(M=\\ell^\\infty\\), \\(I_0=c_0\\); for \\(M=B(H)\\), \\(I_0\\) is the algebra of compact operators.\n\n**Proof.** Let \\(\\psi\\in M^*\\) be singular and positive, and \\(f\\) a minimal projection. The only nonzero projection under \\(f\\) is \\(f\\), so \\(\\psi(f)=0\\) by background fact 23. Let \\(x\\in M\\). Since \\(fx^*xf\\in fMf=\\mathbb Cf\\), \\(fx^*xf=\\mu f\\) with \\(\\mu=\\|xf\\|^2\\). If \\(\\mu>0\\), \\(w=\\mu^{-1/2}xf\\) is a partial isometry with \\(w^*w=f\\), and \\(f'=ww^*\\) is minimal, since \\(f'Mf'=wfMfw^*=\\mathbb Cf'\\); so \\(\\psi(xfx^*)=\\mu\\psi(f')=0\\). This also holds if \\(\\mu=0\\). For \\(y\\in M\\), \\((xfy)(xfy)^*\\le\\|y\\|^2xfx^*\\), and the Cauchy–Schwarz inequality gives \\(|\\psi(xfy)|^2\\le\\psi(1)\\,\\psi((xfy)(xfy)^*)=0\\). So \\(\\psi\\) vanishes on \\(I_0\\). Singular functionals are linear combinations of positive singular ones (background fact 22), so they all vanish on \\(I_0\\).\n\nConversely, let \\(\\varphi(I_0)=0\\). Then \\(\\varphi^{\\rm s}(I_0)=0\\) by the first part, so \\(\\varphi^{\\rm n}(I_0)=0\\). Write \\(1=\\sum_jf_j\\) with mutually orthogonal minimal projections (background fact 26). The finite partial sums \\(e_F=\\sum_{j\\in F}f_j\\) lie in \\(I_0\\) and increase to \\(1\\), so for \\(x\\in M\\), \\(e_Fxe_F\\in I_0\\) and \\(e_Fxe_F\\to x\\) \\(\\sigma\\)-weakly (background facts 15 and 16). Hence \\(\\varphi^{\\rm n}(x)=\\lim_F\\varphi^{\\rm n}(e_Fxe_F)=0\\), and \\(\\varphi\\) is singular. So the singular functionals are the annihilator of \\(I_0\\), which is weak\\(^*\\) closed.\n\nIn \\(\\ell^\\infty\\) the minimal projections are the coordinate projections, and they generate \\(c_0\\). In \\(B(H)\\) they are the rank-one projections, and they generate the closure of the finite-rank operators, which is the algebra of compact operators. \\(\\square\\)\n\n",
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      "id": "OA-FND-PD-15",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "9. The unit ball as a complete metric space",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 9. The unit ball as a complete metric space\n\nWhen \\(M\\) has a faithful normal state, the \\(\\sigma\\)-strong topology on the unit ball comes from a complete metric. This makes the Baire category theorem available in Section 10. The converse holds as well.\n\n**Proposition 9.1.** Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(\\omega\\) be a faithful positive normal functional. Then\n\\[\n\\begin{gathered}\nd(x,y)\\\\\n=\\omega\\big((x-y)^*(x-y)\\big)^{1/2},\\\\\nd^\\#(x,y)\\\\\n=\\omega\\big((x-y)^*(x-y)\\\\\n+(x-y)(x-y)^*\\big)^{1/2}\n\\end{gathered}\n\\]\nare metrics on \\(M_1\\) that define its \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, and \\(M_1\\) is complete for both.\n2. Conversely, if the \\(\\sigma\\)-strong topology, or the \\(\\sigma\\)-strong\\(^*\\) topology, on \\(M_1\\) is metrizable, then \\(M\\) has a faithful positive normal functional.\n\n**Proof.** (1) Let \\((\\pi,H,\\xi)\\) be the cyclic representation of \\(\\omega\\). It is faithful, since \\(\\pi(x)=0\\) gives \\(\\omega(x^*x)=0\\), and normal (background fact 24). So \\(\\pi(M)\\) is a von Neumann algebra, and by background fact 15 we may assume \\(M\\subseteq B(H)\\), \\(\\omega=\\omega_\\xi\\) and \\([M\\xi]=H\\). The vector \\(\\xi\\) is separating: \\(x\\xi=0\\) gives \\(\\omega(x^*x)=0\\), so \\(x=0\\). Hence \\(\\xi\\) is cyclic for \\(M'\\): the projection \\(p\\) onto \\([M'\\xi]\\) lies in \\(M''=M\\), and \\(\\omega(1-p)=\\|(1-p)\\xi\\|^2=0\\) forces \\(p=1\\). Now \\(d(x,y)=\\|(x-y)\\xi\\|\\) is a metric on \\(M\\), as \\(\\xi\\) is separating.\n\nIf \\(x_i\\to x\\) \\(\\sigma\\)-strongly then \\(d(x_i,x)\\to0\\), since \\(d(\\cdot,x)\\) is one of the defining seminorms. Conversely let \\(x_i,x\\in M_1\\) with \\(d(x_i,x)\\to0\\). For \\(a'\\in M'\\), \\[\n\\begin{gathered}\n\\|(x_i-x)a'\\xi\\|\\\\\n=\\|a'(x_i-x)\\xi\\|\\\\\n\\le\\|a'\\|\\,d(x_i,x)\\to0.\n\\end{gathered}\n\\] The vectors \\(a'\\xi\\) are dense and \\(\\|x_i-x\\|\\le2\\), so \\(x_i\\to x\\) strongly, hence \\(\\sigma\\)-strongly (background fact 15). The same argument applied to \\(x_i\\) and \\(x_i^*\\) shows that \\(d^\\#\\) defines the \\(\\sigma\\)-strong\\(^*\\) topology on \\(M_1\\).\n\nCompleteness: \\(x\\mapsto x\\xi\\) is an isometry of \\((M_1,d)\\) onto \\(M_1\\xi\\subseteq H\\). The set \\(M_1\\) is \\(\\sigma\\)-weakly compact and \\(x\\mapsto x\\xi\\) is continuous from the \\(\\sigma\\)-weak topology to the weak topology of \\(H\\). So \\(M_1\\xi\\) is weakly compact and convex, hence norm closed, hence complete. For \\(d^\\#\\) use \\(x\\mapsto(x\\xi,x^*\\xi)\\in H\\oplus H\\), noting that the involution is \\(\\sigma\\)-weakly continuous.\n\n(2) If \\(M=0\\), the zero functional is faithful and the assertion holds. Otherwise suppose the \\(\\sigma\\)-strong topology on \\(M_1\\) has a countable base \\(B_1,B_2,\\dots\\) of neighbourhoods of \\(0\\). Each \\(B_n\\) contains a basic set \\(\\{x\\in M_1:\\sum_{j\\le m_n}\\omega_{n,j}(x^*x)<\\delta_n\\}\\) with \\(\\omega_{n,j}\\in M_*^+\\), which we may take nonzero. Enumerate all \\(\\omega_{n,j}\\) as \\(\\omega_1,\\omega_2,\\dots\\) and put \\(\\omega=\\sum_l2^{-l}\\omega_l/\\|\\omega_l\\|\\). If \\(p\\) is a projection with \\(\\omega(p)=0\\), then \\(\\omega_l(p^*p)=0\\) for all \\(l\\), so \\(p\\) lies in every \\(B_n\\). The topology is Hausdorff, so \\(\\bigcap_nB_n=\\{0\\}\\) and \\(p=0\\). A positive normal functional that vanishes on no nonzero projection is faithful: if \\(\\omega(y)=0\\) with \\(y\\ge0\\), the spectral projections \\(q=1_{[\\varepsilon,\\infty)}(y)\\le\\varepsilon^{-1}y\\) satisfy \\(\\omega(q)=0\\), so \\(q=0\\) for every \\(\\varepsilon>0\\) and \\(y=0\\). For the \\(\\sigma\\)-strong\\(^*\\) topology use the seminorms \\(\\omega(x^*x+xx^*)^{1/2}\\) in the same way. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-16",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "10. Weak compactness in the predual",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 10. Weak compactness in the predual\n\nWe now characterize the relatively weakly compact subsets of a predual. For \\(M=L^\\infty(\\Gamma,\\mu)\\), whose predual is \\(L^1(\\Gamma,\\mu)\\) (Example 2.6), condition (4) of the theorem below, tested on indicator functions \\(a=1_E\\), says that there is one \\(g\\in L^1_+\\) such that the integrals \\(\\int_Eh\\,d\\mu\\), \\(h\\in K\\), are uniformly small whenever \\(\\int_Eg\\,d\\mu\\) is small. In general the control is by one positive normal functional, applied to \\(a^*a+aa^*\\).\n\nWe use the following Banach-space fact twice, here and in Section 14.\n\n**Lemma 10.1.** A bounded subset \\(K\\) of a Banach space \\(X\\) is relatively weakly compact exactly when its \\(\\sigma(X^{**},X^*)\\)-closure in \\(X^{**}\\) lies in \\(X\\).\n\n**Proof.** The weak topology of \\(X\\) is the restriction of \\(\\sigma(X^{**},X^*)\\). If \\(K\\) is relatively weakly compact, its weak closure \\(K'\\) in \\(X\\) is weakly compact, hence \\(\\sigma(X^{**},X^*)\\)-compact and closed in \\(X^{**}\\); so the closure of \\(K\\) in \\(X^{**}\\) lies in \\(K'\\subseteq X\\). Conversely, if the closure \\(\\bar K\\) of \\(K\\) in \\(X^{**}\\) lies in \\(X\\), it is \\(\\sigma(X^{**},X^*)\\)-compact by the Banach–Alaoglu theorem, since \\(K\\) is bounded; so \\(\\bar K\\) is a weakly compact subset of \\(X\\) containing \\(K\\). \\(\\square\\)\n\nFor \\(X=M_*\\) we have \\(X^*=M\\) and \\(X^{**}=M^*\\), so a bounded \\(K\\subseteq M_*\\) is relatively weakly compact exactly when its \\(\\sigma(M^*,M)\\)-closure consists of normal functionals. Relatively weakly compact sets are bounded (background fact 2). For a projection \\(p\\) we write \\((1-p)\\varphi(1-p)\\) for \\(x\\mapsto\\varphi((1-p)x(1-p))\\), as in (0.1).\n\n**Theorem 10.2** (Weak compactness in \\(M_*\\)). For a subset \\(K\\) of the predual of a von Neumann algebra \\(M\\), the following are equivalent.\n\n1. \\(K\\) is relatively weakly compact.\n2. For every abelian von Neumann subalgebra \\(\\mathcal A\\subseteq M\\), the set \\(K|_{\\mathcal A}=\\{\\varphi|_{\\mathcal A}:\\varphi\\in K\\}\\) is relatively weakly compact in \\(\\mathcal A_*\\).\n3. \\(K\\) is bounded, and \\(\\varphi(p_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) whenever the projections \\(p_n\\) decrease to \\(0\\).\n4. \\(K\\) is bounded, and there is \\(\\omega\\in M_*^+\\) with this property: for every \\(\\varepsilon>0\\) there is \\(\\delta>0\\) such that \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\) whenever \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\).\n5. \\(K\\) is bounded, and for every increasing net of projections \\((p_i)\\), \\(\\varphi(p_i)\\) converges uniformly in \\(\\varphi\\in K\\).\n6. \\(K\\) is bounded, and \\(\\|(1-p_i)\\varphi(1-p_i)\\|\\to0\\) uniformly in \\(\\varphi\\in K\\) for every increasing net of projections \\((p_i)\\) with \\(\\sup_ip_i=1\\).\n7. \\(K\\) is bounded, and \\(\\varphi(q_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) for every sequence \\((q_n)\\) of mutually orthogonal projections.\n\n\nWe prove the easy implications first; the implication (1)\\(\\Rightarrow\\)(4) needs two lemmas.\n\n**Proof of (1)\\(\\Rightarrow\\)(2).** Restriction \\(\\varphi\\mapsto\\varphi|_{\\mathcal A}\\) is continuous from \\(\\sigma(M_*,M)\\) to \\(\\sigma(\\mathcal A_*,\\mathcal A)\\), and continuous images of compact sets are compact. \\(\\square\\)\n\n**Proof of (2)\\(\\Rightarrow\\)(1).** Every self-adjoint \\(h\\in M\\) lies in the abelian von Neumann algebra generated by \\(h\\) and \\(1\\). So \\(\\{\\varphi(h):\\varphi\\in K\\}\\) is bounded, hence so is \\(\\{\\varphi(x):\\varphi\\in K\\}\\) for every \\(x=h+ik\\), and \\(K\\) is bounded by uniform boundedness. Let \\(\\varphi\\) lie in the \\(\\sigma(M^*,M)\\)-closure of \\(K\\). For an abelian von Neumann subalgebra \\(\\mathcal A\\), \\(\\varphi|_{\\mathcal A}\\) lies in the \\(\\sigma(\\mathcal A^*,\\mathcal A)\\)-closure of \\(K|_{\\mathcal A}\\), which consists of normal functionals by Lemma 10.1. So \\(\\varphi\\) is normal on every abelian von Neumann subalgebra, hence normal (background fact 23). By Lemma 10.1, \\(K\\) is relatively weakly compact. \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(6)\\(\\Rightarrow\\)(5)\\(\\Rightarrow\\)(3).** (4)\\(\\Rightarrow\\)(6): let \\(\\varepsilon,\\delta\\) be as in (4) and \\(p_i\\uparrow1\\). By normality of \\(\\omega\\) there is \\(i_0\\) with \\(\\omega(1-p_i)<\\delta/2\\) for \\(i\\ge i_0\\). If \\(a\\in M_1\\) and \\(a=(1-p_i)a(1-p_i)\\), then \\(a^*a\\le1-p_i\\) and \\(aa^*\\le1-p_i\\), so \\(\\omega(a^*a+aa^*)<\\delta\\) and \\(|\\varphi(a)|<\\varepsilon\\). Hence \\[\n\\begin{gathered}\n\\|(1-p_i)\\varphi(1-p_i)\\|\\\\\n=\\sup_{a\\in M_1}|\\varphi((1-p_i)a(1-p_i))|\\\\\n\\le\\varepsilon\n\\end{gathered}\n\\] for \\(i\\ge i_0\\) and all \\(\\varphi\\in K\\).\n\n(6)\\(\\Rightarrow\\)(5): let \\((p_i)\\) increase to \\(p\\). Then \\(q_i=p_i+(1-p)\\) increases to \\(1\\), and \\(1-q_i=p-p_i\\). By (6), \\[\n\\begin{gathered}\n|\\varphi(p)-\\varphi(p_i)|\\\\\n=|\\varphi(p-p_i)|\\\\\n\\le\\|(p-p_i)\\varphi(p-p_i)\\|\\to0\n\\end{gathered}\n\\] uniformly in \\(\\varphi\\in K\\).\n\n(5)\\(\\Rightarrow\\)(3): if \\(p_n\\downarrow0\\), then \\(1-p_n\\uparrow1\\). By (5), \\(\\varphi(1-p_n)\\) converges uniformly, and its pointwise limit is \\(\\varphi(1)\\) by normality. So \\(\\varphi(p_n)\\to0\\) uniformly. \\(\\square\\)\n\n**Proof of (3)\\(\\Leftrightarrow\\)(7).** (3)\\(\\Rightarrow\\)(7): for orthogonal \\((q_n)\\), \\(r_m=\\sum_{n\\ge m}q_n\\) decreases to \\(0\\), and \\(\\varphi(q_m)=\\varphi(r_m)-\\varphi(r_{m+1})\\). (7)\\(\\Rightarrow\\)(3): suppose \\(p_n\\downarrow0\\) but \\(\\sup_{\\varphi\\in K}|\\varphi(p_n)|\\not\\to0\\). Then there are \\(\\varepsilon>0\\), indices \\(n_1<n_2<\\cdots\\) and \\(\\varphi_j\\in K\\) with \\(|\\varphi_j(p_{n_j})|\\ge\\varepsilon\\). Since each \\(\\varphi_j\\) is normal, \\(\\varphi_j(p_n)\\to0\\) as \\(n\\to\\infty\\); choosing the indices and functionals inductively, we may assume \\(|\\varphi_j(p_{n_{j+1}})|<\\varepsilon/2\\). The projections \\(q_j=p_{n_j}-p_{n_{j+1}}\\) are mutually orthogonal and \\(|\\varphi_j(q_j)|>\\varepsilon/2\\), which contradicts (7). \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(3).** If \\(p_n\\downarrow0\\), then \\(\\omega(p_n^*p_n+p_np_n^*)=2\\omega(p_n)\\to0\\). \\(\\square\\)\n\n**Proof of (3)\\(\\Rightarrow\\)(1).** By the Eberlein–Šmulian theorem we only need a weak cluster point in \\(M_*\\) for each sequence \\((\\varphi_n)\\) in \\(K\\). Let \\(\\varphi\\in M^*\\) be a \\(\\sigma(M^*,M)\\)-cluster point, which exists by the Banach–Alaoglu theorem. On \\(M_*\\), \\(\\sigma(M^*,M)\\) is the weak topology, so it suffices to show that \\(\\varphi\\) is normal. Put \\(\\omega=\\sum_n2^{-n}\\varphi_n^\\sharp\\in M_*^+\\).\n\n(a) If \\(q\\) is a projection with \\(\\omega(q)=0\\), then \\(\\varphi_n(q)=0\\) for all \\(n\\) (Lemma 8.1(3)), hence \\(\\varphi(q)=0\\).\n\n(b) If projections \\(r_m\\) decrease to \\(0\\), then \\(|\\varphi(r_m)|\\le\\sup_n|\\varphi_n(r_m)|\\to0\\) by (3).\n\nLet \\((p_i)_{i\\in I}\\) be mutually orthogonal with sum \\(p\\). Since \\(\\omega\\) is normal, \\(\\sum_i\\omega(p_i)=\\omega(p)<\\infty\\), so \\(I_0=\\{i:\\omega(p_i)>0\\}\\) is countable. Every projection \\(q\\) under \\(\\sum_{i\\notin I_0}p_i\\) has \\(\\omega(q)=0\\), hence \\(\\varphi(q)=0\\) by (a). For \\(J\\subseteq I_0\\), enumerate \\(J=\\{j_1,j_2,\\dots\\}\\); the projections \\(r_m=\\sum_{k\\ge m}p_{j_k}\\) decrease to \\(0\\), so by (b), \\(\\varphi(\\sum_{i\\in J}p_i)=\\lim_m\\sum_{k<m}\\varphi(p_{j_k})\\). Applying this to \\(J=\\{i\\in I_0:\\operatorname{Re}\\varphi(p_i)\\ge0\\}\\) and to the three analogous sets shows that \\(\\sum_{i\\in I_0}|\\varphi(p_i)|<\\infty\\). Now let \\(F\\subseteq I\\) be finite. Then \\(p-\\sum_{i\\in F}p_i\\) is the sum of a projection under \\(\\sum_{i\\notin I_0}p_i\\) and of \\(\\sum_{i\\in I_0\\setminus F}p_i\\), so\n\\[\n\\begin{gathered}\n\\Big|\\varphi(p)-\\sum_{i\\in F}\\varphi(p_i)\\Big|\\\\\n=\\Big|\\sum_{i\\in I_0\\setminus F}\\varphi(p_i)\\Big|\\\\\n\\le\\sum_{i\\in I_0\\setminus F}|\\varphi(p_i)|,\n\\end{gathered}\n\\]\nwhich tends to \\(0\\) along the finite sets \\(F\\). So \\(\\varphi\\) is completely additive, hence normal (background fact 23). \\(\\square\\)\n\nThe remaining implication rests on the following two lemmas.\n\n**Lemma 10.3.** Let \\((\\varphi_k)\\) be a sequence in \\(M_*\\) converging weakly to \\(\\varphi_0\\in M_*\\), and let \\((a_n)\\) be a sequence in \\(M_1\\) converging \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\). Then \\(\\sup_k|\\varphi_k(a_n)|\\to0\\) as \\(n\\to\\infty\\).\n\n**Proof.** *Reduction.* Let \\(c=\\sup_k\\|\\varphi_k\\|<\\infty\\). Put \\(\\omega=\\varphi_0^\\sharp+\\sum_k2^{-k}\\varphi_k^\\sharp\\in M_*^+\\) and \\(e=s(\\omega)\\). For every \\(k\\ge0\\), \\(s(\\varphi_k^\\sharp)\\le e\\), so \\(\\varphi_k(x)=\\varphi_k(exe)\\) by Lemma 8.1(2). The restriction of \\(\\omega\\) to the von Neumann algebra \\(N=eMe\\) is faithful; the restrictions \\(\\varphi_k|_N\\) converge weakly to \\(\\varphi_0|_N\\); the elements \\(ea_ne\\in N_1\\) converge \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\); and \\(\\varphi_k(a_n)=\\varphi_k(ea_ne)\\). So it is enough to treat a faithful \\(\\omega\\). Let \\(d\\) be the metric of Proposition 9.1 for \\(\\omega\\).\n\n*Baire.* Put \\(\\psi_k=\\varphi_k-\\varphi_0\\), so \\(\\psi_k\\to0\\) weakly and \\(\\|\\psi_k\\|\\le2c\\). Fix \\(\\varepsilon>0\\) and let \\(S_m=\\{a\\in M_1:|\\psi_k(a)|\\le\\varepsilon\\text{ for all }k\\ge m\\}\\). Each \\(\\psi_k\\) is \\(\\sigma\\)-strongly continuous, hence \\(d\\)-continuous on \\(M_1\\), so \\(S_m\\) is \\(d\\)-closed; and \\(\\bigcup_mS_m=M_1\\). By Baire's theorem, some \\(S_{m_0}\\) contains \\(\\{a\\in M_1:d(a,a_0)<\\delta\\}\\) for some \\(a_0\\in M_1\\), \\(\\delta>0\\).\n\n*Cutting \\(a_n\\).* Let \\(h_n=a_n^*a_n+a_na_n^*\\), so \\(0\\le h_n\\le2\\) and \\(\\omega(h_n)\\to0\\). Let \\(p_n=1_{[0,\\varepsilon^2]}(h_n)\\), a spectral projection. Then \\(\\|p_nh_np_n\\|\\le\\varepsilon^2\\), which gives \\(\\|a_np_n\\|\\le\\varepsilon\\) and \\(\\|p_na_n\\|\\le\\varepsilon\\). Also \\(1-p_n\\le\\varepsilon^{-2}h_n\\), so \\(d(p_n,1)^2=\\omega(1-p_n)\\le\\varepsilon^{-2}\\omega(h_n)\\to0\\), and \\(p_n\\to1\\) \\(\\sigma\\)-strongly. Write\n\\[\n\\begin{gathered}\na_n\\\\\n=p_na_n+(1-p_n)a_np_n+c_n,\\\\\nc_n\\\\\n=(1-p_n)a_n(1-p_n).\n\\end{gathered}\n\\]\nThe first two terms have norm at most \\(\\varepsilon\\), so \\(|\\psi_k(a_n)|\\le4c\\varepsilon+|\\psi_k(c_n)|\\).\n\n*Comparing with \\(a_0\\).* Put \\(b_n=p_na_0p_n+c_n\\). It lies in \\(M_1\\): \\(b_n^*b_n=p_na_0^*p_na_0p_n+c_n^*c_n\\) is a sum of two positive contractions with orthogonal supports. As \\(n\\to\\infty\\), \\(p_na_0p_n\\to a_0\\) \\(\\sigma\\)-strongly (joint continuity of multiplication on bounded sets), and \\(d(c_n,0)^2=\\omega(c_n^*c_n)\\le\\omega(1-p_n)\\to0\\). So \\(d(p_na_0p_n,a_0)\\to0\\) and \\(d(b_n,a_0)\\to0\\), and there is \\(n_0\\) with both distances below \\(\\delta\\) for \\(n\\ge n_0\\). For \\(k\\ge m_0\\) and \\(n\\ge n_0\\),\n\\[\n\\begin{gathered}\n|\\psi_k(c_n)|\\\\\n=|\\psi_k(b_n)-\\psi_k(p_na_0p_n)|\\\\\n\\le2\\varepsilon ,\\\\\n\\text{so}\\\\\n|\\psi_k(a_n)|\\\\\n\\le(4c+2)\\varepsilon .\n\\end{gathered}\n\\]\nFor each of the finitely many \\(k<m_0\\), \\(\\psi_k(a_n)\\to0\\), because \\(\\psi_k\\) is normal and a bounded \\(\\sigma\\)-strongly\\(^*\\) null sequence is \\(\\sigma\\)-weakly null. Hence \\(\\limsup_n\\sup_k|\\psi_k(a_n)|\\le(4c+2)\\varepsilon\\) for every \\(\\varepsilon\\). Finally \\(\\varphi_k(a_n)=\\psi_k(a_n)+\\varphi_0(a_n)\\) and \\(\\varphi_0(a_n)\\to0\\). \\(\\square\\)\n\n**Lemma 10.4.** Let \\(K\\subseteq M_*\\) be relatively weakly compact and \\(\\varepsilon>0\\). There are a finite set \\(F\\subseteq K\\) and \\(\\delta>0\\) such that: if \\(a\\in M_1\\) and \\(\\psi^\\sharp(a^*a+aa^*)<\\delta\\) for all \\(\\psi\\in F\\), then \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\).\n\n**Proof.** If \\(K=\\varnothing\\), take \\(F=\\varnothing\\) and any \\(\\delta>0\\). If \\(K=\\{0\\}\\), the conclusion is also immediate. Otherwise \\(K\\) is bounded; scaling, we may assume \\(\\|\\varphi\\|\\le1\\) on \\(K\\). Suppose the claim fails for some \\(\\varepsilon\\). Choose \\(\\varphi_1\\in K\\). Given \\(\\varphi_1,\\dots,\\varphi_n\\), the failure for \\(F=\\{\\varphi_1,\\dots,\\varphi_n\\}\\) and \\(\\delta=2^{-n}\\) gives \\(a_n\\in M_1\\) and \\(\\varphi_{n+1}\\in K\\) with\n\\[\n\\begin{gathered}\n\\varphi_k^\\sharp(a_n^*a_n+a_na_n^*)<2^{-n}\\ (k\\\\\n\\le n)\\\\\n\\text{and}\\\\\n|\\varphi_{n+1}(a_n)|\\\\\n\\ge\\varepsilon .\n\\end{gathered}\n\\]\nLet \\(\\omega=\\sum_k2^{-k}\\varphi_k^\\sharp\\) and \\(e=s(\\omega)\\). Since \\(\\varphi_k^\\sharp(a^*a+aa^*)\\le2\\|\\varphi_k^\\sharp\\|\\le4\\) for \\(a\\in M_1\\),\n\\[\n\\begin{gathered}\n\\omega(a_n^*a_n+a_na_n^*)\\\\\n\\le2^{-n}+\\sum_{k>n}4\\cdot2^{-k}\\\\\n=5\\cdot2^{-n}\\to0 .\n\\end{gathered}\n\\]\nIn \\(N=eMe\\) the functional \\(\\omega\\) is faithful, and \\(b_n=ea_ne\\) has \\(\\omega(b_n^*b_n+b_nb_n^*)\\le\\omega(a_n^*a_n+a_na_n^*)\\), because \\(b_n^*b_n\\le ea_n^*a_ne\\), \\(b_nb_n^*\\le ea_na_n^*e\\) and \\(\\omega(eye)=\\omega(y)\\). By Proposition 9.1, \\(b_n\\to0\\) \\(\\sigma\\)-strongly\\(^*\\) in \\(N\\). By the Eberlein–Šmulian theorem, \\((\\varphi_n)\\) has a weakly convergent subsequence \\((\\varphi_{n_j})\\), and Lemma 10.3 in \\(N\\) gives \\(\\sup_j|\\varphi_{n_j}(b_n)|\\to0\\) as \\(n\\to\\infty\\). But \\(\\varphi_{n_j}(b_n)=\\varphi_{n_j}(a_n)\\) (Lemma 8.1(2)), and \\(|\\varphi_{n_j}(a_{n_j-1})|\\ge\\varepsilon\\) for all \\(j\\) with \\(n_j\\ge2\\). As \\(n_j-1\\to\\infty\\), this is a contradiction. \\(\\square\\)\n\n**Proof of (1)\\(\\Rightarrow\\)(4).** For \\(m\\ge1\\), Lemma 10.4 with \\(\\varepsilon=1/m\\) gives a finite \\(F_m\\subseteq K\\) and \\(\\delta_m>0\\). Put \\(\\gamma_m=\\big(1+\\sum_{\\psi\\in F_m}\\|\\psi^\\sharp\\|\\big)^{-1}\\) and\n\\[\n\\omega=\\sum_m2^{-m}\\gamma_m\\sum_{\\psi\\in F_m}\\psi^\\sharp\\in M_*^+ .\n\\]\nGiven \\(\\varepsilon>0\\), choose \\(m>1/\\varepsilon\\) and \\(\\delta=2^{-m}\\gamma_m\\delta_m\\). If \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\), then \\(\\psi^\\sharp(a^*a+aa^*)<\\delta_m\\) for every \\(\\psi\\in F_m\\), so \\(|\\varphi(a)|<1/m<\\varepsilon\\) for all \\(\\varphi\\in K\\). \\(\\square\\)\n\nThis completes the proof of Theorem 10.2.\n\n**Remark 10.5.** For a \\(C^*\\)-algebra \\(A\\), apply Theorem 10.2 to \\(M=\\tilde A\\), whose predual is \\(A^*\\). It characterizes the relatively weakly compact subsets of \\(A^*\\), in terms of projections of the bidual.\n\nAbsolute values need not stay in a relatively weakly compact set (Example 10.7). One-sided multiples detect exactly when they do.\n\n**Proposition 10.6.** Let \\(K\\subseteq M_*\\) and put \\(|K|=\\{|\\varphi|:\\varphi\\in K\\}\\), \\(|K^*|=\\{|\\varphi^*|:\\varphi\\in K\\}\\), \\(M_1K=\\{a\\varphi:a\\in M_1,\\ \\varphi\\in K\\}\\) and \\(KM_1=\\{\\varphi a:a\\in M_1,\\ \\varphi\\in K\\}\\).\n\n1. \\(M_1K\\) is relatively weakly compact exactly when \\(|K|\\) is.\n2. \\(KM_1\\) is relatively weakly compact exactly when \\(|K^*|\\) is.\n3. In particular, if \\(K\\subseteq M_*^+\\) is relatively weakly compact, so are \\(M_1K\\) and \\(KM_1\\).\n\n**Proof.** (1) If \\(\\varphi=v|\\varphi|\\), then \\(|\\varphi|=v^*\\varphi\\in M_1K\\) and \\(a\\varphi=(av)|\\varphi|\\in M_1|K|\\). So it suffices to show that \\(M_1L\\) is relatively weakly compact when \\(L\\subseteq M_*^+\\) is. It is bounded. If \\(p_n\\downarrow0\\), then for \\(\\omega\\in L\\) and \\(a\\in M_1\\), by the Cauchy–Schwarz inequality,\n\\[\n\\begin{gathered}\n|(a\\omega)(p_n)|\\\\\n=|\\omega(p_na)|\\\\\n\\le\\omega(p_n)^{1/2}\\omega(a^*a)^{1/2}\\\\\n\\le\\|\\omega\\|^{1/2}\\omega(p_n)^{1/2}.\n\\end{gathered}\n\\]\nBy Theorem 10.2 (1)\\(\\Rightarrow\\)(3) for \\(L\\), the right side tends to \\(0\\) uniformly, and (3)\\(\\Rightarrow\\)(1) applies to \\(M_1L\\).\n\n(2) The map \\(\\psi\\mapsto\\psi^*\\) is a conjugate-linear isometry of \\(M_*\\) onto itself and a homeomorphism for the weak topology. Since \\(\\varphi a=(a^*\\varphi^*)^*\\), \\(KM_1=(M_1K^*)^*\\), and (1) for \\(K^*\\) gives (2). (3) For positive \\(\\varphi\\), \\(|\\varphi|=|\\varphi^*|=\\varphi\\). \\(\\square\\)\n\n**Example 10.7** (Absolute values can escape). Let \\(H\\) have an orthonormal sequence \\((\\xi_n)\\), \\(M=B(H)\\), and \\(\\varphi_n=\\omega_{\\xi_1,\\xi_n}\\), so \\(\\varphi_n(x)=\\langle x\\xi_1,\\xi_n\\rangle\\).\n\n1. \\(\\sum_n|\\varphi_n(x)|^2\\le\\|x\\xi_1\\|^2\\), so \\(\\varphi_n\\to0\\) weakly, and \\(K=\\{\\varphi_n\\}\\) is relatively weakly compact. But \\(\\|\\varphi_n\\|=1\\) (Example 1.2).\n2. By (2.5), \\(|\\varphi_n|=\\omega_{\\xi_n}\\). The set \\(|K|\\) is not relatively weakly compact: \\(p_m=\\sum_{j\\ge m}\\theta_{\\xi_j,\\xi_j}\\) decreases to \\(0\\) (strongly), while \\(\\omega_{\\xi_n}(p_m)=1\\) for \\(n\\ge m\\), against Theorem 10.2(3). By Proposition 10.6, \\(M_1K\\) is not relatively weakly compact either.\n3. \\(\\varphi_n^*=\\omega_{\\xi_n,\\xi_1}\\) and \\(|\\varphi_n^*|=\\omega_{\\xi_1}\\) for all \\(n\\). So \\(|K^*|\\) is a single point, and \\(KM_1\\) is relatively weakly compact.\n4. Two-sided multiples of a single positive functional need not form a relatively weakly compact set: \\(\\omega_{\\xi_n}=a_n\\omega_{\\xi_1}a_n^*\\) with \\(a_n=\\theta_{\\xi_n,\\xi_1}\\in M_1\\).\n\nIn the commutative case absolute values cannot escape.\n\n**Proposition 10.8.** If \\(M\\) is abelian and \\(K\\subseteq M_*\\) is relatively weakly compact, then so is \\(|K|\\).\n\n**Proof.** Let \\(\\omega\\) be as in Theorem 10.2(4). For \\(\\varphi=v|\\varphi|\\in K\\) and \\(a\\in M_1\\), \\(|\\varphi|(a)=\\varphi(av^*)\\). In the abelian algebra \\(M\\), with \\(b=av^*\\in M_1\\),\n\\[\nb^*b+bb^*=a^*a\\,vv^*+aa^*\\,v^*v\\le a^*a+aa^* ,\n\\]\nbecause \\(vv^*\\) and \\(v^*v\\) are projections that commute with \\(a^*a\\) and \\(aa^*\\). So \\(\\omega(b^*b+bb^*)<\\delta\\) whenever \\(\\omega(a^*a+aa^*)<\\delta\\), and then \\(||\\varphi|(a)|=|\\varphi(b)|<\\varepsilon\\). Thus \\(|K|\\) satisfies Theorem 10.2(4) with the same \\(\\omega\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-PD-17",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "10. Weak compactness in the predual",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      },
      "full_conditions_and_proof": "## 10. Weak compactness in the predual\n\nWe now characterize the relatively weakly compact subsets of a predual. For \\(M=L^\\infty(\\Gamma,\\mu)\\), whose predual is \\(L^1(\\Gamma,\\mu)\\) (Example 2.6), condition (4) of the theorem below, tested on indicator functions \\(a=1_E\\), says that there is one \\(g\\in L^1_+\\) such that the integrals \\(\\int_Eh\\,d\\mu\\), \\(h\\in K\\), are uniformly small whenever \\(\\int_Eg\\,d\\mu\\) is small. In general the control is by one positive normal functional, applied to \\(a^*a+aa^*\\).\n\nWe use the following Banach-space fact twice, here and in Section 14.\n\n**Lemma 10.1.** A bounded subset \\(K\\) of a Banach space \\(X\\) is relatively weakly compact exactly when its \\(\\sigma(X^{**},X^*)\\)-closure in \\(X^{**}\\) lies in \\(X\\).\n\n**Proof.** The weak topology of \\(X\\) is the restriction of \\(\\sigma(X^{**},X^*)\\). If \\(K\\) is relatively weakly compact, its weak closure \\(K'\\) in \\(X\\) is weakly compact, hence \\(\\sigma(X^{**},X^*)\\)-compact and closed in \\(X^{**}\\); so the closure of \\(K\\) in \\(X^{**}\\) lies in \\(K'\\subseteq X\\). Conversely, if the closure \\(\\bar K\\) of \\(K\\) in \\(X^{**}\\) lies in \\(X\\), it is \\(\\sigma(X^{**},X^*)\\)-compact by the Banach–Alaoglu theorem, since \\(K\\) is bounded; so \\(\\bar K\\) is a weakly compact subset of \\(X\\) containing \\(K\\). \\(\\square\\)\n\nFor \\(X=M_*\\) we have \\(X^*=M\\) and \\(X^{**}=M^*\\), so a bounded \\(K\\subseteq M_*\\) is relatively weakly compact exactly when its \\(\\sigma(M^*,M)\\)-closure consists of normal functionals. Relatively weakly compact sets are bounded (background fact 2). For a projection \\(p\\) we write \\((1-p)\\varphi(1-p)\\) for \\(x\\mapsto\\varphi((1-p)x(1-p))\\), as in (0.1).\n\n**Theorem 10.2** (Weak compactness in \\(M_*\\)). For a subset \\(K\\) of the predual of a von Neumann algebra \\(M\\), the following are equivalent.\n\n1. \\(K\\) is relatively weakly compact.\n2. For every abelian von Neumann subalgebra \\(\\mathcal A\\subseteq M\\), the set \\(K|_{\\mathcal A}=\\{\\varphi|_{\\mathcal A}:\\varphi\\in K\\}\\) is relatively weakly compact in \\(\\mathcal A_*\\).\n3. \\(K\\) is bounded, and \\(\\varphi(p_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) whenever the projections \\(p_n\\) decrease to \\(0\\).\n4. \\(K\\) is bounded, and there is \\(\\omega\\in M_*^+\\) with this property: for every \\(\\varepsilon>0\\) there is \\(\\delta>0\\) such that \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\) whenever \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\).\n5. \\(K\\) is bounded, and for every increasing net of projections \\((p_i)\\), \\(\\varphi(p_i)\\) converges uniformly in \\(\\varphi\\in K\\).\n6. \\(K\\) is bounded, and \\(\\|(1-p_i)\\varphi(1-p_i)\\|\\to0\\) uniformly in \\(\\varphi\\in K\\) for every increasing net of projections \\((p_i)\\) with \\(\\sup_ip_i=1\\).\n7. \\(K\\) is bounded, and \\(\\varphi(q_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) for every sequence \\((q_n)\\) of mutually orthogonal projections.\n\n\nWe prove the easy implications first; the implication (1)\\(\\Rightarrow\\)(4) needs two lemmas.\n\n**Proof of (1)\\(\\Rightarrow\\)(2).** Restriction \\(\\varphi\\mapsto\\varphi|_{\\mathcal A}\\) is continuous from \\(\\sigma(M_*,M)\\) to \\(\\sigma(\\mathcal A_*,\\mathcal A)\\), and continuous images of compact sets are compact. \\(\\square\\)\n\n**Proof of (2)\\(\\Rightarrow\\)(1).** Every self-adjoint \\(h\\in M\\) lies in the abelian von Neumann algebra generated by \\(h\\) and \\(1\\). So \\(\\{\\varphi(h):\\varphi\\in K\\}\\) is bounded, hence so is \\(\\{\\varphi(x):\\varphi\\in K\\}\\) for every \\(x=h+ik\\), and \\(K\\) is bounded by uniform boundedness. Let \\(\\varphi\\) lie in the \\(\\sigma(M^*,M)\\)-closure of \\(K\\). For an abelian von Neumann subalgebra \\(\\mathcal A\\), \\(\\varphi|_{\\mathcal A}\\) lies in the \\(\\sigma(\\mathcal A^*,\\mathcal A)\\)-closure of \\(K|_{\\mathcal A}\\), which consists of normal functionals by Lemma 10.1. So \\(\\varphi\\) is normal on every abelian von Neumann subalgebra, hence normal (background fact 23). By Lemma 10.1, \\(K\\) is relatively weakly compact. \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(6)\\(\\Rightarrow\\)(5)\\(\\Rightarrow\\)(3).** (4)\\(\\Rightarrow\\)(6): let \\(\\varepsilon,\\delta\\) be as in (4) and \\(p_i\\uparrow1\\). By normality of \\(\\omega\\) there is \\(i_0\\) with \\(\\omega(1-p_i)<\\delta/2\\) for \\(i\\ge i_0\\). If \\(a\\in M_1\\) and \\(a=(1-p_i)a(1-p_i)\\), then \\(a^*a\\le1-p_i\\) and \\(aa^*\\le1-p_i\\), so \\(\\omega(a^*a+aa^*)<\\delta\\) and \\(|\\varphi(a)|<\\varepsilon\\). Hence \\[\n\\begin{gathered}\n\\|(1-p_i)\\varphi(1-p_i)\\|\\\\\n=\\sup_{a\\in M_1}|\\varphi((1-p_i)a(1-p_i))|\\\\\n\\le\\varepsilon\n\\end{gathered}\n\\] for \\(i\\ge i_0\\) and all \\(\\varphi\\in K\\).\n\n(6)\\(\\Rightarrow\\)(5): let \\((p_i)\\) increase to \\(p\\). Then \\(q_i=p_i+(1-p)\\) increases to \\(1\\), and \\(1-q_i=p-p_i\\). By (6), \\[\n\\begin{gathered}\n|\\varphi(p)-\\varphi(p_i)|\\\\\n=|\\varphi(p-p_i)|\\\\\n\\le\\|(p-p_i)\\varphi(p-p_i)\\|\\to0\n\\end{gathered}\n\\] uniformly in \\(\\varphi\\in K\\).\n\n(5)\\(\\Rightarrow\\)(3): if \\(p_n\\downarrow0\\), then \\(1-p_n\\uparrow1\\). By (5), \\(\\varphi(1-p_n)\\) converges uniformly, and its pointwise limit is \\(\\varphi(1)\\) by normality. So \\(\\varphi(p_n)\\to0\\) uniformly. \\(\\square\\)\n\n**Proof of (3)\\(\\Leftrightarrow\\)(7).** (3)\\(\\Rightarrow\\)(7): for orthogonal \\((q_n)\\), \\(r_m=\\sum_{n\\ge m}q_n\\) decreases to \\(0\\), and \\(\\varphi(q_m)=\\varphi(r_m)-\\varphi(r_{m+1})\\). (7)\\(\\Rightarrow\\)(3): suppose \\(p_n\\downarrow0\\) but \\(\\sup_{\\varphi\\in K}|\\varphi(p_n)|\\not\\to0\\). Then there are \\(\\varepsilon>0\\), indices \\(n_1<n_2<\\cdots\\) and \\(\\varphi_j\\in K\\) with \\(|\\varphi_j(p_{n_j})|\\ge\\varepsilon\\). Since each \\(\\varphi_j\\) is normal, \\(\\varphi_j(p_n)\\to0\\) as \\(n\\to\\infty\\); choosing the indices and functionals inductively, we may assume \\(|\\varphi_j(p_{n_{j+1}})|<\\varepsilon/2\\). The projections \\(q_j=p_{n_j}-p_{n_{j+1}}\\) are mutually orthogonal and \\(|\\varphi_j(q_j)|>\\varepsilon/2\\), which contradicts (7). \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(3).** If \\(p_n\\downarrow0\\), then \\(\\omega(p_n^*p_n+p_np_n^*)=2\\omega(p_n)\\to0\\). \\(\\square\\)\n\n**Proof of (3)\\(\\Rightarrow\\)(1).** By the Eberlein–Šmulian theorem we only need a weak cluster point in \\(M_*\\) for each sequence \\((\\varphi_n)\\) in \\(K\\). Let \\(\\varphi\\in M^*\\) be a \\(\\sigma(M^*,M)\\)-cluster point, which exists by the Banach–Alaoglu theorem. On \\(M_*\\), \\(\\sigma(M^*,M)\\) is the weak topology, so it suffices to show that \\(\\varphi\\) is normal. Put \\(\\omega=\\sum_n2^{-n}\\varphi_n^\\sharp\\in M_*^+\\).\n\n(a) If \\(q\\) is a projection with \\(\\omega(q)=0\\), then \\(\\varphi_n(q)=0\\) for all \\(n\\) (Lemma 8.1(3)), hence \\(\\varphi(q)=0\\).\n\n(b) If projections \\(r_m\\) decrease to \\(0\\), then \\(|\\varphi(r_m)|\\le\\sup_n|\\varphi_n(r_m)|\\to0\\) by (3).\n\nLet \\((p_i)_{i\\in I}\\) be mutually orthogonal with sum \\(p\\). Since \\(\\omega\\) is normal, \\(\\sum_i\\omega(p_i)=\\omega(p)<\\infty\\), so \\(I_0=\\{i:\\omega(p_i)>0\\}\\) is countable. Every projection \\(q\\) under \\(\\sum_{i\\notin I_0}p_i\\) has \\(\\omega(q)=0\\), hence \\(\\varphi(q)=0\\) by (a). For \\(J\\subseteq I_0\\), enumerate \\(J=\\{j_1,j_2,\\dots\\}\\); the projections \\(r_m=\\sum_{k\\ge m}p_{j_k}\\) decrease to \\(0\\), so by (b), \\(\\varphi(\\sum_{i\\in J}p_i)=\\lim_m\\sum_{k<m}\\varphi(p_{j_k})\\). Applying this to \\(J=\\{i\\in I_0:\\operatorname{Re}\\varphi(p_i)\\ge0\\}\\) and to the three analogous sets shows that \\(\\sum_{i\\in I_0}|\\varphi(p_i)|<\\infty\\). Now let \\(F\\subseteq I\\) be finite. Then \\(p-\\sum_{i\\in F}p_i\\) is the sum of a projection under \\(\\sum_{i\\notin I_0}p_i\\) and of \\(\\sum_{i\\in I_0\\setminus F}p_i\\), so\n\\[\n\\begin{gathered}\n\\Big|\\varphi(p)-\\sum_{i\\in F}\\varphi(p_i)\\Big|\\\\\n=\\Big|\\sum_{i\\in I_0\\setminus F}\\varphi(p_i)\\Big|\\\\\n\\le\\sum_{i\\in I_0\\setminus F}|\\varphi(p_i)|,\n\\end{gathered}\n\\]\nwhich tends to \\(0\\) along the finite sets \\(F\\). So \\(\\varphi\\) is completely additive, hence normal (background fact 23). \\(\\square\\)\n\nThe remaining implication rests on the following two lemmas.\n\n**Lemma 10.3.** Let \\((\\varphi_k)\\) be a sequence in \\(M_*\\) converging weakly to \\(\\varphi_0\\in M_*\\), and let \\((a_n)\\) be a sequence in \\(M_1\\) converging \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\). Then \\(\\sup_k|\\varphi_k(a_n)|\\to0\\) as \\(n\\to\\infty\\).\n\n**Proof.** *Reduction.* Let \\(c=\\sup_k\\|\\varphi_k\\|<\\infty\\). Put \\(\\omega=\\varphi_0^\\sharp+\\sum_k2^{-k}\\varphi_k^\\sharp\\in M_*^+\\) and \\(e=s(\\omega)\\). For every \\(k\\ge0\\), \\(s(\\varphi_k^\\sharp)\\le e\\), so \\(\\varphi_k(x)=\\varphi_k(exe)\\) by Lemma 8.1(2). The restriction of \\(\\omega\\) to the von Neumann algebra \\(N=eMe\\) is faithful; the restrictions \\(\\varphi_k|_N\\) converge weakly to \\(\\varphi_0|_N\\); the elements \\(ea_ne\\in N_1\\) converge \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\); and \\(\\varphi_k(a_n)=\\varphi_k(ea_ne)\\). So it is enough to treat a faithful \\(\\omega\\). Let \\(d\\) be the metric of Proposition 9.1 for \\(\\omega\\).\n\n*Baire.* Put \\(\\psi_k=\\varphi_k-\\varphi_0\\), so \\(\\psi_k\\to0\\) weakly and \\(\\|\\psi_k\\|\\le2c\\). Fix \\(\\varepsilon>0\\) and let \\(S_m=\\{a\\in M_1:|\\psi_k(a)|\\le\\varepsilon\\text{ for all }k\\ge m\\}\\). Each \\(\\psi_k\\) is \\(\\sigma\\)-strongly continuous, hence \\(d\\)-continuous on \\(M_1\\), so \\(S_m\\) is \\(d\\)-closed; and \\(\\bigcup_mS_m=M_1\\). By Baire's theorem, some \\(S_{m_0}\\) contains \\(\\{a\\in M_1:d(a,a_0)<\\delta\\}\\) for some \\(a_0\\in M_1\\), \\(\\delta>0\\).\n\n*Cutting \\(a_n\\).* Let \\(h_n=a_n^*a_n+a_na_n^*\\), so \\(0\\le h_n\\le2\\) and \\(\\omega(h_n)\\to0\\). Let \\(p_n=1_{[0,\\varepsilon^2]}(h_n)\\), a spectral projection. Then \\(\\|p_nh_np_n\\|\\le\\varepsilon^2\\), which gives \\(\\|a_np_n\\|\\le\\varepsilon\\) and \\(\\|p_na_n\\|\\le\\varepsilon\\). Also \\(1-p_n\\le\\varepsilon^{-2}h_n\\), so \\(d(p_n,1)^2=\\omega(1-p_n)\\le\\varepsilon^{-2}\\omega(h_n)\\to0\\), and \\(p_n\\to1\\) \\(\\sigma\\)-strongly. Write\n\\[\n\\begin{gathered}\na_n\\\\\n=p_na_n+(1-p_n)a_np_n+c_n,\\\\\nc_n\\\\\n=(1-p_n)a_n(1-p_n).\n\\end{gathered}\n\\]\nThe first two terms have norm at most \\(\\varepsilon\\), so \\(|\\psi_k(a_n)|\\le4c\\varepsilon+|\\psi_k(c_n)|\\).\n\n*Comparing with \\(a_0\\).* Put \\(b_n=p_na_0p_n+c_n\\). It lies in \\(M_1\\): \\(b_n^*b_n=p_na_0^*p_na_0p_n+c_n^*c_n\\) is a sum of two positive contractions with orthogonal supports. As \\(n\\to\\infty\\), \\(p_na_0p_n\\to a_0\\) \\(\\sigma\\)-strongly (joint continuity of multiplication on bounded sets), and \\(d(c_n,0)^2=\\omega(c_n^*c_n)\\le\\omega(1-p_n)\\to0\\). So \\(d(p_na_0p_n,a_0)\\to0\\) and \\(d(b_n,a_0)\\to0\\), and there is \\(n_0\\) with both distances below \\(\\delta\\) for \\(n\\ge n_0\\). For \\(k\\ge m_0\\) and \\(n\\ge n_0\\),\n\\[\n\\begin{gathered}\n|\\psi_k(c_n)|\\\\\n=|\\psi_k(b_n)-\\psi_k(p_na_0p_n)|\\\\\n\\le2\\varepsilon ,\\\\\n\\text{so}\\\\\n|\\psi_k(a_n)|\\\\\n\\le(4c+2)\\varepsilon .\n\\end{gathered}\n\\]\nFor each of the finitely many \\(k<m_0\\), \\(\\psi_k(a_n)\\to0\\), because \\(\\psi_k\\) is normal and a bounded \\(\\sigma\\)-strongly\\(^*\\) null sequence is \\(\\sigma\\)-weakly null. Hence \\(\\limsup_n\\sup_k|\\psi_k(a_n)|\\le(4c+2)\\varepsilon\\) for every \\(\\varepsilon\\). Finally \\(\\varphi_k(a_n)=\\psi_k(a_n)+\\varphi_0(a_n)\\) and \\(\\varphi_0(a_n)\\to0\\). \\(\\square\\)\n\n**Lemma 10.4.** Let \\(K\\subseteq M_*\\) be relatively weakly compact and \\(\\varepsilon>0\\). There are a finite set \\(F\\subseteq K\\) and \\(\\delta>0\\) such that: if \\(a\\in M_1\\) and \\(\\psi^\\sharp(a^*a+aa^*)<\\delta\\) for all \\(\\psi\\in F\\), then \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\).\n\n**Proof.** If \\(K=\\varnothing\\), take \\(F=\\varnothing\\) and any \\(\\delta>0\\). If \\(K=\\{0\\}\\), the conclusion is also immediate. Otherwise \\(K\\) is bounded; scaling, we may assume \\(\\|\\varphi\\|\\le1\\) on \\(K\\). Suppose the claim fails for some \\(\\varepsilon\\). Choose \\(\\varphi_1\\in K\\). Given \\(\\varphi_1,\\dots,\\varphi_n\\), the failure for \\(F=\\{\\varphi_1,\\dots,\\varphi_n\\}\\) and \\(\\delta=2^{-n}\\) gives \\(a_n\\in M_1\\) and \\(\\varphi_{n+1}\\in K\\) with\n\\[\n\\begin{gathered}\n\\varphi_k^\\sharp(a_n^*a_n+a_na_n^*)<2^{-n}\\ (k\\\\\n\\le n)\\\\\n\\text{and}\\\\\n|\\varphi_{n+1}(a_n)|\\\\\n\\ge\\varepsilon .\n\\end{gathered}\n\\]\nLet \\(\\omega=\\sum_k2^{-k}\\varphi_k^\\sharp\\) and \\(e=s(\\omega)\\). Since \\(\\varphi_k^\\sharp(a^*a+aa^*)\\le2\\|\\varphi_k^\\sharp\\|\\le4\\) for \\(a\\in M_1\\),\n\\[\n\\begin{gathered}\n\\omega(a_n^*a_n+a_na_n^*)\\\\\n\\le2^{-n}+\\sum_{k>n}4\\cdot2^{-k}\\\\\n=5\\cdot2^{-n}\\to0 .\n\\end{gathered}\n\\]\nIn \\(N=eMe\\) the functional \\(\\omega\\) is faithful, and \\(b_n=ea_ne\\) has \\(\\omega(b_n^*b_n+b_nb_n^*)\\le\\omega(a_n^*a_n+a_na_n^*)\\), because \\(b_n^*b_n\\le ea_n^*a_ne\\), \\(b_nb_n^*\\le ea_na_n^*e\\) and \\(\\omega(eye)=\\omega(y)\\). By Proposition 9.1, \\(b_n\\to0\\) \\(\\sigma\\)-strongly\\(^*\\) in \\(N\\). By the Eberlein–Šmulian theorem, \\((\\varphi_n)\\) has a weakly convergent subsequence \\((\\varphi_{n_j})\\), and Lemma 10.3 in \\(N\\) gives \\(\\sup_j|\\varphi_{n_j}(b_n)|\\to0\\) as \\(n\\to\\infty\\). But \\(\\varphi_{n_j}(b_n)=\\varphi_{n_j}(a_n)\\) (Lemma 8.1(2)), and \\(|\\varphi_{n_j}(a_{n_j-1})|\\ge\\varepsilon\\) for all \\(j\\) with \\(n_j\\ge2\\). As \\(n_j-1\\to\\infty\\), this is a contradiction. \\(\\square\\)\n\n**Proof of (1)\\(\\Rightarrow\\)(4).** For \\(m\\ge1\\), Lemma 10.4 with \\(\\varepsilon=1/m\\) gives a finite \\(F_m\\subseteq K\\) and \\(\\delta_m>0\\). Put \\(\\gamma_m=\\big(1+\\sum_{\\psi\\in F_m}\\|\\psi^\\sharp\\|\\big)^{-1}\\) and\n\\[\n\\omega=\\sum_m2^{-m}\\gamma_m\\sum_{\\psi\\in F_m}\\psi^\\sharp\\in M_*^+ .\n\\]\nGiven \\(\\varepsilon>0\\), choose \\(m>1/\\varepsilon\\) and \\(\\delta=2^{-m}\\gamma_m\\delta_m\\). If \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\), then \\(\\psi^\\sharp(a^*a+aa^*)<\\delta_m\\) for every \\(\\psi\\in F_m\\), so \\(|\\varphi(a)|<1/m<\\varepsilon\\) for all \\(\\varphi\\in K\\). \\(\\square\\)\n\nThis completes the proof of Theorem 10.2.\n\n**Remark 10.5.** For a \\(C^*\\)-algebra \\(A\\), apply Theorem 10.2 to \\(M=\\tilde A\\), whose predual is \\(A^*\\). It characterizes the relatively weakly compact subsets of \\(A^*\\), in terms of projections of the bidual.\n\nAbsolute values need not stay in a relatively weakly compact set (Example 10.7). One-sided multiples detect exactly when they do.\n\n**Proposition 10.6.** Let \\(K\\subseteq M_*\\) and put \\(|K|=\\{|\\varphi|:\\varphi\\in K\\}\\), \\(|K^*|=\\{|\\varphi^*|:\\varphi\\in K\\}\\), \\(M_1K=\\{a\\varphi:a\\in M_1,\\ \\varphi\\in K\\}\\) and \\(KM_1=\\{\\varphi a:a\\in M_1,\\ \\varphi\\in K\\}\\).\n\n1. \\(M_1K\\) is relatively weakly compact exactly when \\(|K|\\) is.\n2. \\(KM_1\\) is relatively weakly compact exactly when \\(|K^*|\\) is.\n3. In particular, if \\(K\\subseteq M_*^+\\) is relatively weakly compact, so are \\(M_1K\\) and \\(KM_1\\).\n\n**Proof.** (1) If \\(\\varphi=v|\\varphi|\\), then \\(|\\varphi|=v^*\\varphi\\in M_1K\\) and \\(a\\varphi=(av)|\\varphi|\\in M_1|K|\\). So it suffices to show that \\(M_1L\\) is relatively weakly compact when \\(L\\subseteq M_*^+\\) is. It is bounded. If \\(p_n\\downarrow0\\), then for \\(\\omega\\in L\\) and \\(a\\in M_1\\), by the Cauchy–Schwarz inequality,\n\\[\n\\begin{gathered}\n|(a\\omega)(p_n)|\\\\\n=|\\omega(p_na)|\\\\\n\\le\\omega(p_n)^{1/2}\\omega(a^*a)^{1/2}\\\\\n\\le\\|\\omega\\|^{1/2}\\omega(p_n)^{1/2}.\n\\end{gathered}\n\\]\nBy Theorem 10.2 (1)\\(\\Rightarrow\\)(3) for \\(L\\), the right side tends to \\(0\\) uniformly, and (3)\\(\\Rightarrow\\)(1) applies to \\(M_1L\\).\n\n(2) The map \\(\\psi\\mapsto\\psi^*\\) is a conjugate-linear isometry of \\(M_*\\) onto itself and a homeomorphism for the weak topology. Since \\(\\varphi a=(a^*\\varphi^*)^*\\), \\(KM_1=(M_1K^*)^*\\), and (1) for \\(K^*\\) gives (2). (3) For positive \\(\\varphi\\), \\(|\\varphi|=|\\varphi^*|=\\varphi\\). \\(\\square\\)\n\n**Example 10.7** (Absolute values can escape). Let \\(H\\) have an orthonormal sequence \\((\\xi_n)\\), \\(M=B(H)\\), and \\(\\varphi_n=\\omega_{\\xi_1,\\xi_n}\\), so \\(\\varphi_n(x)=\\langle x\\xi_1,\\xi_n\\rangle\\).\n\n1. \\(\\sum_n|\\varphi_n(x)|^2\\le\\|x\\xi_1\\|^2\\), so \\(\\varphi_n\\to0\\) weakly, and \\(K=\\{\\varphi_n\\}\\) is relatively weakly compact. But \\(\\|\\varphi_n\\|=1\\) (Example 1.2).\n2. By (2.5), \\(|\\varphi_n|=\\omega_{\\xi_n}\\). The set \\(|K|\\) is not relatively weakly compact: \\(p_m=\\sum_{j\\ge m}\\theta_{\\xi_j,\\xi_j}\\) decreases to \\(0\\) (strongly), while \\(\\omega_{\\xi_n}(p_m)=1\\) for \\(n\\ge m\\), against Theorem 10.2(3). By Proposition 10.6, \\(M_1K\\) is not relatively weakly compact either.\n3. \\(\\varphi_n^*=\\omega_{\\xi_n,\\xi_1}\\) and \\(|\\varphi_n^*|=\\omega_{\\xi_1}\\) for all \\(n\\). So \\(|K^*|\\) is a single point, and \\(KM_1\\) is relatively weakly compact.\n4. Two-sided multiples of a single positive functional need not form a relatively weakly compact set: \\(\\omega_{\\xi_n}=a_n\\omega_{\\xi_1}a_n^*\\) with \\(a_n=\\theta_{\\xi_n,\\xi_1}\\in M_1\\).\n\nIn the commutative case absolute values cannot escape.\n\n**Proposition 10.8.** If \\(M\\) is abelian and \\(K\\subseteq M_*\\) is relatively weakly compact, then so is \\(|K|\\).\n\n**Proof.** Let \\(\\omega\\) be as in Theorem 10.2(4). For \\(\\varphi=v|\\varphi|\\in K\\) and \\(a\\in M_1\\), \\(|\\varphi|(a)=\\varphi(av^*)\\). In the abelian algebra \\(M\\), with \\(b=av^*\\in M_1\\),\n\\[\nb^*b+bb^*=a^*a\\,vv^*+aa^*\\,v^*v\\le a^*a+aa^* ,\n\\]\nbecause \\(vv^*\\) and \\(v^*v\\) are projections that commute with \\(a^*a\\) and \\(aa^*\\). So \\(\\omega(b^*b+bb^*)<\\delta\\) whenever \\(\\omega(a^*a+aa^*)<\\delta\\), and then \\(||\\varphi|(a)|=|\\varphi(b)|<\\varepsilon\\). Thus \\(|K|\\) satisfies Theorem 10.2(4) with the same \\(\\omega\\). \\(\\square\\)\n\n",
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      "name": "10. Weak compactness in the predual",
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      "full_conditions_and_proof": "## 10. Weak compactness in the predual\n\nWe now characterize the relatively weakly compact subsets of a predual. For \\(M=L^\\infty(\\Gamma,\\mu)\\), whose predual is \\(L^1(\\Gamma,\\mu)\\) (Example 2.6), condition (4) of the theorem below, tested on indicator functions \\(a=1_E\\), says that there is one \\(g\\in L^1_+\\) such that the integrals \\(\\int_Eh\\,d\\mu\\), \\(h\\in K\\), are uniformly small whenever \\(\\int_Eg\\,d\\mu\\) is small. In general the control is by one positive normal functional, applied to \\(a^*a+aa^*\\).\n\nWe use the following Banach-space fact twice, here and in Section 14.\n\n**Lemma 10.1.** A bounded subset \\(K\\) of a Banach space \\(X\\) is relatively weakly compact exactly when its \\(\\sigma(X^{**},X^*)\\)-closure in \\(X^{**}\\) lies in \\(X\\).\n\n**Proof.** The weak topology of \\(X\\) is the restriction of \\(\\sigma(X^{**},X^*)\\). If \\(K\\) is relatively weakly compact, its weak closure \\(K'\\) in \\(X\\) is weakly compact, hence \\(\\sigma(X^{**},X^*)\\)-compact and closed in \\(X^{**}\\); so the closure of \\(K\\) in \\(X^{**}\\) lies in \\(K'\\subseteq X\\). Conversely, if the closure \\(\\bar K\\) of \\(K\\) in \\(X^{**}\\) lies in \\(X\\), it is \\(\\sigma(X^{**},X^*)\\)-compact by the Banach–Alaoglu theorem, since \\(K\\) is bounded; so \\(\\bar K\\) is a weakly compact subset of \\(X\\) containing \\(K\\). \\(\\square\\)\n\nFor \\(X=M_*\\) we have \\(X^*=M\\) and \\(X^{**}=M^*\\), so a bounded \\(K\\subseteq M_*\\) is relatively weakly compact exactly when its \\(\\sigma(M^*,M)\\)-closure consists of normal functionals. Relatively weakly compact sets are bounded (background fact 2). For a projection \\(p\\) we write \\((1-p)\\varphi(1-p)\\) for \\(x\\mapsto\\varphi((1-p)x(1-p))\\), as in (0.1).\n\n**Theorem 10.2** (Weak compactness in \\(M_*\\)). For a subset \\(K\\) of the predual of a von Neumann algebra \\(M\\), the following are equivalent.\n\n1. \\(K\\) is relatively weakly compact.\n2. For every abelian von Neumann subalgebra \\(\\mathcal A\\subseteq M\\), the set \\(K|_{\\mathcal A}=\\{\\varphi|_{\\mathcal A}:\\varphi\\in K\\}\\) is relatively weakly compact in \\(\\mathcal A_*\\).\n3. \\(K\\) is bounded, and \\(\\varphi(p_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) whenever the projections \\(p_n\\) decrease to \\(0\\).\n4. \\(K\\) is bounded, and there is \\(\\omega\\in M_*^+\\) with this property: for every \\(\\varepsilon>0\\) there is \\(\\delta>0\\) such that \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\) whenever \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\).\n5. \\(K\\) is bounded, and for every increasing net of projections \\((p_i)\\), \\(\\varphi(p_i)\\) converges uniformly in \\(\\varphi\\in K\\).\n6. \\(K\\) is bounded, and \\(\\|(1-p_i)\\varphi(1-p_i)\\|\\to0\\) uniformly in \\(\\varphi\\in K\\) for every increasing net of projections \\((p_i)\\) with \\(\\sup_ip_i=1\\).\n7. \\(K\\) is bounded, and \\(\\varphi(q_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) for every sequence \\((q_n)\\) of mutually orthogonal projections.\n\n\nWe prove the easy implications first; the implication (1)\\(\\Rightarrow\\)(4) needs two lemmas.\n\n**Proof of (1)\\(\\Rightarrow\\)(2).** Restriction \\(\\varphi\\mapsto\\varphi|_{\\mathcal A}\\) is continuous from \\(\\sigma(M_*,M)\\) to \\(\\sigma(\\mathcal A_*,\\mathcal A)\\), and continuous images of compact sets are compact. \\(\\square\\)\n\n**Proof of (2)\\(\\Rightarrow\\)(1).** Every self-adjoint \\(h\\in M\\) lies in the abelian von Neumann algebra generated by \\(h\\) and \\(1\\). So \\(\\{\\varphi(h):\\varphi\\in K\\}\\) is bounded, hence so is \\(\\{\\varphi(x):\\varphi\\in K\\}\\) for every \\(x=h+ik\\), and \\(K\\) is bounded by uniform boundedness. Let \\(\\varphi\\) lie in the \\(\\sigma(M^*,M)\\)-closure of \\(K\\). For an abelian von Neumann subalgebra \\(\\mathcal A\\), \\(\\varphi|_{\\mathcal A}\\) lies in the \\(\\sigma(\\mathcal A^*,\\mathcal A)\\)-closure of \\(K|_{\\mathcal A}\\), which consists of normal functionals by Lemma 10.1. So \\(\\varphi\\) is normal on every abelian von Neumann subalgebra, hence normal (background fact 23). By Lemma 10.1, \\(K\\) is relatively weakly compact. \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(6)\\(\\Rightarrow\\)(5)\\(\\Rightarrow\\)(3).** (4)\\(\\Rightarrow\\)(6): let \\(\\varepsilon,\\delta\\) be as in (4) and \\(p_i\\uparrow1\\). By normality of \\(\\omega\\) there is \\(i_0\\) with \\(\\omega(1-p_i)<\\delta/2\\) for \\(i\\ge i_0\\). If \\(a\\in M_1\\) and \\(a=(1-p_i)a(1-p_i)\\), then \\(a^*a\\le1-p_i\\) and \\(aa^*\\le1-p_i\\), so \\(\\omega(a^*a+aa^*)<\\delta\\) and \\(|\\varphi(a)|<\\varepsilon\\). Hence \\[\n\\begin{gathered}\n\\|(1-p_i)\\varphi(1-p_i)\\|\\\\\n=\\sup_{a\\in M_1}|\\varphi((1-p_i)a(1-p_i))|\\\\\n\\le\\varepsilon\n\\end{gathered}\n\\] for \\(i\\ge i_0\\) and all \\(\\varphi\\in K\\).\n\n(6)\\(\\Rightarrow\\)(5): let \\((p_i)\\) increase to \\(p\\). Then \\(q_i=p_i+(1-p)\\) increases to \\(1\\), and \\(1-q_i=p-p_i\\). By (6), \\[\n\\begin{gathered}\n|\\varphi(p)-\\varphi(p_i)|\\\\\n=|\\varphi(p-p_i)|\\\\\n\\le\\|(p-p_i)\\varphi(p-p_i)\\|\\to0\n\\end{gathered}\n\\] uniformly in \\(\\varphi\\in K\\).\n\n(5)\\(\\Rightarrow\\)(3): if \\(p_n\\downarrow0\\), then \\(1-p_n\\uparrow1\\). By (5), \\(\\varphi(1-p_n)\\) converges uniformly, and its pointwise limit is \\(\\varphi(1)\\) by normality. So \\(\\varphi(p_n)\\to0\\) uniformly. \\(\\square\\)\n\n**Proof of (3)\\(\\Leftrightarrow\\)(7).** (3)\\(\\Rightarrow\\)(7): for orthogonal \\((q_n)\\), \\(r_m=\\sum_{n\\ge m}q_n\\) decreases to \\(0\\), and \\(\\varphi(q_m)=\\varphi(r_m)-\\varphi(r_{m+1})\\). (7)\\(\\Rightarrow\\)(3): suppose \\(p_n\\downarrow0\\) but \\(\\sup_{\\varphi\\in K}|\\varphi(p_n)|\\not\\to0\\). Then there are \\(\\varepsilon>0\\), indices \\(n_1<n_2<\\cdots\\) and \\(\\varphi_j\\in K\\) with \\(|\\varphi_j(p_{n_j})|\\ge\\varepsilon\\). Since each \\(\\varphi_j\\) is normal, \\(\\varphi_j(p_n)\\to0\\) as \\(n\\to\\infty\\); choosing the indices and functionals inductively, we may assume \\(|\\varphi_j(p_{n_{j+1}})|<\\varepsilon/2\\). The projections \\(q_j=p_{n_j}-p_{n_{j+1}}\\) are mutually orthogonal and \\(|\\varphi_j(q_j)|>\\varepsilon/2\\), which contradicts (7). \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(3).** If \\(p_n\\downarrow0\\), then \\(\\omega(p_n^*p_n+p_np_n^*)=2\\omega(p_n)\\to0\\). \\(\\square\\)\n\n**Proof of (3)\\(\\Rightarrow\\)(1).** By the Eberlein–Šmulian theorem we only need a weak cluster point in \\(M_*\\) for each sequence \\((\\varphi_n)\\) in \\(K\\). Let \\(\\varphi\\in M^*\\) be a \\(\\sigma(M^*,M)\\)-cluster point, which exists by the Banach–Alaoglu theorem. On \\(M_*\\), \\(\\sigma(M^*,M)\\) is the weak topology, so it suffices to show that \\(\\varphi\\) is normal. Put \\(\\omega=\\sum_n2^{-n}\\varphi_n^\\sharp\\in M_*^+\\).\n\n(a) If \\(q\\) is a projection with \\(\\omega(q)=0\\), then \\(\\varphi_n(q)=0\\) for all \\(n\\) (Lemma 8.1(3)), hence \\(\\varphi(q)=0\\).\n\n(b) If projections \\(r_m\\) decrease to \\(0\\), then \\(|\\varphi(r_m)|\\le\\sup_n|\\varphi_n(r_m)|\\to0\\) by (3).\n\nLet \\((p_i)_{i\\in I}\\) be mutually orthogonal with sum \\(p\\). Since \\(\\omega\\) is normal, \\(\\sum_i\\omega(p_i)=\\omega(p)<\\infty\\), so \\(I_0=\\{i:\\omega(p_i)>0\\}\\) is countable. Every projection \\(q\\) under \\(\\sum_{i\\notin I_0}p_i\\) has \\(\\omega(q)=0\\), hence \\(\\varphi(q)=0\\) by (a). For \\(J\\subseteq I_0\\), enumerate \\(J=\\{j_1,j_2,\\dots\\}\\); the projections \\(r_m=\\sum_{k\\ge m}p_{j_k}\\) decrease to \\(0\\), so by (b), \\(\\varphi(\\sum_{i\\in J}p_i)=\\lim_m\\sum_{k<m}\\varphi(p_{j_k})\\). Applying this to \\(J=\\{i\\in I_0:\\operatorname{Re}\\varphi(p_i)\\ge0\\}\\) and to the three analogous sets shows that \\(\\sum_{i\\in I_0}|\\varphi(p_i)|<\\infty\\). Now let \\(F\\subseteq I\\) be finite. Then \\(p-\\sum_{i\\in F}p_i\\) is the sum of a projection under \\(\\sum_{i\\notin I_0}p_i\\) and of \\(\\sum_{i\\in I_0\\setminus F}p_i\\), so\n\\[\n\\begin{gathered}\n\\Big|\\varphi(p)-\\sum_{i\\in F}\\varphi(p_i)\\Big|\\\\\n=\\Big|\\sum_{i\\in I_0\\setminus F}\\varphi(p_i)\\Big|\\\\\n\\le\\sum_{i\\in I_0\\setminus F}|\\varphi(p_i)|,\n\\end{gathered}\n\\]\nwhich tends to \\(0\\) along the finite sets \\(F\\). So \\(\\varphi\\) is completely additive, hence normal (background fact 23). \\(\\square\\)\n\nThe remaining implication rests on the following two lemmas.\n\n**Lemma 10.3.** Let \\((\\varphi_k)\\) be a sequence in \\(M_*\\) converging weakly to \\(\\varphi_0\\in M_*\\), and let \\((a_n)\\) be a sequence in \\(M_1\\) converging \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\). Then \\(\\sup_k|\\varphi_k(a_n)|\\to0\\) as \\(n\\to\\infty\\).\n\n**Proof.** *Reduction.* Let \\(c=\\sup_k\\|\\varphi_k\\|<\\infty\\). Put \\(\\omega=\\varphi_0^\\sharp+\\sum_k2^{-k}\\varphi_k^\\sharp\\in M_*^+\\) and \\(e=s(\\omega)\\). For every \\(k\\ge0\\), \\(s(\\varphi_k^\\sharp)\\le e\\), so \\(\\varphi_k(x)=\\varphi_k(exe)\\) by Lemma 8.1(2). The restriction of \\(\\omega\\) to the von Neumann algebra \\(N=eMe\\) is faithful; the restrictions \\(\\varphi_k|_N\\) converge weakly to \\(\\varphi_0|_N\\); the elements \\(ea_ne\\in N_1\\) converge \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\); and \\(\\varphi_k(a_n)=\\varphi_k(ea_ne)\\). So it is enough to treat a faithful \\(\\omega\\). Let \\(d\\) be the metric of Proposition 9.1 for \\(\\omega\\).\n\n*Baire.* Put \\(\\psi_k=\\varphi_k-\\varphi_0\\), so \\(\\psi_k\\to0\\) weakly and \\(\\|\\psi_k\\|\\le2c\\). Fix \\(\\varepsilon>0\\) and let \\(S_m=\\{a\\in M_1:|\\psi_k(a)|\\le\\varepsilon\\text{ for all }k\\ge m\\}\\). Each \\(\\psi_k\\) is \\(\\sigma\\)-strongly continuous, hence \\(d\\)-continuous on \\(M_1\\), so \\(S_m\\) is \\(d\\)-closed; and \\(\\bigcup_mS_m=M_1\\). By Baire's theorem, some \\(S_{m_0}\\) contains \\(\\{a\\in M_1:d(a,a_0)<\\delta\\}\\) for some \\(a_0\\in M_1\\), \\(\\delta>0\\).\n\n*Cutting \\(a_n\\).* Let \\(h_n=a_n^*a_n+a_na_n^*\\), so \\(0\\le h_n\\le2\\) and \\(\\omega(h_n)\\to0\\). Let \\(p_n=1_{[0,\\varepsilon^2]}(h_n)\\), a spectral projection. Then \\(\\|p_nh_np_n\\|\\le\\varepsilon^2\\), which gives \\(\\|a_np_n\\|\\le\\varepsilon\\) and \\(\\|p_na_n\\|\\le\\varepsilon\\). Also \\(1-p_n\\le\\varepsilon^{-2}h_n\\), so \\(d(p_n,1)^2=\\omega(1-p_n)\\le\\varepsilon^{-2}\\omega(h_n)\\to0\\), and \\(p_n\\to1\\) \\(\\sigma\\)-strongly. Write\n\\[\n\\begin{gathered}\na_n\\\\\n=p_na_n+(1-p_n)a_np_n+c_n,\\\\\nc_n\\\\\n=(1-p_n)a_n(1-p_n).\n\\end{gathered}\n\\]\nThe first two terms have norm at most \\(\\varepsilon\\), so \\(|\\psi_k(a_n)|\\le4c\\varepsilon+|\\psi_k(c_n)|\\).\n\n*Comparing with \\(a_0\\).* Put \\(b_n=p_na_0p_n+c_n\\). It lies in \\(M_1\\): \\(b_n^*b_n=p_na_0^*p_na_0p_n+c_n^*c_n\\) is a sum of two positive contractions with orthogonal supports. As \\(n\\to\\infty\\), \\(p_na_0p_n\\to a_0\\) \\(\\sigma\\)-strongly (joint continuity of multiplication on bounded sets), and \\(d(c_n,0)^2=\\omega(c_n^*c_n)\\le\\omega(1-p_n)\\to0\\). So \\(d(p_na_0p_n,a_0)\\to0\\) and \\(d(b_n,a_0)\\to0\\), and there is \\(n_0\\) with both distances below \\(\\delta\\) for \\(n\\ge n_0\\). For \\(k\\ge m_0\\) and \\(n\\ge n_0\\),\n\\[\n\\begin{gathered}\n|\\psi_k(c_n)|\\\\\n=|\\psi_k(b_n)-\\psi_k(p_na_0p_n)|\\\\\n\\le2\\varepsilon ,\\\\\n\\text{so}\\\\\n|\\psi_k(a_n)|\\\\\n\\le(4c+2)\\varepsilon .\n\\end{gathered}\n\\]\nFor each of the finitely many \\(k<m_0\\), \\(\\psi_k(a_n)\\to0\\), because \\(\\psi_k\\) is normal and a bounded \\(\\sigma\\)-strongly\\(^*\\) null sequence is \\(\\sigma\\)-weakly null. Hence \\(\\limsup_n\\sup_k|\\psi_k(a_n)|\\le(4c+2)\\varepsilon\\) for every \\(\\varepsilon\\). Finally \\(\\varphi_k(a_n)=\\psi_k(a_n)+\\varphi_0(a_n)\\) and \\(\\varphi_0(a_n)\\to0\\). \\(\\square\\)\n\n**Lemma 10.4.** Let \\(K\\subseteq M_*\\) be relatively weakly compact and \\(\\varepsilon>0\\). There are a finite set \\(F\\subseteq K\\) and \\(\\delta>0\\) such that: if \\(a\\in M_1\\) and \\(\\psi^\\sharp(a^*a+aa^*)<\\delta\\) for all \\(\\psi\\in F\\), then \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\).\n\n**Proof.** If \\(K=\\varnothing\\), take \\(F=\\varnothing\\) and any \\(\\delta>0\\). If \\(K=\\{0\\}\\), the conclusion is also immediate. Otherwise \\(K\\) is bounded; scaling, we may assume \\(\\|\\varphi\\|\\le1\\) on \\(K\\). Suppose the claim fails for some \\(\\varepsilon\\). Choose \\(\\varphi_1\\in K\\). Given \\(\\varphi_1,\\dots,\\varphi_n\\), the failure for \\(F=\\{\\varphi_1,\\dots,\\varphi_n\\}\\) and \\(\\delta=2^{-n}\\) gives \\(a_n\\in M_1\\) and \\(\\varphi_{n+1}\\in K\\) with\n\\[\n\\begin{gathered}\n\\varphi_k^\\sharp(a_n^*a_n+a_na_n^*)<2^{-n}\\ (k\\\\\n\\le n)\\\\\n\\text{and}\\\\\n|\\varphi_{n+1}(a_n)|\\\\\n\\ge\\varepsilon .\n\\end{gathered}\n\\]\nLet \\(\\omega=\\sum_k2^{-k}\\varphi_k^\\sharp\\) and \\(e=s(\\omega)\\). Since \\(\\varphi_k^\\sharp(a^*a+aa^*)\\le2\\|\\varphi_k^\\sharp\\|\\le4\\) for \\(a\\in M_1\\),\n\\[\n\\begin{gathered}\n\\omega(a_n^*a_n+a_na_n^*)\\\\\n\\le2^{-n}+\\sum_{k>n}4\\cdot2^{-k}\\\\\n=5\\cdot2^{-n}\\to0 .\n\\end{gathered}\n\\]\nIn \\(N=eMe\\) the functional \\(\\omega\\) is faithful, and \\(b_n=ea_ne\\) has \\(\\omega(b_n^*b_n+b_nb_n^*)\\le\\omega(a_n^*a_n+a_na_n^*)\\), because \\(b_n^*b_n\\le ea_n^*a_ne\\), \\(b_nb_n^*\\le ea_na_n^*e\\) and \\(\\omega(eye)=\\omega(y)\\). By Proposition 9.1, \\(b_n\\to0\\) \\(\\sigma\\)-strongly\\(^*\\) in \\(N\\). By the Eberlein–Šmulian theorem, \\((\\varphi_n)\\) has a weakly convergent subsequence \\((\\varphi_{n_j})\\), and Lemma 10.3 in \\(N\\) gives \\(\\sup_j|\\varphi_{n_j}(b_n)|\\to0\\) as \\(n\\to\\infty\\). But \\(\\varphi_{n_j}(b_n)=\\varphi_{n_j}(a_n)\\) (Lemma 8.1(2)), and \\(|\\varphi_{n_j}(a_{n_j-1})|\\ge\\varepsilon\\) for all \\(j\\) with \\(n_j\\ge2\\). As \\(n_j-1\\to\\infty\\), this is a contradiction. \\(\\square\\)\n\n**Proof of (1)\\(\\Rightarrow\\)(4).** For \\(m\\ge1\\), Lemma 10.4 with \\(\\varepsilon=1/m\\) gives a finite \\(F_m\\subseteq K\\) and \\(\\delta_m>0\\). Put \\(\\gamma_m=\\big(1+\\sum_{\\psi\\in F_m}\\|\\psi^\\sharp\\|\\big)^{-1}\\) and\n\\[\n\\omega=\\sum_m2^{-m}\\gamma_m\\sum_{\\psi\\in F_m}\\psi^\\sharp\\in M_*^+ .\n\\]\nGiven \\(\\varepsilon>0\\), choose \\(m>1/\\varepsilon\\) and \\(\\delta=2^{-m}\\gamma_m\\delta_m\\). If \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\), then \\(\\psi^\\sharp(a^*a+aa^*)<\\delta_m\\) for every \\(\\psi\\in F_m\\), so \\(|\\varphi(a)|<1/m<\\varepsilon\\) for all \\(\\varphi\\in K\\). \\(\\square\\)\n\nThis completes the proof of Theorem 10.2.\n\n**Remark 10.5.** For a \\(C^*\\)-algebra \\(A\\), apply Theorem 10.2 to \\(M=\\tilde A\\), whose predual is \\(A^*\\). It characterizes the relatively weakly compact subsets of \\(A^*\\), in terms of projections of the bidual.\n\nAbsolute values need not stay in a relatively weakly compact set (Example 10.7). One-sided multiples detect exactly when they do.\n\n**Proposition 10.6.** Let \\(K\\subseteq M_*\\) and put \\(|K|=\\{|\\varphi|:\\varphi\\in K\\}\\), \\(|K^*|=\\{|\\varphi^*|:\\varphi\\in K\\}\\), \\(M_1K=\\{a\\varphi:a\\in M_1,\\ \\varphi\\in K\\}\\) and \\(KM_1=\\{\\varphi a:a\\in M_1,\\ \\varphi\\in K\\}\\).\n\n1. \\(M_1K\\) is relatively weakly compact exactly when \\(|K|\\) is.\n2. \\(KM_1\\) is relatively weakly compact exactly when \\(|K^*|\\) is.\n3. In particular, if \\(K\\subseteq M_*^+\\) is relatively weakly compact, so are \\(M_1K\\) and \\(KM_1\\).\n\n**Proof.** (1) If \\(\\varphi=v|\\varphi|\\), then \\(|\\varphi|=v^*\\varphi\\in M_1K\\) and \\(a\\varphi=(av)|\\varphi|\\in M_1|K|\\). So it suffices to show that \\(M_1L\\) is relatively weakly compact when \\(L\\subseteq M_*^+\\) is. It is bounded. If \\(p_n\\downarrow0\\), then for \\(\\omega\\in L\\) and \\(a\\in M_1\\), by the Cauchy–Schwarz inequality,\n\\[\n\\begin{gathered}\n|(a\\omega)(p_n)|\\\\\n=|\\omega(p_na)|\\\\\n\\le\\omega(p_n)^{1/2}\\omega(a^*a)^{1/2}\\\\\n\\le\\|\\omega\\|^{1/2}\\omega(p_n)^{1/2}.\n\\end{gathered}\n\\]\nBy Theorem 10.2 (1)\\(\\Rightarrow\\)(3) for \\(L\\), the right side tends to \\(0\\) uniformly, and (3)\\(\\Rightarrow\\)(1) applies to \\(M_1L\\).\n\n(2) The map \\(\\psi\\mapsto\\psi^*\\) is a conjugate-linear isometry of \\(M_*\\) onto itself and a homeomorphism for the weak topology. Since \\(\\varphi a=(a^*\\varphi^*)^*\\), \\(KM_1=(M_1K^*)^*\\), and (1) for \\(K^*\\) gives (2). (3) For positive \\(\\varphi\\), \\(|\\varphi|=|\\varphi^*|=\\varphi\\). \\(\\square\\)\n\n**Example 10.7** (Absolute values can escape). Let \\(H\\) have an orthonormal sequence \\((\\xi_n)\\), \\(M=B(H)\\), and \\(\\varphi_n=\\omega_{\\xi_1,\\xi_n}\\), so \\(\\varphi_n(x)=\\langle x\\xi_1,\\xi_n\\rangle\\).\n\n1. \\(\\sum_n|\\varphi_n(x)|^2\\le\\|x\\xi_1\\|^2\\), so \\(\\varphi_n\\to0\\) weakly, and \\(K=\\{\\varphi_n\\}\\) is relatively weakly compact. But \\(\\|\\varphi_n\\|=1\\) (Example 1.2).\n2. By (2.5), \\(|\\varphi_n|=\\omega_{\\xi_n}\\). The set \\(|K|\\) is not relatively weakly compact: \\(p_m=\\sum_{j\\ge m}\\theta_{\\xi_j,\\xi_j}\\) decreases to \\(0\\) (strongly), while \\(\\omega_{\\xi_n}(p_m)=1\\) for \\(n\\ge m\\), against Theorem 10.2(3). By Proposition 10.6, \\(M_1K\\) is not relatively weakly compact either.\n3. \\(\\varphi_n^*=\\omega_{\\xi_n,\\xi_1}\\) and \\(|\\varphi_n^*|=\\omega_{\\xi_1}\\) for all \\(n\\). So \\(|K^*|\\) is a single point, and \\(KM_1\\) is relatively weakly compact.\n4. Two-sided multiples of a single positive functional need not form a relatively weakly compact set: \\(\\omega_{\\xi_n}=a_n\\omega_{\\xi_1}a_n^*\\) with \\(a_n=\\theta_{\\xi_n,\\xi_1}\\in M_1\\).\n\nIn the commutative case absolute values cannot escape.\n\n**Proposition 10.8.** If \\(M\\) is abelian and \\(K\\subseteq M_*\\) is relatively weakly compact, then so is \\(|K|\\).\n\n**Proof.** Let \\(\\omega\\) be as in Theorem 10.2(4). For \\(\\varphi=v|\\varphi|\\in K\\) and \\(a\\in M_1\\), \\(|\\varphi|(a)=\\varphi(av^*)\\). In the abelian algebra \\(M\\), with \\(b=av^*\\in M_1\\),\n\\[\nb^*b+bb^*=a^*a\\,vv^*+aa^*\\,v^*v\\le a^*a+aa^* ,\n\\]\nbecause \\(vv^*\\) and \\(v^*v\\) are projections that commute with \\(a^*a\\) and \\(aa^*\\). So \\(\\omega(b^*b+bb^*)<\\delta\\) whenever \\(\\omega(a^*a+aa^*)<\\delta\\), and then \\(||\\varphi|(a)|=|\\varphi(b)|<\\varepsilon\\). Thus \\(|K|\\) satisfies Theorem 10.2(4) with the same \\(\\omega\\). \\(\\square\\)\n\n",
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      "full_conditions_and_proof": "## 11. The Mackey topology on bounded sets\n\n**Definition 11.1.** On a von Neumann algebra \\(M\\), let \\(\\tau\\) be the locally convex topology defined by the seminorms\n\\[\np_K(x)=\\sup_{\\varphi\\in K}|\\varphi(x)|,\n\\]\nwhere \\(K\\) runs over the relatively weakly compact subsets of \\(M_*\\), with \\(p_\\varnothing=0\\). It is Hausdorff, since singletons are compact. We also define the *Mackey topology* \\(\\tau(M,M_*)\\) directly as uniform convergence on the absolutely convex weakly compact subsets of \\(M_*\\). Thus its seminorms form a subfamily of these \\(p_K\\). The finest-compatible-topology characterization is not needed.\n\n**Theorem 11.2.** On every bounded subset of \\(M\\), the topology \\(\\tau\\) coincides with the \\(\\sigma\\)-strong\\(^*\\) topology. More precisely, the \\(\\sigma\\)-strong\\(^*\\) topology is coarser than \\(\\tau\\) on all of \\(M\\), and on bounded sets \\(\\tau\\) is coarser than the \\(\\sigma\\)-strong\\(^*\\) topology.\n\n\n**Proof.** *\\(\\sigma\\)-strong\\(^*\\) is coarser than \\(\\tau\\).* Let \\(\\omega\\in M_*^+\\) with cyclic representation \\((\\pi,H,\\xi)\\), which is normal. The sets\n\\[\n\\begin{gathered}\nK_\\omega\\\\\n=\\{x\\mapsto\\langle\\pi(x)\\xi,\\zeta\\rangle:\\ \\|\\zeta\\|\\le1\\},\\\\\nK'_\\omega\\\\\n=\\{x\\mapsto\\langle\\pi(x)\\zeta,\\xi\\rangle:\\ \\|\\zeta\\|\\le1\\}\n\\end{gathered}\n\\]\nare absolutely convex and weakly compact in \\(M_*\\): they are images of the weakly compact unit ball of \\(H\\) under maps that are continuous from the weak topology of \\(H\\) to \\(\\sigma(M_*,M)\\). By Cauchy–Schwarz, with equality at \\(\\zeta=\\pi(x)\\xi/\\|\\pi(x)\\xi\\|\\) when the denominator is nonzero (and both sides zero otherwise),\n\\[\n\\begin{gathered}\n\\omega(x^*x)^{1/2}\\\\\n=\\|\\pi(x)\\xi\\|\\\\\n=p_{K_\\omega}(x),\\\\\n\\omega(xx^*)^{1/2}\\\\\n=\\|\\pi(x^*)\\xi\\|\\\\\n=p_{K'_\\omega}(x).\n\\end{gathered}\n\\]\nSo every \\(\\sigma\\)-strong\\(^*\\) seminorm is dominated by \\(p_{K_\\omega}+p_{K'_\\omega}\\).\n\n*On bounded sets \\(\\tau\\) is coarser.* Let \\(D\\subseteq M\\) be bounded, with \\(\\|x\\|\\le R\\) on \\(D\\), and let \\(x_i\\to x\\) \\(\\sigma\\)-strongly\\(^*\\) in \\(D\\). If \\(R=0\\), then \\(D\\subseteq\\{0\\}\\) and there is nothing to prove. For \\(R>0\\), put \\(a_i=(x_i-x)/(2R)\\in M_1\\). Let \\(K\\) be relatively weakly compact, \\(\\varepsilon>0\\), and \\(\\omega,\\delta\\) as in Theorem 10.2(4). Since \\(\\omega(a_i^*a_i+a_ia_i^*)\\to0\\), eventually \\(\\omega(a_i^*a_i+a_ia_i^*)<\\delta\\), and then \\(p_K(x_i-x)=2Rp_K(a_i)\\le2R\\varepsilon\\). So \\(x_i\\to x\\) in \\(\\tau\\). \\(\\square\\)\n\n**Remark 11.3.** By the direct definition above, \\(\\tau(M,M_*)\\) is coarser than \\(\\tau\\). The absolutely convex weakly compact sets \\(K_\\omega,K'_\\omega\\) in the proof show directly that \\(\\tau(M,M_*)\\) is finer than the \\(\\sigma\\)-strong\\(^*\\) topology. Theorem 11.2 therefore proves, without an additional theorem, that all three topologies coincide on bounded sets. The Mackey–Arens theorem further identifies \\(\\tau(M,M_*)\\) as the finest locally convex topology whose continuous functionals are the normal ones. That compatibility characterization is unproved here, is only further reading (background item 10), and is not used by any proof or solution.\n\nOn unbounded sets the topologies differ, unless \\(M\\) is finite-dimensional.\n\n**Proposition 11.4.** Let \\(M\\) contain an infinite sequence \\((e_n)\\) of mutually orthogonal nonzero projections, and put \\(D=\\{\\sqrt n\\,e_n:n\\ge1\\}\\). Then \\(0\\) lies in the \\(\\sigma\\)-strong\\(^*\\) closure of \\(D\\), but not in its closure for \\(\\tau(M,M_*)\\) or for \\(\\tau\\). Consequently the \\(\\sigma\\)-strong\\(^*\\) topology equals the Mackey topology on all of \\(M\\) exactly when \\(M\\) is finite-dimensional.\n\n**Proof.** A basic \\(\\sigma\\)-strong\\(^*\\) neighbourhood of \\(0\\) contains \\(\\{x:\\omega(x^*x+xx^*)<\\delta\\}\\) for some \\(\\omega\\in M_*^+\\) and \\(\\delta>0\\), since finitely many functionals can be added. As \\(\\sum_n\\omega(e_n)\\le\\omega(1)<\\infty\\), the numbers \\(n\\,\\omega(e_n)\\) are not bounded below by a positive constant, so \\(2n\\,\\omega(e_n)<\\delta\\) for some \\(n\\). Then \\[\n\\begin{gathered}\n\\omega((\\sqrt ne_n)^*(\\sqrt ne_n)+(\\sqrt ne_n)(\\sqrt ne_n)^*)\\\\\n=2n\\,\\omega(e_n)<\\delta.\n\\end{gathered}\n\\]\n\nFor the other claim, choose normal states \\(\\omega_n\\) with \\(\\omega_n(e_n)=1\\) (for instance \\(e_n\\psi e_n/\\psi(e_n)\\) for a normal state \\(\\psi\\) with \\(\\psi(e_n)>0\\)), and put \\(\\varphi_n=n^{-1/2}\\omega_n\\), so \\(\\|\\varphi_n\\|\\to0\\). The operator \\(T:\\ell^1\\to M_*\\), \\(T\\lambda=\\sum_n\\lambda_n\\varphi_n\\), is compact. Indeed its truncations \\(T_N\\) have finite-dimensional bounded images of the unit ball and \\(\\|T-T_N\\|\\le\\sup_{n>N}\\|\\varphi_n\\|\\to0\\). Finite nets for the truncated images give finite nets for \\(T\\) of the unit ball; its closure is complete, since \\(M_*\\) is Banach, and is therefore compact by the metric lemma above. So the closure \\(K\\) of \\(T(\\text{unit ball of }\\ell^1)\\) is absolutely convex and norm compact, hence weakly compact, and it contains every \\(\\varphi_n\\). Now \\(p_K(\\sqrt ne_n)\\ge|\\varphi_n(\\sqrt ne_n)|=\\omega_n(e_n)=1\\), so the neighbourhood \\(\\{x:p_K(x)<1\\}\\) of \\(0\\) misses \\(D\\), for \\(\\tau(M,M_*)\\) and hence for the finer \\(\\tau\\).\n\nIf \\(\\dim M=\\infty\\), such a sequence \\((e_n)\\) exists (background fact 25), so the topologies differ. If \\(\\dim M<\\infty\\), both are Hausdorff vector topologies on a finite-dimensional space and coincide (background fact 8). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    },
    {
      "id": "OA-FND-PD-20",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "12. Norm convergence in atomic algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
      "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "anchor": "oa-fnd-pd-20",
      "proof_locus": {
        "line": 1022,
        "through_line": 1076
      },
      "full_conditions_and_proof": "## 12. Norm convergence in atomic algebras\n\nThe pairs \\(c_0\\subseteq\\ell^\\infty\\) with predual \\(\\ell^1\\), and compact operators inside \\(B(H)\\) with predual \\(B(H)_*\\), are the two basic atomic examples. Schur's theorem says that in \\(\\ell^1\\) weak and norm convergence of sequences agree. In \\(B(H)_*\\) they do not (Example 10.7). The next theorem locates the difference: it disappears once the absolute values are under control.\n\n**Definition 12.1.** A von Neumann algebra is *atomic* if every nonzero projection majorizes a minimal projection (background fact 26). A projection is of *finite rank* if it is a sum of finitely many mutually orthogonal minimal projections.\n\nIn an atomic algebra \\(1=\\sum_jf_j\\) for a family of mutually orthogonal minimal projections (background fact 26). The join of two finite-rank projections is of finite rank (background fact 25), so the finite-rank projections form an increasing net with supremum \\(1\\). For a finite-rank \\(e=f_1+\\dots+f_m\\), \\(\\dim eMe\\le m^2\\), since \\(\\dim f_iMf_j\\le1\\).\n\n**Theorem 12.2.** Let \\(M\\) be atomic, and let \\((\\varphi_i)\\) be a net in \\(M_*\\) converging weakly to \\(\\varphi\\in M_*\\) such that \\(\\{|\\varphi_i|\\}\\) and \\(\\{|\\varphi_i^*|\\}\\) are relatively weakly compact. Then \\(\\|\\varphi_i-\\varphi\\|\\to0\\).\n\n**Proof.** The set \\(K=\\{|\\varphi|,|\\varphi^*|\\}\\cup\\{|\\varphi_i|,|\\varphi_i^*|\\}\\) is relatively weakly compact, and \\(c=\\sup_{\\psi\\in K}\\|\\psi\\|<\\infty\\); note \\(\\|\\varphi_i\\|=\\||\\varphi_i|\\|\\le c\\). Let \\(0<\\varepsilon<1\\). By Theorem 10.2(6) applied to the net of finite-rank projections, there is a finite-rank \\(e\\) with \\(\\|(1-e)\\psi(1-e)\\|<\\varepsilon\\), in particular \\(\\psi(1-e)<\\varepsilon\\), for all \\(\\psi\\in K\\).\n\nSince \\(eMe\\) is finite-dimensional, weak convergence of the restrictions to \\(eMe\\) is norm convergence (background fact 8). The norm of \\(x\\mapsto(\\varphi_i-\\varphi)(exe)\\) on \\(M\\) is the norm of the restriction of \\(\\varphi_i-\\varphi\\) to \\(eMe\\). So there is \\(i_0\\) with \\(\\|e(\\varphi_i-\\varphi)e\\|<\\varepsilon\\) for \\(i\\ge i_0\\).\n\nFor \\(\\psi\\in M_*\\) with \\(\\psi=u|\\psi|\\) and \\(a\\in M_1\\), the Cauchy–Schwarz inequality gives\n\\[\n\\begin{gathered}\n|\\psi((1-e)a)|\\\\\n=||\\psi|((1-e)au)|\\\\\n\\le|\\psi|(1-e)^{1/2}\\,|\\psi|(u^*a^*au)^{1/2}\\\\\n\\le\\big(|\\psi|(1-e)\\,\\|\\psi\\|\\big)^{1/2}.\n\\end{gathered}\n\\tag{12.1}\n\\]\nAlso \\(|\\psi(ea(1-e))|=|\\psi^*((1-e)a^*e)|\\), to which (12.1) applies with \\(\\psi^*\\) and \\(a^*e\\). Now split \\(a=eae+ea(1-e)+(1-e)a\\). For \\(i\\ge i_0\\) and \\(a\\in M_1\\),\n\\[\n\\begin{gathered}\n|(\\varphi-\\varphi_i)(a)|\\\\\n\\le\\varepsilon+|\\varphi(ea(1-e))|\\\\\n+|\\varphi_i(ea(1-e))|\\\\\n+|\\varphi((1-e)a)|\\\\\n+|\\varphi_i((1-e)a)|\\\\\n\\le\\varepsilon+4(c\\varepsilon)^{1/2},\n\\end{gathered}\n\\]\nsince \\(|\\varphi|,|\\varphi^*|,|\\varphi_i|,|\\varphi_i^*|\\in K\\). So \\(\\|\\varphi-\\varphi_i\\|\\le\\varepsilon+4(c\\varepsilon)^{1/2}\\) for \\(i\\ge i_0\\). \\(\\square\\)\n\nFor positive functionals no compactness hypothesis is needed.\n\n**Corollary 12.3.** Let \\(M\\) be atomic and \\((\\varphi_i)\\) a net in \\(M_*^+\\) converging weakly to \\(\\varphi\\). Then \\(\\|\\varphi_i-\\varphi\\|\\to0\\).\n\n**Proof.** \\(\\varphi\\) is positive and \\(\\|\\varphi_i\\|=\\varphi_i(1)\\to\\varphi(1)=\\|\\varphi\\|\\), so eventually \\(\\|\\varphi_i\\|\\le c=\\|\\varphi\\|+1\\). Given \\(\\varepsilon>0\\), normality of \\(\\varphi\\) gives a finite-rank \\(e\\) with \\(\\varphi(1-e)<\\varepsilon\\); then eventually \\(\\varphi_i(1-e)<\\varepsilon\\). Here \\(|\\varphi_i|=|\\varphi_i^*|=\\varphi_i\\), so the estimates in the proof of Theorem 12.2 apply and give \\(\\|\\varphi-\\varphi_i\\|\\le\\varepsilon+4(c\\varepsilon)^{1/2}\\) eventually. \\(\\square\\)\n\n**Corollary 12.4** (Schur's theorem again). A weakly convergent sequence in \\(\\ell^1(\\Gamma)\\) converges in norm.\n\n**Proof.** \\(\\ell^1(\\Gamma)\\) is the predual of the atomic abelian algebra \\(\\ell^\\infty(\\Gamma)\\) (Example 2.6). A weakly convergent sequence together with its limit is weakly compact. By Proposition 10.8 its absolute values form a relatively weakly compact set. Here \\(|\\varphi^*|=|\\varphi|\\), since the adjoint of \\(g\\in\\ell^1(\\Gamma)\\) is \\(\\bar g\\). Theorem 12.2 applies. \\(\\square\\)\n\n**Example 12.5.** The hypotheses of Theorem 12.2 and Corollary 12.3 cannot be dropped.\n\nFor the first example, the Rademacher function \\(r_n\\) has constant value \\(+1\\) or \\(-1\\) on each interval of length \\(2^{-n}\\), alternately. Thus \\(\\|r_n\\|_2=1\\). If \\(m>n\\), each interval on which \\(r_n\\) is constant contains an even number of the alternating intervals for \\(r_m\\), so \\(\\int r_nr_m\\,dt=0\\). Endpoint values affect only a null set. This proves the asserted orthonormality.\n\n1. *Atomicity.* Let \\(M=L^\\infty[0,1]\\) and let \\(r_n(t)=\\operatorname{sgn}\\sin(2^n\\pi t)\\) be the Rademacher functions, an orthonormal sequence in \\(L^2[0,1]\\). For \\(x\\in L^\\infty\\subseteq L^2\\), Bessel's inequality gives \\(\\int xr_n\\,dt\\to0\\). So \\(\\varphi_n=\\varphi_{r_n}\\to0\\) weakly, and by Example 2.6, \\(|\\varphi_n|=|\\varphi_n^*|=\\varphi_{|r_n|}=\\lambda\\), Lebesgue measure, for every \\(n\\). But \\(\\|\\varphi_n\\|=\\int|r_n|=1\\). Likewise the positive functionals \\(\\varphi_{1+r_n}\\) tend weakly to \\(\\lambda\\), while \\(\\|\\varphi_{1+r_n}-\\lambda\\|=1\\).\n2. *Both absolute values.* In Example 10.7, \\(M=B(H)\\) is atomic, \\(\\varphi_n\\to0\\) weakly, and \\(\\{|\\varphi_n^*|\\}\\) is a single point, but \\(\\|\\varphi_n\\|=1\\). Passing to adjoints gives an example where \\(\\{|\\varphi_n|\\}\\) is compact and \\(\\{|\\varphi_n^*|\\}\\) is not.\n3. *Positivity.* The same example shows that Corollary 12.3 fails for sequences that are not positive.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      ]
    },
    {
      "id": "OA-FND-PD-21",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "13. Extending normal functionals from subalgebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
      "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "anchor": "oa-fnd-pd-21",
      "proof_locus": {
        "line": 1077,
        "through_line": 1093
      },
      "full_conditions_and_proof": "## 13. Extending normal functionals from subalgebras\n\nA normal functional on a von Neumann subalgebra extends to a normal functional on the whole algebra with the same norm. For positive functionals this comes from writing them as sums of vector functionals; the polar decomposition then handles the general case.\n\n**Proposition 13.1.** Let \\(M\\) be a von Neumann algebra and \\(N\\subseteq M\\) a \\(\\sigma\\)-weakly closed \\(*\\)-subalgebra.\n\n1. Every \\(\\varphi\\in N_*^+\\) extends to some \\(\\tilde\\varphi\\in M_*^+\\), and \\(\\|\\tilde\\varphi\\|=\\|\\varphi\\|\\).\n2. Every \\(\\varphi\\in N_*\\) extends to some \\(\\tilde\\varphi\\in M_*\\) with \\(\\|\\tilde\\varphi\\|=\\|\\varphi\\|\\).\n\n**Proof.** Represent \\(M\\) faithfully and normally on \\(H\\). The subalgebra \\(N\\) may have a smaller unit. Let \\((u_\\lambda)\\) be its increasing approximate identity of positive contractions, and let \\(q\\) be the projection onto \\([NH]\\). On \\(a\\eta\\), with \\(a\\in N\\), the estimate \\(\\|(u_\\lambda a-a)\\eta\\|\\to0\\), followed by density and the uniform bound, gives \\(u_\\lambda\\to q\\) strongly on \\([NH]\\); on its orthogonal complement every element of \\(N\\) acts as zero. Bounded strong convergence is sigma-weak convergence by fact 15, so the sigma-weak closedness gives \\(q\\in N\\). Thus \\(qa=aq=a\\) for \\(a\\in N\\), and \\(N\\) acts nondegenerately on \\(qH\\). The double commutant theorem makes it a von Neumann algebra there, with unit \\(q\\), and \\(N\\subseteq qMq\\). If \\(N=0\\), take \\(q=0\\) and the assertions are immediate.\n\n(1) By background fact 27 applied to \\(N\\) on \\(qH\\), \\(\\varphi(x)=\\sum_n\\langle x\\xi_n,\\xi_n\\rangle\\) for \\(x\\in N\\), with \\(\\xi_n\\in qH\\) and \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\). The same formula defines a positive normal functional \\(\\tilde\\varphi\\) on \\(M\\), extending \\(\\varphi\\), and \\(\\|\\tilde\\varphi\\|=\\tilde\\varphi(1)=\\sum_n\\|\\xi_n\\|^2=\\varphi(q)=\\|\\varphi\\|\\).\n\n(2) Let \\(\\varphi=v|\\varphi|\\) be the polar decomposition in \\(N\\), so \\(v\\in N\\). By (1), \\(|\\varphi|\\) has a positive normal extension \\(\\psi\\) to \\(M\\) with \\(\\|\\psi\\|=\\|\\varphi\\|\\). Put \\(\\tilde\\varphi=v\\psi\\), that is \\(\\tilde\\varphi(x)=\\psi(xv)\\). For \\(x\\in N\\), \\(xv\\in N\\), so \\(\\tilde\\varphi(x)=|\\varphi|(xv)=\\varphi(x)\\). Also \\(\\|\\tilde\\varphi\\|\\le\\|v\\|\\|\\psi\\|\\le\\|\\varphi\\|\\), and \\(\\|\\tilde\\varphi\\|\\ge\\|\\varphi\\|\\) because \\(\\tilde\\varphi\\) extends \\(\\varphi\\). \\(\\square\\)\n\nThe extension is not unique in general: for \\(N=\\mathbb C1\\subseteq M_2(\\mathbb C)\\) and \\(\\varphi(\\lambda1)=\\lambda\\), every state of \\(M_2(\\mathbb C)\\) is a norm-preserving extension.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      ]
    },
    {
      "id": "OA-FND-PD-22",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "14. Weakly compact and dual C\\*-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
      "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "anchor": "oa-fnd-pd-22",
      "proof_locus": {
        "line": 1094,
        "through_line": 1205
      },
      "full_conditions_and_proof": "## 14. Weakly compact and dual C\\*-algebras\n\nFor a \\(C^*\\)-algebra \\(A\\) and \\(a\\in A\\), consider left multiplication \\(L_a:x\\mapsto ax\\) on \\(A\\). When is it a weakly compact operator? We will see that this happens for all \\(a\\) exactly when \\(A\\) is an ideal of its bidual, exactly when every closed one-sided ideal is recovered from its annihilator, and exactly when \\(A\\) is a \\(c_0\\)-direct sum of algebras of compact operators.\n\nIn this section a *minimal projection* of \\(A\\) is a nonzero projection \\(f\\in A\\) with \\(fAf=\\mathbb Cf\\). For a closed left ideal \\(\\mathfrak m\\) and a closed right ideal \\(\\mathfrak n\\) of \\(A\\) we write\n\\[\n\\begin{gathered}\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=\\{x\\in A\\,:\\,\\mathfrak mx=0\\},\\\\\n\\operatorname{ann}_l(\\mathfrak n)\\\\\n=\\{y\\in A\\,:\\,y\\mathfrak n=0\\}.\n\\end{gathered}\n\\]\nThe first is a closed right ideal and the second a closed left ideal.\n\n**Definition 14.1.** A \\(C^*\\)-algebra \\(A\\) is *weakly compact* if every left multiplication \\(L_a\\), \\(a\\in A\\), is a weakly compact operator, that is, \\(aA_1\\) is relatively weakly compact in \\(A\\). It is *dual* if \\(\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))=\\mathfrak m\\) for every closed left ideal \\(\\mathfrak m\\) and \\(\\operatorname{ann}_r(\\operatorname{ann}_l(\\mathfrak n))=\\mathfrak n\\) for every closed right ideal \\(\\mathfrak n\\).\n\n**Proposition 14.2.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: (a) every left multiplication \\(L_a\\) is weakly compact; (b) every right multiplication \\(R_a:x\\mapsto xa\\) is weakly compact; (c) \\(A\\) is a two-sided ideal of \\(\\tilde A\\).\n\n**Proof.** Fix \\(a\\in A\\). Left multiplication by \\(a\\) on \\(\\tilde A\\) is \\(\\sigma\\)-weakly continuous, and \\(A_1\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A_1\\) (background fact 14). So the \\(\\sigma(\\tilde A,A^*)\\)-closure of \\(aA_1\\) contains \\(a\\tilde A_1\\); and \\(a\\tilde A_1\\) is \\(\\sigma\\)-weakly compact and contains \\(aA_1\\). Hence that closure is \\(a\\tilde A_1\\). By Lemma 10.1 with \\(X=A\\), \\(X^{**}=\\tilde A\\), \\(L_a\\) is weakly compact exactly when \\(a\\tilde A_1\\subseteq A\\), that is \\(a\\tilde A\\subseteq A\\). So (a) says \\(A\\tilde A\\subseteq A\\), and in the same way (b) says \\(\\tilde AA\\subseteq A\\). Since \\(A\\) and \\(\\tilde A\\) are closed under the involution, each inclusion implies the other by taking adjoints, and together they say (c). \\(\\square\\)\n\n**Proposition 14.3.** Let \\(A\\) be an ideal of \\(\\tilde A\\), and \\(\\mathfrak m\\) a closed left ideal of \\(A\\) whose \\(\\sigma\\)-weak closure in \\(\\tilde A\\) is \\(\\tilde Ae\\) (background fact 19). Then\n\\[\n\\begin{gathered}\n\\mathfrak m\\\\\n=Ae\\\\\n=\\{x\\in A:xe=x\\},\\\\\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=(1-e)A,\\\\\n\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))\\\\\n=\\mathfrak m .\n\\end{gathered}\n\\]\nThe symmetric statements hold for closed right ideals. In particular \\(A\\) is dual.\n\n**Proof.** The \\(\\sigma\\)-weak closure of \\(\\mathfrak m\\) is a left ideal of \\(\\tilde A\\), since \\(A\\mathfrak m\\subseteq\\mathfrak m\\), multiplication is separately continuous and \\(A\\) is dense; so it is \\(\\tilde Ae\\) for a projection \\(e\\). If \\(x\\in\\mathfrak m\\), then \\(x\\in\\tilde Ae\\), so \\(x=xe\\). If \\(x\\in A\\), then \\(xe\\in A\\) (as \\(A\\) is an ideal) and \\(xe\\in\\tilde Ae\\), the \\(\\sigma(\\tilde A,A^*)\\)-closure of \\(\\mathfrak m\\). A norm-closed convex subset of \\(A\\) is weakly closed (Mazur), and the weak topology of \\(A\\) is the restriction of \\(\\sigma(\\tilde A,A^*)\\); so \\(xe\\in\\mathfrak m\\). This proves \\(\\mathfrak m=Ae=\\{x\\in A:xe=x\\}\\).\n\nNext, \\(y\\in\\operatorname{ann}_r(\\mathfrak m)\\) means \\(Aey=0\\); the set \\(\\{X\\in\\tilde A:Xey=0\\}\\) is \\(\\sigma\\)-weakly closed and contains \\(A\\), hence contains \\(1\\), so \\(ey=0\\). Conversely \\(ey=0\\) gives \\(\\mathfrak my=Aey=0\\). Thus \\(\\operatorname{ann}_r(\\mathfrak m)=\\{y\\in A:ey=0\\}=(1-e)A\\), using that \\(A\\) is an ideal. In the same way \\(z\\in\\operatorname{ann}_l((1-e)A)\\) exactly when \\(z(1-e)=0\\), that is \\(z\\in\\mathfrak m\\). \\(\\square\\)\n\n**Lemma 14.4.** Let \\(f\\) be a minimal projection of \\(A\\). Then \\(f\\) is minimal in \\(\\tilde A\\), and \\(\\tilde Af=Af\\).\n\n**Proof.** \\(f\\tilde Af\\) is the \\(\\sigma\\)-weak closure of \\(fAf=\\mathbb Cf\\), so it is \\(\\mathbb Cf\\). By background fact 25, \\(\\tilde Af\\) is a Hilbert space for \\(\\langle X,Y\\rangle f=Y^*X\\), with the norm of \\(\\tilde A\\). Let \\(\\omega_f\\) be the normal state with \\(fZf=\\omega_f(Z)f\\); then \\(\\langle X,Y\\rangle=\\omega_f(Y^*X)\\), so each \\(X\\mapsto\\langle X,Y\\rangle\\) is \\(\\sigma\\)-weakly continuous. The subspace \\(Af=\\{x\\in A:xf=x\\}\\) is norm closed, and it is \\(\\sigma\\)-weakly dense in \\(\\tilde Af\\), since \\(X\\mapsto Xf\\) is \\(\\sigma\\)-weakly continuous. So \\(Af\\) is weakly dense in the Hilbert space \\(\\tilde Af\\) and norm closed; closed subspaces of a Hilbert space are weakly closed, so \\(Af=\\tilde Af\\). \\(\\square\\)\n\n**Proposition 14.5** (Structure of dual algebras). Let \\(A\\) be dual.\n\n1. \\(\\mathfrak m\\mapsto\\operatorname{ann}_r(\\mathfrak m)\\) is an inclusion-reversing bijection from the closed left ideals onto the closed right ideals, with inverse \\(\\operatorname{ann}_l\\). A closed left ideal is maximal exactly when its right annihilator is a minimal nonzero closed right ideal; and symmetrically.\n2. Every nonzero closed left ideal contains \\(Af\\) for some minimal projection \\(f\\) of \\(A\\); symmetrically for right ideals.\n3. \\(A\\) is the closed linear span of the sets \\(Af\\), and also of the sets \\(fA\\), where \\(f\\) runs over the minimal projections of \\(A\\).\n4. \\(\\tilde A\\) is atomic, and every finite-rank projection of \\(\\tilde A\\) lies in \\(A\\).\n5. The finite-rank projections of \\(\\tilde A\\), ordered by size, form an increasing approximate unit of \\(A\\).\n6. \\(A\\) is an ideal of \\(\\tilde A\\).\n\n**Proof.** (1) The maps reverse inclusions and are mutually inverse by duality. \\(\\operatorname{ann}_r(0)=A\\), and \\(\\operatorname{ann}_r(A)=0\\) because \\(Ax=0\\) gives \\(x^*x=0\\). So \\(\\mathfrak m\\ne A\\) exactly when \\(\\operatorname{ann}_r(\\mathfrak m)\\ne0\\). An inclusion-reversing bijection between the proper closed left ideals and the nonzero closed right ideals matches maximal elements with minimal ones.\n\n(2) Let \\(\\mathfrak m\\ne0\\). Then \\(\\mathfrak n=\\operatorname{ann}_r(\\mathfrak m)\\) is a proper closed right ideal, since \\(\\mathfrak n=A\\) would give \\(\\mathfrak m=\\operatorname{ann}_l(A)=0\\). By Corollary 6.6 (right-handed), \\(\\mathfrak n\\) lies in a maximal closed right ideal \\(\\mathfrak n_0\\). Then \\(\\operatorname{ann}_l(\\mathfrak n_0)\\subseteq\\operatorname{ann}_l(\\mathfrak n)=\\mathfrak m\\), and \\(\\operatorname{ann}_l(\\mathfrak n_0)\\) is a minimal nonzero closed left ideal by (1). By background fact 13 it is \\(Af\\) for a minimal projection \\(f\\) of \\(A\\). The right-handed statement follows by taking adjoints.\n\n(3) Let \\(\\mathfrak l\\) be the closed span of the \\(Af\\), a closed left ideal. An element \\(x\\) lies in \\(\\operatorname{ann}_r(\\mathfrak l)\\) exactly when \\(fx=0\\) for all such \\(f\\) (use an approximate unit for one direction). If \\(\\operatorname{ann}_r(\\mathfrak l)\\ne0\\), by (2) it contains \\(fA\\) for such an \\(f\\), so \\(f=ff=0\\), which is absurd. So \\(\\operatorname{ann}_r(\\mathfrak l)=0\\) and \\(\\mathfrak l=\\operatorname{ann}_l(0)=A\\). The right-handed statement is symmetric.\n\n(4) Let \\(z\\) be the join in \\(\\tilde A\\) of all minimal projections of \\(A\\). For \\(y\\in A\\) and such \\(f\\), \\(yfz=yf\\); by (3), \\(xz=x\\) for all \\(x\\in A\\), and by density \\(z=1\\). Let \\(e\\ne0\\) be a projection of \\(\\tilde A\\). Some such \\(f\\) has \\(ef\\ne0\\), since otherwise \\(e\\le1-z=0\\). As \\(f\\) is minimal in \\(\\tilde A\\) (Lemma 14.4), \\(fef=\\mu f\\) with \\(\\mu=\\|ef\\|^2>0\\). So \\(w=\\mu^{-1/2}ef\\) is a partial isometry with \\(w^*w=f\\), and \\(q=ww^*\\le e\\) is a projection with \\(q\\tilde Aq=wf\\tilde Afw^*=\\mathbb Cq\\). Thus \\(e\\) majorizes the minimal projection \\(q\\), and \\(\\tilde A\\) is atomic. If \\(e\\) itself is minimal, then \\(q=e\\), and \\(w\\in\\tilde Af=Af\\subseteq A\\) by Lemma 14.4, so \\(e=ww^*\\in A\\). Finite-rank projections are finite sums of minimal ones.\n\n(5) The finite-rank projections form an upward directed set (background fact 25) of positive contractions in \\(A\\), by (4). For \\(x=yf\\) as in (3) and \\(e\\ge f\\) of finite rank, \\(xe=x\\). As such \\(x\\) span a dense subspace and \\(\\|e\\|\\le1\\), \\(\\|xe-x\\|\\to0\\) for all \\(x\\in A\\). Using the \\(fA\\), also \\(\\|ex-x\\|\\to0\\).\n\n(6) For \\(X\\in\\tilde A\\) and \\(x=yf\\) as in (3), \\(Xx=(Xy)f\\in\\tilde Af=Af\\subseteq A\\). By (3) and continuity, \\(XA\\subseteq A\\). So \\(A\\) is a left ideal of \\(\\tilde A\\), and, being self-adjoint, a two-sided ideal. \\(\\square\\)\n\nSo far: \\(A\\) is weakly compact \\(\\Leftrightarrow\\) \\(A\\) is an ideal of \\(\\tilde A\\) \\(\\Leftrightarrow\\) \\(A\\) is dual (Propositions 14.2, 14.3 and 14.5(6)). It remains to identify these algebras.\n\n**Definition 14.6.** For \\(C^*\\)-algebras \\((A_j)_{j\\in J}\\), the \\(c_0\\)*-direct sum* \\(\\bigoplus^0_jA_j\\) is the set of families \\(x=(x_j)\\) with \\(x_j\\in A_j\\) such that \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\) is finite for every \\(\\varepsilon>0\\), with coordinatewise operations and \\(\\|x\\|=\\sup_j\\|x_j\\|\\).\n\n**Lemma 14.7.** Let \\(B=\\bigoplus^0_jA_j\\), and let \\(\\iota_j:A_j\\to B\\) put an element in the \\(j\\)-th coordinate.\n\n1. \\(B\\) is a \\(C^*\\)-algebra, and every \\(x\\in B\\) is the norm limit of its finite truncations \\(\\sum_{j\\in F}\\iota_j(x_j)\\).\n2. The closed left ideals of \\(B\\) are exactly the sets \\(\\bigoplus^0_j\\mathfrak m_j=\\{x\\in B:x_j\\in\\mathfrak m_j\\ \\forall j\\}\\), \\(\\mathfrak m_j\\) a closed left ideal of \\(A_j\\). The same holds for right and two-sided ideals.\n3. If every \\(A_j\\) is dual, \\(B\\) is dual.\n\n**Proof.** (1) The bounded families form a \\(C^*\\)-algebra with the supremum norm, and \\(B\\) is a \\(*\\)-subalgebra of it. It is closed: if \\(\\|x-x'\\|<\\varepsilon/2\\) with \\(x'\\in B\\), then \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\subseteq\\{j:\\|x'_j\\|\\ge\\varepsilon/2\\}\\) is finite. For \\(F=\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\), the truncation differs from \\(x\\) by at most \\(\\varepsilon\\).\n\n(2) Let \\(\\mathfrak m\\) be a closed left ideal and \\(\\mathfrak m_j=\\iota_j^{-1}(\\mathfrak m)\\), a closed left ideal of \\(A_j\\). If \\(x\\in\\mathfrak m\\) and \\((u_\\lambda)\\) is an approximate unit of \\(A_j\\), then \\(\\iota_j(u_\\lambda)x=\\iota_j(u_\\lambda x_j)\\in\\mathfrak m\\) tends to \\(\\iota_j(x_j)\\), so \\(x_j\\in\\mathfrak m_j\\). Conversely, if all \\(x_j\\in\\mathfrak m_j\\), the truncations of \\(x\\) lie in \\(\\mathfrak m\\) and converge to \\(x\\). Right and two-sided ideals are handled in the same way.\n\n(3) By (2), for \\(\\mathfrak m=\\bigoplus^0_j\\mathfrak m_j\\), \\(y\\in\\operatorname{ann}_r(\\mathfrak m)\\) exactly when \\(\\mathfrak m_jy_j=0\\) for all \\(j\\), so \\(\\operatorname{ann}_r(\\mathfrak m)=\\bigoplus^0_j\\operatorname{ann}_r(\\mathfrak m_j)\\); likewise for \\(\\operatorname{ann}_l\\). So \\(\\operatorname{ann}_l\\operatorname{ann}_r(\\mathfrak m)=\\bigoplus^0_j\\mathfrak m_j=\\mathfrak m\\), and symmetrically for right ideals. \\(\\square\\)\n\n**Example 14.8.** \\(K(H)\\) is dual: its bidual is \\(B(H)\\), in which it is an ideal (background fact 14 and Proposition 14.3). By Lemma 14.7(3), every \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) is dual; for instance \\(c_0(\\Gamma)=\\bigoplus^0_{\\gamma\\in\\Gamma}\\mathbb C\\). An infinite-dimensional \\(C^*\\)-algebra with a unit, such as \\(C[0,1]\\) or \\(B(H)\\) with \\(\\dim H=\\infty\\), is never dual. For if \\(A\\) is dual, it is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), and if moreover \\(1\\in A\\), then \\(\\tilde A=\\tilde A\\cdot1\\subseteq A\\), so \\(A\\) would be a von Neumann algebra equal to its bidual, hence reflexive, which an infinite-dimensional von Neumann algebra is not (it contains a copy of \\(\\ell^\\infty\\), background fact 25).\n\nRepresentations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *disjoint* if their central supports are orthogonal: \\(z(\\pi_1)z(\\pi_2)=0\\). Then no nonzero operator \\(T\\) satisfies \\(T\\pi_1(a)=\\pi_2(a)T\\) for all \\(a\\): such a \\(T\\) also intertwines the normal extensions, so \\[\n\\begin{gathered}\nT\\\\\n=T\\bar\\pi_1(z(\\pi_1))\\\\\n=\\bar\\pi_2(z(\\pi_1))T\\\\\n=\\bar\\pi_2(z(\\pi_1)z(\\pi_2))T\\\\\n=0.\n\\end{gathered}\n\\]\n\n**Proposition 14.9.** Let \\(A\\) be dual and \\((\\pi_i)_{i\\in I}\\) a family of mutually disjoint representations, and \\(\\pi=\\bigoplus_i\\pi_i\\). Then \\(\\pi(A)=\\bigoplus^0_i\\pi_i(A)\\) as sets of operators on \\(\\bigoplus_iH_i\\). So \\(\\pi(A)\\) is isomorphic to the \\(c_0\\)-direct sum of the \\(\\pi_i(A)\\).\n\n**Proof.** *Every \\(\\pi(a)\\) lies in the \\(c_0\\)-sum.* For a minimal projection \\(f\\) of \\(\\tilde A\\), the central support \\(c(f)\\) is a minimal central projection. Indeed, let \\(c\\le c(f)\\) be central and nonzero. Then \\(cf\\ne0\\), for otherwise \\(f\\le1-c\\), so \\(c(f)\\le1-c\\) and \\(c=cc(f)=0\\). And \\(cf=fcf\\) is a nonzero projection in \\(f\\tilde Af=\\mathbb Cf\\), so \\(cf=f\\), \\(f\\le c\\) and \\(c(f)\\le c\\). Hence for each \\(i\\), either \\(c(f)\\le z(\\pi_i)\\) or \\(c(f)z(\\pi_i)=0\\), and the first case occurs for at most one \\(i\\). In the second case \\(\\bar\\pi_i(f)=\\bar\\pi_i(fz(\\pi_i))=\\bar\\pi_i(fc(f)z(\\pi_i))=0\\). Now let \\(a\\in A\\) and \\(\\varepsilon>0\\). By Proposition 14.5(5) there is a finite-rank projection \\(e=f_1+\\dots+f_m\\) of \\(\\tilde A\\) with \\(\\|a-ea\\|<\\varepsilon\\). Then \\(\\bar\\pi_i(e)=0\\) for all \\(i\\) outside a set of at most \\(m\\) indices, and for those \\(i\\), \\(\\|\\pi_i(a)\\|=\\|\\pi_i(a-ea)\\|<\\varepsilon\\).\n\n*The \\(c_0\\)-sum lies in \\(\\pi(A)\\).* The normal extension of \\(\\pi\\) is \\(\\bigoplus_i\\bar\\pi_i\\). For \\(i_0\\in I\\), \\(\\bar\\pi_i(z(\\pi_{i_0}))\\) is \\(1\\) for \\(i=i_0\\) and \\(0\\) otherwise, by disjointness. Since \\(A\\) is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), \\(az(\\pi_{i_0})\\in A\\), and \\(\\pi(az(\\pi_{i_0}))\\) is \\(\\pi_{i_0}(a)\\) in the \\(i_0\\)-th place and \\(0\\) elsewhere. So \\(\\pi(A)\\) contains all finite sums of such elements. It is norm closed, being the image of a \\(C^*\\)-algebra, and these finite sums are dense in \\(\\bigoplus^0_i\\pi_i(A)\\) by Lemma 14.7(1). \\(\\square\\)\n\n**Theorem 14.10.** Every dual \\(C^*\\)-algebra is isomorphic to a \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) of compact operators.\n\n**Proof.** Let \\(A\\) be dual. By Proposition 14.5(4), \\(\\tilde A\\) is atomic. Let \\((c_j)_{j\\in J}\\) be the distinct central supports of minimal projections of \\(\\tilde A\\). They are minimal central projections (proof of Proposition 14.9), hence mutually orthogonal, and \\(\\sum_jc_j=1\\), because \\(1\\) is a sum of minimal projections \\(f\\), each under \\(c(f)\\). Each \\(\\tilde Ac_j\\) is a factor containing a minimal projection, so there is an isomorphism \\(\\theta_j:\\tilde Ac_j\\to B(H_j)\\) (background fact 26). Put \\(\\pi_j(a)=\\theta_j(ac_j)\\). Since \\(\\theta_j\\) is normal (every \\(*\\)-isomorphism between von Neumann algebras is, background fact 15), \\(\\bar\\pi_j(X)=\\theta_j(Xc_j)\\), whose kernel is \\(\\tilde A(1-c_j)\\). As \\(\\bar\\pi_j(1)=\\theta_j(c_j)=1\\), normality carries the increasing approximate identity of \\(A\\), whose supremum in \\(\\tilde A\\) is \\(1\\), to a net with supremum \\(1\\). By fact 16 it converges strongly to \\(1\\), and its ranges lie in \\([\\pi_j(A)H_j]\\). Hence \\(\\pi_j\\) is nondegenerate; so \\(z(\\pi_j)=c_j\\), and the \\(\\pi_j\\) are mutually disjoint. The representation \\(\\pi=\\bigoplus_j\\pi_j\\) is faithful: if \\(ac_j=0\\) for all \\(j\\), then \\(a=\\sum_jac_j=0\\). By Proposition 14.9, \\(A\\cong\\pi(A)=\\bigoplus^0_j\\pi_j(A)\\).\n\nIt remains to show \\(\\pi_j(A)=K(H_j)\\). First, \\(\\pi_j(A)=\\theta_j(Ac_j)\\) is an ideal of \\(B(H_j)\\), because \\(Ac_j\\) is an ideal of \\(\\tilde Ac_j\\) (Proposition 14.5(6)). It contains \\(\\theta_j(f)\\) for a minimal projection \\(f\\le c_j\\), which lies in \\(A\\) by Proposition 14.5(4); \\(\\theta_j(f)\\) is a minimal projection of \\(B(H_j)\\), hence of rank one. An ideal of \\(B(H_j)\\) containing a rank-one projection \\(\\theta_{\\zeta,\\zeta}\\) contains every rank-one operator \\(\\theta_{\\xi,\\eta}=\\theta_{\\xi,\\zeta}\\theta_{\\zeta,\\zeta}\\theta_{\\zeta,\\eta}\\) (with \\(\\|\\zeta\\|=1\\)), hence every finite-rank operator, hence \\(K(H_j)\\), as \\(\\pi_j(A)\\) is closed. Conversely, by Proposition 14.5(3), \\(A\\) is the closed span of elements \\(yf\\) with \\(f\\in A\\) minimal in \\(\\tilde A\\). Either \\(c(f)=c_j\\) and \\(fc_j=f\\), or \\(fc_j=0\\). So \\(\\pi_j(yf)=\\pi_j(y)\\theta_j(fc_j)\\) has rank at most one, and \\(\\pi_j(A)\\subseteq K(H_j)\\). \\(\\square\\)\n\n**Corollary 14.11.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: \\(A\\) is weakly compact; every right multiplication is weakly compact; \\(A\\) is an ideal of \\(\\tilde A\\); \\(A\\) is dual; \\(A\\) is isomorphic to a \\(c_0\\)-direct sum of algebras of compact operators.\n\n**Proof.** Proposition 14.2 gives the first three equivalences, Proposition 14.3 gives \"ideal \\(\\Rightarrow\\) dual\", Theorem 14.10 gives \"dual \\(\\Rightarrow\\) \\(c_0\\)-sum\", and Example 14.8 gives \"\\(c_0\\)-sum \\(\\Rightarrow\\) dual \\(\\Rightarrow\\) ideal\" (with Proposition 14.5(6)). \\(\\square\\)\n\n**Corollary 14.12.** Quotients of dual \\(C^*\\)-algebras by closed two-sided ideals are dual.\n\n**Proof.** By Theorem 14.10 we may take \\(B=\\bigoplus^0_{j\\in J}K(H_j)\\). Let \\(\\mathcal I\\) be a closed two-sided ideal of \\(B\\). By Lemma 14.7(2), \\(\\mathcal I=\\bigoplus^0_j\\mathcal I_j\\) with \\(\\mathcal I_j\\) a closed ideal of \\(K(H_j)\\). A nonzero closed ideal of \\(K(H)\\) is all of \\(K(H)\\): if \\(0\\ne x\\in\\mathcal I_j\\) and \\(x\\zeta\\ne0\\), then \\(\\theta_{\\xi,x\\zeta}\\,x\\,\\theta_{\\zeta,\\eta}=\\|x\\zeta\\|^2\\theta_{\\xi,\\eta}\\), so the ideal contains every rank-one operator, hence every compact operator. So \\(\\mathcal I=\\bigoplus^0\\{K(H_j):j\\in S\\}\\) for a set \\(S\\subseteq J\\), and \\(x+\\mathcal I\\mapsto(x_j)_{j\\notin S}\\) is an isomorphism of \\(B/\\mathcal I\\) onto \\(\\bigoplus^0\\{K(H_j):j\\notin S\\}\\): it is a well-defined, surjective \\(*\\)-homomorphism with kernel \\(\\mathcal I\\), and injective \\(*\\)-homomorphisms of \\(C^*\\)-algebras are isometric. This algebra is dual by Example 14.8. \\(\\square\\)\n\n",
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      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "14. Weakly compact and dual C\\*-algebras",
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      "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
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      "full_conditions_and_proof": "## 14. Weakly compact and dual C\\*-algebras\n\nFor a \\(C^*\\)-algebra \\(A\\) and \\(a\\in A\\), consider left multiplication \\(L_a:x\\mapsto ax\\) on \\(A\\). When is it a weakly compact operator? We will see that this happens for all \\(a\\) exactly when \\(A\\) is an ideal of its bidual, exactly when every closed one-sided ideal is recovered from its annihilator, and exactly when \\(A\\) is a \\(c_0\\)-direct sum of algebras of compact operators.\n\nIn this section a *minimal projection* of \\(A\\) is a nonzero projection \\(f\\in A\\) with \\(fAf=\\mathbb Cf\\). For a closed left ideal \\(\\mathfrak m\\) and a closed right ideal \\(\\mathfrak n\\) of \\(A\\) we write\n\\[\n\\begin{gathered}\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=\\{x\\in A\\,:\\,\\mathfrak mx=0\\},\\\\\n\\operatorname{ann}_l(\\mathfrak n)\\\\\n=\\{y\\in A\\,:\\,y\\mathfrak n=0\\}.\n\\end{gathered}\n\\]\nThe first is a closed right ideal and the second a closed left ideal.\n\n**Definition 14.1.** A \\(C^*\\)-algebra \\(A\\) is *weakly compact* if every left multiplication \\(L_a\\), \\(a\\in A\\), is a weakly compact operator, that is, \\(aA_1\\) is relatively weakly compact in \\(A\\). It is *dual* if \\(\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))=\\mathfrak m\\) for every closed left ideal \\(\\mathfrak m\\) and \\(\\operatorname{ann}_r(\\operatorname{ann}_l(\\mathfrak n))=\\mathfrak n\\) for every closed right ideal \\(\\mathfrak n\\).\n\n**Proposition 14.2.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: (a) every left multiplication \\(L_a\\) is weakly compact; (b) every right multiplication \\(R_a:x\\mapsto xa\\) is weakly compact; (c) \\(A\\) is a two-sided ideal of \\(\\tilde A\\).\n\n**Proof.** Fix \\(a\\in A\\). Left multiplication by \\(a\\) on \\(\\tilde A\\) is \\(\\sigma\\)-weakly continuous, and \\(A_1\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A_1\\) (background fact 14). So the \\(\\sigma(\\tilde A,A^*)\\)-closure of \\(aA_1\\) contains \\(a\\tilde A_1\\); and \\(a\\tilde A_1\\) is \\(\\sigma\\)-weakly compact and contains \\(aA_1\\). Hence that closure is \\(a\\tilde A_1\\). By Lemma 10.1 with \\(X=A\\), \\(X^{**}=\\tilde A\\), \\(L_a\\) is weakly compact exactly when \\(a\\tilde A_1\\subseteq A\\), that is \\(a\\tilde A\\subseteq A\\). So (a) says \\(A\\tilde A\\subseteq A\\), and in the same way (b) says \\(\\tilde AA\\subseteq A\\). Since \\(A\\) and \\(\\tilde A\\) are closed under the involution, each inclusion implies the other by taking adjoints, and together they say (c). \\(\\square\\)\n\n**Proposition 14.3.** Let \\(A\\) be an ideal of \\(\\tilde A\\), and \\(\\mathfrak m\\) a closed left ideal of \\(A\\) whose \\(\\sigma\\)-weak closure in \\(\\tilde A\\) is \\(\\tilde Ae\\) (background fact 19). Then\n\\[\n\\begin{gathered}\n\\mathfrak m\\\\\n=Ae\\\\\n=\\{x\\in A:xe=x\\},\\\\\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=(1-e)A,\\\\\n\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))\\\\\n=\\mathfrak m .\n\\end{gathered}\n\\]\nThe symmetric statements hold for closed right ideals. In particular \\(A\\) is dual.\n\n**Proof.** The \\(\\sigma\\)-weak closure of \\(\\mathfrak m\\) is a left ideal of \\(\\tilde A\\), since \\(A\\mathfrak m\\subseteq\\mathfrak m\\), multiplication is separately continuous and \\(A\\) is dense; so it is \\(\\tilde Ae\\) for a projection \\(e\\). If \\(x\\in\\mathfrak m\\), then \\(x\\in\\tilde Ae\\), so \\(x=xe\\). If \\(x\\in A\\), then \\(xe\\in A\\) (as \\(A\\) is an ideal) and \\(xe\\in\\tilde Ae\\), the \\(\\sigma(\\tilde A,A^*)\\)-closure of \\(\\mathfrak m\\). A norm-closed convex subset of \\(A\\) is weakly closed (Mazur), and the weak topology of \\(A\\) is the restriction of \\(\\sigma(\\tilde A,A^*)\\); so \\(xe\\in\\mathfrak m\\). This proves \\(\\mathfrak m=Ae=\\{x\\in A:xe=x\\}\\).\n\nNext, \\(y\\in\\operatorname{ann}_r(\\mathfrak m)\\) means \\(Aey=0\\); the set \\(\\{X\\in\\tilde A:Xey=0\\}\\) is \\(\\sigma\\)-weakly closed and contains \\(A\\), hence contains \\(1\\), so \\(ey=0\\). Conversely \\(ey=0\\) gives \\(\\mathfrak my=Aey=0\\). Thus \\(\\operatorname{ann}_r(\\mathfrak m)=\\{y\\in A:ey=0\\}=(1-e)A\\), using that \\(A\\) is an ideal. In the same way \\(z\\in\\operatorname{ann}_l((1-e)A)\\) exactly when \\(z(1-e)=0\\), that is \\(z\\in\\mathfrak m\\). \\(\\square\\)\n\n**Lemma 14.4.** Let \\(f\\) be a minimal projection of \\(A\\). Then \\(f\\) is minimal in \\(\\tilde A\\), and \\(\\tilde Af=Af\\).\n\n**Proof.** \\(f\\tilde Af\\) is the \\(\\sigma\\)-weak closure of \\(fAf=\\mathbb Cf\\), so it is \\(\\mathbb Cf\\). By background fact 25, \\(\\tilde Af\\) is a Hilbert space for \\(\\langle X,Y\\rangle f=Y^*X\\), with the norm of \\(\\tilde A\\). Let \\(\\omega_f\\) be the normal state with \\(fZf=\\omega_f(Z)f\\); then \\(\\langle X,Y\\rangle=\\omega_f(Y^*X)\\), so each \\(X\\mapsto\\langle X,Y\\rangle\\) is \\(\\sigma\\)-weakly continuous. The subspace \\(Af=\\{x\\in A:xf=x\\}\\) is norm closed, and it is \\(\\sigma\\)-weakly dense in \\(\\tilde Af\\), since \\(X\\mapsto Xf\\) is \\(\\sigma\\)-weakly continuous. So \\(Af\\) is weakly dense in the Hilbert space \\(\\tilde Af\\) and norm closed; closed subspaces of a Hilbert space are weakly closed, so \\(Af=\\tilde Af\\). \\(\\square\\)\n\n**Proposition 14.5** (Structure of dual algebras). Let \\(A\\) be dual.\n\n1. \\(\\mathfrak m\\mapsto\\operatorname{ann}_r(\\mathfrak m)\\) is an inclusion-reversing bijection from the closed left ideals onto the closed right ideals, with inverse \\(\\operatorname{ann}_l\\). A closed left ideal is maximal exactly when its right annihilator is a minimal nonzero closed right ideal; and symmetrically.\n2. Every nonzero closed left ideal contains \\(Af\\) for some minimal projection \\(f\\) of \\(A\\); symmetrically for right ideals.\n3. \\(A\\) is the closed linear span of the sets \\(Af\\), and also of the sets \\(fA\\), where \\(f\\) runs over the minimal projections of \\(A\\).\n4. \\(\\tilde A\\) is atomic, and every finite-rank projection of \\(\\tilde A\\) lies in \\(A\\).\n5. The finite-rank projections of \\(\\tilde A\\), ordered by size, form an increasing approximate unit of \\(A\\).\n6. \\(A\\) is an ideal of \\(\\tilde A\\).\n\n**Proof.** (1) The maps reverse inclusions and are mutually inverse by duality. \\(\\operatorname{ann}_r(0)=A\\), and \\(\\operatorname{ann}_r(A)=0\\) because \\(Ax=0\\) gives \\(x^*x=0\\). So \\(\\mathfrak m\\ne A\\) exactly when \\(\\operatorname{ann}_r(\\mathfrak m)\\ne0\\). An inclusion-reversing bijection between the proper closed left ideals and the nonzero closed right ideals matches maximal elements with minimal ones.\n\n(2) Let \\(\\mathfrak m\\ne0\\). Then \\(\\mathfrak n=\\operatorname{ann}_r(\\mathfrak m)\\) is a proper closed right ideal, since \\(\\mathfrak n=A\\) would give \\(\\mathfrak m=\\operatorname{ann}_l(A)=0\\). By Corollary 6.6 (right-handed), \\(\\mathfrak n\\) lies in a maximal closed right ideal \\(\\mathfrak n_0\\). Then \\(\\operatorname{ann}_l(\\mathfrak n_0)\\subseteq\\operatorname{ann}_l(\\mathfrak n)=\\mathfrak m\\), and \\(\\operatorname{ann}_l(\\mathfrak n_0)\\) is a minimal nonzero closed left ideal by (1). By background fact 13 it is \\(Af\\) for a minimal projection \\(f\\) of \\(A\\). The right-handed statement follows by taking adjoints.\n\n(3) Let \\(\\mathfrak l\\) be the closed span of the \\(Af\\), a closed left ideal. An element \\(x\\) lies in \\(\\operatorname{ann}_r(\\mathfrak l)\\) exactly when \\(fx=0\\) for all such \\(f\\) (use an approximate unit for one direction). If \\(\\operatorname{ann}_r(\\mathfrak l)\\ne0\\), by (2) it contains \\(fA\\) for such an \\(f\\), so \\(f=ff=0\\), which is absurd. So \\(\\operatorname{ann}_r(\\mathfrak l)=0\\) and \\(\\mathfrak l=\\operatorname{ann}_l(0)=A\\). The right-handed statement is symmetric.\n\n(4) Let \\(z\\) be the join in \\(\\tilde A\\) of all minimal projections of \\(A\\). For \\(y\\in A\\) and such \\(f\\), \\(yfz=yf\\); by (3), \\(xz=x\\) for all \\(x\\in A\\), and by density \\(z=1\\). Let \\(e\\ne0\\) be a projection of \\(\\tilde A\\). Some such \\(f\\) has \\(ef\\ne0\\), since otherwise \\(e\\le1-z=0\\). As \\(f\\) is minimal in \\(\\tilde A\\) (Lemma 14.4), \\(fef=\\mu f\\) with \\(\\mu=\\|ef\\|^2>0\\). So \\(w=\\mu^{-1/2}ef\\) is a partial isometry with \\(w^*w=f\\), and \\(q=ww^*\\le e\\) is a projection with \\(q\\tilde Aq=wf\\tilde Afw^*=\\mathbb Cq\\). Thus \\(e\\) majorizes the minimal projection \\(q\\), and \\(\\tilde A\\) is atomic. If \\(e\\) itself is minimal, then \\(q=e\\), and \\(w\\in\\tilde Af=Af\\subseteq A\\) by Lemma 14.4, so \\(e=ww^*\\in A\\). Finite-rank projections are finite sums of minimal ones.\n\n(5) The finite-rank projections form an upward directed set (background fact 25) of positive contractions in \\(A\\), by (4). For \\(x=yf\\) as in (3) and \\(e\\ge f\\) of finite rank, \\(xe=x\\). As such \\(x\\) span a dense subspace and \\(\\|e\\|\\le1\\), \\(\\|xe-x\\|\\to0\\) for all \\(x\\in A\\). Using the \\(fA\\), also \\(\\|ex-x\\|\\to0\\).\n\n(6) For \\(X\\in\\tilde A\\) and \\(x=yf\\) as in (3), \\(Xx=(Xy)f\\in\\tilde Af=Af\\subseteq A\\). By (3) and continuity, \\(XA\\subseteq A\\). So \\(A\\) is a left ideal of \\(\\tilde A\\), and, being self-adjoint, a two-sided ideal. \\(\\square\\)\n\nSo far: \\(A\\) is weakly compact \\(\\Leftrightarrow\\) \\(A\\) is an ideal of \\(\\tilde A\\) \\(\\Leftrightarrow\\) \\(A\\) is dual (Propositions 14.2, 14.3 and 14.5(6)). It remains to identify these algebras.\n\n**Definition 14.6.** For \\(C^*\\)-algebras \\((A_j)_{j\\in J}\\), the \\(c_0\\)*-direct sum* \\(\\bigoplus^0_jA_j\\) is the set of families \\(x=(x_j)\\) with \\(x_j\\in A_j\\) such that \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\) is finite for every \\(\\varepsilon>0\\), with coordinatewise operations and \\(\\|x\\|=\\sup_j\\|x_j\\|\\).\n\n**Lemma 14.7.** Let \\(B=\\bigoplus^0_jA_j\\), and let \\(\\iota_j:A_j\\to B\\) put an element in the \\(j\\)-th coordinate.\n\n1. \\(B\\) is a \\(C^*\\)-algebra, and every \\(x\\in B\\) is the norm limit of its finite truncations \\(\\sum_{j\\in F}\\iota_j(x_j)\\).\n2. The closed left ideals of \\(B\\) are exactly the sets \\(\\bigoplus^0_j\\mathfrak m_j=\\{x\\in B:x_j\\in\\mathfrak m_j\\ \\forall j\\}\\), \\(\\mathfrak m_j\\) a closed left ideal of \\(A_j\\). The same holds for right and two-sided ideals.\n3. If every \\(A_j\\) is dual, \\(B\\) is dual.\n\n**Proof.** (1) The bounded families form a \\(C^*\\)-algebra with the supremum norm, and \\(B\\) is a \\(*\\)-subalgebra of it. It is closed: if \\(\\|x-x'\\|<\\varepsilon/2\\) with \\(x'\\in B\\), then \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\subseteq\\{j:\\|x'_j\\|\\ge\\varepsilon/2\\}\\) is finite. For \\(F=\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\), the truncation differs from \\(x\\) by at most \\(\\varepsilon\\).\n\n(2) Let \\(\\mathfrak m\\) be a closed left ideal and \\(\\mathfrak m_j=\\iota_j^{-1}(\\mathfrak m)\\), a closed left ideal of \\(A_j\\). If \\(x\\in\\mathfrak m\\) and \\((u_\\lambda)\\) is an approximate unit of \\(A_j\\), then \\(\\iota_j(u_\\lambda)x=\\iota_j(u_\\lambda x_j)\\in\\mathfrak m\\) tends to \\(\\iota_j(x_j)\\), so \\(x_j\\in\\mathfrak m_j\\). Conversely, if all \\(x_j\\in\\mathfrak m_j\\), the truncations of \\(x\\) lie in \\(\\mathfrak m\\) and converge to \\(x\\). Right and two-sided ideals are handled in the same way.\n\n(3) By (2), for \\(\\mathfrak m=\\bigoplus^0_j\\mathfrak m_j\\), \\(y\\in\\operatorname{ann}_r(\\mathfrak m)\\) exactly when \\(\\mathfrak m_jy_j=0\\) for all \\(j\\), so \\(\\operatorname{ann}_r(\\mathfrak m)=\\bigoplus^0_j\\operatorname{ann}_r(\\mathfrak m_j)\\); likewise for \\(\\operatorname{ann}_l\\). So \\(\\operatorname{ann}_l\\operatorname{ann}_r(\\mathfrak m)=\\bigoplus^0_j\\mathfrak m_j=\\mathfrak m\\), and symmetrically for right ideals. \\(\\square\\)\n\n**Example 14.8.** \\(K(H)\\) is dual: its bidual is \\(B(H)\\), in which it is an ideal (background fact 14 and Proposition 14.3). By Lemma 14.7(3), every \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) is dual; for instance \\(c_0(\\Gamma)=\\bigoplus^0_{\\gamma\\in\\Gamma}\\mathbb C\\). An infinite-dimensional \\(C^*\\)-algebra with a unit, such as \\(C[0,1]\\) or \\(B(H)\\) with \\(\\dim H=\\infty\\), is never dual. For if \\(A\\) is dual, it is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), and if moreover \\(1\\in A\\), then \\(\\tilde A=\\tilde A\\cdot1\\subseteq A\\), so \\(A\\) would be a von Neumann algebra equal to its bidual, hence reflexive, which an infinite-dimensional von Neumann algebra is not (it contains a copy of \\(\\ell^\\infty\\), background fact 25).\n\nRepresentations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *disjoint* if their central supports are orthogonal: \\(z(\\pi_1)z(\\pi_2)=0\\). Then no nonzero operator \\(T\\) satisfies \\(T\\pi_1(a)=\\pi_2(a)T\\) for all \\(a\\): such a \\(T\\) also intertwines the normal extensions, so \\[\n\\begin{gathered}\nT\\\\\n=T\\bar\\pi_1(z(\\pi_1))\\\\\n=\\bar\\pi_2(z(\\pi_1))T\\\\\n=\\bar\\pi_2(z(\\pi_1)z(\\pi_2))T\\\\\n=0.\n\\end{gathered}\n\\]\n\n**Proposition 14.9.** Let \\(A\\) be dual and \\((\\pi_i)_{i\\in I}\\) a family of mutually disjoint representations, and \\(\\pi=\\bigoplus_i\\pi_i\\). Then \\(\\pi(A)=\\bigoplus^0_i\\pi_i(A)\\) as sets of operators on \\(\\bigoplus_iH_i\\). So \\(\\pi(A)\\) is isomorphic to the \\(c_0\\)-direct sum of the \\(\\pi_i(A)\\).\n\n**Proof.** *Every \\(\\pi(a)\\) lies in the \\(c_0\\)-sum.* For a minimal projection \\(f\\) of \\(\\tilde A\\), the central support \\(c(f)\\) is a minimal central projection. Indeed, let \\(c\\le c(f)\\) be central and nonzero. Then \\(cf\\ne0\\), for otherwise \\(f\\le1-c\\), so \\(c(f)\\le1-c\\) and \\(c=cc(f)=0\\). And \\(cf=fcf\\) is a nonzero projection in \\(f\\tilde Af=\\mathbb Cf\\), so \\(cf=f\\), \\(f\\le c\\) and \\(c(f)\\le c\\). Hence for each \\(i\\), either \\(c(f)\\le z(\\pi_i)\\) or \\(c(f)z(\\pi_i)=0\\), and the first case occurs for at most one \\(i\\). In the second case \\(\\bar\\pi_i(f)=\\bar\\pi_i(fz(\\pi_i))=\\bar\\pi_i(fc(f)z(\\pi_i))=0\\). Now let \\(a\\in A\\) and \\(\\varepsilon>0\\). By Proposition 14.5(5) there is a finite-rank projection \\(e=f_1+\\dots+f_m\\) of \\(\\tilde A\\) with \\(\\|a-ea\\|<\\varepsilon\\). Then \\(\\bar\\pi_i(e)=0\\) for all \\(i\\) outside a set of at most \\(m\\) indices, and for those \\(i\\), \\(\\|\\pi_i(a)\\|=\\|\\pi_i(a-ea)\\|<\\varepsilon\\).\n\n*The \\(c_0\\)-sum lies in \\(\\pi(A)\\).* The normal extension of \\(\\pi\\) is \\(\\bigoplus_i\\bar\\pi_i\\). For \\(i_0\\in I\\), \\(\\bar\\pi_i(z(\\pi_{i_0}))\\) is \\(1\\) for \\(i=i_0\\) and \\(0\\) otherwise, by disjointness. Since \\(A\\) is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), \\(az(\\pi_{i_0})\\in A\\), and \\(\\pi(az(\\pi_{i_0}))\\) is \\(\\pi_{i_0}(a)\\) in the \\(i_0\\)-th place and \\(0\\) elsewhere. So \\(\\pi(A)\\) contains all finite sums of such elements. It is norm closed, being the image of a \\(C^*\\)-algebra, and these finite sums are dense in \\(\\bigoplus^0_i\\pi_i(A)\\) by Lemma 14.7(1). \\(\\square\\)\n\n**Theorem 14.10.** Every dual \\(C^*\\)-algebra is isomorphic to a \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) of compact operators.\n\n**Proof.** Let \\(A\\) be dual. By Proposition 14.5(4), \\(\\tilde A\\) is atomic. Let \\((c_j)_{j\\in J}\\) be the distinct central supports of minimal projections of \\(\\tilde A\\). They are minimal central projections (proof of Proposition 14.9), hence mutually orthogonal, and \\(\\sum_jc_j=1\\), because \\(1\\) is a sum of minimal projections \\(f\\), each under \\(c(f)\\). Each \\(\\tilde Ac_j\\) is a factor containing a minimal projection, so there is an isomorphism \\(\\theta_j:\\tilde Ac_j\\to B(H_j)\\) (background fact 26). Put \\(\\pi_j(a)=\\theta_j(ac_j)\\). Since \\(\\theta_j\\) is normal (every \\(*\\)-isomorphism between von Neumann algebras is, background fact 15), \\(\\bar\\pi_j(X)=\\theta_j(Xc_j)\\), whose kernel is \\(\\tilde A(1-c_j)\\). As \\(\\bar\\pi_j(1)=\\theta_j(c_j)=1\\), normality carries the increasing approximate identity of \\(A\\), whose supremum in \\(\\tilde A\\) is \\(1\\), to a net with supremum \\(1\\). By fact 16 it converges strongly to \\(1\\), and its ranges lie in \\([\\pi_j(A)H_j]\\). Hence \\(\\pi_j\\) is nondegenerate; so \\(z(\\pi_j)=c_j\\), and the \\(\\pi_j\\) are mutually disjoint. The representation \\(\\pi=\\bigoplus_j\\pi_j\\) is faithful: if \\(ac_j=0\\) for all \\(j\\), then \\(a=\\sum_jac_j=0\\). By Proposition 14.9, \\(A\\cong\\pi(A)=\\bigoplus^0_j\\pi_j(A)\\).\n\nIt remains to show \\(\\pi_j(A)=K(H_j)\\). First, \\(\\pi_j(A)=\\theta_j(Ac_j)\\) is an ideal of \\(B(H_j)\\), because \\(Ac_j\\) is an ideal of \\(\\tilde Ac_j\\) (Proposition 14.5(6)). It contains \\(\\theta_j(f)\\) for a minimal projection \\(f\\le c_j\\), which lies in \\(A\\) by Proposition 14.5(4); \\(\\theta_j(f)\\) is a minimal projection of \\(B(H_j)\\), hence of rank one. An ideal of \\(B(H_j)\\) containing a rank-one projection \\(\\theta_{\\zeta,\\zeta}\\) contains every rank-one operator \\(\\theta_{\\xi,\\eta}=\\theta_{\\xi,\\zeta}\\theta_{\\zeta,\\zeta}\\theta_{\\zeta,\\eta}\\) (with \\(\\|\\zeta\\|=1\\)), hence every finite-rank operator, hence \\(K(H_j)\\), as \\(\\pi_j(A)\\) is closed. Conversely, by Proposition 14.5(3), \\(A\\) is the closed span of elements \\(yf\\) with \\(f\\in A\\) minimal in \\(\\tilde A\\). Either \\(c(f)=c_j\\) and \\(fc_j=f\\), or \\(fc_j=0\\). So \\(\\pi_j(yf)=\\pi_j(y)\\theta_j(fc_j)\\) has rank at most one, and \\(\\pi_j(A)\\subseteq K(H_j)\\). \\(\\square\\)\n\n**Corollary 14.11.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: \\(A\\) is weakly compact; every right multiplication is weakly compact; \\(A\\) is an ideal of \\(\\tilde A\\); \\(A\\) is dual; \\(A\\) is isomorphic to a \\(c_0\\)-direct sum of algebras of compact operators.\n\n**Proof.** Proposition 14.2 gives the first three equivalences, Proposition 14.3 gives \"ideal \\(\\Rightarrow\\) dual\", Theorem 14.10 gives \"dual \\(\\Rightarrow\\) \\(c_0\\)-sum\", and Example 14.8 gives \"\\(c_0\\)-sum \\(\\Rightarrow\\) dual \\(\\Rightarrow\\) ideal\" (with Proposition 14.5(6)). \\(\\square\\)\n\n**Corollary 14.12.** Quotients of dual \\(C^*\\)-algebras by closed two-sided ideals are dual.\n\n**Proof.** By Theorem 14.10 we may take \\(B=\\bigoplus^0_{j\\in J}K(H_j)\\). Let \\(\\mathcal I\\) be a closed two-sided ideal of \\(B\\). By Lemma 14.7(2), \\(\\mathcal I=\\bigoplus^0_j\\mathcal I_j\\) with \\(\\mathcal I_j\\) a closed ideal of \\(K(H_j)\\). A nonzero closed ideal of \\(K(H)\\) is all of \\(K(H)\\): if \\(0\\ne x\\in\\mathcal I_j\\) and \\(x\\zeta\\ne0\\), then \\(\\theta_{\\xi,x\\zeta}\\,x\\,\\theta_{\\zeta,\\eta}=\\|x\\zeta\\|^2\\theta_{\\xi,\\eta}\\), so the ideal contains every rank-one operator, hence every compact operator. So \\(\\mathcal I=\\bigoplus^0\\{K(H_j):j\\in S\\}\\) for a set \\(S\\subseteq J\\), and \\(x+\\mathcal I\\mapsto(x_j)_{j\\notin S}\\) is an isomorphism of \\(B/\\mathcal I\\) onto \\(\\bigoplus^0\\{K(H_j):j\\notin S\\}\\): it is a well-defined, surjective \\(*\\)-homomorphism with kernel \\(\\mathcal I\\), and injective \\(*\\)-homomorphisms of \\(C^*\\)-algebras are isometric. This algebra is dual by Example 14.8. \\(\\square\\)\n\n",
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      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "14. Weakly compact and dual C\\*-algebras",
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      "full_conditions_and_proof": "## 14. Weakly compact and dual C\\*-algebras\n\nFor a \\(C^*\\)-algebra \\(A\\) and \\(a\\in A\\), consider left multiplication \\(L_a:x\\mapsto ax\\) on \\(A\\). When is it a weakly compact operator? We will see that this happens for all \\(a\\) exactly when \\(A\\) is an ideal of its bidual, exactly when every closed one-sided ideal is recovered from its annihilator, and exactly when \\(A\\) is a \\(c_0\\)-direct sum of algebras of compact operators.\n\nIn this section a *minimal projection* of \\(A\\) is a nonzero projection \\(f\\in A\\) with \\(fAf=\\mathbb Cf\\). For a closed left ideal \\(\\mathfrak m\\) and a closed right ideal \\(\\mathfrak n\\) of \\(A\\) we write\n\\[\n\\begin{gathered}\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=\\{x\\in A\\,:\\,\\mathfrak mx=0\\},\\\\\n\\operatorname{ann}_l(\\mathfrak n)\\\\\n=\\{y\\in A\\,:\\,y\\mathfrak n=0\\}.\n\\end{gathered}\n\\]\nThe first is a closed right ideal and the second a closed left ideal.\n\n**Definition 14.1.** A \\(C^*\\)-algebra \\(A\\) is *weakly compact* if every left multiplication \\(L_a\\), \\(a\\in A\\), is a weakly compact operator, that is, \\(aA_1\\) is relatively weakly compact in \\(A\\). It is *dual* if \\(\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))=\\mathfrak m\\) for every closed left ideal \\(\\mathfrak m\\) and \\(\\operatorname{ann}_r(\\operatorname{ann}_l(\\mathfrak n))=\\mathfrak n\\) for every closed right ideal \\(\\mathfrak n\\).\n\n**Proposition 14.2.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: (a) every left multiplication \\(L_a\\) is weakly compact; (b) every right multiplication \\(R_a:x\\mapsto xa\\) is weakly compact; (c) \\(A\\) is a two-sided ideal of \\(\\tilde A\\).\n\n**Proof.** Fix \\(a\\in A\\). Left multiplication by \\(a\\) on \\(\\tilde A\\) is \\(\\sigma\\)-weakly continuous, and \\(A_1\\) is \\(\\sigma\\)-weakly dense in \\(\\tilde A_1\\) (background fact 14). So the \\(\\sigma(\\tilde A,A^*)\\)-closure of \\(aA_1\\) contains \\(a\\tilde A_1\\); and \\(a\\tilde A_1\\) is \\(\\sigma\\)-weakly compact and contains \\(aA_1\\). Hence that closure is \\(a\\tilde A_1\\). By Lemma 10.1 with \\(X=A\\), \\(X^{**}=\\tilde A\\), \\(L_a\\) is weakly compact exactly when \\(a\\tilde A_1\\subseteq A\\), that is \\(a\\tilde A\\subseteq A\\). So (a) says \\(A\\tilde A\\subseteq A\\), and in the same way (b) says \\(\\tilde AA\\subseteq A\\). Since \\(A\\) and \\(\\tilde A\\) are closed under the involution, each inclusion implies the other by taking adjoints, and together they say (c). \\(\\square\\)\n\n**Proposition 14.3.** Let \\(A\\) be an ideal of \\(\\tilde A\\), and \\(\\mathfrak m\\) a closed left ideal of \\(A\\) whose \\(\\sigma\\)-weak closure in \\(\\tilde A\\) is \\(\\tilde Ae\\) (background fact 19). Then\n\\[\n\\begin{gathered}\n\\mathfrak m\\\\\n=Ae\\\\\n=\\{x\\in A:xe=x\\},\\\\\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=(1-e)A,\\\\\n\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))\\\\\n=\\mathfrak m .\n\\end{gathered}\n\\]\nThe symmetric statements hold for closed right ideals. In particular \\(A\\) is dual.\n\n**Proof.** The \\(\\sigma\\)-weak closure of \\(\\mathfrak m\\) is a left ideal of \\(\\tilde A\\), since \\(A\\mathfrak m\\subseteq\\mathfrak m\\), multiplication is separately continuous and \\(A\\) is dense; so it is \\(\\tilde Ae\\) for a projection \\(e\\). If \\(x\\in\\mathfrak m\\), then \\(x\\in\\tilde Ae\\), so \\(x=xe\\). If \\(x\\in A\\), then \\(xe\\in A\\) (as \\(A\\) is an ideal) and \\(xe\\in\\tilde Ae\\), the \\(\\sigma(\\tilde A,A^*)\\)-closure of \\(\\mathfrak m\\). A norm-closed convex subset of \\(A\\) is weakly closed (Mazur), and the weak topology of \\(A\\) is the restriction of \\(\\sigma(\\tilde A,A^*)\\); so \\(xe\\in\\mathfrak m\\). This proves \\(\\mathfrak m=Ae=\\{x\\in A:xe=x\\}\\).\n\nNext, \\(y\\in\\operatorname{ann}_r(\\mathfrak m)\\) means \\(Aey=0\\); the set \\(\\{X\\in\\tilde A:Xey=0\\}\\) is \\(\\sigma\\)-weakly closed and contains \\(A\\), hence contains \\(1\\), so \\(ey=0\\). Conversely \\(ey=0\\) gives \\(\\mathfrak my=Aey=0\\). Thus \\(\\operatorname{ann}_r(\\mathfrak m)=\\{y\\in A:ey=0\\}=(1-e)A\\), using that \\(A\\) is an ideal. In the same way \\(z\\in\\operatorname{ann}_l((1-e)A)\\) exactly when \\(z(1-e)=0\\), that is \\(z\\in\\mathfrak m\\). \\(\\square\\)\n\n**Lemma 14.4.** Let \\(f\\) be a minimal projection of \\(A\\). Then \\(f\\) is minimal in \\(\\tilde A\\), and \\(\\tilde Af=Af\\).\n\n**Proof.** \\(f\\tilde Af\\) is the \\(\\sigma\\)-weak closure of \\(fAf=\\mathbb Cf\\), so it is \\(\\mathbb Cf\\). By background fact 25, \\(\\tilde Af\\) is a Hilbert space for \\(\\langle X,Y\\rangle f=Y^*X\\), with the norm of \\(\\tilde A\\). Let \\(\\omega_f\\) be the normal state with \\(fZf=\\omega_f(Z)f\\); then \\(\\langle X,Y\\rangle=\\omega_f(Y^*X)\\), so each \\(X\\mapsto\\langle X,Y\\rangle\\) is \\(\\sigma\\)-weakly continuous. The subspace \\(Af=\\{x\\in A:xf=x\\}\\) is norm closed, and it is \\(\\sigma\\)-weakly dense in \\(\\tilde Af\\), since \\(X\\mapsto Xf\\) is \\(\\sigma\\)-weakly continuous. So \\(Af\\) is weakly dense in the Hilbert space \\(\\tilde Af\\) and norm closed; closed subspaces of a Hilbert space are weakly closed, so \\(Af=\\tilde Af\\). \\(\\square\\)\n\n**Proposition 14.5** (Structure of dual algebras). Let \\(A\\) be dual.\n\n1. \\(\\mathfrak m\\mapsto\\operatorname{ann}_r(\\mathfrak m)\\) is an inclusion-reversing bijection from the closed left ideals onto the closed right ideals, with inverse \\(\\operatorname{ann}_l\\). A closed left ideal is maximal exactly when its right annihilator is a minimal nonzero closed right ideal; and symmetrically.\n2. Every nonzero closed left ideal contains \\(Af\\) for some minimal projection \\(f\\) of \\(A\\); symmetrically for right ideals.\n3. \\(A\\) is the closed linear span of the sets \\(Af\\), and also of the sets \\(fA\\), where \\(f\\) runs over the minimal projections of \\(A\\).\n4. \\(\\tilde A\\) is atomic, and every finite-rank projection of \\(\\tilde A\\) lies in \\(A\\).\n5. The finite-rank projections of \\(\\tilde A\\), ordered by size, form an increasing approximate unit of \\(A\\).\n6. \\(A\\) is an ideal of \\(\\tilde A\\).\n\n**Proof.** (1) The maps reverse inclusions and are mutually inverse by duality. \\(\\operatorname{ann}_r(0)=A\\), and \\(\\operatorname{ann}_r(A)=0\\) because \\(Ax=0\\) gives \\(x^*x=0\\). So \\(\\mathfrak m\\ne A\\) exactly when \\(\\operatorname{ann}_r(\\mathfrak m)\\ne0\\). An inclusion-reversing bijection between the proper closed left ideals and the nonzero closed right ideals matches maximal elements with minimal ones.\n\n(2) Let \\(\\mathfrak m\\ne0\\). Then \\(\\mathfrak n=\\operatorname{ann}_r(\\mathfrak m)\\) is a proper closed right ideal, since \\(\\mathfrak n=A\\) would give \\(\\mathfrak m=\\operatorname{ann}_l(A)=0\\). By Corollary 6.6 (right-handed), \\(\\mathfrak n\\) lies in a maximal closed right ideal \\(\\mathfrak n_0\\). Then \\(\\operatorname{ann}_l(\\mathfrak n_0)\\subseteq\\operatorname{ann}_l(\\mathfrak n)=\\mathfrak m\\), and \\(\\operatorname{ann}_l(\\mathfrak n_0)\\) is a minimal nonzero closed left ideal by (1). By background fact 13 it is \\(Af\\) for a minimal projection \\(f\\) of \\(A\\). The right-handed statement follows by taking adjoints.\n\n(3) Let \\(\\mathfrak l\\) be the closed span of the \\(Af\\), a closed left ideal. An element \\(x\\) lies in \\(\\operatorname{ann}_r(\\mathfrak l)\\) exactly when \\(fx=0\\) for all such \\(f\\) (use an approximate unit for one direction). If \\(\\operatorname{ann}_r(\\mathfrak l)\\ne0\\), by (2) it contains \\(fA\\) for such an \\(f\\), so \\(f=ff=0\\), which is absurd. So \\(\\operatorname{ann}_r(\\mathfrak l)=0\\) and \\(\\mathfrak l=\\operatorname{ann}_l(0)=A\\). The right-handed statement is symmetric.\n\n(4) Let \\(z\\) be the join in \\(\\tilde A\\) of all minimal projections of \\(A\\). For \\(y\\in A\\) and such \\(f\\), \\(yfz=yf\\); by (3), \\(xz=x\\) for all \\(x\\in A\\), and by density \\(z=1\\). Let \\(e\\ne0\\) be a projection of \\(\\tilde A\\). Some such \\(f\\) has \\(ef\\ne0\\), since otherwise \\(e\\le1-z=0\\). As \\(f\\) is minimal in \\(\\tilde A\\) (Lemma 14.4), \\(fef=\\mu f\\) with \\(\\mu=\\|ef\\|^2>0\\). So \\(w=\\mu^{-1/2}ef\\) is a partial isometry with \\(w^*w=f\\), and \\(q=ww^*\\le e\\) is a projection with \\(q\\tilde Aq=wf\\tilde Afw^*=\\mathbb Cq\\). Thus \\(e\\) majorizes the minimal projection \\(q\\), and \\(\\tilde A\\) is atomic. If \\(e\\) itself is minimal, then \\(q=e\\), and \\(w\\in\\tilde Af=Af\\subseteq A\\) by Lemma 14.4, so \\(e=ww^*\\in A\\). Finite-rank projections are finite sums of minimal ones.\n\n(5) The finite-rank projections form an upward directed set (background fact 25) of positive contractions in \\(A\\), by (4). For \\(x=yf\\) as in (3) and \\(e\\ge f\\) of finite rank, \\(xe=x\\). As such \\(x\\) span a dense subspace and \\(\\|e\\|\\le1\\), \\(\\|xe-x\\|\\to0\\) for all \\(x\\in A\\). Using the \\(fA\\), also \\(\\|ex-x\\|\\to0\\).\n\n(6) For \\(X\\in\\tilde A\\) and \\(x=yf\\) as in (3), \\(Xx=(Xy)f\\in\\tilde Af=Af\\subseteq A\\). By (3) and continuity, \\(XA\\subseteq A\\). So \\(A\\) is a left ideal of \\(\\tilde A\\), and, being self-adjoint, a two-sided ideal. \\(\\square\\)\n\nSo far: \\(A\\) is weakly compact \\(\\Leftrightarrow\\) \\(A\\) is an ideal of \\(\\tilde A\\) \\(\\Leftrightarrow\\) \\(A\\) is dual (Propositions 14.2, 14.3 and 14.5(6)). It remains to identify these algebras.\n\n**Definition 14.6.** For \\(C^*\\)-algebras \\((A_j)_{j\\in J}\\), the \\(c_0\\)*-direct sum* \\(\\bigoplus^0_jA_j\\) is the set of families \\(x=(x_j)\\) with \\(x_j\\in A_j\\) such that \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\) is finite for every \\(\\varepsilon>0\\), with coordinatewise operations and \\(\\|x\\|=\\sup_j\\|x_j\\|\\).\n\n**Lemma 14.7.** Let \\(B=\\bigoplus^0_jA_j\\), and let \\(\\iota_j:A_j\\to B\\) put an element in the \\(j\\)-th coordinate.\n\n1. \\(B\\) is a \\(C^*\\)-algebra, and every \\(x\\in B\\) is the norm limit of its finite truncations \\(\\sum_{j\\in F}\\iota_j(x_j)\\).\n2. The closed left ideals of \\(B\\) are exactly the sets \\(\\bigoplus^0_j\\mathfrak m_j=\\{x\\in B:x_j\\in\\mathfrak m_j\\ \\forall j\\}\\), \\(\\mathfrak m_j\\) a closed left ideal of \\(A_j\\). The same holds for right and two-sided ideals.\n3. If every \\(A_j\\) is dual, \\(B\\) is dual.\n\n**Proof.** (1) The bounded families form a \\(C^*\\)-algebra with the supremum norm, and \\(B\\) is a \\(*\\)-subalgebra of it. It is closed: if \\(\\|x-x'\\|<\\varepsilon/2\\) with \\(x'\\in B\\), then \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\subseteq\\{j:\\|x'_j\\|\\ge\\varepsilon/2\\}\\) is finite. For \\(F=\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\), the truncation differs from \\(x\\) by at most \\(\\varepsilon\\).\n\n(2) Let \\(\\mathfrak m\\) be a closed left ideal and \\(\\mathfrak m_j=\\iota_j^{-1}(\\mathfrak m)\\), a closed left ideal of \\(A_j\\). If \\(x\\in\\mathfrak m\\) and \\((u_\\lambda)\\) is an approximate unit of \\(A_j\\), then \\(\\iota_j(u_\\lambda)x=\\iota_j(u_\\lambda x_j)\\in\\mathfrak m\\) tends to \\(\\iota_j(x_j)\\), so \\(x_j\\in\\mathfrak m_j\\). Conversely, if all \\(x_j\\in\\mathfrak m_j\\), the truncations of \\(x\\) lie in \\(\\mathfrak m\\) and converge to \\(x\\). Right and two-sided ideals are handled in the same way.\n\n(3) By (2), for \\(\\mathfrak m=\\bigoplus^0_j\\mathfrak m_j\\), \\(y\\in\\operatorname{ann}_r(\\mathfrak m)\\) exactly when \\(\\mathfrak m_jy_j=0\\) for all \\(j\\), so \\(\\operatorname{ann}_r(\\mathfrak m)=\\bigoplus^0_j\\operatorname{ann}_r(\\mathfrak m_j)\\); likewise for \\(\\operatorname{ann}_l\\). So \\(\\operatorname{ann}_l\\operatorname{ann}_r(\\mathfrak m)=\\bigoplus^0_j\\mathfrak m_j=\\mathfrak m\\), and symmetrically for right ideals. \\(\\square\\)\n\n**Example 14.8.** \\(K(H)\\) is dual: its bidual is \\(B(H)\\), in which it is an ideal (background fact 14 and Proposition 14.3). By Lemma 14.7(3), every \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) is dual; for instance \\(c_0(\\Gamma)=\\bigoplus^0_{\\gamma\\in\\Gamma}\\mathbb C\\). An infinite-dimensional \\(C^*\\)-algebra with a unit, such as \\(C[0,1]\\) or \\(B(H)\\) with \\(\\dim H=\\infty\\), is never dual. For if \\(A\\) is dual, it is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), and if moreover \\(1\\in A\\), then \\(\\tilde A=\\tilde A\\cdot1\\subseteq A\\), so \\(A\\) would be a von Neumann algebra equal to its bidual, hence reflexive, which an infinite-dimensional von Neumann algebra is not (it contains a copy of \\(\\ell^\\infty\\), background fact 25).\n\nRepresentations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *disjoint* if their central supports are orthogonal: \\(z(\\pi_1)z(\\pi_2)=0\\). Then no nonzero operator \\(T\\) satisfies \\(T\\pi_1(a)=\\pi_2(a)T\\) for all \\(a\\): such a \\(T\\) also intertwines the normal extensions, so \\[\n\\begin{gathered}\nT\\\\\n=T\\bar\\pi_1(z(\\pi_1))\\\\\n=\\bar\\pi_2(z(\\pi_1))T\\\\\n=\\bar\\pi_2(z(\\pi_1)z(\\pi_2))T\\\\\n=0.\n\\end{gathered}\n\\]\n\n**Proposition 14.9.** Let \\(A\\) be dual and \\((\\pi_i)_{i\\in I}\\) a family of mutually disjoint representations, and \\(\\pi=\\bigoplus_i\\pi_i\\). Then \\(\\pi(A)=\\bigoplus^0_i\\pi_i(A)\\) as sets of operators on \\(\\bigoplus_iH_i\\). So \\(\\pi(A)\\) is isomorphic to the \\(c_0\\)-direct sum of the \\(\\pi_i(A)\\).\n\n**Proof.** *Every \\(\\pi(a)\\) lies in the \\(c_0\\)-sum.* For a minimal projection \\(f\\) of \\(\\tilde A\\), the central support \\(c(f)\\) is a minimal central projection. Indeed, let \\(c\\le c(f)\\) be central and nonzero. Then \\(cf\\ne0\\), for otherwise \\(f\\le1-c\\), so \\(c(f)\\le1-c\\) and \\(c=cc(f)=0\\). And \\(cf=fcf\\) is a nonzero projection in \\(f\\tilde Af=\\mathbb Cf\\), so \\(cf=f\\), \\(f\\le c\\) and \\(c(f)\\le c\\). Hence for each \\(i\\), either \\(c(f)\\le z(\\pi_i)\\) or \\(c(f)z(\\pi_i)=0\\), and the first case occurs for at most one \\(i\\). In the second case \\(\\bar\\pi_i(f)=\\bar\\pi_i(fz(\\pi_i))=\\bar\\pi_i(fc(f)z(\\pi_i))=0\\). Now let \\(a\\in A\\) and \\(\\varepsilon>0\\). By Proposition 14.5(5) there is a finite-rank projection \\(e=f_1+\\dots+f_m\\) of \\(\\tilde A\\) with \\(\\|a-ea\\|<\\varepsilon\\). Then \\(\\bar\\pi_i(e)=0\\) for all \\(i\\) outside a set of at most \\(m\\) indices, and for those \\(i\\), \\(\\|\\pi_i(a)\\|=\\|\\pi_i(a-ea)\\|<\\varepsilon\\).\n\n*The \\(c_0\\)-sum lies in \\(\\pi(A)\\).* The normal extension of \\(\\pi\\) is \\(\\bigoplus_i\\bar\\pi_i\\). For \\(i_0\\in I\\), \\(\\bar\\pi_i(z(\\pi_{i_0}))\\) is \\(1\\) for \\(i=i_0\\) and \\(0\\) otherwise, by disjointness. Since \\(A\\) is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), \\(az(\\pi_{i_0})\\in A\\), and \\(\\pi(az(\\pi_{i_0}))\\) is \\(\\pi_{i_0}(a)\\) in the \\(i_0\\)-th place and \\(0\\) elsewhere. So \\(\\pi(A)\\) contains all finite sums of such elements. It is norm closed, being the image of a \\(C^*\\)-algebra, and these finite sums are dense in \\(\\bigoplus^0_i\\pi_i(A)\\) by Lemma 14.7(1). \\(\\square\\)\n\n**Theorem 14.10.** Every dual \\(C^*\\)-algebra is isomorphic to a \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) of compact operators.\n\n**Proof.** Let \\(A\\) be dual. By Proposition 14.5(4), \\(\\tilde A\\) is atomic. Let \\((c_j)_{j\\in J}\\) be the distinct central supports of minimal projections of \\(\\tilde A\\). They are minimal central projections (proof of Proposition 14.9), hence mutually orthogonal, and \\(\\sum_jc_j=1\\), because \\(1\\) is a sum of minimal projections \\(f\\), each under \\(c(f)\\). Each \\(\\tilde Ac_j\\) is a factor containing a minimal projection, so there is an isomorphism \\(\\theta_j:\\tilde Ac_j\\to B(H_j)\\) (background fact 26). Put \\(\\pi_j(a)=\\theta_j(ac_j)\\). Since \\(\\theta_j\\) is normal (every \\(*\\)-isomorphism between von Neumann algebras is, background fact 15), \\(\\bar\\pi_j(X)=\\theta_j(Xc_j)\\), whose kernel is \\(\\tilde A(1-c_j)\\). As \\(\\bar\\pi_j(1)=\\theta_j(c_j)=1\\), normality carries the increasing approximate identity of \\(A\\), whose supremum in \\(\\tilde A\\) is \\(1\\), to a net with supremum \\(1\\). By fact 16 it converges strongly to \\(1\\), and its ranges lie in \\([\\pi_j(A)H_j]\\). Hence \\(\\pi_j\\) is nondegenerate; so \\(z(\\pi_j)=c_j\\), and the \\(\\pi_j\\) are mutually disjoint. The representation \\(\\pi=\\bigoplus_j\\pi_j\\) is faithful: if \\(ac_j=0\\) for all \\(j\\), then \\(a=\\sum_jac_j=0\\). By Proposition 14.9, \\(A\\cong\\pi(A)=\\bigoplus^0_j\\pi_j(A)\\).\n\nIt remains to show \\(\\pi_j(A)=K(H_j)\\). First, \\(\\pi_j(A)=\\theta_j(Ac_j)\\) is an ideal of \\(B(H_j)\\), because \\(Ac_j\\) is an ideal of \\(\\tilde Ac_j\\) (Proposition 14.5(6)). It contains \\(\\theta_j(f)\\) for a minimal projection \\(f\\le c_j\\), which lies in \\(A\\) by Proposition 14.5(4); \\(\\theta_j(f)\\) is a minimal projection of \\(B(H_j)\\), hence of rank one. An ideal of \\(B(H_j)\\) containing a rank-one projection \\(\\theta_{\\zeta,\\zeta}\\) contains every rank-one operator \\(\\theta_{\\xi,\\eta}=\\theta_{\\xi,\\zeta}\\theta_{\\zeta,\\zeta}\\theta_{\\zeta,\\eta}\\) (with \\(\\|\\zeta\\|=1\\)), hence every finite-rank operator, hence \\(K(H_j)\\), as \\(\\pi_j(A)\\) is closed. Conversely, by Proposition 14.5(3), \\(A\\) is the closed span of elements \\(yf\\) with \\(f\\in A\\) minimal in \\(\\tilde A\\). Either \\(c(f)=c_j\\) and \\(fc_j=f\\), or \\(fc_j=0\\). So \\(\\pi_j(yf)=\\pi_j(y)\\theta_j(fc_j)\\) has rank at most one, and \\(\\pi_j(A)\\subseteq K(H_j)\\). \\(\\square\\)\n\n**Corollary 14.11.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: \\(A\\) is weakly compact; every right multiplication is weakly compact; \\(A\\) is an ideal of \\(\\tilde A\\); \\(A\\) is dual; \\(A\\) is isomorphic to a \\(c_0\\)-direct sum of algebras of compact operators.\n\n**Proof.** Proposition 14.2 gives the first three equivalences, Proposition 14.3 gives \"ideal \\(\\Rightarrow\\) dual\", Theorem 14.10 gives \"dual \\(\\Rightarrow\\) \\(c_0\\)-sum\", and Example 14.8 gives \"\\(c_0\\)-sum \\(\\Rightarrow\\) dual \\(\\Rightarrow\\) ideal\" (with Proposition 14.5(6)). \\(\\square\\)\n\n**Corollary 14.12.** Quotients of dual \\(C^*\\)-algebras by closed two-sided ideals are dual.\n\n**Proof.** By Theorem 14.10 we may take \\(B=\\bigoplus^0_{j\\in J}K(H_j)\\). Let \\(\\mathcal I\\) be a closed two-sided ideal of \\(B\\). By Lemma 14.7(2), \\(\\mathcal I=\\bigoplus^0_j\\mathcal I_j\\) with \\(\\mathcal I_j\\) a closed ideal of \\(K(H_j)\\). A nonzero closed ideal of \\(K(H)\\) is all of \\(K(H)\\): if \\(0\\ne x\\in\\mathcal I_j\\) and \\(x\\zeta\\ne0\\), then \\(\\theta_{\\xi,x\\zeta}\\,x\\,\\theta_{\\zeta,\\eta}=\\|x\\zeta\\|^2\\theta_{\\xi,\\eta}\\), so the ideal contains every rank-one operator, hence every compact operator. So \\(\\mathcal I=\\bigoplus^0\\{K(H_j):j\\in S\\}\\) for a set \\(S\\subseteq J\\), and \\(x+\\mathcal I\\mapsto(x_j)_{j\\notin S}\\) is an isomorphism of \\(B/\\mathcal I\\) onto \\(\\bigoplus^0\\{K(H_j):j\\notin S\\}\\): it is a well-defined, surjective \\(*\\)-homomorphism with kernel \\(\\mathcal I\\), and injective \\(*\\)-homomorphisms of \\(C^*\\)-algebras are isometric. This algebra is dual by Example 14.8. \\(\\square\\)\n\n",
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      "id": "OA-FND-PD-25",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "name": "15. Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
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      "full_conditions_and_proof": "## 15. Exercises\n\n**Exercise 15.1.** (easy) In \\(M_2(\\mathbb C)\\) let \\(\\varphi(x)=x_{21}\\), the \\((2,1)\\) entry. Find \\(|\\varphi|\\), the partial isometry \\(v\\), \\(s_l(\\varphi)\\), \\(s_r(\\varphi)\\) and \\(|\\varphi^*|\\). Check inequality (3.1), and check that (1.1) is an equality for \\(e=E_{11}\\) and for the projection \\(e\\) onto \\(2^{-1/2}(1,1)\\).\n\n**Solution.** \\(\\varphi=\\omega_{\\epsilon_1,\\epsilon_2}\\) for the standard basis \\(\\epsilon_1,\\epsilon_2\\), since \\(\\langle x\\epsilon_1,\\epsilon_2\\rangle=x_{21}\\). By (2.5), \\(|\\varphi|=\\omega_{\\epsilon_2}\\), that is \\(|\\varphi|(x)=x_{22}\\), and \\(v=\\theta_{\\epsilon_1,\\epsilon_2}=E_{12}\\). Check: \\(|\\varphi|(xE_{12})=(xE_{12})_{22}=x_{21}\\), and \\(E_{12}^*E_{12}=E_{22}=s(|\\varphi|)\\). By Theorem 2.2(3), \\(s_r(\\varphi)=E_{22}\\) and \\(s_l(\\varphi)=E_{11}\\). Also \\(\\varphi^*(x)=x_{12}\\) and \\(|\\varphi^*|(x)=x_{11}\\). Inequality (3.1) reads \\(|x_{21}|^2\\le(xx^*)_{22}=|x_{21}|^2+|x_{22}|^2\\). For \\(e=E_{11}=s_l(\\varphi)\\), \\(e\\varphi=\\varphi\\) and \\((1-e)\\varphi=0\\): \\(1+0=1\\). For \\(e\\) the projection onto \\(2^{-1/2}(1,1)\\), \\(e\\varphi=\\omega_{e\\epsilon_1,\\epsilon_2}\\) has norm \\(\\|e\\epsilon_1\\|=2^{-1/2}\\), and so does \\((1-e)\\varphi\\): \\(\\tfrac12+\\tfrac12=1\\). \\(\\square\\)\n\n**Exercise 15.2.** (medium) Let \\(\\varphi\\in M_*^+\\) and let \\(u\\in M\\) be unitary. Show that \\(|u\\varphi|=\\varphi\\) and \\(|\\varphi u|=u^*\\varphi u\\), where \\((u^*\\varphi u)(x)=\\varphi(uxu^*)\\). Deduce that \\(\\||\\varphi u|-|u\\varphi|\\|\\) can be as large as \\(2\\|\\varphi\\|\\).\n\n**Solution.** \\(u\\varphi=(us(\\varphi))\\varphi\\), and \\(w=us(\\varphi)\\) is a partial isometry with \\(w^*w=s(\\varphi)\\). By uniqueness in Theorem 2.2, \\(|u\\varphi|=\\varphi\\). Next, \\(\\psi=u^*\\varphi u\\) is positive with support \\(u^*s(\\varphi)u\\), since \\(\\psi(x)=\\varphi(uxu^*)\\) and conjugation by \\(u\\) preserves the order of projections. Put \\(w'=s(\\varphi)u\\), so \\(w'^*w'=u^*s(\\varphi)u=s(\\psi)\\). Then \\[\n\\begin{gathered}\n(w'\\psi)(x)\\\\\n=\\psi(xs(\\varphi)u)\\\\\n=\\varphi(uxs(\\varphi))\\\\\n=\\varphi(ux)\\\\\n=(\\varphi u)(x),\n\\end{gathered}\n\\] using \\(\\varphi=s(\\varphi)\\varphi\\). So \\(|\\varphi u|=\\psi\\). For \\(M=B(H)\\), \\(\\varphi=\\omega_\\xi\\) with a unit vector \\(\\xi\\), and a unitary \\(u\\) with \\(u^*\\xi\\perp\\xi\\): \\(|\\varphi u|=\\omega_{u^*\\xi}\\) and \\(|u\\varphi|=\\omega_\\xi\\), and \\(\\|\\omega_{u^*\\xi}-\\omega_\\xi\\|=2\\) by Corollary 2.8(4), as the supports are orthogonal. \\(\\square\\)\n\n**Exercise 15.3.** (medium) Show that a subset \\(K\\) of \\(\\ell^1(\\Gamma)\\) is relatively weakly compact exactly when it is bounded and for every \\(\\varepsilon>0\\) there is a finite \\(F\\subseteq\\Gamma\\) with \\(\\sum_{\\gamma\\notin F}|g(\\gamma)|<\\varepsilon\\) for all \\(g\\in K\\). Deduce that in \\(\\ell^1(\\Gamma)\\) the relatively weakly compact sets are the relatively norm compact sets.\n\n**Solution.** Suppose \\(K\\) is bounded with uniformly small tails. Given \\(\\varepsilon\\), take \\(F\\) as stated. The truncations \\(g1_F\\) form a bounded set in the finite-dimensional space \\(\\mathbb C^F\\), so finitely many \\(\\varepsilon\\)-balls cover them, and the \\(2\\varepsilon\\)-balls with the same centres cover \\(K\\). So \\(K\\) is totally bounded, hence relatively norm compact, hence relatively weakly compact. Conversely let \\(K\\) be relatively weakly compact. By the Eberlein–Šmulian theorem every sequence in \\(K\\) has a weakly convergent subsequence, which converges in norm by Schur's theorem (Corollary 7.2). If \\(K\\) were not totally bounded, an inductively chosen sequence of points at mutual distance at least some \\(\\varepsilon>0\\) would have no norm-convergent subsequence, a contradiction. Its norm closure is complete and totally bounded, hence compact by the metric lemma above. Thus \\(K\\) is relatively norm compact. Cover \\(K\\) by finitely many \\(\\varepsilon\\)-balls with centres \\(g_1,\\dots,g_m\\), and choose a finite \\(F\\) with \\(\\sum_{\\gamma\\notin F}|g_l(\\gamma)|<\\varepsilon\\) for all \\(l\\). Then \\(\\sum_{\\gamma\\notin F}|g(\\gamma)|<2\\varepsilon\\) for every \\(g\\in K\\). The two parts together prove the last statement. \\(\\square\\)\n\n**Exercise 15.4.** (medium) Show that every bounded subset of \\(M_*\\) is relatively weakly compact exactly when \\(M\\) is finite-dimensional.\n\n**Solution.** If \\(\\dim M<\\infty\\), then \\(M_*\\) is finite-dimensional and bounded sets are relatively compact. If \\(\\dim M=\\infty\\), take an infinite sequence \\((e_n)\\) of mutually orthogonal nonzero projections (background fact 25) and normal states \\(\\omega_n\\) with \\(\\omega_n(e_n)=1\\) (as in the proof of Proposition 11.4). The bounded set \\(K=\\{\\omega_n\\}\\) violates condition (7) of Theorem 10.2, since \\(\\sup_{\\varphi\\in K}\\varphi(e_n)\\ge1\\) for all \\(n\\). \\(\\square\\)\n\n**Exercise 15.5.** (medium) Let \\(S\\) be the unilateral shift on \\(\\ell^2=\\ell^2(\\{0,1,2,\\dots\\})\\), \\(S\\delta_k=\\delta_{k+1}\\), and \\(x_n=(S^*)^n\\). Show that \\(x_n\\to0\\) \\(\\sigma\\)-strongly but not in the topology \\(\\tau\\) of Definition 11.1, by exhibiting a relatively weakly compact \\(K\\subseteq B(\\ell^2)_*\\) with \\(p_K(x_n)\\ge1\\) for all \\(n\\). Why does this not contradict Theorem 11.2?\n\n**Solution.** \\((S^*)^n\\delta_k=\\delta_{k-n}\\) for \\(k\\ge n\\) and \\(0\\) otherwise, so \\(\\|x_n\\xi\\|^2=\\sum_{k\\ge n}|\\xi_k|^2\\to0\\): \\(x_n\\to0\\) strongly, hence \\(\\sigma\\)-strongly, as \\(\\|x_n\\|\\le1\\). Let \\(\\psi_n=\\omega_{\\delta_n,\\delta_0}\\), so \\(\\psi_n(x)=\\langle x\\delta_n,\\delta_0\\rangle\\) and \\(\\psi_n(x_n)=\\langle\\delta_0,\\delta_0\\rangle=1\\). By Bessel's inequality, as in Example 10.7(1), \\(\\sum_n|\\langle x\\delta_0,\\delta_n\\rangle|^2\\le\\|x\\delta_0\\|^2\\), so the functionals \\(\\omega_{\\delta_0,\\delta_n}\\) tend weakly to \\(0\\); their adjoints \\(\\psi_n\\) do too, as the adjoint map is weakly continuous. So \\(K=\\{\\psi_n\\}\\cup\\{0\\}\\) is weakly compact, and \\(p_K(x_n)\\ge1\\). There is no contradiction: \\(x_n^*=S^n\\) is an isometry, so \\(\\|x_n^*\\delta_0\\|=1\\) and \\(x_n\\not\\to0\\) \\(\\sigma\\)-strongly\\(^*\\). Theorem 11.2 compares \\(\\tau\\) with the \\(\\sigma\\)-strong\\(^*\\) topology, not with the \\(\\sigma\\)-strong one. \\(\\square\\)\n\n## References\n\n- [Kostecki] R. P. Kostecki, *W\\*-algebras and noncommutative integration*, lecture notes, arXiv:1307.4818 (version 5, 2014), https://arxiv.org/abs/1307.4818v5.\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [Jan Hamhalter; Ondřej F. K. Kalenda; Antonio M. Peralta; Hermann Pfitzner, *Measures of weak non-compactness in preduals of von Neumann algebras and JBW*-triples*, §11, Theorems 11.1 and 11.3](https://arxiv.org/pdf/1901.08056v2) gives a route through modern weak-noncompactness results, including JBW*-triples; these pages cite the foundational criterion, which is proved here. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
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      "id": "OA-FND-IR-01",
      "unit": "integral-representations-of-states",
      "name": "1. Affine functions and envelopes",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 1. Affine functions and envelopes\n\nWe start with three lemmas. Each is proved by separating a point, or a compact convex set, from a closed convex set in \\(E\\times\\mathbb R\\).\n\n**Lemma 1.1** (Affine minorants). Let \\(f\\in\\mathcal L(K)\\). For every \\(x\\in K\\),\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\sup\\{a(x):\\\\\n\\ \\\\\na\\in\\operatorname{Aff}_E(K),\\ a\\\\\n\\leq f\\ \\text{on }K\\}.\n\\end{gathered}\n\\tag{1.1}\n\\]\n\nThe value \\(+\\infty\\) is allowed; Lemma 4.2 needs this case.\n\n**Proof.** The right side is at most \\(f(x)\\). Fix a real \\(\\beta<f(x)\\); we find \\(a\\in\\operatorname{Aff}_E(K)\\) with \\(a\\leq f\\) and \\(a(x)>\\beta\\). First, some affine function lies below \\(f\\): an lsc function on a compact set attains its infimum, which is \\(>-\\infty\\), so a constant works. If \\(f\\equiv+\\infty\\), every constant is a minorant and we are done. Otherwise let\n\\[ M=\\{(y,t)\\in K\\times\\mathbb R:\\ f(y)\\leq t\\}\\subseteq E\\times\\mathbb R . \\]\n\\(M\\) is convex because \\(f\\) is convex. It is closed: if \\((y_i,t_i)\\to(y,t)\\) with \\((y_i,t_i)\\in M\\), then \\(y\\in K\\) and \\(f(y)\\leq\\liminf f(y_i)\\leq t\\). Also \\(M\\neq\\varnothing\\) and \\((x,\\beta)\\notin M\\). The dual of \\(E\\times\\mathbb R\\) consists of the maps \\((y,t)\\mapsto p(y)+\\lambda t\\) with \\(p\\in E^*\\), \\(\\lambda\\in\\mathbb R\\). By the separation theorem, applied to the compact set \\(\\{(x,\\beta)\\}\\) and the closed convex set \\(M\\), there are such \\(p,\\lambda\\) and a real \\(\\gamma\\) with\n\\[\n\\begin{gathered}\np(x)+\\lambda\\beta<\\gamma<p(y)+\\lambda t\\\\\n\\text{for all }(y,t)\\in M .\n\\end{gathered}\n\\]\nSince \\((y,t)\\in M\\) implies \\((y,t+s)\\in M\\) for \\(s\\geq0\\), we get \\(\\lambda\\geq0\\).\n\nIf \\(\\lambda>0\\), put \\(a=(\\gamma-p)/\\lambda\\) on \\(K\\). For \\(y\\) with \\(f(y)<\\infty\\), the point \\((y,f(y))\\) lies in \\(M\\), so \\(f(y)>a(y)\\); where \\(f=+\\infty\\) there is nothing to check. And \\(a(x)>\\beta\\).\n\nIf \\(\\lambda=0\\), then \\(p(x)<\\gamma<p(y)\\) for every \\(y\\) with \\(f(y)<\\infty\\). Take any affine minorant \\(a_0\\in\\operatorname{Aff}_E(K)\\) of \\(f\\), and put \\(a_s=a_0+s(\\gamma-p)\\) for \\(s>0\\). Where \\(f<\\infty\\) we have \\(\\gamma-p<0\\), so \\(a_s\\leq a_0\\leq f\\); elsewhere \\(f=+\\infty\\). Since \\(\\gamma-p(x)>0\\), a large \\(s\\) gives \\(a_s(x)>\\beta\\). \\(\\square\\)\n\n**Lemma 1.2** (Density of \\(\\operatorname{Aff}_E(K)\\)). \\(\\operatorname{Aff}_E(K)\\) is uniformly dense in \\(\\operatorname{Aff}(K)\\).\n\n**Proof.** Let \\(a\\in\\operatorname{Aff}(K)\\) and \\(\\varepsilon>0\\). The graphs \\(G_1=\\{(y,a(y)):y\\in K\\}\\) and \\(G_2=\\{(y,a(y)-\\varepsilon):y\\in K\\}\\) are compact convex subsets of \\(E\\times\\mathbb R\\), since \\(a\\) is continuous and affine, and they are disjoint. Separating them gives \\(p\\in E^*\\), \\(\\lambda\\in\\mathbb R\\) and \\(\\gamma\\) with \\(p(y)+\\lambda(a(y)-\\varepsilon)<\\gamma<p(y)+\\lambda a(y)\\) for all \\(y\\in K\\). Subtracting gives \\(\\lambda\\varepsilon>0\\), so \\(\\lambda>0\\), and then \\(a(y)-\\varepsilon<(\\gamma-p(y))/\\lambda<a(y)\\). So \\(b=(\\gamma-p)/\\lambda\\) on \\(K\\) lies in \\(\\operatorname{Aff}_E(K)\\) and \\(\\|a-b\\|\\leq\\varepsilon\\). \\(\\square\\)\n\n**Lemma 1.3** (Increasing nets of affine functions). Let \\(f:K\\to\\mathbb R\\) be lsc and affine. Then \\(I=\\{g\\in\\operatorname{Aff}_E(K):\\ g<f\\ \\text{on }K\\}\\) is directed upward, and \\(\\sup I=f\\) pointwise. So \\(f\\) is the pointwise limit of an increasing net in \\(\\operatorname{Aff}_E(K)\\).\n\n**Proof.** By Lemma 1.1, for \\(\\varepsilon>0\\) every affine minorant \\(a\\) of \\(f\\) gives \\(a-\\varepsilon\\in I\\); so \\(\\sup I=f\\). Let \\(g_1,g_2\\in I\\). After adding a constant to \\(f,g_1,g_2\\) we may assume that \\(g_1,g_2>0\\). Put\n\\[\n\\begin{gathered}\nM\\\\\n=\\{(y,t):\\\\\n\\ y\\in K,\\ f(y)\\\\\n\\leq t\\},\\\\\nM_i\\\\\n=\\{(y,t):\\\\\n\\ y\\in K,\\ 0\\\\\n\\leq t\\\\\n\\leq g_i(y)\\}\\\\\n(i\\\\\n=1,2).\n\\end{gathered}\n\\]\n\\(M\\) is closed and convex, and \\(M_1,M_2\\) are compact and convex, so the convex hull \\(D\\) of \\(M_1\\cup M_2\\) is compact (it is the image of \\(M_1\\times M_2\\times[0,1]\\)). \\(D\\) misses \\(M\\): a point of \\(D\\) is \\(s(y_1,t_1)+(1-s)(y_2,t_2)\\) with \\(t_i\\leq g_i(y_i)<f(y_i)\\), and since \\(f\\) is affine, \\(st_1+(1-s)t_2<f(sy_1+(1-s)y_2)\\). Separation gives \\(p\\), \\(\\lambda\\), \\(\\gamma\\) with \\(p(y)+\\lambda t<\\gamma\\) on \\(D\\) and \\(p(y)+\\lambda t>\\gamma\\) on \\(M\\). As in the proof of Lemma 1.1, \\(\\lambda\\geq0\\). If \\(\\lambda=0\\), then \\(p<\\gamma\\) on \\(K\\) (use \\((y,0)\\in M_1\\)) and \\(p>\\gamma\\) on \\(K\\) (use \\((y,f(y))\\in M\\)), which is absurd. So \\(\\lambda>0\\), and \\(g=(\\gamma-p)/\\lambda\\) on \\(K\\) satisfies \\(g_1,g_2<g<f\\). \\(\\square\\)\n\n**Definition 1.4** (Envelopes). Let \\(X\\subseteq K\\) be nonempty. For \\(g:X\\to[-\\infty,+\\infty]\\) bounded above, the *upper envelope* is\n\\[\n\\begin{gathered}\n\\overline g(x)\\\\\n=\\inf\\{a(x):\\\\\n\\ a\\in\\operatorname{Aff}(K),\\ a\\\\\n\\geq g\\ \\text{on }X\\}\\\\\n(x\\in K),\n\\end{gathered}\n\\tag{1.2}\n\\]\nand for \\(g\\) bounded below the *lower envelope* is \\(\\underline g(x)=\\sup\\{a(x):\\ a\\in\\operatorname{Aff}(K),\\ a\\leq g\\ \\text{on }X\\}\\). Both are defined on all of \\(K\\). The set \\(X\\) is arbitrary; we mostly use \\(X=K\\). By Lemma 1.2 the same envelopes arise if \\(a\\) runs only through \\(\\operatorname{Aff}_E(K)\\): an \\(a\\in\\operatorname{Aff}(K)\\) with \\(a\\geq g\\) is within \\(\\varepsilon\\) of some \\(b\\in\\operatorname{Aff}_E(K)\\), and then \\(b+\\varepsilon\\geq g\\) and \\(b+\\varepsilon\\leq a+2\\varepsilon\\).\n\n**Proposition 1.5** (Properties of envelopes).\n\n1. \\(\\underline g\\in\\mathcal L(K)\\), and \\(-\\overline g\\in\\mathcal L(K)\\). That is, \\(\\overline g\\) is concave and upper semicontinuous (usc) with values in \\([-\\infty,+\\infty)\\).\n2. For \\(g\\) bounded on \\(X=K\\): \\(\\underline g\\leq g\\leq\\overline g\\). For \\(g\\in C(K)\\): \\(-\\|g\\|\\leq\\underline g\\leq g\\leq\\overline g\\leq\\|g\\|\\).\n3. For bounded \\(g,h\\) on \\(K\\): \\(g\\leq h\\) implies \\(\\overline g\\leq\\overline h\\); \\(\\overline{g+h}\\leq\\overline g+\\overline h\\); \\(\\overline{tg}=t\\overline g\\) for \\(t\\geq0\\); \\(\\overline{g+b}=\\overline g+b\\) for \\(b\\in\\operatorname{Aff}(K)\\); and \\(\\overline{-g}=-\\underline g\\).\n4. If \\(h\\in\\mathcal L(K)\\) then \\(\\underline h=h\\). If \\(g:K\\to[-\\infty,\\infty)\\) and \\(-g\\in\\mathcal L(K)\\), then \\(\\overline g=g\\). In particular \\(\\overline f=f\\) for \\(f\\in-\\mathcal P(K)\\) and \\(\\overline a=a\\) for \\(a\\in\\operatorname{Aff}(K)\\).\n\n**Proof.** (1) A supremum of continuous affine functions is convex and lsc; the family is nonempty because \\(g\\) is bounded below, so \\(\\underline g>-\\infty\\). The statement for \\(\\overline g\\) is the same applied to \\(-g\\), since \\(\\overline g=-\\underline{(-g)}\\) directly from the definitions (\\(a\\geq g\\) on \\(X\\) exactly when \\(-a\\leq-g\\) there). (2) The first claim is immediate from the definitions, and the constants \\(\\pm\\|g\\|\\) are affine. (3) If \\(a\\geq g\\) and \\(b'\\geq h\\) are affine, then \\(a+b'\\geq g+h\\); take infima. For \\(t>0\\), \\(a\\geq g\\) exactly when \\(ta\\geq tg\\). For \\(t=0\\): the constant \\(0\\) is an affine majorant of the function \\(0\\), and every affine majorant of \\(0\\) is \\(\\geq0\\), so \\(\\overline0=0\\). The last two claims are direct. (4) The first claim is Lemma 1.1. For the second apply Lemma 1.1 to \\(-g\\) and use (3). \\(\\square\\)\n\n**Remarks 1.6.**\n\n- (*\\(K\\) inside \\(\\operatorname{Aff}(K)^*\\).*) The map \\(x\\mapsto\\delta_x\\), \\(\\delta_x(a)=a(x)\\), maps \\(K\\) affinely and homeomorphically onto a convex set in \\(\\operatorname{Aff}(K)^*\\) that is weak\\(^*\\)-compact. It is one-to-one because \\(\\operatorname{Aff}_E(K)\\) separates points, continuous because each \\(a\\) is continuous, and a homeomorphism because \\(K\\) is compact. So nothing is lost if \\(E\\) is replaced by \\(\\operatorname{Aff}(K)^*\\) with its weak\\(^*\\) topology.\n- (*Krein–Milman.*) A nonempty compact convex set has an extreme point, and is the closed convex hull of its extreme points.\n\n",
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      "id": "OA-FND-IR-02",
      "unit": "integral-representations-of-states",
      "name": "2. Measures on a compact space",
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      "full_conditions_and_proof": "## 2. Measures on a compact space\n\nThe following facts about measures are used throughout. Here \\(X\\) is a compact Hausdorff space and \\(\\mu\\in M^+(X)\\).\n\n**Proposition 2.1.**\n\n1. (*Integrals of lsc functions.*) For lsc \\(f:X\\to(-\\infty,+\\infty]\\),\n\\(\\int f\\,d\\mu=\\sup\\{\\mu(g):\\ g\\in C(X),\\ g\\leq f\\}\\). Dually, for usc \\(h:X\\to[-\\infty,\\infty)\\), \\(\\int h\\,d\\mu=\\inf\\{\\mu(g):\\ g\\in C(X),\\ g\\geq h\\}\\).\n2. (*Monotone nets.*) If \\((f_i)\\) is an increasing net of lsc functions \\(X\\to(-\\infty,+\\infty]\\) with pointwise supremum \\(f\\), then \\(\\int f\\,d\\mu=\\sup_i\\int f_i\\,d\\mu\\).\n3. (*Support.*) There is a largest open \\(\\mu\\)-null set. Its complement \\(\\operatorname{supp}\\mu\\) is closed, carries \\(\\mu\\), and every open set that meets it has positive measure. If \\(F\\) is closed and \\(\\mu(X\\setminus F)=0\\), then \\(\\operatorname{supp}\\mu\\subseteq F\\).\n4. (*Baire sets.*) A *zero set* is \\(Z(h)=h^{-1}(0)\\) with \\(h\\in C(X)\\); its complement is a *cozero set*. The *Baire* \\(\\sigma\\)-algebra \\(\\mathfrak B_0(X)\\) is generated by the zero sets.\n   - (a) The zero sets are exactly the closed \\(G_\\delta\\) sets (equivalently, the compact \\(G_\\delta\\) sets). Finite unions and countable intersections of zero sets are zero sets. Countable unions of cozero sets are cozero sets.\n   - (b) If \\(\\Phi:X\\to Y\\) is continuous and \\(Y\\) is compact Hausdorff, then \\(\\Phi^{-1}\\) maps Baire sets to Baire sets.\n   - (c) (*Regularity.*) For every Baire set \\(B\\) and \\(\\varepsilon>0\\) there are a zero set \\(Z\\subseteq B\\) and a cozero set \\(V\\supseteq B\\) with \\(\\mu(V\\setminus Z)<\\varepsilon\\).\n   - (d) If \\(X\\) is metrizable, every closed set is a zero set, so the Baire sets are the Borel sets.\n   - (e) Every finite positive measure on \\(\\mathfrak B_0(X)\\) is the restriction of a unique regular Borel measure.\n5. (*Bounded Radon–Nikodym.*) If \\(\\nu\\in M^+(X)\\) and \\(\\nu\\leq c\\mu\\), then \\(\\nu=g\\mu\\) for a Borel \\(g\\) with \\(0\\leq g\\leq c\\).\n6. (*Weak\\(^*\\) density.*) \\(C(X)\\) is weak\\(^*\\)-dense in \\(L^\\infty(X,\\mu)\\) (real or complex scalars).\n7. (*Image measures.*) Let \\(\\Phi:X\\to Y\\) be continuous, \\(Y\\) compact Hausdorff, and \\(\\Phi_*\\mu(g)=\\mu(g\\circ\\Phi)\\). Then \\(\\Phi_*\\mu(N)=\\mu(\\Phi^{-1}(N))\\) for every Borel set \\(N\\subseteq Y\\), so \\(\\int g\\,d\\Phi_*\\mu=\\int g\\circ\\Phi\\,d\\mu\\) for bounded Borel \\(g\\). If \\(\\Phi\\) is one-to-one, then \\(G\\mapsto G\\circ\\Phi\\) is an isometric \\(*\\)-isomorphism of \\(L^\\infty(Y,\\Phi_*\\mu)\\) onto \\(L^\\infty(X,\\mu)\\).\n\n**Proof.** (1) If \\(g\\in C(X)\\) and \\(g\\leq f\\), then \\(\\mu(g)\\leq\\int f\\,d\\mu\\). For the converse, if \\(f\\equiv+\\infty\\), both sides are \\(+\\infty\\) (both \\(0\\) when \\(\\mu=0\\)). Otherwise subtract the minimum of \\(f\\) and assume \\(f\\geq0\\). Fix \\(N\\geq1\\) and \\(0<\\delta<1\\), let \\(m=\\lfloor N/\\delta\\rfloor\\), and let \\(U_k=\\{f>k\\delta\\}\\), an open set. The function \\(s=\\delta\\sum_{k=1}^m1_{U_k}\\) counts the multiples \\(k\\delta\\) below \\(f\\) up to \\(m\\delta\\), so \\(\\min(f,N)-\\delta\\leq s\\leq f\\). Inner regularity on open sets and [Urysohn's lemma](stone-weierstrass-c0.md#oa-fnd-sw-16) give \\(g_k\\in C(X)\\) with \\(0\\leq g_k\\leq1_{U_k}\\) and \\(\\mu(g_k)\\geq\\mu(U_k)-\\delta/m\\). Then \\(g=\\delta\\sum_kg_k\\leq s\\leq f\\) and \\[\n\\begin{gathered}\n\\mu(g)\\\\\n\\geq\\int s\\,d\\mu-\\delta^2\\\\\n\\geq\\int\\min(f,N)\\,d\\mu-\\delta\\mu(X)-\\delta\n\\end{gathered}\n\\]. Let \\(\\delta\\to0\\), then \\(N\\to\\infty\\) (monotone convergence). The usc statement follows by applying this to \\(-h\\).\n\n(2) Clearly \\(\\int f_i\\leq\\int f\\). Let \\(g\\in C(X)\\) with \\(g\\leq f\\), and \\(\\varepsilon>0\\). The sets \\(\\{f_i>g-\\varepsilon\\}\\) are open, increase with \\(i\\), and cover \\(X\\), since \\(\\sup_if_i(x)=f(x)>g(x)-\\varepsilon\\). By compactness one of them is \\(X\\), so \\(\\int f_i\\geq\\mu(g)-\\varepsilon\\mu(X)\\). Now use (1).\n\n(3) Let \\(U\\) be the union of all open null sets. A compact subset of \\(U\\) is covered by finitely many of them, so it is null, and inner regularity gives \\(\\mu(U)=0\\). The rest is immediate.\n\n(4)(a) \\(Z(h)=\\bigcap_n\\{|h|<1/n\\}\\) is a closed \\(G_\\delta\\). Conversely, if \\(C=\\bigcap_nU_n\\) is closed with \\(U_n\\) open, take Urysohn functions \\(h_n:X\\to[0,1]\\) with \\(h_n=0\\) on \\(C\\) and \\(h_n=1\\) off \\(U_n\\); then \\(C=Z(\\sum_n2^{-n}h_n)\\). Also \\(Z(h_1)\\cup Z(h_2)=Z(h_1h_2)\\) and \\(\\bigcap_nZ(h_n)=Z(\\sum_n2^{-n}\\min(|h_n|,1))\\); take complements for cozero sets. (b) \\(\\Phi^{-1}(Z(h))=Z(h\\circ\\Phi)\\), and the sets whose preimage is Baire form a \\(\\sigma\\)-algebra. (c) The sets \\(B\\) with the property form a \\(\\sigma\\)-algebra containing the zero sets. A zero set \\(Z(h)\\) is the decreasing intersection of the cozero sets \\(\\{|h|<1/n\\}\\), so continuity from above works. Complements: if \\(Z\\subseteq B\\subseteq V\\), then \\(V^c\\subseteq B^c\\subseteq Z^c\\) with the same difference. Countable unions: choose \\(Z_k\\subseteq B_k\\subseteq V_k\\) with \\(\\mu(V_k\\setminus Z_k)<\\varepsilon2^{-k-1}\\), put \\(V=\\bigcup V_k\\), and \\(Z=Z_1\\cup\\dots\\cup Z_n\\) with \\(n\\) so large that \\(\\mu(\\bigcup_kZ_k\\setminus Z)<\\varepsilon/2\\). (d) A closed \\(F\\) is \\(Z(\\operatorname{dist}(\\cdot,F))\\). (e) Let \\(\\mu_0\\) be such a measure. Continuous functions are Baire measurable, so \\(g\\mapsto\\int g\\,d\\mu_0\\) lies in \\(M^+(X)\\) and has a Radon measure \\(\\mu\\). For a zero set \\(Z(h)\\), the functions \\(\\max(0,1-n|h|)\\in C(X)\\) decrease to \\(1_{Z(h)}\\), so \\(\\mu(Z)=\\mu_0(Z)\\) by dominated convergence. Zero sets are closed under finite intersections and generate \\(\\mathfrak B_0(X)\\), so \\(\\mu=\\mu_0\\) on \\(\\mathfrak B_0(X)\\) by the \\(\\pi\\)–\\(\\lambda\\) theorem. Uniqueness is part of the Riesz theorem.\n\n(5) The functional \\(g\\mapsto\\int g\\,d\\nu\\) on \\(L^2(X,\\mu)\\) is well defined and bounded, since \\(\\int|g|\\,d\\nu\\leq c\\int|g|\\,d\\mu\\leq c\\,\\mu(X)^{1/2}\\|g\\|_2\\). By the Riesz lemma in Hilbert space it is \\(g\\mapsto\\int g\\bar u\\,d\\mu\\) with \\(u\\in L^2(\\mu)\\). With \\(w=\\bar u\\) and indicator functions, \\(\\nu(B)=\\int_Bw\\,d\\mu\\) for every Borel set \\(B\\). Since \\(0\\leq\\nu(B)\\leq c\\mu(B)\\) for all \\(B\\), we get \\(0\\leq w\\leq c\\) almost everywhere.\n\n(6) The weak\\(^*\\) topology on \\(L^\\infty(\\mu)=L^1(\\mu)^*\\) is locally convex, and its continuous linear functionals are the maps \\(f\\mapsto\\int fu\\,d\\mu\\) with \\(u\\in L^1(\\mu)\\); both facts are recalled in the background section. If some \\(f\\) were outside the weak\\(^*\\) closure of \\(C(X)\\), separation would give \\(u\\in L^1(\\mu)\\) with \\(\\int gu\\,d\\mu=0\\) for all \\(g\\in C(X)\\) and \\(\\int fu\\,d\\mu\\neq0\\). But the measure \\(u\\mu\\) has a finite Radon variation measure \\(|u|\\mu\\), by [the small-set integral lemma and its regularity consequence](abelian-operator-algebras.md#oa-fnd-ao-23), and it annihilates \\(C(X)\\), so \\(u\\mu=0\\) by the uniqueness in the Riesz theorem, and \\(u=0\\) almost everywhere. This is a contradiction.\n\n(7) The Borel measure \\(\\nu(N)=\\mu(\\Phi^{-1}(N))\\) on \\(Y\\) is regular. Given \\(N\\) and \\(\\varepsilon>0\\), regularity of \\(\\mu\\) gives a compact \\(C\\subseteq\\Phi^{-1}(N)\\) with \\(\\mu(\\Phi^{-1}(N)\\setminus C)<\\varepsilon\\). Then \\(\\Phi(C)\\subseteq N\\) is compact, and \\(\\nu(N\\setminus\\Phi(C))\\leq\\mu(\\Phi^{-1}(N)\\setminus C)<\\varepsilon\\) because \\(C\\subseteq\\Phi^{-1}(\\Phi(C))\\). Outer regularity follows by passing to complements. Since \\(\\nu\\) integrates \\(g\\in C(Y)\\) as \\(\\mu(g\\circ\\Phi)\\), the uniqueness in the Riesz theorem gives \\(\\nu=\\Phi_*\\mu\\). The integral formula follows for simple functions, and then for uniform limits of them. For one-to-one \\(\\Phi\\), the map \\(G\\mapsto G\\circ\\Phi\\) is therefore well defined, injective and isometric on classes. It is onto: \\(\\Phi\\) is a homeomorphism of \\(X\\) onto the compact set \\(\\Phi(X)\\), so for a bounded Borel \\(g\\) on \\(X\\) the function \\(G=g\\circ\\Phi^{-1}\\) on \\(\\Phi(X)\\), extended by \\(0\\), is Borel, and \\(G\\circ\\Phi=g\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-IR-04",
      "unit": "integral-representations-of-states",
      "name": "3. Barycentres",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 3. Barycentres\n\nA probability measure on \\(K\\) has a centre of mass in \\(K\\), its barycentre. This section constructs it and collects its basic properties.\n\n**Lemma 3.1** (The barycentre exists). For every \\(\\mu\\in M_1^+(K)\\) there is exactly one \\(y\\in K\\) with\n\\[\np(y)=\\int_Kp\\,d\\mu\\qquad\\text{for all }p\\in E^*.\n\\]\n\n**Proof.** Uniqueness holds because \\(E^*\\) separates points. For a finite set \\(F\\subseteq E^*\\) let \\(K_F=\\{y\\in K:\\ p(y)=\\mu(p)\\text{ for }p\\in F\\}\\), a closed subset of \\(K\\). It is nonempty. Indeed, let \\(T(y)=(p(y))_{p\\in F}\\in\\mathbb R^F\\); \\(T(K)\\) is compact and convex. If \\(m=(\\mu(p))_{p\\in F}\\) were not in \\(T(K)\\), separation in \\(\\mathbb R^F\\) would give \\(c\\in\\mathbb R^F\\) with \\(\\sum_pc_pm_p>\\max_{y\\in K}\\sum_pc_pp(y)\\). But \\(\\sum_pc_pm_p=\\mu(\\sum_pc_pp)\\) is at most that maximum, since \\(\\mu\\) is a probability measure. Since \\(K_{F_1}\\cap K_{F_2}=K_{F_1\\cup F_2}\\), the family \\((K_F)\\) has the finite intersection property, and compactness gives a point in all of them. \\(\\square\\)\n\n**Definition 3.2** (Barycentre). This \\(y\\) is the *barycentre* (or resultant) \\(r(\\mu)\\) of \\(\\mu\\), and \\(\\mu\\) *represents* \\(y\\). We write \\(M_y(K)\\) for the set of probability measures on \\(K\\) that represent \\(y\\).\n\nBoth sides of the next formula are continuous in \\(a\\) for the uniform norm and agree on \\(\\operatorname{Aff}_E(K)\\) (use \\(\\mu(1)=1\\)). By Lemma 1.2 they agree on \\(\\operatorname{Aff}(K)\\):\n\\[\n\\begin{gathered}\na(r(\\mu))\\\\\n=\\int_Ka\\,d\\mu\\\\\n(a\\in\\operatorname{Aff}(K)).\n\\end{gathered}\n\\tag{3.1}\n\\]\n\n**Proposition 3.3** (Properties of barycentres). Let \\(\\mu\\in M_1^+(K)\\) and \\(x=r(\\mu)\\).\n\n1. If \\(C\\subseteq K\\) is closed and convex and \\(\\mu(K\\setminus C)=0\\), then \\(x\\in C\\).\n2. (*Jensen's inequality.*) \\(g(x)\\leq\\int g\\,d\\mu\\) for every \\(g\\in\\mathcal L(K)\\), and \\(h(x)\\geq\\int h\\,d\\mu\\) whenever \\(-h\\in\\mathcal L(K)\\).\n3. (*Bauer's criterion.*) A point \\(x\\in K\\) is extreme if and only if \\(M_x(K)=\\{\\delta_x\\}\\).\n4. \\(r:M_1^+(K)\\to K\\) is affine, continuous and onto.\n\n**Proof.** (1) If \\(x\\notin C\\), separation gives \\(p\\in E^*\\) with \\(p(x)>\\sup_Cp\\). But \\(p(x)=\\int_Cp\\,d\\mu\\leq\\sup_Cp\\).\n\n(2) For each \\(a\\in\\operatorname{Aff}_E(K)\\) with \\(a\\leq g\\), (3.1) gives \\(a(x)=\\mu(a)\\leq\\int g\\,d\\mu\\). Take the supremum and use (1.1). The second claim is the first for \\(-h\\).\n\n(3) If \\(x=ty+(1-t)z\\) with \\(y\\neq z\\) and \\(0<t<1\\), then \\(t\\delta_y+(1-t)\\delta_z\\) is a second measure in \\(M_x(K)\\). Conversely let \\(x\\) be extreme, \\(\\mu\\in M_x(K)\\), and suppose \\(\\mu\\neq\\delta_x\\). The support of \\(\\mu\\) carries \\(\\mu\\) ([Proposition 2.1](#oa-fnd-ir-02)(3)), and \\(\\mu\\) is a probability measure other than \\(\\delta_x\\); so \\(\\operatorname{supp}\\mu\\) is not contained in \\(\\{x\\}\\), and there is \\(y\\in\\operatorname{supp}\\mu\\) with \\(y\\neq x\\). Pick a closed convex neighbourhood \\(W\\) of \\(y\\) in \\(E\\) with \\(x\\notin W\\), and put \\(C=W\\cap K\\) and \\(t=\\mu(C)>0\\). If \\(t=1\\), then \\(x\\in C\\) by (1), which is false. If \\(0<t<1\\), the probability measures \\(\\mu_1=t^{-1}\\mu|_C\\) and \\(\\mu_2=(1-t)^{-1}\\mu|_{K\\setminus C}\\) have barycentres \\(x_1\\in C\\) (by (1)) and \\(x_2\\in K\\), and \\(x=tx_1+(1-t)x_2\\) by (3.1), because \\(E^*\\) separates points. Extremality gives \\(x=x_1\\in C\\), again false.\n\n(4) \\(r\\) is affine by (3.1), and \\(r(\\delta_x)=x\\). For \\(p\\in E^*\\), \\(p\\circ r(\\mu)=\\mu(p|_K)\\) is weak\\(^*\\)-continuous in \\(\\mu\\). So \\(r\\) is continuous into \\(K\\) with the weak topology \\(\\sigma(E,E^*)\\). On the compact set \\(K\\) this topology coincides with the given one, since the identity from \\(K\\) to \\((K,\\sigma(E,E^*))\\) is a continuous bijection from a compact space onto a Hausdorff space. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-IR-03",
      "unit": "integral-representations-of-states",
      "name": "4. The Choquet order",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 4. The Choquet order\n\nThe Choquet order compares two measures through their integrals of continuous convex functions. We first need two facts about convex functions.\n\n**Lemma 4.1** (Differences of convex functions). \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace of \\(C(K)\\), dense for the sup norm, and it is closed under \\(\\vee\\) and \\(\\wedge\\).\n\n**Proof.** \\(\\mathcal P(K)\\) is a convex cone that is closed under \\(\\vee\\). So \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace. If \\(f_i=g_i-h_i\\) with \\(g_i,h_i\\in\\mathcal P(K)\\), then\n\\[\n\\begin{gathered}\nf_1\\vee f_2\\\\\n=\\bigl((g_1+h_2)\\vee(g_2+h_1)\\bigr)\\\\\n-(h_1+h_2)\\\\\n\\in\\mathcal P(K)-\\mathcal P(K),\n\\end{gathered}\n\\]\nand \\(f_1\\wedge f_2=-((-f_1)\\vee(-f_2))\\). The subspace contains the constants and \\(\\operatorname{Aff}_E(K)\\), which separates the points of \\(K\\). By [Stone's lattice lemma](stone-weierstrass-c0.md#oa-fnd-sw-06), a linear subspace of \\(C(K)\\) that is closed under \\(\\vee\\) and \\(\\wedge\\), contains the constants and separates points is dense. \\(\\square\\)\n\n**Lemma 4.2** (Lower semicontinuous convex functions as limits). Let \\(f\\in\\mathcal L(K)\\). The set \\(D_f\\) of functions \\(a_1\\vee\\dots\\vee a_n\\), with \\(n\\geq1\\) and \\(a_i\\in\\operatorname{Aff}_E(K)\\), \\(a_i\\leq f\\), is a subset of \\(\\mathcal P(K)\\) that is directed upward and has pointwise supremum \\(f\\). So \\(f\\) is the pointwise limit of an increasing net in \\(\\mathcal P(K)\\).\n\n**Proof.** \\(D_f\\) is nonempty, because some constant lies below \\(f\\) (see the proof of Lemma 1.1), and it is directed by \\(\\vee\\). Its supremum is \\(f\\) by (1.1). \\(\\square\\)\n\n**Corollary 4.3.** Let \\(\\mu,\\nu\\in M^+(K)\\) with \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). Then \\(\\int g\\,d\\mu\\leq\\int g\\,d\\nu\\) for every \\(g\\in\\mathcal L(K)\\), and \\(\\int h\\,d\\mu\\geq\\int h\\,d\\nu\\) whenever \\(-h\\in\\mathcal L(K)\\).\n\n**Proof.** Let \\(g\\in\\mathcal L(K)\\), and index the increasing net of Lemma 4.2 by \\(D_g\\) itself. Its members are continuous, so the monotone-net property, [Proposition 2.1](#oa-fnd-ir-02)(2), applies to both measures and gives \\(\\int g\\,d\\mu=\\sup_{d\\in D_g}\\mu(d)\\leq\\sup_{d\\in D_g}\\nu(d)=\\int g\\,d\\nu\\). The second claim is the first for \\(g=-h\\). \\(\\square\\)\n\n**Definition 4.4** (The Choquet order). For \\(\\mu,\\nu\\in M(K)\\) we write \\(\\mu\\prec\\nu\\) (\\(\\nu\\) *majorizes* \\(\\mu\\)) if \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). We write \\(\\mu\\sim\\nu\\) if \\(\\mu(a)=\\nu(a)\\) for every \\(a\\in\\operatorname{Aff}(K)\\). A measure is *maximal* if it is maximal for \\(\\prec\\) in \\(M^+(K)\\).\n\n**Proposition 4.5.**\n\n1. \\(\\prec\\) is a partial order on \\(M(K)\\), and \\(\\mu\\prec\\nu\\) implies \\(\\mu\\sim\\nu\\).\n2. If \\(\\mu,\\nu\\in M^+(K)\\) and \\(\\mu\\prec\\nu\\), then \\(\\mu(1)=\\nu(1)\\); if \\(\\mu\\) is a probability measure, so is \\(\\nu\\), and \\(r(\\mu)=r(\\nu)\\).\n3. \\(\\delta_{r(\\mu)}\\prec\\mu\\) for every \\(\\mu\\in M_1^+(K)\\).\n\n**Proof.** (1) Reflexivity and transitivity are clear. If \\(\\mu\\prec\\nu\\prec\\mu\\), the two measures agree on \\(\\mathcal P(K)\\), hence on \\(C(K)\\), because \\(\\mathcal P(K)-\\mathcal P(K)\\) is dense ([Lemma 4.1](#oa-fnd-ir-03)). If \\(\\mu\\prec\\nu\\) and \\(a\\in\\operatorname{Aff}(K)\\), then \\(a\\) and \\(-a\\) are both convex, so \\(\\mu(a)=\\nu(a)\\). (2) \\(\\pm1\\in\\mathcal P(K)\\), so \\(\\mu(1)=\\nu(1)\\); and barycentres are determined by the values on \\(\\operatorname{Aff}(K)\\), which agree by (1). (3) is Jensen's inequality, [Proposition 3.3](#oa-fnd-ir-04)(2). \\(\\square\\)\n\n**Lemma 4.6** (Maximal measures exist). Every chain in \\((M^+(K),\\prec)\\) has an upper bound. Hence every \\(\\nu\\in M^+(K)\\) is majorized by a maximal measure.\n\n**Proof.** Let \\(\\mathcal T\\) be a nonempty chain, indexed by itself as a directed set. All its members have the same mass \\(c\\), by Proposition 4.5(2). For \\(f\\in\\mathcal P(K)\\), the net \\((\\mu(f))_{\\mu\\in\\mathcal T}\\) increases and is bounded by \\(c\\|f\\|\\), so it converges. By linearity the net converges for \\(f\\in\\mathcal P(K)-\\mathcal P(K)\\). For \\(f\\in C(K)\\) and \\(\\varepsilon>0\\) pick such a \\(g\\) with \\(\\|f-g\\|<\\varepsilon\\); then \\(|\\mu(f)-\\mu'(f)|\\leq2c\\varepsilon+|\\mu(g)-\\mu'(g)|\\), so the net is Cauchy. The limit \\(\\lambda(f)\\) is a positive linear functional, so \\(\\lambda\\in M^+(K)\\), and \\(\\mu\\prec\\lambda\\) for every \\(\\mu\\in\\mathcal T\\). Zorn's lemma, applied to \\(\\{\\mu:\\nu\\prec\\mu\\}\\), gives a maximal element there. It is maximal in \\(M^+(K)\\), because anything above it is above \\(\\nu\\). \\(\\square\\)\n\n**Lemma 4.7.** For every \\(\\mu\\in M^+(K)\\) and \\(f\\in C(K)\\) there is \\(\\nu\\in M^+(K)\\) with \\(\\mu\\prec\\nu\\) and \\(\\nu(f)=\\mu(\\overline f)\\).\n\n**Proof.** For \\(g\\in C(K)\\), \\(\\overline g\\) is a bounded usc function ([Proposition 1.5](#oa-fnd-ir-01)), so \\(\\Psi(g)=\\int\\overline g\\,d\\mu\\) is defined. By Proposition 1.5(3), \\(\\Psi\\) is sublinear: \\(\\Psi(g+h)\\leq\\Psi(g)+\\Psi(h)\\) and \\(\\Psi(tg)=t\\Psi(g)\\) for \\(t\\geq0\\). On the line \\(\\mathbb Rf\\) put \\(\\nu_0(tf)=t\\mu(\\overline f)\\). Then \\(\\nu_0\\leq\\Psi\\) there. For \\(t\\geq0\\) this is equality. For \\(t<0\\), \\(0=\\overline{tf+|t|f}\\leq\\overline{tf}+|t|\\overline f\\) gives \\(t\\overline f\\leq\\overline{tf}\\), and we integrate. By the Hahn–Banach extension theorem, \\(\\nu_0\\) extends to a linear \\(\\nu\\) on \\(C(K)\\) with \\(\\nu\\leq\\Psi\\). If \\(g\\leq0\\), then \\(\\overline g\\leq0\\), so \\(\\nu(g)\\leq0\\): \\(\\nu\\) is positive. If \\(g\\in\\mathcal P(K)\\), then \\(\\overline{-g}=-g\\) (Proposition 1.5(4)), so \\(-\\nu(g)=\\nu(-g)\\leq\\mu(\\overline{-g})=-\\mu(g)\\), that is, \\(\\mu(g)\\leq\\nu(g)\\). \\(\\square\\)\n\n**Corollary 4.8** (The upper envelope as a maximum). For \\(x\\in K\\) and \\(f\\in C(K)\\),\n\\[\n\\overline f(x)=\\max\\{\\nu(f):\\ \\nu\\in M_x(K)\\}.\n\\]\nIf \\(f\\in\\mathcal P(K)\\), the maximum is attained at a maximal measure.\n\n**Proof.** For \\(\\nu\\in M_x(K)\\), Jensen's inequality for the usc concave \\(\\overline f\\) gives \\(\\nu(f)\\leq\\nu(\\overline f)\\leq\\overline f(x)\\). Lemma 4.7 with \\(\\mu=\\delta_x\\) gives \\(\\nu\\succ\\delta_x\\) with \\(\\nu(f)=\\overline f(x)\\), and \\(\\nu\\in M_x(K)\\) by Proposition 4.5(2). If \\(f\\) is convex, a maximal \\(\\nu'\\succ\\nu\\) (Lemma 4.6) lies in \\(M_x(K)\\) and has \\(\\nu'(f)\\geq\\nu(f)\\). \\(\\square\\)\n\n**Remark 4.9.** For nonconvex \\(f\\) the maximum need not be attained at a maximal measure. Let \\(K\\) be a square, \\(x\\) its centre, and \\(f\\in C(K)\\) a bump with \\(f(x)=1\\) that vanishes near the four vertices. Then \\(\\overline f(x)\\geq f(x)=1\\), but every maximal measure in \\(M_x(K)\\) lives on the vertices ([Example 7.5](#oa-fnd-ir-27)) and gives \\(f\\) the value \\(0\\).\n\nThe next lemma describes the Choquet order through decompositions of measures. It is used in Sections 14, 15 and 17.\n\n**Lemma 4.10** (The order through decompositions). For \\(\\mu,\\nu\\in M^+(K)\\) the following are equivalent.\n\n1. \\(\\mu\\prec\\nu\\).\n2. Whenever \\(\\mu=\\sum_{i=1}^n\\mu_i\\) with \\(\\mu_i\\in M^+(K)\\), there are \\(\\nu_i\\in M^+(K)\\) with \\(\\nu=\\sum_i\\nu_i\\) and \\(\\nu_i\\sim\\mu_i\\) for each \\(i\\).\n\nIf (1) holds, the \\(\\nu_i\\) in (2) can be chosen with \\(\\mu_i\\prec\\nu_i\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): On \\(C(K)^n\\) with the norm \\(\\|\\vec f\\|=\\max_i\\|f_i\\|\\) put \\(\\Psi(\\vec f)=\\sum_i\\mu_i(\\overline{f_i})\\). It is sublinear by Proposition 1.5(3), as in the proof of Lemma 4.7. On the diagonal \\(\\Delta=\\{(f,\\dots,f)\\}\\) put \\(\\Psi_0(f,\\dots,f)=\\nu(f)\\). Then\n\\[\n\\begin{gathered}\n\\Psi_0(f,\\dots,f)\\\\\n=\\nu(f)\\\\\n\\leq\\nu(\\overline f)\\\\\n\\leq\\mu(\\overline f)\\\\\n=\\Psi(f,\\dots,f),\n\\end{gathered}\n\\]\nwhere the middle step is Corollary 4.3 applied to \\(-\\overline f\\in\\mathcal L(K)\\), and the last step uses \\(\\sum_i\\mu_i=\\mu\\). The Hahn–Banach theorem gives a linear extension \\(\\Lambda\\) of \\(\\Psi_0\\) to \\(C(K)^n\\) with \\(\\Lambda\\leq\\Psi\\). Since \\(\\|\\overline{f_i}\\|_\\infty\\leq\\|f_i\\|\\), we get \\(\\Lambda(\\vec f)\\leq\\sum_i\\|\\mu_i\\|\\|f_i\\|\\leq\\|\\mu\\|\\|\\vec f\\|\\); with \\(-\\vec f\\), \\(|\\Lambda(\\vec f)|\\leq\\|\\mu\\|\\|\\vec f\\|\\). So \\(\\Lambda\\) is bounded, each map \\(f\\mapsto\\Lambda(0,\\dots,f,\\dots,0)\\) is a measure \\(\\nu_i\\in M(K)\\), and \\(\\Lambda(\\vec f)=\\sum_i\\nu_i(f_i)\\). Moreover \\(\\sum_i\\|\\nu_i\\|=\\|\\Lambda\\|\\leq\\|\\mu\\|\\) (choose \\(f_i\\) of norm \\(\\leq1\\) with \\(\\nu_i(f_i)\\) close to \\(\\|\\nu_i\\|\\)). With \\(\\vec1=(1,\\dots,1)\\), and \\(\\nu(1)=\\mu(1)\\) by Proposition 4.5(2): \\[\n\\begin{gathered}\n\\sum_i\\nu_i(1)\\\\\n=\\Lambda(\\vec1)\\\\\n=\\nu(1)\\\\\n=\\mu(1)\\\\\n=\\|\\mu\\|\\\\\n\\geq\\sum_i\\|\\nu_i\\|\n\\end{gathered}\n\\]. Since \\(\\nu_i(1)\\leq\\|\\nu_i\\|\\) for each \\(i\\), all these are equalities, and a real measure with \\(\\nu_i(1)=\\|\\nu_i\\|\\) is positive. For \\(f\\in C(K)\\), \\(\\nu_i(f)=\\Lambda(0,\\dots,f,\\dots,0)\\leq\\mu_i(\\overline f)\\) (note \\(\\overline0=0\\)). For \\(f\\in\\mathcal P(K)\\), applying this to \\(-f\\) and using \\(\\overline{-f}=-f\\) gives \\(\\mu_i(f)\\leq\\nu_i(f)\\). So \\(\\mu_i\\prec\\nu_i\\), hence \\(\\mu_i\\sim\\nu_i\\). Finally \\(\\sum_i\\nu_i=\\nu\\) because \\(\\Lambda\\) extends \\(\\Psi_0\\).\n\n(2)\\(\\Rightarrow\\)(1): Let \\(f\\in\\mathcal P(K)\\) and \\(\\varepsilon>0\\). Each point of \\(K\\) has a closed convex neighbourhood in \\(E\\) on which \\(f\\) varies by less than \\(\\varepsilon\\) (continuity and local convexity). By compactness there are closed convex \\(G_1,\\dots,G_n\\subseteq K\\) that cover \\(K\\) with \\(|f(y)-f(z)|<\\varepsilon\\) for \\(y,z\\in G_i\\). Let \\(A_i=G_i\\setminus\\bigcup_{j<i}G_j\\) and \\(\\mu_i=\\mu|_{A_i}\\). By (2), \\(\\nu=\\sum_i\\nu_i\\) with \\(\\nu_i\\sim\\mu_i\\); so \\(\\nu_i(K)=\\mu_i(K)\\). For \\(\\mu_i\\neq0\\), the normalized \\(\\mu_i\\) and \\(\\nu_i\\) have the same barycentre \\(x_i\\), and \\(x_i\\in G_i\\) by [Proposition 3.3](#oa-fnd-ir-04)(1). Then \\(\\mu_i(f)\\leq\\mu_i(K)(f(x_i)+\\varepsilon)=\\nu_i(K)(f(x_i)+\\varepsilon)\\), and Jensen's inequality gives \\(\\nu_i(K)f(x_i)\\leq\\nu_i(f)\\). Summing, \\(\\mu(f)\\leq\\nu(f)+\\varepsilon\\nu(K)\\). \\(\\square\\)\n\n",
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      "name": "4. The Choquet order",
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      "full_conditions_and_proof": "## 4. The Choquet order\n\nThe Choquet order compares two measures through their integrals of continuous convex functions. We first need two facts about convex functions.\n\n**Lemma 4.1** (Differences of convex functions). \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace of \\(C(K)\\), dense for the sup norm, and it is closed under \\(\\vee\\) and \\(\\wedge\\).\n\n**Proof.** \\(\\mathcal P(K)\\) is a convex cone that is closed under \\(\\vee\\). So \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace. If \\(f_i=g_i-h_i\\) with \\(g_i,h_i\\in\\mathcal P(K)\\), then\n\\[\n\\begin{gathered}\nf_1\\vee f_2\\\\\n=\\bigl((g_1+h_2)\\vee(g_2+h_1)\\bigr)\\\\\n-(h_1+h_2)\\\\\n\\in\\mathcal P(K)-\\mathcal P(K),\n\\end{gathered}\n\\]\nand \\(f_1\\wedge f_2=-((-f_1)\\vee(-f_2))\\). The subspace contains the constants and \\(\\operatorname{Aff}_E(K)\\), which separates the points of \\(K\\). By [Stone's lattice lemma](stone-weierstrass-c0.md#oa-fnd-sw-06), a linear subspace of \\(C(K)\\) that is closed under \\(\\vee\\) and \\(\\wedge\\), contains the constants and separates points is dense. \\(\\square\\)\n\n**Lemma 4.2** (Lower semicontinuous convex functions as limits). Let \\(f\\in\\mathcal L(K)\\). The set \\(D_f\\) of functions \\(a_1\\vee\\dots\\vee a_n\\), with \\(n\\geq1\\) and \\(a_i\\in\\operatorname{Aff}_E(K)\\), \\(a_i\\leq f\\), is a subset of \\(\\mathcal P(K)\\) that is directed upward and has pointwise supremum \\(f\\). So \\(f\\) is the pointwise limit of an increasing net in \\(\\mathcal P(K)\\).\n\n**Proof.** \\(D_f\\) is nonempty, because some constant lies below \\(f\\) (see the proof of Lemma 1.1), and it is directed by \\(\\vee\\). Its supremum is \\(f\\) by (1.1). \\(\\square\\)\n\n**Corollary 4.3.** Let \\(\\mu,\\nu\\in M^+(K)\\) with \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). Then \\(\\int g\\,d\\mu\\leq\\int g\\,d\\nu\\) for every \\(g\\in\\mathcal L(K)\\), and \\(\\int h\\,d\\mu\\geq\\int h\\,d\\nu\\) whenever \\(-h\\in\\mathcal L(K)\\).\n\n**Proof.** Let \\(g\\in\\mathcal L(K)\\), and index the increasing net of Lemma 4.2 by \\(D_g\\) itself. Its members are continuous, so the monotone-net property, [Proposition 2.1](#oa-fnd-ir-02)(2), applies to both measures and gives \\(\\int g\\,d\\mu=\\sup_{d\\in D_g}\\mu(d)\\leq\\sup_{d\\in D_g}\\nu(d)=\\int g\\,d\\nu\\). The second claim is the first for \\(g=-h\\). \\(\\square\\)\n\n**Definition 4.4** (The Choquet order). For \\(\\mu,\\nu\\in M(K)\\) we write \\(\\mu\\prec\\nu\\) (\\(\\nu\\) *majorizes* \\(\\mu\\)) if \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). We write \\(\\mu\\sim\\nu\\) if \\(\\mu(a)=\\nu(a)\\) for every \\(a\\in\\operatorname{Aff}(K)\\). A measure is *maximal* if it is maximal for \\(\\prec\\) in \\(M^+(K)\\).\n\n**Proposition 4.5.**\n\n1. \\(\\prec\\) is a partial order on \\(M(K)\\), and \\(\\mu\\prec\\nu\\) implies \\(\\mu\\sim\\nu\\).\n2. If \\(\\mu,\\nu\\in M^+(K)\\) and \\(\\mu\\prec\\nu\\), then \\(\\mu(1)=\\nu(1)\\); if \\(\\mu\\) is a probability measure, so is \\(\\nu\\), and \\(r(\\mu)=r(\\nu)\\).\n3. \\(\\delta_{r(\\mu)}\\prec\\mu\\) for every \\(\\mu\\in M_1^+(K)\\).\n\n**Proof.** (1) Reflexivity and transitivity are clear. If \\(\\mu\\prec\\nu\\prec\\mu\\), the two measures agree on \\(\\mathcal P(K)\\), hence on \\(C(K)\\), because \\(\\mathcal P(K)-\\mathcal P(K)\\) is dense ([Lemma 4.1](#oa-fnd-ir-03)). If \\(\\mu\\prec\\nu\\) and \\(a\\in\\operatorname{Aff}(K)\\), then \\(a\\) and \\(-a\\) are both convex, so \\(\\mu(a)=\\nu(a)\\). (2) \\(\\pm1\\in\\mathcal P(K)\\), so \\(\\mu(1)=\\nu(1)\\); and barycentres are determined by the values on \\(\\operatorname{Aff}(K)\\), which agree by (1). (3) is Jensen's inequality, [Proposition 3.3](#oa-fnd-ir-04)(2). \\(\\square\\)\n\n**Lemma 4.6** (Maximal measures exist). Every chain in \\((M^+(K),\\prec)\\) has an upper bound. Hence every \\(\\nu\\in M^+(K)\\) is majorized by a maximal measure.\n\n**Proof.** Let \\(\\mathcal T\\) be a nonempty chain, indexed by itself as a directed set. All its members have the same mass \\(c\\), by Proposition 4.5(2). For \\(f\\in\\mathcal P(K)\\), the net \\((\\mu(f))_{\\mu\\in\\mathcal T}\\) increases and is bounded by \\(c\\|f\\|\\), so it converges. By linearity the net converges for \\(f\\in\\mathcal P(K)-\\mathcal P(K)\\). For \\(f\\in C(K)\\) and \\(\\varepsilon>0\\) pick such a \\(g\\) with \\(\\|f-g\\|<\\varepsilon\\); then \\(|\\mu(f)-\\mu'(f)|\\leq2c\\varepsilon+|\\mu(g)-\\mu'(g)|\\), so the net is Cauchy. The limit \\(\\lambda(f)\\) is a positive linear functional, so \\(\\lambda\\in M^+(K)\\), and \\(\\mu\\prec\\lambda\\) for every \\(\\mu\\in\\mathcal T\\). Zorn's lemma, applied to \\(\\{\\mu:\\nu\\prec\\mu\\}\\), gives a maximal element there. It is maximal in \\(M^+(K)\\), because anything above it is above \\(\\nu\\). \\(\\square\\)\n\n**Lemma 4.7.** For every \\(\\mu\\in M^+(K)\\) and \\(f\\in C(K)\\) there is \\(\\nu\\in M^+(K)\\) with \\(\\mu\\prec\\nu\\) and \\(\\nu(f)=\\mu(\\overline f)\\).\n\n**Proof.** For \\(g\\in C(K)\\), \\(\\overline g\\) is a bounded usc function ([Proposition 1.5](#oa-fnd-ir-01)), so \\(\\Psi(g)=\\int\\overline g\\,d\\mu\\) is defined. By Proposition 1.5(3), \\(\\Psi\\) is sublinear: \\(\\Psi(g+h)\\leq\\Psi(g)+\\Psi(h)\\) and \\(\\Psi(tg)=t\\Psi(g)\\) for \\(t\\geq0\\). On the line \\(\\mathbb Rf\\) put \\(\\nu_0(tf)=t\\mu(\\overline f)\\). Then \\(\\nu_0\\leq\\Psi\\) there. For \\(t\\geq0\\) this is equality. For \\(t<0\\), \\(0=\\overline{tf+|t|f}\\leq\\overline{tf}+|t|\\overline f\\) gives \\(t\\overline f\\leq\\overline{tf}\\), and we integrate. By the Hahn–Banach extension theorem, \\(\\nu_0\\) extends to a linear \\(\\nu\\) on \\(C(K)\\) with \\(\\nu\\leq\\Psi\\). If \\(g\\leq0\\), then \\(\\overline g\\leq0\\), so \\(\\nu(g)\\leq0\\): \\(\\nu\\) is positive. If \\(g\\in\\mathcal P(K)\\), then \\(\\overline{-g}=-g\\) (Proposition 1.5(4)), so \\(-\\nu(g)=\\nu(-g)\\leq\\mu(\\overline{-g})=-\\mu(g)\\), that is, \\(\\mu(g)\\leq\\nu(g)\\). \\(\\square\\)\n\n**Corollary 4.8** (The upper envelope as a maximum). For \\(x\\in K\\) and \\(f\\in C(K)\\),\n\\[\n\\overline f(x)=\\max\\{\\nu(f):\\ \\nu\\in M_x(K)\\}.\n\\]\nIf \\(f\\in\\mathcal P(K)\\), the maximum is attained at a maximal measure.\n\n**Proof.** For \\(\\nu\\in M_x(K)\\), Jensen's inequality for the usc concave \\(\\overline f\\) gives \\(\\nu(f)\\leq\\nu(\\overline f)\\leq\\overline f(x)\\). Lemma 4.7 with \\(\\mu=\\delta_x\\) gives \\(\\nu\\succ\\delta_x\\) with \\(\\nu(f)=\\overline f(x)\\), and \\(\\nu\\in M_x(K)\\) by Proposition 4.5(2). If \\(f\\) is convex, a maximal \\(\\nu'\\succ\\nu\\) (Lemma 4.6) lies in \\(M_x(K)\\) and has \\(\\nu'(f)\\geq\\nu(f)\\). \\(\\square\\)\n\n**Remark 4.9.** For nonconvex \\(f\\) the maximum need not be attained at a maximal measure. Let \\(K\\) be a square, \\(x\\) its centre, and \\(f\\in C(K)\\) a bump with \\(f(x)=1\\) that vanishes near the four vertices. Then \\(\\overline f(x)\\geq f(x)=1\\), but every maximal measure in \\(M_x(K)\\) lives on the vertices ([Example 7.5](#oa-fnd-ir-27)) and gives \\(f\\) the value \\(0\\).\n\nThe next lemma describes the Choquet order through decompositions of measures. It is used in Sections 14, 15 and 17.\n\n**Lemma 4.10** (The order through decompositions). For \\(\\mu,\\nu\\in M^+(K)\\) the following are equivalent.\n\n1. \\(\\mu\\prec\\nu\\).\n2. Whenever \\(\\mu=\\sum_{i=1}^n\\mu_i\\) with \\(\\mu_i\\in M^+(K)\\), there are \\(\\nu_i\\in M^+(K)\\) with \\(\\nu=\\sum_i\\nu_i\\) and \\(\\nu_i\\sim\\mu_i\\) for each \\(i\\).\n\nIf (1) holds, the \\(\\nu_i\\) in (2) can be chosen with \\(\\mu_i\\prec\\nu_i\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): On \\(C(K)^n\\) with the norm \\(\\|\\vec f\\|=\\max_i\\|f_i\\|\\) put \\(\\Psi(\\vec f)=\\sum_i\\mu_i(\\overline{f_i})\\). It is sublinear by Proposition 1.5(3), as in the proof of Lemma 4.7. On the diagonal \\(\\Delta=\\{(f,\\dots,f)\\}\\) put \\(\\Psi_0(f,\\dots,f)=\\nu(f)\\). Then\n\\[\n\\begin{gathered}\n\\Psi_0(f,\\dots,f)\\\\\n=\\nu(f)\\\\\n\\leq\\nu(\\overline f)\\\\\n\\leq\\mu(\\overline f)\\\\\n=\\Psi(f,\\dots,f),\n\\end{gathered}\n\\]\nwhere the middle step is Corollary 4.3 applied to \\(-\\overline f\\in\\mathcal L(K)\\), and the last step uses \\(\\sum_i\\mu_i=\\mu\\). The Hahn–Banach theorem gives a linear extension \\(\\Lambda\\) of \\(\\Psi_0\\) to \\(C(K)^n\\) with \\(\\Lambda\\leq\\Psi\\). Since \\(\\|\\overline{f_i}\\|_\\infty\\leq\\|f_i\\|\\), we get \\(\\Lambda(\\vec f)\\leq\\sum_i\\|\\mu_i\\|\\|f_i\\|\\leq\\|\\mu\\|\\|\\vec f\\|\\); with \\(-\\vec f\\), \\(|\\Lambda(\\vec f)|\\leq\\|\\mu\\|\\|\\vec f\\|\\). So \\(\\Lambda\\) is bounded, each map \\(f\\mapsto\\Lambda(0,\\dots,f,\\dots,0)\\) is a measure \\(\\nu_i\\in M(K)\\), and \\(\\Lambda(\\vec f)=\\sum_i\\nu_i(f_i)\\). Moreover \\(\\sum_i\\|\\nu_i\\|=\\|\\Lambda\\|\\leq\\|\\mu\\|\\) (choose \\(f_i\\) of norm \\(\\leq1\\) with \\(\\nu_i(f_i)\\) close to \\(\\|\\nu_i\\|\\)). With \\(\\vec1=(1,\\dots,1)\\), and \\(\\nu(1)=\\mu(1)\\) by Proposition 4.5(2): \\[\n\\begin{gathered}\n\\sum_i\\nu_i(1)\\\\\n=\\Lambda(\\vec1)\\\\\n=\\nu(1)\\\\\n=\\mu(1)\\\\\n=\\|\\mu\\|\\\\\n\\geq\\sum_i\\|\\nu_i\\|\n\\end{gathered}\n\\]. Since \\(\\nu_i(1)\\leq\\|\\nu_i\\|\\) for each \\(i\\), all these are equalities, and a real measure with \\(\\nu_i(1)=\\|\\nu_i\\|\\) is positive. For \\(f\\in C(K)\\), \\(\\nu_i(f)=\\Lambda(0,\\dots,f,\\dots,0)\\leq\\mu_i(\\overline f)\\) (note \\(\\overline0=0\\)). For \\(f\\in\\mathcal P(K)\\), applying this to \\(-f\\) and using \\(\\overline{-f}=-f\\) gives \\(\\mu_i(f)\\leq\\nu_i(f)\\). So \\(\\mu_i\\prec\\nu_i\\), hence \\(\\mu_i\\sim\\nu_i\\). Finally \\(\\sum_i\\nu_i=\\nu\\) because \\(\\Lambda\\) extends \\(\\Psi_0\\).\n\n(2)\\(\\Rightarrow\\)(1): Let \\(f\\in\\mathcal P(K)\\) and \\(\\varepsilon>0\\). Each point of \\(K\\) has a closed convex neighbourhood in \\(E\\) on which \\(f\\) varies by less than \\(\\varepsilon\\) (continuity and local convexity). By compactness there are closed convex \\(G_1,\\dots,G_n\\subseteq K\\) that cover \\(K\\) with \\(|f(y)-f(z)|<\\varepsilon\\) for \\(y,z\\in G_i\\). Let \\(A_i=G_i\\setminus\\bigcup_{j<i}G_j\\) and \\(\\mu_i=\\mu|_{A_i}\\). By (2), \\(\\nu=\\sum_i\\nu_i\\) with \\(\\nu_i\\sim\\mu_i\\); so \\(\\nu_i(K)=\\mu_i(K)\\). For \\(\\mu_i\\neq0\\), the normalized \\(\\mu_i\\) and \\(\\nu_i\\) have the same barycentre \\(x_i\\), and \\(x_i\\in G_i\\) by [Proposition 3.3](#oa-fnd-ir-04)(1). Then \\(\\mu_i(f)\\leq\\mu_i(K)(f(x_i)+\\varepsilon)=\\nu_i(K)(f(x_i)+\\varepsilon)\\), and Jensen's inequality gives \\(\\nu_i(K)f(x_i)\\leq\\nu_i(f)\\). Summing, \\(\\mu(f)\\leq\\nu(f)+\\varepsilon\\nu(K)\\). \\(\\square\\)\n\n",
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      "unit": "integral-representations-of-states",
      "name": "4. The Choquet order",
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      "full_conditions_and_proof": "## 4. The Choquet order\n\nThe Choquet order compares two measures through their integrals of continuous convex functions. We first need two facts about convex functions.\n\n**Lemma 4.1** (Differences of convex functions). \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace of \\(C(K)\\), dense for the sup norm, and it is closed under \\(\\vee\\) and \\(\\wedge\\).\n\n**Proof.** \\(\\mathcal P(K)\\) is a convex cone that is closed under \\(\\vee\\). So \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace. If \\(f_i=g_i-h_i\\) with \\(g_i,h_i\\in\\mathcal P(K)\\), then\n\\[\n\\begin{gathered}\nf_1\\vee f_2\\\\\n=\\bigl((g_1+h_2)\\vee(g_2+h_1)\\bigr)\\\\\n-(h_1+h_2)\\\\\n\\in\\mathcal P(K)-\\mathcal P(K),\n\\end{gathered}\n\\]\nand \\(f_1\\wedge f_2=-((-f_1)\\vee(-f_2))\\). The subspace contains the constants and \\(\\operatorname{Aff}_E(K)\\), which separates the points of \\(K\\). By [Stone's lattice lemma](stone-weierstrass-c0.md#oa-fnd-sw-06), a linear subspace of \\(C(K)\\) that is closed under \\(\\vee\\) and \\(\\wedge\\), contains the constants and separates points is dense. \\(\\square\\)\n\n**Lemma 4.2** (Lower semicontinuous convex functions as limits). Let \\(f\\in\\mathcal L(K)\\). The set \\(D_f\\) of functions \\(a_1\\vee\\dots\\vee a_n\\), with \\(n\\geq1\\) and \\(a_i\\in\\operatorname{Aff}_E(K)\\), \\(a_i\\leq f\\), is a subset of \\(\\mathcal P(K)\\) that is directed upward and has pointwise supremum \\(f\\). So \\(f\\) is the pointwise limit of an increasing net in \\(\\mathcal P(K)\\).\n\n**Proof.** \\(D_f\\) is nonempty, because some constant lies below \\(f\\) (see the proof of Lemma 1.1), and it is directed by \\(\\vee\\). Its supremum is \\(f\\) by (1.1). \\(\\square\\)\n\n**Corollary 4.3.** Let \\(\\mu,\\nu\\in M^+(K)\\) with \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). Then \\(\\int g\\,d\\mu\\leq\\int g\\,d\\nu\\) for every \\(g\\in\\mathcal L(K)\\), and \\(\\int h\\,d\\mu\\geq\\int h\\,d\\nu\\) whenever \\(-h\\in\\mathcal L(K)\\).\n\n**Proof.** Let \\(g\\in\\mathcal L(K)\\), and index the increasing net of Lemma 4.2 by \\(D_g\\) itself. Its members are continuous, so the monotone-net property, [Proposition 2.1](#oa-fnd-ir-02)(2), applies to both measures and gives \\(\\int g\\,d\\mu=\\sup_{d\\in D_g}\\mu(d)\\leq\\sup_{d\\in D_g}\\nu(d)=\\int g\\,d\\nu\\). The second claim is the first for \\(g=-h\\). \\(\\square\\)\n\n**Definition 4.4** (The Choquet order). For \\(\\mu,\\nu\\in M(K)\\) we write \\(\\mu\\prec\\nu\\) (\\(\\nu\\) *majorizes* \\(\\mu\\)) if \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). We write \\(\\mu\\sim\\nu\\) if \\(\\mu(a)=\\nu(a)\\) for every \\(a\\in\\operatorname{Aff}(K)\\). A measure is *maximal* if it is maximal for \\(\\prec\\) in \\(M^+(K)\\).\n\n**Proposition 4.5.**\n\n1. \\(\\prec\\) is a partial order on \\(M(K)\\), and \\(\\mu\\prec\\nu\\) implies \\(\\mu\\sim\\nu\\).\n2. If \\(\\mu,\\nu\\in M^+(K)\\) and \\(\\mu\\prec\\nu\\), then \\(\\mu(1)=\\nu(1)\\); if \\(\\mu\\) is a probability measure, so is \\(\\nu\\), and \\(r(\\mu)=r(\\nu)\\).\n3. \\(\\delta_{r(\\mu)}\\prec\\mu\\) for every \\(\\mu\\in M_1^+(K)\\).\n\n**Proof.** (1) Reflexivity and transitivity are clear. If \\(\\mu\\prec\\nu\\prec\\mu\\), the two measures agree on \\(\\mathcal P(K)\\), hence on \\(C(K)\\), because \\(\\mathcal P(K)-\\mathcal P(K)\\) is dense ([Lemma 4.1](#oa-fnd-ir-03)). If \\(\\mu\\prec\\nu\\) and \\(a\\in\\operatorname{Aff}(K)\\), then \\(a\\) and \\(-a\\) are both convex, so \\(\\mu(a)=\\nu(a)\\). (2) \\(\\pm1\\in\\mathcal P(K)\\), so \\(\\mu(1)=\\nu(1)\\); and barycentres are determined by the values on \\(\\operatorname{Aff}(K)\\), which agree by (1). (3) is Jensen's inequality, [Proposition 3.3](#oa-fnd-ir-04)(2). \\(\\square\\)\n\n**Lemma 4.6** (Maximal measures exist). Every chain in \\((M^+(K),\\prec)\\) has an upper bound. Hence every \\(\\nu\\in M^+(K)\\) is majorized by a maximal measure.\n\n**Proof.** Let \\(\\mathcal T\\) be a nonempty chain, indexed by itself as a directed set. All its members have the same mass \\(c\\), by Proposition 4.5(2). For \\(f\\in\\mathcal P(K)\\), the net \\((\\mu(f))_{\\mu\\in\\mathcal T}\\) increases and is bounded by \\(c\\|f\\|\\), so it converges. By linearity the net converges for \\(f\\in\\mathcal P(K)-\\mathcal P(K)\\). For \\(f\\in C(K)\\) and \\(\\varepsilon>0\\) pick such a \\(g\\) with \\(\\|f-g\\|<\\varepsilon\\); then \\(|\\mu(f)-\\mu'(f)|\\leq2c\\varepsilon+|\\mu(g)-\\mu'(g)|\\), so the net is Cauchy. The limit \\(\\lambda(f)\\) is a positive linear functional, so \\(\\lambda\\in M^+(K)\\), and \\(\\mu\\prec\\lambda\\) for every \\(\\mu\\in\\mathcal T\\). Zorn's lemma, applied to \\(\\{\\mu:\\nu\\prec\\mu\\}\\), gives a maximal element there. It is maximal in \\(M^+(K)\\), because anything above it is above \\(\\nu\\). \\(\\square\\)\n\n**Lemma 4.7.** For every \\(\\mu\\in M^+(K)\\) and \\(f\\in C(K)\\) there is \\(\\nu\\in M^+(K)\\) with \\(\\mu\\prec\\nu\\) and \\(\\nu(f)=\\mu(\\overline f)\\).\n\n**Proof.** For \\(g\\in C(K)\\), \\(\\overline g\\) is a bounded usc function ([Proposition 1.5](#oa-fnd-ir-01)), so \\(\\Psi(g)=\\int\\overline g\\,d\\mu\\) is defined. By Proposition 1.5(3), \\(\\Psi\\) is sublinear: \\(\\Psi(g+h)\\leq\\Psi(g)+\\Psi(h)\\) and \\(\\Psi(tg)=t\\Psi(g)\\) for \\(t\\geq0\\). On the line \\(\\mathbb Rf\\) put \\(\\nu_0(tf)=t\\mu(\\overline f)\\). Then \\(\\nu_0\\leq\\Psi\\) there. For \\(t\\geq0\\) this is equality. For \\(t<0\\), \\(0=\\overline{tf+|t|f}\\leq\\overline{tf}+|t|\\overline f\\) gives \\(t\\overline f\\leq\\overline{tf}\\), and we integrate. By the Hahn–Banach extension theorem, \\(\\nu_0\\) extends to a linear \\(\\nu\\) on \\(C(K)\\) with \\(\\nu\\leq\\Psi\\). If \\(g\\leq0\\), then \\(\\overline g\\leq0\\), so \\(\\nu(g)\\leq0\\): \\(\\nu\\) is positive. If \\(g\\in\\mathcal P(K)\\), then \\(\\overline{-g}=-g\\) (Proposition 1.5(4)), so \\(-\\nu(g)=\\nu(-g)\\leq\\mu(\\overline{-g})=-\\mu(g)\\), that is, \\(\\mu(g)\\leq\\nu(g)\\). \\(\\square\\)\n\n**Corollary 4.8** (The upper envelope as a maximum). For \\(x\\in K\\) and \\(f\\in C(K)\\),\n\\[\n\\overline f(x)=\\max\\{\\nu(f):\\ \\nu\\in M_x(K)\\}.\n\\]\nIf \\(f\\in\\mathcal P(K)\\), the maximum is attained at a maximal measure.\n\n**Proof.** For \\(\\nu\\in M_x(K)\\), Jensen's inequality for the usc concave \\(\\overline f\\) gives \\(\\nu(f)\\leq\\nu(\\overline f)\\leq\\overline f(x)\\). Lemma 4.7 with \\(\\mu=\\delta_x\\) gives \\(\\nu\\succ\\delta_x\\) with \\(\\nu(f)=\\overline f(x)\\), and \\(\\nu\\in M_x(K)\\) by Proposition 4.5(2). If \\(f\\) is convex, a maximal \\(\\nu'\\succ\\nu\\) (Lemma 4.6) lies in \\(M_x(K)\\) and has \\(\\nu'(f)\\geq\\nu(f)\\). \\(\\square\\)\n\n**Remark 4.9.** For nonconvex \\(f\\) the maximum need not be attained at a maximal measure. Let \\(K\\) be a square, \\(x\\) its centre, and \\(f\\in C(K)\\) a bump with \\(f(x)=1\\) that vanishes near the four vertices. Then \\(\\overline f(x)\\geq f(x)=1\\), but every maximal measure in \\(M_x(K)\\) lives on the vertices ([Example 7.5](#oa-fnd-ir-27)) and gives \\(f\\) the value \\(0\\).\n\nThe next lemma describes the Choquet order through decompositions of measures. It is used in Sections 14, 15 and 17.\n\n**Lemma 4.10** (The order through decompositions). For \\(\\mu,\\nu\\in M^+(K)\\) the following are equivalent.\n\n1. \\(\\mu\\prec\\nu\\).\n2. Whenever \\(\\mu=\\sum_{i=1}^n\\mu_i\\) with \\(\\mu_i\\in M^+(K)\\), there are \\(\\nu_i\\in M^+(K)\\) with \\(\\nu=\\sum_i\\nu_i\\) and \\(\\nu_i\\sim\\mu_i\\) for each \\(i\\).\n\nIf (1) holds, the \\(\\nu_i\\) in (2) can be chosen with \\(\\mu_i\\prec\\nu_i\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): On \\(C(K)^n\\) with the norm \\(\\|\\vec f\\|=\\max_i\\|f_i\\|\\) put \\(\\Psi(\\vec f)=\\sum_i\\mu_i(\\overline{f_i})\\). It is sublinear by Proposition 1.5(3), as in the proof of Lemma 4.7. On the diagonal \\(\\Delta=\\{(f,\\dots,f)\\}\\) put \\(\\Psi_0(f,\\dots,f)=\\nu(f)\\). Then\n\\[\n\\begin{gathered}\n\\Psi_0(f,\\dots,f)\\\\\n=\\nu(f)\\\\\n\\leq\\nu(\\overline f)\\\\\n\\leq\\mu(\\overline f)\\\\\n=\\Psi(f,\\dots,f),\n\\end{gathered}\n\\]\nwhere the middle step is Corollary 4.3 applied to \\(-\\overline f\\in\\mathcal L(K)\\), and the last step uses \\(\\sum_i\\mu_i=\\mu\\). The Hahn–Banach theorem gives a linear extension \\(\\Lambda\\) of \\(\\Psi_0\\) to \\(C(K)^n\\) with \\(\\Lambda\\leq\\Psi\\). Since \\(\\|\\overline{f_i}\\|_\\infty\\leq\\|f_i\\|\\), we get \\(\\Lambda(\\vec f)\\leq\\sum_i\\|\\mu_i\\|\\|f_i\\|\\leq\\|\\mu\\|\\|\\vec f\\|\\); with \\(-\\vec f\\), \\(|\\Lambda(\\vec f)|\\leq\\|\\mu\\|\\|\\vec f\\|\\). So \\(\\Lambda\\) is bounded, each map \\(f\\mapsto\\Lambda(0,\\dots,f,\\dots,0)\\) is a measure \\(\\nu_i\\in M(K)\\), and \\(\\Lambda(\\vec f)=\\sum_i\\nu_i(f_i)\\). Moreover \\(\\sum_i\\|\\nu_i\\|=\\|\\Lambda\\|\\leq\\|\\mu\\|\\) (choose \\(f_i\\) of norm \\(\\leq1\\) with \\(\\nu_i(f_i)\\) close to \\(\\|\\nu_i\\|\\)). With \\(\\vec1=(1,\\dots,1)\\), and \\(\\nu(1)=\\mu(1)\\) by Proposition 4.5(2): \\[\n\\begin{gathered}\n\\sum_i\\nu_i(1)\\\\\n=\\Lambda(\\vec1)\\\\\n=\\nu(1)\\\\\n=\\mu(1)\\\\\n=\\|\\mu\\|\\\\\n\\geq\\sum_i\\|\\nu_i\\|\n\\end{gathered}\n\\]. Since \\(\\nu_i(1)\\leq\\|\\nu_i\\|\\) for each \\(i\\), all these are equalities, and a real measure with \\(\\nu_i(1)=\\|\\nu_i\\|\\) is positive. For \\(f\\in C(K)\\), \\(\\nu_i(f)=\\Lambda(0,\\dots,f,\\dots,0)\\leq\\mu_i(\\overline f)\\) (note \\(\\overline0=0\\)). For \\(f\\in\\mathcal P(K)\\), applying this to \\(-f\\) and using \\(\\overline{-f}=-f\\) gives \\(\\mu_i(f)\\leq\\nu_i(f)\\). So \\(\\mu_i\\prec\\nu_i\\), hence \\(\\mu_i\\sim\\nu_i\\). Finally \\(\\sum_i\\nu_i=\\nu\\) because \\(\\Lambda\\) extends \\(\\Psi_0\\).\n\n(2)\\(\\Rightarrow\\)(1): Let \\(f\\in\\mathcal P(K)\\) and \\(\\varepsilon>0\\). Each point of \\(K\\) has a closed convex neighbourhood in \\(E\\) on which \\(f\\) varies by less than \\(\\varepsilon\\) (continuity and local convexity). By compactness there are closed convex \\(G_1,\\dots,G_n\\subseteq K\\) that cover \\(K\\) with \\(|f(y)-f(z)|<\\varepsilon\\) for \\(y,z\\in G_i\\). Let \\(A_i=G_i\\setminus\\bigcup_{j<i}G_j\\) and \\(\\mu_i=\\mu|_{A_i}\\). By (2), \\(\\nu=\\sum_i\\nu_i\\) with \\(\\nu_i\\sim\\mu_i\\); so \\(\\nu_i(K)=\\mu_i(K)\\). For \\(\\mu_i\\neq0\\), the normalized \\(\\mu_i\\) and \\(\\nu_i\\) have the same barycentre \\(x_i\\), and \\(x_i\\in G_i\\) by [Proposition 3.3](#oa-fnd-ir-04)(1). Then \\(\\mu_i(f)\\leq\\mu_i(K)(f(x_i)+\\varepsilon)=\\nu_i(K)(f(x_i)+\\varepsilon)\\), and Jensen's inequality gives \\(\\nu_i(K)f(x_i)\\leq\\nu_i(f)\\). Summing, \\(\\mu(f)\\leq\\nu(f)+\\varepsilon\\nu(K)\\). \\(\\square\\)\n\n",
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    {
      "id": "OA-FND-IR-06",
      "unit": "integral-representations-of-states",
      "name": "5. Boundary measures",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 5. Boundary measures\n\nMaximal measures are the measures that have been \"pushed out\" as far as the Choquet order allows. This section identifies them as the measures carried by the sets on which the upper envelopes of continuous functions agree with the functions.\n\n**Definition 5.1** (Boundary sets). For \\(f\\in C(K)\\) the *boundary set* is \\(B_f=\\{x\\in K:\\ \\overline f(x)=f(x)\\}\\), with the upper envelope \\(\\overline f\\) of (1.2).\n\nSince \\(\\overline f-f\\geq0\\) is usc, \\(B_f=\\bigcap_n\\{\\overline f-f<1/n\\}\\) is a \\(G_\\delta\\) set.\n\n**Lemma 5.2.** \\(\\partial_eK=\\bigcap_{f\\in C(K)}B_f\\).\n\n**Proof.** Let \\(x\\in\\partial_eK\\) and \\(f\\in C(K)\\). Corollary 4.8 gives \\(\\nu\\in M_x(K)\\) with \\(\\nu(f)=\\overline f(x)\\). By Bauer's criterion, [Proposition 3.3](#oa-fnd-ir-04)(3), \\(\\nu=\\delta_x\\); so \\(\\overline f(x)=f(x)\\).\n\nConversely, let \\(x\\) lie in every \\(B_f\\), and let \\(\\nu\\in M_x(K)\\). For \\(f\\in C(K)\\) we have \\(x\\in B_f\\cap B_{-f}\\), and \\(\\overline{-f}=-\\underline f\\), so \\(\\underline f(x)=f(x)=\\overline f(x)\\). Jensen's inequality for the lsc convex \\(\\underline f\\) and the usc concave \\(\\overline f\\) gives\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\underline f(x)\\\\\n\\leq\\nu(\\underline f)\\\\\n\\leq\\nu(f)\\\\\n\\leq\\nu(\\overline f)\\\\\n\\leq\\overline f(x)\\\\\n=f(x).\n\\end{gathered}\n\\]\nSo \\(\\nu(f)=f(x)\\) for all \\(f\\), that is, \\(\\nu=\\delta_x\\). By Bauer's criterion, \\(x\\) is extreme. \\(\\square\\)\n\n**Definition 5.3** (Boundary measures). \\(\\mu\\in M(K)\\) is a *boundary measure* if \\(|\\mu|(K\\setminus B_f)=0\\) for every \\(f\\in C(K)\\).\n\n**Theorem 5.4.**\n\n1. A measure \\(\\mu\\in M^+(K)\\) is maximal if and only if it is a boundary measure.\n2. Every point of \\(K\\) is the barycentre of a boundary probability measure.\n\n**Proof.** (1) Let \\(\\mu\\) be maximal and \\(f\\in C(K)\\). Lemma 4.7 gives \\(\\nu\\succ\\mu\\) with \\(\\nu(f)=\\mu(\\overline f)\\); maximality gives \\(\\nu=\\mu\\). So \\(\\int(\\overline f-f)\\,d\\mu=0\\) with \\(\\overline f-f\\geq0\\), and \\(\\mu(K\\setminus B_f)=0\\).\n\nConversely let \\(\\mu\\) be a boundary measure and \\(\\nu\\in M^+(K)\\) with \\(\\mu\\prec\\nu\\). For \\(f\\in\\mathcal P(K)\\):\n\\[\n\\nu(f)\\leq\\nu(\\overline f)\\leq\\mu(\\overline f)=\\mu(f)\\leq\\nu(f).\n\\]\nThe first step is \\(f\\leq\\overline f\\). The second is Corollary 4.3, applied to \\(-\\overline f\\in\\mathcal L(K)\\). The third holds because \\(\\overline f=f\\) \\(\\mu\\)-almost everywhere, and the fourth is \\(\\mu\\prec\\nu\\). So \\(\\mu\\) and \\(\\nu\\) agree on \\(\\mathcal P(K)\\), and \\(\\mu=\\nu\\) because \\(\\mathcal P(K)-\\mathcal P(K)\\) is dense (Lemma 4.1).\n\n(2) By Lemma 4.6, \\(\\delta_x\\) is majorized by a maximal \\(\\mu\\). By (1) it is a boundary measure, and by Proposition 4.5(2) it represents \\(x\\). \\(\\square\\)\n\n**Proposition 5.5** (The boundary measures).\n\n1. The positive boundary measures form a convex cone. If \\(0\\leq\\nu\\leq\\mu\\) and \\(\\mu\\) is a boundary measure, so is \\(\\nu\\). A measure \\(\\mu\\) is a boundary measure exactly when \\(\\mu^+\\) and \\(\\mu^-\\) are, and the boundary measures form a linear subspace of \\(M(K)\\) that is closed under \\(\\vee\\) and \\(\\wedge\\).\n2. If \\(|\\mu|\\) is concentrated on a Borel set \\(B\\subseteq\\partial_eK\\), then \\(\\mu\\) is a boundary measure.\n\n**Proof.** (1) \\(|\\mu+\\nu|\\leq|\\mu|+|\\nu|\\), \\(|t\\mu|=|t||\\mu|\\), and \\(\\mu^\\pm\\leq|\\mu|=\\mu^++\\mu^-\\). The lattice operations of \\(M(K)\\) satisfy \\(\\mu\\vee\\nu=\\tfrac12(\\mu+\\nu+|\\mu-\\nu|)\\). (2) \\(B\\subseteq\\partial_eK\\subseteq B_f\\) by Lemma 5.2. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-IR-07",
      "unit": "integral-representations-of-states",
      "name": "6. The metrizable case and Baire sets",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
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      "full_conditions_and_proof": "## 6. The metrizable case and Baire sets\n\nA boundary measure is carried by every set \\(B_f\\). If one of these sets equals \\(\\partial_eK\\), boundary measures are carried by the extreme points; this happens when \\(K\\) is metrizable. For general \\(K\\) they still vanish on every Baire set that misses \\(\\partial_eK\\).\n\n**Definition 6.1** (Strictly convex functions). \\(f:K\\to\\mathbb R\\) is *strictly convex* if \\(f(tx+(1-t)y)<tf(x)+(1-t)f(y)\\) whenever \\(x\\neq y\\) and \\(0<t<1\\).\n\n**Lemma 6.2.** If \\(f\\in\\mathcal P(K)\\) is strictly convex, then \\(\\partial_eK=B_f\\). Hence if a strictly convex continuous function exists, \\(\\partial_eK\\) is a \\(G_\\delta\\) set.\n\n**Proof.** \\(\\partial_eK\\subseteq B_f\\) by Lemma 5.2. If \\(x=ty+(1-t)z\\) with \\(y\\neq z\\) and \\(0<t<1\\), then, since \\(\\overline f\\) is concave and \\(\\overline f\\geq f\\),\n\\[\n\\begin{gathered}\n\\overline f(x)\\\\\n\\geq t\\overline f(y)+(1-t)\\overline f(z)\\\\\n\\geq tf(y)+(1-t)f(z)>f(x),\n\\end{gathered}\n\\]\nso \\(x\\notin B_f\\). \\(\\square\\)\n\n**Lemma 6.3.** If \\(K\\) is metrizable, there is a strictly convex \\(f\\in\\mathcal P(K)\\).\n\n**Proof.** A compact metric \\(K\\) has a dense sequence: choose finite \\(1/n\\)-nets and take their countable union, as in [Lemma 5.0 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-10). First, \\(C(K)\\) is separable. Let \\(d\\) be a metric and \\((x_n)\\) a dense sequence. The functions \\(d(\\cdot,x_n)\\) separate points, so the algebra they generate together with \\(1\\) is dense, by [the real Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-07); its combinations with rational coefficients form a countable dense set. A separable metric space has a countable base of balls about a dense sequence with positive rational radii. Intersect these balls with a subset and choose a point in every nonempty intersection; the chosen countable set is dense in that subset. Thus \\(\\mathcal P(K)\\setminus\\{0\\}\\) contains a sequence \\((f_n)\\) that is dense in \\(\\mathcal P(K)\\) (the constant \\(1\\) shows that \\(\\mathcal P(K)\\neq\\{0\\}\\)). Put \\(f=\\sum_n2^{-n}\\|f_n\\|^{-1}f_n\\), a uniformly convergent series of convex functions, so \\(f\\in\\mathcal P(K)\\).\n\nSuppose \\(f(x)=tf(y)+(1-t)f(z)\\) with \\(x=ty+(1-t)z\\), \\(y\\neq z\\), \\(0<t<1\\). Each \\(f_n\\) satisfies \"\\(\\leq\\)\", and the weighted sum is an equality, so \\(f_n(x)=tf_n(y)+(1-t)f_n(z)\\) for every \\(n\\). By density this holds for every \\(g\\in\\mathcal P(K)\\). Take \\(a\\in\\operatorname{Aff}_E(K)\\) with \\(a(y)\\neq a(z)\\). Then \\(a\\) and \\(a^2\\) are in \\(\\mathcal P(K)\\), and\n\\[\n\\begin{gathered}\na(x)^2\\\\\n=ta(y)^2+(1-t)a(z)^2,\\\\\na(x)\\\\\n=ta(y)+(1-t)a(z),\n\\end{gathered}\n\\]\nwhich contradicts the strict convexity of \\(s\\mapsto s^2\\). \\(\\square\\)\n\n**Theorem 6.4** (Choquet's theorem). Let \\(K\\) be metrizable. Then \\(\\partial_eK\\) is a \\(G_\\delta\\) set, and \\(|\\mu|(K\\setminus\\partial_eK)=0\\) for each boundary measure \\(\\mu\\). Hence each point of \\(K\\) has a representing probability measure that is carried by \\(\\partial_eK\\).\n\n\n**Proof.** Take \\(f\\) as in Lemma 6.3. By Lemma 6.2, \\(B_f=\\partial_eK\\), and a boundary measure \\(\\mu\\) has \\(|\\mu|(K\\setminus B_f)=0\\). The second claim follows from Theorem 5.4(2). \\(\\square\\)\n\n**Lemma 6.5.** Let \\(K\\) be any compact convex set, and \\((f_n)\\) a sequence in \\(\\mathcal L(K)\\) with \\(\\sup_n\\sup_Kf_n<\\infty\\). Put \\(f=\\limsup_nf_n\\). Then \\(\\sup_{\\partial_eK}f=\\sup_Kf\\).\n\nHere \\(K\\) need not be metrizable. Metrizability enters the proof only through an auxiliary set \\(K'\\).\n\n**Proof.** Let \\(\\alpha=\\sup_{\\partial_eK}f\\) and \\(x\\in K\\). The \\(f_n\\) are real-valued, since they are bounded above. By Lemma 1.1 there are \\(a_n\\in\\operatorname{Aff}_E(K)\\) with \\(a_n\\leq f_n\\) and \\(a_n(x)>f_n(x)-1/n\\). Each \\(a_n\\) is the restriction of a continuous affine function on \\(E\\), again written \\(a_n\\). Let \\(\\Phi(y)=(a_n(y))_{n\\geq1}\\in\\mathbb R^{\\mathbb N}\\). With the product topology, \\(\\mathbb R^{\\mathbb N}\\) is locally convex and has a compatible metric. The map \\(\\Phi\\) is affine and continuous, so \\(K'=\\Phi(K)\\) is compact, convex and metrizable. Let \\(p_n\\) be the \\(n\\)-th coordinate, so \\(a_n=p_n\\circ\\Phi\\).\n\nLet \\(y'\\in\\partial_eK'\\). The set \\(K\\cap\\Phi^{-1}(y')\\) is a nonempty closed face of \\(K\\) (see the Conventions), so by the Krein–Milman theorem it has an extreme point \\(y\\), and \\(y\\) is extreme in \\(K\\), because an extreme point of a face is an extreme point of \\(K\\). Hence\n\\[\n\\begin{gathered}\n\\limsup_np_n(y')\\\\\n=\\limsup_na_n(y)\\\\\n\\leq\\limsup_nf_n(y)\\\\\n=f(y)\\\\\n\\leq\\alpha .\n\\end{gathered}\n\\]\nBy Theorem 6.4, applied to the metrizable set \\(K'\\), there is \\(\\mu\\in M_1^+(K')\\) with \\(r(\\mu)=\\Phi(x)\\) and \\(\\mu(K'\\setminus\\partial_eK')=0\\). The \\(p_n\\) are bounded above on \\(K'\\), uniformly in \\(n\\), because the \\(a_n\\) are bounded above by the \\(f_n\\). By Fatou's lemma applied to \\(C-p_n\\geq0\\),\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\limsup_na_n(x)\\\\\n=\\limsup_n\\int_{K'}p_n\\,d\\mu\\\\\n\\leq\\int_{\\partial_eK'}\\limsup_np_n\\,d\\mu\\\\\n\\leq\\alpha .\n\\end{gathered}\n\\]\nThe first equality uses \\(f_n(x)-1/n<a_n(x)\\leq f_n(x)\\), and the second uses \\(p_n(\\Phi(x))=\\int p_n\\,d\\mu\\), since \\(p_n\\) is linear. \\(\\square\\)\n\n**Theorem 6.6.** If \\(\\mu\\) is a boundary measure on \\(K\\), then \\(|\\mu|(B)=0\\) for every Baire set \\(B\\subseteq K\\) with \\(B\\cap\\partial_eK=\\varnothing\\).\n\n**Proof.** \\(|\\mu|\\) is a positive boundary measure, since Definition 5.3 depends only on \\(|\\mu|\\). First let \\(C\\) be a zero set disjoint from \\(\\partial_eK\\), say \\(C=Z(h)\\) with \\(h\\geq0\\). The functions \\(f_n=\\max(0,1-nh)\\in C(K)\\) satisfy \\(0\\leq f_n\\leq1\\), \\(f_n=1\\) on \\(C\\), and \\(f_n\\to0\\) pointwise off \\(C\\). The lower envelopes \\(\\underline{f_n}\\) lie in \\(\\mathcal L(K)\\) and are bounded by \\(1\\). Since \\(\\overline{-f_n}=-\\underline{f_n}\\), the boundary set \\(B_{-f_n}\\) is \\(\\{\\underline{f_n}=f_n\\}\\); it contains \\(\\partial_eK\\) (Lemma 5.2) and carries \\(|\\mu|\\). So on \\(\\partial_eK\\) we have \\(\\limsup_n\\underline{f_n}=\\limsup_nf_n=0\\), and Lemma 6.5 gives \\(\\limsup_n\\underline{f_n}\\leq0\\) on \\(K\\). By Fatou's lemma,\n\\[\n\\begin{gathered}\n|\\mu|(C)\\\\\n\\leq\\limsup_n|\\mu|(f_n)\\\\\n=\\limsup_n|\\mu|(\\underline{f_n})\\\\\n\\leq|\\mu|\\bigl(\\limsup_n\\underline{f_n}\\bigr)\\\\\n\\leq0 .\n\\end{gathered}\n\\]\nFor a general Baire set \\(B\\) disjoint from \\(\\partial_eK\\), every zero set \\(Z\\subseteq B\\) is null, and \\(|\\mu|(B)=0\\) by the regularity in [Proposition 2.1](#oa-fnd-ir-02)(4)(c). \\(\\square\\)\n\n**Remark 6.7** (The Bishop–de Leeuw theorem). By Theorems 5.4 and 6.6, every maximal probability measure vanishes on every Baire set that misses \\(\\partial_eK\\); one says that it is *pseudoconcentrated* on \\(\\partial_eK\\). So every point of \\(K\\) is the barycentre of a probability measure that is pseudoconcentrated on \\(\\partial_eK\\). For metrizable \\(K\\) this is the same as being carried by \\(\\partial_eK\\) (Theorem 6.4), because then \\(\\partial_eK\\) is a \\(G_\\delta\\) set and every Borel set is a Baire set. For nonmetrizable \\(K\\) the set \\(\\partial_eK\\) need not be a Borel set, even when \\(K\\) is a simplex, so the Baire form is the natural statement.\n\nFor a non-Borel extreme boundary, the Baire pseudo-support statement above is the precise conclusion.\n\n",
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      "id": "OA-FND-IR-08",
      "unit": "integral-representations-of-states",
      "name": "6. The metrizable case and Baire sets",
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      "full_conditions_and_proof": "## 6. The metrizable case and Baire sets\n\nA boundary measure is carried by every set \\(B_f\\). If one of these sets equals \\(\\partial_eK\\), boundary measures are carried by the extreme points; this happens when \\(K\\) is metrizable. For general \\(K\\) they still vanish on every Baire set that misses \\(\\partial_eK\\).\n\n**Definition 6.1** (Strictly convex functions). \\(f:K\\to\\mathbb R\\) is *strictly convex* if \\(f(tx+(1-t)y)<tf(x)+(1-t)f(y)\\) whenever \\(x\\neq y\\) and \\(0<t<1\\).\n\n**Lemma 6.2.** If \\(f\\in\\mathcal P(K)\\) is strictly convex, then \\(\\partial_eK=B_f\\). Hence if a strictly convex continuous function exists, \\(\\partial_eK\\) is a \\(G_\\delta\\) set.\n\n**Proof.** \\(\\partial_eK\\subseteq B_f\\) by Lemma 5.2. If \\(x=ty+(1-t)z\\) with \\(y\\neq z\\) and \\(0<t<1\\), then, since \\(\\overline f\\) is concave and \\(\\overline f\\geq f\\),\n\\[\n\\begin{gathered}\n\\overline f(x)\\\\\n\\geq t\\overline f(y)+(1-t)\\overline f(z)\\\\\n\\geq tf(y)+(1-t)f(z)>f(x),\n\\end{gathered}\n\\]\nso \\(x\\notin B_f\\). \\(\\square\\)\n\n**Lemma 6.3.** If \\(K\\) is metrizable, there is a strictly convex \\(f\\in\\mathcal P(K)\\).\n\n**Proof.** A compact metric \\(K\\) has a dense sequence: choose finite \\(1/n\\)-nets and take their countable union, as in [Lemma 5.0 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-10). First, \\(C(K)\\) is separable. Let \\(d\\) be a metric and \\((x_n)\\) a dense sequence. The functions \\(d(\\cdot,x_n)\\) separate points, so the algebra they generate together with \\(1\\) is dense, by [the real Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-07); its combinations with rational coefficients form a countable dense set. A separable metric space has a countable base of balls about a dense sequence with positive rational radii. Intersect these balls with a subset and choose a point in every nonempty intersection; the chosen countable set is dense in that subset. Thus \\(\\mathcal P(K)\\setminus\\{0\\}\\) contains a sequence \\((f_n)\\) that is dense in \\(\\mathcal P(K)\\) (the constant \\(1\\) shows that \\(\\mathcal P(K)\\neq\\{0\\}\\)). Put \\(f=\\sum_n2^{-n}\\|f_n\\|^{-1}f_n\\), a uniformly convergent series of convex functions, so \\(f\\in\\mathcal P(K)\\).\n\nSuppose \\(f(x)=tf(y)+(1-t)f(z)\\) with \\(x=ty+(1-t)z\\), \\(y\\neq z\\), \\(0<t<1\\). Each \\(f_n\\) satisfies \"\\(\\leq\\)\", and the weighted sum is an equality, so \\(f_n(x)=tf_n(y)+(1-t)f_n(z)\\) for every \\(n\\). By density this holds for every \\(g\\in\\mathcal P(K)\\). Take \\(a\\in\\operatorname{Aff}_E(K)\\) with \\(a(y)\\neq a(z)\\). Then \\(a\\) and \\(a^2\\) are in \\(\\mathcal P(K)\\), and\n\\[\n\\begin{gathered}\na(x)^2\\\\\n=ta(y)^2+(1-t)a(z)^2,\\\\\na(x)\\\\\n=ta(y)+(1-t)a(z),\n\\end{gathered}\n\\]\nwhich contradicts the strict convexity of \\(s\\mapsto s^2\\). \\(\\square\\)\n\n**Theorem 6.4** (Choquet's theorem). Let \\(K\\) be metrizable. Then \\(\\partial_eK\\) is a \\(G_\\delta\\) set, and \\(|\\mu|(K\\setminus\\partial_eK)=0\\) for each boundary measure \\(\\mu\\). Hence each point of \\(K\\) has a representing probability measure that is carried by \\(\\partial_eK\\).\n\n\n**Proof.** Take \\(f\\) as in Lemma 6.3. By Lemma 6.2, \\(B_f=\\partial_eK\\), and a boundary measure \\(\\mu\\) has \\(|\\mu|(K\\setminus B_f)=0\\). The second claim follows from Theorem 5.4(2). \\(\\square\\)\n\n**Lemma 6.5.** Let \\(K\\) be any compact convex set, and \\((f_n)\\) a sequence in \\(\\mathcal L(K)\\) with \\(\\sup_n\\sup_Kf_n<\\infty\\). Put \\(f=\\limsup_nf_n\\). Then \\(\\sup_{\\partial_eK}f=\\sup_Kf\\).\n\nHere \\(K\\) need not be metrizable. Metrizability enters the proof only through an auxiliary set \\(K'\\).\n\n**Proof.** Let \\(\\alpha=\\sup_{\\partial_eK}f\\) and \\(x\\in K\\). The \\(f_n\\) are real-valued, since they are bounded above. By Lemma 1.1 there are \\(a_n\\in\\operatorname{Aff}_E(K)\\) with \\(a_n\\leq f_n\\) and \\(a_n(x)>f_n(x)-1/n\\). Each \\(a_n\\) is the restriction of a continuous affine function on \\(E\\), again written \\(a_n\\). Let \\(\\Phi(y)=(a_n(y))_{n\\geq1}\\in\\mathbb R^{\\mathbb N}\\). With the product topology, \\(\\mathbb R^{\\mathbb N}\\) is locally convex and has a compatible metric. The map \\(\\Phi\\) is affine and continuous, so \\(K'=\\Phi(K)\\) is compact, convex and metrizable. Let \\(p_n\\) be the \\(n\\)-th coordinate, so \\(a_n=p_n\\circ\\Phi\\).\n\nLet \\(y'\\in\\partial_eK'\\). The set \\(K\\cap\\Phi^{-1}(y')\\) is a nonempty closed face of \\(K\\) (see the Conventions), so by the Krein–Milman theorem it has an extreme point \\(y\\), and \\(y\\) is extreme in \\(K\\), because an extreme point of a face is an extreme point of \\(K\\). Hence\n\\[\n\\begin{gathered}\n\\limsup_np_n(y')\\\\\n=\\limsup_na_n(y)\\\\\n\\leq\\limsup_nf_n(y)\\\\\n=f(y)\\\\\n\\leq\\alpha .\n\\end{gathered}\n\\]\nBy Theorem 6.4, applied to the metrizable set \\(K'\\), there is \\(\\mu\\in M_1^+(K')\\) with \\(r(\\mu)=\\Phi(x)\\) and \\(\\mu(K'\\setminus\\partial_eK')=0\\). The \\(p_n\\) are bounded above on \\(K'\\), uniformly in \\(n\\), because the \\(a_n\\) are bounded above by the \\(f_n\\). By Fatou's lemma applied to \\(C-p_n\\geq0\\),\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\limsup_na_n(x)\\\\\n=\\limsup_n\\int_{K'}p_n\\,d\\mu\\\\\n\\leq\\int_{\\partial_eK'}\\limsup_np_n\\,d\\mu\\\\\n\\leq\\alpha .\n\\end{gathered}\n\\]\nThe first equality uses \\(f_n(x)-1/n<a_n(x)\\leq f_n(x)\\), and the second uses \\(p_n(\\Phi(x))=\\int p_n\\,d\\mu\\), since \\(p_n\\) is linear. \\(\\square\\)\n\n**Theorem 6.6.** If \\(\\mu\\) is a boundary measure on \\(K\\), then \\(|\\mu|(B)=0\\) for every Baire set \\(B\\subseteq K\\) with \\(B\\cap\\partial_eK=\\varnothing\\).\n\n**Proof.** \\(|\\mu|\\) is a positive boundary measure, since Definition 5.3 depends only on \\(|\\mu|\\). First let \\(C\\) be a zero set disjoint from \\(\\partial_eK\\), say \\(C=Z(h)\\) with \\(h\\geq0\\). The functions \\(f_n=\\max(0,1-nh)\\in C(K)\\) satisfy \\(0\\leq f_n\\leq1\\), \\(f_n=1\\) on \\(C\\), and \\(f_n\\to0\\) pointwise off \\(C\\). The lower envelopes \\(\\underline{f_n}\\) lie in \\(\\mathcal L(K)\\) and are bounded by \\(1\\). Since \\(\\overline{-f_n}=-\\underline{f_n}\\), the boundary set \\(B_{-f_n}\\) is \\(\\{\\underline{f_n}=f_n\\}\\); it contains \\(\\partial_eK\\) (Lemma 5.2) and carries \\(|\\mu|\\). So on \\(\\partial_eK\\) we have \\(\\limsup_n\\underline{f_n}=\\limsup_nf_n=0\\), and Lemma 6.5 gives \\(\\limsup_n\\underline{f_n}\\leq0\\) on \\(K\\). By Fatou's lemma,\n\\[\n\\begin{gathered}\n|\\mu|(C)\\\\\n\\leq\\limsup_n|\\mu|(f_n)\\\\\n=\\limsup_n|\\mu|(\\underline{f_n})\\\\\n\\leq|\\mu|\\bigl(\\limsup_n\\underline{f_n}\\bigr)\\\\\n\\leq0 .\n\\end{gathered}\n\\]\nFor a general Baire set \\(B\\) disjoint from \\(\\partial_eK\\), every zero set \\(Z\\subseteq B\\) is null, and \\(|\\mu|(B)=0\\) by the regularity in [Proposition 2.1](#oa-fnd-ir-02)(4)(c). \\(\\square\\)\n\n**Remark 6.7** (The Bishop–de Leeuw theorem). By Theorems 5.4 and 6.6, every maximal probability measure vanishes on every Baire set that misses \\(\\partial_eK\\); one says that it is *pseudoconcentrated* on \\(\\partial_eK\\). So every point of \\(K\\) is the barycentre of a probability measure that is pseudoconcentrated on \\(\\partial_eK\\). For metrizable \\(K\\) this is the same as being carried by \\(\\partial_eK\\) (Theorem 6.4), because then \\(\\partial_eK\\) is a \\(G_\\delta\\) set and every Borel set is a Baire set. For nonmetrizable \\(K\\) the set \\(\\partial_eK\\) need not be a Borel set, even when \\(K\\) is a simplex, so the Baire form is the natural statement.\n\nFor a non-Borel extreme boundary, the Baire pseudo-support statement above is the precise conclusion.\n\n",
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      "id": "OA-FND-IR-23",
      "unit": "integral-representations-of-states",
      "name": "7. Simplices",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 7. Simplices\n\nIn a triangle every point is a unique convex combination of the vertices; in a square it is not. Simplices are the compact convex sets that behave like the triangle, and the Choquet–Meyer theorem characterizes them by the uniqueness of maximal measures.\n\n**Definition 7.1** (Simplex). \\(K\\) is a *simplex* if \\(\\operatorname{Aff}(K)^*\\) is a vector lattice for the dual order: \\(\\psi\\geq0\\) when \\(\\psi(a)\\geq0\\) for every \\(a\\geq0\\) in \\(\\operatorname{Aff}(K)\\).\n\n**Lemma 7.2** (Positive functionals on \\(\\operatorname{Aff}(K)\\)).\n\n1. \\(\\psi\\in\\operatorname{Aff}(K)^*\\) is positive if and only if \\(\\psi=t\\delta_x\\) with \\(t\\geq0\\) and \\(x\\in K\\).\n2. The restriction \\(R(\\mu)=\\mu|_{\\operatorname{Aff}(K)}\\) is a positive linear map of \\(M(K)\\) onto \\(\\operatorname{Aff}(K)^*\\), and \\(R(\\mu)=\\mu(1)\\delta_{r(\\mu)}\\) for \\(0\\neq\\mu\\in M^+(K)\\).\n\n**Proof.** (1) Let \\(\\psi\\geq0\\). From \\(-\\|a\\|\\leq a\\leq\\|a\\|\\), \\(|\\psi(a)|\\leq\\psi(1)\\|a\\|\\). If \\(\\psi(1)=0\\), then \\(\\psi=0\\). Otherwise extend \\(\\psi\\) by the Hahn–Banach theorem to \\(\\tilde\\psi\\in C(K)^*\\) with \\(\\|\\tilde\\psi\\|=\\psi(1)=\\tilde\\psi(1)\\). Such a functional is positive: for \\(0\\leq g\\leq1\\), \\(\\|1-2g\\|\\leq1\\) gives \\(\\tilde\\psi(1)-2\\tilde\\psi(g)\\leq\\tilde\\psi(1)\\). So \\(\\tilde\\psi/\\psi(1)\\) is a probability measure with some barycentre \\(x\\), and \\(\\psi(a)=\\psi(1)a(x)\\) by (3.1). (2) \\(R\\) is onto because every functional on \\(\\operatorname{Aff}(K)\\) extends to \\(C(K)\\) (Hahn–Banach); the formula is (3.1). \\(\\square\\)\n\n**Lemma 7.3** (Riesz decomposition). In a vector lattice, let \\(p_1,\\dots,p_m,q_1,q_2\\geq0\\) with \\(\\sum_jp_j=q_1+q_2\\). Then there are \\(p_{jk}\\geq0\\) with \\(p_j=p_{j1}+p_{j2}\\) and \\(\\sum_jp_{jk}=q_k\\).\n\n**Proof.** Induction on \\(m\\); \\(m=1\\) is trivial. Put \\(P'=\\sum_{j\\geq2}p_j\\), \\(p_{11}=p_1\\wedge q_1\\), \\(p_{12}=p_1-p_{11}\\), \\(r_1=q_1-p_{11}\\) and \\(r_2=q_2-p_{12}\\). Then \\(p_{11},p_{12},r_1\\geq0\\) and \\(r_1+r_2=P'\\). Also \\(p_{12}=(p_1-q_1)\\vee0\\), and \\(p_1-q_1=q_2-P'\\leq q_2\\), so \\(p_{12}\\leq q_2\\vee0=q_2\\) and \\(r_2\\geq0\\). Apply the induction hypothesis to \\(P'=r_1+r_2\\). \\(\\square\\)\n\n**Theorem 7.4** (Choquet–Meyer). The following are equivalent.\n\n- (a) \\(K\\) is a simplex.\n- (b) Every \\(x\\in K\\) is the barycentre of exactly one boundary probability measure (equivalently, \\(M_x(K)\\) has exactly one maximal element).\n- (c) \\(\\overline f\\) is affine for every \\(f\\in\\mathcal P(K)\\).\n\n**Proof.** (a)\\(\\Rightarrow\\)(c). Let \\(f\\in\\mathcal P(K)\\) and \\(x=ty+(1-t)z\\) with \\(0<t<1\\). Since \\(\\overline f\\) is concave, we must show \\(\\overline f(x)\\leq t\\overline f(y)+(1-t)\\overline f(z)\\). By Corollary 4.8, \\(\\overline f(x)=\\sup\\{\\nu(f):\\ \\nu\\in M_x(K)\\}\\). Discrete measures suffice: given \\(\\nu\\in M_x(K)\\) and \\(\\varepsilon>0\\), cover \\(K\\) by closed convex \\(G_1,\\dots,G_n\\) on each of which \\(f\\) varies by less than \\(\\varepsilon\\), put \\(A_i=G_i\\setminus\\bigcup_{j<i}G_j\\), and let \\(x_i\\in G_i\\) be the barycentre of \\(\\nu|_{A_i}/\\nu(A_i)\\) (when \\(\\nu(A_i)>0\\)). Then \\(\\nu'=\\sum_i\\nu(A_i)\\delta_{x_i}\\in M_x(K)\\) and \\(|\\nu(f)-\\nu'(f)|\\leq\\varepsilon\\). So let \\(\\nu'=\\sum_jc_j\\delta_{x_j}\\in M_x(K)\\). In \\(\\operatorname{Aff}(K)^*\\) we have \\(\\sum_jc_j\\delta_{x_j}=\\delta_x=t\\delta_y+(1-t)\\delta_z\\). By the Riesz decomposition (Lemma 7.3) and Lemma 7.2(1), \\(c_j\\delta_{x_j}=c_{j1}\\delta_{x_{j1}}+c_{j2}\\delta_{x_{j2}}\\) with \\(\\sum_jc_{j1}\\delta_{x_{j1}}=t\\delta_y\\) and \\(\\sum_jc_{j2}\\delta_{x_{j2}}=(1-t)\\delta_z\\). Evaluating at \\(1\\) and at affine functions gives \\(c_j=c_{j1}+c_{j2}\\) and \\(c_jx_j=c_{j1}x_{j1}+c_{j2}x_{j2}\\), so convexity of \\(f\\) gives \\(c_jf(x_j)\\leq c_{j1}f(x_{j1})+c_{j2}f(x_{j2})\\). The measure \\(t^{-1}\\sum_jc_{j1}\\delta_{x_{j1}}\\) lies in \\(M_y(K)\\), so \\(\\sum_jc_{j1}f(x_{j1})\\leq t\\overline f(y)\\), and likewise for \\(z\\). Hence \\(\\nu'(f)\\leq t\\overline f(y)+(1-t)\\overline f(z)\\).\n\n(c)\\(\\Rightarrow\\)(b). Existence is Theorem 5.4(2). Let \\(\\mu,\\nu\\) be boundary probability measures with barycentre \\(x\\), and \\(f\\in\\mathcal P(K)\\). Then \\(\\mu(f)=\\mu(\\overline f)\\), since \\(\\mu\\) is carried by \\(B_f\\), where \\(\\overline f=f\\). The function \\(-\\overline f\\) is real-valued, lsc and affine, so by Lemma 1.3 it is the limit of an increasing net \\((a_i)\\) in \\(\\operatorname{Aff}_E(K)\\). By the monotone-net property, [Proposition 2.1](#oa-fnd-ir-02)(2), and by (3.1), \\(\\mu(-\\overline f)=\\sup_i\\mu(a_i)=\\sup_ia_i(x)=-\\overline f(x)\\). So \\(\\mu(f)=\\overline f(x)=\\nu(f)\\), and \\(\\mu=\\nu\\) by Lemma 4.1.\n\n(b)\\(\\Rightarrow\\)(a). Let \\(\\mathcal B\\) be the set of boundary measures, a linear subspace of \\(M(K)\\) closed under \\(\\vee\\) and \\(\\wedge\\) (Proposition 5.5(1)). So \\(\\mathcal B\\) is a vector lattice, and it suffices to show that \\(R|_{\\mathcal B}\\) is an order isomorphism onto \\(\\operatorname{Aff}(K)^*\\). *Onto:* given \\(\\psi=R(\\mu)\\), majorize \\(\\mu^\\pm\\) by maximal measures \\(m^\\pm\\) (Lemma 4.6), which are boundary measures by Theorem 5.4; then \\(R(m^\\pm)=R(\\mu^\\pm)\\) by Proposition 4.5(1), and \\(\\psi=R(m^+-m^-)\\). *One-to-one:* if \\(\\mu\\in\\mathcal B\\) and \\(R(\\mu)=0\\), then \\(\\mu^+\\) and \\(\\mu^-\\) are positive boundary measures with the same mass \\(c\\) and, if \\(c>0\\), the same barycentre; by (b), \\(\\mu^+=\\mu^-\\), so \\(\\mu=0\\). *Order:* \\(R\\) is positive. If \\(\\mu\\in\\mathcal B\\) and \\(R(\\mu)\\geq0\\), then \\(R(\\mu)=c\\delta_x\\) (Lemma 7.2(1)), and \\(c\\,m_x\\), with \\(m_x\\) the boundary probability measure at \\(x\\), satisfies \\(R(cm_x)=R(\\mu)\\); so \\(\\mu=cm_x\\geq0\\). \\(\\square\\)\n\n**Example 7.5** (The square and the triangle). Let \\(K\\) be the square with vertices \\(v_1=(0,0)\\), \\(v_2=(1,0)\\), \\(v_3=(1,1)\\), \\(v_4=(0,1)\\), and \\(c\\) its centre. \\(K\\) is metrizable, so maximal measures live on the four vertices (Theorem 6.4), and every measure on the vertices is a boundary measure (Proposition 5.5(2)). A measure \\(\\sum_ip_i\\delta_{v_i}\\) represents \\(c\\) exactly when \\(p_1=p_3\\), \\(p_2=p_4\\) and \\(p_1+p_2=\\frac12\\). So the maximal measures in \\(M_c(K)\\) form the segment \\(t\\cdot\\frac12(\\delta_{v_1}+\\delta_{v_3})+(1-t)\\cdot\\frac12(\\delta_{v_2}+\\delta_{v_4})\\), \\(t\\in[0,1]\\), and the square is not a simplex. For \\(f(x,y)=(x-y)^2\\), which is convex, \\(\\overline f(v_1)=\\overline f(v_3)=0\\), while \\(\\overline f(c)=\\max_t(1-t)=1\\) by Corollary 4.8; so \\(\\overline f\\) is not affine on the diagonal, as condition (c) of Theorem 7.4 predicts. In a triangle, barycentric coordinates are unique, every point has one maximal measure, and the triangle is a simplex.\n\n**Exercise 7.6** (medium; The Choquet order on an interval). Let \\(K=[0,1]\\). Show that for \\(\\mu,\\nu\\in M_1^+(K)\\), \\(\\mu\\prec\\nu\\) if and only if \\(r(\\mu)=r(\\nu)\\) and \\(\\int(t-s)^+d\\mu(t)\\leq\\int(t-s)^+d\\nu(t)\\) for every \\(s\\in[0,1]\\). Deduce that \\((1-x)\\delta_0+x\\delta_1\\) is the only maximal measure in \\(M_x(K)\\).\n\n*Solution.* Necessity: \\(t\\mapsto(t-s)^+\\) is convex, and \\(\\mu\\prec\\nu\\) forces equal barycentres (Proposition 4.5(2)). Sufficiency: a continuous convex \\(f\\) on \\([0,1]\\) is the uniform limit of its piecewise linear interpolants at the points \\(k/N\\), which are convex and have the form \\(\\alpha+\\beta t+\\sum_kc_k(t-s_k)^+\\) with \\(c_k\\geq0\\) (the jumps of the slope). Integrate: the affine part has the same integral against \\(\\mu\\) and \\(\\nu\\), and each \\(c_k(t-s_k)^+\\) has the larger integral against \\(\\nu\\). For the last claim, \\(\\partial_eK=\\{0,1\\}\\), and the only probability measure on \\(\\{0,1\\}\\) with barycentre \\(x\\) is \\((1-x)\\delta_0+x\\delta_1\\); every maximal measure lives on \\(\\{0,1\\}\\) (Theorem 6.4). So \\([0,1]\\) is a simplex, by Theorem 7.4.\n\n",
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      "id": "OA-FND-IR-27",
      "unit": "integral-representations-of-states",
      "name": "7. Simplices",
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      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 7. Simplices\n\nIn a triangle every point is a unique convex combination of the vertices; in a square it is not. Simplices are the compact convex sets that behave like the triangle, and the Choquet–Meyer theorem characterizes them by the uniqueness of maximal measures.\n\n**Definition 7.1** (Simplex). \\(K\\) is a *simplex* if \\(\\operatorname{Aff}(K)^*\\) is a vector lattice for the dual order: \\(\\psi\\geq0\\) when \\(\\psi(a)\\geq0\\) for every \\(a\\geq0\\) in \\(\\operatorname{Aff}(K)\\).\n\n**Lemma 7.2** (Positive functionals on \\(\\operatorname{Aff}(K)\\)).\n\n1. \\(\\psi\\in\\operatorname{Aff}(K)^*\\) is positive if and only if \\(\\psi=t\\delta_x\\) with \\(t\\geq0\\) and \\(x\\in K\\).\n2. The restriction \\(R(\\mu)=\\mu|_{\\operatorname{Aff}(K)}\\) is a positive linear map of \\(M(K)\\) onto \\(\\operatorname{Aff}(K)^*\\), and \\(R(\\mu)=\\mu(1)\\delta_{r(\\mu)}\\) for \\(0\\neq\\mu\\in M^+(K)\\).\n\n**Proof.** (1) Let \\(\\psi\\geq0\\). From \\(-\\|a\\|\\leq a\\leq\\|a\\|\\), \\(|\\psi(a)|\\leq\\psi(1)\\|a\\|\\). If \\(\\psi(1)=0\\), then \\(\\psi=0\\). Otherwise extend \\(\\psi\\) by the Hahn–Banach theorem to \\(\\tilde\\psi\\in C(K)^*\\) with \\(\\|\\tilde\\psi\\|=\\psi(1)=\\tilde\\psi(1)\\). Such a functional is positive: for \\(0\\leq g\\leq1\\), \\(\\|1-2g\\|\\leq1\\) gives \\(\\tilde\\psi(1)-2\\tilde\\psi(g)\\leq\\tilde\\psi(1)\\). So \\(\\tilde\\psi/\\psi(1)\\) is a probability measure with some barycentre \\(x\\), and \\(\\psi(a)=\\psi(1)a(x)\\) by (3.1). (2) \\(R\\) is onto because every functional on \\(\\operatorname{Aff}(K)\\) extends to \\(C(K)\\) (Hahn–Banach); the formula is (3.1). \\(\\square\\)\n\n**Lemma 7.3** (Riesz decomposition). In a vector lattice, let \\(p_1,\\dots,p_m,q_1,q_2\\geq0\\) with \\(\\sum_jp_j=q_1+q_2\\). Then there are \\(p_{jk}\\geq0\\) with \\(p_j=p_{j1}+p_{j2}\\) and \\(\\sum_jp_{jk}=q_k\\).\n\n**Proof.** Induction on \\(m\\); \\(m=1\\) is trivial. Put \\(P'=\\sum_{j\\geq2}p_j\\), \\(p_{11}=p_1\\wedge q_1\\), \\(p_{12}=p_1-p_{11}\\), \\(r_1=q_1-p_{11}\\) and \\(r_2=q_2-p_{12}\\). Then \\(p_{11},p_{12},r_1\\geq0\\) and \\(r_1+r_2=P'\\). Also \\(p_{12}=(p_1-q_1)\\vee0\\), and \\(p_1-q_1=q_2-P'\\leq q_2\\), so \\(p_{12}\\leq q_2\\vee0=q_2\\) and \\(r_2\\geq0\\). Apply the induction hypothesis to \\(P'=r_1+r_2\\). \\(\\square\\)\n\n**Theorem 7.4** (Choquet–Meyer). The following are equivalent.\n\n- (a) \\(K\\) is a simplex.\n- (b) Every \\(x\\in K\\) is the barycentre of exactly one boundary probability measure (equivalently, \\(M_x(K)\\) has exactly one maximal element).\n- (c) \\(\\overline f\\) is affine for every \\(f\\in\\mathcal P(K)\\).\n\n**Proof.** (a)\\(\\Rightarrow\\)(c). Let \\(f\\in\\mathcal P(K)\\) and \\(x=ty+(1-t)z\\) with \\(0<t<1\\). Since \\(\\overline f\\) is concave, we must show \\(\\overline f(x)\\leq t\\overline f(y)+(1-t)\\overline f(z)\\). By Corollary 4.8, \\(\\overline f(x)=\\sup\\{\\nu(f):\\ \\nu\\in M_x(K)\\}\\). Discrete measures suffice: given \\(\\nu\\in M_x(K)\\) and \\(\\varepsilon>0\\), cover \\(K\\) by closed convex \\(G_1,\\dots,G_n\\) on each of which \\(f\\) varies by less than \\(\\varepsilon\\), put \\(A_i=G_i\\setminus\\bigcup_{j<i}G_j\\), and let \\(x_i\\in G_i\\) be the barycentre of \\(\\nu|_{A_i}/\\nu(A_i)\\) (when \\(\\nu(A_i)>0\\)). Then \\(\\nu'=\\sum_i\\nu(A_i)\\delta_{x_i}\\in M_x(K)\\) and \\(|\\nu(f)-\\nu'(f)|\\leq\\varepsilon\\). So let \\(\\nu'=\\sum_jc_j\\delta_{x_j}\\in M_x(K)\\). In \\(\\operatorname{Aff}(K)^*\\) we have \\(\\sum_jc_j\\delta_{x_j}=\\delta_x=t\\delta_y+(1-t)\\delta_z\\). By the Riesz decomposition (Lemma 7.3) and Lemma 7.2(1), \\(c_j\\delta_{x_j}=c_{j1}\\delta_{x_{j1}}+c_{j2}\\delta_{x_{j2}}\\) with \\(\\sum_jc_{j1}\\delta_{x_{j1}}=t\\delta_y\\) and \\(\\sum_jc_{j2}\\delta_{x_{j2}}=(1-t)\\delta_z\\). Evaluating at \\(1\\) and at affine functions gives \\(c_j=c_{j1}+c_{j2}\\) and \\(c_jx_j=c_{j1}x_{j1}+c_{j2}x_{j2}\\), so convexity of \\(f\\) gives \\(c_jf(x_j)\\leq c_{j1}f(x_{j1})+c_{j2}f(x_{j2})\\). The measure \\(t^{-1}\\sum_jc_{j1}\\delta_{x_{j1}}\\) lies in \\(M_y(K)\\), so \\(\\sum_jc_{j1}f(x_{j1})\\leq t\\overline f(y)\\), and likewise for \\(z\\). Hence \\(\\nu'(f)\\leq t\\overline f(y)+(1-t)\\overline f(z)\\).\n\n(c)\\(\\Rightarrow\\)(b). Existence is Theorem 5.4(2). Let \\(\\mu,\\nu\\) be boundary probability measures with barycentre \\(x\\), and \\(f\\in\\mathcal P(K)\\). Then \\(\\mu(f)=\\mu(\\overline f)\\), since \\(\\mu\\) is carried by \\(B_f\\), where \\(\\overline f=f\\). The function \\(-\\overline f\\) is real-valued, lsc and affine, so by Lemma 1.3 it is the limit of an increasing net \\((a_i)\\) in \\(\\operatorname{Aff}_E(K)\\). By the monotone-net property, [Proposition 2.1](#oa-fnd-ir-02)(2), and by (3.1), \\(\\mu(-\\overline f)=\\sup_i\\mu(a_i)=\\sup_ia_i(x)=-\\overline f(x)\\). So \\(\\mu(f)=\\overline f(x)=\\nu(f)\\), and \\(\\mu=\\nu\\) by Lemma 4.1.\n\n(b)\\(\\Rightarrow\\)(a). Let \\(\\mathcal B\\) be the set of boundary measures, a linear subspace of \\(M(K)\\) closed under \\(\\vee\\) and \\(\\wedge\\) (Proposition 5.5(1)). So \\(\\mathcal B\\) is a vector lattice, and it suffices to show that \\(R|_{\\mathcal B}\\) is an order isomorphism onto \\(\\operatorname{Aff}(K)^*\\). *Onto:* given \\(\\psi=R(\\mu)\\), majorize \\(\\mu^\\pm\\) by maximal measures \\(m^\\pm\\) (Lemma 4.6), which are boundary measures by Theorem 5.4; then \\(R(m^\\pm)=R(\\mu^\\pm)\\) by Proposition 4.5(1), and \\(\\psi=R(m^+-m^-)\\). *One-to-one:* if \\(\\mu\\in\\mathcal B\\) and \\(R(\\mu)=0\\), then \\(\\mu^+\\) and \\(\\mu^-\\) are positive boundary measures with the same mass \\(c\\) and, if \\(c>0\\), the same barycentre; by (b), \\(\\mu^+=\\mu^-\\), so \\(\\mu=0\\). *Order:* \\(R\\) is positive. If \\(\\mu\\in\\mathcal B\\) and \\(R(\\mu)\\geq0\\), then \\(R(\\mu)=c\\delta_x\\) (Lemma 7.2(1)), and \\(c\\,m_x\\), with \\(m_x\\) the boundary probability measure at \\(x\\), satisfies \\(R(cm_x)=R(\\mu)\\); so \\(\\mu=cm_x\\geq0\\). \\(\\square\\)\n\n**Example 7.5** (The square and the triangle). Let \\(K\\) be the square with vertices \\(v_1=(0,0)\\), \\(v_2=(1,0)\\), \\(v_3=(1,1)\\), \\(v_4=(0,1)\\), and \\(c\\) its centre. \\(K\\) is metrizable, so maximal measures live on the four vertices (Theorem 6.4), and every measure on the vertices is a boundary measure (Proposition 5.5(2)). A measure \\(\\sum_ip_i\\delta_{v_i}\\) represents \\(c\\) exactly when \\(p_1=p_3\\), \\(p_2=p_4\\) and \\(p_1+p_2=\\frac12\\). So the maximal measures in \\(M_c(K)\\) form the segment \\(t\\cdot\\frac12(\\delta_{v_1}+\\delta_{v_3})+(1-t)\\cdot\\frac12(\\delta_{v_2}+\\delta_{v_4})\\), \\(t\\in[0,1]\\), and the square is not a simplex. For \\(f(x,y)=(x-y)^2\\), which is convex, \\(\\overline f(v_1)=\\overline f(v_3)=0\\), while \\(\\overline f(c)=\\max_t(1-t)=1\\) by Corollary 4.8; so \\(\\overline f\\) is not affine on the diagonal, as condition (c) of Theorem 7.4 predicts. In a triangle, barycentric coordinates are unique, every point has one maximal measure, and the triangle is a simplex.\n\n**Exercise 7.6** (medium; The Choquet order on an interval). Let \\(K=[0,1]\\). Show that for \\(\\mu,\\nu\\in M_1^+(K)\\), \\(\\mu\\prec\\nu\\) if and only if \\(r(\\mu)=r(\\nu)\\) and \\(\\int(t-s)^+d\\mu(t)\\leq\\int(t-s)^+d\\nu(t)\\) for every \\(s\\in[0,1]\\). Deduce that \\((1-x)\\delta_0+x\\delta_1\\) is the only maximal measure in \\(M_x(K)\\).\n\n*Solution.* Necessity: \\(t\\mapsto(t-s)^+\\) is convex, and \\(\\mu\\prec\\nu\\) forces equal barycentres (Proposition 4.5(2)). Sufficiency: a continuous convex \\(f\\) on \\([0,1]\\) is the uniform limit of its piecewise linear interpolants at the points \\(k/N\\), which are convex and have the form \\(\\alpha+\\beta t+\\sum_kc_k(t-s_k)^+\\) with \\(c_k\\geq0\\) (the jumps of the slope). Integrate: the affine part has the same integral against \\(\\mu\\) and \\(\\nu\\), and each \\(c_k(t-s_k)^+\\) has the larger integral against \\(\\nu\\). For the last claim, \\(\\partial_eK=\\{0,1\\}\\), and the only probability measure on \\(\\{0,1\\}\\) with barycentre \\(x\\) is \\((1-x)\\delta_0+x\\delta_1\\); every maximal measure lives on \\(\\{0,1\\}\\) (Theorem 6.4). So \\([0,1]\\) is a simplex, by Theorem 7.4.\n\n",
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      "id": "OA-FND-IR-09",
      "unit": "integral-representations-of-states",
      "name": "8. The state space",
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      "full_conditions_and_proof": "## 8. The state space\n\nFrom now on \\(A\\) is a nonzero unital C\\(^*\\)-algebra; for a nonunital algebra a nondegenerate representation extends uniquely to a unital representation of its unitization by \\(\\widetilde\\pi(a+\\lambda1)=\\pi(a)+\\lambda I\\). Expanding the product and adjoint identities proves that this is a \\(*\\)-homomorphism; it is bounded by Theorem 4.2 of the continuous-functional-calculus lesson. A unital extension is unique because the adjoined unit must act as \\(I\\). We work with the nonzero unital algebra throughout. We apply Sections 1–7 with \\(E=A^*_h\\), the real space of hermitian functionals, \\(\\psi(a^*)=\\overline{\\psi(a)}\\), with the weak topology \\(\\sigma(A^*_h,A_h)\\). The continuous linear functionals for this topology are the maps \\(\\psi\\mapsto\\psi(h)\\) with \\(h\\in A_h\\), by the duality theorem for weak topologies. States are hermitian, because positive functionals are. For \\(a\\in A\\) we write \\(\\hat a(\\omega)=\\omega(a)\\), a continuous complex function on \\(\\mathfrak S\\), and \\(\\mathcal A_{\\mathbb C}=\\{\\hat a:\\ a\\in A\\}\\).\n\n**Proposition 8.1.**\n\n1. \\(\\mathfrak S\\) is compact and convex in \\(E\\), and \\(\\operatorname{Aff}_E(\\mathfrak S)=\\{\\hat h:\\ h\\in A_h\\}\\). A measure \\(\\mu\\in M_1^+(\\mathfrak S)\\) has barycentre \\(\\varphi\\) exactly when \\(\\varphi(a)=\\int\\omega(a)\\,d\\mu(\\omega)\\) for all \\(a\\in A\\).\n2. (*Kadison's representation.*) For \\(h\\in A_h\\), \\(\\|\\hat h\\|=\\|h\\|\\), and \\(h\\geq0\\) if and only if \\(\\hat h\\geq0\\). The map \\(h\\mapsto\\hat h\\) is an isometric order isomorphism of \\(A_h\\) onto \\(\\operatorname{Aff}(\\mathfrak S)\\).\n3. (*Radon–Nikodym map.*) Let \\(\\varphi\\in\\mathfrak S\\), and write \\(\\pi,H,\\xi\\) for \\(\\pi_\\varphi,H_\\varphi,\\xi_\\varphi\\). Put\n\\[\n\\begin{gathered}\n\\Theta_\\varphi(x)(a)\\\\\n=\\langle\\pi(a)x\\xi,\\xi\\rangle\\\\\n(x\\in\\pi(A)',\\ a\\in A).\n\\end{gathered}\n\\]\nThen \\(\\Theta_\\varphi:\\pi(A)'\\to A^*\\) is linear and one-to-one, \\(\\Theta_\\varphi(x^*)=\\Theta_\\varphi(x)^*\\), and a self-adjoint \\(x\\) is positive if and only if \\(\\Theta_\\varphi(x)\\) is. It maps \\(\\{x\\in\\pi(A)':\\ 0\\leq x\\leq1\\}\\) onto \\(\\{\\psi\\in A^*:\\ 0\\leq\\psi\\leq\\varphi\\}\\).\n4. (*Pure states.*) For \\(\\varphi\\in\\mathfrak S\\) the following are equivalent: (a) \\(\\varphi\\in\\partial_e\\mathfrak S\\); (b) every positive functional \\(\\psi\\leq\\varphi\\) is a multiple of \\(\\varphi\\); (c) \\(\\pi_\\varphi(A)'=\\mathbb C1\\). We call such states *pure* and write \\(P(A)=\\partial_e\\mathfrak S\\).\n5. Every state \\(\\varphi\\) is the barycentre of a maximal probability measure \\(\\mu\\) on \\(\\mathfrak S\\):\n\\[\n\\begin{gathered}\n\\varphi(a)\\\\\n=\\int_{\\mathfrak S}\\omega(a)\\,d\\mu(\\omega)\\\\\n(a\\in A),\n\\end{gathered}\n\\tag{8.1}\n\\]\nand \\(\\mu(B)=0\\) for every Baire set \\(B\\subseteq\\mathfrak S\\) that misses \\(P(A)\\). If \\(A\\) is separable, \\(\\mathfrak S\\) is metrizable, \\(P(A)\\) is a \\(G_\\delta\\) set, and \\(\\mu\\) is concentrated on it:\n\\[\n\\begin{gathered}\n\\varphi(a)\\\\\n=\\int_{P(A)}\\omega(a)\\,d\\mu(\\omega)\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{8.2}\n\\]\n\n**Proof.** (1) \\(\\mathfrak S\\) is a weak\\(^*\\)-closed subset of the unit ball of \\(A^*\\), so it is weak\\(^*\\)-compact by the Banach–Alaoglu theorem. On hermitian functionals the topologies \\(\\sigma(A^*,A)\\) and \\(\\sigma(A^*_h,A_h)\\) agree, since \\(\\psi(h+ik)=\\psi(h)+i\\psi(k)\\). The constant \\(c\\) is \\(\\widehat{c1}\\). The last claim follows by complex linearity from the case \\(a\\in A_h\\).\n\n(2) Let \\(\\pi\\) be a faithful representation of \\(A\\) (the Gelfand–Naimark theorem); it is isometric, being an injective \\(*\\)-homomorphism. The norm of the self-adjoint operator \\(\\pi(h)\\) equals \\(\\sup_{\\|\\eta\\|=1}|\\langle\\pi(h)\\eta,\\eta\\rangle|\\), and each \\(a\\mapsto\\langle\\pi(a)\\eta,\\eta\\rangle\\) is a state. So \\(\\|\\hat h\\|\\geq\\|h\\|\\); the reverse holds because states have norm one. If \\(\\hat h\\geq0\\), then \\(0\\leq\\omega(h)\\leq\\|h\\|\\) for every state, so \\(\\|\\,\\|h\\|1-h\\,\\|=\\sup_\\omega|\\omega(\\|h\\|1-h)|\\leq\\|h\\|\\). The spectrum of \\(h\\) therefore lies in \\([0,2\\|h\\|]\\), and \\(h\\geq0\\). The image \\(\\{\\hat h\\}\\) is complete, hence closed, and it is dense by [Lemma 1.2](#oa-fnd-ir-01). So it is all of \\(\\operatorname{Aff}(\\mathfrak S)\\).\n\n(3) Linearity is clear. If \\(\\Theta_\\varphi(x)=0\\), then \\(\\langle x\\pi(a)\\xi,\\pi(b)\\xi\\rangle=\\Theta_\\varphi(x)(b^*a)=0\\) for all \\(a,b\\), so \\(x=0\\). The adjoint formula is a direct computation. For \\(x\\geq0\\), \\(\\Theta_\\varphi(x)(a^*a)=\\langle x\\pi(a)\\xi,\\pi(a)\\xi\\rangle\\geq0\\), and \\(\\Theta_\\varphi(x)(a^*a)\\leq\\|x\\|\\varphi(a^*a)\\). Conversely, if \\(x=x^*\\) and \\(\\Theta_\\varphi(x)\\geq0\\), then \\(\\langle x\\eta,\\eta\\rangle\\geq0\\) on the dense set \\(\\pi(A)\\xi\\), so \\(x\\geq0\\). Now let \\(0\\leq\\psi\\leq\\varphi\\). By the Cauchy–Schwarz inequality \\(|\\psi(b^*a)|^2\\leq\\psi(a^*a)\\psi(b^*b)\\leq\\|\\pi(a)\\xi\\|^2\\|\\pi(b)\\xi\\|^2\\). So \\((\\pi(a)\\xi,\\pi(b)\\xi)\\mapsto\\psi(b^*a)\\) is a well-defined bounded sesquilinear form on \\(\\pi(A)\\xi\\), and there is an operator \\(x\\) with \\(0\\leq x\\leq1\\) and \\(\\langle x\\pi(a)\\xi,\\pi(b)\\xi\\rangle=\\psi(b^*a)\\). For \\(c\\in A\\), \\[\n\\begin{gathered}\n\\langle x\\pi(c)\\pi(a)\\xi,\\pi(b)\\xi\\rangle\\\\\n=\\psi((c^*b)^*a)\\\\\n=\\langle\\pi(c)x\\pi(a)\\xi,\\pi(b)\\xi\\rangle\n\\end{gathered}\n\\], so \\(x\\in\\pi(A)'\\). With \\(b=1\\) we get \\(\\Theta_\\varphi(x)=\\psi\\).\n\n(4) (a)\\(\\Rightarrow\\)(c): Let \\(x\\in\\pi(A)'\\) with \\(0\\leq x\\leq1\\), and \\(t=\\langle x\\xi,\\xi\\rangle\\). If \\(0<t<1\\), then \\(\\varphi=t\\,\\Theta_\\varphi(x)/t+(1-t)\\,\\Theta_\\varphi(1-x)/(1-t)\\) is a convex combination of states, so \\(\\Theta_\\varphi(x)=t\\varphi=\\Theta_\\varphi(t1)\\) and \\(x=t1\\). If \\(t=0\\), then \\(x^{1/2}\\xi=0\\); as \\(\\xi\\) is separating for \\(\\pi(A)'\\), \\(x=0\\). If \\(t=1\\), apply this to \\(1-x\\). So every positive contraction in \\(\\pi(A)'\\) is a scalar, and \\(\\pi(A)'=\\mathbb C1\\). (c)\\(\\Rightarrow\\)(b) follows from (3). (b)\\(\\Rightarrow\\)(a): if \\(\\varphi=t\\psi_1+(1-t)\\psi_2\\) with states \\(\\psi_i\\) and \\(0<t<1\\), then \\(t\\psi_1\\leq\\varphi\\), so \\(t\\psi_1=\\lambda\\varphi\\), and \\(\\lambda=t\\) by evaluating at \\(1\\).\n\n(5) Apply [Theorem 5.4](#oa-fnd-ir-06)(2) and [Theorem 6.6](#oa-fnd-ir-08) to \\(K=\\mathfrak S\\), and use (1) to write the barycentre condition as (8.1). If \\((a_n)\\) is dense in the unit ball of \\(A\\), then \\(d(\\omega,\\omega')=\\sum_n2^{-n}|\\omega(a_n)-\\omega'(a_n)|\\) is a metric on \\(\\mathfrak S\\) whose topology is weaker than the weak\\(^*\\) topology. A weaker Hausdorff topology on a compact space coincides with it. Now use [Theorem 6.4](#oa-fnd-ir-07). \\(\\square\\)\n\nCondition (c) of part (4) says that \\(\\pi_\\varphi\\) is irreducible. Indeed, a closed subspace is invariant under \\(\\pi_\\varphi(A)\\) exactly when its projection lies in \\(\\pi_\\varphi(A)'\\), and a von Neumann algebra whose only projections are \\(0\\) and \\(1\\) is \\(\\mathbb C1\\), by the spectral theorem. So a state is pure exactly when its GNS representation is irreducible.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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        "measure-and-hilbert-space-tools",
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      ]
    },
    {
      "id": "OA-FND-IR-10",
      "unit": "integral-representations-of-states",
      "name": "9. The operator map of a representing measure",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "anchor": "oa-fnd-ir-10",
      "proof_locus": {
        "line": 507,
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      },
      "full_conditions_and_proof": "## 9. The operator map of a representing measure\n\nA representing measure of a state \\(\\varphi\\) acts on the GNS space of \\(\\varphi\\) through a positive map into the commutant \\(\\pi_\\varphi(A)'\\). We first give a test for the continuity of such maps.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
        "the-stone-weierstrass-theorem-for-functions-vanishing-at-infinity",
        "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "the-double-commutant-theorem",
        "measure-and-hilbert-space-tools",
        "haar-measure-on-locally-compact-groups",
        "abelian-operator-algebras",
        "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
        "hilbert-spaces-and-compact-operators",
        "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "the-spectral-theorem-for-bounded-self-adjoint-operators",
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      ]
    },
    {
      "id": "OA-FND-IR-11",
      "unit": "integral-representations-of-states",
      "name": "10. Orthogonal measures",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
      "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "anchor": "oa-fnd-ir-11",
      "proof_locus": {
        "line": 560,
        "through_line": 638
      },
      "full_conditions_and_proof": "## 10. Orthogonal measures\n\nA representing measure is orthogonal when its operator map is multiplicative. This section gives equivalent conditions and shows that the image of the operator map is then an abelian von Neumann algebra.\n\n**Lemma 10.1** (Positive contractions and projections). Let \\(p,q\\in B(H)\\).\n\n1. If \\(p\\) is a projection and \\(0\\leq x\\leq p\\), \\(0\\leq x\\leq1-p\\), then \\(x=0\\).\n2. If \\(p,q\\) are projections and \\(p+q\\) is a projection, then \\(pq=0\\).\n3. If \\(p,q\\geq0\\) and \\(p+q=1\\), then \\(p\\) and \\(q\\) commute and \\(0\\leq pq\\leq p\\), \\(0\\leq pq\\leq q\\).\n\n**Proof.** (1) \\((1-p)x(1-p)\\leq(1-p)p(1-p)=0\\), so \\(x^{1/2}(1-p)=0\\) and \\(x=xp=px\\). Likewise \\(pxp\\leq p(1-p)p=0\\) gives \\(xp=0\\). So \\(x=0\\). (2) \\((p+q)^2=p+q\\) gives \\(pq+qp=0\\). Multiply by \\(p\\) on the left, and separately on the right: \\(pq+pqp=0=pqp+qp\\). So \\(pq=qp\\), and then \\(2pq=0\\). (3) \\(q=1-p\\) commutes with \\(p\\), and \\(pq=p^{1/2}qp^{1/2}\\geq0\\). Also \\(p-pq=p(1-q)=p^2\\geq0\\) and \\(q-pq=q^2\\geq0\\). \\(\\square\\)\n\n**Theorem 10.2.** Let \\(\\mu\\in M_1^+(\\mathfrak S)\\) with barycentre \\(\\varphi\\). The following are equivalent.\n\n1. \\(\\kappa_\\mu\\) is multiplicative.\n2. \\(\\kappa_\\mu(\\chi_E)\\kappa_\\mu(1-\\chi_E)=0\\) for every Borel set \\(E\\subseteq\\mathfrak S\\).\n3. For every Borel set \\(E\\), the functionals \\(\\varphi_E(a)=\\int_E\\omega(a)\\,d\\mu(\\omega)\\) and \\(\\varphi_{E^c}=\\varphi-\\varphi_E\\) are *orthogonal*: a positive \\(\\psi\\) with \\(\\psi\\leq\\varphi_E\\) and \\(\\psi\\leq\\varphi_{E^c}\\) is \\(0\\).\n\nIn this case \\(\\kappa_\\mu\\) is a \\(*\\)-isomorphism of \\(L^\\infty(\\mathfrak S,\\mu)\\) onto its range, \\(\\|\\kappa_\\mu(f)\\|=\\|f\\|_\\infty\\), and \\(\\|\\kappa_\\mu(f)\\xi_\\varphi\\|^2=\\int|f|^2\\,d\\mu\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): \\(\\chi_E(1-\\chi_E)=0\\).\n\n(2)\\(\\Rightarrow\\)(3): Put \\(p=\\kappa_\\mu(\\chi_E)\\); then \\(\\kappa_\\mu(1-\\chi_E)=1-p\\) and \\(p(1-p)=0\\), so \\(p\\) is a projection. We have \\(\\varphi_E=\\Theta_\\varphi(p)\\) and \\(\\varphi_{E^c}=\\Theta_\\varphi(1-p)\\). If \\(0\\leq\\psi\\leq\\varphi_E\\) and \\(\\psi\\leq\\varphi_{E^c}\\), then \\(\\psi\\leq\\varphi\\), so \\(\\psi=\\Theta_\\varphi(x)\\) with \\(x\\geq0\\) (Proposition 8.1(3)). Since \\(\\Theta_\\varphi\\) reflects order, \\(x\\leq p\\) and \\(x\\leq1-p\\). By Lemma 10.1(1), \\(x=0\\).\n\n(3)\\(\\Rightarrow\\)(1): Fix \\(E\\), and let \\(p=\\kappa_\\mu(\\chi_E)\\), \\(q=\\kappa_\\mu(1-\\chi_E)\\). If \\(x\\in\\pi(A)'\\) and \\(0\\leq x\\leq p\\), \\(0\\leq x\\leq q\\), then \\(\\Theta_\\varphi(x)\\) lies below \\(\\varphi_E\\) and \\(\\varphi_{E^c}\\), so \\(x=0\\). By Lemma 10.1(3), \\(x=pq\\) is such an element, so \\(pq=0\\), and \\(p=p(p+q)=p^2\\). Thus \\(\\kappa_\\mu(\\chi_E)\\) is a projection for every Borel \\(E\\). If \\(G\\cap H=\\varnothing\\), then \\(\\kappa_\\mu(\\chi_G)+\\kappa_\\mu(\\chi_H)=\\kappa_\\mu(\\chi_{G\\cup H})\\) is a projection, so the two are orthogonal (Lemma 10.1(2)). For Borel \\(E,F\\), write \\(\\chi_E=\\chi_{E\\cap F}+\\chi_{E\\setminus F}\\) and \\(\\chi_F=\\chi_{E\\cap F}+\\chi_{F\\setminus E}\\), and expand:\n\\[\n\\kappa_\\mu(\\chi_E)\\kappa_\\mu(\\chi_F)=\\kappa_\\mu(\\chi_{E\\cap F})=\\kappa_\\mu(\\chi_E\\chi_F).\n\\]\nSo \\(\\kappa_\\mu\\) is multiplicative on simple functions, which are norm-dense in \\(L^\\infty\\); and \\(\\kappa_\\mu\\) is bounded.\n\nNow let \\(\\kappa_\\mu\\) be multiplicative. Positivity gives \\(\\kappa_\\mu(\\bar f)=\\kappa_\\mu(f)^*\\), so \\(\\kappa_\\mu\\) is a \\(*\\)-homomorphism, and \\(\\|\\kappa_\\mu(f)\\xi\\|^2=\\langle\\kappa_\\mu(|f|^2)\\xi,\\xi\\rangle=\\int|f|^2d\\mu\\) by (9.1) with \\(a=1\\). Hence \\(\\kappa_\\mu\\) is one-to-one. For the norm: \\(\\|\\kappa_\\mu(f)\\|^2=\\|\\kappa_\\mu(|f|^2)\\|\\leq\\|f\\|_\\infty^2\\). Conversely, for \\(\\varepsilon>0\\) the set \\(E=\\{|f|\\geq\\|f\\|_\\infty-\\varepsilon\\}\\) has positive measure, \\(p=\\kappa_\\mu(\\chi_E)\\) is a nonzero projection, and \\(p\\kappa_\\mu(|f|^2)p=\\kappa_\\mu(\\chi_E|f|^2)\\geq(\\|f\\|_\\infty-\\varepsilon)^2p\\). So \\(\\|\\kappa_\\mu(f)p\\|\\geq\\|f\\|_\\infty-\\varepsilon\\). \\(\\square\\)\n\n**Definition 10.3.** \\(\\mu\\in M_1^+(\\mathfrak S)\\) is *orthogonal* if the conditions of Theorem 10.2 hold. Its *associated abelian algebra* is \\(\\mathcal C_\\mu=\\kappa_\\mu(L^\\infty(\\mathfrak S,\\mu))\\subseteq\\pi_\\varphi(A)'\\).\n\nThe algebra \\(\\mathcal C_\\mu\\) is commutative, because \\(\\kappa_\\mu\\) is a \\(*\\)-homomorphism on the commutative algebra \\(L^\\infty\\). The next proposition shows that it is weakly closed, and it collects tools used later.\n\n**Proposition 10.4** (Structure of an orthogonal measure). Let \\(\\mu\\) be orthogonal with barycentre \\(\\varphi\\), and write \\(\\pi,H,\\xi\\) for its GNS triple.\n\n1. The map \\(V_\\mu:f\\mapsto\\kappa_\\mu(f)\\xi\\), \\(f\\in L^\\infty(\\mu)\\), extends to an isometry \\(V_\\mu:L^2(\\mathfrak S,\\mu)\\to H\\), and \\(\\kappa_\\mu(g)V_\\mu=V_\\mu M_g\\) for \\(g\\in L^\\infty(\\mu)\\), where \\(M_g\\) is multiplication by \\(g\\).\n2. Let \\(e_\\mu\\) be the projection onto \\(V_\\mu(L^2)\\). Then \\(V_\\mu(L^2)=[\\mathcal C_\\mu\\xi]\\), the closed span of \\(\\mathcal C_\\mu\\xi\\), and \\(e_\\mu\\in\\mathcal C_\\mu'\\).\n3. (*The \\(L^2\\) criterion.*) \\(\\mathcal C_\\mu=\\{y\\in\\pi(A)':\\ y\\xi\\in e_\\mu H\\}\\).\n4. \\(\\mathcal C_\\mu\\) is a von Neumann algebra (commutative), and \\(\\xi\\) is separating for it.\n5. For every \\(a\\in A\\),\n\\[\n\\begin{gathered}\n\\kappa_\\mu(\\hat a)\\xi\\\\\n=e_\\mu\\pi(a)\\xi\\\\\n\\text{and}\\\\\ne_\\mu\\pi(a)e_\\mu\\\\\n=\\kappa_\\mu(\\hat a)e_\\mu .\n\\end{gathered}\n\\tag{10.1}\n\\]\n\n**Proof.** (1) The isometry is the last claim of Theorem 10.2. The intertwining holds on \\(L^\\infty\\) by multiplicativity, and both sides are continuous on \\(L^2\\). (2) \\(\\kappa_\\mu(L^\\infty)\\xi\\) is dense in the range of \\(V_\\mu\\), which is closed. The range is invariant under the self-adjoint set \\(\\mathcal C_\\mu\\), so \\(e_\\mu\\) commutes with it.\n\n(3) The inclusion \"\\(\\subseteq\\)\" is clear. Let \\(y\\in\\pi(A)'\\) with \\(y\\xi\\in e_\\mu H\\); so \\(y\\xi=V_\\mu f\\) with \\(f\\in L^2(\\mu)\\). Let \\(E_n=\\{|f|\\leq n\\}\\). By (1), \\(\\kappa_\\mu(\\chi_{E_n})y\\xi=V_\\mu(\\chi_{E_n}f)=\\kappa_\\mu(\\chi_{E_n}f)\\xi\\). Both \\(\\kappa_\\mu(\\chi_{E_n})y\\) and \\(\\kappa_\\mu(\\chi_{E_n}f)\\) lie in \\(\\pi(A)'\\), for which \\(\\xi\\) is separating, so they are equal. Hence \\(\\|\\chi_{E_n}f\\|_\\infty=\\|\\kappa_\\mu(\\chi_{E_n}f)\\|\\leq\\|y\\|\\) for every \\(n\\), so \\(|f|\\leq\\|y\\|\\) almost everywhere. Then \\(f\\in L^\\infty\\), \\(\\kappa_\\mu(f)\\xi=V_\\mu f=y\\xi\\), and \\(y=\\kappa_\\mu(f)\\).\n\n(4) \\(\\mathcal C_\\mu\\) is a commutative \\(*\\)-algebra. Let \\(y\\) be in its weak closure. Then \\(y\\in\\pi(A)'\\), and \\(y\\xi\\) lies in the weak closure of \\(\\mathcal C_\\mu\\xi\\), hence in \\(e_\\mu H\\). By (3), \\(y\\in\\mathcal C_\\mu\\). So \\(\\mathcal C_\\mu\\) is weakly closed and contains \\(1\\); by von Neumann's bicommutant theorem it is a von Neumann algebra. \\(\\xi\\) is separating because \\(\\mathcal C_\\mu\\subseteq\\pi(A)'\\).\n\n(5) For \\(f\\in L^\\infty\\),\n\\[\n\\begin{gathered}\n\\langle\\kappa_\\mu(\\hat a)\\xi,\\kappa_\\mu(f)\\xi\\rangle\\\\\n=\\int\\bar f\\hat a\\,d\\mu\\\\\n=\\langle\\kappa_\\mu(\\bar f)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\langle\\pi(a)\\xi,\\kappa_\\mu(f)\\xi\\rangle .\n\\end{gathered}\n\\]\nSo \\(\\kappa_\\mu(\\hat a)\\xi-\\pi(a)\\xi\\) is orthogonal to \\(e_\\mu H\\), while \\(\\kappa_\\mu(\\hat a)\\xi\\in e_\\mu H\\). This is the first formula. For the second, let \\(x\\in\\mathcal C_\\mu\\). Then \\[\n\\begin{gathered}\ne_\\mu\\pi(a)e_\\mu x\\xi\\\\\n=e_\\mu x\\pi(a)\\xi\\\\\n=x\\,e_\\mu\\pi(a)\\xi\\\\\n=x\\kappa_\\mu(\\hat a)\\xi\\\\\n=\\kappa_\\mu(\\hat a)e_\\mu x\\xi\n\\end{gathered}\n\\]. The vectors \\(x\\xi\\) are dense in \\(e_\\mu H\\), and both sides vanish on \\((1-e_\\mu)H\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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      ]
    },
    {
      "id": "OA-FND-IR-12",
      "unit": "integral-representations-of-states",
      "name": "11. Simplicial measures",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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        "through_line": 663
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      "full_conditions_and_proof": "## 11. Simplicial measures\n\n**Definition 11.1.** A measure \\(\\mu\\in M_x(K)\\) is *simplicial* if it is an extreme point of the convex set \\(M_x(K)\\).\n\n**Proposition 11.2** (Any compact convex set). For \\(\\mu\\in M_x(K)\\), \\(\\mu\\) is simplicial if and only if \\(\\operatorname{Aff}(K)\\) is dense in the real space \\(L^1(K,\\mu)\\).\n\n**Proof.** Suppose \\(\\operatorname{Aff}(K)\\) is dense, and \\(\\mu=\\frac12(\\mu_1+\\mu_2)\\) with \\(\\mu_i\\in M_x(K)\\). Then \\(\\mu_i\\leq2\\mu\\), so \\(\\mu_i=h_i\\mu\\) with \\(0\\leq h_i\\leq2\\) ([Proposition 2.1](#oa-fnd-ir-02)(5)). For \\(a\\in\\operatorname{Aff}(K)\\), (3.1) gives \\(\\int a(h_i-1)\\,d\\mu=a(x)-a(x)=0\\). The functional \\(g\\mapsto\\int g(h_i-1)\\,d\\mu\\) is continuous on \\(L^1\\) and vanishes on a dense set, so \\(h_i=1\\) and \\(\\mu_i=\\mu\\).\n\nSuppose \\(\\mu\\) is simplicial and \\(\\operatorname{Aff}(K)\\) is not dense. By the Hahn–Banach theorem and the duality \\(L^1(\\mu)^*=L^\\infty(\\mu)\\) there is a real \\(h\\in L^\\infty(\\mu)\\), \\(h\\neq0\\), with \\(\\int ah\\,d\\mu=0\\) for all \\(a\\in\\operatorname{Aff}(K)\\); scale it so that \\(\\|h\\|_\\infty\\leq1\\). Then \\(\\mu\\pm h\\mu\\) are positive, and they lie in \\(M_x(K)\\) because \\(\\int a(1\\pm h)\\,d\\mu=a(x)\\). Since \\(\\mu=\\frac12((\\mu+h\\mu)+(\\mu-h\\mu))\\), extremality gives \\(h\\mu=0\\), which is false. \\(\\square\\)\n\n**Proposition 11.3.** Let \\(\\mu\\in M_\\varphi(\\mathfrak S)\\), and consider: (i) \\(\\mu\\) is orthogonal; (ii) \\(\\mathcal A_{\\mathbb C}=\\{\\hat a:\\ a\\in A\\}\\) is dense in \\(L^1(\\mathfrak S,\\mu)\\) (complex scalars); (iii) \\(\\mu\\) is simplicial. Then (i)\\(\\Rightarrow\\)(ii)\\(\\Leftrightarrow\\)(iii). Under (i), \\(\\mathcal A_{\\mathbb C}\\) is even dense in \\(L^2(\\mathfrak S,\\mu)\\).\n\n**Proof.** (i)\\(\\Rightarrow\\)(ii): By (10.1), \\(V_\\mu(\\hat a)=\\kappa_\\mu(\\hat a)\\xi=e_\\mu\\pi(a)\\xi\\). These vectors are dense in \\(e_\\mu H=V_\\mu(L^2)\\), because \\(\\pi(A)\\xi\\) is dense in \\(H\\). \\(V_\\mu\\) is an isometry onto \\(e_\\mu H\\), so \\(\\mathcal A_{\\mathbb C}\\) is dense in \\(L^2(\\mu)\\), and hence in \\(L^1(\\mu)\\), since \\(\\|g\\|_1\\leq\\|g\\|_2\\) for a probability measure.\n\n(ii)\\(\\Leftrightarrow\\)(iii): By Proposition 8.1(2), \\(\\operatorname{Aff}(\\mathfrak S)=\\{\\hat h:\\ h\\in A_h\\}\\), and \\(\\mathcal A_{\\mathbb C}=\\operatorname{Aff}(\\mathfrak S)+i\\operatorname{Aff}(\\mathfrak S)\\). The complex span of a set of real functions is dense in complex \\(L^1\\) exactly when the set is dense in real \\(L^1\\): approximate real and imaginary parts separately, and conversely note \\(\\|g-\\operatorname{Re}u\\|_1\\leq\\|g-u\\|_1\\) for real \\(g\\). Now apply Proposition 11.2. \\(\\square\\)\n\n**Remark 11.4.** The equivalence (ii)\\(\\Leftrightarrow\\)(iii) uses nothing about C\\(^*\\)-algebras: it is Proposition 11.2, which holds for every compact convex set. The implication (i)\\(\\Rightarrow\\)(iii) cannot be reversed. Here is an explicit simplicial measure that is not orthogonal.\n\n**Example 11.5** (Simplicial but not orthogonal). Let \\(A=M_2(\\mathbb C)\\). For \\(v\\in\\mathbb R^3\\) with \\(|v|\\leq1\\) let \\(\\rho_v=\\frac12(1+v_1\\sigma_1+v_2\\sigma_2+v_3\\sigma_3)\\), with the Pauli matrices \\(\\sigma_i\\), and \\(\\omega_v(a)=\\operatorname{tr}(\\rho_va)\\). Every state is some \\(\\omega_v\\), and \\(\\omega_v\\) is pure exactly when \\(|v|=1\\). For density matrices, \\(\\omega_\\rho\\leq\\omega_{\\rho'}\\) if and only if \\(\\rho\\leq\\rho'\\). Let \\(v^{(1)},v^{(2)},v^{(3)}\\) be unit vectors in the \\(v_1v_3\\)-plane at mutual angles of \\(120^\\circ\\), so \\(\\sum_kv^{(k)}=0\\), and let \\(\\mu=\\frac13\\sum_k\\delta_{\\omega_{v^{(k)}}}\\). It represents the tracial state \\(\\tau=\\omega_0\\).\n\n- *\\(\\mu\\) is simplicial.* \\(L^1(\\mu)\\) is \\(\\mathbb C^3\\). The functions \\(\\hat1,\\hat\\sigma_1,\\hat\\sigma_3\\) give the vectors \\((1,1,1)\\), \\((v^{(k)}_1)_k\\), \\((v^{(k)}_3)_k\\), which are independent because the three points are not collinear. So (ii) holds.\n- *\\(\\mu\\) is not orthogonal.* Take \\(E=\\{\\omega_{v^{(1)}}\\}\\). The density of \\(3\\varphi_{E^c}=\\omega_{v^{(2)}}+\\omega_{v^{(3)}}\\) is \\(1-\\frac12v^{(1)}\\cdot\\sigma\\), with eigenvalues \\(\\frac12\\) and \\(\\frac32\\). Let \\(\\psi=\\frac16\\omega_{v^{(1)}}\\), with density \\(\\frac1{12}(1+v^{(1)}\\cdot\\sigma)\\). In the eigenbasis of \\(v^{(1)}\\cdot\\sigma\\) the densities of \\(\\psi\\), \\(\\varphi_E\\) and \\(\\varphi_{E^c}\\) are \\(\\operatorname{diag}(\\frac16,0)\\), \\(\\operatorname{diag}(\\frac13,0)\\) and \\(\\operatorname{diag}(\\frac16,\\frac12)\\). So \\(0\\neq\\psi\\leq\\varphi_E\\) and \\(\\psi\\leq\\varphi_{E^c}\\).\n\nBy contrast, \\(\\frac12(\\delta_{\\omega_v}+\\delta_{\\omega_{-v}})\\) with \\(|v|=1\\) is orthogonal: \\(\\rho_v\\) and \\(\\rho_{-v}\\) are orthogonal rank-one projections, and a positive matrix below multiples of both is \\(0\\). The normalized surface measure on the sphere \\(\\{\\omega_v:|v|=1\\}\\) also represents \\(\\tau\\), by symmetry. It is maximal, since it lives on \\(\\partial_e\\mathfrak S\\) and is therefore a boundary measure (Proposition 5.5(2) and Theorem 5.4). But it is not orthogonal: \\(\\kappa\\) would embed the infinite-dimensional \\(L^\\infty\\) of the sphere into the four-dimensional \\(\\pi_\\tau(A)'\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-IR-13",
      "unit": "integral-representations-of-states",
      "name": "12. Abelian algebras with a cyclic vector",
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      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 12. Abelian algebras with a cyclic vector\n\nThis section proves three facts about commutants that the rest of the lesson uses: an abelian algebra with a cyclic vector has an abelian commutant, compressions of von Neumann algebras by projections behave well, and abelian subalgebras of a commutant match certain projections.\n\n**Lemma 12.1.** Let \\(\\mathfrak A\\subseteq B(H)\\) be a commutative \\(*\\)-algebra (not necessarily closed) with a cyclic vector \\(\\xi\\). Then \\(\\mathfrak A'\\) is commutative and \\(\\mathfrak A'=\\mathfrak A''\\). If \\(\\mathfrak A\\) is a von Neumann algebra, then \\(\\mathfrak A'=\\mathfrak A\\): it is maximal abelian.\n\n**Proof.** For \\(a\\in\\mathfrak A\\), \\(\\|a^*\\xi\\|^2=\\langle aa^*\\xi,\\xi\\rangle=\\langle a^*a\\xi,\\xi\\rangle=\\|a\\xi\\|^2\\). So \\(J(a\\xi)=a^*\\xi\\) is a well-defined conjugate-linear isometry on \\(\\mathfrak A\\xi\\). It extends to a conjugate-linear isometry \\(J\\) of \\(H\\) onto \\(H\\) with \\(J^2=1\\), and \\(\\langle J\\eta,J\\zeta\\rangle=\\langle\\zeta,\\eta\\rangle\\). For \\(c,b\\in\\mathfrak A\\), \\(Jc^*(b\\xi)=b^*c\\xi=cb^*\\xi=cJ(b\\xi)\\), so \\(Jc^*=cJ\\). Let \\(T\\in\\mathfrak A'\\). Then \\(JTJ(b\\xi)=J(b^*T\\xi)=bJT\\xi\\) and \\(T^*(b\\xi)=bT^*\\xi\\). Moreover \\(JT\\xi=T^*\\xi\\), since for \\(c\\in\\mathfrak A\\)\n\\[\n\\begin{gathered}\n\\langle JT\\xi,c\\xi\\rangle\\\\\n=\\langle JT\\xi,J(c^*\\xi)\\rangle\\\\\n=\\langle c^*\\xi,T\\xi\\rangle\\\\\n=\\langle c^*T^*\\xi,\\xi\\rangle\\\\\n=\\langle T^*\\xi,c\\xi\\rangle .\n\\end{gathered}\n\\]\nSo \\(JTJ=T^*\\) on a dense set, hence everywhere. For \\(S,T\\in\\mathfrak A'\\): \\[\n\\begin{gathered}\nT^*S^*\\\\\n=(ST)^*\\\\\n=J(ST)J\\\\\n=(JSJ)(JTJ)\\\\\n=S^*T^*\n\\end{gathered}\n\\], and \\(\\mathfrak A'\\) is commutative. Then \\(\\mathfrak A'\\subseteq\\mathfrak A''\\); and \\(\\mathfrak A\\subseteq\\mathfrak A'\\) gives \\(\\mathfrak A''\\subseteq\\mathfrak A'\\). \\(\\square\\)\n\n**Lemma 12.2** (Reduction). Let \\(\\mathcal N\\) be a von Neumann algebra on \\(H\\), \\(e\\) a projection, and \\(\\mathcal N_e=\\{exe|_{eH}:\\ x\\in\\mathcal N\\}\\subseteq B(eH)\\).\n\n1. If \\(e\\in\\mathcal N'\\), then \\((\\mathcal N_e)'=(\\mathcal N')_e\\), and \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-homomorphism of \\(\\mathcal N\\) onto \\(\\mathcal N_e\\).\n2. If \\(e\\in\\mathcal N\\), then \\((\\mathcal N_e)'=(\\mathcal N')_e\\).\n3. In both cases \\(\\mathcal N_e\\) is a von Neumann algebra on \\(eH\\).\n\n**Proof.** (1) Here \\(exe=xe\\), so \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-homomorphism. If \\(y\\in\\mathcal N'\\), then \\(eye|_{eH}\\) commutes with every \\(xe|_{eH}\\). Conversely, let \\(T\\in B(eH)\\) commute with \\(\\mathcal N_e\\), and let \\(\\tilde T=Te\\) (zero on \\((1-e)H\\)). Each \\(x\\in\\mathcal N\\) leaves \\(eH\\) and \\((1-e)H\\) invariant, so \\(\\tilde T\\in\\mathcal N'\\) and \\(T=e\\tilde Te|_{eH}\\).\n\n(2) The inclusion \"\\(\\supseteq\\)\" is direct. Let \\(T\\in B(eH)\\) commute with every \\(exe|_{eH}\\). For \\(x_1,\\dots,x_n\\in\\mathcal N\\) and \\(\\eta_1,\\dots,\\eta_n\\in eH\\), the matrix \\(X=[ex_j^*x_ie]_{j,i}\\) acts on \\((eH)^n\\) and is positive, since \\(\\langle X\\vec\\eta,\\vec\\eta\\rangle=\\|\\sum_ix_i\\eta_i\\|^2\\). The operator \\(T^{(n)}=T\\oplus\\dots\\oplus T\\) commutes with \\(X\\), hence with \\(X^{1/2}\\). So\n\\[\n\\begin{gathered}\n\\Bigl\\|\\sum_ix_iT\\eta_i\\Bigr\\|^2\\\\\n=\\langle T^{(n)*}T^{(n)}X\\vec\\eta,\\vec\\eta\\rangle\\\\\n=\\|T^{(n)}X^{1/2}\\vec\\eta\\|^2\\\\\n\\leq\\|T\\|^2\\Bigl\\|\\sum_ix_i\\eta_i\\Bigr\\|^2 .\n\\end{gathered}\n\\]\nSo \\(y(\\sum_ix_i\\eta_i)=\\sum_ix_iT\\eta_i\\) defines a bounded operator on the closed span \\([\\mathcal NeH]\\); extend it by \\(0\\) on the orthogonal complement. For \\(z\\in\\mathcal N\\) the formula gives \\(yz=zy\\) on \\([\\mathcal NeH]\\); the complement is \\(\\mathcal N\\)-invariant and \\(y\\) vanishes there. So \\(y\\) commutes with \\(\\mathcal N\\). For \\(\\eta\\in eH\\), \\(y\\eta=y(e\\eta)=eT\\eta=T\\eta\\). So \\(T=eye|_{eH}\\) with \\(y\\in\\mathcal N'\\).\n\n(3) If \\(e\\in\\mathcal N'\\), apply (2) to the von Neumann algebra \\(\\mathcal N'\\), which contains \\(e\\): \\(((\\mathcal N')_e)'=(\\mathcal N'')_e=\\mathcal N_e\\). If \\(e\\in\\mathcal N\\), apply (1) to \\(\\mathcal N'\\) in the same way. So \\(\\mathcal N_e\\) is a commutant. \\(\\square\\)\n\n**Proposition 12.3.** Let \\(\\mathcal M\\) be a unital \\(*\\)-algebra of operators on \\(H\\) with a cyclic vector \\(\\xi_0\\).\n\n- (a) If \\(\\mathcal A\\subseteq\\mathcal M'\\) is a commutative \\(*\\)-algebra and \\(e\\) is the projection onto \\([\\mathcal A\\xi_0]\\), then \\(e\\mathcal Me\\) is commutative, that is, \\(e\\mathcal Me\\subseteq(e\\mathcal Me)'\\).\n- (b) If \\(e\\) is a projection with \\(e\\xi_0=\\xi_0\\) and \\(e\\mathcal Me\\) is commutative, then \\(\\mathcal A=\\mathcal M'\\cap\\{e\\}'\\) is a commutative von Neumann algebra, \\(e\\) is the projection onto \\([\\mathcal A\\xi_0]\\), and \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-isomorphism of \\(\\mathcal A\\) onto the maximal abelian algebra \\((e\\mathcal Me|_{eH})'\\) of \\(eH\\).\n- (c) The maps \\(\\mathcal A\\mapsto[\\mathcal A\\xi_0]\\) and \\(e\\mapsto\\mathcal M'\\cap\\{e\\}'\\) are inverse bijections. One side is the set of commutative von Neumann algebras contained in \\(\\mathcal M'\\); the other is the set of projections \\(e\\) with \\(e\\xi_0=\\xi_0\\) and \\(e\\mathcal Me\\) commutative.\n\n**Proof.** (a) \\(e\\in\\mathcal A'\\), because \\([\\mathcal A\\xi_0]\\) is invariant under the self-adjoint set \\(\\mathcal A\\). The restrictions \\(\\mathcal A_e=\\{a|_{eH}\\}\\) form a commutative \\(*\\)-algebra on \\(eH\\) with cyclic vector \\(e\\xi_0\\): since \\(a(1-e)\\xi_0=(1-e)a\\xi_0=0\\), we have \\(a\\xi_0=ae\\xi_0\\), so \\(\\mathcal A_e(e\\xi_0)=\\mathcal A\\xi_0\\) is dense in \\(eH\\). (If \\(1\\in\\mathcal A\\), then \\(e\\xi_0=\\xi_0\\).) By Lemma 12.1, \\((\\mathcal A_e)'\\) is commutative. For \\(x\\in\\mathcal M\\) and \\(a\\in\\mathcal A\\), \\((exe)(ae)=exae=eaxe=(ae)(exe)\\). So \\(e\\mathcal Me|_{eH}\\subseteq(\\mathcal A_e)'\\), which is commutative; and \\(e\\mathcal Me\\) vanishes on \\((1-e)H\\).\n\n(b) Let \\(S=\\{exe|_{eH}:\\ x\\in\\mathcal M\\}\\), a commuting self-adjoint set that contains \\(1_{eH}\\). The vector \\(\\xi_0\\) is cyclic for \\(S\\) in \\(eH\\), because \\(e\\mathcal M\\xi_0\\) is dense in \\(eH\\) and \\(exe\\xi_0=ex\\xi_0\\). By Lemma 12.1, applied to the algebra generated by \\(S\\), \\(S'\\) is commutative and \\(S'=S''\\); so \\(S'\\) is maximal abelian on \\(eH\\). The set \\(\\mathcal A\\) is the commutant of \\(\\mathcal M\\cup\\{e\\}\\), a von Neumann algebra. Restriction \\(r(x)=x|_{eH}\\) is a \\(*\\)-homomorphism of \\(\\mathcal A\\) into \\(S'\\). It is one-to-one: \\(r(x)=0\\) gives \\(x\\xi_0=0\\), and \\(\\xi_0\\) is separating for \\(\\mathcal M'\\). It is onto \\(S'\\): let \\(y\\in S'\\). For \\(m\\in\\mathcal M\\), the positive operator \\(s=em^*me|_{eH}\\) lies in \\(S\\), so \\(y^*y\\) commutes with \\(s\\) and with \\(s^{1/2}\\); also \\(y\\xi_0\\in eH\\). Hence\n\\[\n\\begin{gathered}\n\\|my\\xi_0\\|^2\\\\\n=\\langle em^*me\\,y\\xi_0,y\\xi_0\\rangle\\\\\n=\\langle y^*y\\,(em^*me)\\xi_0,\\xi_0\\rangle\\\\\n\\leq\\|y\\|^2\\langle em^*me\\xi_0,\\xi_0\\rangle\\\\\n=\\|y\\|^2\\|m\\xi_0\\|^2 .\n\\end{gathered}\n\\]\nSo \\(x_y(m\\xi_0)=my\\xi_0\\) defines a bounded operator on \\(H\\), and \\(x_y\\in\\mathcal M'\\). Also \\(x_{y^*}=x_y^*\\): both sides of \\(\\langle x_ym\\xi_0,m'\\xi_0\\rangle=\\langle m\\xi_0,x_{y^*}m'\\xi_0\\rangle\\) equal \\(\\langle y\\,(em'^*me)\\xi_0,\\xi_0\\rangle\\). For \\(m_1,m_2\\in\\mathcal M\\),\n\\[\n\\begin{gathered}\n\\langle x_ym_1\\xi_0,em_2\\xi_0\\rangle\\\\\n=\\langle(em_2^*e)(em_1e)\\,y\\xi_0,\\xi_0\\rangle\\\\\n=\\langle y\\,em_1\\xi_0,em_2\\xi_0\\rangle ,\n\\end{gathered}\n\\]\nbecause \\(y\\) commutes with the elements \\(em_ie|_{eH}\\) of \\(S\\). So \\(ex_y=ye\\) as operators on \\(H\\). Applying this to \\(y^*\\) and taking adjoints gives \\(x_ye=ey\\), where \\(ey\\) means \\(y\\) on \\(eH\\) and \\(0\\) on \\((1-e)H\\). So \\(x_y\\) commutes with \\(e\\), lies in \\(\\mathcal A\\), and \\(r(x_y)=y\\). Finally \\([\\mathcal A\\xi_0]=[S'\\xi_0]\\supseteq[S\\xi_0]=eH\\), and \\(\\mathcal A\\xi_0=\\mathcal Ae\\xi_0\\subseteq eH\\).\n\n(c) By (a) and (b) both maps land in the right sets, and \\(e\\mapsto\\mathcal M'\\cap\\{e\\}'\\mapsto e\\) is the identity by (b). Let \\(\\mathcal A\\subseteq\\mathcal M'\\) be an abelian von Neumann algebra and \\(e=[\\mathcal A\\xi_0]\\). Clearly \\(\\mathcal A\\subseteq\\mathcal B=\\mathcal M'\\cap\\{e\\}'\\), and \\(\\mathcal B\\) is abelian by (b). By Lemma 12.2(1) and (3), \\(\\mathcal A_e\\) is a von Neumann algebra on \\(eH\\), and it has the cyclic vector \\(\\xi_0\\), so it is maximal abelian (Lemma 12.1). For \\(x\\in\\mathcal B\\), \\(x|_{eH}\\) commutes with \\(\\mathcal A_e\\), so \\(x|_{eH}=a|_{eH}\\) for some \\(a\\in\\mathcal A\\), and \\(x=a\\) because restriction is one-to-one on \\(\\mathcal B\\). \\(\\square\\)\n\n",
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    {
      "id": "OA-FND-IR-14",
      "unit": "integral-representations-of-states",
      "name": "13. Orthogonal measures and abelian subalgebras",
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        "through_line": 768
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      "full_conditions_and_proof": "## 13. Orthogonal measures and abelian subalgebras\n\nThe main structural fact about orthogonal measures: they are in one-to-one correspondence with the abelian von Neumann subalgebras of the commutant.\n\n**Theorem 13.1.** Let \\(\\varphi\\in\\mathfrak S\\), and let \\(\\mathcal C\\subseteq\\pi_\\varphi(A)'\\) be a commutative von Neumann algebra. Then exactly one orthogonal \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) has \\(\\mathcal C_\\mu=\\mathcal C\\). So \\(\\mu\\mapsto\\mathcal C_\\mu\\) maps the orthogonal measures in \\(M_\\varphi(\\mathfrak S)\\) bijectively onto the commutative von Neumann algebras contained in \\(\\pi_\\varphi(A)'\\).\n\n**Proof.** Write \\(\\pi,H,\\xi\\), and let \\(e\\) be the projection onto \\([\\mathcal C\\xi]\\); \\(e\\in\\mathcal C'\\), and \\(\\xi\\) is separating for \\(\\mathcal C\\subseteq\\pi(A)'\\).\n\n*Step 1: the compression.* By Lemma 12.2, \\(\\mathcal C_e\\) is a von Neumann algebra on \\(eH\\), and \\(x\\mapsto x_e=x|_{eH}\\) is a \\(*\\)-homomorphism of \\(\\mathcal C\\) onto it; it is one-to-one because \\(\\xi\\) is separating. \\(\\mathcal C_e\\) is abelian with cyclic vector \\(\\xi\\), hence maximal abelian by Lemma 12.1: \\((\\mathcal C_e)'=\\mathcal C_e\\). Also \\((\\mathcal C_e)'=(\\mathcal C')_e\\) by Lemma 12.2(1). Since \\(\\pi(A)\\subseteq\\mathcal C'\\), each \\(e\\pi(a)e|_{eH}\\) lies in \\(\\mathcal C_e\\). So there is a unique \\(\\theta(a)\\in\\mathcal C\\) with \\(\\theta(a)e=e\\pi(a)e\\); in particular \\(\\theta(a)\\xi=e\\pi(a)\\xi\\). The map \\(\\theta:A\\to\\mathcal C\\) is linear, unital and positive: \\(x\\mapsto x_e\\) is an isometric \\(*\\)-isomorphism of \\(\\mathcal C\\) onto \\(\\mathcal C_e\\), so it reflects positivity.\n\n*Step 2: the measure.* By the Gelfand representation (the commutative Gelfand–Naimark theorem), \\(\\mathcal C\\cong C(\\Omega)\\) for its spectrum \\(\\Omega\\); write \\(\\Gamma\\) for this isomorphism. For a character \\(\\chi\\in\\Omega\\), \\(\\chi\\circ\\theta\\) is a state, and \\(t:\\chi\\mapsto\\chi\\circ\\theta\\) is continuous from \\(\\Omega\\) to \\(\\mathfrak S\\). So \\(\\bar\\theta(g)=\\Gamma^{-1}(g\\circ t)\\) is a unital \\(*\\)-homomorphism \\(C(\\mathfrak S)\\to\\mathcal C\\), and \\(\\bar\\theta(\\hat a)=\\theta(a)\\). Let \\(\\mu\\in M_1^+(\\mathfrak S)\\) be the measure \\(g\\mapsto\\langle\\bar\\theta(g)\\xi,\\xi\\rangle\\). Then \\(\\int\\hat a\\,d\\mu=\\langle\\theta(a)\\xi,\\xi\\rangle=\\langle e\\pi(a)\\xi,\\xi\\rangle=\\varphi(a)\\), so \\(\\mu\\in M_\\varphi(\\mathfrak S)\\).\n\n*Step 3: \\(\\kappa_\\mu=\\bar\\theta\\) on \\(C(\\mathfrak S)\\).* For \\(g\\in C(\\mathfrak S)\\) and \\(a\\in A\\), using \\(\\bar\\theta(\\bar g)\\xi\\in eH\\),\n\\[\n\\begin{gathered}\n\\langle\\bar\\theta(g)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\langle\\pi(a)\\xi,e\\bar\\theta(\\bar g)\\xi\\rangle\\\\\n=\\langle\\theta(a)\\xi,\\bar\\theta(\\bar g)\\xi\\rangle\\\\\n=\\langle\\bar\\theta(g\\hat a)\\xi,\\xi\\rangle\\\\\n=\\int g\\hat a\\,d\\mu .\n\\end{gathered}\n\\]\nBy the uniqueness in Proposition 9.2, \\(\\kappa_\\mu(g)=\\bar\\theta(g)\\). So \\(\\kappa_\\mu\\) is multiplicative on \\(C(\\mathfrak S)\\). It is normal, \\(C(\\mathfrak S)\\) is weak\\(^*\\)-dense in \\(L^\\infty(\\mu)\\) ([Proposition 2.1](#oa-fnd-ir-02)(6)), and multiplication is separately continuous in both topologies; so \\(\\kappa_\\mu\\) is multiplicative on \\(L^\\infty\\). Thus \\(\\mu\\) is orthogonal, and \\(\\mathcal C_\\mu\\subseteq\\mathcal C\\), since \\(\\mathcal C\\) is weakly closed.\n\n*Step 4: \\(\\mathcal C_\\mu=\\mathcal C\\).* By (10.1), \\([\\mathcal C_\\mu\\xi]\\supseteq[\\kappa_\\mu(\\mathcal A_{\\mathbb C})\\xi]=[e\\pi(A)\\xi]=eH\\), so \\(e_\\mu=e\\). For \\(y\\in\\mathcal C\\), \\(y\\xi\\in eH=e_\\mu H\\), and the \\(L^2\\) criterion, Proposition 10.4(3), gives \\(y\\in\\mathcal C_\\mu\\).\n\n*Uniqueness.* Let \\(\\mu,\\nu\\) be orthogonal with \\(\\mathcal C_\\mu=\\mathcal C_\\nu=\\mathcal C\\). Then \\(e_\\mu=e_\\nu=e\\), and (10.1) gives \\(\\kappa_\\mu(\\hat a)\\xi=e\\pi(a)\\xi=\\kappa_\\nu(\\hat a)\\xi\\), so \\(\\kappa_\\mu(\\hat a)=\\kappa_\\nu(\\hat a)\\) (separating vector). The restrictions of \\(\\kappa_\\mu\\) and \\(\\kappa_\\nu\\) to \\(C(\\mathfrak S)\\) are \\(*\\)-homomorphisms that agree on \\(\\mathcal A_{\\mathbb C}\\). This set contains \\(1\\), is closed under conjugation (\\(\\overline{\\hat a}=\\widehat{a^*}\\)), and separates the points of \\(\\mathfrak S\\), so it generates a dense subalgebra, by [the complex Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09). So \\(\\kappa_\\mu=\\kappa_\\nu\\) on \\(C(\\mathfrak S)\\), and \\(\\mu(g)=\\langle\\kappa_\\mu(g)\\xi,\\xi\\rangle=\\nu(g)\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    },
    {
      "id": "OA-FND-IR-16",
      "unit": "integral-representations-of-states",
      "name": "14. Comparing orthogonal measures",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
      "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "anchor": "oa-fnd-ir-16",
      "proof_locus": {
        "line": 769,
        "through_line": 801
      },
      "full_conditions_and_proof": "## 14. Comparing orthogonal measures\n\nUnder the correspondence of Theorem 13.1, the Choquet order between orthogonal measures is inclusion between their abelian algebras.\n\n**Theorem 14.1.** Let \\(\\varphi\\in\\mathfrak S\\) and let \\(\\mu,\\nu\\in M_\\varphi(\\mathfrak S)\\) be orthogonal. The following are equivalent.\n\n1. \\(\\mu\\prec\\nu\\).\n2. \\(\\int\\omega(h)^2\\,d\\mu(\\omega)\\leq\\int\\omega(h)^2\\,d\\nu(\\omega)\\) for every \\(h\\in A_h\\).\n3. \\(\\mathcal C_\\mu\\subseteq\\mathcal C_\\nu\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): \\(\\hat h\\) is real and affine, so \\(\\hat h^2\\) is convex and continuous.\n\n(2)\\(\\Rightarrow\\)(3): For \\(a=h+ik\\) with \\(h,k\\in A_h\\), \\(|\\hat a|^2=\\hat h^2+\\hat k^2\\). By (10.1) and the isometry \\(V_\\mu\\),\n\\[\n\\begin{gathered}\n\\|e_\\mu\\pi(a)\\xi\\|^2\\\\\n=\\|\\kappa_\\mu(\\hat a)\\xi\\|^2\\\\\n=\\int|\\hat a|^2\\,d\\mu\\\\\n\\leq\\int|\\hat a|^2\\,d\\nu\\\\\n=\\|e_\\nu\\pi(a)\\xi\\|^2 .\n\\end{gathered}\n\\]\nSo \\(\\langle e_\\mu\\eta,\\eta\\rangle\\leq\\langle e_\\nu\\eta,\\eta\\rangle\\) on the dense set \\(\\pi(A)\\xi\\), hence \\(e_\\mu\\leq e_\\nu\\). For \\(x\\in\\mathcal C_\\mu\\subseteq\\pi(A)'\\), \\(x\\xi\\in e_\\mu H\\subseteq e_\\nu H\\), and the \\(L^2\\) criterion, Proposition 10.4(3), gives \\(x\\in\\mathcal C_\\nu\\).\n\n(3)\\(\\Rightarrow\\)(1): We check condition (2) of Lemma 4.10. Let \\(\\mu=\\sum_i\\mu_i\\) with \\(\\mu_i\\geq0\\). By [Proposition 2.1](#oa-fnd-ir-02)(5), \\(\\mu_i=f_i\\mu\\) with \\(0\\leq f_i\\leq1\\) and \\(\\sum_if_i=1\\). Since \\(\\kappa_\\mu(f_i)\\in\\mathcal C_\\nu\\) and \\(\\kappa_\\nu\\) is an isomorphism onto \\(\\mathcal C_\\nu\\), there are \\(g_i\\in L^\\infty(\\nu)\\) with \\(\\kappa_\\nu(g_i)=\\kappa_\\mu(f_i)\\); then \\(0\\leq g_i\\leq1\\) and \\(\\sum_ig_i=1\\). Put \\(\\nu_i=g_i\\nu\\). Then \\(\\sum_i\\nu_i=\\nu\\) and \\[\n\\begin{gathered}\n\\nu_i(\\hat a)\\\\\n=\\langle\\kappa_\\nu(g_i)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\langle\\kappa_\\mu(f_i)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\mu_i(\\hat a)\n\\end{gathered}\n\\]. Since \\(\\operatorname{Aff}(\\mathfrak S)=\\{\\hat h\\}\\) (Proposition 8.1(2)), \\(\\nu_i\\sim\\mu_i\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
      "dependencies": [
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    {
      "id": "OA-FND-IR-17",
      "unit": "integral-representations-of-states",
      "name": "15. Multiplicity-free states",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "anchor": "oa-fnd-ir-17",
      "proof_locus": {
        "line": 802,
        "through_line": 847
      },
      "full_conditions_and_proof": "## 15. Multiplicity-free states\n\n**Lemma 15.1** (Extreme positive contractions). In a C\\(^*\\)-algebra \\(\\mathcal B\\) of operators, the extreme points of \\(\\{x\\in\\mathcal B:\\ 0\\leq x\\leq1\\}\\) are exactly the projections in \\(\\mathcal B\\).\n\n**Proof.** Let \\(p\\) be a projection and \\(p=\\frac12(x+y)\\) with \\(0\\leq x,y\\leq1\\). Then \\((1-p)x(1-p)\\leq2(1-p)p(1-p)=0\\), so \\(x=pxp\\leq p\\), and likewise \\(y\\leq p\\). From \\(x+y=2p\\) we get \\(x=p\\). If \\(x\\) is not a projection, then \\(x-x^2\\neq0\\), and \\(x=\\frac12\\bigl((2x-x^2)+x^2\\bigr)\\), where \\(0\\leq x^2\\leq1\\) and \\(0\\leq2x-x^2=1-(1-x)^2\\leq1\\). \\(\\square\\)\n\n**Theorem 15.2.** For \\(\\varphi\\in\\mathfrak S\\) the following are equivalent.\n\n1. \\(\\pi_\\varphi(A)'\\) is abelian.\n2. \\(M_\\varphi(\\mathfrak S)\\) has exactly one maximal measure.\n\nIn that case the maximal measure is the orthogonal measure \\(\\mu\\) with \\(\\mathcal C_\\mu=\\pi_\\varphi(A)'\\), and \\(\\nu\\prec\\mu\\) for every \\(\\nu\\in M_\\varphi(\\mathfrak S)\\).\n\n**Proof.** (1)\\(\\Rightarrow\\)(2): By Theorem 13.1 there is an orthogonal \\(\\mu\\) with \\(\\mathcal C_\\mu=\\pi(A)'\\). Let \\(\\nu\\in M_\\varphi(\\mathfrak S)\\); we check condition (2) of Lemma 4.10 for the pair \\(\\nu,\\mu\\). If \\(\\nu=\\sum_i\\nu_i\\) with \\(\\nu_i\\geq0\\), then \\(\\nu_i=g_i\\nu\\) with \\(0\\leq g_i\\leq1\\) and \\(\\sum_ig_i=1\\). The operators \\(\\kappa_\\nu(g_i)\\) are positive, lie in \\(\\pi(A)'=\\mathcal C_\\mu\\), and add up to \\(1\\). Since \\(\\kappa_\\mu\\) is a \\(*\\)-isomorphism onto \\(\\mathcal C_\\mu\\), \\(\\kappa_\\nu(g_i)=\\kappa_\\mu(f_i)\\) with \\(f_i\\geq0\\) and \\(\\sum_if_i=1\\). With \\(\\mu_i=f_i\\mu\\), \\(\\mu_i(\\hat a)=\\langle\\kappa_\\mu(f_i)\\pi(a)\\xi,\\xi\\rangle=\\nu_i(\\hat a)\\), so \\(\\mu_i\\sim\\nu_i\\). Lemma 4.10 gives \\(\\nu\\prec\\mu\\). So \\(\\mu\\) majorizes all of \\(M_\\varphi(\\mathfrak S)\\). It is maximal: if \\(\\mu\\prec\\lambda\\), then \\(\\lambda\\in M_\\varphi(\\mathfrak S)\\) (Proposition 4.5(2)), so \\(\\lambda\\prec\\mu\\) and \\(\\lambda=\\mu\\). Any maximal \\(\\nu\\in M_\\varphi(\\mathfrak S)\\) satisfies \\(\\nu\\prec\\mu\\), hence \\(\\nu=\\mu\\).\n\n(2)\\(\\Rightarrow\\)(1): Let \\(\\mu\\) be the unique maximal measure in \\(M_\\varphi(\\mathfrak S)\\). Every \\(\\lambda\\in M_\\varphi(\\mathfrak S)\\) is majorized by a maximal measure (Lemma 4.6), which is \\(\\mu\\).\n\n*Step 1: \\(\\kappa_\\mu\\) is one-to-one.* If \\(\\mu=\\frac12(\\mu_1+\\mu_2)\\) in \\(M_\\varphi(\\mathfrak S)\\), then \\(\\mu_1,\\mu_2\\prec\\mu\\), so \\(\\mu(f)\\geq\\mu_i(f)\\) for \\(f\\in\\mathcal P(\\mathfrak S)\\), while \\(\\mu(f)\\) is their average. So \\(\\mu_1=\\mu_2=\\mu\\) on \\(\\mathcal P(\\mathfrak S)\\), and hence everywhere (Lemma 4.1). Thus \\(\\mu\\) is simplicial, and \\(\\mathcal A_{\\mathbb C}\\) is dense in \\(L^1(\\mu)\\) (Proposition 11.3). If \\(\\kappa_\\mu(f)=0\\), then \\(\\int f\\hat a\\,d\\mu=0\\) for all \\(a\\), so \\(f=0\\).\n\n*Step 2: \\(\\kappa_\\mu\\) maps \\(\\{0\\leq f\\leq1\\}\\) onto \\(\\{h\\in\\pi(A)':\\ 0\\leq h\\leq1\\}\\).* Positivity gives \"into\". Let \\(0\\leq h\\leq1\\) in \\(\\pi(A)'\\). The bicommutant \\(\\mathcal C=\\{h\\}''\\) is abelian and lies in \\(\\pi(A)'\\). Let \\(\\nu\\) be the orthogonal measure with \\(\\mathcal C_\\nu=\\mathcal C\\) (Theorem 13.1). For \\(g\\in C(\\mathfrak S)\\) put \\(\\nu_1(g)=\\langle\\kappa_\\nu(g)h\\xi,\\xi\\rangle\\) and \\(\\nu_2(g)=\\langle\\kappa_\\nu(g)(1-h)\\xi,\\xi\\rangle\\). These are positive measures, because \\(\\kappa_\\nu(g)\\geq0\\) commutes with \\(h\\) and \\(1-h\\) when \\(g\\geq0\\); and \\(\\nu=\\nu_1+\\nu_2\\). Since \\(\\nu\\prec\\mu\\), Lemma 4.10 gives \\(\\mu=\\mu_1+\\mu_2\\) with \\(\\mu_i\\sim\\nu_i\\), and \\(\\mu_1=g\\mu\\) with \\(0\\leq g\\leq1\\). For \\(a\\in A\\), using \\(h\\xi\\in[\\mathcal C\\xi]=e_\\nu H\\) and (10.1),\n\\[\n\\begin{gathered}\n\\langle\\kappa_\\mu(g)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\mu_1(\\hat a)\\\\\n=\\nu_1(\\hat a)\\\\\n=\\langle\\kappa_\\nu(\\hat a)e_\\nu h\\xi,\\xi\\rangle\\\\\n=\\langle e_\\nu\\pi(a)e_\\nu h\\xi,\\xi\\rangle\\\\\n=\\langle h\\pi(a)\\xi,\\xi\\rangle .\n\\end{gathered}\n\\]\nSo \\(\\kappa_\\mu(g)=h\\).\n\n*Step 3.* By Steps 1–2, \\(\\kappa_\\mu\\) is an affine bijection of \\(\\{0\\leq f\\leq1\\}\\) onto \\(\\{0\\leq h\\leq1\\}\\subseteq\\pi(A)'\\). The indicator functions \\(\\chi_E\\) are extreme in the first set (if \\(\\chi_E=\\frac12(f_1+f_2)\\) with \\(0\\leq f_i\\leq1\\), then \\(f_i=\\chi_E\\) almost everywhere). So each \\(\\kappa_\\mu(\\chi_E)\\) is extreme in the second set, hence a projection by Lemma 15.1. The proof of (3)\\(\\Rightarrow\\)(1) in Theorem 10.2 used only this, so \\(\\kappa_\\mu\\) is multiplicative. Its range contains every positive contraction of \\(\\pi(A)'\\), and these span \\(\\pi(A)'\\); so \\(\\pi(A)'=\\kappa_\\mu(L^\\infty)\\) is abelian. \\(\\square\\)\n\n**Definition 15.3.** A representation \\(\\pi\\) of \\(A\\) is *multiplicity-free* if \\(\\pi(A)'\\) is abelian.\n\nSo a state is multiplicity-free (its GNS representation is) exactly when its maximal representing measure is unique. That measure is orthogonal and vanishes on Baire sets that miss \\(P(A)\\) (Proposition 8.1(5)).\n\nIf every state is multiplicity-free, the Choquet–Meyer theorem says that \\(\\mathfrak S\\) is a simplex. This happens only for abelian algebras.\n\n**Theorem 15.4.** The state space \\(\\mathfrak S(A)\\) is a simplex exactly when \\(A\\) is abelian.\n\n**Proof.** If \\(A\\) is abelian, \\(A\\cong C(X)\\) by the commutative Gelfand–Naimark theorem, and by Proposition 8.1(2) \\(\\operatorname{Aff}(\\mathfrak S)\\cong A_h=C_{\\mathbb R}(X)\\) as ordered spaces. So \\(\\operatorname{Aff}(\\mathfrak S)^*\\cong C_{\\mathbb R}(X)^*=M(X)\\) with its usual order, which is a vector lattice.\n\nConversely let \\(\\mathfrak S\\) be a simplex. By the Choquet–Meyer theorem (Theorem 7.4), every state has exactly one maximal representing measure, so \\(\\pi_\\varphi(A)'\\) is abelian for every state \\(\\varphi\\) (Theorem 15.2). Suppose \\(A\\) is not abelian. Then some pure state \\(\\omega\\) has \\(\\dim H_\\omega\\geq2\\): otherwise every pure state is multiplicative, so every pure state kills every commutator \\([a,b]\\); by Krein–Milman every state does, and then \\([a,b]=0\\), because the states separate the points of \\(A\\) (Proposition 8.1(2)). Let \\(\\pi=\\pi_\\omega\\) on \\(H\\), irreducible, and let \\(\\eta_1,\\eta_2\\in H\\) be orthonormal. The state \\(\\varphi(a)=\\frac12(\\langle\\pi(a)\\eta_1,\\eta_1\\rangle+\\langle\\pi(a)\\eta_2,\\eta_2\\rangle)\\) is given by \\(\\pi\\oplus\\pi\\) and \\(\\zeta=(\\eta_1,\\eta_2)/\\sqrt2\\). The commutant of \\(\\pi\\oplus\\pi\\) consists of the \\(2\\times2\\) matrices with entries in \\(\\pi(A)'=\\mathbb C1\\), that is, \\(M_2(\\mathbb C)\\otimes1\\). The projection \\(p\\otimes1\\) onto \\([(\\pi\\oplus\\pi)(A)\\zeta]\\) lies there and fixes \\(\\zeta\\), so \\(p_{11}\\eta_1+p_{12}\\eta_2=\\eta_1\\) and \\(p_{21}\\eta_1+p_{22}\\eta_2=\\eta_2\\), which forces \\(p=1\\). So \\(\\zeta\\) is cyclic, \\(\\pi_\\varphi\\cong\\pi\\oplus\\pi\\), and \\(\\pi_\\varphi(A)'\\cong M_2(\\mathbb C)\\) is not abelian. This is a contradiction. \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-IR-24",
      "unit": "integral-representations-of-states",
      "name": "16. The face generated by a state",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "proof_locus": {
        "line": 848,
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      "full_conditions_and_proof": "## 16. The face generated by a state\n\nThe smallest face of \\(\\mathfrak S\\) that contains a state \\(\\varphi\\) is described by the commutant \\(\\pi_\\varphi(A)'\\), through the Radon–Nikodym map.\n\n**Proposition 16.1.** Let \\(\\varphi\\in\\mathfrak S\\) with GNS triple \\(\\pi,H,\\xi\\), and let \\(F_\\varphi\\) be the smallest face of \\(\\mathfrak S\\) that contains \\(\\varphi\\).\n\n1. \\[\n\\begin{gathered}\nF_\\varphi\\\\\n=\\{\\psi\\in\\mathfrak S:\\\\\n\\ t\\psi\\\\\n\\leq\\varphi\\text{ for some }t>0\\}\\\\\n=\\{\\Theta_\\varphi(x):\\\\\n\\ x\\in\\pi(A)'_+,\\ \\\\\n\\langle x\\xi,\\xi\\rangle\\\\\n=1\\}\n\\end{gathered}\n\\]. If \\(\\psi\\in\\mathfrak S\\) has the form \\(\\Theta_\\varphi(x)\\) for some \\(x\\in\\pi(A)'\\), then automatically \\(x\\geq0\\).\n2. \\(F_\\varphi\\) is closed in \\(\\mathfrak S\\) \\(\\iff\\) \\(\\pi(A)'\\) is finite-dimensional \\(\\iff\\) \\(F_\\varphi\\) is finite-dimensional.\n3. The cone \\(\\mathbb R_+F_\\varphi\\) is a lattice in its own order if and only if \\(\\pi(A)'\\) is abelian.\n4. If the closure \\(\\overline{F_\\varphi}\\) is a simplex, then \\(\\pi(A)'\\) is abelian.\n5. There are \\(A\\) and \\(\\varphi\\) with \\(\\pi_\\varphi(A)'\\) abelian and \\(\\overline{F_\\varphi}\\) not a simplex.\n\nParts (3)–(5) distinguish the lattice property of the generated face from the simplex property of its closure. The proof includes an explicit counterexample to the converse of (4).\n\n**Proof.** (1) Call the first set \\(F\\). It is convex and contains \\(\\varphi\\). It is a face: if \\(\\psi=s\\psi_1+(1-s)\\psi_2\\in F\\) with \\(t\\psi\\leq\\varphi\\), then \\(ts\\psi_1\\leq\\varphi\\). It lies in every face \\(F'\\ni\\varphi\\): if \\(t\\psi\\leq\\varphi\\) with \\(0<t<1\\), then \\(\\varphi=t\\psi+(1-t)\\psi'\\) with the state \\(\\psi'=(\\varphi-t\\psi)/(1-t)\\), so \\(\\psi\\in F'\\); and \\(t=1\\) forces \\(\\psi=\\varphi\\). The second description follows from Proposition 8.1(3): \\(t\\psi\\leq\\varphi\\) means \\(t\\psi=\\Theta_\\varphi(y)\\) with \\(0\\leq y\\leq1\\). If \\(\\Theta_\\varphi(x)=\\psi\\) is a state, then \\(\\Theta_\\varphi(x^*)=\\psi^*=\\psi\\), so \\(x=x^*\\), and \\(x\\geq0\\) because \\(\\Theta_\\varphi\\) reflects order.\n\n(2) The linear span of \\(F_\\varphi\\) is \\(\\Theta_\\varphi(\\pi(A)')\\): every \\(x\\) is a combination of positive elements, and a positive \\(x\\neq0\\) has \\(\\langle x\\xi,\\xi\\rangle>0\\) (separating vector). As \\(\\Theta_\\varphi\\) is one-to-one, \\(\\dim F_\\varphi<\\infty\\) exactly when \\(\\dim\\pi(A)'<\\infty\\). If \\(\\dim\\pi(A)'<\\infty\\), the set \\(\\{x\\geq0:\\ \\langle x\\xi,\\xi\\rangle=1\\}\\) is compact: \\(x\\mapsto\\langle x\\xi,\\xi\\rangle\\) is positive on the compact set \\(\\{x\\geq0,\\ \\|x\\|=1\\}\\), hence at least some \\(c>0\\) there, so the set is bounded. Its continuous image \\(F_\\varphi\\) is compact, hence closed. If \\(\\pi(A)'\\) is infinite-dimensional, it has an infinite sequence of nonzero orthogonal projections \\(p_n\\) (Lemma 16.2 below). Put \\(c_n=\\langle p_n\\xi,\\xi\\rangle>0\\); \\(c_n\\to0\\) because \\(\\sum c_n\\leq1\\). Pass to a subsequence with \\(c_{n_k}\\leq4^{-k}\\), and put \\(\\psi=\\sum_k2^{-k}\\Theta_\\varphi(p_{n_k})/c_{n_k}\\), a norm-convergent series of states. \\(\\psi\\) is a norm limit of normalized partial sums, which lie in \\(F_\\varphi\\); so \\(\\psi\\in\\overline{F_\\varphi}\\). If \\(\\psi\\) were in \\(F_\\varphi\\), then \\(\\psi=\\Theta_\\varphi(x)\\) with \\(x\\) bounded, and \\(\\psi\\geq2^{-k}c_{n_k}^{-1}\\Theta_\\varphi(p_{n_k})\\) would give \\(x\\geq2^kp_{n_k}\\), so \\(\\|x\\|\\geq2^k\\) for all \\(k\\). So \\(F_\\varphi\\) is not closed.\n\n(3) The real span of \\(F_\\varphi\\) is \\(\\Theta_\\varphi(\\pi(A)'_h)\\), and \\(\\Theta_\\varphi\\) maps \\(\\pi(A)'_+\\) onto \\(\\mathbb R_+F_\\varphi\\); so it is an order isomorphism. If \\(\\pi(A)'\\) is abelian, \\(\\pi(A)'_h\\cong C_{\\mathbb R}(\\Omega)\\) is a lattice. Conversely, suppose \\(\\mathcal N=\\pi(A)'\\) is not abelian. A von Neumann algebra is generated by its projections (spectral theorem), so some projection \\(p\\in\\mathcal N\\) is not central, and then \\(p\\mathcal N(1-p)\\neq0\\). Take \\(x\\in p\\mathcal N(1-p)\\) with \\(0<\\|x\\|\\leq1\\). Then \\(x^2=0\\), \\((x+x^*)^2=xx^*+x^*x\\), so \\(\\|x+x^*\\|=\\|x\\|\\leq1\\), and \\(u=\\frac12(1+x+x^*)\\) satisfies \\(0\\leq u\\leq1=p+(1-p)\\). If \\(\\mathcal N_h\\) were a lattice, the Riesz decomposition (Lemma 7.3) would give \\(u=u_1+u_2\\) with \\(0\\leq u_1\\leq p\\) and \\(0\\leq u_2\\leq1-p\\). Then \\(u_1=pu_1p\\) and \\(u_2=(1-p)u_2(1-p)\\) (as in the proof of Lemma 10.1(1)), so \\(pu(1-p)=0\\). But \\(pu(1-p)=\\frac12x\\neq0\\).\n\n(4) Suppose \\(\\pi(A)'\\) is not abelian. By Theorem 15.2, \\(M_\\varphi(\\mathfrak S)\\) does not have exactly one maximal measure; it has at least one (Lemma 4.6), so it has two, \\(\\mu_1\\neq\\mu_2\\). Every \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) lives on \\(L=\\overline{F_\\varphi}\\): for \\(\\omega\\in\\operatorname{supp}\\mu\\) and a closed convex neighbourhood \\(W\\) of \\(\\omega\\), \\(C=W\\cap\\mathfrak S\\) has \\(\\mu(C)>0\\), and the barycentre \\(\\psi_C\\) of \\(\\mu|_C/\\mu(C)\\) satisfies \\(\\mu(C)\\psi_C\\leq\\varphi\\), so \\(\\psi_C\\in F_\\varphi\\), and \\(\\psi_C\\in C\\) by Proposition 3.3(1). As \\(W\\) shrinks, \\(\\psi_C\\to\\omega\\). Both \\(\\mu_i\\) are maximal also among measures on \\(L\\): if \\(\\mu_i\\prec\\lambda\\) on \\(L\\), then \\(\\mu_i(f)\\leq\\lambda(f)\\) for every \\(f\\in\\mathcal P(\\mathfrak S)\\) (restrict \\(f\\) to \\(L\\)), so \\(\\lambda=\\mu_i\\). By the Choquet–Meyer theorem, \\(L\\) is not a simplex.\n\n(5) *Counterexample.* Let \\(A=C([0,1],M_2(\\mathbb C))\\), and let \\(v_0=e_1\\), \\(v_1=(e_1+e_2)/\\sqrt2\\), \\(v_2=e_2\\), \\(v_3=(e_1-e_2)/\\sqrt2\\), unit vectors whose Bloch vectors are the vertices of a square. Let \\(J_k=[\\frac12-2^{-k},\\frac12-2^{-k-1})\\) for \\(k\\geq1\\); these intervals cover \\([0,\\frac12)\\). Put \\(\\xi(x)=v_{k\\bmod4}\\) for \\(x\\in J_k\\), \\(\\xi(x)=v_0\\) for \\(x\\geq\\frac12\\), and\n\\[\n\\varphi(a)=\\int_0^1\\langle a(x)\\xi(x),\\xi(x)\\rangle\\,dx .\n\\]\n*GNS.* On \\(\\mathcal H=L^2([0,1];\\mathbb C^2)\\) let \\(\\pi(a)\\) be multiplication by \\(a(\\cdot)\\); then \\(\\varphi(a)=\\langle\\pi(a)\\xi,\\xi\\rangle\\). The vector \\(\\xi\\) is cyclic: if \\(\\eta\\perp\\pi(f\\otimes E_{ij})\\xi\\) for all \\(f\\in C[0,1]\\), then \\(\\xi_j\\overline{\\eta_i}=0\\) almost everywhere for all \\(i,j\\), and since \\(\\xi(x)\\neq0\\), \\(\\eta=0\\). *Commutant.* An operator \\(T\\in\\pi(A)'\\) commutes with \\(1\\otimes E_{ij}\\), so \\(T=t\\otimes1\\) with \\(t\\in B(L^2[0,1])\\). It commutes with \\(\\pi(f\\otimes1)=M_f\\otimes1\\) for \\(f\\in C[0,1]\\); by [Proposition 2.1](#oa-fnd-ir-02)(6) and the normality of \\(g\\mapsto M_g\\) it commutes with every \\(M_g\\), \\(g\\in L^\\infty\\), so \\(t=M_g\\) for some \\(g\\in L^\\infty\\), because the multiplication operators by \\(L^\\infty\\) form a [maximal abelian](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09) algebra on \\(L^2[0,1]\\). So \\(\\pi_\\varphi(A)'=\\{M_g\\otimes1\\}\\) is abelian. *The face.* By (1), \\(F_\\varphi\\) consists of the states \\(a\\mapsto\\int g(x)\\langle a(x)\\xi(x),\\xi(x)\\rangle dx\\) with \\(g\\geq0\\) bounded and \\(\\int g=1\\). Taking \\(g=\\chi_{J_k}/|J_k|\\) with \\(k\\equiv j\\pmod4\\) and letting \\(k\\to\\infty\\) gives, in the closure, the four states \\(\\rho_j(a)=\\langle a(\\frac12)v_j,v_j\\rangle\\). They are pure (vector states of the irreducible representation \\(a\\mapsto a(\\frac12)\\)), hence extreme in \\(\\overline{F_\\varphi}\\). Since \\(|v_0\\rangle\\langle v_0|+|v_2\\rangle\\langle v_2|=1=|v_1\\rangle\\langle v_1|+|v_3\\rangle\\langle v_3|\\), the point \\(\\rho=\\frac12(\\rho_0+\\rho_2)=\\frac12(\\rho_1+\\rho_3)\\) of \\(\\overline{F_\\varphi}\\) is the barycentre of two different measures concentrated on extreme points. Both are boundary measures (Proposition 5.5(2)), hence maximal (Theorem 5.4). By the Choquet–Meyer theorem, \\(\\overline{F_\\varphi}\\) is not a simplex. \\(\\square\\)\n\n**Lemma 16.2.** An infinite-dimensional von Neumann algebra \\(\\mathcal N\\) contains an infinite sequence of nonzero, pairwise orthogonal projections.\n\n**Proof.** Suppose not. Then every nonzero projection dominates a minimal one, since a strictly decreasing sequence \\(p>p_1>p_2>\\dots\\) would give the orthogonal nonzero differences \\(p_k-p_{k+1}\\). Let \\(\\mathcal A\\) be a maximal abelian \\(*\\)-subalgebra; it equals \\(\\mathcal A'\\cap\\mathcal N\\), a von Neumann algebra. A maximal orthogonal family of minimal projections of \\(\\mathcal A\\) is finite, say \\(e_1,\\dots,e_n\\), and \\(\\sum e_i=1\\). Each \\(e_i\\mathcal A=\\mathbb Ce_i\\), by the spectral theorem. A self-adjoint \\(y\\in e_i\\mathcal Ne_i\\) commutes with every \\(e_j\\), hence with \\(\\mathcal A\\), so \\(y\\in\\mathcal A\\cap e_i\\mathcal Ne_i=\\mathbb Ce_i\\). So \\(e_i\\mathcal Ne_i=\\mathbb Ce_i\\). For \\(v,w\\in e_i\\mathcal Ne_j\\), \\(w^*v\\in e_j\\mathcal Ne_j=\\mathbb Ce_j\\) and \\(vv^*\\in\\mathbb Ce_i\\); if \\(w\\neq0\\), scale it so that \\(ww^*=e_i\\), and then \\(v=ww^*v\\in\\mathbb Cw\\). So \\(\\dim e_i\\mathcal Ne_j\\leq1\\), and \\(\\dim\\mathcal N\\leq n^2\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-IR-18",
      "unit": "integral-representations-of-states",
      "name": "17. Invariant states",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 17. Invariant states\n\nLet \\(G\\) be a group, with no topology, and \\(s\\mapsto\\alpha_s\\) a homomorphism of \\(G\\) into the automorphism group of \\(A\\). The *invariant states* are \\(\\mathfrak S^G=\\{\\omega\\in\\mathfrak S:\\ \\omega\\circ\\alpha_s=\\omega\\text{ for all }s\\}\\), and the *ergodic states* are the extreme points \\(\\partial_e\\mathfrak S^G\\). The results of this section are used for the ergodic decomposition in Section 19, and in Sections 23 and 24.\n\n**Proposition 17.1.** Let \\(\\varphi\\in\\mathfrak S^G\\), with GNS triple \\(\\pi,H,\\xi\\).\n\n1. \\(\\mathfrak S^G\\) is compact and convex.\n2. There is exactly one unitary representation \\(U=U_\\varphi\\) of \\(G\\) on \\(H\\) with \\(U_s\\pi(a)\\xi=\\pi(\\alpha_s(a))\\xi\\). It satisfies \\(U_s\\xi=\\xi\\) and \\(U_s\\pi(a)U_s^*=\\pi(\\alpha_s(a))\\).\n3. Let \\(\\mathfrak M_\\varphi=\\pi(A)'\\cap U(G)'\\). Then \\(\\Theta_\\varphi\\) maps \\(\\{x\\in\\mathfrak M_\\varphi:\\ 0\\leq x\\leq1\\}\\) onto the set of invariant functionals \\(\\psi\\) with \\(0\\leq\\psi\\leq\\varphi\\).\n4. \\(\\varphi\\in\\partial_e\\mathfrak S^G\\) if and only if \\(\\mathfrak M_\\varphi=\\mathbb C1\\).\n5. If \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) and \\(\\kappa_\\mu(L^\\infty(\\mu))\\subseteq U(G)'\\), then \\(\\operatorname{supp}\\mu\\subseteq\\mathfrak S^G\\), so \\(\\mu(\\mathfrak S^G)=1\\).\n6. Suppose \\(\\mathfrak M_\\varphi\\) is abelian, and let \\(\\mu\\) be the orthogonal measure with \\(\\mathcal C_\\mu=\\mathfrak M_\\varphi\\) (Theorem 13.1). Then \\(\\mu\\) is concentrated on \\(K=\\mathfrak S^G\\), and, as a measure on \\(K\\), it majorizes every probability measure on \\(K\\) with barycentre \\(\\varphi\\). So it is the unique maximal measure on \\(K\\) that represents \\(\\varphi\\), and \\(\\mu(B)=0\\) for every Baire subset \\(B\\) of the compact space \\(K\\) that misses \\(\\partial_eK\\).\n\n**Proof.** (1) \\(\\mathfrak S^G\\) is the intersection of the closed convex sets \\(\\{\\omega:\\ \\omega(\\alpha_s(a))=\\omega(a)\\}\\).\n\n(2) \\(\\|\\pi(\\alpha_s(a))\\xi\\|^2=\\varphi(\\alpha_s(a^*a))=\\|\\pi(a)\\xi\\|^2\\), so \\(U_s\\) is a well-defined isometry on \\(\\pi(A)\\xi\\) with dense range; it extends to a unitary, and \\(U_sU_t=U_{st}\\). Then \\[\n\\begin{gathered}\nU_s\\pi(a)U_s^*\\pi(b)\\xi\\\\\n=U_s\\pi(a\\,\\alpha_{s^{-1}}(b))\\xi\\\\\n=\\pi(\\alpha_s(a))\\pi(b)\\xi\n\\end{gathered}\n\\], and \\(U_s\\xi=\\pi(\\alpha_s(1))\\xi=\\xi\\).\n\n(3) For \\(x\\in\\pi(A)'\\), \\(U_s^*xU_s\\in\\pi(A)'\\), and since \\(U_s\\xi=\\xi\\),\n\\[\n\\begin{gathered}\n\\Theta_\\varphi(x)(\\alpha_s(a))\\\\\n=\\langle U_s\\pi(a)U_s^*x\\xi,\\xi\\rangle\\\\\n=\\langle\\pi(a)\\,U_s^*xU_s\\,\\xi,\\xi\\rangle\\\\\n=\\Theta_\\varphi(U_s^*xU_s)(a).\n\\end{gathered}\n\\]\nSo \\(\\Theta_\\varphi(x)\\) is invariant exactly when \\(U_s^*xU_s=x\\) for all \\(s\\), because \\(\\Theta_\\varphi\\) is one-to-one. Combine with Proposition 8.1(3).\n\n(4) The proof of Proposition 8.1(4), (a)\\(\\Leftrightarrow\\)(c), works word for word with \\(\\mathfrak S^G\\) in place of \\(\\mathfrak S\\) and \\(\\mathfrak M_\\varphi\\) in place of \\(\\pi(A)'\\), using (3).\n\n(5) For \\(f\\in L^\\infty(\\mu)\\), \\(s\\in G\\) and \\(a\\in A\\), since \\(\\kappa_\\mu(f)\\) commutes with \\(U_s\\) and \\(U_s^*\\xi=\\xi\\),\n\\[\n\\begin{gathered}\n\\int f\\,\\widehat{\\alpha_s(a)}\\,d\\mu\\\\\n=\\langle\\kappa_\\mu(f)U_s\\pi(a)U_s^*\\xi,\\xi\\rangle\\\\\n=\\langle U_s\\kappa_\\mu(f)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\int f\\hat a\\,d\\mu .\n\\end{gathered}\n\\]\nSo \\(\\widehat{\\alpha_s(a)}=\\hat a\\) almost everywhere. The closed set \\(\\{\\omega:\\ \\omega(\\alpha_s(a))=\\omega(a)\\}\\) therefore has a null complement and contains \\(\\operatorname{supp}\\mu\\) ([Proposition 2.1](#oa-fnd-ir-02)(3)). Intersect over \\(s\\) and \\(a\\).\n\n(6) By (5), \\(\\mu\\) is concentrated on the closed convex set \\(K\\), and its restriction is a probability measure on \\(K\\) with barycentre \\(\\varphi\\). Let \\(\\lambda\\in M_1^+(K)\\) have barycentre \\(\\varphi\\), and \\(\\lambda=\\sum_i\\lambda_i\\) with \\(\\lambda_i\\geq0\\). Then \\(\\lambda_i=g_i\\lambda\\) with \\(0\\leq g_i\\leq1\\) and \\(\\sum g_i=1\\). The functional \\(\\psi_i(a)=\\lambda_i(\\hat a)\\) is positive, satisfies \\(\\psi_i\\leq\\varphi\\), and is invariant, because every point of \\(K\\) is. By (3), \\(\\psi_i=\\Theta_\\varphi(x_i)\\) with \\(x_i\\in\\mathfrak M_\\varphi\\), \\(x_i\\geq0\\), and \\(\\sum_ix_i=1\\) since \\(\\sum_i\\psi_i=\\varphi\\). As \\(\\mathfrak M_\\varphi=\\mathcal C_\\mu\\), \\(x_i=\\kappa_\\mu(f_i)\\) with \\(f_i\\geq0\\), \\(\\sum f_i=1\\). Put \\(\\mu_i=f_i\\mu\\). Then \\(\\mu_i(\\hat a)=\\psi_i(a)=\\lambda_i(\\hat a)\\). The functions \\(\\hat h|_K\\), \\(h\\in A_h\\), form \\(\\operatorname{Aff}_E(K)\\), which is dense in \\(\\operatorname{Aff}(K)\\) (Lemma 1.2); so \\(\\mu_i\\sim\\lambda_i\\) on \\(K\\). By Lemma 4.10 on \\(K\\), \\(\\lambda\\prec\\mu\\). If \\(\\mu\\prec\\lambda'\\) on \\(K\\), then \\(\\lambda'\\) represents \\(\\varphi\\), so \\(\\lambda'\\prec\\mu\\) and \\(\\lambda'=\\mu\\): \\(\\mu\\) is maximal, and every maximal \\(\\lambda\\) equals \\(\\mu\\). The last claim is Theorems 5.4 and 6.6 on \\(K\\). \\(\\square\\)\n\nFor \\(G=\\{1\\}\\), part (6) is the implication (1)\\(\\Rightarrow\\)(2) of Theorem 15.2.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-IR-19",
      "unit": "integral-representations-of-states",
      "name": "18. The algebra \\(C(X,A)\\)",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "anchor": "oa-fnd-ir-19",
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        "line": 942,
        "through_line": 983
      },
      "full_conditions_and_proof": "## 18. The algebra \\(C(X,A)\\)\n\nThe decomposition theorem of Section 19 is proved by passing to the algebra \\(C(X,A)\\) of continuous \\(A\\)-valued functions, which plays the role of a tensor product of \\(C(X)\\) and \\(A\\). The facts we need about it are short.\n\n**Proposition 18.1.** Let \\(X\\) be compact Hausdorff and \\(\\mathcal D=C(X,A)\\), the continuous functions \\(X\\to A\\) with pointwise operations and the sup norm. For \\(f\\in C(X)\\) and \\(a\\in A\\) let \\(f\\otimes a\\) be the function \\(x\\mapsto f(x)a\\).\n\n1. \\(\\mathcal D\\) is a unital C\\(^*\\)-algebra, and the span \\(\\mathcal F\\) of the \\(f\\otimes a\\) is dense in it.\n2. Let \\(\\theta:C(X)\\to B(H)\\) be a unital \\(*\\)-homomorphism and \\(\\pi\\) a unital representation of \\(A\\) on \\(H\\) with \\(\\theta(C(X))\\subseteq\\pi(A)'\\). There is exactly one representation \\(\\tilde\\pi\\) of \\(\\mathcal D\\) with \\(\\tilde\\pi(f\\otimes a)=\\theta(f)\\pi(a)\\), and \\(\\tilde\\pi(\\mathcal D)'=\\pi(A)'\\cap\\theta(C(X))'\\).\n3. Let \\(G\\) act on \\(A\\) by \\(\\alpha\\), and on \\(\\mathcal D\\) by \\(\\beta_s(F)=\\alpha_s\\circ F\\); \\(\\beta\\) fixes every \\(f\\otimes1\\). Every \\(\\rho\\in\\partial_e\\mathfrak S^G(\\mathcal D)\\) has the form \\(\\rho(F)=\\omega(F(x))\\) for some \\(x\\in X\\) and some \\(\\omega\\in\\partial_e\\mathfrak S^G(A)\\). For \\(G=\\{1\\}\\): every pure state of \\(\\mathcal D\\) is \\(F\\mapsto\\omega(F(x))\\) with \\(\\omega\\) pure.\n4. Take \\(X=\\mathfrak S(A)\\). Then \\(\\Phi(\\omega)(F)=\\omega(F(\\omega))\\) defines a continuous one-to-one map \\(\\Phi:\\mathfrak S(A)\\to\\mathfrak S(\\mathcal D)\\) with \\(\\Phi(\\mathfrak S^G(A))\\subseteq\\mathfrak S^G(\\mathcal D)\\). And \\(\\Psi(\\rho)(a)=\\rho(1\\otimes a)\\) defines a continuous affine map \\(\\Psi:\\mathfrak S(\\mathcal D)\\to\\mathfrak S(A)\\) with \\(\\Psi\\circ\\Phi=\\mathrm{id}\\), \\(\\Psi(\\mathfrak S^G(\\mathcal D))\\subseteq\\mathfrak S^G(A)\\) and \\(\\Psi(\\partial_e\\mathfrak S^G(\\mathcal D))\\subseteq\\partial_e\\mathfrak S^G(A)\\).\n\n**Proof.** (1) \\(\\|F^*F\\|=\\sup_x\\|F(x)\\|^2\\), and completeness follows from uniform convergence: a sup-norm Cauchy sequence has a pointwise limit in the Banach space \\(A\\), the Cauchy estimate passes to the limit uniformly, and a uniform limit of continuous maps is continuous. Given \\(F\\) and \\(\\varepsilon>0\\), cover \\(X\\) by finitely many open sets \\(U_k\\ni x_k\\) with \\(\\|F(y)-F(x_k)\\|<\\varepsilon\\) on \\(U_k\\), and take a partition of unity \\((u_k)\\) subordinate to this cover (constructed explicitly in the background below from [Urysohn functions](stone-weierstrass-c0.md#oa-fnd-sw-16)). Then \\(\\|F-\\sum_ku_k\\otimes F(x_k)\\|\\leq\\varepsilon\\).\n\n(2) Let \\(F=\\sum_{i=1}^mf_i\\otimes a_i\\) and \\(\\delta>0\\). Choose a partition of unity \\((u_k)_{k\\leq n}\\) and points \\(x_k\\) with \\(|f_i(y)-f_i(x_k)|<\\delta\\) for \\(y\\in\\operatorname{supp}u_k\\) and all \\(i\\). Then\n\\[\n\\begin{gathered}\n\\sum_i\\theta(f_i)\\pi(a_i)\\\\\n=\\sum_k\\theta(u_k)\\pi(F(x_k))\\\\\n+\\sum_i\\theta(g_i)\\pi(a_i),\\\\\ng_i\\\\\n=\\sum_ku_k\\,(f_i-f_i(x_k)),\n\\end{gathered}\n\\]\nand \\(\\|g_i\\|\\leq\\delta\\). The operator \\[\n\\begin{gathered}\nT\\\\\n=\\sum_k\\theta(u_k)\\pi(F(x_k))\\\\\n=\\sum_k\\theta(u_k)^{1/2}\\pi(F(x_k))\\theta(u_k)^{1/2}\n\\end{gathered}\n\\] equals \\(R^*DR\\), where \\(R\\eta=(\\theta(u_k)^{1/2}\\eta)_k\\) maps \\(H\\) into \\(H^n\\) with \\(\\|R\\|\\leq1\\) (because \\(\\sum_k\\theta(u_k)=1\\)), and \\(D=\\operatorname{diag}(\\pi(F(x_k)))\\). So \\(\\|T\\|\\leq\\max_k\\|F(x_k)\\|\\leq\\|F\\|\\), and \\(\\|\\sum_i\\theta(f_i)\\pi(a_i)\\|\\leq\\|F\\|+\\delta\\sum_i\\|a_i\\|\\). Letting \\(\\delta\\to0\\) shows that \\(\\tilde\\pi\\) is well defined and contractive on \\(\\mathcal F\\). It is a \\(*\\)-homomorphism there, because \\(\\theta(f)\\) commutes with \\(\\pi(b)\\); it extends to \\(\\mathcal D\\) by (1). The image is the closed span of \\(\\theta(C(X))\\pi(A)\\), which contains \\(\\pi(A)\\) and \\(\\theta(C(X))\\); this gives the commutant.\n\n(3) Each \\(z=f\\otimes1\\) with \\(0\\leq f\\leq1\\) is central and \\(\\beta\\)-fixed. So \\(F\\mapsto\\rho(zF)\\) and \\(F\\mapsto\\rho((1-z)F)\\) are positive (\\(\\rho(zF^*F)=\\rho(z^{1/2}F^*Fz^{1/2})\\)), invariant, and add up to \\(\\rho\\). If \\(t=\\rho(z)\\in(0,1)\\), extremality gives \\(\\rho(z\\,\\cdot)=t\\rho\\). If \\(t=0\\), the Cauchy–Schwarz inequality gives \\(|\\rho(zF)|^2\\leq\\rho(z^2)\\rho(F^*F)\\leq\\rho(z)\\|F\\|^2=0\\); if \\(t=1\\), apply this to \\(1-z\\). So \\(\\rho(zF)=\\rho(z)\\rho(F)\\) for all such \\(z\\), hence for all \\(z=f\\otimes1\\). Then \\(\\rho\\) is multiplicative on \\(C(X)\\otimes1\\), so it is evaluation at some \\(x\\in X\\) there, and \\(\\rho(f\\otimes a)=f(x)\\rho(1\\otimes a)\\). With \\(\\omega=\\rho(1\\otimes\\cdot)\\), \\(\\rho(F)=\\omega(F(x))\\) on \\(\\mathcal F\\), hence on \\(\\mathcal D\\). \\(\\omega\\) is invariant. If \\(\\omega=t\\omega_1+(1-t)\\omega_2\\) with \\(\\omega_i\\in\\mathfrak S^G(A)\\) and \\(0<t<1\\), then \\(\\rho_i(F)=\\omega_i(F(x))\\) are invariant states with \\(\\rho=t\\rho_1+(1-t)\\rho_2\\); so \\(\\rho_i=\\rho\\) and \\(\\omega_i=\\omega\\).\n\n(4) \\(F\\mapsto F(\\omega)\\) is a unital \\(*\\)-homomorphism, so \\(\\Phi(\\omega)\\) is a state. If \\(\\omega_j\\to\\omega\\), then \\[\n\\begin{gathered}\n|\\omega_j(F(\\omega_j))-\\omega(F(\\omega))|\\\\\n\\leq\\|F(\\omega_j)-F(\\omega)\\|\\\\\n+|\\omega_j(F(\\omega))-\\omega(F(\\omega))|\\\\\n\\to0\n\\end{gathered}\n\\]. \\(\\Psi\\circ\\Phi=\\mathrm{id}\\) is clear, so \\(\\Phi\\) is one-to-one. For invariant \\(\\omega\\), \\(\\Phi(\\omega)(\\beta_sF)=\\omega(\\alpha_s(F(\\omega)))=\\Phi(\\omega)(F)\\). \\(\\Psi\\) is continuous and affine, and it maps invariant states to invariant states because \\(\\beta_s(1\\otimes a)=1\\otimes\\alpha_s(a)\\). The last claim is (3). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-IR-20",
      "unit": "integral-representations-of-states",
      "name": "19. Decomposition into pure and ergodic states",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 19. Decomposition into pure and ergodic states\n\nThe next theorem decomposes an invariant state along a maximal abelian subalgebra of \\(\\mathfrak M_\\varphi\\). For the trivial group it decomposes an arbitrary state into pure states.\n\n**Theorem 19.1.** Let \\(G\\) act on \\(A\\), \\(\\varphi\\in\\mathfrak S^G\\), and let \\(\\mathcal C\\) be a maximal abelian von Neumann subalgebra of \\(\\mathfrak M_\\varphi=\\pi_\\varphi(A)'\\cap U_\\varphi(G)'\\), that is, \\(\\mathcal C'\\cap\\mathfrak M_\\varphi=\\mathcal C\\). Let \\(\\mu\\) be the orthogonal measure with \\(\\mathcal C_\\mu=\\mathcal C\\) (Theorem 13.1). Then:\n\n1. \\(\\mu(\\mathfrak S^G)=1\\);\n2. \\(\\mu(B)=0\\) for every Baire subset \\(B\\) of the compact space \\(\\mathfrak S^G\\) with \\(B\\cap\\partial_e\\mathfrak S^G=\\varnothing\\);\n3. if \\(A\\) is separable, \\(\\mu\\) is concentrated on \\(\\partial_e\\mathfrak S^G\\);\n4. such algebras \\(\\mathcal C\\) exist. So every invariant state is the barycentre of an orthogonal measure that vanishes on each Baire subset of \\(\\mathfrak S^G\\) missing the ergodic states.\n\nFor \\(G=\\{1\\}\\), \\(\\mathfrak S^G=\\mathfrak S\\) and \\(\\partial_e\\mathfrak S^G=P(A)\\): if \\(\\mathcal C_\\mu\\) is maximal abelian in \\(\\pi_\\varphi(A)'\\), then \\(\\mu\\) vanishes on every Baire set that misses the pure states, and it is concentrated on \\(P(A)\\) when \\(A\\) is separable.\n\n**Proof.** (1) is Proposition 17.1(5), since \\(\\mathcal C\\subseteq U(G)'\\). Write \\(\\pi,H,\\xi,U\\) for \\(\\varphi\\), \\(K_A=\\mathfrak S^G(A)\\), and \\(\\mathcal D=C(\\mathfrak S(A),A)\\) with the action \\(\\beta\\) of Proposition 18.1(3).\n\n*The auxiliary state.* \\(\\kappa_\\mu\\) restricted to \\(C(\\mathfrak S)\\) is a unital \\(*\\)-homomorphism into \\(\\mathcal C\\subseteq\\pi(A)'\\). Proposition 18.1(2) gives a representation \\(\\tilde\\pi\\) of \\(\\mathcal D\\) with \\(\\tilde\\pi(f\\otimes a)=\\kappa_\\mu(f)\\pi(a)\\). Since \\(\\kappa_\\mu(f)\\in U(G)'\\), we get \\(\\tilde\\pi(\\beta_sF)=U_s\\tilde\\pi(F)U_s^*\\) on \\(\\mathcal F\\), hence on \\(\\mathcal D\\). So \\(\\tilde\\varphi(F)=\\langle\\tilde\\pi(F)\\xi,\\xi\\rangle\\) is a \\(\\beta\\)-invariant state. Its GNS triple is \\((\\tilde\\pi,H,\\xi)\\), since \\(\\xi\\) is cyclic already for \\(\\pi(A)\\), and its unitary representation is \\(U\\). The algebra \\(\\kappa_\\mu(C(\\mathfrak S))\\) is \\(\\sigma\\)-weakly dense in \\(\\mathcal C\\) ([Proposition 2.1](#oa-fnd-ir-02)(6) and the normality of \\(\\kappa_\\mu\\)), so the two have the same commutant. By Proposition 18.1(2),\n\\[\n\\begin{gathered}\n\\tilde\\pi(\\mathcal D)'\\cap U(G)'\\\\\n=\\pi(A)'\\cap\\mathcal C'\\cap U(G)'\\\\\n=\\mathcal C'\\cap\\mathfrak M_\\varphi\\\\\n=\\mathcal C,\n\\end{gathered}\n\\]\nwhich is abelian. By Proposition 17.1(6) for \\((\\mathcal D,\\beta,\\tilde\\varphi)\\), the orthogonal measure \\(\\tilde\\mu\\) on \\(\\mathfrak S(\\mathcal D)\\) with \\(\\mathcal C_{\\tilde\\mu}=\\mathcal C\\) is concentrated on \\(K_{\\mathcal D}=\\mathfrak S^G(\\mathcal D)\\), and \\(\\tilde\\mu(B')=0\\) for every Baire subset \\(B'\\) of \\(K_{\\mathcal D}\\) that misses \\(\\partial_eK_{\\mathcal D}\\).\n\n*Identification: \\(\\tilde\\mu=\\Phi_*\\mu\\).* Let \\(\\nu=\\Phi_*\\mu\\), with \\(\\Phi\\) from Proposition 18.1(4); so \\(\\nu(N)=\\mu(\\Phi^{-1}(N))\\) for Borel \\(N\\) ([Proposition 2.1](#oa-fnd-ir-02)(7)). For a bounded Borel function \\(G_0\\) on \\(\\mathfrak S(\\mathcal D)\\), \\(f\\in C(\\mathfrak S)\\) and \\(a\\in A\\), with \\(g=G_0\\circ\\Phi\\),\n\\[\n\\begin{gathered}\n\\int G_0\\,\\widehat{f\\otimes a}\\,d\\nu\\\\\n=\\int g(\\omega)f(\\omega)\\omega(a)\\,d\\mu(\\omega)\\\\\n=\\langle\\kappa_\\mu(gf)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\langle\\kappa_\\mu(g)\\tilde\\pi(f\\otimes a)\\xi,\\xi\\rangle .\n\\end{gathered}\n\\]\nBoth ends are continuous in \\(F\\in\\mathcal D\\), so \\(\\int G_0\\hat F\\,d\\nu=\\langle\\kappa_\\mu(G_0\\circ\\Phi)\\tilde\\pi(F)\\xi,\\xi\\rangle\\) for all \\(F\\). With \\(G_0=1\\), \\(\\nu\\) represents \\(\\tilde\\varphi\\). By the uniqueness in Proposition 9.2, \\(\\kappa_\\nu=\\kappa_\\mu\\circ\\Phi^\\#\\), where \\(\\Phi^\\#(G_0)=G_0\\circ\\Phi\\) is a \\(*\\)-isomorphism of \\(L^\\infty(\\nu)\\) onto \\(L^\\infty(\\mu)\\) (Proposition 2.1(7)). So \\(\\nu\\) is orthogonal with \\(\\mathcal C_\\nu=\\mathcal C=\\mathcal C_{\\tilde\\mu}\\), and \\(\\nu=\\tilde\\mu\\) by Theorem 13.1.\n\n*Conclusion.* Let \\(B\\subseteq K_A\\) be a Baire subset of \\(K_A\\) with \\(B\\cap\\partial_eK_A=\\varnothing\\). The map \\(\\Psi\\) of Proposition 18.1(4) sends \\(K_{\\mathcal D}\\) into \\(K_A\\), so \\(B'=\\Psi^{-1}(B)\\cap K_{\\mathcal D}\\) is a Baire subset of \\(K_{\\mathcal D}\\) (Proposition 2.1(4)(b)), and it misses \\(\\partial_eK_{\\mathcal D}\\) because \\(\\Psi(\\partial_eK_{\\mathcal D})\\subseteq\\partial_eK_A\\). Since \\(\\Psi\\circ\\Phi=\\mathrm{id}\\) and \\(\\nu=\\tilde\\mu\\) is concentrated on \\(K_{\\mathcal D}\\),\n\\[\n\\begin{gathered}\n\\mu(B)\\\\\n=\\mu\\bigl(\\Phi^{-1}(\\Psi^{-1}(B))\\bigr)\\\\\n=\\nu(\\Psi^{-1}(B))\\\\\n=\\tilde\\mu(B')\\\\\n=0 .\n\\end{gathered}\n\\]\n\n(3) If \\(A\\) is separable, \\(\\mathfrak S\\) and \\(K_A\\) are metrizable (Proposition 8.1(5)). By Lemmas 6.2 and 6.3, \\(\\partial_eK_A\\) is a \\(G_\\delta\\), and every Borel subset of \\(K_A\\) is a Baire set (Proposition 2.1(4)(d)). Apply (2) to \\(K_A\\setminus\\partial_eK_A\\).\n\n(4) By Zorn's lemma, \\(\\mathfrak M_\\varphi\\) has a maximal commutative \\(*\\)-subalgebra \\(\\mathcal C\\) containing \\(1\\). If \\(x\\in\\mathcal C'\\cap\\mathfrak M_\\varphi\\), its real and imaginary parts lie there too, and each generates with \\(\\mathcal C\\) a commutative \\(*\\)-algebra; so they lie in \\(\\mathcal C\\). Hence \\(\\mathcal C=\\mathcal C'\\cap\\mathfrak M_\\varphi\\), which is weakly closed. \\(\\square\\)\n\n**Remark 19.2** (Non-uniqueness). Different maximal abelian subalgebras give different measures, and they need not be conjugate. For example, let \\(H=\\ell^2\\), \\(A=\\mathbb C1+K(H)\\) (separable and unital), \\((e_n)\\) the standard basis, and \\(\\varphi(a)=\\sum_n2^{-n}\\langle ae_n,e_n\\rangle\\). On \\(H\\otimes\\ell^2\\) the vector \\(\\zeta=\\sum_n2^{-n/2}e_n\\otimes e_n\\) is cyclic for \\(\\{a\\otimes1\\}\\), and gives \\(\\varphi\\); a matrix-unit computation with the operators \\(e_i\\otimes e_j^*\\in K(H)\\) shows that the commutant is \\(1\\otimes B(\\ell^2)\\). In \\(B(\\ell^2)\\) the diagonal algebra \\(\\ell^\\infty\\) is maximal abelian and has minimal projections; the multiplication algebra \\(L^\\infty[0,1]\\) on \\(L^2[0,1]\\cong\\ell^2\\) is [maximal abelian](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09) and has none. So the two are not unitarily conjugate. Their orthogonal measures (Theorem 13.1) are different, and both are concentrated on \\(P(A)\\) by Theorem 19.1(3), since \\(A\\) is separable. The centre \\(\\mathcal Z_\\varphi=\\pi_\\varphi(A)''\\cap\\pi_\\varphi(A)'\\), on the other hand, is determined by \\(\\varphi\\) alone. This leads to the central measure.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-IR-21",
      "unit": "integral-representations-of-states",
      "name": "20. The central measure and factorial states",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 20. The central measure and factorial states\n\n**Definition 20.1.** The *central measure* of \\(\\varphi\\in\\mathfrak S\\) is the orthogonal \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) with \\(\\mathcal C_\\mu=\\mathcal Z_\\varphi=\\pi_\\varphi(A)''\\cap\\pi_\\varphi(A)'\\); it exists and is unique by Theorem 13.1. A state \\(\\varphi\\) is *factorial* (or primary) if \\(\\pi_\\varphi(A)''\\) is a factor. We write \\(\\operatorname{Fac}(A)\\) for the set of factorial states. Pure states are factorial, since \\(\\pi_\\varphi(A)'=\\mathbb C1\\) (Proposition 8.1(4)).\n\n**Lemma 20.2** (Corners of a factor). Let \\(\\mathcal N\\) be a factor on \\(H\\) and \\(e\\in\\mathcal N\\) a nonzero projection. Then \\(\\mathcal N_e=\\{exe|_{eH}:\\ x\\in\\mathcal N\\}\\) is a factor on \\(eH\\).\n\n**Proof.** \\(\\mathcal N_e\\) is a von Neumann algebra (Lemma 12.2(3)). Let \\(z\\) be a projection in its centre, say \\(z=eye|_{eH}\\) with \\(y\\in\\mathcal N\\), and let \\(\\tilde z=eye\\), a projection in \\(\\mathcal N\\) with \\(\\tilde z\\leq e\\). For \\(x\\in\\mathcal N\\), \\(\\tilde z\\,exe=exe\\,\\tilde z\\), since \\(z\\) is central in \\(\\mathcal N_e\\). So \\(\\tilde zx(e-\\tilde z)=\\tilde z(exe)(e-\\tilde z)=(exe)\\tilde z(e-\\tilde z)=0\\). The projection \\(f\\) onto \\([\\mathcal N(e-\\tilde z)H]\\) commutes with \\(\\mathcal N\\), and also with \\(\\mathcal N'\\), because \\(\\mathcal N'\\mathcal N(e-\\tilde z)H=\\mathcal N(e-\\tilde z)\\mathcal N'H\\). So \\(f\\in\\mathcal N\\cap\\mathcal N'=\\mathbb C1\\). If \\(e\\neq\\tilde z\\), then \\(f=1\\), and \\(\\tilde z\\) vanishes on \\([\\mathcal N(e-\\tilde z)H]=H\\), so \\(\\tilde z=0\\). Thus \\(z\\in\\{0,1\\}\\). An abelian von Neumann algebra whose only projections are \\(0\\) and \\(1\\) is \\(\\mathbb C1\\), because the spectral projections of a self-adjoint element lie in it (the spectral theorem). \\(\\square\\)\n\n**Lemma 20.3.** Let \\(C\\) be a C\\(^*\\)-algebra with unit, and let \\(A_1,A_2\\subseteq C\\) be C\\(^*\\)-subalgebras that contain the unit of \\(C\\), commute with each other, and together generate \\(C\\). If \\(\\rho\\) is a factorial state of \\(C\\), then \\(\\rho|_{A_1}\\) is a factorial state of \\(A_1\\).\n\n**Proof.** Let \\(\\pi=\\pi_\\rho\\) on \\(H\\), with cyclic vector \\(\\xi\\), and \\(\\mathcal M=\\pi(C)''\\), a factor. Let \\(\\mathcal Z_1=\\pi(A_1)''\\cap\\pi(A_1)'\\). Since \\(\\pi(A_2)\\subseteq\\pi(A_1)'\\), we have \\(\\pi(A_1)''\\subseteq\\pi(A_2)'\\). So \\(\\mathcal Z_1\\) commutes with \\(\\pi(A_1)\\) and \\(\\pi(A_2)\\), hence with \\(\\pi(C)\\), and \\(\\mathcal Z_1\\subseteq\\mathcal M'\\cap\\mathcal M=\\mathbb C1\\). Thus \\(\\pi(A_1)''\\) is a factor, and so is \\(\\mathcal N=\\pi(A_1)'\\), which has the same centre. Let \\(e\\in\\mathcal N\\) be the projection onto \\([\\pi(A_1)\\xi]\\). By the uniqueness of the GNS construction, the GNS representation of \\(\\psi=\\rho|_{A_1}\\) is \\(a\\mapsto\\pi(a)|_{eH}\\). Restriction to \\(eH\\) is weakly continuous, so \\(\\{\\pi(a)|_{eH}:\\ a\\in A_1\\}\\) and \\((\\pi(A_1)'')_e\\) have the same commutant. By Lemma 12.2(1), applied to the von Neumann algebra \\(\\pi(A_1)''\\), whose commutant \\(\\mathcal N\\) contains \\(e\\), this commutant is \\(\\mathcal N_e\\). By Lemma 20.2, \\(\\mathcal N_e\\) is a factor. So \\(\\pi_\\psi(A_1)'\\), and with it \\(\\pi_\\psi(A_1)''\\), is a factor. \\(\\square\\)\n\n**Theorem 20.4.** The central measure \\(\\mu\\) of \\(\\varphi\\in\\mathfrak S\\) satisfies \\(\\mu(B)=0\\) for every Baire set \\(B\\subseteq\\mathfrak S\\) with \\(B\\cap\\operatorname{Fac}(A)=\\varnothing\\).\n\n\n**Proof.** Write \\(\\pi,H,\\xi\\) for \\(\\varphi\\) and \\(\\mathcal Z=\\mathcal Z_\\varphi\\). Let \\(A_1\\) be the norm closure of \\(\\pi(A)\\), \\(A_2=\\pi(A)'\\), and \\(C\\subseteq B(H)\\) the C\\(^*\\)-algebra they generate. Then \\(C'=\\pi(A)'\\cap\\pi(A)''=\\mathcal Z\\). The vector state \\(\\psi(x)=\\langle x\\xi,\\xi\\rangle\\) on \\(C\\) has GNS triple \\((\\mathrm{id},H,\\xi)\\), because \\(\\xi\\) is cyclic for \\(A_1\\). Its commutant \\(\\mathcal Z\\) is abelian, so by Theorem 15.2 the orthogonal measure \\(\\nu\\) on \\(\\mathfrak S(C)\\) with \\(\\mathcal C_\\nu=\\mathcal Z\\) is the only maximal measure representing \\(\\psi\\). Being maximal, it is a boundary measure (Theorem 5.4), so by Theorem 6.6 it vanishes on every Baire set that misses \\(P(C)\\).\n\nLet \\(\\Phi:\\mathfrak S(C)\\to\\mathfrak S(A)\\), \\(\\Phi(\\rho)=\\rho\\circ\\pi\\), a continuous affine map. If \\(\\rho\\in P(C)\\), then \\(\\rho|_{A_1}\\) is factorial by Lemma 20.3, and so is \\(\\rho\\circ\\pi\\): the GNS representation of \\(\\rho\\circ\\pi\\) is that of \\(\\rho|_{A_1}\\) composed with \\(\\pi\\), and \\(\\pi(A)\\) is norm-dense in \\(A_1\\), so the two bicommutants agree. Thus \\(\\Phi(P(C))\\subseteq\\operatorname{Fac}(A)\\).\n\nWe show \\(\\Phi_*\\nu=\\mu\\). First, \\(\\int\\hat a\\,d\\Phi_*\\nu=\\int\\widehat{\\pi(a)}\\,d\\nu=\\psi(\\pi(a))=\\varphi(a)\\). For \\(g\\in L^\\infty(\\Phi_*\\nu)\\), the class of \\(g\\circ\\Phi\\) in \\(L^\\infty(\\nu)\\) is well defined, since \\(\\Phi_*\\nu(N)=\\nu(\\Phi^{-1}(N))\\) for Borel \\(N\\) ([Proposition 2.1](#oa-fnd-ir-02)(7); here \\(\\Phi\\) need not be one-to-one). The operator \\(\\kappa_\\nu(g\\circ\\Phi)\\) lies in \\(\\mathcal Z\\subseteq\\pi(A)'\\), and\n\\[\n\\begin{gathered}\n\\langle\\kappa_\\nu(g\\circ\\Phi)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\int(g\\circ\\Phi)\\,\\widehat{\\pi(a)}\\,d\\nu\\\\\n=\\int g\\hat a\\,d\\Phi_*\\nu .\n\\end{gathered}\n\\]\nBy the uniqueness in Proposition 9.2, \\(\\kappa_{\\Phi_*\\nu}(g)=\\kappa_\\nu(g\\circ\\Phi)\\). This is multiplicative, so \\(\\Phi_*\\nu\\) is orthogonal. It also shows \\(\\mathcal C_{\\Phi_*\\nu}\\subseteq\\mathcal C_\\nu=\\mathcal Z\\), but not equality, since \\(g\\mapsto g\\circ\\Phi\\) need not map onto \\(L^\\infty(\\nu)\\); so we compare with \\(\\mu\\) directly. By (10.1) for \\(\\nu\\), \\(\\kappa_{\\Phi_*\\nu}(\\hat a)\\xi=\\kappa_\\nu(\\widehat{\\pi(a)})\\xi=e\\pi(a)\\xi\\), where \\(e\\) is the projection onto \\([\\mathcal Z\\xi]\\). By (10.1) for \\(\\mu\\), \\(\\kappa_\\mu(\\hat a)\\xi=e\\pi(a)\\xi\\) as well, since \\(\\mathcal C_\\mu=\\mathcal Z\\). The separating vector \\(\\xi\\) gives \\(\\kappa_{\\Phi_*\\nu}(\\hat a)=\\kappa_\\mu(\\hat a)\\), and the uniqueness part of the proof of Theorem 13.1 gives \\(\\Phi_*\\nu=\\mu\\).\n\nNow let \\(B\\) be a Baire set missing \\(\\operatorname{Fac}(A)\\). Then \\(\\Phi^{-1}(B)\\) is a Baire set (Proposition 2.1(4)(b)) that misses \\(P(C)\\), because \\(\\Phi(P(C))\\subseteq\\operatorname{Fac}(A)\\). So \\(\\mu(B)=\\nu(\\Phi^{-1}(B))=0\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-IR-22",
      "unit": "integral-representations-of-states",
      "name": "21. The factorial states form a Borel set",
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      "full_conditions_and_proof": "## 21. The factorial states form a Borel set\n\nWhen \\(A\\) is separable, the factorial states form a Borel subset of \\(\\mathfrak S\\), so the central measure is concentrated on them. The proof tests factoriality through countably many closed conditions.\n\nLet \\(\\omega\\in\\mathfrak S\\) be a state with GNS triple \\(\\pi,H,\\xi\\). Put\n\n- \\[\n\\begin{gathered}\nI(\\omega)\\\\\n=\\{\\rho\\in A^*:\\\\\n\\ 0\\\\\n\\leq\\rho\\\\\n\\leq\\omega\\}\\\\\n=\\Theta_\\omega(\\{x\\in\\pi(A)':\\\\\n\\ 0\\\\\n\\leq x\\\\\n\\leq1\\})\n\\end{gathered}\n\\] (Proposition 8.1(3));\n- \\(Z(\\omega)=I(\\omega)\\cap\\Theta_\\omega(\\mathcal Z_\\omega)\\).\n\nSince \\(\\Theta_\\omega\\) is one-to-one and preserves order both ways, \\(Z(\\omega)=\\Theta_\\omega(\\{z\\in\\mathcal Z_\\omega:\\ 0\\leq z\\leq1\\})\\). So \\(\\omega\\) is factorial exactly when \\(Z(\\omega)=\\{\\lambda\\omega:\\ 0\\leq\\lambda\\leq1\\}\\).\n\nLet \\(A\\odot A\\) be the algebraic tensor product with the projective (greatest cross) norm \\(\\gamma(u)=\\inf\\{\\sum_i\\|x_i\\|\\|y_i\\|:\\ u=\\sum_ix_i\\otimes y_i\\}\\), let \\(\\operatorname{mul}(\\sum_ix_i\\otimes y_i)=\\sum_ix_iy_i\\) be the multiplication map, and \\(N=\\ker\\operatorname{mul}\\). For \\(T\\in B(H)\\) put\n\\[\n\\Phi_{\\omega,T}\\Bigl(\\sum_ix_i\\otimes y_i\\Bigr)=\\sum_i\\langle\\pi(x_i)T\\pi(y_i)\\xi,\\xi\\rangle ,\n\\]\nand \\(\\Phi_{\\omega,h}=\\Phi_{\\omega,\\pi(h)}\\) for \\(h\\in A\\), so \\(\\Phi_{\\omega,h}(u)=\\omega(\\sum_ix_ihy_i)\\). Also \\(\\Phi_\\rho=\\rho\\circ\\operatorname{mul}\\) for \\(\\rho\\in A^*\\).\n\n**Lemma 21.1.**\n\n1. \\(|\\Phi_{\\omega,T}(u)|\\leq\\|T\\|\\gamma(u)\\).\n2. \\(\\Phi_{\\omega,T}(N)=\\{0\\}\\) if and only if \\(T\\in\\pi(A)'\\).\n3. A linear functional \\(\\Phi\\) on \\(A\\odot A\\) equals \\(\\Phi_\\rho\\) for some \\(\\rho\\in I(\\omega)\\) if and only if \\(\\Phi(N)=\\{0\\}\\) and \\(0\\leq\\Phi(x^*\\otimes x)\\leq\\omega(x^*x)\\) for all \\(x\\in A\\).\n\n**Proof.** (1) \\(|\\langle\\pi(x)T\\pi(y)\\xi,\\xi\\rangle|\\leq\\|x\\|\\|T\\|\\|y\\|\\). (2) If \\(T\\in\\pi(A)'\\), then \\(\\Phi_{\\omega,T}(u)=\\langle T\\pi(\\operatorname{mul}(u))\\xi,\\xi\\rangle\\). Conversely, \\(x\\otimes y-xy\\otimes1\\in N\\) gives \\(\\langle T\\pi(y)\\xi,\\pi(x^*)\\xi\\rangle=\\langle\\pi(y)T\\xi,\\pi(x^*)\\xi\\rangle\\) for all \\(x\\), so \\(T\\pi(y)\\xi=\\pi(y)T\\xi\\). Then \\(T\\pi(y)\\pi(z)\\xi=\\pi(yz)T\\xi=\\pi(y)T\\pi(z)\\xi\\), so \\(T\\pi(y)=\\pi(y)T\\). (3) \\(\\operatorname{mul}\\) is onto (\\(x=\\operatorname{mul}(x\\otimes1)\\)), so \\(\\Phi(N)=0\\) means \\(\\Phi=\\rho\\circ\\operatorname{mul}\\) with \\(\\rho(x)=\\Phi(x\\otimes1)\\) linear. The inequalities say \\(0\\leq\\rho(x^*x)\\leq\\omega(x^*x)\\). A positive functional on a C\\(^*\\)-algebra is bounded, so \\(\\rho\\in I(\\omega)\\). The converse is clear. \\(\\square\\)\n\n**Lemma 21.2.** Let \\(N_0\\) be a \\(\\gamma\\)-dense subset of \\(N\\), and \\(A_0\\) a dense \\(*\\)-subalgebra of \\(A\\) over \\(\\mathbb Q+i\\mathbb Q\\). A state \\(\\omega\\) is factorial exactly when the following holds. For every \\(\\varepsilon>0\\) and \\(x\\in A_0\\) there are \\(\\delta>0\\) and \\(u_1,\\dots,u_n\\in N_0\\) such that, for every \\(h\\in A_0\\) with \\(0\\leq h\\leq1\\),\n\\[\n\\begin{gathered}\n\\max_i|\\Phi_{\\omega,h}(u_i)|<\\delta\\\\\n\\Longrightarrow\\\\\n|\\omega(x^*hx)-\\omega(h)\\omega(x^*x)|<\\varepsilon .\n\\end{gathered}\n\\]\n\n**Proof.** *If the condition fails, \\(\\omega\\) is not factorial.* Then there are \\(\\varepsilon>0\\) and \\(x\\in A_0\\) such that for every pair \\(\\lambda=(\\delta,F)\\), with \\(\\delta>0\\) and \\(F\\subseteq N_0\\) finite, some \\(h_\\lambda\\in A_0\\) with \\(0\\leq h_\\lambda\\leq1\\) has \\(|\\Phi_{\\omega,h_\\lambda}(u)|<\\delta\\) for \\(u\\in F\\) and \\(|\\omega(x^*h_\\lambda x)-\\omega(h_\\lambda)\\omega(x^*x)|\\geq\\varepsilon\\). Direct the pairs by \\((\\delta,F)\\leq(\\delta',F')\\) if \\(\\delta'\\leq\\delta\\) and \\(F\\subseteq F'\\). The operators \\(\\pi(h_\\lambda)\\) lie in the weakly compact set \\(\\{T\\in\\pi(A)'':\\ 0\\leq T\\leq1\\}\\), so a subnet converges weakly to some \\(T\\) there. Each \\(T\\mapsto\\Phi_{\\omega,T}(u)\\) is weakly continuous, so \\(\\Phi_{\\omega,T}(u)=0\\) for \\(u\\in N_0\\), hence for \\(u\\in N\\) by Lemma 21.1(1). By Lemma 21.1(2), \\(T\\in\\pi(A)'\\), so \\(T\\in\\mathcal Z_\\omega\\). But \\(|\\langle T\\pi(x)\\xi,\\pi(x)\\xi\\rangle-\\langle T\\xi,\\xi\\rangle\\|\\pi(x)\\xi\\|^2|\\geq\\varepsilon\\), so \\(T\\) is not a scalar, and \\(\\mathcal Z_\\omega\\neq\\mathbb C1\\).\n\n*If the condition holds, \\(\\omega\\) is factorial.* Let \\(T\\in\\mathcal Z_\\omega\\) with \\(0\\leq T\\leq1\\). By the bicommutant theorem and Kaplansky's density theorem, \\(2T-1\\) is the strong limit of a net of self-adjoint contractions \\(\\pi(b_j)\\), \\(b_j\\in A_h\\). Replace \\(b_j\\) by \\(c_j=g(b_j)\\) with \\(g(t)=\\max(-1,\\min(t,1))\\): then \\(\\|c_j\\|\\leq1\\) and \\(\\pi(c_j)=g(\\pi(b_j))=\\pi(b_j)\\). So \\(h_j=\\frac12(1+c_j)\\) satisfies \\(0\\leq h_j\\leq1\\) and \\(\\pi(h_j)\\to T\\) strongly. Positive contractions of \\(A_0\\) are norm-dense among those of \\(A\\). Indeed, let \\(0\\leq h\\leq1\\), \\(0<\\eta<1/4\\), and let \\(k\\in A_0\\) be self-adjoint with \\(\\|k-h^{1/2}\\|<\\eta\\) (self-adjoint elements of \\(A_0\\) are dense in \\(A_h\\)). Then \\(\\|k^2-h\\|\\leq2\\eta+\\eta^2\\). Choose a rational \\(r\\) with \\((1+\\eta)^{-2}-\\eta\\leq r\\leq(1+\\eta)^{-2}\\). Then \\(rk^2\\in A_0\\), \\(0\\leq rk^2\\leq1\\), and \\(\\|rk^2-h\\|\\leq(2\\eta+\\eta^2)+(1-r)\\leq5\\eta+\\eta^2\\). Replacing \\(h_j\\) by such an element of \\(A_0\\) within norm distance \\(1/m\\), and indexing the new net by the pairs \\((j,m)\\), ordered componentwise, we may take \\(h_j\\in A_0\\). For \\(u=\\sum_ix_i\\otimes y_i\\in N\\), \\[\n\\begin{gathered}\n\\Phi_{\\omega,h_j}(u)\\\\\n=\\sum_i\\langle\\pi(h_j)\\pi(y_i)\\xi,\\pi(x_i)^*\\xi\\rangle\\to\\Phi_{\\omega,T}(u)\\\\\n=0\n\\end{gathered}\n\\]. Given \\(\\varepsilon\\) and \\(x\\in A_0\\), take \\(\\delta\\) and \\(u_1,\\dots,u_n\\) from the condition. For large \\(j\\), \\(|\\omega(x^*h_jx)-\\omega(h_j)\\omega(x^*x)|<\\varepsilon\\), and in the limit \\(|\\langle T\\pi(x)\\xi,\\pi(x)\\xi\\rangle-\\langle T\\xi,\\xi\\rangle\\|\\pi(x)\\xi\\|^2|\\leq\\varepsilon\\). So the quadratic form of \\(S=T-\\langle T\\xi,\\xi\\rangle1\\) vanishes on \\(\\pi(A_0)\\xi\\). This set is dense and closed under combinations with coefficients in \\(\\mathbb Q+i\\mathbb Q\\), which is enough for polarization; so \\(S=0\\). The positive contractions span \\(\\mathcal Z_\\omega\\), so \\(\\mathcal Z_\\omega=\\mathbb C1\\). \\(\\square\\)\n\n**Theorem 21.3.** If \\(A\\) is separable, \\(\\operatorname{Fac}(A)\\) is a Borel subset of \\(\\mathfrak S\\); in fact an \\(F_{\\sigma\\delta}\\) set.\n\nThe quantifier order in (21.1) is essential: Remark 21.4 shows that placing the intersection over \\(h\\) before the unions over \\(m\\) and \\(u\\) can admit states that are not factorial.\n\n**Proof.** Let \\(A_0\\) be a countable dense \\(*\\)-subalgebra over \\(\\mathbb Q+i\\mathbb Q\\) (generate one from a countable dense set). Finite sums of tensors \\(a\\otimes b\\), with \\(a,b\\in A_0\\), form a countable \\(\\gamma\\)-dense set in \\(A\\odot A\\): approximate both factors in every finite representation and use \\(\\gamma(x\\otimes y-a\\otimes b)\\leq\\|x-a\\|\\|y\\|+\\|a\\|\\|y-b\\|\\). Its subset \\(N\\) is therefore separable by the countable-base argument in Lemma 6.3, and has a countable \\(\\gamma\\)-dense subset \\(N_0\\). For \\(m,n\\geq1\\), \\(h\\in A_0\\) with \\(0\\leq h\\leq1\\), \\(u=(u_1,\\dots,u_r)\\in N_0^r\\) and \\(x\\in A_0\\), let\n\\[\n\\begin{gathered}\n\\mathfrak S(m;n;h;u;x)\\\\\n=\\Bigl\\{\\omega\\in\\mathfrak S:\\\\\n\\ \\max_{1\\leq i\\leq r}|\\Phi_{\\omega,h}(u_i)|\\\\\n\\geq\\tfrac1m\\ \\\\\n\\text{ or }\\ |\\omega(x^*hx)-\\omega(h)\\omega(x^*x)|\\\\\n\\leq\\tfrac1n\\Bigr\\}.\n\\end{gathered}\n\\]\nBoth conditions are closed in \\(\\omega\\), because \\(\\Phi_{\\omega,h}(u_i)=\\omega(c_i)\\) for a fixed \\(c_i\\in A\\) (if \\(u_i=\\sum_jx_j\\otimes y_j\\), then \\(c_i=\\sum_jx_jhy_j\\)), so the set is closed. With \\(\\varepsilon=1/n\\) and \\(\\delta=1/m\\), Lemma 21.2 says exactly\n\\[\n\\begin{gathered}\n\\operatorname{Fac}(A)\\\\\n=\\bigcap_{n\\geq1}\\ \\bigcap_{x\\in A_0}\\ \\\\\n\\bigcup_{m\\geq1}\\ \\bigcup_{r\\geq1,\\ u\\in N_0^r}\\ \\\\\n\\bigcap_{h\\in A_0,\\ 0\\leq h\\leq1}\\mathfrak S(m;n;h;u;x).\n\\end{gathered}\n\\tag{21.1}\n\\]\n(The passage between \"\\(<\\varepsilon\\)\" and \"\\(\\leq1/n\\)\", and between arbitrary \\(\\delta>0\\) and \\(1/m\\), is harmless because both are quantified over all values.) The innermost intersection is closed, the countable union is \\(F_\\sigma\\), and the outer countable intersection is \\(F_{\\sigma\\delta}\\). \\(\\square\\)\n\n**Remark 21.4** (The order of the quantifiers). In (21.1) the intersection over \\(h\\) must come after the unions over \\(m\\) and \\(u\\). If it is placed before them, then \\(m\\) and \\(u\\) may depend on \\(h\\), and the set becomes too large. *Example.* Let \\(H=\\ell^2\\), \\(A=\\mathbb C1+K(H)\\), \\(\\xi\\in H\\) a unit vector, \\(\\chi(k+\\lambda1)=\\lambda\\), and \\(\\omega=\\frac12(\\omega_\\xi+\\chi)\\). On \\(H\\oplus\\mathbb C\\) the representation \\(k+\\lambda1\\mapsto(k+\\lambda)\\oplus\\lambda\\) with the cyclic vector \\((\\xi,1)/\\sqrt2\\) is the GNS representation of \\(\\omega\\), and \\(\\pi_\\omega(A)''=B(H)\\oplus\\mathbb C\\). Its centre \\(\\mathbb C\\oplus\\mathbb C\\) is not trivial, so \\(\\omega\\notin\\operatorname{Fac}(A)\\). Yet \\(\\omega\\) lies in the set with the intersection over \\(h\\) placed first. Fix \\(n\\), \\(h\\) and \\(x\\). If \\(\\Phi_{\\omega,h}\\neq0\\) on \\(N\\), it is nonzero at some \\(u\\in N_0\\) (Lemma 21.1(1) and the density of \\(N_0\\)), and a large \\(m\\) puts \\(\\omega\\) in \\(\\mathfrak S(m;n;h;u;x)\\). If \\(\\Phi_{\\omega,h}=0\\) on \\(N\\), then \\(\\pi_\\omega(h)=(k+\\lambda)\\oplus\\lambda\\) is central (Lemma 21.1(2)), so \\(k+\\lambda1\\in\\mathbb C1_H\\), which forces \\(k=0\\) as \\(H\\) is infinite-dimensional; then \\(\\pi_\\omega(h)\\) is a scalar, \\(\\omega(x^*hx)=\\omega(h)\\omega(x^*x)\\), and again \\(\\omega\\in\\mathfrak S(m;n;h;u;x)\\).\n\n**Corollary 21.5.** If \\(A\\) is separable, the central measure of every state is concentrated on \\(\\operatorname{Fac}(A)\\).\n\n**Proof.** \\(\\mathfrak S\\) is metrizable, so the Borel set \\(\\mathfrak S\\setminus\\operatorname{Fac}(A)\\) is a Baire set ([Proposition 2.1](#oa-fnd-ir-02)(4)(d)). Apply Theorem 20.4. \\(\\square\\)\n\n",
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      "id": "OA-FND-IR-28",
      "unit": "integral-representations-of-states",
      "name": "22. Examples and exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
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      "full_conditions_and_proof": "## 22. Examples and exercises\n\n**Example 22.1** (Commutative algebras). Let \\(A=C(X)\\), \\(X\\) compact Hausdorff, so \\(\\mathfrak S=M_1^+(X)\\), and let \\(\\varphi=m\\). The GNS representation is multiplication on \\(L^2(X,m)\\), and \\(\\pi(A)'=\\{M_g:\\ g\\in L^\\infty(m)\\}\\), because \\(C(X)\\) is weak\\(^*\\)-dense in \\(L^\\infty(m)\\) and the multiplication operators by \\(L^\\infty(m)\\) form a [maximal abelian](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09) algebra. This is abelian, so \\(m\\) has exactly one maximal measure (Theorem 15.2). It is \\(\\iota_*m\\), with \\(\\iota(x)=\\delta_x\\): indeed \\(\\int\\hat g\\,d\\iota_*m=m(g)\\), and \\(\\kappa_{\\iota_*m}(G)=M_{G\\circ\\iota}\\) is multiplicative with range \\(\\pi(A)'\\). Since \\(\\pi(A)'=\\pi(A)''\\cap\\pi(A)'\\), it is also the central measure. It is concentrated on the closed set \\(\\iota(X)=P(A)\\), for every \\(X\\), metrizable or not.\n\n**Example 22.2** (The trace on \\(M_n\\)). Let \\(A=M_n(\\mathbb C)\\) and \\(\\tau=\\frac1n\\operatorname{tr}\\). The GNS space is \\(M_n\\) with \\(\\langle X,Y\\rangle=\\tau(Y^*X)\\), \\(\\pi\\) is left multiplication, \\(\\xi=1\\), and \\(\\pi(A)'\\) is the right multiplications \\(R_y\\), a copy of \\(M_n\\). The centre of \\(\\pi(A)''\\) is trivial, so \\(\\tau\\) is factorial and its central measure is \\(\\delta_\\tau\\). For the maximal abelian algebra \\(\\mathcal C=\\{R_d:\\ d\\text{ diagonal}\\}\\), the construction in the proof of Theorem 13.1 gives \\(e\\) = the projection onto the diagonal matrices, \\(\\theta(a)=R_{\\operatorname{diag}(a)}\\), characters \\(\\chi_i(R_d)=d_{ii}\\), \\(\\chi_i\\circ\\theta=\\omega_{e_i}\\), and \\(\\mu=\\frac1n\\sum_i\\delta_{\\omega_{e_i}}\\). It is concentrated on pure states (Theorem 19.1). Conjugating \\(\\mathcal C\\) by a unitary \\(R_w\\) gives the measure \\(\\frac1n\\sum_i\\delta_{\\omega_{w^*e_i}}\\) for another orthonormal basis; since \\(\\pi(A)'\\) is not abelian, \\(\\tau\\) has many maximal measures (Theorem 15.2), including non-orthogonal ones (compare Example 11.5).\n\n**Example 22.3** (Trivial cases). For every state, \\(\\delta_\\varphi\\) is orthogonal with \\(\\mathcal C_{\\delta_\\varphi}=\\mathbb C1\\), since \\(\\kappa_{\\delta_\\varphi}(f)=f(\\varphi)1\\). It is maximal exactly when \\(\\varphi\\) is pure (Lemma 5.2 and Theorem 5.4), and then \\(M_\\varphi(\\mathfrak S)=\\{\\delta_\\varphi\\}\\) (Bauer's criterion, Proposition 3.3(3)). The central measure is \\(\\delta_\\varphi\\) exactly when \\(\\mathcal Z_\\varphi=\\mathbb C1\\), that is, when \\(\\varphi\\) is factorial (Theorem 13.1).\n\n**Remark 22.4** (Further examples). Other small, explicit examples in this lesson: a simplicial measure that is not orthogonal (Example 11.5); a state with abelian \\(\\pi_\\varphi(A)'\\) whose face has a closure that is not a simplex (Proposition 16.1(5)); a state that is not factorial but lies in the set of Remark 21.4, where the intersection over \\(h\\) comes first; and the invariant states in the proof of Theorem 23.2, in Theorem 23.4(3) and in Example 23.6.\n\n**Exercise 22.5** (hard; Orthogonal measures of the trace). In Example 22.2, show that the orthogonal measures of \\(\\tau\\) are exactly the measures \\(\\sum_k\\frac{\\operatorname{tr}p_k}{n}\\delta_{\\omega_k}\\), where \\(p_1,\\dots,p_m\\) are nonzero orthogonal projections with \\(\\sum_kp_k=1\\) and \\(\\omega_k=\\operatorname{tr}(p_k\\,\\cdot)/\\operatorname{tr}p_k\\). Show that such a measure is maximal exactly when every \\(p_k\\) has rank one, and that one of them majorizes another exactly when its partition refines the other's.\n\n*Solution.* An abelian \\(*\\)-subalgebra of \\(\\pi(A)'=\\{R_y\\}\\) is \\(\\{R_d:\\ d\\in D\\}\\) for an abelian \\(*\\)-subalgebra \\(D\\) of \\(M_n\\), and \\(D\\) is the span of the minimal projections \\(p_k\\) of \\(D\\), which add up to \\(1\\). As in Example 22.2, \\(e\\) is the orthogonal projection onto \\(D\\), namely \\(X\\mapsto\\sum_k\\frac{\\operatorname{tr}(p_kX)}{\\operatorname{tr}p_k}p_k\\); so \\(\\theta(a)=R_{d(a)}\\) with \\(d(a)=\\sum_k\\omega_k(a)p_k\\). The characters are \\(R_d\\mapsto\\) (coefficient of \\(p_k\\)), with \\(\\chi_k\\circ\\theta=\\omega_k\\), and \\(\\langle\\xi p_k,\\xi\\rangle=\\operatorname{tr}p_k/n\\). Theorem 13.1 gives the measure, and every orthogonal measure arises this way. If all \\(p_k\\) have rank one, the \\(\\omega_k\\) are pure, and the measure is a boundary measure (Proposition 5.5(2)), hence maximal (Theorem 5.4). If some \\(p_k\\) has rank \\(\\geq2\\), \\(\\omega_k\\) is not pure, and since \\(\\mathfrak S\\) is metrizable a maximal measure would live on \\(P(A)\\) (Theorem 6.4); so the measure is not maximal. The order statement is Theorem 14.1: \\(\\mathcal C_\\mu\\subseteq\\mathcal C_\\nu\\) means that the partition of \\(\\nu\\) refines that of \\(\\mu\\).\n\n**Exercise 22.6** (easy; Pure and factorial states through measures). Show: (a) \\(\\varphi\\) is pure if and only if \\(M_\\varphi(\\mathfrak S)=\\{\\delta_\\varphi\\}\\); (b) \\(\\varphi\\) is factorial exactly when its central measure is \\(\\delta_\\varphi\\); (c) if \\(\\varphi\\) is factorial but not pure, then \\(\\delta_\\varphi\\) is orthogonal but not maximal.\n\n*Solution.* (a) Bauer's criterion (Proposition 3.3(3)) with \\(P(A)=\\partial_e\\mathfrak S\\). (b) The central measure is the orthogonal measure with \\(\\mathcal C_\\mu=\\mathcal Z_\\varphi\\), and \\(\\delta_\\varphi\\) is the orthogonal measure with \\(\\mathcal C=\\mathbb C1\\) (Example 22.3); by the bijection of Theorem 13.1, they coincide exactly when \\(\\mathcal Z_\\varphi=\\mathbb C1\\). (c) \\(\\delta_\\varphi\\) is always orthogonal; by (a) there is another measure in \\(M_\\varphi(\\mathfrak S)\\), and a maximal one majorizing \\(\\delta_\\varphi\\) differs from it, since \\(\\delta_\\varphi\\) is not a boundary measure (Lemma 5.2: \\(\\varphi\\notin\\partial_e\\mathfrak S=\\bigcap_fB_f\\)). Example 22.2 (\\(\\tau\\) on \\(M_n\\), \\(n\\geq2\\)) is such a state.\n\n",
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      "id": "OA-FND-IR-25",
      "unit": "integral-representations-of-states",
      "name": "23. \\(G\\)-abelian systems",
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      "source": "src/integral-representations-of-states.md",
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      "full_conditions_and_proof": "## 23. \\(G\\)-abelian systems\n\nWe keep the setting of Section 17. For \\(\\varphi\\in\\mathfrak S^G\\) let \\(H_0=\\{\\zeta\\in H_\\varphi:\\ U_s\\zeta=\\zeta\\text{ for all }s\\}\\), and let \\(e_0\\) be the projection onto it. \\(\\operatorname{co}U(G)\\) is the set of finite convex combinations \\(\\sum_sc_sU_s\\), and \\(K_G(x)=\\operatorname{co}\\{\\alpha_s(x):\\ s\\in G\\}\\).\n\n**Lemma 23.1** (Mean ergodic lemma).\n\n1. For \\(\\eta\\in H\\), \\(e_0\\eta\\) is the element of minimal norm in the closed convex hull of \\(\\{U_s\\eta\\}\\).\n2. For \\(\\eta_1,\\dots,\\eta_n\\in H\\) and \\(\\varepsilon>0\\) there is a self-adjoint \\(S\\in\\operatorname{co}U(G)\\) with \\(\\|S\\eta_j-e_0\\eta_j\\|<\\varepsilon\\) for all \\(j\\).\n3. \\(Se_0=e_0S=e_0\\) for \\(S\\in\\operatorname{co}U(G)\\), and \\(e_0\\) commutes with \\(U(G)'\\).\n\n**Proof.** (3) \\(U_s\\) fixes \\(H_0\\) pointwise, so \\(U_se_0=e_0\\) and \\(e_0U_s=(U_s^*e_0)^*=e_0\\). An operator commuting with \\(U(G)\\) maps \\(H_0\\) into itself, and so does its adjoint. (1) Let \\(C\\) be the closed convex hull and \\(c\\) its element of minimal norm. \\(U_tC=C\\) and \\(U_t\\) is isometric, so \\(U_tc=c\\), and \\(c\\in H_0\\). By (3), \\(e_0\\) is constant on \\(C\\), equal to \\(e_0\\eta\\); so \\(c=e_0c=e_0\\eta\\). (2) Apply (1) to \\(U\\oplus\\dots\\oplus U\\) on \\(H^n\\), whose fixed vectors are \\(H_0^n\\): some \\(T\\in\\operatorname{co}U(G)\\) has \\(\\|T\\eta_j-e_0\\eta_j\\|<\\varepsilon\\). Then \\(S=T^*T\\in\\operatorname{co}U(G)\\) is self-adjoint and \\(\\|S\\eta_j-e_0\\eta_j\\|=\\|T^*(T\\eta_j-e_0\\eta_j)\\|<\\varepsilon\\), because \\(T^*e_0=e_0\\). \\(\\square\\)\n\n**Theorem 23.2.** For \\(\\varphi\\in\\mathfrak S^G\\), with \\(\\pi=\\pi_\\varphi\\), the following are equivalent.\n\n- (i) \\(e_0\\pi(A)e_0\\) is commutative.\n- (ii\\(^*\\)) For all \\(x,y\\in A\\) and \\(\\zeta\\in H_0\\), \\[\n\\begin{gathered}\n\\inf\\{|\\langle\\pi(x'y-yx')\\zeta,\\zeta\\rangle|:\\\\\n\\ x'\\in K_G(x)\\}\\\\\n=0\n\\end{gathered}\n\\].\n\nConsider also the condition (ii), in which the infimum is taken only over the orbit, \\(x'=\\alpha_s(x)\\) with \\(s\\in G\\). It implies (ii\\(^*\\)), hence (i). But (i) does not imply (ii), as the end of the proof shows.\n\nThe theorem proves the equivalence with the convex-hull condition (ii\\(^*\\)). The end of the proof separates it from the orbit condition (ii), and Theorem 23.4(3) gives a generated face whose closure is not a simplex.\n\n**Proof.** For \\(x'=\\sum_sc_s\\alpha_s(x)\\) put \\(S=\\sum_sc_sU_s\\). Since \\(U_s^*\\zeta=\\zeta\\) for \\(\\zeta\\in H_0\\),\n\\[\n\\begin{gathered}\n\\langle\\pi(x')\\pi(y)\\zeta,\\zeta\\rangle\\\\\n=\\langle\\pi(x)S^*\\pi(y)\\zeta,\\zeta\\rangle,\\\\\n\\langle\\pi(y)\\pi(x')\\zeta,\\zeta\\rangle\\\\\n=\\langle\\pi(y)S\\pi(x)\\zeta,\\zeta\\rangle .\n\\end{gathered}\n\\]\n(i)\\(\\Rightarrow\\)(ii\\(^*\\)): Choose \\(S\\) self-adjoint with \\(S\\approx e_0\\) on \\(\\pi(x)\\zeta\\) and \\(\\pi(y)\\zeta\\) (Lemma 23.1(2)). Then \\[\n\\begin{gathered}\n\\langle\\pi(x'y-yx')\\zeta,\\zeta\\rangle\\\\\n=\\langle S\\pi(y)\\zeta,\\pi(x)^*\\zeta\\rangle\\\\\n-\\langle S\\pi(x)\\zeta,\\pi(y)^*\\zeta\\rangle\n\\end{gathered}\n\\] is close to \\(\\langle\\pi(x)e_0\\pi(y)\\zeta,\\zeta\\rangle-\\langle\\pi(y)e_0\\pi(x)\\zeta,\\zeta\\rangle\\). As \\(\\zeta=e_0\\zeta\\), this is \\(\\langle[e_0\\pi(x)e_0,e_0\\pi(y)e_0]\\zeta,\\zeta\\rangle=0\\).\n\n(ii\\(^*\\))\\(\\Rightarrow\\)(i): Fix \\(x,y\\), \\(\\zeta\\in H_0\\) and \\(\\delta>0\\), and let \\(F=\\langle\\pi(x)e_0\\pi(y)\\zeta,\\zeta\\rangle-\\langle\\pi(y)e_0\\pi(x)\\zeta,\\zeta\\rangle\\). Choose a self-adjoint \\(S'=\\sum_tc'_tU_t\\) with \\(\\|S'\\pi(y)\\zeta-e_0\\pi(y)\\zeta\\|<\\delta\\) and \\(\\|S'\\pi(y)^*\\zeta-e_0\\pi(y)^*\\zeta\\|<\\delta\\), and put \\(y'=\\sum_tc'_t\\alpha_t(y)\\). Apply (ii\\(^*\\)) to the pair \\((x,y')\\): some \\(x'\\in K_G(x)\\), with operator \\(S\\), has \\(|Q|<\\delta\\), where \\(Q=\\langle\\pi(x'y'-y'x')\\zeta,\\zeta\\rangle\\). Expanding both averages as above,\n\\[\n\\begin{gathered}\nQ\\\\\n=\\langle\\pi(x)S^*S'\\pi(y)\\zeta,\\zeta\\rangle\\\\\n-\\langle\\pi(y)S'S\\pi(x)\\zeta,\\zeta\\rangle\\\\\n=\\langle S'\\pi(y)\\zeta,S\\pi(x)^*\\zeta\\rangle\\\\\n-\\langle S\\pi(x)\\zeta,S'\\pi(y)^*\\zeta\\rangle .\n\\end{gathered}\n\\]\nReplacing \\(S'\\pi(y)\\zeta\\) and \\(S'\\pi(y)^*\\zeta\\) by \\(e_0\\pi(y)\\zeta\\) and \\(e_0\\pi(y)^*\\zeta\\) changes \\(Q\\) by at most \\(2\\delta\\|x\\|\\|\\zeta\\|\\). After the replacement, \\(S^*e_0=e_0\\) and \\(e_0S=e_0\\) turn \\(Q\\) into exactly \\(F\\). So \\(|F|\\leq\\delta(1+2\\|x\\|\\|\\zeta\\|)\\) for every \\(\\delta\\), and \\(F=0\\). The operator \\([e_0\\pi(x)e_0,e_0\\pi(y)e_0]\\) on \\(H_0\\) has vanishing quadratic form, so it is \\(0\\).\n\n(ii)\\(\\Rightarrow\\)(ii\\(^*\\)) is trivial. *(i) does not imply (ii).* Let \\(A=M_2(\\mathbb C)\\), \\(G=\\mathbb Z_2\\) acting by \\(\\operatorname{Ad}u\\), \\(u=\\operatorname{diag}(1,-1)\\), and \\(\\varphi=\\tau\\), the normalized trace. The GNS space is \\(M_2\\) with \\(\\langle X,Y\\rangle=\\tau(Y^*X)\\), \\(\\pi\\) is left multiplication, \\(\\xi=1\\), and \\(U_g(X)=uXu^*\\). So \\(H_0\\) is the diagonal matrices, and \\(e_0\\pi(a)e_0\\) acts on \\(H_0\\) as multiplication by \\(\\operatorname{diag}(a_{11},a_{22})\\): (i) holds. For \\(x=E_{12}\\), \\(y=E_{21}\\) and \\(\\zeta=E_{11}\\in H_0\\), \\(\\langle\\pi([x,y])\\zeta,\\zeta\\rangle=\\frac12\\) and \\(\\langle\\pi([\\alpha_g(x),y])\\zeta,\\zeta\\rangle=-\\frac12\\). So the infimum over the orbit is \\(\\frac12\\neq0\\). (Over \\(K_G(x)\\) it is \\(0\\), attained at \\(x'=\\frac12(x+\\alpha_g(x))=0\\).) \\(\\square\\)\n\n**Proposition 23.3.** If (i) holds, then \\(\\mathfrak M_\\varphi=\\pi(A)'\\cap U(G)'\\) is abelian; in fact \\(\\mathfrak M_\\varphi=\\pi(A)'\\cap\\{e_0\\}'\\).\n\n**Proof.** Apply Proposition 12.3(b) to the unital \\(*\\)-algebra \\(\\mathcal M=\\pi(A)\\), with cyclic vector \\(\\xi\\) and \\(e=e_0\\) (\\(e_0\\xi=\\xi\\)): \\(\\mathcal A=\\pi(A)'\\cap\\{e_0\\}'\\) is abelian, and restriction to \\(H_0\\) is one-to-one on it. By Lemma 23.1(3), \\(\\mathfrak M_\\varphi\\subseteq\\mathcal A\\). Conversely, for \\(x\\in\\mathcal A\\) and \\(s\\in G\\), \\(U_s^*xU_s\\in\\mathcal A\\) (since \\(U_s\\) normalizes \\(\\pi(A)\\) and commutes with \\(e_0\\)), and it agrees with \\(x\\) on \\(H_0\\), where \\(U_s\\) acts trivially. So \\(U_s^*xU_s=x\\). \\(\\square\\)\n\n**Theorem 23.4.** Suppose (i) holds for \\(\\varphi\\in\\mathfrak S^G\\).\n\n1. The face \\(F^G_\\varphi\\) of \\(\\mathfrak S^G\\) generated by \\(\\varphi\\) has a lattice cone: \\(\\mathbb R_+F^G_\\varphi\\cong(\\mathfrak M_\\varphi)_+\\).\n2. \\(\\varphi\\) is the barycentre of exactly one maximal (boundary) probability measure on \\(\\mathfrak S^G\\). It is the orthogonal measure \\(\\mu\\) with \\(\\mathcal C_\\mu=\\mathfrak M_\\varphi\\), and \\(\\mu(B)=0\\) for every Baire subset \\(B\\) of \\(\\mathfrak S^G\\) that misses \\(\\partial_e\\mathfrak S^G\\); if \\(A\\) is separable, \\(\\mu\\) is concentrated on \\(\\partial_e\\mathfrak S^G\\).\n3. The closure \\(\\overline{F^G_\\varphi}\\) need not be a simplex.\n\n**Proof.** (1) As in Proposition 16.1(1) and (3), using Proposition 17.1(3): \\(F^G_\\varphi=\\Theta_\\varphi(\\{x\\in(\\mathfrak M_\\varphi)_+:\\ \\langle x\\xi,\\xi\\rangle=1\\})\\), and \\(\\mathfrak M_\\varphi\\) is abelian by Proposition 23.3. (2) Proposition 17.1(6) and Theorem 19.1(3). (3) Take the algebra and the function \\(\\xi(\\cdot)\\) of Proposition 16.1(5). Let \\(Z_0=\\{\\frac12-2^{-k}:\\ k\\geq1\\}\\cup\\{\\frac12\\}\\), a closed countable set, \\(\\theta(x)=\\operatorname{dist}(x,Z_0)\\in[0,\\frac12]\\), \\(\\sigma(x)=2|\\xi(x)\\rangle\\langle\\xi(x)|-1\\), and \\(u(x)=\\cos\\theta(x)\\,1+i\\sin\\theta(x)\\,\\sigma(x)\\). The jumps of \\(\\sigma\\) lie in \\(Z_0\\), where \\(\\sin\\theta=0\\), so \\(u\\in C([0,1],U(2))\\). Let \\(\\mathbb Z\\) act by \\(\\alpha^n\\), \\(\\alpha(a)(x)=u(x)a(x)u(x)^*\\). Since \\(u(x)^*\\xi(x)=e^{-i\\theta(x)}\\xi(x)\\), \\(\\varphi\\) is invariant, and \\(U=U_\\varphi(1)\\) is multiplication by \\(w(x)=e^{-i\\theta(x)}u(x)\\). Its eigenvalues are \\(1\\) on \\(\\xi(x)\\) and \\(e^{-2i\\theta(x)}\\neq1\\) on \\(\\xi(x)^\\perp\\) for almost every \\(x\\). So \\(H_0=\\{g\\xi:\\ g\\in L^2\\}\\), and \\(e_0\\pi(a)e_0\\) is multiplication by the scalar function \\(x\\mapsto\\langle a(x)\\xi(x),\\xi(x)\\rangle\\): (i) holds. Here \\(\\mathfrak M_\\varphi=\\{M_g\\otimes1\\}\\), so \\(F^G_\\varphi=F_\\varphi\\). The four pure states \\(\\rho_j\\) of Proposition 16.1(5) are invariant, since \\(u(\\frac12)=1\\). As there, \\(\\overline{F^G_\\varphi}=\\overline{F_\\varphi}\\) is not a simplex. \\(\\square\\)\n\n**Theorem 23.5.** If (i) holds for every \\(\\varphi\\in\\mathfrak S^G\\) (the system is *\\(G\\)-abelian*), then \\(\\mathfrak S^G\\) is a simplex.\n\n**Proof.** By Theorem 23.4(2) every point of \\(\\mathfrak S^G\\) has exactly one maximal representing measure on \\(\\mathfrak S^G\\). Apply the Choquet–Meyer theorem (Theorem 7.4). \\(\\square\\)\n\n**Example 23.6** (Diagonal unitaries acting on \\(M_n\\)). Let \\(A=M_n(\\mathbb C)\\) with \\(n\\geq2\\), \\(G\\) the diagonal unitary matrices, \\(\\alpha_s=\\operatorname{Ad}s\\), and \\(\\varphi(x)=x_{11}\\).\n\nThis example separates conditions (i) and (ii): the compressed algebra is commutative, while the tested orbit commutator has constant modulus one.\n\n- *\\(\\mathfrak M_\\varphi\\) is one-dimensional.* \\(\\varphi\\) is the vector state of \\(e_1\\) in the identity representation on \\(\\mathbb C^n\\), which is irreducible, so \\(\\pi_\\varphi(A)'=\\mathbb C1\\).\n- *Condition (ii) fails.* \\(U_s=\\bar s_{11}s\\), and for \\(x=E_{12}\\), \\(y=E_{21}\\), \\(\\zeta=e_1\\): \\(\\langle[\\alpha_s(x),y]e_1,e_1\\rangle=s_{11}\\bar s_{22}\\), of modulus \\(1\\) for every \\(s\\).\n- *Condition (i) holds.* \\(H_0=\\{\\zeta:\\ \\bar s_{11}s_{jj}\\zeta_j=\\zeta_j\\ \\forall s\\}=\\mathbb Ce_1\\), so \\(e_0\\pi(A)e_0=\\mathbb Ce_0\\) is commutative.\n\nIn fact every invariant state satisfies (i). An invariant state has a diagonal density matrix \\(D\\); its GNS space is \\(\\{X\\in M_n:\\ X\\text{ vanishes on }\\ker D\\}\\) with \\(\\xi=D^{1/2}\\) and the Hilbert–Schmidt inner product, \\(U_sX=sXs^*\\), \\(H_0\\) is its diagonal part, and \\(e_0\\pi(x)e_0\\) is multiplication by \\(\\operatorname{diag}(x_{11},\\dots,x_{nn})\\). So the system is \\(G\\)-abelian, and \\(\\mathfrak S^G\\), the \\((n-1)\\)-simplex of diagonal states, is a simplex, as Theorem 23.5 predicts. The example shows that (ii) is strictly stronger than (i).\n\n**Exercise 23.7** (medium; An invariant state under a finite group). Let \\(G=\\mathbb Z_2\\) act on \\(M_2(\\mathbb C)\\) by \\(\\operatorname{Ad}\\operatorname{diag}(1,-1)\\), and \\(\\varphi=\\tau\\). Find \\(\\mathfrak S^G\\), \\(\\partial_e\\mathfrak S^G\\), \\(\\mathfrak M_\\tau\\), and the unique maximal measure of \\(\\tau\\) on \\(\\mathfrak S^G\\).\n\n*Solution.* A state \\(\\omega_\\rho\\) is invariant exactly when \\(\\rho\\) commutes with \\(\\operatorname{diag}(1,-1)\\), that is, when \\(\\rho\\) is diagonal. So \\(\\mathfrak S^G\\) is the segment from \\(\\omega_{e_1}\\) to \\(\\omega_{e_2}\\), with these two as extreme points. In the GNS picture used at the end of the proof of Theorem 23.2, \\(U_g=\\operatorname{Ad}u\\) on \\(M_2\\), and \\(R_y\\) commutes with \\(U_g\\) exactly when \\(y\\) is diagonal; so \\(\\mathfrak M_\\tau=\\{R_d:\\ d\\text{ diagonal}\\}\\), which is abelian (condition (i) holds, as shown there). By Proposition 17.1(6) the orthogonal measure with \\(\\mathcal C_\\mu=\\mathfrak M_\\tau\\), which is \\(\\frac12(\\delta_{\\omega_{e_1}}+\\delta_{\\omega_{e_2}})\\) by Exercise 22.5, is the unique maximal measure on \\(\\mathfrak S^G\\). This agrees with the direct computation on the segment.\n\n",
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      "id": "OA-FND-IR-26",
      "unit": "integral-representations-of-states",
      "name": "24. Large groups of automorphisms",
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      "full_conditions_and_proof": "## 24. Large groups of automorphisms\n\nFor \\(\\varphi\\in\\mathfrak S^G\\) and \\(a\\in A\\) let \\(C_a\\) be the weak operator closure of \\(\\pi_\\varphi(K_G(a))\\). It is convex and bounded, hence weakly compact, and it equals the strong closure, because a convex set of operators has the same weak and strong closures. Also \\(C_a\\subseteq\\pi_\\varphi(A)''\\). We use three conditions.\n\n- (S) For every \\(\\varphi\\in\\mathfrak S^G\\) and \\(a\\in A\\), \\(C_a\\cap\\mathcal Z_\\varphi\\neq\\varnothing\\).\n- (L) For every \\(\\varphi\\in\\mathfrak S^G\\), \\(a\\in A\\), and finitely many \\(b_1,\\dots,b_n,x_1,\\dots,x_m\\in A\\): \\[\n\\begin{gathered}\n\\inf_{a'\\in K_G(a)}\\\\\n\\max_{i,j}|\\varphi(x_j(a'b_i-b_ia')x_j^*)|\\\\\n=0\n\\end{gathered}\n\\].\n- (L\\(_p\\)) As (L), but with a single \\(x\\), and with the infimum taken separately for each \\(i\\).\n\nWe call the action *large* if (S) holds.\n\n**Theorem 24.1.**\n\n- (a) (L)\\(\\iff\\)(S), and (S)\\(\\Rightarrow\\)(L\\(_p\\)).\n- (b) If (S) holds, the system is \\(G\\)-abelian, so \\(\\mathfrak S^G\\) is a simplex.\n- (c) If (S) holds, then for \\(\\varphi\\in\\mathfrak S^G\\) the following are equivalent: (i) \\(\\varphi\\) is ergodic; (ii) \\(U_\\varphi(G)'\\cap\\mathcal Z_\\varphi=\\mathbb C1\\); (iii) \\(H_0=\\mathbb C\\xi_\\varphi\\).\n\n\n\n**Proof.** Write \\(\\pi,H,\\xi,U\\) for \\(\\varphi\\). Note that \\(\\varphi(x(a'b-ba')x^*)=\\langle[\\pi(a'),\\pi(b)]\\eta,\\eta\\rangle\\) with \\(\\eta=\\pi(x^*)\\xi\\).\n\n(a) (S)\\(\\Rightarrow\\)(L): take \\(z\\in C_a\\cap\\mathcal Z_\\varphi\\) and a net \\(\\pi(a'_k)\\to z\\) weakly with \\(a'_k\\in K_G(a)\\). For each of the finitely many pairs, \\(\\langle[\\pi(a'_k),\\pi(b_i)]\\eta_j,\\eta_j\\rangle\\to\\langle[z,\\pi(b_i)]\\eta_j,\\eta_j\\rangle=0\\). (L)\\(\\Rightarrow\\)(L\\(_p\\)) is clear. (L)\\(\\Rightarrow\\)(S): the sets \\(Z_{b,x}=\\{z\\in C_a:\\ \\langle[z,\\pi(b)]\\pi(x^*)\\xi,\\pi(x^*)\\xi\\rangle=0\\}\\) are weakly closed. By (L) and compactness, finitely many of them have a common point: a weak cluster point of \\(\\pi(a'_k)\\), where \\(a'_k\\) nearly attains the infimum. So all of them have a common point \\(z\\). Then the quadratic form of \\([z,\\pi(b)]\\) vanishes on the dense subspace \\(\\pi(A)\\xi\\), so \\(z\\in\\pi(A)'\\cap C_a\\subseteq\\mathcal Z_\\varphi\\).\n\n(b) Fix \\(\\varphi\\) and \\(a\\). \\(C_a\\cap\\mathcal Z_\\varphi\\) is nonempty, convex, weakly compact, and invariant under \\(\\operatorname{Ad}U_s\\), because \\(U_s\\pi(K_G(a))U_s^*=\\pi(K_G(a))\\) and \\(U_s\\) normalizes \\(\\pi(A)\\). Take \\(z\\) in it. By the mean ergodic lemma (Lemma 23.1) there are \\(T_k=\\sum_sc^{(k)}_sU_s\\) with \\(T_kz\\xi\\to e_0z\\xi\\). The elements \\(z_k=\\sum_sc^{(k)}_sU_szU_s^*\\) lie in \\(C_a\\cap\\mathcal Z_\\varphi\\) and \\(z_k\\xi=T_kz\\xi\\). A weak cluster point \\(z_\\infty\\) lies in \\(C_a\\cap\\mathcal Z_\\varphi\\), and \\(z_\\infty\\xi=e_0z\\xi\\in H_0\\). Then \\(U_sz_\\infty U_s^*\\in\\mathcal Z_\\varphi\\) and \\(U_sz_\\infty U_s^*\\xi=U_sz_\\infty\\xi=z_\\infty\\xi\\), so \\(U_sz_\\infty U_s^*=z_\\infty\\) (separating vector). Thus \\(z_\\infty\\) commutes with \\(U(G)\\), hence with \\(e_0\\). Since \\(e_0\\pi(\\alpha_s(a))e_0=e_0\\pi(a)e_0\\), weak limits give \\(e_0\\pi(a)e_0=e_0z_\\infty e_0=z_\\infty e_0\\). For \\(b\\in A\\), \\(z_\\infty\\) is central in \\(\\pi(A)''\\), so\n\\[\n\\begin{gathered}\ne_0\\pi(a)e_0\\cdot e_0\\pi(b)e_0\\\\\n=e_0\\pi(b)z_\\infty e_0\\\\\n=e_0\\pi(b)e_0\\cdot e_0\\pi(a)e_0 .\n\\end{gathered}\n\\]\nSo condition (i) of Theorem 23.2 holds for every \\(\\varphi\\), and Theorem 23.5 applies.\n\n(c) (i)\\(\\Leftrightarrow\\)\\(\\mathfrak M_\\varphi=\\mathbb C1\\) by Proposition 17.1(4), and \\(U(G)'\\cap\\mathcal Z_\\varphi\\subseteq\\mathfrak M_\\varphi\\); so (i)\\(\\Rightarrow\\)(ii). (iii)\\(\\Rightarrow\\)(i): for \\(x\\in\\mathfrak M_\\varphi\\), \\(x\\xi\\in H_0=\\mathbb C\\xi\\), so \\(x\\) is a scalar (separating vector). (ii)\\(\\Rightarrow\\)(iii): by the proof of (b), each \\(a\\) has a \\(z_\\infty\\in U(G)'\\cap\\mathcal Z_\\varphi=\\mathbb C1\\) with \\(e_0\\pi(a)e_0=z_\\infty e_0\\). So \\(e_0\\pi(a)\\xi=e_0\\pi(a)e_0\\xi\\in\\mathbb C\\xi\\) for every \\(a\\), and \\(H_0=e_0\\overline{\\pi(A)\\xi}=\\mathbb C\\xi\\). \\(\\square\\)\n\nCondition (L\\(_p\\)) has one \\(x\\) and a separate infimum for each \\(b_i\\), so the compactness argument in (L)\\(\\Rightarrow\\)(S) does not apply to it. We leave open whether (L\\(_p\\)) implies (S). Parts (b) and (c) use only (S).\n\n## Background used without proof\n\n*Functional analysis.*\n\n- (*Hahn–Banach and choice.*) [Theorem 2.1 and Corollary 2.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-02) prove the real sublinear extension theorem and its complex and normed versions. [Theorem 6.3 and Corollary 6.4](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-06) strictly separate a compact convex set from a disjoint closed convex set in a locally convex Hausdorff space; they also give separation of points and annihilation of a proper closed linear subspace. The maximal-measure and maximal-abelian-algebra arguments use [Zorn’s lemma, Theorem 1.1](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-01).\n- (*Weak topologies.*) If \\(X'\\) is a vector space of linear functionals on \\(X\\) that separates points, \\(\\sigma(X,X')\\) is locally convex and Hausdorff, and its continuous linear functionals are exactly \\(X'\\). [Theorem 1.2 of the weak-topology lesson](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-01) gives the full finite-coordinate factorization proof.\n- (*Banach–Alaoglu.*) The closed unit ball of the dual of any normed space is weak\\(^*\\)-compact, by [Theorem 3.1 of the weak-topology lesson](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-03), using its full Tychonoff proof. No separability is assumed.\n- (*Krein–Milman.*) A nonempty compact convex set in a locally convex Hausdorff space has extreme points and is their closed convex hull. [Theorem 6.1 of the weak-topology lesson](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-06) proves both claims by minimal faces and separation.\n\n*Measure theory.*\n\n- (*Riesz representation.*) For a compact Hausdorff space \\(X\\), every bounded complex linear functional on \\(C(X)\\) is integration against a unique finite complex Radon measure, with norm equal to total variation. A real functional has a real measure, and a positive functional has a positive measure. The full positive and complex arguments and regularity are [Theorems 2.2 and 2.4 and Proposition 2.3 of the Haar lesson](haar-measure.md#oa-fnd-hm-01). Restricting the complex theorem to real functions gives the real version used in Sections 1–7: extend a real functional by \\(L(f+ig)=L(f)+iL(g)\\). This extension has the same norm: rotate \\(z\\) by a scalar of modulus one so that \\(L(z)\\) is nonnegative real, and use \\(|L(z)|=L(\\operatorname{Re}z)\\leq\\|L\\|\\|z\\|_\\infty\\) for the rotated function. Uniqueness and conjugation make its measure real. Its positive and negative parts are \\((|\\mu|\\pm\\mu)/2\\), which are positive regular measures.\n- (*Duality of \\(L^1\\) and \\(L^\\infty\\).*) For a sigma-finite measure \\(\\mu\\), the map \\(g\\mapsto(f\\mapsto\\int fg\\,d\\mu)\\) is an isometric isomorphism of \\(L^\\infty(\\mu)\\) onto \\(L^1(\\mu)^*\\). This is [Theorem 4.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#4-densities-and-bounded-functionals); its full proof first treats finite measure by Hilbert representation, then joins finite pieces. In this lesson the reference measures on compact spaces are finite.\n- (*Convergence theorems.*) [Theorems 2.1–2.2 of the measure-tools lesson](measure-and-hilbert-space-tools.md#2-integration-and-convergence-without-countability-assumptions) prove monotone convergence, Fatou and dominated convergence on arbitrary measure spaces: increasing nonnegative sequences have the expected integral limit, the integral of a nonnegative liminf is at most the liminf of the integrals, and almost-everywhere convergence under one integrable bound gives convergence in \\(L^1\\). The monotone-net assertion for lower semicontinuous functions is separately proved in Proposition 2.1(2) above, using compactness and regularity.\n- (*Uniqueness of measures.*) Two finite measures that agree on the whole space and on a family closed under finite intersections agree on the sigma-algebra generated by that family. The [finite-measure uniqueness lemma of the double-commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-18) gives the full Dynkin-system argument; it is used in Proposition 2.1(4)(e).\n- (*Urysohn’s lemma and finite partitions of unity.*) [Lemma 5.1 and Corollary 5.2 of the Stone–Weierstrass lesson](stone-weierstrass-c0.md#oa-fnd-sw-16) separate disjoint closed subsets of a compact Hausdorff space and supply continuous cutoffs with prescribed compact support. Here is the finite-cover consequence used in Section 18. Given a finite open cover \\(U_1,\\dots,U_n\\), each \\(x\\) has an open neighbourhood \\(V_x\\) with \\(\\overline{V_x}\\subseteq U_{i(x)}\\); this is the compact Hausdorff separation proved there. Choose \\(0\\leq h_x\\leq1\\) equal to one on \\(\\overline{V_x}\\), with \\(\\operatorname{supp}h_x\\subseteq U_{i(x)}\\). Finitely many \\(V_{x_j}\\) cover \\(X\\). Set \\(H=\\sum_jh_{x_j}>0\\), and \\(u_i=H^{-1}\\sum_{j:i(x_j)=i}h_{x_j}\\). Then \\(u_i\\geq0\\), \\(\\sum_i u_i=1\\), and \\(\\operatorname{supp}u_i\\subseteq U_i\\).\n- (*Stone–Weierstrass.*) A real subalgebra of \\(C(K)\\) that contains the constants and separates points is dense; so is a linear subspace that is closed under \\(\\vee\\) and \\(\\wedge\\), contains the constants and separates points; and a self-adjoint complex subalgebra that contains the constants and separates points is dense in the complex \\(C(K)\\). All three are proved in [The Stone–Weierstrass theorem for \\(C_0(X)\\)](stone-weierstrass-c0.md).\n\n*Hilbert spaces and operators.*\n\n- [Theorem 2.1, Theorem 2.3, Theorem 3.1 and Corollary 3.2 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-02) prove, respectively, the unique minimum-norm point of a nonempty closed convex set, Riesz–Fréchet representation of bounded functionals, representation of bounded sesquilinear forms by operators, polarization uniqueness, and the quadratic-form formula for the norm of a self-adjoint operator. The same norm formula for a normal operator is [Proposition 8.2 of the spectral lesson](the-spectral-theorem-for-bounded-self-adjoint-operators.md#oa-fnd-st-08), proved there using approximate eigenvectors. The arguments below need only the self-adjoint formula.\n- (*Spectral theorem.*) [Theorem 4.4 and Proposition 5.1 of the spectral lesson](the-spectral-theorem-for-bounded-self-adjoint-operators.md#oa-fnd-st-04) give the unique projection-valued spectral measure of a bounded self-adjoint operator, commutation of its projections with every operator commuting with it, and norm approximation by real combinations of those projections. Each spectral projection lies in every von Neumann algebra containing the operator: it commutes with that algebra’s commutant, so lies in its bicommutant. Thus a von Neumann algebra with only the projections \\(0,1\\) is \\(\\mathbb C1\\). For a normal operator, [Theorem 8.1 there](the-spectral-theorem-for-bounded-self-adjoint-operators.md#oa-fnd-st-08) requires commutation with both \\(T,T^*\\); [Fuglede’s theorem, proved in Section 3.1 of the Kaplansky lesson](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-03), supplies commutation with \\(T^*\\) from commutation with \\(T\\). The proof here consumes only the self-adjoint version.\n- (*Operator topologies.*) Lemma 9.0 above proves weak compactness of bounded weakly closed operator sets, equality of weak and strong closures of convex sets, equality of weak and \\(\\sigma\\)-weak topologies on bounded sets, and the separate multiplication continuity used in Theorem 13.1. Its compactness proof uses Tychonoff and bounded sesquilinear forms; no trace-class duality is needed for this fact.\n- (*Bicommutant theorem.*) If \\(\\mathcal M\\subseteq B(H)\\) is a \\(*\\)-algebra containing \\(1\\), then its weak closure, its strong closure and \\(\\mathcal M''\\) coincide. See [The double commutation theorem](the-double-commutant-theorem.md).\n- (*Kaplansky’s density theorem.*) If a \\(*\\)-algebra \\(\\mathcal M\\subseteq B(H)\\) contains \\(1\\), every self-adjoint contraction of \\(\\mathcal M''\\) is a strong limit of self-adjoint contractions of \\(\\mathcal M\\). This is [Theorem 7.1(2) of the Kaplansky lesson](kaplansky-s-density-theorem-and-its-consequences.md#oa-fnd-kd-07), which proves the stronger strong\\(^*\\) density statement, with the continuity argument in Theorem 5.2.\n- (*Multiplication algebras.*) For a sigma-finite measure space, multiplication by \\(L^\\infty\\) is a maximal abelian von Neumann algebra on \\(L^2\\), by [Theorem 9.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09). The map \\(g\\mapsto M_g\\) is normal: a vector coefficient is integration against \\(\\eta\\overline\\zeta\\in L^1\\), by Hölder, and Lemma 9.1 handles square-summable series of coefficients. This justifies the weak\\(^*\\)-density arguments in the examples.\n\n*C\\(^*\\)-algebras.*\n\n- (*Commutative algebras and functional calculus.*) A unital abelian C\\(^*\\)-algebra is isometrically \\(*\\)-isomorphic to \\(C(\\Omega)\\), where \\(\\Omega\\) is its compact space of characters (the commutative Gelfand–Naimark theorem). A normal element has a continuous functional calculus. An injective \\(*\\)-homomorphism between C\\(^*\\)-algebras is isometric. All this is proved in [C\\(^*\\)-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md).\n- (*Positive functionals and states.*) [Proposition 3.2 of the GNS lesson](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-03) proves \\(|\\varphi(b^*a)|^2\\leq\\varphi(a^*a)\\varphi(b^*b)\\) by polarization and a nonnegative scalar quadratic. [Theorem 4.7 there](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-04) proves that a positive functional on a C\\(^*\\)-algebra is bounded and hermitian, and that \\(|\\varphi(a)|^2\\leq\\|\\varphi\\|\\varphi(a^*a)\\). In the unital case \\(\\|\\varphi\\|=\\varphi(1)\\). A state is a positive functional with this value equal to one; [Lemma 7.1 there](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-07) gives enough positive functionals to detect every nonzero positive element, and hence states of every nonzero unital algebra.\n- (*GNS construction.*) [Construction 5.1, Lemma 5.2 and Theorems 5.3–5.5 of the GNS lesson](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-05) prove the quotient, bounded left action, cyclic vector and unitary uniqueness. For a state \\(\\varphi\\), the unital case has \\(\\xi_\\varphi=1+N_\\varphi\\) and \\(\\|\\xi_\\varphi\\|^2=\\varphi(1)=1\\), with \\(\\varphi(a)=\\langle\\pi_\\varphi(a)\\xi_\\varphi,\\xi_\\varphi\\rangle\\).\n- (*Gelfand–Naimark.*) [Lemma 7.1 and Theorem 7.2 of the GNS lesson](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-07) prove existence of enough positive functionals by separation and then form a faithful direct sum of their GNS representations. Injective \\(*\\)-homomorphisms are isometric by the continuous-functional-calculus lesson, Theorem 4.2.\n\n## Where this leads\n\n- *Direct integrals.* For a separable algebra, an orthogonal measure does more than decompose the state: it writes \\(\\pi_\\varphi\\) as a direct integral of representations over the state space, with \\(\\mathcal C_\\mu\\) as the diagonal algebra; see the following programme lessons. The lessons Measurable fields of Hilbert spaces and their direct integrals and Decomposable operators and the diagonal algebra build the direct integrals that this needs.\n- *Compact convex sets.* Simplices whose extreme boundary is closed (Bauer simplices), split faces and the facial topology continue Section 7; \n- *Group actions.* Here \\(G\\) is an abstract group, and no continuity of the action is used. Invariant states for continuous actions of locally compact groups, and their ergodic decompositions, are further directions beyond the scope proved here.\n- *Two open questions.* This lesson does not settle whether condition (L\\(_p\\)) of Section 24 implies (S), nor whether a system whose invariant states form a simplex must be \\(G\\)-abelian.\n\n## References\n\n- [Blackadar] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n- [van Neerven] J. van Neerven, *Functional Analysis*, [corrected author version, arXiv:2112.11166v7](https://arxiv.org/pdf/2112.11166v7).\n\n*Freely accessible reading:* [D. H. Fremlin, *Measure Theory, Chapter 46*, §461A–§461P](https://www1.essex.ac.uk/maths/people/fremlin/chap46.pdf) gives a route through barycenters, maximal representing measures and simplex uniqueness; Baire pseudo-support is distinguished from mass on the extreme boundary. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
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      "full_conditions_and_proof": "### Operator topology tools\n\nThe weak operator topology tests \\(T\\mapsto\\langle T\\eta,\\zeta\\rangle\\), and the strong operator topology tests \\(T\\mapsto\\|T\\eta\\|\\). We will use the following facts for arbitrary Hilbert spaces.\n\n**Lemma 9.0.**\n\n1. The closed unit ball of \\(B(H)\\) is compact in the weak operator topology. Consequently every bounded weakly closed set of operators is weakly compact.\n2. The weakly continuous and strongly continuous linear functionals on \\(B(H)\\) are exactly the finite sums of vector functionals. Every convex set has the same weak and strong closures.\n3. On a norm-bounded set, the weak and \\(\\sigma\\)-weak topologies agree. Multiplication by a fixed operator on either side is continuous for the \\(\\sigma\\)-weak topology. Multiplication by a fixed \\(g\\in L^\\infty(\\mu)\\) is continuous for \\(\\sigma(L^\\infty,L^1)\\).\n\n**Proof.** (1) Map an operator of norm at most one to all its coefficients \\(\\langle T\\eta,\\zeta\\rangle\\), in the product of the closed discs of radii \\(\\|\\eta\\|\\|\\zeta\\|\\), indexed by \\((\\eta,\\zeta)\\in H^2\\). The product is compact by [Tychonoff's theorem](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-02). Its points belonging to the image are exactly the coefficient arrays that are linear in \\(\\eta\\) and conjugate-linear in \\(\\zeta\\). These identities are closed conditions. Every such array is a bounded sesquilinear form, with bound one supplied by the discs, and is therefore the coefficient array of an operator of norm at most one by [Hilbert representation for bounded forms](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-03), Theorem 3.1. The coefficient map is injective, and the weak operator topology is precisely the topology it induces from the product. Its closed image is compact. A bounded weakly closed set lies in a scalar multiple of this compact ball and is closed there.\n\n(2) By [the duality theorem for weak topologies](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-01), Theorem 1.2, the weakly continuous linear functionals are the finite sums of vector functionals. They are strongly continuous by Cauchy–Schwarz. Conversely, strong continuity of a linear functional \\(F\\) gives finitely many vectors \\(\\eta_1,\\dots,\\eta_n\\) and \\(C>0\\) with \\(|F(T)|\\leq C\\max_i\\|T\\eta_i\\|\\): use a basic neighbourhood on which \\(|F|<1\\) and scale \\(T\\); when all \\(T\\eta_i=0\\), scaling forces \\(F(T)=0\\). Thus \\(F\\) factors as a bounded linear functional on the subspace of \\(H^n\\) consisting of \\(R(T)=(T\\eta_i)_i\\). Its bound for the Hilbert product norm follows from \\(\\max_i\\|T\\eta_i\\|\\leq\\|R(T)\\|\\). Extend it to \\(H^n\\) by [Hahn–Banach](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-02), and apply [the Riesz–Fréchet theorem](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-02), Theorem 2.3, to obtain vectors \\(\\zeta_i\\) with\n\\[\nF(T)=\\sum_{i=1}^n\\langle T\\eta_i,\\zeta_i\\rangle .\n\\]\nThe real continuous linear functionals are the real parts of these: for a real-linear \\(G\\), use the complex-linear functional \\(F(T)=G(T)-iG(iT)\\). Hence the two locally convex topologies also have the same real continuous functionals. [Theorem 4.2 on closures of convex sets](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-04), whose proof separates a point from a closed convex set, gives the closure equality.\n\n(3) A \\(\\sigma\\)-weak test is \\(T\\mapsto\\sum_n\\langle T\\eta_n,\\zeta_n\\rangle\\), with the two vector sequences square summable. On a ball of radius \\(M\\), its tail is bounded uniformly by \\(M\\sum_{n>N}\\|\\eta_n\\|\\|\\zeta_n\\|\\), which tends to zero by Cauchy–Schwarz. It is a uniform limit there of weakly continuous finite sums, proving the topology equality. For fixed \\(a,b\\), the test after \\(T\\mapsto aTb\\) has vectors \\(b\\eta_n,a^*\\zeta_n\\), again square summable; this proves continuity. Finally, \\(\\int (gf)u\\,d\\mu=\\int f(gu)\\,d\\mu\\), with \\(gu\\in L^1\\), proves the last assertion. These are also the bounded-net and multiplication arguments of [Lemma 1.2 of the double-commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-01). \\(\\square\\)\n\n\n**Lemma 9.1** (Normality test). Let \\(\\mu\\in M^+(X)\\) and let \\(\\kappa:L^\\infty(X,\\mu)\\to B(H)\\) be a bounded linear map. Suppose that for \\(\\eta,\\zeta\\) in a dense subspace \\(D\\subseteq H\\) the functional \\(f\\mapsto\\langle\\kappa(f)\\eta,\\zeta\\rangle\\) has the form \\(f\\mapsto\\int fu\\,d\\mu\\) with \\(u\\in L^1(\\mu)\\). Then the same holds for all \\(\\eta,\\zeta\\in H\\), and for all sums \\(f\\mapsto\\sum_n\\langle\\kappa(f)\\eta_n,\\zeta_n\\rangle\\) with \\(\\sum_n\\|\\eta_n\\|^2<\\infty\\) and \\(\\sum_n\\|\\zeta_n\\|^2<\\infty\\). So \\(\\kappa\\) is continuous from the weak\\(^*\\) topology of \\(L^\\infty(\\mu)\\) to the \\(\\sigma\\)-weak topology; we call it *normal*.\n\n**Proof.** The map sending \\(u\\in L^1(\\mu)\\) to the functional \\(f\\mapsto\\int fu\\,d\\mu\\) is isometric (test with \\(f=\\bar u/|u|\\) where \\(u\\neq0\\)), so its range \\(N\\) is norm-closed in \\(L^\\infty(\\mu)^*\\). If \\(\\eta_k\\to\\eta\\) and \\(\\zeta_k\\to\\zeta\\) with \\(\\eta_k,\\zeta_k\\in D\\), then \\[\n\\begin{gathered}\n|\\langle\\kappa(f)\\eta,\\zeta\\rangle\\\\\n-\\langle\\kappa(f)\\eta_k,\\zeta_k\\rangle|\\\\\n\\leq\\|\\kappa\\|\\|f\\|(\\|\\eta-\\eta_k\\|\\|\\zeta\\|\\\\\n+\\|\\eta_k\\|\\|\\zeta-\\zeta_k\\|)\n\\end{gathered}\n\\], a norm convergence of functionals. So the limit lies in \\(N\\). The series converges in norm because \\(\\sum_n\\|\\eta_n\\|\\|\\zeta_n\\|<\\infty\\). \\(\\square\\)\n\n**Proposition 9.2.** Let \\(\\mu\\in M^+_1(\\mathfrak S)\\) have barycentre \\(\\varphi\\), and write \\(\\pi,H,\\xi\\) for its GNS triple. There is exactly one linear map \\(\\kappa_\\mu:L^\\infty(\\mathfrak S,\\mu)\\to\\pi(A)'\\) with\n\\[\n\\begin{gathered}\n\\langle\\kappa_\\mu(f)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\int_{\\mathfrak S}f(\\omega)\\,\\omega(a)\\,d\\mu(\\omega)\\\\\n(f\\in L^\\infty(\\mathfrak S,\\mu),\\ a\\in A).\n\\end{gathered}\n\\tag{9.1}\n\\]\nIt is positive, \\(\\kappa_\\mu(1)=1\\), \\(\\|\\kappa_\\mu(f)\\|\\leq\\|f\\|_\\infty\\) for real \\(f\\), and \\(\\kappa_\\mu\\) is normal.\n\n**Proof.** An operator \\(x\\in\\pi(A)'\\) is determined by the functional \\(\\Theta_\\varphi(x)(a)=\\langle\\pi(a)x\\xi,\\xi\\rangle=\\langle x\\pi(a)\\xi,\\xi\\rangle\\) ([Proposition 8.1](#oa-fnd-ir-09)(3)). This gives uniqueness. For existence let \\(\\kappa'(f)(a)=\\int f\\hat a\\,d\\mu\\). If \\(0\\leq f\\leq c\\), then \\(\\kappa'(f)\\) is a positive functional and \\(\\kappa'(f)\\leq c\\varphi\\), because \\(\\omega(a^*a)\\geq0\\). By Proposition 8.1(3) there is a unique \\(x_f\\in\\pi(A)'\\) with \\(\\Theta_\\varphi(x_f)=\\kappa'(f)\\), and \\(0\\leq x_f\\leq c\\). Every \\(f\\in L^\\infty\\) is a combination \\(\\sum_{k=0}^3i^kf_k\\) with \\(f_k\\geq0\\), so \\(\\kappa'(f)\\) lies in the range of \\(\\Theta_\\varphi\\), and \\(\\kappa_\\mu=\\Theta_\\varphi^{-1}\\circ\\kappa'\\) is linear, positive and satisfies (9.1). \\(\\kappa_\\mu(1)=1\\) because \\(\\kappa'(1)=\\varphi=\\Theta_\\varphi(1)\\). For real \\(f\\), \\(-\\|f\\|\\leq f\\leq\\|f\\|\\) gives \\(-\\|f\\|\\leq\\kappa_\\mu(f)\\leq\\|f\\|\\). Finally, for \\(\\eta=\\pi(a)\\xi\\) and \\(\\zeta=\\pi(b)\\xi\\),\n\\[\n\\langle\\kappa_\\mu(f)\\eta,\\zeta\\rangle=\\langle\\kappa_\\mu(f)\\pi(b^*a)\\xi,\\xi\\rangle=\\int f\\,\\widehat{b^*a}\\,d\\mu ,\n\\]\nand \\(\\widehat{b^*a}\\) is bounded and continuous. The normality test, Lemma 9.1, applies with \\(D=\\pi(A)\\xi\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AF-01",
      "unit": "af-algebras",
      "name": "2. Finite-dimensional C\\*-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 2. Finite-dimensional C\\*-algebras\n\n**Definition 2.1.** A *size vector* is \\(\\mathbf m=(m_1,\\dots,m_r)\\in\\mathbb N^r\\). The *multimatrix algebra* of \\(\\mathbf m\\) is\n\\[\nM_{\\mathbf m}=M_{m_1}\\oplus\\cdots\\oplus M_{m_r}.\n\\]\nWe write its elements as \\(x=(x_1,\\dots,x_r)\\) and call \\(M_{m_i}\\) its \\(i\\)-th *summand*. We allow \\(r=0\\), and then \\(M_{\\mathbf m}=0\\). Let \\(z_i\\) be the unit of the \\(i\\)-th summand and \\(e^{(i)}_{ab}\\) its matrix units. The minimal projections of \\(M_{\\mathbf m}\\) are the rank-one projections of the single summands. The *rank vector* of a projection \\(p=(p_1,\\dots,p_r)\\) is\n\\[\n\\begin{gathered}\n\\operatorname{rk}p\\\\\n=(\\operatorname{Tr}p_1,\\dots,\\operatorname{Tr}p_r)^T\\in\\mathbb Z^r_+,\\\\\n0\\\\\n\\leq\\operatorname{rk}p\\\\\n\\leq\\mathbf m .\n\\end{gathered}\n\\]\n\nThe first aim is to show that every finite-dimensional C\\*-algebra is isomorphic to some \\(M_{\\mathbf m}\\). We do not assume a unit.\n\n**Lemma 2.2.** Let \\(F\\) be a finite-dimensional C\\*-algebra.\n\n1. Every self-adjoint element of \\(F\\) is a real linear combination of mutually orthogonal projections of \\(F\\). In particular \\(F\\) is spanned by its projections.\n2. Mutually orthogonal nonzero projections of \\(F\\) are linearly independent, so there are at most \\(\\dim F\\) of them. Every nonzero projection of \\(F\\) dominates a minimal projection of \\(F\\).\n3. \\(F\\) has a unit.\n\n**Proof.** (1) Let \\(x=x^*\\in F\\) and \\(N=\\dim F\\). The elements \\(x,x^2,\\dots,x^{N+1}\\) are linearly dependent, so \\(Q(x)=0\\) for a nonzero polynomial \\(Q\\) with \\(Q(0)=0\\). By the spectral mapping property (Section 1(c)), \\(Q\\) vanishes on \\(\\sigma(x)\\), so \\(\\sigma(x)\\) is a finite set of real numbers. Let \\(\\lambda_1,\\dots,\\lambda_k\\) be its nonzero points, and let \\(f_j\\) be the function on \\(\\sigma(x)\\) that equals \\(1\\) at \\(\\lambda_j\\) and \\(0\\) elsewhere. It is continuous because \\(\\sigma(x)\\) is finite, and \\(f_j(0)=0\\). So \\(p_j=f_j(x)\\) lies in \\(F\\). Since \\(f_j=f_j^2=\\bar f_j\\) and \\(f_jf_l=0\\) for \\(j\\neq l\\), the \\(p_j\\) are mutually orthogonal projections. Since \\(t=\\sum_j\\lambda_jf_j(t)\\) on \\(\\sigma(x)\\), we get \\(x=\\sum_j\\lambda_jp_j\\). Every \\(y\\in F\\) is the combination \\(\\frac12(y+y^*)+i\\cdot\\frac1{2i}(y-y^*)\\) of two self-adjoint elements.\n\n(2) If \\(\\sum_jc_jp_j=0\\) with mutually orthogonal nonzero projections \\(p_j\\), multiplying by \\(p_l\\) gives \\(c_lp_l=0\\), so \\(c_l=0\\). For a projection \\(q\\) let \\(\\nu(q)\\) be the largest number of mutually orthogonal nonzero subprojections of \\(q\\); so \\(1\\leq\\nu(q)\\leq\\dim F\\) when \\(q\\neq0\\). If \\(q'\\) is a nonzero subprojection of \\(q\\) with \\(q'\\neq q\\), then \\(q-q'\\) is a nonzero subprojection of \\(q\\) orthogonal to \\(q'\\), so \\(\\nu(q)\\geq\\nu(q')+1\\). Hence, among the nonzero subprojections of a nonzero projection \\(p\\), one with the smallest value of \\(\\nu\\) is minimal.\n\n(3) We may assume \\(F\\neq0\\). Choose a projection \\(p\\in F\\) with \\(\\nu(p)\\) as large as possible. Let \\(q\\in F\\) be a projection, \\(y=q-qp\\) and \\(x=y^*y\\). Then \\(yp=0\\), so \\(xp=0\\) and \\(px=(xp)^*=0\\). Suppose \\(x\\neq0\\). By (1), \\(x=\\sum_j\\lambda_jp_j\\) with nonzero \\(\\lambda_j\\) and spectral projections \\(p_j=f_j(x)\\); here \\(p_1\\neq0\\), because its spectrum \\(f_1(\\sigma(x))\\) contains \\(1\\). On the finite set \\(\\sigma(x)\\) the function \\(f_1\\) agrees with a polynomial without constant term, so \\(p_1\\) is a polynomial in \\(x\\) without constant term, and \\(pp_1=0\\). Then \\(p+p_1\\) is a projection with \\(\\nu(p+p_1)\\geq\\nu(p)+1\\), which is impossible. So \\(x=0\\), hence \\(y=0\\) because \\(\\|y\\|^2=\\|x\\|\\). This means \\(q=qp\\), and, taking adjoints, \\(q=pq\\). By (1), \\(pz=zp=z\\) for every \\(z\\in F\\). \\(\\square\\)\n\n**Lemma 2.3** (Full matrix algebras). Let \\(E\\) be a nonzero finite-dimensional C\\*-algebra whose centre is \\(\\mathbb C1_E\\). Let \\(e\\) be a minimal projection of \\(E\\). Then there are \\(v_1=e,v_2,\\dots,v_k\\in E\\) with \\(v_a^*v_a=e\\) for all \\(a\\) and \\(\\sum_av_av_a^*=1_E\\). The elements \\(e_{ab}=v_av_b^*\\) satisfy \\(e_{ab}e_{cd}=\\delta_{bc}e_{ad}\\) and \\(e_{ab}^*=e_{ba}\\), and \\((\\xi_{ab})\\mapsto\\sum_{a,b}\\xi_{ab}e_{ab}\\) is an isomorphism of \\(M_k\\) onto \\(E\\).\n\n**Proof.** The corner \\(eEe\\) is a C\\*-algebra with unit \\(e\\). By Lemma 2.2(1) it is spanned by its projections, and each of them is a subprojection of \\(e\\), hence \\(0\\) or \\(e\\). So\n\\[\neEe=\\mathbb Ce .\n\\tag{2.1}\n\\]\nLet \\(J\\) be the linear span of the products \\(xey\\) with \\(x,y\\in E\\). It is a two-sided ideal, closed under adjoints and finite-dimensional, hence closed. So \\(J\\) is a nonzero C\\*-algebra, and by Lemma 2.2(3) it has a unit \\(c\\), which is a projection. For \\(x\\in E\\), both \\(cx\\) and \\(xc\\) lie in \\(J\\), so \\(cx=cxc=xc\\). Thus \\(c\\) is central and nonzero, so \\(c=1_E\\), and\n\\[\n\\begin{gathered}\n1_E\\\\\n=\\textstyle\\sum_lx_ley_l\\\\\n\\text{for some }x_l,y_l\\in E .\n\\end{gathered}\n\\tag{2.2}\n\\]\nIf \\(v^*v\\) is a projection, then \\(vv^*v=v\\), because \\[\n\\begin{gathered}\n(v-vv^*v)^*(v-vv^*v)\\\\\n=v^*v-2(v^*v)^2+(v^*v)^3\\\\\n=0;\n\\end{gathered}\n\\] hence \\((vv^*)^2=(vv^*v)v^*=vv^*\\), and \\(vv^*\\) is a projection too. Choose \\(v_1=e,v_2,\\dots,v_k\\) in \\(E\\) with \\(v_a^*v_a=e\\) and with the projections \\(v_av_a^*\\) mutually orthogonal, and with \\(k\\) as large as possible; \\(k\\leq\\dim E\\) by Lemma 2.2(2). Put \\(f=1_E-\\sum_av_av_a^*\\), a projection. Suppose \\(f\\neq0\\). By (2.2), \\(f=f1_Ef=\\sum_l(fx_le)(y_lf)\\), so \\(w=fxe\\neq0\\) for some \\(x\\in E\\). By (2.1), \\(w^*w=\\lambda e\\) with \\(\\lambda=\\|w\\|^2>0\\). Then \\(v=\\lambda^{-1/2}w\\) satisfies \\(v^*v=e\\) and \\(fv=v\\), so \\(vv^*\\) is a projection below \\(f\\), orthogonal to every \\(v_av_a^*\\). This contradicts the maximality of \\(k\\). Hence \\(\\sum_av_av_a^*=1_E\\).\n\nFor \\(a\\neq b\\), \\(v_b^*v_a=v_b^*(v_bv_b^*)(v_av_a^*)v_a=0\\), and \\(v_a^*v_a=e\\), \\(v_ae=v_a\\). So \\(e_{ab}e_{cd}=v_a(v_b^*v_c)v_d^*=\\delta_{bc}v_aev_d^*=\\delta_{bc}e_{ad}\\), and clearly \\(e_{ab}^*=e_{ba}\\) and \\(e_{aa}=v_av_a^*\\). The map in the statement is therefore a \\*-homomorphism. For \\(x\\in E\\), the element \\(e_{1a}xe_{b1}\\) lies in \\(eEe\\), so \\(e_{1a}xe_{b1}=\\xi_{ab}(x)e\\) by (2.1), and\n\\[\n\\begin{gathered}\nx\\\\\n=\\sum_{a,b}e_{aa}xe_{bb}\\\\\n=\\sum_{a,b}e_{a1}(e_{1a}xe_{b1})e_{1b}\\\\\n=\\sum_{a,b}\\xi_{ab}(x)e_{ab}.\n\\end{gathered}\n\\]\nSo the map is onto. It is injective, because \\(e_{1c}\\big(\\sum_{a,b}\\xi_{ab}e_{ab}\\big)e_{d1}=\\xi_{cd}e\\). \\(\\square\\)\n\n**Theorem 2.4** (Structure of finite-dimensional C\\*-algebras). Every finite-dimensional C\\*-algebra \\(F\\) is isomorphic to \\(M_{\\mathbf m}\\) for some size vector \\(\\mathbf m\\). The number \\(r\\) of summands is the dimension of the centre of \\(F\\), and the entries of \\(\\mathbf m\\) are determined by \\(F\\) up to their order.\n\n**Proof.** For \\(F=0\\), use the empty size vector; the centre and the number of summands both have dimension zero. Assume \\(F\\neq0\\). Let \\(Z\\) be the centre of \\(F\\). It contains the unit \\(1_F\\) (Lemma 2.2(3)) and is a finite-dimensional commutative C\\*-algebra. Choose mutually orthogonal minimal projections \\(z_1,\\dots,z_r\\) of \\(Z\\) with \\(r\\) as large as possible (Lemma 2.2(2) in \\(Z\\)). Then \\(\\sum_iz_i=1_F\\): otherwise \\(1_F-\\sum_iz_i\\) is a nonzero projection of \\(Z\\), it dominates a minimal projection of \\(Z\\), and that projection is orthogonal to every \\(z_i\\).\n\nEach \\(z_iF\\) is a C\\*-subalgebra with unit \\(z_i\\), \\(z_iF\\cdot z_jF=0\\) for \\(i\\neq j\\), and \\(x=\\sum_iz_ix\\). So \\(x\\mapsto(z_1x,\\dots,z_rx)\\) is an isomorphism of \\(F\\) onto \\(z_1F\\oplus\\cdots\\oplus z_rF\\). If \\(c\\in z_iF\\) commutes with \\(z_iF\\), it commutes with every \\(x=\\sum_jz_jx\\), since \\(cz_jx=0=z_jxc\\) for \\(j\\neq i\\). So \\(c\\in Z\\) and \\(c\\in z_iZ\\). By Lemma 2.2(1), \\(z_iZ\\) is spanned by projections below \\(z_i\\), which are \\(0\\) or \\(z_i\\) by minimality. So the centre of \\(z_iF\\) is \\(\\mathbb Cz_i\\), and Lemma 2.3 gives \\(z_iF\\cong M_{m_i}\\). Hence \\(F\\cong M_{\\mathbf m}\\).\n\nThe centre of \\(M_k\\) is \\(\\mathbb C1_k\\), since a matrix that commutes with all \\(e_{ab}\\) is scalar. So the centre of \\(M_{\\mathbf m}\\) consists of the elements \\((\\lambda_11_{m_1},\\dots,\\lambda_r1_{m_r})\\); it has dimension \\(r\\), its minimal projections are \\(z_1,\\dots,z_r\\), and \\(z_iM_{\\mathbf m}\\cong M_{m_i}\\) has dimension \\(m_i^2\\). An isomorphism carries centre to centre and minimal central projections to minimal central projections. So \\(r\\) and the numbers \\(m_i\\), up to order, are determined by \\(F\\). \\(\\square\\)\n\n**Lemma 2.5** (Projections of a multimatrix algebra). For projections \\(p,q\\in M_{\\mathbf m}\\) the following are equivalent: (i) \\(\\operatorname{rk}p=\\operatorname{rk}q\\); (ii) \\(p\\sim q\\); (iii) \\(q=upu^*\\) for some \\(u\\in U(M_{\\mathbf m})\\). The rank vectors of the projections of \\(M_{\\mathbf m}\\) are exactly the \\(x\\in\\mathbb Z^r\\) with \\(0\\leq x\\leq\\mathbf m\\).\n\n**Proof.** (i)⇒(iii). In each summand, the ranges of \\(p_i\\) and \\(q_i\\) have the same dimension, and so do their orthogonal complements. The unitary \\(u_i\\) of \\(\\mathbb C^{m_i}\\) that carries an orthonormal basis of the range of \\(p_i\\) to one of the range of \\(q_i\\), and an orthonormal basis of the complement of the first range to one of the complement of the second, satisfies \\(u_ip_iu_i^*=q_i\\). (iii)⇒(ii). Put \\(v=up\\): then \\(v^*v=p\\) and \\(vv^*=upu^*=q\\). (ii)⇒(i). If \\(v^*v=p\\) and \\(vv^*=q\\), then \\(v_i\\) maps the range of \\(p_i\\) isometrically onto the range of \\(q_i\\), so the ranks agree. The last statement follows by looking at diagonal projections. \\(\\square\\)\n\n",
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    {
      "id": "OA-FND-AF-02",
      "unit": "af-algebras",
      "name": "3. Homomorphisms between finite-dimensional C\\*-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 3. Homomorphisms between finite-dimensional C\\*-algebras\n\nA reference for this section is [Blackadar 1998, Section 7.2].\n\n**Definition 3.1.** Let \\(\\mathbf m\\in\\mathbb N^r\\), \\(\\mathbf n\\in\\mathbb N^s\\), and let \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) be a \\*-homomorphism. Write \\(\\varphi(x)=(\\varphi_1(x),\\dots,\\varphi_s(x))\\). The *multiplicity matrix* of \\(\\varphi\\) is the \\(s\\times r\\) matrix \\(\\alpha(\\varphi)\\) with entries\n\\[\n\\alpha(\\varphi)_{ji}=\\operatorname{Tr}\\varphi_j\\big(e^{(i)}_{11}\\big),\n\\]\nthe number of times the \\(i\\)-th summand of \\(M_{\\mathbf m}\\) enters the \\(j\\)-th summand of \\(M_{\\mathbf n}\\). Its *defect vector* is \\(d(\\varphi)=\\mathbf n-\\alpha(\\varphi)\\mathbf m\\).\n\n**Theorem 3.2** (Homomorphisms between multimatrix algebras). Let \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) be a \\*-homomorphism with multiplicity matrix \\(\\alpha=\\alpha(\\varphi)\\).\n\n1. For every projection \\(p\\in M_{\\mathbf m}\\),\n\\[\n\\operatorname{rk}\\varphi(p)=\\alpha\\operatorname{rk}p .\n\\tag{3.1}\n\\]\nIn particular \\(\\alpha_{ji}=\\operatorname{Tr}\\varphi_j(e)\\) for every minimal projection \\(e\\) of the \\(i\\)-th summand. If \\(\\psi:M_{\\mathbf n}\\to M_{\\mathbf k}\\) is another \\*-homomorphism, then \\(\\alpha(\\psi\\circ\\varphi)=\\alpha(\\psi)\\alpha(\\varphi)\\).\n2. \\(\\alpha\\mathbf m\\leq\\mathbf n\\), that is, \\(d(\\varphi)\\geq0\\), with equality if and only if \\(\\varphi\\) is unital. The map \\(\\varphi\\) is injective if and only if no column of \\(\\alpha\\) is zero.\n3. Conversely, let \\(\\alpha\\) be any \\(s\\times r\\) matrix with entries in \\(\\mathbb Z_+\\) and \\(\\alpha\\mathbf m\\leq\\mathbf n\\), and put \\(d=\\mathbf n-\\alpha\\mathbf m\\). The *standard homomorphism*\n\\[\n\\begin{gathered}\n\\varphi_\\alpha(x)_j\\\\\n=\\operatorname{diag}\\big(x_1^{(\\alpha_{j1})},x_2^{(\\alpha_{j2})},\\\\\n\\dots,x_r^{(\\alpha_{jr})},0_{d_j}\\big),\\\\\nj=1,\\dots,s,\n\\end{gathered}\n\\tag{3.2}\n\\]\nis a \\*-homomorphism with multiplicity matrix \\(\\alpha\\).\n4. There is \\(u\\in U(M_{\\mathbf n})\\) with \\(\\varphi=\\operatorname{Ad}u\\circ\\varphi_\\alpha\\). Consequently two \\*-homomorphisms \\(M_{\\mathbf m}\\to M_{\\mathbf n}\\) are unitarily equivalent if and only if they have the same multiplicity matrix.\n\n**Proof.** (1) Let \\(e\\) be a minimal projection of the \\(i\\)-th summand. There is a partial isometry \\(w\\) in that summand with \\(w^*w=e^{(i)}_{11}\\) and \\(ww^*=e\\). Then \\(\\varphi_j(w)\\) is a partial isometry from \\(\\varphi_j(e^{(i)}_{11})\\) onto \\(\\varphi_j(e)\\), so these projections have the same rank (Lemma 2.5 in \\(M_{n_j}\\)). A projection \\(p\\) of \\(M_{\\mathbf m}\\) is a sum of mutually orthogonal minimal projections, \\(\\operatorname{Tr}p_i\\) of them in the \\(i\\)-th summand (diagonalize each \\(p_i\\)). Images of orthogonal projections are orthogonal projections, and rank is additive on orthogonal sums. So \\(\\operatorname{Tr}\\varphi_j(p)=\\sum_i\\alpha_{ji}\\operatorname{Tr}p_i\\), which is (3.1). Applying (3.1) twice to \\(p=e^{(i)}_{11}\\) gives \\(\\operatorname{rk}\\psi(\\varphi(p))=\\alpha(\\psi)\\alpha(\\varphi)e_i\\), the \\(i\\)-th column of \\(\\alpha(\\psi)\\alpha(\\varphi)\\).\n\n(2) By (3.1) with \\(p=1\\), \\(\\operatorname{rk}\\varphi(1)=\\alpha\\mathbf m\\). A projection of \\(M_{n_j}\\) has rank at most \\(n_j\\), with equality only for \\(1_{n_j}\\). So \\(\\alpha\\mathbf m\\leq\\mathbf n\\), with equality exactly when \\(\\varphi(1)=1\\). If the \\(i\\)-th column of \\(\\alpha\\) is zero, then \\(\\varphi(e^{(i)}_{11})=0\\) and \\(\\varphi\\) is not injective. Suppose no column is zero, and let \\(\\varphi(x)=0\\). If \\(x\\neq0\\), some \\(x_i\\) has a nonzero entry \\(\\xi\\) in position \\((a,b)\\). Then \\(e^{(i)}_{1a}xe^{(i)}_{b1}=\\xi e^{(i)}_{11}\\), so \\(\\varphi(e^{(i)}_{11})=\\xi^{-1}\\varphi(e^{(i)}_{1a})\\varphi(x)\\varphi(e^{(i)}_{b1})=0\\), and the \\(i\\)-th column of \\(\\alpha\\) is zero, a contradiction.\n\n(3) Each map \\(x\\mapsto x_i^{(b)}\\) is a \\*-homomorphism, and so is a block-diagonal combination of them. The block sizes in (3.2) add up to \\(\\sum_i\\alpha_{ji}m_i+d_j=n_j\\). The image of \\(e^{(i)}_{11}\\) in the \\(j\\)-th summand consists of \\(\\alpha_{ji}\\) copies of a rank-one projection, so its trace is \\(\\alpha_{ji}\\).\n\n(4) Fix \\(j\\) and write \\(\\phi=\\varphi_j:M_{\\mathbf m}\\to M_{n_j}=B(\\mathbb C^{n_j})\\). For each \\(i\\), choose an orthonormal basis \\(\\xi^{(i)}_1,\\dots,\\xi^{(i)}_{\\alpha_{ji}}\\) of the range of \\(\\phi(e^{(i)}_{11})\\), and put\n\\[\n\\eta^{(i)}_{t,b}=\\phi\\big(e^{(i)}_{b1}\\big)\\xi^{(i)}_t,\\qquad 1\\leq t\\leq\\alpha_{ji},\\ 1\\leq b\\leq m_i .\n\\]\nThese vectors are orthonormal: vectors with different \\(i\\) are orthogonal because \\(e^{(i')}_{1c}e^{(i)}_{b1}=0\\) for \\(i\\neq i'\\), and\n\\[\n\\begin{gathered}\n\\big\\langle\\phi(e^{(i)}_{b1})\\xi^{(i)}_t,\\phi(e^{(i)}_{c1})\\xi^{(i)}_u\\big\\rangle\\\\\n=\\big\\langle\\phi(e^{(i)}_{1c}e^{(i)}_{b1})\\xi^{(i)}_t,\\xi^{(i)}_u\\big\\rangle\\\\\n=\\delta_{bc}\\big\\langle\\xi^{(i)}_t,\\xi^{(i)}_u\\big\\rangle\\\\\n=\\delta_{bc}\\delta_{tu}.\n\\end{gathered}\n\\]\nThe range of \\(\\phi(e^{(i)}_{bb})=\\phi(e^{(i)}_{b1})\\phi(e^{(i)}_{11})\\phi(e^{(i)}_{1b})\\) is the image of the range of \\(\\phi(e^{(i)}_{11})\\) under \\(\\phi(e^{(i)}_{b1})\\), so it is spanned by the \\(\\eta^{(i)}_{t,b}\\) with this \\(b\\). Hence the \\(\\eta\\)'s span the range of \\(\\phi(1)=\\sum_{i,b}\\phi(e^{(i)}_{bb})\\). Complete them by an orthonormal basis \\(\\zeta_1,\\dots,\\zeta_{d_j}\\) of the orthogonal complement of that range, where \\(d_j=n_j-\\sum_i\\alpha_{ji}m_i\\). Order the basis as follows: for \\(i=1,\\dots,r\\) and \\(t=1,\\dots,\\alpha_{ji}\\), the block \\(\\eta^{(i)}_{t,1},\\dots,\\eta^{(i)}_{t,m_i}\\); then \\(\\zeta_1,\\dots,\\zeta_{d_j}\\). We have \\(\\phi(e^{(i)}_{bc})\\eta^{(i)}_{t,c'}=\\phi(e^{(i)}_{bc}e^{(i)}_{c'1})\\xi^{(i)}_t=\\delta_{cc'}\\eta^{(i)}_{t,b}\\), \\(\\phi(e^{(i)}_{bc})\\eta^{(i')}_{t,c'}=0\\) for \\(i'\\neq i\\), and \\(\\phi(y)\\zeta_l=\\phi(y)\\phi(1)\\zeta_l=0\\) for every \\(y\\). So in this basis \\(\\phi(x)\\) is the matrix \\(\\varphi_\\alpha(x)_j\\) of (3.2). If \\(U_j\\) is the unitary that carries the standard basis of \\(\\mathbb C^{n_j}\\) to this ordered basis, then \\(\\varphi_j=\\operatorname{Ad}U_j\\circ(\\varphi_\\alpha)_j\\). Put \\(u=(U_1,\\dots,U_s)\\).\n\nIf \\(\\alpha(\\varphi)=\\alpha(\\psi)=\\alpha\\), then \\(\\varphi=\\operatorname{Ad}u\\circ\\varphi_\\alpha\\) and \\(\\psi=\\operatorname{Ad}v\\circ\\varphi_\\alpha\\), so \\(\\psi=\\operatorname{Ad}(vu^*)\\circ\\varphi\\). Conversely, conjugation by a unitary preserves ranks. \\(\\square\\)\n\n**Remark 3.3** (Representations). Take \\(s=1\\): a \\*-homomorphism \\(\\pi:M_{\\mathbf m}\\to M_N=B(\\mathbb C^N)\\) is a representation of \\(M_{\\mathbf m}\\) on \\(\\mathbb C^N\\). Its multiplicity matrix is a row \\((a_1,\\dots,a_r)\\), and by Theorem 3.2(4), after a unitary change of basis,\n\\[\n\\begin{gathered}\n\\mathbb C^N\\\\\n=\\bigoplus_i\\mathbb C^{m_i}\\otimes\\mathbb C^{a_i}\\ \\oplus\\ \\mathbb C^{d},\\\\\n\\pi(x)\\\\\n=\\bigoplus_ix_i\\otimes1\\ \\oplus\\ 0_d ,\n\\end{gathered}\n\\]\nwith \\(d=N-\\sum_ia_im_i\\). The representation is unital exactly when \\(d=0\\), and faithful exactly when every \\(a_i\\geq1\\). A unital representation has no invariant subspaces other than \\(0\\) and \\(\\mathbb C^N\\) exactly when one \\(a_i\\) equals \\(1\\) and the others vanish: every subspace \\(\\mathbb C^{m_i}\\otimes f\\) with \\(f\\in\\mathbb C^{a_i}\\) is invariant, and conversely \\(\\mathbb C^{m_i}\\) with \\(x\\mapsto x_i\\) has no nontrivial invariant subspace. This also accounts for irreducible representations on arbitrary Hilbert spaces. For a nonzero irreducible \\(\\pi\\), the projection \\(\\pi(1)\\) must be \\(1_H\\), since its range and kernel reduce \\(\\pi\\), and the representation is nonzero. The central projections \\(\\pi(z_i)\\) then show that exactly one summand acts. In that summand choose a nonzero \\(\\xi\\in\\pi(e_{11})H\\): some \\(\\pi(e_{aa})\\) has nonzero range, and \\(\\pi(e_{1a})\\) carries it isometrically to \\(\\pi(e_{11})H\\). The vectors \\(\\pi(e_{a1})\\xi/\\|\\xi\\|\\), \\(1\\leq a\\leq m_i\\), are orthonormal by the calculation above. Their finite-dimensional span is invariant under every matrix unit and its adjoint, so irreducibility makes it all of \\(H\\). Thus the irreducible representations of \\(M_{\\mathbf m}\\) are, up to unitary equivalence, the \\(r\\) maps \\(x\\mapsto x_i\\) on \\(\\mathbb C^{m_i}\\).\n\n**Remark 3.4** (Changing the identification). If \\(F\\) is a finite-dimensional C\\*-algebra and \\(\\iota,\\iota':F\\to M_{\\mathbf m}\\) are two isomorphisms, then \\(\\beta=\\iota'\\circ\\iota^{-1}\\) is an automorphism of \\(M_{\\mathbf m}\\). It maps minimal projections to minimal projections, so each column of \\(\\alpha(\\beta)\\) has a single entry \\(1\\) and the others \\(0\\). By Theorem 3.2(1), \\(\\alpha(\\beta^{-1})\\alpha(\\beta)=\\alpha(\\mathrm{id})\\) is the identity matrix, so \\(\\alpha(\\beta)\\) is invertible, and it is a permutation matrix. So rank vectors and multiplicity matrices computed through \\(\\iota\\) and \\(\\iota'\\) differ only by a consistent relabelling of the summands. We may therefore speak of the summands, rank vectors and multiplicity matrices of finite-dimensional C\\*-algebras without naming the identification.\n\n**Definition 3.5** (Bratteli diagram of a homomorphism). The *Bratteli diagram* of \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) is the graph with \\(r\\) vertices on the left, labelled \\(m_1,\\dots,m_r\\), and \\(s\\) vertices on the right, labelled \\(n_1,\\dots,n_s\\), in which the \\(i\\)-th left vertex and the \\(j\\)-th right vertex are joined by \\(\\alpha(\\varphi)_{ji}\\) edges. By Theorem 3.2(4) it determines \\(\\varphi\\) up to unitary equivalence.\n\n**Example 3.6.** (a) Let \\(\\mathbf m=(1,2)\\), \\(\\mathbf n=(5,4)\\) and \\(\\alpha=\\begin{pmatrix}3&1\\\\0&2\\end{pmatrix}\\). Then \\(\\alpha\\mathbf m=(5,4)^T=\\mathbf n\\), so the standard homomorphism\n\\[\n\\begin{gathered}\n\\varphi_\\alpha(\\lambda,y)\\\\\n=\\big(\\operatorname{diag}(\\lambda,\\lambda,\\lambda,y),\\ \\operatorname{diag}(y,y)\\big),\\\\\n\\lambda\\in\\mathbb C,\\ y\\in M_2,\n\\end{gathered}\n\\]\nis unital and injective. In its diagram, the left vertex \\(1\\) is joined to the right vertex \\(5\\) by three edges, and the left vertex \\(2\\) is joined to \\(5\\) by one edge and to \\(4\\) by two edges.\n\n(b) The map \\(M_2\\to M_5\\), \\(x\\mapsto\\operatorname{diag}(x,x,0)\\), has multiplicity matrix \\((2)\\) and defect \\((1)\\): it is injective and not unital. The map \\(\\mathbb C^2\\to\\mathbb C\\), \\((\\lambda,\\mu)\\mapsto\\lambda\\), has multiplicity matrix \\((1\\ \\ 0)\\): its second column is zero, and it is not injective.\n\n(c) The two maps \\(M_2\\to M_4\\) given by \\(x\\mapsto\\operatorname{diag}(x,x)\\) and by \\(x\\mapsto\\begin{pmatrix}x_{11}1_2&x_{12}1_2\\\\x_{21}1_2&x_{22}1_2\\end{pmatrix}\\) both have multiplicity matrix \\((2)\\). By Theorem 3.2(4) they are unitarily equivalent; here the unitary is the permutation matrix that exchanges the second and third basis vectors.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AF-03",
      "unit": "af-algebras",
      "name": "4. Inductive limits and AF-algebras",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 4. Inductive limits and AF-algebras\n\nA reference for inductive limits is [Blackadar 2006, II.8.2].\n\n**Definition 4.1.** An *inductive sequence* \\((A_n,\\varphi_n)_{n\\geq1}\\) consists of C\\*-algebras \\(A_n\\) and \\*-homomorphisms \\(\\varphi_n:A_n\\to A_{n+1}\\). For \\(m>n\\) put \\(\\varphi_{m,n}=\\varphi_{m-1}\\circ\\cdots\\circ\\varphi_n:A_n\\to A_m\\), and \\(\\varphi_{n,n}=\\mathrm{id}\\). An *inductive limit* of the sequence is a C\\*-algebra \\(A\\) with \\*-homomorphisms \\(\\varphi_{\\infty,n}:A_n\\to A\\) such that\n\n1. \\(\\varphi_{\\infty,n+1}\\circ\\varphi_n=\\varphi_{\\infty,n}\\) for all \\(n\\);\n2. \\(\\bigcup_n\\varphi_{\\infty,n}(A_n)\\) is dense in \\(A\\);\n3. \\(\\|\\varphi_{\\infty,n}(a)\\|=\\lim_{m\\to\\infty}\\|\\varphi_{m,n}(a)\\|\\) for all \\(n\\) and \\(a\\in A_n\\).\n\nThe limit in (3) exists, because \\(\\|\\varphi_{m+1,n}(a)\\|=\\|\\varphi_m(\\varphi_{m,n}(a))\\|\\leq\\|\\varphi_{m,n}(a)\\|\\) by Section 1(a).\n\n**Proposition 4.2.** Let \\((A_n,\\varphi_n)\\) be an inductive sequence.\n\n1. It has an inductive limit.\n2. (*Universal property.*) Let \\((A,\\varphi_{\\infty,n})\\) be an inductive limit, \\(B\\) a C\\*-algebra and \\(\\sigma_n:A_n\\to B\\) \\*-homomorphisms with \\(\\sigma_{n+1}\\circ\\varphi_n=\\sigma_n\\). There is exactly one \\*-homomorphism \\(\\sigma:A\\to B\\) with \\(\\sigma\\circ\\varphi_{\\infty,n}=\\sigma_n\\) for all \\(n\\).\n3. If \\((A,\\varphi_{\\infty,n})\\) and \\((A',\\varphi'_{\\infty,n})\\) are inductive limits, there is exactly one isomorphism \\(\\theta:A\\to A'\\) with \\(\\theta\\circ\\varphi_{\\infty,n}=\\varphi'_{\\infty,n}\\) for all \\(n\\).\n4. If every \\(\\varphi_n\\) is injective, then every \\(\\varphi_{\\infty,n}\\) is isometric, and \\(A\\) is the closure of the increasing union of the subalgebras \\(\\varphi_{\\infty,n}(A_n)\\cong A_n\\).\n\n**Proof.** (1) Let \\(\\Pi\\) be the set of bounded sequences \\((a_k)_{k\\geq1}\\) with \\(a_k\\in A_k\\). With coordinatewise operations and the norm \\(\\sup_k\\|a_k\\|\\) it is a C\\*-algebra. To prove completeness, let \\(a^{(n)}\\) be Cauchy in this norm. Each coordinate converges in the complete algebra \\(A_k\\), to \\(a_k\\). The Cauchy estimate, passed to the limit in each coordinate, gives \\(\\sup_k\\|a_k-a_k^{(n)}\\|\\leq\\varepsilon\\) for all sufficiently large \\(n\\). The limit sequence is bounded because one fixed bounded \\(a^{(n)}\\) is within a uniform finite distance of it. Multiplication and adjoints act coordinatewise, and\n\\[\n\\|a^*a\\|=\\sup_k\\|a_k^*a_k\\|=\\sup_k\\|a_k\\|^2=\\|a\\|^2.\n\\]\nThe sequences with \\(\\|a_k\\|\\to0\\) form a two-sided \\(*\\)-ideal \\(J\\): products with bounded sequences still tend to zero. It is closed, since a uniform limit of sequences tending to zero also tends to zero, by first choosing a uniformly close sequence and then its small tail. So \\(Q=\\Pi/J\\) is a C\\*-algebra (Section 1(e)). The quotient norm of the class of \\((b_k)\\) is \\(\\limsup_k\\|b_k\\|\\): for \\((c_k)\\in J\\), \\(\\sup_k\\|b_k+c_k\\|\\geq\\limsup_k\\|b_k\\|\\), and subtracting the first \\(K\\) terms shows that the class has norm at most \\(\\sup_{k\\geq K}\\|b_k\\|\\) for every \\(K\\). Define \\(\\varphi_{\\infty,n}(a)\\) as the class of the sequence whose \\(k\\)-th term is \\(0\\) for \\(k<n\\) and \\(\\varphi_{k,n}(a)\\) for \\(k\\geq n\\). This is a \\*-homomorphism. The sequences for \\(\\varphi_{\\infty,n+1}(\\varphi_n(a))\\) and \\(\\varphi_{\\infty,n}(a)\\) differ only in the \\(n\\)-th term, so condition (1) holds, and the formula for the quotient norm gives (3). The images \\(\\varphi_{\\infty,n}(A_n)\\) increase with \\(n\\) by (1), so their union is a \\*-subalgebra, and its closure \\(A\\) is a C\\*-algebra that satisfies (2).\n\n(2) Define \\(\\sigma\\) on \\(A_0=\\bigcup_n\\varphi_{\\infty,n}(A_n)\\) by \\(\\sigma(\\varphi_{\\infty,n}(a))=\\sigma_n(a)\\). For \\(m\\geq n\\), \\(\\|\\sigma_n(a)\\|=\\|\\sigma_m(\\varphi_{m,n}(a))\\|\\leq\\|\\varphi_{m,n}(a)\\|\\), so \\(\\|\\sigma_n(a)\\|\\leq\\|\\varphi_{\\infty,n}(a)\\|\\). If \\(\\varphi_{\\infty,n}(a)=\\varphi_{\\infty,n'}(a')\\), then at a stage \\(m\\geq n,n'\\) the element \\(b=\\varphi_{m,n}(a)-\\varphi_{m,n'}(a')\\) has \\(\\varphi_{\\infty,m}(b)=0\\), hence \\(\\sigma_m(b)=0\\), that is, \\(\\sigma_n(a)=\\sigma_{n'}(a')\\). So \\(\\sigma\\) is well defined on \\(A_0\\). It is a contractive \\*-homomorphism (compute at a common stage), so it extends by continuity to \\(A\\). Uniqueness holds because \\(A_0\\) is dense.\n\n(3) Part (2) gives \\*-homomorphisms \\(\\theta:A\\to A'\\) and \\(\\theta':A'\\to A\\) compatible with the maps. Then \\(\\theta'\\theta\\) and \\(\\theta\\theta'\\) are the identity on dense subsets, hence everywhere.\n\n(4) Injective \\*-homomorphisms are isometric (Section 1(a)), so \\(\\|\\varphi_{m,n}(a)\\|=\\|a\\|\\) for all \\(m\\), and \\(\\|\\varphi_{\\infty,n}(a)\\|=\\|a\\|\\) by (3) of the definition. \\(\\square\\)\n\nWe write \\(\\varinjlim(A_n,\\varphi_n)\\) for the inductive limit. If \\(A_1\\subseteq A_2\\subseteq\\cdots\\) are C\\*-subalgebras of a C\\*-algebra \\(A\\) with dense union, then \\(A\\), with the inclusion maps, is an inductive limit of the sequence \\((A_n)\\) with the inclusions as connecting maps: condition (3) holds because inclusions are isometric.\n\n**Example 4.3** (Condition (3) matters). Let \\(A_n=C_0([n,\\infty))\\) and let \\(\\varphi_n\\) be restriction to \\([n+1,\\infty)\\). Every \\(\\varphi_n\\) is onto and every \\(A_n\\) is nonzero. For \\(a\\in A_n\\), \\(\\|\\varphi_{m,n}(a)\\|=\\sup_{t\\geq m}|a(t)|\\to0\\), because \\(a\\) vanishes at infinity. So \\(\\varphi_{\\infty,n}=0\\) for every \\(n\\), and the inductive limit is the zero algebra.\n\n**Lemma 4.4** (Ladders). Let \\((A_n,\\varphi_n)\\) and \\((B_n,\\psi_n)\\) be inductive sequences with limits \\(A\\) and \\(B\\), and let \\(\\theta_n:A_n\\to B_n\\) be \\*-homomorphisms with \\(\\theta_{n+1}\\circ\\varphi_n=\\psi_n\\circ\\theta_n\\). There is exactly one \\*-homomorphism \\(\\theta:A\\to B\\) with \\(\\theta\\circ\\varphi_{\\infty,n}=\\psi_{\\infty,n}\\circ\\theta_n\\) for all \\(n\\). If every \\(\\theta_n\\) is an isomorphism, so is \\(\\theta\\).\n\n**Proof.** The maps \\(\\sigma_n=\\psi_{\\infty,n}\\circ\\theta_n\\) satisfy \\(\\sigma_{n+1}\\varphi_n=\\psi_{\\infty,n+1}\\psi_n\\theta_n=\\psi_{\\infty,n}\\theta_n=\\sigma_n\\), so Proposition 4.2(2) gives \\(\\theta\\). If the \\(\\theta_n\\) are isomorphisms, the relations \\(\\varphi_n\\circ\\theta_n^{-1}=\\theta_{n+1}^{-1}\\circ\\psi_n\\) give in the same way \\(\\eta:B\\to A\\) with \\(\\eta\\circ\\psi_{\\infty,n}=\\varphi_{\\infty,n}\\circ\\theta_n^{-1}\\). Then \\(\\eta\\theta\\) and \\(\\theta\\eta\\) are the identity on dense subsets, hence everywhere. \\(\\square\\)\n\nThe next theorem says that conjugating the connecting maps by unitaries does not change the limit. We state it without assuming units or injectivity.\n\n**Theorem 4.5** (Unitary perturbation of the connecting maps). Let \\((A_n,\\varphi_n)\\) be an inductive sequence, and for each \\(n\\geq2\\) let \\(u_n\\) be a unitary of \\(A_n^+\\). Put \\(\\varphi_n'=\\operatorname{Ad}u_{n+1}\\circ\\varphi_n\\). Define unitaries \\(w_n\\in A_n^+\\) by\n\\[\n\\begin{gathered}\nw_1\\\\\n=1,\\\\\nw_{n+1}\\\\\n=u_{n+1}\\,\\varphi_n^+(w_n).\n\\end{gathered}\n\\tag{4.1}\n\\]\nThen \\(\\operatorname{Ad}w_{n+1}\\circ\\varphi_n=\\varphi_n'\\circ\\operatorname{Ad}w_n\\) for every \\(n\\). Consequently there is an isomorphism \\(\\theta:\\varinjlim(A_n,\\varphi_n)\\to\\varinjlim(A_n,\\varphi_n')\\) with \\(\\theta\\circ\\varphi_{\\infty,n}=\\varphi'_{\\infty,n}\\circ\\operatorname{Ad}w_n\\).\n\n**Proof.** The element \\(w_{n+1}\\) is a product of unitaries of \\(A_{n+1}^+\\), because \\(\\varphi_n^+\\) is a unital \\*-homomorphism. For \\(x\\in A_n\\), the element \\(w_nxw_n^*\\) lies in the ideal \\(A_n\\), and\n\\[\n\\begin{gathered}\n\\varphi_n'(w_nxw_n^*)\\\\\n=u_{n+1}\\varphi_n^+(w_n)\\varphi_n(x)\\varphi_n^+(w_n)^*u_{n+1}^*\\\\\n=w_{n+1}\\varphi_n(x)w_{n+1}^* .\n\\end{gathered}\n\\]\nSo the automorphisms \\(\\theta_n=\\operatorname{Ad}w_n\\) of \\(A_n\\) satisfy \\(\\theta_{n+1}\\circ\\varphi_n=\\varphi_n'\\circ\\theta_n\\), and Lemma 4.4 gives \\(\\theta\\). \\(\\square\\)\n\n**Remark 4.6** (The unital case). Suppose every \\(A_n\\) has a unit \\(1_n\\) and \\(u_{n+1}\\in U(A_{n+1})\\); the maps \\(\\varphi_n\\) need not be unital. A unitary \\(u\\) of \\(A_{n+1}\\) and the unitary \\(u+(1-1_{n+1})\\) of \\(A_{n+1}^+\\) induce the same automorphism of \\(A_{n+1}\\). Use the latter in (4.1). By induction, the unitaries of (4.1) then have the form \\(w_n+(1-1_n)\\), where \\(w_n\\in U(A_n)\\) is given by\n\\[\n\\begin{gathered}\nw_1\\\\\n=1_1,\\\\\nw_{n+1}\\\\\n=u_{n+1}\\big(\\varphi_n(w_n)+1_{n+1}-\\varphi_n(1_n)\\big);\n\\end{gathered}\n\\]\nto see this, expand \\(\\big(u_{n+1}+1-1_{n+1}\\big)\\big(\\varphi_n(w_n)+1-\\varphi_n(1_n)\\big)\\), using \\(u_{n+1}(1-1_{n+1})=0\\), \\((1-1_{n+1})\\varphi_n(w_n)=0\\) and \\(\\varphi_n(1_n)\\leq1_{n+1}\\). One can also check directly that \\(v=\\varphi_n(w_n)+1_{n+1}-\\varphi_n(1_n)\\) is a unitary of \\(A_{n+1}\\) (the sum of a unitary of the corner \\(\\varphi_n(1_n)A_{n+1}\\varphi_n(1_n)\\) and the complementary projection) and that \\(v\\varphi_n(x)v^*=\\varphi_n(w_nxw_n^*)\\). So conjugating by \\(w_n\\) intertwines \\(\\varphi_n\\) and \\(\\varphi_n'\\) inside the unital algebras \\(A_n\\).\n\n**Definition 4.7.** A C\\*-algebra \\(A\\) is an *AF-algebra* if it contains an increasing sequence \\(A_1\\subseteq A_2\\subseteq\\cdots\\) of finite-dimensional C\\*-subalgebras whose union is dense. Such a sequence is a *generating sequence* of \\(A\\), and the \\*-subalgebra \\(A_\\infty=\\bigcup_kA_k\\) is its *local algebra*.\n\n\n**Proposition 4.8.**\n\n1. The inductive limit of any inductive sequence of finite-dimensional C\\*-algebras is an AF-algebra.\n2. An AF-algebra is separable.\n3. If \\(A\\) is a unital AF-algebra with generating sequence \\((A_k)\\), then \\(1_A\\in A_k\\) for all large \\(k\\). So every unital AF-algebra has a generating sequence of C\\*-subalgebras that contain \\(1_A\\).\n\n**Proof.** (1) The images \\(\\varphi_{\\infty,n}(A_n)\\) are finite-dimensional \\*-subalgebras, hence closed; they increase, and their union is dense. (2) Rational combinations of bases of the \\(A_k\\) form a countable dense set. (3) Choose \\(k\\) and a self-adjoint \\(a\\in A_k\\) with \\(\\|1_A-a\\|<1\\) (replace an approximant \\(b\\) by \\((b+b^*)/2\\)). Let \\(1_k\\) be the unit of \\(A_k\\) (Lemma 2.2(3)). Since \\(a(1_A-1_k)=0\\),\n\\[\n1_A-1_k=(1_A-1_k)(1_A-a)(1_A-1_k),\n\\]\nso \\(\\|1_A-1_k\\|<1\\). A nonzero projection has norm one, so \\(1_A=1_k\\in A_k\\), and then \\(1_A\\in A_l\\) for all \\(l\\geq k\\). \\(\\square\\)\n\nBy Theorem 2.4, each member of a generating sequence is isomorphic to a multimatrix algebra. So an AF-algebra is described by a sequence of size vectors and multiplicity matrices.\n\n**Definition 4.9** (Bratteli diagram of a sequence). Let \\((A_k)\\) be a generating sequence of an AF-algebra, with \\(A_k\\cong M_{\\mathbf m(k)}\\), \\(\\mathbf m(k)\\in\\mathbb N^{r_k}\\), and let \\(\\alpha_k\\) be the multiplicity matrix of the inclusion \\(A_k\\subseteq A_{k+1}\\) (an \\(r_{k+1}\\times r_k\\) matrix). The *Bratteli diagram* of the sequence has, at level \\(k\\), one vertex for each summand of \\(A_k\\), labelled by its size, and \\((\\alpha_k)_{ji}\\) edges between the \\(i\\)-th vertex at level \\(k\\) and the \\(j\\)-th vertex at level \\(k+1\\). The same definition applies to any inductive sequence of multimatrix algebras. We write \\(\\alpha_{l,k}=\\alpha_{l-1}\\cdots\\alpha_k\\) for \\(l>k\\) and \\(\\alpha_{k,k}=1\\).\n\n**Corollary 4.10** (The diagram determines the algebra). Let \\((M_{\\mathbf m(k)},\\varphi_k)\\) and \\((M_{\\mathbf m(k)},\\varphi'_k)\\) be inductive sequences with the same size vectors and with \\(\\alpha(\\varphi_k)=\\alpha(\\varphi'_k)\\) for all \\(k\\). Then their inductive limits are isomorphic. Conversely, for every sequence of size vectors \\(\\mathbf m(k)\\) and matrices \\(\\alpha_k\\) with entries in \\(\\mathbb Z_+\\) and \\(\\alpha_k\\mathbf m(k)\\leq\\mathbf m(k+1)\\), the standard homomorphisms \\(\\varphi_{\\alpha_k}\\) form an inductive sequence with these multiplicity matrices; its limit is an AF-algebra, and the connecting maps are injective exactly when no \\(\\alpha_k\\) has a zero column.\n\n**Proof.** By Theorem 3.2(4), \\(\\varphi'_k=\\operatorname{Ad}u_{k+1}\\circ\\varphi_k\\) with \\(u_{k+1}\\in U(M_{\\mathbf m(k+1)})\\). By Remark 4.6 and Theorem 4.5 the limits are isomorphic. The converse follows from Theorem 3.2(2) and (3) and Proposition 4.8(1). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AF-04",
      "unit": "af-algebras",
      "name": "5. First examples",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 5. First examples\n\n**Example 5.1** (Compact operators). Let \\(H\\) be a Hilbert space with a countably infinite orthonormal basis \\((\\varepsilon_i)_{i\\geq1}\\), let \\(P_n\\) be the projection onto the span of \\(\\varepsilon_1,\\dots,\\varepsilon_n\\), and let \\(\\mathcal K\\) be the C\\*-algebra of compact operators on \\(H\\). The algebras \\(A_n=P_nB(H)P_n\\cong M_n\\) increase. Their union is dense in \\(\\mathcal K\\): for compact \\(T\\), Section 1(h) gives\n\\[\n\\begin{gathered}\n\\|T-P_nTP_n\\|\\\\\n\\leq\\|T-P_nT\\|+\\|P_n\\|\\,\\|(T^*-P_nT^*)^*\\|\\to0 .\n\\end{gathered}\n\\]\nSo \\(\\mathcal K\\) is an AF-algebra. The inclusion \\(A_n\\subseteq A_{n+1}\\) is \\(x\\mapsto\\operatorname{diag}(x,0)\\): its multiplicity matrix is \\((1)\\) and its defect is \\((1)\\). The Bratteli diagram has one vertex at each level, labelled \\(n\\) at level \\(n\\), with single edges between consecutive levels.\n\n**Example 5.2** (The unitization of the compact operators). The C\\*-algebra \\(\\mathcal K+\\mathbb C1\\subseteq B(H)\\) is isomorphic to \\(\\mathcal K^+\\): the map \\(k+\\lambda1\\mapsto k+\\lambda1_H\\) is an injective \\*-homomorphism (\\(1_H\\notin\\mathcal K\\)), hence an isomorphism onto its image (Section 1(a)). The algebras\n\\[\nA_n=P_nB(H)P_n+\\mathbb C(1-P_n)\\cong M_n\\oplus\\mathbb C\n\\]\nincrease and contain \\(1_H\\), and their union is dense in \\(\\mathcal K+\\mathbb C1\\), since \\(k+\\lambda1\\) is the limit of \\(P_nkP_n+\\lambda1\\in A_n\\). The inclusion \\(A_n\\subseteq A_{n+1}\\) is \\((x,\\lambda)\\mapsto(\\operatorname{diag}(x,\\lambda),\\lambda)\\). With the summands ordered as \\((M_n,\\mathbb C)\\), its multiplicity matrix is\n\\[\n\\alpha_n=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix},\\qquad \\alpha_n\\binom n1=\\binom{n+1}1 ,\n\\]\nso the inclusions are unital. In the Bratteli diagram, level \\(n\\) has two vertices, labelled \\(n\\) and \\(1\\); the vertex \\(n\\) is joined to the vertex \\(n+1\\) of the next level, and the vertex \\(1\\) is joined both to the vertex \\(1\\) and to the vertex \\(n+1\\) of the next level.\n\n**Example 5.3** (UHF algebras). Let \\(k_1,k_2,\\dots\\) be integers \\(\\geq2\\), put \\(d_1=1\\) and \\(d_{n+1}=k_1k_2\\cdots k_n\\), and let \\(A_n=M_{d_n}\\) with the unital connecting maps \\(x\\mapsto x^{(k_n)}\\) of multiplicity \\(k_n\\). The limit is the *UHF algebra of type \\((k_n)\\)*; its Bratteli diagram has one vertex at each level, labelled \\(d_n\\), and \\(k_n\\) edges between levels \\(n\\) and \\(n+1\\). When every \\(k_n=2\\) we get the *CAR algebra*. Identify \\(M_{d_n}=M_{2^{n-1}}\\) with the tensor product of \\(n-1\\) copies of \\(M_2\\) (for \\(n=1\\), with \\(\\mathbb C\\)). The map \\(x\\mapsto x\\otimes1\\) into the tensor product of \\(n\\) copies has multiplicity \\(2\\), so by Corollary 4.10 the CAR algebra is also the limit of the tensor powers of \\(M_2\\) under \\(x\\mapsto x\\otimes1\\): it is the infinite tensor product \\(M_2\\otimes M_2\\otimes\\cdots\\).\n\nFor a C\\*-algebra \\(A\\), let \\(M_\\infty(A)=\\bigcup_nM_n(A)\\), where \\(M_n(A)\\) sits in \\(M_{n+1}(A)\\) as the upper left corner, \\(x\\mapsto\\operatorname{diag}(x,0)\\). This inclusion is an injective \\*-homomorphism of C\\*-algebras (Section 1(f)), hence isometric, so \\(M_\\infty(A)\\) carries a norm. The completion lemma in Section 1(i) gives its Banach completion. Products and adjoints extend to it: for Cauchy sequences \\(a_n,b_n\\), their boundedness and\n\\[\n\\begin{gathered}\n\\|a_nb_n-a_mb_m\\|\\\\\n\\leq\\|a_n\\|\\,\\|b_n-b_m\\|+\\|a_n-a_m\\|\\,\\|b_m\\|\n\\end{gathered}\n\\]\nshow that \\(a_nb_n\\) is Cauchy and its class is independent of the approximants; adjoints are isometric. Associativity and the \\(C^*\\)-identity pass to the limit. Thus the completion is a \\(C^*\\)-algebra, called the *stable algebra* \\(A\\otimes\\mathcal K\\) [Blackadar 2006, II.6.6.11]; it is an inductive limit of the sequence \\((M_n(A))\\) under the corner inclusions.\n\n**Proposition 5.4** (Matrices and stabilization). Let \\(A\\) be an AF-algebra with generating sequence \\((A_k)\\), \\(A_k\\cong M_{\\mathbf m(k)}\\), and multiplicity matrices \\(\\alpha_k\\).\n\n1. For each \\(n\\), \\(M_n(A)\\) is an AF-algebra with generating sequence \\((M_n(A_k))_k\\), \\(M_n(A_k)\\cong M_{n\\mathbf m(k)}\\), and the multiplicity matrices are again the \\(\\alpha_k\\).\n2. \\(A\\otimes\\mathcal K\\) is an AF-algebra with generating sequence \\((M_k(A_k))_k\\), \\(M_k(A_k)\\cong M_{k\\mathbf m(k)}\\), and the multiplicity matrices are again the \\(\\alpha_k\\).\n\n**Proof.** \\(M_n(M_{\\mathbf m})=\\bigoplus_iM_n(M_{m_i})\\cong M_{n\\mathbf m}\\), with the same summands. A minimal projection of the \\(i\\)-th summand of \\(M_n(A_k)\\) is equivalent to \\(\\operatorname{diag}(e,0,\\dots,0)\\) with \\(e\\) minimal in the \\(i\\)-th summand of \\(A_k\\), and the rank vector of \\(\\operatorname{diag}(e,0,\\dots,0)\\) in \\(M_n(A_{k+1})\\), or in \\(M_{k+1}(A_{k+1})\\), is the rank vector of \\(e\\) in \\(A_{k+1}\\), the \\(i\\)-th column of \\(\\alpha_k\\). This gives the multiplicity matrices. Density: by the estimate in Section 1(f), every \\(x\\in M_n(A)\\) is a limit of matrices with entries in \\(A_k\\), \\(k\\to\\infty\\); and \\(M_n(A_k)\\subseteq M_k(A_k)\\) for \\(k\\geq n\\), while \\(\\bigcup_nM_n(A)\\) is dense in \\(A\\otimes\\mathcal K\\). \\(\\square\\)\n\n**Remark 5.5** (Spatial tensor products). Let \\(A\\subseteq B(H_A)\\) be faithfully represented (Section 1(g)), and let \\(\\mathcal K=\\mathcal K(\\ell^2)\\) with matrix units \\(e_{ij}\\) for the standard basis. On \\(H_A\\otimes\\ell^2=\\bigoplus_{i\\geq1}H_A\\), the operator \\(a\\otimes e_{ij}\\) is the infinite matrix with the single entry \\(a\\) in position \\((i,j)\\). The spatial tensor product \\(A\\otimes_{\\min}\\mathcal K\\) is the closed linear span of the operators \\(a\\otimes k\\), \\(a\\in A\\), \\(k\\in\\mathcal K\\). The direct-sum identification follows from the tensor basis in Section 1(i). Under it, \\(a\\otimes1\\) acts by \\(a\\) on every coordinate, so it is bounded with norm at most \\(\\|a\\|\\). Exchanging the two factors gives a unitary, because it sends the tensor orthonormal basis to the exchanged basis; in that identification \\(1\\otimes k\\) acts by \\(k\\) on every coordinate, with norm at most \\(\\|k\\|\\). On elementary tensors their product is \\(a\\otimes k\\), so density gives the same identity everywhere. Since \\(\\|a\\otimes k\\|\\leq\\|a\\|\\,\\|k\\|\\) and the \\(e_{ij}\\) span a dense subspace of \\(\\mathcal K\\) (Example 5.1), this closed span is the closure of the span of the \\(a\\otimes e_{ij}\\), that is, the closure of \\(\\bigcup_nM_n(A)\\) acting on \\(H_A^n\\subseteq H_A\\otimes\\ell^2\\). On \\(M_n(A)\\) the operator norm is its C\\*-norm, the only one (Section 1(a)). So \\(A\\otimes_{\\min}\\mathcal K\\) is an inductive limit of the sequence \\((M_n(A))\\), that is, it is \\(A\\otimes\\mathcal K\\), whatever the faithful representation. The same holds with \\(\\mathcal K(H)\\) whenever \\(H\\) has a countably infinite orthonormal basis, since then \\(H\\cong\\ell^2\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AF-05",
      "unit": "af-algebras",
      "name": "6. The scaled dimension group",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 6. The scaled dimension group\n\n**Definition 6.1.** An *ordered abelian group* \\((G,G^+)\\) is an abelian group \\(G\\) with a subset \\(G^+\\) such that \\(G^++G^+\\subseteq G^+\\), \\(G^+\\cap(-G^+)=\\{0\\}\\) and \\(G=G^+-G^+\\). We write \\(g\\leq h\\) if \\(h-g\\in G^+\\). A *scaled ordered group* \\((G,G^+,\\Sigma)\\) is an ordered abelian group with a subset \\(\\Sigma\\subseteq G^+\\), the *scale*. A *homomorphism* \\(h:(G,G^+,\\Sigma)\\to(H,H^+,\\Sigma')\\) is a group homomorphism with \\(h(G^+)\\subseteq H^+\\) and \\(h(\\Sigma)\\subseteq\\Sigma'\\); an *isomorphism* is a bijective homomorphism whose inverse is a homomorphism, that is, \\(h(G^+)=H^+\\) and \\(h(\\Sigma)=\\Sigma'\\).\n\nThe multimatrix algebra \\(M_{\\mathbf m}\\) gives the scaled ordered group \\((\\mathbb Z^r,\\mathbb Z^r_+,\\Sigma_{\\mathbf m})\\) with\n\\[\n\\Sigma_{\\mathbf m}=\\{x\\in\\mathbb Z^r:0\\leq x\\leq\\mathbf m\\},\n\\]\nthe set of rank vectors of projections of \\(M_{\\mathbf m}\\) (Lemma 2.5). A \\*-homomorphism \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) gives the homomorphism \\(x\\mapsto\\alpha(\\varphi)x\\) of scaled ordered groups: it maps \\(\\mathbb Z^r_+\\) into \\(\\mathbb Z^s_+\\), and \\(\\Sigma_{\\mathbf m}\\) into \\(\\Sigma_{\\mathbf n}\\) because \\(\\alpha(\\varphi)\\mathbf m\\leq\\mathbf n\\). By (3.1) it sends the rank vector of \\(p\\) to the rank vector of \\(\\varphi(p)\\).\n\n**Definition 6.2** (Scaled dimension group of a generating sequence). Let \\(A\\) be an AF-algebra with a generating sequence \\((A_k)\\), \\(A_k\\cong M_{\\mathbf m(k)}\\) with \\(\\mathbf m(k)\\in\\mathbb N^{r_k}\\), and multiplicity matrices \\(\\alpha_k\\). Let \\(G\\) be the direct limit of the groups \\(\\mathbb Z^{r_k}\\) under the maps \\(\\alpha_k\\): its elements are the classes \\(\\alpha_{\\infty,k}(x)\\) of pairs \\((k,x)\\), \\(x\\in\\mathbb Z^{r_k}\\), where \\((k,x)\\) and \\((l,y)\\) define the same class if \\(\\alpha_{N,k}x=\\alpha_{N,l}y\\) for some \\(N\\geq k,l\\); addition is computed at a common stage. Put\n\\[\nG^+=\\bigcup_k\\alpha_{\\infty,k}\\big(\\mathbb Z^{r_k}_+\\big),\\qquad \\Sigma=\\bigcup_k\\alpha_{\\infty,k}\\big(\\Sigma_{\\mathbf m(k)}\\big).\n\\]\nThe triple \\((G,G^+,\\Sigma)\\) is the *scaled dimension group* of the sequence. For a projection \\(p\\in A_k\\) we write \\([p]=\\alpha_{\\infty,k}(\\operatorname{rk}p)\\); by (3.1) this does not depend on \\(k\\).\n\nBy Remark 3.4, other identifications \\(A_k\\cong M_{\\mathbf m(k)}\\) change the triple only by an isomorphism. Section 7 shows that it does not depend on the generating sequence either.\n\n**Lemma 6.3.** In the setting of Definition 6.2:\n\n1. \\((G,G^+)\\) is an ordered abelian group.\n2. If \\(0\\neq x\\in\\mathbb Z^{r_k}_+\\), then \\(\\alpha_{\\infty,k}(x)\\neq0\\).\n3. \\(\\alpha_{\\infty,k}(x)\\in G^+\\) if and only if \\(\\alpha_{l,k}x\\geq0\\) for some \\(l\\geq k\\); and \\(\\alpha_{\\infty,k}(x)\\in\\Sigma\\) if and only if \\(\\alpha_{l,k}x\\in\\Sigma_{\\mathbf m(l)}\\) for some \\(l\\geq k\\).\n4. The scale is *hereditary*: if \\(g\\in G\\), \\(h\\in\\Sigma\\) and \\(0\\leq g\\leq h\\), then \\(g\\in\\Sigma\\).\n5. If \\(A\\) is unital and \\(1_A\\in A_k\\) for all \\(k\\), then \\([1_A]=\\alpha_{\\infty,k}(\\mathbf m(k))\\) for every \\(k\\), and \\(\\Sigma=\\{g\\in G:0\\leq g\\leq[1_A]\\}\\).\n6. For the generating sequence \\((M_k(A_k))\\) of \\(A\\otimes\\mathcal K\\) (Proposition 5.4), the group and positive cone are those of \\((A_k)\\), and the scale is the whole positive cone.\n\n**Proof.** The matrices \\(\\alpha_k\\) have entries in \\(\\mathbb Z_+\\), so they preserve \\(\\geq0\\), and they map \\(\\Sigma_{\\mathbf m(k)}\\) into \\(\\Sigma_{\\mathbf m(k+1)}\\).\n\n(3) If \\(\\alpha_{\\infty,k}(x)=\\alpha_{\\infty,k'}(y)\\) with \\(y\\geq0\\) (respectively \\(y\\in\\Sigma_{\\mathbf m(k')}\\)), then \\(\\alpha_{N,k}x=\\alpha_{N,k'}y\\) for some \\(N\\), and the right side is \\(\\geq0\\) (respectively in \\(\\Sigma_{\\mathbf m(N)}\\)). The converse is clear.\n\n(1) Sums of positive elements are positive (add at a common stage), and \\(x=x_+-x_-\\) with \\(x_\\pm\\geq0\\) at every stage, so \\(G=G^+-G^+\\). If \\(g=\\alpha_{\\infty,k}(x)\\) and \\(-g\\) are both positive, then by (3) there is a stage \\(l\\) with \\(\\alpha_{l,k}x\\geq0\\) and \\(-\\alpha_{l,k}x\\geq0\\); so \\(\\alpha_{l,k}x=0\\) and \\(g=0\\).\n\n(2) The inclusions are injective, so no \\(\\alpha_k\\) has a zero column (Theorem 3.2(2)). If \\(y\\geq0\\) and \\(y_i>0\\), choose \\(j\\) with \\((\\alpha_k)_{ji}>0\\); then \\((\\alpha_ky)_j\\geq(\\alpha_k)_{ji}y_i>0\\). By induction \\(\\alpha_{l,k}x\\neq0\\) for all \\(l\\geq k\\), so \\((k,x)\\) is not equivalent to \\((k,0)\\).\n\n(4) Write \\(g=\\alpha_{\\infty,k}(x)\\) and \\(h=\\alpha_{\\infty,k}(y)\\) at a common stage. By (3), after moving to a later stage \\(l\\), we have \\(\\alpha_{l,k}y\\in\\Sigma_{\\mathbf m(l)}\\), \\(\\alpha_{l,k}x\\geq0\\) and \\(\\alpha_{l,k}(y-x)\\geq0\\). Then \\(0\\leq\\alpha_{l,k}x\\leq\\alpha_{l,k}y\\leq\\mathbf m(l)\\), so \\(g\\in\\Sigma\\).\n\n(5) \\(\\operatorname{rk}1_A=\\mathbf m(k)\\) in \\(A_k\\), and the inclusions are unital, so \\(\\alpha_k\\mathbf m(k)=\\mathbf m(k+1)\\). Every \\(x\\in\\Sigma_{\\mathbf m(k)}\\) satisfies \\(0\\leq x\\leq\\mathbf m(k)\\), so \\(\\Sigma\\) lies in the order interval \\(\\{g:0\\leq g\\leq[1_A]\\}\\). The reverse inclusion is (4) with \\(h=[1_A]\\).\n\n(6) The multiplicity matrices are the same (Proposition 5.4), so the group and the cone are the same. Let \\(x\\in\\mathbb Z^{r_k}_+\\) and \\(c=\\max_ix_i\\). Since \\(\\mathbf m(k)\\geq(1,\\dots,1)^T\\), we have \\(x\\leq c\\,\\mathbf m(k)\\), and so \\(\\alpha_{l,k}x\\leq c\\,\\alpha_{l,k}\\mathbf m(k)\\leq c\\,\\mathbf m(l)\\leq l\\,\\mathbf m(l)\\) for \\(l\\geq\\max(k,c)\\). So \\(\\alpha_{l,k}x\\) lies in the scale \\(\\Sigma_{l\\mathbf m(l)}\\) of \\(M_l(A_l)\\). \\(\\square\\)\n\n**Example 6.4** (Computations).\n\n(a) *Compact operators* (Example 5.1). All the matrices are \\((1)\\), so \\(G=\\mathbb Z\\) and \\(G^+=\\mathbb Z_+\\). The scale of \\(M_n\\) is \\(\\{0,1,\\dots,n\\}\\), so \\(\\Sigma=\\mathbb Z_+\\): the classes of the projections of rank \\(0,1,2,\\dots\\) of \\(\\mathcal K\\).\n\nThe finite-rank projections constructed here prove directly that the scale is all of \\(\\mathbb Z_+\\).\n\n(b) *Unitization of the compact operators* (Example 5.2). Define \\(h_n:\\mathbb Z^2\\to\\mathbb Z^2\\) by \\(h_n(x)=(x_1-nx_2,\\,x_2)\\). Then\n\\[\n\\begin{gathered}\nh_{n+1}(\\alpha_nx)\\\\\n=\\big(x_1+x_2-(n+1)x_2,\\ x_2\\big)\\\\\n=h_n(x),\n\\end{gathered}\n\\]\nso the \\(h_n\\) induce a homomorphism \\(G\\to\\mathbb Z^2\\), which is bijective because each \\(h_n\\) is. Under this identification,\n\\[\n\\begin{gathered}\nG^+\\\\\n=\\{(a,b):b\\geq1\\}\\cup\\{(a,0):a\\geq0\\},\\\\\n\\Sigma\\\\\n=\\{(a,0):a\\geq0\\}\\cup\\{(a,1):a\\leq0\\},\\\\\n[1]\\\\\n=(0,1).\n\\end{gathered}\n\\]\nIndeed, for \\(b\\geq1\\) the first coordinate \\(x_1-nb\\) of \\(h_n(x_1,b)\\) runs through all integers \\(\\geq-nb\\), and for \\(b=1\\) and \\(0\\leq x_1\\leq n\\) it runs through \\(-n,\\dots,0\\). In words: \\((a,0)\\) is the class of a projection of rank \\(a\\) in \\(\\mathcal K\\), and \\((-a,1)\\) is the class of \\(1-p\\) for a projection \\(p\\in\\mathcal K\\) of rank \\(a\\). The cone \\(G^+\\) is not finitely generated as a monoid. Suppose finitely many elements generated it. A sum of generators with second coordinate \\(1\\) contains exactly one generator \\((a,1)\\) and otherwise generators \\((c,0)\\) with \\(c\\geq0\\), so the first coordinates of the elements \\((a,1)\\) of \\(G^+\\) would be bounded below; but every \\((a,1)\\), \\(a\\in\\mathbb Z\\), lies in \\(G^+\\). Since \\(\\mathbb Z^2_+\\) is generated by two elements, \\((G,G^+)\\) is not isomorphic to \\((\\mathbb Z^2,\\mathbb Z^2_+)\\).\n\nThe explicit connecting maps and the non-finite-generation argument above distinguish this cone from \\(\\mathbb Z^2_+\\).\n\n(c) *UHF algebras* (Example 5.3). The maps \\(\\mathbb Z\\to\\mathbb Q\\), \\(x\\mapsto x/d_n\\), are compatible with multiplication by \\(k_n\\), because \\(k_nx/d_{n+1}=x/d_n\\). They identify \\(G\\) with the subgroup \\(\\bigcup_nd_n^{-1}\\mathbb Z\\) of \\(\\mathbb Q\\), with\n\\[\n\\begin{gathered}\nG^+\\\\\n=G\\cap[0,\\infty),\\\\\n\\Sigma\\\\\n=G\\cap[0,1],\\\\\n[1]\\\\\n=1 .\n\\end{gathered}\n\\]\nFor the CAR algebra, \\(G=\\mathbb Z[\\tfrac12]\\), the dyadic rationals.\n\n(d) *Stabilization.* By Lemma 6.3(6), \\(A\\otimes\\mathcal K\\) has the group and cone of \\(A\\) and the scale \\(G^+\\). For instance, \\(\\mathcal K\\otimes\\mathcal K\\) has the invariant \\((\\mathbb Z,\\mathbb Z_+,\\mathbb Z_+)\\), the same as \\(\\mathcal K\\).\n\n",
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    {
      "id": "OA-FND-AF-06",
      "unit": "af-algebras",
      "name": "7. Projections and the dimension group as an invariant",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 7. Projections and the dimension group as an invariant\n\nIn this section we show that the scaled dimension group is the ordered \\(K_0\\)-group of the algebra with the classes of its projections as scale. The main tool is that projections in the closure of an increasing union can be moved into the union.\n\n**Lemma 7.1** (Close projections are equivalent). Let \\(p,q\\) be projections of a C\\*-algebra \\(A\\) with \\(\\|p-q\\|<1\\). There is a unitary \\(u\\) in the C\\*-subalgebra of \\(A^+\\) generated by \\(1,p,q\\) with \\(upu^*=q\\). In particular \\(v=up\\in A\\) satisfies \\(v^*v=p\\) and \\(vv^*=q\\).\n\n**Proof.** Put \\(z=qp+(1-q)(1-p)\\in A^+\\). Expanding,\n\\[\n\\begin{gathered}\nz^*z\\\\\n=pqp+(1-p)(1-q)(1-p)\\\\\n=1-(p-q)^2,\\\\\nzz^*\\\\\n=qpq+(1-q)(1-p)(1-q)\\\\\n=1-(p-q)^2 .\n\\end{gathered}\n\\]\nSince \\(\\|(p-q)^2\\|=\\|p-q\\|^2<1\\), both \\(z^*z\\) and \\(zz^*\\) are invertible (Section 1(b)), so \\(z\\) has the left inverse \\((z^*z)^{-1}z^*\\) and the right inverse \\(z^*(zz^*)^{-1}\\), and is invertible. Also \\(zp=qp=qz\\). Taking adjoints, \\(z^*q=pz^*\\), hence \\(z^*zp=z^*qz=pz^*z\\): the element \\(p\\) commutes with \\(z^*z\\), and hence with \\(h=(z^*z)^{-1/2}\\), which is a limit of polynomials in \\(z^*z\\) (Section 1(c)). Put \\(u=zh\\). Then \\(u^*u=hz^*zh=1\\), and \\(u\\) is invertible, so \\(uu^*=1\\) as well. Finally\n\\[\n\\begin{gathered}\nupu^*\\\\\n=zhphz^*\\\\\n=zph^2z^*\\\\\n=zp(z^*z)^{-1}z^*\\\\\n=qz(z^*z)^{-1}z^*\\\\\n=q,\n\\end{gathered}\n\\]\nbecause \\(z(z^*z)^{-1}z^*=zz^{-1}(z^*)^{-1}z^*=1\\). Since \\(A\\) is an ideal of \\(A^+\\), \\(v=up\\in A\\), and \\(v^*v=pu^*up=p\\), \\(vv^*=upu^*=q\\). \\(\\square\\)\n\n**Lemma 7.2** (Projections near a subalgebra). Let \\(B\\) be a C\\*-subalgebra of a C\\*-algebra \\(A\\), and \\(p\\in A\\) a projection whose distance to \\(B\\) is less than \\(\\frac14\\). Then there is a projection \\(q\\in B\\) with \\(\\|p-q\\|<\\frac12\\); by Lemma 7.1, \\(p\\) and \\(q\\) are equivalent in \\(A\\).\n\n**Proof.** Choose \\(b\\in B\\) with \\(\\|p-b\\|<\\frac14\\) and put \\(a=\\frac12(b+b^*)\\in B\\); then \\(a\\) is self-adjoint and \\(\\|p-a\\|<\\frac14\\). Let \\(\\lambda\\) be real with distance at least \\(\\frac14\\) from \\(\\{0,1\\}\\). In \\(A^+\\), \\(p-\\lambda\\) is invertible, and its inverse is the function \\(t\\mapsto(t-\\lambda)^{-1}\\) of \\(p\\), of norm at most \\(4\\) because \\(\\sigma(p)\\subseteq\\{0,1\\}\\) (Section 1(c)). So\n\\[\na-\\lambda=(p-\\lambda)\\big(1+(p-\\lambda)^{-1}(a-p)\\big)\n\\]\nis invertible by Section 1(b), since \\(\\|(p-\\lambda)^{-1}(a-p)\\|<4\\cdot\\frac14=1\\). Hence \\(\\sigma(a)\\subseteq(-\\frac14,\\frac14)\\cup(\\frac34,\\frac54)\\). Let \\(f=0\\) on \\((-\\infty,\\frac12]\\) and \\(f=1\\) on \\((\\frac12,\\infty)\\). It is continuous on \\(\\sigma(a)\\) and \\(f(0)=0\\), so \\(q=f(a)\\in B\\). Since \\(f=f^2=\\bar f\\) on \\(\\sigma(a)\\), \\(q\\) is a projection, and \\(\\|q-a\\|=\\max_{t\\in\\sigma(a)}|f(t)-t|<\\frac14\\). So \\(\\|p-q\\|<\\frac12\\). \\(\\square\\)\n\n**Proposition 7.3** (Projections in the closure of a union). Let \\(A\\) be a C\\*-algebra and \\(A_1\\subseteq A_2\\subseteq\\cdots\\) C\\*-subalgebras with dense union.\n\n1. Every projection \\(p\\in A\\) is equivalent in \\(A\\) to a projection of some \\(A_k\\).\n2. If \\(p,q\\in A_k\\) are projections that are equivalent in \\(A\\), they are equivalent in \\(A_l\\) for some \\(l\\geq k\\).\n\n**Proof.** (1) The distance from \\(p\\) to \\(A_k\\) decreases to \\(0\\); apply Lemma 7.2 once it is below \\(\\frac14\\).\n\n(2) If \\(p=0\\), any implementing partial isometry has norm zero, so \\(q=0\\) and the assertion is immediate. Assume \\(p\\neq0\\). Let \\(v\\in A\\) with \\(v^*v=p\\) and \\(vv^*=q\\); then \\(v=qvp\\). Choose \\(y_l\\in A_l\\) with \\(y_l\\to v\\), and put \\(w_l=qy_lp\\in A_l\\) for \\(l\\geq k\\); then \\(w_l\\to qvp=v\\). The elements \\(x_l=w_l^*w_l\\) lie in the corner \\(pA_lp\\), a C\\*-algebra with unit \\(p\\), and \\(x_l\\to v^*v=p\\). For large \\(l\\), \\(\\|x_l-p\\|<\\frac12\\), so \\(x_l\\) is invertible in \\(pA_lp\\) with spectrum in \\([\\frac12,\\frac32]\\) (Section 1(b), applied to \\(x_l-\\lambda p\\)). Let \\(h_l=x_l^{-1/2}\\), computed in \\(pA_lp\\); by Section 1(c) it is the same element when computed in the unital C\\*-algebra \\(pAp\\), and \\(h_l\\to p^{-1/2}=p\\). Put \\(v_l=w_lh_l\\in A_l\\). Then \\(v_l^*v_l=h_lx_lh_l=p\\), so \\(e_l=v_lv_l^*\\) is a projection, and \\(qe_l=e_l\\) because \\(qw_l=w_l\\). Moreover \\(v_l\\to vp=v\\), so \\(e_l\\to vv^*=q\\). For large \\(l\\), \\(q-e_l\\) is a projection of norm less than one, hence zero. So \\(p\\sim q\\) in \\(A_l\\). \\(\\square\\)\n\n**Definition 7.4.** Let \\(A\\) be a C\\*-algebra. Two projections \\(p\\in M_n(A)\\) and \\(q\\in M_{n'}(A)\\) are *equivalent* if they are equivalent in \\(M_N(A)\\), \\(N=\\max(n,n')\\), where both sit as upper left corners; this does not depend on \\(N\\), because a partial isometry \\(v\\) with \\(v^*v=p\\) and \\(vv^*=q\\) satisfies \\(v=qvp\\). Let \\(V(A)\\) be the set of equivalence classes \\([p]\\) of projections in \\(M_\\infty(A)\\). For projections \\(p,q\\), view both in some \\(M_N(A)\\) and put\n\\[\n\\begin{gathered}\n{}[p]+[q]\\\\\n=[\\operatorname{diag}(p,q)],\\\\\n\\operatorname{diag}(p,q)\\in M_{2N}(A).\n\\end{gathered}\n\\]\nEnlarging \\(N\\) changes \\(\\operatorname{diag}(p,q)\\) by a permutation of the basis, which is implemented by a permutation matrix, so the class does not depend on \\(N\\). The addition is well defined and makes \\(V(A)\\) an abelian monoid with zero \\([0]\\): if \\(v,w\\) implement \\(p\\sim p'\\) and \\(q\\sim q'\\), then \\(\\operatorname{diag}(v,w)\\) implements \\(\\operatorname{diag}(p,q)\\sim\\operatorname{diag}(p',q')\\); and \\(\\begin{pmatrix}0&q\\\\p&0\\end{pmatrix}\\) implements \\(\\operatorname{diag}(p,q)\\sim\\operatorname{diag}(q,p)\\). A \\*-homomorphism \\(\\psi:A\\to B\\) induces the monoid homomorphism \\(\\psi_*[p]=[\\psi(p)]\\), where \\(\\psi\\) acts entrywise.\n\nFor orthogonal projections \\(p,q\\in A\\), \\(p+q\\sim\\operatorname{diag}(p,q)\\) through \\(\\begin{pmatrix}p&q\\\\0&0\\end{pmatrix}\\), so \\([p+q]=[p]+[q]\\).\n\n**Theorem 7.5** (The dimension group is an invariant). Let \\(A\\) be an AF-algebra with generating sequence \\((A_k)\\) and scaled dimension group \\((G,G^+,\\Sigma)\\). For a projection \\(p\\in M_N(A_k)\\cong M_{N\\mathbf m(k)}\\), let \\(\\operatorname{rk}p\\in\\mathbb Z^{r_k}_+\\) be its rank vector.\n\n1. The rule \\(\\alpha_{\\infty,k}(\\operatorname{rk}p)\\mapsto[p]\\) is a well-defined isomorphism of monoids \\(\\Phi:G^+\\to V(A)\\). In particular \\(V(A)\\) has cancellation: \\(a+c=b+c\\) implies \\(a=b\\).\n2. \\(\\Phi(\\Sigma)\\) is the set of classes of projections of \\(A\\) itself.\n3. \\(G\\) is the Grothendieck group of \\(V(A)\\): every monoid homomorphism from \\(V(A)\\) to an abelian group extends uniquely to a group homomorphism on \\(G\\), through \\(\\Phi^{-1}\\).\n4. Every \\*-homomorphism \\(\\psi:A\\to B\\) between AF-algebras induces a homomorphism of scaled dimension groups \\(\\psi_*\\) with \\(\\psi_*[p]=[\\psi(p)]\\) for projections \\(p\\in M_\\infty(A)\\); \\((\\psi\\circ\\psi')_*=\\psi_*\\psi'_*\\) and \\(\\mathrm{id}_*=\\mathrm{id}\\). Isomorphic AF-algebras have isomorphic scaled dimension groups, whatever generating sequences are used.\n\n**Proof.** By Proposition 5.4(1), the inclusion \\(M_N(A_k)\\subseteq M_N(A_l)\\) has multiplicity matrix \\(\\alpha_{l,k}\\), and the corner inclusion \\(M_N(A_k)\\subseteq M_{N'}(A_k)\\) preserves rank vectors. So by (3.1), the rank vector of \\(p\\) at stage \\(l\\) is \\(\\alpha_{l,k}\\operatorname{rk}p\\).\n\n(1) *Well defined and injective.* Let \\(p\\in M_N(A_k)\\) and \\(p'\\in M_{N'}(A_{k'})\\), and view both in \\(M_{N''}(A_l)\\) with \\(N''=\\max(N,N')\\) and \\(l\\geq k,k'\\). If \\(\\alpha_{\\infty,k}\\operatorname{rk}p=\\alpha_{\\infty,k'}\\operatorname{rk}p'\\), then at some later stage the rank vectors agree, so \\(p\\sim p'\\) there by Lemma 2.5, and \\([p]=[p']\\). Conversely, if \\([p]=[p']\\) in \\(V(A)\\), Proposition 7.3(2), applied in \\(M_{N''}(A)\\) with the subalgebras \\(M_{N''}(A_l)\\), gives a stage at which \\(p\\sim p'\\); there the rank vectors agree (Lemma 2.5), so \\(\\alpha_{\\infty,k}\\operatorname{rk}p=\\alpha_{\\infty,k'}\\operatorname{rk}p'\\). Every element \\(\\alpha_{\\infty,k}(x)\\) of \\(G^+\\), \\(x\\geq0\\), is of the form \\(\\alpha_{\\infty,k}\\operatorname{rk}p\\) for a diagonal projection \\(p\\in M_N(A_k)\\) with \\(N\\geq\\max(1,\\max_ix_i)\\). *Onto.* The union of the \\(M_N(A_k)\\) is dense in \\(M_N(A)\\) (Section 1(f)), so by Proposition 7.3(1) every projection of \\(M_N(A)\\) is equivalent to one in some \\(M_N(A_k)\\). *Additive.* \\(\\operatorname{rk}\\operatorname{diag}(p,q)=\\operatorname{rk}p+\\operatorname{rk}q\\). Cancellation holds because \\(G^+\\) sits in the group \\(G\\).\n\n(2) If \\(p\\in A_k\\), then \\(\\operatorname{rk}p\\in\\Sigma_{\\mathbf m(k)}\\). Conversely, every projection of \\(A\\) is equivalent to a projection of some \\(A_k\\) (Proposition 7.3(1)), and every element of \\(\\Sigma_{\\mathbf m(k)}\\) is the rank vector of a projection of \\(A_k\\).\n\n(3) \\(G=G^+-G^+\\). If \\(\\chi:V(A)\\to H\\) is a monoid homomorphism into an abelian group, put \\(\\tilde\\chi(g_1-g_2)=\\chi\\Phi(g_1)-\\chi\\Phi(g_2)\\) for \\(g_1,g_2\\in G^+\\). If \\(g_1-g_2=g_1'-g_2'\\), then \\(g_1+g_2'=g_1'+g_2\\) in \\(G^+\\), and applying \\(\\chi\\Phi\\) shows that \\(\\tilde\\chi\\) is well defined. It is a homomorphism, and it is the only one extending \\(\\chi\\Phi\\).\n\n(4) \\(\\psi\\) maps equivalent projections to equivalent projections and respects \\(\\operatorname{diag}\\), so \\(\\psi_*:V(A)\\to V(B)\\) is a monoid homomorphism. By (1) and (3) it gives a group homomorphism \\(G(A)\\to G(B)\\), which maps \\(G(A)^+\\) into \\(G(B)^+\\) and, by (2), \\(\\Sigma(A)\\) into \\(\\Sigma(B)\\). Functoriality is clear, and an isomorphism has an inverse, so it induces an isomorphism. \\(\\square\\)\n\n**Definition 7.6.** For an AF-algebra \\(A\\) we write \\((K_0(A),K_0(A)^+,\\Sigma(A))\\) for its scaled dimension group, computed from any generating sequence: \\(K_0(A)\\) is the Grothendieck group of \\(V(A)\\), \\(K_0(A)^+\\) the image of \\(V(A)\\), and \\(\\Sigma(A)\\) the set of classes of projections of \\(A\\).\n\nFor unital C\\*-algebras, \\(K_0(A)\\) is defined in K-theory as the Grothendieck group of \\(V(A)\\), so the two notions agree. For a nonunital C\\*-algebra, K-theory defines \\(K_0(A)\\) as the kernel of the map \\(K_0(A^+)\\to K_0(\\mathbb C)=\\mathbb Z\\) induced by the quotient map \\(A^+\\to\\mathbb C\\) that kills \\(A\\), with positive cone the image of \\(V(A)\\) [Blackadar 2006, V.1.1.15–V.1.1.17]. The next proposition shows that this also agrees with Definition 7.6.\n\n**Proposition 7.7** (Nonunital AF-algebras). Let \\(A\\) be a nonunital AF-algebra. Then \\(A^+\\) is an AF-algebra, and the map \\(V(A)\\to V(A^+)\\) induced by the inclusion extends to an isomorphism of \\(K_0(A)\\) onto the kernel of \\(K_0(A^+)\\to\\mathbb Z\\), carrying \\(K_0(A)^+\\) onto the image of \\(V(A)\\).\n\n**Proof.** Let \\((A_k)\\) be a generating sequence with units \\(1_k\\). Since \\(A\\) has no unit, \\(1_k\\neq1\\), and \\[\n\\begin{gathered}\nA_k^+:\\\\\n=A_k+\\mathbb C1\\\\\n=A_k\\oplus\\mathbb C(1-1_k)\\cong M_{\\mathbf m(k)}\\oplus\\mathbb C\n\\end{gathered}\n\\] is a finite-dimensional C\\*-subalgebra of \\(A^+\\). These algebras increase and their union is dense in \\(A^+\\), so \\(A^+\\) is an AF-algebra. In the inclusion \\(A_k^+\\subseteq A_{k+1}^+\\), the summands of \\(A_k\\) enter those of \\(A_{k+1}\\) as before, and the minimal projection \\(1-1_k\\) of the extra summand splits as \\((1-1_{k+1})+(1_{k+1}-1_k)\\), where \\(1_{k+1}-1_k\\in A_{k+1}\\) has rank vector \\(d_k=\\mathbf m(k+1)-\\alpha_k\\mathbf m(k)\\). So the multiplicity matrix is\n\\[\n\\begin{pmatrix}\\alpha_k&d_k\\\\0&1\\end{pmatrix}.\n\\]\nThe quotient map \\(A^+\\to\\mathbb C\\) kills \\(A_k\\) and sends \\(1-1_k\\) to \\(1\\), so on \\(\\mathbb Z^{r_k}\\oplus\\mathbb Z\\) it induces the projection \\((x,t)\\mapsto t\\) at every stage. An element of \\(K_0(A^+)\\), represented by \\((x,t)\\) at stage \\(k\\), is in the kernel exactly when \\(t=0\\). The connecting maps send \\((x,0)\\) to \\((\\alpha_kx,0)\\), so \\((x,0)\\) represents \\(0\\) in \\(K_0(A^+)\\) exactly when \\(\\alpha_{l,k}x=0\\) for some \\(l\\). So the kernel is the direct limit of the subgroups \\(\\mathbb Z^{r_k}\\oplus0\\) under the maps \\(\\alpha_k\\), which is \\(K_0(A)\\), and the element \\((x,0)\\) with \\(x\\geq0\\) is the image of the class of a projection in some \\(M_N(A_k)\\). \\(\\square\\)\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AF-07",
      "unit": "af-algebras",
      "name": "8. Elliott's classification theorem",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
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      "full_conditions_and_proof": "## 8. Elliott's classification theorem\n\nA reference for this section is [Blackadar 1998, Section 7.3]. In the two lemmas, \\(B\\) is any AF-algebra with generating sequence \\((B_l)\\), \\(B_l\\cong M_{\\mathbf n(l)}\\) with \\(\\mathbf n(l)\\in\\mathbb N^{s_l}\\), and multiplicity matrices \\(\\beta_l\\); its classes are \\(\\beta_{\\infty,l}(y)\\). In the proof of the theorem they are applied both to \\(B\\) and to \\(A\\).\n\n**Lemma 8.1** (Existence). Let \\(\\mathbf m\\in\\mathbb N^r\\), and let \\(h:\\mathbb Z^r\\to K_0(B)\\) be a group homomorphism with \\(h(\\mathbb Z^r_+)\\subseteq K_0(B)^+\\) and \\(h(\\mathbf m)\\in\\Sigma(B)\\). Then there are \\(l\\) and a \\*-homomorphism \\(\\varphi:M_{\\mathbf m}\\to B_l\\) with\n\\[\n\\beta_{\\infty,l}\\big(\\alpha(\\varphi)x\\big)=h(x)\\qquad(x\\in\\mathbb Z^r),\n\\]\nthat is, \\([\\varphi(p)]=h(\\operatorname{rk}p)\\) for every projection \\(p\\in M_{\\mathbf m}\\). If \\(h(e_i)\\neq0\\) for every \\(i\\), then \\(\\varphi\\) is injective.\n\n**Proof.** Each \\(h(e_i)\\) lies in \\(K_0(B)^+\\), so \\(h(e_i)=\\beta_{\\infty,l}(y_i)\\) with \\(y_i\\geq0\\) at a common stage \\(l\\) (Lemma 6.3(3)). The element \\(\\sum_im_iy_i\\) represents \\(h(\\mathbf m)\\in\\Sigma(B)\\), so by Lemma 6.3(3) it lies in the scale at some later stage; moving there keeps the \\(y_i\\geq0\\). So we may assume that \\(\\sum_im_iy_i\\leq\\mathbf n(l)\\). Let \\(\\alpha\\) be the \\(s_l\\times r\\) matrix with columns \\(y_1,\\dots,y_r\\). Its entries lie in \\(\\mathbb Z_+\\) and \\(\\alpha\\mathbf m=\\sum_im_iy_i\\leq\\mathbf n(l)\\), so Theorem 3.2(3) gives \\(\\varphi=\\varphi_\\alpha:M_{\\mathbf m}\\to M_{\\mathbf n(l)}\\cong B_l\\) with \\(\\alpha(\\varphi)=\\alpha\\), and \\(\\beta_{\\infty,l}(\\alpha x)=\\sum_ix_i\\beta_{\\infty,l}(y_i)=h(x)\\). If \\(h(e_i)\\neq0\\), then \\(y_i\\neq0\\), so no column of \\(\\alpha\\) vanishes and \\(\\varphi\\) is injective (Theorem 3.2(2)). \\(\\square\\)\n\n**Lemma 8.2** (Uniqueness). Let \\(\\varphi,\\psi:M_{\\mathbf m}\\to B_l\\) be \\*-homomorphisms with \\(\\beta_{\\infty,l}\\alpha(\\varphi)=\\beta_{\\infty,l}\\alpha(\\psi)\\). Then there are \\(l'\\geq l\\) and \\(u\\in U(B_{l'})\\) with \\(u\\varphi(x)u^*=\\psi(x)\\) for all \\(x\\in M_{\\mathbf m}\\).\n\n**Proof.** For each \\(i\\), the \\(i\\)-th columns of \\(\\alpha(\\varphi)\\) and \\(\\alpha(\\psi)\\) have the same image in \\(K_0(B)\\), so they become equal at some later stage. Take \\(l'\\) beyond all these stages. As maps into \\(B_{l'}\\), \\(\\varphi\\) and \\(\\psi\\) have the multiplicity matrices \\(\\beta_{l',l}\\alpha(\\varphi)=\\beta_{l',l}\\alpha(\\psi)\\) (Theorem 3.2(1)), and Theorem 3.2(4) gives \\(u\\). \\(\\square\\)\n\n**Theorem 8.3** (Classification of AF-algebras). Let \\(A\\) and \\(B\\) be AF-algebras with generating sequences \\((A_k)\\) and \\((B_l)\\) and local algebras \\(A_\\infty\\) and \\(B_\\infty\\).\n\n1. For every homomorphism \\[\n\\begin{gathered}\nh:(K_0(A),K_0(A)^+,\\Sigma(A))\\\\\n\\to(K_0(B),K_0(B)^+,\\Sigma(B))\n\\end{gathered}\n\\] of scaled ordered groups there is a \\*-homomorphism \\(\\psi:A\\to B\\) with \\(\\psi(A_\\infty)\\subseteq B_\\infty\\) and \\(\\psi_*=h\\).\n2. For every isomorphism \\(\\theta\\) of the scaled dimension groups there is an isomorphism \\(\\Phi:A\\to B\\) with \\(\\Phi(A_\\infty)=B_\\infty\\) and \\(\\Phi_*=\\theta\\).\n3. The following are equivalent: (i) \\(A\\cong B\\); (ii) \\(A_\\infty\\) and \\(B_\\infty\\) are isomorphic \\*-algebras; (iii) the scaled dimension groups of \\(A\\) and \\(B\\) are isomorphic.\n\n\n**Proof.** Write \\(\\alpha_k\\) for the multiplicity matrices of \\((A_k)\\). The classes \\([e]\\) of minimal projections \\(e\\) of the \\(A_k\\) generate \\(K_0(A)\\) as a group, so two homomorphisms from \\(K_0(A)\\) that agree on them are equal.\n\n(1) We build \\(l_1<l_2<\\cdots\\) and \\*-homomorphisms \\(\\varphi_n:A_n\\to B_{l_n}\\) such that \\([\\varphi_n(p)]=h[p]\\) for projections \\(p\\in A_n\\), and such that \\(\\varphi_{n+1}\\) extends \\(\\varphi_n\\) as a map into \\(B_{l_{n+1}}\\). The homomorphism \\(h\\circ\\alpha_{\\infty,1}:\\mathbb Z^{r_1}\\to K_0(B)\\) is positive and maps \\(\\mathbf m(1)=\\operatorname{rk}1_{A_1}\\) to \\(h[1_{A_1}]\\in\\Sigma(B)\\); Lemma 8.1 gives \\(\\varphi_1\\). Suppose \\(\\varphi_n\\) is built. Lemma 8.1, applied to \\(h\\circ\\alpha_{\\infty,n+1}\\) (which satisfies its hypotheses for the same reason), gives \\(\\varphi':A_{n+1}\\to B_{l'}\\) with \\([\\varphi'(p)]=h[p]\\); we may take \\(l'>l_n\\). The restriction of \\(\\varphi'\\) to \\(A_n\\) and the map \\(\\varphi_n\\), viewed in \\(B_{l'}\\), induce the same map \\(h\\circ\\alpha_{\\infty,n}\\). By Lemma 8.2 there are \\(l_{n+1}\\geq l'\\) and \\(u\\in U(B_{l_{n+1}})\\) with \\(u\\varphi'(x)u^*=\\varphi_n(x)\\) for \\(x\\in A_n\\). Put \\(\\varphi_{n+1}=\\operatorname{Ad}u\\circ\\varphi'\\); conjugation does not change classes. The \\(\\varphi_n\\) define a \\*-homomorphism \\(A_\\infty\\to B_\\infty\\), which is contractive on each \\(A_n\\) (Section 1(a)) and extends to \\(\\psi:A\\to B\\). By construction \\(\\psi_*[e]=h[e]\\) for minimal projections \\(e\\) of the \\(A_n\\), so \\(\\psi_*=h\\).\n\n(2) We build \\(k_1<k_2<\\cdots\\), \\(l_1<l_2<\\cdots\\) and injective \\*-homomorphisms\n\\[\n\\varphi_n:A_{k_n}\\to B_{l_n},\\qquad\\psi_n:B_{l_n}\\to A_{k_{n+1}}\n\\]\nsuch that \\(\\psi_n\\circ\\varphi_n\\) is the inclusion \\(A_{k_n}\\subseteq A_{k_{n+1}}\\), \\(\\varphi_{n+1}\\circ\\psi_n\\) is the inclusion \\(B_{l_n}\\subseteq B_{l_{n+1}}\\), \\([\\varphi_n(p)]=\\theta[p]\\) and \\([\\psi_n(q)]=\\theta^{-1}[q]\\) for projections \\(p\\in A_{k_n}\\), \\(q\\in B_{l_n}\\).\n\nPut \\(k_1=1\\). The map \\(\\theta\\circ\\alpha_{\\infty,1}\\) is positive, sends \\(\\mathbf m(1)\\) into \\(\\theta(\\Sigma(A))=\\Sigma(B)\\), and sends each \\(e_i\\) to a nonzero element (Lemma 6.3(2) and injectivity of \\(\\theta\\)). Lemma 8.1 gives an injective \\(\\varphi_1:A_1\\to B_{l_1}\\). Suppose \\(\\varphi_n\\) is built. Lemma 8.1, applied to \\(\\theta^{-1}\\circ\\beta_{\\infty,l_n}\\) in the same way, gives an injective \\(\\psi':B_{l_n}\\to A_{k'}\\) with \\([\\psi'(q)]=\\theta^{-1}[q]\\); composing with an inclusion we may take \\(k'>k_n\\). Then \\(\\psi'\\circ\\varphi_n\\) and the inclusion \\(A_{k_n}\\subseteq A_{k'}\\) both send \\([p]\\) to \\([p]\\). By Lemma 8.2 there are \\(k_{n+1}\\geq k'\\) and \\(u\\in U(A_{k_{n+1}})\\) with \\(u\\psi'(\\varphi_n(x))u^*=x\\) for \\(x\\in A_{k_n}\\). Put \\(\\psi_n=\\operatorname{Ad}u\\circ\\psi'\\), as a map into \\(A_{k_{n+1}}\\). Exchanging the roles of \\(A\\) and \\(B\\) and of \\(\\theta\\) and \\(\\theta^{-1}\\), the same step produces \\(\\varphi_{n+1}:A_{k_{n+1}}\\to B_{l_{n+1}}\\) with \\(\\varphi_{n+1}\\circ\\psi_n\\) equal to the inclusion \\(B_{l_n}\\subseteq B_{l_{n+1}}\\).\n\nOn \\(A_{k_n}\\) we get \\(\\varphi_{n+1}=\\varphi_{n+1}\\circ\\psi_n\\circ\\varphi_n=\\varphi_n\\). So the \\(\\varphi_n\\) define a \\*-homomorphism \\(\\Phi_0:A_\\infty\\to B_\\infty\\), and likewise the \\(\\psi_n\\) define \\(\\Psi_0:B_\\infty\\to A_\\infty\\), with \\(\\Psi_0\\Phi_0=\\mathrm{id}\\) and \\(\\Phi_0\\Psi_0=\\mathrm{id}\\). Each \\(\\varphi_n\\) is isometric (Section 1(a)), so \\(\\Phi_0\\) extends to an isometric \\*-homomorphism \\(\\Phi:A\\to B\\); similarly \\(\\Psi\\), and \\(\\Psi\\Phi=\\mathrm{id}\\), \\(\\Phi\\Psi=\\mathrm{id}\\) by continuity. By construction \\(\\Phi(A_\\infty)=B_\\infty\\) and \\(\\Phi_*=\\theta\\).\n\n(3) (i)⇒(iii) is Theorem 7.5(4), and (iii)⇒(ii) is (2). (ii)⇒(i): let \\(\\Phi_0:A_\\infty\\to B_\\infty\\) be an isomorphism of \\*-algebras. The finite-dimensional subspace \\(\\Phi_0(A_k)\\) lies in some \\(B_l\\), and \\(\\Phi_0\\) restricted to \\(A_k\\) is an injective \\*-homomorphism of C\\*-algebras into \\(B_l\\), hence isometric. So \\(\\Phi_0\\) is isometric on \\(A_\\infty\\) and extends to an isometric \\*-homomorphism \\(\\Phi:A\\to B\\). Its range is closed and contains \\(B_\\infty\\), so \\(\\Phi\\) is onto. \\(\\square\\)\n\n**Corollary 8.4.** Let \\(A\\) be an AF-algebra. (1) Any two generating sequences of \\(A\\) have isomorphic local algebras. (2) Every automorphism of the scaled dimension group of \\(A\\) is induced by an automorphism of \\(A\\).\n\n**Proof.** Apply Theorem 8.3(2) with \\(B=A\\): to the identity of \\(K_0(A)\\) and two generating sequences for (1), and to the given automorphism for (2). \\(\\square\\)\n\n**Theorem 8.5** (Stable isomorphism). For AF-algebras \\(A\\) and \\(B\\), \\(A\\otimes\\mathcal K\\cong B\\otimes\\mathcal K\\) if and only if the ordered groups \\((K_0(A),K_0(A)^+)\\) and \\((K_0(B),K_0(B)^+)\\) are isomorphic.\n\n**Proof.** By Proposition 5.4 and Lemma 6.3(6), \\(A\\otimes\\mathcal K\\) is an AF-algebra whose scaled dimension group is \\((K_0(A),K_0(A)^+,K_0(A)^+)\\), and similarly for \\(B\\). If \\(A\\otimes\\mathcal K\\cong B\\otimes\\mathcal K\\), Theorem 7.5(4) gives an isomorphism of these triples, in particular of the ordered groups. Conversely, an isomorphism of ordered groups maps \\(K_0(A)^+\\) onto \\(K_0(B)^+\\), that is, scale onto scale, and Theorem 8.3(3) gives \\(A\\otimes\\mathcal K\\cong B\\otimes\\mathcal K\\). \\(\\square\\)\n\nFor a UHF algebra of type \\((k_n)\\) (Example 5.3), its *supernatural number* \\(q\\) records, for each prime \\(p\\), the exponent \\(\\nu_p(q)=\\sum_n\\nu_p(k_n)\\in\\{0,1,2,\\dots,\\infty\\}\\), where \\(\\nu_p(k)\\) is the exponent of \\(p\\) in \\(k\\). Let \\(\\mathbb Z(q)\\) be the set of rationals \\(a/b\\) (\\(a\\in\\mathbb Z\\), \\(b\\in\\mathbb N\\)) with \\(\\nu_p(b)\\leq\\nu_p(q)\\) for every prime \\(p\\).\n\n**Corollary 8.6** (Classification of UHF algebras). The UHF algebra of type \\((k_n)\\) has the scaled dimension group \\((\\mathbb Z(q),\\mathbb Z(q)\\cap[0,\\infty),\\mathbb Z(q)\\cap[0,1])\\). Two UHF algebras are isomorphic if and only if their supernatural numbers are equal.\n\n\n**Proof.** By Example 6.4(c), the group is \\(\\bigcup_nd_n^{-1}\\mathbb Z\\). Each \\(d_n\\) satisfies \\(\\nu_p(d_n)\\leq\\nu_p(q)\\), so this union lies in \\(\\mathbb Z(q)\\). Conversely, let \\(a/b\\in\\mathbb Z(q)\\). Only finitely many primes divide \\(b\\), and \\(\\nu_p(d_n)\\) increases to \\(\\nu_p(q)\\geq\\nu_p(b)\\), so \\(b\\) divides \\(d_n\\) for large \\(n\\), and \\(a/b\\in d_n^{-1}\\mathbb Z\\).\n\nIf two UHF algebras have the same supernatural number, their scaled dimension groups are the same subsets of \\(\\mathbb Q\\), and Theorem 8.3(3) shows that the algebras are isomorphic. Conversely, an isomorphism of the algebras gives an isomorphism \\(\\theta:\\mathbb Z(q)\\to\\mathbb Z(q')\\) of scaled ordered groups (Theorem 7.5(4)). The largest element of the scale \\(\\mathbb Z(q)\\cap[0,1]\\) is \\(1\\), so \\(\\theta(1)=1\\). For \\(a/b\\in\\mathbb Z(q)\\), \\(b\\,\\theta(a/b)=\\theta(a)=a\\), so \\(\\theta(a/b)=a/b\\). Hence \\(\\mathbb Z(q)=\\mathbb Z(q')\\), and \\(\\nu_p(q)=\\sup\\{\\nu_p(b):1/b\\in\\mathbb Z(q)\\}\\) shows \\(q=q'\\). \\(\\square\\)\n\n**Example 8.7** (The scale cannot be dropped). Let \\(A\\) be the CAR algebra. By Proposition 5.4(1), \\(M_3(A)\\) is an AF-algebra with the same group and cone as \\(A\\), namely \\(\\mathbb Z[\\frac12]\\) with the usual order, but with unit class \\(3\\) and scale \\([0,3]\\cap\\mathbb Z[\\frac12]\\). A group homomorphism \\(\\theta\\) of \\(\\mathbb Z[\\frac12]\\) satisfies \\(2^n\\theta(2^{-n})=\\theta(1)\\), so \\(\\theta(x)=\\theta(1)x\\). If \\(\\theta\\) is bijective and positive, then \\(\\theta(1)\\) is a positive unit of the ring \\(\\mathbb Z[\\frac12]\\), that is, \\(\\theta(1)=2^j\\) for some \\(j\\in\\mathbb Z\\). Then \\(\\theta([0,1])=[0,2^j]\\neq[0,3]\\). So \\(M_3(A)\\not\\cong A\\), although \\(M_3(A)\\otimes\\mathcal K\\cong A\\otimes\\mathcal K\\) by Theorem 8.5. This agrees with Corollary 8.6: \\(M_3(A)\\) is the UHF algebra of type \\((3,2,2,\\dots)\\), with supernatural number \\(3\\cdot2^\\infty\\neq2^\\infty\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AF-08",
      "unit": "af-algebras",
      "name": "9. The gauge-invariant CAR algebra and Pascal's triangle",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 9. The gauge-invariant CAR algebra and Pascal's triangle\n\n**9.1. The algebra.** For \\(n\\geq0\\) let \\(H_n=(\\mathbb C^2)^{\\otimes n}\\) (\\(H_0=\\mathbb C\\)), with the orthonormal basis \\(\\varepsilon_\\xi=\\varepsilon_{\\xi_1}\\otimes\\cdots\\otimes\\varepsilon_{\\xi_n}\\), \\(\\xi\\in\\{0,1\\}^n\\), where \\(\\varepsilon_0,\\varepsilon_1\\) is the standard basis of \\(\\mathbb C^2\\). Let \\(|\\xi|\\) be the number of ones in \\(\\xi\\). Put \\(B_n=B(H_n)\\cong M_{2^n}\\), identify \\(H_{n+1}=H_n\\otimes\\mathbb C^2\\), and let \\(B_n\\to B_{n+1}\\) be \\(x\\mapsto x\\otimes1\\). By Example 5.3, the limit \\(B\\) is the CAR algebra. For \\(\\lambda\\) in the unit circle \\(\\mathbb T\\), let \\(U_n(\\lambda)\\) be the unitary of \\(H_n\\) with\n\\[\nU_n(\\lambda)\\varepsilon_\\xi=\\lambda^{\\,n-2|\\xi|}\\varepsilon_\\xi ;\n\\]\nthus \\(U_n(\\lambda)=u(\\lambda)^{\\otimes n}\\) with \\(u(\\lambda)=\\operatorname{diag}(\\lambda,\\bar\\lambda)\\), and \\(U_{n+1}(\\lambda)=U_n(\\lambda)\\otimes u(\\lambda)\\). Hence \\(\\operatorname{Ad}U_{n+1}(\\lambda)(x\\otimes1)=(\\operatorname{Ad}U_n(\\lambda)(x))\\otimes1\\). By Lemma 4.4 there is an automorphism \\(\\sigma_\\lambda\\) of \\(B\\) that equals \\(\\operatorname{Ad}U_n(\\lambda)\\) on \\(B_n\\), and \\(\\sigma_\\lambda\\sigma_\\mu=\\sigma_{\\lambda\\mu}\\) (check on each \\(B_n\\) and extend by continuity). We call the fixed-point algebra\n\\[\nA=\\{x\\in B\\colon\\sigma_\\lambda(x)=x\\text{ for all }\\lambda\\in\\mathbb T\\}\n\\]\nthe *gauge-invariant CAR algebra*. It is a C\\*-subalgebra of \\(B\\) containing \\(1\\). Put \\(A_n=A\\cap B_n\\).\n\n**Lemma 9.2.** Let \\(H_{n,j}\\) be the span of the \\(\\varepsilon_\\xi\\) with \\(|\\xi|=j\\), of dimension \\(\\binom nj\\), and \\(P_{n,j}\\) the projection onto it. Then\n\\[\n\\begin{gathered}\nA_n\\\\\n=\\{x\\in B_n:xH_{n,j}\\subseteq H_{n,j}\\text{ for }j=0,\\dots,n\\}\\\\\n=\\bigoplus_{j=0}^nB(H_{n,j})\\cong\\bigoplus_{j=0}^nM_{\\binom nj}.\n\\end{gathered}\n\\]\nMoreover, for \\(x\\in B_n\\) and every integer \\(N>2n\\),\n\\[\n\\sum_{j=0}^nP_{n,j}\\,x\\,P_{n,j}=\\frac1N\\sum_{\\omega^N=1}\\sigma_\\omega(x).\n\\tag{9.1}\n\\]\n\n**Proof.** Let \\(e_{\\xi\\eta}\\) be the operator \\(\\zeta\\mapsto\\langle\\zeta,\\varepsilon_\\eta\\rangle\\varepsilon_\\xi\\). Since \\(\\bar\\lambda=\\lambda^{-1}\\),\n\\[\n\\begin{gathered}\n\\sigma_\\lambda(e_{\\xi\\eta})\\\\\n=U_n(\\lambda)e_{\\xi\\eta}U_n(\\lambda)^*\\\\\n=\\lambda^{\\,n-2|\\xi|}\\,\\overline{\\lambda^{\\,n-2|\\eta|}}\\,e_{\\xi\\eta}\\\\\n=\\lambda^{\\,2(|\\eta|-|\\xi|)}e_{\\xi\\eta}.\n\\end{gathered}\n\\]\nWrite \\(x=\\sum x_{\\xi\\eta}e_{\\xi\\eta}\\). The exponents \\(2(|\\eta|-|\\xi|)\\) lie between \\(-2n\\) and \\(2n\\). For \\(N>2n\\), the average of \\(\\omega^k\\) over the \\(N\\)-th roots of unity is \\(1\\) for \\(k=0\\) and \\(0\\) for \\(0<|k|\\leq2n\\). So the right side of (9.1) is \\(\\sum_{|\\xi|=|\\eta|}x_{\\xi\\eta}e_{\\xi\\eta}\\), which is the left side. If \\(x\\) is fixed by every \\(\\sigma_\\lambda\\), it equals this average, so it maps each \\(H_{n,j}\\) into itself. Conversely, an operator that maps each \\(H_{n,j}\\) into itself is a combination of the \\(e_{\\xi\\eta}\\) with \\(|\\xi|=|\\eta|\\), and these are fixed. \\(\\square\\)\n\n**Proposition 9.3** (Pascal's triangle). The algebras \\(A_0\\subseteq A_1\\subseteq\\cdots\\) form a generating sequence of \\(A\\), so \\(A\\) is a unital AF-algebra. Index the summands of \\(A_n\\) by \\(j=0,\\dots,n\\), the \\(j\\)-th being \\(B(H_{n,j})\\) of size \\(\\binom nj\\). The inclusion \\(A_n\\subseteq A_{n+1}\\) is unital, and the \\(j\\)-th summand of \\(A_n\\) enters the summands \\(j\\) and \\(j+1\\) of \\(A_{n+1}\\), each with multiplicity one. So the Bratteli diagram is Pascal's triangle:\n\n```\nlevel 0                         1\nlevel 1                      1     1\nlevel 2                   1     2     1\nlevel 3                1     3     3     1\nlevel 4             1     4     6     4     1\nlevel 5          1     5    10    10     5     1\nlevel 6       1     6    15    20    15     6     1\n```\n\nHere each vertex is joined to the two vertices just below it, and the labels are the sizes \\(\\binom nj\\).\n\n**Proof.** \\(A_n=A\\cap B_n\\subseteq A\\cap B_{n+1}=A_{n+1}\\). Let \\(x\\in A\\) and \\(\\varepsilon>0\\). Choose \\(n\\) and \\(y\\in B_n\\) with \\(\\|x-y\\|<\\varepsilon\\), and \\(N>2n\\). By Lemma 9.2 the element \\(E(y)=\\frac1N\\sum_{\\omega^N=1}\\sigma_\\omega(y)\\) lies in \\(A_n\\). Since \\(\\sigma_\\omega(x)=x\\) and each \\(\\sigma_\\omega\\) is isometric,\n\\[\n\\begin{gathered}\n\\|x-E(y)\\|\\\\\n=\\Big\\|\\frac1N\\sum_{\\omega^N=1}\\sigma_\\omega(x-y)\\Big\\|\\\\\n\\leq\\|x-y\\|<\\varepsilon .\n\\end{gathered}\n\\]\nSo \\(\\bigcup_nA_n\\) is dense in \\(A\\). Next, \\(H_{n,j}\\otimes\\mathbb C^2=(H_{n,j}\\otimes\\varepsilon_0)\\oplus(H_{n,j}\\otimes\\varepsilon_1)\\) with \\(H_{n,j}\\otimes\\varepsilon_0\\subseteq H_{n+1,j}\\) and \\(H_{n,j}\\otimes\\varepsilon_1\\subseteq H_{n+1,j+1}\\). A minimal projection \\(e\\) of \\(B(H_{n,j})\\), the projection onto a unit vector \\(v\\), goes to \\(e\\otimes1\\), the projection onto the span of \\(v\\otimes\\varepsilon_0\\) and \\(v\\otimes\\varepsilon_1\\); its components in \\(B(H_{n+1,j})\\) and \\(B(H_{n+1,j+1})\\) have rank one, and the others vanish. Finally \\(1\\otimes1=1\\). \\(\\square\\)\n\nTo compute the ordered group we need a classical theorem on positive polynomials.\n\n**Theorem 9.4** (Pólya's theorem). Let \\(F(x,y)=\\sum_{j=0}^dc_jx^jy^{d-j}\\) be a homogeneous polynomial with real coefficients such that \\(F(x,y)>0\\) whenever \\(x,y\\geq0\\) and \\(x+y=1\\). Then for all sufficiently large \\(N\\), every coefficient of \\((x+y)^NF(x,y)\\) is strictly positive.\n\n\n**Proof.** If \\(d=0\\) there is nothing to prove, so let \\(d\\geq1\\). Put \\(M=N+d\\) and write \\((x+y)^NF(x,y)=\\sum_{k=0}^Mb_kx^ky^{M-k}\\). Then \\(b_k=\\sum_jc_j\\binom N{k-j}\\), where \\(\\binom Ni=0\\) for \\(i<0\\) or \\(i>N\\). For integers \\(a\\geq0\\) and \\(i\\geq0\\) let \\((a)_i=a(a-1)\\cdots(a-i+1)\\), with \\((a)_0=1\\); note that \\((a)_i=0\\) when \\(0\\leq a<i\\). For \\(0\\leq j\\leq d\\) and \\(0\\leq k\\leq M\\),\n\\[\n\\begin{gathered}\n\\binom N{k-j}\\\\\n=\\binom Mk\\frac{(k)_j\\,(M-k)_{d-j}}{(M)_d}.\n\\end{gathered}\n\\tag{9.2}\n\\]\nIndeed, if \\(j\\leq k\\) and \\(k-j\\leq N\\), then, using \\(N-k+j=M-k-(d-j)\\) and \\(M!/N!=(M)_d\\),\n\\[\n\\begin{gathered}\n\\binom N{k-j}\\Big/\\binom Mk\\\\\n=\\frac{k!}{(k-j)!}\\cdot\\frac{(M-k)!}{(M-k-(d-j))!}\\cdot\\frac{N!}{M!}\\\\\n=\\frac{(k)_j(M-k)_{d-j}}{(M)_d};\n\\end{gathered}\n\\]\nif \\(k<j\\), both sides of (9.2) vanish because \\((k)_j=0\\); and if \\(k-j>N\\), then \\(M-k<d-j\\) and both sides vanish because \\((M-k)_{d-j}=0\\). Put \\(\\delta=1/M\\) and \\(t=k/M\\in[0,1]\\). Dividing the numerator and the denominator of (9.2) by \\(M^d\\) gives\n\\[\n\\begin{gathered}\nb_k\\\\\n=\\binom Mk\\frac{F_\\delta(t,1-t)}{\\prod_{i=0}^{d-1}(1-i\\delta)},\\\\\nF_\\delta(x,y)\\\\\n=\\sum_{j=0}^dc_j\\prod_{i=0}^{j-1}(x-i\\delta)\\prod_{i=0}^{d-j-1}(y-i\\delta).\n\\end{gathered}\n\\tag{9.3}\n\\]\nThe function \\((x,y,\\delta)\\mapsto F_\\delta(x,y)\\) is a polynomial, and \\(F_0=F\\). On the compact set \\(\\Delta=\\{(x,y):x,y\\geq0,\\ x+y=1\\}\\), \\(F\\) has a positive minimum \\(\\mu\\). Here compactness and the minimum assertion follow from Section 1(j). Expanding the finite products shows that \\(F_\\delta-F_0=\\delta R(x,y,\\delta)\\) for a polynomial \\(R\\). On \\(\\Delta\\times[0,1]\\), all variables have absolute value at most one, so the sum \\(C\\) of the absolute values of the coefficients of \\(R\\) bounds \\(|R|\\). Take \\(\\delta_0=\\min(1/d,\\mu/(2(C+1)))\\). Then \\(|F_\\delta-F|\\leq C\\delta\\leq\\mu/2\\) and \\(F_\\delta\\geq\\mu/2\\) for \\(0\\leq\\delta\\leq\\delta_0\\). If \\(M\\geq1/\\delta_0\\), then \\(\\delta\\leq\\delta_0\\) and \\(i\\delta\\leq(d-1)/M<1\\) for \\(i<d\\), so every \\(b_k\\) in (9.3) is positive. \\(\\square\\)\n\n**Theorem 9.5** (Dimension group of the gauge-invariant CAR algebra). For a minimal projection \\(e\\) of the \\(j\\)-th summand of \\(A_n\\), put \\(\\rho[e]=t^j(1-t)^{n-j}\\in\\mathbb Z[t]\\). This rule extends to an isomorphism of scaled ordered groups\n\\[\n\\rho:(K_0(A),K_0(A)^+,\\Sigma(A))\\longrightarrow(\\mathbb Z[t],\\mathcal P,\\mathcal S),\n\\]\nwhere\n\\[\n\\begin{gathered}\n\\mathcal P\\\\\n=\\{0\\}\\cup\\{f\\in\\mathbb Z[t]:f(t)>0\\\\\n\\text{ for all }0<t<1\\},\\\\\n\\mathcal S\\\\\n=\\{0,1\\}\\cup\\{f\\in\\mathbb Z[t]:0<f(t)<1\\\\\n\\text{ for all }0<t<1\\}.\n\\end{gathered}\n\\]\nThe class of the unit is the constant polynomial \\(1\\).\n\n**Proof.** *The group.* Let \\(\\rho_n:\\mathbb Z^{n+1}\\to\\mathbb Z[t]\\), \\(\\rho_n(x)=\\sum_{j=0}^nx_jt^j(1-t)^{n-j}\\). By Proposition 9.3, \\((\\alpha_nx)_j=x_j+x_{j-1}\\) (with \\(x_{-1}=x_{n+1}=0\\)), so\n\\[\n\\begin{gathered}\n\\rho_{n+1}(\\alpha_nx)\\\\\n=\\sum_jx_j\\big(t^j(1-t)^{n+1-j}+t^{j+1}(1-t)^{n-j}\\big)\\\\\n=\\sum_jx_jt^j(1-t)^{n-j}\\\\\n=\\rho_n(x).\n\\end{gathered}\n\\]\nSo the \\(\\rho_n\\) induce a homomorphism \\(\\rho:K_0(A)\\to\\mathbb Z[t]\\). Each \\(\\rho_n\\) is injective: under the substitution \\(t=s/(1+s)\\), the polynomials \\(t^j(1-t)^{n-j}\\) become \\(s^j/(1+s)^n\\), which are linearly independent. Hence \\(\\rho\\) is injective. It is onto: a polynomial of degree at most \\(n\\) with integer coefficients lies in \\(\\rho_n(\\mathbb Z^{n+1})\\), because\n\\[\n\\begin{gathered}\nt^i\\\\\n=t^i\\big(t+(1-t)\\big)^{n-i}\\\\\n=\\sum_{l=0}^{n-i}\\binom{n-i}l\\,t^{i+l}(1-t)^{n-i-l}.\n\\end{gathered}\n\\]\n\n*The cone.* \\(\\rho(K_0(A)^+)=\\bigcup_n\\rho_n(\\mathbb Z^{n+1}_+)\\). A nonzero \\(\\sum_jx_jt^j(1-t)^{n-j}\\) with all \\(x_j\\geq0\\) is positive on \\((0,1)\\), so this set lies in \\(\\mathcal P\\). Conversely, let \\(0\\neq f\\in\\mathcal P\\). Divide \\(f\\) by \\(t\\) as long as its value at \\(0\\) vanishes, and by \\(t-1\\) as long as its value at \\(1\\) vanishes; division by these monic polynomials keeps integer coefficients. This gives \\(f=t^a(1-t)^bg\\) with \\(a,b\\geq0\\), \\(g\\in\\mathbb Z[t]\\), \\(g(0)\\neq0\\) and \\(g(1)\\neq0\\). On \\((0,1)\\), \\(g=f/(t^a(1-t)^b)>0\\); by continuity \\(g(0),g(1)\\geq0\\), hence \\(g>0\\) on \\([0,1]\\). Let \\(e=\\deg g\\) and write \\(g=\\sum_{j=0}^ec_jt^j(1-t)^{e-j}\\) with \\(c_j\\in\\mathbb Z\\), as above. The form \\(G(x,y)=\\sum_jc_jx^jy^{e-j}\\) satisfies \\(G(t,1-t)=g(t)>0\\) for \\(t\\in[0,1]\\). By Theorem 9.4, for some \\(N\\) all coefficients \\(b_k\\) of \\((x+y)^NG(x,y)=\\sum_kb_kx^ky^{N+e-k}\\) are positive; they are integers. Putting \\(x=t\\) and \\(y=1-t\\),\n\\[\nf=\\sum_kb_k\\,t^{k+a}(1-t)^{N+e-k+b},\n\\]\nwhich is \\(\\rho_{N+e+a+b}\\) of a vector with entries in \\(\\mathbb Z_+\\). So \\(f\\in\\rho(K_0(A)^+)\\).\n\n*The unit and the scale.* The unit of \\(A_n\\) has rank vector \\(\\big(\\binom n0,\\dots,\\binom nn\\big)\\), and \\(\\sum_j\\binom njt^j(1-t)^{n-j}=1\\). Since \\(1\\in A_n\\) for all \\(n\\), Lemma 6.3(5) gives \\(\\Sigma(A)=\\{g:0\\leq g\\leq[1]\\}\\), whose image is \\(\\{f:f\\in\\mathcal P,\\ 1-f\\in\\mathcal P\\}\\). An element of this set is \\(0\\), or \\(1\\), or is positive on \\((0,1)\\) together with \\(1-f\\); this is \\(\\mathcal S\\). \\(\\square\\)\n\n**Example 9.6** (Edge cases). (a) The polynomial \\(t(1-t)\\) vanishes at both ends of \\([0,1]\\) and lies in \\(\\mathcal P\\): it is the class of a minimal projection of the middle summand of \\(A_2\\). (b) The polynomial \\((2t-1)^2\\) is \\(\\geq0\\) on \\([0,1]\\) but vanishes at \\(\\frac12\\), so it is not in \\(\\mathcal P\\); neither is its negative. So the order of \\(K_0(A)\\) is not the pointwise order of functions on \\([0,1]\\), and \\((2t-1)^2\\) is not the class of any projection in any matrix algebra over \\(A\\). (c) Neither \\(2t-1\\) nor \\(1-2t\\) lies in \\(\\mathcal P\\), so \\(K_0(A)\\) is not totally ordered, unlike the dimension groups of UHF algebras.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-AF-09",
      "unit": "af-algebras",
      "name": "10. Traces",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 10. Traces\n\n**Definition 10.1.** A *trace* on a C\\*-algebra \\(A\\) is a bounded linear functional \\(\\tau\\) with \\(\\tau(a)\\geq0\\) for \\(a\\geq0\\) and \\(\\tau(xy)=\\tau(yx)\\) for all \\(x,y\\in A\\). A *tracial state* is a trace of norm one. Let \\(T(A)\\) be the set of tracial states and \\(T_{\\leq1}(A)\\) the set of traces of norm at most one, both with the weak\\* topology (pointwise convergence on \\(A\\)).\n\n**Lemma 10.2** (Traces on \\(M_{\\mathbf m}\\)). For \\(t\\in\\mathbb R^r_+\\) put \\(\\tau_t(x)=\\sum_it_i\\operatorname{Tr}(x_i)\\). The traces of \\(M_{\\mathbf m}\\) are exactly the \\(\\tau_t\\), \\(t\\in\\mathbb R^r_+\\), and \\(t_i\\) is the value of \\(\\tau_t\\) on any minimal projection of the \\(i\\)-th summand. Moreover \\(\\|\\tau_t\\|=\\tau_t(1)=\\mathbf m^Tt\\). So \\(T(M_{\\mathbf m})\\) is identified with the simplex \\(\\Delta(\\mathbf m)=\\{t\\in\\mathbb R^r_+:\\mathbf m^Tt=1\\}\\), whose vertices are the \\(e_i/m_i\\).\n\n**Proof.** Let \\(\\tau\\) be a trace, and \\(\\tau_i\\) its restriction to the \\(i\\)-th summand. Then \\(\\tau_i(e_{ab})=\\tau_i(e_{a1}e_{1b})=\\tau_i(e_{1b}e_{a1})=\\delta_{ab}\\tau_i(e_{11})\\), so \\(\\tau_i=\\tau_i(e_{11})\\operatorname{Tr}\\), and \\(t_i=\\tau(e^{(i)}_{11})\\geq0\\) by positivity. Conversely \\(\\tau_t\\) is linear and tracial, and positive since \\(\\operatorname{Tr}(y^*y)\\geq0\\). Since \\(|\\operatorname{Tr}y|\\leq k\\|y\\|\\) for \\(y\\in M_k\\), \\(|\\tau_t(x)|\\leq\\sum_it_im_i\\|x_i\\|\\leq(\\mathbf m^Tt)\\|x\\|\\), with equality for \\(x=1\\). \\(\\square\\)\n\n**Lemma 10.3** (Pulling back traces). If \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) has multiplicity matrix \\(\\alpha\\), then \\(\\tau_y\\circ\\varphi=\\tau_{\\alpha^Ty}\\) for \\(y\\in\\mathbb R^s_+\\), and \\(\\|\\tau_y\\circ\\varphi\\|\\leq\\|\\tau_y\\|\\), with equality when \\(\\varphi\\) is unital.\n\n**Proof.** \\(\\tau_y\\circ\\varphi\\) is a trace, and on a minimal projection \\(e\\) of the \\(i\\)-th summand it takes the value \\(\\sum_jy_j\\operatorname{Tr}\\varphi_j(e)=\\sum_jy_j\\alpha_{ji}=(\\alpha^Ty)_i\\). Its norm is \\(\\mathbf m^T\\alpha^Ty=(\\alpha\\mathbf m)^Ty\\leq\\mathbf n^Ty=\\|\\tau_y\\|\\), with equality when \\(\\alpha\\mathbf m=\\mathbf n\\). \\(\\square\\)\n\n**Theorem 10.4** (Traces as a projective limit). Let \\(A\\) be an AF-algebra with generating sequence \\((A_k)\\), \\(A_k\\cong M_{\\mathbf m(k)}\\), and multiplicity matrices \\(\\alpha_k\\). For a trace \\(\\tau\\) on \\(A\\), let \\(t^{(k)}(\\tau)\\in\\mathbb R^{r_k}_+\\) be the vector with \\(\\tau|_{A_k}=\\tau_{t^{(k)}(\\tau)}\\).\n\n1. The map \\(\\tau\\mapsto(t^{(k)}(\\tau))_k\\) is an affine bijection of \\(T_{\\leq1}(A)\\) onto the set of sequences \\((t^{(k)})\\) with \\(t^{(k)}\\in\\mathbb R^{r_k}_+\\), \\(\\alpha_k^Tt^{(k+1)}=t^{(k)}\\) and \\(\\mathbf m(k)^Tt^{(k)}\\leq1\\) for all \\(k\\). It is a homeomorphism for the weak\\* topology and the product topology. Moreover \\(\\|\\tau\\|=\\lim_k\\mathbf m(k)^Tt^{(k)}(\\tau)\\), and the sequence \\(\\mathbf m(k)^Tt^{(k)}(\\tau)\\) is nondecreasing.\n2. A trace \\(\\tau\\) has norm one exactly when \\(\\lim_k\\mathbf m(k)^Tt^{(k)}(\\tau)=1\\).\n3. If \\(A\\) is nonzero and unital and \\(1_A\\in A_k\\) for all \\(k\\), then \\(\\mathbf m(k)^Tt^{(k)}(\\tau)=\\tau(1)\\) for every \\(k\\). So \\(T(A)\\) is affinely homeomorphic to the projective limit of the simplices \\(\\Delta(\\mathbf m(k))\\) under the maps \\(t\\mapsto\\alpha_k^Tt\\), and \\(T(A)\\) is not empty.\n\n**Proof.** (1) The restriction of a trace \\(\\tau\\) of norm at most one to \\(A_k\\) is a trace of norm at most one, so \\(\\mathbf m(k)^Tt^{(k)}\\leq1\\) (Lemma 10.2), and \\(\\alpha_k^Tt^{(k+1)}=t^{(k)}\\) by Lemma 10.3 applied to the inclusion. A trace is determined by its values on the dense set \\(A_\\infty\\), so the map is injective. Let \\((t^{(k)})\\) be a sequence as in the statement. Define \\(\\tau_0\\) on \\(A_\\infty\\) by \\(\\tau_0(x)=\\tau_{t^{(k)}}(x)\\) for \\(x\\in A_k\\); by Lemma 10.3 this is consistent. It is linear and tracial, and \\(|\\tau_0(x)|\\leq\\|x\\|\\) by Lemma 10.2. So it extends to a bounded linear functional \\(\\tau\\) of norm at most one on \\(A\\), tracial by continuity. It is positive: if \\(a\\geq0\\), then \\(a=b^*b\\) (Section 1(d)); choose \\(b_n\\in A_\\infty\\) with \\(b_n\\to b\\); then \\(b_n^*b_n\\to a\\), and \\(\\tau(b_n^*b_n)\\geq0\\) because \\(b_n^*b_n\\) is positive in the finite-dimensional algebra that contains \\(b_n\\). The map is clearly affine. It is continuous because each coordinate \\(t^{(k)}_i(\\tau)\\) is the value of \\(\\tau\\) at a fixed element. Its inverse is continuous: if \\(t_\\lambda\\to t\\) coordinatewise, the corresponding traces converge at every \\(x\\in A_\\infty\\), and since all have norm at most one, an \\(\\varepsilon/3\\) argument gives convergence at every \\(x\\in A\\). Finally, \\(\\|\\tau\\|\\) is the supremum of \\(|\\tau(x)|\\) over the unit ball of the dense subalgebra \\(A_\\infty\\), so \\(\\|\\tau\\|=\\sup_k\\|\\tau|_{A_k}\\|=\\sup_k\\mathbf m(k)^Tt^{(k)}\\), and \\(\\|\\tau|_{A_k}\\|\\) increases with \\(k\\).\n\n(2) is (1) with \\(\\|\\tau\\|=1\\).\n\n(3) The units of the \\(A_k\\) all equal \\(1_A\\), so \\(\\mathbf m(k)^Tt^{(k)}=\\tau(1_A)\\), and \\(\\tau\\) is a tracial state exactly when every \\(t^{(k)}\\) lies in \\(\\Delta(\\mathbf m(k))\\). The inclusions are unital, so \\(t\\mapsto\\alpha_k^Tt\\) maps \\(\\Delta(\\mathbf m(k+1))\\) into \\(\\Delta(\\mathbf m(k))\\) (Lemma 10.3). To see that \\(T(A)\\neq\\varnothing\\), choose for each \\(N\\) a point \\(s_N\\in\\Delta(\\mathbf m(N))\\) (for instance \\(e_1/m(N)_1\\)), and define a sequence \\(t^{[N]}\\) by \\(t^{[N],(k)}=\\alpha_k^T\\cdots\\alpha_{N-1}^Ts_N\\in\\Delta(\\mathbf m(k))\\) for \\(k\\leq N\\) and \\(t^{[N],(k)}\\) any point of \\(\\Delta(\\mathbf m(k))\\) for \\(k>N\\). Each simplex is compact. For clarity, the diagonal argument uses only Section 1(j): successively take infinite nested subsequences of the indices \\(N\\), on the \\(k\\)-th of which the \\(k\\)-th coordinate converges. Choose the \\(\\nu\\)-th diagonal index from the \\(\\nu\\)-th subsequence, larger than the preceding index. For every fixed \\(k\\), its tail lies in the \\(k\\)-th subsequence. Thus there are \\(N_1<N_2<\\cdots\\) such that \\(t^{[N_\\nu],(k)}\\) converges, to \\(t^{(k)}\\in\\Delta(\\mathbf m(k))\\) say, for every \\(k\\). The relation \\(\\alpha_k^Tt^{[N_\\nu],(k+1)}=t^{[N_\\nu],(k)}\\) holds once \\(N_\\nu>k\\), so it passes to the limit. By (1), \\((t^{(k)})\\) defines a tracial state. \\(\\square\\)\n\n**Example 10.5.** (a) *Compact operators.* All matrices are \\((1)\\), so a compatible sequence is constant, \\(t^{(n)}=t\\), with \\(nt\\leq1\\) for all \\(n\\). So \\(t=0\\): the only trace of norm at most one is \\(0\\), and \\(T(\\mathcal K)=\\varnothing\\). So a nonunital AF-algebra may have no tracial state.\n\n(b) *Unitization of the compact operators.* With \\(t^{(n)}=(a_n,b_n)\\) and \\(\\alpha_n^T=\\begin{pmatrix}1&0\\\\1&1\\end{pmatrix}\\), compatibility says \\(a_n=a_{n+1}\\) and \\(b_n=a_{n+1}+b_{n+1}\\), and normalization says \\(na_n+b_n=1\\). So \\(a_n=a\\) is constant with \\(na\\leq1\\) for all \\(n\\), hence \\(a=0\\) and \\(b_n=1\\). So \\(\\mathcal K+\\mathbb C1\\) has exactly one tracial state, \\(\\tau(k+\\lambda1)=\\lambda\\).\n\nThe explicit quotient trace and Theorem 10.4(3) prove that every nonzero unital AF-algebra has a tracial state. The zero algebra has only the zero functional and no norm-one state.\n\n(c) *UHF algebras.* Compatibility forces \\(t^{(n)}=1/d_n\\): a UHF algebra has exactly one tracial state. By Exercise 3 below, the gauge-invariant CAR algebra has infinitely many.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AF-10",
      "unit": "af-algebras",
      "name": "11. Commutants and the reflected Bratteli diagram",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## 11. Commutants and the reflected Bratteli diagram\n\nIn this section \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) is a \\*-homomorphism with multiplicity matrix \\(\\alpha\\) (an \\(s\\times r\\) matrix) and defect \\(d=\\mathbf n-\\alpha\\mathbf m\\), and \\(\\pi:M_{\\mathbf n}\\to B(H)\\) is a unital representation on a finite-dimensional Hilbert space. By Remark 3.3, \\(\\pi\\) has a *multiplicity vector* \\(\\mathbf n'\\in\\mathbb Z^s_+\\) with \\(\\sum_jn_jn_j'=\\dim H\\), and after a unitary change of basis \\(H=\\bigoplus_j\\mathbb C^{n_j}\\otimes\\mathbb C^{n_j'}\\) and \\(\\pi(y)=\\bigoplus_jy_j\\otimes1\\). For a set \\(S\\subseteq B(H)\\), \\(S'\\) is its commutant.\n\n**Lemma 11.1.** Let \\(H=\\bigoplus_i(\\mathbb C^{k_i}\\otimes K_i)\\oplus L\\) with finite-dimensional \\(K_i\\) and \\(L\\), and let \\(Q\\subseteq B(H)\\) consist of the operators \\(\\bigoplus_i(y_i\\otimes1_{K_i})\\oplus0_L\\), \\(y_i\\in M_{k_i}\\). Then\n\\[\n\\begin{gathered}\nQ'\\\\\n=\\bigoplus_i\\big(1_{k_i}\\otimes B(K_i)\\big)\\oplus B(L)\\\\\n\\cong\\bigoplus_{i:K_i\\neq0}B(K_i)\\ \\oplus\\ B(L).\n\\end{gathered}\n\\]\n\n**Proof.** An operator \\(T\\in Q'\\) commutes with the projections onto the spaces \\(\\mathbb C^{k_i}\\otimes K_i\\), which are images of units of summands, and hence with the projection onto \\(L\\). So \\(T=\\bigoplus_iT_i\\oplus T_L\\). Write \\(T_i\\) as a \\(k_i\\times k_i\\) block matrix \\((T_{ab})\\) with blocks in \\(B(K_i)\\), using \\(\\mathbb C^{k_i}\\otimes K_i=\\bigoplus_a\\varepsilon_a\\otimes K_i\\). Commuting with \\(e_{cd}\\otimes1\\) means \\(\\delta_{ac}T_{db}=T_{ac}\\delta_{db}\\) for all \\(a,b,c,d\\). With \\(a=c\\) and \\(b=d\\) this gives \\(T_{bb}=T_{cc}\\); with \\(a=c\\) and \\(b\\neq d\\) it gives \\(T_{db}=0\\). So \\(T_i=1\\otimes t_i\\). There is no condition on \\(T_L\\). The converse inclusion is clear. \\(\\square\\)\n\n**Theorem 11.2** (Commutants reflect the diagram). Put \\(P=\\pi(M_{\\mathbf n})\\) and \\(Q=\\pi(\\varphi(M_{\\mathbf m}))\\subseteq P\\), so that \\(P'\\subseteq Q'\\). Let \\(\\mathbf m'=\\alpha^T\\mathbf n'\\) (so \\(m_i'=\\sum_j\\alpha_{ji}n_j'\\)) and \\(d'=\\sum_jd_jn_j'\\). Then:\n\n1. \\(P'\\cong\\bigoplus_{j:n_j'>0}M_{n_j'}\\);\n2. \\(Q'\\cong\\bigoplus_{i:m_i'>0}M_{m_i'}\\oplus M_{d'}\\), where the last summand is present only when \\(d'>0\\);\n3. the inclusion \\(P'\\subseteq Q'\\) is unital, the summand \\(M_{n_j'}\\) of \\(P'\\) enters the summand \\(M_{m_i'}\\) of \\(Q'\\) with multiplicity \\(\\alpha_{ji}\\), and it enters \\(M_{d'}\\) with multiplicity \\(d_j\\).\n\nIn particular, if \\(\\varphi\\) is unital and injective and \\(\\pi\\) is faithful, then \\(P'\\cong M_{\\mathbf n'}\\), \\(Q'\\cong M_{\\alpha^T\\mathbf n'}\\), and the multiplicity matrix of \\(P'\\subseteq Q'\\) is the transpose \\(\\alpha^T\\): the Bratteli diagram of \\(P'\\subseteq Q'\\) is the diagram of \\(\\varphi\\) read from right to left, with the new sizes \\(\\mathbf n'\\) and \\(\\alpha^T\\mathbf n'\\).\n\n**Proof.** A unitary change of basis in \\(H\\) conjugates \\(P\\), \\(Q\\), \\(P'\\) and \\(Q'\\) simultaneously, so we may assume \\(\\pi\\) has the form \\(\\bigoplus_jy_j\\otimes1\\). By Theorem 3.2(4), \\(\\varphi=\\operatorname{Ad}U\\circ\\varphi_\\alpha\\) for a unitary \\(U\\in M_{\\mathbf n}\\). Replacing \\(\\varphi\\) by \\(\\varphi_\\alpha\\) replaces \\(Q\\) and \\(Q'\\) by their conjugates under \\(\\pi(U)^*\\); since \\(\\pi(U)\\in P\\) commutes with every element of \\(P'\\), this conjugation fixes \\(P'\\) pointwise and does not change the inclusion. So we may assume \\(\\varphi=\\varphi_\\alpha\\). Then \\(\\mathbb C^{n_j}=\\bigoplus_i\\mathbb C^{m_i}\\otimes\\mathbb C^{\\alpha_{ji}}\\oplus\\mathbb C^{d_j}\\), with \\(\\varphi(x)_j=\\bigoplus_ix_i\\otimes1\\oplus0\\). Regrouping,\n\\[\n\\begin{gathered}\nH\\\\\n=\\bigoplus_i\\big(\\mathbb C^{m_i}\\otimes K_i\\big)\\oplus L,\\\\\nK_i\\\\\n=\\bigoplus_j\\mathbb C^{\\alpha_{ji}}\\otimes\\mathbb C^{n_j'},\\\\\nL\\\\\n=\\bigoplus_j\\mathbb C^{d_j}\\otimes\\mathbb C^{n_j'},\n\\end{gathered}\n\\]\nand \\(\\pi(\\varphi(x))=\\bigoplus_ix_i\\otimes1_{K_i}\\oplus0_L\\), with \\(\\dim K_i=m_i'\\) and \\(\\dim L=d'\\). Lemma 11.1 gives (2), and, applied to \\(P\\) itself, (1): \\(P'=\\bigoplus_j1_{n_j}\\otimes B(\\mathbb C^{n_j'})\\). An element \\((t_j)_j\\) of \\(P'\\) acts on \\(\\mathbb C^{n_j}\\otimes\\mathbb C^{n_j'}=\\bigoplus_i\\mathbb C^{m_i}\\otimes\\mathbb C^{\\alpha_{ji}}\\otimes\\mathbb C^{n_j'}\\oplus\\mathbb C^{d_j}\\otimes\\mathbb C^{n_j'}\\) as \\(1\\otimes1\\otimes t_j\\) on each piece. So in \\(B(K_i)\\) it is \\(\\bigoplus_j1_{\\alpha_{ji}}\\otimes t_j\\), and in \\(B(L)\\) it is \\(\\bigoplus_j1_{d_j}\\otimes t_j\\). A minimal projection of \\(M_{n_j'}\\) therefore has rank \\(\\alpha_{ji}\\) in \\(B(K_i)\\) and rank \\(d_j\\) in \\(B(L)\\), which is (3). Both algebras contain \\(1_H\\). If \\(\\varphi\\) is unital, \\(d=0\\); if \\(\\pi\\) is faithful, every \\(n_j'\\geq1\\); and if \\(\\varphi\\) is also injective, every column of \\(\\alpha\\) is nonzero, so every \\(m_i'\\geq1\\). \\(\\square\\)\n\n**Example 11.3.** (a) Take \\(\\varphi_\\alpha:\\mathbb C\\oplus M_2\\to M_5\\oplus M_4\\) from Example 3.6(a), with \\(\\alpha=\\begin{pmatrix}3&1\\\\0&2\\end{pmatrix}\\), and the identity representation of \\(M_5\\oplus M_4\\) on \\(\\mathbb C^5\\oplus\\mathbb C^4\\), with \\(\\mathbf n'=(1,1)\\). Then \\(P'=\\mathbb C\\oplus\\mathbb C\\) is the centre of \\(M_5\\oplus M_4\\), and \\(\\mathbf m'=\\alpha^T\\mathbf n'=(3,3)\\), so \\(Q'\\cong M_3\\oplus M_3\\). Directly: \\(Q\\) acts on \\(\\mathbb C^9\\) as \\(\\lambda\\) on a three-dimensional subspace and as \\(y\\otimes1\\) on \\(\\mathbb C^2\\otimes\\mathbb C^3\\), so its commutant is \\(M_3\\oplus(1_2\\otimes M_3)\\). The pair \\((t_1,t_2)\\in P'\\) becomes \\((\\operatorname{diag}(t_1,t_1,t_1),\\operatorname{diag}(t_1,t_2,t_2))\\), with multiplicity matrix \\(\\begin{pmatrix}3&0\\\\1&2\\end{pmatrix}=\\alpha^T\\).\n\n(b) *A nonunital inclusion.* Take \\(\\varphi:M_2\\to M_5\\), \\(x\\mapsto\\operatorname{diag}(x,x,0)\\), with \\(\\alpha=(2)\\) and \\(d=(1)\\), and \\(\\pi\\) the identity on \\(\\mathbb C^5\\). Then \\(P'=\\mathbb C1\\), and \\(Q'=(1_2\\otimes M_2)\\oplus\\mathbb C\\cong M_2\\oplus\\mathbb C\\): the defect produces the extra summand \\(M_{d'}=\\mathbb C\\). The scalar \\(t\\in P'\\) becomes \\((t1_2,t)\\), with multiplicities \\(2=\\alpha\\) and \\(1=d\\). Without unitality of \\(\\varphi\\), the commutant \\(Q'\\) is not \\(M_{\\alpha^T\\mathbf n'}=M_2\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AF-11",
      "unit": "af-algebras",
      "name": "12. Group algebras of locally finite groups",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "anchor": "oa-fnd-af-11",
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        "line": 799,
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      "full_conditions_and_proof": "## 12. Group algebras of locally finite groups\n\nFor a finite group \\(\\Gamma\\), the group algebra \\(\\mathbb C[\\Gamma]\\) is the space of functions \\(\\Gamma\\to\\mathbb C\\) with the convolution \\(f*g(x)=\\sum_yf(y)g(y^{-1}x)\\) and the involution \\(f^*(x)=\\overline{f(x^{-1})}\\). Let \\(\\delta_y\\) be the function equal to \\(1\\) at \\(y\\) and \\(0\\) elsewhere; then \\(\\delta_y*\\delta_z=\\delta_{yz}\\) and \\(\\delta_y^*=\\delta_{y^{-1}}\\). The *left regular representation* \\(\\lambda\\) of \\(\\mathbb C[\\Gamma]\\) on \\(\\ell^2(\\Gamma)\\) sends \\(\\delta_y\\) to the unitary \\(\\lambda_y\\), \\((\\lambda_y\\xi)(x)=\\xi(y^{-1}x)\\), and \\(f\\) to \\(\\sum_yf(y)\\lambda_y\\); since \\(\\lambda_y\\lambda_z=\\lambda_{yz}\\) and \\(\\lambda_y^*=\\lambda_{y^{-1}}\\), it is a \\*-homomorphism, and it is injective because \\(\\lambda(f)\\delta_e=f\\). So \\(\\mathbb C[\\Gamma]\\), with the norm \\(\\|\\lambda(f)\\|\\), is a finite-dimensional C\\*-algebra, and by Section 1(a) this is its only C\\*-norm.\n\n**Proposition 12.1.** Let \\(\\Gamma\\) be a finite group.\n\n1. The centre of \\(\\mathbb C[\\Gamma]\\) consists of the functions that are constant on conjugacy classes, so the number of summands of \\(\\mathbb C[\\Gamma]\\) is the number of conjugacy classes.\n2. Unitary representations of \\(\\Gamma\\) on finite-dimensional Hilbert spaces correspond to unital representations of \\(\\mathbb C[\\Gamma]\\) by \\(\\rho(f)=\\sum_yf(y)\\rho(y)\\), with the same invariant subspaces. So the summands of \\(\\mathbb C[\\Gamma]\\) correspond to the unitary equivalence classes of irreducible representations of \\(\\Gamma\\), and the size of a summand is the dimension of the representation.\n3. If \\(\\Gamma_0\\subseteq\\Gamma\\) is a subgroup, the inclusion \\(\\mathbb C[\\Gamma_0]\\subseteq\\mathbb C[\\Gamma]\\) (extension by zero) is a unital embedding, and its multiplicity matrix has, in the row of an irreducible representation \\(\\rho\\) of \\(\\Gamma\\) and the column of an irreducible representation \\(\\sigma\\) of \\(\\Gamma_0\\), the multiplicity of \\(\\sigma\\) in the restriction of \\(\\rho\\) to \\(\\Gamma_0\\).\n\n**Proof.** (1) \\(f\\) is central if and only if \\(\\delta_y*f*\\delta_{y^{-1}}=f\\) for all \\(y\\), and \\((\\delta_y*f*\\delta_{y^{-1}})(x)=f(y^{-1}xy)\\). The centre has one basis element for each conjugacy class; apply Theorem 2.4. (2) The correspondence is inverse to \\(\\rho\\mapsto(y\\mapsto\\rho(\\delta_y))\\), and a subspace is invariant under all \\(\\rho(y)\\) if and only if it is invariant under all \\(\\rho(f)\\). By Remark 3.3, the irreducible representations of \\(M_{\\mathbf m}\\) are the maps to its summands. (3) Restricting \\(\\rho\\) to \\(\\Gamma_0\\) is the same as composing the corresponding representation of \\(\\mathbb C[\\Gamma]\\) with the inclusion; by Remark 3.3, the multiplicities of this composite are the multiplicities of the irreducible representations of \\(\\Gamma_0\\) in it. \\(\\square\\)\n\n**Proposition 12.2** (Locally finite groups). Let \\(\\Gamma\\) be the union of an increasing sequence \\(\\Gamma_1\\subseteq\\Gamma_2\\subseteq\\cdots\\) of finite subgroups, and let \\(\\mathbb C[\\Gamma]\\) be the \\*-algebra of finitely supported functions on \\(\\Gamma\\), with convolution and involution as above. Then \\(\\mathbb C[\\Gamma]\\) has exactly one C\\*-norm, namely \\(\\|\\lambda(f)\\|\\) for the left regular representation on \\(\\ell^2(\\Gamma)\\). Its completion \\(C^*(\\Gamma)\\) is an AF-algebra with generating sequence \\((\\mathbb C[\\Gamma_n])\\), and the multiplicities in its Bratteli diagram are the restriction multiplicities of Proposition 12.1(3).\n\n**Proof.** \\(\\mathbb C[\\Gamma]=\\bigcup_n\\mathbb C[\\Gamma_n]\\), and the inclusions are \\*-homomorphisms. The left regular representation of \\(\\mathbb C[\\Gamma]\\) on \\(\\ell^2(\\Gamma)\\) is injective (\\(\\lambda(f)\\delta_e=f\\)), so \\(\\|\\lambda(f)\\|\\) is a C\\*-norm. Any C\\*-norm on \\(\\mathbb C[\\Gamma]\\) restricts to a C\\*-norm on each \\(\\mathbb C[\\Gamma_n]\\), which is unique; so any two C\\*-norms agree on each \\(\\mathbb C[\\Gamma_n]\\), hence on \\(\\mathbb C[\\Gamma]\\). The completion is the closure of the increasing union of the finite-dimensional C\\*-algebras \\(\\mathbb C[\\Gamma_n]\\). The last statement is Proposition 12.1(3). \\(\\square\\)\n\n**Example 12.3** (The infinite symmetric group). Let \\(S_n\\) be the group of permutations of \\(\\{1,\\dots,n\\}\\), embedded in \\(S_{n+1}\\) as the permutations fixing \\(n+1\\). The union \\(S_\\infty\\) is the group of permutations of \\(\\mathbb N\\) that move only finitely many points, and \\(C^*(S_\\infty)\\) is an AF-algebra by Proposition 12.2. We compute the first levels of its diagram.\n\n\\(\\mathbb C[S_1]=\\mathbb C\\). The group \\(S_2\\) has two conjugacy classes and order \\(2\\), so \\(\\mathbb C[S_2]\\cong\\mathbb C\\oplus\\mathbb C\\); the two summands are the trivial and the sign representation. The group \\(S_3\\) has three conjugacy classes and order \\(6\\), and the only way to write \\(6\\) as a sum of three squares of positive integers is \\(1+1+4\\). So \\(\\mathbb C[S_3]\\cong\\mathbb C\\oplus\\mathbb C\\oplus M_2\\): the trivial representation, the sign representation and a two-dimensional irreducible representation \\(\\rho\\).\n\nWe identify \\(\\rho\\). Let \\(S_3\\) permute the coordinates of \\(V=\\{v\\in\\mathbb C^3:v_1+v_2+v_3=0\\}\\), a two-dimensional unitary representation. A one-dimensional invariant subspace would be spanned by a common eigenvector \\(v\\) of the transpositions \\((12)\\) and \\((23)\\), which generate \\(S_3\\); since they are involutions, each acts on \\(v\\) by \\(+1\\) or \\(-1\\). If both eigenvalues are \\(+1\\), the coordinates of \\(v\\) are equal and their sum zero forces \\(v=0\\). If \\((12)\\) has eigenvalue \\(-1\\), then \\(v=(a,-a,0)\\). For \\((23)\\) to have eigenvalue \\(+1\\) requires \\(-a=0\\), and eigenvalue \\(-1\\) requires \\(a=0\\). In the remaining case, \\((12)\\) has eigenvalue \\(+1\\) and \\((23)\\) has eigenvalue \\(-1\\); then \\(v=(0,a,-a)\\) and \\(v_1=v_2\\) again forces \\(a=0\\). So \\(V\\) is irreducible, and since \\(\\mathbb C[S_3]\\) has only one summand of size \\(2\\), \\(V\\) is \\(\\rho\\). On \\(V\\), the transposition \\((12)\\) fixes \\((1,1,-2)\\) and negates \\((1,-1,0)\\), so \\(\\rho\\) restricted to \\(S_2\\) is the sum of the trivial and the sign representations. The trivial and sign representations of \\(S_3\\) restrict to those of \\(S_2\\). With the summands ordered as (trivial, sign, \\(\\rho\\)) and (trivial, sign), the multiplicity matrices of \\(\\mathbb C[S_1]\\subseteq\\mathbb C[S_2]\\subseteq\\mathbb C[S_3]\\) are\n\\[\n\\begin{pmatrix}1\\\\1\\end{pmatrix},\\qquad\\begin{pmatrix}1&0\\\\0&1\\\\1&1\\end{pmatrix}.\n\\]\n*Unused extension, not proved here.* The general Young-diagram branching rule is not used in the finite-level computations, the locally finite group theorem, or any subsequent proof of this lesson. In general, the irreducible representations of \\(S_n\\) are indexed by the partitions of \\(n\\), drawn as Young diagrams, and the restriction of the representation of a diagram to \\(S_{n-1}\\) is the sum, with multiplicity one each, of the representations of the diagrams obtained by removing one box . So the Bratteli diagram of \\(C^*(S_\\infty)\\) is the graph of Young diagrams ordered by adding one box, and all its multiplicities are \\(0\\) or \\(1\\). The case \\(n\\leq3\\) above agrees: \\((2)\\) and \\((1,1)\\) come from \\((1)\\), and the diagram \\((2,1)\\) of \\(\\rho\\) contains both \\((2)\\) and \\((1,1)\\).\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AF-12",
      "unit": "af-algebras",
      "name": "13. Exercises",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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        "line": 825,
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      "full_conditions_and_proof": "## 13. Exercises\n\n**Exercise 1** (medium; The Cantor set). Let \\(X=\\{0,1\\}^{\\mathbb N}\\) with the product topology. For a word \\(w\\in\\{0,1\\}^n\\) let \\([w]\\) be the set of sequences that begin with \\(w\\), and let \\(A_n\\subseteq C(X)\\) be the span of the indicator functions \\(\\chi_{[w]}\\), \\(w\\in\\{0,1\\}^n\\).\n(a) Show that \\(C(X)\\) is an AF-algebra with generating sequence \\((A_n)\\) and that its Bratteli diagram is the binary tree: all sizes are \\(1\\), and each vertex at level \\(n\\) is joined by one edge to each of two vertices at level \\(n+1\\).\n(b) Show that the scaled dimension group of \\(C(X)\\) is isomorphic to \\((C(X,\\mathbb Z),C(X,\\mathbb Z_+),\\{\\chi_U:U\\subseteq X\\text{ clopen}\\})\\).\n\n*Solution.* (a) The \\(\\chi_{[w]}\\), \\(w\\in\\{0,1\\}^n\\), are mutually orthogonal projections with sum \\(1\\), so \\(A_n\\cong\\mathbb C^{2^n}\\); and \\(\\chi_{[w]}=\\chi_{[w0]}+\\chi_{[w1]}\\), so \\(A_n\\subseteq A_{n+1}\\), and the minimal projection \\(\\chi_{[w]}\\) of \\(A_n\\) enters exactly the two summands of \\(A_{n+1}\\) indexed by \\(w0\\) and \\(w1\\), once each. The union \\(\\bigcup_nA_n\\) is a \\*-subalgebra of \\(C(X)\\) that contains the constants and separates points: two different sequences first differ at some place \\(n\\), and \\(\\chi_{[w]}\\) for the first \\(n\\) terms of one of them separates them. By the Stone–Weierstrass theorem ([The Stone–Weierstrass theorem for functions vanishing at infinity](stone-weierstrass-c0.md)), it is dense.\n(b) Let \\(\\rho_n(x)=\\sum_wx_w\\chi_{[w]}\\) for \\(x\\in\\mathbb Z^{\\{0,1\\}^n}\\). These maps are compatible with the connecting maps and injective, so they define an injective homomorphism \\(\\rho\\) from the dimension group into \\(C(X,\\mathbb Z)\\), whose image consists of the functions that depend on finitely many coordinates. Every continuous \\(f:X\\to\\mathbb Z\\) is of this kind: each set \\(f^{-1}(c)\\) is clopen, only finitely many are nonempty because \\(X\\) is compact, and a clopen set is a union of sets \\([w]\\) (it is open) and hence a finite union (it is compact); taking all words of a common length \\(n\\), \\(f\\in\\rho_n(\\mathbb Z^{\\{0,1\\}^n})\\). A function in this image is \\(\\geq0\\) exactly when its coefficients at a stage where it is defined are \\(\\geq0\\), and it takes only the values \\(0,1\\) exactly when its coefficients lie in \\(\\{0,1\\}\\), that is, in the scale \\(\\Sigma_{\\mathbf m(n)}\\) with \\(\\mathbf m(n)=(1,\\dots,1)\\). The \\(\\{0,1\\}\\)-valued continuous functions are the \\(\\chi_U\\) with \\(U\\) clopen.\n\n**Exercise 2** (medium; Two diagrams, one algebra). Let \\(A\\) be the limit of \\(A_1=\\mathbb C^2\\to A_2\\to\\cdots\\), where \\(A_k=M_{2^{k-1}}\\oplus M_{2^{k-1}}\\) and every connecting map has multiplicity matrix \\(\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\). Show that \\(A\\) is isomorphic to the CAR algebra, although no \\(A_k\\) is a full matrix algebra.\n\n*Solution.* The sizes satisfy \\(\\alpha\\mathbf m(k)=(2^k,2^k)^T=\\mathbf m(k+1)\\), so the standard maps are unital and injective (Corollary 4.10). Let \\(\\rho_k(x)=(x_1+x_2)/2^k\\in\\mathbb Z[\\frac12]\\). Since \\(\\alpha x=(x_1+x_2,x_1+x_2)^T\\), we get \\(\\rho_{k+1}(\\alpha x)=2(x_1+x_2)/2^{k+1}=\\rho_k(x)\\), so the \\(\\rho_k\\) define \\(\\rho:K_0(A)\\to\\mathbb Z[\\frac12]\\). It is injective: if \\(\\rho_k(x)=0\\), then \\(x_1+x_2=0\\), so \\(\\alpha x=0\\) and \\(\\alpha_{\\infty,k}(x)=\\alpha_{\\infty,k+1}(\\alpha x)=0\\). It is onto, since \\(a/2^k=\\rho_k(a,0)\\). Nonnegative vectors go to nonnegative numbers, and \\(a/2^k\\geq0\\) is \\(\\rho_k(a,0)\\) with \\((a,0)\\geq0\\). The scale of \\(A_k\\), the vectors with \\(0\\leq x_1,x_2\\leq2^{k-1}\\), goes onto \\([0,1]\\cap2^{-k}\\mathbb Z\\). So the scaled dimension group of \\(A\\) is \\((\\mathbb Z[\\frac12],\\mathbb Z[\\frac12]\\cap[0,\\infty),\\mathbb Z[\\frac12]\\cap[0,1])\\), the same as for the CAR algebra (Example 6.4(c)). By Theorem 8.3, \\(A\\) is isomorphic to the CAR algebra, and by Theorem 8.3(3) their local algebras are isomorphic too.\n\n**Exercise 3** (easy; Traces of the gauge-invariant CAR algebra). For \\(s\\in[0,1]\\) define vectors \\(t^{(n)}\\in\\mathbb R^{n+1}_+\\) by \\(t^{(n)}_j=s^j(1-s)^{n-j}\\). Show that they define a tracial state \\(\\tau_s\\) of the gauge-invariant CAR algebra \\(A\\), that \\(\\tau_s\\neq\\tau_{s'}\\) for \\(s\\neq s'\\), and that \\(\\tau_s(p)=\\rho[p](s)\\) for every projection \\(p\\in A\\), with \\(\\rho\\) as in Theorem 9.5.\n\n*Solution.* By Proposition 9.3, \\[\n\\begin{gathered}\n(\\alpha_n^Tt^{(n+1)})_j\\\\\n=t^{(n+1)}_j+t^{(n+1)}_{j+1}\\\\\n=s^j(1-s)^{n+1-j}+s^{j+1}(1-s)^{n-j}\\\\\n=s^j(1-s)^{n-j}\\\\\n=t^{(n)}_j,\n\\end{gathered}\n\\] and \\(\\mathbf m(n)^Tt^{(n)}=\\sum_j\\binom njs^j(1-s)^{n-j}=1\\). By Theorem 10.4(3), these vectors define a tracial state. Its value on a minimal projection of the summand \\(j=1\\) of \\(A_1\\) is \\(s\\), so different \\(s\\) give different traces. For a projection \\(p\\in A_n\\) with rank vector \\(x\\), \\[\n\\begin{gathered}\n\\tau_s(p)\\\\\n=\\sum_jx_js^j(1-s)^{n-j}\\\\\n=\\rho_n(x)(s)\\\\\n=\\rho[p](s).\n\\end{gathered}\n\\] A projection \\(p\\in A\\) is equivalent to a projection \\(q\\) of some \\(A_n\\) (Proposition 7.3(1)), say \\(p=v^*v\\) and \\(q=vv^*\\). Then \\([p]=[q]\\), and \\(\\tau_s(p)=\\tau_s(v^*v)=\\tau_s(vv^*)=\\tau_s(q)\\) because \\(\\tau_s\\) is tracial, so the formula holds for \\(p\\) as well.\n\n**Exercise 4** (medium; AF-algebras are finite). (a) Show that a projection \\(q\\) in an AF-algebra with \\([q]=0\\) in \\(K_0\\) is zero. (b) Deduce that in a unital AF-algebra every \\(v\\) with \\(v^*v=1\\) satisfies \\(vv^*=1\\). (c) Show that the C\\*-subalgebra of \\(B(\\ell^2)\\) generated by the unilateral shift is not an AF-algebra.\n\n*Solution.* (a) By Theorem 7.5(1), \\(\\Phi\\) is injective, so \\([q]=[0]\\) in \\(V(A)\\): \\(q=w^*w\\) with \\(ww^*=0\\). Then \\(w=0\\), since \\(\\|w\\|^2=\\|ww^*\\|\\), and \\(q=0\\). (b) \\(vv^*\\) is a projection equivalent to \\(v^*v=1\\). The projections \\(vv^*\\) and \\(1-vv^*\\) are orthogonal, so \\([vv^*]+[1-vv^*]=[1]=[vv^*]\\), and \\([1-vv^*]=0\\) by cancellation (Theorem 7.5(1)). By (a), \\(vv^*=1\\). (c) The shift \\(S\\) satisfies \\(S^*S=1\\neq SS^*\\). The algebra it generates is unital, since it contains \\(S^*S=1\\), so by (b) it is not an AF-algebra.\n\n**Exercise 5** (easy; Stable isomorphism of UHF algebras). Show that the CAR algebra and the UHF algebra of type \\((3,3,3,\\dots)\\) are not stably isomorphic, while the CAR algebra and the UHF algebra of type \\((4,4,4,\\dots)\\) are isomorphic.\n\n*Solution.* By Theorem 8.5 we must compare \\((\\mathbb Z[\\frac12],\\geq)\\) and \\((\\mathbb Z[\\frac13],\\geq)\\). A homomorphism \\(\\theta:\\mathbb Z[\\frac12]\\to\\mathbb Z[\\frac13]\\) satisfies \\(2^n\\theta(2^{-n})=\\theta(1)\\), so \\(\\theta(x)=rx\\) with \\(r=\\theta(1)\\in\\mathbb Z[\\frac13]\\). If \\(\\theta\\) is an order isomorphism, then \\(r>0\\), and \\(r/2^n\\in\\mathbb Z[\\frac13]\\) for all \\(n\\). Write \\(r=b/3^j\\) with \\(b\\in\\mathbb N\\); then \\(b/(3^j2^n)\\in\\mathbb Z[\\frac13]\\) forces \\(2^n\\) to divide \\(b\\) for every \\(n\\), which is impossible. For the second claim, both supernatural numbers are \\(2^\\infty\\), and Corollary 8.6 applies.\n\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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    {
      "id": "OA-FND-AF-13",
      "unit": "af-algebras",
      "name": "Where this leads",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "source": "src/af-algebras.md",
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      "full_conditions_and_proof": "## Where this leads\n\nThe following four extension results are further directions, not proved here and unused in the proofs and solutions above.\n\n- *Which groups occur.* The scaled ordered groups that are dimension groups of AF-algebras are exactly those that are countable, unperforated (\\(ng\\geq0\\) with \\(n\\geq1\\) implies \\(g\\geq0\\)) and have the Riesz interpolation property, with a scale that generates the group and is hereditary and upward directed; see also [Blackadar 2006, V.2.4.20].\n- *A local criterion.* A separable C\\*-algebra is an AF-algebra as soon as every finite subset lies within any given distance of some finite-dimensional C\\*-subalgebra . The proof perturbs finite-dimensional subalgebras by unitaries close to \\(1\\), in the spirit of Lemma 7.1.\n- *Traces and states.* For a unital AF-algebra, the tracial states correspond exactly to the positive homomorphisms \\(f:K_0(A)\\to\\mathbb R\\) with \\(f[1]=1\\) [Blackadar 1998, Section 7.3]; Exercise 3 exhibits a family of such states for the gauge-invariant CAR algebra.\n- *Subfactors.* For a subfactor \\(N\\) of a factor \\(M\\) of type II\\(_1\\), the index \\([M:N]\\) is the coupling constant of \\(N\\) on \\(L^2(M)\\), a number in \\([1,\\infty]\\). Jones proved that the index lies in \\(\\{4\\cos^2(\\pi/n):n\\geq3\\}\\cup[4,\\infty]\\) and that each of these values occurs. Subfactors are studied through the tower \\(N\\subseteq M\\subseteq M_1\\subseteq M_2\\subseteq\\cdots\\) obtained by iterating Jones's basic construction, and through inclusions of finite-dimensional algebras and their Bratteli diagrams; Theorem 11.2 describes how such diagrams change when one passes to commutants. \n\nThe alternative expression \\(4\\cos^2(2\\pi/n)\\) would give \\(0\\) at \\(n=4\\), and therefore cannot give only values satisfying the stated lower bound \\(1\\).\n\n## References\n\n- [Blackadar 1998] B. Blackadar, *K-Theory for Operator Algebras*, second edition (1998), corrected [author-hosted version](https://www.bruceblackadar.com/Mathematics/book6.pdf).\n- [Blackadar 2006] B. Blackadar, *Operator Algebras: Theory of C*-Algebras and von Neumann Algebras*, [revised author edition, 8 February 2017](https://www.bruceblackadar.com/Mathematics/Cycr.pdf).\n\n*Freely accessible reading:* [B. Blackadar, *K-Theory for Operator Algebras*, §§7.2–7.3, printed pp. 48–54](https://www.bruceblackadar.com/Mathematics/book6.pdf) gives a route through scaled AF classification and Elliott intertwining, including nonunital algebras. The lesson includes its own complete proofs at the stated hypotheses; references to human sources do not imply permission to adapt their expression.\n",
      "status": "Full proof at the stated hypotheses; author self-checked; independent review and formalization not asserted.",
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      "id": "OA-FND-GN-17",
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      "name": "Self-adjoint square roots without continuous involution",
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        "through_line": 203
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      "full_conditions_and_proof": "**Lemma 4.4a** (Self-adjoint roots by spectral separation). Let \\(A\\ne\\{0\\}\\) be a unital complex Banach algebra with an algebraic involution. If \\(a=a^*\\) and\n\\[\n\\sigma_A(a)\\cap(-\\infty,0]=\\varnothing,\n\\]\nthen there is a self-adjoint \\(b\\) with \\(b^2=a\\). It is the unique square root of \\(a\\) whose spectrum lies in the open right half-plane, and it commutes with every element commuting with \\(a\\).\n\n**Proof.** First \\(1^*=1\\): taking adjoints of the two identity laws shows that \\(1^*\\) is an identity. If \\(x\\) is invertible, then \\((x^{-1})^*\\) is the inverse of \\(x^*\\), by taking adjoints of both inverse identities. Therefore\n\\[\n\\sigma_A(x^*)=\\overline{\\sigma_A(x)}.\n\\tag{4.4a}\n\\]\nThese are algebraic statements, not continuity statements.\n\nOn \\(U=\\mathbb C\\setminus(-\\infty,0]\\), define\n\\[\n\\begin{gathered}\nu(z)=\\sqrt{\\frac{|z|+\\operatorname{Re}z}{2}},\\\\\nf(z)=u(z)+\\frac{i\\operatorname{Im}z}{2u(z)}.\n\\end{gathered}\n\\]\nHere \\(u(z)>0\\). Direct multiplication gives \\(f(z)^2=z\\), and \\(f\\) is continuous with positive real part. For nonzero \\(t\\) small enough that \\(z+t\\in U\\),\n\\[\n\\begin{aligned}\n\\frac{f(z+t)-f(z)}{t}\n&=\\frac{1}{f(z+t)+f(z)}\\\\\n&\\longrightarrow\\frac{1}{2f(z)}.\n\\end{aligned}\n\\]\nThus \\(f\\) is holomorphic on \\(U\\). Apply the programme's [holomorphic functional calculus and spectral mapping theorem](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-12). The element \\(b=f(a)\\) satisfies \\(b^2=a\\) and \\(\\sigma_A(b)=f(\\sigma_A(a))\\), which lies in the open right half-plane. By [Proposition 6.5(4) of the Banach-algebra lesson](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-10), \\(b\\) commutes with everything commuting with \\(a\\).\n\nWe check uniqueness without taking an adjoint through the calculus. Suppose \\(c^2=a\\) and \\(\\sigma_A(c)\\) also lies in that half-plane. Since \\(c\\) commutes with its square \\(a\\), it commutes with \\(b\\). Let \\(B\\) be the closed unital subalgebra generated by \\(b,c\\) and all the resolvents\n\\[\n\\begin{gathered}\n(\\lambda1-b)^{-1},\\quad(\\mu1-c)^{-1},\\\\\n\\lambda\\notin\\sigma_A(b),\\quad\\mu\\notin\\sigma_A(c).\n\\end{gathered}\n\\]\nAll these generators commute. Indeed, an element commuting with an invertible element also commutes with its inverse, by multiplying the commutation identity on both sides by that inverse. Thus \\(B\\) is a commutative Banach algebra. Its spectra of \\(b\\) and \\(c\\) are exactly their spectra in \\(A\\): invertibility in \\(B\\) implies invertibility in \\(A\\), and every required resolvent in the reverse direction was included among the generators.\n\nFor each character \\(\\chi\\) of \\(B\\), both \\(\\chi(b)\\) and \\(\\chi(c)\\) have positive real part. Hence \\(\\chi(b+c)\\ne0\\). The [character criterion for invertibility](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-18) makes \\(b+c\\) invertible in \\(B\\), and therefore in \\(A\\). Consequently\n\\[\n(b-c)(b+c)=b^2-c^2=0\n\\quad\\Longrightarrow\\quad b=c.\n\\]\nNow \\((b^*)^2=a^*=a\\), and (4.4a) places \\(\\sigma_A(b^*)\\) in the same half-plane. Uniqueness gives \\(b^*=b\\). No continuity of the involution has been used. \\(\\square\\)\n\n",
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      "local_numbered_label": "Lemma 4.4a",
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    {
      "id": "OA-FND-GN-18",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "Automatic continuity of positive functionals",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
      "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "anchor": "oa-fnd-gn-18",
      "proof_locus": {
        "line": 204,
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      },
      "full_conditions_and_proof": "**Theorem 4.4b** (Automatic continuity with an arbitrary involution). Let \\(A\\) be a unital complex Banach algebra with an algebraic involution, and let \\(\\omega:A\\to\\mathbb C\\) be complex linear with \\(\\omega(x^*x)\\ge0\\) for every \\(x\\in A\\). Then \\(\\omega\\) is continuous. More precisely, there is a finite constant \\(C\\), depending only on the normed algebra and its involution, such that\n\\[\n\\begin{gathered}\n|\\omega(a)|\\le C\\omega(1)\\|a\\|\\\\\n(a\\in A)\n\\end{gathered}\n\\tag{4.4b}\n\\]\nfor every such \\(\\omega\\). If \\(\\omega(1)=0\\), then \\(\\omega=0\\).\n\n**Proof.** The zero algebra is immediate; assume \\(A\\ne\\{0\\}\\). The algebraic proof of Proposition 3.2 gives the hermitian property and Cauchy–Schwarz without any topological assumption. In particular \\(\\omega(1)\\ge0\\) and\n\\[\n|\\omega(a)|^2\\le\\omega(1)\\omega(a^*a).\n\\]\nIf \\(\\omega(1)=0\\), this proves the assertion. Otherwise put \\(F=\\omega/\\omega(1)\\), so \\(F(1)=1\\).\n\nFor \\(h=h^*\\) and real \\(R>r(h)\\), the spectra of \\(R1+h\\) and \\(R1-h\\) lie in the open right half-plane. Lemma 4.4a writes each as a self-adjoint square, so positivity gives\n\\[\n-R\\le F(h)\\le R.\n\\]\nLetting \\(R\\downarrow r(h)\\) yields\n\\[\n\\begin{gathered}\nF(h)\\in\\mathbb R,\\\\\n|F(h)|\\le r(h)\\le\\|h\\|.\n\\end{gathered}\n\\tag{4.4c}\n\\]\n\nLet \\(H=\\{h\\in A:h=h^*\\}\\), a real linear subspace, and let \\(X=\\overline H\\), with closure in the given norm. Estimate (4.4c) extends \\(F|_H\\) uniquely to a bounded real-linear map \\(\\Phi:X\\to\\mathbb R\\), of norm at most one. Explicitly, for \\(h_n\\to x\\) in \\(H\\), the estimate on \\(h_n-h_m\\) makes \\(F(h_n)\\) Cauchy; its limit is independent of the approximating sequence and defines \\(\\Phi(x)\\). We have not yet proved that \\(\\Phi\\) agrees with \\(F\\) at points of \\(X\\setminus H\\).\n\nThe key fact is that \\(\\Phi\\) vanishes on \\(X\\cap iX\\). For \\(z\\) in this intersection choose \\(u_n,v_n\\in H\\) with \\(u_n\\to z\\) and \\(v_n\\to-iz\\). Continuity of multiplication gives\n\\[\nu_n^2+v_n^2\\longrightarrow z^2+(-iz)^2=0.\n\\]\nEach summand has nonnegative \\(F\\)-value. Cauchy–Schwarz and (4.4c), applied to the self-adjoint sum, give\n\\[\n\\begin{aligned}\n|F(u_n)|^2\n&\\le F(u_n^2)\\\\\n&\\le F(u_n^2+v_n^2)\\\\\n&\\le\\|u_n^2+v_n^2\\|\\longrightarrow0.\n\\end{aligned}\n\\]\nBy the definition of \\(\\Phi\\), this proves \\(\\Phi(z)=0\\).\n\nConsider the bounded real-linear map\n\\[\n\\begin{aligned}\nS:X\\oplus_1X&\\longrightarrow A_{\\mathbb R},\\\\\nS(u,v)&=u+iv,\\\\\n\\|(u,v)\\|_1&=\\|u\\|+\\|v\\|.\n\\end{aligned}\n\\]\nIts domain is Banach: a Cauchy sequence is Cauchy in both closed-subspace coordinates, and the two coordinate limits give convergence in the sum norm. Its codomain is the underlying real Banach space of \\(A\\). It is onto, since the algebraic decomposition\n\\[\na=\\frac{a+a^*}{2}\n+i\\frac{a-a^*}{2i}\n\\]\nhas both components in \\(H\\subseteq X\\). The programme's [open mapping theorem](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-05) supplies \\(\\delta>0\\) such that \\(S\\) maps its open unit ball onto a set containing the ball of radius \\(\\delta\\). Scaling by \\(2\\|a\\|/\\delta\\), for \\(a\\ne0\\), gives some decomposition \\(a=u+iv\\) with\n\\[\n\\begin{gathered}\n\\|u\\|+\\|v\\|\\le C\\|a\\|,\\\\\nC=2/\\delta.\n\\end{gathered}\n\\tag{4.4d}\n\\]\nFor \\(a=0\\), take both components zero. The constant depends on \\(S\\), not on \\(F\\).\n\nDefine \\(T(u,v)=\\Phi(u)+i\\Phi(v)\\). If \\(S(u,v)=0\\), then \\(u=-iv\\) and \\(v=iu\\), so both \\(u\\) and \\(v\\) lie in \\(X\\cap iX\\). Thus \\(T(u,v)=0\\). This proves that\n\\[\nL(a)=\\Phi(u)+i\\Phi(v),\\qquad a=u+iv,\n\\]\nis independent of the chosen decomposition. It is real-linear, and replacing \\((u,v)\\) by \\((-v,u)\\) shows \\(L(ia)=iL(a)\\); hence it is complex-linear. Formula (4.4d) implies \\(|L(a)|\\le C\\|a\\|\\). Finally choose the original algebraic decomposition in \\(H+iH\\). Since \\(\\Phi|_H=F|_H\\), complex linearity of \\(F\\) gives \\(L(a)=F(a)\\). Therefore \\(F\\), and then \\(\\omega\\), is continuous, with (4.4b). \\(\\square\\)\n\n![A bounded map on two closed real subspaces descends through their sum because it vanishes on the kernel.](assets/positive-functional-continuity.png)\n\n*The descent in Theorem 4.4b.* The arrows are real-linear during the construction. The condition \\(\\ker S\\subseteq\\ker T\\) defines \\(L\\) independently of a decomposition; the open mapping estimate makes it bounded. The proof then checks complex linearity and identifies \\(L\\) with \\(F\\). This diagram depicts maps, not an orthogonal decomposition of \\(A\\).\n\n",
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      "status": "Complete programme proof; author self-check preserved. GN results independently replayed by GPT-6.1 Sol (OpenAI), Ultra in bounded review; CF power proof replayed by course owner. No formal verification or whole-programme certificate.",
      "local_numbered_label": "Theorem 4.4b",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
    },
    {
      "id": "OA-FND-GN-19",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "name": "Spectral estimates for positive functionals",
      "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
      "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "anchor": "oa-fnd-gn-19",
      "proof_locus": {
        "line": 283,
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      },
      "full_conditions_and_proof": "**Corollary 4.4c** (Spectral estimates and familiar special cases). Under the assumptions of Theorem 4.4b,\n\\[\n\\begin{aligned}\n|\\omega(x)|^2\n&\\le\\omega(1)\\omega(x^*x)\\\\\n&\\le\\omega(1)^2 r(x^*x).\n\\end{aligned}\n\\]\nIf \\(x\\) is normal, then \\(|\\omega(x)|\\le\\omega(1)r(x)\\). If \\(\\|x^*\\|\\le\\beta\\|x\\|\\) for all \\(x\\), then \\(\\|\\omega\\|\\le\\sqrt\\beta\\,\\omega(1)\\). If \\(A\\) is commutative, then\n\\[\n\\|\\omega\\|\\le\\omega(1)\\le\\|1\\|\\,\\|\\omega\\|;\n\\]\nin particular \\(\\|\\omega\\|=\\omega(1)\\) when \\(\\|1\\|=1\\), even without a continuous involution.\n\n**Proof.** Apply (4.4c), scaled back from \\(F\\) to \\(\\omega\\), to the self-adjoint element \\(x^*x\\); combine it with algebraic Cauchy–Schwarz. If \\(x\\) commutes with \\(x^*\\), form the closed commutative algebra generated by both elements and their resolvents, exactly as in Lemma 4.4a. For every character \\(\\chi\\),\n\\[\n\\begin{aligned}\n|\\chi(x^*x)|&=|\\chi(x^*)\\chi(x)|\\\\\n&\\le r_A(x^*)r_A(x)=r_A(x)^2.\n\\end{aligned}\n\\]\nThe character criterion identifies the spectrum of \\(x^*x\\) in this commutative algebra; its spectrum in \\(A\\) is contained in that spectrum. Thus \\(r_A(x^*x)\\le r_A(x)^2\\), which proves the normal-element estimate. The bounded-involution estimate follows instead from \\(r(x^*x)\\le\\|x^*\\|\\|x\\|\\le\\beta\\|x\\|^2\\). In a commutative algebra every element is normal, so the normal-element estimate gives \\(\\|\\omega\\|\\le\\omega(1)\\); evaluation at the identity gives the other inequality. \\(\\square\\)\n\n**Checkpoint (with solution).** Why would either of the following shortcuts be invalid: declaring \\(H\\) closed because it consists of self-adjoint elements, or identifying \\(\\Phi(z)\\) with \\(F(z)\\) before constructing \\(L\\)?\n\n*Solution.* The first would use continuity of the involution, which is not assumed. The second would use continuity of \\(F\\), which is the conclusion. We only extend the already bounded restriction \\(F|_H\\). The vanishing argument on \\(X\\cap iX\\) and the open mapping theorem then prove the needed agreement on all of \\(A\\). Without an identity the theorem need not hold: in the infinite-dimensional zero-product example 3.4(4), every linear functional is positive, including discontinuous ones.\n\nFrom the next paragraph onward, “involutive Banach algebra” again has the isometric-involution meaning fixed in the conventions. The broader theorem does not silently change the hypotheses of the GNS estimates that follow.\n\n",
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      "status": "Complete programme proof; author self-check preserved. GN results independently replayed by GPT-6.1 Sol (OpenAI), Ultra in bounded review; CF power proof replayed by course owner. No formal verification or whole-programme certificate.",
      "local_numbered_label": "Corollary 4.4c",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
    },
    {
      "id": "OA-FND-CF-POWER-RANGE",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "name": "The full range of monotone powers",
      "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
      "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "anchor": "monotone-powers-complete-range",
      "proof_locus": {
        "line": 556,
        "through_line": 615
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      "full_conditions_and_proof": "**Proposition 9.4** (The full range of monotone powers). On positive definite matrices of every size, \\(x\\mapsto x^\\alpha\\), for real \\(\\alpha\\), preserves order exactly when \\(0\\leq\\alpha\\leq1\\). For every \\(\\alpha>1\\) there are already positive definite two-by-two matrices \\(0<b\\leq a\\) for which \\(b^\\alpha\\not\\leq a^\\alpha\\). For \\(0<\\alpha\\leq1\\), the order-preserving statement holds on the entire positive cone of every C\\*-algebra, including nonunital algebras, by Theorem 9.1. At \\(\\alpha=0\\) on positive definite matrices the function is the constant identity.\n\n**Proof.** The affirmative cases are Theorem 9.1 and the constant function. If \\(\\alpha<0\\), the scalar inequality \\(1<2\\) gives \\(1^\\alpha>2^\\alpha\\), so even scalar monotonicity fails.\n\nFix \\(\\alpha>1\\), and choose\n\\[\n\\begin{gathered}\ns=(2\\alpha^2)^{-1/(\\alpha-1)},\\\\\nb=\\begin{pmatrix}s&0\\\\0&1\\end{pmatrix},\\qquad\nh=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix},\\\\\na_t=b+th .\n\\end{gathered}\n\\]\nHere \\(0<s<1\\), \\(h\\geq0\\), and \\(a_t\\geq b>0\\) for every \\(t>0\\). We calculate the first-order change of \\(a_t^\\alpha\\) directly, rather than assuming a criterion for matrix monotonicity.\n\nPut \\(d=1-s>0\\). The two distinct eigenvalues of \\(a_t\\) are\n\\[\n\\lambda_\\pm(t)=\\frac{1+s+2t\\pm\\sqrt{d^2+4t^2}}2 .\n\\]\nSince\n\\[\n0\\leq\\sqrt{d^2+4t^2}-d\n=\\frac{4t^2}{\\sqrt{d^2+4t^2}+d}\\leq\\frac{2t^2}{d},\n\\]\nwe have \\(\\lambda_-(t)=s+t+O(t^2)\\) and \\(\\lambda_+(t)=1+t+O(t^2)\\). For sufficiently small \\(t\\), all these numbers stay in a fixed compact subinterval of \\((0,\\infty)\\).\n\nFor \\(f(u)=u^\\alpha\\), set\n\\[\nc_t=\\frac{f(\\lambda_+(t))-f(\\lambda_-(t))}\n{\\lambda_+(t)-\\lambda_-(t)} .\n\\]\nThe affine polynomial \\(f(\\lambda_-)+c_t(u-\\lambda_-)\\) agrees with \\(f\\) on the two-point spectrum of \\(a_t\\). Functional calculus therefore gives\n\\[\nf(a_t)=f(\\lambda_-(t))I+\nc_t\\bigl(a_t-\\lambda_-(t)I\\bigr).\n\\]\nThis is precisely Theorem 5.1 applied to two functions with equal values on the spectrum. As \\(t\\to0\\),\n\\[\nc_t\\longrightarrow c=\\frac{1-s^\\alpha}{1-s}>1.\n\\]\nThe upper-left entry is \\(f(s)+t f'(s)+o(t)\\): indeed \\(s+t-\\lambda_-(t)=O(t^2)\\), \\(c_t\\) stays bounded, and \\(f(\\lambda_-(t))=f(s)+t f'(s)+o(t)\\). Using the equivalent formula\n\\(f(a_t)=f(\\lambda_+(t))I+c_t(a_t-\\lambda_+(t)I)\\)\ngives the lower-right entry \\(f(1)+t f'(1)+o(t)\\). The off-diagonal entries are \\(t c_t\\). Thus, entrywise and hence in matrix norm,\n\\[\n\\frac{a_t^\\alpha-b^\\alpha}{t}\n\\longrightarrow\nL=\\begin{pmatrix}\\alpha s^{\\alpha-1}&c\\\\c&\\alpha\\end{pmatrix}.\n\\]\nBut\n\\[\n\\det L=\\alpha^2s^{\\alpha-1}-c^2\n=\\frac12-c^2<0.\n\\]\nBy continuity of the determinant, the self-adjoint matrix\n\\((a_t^\\alpha-b^\\alpha)/t\\) has negative determinant for all sufficiently small positive \\(t\\). It cannot be positive: a positive two-by-two matrix has nonnegative eigenvalues and therefore nonnegative determinant, by Theorem 8.5(11) and the finite-dimensional spectral calculus. Consequently \\(a_t^\\alpha-b^\\alpha\\not\\geq0\\), although \\(a_t\\geq b>0\\). This supplies a counterexample for each \\(\\alpha>1\\), and completes the classification. \\(\\square\\)\n\nThe matrices of divided differences that appear here are studied more generally in [Hiai and Sano's freely accessible paper](https://arxiv.org/html/1007.2478v2). Its Proposition 3.1 treats power functions; the argument above proves the monotonicity range within this programme and does not require the paper's external prerequisites.\n\n*Proposition 9.4 written and self-checked by GPT-6 Astra (OpenAI), Ultra, in Codex.*\n\n",
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      "status": "Complete programme proof; author self-check preserved. GN results independently replayed by GPT-6.1 Sol (OpenAI), Ultra in bounded review; CF power proof replayed by course owner. No formal verification or whole-programme certificate.",
      "local_numbered_label": "Proposition 9.4",
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      "local_label": "Theorem 1.1",
      "kind": "theorem",
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      "statement_and_full_conditions": "**Theorem 1.1** (Zorn's lemma). Let \\((P,\\le)\\) be a nonempty partially ordered set in which every chain has an upper bound. Then \\(P\\) has a maximal element.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
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      "local_label": "Theorem 2.1",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-02::Theorem 2.1",
      "anchor": "oa-fnd-hb-02",
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      "statement_and_full_conditions": "**Theorem 2.1** (Hahn–Banach, real form). Let \\(V\\) be a real vector space, \\(p\\) a sublinear functional on \\(V\\), \\(M\\) a subspace, and \\(f:M\\to\\mathbb R\\) linear with \\(f\\le p\\) on \\(M\\). Then \\(f\\) extends to a linear \\(F:V\\to\\mathbb R\\) with \\(F\\le p\\) on \\(V\\).",
      "proof_locus": {
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          "source_locus": "reader-work-selected.tex, CH03 RW001",
          "role": "proof comparison",
          "correspondence": "real dominated extension exact; complex extension internal",
          "explanation": "Selected Zorn/one-dimensional companion proof read; the course keeps the full complex norm-preserving proof."
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              "heading": "2. The Hahn–Banach theorem",
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "3. The Baire category theorem",
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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            {
              "heading": "4. Uniform boundedness",
              "line": 131,
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
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                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.2",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-02::Theorem 2.2",
      "anchor": "oa-fnd-hb-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 2.2** (Hahn–Banach, seminorm form). Let \\(V\\) be a vector space over \\(\\mathbb K\\), \\(p\\) a seminorm on \\(V\\), \\(M\\) a subspace, and \\(f:M\\to\\mathbb K\\) linear with \\(|f|\\le p\\) on \\(M\\). Then \\(f\\) extends to a linear \\(F:V\\to\\mathbb K\\) with \\(|F|\\le p\\) on \\(V\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 74,
        "through_line": 97,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
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              "heading": "2. The Hahn–Banach theorem",
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                "OA-FND-HB-02"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
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                "OA-FND-HB-03"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "4. Uniform boundedness",
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                "OA-FND-HB-04"
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            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
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                "OA-FND-HB-05"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
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                "OA-FND-HB-06"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.3",
      "kind": "corollary",
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      "anchor": "oa-fnd-hb-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Corollary 2.3** (normed spaces). Let \\(E\\) be a normed space.\n1. Every \\(\\varphi\\in M^*\\), for a subspace \\(M\\subseteq E\\), extends to some \\(\\Phi\\in E^*\\) with \\(\\|\\Phi\\|=\\|\\varphi\\|\\).\n2. For every \\(x\\in E\\) there is \\(\\varphi\\in E^*\\) with \\(\\|\\varphi\\|\\le1\\) and \\(\\varphi(x)=\\|x\\|\\); if \\(x\\ne0\\), then \\(\\|\\varphi\\|=1\\). Hence \\(\\|x\\|=\\max_{\\|\\varphi\\|\\le1}|\\varphi(x)|\\), and \\(E^*\\) separates the points of \\(E\\).\n3. Let \\(M\\) be a closed subspace and \\(x\\notin M\\), with \\(d=\\operatorname{dist}(x,M)>0\\). Then some \\(\\varphi\\in E^*\\) has \\(\\varphi|_M=0\\), \\(\\varphi(x)=d\\) and \\(\\|\\varphi\\|=1\\). Consequently a subspace is dense exactly when the only bounded functional vanishing on it is \\(0\\).\n4. The canonical map \\(j:E\\to E^{**}\\), \\(j(x)(\\varphi)=\\varphi(x)\\), is a linear isometry.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 98,
        "through_line": 118,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "3. The Baire category theorem",
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "4. Uniform boundedness",
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                "OA-FND-HB-04"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
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              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
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              "heading": "6. Topological vector spaces and separation of convex sets",
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                "OA-FND-HB-06"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-03::Theorem 3.1",
      "anchor": "oa-fnd-hb-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 3.1** (Baire). Let \\((X,d)\\) be a nonempty complete metric space, and \\((U_n)_{n\\ge1}\\) a sequence of dense open subsets. Then \\(\\bigcap_nU_n\\) is dense. Equivalently, \\(X\\) is not a countable union of closed sets with empty interior.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 121,
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        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "3. The Baire category theorem",
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              "heading": "4. Uniform boundedness",
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                "OA-FND-HB-04"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
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              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
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              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.1",
      "kind": "lemma",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-04::Lemma 4.1",
      "anchor": "oa-fnd-hb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Lemma 4.1.** If \\(E\\) is a normed space and \\(F\\) a Banach space, then \\(B(E,F)\\) is a Banach space. In particular \\(E^*\\) is a Banach space for every normed space \\(E\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 133,
        "through_line": 136,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "3. The Baire category theorem",
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                "OA-FND-HB-03"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "4. Uniform boundedness",
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                "OA-FND-HB-04"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
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              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
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              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
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                "OA-FND-HB-06"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.2",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-04::Theorem 4.2",
      "anchor": "oa-fnd-hb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 4.2** (uniform boundedness principle). Let \\(E\\) be a Banach space, \\(F\\) a normed space, and \\(\\mathcal T\\subseteq B(E,F)\\) a family with \\(\\sup_{T\\in\\mathcal T}\\|Tx\\|<\\infty\\) for every \\(x\\in E\\). Then \\(\\sup_{T\\in\\mathcal T}\\|T\\|<\\infty\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 137,
        "through_line": 140,
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      },
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "3. The Baire category theorem",
              "line": 119,
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                "OA-FND-HB-03"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "4. Uniform boundedness",
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                "OA-FND-HB-04"
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              "heading": "5. The open mapping and closed graph theorems",
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                "OA-FND-HB-05"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "6. Topological vector spaces and separation of convex sets",
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                "OA-FND-HB-06"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.3",
      "kind": "corollary",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-04::Corollary 4.3",
      "anchor": "oa-fnd-hb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Corollary 4.3.**\n1. Let \\(E\\) be a Banach space. If \\(\\varphi_n\\in E^*\\) and \\(\\varphi_n(x)\\) converges for every \\(x\\in E\\), then \\(\\sup_n\\|\\varphi_n\\|<\\infty\\), and \\(\\varphi(x)=\\lim\\varphi_n(x)\\) defines \\(\\varphi\\in E^*\\).\n2. Let \\(E\\) be a normed space and \\(S\\subseteq E\\) with \\(\\sup_{x\\in S}|\\varphi(x)|<\\infty\\) for every \\(\\varphi\\in E^*\\). Then \\(S\\) is norm bounded.\n3. Let \\(E\\) be a Banach space and \\(F\\) a normed space. If \\(T_n\\in B(E,F)\\) and \\(T_nx\\) converges for every \\(x\\), then \\(\\sup_n\\|T_n\\|<\\infty\\), and \\(Tx=\\lim T_nx\\) defines \\(T\\in B(E,F)\\) with \\(\\|T\\|\\le\\liminf_n\\|T_n\\|\\). In particular, a strongly convergent sequence of operators on a Hilbert space is norm bounded.",
      "proof_locus": {
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        "line": 141,
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
              "through_line": 130,
              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.4",
      "kind": "example",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-04::Example 4.4",
      "anchor": "oa-fnd-hb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Example 4.4** (completeness cannot be dropped). Let \\(c_{00}\\) be the space of finitely supported sequences, with the supremum norm, and \\(\\varphi_n(x)=nx_n\\). For each \\(x\\), \\(\\varphi_n(x)=0\\) once \\(n\\) exceeds the support of \\(x\\), so the family is pointwise bounded. But \\(\\|\\varphi_n\\|=n\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 150,
        "through_line": 151,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [
        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
          "local_scopes": [
            {
              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
              "through_line": 118,
              "anchors": [
                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
              "through_line": 130,
              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
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              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-05::Theorem 5.1",
      "anchor": "oa-fnd-hb-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 5.1** (open mapping). Let \\(E,F\\) be Banach spaces and \\(T\\in B(E,F)\\) surjective. Then \\(T\\) maps open sets to open sets. More precisely, there is \\(\\delta>0\\) with \\(T(B_E(0,1))\\supseteq B_F(0,\\delta)\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 154,
        "through_line": 173,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
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        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
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              "heading": "2. The Hahn–Banach theorem",
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                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
              "through_line": 130,
              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.2",
      "kind": "corollary",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-05::Corollary 5.2",
      "anchor": "oa-fnd-hb-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Corollary 5.2** (inverse mapping). A bijective \\(T\\in B(E,F)\\) between Banach spaces has a bounded inverse. If two complete norms on one vector space satisfy \\(\\|\\cdot\\|_1\\le C\\|\\cdot\\|_2\\), they are equivalent.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 174,
        "through_line": 177,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
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        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
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            {
              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
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                "OA-FND-HB-02"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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            {
              "heading": "3. The Baire category theorem",
              "line": 119,
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                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
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              "heading": "4. Uniform boundedness",
              "line": 131,
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                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.3",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-05::Theorem 5.3",
      "anchor": "oa-fnd-hb-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 5.3** (closed graph). Let \\(T:E\\to F\\) be linear between Banach spaces, with closed graph \\(G=\\{(x,Tx):x\\in E\\}\\subseteq E\\oplus F\\). Then \\(T\\) is bounded.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 178,
        "through_line": 183,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [
        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
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              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
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              "anchors": [
                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
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              "heading": "3. The Baire category theorem",
              "line": 119,
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              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
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              "heading": "4. Uniform boundedness",
              "line": 131,
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              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.1",
      "kind": "lemma",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-06::Lemma 6.1",
      "anchor": "oa-fnd-hb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Lemma 6.1** (Minkowski functionals). Let \\(U\\) be a convex open neighbourhood of \\(0\\) in a topological vector space \\(X\\), and \\(p_U(x)=\\inf\\{t>0:x\\in tU\\}\\).\n1. \\(p_U\\) is finite and sublinear.\n2. \\(U=\\{x:p_U(x)<1\\}\\).\n3. \\[\n\\begin{gathered}\n|p_U(x)-p_U(y)|\\\\\n\\le\\max(p_U(x-y),p_U(y-x)),\n\\end{gathered}\n\\] and \\(p_U<\\varepsilon\\) on \\(\\varepsilon U\\). So \\(p_U\\) is continuous.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 188,
        "through_line": 209,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [
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              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
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                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
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              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-06::Theorem 6.2",
      "anchor": "oa-fnd-hb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 6.2** (separation from an open convex set). Let \\(X\\) be a topological vector space, and \\(A,B\\subseteq X\\) nonempty disjoint convex sets with \\(A\\) open. Then there are a continuous linear functional \\(\\varphi\\) on \\(X\\) and \\(t\\in\\mathbb R\\) with\n\\[\n\\operatorname{Re}\\varphi(a)<t\\le\\operatorname{Re}\\varphi(b)\\qquad(a\\in A,\\ b\\in B).\n\\]",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 210,
        "through_line": 234,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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              "heading": "3. The Baire category theorem",
              "line": 119,
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              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.3",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-06::Theorem 6.3",
      "anchor": "oa-fnd-hb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 6.3** (strict separation). Let \\(X\\) be a locally convex space, \\(K\\subseteq X\\) compact and convex, and \\(C\\subseteq X\\) closed and convex, both nonempty, with \\(K\\cap C=\\varnothing\\). Then there are a continuous linear functional \\(\\varphi\\) and numbers \\(t_1<t_2\\) with \\(\\operatorname{Re}\\varphi<t_1\\) on \\(K\\) and \\(\\operatorname{Re}\\varphi>t_2\\) on \\(C\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 235,
        "through_line": 240,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [
        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
          "local_scopes": [
            {
              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
              "through_line": 118,
              "anchors": [
                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
              "through_line": 130,
              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 6.4",
      "kind": "corollary",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-06::Corollary 6.4",
      "anchor": "oa-fnd-hb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Corollary 6.4.** Let \\(X\\) be a locally convex space.\n1. If \\(X\\) is Hausdorff, its continuous linear functionals separate points.\n2. A point \\(x\\) outside a closed convex set \\(C\\) is strictly separated from \\(C\\) by a continuous linear functional.\n3. Every continuous linear functional on a subspace \\(M\\subseteq X\\) extends to a continuous linear functional on \\(X\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 241,
        "through_line": 249,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [
        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
          "local_scopes": [
            {
              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
              "through_line": 118,
              "anchors": [
                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
              "through_line": 130,
              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.5",
      "kind": "lemma",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-06::Lemma 6.5",
      "anchor": "oa-fnd-hb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Lemma 6.5** (Closed kernels). A linear functional \\(f\\) on a topological vector space \\(X\\) is continuous if and only if its kernel is closed.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 250,
        "through_line": 257,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [
        {
          "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
          "local_scopes": [
            {
              "heading": "2. The Hahn–Banach theorem",
              "line": 47,
              "through_line": 118,
              "anchors": [
                "OA-FND-HB-02"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "3. The Baire category theorem",
              "line": 119,
              "through_line": 130,
              "anchors": [
                "OA-FND-HB-03"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "4. Uniform boundedness",
              "line": 131,
              "through_line": 151,
              "anchors": [
                "OA-FND-HB-04"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "5. The open mapping and closed graph theorems",
              "line": 152,
              "through_line": 183,
              "anchors": [
                "OA-FND-HB-05"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            },
            {
              "heading": "6. Topological vector spaces and separation of convex sets",
              "line": 184,
              "through_line": 257,
              "anchors": [
                "OA-FND-HB-06"
              ],
              "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
            }
          ],
          "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
          "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
          "role": "proof comparison",
          "correspondence": "Classical Banach-space scope",
          "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.1",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-07::Theorem 7.1",
      "anchor": "oa-fnd-hb-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 7.1.** Let \\(X\\) be a Hausdorff topological vector space of finite dimension \\(n\\). Then every linear bijection \\(f:\\mathbb K^n\\to X\\) is a homeomorphism, where \\(\\mathbb K^n\\) has its Euclidean topology. Consequently:\n1. all Hausdorff vector topologies on a finite-dimensional space coincide;\n2. every finite-dimensional subspace of a Hausdorff topological vector space is closed;\n3. every linear map from a finite-dimensional Hausdorff topological vector space to a topological vector space is continuous.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 260,
        "through_line": 276,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.1",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-08::Theorem 8.1",
      "anchor": "oa-fnd-hb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 8.1** (Cantor–Schröder–Bernstein). If \\(|X|\\leq|Y|\\) and \\(|Y|\\leq|X|\\), then \\(|X|=|Y|\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 281,
        "through_line": 286,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-08::Theorem 8.2",
      "anchor": "oa-fnd-hb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 8.2** (Cantor). For every set \\(X\\), \\(|X|\\leq|\\mathcal P(X)|\\) and \\(|X|\\neq|\\mathcal P(X)|\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 287,
        "through_line": 290,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.3",
      "kind": "proposition",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-08::Proposition 8.3",
      "anchor": "oa-fnd-hb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Proposition 8.3** (Exponents). For sets \\(X,Y,Z\\), the map that sends \\(F:Y\\times Z\\to X\\) to \\(z\\mapsto F(\\cdot,z)\\) is a bijection from \\(X^{Y\\times Z}\\) onto \\((X^Y)^Z\\). The map \\((m,n)\\mapsto2^m(2n+1)-1\\) is a bijection \\(\\mathbb N\\times\\mathbb N\\to\\mathbb N\\). Consequently \\(|(\\{0,1\\}^{\\mathbb N})^{\\mathbb N}|=|\\{0,1\\}^{\\mathbb N}|\\), that is, \\((2^{\\aleph_0})^{\\aleph_0}=2^{\\aleph_0}\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 291,
        "through_line": 294,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.4",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-08::Theorem 8.4",
      "anchor": "oa-fnd-hb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 8.4.** Let \\(X\\) be an infinite set. Then:\n1. \\(|X\\times\\mathbb N|=|X|\\);\n2. \\(|X\\times\\{0,1\\}|=|X|\\);\n3. if \\((A_i)_{i\\in I}\\) is a family of countable sets indexed by an infinite set \\(I\\), then \\(|\\bigcup_iA_i|\\leq|I|\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 295,
        "through_line": 312,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.5",
      "kind": "theorem",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Theorem 8.5",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Theorem 8.5.** Every set can be well-ordered. If \\(X\\) is infinite, then \\(|X\\times X|=|X|\\). Consequently, if \\(0<|Y|\\leq|X|\\), then \\(|X\\times Y|=|X|\\). In particular an uncountable set can be partitioned into uncountably many subsets, each in bijection with the whole set.\n\n*Proof.* A *well-order* is a linear order in which every nonempty subset has a least member. Order the well-orders on subsets of \\(X\\) by extension as an initial segment: an extension may append elements, but may not insert elements before an old element. The union of a chain is again a well-order. To see this, take a member \\(a\\) of a nonempty subset \\(S\\) of the union and a chain member containing \\(a\\). All predecessors of \\(a\\) in the union already lie in that member. The nonempty set of elements of \\(S\\) at or before \\(a\\) therefore has a least member there, which is least in all of \\(S\\). Zorn's lemma gives a maximal such well-order. If its domain omitted a point of \\(X\\), appending that point would extend it. Its domain is therefore \\(X\\).\n\nHere are the order-type facts needed to count the square. Two well-orders have at most one order isomorphism between initial segments: if two such maps first differ at \\(a\\), their common image of the predecessors of \\(a\\) determines the least unused image of \\(a\\), a contradiction. Take the union of all these initial-segment isomorphisms for two well-orders. They are compatible by uniqueness, and their union has initial-segment domains and ranges. If neither order were exhausted, mapping the least remaining point of one to the least remaining point of the other would extend the union. Thus one order is isomorphic to an initial segment of the other.\n\nA well-order cannot be isomorphic to a proper initial segment of itself. Indeed, suppose \\(f\\) is such an isomorphism and \\(a\\) is the first point with \\(f(a)\\ne a\\). The predecessors of \\(a\\) are fixed, so order preservation forces \\(f(a)\\geq a\\). Equality is excluded, and \\(f(a)>a\\) would omit \\(a\\) from the initial-segment range. If there is no such \\(a\\), the range is the whole order. These contradictions prove the assertion. Hence order types are linearly ordered by proper initial-segment inclusion.\n\nEvery nonempty set of order types has a least one. Pick a type \\(\\tau\\) in it. If there are types below \\(\\tau\\), identify them with proper initial segments of a representative of \\(\\tau\\). Their endpoint set has a least member, giving the least type below \\(\\tau\\). If there are none, \\(\\tau\\) itself is least. An *initial order type*, or cardinal, is the least type of a well-order on a set of its given cardinality. Such a least type exists, since the orders on that set form a set. Injections respect cardinal types: if \\(A\\) injects into \\(B\\) but its cardinal type were larger, the type of \\(B\\) would be an initial segment of that of \\(A\\), giving an injection in the other direction. Cantor–Schröder–Bernstein would make their cardinalities, and hence their least types, equal. Every proper initial segment of an initial type \\(\\kappa\\) therefore has cardinality less than \\(\\kappa\\): equality would give a smaller well-order of the same set. No infinite initial type has a last point, because an infinite set absorbs one extra point by the explicit shifting bijection in the proof of Theorem 8.4.\n\nSuppose now that an infinite cardinal fails the square identity, and take the least such \\(\\kappa\\). This choice is legitimate within a set: below any proposed counterexample, all smaller cardinal types are represented by well-orders on subsets of that counterexample. Represent \\(\\kappa\\) by its initial well-order. Order pairs \\((\\alpha,\\beta)\\) first by \\(\\max(\\alpha,\\beta)\\), then by \\(\\alpha\\), then by \\(\\beta\\). This is a well-order: a nonempty collection of pairs has a least maximum, a least first coordinate among pairs with that maximum, and then a least second coordinate.\n\nThe predecessors of a pair with maximum \\(\\gamma\\) are contained in the square of the initial segment through \\(\\gamma\\). That segment is proper, since \\(\\kappa\\) has no last point. Its cardinal \\(\\mu\\) is less than \\(\\kappa\\). If \\(\\mu\\) is finite, its square is finite; if \\(\\mu\\) is infinite, minimality of \\(\\kappa\\) gives \\(\\mu^2=\\mu\\). Thus every predecessor set in the pair order has cardinality less than \\(\\kappa\\).\n\nLet \\(\\tau\\) be the type of this pair order. If \\(\\tau>\\kappa\\), comparison of well-orders embeds \\(\\kappa\\) as a proper initial segment of it. The least point outside that segment then has \\(\\kappa\\) predecessors, a contradiction. Thus \\(\\tau\\leq\\kappa\\), giving \\(|\\kappa\\times\\kappa|\\leq\\kappa\\). The map \\(\\alpha\\mapsto(\\alpha,\\alpha)\\) gives the reverse inequality, so Cantor–Schröder–Bernstein proves the square identity. This contradicts the choice of \\(\\kappa\\), and proves the identity for every infinite set.\n\nFor nonempty \\(Y\\) with an injection into \\(X\\), choose \\(y_0\\in Y\\). The maps \\(x\\mapsto(x,y_0)\\) and the coordinate injection into \\(X\\times X\\) give\n\\[\n|X|\\leq|X\\times Y|\\leq|X\\times X|=|X|.\n\\]\nApply Cantor–Schröder–Bernstein again. Finally, choose a bijection \\(b:X\\times X\\to X\\). The sets \\(b(X\\times\\{y\\})\\), \\(y\\in X\\), partition \\(X\\) and each is in bijection with \\(X\\). If \\(X\\) is uncountable, their index set is uncountable. \\(\\square\\)",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 315,
        "through_line": 337,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Exercise 1",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Exercise 1** (medium; two complete norms). Let \\(\\|\\cdot\\|_1\\) and \\(\\|\\cdot\\|_2\\) be complete norms on a vector space \\(V\\) with \\(\\|v\\|_1\\le C\\|v\\|_2\\) for all \\(v\\). Show that the norms are equivalent. Show by example that completeness of both norms is needed.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 340,
        "through_line": 343,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Exercise 2",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Exercise 2** (medium; Banach limits). Let \\(\\ell^\\infty_{\\mathbb R}\\) be the real space of bounded real sequences and \\(S\\) the shift, \\((Sx)_n=x_{n+1}\\). Show that there is a linear \\(L:\\ell^\\infty_{\\mathbb R}\\to\\mathbb R\\) with \\(\\liminf_nx_n\\le L(x)\\le\\limsup_nx_n\\) and \\(L(Sx)=L(x)\\) for all \\(x\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 344,
        "through_line": 351,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Exercise 3",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Exercise 3** (easy; closed convex sets are intersections of half-spaces). Let \\(E\\) be a normed space and \\(C\\subseteq E\\) closed and convex. Show that \\(C\\) is the intersection of the sets \\(\\{x:\\operatorname{Re}\\varphi(x)\\le t\\}\\), over all \\(\\varphi\\in E^*\\) and \\(t\\in\\mathbb R\\) with \\(C\\subseteq\\{\\operatorname{Re}\\varphi\\le t\\}\\).",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 352,
        "through_line": 355,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Exercise 4",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Exercise 4** (medium; weakly continuous implies bounded). Let \\(T:E\\to F\\) be linear between Banach spaces, with \\(\\psi\\circ T\\in E^*\\) for every \\(\\psi\\in F^*\\). Show that \\(T\\) is bounded.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 356,
        "through_line": 359,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5",
      "kind": "exercise",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Exercise 5",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Exercise 5** (medium; quotients). Let \\(E\\) be a Banach space and \\(M\\) a closed subspace. Show that \\(\\|x+M\\|=\\operatorname{dist}(x,M)\\) is a complete norm on \\(E/M\\). Deduce that a bounded linear surjection \\(T:E\\to F\\) onto a normed space \\(F\\) that is open forces \\(F\\) to be complete.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 360,
        "through_line": 376,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 6",
      "kind": "exercise",
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces#oa-fnd-hb-09::Exercise 6",
      "anchor": "oa-fnd-hb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
      "statement_and_full_conditions": "**Exercise 6** (easy; an indiscrete example). Give a vector space with a vector topology in which a one-dimensional subspace is not closed, and explain which step of Theorem 7.1 fails.",
      "proof_locus": {
        "source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "line": 377,
        "through_line": 380,
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.1",
      "kind": "lemma",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-01::Lemma 1.1",
      "anchor": "oa-fnd-wt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Lemma 1.1.** Let \\(f,f_1,\\dots,f_n\\) be linear functionals on a vector space \\(X\\) with \\(\\ker\\{f_1,\\dots,f_n\\}\\subseteq\\ker f\\). Then \\(f\\) is a linear combination of \\(f_1,\\dots,f_n\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 30,
        "through_line": 33,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 1.2",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-01::Theorem 1.2",
      "anchor": "oa-fnd-wt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 1.2.** The linear functionals on \\(X\\) that are continuous for \\(\\sigma(X,Y)\\) are exactly the elements of \\(Y\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 34,
        "through_line": 37,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 1.3",
      "kind": "corollary",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-01::Corollary 1.3",
      "anchor": "oa-fnd-wt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Corollary 1.3.** The weak topology of a normed space has continuous dual \\(E^*\\). The weak\\* topology of \\(E^*\\) has continuous dual \\(j(E)\\). Both topologies are Hausdorff: \\(E^*\\) separates points of \\(E\\) by the previous lesson, Corollary 2.3, and \\(j(E)\\) separates points of \\(E^*\\) trivially.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 38,
        "through_line": 39,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.1",
      "kind": "lemma",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-02::Lemma 2.1",
      "anchor": "oa-fnd-wt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Lemma 2.1.**\n1. Every filter is contained in an ultrafilter.\n2. A filter \\(\\mathcal U\\) is an ultrafilter iff for every \\(A\\subseteq S\\), either \\(A\\in\\mathcal U\\) or \\(S\\setminus A\\in\\mathcal U\\).\n3. If \\(f:S\\to T\\) and \\(\\mathcal U\\) is an ultrafilter on \\(S\\), then \\(f_*\\mathcal U=\\{B\\subseteq T:f^{-1}(B)\\in\\mathcal U\\}\\) is an ultrafilter on \\(T\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 49,
        "through_line": 59,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.2",
      "kind": "lemma",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-02::Lemma 2.2",
      "anchor": "oa-fnd-wt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Lemma 2.2.** A topological space \\(K\\) is compact iff every ultrafilter on \\(K\\) converges.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 60,
        "through_line": 65,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.3",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-02::Theorem 2.3",
      "anchor": "oa-fnd-wt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 2.3** (Tychonoff). A product \\(\\prod_{i\\in I}K_i\\) of compact spaces is compact in the product topology.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 66,
        "through_line": 71,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-03::Theorem 3.1",
      "anchor": "oa-fnd-wt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 3.1** (Banach–Alaoglu). For every normed space \\(E\\), the closed unit ball \\(B^*\\) of \\(E^*\\) is weak\\* compact.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 74,
        "through_line": 83,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.2",
      "kind": "proposition",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-03::Proposition 3.2",
      "anchor": "oa-fnd-wt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Proposition 3.2.** If \\(E\\) is separable, the weak\\* topology on \\(B^*\\) is metrizable.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 84,
        "through_line": 90,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.1",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-04::Theorem 4.1",
      "anchor": "oa-fnd-wt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 4.1** (Mazur). A convex subset \\(C\\) of a normed space has the same closure in the weak and in the norm topology. In particular, norm-closed convex sets are weakly closed. If \\(x_n\\to x\\) weakly, then some sequence of convex combinations of the \\(x_n\\) converges to \\(x\\) in norm.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 93,
        "through_line": 100,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.2",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-04::Theorem 4.2",
      "anchor": "oa-fnd-wt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 4.2** (Closures of convex sets). Let \\((X,\\tau)\\) be a locally convex space with continuous dual \\(X^*\\). A convex set \\(C\\subseteq X\\) has the same closure for \\(\\tau\\) and for \\(\\sigma(X,X^*)\\). Consequently, two locally convex topologies on \\(X\\) with the same continuous linear functionals have the same closed convex sets.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 101,
        "through_line": 104,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-05::Theorem 5.1",
      "anchor": "oa-fnd-wt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 5.1.** Let \\(E\\) be a normed space.\n1. For a subspace \\(L\\subseteq E\\), the norm closure of \\(L\\) is \\((L^\\perp)_\\perp\\).\n2. For a subspace \\(N\\subseteq E^*\\), the weak\\* closure of \\(N\\) is \\((N_\\perp)^\\perp\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 109,
        "through_line": 120,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.2",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-05::Theorem 5.2",
      "anchor": "oa-fnd-wt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 5.2** (Goldstine). The image \\(j(B)\\) of the closed unit ball \\(B\\) of \\(E\\) is weak\\* dense in the closed unit ball of \\(E^{**}\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 121,
        "through_line": 128,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.3",
      "kind": "proposition",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-05::Proposition 5.3",
      "anchor": "oa-fnd-wt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Proposition 5.3** (second adjoints). For \\(T\\in B(E,F)\\) let \\(T^*\\in B(F^*,E^*)\\), \\(T^*\\psi=\\psi\\circ T\\), and \\(T^{**}=(T^*)^*\\).\n1. \\(\\|T^*\\|=\\|T\\|\\) and \\(\\|T^{**}\\|=\\|T\\|\\).\n2. \\(T^{**}\\circ j_E=j_F\\circ T\\).\n3. \\((ST)^{**}=S^{**}T^{**}\\).\n4. \\(T^{**}\\) is weak\\*–weak\\* continuous.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 129,
        "through_line": 155,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.4",
      "kind": "proposition",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-05::Proposition 5.4",
      "anchor": "oa-fnd-wt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Proposition 5.4** (The dual of a quotient). Let \\(M\\) be a closed subspace of a normed space \\(E\\), with quotient map \\(q:E\\to E/M\\) and quotient norm \\(\\|x+M\\|=\\operatorname{dist}(x,M)\\). Then \\(g\\mapsto g\\circ q\\) is an isometric isomorphism of \\((E/M)^*\\) onto \\(M^\\perp\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 156,
        "through_line": 162,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-06::Theorem 6.1",
      "anchor": "oa-fnd-wt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 6.1** (Krein–Milman). Let \\(X\\) be a Hausdorff locally convex space and \\(K\\subseteq X\\) a nonempty compact convex set. Then \\(K\\) has an extreme point, and \\(K\\) is the closed convex hull of its extreme points.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 167,
        "through_line": 181,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-06::Theorem 6.2",
      "anchor": "oa-fnd-wt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 6.2** (Milman). Let \\(X\\) be a Hausdorff locally convex space, \\(Q\\subseteq X\\) compact, and suppose that the closed convex hull \\(K\\) of \\(Q\\) is compact. Then every extreme point of \\(K\\) lies in \\(Q\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 182,
        "through_line": 194,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.1",
      "kind": "lemma",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-07::Lemma 7.1",
      "anchor": "oa-fnd-wt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Lemma 7.1.** Let \\(E\\) be a normed space and \\(M\\subseteq E^{**}\\) a finite-dimensional subspace. Then there are finitely many \\(\\varphi_1,\\dots,\\varphi_m\\in E^*\\) of norm \\(1\\) with \\(\\max\\{0,|\\Phi(\\varphi_1)|,\\dots,|\\Phi(\\varphi_m)|\\}\\ge\\frac12\\|\\Phi\\|\\) for every \\(\\Phi\\in M\\). The family may be empty when \\(M=\\{0\\}\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 197,
        "through_line": 200,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [
        {
          "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
          "local_scopes": [
            {
              "heading": "7. The Eberlein–Šmulian theorem",
              "line": 195,
              "through_line": 240,
              "anchors": [
                "OA-FND-WT-07"
              ],
              "source_sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
            }
          ],
          "source_key": "human:vogt@9534B2B9478680458400CDC966BE6B17E0260E4B8916EEF7A74191910706B0A7",
          "source_locus": "Full four-page paper, Theorems 3–4",
          "role": "proof comparison",
          "correspondence": "Related weak-star theorem with additional hypothesis",
          "explanation": "Tail closed convex hulls require the extra intersection condition; not an unrestricted weak-star version of Eberlein–Šmulian."
        },
        {
          "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
          "local_labels": [
            "Lemma 7.1",
            "Theorem 7.2"
          ],
          "source_key": "human:whitley@2C2AD4D549E69E8D7E0A2DCE93340EFB8F31283AEE6BEA3773AB64A9DD83F0F0",
          "source_locus": "Printed 116–118; IIIF canvases 126–128",
          "role": "proof comparison",
          "correspondence": "Exact Banach-space scope; independently worded proof",
          "explanation": "Read all three page images. Countable norming sets on a separable closed span and the bounded bidual finite-norming induction match the lesson’s two directions. No separability of the ambient Banach space is assumed."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.2",
      "kind": "theorem",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-07::Theorem 7.2",
      "anchor": "oa-fnd-wt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Theorem 7.2** (Eberlein–Šmulian). For a subset \\(A\\) of a Banach space \\(E\\), the following are equivalent:\n1. \\(A\\) is relatively weakly compact;\n2. every sequence in \\(A\\) has a subsequence that converges weakly to a point of \\(E\\);\n3. every sequence in \\(A\\) has a weak cluster point in \\(E\\).",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 201,
        "through_line": 240,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [
        {
          "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
          "local_scopes": [
            {
              "heading": "7. The Eberlein–Šmulian theorem",
              "line": 195,
              "through_line": 240,
              "anchors": [
                "OA-FND-WT-07"
              ],
              "source_sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
            }
          ],
          "source_key": "human:vogt@9534B2B9478680458400CDC966BE6B17E0260E4B8916EEF7A74191910706B0A7",
          "source_locus": "Full four-page paper, Theorems 3–4",
          "role": "proof comparison",
          "correspondence": "Related weak-star theorem with additional hypothesis",
          "explanation": "Tail closed convex hulls require the extra intersection condition; not an unrestricted weak-star version of Eberlein–Šmulian."
        },
        {
          "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
          "local_labels": [
            "Lemma 7.1",
            "Theorem 7.2"
          ],
          "source_key": "human:whitley@2C2AD4D549E69E8D7E0A2DCE93340EFB8F31283AEE6BEA3773AB64A9DD83F0F0",
          "source_locus": "Printed 116–118; IIIF canvases 126–128",
          "role": "proof comparison",
          "correspondence": "Exact Banach-space scope; independently worded proof",
          "explanation": "Read all three page images. Countable norming sets on a separable closed span and the bounded bidual finite-norming induction match the lesson’s two directions. No separability of the ambient Banach space is assumed."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-07::Exercise 1",
      "anchor": "oa-fnd-wt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Exercise 1** (medium). Show that in an infinite-dimensional normed space the weak closure of the unit sphere \\(\\{\\|x\\|=1\\}\\) is the closed unit ball.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 243,
        "through_line": 249,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-07::Exercise 2",
      "anchor": "oa-fnd-wt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Exercise 2** (medium). Let \\(E=c_0\\), the null sequences with the supremum norm. Show that \\(\\delta_n\\to0\\) weakly but not in norm, and find convex combinations of the \\(\\delta_n\\) that converge to \\(0\\) in norm.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 250,
        "through_line": 256,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-07::Exercise 3",
      "anchor": "oa-fnd-wt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Exercise 3** (medium). Show that the closed unit ball of \\(\\ell^1\\) has extreme points \\(\\{\\lambda\\delta_n:|\\lambda|=1\\}\\), but that the closed unit ball of \\(c_0\\) has none. Deduce that \\(c_0\\) is not isometrically isomorphic to the dual of any normed space.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 257,
        "through_line": 263,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian#oa-fnd-wt-07::Exercise 4",
      "anchor": "oa-fnd-wt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
      "statement_and_full_conditions": "**Exercise 4** (easy). Show that a reflexive Banach space (one with \\(j(E)=E^{**}\\)) has a weakly compact closed unit ball, and deduce that every bounded sequence in it has a weakly convergent subsequence.",
      "proof_locus": {
        "source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "line": 264,
        "through_line": 269,
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.1",
      "kind": "proposition",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-01::Proposition 1.1",
      "anchor": "oa-fnd-hs-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Proposition 1.1.**\n1. (Polarization) \\(4B(\\xi,\\eta)=\\sum_{k=0}^3i^kB(\\xi+i^k\\eta,\\xi+i^k\\eta)\\).\n2. A form with \\(B(\\xi,\\xi)\\in\\mathbb R\\) for all \\(\\xi\\) is Hermitian.\n3. (Cauchy–Schwarz) If \\(B\\) is positive semidefinite, then \\(|B(\\xi,\\eta)|^2\\le B(\\xi,\\xi)B(\\eta,\\eta)\\). Consequently \\(\\{\\xi:B(\\xi,\\xi)=0\\}\\) is a subspace, and \\(\\xi\\mapsto B(\\xi,\\xi)^{1/2}\\) is a seminorm.",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 26,
        "through_line": 47,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-02::Theorem 2.1",
      "anchor": "oa-fnd-hs-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 2.1.** Let \\(C\\) be a nonempty closed convex subset of a real or complex Hilbert space \\(H\\), and \\(\\xi\\in H\\). There is exactly one \\(c_0\\in C\\) with \\(\\|\\xi-c_0\\|=\\operatorname{dist}(\\xi,C)\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 86,
        "through_line": 98,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.2",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-02::Theorem 2.2",
      "anchor": "oa-fnd-hs-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 2.2** (projection theorem). Let \\(L\\) be a closed subspace of a real or complex Hilbert space \\(H\\). Then:\n1. \\(H=L\\oplus L^\\perp\\);\n2. the map \\(P:\\xi\\mapsto c_0\\) of Theorem 2.1 is the orthogonal projection onto \\(L\\), a bounded self-adjoint idempotent with \\(\\|P\\|\\le1\\);\n3. for every subspace \\(Y\\), \\(\\overline Y=(Y^\\perp)^\\perp\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 99,
        "through_line": 109,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.3",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-02::Theorem 2.3",
      "anchor": "oa-fnd-hs-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 2.3** (Riesz–Fréchet). Every bounded linear functional \\(f\\) on \\(H\\) is \\(f(\\xi)=\\langle\\xi,\\eta\\rangle\\) for exactly one \\(\\eta\\in H\\), and \\(\\|f\\|=\\|\\eta\\|\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 110,
        "through_line": 115,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_labels": [
            "Theorem 2.3"
          ],
          "source_key": "programme:D20-companion@059bda086dfd6e6aa80f2077b2338c5d15039057",
          "source_locus": "reader-work-selected.tex, CH04 RW001",
          "role": "proof comparison",
          "correspondence": "exact Hilbert duality mechanism",
          "explanation": "Riesz–Fréchet proof read with first-variable linear convention. Actual own statement is bound through its section source, not inferred from the core title."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-03::Theorem 3.1",
      "anchor": "oa-fnd-hs-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 3.1.** Let \\(B\\) be a sesquilinear form on \\(H\\) with \\(|B(\\xi,\\eta)|\\le C\\|\\xi\\|\\|\\eta\\|\\). There is exactly one \\(t\\in B(H)\\) with \\(B(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\), and \\(\\|t\\|=\\sup\\{|B(\\xi,\\eta)|:\\|\\xi\\|,\\|\\eta\\|\\le1\\}\\le C\\). If \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\), then \\(\\langle t\\xi,\\xi\\rangle\\ge0\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 118,
        "through_line": 121,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.2",
      "kind": "corollary",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-03::Corollary 3.2",
      "anchor": "oa-fnd-hs-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Corollary 3.2.** Every \\(T\\in B(H)\\) has a unique adjoint \\(T^*\\in B(H)\\) with \\(\\langle T\\xi,\\eta\\rangle=\\langle\\xi,T^*\\eta\\rangle\\). Moreover:\n- \\(T^{**}=T\\), \\(\\|T^*\\|=\\|T\\|\\) and \\(\\|T^*T\\|=\\|T\\|^2\\);\n- \\((ST)^*=T^*S^*\\), and \\(T\\mapsto T^*\\) is conjugate-linear;\n- (complex scalars) if \\(\\langle T\\xi,\\xi\\rangle=0\\) for all \\(\\xi\\), then \\(T=0\\);\n- (complex scalars) if \\(\\langle T\\xi,\\xi\\rangle\\in\\mathbb R\\) for all \\(\\xi\\), then \\(T=T^*\\);\n- for self-adjoint \\(T\\), \\(\\|T\\|=\\sup_{\\|\\xi\\|=1}|\\langle T\\xi,\\xi\\rangle|\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 122,
        "through_line": 142,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-04::Theorem 4.1",
      "anchor": "oa-fnd-hs-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 4.1.** Let \\((e_i)_{i\\in I}\\) be an orthonormal family in \\(H\\).\n1. (Bessel) \\(\\sum_i|\\langle\\xi,e_i\\rangle|^2\\le\\|\\xi\\|^2\\). In particular \\(\\langle\\xi,e_i\\rangle\\ne0\\) for at most countably many \\(i\\).\n2. For \\((c_i)\\in\\ell^2(I)\\), the sum \\(\\sum_ic_ie_i\\) converges, as the net of finite partial sums, and \\(\\|\\sum_ic_ie_i\\|^2=\\sum_i|c_i|^2\\).\n3. The following are equivalent:\n   - the family is a basis;\n   - it is maximal among orthonormal families;\n   - \\(\\xi=\\sum_i\\langle\\xi,e_i\\rangle e_i\\) for every \\(\\xi\\);\n   - Parseval's identity \\(\\langle\\xi,\\eta\\rangle=\\sum_i\\langle\\xi,e_i\\rangle\\langle e_i,\\eta\\rangle\\) holds for all \\(\\xi,\\eta\\).\n4. Every orthonormal family is contained in an orthonormal basis. A separable Hilbert space has a finite or countable orthonormal basis.\n5. Any two orthonormal bases of \\(H\\) have the same cardinality, the *Hilbert dimension* of \\(H\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 147,
        "through_line": 171,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.0",
      "kind": "lemma",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-10::Lemma 5.0",
      "anchor": "oa-fnd-hs-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Lemma 5.0.** A metric space is compact if and only if it is complete and totally bounded. A compact metric space is sequentially compact and separable.\n\n*Proof.* In a compact space, every sequence has a point whose every neighbourhood contains infinitely many sequence terms, counted with their indices: otherwise a finite subcover of neighbourhoods containing only finitely many terms gives a contradiction. Choosing increasing indices in the balls of radius \\(1/k\\) at that point gives a convergent subsequence. A Cauchy sequence with a convergent subsequence converges to the same limit, so the space is complete. Compactness also gives finite covers by balls of any prescribed radius.\n\nConversely, in a totally bounded space, repeatedly choose an infinite set of remaining sequence indices lying in one ball of radius \\(2^{-k}\\). Take these sets nested and choose increasing indices from them. The resulting subsequence is Cauchy, since its terms from stage \\(k\\) onward are at mutual distance at most \\(2^{1-k}\\). Completeness gives its limit.\n\nTo obtain compactness, consider an open cover. There is a number \\(\\delta>0\\) such that every ball of radius \\(\\delta\\) is contained in a member of the cover. If there were no such number, choose \\(x_n\\) whose ball of radius \\(1/n\\) is contained in no member. A convergent subsequence tends to some \\(x\\) in a member \\(U\\). Choose \\(r>0\\) with \\(B(x,2r)\\subseteq U\\). For large subsequence indices, \\(d(x_n,x)<r\\) and \\(1/n<r\\), so \\(B(x_n,1/n)\\subseteq U\\), a contradiction. A finite cover by \\(\\delta\\)-balls now gives a finite subcover of the original cover. Finally choose finite \\(1/n\\)-nets for all \\(n\\); their union is countable and dense. \\(\\square\\)\n\nIn a finite-dimensional Hilbert space, bounded sets are totally bounded: express them in an orthonormal basis, bound each coordinate by Cauchy–Schwarz, and use a finite grid in the resulting bounded coordinate box. A finite-dimensional subspace is closed, because the coordinates of a convergent sequence converge and its limit is their finite basis expansion. Its bounded closed subsets are therefore complete and compact by the lemma.",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 180,
        "through_line": 189,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_scopes": [
            {
              "heading": "5. Compact operators",
              "line": 172,
              "through_line": 225,
              "anchors": [
                "OA-FND-HS-05",
                "OA-FND-HS-10"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            },
            {
              "heading": "6. The spectral theorem for compact self-adjoint operators",
              "line": 226,
              "through_line": 255,
              "anchors": [
                "OA-FND-HS-06"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.3; PDF 47–49",
          "role": "proof comparison",
          "correspondence": "Compact-operator proof mechanisms",
          "explanation": "Finite-rank approximation on the unit ball, compact approximate eigenvector argument, finite eigenspaces and norm-convergent spectral sum; the lesson supplies its own complete operators-between-two-spaces arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-10::Theorem 5.1",
      "anchor": "oa-fnd-hs-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 5.1.**\n1. \\(K(H)\\) is a norm-closed two-sided ideal of \\(B(H)\\) that contains \\(F(H)\\).\n2. \\(T\\) is compact iff \\(T^*\\) is compact.\n3. \\(K(H)\\) is the norm closure of \\(F(H)\\).\n4. A compact operator maps weakly convergent sequences to norm convergent sequences.\n5. Let \\((P_i)\\) be a net in \\(B(H)\\) with \\(\\|P_i\\|\\le1\\) and \\(P_i\\xi\\to\\xi\\) for every \\(\\xi\\in H\\). Then \\(\\|T-P_iT\\|\\to0\\) for every compact \\(T\\). This applies to the projections \\(P_n\\) onto \\(\\operatorname{span}\\{e_1,\\dots,e_n\\}\\) for an orthonormal basis \\((e_k)\\) of a separable \\(H\\) (Theorem 4.1(3)).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 192,
        "through_line": 225,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_scopes": [
            {
              "heading": "5. Compact operators",
              "line": 172,
              "through_line": 225,
              "anchors": [
                "OA-FND-HS-05",
                "OA-FND-HS-10"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            },
            {
              "heading": "6. The spectral theorem for compact self-adjoint operators",
              "line": 226,
              "through_line": 255,
              "anchors": [
                "OA-FND-HS-06"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.3; PDF 47–49",
          "role": "proof comparison",
          "correspondence": "Compact-operator proof mechanisms",
          "explanation": "Finite-rank approximation on the unit ball, compact approximate eigenvector argument, finite eigenspaces and norm-convergent spectral sum; the lesson supplies its own complete operators-between-two-spaces arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.1",
      "kind": "lemma",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-06::Lemma 6.1",
      "anchor": "oa-fnd-hs-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Lemma 6.1.** Let \\(T\\) be compact and self-adjoint, \\(T\\ne0\\). Then \\(\\|T\\|\\) or \\(-\\|T\\|\\) is an eigenvalue of \\(T\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 228,
        "through_line": 239,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_labels": [
            "Lemma 6.1",
            "Theorem 6.2"
          ],
          "source_key": "programme:D20-companion@059bda086dfd6e6aa80f2077b2338c5d15039057",
          "source_locus": "compact-spectral-svd.tex, O008 bridge LEM001 and THM002",
          "role": "proof comparison",
          "correspondence": "same compact Hilbert scope",
          "explanation": "Selected full proof passages read; the core Riesz–Schauder step consumes its earlier Fredholm alternative. Own source keeps its complete prerequisite chain. The subsequently read core SVD is related context, not asserted as a new HS theorem."
        },
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_scopes": [
            {
              "heading": "5. Compact operators",
              "line": 172,
              "through_line": 225,
              "anchors": [
                "OA-FND-HS-05",
                "OA-FND-HS-10"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            },
            {
              "heading": "6. The spectral theorem for compact self-adjoint operators",
              "line": 226,
              "through_line": 255,
              "anchors": [
                "OA-FND-HS-06"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.3; PDF 47–49",
          "role": "proof comparison",
          "correspondence": "Compact-operator proof mechanisms",
          "explanation": "Finite-rank approximation on the unit ball, compact approximate eigenvector argument, finite eigenspaces and norm-convergent spectral sum; the lesson supplies its own complete operators-between-two-spaces arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-06::Theorem 6.2",
      "anchor": "oa-fnd-hs-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 6.2.** Let \\(T\\) be compact and self-adjoint, \\(T\\ne0\\). There are an orthonormal family \\((e_n)_{n\\in N}\\), with \\(N=\\{1,\\dots,r\\}\\) or \\(N=\\mathbb N\\), and real numbers \\(\\lambda_n\\ne0\\) with \\(|\\lambda_1|\\ge|\\lambda_2|\\ge\\dots\\), tending to \\(0\\) if \\(N=\\mathbb N\\), such that\n\\[\nT=\\sum_n\\lambda_n\\theta_{e_n,e_n},\n\\]\nwith convergence in norm. The nonzero eigenvalues of \\(T\\) are the \\(\\lambda_n\\). The eigenspace of an eigenvalue \\(\\mu\\ne0\\) is spanned by the \\(e_n\\) with \\(\\lambda_n=\\mu\\) and is finite-dimensional. \\(T\\ge0\\) iff all \\(\\lambda_n>0\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 240,
        "through_line": 255,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_labels": [
            "Lemma 6.1",
            "Theorem 6.2"
          ],
          "source_key": "programme:D20-companion@059bda086dfd6e6aa80f2077b2338c5d15039057",
          "source_locus": "compact-spectral-svd.tex, O008 bridge LEM001 and THM002",
          "role": "proof comparison",
          "correspondence": "same compact Hilbert scope",
          "explanation": "Selected full proof passages read; the core Riesz–Schauder step consumes its earlier Fredholm alternative. Own source keeps its complete prerequisite chain. The subsequently read core SVD is related context, not asserted as a new HS theorem."
        },
        {
          "unit": "hilbert-spaces-and-compact-operators",
          "local_scopes": [
            {
              "heading": "5. Compact operators",
              "line": 172,
              "through_line": 225,
              "anchors": [
                "OA-FND-HS-05",
                "OA-FND-HS-10"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            },
            {
              "heading": "6. The spectral theorem for compact self-adjoint operators",
              "line": 226,
              "through_line": 255,
              "anchors": [
                "OA-FND-HS-06"
              ],
              "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.3; PDF 47–49",
          "role": "proof comparison",
          "correspondence": "Compact-operator proof mechanisms",
          "explanation": "Finite-rank approximation on the unit ball, compact approximate eigenvector argument, finite eigenspaces and norm-convergent spectral sum; the lesson supplies its own complete operators-between-two-spaces arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-07::Theorem 7.1",
      "anchor": "oa-fnd-hs-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 7.1.** Let \\(T\\in K(H)\\) and \\(\\lambda\\in\\mathbb C\\setminus\\{0\\}\\), and put \\(S=T-\\lambda\\).\n1. \\(\\ker S\\) is finite-dimensional.\n2. \\(S(H)\\) is closed.\n3. If \\(S\\) is injective, it is surjective, hence invertible in \\(B(H)\\).\n4. Consequently, every nonzero point of the spectrum \\(\\sigma(T)\\) is an eigenvalue of finite multiplicity.\n5. The nonzero eigenvalues of \\(T\\) have no accumulation point other than \\(0\\). Each nonzero point of \\(\\sigma(T)\\) is isolated.",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 258,
        "through_line": 291,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-08::Theorem 8.1",
      "anchor": "oa-fnd-hs-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 8.1** (Hilbert tensor products). Let \\(H\\) and \\(K\\) be Hilbert spaces.\n1. There are a Hilbert space \\(H\\otimes K\\) and a bilinear map \\((\\xi,\\eta)\\mapsto\\xi\\otimes\\eta\\) from \\(H\\times K\\) to \\(H\\otimes K\\) with\n\\[\n\\langle\\xi\\otimes\\eta,\\xi'\\otimes\\eta'\\rangle=\\langle\\xi,\\xi'\\rangle\\langle\\eta,\\eta'\\rangle ,\n\\]\nsuch that the elementary tensors \\(\\xi\\otimes\\eta\\) span a dense subspace.\n2. If \\((e_i)_{i\\in I}\\) and \\((f_j)_{j\\in J}\\) are orthonormal bases of \\(H\\) and \\(K\\), then \\((e_i\\otimes f_j)\\) is an orthonormal basis of \\(H\\otimes K\\).\n3. (*Universal property.*) Let \\(B:H\\times K\\to\\mathcal K\\) be a bilinear map into a Hilbert space with \\(\\langle B(\\xi,\\eta),B(\\xi',\\eta')\\rangle=\\langle\\xi,\\xi'\\rangle\\langle\\eta,\\eta'\\rangle\\). There is exactly one isometry \\(U:H\\otimes K\\to\\mathcal K\\) with \\(U(\\xi\\otimes\\eta)=B(\\xi,\\eta)\\). It is unitary if the vectors \\(B(\\xi,\\eta)\\) span a dense subspace. So \\(H\\otimes K\\) is unique up to a unitary that matches the elementary tensors.",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 302,
        "through_line": 332,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.1",
      "kind": "theorem",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-09::Theorem 9.1",
      "anchor": "oa-fnd-hs-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Theorem 9.1.**\n1. \\(f\\mapsto m_f\\) is an isometric unital \\(*\\)-homomorphism from \\(L^\\infty(\\nu)\\) into \\(B(L^2(\\nu))\\).\n2. An operator \\(T\\in B(L^2(\\nu))\\) commutes with every \\(m_f\\) if and only if \\(T=m_g\\) for some \\(g\\in L^\\infty(\\nu)\\). So the algebra \\(\\{m_f:f\\in L^\\infty(\\nu)\\}\\) equals its own commutant: it is *maximal abelian*.",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 337,
        "through_line": 358,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-09::Exercise 1",
      "anchor": "oa-fnd-hs-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Exercise 1** (easy; diagonal operators). Let \\((e_n)\\) be an orthonormal basis of \\(H\\) and \\(T=\\sum_n\\mu_n\\theta_{e_n,e_n}\\) for a bounded sequence \\((\\mu_n)\\). Show that \\(T\\) is compact iff \\(\\mu_n\\to0\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 361,
        "through_line": 364,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-09::Exercise 2",
      "anchor": "oa-fnd-hs-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Exercise 2** (medium; Hilbert–Schmidt operators). Show that an operator \\(T\\) with \\(\\sum_n\\|Te_n\\|^2<\\infty\\) for some orthonormal basis \\((e_n)\\) is compact.",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 365,
        "through_line": 374,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-09::Exercise 3",
      "anchor": "oa-fnd-hs-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Exercise 3** (hard; the Volterra operator). On \\(L^2[0,1]\\), let \\((V\\xi)(s)=\\int_0^s\\xi(t)\\,dt\\). Show that \\(V\\) is compact and that \\(\\sigma(V)=\\{0\\}\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 375,
        "through_line": 387,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "hilbert-spaces-and-compact-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators#oa-fnd-hs-09::Exercise 4",
      "anchor": "oa-fnd-hs-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
      "statement_and_full_conditions": "**Exercise 4** (medium; closed range of \\(1-T\\)). Let \\(T\\) be compact. Show that \\(\\dim\\ker(1-T)=\\dim\\ker(1-T^*)\\).",
      "proof_locus": {
        "source": "src/hilbert-spaces-and-compact-operators.md",
        "line": 388,
        "through_line": 395,
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 0.1",
      "kind": "lemma",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-07::Lemma 0.1",
      "anchor": "oa-fnd-ct-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Lemma 0.1** (Intervals, rectangles and parameters).\n\n1. A continuous \\(F:[a,b]\\to X\\) has a Riemann integral. It is linear, commutes with bounded linear maps, satisfies \\(\\|\\int_a^b F\\|\\leq\\int_a^b\\|F\\|\\), and respects uniform limits.\n2. The function \\(P(t)=\\int_a^t F(s)\\,ds\\) has derivative \\(P'(t)=F(t)\\) in the interior, with one-sided derivatives at the endpoints. If \\(G\\) is continuously differentiable, then \\(\\int_a^bG'(t)\\,dt=G(b)-G(a)\\). The chain rule and substitution hold for continuously differentiable real paths; a holomorphic primitive with continuous derivative satisfies the same chain rule along a piecewise continuously differentiable complex path.\n3. For continuous \\(F:[a,b]\\times[c,d]\\to X\\), the two iterated Riemann integrals agree.\n4. Suppose \\(K(t,s)\\) and its partial derivative \\(\\partial_tK(t,s)\\) are continuous on a rectangle. Then \\(t\\mapsto\\int_c^dK(t,s)\\,ds\\) is differentiable, with derivative \\(\\int_c^d\\partial_tK(t,s)\\,ds\\). When both variables range over \\([a,b]\\), the moving-endpoint integral satisfies\n\\[\n\\begin{gathered}\n\\frac{d}{dt}\\int_a^tK(t,s)\\,ds\n\\\\\n=K(t,t)+\\int_a^t\\partial_tK(t,s)\\,ds .\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 39,
        "through_line": 99,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.1",
      "kind": "lemma",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-01::Lemma 1.1",
      "anchor": "oa-fnd-ct-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Lemma 1.1.**\n1. \\(\\operatorname{Ind}_\\Gamma\\) takes integer values on \\(\\mathbb C\\setminus\\Gamma^*\\), is constant on each connected component, and is \\(0\\) on the unbounded component.\n2. For the circle \\(\\partial D(c,r)\\), the index is \\(1\\) on \\(D(c,r)\\) and \\(0\\) outside \\(\\bar D(c,r)\\).\n3. For the positively oriented boundary \\(\\partial Q\\) of a closed square \\(Q\\), the index is \\(1\\) on the interior of \\(Q\\) and \\(0\\) outside \\(Q\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 107,
        "through_line": 147,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.1",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-02::Theorem 2.1",
      "anchor": "oa-fnd-ct-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 2.1** (Goursat). Let \\(V\\) be open, \\(p\\in V\\), and \\(f:V\\to\\mathbb C\\) continuous on \\(V\\) and holomorphic on \\(V\\setminus\\{p\\}\\). Then \\(\\int_{\\partial\\Delta}f=0\\) for every closed triangle \\(\\Delta\\subseteq V\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 150,
        "through_line": 181,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.2",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-02::Theorem 2.2",
      "anchor": "oa-fnd-ct-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 2.2** (Cauchy's theorem in a convex set). Let \\(V\\) be convex and open, \\(p\\in V\\), and \\(f\\) continuous on \\(V\\) and holomorphic on \\(V\\setminus\\{p\\}\\). Fix \\(a\\in V\\) and put \\(F(z)=\\int_{[a,z]}f\\). Then \\(F\\) is holomorphic on \\(V\\) with \\(F'=f\\). Hence \\(\\int_\\gamma f=0\\) for every closed path \\(\\gamma\\) in \\(V\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 182,
        "through_line": 193,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.3",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-02::Theorem 2.3",
      "anchor": "oa-fnd-ct-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 2.3** (Cauchy's formula in a convex set). Let \\(V\\) be convex and open, \\(f\\) holomorphic on \\(V\\), \\(\\gamma\\) a closed path in \\(V\\), and \\(z\\in V\\setminus\\gamma^*\\). Then\n\\[\nf(z)\\operatorname{Ind}_\\gamma(z)=\\frac1{2\\pi i}\\int_\\gamma\\frac{f(w)}{w-z}\\,dw .\n\\]",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 194,
        "through_line": 200,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.1",
      "kind": "lemma",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Lemma 3.1",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Lemma 3.1** (Power series). Let \\(c\\in\\mathbb C\\), \\(0<R\\leq\\infty\\), and let \\(a_n\\in\\mathbb C\\) satisfy \\(\\sum_n|a_n|\\rho^n<\\infty\\) for every \\(0<\\rho<R\\). Then:\n- \\(f(z)=\\sum_na_n(z-c)^n\\) is holomorphic on \\(D(c,R)\\), with \\(f'(z)=\\sum_{n\\geq1}na_n(z-c)^{n-1}\\);\n- the coefficients of \\(f'\\) satisfy the same hypothesis.\n\nConsequently \\(f\\) has derivatives of all orders, and \\(a_n=f^{(n)}(c)/n!\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 203,
        "through_line": 228,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.2",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Theorem 3.2",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 3.2** (Holomorphic functions are analytic). Let \\(f\\) be holomorphic on \\(U\\supseteq D(c,R)\\). Then\n\\[\nf(z)=\\sum_na_n(z-c)^n\\qquad(z\\in D(c,R)),\n\\]\nwith \\(\\sum_n|a_n|\\rho^n<\\infty\\) for every \\(0<\\rho<R\\). For every \\(0<r<R\\),\n\\[\n\\begin{gathered}\na_n\\\\\n=\\frac{f^{(n)}(c)}{n!}\\\\\n=\\frac1{2\\pi i}\\int_{\\partial D(c,r)}\\frac{f(w)}{(w-c)^{n+1}}\\,dw,\\\\\n|a_n|\\\\\n\\leq\\frac{M(r)}{r^n},\n\\end{gathered}\n\\]\nwhere \\(M(r)=\\max_{|w-c|=r}|f(w)|\\) (*Cauchy's estimates*). In particular \\(f'\\) is holomorphic, and \\(f\\) has derivatives of all orders.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 229,
        "through_line": 254,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.3",
      "kind": "corollary",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Corollary 3.3",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Corollary 3.3** (Liouville). A bounded holomorphic function \\(f:\\mathbb C\\to\\mathbb C\\) is constant.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 255,
        "through_line": 258,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.4",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Theorem 3.4",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 3.4** (Morera). Let \\(f\\) be continuous on \\(U\\), with \\(\\int_{\\partial\\Delta}f=0\\) for every closed triangle \\(\\Delta\\subseteq U\\). Then \\(f\\) is holomorphic.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 259,
        "through_line": 262,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.5",
      "kind": "corollary",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Corollary 3.5",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Corollary 3.5** (Weierstrass). Let \\(f_n\\) be holomorphic on \\(U\\), and let \\(f_n\\to f\\) uniformly on every compact subset of \\(U\\). Then \\(f\\) is holomorphic.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 263,
        "through_line": 266,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.6",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Theorem 3.6",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 3.6** (Maximum modulus). Let \\(f\\) be continuous on the closed unit disc \\(\\bar D=\\bar D(0,1)\\) and holomorphic on \\(D(0,1)\\). Then\n\\[\n\\max_{\\bar D}|f|=\\max_{|z|=1}|f(z)| .\n\\]\nIn particular, if \\(f=0\\) on the unit circle, then \\(f=0\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 267,
        "through_line": 280,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.7",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-03::Theorem 3.7",
      "anchor": "oa-fnd-ct-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 3.7** (Identity theorem). Let \\(U\\) be connected and \\(f\\) holomorphic on \\(U\\). If the zeros of \\(f\\) have an accumulation point in \\(U\\), then \\(f=0\\). In particular, two holomorphic functions on \\(U\\) that agree on a nonempty open subset agree everywhere.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 281,
        "through_line": 290,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.1",
      "kind": "lemma",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-04::Lemma 4.1",
      "anchor": "oa-fnd-ct-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Lemma 4.1.** The image of a path has empty interior. So does \\(\\Gamma^*\\) for a cycle \\(\\Gamma\\). Hence \\(U\\setminus\\Gamma^*\\neq\\varnothing\\) for every nonempty open \\(U\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 293,
        "through_line": 301,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [
        {
          "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
          "local_scopes": [
            {
              "heading": "4. Cauchy's theorem for cycles",
              "line": 291,
              "through_line": 378,
              "anchors": [
                "OA-FND-CT-04"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            }
          ],
          "source_key": "human:mit-dixon@28860604809CA6E48B2EF1A1F946DB9DD92B8B8F0CD801FA3D2917EF8CF7A71C",
          "source_locus": "PDF 2–5, printed 1–4",
          "role": "proof comparison",
          "correspondence": "Closed-curve treatment; finite cycles supplied internally",
          "explanation": "Divided-difference continuity, Morera, holomorphic gluing and Liouville. Original Dixon attribution retained; actual reading is the MIT exposition."
        },
        {
          "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
          "local_scopes": [
            {
              "heading": "4. Cauchy's theorem for cycles",
              "line": 291,
              "through_line": 378,
              "anchors": [
                "OA-FND-CT-04"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            },
            {
              "heading": "5. Cycles that surround a compact set",
              "line": 379,
              "through_line": 404,
              "anchors": [
                "OA-FND-CT-05"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            }
          ],
          "source_key": "human:cerny@1940CC2DCFFBFEA58C26CDDA70F2B38613960A4EA039FED5875726199B587174",
          "source_locus": "Main proof pp.1–5; comments pp.6–7 also read",
          "role": "proof comparison",
          "correspondence": "Alternative finite-cycle proof",
          "explanation": "Grid side cancellation and contour interchange; source assumes rectangular Cauchy formula, supplied by the lesson’s full local proof."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.2",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-04::Theorem 4.2",
      "anchor": "oa-fnd-ct-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 4.2** (Cauchy's theorem for cycles). Let \\(\\Gamma\\) be a cycle in \\(U\\) with \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for every \\(\\alpha\\notin U\\), and let \\(f\\) be holomorphic on \\(U\\). Then\n\\[\n\\begin{gathered}\nf(z)\\operatorname{Ind}_\\Gamma(z)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{f(w)}{w-z}\\,dw\\\\\n(z\\in U\\setminus\\Gamma^*),\n\\\\\n\\text{and}\\\\\n\\int_\\Gamma f(w)\\,dw\\\\\n=0 .\n\\end{gathered}\n\\]\n\n*Reference:* The Dixon argument for a closed curve is presented in [MIT OCW Lecture 13, by Zuoqin Wang for Sigurdur Helgason’s course](https://ocw.mit.edu/courses/18-112-functions-of-a-complex-variable-fall-2008/8793d412fa0da2a4183539f5d8f7e3fd_lecture13.pdf). The proof below includes the finite-cycle formulation.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 302,
        "through_line": 378,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [
        {
          "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
          "local_scopes": [
            {
              "heading": "4. Cauchy's theorem for cycles",
              "line": 291,
              "through_line": 378,
              "anchors": [
                "OA-FND-CT-04"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            }
          ],
          "source_key": "human:mit-dixon@28860604809CA6E48B2EF1A1F946DB9DD92B8B8F0CD801FA3D2917EF8CF7A71C",
          "source_locus": "PDF 2–5, printed 1–4",
          "role": "proof comparison",
          "correspondence": "Closed-curve treatment; finite cycles supplied internally",
          "explanation": "Divided-difference continuity, Morera, holomorphic gluing and Liouville. Original Dixon attribution retained; actual reading is the MIT exposition."
        },
        {
          "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
          "local_scopes": [
            {
              "heading": "4. Cauchy's theorem for cycles",
              "line": 291,
              "through_line": 378,
              "anchors": [
                "OA-FND-CT-04"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            },
            {
              "heading": "5. Cycles that surround a compact set",
              "line": 379,
              "through_line": 404,
              "anchors": [
                "OA-FND-CT-05"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            }
          ],
          "source_key": "human:cerny@1940CC2DCFFBFEA58C26CDDA70F2B38613960A4EA039FED5875726199B587174",
          "source_locus": "Main proof pp.1–5; comments pp.6–7 also read",
          "role": "proof comparison",
          "correspondence": "Alternative finite-cycle proof",
          "explanation": "Grid side cancellation and contour interchange; source assumes rectangular Cauchy formula, supplied by the lesson’s full local proof."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-05::Theorem 5.1",
      "anchor": "oa-fnd-ct-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 5.1.** Let \\(K\\subseteq U\\) with \\(K\\) compact and \\(U\\) open. Some cycle made of finitely many oriented segments surrounds \\(K\\) in \\(U\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 386,
        "through_line": 404,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [
        {
          "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
          "local_scopes": [
            {
              "heading": "4. Cauchy's theorem for cycles",
              "line": 291,
              "through_line": 378,
              "anchors": [
                "OA-FND-CT-04"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            },
            {
              "heading": "5. Cycles that surround a compact set",
              "line": 379,
              "through_line": 404,
              "anchors": [
                "OA-FND-CT-05"
              ],
              "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
            }
          ],
          "source_key": "human:cerny@1940CC2DCFFBFEA58C26CDDA70F2B38613960A4EA039FED5875726199B587174",
          "source_locus": "Main proof pp.1–5; comments pp.6–7 also read",
          "role": "proof comparison",
          "correspondence": "Alternative finite-cycle proof",
          "explanation": "Grid side cancellation and contour interchange; source assumes rectangular Cauchy formula, supplied by the lesson’s full local proof."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-06::Theorem 6.1",
      "anchor": "oa-fnd-ct-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Theorem 6.1.** Let \\(X\\) be a Banach space and \\(g:U\\to X\\) a map such that \\(\\varphi\\circ g\\) is holomorphic for every \\(\\varphi\\in X^*\\).\n1. \\(g\\) is continuous; it is even Lipschitz on a neighbourhood of each point.\n2. For every cycle \\(\\Gamma\\) in \\(U\\) with \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for all \\(\\alpha\\notin U\\),\n\\[\n\\begin{gathered}\n\\int_\\Gamma g(w)\\,dw\\\\\n=0,\\\\\ng(z)\\operatorname{Ind}_\\Gamma(z)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{g(w)}{w-z}\\,dw\\\\\n(z\\in U\\setminus\\Gamma^*).\n\\end{gathered}\n\\]\n3. On each disc \\(D(c,R)\\subseteq U\\), \\(g(z)=\\sum_na_n(z-c)^n\\) with \\(a_n\\in X\\) and \\(\\sum_n\\|a_n\\|\\rho^n<\\infty\\) for \\(\\rho<R\\). In particular the difference quotients \\((g(z+h)-g(z))/h\\) converge in norm as \\(h\\to0\\).\n4. (*Liouville.*) If \\(U=\\mathbb C\\) and \\(g\\) is bounded, then \\(g\\) is constant.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 407,
        "through_line": 449,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-06::Exercise 1",
      "anchor": "oa-fnd-ct-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Exercise 1** (medium; The fundamental theorem of algebra). Show that every nonconstant polynomial \\(p\\) has a complex root.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 452,
        "through_line": 455,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-06::Exercise 2",
      "anchor": "oa-fnd-ct-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Exercise 2** (easy; The condition on the index). Let \\(U=\\mathbb C\\setminus\\{0\\}\\), \\(f(z)=1/z\\), and let \\(\\Gamma\\) be the unit circle. Compute \\(\\int_\\Gamma f\\), and say which hypothesis of Theorem 4.2 fails.",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 456,
        "through_line": 459,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-06::Exercise 3",
      "anchor": "oa-fnd-ct-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Exercise 3** (medium; Schwarz's lemma). Let \\(f\\) be holomorphic on \\(D(0,1)\\) with \\(|f|\\leq1\\) and \\(f(0)=0\\). Show that \\(|f(z)|\\leq|z|\\) for all \\(z\\), and \\(|f'(0)|\\leq1\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 460,
        "through_line": 466,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences#oa-fnd-ct-06::Exercise 4",
      "anchor": "oa-fnd-ct-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/cauchy-s-theorem-for-cycles-and-its-consequences@B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A",
      "statement_and_full_conditions": "**Exercise 4** (medium; Polynomials converging on the circle). Let \\(p_n\\) be polynomials that converge uniformly on the unit circle \\(\\mathbb T\\). Show that they converge uniformly on \\(\\bar D(0,1)\\), and that the limit \\(G\\) is continuous on \\(\\bar D(0,1)\\) and holomorphic on \\(D(0,1)\\). Show that \\(G=0\\) if \\(G=0\\) on \\(\\mathbb T\\).",
      "proof_locus": {
        "source": "src/cauchy-s-theorem-for-cycles-and-its-consequences.md",
        "line": 467,
        "through_line": 473,
        "sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 0.1",
      "kind": "lemma",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-23::Lemma 0.1",
      "anchor": "oa-fnd-bn-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Lemma 0.1** (Products and regrouping). In a Banach algebra, the product of two absolutely convergent series is the sum of the products of their terms, and that double series may be regrouped in any countable partition of its index set. In particular, for series indexed by nonnegative integers,\n\\[\n\\left(\\sum_{j\\geq0}a_j\\right)\n\\left(\\sum_{k\\geq0}b_k\\right)\n=\\sum_{n\\geq0}\\sum_{j+k=n}a_jb_k .\n\\]\nIf \\(xy=yx\\) in a unital algebra, then\n\\((x+y)^n=\\sum_{j=0}^n\\binom njx^jy^{n-j}\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 46,
        "through_line": 64,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.1",
      "kind": "definition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-01::Definition 1.1",
      "anchor": "oa-fnd-bn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Definition 1.1.**\n1. An *involution* on an algebra \\(A\\) is a map \\(x\\mapsto x^*\\) with \\((x^*)^*=x\\), \\((x+y)^*=x^*+y^*\\), \\((\\lambda x)^*=\\bar\\lambda x^*\\) and \\((xy)^*=y^*x^*\\).\n2. An *involutive Banach algebra* is a Banach algebra together with an isometric involution: \\(\\|x^*\\|=\\|x\\|\\).\n3. A *C\\*-algebra* is an involutive Banach algebra in which \\(\\|x^*x\\|=\\|x^*\\|\\|x\\|\\) for every \\(x\\). Because the involution is isometric, this says \\(\\|x^*x\\|=\\|x\\|^2\\) (the *C\\*-identity*).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 67,
        "through_line": 71,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.2",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-01::Proposition 1.2",
      "anchor": "oa-fnd-bn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 1.2.**\n1. Multiplication is jointly continuous: \\[\n\\begin{gathered}\n\\|x_1y_1-x_2y_2\\|\\\\\n\\leq\\|x_1\\|\\|y_1-y_2\\|+\\|x_1-x_2\\|\\|y_2\\|.\n\\end{gathered}\n\\]\n2. For a Banach space \\(E\\), \\(B(E)\\) with the operator norm is a Banach algebra.\n3. (*The C\\*-identity alone is enough.*) Let \\(A\\) be an algebra with an involution and a complete submultiplicative norm such that \\(\\|x^*x\\|=\\|x\\|^2\\) for all \\(x\\). Then \\(\\|x^*\\|=\\|x\\|\\), so \\(A\\) is a C\\*-algebra.\n4. For an LCH space \\(\\Omega\\), \\(C_0(\\Omega)\\) is a commutative C\\*-algebra under pointwise operations, with \\(x^*=\\bar x\\). It is unital exactly when \\(\\Omega\\) is compact; then it is written \\(C(\\Omega)\\). This includes \\(\\Omega=\\varnothing\\): \\(C_0(\\varnothing)=\\{0\\}\\) is unital, and \\(\\varnothing\\) is compact.\n5. For a Hilbert space \\(H\\), \\(B(H)\\) with the operator adjoint is a C\\*-algebra. It is commutative exactly when \\(\\dim H\\leq1\\). Every norm-closed \\(*\\)-subalgebra of \\(B(H)\\) is a C\\*-algebra.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 72,
        "through_line": 95,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.1",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-02::Proposition 2.1",
      "anchor": "oa-fnd-bn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 2.1** (Neumann series). Let \\(A\\) be a unital Banach algebra and \\(\\|1-x\\|<1\\). Then \\(x\\) is invertible, and\n\\[\nx^{-1}=\\sum_{n=0}^\\infty(1-x)^n,\n\\tag{2.1}\n\\]\nwhere \\((1-x)^0=1\\) and the series converges in norm. Moreover \\(\\|x^{-1}\\|\\leq\\|1\\|+\\|1-x\\|/(1-\\|1-x\\|)\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 100,
        "through_line": 108,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.2",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-02::Proposition 2.2",
      "anchor": "oa-fnd-bn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 2.2** (the invertible group is open, and inversion is continuous). Let \\(A\\) be a nontrivial unital Banach algebra, \\(x_0\\in G(A)\\), and \\(\\|x-x_0\\|<1/\\|x_0^{-1}\\|\\). Then \\(x\\) is invertible,\n\\[\nx^{-1}=\\sum_{n=0}^\\infty\\big[x_0^{-1}(x_0-x)\\big]^n\\,x_0^{-1},\n\\tag{2.2}\n\\]\n\\[\n\\begin{gathered}\n\\|x^{-1}-x_0^{-1}\\|\\\\\n\\leq\\frac{\\|x_0^{-1}\\|^2\\,\\|x-x_0\\|}{1-\\|x_0^{-1}\\|\\,\\|x-x_0\\|}.\n\\end{gathered}\n\\tag{2.3}\n\\]\nConsequently \\(G(A)\\) is open, and \\(x\\mapsto x^{-1}\\) is continuous on \\(G(A)\\). The distance from \\(x_0\\in G(A)\\) to the set of noninvertible elements is at least \\(1/\\|x_0^{-1}\\|\\). If \\(x_n\\in G(A)\\) converge to a noninvertible element, then \\(\\|x_n^{-1}\\|\\to\\infty\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 109,
        "through_line": 126,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.1",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-03::Proposition 3.1",
      "anchor": "oa-fnd-bn-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 3.1.** Let \\(A\\) be a nontrivial unital Banach algebra.\n1. \\(\\|1\\|\\geq1\\).\n2. Put \\(\\|x\\|_\\ell=\\sup\\{\\|xy\\|:\\|y\\|\\leq1\\}\\), the operator norm of left multiplication \\(L_x:y\\mapsto xy\\). This is an algebra norm on \\(A\\), with\n\\[\n\\|x\\|/\\|1\\|\\leq\\|x\\|_\\ell\\leq\\|x\\|\\quad\\text{and}\\quad\\|1\\|_\\ell=1 .\n\\]\nSo \\((A,\\|\\cdot\\|_\\ell)\\) is a Banach algebra whose norm is equivalent to the given one. If \\(\\|1\\|=1\\), the two norms coincide.\n3. If \\(A\\) has an isometric involution, then \\(1^*=1\\), and \\(N(x)=\\max(\\|x\\|_\\ell,\\|x^*\\|_\\ell)\\) is an equivalent Banach algebra norm with \\(N(x^*)=N(x)\\) and \\(N(1)=1\\).\n4. If \\(A\\) is a C\\*-algebra, then \\(\\|1\\|=1\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 133,
        "through_line": 149,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Construction 3.2",
      "kind": "construction",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-04::Construction 3.2",
      "anchor": "oa-fnd-bn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Construction 3.2.** For any algebra \\(A\\), let \\(A_1=A\\oplus\\mathbb C\\) with the linear structure of the direct sum and the product\n\\[\n\\begin{gathered}\n(a,\\lambda)(b,\\mu)\\\\\n=(ab+\\lambda b+\\mu a,\\ \\lambda\\mu).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A\\) has an involution, put \\((a,\\lambda)^*=(a^*,\\bar\\lambda)\\). Write \\(j(a)=(a,0)\\), \\(q(a,\\lambda)=\\lambda\\), and \\(1=(0,1)\\). If \\(A\\) is normed, put \\(\\|(a,\\lambda)\\|_1=\\|a\\|+|\\lambda|\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 152,
        "through_line": 161,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.3",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-04::Proposition 3.3",
      "anchor": "oa-fnd-bn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 3.3.**\n1. \\(A_1\\) is a unital algebra with identity \\((0,1)\\). The map \\(j\\) is an injective homomorphism onto an ideal \\(j(A)\\) of codimension one, and \\(q\\) is a unital homomorphism onto \\(\\mathbb C\\) with kernel \\(j(A)\\). \\(A_1\\) is commutative exactly when \\(A\\) is. If \\(A\\) has an involution, so does \\(A_1\\), and \\(j\\) and \\(q\\) preserve it.\n2. If \\(A\\) is a normed algebra, so is \\((A_1,\\|\\cdot\\|_1)\\), with \\(\\|1\\|_1=1\\). The map \\(j\\) is isometric and \\(|q(u)|\\leq\\|u\\|_1\\). If \\(A\\) is a Banach algebra, so is \\(A_1\\), and \\(j(A)\\) is closed. If the involution of \\(A\\) is isometric, so is that of \\(A_1\\).\n3. If \\(A\\) has an identity \\(1_A\\), then \\((a,\\lambda)\\mapsto(a+\\lambda1_A,\\lambda)\\) is an algebra isomorphism of \\(A_1\\) onto the product algebra \\(A\\times\\mathbb C\\). In particular \\(j(1_A)\\) is an idempotent of \\(A_1\\) different from the new identity \\((0,1)\\). The construction adds a new identity even when \\(A\\) has one.\n4. If \\(A\\) is a C\\*-algebra with a nonzero projection \\(p=p^*=p^2\\) (for example \\(A=\\mathbb C\\)), then \\((A_1,\\|\\cdot\\|_1)\\) is not a C\\*-algebra.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 162,
        "through_line": 180,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.4",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-05::Proposition 3.4",
      "anchor": "oa-fnd-bn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 3.4.**\n1. \\(p\\) is a submultiplicative seminorm on \\(A_1\\), \\(p(u)\\leq\\|u\\|_1\\), and \\(p(a)=\\|a\\|\\) for \\(a\\in A\\).\n2. \\(p(u)^2\\leq p(u^*u)\\leq p(u^*)p(u)\\). Consequently \\(p(u^*)=p(u)\\) and \\(p(u^*u)=p(u)^2\\).\n3. If \\(A\\) is not unital, then \\(p(u)=0\\) only for \\(u=0\\). If \\(A\\) has an identity \\(1_A\\), then \\(p(u)=0\\) exactly for \\(u\\in\\mathbb C(1_A-1)\\).\n4. If \\(A\\) is not unital, then \\((A_1,p)\\) is a unital C\\*-algebra containing \\(A\\) isometrically as a closed ideal of codimension one. Moreover \\(|\\lambda|\\leq p(a+\\lambda)\\), so \\(p\\) equals the norm \\(N(a,\\lambda)=\\max\\big(p(a+\\lambda),|\\lambda|\\big)\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 188,
        "through_line": 209,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.5",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-05::Example 3.5",
      "anchor": "oa-fnd-bn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 3.5.** Let \\(\\Omega\\) be LCH and not compact, and \\(A=C_0(\\Omega)\\), which is not unital by [Proposition 1.2(4)](#oa-fnd-bn-01). For \\(u=f+\\lambda\\), \\(p(u)=\\sup_{\\omega\\in\\Omega}|f(\\omega)+\\lambda|\\). Indeed, \\(\\|ub\\|_\\infty\\leq\\sup|f+\\lambda|\\) when \\(\\|b\\|_\\infty\\leq1\\). Conversely, for \\(\\omega_0\\in\\Omega\\), Urysohn's lemma gives \\(b\\in C_c(\\Omega)\\) with \\(0\\leq b\\leq1\\) and \\(b(\\omega_0)=1\\), and then \\(\\|ub\\|_\\infty\\geq|f(\\omega_0)+\\lambda|\\). Since \\(f\\) vanishes at infinity, \\(|\\lambda|\\leq p(u)\\), as (4) predicts. If \\(\\Omega\\) is compact, \\(A\\) is unital with \\(1_A\\equiv1\\), and \\(p(1_A-1)=\\sup|1-1|=0\\), as (3) predicts.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 210,
        "through_line": 211,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-06::Definition 4.1",
      "anchor": "oa-fnd-bn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Definition 4.1.** Let \\(A\\) be unital and \\(x\\in A\\). The *spectrum* of \\(x\\) is\n\\[\n\\sigma_A(x)=\\{\\lambda\\in\\mathbb C :\\ \\lambda-x\\ \\text{is not invertible}\\},\n\\]\nand its complement \\(\\rho_A(x)\\) is the *resolvent set*. For an arbitrary algebra \\(A\\) and \\(x\\in A\\), the *quasi-spectrum* is \\(\\sigma'_A(x)=\\sigma_{A_1}(j(x))\\), computed in the unitization of [Construction 3.2](#oa-fnd-bn-04). The *spectral radius* is \\(r(x)=\\sup\\{|\\lambda|:\\lambda\\in\\sigma'_A(x)\\}\\in[0,\\infty]\\).\n\nThe quasi-spectrum matters mainly when \\(A\\) has no identity, but we define it for every \\(A\\). When \\(A\\) is unital, part (1) below shows that it is \\(\\sigma_A(x)\\cup\\{0\\}\\), which has the same radius as \\(\\sigma_A(x)\\) whenever \\(\\sigma_A(x)\\) is not empty.",
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              "heading": "11. The Gelfand representation",
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
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      "local_label": "Proposition 4.2",
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      "statement_and_full_conditions": "**Proposition 4.2.**\n1. \\(0\\in\\sigma'_A(x)\\). If \\(A\\) is unital, \\(\\sigma'_A(x)=\\sigma_A(x)\\cup\\{0\\}\\). So \\(r(x)=\\sup\\{|\\lambda|:\\lambda\\in\\sigma_A(x)\\}\\) when \\(A\\) is unital and \\(\\sigma_A(x)\\neq\\varnothing\\).\n2. Let \\(A\\) be unital, \\(x,y\\in A\\) and \\(\\lambda\\neq0\\). If \\(\\lambda-xy\\) has inverse \\(u\\), then \\(\\lambda-yx\\) has inverse \\(\\lambda^{-1}(1+yux)\\). Hence \\(\\sigma_A(xy)\\cup\\{0\\}=\\sigma_A(yx)\\cup\\{0\\}\\). For every algebra, \\(\\sigma'_A(xy)=\\sigma'_A(yx)\\) and \\(r(xy)=r(yx)\\).\n3. If \\(\\pi:A\\to B\\) is a unital homomorphism, then \\(\\sigma_B(\\pi(x))\\subseteq\\sigma_A(x)\\). If \\(B\\) is a subalgebra of the unital algebra \\(A\\) with \\(1_A\\in B\\), then \\(\\sigma_A(x)\\subseteq\\sigma_B(x)\\) for \\(x\\in B\\).\n4. If \\(x\\in G(A)\\), then \\(0\\notin\\sigma_A(x)\\) and \\(\\sigma_A(x^{-1})=\\{\\lambda^{-1}:\\lambda\\in\\sigma_A(x)\\}\\).\n5. \\(\\sigma_A(x-c)=\\sigma_A(x)-c\\) for \\(c\\in\\mathbb C\\), and \\(\\sigma_A(cx)=c\\,\\sigma_A(x)\\) for \\(c\\neq0\\).",
      "proof_locus": {
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              "heading": "11. The Gelfand representation",
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
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      "local_label": "Examples 4.3",
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      "statement_and_full_conditions": "**Examples 4.3.**\n- *The \\(\\{0\\}\\) in (2) is needed.* On \\(\\ell^2(\\mathbb N)\\) let \\(Se_n=e_{n+1}\\). Then \\(S^*e_0=0\\) and \\(S^*e_{n+1}=e_n\\), so \\(S^*S=1\\), while \\(SS^*\\) is the orthogonal projection \\(P\\) onto the closed span of \\(e_1,e_2,\\dots\\). Since \\(0\\neq P\\neq1\\), \\(\\sigma(P)=\\{0,1\\}\\): for \\(\\lambda\\notin\\{0,1\\}\\) the inverse of \\(\\lambda-P\\) is \\(\\lambda^{-1}(1-P)+(\\lambda-1)^{-1}P\\), while \\(P\\) and \\(1-P\\) have nonzero kernels. So \\(\\sigma(SS^*)=\\{0,1\\}\\) and \\(\\sigma(S^*S)=\\{1\\}\\).\n- *Matrices.* In \\(M_n(\\mathbb C)\\), \\(xy\\) is invertible exactly when \\(yx\\) is, because \\(\\det(xy)=\\det(yx)\\). So there \\(\\sigma(xy)=\\sigma(yx)\\).\n- *The spectrum depends on the algebra.* Example 4.6 below gives an element whose spectrum is the unit circle in one algebra and the closed unit disc in a closed subalgebra.",
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
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              "heading": "10. Characters",
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              "heading": "11. The Gelfand representation",
              "line": 739,
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
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      "local_label": "Proposition 4.4",
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      "statement_and_full_conditions": "**Proposition 4.4.**\n1. If \\(|\\lambda|>\\|x\\|\\), then \\(\\lambda\\in\\rho_A(x)\\),\n\\[\nR_x(\\lambda)=\\sum_{n=0}^\\infty\\frac{x^n}{\\lambda^{n+1}},\n\\tag{4.1}\n\\]\nand \\(\\|R_x(\\lambda)\\|\\leq c_1/(|\\lambda|-\\|x\\|)\\).\n2. If \\(\\lambda_0\\in\\rho_A(x)\\) and \\(|\\lambda-\\lambda_0|<1/\\|R_x(\\lambda_0)\\|\\), then \\(\\lambda\\in\\rho_A(x)\\) and\n\\[\n\\begin{gathered}\nR_x(\\lambda)\\\\\n=\\sum_{n=0}^\\infty(\\lambda_0-\\lambda)^n\\,R_x(\\lambda_0)^{n+1}.\n\\end{gathered}\n\\tag{4.2}\n\\]\nHence \\(\\rho_A(x)\\) is open, and \\(\\operatorname{dist}(\\lambda_0,\\sigma_A(x))\\geq1/\\|R_x(\\lambda_0)\\|\\).\n3. \\(\\sigma_A(x)\\) is compact and lies in the closed disc of radius \\(\\|x\\|\\). For every Banach algebra \\(A\\) and \\(x\\in A\\), \\(\\sigma'_A(x)\\) is compact and not empty, and \\(r(x)\\leq\\|x\\|\\).\n4. (*Resolvent identity.*) For \\(\\lambda,\\mu\\in\\rho_A(x)\\), \\(R_x(\\lambda)-R_x(\\mu)=(\\mu-\\lambda)R_x(\\lambda)R_x(\\mu)\\). All resolvents \\(R_x(\\lambda)\\) commute with each other and with every element that commutes with \\(x\\).\n5. \\(R_x\\) is continuous on \\(\\rho_A(x)\\). For each \\(\\varphi\\in A^*\\), \\(\\varphi\\circ R_x\\) is given near each \\(\\lambda_0\\in\\rho_A(x)\\) by a power series in \\(\\lambda-\\lambda_0\\); so it is holomorphic, with a continuous derivative.\n6. \\(\\|R_x(\\lambda)\\|\\to0\\) as \\(|\\lambda|\\to\\infty\\), and \\(\\|R_x(\\lambda)\\|\\to\\infty\\) as \\(\\lambda\\in\\rho_A(x)\\) approaches a point of \\(\\sigma_A(x)\\).",
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              "heading": "10. Characters",
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              "heading": "11. The Gelfand representation",
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
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        }
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      "local_label": "Examples 4.5",
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      "statement_and_full_conditions": "**Examples 4.5** (completeness is needed).\n- Give \\(\\mathbb C[z]\\) the norm \\(\\|p\\|=\\max_{|z|\\leq1}|p(z)|\\). It is a unital normed algebra, but not complete. A nonzero polynomial multiple of \\(z-\\lambda\\) has degree at least one, so \\(z-\\lambda\\) is never invertible: \\(\\sigma(z)=\\mathbb C\\), which is not bounded.\n- In the field \\(\\mathbb C(z)\\) of rational functions, every \\(z-\\lambda\\) is invertible, so \\(\\sigma(z)=\\varnothing\\). By the Gelfand–Mazur theorem ([Corollary 5.3](#oa-fnd-bn-08)), \\(\\mathbb C(z)\\) carries no algebra norm at all.",
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              "heading": "10. Characters",
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              "heading": "11. The Gelfand representation",
              "line": 739,
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
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    {
      "local_label": "Example 4.6",
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      "statement_and_full_conditions": "**Example 4.6.** Let \\(B=C(\\mathbb T)\\), and let \\(A\\) be the closure in \\(B\\) of the polynomials in \\(z\\) (the disc algebra, seen on the circle). Then \\(\\sigma_B(z)=\\mathbb T\\), while \\(\\sigma_A(z)\\) is the closed unit disc. In general, if \\(A\\) is a closed subalgebra of a unital Banach algebra \\(B\\) with \\(1_B\\in A\\), then \\(\\sigma_B(x)\\subseteq\\sigma_A(x)\\), and every boundary point of \\(\\sigma_A(x)\\) lies in \\(\\sigma_B(x)\\).",
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
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              "heading": "10. Characters",
              "line": 700,
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              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.1",
      "kind": "lemma",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-08::Lemma 5.1",
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      "statement_and_full_conditions": "**Lemma 5.1** (circle means). Let \\(0\\leq r_1<r_2\\leq\\infty\\), and let \\(g\\) be holomorphic with a continuous derivative on the annulus \\(\\{r_1<|\\lambda|<r_2\\}\\). Then\n\\[\nM(\\rho)=\\frac1{2\\pi}\\int_0^{2\\pi}g(\\rho e^{i\\theta})\\,d\\theta\n\\]\ndoes not depend on \\(\\rho\\in(r_1,r_2)\\). If \\(g\\) is holomorphic with a continuous derivative on the disc \\(\\{|\\lambda|<r_2\\}\\), then \\(M(\\rho)=g(0)\\) for \\(0<\\rho<r_2\\).",
      "proof_locus": {
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              "line": 212,
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
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              "heading": "10. Characters",
              "line": 700,
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                "OA-FND-BN-17"
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            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.2",
      "kind": "theorem",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-08::Theorem 5.2",
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      "statement_and_full_conditions": "**Theorem 5.2** (the spectrum is not empty). If \\(A\\) is a nontrivial unital Banach algebra, then \\(\\sigma_A(x)\\neq\\varnothing\\) for every \\(x\\in A\\).",
      "proof_locus": {
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        "line": 328,
        "through_line": 337,
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              "line": 212,
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              ],
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
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                "OA-FND-BN-09"
              ],
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            },
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              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
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                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
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                "OA-FND-BN-19"
              ],
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.3",
      "kind": "corollary",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-08::Corollary 5.3",
      "anchor": "oa-fnd-bn-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Corollary 5.3** (Gelfand–Mazur theorem, for normed algebras). Let \\(A\\) be a nontrivial unital normed algebra, not necessarily complete, in which every nonzero element is invertible. Then \\(A=\\mathbb C1\\), and \\(\\lambda\\mapsto\\lambda1\\) is the only unital algebra isomorphism of \\(\\mathbb C\\) onto \\(A\\).",
      "proof_locus": {
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        "line": 338,
        "through_line": 353,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
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              "heading": "10. Characters",
              "line": 700,
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                "OA-FND-BN-17"
              ],
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            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
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    {
      "local_label": "Theorem 5.4",
      "kind": "theorem",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-09::Theorem 5.4",
      "anchor": "oa-fnd-bn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 5.4** (spectral radius formula). For every element \\(x\\) of a Banach algebra \\(A\\),\n\\[\n\\begin{gathered}\nr(x)\\\\\n=\\lim_{n\\to\\infty}\\|x^n\\|^{1/n}\\\\\n=\\inf_{n\\geq1}\\|x^n\\|^{1/n}.\n\\end{gathered}\n\\tag{5.1}\n\\]\n\nStep 1 below gives the lower spectral-radius bound directly, before any limit of roots is known to exist. Remark 5.5 supplies a second proof using submultiplicativity.",
      "proof_locus": {
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        "line": 356,
        "through_line": 403,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
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              ],
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
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              "heading": "10. Characters",
              "line": 700,
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              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
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              "heading": "11. The Gelfand representation",
              "line": 739,
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
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      "global_tag_allocated": false
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    {
      "local_label": "Corollary 5.6",
      "kind": "corollary",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-09::Corollary 5.6",
      "anchor": "oa-fnd-bn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Corollary 5.6.**\n1. \\(r(x^k)=r(x)^k\\) and \\(r(cx)=|c|\\,r(x)\\).\n2. \\(r(xy)=r(yx)\\) for all \\(x,y\\).\n3. If \\(xy=yx\\), then \\(r(xy)\\leq r(x)r(y)\\) and \\(r(x+y)\\leq r(x)+r(y)\\).\n4. If \\(A\\) is unital and \\(r(1-x)<1\\), then \\(x\\) is invertible, and the series (2.1) converges absolutely.",
      "proof_locus": {
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              ],
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
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              "heading": "10. Characters",
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                "OA-FND-BN-17"
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              "heading": "11. The Gelfand representation",
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          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
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    },
    {
      "local_label": "Examples 5.7",
      "kind": "examples",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-09::Examples 5.7",
      "anchor": "oa-fnd-bn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Examples 5.7.**\n- A nonzero nilpotent element \\(N\\) (\\(N^m=0\\)) has \\(r(N)=0<\\|N\\|\\).\n- *The Volterra operator.* On \\(C([0,1])\\) with the supremum norm, let \\((Vf)(t)=\\int_0^tf(s)\\,ds\\). Put \\(W_n(t)=\\int_0^t(t-s)^{n-1}f(s)/(n-1)!\\,ds\\). The fundamental theorem gives \\(W_1=Vf\\). For \\(n\\geq2\\), the moving-endpoint formula in [Lemma 0.1(4) of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) gives \\(W_n'=W_{n-1}\\) and \\(W_n(0)=0\\). Thus \\(W_n=VW_{n-1}\\) by the fundamental theorem, and induction gives \\(V^nf=W_n\\). So \\(\\|V^n\\|=1/n!\\), with equality at \\(f\\equiv1\\) and \\(t=1\\). Since at least half of the factors of \\(n!\\) are at least \\(n/2\\), \\((n!)^{1/n}\\geq(n/2)^{1/2}\\to\\infty\\). So \\(r(V)=0\\). The spectrum is not empty (Theorem 5.2) and lies in \\(\\{0\\}\\), so \\(\\sigma(V)=\\{0\\}\\), although \\(V\\neq0\\).\n- *Commutativity is needed in (3).* In \\(M_2(\\mathbb C)\\), \\(E_{12}\\) and \\(E_{21}\\) square to \\(0\\), so each has spectral radius \\(0\\). But \\(E_{12}+E_{21}\\) has eigenvalues \\(\\pm1\\), and \\(E_{12}E_{21}=E_{11}\\) has eigenvalues \\(0,1\\); both have spectral radius \\(1\\).\n- *Self-adjoint elements of a C\\*-algebra.* If \\(x=x^*\\), then \\(\\|x^2\\|=\\|x^*x\\|=\\|x\\|^2\\), so \\(\\|x^{2^k}\\|=\\|x\\|^{2^k}\\) for all \\(k\\), and (5.1) along the subsequence \\(2^k\\) gives \\(r(x)=\\|x\\|\\). The same holds for normal elements; this is proved in the [next lesson, on C\\*-algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-02).\n\n**Remark 5.8** (at most one C\\*-norm). A \\(*\\)-algebra carries at most one norm that makes it a C\\*-algebra. Indeed, let \\(N\\) be such a norm and \\(u\\) an element. Since \\(u^*u\\) is self-adjoint, the last example gives \\(N(u)^2=N(u^*u)=r(u^*u)\\). The spectral radius \\(r(u^*u)\\) is determined by the quasi-spectrum of \\(u^*u\\), which is defined algebraically and does not depend on the norm. In particular, for a nonunital C\\*-algebra \\(A\\), the C\\*-norm \\(p\\) on \\(A_1\\) of [Proposition 3.4](#oa-fnd-bn-05) is the only norm that makes \\(A_1\\) a C\\*-algebra.",
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          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
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                "OA-FND-BN-07"
              ],
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
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                "OA-FND-BN-09"
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            },
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              "heading": "10. Characters",
              "line": 700,
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                "OA-FND-BN-17"
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            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
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                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
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          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-09::Theorem 6.1",
      "anchor": "oa-fnd-bn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 6.1** (Cauchy's theorem). Let \\(U\\subseteq\\mathbb C\\) be open and let \\(\\Gamma\\) be a cycle in \\(U\\) with \\(\\operatorname{Ind}_\\Gamma(\\alpha)=0\\) for every \\(\\alpha\\notin U\\). Let \\(g:U\\to X\\) be a map into a Banach space such that \\(\\varphi\\circ g\\) is holomorphic for every \\(\\varphi\\in X^*\\). Then \\(g\\) is continuous, \\(\\int_\\Gamma g(\\lambda)\\,d\\lambda=0\\), and\n\\[\n\\begin{gathered}\ng(z)\\operatorname{Ind}_\\Gamma(z)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma\\frac{g(\\lambda)}{\\lambda-z}\\,d\\lambda\\\\\n(z\\in U\\setminus\\Gamma^*).\n\\end{gathered}\n\\]\n\nThis is Theorem 6.1(1)–(2) of the lesson on Cauchy's theorem. Its scalar case is Theorem 4.2 there. Applying a functional \\(\\varphi\\in X^*\\) reduces both formulas to the scalar case, and the Hahn–Banach theorem then gives them in \\(X\\).\n\nWe use the formula only at points of index \\(1\\) or \\(0\\). At a point \\(z\\) of index \\(0\\) it is the first conclusion, applied to \\(\\lambda\\mapsto g(\\lambda)/(\\lambda-z)\\) on \\(U\\setminus\\{z\\}\\). Applied on a disc, the formula gives the Taylor expansion of a holomorphic function at each point, in the usual way.",
      "proof_locus": {
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        "line": 438,
        "through_line": 450,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
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          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
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                "OA-FND-BN-11",
                "OA-FND-BN-12",
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          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-09::Theorem 6.2",
      "anchor": "oa-fnd-bn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 6.2** (surrounding cycles). If \\(K\\subseteq U\\subseteq\\mathbb C\\) with \\(K\\) compact and \\(U\\) open, some cycle made of finitely many oriented segments surrounds \\(K\\) in \\(U\\).\n\nThis is Theorem 5.1 of the lesson on Cauchy's theorem.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 451,
        "through_line": 454,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
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                "OA-FND-BN-11",
                "OA-FND-BN-12",
                "OA-FND-BN-22"
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            }
          ],
          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.3",
      "kind": "lemma",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-09::Lemma 6.3",
      "anchor": "oa-fnd-bn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Lemma 6.3** (index). Let \\(\\Gamma\\) be a cycle. On \\(\\mathbb C\\setminus\\Gamma^*\\), the function \\(\\operatorname{Ind}_\\Gamma\\) takes integer values, is constant on each connected component, and is \\(0\\) on the unbounded component. For the positively oriented circle \\(|\\lambda-c|=\\rho\\), the index is \\(1\\) on the open disc and \\(0\\) outside the closed disc.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 455,
        "through_line": 473,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
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          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
                "OA-FND-BN-10",
                "OA-FND-BN-11",
                "OA-FND-BN-12",
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              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.4",
      "kind": "definition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-10::Definition 6.4",
      "anchor": "oa-fnd-bn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Definition 6.4** (holomorphic functional calculus). Let \\(A\\) be a unital Banach algebra, \\(x\\in A\\), \\(U\\subseteq\\mathbb C\\) open with \\(\\sigma_A(x)\\subseteq U\\), and \\(f\\in H(U)\\). Choose a cycle \\(\\Gamma\\) that surrounds \\(\\sigma_A(x)\\) in \\(U\\), which exists by Theorem 6.2, and put\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma f(\\lambda)\\,(\\lambda-x)^{-1}\\,d\\lambda\\\\\n=\\frac1{2\\pi i}\\int_\\Gamma f(\\lambda)\\,R_x(\\lambda)\\,d\\lambda .\n\\end{gathered}\n\\tag{6.1}\n\\]\nThe integrand is continuous on \\(\\Gamma^*\\), because \\(\\Gamma^*\\subseteq\\rho_A(x)\\) and the resolvent is continuous there ([Proposition 4.4(5)](#oa-fnd-bn-07)).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 476,
        "through_line": 486,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
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          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
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          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 6.5",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-10::Proposition 6.5",
      "anchor": "oa-fnd-bn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 6.5.**\n1. Every cycle that surrounds \\(\\sigma_A(x)\\) in \\(U\\) gives the same element \\(f(x)\\).\n2. If \\(f\\in H(U)\\) and \\(g\\in H(V)\\) agree on an open set \\(W\\) with \\(\\sigma_A(x)\\subseteq W\\subseteq U\\cap V\\), then \\(f(x)=g(x)\\). So \\(f(x)\\) depends only on the germ of \\(f\\) at \\(\\sigma_A(x)\\), and (6.1) defines \\(f(x)\\) for every \\(f\\) holomorphic on some neighbourhood of \\(\\sigma_A(x)\\). These functions form an algebra, with the operations taken on the intersection of the domains.\n3. For \\(\\varphi\\in A^*\\), \\(\\varphi(f(x))=\\frac1{2\\pi i}\\int_\\Gamma f(\\lambda)\\,\\varphi(R_x(\\lambda))\\,d\\lambda\\).\n4. \\(f(x)\\) commutes with every element of \\(A\\) that commutes with \\(x\\).\n5. If \\(\\chi:A\\to\\mathbb C\\) is a unital homomorphism, then \\(\\chi(f(x))=f(\\chi(x))\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 487,
        "through_line": 515,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
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                "OA-FND-BN-11",
                "OA-FND-BN-12",
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              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
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          ],
          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.7",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-11::Theorem 6.7",
      "anchor": "oa-fnd-bn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 6.7** (the calculus is a homomorphism). Let \\(A\\) be a unital Banach algebra, \\(x\\in A\\), and \\(U\\supseteq\\sigma_A(x)\\) open. The map \\(f\\mapsto f(x)\\) from \\(H(U)\\) to \\(A\\) is a unital algebra homomorphism: it is linear, \\((fg)(x)=f(x)g(x)\\), \\(1(x)=1\\), and \\(u(x)=x\\) for \\(u(\\lambda)=\\lambda\\). So \\(p(x)\\) has its usual meaning for every polynomial \\(p\\). The map is continuous: if \\(f_n\\to f\\) uniformly on compact subsets of \\(U\\), then \\(f_n(x)\\to f(x)\\). Any two elements \\(f(x),g(x)\\) commute.\n\nThe proof below uses cycles satisfying the exact index conditions. Remark 6.6 explains the conditions required for a single curve, and Example 6.12 shows why cycles are useful for disconnected spectra.",
      "proof_locus": {
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        "line": 518,
        "through_line": 558,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
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          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
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                "OA-FND-BN-11",
                "OA-FND-BN-12",
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          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 6.8",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-11::Proposition 6.8",
      "anchor": "oa-fnd-bn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 6.8** (power series). Let \\(f(\\lambda)=\\sum_nc_n(\\lambda-c)^n\\) converge for \\(|\\lambda-c|<\\rho_0\\), and let \\(r(x-c)<\\rho_0\\). Then \\(\\sigma_A(x)\\) lies in the disc \\(|\\lambda-c|<\\rho_0\\), and \\(f(x)=\\sum_nc_n(x-c)^n\\), with the series converging in norm. In particular, the calculus of \\(\\lambda\\mapsto e^\\lambda\\) is \\(\\exp x=\\sum_nx^n/n!\\) ([Section 7](#oa-fnd-bn-14)).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 559,
        "through_line": 562,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
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          "local_scopes": [
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
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                "OA-FND-BN-11",
                "OA-FND-BN-12",
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              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
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          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.9",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-11::Example 6.9",
      "anchor": "oa-fnd-bn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 6.9.** Let \\(x=\\alpha+N\\) with \\(N^2=0\\) and \\(N\\neq0\\), for instance a \\(2\\times2\\) Jordan block. Then \\(\\sigma_A(x)=\\{\\alpha\\}\\) (the spectrum of \\(N\\) is not empty and \\(r(N)=0\\)), and by the proposition \\(f(x)=f(\\alpha)+f'(\\alpha)N\\) for every \\(f\\) holomorphic near \\(\\alpha\\). So \\(f(x)\\) is not determined by the values of \\(f\\) on \\(\\sigma_A(x)\\).",
      "proof_locus": {
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        "line": 563,
        "through_line": 564,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
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              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
                "OA-FND-BN-10",
                "OA-FND-BN-11",
                "OA-FND-BN-12",
                "OA-FND-BN-22"
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            }
          ],
          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.10",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-12::Theorem 6.10",
      "anchor": "oa-fnd-bn-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 6.10** (invertibility, spectral mapping and composition). Let \\(A\\) be a unital Banach algebra, \\(x\\in A\\), \\(U\\supseteq\\sigma_A(x)\\) open, and \\(f\\in H(U)\\).\n1. \\(f(x)\\) is invertible if and only if \\(f\\) has no zero on \\(\\sigma_A(x)\\). Then \\(f(x)^{-1}=(1/f)(x)\\), where \\(1/f\\) is holomorphic on the open set \\(\\{\\lambda\\in U:f(\\lambda)\\neq0\\}\\supseteq\\sigma_A(x)\\).\n2. (*Spectral mapping theorem.*) \\(\\sigma_A(f(x))=f(\\sigma_A(x))\\).\n3. (*Composition.*) Let \\(V\\supseteq f(\\sigma_A(x))\\) be open and \\(g\\in H(V)\\). Then \\(g\\circ f\\) is holomorphic on \\(U\\cap f^{-1}(V)\\), which contains \\(\\sigma_A(x)\\), and \\((g\\circ f)(x)=g(f(x))\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 567,
        "through_line": 604,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
                "OA-FND-BN-10",
                "OA-FND-BN-11",
                "OA-FND-BN-12",
                "OA-FND-BN-22"
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          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.12",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-22::Example 6.12",
      "anchor": "oa-fnd-bn-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 6.12.** Let \\(A\\) be a unital Banach algebra, and let \\(x\\in A\\) have spectrum \\(\\sigma_A(x)=K_0\\cup K_1\\), where \\(K_0\\) and \\(K_1\\) are disjoint, compact and nonempty. Choose disjoint open sets \\(U_0\\supseteq K_0\\) and \\(U_1\\supseteq K_1\\), and let \\(f\\) be \\(0\\) on \\(U_0\\) and \\(1\\) on \\(U_1\\). Then \\(f\\in H(U_0\\cup U_1)\\) and \\(f^2=f\\). The element \\(e=f(x)\\) satisfies \\(e^2=e\\) (Theorem 6.7), \\(ex=xe\\) (Proposition 6.5(4)), and \\(\\sigma_A(e)=f(\\sigma_A(x))=\\{0,1\\}\\) (Theorem 6.10); so \\(e\\neq0\\) and \\(e\\neq1\\). No single closed curve whose closed inside lies in \\(U_0\\cup U_1\\) encloses \\(\\sigma_A(x)\\): the closed region bounded by it is connected, so it lies in \\(U_0\\) or in \\(U_1\\). So \\(e\\) is out of reach of a calculus built on one closed curve. For the matrix \\(x=\\begin{pmatrix}1&1\\\\0&2\\end{pmatrix}\\), with \\(K_0=\\{1\\}\\) and \\(K_1=\\{2\\}\\), Exercise 3 at the end of the lesson gives \\(e=x-1=\\begin{pmatrix}0&1\\\\0&1\\end{pmatrix}\\), which is indeed idempotent.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 607,
        "through_line": 608,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "6. The holomorphic functional calculus",
              "line": 420,
              "through_line": 608,
              "anchors": [
                "OA-FND-BN-10",
                "OA-FND-BN-11",
                "OA-FND-BN-12",
                "OA-FND-BN-22"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
          "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
          "role": "proof comparison",
          "correspondence": "Source contour formulation narrower",
          "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.1",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-14::Proposition 7.1",
      "anchor": "oa-fnd-bn-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 7.1.**\n1. \\(\\exp x=\\sum_{n\\geq0}x^n/n!\\) converges absolutely, and it is the calculus of \\(\\lambda\\mapsto e^\\lambda\\). If \\(xy=yx\\), then \\(\\exp(x+y)=\\exp x\\exp y\\). Hence \\(\\exp(-x)=(\\exp x)^{-1}\\), \\(\\exp(A)\\subseteq G(A)\\), and \\(t\\mapsto\\exp(tx)\\) is a norm-continuous homomorphism from \\((\\mathbb R,+)\\) into \\(G(A)\\).\n2. Let \\(G_0(A)\\) be the connected component of \\(1\\) in \\(G(A)\\), the *principal component*. It is an open and closed normal subgroup of \\(G(A)\\), and it consists of the elements that can be joined to \\(1\\) by a continuous path in \\(G(A)\\). It contains \\(\\exp(A)\\). The quotient \\(G(A)/G_0(A)\\) is the *index group*.\n3. If \\(\\sigma_A(x)\\subseteq D\\), put \\(\\log x=\\operatorname{Log}(x)\\), the calculus of \\(\\operatorname{Log}\\). Then \\(\\exp(\\log x)=x\\). If \\(r(x-1)<1\\), then \\(\\log x=-\\sum_{n\\geq1}(1-x)^n/n\\).\n4. If \\(\\sigma_A(x)\\) lies in the strip \\(S=\\{|\\operatorname{Im}\\lambda|<\\pi\\}\\), then \\(\\sigma_A(\\exp x)\\subseteq D\\) and \\(\\log(\\exp x)=x\\).\n5. \\(G_0(A)\\) is the subgroup generated by \\(\\exp(A)\\). If \\(A\\) is commutative, \\(G_0(A)=\\exp(A)\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 621,
        "through_line": 641,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.2",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-14::Example 7.2",
      "anchor": "oa-fnd-bn-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 7.2** (a nontrivial index group). Let \\(A=C(\\mathbb T)\\), the continuous functions on the unit circle, and let \\(z\\) be the identity function. It is invertible, with inverse \\(\\bar z\\). By (5), \\(G_0(A)=\\exp(A)\\), and \\(\\exp g=e^{g}\\) pointwise. Suppose \\(z=e^{g}\\) with \\(g\\in C(\\mathbb T)\\), and put \\(h(t)=g(e^{it})-it\\) for \\(t\\in[0,2\\pi]\\). Then \\(e^{h(t)}=1\\), so \\(h\\) takes values in \\(2\\pi i\\mathbb Z\\); being continuous, it is constant. But \\(h(2\\pi)=g(1)-2\\pi i\\neq g(1)=h(0)\\). So \\(z\\notin G_0(A)\\), and the index group of \\(C(\\mathbb T)\\) is not trivial.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 642,
        "through_line": 643,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.1",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-13::Theorem 8.1",
      "anchor": "oa-fnd-bn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 8.1** (upper semicontinuity of the spectrum). Let \\(A\\) be a unital Banach algebra and \\(U\\subseteq\\mathbb C\\) open. The set \\(E_U=\\{x\\in A:\\sigma_A(x)\\subseteq U\\}\\) is open. More precisely, if \\(\\sigma_A(x_0)\\subseteq U\\), then \\(M=\\sup_{\\lambda\\notin U}\\|R_{x_0}(\\lambda)\\|\\) is finite, and \\(\\sigma_A(x)\\subseteq U\\) whenever \\(M\\|x-x_0\\|<1\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 646,
        "through_line": 654,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-13::Theorem 8.2",
      "anchor": "oa-fnd-bn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 8.2** (continuity of the calculus). Let \\(f\\in H(U)\\). The map \\(x\\mapsto f(x)\\) is continuous on the open set \\(E_U\\), and it is locally Lipschitz: each \\(x_0\\in E_U\\) has \\(\\delta>0\\) and \\(L\\) with \\(\\|f(x)-f(x_0)\\|\\leq L\\|x-x_0\\|\\) for \\(\\|x-x_0\\|<\\delta\\). In particular, for a compact \\(K\\subseteq U\\), the map is continuous on \\(A_K=\\{x:\\sigma_A(x)\\subseteq K\\}\\), which is contained in \\(E_U\\).\n\nThe set \\(A_K\\) need not be open: in a nontrivial algebra, for \\(K=\\{0\\}\\), it contains \\(0\\) but no \\(\\varepsilon1\\) with \\(\\varepsilon\\neq0\\). So continuity on the open set \\(E_U\\) is the stronger statement.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 655,
        "through_line": 668,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.3",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-13::Example 8.3",
      "anchor": "oa-fnd-bn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 8.3** (the spectrum is not continuous). On \\(\\ell^2(\\mathbb Z)\\) with orthonormal basis \\((e_n)\\), a bounded sequence \\(w=(w_n)\\) defines the weighted shift \\(W_we_n=w_ne_{n+1}\\). It maps the basis to orthogonal vectors, so \\(\\|W_w\\|=\\sup_n|w_n|\\). Also \\(W_w^ke_n=w_nw_{n+1}\\cdots w_{n+k-1}e_{n+k}\\), so \\(\\|W_w^k\\|\\) is the supremum of the products of \\(k\\) consecutive weights. For \\(t\\in\\mathbb C\\) let \\(W_t\\) have weights \\(w_0=t\\) and \\(w_n=1\\) for \\(n\\neq0\\). Then \\(\\|W_t-W_0\\|=|t|\\).\n- *\\(\\sigma(W_0)\\) is the closed unit disc.* \\(\\|W_0\\|=1\\). For \\(|\\lambda|<1\\), the vector \\(v=\\sum_{n\\leq0}\\lambda^{-n}e_n\\) lies in \\(\\ell^2(\\mathbb Z)\\), and \\(W_0v=\\sum_{n\\leq-1}\\lambda^{-n}e_{n+1}=\\lambda v\\). So every \\(\\lambda\\) with \\(|\\lambda|<1\\) is an eigenvalue, and the closed spectrum contains the closed disc.\n- *For \\(t\\neq0\\), \\(\\sigma(W_t)\\) lies in the unit circle.* Every product of \\(k\\) consecutive weights contains the weight \\(t\\) at most once, so \\(\\|W_t^k\\|\\leq\\max(1,|t|)\\) and \\(r(W_t)\\leq1\\) by the spectral radius formula (5.1). \\(W_t\\) is invertible, with \\(W_t^{-1}e_{n+1}=w_n^{-1}e_n\\), and the same count gives \\(\\|W_t^{-k}\\|\\leq\\max(1,1/|t|)\\), so \\(r(W_t^{-1})\\leq1\\). By [Proposition 4.2(4)](#oa-fnd-bn-06), \\(\\sigma(W_t)\\) lies in \\(\\{|\\lambda|\\geq1\\}\\), hence in the circle.\n\nSo \\(W_t\\to W_0\\) in norm while every \\(\\sigma(W_t)\\), \\(t\\neq0\\), stays in the circle and \\(\\sigma(W_0)\\) is the whole disc. A small perturbation can make the spectrum much smaller; by Theorem 8.1, it can never make it much larger. Theorem 8.1 cannot be improved to continuity.",
      "proof_locus": {
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        "line": 669,
        "through_line": 674,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 9.1",
      "kind": "definition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-15::Definition 9.1",
      "anchor": "oa-fnd-bn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Definition 9.1.** Let \\(A\\) be an algebra.\n1. A left ideal \\(L\\) is *modular* if some \\(u\\in A\\) satisfies \\(x-xu\\in L\\) for all \\(x\\in A\\); such a \\(u\\) is a *right unit modulo \\(L\\)*. A two-sided ideal \\(I\\) is *modular* if \\(A/I\\) is unital, that is, if some \\(u\\) satisfies \\(x-xu\\in I\\) and \\(x-ux\\in I\\) for all \\(x\\). In a commutative algebra the two notions agree for ideals. Modular ideals are also called *regular*, and \\(u\\) is then called an *identity modulo* the ideal. Every left ideal of a unital algebra is modular, with \\(u=1\\).\n2. A proper left ideal is *maximal* if no proper left ideal strictly contains it. Maximal two-sided ideals are defined in the same way among two-sided ideals.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 677,
        "through_line": 680,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.2",
      "kind": "lemma",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-15::Lemma 9.2",
      "anchor": "oa-fnd-bn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Lemma 9.2** (no norm is needed). Let \\(L\\) be a left ideal of an algebra \\(A\\), with a right unit \\(u\\) modulo \\(L\\).\n1. Every left ideal that contains \\(L\\) is modular, with the same \\(u\\).\n2. \\(L=A\\) if and only if \\(u\\in L\\).\n3. If \\(L\\) is proper, it lies in a maximal left ideal, and that ideal is modular.\n\nThe same holds for two-sided modular ideals, with maximality among two-sided ideals.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 681,
        "through_line": 689,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.3",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-15::Proposition 9.3",
      "anchor": "oa-fnd-bn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 9.3** (modular ideals stay away from the unit). Let \\(A\\) be a Banach algebra, not necessarily commutative, and \\(L\\) a proper modular left ideal with right unit \\(u\\). Then \\(\\|u-\\ell\\|\\geq1\\) for every \\(\\ell\\in L\\). Consequently the closure of \\(L\\) is again a proper modular left ideal, and every maximal modular left ideal is closed. The same holds for two-sided ideals.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 690,
        "through_line": 693,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.4",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-15::Proposition 9.4",
      "anchor": "oa-fnd-bn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 9.4** (quotient algebras). Let \\(I\\) be a closed ideal of a Banach algebra \\(A\\). With the quotient norm \\(\\|x+I\\|=\\inf\\{\\|x+k\\|:k\\in I\\}\\), the algebra \\(A/I\\) is a Banach algebra. If \\(I\\) is proper and modular, with unit \\(u\\) modulo \\(I\\), then \\(u+I\\) is the identity of \\(A/I\\), and \\(\\|u+I\\|\\geq1\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 694,
        "through_line": 697,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.5",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-15::Example 9.5",
      "anchor": "oa-fnd-bn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 9.5** (completeness is needed). In the normed algebra \\(\\mathbb C[z]\\) of [Examples 4.5](#oa-fnd-bn-07) (norm \\(\\max_{|z|\\leq1}|p(z)|\\)), the ideal \\(I=(z-2)\\mathbb C[z]\\) is maximal, since \\(\\mathbb C[z]/I\\cong\\mathbb C\\) through \\(p\\mapsto p(2)\\), and it is modular because \\(\\mathbb C[z]\\) is unital. But it is dense. With \\(p_N=-\\frac12\\sum_{n=0}^N(z/2)^n\\) we get \\((z-2)p_N-1=-(z/2)^{N+1}\\), of norm \\(2^{-N-1}\\). So the distance from \\(1\\) to \\(I\\) is \\(0\\), and this maximal ideal is not closed. The character \\(p\\mapsto p(2)\\) is unbounded, since the polynomials \\(z^n\\) have norm \\(1\\) and value \\(2^n\\). Compare [Proposition 10.3(1)](#oa-fnd-bn-17).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 698,
        "through_line": 699,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.1",
      "kind": "definition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-16::Definition 10.1",
      "anchor": "oa-fnd-bn-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Definition 10.1.** A *character* of an algebra \\(A\\) is a nonzero algebra homomorphism \\(A\\to\\mathbb C\\), and \\(\\operatorname{Ch}(A)\\) is the set of characters. A nonzero homomorphism into \\(\\mathbb C\\) is automatically onto, because its image is a nonzero subspace of \\(\\mathbb C\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 704,
        "through_line": 705,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.2",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-16::Proposition 10.2",
      "anchor": "oa-fnd-bn-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 10.2.**\n1. In a unital commutative algebra, every noninvertible element lies in a maximal ideal. No norm is needed.\n2. Let \\(A\\) be a commutative Banach algebra and \\(\\mathfrak m\\) a maximal modular ideal, with unit \\(u\\) modulo \\(\\mathfrak m\\). Then \\(\\mathfrak m\\) is closed, and \\(A/\\mathfrak m\\) is a field. The map \\(\\lambda\\mapsto\\lambda(u+\\mathfrak m)\\) is the only unital algebra isomorphism of \\(\\mathbb C\\) onto \\(A/\\mathfrak m\\). Let \\(\\omega_{\\mathfrak m}(x)\\) be the number \\(\\lambda\\) with \\(x+\\mathfrak m=\\lambda(u+\\mathfrak m)\\). Then \\(\\omega_{\\mathfrak m}\\) is a character with kernel \\(\\mathfrak m\\).\n3. For a commutative Banach algebra \\(A\\), \\(\\mathfrak m\\mapsto\\omega_{\\mathfrak m}\\) is a bijection from the set \\(\\mathcal M(A)\\) of maximal modular ideals onto \\(\\operatorname{Ch}(A)\\). Its inverse is \\(\\omega\\mapsto\\ker\\omega\\).\n4. In any algebra, two characters with the same kernel are equal, and the kernel of a character is a maximal modular ideal of codimension one.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 706,
        "through_line": 718,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
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          "local_scopes": [
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              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.3",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-17::Proposition 10.3",
      "anchor": "oa-fnd-bn-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 10.3.** Let \\(A\\) be a Banach algebra, not necessarily commutative.\n1. For every character \\(\\omega\\) and every \\(x\\in A\\), \\(\\omega(x)\\in\\sigma'_A(x)\\), and \\(|\\omega(x)|\\leq r(x)\\leq\\|x\\|\\). So every character is continuous, with \\(\\|\\omega\\|\\leq1\\). If \\(A\\) is unital, \\(\\omega(1)=1\\).\n2. With the weak\\* topology, \\(\\operatorname{Ch}(A)\\cup\\{0\\}\\) is compact, and \\(\\operatorname{Ch}(A)\\) is locally compact Hausdorff. If \\(A\\) is unital, \\(\\operatorname{Ch}(A)\\) is compact.\n3. For \\(x\\in A\\), the function \\(\\hat x(\\omega)=\\omega(x)\\) lies in \\(C_0(\\operatorname{Ch}(A))\\).\n4. The characters of \\(A_1\\) are \\(q\\) and the maps \\(\\omega_1(a+\\lambda)=\\omega(a)+\\lambda\\) with \\(\\omega\\in\\operatorname{Ch}(A)\\). The map \\(\\omega\\mapsto\\omega_1\\) is a homeomorphism of \\(\\operatorname{Ch}(A)\\) onto \\(\\operatorname{Ch}(A_1)\\setminus\\{q\\}\\), which is open in the compact space \\(\\operatorname{Ch}(A_1)\\).",
      "proof_locus": {
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        "line": 721,
        "through_line": 733,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 10.4",
      "kind": "examples",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-17::Examples 10.4",
      "anchor": "oa-fnd-bn-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Examples 10.4.**\n- For \\(n\\geq2\\), \\(M_n(\\mathbb C)\\) has no characters. A character vanishes on \\(E_{ij}\\) for \\(i\\neq j\\), since \\(E_{ij}^2=0\\). Then \\(\\omega(E_{ii})=\\omega(E_{ij}E_{ji})=0\\) for every \\(i\\), and \\(\\omega(1)=\\sum_i\\omega(E_{ii})=0\\), which contradicts (1).\n- A nonzero Banach space with the zero product is a commutative Banach algebra without characters, since \\(\\omega(x)^2=\\omega(x^2)=0\\).\n- For normed algebras that are not complete, (1) fails: see Example 9.5.",
      "proof_locus": {
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        "line": 734,
        "through_line": 738,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
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          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
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              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.1",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-18::Theorem 11.1",
      "anchor": "oa-fnd-bn-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 11.1** (Gelfand representation). Let \\(A\\) be a commutative Banach algebra, and define \\(\\mathcal G:A\\to C_0(\\operatorname{Ch}(A))\\) by \\(\\mathcal G(x)=\\hat x\\).\n1. \\(\\mathcal G\\) is an algebra homomorphism, and \\(\\|\\hat x\\|_\\infty=r(x)\\leq\\|x\\|\\).\n2. If \\(A\\) is unital, then \\(\\operatorname{Ch}(A)\\) is compact and \\(\\sigma_A(x)=\\hat x(\\operatorname{Ch}(A))\\).\n3. In general, \\(\\sigma'_A(x)=\\hat x(\\operatorname{Ch}(A))\\cup\\{0\\}\\).",
      "proof_locus": {
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        "line": 741,
        "through_line": 757,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
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          "local_scopes": [
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              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
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              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 11.2",
      "kind": "definition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
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      "anchor": "oa-fnd-bn-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Definition 11.2.** Let \\(A\\) be a commutative Banach algebra. The map \\(\\mathcal G\\) of Theorem 11.1 is the *Gelfand representation*; \\(\\operatorname{Ch}(A)\\) is the *spectrum* (or character space) of \\(A\\), and its members are the *characters*. The kernel of \\(\\mathcal G\\) is the *radical* \\(\\operatorname{rad}(A)\\). The algebra is *semisimple* if \\(\\operatorname{rad}(A)=\\{0\\}\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 760,
        "through_line": 761,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.3",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-19::Proposition 11.3",
      "anchor": "oa-fnd-bn-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 11.3.** Let \\(A\\) be a commutative Banach algebra.\n1. \\(\\operatorname{rad}(A)=\\{x:r(x)=0\\}=\\bigcap_{\\mathfrak m\\in\\mathcal M(A)}\\mathfrak m\\), which is \\(A\\) when \\(\\mathcal M(A)\\) is empty. It is a closed ideal.\n2. If \\(A\\) is semisimple, then \\(\\mathcal G\\) is an injective homomorphism of \\(A\\) onto a subalgebra of \\(C_0(\\operatorname{Ch}(A))\\), with \\(\\|\\hat x\\|_\\infty\\leq\\|x\\|\\). This subalgebra separates the points of \\(\\operatorname{Ch}(A)\\) and vanishes at no point.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 762,
        "through_line": 770,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.4",
      "kind": "example",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-19::Example 11.4",
      "anchor": "oa-fnd-bn-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Example 11.4** (the characters of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be LCH. Every character \\(\\omega\\) of \\(C_0(\\Omega)\\) is an evaluation \\(f\\mapsto f(p)\\), and \\(p\\mapsto\\text{(evaluation at }p)\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\). So \\(\\hat f\\) is \\(f\\) itself, and \\(\\mathcal G\\) is isometric.\n\n*Proof.* First we find a point \\(p\\) at which every \\(f\\in\\ker\\omega\\) vanishes. Suppose, to the contrary, that for every \\(p\\in\\Omega\\) some \\(f_p\\in\\ker\\omega\\) has \\(f_p(p)\\neq0\\). Choose \\(g\\) with \\(\\omega(g)=1\\). The set \\(C=\\{|g|\\geq1/2\\}\\) is compact, and it is not empty, since \\(\\|g\\|_\\infty\\geq|\\omega(g)|=1\\) by [Proposition 10.3(1)](#oa-fnd-bn-17). So finitely many sets \\(\\{f_{p_i}\\neq0\\}\\) cover it, and \\(h=\\sum_i\\bar f_{p_i}f_{p_i}\\) lies in the ideal \\(\\ker\\omega\\) and is positive on \\(C\\). Let \\(m=\\min_Ch>0\\) and \\(s=1/\\max(h,m)\\), a bounded continuous function. Then \\(gs\\in C_0(\\Omega)\\), so \\(ghs=(gs)h\\in\\ker\\omega\\). On \\(C\\), \\(hs=1\\), so \\(g-ghs=0\\); off \\(C\\), \\(0\\leq hs\\leq1\\) gives \\(|g-ghs|\\leq|g|<1/2\\). So \\(\\|g-ghs\\|_\\infty\\leq1/2\\), while \\(\\omega(g-ghs)=1\\). This contradicts the bound \\(|\\omega(x)|\\leq\\|x\\|\\) of Proposition 10.3(1). Hence some \\(p\\) has \\(f(p)=0\\) for all \\(f\\in\\ker\\omega\\).\n\nNext, \\(\\omega\\) is evaluation at this \\(p\\). Evaluation at \\(p\\) is nonzero (Urysohn), and its kernel contains \\(\\ker\\omega\\); both kernels have codimension one, so they are equal, and [Proposition 10.2(4)](#oa-fnd-bn-16) gives \\(\\omega=\\) evaluation at \\(p\\).\n\nFinally, the map is a homeomorphism. Distinct points give distinct evaluations (Urysohn). The map is weak\\* continuous, since \\(p\\mapsto f(p)\\) is continuous for each \\(f\\). Its inverse is continuous too: if evaluations at \\(p_i\\) converge to evaluation at \\(p\\) but \\(p_i\\) stays outside a neighbourhood \\(U\\) of \\(p\\) along a subnet, a Urysohn function \\(f\\) with \\(f(p)=1\\) and support in \\(U\\) gives \\(0=f(p_i)\\to1\\), a contradiction.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 771,
        "through_line": 778,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 11.5",
      "kind": "examples",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-19::Examples 11.5",
      "anchor": "oa-fnd-bn-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Examples 11.5.**\n- *Radicals.* A Banach space with the zero product is its own radical. The dual numbers \\(\\mathbb C[\\varepsilon]\\), with \\(\\varepsilon^2=0\\) and norm \\(|a|+|b|\\) for \\(a+b\\varepsilon\\), are unital with radical \\(\\mathbb C\\varepsilon\\).\n- *The Wiener algebra* ([Section 13](#oa-fnd-bn-21)) is semisimple, and its Gelfand representation is injective but not isometric.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 779,
        "through_line": 782,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [
        {
          "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
          "local_scopes": [
            {
              "heading": "4. The spectrum",
              "line": 212,
              "through_line": 304,
              "anchors": [
                "OA-FND-BN-06",
                "OA-FND-BN-07"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "5. Nonempty spectrum and the spectral radius formula",
              "line": 305,
              "through_line": 419,
              "anchors": [
                "OA-FND-BN-08",
                "OA-FND-BN-09"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "10. Characters",
              "line": 700,
              "through_line": 738,
              "anchors": [
                "OA-FND-BN-16",
                "OA-FND-BN-17"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            },
            {
              "heading": "11. The Gelfand representation",
              "line": 739,
              "through_line": 782,
              "anchors": [
                "OA-FND-BN-18",
                "OA-FND-BN-19"
              ],
              "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
          "role": "proof comparison",
          "correspondence": "Partial textbook treatment",
          "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.1",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-20::Proposition 12.1",
      "anchor": "oa-fnd-bn-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 12.1** (the transformation-group algebra). (12.2) makes \\(K\\) a \\(*\\)-algebra with a submultiplicative norm and an isometric involution. For \\(x,y,z\\in K\\):\n\\[\n\\begin{gathered}\nx\\star y\\in K,\\\\\n(x\\star y)\\star z\\\\\n=x\\star(y\\star z),\\\\\nx^\\sharp\\in K,\\\\\nx^{\\sharp\\sharp}\\\\\n=x,\\\\\n(x\\star y)^\\sharp\\\\\n=y^\\sharp\\star x^\\sharp,\\\\\n\\|x\\star y\\|_1\\\\\n\\leq\\|x\\|_1\\|y\\|_1,\\\\\n\\|x^\\sharp\\|_1\\\\\n=\\|x\\|_1,\n\\end{gathered}\n\\]\nand \\(\\star\\) is bilinear and \\(\\sharp\\) conjugate-linear. So the completion \\(\\mathfrak A(\\Omega,G)\\) of \\(K\\) is an involutive Banach algebra. The isometric identification \\(\\mathfrak A(\\Omega,G)\\cong L^1(G,C_0(\\Omega))\\) and density of \\(K\\) are proved just after the algebra laws.",
      "proof_locus": {
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        "line": 813,
        "through_line": 884,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.3",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-20::Proposition 12.3",
      "anchor": "oa-fnd-bn-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 12.3** (the group algebra). \\(L^1(G)\\), with \\((xy)(t)=\\int x(s)y(s^{-1}t)\\,ds\\) and \\(x^*(t)=\\Delta(t)^{-1}\\overline{x(t^{-1})}\\), is an involutive Banach algebra. It is \\(\\mathfrak A(\\Omega,G)\\) for \\(\\Omega\\) a single point: \\(C_c(G)\\) is dense in \\(L^1(G)\\) ([density of compactly supported functions](haar-measure.md#oa-fnd-hm-02)), \\(L^1(G)\\) is complete, and the \\(L^1\\) convolution extends the product of \\(C_c(G)\\) continuously ([convolution](haar-measure.md#oa-fnd-hm-12)). The algebra laws on all of \\(L^1(G)\\) are also proved directly in the lesson on Haar measure, in the section on [convolution](haar-measure.md#oa-fnd-hm-12).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 885,
        "through_line": 886,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 12.4",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-20::Theorem 12.4",
      "anchor": "oa-fnd-bn-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 12.4** (when the transformation-group algebra has an identity). Let \\(\\Omega\\) be nonempty. Then \\(\\mathfrak A(\\Omega,G)\\) has an identity exactly when \\(\\Omega\\) is compact and \\(G\\) is discrete. In that case the identity is \\(\\varepsilon=c^{-1}1_{\\Omega\\times\\{e\\}}\\), where \\(c=\\mu(\\{e\\})>0\\); moreover \\(\\|\\varepsilon\\|_1=1\\) and \\(\\varepsilon^\\sharp=\\varepsilon\\).\n\nThe hypothesis that \\(\\Omega\\) is nonempty is needed. If \\(\\Omega=\\varnothing\\), then \\(K=\\{0\\}\\) and \\(\\mathfrak A=\\{0\\}\\), and the equivalence fails whichever convention is used for the zero algebra. If \\(\\{0\\}\\) counts as unital, the left side holds for every \\(G\\); if it does not, the left side fails for every \\(G\\). The right side holds exactly when \\(G\\) is discrete.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 887,
        "through_line": 943,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 12.5",
      "kind": "corollary",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-20::Corollary 12.5",
      "anchor": "oa-fnd-bn-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Corollary 12.5** (the identity of \\(L^1(G)\\)). \\(L^1(G)\\) has an identity exactly when \\(G\\) is discrete. The identity is then \\(c^{-1}1_{\\{e\\}}\\) with \\(c=\\mu(\\{e\\})\\). This is Theorem 12.4 with \\(\\Omega\\) a point, together with Proposition 12.3. For counting measure (\\(c=1\\)), the lesson on Haar measure proves the same by a different argument, in the section on [approximate identities](haar-measure.md#oa-fnd-hm-13).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 944,
        "through_line": 945,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 12.6",
      "kind": "theorem",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-20::Theorem 12.6",
      "anchor": "oa-fnd-bn-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Theorem 12.6** (commutativity of \\(L^1(G)\\)). \\(L^1(G)\\) is commutative if and only if \\(G\\) is abelian.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 946,
        "through_line": 956,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 13.1",
      "kind": "lemma",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Lemma 13.1",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Lemma 13.1** (uniqueness of Fourier coefficients). A continuous \\(1\\)-periodic function \\(h\\) with \\(\\hat h(n)=0\\) for all \\(n\\) is zero.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 963,
        "through_line": 974,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.2",
      "kind": "proposition",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Proposition 13.2",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Proposition 13.2.**\n1. For \\(f\\in W\\), \\(f(s)=\\sum_n\\hat f(n)e_n(s)\\), uniformly in \\(s\\), and \\(\\|f\\|_\\infty\\leq\\|f\\|_W\\). \\(W\\) is a commutative unital Banach algebra under pointwise multiplication, with \\(\\widehat{fg}(n)=\\sum_k\\hat f(k)\\hat g(n-k)\\). The map \\(f\\mapsto(\\hat f(n))_n\\) is an isometric algebra isomorphism of \\(W\\) onto \\(\\ell^1(\\mathbb Z)\\) with convolution.\n2. For each \\(t\\in\\mathbb R/\\mathbb Z\\), \\(\\omega_t(f)=f(t)\\) is a character of \\(W\\).\n3. \\(e_1\\) is invertible in \\(W\\), and \\(\\|e_1\\|_W=\\|e_1^{-1}\\|_W=1\\). For every character \\(\\omega\\), \\(|\\omega(e_1)|=1\\), so \\(\\omega(e_1)=e^{2\\pi it}\\) for exactly one \\(t=t_\\omega\\in[0,1)\\).\n4. \\(\\omega(f)=f(t_\\omega)\\) for every \\(f\\in W\\).\n5. \\(t\\mapsto\\omega_t\\) is a homeomorphism of the circle \\(\\mathbb R/\\mathbb Z\\) onto \\(\\operatorname{Ch}(W)\\). The Gelfand transform of \\(f\\) is \\(f\\) itself, seen on the circle. \\(W\\) is semisimple, and its Gelfand representation is not isometric.\n6. (*Wiener's lemma.*) If \\(f\\in W\\) has no zero, then \\(1/f\\in W\\): the Fourier coefficients of \\(1/f\\) are absolutely summable.\n\nPart (5) identifies the character space with the circle. This is different from the closed interval: removing an interior point disconnects the interval, whereas removing any point leaves the circle connected.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 975,
        "through_line": 998,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Exercise 1",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Exercise 1** (medium; the algebra \\(\\ell^1(\\mathbb Z_{\\geq0})\\)). Let \\(A\\) be the space of sequences \\((x_n)_{n\\geq0}\\) with \\(\\|x\\|=\\sum_n|x_n|<\\infty\\), with the product \\((xy)_n=\\sum_{k=0}^nx_ky_{n-k}\\). Let \\(\\delta_m\\) be the sequence with \\(1\\) in place \\(m\\) and \\(0\\) elsewhere. Show that \\(\\operatorname{Ch}(A)\\) is homeomorphic to the closed unit disc \\(\\bar{\\mathbb D}\\), with Gelfand transform \\(\\hat x(z)=\\sum_nx_nz^n\\); that \\(A\\) is semisimple; and that \\(\\sigma_A(\\delta_1)=\\bar{\\mathbb D}\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 1001,
        "through_line": 1004,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Exercise 2",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Exercise 2** (medium; spectra of commuting elements). Let \\(A\\) be a unital Banach algebra and \\(x,y\\in A\\) with \\(xy=yx\\). Show that \\(\\sigma_A(x+y)\\subseteq\\sigma_A(x)+\\sigma_A(y)\\) and \\(\\sigma_A(xy)\\subseteq\\sigma_A(x)\\sigma_A(y)\\), and that commutativity cannot be dropped.",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 1005,
        "through_line": 1016,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Exercise 3",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Exercise 3** (medium; polynomial identities). Let \\(P(\\lambda)=(\\lambda-\\alpha_1)\\cdots(\\lambda-\\alpha_n)\\) with distinct \\(\\alpha_i\\), and let \\(x\\) be an element of a nontrivial unital Banach algebra with \\(P(x)=0\\). Show that \\(\\sigma_A(x)\\subseteq\\{\\alpha_1,\\dots,\\alpha_n\\}\\), and that \\(f(x)=Q(x)\\) for every \\(f\\) holomorphic near these points, where \\(Q\\) is the polynomial of degree less than \\(n\\) with \\(Q(\\alpha_i)=f(\\alpha_i)\\). Compute the idempotent \\(e\\) of [Example 6.12](#oa-fnd-bn-22).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 1017,
        "through_line": 1020,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Exercise 4",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Exercise 4** (hard; the index group has no torsion). Let \\(A\\) be a commutative nontrivial unital Banach algebra. Show that if \\(x\\in G(A)\\) and \\(x^n\\in G_0(A)\\) for some \\(n\\geq1\\), then \\(x\\in G_0(A)\\).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 1021,
        "through_line": 1024,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5",
      "kind": "exercise",
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory#oa-fnd-bn-21::Exercise 5",
      "anchor": "oa-fnd-bn-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory@C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4",
      "statement_and_full_conditions": "**Exercise 5** (easy; the unitization of a unital C\\*-algebra). Let \\(A\\) be a C\\*-algebra with identity \\(1_A\\), and let \\(N(a+\\lambda)=\\max\\big(p(a+\\lambda),|\\lambda|\\big)\\) on \\(A_1\\), as in [Proposition 3.4](#oa-fnd-bn-05). Show that \\(p(a+\\lambda)=\\|a+\\lambda1_A\\|\\), and that \\((a,\\lambda)\\mapsto(a+\\lambda1_A,\\lambda)\\) is an isometric \\(*\\)-isomorphism of \\((A_1,N)\\) onto \\(A\\oplus\\mathbb C\\) with the norm \\(\\max(\\|b\\|,|\\mu|)\\). Conclude that \\((A_1,N)\\) is a C\\*-algebra, without using the inequality of Proposition 3.4(2).",
      "proof_locus": {
        "source": "src/banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md",
        "line": 1025,
        "through_line": 1028,
        "sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.1",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-01::Definition 1.1",
      "anchor": "oa-fnd-cf-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 1.1.** In an involutive Banach algebra \\(A\\), call \\(x\\) *self-adjoint* (or *hermitian*) if \\(x^*=x\\), and *normal* if \\(x^*x=xx^*\\). If \\(A\\) is unital, \\(x\\) is *unitary* if \\(x^*x=xx^*=1\\). A *projection* is an element \\(p\\) with \\(p=p^*=p^2\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 35,
        "through_line": 36,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.2",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-01::Proposition 1.2",
      "anchor": "oa-fnd-cf-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 1.2.**\n1. (*Cartesian decomposition.*) Every \\(x\\in A\\) can be written in exactly one way as \\(x=x_1+ix_2\\) with \\(x_1,x_2\\in A_h\\), namely \\(x_1=\\frac12(x+x^*)\\) and \\(x_2=\\frac1{2i}(x-x^*)\\). Moreover \\(\\|x_1\\|\\leq\\|x\\|\\) and \\(\\|x_2\\|\\leq\\|x\\|\\). The set \\(A_h\\) is a closed real subspace, and \\(A=A_h\\oplus iA_h\\) as real Banach spaces.\n2. \\(x\\) is normal exactly when \\(x_1\\) and \\(x_2\\) commute. Self-adjoint and unitary elements are normal.\n3. If \\(A\\) is unital, then \\(1\\in A_h\\), and \\(U(A)\\) is a closed subgroup of \\(G(A)\\) with \\(u^{-1}=u^*\\in U(A)\\). If \\(A\\) is a nontrivial unital C\\*-algebra, then \\(\\|u\\|=1\\) for every unitary \\(u\\).\n4. In a C\\*-algebra, every projection has norm \\(0\\) or \\(1\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 37,
        "through_line": 58,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 1.3",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-01::Theorem 1.3",
      "anchor": "oa-fnd-cf-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 1.3** (The norm of a normal element is its spectral radius). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\), \\(\\|h^{2^k}\\|=\\|h\\|^{2^k}\\) for all \\(k\\geq0\\), and \\(r(h)=\\|h\\|\\).\n2. For every normal \\(x\\in A\\), \\(r(x)=\\|x\\|\\).\n3. For every \\(x\\in A\\), \\(\\|x\\|^2=\\|x^*x\\|=r(x^*x)\\).\n4. (*The norm is determined by the algebra.*) If an algebra with involution is a C\\*-algebra for two norms, the two norms are equal.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 59,
        "through_line": 69,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 1.4",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-01::Example 1.4",
      "anchor": "oa-fnd-cf-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 1.4** (Normality is needed). In \\(M_2(\\mathbb C)\\), \\(E_{12}^2=0\\), so \\(r(E_{12})=0\\), while \\(\\|E_{12}\\|=1\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 70,
        "through_line": 71,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.5",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-01::Proposition 1.5",
      "anchor": "oa-fnd-cf-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 1.5** (Spectra of adjoints, unitaries and self-adjoint elements). Let \\(A\\) be a C\\*-algebra.\n1. For every \\(x\\in A\\), \\(\\sigma'_A(x^*)=\\{\\bar\\lambda:\\lambda\\in\\sigma'_A(x)\\}\\).\n2. If \\(A\\) is unital and \\(u\\in U(A)\\), then \\(\\sigma_A(u)\\subseteq\\mathbb T=\\{\\lambda:|\\lambda|=1\\}\\).\n3. For \\(h\\in A_h\\), \\(\\sigma'_A(h)\\subseteq[-\\|h\\|,\\|h\\|]\\), and \\(\\sigma'_A(h)\\) contains \\(\\|h\\|\\) or \\(-\\|h\\|\\). If \\(A\\) is unital and nontrivial, the same holds for \\(\\sigma_A(h)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 72,
        "through_line": 84,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1.6",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-01::Exercise 1.6",
      "anchor": "oa-fnd-cf-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 1.6** (easy; Powers of normal elements). Show that \\(\\|x^n\\|=\\|x\\|^n\\) for normal \\(x\\) and all \\(n\\geq1\\). Show by examples that the equality can fail for nonnormal \\(x\\), and that it can hold for all \\(n\\) without \\(x\\) being normal.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 85,
        "through_line": 88,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.1",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-04::Theorem 2.1",
      "anchor": "oa-fnd-cf-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 2.1** (Commutative Gelfand–Naimark theorem). For an abelian C\\*-algebra \\(A\\), put \\(\\Omega=\\operatorname{Ch}(A)\\).\n1. Every character of \\(A\\) is a \\(*\\)-homomorphism: \\(\\omega(x^*)=\\overline{\\omega(x)}\\).\n2. The Gelfand representation \\(\\mathcal G(x)=\\hat x\\) is an isometric \\(*\\)-isomorphism of \\(A\\) onto \\(C_0(\\Omega)\\). In particular \\(A\\) is semisimple.\n3. If \\(A\\) is unital, \\(\\Omega\\) is compact and \\(A\\cong C(\\Omega)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 91,
        "through_line": 100,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.2",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-04::Proposition 2.2",
      "anchor": "oa-fnd-cf-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 2.2** (The characters of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be an LCH space, and let \\(\\operatorname{ev}_p(f)=f(p)\\) for \\(p\\in\\Omega\\) and \\(f\\in C_0(\\Omega)\\).\n1. \\(p\\mapsto\\operatorname{ev}_p\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\operatorname{Ch}(C_0(\\Omega))\\) with the weak\\* topology. Under it, the Gelfand transform of \\(f\\) is \\(f\\) itself.\n2. Two LCH spaces \\(\\Omega\\) and \\(\\Omega'\\) are homeomorphic if and only if \\(C_0(\\Omega)\\) and \\(C_0(\\Omega')\\) are isomorphic as algebras; a \\(*\\)-isomorphism is not needed.\n3. Every abelian C\\*-algebra \\(A\\) is \\(*\\)-isomorphic to \\(C_0(\\Omega)\\) for an LCH space \\(\\Omega\\), unique up to homeomorphism; one choice is \\(\\Omega=\\operatorname{Ch}(A)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 101,
        "through_line": 117,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2.4",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-04::Exercise 2.4",
      "anchor": "oa-fnd-cf-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 2.4** (hard; The Stone–Čech compactification). Let \\(\\Gamma\\) be a completely regular Hausdorff space and \\(A=C_b(\\Gamma)\\) with the supremum norm. For \\(\\gamma\\in\\Gamma\\) let \\(\\omega_\\gamma(x)=x(\\gamma)\\), and let \\(\\iota:\\Gamma\\to\\operatorname{Ch}(A)\\), \\(\\iota(\\gamma)=\\omega_\\gamma\\).\n(a) \\(A\\) is a C\\*-algebra with identity.\n(b) \\(\\iota\\) is a homeomorphism of \\(\\Gamma\\) onto a dense subset of \\(\\operatorname{Ch}(A)\\).\n(c) \\(\\iota(\\Gamma)\\) is open in \\(\\operatorname{Ch}(A)\\) if and only if \\(\\Gamma\\) is locally compact.\n(d) Every continuous map \\(f\\) of \\(\\Gamma\\) into a compact Hausdorff space \\(K\\) is \\(g\\circ\\iota\\) for exactly one continuous \\(g:\\operatorname{Ch}(A)\\to K\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 118,
        "through_line": 130,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2.5",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-04::Exercise 2.5",
      "anchor": "oa-fnd-cf-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 2.5** (medium; Separability). Let \\(A\\) be an abelian C\\*-algebra with character space \\(\\Omega\\). Show that \\(A\\) has a countable dense subset exactly when \\(\\Omega\\) has a countable base.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 131,
        "through_line": 134,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.1",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-06::Definition 3.1",
      "anchor": "oa-fnd-cf-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 3.1.** In a C\\*-algebra \\(A\\), a *C\\*-subalgebra* is a norm-closed subalgebra \\(B\\) with \\(B^*=B\\); with the inherited structure, \\(B\\) is itself a C\\*-algebra. If \\(A\\) is unital, \\(B\\) is a *unital C\\*-subalgebra* when \\(1_A\\in B\\). Intersections of C\\*-subalgebras are C\\*-subalgebras. So every \\(E\\subseteq A\\) lies in a smallest C\\*-subalgebra \\(C^*(E)\\), the C\\*-subalgebra *generated* by \\(E\\); for unital \\(A\\), \\(C^*(1,E)\\) is the smallest unital one. \\(C^*(E)\\) is the closure of the set of noncommutative polynomials without constant term in the elements of \\(E\\cup E^*\\). If \\(x\\) is normal, \\(C^*(x)\\) and \\(C^*(1,x)\\) are commutative, because \\(x\\), \\(x^*\\) and \\(1\\) commute and closures of commutative algebras are commutative.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 137,
        "through_line": 138,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.2",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-06::Theorem 3.2",
      "anchor": "oa-fnd-cf-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 3.2** (Spectral permanence). Let \\(A\\) be a C\\*-algebra, \\(B\\subseteq A\\) a C\\*-subalgebra, and \\(x\\in B\\).\n1. If \\(A\\) is unital and \\(1_A\\in B\\), then \\(\\sigma_B(x)=\\sigma_A(x)\\). Equivalently, an element of \\(B\\) that is invertible in \\(A\\) has its inverse in \\(B\\).\n2. \\(\\sigma'_B(x)=\\sigma'_A(x)\\).\n3. If \\(A\\) and \\(B\\) are both unital, possibly with different identities, then \\(\\sigma_B(x)\\cup\\{0\\}=\\sigma_A(x)\\cup\\{0\\}\\).\n\nExample 3.3 explains why spectra in algebras with different identities must be compared after adjoining zero, as in (3).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 139,
        "through_line": 154,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.3",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-06::Example 3.3",
      "anchor": "oa-fnd-cf-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 3.3** (Why an identity is adjoined to every algebra). Part (2) needs the quasi-spectrum of a unital algebra to be computed after adjoining a new identity, as in the Conventions. If instead \\(\\sigma'_B=\\sigma_B\\) for unital \\(B\\), then (2) fails. Take \\(A=c_0\\), the null sequences, which is not unital, \\(B=\\mathbb Ce_1\\), which is unital with identity \\(e_1\\), and \\(x=e_1\\). Then \\(\\sigma'_A(e_1)=\\{0,1\\}\\) but \\(\\sigma_B(e_1)=\\{1\\}\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 155,
        "through_line": 156,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.4",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-06::Example 3.4",
      "anchor": "oa-fnd-cf-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 3.4** (Closure under the involution is needed). Let \\(A=C(\\mathbb T)\\), and let \\(B\\) be the closure in \\(A\\) of the polynomials in \\(z\\). It is a closed unital subalgebra, but not \\(*\\)-closed. By the maximum modulus principle (Theorem 3.6 and Exercise 4 of [Cauchy's theorem for cycles and its consequences](cauchy-s-theorem-for-cycles-and-its-consequences.md)), a sequence of polynomials that converges uniformly on \\(\\mathbb T\\) converges uniformly on the closed disc \\(\\bar{\\mathbb D}\\); so each \\(g\\in B\\) extends to a function \\(G\\) continuous on \\(\\bar{\\mathbb D}\\) and holomorphic inside. If \\(|\\lambda|<1\\) and \\((z-\\lambda)g=1\\) on \\(\\mathbb T\\), then \\((w-\\lambda)G(w)-1\\) is holomorphic in \\(\\mathbb D\\), continuous on \\(\\bar{\\mathbb D}\\) and zero on \\(\\mathbb T\\), hence zero; at \\(w=\\lambda\\) this says \\(-1=0\\). So \\(\\sigma_B(z)\\) contains the open disc, and \\(\\sigma_B(z)=\\bar{\\mathbb D}\\), while \\(\\sigma_A(z)=\\mathbb T\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 157,
        "through_line": 158,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Definition 4.1",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 4.1.** A *\\(*\\)-homomorphism* between algebras with involution is an algebra homomorphism \\(\\pi\\) with \\(\\pi(x^*)=\\pi(x)^*\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 161,
        "through_line": 162,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.2",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Theorem 4.2",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 4.2** (\\(*\\)-homomorphisms are contractive). Let \\(A\\) be a Banach algebra with an involution that is not assumed to be continuous, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) a \\(*\\)-homomorphism. Then for every \\(x\\in A\\)\n\\[\n\\begin{gathered}\n\\sigma'_B(\\pi(x))\\\\\n\\subseteq\\sigma'_A(x),\\\\\n\\|\\pi(x)\\|^2\\\\\n\\leq r_A(x^*x)\\\\\n\\leq\\|x^*x\\|\\\\\n\\leq\\|x^*\\|\\,\\|x\\| .\n\\end{gathered}\n\\tag{4.1}\n\\]\nConsequently:\n1. if \\(\\|x^*\\|\\leq C\\|x\\|\\) for all \\(x\\), then \\(\\|\\pi(x)\\|\\leq C^{1/2}\\|x\\|\\);\n2. if the involution is isometric, then \\(\\|\\pi(x)\\|\\leq\\|x\\|\\);\n3. in particular every \\(*\\)-homomorphism between C\\*-algebras is contractive, whatever is assumed about its continuity.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 163,
        "through_line": 191,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.3",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Example 4.3",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 4.3** (An isometric involution is needed for contractivity). Let \\(t>1\\), \\(S=\\operatorname{diag}(1,t)\\in M_2(\\mathbb C)\\), and \\(\\|x\\|_S=\\|SxS^{-1}\\|\\) (operator norm). This is a complete algebra norm on \\(M_2(\\mathbb C)\\), and the usual adjoint is an involution for it, continuous but not isometric. The identity map \\(\\pi\\) from \\((M_2(\\mathbb C),\\|\\cdot\\|_S)\\) to the C\\*-algebra \\(M_2(\\mathbb C)\\) is a \\(*\\)-homomorphism. Since \\(SE_{12}S^{-1}=t^{-1}E_{12}\\) and \\(SE_{21}S^{-1}=tE_{21}\\), we get \\(\\|\\pi(E_{12})\\|=1>t^{-1}=\\|E_{12}\\|_S\\). The bound (4.1) is sharp here: \\(\\|E_{12}^*\\|_S\\|E_{12}\\|_S=t\\cdot t^{-1}=1\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 192,
        "through_line": 193,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.4",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Theorem 4.4",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 4.4** (Injective homomorphisms do not shrink normal elements). Let \\(A\\) be a C\\*-algebra, \\(B\\) a Banach algebra and \\(\\pi:A\\to B\\) an injective algebra homomorphism. Neither continuity nor compatibility with an involution is assumed.\n1. \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\in A\\).\n2. If \\(B\\) has an involution and \\(\\pi\\) is a \\(*\\)-homomorphism, then \\(\\|x\\|^2\\leq\\|\\pi(x)^*\\|\\,\\|\\pi(x)\\|\\) for all \\(x\\). If moreover the involution of \\(B\\) is isometric, then \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for all \\(x\\).\n3. If, in (2), \\(\\pi(A)\\) is closed, the last inequality also follows from Theorem 4.2 applied to \\(\\pi^{-1}:\\pi(A)\\to A\\).\n\nThe proof below uses the closure of \\(\\pi(C^*(h))\\), so it makes no continuity assumption on \\(\\pi\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 194,
        "through_line": 224,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
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          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.5",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Example 4.5",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 4.5** (Normality is needed in (1)). With \\(S=\\operatorname{diag}(1,t)\\), \\(t>1\\), the map \\(\\pi(x)=SxS^{-1}\\) is an algebra automorphism of the C\\*-algebra \\(M_2(\\mathbb C)\\), not \\(*\\)-preserving. It satisfies \\(\\|\\pi(E_{12})\\|=t^{-1}<1=\\|E_{12}\\|\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 225,
        "through_line": 226,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.6",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Corollary 4.6",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 4.6.** An injective \\(*\\)-homomorphism from a C\\*-algebra into a C\\*-algebra is isometric. Its range is closed.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 227,
        "through_line": 230,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.7",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Theorem 4.7",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 4.7** (Automatic continuity). Let \\(A\\) be a Banach algebra, \\(B\\) a C\\*-algebra, and \\(\\pi:A\\to B\\) an injective algebra homomorphism whose range is self-adjoint: \\(\\pi(A)^*=\\pi(A)\\). Then \\(\\pi\\) is continuous.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 231,
        "through_line": 248,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
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              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
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                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.8",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Corollary 4.8",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 4.8.** Let \\(\\pi\\) be an algebra isomorphism of a C\\*-algebra \\(A\\) onto a C\\*-algebra \\(B\\), not assumed to preserve the involution. Then \\(\\pi\\) and \\(\\pi^{-1}\\) are continuous: \\(c\\|x\\|\\leq\\|\\pi(x)\\|\\leq C\\|x\\|\\) with constants \\(c,C>0\\). Moreover \\(\\|\\pi(x)\\|\\geq\\|x\\|\\) for every normal \\(x\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 249,
        "through_line": 252,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
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              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.9",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-12::Example 4.9",
      "anchor": "oa-fnd-cf-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 4.9** (A self-adjoint range is needed). Let \\(E=\\ell^2(\\mathbb N)\\) with the zero product; it is a commutative Banach algebra. Choose a discontinuous linear functional \\(\\varphi\\) on \\(E\\). One exists by the axiom of choice: extend the linearly independent unit vectors \\(e_1,e_2,\\dots\\) to a Hamel basis, and put \\(\\varphi(e_n)=n\\) and \\(\\varphi=0\\) on the other basis vectors. Let \\(D(\\xi)\\) be the diagonal operator on \\(\\ell^2(\\mathbb N)\\) with entries \\(\\xi_n\\); it is bounded, and \\(\\xi\\mapsto D(\\xi)\\) is injective. Put \\(T(\\xi)=D(\\xi)+\\varphi(\\xi)E_{12}\\), where \\(E_{12}e_2=e_1\\), and on \\(\\ell^2\\oplus\\ell^2\\) put\n\\[\n\\pi(\\xi)=\\begin{pmatrix}0&T(\\xi)\\\\0&0\\end{pmatrix}.\n\\]\nThen \\(\\pi(\\xi)\\pi(\\eta)=0=\\pi(\\xi\\eta)\\), so \\(\\pi\\) is a homomorphism into \\(B(\\ell^2\\oplus\\ell^2)\\). It is injective, since the diagonal of \\(T(\\xi)\\) is \\(\\xi\\). It is not continuous, since the matrix entry \\(\\langle T(\\xi)e_2,e_1\\rangle=\\varphi(\\xi)\\) is not. Its range consists of nonzero nilpotents and zero, and is not self-adjoint.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 253,
        "through_line": 258,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
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          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
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              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-07::Theorem 5.1",
      "anchor": "oa-fnd-cf-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 5.1** (The continuous functional calculus). Let \\(A\\) be a unital C\\*-algebra, \\(x\\in A\\) normal, \\(S=\\sigma_A(x)\\), and \\(\\iota\\in C(S)\\) the function \\(\\iota(\\lambda)=\\lambda\\).\n1. There is exactly one unital \\(*\\)-homomorphism \\(\\Phi_x:C(S)\\to A\\) with \\(\\Phi_x(\\iota)=x\\). It is an isometric \\(*\\)-isomorphism of \\(C(S)\\) onto \\(C^*(1,x)\\). We write \\(f(x)=\\Phi_x(f)\\).\n2. For \\(f,g\\in C(S)\\) and scalars \\(\\alpha,\\beta\\): \\((\\alpha f+\\beta g)(x)=\\alpha f(x)+\\beta g(x)\\), \\((fg)(x)=f(x)g(x)\\), \\(\\bar f(x)=f(x)^*\\), \\(1(x)=1\\), and \\(p(x)=\\sum c_{jk}x^j(x^*)^k\\) for \\(p(\\lambda)=\\sum c_{jk}\\lambda^j\\bar\\lambda^k\\).\n3. (*Spectral mapping.*) \\(\\sigma_A(f(x))=f(S)\\), and \\(f(x)\\) is normal.\n4. \\(\\|f(x)\\|=\\|f\\|_S\\).\n5. (*Composition.*) If \\(g\\in C(f(S))\\), then \\((g\\circ f)(x)=g(f(x))\\).\n6. \\(f(x)\\) commutes with every \\(y\\in A\\) that commutes with \\(x\\) and \\(x^*\\).\n7. (*Independence of the algebra.*) If \\(B\\) is a unital C\\*-subalgebra of \\(A\\) containing \\(x\\), the calculus of \\(x\\) in \\(B\\) is the same map.\n8. (*Characters.*) If \\(D\\) is a commutative unital C\\*-subalgebra containing \\(x\\) and \\(\\chi\\in\\operatorname{Ch}(D)\\), then \\(\\chi(f(x))=f(\\chi(x))\\).\n9. (*Real form.*) If \\(f\\) is real, \\(f(x)\\) is self-adjoint. For \\(x\\in A_h\\), \\(f\\mapsto f(x)\\) maps \\(C(S;\\mathbb R)\\) isometrically onto the self-adjoint part of \\(C^*(1,x)\\).\n10. (*Agreement with the holomorphic calculus.*) If \\(F\\) is holomorphic on an open set \\(U\\supseteq S\\), the element \\(F(x)\\) given by the [holomorphic functional calculus](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-10) equals \\(\\Phi_x(F|_S)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 261,
        "through_line": 296,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
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          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.3",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-31::Theorem 5.3",
      "anchor": "oa-fnd-cf-31",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 5.3** (The calculus without an identity). Let \\(A\\) be a C\\*-algebra, \\(x\\in A\\) normal and \\(S'=\\sigma'_A(x)\\), a compact set containing \\(0\\). For \\(f\\in C(S')\\), let \\(f(x)\\in\\widetilde A\\) be given by Theorem 5.1 in \\(\\widetilde A\\). Identify \\(C_0(S'\\setminus\\{0\\})\\) with \\(\\{f\\in C(S'):f(0)=0\\}\\).\n1. \\(q(f(x))=f(0)\\). So \\(f(x)\\in A\\) if and only if \\(f(0)=0\\).\n2. \\(f\\mapsto f(x)\\) is an isometric \\(*\\)-isomorphism of \\(C_0(S'\\setminus\\{0\\})\\) onto \\(C^*(x)\\).\n3. If \\(A\\) is unital and \\(f(0)=0\\), then \\(f(x)\\) equals the unital calculus \\((f|_{\\sigma_A(x)})(x)\\) of Theorem 5.1 in \\(A\\). So the two calculi never disagree.\n4. If \\(D\\subseteq A\\) is a commutative C\\*-subalgebra containing \\(x\\), \\(\\chi\\in\\operatorname{Ch}(D)\\) and \\(f(0)=0\\), then \\(\\chi(f(x))=f(\\chi(x))\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 328,
        "through_line": 338,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.4",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-31::Corollary 5.4",
      "anchor": "oa-fnd-cf-31",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 5.4** (The calculus commutes with \\(*\\)-homomorphisms). Let \\(\\pi:A\\to B\\) be a \\(*\\)-homomorphism of C\\*-algebras, and \\(a\\in A\\) normal. Let \\(\\tilde\\pi:\\widetilde A\\to\\widetilde B\\), \\(\\tilde\\pi(x+\\lambda)=\\pi(x)+\\lambda\\), be its unital extension. In each statement, \\(f(\\pi(a))\\) is the calculus of the restriction of \\(f\\) to the smaller spectrum.\n1. \\(\\sigma'_B(\\pi(a))\\subseteq\\sigma'_A(a)\\).\n2. \\(\\tilde\\pi(f(a))=f(\\pi(a))\\) for every \\(f\\in C(\\sigma'_A(a))\\).\n3. \\(\\pi(f(a))=f(\\pi(a))\\) for every \\(f\\in C(\\sigma'_A(a))\\) with \\(f(0)=0\\).\n4. If \\(A\\) and \\(B\\) are unital and \\(\\pi(1)=1\\), then \\(\\sigma_B(\\pi(a))\\subseteq\\sigma_A(a)\\), and \\(\\pi(f(a))=f(\\pi(a))\\) for every \\(f\\in C(\\sigma_A(a))\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 339,
        "through_line": 346,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
              "line": 159,
              "through_line": 258,
              "anchors": [
                "OA-FND-CF-12",
                "OA-FND-CF-13",
                "OA-FND-CF-14"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "5. The continuous functional calculus",
              "line": 259,
              "through_line": 346,
              "anchors": [
                "OA-FND-CF-07",
                "OA-FND-CF-08",
                "OA-FND-CF-31"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
          "role": "proof comparison",
          "correspondence": "Classical C*-algebra mechanisms",
          "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-11::Theorem 6.1",
      "anchor": "oa-fnd-cf-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 6.1** (Continuity of the calculus).\n1. (*Uniform continuity.*) For \\(f\\in C(K)\\), the map \\(x\\mapsto f(x)\\) is uniformly continuous on \\(A_K\\).\n2. (*Joint continuity.*) For \\(f,g\\in C(K)\\) and \\(x,y\\in A_K\\), \\[\n\\begin{gathered}\n\\|f(x)-g(y)\\|\\\\\n\\leq\\|f-g\\|_K+\\|g(x)-g(y)\\|.\n\\end{gathered}\n\\]\n3. (*Spectra move little.*) If \\(x_0\\in A\\) is normal and \\(y\\in A\\) is arbitrary, then every point of \\(\\sigma'_A(y)\\) lies within \\(\\|y-x_0\\|\\) of \\(\\sigma'_A(x_0)\\).\n4. (*Open sets.*) Let \\(U\\subseteq\\mathbb C\\) be open and \\(f\\in C(U)\\). Then \\(x\\mapsto f(x)\\) is continuous on the set of normal \\(x\\) with \\(\\sigma'_A(x)\\subseteq U\\), which is relatively open in the set of normal elements.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 351,
        "through_line": 374,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.1",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-09::Definition 7.1",
      "anchor": "oa-fnd-cf-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 7.1.** For \\(h\\in A_h\\) put \\(|h|=(h^2)^{1/2}\\), \\(h_+=\\frac12(|h|+h)\\) and \\(h_-=\\frac12(|h|-h)\\). By the composition rule (Theorem 5.1(5)), these are the calculi of \\(t\\mapsto|t|\\), \\(t\\mapsto\\max(t,0)\\) and \\(t\\mapsto\\max(-t,0)\\) at \\(h\\). They vanish at \\(0\\), so they lie in \\(C^*(h)\\subseteq A\\) (Theorem 5.3). They are the *absolute value*, the *positive part* and the *negative part* of \\(h\\), and \\(h=h_+-h_-\\) is its *Jordan decomposition*.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 377,
        "through_line": 378,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.2",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-09::Proposition 7.2",
      "anchor": "oa-fnd-cf-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 7.2.** Let \\(h\\in A_h\\).\n1. \\(h=h_+-h_-\\), \\(|h|=h_++h_-\\), \\(h_+h_-=h_-h_+=0\\), and the quasi-spectra of \\(h_+\\), \\(h_-\\) and \\(|h|\\) lie in \\([0,\\infty)\\). Also \\(\\|h_\\pm\\|\\leq\\|h\\|=\\||h|\\|=\\max(\\|h_+\\|,\\|h_-\\|)\\).\n2. (*Uniqueness.*) If \\(h=a-b\\) with \\(a,b\\in A_h\\), \\(\\sigma'(a)\\cup\\sigma'(b)\\subseteq[0,\\infty)\\) and \\(ab=0\\), then \\(a=h_+\\) and \\(b=h_-\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 379,
        "through_line": 387,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.3",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-09::Proposition 7.3",
      "anchor": "oa-fnd-cf-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 7.3** (Unitaries span a unital C\\*-algebra). Let \\(A\\) be a C\\*-algebra with identity.\n1. If \\(h\\in A_h\\) and \\(\\|h\\|\\leq1\\), then \\(u=h+i(1-h^2)^{1/2}\\) is unitary and \\(h=\\frac12(u+u^*)\\).\n2. If \\(\\|x\\|\\leq1\\), then \\(x=\\frac12(u_1+u_2)+\\frac i2(u_3+u_4)\\) with unitaries \\(u_j\\). Every \\(x\\) is a combination \\(\\sum_{j=1}^4c_ju_j\\) of four unitaries with \\(\\sum_j|c_j|\\leq2\\|x\\|\\).\n3. If \\(A\\) is not unital, every element of \\(A\\) is such a combination of unitaries of \\(\\widetilde A\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 388,
        "through_line": 396,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.4",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-09::Exercise 7.4",
      "anchor": "oa-fnd-cf-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 7.4** (medium; Unitaries with a gap in the spectrum). In a C\\*-algebra \\(A\\) with identity, let \\(u\\in U(A)\\) with \\(\\sigma(u)\\neq\\mathbb T\\). Prove that \\(u=\\exp(ih)\\) for some \\(h\\in A_h\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 397,
        "through_line": 400,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.5",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-09::Exercise 7.5",
      "anchor": "oa-fnd-cf-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 7.5** (medium; Not every unitary is an exponential). In \\(C(\\mathbb T)\\), the unitary \\(u(\\lambda)=\\lambda\\) is not \\(\\exp(ih)\\) for any \\(h\\in C(\\mathbb T)\\), self-adjoint or not.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 401,
        "through_line": 404,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.6",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-09::Exercise 7.6",
      "anchor": "oa-fnd-cf-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 7.6** (medium; Self-adjointness through the norm). In a nontrivial C\\*-algebra \\(A\\) with identity, prove that \\(x\\in A\\) is self-adjoint exactly when \\(\\lim_{t\\to0}\\frac1t(\\|1+itx\\|-1)=0\\). (In the zero algebra \\(\\|1+itx\\|=0\\), so the quotient is \\(-1/t\\) and the statement fails.)",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 405,
        "through_line": 415,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-15::Lemma 8.1",
      "anchor": "oa-fnd-cf-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Lemma 8.1.** Let \\(h\\in A_h\\) and \\(t\\geq\\|h\\|\\). Then \\(\\sigma'_A(h)\\subseteq[0,\\infty)\\) if and only if \\(\\|t-h\\|\\leq t\\), the norm taken in \\(\\widetilde A\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 418,
        "through_line": 427,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "8. The positive cone and the order",
              "line": 416,
              "through_line": 483,
              "anchors": [
                "OA-FND-CF-15",
                "OA-FND-CF-16"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "9. Operator monotone powers and a Hölder estimate",
              "line": 484,
              "through_line": 615,
              "anchors": [
                "OA-FND-CF-17"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-15::Theorem 8.2",
      "anchor": "oa-fnd-cf-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 8.2** (The positive cone). For \\(h\\in A_h\\) the following are equivalent:\n- (i) \\(\\sigma'_A(h)\\subseteq[0,\\infty)\\);\n- (ii) \\(h=y^*y\\) for some \\(y\\in A\\);\n- (iii) \\(h=k^2\\) for some \\(k\\in A_h\\).\n\nThese elements form a closed convex cone \\(A_+\\subseteq A_h\\), and \\(A_+\\cap(-A_+)=\\{0\\}\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 428,
        "through_line": 450,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "8. The positive cone and the order",
              "line": 416,
              "through_line": 483,
              "anchors": [
                "OA-FND-CF-15",
                "OA-FND-CF-16"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            },
            {
              "heading": "9. Operator monotone powers and a Hölder estimate",
              "line": 484,
              "through_line": 615,
              "anchors": [
                "OA-FND-CF-17"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 8.3",
      "kind": "definition",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-15::Definition 8.3",
      "anchor": "oa-fnd-cf-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 8.3.** An element of \\(A_+\\) is *positive*, written \\(h\\geq0\\). For \\(h,k\\in A_h\\), \\(h\\leq k\\) means \\(k-h\\in A_+\\); by the theorem this is a partial order on \\(A_h\\), compatible with sums and with multiplication by nonnegative scalars. The *absolute value* of \\(x\\in A\\) is \\(|x|=(x^*x)^{1/2}\\). For \\(h\\in A_h\\) it agrees with Definition 7.1, by the composition rule (Theorem 5.1(5)).\n\n**Remark 8.4** (Where positivity is computed). Condition (i) uses the quasi-spectrum, which is the same in every C\\*-subalgebra containing \\(h\\) and in \\(\\widetilde A\\) (Theorem 3.2(2)). So \\(A_+=A\\cap\\widetilde A_+\\), and \\(B_+=B\\cap A_+\\) for a C\\*-subalgebra \\(B\\).\n\nThe next proposition collects the rules for working with the order; they are used constantly below. Inequalities involving scalars are read in \\(\\widetilde A\\).",
      "proof_locus": {
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        "line": 451,
        "through_line": 456,
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            },
            {
              "heading": "9. Operator monotone powers and a Hölder estimate",
              "line": 484,
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                "OA-FND-CF-17"
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          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
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    {
      "local_label": "Proposition 8.5",
      "kind": "proposition",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-15::Proposition 8.5",
      "anchor": "oa-fnd-cf-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 8.5** (Working with the order). Let \\(A\\) be a C\\*-algebra.\n1. For \\(h\\in A_h\\) and \\(t\\in\\mathbb R\\), \\(h\\leq t\\) in the forced unitization if and only if \\(\\sigma'(h)\\subseteq(-\\infty,t]\\). If \\(A\\) is unital, comparison with its own identity instead gives \\(h\\leq t1_A\\) if and only if \\(\\sigma_A(h)\\subseteq(-\\infty,t]\\). In particular \\(-\\|h\\|\\leq h\\leq\\|h\\|\\). If \\(b\\geq0\\) and \\(-b\\leq h\\leq b\\), then \\(\\|h\\|\\leq\\|b\\|\\).\n2. \\(x^*x\\leq\\|x\\|^2\\) for \\(x\\in\\widetilde A\\).\n3. If \\(h\\leq k\\), then \\(c^*hc\\leq c^*kc\\) for every \\(c\\in\\widetilde A\\).\n4. If \\(0\\leq a\\leq b\\), then \\(\\|a\\|\\leq\\|b\\|\\). For \\(a\\geq0\\): \\(a\\leq1\\) if and only if \\(\\|a\\|\\leq1\\).\n5. If \\(a\\geq0\\), \\(c\\in\\widetilde A\\) and \\(c^*ac=0\\), then \\(a^{1/2}c=0\\) and \\(ac=0\\).\n6. (*Support.*) If \\(0\\leq v\\leq a\\) and \\(ab=0\\) for some \\(b\\in\\widetilde A\\), then \\(vb=0\\).\n7. Let \\(A\\) be unital and nontrivial, \\(\\varepsilon>0\\), and \\(\\varepsilon1_A\\leq a\\leq b\\). Then \\(a,b\\) are invertible in \\(A\\) and \\(b^{-1}\\leq a^{-1}\\). In this item all scalar bounds and inverses use \\(A\\) and its own identity \\(1_A\\), not a second forced unitization.\n8. Every \\(h\\in A_h\\) is \\(h_+-h_-\\) with \\(h_\\pm\\in A_+\\) and \\(\\|h_\\pm\\|\\leq\\|h\\|\\). Every \\(x\\in A\\) is a combination of four positive elements of norm at most \\(\\|x\\|\\).\n9. If \\(a,b\\geq0\\) commute, then \\(ab\\geq0\\).\n10. (*Positive square roots.*) Every \\(a\\in A_+\\) has exactly one \\(b\\in A_+\\) with \\(b^2=a\\), namely \\(b=a^{1/2}\\). It lies in \\(C^*(a)\\) and commutes with every element that commutes with \\(a\\). No unit and no invertibility are needed; for \\(A=\\{0\\}\\), \\(b=0\\).\n11. (*Positivity in \\(B(H)\\).*) For \\(a\\in B(H)\\), \\(a\\geq0\\) if and only if \\(\\langle a\\xi,\\xi\\rangle\\geq0\\) for every \\(\\xi\\in H\\); in that case \\(a=a^*\\) automatically. The same holds in every C\\*-subalgebra of \\(B(H)\\).\n12. A \\(*\\)-homomorphism \\(\\pi:A\\to B\\) of C\\*-algebras maps \\(A_+\\) into \\(B_+\\). If \\(\\pi\\) is injective and \\(h\\in A_h\\) has \\(\\pi(h)\\geq0\\), then \\(h\\geq0\\).",
      "proof_locus": {
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        "line": 457,
        "through_line": 483,
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            },
            {
              "heading": "9. Operator monotone powers and a Hölder estimate",
              "line": 484,
              "through_line": 615,
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          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.1",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-17::Theorem 9.1",
      "anchor": "oa-fnd-cf-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 9.1** (Löwner–Heinz inequality). Let \\(A\\) be a C\\*-algebra, \\(0\\leq b\\leq a\\) and \\(0<\\alpha\\leq1\\). Then \\(b^\\alpha\\leq a^\\alpha\\).\n\nHere \\(t^\\alpha\\) vanishes at \\(0\\), so \\(a^\\alpha,b^\\alpha\\in A\\). The statement also makes sense for \\(\\alpha=0\\) when \\(a\\) and \\(b\\) are invertible, with \\(a^0=b^0=1\\).",
      "proof_locus": {
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        "line": 486,
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              "heading": "9. Operator monotone powers and a Hölder estimate",
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          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 9.2",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-17::Corollary 9.2",
      "anchor": "oa-fnd-cf-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 9.2** (A Hölder estimate). For all \\(a,b\\in A_+\\) and \\(0<\\alpha\\leq1\\),\n\\[\n\\|a^\\alpha-b^\\alpha\\|\\leq\\|a-b\\|^\\alpha .\n\\tag{9.1}\n\\]\nIn particular \\(\\|a^{1/2}-b^{1/2}\\|\\leq\\|a-b\\|^{1/2}\\), and \\(a\\mapsto a^\\alpha\\) is uniformly continuous on the whole cone \\(A_+\\).",
      "proof_locus": {
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        "line": 502,
        "through_line": 510,
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            },
            {
              "heading": "9. Operator monotone powers and a Hölder estimate",
              "line": 484,
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                "OA-FND-CF-17"
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          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.3",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-17::Example 9.3",
      "anchor": "oa-fnd-cf-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 9.3** (No exponent greater than 1 is allowed). In \\(M_2(\\mathbb C)\\) let \\(a=\\begin{pmatrix}2&1\\\\1&1\\end{pmatrix}\\) and \\(b=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\\). Then \\(a-b=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\geq0\\), so \\(0\\leq b\\leq a\\). But \\(a^2-b^2=\\begin{pmatrix}4&3\\\\3&2\\end{pmatrix}\\) has determinant \\(-1\\), so it is not positive.\n\nThe same obstruction occurs for every \\(\\alpha>1\\). Set\n\\[\n\\begin{gathered}\nt=(\\alpha^2+1)^{1/(\\alpha-1)}>1,\\qquad\nD=\\begin{pmatrix}t&0\\\\0&1\\end{pmatrix},\\\\\nJ=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\geq0 .\n\\end{gathered}\n\\]\nWe compute the first-order change of the power using only the two-dimensional spectral calculus. The eigenvalues of \\(D+\\varepsilon J\\) are\n\\[\n\\lambda_\\pm(\\varepsilon)\n=\\frac{t+1+2\\varepsilon\n\\ \\pm\\sqrt{(t-1)^2+4\\varepsilon^2}}2 .\n\\]\nAt zero they are \\(t,1\\), respectively, and both have derivative \\(1\\). Its upper spectral projection is\n\\[\nP_+(\\varepsilon)\n=\\frac{D+\\varepsilon J-\\lambda_-(\\varepsilon)1}\n{\\lambda_+(\\varepsilon)-\\lambda_-(\\varepsilon)} .\n\\]\nThis formula follows by applying the scalar function that equals \\(1\\) at \\(\\lambda_+\\) and \\(0\\) at \\(\\lambda_-\\); thus \\(P_-=1-P_+\\). At zero the diagonal entries of \\(P_+\\) have derivative \\(0\\), and both off-diagonal entries have derivative \\(1/(t-1)\\). Differentiating\n\\((D+\\varepsilon J)^\\alpha=\\lambda_+^\\alpha P_++\\lambda_-^\\alpha P_-\\)\ntherefore gives the norm limit\n\\[\n\\begin{gathered}\n\\frac{(D+\\varepsilon J)^\\alpha-D^\\alpha}{\\varepsilon}\n\\longrightarrow\nL=\\begin{pmatrix}\\alpha t^{\\alpha-1}&q\\\\q&\\alpha\\end{pmatrix},\\\\\nq=\\frac{t^\\alpha-1}{t-1}.\n\\end{gathered}\n\\]\nSince \\(t>1\\), \\(t^\\alpha-1\\geq t^{\\alpha-1}(t-1)\\), so \\(q\\geq t^{\\alpha-1}\\). Consequently\n\\[\n\\det L\n=\\alpha^2t^{\\alpha-1}-q^2\n\\leq t^{\\alpha-1}(\\alpha^2-t^{\\alpha-1})<0 .\n\\]\nThe two eigenvalues of the self-adjoint matrix \\(L\\) have opposite signs. Choose a unit eigenvector for its negative eigenvalue. The displayed norm limit shows that the quadratic form of \\((D+\\varepsilon J)^\\alpha-D^\\alpha\\) on that vector is negative for every sufficiently small positive \\(\\varepsilon\\). Yet \\(D\\leq D+\\varepsilon J\\), and both matrices are positive. Thus \\(s\\mapsto s^\\alpha\\) fails to be operator monotone on \\([0,\\infty)\\) for every \\(\\alpha>1\\).\n\nThe Löwner–Heinz theorem for \\(0<\\alpha\\leq1\\), Theorem 9.1 above, proves the complementary range in full. Theorem 10.2 shows that in a C*-algebra where \\(t^2\\) is monotone, all elements commute.\n\nThe next proposition gives a second two-by-two argument, using an affine interpolant on the spectrum, and records the complete range for real exponents.",
      "proof_locus": {
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        "line": 511,
        "through_line": 555,
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              "heading": "9. Operator monotone powers and a Hölder estimate",
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          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
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    {
      "local_label": "Proposition 9.4",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-17::Proposition 9.4",
      "anchor": "oa-fnd-cf-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 9.4** (The full range of monotone powers). On positive definite matrices of every size, \\(x\\mapsto x^\\alpha\\), for real \\(\\alpha\\), preserves order exactly when \\(0\\leq\\alpha\\leq1\\). For every \\(\\alpha>1\\) there are already positive definite two-by-two matrices \\(0<b\\leq a\\) for which \\(b^\\alpha\\not\\leq a^\\alpha\\). For \\(0<\\alpha\\leq1\\), the order-preserving statement holds on the entire positive cone of every C\\*-algebra, including nonunital algebras, by Theorem 9.1. At \\(\\alpha=0\\) on positive definite matrices the function is the constant identity.",
      "proof_locus": {
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        "line": 556,
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          "source_locus": "II.3.1; PDF 73–76",
          "role": "proof comparison",
          "correspondence": "Classical positive-cone and power-order mechanisms",
          "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.1",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Definition 10.1",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 10.1.** \\(A_h\\) is a *lattice* if every pair \\(h,k\\in A_h\\) has a least upper bound \\(h\\vee k\\) for \\(\\leq\\); then \\(h\\wedge k=-((-h)\\vee(-k))\\) is a greatest lower bound. \\(A_h\\) has the *Riesz decomposition property* (RDP) if \\(0\\leq x\\leq y_1+y_2\\) with \\(y_1,y_2\\geq0\\) implies \\(x=x_1+x_2\\) with \\(0\\leq x_j\\leq y_j\\). It has the *interpolation property* (RIP) if, whenever \\(u_1,u_2\\leq v_1,v_2\\), some \\(w\\) has \\(u_1,u_2\\leq w\\leq v_1,v_2\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 620,
        "through_line": 621,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.2",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Theorem 10.2",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 10.2** (Order and commutativity). For a C\\*-algebra \\(A\\), the following are equivalent:\n- (a) \\(A\\) is abelian;\n- (b) \\(A_h\\) is a lattice;\n- (c) \\(A_h\\) has the RDP;\n- (c′) \\(A_h\\) has the RIP;\n- (d) \\(0\\leq b\\leq a\\) implies \\(b^2\\leq a^2\\);\n- (e) \\(ab+ba\\geq0\\) for all \\(a,b\\in A_+\\);\n- (f) \\(azb=0\\) whenever \\(a,b\\in A_+\\), \\(ab=0\\) and \\(z\\in A\\).\n\nIf \\(A\\) is abelian, the real space \\(A^*_h\\) of bounded hermitian functionals, ordered by \\(\\varphi\\leq\\psi\\) when \\(\\psi-\\varphi\\) is positive, is a lattice.\n\nThe proof uses three lemmas.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 622,
        "through_line": 634,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.3",
      "kind": "lemma",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Lemma 10.3",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Lemma 10.3** (Orthogonal sums). Let \\(p_1,\\dots,p_n\\) and \\(q_1,\\dots,q_n\\) be positive elements of \\(\\widetilde A\\) of norm at most \\(1\\), with \\(p_kp_l=0\\) and \\(q_kq_l=0\\) for \\(k\\neq l\\). Then \\(\\|\\sum_kp_kyq_k\\|\\leq\\|y\\|\\) for every \\(y\\in\\widetilde A\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 635,
        "through_line": 638,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.4",
      "kind": "lemma",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Lemma 10.4",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Lemma 10.4** ((f) implies (a)). Assume (f).\n1. If \\(a\\in A_+\\), \\(c\\in\\widetilde A\\) and \\(ca=0\\), then \\(azc=0\\) for all \\(z\\in A\\). If \\(ac=0\\), then \\(cza=0\\) for all \\(z\\in A\\).\n2. Every \\(x\\in A_+\\) commutes with every \\(y\\in A\\). Hence \\(A\\) is abelian.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 639,
        "through_line": 653,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.5",
      "kind": "lemma",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Lemma 10.5",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Lemma 10.5** ((c′) implies (f)). Assume (c′). Let \\(p,q\\in A_+\\) with \\(pq=0\\), and \\(z\\in A\\). Then \\(p^2zq^2=0\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 654,
        "through_line": 733,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.6",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Example 10.6",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 10.6** (\\(2\\times2\\) matrices violate every condition of Theorem 10.2). In \\(M_2(\\mathbb C)\\):\n- (f) fails: \\(E_{11}E_{22}=0\\), but \\(E_{11}E_{12}E_{22}=E_{12}\\neq0\\).\n- (e) fails: for \\(p=E_{11}\\) and \\(q=\\frac12\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\), \\(pq+qp=\\frac12\\begin{pmatrix}2&1\\\\1&0\\end{pmatrix}\\), whose determinant is \\(-\\frac14\\).\n- (d) fails: see Example 9.3.\n- (c) fails: \\(q\\leq E_{11}+E_{22}=1\\), but if \\(q=q_1+q_2\\) with \\(0\\leq q_1\\leq E_{11}\\) and \\(0\\leq q_2\\leq E_{22}\\), then \\(q_1E_{22}=0\\) and \\(q_2E_{11}=0\\) by Proposition 8.5(6), so \\(q_1\\) and \\(q_2\\) are diagonal, and so is \\(q\\), which it is not.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 734,
        "through_line": 739,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 10.7",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-18::Exercise 10.7",
      "anchor": "oa-fnd-cf-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 10.7** (medium; The positive part is not monotone). In \\(M_2(\\mathbb C)\\), find self-adjoint \\(h\\leq k\\) with \\(h_+\\not\\leq k_+\\). Explain why this cannot happen in an abelian C\\*-algebra.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 740,
        "through_line": 743,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 11.1",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Definition 11.1",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 11.1.** Let \\(A\\) be a Banach algebra and \\(J\\subseteq A\\). A net \\((u_i)\\) in \\(A\\) is a *right approximate identity for \\(J\\)* if \\(\\|xu_i-x\\|\\to0\\) for every \\(x\\in J\\), a *left* one if \\(\\|u_ix-x\\|\\to0\\), and an *approximate identity for \\(J\\)* if both hold; for \\(J=A\\) we say \"of \\(A\\)\". It is *bounded* if \\(\\sup_i\\|u_i\\|<\\infty\\). In a C\\*-algebra one often also asks that \\(0\\leq u_i\\leq u_j\\) for \\(i\\leq j\\) and \\(\\|u_i\\|\\leq1\\); we call such a net *increasing and contractive*.\n\nFor \\(h\\in A_+\\), the *closed right ideal generated by \\(h\\)* is the smallest closed right ideal of \\(A\\) containing \\(h\\), the closure of \\(hA+\\mathbb Ch\\); similarly on the left. For \\(\\varepsilon>0\\) and \\(t\\geq0\\) put \\(f_\\varepsilon(t)=t/(t+\\varepsilon)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 754,
        "through_line": 757,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.2",
      "kind": "lemma",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Lemma 11.2",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Lemma 11.2.** Let \\(A\\) be a C\\*-algebra and \\(\\varepsilon>0\\).\n1. If \\(0\\leq h\\leq k\\), then \\(f_\\varepsilon(h)\\leq f_\\varepsilon(k)\\).\n2. Let \\(h\\in A_+\\). Then \\(f_\\varepsilon(h)\\in A_+\\), \\(\\|f_\\varepsilon(h)\\|<1\\), and \\(f_\\varepsilon(h)\\) increases as \\(\\varepsilon\\) decreases. For every \\(x\\) in the closed right ideal generated by \\(h\\), \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\) as \\(\\varepsilon\\to0\\). For every \\(x\\) in the closed left ideal generated by \\(h\\), \\(\\|xf_\\varepsilon(h)-x\\|\\to0\\). For \\(x=hy+\\mu h\\) with \\(y\\in A\\), \\(\\|f_\\varepsilon(h)x-x\\|\\leq\\varepsilon(\\|y\\|+|\\mu|)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 758,
        "through_line": 772,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Lemma 11.2",
            "Theorem 11.4",
            "Corollary 11.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.4.1.1–4",
          "role": "proof comparison",
          "correspondence": "weaker",
          "explanation": "Blackadar proves positive increasing units from the open positive ball for a dense two-sided ideal. The retained theorem treats arbitrary, possibly nonclosed one-sided ideals and their closures; its full proof stays internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.3",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Corollary 11.3",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 11.3** (The closed one-sided ideal generated by a positive element). Let \\(h\\in A_+\\) and let \\(\\mathfrak m\\) be the closed right ideal generated by \\(h\\).\n1. \\(x\\in\\mathfrak m\\) if and only if \\(\\|f_\\varepsilon(h)x-x\\|\\to0\\).\n2. If \\(xx^*\\leq\\lambda h\\) for some \\(\\lambda>0\\), then \\(x\\in\\mathfrak m\\), and \\(\\|x-f_\\varepsilon(h)x\\|\\leq\\frac12(\\lambda\\varepsilon)^{1/2}\\).\n3. \\(\\mathfrak m\\) is the closure of \\(hA\\), and it is also the closed right ideal generated by \\(h^\\alpha\\), for every \\(\\alpha>0\\).\n\nThe mirror statements hold for left ideals, with \\(x^*x\\leq\\lambda h\\) in (2).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 773,
        "through_line": 797,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.4",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Theorem 11.4",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 11.4** (Approximate identities of one-sided ideals). Let \\(A\\) be a C\\*-algebra, \\(S_0\\) its open unit ball, and \\(\\mathfrak m\\) a left ideal of \\(A\\), not necessarily closed, with \\(\\mathfrak m_+=\\mathfrak m\\cap A_+\\). Order \\(\\Lambda=\\mathfrak m_+\\cap S_0\\) by \\(\\leq\\).\n1. \\(\\Lambda\\) is upward directed.\n2. The net \\((u)_{u\\in\\Lambda}\\), which is increasing and contractive, satisfies \\(\\|x-xu\\|\\to0\\) for every \\(x\\) in the closure \\(\\overline{\\mathfrak m}\\); that is, it is a right approximate identity for \\(\\overline{\\mathfrak m}\\).\n3. If \\(\\mathfrak m\\) is a right ideal, the same set \\(\\Lambda\\) is upward directed and is a left approximate identity for \\(\\overline{\\mathfrak m}\\). If \\(\\mathfrak m\\) is a two-sided ideal, \\(\\Lambda\\) is an increasing contractive approximate identity for \\(\\overline{\\mathfrak m}\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 798,
        "through_line": 828,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Lemma 11.2",
            "Theorem 11.4",
            "Corollary 11.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.4.1.1–4",
          "role": "proof comparison",
          "correspondence": "weaker",
          "explanation": "Blackadar proves positive increasing units from the open positive ball for a dense two-sided ideal. The retained theorem treats arbitrary, possibly nonclosed one-sided ideals and their closures; its full proof stays internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.5",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Corollary 11.5",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 11.5** (Existence; the separable case).\n1. Every C\\*-algebra has an increasing contractive approximate identity, for instance \\(A_+\\cap S_0\\) itself.\n2. If \\(A\\) is separable, it has one that is an increasing sequence \\((u_n)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 829,
        "through_line": 844,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Lemma 11.2",
            "Theorem 11.4",
            "Corollary 11.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.4.1.1–4",
          "role": "proof comparison",
          "correspondence": "weaker",
          "explanation": "Blackadar proves positive increasing units from the open positive ball for a dense two-sided ideal. The retained theorem treats arbitrary, possibly nonclosed one-sided ideals and their closures; its full proof stays internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 11.6",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Exercise 11.6",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 11.6** (medium; Closed one-sided ideals are hereditary). If \\(\\mathfrak m\\) is a closed ideal of \\(A\\), \\(0\\leq x\\leq y\\) and \\(y\\in\\mathfrak m\\), then \\(x\\in\\mathfrak m\\). In fact this holds for every closed left ideal \\(\\mathfrak m\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 845,
        "through_line": 855,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.7",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-19::Example 11.7",
      "anchor": "oa-fnd-cf-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 11.7** (A nonclosed ideal need not be self-adjoint). In \\(A=C([-1,1])\\), put \\(f(t)=t+i|t|\\) and \\(I=fA\\). This is a two-sided algebraic ideal. For \\(t>0\\) the ratio \\(\\overline{f(t)}/f(t)\\) is \\(-i\\), and for \\(t<0\\) it is \\(i\\). A continuous function cannot have these two one-sided limits at \\(0\\), so \\(\\bar f\\notin I\\), although \\(f\\in I\\). Thus \\(I^*\\ne I\\).\n\nIts closure is precisely \\(J=\\{h\\in A:h(0)=0\\}\\). Certainly \\(I\\subseteq J\\). Conversely, for \\(h\\in J\\) and \\(\\varepsilon>0\\), set\n\\[\nh_\\varepsilon(t)=\\frac{h(t)|f(t)|^2}{|f(t)|^2+\\varepsilon}\n=f(t)\\frac{h(t)\\overline{f(t)}}{|f(t)|^2+\\varepsilon}\\in I.\n\\]\nGiven \\(\\delta>0\\), continuity of \\(h\\) makes \\(|h|<\\delta\\) on some interval about \\(0\\). There \\(|h-h_\\varepsilon|\\leq\\delta\\); on the compact complement, \\(|f|^2\\) has a positive minimum, so \\(h_\\varepsilon\\to h\\) uniformly. Hence \\(\\overline I=J\\). In particular \\(\\bar f\\in\\overline I\\setminus I\\), so \\(I\\) is not closed.\n\nTheorem 11.4 still supplies positive approximate identities inside \\(I\\) for its closure. It does not assert that \\(I\\) is self-adjoint. The closedness hypothesis in Theorem 15.1 is what permits a C*-algebra quotient. For related examples and the closed-ideal proof, see [Blackadar, *Operator Algebras*, II.5.2.1 and II.5.1.1, corrected author version](https://bruceblackadar.com/Mathematics/Cycr.pdf).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 856,
        "through_line": 866,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Example 11.7"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.5.2.1(i)",
          "role": "example",
          "correspondence": "same phenomenon, independent calculation",
          "explanation": "The new profile f(t)=t+i|t| and its exact closure are independently calculated; no PDF wording is imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 15.1",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-25::Theorem 15.1",
      "anchor": "oa-fnd-cf-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 15.1** (Quotients of C\\*-algebras). Let \\(A\\) be a C\\*-algebra and \\(\\mathfrak m\\subseteq A\\) a closed ideal.\n1. \\(\\mathfrak m^*=\\mathfrak m\\), so \\(\\mathfrak m\\) is a C\\*-subalgebra.\n2. \\(A/\\mathfrak m\\), with the quotient norm and \\((x+\\mathfrak m)^*=x^*+\\mathfrak m\\), is a C\\*-algebra, and the quotient map is a \\(*\\)-homomorphism.\n3. (*Quotient norm.*) Let \\((e_i)\\) be any net of positive contractions in \\(\\mathfrak m\\) with \\(\\|y-ye_i\\|\\to0\\) for every \\(y\\in\\mathfrak m\\), for instance the net of Theorem 11.4. Then for every \\(x\\in A\\),\n\\[\n\\|x+\\mathfrak m\\|=\\lim_i\\|x-xe_i\\| .\n\\tag{15.1}\n\\]",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 869,
        "through_line": 900,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Theorem 15.1",
            "Corollary 15.4"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.5.1.1–3",
          "role": "proof construction comparison",
          "correspondence": "exact",
          "explanation": "Closed two-sided ideals are self-adjoint; cutoff units compute the quotient norm and prove the quotient C*-identity. Complete independent internal proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 15.2",
      "kind": "examples",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-25::Examples 15.2",
      "anchor": "oa-fnd-cf-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Examples 15.2** (The two hypotheses of (1) are needed).\n- *A closed left ideal need not be self-adjoint.* In \\(M_2(\\mathbb C)\\), \\(L=\\{x:xE_{11}=x\\}\\), the matrices with zero second column, is a closed left ideal. It contains \\(E_{21}\\) but not \\(E_{21}^*=E_{12}\\).\n- *An ideal that is not closed need not be self-adjoint.* In \\(C([0,1])\\), let \\(h(t)=te^{i/t}\\) for \\(t>0\\) and \\(h(0)=0\\), and let \\(I=hC([0,1])\\), an ideal. If \\(\\bar h=hg\\) with \\(g\\) continuous, then \\(g(t)=e^{-2i/t}\\) for \\(t>0\\), which has no limit at \\(0\\). So \\(\\bar h\\notin I\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 901,
        "through_line": 904,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 15.3",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-25::Example 15.3",
      "anchor": "oa-fnd-cf-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 15.3** (The quotient norm formula). In \\(A=C([0,1])\\), let \\(\\mathfrak m=\\{g:g(0)=0\\}\\) and \\(e_n(t)=\\min(1,nt)\\), a positive contractive (and increasing) approximate identity of \\(\\mathfrak m\\). For \\(f\\in A\\), \\[\n\\begin{gathered}\n\\|f-fe_n\\|\\\\\n=\\sup_{t\\leq1/n}|f(t)|(1-nt)\\to|f(0)|\\\\\n=\\|f+\\mathfrak m\\|,\n\\end{gathered}\n\\] as (15.1) says.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 905,
        "through_line": 912,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 15.4",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-25::Corollary 15.4",
      "anchor": "oa-fnd-cf-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 15.4** (Kernels and ranges of \\(*\\)-homomorphisms). Let \\(\\pi:A\\to B\\) be a \\(*\\)-homomorphism of C\\*-algebras. Then \\(\\ker\\pi\\) is a closed ideal, the range \\(\\pi(A)\\) is closed, hence a C\\*-subalgebra of \\(B\\), and \\(\\tilde\\pi(x+\\ker\\pi)=\\pi(x)\\) is an isometric \\(*\\)-isomorphism of \\(A/\\ker\\pi\\) onto \\(\\pi(A)\\). In particular \\(\\|\\pi(x)\\|=\\|x+\\ker\\pi\\|\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 913,
        "through_line": 916,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Theorem 15.1",
            "Corollary 15.4"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.5.1.1–3",
          "role": "proof construction comparison",
          "correspondence": "exact",
          "explanation": "Closed two-sided ideals are self-adjoint; cutoff units compute the quotient norm and prove the quotient C*-identity. Complete independent internal proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.5",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-25::Exercise 15.5",
      "anchor": "oa-fnd-cf-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 15.5** (easy; Ideals of ideals). Let \\(I\\) be a closed ideal of \\(A\\) and \\(J\\) a closed ideal of the C\\*-algebra \\(I\\). Show that \\(J\\) is an ideal of \\(A\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 917,
        "through_line": 920,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.6",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-25::Exercise 15.6",
      "anchor": "oa-fnd-cf-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 15.6** (easy; A C\\*-subalgebra plus a closed ideal). If \\(B\\) is a C\\*-subalgebra and \\(\\mathfrak m\\) a closed ideal of \\(A\\), then \\(B+\\mathfrak m\\) is a C\\*-subalgebra.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 921,
        "through_line": 924,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 16.1",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-27::Proposition 16.1",
      "anchor": "oa-fnd-cf-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 16.1** (Hull and kernel). Let \\(A\\) be abelian, with character space \\(\\Omega\\). For a closed set \\(\\Gamma\\subseteq\\Omega\\) and a closed ideal \\(\\mathfrak m\\) put\n\\[\n\\begin{gathered}\n\\mathfrak m_\\Gamma\\\\\n=\\{x\\in A: \\ \\omega(x)=0\\ \\text{for all }\\omega\\in\\Gamma\\},\\\\\n\\Gamma_{\\mathfrak m}\\\\\n=\\{\\omega\\in\\Omega: \\ \\omega(x)=0\\ \\text{for all }x\\in\\mathfrak m\\}.\n\\end{gathered}\n\\]\n1. \\(\\Gamma\\mapsto\\mathfrak m_\\Gamma\\) and \\(\\mathfrak m\\mapsto\\Gamma_{\\mathfrak m}\\) are mutually inverse bijections, reversing inclusion, between the closed subsets of \\(\\Omega\\) and the closed ideals of \\(A\\).\n2. If \\(\\rho:A\\to A/\\mathfrak m\\) is the quotient map, then \\(\\omega'\\mapsto\\omega'\\circ\\rho\\) is a homeomorphism of \\(\\operatorname{Ch}(A/\\mathfrak m)\\) onto \\(\\Gamma_{\\mathfrak m}\\).\n3. Restriction \\(\\omega\\mapsto\\omega|_{\\mathfrak m}\\) is a homeomorphism of \\(\\Omega\\setminus\\Gamma_{\\mathfrak m}\\) onto \\(\\operatorname{Ch}(\\mathfrak m)\\).\n\nIdentifying \\(A\\) with \\(C_0(\\Omega)\\) (Theorem 2.1 and Proposition 2.2): \\(\\mathfrak m_\\Gamma\\) is the set of functions vanishing on \\(\\Gamma\\); restriction to \\(\\Gamma\\) identifies \\(A/\\mathfrak m_\\Gamma\\) with \\(C_0(\\Gamma)\\); and \\(\\mathfrak m_\\Gamma\\) is \\(C_0(\\Omega\\setminus\\Gamma)\\). The ideal \\(\\mathfrak m_\\Gamma\\) is called the *kernel* of \\(\\Gamma\\), and \\(\\Gamma_{\\mathfrak m}\\) the *hull* of \\(\\mathfrak m\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 927,
        "through_line": 951,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 16.2",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-27::Example 16.2",
      "anchor": "oa-fnd-cf-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 16.2** (\\(C_0(\\mathbb R)\\)). Let \\(A=C_0(\\mathbb R)\\). By Proposition 2.2, the characters are the evaluations, so \\(\\sigma'(x)=x(\\mathbb R)\\cup\\{0\\}\\), and by Theorem 5.3(4), \\(f(x)=f\\circ x\\) for \\(f(0)=0\\). Positivity is pointwise. The function \\(a(t)=e^{-t^2}\\) is strictly positive: \\(f_\\varepsilon(a)=a/(a+\\varepsilon)\\) tends to \\(1\\) uniformly on compact sets, and for \\(g\\in C_0(\\mathbb R)\\) and \\(\\eta>0\\), \\(|g|<\\eta\\) off a compact set \\(C\\), so \\[\n\\begin{gathered}\n\\|g-f_\\varepsilon(a)g\\|\\\\\n\\leq\\max(\\eta,\\|g\\|\\max_C|1-f_\\varepsilon(a)|)\\to\\eta;\n\\end{gathered}\n\\] this is Proposition 13.3(3) seen directly. The closed ideals correspond to closed \\(F\\subseteq\\mathbb R\\) (Proposition 16.1): for \\(F=\\{0\\}\\), \\(\\mathfrak m_F=\\{g:g(0)=0\\}\\cong C_0(\\mathbb R\\setminus\\{0\\})\\), and \\(A/\\mathfrak m_F\\cong\\mathbb C\\) through \\(g\\mapsto g(0)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 952,
        "through_line": 958,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 17.1",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-28::Proposition 17.1",
      "anchor": "oa-fnd-cf-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 17.1** (Lifting self-adjoint, positive and arbitrary elements). Let \\(\\pi:A\\to B\\) be a surjective \\(*\\)-homomorphism of C\\*-algebras.\n1. Every \\(k\\in B_h\\) is \\(\\pi(h)\\) for some \\(h\\in A_h\\) with \\(\\|h\\|=\\|k\\|\\). If \\(k\\geq0\\), \\(h\\) can be chosen with \\(h\\geq0\\) and \\(\\|h\\|=\\|k\\|\\).\n2. Every \\(k\\in B\\) is \\(\\pi(x)\\) for some \\(x\\in A\\) with \\(\\|x\\|=\\|k\\|\\).\n3. \\(\\pi(A_+)=B_+\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 961,
        "through_line": 980,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Proposition 17.1"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.5.1.5",
          "role": "alternative proof comparison",
          "correspondence": "exact",
          "explanation": "Norm-attaining lifts and positive/self-adjoint lifts compared at the stated scope; internal proof remains the provider."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 17.2",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-28::Proposition 17.2",
      "anchor": "oa-fnd-cf-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 17.2** (Lifting \\(\\{y^*y\\leq\\pi(a)\\}\\)). Let \\(\\pi:A\\to B\\) be a surjective \\(*\\)-homomorphism of C\\*-algebras and \\(a\\in A_+\\). Then\n\\[\n\\begin{gathered}\n\\pi\\big(\\{x\\in A: \\ x^*x\\leq a\\}\\big)\\\\\n=\\{y\\in B: \\ y^*y\\leq\\pi(a)\\}.\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 981,
        "through_line": 1006,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Proposition 17.2"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.5.1.6",
          "role": "statement comparison",
          "correspondence": "exact statement only",
          "explanation": "The free source states constrained lifting without a proof. It cannot replace the complete internal argument."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 17.3",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-28::Corollary 17.3",
      "anchor": "oa-fnd-cf-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 17.3** (Order intervals; sums of closed ideals).\n1. (*Order intervals.*) \\[\n\\begin{gathered}\n\\pi(\\{p\\in A:0\\leq p\\leq a\\})\\\\\n=\\{q\\in B:0\\leq q\\leq\\pi(a)\\}.\n\\end{gathered}\n\\]\n2. For closed ideals \\(\\mathfrak m\\) and \\(\\mathfrak n\\) of \\(A\\), \\(\\mathfrak m+\\mathfrak n\\) is a closed ideal and \\((\\mathfrak m+\\mathfrak n)_+=\\mathfrak m_++\\mathfrak n_+\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1007,
        "through_line": 1021,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 17.4",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-28::Exercise 17.4",
      "anchor": "oa-fnd-cf-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 17.4** (medium; Lifting unitaries). Let \\(A\\) be unital, \\(\\mathfrak m\\) a closed ideal and \\(\\pi:A\\to A/\\mathfrak m\\). (a) If \\(\\dot u\\in U(A/\\mathfrak m)\\) and \\(\\sigma(\\dot u)\\neq\\mathbb T\\), then \\(\\dot u=\\pi(u)\\) for a unitary \\(u\\in A\\). (b) More generally, every unitary in the connected component \\(U_0(A/\\mathfrak m)\\) of \\(1\\) in \\(U(A/\\mathfrak m)\\) is the image of a unitary in \\(U_0(A)\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1022,
        "through_line": 1028,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 17.5",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-28::Exercise 17.5",
      "anchor": "oa-fnd-cf-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 17.5** (medium; A unitary that does not lift). Let \\(\\bar{\\mathbb D}\\) be the closed unit disc, \\(A=C(\\bar{\\mathbb D})\\), and \\(\\mathfrak m\\) the closed ideal of functions vanishing on \\(\\mathbb T\\) (Proposition 16.1), so that \\(A/\\mathfrak m=C(\\mathbb T)\\) by restriction. The unitary \\(\\dot u(\\lambda)=\\lambda\\) of \\(C(\\mathbb T)\\) is not the image of a unitary of \\(A\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1029,
        "through_line": 1036,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 13.1",
      "kind": "lemma",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-22::Lemma 13.1",
      "anchor": "oa-fnd-cf-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Lemma 13.1** (Positive linear functionals). Let \\(A\\) be a C\\*-algebra and \\(\\varphi\\) a positive linear functional on \\(A\\).\n1. \\(\\varphi\\) is bounded.\n2. \\(\\varphi\\) is hermitian, and \\(|\\varphi(y^*x)|^2\\leq\\varphi(x^*x)\\varphi(y^*y)\\) for all \\(x,y\\in A\\).\n3. \\(|\\varphi(x)|^2\\leq\\|\\varphi\\|\\varphi(x^*x)\\) for all \\(x\\in A\\).\n4. The set \\(Q\\) of positive linear functionals of norm at most one is weak\\* compact.\n5. If \\(A\\) is unital and nontrivial, then \\(\\|\\varphi\\|=\\varphi(1)\\).\n6. (*States.*) If \\(A\\) is unital and nontrivial, \\(h\\in A_h\\) and \\(\\lambda\\in\\sigma(h)\\), some state \\(\\omega\\) has \\(\\omega(h)=\\lambda\\).\n7. (*States.*) If \\(A\\neq0\\) and \\(a\\in A_+\\), some state \\(\\omega\\) has \\(\\omega(a)=\\|a\\|\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1047,
        "through_line": 1078,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 13.2",
      "kind": "definition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-22::Definition 13.2",
      "anchor": "oa-fnd-cf-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Definition 13.2.** An element \\(a\\in A_+\\) is *strictly positive* if \\(\\varphi(a)>0\\) for every nonzero positive linear functional \\(\\varphi\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1079,
        "through_line": 1080,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.3",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-22::Proposition 13.3",
      "anchor": "oa-fnd-cf-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 13.3** (Strictly positive elements). Let \\(A\\) be a C\\*-algebra, \\(a\\in A_+\\), and \\(f_t(\\lambda)=\\lambda/(\\lambda+t)\\).\n1. If \\(A\\) is unital and nontrivial, \\(a\\) is strictly positive if and only if it is invertible.\n2. Every separable C\\*-algebra has a strictly positive element.\n3. \\(a\\) is strictly positive if and only if \\(\\|f_t(a)x-x\\|\\to0\\) as \\(t\\to0\\), for every \\(x\\in A\\).\n4. Let \\(a\\) be strictly positive, and let \\(\\pi:A\\to B(H)\\) be a *nondegenerate representation*: a \\(*\\)-homomorphism such that \\(\\pi(A)H\\) spans a dense subspace. Then \\(\\pi(f_t(a))\\xi\\to\\xi\\) for every \\(\\xi\\in H\\).\n5. If \\(a\\) is strictly positive, \\((f_{1/n}(a))_{n\\geq1}\\) is an increasing, contractive, commuting approximate identity.\n6. (*Converse of (5).*) If some sequence \\((e_n)\\) in \\(A\\) satisfies \\(\\|e_nx-x\\|\\to0\\) for every \\(x\\in A\\), for instance a sequential approximate identity, then \\(a=\\sum_n2^{-n}(1+\\|e_n\\|^2)^{-1}e_ne_n^*\\) is strictly positive. If the \\(e_n\\) are positive contractions, \\(a=\\sum_n2^{-n}e_n\\) is strictly positive as well.\n7. Every separable C\\*-algebra has an increasing, contractive approximate identity made of a commuting sequence.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1081,
        "through_line": 1111,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Proposition 13.3"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.4.2.1–5",
          "role": "proof comparison",
          "correspondence": "partial",
          "explanation": "Strict positivity and sequential units are compared. The free text/errata give complementary constructions; the lesson retains its complete functional-calculus/compactness proof."
        },
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Proposition 13.3"
          ],
          "source_key": "blackadar-operator-algebras-errata@7FA0252E0A61D00400737947869400637088B25399457FCAB41E0618AB4CE472",
          "source_locus": "PDF 6–7, correction to II.4.2.3",
          "role": "correction comparison",
          "correspondence": "partial",
          "explanation": "The missing original converse proof is supplied in the errata. The current internal proof already includes the relevant compactness argument; the unrelated nonunital state-space claim is not imported."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.4",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-22::Proposition 13.4",
      "anchor": "oa-fnd-cf-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 13.4** (A C\\*-algebra without a strictly positive element). Let \\((\\Gamma,\\Sigma,\\mu)\\) be a \\(\\sigma\\)-finite measure space without atoms, with \\(\\mu(\\Gamma)=\\infty\\). Let \\(A\\subseteq L^\\infty(\\Gamma,\\mu)\\) be the C\\*-subalgebra generated by the projections \\(1_E\\) with \\(\\mu(E)<\\infty\\), that is, the closed linear span of these projections. Then:\n1. for \\(h\\in A\\) and \\(\\eta>0\\), \\(\\mu(\\{|h|>\\eta\\})<\\infty\\); in particular \\(A\\) is not unital;\n2. for every \\(h\\in A_+\\) there is a state \\(\\varphi\\) of \\(A\\) with \\(\\varphi(h)=0\\);\n3. hence \\(A\\) has no strictly positive element and no sequential approximate identity (Proposition 13.3(6)), although it has the approximate identity of Theorem 11.4.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1112,
        "through_line": 1121,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 13.5",
      "kind": "example",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-22::Example 13.5",
      "anchor": "oa-fnd-cf-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Example 13.5** (\\(c_0\\)). In \\(c_0\\), the sequences \\(e_n=(1,\\dots,1,0,0,\\dots)\\) (\\(n\\) ones) form an increasing contractive sequential approximate identity, and \\(a=(1,\\frac12,\\frac13,\\dots)\\) is strictly positive. Indeed, a positive functional is \\(\\varphi(x)=\\sum_k\\varphi_kx_k\\) with \\(\\varphi_k=\\varphi(\\delta_k)\\geq0\\) and \\(\\sum_k\\varphi_k<\\infty\\) (by Lemma 13.1(1) and \\(\\|e_n\\|=1\\)), and \\(\\varphi(a)=\\sum_k\\varphi_k/k>0\\) unless all \\(\\varphi_k=0\\). Here \\(\\varphi(x)=\\lim_n\\varphi(e_nx)=\\sum_k\\varphi_kx_k\\) uses that \\((e_n)\\) is an approximate identity.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1122,
        "through_line": 1123,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 13.6",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-22::Exercise 13.6",
      "anchor": "oa-fnd-cf-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 13.6** (medium; \\(\\sigma\\)-unital commutative algebras). Let \\(X\\) be an LCH space. Show that \\(C_0(X)\\) has a strictly positive element if and only if \\(X\\) is \\(\\sigma\\)-compact. Deduce that \\(C_0(X)\\) has a sequential approximate identity exactly when \\(X\\) is \\(\\sigma\\)-compact.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1124,
        "through_line": 1127,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.1",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-23::Proposition 12.1",
      "anchor": "oa-fnd-cf-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 12.1** (Approximate identities of \\(L^1(G)\\)). For each neighbourhood \\(V\\) of \\(e\\), let \\(u_V\\in L^1(G)\\) satisfy \\(u_V\\geq0\\), \\(u_V=0\\) off \\(V\\) and \\(\\int u_V=1\\); continuity, compact support and symmetry are not required. Direct the neighbourhoods by reverse inclusion. Then \\((u_V)\\) is an approximate identity of \\(L^1(G)\\) with \\(\\|u_V\\|_1=1\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1132,
        "through_line": 1146,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.2",
      "kind": "proposition",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-23::Proposition 12.2",
      "anchor": "oa-fnd-cf-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Proposition 12.2** (Approximate identities of the transformation-group algebra). Let \\(\\Omega\\) be an LCH space on which \\(G\\) acts continuously from the right, \\((\\omega,s)\\mapsto\\omega s\\). Let \\(A\\) be the completion of \\(C_c(\\Omega\\times G)\\) for \\(\\|x\\|=\\int_G\\sup_\\omega|x(\\omega,s)|\\,ds\\), with the product \\((xy)(\\omega,s)=\\int_Gx(\\omega,t)\\,y(\\omega t,t^{-1}s)\\,dt\\). For each compact \\(K\\subseteq\\Omega\\) let \\(f_K\\in C_c(\\Omega)\\) with \\(0\\leq f_K\\leq1\\) and \\(f_K=1\\) on \\(K\\). For each neighbourhood \\(V\\) of \\(e\\) inside a fixed compact neighbourhood \\(V_0\\), let \\(u_V\\) be continuous, \\(u_V\\geq0\\), \\(u_V=0\\) off \\(V\\), \\(\\int u_V=1\\). Put \\(u_{K,V}(\\omega,s)=f_K(\\omega)u_V(s)\\), and direct the pairs by \\(K\\) increasing and \\(V\\) decreasing. Then \\((u_{K,V})\\) is an approximate identity of \\(A\\) with \\(\\|u_{K,V}\\|\\leq1\\).\n\nRestricting to \\(V\\subseteq V_0\\) only makes \\(u_{K,V}\\) compactly supported; the directed set is unchanged far out, which is all that matters for convergence.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1147,
        "through_line": 1186,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 14.1",
      "kind": "theorem",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-21::Theorem 14.1",
      "anchor": "oa-fnd-cf-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Theorem 14.1** (Asymmetric Riesz decomposition). Let \\((x_i)_{i\\in I}\\) and \\((y_j)_{j\\in J}\\) be families in a C\\*-algebra \\(A\\) such that the sums \\(\\sum_ix_i^*x_i\\) and \\(\\sum_jy_j^*y_j\\) converge unconditionally in norm (the nets of finite partial sums converge) to the same element \\(a\\). Then there are \\(z_{ij}\\in A\\) with\n\\[\n\\begin{gathered}\nx_ix_i^*\\\\\n=\\sum_jz_{ij}^*z_{ij}\\\\\n(i\\in I),\\\\\ny_jy_j^*\\\\\n=\\sum_iz_{ij}z_{ij}^*\\\\\n(j\\in J),\n\\end{gathered}\n\\tag{14.1}\n\\]\nboth sums converging unconditionally in norm.",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1191,
        "through_line": 1241,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "14. The asymmetric Riesz decomposition",
              "line": 1187,
              "through_line": 1261,
              "anchors": [
                "OA-FND-CF-21"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.3.2.1–II.3.2.7; PDF 77–79",
          "role": "proof comparison",
          "correspondence": "Partial external proof; asymmetric conclusion preserved",
          "explanation": "Soft polar approximation is outlined; asymmetric Riesz decomposition II.3.2.7 refers to Pedersen for the proof. The lesson’s full proof and exact distinction between u*u and uu* remain the provider."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 14.2",
      "kind": "corollary",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-21::Corollary 14.2",
      "anchor": "oa-fnd-cf-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Corollary 14.2.** If \\(\\sum_jy_j^*y_j\\leq\\sum_ix_i^*x_i\\), with both sums as in the theorem, there are \\(z_{ij}\\) with \\(y_jy_j^*=\\sum_iz_{ij}z_{ij}^*\\) and \\(x_ix_i^*\\geq\\sum_jz_{ij}^*z_{ij}\\).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1242,
        "through_line": 1245,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "14. The asymmetric Riesz decomposition",
              "line": 1187,
              "through_line": 1261,
              "anchors": [
                "OA-FND-CF-21"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.3.2.1–II.3.2.7; PDF 77–79",
          "role": "proof comparison",
          "correspondence": "Partial external proof; asymmetric conclusion preserved",
          "explanation": "Soft polar approximation is outlined; asymmetric Riesz decomposition II.3.2.7 refers to Pedersen for the proof. The lesson’s full proof and exact distinction between u*u and uu* remain the provider."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.3",
      "kind": "exercise",
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones#oa-fnd-cf-21::Exercise 14.3",
      "anchor": "oa-fnd-cf-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
      "statement_and_full_conditions": "**Exercise 14.3** (medium; Factorization of operators). (a) Let \\(x:H\\to K_1\\) and \\(y:H\\to K_2\\) be bounded operators between Hilbert spaces with \\(x^*x\\leq y^*y\\). Then \\(x=cy\\) for some \\(c:K_2\\to K_1\\) with \\(\\|c\\|\\leq1\\), and \\(c\\) can be taken to vanish on the orthogonal complement of the range of \\(y\\). (b) Prove Theorem 14.1 for finite families in \\(A=B(H)\\) from (a).",
      "proof_locus": {
        "source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "line": 1246,
        "through_line": 1261,
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
      },
      "source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_scopes": [
            {
              "heading": "14. The asymmetric Riesz decomposition",
              "line": 1187,
              "through_line": 1261,
              "anchors": [
                "OA-FND-CF-21"
              ],
              "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.3.2.1–II.3.2.7; PDF 77–79",
          "role": "proof comparison",
          "correspondence": "Partial external proof; asymmetric conclusion preserved",
          "explanation": "Soft polar approximation is outlined; asymmetric Riesz decomposition II.3.2.7 refers to Pedersen for the proof. The lesson’s full proof and exact distinction between u*u and uu* remain the provider."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.1",
      "kind": "definition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-01::Definition 1.1",
      "anchor": "oa-fnd-gn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Definition 1.1.** A *representation* of \\(A\\) on a Hilbert space \\(H\\) is a \\(*\\)-homomorphism \\(\\pi:A\\to B(H)\\), that is, a linear map with \\(\\pi(xy)=\\pi(x)\\pi(y)\\) and \\(\\pi(x^*)=\\pi(x)^*\\). We write \\((\\pi,H)\\), or just \\(\\pi\\). A bounded operator \\(T:H_1\\to H_2\\) *intertwines* representations \\((\\pi_1,H_1)\\) and \\((\\pi_2,H_2)\\) if \\(T\\pi_1(x)=\\pi_2(x)T\\) for all \\(x\\in A\\). The two are *unitarily equivalent*, written \\(\\pi_1\\cong\\pi_2\\), if some unitary operator intertwines them. A representation is *faithful* if it is injective. The *zero representation* on \\(H\\) sends every element to \\(0\\).\n\nWhen \\(A\\) is an involutive Banach algebra, every representation satisfies \\(\\|\\pi(x)\\|\\le\\|x\\|\\), and no continuity has to be assumed ([\\(*\\)-homomorphisms are contractive](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-12)). The proof there uses that the involution is isometric.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 34,
        "through_line": 37,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.2",
      "kind": "definition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-01::Definition 1.2",
      "anchor": "oa-fnd-gn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Definition 1.2.** Let \\((\\pi,H)\\) be a representation.\n1. A closed subspace \\(M\\subseteq H\\) is *invariant* if \\(\\pi(x)M\\subseteq M\\) for all \\(x\\in A\\). Then \\(\\pi_M(x)=\\pi(x)|_M\\) defines a representation on \\(M\\), a *subrepresentation* of \\(\\pi\\).\n2. The *essential subspace* \\(E_\\pi\\) is the closed linear span of all vectors \\(\\pi(x)\\xi\\) with \\(x\\in A\\) and \\(\\xi\\in H\\). The *null space* is \\(N_\\pi=\\{\\xi\\in H:\\pi(x)\\xi=0\\text{ for all }x\\in A\\}\\). The representation is *nondegenerate* if \\(E_\\pi=H\\).\n3. A vector \\(\\xi\\in H\\) is *cyclic* if the subspace \\(\\pi(A)\\xi=\\{\\pi(x)\\xi:x\\in A\\}\\) is dense in \\(H\\). A representation with a cyclic vector is *cyclic*.\n4. Let \\((\\pi_i,H_i)_{i\\in I}\\) be representations with \\(\\sup_i\\|\\pi_i(x)\\|<\\infty\\) for every \\(x\\); for involutive Banach algebras this holds automatically. Their *direct sum* \\(\\bigoplus_i\\pi_i\\) acts on the Hilbert space direct sum \\(\\bigoplus_iH_i\\) by \\((\\xi_i)_i\\mapsto(\\pi_i(x)\\xi_i)_i\\). Each such operator has norm at most \\(\\sup_i\\|\\pi_i(x)\\|\\), and the rules of a representation hold coordinate by coordinate. A vector \\((\\xi_i)_i\\) is killed by every \\(\\bigoplus_i\\pi_i(x)\\) exactly when each \\(\\xi_i\\) is killed by every \\(\\pi_i(x)\\), so the null space of the direct sum is the direct sum of the null spaces, and by Proposition 1.4(2) below the direct sum is nondegenerate exactly when every summand is.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 38,
        "through_line": 43,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.3",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-01::Lemma 1.3",
      "anchor": "oa-fnd-gn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 1.3** (Invariant subspaces and the commutant). Let \\(\\mathcal S\\subseteq B(H)\\) be a self-adjoint set of operators, that is, \\(T^*\\in\\mathcal S\\) whenever \\(T\\in\\mathcal S\\). Let \\(M\\) be a closed subspace with projection \\(p\\). The following are equivalent:\n- (a) \\(TM\\subseteq M\\) for every \\(T\\in\\mathcal S\\);\n- (b) \\(TM^\\perp\\subseteq M^\\perp\\) for every \\(T\\in\\mathcal S\\);\n- (c) \\(p\\in\\mathcal S'\\).\n\nIn particular, if \\(M\\) is invariant for a representation \\(\\pi\\), so is \\(M^\\perp\\), and the unitary \\(M\\oplus M^\\perp\\to H\\), \\((\\xi,\\eta)\\mapsto\\xi+\\eta\\), intertwines \\(\\pi_M\\oplus\\pi_{M^\\perp}\\) with \\(\\pi\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 44,
        "through_line": 56,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.4",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-01::Proposition 1.4",
      "anchor": "oa-fnd-gn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 1.4** (The nondegenerate part). Let \\((\\pi,H)\\) be a representation.\n1. \\(N_\\pi=E_\\pi^\\perp\\). Both subspaces are invariant, \\(\\pi\\) is unitarily equivalent to \\(\\pi_{E_\\pi}\\oplus0\\), where \\(0\\) is the zero representation on \\(N_\\pi\\), and \\(\\pi_{E_\\pi}\\) is nondegenerate.\n2. \\(\\pi\\) is nondegenerate exactly when \\(N_\\pi=\\{0\\}\\), that is, when every nonzero \\(\\xi\\) has \\(\\pi(x)\\xi\\ne0\\) for some \\(x\\).\n3. If \\(\\pi\\) is nondegenerate, then every \\(\\xi\\in H\\) lies in the closure of \\(\\pi(A)\\xi\\).\n4. Suppose that \\((u_i)\\) is a left approximate identity of the involutive Banach algebra \\(A\\), bounded by \\(\\gamma\\). Then \\(\\pi(u_i)\\) converges strongly to the projection onto \\(E_\\pi\\). So nondegeneracy of \\(\\pi\\) amounts to \\(\\pi(u_i)\\xi\\to\\xi\\) for every \\(\\xi\\in H\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 57,
        "through_line": 67,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.5",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-01::Proposition 1.5",
      "anchor": "oa-fnd-gn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 1.5** (Decomposition into cyclic representations). Let \\((\\pi,H)\\) be nondegenerate. There is a family \\((\\xi_j)_{j\\in J}\\) of nonzero vectors such that the closures \\(H_j\\) of \\(\\pi(A)\\xi_j\\) are pairwise orthogonal and \\(H=\\bigoplus_jH_j\\). Each \\(H_j\\) is invariant and \\(\\xi_j\\) is a cyclic vector for \\(\\pi_{H_j}\\), so \\(\\pi\\cong\\bigoplus_j\\pi_{H_j}\\) splits into cyclic summands. If \\(H\\) is separable, \\(J\\) is countable. An arbitrary representation is the direct sum of a nondegenerate one, which decomposes in this way, and a zero representation.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 68,
        "through_line": 71,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 1.6",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-01::Examples 1.6",
      "anchor": "oa-fnd-gn-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 1.6.**\n1. For \\(H\\ne\\{0\\}\\), the identity representation of \\(B(H)\\) on \\(H\\) is nondegenerate, and every nonzero vector \\(\\xi\\) is cyclic, since \\(\\theta_{\\eta,\\xi}\\xi=\\|\\xi\\|^2\\eta\\) for every \\(\\eta\\).\n2. Let \\(X\\) be a locally compact Hausdorff space, and let \\(C_0(X)\\) act on \\(\\ell^2(X)\\) by \\((\\pi(f)\\xi)(x)=f(x)\\xi(x)\\). Each line \\(\\mathbb C\\delta_x\\) is invariant, and \\(\\pi\\) is the direct sum of the one-dimensional representations \\(f\\mapsto f(x)\\). It is nondegenerate, because by Urysohn's lemma each \\(x\\) has some \\(f\\) with \\(f(x)\\ne0\\), and it is faithful, because \\(\\pi(f)\\delta_x=f(x)\\delta_x\\). If \\(X\\) is uncountable, \\(\\pi\\) is not cyclic: a vector \\(\\xi\\in\\ell^2(X)\\) is supported on a countable set, and so is every \\(\\pi(f)\\xi\\).\n3. The representation \\(x\\mapsto x\\oplus0\\) of \\(M_n(\\mathbb C)\\) on \\(\\mathbb C^n\\oplus\\mathbb C\\) is degenerate, with null space \\(0\\oplus\\mathbb C\\). Here the image of the identity is the projection onto the essential subspace, as in Proposition 1.4(4).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 72,
        "through_line": 76,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.1",
      "kind": "definition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-03::Definition 3.1",
      "anchor": "oa-fnd-gn-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Definition 3.1.** A linear functional \\(\\omega\\) on \\(A\\) is *positive* if \\(\\omega(x^*x)\\ge0\\) for every \\(x\\in A\\). The *adjoint* of a linear functional \\(f\\) is \\(f^*(x)=\\overline{f(x^*)}\\), and \\(f\\) is *hermitian* if \\(f^*=f\\). For positive \\(\\psi,\\varphi\\) we write \\(\\psi\\le\\varphi\\) if \\(\\varphi-\\psi\\) is positive. A positive \\(\\omega\\) is *faithful* if \\(\\omega(x^*x)=0\\) only for \\(x=0\\). If \\(A\\) is normed, a *state* is a bounded positive functional of norm one.\n\nFor a representation \\((\\pi,H)\\) and \\(\\xi,\\eta\\in H\\), the *coefficient functional* \\(\\omega_{\\xi,\\eta}(x)=\\langle\\pi(x)\\xi,\\eta\\rangle\\) satisfies \\(\\omega_{\\xi,\\eta}^*=\\omega_{\\eta,\\xi}\\), because \\(\\overline{\\langle\\pi(x^*)\\xi,\\eta\\rangle}=\\overline{\\langle\\xi,\\pi(x)\\eta\\rangle}=\\langle\\pi(x)\\eta,\\xi\\rangle\\). The *vector functional* \\(\\omega_\\xi=\\omega_{\\xi,\\xi}\\) is positive, since \\(\\omega_\\xi(x^*x)=\\|\\pi(x)\\xi\\|^2\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 81,
        "through_line": 84,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.2",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-03::Proposition 3.2",
      "anchor": "oa-fnd-gn-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 3.2** (Cauchy–Schwarz). Let \\(\\omega\\) be a positive functional on \\(A\\). For all \\(x,y\\in A\\),\n\\[\n\\begin{gathered}\n\\omega(y^*x)\\\\\n=\\overline{\\omega(x^*y)},\\\\\n|\\omega(y^*x)|^2\\\\\n\\le\\omega(x^*x)\\,\\omega(y^*y).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A\\) has an identity, then \\(\\omega(x^*)=\\overline{\\omega(x)}\\) and \\(|\\omega(x)|^2\\le\\omega(1)\\,\\omega(x^*x)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 85,
        "through_line": 104,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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      "source_uses": [
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          "local_labels": [
            "Proposition 3.2",
            "Theorem 4.7"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.2.1–5",
          "role": "proof comparison",
          "correspondence": "weaker for 3.2; exact C*-scope for 4.7",
          "explanation": "The free source supplies the C*-positive functional norm and Schwarz arguments; the retained algebraic and involutive Banach generality is not replaced."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.3",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-03::Lemma 3.3",
      "anchor": "oa-fnd-gn-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 3.3** (The left kernel). Let \\(\\omega\\) be positive and \\(N_\\omega=\\{x\\in A:\\omega(x^*x)=0\\}\\). Then \\(N_\\omega=\\{x\\in A:\\omega(y^*x)=0\\text{ for all }y\\in A\\}\\), and \\(N_\\omega\\) is a left ideal of \\(A\\). Likewise \\(\\{x:\\omega(xx^*)=0\\}=N_\\omega^*\\) is a right ideal.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 105,
        "through_line": 110,
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      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
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              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 3.4",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-03::Examples 3.4",
      "anchor": "oa-fnd-gn-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 3.4.**\n1. Vector functionals are positive.\n2. Let \\(X\\) be a locally compact Hausdorff space and \\(\\mu\\) a finite positive Radon measure on \\(X\\). Then \\(\\omega_\\mu(f)=\\int f\\,d\\mu\\) is a positive functional on \\(C_0(X)\\), since \\(\\omega_\\mu(\\bar ff)=\\int|f|^2d\\mu\\). It is faithful exactly when every nonempty open set has positive measure. If some nonempty open \\(U\\) has \\(\\mu(U)=0\\), Urysohn's lemma gives \\(f\\ne0\\) vanishing outside \\(U\\), with \\(\\omega_\\mu(\\bar ff)=0\\). Conversely, if \\(f\\ne0\\), then \\(|f|^2>c>0\\) on some nonempty open set, which has positive measure.\n3. Every linear functional on \\(M_n(\\mathbb C)\\) has the form \\(\\omega_\\rho(x)=\\operatorname{Tr}(\\rho x)\\) for exactly one matrix \\(\\rho\\), because the pairing \\((\\rho,x)\\mapsto\\operatorname{Tr}(\\rho x)\\) is nondegenerate. It is positive if and only if \\(\\rho\\ge0\\). If \\(\\rho\\ge0\\), then \\(\\operatorname{Tr}(\\rho x^*x)=\\operatorname{Tr}(x\\rho x^*)\\ge0\\). Conversely, for \\(\\xi\\ne0\\) the operator \\(\\theta_{\\xi,\\xi}=\\|\\xi\\|^{-2}\\theta_{\\xi,\\xi}^*\\theta_{\\xi,\\xi}\\) is of the form \\(x^*x\\), and \\(\\omega_\\rho(\\theta_{\\xi,\\xi})=\\langle\\rho\\xi,\\xi\\rangle\\). A positive \\(\\omega_\\rho\\) is faithful exactly when \\(\\rho\\) is invertible: if \\(\\rho\\ge c1\\) with \\(c>0\\), then \\(\\operatorname{Tr}(x\\rho x^*)\\ge c\\operatorname{Tr}(xx^*)\\), and if \\(\\rho\\xi=0\\) with \\(\\xi\\ne0\\), then \\(\\omega_\\rho(\\theta_{\\xi,\\xi}^*\\theta_{\\xi,\\xi})=\\|\\xi\\|^2\\langle\\rho\\xi,\\xi\\rangle=0\\).\n4. (Zero products.) Let \\(E\\) be a Banach space with an isometric conjugate-linear map \\(x\\mapsto x^*\\) satisfying \\(x^{**}=x\\), and make it an involutive Banach algebra with the zero product \\(xy=0\\). Every \\(x^*x\\) is \\(0\\), so every linear functional is positive. On \\(E=\\mathbb C\\) with complex conjugation, \\(\\omega(x)=ix\\) is positive but not hermitian: \\(\\omega(1^*)=i\\) while \\(\\overline{\\omega(1)}=-i\\). On an infinite-dimensional \\(E\\), such as \\(\\ell^2(\\mathbb N)\\) with coordinatewise conjugation, any linear functional that is not continuous (Example 6.5) is positive. So without an identity or an approximate identity, positivity implies neither the hermitian property nor continuity.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 111,
        "through_line": 116,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
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              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.1",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Lemma 4.1",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 4.1** (Square roots near the identity). Let \\(A\\) be unital and \\(a\\in A\\) with \\(r(1-a)<1\\). Put \\(c_0=1\\) and \\(c_n=\\binom{1/2}{n}=\\frac{(1/2)(1/2-1)\\cdots(1/2-n+1)}{n!}\\) for \\(n\\geq1\\); the coefficient identity needed below is proved in the final background list. The series \\(b=\\sum_{n\\ge0}c_n(a-1)^n\\) converges absolutely in \\(A\\), \\(b^2=a\\), and \\(b\\) commutes with every element that commutes with \\(a\\). If \\(a=a^*\\), then \\(b=b^*\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 123,
        "through_line": 133,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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      "source_uses": [
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          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
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              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.2",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Proposition 4.2",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 4.2** (Positive functionals on unital algebras). Let \\(A\\) be unital and \\(\\omega\\) positive.\n1. For \\(h\\in A_h\\), \\(\\omega(h)\\) is real and \\(-r(h)\\,\\omega(1)\\le\\omega(h)\\le r(h)\\,\\omega(1)\\).\n2. For \\(x\\in A\\), \\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\omega(1)\\,\\omega(x^*x)\\\\\n\\le\\omega(1)^2\\,r(x^*x)\\\\\n\\le\\omega(1)^2\\|x\\|^2.\n\\end{gathered}\n\\]\n3. \\(\\omega\\) is bounded, and \\(\\|\\omega\\|\\le\\omega(1)\\le\\|1\\|\\,\\|\\omega\\|\\). In particular \\(\\|\\omega\\|=\\omega(1)\\) when \\(\\|1\\|=1\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 134,
        "through_line": 149,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
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              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.3",
      "kind": "example",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Example 4.3",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Example 4.3** (The norm of the identity matters). Let \\(A=\\mathbb C^2\\) with coordinatewise product and conjugation and the norm \\(\\|(a,b)\\|=|a|+|b|\\). It is an involutive Banach algebra whose identity \\((1,1)\\) has norm \\(2\\). The functional \\(\\omega(a,b)=a+b\\) is positive, since \\(\\omega(|a|^2,|b|^2)=|a|^2+|b|^2\\). But \\(\\|\\omega\\|=\\sup|a+b|/(|a|+|b|)=1\\), while \\(\\omega(1)=2\\). So \\(\\|\\omega\\|<\\omega(1)\\) can happen when \\(\\|1\\|>1\\), and the second inequality in Proposition 4.2(3) is attained here.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 150,
        "through_line": 151,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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      "source_uses": [
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          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
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              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
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                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.4a",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Lemma 4.4a",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 4.4a** (Self-adjoint roots by spectral separation). Let \\(A\\ne\\{0\\}\\) be a unital complex Banach algebra with an algebraic involution. If \\(a=a^*\\) and\n\\[\n\\sigma_A(a)\\cap(-\\infty,0]=\\varnothing,\n\\]\nthen there is a self-adjoint \\(b\\) with \\(b^2=a\\). It is the unique square root of \\(a\\) whose spectrum lies in the open right half-plane, and it commutes with every element commuting with \\(a\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 158,
        "through_line": 203,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.4b",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Theorem 4.4b",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 4.4b** (Automatic continuity with an arbitrary involution). Let \\(A\\) be a unital complex Banach algebra with an algebraic involution, and let \\(\\omega:A\\to\\mathbb C\\) be complex linear with \\(\\omega(x^*x)\\ge0\\) for every \\(x\\in A\\). Then \\(\\omega\\) is continuous. More precisely, there is a finite constant \\(C\\), depending only on the normed algebra and its involution, such that\n\\[\n\\begin{gathered}\n|\\omega(a)|\\le C\\omega(1)\\|a\\|\\\\\n(a\\in A)\n\\end{gathered}\n\\tag{4.4b}\n\\]\nfor every such \\(\\omega\\). If \\(\\omega(1)=0\\), then \\(\\omega=0\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 204,
        "through_line": 282,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.4c",
      "kind": "corollary",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Corollary 4.4c",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Corollary 4.4c** (Spectral estimates and familiar special cases). Under the assumptions of Theorem 4.4b,\n\\[\n\\begin{aligned}\n|\\omega(x)|^2\n&\\le\\omega(1)\\omega(x^*x)\\\\\n&\\le\\omega(1)^2 r(x^*x).\n\\end{aligned}\n\\]\nIf \\(x\\) is normal, then \\(|\\omega(x)|\\le\\omega(1)r(x)\\). If \\(\\|x^*\\|\\le\\beta\\|x\\|\\) for all \\(x\\), then \\(\\|\\omega\\|\\le\\sqrt\\beta\\,\\omega(1)\\). If \\(A\\) is commutative, then\n\\[\n\\|\\omega\\|\\le\\omega(1)\\le\\|1\\|\\,\\|\\omega\\|;\n\\]\nin particular \\(\\|\\omega\\|=\\omega(1)\\) when \\(\\|1\\|=1\\), even without a continuous involution.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 283,
        "through_line": 311,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.5",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Lemma 4.5",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 4.5.** Let \\(\\omega\\) be a positive functional on \\(A\\); no identity and no continuity are assumed. For every \\(a\\in A\\) the functional \\(\\omega_a(x)=\\omega(axa^*)\\) is positive and bounded, with \\(\\|\\omega_a\\|\\le\\omega(aa^*)\\). Consequently\n\\[\n\\begin{gathered}\n\\omega(x^*a^*ax)\\\\\n\\le\\|a\\|^2\\,\\omega(x^*x)\\\\\n(a,x\\in A).\n\\end{gathered}\n\\tag{4.1}\n\\]",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 314,
        "through_line": 325,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.6",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Proposition 4.6",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 4.6.** Let \\(A\\) have an approximate identity \\((u_i)\\) bounded by \\(\\gamma\\), and let \\(\\omega\\) be positive and continuous. Then \\(\\omega\\) is hermitian, and\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\gamma^2\\|\\omega\\|\\,\\omega(x^*x)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{4.2}\n\\]",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 328,
        "through_line": 341,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.7",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-04::Theorem 4.7",
      "anchor": "oa-fnd-gn-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 4.7.** Let \\(A\\) be a C\\*-algebra and \\(\\omega\\) a positive linear functional on \\(A\\).\n1. \\(\\omega\\) is bounded. More precisely, \\(M=\\sup\\{\\omega(a):a\\in A_+,\\ \\|a\\|\\le1\\}\\) is finite and \\(\\|\\omega\\|\\le2M\\).\n2. If \\((u_i)\\) is a net of positive contractions with \\(u_ix\\to x\\) for every \\(x\\), then \\(\\|\\omega\\|=\\lim_i\\omega(u_i)=M\\). If \\(A\\) is unital and nonzero, \\(\\|\\omega\\|=\\omega(1)\\).\n3. \\(\\omega\\) is hermitian, and \\(|\\omega(x)|^2\\le\\|\\omega\\|\\,\\omega(x^*x)\\).\n4. If \\(\\psi\\) is also positive, then \\(\\|\\omega+\\psi\\|=\\|\\omega\\|+\\|\\psi\\|\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 344,
        "through_line": 356,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Proposition 3.2",
            "Theorem 4.7"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.2.1–5",
          "role": "proof comparison",
          "correspondence": "weaker for 3.2; exact C*-scope for 4.7",
          "explanation": "The free source supplies the C*-positive functional norm and Schwarz arguments; the retained algebraic and involutive Banach generality is not replaced."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Construction 5.1",
      "kind": "construction",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-05::Construction 5.1",
      "anchor": "oa-fnd-gn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Construction 5.1.** By Lemma 3.3, the left kernel \\(N_\\omega\\) is a left ideal, and \\(\\omega(y^*x)=0\\) whenever \\(x\\in N_\\omega\\) or \\(y\\in N_\\omega\\); for \\(y\\in N_\\omega\\) use the first identity in (3.1). So\n\\[ \\langle x+N_\\omega,\\,y+N_\\omega\\rangle=\\omega(y^*x) \\tag{5.1} \\]\nis a well-defined inner product on the quotient space \\(A/N_\\omega\\); it is positive definite by the definition of \\(N_\\omega\\). The [completion lemma in Section 1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#completing-normed-and-inner-product-spaces) constructs the Hilbert completion \\(H_\\omega\\). Let \\(\\Lambda_\\omega:A\\to H_\\omega\\), \\(\\Lambda_\\omega(x)=x+N_\\omega\\). Then \\(\\Lambda_\\omega\\) is linear with dense range, and \\(\\langle\\Lambda_\\omega(x),\\Lambda_\\omega(y)\\rangle=\\omega(y^*x)\\).",
      "proof_locus": {
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        "line": 381,
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          "role": "proof construction comparison",
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          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
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          "role": "statement comparison",
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          "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
        },
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          "role": "permitted adaptation / proof comparison",
          "correspondence": "weaker outside the unital C*-case",
          "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.2",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-05::Lemma 5.2",
      "anchor": "oa-fnd-gn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 5.2.** For each \\(a\\in A\\) there is exactly one \\(\\pi_\\omega(a)\\in B(H_\\omega)\\) with \\(\\pi_\\omega(a)\\Lambda_\\omega(x)=\\Lambda_\\omega(ax)\\) for all \\(x\\in A\\). It satisfies \\(\\|\\pi_\\omega(a)\\|\\le\\|a\\|\\), and \\(\\pi_\\omega\\) is a representation of \\(A\\) on \\(H_\\omega\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 385,
        "through_line": 397,
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          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
          "correspondence": "weaker",
          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
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          "source_key": "erdman-faoa-2015@F320B16AF7448FBB43582C21569840FE657FCCF6F31D97F176913FDD0E1EB823",
          "source_locus": "GNS_construction.tex, theorem label 0029",
          "role": "statement comparison",
          "correspondence": "weaker",
          "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
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          "source_locus": "cstar.tex, omega-norm-basic, gns, rho-omega-miu and proto-gelfand-naimark",
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          "correspondence": "weaker outside the unital C*-case",
          "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
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                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "4. Continuity and norms of positive functionals",
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              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.3",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-05::Theorem 5.3",
      "anchor": "oa-fnd-gn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 5.3** (When the construction has a cyclic vector). The following are equivalent:\n- (i) there are a representation \\((\\pi,H)\\) of \\(A\\) and a vector \\(\\xi\\in H\\) with \\(\\omega(x)=\\langle\\pi(x)\\xi,\\xi\\rangle\\) for all \\(x\\);\n- (ii) there is a constant \\(C\\) with \\(|\\omega(x)|^2\\le C\\,\\omega(x^*x)\\) for all \\(x\\);\n- (iii) there is a vector \\(\\xi_\\omega\\in H_\\omega\\) with \\(\\omega(x)=\\langle\\Lambda_\\omega(x),\\xi_\\omega\\rangle\\) for all \\(x\\).\n\nWhen they hold, \\(\\xi_\\omega\\) is unique, \\(\\pi_\\omega(x)\\xi_\\omega=\\Lambda_\\omega(x)\\) for all \\(x\\), \\(\\xi_\\omega\\) is a cyclic vector for \\(\\pi_\\omega\\), and \\(\\omega(x)=\\langle\\pi_\\omega(x)\\xi_\\omega,\\xi_\\omega\\rangle\\). The smallest possible \\(C\\) in (ii) is \\(\\|\\xi_\\omega\\|^2\\). Moreover \\(\\omega\\) is hermitian and bounded, with \\(\\|\\omega\\|\\le\\|\\xi_\\omega\\|^2\\).\n\nWe call \\(\\omega\\) *representable* when these conditions hold.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 398,
        "through_line": 427,
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          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
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          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.4",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-05::Theorem 5.4",
      "anchor": "oa-fnd-gn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 5.4** (GNS with an approximate identity). Let \\(A\\) have an approximate identity \\((u_i)\\) bounded by \\(\\gamma\\), and let \\(\\omega\\) be positive and continuous. Then \\(\\omega\\) is representable, \\(\\|\\omega\\|\\le\\|\\xi_\\omega\\|^2\\le\\gamma^2\\|\\omega\\|\\), and \\(\\xi_\\omega=\\lim_i\\Lambda_\\omega(u_i)\\). If \\(\\gamma=1\\), in particular if \\(A\\) is a C\\*-algebra, then \\(\\|\\xi_\\omega\\|^2=\\|\\omega\\|\\). If \\(A\\) is unital, every positive \\(\\omega\\) is representable, with \\(\\xi_\\omega=\\Lambda_\\omega(1)\\) and \\(\\|\\xi_\\omega\\|^2=\\omega(1)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 428,
        "through_line": 431,
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          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
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          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_scopes": [
            {
              "heading": "3. Positive functionals",
              "line": 77,
              "through_line": 116,
              "anchors": [
                "OA-FND-GN-03"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "4. Continuity and norms of positive functionals",
              "line": 117,
              "through_line": 356,
              "anchors": [
                "OA-FND-GN-04"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
              "through_line": 452,
              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.5",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-05::Theorem 5.5",
      "anchor": "oa-fnd-gn-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 5.5** (Uniqueness). Let \\(\\omega\\) be representable, and let \\((\\pi,H)\\) be a representation with a cyclic vector \\(\\xi\\) such that \\(\\omega(x)=\\langle\\pi(x)\\xi,\\xi\\rangle\\) for all \\(x\\). There is exactly one unitary \\(U:H_\\omega\\to H\\) with \\(U\\Lambda_\\omega(x)=\\pi(x)\\xi\\) for all \\(x\\). It intertwines \\(\\pi_\\omega\\) and \\(\\pi\\), and \\(U\\xi_\\omega=\\xi\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 432,
        "through_line": 442,
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          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
          "correspondence": "weaker",
          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
        },
        {
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            "Construction 5.1",
            "Lemma 5.2",
            "Theorem 5.5"
          ],
          "source_key": "erdman-faoa-2015@F320B16AF7448FBB43582C21569840FE657FCCF6F31D97F176913FDD0E1EB823",
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          "role": "statement comparison",
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          "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
        },
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "5. The Gelfand–Naimark–Segal construction",
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
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                "OA-FND-GN-07"
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          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
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          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.6",
      "kind": "definition",
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      "statement_and_full_conditions": "**Definition 5.6.** For a representable \\(\\omega\\), the triple \\((\\pi_\\omega,H_\\omega,\\xi_\\omega)\\) is the *GNS triple* of \\(\\omega\\), and \\(\\pi_\\omega\\) is its *GNS representation*; we also call it the cyclic representation defined by \\(\\omega\\).\n\nTheorems 5.3–5.5 give a bijection between representable positive functionals and unitary equivalence classes of representations with a distinguished cyclic vector: \\(\\omega\\) goes to its GNS triple, and a triple \\((\\pi,H,\\xi)\\) goes back to the vector functional \\(\\omega_\\xi\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 443,
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "5. The Gelfand–Naimark–Segal construction",
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                "OA-FND-GN-05"
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
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                "OA-FND-GN-07"
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          "source_locus": "§7.8–§7.12; PDF 39–41",
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        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Examples 5.7",
      "kind": "examples",
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      "statement_and_full_conditions": "**Examples 5.7.**\n1. If \\((\\pi,H)\\) has a cyclic vector \\(\\xi\\), the GNS triple of \\(\\omega_\\xi\\) is \\((\\pi,H,\\xi)\\), up to the unitary of Theorem 5.5.\n2. Let \\(\\mu\\) be a finite positive Radon measure on a locally compact Hausdorff space \\(X\\) and \\(\\omega(f)=\\int f\\,d\\mu\\) on \\(C_0(X)\\). Then \\(\\|\\Lambda_\\omega(f)\\|^2=\\int|f|^2d\\mu\\), so \\(\\Lambda_\\omega(f)\\mapsto f\\) is an isometry into \\(L^2(\\mu)\\). Its range contains \\(C_c(X)\\), which is dense in \\(L^2(\\mu)\\) ([approximation by continuous functions](haar-measure.md#oa-fnd-hm-02)). So \\(H_\\omega=L^2(\\mu)\\), \\(\\pi_\\omega(f)\\) is multiplication by \\(f\\), and \\(\\xi_\\omega\\) is the constant function \\(1\\), which lies in \\(L^2(\\mu)\\) because \\(\\mu\\) is finite: indeed \\(\\langle f,1\\rangle=\\int f\\,d\\mu=\\omega(f)\\). The left kernel consists of the functions that vanish on the support of \\(\\mu\\). Indeed, a continuous function nonzero at a support point is bounded away from zero on a neighborhood of positive measure. Conversely, the complement of the support is the union of open null sets; each compact subset of that union has a finite null cover, so Radon inner regularity makes the union null. For a finitely supported \\(\\mu\\), \\(\\dim H_\\omega\\) is the number of points in the support.\n3. On \\(M_n(\\mathbb C)\\), let \\(\\tau=\\frac1n\\operatorname{Tr}\\). Then \\(H_\\tau=M_n(\\mathbb C)\\) with \\(\\langle x,y\\rangle=\\frac1n\\operatorname{Tr}(y^*x)\\), \\(\\pi_\\tau(a)\\) is left multiplication by \\(a\\), and \\(\\xi_\\tau=1\\). Right multiplications commute with \\(\\pi_\\tau\\), so for \\(n\\ge2\\) the commutant is not \\(\\mathbb C1\\) and \\(\\pi_\\tau\\) is reducible. For \\(\\omega(x)=\\langle xe_1,e_1\\rangle\\), the left kernel is \\(\\{x:xe_1=0\\}\\), the map \\(\\Lambda_\\omega(x)\\mapsto xe_1\\) identifies \\(H_\\omega\\) with \\(\\mathbb C^n\\), \\(\\pi_\\omega\\) becomes the identity representation, and \\(\\xi_\\omega=e_1\\). Exercise 12.1 treats all positive functionals on \\(M_n(\\mathbb C)\\).\n4. Continuity is not enough without an approximate identity. On the algebra \\(\\mathbb C\\) with the zero product (Example 3.4(4)), \\(\\omega(x)=x\\) is positive and continuous. Its left kernel is everything, so \\(H_\\omega=\\{0\\}\\), and \\(\\omega\\) is not representable. In fact every representation of this algebra is zero, since \\(\\|\\pi(x)\\|^2=\\|\\pi(x^*x)\\|=0\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 447,
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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                "OA-FND-GN-04"
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
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              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
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              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
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          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.1",
      "kind": "lemma",
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      "anchor": "oa-fnd-gn-07",
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      "statement_and_full_conditions": "**Lemma 7.1** (Enough positive functionals). Let \\(A\\) be a C\\*-algebra and \\(a\\in A_+\\), \\(a\\ne0\\). Some positive linear functional \\(f\\) on \\(A\\) has \\(\\|f\\|\\le1\\) and \\(f(a)>0\\).",
      "proof_locus": {
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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            {
              "heading": "5. The Gelfand–Naimark–Segal construction",
              "line": 377,
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              "anchors": [
                "OA-FND-GN-05"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            },
            {
              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
              "line": 453,
              "through_line": 488,
              "anchors": [
                "OA-FND-GN-07"
              ],
              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.1a",
      "kind": "proposition",
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      "anchor": "oa-fnd-gn-07",
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      "statement_and_full_conditions": "**Proposition 7.1a** (A state that attains a positive norm). If \\(0\\ne a\\in A_+\\), there is a state \\(\\omega\\) on \\(A\\) with \\(\\omega(a)=\\|a\\|\\). More generally, a state on a unital C*-subalgebra with the same identity extends to a state on the containing unital C*-algebra.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 468,
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          "role": "alternative proof and strengthening",
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          "explanation": "Added norm-at-one positivity criterion, common-unit state extension and a norming state for any nonzero positive element, with the full independent proof. The old separating-functional argument remains."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
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              "heading": "3. Positive functionals",
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
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              "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
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          "source_locus": "§7.8–§7.12; PDF 39–41",
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        }
      ],
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      "global_tag_allocated": false
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    {
      "local_label": "Theorem 7.2",
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      "statement_and_full_conditions": "**Theorem 7.2** (Gelfand–Naimark). Every C\\*-algebra \\(A\\) has a faithful representation. Hence \\(A\\) is isometrically \\(*\\)-isomorphic to a norm-closed \\(*\\)-subalgebra of \\(B(H)\\) for some Hilbert space \\(H\\).",
      "proof_locus": {
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                "OA-FND-GN-07"
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              "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§7.8–§7.12; PDF 39–41",
          "role": "proof comparison",
          "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
          "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.1",
      "kind": "definition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-02::Definition 2.1",
      "anchor": "oa-fnd-gn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Definition 2.1.** A representation \\((\\pi,H)\\) of a \\(*\\)-algebra is *irreducible* if it is not the zero representation and its only closed invariant subspaces are \\(\\{0\\}\\) and \\(H\\).\n\nAn irreducible representation \\(\\pi\\) is nondegenerate, because its essential subspace is invariant and not \\(\\{0\\}\\). By Proposition 1.4, every nonzero vector \\(\\xi\\) then has \\(\\pi(A)\\xi\\ne\\{0\\}\\), and the closure of \\(\\pi(A)\\xi\\) is a nonzero invariant subspace. So every nonzero vector is cyclic. The zero representation on a one-dimensional space also has no invariant subspaces other than \\(\\{0\\}\\) and \\(H\\); the definition excludes it on purpose.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 497,
        "through_line": 500,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.2",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-02::Theorem 2.2",
      "anchor": "oa-fnd-gn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 2.2** (Schur's lemma). Let \\(H\\ne\\{0\\}\\) and let \\(\\mathcal S\\subseteq B(H)\\) be a self-adjoint set of operators. The following are equivalent:\n- (i) the only closed subspaces \\(M\\) with \\(TM\\subseteq M\\) for all \\(T\\in\\mathcal S\\) are \\(\\{0\\}\\) and \\(H\\);\n- (ii) \\(\\mathcal S'=\\mathbb C1\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 501,
        "through_line": 512,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.3",
      "kind": "corollary",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-02::Corollary 2.3",
      "anchor": "oa-fnd-gn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Corollary 2.3.** A representation \\(\\pi\\) of a \\(*\\)-algebra is irreducible if and only if \\(\\pi\\ne0\\) and \\(\\pi(A)'=\\mathbb C1\\). Unitarily equivalent representations are irreducible together.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 513,
        "through_line": 516,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 2.4",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-02::Examples 2.4",
      "anchor": "oa-fnd-gn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 2.4.**\n1. For \\(H\\ne\\{0\\}\\), both \\(B(H)\\) and \\(K(H)\\) act irreducibly on \\(H\\). Indeed, let \\(M\\ne\\{0\\}\\) be invariant under all rank-one operators and pick \\(\\xi\\ne0\\) in \\(M\\). Then \\(\\theta_{\\eta,\\xi}\\xi=\\|\\xi\\|^2\\eta\\in M\\) for every \\(\\eta\\), so \\(M=H\\).\n2. An irreducible representation \\(\\pi\\) of a commutative \\(*\\)-algebra is one-dimensional. Indeed \\(\\pi(A)\\subseteq\\pi(A)'=\\mathbb C1\\), so \\(\\pi(x)=\\chi(x)1\\) for a nonzero \\(*\\)-homomorphism \\(\\chi:A\\to\\mathbb C\\). Then every closed subspace is invariant, which forces \\(\\dim H=1\\). Conversely, every nonzero \\(*\\)-homomorphism \\(\\chi:A\\to\\mathbb C\\) is an irreducible representation on \\(\\mathbb C\\). For \\(A=C_0(X)\\) these are the point evaluations \\(f\\mapsto f(x)\\) ([the characters of \\(C_0(X)\\)](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04)).\n3. The representation \\(x\\mapsto x\\oplus x\\) of \\(M_n(\\mathbb C)\\) on \\(\\mathbb C^n\\oplus\\mathbb C^n\\) is not irreducible: \\(\\mathbb C^n\\oplus0\\) is invariant. Its commutant consists of the block matrices \\(\\begin{pmatrix}a1&b1\\\\c1&d1\\end{pmatrix}\\) with scalars \\(a,b,c,d\\), because each block must commute with all of \\(M_n(\\mathbb C)\\), which acts irreducibly on \\(\\mathbb C^n\\) by (1). So the commutant is a copy of \\(M_2(\\mathbb C)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 517,
        "through_line": 521,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.5",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-02::Proposition 2.5",
      "anchor": "oa-fnd-gn-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 2.5** (Restriction to an ideal). Let \\(J\\) be a two-sided ideal of a \\(*\\)-algebra \\(A\\) with \\(J^*=J\\), and let \\((\\pi,H)\\) be an irreducible representation of \\(A\\). Then the restriction \\(\\pi|_J\\) vanishes identically or is irreducible. In particular, for every closed two-sided ideal \\(J\\) of a C\\*-algebra, \\(\\pi|_J\\) is zero or irreducible, since such ideals are self-adjoint ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 522,
        "through_line": 527,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Theorem 6.1",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 6.1** (Cohen–Hewitt factorization). Let \\(B\\) be a Banach algebra with a left approximate identity \\((e_i)\\) bounded by \\(\\gamma\\), and let \\(X\\) be a left Banach \\(B\\)-module. Let \\(v\\in X\\) satisfy \\(e_i\\cdot v\\to v\\). For every \\(\\delta>0\\) there are \\(b\\in B\\) and \\(w\\in X\\) with\n\\[ v=b\\cdot w,\\qquad\\|b\\|\\le\\gamma,\\qquad\\|w-v\\|\\le\\delta. \\]",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 532,
        "through_line": 596,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 6.2",
      "kind": "corollary",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Corollary 6.2",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Corollary 6.2.** Suppose the Banach algebra \\(A\\) has a bounded approximate identity.\n1. Every \\(x\\in A\\) is a product \\(x=yz\\) of two elements of \\(A\\).\n2. If \\(x_n\\to0\\) in \\(A\\), there are \\(a,b\\in A\\) and a sequence \\(y_n\\to0\\) in \\(A\\) with \\(x_n=ay_nb\\) for every \\(n\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 597,
        "through_line": 604,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.3",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Theorem 6.3",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 6.3** (Varopoulos). If an involutive Banach algebra \\(A\\) has a bounded approximate identity, then every positive linear functional on \\(A\\) is continuous.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 605,
        "through_line": 617,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 6.4",
      "kind": "corollary",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Corollary 6.4",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Corollary 6.4.** Suppose the involutive Banach algebra \\(A\\) has an approximate identity bounded by \\(\\gamma\\). Then every positive linear functional \\(\\omega\\) on \\(A\\) is continuous, hermitian and representable, and \\(|\\omega(x)|^2\\le\\gamma^2\\|\\omega\\|\\,\\omega(x^*x)\\). In particular Theorems 5.4 and 5.5 apply to every positive functional on \\(A\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 618,
        "through_line": 621,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.5",
      "kind": "example",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Example 6.5",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Example 6.5** (The approximate identity is needed). Let \\(E=\\ell^2(\\mathbb N)\\) with the zero product and coordinatewise complex conjugation, as in Example 3.4(4). The algebraic basis extension used here follows directly from [Zorn’s lemma](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-01): partially order linearly independent sets containing a given independent family by inclusion. A chain union is independent because every finite relation lies in one chain member. A maximal such set spans the space, since a vector outside its span could be adjoined while preserving independence. Extend the unit vectors \\(e_1,e_2,\\dots\\) to a basis of \\(E\\) as a vector space (a Hamel basis, which exists by Zorn's lemma), and let \\(f\\) be the linear functional with \\(f(e_n)=n\\) and \\(f=0\\) on the remaining basis vectors. Then \\(f\\) is not bounded, and it is positive because every \\(x^*x\\) is \\(0\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 622,
        "through_line": 623,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.3",
      "kind": "definition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Definition 7.3",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Definition 7.3.** An involutive Banach algebra is an *A\\*-algebra* if it has a faithful representation. For an involutive Banach algebra \\(A\\) and \\(x\\in A\\) put\n\\[ \\|x\\|_{\\rm u}=\\sup\\{\\|\\pi(x)\\|:\\pi\\text{ a representation of }A\\}. \\]\nThe possible numbers \\(\\|\\pi(x)\\|\\) form a subset of the set \\([0,\\|x\\|]\\), defined by the condition that a representation attains that number. The set is nonempty because it contains zero, so its supremum is well defined. No direct sum over a proper class of representations is formed.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 634,
        "through_line": 637,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.4",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Proposition 7.4",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 7.4** (The enveloping C\\*-algebra). Let \\(A\\) be an involutive Banach algebra.\n1. \\(\\|\\cdot\\|_{\\rm u}\\) is a seminorm with \\(\\|x\\|_{\\rm u}\\le\\|x\\|\\), \\(\\|xy\\|_{\\rm u}\\le\\|x\\|_{\\rm u}\\|y\\|_{\\rm u}\\), \\(\\|x^*\\|_{\\rm u}=\\|x\\|_{\\rm u}\\) and \\(\\|x^*x\\|_{\\rm u}=\\|x\\|_{\\rm u}^2\\).\n2. \\(I=\\{x:\\|x\\|_{\\rm u}=0\\}\\) is the intersection of the kernels of all representations. It is a closed self-adjoint two-sided ideal. The completion \\(C^*(A)\\) of \\(A/I\\) for the norm \\(\\|x+I\\|=\\|x\\|_{\\rm u}\\) is a C\\*-algebra, the *enveloping C\\*-algebra* of \\(A\\). The map \\(j:A\\to C^*(A)\\), \\(j(x)=x+I\\), is a contractive \\(*\\)-homomorphism with dense range. It is injective exactly when \\(A\\) is an A\\*-algebra.\n3. (Universal property.) For every representation \\(\\pi\\) of \\(A\\) there is exactly one representation \\(\\tilde\\pi\\) of \\(C^*(A)\\) with \\(\\pi=\\tilde\\pi\\circ j\\), and every representation \\(\\sigma\\) of \\(C^*(A)\\) is of this form, with \\(\\pi=\\sigma\\circ j\\). The representations \\(\\pi\\) and \\(\\tilde\\pi\\) have the same closed invariant subspaces, the same essential subspace and the same commutant, so one is irreducible, or nondegenerate, exactly when the other is.\n4. If \\(A\\) is a C\\*-algebra, then \\(\\|x\\|_{\\rm u}=\\|x\\|\\), and \\(j\\) is an isometric \\(*\\)-isomorphism of \\(A\\) onto \\(C^*(A)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 638,
        "through_line": 648,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 7.5",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-06::Examples 7.5",
      "anchor": "oa-fnd-gn-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 7.5.**\n1. For the algebra of Example 3.4(4) with the zero product, every representation vanishes, so \\(C^*(A)=\\{0\\}\\).\n2. Let \\(E\\) be such a zero-product algebra and \\(A=E\\oplus\\mathbb C\\) its unitization. Every representation \\(\\pi\\) vanishes on \\(E\\), because \\(\\|\\pi(e)\\|^2=\\|\\pi(e^*e)\\|=0\\). So \\(C^*(A)=\\mathbb C\\) and \\(j(e+\\lambda)=\\lambda\\): an involutive Banach algebra with an identity need not be an A\\*-algebra.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 649,
        "through_line": 652,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 0.1",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-16::Lemma 0.1",
      "anchor": "oa-fnd-gn-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 0.1** (Vector integration and Hilbert-valued sections). Let \\(E\\) be a Banach space and let \\(\\mu\\) be Haar measure on a locally compact Hausdorff group \\(G\\).\n\n1. Finite-valued simple \\(E\\)-valued functions supported on sets of finite measure have an integral satisfying \\(\\|\\int f\\,d\\mu\\|\\leq\\int\\|f\\|\\,d\\mu\\). Their completion in this last norm is \\(L^1(G,E)\\). Integration and every bounded linear map extend to the completion, and bounded linear maps commute with integration. The space \\(C_c(G,E)\\) is dense there.\n2. For an \\(E\\)-valued integrand on a Radon product that is a norm limit in \\(L^1\\) of simple functions, with sigma-finite support, the two iterated Bochner integrals exist almost everywhere and equal the product integral. In particular this holds for the compactly supported continuous integrands and their \\(L^1\\) limits used below.\n3. For a Hilbert space \\(K\\), finite-valued simple functions complete in the norm \\((\\int\\|f\\|^2)^{1/2}\\) to a Hilbert space \\(L^2(G,K)\\). The finite sums \\(\\sum_jh_j\\eta_j\\), with \\(h_j\\in C_c(G)\\) and \\(\\eta_j\\in K\\), are dense. The map \\(h\\otimes\\eta\\mapsto[s\\mapsto h(s)\\eta]\\) extends to a unitary \\(L^2(G)\\otimes K\\to L^2(G,K)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 657,
        "through_line": 699,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.6",
      "kind": "example",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-08::Example 7.6",
      "anchor": "oa-fnd-gn-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Example 7.6** (Group algebras). Let \\(G\\) be a locally compact group with left Haar measure \\(ds\\) and modular function \\(\\Delta\\), and let \\(L^1(G)\\) be the involutive Banach algebra with convolution \\((f*g)(t)=\\int f(s)g(s^{-1}t)\\,ds\\) and involution \\(f^*(t)=\\Delta(t)^{-1}\\overline{f(t^{-1})}\\) ([group algebras and transformation-group algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-20)). It has an approximate identity of norm one ([approximate identities of convolution algebras](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-23)), so by Corollary 6.4 all its positive functionals are continuous and representable. The fully proved [group-representation lesson](24-group-representation-completions.md#oa-flow.grp.integration), in *Crossed products and the flow of weights*, supplies the following for every locally compact group. Its exact Haar and operator inputs are the full programme providers listed at the end of this lesson, together with Lemma 0.1 above.\n- The left regular representation \\(\\lambda(f)\\xi=f*\\xi\\) on \\(L^2(G)\\) is faithful and nondegenerate. So \\(L^1(G)\\) is an A\\*-algebra.\n- For a strongly continuous unitary representation \\(U\\) of \\(G\\), the formula \\(\\langle\\pi_U(f)\\xi,\\eta\\rangle=\\int f(s)\\langle U(s)\\xi,\\eta\\rangle\\,ds\\) defines a nondegenerate representation \\(\\pi_U\\) of \\(L^1(G)\\). Every nondegenerate representation \\(\\pi\\) of \\(L^1(G)\\) equals \\(\\pi_U\\) for exactly one such \\(U\\), recovered from \\(U(s)\\pi(f)\\xi=\\pi(\\lambda_sf)\\xi\\) with \\((\\lambda_sf)(t)=f(s^{-1}t)\\). A bounded operator intertwines \\(U\\) and \\(V\\) exactly when it intertwines \\(\\pi_U\\) and \\(\\pi_V\\); so the correspondence preserves unitary equivalence and irreducibility.\n\nThe enveloping C\\*-algebra \\(C^*(G)=C^*(L^1(G))\\) is the *group C\\*-algebra*. By Proposition 1.4, a degenerate representation of \\(L^1(G)\\) is a nondegenerate one plus a zero summand, so \\(\\|f\\|_{\\rm u}=\\sup_U\\|\\pi_U(f)\\|\\) over strongly continuous unitary representations \\(U\\). The norm closure of \\(\\lambda(L^1(G))\\) is the *reduced group C\\*-algebra* \\(C^*_r(G)\\). By Proposition 7.4(3), \\(\\lambda\\) extends to a \\(*\\)-homomorphism of \\(C^*(G)\\) onto \\(C^*_r(G)\\): its range is closed ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)) and contains the dense set \\(\\lambda(L^1(G))\\). The [four-neighbor-tree calculation in that lesson](24-group-representation-completions.md#oa-flow.grp.models) proves noninjectivity for the free group on two generators: the sum of the four generator and inverse point masses has full norm \\(4\\) and reduced norm \\(2\\sqrt3\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 702,
        "through_line": 707,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.7",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-08::Proposition 7.7",
      "anchor": "oa-fnd-gn-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 7.7** (Transformation-group algebras). Let a locally compact group \\(G\\) act continuously on the right of a locally compact Hausdorff space \\(\\Omega\\), \\((\\omega,s)\\mapsto\\omega s\\), and put \\((\\alpha_sf)(\\omega)=f(\\omega s)\\) for \\(f\\in C_0(\\Omega)\\). Let \\(\\mathfrak A(\\Omega,G)\\) be the involutive Banach algebra obtained by completing \\(C_c(\\Omega\\times G)\\) for \\(\\|x\\|_1=\\int_G\\sup_\\omega|x(\\omega,s)|\\,ds\\), with\n\\[\n\\begin{gathered}\n(x\\star y)(\\omega,s)\\\\\n=\\int_Gx(\\omega,t)\\,y(\\omega t,t^{-1}s)\\,dt,\\\\\nx^\\sharp(\\omega,s)\\\\\n=\\Delta(s)^{-1}\\overline{x(\\omega s,s^{-1})}\n\\end{gathered}\n\\]\n([group algebras and transformation-group algebras](banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory.md#oa-fnd-bn-20)). A *covariant pair* \\((\\rho,U)\\) on \\(H\\) consists of a nondegenerate representation \\(\\rho\\) of \\(C_0(\\Omega)\\) and a strongly continuous unitary representation \\(U\\) of \\(G\\) on \\(H\\) with\n\\[\n\\begin{gathered}\nU(s)\\rho(f)U(s)^*\\\\\n=\\rho(\\alpha_sf)\\\\\n(f\\in C_0(\\Omega),\\ s\\in G).\n\\end{gathered}\n\\tag{7.1}\n\\]\nIts *integrated form* on \\(x\\in C_c(\\Omega\\times G)\\) is defined weakly by \\(\\langle\\pi(x)\\xi,\\eta\\rangle=\\int_G\\langle\\rho(x(\\cdot,s))U(s)\\xi,\\eta\\rangle\\,ds\\), where \\(x(\\cdot,s)\\in C_c(\\Omega)\\). Then, for \\(f\\in C_0(\\Omega)\\), \\(t\\in G\\) and \\(x\\in C_c(\\Omega\\times G)\\),\n\\[\n\\begin{gathered}\n\\rho(f)\\pi(x)\\\\\n=\\pi(L_fx),\\\\\nU(t)\\pi(x)\\\\\n=\\pi(M_tx),\\\\\n(L_fx)(\\omega,s)\\\\\n=f(\\omega)x(\\omega,s),\\\\\n(M_tx)(\\omega,s)\\\\\n=x(\\omega t,t^{-1}s).\n\\end{gathered}\n\\tag{7.2}\n\\]\n\nThe identities (7.2) follow directly from the integrated convention (7.1); the [three-moving-points model](25-cstar-covariant-representations.md#oa-flow.ccov.transformations) gives an explicit finite check of both coordinate shifts.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 708,
        "through_line": 760,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.8",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-09::Proposition 7.8",
      "anchor": "oa-fnd-gn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 7.8.**\n1. Every cyclic representation of a separable involutive Banach algebra acts on a separable Hilbert space.\n2. A separable C\\*-algebra \\(A\\ne0\\) has a faithful state. More generally, a separable A\\*-algebra has a faithful continuous positive functional.\n3. Every separable A\\*-algebra, and in particular every separable C\\*-algebra, can be represented faithfully on some separable Hilbert space.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 763,
        "through_line": 779,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 7.9",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-09::Examples 7.9",
      "anchor": "oa-fnd-gn-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 7.9** (Separability is not necessary, but some countability is).\n1. \\(B(\\ell^2(\\mathbb N))\\) is not separable: the diagonal operators with entries in \\(\\{0,1\\}\\) form an uncountable family at pairwise norm distance one, whereas a countable dense set would give distinct approximating points at distance less than \\(1/3\\) for every member of that family. Nevertheless, for an orthonormal basis \\((e_n)_{n\\ge1}\\) of \\(\\ell^2(\\mathbb N)\\), \\(\\omega=\\sum_{n\\ge1}2^{-n}\\omega_{e_n}\\) is a faithful state, since \\(\\omega(x^*x)=\\sum_n2^{-n}\\|xe_n\\|^2\\), and the identity representation is faithful on a separable space.\n2. Let \\(\\Gamma\\) be an uncountable set and \\(A=c_0(\\Gamma)\\), the functions on \\(\\Gamma\\) that tend to zero at infinity. Suppose \\(\\omega\\) were a faithful positive functional. The numbers \\(\\omega(\\delta_\\gamma)\\), \\(\\gamma\\in\\Gamma\\), would all be positive, so some \\(m\\) would have \\(\\omega(\\delta_\\gamma)>1/m\\) for infinitely many \\(\\gamma\\). For a finite set \\(F\\) of such \\(\\gamma\\), the element \\(\\sum_{\\gamma\\in F}\\delta_\\gamma\\) has norm one, while \\(\\omega(\\sum_{\\gamma\\in F}\\delta_\\gamma)>|F|/m\\), which exceeds \\(\\|\\omega\\|\\) for large \\(|F|\\). And a faithful representation \\(\\pi\\) would give uncountably many nonzero, mutually orthogonal projections \\(\\pi(\\delta_\\gamma)\\), which a separable Hilbert space cannot carry.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 780,
        "through_line": 783,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-10::Lemma 8.1",
      "anchor": "oa-fnd-gn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 8.1** (Radon–Nikodym lemma for positive functionals). Let \\(\\varphi\\) be a representable positive functional on \\(A\\), where \\(A\\) is an involutive Banach algebra; write \\((\\pi,H,\\xi)\\) for its GNS triple.\n1. For \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\), the functional \\(\\varphi_T(x)=\\langle\\pi(x)T\\xi,\\xi\\rangle\\) is positive, \\(\\varphi_T\\le\\varphi\\), and \\(\\varphi_T(y^*x)=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\). The map \\(T\\mapsto\\varphi_T\\) is affine and injective.\n2. If \\(\\psi\\) is positive and \\(\\psi\\le\\varphi\\), there is exactly one \\(T\\in\\pi(A)'\\) with \\(0\\le T\\le1\\) and \\(\\psi(y^*x)=\\langle T\\pi(x)\\xi,\\pi(y)\\xi\\rangle\\) for all \\(x,y\\in A\\).\n3. If every element of \\(A\\) is a product of two elements, then \\(\\psi=\\varphi_T\\) in (2), and \\(T\\mapsto\\varphi_T\\) is a bijection from \\(\\{T\\in\\pi(A)':0\\le T\\le1\\}\\) onto \\(\\{\\psi\\text{ positive}:\\psi\\le\\varphi\\}\\). This applies when \\(A\\) has a bounded approximate identity (Corollary 6.2), in particular to C\\*-algebras.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 788,
        "through_line": 812,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Lemma 8.1",
            "Theorem 8.3",
            "Theorem 8.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.4.6, II.6.4.8–10",
          "role": "proof comparison",
          "correspondence": "weaker outside C*-algebras",
          "explanation": "Free commutant Radon–Nikodym and pure-state proofs match the C*-case. The retained Banach version uses its proved factorization and represents dominated functionals on products before extending equality to all elements."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 8.2",
      "kind": "definition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-10::Definition 8.2",
      "anchor": "oa-fnd-gn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Definition 8.2.** A positive functional \\(\\varphi\\) is *pure* if every positive \\(\\psi\\le\\varphi\\) is a scalar multiple of \\(\\varphi\\). If \\(\\varphi(x^*x)>0\\) for some \\(x\\), the multiple is \\(\\lambda\\varphi\\) with \\(0\\le\\lambda\\le1\\), since both \\(\\lambda\\varphi\\) and \\((1-\\lambda)\\varphi\\) are positive. This holds for every representable \\(\\varphi\\ne0\\): then \\(\\xi_\\varphi\\ne0\\), and \\(\\varphi(x^*x)=\\|\\pi_\\varphi(x)\\xi_\\varphi\\|^2\\) cannot vanish for all \\(x\\), because \\(\\pi_\\varphi(A)\\xi_\\varphi\\) is dense in \\(H_\\varphi\\). Without such an \\(x\\) the multiple need not lie in \\([0,1]\\): on \\(\\mathbb C\\) with the zero product (Example 3.4(4)), \\(\\varphi(x)=x\\) is pure, and \\(2\\varphi\\le\\varphi\\). A *pure state* is a state that is pure. \\(P(A)\\) denotes the set of pure states.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 815,
        "through_line": 816,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.3",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-10::Theorem 8.3",
      "anchor": "oa-fnd-gn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 8.3.** Suppose the involutive Banach algebra \\(A\\) has a bounded approximate identity (for instance, \\(A\\) is a C\\*-algebra), and let \\(\\varphi\\ne0\\) be a positive functional on \\(A\\). Then \\(\\varphi\\) is pure if and only if its GNS representation \\(\\pi_\\varphi\\) is irreducible.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 817,
        "through_line": 822,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Lemma 8.1",
            "Theorem 8.3",
            "Theorem 8.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.4.6, II.6.4.8–10",
          "role": "proof comparison",
          "correspondence": "weaker outside C*-algebras",
          "explanation": "Free commutant Radon–Nikodym and pure-state proofs match the C*-case. The retained Banach version uses its proved factorization and represents dominated functionals on products before extending equality to all elements."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.4",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-10::Proposition 8.4",
      "anchor": "oa-fnd-gn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 8.4** (Pure states as extreme points). Let \\(A\\) be a C\\*-algebra, and let \\(Q(A)\\) be the set of positive functionals of norm at most one, with the weak\\* topology.\n1. \\(Q(A)\\) is compact and convex.\n2. The extreme points of \\(Q(A)\\) are \\(0\\) and the pure states.\n3. A state is pure if and only if it is an extreme point of the set \\(S(A)\\) of all states.\n4. \\(Q(A)\\) is the weak\\*-closed convex hull of \\(\\{0\\}\\cup P(A)\\). If \\(A\\) is unital, \\(S(A)\\) is weak\\*-compact and is the weak\\*-closed convex hull of \\(P(A)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 823,
        "through_line": 836,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.5",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-10::Theorem 8.5",
      "anchor": "oa-fnd-gn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 8.5.**\n1. Let \\(A\\) be an A\\*-algebra. For every nonzero \\(x\\in A\\) there is an irreducible representation \\(\\pi\\) of \\(A\\) with \\(\\pi(x)\\ne0\\).\n2. Let \\(A\\ne0\\) be a C\\*-algebra. For every \\(x\\in A\\) there is a pure state \\(\\varphi\\) with \\(\\varphi(x^*x)=\\|x\\|^2\\), and then \\(\\pi_\\varphi\\) is irreducible with \\(\\|\\pi_\\varphi(x)\\|=\\|x\\|\\). The direct sum of the GNS representations of all pure states of \\(A\\) is faithful.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 839,
        "through_line": 846,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Lemma 8.1",
            "Theorem 8.3",
            "Theorem 8.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.4.6, II.6.4.8–10",
          "role": "proof comparison",
          "correspondence": "weaker outside C*-algebras",
          "explanation": "Free commutant Radon–Nikodym and pure-state proofs match the C*-case. The retained Banach version uses its proved factorization and represents dominated functionals on products before extending equality to all elements."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 8.6",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-10::Examples 8.6",
      "anchor": "oa-fnd-gn-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 8.6.**\n1. The pure states of \\(C_0(X)\\) are the point evaluations. If \\(\\varphi\\) is a pure state, \\(\\pi_\\varphi\\) is irreducible, hence one-dimensional (Example 2.4(2)): \\(\\pi_\\varphi(f)=f(x)1\\) for some \\(x\\in X\\), and \\[\n\\begin{gathered}\n\\varphi(f)\\\\\n=\\langle\\pi_\\varphi(f)\\xi_\\varphi,\\xi_\\varphi\\rangle\\\\\n=f(x)\\|\\xi_\\varphi\\|^2\\\\\n=f(x).\n\\end{gathered}\n\\] Conversely, a point evaluation has a one-dimensional GNS space, so its GNS representation is irreducible and it is pure.\n2. The pure states of \\(M_n(\\mathbb C)\\) are the vector states \\(x\\mapsto\\langle x\\xi,\\xi\\rangle\\) with \\(\\|\\xi\\|=1\\); see Exercise 12.1. The tracial state \\(\\frac1n\\operatorname{Tr}=\\frac1n\\sum_k\\omega_{e_k}\\) is not pure when \\(n\\ge2\\).\n3. On a nonunital C\\*-algebra such as \\(C_0(\\mathbb R)\\), the zero functional lies in the weak\\*-closure of \\(P(A)\\): the point evaluations at \\(n\\) tend to \\(0\\) as \\(n\\to\\infty\\). This is why \\(0\\) appears in Proposition 8.4(4).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 847,
        "through_line": 858,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.1",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-11::Lemma 9.1",
      "anchor": "oa-fnd-gn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 9.1.** Let \\(J\\subseteq K(H)\\) be a C\\*-subalgebra.\n1. If \\(k\\in J\\) is self-adjoint and \\(\\lambda\\ne0\\) is a point of \\(\\sigma(k)\\), then \\(J\\) contains a nonzero projection \\(q\\) of finite rank with \\(kq=\\lambda q\\), whose range lies in the range of \\(k\\).\n2. If \\(J\\ne\\{0\\}\\), then \\(J\\) contains a nonzero projection of finite rank.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 869,
        "through_line": 875,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.2",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-11::Theorem 9.2",
      "anchor": "oa-fnd-gn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 9.2** (Irreducible algebras of compact operators). Let \\(J\\subseteq K(H)\\) be a C\\*-subalgebra with \\(J\\ne\\{0\\}\\) whose only invariant closed subspaces are \\(\\{0\\}\\) and \\(H\\). Then \\(J=K(H)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 876,
        "through_line": 883,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 9.3",
      "kind": "corollary",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-11::Corollary 9.3",
      "anchor": "oa-fnd-gn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Corollary 9.3.** Let \\(A\\) be a C\\*-algebra and \\(\\pi\\) an irreducible representation of \\(A\\) on \\(H\\). Then either \\(\\pi(A)\\supseteq K(H)\\) or \\(\\pi(A)\\cap K(H)=\\{0\\}\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 884,
        "through_line": 887,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.3a",
      "kind": "lemma",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-11::Lemma 9.3a",
      "anchor": "oa-fnd-gn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Lemma 9.3a** (Closed ideals of \\(B(H)\\)). If \\(H\\) is separable and infinite-dimensional, the only norm-closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 888,
        "through_line": 911,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 9.4",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-11::Examples 9.4",
      "anchor": "oa-fnd-gn-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 9.4.**\n1. Any C\\*-algebra \\(A\\) with \\(K(H)\\subseteq A\\subseteq B(H)\\), such as \\(K(H)+\\mathbb C1\\), acts irreducibly on \\(H\\) (Example 2.4(1)), with the first alternative. So does the extension of the representation of Theorem 9.5 below to the transformation-group C\\*-algebra (Remark 9.6).\n2. Let \\(H\\) be separable and infinite-dimensional. The only closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\) (Lemma 9.3a above). So the Calkin algebra \\(\\mathcal Q=B(H)/K(H)\\), a unital C\\*-algebra ([closed ideals and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-25)), has no closed two-sided ideals except \\(\\{0\\}\\) and \\(\\mathcal Q\\), because a closed ideal of \\(\\mathcal Q\\) pulls back to a closed ideal of \\(B(H)\\) containing \\(K(H)\\). It is infinite-dimensional: splitting \\(H\\) into infinitely many orthogonal infinite-dimensional subspaces gives infinitely many orthogonal projections that are not compact; their images in \\(\\mathcal Q\\) are nonzero and orthogonal, hence linearly independent. By Theorem 8.5, \\(\\mathcal Q\\) has an irreducible representation \\(\\sigma\\) on some \\(H_\\sigma\\). Its kernel is a proper closed ideal, so \\(\\sigma\\) is faithful. If \\(\\sigma(\\mathcal Q)\\) met \\(K(H_\\sigma)\\), then \\(\\sigma(\\mathcal Q)\\cap K(H_\\sigma)\\) would be a nonzero closed ideal of \\(\\sigma(\\mathcal Q)\\cong\\mathcal Q\\), hence all of \\(\\sigma(\\mathcal Q)\\). Then \\(\\sigma(1)\\), which is the identity operator because \\(\\sigma\\) is nondegenerate, would be compact, and \\(H_\\sigma\\) and \\(\\mathcal Q\\) would be finite-dimensional. So \\(\\sigma\\), composed with the quotient map, is an irreducible representation of \\(B(H)\\) with the second alternative.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 912,
        "through_line": 915,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.5",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-12::Theorem 9.5",
      "anchor": "oa-fnd-gn-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 9.5.**\n1. For \\(x\\in C_c(\\mathbb R^2)\\), \\(\\pi(x)\\) is the integral operator with kernel \\(k_x(s,r)=x(s,r-s)\\), that is, \\((\\pi(x)\\xi)(s)=\\int k_x(s,r)\\xi(r)\\,dr\\). It is a Hilbert–Schmidt operator: for every orthonormal basis \\((e_n)\\) of \\(L^2(\\mathbb R)\\), \\(\\sum_n\\|\\pi(x)e_n\\|^2=\\|k_x\\|_{L^2(\\mathbb R^2)}^2=\\|x\\|_{L^2(\\mathbb R^2)}^2\\). In particular \\(\\|\\pi(x)\\|\\le\\|x\\|_{L^2(\\mathbb R^2)}\\), and \\(\\pi(x)\\) is compact.\n2. The map \\(x\\mapsto k_x\\) is a linear bijection of \\(C_c(\\mathbb R^2)\\) onto itself. Moreover \\(\\pi(x\\star y)=\\pi(x)\\pi(y)\\), \\(\\pi(x^\\sharp)=\\pi(x)^*\\) and \\(\\|\\pi(x)\\|\\le\\|x\\|_1\\). So \\(\\pi\\) extends to a representation of \\(\\mathfrak A\\), and \\(\\pi(\\mathfrak A)\\) consists of compact operators.\n3. The norm closure of \\(\\pi(\\mathfrak A)\\) is \\(K(L^2(\\mathbb R))\\). In particular \\(\\pi\\) is irreducible.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 939,
        "through_line": 968,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.1",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-13::Theorem 10.1",
      "anchor": "oa-fnd-gn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 10.1** (Implementing an invariant state). Let \\(\\varphi\\in S^\\alpha\\) have GNS triple \\((\\pi,H,\\xi)\\).\n1. For each \\(s\\in G\\) there is a unique unitary \\(U(s)\\) on \\(H\\) with \\(U(s)\\pi(x)\\xi=\\pi(\\alpha_s(x))\\xi\\) for all \\(x\\). The map \\(s\\mapsto U(s)\\) is a homomorphism into the unitary group, and\n\\[\n\\begin{gathered}\nU(s)\\xi\\\\\n=\\xi,\\\\\nU(s)\\pi(x)U(s)^*\\\\\n=\\pi(\\alpha_s(x))\\\\\n(s\\in G,\\ x\\in A).\n\\end{gathered}\n\\tag{10.2}\n\\]\nA unitary representation satisfying (10.2) is necessarily this \\(U\\).\n2. The following are equivalent: (a) \\(\\varphi\\in S_\\alpha\\); (b) \\(\\varphi(x^*\\alpha_s(x))\\to\\varphi(x^*x)\\) as \\(s\\to e\\), for every \\(x\\in A\\); (c) \\(U\\) is strongly continuous.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 982,
        "through_line": 1026,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.2",
      "kind": "example",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-13::Example 10.2",
      "anchor": "oa-fnd-gn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Example 10.2** (An invariant state without continuity). The numbers \\(1\\) and \\(\\sqrt2\\) are linearly independent over \\(\\mathbb Q\\). Extend them to a basis of \\(\\mathbb R\\) as a vector space over \\(\\mathbb Q\\), using the basis-extension proof in Example 6.5, and let \\(\\vartheta:\\mathbb R\\to\\mathbb R\\) be the \\(\\mathbb Q\\)-linear map with \\(\\vartheta(1)=0\\), \\(\\vartheta(\\sqrt2)=1\\) and \\(\\vartheta=0\\) on the other basis elements. Then \\(\\vartheta\\) is additive, so \\(\\alpha_t(f)(z)=f(e^{i\\vartheta(t)}z)\\) defines a homomorphism of the usual topological group \\(\\mathbb R\\) into \\(\\operatorname{Aut}(C(\\mathbb T))\\). The state \\(\\varphi(f)=\\frac1{2\\pi}\\int_0^{2\\pi}f(e^{iu})\\,du\\) is invariant. For the coordinate function \\(x(z)=z\\), \\(\\varphi(x^*\\alpha_t(x))=e^{i\\vartheta(t)}\\). Choose rational numbers \\(r_n\\to\\sqrt2\\) and put \\(t_n=\\sqrt2-r_n\\), so \\(t_n\\to0\\). Then \\(\\vartheta(t_n)=1\\), so \\(\\varphi(x^*\\alpha_{t_n}(x))=e^{i}\\) does not tend to \\(\\varphi(x^*x)=1\\). So \\(\\varphi\\in S^\\alpha\\setminus S_\\alpha\\): the unitaries of Theorem 10.1(1) exist, but they do not depend continuously on \\(t\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1027,
        "through_line": 1028,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.3",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-13::Theorem 10.3",
      "anchor": "oa-fnd-gn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 10.3** (Ergodic states). Let \\(\\varphi\\in S_\\alpha\\) with GNS triple \\((\\pi,H,\\xi)\\), and let \\(U\\) be as in Theorem 10.1. The following are equivalent:\n- (i) the only closed subspaces of \\(H\\) invariant under both \\(\\pi(A)\\) and \\(U(G)\\) are \\(\\{0\\}\\) and \\(H\\);\n- (ii) \\(\\pi(A)'\\cap U(G)'=\\mathbb C1\\);\n- (iii) \\(\\varphi\\) is an extreme point of \\(S_\\alpha\\);\n- (iv) \\(\\varphi\\) is an extreme point of \\(S^\\alpha\\).\n\nA state with these properties is called *ergodic*.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1029,
        "through_line": 1075,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.4",
      "kind": "theorem",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-13::Theorem 10.4",
      "anchor": "oa-fnd-gn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Theorem 10.4** (Existence of ergodic states). Suppose that \\(\\alpha\\) is pointwise norm continuous: \\(\\|\\alpha_s(x)-x\\|\\to0\\) as \\(s\\to e\\), for every \\(x\\in A\\).\n1. \\(S_\\alpha=S^\\alpha\\).\n2. If \\(A\\) is unital, \\(S^\\alpha\\) is weak\\*-compact, so it has extreme points whenever it is not empty.\n3. Without an identity, \\(S^\\alpha\\) need not be compact, but it still has extreme points whenever it is not empty.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1076,
        "through_line": 1090,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Examples 10.5",
      "kind": "examples",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-13::Examples 10.5",
      "anchor": "oa-fnd-gn-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Examples 10.5.**\n1. Let \\(A=C_0(\\mathbb R)\\) and let \\(G\\) be the trivial group. Then \\(S^\\alpha=S(A)\\) contains the point evaluations \\(\\delta_n\\), which converge weak\\* to \\(0\\notin S(A)\\), because every \\(f\\in C_0(\\mathbb R)\\) has \\(f(n)\\to0\\). So \\(S^\\alpha\\) is not compact; its extreme points are the point evaluations (Example 8.6(1)).\n2. Let \\(\\mathbb R\\) act on \\(C_0(\\mathbb R)\\) by translation, \\(\\alpha_t(f)(s)=f(s+t)\\). This action is pointwise norm continuous, by the point-norm continuity argument after Lemma 0.1. It has no invariant state. Indeed, let \\(\\varphi\\) be an invariant positive functional and \\(f\\in C_c(\\mathbb R)\\) with \\(0\\le f\\le1\\), supported in \\([-N,N]\\), with \\(N\\geq1\\). The translates \\(\\alpha_{(2N+1)k}(f)\\), \\(k=1,\\dots,m\\), have disjoint supports, so their sum has norm at most \\(1\\), and \\(m\\varphi(f)=\\varphi\\big(\\sum_k\\alpha_{(2N+1)k}(f)\\big)\\le\\|\\varphi\\|\\). So \\(\\varphi(f)=0\\). Such \\(f\\), and their positive multiples, are dense in the positive cone, so \\(\\varphi=0\\) on \\(A_+\\), and \\(\\varphi=0\\).\n3. Let the circle group \\(\\mathbb T\\) act on \\(C(\\mathbb T)\\) by rotation, \\(\\alpha_w(f)(z)=f(wz)\\); the action is pointwise norm continuous. An invariant state \\(\\varphi\\) satisfies \\(\\varphi(z^n)=\\varphi(\\alpha_w(z^n))=w^n\\varphi(z^n)\\) for all \\(w\\in\\mathbb T\\), so \\(\\varphi(z^n)=0\\) for \\(n\\ne0\\), and \\(\\varphi(1)=1\\). The trigonometric polynomials are dense in \\(C(\\mathbb T)\\) ([the complex Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09)), so \\(\\varphi(f)=\\frac1{2\\pi}\\int_0^{2\\pi}f(e^{iu})\\,du\\) is the only invariant state. It is extreme in the one-point set \\(S^\\alpha\\), so it is ergodic. By Theorem 10.3, only the scalars commute with all multiplications and all rotations on \\(L^2(\\mathbb T)\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1091,
        "through_line": 1095,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.1",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-14::Proposition 11.1",
      "anchor": "oa-fnd-gn-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 11.1** (Gram determinants). For \\(x_1,\\dots,x_n\\in A\\), the vectors \\(\\Lambda_\\omega(x_1),\\dots,\\Lambda_\\omega(x_n)\\) are linearly independent exactly when the Gram determinant \\(\\det\\big[\\omega(x_j^*x_i)\\big]_{i,j=1}^n\\) is nonzero.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1100,
        "through_line": 1103,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.2",
      "kind": "proposition",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-14::Proposition 11.2",
      "anchor": "oa-fnd-gn-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Proposition 11.2** (Lower semicontinuity). Give the set of positive functionals on \\(A\\) the topology of pointwise convergence on \\(A\\); on bounded positive functionals on a normed algebra, this is the weak\\* topology. For every integer \\(m\\ge0\\), the set \\(\\{\\omega:d(\\omega)>m\\}\\) is open. So \\(d\\) is lower semicontinuous.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1104,
        "through_line": 1114,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.3",
      "kind": "example",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-14::Example 11.3",
      "anchor": "oa-fnd-gn-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Example 11.3** (Why all infinite dimensions count as \\(\\infty\\)). Let \\(\\Gamma\\) be an uncountable set and \\(K=\\{0,1\\}^\\Gamma\\), a compact abelian group under coordinatewise addition modulo \\(2\\) with the product topology. Compactness is [Tychonoff’s theorem](weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md#oa-fnd-wt-02); the finite discrete factors are compact, the product is Hausdorff because distinct points differ in a coordinate, and continuity of each coordinate proves continuity of the group operations. Let \\(\\mu\\) be its normalized Haar measure ([existence of Haar measure](haar-measure.md#oa-fnd-hm-05)) and \\(\\omega(f)=\\int f\\,d\\mu\\) on \\(A=C(K)\\). For a finite \\(F\\subseteq\\Gamma\\), the function \\(w_F(\\varepsilon)=\\prod_{\\gamma\\in F}(-1)^{\\varepsilon_\\gamma}\\) is a continuous character of \\(K\\), and \\(w_Fw_{F'}=w_{F\\triangle F'}\\). For \\(F\\ne\\emptyset\\) pick \\(h\\in K\\) with \\(w_F(h)=-1\\); invariance of \\(\\mu\\) gives \\(\\int w_F\\,d\\mu=\\int w_F(\\varepsilon+h)\\,d\\mu(\\varepsilon)=-\\int w_F\\,d\\mu\\), so \\(\\int w_F\\,d\\mu=0\\). Hence the \\(w_F\\) are orthonormal in \\(H_\\omega=L^2(\\mu)\\) (Example 5.7(2)), and there are uncountably many of them.\n\nOn the other hand, for a finite \\(F\\subseteq\\Gamma\\) let \\(\\mu_F\\) be the average of the point masses at the \\(2^{|F|}\\) points \\(\\varepsilon\\) with \\(\\varepsilon_\\gamma=0\\) for \\(\\gamma\\notin F\\). Then \\(d(\\mu_F)=2^{|F|}\\). For \\(S\\subseteq F\\), \\(\\int w_S\\,d\\mu_F\\) is \\(1\\) if \\(S=\\emptyset\\) and \\(0\\) otherwise, which is also \\(\\int w_S\\,d\\mu\\). The linear combinations of the \\(w_S\\) form a unital self-adjoint subalgebra of \\(C(K)\\) that separates points, so they are dense ([the complex Stone–Weierstrass theorem](stone-weierstrass-c0.md#oa-fnd-sw-09)). Since \\(\\|\\mu_F\\|=\\|\\mu\\|=1\\), it follows that \\(\\int f\\,d\\mu_F\\to\\int f\\,d\\mu\\) for every \\(f\\in C(K)\\), along the finite subsets \\(F\\) directed by inclusion. So the functionals \\(\\mu_F\\), each with finite \\(d\\), converge to \\(\\omega\\), whose GNS space has an uncountable orthonormal family. The set of positive functionals whose GNS space has an uncountable orthonormal family is therefore not open: Proposition 11.2 would fail if uncountable dimensions were distinguished from countable ones.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1115,
        "through_line": 1118,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.1",
      "kind": "exercise",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-15::Exercise 12.1",
      "anchor": "oa-fnd-gn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Exercise 12.1** (medium; Positive functionals on matrices). Let \\(\\rho\\in M_n(\\mathbb C)\\) be positive with \\(\\operatorname{Tr}\\rho=1\\) and rank \\(r\\), and let \\(\\omega(x)=\\operatorname{Tr}(\\rho x)\\). (a) Show that \\(N_\\omega=\\{x:x\\rho=0\\}\\) and \\(d(\\omega)=nr\\). (b) Show that \\(\\pi_\\omega\\) is unitarily equivalent to the direct sum of \\(r\\) copies of the identity representation of \\(M_n(\\mathbb C)\\) on \\(\\mathbb C^n\\). (c) Deduce that \\(\\omega\\) is pure exactly when \\(r=1\\), that is, when \\(\\omega(x)=\\langle x\\xi,\\xi\\rangle\\) for a unit vector \\(\\xi\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1121,
        "through_line": 1132,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.2",
      "kind": "exercise",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-15::Exercise 12.2",
      "anchor": "oa-fnd-gn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Exercise 12.2** (medium; The trace on a finite group algebra). Let \\(G\\) be a finite group and \\(A=\\mathbb C[G]\\), the functions on \\(G\\) with \\((f*g)(t)=\\sum_sf(s)g(s^{-1}t)\\), \\(f^*(t)=\\overline{f(t^{-1})}\\) and \\(\\|f\\|=\\sum_t|f(t)|\\). Let \\(\\tau(f)=f(e)\\). Show that \\(\\tau\\) is a faithful positive functional, that its GNS representation is the left regular representation \\(\\lambda(f)\\xi=f*\\xi\\) on \\(\\ell^2(G)\\) with cyclic vector \\(\\delta_e\\), and that \\(\\pi_\\tau\\) is irreducible only for the trivial group.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1133,
        "through_line": 1149,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.3",
      "kind": "exercise",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-15::Exercise 12.3",
      "anchor": "oa-fnd-gn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Exercise 12.3** (medium; Ergodic states of a permutation action). Let a group \\(G\\), with the discrete topology, act on a finite set \\(X\\), and let it act on \\(A=C(X)\\) by \\(\\alpha_s(f)(x)=f(s^{-1}x)\\). (a) Show that the invariant states are the functionals \\(f\\mapsto\\sum_xp(x)f(x)\\) given by probability vectors \\(p\\) that are constant on each orbit. (b) Show that the ergodic states are the uniform distributions on single orbits. (c) For the uniform distribution on an orbit \\(O\\), describe \\((\\pi,H,\\xi,U)\\) and check condition (ii) of Theorem 10.3 directly.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1150,
        "through_line": 1155,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.4",
      "kind": "exercise",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-15::Exercise 12.4",
      "anchor": "oa-fnd-gn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Exercise 12.4** (easy; Ideals that are not closed). Let \\(A\\) be a C\\*-algebra, \\(\\pi\\) an irreducible representation of \\(A\\) on \\(H\\), and \\(J\\) a two-sided ideal of \\(A\\) that is not assumed to be closed or self-adjoint. Show that either \\(\\pi(J)=\\{0\\}\\), or the only closed subspaces invariant under \\(\\pi(J)\\) are \\(\\{0\\}\\) and \\(H\\).",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1156,
        "through_line": 1159,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.5",
      "kind": "exercise",
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark#oa-fnd-gn-15::Exercise 12.5",
      "anchor": "oa-fnd-gn-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
      "statement_and_full_conditions": "**Exercise 12.5** (medium; Extending to the unitization). Let \\(A\\) be a C\\*-algebra, \\(\\tilde A=A+\\mathbb C1\\) its C\\*-unitization (see the conventions of [C\\*-algebras: continuous functional calculus, automatic continuity, positive cones, approximate identities and quotients](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md)), and \\(\\omega\\) a positive functional on \\(A\\). Show that \\(\\tilde\\omega(x+\\lambda)=\\omega(x)+\\lambda\\|\\omega\\|\\) is positive on \\(\\tilde A\\), that \\(\\|\\tilde\\omega\\|=\\|\\omega\\|\\), and that it is the only positive extension of \\(\\omega\\) with this norm.",
      "proof_locus": {
        "source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "line": 1160,
        "through_line": 1168,
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.1",
      "kind": "lemma",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-01::Lemma 1.1",
      "anchor": "oa-fnd-st-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Lemma 1.1.** Let \\(K\\) be a metric space. Let \\(\\mathcal M\\) be a set of bounded complex functions on \\(K\\) with two properties:\n- \\(\\mathcal M\\) contains every bounded continuous function;\n- \\(\\mathcal M\\) contains the limit of every boundedly convergent sequence in \\(\\mathcal M\\).\n\nThen \\(\\mathcal M\\) contains every bounded Borel function on \\(K\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 41,
        "through_line": 72,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 1.2",
      "kind": "corollary",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-01::Corollary 1.2",
      "anchor": "oa-fnd-st-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Corollary 1.2** (Measures are determined by continuous functions). Let \\(K\\) be a metric space, let \\(\\mu_1,\\dots,\\mu_m\\) be finite measures on the Borel sets of \\(K\\), and let \\(c_1,\\dots,c_m\\in\\mathbb C\\). Suppose that\n\\[\n\\sum_jc_j\\int f\\,d\\mu_j=0\n\\]\nfor every bounded continuous \\(f\\). Then the same holds for every bounded Borel \\(f\\). In particular, two finite Borel measures on \\(K\\) with the same integrals of bounded continuous functions are equal.",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 73,
        "through_line": 80,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.1",
      "kind": "proposition",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-02::Proposition 2.1",
      "anchor": "oa-fnd-st-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Proposition 2.1.** For every \\(\\xi\\in H\\) there is exactly one Radon measure \\(\\mu_\\xi\\) on \\(S\\) with\n\\[\n\\langle f(h)\\xi,\\xi\\rangle=\\int_Sf\\,d\\mu_\\xi\\qquad(f\\in C(S)).\n\\]\nIt is finite, with \\(\\mu_\\xi(S)=\\|\\xi\\|^2\\). For \\(c\\in\\mathbb C\\), \\(\\mu_{c\\xi}=|c|^2\\mu_\\xi\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 85,
        "through_line": 98,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-03::Theorem 3.1",
      "anchor": "oa-fnd-st-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Theorem 3.1.** Let \\(h\\in B(H)\\) be self-adjoint with spectrum \\(S\\). For every \\(f\\in B_b(S)\\) there is exactly one operator \\(f(h)\\in B(H)\\) with\n\\[\n\\begin{gathered}\n\\langle f(h)\\xi,\\xi\\rangle\\\\\n=\\int_Sf\\,d\\mu_\\xi\\\\\n(\\xi\\in H).\n\\end{gathered}\n\\tag{3.1}\n\\]\nThe map \\(f\\mapsto f(h)\\) has the following properties.\n1. For continuous \\(f\\) it is the continuous functional calculus.\n2. It is a unital \\(*\\)-homomorphism: it is linear, \\((fg)(h)=f(h)g(h)\\), \\(\\bar f(h)=f(h)^*\\) and \\(1(h)=1\\).\n3. \\(\\|f(h)\\xi\\|^2=\\int_S|f|^2\\,d\\mu_\\xi\\). Hence \\(\\|f(h)\\|\\leq\\|f\\|_S\\). If \\(f\\geq0\\), then \\(f(h)\\geq0\\). If \\(f\\) is real, then \\(f(h)\\) is self-adjoint.\n4. (*Bounded convergence.*) If \\(f_n\\to f\\) boundedly on \\(S\\), then \\(f_n(h)\\xi\\to f(h)\\xi\\) for every \\(\\xi\\in H\\).\n5. \\(f(h)\\) commutes with every operator that commutes with \\(h\\).\n6. \\(\\sigma(f(h))\\) is contained in the closure of \\(f(S)\\).\n\nFor a bounded Borel function \\(f\\) on \\(\\mathbb R\\) we write \\(f(h)\\) for \\((f|_S)(h)\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 101,
        "through_line": 202,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-04::Definition 4.1",
      "anchor": "oa-fnd-st-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Definition 4.1.** Let \\(X\\) be a metric space; we use \\(X=\\mathbb R\\) and \\(X=\\mathbb C\\). A *projection-valued measure* on \\(X\\), acting on \\(H\\), is a map \\(E\\) from the Borel sets of \\(X\\) to the projections of \\(B(H)\\) with two properties:\n- \\(E(X)=1\\);\n- for every \\(\\xi\\in H\\), the function \\(\\mu^E_\\xi(\\Delta)=\\langle E(\\Delta)\\xi,\\xi\\rangle=\\|E(\\Delta)\\xi\\|^2\\) is a measure.\n\n\\(E\\) is *supported* by a Borel set \\(Y\\) if \\(E(Y)=1\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 205,
        "through_line": 210,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.2",
      "kind": "lemma",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-04::Lemma 4.2",
      "anchor": "oa-fnd-st-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Lemma 4.2.** Let \\(E\\) be a projection-valued measure.\n1. \\(E(\\varnothing)=0\\). If \\(\\Delta\\cap\\Delta'=\\varnothing\\), then \\(E(\\Delta\\cup\\Delta')=E(\\Delta)+E(\\Delta')\\) and \\(E(\\Delta)E(\\Delta')=0\\).\n2. \\(E(\\Delta\\cap\\Delta')=E(\\Delta)E(\\Delta')=E(\\Delta')E(\\Delta)\\) for all Borel sets \\(\\Delta,\\Delta'\\).\n3. If \\(\\Delta\\) is the union of disjoint Borel sets \\(\\Delta_n\\), then \\(E(\\Delta)\\xi=\\sum_nE(\\Delta_n)\\xi\\) for every \\(\\xi\\in H\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 211,
        "through_line": 240,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.3",
      "kind": "proposition",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-04::Proposition 4.3",
      "anchor": "oa-fnd-st-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Proposition 4.3** (Integration against a projection-valued measure). Let \\(E\\) be a projection-valued measure on \\(X\\). There is exactly one linear map \\(f\\mapsto\\int f\\,dE\\) from \\(B_b(X)\\) to \\(B(H)\\) with\n\\[\n\\begin{gathered}\n\\int1_\\Delta\\,dE\\\\\n=E(\\Delta)\\\\\n\\text{for every Borel set }\\Delta,\\\\\n\\Big\\|\\int f\\,dE\\Big\\|\\\\\n\\leq\\|f\\|_X .\n\\end{gathered}\n\\]\nIt is a unital \\(*\\)-homomorphism, and \\(\\big\\langle\\big(\\int f\\,dE\\big)\\xi,\\xi\\big\\rangle=\\int f\\,d\\mu^E_\\xi\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 241,
        "through_line": 266,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.4",
      "kind": "theorem",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-04::Theorem 4.4",
      "anchor": "oa-fnd-st-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Theorem 4.4** (Spectral theorem). Let \\(h\\in B(H)\\) be self-adjoint with spectrum \\(S\\).\n1. \\(E_h(\\Delta)=1_\\Delta(h)\\), for Borel sets \\(\\Delta\\subseteq\\mathbb R\\), is a projection-valued measure on \\(\\mathbb R\\), supported by \\(S\\subseteq[-\\|h\\|,\\|h\\|]\\). For every bounded Borel function \\(f\\) on \\(\\mathbb R\\), \\(\\int f\\,dE_h=f(h)\\). In particular\n\\[\nh=\\int\\iota_S\\,dE_h ,\n\\]\nwhere \\(\\iota_S(\\lambda)=\\lambda\\) on \\(S\\) and \\(\\iota_S=0\\) off \\(S\\). This is usually written \\(h=\\int\\lambda\\,dE_h(\\lambda)\\).\n2. (*Uniqueness.*) Let \\(E\\) be a projection-valued measure on \\(\\mathbb R\\), supported by a compact set \\(L\\), with \\(h=\\int\\iota_L\\,dE\\). Then \\(E=E_h\\).\n3. For \\(\\xi\\in H\\), the spectral measure satisfies \\(\\mu_\\xi(\\Delta)=\\langle E_h(\\Delta)\\xi,\\xi\\rangle=\\|E_h(\\Delta)\\xi\\|^2\\), \\(\\mu_\\xi(\\mathbb R)=\\|\\xi\\|^2\\), and \\(\\langle f(h)\\xi,\\xi\\rangle=\\int f\\,d\\mu_\\xi\\).\n4. Each \\(E_h(\\Delta)\\) commutes with every operator that commutes with \\(h\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 267,
        "through_line": 297,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.1",
      "kind": "proposition",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-05::Proposition 5.1",
      "anchor": "oa-fnd-st-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Proposition 5.1.** Let \\(h\\in B(H)\\) be self-adjoint, with spectrum \\(S\\) and \\(E=E_h\\).\n1. For every \\(n\\geq1\\) there are disjoint Borel sets \\(\\Delta_1,\\dots,\\Delta_m\\) and real numbers \\(c_1,\\dots,c_m\\) with \\(\\|h-\\sum_jc_jE(\\Delta_j)\\|\\leq1/n\\). So \\(h\\) is a norm limit of real linear combinations of its spectral projections.\n2. If \\(h\\geq0\\), the \\(c_j\\) can be taken \\(\\geq0\\) and the \\(\\Delta_j\\) inside \\([1/n,\\infty)\\). So \\(h\\) is a norm limit of nonnegative combinations of spectral projections of Borel sets bounded away from \\(0\\).\n3. \\(\\ker h=E(\\{0\\})H\\).\n4. Let \\(h\\geq0\\). Then \\(E((0,\\infty))\\) is the projection onto the closure of \\(hH\\). As \\(\\varepsilon\\downarrow0\\), the operators \\(h(h+\\varepsilon)^{-1}\\) increase and converge strongly to \\(E((0,\\infty))\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 302,
        "through_line": 341,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.2",
      "kind": "proposition",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-05::Proposition 5.2",
      "anchor": "oa-fnd-st-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Proposition 5.2** (Polar decomposition). Let \\(x\\in B(H)\\) and \\(|x|=(x^*x)^{1/2}\\). There is exactly one operator \\(u\\) with \\(x=u|x|\\) and \\(\\ker u=\\ker x\\). It is a partial isometry with initial space \\((\\ker x)^\\perp\\), the closure of \\(|x|H\\), and final space the closure of \\(xH\\). Moreover \\(u^*u\\) and \\(uu^*\\) are the projections onto these two spaces, and \\(u^*x=|x|\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 342,
        "through_line": 350,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-06::Theorem 6.1",
      "anchor": "oa-fnd-st-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Theorem 6.1** (Vigier). Let \\((a_i)_{i\\in I}\\) be an increasing net of self-adjoint operators on \\(H\\) with \\(C=\\sup_i\\|a_i\\|<\\infty\\). Then:\n- there is a self-adjoint \\(a\\in B(H)\\) with \\(a_i\\xi\\to a\\xi\\) for every \\(\\xi\\);\n- \\(a\\) is the least upper bound of the \\(a_i\\) among self-adjoint operators.\n\nThe same holds for decreasing nets, with the greatest lower bound.",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 353,
        "through_line": 381,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [
        {
          "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
          "local_scopes": [
            {
              "heading": "2. The spectral measure of a vector",
              "line": 81,
              "through_line": 98,
              "anchors": [
                "OA-FND-ST-02"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "3. The Borel functional calculus",
              "line": 99,
              "through_line": 202,
              "anchors": [
                "OA-FND-ST-03"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "4. Projection-valued measures and the spectral theorem",
              "line": 203,
              "through_line": 297,
              "anchors": [
                "OA-FND-ST-04"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "5. Spectral projections and polar decomposition",
              "line": 298,
              "through_line": 350,
              "anchors": [
                "OA-FND-ST-05"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            },
            {
              "heading": "6. Monotone convergence of operators",
              "line": 351,
              "through_line": 381,
              "anchors": [
                "OA-FND-ST-06"
              ],
              "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.7; PDF 58–63",
          "role": "proof comparison",
          "correspondence": "Spectral-measure construction compared",
          "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.1",
      "kind": "theorem",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-07::Theorem 7.1",
      "anchor": "oa-fnd-st-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Theorem 7.1** (Calkin). Let \\(H\\) be a separable infinite-dimensional Hilbert space. The closed two-sided ideals of \\(B(H)\\) are \\(\\{0\\}\\), \\(K(H)\\) and \\(B(H)\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 384,
        "through_line": 409,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.1",
      "kind": "theorem",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Theorem 8.1",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Theorem 8.1.** Let \\(n\\in B(H)\\) be normal, with spectrum \\(S\\subseteq\\mathbb C\\). Sections 2–5 hold with \\(h\\) replaced by \\(n\\) and \\(\\mathbb R\\) by \\(\\mathbb C\\), with the following changes. These conditions give the normal version without invoking Fuglede’s theorem:\n- in Theorem 3.1(5) and Theorem 4.4(4), the operator must commute with \\(n\\) and \\(n^*\\);\n- in Proposition 5.1(1), the coefficients \\(c_j\\) are complex, and a grid of mesh \\(1/(2n)\\) gives the same \\(1/n\\) error bound;\n- the positive-coefficient and positive-support assertions, Proposition 5.1(2) and (4), still assume that the operator is positive; such an operator is self-adjoint. The kernel formula in (3) holds for every normal operator, using \\(|z|^2\\) in its proof.\n\nIn particular, \\(E_n(\\Delta)=1_\\Delta(n)\\) is the only projection-valued measure on \\(\\mathbb C\\) that is supported by a compact set \\(L\\) and has \\(n=\\int\\iota_L\\,dE_n\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 412,
        "through_line": 425,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.2",
      "kind": "proposition",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Proposition 8.2",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Proposition 8.2.** Let \\(n\\in B(H)\\) be normal.\n1. Every \\(\\lambda\\in\\sigma(n)\\) is an approximate eigenvalue: there are unit vectors \\(\\xi_k\\) with \\(\\|(n-\\lambda)\\xi_k\\|\\to0\\).\n2. \\(\\|n\\|=\\sup_{\\|\\xi\\|=1}|\\langle n\\xi,\\xi\\rangle|\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 426,
        "through_line": 442,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Exercise 1",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Exercise 1** (easy; Multiplication operators). Let \\(\\mu\\) be a finite Borel measure on a compact set \\(K\\subseteq\\mathbb R\\), and let \\(h\\) be multiplication by \\(\\lambda\\) on \\(L^2(K,\\mu)\\). Let \\(\\operatorname{supp}\\mu\\) be the set of points all of whose neighbourhoods have positive measure. Show:\n- \\(\\sigma(h)=\\operatorname{supp}\\mu\\);\n- \\(f(h)\\) is multiplication by \\(f\\), for every bounded Borel \\(f\\);\n- \\(E_h(\\Delta)\\) is multiplication by \\(1_\\Delta\\);\n- the spectral measure of the constant function \\(1\\) is \\(\\mu\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 445,
        "through_line": 459,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Exercise 2",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Exercise 2** (medium; A dense set of eigenvalues). Let \\((q_k)\\) enumerate \\(\\mathbb Q\\cap[0,1]\\) without repetitions, and let \\(h\\) be the operator on \\(\\ell^2(\\mathbb N)\\) with \\(he_k=q_ke_k\\). Show:\n- \\(\\sigma(h)=[0,1]\\);\n- \\(E_h(\\{q_k\\})\\) is the projection onto \\(\\mathbb Ce_k\\);\n- \\(E_h([0,1]\\setminus\\mathbb Q)=0\\).\n\nConclude that \\(f=1_{[0,1]\\setminus\\mathbb Q}\\) has \\(f(h)=0\\), although the closure of \\(f(\\sigma(h))\\) is \\(\\{0,1\\}\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 460,
        "through_line": 471,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Exercise 3",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Exercise 3** (medium; Unitaries are exponentials). Let \\(u\\in B(H)\\) be unitary. Show that \\(u=e^{ih}\\) for a self-adjoint \\(h\\) with \\(\\|h\\|\\leq\\pi\\) that commutes with every operator commuting with \\(u\\) and \\(u^*\\). Deduce that the unitary group of \\(B(H)\\) is path-connected in the norm topology.",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 472,
        "through_line": 485,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Exercise 4",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Exercise 4** (easy; The Calkin algebra). Let \\(H\\) be separable and infinite-dimensional. Show that \\(B(H)/K(H)\\) has no closed two-sided ideals other than \\(\\{0\\}\\) and itself.",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 486,
        "through_line": 491,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5",
      "kind": "exercise",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Exercise 5",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Exercise 5** (hard; Without separability). Let \\(H\\) be a Hilbert space that is not separable. Let \\(J\\) be the norm closure of the set of operators whose range is separable. Show that \\(J\\) is a closed two-sided ideal with \\(K(H)\\subsetneq J\\subsetneq B(H)\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 492,
        "through_line": 499,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 6",
      "kind": "exercise",
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators#oa-fnd-st-08::Exercise 6",
      "anchor": "oa-fnd-st-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-spectral-theorem-for-bounded-self-adjoint-operators@5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
      "statement_and_full_conditions": "**Exercise 6** (easy; The support of a positive operator). Let \\(h\\geq0\\). Show that \\(E_h((0,\\infty))\\) is the smallest projection \\(p\\) with \\(ph=h\\).",
      "proof_locus": {
        "source": "src/the-spectral-theorem-for-bounded-self-adjoint-operators.md",
        "line": 500,
        "through_line": 506,
        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.1",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-02::Definition 1.1",
      "anchor": "oa-fnd-lt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 1.1.** A *sesquilinear form* on \\(H\\) is a map \\(B:H\\times H\\to\\mathbb C\\) that is linear in the first variable and conjugate linear in the second. (The same words are used for forms on any complex vector space, and for maps with values in any complex vector space.) The form is *hermitian* if \\(B(\\eta,\\xi)=\\overline{B(\\xi,\\eta)}\\) for all \\(\\xi,\\eta\\), *positive* if \\(B(\\xi,\\xi)\\ge0\\) for all \\(\\xi\\), and *bounded* if\n\\[\n\\|B\\|=\\sup\\{|B(\\xi,\\eta)|:\\ \\|\\xi\\|\\le1,\\ \\|\\eta\\|\\le1\\}\n\\]\nis finite. The bounded forms make up a normed space \\(\\operatorname{Sesq}(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 173,
        "through_line": 178,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.2",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-02::Lemma 1.2",
      "anchor": "oa-fnd-lt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 1.2** (polarization). For every sesquilinear form \\(B\\) on \\(H\\), or on any complex vector space, and all vectors \\(\\xi,\\eta\\),\n\\[\n\\begin{gathered}\nB(\\xi,\\eta)\\\\\n=\\frac14\\sum_{k=0}^{3}i^k\\,B(\\xi+i^k\\eta,\\ \\xi+i^k\\eta).\n\\end{gathered}\n\\tag{1.1}\n\\]\nSo \\(B\\) is determined by its values \\(B(\\zeta,\\zeta)\\). The form \\(B\\) is hermitian exactly when every value \\(B(\\zeta,\\zeta)\\) is real; in particular every positive form is hermitian.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 179,
        "through_line": 207,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 1.3",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-02::Theorem 1.3",
      "anchor": "oa-fnd-lt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 1.3.** For every bounded sesquilinear form \\(B\\) on \\(H\\) there is a unique \\(t\\in B(H)\\) with \\(B(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\) for all \\(\\xi,\\eta\\). The map \\(t\\mapsto B_t\\), \\(B_t(\\xi,\\eta)=\\langle t\\xi,\\eta\\rangle\\), is a linear isometry of \\(B(H)\\) onto \\(\\operatorname{Sesq}(H)\\); in particular \\(\\operatorname{Sesq}(H)\\) is a Banach space. The operator \\(t\\) is self-adjoint if and only if \\(B_t\\) is hermitian, and positive if and only if \\(B_t\\) is positive.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 208,
        "through_line": 215,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.4",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-02::Lemma 1.4",
      "anchor": "oa-fnd-lt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 1.4** (rank-one operators and vector functionals). Let \\(\\xi,\\eta,\\zeta,\\upsilon\\in H\\) and \\(a,b\\in B(H)\\).\n\n(a) \\(\\|\\theta_{\\xi,\\eta}\\|=\\|\\xi\\|\\,\\|\\eta\\|\\) and \\(\\theta_{\\xi,\\eta}^*=\\theta_{\\eta,\\xi}\\).\n\n(b) \\(a\\,\\theta_{\\xi,\\eta}\\,b=\\theta_{a\\xi,\\,b^*\\eta}\\) and \\(\\theta_{\\xi,\\eta}\\theta_{\\zeta,\\upsilon}=\\langle\\zeta,\\eta\\rangle\\,\\theta_{\\xi,\\upsilon}\\).\n\n(c) For a unit vector \\(\\xi\\), \\(\\theta_{\\xi,\\xi}\\) is the orthogonal projection onto \\(\\mathbb C\\xi\\).\n\n(d) Every operator of rank one equals \\(\\theta_{\\xi,\\eta}\\) for some nonzero \\(\\xi,\\eta\\). If \\(x\\in F(H)\\) and \\(\\varepsilon_1,\\dots,\\varepsilon_r\\) is an orthonormal basis of \\(xH\\), then \\(x=\\sum_{j=1}^r\\theta_{\\varepsilon_j,\\,x^*\\varepsilon_j}\\). So \\(F(H)\\) is the linear span of the rank-one operators, and it is a two-sided ideal of \\(B(H)\\) closed under adjoints.\n\n(e) \\(\\omega_{\\xi,\\eta}(\\theta_{\\zeta,\\upsilon})=\\langle\\zeta,\\eta\\rangle\\langle\\xi,\\upsilon\\rangle\\), and \\(\\|\\omega_{\\xi,\\eta}\\|=\\|\\xi\\|\\,\\|\\eta\\|\\), both as a functional on \\(B(H)\\) and on \\(K(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 216,
        "through_line": 244,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.1",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-03::Lemma 2.1",
      "anchor": "oa-fnd-lt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 2.1.** Let \\(x\\in K(H)\\) and let \\((\\xi_n)\\) be an orthonormal sequence. Then \\(\\|x\\xi_n\\|\\to0\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 249,
        "through_line": 252,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.2",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-03::Example 2.2",
      "anchor": "oa-fnd-lt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 2.2** (diagonal operators). Let \\((\\varepsilon_i)_{i\\in I}\\) be an orthonormal basis of \\(H\\) and \\(\\lambda\\in\\ell^\\infty(I)\\). Put\n\\[\nd_\\lambda\\zeta=\\sum_i\\lambda_i\\langle\\zeta,\\varepsilon_i\\rangle\\varepsilon_i .\n\\]\nThe series converges because its coefficients are square summable, and \\(\\|d_\\lambda\\zeta\\|^2=\\sum_i|\\lambda_i|^2|\\langle\\zeta,\\varepsilon_i\\rangle|^2\\le\\|\\lambda\\|_\\infty^2\\|\\zeta\\|^2\\). Since \\(d_\\lambda\\varepsilon_i=\\lambda_i\\varepsilon_i\\), \\(\\|d_\\lambda\\|=\\|\\lambda\\|_\\infty\\). One checks \\(d_\\lambda d_\\mu=d_{\\lambda\\mu}\\) and \\(d_\\lambda^*=d_{\\bar\\lambda}\\). The operator \\(d_\\lambda\\) is compact if and only if \\(\\lambda\\in c_0(I)\\). Indeed, if \\(\\lambda\\in c_0(I)\\) and \\(F\\subseteq I\\) is a finite set outside of which \\(|\\lambda_i|<\\varepsilon\\), then \\(d_{\\lambda1_F}\\) has finite rank and \\(\\|d_\\lambda-d_{\\lambda1_F}\\|\\le\\varepsilon\\). If \\(\\lambda\\notin c_0(I)\\), there are \\(\\varepsilon>0\\) and distinct \\(i_1,i_2,\\dots\\) with \\(|\\lambda_{i_n}|\\ge\\varepsilon\\), so \\(\\|d_\\lambda\\varepsilon_{i_n}\\|\\ge\\varepsilon\\), and \\(d_\\lambda\\) is not compact by Lemma 2.1. In this way \\(\\ell^\\infty(I)\\) and \\(c_0(I)\\) sit inside \\(B(H)\\) and \\(K(H)\\) as algebras of diagonal operators.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 253,
        "through_line": 258,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.3",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-03::Theorem 2.3",
      "anchor": "oa-fnd-lt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 2.3** (Schmidt decomposition). Let \\(x\\in K(H)\\), \\(x\\ne0\\). There are orthonormal families \\((e_n)_{n\\in N}\\) and \\((f_n)_{n\\in N}\\), where \\(N=\\{1,\\dots,r\\}\\) or \\(N=\\mathbb N\\), and numbers \\(s_1\\ge s_2\\ge\\cdots>0\\), tending to \\(0\\) when \\(N=\\mathbb N\\), such that\n\\[\n\\begin{gathered}\nx\\\\\n=\\sum_{n\\in N}s_n\\,\\theta_{f_n,e_n},\\\\\n\\text{that is,}\\\\\nx\\zeta\\\\\n=\\sum_n s_n\\langle\\zeta,e_n\\rangle f_n,\n\\end{gathered}\n\\tag{2.1}\n\\]\nwith convergence in norm. For every decomposition of this form:\n\n(a) \\(\\|x\\|=s_1\\), the kernel of \\(x\\) is \\(\\{e_n:n\\in N\\}^\\perp\\), and \\(xe_n=s_nf_n\\), \\(x^*f_n=s_ne_n\\);\n\n(b) \\(x^*=\\sum_ns_n\\theta_{e_n,f_n}\\), \\(|x|=\\sum_ns_n\\theta_{e_n,e_n}\\) and \\(|x^*|=\\sum_ns_n\\theta_{f_n,f_n}\\);\n\n(c) the numbers \\(s_n\\), listed with repetitions, are the nonzero eigenvalues of \\(|x|\\) counted with multiplicity; so they do not depend on the decomposition;\n\n(d) the operator \\(v\\zeta=\\sum_n\\langle\\zeta,f_n\\rangle e_n\\) satisfies \\(\\|v\\|\\le1\\), \\(vx=|x|\\) and \\(v^*|x|=x\\);\n\n(e) if \\(x\\) is self-adjoint, a decomposition exists with \\(f_n=\\pm e_n\\) for every \\(n\\), and if \\(x\\ge0\\), one with \\(f_n=e_n\\);\n\n(f) if \\(x\\) is self-adjoint and \\(x=\\sum_n\\lambda_n\\theta_{e_n,e_n}\\) as in (e), then the spectrum of \\(x\\) contains every \\(\\lambda_n\\), and \\(\\sigma(x)\\setminus\\{0\\}=\\{\\lambda_n:n\\in N\\}\\); if \\(H\\) is infinite-dimensional, then \\(0\\in\\sigma(x)\\), so \\(\\sigma(x)=\\{\\lambda_n:n\\in N\\}\\cup\\{0\\}\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 259,
        "through_line": 316,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.4",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-03::Definition 2.4",
      "anchor": "oa-fnd-lt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 2.4.** The *singular values* of \\(x\\in K(H)\\) are the numbers \\(s_n(x)=s_n\\), \\(n\\in N\\), of any decomposition (2.1), followed by \\(s_n(x)=0\\) for \\(n>r\\) when \\(N=\\{1,\\dots,r\\}\\). For \\(x=0\\) all \\(s_n(x)=0\\). We write \\(s(x)=(s_n(x))_{n\\ge1}\\); it is a decreasing sequence in \\(c_0\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 317,
        "through_line": 318,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.5",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-03::Proposition 2.5",
      "anchor": "oa-fnd-lt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 2.5** (approximation numbers). For \\(x\\in K(H)\\) and \\(n\\ge1\\),\n\\[\n\\begin{gathered}\ns_n(x)\\\\\n=\\min\\{\\|x-f\\|:\\\\\n\\ f\\in F(H),\\ \\operatorname{rank}f<n\\}.\n\\end{gathered}\n\\tag{2.2}\n\\]\nConsequently, for \\(x,y\\in K(H)\\), \\(a,b\\in B(H)\\) and \\(m,n\\ge1\\):\n\n(a) \\(s_n(axb)\\le\\|a\\|\\,s_n(x)\\,\\|b\\|\\);\n\n(b) \\(s_{m+n-1}(x+y)\\le s_m(x)+s_n(y)\\);\n\n(c) \\(s_n(x^*)=s_n(|x|)=s_n(x)\\);\n\n(d) \\(|s_n(x)-s_n(y)|\\le\\|x-y\\|\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 319,
        "through_line": 347,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.6",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-03::Example 2.6",
      "anchor": "oa-fnd-lt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 2.6.** For \\(\\lambda\\in c_0(I)\\), the diagonal operator \\(d_\\lambda\\) of Example 2.2 satisfies \\(|d_\\lambda|=d_{|\\lambda|}=\\sum_i|\\lambda_i|\\theta_{\\varepsilon_i,\\varepsilon_i}\\). By Theorem 2.3(c), \\(s(d_\\lambda)\\) lists the nonzero numbers \\(|\\lambda_i|\\) in decreasing order, with repetitions, followed by zeros. Only countably many \\(\\lambda_i\\) are nonzero because \\(\\lambda\\in c_0(I)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 348,
        "through_line": 349,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.1",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-04::Lemma 3.1",
      "anchor": "oa-fnd-lt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 3.1.** Let \\(x\\in B(H)\\), and let \\((\\varepsilon_i)_{i\\in I}\\) and \\((\\varepsilon'_j)_{j\\in J}\\) be orthonormal bases. Then, in \\([0,\\infty]\\),\n\\[\n\\begin{gathered}\n\\sum_i\\|x\\varepsilon_i\\|^2\\\\\n=\\sum_{i,j}|\\langle x\\varepsilon_i,\\varepsilon'_j\\rangle|^2\\\\\n=\\sum_j\\|x^*\\varepsilon'_j\\|^2 .\n\\end{gathered}\n\\tag{3.1}\n\\]\nIn particular \\(\\sum_i\\|x\\varepsilon_i\\|^2\\) does not depend on the basis, and it does not change when \\(x\\) is replaced by \\(x^*\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 354,
        "through_line": 366,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.2",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-04::Definition 3.2",
      "anchor": "oa-fnd-lt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 3.2.** The *Hilbert–Schmidt norm* of \\(x\\in B(H)\\) is \\(\\|x\\|_2=(\\sum_i\\|x\\varepsilon_i\\|^2)^{1/2}\\in[0,\\infty]\\) for any orthonormal basis \\((\\varepsilon_i)\\). The operators with \\(\\|x\\|_2<\\infty\\) are the *Hilbert–Schmidt operators*; they form the set \\(\\mathcal L^2(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 367,
        "through_line": 368,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.3",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-04::Theorem 3.3",
      "anchor": "oa-fnd-lt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 3.3.**\n\n(a) \\(\\|x\\|\\le\\|x\\|_2=\\|x^*\\|_2\\) and \\(\\|axb\\|_2\\le\\|a\\|\\,\\|x\\|_2\\,\\|b\\|\\) for \\(a,b\\in B(H)\\). So \\(\\mathcal L^2(H)\\) is a two-sided ideal of \\(B(H)\\), closed under adjoints.\n\n(b) \\(\\mathcal L^2(H)\\subseteq K(H)\\). A compact operator \\(x\\) is Hilbert–Schmidt if and only if \\(\\sum_ns_n(x)^2<\\infty\\), and then \\(\\|x\\|_2^2=\\sum_ns_n(x)^2\\).\n\n(c) For \\(x,y\\in\\mathcal L^2(H)\\), the sum \\(\\langle x,y\\rangle_2=\\sum_i\\langle x\\varepsilon_i,y\\varepsilon_i\\rangle\\) converges absolutely and does not depend on the basis. With this inner product \\(\\mathcal L^2(H)\\) is a Hilbert space with norm \\(\\|\\cdot\\|_2\\), and \\(F(H)\\) is dense in it.\n\n(d) \\(\\|\\theta_{\\xi,\\eta}\\|_2=\\|\\xi\\|\\,\\|\\eta\\|\\), and \\(\\langle\\theta_{\\xi,\\eta},\\theta_{\\xi',\\eta'}\\rangle_2=\\langle\\xi,\\xi'\\rangle\\langle\\eta',\\eta\\rangle\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 369,
        "through_line": 406,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.4",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-04::Example 3.4",
      "anchor": "oa-fnd-lt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 3.4** (Hilbert–Schmidt operators as integral operators). Let \\((X,\\mu)\\) be a \\(\\sigma\\)-finite measure space and \\(H=L^2(X,\\mu)\\). For \\(k\\in L^2(X\\times X,\\mu\\otimes\\mu)\\) define\n\\[\n(x_kf)(s)=\\int_Xk(s,t)f(t)\\,d\\mu(t).\n\\]\nThen \\(x_k\\in\\mathcal L^2(H)\\), \\(\\|x_k\\|_2=\\|k\\|_{L^2}\\), and \\(k\\mapsto x_k\\) is a unitary operator of \\(L^2(X\\times X,\\mu\\otimes\\mu)\\) onto \\(\\mathcal L^2(H)\\). So the Hilbert–Schmidt operators on \\(L^2(X,\\mu)\\) are exactly the integral operators with square-integrable kernels.\n\n*Proof.* By Tonelli's theorem \\(\\int\\!\\!\\int|k(s,t)|^2d\\mu(t)\\,d\\mu(s)=\\|k\\|^2<\\infty\\), so \\(k(s,\\cdot)\\in L^2(X,\\mu)\\) for almost every \\(s\\). For such \\(s\\) the integral defining \\((x_kf)(s)\\) converges absolutely, and \\(|(x_kf)(s)|\\le\\|k(s,\\cdot)\\|\\,\\|f\\|\\) by Cauchy–Schwarz. If \\(A\\subseteq X\\) has finite measure, then \\[\n\\begin{gathered}\n\\int_A\\int|k(s,t)f(t)|\\,d\\mu(t)\\,d\\mu(s)\\\\\n\\le\\mu(A)^{1/2}\\|k\\|\\,\\|f\\|<\\infty,\n\\end{gathered}\n\\] so Fubini's theorem shows that \\(x_kf\\) is measurable on \\(A\\); as \\(X\\) is a countable union of such sets, \\(x_kf\\) is measurable. Then \\(\\|x_kf\\|^2\\le\\|k\\|^2\\|f\\|^2\\). So \\(x_k\\in B(H)\\), \\(\\|x_k\\|\\le\\|k\\|\\), and \\(k\\mapsto x_k\\) is linear.\n\nFor a product kernel \\(k(s,t)=\\varphi(s)\\overline{\\psi(t)}\\) with \\(\\varphi,\\psi\\in L^2(X,\\mu)\\) we get \\(x_k=\\theta_{\\varphi,\\psi}\\). By Theorem 3.3(d),\n\\[\n\\begin{gathered}\n\\langle\\theta_{\\varphi,\\psi},\\theta_{\\varphi',\\psi'}\\rangle_2\\\\\n=\\langle\\varphi,\\varphi'\\rangle\\langle\\psi',\\psi\\rangle\\\\\n=\\langle\\varphi\\otimes\\bar\\psi,\\ \\varphi'\\otimes\\bar\\psi'\\rangle_{L^2(X\\times X)} .\n\\end{gathered}\n\\]\nSo \\(k\\mapsto x_k\\) is isometric from the span \\(P\\) of the product kernels into \\(\\mathcal L^2(H)\\). The span \\(P\\) is dense in \\(L^2(X\\times X)\\). Indeed, if \\(g\\in L^2(X\\times X)\\) is orthogonal to \\(P\\), let \\(X=\\bigcup_nA_n\\) with \\(A_n\\) increasing of finite measure. The finite signed measures \\(E\\mapsto\\int_E\\operatorname{Re}g\\) and \\(E\\mapsto\\int_E\\operatorname{Im}g\\) on \\(A_n\\times A_n\\) vanish on the rectangles \\(A\\times B\\subseteq A_n\\times A_n\\), which form a family closed under intersections that generates the product \\(\\sigma\\)-algebra; by Dynkin's lemma they vanish identically, so \\(g=0\\) almost everywhere on each \\(A_n\\times A_n\\), and hence almost everywhere on \\(X\\times X\\).\n\nConsequently \\(k\\mapsto x_k\\) extends from \\(P\\) to an isometry \\(U\\) of \\(L^2(X\\times X)\\) into \\(\\mathcal L^2(H)\\). If \\(k_n\\in P\\) and \\(k_n\\to k\\) in \\(L^2\\), then \\(x_{k_n}\\to x_k\\) in operator norm (since \\(\\|x_k-x_{k_n}\\|\\le\\|k-k_n\\|\\)), and \\(x_{k_n}\\to Uk\\) in \\(\\|\\cdot\\|_2\\), hence in operator norm. So \\(Uk=x_k\\). The range of \\(U\\) is closed and contains every \\(\\theta_{\\varphi,\\psi}\\), hence \\(F(H)\\), which is dense in \\(\\mathcal L^2(H)\\) by Theorem 3.3(c). So \\(U\\) is onto. \\(\\square\\)",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 407,
        "through_line": 431,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.5",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-04::Example 3.5",
      "anchor": "oa-fnd-lt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 3.5** (an operator that is Hilbert–Schmidt but not of trace class). On \\(H=L^2(0,1)\\) let \\((Vf)(s)=\\int_0^sf(t)\\,dt\\), the integral operator with kernel \\(k(s,t)=1\\) for \\(t<s\\) and \\(0\\) otherwise. By Example 3.4, \\(V\\in\\mathcal L^2(H)\\) and \\(\\|V\\|_2^2=\\int_0^1\\!\\int_0^1k^2=\\frac12\\). We compute its singular values. The adjoint is \\((V^*g)(t)=\\int_t^1g(s)\\,ds\\). Let \\(\\mu>0\\) and \\(f\\ne0\\) with \\(V^*Vf=\\mu f\\). The function \\(u=Vf\\) is continuous with \\(u(0)=0\\): Cauchy–Schwarz gives \\(|u(t)-u(s)|\\leq|t-s|^{1/2}\\|f\\|_2\\). The continuous-integrand fundamental theorem in [Lemma 0.1 of the Cauchy lesson](cauchy-s-theorem-for-cycles-and-its-consequences.md#oa-fnd-ct-07) justifies the derivatives below. Then \\(\\mu f=V^*u\\) is continuous, so \\(u\\) is continuously differentiable with \\(u'=f\\). Next \\(\\mu f=V^*u\\) is continuously differentiable with \\((\\mu f)'=-u\\) and \\(\\mu f(1)=0\\). Thus \\(\\mu u''=-u\\), \\(u(0)=0\\) and \\(u'(1)=0\\). Put \\(a=\\mu^{-1/2}\\). The general solution of \\(u''+a^2u=0\\) is \\(A\\cos(at)+B\\sin(at)\\). To verify completeness of this list, choose \\(A=u(0)\\), \\(B=u'(0)/a\\), and subtract that solution. The difference \\(w\\) has \\(w(0)=w'(0)=0\\); differentiating \\(|w'|^2+a^2|w|^2\\) gives zero, so \\(w=0\\). Thus the boundary conditions leave precisely the multiples of \\(\\sin(t/\\sqrt\\mu)\\) with \\(\\cos(1/\\sqrt\\mu)=0\\). So the nonzero eigenvalues of \\(V^*V\\) are \\(\\mu_n=((n-\\frac12)\\pi)^{-2}\\), \\(n\\ge1\\), each with a one-dimensional eigenspace spanned by \\(\\cos((n-\\frac12)\\pi t)\\). One checks directly that these functions are eigenvectors. By Theorem 2.3(c),\n\\[\ns_n(V)=\\frac{1}{(n-\\frac12)\\pi},\\qquad n\\ge1 .\n\\]\nTheorem 3.3(b) and the independently computed kernel norm now give \\(\\frac4{\\pi^2}\\sum_n(2n-1)^{-2}=\\sum_ns_n(V)^2=\\|V\\|_2^2=\\frac12\\). In particular this calculation proves \\(\\sum_n(2n-1)^{-2}=\\pi^2/8\\), rather than assuming that sum. But \\(\\sum_ns_n(V)=\\infty\\), so \\(V\\) is not of trace class (Theorem 4.3 below).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 432,
        "through_line": 437,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Definition 4.1",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 4.1.** For \\(x\\in B(H)\\) let\n\\[\n\\begin{gathered}\n\\|x\\|_1\\\\\n=\\sup\\Big\\{\\sum_{j\\in J}|\\langle x\\xi_j,\\eta_j\\rangle|\\Big\\}\\in[0,\\infty],\n\\end{gathered}\n\\tag{4.1}\n\\]\nthe supremum over all finite sets \\(J\\) and all orthonormal families \\((\\xi_j)_{j\\in J}\\) and \\((\\eta_j)_{j\\in J}\\) in \\(H\\). The operators with \\(\\|x\\|_1<\\infty\\) are the *trace-class* (or *nuclear*) operators; they form the set \\(\\mathcal L^1(H)\\). Since the terms are nonnegative, the supremum does not change if infinite index sets \\(J\\) are allowed.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 442,
        "through_line": 451,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
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        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.2",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Lemma 4.2",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 4.2.**\n\n(a) \\(\\mathcal L^1(H)\\) is a linear subspace of \\(B(H)\\), \\(\\|\\cdot\\|_1\\) is a norm on it, \\(\\|x\\|\\le\\|x\\|_1\\), and \\(\\|x^*\\|_1=\\|x\\|_1\\).\n\n(b) \\(\\|\\theta_{\\xi,\\eta}\\|_1=\\|\\xi\\|\\,\\|\\eta\\|\\).\n\n(c) If \\((x_\\alpha)\\) is a net with \\(\\langle x_\\alpha\\xi,\\eta\\rangle\\to\\langle x\\xi,\\eta\\rangle\\) for all \\(\\xi,\\eta\\), then \\(\\|x\\|_1\\le\\liminf_\\alpha\\|x_\\alpha\\|_1\\).\n\n(d) \\((\\mathcal L^1(H),\\|\\cdot\\|_1)\\) is a Banach space.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 452,
        "through_line": 484,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
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              "heading": "4. Trace-class operators and the trace",
              "line": 438,
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                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
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              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
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                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.3",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Theorem 4.3",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 4.3.**\n\n(a) \\(\\|x\\|_2^2\\le\\|x\\|\\,\\|x\\|_1\\) for every \\(x\\in B(H)\\). Hence \\(\\mathcal L^1(H)\\subseteq\\mathcal L^2(H)\\subseteq K(H)\\).\n\n(b) For \\(x\\in K(H)\\), \\(\\|x\\|_1=\\sum_ns_n(x)\\) in \\([0,\\infty]\\). So \\(x\\in\\mathcal L^1(H)\\) if and only if \\(x\\) is compact and \\(\\sum_ns_n(x)<\\infty\\).\n\n(c) If \\(x\\in\\mathcal L^1(H)\\), every decomposition (2.1) of \\(x\\) converges in \\(\\|\\cdot\\|_1\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 485,
        "through_line": 516,
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      },
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              "heading": "4. Trace-class operators and the trace",
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                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
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              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.4",
      "kind": "corollary",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Corollary 4.4",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Corollary 4.4.**\n\n(a) If \\(x\\in\\mathcal L^1(H)\\) and \\(a,b\\in B(H)\\), then \\(axb\\in\\mathcal L^1(H)\\) and \\(\\|axb\\|_1\\le\\|a\\|\\,\\|x\\|_1\\,\\|b\\|\\). So \\(\\mathcal L^1(H)\\) is a two-sided ideal of \\(B(H)\\), closed under adjoints.\n\n(b) \\(x\\in\\mathcal L^1(H)\\) if and only if \\(|x|\\in\\mathcal L^1(H)\\), if and only if \\(x^*\\in\\mathcal L^1(H)\\), and \\(\\|x\\|_1=\\||x|\\|_1=\\|x^*\\|_1\\).\n\n(c) \\(F(H)\\subseteq\\mathcal L^1(H)\\), and \\(F(H)\\) is dense in \\(\\mathcal L^1(H)\\) for \\(\\|\\cdot\\|_1\\). For \\(x\\in\\mathcal L^1(H)\\), \\(\\|x\\|\\le\\|x\\|_2\\le\\|x\\|_1\\).\n\n(d) If \\(\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\), the series \\(\\sum_n\\theta_{\\xi_n,\\eta_n}\\) converges in \\(\\|\\cdot\\|_1\\), and its sum \\(t\\) satisfies \\(\\|t\\|_1\\le\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 517,
        "through_line": 528,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.5",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Example 4.5",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 4.5** (three different norms). Let \\(\\xi\\perp\\eta\\) be unit vectors and \\(x=\\theta_{\\xi,\\eta}+\\theta_{\\eta,\\xi}\\). Then \\(x(\\xi\\pm\\eta)=\\pm(\\xi\\pm\\eta)\\) and \\(x\\) vanishes on \\(\\{\\xi,\\eta\\}^\\perp\\). So \\(x\\) is self-adjoint with eigenvalues \\(\\pm1\\), and \\(s(x)=(1,1,0,0,\\dots)\\). Hence \\(\\|x\\|=1\\), \\(\\|x\\|_2=\\sqrt2\\), \\(\\|x\\|_1=2\\), and \\(\\operatorname{Tr}(x)=0\\) (Theorem 4.6). For the diagonal operators of Example 2.2, Example 2.6 shows that \\(d_\\lambda\\in\\mathcal L^2(H)\\) exactly when \\(\\lambda\\in\\ell^2(I)\\) and \\(d_\\lambda\\in\\mathcal L^1(H)\\) exactly when \\(\\lambda\\in\\ell^1(I)\\), with \\(\\|d_\\lambda\\|_2=\\|\\lambda\\|_2\\) and \\(\\|d_\\lambda\\|_1=\\|\\lambda\\|_1\\). The Volterra operator of Example 3.5 lies in \\(\\mathcal L^2(H)\\) but not in \\(\\mathcal L^1(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 529,
        "through_line": 530,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
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          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.6",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Theorem 4.6",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 4.6** (the trace). Let \\(x\\in\\mathcal L^1(H)\\) and let \\((\\varepsilon_i)_{i\\in I}\\) be an orthonormal basis. Then \\(\\sum_i\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\) converges absolutely, \\(\\sum_i|\\langle x\\varepsilon_i,\\varepsilon_i\\rangle|\\le\\|x\\|_1\\), and the sum, denoted \\(\\operatorname{Tr}(x)\\), does not depend on the basis. More precisely, if \\(x=\\sum_n\\theta_{\\xi_n,\\eta_n}\\) with \\(\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\), as in (2.1), then\n\\[\n\\operatorname{Tr}(x)=\\sum_n\\langle\\xi_n,\\eta_n\\rangle .\n\\tag{4.2}\n\\]\nThe trace is a linear functional on \\(\\mathcal L^1(H)\\) with \\(|\\operatorname{Tr}(x)|\\le\\|x\\|_1\\), \\(\\operatorname{Tr}(x^*)=\\overline{\\operatorname{Tr}(x)}\\), \\(\\operatorname{Tr}(x)\\ge0\\) for \\(x\\ge0\\), and \\(\\operatorname{Tr}(\\theta_{\\xi,\\eta})=\\langle\\xi,\\eta\\rangle\\). For \\(a\\in B(H)\\),\n\\[\n\\begin{gathered}\n\\operatorname{Tr}(ax)\\\\\n=\\operatorname{Tr}(xa),\\\\\n|\\operatorname{Tr}(ax)|\\\\\n\\le\\|a\\|\\,\\|x\\|_1,\n\\end{gathered}\n\\tag{4.3}\n\\]\nand in particular\n\\[\n\\operatorname{Tr}(a\\,\\theta_{\\xi,\\eta})=\\langle a\\xi,\\eta\\rangle=\\omega_{\\xi,\\eta}(a).\n\\tag{4.4}\n\\]\nFinally \\(\\|x\\|_1=\\operatorname{Tr}(|x|)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 531,
        "through_line": 576,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
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              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.7",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Proposition 4.7",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 4.7** (testing on one basis). Let \\(h\\in B(H)_+\\). The sum \\(\\sum_i\\langle h\\varepsilon_i,\\varepsilon_i\\rangle\\in[0,\\infty]\\) is the same for all orthonormal bases. It is finite if and only if \\(h\\in\\mathcal L^1(H)\\), and then it equals \\(\\operatorname{Tr}(h)=\\|h\\|_1\\). Consequently, an operator \\(x\\) is of trace class if and only if \\(\\sum_i\\langle|x|\\varepsilon_i,\\varepsilon_i\\rangle<\\infty\\) for one orthonormal basis; and then \\(\\operatorname{Tr}(x)=\\sum_i\\langle x\\varepsilon_i,\\varepsilon_i\\rangle\\) for every orthonormal basis, with absolute convergence.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 577,
        "through_line": 580,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
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        {
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          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
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                "OA-FND-LT-05"
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              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
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              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
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              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.8",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Proposition 4.8",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 4.8** (positive parts). Every \\(x\\in\\mathcal L^1(H)\\) can be written \\(x=h_1-h_2+i(h_3-h_4)\\) with positive trace-class operators \\(h_1,\\dots,h_4\\). A self-adjoint \\(x\\in\\mathcal L^1(H)\\) can be written \\(x=h_1-h_2\\) with \\(h_1,h_2\\ge0\\) of trace class and \\(\\|x\\|_1=\\operatorname{Tr}(h_1)+\\operatorname{Tr}(h_2)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 581,
        "through_line": 584,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
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              "line": 438,
              "through_line": 611,
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              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
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              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.9",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-05::Proposition 4.9",
      "anchor": "oa-fnd-lt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 4.9** (products of Hilbert–Schmidt operators).\n\n(a) An operator \\(x\\) is Hilbert–Schmidt if and only if \\(x^*x\\) is of trace class, and then \\(\\operatorname{Tr}(x^*x)=\\|x\\|_2^2\\).\n\n(b) If \\(x,y\\in\\mathcal L^2(H)\\), then \\(xy\\in\\mathcal L^1(H)\\), \\(\\|xy\\|_1\\le\\|x\\|_2\\,\\|y\\|_2\\), and \\(\\operatorname{Tr}(y^*x)=\\langle x,y\\rangle_2\\).\n\n(c) Every \\(t\\in\\mathcal L^1(H)\\) is a product \\(t=xy\\) of two Hilbert–Schmidt operators, with \\(\\|x\\|_2^2=\\|y\\|_2^2=\\|t\\|_1\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 585,
        "through_line": 611,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
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            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.1",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-06::Example 5.1",
      "anchor": "oa-fnd-lt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 5.1** (the commutative model). Let \\(I\\) be a set and \\(\\langle\\lambda,\\alpha\\rangle=\\sum_i\\lambda_i\\alpha_i\\) for \\(\\lambda\\in\\ell^\\infty(I)\\), \\(\\alpha\\in\\ell^1(I)\\).\n\n(a) \\(\\alpha\\mapsto\\langle\\cdot,\\alpha\\rangle\\) is an isometric isomorphism of \\(\\ell^1(I)\\) onto \\(c_0(I)^*\\).\n\n(b) \\(\\lambda\\mapsto\\langle\\lambda,\\cdot\\rangle\\) is an isometric isomorphism of \\(\\ell^\\infty(I)\\) onto \\(\\ell^1(I)^*\\).\n\n*Proof.* In both cases \\(|\\langle\\lambda,\\alpha\\rangle|\\le\\|\\lambda\\|_\\infty\\|\\alpha\\|_1\\). (a) Let \\(f\\in c_0(I)^*\\) and \\(\\alpha_i=f(\\delta_i)\\). For a finite \\(F\\subseteq I\\) choose \\(c_i\\) with \\(|c_i|=1\\) and \\(c_i\\alpha_i=|\\alpha_i|\\); then \\(\\sum_{i\\in F}|\\alpha_i|=f(\\sum_{i\\in F}c_i\\delta_i)\\le\\|f\\|\\). So \\(\\alpha\\in\\ell^1(I)\\) with \\(\\|\\alpha\\|_1\\le\\|f\\|\\). The finite-cutoff vectors are dense in \\(c_0(I)\\): retaining \\(\\{i:|\\lambda_i|\\geq\\varepsilon\\}\\) leaves supremum-norm error at most \\(\\varepsilon\\). Thus the functionals \\(f\\) and \\(\\langle\\cdot,\\alpha\\rangle\\), agreeing on \\(c_{00}(I)\\), are equal. The bound \\(\\|\\alpha\\|_1\\leq\\|f\\|\\) together with the first inequality proves the isometry. (b) Let \\(g\\in\\ell^1(I)^*\\) and \\(\\lambda_i=g(\\delta_i)\\). Then \\(|\\lambda_i|\\le\\|g\\|\\), and finite cutoffs are dense in \\(\\ell^1(I)\\): a finite partial absolute sum within \\(\\varepsilon\\) of \\(\\|\\alpha\\|_1\\) leaves tail norm at most \\(\\varepsilon\\). Hence \\(g\\) and \\(\\langle\\lambda,\\cdot\\rangle\\), agreeing on \\(c_{00}(I)\\), are equal. Testing on \\(\\delta_i\\) shows \\(\\|\\langle\\lambda,\\cdot\\rangle\\|\\ge\\|\\lambda\\|_\\infty\\). \\(\\square\\)\n\nSo \\(c_0(I)^{**}=\\ell^\\infty(I)\\). Through the diagonal operators of Example 2.2 this model sits inside the operator picture: for \\(\\lambda\\in\\ell^\\infty(I)\\) and \\(\\alpha\\in\\ell^1(I)\\), \\(d_\\lambda d_\\alpha=d_{\\lambda\\alpha}\\) is of trace class and \\(\\operatorname{Tr}(d_\\lambda d_\\alpha)=\\sum_i\\lambda_i\\alpha_i\\) by (4.2).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 616,
        "through_line": 625,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.2",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-06::Theorem 5.2",
      "anchor": "oa-fnd-lt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 5.2** (the dual of \\(K(H)\\)). For \\(t\\in\\mathcal L^1(H)\\) let \\(\\varphi_t(x)=\\operatorname{Tr}(xt)\\), \\(x\\in K(H)\\). Then \\(t\\mapsto\\varphi_t\\) is an isometric linear isomorphism of \\(\\mathcal L^1(H)\\) onto \\(K(H)^*\\). Its inverse sends \\(\\omega\\in K(H)^*\\) to the operator \\(t(\\omega)\\) determined by\n\\[\n\\begin{gathered}\n\\langle t(\\omega)\\xi,\\eta\\rangle\\\\\n=\\omega(\\theta_{\\xi,\\eta})\\\\\n(\\xi,\\eta\\in H).\n\\end{gathered}\n\\tag{5.1}\n\\]\nConsequently every \\(\\omega\\in K(H)^*\\) can be written\n\\[\n\\begin{gathered}\n\\omega\\\\\n=\\sum_ns_n\\,\\omega_{f_n,e_n}\\ \\ \\\\\n(\\text{norm convergent}),\\\\\n\\|\\omega\\|\\\\\n=\\sum_ns_n,\n\\end{gathered}\n\\tag{5.2}\n\\]\nwith orthonormal families \\((e_n)\\), \\((f_n)\\) and numbers \\(s_n>0\\) with \\(\\sum s_n<\\infty\\); here \\(t(\\omega)=\\sum_ns_n\\theta_{f_n,e_n}\\). Conversely, for orthonormal sequences \\((\\xi_n)\\), \\((\\eta_n)\\) and \\(\\alpha\\in\\ell^1\\), the series \\(\\omega=\\sum_n\\alpha_n\\omega_{\\xi_n,\\eta_n}\\) converges in \\(K(H)^*\\), \\(t(\\omega)=\\sum_n\\alpha_n\\theta_{\\xi_n,\\eta_n}\\), and \\(\\|\\omega\\|=\\|\\alpha\\|_1\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 626,
        "through_line": 680,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.3",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-06::Proposition 5.3",
      "anchor": "oa-fnd-lt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 5.3** (the module structure). For \\(a\\in B(H)\\) and \\(\\omega\\in K(H)^*\\) define \\((a\\omega)(x)=\\omega(xa)\\) and \\((\\omega a)(x)=\\omega(ax)\\), \\(x\\in K(H)\\). Then \\(t(a\\omega)=a\\,t(\\omega)\\) and \\(t(\\omega a)=t(\\omega)\\,a\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 681,
        "through_line": 686,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
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        {
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          "local_scopes": [
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              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.4",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-06::Theorem 5.4",
      "anchor": "oa-fnd-lt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 5.4** (the dual of \\(\\mathcal L^1(H)\\)). For \\(a\\in B(H)\\) let \\(\\psi_a(t)=\\operatorname{Tr}(at)\\), \\(t\\in\\mathcal L^1(H)\\). Then \\(a\\mapsto\\psi_a\\) is an isometric linear isomorphism of \\(B(H)\\) onto \\(\\mathcal L^1(H)^*\\). Consequently \\(B(H)\\) is isometrically isomorphic to \\(K(H)^{**}\\): the operator \\(a\\) corresponds to the functional \\(\\omega\\mapsto\\operatorname{Tr}(a\\,t(\\omega))\\) on \\(K(H)^*\\), whose value at \\(\\omega_{\\xi,\\eta}\\) is \\(\\langle a\\xi,\\eta\\rangle\\). Under this identification the canonical embedding of \\(K(H)\\) into \\(K(H)^{**}\\) is the inclusion \\(K(H)\\subseteq B(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 687,
        "through_line": 692,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
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          "local_scopes": [
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              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
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                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
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              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.1",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-07::Definition 6.1",
      "anchor": "oa-fnd-lt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 6.1.** A linear functional \\(\\omega\\) on \\(B(H)\\) is *normal* if \\(\\omega(x)=\\operatorname{Tr}(xt)\\) for some \\(t\\in\\mathcal L^1(H)\\). By Theorem 5.4 the operator \\(t\\) is unique; we write \\(t=t_\\omega\\). The normal functionals form a subspace \\(B(H)_*\\) of \\(B(H)^*\\), with the norm of \\(B(H)^*\\); it is the *predual* of \\(B(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 695,
        "through_line": 696,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-07::Theorem 6.2",
      "anchor": "oa-fnd-lt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 6.2.**\n\n(a) \\(t\\mapsto\\operatorname{Tr}(\\cdot\\,t)\\) is an isometric isomorphism of \\(\\mathcal L^1(H)\\) onto \\(B(H)_*\\). In particular \\(B(H)_*\\) is a norm-closed subspace of \\(B(H)^*\\).\n\n(b) Restriction to \\(K(H)\\) is an isometric isomorphism of \\(B(H)_*\\) onto \\(K(H)^*\\). So every bounded functional on \\(K(H)\\) has exactly one normal extension to \\(B(H)\\), and it has the same norm.\n\n(c) The map sending \\(a\\in B(H)\\) to the functional \\(\\omega\\mapsto\\omega(a)\\) is an isometric isomorphism of \\(B(H)\\) onto \\((B(H)_*)^*\\).\n\n(d) Every \\(\\omega\\in B(H)_*\\) can be written \\(\\omega=\\sum_ns_n\\omega_{f_n,e_n}\\) with orthonormal families \\((e_n)\\), \\((f_n)\\) and \\(\\sum s_n=\\|\\omega\\|\\), or equivalently \\(\\omega=\\sum_n\\omega_{\\xi_n,\\eta_n}\\) with \\(\\sum\\|\\xi_n\\|^2=\\sum\\|\\eta_n\\|^2=\\|\\omega\\|\\), the series converging in norm. Conversely, if \\(\\sum_n\\|\\xi_n\\|\\,\\|\\eta_n\\|<\\infty\\), then \\(\\sum_n\\omega_{\\xi_n,\\eta_n}\\) converges in norm to an element of \\(B(H)_*\\) of norm at most \\(\\sum\\|\\xi_n\\|\\,\\|\\eta_n\\|\\). In particular the finite sums of vector functionals are norm dense in \\(B(H)_*\\).\n\n(e) For \\(\\omega\\in B(H)_*\\), \\(\\omega(x)=\\operatorname{Tr}(xt_\\omega)=\\operatorname{Tr}(t_\\omega x)\\) and \\(\\omega(1)=\\operatorname{Tr}(t_\\omega)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 697,
        "through_line": 710,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 6.3",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-07::Proposition 6.3",
      "anchor": "oa-fnd-lt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 6.3** (positive and hermitian normal functionals). Let \\(\\omega\\in B(H)_*\\).\n\n(a) \\(\\omega\\) is positive, that is \\(\\omega(x^*x)\\ge0\\) for all \\(x\\), if and only if \\(t_\\omega\\ge0\\). Then \\(\\omega=\\sum_ns_n\\omega_{e_n}\\) for an orthonormal family \\((e_n)\\) and numbers \\(s_n>0\\) with \\(\\sum s_n=\\omega(1)=\\|\\omega\\|\\); equivalently \\(\\omega=\\sum_n\\omega_{\\zeta_n}\\) with \\(\\sum\\|\\zeta_n\\|^2=\\omega(1)\\).\n\n(b) \\(\\omega\\) is hermitian, that is \\(\\omega(x^*)=\\overline{\\omega(x)}\\) for all \\(x\\), if and only if \\(t_\\omega\\) is self-adjoint. A hermitian \\(\\omega\\) is a difference \\(\\omega=\\omega_1-\\omega_2\\) of positive normal functionals with \\(\\|\\omega\\|=\\|\\omega_1\\|+\\|\\omega_2\\|\\).\n\n(c) Every normal functional can be written \\(\\omega_1-\\omega_2+i(\\omega_3-\\omega_4)\\) with positive normal functionals \\(\\omega_1,\\dots,\\omega_4\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 711,
        "through_line": 726,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.1",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Proposition 7.1",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 7.1.** Every nonzero ideal \\(J\\) of \\(B(H)\\) contains \\(F(H)\\). So \\(F(H)\\) is the smallest nonzero ideal, and \\(K(H)\\) is the smallest nonzero closed ideal.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 731,
        "through_line": 734,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.2",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Lemma 7.2",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 7.2.** Let \\(h\\in B(H)_+\\) be noncompact. There are \\(\\varepsilon>0\\) and a projection \\(e\\) of infinite rank that commutes with \\(h\\) and satisfies \\(\\varepsilon e\\le he\\le h\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 735,
        "through_line": 738,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.3",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Proposition 7.3",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 7.3.** Let \\(H\\) be separable and infinite-dimensional, and let \\(a\\in B(H)\\) be noncompact. There are \\(b,c\\in B(H)\\) with \\(bac=1\\). Consequently every proper ideal of \\(B(H)\\) is contained in \\(K(H)\\); so \\(K(H)\\) is the largest proper ideal, and the closed ideals of \\(B(H)\\) are exactly \\(0\\), \\(K(H)\\) and \\(B(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 739,
        "through_line": 750,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.4",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Example 7.4",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 7.4** (separability is needed). Let \\(H\\) be nonseparable, and let \\(J\\) be the set of operators whose range lies in a separable closed subspace. If \\(x,y\\in J\\), then \\((x+y)H\\) lies in the closed span of two separable subspaces, which is separable; \\(axH\\) lies in the closure of \\(a\\) applied to a separable subspace, which is separable; and \\(xaH\\subseteq xH\\). So \\(J\\) is an ideal. It does not contain \\(1\\), and it contains the projection onto any separable infinite-dimensional subspace, which is not compact. So \\(K(H)\\) is not the largest proper ideal, and Proposition 7.3 fails without separability.\n\nFrom now on in this section, \\(H\\) is separable and infinite-dimensional. For \\(\\lambda\\in\\ell^\\infty\\) and an orthonormal sequence \\(\\xi=(\\xi_n)\\) in \\(H\\) let\n\\[\nd^\\xi_\\lambda=\\sum_n\\lambda_n\\theta_{\\xi_n,\\xi_n},\n\\]\nthe series converging strongly. It is the diagonal operator of Example 2.2 on the closed span of the \\(\\xi_n\\), extended by \\(0\\); so \\(\\|d^\\xi_\\lambda\\|=\\|\\lambda\\|_\\infty\\), \\(d^\\xi_\\lambda d^\\xi_\\mu=d^\\xi_{\\lambda\\mu}\\), and \\(d^\\xi_\\lambda\\) is compact exactly when \\(\\lambda\\in c_0\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 751,
        "through_line": 758,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.5",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Definition 7.5",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 7.5.** For an ideal \\(J\\) of \\(B(H)\\) let \\(E(J)\\) be the set of \\(\\lambda\\in\\ell^\\infty\\) such that \\(d^\\xi_\\lambda\\in J\\) for some orthonormal sequence \\(\\xi\\). A linear subspace \\(E\\) of \\(c_0\\) is *solid* if \\(\\mu\\in E\\) whenever \\(\\lambda\\in E\\) and \\(|\\mu_n|\\le|\\lambda_n|\\) for all \\(n\\); solid subspaces are the ideals of \\(c_0\\) in the sense of ordered vector spaces. A solid subspace is *symmetric* if \\((\\lambda_{\\pi(n)})_n\\in E\\) for every \\(\\lambda\\in E\\) and every bijection \\(\\pi\\) of \\(\\mathbb N\\). For a symmetric solid subspace \\(E\\) let\n\\[\nJ(E)=\\{x\\in K(H):\\ s(x)\\in E\\}.\n\\]",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 759,
        "through_line": 763,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.6",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Lemma 7.6",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 7.6** (rearrangements). Let \\(E\\ne0\\) be a symmetric solid subspace of \\(c_0\\).\n\n(a) \\(E\\) contains \\(c_{00}\\).\n\n(b) Let \\(\\lambda\\in E\\) and \\(\\mu\\in c_0\\), and suppose there is an injective map \\(\\sigma\\) from \\(S=\\{n:\\mu_n\\ne0\\}\\) into \\(\\mathbb N\\) with \\(|\\mu_n|\\le|\\lambda_{\\sigma(n)}|\\) for \\(n\\in S\\). Then \\(\\mu\\in E\\).\n\nIn particular, \\(E\\) contains every subsequence of each of its elements, every sequence obtained from one of its elements by inserting zeros or by prefixing finitely many terms, and the sequence \\((\\lambda_1,\\lambda_1,\\lambda_2,\\lambda_2,\\dots)\\) for each \\(\\lambda\\in E\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 764,
        "through_line": 777,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.7",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Theorem 7.7",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 7.7** (Calkin). Let \\(H\\) be separable and infinite-dimensional. For every nonzero proper ideal \\(J\\) of \\(B(H)\\), \\(E(J)\\) is a nonzero symmetric solid subspace of \\(c_0\\); for every nonzero symmetric solid subspace \\(E\\) of \\(c_0\\), \\(J(E)\\) is a nonzero proper ideal; and the maps \\(J\\mapsto E(J)\\) and \\(E\\mapsto J(E)\\) are inverse to each other. Moreover, \\(\\lambda\\in E(J)\\) if and only if \\(d^\\xi_\\lambda\\in J\\) for every orthonormal sequence \\(\\xi\\), and \\(x\\in J\\) if and only if \\(x\\) is compact and \\(s(x)\\in E(J)\\).\n\n\nUnder this correspondence \\(c_{00}\\), \\(\\ell^1\\), \\(\\ell^2\\) and \\(c_0\\) correspond to \\(F(H)\\), \\(\\mathcal L^1(H)\\), \\(\\mathcal L^2(H)\\) and \\(K(H)\\), by Theorems 3.3(b) and 4.3(b).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 778,
        "through_line": 803,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.8",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Example 7.8",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 7.8** (why the doubling operation matters). One may try to describe an ideal \\(J\\) by the set \\(\\Sigma(J)=\\{s(x):x\\in J\\}\\) of decreasing sequences alone. This set is hereditary (a decreasing sequence below a member is a member) and closed under addition. These two properties do not suffice. Let \\(\\Sigma\\) be the set of decreasing nonnegative sequences \\(\\alpha\\) with \\(\\alpha_n\\le C2^{-n}\\) for some \\(C\\). It is hereditary and closed under addition, but \\(\\{x\\in K(H):s(x)\\in\\Sigma\\}\\) is not closed under addition. Indeed, let \\(\\xi\\) and \\(\\eta\\) be orthonormal sequences with \\(\\xi_n\\perp\\eta_m\\) for all \\(n,m\\), and \\(\\lambda_n=2^{-n}\\). Then \\(s(d^\\xi_\\lambda)=s(d^\\eta_\\lambda)=\\lambda\\in\\Sigma\\), while \\(d^\\xi_\\lambda+d^\\eta_\\lambda\\) has each eigenvalue \\(2^{-n}\\) twice, so \\(s_{2n}(d^\\xi_\\lambda+d^\\eta_\\lambda)=2^{-n}\\), which is not \\(O(2^{-2n})\\). The missing condition is closure under \\(D\\), which Theorem 7.7 obtains from symmetry and solidity (Lemma 7.6).\n\n*Human reference:* [Blackadar]. The counterexample above directly establishes the need for doubling; Theorem 7.7 includes it through symmetry and solidity.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 804,
        "through_line": 807,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.9",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-08::Proposition 7.9",
      "anchor": "oa-fnd-lt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 7.9** (the Calkin algebra). Let \\(H\\) be infinite-dimensional, \\(Q(H)=B(H)/K(H)\\) the Calkin algebra, and \\(q:B(H)\\to Q(H)\\) the quotient map. \\(Q(H)\\) is a C\\*-algebra, as a quotient of a C\\*-algebra by a closed ideal (lesson on C\\*-algebras, Section 15).\n\n(a) Fix an orthonormal sequence \\(\\xi\\). The map \\(\\lambda\\mapsto q(d^\\xi_\\lambda)\\) induces an isometric \\(*\\)-isomorphism of \\(\\ell^\\infty/c_0\\) onto a C\\*-subalgebra of \\(Q(H)\\).\n\n(b) Every representation of \\(Q(H)\\) on a separable Hilbert space is zero. If \\(H\\) is separable, \\(Q(H)\\) is simple: its only closed ideals are \\(0\\) and \\(Q(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 808,
        "through_line": 821,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 8.1",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-09::Definition 8.1",
      "anchor": "oa-fnd-lt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 8.1.** Each topology below is the locally convex topology on \\(B(H)\\) generated by the listed seminorms. Here \\(\\xi,\\eta\\in H\\), and \\(\\omega\\) runs through \\(B(H)_*\\), or through the positive elements \\(B(H)_*^+\\) of \\(B(H)_*\\) where stated.\n\n| topology | seminorms |\n|---|---|\n| weak | \\(x\\mapsto\\lvert\\langle x\\xi,\\eta\\rangle\\rvert\\) |\n| strong | \\(x\\mapsto\\lVert x\\xi\\rVert\\) |\n| strong\\(^*\\) | \\(x\\mapsto(\\lVert x\\xi\\rVert^2+\\lVert x^*\\xi\\rVert^2)^{1/2}\\) |\n| \\(\\sigma\\)-weak | \\(x\\mapsto\\lvert\\omega(x)\\rvert\\), \\(\\omega\\in B(H)_*\\) |\n| \\(\\sigma\\)-strong | \\(x\\mapsto p_\\omega(x)=\\omega(x^*x)^{1/2}\\), \\(\\omega\\in B(H)_*^+\\) |\n| \\(\\sigma\\)-strong\\(^*\\) | \\(x\\mapsto(p_\\omega(x)^2+p_\\omega(x^*)^2)^{1/2}\\), \\(\\omega\\in B(H)_*^+\\) |\n| norm | \\(x\\mapsto\\lVert x\\rVert\\) |\n\nThe norm topology is also called the uniform topology, the \\(\\sigma\\)-weak topology the ultraweak topology, and the \\(\\sigma\\)-strong topology the ultrastrong topology. By definition the \\(\\sigma\\)-weak topology is \\(\\sigma(B(H),B(H)_*)\\), the weak\\(^*\\) topology of \\(B(H)\\) as the dual of \\(B(H)_*\\), or equivalently of \\(\\mathcal L^1(H)\\). The weak topology is \\(\\sigma(B(H),F_*)\\), where \\(F_*\\) denotes the space of finite sums of vector functionals \\(\\omega_{\\xi,\\eta}\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 826,
        "through_line": 839,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.2",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-09::Lemma 8.2",
      "anchor": "oa-fnd-lt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 8.2** (equivalent seminorms).\n\n(a) For \\(\\omega\\in B(H)_*^+\\) write \\(\\omega=\\sum_n\\omega_{\\zeta_n}\\) with \\(\\sum\\|\\zeta_n\\|^2<\\infty\\) (Proposition 6.3(a)). Then \\(p_\\omega(x)=(\\sum_n\\|x\\zeta_n\\|^2)^{1/2}\\), and \\(p_\\omega\\) is a seminorm. Conversely, for every square-summable sequence \\((\\xi_n)\\), \\((\\sum_n\\|x\\xi_n\\|^2)^{1/2}=p_\\omega(x)\\) with \\(\\omega=\\sum_n\\omega_{\\xi_n}\\in B(H)_*^+\\). So the \\(\\sigma\\)-strong topology is generated by the seminorms \\(x\\mapsto(\\sum_n\\|x\\xi_n\\|^2)^{1/2}\\), and the \\(\\sigma\\)-strong\\(^*\\) topology by \\(x\\mapsto(\\sum_n\\|x\\xi_n\\|^2+\\|x^*\\xi_n\\|^2)^{1/2}\\), with \\((\\xi_n)\\) square summable.\n\n(b) The \\(\\sigma\\)-weak topology is generated by the seminorms \\(x\\mapsto|\\sum_n\\langle x\\xi_n,\\eta_n\\rangle|\\) with \\((\\xi_n)\\), \\((\\eta_n)\\) square summable.\n\n(c) In each of the strong, strong\\(^*\\), \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, finitely many of the defining seminorms are dominated by a single seminorm of the same kind. So every neighbourhood of \\(x_0\\) contains a set \\(\\{x:p(x-x_0)<\\varepsilon\\}\\) with one seminorm \\(p\\) of that kind; for the strong topology \\(p\\) can be taken as \\((\\sum_{j\\le m}\\|x\\xi_j\\|^2)^{1/2}\\) for finitely many vectors, and similarly for the strong\\(^*\\) topology.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 840,
        "through_line": 856,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.3",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-09::Proposition 8.3",
      "anchor": "oa-fnd-lt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 8.3** (comparison). In the table\n\n| | weak type | strong type | strong\\(^*\\) type |\n|---|---|---|---|\n| plain | weak | strong | strong\\(^*\\) |\n| \\(\\sigma\\) | \\(\\sigma\\)-weak | \\(\\sigma\\)-strong | \\(\\sigma\\)-strong\\(^*\\) |\n\neach topology is finer than the topologies to its left and than the topology above it, and the norm topology is finer than all six. So the weak topology is the coarsest and the \\(\\sigma\\)-strong\\(^*\\) topology the finest of the six.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 857,
        "through_line": 872,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.4",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-09::Example 8.4",
      "anchor": "oa-fnd-lt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 8.4** (the comparisons are strict). Let \\(H\\) be infinite-dimensional and \\((\\xi_n)\\) an orthonormal sequence.\n\n(a) *Norm and \\(\\sigma\\)-strong\\(^*\\).* The projections \\(e_n=\\theta_{\\xi_n,\\xi_n}\\) tend to \\(0\\) \\(\\sigma\\)-strongly\\(^*\\), although \\(\\|e_n\\|=1\\). Indeed \\(p_\\omega(e_n)^2=\\omega(e_n)\\), and \\(\\sum_n\\omega(e_n)\\le\\omega(1)\\) for \\(\\omega\\in B(H)_*^+\\) because \\(e_1+\\dots+e_m\\le1\\).\n\n(b) *Starred and unstarred.* \\(\\theta_{\\xi_1,\\xi_n}\\to0\\) \\(\\sigma\\)-strongly: for square-summable \\((\\zeta_k)\\), \\(\\sum_k\\|\\theta_{\\xi_1,\\xi_n}\\zeta_k\\|^2=\\sum_k|\\langle\\zeta_k,\\xi_n\\rangle|^2\\to0\\) by dominated convergence, since each term tends to \\(0\\) by Bessel's inequality and is at most \\(\\|\\zeta_k\\|^2\\). But \\(\\theta_{\\xi_1,\\xi_n}^*=\\theta_{\\xi_n,\\xi_1}\\) and \\(\\|\\theta_{\\xi_n,\\xi_1}\\xi_1\\|=1\\), so the sequence does not tend to \\(0\\) strongly\\(^*\\). So the strong\\(^*\\) topology is finer than the strong one without being equal to it, and the same holds for the \\(\\sigma\\)-strong\\(^*\\) and \\(\\sigma\\)-strong topologies.\n\n(c) *Strong type and weak type.* \\(\\theta_{\\xi_n,\\xi_1}\\to0\\) \\(\\sigma\\)-weakly: for \\(\\omega\\in B(H)_*\\), (4.4) gives \\(\\omega(\\theta_{\\xi_n,\\xi_1})=\\operatorname{Tr}(\\theta_{\\xi_n,\\xi_1}t_\\omega)=\\langle\\xi_n,t_\\omega^*\\xi_1\\rangle\\to0\\). But \\(\\|\\theta_{\\xi_n,\\xi_1}\\xi_1\\|=1\\), so there is no strong convergence to \\(0\\).\n\nExample 9.2 below shows that each \\(\\sigma\\)-topology is strictly finer than its plain version. If \\(H\\) is finite-dimensional, all seven topologies coincide: for an orthonormal basis \\(\\varepsilon_1,\\dots,\\varepsilon_d\\), \\(x=\\sum_{i,j}\\langle x\\varepsilon_j,\\varepsilon_i\\rangle\\theta_{\\varepsilon_i,\\varepsilon_j}\\), so \\(\\|x\\|\\le\\sum_{i,j}|\\langle x\\varepsilon_j,\\varepsilon_i\\rangle|\\), and the weak topology is already the norm topology.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 873,
        "through_line": 882,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.5",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-09::Lemma 8.5",
      "anchor": "oa-fnd-lt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 8.5** (bounded sets). On every bounded subset of \\(B(H)\\), the weak and \\(\\sigma\\)-weak topologies coincide, the strong and \\(\\sigma\\)-strong topologies coincide, and the strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topologies coincide.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 883,
        "through_line": 893,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.6",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-09::Proposition 8.6",
      "anchor": "oa-fnd-lt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 8.6** (continuity of the operations).\n\n(a) For fixed \\(a,b\\in B(H)\\), the map \\(x\\mapsto axb\\) is continuous for all seven topologies. In particular multiplication is separately continuous in each of them.\n\n(b) The adjoint \\(x\\mapsto x^*\\) is continuous for the norm, weak, \\(\\sigma\\)-weak, strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topologies. If \\(H\\) is infinite-dimensional, it is not continuous for the strong or the \\(\\sigma\\)-strong topology, not even on the unit ball.\n\n(c) If \\(x_\\alpha\\to x\\) and \\(y_\\alpha\\to y\\) strongly (resp. \\(\\sigma\\)-strongly) and \\(\\sup_\\alpha\\|x_\\alpha\\|<\\infty\\), then \\(x_\\alpha y_\\alpha\\to xy\\) strongly (resp. \\(\\sigma\\)-strongly). If both nets are bounded and converge strongly\\(^*\\) (resp. \\(\\sigma\\)-strongly\\(^*\\)), then \\(x_\\alpha y_\\alpha\\to xy\\) strongly\\(^*\\) (resp. \\(\\sigma\\)-strongly\\(^*\\)).\n\n(d) If \\(H\\) is infinite-dimensional, multiplication is not jointly continuous for the weak or the \\(\\sigma\\)-weak topology, not even on the unit ball.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 894,
        "through_line": 916,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.1",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Theorem 9.1",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 9.1** (continuous functionals). Let \\(M\\subseteq B(H)\\) be a linear subspace and \\(\\omega\\) a linear functional on \\(M\\). Continuity refers to the topologies restricted to \\(M\\).\n\n(i) The following are equivalent: (1) \\(\\omega\\) is weakly continuous; (2) \\(\\omega\\) is strongly continuous; (3) \\(\\omega\\) is strongly\\(^*\\) continuous; (4) there are finitely many vectors \\(\\xi_k,\\eta_k\\) with \\(\\omega(x)=\\sum_k\\langle x\\xi_k,\\eta_k\\rangle\\) for \\(x\\in M\\).\n\n(ii) The following are equivalent: (1) \\(\\omega\\) is \\(\\sigma\\)-weakly continuous; (2) \\(\\omega\\) is \\(\\sigma\\)-strongly continuous; (3) \\(\\omega\\) is \\(\\sigma\\)-strongly\\(^*\\) continuous; (4) there are square-summable sequences \\((\\xi_n)\\), \\((\\eta_n)\\) with \\(\\omega(x)=\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\) for \\(x\\in M\\); (5) \\(\\omega\\) is the restriction of a normal functional on \\(B(H)\\).\n\nIn particular, for \\(M=B(H)\\), the continuous linear functionals of the weak, strong and strong\\(^*\\) topologies are the elements of \\(F_*\\), and those of the three \\(\\sigma\\)-topologies are the normal functionals.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 921,
        "through_line": 945,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.2",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Example 9.2",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 9.2** (the \\(\\sigma\\)-topologies are strictly finer). Let \\(H\\) be infinite-dimensional, \\((\\xi_n)\\) orthonormal, and \\(\\omega=\\sum_n2^{-n}\\omega_{\\xi_n}\\). Then \\(\\omega\\) is normal, so it is \\(\\sigma\\)-weakly continuous. It is not strongly\\(^*\\) continuous: otherwise, by Theorem 9.1(i), \\(\\omega=\\sum_{k\\le m}\\omega_{\\xi'_k,\\eta'_k}\\), and then \\(t_\\omega=\\sum_{k\\le m}\\theta_{\\xi'_k,\\eta'_k}\\) would have finite rank (Theorem 6.2(a)), whereas \\(t_\\omega=\\sum_n2^{-n}\\theta_{\\xi_n,\\xi_n}\\) has infinite rank. Since each \\(\\sigma\\)-topology is finer than its plain version (Proposition 8.3) and \\(\\omega\\) is continuous for the \\(\\sigma\\)-weak topology but not for the strong\\(^*\\) topology, no \\(\\sigma\\)-topology equals its plain version: the \\(\\sigma\\)-weak topology differs from the weak one, the \\(\\sigma\\)-strong from the strong, and the \\(\\sigma\\)-strong\\(^*\\) from the strong\\(^*\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 946,
        "through_line": 947,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.3",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Theorem 9.3",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 9.3** (Krein–Šmulian). Let \\(X\\) be a Banach space and \\(K\\subseteq X^*\\) a convex set. If \\(K\\cap rB_{X^*}\\) is weak\\(^*\\) closed for every \\(r>0\\), then \\(K\\) is weak\\(^*\\) closed. Here \\(B_{X^*}\\) is the closed unit ball of \\(X^*\\) and weak\\(^*\\) means \\(\\sigma(X^*,X)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 948,
        "through_line": 983,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.4",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Theorem 9.4",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 9.4** (preduals of subspaces). Let \\(M\\subseteq B(H)\\) be a \\(\\sigma\\)-weakly closed linear subspace. Let \\(M_*\\) be the space of \\(\\sigma\\)-weakly continuous linear functionals on \\(M\\) and \\(M_\\sim\\) the space of weakly continuous ones, both with the norm of \\(M^*\\), and let \\(M_\\perp=\\{\\omega\\in B(H)_*:\\omega|_M=0\\}\\).\n\n(a) Restriction \\(\\omega\\mapsto\\omega|_M\\) maps \\(B(H)_*\\) onto \\(M_*\\), and it induces an isometric isomorphism of \\(B(H)_*/M_\\perp\\) onto \\(M_*\\). In particular \\(M_*\\) is complete, and it is a norm-closed subspace of \\(M^*\\).\n\n(b) The map sending \\(x\\in M\\) to the functional \\(\\varphi\\mapsto\\varphi(x)\\) is an isometric isomorphism of \\(M\\) onto \\((M_*)^*\\). Under it, \\(\\sigma(M,M_*)\\) is the \\(\\sigma\\)-weak topology restricted to \\(M\\).\n\n(c) \\(M_\\sim\\) is norm dense in \\(M_*\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 984,
        "through_line": 1010,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.5",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Theorem 9.5",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 9.5** (continuity on the unit ball). Let \\(M\\subseteq B(H)\\) be a \\(\\sigma\\)-weakly closed subspace, \\(M_1=M\\cap B(H)_1\\), and \\(\\omega\\) a linear functional on \\(M\\). The following are equivalent: (1) \\(\\omega\\) is \\(\\sigma\\)-weakly continuous on \\(M\\); (2) \\(\\omega\\) is weakly continuous on \\(M_1\\); (3) \\(\\omega\\) is strongly continuous on \\(M_1\\); (4) \\(\\omega\\) is strongly\\(^*\\) continuous on \\(M_1\\). No boundedness of \\(\\omega\\) is assumed.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1011,
        "through_line": 1018,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.6",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Theorem 9.6",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 9.6** (closed convex sets). Let \\(C\\subseteq B(H)\\) be convex.\n\n(a) \\(C\\) has the same closure in the \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, and the same closure in the weak, strong and strong\\(^*\\) topologies.\n\n(b) The following are equivalent: (1) \\(C\\) is \\(\\sigma\\)-weakly closed; (2) \\(C\\) is \\(\\sigma\\)-strongly closed; (3) \\(C\\) is \\(\\sigma\\)-strongly\\(^*\\) closed; (4) \\(C\\cap rB(H)_1\\) is weakly closed for every \\(r>0\\); (5) \\(C\\cap rB(H)_1\\) is strongly closed for every \\(r>0\\); (6) \\(C\\cap rB(H)_1\\) is strongly\\(^*\\) closed for every \\(r>0\\).\n\n(c) The same holds for convex subsets of a \\(\\sigma\\)-weakly closed subspace \\(M\\), and for convex subsets of \\(B(H)_{\\mathrm{sa}}\\), with the relative topologies.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1019,
        "through_line": 1032,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 9.7",
      "kind": "corollary",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-10::Corollary 9.7",
      "anchor": "oa-fnd-lt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Corollary 9.7.**\n\n(a) The unit ball \\(B(H)_1\\) is \\(\\sigma\\)-weakly compact and weakly compact, and so are \\(\\{x\\in B(H)_{\\mathrm{sa}}:\\|x\\|\\le1\\}\\) and \\(\\{x:0\\le x\\le1\\}\\).\n\n(b) A linear subspace of \\(B(H)\\) is \\(\\sigma\\)-weakly closed if and only if it is \\(\\sigma\\)-strongly\\(^*\\) closed, if and only if its intersection with the unit ball is weakly closed.\n\n(c) On \\(B(H)_{\\mathrm{sa}}\\) the \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies coincide. The real-valued real-linear functionals on \\(B(H)_{\\mathrm{sa}}\\) that are continuous for this topology, or for the \\(\\sigma\\)-weak topology, are exactly the restrictions of hermitian normal functionals.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1033,
        "through_line": 1046,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [
        {
          "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
          "local_scopes": [
            {
              "heading": "4. Trace-class operators and the trace",
              "line": 438,
              "through_line": 611,
              "anchors": [
                "OA-FND-LT-05"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "5. Duality between compact, trace-class and bounded operators",
              "line": 612,
              "through_line": 692,
              "anchors": [
                "OA-FND-LT-06"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "6. The predual of B(H)",
              "line": 693,
              "through_line": 726,
              "anchors": [
                "OA-FND-LT-07"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "8. Seven topologies on B(H)",
              "line": 822,
              "through_line": 916,
              "anchors": [
                "OA-FND-LT-09"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            },
            {
              "heading": "9. Continuous functionals and closed convex sets",
              "line": 917,
              "through_line": 1046,
              "anchors": [
                "OA-FND-LT-10"
              ],
              "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§§3.1–3.4; PDF 40–52",
          "role": "proof comparison",
          "correspondence": "Trace duality and topological mechanisms",
          "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.1",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-11::Proposition 10.1",
      "anchor": "oa-fnd-lt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 10.1** (metrizability on bounded sets).\n\n(a) If \\(H\\) is separable, each of the six topologies weak, strong, strong\\(^*\\), \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) is metrizable on every bounded subset of \\(B(H)\\). Explicitly, let \\((\\zeta_n)\\) be dense in the unit ball of \\(H\\) and put\n\\[\n\\begin{gathered}\nd_w(x,y)\\\\\n=\\sum_{n,m}2^{-n-m}|\\langle(x-y)\\zeta_n,\\zeta_m\\rangle|,\\\\\nd_s(x,y)\\\\\n=\\sum_n2^{-n}\\|(x-y)\\zeta_n\\|,\\\\\nd_{s^*}(x,y)\\\\\n=d_s(x,y)+d_s(x^*,y^*).\n\\end{gathered}\n\\]\nOn each ball \\(rB(H)_1\\), \\(d_w\\) induces the weak and \\(\\sigma\\)-weak topology, \\(d_s\\) the strong and \\(\\sigma\\)-strong topology, and \\(d_{s^*}\\) the strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topology.\n\n(b) If \\(H\\) is not separable, none of the six topologies is metrizable on \\(B(H)_1\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1049,
        "through_line": 1075,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.2",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-11::Proposition 10.2",
      "anchor": "oa-fnd-lt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 10.2** (the six topologies are not metrizable on B(H)). Let \\(H\\) be infinite-dimensional, let \\((e_n)\\) be a sequence of nonzero mutually orthogonal projections, let \\(c_n>0\\), and let \\(A=\\{c_ne_n:n\\ge1\\}\\).\n\n(a) If \\(\\sum_nc_n^{-2}=\\infty\\), then \\(0\\) is an accumulation point of \\(A\\) for the \\(\\sigma\\)-strong\\(^*\\) topology, and hence for all six topologies.\n\n(b) If \\(\\sum_nc_n^{-1}=\\infty\\), then \\(0\\) is an accumulation point of \\(A\\) for the \\(\\sigma\\)-weak topology.\n\n(c) Let \\(e_n=\\theta_{\\xi_n,\\xi_n}\\) for an orthonormal sequence \\((\\xi_n)\\). If \\(\\sum_nc_n^{-2}<\\infty\\), then \\(0\\) is not in the strong closure of \\(A\\); if \\(\\sum_nc_n^{-1}<\\infty\\), then \\(0\\) is not in the weak closure of \\(A\\).\n\n(d) If \\(c_n\\to\\infty\\), no subsequence of \\((c_ne_n)\\) converges weakly.\n\n(e) None of the six topologies is metrizable on \\(B(H)\\); indeed none of them has a countable base of neighbourhoods at \\(0\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1076,
        "through_line": 1101,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.3",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-11::Definition 10.3",
      "anchor": "oa-fnd-lt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 10.3.** A *state* of \\(B(H)\\) is a positive linear functional \\(\\varphi\\), that is \\(\\varphi(x^*x)\\ge0\\) for all \\(x\\), with \\(\\varphi(1)=1\\). The states form a convex set \\(S(H)\\). A *pure state* is an extreme point of \\(S(H)\\). A *vector state* is a state \\(\\omega_\\xi\\) with \\(\\|\\xi\\|=1\\); let \\(V\\) be the set of vector states and \\(\\overline V\\) its closure in the weak\\(^*\\) topology \\(\\sigma(B(H)^*,B(H))\\). Let \\(S_0(H)\\) be the set of states that vanish on \\(K(H)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1102,
        "through_line": 1103,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.4",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-11::Lemma 10.4",
      "anchor": "oa-fnd-lt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 10.4.** (a) Every positive linear functional \\(\\varphi\\) on \\(B(H)\\) is bounded, with \\(\\|\\varphi\\|=\\varphi(1)\\), and hermitian: \\(\\varphi(x^*)=\\overline{\\varphi(x)}\\). (b) \\(S(H)\\) is weak\\(^*\\) compact, and \\(S_0(H)\\) is a weak\\(^*\\) closed convex subset of it.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1104,
        "through_line": 1107,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.5",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-11::Proposition 10.5",
      "anchor": "oa-fnd-lt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 10.5** (vector states).\n\n(a) Every pure state of \\(B(H)\\) lies in \\(\\overline V\\).\n\n(b) Let \\(H\\) be infinite-dimensional. Then \\(S_0(H)\\cap\\overline V\\ne\\emptyset\\). If \\(f\\in\\overline V\\cap S_0(H)\\) and \\(L\\subseteq H\\) is a finite-dimensional subspace, then \\(f\\) lies in the weak\\(^*\\) closure of \\(\\{\\omega_\\eta:\\eta\\in L^\\perp,\\ \\|\\eta\\|=1\\}\\).\n\n(c) \\(\\overline V\\cap S_0(H)\\) is convex.\n\n(d) \\(S_0(H)\\subseteq\\overline V\\): every state that vanishes on the compact operators is a weak\\(^*\\) limit of vector states.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1108,
        "through_line": 1127,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.1",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-12::Lemma 11.1",
      "anchor": "oa-fnd-lt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.1.** Let \\(h\\in B(H)_{\\mathrm{sa}}\\), \\(\\xi\\in H\\) and \\(m\\ge1\\). There is a projection \\(p\\) of rank at most \\(m\\) with \\(p\\xi=\\xi\\) and \\(\\|(1-p)hp\\|_2\\le\\|h\\|\\,m^{-1/2}\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1134,
        "through_line": 1148,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.2",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-12::Theorem 11.2",
      "anchor": "oa-fnd-lt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 11.2** (Weyl–von Neumann). Let \\(H\\) be separable, \\(h\\in B(H)_{\\mathrm{sa}}\\) and \\(\\varepsilon>0\\). Then \\(h=k+a\\), where \\(a\\) is a self-adjoint Hilbert–Schmidt operator with \\(\\|a\\|_2<\\varepsilon\\) and \\(k\\) is a self-adjoint operator for which \\(H\\) has an orthonormal basis of eigenvectors.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1149,
        "through_line": 1170,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.3",
      "kind": "example",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-12::Example 11.3",
      "anchor": "oa-fnd-lt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Example 11.3** (separability cannot be dropped). Let \\(\\Gamma\\) be an uncountable set, \\(H=\\ell^2(\\Gamma;L^2(0,1))\\), and let \\(h\\) act on each coordinate as the operator \\(m\\) of multiplication by the variable, \\((m g)(t)=tg(t)\\). Then \\(h\\ne k+a\\) for every Hilbert–Schmidt operator \\(a\\) and every operator \\(k\\) for which \\(H\\) has an orthonormal basis of eigenvectors. Suppose otherwise. Since \\(a\\) is compact, the closures of the ranges of \\(a\\) and \\(a^*\\) are separable, and so is the closed span \\(L\\) of the vectors \\(h^j v\\), \\(j\\ge0\\), with \\(v\\) in these ranges. \\(L\\) is invariant under \\(h\\), hence reduces \\(h\\). It contains the ranges of \\(a\\) and \\(a^*\\), so \\(L^\\perp\\) lies in the kernels of \\(a^*\\) and \\(a\\): both \\(a\\) and \\(a^*\\) vanish on \\(L^\\perp\\). Every vector of \\(H\\) has countably many nonzero coordinates, so a countable dense subset of \\(L\\) is supported in a countable set \\(\\Gamma_0\\subseteq\\Gamma\\), and \\(L\\) lies in the coordinates \\(\\Gamma_0\\). For \\(\\gamma\\notin\\Gamma_0\\), the subspace \\(X_\\gamma\\) of vectors supported at \\(\\gamma\\) lies in \\(L^\\perp\\) and reduces \\(h\\), \\(a\\) (which is \\(0\\) there) and hence \\(k\\). The projection \\(P\\) onto \\(X_\\gamma\\) commutes with \\(k\\), so it maps each eigenvector of \\(k\\) to an eigenvector or to \\(0\\). These images span a dense subspace of \\(X_\\gamma\\), so \\(k\\) has an eigenvector in \\(X_\\gamma\\). But on \\(X_\\gamma\\), \\(k=h\\) acts as \\(m\\), which has no eigenvectors: \\((t-c)g(t)=0\\) almost everywhere forces \\(g=0\\). This contradiction shows that Theorem 11.2 needs separability, even for Hilbert–Schmidt perturbations of arbitrary size.\n\n**Remark 11.4.** The Hilbert–Schmidt norm enters only through Lemma 11.1: a block of rank at most \\(m\\) whose pieces have norm at most \\(\\|h\\|/m\\) has Hilbert–Schmidt norm at most \\(\\|h\\|m^{-1/2}\\), which tends to \\(0\\). The trace norm of the same block is only bounded by \\(\\|h\\|\\). Theorem 11.12 and Remark 11.17 show that this is not a weakness of the proof: the theorem is false for trace-class perturbations.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1171,
        "through_line": 1174,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 11.5",
      "kind": "definition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Definition 11.5",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Definition 11.5.** For \\(h\\in B(H)_{\\mathrm{sa}}\\) let \\(H_{\\mathrm{ac}}(h)\\) be the set of vectors \\(\\xi\\) whose spectral measure \\(\\mu_\\xi\\) vanishes on null sets, and \\(H_{\\mathrm s}(h)\\) the set of vectors whose spectral measure is concentrated on a null Borel set. The operator \\(h\\) is *absolutely continuous* if \\(H_{\\mathrm{ac}}(h)=H\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1177,
        "through_line": 1178,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.6",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Proposition 11.6",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 11.6.** \\(H_{\\mathrm{ac}}(h)\\) and \\(H_{\\mathrm s}(h)\\) are closed subspaces, \\(H_{\\mathrm s}(h)=H_{\\mathrm{ac}}(h)^\\perp\\), and both are invariant under every \\(f(h)\\), \\(f\\) a bounded Borel function; in particular they reduce \\(h\\). Every eigenvector of \\(h\\) lies in \\(H_{\\mathrm s}(h)\\). We write \\(P_{\\mathrm{ac}}(h)\\) for the projection onto \\(H_{\\mathrm{ac}}(h)\\), and \\(h_{\\mathrm{ac}}\\) and \\(h_{\\mathrm s}\\) for the restrictions of \\(h\\) to the two subspaces; the spectral measures of \\(h_{\\mathrm{ac}}\\) are the \\(\\mu_\\xi\\), \\(\\xi\\in H_{\\mathrm{ac}}(h)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1179,
        "through_line": 1184,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.7",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Lemma 11.7",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.7** (cyclic subspaces). For \\(\\xi\\in H\\) let \\(Z(\\xi)\\) be the closure of \\(\\{f(h)\\xi:f\\ \\text{bounded Borel}\\}\\). Then \\(Z(\\xi)\\) reduces \\(h\\), and there is a unitary \\(U:Z(\\xi)\\to L^2(\\mathbb R,\\mu_\\xi)\\) with \\(U(f(h)\\xi)=f\\). Under \\(U\\), each \\(g(h)\\) restricted to \\(Z(\\xi)\\) becomes multiplication by \\(g\\); in particular \\(h\\) becomes multiplication by the variable \\(\\lambda\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1185,
        "through_line": 1188,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.8",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Proposition 11.8",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 11.8** (the model). The operator \\(h_{\\mathrm{ac}}\\) is unitarily equivalent to a direct sum \\(\\bigoplus_{i\\in I}m_{S_i}\\), where each \\(S_i\\subseteq[-\\|h\\|,\\|h\\|]\\) is a Borel set and \\(m_S\\) is multiplication by \\(\\lambda\\) on \\(L^2(S,d\\lambda)\\). If \\(H\\) is separable, \\(I\\) is countable. Conversely, every such direct sum is absolutely continuous. So \\(h\\) is absolutely continuous exactly when it is unitarily equivalent to such a direct sum.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1189,
        "through_line": 1192,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.9",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Proposition 11.9",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 11.9** (multiplication operators). Let \\((X,\\mu)\\) be a \\(\\sigma\\)-finite measure space, \\(f:X\\to\\mathbb R\\) a bounded measurable function, and \\(m_f\\) multiplication by \\(f\\) on \\(L^2(X,\\mu)\\).\n\n(a) \\(m_f\\) is absolutely continuous exactly when \\(\\mu(f^{-1}(N))=0\\) for every null Borel set \\(N\\subseteq\\mathbb R\\).\n\n(b) Let \\(X\\) be a bounded open interval with Lebesgue measure and \\(f\\) continuously differentiable and bounded. Then \\(m_f\\) is absolutely continuous exactly when \\(\\{t:f'(t)=0\\}\\) is a null set.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1193,
        "through_line": 1206,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.10",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Lemma 11.10",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.10** (square-integrable Fourier transforms). Let \\(\\mu\\) be a finite positive Borel measure on \\(\\mathbb R\\) and \\(\\hat\\mu(s)=\\int e^{is\\lambda}d\\mu(\\lambda)\\).\n\n(a) If \\(\\hat\\mu\\in L^2(\\mathbb R)\\), then \\(\\mu=g\\,d\\lambda\\) with \\(g\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\).\n\n(b) If \\(\\mu=g\\,d\\lambda\\) with \\(g\\in L^1(\\mathbb R)\\cap L^2(\\mathbb R)\\), then \\(\\hat\\mu\\in L^2(\\mathbb R)\\) and \\(\\int|\\hat\\mu|^2ds=2\\pi\\int g^2d\\lambda\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1207,
        "through_line": 1222,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.11",
      "kind": "proposition",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-13::Proposition 11.11",
      "anchor": "oa-fnd-lt-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Proposition 11.11** (a Fourier criterion). Let \\(u(t)=e^{ith}\\), so that \\(\\langle u(t)\\xi,\\xi\\rangle=\\hat\\mu_\\xi(t)\\). Let \\(D\\) be the set of \\(\\xi\\in H\\) with \\(\\int_{\\mathbb R}|\\langle u(t)\\xi,\\xi\\rangle|^2dt<\\infty\\). Then \\(D\\subseteq H_{\\mathrm{ac}}(h)\\), and the set of \\(\\xi\\in H_{\\mathrm{ac}}(h)\\) whose spectral measure has a bounded density is contained in \\(D\\) and dense in \\(H_{\\mathrm{ac}}(h)\\). Consequently \\(h\\) is absolutely continuous exactly when \\(D\\) is dense in \\(H\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1223,
        "through_line": 1226,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.12",
      "kind": "theorem",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-14::Theorem 11.12",
      "anchor": "oa-fnd-lt-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Theorem 11.12** (Kato–Rosenblum). Let \\(h,k\\in B(H)_{\\mathrm{sa}}\\) with \\(h-k\\in\\mathcal L^1(H)\\). Then the strong limit\n\\[\nW_+=\\lim_{t\\to\\infty}e^{ith}e^{-itk}P_{\\mathrm{ac}}(k)\n\\]\nexists. It is a partial isometry with initial space \\(H_{\\mathrm{ac}}(k)\\) and final space \\(H_{\\mathrm{ac}}(h)\\), and \\(hW_+=W_+k\\). In particular \\(h_{\\mathrm{ac}}\\) and \\(k_{\\mathrm{ac}}\\) are unitarily equivalent. The same holds for \\(t\\to-\\infty\\).\n\nThe proof takes the rest of this section. Its argument uses the following elementary stages: an estimate for the rate of convergence is proved first for perturbations for which convergence is easy, in a form that does not involve the limit, and then carried over to all trace-class perturbations by approximation. The operator \\(W_+\\) is called a *wave operator*. Nothing in the proof uses separability of \\(H\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1239,
        "through_line": 1246,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.13",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-14::Lemma 11.13",
      "anchor": "oa-fnd-lt-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.13** (an \\(L^2\\) estimate). Let \\(k\\in B(H)_{\\mathrm{sa}}\\) and let \\(\\zeta\\in H_{\\mathrm{ac}}(k)\\) have spectral density \\(\\rho\\le M\\) almost everywhere. Then for every \\(\\eta\\in H\\),\n\\[\n\\int_{\\mathbb R}|\\langle e^{-isk}\\zeta,\\eta\\rangle|^2ds\\le2\\pi M\\,\\|\\eta\\|^2 .\n\\]",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1247,
        "through_line": 1272,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.14",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-14::Lemma 11.14",
      "anchor": "oa-fnd-lt-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.14** (Cook's criterion). Let \\(a,b\\in B(H)_{\\mathrm{sa}}\\), and let \\(D\\subseteq H_{\\mathrm{ac}}(b)\\) be a set whose linear span is dense in \\(H_{\\mathrm{ac}}(b)\\). If \\(\\int_0^\\infty\\|(a-b)e^{-isb}\\zeta\\|\\,ds<\\infty\\) for every \\(\\zeta\\in D\\), then \\(\\lim_{t\\to\\infty}W_{a,b}(t)\\zeta\\) exists for every \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1273,
        "through_line": 1282,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.15",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-14::Lemma 11.15",
      "anchor": "oa-fnd-lt-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.15** (properties of wave operators). Let \\(a,b\\in B(H)_{\\mathrm{sa}}\\), and suppose that \\(\\lim_{t\\to\\infty}W(t)\\zeta\\) exists for every \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\), where \\(W=W_{a,b}\\). Let \\(W_+\\) equal this limit on \\(H_{\\mathrm{ac}}(b)\\) and \\(0\\) on \\(H_{\\mathrm{ac}}(b)^\\perp\\).\n\n(a) \\(\\|W_+\\zeta\\|=\\|\\zeta\\|\\) for \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\), and \\(\\|W_+\\|\\le1\\).\n\n(b) \\(e^{isa}W_+=W_+e^{isb}\\) for all \\(s\\in\\mathbb R\\), and \\(aW_+=W_+b\\).\n\n(c) For \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\), the spectral measure of \\(W_+\\zeta\\) for \\(a\\) equals the spectral measure of \\(\\zeta\\) for \\(b\\). In particular \\(W_+H_{\\mathrm{ac}}(b)\\subseteq H_{\\mathrm{ac}}(a)\\).\n\n(d) Suppose \\(a-b=\\sum_{n=1}^Nc_n\\theta_{f_n,f_n}\\) with real \\(c_n\\) and vectors \\(f_n\\in H\\), and let \\(\\zeta\\in H_{\\mathrm{ac}}(b)\\) have spectral density bounded by \\(M\\). Put \\(\\eta_n(t)=\\int_t^\\infty|\\langle e^{-isb}\\zeta,f_n\\rangle|^2ds\\), finite by Lemma 11.13. Then for every \\(t\\),\n\\[\n\\begin{gathered}\n\\|W_+\\zeta-W(t)\\zeta\\|^2\\\\\n\\le2(2\\pi M)^{1/2}\\\\\n{}\\cdot\\Big(\\sum_n|c_n|\\,\\|f_n\\|^2\\Big)^{1/2}\\\\\n{}\\cdot\\Big(\\sum_n|c_n|\\,\\eta_n(t)\\Big)^{1/2}.\n\\end{gathered}\n\\tag{11.2}\n\\]",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1283,
        "through_line": 1350,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.16",
      "kind": "lemma",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-14::Lemma 11.16",
      "anchor": "oa-fnd-lt-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Lemma 11.16** (a model for the absolutely continuous part). Let \\(k\\in B(H)_{\\mathrm{sa}}\\). There are a Hilbert space \\(H'\\), an absolutely continuous \\(k'\\in B(H')_{\\mathrm{sa}}\\), a bounded open interval \\(I\\), a set \\(J\\), and a unitary \\(\\Phi\\) of \\(H_{\\mathrm{ac}}(k)\\oplus H'\\) onto the Hilbert space \\(L^2(I;\\ell^2(J))\\) of families \\((g_j)_{j\\in J}\\) in \\(L^2(I)\\) with \\(\\sum_j\\|g_j\\|^2<\\infty\\), such that \\(H_{\\mathrm{ac}}(k\\oplus k')=H_{\\mathrm{ac}}(k)\\oplus H'\\) and \\(\\Phi(k\\oplus k')\\Phi^{-1}\\) is multiplication by \\(\\lambda\\) in each entry. Let \\(D\\) be the set of \\(\\zeta\\in H_{\\mathrm{ac}}(k\\oplus k')\\) such that \\(\\Phi\\zeta\\) has finitely many nonzero entries, each in \\(C_c^\\infty(I)\\). Then \\(D\\) is a dense subspace of \\(H_{\\mathrm{ac}}(k\\oplus k')\\), and for \\(\\zeta,\\eta\\in D\\):\n\n(i) the spectral density of \\(\\zeta\\) for \\(k\\oplus k'\\) is the bounded function \\(\\lambda\\mapsto\\sum_j|(\\Phi\\zeta)_j(\\lambda)|^2\\);\n\n(ii) the function \\(s\\mapsto\\langle e^{-is(k\\oplus k')}\\zeta,\\eta\\rangle\\) is integrable on \\(\\mathbb R\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1351,
        "through_line": 1428,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.1",
      "kind": "exercise",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-15::Exercise 12.1",
      "anchor": "oa-fnd-lt-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Exercise 12.1** (medium; the adjoint on normal operators). Show that the adjoint is strongly continuous on the set of normal operators: if \\(x_\\alpha\\) and \\(x\\) are normal and \\(x_\\alpha\\to x\\) strongly, then \\(x_\\alpha^*\\to x^*\\) strongly. Compare with Proposition 8.6(b).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1486,
        "through_line": 1497,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.2",
      "kind": "exercise",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-15::Exercise 12.2",
      "anchor": "oa-fnd-lt-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Exercise 12.2** (medium; cyclicity of the trace for Hilbert–Schmidt operators). Let \\(x,y\\in\\mathcal L^2(H)\\). Show that \\(xy\\) and \\(yx\\) are of trace class and that \\(\\operatorname{Tr}(xy)=\\operatorname{Tr}(yx)\\), although neither \\(x\\) nor \\(y\\) need be of trace class.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1498,
        "through_line": 1514,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.3",
      "kind": "exercise",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-15::Exercise 12.3",
      "anchor": "oa-fnd-lt-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Exercise 12.3** (medium; the diagonal algebra and its predual). Let \\((\\varepsilon_i)_{i\\in I}\\) be an orthonormal basis and \\(\\mathcal D=\\{d_\\lambda:\\lambda\\in\\ell^\\infty(I)\\}\\) (Example 2.2). Show that \\(\\mathcal D\\) is \\(\\sigma\\)-weakly closed, that \\(\\omega\\mapsto(\\omega(d_{\\delta_i}))_i\\) is an isometric isomorphism of \\(\\mathcal D_*\\) onto \\(\\ell^1(I)\\), and that on the unit ball of \\(\\mathcal D\\) the \\(\\sigma\\)-weak topology is the topology of coordinatewise convergence of \\(\\lambda\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1515,
        "through_line": 1518,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.4",
      "kind": "exercise",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-15::Exercise 12.4",
      "anchor": "oa-fnd-lt-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Exercise 12.4** (medium; the shift). Let \\(S\\) be the unilateral shift on \\(\\ell^2\\). Show that \\(S\\) is a strong limit of unitary operators, so that the unitary group is not strongly closed, but that \\(\\|S-u\\|\\ge1\\) for every unitary \\(u\\).",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1519,
        "through_line": 1522,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.5",
      "kind": "exercise",
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies#oa-fnd-lt-15::Exercise 12.5",
      "anchor": "oa-fnd-lt-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
      "statement_and_full_conditions": "**Exercise 12.5** (easy; the trace is not weak\\(^*\\) continuous). Let \\((\\xi_n)\\) be an orthonormal sequence. Show that \\(\\theta_{\\xi_n,\\xi_n}\\to0\\) in the weak\\(^*\\) topology \\(\\sigma(\\mathcal L^1(H),K(H))\\) that comes from Theorem 5.2, while \\(\\operatorname{Tr}(\\theta_{\\xi_n,\\xi_n})=1\\). Conclude that the trace is not continuous for this topology on the unit ball of \\(\\mathcal L^1(H)\\), although this ball is compact for it.",
      "proof_locus": {
        "source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "line": 1523,
        "through_line": 1526,
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.1",
      "kind": "definition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-01::Definition 1.1",
      "anchor": "oa-fnd-bi-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Definition 1.1.** Each of the six topologies below is the locally convex topology on \\(B(H)\\) given by a family of seminorms. In the table, \\((\\xi_n)\\) and \\((\\eta_n)\\) are sequences in \\(H\\) with \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\) and \\(\\sum_n\\|\\eta_n\\|^2<\\infty\\).\n\n| topology | seminorms |\n|---|---|\n| weak | \\(x\\mapsto\\lvert\\langle x\\xi,\\eta\\rangle\\rvert\\) |\n| strong | \\(x\\mapsto\\lVert x\\xi\\rVert\\) |\n| strong\\(^*\\) | \\(x\\mapsto(\\lVert x\\xi\\rVert^2+\\lVert x^*\\xi\\rVert^2)^{1/2}\\) |\n| \\(\\sigma\\)-weak | \\(x\\mapsto\\lvert\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\rvert\\) |\n| \\(\\sigma\\)-strong | \\(x\\mapsto(\\sum_n\\lVert x\\xi_n\\rVert^2)^{1/2}\\) |\n| \\(\\sigma\\)-strong\\(^*\\) | \\(x\\mapsto(\\sum_n\\lVert x\\xi_n\\rVert^2+\\lVert x^*\\xi_n\\rVert^2)^{1/2}\\) |\n\nThe \\(\\sigma\\)-weak topology is also called the *ultraweak* topology. With \\(\\omega=\\sum_n\\omega_{\\xi_n}\\), the \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) seminorms read \\(\\omega(x^*x)^{1/2}\\) and \\((\\omega(x^*x)+\\omega(xx^*))^{1/2}\\). By Theorem 10.1, every positive \\(\\sigma\\)-weakly continuous functional on \\(B(H)\\) has this form \\(\\omega\\). For a set \\(S\\subseteq B(H)\\) we write \\(\\overline S^{\\,w}\\), \\(\\overline S^{\\,s}\\), \\(\\overline S^{\\,s*}\\), \\(\\overline S^{\\,\\sigma w}\\), \\(\\overline S^{\\,\\sigma s}\\), \\(\\overline S^{\\,\\sigma s*}\\) for its closures.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 43,
        "through_line": 55,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.2",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-01::Lemma 1.2",
      "anchor": "oa-fnd-bi-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 1.2** (elementary facts).\n\n(a) *One seminorm is enough.* In the \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, finitely many seminorms are dominated by a single seminorm of the same kind: concatenate the sequences. In the strong and strong\\(^*\\) topologies, the seminorms of finitely many vectors \\(\\xi_1,\\dots,\\xi_m\\) are dominated by \\((\\sum_{j=1}^m\\lVert x\\xi_j\\rVert^2)^{1/2}\\), respectively \\((\\sum_{j=1}^m\\lVert x\\xi_j\\rVert^2+\\lVert x^*\\xi_j\\rVert^2)^{1/2}\\), the \\(\\sigma\\)-type seminorm of a finite sequence. So in each of the four strong-type topologies, every neighbourhood of \\(a\\) contains a set \\(\\{x:p(x-a)<\\varepsilon\\}\\) for one such seminorm \\(p\\).\n\n(b) *Comparisons.* A sequence with one nonzero term shows that each \\(\\sigma\\)-topology is finer than its plain counterpart. The Cauchy–Schwarz inequality \\(\\lvert\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\rvert\\le(\\sum_n\\|x\\xi_n\\|^2)^{1/2}(\\sum_n\\|\\eta_n\\|^2)^{1/2}\\) shows that the \\(\\sigma\\)-strong topology is finer than the \\(\\sigma\\)-weak one; in the same way the strong topology is finer than the weak one. Each starred topology is finer than its unstarred version. So the weak topology is the coarsest of the six and the \\(\\sigma\\)-strong\\(^*\\) topology is the finest, and all six are coarser than the norm topology. A finer topology has smaller closures:\n\\[\n\\begin{gathered}\n\\overline S^{\\,\\sigma s*}\\subseteq\\overline S^{\\,\\sigma s}\\subseteq\\overline S^{\\,\\sigma w}\\subseteq\\overline S^{\\,w},\\\\\n\\overline S^{\\,\\sigma s*}\\subseteq\\overline S^{\\,s*}\\subseteq\\overline S^{\\,s}\\subseteq\\overline S^{\\,w},\\\\\n\\overline S^{\\,\\sigma s}\\subseteq\\overline S^{\\,s}.\n\\end{gathered}\n\\]\n\n(c) *Continuity of the operations.* Addition and scalar multiplication are continuous. For fixed \\(a,b\\in B(H)\\), the map \\(x\\mapsto axb\\) is continuous in all six topologies. Indeed \\(\\|axb\\xi_n\\|\\le\\|a\\|\\,\\|x(b\\xi_n)\\|\\), \\(\\langle axb\\xi_n,\\eta_n\\rangle=\\langle x(b\\xi_n),a^*\\eta_n\\rangle\\), and \\(\\|(axb)^*\\xi_n\\|\\le\\|b\\|\\,\\|x^*(a^*\\xi_n)\\|\\), where \\((b\\xi_n)\\) and \\((a^*\\eta_n)\\) are again square summable. The adjoint \\(x\\mapsto x^*\\) is continuous for the weak, \\(\\sigma\\)-weak, strong\\(^*\\) and \\(\\sigma\\)-strong\\(^*\\) topologies, because it maps their seminorms to seminorms of the same kind. (When \\(\\dim H=\\infty\\) it is not \\(\\sigma\\)-strongly continuous. For an orthonormal sequence \\((\\xi_n)\\), \\(\\theta_{\\xi_1,\\xi_n}\\to0\\) \\(\\sigma\\)-strongly, while \\(\\|\\theta_{\\xi_1,\\xi_n}^*\\xi_1\\|=1\\). The same example shows that it is not strongly continuous. We do not need this.)\n\n(d) *Bounded nets.* Let \\(x_\\lambda\\to x\\) strongly, with \\(C=\\sup_\\lambda\\|x_\\lambda\\|+\\|x\\|<\\infty\\). Then \\(x_\\lambda\\to x\\) \\(\\sigma\\)-strongly. Given \\((\\xi_n)\\) and \\(\\varepsilon>0\\), choose \\(N\\) with \\(C^2\\sum_{n>N}\\|\\xi_n\\|^2<\\varepsilon^2/2\\); then \\(\\sum_n\\|(x_\\lambda-x)\\xi_n\\|^2<\\varepsilon^2\\) as soon as the finitely many terms with \\(n\\le N\\) add up to less than \\(\\varepsilon^2/2\\). If moreover \\(x_\\lambda^*\\to x^*\\) strongly, the same argument gives \\(\\sigma\\)-strong\\(^*\\) convergence. A bounded weakly convergent net converges \\(\\sigma\\)-weakly, by the same tail estimate.\n\n(e) *Increasing nets.* If \\(0\\le x_\\lambda\\) increases and \\(\\sup_\\lambda\\|x_\\lambda\\|<\\infty\\), then \\(x_\\lambda\\) converges strongly to its least upper bound, by Vigier's theorem (V). The operators are self-adjoint, so by (d) the convergence is also \\(\\sigma\\)-strong\\(^*\\) and \\(\\sigma\\)-weak.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 56,
        "through_line": 74,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.3",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-01::Lemma 1.3",
      "anchor": "oa-fnd-bi-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 1.3** (closures of subspaces). Let \\(S\\subseteq B(H)\\) be a complex linear subspace. Then\n\n1. \\(\\overline S^{\\,w}=\\overline S^{\\,s}=\\overline S^{\\,s*}\\), and\n2. \\(\\overline S^{\\,\\sigma w}=\\overline S^{\\,\\sigma s}=\\overline S^{\\,\\sigma s*}\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 75,
        "through_line": 97,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1.5",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-01::Exercise 1.5",
      "anchor": "oa-fnd-bi-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 1.5** (medium; the two triples of closures differ for subspaces). On \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_n)\\), let \\(\\omega(x)=\\sum_n2^{-n}\\langle x\\delta_n,\\delta_n\\rangle\\). Show that \\(\\ker\\omega\\) is \\(\\sigma\\)-weakly closed but not weakly closed. So Lemma 1.3 cannot be extended to all six topologies for subspaces, while [Theorem 4.4](#oa-fnd-bi-07) does this for \\(*\\)-subalgebras.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 98,
        "through_line": 105,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.1",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-02::Proposition 2.1",
      "anchor": "oa-fnd-bi-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 2.1.** Let \\(S,T\\subseteq B(H)\\).\n\n1. \\(S'\\) is a subalgebra of \\(B(H)\\) that contains \\(1\\). It is closed in the weak topology, hence in each of the six topologies of Section 1 and in norm.\n2. \\(S\\subseteq S''\\). If \\(S\\subseteq T\\), then \\(T'\\subseteq S'\\). Moreover \\(S'=S'''\\). So all commutants of odd order equal \\(S'\\), and all commutants of even order, from the second on, equal \\(S''\\).\n3. \\((S\\cup T)'=S'\\cap T'\\).\n4. If \\(S^*=S\\), then \\(S'\\) is a \\(*\\)-subalgebra with \\((S')''=S'\\). In particular \\(S''\\) is a \\(*\\)-subalgebra with \\((S'')''=S''\\) that contains \\(S\\). Every \\(*\\)-subalgebra \\(N\\) with \\(S\\subseteq N=N''\\) contains \\(S''\\).\n5. (*Invariant subspaces.*) Let \\(S^*=S\\), let \\(L\\subseteq H\\) be a closed subspace, and let \\(p\\) be its projection. Then \\(SL\\subseteq L\\) if and only if \\(p\\in S'\\). In that case \\(SL^\\perp\\subseteq L^\\perp\\) as well.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 112,
        "through_line": 140,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.3",
      "kind": "definition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-02::Definition 2.3",
      "anchor": "oa-fnd-bi-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Definition 2.3.** A *\\(*\\)-subalgebra* \\(M\\) of \\(B(H)\\) is a subalgebra closed under adjoints. It is *nondegenerate* if \\([MH]=H\\). We call a \\(*\\)-subalgebra \\(M\\) with \\(M=M''\\) a *von Neumann algebra*, and write \\(\\{M,H\\}\\) when the Hilbert space matters. The *centre* of \\(M\\) is \\(Z(M)=M\\cap M'\\), and \\(M\\) is a *factor* if \\(Z(M)=\\mathbb C1\\). For a self-adjoint set \\(S\\subseteq B(H)\\) we say that \\(S''\\) is *generated* by \\(S\\). By [Proposition 2.1](#oa-fnd-bi-02)(4), \\(S''\\) is a von Neumann algebra, and every von Neumann algebra that contains \\(S\\) also contains \\(S''\\).\n\nTwo von Neumann algebras \\(\\{M_1,H_1\\}\\) and \\(\\{M_2,H_2\\}\\) are *spatially isomorphic* if there is a unitary \\(U:H_1\\to H_2\\) (an isometry of \\(H_1\\) onto \\(H_2\\)) with \\(UM_1U^*=M_2\\); the map \\(x\\mapsto UxU^*\\) is then a *spatial isomorphism*. They are *isomorphic* if there is a bijective linear map \\(\\pi:M_1\\to M_2\\) that is multiplicative and satisfies \\(\\pi(x^*)=\\pi(x)^*\\). A spatial isomorphism is an isomorphism, but not conversely ([Example 5.5](#oa-fnd-bi-15)).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 141,
        "through_line": 144,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.4",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-02::Proposition 2.4",
      "anchor": "oa-fnd-bi-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 2.4.**\n\n1. For a \\(*\\)-subalgebra \\(M\\), \\([MH]^\\perp=\\{\\xi\\in H:x\\xi=0\\text{ for all }x\\in M\\}\\). So \\(M\\) is nondegenerate exactly when no nonzero vector is killed by all of \\(M\\). Every \\(*\\)-subalgebra that contains \\(1\\) is nondegenerate.\n2. A von Neumann algebra \\(M\\) contains \\(1\\), is nondegenerate, and is closed in norm and in all six topologies. So it is a nondegenerate norm-closed \\(*\\)-algebra. \\(M'\\) is a von Neumann algebra, \\(Z(M)\\) is a commutative von Neumann algebra, and \\(Z(M')=Z(M)\\). In particular \\(M\\) is a factor if and only if \\(M'\\) is.\n3. A spatial isomorphism carries commutants to commutants and centres to centres: \\(UM_1'U^*=M_2'\\) and \\(UZ(M_1)U^*=Z(M_2)\\).\n4. \\(B(H)'=\\mathbb C1\\). So \\(B(H)\\) and \\(\\mathbb C1\\) are factors, each the commutant of the other.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 145,
        "through_line": 159,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.1",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-05::Lemma 3.1",
      "anchor": "oa-fnd-bi-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 3.1** (matrices).\n\n1. \\(X\\) is determined by its entries, and \\(\\|X_{ij}\\|\\le\\|X\\|\\).\n2. \\((\\alpha X+\\beta Y)_{ij}=\\alpha X_{ij}+\\beta Y_{ij}\\) and \\((X^*)_{ij}=(X_{ji})^*\\). For each \\(\\xi\\in H\\),\n\\[\n(XY)_{ij}\\xi=\\sum_kX_{ik}Y_{kj}\\xi ,\n\\tag{3.2}\n\\]\nwith unconditional convergence in norm.\n3. The operators \\(E_{ij}=U_iU_j^*\\) (the *matrix units*) have entries \\((E_{ij})_{kl}=\\delta_{ki}\\delta_{jl}1_H\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 170,
        "through_line": 205,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.2",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-05::Proposition 3.2",
      "anchor": "oa-fnd-bi-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 3.2** (the range of the amplification). An operator \\(X\\in B(\\tilde H)\\) lies in \\(\\pi(B(H))\\) if and only if it commutes with every matrix unit \\(E_{ij}\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 206,
        "through_line": 213,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.3",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-05::Proposition 3.3",
      "anchor": "oa-fnd-bi-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 3.3** (the commutant of an amplified set). For every subset \\(S\\subseteq B(H)\\),\n\\[\n\\begin{gathered}\n\\pi(S)'\\\\\n=\\{X\\in B(\\tilde H):\\\\\nX_{ij}\\in S'\\text{ for all }i,j\\in I\\}.\n\\end{gathered}\n\\tag{3.4}\n\\]",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 214,
        "through_line": 225,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.4",
      "kind": "corollary",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-05::Corollary 3.4",
      "anchor": "oa-fnd-bi-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Corollary 3.4.** \\(\\pi(S)''=\\pi(S'')\\) for every subset \\(S\\subseteq B(H)\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 226,
        "through_line": 229,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.5",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-05::Lemma 3.5",
      "anchor": "oa-fnd-bi-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 3.5** (nondegeneracy survives amplification). If \\(M\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\), then \\(\\pi(M)\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(\\tilde H)\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 230,
        "through_line": 233,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.1",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-06::Lemma 4.1",
      "anchor": "oa-fnd-bi-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 4.1.** Let \\(S\\subseteq B(H)\\) be closed under products and adjoints, and suppose no nonzero vector is killed by every element of \\(S\\). Then \\(\\xi\\in[S\\xi]\\) for every \\(\\xi\\in H\\). By [Proposition 2.4](#oa-fnd-bi-03)(1), this applies to every nondegenerate \\(*\\)-subalgebra.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 238,
        "through_line": 243,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [
        {
          "unit": "the-double-commutant-theorem",
          "local_scopes": [
            {
              "heading": "4. The double commutant theorem",
              "line": 234,
              "through_line": 319,
              "anchors": [
                "OA-FND-BI-06",
                "OA-FND-BI-07"
              ],
              "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.5; PDF 53–55",
          "role": "proof comparison",
          "correspondence": "Bicommutant proof at arbitrary Hilbert dimension",
          "explanation": "One-vector invariant subspace followed by finite diagonal amplification; own degenerate essential-subspace and all-six-closure statements retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.2",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-06::Example 4.2",
      "anchor": "oa-fnd-bi-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 4.2** (Lemma 4.1 needs adjoints and nondegeneracy). On \\(\\mathbb C^2\\), let \\(S\\) be the set of matrices whose second column is zero. It is closed under products, and \\([S\\mathbb C^2]=\\mathbb C^2\\), since \\(S\\xi=\\mathbb C^2\\) whenever the first coordinate of \\(\\xi\\) is nonzero. For \\(\\xi=(0,1)\\), however, \\(S\\xi=\\{0\\}\\), so \\(\\xi\\notin[S\\xi]\\). Here \\(S\\) is not closed under adjoints, and some nonzero vectors are killed by all of \\(S\\): for non-self-adjoint algebras, the spanning condition and the kernel condition of [Proposition 2.4](#oa-fnd-bi-03)(1) differ. Adjoints are needed even when the kernel condition holds. The set \\(T\\) of matrices whose second row is zero is closed under products and kills no nonzero vector, but for \\(\\xi=(0,1)\\), \\([T\\xi]=\\mathbb C(1,0)\\) does not contain \\(\\xi\\). For the \\(*\\)-algebra \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\([M\\xi]=\\{0\\}\\) for every \\(\\xi\\), so nondegeneracy is needed too.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 244,
        "through_line": 245,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [
        {
          "unit": "the-double-commutant-theorem",
          "local_scopes": [
            {
              "heading": "4. The double commutant theorem",
              "line": 234,
              "through_line": 319,
              "anchors": [
                "OA-FND-BI-06",
                "OA-FND-BI-07"
              ],
              "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
            }
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          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.5; PDF 53–55",
          "role": "proof comparison",
          "correspondence": "Bicommutant proof at arbitrary Hilbert dimension",
          "explanation": "One-vector invariant subspace followed by finite diagonal amplification; own degenerate essential-subspace and all-six-closure statements retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.3",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-06::Lemma 4.3",
      "anchor": "oa-fnd-bi-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 4.3.** Let \\(M\\) be a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\). Then \\([M\\xi]=[M''\\xi]\\) for every \\(\\xi\\in H\\). Consequently, for \\(a\\in M''\\), \\(\\xi\\in H\\) and \\(\\varepsilon>0\\) there is \\(b\\in M\\) with \\(\\|(a-b)\\xi\\|<\\varepsilon\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 246,
        "through_line": 251,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [
        {
          "unit": "the-double-commutant-theorem",
          "local_scopes": [
            {
              "heading": "4. The double commutant theorem",
              "line": 234,
              "through_line": 319,
              "anchors": [
                "OA-FND-BI-06",
                "OA-FND-BI-07"
              ],
              "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.5; PDF 53–55",
          "role": "proof comparison",
          "correspondence": "Bicommutant proof at arbitrary Hilbert dimension",
          "explanation": "One-vector invariant subspace followed by finite diagonal amplification; own degenerate essential-subspace and all-six-closure statements retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.4",
      "kind": "theorem",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-06::Theorem 4.4",
      "anchor": "oa-fnd-bi-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Theorem 4.4** (the double commutant theorem). Let \\(M\\) be a \\(*\\)-subalgebra of \\(B(H)\\), and let \\(e\\) be the projection onto \\([MH]\\).\n\n1. \\(e\\in M'\\cap M''\\), and \\(x=ex=xe\\) for every \\(x\\in M\\).\n2. The closures of \\(M\\) in the weak, strong, strong\\(^*\\), \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies coincide. Call this common closure \\(\\overline M\\). It is a \\(*\\)-subalgebra. The projection \\(e\\) lies in \\(\\overline M\\), is its unit (\\(x=ex=xe\\) for \\(x\\in\\overline M\\)), and is the greatest projection in \\(\\overline M\\).\n3. \\(M''\\) consists of the operators \\(x+\\alpha1\\) with \\(x\\in\\overline M\\) and \\(\\alpha\\in\\mathbb C\\), and\n\\[\n\\begin{gathered}\nM''\\\\\n=\\overline M+\\mathbb C1,\\\\\n\\overline M\\\\\n=eM''\\\\\n=M''e .\n\\end{gathered}\n\\tag{4.1}\n\\]\n4. \\(\\overline M=M''\\) if and only if \\(M\\) is nondegenerate.\n5. \\(M\\) is a von Neumann algebra if and only if \\(M\\) is nondegenerate and closed in at least one of the six topologies. It is then closed in all of them.\n\n\n*Remark 4.5.* Suppose that \\(M\\) is closed in one of the six topologies. Since the \\(\\sigma\\)-strong\\(^*\\) topology is the finest of the six, closedness for it is the weakest hypothesis of this kind. Then \\(\\overline M=M\\), and parts (2)–(4) say: \\(e\\) is the greatest projection of \\(M\\) and its unit; \\(M''=M+\\mathbb C1\\); and \\(M=M''\\) when \\(M\\) is nondegenerate. Part (2) is von Neumann's density theorem: a nondegenerate \\(*\\)-algebra is dense in its bicommutant, even \\(\\sigma\\)-strongly\\(^*\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 252,
        "through_line": 317,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [
        {
          "unit": "the-double-commutant-theorem",
          "local_scopes": [
            {
              "heading": "4. The double commutant theorem",
              "line": 234,
              "through_line": 319,
              "anchors": [
                "OA-FND-BI-06",
                "OA-FND-BI-07"
              ],
              "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.5; PDF 53–55",
          "role": "proof comparison",
          "correspondence": "Bicommutant proof at arbitrary Hilbert dimension",
          "explanation": "One-vector invariant subspace followed by finite diagonal amplification; own degenerate essential-subspace and all-six-closure statements retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.6",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-06::Example 4.6",
      "anchor": "oa-fnd-bi-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 4.6** (compact operators). Let \\(\\dim H=\\infty\\). Then \\(K(H)\\) is a norm-closed \\(*\\)-subalgebra that contains every \\(\\theta_{\\xi,\\xi}\\), so it is nondegenerate. The proof of [Proposition 2.4](#oa-fnd-bi-03)(4) uses only rank-one operators, so \\(K(H)'=\\mathbb C1\\) and \\(K(H)''=B(H)\\). By [Theorem 4.4](#oa-fnd-bi-07), \\(K(H)\\) is dense in \\(B(H)\\) for all six topologies; for instance, the finite-rank projections onto spans of finitely many vectors of an orthonormal basis increase strongly to \\(1\\). But \\(1\\notin K(H)\\), so \\(K(H)\\) is not a von Neumann algebra. Being norm closed is not enough.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 318,
        "through_line": 319,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [
        {
          "unit": "the-double-commutant-theorem",
          "local_scopes": [
            {
              "heading": "4. The double commutant theorem",
              "line": 234,
              "through_line": 319,
              "anchors": [
                "OA-FND-BI-06",
                "OA-FND-BI-07"
              ],
              "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.5; PDF 53–55",
          "role": "proof comparison",
          "correspondence": "Bicommutant proof at arbitrary Hilbert dimension",
          "explanation": "One-vector invariant subspace followed by finite diagonal amplification; own degenerate essential-subspace and all-six-closure statements retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.1",
      "kind": "definition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Definition 5.1",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Definition 5.1.** The *direct sum* \\(\\sum^\\oplus_iM_i\\), also written \\(\\sum^\\oplus_i\\{M_i,H_i\\}\\), is the set of operators \\(\\bigoplus_ix_i\\) with \\(x_i\\in M_i\\) and \\(\\sup_i\\|x_i\\|<\\infty\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 326,
        "through_line": 327,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.2",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Proposition 5.2",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 5.2.**\n\n1. \\(T\\in B(H)\\) commutes with every \\(P_i\\) if and only if \\(T=\\bigoplus_iT_i\\) for a bounded family \\(T_i\\in B(H_i)\\).\n2. \\(\\big(\\sum^\\oplus_iM_i\\big)'=\\sum^\\oplus_iM_i'\\).\n3. \\(\\sum^\\oplus_iM_i\\) is itself a von Neumann algebra, and its centre is \\(\\sum^\\oplus_iZ(M_i)\\).\n4. If at least two of the spaces \\(H_i\\) are nonzero, \\(\\sum^\\oplus_iM_i\\) is not a factor. If exactly one, \\(H_{i_0}\\), is nonzero, then \\(\\sum^\\oplus_iM_i\\) is spatially isomorphic to \\(M_{i_0}\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 328,
        "through_line": 344,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.4",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Example 5.4",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 5.4** (a degenerate algebra). Let \\(p\\) be a projection with \\(p\\neq0\\) and \\(p\\neq1\\), and \\(M=\\mathbb Cp\\). This is a closed \\(*\\)-subalgebra, \\([MH]=pH\\), and \\(e=p\\). An operator commutes with \\(p\\) exactly when it is block diagonal for \\(H=pH\\oplus(1-p)H\\), so \\(M'=B(pH)\\oplus B((1-p)H)\\). By [Proposition 5.2](#oa-fnd-bi-04)(2) and [Proposition 2.4](#oa-fnd-bi-03)(4), \\(M''=\\{\\alpha p+\\beta(1-p)\\}=M+\\mathbb C1\\). So \\(M''\\neq M=\\overline M=eM''\\), as (4.1) and [Theorem 4.4](#oa-fnd-bi-07)(4) predict.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 345,
        "through_line": 346,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.5",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Example 5.5",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 5.5** (an isomorphism that is not spatial). On \\(\\mathbb C^4\\) let \\(M_1=\\{\\operatorname{diag}(a,b,b,b)\\}\\) and \\(M_2=\\{\\operatorname{diag}(a,a,b,b)\\}\\), with \\(a,b\\in\\mathbb C\\). By [Proposition 5.2](#oa-fnd-bi-04) and [Proposition 2.4](#oa-fnd-bi-03)(4), these are von Neumann algebras: \\(M_1=\\mathbb C1_{\\mathbb C}\\oplus\\mathbb C1_{\\mathbb C^3}\\) and \\(M_2=\\mathbb C1_{\\mathbb C^2}\\oplus\\mathbb C1_{\\mathbb C^2}\\). The map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism. It is not spatial. A unitary \\(U\\) with \\(UM_1U^*=M_2\\) would carry the rank-one projection \\(\\operatorname{diag}(1,0,0,0)\\in M_1\\) to a rank-one projection in \\(M_2\\), but the projections of \\(M_2\\) have rank \\(0\\), \\(2\\) or \\(4\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 347,
        "through_line": 348,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.6",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Example 5.6",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 5.6** (commutants of direct sums with a common summand). Suppose \\(H\\neq\\{0\\}\\), and let \\(M=\\{x\\oplus x:x\\in B(H)\\}\\) on \\(H\\oplus H\\). This is \\(\\pi(B(H))\\) for \\(I=\\{1,2\\}\\) in [Section 3](#oa-fnd-bi-05), so \\(M''=M\\) by Corollary 3.4, and by (3.4) \\(M'\\) consists of the \\(2\\times2\\) scalar matrices tensored with \\(1_H\\). An operator in both \\(M\\) and \\(M'\\) is \\(\\pi(x)\\) with \\(x\\) scalar. So \\(M\\) is a factor, while the direct sum \\(B(H)\\oplus B(H)\\) of [Definition 5.1](#oa-fnd-bi-04) is not: its central projection \\(1_H\\oplus0\\) is neither zero nor the identity, by \\(H\\neq\\{0\\}\\). The two algebras are different subalgebras of \\(B(H\\oplus H)\\) built from the same summands. If \\(H=\\{0\\}\\), both constructions give the same zero algebra; its centre is \\(\\mathbb C1_H=\\{0\\}\\), so it satisfies this lesson's centre-based factor convention. Thus the contrast between the two constructions requires a nonzero summand.\n\nNow fix a von Neumann algebra \\(\\{M,H\\}\\) and a projection \\(e\\in M\\), and put \\(K=eH\\). For \\(x\\in M\\), the operator \\(exe\\) maps \\(K\\) into \\(K\\); write \\(x_e\\) for its restriction to \\(K\\). Each \\(x'\\in M'\\) commutes with \\(e\\), so it maps \\(K\\) into \\(K\\); write \\(x'_e\\) for its restriction. Put \\(M_e=\\{x_e:x\\in M\\}\\) and \\(M'_e=\\{x'_e:x'\\in M'\\}\\). Both are \\(*\\)-subalgebras of \\(B(K)\\) containing \\(1_K\\). The map \\(x'\\mapsto x'_e\\) is a unital \\(*\\)-homomorphism from \\(M'\\) onto \\(M'_e\\), because \\(e\\) commutes with \\(M'\\). Let \\(z\\) be the projection onto \\([MK]=[MeH]\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 349,
        "through_line": 352,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.7",
      "kind": "definition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Definition 5.7",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Definition 5.7.** On \\(K\\), the algebra \\(M_e\\) is called *reduced* (from \\(M\\)) and \\(M'_e\\) is called *induced* (from \\(M'\\)). The map \\(x'\\mapsto x'_e\\) from \\(M'\\) onto \\(M'_e\\) is the *induction*. (Part (3) of Theorem 5.8 shows that both algebras are von Neumann algebras.)",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 353,
        "through_line": 354,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.8",
      "kind": "theorem",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Theorem 5.8",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Theorem 5.8.**\n\n1. \\((M'_e)'=M_e\\).\n2. \\((M_e)'=M'_e\\).\n3. \\(M_e\\) and \\(M'_e\\) are von Neumann algebras on \\(K\\).\n4. \\(z\\in Z(M)\\), \\(e\\le z\\), and \\(z\\) is the smallest projection of \\(Z(M)\\) that majorizes \\(e\\). The kernel of the induction is \\(M'(1-z)\\). So the induction is injective if and only if \\([MeH]=H\\).\n5. \\(Z(M_e)=\\{c_e:c\\in Z(M)\\}\\). In particular, if \\(M\\) is a factor, so are \\(M_e\\) and \\(M'_e\\).\n\nBy symmetry, the same statements hold for a projection \\(e'\\in M'\\), with the roles of \\(M\\) and \\(M'\\) exchanged: apply the theorem to the von Neumann algebra \\(M'\\), whose commutant is \\(M\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 355,
        "through_line": 400,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5.10",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-04::Exercise 5.10",
      "anchor": "oa-fnd-bi-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 5.10** (easy; the induction need not be injective). Let \\(H=H_1\\oplus H_2\\) with \\(H_1,H_2\\neq\\{0\\}\\), \\(M=B(H_1)\\oplus B(H_2)\\), and \\(e=1\\oplus0\\). Compute \\(M'\\), \\(M_e\\), \\(M'_e\\), the central projection \\(z\\) of Theorem 5.8(4), and the kernel of the induction.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 401,
        "through_line": 404,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.1",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-16::Example 6.1",
      "anchor": "oa-fnd-bi-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 6.1** (commutants taken inside \\(K(H)\\)). Let \\(H=\\ell^2\\oplus\\ell^2\\) and \\(A=K(\\ell^2)\\oplus K(\\ell^2)=\\{a\\oplus b\\}\\). This is a norm-closed nondegenerate \\(*\\)-subalgebra contained in \\(K(H)\\). An operator that commutes with every \\(a\\oplus0\\) and every \\(0\\oplus b\\) has zero off-diagonal blocks, because \\(K(\\ell^2)\\) is nondegenerate, and its diagonal blocks lie in \\(K(\\ell^2)'=\\mathbb C1\\) (Example 4.6). So \\(A'=\\{\\alpha1\\oplus\\beta1\\}\\). ([Proposition 5.2](#oa-fnd-bi-04)(2) does not apply directly, since \\(1\\notin K(\\ell^2)\\); nondegeneracy replaces the projections \\(P_i\\) used there.) None of these operators is compact unless \\(\\alpha=\\beta=0\\), so \\(A'\\cap K(H)=\\{0\\}\\), and \\(\\{0\\}'\\cap K(H)=K(H)\\). But \\(K(H)\\) contains the rank-one operator \\(\\theta_{0\\oplus\\delta_1,\\,\\delta_1\\oplus0}\\), which maps the first summand into the second and so is not in \\(A\\). Hence \\(\\{A'\\cap K(H)\\}'\\cap K(H)\\neq A\\): commutants taken inside \\(K(H)\\) do not recover \\(A\\). When \\(\\dim H<\\infty\\) this identity does hold. Then \\(K(H)=B(H)\\), and \\(A\\) is finite dimensional, hence closed in every topology, so the identity reads \\(A''=A\\), which is [Theorem 4.4](#oa-fnd-bi-07)(5).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 409,
        "through_line": 410,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 6.2",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-16::Exercise 6.2",
      "anchor": "oa-fnd-bi-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 6.2** (hard; compact operators in the bicommutant). Let \\(A\\subseteq K(H)\\) be a nondegenerate norm-closed \\(*\\)-subalgebra. Show that \\(A''\\cap K(H)=A\\).\n\nExample 6.1 proves directly that taking the first commutant inside \\(K(H)\\) can fail to recover the algebra.\n\nNondegeneracy is necessary here. For the degenerate algebra \\(A=\\mathbb CE_{11}\\) on \\(\\mathbb C^2\\), Example 5.4 gives \\(A''\\cap K(\\mathbb C^2)=A''=\\{\\operatorname{diag}(\\alpha,\\beta)\\}\\neq A\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 411,
        "through_line": 426,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 6.3",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-16::Exercise 6.3",
      "anchor": "oa-fnd-bi-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 6.3** (medium; minimal projections and simple C\\(^*\\)-algebras). Here a projection \\(e\\) in a C\\(^*\\)-algebra \\(A\\) is *minimal* if \\(eAe=\\mathbb Ce\\). A representation \\(\\{\\pi,H\\}\\) is *irreducible* if \\(H\\) and \\(\\{0\\}\\) are its only closed invariant subspaces. \\(A\\) is *simple* if its only closed two-sided ideals are \\(\\{0\\}\\) and \\(A\\). Let \\(e\\neq0\\) be a minimal projection of \\(A\\). Prove (a) and (b).\n\n(a) *If \\(\\pi\\) is irreducible and \\(\\pi(e)\\neq0\\), then \\(\\pi(e)\\) is a projection of rank one.* It is a projection because \\(\\pi\\) is a \\(*\\)-homomorphism. Take a unit vector \\(\\xi\\in\\pi(e)H\\). The subspace \\([\\pi(A)\\xi]\\) is closed, invariant and contains \\(\\xi=\\pi(e)\\xi\\), so it is \\(H\\). Let \\(\\eta\\in\\pi(e)H\\) and choose \\(a_k\\in A\\) with \\(\\pi(a_k)\\xi\\to\\eta\\). Write \\(ea_ke=\\lambda_ke\\). Then \\(\\eta=\\pi(e)\\eta=\\lim_k\\pi(e)\\pi(a_k)\\pi(e)\\xi=\\lim_k\\lambda_k\\xi\\). So \\(\\eta\\in\\mathbb C\\xi\\), and \\(\\pi(e)H=\\mathbb C\\xi\\). \\(\\square\\)\n\n(b) *If \\(A\\) is simple, then \\(A\\cong K(H)\\) for some Hilbert space \\(H\\).* Let \\(H=Ae=\\{x\\in A:xe=x\\}\\), a norm-closed subspace of \\(A\\). For \\(\\xi,\\eta\\in H\\), \\(\\eta^*\\xi=e\\eta^*\\xi e\\in eAe\\), so \\(\\eta^*\\xi=\\langle\\xi,\\eta\\rangle e\\) for a unique scalar \\(\\langle\\xi,\\eta\\rangle\\). This is linear in \\(\\xi\\) and conjugate linear in \\(\\eta\\). Since \\(\\xi^*\\xi\\ge0\\) and \\(e\\neq0\\), \\(\\langle\\xi,\\xi\\rangle\\ge0\\), and \\(\\|\\xi\\|_A^2=\\|\\xi^*\\xi\\|=\\langle\\xi,\\xi\\rangle\\|e\\|=\\langle\\xi,\\xi\\rangle\\). So the Hilbert norm equals the C\\(^*\\)-norm, and \\(H\\) is complete: a Hilbert space, nonzero because \\(e\\in H\\). Put \\(\\pi(x)\\xi=x\\xi\\). Then \\(\\|\\pi(x)\\|\\le\\|x\\|\\), \\(\\pi\\) is multiplicative, and \\(\\langle\\pi(x)\\xi,\\eta\\rangle e=\\eta^*x\\xi=(x^*\\eta)^*\\xi=\\langle\\xi,\\pi(x^*)\\eta\\rangle e\\). So \\(\\pi\\) is a representation.\n\nFor \\(\\zeta,\\eta\\in H\\), \\(\\pi(\\zeta\\eta^*)\\xi=\\zeta\\eta^*\\xi=\\langle\\xi,\\eta\\rangle\\zeta e=\\theta_{\\zeta,\\eta}\\xi\\). So \\(\\pi(A)\\) contains every rank-one operator, hence every finite-rank operator. In particular \\(\\pi(e)=\\theta_{e,e}\\) has rank one. The kernel of \\(\\pi\\) is a closed two-sided ideal that does not contain \\(e\\), so it is \\(\\{0\\}\\) by simplicity. The set \\(J=\\{x\\in A:\\pi(x)\\in K(H)\\}\\) is a closed two-sided ideal, since \\(\\pi\\) is continuous and \\(K(H)\\) is a closed ideal of \\(B(H)\\). It contains \\(e\\), so \\(J=A\\) and \\(\\pi(A)\\subseteq K(H)\\). An injective \\(*\\)-homomorphism between C\\(^*\\)-algebras is isometric (C), so \\(\\pi(A)\\) is norm closed. It contains the finite-rank operators, whose closure is \\(K(H)\\) (K). So \\(\\pi(A)=K(H)\\), and \\(\\pi:A\\to K(H)\\) is a \\(*\\)-isomorphism. \\(\\square\\)\n\n*Remarks.* (i) The definition \\(eAe=\\mathbb Ce\\) matters. A minimal projection in this sense majorizes no nonzero projection other than itself. The converse fails in C\\(^*\\)-algebras, and with that weaker definition (a) is false. The algebra \\(B=\\{f\\in C([0,1],M_2(\\mathbb C)):f(0),f(1)\\in\\mathbb C1\\}\\) has only the projections \\(0\\) and \\(1\\): the rank of a continuous projection-valued function is constant on \\([0,1]\\), and at \\(t=0\\) it is \\(0\\) or \\(2\\). Evaluation at \\(t=1/2\\) is an irreducible representation on \\(\\mathbb C^2\\), since every matrix is \\(f(1/2)\\) for some \\(f\\in B\\). But it sends the projection \\(1\\) to a projection of rank two. (ii) If \"simple\" is read as having no two-sided ideals other than \\(\\{0\\}\\) and \\(A\\), closed or not, then the ideal \\(AeA\\) of finite sums \\(\\sum x_iey_i\\) equals \\(A\\). Since \\(\\pi(xey)\\) has rank at most one, \\(\\pi(A)\\) then consists of finite-rank operators, so \\(K(H)\\) consists of finite-rank operators and \\(\\dim H<\\infty\\): \\(A\\cong M_n(\\mathbb C)\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 427,
        "through_line": 436,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.1",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-09::Lemma 7.1",
      "anchor": "oa-fnd-bi-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 7.1** (uniqueness of the polar decomposition). Let \\(x=wk\\), where \\(k\\ge0\\) and \\(w\\) is a partial isometry with \\(\\ker w=\\ker k\\). Then \\(k=|x|\\), and \\(w\\) is the partial isometry \\(u\\) of (P).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 441,
        "through_line": 444,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.2",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-09::Proposition 7.2",
      "anchor": "oa-fnd-bi-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 7.2.** Let \\(x\\) be an element of a von Neumann algebra \\(M\\), with polar decomposition \\(x=u|x|\\). Then \\(u\\in M\\) and \\(|x|\\in M\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 445,
        "through_line": 466,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 7.3",
      "kind": "corollary",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-09::Corollary 7.3",
      "anchor": "oa-fnd-bi-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Corollary 7.3** (ideals and the polar decomposition). Fix a von Neumann algebra \\(M\\) and \\(x\\in M\\) with \\(x=u|x|\\). No closure assumption is made on the ideals below.\n\n1. If \\(l\\) is a left ideal of \\(M\\) (a linear subspace with \\(Ml\\subseteq l\\)) and \\(x\\in l\\), then \\(|x|=u^*x\\in l\\).\n2. If \\(r\\) is a right ideal (\\(rM\\subseteq r\\)) and \\(x\\in r\\), then \\(|x^*|=xu^*\\in r\\).\n3. Every two-sided ideal \\(m\\) of \\(M\\) is self-adjoint.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 467,
        "through_line": 476,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.4",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-09::Example 7.4",
      "anchor": "oa-fnd-bi-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 7.4** (a two-sided ideal that is not self-adjoint, outside von Neumann algebras). In the C\\(^*\\)-algebra \\(C[0,1]\\), let \\(g(t)=te^{i/t}\\) for \\(t>0\\) and \\(g(0)=0\\); \\(g\\) is continuous. The ideal \\(gC[0,1]\\) is two-sided, but it does not contain \\(\\bar g\\). Indeed, \\(\\bar g=gk\\) would force \\(k(t)=e^{-2i/t}\\) for \\(t>0\\), which has no limit as \\(t\\downarrow0\\): it equals \\(1\\) at \\(t=1/(\\ell\\pi)\\) and \\(-i\\) at \\(t=1/(\\ell\\pi+\\pi/4)\\) for every integer \\(\\ell\\ge1\\). In the von Neumann algebra \\(L^\\infty[0,1]\\) of multiplication operators on \\(L^2[0,1]\\) (its full multiplication-commutant proof is [Theorem 9.1 of the Hilbert-space lesson](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09)), the same ideal does contain \\(\\bar g\\), because \\(e^{-2i/t}\\) is bounded and measurable. This matches [Corollary 7.3](#oa-fnd-bi-09)(3).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 477,
        "through_line": 478,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.5",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-09::Exercise 7.5",
      "anchor": "oa-fnd-bi-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 7.5** (hard; asymmetric Riesz decomposition). In a von Neumann algebra \\(M\\), suppose that \\(\\sum_{i\\in I}x_i^*x_i=\\sum_{j\\in J}y_j^*y_j\\), the sums converging \\(\\sigma\\)-strongly. Show that there are \\(z_{ij}\\in M\\) with\n\\[\n\\begin{gathered}\n\\sum_jz_{ij}^*z_{ij}\\\\\n=x_ix_i^*\\quad(i\\in I),\\\\\n\\sum_iz_{ij}z_{ij}^*\\\\\n=y_jy_j^*\\quad(j\\in J).\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 479,
        "through_line": 535,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-10::Lemma 8.1",
      "anchor": "oa-fnd-bi-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Lemma 8.1** (approximate units for right ideals). Let \\(M\\) be a von Neumann algebra on \\(H\\) and \\(r\\subseteq M\\) a right ideal (a linear subspace with \\(rM\\subseteq r\\)); \\(r\\) need not be closed. Let \\(f\\) be the projection onto \\([rH]\\).\n\n1. \\(f\\in M\\), and \\(fa=a\\) for every \\(a\\in r\\).\n2. There is an increasing net \\((u_\\lambda)\\) in \\(r\\), with \\(0\\le u_\\lambda\\le f\\), such that \\(\\|(1-u_\\lambda)a\\|\\to0\\) for every \\(a\\in r\\).\n3. \\(u_\\lambda\\to f\\) strongly, \\(\\sigma\\)-strongly\\(^*\\) and \\(\\sigma\\)-weakly.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 540,
        "through_line": 578,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.3",
      "kind": "theorem",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-10::Theorem 8.3",
      "anchor": "oa-fnd-bi-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Theorem 8.3** (closures of one-sided ideals). Let \\(M\\) be a von Neumann algebra on \\(H\\).\n\n1. Let \\(l\\) be a left ideal of \\(M\\), not necessarily closed, and let \\(f\\) be the projection onto \\([l^*H]\\), where \\(l^*=\\{x^*:x\\in l\\}\\). Then the closure of \\(l\\) in each of the six topologies is \\(Mf=\\{y\\in M:y=yf\\}\\).\n2. Let \\(r\\) be a right ideal, and \\(f\\) the projection onto \\([rH]\\). Then the closure of \\(r\\) in each of the six topologies is \\(fM\\).\n3. A left ideal that is closed in one of the six topologies is \\(Me\\) for exactly one projection \\(e\\in M\\). A closed right ideal is \\(eM\\) for exactly one projection \\(e\\in M\\).\n4. Let \\(m\\) be a two-sided ideal and \\(\\overline m\\) its closure, which is the same in all six topologies. Let \\(e\\) be the projection onto \\([mH]\\). Then \\(\\overline m=Me=eM\\), and \\(e\\in Z(M)\\).\n5. If \\(M\\) is a factor, every two-sided ideal other than \\(\\{0\\}\\) is dense in \\(M\\) for all six topologies.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 579,
        "through_line": 598,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.5",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-10::Proposition 8.5",
      "anchor": "oa-fnd-bi-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 8.5** (monotone approximation in two-sided ideals). Take a two-sided ideal \\(m\\) in a von Neumann algebra \\(M\\), not assumed closed. Let \\(e\\) be the projection onto \\([mH]\\), so that \\(\\overline m=Me\\) by Theorem 8.3, and let \\((u_\\lambda)\\) be the net of Lemma 8.1 for the right ideal \\(m\\). Let \\(x\\in\\overline m\\) be positive, that is \\(x\\in M_+\\) with \\(x=xe\\). Put \\(x_\\lambda=x^{1/2}u_\\lambda x^{1/2}\\). Then \\(x_\\lambda\\in m\\), \\(0\\le x_\\lambda\\le x\\), the net \\((x_\\lambda)\\) increases, and \\(x_\\lambda\\to x\\) strongly, \\(\\sigma\\)-strongly\\(^*\\) and \\(\\sigma\\)-weakly.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 599,
        "through_line": 604,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.6",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-10::Example 8.6",
      "anchor": "oa-fnd-bi-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 8.6** (monotone approximation fails for one-sided ideals). Let \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_k)\\), let \\(P_n\\) be the projection onto the span of \\(\\delta_1,\\dots,\\delta_n\\), and let \\(l=\\{x\\in B(H):x=xP_n\\text{ for some }n\\}\\). This is a left ideal. Its adjoint set \\(l^*\\) consists of the operators with range in some \\(P_nH\\), so \\([l^*H]=H\\), and by [Theorem 8.3](#oa-fnd-bi-11)(1) the closure of \\(l\\) is \\(B(H)\\). Let \\(v=\\sum_k2^{-k}\\delta_k\\) and let \\(q\\) be the projection onto \\(\\mathbb Cv\\). Suppose \\(y\\in l\\) and \\(0\\le y\\le q\\). Then \\((1-q)y(1-q)\\le0\\), so \\(y^{1/2}(1-q)=0\\), and \\(y=qyq=cq\\) for some \\(c\\ge0\\). Also \\(y=yP_n\\) for some \\(n\\). If \\(c>0\\), then \\(q=qP_n\\), hence \\(q=P_nq\\) and \\(v\\in P_nH\\), which is false. So \\(y=0\\). An increasing net in \\(l_+\\) that converged strongly to \\(q\\) would lie below \\(q\\), since \\(\\langle y_\\lambda\\xi,\\xi\\rangle\\le\\lim_\\mu\\langle y_\\mu\\xi,\\xi\\rangle=\\langle q\\xi,\\xi\\rangle\\). So it would be zero, and could not converge to \\(q\\). The positive element \\(q\\) of the closure of \\(l\\) is therefore not the limit of any increasing net in \\(l_+\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 605,
        "through_line": 606,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 9.1",
      "kind": "definition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Definition 9.1",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Definition 9.1.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\). The set \\(\\mathfrak A\\) is *separating* for \\(S\\) if \\(x\\in S\\) and \\(x\\mathfrak A=\\{0\\}\\) imply \\(x=0\\). It is *cyclic* for \\(S\\) if \\([S\\mathfrak A]=H\\). A vector \\(\\xi\\) is separating or cyclic if \\(\\{\\xi\\}\\) is.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 611,
        "through_line": 612,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.2",
      "kind": "proposition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Proposition 9.2",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Proposition 9.2.** Let \\(S\\subseteq B(H)\\) and \\(\\mathfrak A\\subseteq H\\).\n\n1. If \\(\\mathfrak A\\) is cyclic for \\(S\\), then \\(\\mathfrak A\\) is separating for \\(S'\\).\n2. If \\(M\\) is a nondegenerate \\(*\\)-subalgebra of \\(B(H)\\), closed or not, and \\(\\mathfrak A\\) is separating for \\(M'\\), then \\(\\mathfrak A\\) is cyclic for \\(M\\).\n3. For a von Neumann algebra \\(M\\): \\(\\mathfrak A\\) is cyclic for \\(M\\) if and only if it is separating for \\(M'\\), and \\(\\mathfrak A\\) is separating for \\(M\\) if and only if it is cyclic for \\(M'\\).\n4. Nondegeneracy cannot be dropped in (2). For \\(M=\\{0\\}\\) on \\(H\\neq\\{0\\}\\), \\(M'=B(H)\\), and \\(\\mathfrak A=H\\) is separating for \\(M'\\) but not cyclic for \\(M\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 613,
        "through_line": 629,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 9.3",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Exercise 9.3",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 9.3** (medium; separating and cyclic vectors for \\(M_n\\otimes1_k\\)). Let \\(n,k\\ge1\\), \\(I=\\{1,\\dots,k\\}\\), and \\(M=\\pi(B(\\mathbb C^n))\\) on \\(\\tilde H=\\ell^2(I;\\mathbb C^n)\\) as in [Section 3](#oa-fnd-bi-05); this is \\(M_n(\\mathbb C)\\otimes1_k\\). Write a vector as \\(\\xi=(\\xi_1,\\dots,\\xi_k)\\), or as the \\(n\\times k\\) matrix \\(X\\) with columns \\(\\xi_j\\). Show that \\(M''=M\\) and that \\(M'\\) consists of the scalar \\(k\\times k\\) matrices tensored with \\(1_{\\mathbb C^n}\\). Show that \\(\\xi\\) is separating for \\(M\\) if and only if \\(\\operatorname{rank}X=n\\), and cyclic for \\(M\\) if and only if \\(\\operatorname{rank}X=k\\). Deduce that \\(M\\) has a separating vector if and only if \\(k\\ge n\\), a cyclic vector if and only if \\(k\\le n\\), and a vector that is both if and only if \\(k=n\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 630,
        "through_line": 633,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 9.4",
      "kind": "definition",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Definition 9.4",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Definition 9.4.** A von Neumann algebra \\(M\\) is *\\(\\sigma\\)-finite* (also called *countably decomposable*) if every family of mutually orthogonal nonzero projections in \\(M\\) is countable.\n\nA linear functional \\(\\varphi\\) on \\(M\\) is *positive* if \\(\\varphi(x^*x)\\ge0\\) for all \\(x\\), and a positive \\(\\varphi\\) is *faithful* if \\(\\varphi(x^*x)=0\\) implies \\(x=0\\). A *state* is a positive functional with \\(\\varphi(1)=1\\). A linear functional on \\(M\\) is *normal* if it is \\(\\sigma\\)-weakly continuous.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 634,
        "through_line": 637,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.5",
      "kind": "theorem",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Theorem 9.5",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Theorem 9.5** (\\(\\sigma\\)-finite von Neumann algebras). For a von Neumann algebra \\(M\\) on \\(H\\), the following are equivalent.\n\n1. \\(M\\) is \\(\\sigma\\)-finite.\n2. \\(H\\) contains a countable set that is separating for \\(M\\).\n3. \\(H\\) contains a countable set that is cyclic for \\(M'\\).\n4. There is a faithful positive \\(\\sigma\\)-weakly continuous linear functional on \\(M\\).\n5. There is a faithful positive linear functional on \\(M\\), with no continuity assumed.\n\nIf \\(H\\neq\\{0\\}\\), the functional in (4) can be taken to be a state, that is, with \\(\\varphi(1)=1\\).",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 638,
        "through_line": 664,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.7",
      "kind": "example",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Example 9.7",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Example 9.7** (separating vectors and \\(\\sigma\\)-finiteness). \\(M=B(\\mathbb C^2)\\) is \\(\\sigma\\)-finite, and a basis is a separating set. No single vector separates: for \\(\\xi\\neq0\\), choose a nonzero \\(\\eta\\perp\\xi\\); then \\(\\theta_{\\eta,\\eta}\\xi=0\\). Every nonzero vector is cyclic for \\(M\\). Now let \\(\\Gamma\\) be uncountable and \\(M=\\mathbb C1\\) on \\(\\ell^2(\\Gamma)\\). It is \\(\\sigma\\)-finite, and every nonzero vector separates it. But \\(M'=B(\\ell^2(\\Gamma))\\) contains the uncountable family of mutually orthogonal rank-one projections onto the basis vectors, so \\(M'\\) is not \\(\\sigma\\)-finite. By [Theorem 9.5](#oa-fnd-bi-13), (5)\\(\\Rightarrow\\)(1), \\(B(\\ell^2(\\Gamma))\\) has no faithful positive functional at all.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 665,
        "through_line": 666,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 9.8",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-12::Exercise 9.8",
      "anchor": "oa-fnd-bi-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 9.8** (easy; no faithful functional on a large \\(B(H)\\)). For an uncountable set \\(\\Gamma\\), show that \\(B(\\ell^2(\\Gamma))\\) has no faithful positive linear functional, normal or not, while its commutant \\(\\mathbb C1\\) has a faithful normal state.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 667,
        "through_line": 670,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 9.9",
      "kind": "exercise",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-18::Exercise 9.9",
      "anchor": "oa-fnd-bi-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Exercise 9.9** (medium; a cyclic and separating vector in a large Hilbert space). Let \\((\\Gamma_i,\\mu_i)_{i\\in I}\\) be an uncountable family of probability spaces, let \\((\\Gamma,\\mu)\\) be the product space, \\(H=L^2(\\Gamma,\\mu)\\), and let \\(M\\) be the von Neumann algebra generated by the multiplication operators \\(m_f\\), for bounded measurable \\(f\\). Show that the function equal to \\(1\\) everywhere is cyclic and separating for \\(M\\). Suppose moreover that uncountably many factors are *nontrivial*: for uncountably many \\(i\\) there is a measurable \\(B_i\\subseteq\\Gamma_i\\) with \\(0<\\mu_i(B_i)<1\\). Show that \\(H\\) is then not separable. Some hypothesis of this kind is needed: if every \\(\\Gamma_i\\) is a one-point space, then \\(\\Gamma\\) is one point and \\(H=\\mathbb C\\) is separable.\n\nThe one-point example in the question proves that an uncountable index set alone does not imply nonseparability.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 760,
        "through_line": 775,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.1",
      "kind": "theorem",
      "unit": "the-double-commutant-theorem",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-double-commutant-theorem#oa-fnd-bi-14::Theorem 10.1",
      "anchor": "oa-fnd-bi-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
      "statement_and_full_conditions": "**Theorem 10.1.** Let \\(M\\) be a \\(*\\)-subalgebra of \\(B(H)\\) that contains \\(1\\), closed or not. Let \\(\\varphi:M\\to\\mathbb C\\) be linear and positive, and continuous for the \\(\\sigma\\)-strong topology restricted to \\(M\\). Then there is a sequence \\((\\xi_n)\\) in \\(H\\) with \\(\\sum_n\\|\\xi_n\\|^2<\\infty\\) and\n\\[\n\\begin{gathered}\n\\varphi(x)\\\\\n=\\sum_n\\langle x\\xi_n,\\xi_n\\rangle\\\\\n(x\\in M).\n\\end{gathered}\n\\tag{10.1}\n\\]\nIf \\(\\varphi\\) is continuous for the strong topology, finitely many vectors suffice. Every \\(\\sigma\\)-weakly continuous functional is \\(\\sigma\\)-strongly continuous ([Lemma 1.2](#oa-fnd-bi-01)(b)). So the theorem applies to every positive normal functional on a von Neumann algebra.",
      "proof_locus": {
        "source": "src/the-double-commutant-theorem.md",
        "line": 780,
        "through_line": 833,
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.1",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-01::Lemma 1.1",
      "anchor": "oa-fnd-kd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 1.1** (bounded nets). Let \\((x_i)\\) and \\((y_i)\\) be nets in \\(B(H)\\) over the same directed set, with \\(\\sup_i\\|x_i\\|<\\infty\\).\n\n1. If \\(x_i\\to x\\) and \\(y_i\\to y\\) strongly, then \\(x_iy_i\\to xy\\) strongly.\n2. If moreover \\(\\sup_i\\|y_i\\|<\\infty\\), and \\(x_i\\to x\\) and \\(y_i\\to y\\) strongly\\(^*\\), then \\(x_iy_i\\to xy\\) strongly\\(^*\\).\n3. A bounded net that converges strongly also converges \\(\\sigma\\)-strongly and \\(\\sigma\\)-weakly, and a bounded net that converges strongly\\(^*\\) also converges \\(\\sigma\\)-strongly\\(^*\\). In particular \\(\\varphi(x_i)\\to\\varphi(x)\\) for every normal functional \\(\\varphi\\) on a von Neumann algebra that contains the net.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 39,
        "through_line": 53,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.2",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-01::Lemma 1.2",
      "anchor": "oa-fnd-kd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 1.2** (closed sets). For \\(r\\ge0\\), the ball \\(rS\\), the set \\(B(H)_h\\) and the cone \\(B(H)_+\\) are closed in each of the six topologies. So is the set \\(\\{x:x\\ge c\\}\\) for a fixed self-adjoint \\(c\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 54,
        "through_line": 57,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 1.3",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-01::Theorem 1.3",
      "anchor": "oa-fnd-kd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 1.3** (Vigier's theorem). Let \\((x_i)\\) be an increasing net of self-adjoint operators with \\(\\sup_i\\|x_i\\|<\\infty\\). Then \\((x_i)\\) converges strongly to a self-adjoint operator \\(x\\), and \\(x\\) is the least upper bound of the net: \\(x_i\\le x\\) for all \\(i\\), and \\(x\\le y\\) for every self-adjoint \\(y\\) with \\(x_i\\le y\\) for all \\(i\\). If all \\(x_i\\) lie in a von Neumann algebra \\(M\\), so does \\(x\\). Decreasing nets behave in the same way, with greatest lower bounds.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 58,
        "through_line": 67,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 1.4",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-01::Corollary 1.4",
      "anchor": "oa-fnd-kd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 1.4** (monotone nets of projections). An increasing net of projections \\((p_i)\\) converges strongly to \\(\\bigvee_ip_i\\), and a decreasing net of projections converges strongly to \\(\\bigwedge_ip_i\\). The limits lie in every von Neumann algebra that contains the net.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 68,
        "through_line": 71,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.1",
      "kind": "definition",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-02::Definition 2.1",
      "anchor": "oa-fnd-kd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Definition 2.1.** An operator \\(a\\) is *hyponormal* if \\(\\|a^*\\xi\\|\\le\\|a\\xi\\|\\) for every \\(\\xi\\in H\\). Normal operators are hyponormal, and so are isometries, since \\(\\|v^*\\xi\\|\\le\\|\\xi\\|=\\|v\\xi\\|\\). An operator \\(a\\) is normal exactly when \\(\\|a^*\\xi\\|=\\|a\\xi\\|\\) for all \\(\\xi\\): by polarization, this says \\(\\langle a^*a\\xi,\\xi\\rangle=\\langle aa^*\\xi,\\xi\\rangle\\) for all \\(\\xi\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 76,
        "through_line": 77,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.2",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-02::Lemma 2.2",
      "anchor": "oa-fnd-kd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 2.2.** For all \\(a,b\\in B(H)\\) and \\(\\xi\\in H\\),\n\\[\n\\begin{gathered}\n\\|(a^*-b^*)\\xi\\|^2\\\\\n=\\|a^*\\xi\\|^2-\\|b^*\\xi\\|^2\\\\\n-2\\operatorname{Re}\\langle(a-b)b^*\\xi,\\xi\\rangle .\n\\end{gathered}\n\\tag{2.1}\n\\]",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 78,
        "through_line": 94,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.3",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-02::Theorem 2.3",
      "anchor": "oa-fnd-kd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 2.3.** Let \\((a_i)\\) be a net in \\(B(H)\\) that converges strongly to \\(b\\).\n\n1. If every \\(a_i\\) is hyponormal, then \\(b\\) is hyponormal.\n2. If every \\(a_i\\) is hyponormal and \\(b\\) is normal, then \\(a_i^*\\to b^*\\) strongly. So \\(a_i\\to b\\) strongly\\(^*\\).\n3. If every \\(a_i\\) is normal, then \\(a_i^*\\to b^*\\) strongly if and only if \\(b\\) is normal.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 95,
        "through_line": 106,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.4",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-02::Corollary 2.4",
      "anchor": "oa-fnd-kd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 2.4.** On the set of normal operators, the strong and the strong\\(^*\\) topologies coincide, and the adjoint is strongly continuous there. A net of isometries that converges strongly to a unitary converges strongly\\(^*\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 107,
        "through_line": 110,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.5",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-02::Example 2.5",
      "anchor": "oa-fnd-kd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 2.5** (the limit must be normal). Let \\((\\xi_k)_{k\\ge1}\\) be an orthonormal sequence in \\(H\\). For \\(n\\ge1\\) let \\(u_n\\) be the unitary with \\(u_n\\xi_k=\\xi_{k+1}\\) for \\(k<n\\), \\(u_n\\xi_n=\\xi_1\\), and \\(u_n=1\\) on the orthogonal complement of \\(\\xi_1,\\dots,\\xi_n\\). Let \\(v\\) be the isometry with \\(v\\xi_k=\\xi_{k+1}\\) for all \\(k\\) and \\(v=1\\) on the orthogonal complement of all \\(\\xi_k\\). For fixed \\(k\\), \\(u_n\\xi_k=v\\xi_k\\) as soon as \\(n>k\\), and \\(u_n=v\\) on the complement. The net is bounded, so \\(u_n\\to v\\) strongly. The limit is not normal, since \\(\\|v^*\\xi_1\\|=0\\ne1=\\|v\\xi_1\\|\\). As Theorem 2.3(3) predicts, the adjoints do not converge strongly: \\(u_n^*\\xi_1=\\xi_n\\), and these vectors are at mutual distance \\(\\sqrt2\\). So a strong limit of unitaries need not be unitary, and a strong limit of normal operators need not be normal.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 111,
        "through_line": 112,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-03::Theorem 3.1",
      "anchor": "oa-fnd-kd-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 3.1** (Fuglede's theorem). Let \\(a\\in B(H)\\) be normal and let \\(b\\in B(H)\\) commute with \\(a\\). Then \\(b\\) commutes with \\(a^*\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 117,
        "through_line": 138,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.2",
      "kind": "definition",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-03::Definition 3.2",
      "anchor": "oa-fnd-kd-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Definition 3.2.** A *commuting normal tuple* is an \\(n\\)-tuple \\(a=(a_1,\\dots,a_n)\\) of normal operators on \\(H\\) with \\(a_ja_k=a_ka_j\\) for all \\(j,k\\). By Fuglede's theorem, each \\(a_j\\) also commutes with each \\(a_k^*\\). So the C\\(^*\\)-algebra \\(C^*(1,a)\\) generated by \\(1,a_1,\\dots,a_n\\) is commutative: it is the norm closure of the polynomials in the pairwise commuting operators \\(a_j,a_j^*\\). Its characters form a compact space \\(\\Omega\\), and the *joint spectrum* of \\(a\\) is\n\\[\n\\sigma(a)=\\{(\\chi(a_1),\\dots,\\chi(a_n)):\\chi\\in\\Omega\\}\\subseteq\\mathbb C^n .\n\\]",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 139,
        "through_line": 143,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.3",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-03::Theorem 3.3",
      "anchor": "oa-fnd-kd-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 3.3** (the joint functional calculus). Let \\(a\\) be a commuting normal tuple on \\(H\\ne\\{0\\}\\), and let \\(\\iota_j\\) be the \\(j\\)-th coordinate function on \\(\\mathbb C^n\\).\n\n1. The map \\(\\Psi(\\chi)=(\\chi(a_1),\\dots,\\chi(a_n))\\) is a homeomorphism of \\(\\Omega\\) onto \\(\\sigma(a)\\). So \\(\\sigma(a)\\) is a nonempty compact set. It lies in \\(\\sigma(a_1)\\times\\dots\\times\\sigma(a_n)\\), and its projection to the \\(j\\)-th coordinate is \\(\\sigma(a_j)\\).\n2. There is exactly one unital \\(*\\)-homomorphism \\(f\\mapsto f(a)\\) from \\(C(\\sigma(a))\\) into \\(B(H)\\) with \\(\\iota_j(a)=a_j\\) for all \\(j\\). It is isometric, \\(\\|f(a)\\|=\\max_{\\sigma(a)}|f|\\), and its range is \\(C^*(1,a)\\). Each \\(f(a)\\) is normal.\n3. \\(f(a)\\) commutes with every operator that commutes with \\(a_1,\\dots,a_n\\).\n4. For \\(n=1\\) this is the continuous functional calculus of a normal operator.\n\nFor a closed set \\(G\\supseteq\\sigma(a)\\) and a continuous \\(f\\) on \\(G\\) we write \\(f(a)\\) for \\((f|_{\\sigma(a)})(a)\\). If \\(f\\) is bounded, then \\(\\|f(a)\\|\\le\\sup_G|f|\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 144,
        "through_line": 162,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.4",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-03::Example 3.4",
      "anchor": "oa-fnd-kd-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 3.4.** (a) Let \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_k)\\), and let \\(a_j\\) be the diagonal operator \\(a_j\\delta_k=\\lambda^{(k)}_j\\delta_k\\) for bounded sequences \\((\\lambda^{(k)}_j)_k\\), \\(j=1,\\dots,n\\). Put \\(\\lambda^{(k)}=(\\lambda^{(k)}_1,\\dots,\\lambda^{(k)}_n)\\), and let \\(K\\) be the closure of \\(\\{\\lambda^{(k)}:k\\in\\mathbb N\\}\\). Then \\(\\sigma(a)=K\\). Indeed, \\(f\\mapsto\\) the diagonal operator with entries \\(f(\\lambda^{(k)})\\) is a unital \\(*\\)-homomorphism from \\(C(K)\\) into \\(B(H)\\). It is isometric, because the points \\(\\lambda^{(k)}\\) are dense in \\(K\\), and it sends \\(\\iota_j\\) to \\(a_j\\). Its range is therefore \\(C^*(1,a)\\), isomorphic to \\(C(K)\\), and the characters of \\(C(K)\\) are the evaluations at points of \\(K\\) ([the characters of \\(C_0(\\Omega)\\)](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-04)). (b) For a projection \\(p\\ne0,1\\), the pair \\((p,1-p)\\) has joint spectrum \\(\\{(1,0),(0,1)\\}\\), a proper subset of \\(\\sigma(p)\\times\\sigma(1-p)=\\{0,1\\}^2\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 163,
        "through_line": 164,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-04::Definition 4.1",
      "anchor": "oa-fnd-kd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Definition 4.1.** Let \\(K\\subseteq\\mathbb C\\) be compact. We write \\(\\mathcal B_1(K)\\) for the set of functions \\(\\varphi:K\\to\\mathbb C\\) for which there are continuous functions \\(\\varphi_k\\) on \\(K\\) with \\(\\sup_k\\max_K|\\varphi_k|<\\infty\\) and \\(\\varphi_k(\\lambda)\\to\\varphi(\\lambda)\\) for every \\(\\lambda\\in K\\). This set is a \\(*\\)-algebra under pointwise operations, and it contains \\(C(K)\\). Corollaries 4.3, 4.5 and 4.6 below show that it contains indicator functions of intervals, the argument function on the unit circle and binary digits.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 169,
        "through_line": 170,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.2",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-04::Theorem 4.2",
      "anchor": "oa-fnd-kd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 4.2** (bounded pointwise limits). Let \\(a\\) be a normal operator with spectrum \\(K\\). For \\(\\varphi\\in\\mathcal B_1(K)\\) and any sequence \\((\\varphi_k)\\) as in Definition 4.1, the operators \\(\\varphi_k(a)\\) converge strongly to an operator \\(\\varphi(a)\\) that depends only on \\(\\varphi\\). The map \\(\\varphi\\mapsto\\varphi(a)\\) is a \\(*\\)-homomorphism from \\(\\mathcal B_1(K)\\) into \\(B(H)\\) that extends the continuous functional calculus. It satisfies \\(\\|\\varphi(a)\\|\\le\\sup_K|\\varphi|\\), and \\(\\varphi\\ge0\\) implies \\(\\varphi(a)\\ge0\\). Every \\(\\varphi(a)\\) lies in every von Neumann algebra that contains \\(a\\), and commutes with every operator that commutes with \\(a\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 171,
        "through_line": 212,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.3",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-04::Corollary 4.3",
      "anchor": "oa-fnd-kd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 4.3** (spectral projections). Let \\(x\\) be a self-adjoint element of a von Neumann algebra \\(N\\), let \\(c\\in\\mathbb R\\), and let \\(p=1_{(c,\\infty)}(x)\\).\n\n1. \\(p\\) is a projection in \\(N\\), and it commutes with every operator that commutes with \\(x\\).\n2. \\((x-c)p\\ge0\\) and \\((x-c)(1-p)\\le0\\). In particular \\(cp\\le xp\\) and \\(x(1-p)\\le c(1-p)\\).\n3. If \\(c\\ge0\\) and \\(x=exe\\) for a projection \\(e\\), then \\(p\\le e\\).\n4. If \\(c<\\max\\sigma(x)\\), then \\(p\\ne0\\). If \\(c>\\min\\sigma(x)\\), then \\(p\\ne1\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 213,
        "through_line": 221,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.4",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-04::Corollary 4.4",
      "anchor": "oa-fnd-kd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 4.4.** A von Neumann algebra \\(N\\) on \\(H\\ne\\{0\\}\\) whose only projections are \\(0\\) and \\(1\\) is \\(\\mathbb C1\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 222,
        "through_line": 225,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.5",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-04::Corollary 4.5",
      "anchor": "oa-fnd-kd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 4.5** (unitaries are exponentials). Let \\(u\\) be a unitary in a von Neumann algebra \\(N\\).\n\n1. There is a self-adjoint \\(h\\in N\\) with \\(\\|h\\|\\le\\pi\\) and \\(u=\\exp(ih)\\).\n2. If \\(\\|u-1\\|<2\\), put \\(\\alpha=2\\arcsin(\\|u-1\\|/2)<\\pi\\). Then \\(h\\) can be chosen in \\(C^*(1,u)\\), with \\(\\|h\\|\\le\\alpha\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 226,
        "through_line": 236,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.6",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-04::Corollary 4.6",
      "anchor": "oa-fnd-kd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 4.6** (dyadic expansion). Every \\(x\\) in a von Neumann algebra \\(N\\) with \\(0\\le x\\le1\\) is a norm-convergent sum \\(x=\\sum_{k\\ge1}2^{-k}p_k\\) of projections \\(p_k\\in N\\), each a spectral projection of \\(x\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 237,
        "through_line": 240,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.1",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-05::Lemma 5.1",
      "anchor": "oa-fnd-kd-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 5.1** (resolvent estimates). Let \\(a\\) and \\(b\\) be \\(n\\)-tuples of operators.\n\n1. \\(0\\le R(a)\\le1\\), \\(\\|a_jR(a)\\|\\le\\frac12\\), \\(\\|R(a)a_j^*\\|\\le\\frac12\\) and \\(\\|a_jR(a)a_k^*\\|\\le1\\) for all \\(j,k\\).\n2. \\[\n\\begin{gathered}\nR(a)-R(b)\\\\\n=\\sum_kR(a)\\big[(b_k^*-a_k^*)b_k\\\\\n+a_k^*(b_k-a_k)\\big]R(b).\n\\end{gathered}\n\\]\n3. If \\((a^{(i)})\\) is a net of \\(n\\)-tuples with \\(a^{(i)}_k\\to b_k\\) and \\(a^{(i)*}_k\\to b_k^*\\) strongly for every \\(k\\), then \\(R(a^{(i)})\\to R(b)\\) and \\(a^{(i)}_jR(a^{(i)})\\to b_jR(b)\\) strongly for every \\(j\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 250,
        "through_line": 301,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.2",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-05::Theorem 5.2",
      "anchor": "oa-fnd-kd-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 5.2** (Kaplansky's continuity theorem). Let \\(G\\subseteq\\mathbb C^n\\) be closed, and let \\(f:G\\to\\mathbb C\\) be continuous with\n\\[\n\\sup_{\\lambda\\in G}\\frac{|f(\\lambda)|}{1+|\\lambda|}<\\infty .\n\\tag{5.1}\n\\]\nIf a net \\((a^{(i)})\\) in \\(\\mathcal N_G\\) converges strongly to \\(a\\in\\mathcal N_G\\), then \\(f(a^{(i)})\\to f(a)\\) strongly and strongly\\(^*\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 302,
        "through_line": 348,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.3",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-05::Corollary 5.3",
      "anchor": "oa-fnd-kd-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 5.3.** Let \\(G\\subseteq\\mathbb C^n\\) be closed and \\(f:G\\to\\mathbb C\\) continuous.\n\n1. If \\(G\\) is compact, the calculus of \\(f\\) is strongly continuous on \\(\\mathcal N_G\\).\n2. For every \\(c>0\\), the calculus of \\(f\\) is strongly continuous on the set of \\(a\\in\\mathcal N_G\\) with \\(\\|a_j\\|\\le c\\) for all \\(j\\).\n3. If a sequence \\((a^{(m)})\\) in \\(\\mathcal N_G\\) converges strongly to \\(a\\in\\mathcal N_G\\), then \\(f(a^{(m)})\\to f(a)\\) strongly.\n4. For self-adjoint operators (\\(n=1\\), \\(G=\\mathbb R\\)), every continuous \\(f\\) with \\(|f(t)|\\le C(1+|t|)\\) acts strongly continuously. Examples are \\(|t|\\), \\(\\max(t,0)\\), \\(e^{it}\\), \\(2t/(1+t^2)\\), and any continuous function on a compact interval, applied to self-adjoint operators with spectrum in that interval.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 349,
        "through_line": 357,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.4",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-05::Example 5.4",
      "anchor": "oa-fnd-kd-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 5.4** (the Cayley transform). For a self-adjoint \\(h\\) the *Cayley transform* \\(\\kappa(h)=(h-i)(h+i)^{-1}\\) is the calculus of the bounded continuous function \\((t-i)/(t+i)\\), which has modulus \\(1\\). So \\(\\kappa(h)\\) is unitary, and \\(\\kappa\\) is strongly continuous on self-adjoint operators by Corollary 5.3(4). A direct estimate is also available. Since \\(\\kappa(h)=1-2i(h+i)^{-1}\\), the resolvent identity gives\n\\[\n\\begin{gathered}\n\\kappa(h)-\\kappa(k)\\\\\n=2i\\,(h+i)^{-1}(h-k)(k+i)^{-1},\\\\\n\\text{so}\\\\\n\\|(\\kappa(h)-\\kappa(k))\\xi\\|\\\\\n\\le2\\|(h-k)(k+i)^{-1}\\xi\\|,\n\\end{gathered}\n\\]\nbecause \\(\\|(h+i)^{-1}\\|\\le1\\). For fixed \\(k\\) and \\(\\xi\\) the right side tends to \\(0\\) as \\(h\\to k\\) strongly.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 358,
        "through_line": 369,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-06::Theorem 6.1",
      "anchor": "oa-fnd-kd-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 6.1.** Let \\(H\\) be infinite-dimensional, \\(G\\subseteq\\mathbb C^n\\) closed, and \\(f:G\\to\\mathbb C\\) continuous. Suppose that for some \\(g_0\\in G\\) the calculus of \\(f\\) is strongly continuous on \\(\\mathcal N_G\\) at the constant tuple \\(g_0\\cdot1=(g_{0,1}1,\\dots,g_{0,n}1)\\). Then \\(f\\) satisfies (5.1). If \\(\\dim H<\\infty\\), every continuous \\(f\\) acts strongly continuously on \\(\\mathcal N_G\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 374,
        "through_line": 404,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.2",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-06::Example 6.2",
      "anchor": "oa-fnd-kd-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 6.2** (squaring). On an infinite-dimensional \\(H\\), the map \\(h\\mapsto h^2\\) on self-adjoint operators is not strongly continuous at \\(0\\). In the construction above take \\(f(t)=t^2\\), \\(g_0=0\\), \\(t_k=k\\) and \\(c_k=k^{-2}\\). Then \\(h_\\lambda=kp_\\lambda\\to0\\) strongly, but \\(\\|h_\\lambda^2\\xi\\|=k^2\\,|\\langle\\xi,v_\\lambda\\rangle|=1\\). By Corollary 5.3(3), no sequence can show this.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 405,
        "through_line": 406,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.1",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-07::Theorem 7.1",
      "anchor": "oa-fnd-kd-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 7.1** (Kaplansky's density theorem). Let \\(A\\) be a \\(*\\)-subalgebra of \\(B(H)\\), not necessarily norm closed or nondegenerate, and let \\(M\\) be its weak closure.\n\n1. \\(A\\cap S\\) is strongly\\(^*\\) dense in \\(M\\cap S\\).\n2. \\(A_h\\cap S\\) is strongly\\(^*\\) dense in \\(M_h\\cap S\\).\n3. \\(A_+\\cap S\\) is strongly\\(^*\\) dense in \\(M_+\\cap S\\).\n4. In each of (1)–(3), the closure of the smaller set in any of the six operator topologies is the larger set.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 409,
        "through_line": 428,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [
        {
          "unit": "kaplansky-s-density-theorem-and-its-consequences",
          "local_scopes": [
            {
              "heading": "7. Kaplansky's density theorem",
              "line": 407,
              "through_line": 465,
              "anchors": [
                "OA-FND-KD-07"
              ],
              "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.6; PDF 56–57",
          "role": "proof comparison",
          "correspondence": "Standard bounded-density proof",
          "explanation": "Self-adjoint clamping and matrix amplification compared; own strong* conclusion and nonclosed *-algebra treatment retained."
        },
        {
          "unit": "kaplansky-s-density-theorem-and-its-consequences",
          "local_scopes": [
            {
              "heading": "7. Kaplansky's density theorem",
              "line": 407,
              "through_line": 465,
              "anchors": [
                "OA-FND-KD-07"
              ],
              "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
            }
          ],
          "source_key": "human:elliott-griffin@DA2CCC95ADD8212265B23ABDABC18DE6CE5B8E9DCE1BBFC629286C51BC8C66ED",
          "source_locus": "Entire main argument and concluding remark",
          "role": "proof comparison",
          "correspondence": "New full proof of bounded-net method",
          "explanation": "Adjoint closure via convex balls, bounded continuous calculus with f(0)=0, M2 strong* control, trace duality and Krein–Šmulian. BI Theorem 4.4 justifies the final ultraweak-to-strong algebra closure step.",
          "local_labels": [
            "Theorem 7.2"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.3",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-07::Example 7.3",
      "anchor": "oa-fnd-kd-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 7.3** (the algebraic structure is needed). Kaplansky's theorem fails for self-adjoint subspaces that are not algebras. On \\(H=\\ell^2(\\mathbb N)\\) with basis \\((\\delta_m)\\), let \\(\\omega(x)=\\sum_m2^{-m}\\langle x\\delta_m,\\delta_m\\rangle\\) and \\(V=\\ker\\omega\\). Then \\(V^*=V\\). By [Exercise 1.5 of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-01), \\(V\\) is not weakly closed. Its weak closure is a weakly closed subspace that strictly contains a subspace of codimension one, so it is \\(B(H)\\). But \\(V\\cap S\\) is weakly closed. Indeed, a bounded weakly convergent net converges \\(\\sigma\\)-weakly ([Lemma 1.2(d) of the double commutant lesson](the-double-commutant-theorem.md#oa-fnd-bi-01)), and \\(\\omega\\) is \\(\\sigma\\)-weakly continuous, so \\(\\omega\\) vanishes at every weak limit of a net in \\(V\\cap S\\). So the weak closure of \\(V\\cap S\\) is \\(V\\cap S\\), which does not contain \\(1\\in B(H)\\cap S\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 429,
        "through_line": 430,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [
        {
          "unit": "kaplansky-s-density-theorem-and-its-consequences",
          "local_scopes": [
            {
              "heading": "7. Kaplansky's density theorem",
              "line": 407,
              "through_line": 465,
              "anchors": [
                "OA-FND-KD-07"
              ],
              "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.6; PDF 56–57",
          "role": "proof comparison",
          "correspondence": "Standard bounded-density proof",
          "explanation": "Self-adjoint clamping and matrix amplification compared; own strong* conclusion and nonclosed *-algebra treatment retained."
        },
        {
          "unit": "kaplansky-s-density-theorem-and-its-consequences",
          "local_scopes": [
            {
              "heading": "7. Kaplansky's density theorem",
              "line": 407,
              "through_line": 465,
              "anchors": [
                "OA-FND-KD-07"
              ],
              "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
            }
          ],
          "source_key": "human:elliott-griffin@DA2CCC95ADD8212265B23ABDABC18DE6CE5B8E9DCE1BBFC629286C51BC8C66ED",
          "source_locus": "Entire main argument and concluding remark",
          "role": "proof comparison",
          "correspondence": "New full proof of bounded-net method",
          "explanation": "Adjoint closure via convex balls, bounded continuous calculus with f(0)=0, M2 strong* control, trace duality and Krein–Šmulian. BI Theorem 4.4 justifies the final ultraweak-to-strong algebra closure step.",
          "local_labels": [
            "Theorem 7.2"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.2",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-07::Theorem 7.2",
      "anchor": "oa-fnd-kd-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 7.2** (The bounded-net algebra). Let \\(A\\subseteq B(H)\\) be a norm-closed *-subalgebra, with no nondegeneracy or identity assumption. Let \\(D\\) consist of the strong limits of norm-bounded nets in \\(A\\). Then \\(D\\) is the strong closure of \\(A\\), and\n\\[\nD\\cap B(H)_1=\\overline{A\\cap B(H)_1}^{\\,s}.\n\\]\nThe same statement holds for self-adjoint and positive unit balls. In the unrestricted unit ball, approximation can also be chosen strongly* convergent.\n\nThis is the bounded-net route of [G. A. Elliott and C. J. K. Griffin, *On a question of Kaplansky concerning his density theorem* (2024)](https://arxiv.org/html/2410.03668v1). It explains how bounded-set functional calculus suffices for density. The argument below uses the complete trace-duality and Krein–Šmulian proofs in the operator-topologies lesson: [trace duality, Theorem 5.4](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md#oa-fnd-lt-06), and [Krein–Šmulian and its bounded-ball consequence, Theorems 9.3 and 9.6](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md#oa-fnd-lt-10), together with [the double commutant lesson, Theorem 4.4](the-double-commutant-theorem.md#oa-fnd-bi-06). Those results do not use Kaplansky density. The stronger continuity theorem on unbounded normal nets remains Theorem 5.2 above.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 431,
        "through_line": 465,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [
        {
          "unit": "kaplansky-s-density-theorem-and-its-consequences",
          "local_scopes": [
            {
              "heading": "7. Kaplansky's density theorem",
              "line": 407,
              "through_line": 465,
              "anchors": [
                "OA-FND-KD-07"
              ],
              "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.6; PDF 56–57",
          "role": "proof comparison",
          "correspondence": "Standard bounded-density proof",
          "explanation": "Self-adjoint clamping and matrix amplification compared; own strong* conclusion and nonclosed *-algebra treatment retained."
        },
        {
          "unit": "kaplansky-s-density-theorem-and-its-consequences",
          "local_scopes": [
            {
              "heading": "7. Kaplansky's density theorem",
              "line": 407,
              "through_line": 465,
              "anchors": [
                "OA-FND-KD-07"
              ],
              "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
            }
          ],
          "source_key": "human:elliott-griffin@DA2CCC95ADD8212265B23ABDABC18DE6CE5B8E9DCE1BBFC629286C51BC8C66ED",
          "source_locus": "Entire main argument and concluding remark",
          "role": "proof comparison",
          "correspondence": "New full proof of bounded-net method",
          "explanation": "Adjoint closure via convex balls, bounded continuous calculus with f(0)=0, M2 strong* control, trace duality and Krein–Šmulian. BI Theorem 4.4 justifies the final ultraweak-to-strong algebra closure step.",
          "local_labels": [
            "Theorem 7.2"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.1",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-08::Theorem 8.1",
      "anchor": "oa-fnd-kd-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 8.1.** Let \\(U(H)\\) be the unitary group of \\(B(H)\\).\n\n1. \\(U(H)\\) is closed for the strong\\(^*\\) topology.\n2. On the set of isometries, the weak and the strong topologies coincide. On \\(U(H)\\), all six operator topologies coincide.\n3. The strong closure of \\(U(H)\\) is the set of all isometries.\n4. If \\(\\dim H=\\infty\\), the weak closure of \\(U(H)\\) is the unit ball \\(S\\). In particular, it contains all projections and all partial isometries. If \\(\\dim H<\\infty\\), \\(U(H)\\) is compact.\n5. \\(U(H)\\) is complete for the strong\\(^*\\) uniform structure: if \\((u_i)\\) is a net of unitaries such that \\((u_i\\xi)\\) and \\((u_i^*\\xi)\\) are Cauchy nets for every \\(\\xi\\), then it converges strongly\\(^*\\) to a unitary. If \\(\\dim H=\\infty\\), \\(U(H)\\) is not complete for the strong uniform structure.\n6. \\(U(H)\\) is relatively compact in the weak topology. If \\(\\dim H=\\infty\\), it is not weakly compact.\n7. The weak closure of \\(U(H)\\) is closed under products and adjoints.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 468,
        "through_line": 521,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.2",
      "kind": "proposition",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-08::Proposition 8.2",
      "anchor": "oa-fnd-kd-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Proposition 8.2** (products of two projections). Let \\(0\\le h\\le1\\), and let \\(R\\) be the closure of the range of \\(h-h^2\\). The following are equivalent.\n\n1. \\(h=efe\\) for two projections \\(e\\) and \\(f\\).\n2. There is a linear isometry of \\(R\\) into \\(\\ker h\\).\n\nCondition (2) holds, for example, if \\(H\\) is separable and \\(\\ker h\\) is infinite-dimensional, or if \\(\\ker h\\) contains a closed subspace isometric to \\(H\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 522,
        "through_line": 564,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.3",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-08::Example 8.3",
      "anchor": "oa-fnd-kd-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 8.3** (an infinite-dimensional kernel is not enough). Let \\(K\\) be a nonseparable Hilbert space, for instance \\(\\ell^2(\\Gamma)\\) for an uncountable set \\(\\Gamma\\), let \\(N\\) be a separable infinite-dimensional Hilbert space, and let \\(h=\\frac12\\cdot1_K\\oplus0_N\\) on \\(H=K\\oplus N\\). Then \\(0\\le h\\le1\\), and \\(\\ker h=N\\) is infinite-dimensional. But \\(h-h^2=\\frac14\\cdot1_K\\oplus0\\), so \\(R=K\\) is nonseparable. An isometric image of a nonseparable space is nonseparable, while every subset of the separable space \\(N\\) is separable. So there is no isometry of \\(R\\) into \\(\\ker h\\), and \\(h\\) is not a product \\(efe\\) of two projections.\n\nProposition 8.2 proves the exact dimension criterion; Example 8.3 proves directly that infinite-dimensional null space alone does not suffice on a nonseparable space.\n\nIn finite dimensions, Proposition 8.2 says that \\(h=efe\\) exactly when the rank of \\(h-h^2\\) is at most \\(\\dim\\ker h\\). For example, \\(\\operatorname{diag}(\\frac12,0)\\) on \\(\\mathbb C^2\\) equals \\(efe\\) with \\(e=\\operatorname{diag}(1,0)\\) and \\(f\\) the projection onto \\(\\mathbb C(1,1)\\), while \\(\\frac12\\) on \\(\\mathbb C\\) is not a product of two projections of \\(\\mathbb C\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 565,
        "through_line": 570,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.1",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-10::Theorem 9.1",
      "anchor": "oa-fnd-kd-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 9.1** (density of unitary groups). Let \\(A\\) be a concrete C\\(^*\\)-algebra on \\(H\\) with \\(1\\in A\\), and let \\(M=A''\\), its weak closure. For \\(\\lambda>0\\) put\n\\[\n\\begin{gathered}\nU(A,\\lambda)\\\\\n=\\{u\\in U(A):\\|u-1\\|\\le\\lambda\\},\\\\\nU(M,\\lambda)\\\\\n=\\{u\\in U(M):\\|u-1\\|\\le\\lambda\\}.\n\\end{gathered}\n\\]\nThen \\(U(M,\\lambda)\\) is the strong\\(^*\\) closure of \\(U(A,\\lambda)\\). In particular, \\(U(A)\\) is strongly\\(^*\\) dense in \\(U(M)\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 573,
        "through_line": 601,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.1",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Lemma 10.1",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 10.1** (cutting down a null net). Let \\(e\\in M\\) be a projection and \\((x_i)\\) a bounded net in \\(M\\) with \\(x_ie\\to0\\) strongly, and let \\(\\varepsilon>0\\). Then there are projections \\(e_i\\le e\\) in \\(M\\) with \\(e_i\\to e\\) strongly and \\(\\|x_ie_i\\|\\le\\varepsilon\\) for every \\(i\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 606,
        "through_line": 619,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.2",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Theorem 10.2",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 10.2** (noncommutative Egoroff theorem). Let \\(B\\subseteq M\\) be a bounded set, \\(x\\) an element of its strong closure, \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection and \\(\\varepsilon>0\\). Then there are a projection \\(f\\le e\\) in \\(M\\) and a sequence \\((b_k)\\) in \\(B\\) with\n\\[\n\\begin{gathered}\n\\lim_{k\\to\\infty}\\|(b_k-x)f\\|\\\\\n=0\\\\\n\\text{and}\\\\\n\\varphi(e-f)<\\varepsilon .\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 620,
        "through_line": 647,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.3",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Lemma 10.3",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 10.3** (self-adjoint completion). Let \\(a\\in M\\) be self-adjoint, \\(e\\in M\\) a projection and \\(c=\\|ae\\|\\). There is a self-adjoint \\(b\\in M\\) with \\(be=ae\\) and \\(\\|b\\|=c\\).\n\nThe proof below gives the required norm-preserving completion explicitly.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 648,
        "through_line": 678,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 10.4",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Corollary 10.4",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 10.4** (approximation with norm control). Let \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection, \\(\\varepsilon>0\\) and \\(\\delta>0\\).\n\n1. For every \\(x\\in M\\) there are \\(a\\in A\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(\\|(x-a)f\\|<\\delta\\), \\(\\|a\\|\\le\\|xe\\|\\) and \\(\\varphi(e-f)<\\varepsilon\\).\n2. If \\(x\\) is self-adjoint, \\(a\\) can be chosen self-adjoint, with the same three properties.\n3. If \\(1\\in A\\) and \\(x\\in U(M)\\), \\(a\\) can be chosen in \\(U(A)\\) with \\(\\|(x-a)f\\|<\\delta\\), \\(\\|a-1\\|\\le\\|x-1\\|\\) and \\(\\varphi(e-f)<\\varepsilon\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 679,
        "through_line": 692,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.5",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Lemma 10.5",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 10.5** (correcting a unitary). Let \\(w\\in U(M)\\) and let \\(e\\in M\\) be a projection with \\(\\|(1-w)e\\|\\le\\frac18\\). There is \\(v\\in U(M)\\) with \\(ve=we\\) and \\(\\|1-v\\|\\le6\\|(1-w)e\\|\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 693,
        "through_line": 719,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.6",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Theorem 10.6",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 10.6** (noncommutative Lusin theorem). Let \\(\\varphi\\in M_*^+\\), \\(e\\in M\\) a projection, \\(\\varepsilon>0\\) and \\(\\delta>0\\).\n\n1. For every \\(x\\in M\\) there are \\(a\\in A\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(xf=af\\), \\(\\varphi(e-f)<\\varepsilon\\) and \\(\\|a\\|\\le(1+\\delta)\\|xf\\|\\).\n2. If \\(x\\) is self-adjoint, \\(a\\) can be chosen self-adjoint, with the same properties.\n3. If \\(1\\in A\\) and \\(x\\in U(M)\\), there are \\(a\\in U(A)\\) and a projection \\(f\\le e\\) in \\(M\\) with \\(xf=af\\), \\(\\varphi(e-f)<\\varepsilon\\) and \\(\\|a-1\\|\\le\\|x-1\\|+\\delta\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 720,
        "through_line": 800,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.7",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-11::Example 10.7",
      "anchor": "oa-fnd-kd-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 10.7** (the commutative case). Let \\(\\mu\\) be Lebesgue measure on \\([0,1]\\), \\(M\\) the algebra of multiplication operators \\(m_g\\) by bounded measurable \\(g\\) on \\(L^2[0,1]\\), \\(A\\) the multiplication operators by continuous functions, and \\(\\varphi(m_g)=\\int g\\,d\\mu\\), the vector functional of the constant function \\(1\\). The projections of \\(M\\) are the operators \\(m_{1_E}\\), and \\(\\varphi(1-m_{1_E})=\\mu([0,1]\\setminus E)\\). A bounded sequence \\(g_k\\to g\\) almost everywhere gives \\(m_{g_k}\\to m_g\\) strongly, by dominated convergence. Theorem 10.2 then yields a set \\(E\\) with small complement on which a sequence of the \\(g_k\\) converges uniformly, up to a null set: a form of Egoroff's theorem. Theorem 10.6(1) yields a continuous \\(a\\) that agrees with a bounded measurable \\(g\\) almost everywhere on a set \\(E\\) with small complement, with \\(\\max|a|\\le(1+\\delta)\\operatorname{ess\\,sup}_E|g|\\): this is Lusin's theorem with control of the norm. (That \\(M\\) is the weak closure of \\(A\\) is shown in Exercise 14.2.)",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 801,
        "through_line": 802,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.1",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-13::Theorem 11.1",
      "anchor": "oa-fnd-kd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 11.1** (Kadison's transitivity theorem). Let \\(A\\) be an irreducible concrete C\\(^*\\)-algebra on \\(H\\), let \\(e\\) be a projection of finite rank on \\(H\\), and let \\(\\varepsilon>0\\).\n\n1. For every \\(b\\in B(H)\\) there is \\(a\\in A\\) with \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|be\\|\\). In particular \\(Ae=B(H)e\\).\n2. If \\(b\\) is self-adjoint, \\(a\\) can be chosen self-adjoint with \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|be\\|\\).\n3. If \\(b\\ge0\\), \\(a\\) can be chosen with \\(a\\ge0\\), \\(ae=be\\) and \\(\\|a\\|\\le(1+\\varepsilon)\\|b\\|\\).\n4. For every unitary \\(u\\in B(H)\\) there is a unitary \\(v\\) in the C\\(^*\\)-algebra \\(A+\\mathbb C1\\) with \\(ve=ue\\) and \\(\\|v-1\\|\\le\\|u-1\\|+\\varepsilon\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 807,
        "through_line": 828,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.2",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-13::Corollary 11.2",
      "anchor": "oa-fnd-kd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 11.2.** Let \\(A\\) be an irreducible concrete C\\(^*\\)-algebra on \\(H\\).\n\n1. If \\(\\xi_1,\\dots,\\xi_n\\in H\\) are linearly independent and \\(\\eta_1,\\dots,\\eta_n\\in H\\) are arbitrary, there is \\(a\\in A\\) with \\(a\\xi_j=\\eta_j\\) for all \\(j\\).\n2. For every \\(\\xi\\ne0\\), \\(A\\xi=H\\). So \\(H\\) has no subspaces invariant under \\(A\\) except \\(\\{0\\}\\) and \\(H\\), closed or not: \\(A\\) acts *algebraically irreducibly*.\n3. Let \\(\\pi\\) be a representation of a C\\(^*\\)-algebra \\(B\\) on \\(H\\) such that \\(\\pi(B)\\) is irreducible. Then \\(\\pi(B)\\xi=H\\) for every nonzero \\(\\xi\\in H\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 829,
        "through_line": 836,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.3",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-13::Example 11.3",
      "anchor": "oa-fnd-kd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 11.3** (norm closedness is needed). Let \\(s\\) be the unilateral shift on \\(\\ell^2(\\{0,1,2,\\dots\\})\\), \\(s\\delta_m=\\delta_{m+1}\\), and let \\(P\\) be the \\(*\\)-algebra of polynomials in \\(s\\) and \\(s^*\\). Then \\(1-ss^*=\\theta_{\\delta_0,\\delta_0}\\), and \\(s^i(1-ss^*)s^{*j}=\\theta_{\\delta_i,\\delta_j}\\), so \\(P\\) contains all matrix units. If a closed subspace \\(W\\ne\\{0\\}\\) is invariant under \\(P\\), pick \\(w\\in W\\) and \\(j\\) with \\(\\langle w,\\delta_j\\rangle\\ne0\\); then \\(\\theta_{\\delta_i,\\delta_j}w=\\langle w,\\delta_j\\rangle\\delta_i\\in W\\) for all \\(i\\), so \\(W\\) is the whole space. Thus \\(P\\) is irreducible in the topological sense. But \\(s\\) and \\(s^*\\) map finitely supported vectors to finitely supported vectors, so \\(P\\delta_0\\) consists of finitely supported vectors and is not the whole space. So the \\(*\\)-algebra \\(P\\) is not algebraically irreducible, while its norm closure is, by Corollary 11.2.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 837,
        "through_line": 838,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 12.1",
      "kind": "definition",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Definition 12.1",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Definition 12.1.** Let \\(X\\) be a set of self-adjoint operators on \\(H\\). Then \\(X^{\\nearrow}\\) consists of all strong limits of increasing *sequences* in \\(X\\) that are bounded in norm, and \\(X^{\\searrow}\\) of all strong limits of such decreasing sequences. The sets \\(X^{\\uparrow}\\) and \\(X^{\\downarrow}\\) are defined in the same way with *nets* in place of sequences. By Vigier's theorem 1.3 such limits exist, and they are least upper, respectively greatest lower, bounds.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 843,
        "through_line": 844,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.2",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Lemma 12.2",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 12.2.** Let \\(X,Y\\) be sets of self-adjoint operators.\n\n1. \\(X\\subseteq X^{\\nearrow}\\subseteq X^{\\uparrow}\\) and \\(X\\subseteq X^{\\searrow}\\subseteq X^{\\downarrow}\\). If \\(X\\subseteq Y\\), then \\(X^{\\uparrow}\\subseteq Y^{\\uparrow}\\), and likewise for the other three operations.\n2. If \\(X\\) is convex, or closed under sums, or closed under multiplication by nonnegative numbers, then so are \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\).\n3. \\(-(X^{\\uparrow})=(-X)^{\\downarrow}\\) and \\(-(X^{\\nearrow})=(-X)^{\\searrow}\\). More generally, for the order-reversing map \\(t\\mapsto1-t\\): \\(1-X^{\\uparrow}=(1-X)^{\\downarrow}\\) and \\(1-X^{\\nearrow}=(1-X)^{\\searrow}\\), and with the roles of increasing and decreasing exchanged.\n4. Let \\(X\\subseteq B(H)_+\\cap S\\), and let \\(g:[0,1]\\to[0,1]\\) be continuous and operator monotone on \\([0,1]\\) (Definition 12.3) with \\(g(X)\\subseteq X\\). Then \\(g(Y)\\subseteq Y\\) for \\(Y\\) each of \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\).\n5. If \\(X\\subseteq M_+\\cap S\\) for a von Neumann algebra \\(M\\), then \\(X^{\\nearrow},X^{\\searrow},X^{\\uparrow},X^{\\downarrow}\\subseteq M_+\\cap S\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 845,
        "through_line": 854,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 12.3",
      "kind": "definition",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Definition 12.3",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Definition 12.3.** A real continuous function \\(g\\) on an interval \\(J\\) is *operator monotone on \\(J\\)* if \\(g(x)\\le g(y)\\) whenever \\(x\\le y\\) are self-adjoint operators with spectra in \\(J\\).\n\nEvery operator monotone function is increasing, as one sees on multiples of \\(1\\). The converse fails: \\(t\\mapsto t^2\\) is not operator monotone on \\([0,\\infty)\\), since \\(x=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix}\\le y=\\begin{pmatrix}2&1\\\\1&1\\end{pmatrix}\\), while \\(y^2-x^2=\\begin{pmatrix}4&3\\\\3&2\\end{pmatrix}\\) has determinant \\(-1\\) and so is not positive. By the [Löwner–Heinz inequality](c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md#oa-fnd-cf-17), \\(t\\mapsto t^\\alpha\\) is operator monotone on \\([0,\\infty)\\) for \\(0\\le\\alpha\\le1\\) (for \\(\\alpha=0\\) the function is constant). We need two simpler families.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 855,
        "through_line": 858,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.4",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Lemma 12.4",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 12.4** (resolvent-type operator monotone functions). Let \\(c>0\\).\n\n1. \\(g_c(t)=t/(c+t)\\) is operator monotone on \\([0,\\infty)\\). It takes values in \\([0,1)\\), \\(g_c(0)=0\\), and \\(g_c\\) increases as \\(c\\) decreases.\n2. \\(\\phi_c(t)=t/(1+c(1-t))\\) is operator monotone on \\([0,1]\\). It maps \\([0,1]\\) onto \\([0,1]\\), \\(\\phi_c(0)=0\\), \\(\\phi_c(1)=1\\), \\(\\phi_c\\) decreases as \\(c\\) increases, and \\(\\phi_c(t)\\to0\\) as \\(c\\to\\infty\\) for \\(t<1\\).\n3. For \\(\\alpha>0\\), the function \\(t/(\\alpha+(1-\\alpha)t)\\) is operator monotone on \\([0,1]\\), and for \\(0<\\alpha\\le1\\) also on \\([0,\\infty)\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 859,
        "through_line": 866,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.5",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Lemma 12.5",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 12.5** (a projection on countably many vectors). Let \\(A\\) be a concrete C\\(^*\\)-algebra on \\(H\\) with weak closure \\(M\\), let \\(p\\in M\\) be a projection, and let \\(\\xi_1,\\xi_2,\\dots\\in H\\). There are \\(y\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\) and a projection \\(q\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\) such that, for all \\(k\\),\n\\[\n\\begin{gathered}\ny(1-p)\\xi_k\\\\\n=0,\\\\\nyp\\xi_k\\\\\n=p\\xi_k,\\\\\nq(1-p)\\xi_k\\\\\n=0,\\\\\nqp\\xi_k\\\\\n=p\\xi_k .\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 867,
        "through_line": 930,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 12.6",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Theorem 12.6",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 12.6** (the up-down theorem). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and suppose that \\(M=A''\\) is \\(\\sigma\\)-finite. Then\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\\\\n\\text{and}\\\\\nM_h\\\\\n=\\big((A_h)^{\\nearrow}\\big)^{\\searrow}.\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 931,
        "through_line": 974,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.7",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Lemma 12.7",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 12.7** (adding a unit). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and \\(\\tilde A=A+\\mathbb C1\\).\n\n1. For \\(\\varepsilon>0\\) and \\(x\\in(\\tilde A_+\\cap S)^{\\nearrow}\\), \\((1+\\varepsilon)^{-1}(x+\\varepsilon)\\in(A_+\\cap S)^{\\uparrow}\\).\n2. \\(\\big((\\tilde A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\subseteq\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 975,
        "through_line": 1028,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.8",
      "kind": "lemma",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Lemma 12.8",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Lemma 12.8** (suprema of projections). Let \\(Y\\subseteq B(H)_+\\cap S\\) be convex, with \\(0\\in Y\\) and \\(g_\\beta(Y)\\subseteq Y\\) for all \\(\\beta\\in(0,1]\\). If \\((p_i)_{i\\in I}\\) is any family of projections in \\(Y^{\\uparrow}\\), then \\(\\bigvee_ip_i\\in Y^{\\uparrow}\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1029,
        "through_line": 1046,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 12.9",
      "kind": "theorem",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Theorem 12.9",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Theorem 12.9** (the up-down-up theorem). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\), and \\(M=A''\\). Then\n\\[\n\\begin{gathered}\nM_+\\cap S\\\\\n=\\Big(\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\\\\n\\text{and}\\\\\nM_h\\\\\n=\\Big(\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}.\n\\end{gathered}\n\\]\nIf \\(1\\in A\\), then moreover \\(M_+\\cap S=\\Big(\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\downarrow}\\Big)^{\\uparrow}\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1047,
        "through_line": 1080,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 12.10",
      "kind": "corollary",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Corollary 12.10",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Corollary 12.10** (the monotone closure criterion). Let \\(A\\) be a concrete C\\(^*\\)-algebra acting nondegenerately on \\(H\\).\n\n1. \\(A\\) is a von Neumann algebra if and only if the strong limit of every norm-bounded increasing net in \\(A_h\\) lies in \\(A\\).\n2. If \\(A''\\) is \\(\\sigma\\)-finite, for instance if \\(H\\) is separable, then \\(A\\) is a von Neumann algebra if and only if the strong limit of every norm-bounded increasing sequence in \\(A_h\\) lies in \\(A\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1081,
        "through_line": 1087,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 12.11",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Example 12.11",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 12.11** (sequences are not enough without \\(\\sigma\\)-finiteness). Let \\(H=\\ell^2([0,1])\\), for counting measure on \\([0,1]\\), with basis \\((\\delta_t)\\), and let \\(A\\) be the set of multiplication operators \\(m_g\\) by bounded Borel functions \\(g\\) on \\([0,1]\\). Since \\(\\langle m_g\\delta_t,\\delta_t\\rangle=g(t)\\), \\(\\|m_g\\|=\\sup|g|\\), and \\(g\\) is determined by \\(m_g\\). Uniform limits of Borel functions are Borel, so \\(A\\) is a C\\(^*\\)-algebra; it contains \\(1\\). If \\(g_k\\) is a bounded increasing sequence of real Borel functions with pointwise limit \\(g\\), then \\(g\\) is Borel, and \\(m_{g_k}\\to m_g\\) strongly by dominated convergence (for counting measure). So \\(A\\) contains the strong limits of its bounded increasing sequences of self-adjoint elements. But \\(A\\) is not a von Neumann algebra. There are subsets \\(T\\subseteq[0,1]\\) that are not Borel: [Corollary 2.8 and Lemma 1.3 of the Polish-space lesson](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-12) prove that this uncountable Polish space has \\(2^{\\aleph_0}\\) points but at most that many Borel sets, whereas Cantor’s diagonal theorem gives strictly more subsets. The cardinal identities and diagonal theorem have full proofs in [Theorem 8.2 and Proposition 8.3 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08). The operators \\(m_{1_{T\\cap F}}\\), for finite \\(F\\subseteq[0,1]\\), lie in \\(A\\) and increase strongly to \\(m_{1_T}\\), which lies in the weak closure \\(A''\\) but not in \\(A\\). Here \\(A''\\) is not \\(\\sigma\\)-finite: it contains the uncountably many orthogonal projections \\(m_{1_{\\{t\\}}}\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1088,
        "through_line": 1089,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 12.12",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-14::Example 12.12",
      "anchor": "oa-fnd-kd-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 12.12** (nondegeneracy is needed). Let \\(H\\ne\\{0\\}\\) and \\(A=\\{0\\}\\). Then \\(A'=B(H)\\) and \\(A''=\\mathbb C1\\), but every monotone limit of elements of \\(A\\) is \\(0\\). So \\(1\\in A''_+\\cap S\\) is not in any of the sets built from \\(A\\) in Theorems 12.6 and 12.9. The same happens for every \\(A\\) with \\([AH]\\ne H\\): all monotone limits \\(x\\) of elements of \\(A\\) satisfy \\(x=exe\\) for the projection \\(e\\) onto \\([AH]\\), while \\(1\\in A''\\). Restricting to \\([AH]\\) shows that both theorems hold for every concrete C\\(^*\\)-algebra if \\(A''\\) is replaced by the weak closure of \\(A\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1090,
        "through_line": 1091,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 13.1",
      "kind": "example",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-16::Example 13.1",
      "anchor": "oa-fnd-kd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Example 13.1.** Let \\(H_d=\\ell^2([0,1])\\) for counting measure, with basis \\((\\delta_t)_{t\\in[0,1]}\\), and \\(H_c=L^2[0,1]\\) for Lebesgue measure. For \\(g\\in C[0,1]\\) let \\(m_g\\) denote multiplication by \\(g\\) on either space, and let\n\\[\n\\begin{gathered}\nA\\\\\n=\\{m_g\\oplus m_g:g\\in C[0,1]\\}\\\\\n\\subseteq B(H_d\\oplus H_c),\\\\\nM\\\\\n=A'',\\\\\nz\\\\\n=1\\oplus0 .\n\\end{gathered}\n\\]\nSince \\(\\|m_g\\oplus m_g\\|=\\max|g|\\), \\(A\\) is a unital concrete C\\(^*\\)-algebra. We show:\n\n(a) Every element of \\((A_h)^{\\uparrow}\\) is \\(m_f\\oplus m_f\\) for a unique bounded lower semicontinuous \\(f\\) on \\([0,1]\\), namely the pointwise supremum of the net. Conversely every bounded lower semicontinuous \\(f\\) arises.\n\n(b) \\(z\\in M\\).\n\n(c) If \\(a\\in(A_h)^{\\uparrow}\\) and \\(a\\ge z\\), then \\(a\\ge1\\). Consequently \\(z\\notin\\big((A_h)^{\\uparrow}\\big)^{\\downarrow}\\), and in particular \\(z\\notin\\big((A_+\\cap S)^{\\uparrow}\\big)^{\\downarrow}\\).\n\n(d) \\(z\\in\\big((A_+\\cap S)^{\\searrow}\\big)^{\\uparrow}\\), and \\(M\\) is not \\(\\sigma\\)-finite.\n\nSo \\(z\\in M_+\\cap S\\) is reached by three monotone limits, as Theorem 12.9 asserts, but not by the first two.\n\n*Proof of (a).* Let \\((g_i)\\) be an increasing net of real continuous functions with \\(|g_i|\\le C\\), and \\(f=\\sup_ig_i\\) pointwise. Then \\(f\\) is bounded and lower semicontinuous. For each rational \\(r\\), the open set \\(\\{f>r\\}\\) is the union of the open sets \\(\\{g_i>r\\}\\). For every rational-interval basic open set contained in some \\(\\{g_i>r\\}\\), choose one such index. There are countably many choices, and they cover \\(\\{f>r\\}\\), since each point has such a basic neighborhood inside one member of the cover. Collecting these indices over all rational \\(r\\) gives a sequence of indices, and since the index set is directed we can choose an increasing sequence \\(i_1\\le i_2\\le\\cdots\\) that eventually lies above each of them. Then \\(g_{i_n}\\to f\\) pointwise: if \\(r<f(t)\\) is rational, some chosen index \\(j\\) has \\(g_j(t)>r\\), and \\(g_{i_n}(t)\\ge g_j(t)\\) for large \\(n\\). For \\(\\xi\\) in \\(H_d\\) or \\(H_c\\) and \\(i\\ge i_n\\),\n\\[\n\\begin{gathered}\n\\|(m_f-m_{g_i})\\xi\\|^2\\\\\n=\\int(f-g_i)^2|\\xi|^2\\\\\n\\le2C\\int(f-g_{i_n})|\\xi|^2 ,\n\\end{gathered}\n\\]\nthe integral taken for counting measure or Lebesgue measure. The right side tends to \\(0\\) as \\(n\\to\\infty\\), by dominated convergence. So \\(m_{g_i}\\oplus m_{g_i}\\to m_f\\oplus m_f\\) strongly. Uniqueness holds because \\(f(t)=\\langle a\\delta_t,\\delta_t\\rangle\\). Conversely, let \\(f\\) be bounded and lower semicontinuous, with \\(|f|\\le C\\). The continuous functions \\(g\\) with \\(-C\\le g\\le f\\) form an upward directed family, since \\(\\max(g_1,g_2)\\) is again one, and it is bounded in norm. Its pointwise supremum is \\(f\\): put \\(g_n(t)=\\inf_{s\\in[0,1]}(f(s)+n|s-t|)\\). The triangle inequality gives \\(|g_n(t)-g_n(t')|\\leq n|t-t'|\\), so \\(g_n\\) is continuous. Also \\(-C\\leq g_n\\leq f\\leq C\\), and \\(g_n\\) increases in \\(n\\). Given \\(t\\) and \\(\\varepsilon>0\\), lower semicontinuity gives \\(\\rho>0\\) with \\(f(s)>f(t)-\\varepsilon\\) for \\(|s-t|<\\rho\\). Outside that neighborhood, \\(f(s)+n|s-t|\\geq-C+n\\rho\\geq f(t)-\\varepsilon\\) once \\(n\\) is large. Hence \\(g_n(t)\\geq f(t)-\\varepsilon\\), proving \\(g_n(t)\\uparrow f(t)\\). Indexed by itself, this family is an increasing net, and the corresponding net in \\(A_h\\) converges strongly to \\(m_f\\oplus m_f\\).\n\n*Proof of (b).* It suffices to show that every \\(T\\in A'\\) commutes with \\(z\\), that is, has no off-diagonal parts. Let \\(Q\\) be the part of \\(T\\) that maps \\(H_d\\) to \\(H_c\\). Compressing \\(T(m_g\\oplus m_g)=(m_g\\oplus m_g)T\\) gives \\(Qm_g=m_gQ\\) for all \\(g\\). For \\(t\\in[0,1]\\), \\(m_g\\delta_t=g(t)\\delta_t\\), so the function \\(\\eta_t=Q\\delta_t\\in L^2[0,1]\\) satisfies \\((g-g(t))\\eta_t=0\\) almost everywhere. With \\(g(s)=s\\) this says \\((s-t)\\eta_t(s)=0\\) for almost every \\(s\\), so \\(\\eta_t=0\\) almost everywhere. Hence \\(Q=0\\) on the dense span of the \\(\\delta_t\\), and \\(Q=0\\). The part of \\(T\\) from \\(H_c\\) to \\(H_d\\) is the adjoint of the corresponding part \\(Q'\\) of \\(T^*\\in A'\\), and \\(Q'=0\\) by the same argument. So \\(z\\in A''=M\\).\n\n*Proof of (c).* By (a), \\(a=m_f\\oplus m_f\\) with \\(f\\) bounded. From \\(a\\ge z\\), \\(f(t)=\\langle a\\delta_t,\\delta_t\\rangle\\ge\\langle z\\delta_t,\\delta_t\\rangle=1\\) for every \\(t\\). So \\(f\\ge1\\) everywhere, and \\(a\\ge1\\) on both summands. If \\(z\\) were the limit of a decreasing net \\((a_j)\\) in \\((A_h)^{\\uparrow}\\), then \\(a_j\\ge z\\), so \\(a_j\\ge1\\) for all \\(j\\), and \\(z\\ge1\\), which is false because \\(H_c\\ne\\{0\\}\\).\n\n*Proof of (d).* For \\(t\\in[0,1]\\) let \\(h_{t,n}(s)=\\max(0,1-n|s-t|)\\). As \\(n\\to\\infty\\), \\(h_{t,n}\\) decreases to \\(1_{\\{t\\}}\\). By dominated convergence, \\(m_{h_{t,n}}\\to m_{1_{\\{t\\}}}\\) strongly on \\(H_d\\) and \\(m_{h_{t,n}}\\to0\\) strongly on \\(H_c\\), since \\(\\{t\\}\\) is a null set. For a finite set \\(F\\subseteq[0,1]\\), the sums \\(\\sum_{t\\in F}h_{t,n}\\) have disjoint supports and values in \\([0,1]\\) once \\(n\\) is large, and they decrease to \\(1_F\\). So \\(z_F=m_{1_F}\\oplus0\\in(A_+\\cap S)^{\\searrow}\\). As \\(F\\) increases, \\(z_F\\) increases strongly to \\(z\\). So \\(z\\in\\big((A_+\\cap S)^{\\searrow}\\big)^{\\uparrow}\\). The projections \\(z_{\\{t\\}}\\), \\(t\\in[0,1]\\), lie in \\(M\\) and are mutually orthogonal and nonzero, so \\(M\\) is not \\(\\sigma\\)-finite. \\(\\square\\)",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1096,
        "through_line": 1135,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.1",
      "kind": "exercise",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-18::Exercise 14.1",
      "anchor": "oa-fnd-kd-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Exercise 14.1** (hard; a second proof of the density theorem). For \\(x\\in B(H)\\) put \\(F(x)=2x(1+x^*x)^{-1}\\) and, for \\(y\\in S\\), \\(G(y)=y\\big(1+(1-y^*y)^{1/2}\\big)^{-1}\\).\n\n(a) Show that \\(\\|F(x)\\|\\le1\\), \\(F(x)^*=F(x^*)\\), \\(G(F(x))=x\\) for \\(x\\in S\\), and \\(F(G(y))=y\\) for \\(y\\in S\\).\n\n(b) Show that \\(F\\) is continuous from \\(B(H)\\) to \\(B(H)\\) for the strong\\(^*\\) topology, along arbitrary nets.\n\n(c) Deduce Theorem 7.1(1) for a unital concrete C\\(^*\\)-algebra \\(A\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1138,
        "through_line": 1192,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.2",
      "kind": "exercise",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-18::Exercise 14.2",
      "anchor": "oa-fnd-kd-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Exercise 14.2** (medium; continuous functions on \\(L^2[0,1]\\)). Let \\(\\mu\\) be Lebesgue measure on \\([0,1]\\), that is, the Radon measure that represents the Riemann integral, let \\(A=\\{m_g:g\\in C[0,1]\\}\\) act on \\(L^2[0,1]\\) by multiplication, and let \\(L=\\{m_f:f\\in L^\\infty[0,1]\\}\\).\n\n(a) For open \\(U\\subseteq[0,1]\\) show \\(m_{1_U}\\in(A_+\\cap S)^{\\nearrow}\\), and for closed \\(F\\) show \\(m_{1_F}\\in(A_+\\cap S)^{\\searrow}\\).\n\n(b) For every Borel set \\(E\\) show \\(m_{1_E}\\in\\big((A_+\\cap S)^{\\nearrow}\\big)^{\\searrow}\\).\n\n(c) Deduce that \\(A''=L\\), and compare with Theorem 12.6.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1193,
        "through_line": 1206,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.3",
      "kind": "exercise",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-18::Exercise 14.3",
      "anchor": "oa-fnd-kd-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Exercise 14.3** (easy; closedness of some classes of operators). Show that the self-adjoint operators and the positive operators form strongly closed sets, and that the normal operators form a strongly\\(^*\\) closed set. Show that if \\(\\dim H=\\infty\\), the normal operators and the unitaries do not form strongly closed sets.",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1207,
        "through_line": 1210,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.4",
      "kind": "exercise",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-18::Exercise 14.4",
      "anchor": "oa-fnd-kd-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Exercise 14.4** (medium; finite-rank perturbations of the identity). Let \\(\\dim H=\\infty\\) and \\(A=K(H)+\\mathbb C1\\), the compact operators with the identity adjoined.\n\n(a) Show directly that every unitary \\(u\\in B(H)\\) is the strong limit of unitaries \\(u_F\\) such that \\(u_F-1\\) has finite rank.\n\n(b) Show that if \\(\\|u-1\\|\\le\\lambda\\), then \\(u\\) is a strong\\(^*\\) limit of unitaries \\(v\\in A\\) with \\(\\|v-1\\|\\le\\lambda\\).\n\n(c) Show that the construction of (a) does not control \\(\\|u_F-1\\|\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1211,
        "through_line": 1224,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.5",
      "kind": "exercise",
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences#oa-fnd-kd-18::Exercise 14.5",
      "anchor": "oa-fnd-kd-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/kaplansky-s-density-theorem-and-its-consequences@7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
      "statement_and_full_conditions": "**Exercise 14.5** (medium; Lusin's theorem with a norm bound). With the notation of Exercise 14.2, let \\(f\\) be a bounded real Borel function on \\([0,1]\\), and \\(\\varepsilon,\\delta>0\\). Show that there are a Borel set \\(E\\) with \\(\\mu([0,1]\\setminus E)<\\varepsilon\\) and a real \\(g\\in C[0,1]\\) with \\(g=f\\) almost everywhere on \\(E\\) and \\(\\max|g|\\le(1+\\delta)\\operatorname{ess\\,sup}_E|f|\\).",
      "proof_locus": {
        "source": "src/kaplansky-s-density-theorem-and-its-consequences.md",
        "line": 1225,
        "through_line": 1228,
        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.1",
      "kind": "proposition",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-01::Proposition 2.1",
      "anchor": "oa-fnd-cm-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Proposition 2.1** (The C\\(^*\\)-algebra \\(M_n(A)\\)).\n\n1. \\(\\pi^{(n)}\\) is a \\(*\\)-homomorphism of \\(M_n(A)\\) into \\(B(H^n)\\), with entries \\(R_i^*\\pi^{(n)}(a)R_j=\\pi(a_{ij})\\), and\n\\[\n\\begin{gathered}\n\\max_{i,j}\\|\\pi(a_{ij})\\| \\\\\n\\le\\|\\pi^{(n)}(a)\\| \\\\\n\\le\\sum_{i,j}\\|\\pi(a_{ij})\\|.\n\\end{gathered}\n\\tag{2.2}\n\\]\nIt is injective if and only if \\(\\pi\\) is. Its range is the set of operators on \\(H^n\\) whose matrix entries all lie in \\(\\pi(A)\\).\n2. \\(M_n(A)\\) has exactly one norm in which it is a C\\(^*\\)-algebra. For every faithful \\(\\pi\\) it is \\(\\|a\\|=\\|\\pi^{(n)}(a)\\|\\), and \\(\\max_{i,j}\\|a_{ij}\\|\\le\\|a\\|\\le\\sum_{i,j}\\|a_{ij}\\|\\).\n3. For every representation \\(\\pi\\), \\(\\pi^{(n)}\\) is a representation of the C\\(^*\\)-algebra \\(M_n(A)\\). For a \\(*\\)-homomorphism \\(\\sigma:A\\to B\\), the entrywise map \\(\\sigma^{(n)}:M_n(A)\\to M_n(B)\\) is a \\(*\\)-homomorphism, injective when \\(\\sigma\\) is.\n4. If \\(B\\subseteq A\\) is a C\\(^*\\)-subalgebra, then \\(M_n(B)\\) is a C\\(^*\\)-subalgebra of \\(M_n(A)\\), and \\(M_n(B)_+=M_n(B)\\cap M_n(A)_+\\).\n5. \\(X\\mapsto\\sum_{i,j}R_iX_{ij}R_j^*\\) is a \\(*\\)-isomorphism of \\(M_n(B(H))\\) onto \\(B(H^n)\\). So \\(X\\in M_n(B(H))\\) is positive exactly when \\(\\sum_{i,j}\\langle X_{ij}\\zeta_j,\\zeta_i\\rangle\\ge0\\) for every \\(\\zeta\\in H^n\\).\n6. Let \\(\\Omega\\) be a locally compact Hausdorff space. Identify \\(F\\in M_n(C_0(\\Omega))\\) with the function \\(\\omega\\mapsto F(\\omega)=[F_{ij}(\\omega)]\\in M_n(\\mathbb C)\\). Then \\(\\|F\\|=\\sup_\\omega\\|F(\\omega)\\|\\), and \\(F\\ge0\\) if and only if \\(F(\\omega)\\ge0\\) for every \\(\\omega\\).\n7. (*Scalar compressions.*) For \\(a\\in M_n(A)\\) and a scalar \\(n\\times m\\) matrix \\(\\alpha\\), let \\(\\alpha^*a\\alpha\\in M_m(A)\\) have entries \\((\\alpha^*a\\alpha)_{pq}=\\sum_{i,j}\\overline{\\alpha_{ip}}\\,a_{ij}\\,\\alpha_{jq}\\). If \\(a\\ge0\\), then \\(\\alpha^*a\\alpha\\ge0\\). In particular, square corners of positive matrices are positive, so are the diagonal entries, so is \\(a\\oplus0\\), and so is every relabelling \\([a_{\\tau(i)\\tau(j)}]\\) by a permutation \\(\\tau\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 65,
        "through_line": 103,
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.2",
      "kind": "lemma",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-01::Lemma 2.2",
      "anchor": "oa-fnd-cm-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Lemma 2.2** (The positive cone of \\(M_n(A)\\)).\n\n1. For \\(a_1,\\dots,a_n\\in A\\), the matrix \\([a_i^*a_j]\\) is positive. Each positive element of \\(M_n(A)\\) can be written as the sum of \\(n\\) such matrices.\n2. \\(a\\in M_n(A)\\) is positive exactly when \\(\\sum_{i,j}x_i^*a_{ij}x_j\\ge0\\) in \\(A\\) for all \\(x_1,\\dots,x_n\\in A\\).\n3. Let \\(\\pi\\) be a representation of \\(A\\) on \\(H\\). If \\(a\\ge0\\), then \\(\\sum_{i,j}\\langle\\pi(a_{ij})\\zeta_j,\\zeta_i\\rangle\\ge0\\) for all \\(\\zeta\\in H^n\\). If \\(\\pi\\) is faithful, the converse holds.\n4. (*Two-by-two test.*) Let \\(x,y\\in B(H)\\) and \\(t>0\\). The operator \\(\\begin{bmatrix}t1&x\\\\x^*&y\\end{bmatrix}\\) on \\(H\\oplus H\\) is positive if and only if \\(y\\ge t^{-1}x^*x\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 104,
        "through_line": 130,
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.1",
      "kind": "definition",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-03::Definition 3.1",
      "anchor": "oa-fnd-cm-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Definition 3.1.** Let \\(A,B\\) be C\\(^*\\)-algebras and \\(\\varphi:A\\to B\\) linear. Its \\(n\\)-th amplification \\(\\varphi^{(n)}:M_n(A)\\to M_n(B)\\) applies \\(\\varphi\\) to every entry. The map \\(\\varphi\\) is *\\(n\\)-positive* if \\(\\varphi^{(n)}\\) maps positive elements to positive elements, and *completely positive* (CP) if it is \\(n\\)-positive for every \\(n\\). *Positive* means 1-positive.",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 133,
        "through_line": 134,
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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      "source_uses": [
        {
          "unit": "completely-positive-maps",
          "local_scopes": [
            {
              "heading": "3. Completely positive maps",
              "line": 131,
              "through_line": 184,
              "anchors": [
                "OA-FND-CM-03"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
              "through_line": 249,
              "anchors": [
                "OA-FND-CM-07"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "5. When positivity implies complete positivity",
              "line": 250,
              "through_line": 336,
              "anchors": [
                "OA-FND-CM-04",
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
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    {
      "local_label": "Proposition 3.2",
      "kind": "proposition",
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      "anchor": "oa-fnd-cm-03",
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      "statement_and_full_conditions": "**Proposition 3.2.**\n\n1. (*A criterion.*) \\(\\varphi\\) is \\(n\\)-positive exactly when\n\\[\n\\begin{gathered}\n\\sum_{i,j=1}^ny_i^*\\,\\varphi(x_i^*x_j)\\,y_j\\ge0, \\\\\n\\text{for all }x_1,\\dots,x_n\\in A, \\\\\ny_1,\\dots,y_n\\in B.\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(B\\subseteq B(H)\\), this holds if and only if \\(\\sum_{i,j}\\langle\\varphi(x_i^*x_j)\\zeta_j,\\zeta_i\\rangle\\ge0\\) for all \\(x\\in A^n\\) and \\(\\zeta\\in H^n\\).\n2. An \\(n\\)-positive map is \\(k\\)-positive for every \\(k\\le n\\). A positive map is hermitian: \\(\\varphi(x^*)=\\varphi(x)^*\\). If \\(\\psi:B\\to C\\) is \\(n\\)-positive too, so is \\(\\psi\\circ\\varphi\\), because \\((\\psi\\circ\\varphi)^{(n)}=\\psi^{(n)}\\circ\\varphi^{(n)}\\).\n3. (*Examples.*) Every \\(*\\)-homomorphism is CP. For \\(V\\in B(H,K)\\), the map \\(y\\mapsto V^*yV\\) from \\(B(K)\\) to \\(B(H)\\) is CP. The transpose \\(t(x)=x^{\\mathsf T}\\) on \\(M_d(\\mathbb C)\\), \\(d\\ge2\\), is positive but not 2-positive.\n4. (*Automatic boundedness.*) A positive map \\(\\varphi:A\\to B\\) is bounded, and\n\\[\n\\|\\varphi\\|\\le4\\sup\\{\\|\\varphi(a)\\|:\\ a\\in A_+,\\ \\|a\\|\\le1\\}.\n\\]",
      "proof_locus": {
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
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    {
      "local_label": "Exercise 3.3",
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      "statement_and_full_conditions": "**Exercise 3.3** (medium; Choi's criterion). Let \\(\\varphi:M_k(\\mathbb C)\\to B\\) be linear. Show that the following are equivalent: (a) \\(\\varphi\\) is CP; (b) \\(\\varphi\\) is \\(k\\)-positive; (c) the matrix \\(C_\\varphi=[\\varphi(e_{ij})]_{i,j=1}^k\\in M_k(B)\\) is positive. Compute \\(C_t\\) for the transpose.",
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          "source_locus": "§10; PDF 54–61",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
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        }
      ],
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      "global_tag_allocated": false
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    {
      "local_label": "Theorem 4.1",
      "kind": "theorem",
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      "statement_and_full_conditions": "**Theorem 4.1.** Let \\(\\varphi:A\\to B\\) be 2-positive.\n\n1. (*The Kadison–Schwarz inequality.*) \\(\\varphi(a)^*\\varphi(a)\\le\\|\\varphi\\|\\,\\varphi(a^*a)\\) for all \\(a\\in A\\).\n2. \\[\n\\begin{gathered}\n\\|\\varphi\\|\\\\\n=\\sup\\{\\|\\varphi(b)\\|:\\ b\\in A_+,\\ \\|b\\|\\le1\\}\\\\\n=\\lim_i\\|\\varphi(u_i)\\|\n\\end{gathered}\n\\] for every approximate identity \\((u_i)\\). If \\(A\\) has a unit, \\(\\|\\varphi\\|=\\|\\varphi(1)\\|\\).\n3. (*Multiplicative domain.*) Let \\(\\|\\varphi\\|\\le1\\). If \\(\\varphi(a^*a)=\\varphi(a)^*\\varphi(a)\\), then \\(\\varphi(xa)=\\varphi(x)\\varphi(a)\\) for all \\(x\\in A\\). If \\(\\varphi(aa^*)=\\varphi(a)\\varphi(a)^*\\), then \\(\\varphi(ax)=\\varphi(a)\\varphi(x)\\) for all \\(x\\in A\\).\n4. Positivity alone does not give (1). For the transpose \\(t\\) on \\(M_2(\\mathbb C)\\), \\(\\|t\\|=1\\), but \\(t(e_{12})^*t(e_{12})=e_{11}\\) is not below \\(t(e_{12}^*e_{12})=e_{22}\\).",
      "proof_locus": {
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
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          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
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    {
      "local_label": "Exercise 4.2",
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      "statement_and_full_conditions": "**Exercise 4.2** (easy; Amplification detects multiplicativity). Call a linear map \\(\\pi:A\\to B\\) between C\\(^*\\)-algebras a *Jordan \\(*\\)-homomorphism* if \\(\\pi(x^*)=\\pi(x)^*\\) for all \\(x\\) and \\(\\pi(h^2)=\\pi(h)^2\\) for self-adjoint \\(h\\). Let \\(\\pi:A\\to B\\) be linear and \\(n\\ge2\\), and suppose that \\(\\pi^{(n)}=\\pi\\otimes\\mathrm{id}_{M_n}:M_n(A)\\to M_n(B)\\) is a Jordan \\(*\\)-homomorphism. Show that \\(\\pi\\) is a \\(*\\)-homomorphism; in particular, if \\(\\pi\\) is bijective, it is a \\(*\\)-isomorphism. Show that the statement fails for \\(n=1\\).",
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          "role": "proof comparison",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4.3",
      "kind": "exercise",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-07::Exercise 4.3",
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      "statement_and_full_conditions": "**Exercise 4.3** (medium; Jordan maps that are 2-positive). Let \\(\\pi:A\\to B\\) be a Jordan \\(*\\)-homomorphism (Exercise 4.2). Show that \\(\\pi\\) is positive and contractive on self-adjoint elements, and that \\(\\pi\\) is a \\(*\\)-homomorphism exactly when it is 2-positive.",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
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        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
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      "anchor": "oa-fnd-cm-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Theorem 5.1.**\n\n1. If \\(B\\) is abelian, every positive linear map \\(\\varphi:A\\to B\\) is completely positive. In particular, every positive linear functional is completely positive.\n2. A \\(k\\)-positive map \\(\\psi:A\\to M_k(\\mathbb C)\\) is completely positive.\n3. Let \\(n\\ge1\\), and let \\(B\\) have a separating family of representations \\(\\sigma_\\alpha\\) on spaces of dimension at most \\(n\\) (for every \\(b\\ne0\\) some \\(\\sigma_\\alpha(b)\\ne0\\)). Then every \\(n\\)-positive map \\(\\varphi:A\\to B\\) is completely positive. Since the irreducible representations of \\(B\\) separate its points (B9), this applies whenever every irreducible representation of \\(B\\) has dimension at most \\(n\\).\n\n*Further reading:* [Blackadar, II.6.9.10]. The proof below needs only a separating family of representations of dimension at most \\(n\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 256,
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
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    {
      "local_label": "Lemma 5.2",
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      "statement_and_full_conditions": "**Lemma 5.2** (Partitions of unity). Let \\(C\\subseteq\\Omega\\) be compact and \\(U_1,\\dots,U_L\\) open sets covering \\(C\\). There are \\(g_1,\\dots,g_L\\in C_c(\\Omega)\\) with \\(g_l\\ge0\\), \\(g_l=0\\) off \\(U_l\\), \\(\\sum_lg_l\\le1\\) on \\(\\Omega\\), and \\(\\sum_lg_l=1\\) on \\(C\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 288,
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
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              "heading": "4. The Kadison–Schwarz inequality",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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            {
              "heading": "5. When positivity implies complete positivity",
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              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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            },
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              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
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          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.3",
      "kind": "lemma",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-08::Lemma 5.3",
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      "statement_and_full_conditions": "**Lemma 5.3** (Approximation). Let \\(F:\\Omega\\to M_N(\\mathbb C)\\) be continuous and vanish at infinity, and let \\(\\varepsilon>0\\). There are points \\(\\omega_1,\\dots,\\omega_L\\) and functions \\(g_1,\\dots,g_L\\) as in Lemma 5.2 with\n\\[\n\\sup_{\\omega\\in\\Omega}\\Bigl\\|F(\\omega)-\\sum_lg_l(\\omega)F(\\omega_l)\\Bigr\\|\\le2\\varepsilon .\n\\]",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 292,
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              "heading": "5. When positivity implies complete positivity",
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
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              "heading": "5. When positivity implies complete positivity",
              "line": 250,
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
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                "OA-FND-CM-09",
                "OA-FND-CM-10",
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          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.4",
      "kind": "theorem",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-08::Theorem 5.4",
      "anchor": "oa-fnd-cm-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Theorem 5.4.**\n\n1. Let \\(k\\ge1\\). Every \\(k\\)-positive map \\(\\varphi\\) from \\(A=M_k(C_0(\\Omega))=C_0(\\Omega,M_k)\\) into a C\\(^*\\)-algebra \\(B\\) is completely positive.\n2. In particular, every positive map from an abelian C\\(^*\\)-algebra into a C\\(^*\\)-algebra is completely positive (\\(k=1\\)), and every \\(k\\)-positive map from \\(M_k(\\mathbb C)\\) is completely positive (\\(\\Omega\\) a point).\n\nThe proof below uses finite partitions of unity and scalar matrix compressions; it makes no point-dependent choice outside exceptional null sets.",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 307,
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "4. The Kadison–Schwarz inequality",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "5. When positivity implies complete positivity",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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                "OA-FND-CM-05",
                "OA-FND-CM-06",
                "OA-FND-CM-12"
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            },
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              "heading": "7. Completely positive maps and dual spaces",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.5",
      "kind": "corollary",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-08::Corollary 5.5",
      "anchor": "oa-fnd-cm-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Corollary 5.5** (Kadison's inequality). If \\(\\varphi:A\\to B\\) is positive and \\(a\\in A\\) is normal, then \\(\\varphi(a)^*\\varphi(a)\\le\\|\\varphi\\|\\,\\varphi(a^*a)\\). In particular \\(\\varphi(h)^2\\le\\|\\varphi\\|\\,\\varphi(h^2)\\) for \\(h\\in A_h\\).",
      "proof_locus": {
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        "line": 331,
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "4. The Kadison–Schwarz inequality",
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              "heading": "5. When positivity implies complete positivity",
              "line": 250,
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                "OA-FND-CM-04",
                "OA-FND-CM-08"
              ],
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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                "OA-FND-CM-12"
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
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              "heading": "6. Stinespring's dilation theorem",
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              "line": 566,
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                "OA-FND-CM-10",
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          ],
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.6",
      "kind": "example",
      "unit": "completely-positive-maps",
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      "anchor": "oa-fnd-cm-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Example 5.6** (How far positivity goes). The transpose \\(t\\) on \\(M_2(\\mathbb C)\\) is positive, isometric and unital. It is not 2-positive (Proposition 3.2(3)), and it breaks the Kadison–Schwarz inequality (Theorem 4.1(4)). It still satisfies Kadison's inequality for normal elements (Corollary 5.5), for instance \\(t(h)^2=(h^2)^{\\mathsf T}=t(h^2)\\) for self-adjoint \\(h\\). Its restriction to any abelian C\\(^*\\)-subalgebra, such as the diagonal matrices, is CP, by Theorem 5.4(2). And its composition with the identification \\(\\Phi_1\\) of Proposition 7.5 is \\(\\Phi_2\\), which is not 2-positive.",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 335,
        "through_line": 336,
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              "line": 250,
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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              "heading": "7. Completely positive maps and dual spaces",
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          "source_locus": "§10; PDF 54–61",
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          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "II.6.9; PDF 141–145",
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          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
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    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
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      "statement_and_full_conditions": "**Theorem 6.1** (Stinespring). Fix a Hilbert space \\(H\\) and a C\\(^*\\)-algebra \\(A\\).\n\n1. If \\(\\pi\\) is a representation of \\(A\\) on \\(K\\) and \\(V\\in B(H,K)\\), then \\(\\varphi(a)=V^*\\pi(a)V\\) is completely positive, and \\(\\|\\varphi\\|\\le\\|V\\|^2\\).\n2. Let \\(\\varphi:A\\to B(H)\\) be completely positive, and put \\(N=\\varphi(A)'\\). This is a von Neumann algebra, because \\(\\varphi(A)\\) is self-adjoint: positive maps are hermitian (Proposition 3.2(2)). There are a Hilbert space \\(K\\), a representation \\(\\pi\\) of \\(A\\) on \\(K\\), an operator \\(V\\in B(H,K)\\) and a unital normal representation \\(\\rho\\) of \\(N\\) on \\(K\\) such that\n\\[\n\\begin{gathered}\n\\varphi(a)=V^*\\pi(a)V, \\\\\nK=\\overline{\\operatorname{span}}\\,\\pi(A)VH, \\\\\n\\rho(N)\\subseteq\\pi(A)', \\\\\n\\rho(x)V=Vx, \\\\\na\\in A,\\quad x\\in N.\n\\end{gathered}\n\\tag{6.1}\n\\]\n3. Let \\((u_i)\\) be any approximate identity of \\(A\\). The increasing net \\((\\varphi(u_i))\\) converges weakly to \\(a_0:=V^*V\\), which is its least upper bound, and\n\\[\n\\begin{gathered}\n\\|\\varphi\\|=\\|V\\|^2=\\|a_0\\| \\\\\n=\\lim_i\\|\\varphi(u_i)\\|.\n\\end{gathered}\n\\tag{6.2}\n\\]\nIf \\(A\\) has a unit, then \\(a_0=\\varphi(1)\\) and \\(\\|\\varphi\\|=\\|\\varphi(1)\\|\\).\n\n*Further reading:* [Blackadar, II.6.9.7]. The norm equality is proved in (3).",
      "proof_locus": {
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
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      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
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      "statement_and_full_conditions": "**Theorem 6.2** (Uniqueness of the minimal dilation, and when \\(\\rho\\) is faithful). Let \\(\\varphi\\), \\(N\\), \\(K\\), \\(\\pi\\), \\(V\\), \\(\\rho\\) and \\(a_0\\) be as in Theorem 6.1.\n\n1. Let \\(\\pi'\\) be a representation of \\(A\\) on \\(K'\\) and \\(V'\\in B(H,K')\\) with \\(\\varphi=V'^*\\pi'(\\cdot)V'\\) and \\(K'=\\overline{\\operatorname{span}}\\,\\pi'(A)V'H\\). There is exactly one unitary \\(U:K\\to K'\\) with \\(U\\pi(a)=\\pi'(a)U\\) for all \\(a\\in A\\) and \\(UV=V'\\). If a map \\(\\rho':N\\to B(K')\\) satisfies \\(\\rho'(N)\\subseteq\\pi'(A)'\\) and \\(\\rho'(x)V'=V'x\\) for all \\(x\\), then \\(\\rho'(x)=U\\rho(x)U^*\\). In particular \\(\\rho\\) is determined by (6.1), and any such \\(\\rho'\\) is automatically a unital normal representation.\n2. \\(\\ker\\rho=\\{x\\in N:\\ a_0x=0\\}=N(1-z)\\), where \\(z\\) is the projection onto the closed span of \\(N'a_0H\\). The projection \\(z\\) lies in the centre \\(N\\cap N'\\). So \\(\\rho\\) is faithful if and only if \\(z=1\\). This holds when \\(a_0\\) has zero kernel, in particular when \\(a_0=1\\).\n3. There are completely positive maps for which \\(\\rho\\) is not faithful. So a triple as in (6.1) with a faithful \\(\\rho\\) need not exist, and the uniqueness in (1) must be stated without faithfulness.\n4. If \\(A\\) has a unit, then \\(V\\xi=q(1\\otimes\\xi)\\) and \\(V^*V=\\varphi(1)\\). \\(V\\) is an isometry if and only if \\(\\varphi(1)=1\\); for general \\(A\\), if and only if \\(a_0=1\\). Then \\(\\rho\\) is faithful, and \\(\\varphi(a)\\) is the compression of \\(\\pi(a)\\) to the subspace \\(VH\\), identified with \\(H\\).\n\nThe explicit example in (3) proves that faithfulness cannot be required for every minimal dilation; (2) gives the exact condition.",
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
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      "global_tag_allocated": false
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    {
      "local_label": "Example 6.3",
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      "statement_and_full_conditions": "**Example 6.3** (A state is its own dilation). Let \\(f\\) be a positive functional on \\(A\\), viewed as a CP map into \\(B(\\mathbb C)=\\mathbb C\\) (Theorem 5.1(1)). In the proof of Theorem 6.1, \\(A\\odot\\mathbb C=A\\) and the form (6.3) is \\(\\langle x,y\\rangle_f=f(y^*x)\\), so \\(K\\) is the GNS space \\(H_f\\), \\(\\pi=\\pi_f\\), and \\(V1=\\xi_f\\), the limit of the classes of \\(u_i\\). Here \\(N=\\mathbb C\\) and \\(\\rho(\\lambda)=\\lambda1_{H_f}\\). If \\(f=0\\), then \\(H_f=\\{0\\}\\), and \\(\\pi,V,\\rho\\) are the zero operators, as the same formulas require. The identity \\(\\|f\\|=\\|V\\|^2=\\lim_if(u_i)\\) of (6.2) is the familiar formula for the norm of a positive functional, and \\(a_0=\\|f\\|\\).",
      "proof_locus": {
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              "heading": "6. Stinespring's dilation theorem",
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            },
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              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
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                "OA-FND-CM-10",
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          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.4",
      "kind": "example",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-12::Example 6.4",
      "anchor": "oa-fnd-cm-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Example 6.4** (Compressions and minimality). Let \\(\\sigma\\) be a representation of \\(A\\) on \\(L\\), and \\(V\\in B(H,L)\\). Then \\(\\varphi=V^*\\sigma(\\cdot)V\\) is CP (Theorem 6.1(1)). The closed span \\(K_1\\) of \\(\\sigma(A)VH\\) reduces \\(\\sigma\\): multiplication and adjoints preserve this span. Let \\(P_1\\) be its orthogonal projection, \\(\\pi=\\sigma|_{K_1}\\) and \\(V_1=P_1V\\). For every \\(a\\in A\\), \\(\\sigma(a)VH\\subseteq K_1\\). Since \\(K_1\\) reduces \\(\\sigma\\), the vector \\(\\sigma(a)(1-P_1)V\\xi\\) also belongs to \\(K_1^\\perp\\), hence is zero. Thus \\(\\sigma(a)V=P_1\\sigma(a)P_1V\\), and\n\\[\nV_1^*\\pi(a)V_1=V^*\\sigma(a)V=\\varphi(a).\n\\]\nMoreover \\(\\pi(A)V_1H=\\sigma(A)VH\\), so this triple is minimal and Theorem 6.2(1) identifies it with the Stinespring triple. If \\(\\sigma\\) is nondegenerate, \\(\\sigma(u_i)V\\xi\\to V\\xi\\) shows \\(VH\\subseteq K_1\\), and \\(V_1=V\\). Without nondegeneracy, \\(VH\\) may leave \\(K_1\\): for \\(A=\\mathbb C\\), \\(L=\\mathbb C^2\\), \\(\\sigma(\\lambda)=\\lambda e_{11}\\) and \\(V=1\\), the map \\(\\varphi(\\lambda)=\\lambda e_{11}\\) of Theorem 6.2(3) appears, with \\(K_1=\\mathbb C\\varepsilon_1\\), and the minimal dilation replaces \\(V\\) by the compression \\(e_{11}V\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 525,
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              "heading": "4. The Kadison–Schwarz inequality",
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              "heading": "5. When positivity implies complete positivity",
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Example 6.5",
      "kind": "example",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-12::Example 6.5",
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      "statement_and_full_conditions": "**Example 6.5** (A faithful \\(\\rho\\) with \\(a_0\\ne1\\)). Let \\(A=c_0\\), the sequences tending to \\(0\\), acting diagonally on \\(H=\\ell^2\\), and \\(\\varphi(x)=\\operatorname{diag}(x_n/n)\\). With \\(\\pi(x)=\\operatorname{diag}(x_n)\\) and \\(V=\\operatorname{diag}(n^{-1/2})\\), \\(\\varphi=V^*\\pi(\\cdot)V\\), and \\(\\pi(A)VH\\) contains every finitely supported sequence, so the triple is minimal. Here \\(N=\\varphi(A)'\\) is the algebra of diagonal operators, \\(\\rho(b)=b\\), and \\(a_0=V^*V=\\operatorname{diag}(1/n)\\). It is not \\(1\\) and not invertible, but it has zero kernel, so \\(\\rho\\) is faithful, as Theorem 6.2(2) predicts, although the condition \\(a_0=1\\) of Theorem 6.2(4) fails. The norm formula gives \\(\\|\\varphi\\|=\\|a_0\\|=1\\), attained on the approximate identity \\(u_i=1_{\\{1,\\dots,i\\}}\\).",
      "proof_locus": {
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        "line": 531,
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
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    {
      "local_label": "Example 6.6",
      "kind": "example",
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      "statement_and_full_conditions": "**Example 6.6** (The diagonal compression). On \\(M_d(\\mathbb C)\\) let \\(E(x)=\\sum_ie_{ii}xe_{ii}\\), which keeps the diagonal of \\(x\\). With \\(\\pi(x)=x\\oplus\\cdots\\oplus x\\) on \\((\\mathbb C^d)^d\\) and \\(V\\xi=(e_{11}\\xi,\\dots,e_{dd}\\xi)\\), \\(V^*\\pi(x)V=\\sum_ie_{ii}xe_{ii}=E(x)\\). The triple is minimal: \\(\\pi(x)V\\varepsilon_i\\) has \\(x\\varepsilon_i\\) in the \\(i\\)-th place and \\(0\\) elsewhere, and \\(x\\varepsilon_i\\) runs through \\(\\mathbb C^d\\). Here \\(N=E(M_d)'\\) is the diagonal algebra, and \\(\\rho(b)=b_{11}1\\oplus\\cdots\\oplus b_{dd}1\\) satisfies \\(\\rho(b)V=Vb\\). Since \\(E(1)=1\\), \\(V\\) is an isometry and \\(\\rho\\) is faithful (Theorem 6.2(4)). \\(E\\) is a unital CP map that is not multiplicative, and \\(E(x)^*E(x)\\le E(x^*x)\\) is the Kadison–Schwarz inequality of Theorem 4.1(1) with \\(\\|E\\|=1\\).",
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              "heading": "6. Stinespring's dilation theorem",
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          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
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          "source_locus": "II.6.9; PDF 141–145",
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        }
      ],
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    {
      "local_label": "Exercise 6.7",
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      "statement_and_full_conditions": "**Exercise 6.7** (hard; Completely positive definite functions). Let \\(G\\) be a topological group, \\(H\\) a Hilbert space, and \\(x:G\\to B(H)\\) a function that is *completely positive definite*: for all \\(s_1,\\dots,s_n\\in G\\), the operator matrix \\([x(s_i^{-1}s_j)]_{i,j}\\) is positive on \\(H^n\\). Suppose \\(x\\) is weakly continuous at the identity \\(e\\). Show that there are a strongly continuous unitary representation \\(U\\) of \\(G\\) on a Hilbert space \\(K\\) and \\(T\\in B(H,K)\\) with \\(x(s)=T^*U(s)T\\) for all \\(s\\), and \\(K=\\overline{\\operatorname{span}}\\,U(G)TH\\). Show that such a pair is unique up to a unitary \\(W\\) with \\(WU(s)W^*=U'(s)\\) and \\(WT=T'\\), and that \\(x\\) is then strongly continuous on all of \\(G\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "4. The Kadison–Schwarz inequality",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "5. When positivity implies complete positivity",
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
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              "line": 131,
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
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              "anchors": [
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "5. When positivity implies complete positivity",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
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              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
              "through_line": 565,
              "anchors": [
                "OA-FND-CM-05",
                "OA-FND-CM-06",
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            },
            {
              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.1",
      "kind": "proposition",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-09::Proposition 7.1",
      "anchor": "oa-fnd-cm-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Proposition 7.1.**\n\n1. (7.1) is a linear bijection of \\(M_n(A^*)\\) onto \\(M_n(A)^*\\), with \\(\\max_{i,j}\\|f_{ij}\\|\\le\\|f\\|\\le\\sum_{i,j}\\|f_{ij}\\|\\). The cone \\(M_n(A^*)_+\\) is weak\\(^*\\)-closed, hence norm-closed.\n2. \\(f\\in M_n(A^*)\\) is positive exactly when \\(\\sum_{i,j}f_{ij}(a_i^*a_j)\\ge0\\) for all \\(a_1,\\dots,a_n\\in A\\).\n3. If \\(f\\in M_n(A^*)_+\\) and \\(\\alpha\\in M_{n,m}(\\mathbb C)\\), then \\(\\alpha^*f\\alpha:=\\bigl[\\sum_{i,j}\\overline{\\alpha_{ip}}f_{ij}\\alpha_{jq}\\bigr]_{p,q}\\) lies in \\(M_m(A^*)_+\\).\n4. A composition of \\(n\\)-positive maps between such subspaces is \\(n\\)-positive; so a composition of CP maps is CP.\n5. A positive linear map from a C\\(^*\\)-algebra \\(A\\) into a subspace \\(F\\) of a C\\(^*\\)-algebra \\(B\\) or of a dual \\(B^*\\) is bounded.\n6. Let \\(\\varphi:A\\to B\\) be a bounded linear map of C\\(^*\\)-algebras. Under (7.1), \\((\\varphi^{(n)})^\\sharp=(\\varphi^\\sharp)^{(n)}\\). The map \\(\\varphi\\) is positive exactly when \\(\\varphi^\\sharp\\) is. Hence \\(\\varphi\\) and \\(\\varphi^\\sharp\\) are \\(n\\)-positive together, and CP together.\n7. Theorem 5.4(1) holds for maps into a subspace \\(F\\) of a dual \\(B^*\\): every \\(k\\)-positive map from \\(C_0(\\Omega,M_k)\\) into \\(F\\) is completely positive.",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 582,
        "through_line": 621,
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "4. The Kadison–Schwarz inequality",
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              "heading": "6. Stinespring's dilation theorem",
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          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
        {
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              "heading": "3. Completely positive maps",
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "4. The Kadison–Schwarz inequality",
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              "anchors": [
                "OA-FND-CM-07"
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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            {
              "heading": "5. When positivity implies complete positivity",
              "line": 250,
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              "anchors": [
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                "OA-FND-CM-08"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
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                "OA-FND-CM-05",
                "OA-FND-CM-06",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Lemma 7.2",
      "kind": "lemma",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-10::Lemma 7.2",
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      "statement_and_full_conditions": "**Lemma 7.2** (Few orthogonal projections force finite dimension). Let \\(M\\) be a von Neumann algebra in which every family of pairwise orthogonal nonzero projections has at most \\(N\\) members. Then \\(\\dim M\\le N^2\\).",
      "proof_locus": {
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              "heading": "4. The Kadison–Schwarz inequality",
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              "heading": "6. Stinespring's dilation theorem",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
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              "heading": "5. When positivity implies complete positivity",
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              "heading": "6. Stinespring's dilation theorem",
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              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
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                "OA-FND-CM-10",
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.3",
      "kind": "theorem",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-10::Theorem 7.3",
      "anchor": "oa-fnd-cm-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Theorem 7.3.**\n\n1. \\(\\theta_\\omega\\) is an injective CP map of \\(\\pi(A)'\\) onto \\(C_\\omega\\). It maps \\(\\pi(A)'_+\\) onto \\(C_\\omega^+\\), \\(C_\\omega\\cap A^*_+=C_\\omega^+\\), and \\(\\theta_\\omega(1)=\\omega\\). The inverse \\(\\theta_\\omega^{-1}:C_\\omega\\to\\pi(A)'\\) is CP, with \\(C_\\omega\\subseteq A^*\\) ordered as in Proposition 7.1.\n2. \\(\\|\\theta_\\omega(x)\\|\\le\\|\\omega\\|\\,\\|x\\|\\). In particular \\(\\theta_\\omega\\) is contractive when \\(\\omega\\) is a state.\n3. The following are equivalent: (a) \\(\\theta_\\omega^{-1}\\) is bounded for the norm of \\(A^*\\); (b) \\(C_\\omega\\) is norm-closed in \\(A^*\\); (c) \\(\\pi(A)'\\) is finite-dimensional.\n4. Let \\(B\\) be a unital C\\(^*\\)-algebra and \\(\\varphi:B\\to A^*\\) positive with \\(\\varphi(1)\\le\\alpha\\omega\\). Then \\(\\varphi(B)\\subseteq C_\\omega\\), and \\(\\psi=\\theta_\\omega^{-1}\\circ\\varphi:B\\to\\pi(A)'\\) is positive, hence bounded. If \\(\\varphi\\) is \\(n\\)-positive (CP), so is \\(\\psi\\). If \\(\\varphi\\) is 2-positive, then \\(\\|\\psi\\|=\\|\\psi(1)\\|\\le\\alpha\\), and \\(\\psi\\) is unital when \\(\\varphi(1)=\\omega\\).\n\nPart (3) characterizes boundedness of \\(\\theta_\\omega^{-1}\\), and (4) needs only positivity and domination by a multiple of \\(\\omega\\); complete positivity and equality at the identity are not needed for its factorization.",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 643,
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              "heading": "6. Stinespring's dilation theorem",
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            }
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          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
        {
          "unit": "completely-positive-maps",
          "local_scopes": [
            {
              "heading": "3. Completely positive maps",
              "line": 131,
              "through_line": 184,
              "anchors": [
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              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
              "through_line": 249,
              "anchors": [
                "OA-FND-CM-07"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "5. When positivity implies complete positivity",
              "line": 250,
              "through_line": 336,
              "anchors": [
                "OA-FND-CM-04",
                "OA-FND-CM-08"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
              "through_line": 565,
              "anchors": [
                "OA-FND-CM-05",
                "OA-FND-CM-06",
                "OA-FND-CM-12"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
              "through_line": 757,
              "anchors": [
                "OA-FND-CM-09",
                "OA-FND-CM-10",
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            }
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          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.4",
      "kind": "example",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-10::Example 7.4",
      "anchor": "oa-fnd-cm-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Example 7.4** (An unbounded inverse). Let \\(A=C[0,1]\\) and \\(\\omega(a)=\\int_0^1a(t)\\,dt\\). The GNS space is \\(L^2[0,1]\\), \\(\\pi(a)\\) is multiplication by \\(a\\), and \\(\\xi=1\\). Multiplication by the indicator of an interval commutes with \\(\\pi(A)\\). The indicators of \\(I_k=[2^{-k},2^{-k+1})\\), \\(k\\ge1\\), give infinitely many pairwise orthogonal nonzero projections in \\(\\pi(A)'\\), so \\(\\theta_\\omega^{-1}\\) is unbounded by Theorem 7.3(3). Concretely, \\(\\theta_\\omega\\) sends multiplication by \\(1_{I_k}\\), an operator of norm \\(1\\), to the functional \\(a\\mapsto\\int_{I_k}a\\,dt\\), of norm \\(2^{-k}\\).",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 707,
        "through_line": 708,
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
      },
      "source_uses": [
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          "unit": "completely-positive-maps",
          "local_scopes": [
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              "heading": "3. Completely positive maps",
              "line": 131,
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              "anchors": [
                "OA-FND-CM-03"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
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              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "5. When positivity implies complete positivity",
              "line": 250,
              "through_line": 336,
              "anchors": [
                "OA-FND-CM-04",
                "OA-FND-CM-08"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
              "through_line": 565,
              "anchors": [
                "OA-FND-CM-05",
                "OA-FND-CM-06",
                "OA-FND-CM-12"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
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                "OA-FND-CM-10",
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          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
        {
          "unit": "completely-positive-maps",
          "local_scopes": [
            {
              "heading": "3. Completely positive maps",
              "line": 131,
              "through_line": 184,
              "anchors": [
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              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
              "through_line": 249,
              "anchors": [
                "OA-FND-CM-07"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "5. When positivity implies complete positivity",
              "line": 250,
              "through_line": 336,
              "anchors": [
                "OA-FND-CM-04",
                "OA-FND-CM-08"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
              "through_line": 565,
              "anchors": [
                "OA-FND-CM-05",
                "OA-FND-CM-06",
                "OA-FND-CM-12"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
              "through_line": 757,
              "anchors": [
                "OA-FND-CM-09",
                "OA-FND-CM-10",
                "OA-FND-CM-11"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.5",
      "kind": "proposition",
      "unit": "completely-positive-maps",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/completely-positive-maps#oa-fnd-cm-11::Proposition 7.5",
      "anchor": "oa-fnd-cm-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
      "statement_and_full_conditions": "**Proposition 7.5.**\n\n1. \\(\\Phi_1\\) and \\(\\Phi_2\\) are positive linear bijections of \\(A\\) onto \\(A^*\\).\n2. \\(\\Phi_1\\) is completely positive, and so is \\(\\Phi_1^{-1}\\).\n3. \\(\\Phi_2\\) is not 2-positive.\n4. Under the identification (7.1) for the algebra \\(\\mathbb C\\), a functional \\(f\\) on \\(M_d(\\mathbb C)\\) corresponds to the matrix \\([f(e_{ij})]\\), because \\(f(x)=\\sum_{i,j}f(e_{ij})x_{ij}\\). Then \\(\\Phi_1(a)\\) corresponds to \\([a_{ij}]\\) and \\(\\Phi_2(a)\\) to \\([a_{ji}]\\). Under the trace pairing, \\(f\\) corresponds instead to the matrix \\(D_f\\) with \\(f(x)=\\mathrm{Tr}(D_fx)\\); then \\(\\Phi_2(a)\\) corresponds to \\(a\\) and \\(\\Phi_1(a)\\) to \\(a^{\\mathsf T}\\).\n\nThe explicit computations in (2)–(4) show how switching between the entrywise pairing (7.1) and the trace pairing introduces a transpose.",
      "proof_locus": {
        "source": "src/completely-positive-maps.md",
        "line": 722,
        "through_line": 757,
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
      },
      "source_uses": [
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          "unit": "completely-positive-maps",
          "local_scopes": [
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              "heading": "3. Completely positive maps",
              "line": 131,
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              "anchors": [
                "OA-FND-CM-03"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
              "through_line": 249,
              "anchors": [
                "OA-FND-CM-07"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "5. When positivity implies complete positivity",
              "line": 250,
              "through_line": 336,
              "anchors": [
                "OA-FND-CM-04",
                "OA-FND-CM-08"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
              "through_line": 565,
              "anchors": [
                "OA-FND-CM-05",
                "OA-FND-CM-06",
                "OA-FND-CM-12"
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
              "through_line": 757,
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                "OA-FND-CM-09",
                "OA-FND-CM-10",
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          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§10; PDF 54–61",
          "role": "proof comparison",
          "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
          "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
        },
        {
          "unit": "completely-positive-maps",
          "local_scopes": [
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              "heading": "3. Completely positive maps",
              "line": 131,
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              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "4. The Kadison–Schwarz inequality",
              "line": 185,
              "through_line": 249,
              "anchors": [
                "OA-FND-CM-07"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
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              "heading": "5. When positivity implies complete positivity",
              "line": 250,
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              "anchors": [
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                "OA-FND-CM-08"
              ],
              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
            {
              "heading": "6. Stinespring's dilation theorem",
              "line": 337,
              "through_line": 565,
              "anchors": [
                "OA-FND-CM-05",
                "OA-FND-CM-06",
                "OA-FND-CM-12"
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              "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
            },
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              "heading": "7. Completely positive maps and dual spaces",
              "line": 566,
              "through_line": 757,
              "anchors": [
                "OA-FND-CM-09",
                "OA-FND-CM-10",
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            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.9; PDF 141–145",
          "role": "proof comparison",
          "correspondence": "Classical CP treatment with outlines",
          "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.1",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-01::Proposition 1.1",
      "anchor": "oa-fnd-wa-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 1.1.** Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. Every hermitian \\(f\\in A^*\\) is a difference \\(f=f_+-f_-\\) of positive functionals with \\(\\|f\\|=\\|f_+\\|+\\|f_-\\|\\).\n2. Every \\(f\\in A^*\\) has the form \\(f=\\omega_{\\pi_\\psi;\\xi_\\psi,\\eta}\\), where \\(\\psi\\in A^*_+\\), \\((\\pi_\\psi,H_\\psi,\\xi_\\psi)\\) is its cyclic representation and \\(\\eta\\in H_\\psi\\). One can arrange \\(\\|\\xi_\\psi\\|\\,\\|\\eta\\|\\le4\\|f\\|\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 121,
        "through_line": 161,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
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          "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
          "local_scopes": [
            {
              "heading": "1. Functionals as vector coefficients",
              "line": 106,
              "through_line": 189,
              "anchors": [
                "OA-FND-WA-01"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
              "through_line": 243,
              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
          ],
          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.1",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-02::Lemma 2.1",
      "anchor": "oa-fnd-wa-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 2.1.** Let \\(\\pi:A\\to B(H)\\) be a possibly degenerate representation. Let \\(q\\) be the projection onto the essential space \\(H_0\\), the closed span of \\(\\pi(A)H\\), and let \\(N(\\pi)\\) be the \\(\\sigma\\)-weak closure of \\(\\pi(A)\\).\n\n1. \\(q\\) commutes with \\(\\pi(A)\\), \\(\\pi(a)=q\\pi(a)q\\), and \\(\\pi(e_i)\\to q\\) strongly for every approximate unit \\((e_i)\\) of \\(A\\). \\(N(\\pi)\\) acts on \\(H_0\\) as the von Neumann algebra \\((\\pi(A)|_{H_0})''\\) and is zero on \\(H_0^\\perp\\). Moreover\n\\[\n\\pi(A)''=N(\\pi)+\\mathbb C(1-q),\n\\tag{2.1}\n\\]\nand the sum is direct when \\(q\\neq1\\). So \\(N(\\pi)=\\pi(A)''\\) exactly when \\(\\pi\\) is nondegenerate.\n2. There is exactly one linear map \\(\\bar\\pi:A^{**}\\to B(H)\\) that is continuous from \\(\\sigma(A^{**},A^*)\\) to the \\(\\sigma\\)-weak topology and satisfies \\(\\bar\\pi\\circ j=\\pi\\). It is a normal \\(*\\)-homomorphism with \\(\\bar\\pi(1)=q\\) and \\(\\bar\\pi(A^{**})=N(\\pi)\\).\n3. \\(\\bar\\pi\\) maps the closed unit ball of \\(A^{**}\\) onto the closed unit ball of \\(N(\\pi)\\).\n4. There is a unique central projection \\(z(\\pi)\\in A^{**}\\) with \\(\\ker\\bar\\pi=A^{**}(1-z(\\pi))\\). The restriction of \\(\\bar\\pi\\) to \\(A^{**}z(\\pi)\\) is a \\(*\\)-isomorphism onto \\(N(\\pi)\\). It is normal, and so is its inverse.\n\nFor nondegenerate \\(\\pi\\) we get \\(q=1\\) and \\(N(\\pi)=\\mathcal M(\\pi)\\): then \\(\\bar\\pi\\) maps \\(A^{**}\\) onto \\(\\mathcal M(\\pi)\\), and the unit ball onto the unit ball.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 194,
        "through_line": 227,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
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              "heading": "1. Functionals as vector coefficients",
              "line": 106,
              "through_line": 189,
              "anchors": [
                "OA-FND-WA-01"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
              "through_line": 243,
              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
          ],
          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.2",
      "kind": "example",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-02::Example 2.2",
      "anchor": "oa-fnd-wa-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Example 2.2** (Degenerate representations). Suppose \\(q\\neq1\\). By (2.1), \\(1_H\\in\\pi(A)''\\) but \\(1_H\\notin\\bar\\pi(A^{**})\\subseteq qB(H)q\\). So \\(\\bar\\pi\\) is not onto \\(\\mathcal M(\\pi)\\), and the unit ball of \\(\\mathcal M(\\pi)\\) is not the image of the unit ball of \\(A^{**}\\). The smallest example is \\(A=\\mathbb C\\) with \\(\\pi(\\lambda)=\\lambda\\oplus0\\) on \\(\\mathbb C^2\\). Then \\(A^{**}=\\mathbb C\\), \\(N(\\pi)=\\mathbb C\\oplus0\\) and \\(\\mathcal M(\\pi)=\\mathbb C\\oplus\\mathbb C\\). This is why Lemma 2.1 is stated with \\(N(\\pi)\\) in place of \\(\\mathcal M(\\pi)\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 242,
        "through_line": 243,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [
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          "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
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              "line": 106,
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                "OA-FND-WA-01"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
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              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
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          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.1",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-03::Definition 3.1",
      "anchor": "oa-fnd-wa-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 3.1.** A representation \\(\\pi\\) of \\(A\\) is *universal* if for every representation \\(\\rho\\) of \\(A\\) there is a normal \\(*\\)-homomorphism \\(\\tilde\\rho\\) of \\(\\mathcal M(\\pi)\\) onto \\(\\mathcal M(\\rho)\\) with \\(\\tilde\\rho\\circ\\pi=\\rho\\). Then \\(\\mathcal M(\\pi)\\) is called a *universal enveloping von Neumann algebra* for \\(A\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 248,
        "through_line": 249,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
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              "line": 106,
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              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
              "through_line": 243,
              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
          ],
          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.2",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-03::Proposition 3.2",
      "anchor": "oa-fnd-wa-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 3.2** (Uniqueness). Let \\(\\pi_1,\\pi_2\\) be universal. There is exactly one normal \\(*\\)-homomorphism \\(\\theta:\\mathcal M(\\pi_1)\\to\\mathcal M(\\pi_2)\\) with \\(\\theta\\circ\\pi_1=\\pi_2\\). It is a \\(*\\)-isomorphism, and its inverse is the unique normal \\(\\theta'\\) with \\(\\theta'\\circ\\pi_2=\\pi_1\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 250,
        "through_line": 253,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
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          "local_scopes": [
            {
              "heading": "1. Functionals as vector coefficients",
              "line": 106,
              "through_line": 189,
              "anchors": [
                "OA-FND-WA-01"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
              "through_line": 243,
              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
          ],
          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.3",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-03::Theorem 3.3",
      "anchor": "oa-fnd-wa-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 3.3** (The universal representation). Put\n\\[\n\\begin{gathered}\n\\pi_u\\\\\n=\\bigoplus_{\\omega\\in A^*_+}\\pi_\\omega\\\\\n\\text{on}\\\\\nH_u\\\\\n=\\bigoplus_{\\omega\\in A^*_+}H_\\omega ,\n\\end{gathered}\n\\tag{3.1}\n\\]\nwhere the summand for \\(\\omega=0\\) is the zero space. Then \\(\\pi_u\\) is universal. The extension \\(\\bar\\pi_u:A^{**}\\to\\mathcal M(\\pi_u)\\) is an isometric \\(*\\)-isomorphism, and it is a homeomorphism for \\(\\sigma(A^{**},A^*)\\) and the \\(\\sigma\\)-weak topology. The same holds for the direct sum over the states only. We call \\(\\pi_u\\) the *universal representation* of \\(A\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 254,
        "through_line": 274,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
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          "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
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              "heading": "1. Functionals as vector coefficients",
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              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
              "through_line": 243,
              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
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          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.4",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-03::Proposition 3.4",
      "anchor": "oa-fnd-wa-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 3.4** (The norm-additive decomposition is unique). Let \\(f\\in A^*\\) be hermitian. There is a self-adjoint \\(X\\) in the unit ball of \\(\\tilde A\\) with \\(\\hat f(X)=\\|f\\|\\). For any such \\(X\\) let \\(E_+\\) and \\(E_-\\) be the projections onto the kernels of \\(1-X\\) and \\(1+X\\), that is \\(E_\\pm=1-s(1\\mp X)\\). Then \\(E_+E_-=0\\), and every decomposition \\(f=f_+-f_-\\) with \\(f_\\pm\\in A^*_+\\) and \\(\\|f\\|=\\|f_+\\|+\\|f_-\\|\\) satisfies\n\\[\n\\begin{gathered}\nf_+(a)\\\\\n=\\hat f(aE_+),\\\\\nf_-(a)\\\\\n=-\\hat f(aE_-)\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{3.2}\n\\]\nIn particular the decomposition of [Proposition 1.1](#oa-fnd-wa-01)(1) is unique, and the supports of \\(\\hat f_+\\) and \\(\\hat f_-\\) are orthogonal, lying under \\(E_+\\) and \\(E_-\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 275,
        "through_line": 327,
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              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "2. Extending a representation to the bidual",
              "line": 190,
              "through_line": 243,
              "anchors": [
                "OA-FND-WA-02"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            },
            {
              "heading": "3. The universal enveloping von Neumann algebra",
              "line": 244,
              "through_line": 327,
              "anchors": [
                "OA-FND-WA-03",
                "OA-FND-WA-04"
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          "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
          "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
          "role": "proof comparison",
          "correspondence": "Universal bidual mechanism; own dependency order retained",
          "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-05::Definition 4.1",
      "anchor": "oa-fnd-wa-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 4.1.** Let \\(A\\) be a \\(C^*\\)-algebra, acting on \\(A^*\\) by (0.1) through \\(j\\): \\((af)(x)=f(xa)\\) and \\((fa)(x)=f(ax)\\). A subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), *right invariant* if \\(Va\\subseteq V\\) for all \\(a\\in A\\), and *invariant* if it is both.\n\nTwo facts organise what follows.\n\n1. Let \\(M\\) be a von Neumann algebra, regarded as a \\(C^*\\)-algebra. Then \\(M_*\\) is a norm-closed invariant subspace of \\(M^*\\). It is invariant because multiplication by a fixed element is \\(\\sigma\\)-weakly continuous, and it is norm closed by the background fact on preduals.\n2. The algebra \\(\\tilde A=A^{**}\\) also acts on \\(A^*\\), since \\(A^*\\) is its predual. For a norm-closed subspace \\(V\\subseteq A^*\\), invariance under \\(A\\) and invariance under \\(\\tilde A\\) are the same thing. This is [Theorem 4.3](#oa-fnd-wa-07)(1) with \\(S=j(A)\\). So it makes no difference whether \"invariant\" refers to \\(A\\) or to \\(\\tilde A\\).\n\nThe correspondence rests on a description of the weak\\(^*\\) closed one-sided ideals of a von Neumann algebra.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 332,
        "through_line": 340,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
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      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.2",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-05::Lemma 4.2",
      "anchor": "oa-fnd-wa-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 4.2.** Let \\(J\\) be a right ideal of a von Neumann algebra \\(M\\), closed in the \\(\\sigma\\)-weak topology. There is a unique projection \\(p\\in M\\) with \\(J=pM\\). It lies in \\(J\\) and is a left identity for \\(J\\). Symmetrically, a \\(\\sigma\\)-weakly closed left ideal is \\(Mp\\) for a unique projection \\(p\\), and a \\(\\sigma\\)-weakly closed two-sided ideal is \\(Mp\\) with \\(p\\) central.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 341,
        "through_line": 348,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.3",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-05::Theorem 4.3",
      "anchor": "oa-fnd-wa-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 4.3** (Invariant subspaces and weak\\(^*\\) closed ideals). Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(S\\subseteq M\\) have \\(\\sigma\\)-weakly dense linear span, and let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(sV\\subseteq V\\) (resp. \\(Vs\\subseteq V\\)) for every \\(s\\in S\\). Then \\(V\\) is left (resp. right) invariant under all of \\(M\\).\n2. Taking polars, \\(V\\mapsto V^\\circ\\), is a bijection from the norm-closed subspaces \\(V\\subseteq M_*\\) that are invariant on the left (resp. right) onto the right (resp. left) ideals \\(J\\) of \\(M\\) that are \\(\\sigma\\)-weakly closed; its inverse is \\(J\\mapsto J^\\circ\\).\n3. A norm-closed subspace \\(V\\) invariant on the left (resp. right) has the form \\(V=M_*e\\) (resp. \\(V=eM_*\\)) for a unique projection \\(e\\in M\\), with \\(V^\\circ=(1-e)M\\) (resp. \\(M(1-e)\\)). This \\(e\\) is called the *support* of \\(V\\). A norm-closed \\(V\\) invariant on the left is invariant on both sides exactly when \\(e\\) is central, and then \\(M_*e=eM_*\\).\n\nFor a \\(C^*\\)-algebra \\(A\\), apply this to \\(M=\\tilde A\\), \\(M_*=A^*\\) and \\(S=j(A)\\). Then the norm-closed subspaces of \\(A^*\\) that are invariant on the left (resp. right) under \\(A\\), as in [Definition 4.1](#oa-fnd-wa-05), match the right (resp. left) ideals of \\(\\tilde A\\) that are \\(\\sigma\\)-weakly closed, and each such subspace is \\(A^*e\\) (resp. \\(eA^*\\)) for a unique projection \\(e\\in\\tilde A\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 349,
        "through_line": 380,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.4",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-05::Proposition 4.4",
      "anchor": "oa-fnd-wa-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 4.4** (Meets and joins through invariant subspaces). Let \\(M\\) be a von Neumann algebra with projections \\((e_i)_{i\\in I}\\), and put \\(p=\\bigwedge_ie_i\\) and \\(q=\\bigvee_ie_i\\). Then\n\\[\n\\begin{gathered}\nM_*p\\\\\n=\\bigcap_iM_*e_i,\\\\\npM_*\\\\\n=\\bigcap_ie_iM_*,\\\\\nM_*q\\\\\n=\\overline{\\operatorname{span}}\\bigcup_iM_*e_i,\\\\\nqM_*\\\\\n=\\overline{\\operatorname{span}}\\bigcup_ie_iM_* .\n\\end{gathered}\n\\tag{4.2}\n\\]",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 381,
        "through_line": 401,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.1",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Definition 5.1",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 5.1.** Representations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *quasi-equivalent*, written \\(\\pi_1\\sim\\pi_2\\), if there is a \\(*\\)-isomorphism \\(\\theta\\) of \\(\\mathcal M(\\pi_1)\\) onto \\(\\mathcal M(\\pi_2)\\) with \\(\\theta\\circ\\pi_1=\\pi_2\\). For a representation \\(\\pi\\), the *support* of \\(\\pi\\) is the central projection \\(z(\\pi)\\in\\tilde A\\) of [Lemma 2.1](#oa-fnd-wa-02)(4), so that \\(\\ker\\bar\\pi=\\tilde A(1-z(\\pi))\\). The subspace *associated with* \\(\\pi\\) is\n\\[\nV(\\pi)=\\{\\psi\\circ\\pi:\\psi\\in\\mathcal M(\\pi)_*\\}\\subseteq A^* .\n\\]\n\nWe do not ask \\(\\theta\\) to be \\(\\sigma\\)-weakly bicontinuous: every \\(*\\)-isomorphism between von Neumann algebras is normal with normal inverse (background), so that requirement would not change the relation.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 406,
        "through_line": 412,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.2",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Proposition 5.2",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 5.2.**\n\n1. \\(\\psi\\mapsto\\psi\\circ\\pi\\) is an isometry of \\(\\mathcal M(\\pi)_*\\) onto \\(V(\\pi)\\).\n2. \\(V(\\pi)=A^*z(\\pi)\\): a functional \\(f\\) lies in \\(V(\\pi)\\) exactly when \\(\\hat f=\\hat f z(\\pi)\\). In particular \\(V(\\pi)\\) is a norm-closed invariant subspace, and its support in the sense of [Theorem 4.3](#oa-fnd-wa-07)(3) is \\(z(\\pi)\\).\n3. Every norm-closed invariant subspace \\(V\\subseteq A^*\\) equals \\(V(\\rho)\\) for some representation \\(\\rho\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 413,
        "through_line": 432,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.3",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Theorem 5.3",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 5.3** (Quasi-equivalence and supports). For representations \\(\\pi_1,\\pi_2\\) of \\(A\\) the following are equivalent: (i) \\(\\pi_1\\sim\\pi_2\\); (ii) \\(V(\\pi_1)=V(\\pi_2)\\); (iii) \\(z(\\pi_1)=z(\\pi_2)\\). When they hold, the isomorphism \\(\\theta\\) in (i) is unique, and \\(\\theta\\) and \\(\\theta^{-1}\\) are normal.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 433,
        "through_line": 446,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.5",
      "kind": "example",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Example 5.5",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Example 5.5** (\\(c_0^{**}=\\ell^\\infty\\)). The identity representation of \\(c_0\\) on \\(\\ell^2\\) is nondegenerate, and \\(\\mathcal M(\\mathrm{id})=\\ell^\\infty\\) (diagonal operators). Its normal functionals are \\(\\ell^1\\) (background fact on preduals), and their restrictions to \\(c_0\\) are all of \\(c_0^*=\\ell^1\\). So \\(V(\\mathrm{id})=c_0^*\\), \\(z(\\mathrm{id})=1\\), and \\(\\overline{\\mathrm{id}}:\\tilde{c_0}\\to\\ell^\\infty\\) is an isomorphism ([Proposition 5.2](#oa-fnd-wa-09) and Lemma 2.1(4)). Here \\(j(c_0)\\) is \\(\\sigma\\)-weakly dense but far from closed: the unit \\(1\\in\\ell^\\infty\\) is a weak\\(^*\\) limit of the finite sequences \\(1_{\\{1,\\dots,n\\}}\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 447,
        "through_line": 448,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.6",
      "kind": "example",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Example 5.6",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Example 5.6** (\\(\\mathcal K(H)^{**}=B(H)\\)). The identity representation of \\(\\mathcal K(H)\\) is nondegenerate and irreducible, and \\(\\mathcal K(H)''=B(H)\\). We show that every positive \\(f\\in\\mathcal K(H)^*\\) is normal on \\(B(H)\\). For a finite-rank projection \\(P\\), the compression of \\(f\\) to \\(P\\mathcal K(H)P=B(PH)\\) is \\(x\\mapsto\\operatorname{Tr}(\\rho_Px)\\) for a unique \\(\\rho_P\\ge0\\) on \\(PH\\), with \\(\\operatorname{Tr}\\rho_P=f(P)\\le\\|f\\|\\), and \\(\\rho_P=P\\rho_QP\\) for \\(P\\le Q\\). So there is a positive trace-class \\(\\rho\\) with \\(P\\rho P=\\rho_P\\) and \\(\\operatorname{Tr}\\rho\\le\\|f\\|\\): put \\(\\langle\\rho\\xi,\\eta\\rangle=\\langle\\rho_P\\xi,\\eta\\rangle\\) for any \\(P\\) whose range contains \\(\\xi\\) and \\(\\eta\\), which the compatibility makes independent of \\(P\\), and note \\(\\sum_k\\langle\\rho e_k,e_k\\rangle=\\sup_P\\operatorname{Tr}\\rho_P\\le\\|f\\|\\) for an orthonormal basis \\((e_k)\\). For compact \\(x\\), \\(PxP\\to x\\) in norm along the finite-rank projections, so \\(f(x)=\\lim f(PxP)=\\lim\\operatorname{Tr}(\\rho PxP)=\\operatorname{Tr}(\\rho x)\\), and \\(x\\mapsto\\operatorname{Tr}(\\rho x)\\) is a vector series, hence normal. As positive functionals span \\(\\mathcal K(H)^*\\) (see the proof of Proposition 1.1(2)), \\(V(\\mathrm{id})=\\mathcal K(H)^*\\), so \\(z(\\mathrm{id})=1\\) and \\(\\mathcal K(H)^{**}\\cong B(H)\\).\n\nA representation of an ideal extends to the whole algebra, and the bidual shows why.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 449,
        "through_line": 452,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.7",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Proposition 5.7",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 5.7** (Extending representations from an ideal). Let \\(J\\) be a closed two-sided ideal of \\(A\\) and \\(\\pi\\) a representation of \\(J\\) on \\(H\\). There is exactly one representation \\(\\pi^A\\) of \\(A\\) on \\(H\\) with \\(\\pi^A|_J=\\pi\\). It satisfies \\(\\pi^A(a)\\pi(x)\\zeta=\\pi(ax)\\zeta\\) for \\(a\\in A\\), \\(x\\in J\\), \\(\\zeta\\in H\\), and \\(\\pi^A(A)''=\\pi(J)''\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 453,
        "through_line": 460,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5.8",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-09::Exercise 5.8",
      "anchor": "oa-fnd-wa-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 5.8.** (easy) Let \\(A\\ne0\\) and let \\(\\pi=0\\) on a Hilbert space \\(H\\ne0\\). Compute \\(N(\\pi)\\), \\(\\mathcal M(\\pi)\\), \\(\\bar\\pi\\), \\(z(\\pi)\\) and \\(V(\\pi)\\), and say which statements of Lemma 2.1 would fail with \\(\\mathcal M(\\pi)\\) in place of \\(N(\\pi)\\).\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 461,
        "through_line": 464,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.1",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-13::Lemma 6.1",
      "anchor": "oa-fnd-wa-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 6.1** (Minimal projections). Let \\(N\\) be a \\(C^*\\)-algebra and \\(f,g\\) minimal projections.\n\n1. \\(Nf\\) is a Hilbert space for \\(\\langle x,y\\rangle f=y^*x\\), and its Hilbert norm is the norm of \\(N\\).\n2. \\(\\dim fNg\\le1\\).\n3. Left multiplication \\(\\lambda(a)x=ax\\) is an irreducible representation of \\(N\\) on \\(Nf\\).\n4. If \\(N\\) is a von Neumann algebra, every bounded linear functional on \\(Nf\\) is the restriction of a normal functional on \\(N\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 469,
        "through_line": 477,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.2",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-13::Lemma 6.2",
      "anchor": "oa-fnd-wa-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 6.2** (Infinite dimension). A von Neumann algebra \\(N\\) of infinite dimension contains an infinite sequence \\(p_1,p_2,\\dots\\) of mutually orthogonal nonzero projections. Then \\(c\\mapsto\\sum_nc_np_n\\) (a strong sum) is an isometric \\(*\\)-homomorphism of \\(\\ell^\\infty\\) into \\(N\\). Consequently \\(N\\) is neither reflexive nor norm separable.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 478,
        "through_line": 483,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.3",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-13::Lemma 6.3",
      "anchor": "oa-fnd-wa-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 6.3** (Finite dimension). A finite-dimensional von Neumann algebra \\(N\\) with centre \\(\\mathbb C1\\) is isomorphic to \\(M_n(\\mathbb C)\\). A finite-dimensional von Neumann algebra is the finite direct sum of the algebras \\(Nc\\), \\(c\\) running over its minimal central projections, and each \\(Nc\\) is a matrix algebra.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 484,
        "through_line": 487,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.4",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-13::Lemma 6.4",
      "anchor": "oa-fnd-wa-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 6.4** (Isomorphisms of \\(B(H)\\) are spatial). Every \\(*\\)-isomorphism \\(\\theta:B(H_1)\\to B(H_2)\\) is \\(x\\mapsto UxU^*\\) for a unitary \\(U:H_1\\to H_2\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 488,
        "through_line": 491,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.5",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-13::Lemma 6.5",
      "anchor": "oa-fnd-wa-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 6.5** (Finite joins of minimal projections). Let \\(N\\) be a von Neumann algebra containing minimal projections \\(e_1,\\dots,e_n\\). Then \\(e=\\bigvee_ke_k\\) is a sum of \\(m\\le n\\) mutually orthogonal minimal projections, and \\(\\dim eNe\\le m^2\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 492,
        "through_line": 495,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 6.6",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-13::Exercise 6.6",
      "anchor": "oa-fnd-wa-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 6.6** (hard) (Compact unitary groups). For a von Neumann algebra \\(M\\) with unitary group \\(\\mathcal U_M\\), prove that \\(\\mathcal U_M\\) is compact exactly when \\(M\\) is \\(*\\)-isomorphic to a direct sum of full matrix algebras \\(M_{n_i}(\\mathbb C)\\). Deduce compactness of \\(\\mathcal U_M\\) for every abelian \\(M\\) in which each nonzero projection majorises a minimal one.\n\nWe use the \\(\\sigma\\)-weak topology. On \\(\\mathcal U_M\\) it coincides with the weak, strong and strong\\(^*\\) topologies of any faithful normal representation: on bounded sets \\(\\sigma\\)-weak and weak agree, and for unitaries \\(\\|(u_\\alpha-u)\\zeta\\|^2=2\\|\\zeta\\|^2-2\\operatorname{Re}\\langle u_\\alpha\\zeta,u\\zeta\\rangle\\) and \\(\\|(u_\\alpha^*-u^*)\\zeta\\|^2=2\\|\\zeta\\|^2-2\\operatorname{Re}\\langle u_\\alpha u^*\\zeta,\\zeta\\rangle\\). A \\(*\\)-isomorphism between von Neumann algebras is \\(\\sigma\\)-weakly bicontinuous (background; see also [Corollary 11.4](#oa-fnd-wa-24)), so this topology does not depend on the representation. Multiplication is jointly strongly continuous on bounded sets and \\(u\\mapsto u^*\\) is strong\\(^*\\) continuous, so \\(\\mathcal U_M\\) is a Hausdorff topological group. \"Direct sum\" means an \\(\\ell^\\infty\\)-direct sum over an arbitrary index set \\(I\\), each \\(n_i\\) finite. (In the norm topology, \\(\\mathcal U_M\\) is compact exactly when \\(\\dim M<\\infty\\): by Lemma 6.2 an infinite-dimensional \\(M\\) contains unitaries \\(1-2p_n\\) at mutual distance \\(2\\).)\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 496,
        "through_line": 511,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.1",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-11::Lemma 7.1",
      "anchor": "oa-fnd-wa-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 7.1.** Let \\(T:X\\to Y\\) be a bounded linear map from a Banach space to a normed space. Write \\(B_X,B_Y\\) for the closed unit balls. If \\(T(B_X)\\) is norm dense in \\(B_Y\\), then every \\(y\\in Y\\) with \\(\\|y\\|<1\\) is \\(Tx\\) for some \\(x\\in X\\) with \\(\\|x\\|<1\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 516,
        "through_line": 521,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.2",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-11::Lemma 7.2",
      "anchor": "oa-fnd-wa-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 7.2** (Pure states and irreducibility). A state \\(\\omega\\) is pure exactly when its cyclic representation is irreducible, that is \\(\\pi_\\omega(A)'=\\mathbb C1\\); then \\(\\pi_\\omega(A)''=B(H_\\omega)\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 522,
        "through_line": 525,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.3",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-11::Proposition 7.3",
      "anchor": "oa-fnd-wa-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 7.3** (Hilbert space quotients). Let \\(\\omega\\) be a pure state of \\(A\\) with left kernel \\(N_\\omega=\\{a:\\omega(a^*a)=0\\}\\). Then the quotient norm of \\(A/N_\\omega\\) is\n\\[\n\\begin{gathered}\n\\|a+N_\\omega\\|\\\\\n=\\omega(a^*a)^{1/2}\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{7.1}\n\\]\nConsequently the quotient norm comes from the inner product \\(\\langle a+N_\\omega,b+N_\\omega\\rangle=\\omega(b^*a)\\), so the complete space \\(A/N_\\omega\\) is a Hilbert space, and \\(a+N_\\omega\\mapsto\\pi_\\omega(a)\\xi_\\omega\\) is a unitary map onto \\(H_\\omega\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 526,
        "through_line": 548,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.5",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-11::Exercise 7.5",
      "anchor": "oa-fnd-wa-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 7.5** (medium) (Minimal left ideals). Let \\(\\mathfrak m\\ne\\{0\\}\\) be a closed left ideal of a \\(C^*\\)-algebra \\(A\\) that contains no smaller nonzero closed left ideal. Show: (a) \\(\\dim(\\mathfrak m\\cap\\mathfrak m^*)=1\\); (b) \\(\\mathfrak m=Ae\\) for a minimal projection \\(e\\in A\\); (c) \\(\\mathfrak m\\) is a Hilbert space on which left multiplication is an irreducible representation of \\(A\\).\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 549,
        "through_line": 560,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.6",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-11::Exercise 7.6",
      "anchor": "oa-fnd-wa-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 7.6** (hard) (Reflexive quotients). Let \\(\\omega\\) be a state of \\(A\\). Prove that the quotient \\(A/N_\\omega\\), with its quotient norm, is reflexive exactly when \\(\\omega=\\sum_{k=1}^n\\lambda_k\\omega_k\\) for finitely many pure states \\(\\omega_k\\) and weights \\(\\lambda_k>0\\) with sum one.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 561,
        "through_line": 576,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-20::Lemma 8.1",
      "anchor": "oa-fnd-wa-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 8.1** (Positivity from a norming net). Let \\(\\omega\\in A^*\\). Suppose there is a net \\((a_\\lambda)\\) in \\(A_+\\) with \\(\\|a_\\lambda\\|\\le1\\) and \\(\\omega(a_\\lambda)\\to\\|\\omega\\|\\). Then \\(\\omega\\) is positive. In particular this holds when \\(\\omega(a)=\\|\\omega\\|\\) for a single \\(a\\in A_+\\) with \\(\\|a\\|\\le1\\).\n\nThe net form matters when the subalgebra has no unit: a positive functional may have no single norming element there (Example 8.3). The proof below establishes the form needed for Corollary 8.2 directly.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 581,
        "through_line": 588,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 8.2",
      "kind": "corollary",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-20::Corollary 8.2",
      "anchor": "oa-fnd-wa-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Corollary 8.2** (Norm-preserving extensions are positive). Let \\(B\\) be a \\(C^*\\)-subalgebra of \\(A\\) and \\(\\varphi\\in B^*_+\\). Every \\(\\omega\\in A^*\\) with \\(\\omega|_B=\\varphi\\) and \\(\\|\\omega\\|=\\|\\varphi\\|\\) is positive.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 589,
        "through_line": 592,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.3",
      "kind": "example",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-20::Example 8.3",
      "anchor": "oa-fnd-wa-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Example 8.3** (No single norming element). If \\(B\\) has a unit, the single element \\(a=1_B\\) norms \\(\\varphi\\) in Corollary 8.2. Without a unit there may be no such element: for \\(B=A=c_0\\) and \\(\\varphi(x)=\\sum_n2^{-n}x_n\\), no \\(a\\in c_0\\) with \\(0\\le a\\le1\\) has \\(\\varphi(a)=1\\). This is why Lemma 8.1 is stated for nets.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 593,
        "through_line": 594,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 8.4",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-20::Definition 8.4",
      "anchor": "oa-fnd-wa-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 8.4.** Let \\(B\\) be a \\(C^*\\)-subalgebra of \\(A\\). A *projection of norm one* of \\(A\\) onto \\(B\\) is a linear map \\(E:A\\to B\\) with \\(E(b)=b\\) for \\(b\\in B\\) and \\(\\|E(x)\\|\\le\\|x\\|\\) for \\(x\\in A\\). (If \\(B\\ne0\\) then \\(\\|E\\|=1\\); if \\(B=0\\) then \\(E=0\\). Such a map is also called a contractive retraction.)",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 595,
        "through_line": 596,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.5",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-20::Theorem 8.5",
      "anchor": "oa-fnd-wa-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 8.5** (Tomiyama's theorem). Let \\(E\\) be a projection of norm one of \\(A\\) onto \\(B\\). Then\n\n1. \\(E\\) is positive, \\(E(x^*x)\\ge0\\), and hence \\(E(x^*)=E(x)^*\\);\n2. \\(E(bxc)=bE(x)c\\) for \\(b,c\\in B\\) and \\(x\\in A\\);\n3. \\(E(x)^*E(x)\\le E(x^*x)\\) for \\(x\\in A\\);\n4. if \\(A\\) has a unit and \\(B\\ne0\\), then \\(B\\) has a unit and \\(E(1_A)=1_B\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 597,
        "through_line": 667,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 8.7",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-20::Exercise 8.7",
      "anchor": "oa-fnd-wa-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 8.7.** (easy) Show that there is no projection of norm one of \\(\\ell^\\infty\\) onto \\(c_0\\), and none onto the space \\(c\\) of convergent sequences.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 668,
        "through_line": 679,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 9.1",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-22::Definition 9.1",
      "anchor": "oa-fnd-wa-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 9.1.** A \\(C^*\\)-algebra \\(A\\) is a *\\(W^*\\)-algebra* if it has a faithful representation \\(\\pi\\) with \\(\\pi(A)=\\pi(A)''\\). Equivalently, some von Neumann algebra is the image of \\(A\\) under a \\(*\\)-isomorphism. Every \\(W^*\\)-algebra has a unit. The zero algebra is a \\(W^*\\)-algebra, acting on the zero space.\n\nA *predual* of a \\(C^*\\)-algebra \\(A\\) is a Banach space \\(F\\) with an isometric linear bijection \\(\\theta:A\\to F^*\\). It gives an isometry \\(\\iota_F:F\\to A^*\\), \\(\\iota_F(f)(a)=\\theta(a)(f)\\); it is isometric because \\(\\sup_{\\|a\\|\\le1}|\\theta(a)(f)|=\\|f\\|\\) by the Hahn–Banach theorem. We write \\(\\sigma(A,F)\\) for the weak\\(^*\\) topology carried over by \\(\\theta\\); its continuous functionals are exactly \\(\\iota_F(F)\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 684,
        "through_line": 687,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [
        {
          "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
          "local_scopes": [
            {
              "heading": "9. \\(W^*\\)-algebras and dual \\(C^*\\)-algebras",
              "line": 680,
              "through_line": 706,
              "anchors": [
                "OA-FND-WA-22"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
          ],
          "source_key": "human:sakai1956@7769A0DF4B803AFBAF405DF19E431CAA0B229DCAD461CC9BE69AD4BA2C5A8633",
          "source_locus": "Complete article, printed 763–773: Lemmas 1–8, §4 and appendix",
          "role": "proof comparison",
          "correspondence": "Original representation proof differs from the lesson’s Tomiyama route",
          "explanation": "Weak-star closed self-adjoint/positive cones, separating positive functionals, monotone projection limits and normal-state GNS direct sum yield a weakly closed faithful representation. The appendix removes the unit hypothesis using extreme points. Krein–Šmulian, separation, GNS and the final convex closure criterion are outside source inputs; full course providers supply them. The existing complete projection-of-norm-one/bidual proof is retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.2",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-22::Theorem 9.2",
      "anchor": "oa-fnd-wa-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 9.2** (Sakai's theorem). For a \\(C^*\\)-algebra \\(A\\) the following are equivalent:\n\n1. \\(A\\) is a \\(W^*\\)-algebra;\n2. \\(A\\) has a predual \\(F\\).\n\nWhen (2) holds, there is a faithful representation \\(\\pi\\) with \\(\\pi(A)\\) a von Neumann algebra that is a homeomorphism from \\(\\sigma(A,F)\\) onto the \\(\\sigma\\)-weak topology; and \\(\\psi\\mapsto\\psi\\circ\\pi\\) maps \\(\\pi(A)_*\\) isometrically onto \\(\\iota_F(F)\\).\n\n*Reference:* [Sakai 1956] gives the original characterization using normal-state representations. The complete proof below uses the norm-one projection and bidual route.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 688,
        "through_line": 706,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [
        {
          "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
          "local_scopes": [
            {
              "heading": "9. \\(W^*\\)-algebras and dual \\(C^*\\)-algebras",
              "line": 680,
              "through_line": 706,
              "anchors": [
                "OA-FND-WA-22"
              ],
              "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
            }
          ],
          "source_key": "human:sakai1956@7769A0DF4B803AFBAF405DF19E431CAA0B229DCAD461CC9BE69AD4BA2C5A8633",
          "source_locus": "Complete article, printed 763–773: Lemmas 1–8, §4 and appendix",
          "role": "proof comparison",
          "correspondence": "Original representation proof differs from the lesson’s Tomiyama route",
          "explanation": "Weak-star closed self-adjoint/positive cones, separating positive functionals, monotone projection limits and normal-state GNS direct sum yield a weakly closed faithful representation. The appendix removes the unit hypothesis using extreme points. Krein–Šmulian, separation, GNS and the final convex closure criterion are outside source inputs; full course providers supply them. The existing complete projection-of-norm-one/bidual proof is retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.1",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-10::Definition 10.1",
      "anchor": "oa-fnd-wa-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 10.1.** The elements of \\(M_*\\) are the *normal* functionals, and \\(M_*\\) is the *predual* of \\(M\\). The elements of \\(M_*^{\\perp}:=M^*(1-z_0)\\) are the *singular* functionals. (The symbol \\(M_*^\\perp\\) is only a name here; it is not an annihilator.)\n\nFor a \\(W^*\\)-algebra \\(A\\) with predual \\(F\\), normal and singular functionals are defined in the same way through the representation \\(\\pi\\) of [Theorem 9.2](#oa-fnd-wa-22). Since \\(\\mathcal M(\\pi)=\\pi(A)\\), [Proposition 5.2](#oa-fnd-wa-09)(2) gives the normal ones as \\(\\iota_F(F)=V(\\pi)=A^*z(\\pi)\\), and the singular ones are \\(A^*(1-z(\\pi))\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 717,
        "through_line": 720,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.2",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-10::Lemma 10.2",
      "anchor": "oa-fnd-wa-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 10.2** (Norms split along central projections). Let \\(N\\) be a von Neumann algebra, \\(z\\) a central projection and \\(\\varphi\\in N_*\\). Then \\(\\|\\varphi\\|=\\|\\varphi z\\|+\\|\\varphi(1-z)\\|\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 721,
        "through_line": 730,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.3",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-10::Theorem 10.3",
      "anchor": "oa-fnd-wa-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 10.3** (The normal–singular splitting).\n\n1. Every norm-closed left or right invariant subspace \\(V\\subseteq M^*\\) splits as\n\\[\n\\begin{gathered}\nV\\\\\n=(V\\cap M_*)\\\\\n\\oplus_1(V\\cap M_*^\\perp),\\\\\nV\\cap M_*\\\\\n=Vz_0,\\\\\nV\\cap M_*^\\perp\\\\\n=V(1-z_0),\n\\end{gathered}\n\\tag{10.2}\n\\]\nwhere \\(\\oplus_1\\) means \\(\\|\\varphi+\\psi\\|=\\|\\varphi\\|+\\|\\psi\\|\\) for \\(\\varphi\\in V\\cap M_*\\) and \\(\\psi\\in V\\cap M_*^\\perp\\).\n2. Let \\(\\pi\\) be a representation of the \\(C^*\\)-algebra \\(M\\) on \\(H_\\pi\\), and put \\(z=\\bar\\pi(z_0)\\), a central projection of \\(\\mathcal M(\\pi)\\). Then \\(\\pi_n(x)=\\pi(x)z\\) is a normal representation of \\(M\\) on \\(zH_\\pi\\), and \\(\\pi_s(x)=\\pi(x)(1-z)\\) is a representation of \\(M\\) on \\((1-z)H_\\pi\\) whose coefficient functionals \\(\\omega_{\\pi_s;\\xi,\\eta}\\) are all singular.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 731,
        "through_line": 754,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.4",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-10::Definition 10.4",
      "anchor": "oa-fnd-wa-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 10.4.** A bounded linear map \\(T:M\\to N\\) between von Neumann algebras is *normal* if \\(\\psi\\circ T\\in M_*\\) for every \\(\\psi\\in N_*\\), and *singular* if \\(\\psi\\circ T\\in M_*^\\perp\\) for every \\(\\psi\\in N_*\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 755,
        "through_line": 756,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.5",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-10::Proposition 10.5",
      "anchor": "oa-fnd-wa-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 10.5** (Splitting of maps). Every bounded linear \\(T:M\\to N\\) is uniquely \\(T=T_n+T_s\\) with \\(T_n\\) normal and \\(T_s\\) singular. If \\(T\\) is positive, so are \\(T_n\\) and \\(T_s\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 757,
        "through_line": 760,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 10.6",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-10::Exercise 10.6",
      "anchor": "oa-fnd-wa-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 10.6** (medium) (Nearby pure states). Let \\(\\varphi,\\psi\\) be pure states of \\(A\\) with \\(\\|\\varphi-\\psi\\|<2\\). Show that \\(\\pi_\\varphi\\) and \\(\\pi_\\psi\\) are unitarily equivalent.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 761,
        "through_line": 777,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.1",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Lemma 11.1",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 11.1** (Supports of normal positive functionals). Let \\(A\\) be a \\(W^*\\)-algebra with a predual \\(A_*\\), and \\(\\omega\\) a nonzero normal positive functional. There is a unique nonzero projection \\(e\\in A\\) such that \\(\\omega=e\\omega=\\omega e\\) (so \\(\\omega(x)=\\omega(exe)\\)) and \\(\\omega\\) is faithful on \\(eAe\\). It is the *support* \\(s(\\omega)\\). Also \\(1-s(\\omega)\\) is the largest projection on which \\(\\omega\\) vanishes, so this is the support recalled in the background.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 782,
        "through_line": 793,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.2",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Theorem 11.2",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 11.2** (Singular functionals). For a positive \\(\\omega\\in A^*\\), the following are equivalent:\n\n1. \\(\\omega\\) is singular;\n2. every nonzero projection \\(e\\in A\\) majorises a nonzero projection \\(e_0\\) with \\(\\omega(e_0)=0\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 794,
        "through_line": 802,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.3",
      "kind": "corollary",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Corollary 11.3",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Corollary 11.3** (Uniqueness of the predual). If \\(F_1,F_2\\) are preduals of a \\(C^*\\)-algebra \\(A\\), then \\(\\iota_{F_1}(F_1)=\\iota_{F_2}(F_2)\\) in \\(A^*\\); equivalently \\(\\sigma(A,F_1)=\\sigma(A,F_2)\\). We write \\(A_*\\) for this space.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 803,
        "through_line": 806,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.4",
      "kind": "corollary",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Corollary 11.4",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Corollary 11.4** (Isomorphisms are normal). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) between von Neumann algebras is a homeomorphism for the \\(\\sigma\\)-weak topologies, and also for the \\(\\sigma\\)-strong and the \\(\\sigma\\)-strong\\(^*\\) topologies.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 807,
        "through_line": 812,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.5",
      "kind": "corollary",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Corollary 11.5",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Corollary 11.5** (Complete additivity). For \\(\\omega\\in A^*\\) on a \\(W^*\\)-algebra \\(A\\), the following are equivalent:\n\n1. \\(\\omega\\) is normal;\n2. \\(\\omega\\) is completely additive: for every family \\((e_i)_{i\\in I}\\) of mutually orthogonal projections, the finite partial sums of \\(\\sum_i\\omega(e_i)\\) converge to \\(\\omega(\\sum_ie_i)\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 813,
        "through_line": 821,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.7",
      "kind": "example",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Example 11.7",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Example 11.7** (Which topologies are intrinsic). The \\(\\sigma\\)-weak, \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies are (Corollary 11.4). The weak, strong and strong\\(^*\\) operator topologies are not. Let \\(M=B(H)\\) with \\(\\dim H=\\infty\\), \\(\\pi_1\\) the identity representation and \\(\\pi_2(x)=x\\oplus x\\oplus\\cdots\\) on \\(H\\oplus H\\oplus\\cdots\\). Both are faithful normal representations with von Neumann algebra images. The weak operator topology of \\(\\pi_2(M)\\), pulled back to \\(M\\), is the \\(\\sigma\\)-weak topology, since the vector functionals of \\(\\pi_2\\) are exactly the square-summable series \\(\\sum_n\\langle x\\xi_n,\\eta_n\\rangle\\). It is strictly finer than the weak operator topology of \\(\\pi_1\\). Indeed, with an orthonormal sequence \\((e_n)\\), the functional \\(\\varphi(x)=\\sum_nn^{-2}\\langle xe_n,e_n\\rangle\\) is \\(\\sigma\\)-weakly continuous. It is not weakly continuous: a weakly continuous functional has the form \\(\\sum_{k=1}^m\\langle x\\xi_k,\\eta_k\\rangle\\) (background fact on preduals). Choose a unit vector \\(\\zeta\\in\\operatorname{span}\\{e_1,\\dots,e_{m+1}\\}\\) orthogonal to \\(\\xi_1,\\dots,\\xi_m\\), and let \\(x\\) be the projection onto \\(\\mathbb C\\zeta\\). Then the finite sum vanishes at \\(x\\), but \\(\\varphi(x)=\\sum_nn^{-2}|\\langle\\zeta,e_n\\rangle|^2>0\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 832,
        "through_line": 833,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.8",
      "kind": "example",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Example 11.8",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Example 11.8** (Normal and singular parts on \\(\\ell^\\infty\\)). In \\(\\ell^\\infty\\) every nonzero projection majorises some coordinate projection \\(e_n\\), and the \\(e_n\\) are minimal. By Theorem 11.2, a positive functional is singular exactly when it vanishes on every \\(e_n\\), that is on \\(c_0\\). Hence for any \\(\\varphi\\in(\\ell^\\infty)^*\\), \\(\\varphi_n(x)=\\sum_k\\varphi(e_k)x_k\\) and \\(\\varphi_s=\\varphi-\\varphi_n\\). A state that is a limit along a free ultrafilter is singular, and it has no support in the sense of [Lemma 11.1](#oa-fnd-wa-23).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 834,
        "through_line": 835,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 11.9",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Exercise 11.9",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 11.9.** (easy) Let \\(\\mathcal V\\) be a free ultrafilter on \\(\\mathbb N\\), \\(\\chi(x)=\\lim_{n\\to\\mathcal V}x_n\\) on \\(\\ell^\\infty\\), and \\(\\psi=\\tfrac12(\\delta_1+\\chi)\\), where \\(\\delta_1(x)=x_1\\). Find \\(\\psi_n\\), \\(\\psi_s\\), the cyclic representation \\(\\pi_\\psi\\), and its normal and singular parts.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 836,
        "through_line": 839,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 11.10",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-18::Exercise 11.10",
      "anchor": "oa-fnd-wa-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 11.10** (medium) (When every state is normal). Prove that all states of a von Neumann algebra \\(M\\) are normal exactly when \\(\\dim M<\\infty\\).\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 840,
        "through_line": 843,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.1",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-19::Proposition 12.1",
      "anchor": "oa-fnd-wa-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 12.1.** Let \\(A\\) be a \\(W^*\\)-algebra and \\(\\pi\\) a normal representation of \\(A\\) on \\(H\\). Then \\(\\ker\\pi=A(1-z)\\) for a central projection \\(z\\in A\\); the image satisfies \\(\\pi(A)=\\pi(A)''\\); and \\(\\pi\\) restricts to a \\(*\\)-isomorphism of \\(Az\\) onto \\(\\pi(A)\\), which is a homeomorphism for the \\(\\sigma\\)-weak topologies. The cyclic representation of a normal positive functional is normal.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 848,
        "through_line": 861,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.2",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-19::Exercise 12.2",
      "anchor": "oa-fnd-wa-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 12.2** (hard) (Fixed points of a unitary group). Let \\(G\\) be a group of unitaries on \\(H\\), \\(M=G''\\), and \\(e_0\\) the projection onto \\(H_0=\\{\\xi:u\\xi=\\xi\\ \\forall u\\in G\\}\\). Show: (a) \\(e_0\\) is a central projection of \\(M\\) with \\(Me_0=\\mathbb Ce_0\\), so \\(M_{e_0}\\cong\\mathbb C\\) when \\(e_0\\ne0\\); (b) a nonempty closed convex \\(G\\)-invariant set \\(\\mathfrak L\\subseteq H\\) meets \\(H_0\\); (c) if \\(\\mathcal K\\) is the weakly (equivalently strongly) closed convex hull of \\(G\\) and \\(\\pi\\) is a normal representation of \\(M\\), then \\(\\pi(e_0)\\xi\\in\\pi(\\mathcal K)\\xi\\) for every vector \\(\\xi\\) of \\(\\pi\\); (d) \\(e_0\\in\\mathcal K\\).\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 862,
        "through_line": 877,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.3",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-19::Exercise 12.3",
      "anchor": "oa-fnd-wa-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 12.3** (hard) (Faithfulness seen on a separable subalgebra). Let \\(A\\) be a separable \\(C^*\\)-algebra acting nondegenerately on \\(H\\ne0\\), and \\(M=A''\\). Assume that every normal state of \\(M\\) whose restriction to \\(A\\) is faithful is itself faithful. Show:\n(a) \\(M\\) has a faithful normal state \\(\\varphi\\);\n(b) every nonzero projection \\(e\\in M\\) majorises a nonzero positive element of \\(A\\), that is \\(eMe\\cap A\\ne\\{0\\}\\);\n(c) no singular state of \\(M\\) is faithful on \\(A\\);\n(d) if a state \\(\\omega\\) of \\(M\\) is faithful on \\(A\\), its normal part \\(\\omega_n\\) is faithful on \\(M\\);\n(e) \\(A\\) has a faithful state \\(\\omega=\\sum_n\\lambda_n\\omega_n\\) with pure states \\(\\omega_n\\), \\(\\lambda_n\\ge0\\), \\(\\sum_n\\lambda_n=1\\);\n(f) if \\(\\varphi\\) is the normal part of a Hahn–Banach extension \\(\\bar\\omega\\) of this \\(\\omega\\) to \\(M\\), then the cyclic representation \\(\\pi_\\varphi\\) of \\(M\\) is faithful and normal and \\(\\pi_\\varphi(M)\\) is atomic;\n(g) \\(M\\) is generated by its minimal projections, and \\(A\\) contains all of them.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 878,
        "through_line": 904,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 13.1",
      "kind": "definition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Definition 13.1",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Definition 13.1.** We call a \\(C^*\\)-algebra \\(A\\) *monotone closed* when each increasing, norm-bounded net of self-adjoint elements has a least upper bound among the self-adjoint elements of \\(A\\). A positive functional \\(\\omega\\) on such an \\(A\\) is *normal* if \\(\\omega(\\sup_ix_i)=\\sup_i\\omega(x_i)\\) for each such net \\((x_i)\\). We say \\(A\\) has *sufficiently many* normal positive functionals if for every nonzero \\(x\\in A_+\\) some normal positive \\(\\omega\\) has \\(\\omega(x)>0\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 909,
        "through_line": 910,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.2",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Proposition 13.2",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 13.2.**\n\n1. A \\(W^*\\)-algebra is monotone closed, and for it the two meanings of *normal positive functional* agree.\n2. A monotone closed \\(C^*\\)-algebra has a unit.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 911,
        "through_line": 921,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 13.4",
      "kind": "lemma",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Lemma 13.4",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Lemma 13.4.** Let \\(A\\) be monotone closed and \\(\\omega\\in A^*_+\\). If \\(\\omega\\) lies in the norm closure of the set of normal positive functionals, then \\(\\omega\\) is normal. In particular this holds if \\(\\|\\omega-\\omega_n\\|\\to0\\) for a sequence of normal positive \\(\\omega_n\\).",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 922,
        "through_line": 936,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.6",
      "kind": "proposition",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Proposition 13.6",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Proposition 13.6.** Let \\(A\\) be monotone closed and \\(\\omega\\) a normal positive functional with cyclic representation \\((\\pi,H,\\xi)\\). Then \\(\\pi(A)=\\pi(A)''\\).\n\nThe representation need not be faithful: a vector state on the first summand of \\(B(H_1)\\oplus B(H_2)\\) kills the second summand. Steps 2 and 3 prove that every bounded increasing net in the image lifts to a bounded increasing net in the appropriate central corner. This lifting justifies the use of the full monotone closure criterion.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 937,
        "through_line": 959,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 13.7",
      "kind": "theorem",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Theorem 13.7",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Theorem 13.7** (Kadison's characterization). A \\(C^*\\)-algebra is a \\(W^*\\)-algebra exactly when it is monotone closed and has sufficiently many normal positive functionals.",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 960,
        "through_line": 966,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 13.9",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Exercise 13.9",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 13.9.** (easy) Show that a von Neumann algebra that is separable in the norm topology is finite-dimensional.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 967,
        "through_line": 970,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 13.10",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Exercise 13.10",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 13.10** (hard) (Ranges of projections of norm one). Let \\(A\\) be a \\(C^*\\)-algebra acting on \\(H\\), let \\(M\\supseteq A\\) be a von Neumann algebra acting on \\(H\\), and let \\(\\varepsilon:M\\to A\\) be a projection of norm one. (a) Prove that each increasing, norm-bounded net \\((h_i)\\) of positive elements of \\(A\\) has a least upper bound in \\(A\\), and that this bound need not be the strong limit. (b) Prove that a separable such \\(A\\) is finite-dimensional.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 971,
        "through_line": 993,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 13.11",
      "kind": "exercise",
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras#oa-fnd-wa-27::Exercise 13.11",
      "anchor": "oa-fnd-wa-27",
      "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
      "statement_and_full_conditions": "**Exercise 13.11** (medium) (Algebras that are not dual spaces). Show that neither \\(c_0\\) nor \\(C[0,1]\\) is isometrically isomorphic to the dual of a Banach space.\n\n*",
      "proof_locus": {
        "source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "line": 994,
        "through_line": 997,
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 1.1",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-01::Theorem 1.1",
      "anchor": "oa-fnd-pb-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 1.1** (Continuous images of the Baire space). Let \\(X\\) be a nonempty Polish space with a complete compatible metric \\(d\\). There are nonempty closed sets \\(F_s\\subseteq X\\), \\(s\\in\\mathbb N^{<\\mathbb N}\\), with\n\\[\n\\begin{gathered}\nF_\\varnothing=X,\\\\\nF_s=\\bigcup_{n\\in\\mathbb N}F_{sn},\\\\\n\\operatorname{diam}F_s\\leq2^{-|s|}\\ \\ (|s|\\geq1).\n\\end{gathered}\n\\tag{1}\n\\]\nFor every family as in (1), \\(\\{\\varphi(t)\\}=\\bigcap_kF_{t|k}\\) defines a continuous map \\(\\varphi:\\Lambda\\to X\\) with \\(\\varphi(\\Lambda_s)=F_s\\) for all \\(s\\). In particular \\(\\varphi\\) maps \\(\\Lambda\\) onto \\(X\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 71,
        "through_line": 87,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.3",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-01::Lemma 1.3",
      "anchor": "oa-fnd-pb-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 1.3** (Perfect set lemma). Every uncountable Polish space \\(X\\) contains a compact subset homeomorphic to \\(\\mathcal C\\). Hence \\(|X|=2^{\\aleph_0}\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 88,
        "through_line": 95,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.1",
      "kind": "definition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-02::Definition 2.1",
      "anchor": "oa-fnd-pb-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Definition 2.1.** A *Souslin space*, also called an analytic space, is a metrizable space onto which some Polish space maps continuously. A *Souslin set* in a topological space \\(Z\\) is a subset of \\(Z\\) that, with the relative topology, is a Souslin space.\n\nSo the Souslin sets of a metrizable space are exactly its Polish images. The empty space is a Souslin space. A nonempty Souslin space is the image of \\(\\Lambda\\) under a continuous map: compose a continuous map of \\(\\Lambda\\) onto the Polish space ([Theorem 1.1](#oa-fnd-pb-01)) with the given map. Whether a set is a Souslin set depends only on the set with its own topology. Hence a Souslin set of a subspace \\(Y\\subseteq Z\\) is a Souslin set of \\(Z\\), and a subset of \\(Y\\) that is a Souslin set of \\(Z\\) is one of \\(Y\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 102,
        "through_line": 105,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.2",
      "kind": "proposition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-02::Proposition 2.2",
      "anchor": "oa-fnd-pb-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Proposition 2.2.**\n\n1. A Souslin space is separable, and its Borel structure is countably generated and countably separated.\n2. If \\(X\\) is a Souslin space, \\(Y\\) is metrizable and \\(f:X\\to Y\\) is continuous, then the subset \\(f(X)\\) of \\(Y\\) is a Souslin set.\n3. A countable product of Souslin spaces is a Souslin space.\n4. Let \\(Z\\) be a Hausdorff space and \\(A_1,A_2,\\ldots\\) Polish images in \\(Z\\). Then \\(\\bigcup_nA_n\\) and \\(\\bigcap_nA_n\\) are Polish images in \\(Z\\). In particular, in a metrizable space countable unions and countable intersections of Souslin sets are Souslin sets.\n5. Open subsets and closed subsets of a Souslin space are Souslin sets.\n6. Let \\(f:X\\to Y\\) be a continuous map between metrizable spaces, and \\(A\\subseteq X\\), \\(A'\\subseteq Y\\) Souslin sets. Then \\(f(A)\\) and \\(A\\cap f^{-1}(A')\\) are Souslin sets.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 106,
        "through_line": 135,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.3",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-03::Lemma 2.3",
      "anchor": "oa-fnd-pb-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 2.3.** Let \\(E_n,E'_m\\) (\\(n,m\\in\\mathbb N\\)) be subsets of a Borel space, and for all \\(n\\) and \\(m\\) let \\(B_{n,m}\\) be a Borel set that contains \\(E_n\\) and misses \\(E'_m\\). Then \\(\\bigcup_nE_n\\) and \\(\\bigcup_mE'_m\\) are Borel-separated by\n\\[\nB=\\bigcup_n\\bigcap_mB_{n,m}.\n\\tag{2}\n\\]",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 140,
        "through_line": 149,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.4",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-03::Theorem 2.4",
      "anchor": "oa-fnd-pb-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 2.4** (Lusin's separation theorem). Let \\(X\\) be a Hausdorff space and \\((X_n)_{n\\in I}\\), with \\(I\\) countable, pairwise disjoint Polish images in \\(X\\). There are pairwise disjoint Borel sets \\(B_n\\) with \\(X_n\\subseteq B_n\\). In particular this holds for pairwise disjoint Souslin sets of a metrizable space.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 150,
        "through_line": 158,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.5",
      "kind": "corollary",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-03::Corollary 2.5",
      "anchor": "oa-fnd-pb-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Corollary 2.5** (Souslin's theorem). Let \\(X\\) be a Hausdorff space and \\(A\\subseteq X\\). If \\(A\\) and \\(X\\setminus A\\) are both Polish images, then \\(A\\) is Borel. In particular, in a metrizable space a Souslin set whose complement is a Souslin set is Borel.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 159,
        "through_line": 163,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.6",
      "kind": "corollary",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-12::Corollary 2.6",
      "anchor": "oa-fnd-pb-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Corollary 2.6** (Borel subsets of Souslin spaces). Every Borel set in a Souslin space is itself a Souslin set. In particular, all Borel subsets of Polish spaces are Souslin sets.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 166,
        "through_line": 171,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.8",
      "kind": "corollary",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-12::Corollary 2.8",
      "anchor": "oa-fnd-pb-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Corollary 2.8** (Counting). A Polish space has at most \\(2^{\\aleph_0}\\) Souslin sets, and so at most \\(2^{\\aleph_0}\\) Borel sets. Consequently \\(\\mathcal C\\) has subsets that are not Borel.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 172,
        "through_line": 175,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.9",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-12::Example 2.9",
      "anchor": "oa-fnd-pb-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 2.9** (A Souslin set that is not Borel). Let \\(Z=\\mathcal C\\times\\Lambda\\), a Polish space, with a countable base \\((V_n)_{n\\geq1}\\).\n\n(a) The set \\[\n\\begin{gathered}\nO\\\\\n=\\{(y,z)\\in\\mathcal C\\times Z:\\ y_n=1\\text{ and }z\\in V_n\\\\\n\\text{ for some }n\\}\n\\end{gathered}\n\\] is open, being the union of the open sets \\(\\{y:y_n=1\\}\\times V_n\\). For an open \\(W\\subseteq Z\\), let \\(y_n=1\\) exactly when \\(V_n\\subseteq W\\); then the section \\(O_y=\\{z:(y,z)\\in O\\}\\) equals \\(W\\). So \\(F=(\\mathcal C\\times Z)\\setminus O\\) is a closed set whose sections \\(F_y\\) run through all closed subsets of \\(Z\\).\n\n(b) Let \\[\n\\begin{gathered}\nU\\\\\n=\\{(y,x)\\in\\mathcal C\\times\\mathcal C:\\ (y,x,w)\\in F\\\\\n\\text{ for some }w\\in\\Lambda\\},\n\\end{gathered}\n\\] the projection of \\(F\\) along \\(\\Lambda\\). It is a Souslin set of \\(\\mathcal C\\times\\mathcal C\\), being the image of the Polish space \\(F\\) under a continuous map. Every Souslin set \\(A\\subseteq\\mathcal C\\) is a section \\(U_y\\). If \\(A=\\varnothing\\), take \\(y\\) with \\(F_y=\\varnothing\\). Otherwise \\(A=g(\\Lambda)\\) with \\(g\\) continuous, as noted after Definition 2.1; the set \\(K=\\{(g(w),w):w\\in\\Lambda\\}\\) is closed in \\(Z\\) because \\(\\mathcal C\\) is Hausdorff, its projection to \\(\\mathcal C\\) is \\(A\\), and we take \\(y\\) with \\(F_y=K\\).\n\n(c) Let \\(D=\\{x\\in\\mathcal C:(x,x)\\in U\\}\\). It is the projection to \\(\\mathcal C\\) of the closed set \\(\\{(x,w)\\in\\mathcal C\\times\\Lambda:(x,x,w)\\in F\\}\\), so \\(D\\) is a Souslin set. If \\(D\\) were Borel, \\(\\mathcal C\\setminus D\\) would be Borel, hence a Souslin set (Corollary 2.6), hence \\(\\mathcal C\\setminus D=U_{y_0}\\) for some \\(y_0\\). Then \\(y_0\\in D\\) exactly when \\((y_0,y_0)\\in U\\), that is, exactly when \\(y_0\\in U_{y_0}=\\mathcal C\\setminus D\\). This is absurd. So \\(D\\) is a Souslin set that is not Borel, and by Souslin's theorem its complement is not a Souslin set.\n\nThe argument in (c) is Cantor's diagonal argument, applied to the universal set \\(U\\). It shows that the image of a closed set under a continuous map need not be Borel: \\(D\\) is the projection of a closed subset of \\(\\mathcal C\\times\\Lambda\\). This is a strong form of the fact that Borel maps need not carry Borel sets to Borel sets. Examples 5.9 and 6.4 return to \\(D\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 176,
        "through_line": 197,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
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              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
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                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
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            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
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                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
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              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
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          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.1",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-04::Lemma 3.1",
      "anchor": "oa-fnd-pb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 3.1** (Zero-dimensional refinement). Let \\((X,\\tau)\\) be a Polish space and \\(B_1,B_2,\\ldots\\) Borel sets. There is a zero-dimensional Polish topology \\(\\tau'\\supseteq\\tau\\) with the same Borel sets as \\(\\tau\\), in which every \\(B_n\\) is open and closed.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 204,
        "through_line": 211,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
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              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.2",
      "kind": "definition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-04::Definition 3.2",
      "anchor": "oa-fnd-pb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Definition 3.2.** A metrizable space \\(X\\) is a *Lusin space* if some zero-dimensional Polish space maps continuously and bijectively onto \\(X\\). A *Lusin set* in a topological space \\(Z\\) is a subset of \\(Z\\) that, with the relative topology, is a Lusin space.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 212,
        "through_line": 213,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
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          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
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          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.3",
      "kind": "proposition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-04::Proposition 3.3",
      "anchor": "oa-fnd-pb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Proposition 3.3.**\n\n1. A metrizable space is a Lusin space if and only if some Polish space, zero-dimensional or not, maps continuously and bijectively onto it. So zero-dimensionality in the definition costs nothing.\n2. Polish spaces are Lusin spaces, and Lusin spaces are Souslin spaces.\n3. In a Hausdorff space, a countable intersection of Lusin sets is a Lusin set.\n4. A countable union of pairwise disjoint Lusin sets of a metrizable space is a Lusin set.\n5. Open subsets, closed subsets and \\(G_\\delta\\) subsets of a Lusin space are Lusin sets.\n6. A countable product of Lusin spaces is a Lusin space.\n\nExample 3.8 proves that the intersection assertion (3) needs the Hausdorff hypothesis on the ambient space.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 214,
        "through_line": 238,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
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          "local_scopes": [
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              "heading": "1. Trees in Polish spaces",
              "line": 67,
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              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
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            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.4",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-04::Theorem 3.4",
      "anchor": "oa-fnd-pb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 3.4** (Closed subsets of the Baire space).\n\n1. Every zero-dimensional Polish space is homeomorphic to a closed subset of \\(\\Lambda\\).\n2. For every Lusin space \\(X\\), in particular for every Polish space, there are a closed set \\(T\\subseteq\\Lambda\\) and a continuous bijection of \\(T\\) onto \\(X\\).\n3. The closed set \\(T\\) in (2) cannot in general be a product \\(\\prod_kN_k\\) in which each \\(N_k\\) is \\(\\mathbb N\\) or a finite set \\(\\{1,\\dots,N_k\\}\\): the Polish space \\(X=[0,1]\\cup\\{2\\}\\subseteq\\mathbb R\\) is not the image of any such product under a continuous bijection.\n\nThe explicit isolated-point argument in (3) proves that the closed tree in (2) cannot always be replaced by a product.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 239,
        "through_line": 258,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
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              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
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              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
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              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
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              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
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              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
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              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
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          ],
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          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.5",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-04::Example 3.5",
      "anchor": "oa-fnd-pb-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 3.5** (A tree, not a product). By Theorem 3.4(3), no product \\(\\prod_kN_k\\) maps continuously and bijectively onto \\(X=[0,1]\\cup\\{2\\}\\). A closed subset of \\(\\Lambda\\) does. Take a closed \\(T_1\\subseteq\\Lambda\\) and a continuous bijection \\(\\chi:T_1\\to[0,1]\\) (Theorem 3.4(2)), and let\n\\[\n\\begin{gathered}\nT=\\{(1,t_1,t_2,\\dots):t\\in T_1\\}\\\\\n\\cup\\{(2,1,1,1,\\dots)\\}.\n\\end{gathered}\n\\]\n\\(T\\) is closed, \\((2,1,1,\\dots)\\) is an isolated point of \\(T\\), and the map sending \\((1,t_1,t_2,\\dots)\\) to \\(\\chi(t)\\) and \\((2,1,1,\\dots)\\) to \\(2\\) is a continuous bijection of \\(T\\) onto \\(X\\). Below the node \\((2)\\) the tree has a single branch, while below \\((1)\\) it has uncountably many. A product cannot express branching that depends on the node. This is why the partitions in the proof of Theorem 3.4 produce a closed subset of \\(\\Lambda\\) and not a product: the number of nonempty pieces depends on the piece being divided, not only on the level.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 259,
        "through_line": 267,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
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              "heading": "1. Trees in Polish spaces",
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              "anchors": [
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              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
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              "heading": "2. Souslin sets",
              "line": 96,
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                "OA-FND-PB-03",
                "OA-FND-PB-12"
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            },
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              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
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                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
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            },
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              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
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                "OA-FND-PB-06"
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              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
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              "heading": "5. Standard Borel spaces",
              "line": 354,
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                "OA-FND-PB-08"
              ],
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            },
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              "line": 685,
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                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
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          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
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          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.6",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-05::Theorem 3.6",
      "anchor": "oa-fnd-pb-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 3.6** (Lusin–Souslin). Let \\(P\\) be a Polish space, \\(X\\) a Hausdorff space and \\(g:P\\to X\\) continuous and injective. Then \\(g(B)\\) is Borel in \\(X\\) for every Borel set \\(B\\subseteq P\\). In particular \\(g(P)\\) is Borel, and \\(g\\) is a Borel isomorphism of \\(P\\) onto \\(g(P)\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 270,
        "through_line": 284,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.7",
      "kind": "corollary",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-05::Corollary 3.7",
      "anchor": "oa-fnd-pb-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Corollary 3.7.**\n\n1. Every Borel set in a Lusin space is itself a Lusin set.\n2. A Lusin set in a Hausdorff space, in particular in a metrizable space, is Borel.\n3. If \\(L\\) is a Lusin space and \\(h:L\\to X\\) is a continuous injection into a Hausdorff space, then \\(h(L)\\) is Borel and \\(h\\) is a Borel isomorphism of \\(L\\) onto \\(h(L)\\).\n4. A Lusin space, with its Borel sets, is a standard Borel space.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 285,
        "through_line": 295,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.8",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-05::Example 3.8",
      "anchor": "oa-fnd-pb-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 3.8** (Lusin sets in a space that is not Hausdorff). Let \\(A\\subseteq[0,1]\\) be a set that is not Borel. It exists by the counting argument ([Corollary 2.8](#oa-fnd-pb-12)), because \\([0,1]\\) is Borel isomorphic to \\(\\mathcal C\\) ([Theorem 5.2](#oa-fnd-pb-07)). Glue two copies of \\([0,1]\\) along \\(A\\): let \\(X\\) be the quotient of \\([0,1]\\times\\{1,2\\}\\) by the relation \\((a,1)\\sim(a,2)\\) for \\(a\\in A\\), with quotient map \\(q\\) and the quotient topology. Put \\(L_i=q([0,1]\\times\\{i\\})\\).\n\n- For each \\(i\\), the map \\(t\\mapsto q(t,i)\\) is a homeomorphism of \\([0,1]\\) onto \\(L_i\\) with its relative topology. It is continuous and injective. For \\(V\\subseteq[0,1]\\) open, the preimage of \\(q(V\\times\\{1,2\\})\\) is \\(V\\times\\{1,2\\}\\), so \\(q(V\\times\\{1,2\\})\\) is open in \\(X\\). Its intersection with \\(L_1\\) is \\(q(V\\times\\{1\\})\\), because \\(q(t,2)\\) lies in \\(L_1\\) only when \\(t\\in A\\), and then \\(q(t,2)=q(t,1)\\). So the map is open onto \\(L_1\\), and likewise onto \\(L_2\\).\n- Hence \\(L_1\\) and \\(L_2\\) are Lusin sets (Proposition 3.3(2)).\n- \\(L_1\\cap L_2=q(A\\times\\{1\\})\\) is homeomorphic to \\(A\\subseteq[0,1]\\). If \\(A\\) were a Lusin space, it would be Borel in \\([0,1]\\) by Corollary 3.7(2). So \\(L_1\\cap L_2\\) is not a Lusin set.\n- \\(X\\) is not Hausdorff. \\(A\\) is not closed, so some \\(a\\notin A\\) is a limit of points \\(a_n\\in A\\). Then \\(q(a,1)\\neq q(a,2)\\), but every neighbourhood of either point contains \\(q(a_n,1)=q(a_n,2)\\) for large \\(n\\).\n\nSo the intersection rule of Proposition 3.3(3) fails here. In its proof, the fibre product \\(P\\) is closed only because the diagonal of a Hausdorff space is closed.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 296,
        "through_line": 304,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.1",
      "kind": "proposition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-06::Proposition 4.1",
      "anchor": "oa-fnd-pb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Proposition 4.1** (Graphs of Borel maps). Let \\((X,\\Sigma)\\) be a measurable space and \\((Y,\\mathcal B_Y)\\) a Borel space whose points are separated by a sequence \\((S_n)\\) of Borel sets. For every measurable \\(f:X\\to Y\\), with graph \\(\\operatorname{Gr}f=\\{(x,f(x)):x\\in X\\}\\),\n\\[\n\\begin{gathered}\n(X\\times Y)\\setminus\\operatorname{Gr}f\\\\\n=\\bigcup_n\\Big(\\big(f^{-1}(S_n)\\times(Y\\setminus S_n)\\big)\\\\\n\\cup\\big(f^{-1}(Y\\setminus S_n)\\times S_n\\big)\\Big),\n\\end{gathered}\n\\tag{4}\n\\]\nso \\(\\operatorname{Gr}f\\) belongs to \\(\\Sigma\\otimes\\mathcal B_Y\\). Consequently:\n\n1. If \\(X\\) and \\(Y\\) are topological spaces, \\(f\\) is Borel, and countably many Borel sets separate the points of \\(Y\\) (for example, \\(Y\\) separable metrizable), then \\(\\operatorname{Gr}f\\) is a Borel subset of \\(X\\times Y\\).\n2. If \\(Y\\) is countably separated, the points fixed by a Borel map \\(h:Y\\to Y\\) form a Borel set.\n3. Conversely, if the diagonal \\(\\Delta_Y=\\{(y,y):y\\in Y\\}\\) lies in \\(\\mathcal B_Y\\otimes\\mathcal B_Y\\), then \\(Y\\) is countably separated.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 309,
        "through_line": 333,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.3",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-06::Theorem 4.3",
      "anchor": "oa-fnd-pb-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 4.3** (Images and preimages under Borel maps). Let \\(X\\) be a Souslin space, \\(Y\\) a separable metrizable space and \\(f:X\\to Y\\) a Borel map.\n\n1. \\(f(A)\\) is a Souslin set in \\(Y\\) for every Souslin set \\(A\\subseteq X\\), in particular for every Borel set.\n2. \\(f^{-1}(A')\\) is a Souslin set in \\(X\\) for every Souslin set \\(A'\\subseteq Y\\).\n3. If \\(f\\) is injective, it is a Borel isomorphism of \\(X\\) onto \\(f(X)\\).\n4. If \\(X\\) is a Lusin space and \\(f\\) is injective, then \\(f(B)\\) is Borel in \\(Y\\) for every Borel \\(B\\subseteq X\\).\n5. Let \\(X'\\) be a standard Borel space and \\(f':X'\\to Y\\) an injective Borel map. Then \\(f'(X')\\) is Borel in \\(Y\\), and \\(f'\\) is a Borel isomorphism of \\(X'\\) onto \\(f'(X')\\). In particular, a Borel bijection between standard Borel spaces is a Borel isomorphism.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 334,
        "through_line": 353,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.1",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-07::Lemma 5.1",
      "anchor": "oa-fnd-pb-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 5.1** (Borel Schröder–Bernstein). Let \\(X\\) and \\(Y\\) be Borel spaces, and let \\(f:X\\to Y\\) and \\(g:Y\\to X\\) be injective Borel maps that carry Borel sets to Borel sets. Then \\(X\\) and \\(Y\\) are isomorphic.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 360,
        "through_line": 365,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.2",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-07::Theorem 5.2",
      "anchor": "oa-fnd-pb-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 5.2** (Isomorphism theorem). Consider a standard Borel space \\(X\\).\n\n1. When \\(X\\) is countable, every subset of \\(X\\) is Borel.\n2. If \\(X\\) is uncountable, \\(X\\) is isomorphic to \\(\\mathcal C\\), and therefore to \\([0,1]\\), to \\(\\mathbb R\\), to \\(\\Lambda\\), and to every other uncountable standard Borel space.\n3. Two standard Borel spaces are isomorphic if and only if they have the same cardinality. The possible cardinalities are the finite ones, \\(\\aleph_0\\) and \\(2^{\\aleph_0}\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 366,
        "through_line": 379,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.3",
      "kind": "definition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Definition 5.3",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Definition 5.3.** A *Souslin–Borel space* is a countably separated Borel space that is the image of some standard Borel space under a Borel map.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 382,
        "through_line": 383,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.4",
      "kind": "definition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Definition 5.4",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Definition 5.4.** In a standard Borel space \\(X\\), call \\(A\\subseteq X\\) a *Souslin set* if \\(A=\\varnothing\\) or \\(A=f(Z)\\) for some standard Borel space \\(Z\\) and some Borel map \\(f:Z\\to X\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 384,
        "through_line": 385,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
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              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
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              "heading": "8. Topological quotients and open homomorphisms",
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              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
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              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
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          ],
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          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
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              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.5",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Lemma 5.5",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 5.5.**\n\n1. For a Polish space \\(X\\) and a set \\(A\\subseteq X\\), \\(A\\) is a Souslin set in the topological sense of Definition 2.1 exactly when it is one in the sense of Definition 5.4. So being a Souslin set of a standard Borel space does not depend on the Polish topology used to describe its Borel sets.\n2. With its relative Borel structure, a Souslin set \\(A\\) of a standard Borel space is a Souslin–Borel space.\n3. For a Souslin–Borel space \\(X\\) and a sequence \\((B_n)\\) of Borel sets that separates points, the formula \\(\\varphi(x)=(1_{B_n}(x))_n\\) defines an injective Borel map \\(\\varphi:X\\to\\mathcal C\\) whose image \\(\\varphi(X)\\) is a Souslin set in \\(\\mathcal C\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 386,
        "through_line": 397,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.6",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Theorem 5.6",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 5.6** (Blackwell). Let \\(X\\) be a Souslin–Borel space, \\((B_n)\\) a countable family of Borel sets, and \\(\\mathcal B_0\\) the \\(\\sigma\\)-algebra they generate. A Borel set \\(B\\subseteq X\\) belongs to \\(\\mathcal B_0\\) exactly when it is a union of atoms of \\(\\mathcal B_0\\), that is: whenever \\(x\\in B\\) and \\(y\\notin B\\), some \\(B_n\\) contains exactly one of \\(x\\) and \\(y\\). In particular:\n\n1. every sequence of Borel sets that separates points generates the Borel structure of \\(X\\);\n2. the map \\(\\varphi\\) of Lemma 5.5(3) is a Borel isomorphism of \\(X\\) onto the Souslin set \\(\\varphi(X)\\subseteq\\mathcal C\\).\n\nExample 5.7 proves that the countability hypothesis in (1) is essential.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 398,
        "through_line": 414,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.7",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Example 5.7",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 5.7** (Countability is needed in (1)). The singletons of \\([0,1]\\) are Borel sets, and they separate points. The countable sets and their complements form a \\(\\sigma\\)-algebra that contains every singleton, and each of its members lies in the \\(\\sigma\\)-algebra generated by the singletons; so it is the \\(\\sigma\\)-algebra they generate. It does not contain \\([0,\\tfrac12]\\), which is uncountable and has an uncountable complement.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 415,
        "through_line": 416,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.8",
      "kind": "corollary",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Corollary 5.8",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Corollary 5.8.**\n\n1. Let \\((X,\\mathcal B)\\) be a Souslin–Borel space and \\(\\mathcal B'\\subseteq\\mathcal B\\) a \\(\\sigma\\)-algebra that is countably generated and separates points. Then \\(\\mathcal B'=\\mathcal B\\).\n2. Let \\(f\\) be a Borel bijection of a Souslin–Borel space onto a Borel space whose points are separated by countably many Borel sets. Then \\(f\\) is a Borel isomorphism.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 417,
        "through_line": 425,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.9",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-08::Example 5.9",
      "anchor": "oa-fnd-pb-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 5.9** (A Souslin–Borel space that is not standard). The Souslin set \\(D\\subseteq\\mathcal C\\) of [Example 2.9](#oa-fnd-pb-12), with its relative Borel structure, is a Souslin–Borel space, by Lemma 5.5(1) and (2). It is not standard. If it were, the inclusion \\(D\\to\\mathcal C\\) would be an injective Borel map on a standard Borel space, and \\(D\\) would be Borel by Theorem 4.3(5).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 426,
        "through_line": 427,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        },
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.1",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-09::Lemma 6.1",
      "anchor": "oa-fnd-pb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 6.1** (Outer measure).\n\n1. The function \\(\\mu^*\\) of (5) is countably subadditive, and every \\(E\\) has a Borel \\(H\\supseteq E\\) with \\(\\mu(H)=\\mu^*(E)\\).\n2. If \\(E_1\\subseteq E_2\\subseteq\\cdots\\), then \\(\\mu^*(\\bigcup_kE_k)=\\lim_k\\mu^*(E_k)\\).\n3. Borel sets are \\(\\mu^*\\)-measurable.\n4. If \\(\\mu\\) is \\(\\sigma\\)-finite, some finite measure \\(\\nu\\) has the same null sets, and a set is \\(\\mu\\)-measurable exactly when it is \\(\\nu\\)-measurable.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 448,
        "through_line": 475,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.2",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-09::Theorem 6.2",
      "anchor": "oa-fnd-pb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 6.2** (Souslin sets are measurable). Let \\(X\\) be a Hausdorff space, \\(\\mu\\) a measure on \\(\\mathcal B(X)\\), and \\(A\\) a Polish image in \\(X\\), for instance a Souslin set of a metrizable \\(X\\).\n\n1. If \\(\\mu\\) is finite, then \\(\\mu^*(A)=\\sup\\{\\mu(K):K\\subseteq A\\text{ compact}\\}\\), and \\(A\\) is \\(\\mu\\)-measurable.\n2. If \\(\\mu\\) is \\(\\sigma\\)-finite, \\(A\\) is \\(\\mu\\)-measurable.\n3. For every measure \\(\\mu\\), \\(A\\) is \\(\\mu^*\\)-measurable.\n\nIn particular, in a standard Borel space every Souslin set is \\(\\mu\\)-measurable for every \\(\\sigma\\)-finite \\(\\mu\\). By [Lemma 5.5](#oa-fnd-pb-08)(1), it does not matter which Polish topology is used to describe the Borel sets.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 476,
        "through_line": 526,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 6.3",
      "kind": "corollary",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-09::Corollary 6.3",
      "anchor": "oa-fnd-pb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Corollary 6.3** (Standard measures). Let \\(X\\) be a Souslin–Borel space and \\(\\mu\\) a \\(\\sigma\\)-finite measure on it. Then \\(\\mu\\) is standard: some Borel set \\(N\\subseteq X\\) with \\(\\mu(N)=0\\) has \\(X\\setminus N\\) standard. One can take \\(X\\setminus N\\) isomorphic to a \\(\\sigma\\)-compact subset of \\(\\mathcal C\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 527,
        "through_line": 530,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.4",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-09::Example 6.4",
      "anchor": "oa-fnd-pb-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 6.4** (\\(\\sigma\\)-finiteness is needed). Let \\(D\\subseteq\\mathcal C\\) be the Souslin set of [Example 2.9](#oa-fnd-pb-12). The counting measure on \\(D\\), which gives each subset its number of points, is a measure that is not standard: its only null set is \\(\\varnothing\\), and \\(D\\) is not standard (Example 5.9). \\(D\\) is uncountable, since countable sets are Borel, so this measure is not \\(\\sigma\\)-finite. Hence Corollary 6.3 needs \\(\\sigma\\)-finiteness. On the other hand, every \\(\\sigma\\)-finite Borel measure on \\(\\mathcal C\\) still measures \\(D\\), by Theorem 6.2.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 531,
        "through_line": 532,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.1",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-10::Theorem 7.1",
      "anchor": "oa-fnd-pb-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 7.1** (Borel transversals). Let \\(X\\) be a separable metrizable space with a compatible metric \\(d\\), and \\(R\\) an equivalence relation on \\(X\\) whose classes are complete for \\(d\\); for example, \\(X\\) Polish, \\(d\\) complete and every class closed. For \\(E\\subseteq X\\) let \\(R(E)\\) be the set of points equivalent to a point of \\(E\\), and let \\([x]\\) be the class of \\(x\\). Assume that \\(R(U)\\) is Borel for every open \\(U\\subseteq X\\). Then:\n\n1. There is a Borel map \\(\\eta:X\\to X\\) with \\(\\eta(x)\\in[x]\\) for all \\(x\\), and \\(\\eta(x)=\\eta(y)\\) whenever \\([x]=[y]\\).\n2. The set \\(S=\\{x:\\eta(x)=x\\}\\) is Borel and meets every class exactly once.\n\nIf \\(R(F)\\) is Borel for every closed \\(F\\subseteq X\\), then \\(R(U)\\) is Borel for every open \\(U\\), and the same conclusions hold.\n\nExample 7.2 proves that replacing the open sets by their closures can lose a class. The closed-saturation hypothesis is therefore reduced to the open-saturation case before using the construction.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 543,
        "through_line": 592,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.2",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-10::Example 7.2",
      "anchor": "oa-fnd-pb-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 7.2** (Closures instead of open sets). Let \\(X=[-1,1]\\) with \\(d(x,y)=|x-y|\\), and let \\(x\\mathrel Ry\\) mean \\(|x|=|y|\\). The classes \\(\\{x,-x\\}\\) are closed, and \\(R(E)=E\\cup(-E)\\) is open for open \\(E\\) and closed for closed \\(E\\). So both forms of the hypothesis of Theorem 7.1 hold, and the theorem gives a Borel transversal, for instance \\([0,1]\\). Choose a system (7) with\n\\[\n\\begin{gathered}\nX(1)=(0,\\tfrac14),\\\\\nX(2)=(-\\tfrac38,-\\tfrac18),\\\\\nX(1,j)=(2^{-j-3},\\tfrac14-2^{-j-3}),\\\\\nX(2,j)=(-\\tfrac38+2^{-j-3},-\\tfrac18-2^{-j-3})\\\\\n(j\\geq1),\n\\end{gathered}\n\\]\nand the remaining sets chosen in any way that satisfies (7). These sets do satisfy (7): the \\(X(1,j)\\) increase to \\(X(1)\\), the \\(X(2,j)\\) increase to \\(X(2)\\), their closures lie inside, and all their diameters are at most \\(\\tfrac14\\). Now run the construction with closures. At level \\(k\\) it keeps, from each class, the points that lie in \\(\\overline{X(s)}\\) for the first \\(s\\in\\mathbb N^k\\) whose closure meets the class. Take the class \\(H=\\{\\tfrac14,-\\tfrac14\\}\\).\n\n- Level 1: \\(\\overline{X(1)}=[0,\\tfrac14]\\) meets \\(H\\), so the level-1 set keeps only the point \\(\\tfrac14\\) of \\(H\\).\n- Level 2: every \\(\\overline{X(1,j)}\\) lies in \\((0,\\tfrac14)\\) and misses \\(H\\), while \\(\\overline{X(2,1)}=[-\\tfrac5{16},-\\tfrac3{16}]\\) contains \\(-\\tfrac14\\). So the level-2 set keeps only the point \\(-\\tfrac14\\) of \\(H\\).\n\nThe final set is the intersection of the sets of all levels, so it misses \\(H\\) entirely, and it is not a transversal. Writing \\(S_k\\) for the set kept at level \\(k\\), what fails is the inclusion of the closure of \\(H\\cap S_{k+1}\\) in \\(H\\cap S_k\\). With the open sets \\(X(s)\\), step (a) of the proof of Theorem 7.1 rules this out.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 593,
        "through_line": 609,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.3",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-10::Example 7.3",
      "anchor": "oa-fnd-pb-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 7.3** (Both hypotheses are needed).\n\n(a) *Closed classes.* On \\(X=\\mathbb R\\) let \\(x\\mathrel Ry\\) mean \\(x-y\\in\\mathbb Q\\). The classes \\(x+\\mathbb Q\\) are countable and dense, so not closed, and \\(R(U)=U+\\mathbb Q\\) is open for open \\(U\\). No Lebesgue measurable set \\(S\\) meets every class exactly once. If one did, \\(\\mathbb R\\) would be the disjoint union of the translates \\(S+q\\), \\(q\\in\\mathbb Q\\). If \\(\\lambda(S)=0\\), then \\(\\lambda(\\mathbb R)=0\\). If \\(\\lambda(S)>0\\), some \\(S_0=S\\cap[n,n+1]\\) has \\(\\lambda(S_0)>0\\), and the disjoint translates \\(S_0+q\\), \\(q\\in\\mathbb Q\\cap[0,1]\\), all lie in \\([n,n+2]\\) and all have measure \\(\\lambda(S_0)\\); infinitely many of them give infinite measure inside \\([n,n+2]\\). Both cases are absurd. In particular no Borel transversal exists.\n\n(b) *Borel saturations.* Let \\(A\\subseteq\\mathcal C\\) be a set that is not Borel ([Corollary 2.8](#oa-fnd-pb-12)). On \\(X=\\mathcal C\\times\\{0,1\\}\\) let \\((c,i)\\mathrel R(c',i')\\) mean that \\(c=c'\\) and either \\(i=i'\\) or \\(c\\in A\\). The classes have one or two points, so they are closed. The saturation of the open set \\(\\mathcal C\\times\\{1\\}\\) is \\((\\mathcal C\\times\\{1\\})\\cup(A\\times\\{0\\})\\), which is not Borel, since its preimage under \\(c\\mapsto(c,0)\\) is \\(A\\). Suppose \\(S\\) were a Borel transversal. For \\(c\\notin A\\), both one-point classes \\(\\{(c,0)\\}\\) and \\(\\{(c,1)\\}\\) must meet \\(S\\); for \\(c\\in A\\), exactly one of \\((c,0),(c,1)\\) lies in \\(S\\). So \\(\\{c:(c,0)\\in S\\text{ and }(c,1)\\in S\\}=\\mathcal C\\setminus A\\), which would be Borel. This is a contradiction.\n\nSo countable classes do not suffice, and neither do finite classes with a saturation that is not Borel.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 610,
        "through_line": 617,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.4",
      "kind": "proposition",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-10::Proposition 7.4",
      "anchor": "oa-fnd-pb-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Proposition 7.4** (Borel sections for coset spaces). Let \\(G\\) be a Polish group, that is, a topological group whose topology is Polish, and \\(H\\) a closed subgroup. Let \\(q:G\\to G/H\\) be the map onto the space of left cosets, and give \\(G/H\\) the quotient topology and its Borel sets.\n\n1. \\(q\\) is continuous and open, and \\(G/H\\) is Hausdorff.\n2. There is a Borel set \\(S\\subseteq G\\) that meets every left coset exactly once and contains \\(1\\).\n3. The map \\(\\sigma:G/H\\to G\\) that sends each coset to its point in \\(S\\) is Borel, satisfies \\(q\\circ\\sigma=\\mathrm{id}\\) and \\(\\sigma(H)=1\\), and is a Borel isomorphism of \\(G/H\\) onto \\(S\\). In particular \\(G/H\\) is a standard Borel space.\n\nThe same holds for right cosets.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 618,
        "through_line": 631,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.5",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-10::Example 7.5",
      "anchor": "oa-fnd-pb-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 7.5** (Borel is the best one can ask). For \\(G=\\mathbb R\\) and \\(H=\\mathbb Z\\), \\(S=[0,1)\\) is a Borel transversal, and \\(\\sigma(x+\\mathbb Z)\\) is the fractional part of \\(x\\). No continuous section exists. Here \\(G/H\\) is a circle, and a continuous injective map \\(\\sigma\\) from a circle into \\(\\mathbb R\\) is impossible: it takes its largest and smallest values at two different points, each of the two arcs joining these points is mapped onto the whole interval between the two values, and so the values strictly between them are taken twice.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 632,
        "through_line": 633,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.6",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-11::Lemma 7.6",
      "anchor": "oa-fnd-pb-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 7.6** (Lexicographic order on \\(\\Lambda\\)). For \\(t\\neq u\\) in \\(\\Lambda\\) write \\(t\\prec u\\) if \\(t_i<u_i\\) at the first index \\(i\\) with \\(t_i\\neq u_i\\), and \\(t\\preceq u\\) if \\(t\\prec u\\) or \\(t=u\\).\n\n1. \\(\\prec\\) is a strict total order on \\(\\Lambda\\).\n2. For every \\(u\\in\\Lambda\\), the set \\(\\{t:t\\prec u\\}\\) is open.\n3. Every nonempty closed set \\(F\\subseteq\\Lambda\\) has a least element.\n4. For \\(s=(s_1,\\dots,s_k)\\) with \\(k\\geq1\\), put \\(a_s=(s_1,\\dots,s_k,1,1,\\dots)\\) and \\(b_s=(s_1,\\dots,s_{k-1},s_k+1,1,1,\\dots)\\). Then\n\\[\n\\Lambda_s=\\{t\\in\\Lambda: \\ a_s\\preceq t\\prec b_s\\}.\n\\tag{8}\n\\]",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 636,
        "through_line": 656,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.7",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-11::Theorem 7.7",
      "anchor": "oa-fnd-pb-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 7.7** (Measurable sections; Jankov–von Neumann). Let \\(X\\) be a Souslin space, \\(Y\\) a separable metrizable space and \\(f:X\\to Y\\) a Borel map. Then \\(f(X)\\) is a Souslin set, and there is a map \\(\\varphi:f(X)\\to X\\) with \\(f(\\varphi(y))=y\\) for every \\(y\\in f(X)\\), such that for every Borel set \\(B\\subseteq X\\) the set \\(\\varphi^{-1}(B)\\) belongs to the \\(\\sigma\\)-algebra generated by the Souslin subsets of \\(f(X)\\). Consequently \\(\\varphi^{-1}(B)\\) is \\(\\mu^*\\)-measurable for every measure \\(\\mu\\) on \\(\\mathcal B(Y)\\), and \\(\\mu\\)-measurable for every \\(\\sigma\\)-finite \\(\\mu\\). One map \\(\\varphi\\) serves all measures at once.\n\nIn particular, if \\(X\\) and \\(Y\\) are Souslin spaces, \\(f\\) maps \\(X\\) onto \\(Y\\), and \\(\\mu\\) is a \\(\\sigma\\)-finite measure on \\(Y\\), then there is a \\(\\mu\\)-measurable map \\(\\varphi:Y\\to X\\) with \\(f\\circ\\varphi=\\mathrm{id}_Y\\).\n\nThe two-point example after Lemma 7.6 proves that independent coordinate minima can leave a closed fibre; the proof below minimizes within the previously chosen prefix.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 657,
        "through_line": 680,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "7. Choosing points in a Borel way",
              "line": 537,
              "through_line": 680,
              "anchors": [
                "OA-FND-PB-10",
                "OA-FND-PB-11"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
          "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
          "role": "proof comparison",
          "correspondence": "Exact countable-generator and prefix-selector scope",
          "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-14::Lemma 8.1",
      "anchor": "oa-fnd-pb-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 8.1** (The Baire property of Polish images). A continuous image of a Polish space in a Polish space has the Baire property.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 695,
        "through_line": 706,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-14::Theorem 8.2",
      "anchor": "oa-fnd-pb-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 8.2** (Pettis). Let \\(G\\) be a Polish group and \\(A\\subseteq G\\) have the Baire property and be nonmeagre. Both \\(A^{-1}A\\) and \\(AA^{-1}\\) contain an open neighbourhood of the identity.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 707,
        "through_line": 710,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.3",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-14::Theorem 8.3",
      "anchor": "oa-fnd-pb-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 8.3** (Open mapping for Polish groups). A continuous surjective homomorphism \\(\\phi:G\\to L\\) between Polish groups is open. It therefore induces a topological group isomorphism \\(G/\\ker\\phi\\cong L\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 711,
        "through_line": 718,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.4",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-15::Lemma 8.4",
      "anchor": "oa-fnd-pb-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 8.4** (Birkhoff–Kakutani invariant metrization). Every first countable Hausdorff topological group has a compatible right invariant metric. The metric is not asserted to be complete.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 721,
        "through_line": 740,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.5",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-15::Lemma 8.5",
      "anchor": "oa-fnd-pb-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 8.5** (Metrizing cosets). If \\(H\\) is a closed subgroup of a first countable Hausdorff group \\(G\\), the space of left cosets \\(G/H\\) has a compatible metric\n\\[\n\\begin{aligned}\nD(gH,kH)&=\\inf_{h,l\\in H}d(gh,kl)\\\\\n&=\\inf_{h\\in H}d(gh,k),\n\\end{aligned}\n\\]\nwhere \\(d\\) is a compatible right invariant metric on \\(G\\). The quotient map is open. If \\(G\\) is separable, so is \\(G/H\\).",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 741,
        "through_line": 755,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.6",
      "kind": "lemma",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Lemma 8.6",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Lemma 8.6** (Completeness criterion). A second countable metrizable space with a winning strong Choquet strategy is Polish.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 787,
        "through_line": 825,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.7",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Theorem 8.7",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 8.7** (Open images). Let \\(P\\) be Polish, \\(Y\\) metrizable, and \\(f:P\\to Y\\) continuous, open and surjective. Then \\(Y\\) is Polish.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 826,
        "through_line": 831,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.8",
      "kind": "theorem",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Theorem 8.8",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Theorem 8.8** (Polish coset spaces). If \\(G\\) is a Polish group and \\(H\\subseteq G\\) is a closed subgroup, \\(G/H\\), with the quotient topology, is Polish. The same holds for \\(H\\backslash G\\). No normality is required. When \\(H\\) is normal, the quotient is a Polish topological group.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 832,
        "through_line": 837,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.9",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Example 8.9",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 8.9** (A quotient and its section). The quotient \\(\\mathbb R/\\mathbb Z\\) is homeomorphic to the circle through \\(x+\\mathbb Z\\mapsto e^{2\\pi ix}\\). The displayed map is well defined, continuous and bijective: two real numbers have the same exponential exactly when their difference is an integer. The quotient is compact since it is the image of \\([0,1]\\), and the circle is Hausdorff. The compact-to-Hausdorff argument (B3) therefore proves the homeomorphism. It is Polish by Theorem 8.8. The Borel section from Example 7.5 takes values in \\([0,1)\\), but is not continuous. Polish topology on the quotient does not make every Borel choice continuous.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 838,
        "through_line": 839,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.10",
      "kind": "example",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Example 8.10",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Example 8.10** (Why both Polish hypotheses matter for open mapping). Give \\(\\mathbb R\\) its discrete topology. The identity homomorphism to \\(\\mathbb R\\) with its usual topology is continuous and onto, but is not open, since the image of a singleton is not open. The discrete source is complete but is not separable. Separability cannot simply be omitted from Theorem 8.3.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 840,
        "through_line": 841,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [
        {
          "unit": "polish-spaces-and-standard-borel-spaces",
          "local_scopes": [
            {
              "heading": "1. Trees in Polish spaces",
              "line": 67,
              "through_line": 95,
              "anchors": [
                "OA-FND-PB-01"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "2. Souslin sets",
              "line": 96,
              "through_line": 197,
              "anchors": [
                "OA-FND-PB-02",
                "OA-FND-PB-03",
                "OA-FND-PB-12"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "3. Lusin spaces and one-to-one images",
              "line": 198,
              "through_line": 304,
              "anchors": [
                "OA-FND-PB-04",
                "OA-FND-PB-05"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "4. Borel maps between Souslin spaces",
              "line": 305,
              "through_line": 353,
              "anchors": [
                "OA-FND-PB-06"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "5. Standard Borel spaces",
              "line": 354,
              "through_line": 427,
              "anchors": [
                "OA-FND-PB-07",
                "OA-FND-PB-08"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            },
            {
              "heading": "8. Topological quotients and open homomorphisms",
              "line": 685,
              "through_line": 841,
              "anchors": [
                "OA-FND-PB-13",
                "OA-FND-PB-14",
                "OA-FND-PB-15",
                "OA-FND-PB-16"
              ],
              "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
          "role": "proof comparison",
          "correspondence": "Polish and standard Borel proofs; exercises identified",
          "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 1",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 1** (medium; Borel graphs). Let \\(X\\) and \\(Y\\) be standard Borel spaces and \\(f:X\\to Y\\) a map. Show that \\(f\\) is Borel if and only if its graph is a Borel subset of \\(X\\times Y\\), with the product Borel structure.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 844,
        "through_line": 856,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 2",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 2** (medium; Quotients by Borel surjections). Let \\(Z\\) be a standard Borel space, \\(Y\\) a Borel space whose points are separated by countably many Borel sets, and \\(f:Z\\to Y\\) a Borel surjection. Show that \\(E\\subseteq Y\\) is Borel if and only if \\(f^{-1}(E)\\) is Borel in \\(Z\\). Show by an example that the hypothesis \"countably separated\" cannot be dropped.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 857,
        "through_line": 862,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 3",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 3** (medium; Orbits of a compact group). Let a compact metrizable group \\(K\\) act continuously on a Polish space \\(X\\), that is, \\((k,x)\\mapsto k\\cdot x\\) is a continuous action. Show that some Borel set meets every orbit exactly once, and that the orbit space \\(X/K\\), with the quotient topology and its Borel sets, is a standard Borel space.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 863,
        "through_line": 870,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 4",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 4** (medium; Product \\(\\sigma\\)-algebras). Let \\(D\\) be a set of cardinality greater than \\(2^{\\aleph_0}\\), with the discrete topology. Show that the diagonal of \\(D\\times D\\) is Borel in the discrete space \\(D\\times D\\) but does not belong to \\(\\mathcal B(D)\\otimes\\mathcal B(D)\\). Conclude that the open boxes of \\(D\\times D\\) generate a strictly smaller \\(\\sigma\\)-algebra than its open sets.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 871,
        "through_line": 874,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 5",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 5** (medium; An explicit isomorphism). Build a Borel isomorphism of \\(\\mathcal C\\) onto \\([0,1]\\) by hand, using binary expansions and moving a countable set. Show that no homeomorphism exists.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 875,
        "through_line": 894,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 6",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 6",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 6** (easy; First application). Let \\(\\phi:G\\to L\\) be a continuous bijective homomorphism of Polish groups. Prove that its inverse is continuous. Apply this to show that two Polish group topologies on the same group coincide whenever one is finer than the other.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 895,
        "through_line": 898,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 7",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 7** (easy; Closedness and quotient topology). Show that \\(\\mathbb R/\\mathbb Q\\), with the quotient topology, is not Hausdorff. In fact show that its only open sets are the empty set and the entire quotient. Explain why a standard Borel structure on a set, even if one is chosen separately, would not repair this topology.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 899,
        "through_line": 902,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 8",
      "kind": "exercise",
      "unit": "polish-spaces-and-standard-borel-spaces",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces#oa-fnd-pb-16::Exercise 8",
      "anchor": "oa-fnd-pb-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
      "statement_and_full_conditions": "**Exercise 8** (medium; Compact orbits, a stronger conclusion). In Exercise 3, prove that the quotient \\(X/K\\) is Polish, rather than merely standard Borel.",
      "proof_locus": {
        "source": "src/polish-spaces-and-standard-borel-spaces.md",
        "line": 903,
        "through_line": 906,
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.1",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-01::Definition 1.1",
      "anchor": "oa-fnd-dt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 1.1.** A functional \\(\\omega\\in A^*\\) is *unital* if \\(\\omega(1)=1=\\|\\omega\\|\\). Let \\(V(A)\\) be the set of unital functionals. The *numerical range* of \\(a\\in A\\) is\n\\[\nW(a)=W_A(a)=\\{\\omega(a):\\ \\omega\\in V(A)\\},\n\\]\nand the *numerical radius* of \\(a\\) is \\(v(a)=\\max\\{|z|:z\\in W(a)\\}\\).\n\nFor example, let \\(A=B(H)\\) for a Hilbert space \\(H\\neq0\\), and let \\(\\xi\\in H\\) be a unit vector. The functional \\(x\\mapsto\\langle x\\xi,\\xi\\rangle\\) takes the value \\(\\|\\xi\\|^2=1\\) at \\(1\\), and \\(|\\langle x\\xi,\\xi\\rangle|\\leq\\|x\\|\\). So it is unital, and \\(W(x)\\) contains every value of the quadratic form \\(\\xi\\mapsto\\langle x\\xi,\\xi\\rangle\\) on the unit sphere. For a general C\\*-algebra, Corollary 3.6 shows that the unital functionals are exactly the states.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 70,
        "through_line": 77,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.2",
      "kind": "proposition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-01::Proposition 1.2",
      "anchor": "oa-fnd-dt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Proposition 1.2.** Let \\(a,b\\in A\\).\n1. \\(V(A)=\\{\\omega\\in A^*:\\ \\|\\omega\\|\\leq1,\\ \\omega(1)=1\\}\\). This set is nonempty, convex and weak\\*-compact.\n2. \\(W(a)\\) is a nonempty compact convex subset of the closed disc of radius \\(\\|a\\|\\).\n3. \\(W(\\alpha+\\beta a)=\\alpha+\\beta W(a)\\) for \\(\\alpha,\\beta\\in\\mathbb C\\), and \\(W(a+b)\\subseteq W(a)+W(b)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 78,
        "through_line": 90,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 1.3",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-01::Theorem 1.3",
      "anchor": "oa-fnd-dt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 1.3.** Let \\(a\\in A\\) and \\(\\alpha\\in\\mathbb C\\). The following are equivalent.\n1. \\(\\alpha\\in W(a)\\).\n2. The formula \\(f(\\lambda+\\mu a)=\\lambda+\\mu\\alpha\\), for \\(\\lambda,\\mu\\in\\mathbb C\\), defines a linear functional \\(f\\) on \\(\\operatorname{span}\\{1,a\\}\\) with \\(|f(y)|\\leq\\|y\\|\\).\n3. \\(|\\lambda+\\mu\\alpha|\\leq\\|\\lambda+\\mu a\\|\\) for all \\(\\lambda,\\mu\\in\\mathbb C\\).\n4. \\(|\\lambda+\\alpha|\\leq\\|\\lambda+a\\|\\) for all \\(\\lambda\\in\\mathbb C\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 91,
        "through_line": 104,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 1.4",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-01::Corollary 1.4",
      "anchor": "oa-fnd-dt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 1.4.** Let \\(a\\in A\\).\n1. If \\(B\\subseteq A\\) is a closed subalgebra that contains \\(1\\) and \\(a\\), with the norm of \\(A\\), then \\(W_B(a)=W_A(a)\\).\n2. \\(\\sigma(a)\\subseteq W(a)\\). Hence \\(W(a)\\) contains the convex hull of \\(\\sigma(a)\\), and \\(r(a)\\leq v(a)\\leq\\|a\\|\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 105,
        "through_line": 114,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.4a",
      "kind": "proposition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-01::Proposition 1.4a",
      "anchor": "oa-fnd-dt-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Proposition 1.4a** (Stability under norm perturbation). For \\(a,b\\in A\\), the Hausdorff distance between their numerical ranges is at most \\(\\|a-b\\|\\). In particular, both the numerical radius and the largest real part of the numerical range change by at most \\(\\|a-b\\|\\).\n\nHere the Hausdorff distance of nonempty compact sets \\(K,L\\subseteq\\mathbb C\\) is the larger of \\(\\sup_{z\\in K}\\inf_{w\\in L}|z-w|\\) and \\(\\sup_{w\\in L}\\inf_{z\\in K}|z-w|\\). The numerical-range estimate is Lemma 2.1 of [H. Blazhko, D. Homza, F. L. Schwenninger, J. de Vries and M. Wojtylak, *The algebraic numerical range as a spectral set in Banach algebras* (2025)](https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/algebraic-numerical-range-as-a-spectral-set-in-banach-algebras/96536D155B032F6C67B750582F5DBF07#sec2).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 115,
        "through_line": 126,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.1",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-02::Lemma 2.1",
      "anchor": "oa-fnd-dt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 2.1** (support functions). Let \\(K\\subseteq\\mathbb C\\) be nonempty, compact and convex, and put \\(h_K(\\theta)=\\max_{z\\in K}\\operatorname{Re}(e^{i\\theta}z)\\) for \\(\\theta\\in\\mathbb R\\). Then\n\\[\n\\begin{gathered}\nK\\\\\n=\\{z\\in\\mathbb C :\\ \\operatorname{Re}(e^{i\\theta}z)\\leq h_K(\\theta)\\text{ for all }\\theta\\in\\mathbb R\\}.\n\\end{gathered}\n\\]\nHence two nonempty compact convex sets with the same function \\(h\\) are equal. If \\(h_K(\\theta)=R\\) for every \\(\\theta\\), then \\(K\\) is the closed disc \\(\\{|z|\\leq R\\}\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 135,
        "through_line": 154,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.2",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-02::Lemma 2.2",
      "anchor": "oa-fnd-dt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 2.2** (subadditive functions near zero). Let \\(F:[0,\\infty)\\to\\mathbb R\\) satisfy \\(F(0)=0\\), \\(F(s+t)\\leq F(s)+F(t)\\) for \\(s,t\\geq0\\), and \\(F(t)\\to0\\) as \\(t\\to0^+\\). Suppose that \\(\\alpha=\\sup_{t>0}F(t)/t\\) is finite. Then \\(F(t)/t\\to\\alpha\\) as \\(t\\to0^+\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 155,
        "through_line": 163,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.3",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-02::Theorem 2.3",
      "anchor": "oa-fnd-dt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 2.3.** Let \\(a\\in A\\).\n1. The function \\(g(u)=\\|u+a\\|-u\\), \\(u\\in\\mathbb R\\), is nonincreasing, and\n\\[\n\\begin{gathered}\nm(a)\\\\\n=\\inf_{u\\in\\mathbb R}g(u)\\\\\n=\\lim_{u\\to+\\infty}\\big(\\|u+a\\|-u\\big)\\\\\n=\\inf_{t>0}\\frac{\\|1+ta\\|-1}{t}\\\\\n=\\lim_{t\\to0^+}\\frac{\\|1+ta\\|-1}{t}.\n\\end{gathered}\n\\tag{2.2}\n\\]\n2. \\(\\operatorname{Re}W(a)=[-m(-a),\\,m(a)]\\).\n3. We have\n\\[\n\\begin{gathered}\nm(a)\\\\\n=\\lim_{t\\to0^+}\\frac{\\|e^{ta}\\|-1}{t}\\\\\n=\\lim_{t\\to0^+}\\frac{\\log\\|e^{ta}\\|}{t}\\\\\n=\\sup_{t>0}\\frac{\\log\\|e^{ta}\\|}{t}.\n\\end{gathered}\n\\tag{2.3}\n\\]\nIn particular \\(\\|e^{ta}\\|\\leq e^{t\\,m(a)}\\) for all \\(t\\geq0\\), and \\(m(a)\\) is the smallest real number \\(w\\) with \\(\\|e^{ta}\\|\\leq e^{tw}\\) for all \\(t\\geq0\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 164,
        "through_line": 235,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
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              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.4",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-02::Corollary 2.4",
      "anchor": "oa-fnd-dt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 2.4** (dissipative elements). \\(\\operatorname{Re}W(a)\\subseteq(-\\infty,0]\\) if and only if \\(\\|e^{ta}\\|\\leq1\\) for all \\(t\\geq0\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 236,
        "through_line": 241,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
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              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
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              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.5",
      "kind": "example",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-02::Example 2.5",
      "anchor": "oa-fnd-dt-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Example 2.5** (a nilpotent matrix and two norms). Let \\(N=\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\in M_2(\\mathbb C)\\). Then \\(N^2=0\\), \\(\\sigma(N)=\\{0\\}\\) and \\(r(N)=0\\). We compute \\(W(N)\\) for two algebra norms on \\(M_2(\\mathbb C)\\) with \\(\\|1\\|=1\\), using Theorem 2.3(1) and Lemma 2.1. For \\(z\\in\\mathbb C\\), \\(1+zN=\\begin{pmatrix}1&z\\\\0&1\\end{pmatrix}\\).\n\n(a) *The operator norm for the Euclidean norm on \\(\\mathbb C^2\\).* Then \\(M_2(\\mathbb C)=B(\\mathbb C^2)\\) is a C\\*-algebra, and by (C1), \\(\\|1+zN\\|^2=\\|(1+zN)^*(1+zN)\\|\\) is the largest eigenvalue of the self-adjoint matrix \\(\\begin{pmatrix}1&z\\\\\\bar z&1+|z|^2\\end{pmatrix}\\). This matrix has trace \\(2+|z|^2\\) and determinant \\(1\\), so its eigenvalues are \\(\\big(2+|z|^2\\pm|z|\\sqrt{|z|^2+4}\\big)/2\\), and the larger one equals \\(\\big((|z|+\\sqrt{|z|^2+4})/2\\big)^2\\). For \\(t>0\\) and real \\(\\theta\\),\n\\[\n\\begin{gathered}\n\\frac{\\|1+te^{i\\theta}N\\|-1}{t}\\\\\n=\\frac{t+\\sqrt{t^2+4}-2}{2t}\\\\\n=\\frac12+\\frac{t}{2\\big(\\sqrt{t^2+4}+2\\big)}\\longrightarrow\\frac12 .\n\\end{gathered}\n\\]\nSo \\(m(e^{i\\theta}N)=1/2\\) for every \\(\\theta\\), and \\(W(N)=\\{|z|\\leq1/2\\}\\).\n\n(b) *The operator norm for the maximum norm on \\(\\mathbb C^2\\)*, which is the largest row sum of absolute values. Here \\(\\|1+zN\\|=1+|z|\\), so \\(m(e^{i\\theta}N)=1\\) for every \\(\\theta\\), and \\(W(N)=\\{|z|\\leq1\\}\\).\n\nIn both cases \\(\\|N\\|=1\\). In (a), \\(r(N)=0<v(N)=\\tfrac12<\\|N\\|\\); in (b), \\(v(N)=\\|N\\|\\). So the numerical range depends on the norm, and not only on the algebra.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 242,
        "through_line": 257,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.1",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Definition 3.1",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 3.1.** An element \\(h\\in A\\) is *hermitian* if \\(W(h)\\subseteq\\mathbb R\\).\n\nBy Proposition 1.2(3), the hermitian elements form a real vector space that contains \\(\\mathbb R1\\). It is closed, since \\(\\omega(h_n)\\to\\omega(h)\\) for every \\(\\omega\\in V(A)\\) when \\(h_n\\to h\\). By Corollary 1.4(2), a hermitian element has real spectrum.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 260,
        "through_line": 263,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.2",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Theorem 3.2",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 3.2.** For \\(h\\in A\\) the following are equivalent.\n1. \\(h\\) is hermitian.\n2. \\(\\|e^{ith}\\|=1\\) for every \\(t\\in\\mathbb R\\).\n3. \\(\\|e^{ith}\\|\\leq1\\) for every \\(t\\in\\mathbb R\\).\n4. \\(\\|1+ith\\|=1+o(t)\\) as \\(t\\to0\\) through real values; that is, \\((\\|1+ith\\|-1)/t\\to0\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 264,
        "through_line": 279,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.3",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Lemma 3.3",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 3.3** (the arcsine series). Let \\(c_n=\\binom{2n}{n}4^{-n}/(2n+1)\\) for \\(n\\geq0\\), and \\(S(z)=\\sum_{n\\geq0}c_nz^{2n+1}\\) for \\(|z|<1\\).\n1. \\(c_n>0\\), the series converges for \\(|z|<1\\), and \\(\\sum_nc_n\\leq\\pi/2\\).\n2. Let \\(0<c'<\\pi/2\\), choose \\(\\delta>0\\) with \\(\\sin^2c'+\\sinh^2\\delta<1\\), and let \\(R=\\{x+iy:\\ |x|<c',\\ |y|<\\delta\\}\\). Then \\(|\\sin z|<1\\) and \\(S(\\sin z)=z\\) for every \\(z\\in R\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 280,
        "through_line": 319,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.4",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Theorem 3.4",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 3.4** (Sinclair). If \\(h\\in A\\) is hermitian, then \\(\\|h\\|=r(h)\\), and\n\\[\nW(h)=[\\min\\sigma(h),\\ \\max\\sigma(h)],\n\\]\nthe convex hull of \\(\\sigma(h)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 320,
        "through_line": 354,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.5",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Corollary 3.5",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 3.5** (bounded groups). Let \\(a\\in A\\), and suppose \\(M=\\sup_{t\\in\\mathbb R}\\|e^{ita}\\|\\) is finite. Then \\(\\|a\\|\\leq M\\,r(a)\\). The case \\(M=1\\) is the first statement of Theorem 3.4.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 355,
        "through_line": 377,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.6",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Corollary 3.6",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 3.6** (C\\*-algebras). Let \\(A\\) be a unital C\\*-algebra with \\(1\\neq0\\).\n1. The hermitian elements of \\(A\\) are exactly its self-adjoint elements.\n2. The unital functionals are exactly the states. So \\(W(a)=\\{\\varphi(a):\\ \\varphi\\text{ a state}\\}\\).\n3. For \\(h=h^*\\), \\(W(h)=[\\min\\sigma(h),\\max\\sigma(h)]\\), and \\(v(h)=\\|h\\|\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 378,
        "through_line": 408,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
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          "local_scopes": [
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              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.7",
      "kind": "example",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-03::Example 3.7",
      "anchor": "oa-fnd-dt-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Example 3.7** (group algebras). Let \\(G\\) be a discrete group with identity \\(e\\), and \\(A=\\ell^1(G)\\) as in (B4). Write \\(a\\in A\\) as \\(a=a(e)\\delta_e+b\\) with \\(b(e)=0\\). Then\n\\[\nW(a)=\\{z\\in\\mathbb C :\\ |z-a(e)|\\leq\\|b\\|_1\\}.\n\\]\nIndeed, for \\(z\\in\\mathbb C\\), \\(\\delta_e\\) and \\(zb\\) have disjoint supports, so \\(\\|\\delta_e+zb\\|_1=1+|z|\\,\\|b\\|_1\\). For \\(t>0\\) and real \\(\\theta\\) this gives \\((\\|1+te^{i\\theta}b\\|-1)/t=\\|b\\|_1\\). By Theorem 2.3(1), \\(m(e^{i\\theta}b)=\\|b\\|_1\\) for all \\(\\theta\\), and Lemma 2.1 gives \\(W(b)=\\{|z|\\leq\\|b\\|_1\\}\\). Then \\(W(a)=a(e)+W(b)\\) by Proposition 1.2(3).\n\nSo the hermitian elements of \\(\\ell^1(G)\\) are exactly the real multiples of \\(\\delta_e\\). The self-adjoint elements, those with \\(a(g^{-1})=\\overline{a(g)}\\), form a much larger set when \\(G\\neq\\{e\\}\\). For \\(G=\\mathbb Z\\) and \\(a=\\delta_1+\\delta_{-1}\\): \\(a\\) is self-adjoint, \\(\\sigma(a)=[-2,2]\\) by (B4), and \\(\\|a\\|_1=2=r(a)\\); but \\(W(a)\\) is the closed disc of radius \\(2\\). Exercise 2 gives a self-adjoint element whose spectrum is not even real.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 409,
        "through_line": 416,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "1. Unital functionals and the numerical range",
              "line": 66,
              "through_line": 126,
              "anchors": [
                "OA-FND-DT-01"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "2. The numerical range from the norm",
              "line": 127,
              "through_line": 257,
              "anchors": [
                "OA-FND-DT-02"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "3. Hermitian elements",
              "line": 258,
              "through_line": 416,
              "anchors": [
                "OA-FND-DT-03"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
          "source_locus": "Complete §2, especially Lemma 2.1",
          "role": "proof comparison",
          "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
          "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
          "local_labels": [
            "Proposition 1.4a"
          ]
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.1",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Lemma 4.1",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 4.1.** Let \\(T\\in M_n(\\mathbb C)\\) and \\(x,y,u\\in\\mathbb C^n\\).\n1. If \\(T\\geq0\\), then \\(|Tx|\\leq T|x|\\), and \\(0\\leq x\\leq y\\) implies \\(0\\leq Tx\\leq Ty\\).\n2. If \\(T\\gg0\\), \\(x\\geq0\\) and \\(x\\neq0\\), then \\(Tx\\gg0\\).\n3. If \\(|x|\\leq y\\), then \\(\\|x\\|\\leq\\|y\\|\\). If \\(T\\geq0\\), then \\(\\|T\\|=\\|T\\mathbf 1\\|\\).\n4. If \\(u\\gg0\\), \\(x\\geq0\\) and \\(x\\neq0\\), then \\(u^{\\top}x>0\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 421,
        "through_line": 428,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.2",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Theorem 4.2",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 4.2** (the spectral radius is an eigenvalue). Let \\(T\\geq0\\) and \\(\\rho=r(T)\\). Then \\(\\rho\\) is an eigenvalue of \\(T\\) with an eigenvector \\(x\\geq0\\), \\(x\\neq0\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 429,
        "through_line": 443,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.3",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Definition 4.3",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 4.3.** A matrix \\(T\\geq0\\) is *primitive* if \\(T^m\\gg0\\) for some \\(m\\geq1\\). Every \\(T\\gg0\\) is primitive. For a primitive \\(T\\), \\(\\rho=r(T)\\) is the *Perron–Frobenius eigenvalue* of \\(T\\), and a vector \\(v\\gg0\\) with \\(Tv=\\rho v\\) is a *Perron–Frobenius eigenvector*. By the next theorem such \\(v\\) exists and is unique up to a positive factor.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 444,
        "through_line": 445,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.4",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Theorem 4.4",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 4.4** (Perron–Frobenius, primitive matrices). Let \\(T\\geq0\\) be primitive and \\(\\rho=r(T)\\).\n1. \\(\\rho>0\\), and there is \\(v\\gg0\\) with \\(Tv=\\rho v\\).\n2. Every \\(y\\in\\mathbb C^n\\) with \\(Ty=\\rho y\\) is a multiple of \\(v\\).\n3. Every eigenvalue \\(\\lambda\\neq\\rho\\) of \\(T\\) has \\(|\\lambda|<\\rho\\).\n4. There is \\(u\\gg0\\) with \\(u^{\\top}T=\\rho u^{\\top}\\). The space \\(\\mathbb C^n\\) is the direct sum of \\(\\mathbb Cv\\) and \\(\\ker u^{\\top}\\), both invariant under \\(T\\), and \\(\\rho\\) is a simple root of the characteristic polynomial of \\(T\\).\n5. If \\(x\\geq0\\), \\(x\\neq0\\), is an eigenvector of \\(T\\) for any eigenvalue, then \\(x\\) is a positive multiple of \\(v\\), and \\(Tx=\\rho x\\).\n6. (Max–min formula)\n\\[\n\\begin{gathered}\n\\rho\\\\\n=\\max_{x\\geq0,\\ x\\neq0}\\ \\min_{i:\\,x_i>0}\\frac{(Tx)_i}{x_i}\\\\\n=\\min_{x\\gg0}\\ \\max_i\\frac{(Tx)_i}{x_i}.\n\\end{gathered}\n\\]\n7. \\((T/\\rho)^k\\to P=vu^{\\top}/(u^{\\top}v)\\) as \\(k\\to\\infty\\), where \\(P\\) is the projection onto \\(\\mathbb Cv\\) along \\(\\ker u^{\\top}\\). Consequently \\(T^kw/\\|T^kw\\|\\to v/\\|v\\|\\) for every \\(w\\geq0\\), \\(w\\neq0\\).\n\nThe strict-positivity case and the max–min formula are both proved below.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 446,
        "through_line": 485,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.5",
      "kind": "proposition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Proposition 4.5",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Proposition 4.5** (irreducible matrices). Let \\(T\\geq0\\) be irreducible and \\(\\rho=r(T)\\).\n1. \\(1+T\\) is primitive, and \\(r(1+T)=1+\\rho\\).\n2. There is \\(v\\gg0\\) with \\(Tv=\\rho v\\). Parts (2), (4), (5) and (6) of Theorem 4.4 hold for \\(T\\), and \\(\\rho>0\\) when \\(n\\geq2\\).\n3. Parts (3) and (7) of Theorem 4.4 can fail: for \\(T=\\begin{pmatrix}0&1\\\\1&0\\end{pmatrix}\\), which is irreducible, \\(\\rho=1\\), \\(-1\\) is an eigenvalue, and \\(T^k\\) alternates between \\(1\\) and \\(T\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 486,
        "through_line": 497,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.6",
      "kind": "example",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Example 4.6",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Example 4.6** (the hypotheses matter). For the \\(2\\times2\\) identity matrix, \\(\\rho=1\\) has a two-dimensional eigenspace, and \\((1,0)\\) is a nonnegative eigenvector that is not \\(\\gg0\\). For \\(\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix}\\), \\(\\rho=1\\) is a double root, and the only eigenvectors are the multiples of \\((1,0)\\). For \\(\\begin{pmatrix}0&1\\\\0&0\\end{pmatrix}\\), \\(\\rho=0\\). None of these three matrices is irreducible. On the other hand, \\(T=\\begin{pmatrix}1&2\\\\3&4\\end{pmatrix}\\gg0\\) has eigenvalues \\((5\\pm\\sqrt{33})/2\\); here \\(\\rho=(5+\\sqrt{33})/2\\), the other eigenvalue has modulus less than \\(1\\), and \\(v=(2,\\rho-1)\\gg0\\) is a Perron–Frobenius eigenvector, since its first coordinate gives \\(2+2(\\rho-1)=2\\rho\\) and its second \\(6+4(\\rho-1)=\\rho(\\rho-1)\\), which is the equation \\(\\rho^2=5\\rho+2\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 498,
        "through_line": 499,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 0.1",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-04::Theorem 0.1",
      "anchor": "oa-fnd-dt-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 0.1** (Well-orders and ordinals). If a strictly increasing map \\(f:W\\to W\\) acts on a well-ordered set, then \\(f(w)\\geq w\\) for every \\(w\\). No well-order is isomorphic to a proper initial segment of itself. Every well-order is isomorphic to exactly one ordinal. Ordinals are linearly ordered by membership; every nonempty set of ordinals has a least member, and an ordinal is exactly the set of its predecessors. Consequently two well-orders are either isomorphic or exactly one is isomorphic to a proper initial segment of the other. A countable well-order has a countable ordinal as its order type.\n\n*Proof.* **Induction and increasing maps.** A property holds throughout a well-order if, whenever it holds at all predecessors of a point, it holds at that point: a nonempty set of failures would have a least member. If \\(w\\) is the least point with \\(f(w)<w\\), put \\(v=f(w)\\). Then \\(v<w\\), so \\(f(v)\\geq v=f(w)\\), whereas strict increase gives \\(f(v)<f(w)\\), a contradiction. Thus \\(f(w)\\geq w\\). An isomorphism of \\(W\\) onto the predecessors of \\(q\\in W\\) would give \\(f(q)<q\\), which is impossible.\n\n**Comparison of well-orders.** Consider all order isomorphisms between initial segments of two well-orders \\(W,V\\). Any two agree on their common domain. Otherwise take the first point where they disagree. Their values at all preceding points agree, and each value at the new point must be the least element of \\(V\\) outside that common predecessor image. The values therefore agree, a contradiction. These maps are subsets of the set \\(W\\times V\\). Their union is an order isomorphism between initial segments: domains are nested initial segments, compatibility makes the union a function, and the ranges are initial segments too. If both domain and range were proper, map the least unused point of \\(W\\) to the least unused point of \\(V\\). This would give another initial-segment isomorphism strictly extending the union, a contradiction. Thus one order is exhausted. The alternatives cannot overlap, since composing competing comparisons would make a well-order isomorphic to a proper initial segment of itself.\n\n**Ordinals.** A set \\(\\alpha\\) is an ordinal if it is transitive (\\(x\\in\\alpha\\) implies \\(x\\subseteq\\alpha\\)) and membership is a strict well-order on \\(\\alpha\\). If \\(x\\in\\alpha\\), its predecessors in \\(\\alpha\\) are precisely the members of \\(x\\): transitivity of \\(\\alpha\\) puts those members in \\(\\alpha\\). Also \\(x\\) is transitive, because \\(z\\in y\\in x\\), with all three in \\(\\alpha\\), implies \\(z\\in x\\) by transitivity of the membership order. Thus \\(x\\) is itself an ordinal. In particular the proper initial segments of an ordinal are its members. No ordinal belongs to itself: if \\(\\alpha\\in\\alpha\\), membership restricted to \\(\\alpha\\) would have the forbidden loop \\(\\alpha\\in\\alpha\\).\n\nAn isomorphism between initial segments of ordinals is the identity on its domain. Induct on \\(x\\) in that domain. The predecessors of its image are the images of the predecessors of \\(x\\), so\n\\[\nf(x)=\\{f(y):y\\in x\\}=\\{y:y\\in x\\}=x.\n\\]\nThe preceding comparison theorem now implies that for any two ordinals \\(\\alpha,\\beta\\), exactly one of \\(\\alpha=\\beta\\), \\(\\alpha\\in\\beta\\), \\(\\beta\\in\\alpha\\) holds. Moreover \\(\\beta+1:=\\beta\\cup\\{\\beta\\}\\) is an ordinal: it is transitive, its old elements retain their membership order, and \\(\\beta\\) is the new last element. Given a nonempty set \\(S\\) of ordinals, choose \\(\\beta\\in S\\). The nonempty subset \\(S\\cap(\\beta+1)\\) has a least member \\(\\gamma\\) in this well-order. Every member of \\(S\\) outside \\(\\beta+1\\) is larger than \\(\\beta\\), so \\(\\gamma\\) is least in all of \\(S\\). Since all members of an ordinal are smaller ordinals and comparison identifies every smaller ordinal with a proper initial segment, an ordinal is exactly the set of smaller ordinals.\n\n**Constructing the order type.** We justify the recursion\n\\[\nF(w)=\\{F(u):u<w\\}\n\\]\non a well-order \\(W\\). Call a function on an initial segment *compatible* if it satisfies this formula at every point of its domain. Two compatible functions agree on the intersection of their domains: the first disagreement would have exactly the same predecessor values on both sides. By Separation form the initial segment \\(D\\subseteq W\\) consisting of points contained in some compatible domain. At each point of \\(D\\) its compatible value exists and is unique. Replacement therefore supplies the function \\(F_D\\) with those values. It obeys the recursion, because any compatible domain containing a point contains all its predecessors. If \\(D\\ne W\\), let \\(w\\) be its least omitted point. Then \\(D=\\{u:u<w\\}\\), and extending \\(F_D\\) by the value \\(\\{F_D(u):u<w\\}\\) at \\(w\\) gives a compatible function containing \\(w\\), a contradiction. Hence \\(D=W\\), and the recursion defines a unique function on all of \\(W\\).\n\nInductively \\(F(w)\\) is an ordinal and \\(u\\mapsto F(u)\\), for \\(u<w\\), is an isomorphism onto \\(F(w)\\). Indeed, predecessor values are ordinals, and \\(u<v<w\\) implies \\(F(u)\\in F(v)\\) by the recursion. They are distinct, since an ordinal cannot belong to itself. A membership in the reverse direction would give \\(F(v)\\in F(u)\\in F(v)\\), and transitivity of the ordinal \\(F(v)\\) would force \\(F(v)\\in F(v)\\), a contradiction. Thus membership agrees exactly with the predecessor order. The set of predecessor values is transitive, since every member of \\(F(u)\\) is an earlier value, and it is well-ordered by pulling nonempty subsets back to the predecessor order. This proves the induction assertion. The same argument applied to all values shows that\n\\[\n\\alpha=\\{F(w):w\\in W\\}\n\\]\nis an ordinal and \\(F:W\\to\\alpha\\) is an order isomorphism. Two ordinals isomorphic to \\(W\\) are isomorphic to each other; the identity result above forces equality. Finally this isomorphism is a bijection, so if \\(W\\) is countable, so is \\(\\alpha\\). \\(\\square\\)",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 529,
        "through_line": 554,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.1",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-05::Definition 5.1",
      "anchor": "oa-fnd-dt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 5.1.** A subset of \\(X\\) is *co-Souslin* if its complement is a Souslin set.\n\nWhen a Polish topology generating the Borel sets is fixed, the Souslin sets are the analytic sets, and the co-Souslin sets are the sets denoted by \\(\\Pi^1_1\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 559,
        "through_line": 562,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
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          "local_scopes": [
            {
              "heading": "5. Souslin and co-Souslin sets",
              "line": 555,
              "through_line": 571,
              "anchors": [
                "OA-FND-DT-05"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§12A–§12C; PDF 46–49",
          "role": "proof comparison",
          "correspondence": "Analytic/coanalytic notation and first separation",
          "explanation": "Closed projections, countable closure and analytic separation are fully compared; this is not a proof of the second separation theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.2",
      "kind": "proposition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-05::Proposition 5.2",
      "anchor": "oa-fnd-dt-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Proposition 5.2.**\n1. Borel sets are both Souslin and co-Souslin. A set that is both Souslin and co-Souslin is Borel.\n2. Countable unions and countable intersections of co-Souslin sets are co-Souslin.\n3. If \\(f:X\\to Y\\) is Borel and \\(C\\subseteq Y\\) is co-Souslin, then \\(f^{-1}(C)\\) is co-Souslin.\n4. If \\(S\\subseteq X\\times Y\\) is Souslin, its projection \\(\\{x:(x,y)\\in S\\text{ for some }y\\}\\) is Souslin. If \\(C\\subseteq X\\times Y\\) is co-Souslin, then \\(\\{x:(x,y)\\in C\\text{ for all }y\\}\\) is co-Souslin.\n5. Some co-Souslin set in \\(\\mathcal C\\) is not a Souslin set, namely \\(\\mathcal C\\setminus D\\) for the set \\(D\\) of (D4).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 563,
        "through_line": 571,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "5. Souslin and co-Souslin sets",
              "line": 555,
              "through_line": 571,
              "anchors": [
                "OA-FND-DT-05"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
          "source_locus": "§12A–§12C; PDF 46–49",
          "role": "proof comparison",
          "correspondence": "Analytic/coanalytic notation and first separation",
          "explanation": "Closed projections, countable closure and analytic separation are fully compared; this is not a proof of the second separation theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.1",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Lemma 6.1",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 6.1.** Let \\(Z\\) be a countable set with the discrete topology. On the set of nonempty subsets of \\(Z\\), which are the nonempty closed subsets, the Effros Borel structure equals the Borel structure of \\(2^Z\\setminus\\{\\varnothing\\}\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 578,
        "through_line": 581,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.2",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Definition 6.2",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 6.2.** A *tree* on \\(Q\\) is a set \\(T\\subseteq Q^{<\\mathbb N}\\) such that \\(s\\in T\\) whenever \\(s\\sqsubseteq t\\in T\\). Its *body* is\n\\[\n[T]=\\{x\\in Q^{\\mathbb N}:\\ x|k\\in T\\text{ for all }k\\geq0\\}.\n\\]\nThe tree \\(T\\) is *well-founded* if \\([T]=\\varnothing\\) and *ill-founded* otherwise, and *pruned* if every \\(s\\in T\\) has an extension \\(s{}^\\frown a\\in T\\). We write \\(\\mathrm{Tree}(Q)\\) and \\(\\mathrm{WF}(Q)\\) for the sets of trees and of well-founded trees on \\(Q\\), and \\(\\mathrm{PT}(Q)\\) for the set of nonempty pruned trees.\n\nA tree is well-founded exactly when it contains no infinite chain \\(s^1\\sqsubset s^2\\sqsubset\\cdots\\). Indeed, the lengths along such a chain tend to infinity, and there is a unique \\(x\\in Q^{\\mathbb N}\\) with \\(x|\\,|s^k|=s^k\\) for all \\(k\\). Each \\(x|j\\) is an initial segment of some \\(s^k\\), so it lies in \\(T\\), and \\(x\\in[T]\\). Conversely, \\(x\\in[T]\\) gives the chain \\(x|1\\sqsubset x|2\\sqsubset\\cdots\\). The empty set is a well-founded tree, and every nonempty tree contains \\(\\varnothing\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 582,
        "through_line": 589,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 6.3",
      "kind": "proposition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Proposition 6.3",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Proposition 6.3.**\n1. \\(\\mathrm{Tree}(Q)\\) is closed in \\(2^{Q^{<\\mathbb N}}\\), and \\([T]\\) is closed in \\(Q^{\\mathbb N}\\) for every tree \\(T\\).\n2. For a closed set \\(F\\subseteq Q^{\\mathbb N}\\), the set \\(T_F=\\{x|k:\\ x\\in F,\\ k\\geq0\\}\\) is a pruned tree with \\([T_F]=F\\). The map \\(F\\mapsto T_F\\) is a Borel isomorphism of the Effros space \\(\\mathcal C_0(Q^{\\mathbb N})\\) onto \\(\\mathrm{PT}(Q)\\), which is a \\(G_\\delta\\) subset of \\(2^{Q^{<\\mathbb N}}\\). The inverse map is \\(T\\mapsto[T]\\).\n3. The set \\(\\mathrm{WF}(Q)\\) is co-Souslin in \\(2^{Q^{<\\mathbb N}}\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 590,
        "through_line": 613,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.4",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Definition 6.4",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 6.4** (the Kleene–Brouwer order). For distinct \\(s,t\\in\\mathbb N^{<\\mathbb N}\\), put \\(s<_{\\mathrm{KB}}t\\) if either \\(t\\sqsubset s\\), or there is an index \\(i\\leq\\min(|s|,|t|)\\) with \\(s_j=t_j\\) for \\(j<i\\) and \\(s_i<t_i\\). Write \\(s\\leq_{\\mathrm{KB}}t\\) if \\(s<_{\\mathrm{KB}}t\\) or \\(s=t\\).\n\nSo a proper extension of \\(t\\) comes before \\(t\\), and sequences that branch apart are compared at the first place where they differ.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 614,
        "through_line": 617,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.5",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Lemma 6.5",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 6.5.** \\(<_{\\mathrm{KB}}\\) is a strict linear order on \\(\\mathbb N^{<\\mathbb N}\\). The empty sequence is its largest element, and \\(s{}^\\frown a<_{\\mathrm{KB}}s\\) for all \\(s\\) and \\(a\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 618,
        "through_line": 627,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.6",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Theorem 6.6",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 6.6.** A tree \\(T\\) on \\(\\mathbb N\\) is well-founded if and only if \\(<_{\\mathrm{KB}}\\) restricted to \\(T\\) is a well-order.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 628,
        "through_line": 641,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.7",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-06::Lemma 6.7",
      "anchor": "oa-fnd-dt-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 6.7** (sections). Let \\(T\\) be a tree on \\(Q\\times P\\).\n1. \\([T(x)]=\\{y\\in P^{\\mathbb N}:(x,y)\\in[T]\\}\\) for every \\(x\\in Q^{\\mathbb N}\\).\n2. The map \\((T,x)\\mapsto T(x)\\), from \\(\\mathrm{Tree}(Q\\times P)\\times Q^{\\mathbb N}\\) to \\(\\mathrm{Tree}(P)\\), is continuous.\n3. Let \\(C\\subseteq Q^{\\mathbb N}\\times P^{\\mathbb N}\\) be closed and \\(T_C\\) its tree (Proposition 6.3(2)). Then \\(x\\mapsto T_C(x)\\) is continuous, and \\(x\\) lies outside the projection \\(\\{x:(x,y)\\in C\\text{ for some }y\\}\\) if and only if \\(T_C(x)\\in\\mathrm{WF}(P)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 642,
        "through_line": 652,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.1",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-07::Definition 7.1",
      "anchor": "oa-fnd-dt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 7.1.** A relation \\(R\\) is an *order* if\n1. \\(q\\leq_Rq'\\) implies \\(q,q'\\in D(R)\\);\n2. \\(q\\leq_Rq'\\) and \\(q'\\leq_Rq\\) imply \\(q=q'\\);\n3. \\(q\\leq_Rq'\\leq_Rq''\\) implies \\(q\\leq_Rq''\\).\n\nIt is a *linear order* if moreover any \\(q,q'\\in D(R)\\) satisfy \\(q\\leq_Rq'\\) or \\(q'\\leq_Rq\\), and a *well-order* if moreover every nonempty subset of \\(D(R)\\) has an \\(R\\)-least element. We write \\(\\mathrm{PO}(Q)\\supseteq\\mathrm{LO}(Q)\\supseteq\\mathrm{WO}(Q)\\) for these sets of relations.\n\nSo an order is a reflexive partial order on its domain that relates nothing outside the domain. Since \\(Q\\) is countable, a linear order \\(R\\) is a well-order exactly when there is no sequence \\((z_k)_{k\\geq1}\\) in \\(D(R)\\) with \\(z_{k+1}<_Rz_k\\) for all \\(k\\). Indeed, such a sequence has no least element. Conversely, if a nonempty \\(S\\subseteq D(R)\\) has no least element, fix an enumeration of \\(Q\\), pick \\(z_1\\in S\\), and let \\(z_{k+1}\\) be the first element of \\(S\\) in the enumeration with \\(z_{k+1}<_Rz_k\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 657,
        "through_line": 665,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.2",
      "kind": "proposition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-07::Proposition 7.2",
      "anchor": "oa-fnd-dt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Proposition 7.2.**\n1. \\(\\mathrm{PO}(Q)\\) and \\(\\mathrm{LO}(Q)\\) are closed subsets of \\(2^{Q\\times Q}\\). So they are compact metrizable spaces and standard Borel spaces.\n2. \\(\\mathrm{WO}(Q)\\) is co-Souslin, in \\(\\mathrm{LO}(Q)\\) and in \\(2^{Q\\times Q}\\).\n3. For \\(R\\in\\mathrm{LO}(Q)\\), let \\(T_R\\) consist of the finite sequences \\((z_1,\\dots,z_k)\\) with entries in \\(D(R)\\) and \\(z_{i+1}<_Rz_i\\) for \\(i<k\\). Then \\(T_R\\) is a tree, \\(R\\mapsto T_R\\) is continuous from \\(\\mathrm{LO}(Q)\\) to \\(2^{Q^{<\\mathbb N}}\\), and \\(R\\in\\mathrm{WO}(Q)\\) if and only if \\(T_R\\in\\mathrm{WF}(Q)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 666,
        "through_line": 680,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.3",
      "kind": "definition",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-07::Definition 7.3",
      "anchor": "oa-fnd-dt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Definition 7.3.** For \\(R,S\\in\\mathrm{LO}(\\mathbb N)\\), write \\(R\\preceq S\\) if some injective map \\(\\varphi:D(R)\\to D(S)\\) satisfies \\(m\\leq_Rn\\iff\\varphi(m)\\leq_S\\varphi(n)\\) for \\(m,n\\in D(R)\\). Write \\(R\\prec S\\) if such a \\(\\varphi\\) can be chosen with values in \\(\\{n\\in D(S):\\ n<_Sq\\}\\) for some \\(q\\in D(S)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 681,
        "through_line": 682,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.4",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-07::Lemma 7.4",
      "anchor": "oa-fnd-dt-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 7.4.**\n1. The sets \\(\\{(R,S):R\\preceq S\\}\\) and \\(\\{(R,S):R\\prec S\\}\\) are Souslin subsets of \\(\\mathrm{LO}(\\mathbb N)\\times\\mathrm{LO}(\\mathbb N)\\).\n2. If \\(R\\preceq S\\) and \\(S\\in\\mathrm{WO}(\\mathbb N)\\), then \\(R\\in\\mathrm{WO}(\\mathbb N)\\).\n3. Let \\(R,S\\in\\mathrm{WO}(\\mathbb N)\\) have order types \\(|R|\\) and \\(|S|\\) (S2). Then \\(R\\preceq S\\) if and only if \\(|R|\\leq|S|\\), and \\(R\\prec S\\) if and only if \\(|R|<|S|\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 683,
        "through_line": 693,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-08::Lemma 8.1",
      "anchor": "oa-fnd-dt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 8.1.** For \\(T\\subseteq\\mathbb N^{<\\mathbb N}\\) let\n\\[\n\\begin{gathered}\nK(T)\\\\\n=\\{(m,n)\\in\\mathbb N\\times\\mathbb N :\\\\\n\\ e(m)\\in T,\\ e(n)\\in T,\\\\\n\\ e(m)\\leq_{\\mathrm{KB}}e(n)\\}.\n\\end{gathered}\n\\]\nThen \\(K\\) is a continuous map of \\(2^{\\mathbb N^{<\\mathbb N}}\\) into \\(\\mathrm{LO}(\\mathbb N)\\), the domain of \\(K(T)\\) is \\(e^{-1}(T)\\), and \\(e\\) is an order isomorphism of \\(\\big(e^{-1}(T),\\leq_{K(T)}\\big)\\) onto \\((T,\\leq_{\\mathrm{KB}})\\). In particular, when \\(T\\) is a tree, \\(K(T)\\) is a well-order exactly when \\(T\\) is well-founded.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 698,
        "through_line": 710,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-08::Theorem 8.2",
      "anchor": "oa-fnd-dt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 8.2** (universality of the well-orders). Let \\(X\\) be a standard Borel space and \\(A\\subseteq X\\) a co-Souslin set.\n1. There is a Borel map \\(F:X\\to\\mathrm{LO}(\\mathbb N)\\) with \\(A=F^{-1}(\\mathrm{WO}(\\mathbb N))\\).\n2. If \\(X=\\Lambda\\), the map \\(F\\) can be chosen continuous.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 711,
        "through_line": 726,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 8.3",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-08::Corollary 8.3",
      "anchor": "oa-fnd-dt-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 8.3.** \\(\\mathrm{WO}(\\mathbb N)\\) and \\(\\mathrm{WF}(\\mathbb N)\\) are co-Souslin but not Souslin. In particular they are not Borel.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 727,
        "through_line": 731,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.1",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-09::Theorem 9.1",
      "anchor": "oa-fnd-dt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 9.1** (reduction). Let \\(X\\) be a standard Borel space and \\(A_n\\), \\(n\\in I\\), countably many co-Souslin subsets of \\(X\\). There are pairwise disjoint co-Souslin sets \\(C_n\\subseteq A_n\\) with \\(\\bigcup_nC_n=\\bigcup_nA_n\\).\n\nMarker’s freely readable notes give the two-set rank argument; the proof here establishes the stated countable reduction.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 734,
        "through_line": 749,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 9.2",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-09::Corollary 9.2",
      "anchor": "oa-fnd-dt-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 9.2** (second separation theorem). Let \\(X\\) be a standard Borel space.\n1. If \\(A_n\\), \\(n\\in I\\), are countably many Souslin subsets of \\(X\\), there are pairwise disjoint co-Souslin sets \\(C_n\\) with \\(A_n\\setminus\\bigcup_{m\\neq n}A_m\\subseteq C_n\\).\n2. In particular, for Souslin sets \\(A\\) and \\(B\\) there are disjoint co-Souslin sets \\(C\\) and \\(D\\) with \\(A\\setminus B\\subseteq C\\) and \\(B\\setminus A\\subseteq D\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 750,
        "through_line": 757,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [
        {
          "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
          "local_scopes": [
            {
              "heading": "6. Trees on a countable set",
              "line": 572,
              "through_line": 652,
              "anchors": [
                "OA-FND-DT-06"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "7. Countable orders and well-orders",
              "line": 653,
              "through_line": 693,
              "anchors": [
                "OA-FND-DT-07"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "8. Every co-Souslin set comes from the well-orders",
              "line": 694,
              "through_line": 731,
              "anchors": [
                "OA-FND-DT-08"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            },
            {
              "heading": "9. Reduction and the second separation theorem",
              "line": 732,
              "through_line": 757,
              "anchors": [
                "OA-FND-DT-09"
              ],
              "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
            }
          ],
          "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
          "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
          "role": "proof comparison",
          "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
          "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.1",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-10::Theorem 10.1",
      "anchor": "oa-fnd-dt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 10.1** (sets of uniqueness). Let \\(X,Y\\) be standard Borel spaces and \\(G\\subseteq X\\times Y\\) a Borel set. The set of \\(y\\in Y\\) for which exactly one \\(x\\in X\\) has \\((x,y)\\in G\\) is co-Souslin. In particular, for a Borel map \\(f:X\\to Y\\), the set\n\\[\nZ_f=\\{y\\in Y :\\ f^{-1}(y)\\text{ has exactly one point}\\}\n\\]\nis co-Souslin.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 760,
        "through_line": 791,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.2",
      "kind": "example",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-10::Example 10.2",
      "anchor": "oa-fnd-dt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Example 10.2** (the theorem is sharp). Let \\(D\\subseteq\\mathcal C\\) be the Souslin set of (D4) that is not Borel, and write \\(D=g(\\Lambda)\\) with \\(g\\) continuous (D3).\n\n(a) *\\(Z_f\\) need not be Borel.* Let \\(X\\) be the disjoint union of \\(\\Lambda\\) and \\(\\mathcal C\\), a Polish space, and let \\(f:X\\to\\mathcal C\\) be \\(g\\) on \\(\\Lambda\\) and the identity on \\(\\mathcal C\\). Then \\(f^{-1}(y)\\) consists of \\(y\\) and the points of \\(g^{-1}(y)\\), so it is a singleton exactly when \\(y\\notin D\\). Hence \\(Z_f=\\mathcal C\\setminus D\\), which is co-Souslin and not Borel.\n\n(b) *The domain must be standard.* Give \\(D\\) its relative Borel structure and let \\(f:D\\to\\mathcal C\\) be the inclusion. Every fibre has at most one point, and \\(Z_f=D\\). This set is not co-Souslin: otherwise it would be Borel by (D2).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 792,
        "through_line": 797,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.3",
      "kind": "lemma",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-10::Lemma 10.3",
      "anchor": "oa-fnd-dt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Lemma 10.3** (isolated points). Let \\(L\\) be a nonempty countable closed subset of a Polish space. Then every nonempty \\(S\\subseteq L\\) contains a point \\(q\\) with an open set \\(V\\) such that \\(V\\cap S=\\{q\\}\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 798,
        "through_line": 803,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.4",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-10::Theorem 10.4",
      "anchor": "oa-fnd-dt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 10.4** (countable-to-one maps). Let \\(X,Y\\) be standard Borel spaces and \\(f:X\\to Y\\) a Borel map such that every fibre \\(f^{-1}(y)\\) is countable.\n1. \\(f(B)\\) is Borel for every Borel set \\(B\\subseteq X\\); in particular \\(f(X)\\) is Borel.\n2. There is a Borel map \\(g:f(X)\\to X\\) with \\(f(g(y))=y\\) for all \\(y\\in f(X)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 804,
        "through_line": 819,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.5",
      "kind": "example",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-10::Example 10.5",
      "anchor": "oa-fnd-dt-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Example 10.5** (the hypotheses are needed). (a) *Countable fibres.* The set \\(D\\) of (D4) is not Borel, but it is \\(g(\\Lambda)\\) for a continuous \\(g\\) on the standard Borel space \\(\\Lambda\\). By Theorem 10.4, some fibres of \\(g\\) are uncountable. (b) *A standard domain.* The inclusion of \\(D\\) into \\(\\mathcal C\\), as in Example 10.2(b), is injective, and its image \\(D\\) is not Borel.\n\nPart (2) can also be derived from the Borel transversal theorem of [Polish spaces and standard Borel spaces](polish-spaces-and-standard-borel-spaces.md#oa-fnd-pb-10): after the reduction the fibres of \\(f\\) are closed, and \\(f^{-1}(f(U))\\) is Borel for every open \\(U\\) by part (1).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 820,
        "through_line": 823,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.1",
      "kind": "theorem",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-11::Theorem 11.1",
      "anchor": "oa-fnd-dt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Theorem 11.1** (Kuratowski–Ryll-Nardzewski). Let \\((\\Omega,\\mathcal S)\\) be a measurable space, \\(X\\) a Polish space, and \\(\\Phi\\) a map that assigns to each \\(\\omega\\in\\Omega\\) a nonempty closed set \\(\\Phi(\\omega)\\subseteq X\\), such that\n\\[\n\\begin{gathered}\n\\{\\omega:\\ \\Phi(\\omega)\\cap U\\neq\\varnothing\\}\\in\\mathcal S\\\\\n\\text{for every open }U\\\\\n\\subseteq X.\n\\end{gathered}\n\\tag{11.1}\n\\]\n1. For every open \\(U\\subseteq X\\) there is an \\(\\mathcal S\\)-measurable \\(f:\\Omega\\to X\\) with \\(f(\\omega)\\in\\Phi(\\omega)\\) for all \\(\\omega\\), and \\(f(\\omega)\\in U\\) whenever \\(\\Phi(\\omega)\\cap U\\neq\\varnothing\\).\n2. There are \\(\\mathcal S\\)-measurable maps \\(f_1,f_2,\\ldots:\\Omega\\to X\\) such that, for every \\(\\omega\\), all \\(f_n(\\omega)\\) lie in \\(\\Phi(\\omega)\\) and \\(\\{f_n(\\omega):n\\geq1\\}\\) is dense in \\(\\Phi(\\omega)\\).\n3. Conversely, if maps as in (2) exist, then \\(\\Phi\\) satisfies (11.1).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 828,
        "through_line": 872,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.2",
      "kind": "corollary",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-11::Corollary 11.2",
      "anchor": "oa-fnd-dt-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Corollary 11.2** (Borel selectors on the Effros space). Let \\(X\\) be a Polish space and \\(\\mathcal C_0(X)\\) the standard Borel space of its nonempty closed subsets (D8).\n1. For every open \\(U\\subseteq X\\) there is a Borel map \\(f:\\mathcal C_0(X)\\to X\\) with \\(f(F)\\in F\\) for all \\(F\\), and \\(f(F)\\in U\\) whenever \\(F\\cap U\\neq\\varnothing\\).\n2. There are Borel maps \\(f_n:\\mathcal C_0(X)\\to X\\) such that \\(\\{f_n(F):n\\geq1\\}\\) is a dense subset of \\(F\\) for every \\(F\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 873,
        "through_line": 880,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-12::Exercise 1",
      "anchor": "oa-fnd-dt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Exercise 1** (medium; commutative case). Let \\(K\\) be a nonempty compact Hausdorff space and \\(A=C(K)\\) with the supremum norm. Show that for every \\(f\\in A\\), \\(W(f)\\) is the closed convex hull of \\(f(K)=\\sigma(f)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 883,
        "through_line": 903,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-12::Exercise 2",
      "anchor": "oa-fnd-dt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Exercise 2** (easy; a self-adjoint element with nonreal spectrum). On \\(A=\\mathbb C^2\\), with coordinatewise operations and the norm \\(\\|(z,w)\\|=\\max(|z|,|w|)\\), put \\((z,w)^*=(\\bar w,\\bar z)\\). Show that this is an isometric involution, that \\(A\\) is not a C\\*-algebra, and that \\(a=(i,-i)\\) is self-adjoint and not hermitian. Compute \\(\\sigma(a)\\) and \\(W(a)\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 904,
        "through_line": 907,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-12::Exercise 3",
      "anchor": "oa-fnd-dt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Exercise 3** (easy; stochastic matrices). Let \\(T\\gg0\\) be an \\(n\\times n\\) matrix whose rows sum to \\(1\\). Show that \\(r(T)=1\\), and that \\(T^k\\) converges to the matrix all of whose rows equal the vector \\(u\\gg0\\) determined by \\(u^{\\top}T=u^{\\top}\\) and \\(\\sum_iu_i=1\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 908,
        "through_line": 911,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-12::Exercise 4",
      "anchor": "oa-fnd-dt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Exercise 4** (medium; operations on relations). Let \\(Q\\) be countable. For relations \\(R,S\\) on \\(Q\\) put \\[\n\\begin{gathered}\nR\\ast S\\\\\n=\\{(q,q',q''):(q,q')\\in R,\\ (q',q'')\\in S\\}\\\\\n\\subseteq Q^3\n\\end{gathered}\n\\] and \\[\n\\begin{gathered}\nR\\circ S\\\\\n=\\{(q,q''):(q,q',q'')\\in R\\ast S\\text{ for some }q'\\}.\n\\end{gathered}\n\\] Show that \\((R,S)\\mapsto R\\ast S\\) is continuous, that \\((R,S)\\mapsto R\\circ S\\) is Borel, and that it is not continuous when \\(Q\\) is infinite. Use this to show once more that \\(\\mathrm{PO}(Q)\\) is a Borel set, from the description: \\(R\\in\\mathrm{PO}(Q)\\) if and only if \\(R\\cap R^{-1}\\) is the diagonal of \\(D(R)\\), both projections of \\(R\\) equal \\(D(R)\\), and \\(R\\circ R\\subseteq R\\).",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 912,
        "through_line": 926,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5",
      "kind": "exercise",
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets#oa-fnd-dt-12::Exercise 5",
      "anchor": "oa-fnd-dt-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/numerical-ranges-positive-matrices-and-co-souslin-sets@78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682",
      "statement_and_full_conditions": "**Exercise 5** (hard; the second separation theorem cannot use Borel sets). Using the universal Souslin set \\(U\\subseteq\\mathcal C\\times\\mathcal C\\) of (D4), find disjoint co-Souslin sets in \\(\\mathcal C\\times\\mathcal C\\) that no Borel set separates. Conclude that in Corollary 9.2(2) the sets \\(C\\) and \\(D\\) cannot always be chosen Borel.",
      "proof_locus": {
        "source": "src/numerical-ranges-positive-matrices-and-co-souslin-sets.md",
        "line": 927,
        "through_line": 941,
        "sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 1.1",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-01::Example 1.1",
      "anchor": "oa-fnd-ao-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 1.1** (Diagonal operators). Let \\(I\\) be a set, \\(H=\\ell^2(I)\\) with its standard basis \\((\\delta_i)_{i\\in I}\\), and for a bounded function \\(f\\) on \\(I\\) let \\(M_f\\) be the diagonal operator \\(M_f\\delta_i=f(i)\\delta_i\\). The algebra \\(\\mathcal A=\\{M_f:f\\in\\ell^\\infty(I)\\}\\) is abelian, and it is its own commutant. Indeed, if \\(T\\) commutes with each rank-one projection \\(M_{1_{\\{i\\}}}\\), then \\(T\\) maps \\(\\mathbb C\\delta_i\\) into itself, so \\(T\\delta_i=t_i\\delta_i\\) with \\(|t_i|\\le\\|T\\|\\), and \\(T=M_t\\). So \\(\\mathcal A\\) is a maximal abelian von Neumann algebra. In \\(\\ell^\\infty(I)\\) a bounded increasing net of real functions \\(f_\\alpha\\) has its pointwise supremum \\(f\\) as least upper bound, and \\(M_{f_\\alpha}\\to M_f\\) strongly. Indeed, for a fixed vector, choose a finite set of coordinates with arbitrarily small squared-norm tail. Pointwise convergence handles the finite set, and the uniform bound on \\(f_\\alpha-f\\) controls the tail.\n\nIf \\(I=\\{i_1,i_2,\\dots\\}\\) is countable (listed without repetitions, with a finite sum when \\(I\\) is finite), the vector \\(\\xi=\\sum_k2^{-k}\\delta_{i_k}\\) is cyclic for \\(\\mathcal A\\), because \\(M_{1_{\\{i_k\\}}}\\xi=2^{-k}\\delta_{i_k}\\). For \\(I=\\varnothing\\), \\(H=\\{0\\}\\) and the zero vector is cyclic. If \\(I\\) is uncountable, no vector is cyclic: a vector \\(\\xi\\in\\ell^2(I)\\) has only countably many nonzero coordinates (the sets where \\(|\\xi_i|\\geq1/n\\) are finite), and \\(\\mathcal A\\xi\\) lies in the closed subspace spanned by the corresponding basis vectors. The spectrum of \\(\\ell^\\infty(I)\\) is the Stone–Čech compactification \\(\\beta I\\) of the discrete space \\(I\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 27,
        "through_line": 30,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 1.2",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-01::Example 1.2",
      "anchor": "oa-fnd-ao-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 1.2** (Multiplication operators on the unit interval). Let \\(m\\) be Lebesgue measure on \\([0,1]\\) and \\(\\mathcal A=\\{M_f:f\\in L^\\infty[0,1]\\}\\) on \\(L^2[0,1]\\), where \\(M_f\\xi=f\\xi\\). This is a maximal abelian von Neumann algebra, and the constant function \\(1\\) is a cyclic vector (Theorem 3.1). The multiplications by continuous functions form a C\\*-subalgebra that is not a von Neumann algebra. For instance, \\(g_n(t)=\\min\\{1,\\max\\{0,n(t-\\frac12)\\}\\}\\) increases to the indicator function of \\((\\frac12,1]\\), and \\(M_{g_n}\\) converges strongly to multiplication by that indicator. In \\(C[0,1]\\) itself the sequence \\((g_n)\\) has no least upper bound. An upper bound \\(g\\) satisfies \\(g\\ge1\\) on \\((\\frac12,1]\\), so \\(g(\\frac12)\\ge1\\) by continuity, and \\(g\\ge0\\). Hence \\(g>\\frac12\\) on some interval \\([\\frac12-\\delta,\\frac12]\\). Subtracting a continuous bump \\(b\\) with \\(0\\le b\\le\\frac12\\), supported in \\((\\frac12-\\delta,\\frac12)\\) and not identically zero, gives a smaller upper bound, because every \\(g_n\\) vanishes on \\([0,\\frac12]\\). So \\(C[0,1]\\) lacks least upper bounds that \\(L^\\infty[0,1]\\) has. The reason is topological: \\([0,1]\\) is connected, while spectra of von Neumann algebras are extremally disconnected (Sections 4 and 7).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 31,
        "through_line": 32,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 1.3",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-01::Example 1.3",
      "anchor": "oa-fnd-ao-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 1.3** (Multiplicity two). On \\(L^2[0,1]\\oplus L^2[0,1]\\) let \\(\\mathcal A_2=\\{M_f\\oplus M_f:f\\in L^\\infty[0,1]\\}\\). This abelian von Neumann algebra is isomorphic to the algebra of Example 1.2 by \\(M_f\\mapsto M_f\\oplus M_f\\). But its commutant contains the operators \\(\\begin{pmatrix}\\alpha&\\beta\\\\\\gamma&\\delta\\end{pmatrix}\\) with scalar entries, which do not commute with each other. So \\(\\mathcal A_2\\) is not maximal abelian, and by Corollary 3.3 it has no cyclic vector. The two algebras are isomorphic but not spatially isomorphic, since their commutants differ (see Exercise 12.4).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 33,
        "through_line": 34,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 1.4",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-01::Example 1.4",
      "anchor": "oa-fnd-ao-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 1.4** (An \\(L^\\infty\\) space that is not a von Neumann algebra on its \\(L^2\\) space). Let \\(X\\) be an uncountable set, \\(\\Sigma\\) the \\(\\sigma\\)-algebra of sets that are countable or have countable complement, and \\(\\nu(E)\\) the number of points of \\(E\\) if \\(E\\) is countable and \\(\\nu(E)=\\infty\\) otherwise. This is a measure, and only the empty set has measure zero.\n\n*Measurable functions are constant off a countable set.* Let \\(f:X\\to\\mathbb R\\) be \\(\\Sigma\\)-measurable. If \\(\\{f>r\\}\\) had countable complement for every rational \\(r\\), then \\(\\bigcap_r\\{f>r\\}=\\varnothing\\) would have countable complement, which is absurd. If \\(\\{f>r\\}\\) were countable for every rational \\(r\\), then \\(X=\\bigcup_r\\{f>r\\}\\) would be countable. The sets \\(\\{f>r\\}\\) decrease in \\(r\\), so there is a number \\(c\\) with \\(\\{f>r\\}\\) co-countable for rational \\(r<c\\) and countable for rational \\(r>c\\). Then \\(\\{f\\ne c\\}\\) is contained in \\(\\bigcup_{r<c}\\{f\\le r\\}\\cup\\bigcup_{r>c}\\{f>r\\}\\), a countable set. The same holds for complex functions, by taking real and imaginary parts.\n\n*The spaces.* If \\(\\int|f|^2d\\nu<\\infty\\), the constant \\(c\\) must be \\(0\\), since the co-countable set \\(\\{f=c\\}\\) has infinite measure. Conversely, every function with countable support is measurable. So \\(L^2(X,\\Sigma,\\nu)=\\ell^2(X)\\), with the same norm. Since only \\(\\varnothing\\) is null, \\(L^\\infty(X,\\Sigma,\\nu)\\) consists of the bounded functions that are constant off a countable set, acting on \\(\\ell^2(X)\\) by diagonal operators.\n\n*The algebra is too small.* Every singleton is measurable, so an operator that commutes with \\(L^\\infty(X,\\Sigma,\\nu)\\) commutes with the projections onto the basis vectors, and it is diagonal, as in Example 1.1. So the commutant of the multiplication algebra is the algebra of all diagonal operators \\(M_t\\), \\(t\\in\\ell^\\infty(X)\\), and its bicommutant is again this algebra. [Proposition 8.5 of the Hahn–Banach lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-08) supplies a partition of \\(X\\) into two uncountable sets. The multiplication operator by the indicator of one of them lies in the bicommutant but not in the multiplication algebra. So the multiplication algebra of \\(L^\\infty(X,\\Sigma,\\nu)\\) is not a von Neumann algebra.\n\nWhether \\(L^\\infty\\) acts on \\(L^2\\) as a von Neumann algebra is therefore a property of the measure. It holds for finite measures (Remark 3.2), for sigma-finite measures and for Radon measures on locally compact spaces with the local conventions (Proposition 3.2a below). Proposition 3.2b proves the finite-piece gluing criterion used here; Example 3.2c shows why a local-null convention matters.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 35,
        "through_line": 44,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.1",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-23::Lemma 2.1",
      "anchor": "oa-fnd-ao-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 2.1** (Rare and meager sets). Let \\(X\\) be a topological space.\n\n1. A set is rare exactly when its closure is rare. Subsets of rare sets and finite unions of rare sets are rare. For an open set \\(G\\) the set \\(\\overline G\\setminus G\\) is rare, and for a closed set \\(F\\) the set \\(F\\setminus F^\\circ\\) is rare.\n2. (*Baire category theorem for compact spaces.*) Every compact space is a Baire space. Equivalently, a countable intersection of dense open subsets is dense, and the complement of a meager set is dense.\n3. Let \\((G_j)_{j\\in J}\\) be pairwise disjoint open sets and \\(R_j\\subseteq G_j\\). If every \\(R_j\\) is rare, then \\(\\bigcup_jR_j\\) is rare. If every \\(R_j\\) is meager, then \\(\\bigcup_jR_j\\) is meager.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 117,
        "through_line": 128,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.2",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-23::Lemma 2.2",
      "anchor": "oa-fnd-ao-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 2.2** (Regularizations). Let \\(f\\) be a bounded real function on a topological space \\(X\\).\n\n1. \\(f_*\\le f\\le f^*\\). The function \\(f_*\\) is lsc, and it is the largest lsc function below \\(f\\); likewise \\(f^*\\) is the smallest usc function above \\(f\\). If \\(f\\) is continuous at \\(x\\), then \\(f_*(x)=f(x)=f^*(x)\\).\n2. If \\(f\\) is lsc, then \\(\\{f^*\\ne f\\}\\) is meager. If \\(f\\) is usc, then \\(\\{f_*\\ne f\\}\\) is meager.\n\nNo separation axiom is needed.",
      "proof_locus": {
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      },
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    {
      "local_label": "Lemma 2.3",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-23::Lemma 2.3",
      "anchor": "oa-fnd-ao-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 2.3** (Radon measures and increasing nets). Let \\(\\mu\\) be a Radon measure on a compact space \\(\\Omega\\).\n\n1. If \\((U_i)\\) is an increasing net of open sets, then \\(\\mu(\\bigcup_iU_i)=\\sup_i\\mu(U_i)\\).\n2. Let \\((f_i)\\) be an increasing net of real continuous functions with \\(\\sup_i\\|f_i\\|<\\infty\\), and let \\(f\\) be its pointwise supremum. Then \\(f\\) is lsc and \\(\\int f\\,d\\mu=\\sup_i\\mu(f_i)\\).\n3. \\(C(\\Omega)\\) is dense in \\(L^2(\\Omega,\\mu)\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
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      },
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      "proof_basis": "Complete argument at the identified source locus.",
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    {
      "local_label": "Lemma 2.4",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-23::Lemma 2.4",
      "anchor": "oa-fnd-ao-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 2.4** (Increasing nets of operators). Let \\((x_i)\\) be an increasing net of self-adjoint operators on a Hilbert space \\(H\\) with \\(\\sup_i\\|x_i\\|<\\infty\\). Then \\((x_i)\\) converges strongly to a self-adjoint operator \\(x\\), and among the self-adjoint operators on \\(H\\) the net has \\(x\\) as its least upper bound. If all \\(x_i\\) lie in a von Neumann algebra \\(M\\), then \\(x\\in M\\), and \\(x\\) is also its least upper bound in \\(M_h\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
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      },
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    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Theorem 3.1",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 3.1** (Cyclic representations of \\(C_0(\\Omega)\\)). Let \\(\\Omega\\) be locally compact, with a finite positive Radon measure \\(\\mu\\).\n\n1. The constant function \\(1\\) is a cyclic vector for \\(\\pi_\\mu\\), and \\(\\langle\\pi_\\mu(x)1,1\\rangle=\\int x\\,d\\mu\\) for \\(x\\in C_0(\\Omega)\\).\n2. Let \\(\\pi\\) be a representation of \\(C_0(\\Omega)\\) on \\(H\\) with a cyclic vector \\(\\xi\\) such that \\(\\langle\\pi(x)\\xi,\\xi\\rangle=\\int x\\,d\\mu\\) for all \\(x\\). Then there is exactly one unitary \\(U:L^2(\\Omega,\\mu)\\to H\\) with \\(U\\pi_\\mu(x)=\\pi(x)U\\) for all \\(x\\) and \\(U1=\\xi\\).\n3. \\[\n\\begin{gathered}\n\\pi_\\mu(C_0(\\Omega))'\\\\\n=\\{M_f:f\\in L^\\infty(\\Omega,\\mu)\\}\\\\\n=\\pi_\\mu(C_0(\\Omega))''.\n\\end{gathered}\n\\] So the von Neumann algebra generated by \\(\\pi_\\mu(C_0(\\Omega))\\) is the algebra of multiplications by \\(L^\\infty(\\Omega,\\mu)\\), it is maximal abelian, and \\(f\\mapsto M_f\\) is an isometric \\(*\\)-isomorphism of \\(L^\\infty(\\Omega,\\mu)\\) onto it.\n\nPart (2) identifies \\(\\pi_\\mu\\) with the GNS representation of the positive functional \\(x\\mapsto\\int x\\,d\\mu\\). Its cyclic vector has squared norm \\(\\mu(\\Omega)\\); the functional is a state when this mass is one. The nonunital construction and its uniqueness are proved in [the GNS lesson, Theorems 5.4–5.5](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-05).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
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              "heading": "8. Countably generated abelian von Neumann algebras",
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                "OA-FND-AO-14",
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          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.2a",
      "kind": "proposition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Proposition 3.2a",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Proposition 3.2a** (Multiplication beyond finite measures). Multiplication gives an isometric unital \\(*\\)-isomorphism of \\(L^\\infty\\) onto a maximal abelian von Neumann algebra on \\(L^2\\) in either of the following settings: an arbitrary sigma-finite measure space, or a positive Radon measure on a locally compact Hausdorff space, with measurable functions and equality understood locally in the second setting.\n\n*Proof.* For a sigma-finite measure space, partition the space into countably many measurable sets \\(E_i\\) of finite measure. The projections \\(P_i=M_{1_{E_i}}\\) are pairwise orthogonal and their sum is \\(1\\) strongly, because the squared norm of the omitted tail is the tail of the convergent series \\(\\sum_i\\int_{E_i}|\\xi|^2\\). If \\(T\\) commutes with every bounded multiplication, it commutes with these projections. On \\(P_iL^2=L^2(E_i)\\), Remark 3.2 gives \\(T|_{P_iL^2}=M_{g_i}\\) with \\(\\|g_i\\|_\\infty\\leq\\|T\\|\\). Choose measurable representatives bounded everywhere by \\(\\|T\\|\\), and glue them on the \\(E_i\\). The resulting bounded measurable \\(g\\) satisfies \\(T=M_g\\), first on finite sums of the subspaces and then on all of \\(L^2\\) by continuity. The multiplication algebra is therefore its own commutant. Its operator norm equals the essential supremum norm: a level set of positive measure meets some \\(E_i\\) in a set of positive finite measure, and the indicator of that intersection tests the required lower bound.\n\nFor a Radon measure, [Lemma 2.1 of the vector-valued-functions lesson](vector-valued-integration-and-preduals.md#oa-fnd-vv-03) proves a decomposition into pairwise disjoint compact pieces \\(K_i\\), with locally null complement \\(N\\), such that every compact set meets only countably many pieces. Its gluing assertion says that scalar functions measurable on the pieces are locally measurable after being set to zero on \\(N\\); Lemma 1.2(4) identifies measurability on each compact piece with completed Borel measurability. These conclusions use only Radon regularity and Zorn's lemma, and their full proofs are given there. [Lemma 2.3(2)–(3) of the same lesson](vector-valued-integration-and-preduals.md#oa-fnd-vv-03) proves\n\\[\n\\|\\xi\\|_2^2=\\sum_i\\int_{K_i}|\\xi|^2\\,d\\mu.\n\\]\nThus \\(L^2\\) is the Hilbert sum of the finite-measure spaces \\(L^2(K_i)\\). The identification is onto: a square-summable family of piecewise \\(L^2\\) classes has only countably many nonzero components; choose completed measurable representatives on those pieces, put zero on all others and on \\(N\\), and apply the scalar gluing assertion. The displayed integral identity gives its squared norm and its prescribed components. In the compact-set approximation used by that gluing proof, a test set of measure zero needs no pieces: take the approximating compact subset to be empty, with error zero. For a test set of positive measure, the list of meeting pieces is nonempty, and the finite-tail approximation uses an integer \\(J\\geq1\\). In particular every vector has only countably many nonzero components, and the finite partial sums of \\(P_i=M_{1_{K_i}}\\) converge strongly to \\(1\\).\n\nAgain an operator commuting with every multiplication restricts on each piece to \\(M_{g_i}\\), by Remark 3.2. Truncate representatives on each piece so that \\(|g_i|\\leq\\|T\\|\\) everywhere. The gluing assertion gives a locally measurable bounded function \\(g\\), zero on \\(N\\), and the Hilbert-sum identity gives \\(T=M_g\\). Conversely all scalar multiplications commute, so the multiplication algebra equals its commutant and is a von Neumann algebra. If \\(0<c<\\|g\\|_\\infty\\), the level set \\(\\{|g|>c\\}\\) is not locally null. It therefore meets a compact set in a completed measurable set \\(F\\) with \\(0<\\mu(F)<\\infty\\); testing on \\(1_F\\) gives \\(\\|M_g\\|\\geq c\\). Together with the upper bound this proves isometry and faithfulness. If the norm is zero, the assertion is immediate. No lifting or general decomposable-operator theorem is used. \\(\\square\\)\n\nThe local integral used here is the supremum of integrals over compact subsets, as defined and justified in [Definitions 2.2–2.4 of the same provider](vector-valued-integration-and-preduals.md#oa-fnd-vv-03). It ignores locally null sets even when their outer-regular Borel measure is not zero. For a finite measure on a compact space it is the usual integral.",
      "proof_locus": {
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          "source_key": "fremlin-measure-theory-vol4-2013-source",
          "source_locus": "415A, 416A–B",
          "role": "context / statement",
          "correspondence": "different measure conventions",
          "explanation": "Fremlin uses complete, locally determined Radon measures. The lesson’s outer-regular Radon local-integral convention is instead supplied by the exact programme compact-piece proof. No unproved convention interchange is used."
        },
        {
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          "local_scopes": [
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                "OA-FND-AO-14",
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          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.2b",
      "kind": "proposition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Proposition 3.2b",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Proposition 3.2b** (Finite-piece gluing beyond sigma-finiteness). Let \\((X,\\Sigma,\\mu)\\) admit a partition \\((E_i)_{i\\in I}\\) into measurable sets of finite measure such that a set \\(S\\subseteq X\\) belongs to \\(\\Sigma\\) exactly when each \\(S\\cap E_i\\) is measurable, and then\n\\[\n\\mu(S)=\\sum_{i\\in I}\\mu(S\\cap E_i).\n\\]\nThe sum means the supremum of the finite partial sums. Such a measure space is called *strictly localizable*. No countability of \\(I\\) is required. Multiplication is an isometric unital *-isomorphism of \\(L^\\infty(X,\\mu)\\) onto a maximal abelian von Neumann algebra on \\(L^2(X,\\mu)\\).",
      "proof_locus": {
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        "through_line": 240,
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          "source_key": "bouafia-de-pauw-2105.11331v1",
          "source_locus": "§4.7 and §6.1–4; measure-space definition in §6.11 footnote 1",
          "role": "generalization / proof mechanism",
          "correspondence": "partial source, complete independent operator proof",
          "explanation": "Finite-measure pieces glue over arbitrary index sets. The paper studies more general negligible-set categories; no claim equating all its hypotheses with the lesson is made."
        },
        {
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          "source_locus": "415A, 416A–B",
          "role": "context / statement",
          "correspondence": "different measure conventions",
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        },
        {
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          "local_scopes": [
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              "heading": "3. Measures and cyclic representations",
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              "heading": "8. Countably generated abelian von Neumann algebras",
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                "OA-FND-AO-14",
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          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.2c",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Example 3.2c",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 3.2c** (A measure whose multiplication representation loses the identity). Let \\(X\\) be uncountable, let \\(\\Sigma\\) consist of the countable and co-countable subsets, and put \\(\\mu(S)=0\\) for countable \\(S\\), \\(\\mu(S)=\\infty\\) otherwise. This is countably additive: a disjoint sequence contains at most one co-countable member; if it contains none, its union is countable. Every integrable squared modulus has null positive level sets, since \\(\\mu(\\{|\\xi|>1/n\\})\\leq n^2\\int|\\xi|^2<\\infty\\), and every finite-measure set here is null. Thus \\(L^2(X,\\mu)=\\{0\\}\\), whereas the constant function \\(1\\) has \\(L^\\infty\\)-norm one. Multiplication is not faithful. The measure is not semi-finite: its positive-measure sets contain no subsets of positive finite measure. Finite-piece gluing in Proposition 3.2b prevents this loss, and Proposition 3.2a already specifies conventions that prevent it in the Radon model.",
      "proof_locus": {
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          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Corollary 3.3",
      "kind": "corollary",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Corollary 3.3",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Corollary 3.3** (Abelian algebras with a cyclic vector). Suppose that the abelian von Neumann algebra \\(M\\subseteq B(H)\\) has a cyclic vector \\(\\xi\\). Let \\(K\\) be the spectrum of \\(M\\), write \\(x\\mapsto\\hat x\\) for the Gelfand isomorphism of \\(M\\) onto \\(C(K)\\), and let \\(\\mu\\) be the Radon measure on \\(K\\) with \\(\\int\\hat x\\,d\\mu=\\langle x\\xi,\\xi\\rangle\\). Then:\n\n1. there is a unitary \\(U:L^2(K,\\mu)\\to H\\) with \\(U1=\\xi\\) and \\(UM_{\\hat x}U^*=x\\) for \\(x\\in M\\);\n2. \\(M\\) is maximal abelian, \\(M'=M\\);\n3. every bounded Borel function on \\(K\\) agrees \\(\\mu\\)-almost everywhere with a continuous function.",
      "proof_locus": {
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            }
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          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.4",
      "kind": "corollary",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Corollary 3.4",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Corollary 3.4** (Equivalent measures give equivalent representations). Let \\(\\Omega\\) be locally compact, and let \\(\\mu\\) and \\(\\nu\\) be finite positive Radon measures on it. The representations \\(\\pi_\\mu\\) and \\(\\pi_\\nu\\) are unitarily equivalent if and only if \\(\\mu\\) and \\(\\nu\\) have the same null sets.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 260,
        "through_line": 275,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.5",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-03::Example 3.5",
      "anchor": "oa-fnd-ao-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 3.5.** On \\([0,1]\\) let \\(m\\) be Lebesgue measure and \\(\\delta_{1/2}\\) the point mass at \\(\\frac12\\). The measures \\(m+\\delta_{1/2}\\) and \\(2m+3\\delta_{1/2}\\) have the same null sets, so their multiplication representations are equivalent. The measures \\(m\\) and \\(m+\\delta_{1/2}\\) do not, and indeed \\(\\pi_{m+\\delta_{1/2}}(C[0,1])''\\) contains a rank-one projection, multiplication by \\(1_{\\{1/2\\}}\\), while \\(\\pi_m(C[0,1])''=L^\\infty[0,1]\\) contains none.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 276,
        "through_line": 277,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.1",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-18::Lemma 10.1",
      "anchor": "oa-fnd-ao-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 10.1.** Let \\(\\nu\\in\\operatorname{ba}(\\Sigma,\\mathcal I)\\). Then \\(|\\nu|(X)\\le4\\sup_E|\\nu(E)|\\). If \\(E_1,\\dots,E_n\\in\\Sigma\\) are disjoint, then \\(\\sum_k|\\nu|(E_k)=|\\nu|(\\bigcup_kE_k)\\). So \\(|\\nu|\\) is a bounded, positive, finitely additive function on \\(\\Sigma\\), and \\(\\|\\nu\\|=|\\nu|(X)\\) is a norm on \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 294,
        "through_line": 309,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.2",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-18::Theorem 10.2",
      "anchor": "oa-fnd-ao-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 10.2** (The dual of \\(L^\\infty\\)). Let \\(\\nu\\in\\operatorname{ba}(\\Sigma,\\mathcal I)\\). For a simple function \\(s=\\sum_kc_k1_{E_k}\\), with \\((E_k)\\) a finite partition of \\(X\\) into sets of \\(\\Sigma\\), put \\(\\int s\\,d\\nu=\\sum_kc_k\\nu(E_k)\\). This extends to a bounded linear functional \\(\\varphi_\\nu\\) on \\(L^\\infty(\\Sigma,\\mathcal I)\\). The map \\(\\nu\\mapsto\\varphi_\\nu\\) is a linear bijection of \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) onto the dual of \\(L^\\infty(\\Sigma,\\mathcal I)\\), and it is isometric for the norm \\(\\|\\nu\\|=|\\nu|(X)\\); in particular \\(\\operatorname{ba}(\\Sigma,\\mathcal I)\\) is a Banach space. The functional \\(\\varphi_\\nu\\) is positive if and only if \\(\\nu\\ge0\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 310,
        "through_line": 323,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.3",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-18::Theorem 10.3",
      "anchor": "oa-fnd-ao-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 10.3** (Phillips' lemma). Let \\(\\Gamma\\) be a set and \\((\\nu_n)\\) a sequence in \\(\\operatorname{ba}(2^\\Gamma,\\{\\varnothing\\})=\\ell^\\infty(\\Gamma)^*\\) with \\(\\sup_n|\\nu_n|(\\Gamma)<\\infty\\). If \\(\\nu_n(E)\\to0\\) for every \\(E\\subseteq\\Gamma\\), then\n\\[\n\\lim_{n\\to\\infty}\\sum_{\\gamma\\in\\Gamma}|\\nu_n(\\{\\gamma\\})|=0 .\n\\]",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 324,
        "through_line": 360,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 10.4",
      "kind": "corollary",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-18::Corollary 10.4",
      "anchor": "oa-fnd-ao-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Corollary 10.4** (Weak and norm convergence in \\(\\ell^1\\)). Let \\(\\Gamma\\) be a set. Every weakly convergent sequence in \\(\\ell^1(\\Gamma)\\) converges in norm. More generally, every weakly Cauchy sequence in \\(\\ell^1(\\Gamma)\\) converges in norm.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 361,
        "through_line": 365,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.5",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-18::Example 10.5",
      "anchor": "oa-fnd-ao-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 10.5** (The hypotheses matter). The unit vectors \\(e_n\\in\\ell^1(\\mathbb N)\\) have norm one. They converge to \\(0\\) against every element of \\(c_0\\), the predual of \\(\\ell^1\\), but not weakly: pairing with the constant function \\(1\\in\\ell^\\infty\\) gives \\(1\\). Correspondingly, the point evaluations \\(\\nu_n=\\delta_n\\in\\ell^\\infty(\\mathbb N)^*\\) satisfy \\(\\nu_n(E)\\to0\\) for every finite set \\(E\\), but not for \\(E=\\mathbb N\\), and \\(\\sum_k|\\nu_n(\\{k\\})|=1\\) for all \\(n\\). So in Theorem 10.3 the hypothesis is needed for all subsets \\(E\\), not only for the finite ones.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 366,
        "through_line": 367,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-05::Definition 4.1",
      "anchor": "oa-fnd-ao-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Definition 4.1.** A Hausdorff space \\(X\\) is called *extremally disconnected* when each open set \\(U\\subseteq X\\) has an open closure \\(\\overline U\\). A compact extremally disconnected space is called *stonean*.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 378,
        "through_line": 379,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.2",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-05::Lemma 4.2",
      "anchor": "oa-fnd-ao-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 4.2.** Let \\(X\\) be a Hausdorff space.\n\n1. \\(X\\) is extremally disconnected if and only if any two disjoint open subsets have disjoint closures. In that case the interior of every closed set is clopen.\n2. An open subspace of an extremally disconnected space is extremally disconnected. A clopen subset of a stonean space is stonean.\n3. In a stonean space the clopen sets form a base of the topology. So every open set is the union of the clopen sets it contains, and these form an increasing net under inclusion.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 380,
        "through_line": 391,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.3",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-05::Theorem 4.3",
      "anchor": "oa-fnd-ao-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 4.3** (Order completeness and stonean spaces). For a compact space \\(\\Omega\\) the following are equivalent.\n\n1. \\(\\Omega\\) is stonean.\n2. Every nonempty subset of \\(C_{\\mathbb R}(\\Omega)\\) that is bounded above has a supremum in \\(C_{\\mathbb R}(\\Omega)\\), that is, a least upper bound for the pointwise order.\n3. Every bounded lsc function on \\(\\Omega\\) agrees with a continuous function outside a meager set.\n\nWhen they hold, the following is true as well. For every bounded lsc function \\(g\\), the continuous function in (3) is unique, it is the upper regularization \\(g^*\\), and \\(g^*\\ge g\\). For every nonempty set \\(S\\subseteq C_{\\mathbb R}(\\Omega)\\) that is bounded above, the supremum of \\(S\\) in \\(C_{\\mathbb R}(\\Omega)\\) is \\(g^*\\), where \\(g=\\sup_{s\\in S}s\\) is the pointwise supremum; so it agrees with the pointwise supremum outside a meager set.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 392,
        "through_line": 416,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.4",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-05::Theorem 4.4",
      "anchor": "oa-fnd-ao-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 4.4** (Extending continuous functions). Let \\(\\Omega\\) be a stonean space and \\(D\\subseteq\\Omega\\) a subset that is dense or open. Every bounded continuous function on \\(D\\) extends to a continuous function on \\(\\Omega\\). If \\(D\\) is dense, the extension is unique, and \\(x\\mapsto x|_D\\) is an isometric \\(*\\)-isomorphism of \\(C(\\Omega)\\) onto \\(C_b(D)\\). So \\(\\Omega\\) is the Stone–Čech compactification of every dense subset.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 417,
        "through_line": 436,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.5",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-05::Example 4.5",
      "anchor": "oa-fnd-ao-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 4.5.** (a) For every set \\(I\\) the Stone–Čech compactification \\(\\beta I\\) of the discrete space \\(I\\) is stonean. Indeed \\(C(\\beta I)\\cong\\ell^\\infty(I)\\), and every nonempty set of real functions on \\(I\\) that is bounded above has a pointwise supremum, which is its least upper bound in \\(\\ell^\\infty(I)\\); Theorem 4.3 applies.\n\n(b) \\([0,1]\\) is not stonean: the closure of \\([0,\\frac12)\\) is not open. In fact no infinite compact metrizable space is stonean (Exercise 12.1), and so the Cantor set, although it has a base of clopen sets, is not stonean.\n\n(c) If \\(\\Omega\\) is stonean and \\(\\omega\\) is not an isolated point, then \\(\\Omega\\setminus\\{\\omega\\}\\) is open and dense, and by Theorem 4.4 it has \\(\\Omega\\) as its Stone–Čech compactification.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 437,
        "through_line": 442,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.1",
      "kind": "definition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Definition 5.1",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Definition 5.1.** Let \\(\\Omega\\) be stonean. A Radon measure \\(\\mu\\) on \\(\\Omega\\) is *normal* if \\(\\mu(f)=\\sup_i\\mu(f_i)\\) for every increasing net \\((f_i)\\) in \\(C_{\\mathbb R}(\\Omega)\\) with \\(\\sup_i\\|f_i\\|<\\infty\\), where \\(f\\) is its least upper bound in \\(C_{\\mathbb R}(\\Omega)\\). A complex Radon measure is called *normal* when it is a finite sum of complex multiples of positive normal measures.\n\nThe least upper bound \\(f\\) exists by Theorem 4.3. Normality asks that the integral see \\(f\\), and not only the pointwise supremum, which may be smaller on a meager set.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 445,
        "through_line": 448,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.2",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Theorem 5.2",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 5.2** (Normal measures kill rare sets). Let \\(\\Omega\\) be stonean. For a Radon measure \\(\\mu\\) on \\(\\Omega\\) the following are equivalent.\n\n1. \\(\\mu\\) is normal.\n2. Every rare set is \\(\\mu\\)-null.\n3. Every meager set is \\(\\mu\\)-null.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 449,
        "through_line": 460,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.3",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Lemma 5.3",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 5.3** (Open kernels of measurable sets). Let \\(\\Omega\\) be stonean, \\(\\mu\\) a normal measure on \\(\\Omega\\), and \\(E\\) a \\(\\mu\\)-measurable set. There is an open set \\(G\\subseteq E\\) with \\(\\mu(E\\setminus G)=0\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 461,
        "through_line": 464,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.4",
      "kind": "corollary",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Corollary 5.4",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Corollary 5.4.** Let \\(\\Omega\\) be stonean and \\(\\mu\\) a normal measure on \\(\\Omega\\).\n\n1. The closure of a \\(\\mu\\)-null set is \\(\\mu\\)-null.\n2. If \\(E\\) is \\(\\mu\\)-measurable, the sets \\(E\\), \\(\\overline E\\), \\(E^\\circ\\), \\((\\overline E)^\\circ\\) and \\(\\overline{E^\\circ}\\) differ from each other by null sets. The last two are clopen. So every measurable set agrees up to a null set with a clopen set.\n3. The support of \\(\\mu\\) is clopen.\n4. Let \\(S\\) be the support of \\(\\mu\\). A \\(\\mu\\)-measurable set \\(E\\) is null if and only if \\(E\\cap S\\) is rare. In particular, two normal measures with the same support have the same null sets, and if \\(S=\\Omega\\), the measurable null sets are exactly the measurable rare sets.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 465,
        "through_line": 479,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.5",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Theorem 5.5",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 5.5** (Measurable functions are almost continuous). Let \\(\\Omega\\) be stonean, \\(\\mu\\) a normal measure on \\(\\Omega\\), and \\(f\\) a bounded real \\(\\mu\\)-measurable function. Then \\(f=f_*=f^*\\) almost everywhere, the functions \\((f_*)^*\\) and \\((f^*)_*\\) are continuous, and both agree with \\(f\\) almost everywhere. Consequently, every element of \\(L^\\infty(\\Omega,\\mu)\\) contains a continuous function. Let \\(S\\) be the support of \\(\\mu\\). The map that sends \\(x\\in C(S)\\) to the class of its extension by \\(0\\) is an isometric \\(*\\)-isomorphism\n\\[\nC(S)\\longrightarrow L^\\infty(\\Omega,\\mu).\n\\tag{5.1}\n\\]\nIf \\(S=\\Omega\\), the map \\(x\\mapsto[x]\\) from \\(C(\\Omega)\\) onto \\(L^\\infty(\\Omega,\\mu)\\) is an isometric \\(*\\)-isomorphism.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 480,
        "through_line": 499,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.6",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Theorem 5.6",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 5.6** (Normal and singular parts). Let \\(\\Omega\\) be stonean. Every Radon measure \\(\\mu\\) on \\(\\Omega\\) can be written in exactly one way as \\(\\mu=\\mu_n+\\mu_s\\), where \\(\\mu_n\\) is a normal measure and \\(\\mu_s\\) is a Radon measure concentrated on a meager set, that is, \\(\\mu_s(\\Omega\\setminus M)=0\\) for some meager Borel set \\(M\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 500,
        "through_line": 511,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.7",
      "kind": "corollary",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Corollary 5.7",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Corollary 5.7.** If a stonean space carries no normal measure other than \\(0\\), every Radon measure on it is concentrated on a meager set.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 512,
        "through_line": 515,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.8",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-07::Example 5.8",
      "anchor": "oa-fnd-ao-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 5.8** (The space \\(\\beta I\\)). Let \\(I\\) be a set and \\(\\Omega=\\beta I\\), which is stonean by Example 4.5. The points of \\(I\\) are isolated in \\(\\beta I\\). Indeed, the indicator \\(1_{\\{i\\}}\\in\\ell^\\infty(I)=C(\\beta I)\\) is the indicator of a clopen set, and a character \\(\\omega\\) of \\(\\ell^\\infty(I)\\) with \\(\\omega(1_{\\{i\\}})=1\\) satisfies \\(\\omega(f)=\\omega(f1_{\\{i\\}})=f(i)\\) for all \\(f\\), so this clopen set is \\(\\{i\\}\\). So a set that contains a point of \\(I\\) is not rare. A set \\(R\\subseteq\\beta I\\setminus I\\) is rare: its closure lies in the closed set \\(\\beta I\\setminus I\\), which has empty interior because \\(I\\) is dense. So the rare sets are exactly the subsets of \\(\\beta I\\setminus I\\), and by Theorem 5.2 the normal measures are the Radon measures \\(\\mu\\) with \\(\\mu(\\beta I\\setminus I)=0\\), that is, \\(\\mu=\\sum_{i\\in I}c_i\\delta_i\\) with \\(c_i\\ge0\\) and \\(\\sum_ic_i<\\infty\\). The decomposition of Theorem 5.6 is \\(\\mu_n=\\mu(\\,\\cdot\\cap I)\\) and \\(\\mu_s=\\mu(\\,\\cdot\\cap(\\beta I\\setminus I))\\). For instance, a point mass at a point of \\(\\beta I\\setminus I\\) is purely singular.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 516,
        "through_line": 517,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.1",
      "kind": "definition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-09::Definition 6.1",
      "anchor": "oa-fnd-ao-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Definition 6.1.** Let \\(\\Omega\\) be stonean. A family \\(\\mathfrak F\\) of normal measures is *sufficient* if for every nonzero \\(x\\in C(\\Omega)\\) with \\(x\\ge0\\) some \\(\\mu\\in\\mathfrak F\\) has \\(\\mu(x)>0\\). The space \\(\\Omega\\) is *hyperstonean* if the family of all normal measures on \\(\\Omega\\) is sufficient.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 520,
        "through_line": 521,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.2",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-09::Lemma 6.2",
      "anchor": "oa-fnd-ao-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 6.2.** A family \\(\\mathfrak F\\) of normal measures is sufficient exactly when the supports of its members have a dense union.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 522,
        "through_line": 525,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 6.3",
      "kind": "proposition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-09::Proposition 6.3",
      "anchor": "oa-fnd-ao-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Proposition 6.3** (Rare sets in a hyperstonean space). Let \\(\\Omega\\) be hyperstonean and \\(\\mathfrak F\\) a sufficient family of normal measures.\n\n1. A set \\(A\\subseteq\\Omega\\) is rare exactly when it is \\(\\mu\\)-null for all \\(\\mu\\in\\mathfrak F\\). In particular every meager set is rare.\n2. If \\(E\\) is \\(\\mu\\)-measurable for every \\(\\mu\\in\\mathfrak F\\), then \\((\\overline E)^\\circ=\\overline{E^\\circ}\\), and the sets \\(E\\), \\(\\overline E\\), \\(E^\\circ\\) and \\((\\overline E)^\\circ\\) differ from each other by rare sets.\n3. If \\(f\\) is bounded, real and \\(\\mu\\)-measurable for every \\(\\mu\\in\\mathfrak F\\), then \\((f_*)^*=(f^*)_*\\). This function is continuous, and it agrees with \\(f\\), \\(f_*\\) and \\(f^*\\) outside a rare set.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 526,
        "through_line": 537,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.1",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-12::Theorem 7.1",
      "anchor": "oa-fnd-ao-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 7.1** (Abelian von Neumann algebras and hyperstonean spaces). Let \\(A\\) be an abelian C\\*-algebra and \\(\\Omega\\) its spectrum. The following are equivalent.\n\n1. \\(\\Omega\\) is hyperstonean.\n2. \\(A\\) is \\(*\\)-isomorphic to some von Neumann algebra: it has a faithful representation \\(\\pi\\) such that \\(\\pi(A)\\) is a von Neumann algebra.\n3. \\(A\\cong L^\\infty(\\Gamma,\\mu)\\) for a locally compact space \\(\\Gamma\\) and a positive Radon measure \\(\\mu\\) on \\(\\Gamma\\).\n4. \\(A\\cong L^\\infty(\\Gamma,\\mu)\\), where \\(\\Gamma\\) is the union of pairwise disjoint compact open sets \\(\\Gamma_i\\), \\(i\\in I\\), each of them hyperstonean, and \\(\\mu\\) is a Radon measure on \\(\\Gamma\\) whose restriction \\(\\mu_i\\) to each \\(\\Gamma_i\\) is a normal measure with support \\(\\Gamma_i\\).\n\nIn (4) one can take \\(\\Gamma\\) to be an open dense subset of \\(\\Omega\\), and then \\(A\\cong C(\\Omega)\\cong\\prod_iC(\\Gamma_i)\\cong\\prod_iL^\\infty(\\Gamma_i,\\mu_i)\\).\n\n*Reference:* [Dixmier, “Sur certains espaces considérés par M. H. Stone” (1951), §5 Theorem 1 and §6 Theorem 2](https://dmitripavlov.org/scans/dixmier.pdf).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 550,
        "through_line": 580,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_labels": [
            "Theorem 7.1"
          ],
          "source_key": "dixmier-stone-1951@931EBC486E9A20F65E7C1A976F64F4F2D6809D5621298BE27B959A81D65E7A2A",
          "source_locus": "§5 Theorem 1 and §6 Theorem 2, printed 169–174",
          "role": "proof construction comparison",
          "correspondence": "exact at stated abstract/spatial scope",
          "explanation": "Arbitrary Radon model uses local null sets, not a sigma-finite restriction. Full own normal-measure, gluing and operator proof/dependencies remain internal; external preceding lemmas are not claimed freshly audited."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.3",
      "kind": "proposition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-12::Proposition 7.3",
      "anchor": "oa-fnd-ao-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Proposition 7.3** (Isomorphisms preserve monotone limits). Let \\(\\theta:M\\to N\\) be a \\(*\\)-isomorphism between von Neumann algebras on Hilbert spaces \\(H\\) and \\(K\\). Then \\(\\theta\\) maps \\(M_+\\) onto \\(N_+\\), and for every bounded increasing net \\((x_i)\\) in \\(M_h\\) with strong limit \\(x\\), the net \\((\\theta(x_i))\\) converges strongly to \\(\\theta(x)\\). In particular \\(\\omega_\\eta\\circ\\theta\\) preserves the limits of such nets, for every \\(\\eta\\in K\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 581,
        "through_line": 586,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.4",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-12::Theorem 7.4",
      "anchor": "oa-fnd-ao-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 7.4** (Cyclic vectors and spatial isomorphisms).\n\n1. Let \\(M\\subseteq B(H)\\) and \\(N\\subseteq B(K)\\) be abelian von Neumann algebras with cyclic vectors \\(\\xi\\) and \\(\\eta\\). Every \\(*\\)-isomorphism \\(\\theta:M\\to N\\) is spatial: there is a unitary \\(U:H\\to K\\) with \\(\\theta(x)=UxU^*\\) for all \\(x\\in M\\).\n2. An abelian von Neumann algebra has a cyclic vector exactly when it is both maximal abelian and \\(\\sigma\\)-finite.\n3. Every \\(*\\)-isomorphism between maximal abelian von Neumann algebras is spatial.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 587,
        "through_line": 617,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.5",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-12::Example 7.5",
      "anchor": "oa-fnd-ao-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 7.5.** (a) The algebra \\(\\ell^\\infty(I)\\) on \\(\\ell^2(I)\\), for uncountable \\(I\\), is maximal abelian but not \\(\\sigma\\)-finite, and it has no cyclic vector (Example 1.1). (b) The algebra \\(\\mathcal A_2\\) of Example 1.3 is \\(\\sigma\\)-finite but not maximal abelian, and the isomorphism \\(M_f\\mapsto M_f\\oplus M_f\\) from the algebra of Example 1.2 onto it is not spatial. So maximality cannot be dropped in Theorem 7.4(3). (c) On \\(\\mathbb C^4\\) the map \\(\\operatorname{diag}(a,b,b,b)\\mapsto\\operatorname{diag}(a,a,b,b)\\) is a \\(*\\)-isomorphism of abelian von Neumann algebras that is not spatial, since it sends a projection of rank one to a projection of rank two.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 618,
        "through_line": 619,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 6.5",
      "kind": "proposition",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-11::Proposition 6.5",
      "anchor": "oa-fnd-ao-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Proposition 6.5** (Atomic and diffuse parts). Let \\(\\Omega\\) be stonean, let \\(I\\) be the set of its isolated points, and put \\(\\Omega_d=\\overline I\\) and \\(\\Omega_c=\\Omega\\setminus\\Omega_d\\).\n\n1. \\(\\Omega_d\\) and \\(\\Omega_c\\) are clopen. \\(I\\) is an open discrete subset of \\(\\Omega_d\\) that is dense in \\(\\Omega_d\\), and \\(\\Omega_c\\) has no isolated points. The space \\(\\Omega_d\\) is the Stone–Čech compactification of the discrete space \\(I\\), so \\(C(\\Omega_d)\\cong\\ell^\\infty(I)\\).\n2. If \\(\\Omega=A\\cup B\\) is a partition into clopen sets such that some open discrete subset of \\(A\\) is dense in \\(A\\) and \\(B\\) has no isolated points, then \\(A=\\Omega_d\\) and \\(B=\\Omega_c\\).\n3. If \\(\\Omega\\) is hyperstonean, so are \\(\\Omega_d\\) and \\(\\Omega_c\\).\n\nThe clopen hypothesis in (2) is essential, as the complete counterexample in Example 6.6 proves.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 622,
        "through_line": 635,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.6",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-11::Example 6.6",
      "anchor": "oa-fnd-ao-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 6.6** (The parts must be clopen). In \\(\\beta\\mathbb N\\) the isolated points are the points of \\(\\mathbb N\\), so \\(\\Omega_d=\\beta\\mathbb N\\) and \\(\\Omega_c=\\varnothing\\). The partition into \\(A=\\mathbb N\\) and \\(B=\\beta\\mathbb N\\setminus\\mathbb N\\) also has the two properties of Proposition 6.5(2), except that the parts are not clopen. Clearly \\(\\mathbb N\\) is open, discrete and dense in itself. To see that \\(B\\) has no isolated points, let \\(p\\in B\\) and let \\(C\\) be a clopen neighbourhood of \\(p\\) in \\(\\beta\\mathbb N\\). The set \\(C\\cap\\mathbb N\\) is infinite: otherwise \\(C=\\overline{C\\cap\\mathbb N}\\), which holds because \\(\\mathbb N\\) is dense and \\(C\\) is open, would be a finite subset of \\(\\mathbb N\\) and could not contain \\(p\\). Split \\(C\\cap\\mathbb N\\) into two disjoint infinite sets \\(N_1,N_2\\). They are open, so their closures are disjoint (Lemma 4.2(1)) and lie in \\(C\\). The closure of an infinite subset of \\(\\mathbb N\\) is compact and so is not a subset of the discrete set \\(\\mathbb N\\); hence each closure contains a point of \\(B\\). So \\(C\\cap B\\) has at least two points, and \\(p\\) is not isolated in \\(B\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 636,
        "through_line": 637,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-14::Lemma 8.1",
      "anchor": "oa-fnd-ao-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 8.1** (Singly generated algebras of functions). Let \\(\\Gamma\\) be a compact space.\n\n1. \\(C(\\Gamma)\\) is generated as a C\\*-algebra by a single self-adjoint element if and only if \\(\\Gamma\\) is homeomorphic to a compact subset of \\(\\mathbb R\\).\n2. Suppose a sequence \\((E_n)\\) of clopen sets separates the points of \\(\\Gamma\\): for \\(\\gamma\\ne\\gamma'\\) some \\(E_n\\) contains exactly one of them. Then \\(a=\\sum_n3^{-n}(2\\cdot1_{E_n}-1)\\) generates \\(C(\\Gamma)\\) as a C\\*-algebra.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 642,
        "through_line": 658,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-14::Theorem 8.2",
      "anchor": "oa-fnd-ao-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 8.2** (Countable generation). For an abelian von Neumann algebra \\(M\\) the following are equivalent.\n\n1. \\(M\\) is generated by countably many elements.\n2. \\(M\\) is generated by countably many projections.\n3. \\(M\\) is generated by one self-adjoint element.\n\nOn a separable Hilbert space, every abelian von Neumann algebra has these properties.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 659,
        "through_line": 678,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.3",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-14::Lemma 8.3",
      "anchor": "oa-fnd-ao-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 8.3** (Diffuse measures on the unit interval). Let \\(\\nu\\) be a Radon probability measure on \\([0,1]\\) with no atoms, let \\(m\\) be Lebesgue measure, and put \\(F(t)=\\nu([0,t])\\). Then \\(V\\varphi=\\varphi\\circ F\\) defines a unitary \\(V:L^2([0,1],m)\\to L^2([0,1],\\nu)\\), and\n\\[\n\\begin{gathered}\nV\\{M_\\varphi:\\varphi\\in L^\\infty(m)\\}V^*\\\\\n=\\{M_f:f\\in L^\\infty(\\nu)\\},\\\\\nVM_\\varphi V^*\\\\\n=M_{\\varphi\\circ F}.\n\\end{gathered}\n\\tag{8.1}\n\\]\nIn particular \\(L^\\infty([0,1],\\nu)\\) is isomorphic to \\(L^\\infty[0,1]\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 679,
        "through_line": 698,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.4",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-14::Theorem 8.4",
      "anchor": "oa-fnd-ao-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 8.4** (Diffuse countably generated algebras). For an abelian von Neumann algebra \\(M\\) the following are equivalent.\n\n1. \\(M\\) is isomorphic to \\(L^\\infty[0,1]\\), with Lebesgue measure.\n2. \\(M\\ne0\\), and \\(M\\) is countably generated, \\(\\sigma\\)-finite, and without minimal projections.\n\nIn particular, if a nonzero separable Hilbert space carries an abelian von Neumann algebra without minimal projections, that algebra is isomorphic to \\(L^\\infty[0,1]\\). The condition \\(M\\ne0\\) excludes the zero algebra on the zero space, which has the other three properties.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 699,
        "through_line": 718,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.5",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-14::Example 8.5",
      "anchor": "oa-fnd-ao-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 8.5** (Each hypothesis is needed).\n\n(a) *\\(\\sigma\\)-finiteness.* Let \\(M=\\prod_{t\\in[0,1]}L^\\infty[0,1]\\), acting on the Hilbert sum \\(\\bigoplus_{t\\in[0,1]}L^2[0,1]\\) by \\((f_t)_t\\mapsto\\bigoplus_tM_{f_t}\\). It is a von Neumann algebra, because each summand is maximal abelian and the commutant of a direct sum is the direct sum of the commutants ([The double commutant theorem](the-double-commutant-theorem.md), Proposition 5.2). It has no minimal projection: a nonzero projection of \\(M\\) has a coordinate \\(1_E\\) with \\(m(E)>0\\), and \\(E\\) splits into two sets of positive measure, as in the proof of Theorem 8.4. It is not \\(\\sigma\\)-finite: the projections onto the summands are uncountably many. It is countably generated. Let \\(a\\) act as the scalar \\(t\\) on the summand with index \\(t\\), and \\(b\\) as \\(M_\\iota\\) on every summand. An operator commuting with \\(a\\) maps each eigenspace \\(\\ker(a-t)\\), which is the summand with index \\(t\\), into itself, so the projections onto the summands lie in the von Neumann algebra generated by \\(a\\). An operator commuting with \\(a\\) and \\(b\\) is therefore a bounded family of operators that commute with \\(M_\\iota\\), hence of multiplication operators (Theorem 3.1(3)). So the commutant of \\(\\{a,b\\}\\) is \\(M\\), and \\(\\{a,b\\}''=M'=M\\). So \\(M\\) is not isomorphic to \\(L^\\infty[0,1]\\).\n\n(b) *Countable generation.* Let \\(J\\) be uncountable, \\(P\\) the product of the fair coin measures on \\(\\{0,1\\}^J\\), defined on the product \\(\\sigma\\)-algebra, and \\(M=L^\\infty(P)\\) on \\(L^2(P)\\). By Remark 3.2 it is maximal abelian, and \\(1\\) is a cyclic vector because \\(L^\\infty(P)\\) is dense in \\(L^2(P)\\); so it is \\(\\sigma\\)-finite (Theorem 7.4(2)). It has no minimal projection. Every measurable set \\(E\\) depends only on the coordinates in some countable set \\(J_E\\), because the sets with this property form a \\(\\sigma\\)-algebra containing the cylinder sets. For \\(j\\notin J_E\\), independence of the coordinates gives \\(P(E\\cap\\{\\gamma_j=0\\})=\\frac12P(E)\\), which splits \\(E\\). Finally, \\(M\\) is not countably generated. Otherwise \\(M\\) is generated by one self-adjoint \\(a\\) (Theorem 8.2), and \\(L^2(P)=[M1]\\) is the closure of the set of vectors \\(p(a)1\\), with \\(p\\) a polynomial whose coefficients have rational real and imaginary parts; so \\(L^2(P)\\) would be separable. But it is not ([The double commutant theorem](the-double-commutant-theorem.md), Exercise 9.9).\n\n(c) *No minimal projections.* \\(\\ell^\\infty(\\mathbb N)\\) is countably generated and \\(\\sigma\\)-finite, but each point gives a minimal projection.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 719,
        "through_line": 726,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_scopes": [
            {
              "heading": "3. Measures and cyclic representations",
              "line": 172,
              "through_line": 277,
              "anchors": [
                "OA-FND-AO-03",
                "OA-FND-AO-04"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            },
            {
              "heading": "8. Countably generated abelian von Neumann algebras",
              "line": 638,
              "through_line": 726,
              "anchors": [
                "OA-FND-AO-14",
                "OA-FND-AO-15"
              ],
              "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.8; PDF 63–66",
          "role": "proof comparison",
          "correspondence": "Cyclic model; arbitrary decomposition sketched",
          "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.4",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-10::Theorem 6.4",
      "anchor": "oa-fnd-ao-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 6.4** (Decomposition of a stonean space). Let \\(\\Omega\\) be stonean. There is exactly one partition of \\(\\Omega\\) into three clopen sets \\(\\Omega_1\\), \\(\\Omega_2\\), \\(\\Omega_3\\) with the following properties.\n\n1. \\(\\Omega_1\\) is hyperstonean.\n2. \\(\\Omega_2\\) contains a dense meager subset.\n3. Every meager subset of \\(\\Omega_3\\) is rare, and every Radon measure on \\(\\Omega_3\\) is concentrated on a closed rare subset of \\(\\Omega_3\\).\n\nMoreover, \\(\\Omega_1\\) is the closure of the union of all supports of normal measures on \\(\\Omega\\), and \\(\\Omega_2\\cup\\Omega_3\\) carries no normal measure other than \\(0\\). Any of the three parts may be empty; Section 9 gives stonean spaces with \\(\\Omega=\\Omega_2\\) and with \\(\\Omega=\\Omega_3\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 735,
        "through_line": 756,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [
        {
          "unit": "abelian-operator-algebras",
          "local_labels": [
            "Theorem 6.4"
          ],
          "source_key": "dixmier-stone-1951@931EBC486E9A20F65E7C1A976F64F4F2D6809D5621298BE27B959A81D65E7A2A",
          "source_locus": "§4 Proposition 9, printed 165–166",
          "role": "proof comparison",
          "correspondence": "exact after relabelling",
          "explanation": "Dixmier E1 is the dense-meager part and E2 hyperstonean; the lesson uses Ω2 and Ω1 respectively. Three clopen parts, uniqueness and rare-supported measures are retained with full internal proof."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.1",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Lemma 9.1",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 9.1** (The Baire property). Every bounded real Borel function on \\(\\Gamma\\) agrees outside a meager set with a bounded lsc function, and also with a bounded usc function. In particular, for every Borel set \\(E\\) there is an open set \\(U\\) such that the symmetric difference \\(E\\triangle U\\) is meager.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 769,
        "through_line": 780,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.2",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Lemma 9.2",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 9.2** (Clopen subsets of \\(\\Omega_\\Gamma\\)). For a Borel set \\(E\\subseteq\\Gamma\\), the class \\(\\tilde1_E\\) is a projection; let \\(\\widehat E\\subseteq\\Omega_\\Gamma\\) be the clopen set on which its Gelfand transform equals \\(1\\).\n\n1. \\(\\widehat E=\\varnothing\\) if and only if \\(E\\) is meager, and \\(\\widehat E\\subseteq\\widehat F\\) if and only if \\(E\\setminus F\\) is meager. Moreover \\(\\widehat{E\\cap F}=\\widehat E\\cap\\widehat F\\) and \\(\\widehat{\\Gamma\\setminus E}=\\Omega_\\Gamma\\setminus\\widehat E\\).\n2. Every clopen subset of \\(\\Omega_\\Gamma\\) is \\(\\widehat U\\) for some open set \\(U\\subseteq\\Gamma\\).\n3. If \\(\\Gamma\\) is a Baire space, then \\(\\widehat U\\ne\\varnothing\\) for every nonempty open set \\(U\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 781,
        "through_line": 788,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.3",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Theorem 9.3",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 9.3** (Every stonean space is a space of this kind).\n\n1. For every topological space \\(\\Gamma\\), the space \\(\\Omega_\\Gamma\\) is stonean. More precisely, let \\((\\tilde f_i)\\) be an increasing net in \\(\\mathcal D(\\Gamma)_h\\) that is bounded in norm, and let the \\(f_i\\) be lsc representatives with a common bound. Then the supremum of \\((\\tilde f_i)\\) in \\(\\mathcal D(\\Gamma)_h\\) is the class of the pointwise supremum \\(\\sup_if_i\\).\n2. If \\(\\Omega\\) is stonean, then \\(x\\mapsto\\tilde x\\) is a \\(*\\)-isomorphism of \\(C(\\Omega)\\) onto \\(\\mathcal D(\\Omega)\\). So every stonean space \\(\\Omega\\) is homeomorphic to \\(\\Omega_\\Omega\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 789,
        "through_line": 803,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.5",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Theorem 9.5",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 9.5** (A stonean space with a dense meager subset). Let \\(\\Gamma\\) be a regular Baire space that contains a dense meager subset; here *regular* means that a point and a closed set not containing it have disjoint open neighbourhoods. Then \\(\\Omega_\\Gamma\\) contains a dense meager subset. So \\(\\Omega_\\Gamma\\) is its own second part in Theorem 6.4, and it carries no normal measure other than \\(0\\). If moreover \\(\\Gamma\\) is Hausdorff, has no isolated points and has a countable family of nonempty open sets such that every nonempty open set contains one of them, then \\(\\Omega_\\Gamma\\) has no isolated points and has a countable dense subset. All of this applies to \\(\\Gamma=[0,1]\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 804,
        "through_line": 819,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.6",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Theorem 9.6",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 9.6** (Stonean spaces on which every measure lives on a rare set). Let \\(\\Gamma\\) be a topological space with a base \\(\\mathfrak B\\) of nonempty open sets such that\n\n\\((\\ast)\\) for every decreasing sequence \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\), the intersection \\(\\bigcap_nB_n\\) contains a member of \\(\\mathfrak B\\).\n\n1. If \\(G_1,G_2,\\dots\\) are dense open subsets of \\(\\Gamma\\), the interior of \\(\\bigcap_nG_n\\) is dense. So \\(\\Gamma\\) is a Baire space, and every meager subset of \\(\\Gamma\\) is rare.\n2. The sets \\(\\widehat B\\), \\(B\\in\\mathfrak B\\), are nonempty clopen sets, and every nonempty open subset of \\(\\Omega_\\Gamma\\) contains one of them. Every nonempty clopen subset of \\(\\widehat B\\) contains \\(\\widehat C\\) for some \\(C\\in\\mathfrak B\\) with \\(C\\subseteq B\\). For every decreasing sequence \\(B_1\\supseteq B_2\\supseteq\\cdots\\) in \\(\\mathfrak B\\) there is \\(C\\in\\mathfrak B\\) with \\(\\widehat C\\subseteq\\bigcap_n\\widehat{B_n}\\).\n3. Every meager subset of \\(\\Omega_\\Gamma\\) is rare.\n4. If \\(\\Gamma\\) is Hausdorff and has no isolated points, every nonempty clopen subset of \\(\\Omega_\\Gamma\\) is uncountable, and every Radon measure on \\(\\Omega_\\Gamma\\) has a rare support.\n\nIf \\(\\Gamma\\) is a nonempty Hausdorff space without isolated points, \\(\\Omega_\\Gamma\\) is a nonempty stonean space that is its own third part in Theorem 6.4.\n\nExample 9.8 proves that for the space of Example 9.7 no base of \\(\\Omega_\\Gamma\\) satisfies \\((\\ast)\\), so (2) is stated for sequences that decrease in \\(\\Gamma\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 820,
        "through_line": 844,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.7",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Example 9.7",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 9.7** (A space satisfying \\((\\ast)\\)). Let \\(I\\) be an uncountable set and \\(\\Gamma=\\{0,1\\}^I\\). For a countable set \\(J\\subseteq I\\) and \\(\\alpha\\in\\{0,1\\}^J\\) put \\(U(J,\\alpha)=\\{\\gamma\\in\\Gamma:\\gamma_j=\\alpha_j\\text{ for all }j\\in J\\}\\). The intersection of two such sets is empty or again of this form, so they form a base \\(\\mathfrak B\\) of a topology. Two different points differ at some coordinate \\(i\\), and the sets \\(U(\\{i\\},\\cdot)\\) through them are disjoint; so \\(\\Gamma\\) is Hausdorff. Each \\(U(J,\\alpha)\\) is also closed, since its complement is the union of the sets \\(U(\\{j\\},1-\\alpha_j)\\), \\(j\\in J\\). So \\(\\Gamma\\) has a base of clopen sets, and it is completely regular, because the indicators of these sets are continuous. No point is isolated, because a basic set fixes only countably many of the uncountably many coordinates. If \\(U(J_1,\\alpha_1)\\supseteq U(J_2,\\alpha_2)\\supseteq\\cdots\\), then \\(J_1\\subseteq J_2\\subseteq\\cdots\\) and each \\(\\alpha_{n+1}\\) extends \\(\\alpha_n\\), so the intersection is \\(U(\\bigcup_nJ_n,\\bigcup_n\\alpha_n)\\), which lies in \\(\\mathfrak B\\). So Theorem 9.6 applies: \\(\\Omega_\\Gamma\\) is a stonean space in which every meager set is rare and every Radon measure has a rare support.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 845,
        "through_line": 846,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.8",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-16::Example 9.8",
      "anchor": "oa-fnd-ao-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 9.8** (No base of \\(\\Omega_\\Gamma\\) satisfies \\((\\ast)\\)). Keep \\(\\Gamma\\) of Example 9.7, and choose distinct indices \\(j_1,j_2,\\dots\\) in \\(I\\). Let \\(V_k\\) be the set of \\(\\gamma\\) with \\(\\gamma_{j_1}=\\dots=\\gamma_{j_{k-1}}=0\\) and \\(\\gamma_{j_k}=1\\); these are disjoint basic sets. Put \\(R_n=\\bigcup_{k\\ge n}V_k\\). It is the intersection of \\(U(\\{j_1,\\dots,j_{n-1}\\},0)\\) with the complement of the basic set \\(Z=U(\\{j_k:k\\ge1\\},0)\\), so it is clopen. The sets \\(R_n\\) decrease, they are nonempty, and \\(\\bigcap_nR_n=\\varnothing\\), because a point of \\(R_n\\) lies in exactly one \\(V_k\\), and then \\(k\\ge n\\).\n\nIn \\(\\Omega_\\Gamma\\) the clopen sets \\(\\widehat{R_n}\\) decrease and are nonempty, by Lemma 9.2(3), since \\(\\Gamma\\) is a Baire space by Theorem 9.6(1). By compactness they have a common point \\(\\omega\\). Their intersection has empty interior. Otherwise it contains a nonempty clopen set \\(\\widehat E\\), with \\(E\\) not meager, and \\(E\\setminus R_n\\) is meager for every \\(n\\); then \\(E\\subseteq\\bigcup_n(E\\setminus R_n)\\cup\\bigcap_nR_n\\) is meager. Now let \\(\\mathfrak G\\) be any base of \\(\\Omega_\\Gamma\\). Choose \\(W_1\\in\\mathfrak G\\) with \\(\\omega\\in W_1\\subseteq\\widehat{R_1}\\), and inductively \\(W_{n+1}\\in\\mathfrak G\\) with \\(\\omega\\in W_{n+1}\\subseteq W_n\\cap\\widehat{R_{n+1}}\\). Then \\((W_n)\\) decreases in \\(\\mathfrak G\\), and \\(\\bigcap_nW_n\\subseteq\\bigcap_n\\widehat{R_n}\\) has empty interior, so it contains no nonempty member of \\(\\mathfrak G\\). This is why Theorem 9.6(2) speaks of the sets \\(\\widehat B\\), which form a base for the nonempty open sets in the weaker sense that every nonempty open set contains one of them, and of sequences that decrease in \\(\\Gamma\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 847,
        "through_line": 850,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.1",
      "kind": "lemma",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-20::Lemma 11.1",
      "anchor": "oa-fnd-ao-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Lemma 11.1** (Extending into \\(C_{\\mathbb R}(\\Omega)\\)). Let \\(\\Omega\\) be stonean, \\(V\\) a real vector space, \\(W\\subseteq V\\) a subspace, and \\(p:V\\to C_{\\mathbb R}(\\Omega)\\) sublinear, that is, \\(p(v+w)\\le p(v)+p(w)\\) and \\(p(tv)=tp(v)\\) for \\(t\\ge0\\). If \\(T_0:W\\to C_{\\mathbb R}(\\Omega)\\) is linear and \\(T_0\\le p\\) on \\(W\\), then \\(T_0\\) has a linear extension \\(T:V\\to C_{\\mathbb R}(\\Omega)\\) with \\(T\\le p\\) on \\(V\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 855,
        "through_line": 869,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.2",
      "kind": "theorem",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-20::Theorem 11.2",
      "anchor": "oa-fnd-ao-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Theorem 11.2** (A positive projection onto \\(C(\\Omega)\\), and injectivity). Let \\(\\Omega\\) be stonean. For a bounded real function \\(f\\) on \\(\\Omega\\) let \\(m(f)\\) be the supremum in \\(C_{\\mathbb R}(\\Omega)\\) of \\(\\{g\\in C_{\\mathbb R}(\\Omega):g\\le f\\}\\), and \\(M(f)\\) the infimum there of \\(\\{g\\in C_{\\mathbb R}(\\Omega):g\\ge f\\}\\); both exist by Theorem 4.3, and \\(M(f)=-m(-f)\\).\n\n1. \\(m(f)\\le M(f)\\). The map \\(m\\) is superadditive and \\(M\\) is subadditive; both are positively homogeneous and equal to the identity on \\(C_{\\mathbb R}(\\Omega)\\). Moreover \\(m(f)=(f_*)^*\\), which agrees with \\(f_*\\) outside a meager set.\n2. There is a linear map \\(\\varepsilon:\\ell^\\infty(\\Omega)\\to C(\\Omega)\\) such that \\(\\varepsilon(g)=g\\) for \\(g\\in C(\\Omega)\\), \\(m(f)\\le\\varepsilon(f)\\le M(f)\\) for real \\(f\\), \\(\\varepsilon(f)\\ge0\\) for \\(f\\ge0\\), \\(\\|\\varepsilon\\|\\leq1\\), with equality when \\(\\Omega\\ne\\varnothing\\), and \\(\\varepsilon(gf)=g\\,\\varepsilon(f)\\) for \\(g\\in C(\\Omega)\\) and \\(f\\in\\ell^\\infty(\\Omega)\\).\n3. \\(C(\\Omega)\\) is injective: for every complex Banach space \\(E\\), every subspace \\(F\\subseteq E\\) and every bounded linear \\(T:F\\to C(\\Omega)\\) there is a linear \\(\\tilde T:E\\to C(\\Omega)\\) extending \\(T\\) with \\(\\|\\tilde T\\|=\\|T\\|\\).\n\nThe real and complex cases are both proved below. No converse is used in the subsequent arguments.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 870,
        "through_line": 887,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.3",
      "kind": "example",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-20::Example 11.3",
      "anchor": "oa-fnd-ao-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Example 11.3** (The map \\(m\\) is not additive). Let \\(\\Omega=\\Omega_{[0,1]}\\), the stonean space of Theorem 9.5. It has a countable dense subset \\(D\\) and no isolated points. Each point of \\(D\\) is a rare set, so \\(D\\) is meager and \\(\\Omega\\setminus D\\) is dense (Lemma 2.1(2)). A continuous \\(g\\le1_D\\) satisfies \\(g\\le0\\) on the dense set \\(\\Omega\\setminus D\\), hence everywhere; so \\(m(1_D)=0\\). In the same way \\(m(1_{\\Omega\\setminus D})=0\\), because \\(D\\) is dense. But \\(m(1_D+1_{\\Omega\\setminus D})=m(1)=1\\). So \\(m\\) is not additive, and the linear projection \\(\\varepsilon\\) of Theorem 11.2 cannot be taken to be \\(m\\). For this \\(\\varepsilon\\), \\(\\varepsilon(1_D)+\\varepsilon(1_{\\Omega\\setminus D})=1\\), so at least one of the two values differs from the value of \\(m\\).\n\nExample 11.3 proves that \\(m\\) need not be linear; the linear projection \\(\\varepsilon\\) therefore comes from Lemma 11.1.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 888,
        "through_line": 891,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.1",
      "kind": "exercise",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-21::Exercise 12.1",
      "anchor": "oa-fnd-ao-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Exercise 12.1** (medium; No convergent sequences). Let \\(X\\) be an extremally disconnected Hausdorff space.\n\n(a) Show that every convergent sequence in \\(X\\) is eventually constant.\n\n(b) Deduce that an infinite stonean space is not metrizable, and that an abelian von Neumann algebra of infinite dimension is never norm separable.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 894,
        "through_line": 903,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.2",
      "kind": "exercise",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-21::Exercise 12.2",
      "anchor": "oa-fnd-ao-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Exercise 12.2** (easy; States of \\(\\ell^\\infty\\) that vanish on \\(c_0\\)). Let \\(\\varphi\\) be a state of \\(\\ell^\\infty(\\mathbb N)\\) with \\(\\varphi(x)=0\\) for all \\(x\\in c_0\\), and let \\(\\mu\\) be the Radon measure on \\(\\beta\\mathbb N\\) with \\(\\varphi(x)=\\int\\hat x\\,d\\mu\\).\n\n(a) Show that \\(\\mu(\\mathbb N)=0\\). Conclude that \\(\\mu\\) is purely singular in the sense of Theorem 5.6, and that \\(\\varphi\\) is not of the form \\(x\\mapsto\\sum_nc_nx_n\\) with \\(c\\in\\ell^1\\).\n\n(b) Show that such states exist.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 904,
        "through_line": 913,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.3",
      "kind": "exercise",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-21::Exercise 12.3",
      "anchor": "oa-fnd-ao-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Exercise 12.3** (medium; Minimal projections and isolated points). Let \\(M\\) be a commutative von Neumann algebra, with spectrum \\(\\Omega\\). For a projection \\(e\\in M\\) let \\(C_e\\) be the clopen set on which its transform is \\(1\\).\n\n(a) Show that \\(e\\ne0\\) is a minimal projection if and only if \\(C_e\\) is a single point, which is then isolated.\n\n(b) Show that \\(M\\) is *atomic*, that is, every nonzero projection majorizes a minimal projection, if and only if \\(\\Omega=\\Omega_d\\) in Proposition 6.5, and that then \\(M\\cong\\ell^\\infty(I)\\), where \\(I\\) is the set of minimal projections.\n\n(c) Show that \\(M\\cong\\ell^\\infty(I)\\oplus M_c\\), where the abelian von Neumann algebra \\(M_c\\) has no minimal projections; \\(I\\) may be empty and \\(M_c\\) may be \\(0\\).\n\n(d) Show that the spectrum of \\(L^\\infty[0,1]\\) has no isolated points.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 914,
        "through_line": 931,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.4",
      "kind": "exercise",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-21::Exercise 12.4",
      "anchor": "oa-fnd-ao-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Exercise 12.4** (easy; Multiplicity). Let \\(M_1=\\{M_f:f\\in L^\\infty[0,1]\\}\\) on \\(L^2[0,1]\\), and let \\(M_2=\\mathcal A_2\\) be the algebra of Example 1.3 on \\(L^2[0,1]\\oplus L^2[0,1]\\).\n\n(a) Show that \\(M_2'\\) consists of the operator matrices \\(\\begin{pmatrix}M_{f_{11}}&M_{f_{12}}\\\\M_{f_{21}}&M_{f_{22}}\\end{pmatrix}\\) with \\(f_{ij}\\in L^\\infty[0,1]\\).\n\n(b) Show that \\(M_1\\) and \\(M_2\\) are isomorphic but not spatially isomorphic, and name the hypothesis of Theorem 7.4(3) that fails.",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 932,
        "through_line": 939,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 12.5",
      "kind": "exercise",
      "unit": "abelian-operator-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/abelian-operator-algebras#oa-fnd-ao-21::Exercise 12.5",
      "anchor": "oa-fnd-ao-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/abelian-operator-algebras@46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
      "statement_and_full_conditions": "**Exercise 12.5** (easy; A sliding hump without Phillips' lemma). Let \\((g_n)\\) be a sequence in \\(\\ell^1(\\mathbb N)\\) that converges weakly to \\(0\\), and suppose the \\(g_n\\) have pairwise disjoint finite supports \\(F_n\\). Show directly that \\(\\|g_n\\|_1\\to0\\).",
      "proof_locus": {
        "source": "src/abelian-operator-algebras.md",
        "line": 940,
        "through_line": 943,
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.1",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-01::Proposition 3.1",
      "anchor": "oa-fnd-ty-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 3.1** (The projection lattice). Let \\(\\{e_i\\}_{i\\in I}\\) be any family in \\(\\mathcal P(M)\\). The projection \\(\\bigwedge_ie_i\\) onto \\(\\bigcap_ie_iH\\) and the projection \\(\\bigvee_ie_i\\) onto \\([\\bigcup_ie_iH]\\) lie in \\(M\\). They are the greatest lower bound and the least upper bound of the family, both in \\(\\mathcal P(M)\\) and in \\(\\mathcal P(B(H))\\). So \\(\\mathcal P(M)\\) is a complete lattice whose meets and joins are those of \\(B(H)\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 51,
        "through_line": 56,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.2",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-01::Definition 3.2",
      "anchor": "oa-fnd-ty-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 3.2** (Equivalence of projections). Let \\(e,f\\in\\mathcal P(M)\\).\n\n1. \\(e\\) and \\(f\\) are *equivalent*, \\(e\\sim f\\), if some \\(u\\in M\\) satisfies \\(u^*u=e\\) and \\(uu^*=f\\). Then \\(u\\) is a partial isometry with *initial projection* \\(e\\) and *final projection* \\(f\\), and we say that \\(u\\) implements \\(e\\sim f\\).\n2. \\(e\\precsim f\\), also written \\(f\\succsim e\\), if \\(e\\sim f_1\\) for some \\(f_1\\in\\mathcal P(M)\\) with \\(f_1\\le f\\).\n3. \\(e\\prec f\\) if \\(e\\precsim f\\) and \\(e\\) is not equivalent to \\(f\\).\n\nThese relations are also applied to closed subspaces whose projections lie in \\(M\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 57,
        "through_line": 64,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.3",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-01::Lemma 3.3",
      "anchor": "oa-fnd-ty-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 3.3.**\n\n1. If \\(u\\in M\\) and \\(u^*u=e\\) is a projection, then \\(u=ue\\), \\(uu^*\\) is a projection, and \\(u\\) maps \\(eH\\) isometrically onto \\(uu^*H\\).\n2. \\(\\sim\\) is an equivalence relation, and \\(\\precsim\\) is reflexive and transitive.\n3. (*Additivity*) Let \\(\\{e_i\\}_{i\\in I}\\) and \\(\\{f_i\\}_{i\\in I}\\) be families of mutually orthogonal projections. If \\(e_i\\sim f_i\\) for every \\(i\\), then \\(\\sum_ie_i\\sim\\sum_if_i\\). The same holds with \\(\\precsim\\) in place of \\(\\sim\\).\n4. (*Central cuts*) If \\(e\\sim f\\) (or \\(e\\precsim f\\)) and \\(z\\) is a central projection, then \\(ze\\sim zf\\) (or \\(ze\\precsim zf\\)).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 65,
        "through_line": 79,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
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          "unit": "projections-and-types-of-von-neumann-algebras",
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              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.4",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-01::Definition 3.4",
      "anchor": "oa-fnd-ty-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 3.4** (Central support). For \\(e\\in\\mathcal P(M)\\), the *central support* \\(c(e)\\) is the least central projection that majorizes \\(e\\). It exists: \\(Z=(M\\cup M')'\\) is a von Neumann algebra whose projections are the central projections, so by [Proposition 3.1](#oa-fnd-ty-01), applied to \\(Z\\), the meet of all central projections that majorize \\(e\\) is again central. It majorizes \\(e\\), and it is the least such projection.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 80,
        "through_line": 81,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.5",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-01::Proposition 3.5",
      "anchor": "oa-fnd-ty-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 3.5** (Central supports).\n\n1. \\(c(e)\\) is the projection onto \\([MeH]\\). For a central projection \\(z\\), \\(ze=0\\) if and only if \\(zc(e)=0\\). Moreover \\(c(ze)=zc(e)\\).\n2. If \\(e\\sim f\\) then \\(c(e)=c(f)\\). If \\(e\\precsim f\\) then \\(c(e)\\le c(f)\\).\n3. (*Induction*) The map \\(x'\\mapsto x'e|_{eH}\\) is a \\(*\\)-homomorphism of \\(M'\\) onto the induced algebra \\(M'e\\), with kernel \\(M'(1-c(e))\\). Likewise, for \\(e'\\in\\mathcal P(M')\\), the map \\(x\\mapsto xe'|_{e'H}\\) is a \\(*\\)-homomorphism of \\(M\\) onto \\(Me'\\) with kernel \\(M(1-c(e'))\\); here \\(c(e')\\) is the central support of \\(e'\\) in \\(M'\\), the projection onto \\([M'e'H]\\).\n4. (*Centre of a reduced algebra*) The centre of \\(eMe\\) (on \\(eH\\)) is \\(Ze=\\{ae:a\\in Z\\}\\), and \\(a\\mapsto ae|_{eH}\\) is an isomorphism of \\(Zc(e)\\) onto it. In particular every projection in the centre of \\(eMe\\) is \\(ze\\) for a central projection \\(z\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 82,
        "through_line": 96,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
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              "heading": "3. The projection lattice and equivalence of projections",
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              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Definition 4.1",
      "anchor": "oa-fnd-ty-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 4.1** (Left and right supports). For \\(x\\in M\\), the *left support* \\(\\mathrm l(x)\\) is the least projection \\(p\\in M\\) with \\(px=x\\), and the *right support* \\(\\mathrm r(x)\\) is the least projection \\(q\\in M\\) with \\(xq=x\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 101,
        "through_line": 102,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
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              "heading": "3. The projection lattice and equivalence of projections",
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.2",
      "kind": "lemma",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Lemma 4.2",
      "anchor": "oa-fnd-ty-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 4.2.** \\(\\mathrm l(x)\\) is the projection onto \\([xH]\\), and \\(\\mathrm r(x)=\\mathrm l(x^*)\\) is the projection onto \\([x^*H]=(\\ker x)^\\perp\\). Both lie in \\(M\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 103,
        "through_line": 106,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
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              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
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                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.3",
      "kind": "proposition",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Proposition 4.3",
      "anchor": "oa-fnd-ty-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 4.3.** \\(\\mathrm l(x)\\sim\\mathrm r(x)\\) for every \\(x\\in M\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 107,
        "through_line": 110,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.4",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Proposition 4.4",
      "anchor": "oa-fnd-ty-04",
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      "statement_and_full_conditions": "**Proposition 4.4** (The parallelogram law). For \\(e,f\\in\\mathcal P(M)\\), \\((e\\vee f)-e\\sim f-(e\\wedge f)\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 111,
        "through_line": 116,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
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        {
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              "heading": "3. The projection lattice and equivalence of projections",
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              "anchors": [
                "OA-FND-TY-01",
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              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
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              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
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                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.5",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Definition 4.5",
      "anchor": "oa-fnd-ty-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 4.5** (Cyclic projections). For \\(\\xi\\in H\\), \\(p_\\xi\\) is the projection onto \\([M'\\xi]\\) and \\(p'_\\xi\\) the projection onto \\([M\\xi]\\). These are the *cyclic projections* of \\(M\\) and of \\(M'\\) defined by \\(\\xi\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 117,
        "through_line": 118,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.6",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Lemma 4.6",
      "anchor": "oa-fnd-ty-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 4.6.** \\(p_\\xi\\in M\\), and \\(p_\\xi\\) is the least projection \\(p\\in M\\) with \\(p\\xi=\\xi\\). Equivalently, \\(p_\\xi\\) is the support of the restriction of \\(\\omega_\\xi\\) to \\(M\\). The same holds for \\(p'_\\xi\\) with \\(M'\\) in place of \\(M\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 119,
        "through_line": 122,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
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              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4.7",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-04::Exercise 4.7",
      "anchor": "oa-fnd-ty-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 4.7.** (medium) Let \\(\\mathfrak m\\) be a left ideal of \\(M\\), not necessarily closed, and \\(\\mathcal P=\\mathcal P(M)\\).\n\n- (a) \\(\\mathfrak m=\\{xh:\\ x\\in M,\\ h\\in\\mathfrak m\\cap M_+\\}\\).\n- (b) The linear span of \\(\\{xe:\\ x\\in M,\\ e\\in\\mathcal P\\cap\\mathfrak m\\}\\) is norm dense in \\(\\mathfrak m\\). So a norm-closed left ideal is determined by its projections.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 123,
        "through_line": 131,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
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      "source_uses": [
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              "heading": "3. The projection lattice and equivalence of projections",
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              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
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              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.1",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-03::Proposition 5.1",
      "anchor": "oa-fnd-ty-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 5.1** (Schröder–Bernstein for projections). If \\(e\\precsim f\\) and \\(f\\precsim e\\), then \\(e\\sim f\\).\n\nThe proof makes one \"Hilbert hotel\" shift inside \\(e\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 136,
        "through_line": 162,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
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          "local_scopes": [
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              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.2",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-03::Definition 5.2",
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      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 5.2.** Projections \\(e,f\\in\\mathcal P(M)\\) are *centrally orthogonal* if \\(c(e)c(f)=0\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 163,
        "through_line": 164,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
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              "heading": "3. The projection lattice and equivalence of projections",
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.3",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
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      "anchor": "oa-fnd-ty-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 5.3** (Central orthogonality). For \\(e,f\\in\\mathcal P(M)\\) the following are equivalent.\n\n1. \\(c(e)c(f)\\ne0\\).\n2. \\(eMf\\ne\\{0\\}\\).\n3. There are nonzero projections \\(e_1\\le e\\) and \\(f_1\\le f\\) in \\(M\\) with \\(e_1\\sim f_1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 165,
        "through_line": 178,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
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              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
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              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 5.4",
      "kind": "corollary",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-03::Corollary 5.4",
      "anchor": "oa-fnd-ty-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Corollary 5.4.** If \\(e\\ne0\\) and \\(c(e)\\le c(f)\\), there are nonzero \\(e_1\\le e\\) and \\(f_1\\le f\\) with \\(e_1\\sim f_1\\). Indeed \\(c(e)c(f)=c(e)\\ne0\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 179,
        "through_line": 180,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
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              "heading": "3. The projection lattice and equivalence of projections",
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              "anchors": [
                "OA-FND-TY-01",
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
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              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.5",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-03::Theorem 5.5",
      "anchor": "oa-fnd-ty-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 5.5** (The comparison theorem). For \\(e,f\\in\\mathcal P(M)\\) there is a central projection \\(z\\) with\n\\[\nze\\precsim zf\\qquad\\text{and}\\qquad(1-z)f\\precsim(1-z)e .\n\\]\nIf \\(M\\) is a factor, exactly one of \\(e\\prec f\\), \\(e\\sim f\\), \\(f\\prec e\\) holds.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 181,
        "through_line": 210,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
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            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
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                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
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              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5.6",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-03::Exercise 5.6",
      "anchor": "oa-fnd-ty-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 5.6.** (hard) Let \\(\\alpha\\) be an automorphism of \\(M\\). Suppose that some \\(e\\in\\mathcal P(M)\\) with \\(c(e)=1\\) and some \\(u\\in M\\) satisfy \\(\\alpha(x)=uxu^*\\) for all \\(x\\in eMe\\). Show that \\(\\alpha\\) is inner.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 211,
        "through_line": 237,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
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          "unit": "projections-and-types-of-von-neumann-algebras",
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              "heading": "3. The projection lattice and equivalence of projections",
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              "anchors": [
                "OA-FND-TY-01",
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              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
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              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.1",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-09::Definition 6.1",
      "anchor": "oa-fnd-ty-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 6.1.** A projection \\(e\\in\\mathcal P(M)\\) is\n\n- *finite* if \\(e\\sim f\\le e\\) implies \\(f=e\\), and *infinite* otherwise;\n- *purely infinite* if \\(0\\) is its only finite subprojection in \\(M\\);\n- *properly infinite* if each nonzero central cut \\(ze\\) (\\(z\\) a central projection) is infinite;\n- *abelian* if \\(eMe\\) is commutative.\n\nThe same four words are applied to \\(M\\) itself when the projection \\(1\\) has the property. By these definitions, \\(0\\) is finite, abelian, properly infinite and purely infinite.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 242,
        "through_line": 250,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
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          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
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              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.2",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-09::Lemma 6.2",
      "anchor": "oa-fnd-ty-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 6.2.**\n\n1. A subprojection of a finite (or abelian) projection is finite (or abelian). A projection equivalent to a finite (or abelian) projection is finite (or abelian). A projection that is subequivalent to a finite projection is finite.\n2. Abelian projections are finite. Minimal projections, meaning nonzero projections whose only subprojections are \\(0\\) and themselves, are abelian.\n3. \\(e\\) is finite (or abelian) in \\(M\\) if and only if \\(eMe\\) is a finite (or commutative) algebra. A nonzero central cut \\(ze\\) of a properly infinite \\(e\\) is properly infinite, and a central cut of \\(e\\) that is finite is \\(0\\). A purely infinite projection is properly infinite.\n4. If \\(e\\) is abelian, then \\(eMe=Ze\\), and every subprojection \\(f\\) of \\(e\\) satisfies \\(f=c(f)e\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 251,
        "through_line": 265,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.3",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-09::Lemma 6.3",
      "anchor": "oa-fnd-ty-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 6.3** (Centrally orthogonal sums). The sum of a family of mutually centrally orthogonal abelian (or finite) projections is abelian (or finite).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 266,
        "through_line": 271,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
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              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.1",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Definition 7.1",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 7.1** (Types). \\(M\\) is\n\n- *of type I* if below each nonzero central projection there is a nonzero abelian projection;\n- *of type II* if \\(0\\) is its only abelian projection, while below each nonzero central projection there is a nonzero finite projection;\n- *of type III* if \\(M\\) is purely infinite: \\(0\\) is its only finite projection;\n- *of type II\\(_1\\)* if it is of type II and finite, and *of type II\\(_\\infty\\)* if it is of type II and \\(0\\) is its only finite central projection;\n- *semifinite* if it has no nonzero central summand of type III.\n\nFor a central projection \\(z\\), the projections of the central summand \\(Mz\\) (on \\(zH\\)) are the projections of \\(M\\) below \\(z\\), its central projections are the central projections of \\(M\\) below \\(z\\), and equivalence among them is the same as in \\(M\\). So each type passes to central summands. Each type is also invariant under isomorphism.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 274,
        "through_line": 283,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.2",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Theorem 7.2",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 7.2** (The type decomposition). There are unique mutually orthogonal central projections \\(z_{\\rm I}\\), \\(z_{{\\rm II}_1}\\), \\(z_{{\\rm II}_\\infty}\\), \\(z_{\\rm III}\\) with sum \\(1\\) such that \\(Mz_{\\rm I}\\), \\(Mz_{{\\rm II}_1}\\), \\(Mz_{{\\rm II}_\\infty}\\) and \\(Mz_{\\rm III}\\) are of types I, II\\(_1\\), II\\(_\\infty\\) and III. Every projection \\(e\\in M\\) can be written in exactly one way as \\(e=e_1+e_2\\) with \\(e_1\\) and \\(e_2\\) centrally orthogonal, \\(e_1\\) finite and \\(e_2\\) properly infinite.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 284,
        "through_line": 295,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
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          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
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              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 7.3",
      "kind": "corollary",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Corollary 7.3",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Corollary 7.3.** A nonzero factor is of exactly one of the types I, II\\(_1\\), II\\(_\\infty\\), III.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 296,
        "through_line": 299,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.4",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Lemma 7.4",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 7.4** (Good projections).\n\n1. In a type I algebra, every nonzero projection majorizes a nonzero abelian projection, and some abelian projection has central support \\(1\\).\n2. In a semifinite algebra, below each nonzero central projection there is a nonzero finite projection, and some finite projection has central support \\(1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 300,
        "through_line": 306,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.5",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Example 7.5",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 7.5** (\\(B(H)\\)). Assume \\(H\\ne0\\); the zero algebra has only its zero projection and is covered separately by the conventions. Here \\(M'=\\mathbb C1\\). Two projections are equivalent iff their ranges have the same Hilbert dimension, since a partial isometry is a unitary between its initial and final spaces. A projection is finite iff it has finite rank: an infinite-dimensional \\(eH\\) is unitarily equivalent to a proper subspace of itself (shift a countable part of an orthonormal basis), while in finite dimension \\(e\\sim f\\le e\\) forces \\(f=e\\). The abelian projections are those of rank at most \\(1\\), and \\(B(H)\\) is of type I\\(_{\\dim H}\\) in the sense of Definition 10.1.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 307,
        "through_line": 308,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.6",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Example 7.6",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 7.6** (Atomic algebras). For \\(H\\ne0\\), \\(B(H)\\) is a type I factor: its commutant is \\(\\mathbb C1\\), and a rank-one projection is minimal, hence abelian (Lemma 6.2(2)). More generally, call \\(M\\) *atomic* if every nonzero projection majorizes a minimal projection. An atomic algebra is of type I, because minimal projections are abelian. Equivalently, \\(1\\) is a sum of minimal projections. Indeed, if \\(M\\) is atomic, a maximal orthogonal family of minimal projections has sum \\(1\\). Conversely, let \\(1=\\sum_ip_i\\) with minimal \\(p_i\\), and let \\(e\\ne0\\). Then \\(ep_i\\ne0\\) for some \\(i\\), and Lemma 5.3 gives nonzero \\(e_1\\le e\\) and \\(f_1\\le p_i\\) with \\(e_1\\sim f_1\\). Then \\(f_1=p_i\\). If \\(u\\) implements \\(p_i\\sim e_1\\) and \\(g\\le e_1\\), then \\(u^*gu\\le p_i\\) is \\(0\\) or \\(p_i\\), so \\(g=u(u^*gu)u^*\\) is \\(0\\) or \\(e_1\\): the projection \\(e_1\\) is minimal.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 309,
        "through_line": 310,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.7",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-10::Example 7.7",
      "anchor": "oa-fnd-ty-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 7.7** (Commutative algebras). In a commutative \\(A\\), \\(e\\sim f\\) iff \\(e=f\\) (since \\(u^*u=uu^*\\)), every projection is abelian and finite, \\(c(e)=e\\), and \\(A\\) is of type I\\(_1\\) in the sense of Definition 10.1.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 311,
        "through_line": 312,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "3. The projection lattice and equivalence of projections",
              "line": 49,
              "through_line": 96,
              "anchors": [
                "OA-FND-TY-01",
                "OA-FND-TY-02"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "4. Supports, cyclic projections and the parallelogram law",
              "line": 97,
              "through_line": 131,
              "anchors": [
                "OA-FND-TY-04",
                "OA-FND-TY-07"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "5. The comparison theorem",
              "line": 132,
              "through_line": 237,
              "anchors": [
                "OA-FND-TY-03",
                "OA-FND-TY-05",
                "OA-FND-TY-06"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "6. Finite, infinite and abelian projections",
              "line": 238,
              "through_line": 271,
              "anchors": [
                "OA-FND-TY-09"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            },
            {
              "heading": "7. The type decomposition",
              "line": 272,
              "through_line": 312,
              "anchors": [
                "OA-FND-TY-10"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.1–III.1.4; PDF 244–254",
          "role": "proof comparison",
          "correspondence": "Projection comparison and type mechanisms",
          "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.2",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-11::Lemma 8.2",
      "anchor": "oa-fnd-ty-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 8.2** (Matrix algebras).\n\n1. \\((N\\otimes1)'=\\mathbb M_I(N')\\).\n2. \\(\\mathbb M_I(N)'=N'\\otimes1\\).\n3. \\(\\mathbb M_I(N)\\) and \\(N\\otimes1\\) are von Neumann algebras, and \\(N\\otimes1\\) together with \\(1\\otimes B(\\ell^2(I))\\) generates \\(\\mathbb M_I(N)\\) as a von Neumann algebra. We also write \\(N\\bar\\otimes B(\\ell^2(I))\\) for it.\n4. The centre of \\(\\mathbb M_I(N)\\) is \\((N\\cap N')\\otimes1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 330,
        "through_line": 344,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 8.3",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-11::Definition 8.3",
      "anchor": "oa-fnd-ty-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 8.3** (Matrix units). A *matrix unit* in \\(M\\) indexed by \\(I\\) is a family \\(\\{w_{ij}\\}_{i,j\\in I}\\subseteq M\\) with \\(w_{ij}^*=w_{ji}\\), \\(w_{ij}w_{kl}=\\delta_{jk}w_{il}\\), and \\(\\sum_iw_{ii}=1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 345,
        "through_line": 346,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.4",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-11::Proposition 8.4",
      "anchor": "oa-fnd-ty-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 8.4** (Splitting off a matrix algebra). Let \\(\\{e_i\\}_{i\\in I}\\) be mutually orthogonal, mutually equivalent projections with \\(\\sum_ie_i=1\\). Fix \\(i_0\\in I\\), put \\(e=e_{i_0}\\), and choose \\(u_i\\) implementing \\(e\\sim e_i\\), with \\(u_{i_0}=e\\). Then \\(w_{ij}=u_iu_j^*\\) is a matrix unit, and \\(W:eH\\otimes\\ell^2(I)\\to H\\), \\(W(\\zeta\\otimes\\delta_i)=u_i\\zeta\\), is a unitary with\n\\[\n\\begin{gathered}\nW^*MW\\\\\n=\\mathbb M_I(eMe)\\\\\n=eMe\\bar\\otimes B(\\ell^2(I)),\\\\\nW^*M'W\\\\\n=M'e\\otimes1 .\n\\end{gathered}\n\\]\nSo \\(\\{M,H\\}\\cong\\{eMe,eH\\}\\otimes\\{B(\\ell^2(I)),\\ell^2(I)\\}\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 347,
        "through_line": 367,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.5",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-11::Lemma 8.5",
      "anchor": "oa-fnd-ty-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 8.5** (Equivalent projections have isomorphic corners). If \\(u\\) implements \\(e\\sim f\\), then \\(x\\mapsto uxu^*\\) is an isomorphism of \\(eMe\\) onto \\(fMf\\), implemented by the unitary \\(u|_{eH}:eH\\to fH\\). So \\(\\{eMe,eH\\}\\cong\\{fMf,fH\\}\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 368,
        "through_line": 371,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.1",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-12::Proposition 9.1",
      "anchor": "oa-fnd-ty-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 9.1** (Matrices over a commutative algebra). Let \\(A\\) be a commutative von Neumann algebra acting on \\(K\\).\n\n1. For every nonzero Hilbert space \\(L\\), \\(A\\bar\\otimes B(L)\\) is of type I. For a rank-one projection \\(q\\) on \\(L\\), \\(1\\otimes q\\) is abelian with central support \\(1\\).\n2. For finite \\(n\\), the matrix algebra \\(\\mathbb M_n(A)\\) of (8.1) is finite and of type I. The linear map \\(\\Phi(x)=\\frac1n\\sum_ix_{ii}\\), with values in \\(A\\) (identified with the diagonal copy \\(A\\otimes1\\)), satisfies\n\\[\n\\begin{gathered}\n\\Phi(a)\\\\\n=a,\\\\\n\\Phi(axb)\\\\\n=a\\Phi(x)b\\ \\ (a,b\\in A),\\\\\n\\Phi(x^*x)\\\\\n=\\Phi(xx^*)\\\\\n\\ge0,\n\\end{gathered}\n\\]\nand \\(\\Phi(x^*x)=0\\) only if \\(x=0\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 376,
        "through_line": 396,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 9.2",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-12::Definition 9.2",
      "anchor": "oa-fnd-ty-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 9.2** (Algebra-valued traces). Let \\(A\\) be a \\(*\\)-subalgebra of \\(M\\). An *\\(A\\)-valued trace* on \\(M\\) is a linear map \\(\\Phi:M\\to A\\) with \\(\\Phi(a)=a\\), \\(\\Phi(axb)=a\\Phi(x)b\\) for \\(a,b\\in A\\), and \\(\\Phi(x^*x)=\\Phi(xx^*)\\ge0\\) for \\(x\\in M\\). It is *faithful* if \\(\\Phi(x^*x)=0\\) only for \\(x=0\\). When \\(A\\) is the centre of \\(M\\), \\(\\Phi\\) is a *centre-valued trace*.\n\nAn \\(A\\)-valued trace is positive, since every positive element of \\(M\\) has the form \\(x^*x\\); hence it is monotone. It is constant on equivalence classes: if \\(u\\) implements \\(e\\sim f\\), then \\(\\Phi(e)=\\Phi(u^*u)=\\Phi(uu^*)=\\Phi(f)\\). Centre-valued traces on finite algebras are the subject of the lesson Traces on von Neumann algebras.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 397,
        "through_line": 400,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.3",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-12::Lemma 9.3",
      "anchor": "oa-fnd-ty-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 9.3** (Comparing abelian projections). Let \\(e\\) be abelian and \\(f\\in\\mathcal P(M)\\) with \\(c(e)\\le c(f)\\) (equivalently, \\(e\\le c(f)\\)). Then \\(e\\precsim f\\). If \\(f\\) is also abelian and \\(c(e)=c(f)\\), then \\(e\\sim f\\).\n\nThe direction of subequivalence matters: Remark 9.4 gives the rank-one matrix test.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 401,
        "through_line": 415,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.5",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-12::Lemma 9.5",
      "anchor": "oa-fnd-ty-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 9.5** (Counting abelian projections). Let \\(c\\ne0\\) be a central projection. Let \\(\\{e_i\\}_{i\\in I}\\) be mutually orthogonal abelian projections with \\(c(e_i)=c\\) and \\(\\sum_ie_i=c\\). Let \\(\\{f_j\\}_{j\\in J}\\) be mutually orthogonal abelian projections with \\(c(f_j)=c\\) and \\(\\sum_jf_j\\le c\\). Then \\(|J|\\le|I|\\). Consequently, if also \\(\\sum_jf_j=c\\), then \\(|I|=|J|\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 416,
        "through_line": 425,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.1",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-14::Definition 10.1",
      "anchor": "oa-fnd-ty-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 10.1** (Homogeneous algebras). Let \\(\\alpha\\) be a nonzero cardinal. A central projection \\(z\\) is *\\(\\alpha\\)-homogeneous* if \\(z=\\sum_{i\\in I}e_i\\) for mutually orthogonal abelian projections \\(e_i\\) with \\(c(e_i)=z\\) and \\(|I|=\\alpha\\). A type I algebra is *of type I\\(_\\alpha\\)* if \\(1\\) is \\(\\alpha\\)-homogeneous.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 428,
        "through_line": 429,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.2",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-14::Lemma 10.2",
      "anchor": "oa-fnd-ty-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 10.2.**\n\n1. A central projection below an \\(\\alpha\\)-homogeneous one is \\(\\alpha\\)-homogeneous. The sum of mutually orthogonal \\(\\alpha\\)-homogeneous central projections is \\(\\alpha\\)-homogeneous.\n2. A nonzero central projection is \\(\\alpha\\)-homogeneous for at most one \\(\\alpha\\).\n3. If \\(M\\) is of type I, then below each nonzero central projection there is a nonzero central projection that is \\(\\alpha\\)-homogeneous for some \\(\\alpha\\).\n4. If \\(M\\cong A\\bar\\otimes B(L)\\) with \\(A\\) abelian and \\(\\dim L=\\alpha\\), then \\(1\\) is \\(\\alpha\\)-homogeneous in \\(M\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 430,
        "through_line": 444,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.3",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-14::Theorem 10.3",
      "anchor": "oa-fnd-ty-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 10.3** (Structure of type I algebras). Let \\(M\\) be of type I. For each nonzero cardinal \\(\\alpha\\) there is a largest \\(\\alpha\\)-homogeneous central projection \\(z_\\alpha\\). The \\(z_\\alpha\\) are mutually orthogonal, \\(\\sum_\\alpha z_\\alpha=1\\), and \\(z_\\alpha=0\\) unless \\(\\alpha\\le\\dim H\\). For each \\(\\alpha\\) with \\(z_\\alpha\\ne0\\),\n\\[\n\\{Mz_\\alpha,z_\\alpha H\\}\\cong\\{A_\\alpha\\bar\\otimes B(\\ell^2(\\alpha)),\\,K_\\alpha\\otimes\\ell^2(\\alpha)\\},\n\\]\nwhere \\(A_\\alpha\\) is commutative and isomorphic to \\(Zz_\\alpha\\). Hence \\(M\\cong\\bigoplus_\\alpha A_\\alpha\\bar\\otimes B(\\ell^2(\\alpha))\\). The family \\(\\{z_\\alpha\\}\\) is unique: if \\(\\{c_\\alpha\\}\\) are mutually orthogonal central projections with \\(\\sum_\\alpha c_\\alpha=1\\) and \\(Mc_\\alpha\\cong B_\\alpha\\bar\\otimes B(\\ell^2(\\alpha))\\) with \\(B_\\alpha\\) abelian, then \\(c_\\alpha=z_\\alpha\\) for every \\(\\alpha\\). Finally, \\(M\\) is finite if and only if \\(z_\\alpha=0\\) for every infinite \\(\\alpha\\).\n\nThe sum and uniqueness are proved explicitly below.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 445,
        "through_line": 464,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 10.4",
      "kind": "corollary",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-14::Corollary 10.4",
      "anchor": "oa-fnd-ty-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Corollary 10.4** (Type I factors). Every type I factor is isomorphic to \\(B(L)\\) for some Hilbert space \\(L\\), and \\(B(L_1)\\cong B(L_2)\\) if and only if \\(\\dim L_1=\\dim L_2\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 465,
        "through_line": 470,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.6",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-14::Example 10.6",
      "anchor": "oa-fnd-ty-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 10.6** (A type decomposition). Let \\(M=\\mathbb C\\oplus M_2(\\mathbb C)\\oplus B(\\ell^2(\\mathbb N))\\oplus\\mathbb M_3(L^\\infty[0,1])\\) on \\(\\mathbb C\\oplus\\mathbb C^2\\oplus\\ell^2(\\mathbb N)\\oplus(L^2[0,1]\\otimes\\mathbb C^3)\\). It is of type I, with \\(z_1=1\\oplus0\\oplus0\\oplus0\\), \\(z_2=0\\oplus1\\oplus0\\oplus0\\), \\(z_3=0\\oplus0\\oplus0\\oplus1\\) and \\(z_{\\aleph_0}=0\\oplus0\\oplus1\\oplus0\\) in Theorem 10.3. It is not finite, and \\(z_{\\aleph_0}\\) is the properly infinite part of \\(1\\). For \\(e=0\\oplus E_{11}\\oplus p\\oplus0\\) with \\(p\\) of infinite rank, the decomposition of Theorem 7.2 is \\(e_1=0\\oplus E_{11}\\oplus0\\oplus0\\) (finite) and \\(e_2=0\\oplus0\\oplus p\\oplus0\\) (properly infinite). If \\(p\\) has finite rank, \\(e\\) is finite.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 471,
        "through_line": 472,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 10.7",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-14::Exercise 10.7",
      "anchor": "oa-fnd-ty-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 10.7.** (medium) Let \\(M\\) be of type I. Show that an automorphism \\(\\alpha\\) of \\(M\\) that fixes every element of the centre is inner.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 473,
        "through_line": 476,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.1",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-15::Lemma 11.1",
      "anchor": "oa-fnd-ty-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 11.1** (Cyclic commutative algebras are maximal abelian). Let \\(A\\) be a commutative von Neumann algebra on \\(K\\), and \\(\\xi\\in K\\) a cyclic vector for it. Then \\(A\\) is maximal abelian: \\(A'=A\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 481,
        "through_line": 499,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.2",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-15::Proposition 11.2",
      "anchor": "oa-fnd-ty-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 11.2** (Type I through the commutant). \\(M\\) is of type I exactly when some faithful normal representation \\(\\pi\\) of \\(M\\) has a commutative commutant \\(\\pi(M)'\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 500,
        "through_line": 505,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.3",
      "kind": "corollary",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-15::Corollary 11.3",
      "anchor": "oa-fnd-ty-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Corollary 11.3.** If \\(M\\) is of type I, so is \\(M'\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 506,
        "through_line": 509,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.4",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-15::Theorem 11.4",
      "anchor": "oa-fnd-ty-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 11.4** (The spatial form of a type I algebra). Let \\(M\\) be of type I on \\(H\\). By Corollary 11.3, \\(M'\\) is of type I as well, and its centre is \\(Z\\), so Theorem 10.3 applies to both algebras. Let \\(z_\\alpha\\) be the largest \\(\\alpha\\)-homogeneous central projection of \\(M\\) and \\(z'_\\beta\\) the largest \\(\\beta\\)-homogeneous central projection of \\(M'\\). Put \\(z_{\\alpha,\\beta}=z_\\alpha z'_\\beta\\). These central projections are mutually orthogonal with sum \\(1\\). For each \\((\\alpha,\\beta)\\) with \\(z_{\\alpha,\\beta}\\ne0\\) there are a Hilbert space \\(K_{\\alpha,\\beta}\\), a maximal abelian algebra \\(A_{\\alpha,\\beta}\\) on it, and a unitary \\(W:K_{\\alpha,\\beta}\\otimes\\ell^2(\\alpha)\\otimes\\ell^2(\\beta)\\to z_{\\alpha,\\beta}H\\) such that \\(W^*(Mz_{\\alpha,\\beta})W=\\mathbb M_\\alpha(A_{\\alpha,\\beta})\\otimes1_{\\ell^2(\\beta)}\\), and \\(W^*(M'z_{\\alpha,\\beta})W\\) is the algebra of operators whose matrix entries with respect to \\(\\ell^2(\\beta)\\) lie in \\(A_{\\alpha,\\beta}\\otimes1_{\\ell^2(\\alpha)}\\). That is,\n\\[\n\\begin{gathered}\n\\{M,H\\}\\\\\n\\cong\\bigoplus_{\\alpha,\\beta}\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\\\\n\\otimes\\{B(\\ell^2(\\alpha)),\\ell^2(\\alpha)\\}\\\\\n\\otimes\\{\\mathbb C,\\ell^2(\\beta)\\},\n\\end{gathered}\n\\]\n\\[\n\\begin{gathered}\n\\{M',H\\}\\\\\n\\cong\\bigoplus_{\\alpha,\\beta}\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\\\\n\\otimes\\{\\mathbb C,\\ell^2(\\alpha)\\}\\\\\n\\otimes\\{B(\\ell^2(\\beta)),\\ell^2(\\beta)\\},\n\\end{gathered}\n\\]\nwith the same unitary. The decomposition is unique: the projections \\(z_{\\alpha,\\beta}\\) are determined by \\(M\\), and each \\(\\{A_{\\alpha,\\beta},K_{\\alpha,\\beta}\\}\\) is determined up to spatial isomorphism.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 510,
        "through_line": 560,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 11.5",
      "kind": "corollary",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-15::Corollary 11.5",
      "anchor": "oa-fnd-ty-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Corollary 11.5** (Commutative von Neumann algebras and multiplicity). A commutative von Neumann algebra \\(\\{A,H\\}\\) is spatially isomorphic to \\(\\bigoplus_\\beta\\{A_\\beta\\otimes1,K_\\beta\\otimes\\ell^2(\\beta)\\}\\) with each \\(\\{A_\\beta,K_\\beta\\}\\) maximal abelian. The central projections of the summands are unique, and each \\(\\{A_\\beta,K_\\beta\\}\\) is unique up to spatial isomorphism.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 561,
        "through_line": 564,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.6",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-15::Example 11.6",
      "anchor": "oa-fnd-ty-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 11.6** (Multiplicity \\(n\\)). For \\(A=L^\\infty[0,1]\\otimes1\\) on \\(L^2[0,1]\\otimes\\mathbb C^n\\), [Lemma 8.2(1)](#oa-fnd-ty-11) gives \\(A'=\\mathbb M_n(L^\\infty[0,1])\\), because \\(L^\\infty[0,1]\\) is maximal abelian on \\(L^2[0,1]\\) (Lemma 11.1, with the cyclic vector \\(1\\)). This algebra is of type I\\(_n\\) (Proposition 9.1(1) and Lemma 10.2(4)). So in Corollary 11.5 only \\(\\beta=n\\) occurs: the multiplicity is \\(n\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 565,
        "through_line": 566,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.1",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-21::Lemma 12.1",
      "anchor": "oa-fnd-ty-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 12.1** (The diagonal part). The four projections\n\\[\n\\begin{gathered}\np_{11}\\\\\n=e\\wedge f,\\\\\np_{10}\\\\\n=e\\wedge f^\\perp,\\\\\np_{01}\\\\\n=e^\\perp\\wedge f,\\\\\np_{00}\\\\\n=e^\\perp\\wedge f^\\perp\n\\end{gathered}\n\\]\nlie in \\(M\\), are mutually orthogonal, and are central in \\(M\\). Let \\(z=1-(p_{11}+p_{10}+p_{01}+p_{00})\\), \\(H_0=zH\\), \\(e_0=ez\\) and \\(f_0=fz\\).\n\n1. \\(e=p_{11}+p_{10}+e_0\\) and \\(f=p_{11}+p_{01}+f_0\\). In particular \\(e\\) commutes with \\(e\\wedge f+e^\\perp\\wedge f\\), and \\(f\\) commutes with \\(e\\wedge f+e\\wedge f^\\perp\\).\n2. (*Generic position*) On \\(H_0\\), with complements taken in \\(H_0\\): \\[\n\\begin{gathered}\ne_0\\wedge f_0\\\\\n=e_0\\wedge f_0^\\perp\\\\\n=e_0^\\perp\\wedge f_0\\\\\n=e_0^\\perp\\wedge f_0^\\perp\\\\\n=0.\n\\end{gathered}\n\\] Hence all four joins \\(e_0\\vee f_0\\), \\(e_0^\\perp\\vee f_0\\), \\(e_0\\vee f_0^\\perp\\), \\(e_0^\\perp\\vee f_0^\\perp\\) equal \\(z\\).\n3. \\(M(1-z)\\), on \\((1-z)H\\), is the linear span of the nonzero \\(p_{ab}\\); it is commutative and has dimension at most \\(4\\). And \\(Mz\\), on \\(H_0\\), equals \\(\\{e_0,f_0\\}''\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 571,
        "through_line": 611,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.2",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-21::Proposition 12.2",
      "anchor": "oa-fnd-ty-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 12.2** (A pair in generic position). Suppose \\(e\\wedge f=e\\wedge f^\\perp=e^\\perp\\wedge f=e^\\perp\\wedge f^\\perp=0\\) (so \\(z=1\\) above) and \\(H\\ne0\\). Let \\(M=\\{e,f\\}''\\).\n\n1. \\(e\\sim f\\sim e^\\perp\\sim f^\\perp\\) in \\(M\\).\n2. The operator \\(a=e^\\perp fe\\) is injective on \\(eH\\) and has dense range in \\(e^\\perp H\\). In its polar decomposition \\(a=u|a|\\), \\(u\\in M\\), \\(u^*u=e\\) and \\(uu^*=e^\\perp\\).\n3. The map \\(W:eH\\otimes\\mathbb C^2\\to H\\), \\(W(\\zeta_1\\otimes\\delta_1+\\zeta_2\\otimes\\delta_2)=\\zeta_1+u\\zeta_2\\), is unitary, and\n\\[\n\\begin{gathered}\nW^*eW\\\\\n=\\begin{pmatrix}1&0\\\\0&0\\end{pmatrix},\\\\\nW^*fW\\\\\n=\\begin{pmatrix}C^2&CS\\\\CS&S^2\\end{pmatrix},\n\\end{gathered}\n\\tag{12.3}\n\\]\nwhere \\(C=(efe|_{eH})^{1/2}\\) and \\(S=(1-C^2)^{1/2}\\) are commuting, injective, positive contractions on \\(eH\\) with \\(C^2+S^2=1\\).\n4. \\(W^*MW=\\mathbb M_2(A)\\) with \\(A=\\{C^2\\}''\\) on \\(eH\\), a commutative von Neumann algebra. So \\(M\\) is of type I\\(_2\\).\n5. \\(W^*|e-f|W=S\\otimes1\\) and \\(W^*|e-f^\\perp|W=C\\otimes1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 612,
        "through_line": 657,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 12.4",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-21::Theorem 12.4",
      "anchor": "oa-fnd-ty-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 12.4** (The algebra of two projections). Let \\(e\\) and \\(f\\) be projections on \\(H\\), and \\(M=\\{e,f\\}''\\).\n\n1. \\(M\\) is of type I.\n2. There is exactly one central projection \\(z\\) of \\(M\\) such that \\(Mz\\) is of type I\\(_2\\) and \\(M(1-z)\\) is commutative. Moreover \\(\\dim M(1-z)\\le4\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 658,
        "through_line": 664,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 12.5",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-21::Definition 12.5",
      "anchor": "oa-fnd-ty-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 12.5** (Sine and cosine). For projections \\(e\\) and \\(f\\), we call \\(\\sin(e,f)=|e-f|\\) the *sine* of the pair and \\(\\cos(e,f)=|e-f^\\perp|\\) its *cosine*.\n\n**Remark 12.6.** The identity \\((e-f)^2+(e+f-1)^2=1\\), which holds for any two projections, gives \\(\\sin(e,f)^2+\\cos(e,f)^2=1\\). Moreover \\((e-f)^2\\) commutes with \\(e\\) and with \\(f\\) (both products equal \\(e-efe\\) for \\(e\\), and similarly for \\(f\\)), so \\(\\sin(e,f)\\) and \\(\\cos(e,f)\\) are central in \\(\\{e,f\\}''\\). On the generic part they are \\(S\\otimes1\\) and \\(C\\otimes1\\), by Proposition 12.2(5). On \\((1-z)H\\) they are \\(p_{10}+p_{01}\\) and \\(p_{11}+p_{00}\\). If \\(H=\\mathbb C^2\\) and \\(eH\\), \\(fH\\) are lines at an angle \\(\\theta\\) with \\(0<\\theta<\\pi/2\\), the pair is in generic position, \\(C=\\cos\\theta\\) and \\(S=\\sin\\theta\\). So \\(C\\) and \\(S\\) generalize \\(\\cos\\theta\\) and \\(\\sin\\theta\\) for a pair of lines.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 665,
        "through_line": 668,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 12.7",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-21::Example 12.7",
      "anchor": "oa-fnd-ty-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 12.7** (Two lines in \\(\\mathbb C^2\\)). Let \\(eH=\\mathbb C\\delta_1\\) and \\(fH=\\mathbb C(\\cos\\theta\\,\\delta_1+\\sin\\theta\\,\\delta_2)\\) with \\(0<\\theta<\\pi/2\\). The four corner projections vanish, so \\(z=1\\) and \\(M=M_2(\\mathbb C)\\), of type I\\(_2\\). Here \\(efe=\\cos^2\\theta\\,e\\), so in (12.3) \\(C=\\cos\\theta\\) and \\(S=\\sin\\theta\\) (scalars on the line \\(eH\\)), \\(|e-f|=\\sin\\theta\\cdot1\\) and \\(|e-f^\\perp|=\\cos\\theta\\cdot1\\). For \\(\\theta=0\\) or \\(\\theta=\\pi/2\\) the projections commute, \\(z=0\\), and \\(M\\) is commutative.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 669,
        "through_line": 670,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.1",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-17::Proposition 13.1",
      "anchor": "oa-fnd-ty-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 13.1** (Orthogonal families of equivalent projections). Let \\(\\{e_i\\}_{i\\in I}\\), with \\(I\\ne\\emptyset\\), be mutually orthogonal, mutually equivalent, nonzero projections. There are a nonzero central projection \\(z\\) and mutually orthogonal, mutually equivalent projections \\(\\{f_j\\}_{j\\in J}\\), with \\(J\\supseteq I\\), such that \\(f_i=ze_i\\) for \\(i\\in I\\) and\n\\[\nf_0:=z-\\sum_{j\\in J}f_j\\prec f_j\\qquad\\text{for every }j\\in J .\n\\]\nIf \\(J\\) is infinite, the \\(f_j\\) can be replaced by mutually orthogonal, mutually equivalent projections with \\(f_i\\sim ze_i\\) for \\(i\\in I\\) and \\(f_0=0\\). The \\(e_i\\) must be nonzero: for a family of zero projections the conclusion fails ([Example 16.1](#oa-fnd-ty-22)).\n\nNonzero projections are essential for the strict comparison conclusion, as Example 16.1 verifies.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 675,
        "through_line": 696,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.3",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-17::Proposition 13.3",
      "anchor": "oa-fnd-ty-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 13.3** (Algebras without a type I part). The following are equivalent.\n\n1. \\(M\\) has no nonzero central summand of type I, that is, \\(z_{\\rm I}=0\\) in Theorem 7.2.\n2. \\(M\\) has no nonzero abelian projection.\n3. Every projection of \\(M\\) is the sum of two orthogonal, equivalent projections.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 697,
        "through_line": 708,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.4",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-17::Proposition 13.4",
      "anchor": "oa-fnd-ty-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 13.4** (Properly infinite algebras halve). If \\(M\\) is properly infinite, there is \\(e\\in\\mathcal P(M)\\) with \\(e\\sim1-e\\sim1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 709,
        "through_line": 720,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 13.5",
      "kind": "corollary",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-17::Corollary 13.5",
      "anchor": "oa-fnd-ty-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Corollary 13.5** (Division by \\(\\aleph_0\\)). If \\(e\\) is properly infinite, then \\(e=\\sum_{n\\ge1}e_n\\) with mutually orthogonal \\(e_n\\sim e\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 721,
        "through_line": 724,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 14.1",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-18::Theorem 14.1",
      "anchor": "oa-fnd-ty-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 14.1** (Finite projections form a modular lattice). If \\(e\\) and \\(f\\) are finite, then \\(e\\vee f\\) and \\(e\\wedge f\\) are finite, so the finite projections form a sublattice of \\(\\mathcal P(M)\\). This sublattice is modular. More generally, the modular law\n\\[\n(e\\vee f)\\wedge g=e\\vee(f\\wedge g)\\qquad(e\\le g)\n\\]\nholds for all \\(e,f,g\\in\\mathcal P(M)\\) with \\(e\\le g\\) as soon as \\(g\\) is finite.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 729,
        "through_line": 766,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 14.2",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-18::Proposition 14.2",
      "anchor": "oa-fnd-ty-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 14.2** (Unitary equivalence of finite projections). If \\(e\\sim f\\) and \\(e\\) is finite, then \\(1-e\\sim1-f\\), and \\(ueu^*=f\\) for some unitary \\(u\\in M\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 767,
        "through_line": 770,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.3",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-18::Exercise 14.3",
      "anchor": "oa-fnd-ty-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 14.3.** (easy) Show that \\(M\\) is finite if and only if, for all \\(e,f\\in\\mathcal P(M)\\), \\(e\\precsim f\\) implies \\(1-f\\precsim1-e\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 771,
        "through_line": 774,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 14.4",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-18::Exercise 14.4",
      "anchor": "oa-fnd-ty-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 14.4.** (hard) Projections \\(e,f\\) are *unitarily equivalent* if \\(ueu^*=f\\) for a unitary \\(u\\in M\\); then \\(e\\sim f\\). Show that if \\(e\\sim f\\), there are orthogonal decompositions \\(e=\\sum_ie_i\\) and \\(f=\\sum_if_i\\) such that \\(e_i\\) and \\(f_i\\) are unitarily equivalent for each \\(i\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 775,
        "through_line": 783,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 15.1",
      "kind": "definition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-19::Definition 15.1",
      "anchor": "oa-fnd-ty-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Definition 15.1** (Locally \\(\\sigma\\)-finite projections). A projection \\(f\\) is *locally \\(\\sigma\\)-finite* if every central projection \\(z\\) with \\(zf\\ne0\\) majorizes a central projection \\(z'\\) such that \\(z'f\\) is nonzero and \\(\\sigma\\)-finite. \\(M\\) is *locally \\(\\sigma\\)-finite* if \\(1\\) is. Every \\(\\sigma\\)-finite projection is locally \\(\\sigma\\)-finite.\n\nBy Zorn's lemma, \\(M\\) is locally \\(\\sigma\\)-finite exactly when \\(1\\) is a sum of mutually orthogonal central projections \\(z_k\\) with every \\(Mz_k\\) \\(\\sigma\\)-finite. Indeed, if \\(M\\) is locally \\(\\sigma\\)-finite, a maximal family of mutually orthogonal nonzero central projections \\(z_k\\) with \\(Mz_k\\) \\(\\sigma\\)-finite has sum \\(1\\), since otherwise \\(1-\\sum_kz_k\\) would majorize a further member. Conversely, if \\(1=\\sum_kz_k\\) with every \\(Mz_k\\) \\(\\sigma\\)-finite, and \\(z\\) is a nonzero central projection, then \\(zz_k\\ne0\\) for some \\(k\\), and \\(Mzz_k\\) is \\(\\sigma\\)-finite, since orthogonal families in \\(Mzz_k\\) are orthogonal families in \\(Mz_k\\). Such algebras are also called *locally countably decomposable*.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 788,
        "through_line": 791,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "15. Countable sums and properly infinite algebras",
              "line": 784,
              "through_line": 879,
              "anchors": [
                "OA-FND-TY-19",
                "OA-FND-TY-20"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.3.6; PDF 251–252",
          "role": "proof comparison",
          "correspondence": "Locally countably decomposable hypothesis is essential",
          "explanation": "Countable decomposition on central pieces compared; course retains its explicit nonseparable rank obstruction and does not extend the conclusion to arbitrary algebras."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 15.2",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-19::Proposition 15.2",
      "anchor": "oa-fnd-ty-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 15.2** (Properly infinite projections absorb countable sums). Let \\(e\\) be a properly infinite projection.\n\n1. If \\(f_1,f_2,\\ldots\\) are mutually orthogonal projections with \\(f_n\\precsim e\\), then \\(\\sum_nf_n\\precsim e\\).\n2. If \\(f\\) is \\(\\sigma\\)-finite and \\(c(f)\\le c(e)\\), then \\(f\\precsim e\\).\n3. If \\(f\\) is locally \\(\\sigma\\)-finite and \\(c(f)\\le c(e)\\), then \\(f\\precsim e\\).\n4. If \\(M\\) is locally \\(\\sigma\\)-finite and \\(c(e)=1\\), then \\(e\\sim1\\). In particular, in a \\(\\sigma\\)-finite factor any two infinite projections are equivalent.\n\nThe countability conditions cannot be dropped ([Example 16.4](#oa-fnd-ty-22)).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 792,
        "through_line": 808,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "15. Countable sums and properly infinite algebras",
              "line": 784,
              "through_line": 879,
              "anchors": [
                "OA-FND-TY-19",
                "OA-FND-TY-20"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.3.6; PDF 251–252",
          "role": "proof comparison",
          "correspondence": "Locally countably decomposable hypothesis is essential",
          "explanation": "Countable decomposition on central pieces compared; course retains its explicit nonseparable rank obstruction and does not extend the conclusion to arbitrary algebras."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 15.3",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-19::Proposition 15.3",
      "anchor": "oa-fnd-ty-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 15.3** (Properly infinite semifinite algebras). Let \\(M\\) be properly infinite and semifinite, and let \\(f\\) be a finite projection with \\(c(f)=1\\) ([Lemma 7.4(2)](#oa-fnd-ty-10)).\n\n1. There are mutually orthogonal central projections \\(z_\\alpha\\), indexed by the infinite cardinals \\(\\alpha\\le|M|\\), some possibly \\(0\\), with \\(\\sum_\\alpha z_\\alpha=1\\) and \\[\n\\begin{gathered}\n\\{Mz_\\alpha,z_\\alpha H\\}\\\\\n\\cong\\{N_\\alpha\\bar\\otimes B(\\ell^2(\\alpha)),\\,fz_\\alpha H\\otimes\\ell^2(\\alpha)\\},\n\\end{gathered}\n\\] where \\(N_\\alpha=fz_\\alpha Mfz_\\alpha\\) is finite. The \\(N_\\alpha\\) are not unique ([Example 16.5](#oa-fnd-ty-22)).\n2. If \\(M\\) is locally \\(\\sigma\\)-finite, then \\(M\\cong fMf\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n3. More generally, if \\(M\\) is properly infinite and locally \\(\\sigma\\)-finite (not necessarily semifinite) and \\(p\\) is any projection with \\(c(p)=1\\), then \\(1=\\sum_{n\\in\\mathbb N}p_n\\) with mutually orthogonal \\(p_n\\sim p\\), and \\(M\\cong pMp\\bar\\otimes B(\\ell^2(\\mathbb N))\\).\n\n*Reference:* The decomposition of \\(1\\) in part (3) is [Blackadar, III.1.3.6].\n\nThe further trace theory studies uniqueness of the family \\(\\{z_\\alpha\\}\\) in (1); see Traces on von Neumann algebras. That extension is not proved or used here. The present proof establishes existence; when the countability hypothesis in (2) holds, it constructs a single amplification by \\(\\ell^2(\\mathbb N)\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 809,
        "through_line": 843,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "15. Countable sums and properly infinite algebras",
              "line": 784,
              "through_line": 879,
              "anchors": [
                "OA-FND-TY-19",
                "OA-FND-TY-20"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.3.6; PDF 251–252",
          "role": "proof comparison",
          "correspondence": "Locally countably decomposable hypothesis is essential",
          "explanation": "Countable decomposition on central pieces compared; course retains its explicit nonseparable rank obstruction and does not extend the conclusion to arbitrary algebras."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.4",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-19::Exercise 15.4",
      "anchor": "oa-fnd-ty-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 15.4.** (hard) Let \\(M\\) on \\(H\\) and \\(N\\) on \\(K\\) be von Neumann algebras, with \\(H\\) and \\(K\\) separable, and let \\(\\pi:M\\to N\\) be an isomorphism. Show that \\(x\\otimes1\\mapsto\\pi(x)\\otimes1\\) is a spatial isomorphism of \\(\\{M\\otimes1,H\\otimes\\ell^2(\\mathbb N)\\}\\) onto \\(\\{N\\otimes1,K\\otimes\\ell^2(\\mathbb N)\\}\\). Show that separability can be replaced by \\(\\sigma\\)-finiteness of \\(M'\\) and \\(N'\\), and that it cannot simply be dropped.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 844,
        "through_line": 855,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "15. Countable sums and properly infinite algebras",
              "line": 784,
              "through_line": 879,
              "anchors": [
                "OA-FND-TY-19",
                "OA-FND-TY-20"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.3.6; PDF 251–252",
          "role": "proof comparison",
          "correspondence": "Locally countably decomposable hypothesis is essential",
          "explanation": "Countable decomposition on central pieces compared; course retains its explicit nonseparable rank obstruction and does not extend the conclusion to arbitrary algebras."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.5",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-19::Exercise 15.5",
      "anchor": "oa-fnd-ty-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 15.5.** (hard) Let \\(\\mathcal J\\) be a two-sided ideal of \\(M\\), not necessarily closed, and \\(\\mathcal P=\\mathcal P(M)\\).\n\n- (a) If \\(e,f\\in\\mathcal P\\), \\(e\\precsim f\\) and \\(f\\in\\mathcal J\\), then \\(e\\in\\mathcal J\\).\n- (b) \\(\\mathcal P\\cap\\mathcal J\\) is a sublattice of \\(\\mathcal P\\).\n- (c) If \\(\\mathcal P_0\\) is a nonempty sublattice of \\(\\mathcal P\\) such that \\(e\\precsim f\\in\\mathcal P_0\\) implies \\(e\\in\\mathcal P_0\\), then \\(\\mathcal J_0=\\{x\\in M:\\mathrm l(x)\\in\\mathcal P_0\\}\\) is a two-sided ideal and \\(\\mathcal J_0\\cap\\mathcal P=\\mathcal P_0\\).\n- (d) For every two-sided ideal \\(\\mathcal J\\), the ideal \\(\\mathcal J_0=\\{x:\\mathrm l(x)\\in\\mathcal P\\cap\\mathcal J\\}\\) satisfies \\(\\mathcal J_0\\subseteq\\mathcal J\\subseteq\\overline{\\mathcal J_0}\\) (norm closure).\n- (e) The norm-closed two-sided ideals of a factor are totally ordered by inclusion.\n- (f) A factor that is finite, or \\(\\sigma\\)-finite and of type III, has no two-sided ideals other than \\(\\{0\\}\\) and itself; in particular it is simple.\n- (g) Conversely, a factor that is infinite and semifinite, or of type III and not \\(\\sigma\\)-finite, has a proper nonzero norm-closed two-sided ideal. So a factor is simple exactly when it is finite, or \\(\\sigma\\)-finite of type III.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 856,
        "through_line": 879,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [
        {
          "unit": "projections-and-types-of-von-neumann-algebras",
          "local_scopes": [
            {
              "heading": "15. Countable sums and properly infinite algebras",
              "line": 784,
              "through_line": 879,
              "anchors": [
                "OA-FND-TY-19",
                "OA-FND-TY-20"
              ],
              "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
            }
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "III.1.3.6; PDF 251–252",
          "role": "proof comparison",
          "correspondence": "Locally countably decomposable hypothesis is essential",
          "explanation": "Countable decomposition on central pieces compared; course retains its explicit nonseparable rank obstruction and does not extend the conclusion to arbitrary algebras."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 16.1",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Example 16.1",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 16.1** (A family of zero projections). If \\(I\\ne\\emptyset\\) and all \\(e_i=0\\), then \\(f_i\\sim ze_i=0\\), so every \\(f_j=0\\), and \\(f_0=z\\prec0\\) is impossible (\\(z\\precsim0\\) forces \\(z=0\\sim0\\)). This is why Proposition 13.1 asks for nonzero \\(e_i\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 884,
        "through_line": 885,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 16.2",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Example 16.2",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 16.2** (Modularity needs finiteness). Let \\(K=\\ell^2(\\mathbb N)\\) and \\(T\\delta_n=\\delta_n/(n+1)\\). Then \\(T\\) is injective with dense range, and \\(w=\\sum_n\\delta_n/(n+1)\\) is not in the range of \\(T\\) (its preimage would be \\(\\sum_n\\delta_n\\notin K\\)). On \\(H=K\\oplus K\\), let \\(e\\) be the projection onto \\(K\\oplus0\\), \\(f\\) the projection onto the graph \\(\\{(\\xi,T\\xi)\\}\\), and \\(g\\) the projection onto \\((K\\oplus0)+\\mathbb C(0,w)\\); these subspaces are closed, and \\(e\\le g\\). If \\((\\xi,T\\xi)=(x,\\lambda w)\\), then \\(T\\xi=\\lambda w\\) forces \\(\\lambda=0\\) and \\(\\xi=0\\); so \\(f\\wedge g=0\\) and \\(e\\vee(f\\wedge g)=e\\). But \\((K\\oplus0)+\\text{graph}=K\\oplus T(K)\\) is dense in \\(H\\), so \\(e\\vee f=1\\) and \\((e\\vee f)\\wedge g=g\\ne e\\). So the modular law fails in \\(B(H)\\), with \\(g\\) infinite. Exercise 16.6 extends this to every algebra that is not finite.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 886,
        "through_line": 887,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 16.3",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Example 16.3",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 16.3** (The shift). On \\(\\ell^2(\\mathbb N)\\) the unilateral shift \\(s\\) satisfies \\(s^*s=1\\) and \\(ss^*=1-p_0\\), where \\(p_0\\) is the projection onto \\(\\mathbb C\\delta_0\\). So \\(1\\sim1-p_0\\), and \\(1\\) is infinite. The projections \\(1\\) and \\(1-p_0\\) are equivalent but not unitarily equivalent, since their complements \\(0\\) and \\(p_0\\) are not equivalent. So Proposition 14.2 needs finiteness.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 888,
        "through_line": 889,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 16.4",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Example 16.4",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 16.4** (Countability in Propositions 15.2 and 15.3). Let \\(H\\) have Hilbert dimension \\(\\aleph_1\\), \\(M=B(H)\\), and \\(e\\) the projection onto a separable infinite-dimensional subspace. Then \\(e\\) is properly infinite (\\(M\\) is a factor and \\(e\\) is infinite), and \\(c(e)=1=c(1)\\). But \\(1\\not\\precsim e\\), since an isometry of \\(H\\) into \\(eH\\) would give \\(\\dim H\\le\\aleph_0\\). So the conclusions of Proposition 15.2(2)–(3) fail for \\(f=1\\), which is neither \\(\\sigma\\)-finite nor locally \\(\\sigma\\)-finite. Also \\(M\\) is properly infinite and semifinite, and a rank-one \\(f\\) is finite with \\(c(f)=1\\), but \\(M\\not\\cong fMf\\bar\\otimes B(\\ell^2(\\mathbb N))=B(\\ell^2(\\mathbb N))\\) by Corollary 10.4. So Proposition 15.3(2) needs its countability hypothesis. In Proposition 15.3(1), \\(z_{\\aleph_1}=1\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 890,
        "through_line": 891,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 16.5",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Example 16.5",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 16.5** (Non-uniqueness in Proposition 15.3). \\(B(\\ell^2(\\mathbb N))\\cong\\mathbb C\\bar\\otimes B(\\ell^2(\\mathbb N))\\cong M_2(\\mathbb C)\\bar\\otimes B(\\ell^2(\\mathbb N))\\), because \\(\\mathbb C^2\\otimes\\ell^2(\\mathbb N)\\cong\\ell^2(\\mathbb N)\\). Both \\(\\mathbb C\\) and \\(M_2(\\mathbb C)\\) are finite; they are the corners \\(fMf\\) for an \\(f\\) of rank one and of rank two.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 892,
        "through_line": 893,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 16.6",
      "kind": "exercise",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Exercise 16.6",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Exercise 16.6.** (hard) Show that the projection lattice \\(\\mathcal P(M)\\) is modular if and only if \\(M\\) is finite.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 894,
        "through_line": 897,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 17.1",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Lemma 17.1",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 17.1** (A positivity test). Let \\(A\\) be a unital C\\(^*\\)-algebra and \\(\\omega\\) a bounded linear functional on \\(A\\). If \\(\\omega(h)=\\|\\omega\\|\\) for some \\(h\\in A\\) with \\(0\\le h\\le1\\), then \\(\\omega\\) is positive.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 902,
        "through_line": 907,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 17.2",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Lemma 17.2",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 17.2** (Dropping a projection). Let \\(\\varphi\\) be a weakly continuous linear functional on \\(M\\), let \\(e\\in\\mathcal P(M)\\), and put \\(\\varphi_e(x)=\\varphi(xe)\\). If \\(\\|\\varphi_e\\|=\\|\\varphi\\|\\), then \\(\\varphi_e=\\varphi\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 908,
        "through_line": 911,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 17.3",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Lemma 17.3",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 17.3** (Rotating a vector functional). Let \\(\\xi,\\eta\\in H\\) and \\(\\varphi(x)=\\langle x\\eta,\\xi\\rangle\\) for \\(x\\in M\\). There is a partial isometry \\(u\\in M\\) such that\n\\[\n\\begin{gathered}\n\\varphi(xuu^*)\\\\\n=\\varphi(x)\\\\\n(x\\in M)\\\\\n\\text{and}\\\\\nx\\mapsto\\varphi(xu^*)\\ \\text{is positive on }M.\n\\end{gathered}\n\\tag{17.4}\n\\]\nIf \\(\\eta\\in[M\\xi]\\), then \\(uu^*\\eta=\\eta\\), and the vector \\(\\eta_2=u^*\\eta\\) lies in \\([M'\\xi]\\cap[M\\xi]\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 912,
        "through_line": 932,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 17.5",
      "kind": "theorem",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Theorem 17.5",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Theorem 17.5** (The transfer theorem). For \\(\\xi,\\eta\\in H\\),\n\\[\np_\\xi\\succsim p_\\eta\\ \\text{ in } M\\quad\\Longleftrightarrow\\quad p'_\\xi\\succsim p'_\\eta\\ \\text{ in } M'.\n\\]",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 933,
        "through_line": 943,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 17.6",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-22::Example 17.6",
      "anchor": "oa-fnd-ty-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 17.6** (\\(B(H)\\) and the scalars). For \\(M=B(H)\\) and \\(\\xi\\ne0\\), \\(p_\\xi\\) is the projection onto \\(\\mathbb C\\xi\\) and \\(p'_\\xi=1\\); Theorem 17.5 reads: \\(p_\\xi\\succsim p_\\eta\\) iff \\(\\xi\\ne0\\) or \\(\\eta=0\\), iff \\(p'_\\xi\\succsim p'_\\eta\\). The algebra \\(M=\\mathbb C1\\) on \\(H\\ne0\\) is this example with \\(M\\) and \\(M'\\) exchanged: for \\(\\xi\\ne0\\), \\(p_\\xi=1\\) and \\(p'_\\xi\\) is the projection onto \\(\\mathbb C\\xi\\). Theorem 17.5 is symmetric under this exchange.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 944,
        "through_line": 945,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 18.1",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-08::Proposition 18.1",
      "anchor": "oa-fnd-ty-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 18.1** (Subrepresentations and cyclic projections). Let \\(A\\) be a C\\(^*\\)-algebra, \\(\\{\\pi,H\\}\\) a nondegenerate representation of \\(A\\), \\(\\xi,\\eta\\in H\\), \\(\\varphi=\\omega_\\xi\\circ\\pi\\), \\(\\psi=\\omega_\\eta\\circ\\pi\\), and \\(N=\\pi(A)''\\). The following are equivalent.\n\n1. The cyclic representation \\(\\pi_\\varphi\\) of \\(\\varphi\\) *embeds* in \\(\\pi_\\psi\\): some isometry \\(H_\\varphi\\to H_\\psi\\) intertwines \\(\\pi_\\varphi\\) with \\(\\pi_\\psi\\). (Its range is then an invariant subspace, so this says that \\(\\pi_\\varphi\\) is carried by a unitary onto a subrepresentation of \\(\\pi_\\psi\\).)\n2. \\(p'_\\xi\\precsim p'_\\eta\\) in \\(N'=\\pi(A)'\\).\n3. \\(p_\\xi\\precsim p_\\eta\\) in \\(N\\).\n\nIn the universal representation (Fact 2.7), \\(p_\\xi=s(\\varphi)\\) and \\(p_\\eta=s(\\psi)\\). So \\(\\pi_\\varphi\\) embeds in \\(\\pi_\\psi\\) exactly when \\(s(\\varphi)\\precsim s(\\psi)\\) in \\(\\tilde A\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 950,
        "through_line": 967,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 18.2",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-08::Proposition 18.2",
      "anchor": "oa-fnd-ty-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 18.2** (Which normal functionals are vector functionals). Let \\(\\xi_0\\in H\\) and let \\(\\varphi\\) be a positive normal functional on \\(M\\). Then \\(\\varphi=\\omega_\\xi|_M\\) for some \\(\\xi\\in[M\\xi_0]\\) if and only if \\(s(\\varphi)\\precsim p_{\\xi_0}\\). In particular, if \\(M\\) has a separating vector \\(\\xi_0\\), then \\(p_{\\xi_0}=1\\), and every positive normal functional on \\(M\\) is a vector functional \\(\\omega_\\xi\\) with \\(\\xi\\in[M\\xi_0]\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 968,
        "through_line": 983,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 18.3",
      "kind": "example",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-08::Example 18.3",
      "anchor": "oa-fnd-ty-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Example 18.3** (A normal state that is not a vector state). Let \\(M=M_2(\\mathbb C)\\) on \\(\\mathbb C^2\\) and \\(\\varphi=\\frac12\\mathrm{tr}\\). Its support is \\(1\\). Each vector functional \\(\\omega_\\xi\\) has support \\(p_\\xi\\), the rank-one projection onto \\(\\mathbb C\\xi\\) (here \\(M'=\\mathbb C1\\)). So \\(\\varphi\\) is not a vector functional, in line with Proposition 18.2: \\(s(\\varphi)=1\\not\\precsim p_{\\xi_0}\\) for every \\(\\xi_0\\). Accordingly \\(M\\) has no separating vector. On \\(\\mathbb C^2\\otimes\\mathbb C^2\\), with \\(M\\) acting as \\(M\\otimes1\\), the vector \\((\\delta_1\\otimes\\delta_1+\\delta_2\\otimes\\delta_2)/\\sqrt2\\) is separating and gives \\(\\varphi\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 984,
        "through_line": 985,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 18.4",
      "kind": "lemma",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-08::Lemma 18.4",
      "anchor": "oa-fnd-ty-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Lemma 18.4** (Interpolating two vectors). Let \\(\\xi_1,\\xi_2\\in H\\) with \\(p'_{\\xi_1}\\sim p'_{\\xi_2}\\) in \\(M'\\); by Theorem 17.5 and Proposition 5.1 this is the same as \\(p_{\\xi_1}\\sim p_{\\xi_2}\\) in \\(M\\). Then some \\(\\xi_0\\in H\\) has \\(p'_{\\xi_0}=p'_{\\xi_1}\\) and \\(p_{\\xi_0}=p_{\\xi_2}\\).",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 986,
        "through_line": 989,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 18.5",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-08::Proposition 18.5",
      "anchor": "oa-fnd-ty-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 18.5** (A cyclic and separating vector). If some vector is cyclic for \\(M\\) and some vector is separating for \\(M\\), then a single vector is both cyclic and separating.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 990,
        "through_line": 993,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 18.6",
      "kind": "proposition",
      "unit": "projections-and-types-of-von-neumann-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras#oa-fnd-ty-08::Proposition 18.6",
      "anchor": "oa-fnd-ty-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
      "statement_and_full_conditions": "**Proposition 18.6** (Isomorphisms are spatial). Let \\(\\{M_1,H_1\\}\\) and \\(\\{M_2,H_2\\}\\) be von Neumann algebras with cyclic and separating vectors \\(\\xi_1\\) and \\(\\xi_2\\). Every isomorphism \\(\\pi:M_1\\to M_2\\) is spatial.",
      "proof_locus": {
        "source": "src/projections-and-types-of-von-neumann-algebras.md",
        "line": 994,
        "through_line": 999,
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.1",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-01::Lemma 1.1",
      "anchor": "oa-fnd-pd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 1.1.** Let \\(B\\) be a \\(C^*\\)-algebra, \\(\\varphi\\in B^*\\) and \\(e\\in B\\) a projection. Put \\(f=1-e\\), computed in the unitization if \\(B\\) has no unit, so that \\(xf=x-xe\\in B\\) for \\(x\\in B\\). Then\n\\[\n\\begin{gathered}\n\\|e\\varphi\\|^2+\\|f\\varphi\\|^2\\\\\n\\le\\|\\varphi\\|^2\\\\\n\\text{and}\\\\\n\\|\\varphi e\\|^2+\\|\\varphi f\\|^2\\\\\n\\le\\|\\varphi\\|^2 .\n\\end{gathered}\n\\tag{1.1}\n\\]\nIn particular, if \\(\\|e\\varphi\\|=\\|\\varphi\\|\\) then \\(e\\varphi=\\varphi\\), and if \\(\\|\\varphi e\\|=\\|\\varphi\\|\\) then \\(\\varphi e=\\varphi\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 98,
        "through_line": 124,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 1.2",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-01::Example 1.2",
      "anchor": "oa-fnd-pd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 1.2** (The two extremes). Let \\(B=B(H)\\), \\(\\xi,\\eta\\in H\\) and \\(\\varphi=\\omega_{\\xi,\\eta}\\). Then \\(\\|\\omega_{\\xi,\\eta}\\|=\\|\\xi\\|\\|\\eta\\|\\): the inequality \\(\\le\\) is Cauchy–Schwarz, and for nonzero \\(\\xi,\\eta\\), \\(x=\\theta_{\\eta,\\xi}/(\\|\\xi\\|\\|\\eta\\|)\\) attains it. If either vector is zero, the functional and both sides are zero. Since \\(e\\omega_{\\xi,\\eta}=\\omega_{e\\xi,\\eta}\\),\n\\[\n\\begin{gathered}\n\\|e\\varphi\\|^2+\\|(1-e)\\varphi\\|^2\\\\\n=\\big(\\|e\\xi\\|^2+\\|(1-e)\\xi\\|^2\\big)\\|\\eta\\|^2\\\\\n=\\|\\varphi\\|^2 .\n\\end{gathered}\n\\]\nSo (1.1) is an equality for vector functionals, and it cannot be improved. At the other extreme, for a central projection \\(e\\) of a von Neumann algebra and a normal \\(\\varphi\\), the norm is additive: \\(\\|\\varphi\\|=\\|e\\varphi\\|+\\|(1-e)\\varphi\\|\\) ([Lemma 10.2](the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md#oa-fnd-wa-10)). Every functional lies between the two: \\[\n\\begin{gathered}\n(\\|e\\varphi\\|^2+\\|(1-e)\\varphi\\|^2)^{1/2}\\\\\n\\le\\|\\varphi\\|\\\\\n\\le\\|e\\varphi\\|+\\|(1-e)\\varphi\\|.\n\\end{gathered}\n\\]\n\nA normal functional determines two projections. Let \\(\\varphi\\in M_*\\). The set \\(R_\\varphi=\\{y\\in M:\\varphi(yx)=0\\text{ for all }x\\in M\\}\\) is a \\(\\sigma\\)-weakly closed right ideal, because \\(y\\mapsto\\varphi(yx)\\) is \\(\\sigma\\)-weakly continuous for each \\(x\\). By background fact 19, \\(R_\\varphi=pM\\) for a unique projection \\(p\\). Likewise \\[\n\\begin{gathered}\nL_\\varphi\\\\\n=\\{y\\in M:\\varphi(xy)=0\\text{ for all }x\\in M\\}\\\\\n=Mq\n\\end{gathered}\n\\] for a unique projection \\(q\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 125,
        "through_line": 148,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.3",
      "kind": "definition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-01::Definition 1.3",
      "anchor": "oa-fnd-pd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Definition 1.3.** The *right support* of \\(\\varphi\\in M_*\\) is \\(s_r(\\varphi)=1-p\\) and the *left support* is \\(s_l(\\varphi)=1-q\\), with \\(p,q\\) as above.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 149,
        "through_line": 150,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.4",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-01::Lemma 1.4",
      "anchor": "oa-fnd-pd-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 1.4.** Let \\(\\varphi\\in M_*\\).\n\n1. For a projection \\(e\\), \\(\\varphi=\\varphi e\\) exactly when \\(e\\ge s_r(\\varphi)\\), and \\(\\varphi=e\\varphi\\) exactly when \\(e\\ge s_l(\\varphi)\\). So \\(s_r(\\varphi)\\) is the least projection \\(e\\) with \\(\\varphi(ex)=\\varphi(x)\\) for all \\(x\\), and \\(s_l(\\varphi)\\) is the least projection \\(f\\) with \\(\\varphi(xf)=\\varphi(x)\\) for all \\(x\\).\n2. \\(\\varphi(x)=\\varphi(s_r(\\varphi)\\,x\\,s_l(\\varphi))\\) for all \\(x\\in M\\).\n3. \\(s_l(\\varphi^*)=s_r(\\varphi)\\) and \\(s_r(\\varphi^*)=s_l(\\varphi)\\).\n4. If \\(\\omega\\) is positive, \\(s_l(\\omega)=s_r(\\omega)=s(\\omega)\\). If \\(\\varphi\\) is hermitian, \\(s_l(\\varphi)=s_r(\\varphi)\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 151,
        "through_line": 167,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.1",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Lemma 2.1",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 2.1** (Comparison of positive functionals). Let \\(A\\) be a \\(C^*\\)-algebra and \\(\\omega,\\omega_1\\in A^*_+\\) with \\(\\|\\omega\\|=\\|\\omega_1\\|\\) and\n\\[\n\\begin{gathered}\n|\\omega(x)|^2\\\\\n\\le\\|\\omega_1\\|\\,\\omega_1(x^*x)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{2.1}\n\\]\nThen \\(\\omega=\\omega_1\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 172,
        "through_line": 193,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.2",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Theorem 2.2",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 2.2** (Polar decomposition). Let \\(M\\) be a von Neumann algebra and \\(\\varphi\\in M_*\\). There is exactly one pair \\((v,\\omega)\\) of a partial isometry \\(v\\in M\\) and a positive normal functional \\(\\omega\\) with\n\\[\n\\varphi=v\\omega\\qquad\\text{and}\\qquad v^*v=s(\\omega).\n\\tag{2.2}\n\\]\nFor this pair:\n\n1. \\(\\omega=v^*\\varphi\\), and \\(\\|\\omega\\|=\\|\\varphi\\|=\\varphi(v^*)\\);\n2. \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\,\\omega(xx^*)\\) for all \\(x\\in M\\);\n3. \\(v^*v=s_r(\\varphi)\\) and \\(vv^*=s_l(\\varphi)\\);\n4. \\(\\varphi^*=v^*\\psi\\) with \\(\\psi=v\\omega v^*\\), that is \\(\\psi(x)=\\omega(v^*xv)\\), and \\((v^*,\\psi)\\) is the pair (2.2) for \\(\\varphi^*\\). Moreover \\(\\varphi=\\psi v\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 194,
        "through_line": 276,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.3",
      "kind": "definition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Definition 2.3",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Definition 2.3.** The functional \\(\\omega\\) of Theorem 2.2 is the *absolute value* \\(|\\varphi|\\) of \\(\\varphi\\), and \\(\\varphi=v|\\varphi|\\) is the *polar decomposition* of \\(\\varphi\\). By Theorem 2.2,\n\\[\n\\begin{gathered}\n\\||\\varphi|\\|\\\\\n=\\||\\varphi^*|\\|\\\\\n=\\|\\varphi\\|,\\\\\n|\\varphi|\\\\\n=v^*\\varphi,\\\\\n|\\varphi^*|\\\\\n=v|\\varphi|v^*,\\\\\n\\varphi\\\\\n=v|\\varphi|\\\\\n=|\\varphi^*|v .\n\\end{gathered}\n\\tag{2.4}\n\\]\nAlso \\(v^*|\\varphi^*|v=|\\varphi|\\), since \\(\\omega(v^*vxv^*v)=\\omega(x)\\). If \\(\\varphi\\) is positive, then \\(\\varphi=s(\\varphi)\\varphi\\) is its polar decomposition, so \\(|\\varphi|=\\varphi\\) and \\(v=s(\\varphi)\\).\n\nThree examples show what the absolute value is in familiar cases.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 277,
        "through_line": 296,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.4",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Example 2.4",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 2.4** (Matrices). Let \\(M=M_n(\\mathbb C)\\). Every functional is \\(\\varphi(x)=\\operatorname{Tr}(\\rho x)\\) for a unique matrix \\(\\rho\\). Let \\(\\rho=w|\\rho|\\) be the polar decomposition of the matrix \\(\\rho\\). Then \\(|\\varphi|=\\operatorname{Tr}(|\\rho|\\,\\cdot\\,)\\) and \\(v=w\\). Indeed, \\((w|\\varphi|)(x)=\\operatorname{Tr}(|\\rho|xw)=\\operatorname{Tr}(w|\\rho|x)=\\varphi(x)\\). And the support of \\(\\operatorname{Tr}(\\sigma\\,\\cdot\\,)\\), for \\(\\sigma\\ge0\\), is the range projection of \\(\\sigma\\): \\(\\operatorname{Tr}(\\sigma(1-p))=\\operatorname{Tr}((1-p)\\sigma(1-p))\\) vanishes exactly when \\(\\sigma^{1/2}(1-p)=0\\), that is when \\(p\\) majorizes the range projection of \\(\\sigma\\). For \\(\\sigma=|\\rho|\\) this range projection is \\(w^*w\\). So (2.2) holds. By Theorem 2.2(3), \\(s_l(\\varphi)=ww^*\\) is the range projection of \\(\\rho\\), and \\(s_r(\\varphi)=w^*w\\) is the projection onto \\((\\ker\\rho)^\\perp\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 297,
        "through_line": 298,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.5",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Example 2.5",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 2.5** (Vector functionals). Let \\(M=B(H)\\) and \\(\\xi,\\eta\\ne0\\). Then\n\\[\n\\begin{gathered}\n|\\omega_{\\xi,\\eta}|\\\\\n=\\frac{\\|\\xi\\|}{\\|\\eta\\|}\\,\\omega_\\eta,\\\\\nv\\\\\n=\\frac{\\theta_{\\xi,\\eta}}{\\|\\xi\\|\\|\\eta\\|} .\n\\end{gathered}\n\\tag{2.5}\n\\]\nIndeed, \\(v^*v=\\theta_{\\eta,\\eta}/\\|\\eta\\|^2\\) is the projection onto \\(\\mathbb C\\eta\\), which is the support of \\(\\omega_\\eta\\); and \\(v\\eta=(\\|\\eta\\|/\\|\\xi\\|)\\xi\\), so \\[\n\\begin{gathered}\n(\\|\\xi\\|/\\|\\eta\\|)\\,\\omega_\\eta(xv)\\\\\n=(\\|\\xi\\|/\\|\\eta\\|)\\langle xv\\eta,\\eta\\rangle\\\\\n=\\langle x\\xi,\\eta\\rangle.\n\\end{gathered}\n\\] So the absolute value of \\(\\omega_{\\xi,\\eta}\\) lives on \\(\\eta\\), and that of \\(\\omega_{\\xi,\\eta}^*=\\omega_{\\eta,\\xi}\\) lives on \\(\\xi\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 299,
        "through_line": 316,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 2.6",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Example 2.6",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 2.6** (Commutative algebras). Let \\((\\Gamma,\\mu)\\) be a \\(\\sigma\\)-finite measure space and \\(M=L^\\infty(\\Gamma,\\mu)\\) acting on \\(L^2(\\Gamma,\\mu)\\) (background fact 28). The normal functionals are exactly \\(\\varphi_h(x)=\\int xh\\,d\\mu\\) with \\(h\\in L^1(\\Gamma,\\mu)\\), and \\(\\|\\varphi_h\\|=\\|h\\|_1\\). Indeed, \\(\\varphi_h=\\omega_{f,g}\\) for \\(f=|h|^{1/2}\\operatorname{sgn}h\\) and \\(g=|h|^{1/2}\\), so \\(\\varphi_h\\) is normal. Conversely, a \\(\\sigma\\)-weakly continuous functional on \\(M\\) extends to a \\(\\sigma\\)-weakly continuous functional on \\(B(L^2)\\) (background fact 3, since the \\(\\sigma\\)-weak topology of \\(M\\) is the relative one, and the \\(\\sigma\\)-weak topology is locally convex). That extension is \\(\\sum_n\\omega_{\\xi_n,\\eta_n}\\) with \\(\\sum_n\\|\\xi_n\\|\\|\\eta_n\\|<\\infty\\) (background fact 27), so on \\(M\\) it is \\(\\varphi_h\\) with \\(h=\\sum_n\\xi_n\\bar\\eta_n\\in L^1\\). Finally \\(\\|\\varphi_h\\|\\le\\|h\\|_1\\), and \\(x=\\overline{\\operatorname{sgn}h}\\) gives equality. Now put \\(v=\\operatorname{sgn}h\\), with \\(\\operatorname{sgn}h=h/|h|\\) where \\(h\\ne0\\) and \\(0\\) elsewhere. Then \\(v\\varphi_{|h|}=\\varphi_h\\), and \\(v^*v=1_{\\{h\\ne0\\}}\\) is the support of \\(\\varphi_{|h|}\\). So\n\\[\n|\\varphi_h|=\\varphi_{|h|} ,\n\\]\nthe familiar total variation. The same argument works for counting measure on any set \\(\\Gamma\\), even when \\(\\Gamma\\) is uncountable: \\(\\ell^\\infty(\\Gamma)\\), acting diagonally on \\(\\ell^2(\\Gamma)\\), is a von Neumann algebra, because an operator that commutes with every coordinate projection is diagonal, so \\(\\ell^\\infty(\\Gamma)'=\\ell^\\infty(\\Gamma)\\). Thus \\(\\ell^\\infty(\\Gamma)_*=\\ell^1(\\Gamma)\\), and the absolute value is taken coordinatewise.\n\nFor a \\(C^*\\)-algebra the partial isometry lives in the bidual.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 317,
        "through_line": 324,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
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          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
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                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
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          ],
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          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.7",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Theorem 2.7",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 2.7** (Polar decomposition on a \\(C^*\\)-algebra). Let \\(A\\) be a \\(C^*\\)-algebra and \\(f\\in A^*\\). There is exactly one pair of a partial isometry \\(v\\in\\tilde A\\) and a positive \\(\\omega\\in A^*\\) with \\(f=v\\omega\\) and \\(v^*v=s(\\omega)\\), the support of \\(\\omega\\) in \\(\\tilde A\\). We write \\(|f|=\\omega\\). Then \\(\\||f|\\|=\\|f\\|\\) and \\(|f(x)|^2\\le\\|f\\|\\,|f|(xx^*)\\) for all \\(x\\in\\tilde A\\), in particular for all \\(x\\in A\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 325,
        "through_line": 332,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
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          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
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          ],
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          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 2.8",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-02::Corollary 2.8",
      "anchor": "oa-fnd-pd-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 2.8** (Hermitian functionals). Let \\(\\varphi\\in M_*\\) be hermitian, with polar decomposition \\(\\varphi=v|\\varphi|\\), and put \\(s=s(|\\varphi|)\\).\n\n1. \\(v=v^*\\), \\(|\\varphi^*|=|\\varphi|\\), and \\(v|\\varphi|=|\\varphi|v\\).\n2. \\(e=\\tfrac12(s+v)\\) and \\(f=\\tfrac12(s-v)\\) are orthogonal projections with \\(e+f=s\\) and \\(e-f=v\\). The functionals \\(\\varphi_+=e|\\varphi|=\\tfrac12(|\\varphi|+\\varphi)\\) and \\(\\varphi_-=f|\\varphi|=\\tfrac12(|\\varphi|-\\varphi)\\) are positive and normal, \\(s(\\varphi_+)=e\\), \\(s(\\varphi_-)=f\\), \\(\\varphi=\\varphi_+-\\varphi_-\\) and \\(\\|\\varphi\\|=\\|\\varphi_+\\|+\\|\\varphi_-\\|\\).\n3. If \\(\\varphi=\\psi_1-\\psi_2\\) with \\(\\psi_1,\\psi_2\\in M_*^+\\) and \\(\\|\\varphi\\|=\\|\\psi_1\\|+\\|\\psi_2\\|\\), then \\(\\psi_1=\\varphi_+\\) and \\(\\psi_2=\\varphi_-\\).\n4. For \\(\\psi_1,\\psi_2\\in M_*^+\\): \\(\\|\\psi_1-\\psi_2\\|=\\|\\psi_1\\|+\\|\\psi_2\\|\\) exactly when \\(s(\\psi_1)s(\\psi_2)=0\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 333,
        "through_line": 369,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
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              "heading": "2. The polar decomposition",
              "line": 168,
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                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
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              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
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          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.1",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-04::Theorem 3.1",
      "anchor": "oa-fnd-pd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 3.1** (Characterization of the absolute value). Let \\(A\\) be a \\(C^*\\)-algebra and \\(f\\in A^*\\). Then \\(|f|\\) is the only positive functional \\(\\omega\\) on \\(A\\) with \\(\\|\\omega\\|\\le\\|f\\|\\) and\n\\[\n\\begin{gathered}\n|f(x)|^2\\\\\n\\le\\|f\\|\\,\\omega(xx^*)\\\\\n(x\\in A).\n\\end{gathered}\n\\tag{3.1}\n\\]\nIf \\(A=M\\) is a von Neumann algebra and \\(f\\in M_*\\), then the \\(|f|\\) of Theorem 2.2 is the only positive functional on \\(M\\), normal or not, with these two properties. In both cases such an \\(\\omega\\) has \\(\\|\\omega\\|=\\|f\\|\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 374,
        "through_line": 406,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
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          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
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              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
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          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 3.2",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-04::Corollary 3.2",
      "anchor": "oa-fnd-pd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 3.2.** Let \\(M\\) be a von Neumann algebra and \\(\\varphi\\in M_*\\). The absolute value of \\(\\varphi\\) as a functional on the \\(C^*\\)-algebra \\(M\\) (Theorem 2.7, through \\(M^{**}\\)) is the absolute value of Theorem 2.2. In particular it is normal.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 407,
        "through_line": 410,
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              "heading": "2. The polar decomposition",
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                "OA-FND-PD-02",
                "OA-FND-PD-03"
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            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
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              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
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          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.3",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-04::Example 3.3",
      "anchor": "oa-fnd-pd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 3.3** (Where the partial isometry lives). Let \\(A=C[0,1]\\) and \\(\\lambda(x)=\\int_0^1x\\,dt\\).\n\n1. For \\(s\\ne t\\) in \\([0,1]\\), \\(|\\delta_s-\\delta_t|=\\delta_s+\\delta_t\\). Indeed, \\(\\omega=\\delta_s+\\delta_t\\) is positive with \\(\\|\\omega\\|=2=\\|\\delta_s-\\delta_t\\|\\) (test on a continuous \\(x\\) with \\(|x|\\le1\\), \\(x(s)=1\\), \\(x(t)=-1\\)), and \\[\n\\begin{gathered}\n|x(s)-x(t)|^2\\\\\n\\le2(|x(s)|^2+|x(t)|^2)\\\\\n=2\\,\\omega(xx^*).\n\\end{gathered}\n\\] Theorem 3.1 applies.\n2. Let \\(\\sigma=1_{[0,1/2)}-1_{[1/2,1]}\\) and \\(f(x)=\\int_0^1x\\sigma\\,dt\\). Then \\(\\|f\\|=1\\): the bound \\(\\le\\) is clear, and continuous functions \\(x_n\\) with \\(|x_n|\\le1\\) that equal \\(\\sigma\\) outside an interval of length \\(1/n\\) give \\(f(x_n)\\ge1-2/n\\). Also \\(\\|\\lambda\\|=1\\) and \\(|f(x)|^2\\le\\big(\\int|x|\\big)^2\\le\\int|x|^2=\\lambda(xx^*)\\). By Theorem 3.1, \\(|f|=\\lambda\\). Now suppose the partial isometry \\(v\\) of \\(f\\) lay in \\(A\\), say \\(v=a\\). Then \\(f(x)=\\lambda(xa)=\\int_0^1xa\\,dt\\) for all \\(x\\in A\\). Take \\(x\\ge0\\) continuous, supported in \\([\\tfrac12-\\delta,\\tfrac12]\\), with \\(\\int x=1\\). Then \\(1=f(x)=\\int xa\\), and as \\(\\delta\\to0\\) this tends to \\(a(\\tfrac12)\\), by continuity of \\(a\\). So \\(a(\\tfrac12)=1\\). The same test with support in \\([\\tfrac12,\\tfrac12+\\delta]\\) gives \\(a(\\tfrac12)=-1\\), a contradiction. So \\(v\\notin A\\).\n3. Even for positive functionals, \\(v=s(f)\\) need not lie in \\(A\\). Let \\(A=K(H)\\), whose bidual is \\(B(H)\\) (background fact 14), let \\((e_n)\\) be an infinite orthonormal sequence, and let \\(f=\\sum_n2^{-n}\\omega_{e_n}\\). Its normal extension to \\(B(H)\\) is given by the same formula. Its support is the projection \\(P\\) onto the closed span of the \\(e_n\\): \\(f(1-P)=0\\), and a projection \\(q\\le P\\) with \\(f(q)=0\\) has \\(qe_n=0\\) for all \\(n\\), so \\(q=0\\). So \\(v=P\\) has infinite rank and is not compact.\n\nThe next result replaces the triangle inequality, which fails for absolute values (Example 3.5).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 411,
        "through_line": 424,
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              "heading": "2. The polar decomposition",
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                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
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              "heading": "3. How the absolute value is determined",
              "line": 370,
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                "OA-FND-PD-05"
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          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
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    {
      "local_label": "Proposition 3.4",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-04::Proposition 3.4",
      "anchor": "oa-fnd-pd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 3.4** (A weak triangle inequality). Let \\(A\\) be a \\(C^*\\)-algebra and \\((f_k)\\) a finite or infinite sequence in \\(A^*\\) with \\(\\sum_k\\|f_k\\|<\\infty\\). Put \\(f=\\sum_kf_k\\). Then for every \\(x\\in\\tilde A\\),\n\\[\n\\begin{gathered}\n||f|(x)|^2\\\\\n\\le\\Big(\\sum_k\\|f_k\\|\\Big)\\Big(\\sum_k|f_k|(xx^*)\\Big).\n\\end{gathered}\n\\tag{3.2}\n\\]",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 425,
        "through_line": 445,
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              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.5",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-04::Example 3.5",
      "anchor": "oa-fnd-pd-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 3.5** (No triangle inequality). In \\(M_2(\\mathbb C)\\) let \\(\\varphi(x)=\\operatorname{Tr}(E_{11}x)\\) and \\(\\psi(x)=\\operatorname{Tr}(E_{12}x)\\), with matrix units \\(E_{ij}\\). By Example 2.4, \\(|\\varphi|=\\operatorname{Tr}(E_{11}\\,\\cdot\\,)\\) and \\(|\\psi|=\\operatorname{Tr}(E_{22}\\,\\cdot\\,)\\), because \\((E_{12}^*E_{12})^{1/2}=E_{22}\\). For \\(\\rho=E_{11}+E_{12}\\) we have \\(\\rho^*\\rho=\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\), so \\(|\\rho|=2^{-1/2}\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\) and \\(|\\varphi+\\psi|=\\operatorname{Tr}(|\\rho|\\,\\cdot\\,)\\). Let \\(P\\) be the projection onto \\(2^{-1/2}(1,1)\\). Then \\(|\\varphi+\\psi|(P)=\\sqrt2\\), while \\((|\\varphi|+|\\psi|)(P)=\\operatorname{Tr}(P)=1\\). So \\(|\\varphi+\\psi|\\le|\\varphi|+|\\psi|\\) fails. Inequality (3.2) holds here with equality at \\(x=P\\): \\(2\\le(1+1)\\cdot1\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 446,
        "through_line": 447,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "2. The polar decomposition",
              "line": 168,
              "through_line": 369,
              "anchors": [
                "OA-FND-PD-02",
                "OA-FND-PD-03"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "3. How the absolute value is determined",
              "line": 370,
              "through_line": 447,
              "anchors": [
                "OA-FND-PD-04",
                "OA-FND-PD-05"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
          "source_locus": "§3.11; PDF 78–81",
          "role": "proof comparison",
          "correspondence": "Polar and Jordan mechanisms",
          "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.1",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-06::Proposition 4.1",
      "anchor": "oa-fnd-pd-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 4.1.** Let \\(B\\) be a \\(C^*\\)-algebra, \\(\\varphi\\in B^*_+\\) and \\(a\\in B\\), and suppose that \\(a\\varphi\\) is hermitian, that is \\(\\varphi(ya)=\\varphi(a^*y)\\) for all \\(y\\in B\\). Then\n\\[\n\\begin{gathered}\n|\\varphi(ha)|\\\\\n\\le r(a)\\,\\varphi(h)\\\\\n(h\\in B_+),\n\\end{gathered}\n\\tag{4.1}\n\\]\nthat is, \\(-r(a)\\varphi\\le a\\varphi\\le r(a)\\varphi\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 452,
        "through_line": 492,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.2",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-06::Corollary 4.2",
      "anchor": "oa-fnd-pd-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 4.2.** Let \\(M\\) be a von Neumann algebra, \\(\\varphi\\in M_*^+\\) and \\(a\\in M\\). Then\n\\[\n|a\\varphi|\\le\\|as(\\varphi)\\|\\,\\varphi\\le\\|a\\|\\,\\varphi .\n\\tag{4.2}\n\\]\nThe same holds for \\(\\varphi\\in A^*_+\\) and \\(a\\in\\tilde A\\), for any \\(C^*\\)-algebra \\(A\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 493,
        "through_line": 508,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.3",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-06::Example 4.3",
      "anchor": "oa-fnd-pd-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 4.3.** Let \\(M=B(H)\\), \\(\\xi\\) a unit vector and \\(\\varphi=\\omega_\\xi\\).\n\n1. Left multiplication: \\(a\\omega_\\xi=\\omega_{a\\xi,\\xi}\\), so \\(|a\\omega_\\xi|=\\|a\\xi\\|\\,\\omega_\\xi\\) by (2.5). Since \\(s(\\omega_\\xi)=\\theta_{\\xi,\\xi}\\), \\(\\|as(\\omega_\\xi)\\|=\\|a\\xi\\|\\): the first inequality in (4.2) is an equality.\n2. Right multiplication has no such bound. \\(\\omega_\\xi a=\\omega_{\\xi,a^*\\xi}\\), so \\(|\\omega_\\xi a|=\\|a^*\\xi\\|^{-1}\\omega_{a^*\\xi}\\) when \\(a^*\\xi\\ne0\\). If \\(a^*\\xi\\notin\\mathbb C\\xi\\), the support of \\(|\\omega_\\xi a|\\) is not under \\(s(\\omega_\\xi)\\), so \\(|\\omega_\\xi a|\\le C\\omega_\\xi\\) fails for every \\(C\\).\n3. The spectral radius matters. In \\(M_2(\\mathbb C)\\) let \\(a=E_{12}\\), which has \\(r(a)=0\\), and \\(\\varphi=\\operatorname{Tr}(\\rho\\,\\cdot\\,)\\) with \\(\\rho=\\begin{pmatrix}p&q\\\\\\bar q&t\\end{pmatrix}\\ge0\\). Since \\((a\\varphi)(x)=\\operatorname{Tr}(a\\rho x)\\), \\(a\\varphi\\) is hermitian exactly when \\(a\\rho=\\rho a^*\\), that is \\(\\begin{pmatrix}\\bar q&t\\\\0&0\\end{pmatrix}=\\begin{pmatrix}q&0\\\\t&0\\end{pmatrix}\\). This forces \\(t=0\\), hence \\(q=0\\) by positivity, and then \\(a\\rho=0\\). So \\(a\\varphi=0\\), as (4.1) predicts.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 509,
        "through_line": 514,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.1",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-07::Theorem 5.1",
      "anchor": "oa-fnd-pd-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 5.1.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(\\varphi,\\psi\\in A^*\\) (for instance \\(A=M\\) and \\(\\varphi,\\psi\\in M_*\\)). Then\n\\[\n\\begin{gathered}\n\\||\\varphi|-|\\psi|\\|\\\\\n\\le\\|\\varphi-\\psi\\|+2\\big(\\|\\varphi\\|\\,\\|\\varphi-\\psi\\|\\big)^{1/2}.\n\\end{gathered}\n\\tag{5.1}\n\\]\nThe same bound holds for \\(\\||\\varphi^*|-|\\psi^*|\\|\\). So \\(\\varphi\\mapsto|\\varphi|\\) is norm continuous, uniformly on bounded sets.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 517,
        "through_line": 558,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.3",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-07::Theorem 5.3",
      "anchor": "oa-fnd-pd-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 5.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\((f_i)\\) a net in \\(A^*\\) with \\(f_i\\to f\\) in \\(\\sigma(A^*,A)\\) and \\(\\|f_i\\|\\to\\|f\\|\\). Then \\(|f_i|\\to|f|\\) in \\(\\sigma(A^*,A)\\). Likewise, if \\((\\varphi_i)\\) is a net in the predual of a von Neumann algebra \\(M\\) with \\(\\varphi_i\\to\\varphi\\) weakly and \\(\\|\\varphi_i\\|\\to\\|\\varphi\\|\\), then \\(|\\varphi_i|\\to|\\varphi|\\) weakly.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 559,
        "through_line": 564,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.4",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-07::Example 5.4",
      "anchor": "oa-fnd-pd-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 5.4.** On \\(A=C[0,1]\\), \\(f_n=\\delta_{1/n}-\\delta_0\\) tends to \\(0\\) weak\\(^*\\). By Example 3.3(1), \\(|f_n|=\\delta_{1/n}+\\delta_0\\), which tends to \\(2\\delta_0\\ne|0|\\). Here \\(\\|f_n\\|=2\\) does not tend to \\(\\|0\\|\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 565,
        "through_line": 566,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.1",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-09::Theorem 6.1",
      "anchor": "oa-fnd-pd-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 6.1.** Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(V\\subseteq M_*\\) be a norm-closed subspace with \\(MV\\subseteq V\\), and \\(V=M_*e\\) as in background fact 20. Then \\[\n\\begin{gathered}\nV_+\\\\\n=V\\cap M_*^+\\\\\n=\\{\\omega\\in M_*^+:\\omega(1-e)=0\\}\n\\end{gathered}\n\\] is a norm-closed hereditary cone, and \\(V=MV_+=\\{a\\omega:a\\in M,\\ \\omega\\in V_+\\}\\).\n2. Let \\(C\\subseteq M_*^+\\) be a norm-closed hereditary cone. Then \\(V=MC\\) is a norm-closed subspace with \\(MV\\subseteq V\\) and \\(V\\cap M_*^+=C\\).\n3. Hence \\(V\\mapsto V\\cap M_*^+\\) is a bijection from the norm-closed left invariant subspaces of \\(M_*\\) onto the norm-closed hereditary cones in \\(M_*^+\\), with inverse \\(C\\mapsto MC\\). Both correspond to projections \\(e\\): \\(V=M_*e\\) and \\(C=\\{\\omega\\ge0:\\omega(1-e)=0\\}\\).\n\nThe right-handed statements (subspaces with \\(VM\\subseteq V\\), and \\(CM\\)) follow by applying \\(\\varphi\\mapsto\\varphi^*\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 571,
        "through_line": 598,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.2",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-09::Example 6.2",
      "anchor": "oa-fnd-pd-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 6.2** (Convexity is needed). In \\(M=\\mathbb C^2\\), \\(M_*^+\\) is the quadrant \\([0,\\infty)^2\\). The union of the two axes is a closed cone, and it is hereditary, but it is not convex. It is not the positive part of any subspace, because such a positive part is convex.\n\nFor a \\(C^*\\)-algebra \\(A\\), a subset \\(V\\subseteq A^*\\) is *left invariant* if \\(aV\\subseteq V\\) for all \\(a\\in A\\), where \\((af)(x)=f(xa)\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 599,
        "through_line": 602,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 6.3",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-09::Corollary 6.3",
      "anchor": "oa-fnd-pd-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 6.3.** Let \\(A\\) be a \\(C^*\\)-algebra and \\(V\\subseteq A^*\\) a norm-closed left invariant subspace, and put \\(V_+=V\\cap A^*_+\\). Then \\(V\\) is invariant under \\(\\tilde A\\), \\(V_+\\) is a norm-closed hereditary cone, \\(V=\\tilde AV_+\\), and \\(V\\) is the norm closure of the set \\(AV_+=\\{af:a\\in A,\\ f\\in V_+\\}\\). In particular a nonzero \\(V\\) contains a nonzero positive functional, and \\(V\\) is determined by \\(V_+\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 603,
        "through_line": 608,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.4",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-09::Lemma 6.4",
      "anchor": "oa-fnd-pd-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 6.4** (Krein–Šmulian, subspace case). Let \\(X\\) be a Banach space and \\(V\\subseteq X^*\\) a linear subspace whose intersection with the closed unit ball \\(B\\) of \\(X^*\\) is weak\\(^*\\) closed. Then \\(V\\) is weak\\(^*\\) closed.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 609,
        "through_line": 633,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.5",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-09::Theorem 6.5",
      "anchor": "oa-fnd-pd-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 6.5.** Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. If \\(V\\subseteq A^*\\) is a weak\\(^*\\) closed left invariant subspace, then \\(V\\cap A^*_+\\) is a weak\\(^*\\) closed hereditary cone.\n2. If \\(C\\subseteq A^*_+\\) is a weak\\(^*\\) closed hereditary cone, then \\(\\tilde AC\\) is a weak\\(^*\\) closed left invariant subspace with \\(\\tilde AC\\cap A^*_+=C\\).\n3. The map \\(\\mathfrak r\\mapsto\\mathfrak r^\\perp\\cap A^*_+\\), where \\(\\mathfrak r^\\perp=\\{f\\in A^*:f(\\mathfrak r)=0\\}\\), is a bijection from the closed right ideals of \\(A\\) onto the weak\\(^*\\) closed hereditary cones in \\(A^*_+\\). Its inverse is \\(C\\mapsto\\{x\\in A:\\omega(xx^*)=0\\ \\forall\\omega\\in C\\}\\), and \\(\\mathfrak r^\\perp=\\tilde A(\\mathfrak r^\\perp\\cap A^*_+)\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 634,
        "through_line": 656,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 6.6",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-09::Corollary 6.6",
      "anchor": "oa-fnd-pd-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 6.6** (Left ideals and pure states). Let \\(A\\) be a \\(C^*\\)-algebra.\n\n1. Every closed left ideal \\(\\mathfrak m\\) is the intersection of the left kernels \\(N_\\omega\\) of the pure states \\(\\omega\\) with \\(\\mathfrak m\\subseteq N_\\omega\\). (For \\(\\mathfrak m=A\\) the family is empty and the intersection is \\(A\\).)\n2. For a pure state \\(\\omega\\), \\(N_\\omega\\) is a maximal closed left ideal. Every proper closed left ideal is contained in some \\(N_\\omega\\), and every maximal closed left ideal is some \\(N_\\omega\\).\n\nThe same holds for closed right ideals with right kernels \\(\\{x:\\omega(xx^*)=0\\}\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 657,
        "through_line": 672,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.1",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-12::Lemma 7.1",
      "anchor": "oa-fnd-pd-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 7.1** (Phillips's lemma). Let \\((\\mu_n)\\) be a bounded sequence in \\(\\ell^\\infty(\\Gamma)^*\\) with \\(\\mu_n(E)\\to0\\) for every \\(E\\subseteq\\Gamma\\). Then\n\\[\n\\sum_{\\gamma\\in\\Gamma}|\\mu_n(\\{\\gamma\\})|\\to0 .\n\\]",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 677,
        "through_line": 698,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 7.2",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-12::Corollary 7.2",
      "anchor": "oa-fnd-pd-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 7.2** (Schur's theorem). In \\(\\ell^1(\\Gamma)\\) every weakly convergent sequence converges in norm.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 699,
        "through_line": 705,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Lemma 8.1",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 8.1.** For \\(\\varphi\\in M^*\\) put \\(\\varphi^\\sharp=|\\varphi|+|\\varphi^*|\\), the absolute values being taken in the \\(C^*\\)-algebra \\(M\\) (Theorem 2.7).\n\n1. \\(\\varphi^\\sharp\\ge0\\), \\(\\|\\varphi^\\sharp\\|=2\\|\\varphi\\|\\), and \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(xx^*)\\), \\(|\\varphi(x)|^2\\le\\|\\varphi\\|\\varphi^\\sharp(x^*x)\\).\n2. If \\(\\varphi\\) is normal, so is \\(\\varphi^\\sharp\\), and \\(\\varphi(x)=\\varphi(gxg)\\) with \\(g=s(\\varphi^\\sharp)\\). If \\(\\varphi\\) is singular, so is \\(\\varphi^\\sharp\\).\n3. If \\(p\\) is a projection with \\(\\varphi^\\sharp(p)=0\\), then \\(\\varphi(p)=0\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 710,
        "through_line": 724,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.2",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Theorem 8.2",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 8.2.** Let \\((\\varphi_k)\\) be a sequence in \\(M^*\\) that converges to \\(\\varphi\\) in \\(\\sigma(M^*,M)\\). Then \\(\\varphi_k^{\\rm n}\\to\\varphi^{\\rm n}\\) and \\(\\varphi_k^{\\rm s}\\to\\varphi^{\\rm s}\\) in \\(\\sigma(M^*,M)\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 725,
        "through_line": 736,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 8.3",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Corollary 8.3",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 8.3** (Weak sequential completeness).\n\n1. The predual \\(M_*\\) of a von Neumann algebra is weakly sequentially complete: every weakly Cauchy sequence in \\(M_*\\) converges weakly to an element of \\(M_*\\).\n2. The dual \\(A^*\\) of a \\(C^*\\)-algebra is weakly sequentially complete.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 737,
        "through_line": 743,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.4",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Example 8.4",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 8.4** (Sequences cannot be replaced by nets). Let \\(M\\) be infinite-dimensional. It has a non-normal state (background fact 23), whose singular part, normalized, is a singular state \\(\\chi\\). The normal states are weak\\(^*\\) dense in the state space. Indeed, otherwise the separation theorem in \\((M^*,\\sigma(M^*,M))\\), whose dual is \\(M\\), gives a self-adjoint \\(k\\in M\\) and a state \\(\\chi'\\) with \\(\\chi'(k)>\\sup\\{\\omega(k):\\omega\\text{ a normal state}\\}\\). The right side is at least \\(\\sup_{\\|\\zeta\\|=1}\\langle k\\zeta,\\zeta\\rangle=\\max\\sigma(k)\\), computed in a faithful normal representation; and \\(\\chi'(k)\\le\\max\\sigma(k)\\) because \\(k\\le\\max\\sigma(k)1\\). This is a contradiction. So there is a net of normal states \\(\\omega_i\\to\\chi\\). Their normal parts are \\(\\omega_i\\), which tend to \\(\\chi\\), not to \\(\\chi^{\\rm n}=0\\). In particular the net \\((\\omega_i)\\) is weakly Cauchy in \\(M_*\\) without a weak limit in \\(M_*\\).\n\nThe singular functionals form a norm-closed subspace. Its weak\\(^*\\) closure can be large, but countable subsets cannot escape from it.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 744,
        "through_line": 747,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.5",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Proposition 8.5",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 8.5.** The \\(\\sigma(M^*,M)\\)-closure of any countable set of singular functionals consists of singular functionals.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 748,
        "through_line": 751,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.6",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Example 8.6",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 8.6** (Countability is needed). Let \\(M=L^\\infty[0,1]\\) with Lebesgue measure. We show that the singular functionals are weak\\(^*\\) dense in \\(M^*\\). By background fact 7 it suffices to show that no \\(x\\ne0\\) in \\(M\\) is annihilated by all singular functionals.\n\nFirst, for every measurable \\(E\\) of positive measure there is a singular state \\(\\psi\\) of \\(M\\) with \\(\\psi(1_E)=1\\). The algebra \\(N=1_EM\\cong L^\\infty(E)\\) is infinite-dimensional: \\(t\\mapsto\\mu(E\\cap[0,t])\\) is continuous because its increments have absolute value at most the length of the interval. Bisect its positive total mass repeatedly by the intermediate value property, obtaining infinitely many disjoint subsets of positive measure and hence independent indicator functions; so it has a singular state \\(\\psi_1\\) (as in Example 8.4). Put \\(\\psi(y)=\\psi_1(y1_E)\\). This is a state of \\(M\\) with \\(\\psi(1_E)=1\\). It is singular by background fact 23: if \\(q\\ne0\\) is a projection of \\(M\\) and \\(q1_E=0\\), then \\(\\psi(q)=0\\); otherwise \\(q1_E\\) is a nonzero projection of \\(N\\), which majorizes a nonzero projection \\(q_0\\) of \\(N\\) with \\(\\psi_1(q_0)=0\\), and \\(q_0\\le q\\), \\(\\psi(q_0)=0\\).\n\nNow let \\(x\\ne0\\). Choose \\(\\varepsilon>0\\) such that \\(\\{|x|\\ge\\varepsilon\\}\\) has positive measure. Cover the bounded annulus \\(\\{\\varepsilon\\le|z|\\le\\|x\\|_\\infty\\}\\) by finitely many disks of radius \\(\\varepsilon/4\\), with centres \\(c\\) in the annulus. At least one preimage of a disk meets that set in positive measure. For its nonzero centre \\(c\\), \\(E=\\{t:|x(t)-c|\\le|c|/2\\}\\) therefore has positive measure. With \\(\\psi\\) as above, \\(\\psi(x)=\\psi(x1_E)\\) (Cauchy–Schwarz, as \\(\\psi(1-1_E)=0\\)), and \\(|\\psi(x1_E)-c|=|\\psi((x-c)1_E)|\\le|c|/2\\). So \\(\\psi(x)\\ne0\\). Hence Lebesgue measure, a normal state, is a weak\\(^*\\) limit of a net of singular functionals, and Proposition 8.5 fails for uncountable sets.\n\nIn atomic algebras, such as \\(\\ell^\\infty\\) and \\(B(H)\\), the singular functionals do form a weak\\(^*\\) closed subspace.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 752,
        "through_line": 759,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.7",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-13::Proposition 8.7",
      "anchor": "oa-fnd-pd-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 8.7.** Let \\(M\\) be atomic, that is, every nonzero projection majorizes a minimal projection, and let \\(I_0\\) be the norm-closed two-sided ideal generated by the minimal projections. Then \\(\\varphi\\in M^*\\) is singular exactly when \\(\\varphi(I_0)=0\\). So the singular functionals form a weak\\(^*\\) closed subspace, and Proposition 8.5 holds for every set of singular functionals. For \\(M=\\ell^\\infty\\), \\(I_0=c_0\\); for \\(M=B(H)\\), \\(I_0\\) is the algebra of compact operators.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 760,
        "through_line": 767,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.1",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-15::Proposition 9.1",
      "anchor": "oa-fnd-pd-15",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 9.1.** Let \\(M\\) be a von Neumann algebra.\n\n1. Let \\(\\omega\\) be a faithful positive normal functional. Then\n\\[\n\\begin{gathered}\nd(x,y)\\\\\n=\\omega\\big((x-y)^*(x-y)\\big)^{1/2},\\\\\nd^\\#(x,y)\\\\\n=\\omega\\big((x-y)^*(x-y)\\\\\n+(x-y)(x-y)^*\\big)^{1/2}\n\\end{gathered}\n\\]\nare metrics on \\(M_1\\) that define its \\(\\sigma\\)-strong and \\(\\sigma\\)-strong\\(^*\\) topologies, and \\(M_1\\) is complete for both.\n2. Conversely, if the \\(\\sigma\\)-strong topology, or the \\(\\sigma\\)-strong\\(^*\\) topology, on \\(M_1\\) is metrizable, then \\(M\\) has a faithful positive normal functional.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 772,
        "through_line": 800,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.1",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Lemma 10.1",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 10.1.** A bounded subset \\(K\\) of a Banach space \\(X\\) is relatively weakly compact exactly when its \\(\\sigma(X^{**},X^*)\\)-closure in \\(X^{**}\\) lies in \\(X\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 807,
        "through_line": 812,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.2",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Theorem 10.2",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 10.2** (Weak compactness in \\(M_*\\)). For a subset \\(K\\) of the predual of a von Neumann algebra \\(M\\), the following are equivalent.\n\n1. \\(K\\) is relatively weakly compact.\n2. For every abelian von Neumann subalgebra \\(\\mathcal A\\subseteq M\\), the set \\(K|_{\\mathcal A}=\\{\\varphi|_{\\mathcal A}:\\varphi\\in K\\}\\) is relatively weakly compact in \\(\\mathcal A_*\\).\n3. \\(K\\) is bounded, and \\(\\varphi(p_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) whenever the projections \\(p_n\\) decrease to \\(0\\).\n4. \\(K\\) is bounded, and there is \\(\\omega\\in M_*^+\\) with this property: for every \\(\\varepsilon>0\\) there is \\(\\delta>0\\) such that \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\) whenever \\(a\\in M_1\\) and \\(\\omega(a^*a+aa^*)<\\delta\\).\n5. \\(K\\) is bounded, and for every increasing net of projections \\((p_i)\\), \\(\\varphi(p_i)\\) converges uniformly in \\(\\varphi\\in K\\).\n6. \\(K\\) is bounded, and \\(\\|(1-p_i)\\varphi(1-p_i)\\|\\to0\\) uniformly in \\(\\varphi\\in K\\) for every increasing net of projections \\((p_i)\\) with \\(\\sup_ip_i=1\\).\n7. \\(K\\) is bounded, and \\(\\varphi(q_n)\\to0\\) uniformly in \\(\\varphi\\in K\\) for every sequence \\((q_n)\\) of mutually orthogonal projections.\n\n\nWe prove the easy implications first; the implication (1)\\(\\Rightarrow\\)(4) needs two lemmas.\n\n**Proof of (1)\\(\\Rightarrow\\)(2).** Restriction \\(\\varphi\\mapsto\\varphi|_{\\mathcal A}\\) is continuous from \\(\\sigma(M_*,M)\\) to \\(\\sigma(\\mathcal A_*,\\mathcal A)\\), and continuous images of compact sets are compact. \\(\\square\\)\n\n**Proof of (2)\\(\\Rightarrow\\)(1).** Every self-adjoint \\(h\\in M\\) lies in the abelian von Neumann algebra generated by \\(h\\) and \\(1\\). So \\(\\{\\varphi(h):\\varphi\\in K\\}\\) is bounded, hence so is \\(\\{\\varphi(x):\\varphi\\in K\\}\\) for every \\(x=h+ik\\), and \\(K\\) is bounded by uniform boundedness. Let \\(\\varphi\\) lie in the \\(\\sigma(M^*,M)\\)-closure of \\(K\\). For an abelian von Neumann subalgebra \\(\\mathcal A\\), \\(\\varphi|_{\\mathcal A}\\) lies in the \\(\\sigma(\\mathcal A^*,\\mathcal A)\\)-closure of \\(K|_{\\mathcal A}\\), which consists of normal functionals by Lemma 10.1. So \\(\\varphi\\) is normal on every abelian von Neumann subalgebra, hence normal (background fact 23). By Lemma 10.1, \\(K\\) is relatively weakly compact. \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(6)\\(\\Rightarrow\\)(5)\\(\\Rightarrow\\)(3).** (4)\\(\\Rightarrow\\)(6): let \\(\\varepsilon,\\delta\\) be as in (4) and \\(p_i\\uparrow1\\). By normality of \\(\\omega\\) there is \\(i_0\\) with \\(\\omega(1-p_i)<\\delta/2\\) for \\(i\\ge i_0\\). If \\(a\\in M_1\\) and \\(a=(1-p_i)a(1-p_i)\\), then \\(a^*a\\le1-p_i\\) and \\(aa^*\\le1-p_i\\), so \\(\\omega(a^*a+aa^*)<\\delta\\) and \\(|\\varphi(a)|<\\varepsilon\\). Hence \\[\n\\begin{gathered}\n\\|(1-p_i)\\varphi(1-p_i)\\|\\\\\n=\\sup_{a\\in M_1}|\\varphi((1-p_i)a(1-p_i))|\\\\\n\\le\\varepsilon\n\\end{gathered}\n\\] for \\(i\\ge i_0\\) and all \\(\\varphi\\in K\\).\n\n(6)\\(\\Rightarrow\\)(5): let \\((p_i)\\) increase to \\(p\\). Then \\(q_i=p_i+(1-p)\\) increases to \\(1\\), and \\(1-q_i=p-p_i\\). By (6), \\[\n\\begin{gathered}\n|\\varphi(p)-\\varphi(p_i)|\\\\\n=|\\varphi(p-p_i)|\\\\\n\\le\\|(p-p_i)\\varphi(p-p_i)\\|\\to0\n\\end{gathered}\n\\] uniformly in \\(\\varphi\\in K\\).\n\n(5)\\(\\Rightarrow\\)(3): if \\(p_n\\downarrow0\\), then \\(1-p_n\\uparrow1\\). By (5), \\(\\varphi(1-p_n)\\) converges uniformly, and its pointwise limit is \\(\\varphi(1)\\) by normality. So \\(\\varphi(p_n)\\to0\\) uniformly. \\(\\square\\)\n\n**Proof of (3)\\(\\Leftrightarrow\\)(7).** (3)\\(\\Rightarrow\\)(7): for orthogonal \\((q_n)\\), \\(r_m=\\sum_{n\\ge m}q_n\\) decreases to \\(0\\), and \\(\\varphi(q_m)=\\varphi(r_m)-\\varphi(r_{m+1})\\). (7)\\(\\Rightarrow\\)(3): suppose \\(p_n\\downarrow0\\) but \\(\\sup_{\\varphi\\in K}|\\varphi(p_n)|\\not\\to0\\). Then there are \\(\\varepsilon>0\\), indices \\(n_1<n_2<\\cdots\\) and \\(\\varphi_j\\in K\\) with \\(|\\varphi_j(p_{n_j})|\\ge\\varepsilon\\). Since each \\(\\varphi_j\\) is normal, \\(\\varphi_j(p_n)\\to0\\) as \\(n\\to\\infty\\); choosing the indices and functionals inductively, we may assume \\(|\\varphi_j(p_{n_{j+1}})|<\\varepsilon/2\\). The projections \\(q_j=p_{n_j}-p_{n_{j+1}}\\) are mutually orthogonal and \\(|\\varphi_j(q_j)|>\\varepsilon/2\\), which contradicts (7). \\(\\square\\)\n\n**Proof of (4)\\(\\Rightarrow\\)(3).** If \\(p_n\\downarrow0\\), then \\(\\omega(p_n^*p_n+p_np_n^*)=2\\omega(p_n)\\to0\\). \\(\\square\\)\n\n**Proof of (3)\\(\\Rightarrow\\)(1).** By the Eberlein–Šmulian theorem we only need a weak cluster point in \\(M_*\\) for each sequence \\((\\varphi_n)\\) in \\(K\\). Let \\(\\varphi\\in M^*\\) be a \\(\\sigma(M^*,M)\\)-cluster point, which exists by the Banach–Alaoglu theorem. On \\(M_*\\), \\(\\sigma(M^*,M)\\) is the weak topology, so it suffices to show that \\(\\varphi\\) is normal. Put \\(\\omega=\\sum_n2^{-n}\\varphi_n^\\sharp\\in M_*^+\\).\n\n(a) If \\(q\\) is a projection with \\(\\omega(q)=0\\), then \\(\\varphi_n(q)=0\\) for all \\(n\\) (Lemma 8.1(3)), hence \\(\\varphi(q)=0\\).\n\n(b) If projections \\(r_m\\) decrease to \\(0\\), then \\(|\\varphi(r_m)|\\le\\sup_n|\\varphi_n(r_m)|\\to0\\) by (3).\n\nLet \\((p_i)_{i\\in I}\\) be mutually orthogonal with sum \\(p\\). Since \\(\\omega\\) is normal, \\(\\sum_i\\omega(p_i)=\\omega(p)<\\infty\\), so \\(I_0=\\{i:\\omega(p_i)>0\\}\\) is countable. Every projection \\(q\\) under \\(\\sum_{i\\notin I_0}p_i\\) has \\(\\omega(q)=0\\), hence \\(\\varphi(q)=0\\) by (a). For \\(J\\subseteq I_0\\), enumerate \\(J=\\{j_1,j_2,\\dots\\}\\); the projections \\(r_m=\\sum_{k\\ge m}p_{j_k}\\) decrease to \\(0\\), so by (b), \\(\\varphi(\\sum_{i\\in J}p_i)=\\lim_m\\sum_{k<m}\\varphi(p_{j_k})\\). Applying this to \\(J=\\{i\\in I_0:\\operatorname{Re}\\varphi(p_i)\\ge0\\}\\) and to the three analogous sets shows that \\(\\sum_{i\\in I_0}|\\varphi(p_i)|<\\infty\\). Now let \\(F\\subseteq I\\) be finite. Then \\(p-\\sum_{i\\in F}p_i\\) is the sum of a projection under \\(\\sum_{i\\notin I_0}p_i\\) and of \\(\\sum_{i\\in I_0\\setminus F}p_i\\), so\n\\[\n\\begin{gathered}\n\\Big|\\varphi(p)-\\sum_{i\\in F}\\varphi(p_i)\\Big|\\\\\n=\\Big|\\sum_{i\\in I_0\\setminus F}\\varphi(p_i)\\Big|\\\\\n\\le\\sum_{i\\in I_0\\setminus F}|\\varphi(p_i)|,\n\\end{gathered}\n\\]\nwhich tends to \\(0\\) along the finite sets \\(F\\). So \\(\\varphi\\) is completely additive, hence normal (background fact 23). \\(\\square\\)\n\nThe remaining implication rests on the following two lemmas.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 813,
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          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
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          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.3",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Lemma 10.3",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 10.3.** Let \\((\\varphi_k)\\) be a sequence in \\(M_*\\) converging weakly to \\(\\varphi_0\\in M_*\\), and let \\((a_n)\\) be a sequence in \\(M_1\\) converging \\(\\sigma\\)-strongly\\(^*\\) to \\(0\\). Then \\(\\sup_k|\\varphi_k(a_n)|\\to0\\) as \\(n\\to\\infty\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 870,
        "through_line": 899,
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      },
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        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
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          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.4",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Lemma 10.4",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 10.4.** Let \\(K\\subseteq M_*\\) be relatively weakly compact and \\(\\varepsilon>0\\). There are a finite set \\(F\\subseteq K\\) and \\(\\delta>0\\) such that: if \\(a\\in M_1\\) and \\(\\psi^\\sharp(a^*a+aa^*)<\\delta\\) for all \\(\\psi\\in F\\), then \\(|\\varphi(a)|<\\varepsilon\\) for all \\(\\varphi\\in K\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 900,
        "through_line": 933,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.6",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Proposition 10.6",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 10.6.** Let \\(K\\subseteq M_*\\) and put \\(|K|=\\{|\\varphi|:\\varphi\\in K\\}\\), \\(|K^*|=\\{|\\varphi^*|:\\varphi\\in K\\}\\), \\(M_1K=\\{a\\varphi:a\\in M_1,\\ \\varphi\\in K\\}\\) and \\(KM_1=\\{\\varphi a:a\\in M_1,\\ \\varphi\\in K\\}\\).\n\n1. \\(M_1K\\) is relatively weakly compact exactly when \\(|K|\\) is.\n2. \\(KM_1\\) is relatively weakly compact exactly when \\(|K^*|\\) is.\n3. In particular, if \\(K\\subseteq M_*^+\\) is relatively weakly compact, so are \\(M_1K\\) and \\(KM_1\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 934,
        "through_line": 952,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.7",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Example 10.7",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 10.7** (Absolute values can escape). Let \\(H\\) have an orthonormal sequence \\((\\xi_n)\\), \\(M=B(H)\\), and \\(\\varphi_n=\\omega_{\\xi_1,\\xi_n}\\), so \\(\\varphi_n(x)=\\langle x\\xi_1,\\xi_n\\rangle\\).\n\n1. \\(\\sum_n|\\varphi_n(x)|^2\\le\\|x\\xi_1\\|^2\\), so \\(\\varphi_n\\to0\\) weakly, and \\(K=\\{\\varphi_n\\}\\) is relatively weakly compact. But \\(\\|\\varphi_n\\|=1\\) (Example 1.2).\n2. By (2.5), \\(|\\varphi_n|=\\omega_{\\xi_n}\\). The set \\(|K|\\) is not relatively weakly compact: \\(p_m=\\sum_{j\\ge m}\\theta_{\\xi_j,\\xi_j}\\) decreases to \\(0\\) (strongly), while \\(\\omega_{\\xi_n}(p_m)=1\\) for \\(n\\ge m\\), against Theorem 10.2(3). By Proposition 10.6, \\(M_1K\\) is not relatively weakly compact either.\n3. \\(\\varphi_n^*=\\omega_{\\xi_n,\\xi_1}\\) and \\(|\\varphi_n^*|=\\omega_{\\xi_1}\\) for all \\(n\\). So \\(|K^*|\\) is a single point, and \\(KM_1\\) is relatively weakly compact.\n4. Two-sided multiples of a single positive functional need not form a relatively weakly compact set: \\(\\omega_{\\xi_n}=a_n\\omega_{\\xi_1}a_n^*\\) with \\(a_n=\\theta_{\\xi_n,\\xi_1}\\in M_1\\).\n\nIn the commutative case absolute values cannot escape.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 953,
        "through_line": 961,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.8",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-16::Proposition 10.8",
      "anchor": "oa-fnd-pd-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 10.8.** If \\(M\\) is abelian and \\(K\\subseteq M_*\\) is relatively weakly compact, then so is \\(|K|\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 962,
        "through_line": 969,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
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              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 11.1",
      "kind": "definition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-19::Definition 11.1",
      "anchor": "oa-fnd-pd-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Definition 11.1.** On a von Neumann algebra \\(M\\), let \\(\\tau\\) be the locally convex topology defined by the seminorms\n\\[\np_K(x)=\\sup_{\\varphi\\in K}|\\varphi(x)|,\n\\]\nwhere \\(K\\) runs over the relatively weakly compact subsets of \\(M_*\\), with \\(p_\\varnothing=0\\). It is Hausdorff, since singletons are compact. We also define the *Mackey topology* \\(\\tau(M,M_*)\\) directly as uniform convergence on the absolutely convex weakly compact subsets of \\(M_*\\). Thus its seminorms form a subfamily of these \\(p_K\\). The finest-compatible-topology characterization is not needed.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 972,
        "through_line": 977,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.2",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-19::Theorem 11.2",
      "anchor": "oa-fnd-pd-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 11.2.** On every bounded subset of \\(M\\), the topology \\(\\tau\\) coincides with the \\(\\sigma\\)-strong\\(^*\\) topology. More precisely, the \\(\\sigma\\)-strong\\(^*\\) topology is coarser than \\(\\tau\\) on all of \\(M\\), and on bounded sets \\(\\tau\\) is coarser than the \\(\\sigma\\)-strong\\(^*\\) topology.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 978,
        "through_line": 1008,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.4",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-19::Proposition 11.4",
      "anchor": "oa-fnd-pd-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 11.4.** Let \\(M\\) contain an infinite sequence \\((e_n)\\) of mutually orthogonal nonzero projections, and put \\(D=\\{\\sqrt n\\,e_n:n\\ge1\\}\\). Then \\(0\\) lies in the \\(\\sigma\\)-strong\\(^*\\) closure of \\(D\\), but not in its closure for \\(\\tau(M,M_*)\\) or for \\(\\tau\\). Consequently the \\(\\sigma\\)-strong\\(^*\\) topology equals the Mackey topology on all of \\(M\\) exactly when \\(M\\) is finite-dimensional.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1009,
        "through_line": 1021,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [
        {
          "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
          "local_scopes": [
            {
              "heading": "10. Weak compactness in the predual",
              "line": 801,
              "through_line": 969,
              "anchors": [
                "OA-FND-PD-16",
                "OA-FND-PD-17",
                "OA-FND-PD-18"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            },
            {
              "heading": "11. The Mackey topology on bounded sets",
              "line": 970,
              "through_line": 1021,
              "anchors": [
                "OA-FND-PD-19"
              ],
              "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
            }
          ],
          "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
          "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
          "role": "proof comparison",
          "correspondence": "Further reading, not complete foundation",
          "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 12.1",
      "kind": "definition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-20::Definition 12.1",
      "anchor": "oa-fnd-pd-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Definition 12.1.** A von Neumann algebra is *atomic* if every nonzero projection majorizes a minimal projection (background fact 26). A projection is of *finite rank* if it is a sum of finitely many mutually orthogonal minimal projections.\n\nIn an atomic algebra \\(1=\\sum_jf_j\\) for a family of mutually orthogonal minimal projections (background fact 26). The join of two finite-rank projections is of finite rank (background fact 25), so the finite-rank projections form an increasing net with supremum \\(1\\). For a finite-rank \\(e=f_1+\\dots+f_m\\), \\(\\dim eMe\\le m^2\\), since \\(\\dim f_iMf_j\\le1\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1026,
        "through_line": 1029,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 12.2",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-20::Theorem 12.2",
      "anchor": "oa-fnd-pd-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 12.2.** Let \\(M\\) be atomic, and let \\((\\varphi_i)\\) be a net in \\(M_*\\) converging weakly to \\(\\varphi\\in M_*\\) such that \\(\\{|\\varphi_i|\\}\\) and \\(\\{|\\varphi_i^*|\\}\\) are relatively weakly compact. Then \\(\\|\\varphi_i-\\varphi\\|\\to0\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1030,
        "through_line": 1060,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 12.3",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-20::Corollary 12.3",
      "anchor": "oa-fnd-pd-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 12.3.** Let \\(M\\) be atomic and \\((\\varphi_i)\\) a net in \\(M_*^+\\) converging weakly to \\(\\varphi\\). Then \\(\\|\\varphi_i-\\varphi\\|\\to0\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1061,
        "through_line": 1064,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 12.4",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-20::Corollary 12.4",
      "anchor": "oa-fnd-pd-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 12.4** (Schur's theorem again). A weakly convergent sequence in \\(\\ell^1(\\Gamma)\\) converges in norm.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1065,
        "through_line": 1068,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 12.5",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-20::Example 12.5",
      "anchor": "oa-fnd-pd-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 12.5.** The hypotheses of Theorem 12.2 and Corollary 12.3 cannot be dropped.\n\nFor the first example, the Rademacher function \\(r_n\\) has constant value \\(+1\\) or \\(-1\\) on each interval of length \\(2^{-n}\\), alternately. Thus \\(\\|r_n\\|_2=1\\). If \\(m>n\\), each interval on which \\(r_n\\) is constant contains an even number of the alternating intervals for \\(r_m\\), so \\(\\int r_nr_m\\,dt=0\\). Endpoint values affect only a null set. This proves the asserted orthonormality.\n\n1. *Atomicity.* Let \\(M=L^\\infty[0,1]\\) and let \\(r_n(t)=\\operatorname{sgn}\\sin(2^n\\pi t)\\) be the Rademacher functions, an orthonormal sequence in \\(L^2[0,1]\\). For \\(x\\in L^\\infty\\subseteq L^2\\), Bessel's inequality gives \\(\\int xr_n\\,dt\\to0\\). So \\(\\varphi_n=\\varphi_{r_n}\\to0\\) weakly, and by Example 2.6, \\(|\\varphi_n|=|\\varphi_n^*|=\\varphi_{|r_n|}=\\lambda\\), Lebesgue measure, for every \\(n\\). But \\(\\|\\varphi_n\\|=\\int|r_n|=1\\). Likewise the positive functionals \\(\\varphi_{1+r_n}\\) tend weakly to \\(\\lambda\\), while \\(\\|\\varphi_{1+r_n}-\\lambda\\|=1\\).\n2. *Both absolute values.* In Example 10.7, \\(M=B(H)\\) is atomic, \\(\\varphi_n\\to0\\) weakly, and \\(\\{|\\varphi_n^*|\\}\\) is a single point, but \\(\\|\\varphi_n\\|=1\\). Passing to adjoints gives an example where \\(\\{|\\varphi_n|\\}\\) is compact and \\(\\{|\\varphi_n^*|\\}\\) is not.\n3. *Positivity.* The same example shows that Corollary 12.3 fails for sequences that are not positive.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1069,
        "through_line": 1076,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 13.1",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-21::Proposition 13.1",
      "anchor": "oa-fnd-pd-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 13.1.** Let \\(M\\) be a von Neumann algebra and \\(N\\subseteq M\\) a \\(\\sigma\\)-weakly closed \\(*\\)-subalgebra.\n\n1. Every \\(\\varphi\\in N_*^+\\) extends to some \\(\\tilde\\varphi\\in M_*^+\\), and \\(\\|\\tilde\\varphi\\|=\\|\\varphi\\|\\).\n2. Every \\(\\varphi\\in N_*\\) extends to some \\(\\tilde\\varphi\\in M_*\\) with \\(\\|\\tilde\\varphi\\|=\\|\\varphi\\|\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1081,
        "through_line": 1093,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 14.1",
      "kind": "definition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Definition 14.1",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Definition 14.1.** A \\(C^*\\)-algebra \\(A\\) is *weakly compact* if every left multiplication \\(L_a\\), \\(a\\in A\\), is a weakly compact operator, that is, \\(aA_1\\) is relatively weakly compact in \\(A\\). It is *dual* if \\(\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))=\\mathfrak m\\) for every closed left ideal \\(\\mathfrak m\\) and \\(\\operatorname{ann}_r(\\operatorname{ann}_l(\\mathfrak n))=\\mathfrak n\\) for every closed right ideal \\(\\mathfrak n\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1109,
        "through_line": 1110,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 14.2",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Proposition 14.2",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 14.2.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: (a) every left multiplication \\(L_a\\) is weakly compact; (b) every right multiplication \\(R_a:x\\mapsto xa\\) is weakly compact; (c) \\(A\\) is a two-sided ideal of \\(\\tilde A\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1111,
        "through_line": 1114,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 14.3",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Proposition 14.3",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 14.3.** Let \\(A\\) be an ideal of \\(\\tilde A\\), and \\(\\mathfrak m\\) a closed left ideal of \\(A\\) whose \\(\\sigma\\)-weak closure in \\(\\tilde A\\) is \\(\\tilde Ae\\) (background fact 19). Then\n\\[\n\\begin{gathered}\n\\mathfrak m\\\\\n=Ae\\\\\n=\\{x\\in A:xe=x\\},\\\\\n\\operatorname{ann}_r(\\mathfrak m)\\\\\n=(1-e)A,\\\\\n\\operatorname{ann}_l(\\operatorname{ann}_r(\\mathfrak m))\\\\\n=\\mathfrak m .\n\\end{gathered}\n\\]\nThe symmetric statements hold for closed right ideals. In particular \\(A\\) is dual.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1115,
        "through_line": 1132,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 14.4",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Lemma 14.4",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 14.4.** Let \\(f\\) be a minimal projection of \\(A\\). Then \\(f\\) is minimal in \\(\\tilde A\\), and \\(\\tilde Af=Af\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1133,
        "through_line": 1136,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 14.5",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Proposition 14.5",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 14.5** (Structure of dual algebras). Let \\(A\\) be dual.\n\n1. \\(\\mathfrak m\\mapsto\\operatorname{ann}_r(\\mathfrak m)\\) is an inclusion-reversing bijection from the closed left ideals onto the closed right ideals, with inverse \\(\\operatorname{ann}_l\\). A closed left ideal is maximal exactly when its right annihilator is a minimal nonzero closed right ideal; and symmetrically.\n2. Every nonzero closed left ideal contains \\(Af\\) for some minimal projection \\(f\\) of \\(A\\); symmetrically for right ideals.\n3. \\(A\\) is the closed linear span of the sets \\(Af\\), and also of the sets \\(fA\\), where \\(f\\) runs over the minimal projections of \\(A\\).\n4. \\(\\tilde A\\) is atomic, and every finite-rank projection of \\(\\tilde A\\) lies in \\(A\\).\n5. The finite-rank projections of \\(\\tilde A\\), ordered by size, form an increasing approximate unit of \\(A\\).\n6. \\(A\\) is an ideal of \\(\\tilde A\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1137,
        "through_line": 1159,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 14.6",
      "kind": "definition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Definition 14.6",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Definition 14.6.** For \\(C^*\\)-algebras \\((A_j)_{j\\in J}\\), the \\(c_0\\)*-direct sum* \\(\\bigoplus^0_jA_j\\) is the set of families \\(x=(x_j)\\) with \\(x_j\\in A_j\\) such that \\(\\{j:\\|x_j\\|\\ge\\varepsilon\\}\\) is finite for every \\(\\varepsilon>0\\), with coordinatewise operations and \\(\\|x\\|=\\sup_j\\|x_j\\|\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1160,
        "through_line": 1161,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 14.7",
      "kind": "lemma",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Lemma 14.7",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Lemma 14.7.** Let \\(B=\\bigoplus^0_jA_j\\), and let \\(\\iota_j:A_j\\to B\\) put an element in the \\(j\\)-th coordinate.\n\n1. \\(B\\) is a \\(C^*\\)-algebra, and every \\(x\\in B\\) is the norm limit of its finite truncations \\(\\sum_{j\\in F}\\iota_j(x_j)\\).\n2. The closed left ideals of \\(B\\) are exactly the sets \\(\\bigoplus^0_j\\mathfrak m_j=\\{x\\in B:x_j\\in\\mathfrak m_j\\ \\forall j\\}\\), \\(\\mathfrak m_j\\) a closed left ideal of \\(A_j\\). The same holds for right and two-sided ideals.\n3. If every \\(A_j\\) is dual, \\(B\\) is dual.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1162,
        "through_line": 1173,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 14.8",
      "kind": "example",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Example 14.8",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Example 14.8.** \\(K(H)\\) is dual: its bidual is \\(B(H)\\), in which it is an ideal (background fact 14 and Proposition 14.3). By Lemma 14.7(3), every \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) is dual; for instance \\(c_0(\\Gamma)=\\bigoplus^0_{\\gamma\\in\\Gamma}\\mathbb C\\). An infinite-dimensional \\(C^*\\)-algebra with a unit, such as \\(C[0,1]\\) or \\(B(H)\\) with \\(\\dim H=\\infty\\), is never dual. For if \\(A\\) is dual, it is an ideal of \\(\\tilde A\\) (Proposition 14.5(6)), and if moreover \\(1\\in A\\), then \\(\\tilde A=\\tilde A\\cdot1\\subseteq A\\), so \\(A\\) would be a von Neumann algebra equal to its bidual, hence reflexive, which an infinite-dimensional von Neumann algebra is not (it contains a copy of \\(\\ell^\\infty\\), background fact 25).\n\nRepresentations \\(\\pi_1,\\pi_2\\) of \\(A\\) are *disjoint* if their central supports are orthogonal: \\(z(\\pi_1)z(\\pi_2)=0\\). Then no nonzero operator \\(T\\) satisfies \\(T\\pi_1(a)=\\pi_2(a)T\\) for all \\(a\\): such a \\(T\\) also intertwines the normal extensions, so \\[\n\\begin{gathered}\nT\\\\\n=T\\bar\\pi_1(z(\\pi_1))\\\\\n=\\bar\\pi_2(z(\\pi_1))T\\\\\n=\\bar\\pi_2(z(\\pi_1)z(\\pi_2))T\\\\\n=0.\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1174,
        "through_line": 1185,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 14.9",
      "kind": "proposition",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Proposition 14.9",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Proposition 14.9.** Let \\(A\\) be dual and \\((\\pi_i)_{i\\in I}\\) a family of mutually disjoint representations, and \\(\\pi=\\bigoplus_i\\pi_i\\). Then \\(\\pi(A)=\\bigoplus^0_i\\pi_i(A)\\) as sets of operators on \\(\\bigoplus_iH_i\\). So \\(\\pi(A)\\) is isomorphic to the \\(c_0\\)-direct sum of the \\(\\pi_i(A)\\).",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1186,
        "through_line": 1191,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 14.10",
      "kind": "theorem",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Theorem 14.10",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Theorem 14.10.** Every dual \\(C^*\\)-algebra is isomorphic to a \\(c_0\\)-direct sum of algebras \\(K(H_j)\\) of compact operators.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1192,
        "through_line": 1197,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 14.11",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Corollary 14.11",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 14.11.** For a \\(C^*\\)-algebra \\(A\\) the following are equivalent: \\(A\\) is weakly compact; every right multiplication is weakly compact; \\(A\\) is an ideal of \\(\\tilde A\\); \\(A\\) is dual; \\(A\\) is isomorphic to a \\(c_0\\)-direct sum of algebras of compact operators.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1198,
        "through_line": 1201,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 14.12",
      "kind": "corollary",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-22::Corollary 14.12",
      "anchor": "oa-fnd-pd-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Corollary 14.12.** Quotients of dual \\(C^*\\)-algebras by closed two-sided ideals are dual.",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1202,
        "through_line": 1205,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.1",
      "kind": "exercise",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-25::Exercise 15.1",
      "anchor": "oa-fnd-pd-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Exercise 15.1.** (easy) In \\(M_2(\\mathbb C)\\) let \\(\\varphi(x)=x_{21}\\), the \\((2,1)\\) entry. Find \\(|\\varphi|\\), the partial isometry \\(v\\), \\(s_l(\\varphi)\\), \\(s_r(\\varphi)\\) and \\(|\\varphi^*|\\). Check inequality (3.1), and check that (1.1) is an equality for \\(e=E_{11}\\) and for the projection \\(e\\) onto \\(2^{-1/2}(1,1)\\).\n\n*",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1208,
        "through_line": 1211,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.2",
      "kind": "exercise",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-25::Exercise 15.2",
      "anchor": "oa-fnd-pd-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Exercise 15.2.** (medium) Let \\(\\varphi\\in M_*^+\\) and let \\(u\\in M\\) be unitary. Show that \\(|u\\varphi|=\\varphi\\) and \\(|\\varphi u|=u^*\\varphi u\\), where \\((u^*\\varphi u)(x)=\\varphi(uxu^*)\\). Deduce that \\(\\||\\varphi u|-|u\\varphi|\\|\\) can be as large as \\(2\\|\\varphi\\|\\).\n\n*",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1212,
        "through_line": 1223,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.3",
      "kind": "exercise",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-25::Exercise 15.3",
      "anchor": "oa-fnd-pd-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Exercise 15.3.** (medium) Show that a subset \\(K\\) of \\(\\ell^1(\\Gamma)\\) is relatively weakly compact exactly when it is bounded and for every \\(\\varepsilon>0\\) there is a finite \\(F\\subseteq\\Gamma\\) with \\(\\sum_{\\gamma\\notin F}|g(\\gamma)|<\\varepsilon\\) for all \\(g\\in K\\). Deduce that in \\(\\ell^1(\\Gamma)\\) the relatively weakly compact sets are the relatively norm compact sets.\n\n*",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1224,
        "through_line": 1227,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.4",
      "kind": "exercise",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-25::Exercise 15.4",
      "anchor": "oa-fnd-pd-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Exercise 15.4.** (medium) Show that every bounded subset of \\(M_*\\) is relatively weakly compact exactly when \\(M\\) is finite-dimensional.\n\n*",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1228,
        "through_line": 1231,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 15.5",
      "kind": "exercise",
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals#oa-fnd-pd-25::Exercise 15.5",
      "anchor": "oa-fnd-pd-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
      "statement_and_full_conditions": "**Exercise 15.5.** (medium) Let \\(S\\) be the unilateral shift on \\(\\ell^2=\\ell^2(\\{0,1,2,\\dots\\})\\), \\(S\\delta_k=\\delta_{k+1}\\), and \\(x_n=(S^*)^n\\). Show that \\(x_n\\to0\\) \\(\\sigma\\)-strongly but not in the topology \\(\\tau\\) of Definition 11.1, by exhibiting a relatively weakly compact \\(K\\subseteq B(\\ell^2)_*\\) with \\(p_K(x_n)\\ge1\\) for all \\(n\\). Why does this not contradict Theorem 11.2?\n\n*",
      "proof_locus": {
        "source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "line": 1232,
        "through_line": 1235,
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-01::Lemma 1.1",
      "anchor": "oa-fnd-ir-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 1.1** (Affine minorants). Let \\(f\\in\\mathcal L(K)\\). For every \\(x\\in K\\),\n\\[\n\\begin{gathered}\nf(x)\\\\\n=\\sup\\{a(x):\\\\\n\\ \\\\\na\\in\\operatorname{Aff}_E(K),\\ a\\\\\n\\leq f\\ \\text{on }K\\}.\n\\end{gathered}\n\\tag{1.1}\n\\]\n\nThe value \\(+\\infty\\) is allowed; Lemma 4.2 needs this case.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 40,
        "through_line": 68,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-01::Lemma 1.2",
      "anchor": "oa-fnd-ir-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 1.2** (Density of \\(\\operatorname{Aff}_E(K)\\)). \\(\\operatorname{Aff}_E(K)\\) is uniformly dense in \\(\\operatorname{Aff}(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 69,
        "through_line": 72,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 1.3",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-01::Lemma 1.3",
      "anchor": "oa-fnd-ir-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 1.3** (Increasing nets of affine functions). Let \\(f:K\\to\\mathbb R\\) be lsc and affine. Then \\(I=\\{g\\in\\operatorname{Aff}_E(K):\\ g<f\\ \\text{on }K\\}\\) is directed upward, and \\(\\sup I=f\\) pointwise. So \\(f\\) is the pointwise limit of an increasing net in \\(\\operatorname{Aff}_E(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 73,
        "through_line": 92,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 1.4",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-01::Definition 1.4",
      "anchor": "oa-fnd-ir-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 1.4** (Envelopes). Let \\(X\\subseteq K\\) be nonempty. For \\(g:X\\to[-\\infty,+\\infty]\\) bounded above, the *upper envelope* is\n\\[\n\\begin{gathered}\n\\overline g(x)\\\\\n=\\inf\\{a(x):\\\\\n\\ a\\in\\operatorname{Aff}(K),\\ a\\\\\n\\geq g\\ \\text{on }X\\}\\\\\n(x\\in K),\n\\end{gathered}\n\\tag{1.2}\n\\]\nand for \\(g\\) bounded below the *lower envelope* is \\(\\underline g(x)=\\sup\\{a(x):\\ a\\in\\operatorname{Aff}(K),\\ a\\leq g\\ \\text{on }X\\}\\). Both are defined on all of \\(K\\). The set \\(X\\) is arbitrary; we mostly use \\(X=K\\). By Lemma 1.2 the same envelopes arise if \\(a\\) runs only through \\(\\operatorname{Aff}_E(K)\\): an \\(a\\in\\operatorname{Aff}(K)\\) with \\(a\\geq g\\) is within \\(\\varepsilon\\) of some \\(b\\in\\operatorname{Aff}_E(K)\\), and then \\(b+\\varepsilon\\geq g\\) and \\(b+\\varepsilon\\leq a+2\\varepsilon\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 93,
        "through_line": 105,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 1.5",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-01::Proposition 1.5",
      "anchor": "oa-fnd-ir-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 1.5** (Properties of envelopes).\n\n1. \\(\\underline g\\in\\mathcal L(K)\\), and \\(-\\overline g\\in\\mathcal L(K)\\). That is, \\(\\overline g\\) is concave and upper semicontinuous (usc) with values in \\([-\\infty,+\\infty)\\).\n2. For \\(g\\) bounded on \\(X=K\\): \\(\\underline g\\leq g\\leq\\overline g\\). For \\(g\\in C(K)\\): \\(-\\|g\\|\\leq\\underline g\\leq g\\leq\\overline g\\leq\\|g\\|\\).\n3. For bounded \\(g,h\\) on \\(K\\): \\(g\\leq h\\) implies \\(\\overline g\\leq\\overline h\\); \\(\\overline{g+h}\\leq\\overline g+\\overline h\\); \\(\\overline{tg}=t\\overline g\\) for \\(t\\geq0\\); \\(\\overline{g+b}=\\overline g+b\\) for \\(b\\in\\operatorname{Aff}(K)\\); and \\(\\overline{-g}=-\\underline g\\).\n4. If \\(h\\in\\mathcal L(K)\\) then \\(\\underline h=h\\). If \\(g:K\\to[-\\infty,\\infty)\\) and \\(-g\\in\\mathcal L(K)\\), then \\(\\overline g=g\\). In particular \\(\\overline f=f\\) for \\(f\\in-\\mathcal P(K)\\) and \\(\\overline a=a\\) for \\(a\\in\\operatorname{Aff}(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 106,
        "through_line": 119,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 2.1",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-02::Proposition 2.1",
      "anchor": "oa-fnd-ir-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 2.1.**\n\n1. (*Integrals of lsc functions.*) For lsc \\(f:X\\to(-\\infty,+\\infty]\\),\n\\(\\int f\\,d\\mu=\\sup\\{\\mu(g):\\ g\\in C(X),\\ g\\leq f\\}\\). Dually, for usc \\(h:X\\to[-\\infty,\\infty)\\), \\(\\int h\\,d\\mu=\\inf\\{\\mu(g):\\ g\\in C(X),\\ g\\geq h\\}\\).\n2. (*Monotone nets.*) If \\((f_i)\\) is an increasing net of lsc functions \\(X\\to(-\\infty,+\\infty]\\) with pointwise supremum \\(f\\), then \\(\\int f\\,d\\mu=\\sup_i\\int f_i\\,d\\mu\\).\n3. (*Support.*) There is a largest open \\(\\mu\\)-null set. Its complement \\(\\operatorname{supp}\\mu\\) is closed, carries \\(\\mu\\), and every open set that meets it has positive measure. If \\(F\\) is closed and \\(\\mu(X\\setminus F)=0\\), then \\(\\operatorname{supp}\\mu\\subseteq F\\).\n4. (*Baire sets.*) A *zero set* is \\(Z(h)=h^{-1}(0)\\) with \\(h\\in C(X)\\); its complement is a *cozero set*. The *Baire* \\(\\sigma\\)-algebra \\(\\mathfrak B_0(X)\\) is generated by the zero sets.\n   - (a) The zero sets are exactly the closed \\(G_\\delta\\) sets (equivalently, the compact \\(G_\\delta\\) sets). Finite unions and countable intersections of zero sets are zero sets. Countable unions of cozero sets are cozero sets.\n   - (b) If \\(\\Phi:X\\to Y\\) is continuous and \\(Y\\) is compact Hausdorff, then \\(\\Phi^{-1}\\) maps Baire sets to Baire sets.\n   - (c) (*Regularity.*) For every Baire set \\(B\\) and \\(\\varepsilon>0\\) there are a zero set \\(Z\\subseteq B\\) and a cozero set \\(V\\supseteq B\\) with \\(\\mu(V\\setminus Z)<\\varepsilon\\).\n   - (d) If \\(X\\) is metrizable, every closed set is a zero set, so the Baire sets are the Borel sets.\n   - (e) Every finite positive measure on \\(\\mathfrak B_0(X)\\) is the restriction of a unique regular Borel measure.\n5. (*Bounded Radon–Nikodym.*) If \\(\\nu\\in M^+(X)\\) and \\(\\nu\\leq c\\mu\\), then \\(\\nu=g\\mu\\) for a Borel \\(g\\) with \\(0\\leq g\\leq c\\).\n6. (*Weak\\(^*\\) density.*) \\(C(X)\\) is weak\\(^*\\)-dense in \\(L^\\infty(X,\\mu)\\) (real or complex scalars).\n7. (*Image measures.*) Let \\(\\Phi:X\\to Y\\) be continuous, \\(Y\\) compact Hausdorff, and \\(\\Phi_*\\mu(g)=\\mu(g\\circ\\Phi)\\). Then \\(\\Phi_*\\mu(N)=\\mu(\\Phi^{-1}(N))\\) for every Borel set \\(N\\subseteq Y\\), so \\(\\int g\\,d\\Phi_*\\mu=\\int g\\circ\\Phi\\,d\\mu\\) for bounded Borel \\(g\\). If \\(\\Phi\\) is one-to-one, then \\(G\\mapsto G\\circ\\Phi\\) is an isometric \\(*\\)-isomorphism of \\(L^\\infty(Y,\\Phi_*\\mu)\\) onto \\(L^\\infty(X,\\mu)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 124,
        "through_line": 159,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 3.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-04::Lemma 3.1",
      "anchor": "oa-fnd-ir-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 3.1** (The barycentre exists). For every \\(\\mu\\in M_1^+(K)\\) there is exactly one \\(y\\in K\\) with\n\\[\np(y)=\\int_Kp\\,d\\mu\\qquad\\text{for all }p\\in E^*.\n\\]",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 164,
        "through_line": 170,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.2",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-04::Definition 3.2",
      "anchor": "oa-fnd-ir-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 3.2** (Barycentre). This \\(y\\) is the *barycentre* (or resultant) \\(r(\\mu)\\) of \\(\\mu\\), and \\(\\mu\\) *represents* \\(y\\). We write \\(M_y(K)\\) for the set of probability measures on \\(K\\) that represent \\(y\\).\n\nBoth sides of the next formula are continuous in \\(a\\) for the uniform norm and agree on \\(\\operatorname{Aff}_E(K)\\) (use \\(\\mu(1)=1\\)). By Lemma 1.2 they agree on \\(\\operatorname{Aff}(K)\\):\n\\[\n\\begin{gathered}\na(r(\\mu))\\\\\n=\\int_Ka\\,d\\mu\\\\\n(a\\in\\operatorname{Aff}(K)).\n\\end{gathered}\n\\tag{3.1}\n\\]",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 171,
        "through_line": 182,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 3.3",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-04::Proposition 3.3",
      "anchor": "oa-fnd-ir-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 3.3** (Properties of barycentres). Let \\(\\mu\\in M_1^+(K)\\) and \\(x=r(\\mu)\\).\n\n1. If \\(C\\subseteq K\\) is closed and convex and \\(\\mu(K\\setminus C)=0\\), then \\(x\\in C\\).\n2. (*Jensen's inequality.*) \\(g(x)\\leq\\int g\\,d\\mu\\) for every \\(g\\in\\mathcal L(K)\\), and \\(h(x)\\geq\\int h\\,d\\mu\\) whenever \\(-h\\in\\mathcal L(K)\\).\n3. (*Bauer's criterion.*) A point \\(x\\in K\\) is extreme if and only if \\(M_x(K)=\\{\\delta_x\\}\\).\n4. \\(r:M_1^+(K)\\to K\\) is affine, continuous and onto.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 183,
        "through_line": 197,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Lemma 4.1",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 4.1** (Differences of convex functions). \\(\\mathcal P(K)-\\mathcal P(K)\\) is a linear subspace of \\(C(K)\\), dense for the sup norm, and it is closed under \\(\\vee\\) and \\(\\wedge\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 202,
        "through_line": 214,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Lemma 4.2",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 4.2** (Lower semicontinuous convex functions as limits). Let \\(f\\in\\mathcal L(K)\\). The set \\(D_f\\) of functions \\(a_1\\vee\\dots\\vee a_n\\), with \\(n\\geq1\\) and \\(a_i\\in\\operatorname{Aff}_E(K)\\), \\(a_i\\leq f\\), is a subset of \\(\\mathcal P(K)\\) that is directed upward and has pointwise supremum \\(f\\). So \\(f\\) is the pointwise limit of an increasing net in \\(\\mathcal P(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 215,
        "through_line": 218,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.3",
      "kind": "corollary",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Corollary 4.3",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Corollary 4.3.** Let \\(\\mu,\\nu\\in M^+(K)\\) with \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). Then \\(\\int g\\,d\\mu\\leq\\int g\\,d\\nu\\) for every \\(g\\in\\mathcal L(K)\\), and \\(\\int h\\,d\\mu\\geq\\int h\\,d\\nu\\) whenever \\(-h\\in\\mathcal L(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 219,
        "through_line": 222,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.4",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Definition 4.4",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 4.4** (The Choquet order). For \\(\\mu,\\nu\\in M(K)\\) we write \\(\\mu\\prec\\nu\\) (\\(\\nu\\) *majorizes* \\(\\mu\\)) if \\(\\mu(f)\\leq\\nu(f)\\) for every \\(f\\in\\mathcal P(K)\\). We write \\(\\mu\\sim\\nu\\) if \\(\\mu(a)=\\nu(a)\\) for every \\(a\\in\\operatorname{Aff}(K)\\). A measure is *maximal* if it is maximal for \\(\\prec\\) in \\(M^+(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 223,
        "through_line": 224,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.5",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Proposition 4.5",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 4.5.**\n\n1. \\(\\prec\\) is a partial order on \\(M(K)\\), and \\(\\mu\\prec\\nu\\) implies \\(\\mu\\sim\\nu\\).\n2. If \\(\\mu,\\nu\\in M^+(K)\\) and \\(\\mu\\prec\\nu\\), then \\(\\mu(1)=\\nu(1)\\); if \\(\\mu\\) is a probability measure, so is \\(\\nu\\), and \\(r(\\mu)=r(\\nu)\\).\n3. \\(\\delta_{r(\\mu)}\\prec\\mu\\) for every \\(\\mu\\in M_1^+(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 225,
        "through_line": 232,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.6",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Lemma 4.6",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 4.6** (Maximal measures exist). Every chain in \\((M^+(K),\\prec)\\) has an upper bound. Hence every \\(\\nu\\in M^+(K)\\) is majorized by a maximal measure.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 233,
        "through_line": 236,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.7",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Lemma 4.7",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 4.7.** For every \\(\\mu\\in M^+(K)\\) and \\(f\\in C(K)\\) there is \\(\\nu\\in M^+(K)\\) with \\(\\mu\\prec\\nu\\) and \\(\\nu(f)=\\mu(\\overline f)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 237,
        "through_line": 240,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.8",
      "kind": "corollary",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Corollary 4.8",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Corollary 4.8** (The upper envelope as a maximum). For \\(x\\in K\\) and \\(f\\in C(K)\\),\n\\[\n\\overline f(x)=\\max\\{\\nu(f):\\ \\nu\\in M_x(K)\\}.\n\\]\nIf \\(f\\in\\mathcal P(K)\\), the maximum is attained at a maximal measure.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 241,
        "through_line": 252,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.10",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-03::Lemma 4.10",
      "anchor": "oa-fnd-ir-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 4.10** (The order through decompositions). For \\(\\mu,\\nu\\in M^+(K)\\) the following are equivalent.\n\n1. \\(\\mu\\prec\\nu\\).\n2. Whenever \\(\\mu=\\sum_{i=1}^n\\mu_i\\) with \\(\\mu_i\\in M^+(K)\\), there are \\(\\nu_i\\in M^+(K)\\) with \\(\\nu=\\sum_i\\nu_i\\) and \\(\\nu_i\\sim\\mu_i\\) for each \\(i\\).\n\nIf (1) holds, the \\(\\nu_i\\) in (2) can be chosen with \\(\\mu_i\\prec\\nu_i\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 253,
        "through_line": 282,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.1",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-06::Definition 5.1",
      "anchor": "oa-fnd-ir-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 5.1** (Boundary sets). For \\(f\\in C(K)\\) the *boundary set* is \\(B_f=\\{x\\in K:\\ \\overline f(x)=f(x)\\}\\), with the upper envelope \\(\\overline f\\) of (1.2).\n\nSince \\(\\overline f-f\\geq0\\) is usc, \\(B_f=\\bigcap_n\\{\\overline f-f<1/n\\}\\) is a \\(G_\\delta\\) set.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 287,
        "through_line": 290,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 5.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-06::Lemma 5.2",
      "anchor": "oa-fnd-ir-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 5.2.** \\(\\partial_eK=\\bigcap_{f\\in C(K)}B_f\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 291,
        "through_line": 308,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 5.3",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-06::Definition 5.3",
      "anchor": "oa-fnd-ir-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 5.3** (Boundary measures). \\(\\mu\\in M(K)\\) is a *boundary measure* if \\(|\\mu|(K\\setminus B_f)=0\\) for every \\(f\\in C(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 309,
        "through_line": 310,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 5.4",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-06::Theorem 5.4",
      "anchor": "oa-fnd-ir-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 5.4.**\n\n1. A measure \\(\\mu\\in M^+(K)\\) is maximal if and only if it is a boundary measure.\n2. Every point of \\(K\\) is the barycentre of a boundary probability measure.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 311,
        "through_line": 325,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.5",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-06::Proposition 5.5",
      "anchor": "oa-fnd-ir-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 5.5** (The boundary measures).\n\n1. The positive boundary measures form a convex cone. If \\(0\\leq\\nu\\leq\\mu\\) and \\(\\mu\\) is a boundary measure, so is \\(\\nu\\). A measure \\(\\mu\\) is a boundary measure exactly when \\(\\mu^+\\) and \\(\\mu^-\\) are, and the boundary measures form a linear subspace of \\(M(K)\\) that is closed under \\(\\vee\\) and \\(\\wedge\\).\n2. If \\(|\\mu|\\) is concentrated on a Borel set \\(B\\subseteq\\partial_eK\\), then \\(\\mu\\) is a boundary measure.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 326,
        "through_line": 332,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.1",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-07::Definition 6.1",
      "anchor": "oa-fnd-ir-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 6.1** (Strictly convex functions). \\(f:K\\to\\mathbb R\\) is *strictly convex* if \\(f(tx+(1-t)y)<tf(x)+(1-t)f(y)\\) whenever \\(x\\neq y\\) and \\(0<t<1\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 337,
        "through_line": 338,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.2",
      "kind": "lemma",
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      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-07::Lemma 6.2",
      "anchor": "oa-fnd-ir-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 6.2.** If \\(f\\in\\mathcal P(K)\\) is strictly convex, then \\(\\partial_eK=B_f\\). Hence if a strictly convex continuous function exists, \\(\\partial_eK\\) is a \\(G_\\delta\\) set.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 339,
        "through_line": 350,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.3",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-07::Lemma 6.3",
      "anchor": "oa-fnd-ir-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 6.3.** If \\(K\\) is metrizable, there is a strictly convex \\(f\\in\\mathcal P(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 351,
        "through_line": 365,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.4",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-07::Theorem 6.4",
      "anchor": "oa-fnd-ir-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 6.4** (Choquet's theorem). Let \\(K\\) be metrizable. Then \\(\\partial_eK\\) is a \\(G_\\delta\\) set, and \\(|\\mu|(K\\setminus\\partial_eK)=0\\) for each boundary measure \\(\\mu\\). Hence each point of \\(K\\) has a representing probability measure that is carried by \\(\\partial_eK\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 366,
        "through_line": 370,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.5",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-07::Lemma 6.5",
      "anchor": "oa-fnd-ir-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 6.5.** Let \\(K\\) be any compact convex set, and \\((f_n)\\) a sequence in \\(\\mathcal L(K)\\) with \\(\\sup_n\\sup_Kf_n<\\infty\\). Put \\(f=\\limsup_nf_n\\). Then \\(\\sup_{\\partial_eK}f=\\sup_Kf\\).\n\nHere \\(K\\) need not be metrizable. Metrizability enters the proof only through an auxiliary set \\(K'\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 371,
        "through_line": 398,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 6.6",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-07::Theorem 6.6",
      "anchor": "oa-fnd-ir-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 6.6.** If \\(\\mu\\) is a boundary measure on \\(K\\), then \\(|\\mu|(B)=0\\) for every Baire set \\(B\\subseteq K\\) with \\(B\\cap\\partial_eK=\\varnothing\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 399,
        "through_line": 416,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.1",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-23::Definition 7.1",
      "anchor": "oa-fnd-ir-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 7.1** (Simplex). \\(K\\) is a *simplex* if \\(\\operatorname{Aff}(K)^*\\) is a vector lattice for the dual order: \\(\\psi\\geq0\\) when \\(\\psi(a)\\geq0\\) for every \\(a\\geq0\\) in \\(\\operatorname{Aff}(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 421,
        "through_line": 422,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-23::Lemma 7.2",
      "anchor": "oa-fnd-ir-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 7.2** (Positive functionals on \\(\\operatorname{Aff}(K)\\)).\n\n1. \\(\\psi\\in\\operatorname{Aff}(K)^*\\) is positive if and only if \\(\\psi=t\\delta_x\\) with \\(t\\geq0\\) and \\(x\\in K\\).\n2. The restriction \\(R(\\mu)=\\mu|_{\\operatorname{Aff}(K)}\\) is a positive linear map of \\(M(K)\\) onto \\(\\operatorname{Aff}(K)^*\\), and \\(R(\\mu)=\\mu(1)\\delta_{r(\\mu)}\\) for \\(0\\neq\\mu\\in M^+(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 423,
        "through_line": 429,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.3",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-23::Lemma 7.3",
      "anchor": "oa-fnd-ir-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 7.3** (Riesz decomposition). In a vector lattice, let \\(p_1,\\dots,p_m,q_1,q_2\\geq0\\) with \\(\\sum_jp_j=q_1+q_2\\). Then there are \\(p_{jk}\\geq0\\) with \\(p_j=p_{j1}+p_{j2}\\) and \\(\\sum_jp_{jk}=q_k\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 430,
        "through_line": 433,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.4",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-23::Theorem 7.4",
      "anchor": "oa-fnd-ir-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 7.4** (Choquet–Meyer). The following are equivalent.\n\n- (a) \\(K\\) is a simplex.\n- (b) Every \\(x\\in K\\) is the barycentre of exactly one boundary probability measure (equivalently, \\(M_x(K)\\) has exactly one maximal element).\n- (c) \\(\\overline f\\) is affine for every \\(f\\in\\mathcal P(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 434,
        "through_line": 445,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 7.5",
      "kind": "example",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-23::Example 7.5",
      "anchor": "oa-fnd-ir-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Example 7.5** (The square and the triangle). Let \\(K\\) be the square with vertices \\(v_1=(0,0)\\), \\(v_2=(1,0)\\), \\(v_3=(1,1)\\), \\(v_4=(0,1)\\), and \\(c\\) its centre. \\(K\\) is metrizable, so maximal measures live on the four vertices (Theorem 6.4), and every measure on the vertices is a boundary measure (Proposition 5.5(2)). A measure \\(\\sum_ip_i\\delta_{v_i}\\) represents \\(c\\) exactly when \\(p_1=p_3\\), \\(p_2=p_4\\) and \\(p_1+p_2=\\frac12\\). So the maximal measures in \\(M_c(K)\\) form the segment \\(t\\cdot\\frac12(\\delta_{v_1}+\\delta_{v_3})+(1-t)\\cdot\\frac12(\\delta_{v_2}+\\delta_{v_4})\\), \\(t\\in[0,1]\\), and the square is not a simplex. For \\(f(x,y)=(x-y)^2\\), which is convex, \\(\\overline f(v_1)=\\overline f(v_3)=0\\), while \\(\\overline f(c)=\\max_t(1-t)=1\\) by Corollary 4.8; so \\(\\overline f\\) is not affine on the diagonal, as condition (c) of Theorem 7.4 predicts. In a triangle, barycentric coordinates are unique, every point has one maximal measure, and the triangle is a simplex.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 446,
        "through_line": 447,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 7.6",
      "kind": "exercise",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-23::Exercise 7.6",
      "anchor": "oa-fnd-ir-23",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Exercise 7.6** (medium; The Choquet order on an interval). Let \\(K=[0,1]\\). Show that for \\(\\mu,\\nu\\in M_1^+(K)\\), \\(\\mu\\prec\\nu\\) if and only if \\(r(\\mu)=r(\\nu)\\) and \\(\\int(t-s)^+d\\mu(t)\\leq\\int(t-s)^+d\\nu(t)\\) for every \\(s\\in[0,1]\\). Deduce that \\((1-x)\\delta_0+x\\delta_1\\) is the only maximal measure in \\(M_x(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 448,
        "through_line": 451,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [
        {
          "unit": "integral-representations-of-states",
          "local_scopes": [
            {
              "heading": "3. Barycentres",
              "line": 160,
              "through_line": 197,
              "anchors": [
                "OA-FND-IR-04"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "4. The Choquet order",
              "line": 198,
              "through_line": 282,
              "anchors": [
                "OA-FND-IR-03",
                "OA-FND-IR-05",
                "OA-FND-IR-15"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "5. Boundary measures",
              "line": 283,
              "through_line": 332,
              "anchors": [
                "OA-FND-IR-06"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "6. The metrizable case and Baire sets",
              "line": 333,
              "through_line": 416,
              "anchors": [
                "OA-FND-IR-07",
                "OA-FND-IR-08"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            },
            {
              "heading": "7. Simplices",
              "line": 417,
              "through_line": 451,
              "anchors": [
                "OA-FND-IR-23",
                "OA-FND-IR-27"
              ],
              "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
            }
          ],
          "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
          "source_locus": "§461A–§461P; PDF 1–9",
          "role": "proof comparison",
          "correspondence": "General compact convex scope, not only metrizable",
          "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 8.1",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-09::Proposition 8.1",
      "anchor": "oa-fnd-ir-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 8.1.**\n\n1. \\(\\mathfrak S\\) is compact and convex in \\(E\\), and \\(\\operatorname{Aff}_E(\\mathfrak S)=\\{\\hat h:\\ h\\in A_h\\}\\). A measure \\(\\mu\\in M_1^+(\\mathfrak S)\\) has barycentre \\(\\varphi\\) exactly when \\(\\varphi(a)=\\int\\omega(a)\\,d\\mu(\\omega)\\) for all \\(a\\in A\\).\n2. (*Kadison's representation.*) For \\(h\\in A_h\\), \\(\\|\\hat h\\|=\\|h\\|\\), and \\(h\\geq0\\) if and only if \\(\\hat h\\geq0\\). The map \\(h\\mapsto\\hat h\\) is an isometric order isomorphism of \\(A_h\\) onto \\(\\operatorname{Aff}(\\mathfrak S)\\).\n3. (*Radon–Nikodym map.*) Let \\(\\varphi\\in\\mathfrak S\\), and write \\(\\pi,H,\\xi\\) for \\(\\pi_\\varphi,H_\\varphi,\\xi_\\varphi\\). Put\n\\[\n\\begin{gathered}\n\\Theta_\\varphi(x)(a)\\\\\n=\\langle\\pi(a)x\\xi,\\xi\\rangle\\\\\n(x\\in\\pi(A)',\\ a\\in A).\n\\end{gathered}\n\\]\nThen \\(\\Theta_\\varphi:\\pi(A)'\\to A^*\\) is linear and one-to-one, \\(\\Theta_\\varphi(x^*)=\\Theta_\\varphi(x)^*\\), and a self-adjoint \\(x\\) is positive if and only if \\(\\Theta_\\varphi(x)\\) is. It maps \\(\\{x\\in\\pi(A)':\\ 0\\leq x\\leq1\\}\\) onto \\(\\{\\psi\\in A^*:\\ 0\\leq\\psi\\leq\\varphi\\}\\).\n4. (*Pure states.*) For \\(\\varphi\\in\\mathfrak S\\) the following are equivalent: (a) \\(\\varphi\\in\\partial_e\\mathfrak S\\); (b) every positive functional \\(\\psi\\leq\\varphi\\) is a multiple of \\(\\varphi\\); (c) \\(\\pi_\\varphi(A)'=\\mathbb C1\\). We call such states *pure* and write \\(P(A)=\\partial_e\\mathfrak S\\).\n5. Every state \\(\\varphi\\) is the barycentre of a maximal probability measure \\(\\mu\\) on \\(\\mathfrak S\\):\n\\[\n\\begin{gathered}\n\\varphi(a)\\\\\n=\\int_{\\mathfrak S}\\omega(a)\\,d\\mu(\\omega)\\\\\n(a\\in A),\n\\end{gathered}\n\\tag{8.1}\n\\]\nand \\(\\mu(B)=0\\) for every Baire set \\(B\\subseteq\\mathfrak S\\) that misses \\(P(A)\\). If \\(A\\) is separable, \\(\\mathfrak S\\) is metrizable, \\(P(A)\\) is a \\(G_\\delta\\) set, and \\(\\mu\\) is concentrated on it:\n\\[\n\\begin{gathered}\n\\varphi(a)\\\\\n=\\int_{P(A)}\\omega(a)\\,d\\mu(\\omega)\\\\\n(a\\in A).\n\\end{gathered}\n\\tag{8.2}\n\\]",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 456,
        "through_line": 506,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.0",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-29::Lemma 9.0",
      "anchor": "oa-fnd-ir-29",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 9.0.**\n\n1. The closed unit ball of \\(B(H)\\) is compact in the weak operator topology. Consequently every bounded weakly closed set of operators is weakly compact.\n2. The weakly continuous and strongly continuous linear functionals on \\(B(H)\\) are exactly the finite sums of vector functionals. Every convex set has the same weak and strong closures.\n3. On a norm-bounded set, the weak and \\(\\sigma\\)-weak topologies agree. Multiplication by a fixed operator on either side is continuous for the \\(\\sigma\\)-weak topology. Multiplication by a fixed \\(g\\in L^\\infty(\\mu)\\) is continuous for \\(\\sigma(L^\\infty,L^1)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 515,
        "through_line": 531,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-29::Lemma 9.1",
      "anchor": "oa-fnd-ir-29",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 9.1** (Normality test). Let \\(\\mu\\in M^+(X)\\) and let \\(\\kappa:L^\\infty(X,\\mu)\\to B(H)\\) be a bounded linear map. Suppose that for \\(\\eta,\\zeta\\) in a dense subspace \\(D\\subseteq H\\) the functional \\(f\\mapsto\\langle\\kappa(f)\\eta,\\zeta\\rangle\\) has the form \\(f\\mapsto\\int fu\\,d\\mu\\) with \\(u\\in L^1(\\mu)\\). Then the same holds for all \\(\\eta,\\zeta\\in H\\), and for all sums \\(f\\mapsto\\sum_n\\langle\\kappa(f)\\eta_n,\\zeta_n\\rangle\\) with \\(\\sum_n\\|\\eta_n\\|^2<\\infty\\) and \\(\\sum_n\\|\\zeta_n\\|^2<\\infty\\). So \\(\\kappa\\) is continuous from the weak\\(^*\\) topology of \\(L^\\infty(\\mu)\\) to the \\(\\sigma\\)-weak topology; we call it *normal*.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 532,
        "through_line": 542,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.2",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-29::Proposition 9.2",
      "anchor": "oa-fnd-ir-29",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 9.2.** Let \\(\\mu\\in M^+_1(\\mathfrak S)\\) have barycentre \\(\\varphi\\), and write \\(\\pi,H,\\xi\\) for its GNS triple. There is exactly one linear map \\(\\kappa_\\mu:L^\\infty(\\mathfrak S,\\mu)\\to\\pi(A)'\\) with\n\\[\n\\begin{gathered}\n\\langle\\kappa_\\mu(f)\\pi(a)\\xi,\\xi\\rangle\\\\\n=\\int_{\\mathfrak S}f(\\omega)\\,\\omega(a)\\,d\\mu(\\omega)\\\\\n(f\\in L^\\infty(\\mathfrak S,\\mu),\\ a\\in A).\n\\end{gathered}\n\\tag{9.1}\n\\]\nIt is positive, \\(\\kappa_\\mu(1)=1\\), \\(\\|\\kappa_\\mu(f)\\|\\leq\\|f\\|_\\infty\\) for real \\(f\\), and \\(\\kappa_\\mu\\) is normal.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 543,
        "through_line": 559,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-11::Lemma 10.1",
      "anchor": "oa-fnd-ir-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 10.1** (Positive contractions and projections). Let \\(p,q\\in B(H)\\).\n\n1. If \\(p\\) is a projection and \\(0\\leq x\\leq p\\), \\(0\\leq x\\leq1-p\\), then \\(x=0\\).\n2. If \\(p,q\\) are projections and \\(p+q\\) is a projection, then \\(pq=0\\).\n3. If \\(p,q\\geq0\\) and \\(p+q=1\\), then \\(p\\) and \\(q\\) commute and \\(0\\leq pq\\leq p\\), \\(0\\leq pq\\leq q\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 564,
        "through_line": 571,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.2",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-11::Theorem 10.2",
      "anchor": "oa-fnd-ir-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 10.2.** Let \\(\\mu\\in M_1^+(\\mathfrak S)\\) with barycentre \\(\\varphi\\). The following are equivalent.\n\n1. \\(\\kappa_\\mu\\) is multiplicative.\n2. \\(\\kappa_\\mu(\\chi_E)\\kappa_\\mu(1-\\chi_E)=0\\) for every Borel set \\(E\\subseteq\\mathfrak S\\).\n3. For every Borel set \\(E\\), the functionals \\(\\varphi_E(a)=\\int_E\\omega(a)\\,d\\mu(\\omega)\\) and \\(\\varphi_{E^c}=\\varphi-\\varphi_E\\) are *orthogonal*: a positive \\(\\psi\\) with \\(\\psi\\leq\\varphi_E\\) and \\(\\psi\\leq\\varphi_{E^c}\\) is \\(0\\).\n\nIn this case \\(\\kappa_\\mu\\) is a \\(*\\)-isomorphism of \\(L^\\infty(\\mathfrak S,\\mu)\\) onto its range, \\(\\|\\kappa_\\mu(f)\\|=\\|f\\|_\\infty\\), and \\(\\|\\kappa_\\mu(f)\\xi_\\varphi\\|^2=\\int|f|^2\\,d\\mu\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 572,
        "through_line": 591,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.3",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-11::Definition 10.3",
      "anchor": "oa-fnd-ir-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 10.3.** \\(\\mu\\in M_1^+(\\mathfrak S)\\) is *orthogonal* if the conditions of Theorem 10.2 hold. Its *associated abelian algebra* is \\(\\mathcal C_\\mu=\\kappa_\\mu(L^\\infty(\\mathfrak S,\\mu))\\subseteq\\pi_\\varphi(A)'\\).\n\nThe algebra \\(\\mathcal C_\\mu\\) is commutative, because \\(\\kappa_\\mu\\) is a \\(*\\)-homomorphism on the commutative algebra \\(L^\\infty\\). The next proposition shows that it is weakly closed, and it collects tools used later.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 592,
        "through_line": 595,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 10.4",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-11::Proposition 10.4",
      "anchor": "oa-fnd-ir-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 10.4** (Structure of an orthogonal measure). Let \\(\\mu\\) be orthogonal with barycentre \\(\\varphi\\), and write \\(\\pi,H,\\xi\\) for its GNS triple.\n\n1. The map \\(V_\\mu:f\\mapsto\\kappa_\\mu(f)\\xi\\), \\(f\\in L^\\infty(\\mu)\\), extends to an isometry \\(V_\\mu:L^2(\\mathfrak S,\\mu)\\to H\\), and \\(\\kappa_\\mu(g)V_\\mu=V_\\mu M_g\\) for \\(g\\in L^\\infty(\\mu)\\), where \\(M_g\\) is multiplication by \\(g\\).\n2. Let \\(e_\\mu\\) be the projection onto \\(V_\\mu(L^2)\\). Then \\(V_\\mu(L^2)=[\\mathcal C_\\mu\\xi]\\), the closed span of \\(\\mathcal C_\\mu\\xi\\), and \\(e_\\mu\\in\\mathcal C_\\mu'\\).\n3. (*The \\(L^2\\) criterion.*) \\(\\mathcal C_\\mu=\\{y\\in\\pi(A)':\\ y\\xi\\in e_\\mu H\\}\\).\n4. \\(\\mathcal C_\\mu\\) is a von Neumann algebra (commutative), and \\(\\xi\\) is separating for it.\n5. For every \\(a\\in A\\),\n\\[\n\\begin{gathered}\n\\kappa_\\mu(\\hat a)\\xi\\\\\n=e_\\mu\\pi(a)\\xi\\\\\n\\text{and}\\\\\ne_\\mu\\pi(a)e_\\mu\\\\\n=\\kappa_\\mu(\\hat a)e_\\mu .\n\\end{gathered}\n\\tag{10.1}\n\\]",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 596,
        "through_line": 638,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 11.1",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-12::Definition 11.1",
      "anchor": "oa-fnd-ir-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 11.1.** A measure \\(\\mu\\in M_x(K)\\) is *simplicial* if it is an extreme point of the convex set \\(M_x(K)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 641,
        "through_line": 642,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.2",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-12::Proposition 11.2",
      "anchor": "oa-fnd-ir-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 11.2** (Any compact convex set). For \\(\\mu\\in M_x(K)\\), \\(\\mu\\) is simplicial if and only if \\(\\operatorname{Aff}(K)\\) is dense in the real space \\(L^1(K,\\mu)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 643,
        "through_line": 648,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 11.3",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-12::Proposition 11.3",
      "anchor": "oa-fnd-ir-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 11.3.** Let \\(\\mu\\in M_\\varphi(\\mathfrak S)\\), and consider: (i) \\(\\mu\\) is orthogonal; (ii) \\(\\mathcal A_{\\mathbb C}=\\{\\hat a:\\ a\\in A\\}\\) is dense in \\(L^1(\\mathfrak S,\\mu)\\) (complex scalars); (iii) \\(\\mu\\) is simplicial. Then (i)\\(\\Rightarrow\\)(ii)\\(\\Leftrightarrow\\)(iii). Under (i), \\(\\mathcal A_{\\mathbb C}\\) is even dense in \\(L^2(\\mathfrak S,\\mu)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 649,
        "through_line": 656,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.5",
      "kind": "example",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-12::Example 11.5",
      "anchor": "oa-fnd-ir-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Example 11.5** (Simplicial but not orthogonal). Let \\(A=M_2(\\mathbb C)\\). For \\(v\\in\\mathbb R^3\\) with \\(|v|\\leq1\\) let \\(\\rho_v=\\frac12(1+v_1\\sigma_1+v_2\\sigma_2+v_3\\sigma_3)\\), with the Pauli matrices \\(\\sigma_i\\), and \\(\\omega_v(a)=\\operatorname{tr}(\\rho_va)\\). Every state is some \\(\\omega_v\\), and \\(\\omega_v\\) is pure exactly when \\(|v|=1\\). For density matrices, \\(\\omega_\\rho\\leq\\omega_{\\rho'}\\) if and only if \\(\\rho\\leq\\rho'\\). Let \\(v^{(1)},v^{(2)},v^{(3)}\\) be unit vectors in the \\(v_1v_3\\)-plane at mutual angles of \\(120^\\circ\\), so \\(\\sum_kv^{(k)}=0\\), and let \\(\\mu=\\frac13\\sum_k\\delta_{\\omega_{v^{(k)}}}\\). It represents the tracial state \\(\\tau=\\omega_0\\).\n\n- *\\(\\mu\\) is simplicial.* \\(L^1(\\mu)\\) is \\(\\mathbb C^3\\). The functions \\(\\hat1,\\hat\\sigma_1,\\hat\\sigma_3\\) give the vectors \\((1,1,1)\\), \\((v^{(k)}_1)_k\\), \\((v^{(k)}_3)_k\\), which are independent because the three points are not collinear. So (ii) holds.\n- *\\(\\mu\\) is not orthogonal.* Take \\(E=\\{\\omega_{v^{(1)}}\\}\\). The density of \\(3\\varphi_{E^c}=\\omega_{v^{(2)}}+\\omega_{v^{(3)}}\\) is \\(1-\\frac12v^{(1)}\\cdot\\sigma\\), with eigenvalues \\(\\frac12\\) and \\(\\frac32\\). Let \\(\\psi=\\frac16\\omega_{v^{(1)}}\\), with density \\(\\frac1{12}(1+v^{(1)}\\cdot\\sigma)\\). In the eigenbasis of \\(v^{(1)}\\cdot\\sigma\\) the densities of \\(\\psi\\), \\(\\varphi_E\\) and \\(\\varphi_{E^c}\\) are \\(\\operatorname{diag}(\\frac16,0)\\), \\(\\operatorname{diag}(\\frac13,0)\\) and \\(\\operatorname{diag}(\\frac16,\\frac12)\\). So \\(0\\neq\\psi\\leq\\varphi_E\\) and \\(\\psi\\leq\\varphi_{E^c}\\).\n\nBy contrast, \\(\\frac12(\\delta_{\\omega_v}+\\delta_{\\omega_{-v}})\\) with \\(|v|=1\\) is orthogonal: \\(\\rho_v\\) and \\(\\rho_{-v}\\) are orthogonal rank-one projections, and a positive matrix below multiples of both is \\(0\\). The normalized surface measure on the sphere \\(\\{\\omega_v:|v|=1\\}\\) also represents \\(\\tau\\), by symmetry. It is maximal, since it lives on \\(\\partial_e\\mathfrak S\\) and is therefore a boundary measure (Proposition 5.5(2) and Theorem 5.4). But it is not orthogonal: \\(\\kappa\\) would embed the infinite-dimensional \\(L^\\infty\\) of the sphere into the four-dimensional \\(\\pi_\\tau(A)'\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 657,
        "through_line": 663,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-13::Lemma 12.1",
      "anchor": "oa-fnd-ir-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 12.1.** Let \\(\\mathfrak A\\subseteq B(H)\\) be a commutative \\(*\\)-algebra (not necessarily closed) with a cyclic vector \\(\\xi\\). Then \\(\\mathfrak A'\\) is commutative and \\(\\mathfrak A'=\\mathfrak A''\\). If \\(\\mathfrak A\\) is a von Neumann algebra, then \\(\\mathfrak A'=\\mathfrak A\\): it is maximal abelian.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 668,
        "through_line": 689,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 12.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-13::Lemma 12.2",
      "anchor": "oa-fnd-ir-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 12.2** (Reduction). Let \\(\\mathcal N\\) be a von Neumann algebra on \\(H\\), \\(e\\) a projection, and \\(\\mathcal N_e=\\{exe|_{eH}:\\ x\\in\\mathcal N\\}\\subseteq B(eH)\\).\n\n1. If \\(e\\in\\mathcal N'\\), then \\((\\mathcal N_e)'=(\\mathcal N')_e\\), and \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-homomorphism of \\(\\mathcal N\\) onto \\(\\mathcal N_e\\).\n2. If \\(e\\in\\mathcal N\\), then \\((\\mathcal N_e)'=(\\mathcal N')_e\\).\n3. In both cases \\(\\mathcal N_e\\) is a von Neumann algebra on \\(eH\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 690,
        "through_line": 710,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.3",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-13::Proposition 12.3",
      "anchor": "oa-fnd-ir-13",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 12.3.** Let \\(\\mathcal M\\) be a unital \\(*\\)-algebra of operators on \\(H\\) with a cyclic vector \\(\\xi_0\\).\n\n- (a) If \\(\\mathcal A\\subseteq\\mathcal M'\\) is a commutative \\(*\\)-algebra and \\(e\\) is the projection onto \\([\\mathcal A\\xi_0]\\), then \\(e\\mathcal Me\\) is commutative, that is, \\(e\\mathcal Me\\subseteq(e\\mathcal Me)'\\).\n- (b) If \\(e\\) is a projection with \\(e\\xi_0=\\xi_0\\) and \\(e\\mathcal Me\\) is commutative, then \\(\\mathcal A=\\mathcal M'\\cap\\{e\\}'\\) is a commutative von Neumann algebra, \\(e\\) is the projection onto \\([\\mathcal A\\xi_0]\\), and \\(x\\mapsto x|_{eH}\\) is a \\(*\\)-isomorphism of \\(\\mathcal A\\) onto the maximal abelian algebra \\((e\\mathcal Me|_{eH})'\\) of \\(eH\\).\n- (c) The maps \\(\\mathcal A\\mapsto[\\mathcal A\\xi_0]\\) and \\(e\\mapsto\\mathcal M'\\cap\\{e\\}'\\) are inverse bijections. One side is the set of commutative von Neumann algebras contained in \\(\\mathcal M'\\); the other is the set of projections \\(e\\) with \\(e\\xi_0=\\xi_0\\) and \\(e\\mathcal Me\\) commutative.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 711,
        "through_line": 740,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 13.1",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-14::Theorem 13.1",
      "anchor": "oa-fnd-ir-14",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 13.1.** Let \\(\\varphi\\in\\mathfrak S\\), and let \\(\\mathcal C\\subseteq\\pi_\\varphi(A)'\\) be a commutative von Neumann algebra. Then exactly one orthogonal \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) has \\(\\mathcal C_\\mu=\\mathcal C\\). So \\(\\mu\\mapsto\\mathcal C_\\mu\\) maps the orthogonal measures in \\(M_\\varphi(\\mathfrak S)\\) bijectively onto the commutative von Neumann algebras contained in \\(\\pi_\\varphi(A)'\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 745,
        "through_line": 768,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 14.1",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-16::Theorem 14.1",
      "anchor": "oa-fnd-ir-16",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 14.1.** Let \\(\\varphi\\in\\mathfrak S\\) and let \\(\\mu,\\nu\\in M_\\varphi(\\mathfrak S)\\) be orthogonal. The following are equivalent.\n\n1. \\(\\mu\\prec\\nu\\).\n2. \\(\\int\\omega(h)^2\\,d\\mu(\\omega)\\leq\\int\\omega(h)^2\\,d\\nu(\\omega)\\) for every \\(h\\in A_h\\).\n3. \\(\\mathcal C_\\mu\\subseteq\\mathcal C_\\nu\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 773,
        "through_line": 801,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 15.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-17::Lemma 15.1",
      "anchor": "oa-fnd-ir-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 15.1** (Extreme positive contractions). In a C\\(^*\\)-algebra \\(\\mathcal B\\) of operators, the extreme points of \\(\\{x\\in\\mathcal B:\\ 0\\leq x\\leq1\\}\\) are exactly the projections in \\(\\mathcal B\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 804,
        "through_line": 807,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 15.2",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-17::Theorem 15.2",
      "anchor": "oa-fnd-ir-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 15.2.** For \\(\\varphi\\in\\mathfrak S\\) the following are equivalent.\n\n1. \\(\\pi_\\varphi(A)'\\) is abelian.\n2. \\(M_\\varphi(\\mathfrak S)\\) has exactly one maximal measure.\n\nIn that case the maximal measure is the orthogonal measure \\(\\mu\\) with \\(\\mathcal C_\\mu=\\pi_\\varphi(A)'\\), and \\(\\nu\\prec\\mu\\) for every \\(\\nu\\in M_\\varphi(\\mathfrak S)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 808,
        "through_line": 835,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 15.3",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-17::Definition 15.3",
      "anchor": "oa-fnd-ir-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 15.3.** A representation \\(\\pi\\) of \\(A\\) is *multiplicity-free* if \\(\\pi(A)'\\) is abelian.\n\nSo a state is multiplicity-free (its GNS representation is) exactly when its maximal representing measure is unique. That measure is orthogonal and vanishes on Baire sets that miss \\(P(A)\\) (Proposition 8.1(5)).\n\nIf every state is multiplicity-free, the Choquet–Meyer theorem says that \\(\\mathfrak S\\) is a simplex. This happens only for abelian algebras.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 836,
        "through_line": 841,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 15.4",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-17::Theorem 15.4",
      "anchor": "oa-fnd-ir-17",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 15.4.** The state space \\(\\mathfrak S(A)\\) is a simplex exactly when \\(A\\) is abelian.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 842,
        "through_line": 847,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 16.1",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-24::Proposition 16.1",
      "anchor": "oa-fnd-ir-24",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 16.1.** Let \\(\\varphi\\in\\mathfrak S\\) with GNS triple \\(\\pi,H,\\xi\\), and let \\(F_\\varphi\\) be the smallest face of \\(\\mathfrak S\\) that contains \\(\\varphi\\).\n\n1. \\[\n\\begin{gathered}\nF_\\varphi\\\\\n=\\{\\psi\\in\\mathfrak S:\\\\\n\\ t\\psi\\\\\n\\leq\\varphi\\text{ for some }t>0\\}\\\\\n=\\{\\Theta_\\varphi(x):\\\\\n\\ x\\in\\pi(A)'_+,\\ \\\\\n\\langle x\\xi,\\xi\\rangle\\\\\n=1\\}\n\\end{gathered}\n\\]. If \\(\\psi\\in\\mathfrak S\\) has the form \\(\\Theta_\\varphi(x)\\) for some \\(x\\in\\pi(A)'\\), then automatically \\(x\\geq0\\).\n2. \\(F_\\varphi\\) is closed in \\(\\mathfrak S\\) \\(\\iff\\) \\(\\pi(A)'\\) is finite-dimensional \\(\\iff\\) \\(F_\\varphi\\) is finite-dimensional.\n3. The cone \\(\\mathbb R_+F_\\varphi\\) is a lattice in its own order if and only if \\(\\pi(A)'\\) is abelian.\n4. If the closure \\(\\overline{F_\\varphi}\\) is a simplex, then \\(\\pi(A)'\\) is abelian.\n5. There are \\(A\\) and \\(\\varphi\\) with \\(\\pi_\\varphi(A)'\\) abelian and \\(\\overline{F_\\varphi}\\) not a simplex.\n\nParts (3)–(5) distinguish the lattice property of the generated face from the simplex property of its closure. The proof includes an explicit counterexample to the converse of (4).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 852,
        "through_line": 886,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 16.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-24::Lemma 16.2",
      "anchor": "oa-fnd-ir-24",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 16.2.** An infinite-dimensional von Neumann algebra \\(\\mathcal N\\) contains an infinite sequence of nonzero, pairwise orthogonal projections.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 887,
        "through_line": 890,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 17.1",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-18::Proposition 17.1",
      "anchor": "oa-fnd-ir-18",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 17.1.** Let \\(\\varphi\\in\\mathfrak S^G\\), with GNS triple \\(\\pi,H,\\xi\\).\n\n1. \\(\\mathfrak S^G\\) is compact and convex.\n2. There is exactly one unitary representation \\(U=U_\\varphi\\) of \\(G\\) on \\(H\\) with \\(U_s\\pi(a)\\xi=\\pi(\\alpha_s(a))\\xi\\). It satisfies \\(U_s\\xi=\\xi\\) and \\(U_s\\pi(a)U_s^*=\\pi(\\alpha_s(a))\\).\n3. Let \\(\\mathfrak M_\\varphi=\\pi(A)'\\cap U(G)'\\). Then \\(\\Theta_\\varphi\\) maps \\(\\{x\\in\\mathfrak M_\\varphi:\\ 0\\leq x\\leq1\\}\\) onto the set of invariant functionals \\(\\psi\\) with \\(0\\leq\\psi\\leq\\varphi\\).\n4. \\(\\varphi\\in\\partial_e\\mathfrak S^G\\) if and only if \\(\\mathfrak M_\\varphi=\\mathbb C1\\).\n5. If \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) and \\(\\kappa_\\mu(L^\\infty(\\mu))\\subseteq U(G)'\\), then \\(\\operatorname{supp}\\mu\\subseteq\\mathfrak S^G\\), so \\(\\mu(\\mathfrak S^G)=1\\).\n6. Suppose \\(\\mathfrak M_\\varphi\\) is abelian, and let \\(\\mu\\) be the orthogonal measure with \\(\\mathcal C_\\mu=\\mathfrak M_\\varphi\\) (Theorem 13.1). Then \\(\\mu\\) is concentrated on \\(K=\\mathfrak S^G\\), and, as a measure on \\(K\\), it majorizes every probability measure on \\(K\\) with barycentre \\(\\varphi\\). So it is the unique maximal measure on \\(K\\) that represents \\(\\varphi\\), and \\(\\mu(B)=0\\) for every Baire subset \\(B\\) of the compact space \\(K\\) that misses \\(\\partial_eK\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 895,
        "through_line": 941,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 18.1",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-19::Proposition 18.1",
      "anchor": "oa-fnd-ir-19",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 18.1.** Let \\(X\\) be compact Hausdorff and \\(\\mathcal D=C(X,A)\\), the continuous functions \\(X\\to A\\) with pointwise operations and the sup norm. For \\(f\\in C(X)\\) and \\(a\\in A\\) let \\(f\\otimes a\\) be the function \\(x\\mapsto f(x)a\\).\n\n1. \\(\\mathcal D\\) is a unital C\\(^*\\)-algebra, and the span \\(\\mathcal F\\) of the \\(f\\otimes a\\) is dense in it.\n2. Let \\(\\theta:C(X)\\to B(H)\\) be a unital \\(*\\)-homomorphism and \\(\\pi\\) a unital representation of \\(A\\) on \\(H\\) with \\(\\theta(C(X))\\subseteq\\pi(A)'\\). There is exactly one representation \\(\\tilde\\pi\\) of \\(\\mathcal D\\) with \\(\\tilde\\pi(f\\otimes a)=\\theta(f)\\pi(a)\\), and \\(\\tilde\\pi(\\mathcal D)'=\\pi(A)'\\cap\\theta(C(X))'\\).\n3. Let \\(G\\) act on \\(A\\) by \\(\\alpha\\), and on \\(\\mathcal D\\) by \\(\\beta_s(F)=\\alpha_s\\circ F\\); \\(\\beta\\) fixes every \\(f\\otimes1\\). Every \\(\\rho\\in\\partial_e\\mathfrak S^G(\\mathcal D)\\) has the form \\(\\rho(F)=\\omega(F(x))\\) for some \\(x\\in X\\) and some \\(\\omega\\in\\partial_e\\mathfrak S^G(A)\\). For \\(G=\\{1\\}\\): every pure state of \\(\\mathcal D\\) is \\(F\\mapsto\\omega(F(x))\\) with \\(\\omega\\) pure.\n4. Take \\(X=\\mathfrak S(A)\\). Then \\(\\Phi(\\omega)(F)=\\omega(F(\\omega))\\) defines a continuous one-to-one map \\(\\Phi:\\mathfrak S(A)\\to\\mathfrak S(\\mathcal D)\\) with \\(\\Phi(\\mathfrak S^G(A))\\subseteq\\mathfrak S^G(\\mathcal D)\\). And \\(\\Psi(\\rho)(a)=\\rho(1\\otimes a)\\) defines a continuous affine map \\(\\Psi:\\mathfrak S(\\mathcal D)\\to\\mathfrak S(A)\\) with \\(\\Psi\\circ\\Phi=\\mathrm{id}\\), \\(\\Psi(\\mathfrak S^G(\\mathcal D))\\subseteq\\mathfrak S^G(A)\\) and \\(\\Psi(\\partial_e\\mathfrak S^G(\\mathcal D))\\subseteq\\partial_e\\mathfrak S^G(A)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 946,
        "through_line": 983,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 19.1",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-20::Theorem 19.1",
      "anchor": "oa-fnd-ir-20",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 19.1.** Let \\(G\\) act on \\(A\\), \\(\\varphi\\in\\mathfrak S^G\\), and let \\(\\mathcal C\\) be a maximal abelian von Neumann subalgebra of \\(\\mathfrak M_\\varphi=\\pi_\\varphi(A)'\\cap U_\\varphi(G)'\\), that is, \\(\\mathcal C'\\cap\\mathfrak M_\\varphi=\\mathcal C\\). Let \\(\\mu\\) be the orthogonal measure with \\(\\mathcal C_\\mu=\\mathcal C\\) (Theorem 13.1). Then:\n\n1. \\(\\mu(\\mathfrak S^G)=1\\);\n2. \\(\\mu(B)=0\\) for every Baire subset \\(B\\) of the compact space \\(\\mathfrak S^G\\) with \\(B\\cap\\partial_e\\mathfrak S^G=\\varnothing\\);\n3. if \\(A\\) is separable, \\(\\mu\\) is concentrated on \\(\\partial_e\\mathfrak S^G\\);\n4. such algebras \\(\\mathcal C\\) exist. So every invariant state is the barycentre of an orthogonal measure that vanishes on each Baire subset of \\(\\mathfrak S^G\\) missing the ergodic states.\n\nFor \\(G=\\{1\\}\\), \\(\\mathfrak S^G=\\mathfrak S\\) and \\(\\partial_e\\mathfrak S^G=P(A)\\): if \\(\\mathcal C_\\mu\\) is maximal abelian in \\(\\pi_\\varphi(A)'\\), then \\(\\mu\\) vanishes on every Baire set that misses the pure states, and it is concentrated on \\(P(A)\\) when \\(A\\) is separable.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 988,
        "through_line": 1037,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 20.1",
      "kind": "definition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-21::Definition 20.1",
      "anchor": "oa-fnd-ir-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Definition 20.1.** The *central measure* of \\(\\varphi\\in\\mathfrak S\\) is the orthogonal \\(\\mu\\in M_\\varphi(\\mathfrak S)\\) with \\(\\mathcal C_\\mu=\\mathcal Z_\\varphi=\\pi_\\varphi(A)''\\cap\\pi_\\varphi(A)'\\); it exists and is unique by Theorem 13.1. A state \\(\\varphi\\) is *factorial* (or primary) if \\(\\pi_\\varphi(A)''\\) is a factor. We write \\(\\operatorname{Fac}(A)\\) for the set of factorial states. Pure states are factorial, since \\(\\pi_\\varphi(A)'=\\mathbb C1\\) (Proposition 8.1(4)).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1040,
        "through_line": 1041,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 20.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-21::Lemma 20.2",
      "anchor": "oa-fnd-ir-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 20.2** (Corners of a factor). Let \\(\\mathcal N\\) be a factor on \\(H\\) and \\(e\\in\\mathcal N\\) a nonzero projection. Then \\(\\mathcal N_e=\\{exe|_{eH}:\\ x\\in\\mathcal N\\}\\) is a factor on \\(eH\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1042,
        "through_line": 1045,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 20.3",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-21::Lemma 20.3",
      "anchor": "oa-fnd-ir-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 20.3.** Let \\(C\\) be a C\\(^*\\)-algebra with unit, and let \\(A_1,A_2\\subseteq C\\) be C\\(^*\\)-subalgebras that contain the unit of \\(C\\), commute with each other, and together generate \\(C\\). If \\(\\rho\\) is a factorial state of \\(C\\), then \\(\\rho|_{A_1}\\) is a factorial state of \\(A_1\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1046,
        "through_line": 1049,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 20.4",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-21::Theorem 20.4",
      "anchor": "oa-fnd-ir-21",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 20.4.** The central measure \\(\\mu\\) of \\(\\varphi\\in\\mathfrak S\\) satisfies \\(\\mu(B)=0\\) for every Baire set \\(B\\subseteq\\mathfrak S\\) with \\(B\\cap\\operatorname{Fac}(A)=\\varnothing\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1050,
        "through_line": 1068,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 21.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-22::Lemma 21.1",
      "anchor": "oa-fnd-ir-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 21.1.**\n\n1. \\(|\\Phi_{\\omega,T}(u)|\\leq\\|T\\|\\gamma(u)\\).\n2. \\(\\Phi_{\\omega,T}(N)=\\{0\\}\\) if and only if \\(T\\in\\pi(A)'\\).\n3. A linear functional \\(\\Phi\\) on \\(A\\odot A\\) equals \\(\\Phi_\\rho\\) for some \\(\\rho\\in I(\\omega)\\) if and only if \\(\\Phi(N)=\\{0\\}\\) and \\(0\\leq\\Phi(x^*\\otimes x)\\leq\\omega(x^*x)\\) for all \\(x\\in A\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1098,
        "through_line": 1105,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 21.2",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-22::Lemma 21.2",
      "anchor": "oa-fnd-ir-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 21.2.** Let \\(N_0\\) be a \\(\\gamma\\)-dense subset of \\(N\\), and \\(A_0\\) a dense \\(*\\)-subalgebra of \\(A\\) over \\(\\mathbb Q+i\\mathbb Q\\). A state \\(\\omega\\) is factorial exactly when the following holds. For every \\(\\varepsilon>0\\) and \\(x\\in A_0\\) there are \\(\\delta>0\\) and \\(u_1,\\dots,u_n\\in N_0\\) such that, for every \\(h\\in A_0\\) with \\(0\\leq h\\leq1\\),\n\\[\n\\begin{gathered}\n\\max_i|\\Phi_{\\omega,h}(u_i)|<\\delta\\\\\n\\Longrightarrow\\\\\n|\\omega(x^*hx)-\\omega(h)\\omega(x^*x)|<\\varepsilon .\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1106,
        "through_line": 1124,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 21.3",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-22::Theorem 21.3",
      "anchor": "oa-fnd-ir-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 21.3.** If \\(A\\) is separable, \\(\\operatorname{Fac}(A)\\) is a Borel subset of \\(\\mathfrak S\\); in fact an \\(F_{\\sigma\\delta}\\) set.\n\nThe quantifier order in (21.1) is essential: Remark 21.4 shows that placing the intersection over \\(h\\) before the unions over \\(m\\) and \\(u\\) can admit states that are not factorial.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1125,
        "through_line": 1153,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 21.5",
      "kind": "corollary",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-22::Corollary 21.5",
      "anchor": "oa-fnd-ir-22",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Corollary 21.5.** If \\(A\\) is separable, the central measure of every state is concentrated on \\(\\operatorname{Fac}(A)\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1154,
        "through_line": 1157,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 22.1",
      "kind": "example",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-28::Example 22.1",
      "anchor": "oa-fnd-ir-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Example 22.1** (Commutative algebras). Let \\(A=C(X)\\), \\(X\\) compact Hausdorff, so \\(\\mathfrak S=M_1^+(X)\\), and let \\(\\varphi=m\\). The GNS representation is multiplication on \\(L^2(X,m)\\), and \\(\\pi(A)'=\\{M_g:\\ g\\in L^\\infty(m)\\}\\), because \\(C(X)\\) is weak\\(^*\\)-dense in \\(L^\\infty(m)\\) and the multiplication operators by \\(L^\\infty(m)\\) form a [maximal abelian](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-09) algebra. This is abelian, so \\(m\\) has exactly one maximal measure (Theorem 15.2). It is \\(\\iota_*m\\), with \\(\\iota(x)=\\delta_x\\): indeed \\(\\int\\hat g\\,d\\iota_*m=m(g)\\), and \\(\\kappa_{\\iota_*m}(G)=M_{G\\circ\\iota}\\) is multiplicative with range \\(\\pi(A)'\\). Since \\(\\pi(A)'=\\pi(A)''\\cap\\pi(A)'\\), it is also the central measure. It is concentrated on the closed set \\(\\iota(X)=P(A)\\), for every \\(X\\), metrizable or not.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1160,
        "through_line": 1161,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 22.2",
      "kind": "example",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-28::Example 22.2",
      "anchor": "oa-fnd-ir-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Example 22.2** (The trace on \\(M_n\\)). Let \\(A=M_n(\\mathbb C)\\) and \\(\\tau=\\frac1n\\operatorname{tr}\\). The GNS space is \\(M_n\\) with \\(\\langle X,Y\\rangle=\\tau(Y^*X)\\), \\(\\pi\\) is left multiplication, \\(\\xi=1\\), and \\(\\pi(A)'\\) is the right multiplications \\(R_y\\), a copy of \\(M_n\\). The centre of \\(\\pi(A)''\\) is trivial, so \\(\\tau\\) is factorial and its central measure is \\(\\delta_\\tau\\). For the maximal abelian algebra \\(\\mathcal C=\\{R_d:\\ d\\text{ diagonal}\\}\\), the construction in the proof of Theorem 13.1 gives \\(e\\) = the projection onto the diagonal matrices, \\(\\theta(a)=R_{\\operatorname{diag}(a)}\\), characters \\(\\chi_i(R_d)=d_{ii}\\), \\(\\chi_i\\circ\\theta=\\omega_{e_i}\\), and \\(\\mu=\\frac1n\\sum_i\\delta_{\\omega_{e_i}}\\). It is concentrated on pure states (Theorem 19.1). Conjugating \\(\\mathcal C\\) by a unitary \\(R_w\\) gives the measure \\(\\frac1n\\sum_i\\delta_{\\omega_{w^*e_i}}\\) for another orthonormal basis; since \\(\\pi(A)'\\) is not abelian, \\(\\tau\\) has many maximal measures (Theorem 15.2), including non-orthogonal ones (compare Example 11.5).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1162,
        "through_line": 1163,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 22.3",
      "kind": "example",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-28::Example 22.3",
      "anchor": "oa-fnd-ir-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Example 22.3** (Trivial cases). For every state, \\(\\delta_\\varphi\\) is orthogonal with \\(\\mathcal C_{\\delta_\\varphi}=\\mathbb C1\\), since \\(\\kappa_{\\delta_\\varphi}(f)=f(\\varphi)1\\). It is maximal exactly when \\(\\varphi\\) is pure (Lemma 5.2 and Theorem 5.4), and then \\(M_\\varphi(\\mathfrak S)=\\{\\delta_\\varphi\\}\\) (Bauer's criterion, Proposition 3.3(3)). The central measure is \\(\\delta_\\varphi\\) exactly when \\(\\mathcal Z_\\varphi=\\mathbb C1\\), that is, when \\(\\varphi\\) is factorial (Theorem 13.1).\n\n**Remark 22.4** (Further examples). Other small, explicit examples in this lesson: a simplicial measure that is not orthogonal (Example 11.5); a state with abelian \\(\\pi_\\varphi(A)'\\) whose face has a closure that is not a simplex (Proposition 16.1(5)); a state that is not factorial but lies in the set of Remark 21.4, where the intersection over \\(h\\) comes first; and the invariant states in the proof of Theorem 23.2, in Theorem 23.4(3) and in Example 23.6.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1164,
        "through_line": 1167,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 22.5",
      "kind": "exercise",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-28::Exercise 22.5",
      "anchor": "oa-fnd-ir-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Exercise 22.5** (hard; Orthogonal measures of the trace). In Example 22.2, show that the orthogonal measures of \\(\\tau\\) are exactly the measures \\(\\sum_k\\frac{\\operatorname{tr}p_k}{n}\\delta_{\\omega_k}\\), where \\(p_1,\\dots,p_m\\) are nonzero orthogonal projections with \\(\\sum_kp_k=1\\) and \\(\\omega_k=\\operatorname{tr}(p_k\\,\\cdot)/\\operatorname{tr}p_k\\). Show that such a measure is maximal exactly when every \\(p_k\\) has rank one, and that one of them majorizes another exactly when its partition refines the other's.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1168,
        "through_line": 1171,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 22.6",
      "kind": "exercise",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-28::Exercise 22.6",
      "anchor": "oa-fnd-ir-28",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Exercise 22.6** (easy; Pure and factorial states through measures). Show: (a) \\(\\varphi\\) is pure if and only if \\(M_\\varphi(\\mathfrak S)=\\{\\delta_\\varphi\\}\\); (b) \\(\\varphi\\) is factorial exactly when its central measure is \\(\\delta_\\varphi\\); (c) if \\(\\varphi\\) is factorial but not pure, then \\(\\delta_\\varphi\\) is orthogonal but not maximal.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1172,
        "through_line": 1175,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 23.1",
      "kind": "lemma",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Lemma 23.1",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Lemma 23.1** (Mean ergodic lemma).\n\n1. For \\(\\eta\\in H\\), \\(e_0\\eta\\) is the element of minimal norm in the closed convex hull of \\(\\{U_s\\eta\\}\\).\n2. For \\(\\eta_1,\\dots,\\eta_n\\in H\\) and \\(\\varepsilon>0\\) there is a self-adjoint \\(S\\in\\operatorname{co}U(G)\\) with \\(\\|S\\eta_j-e_0\\eta_j\\|<\\varepsilon\\) for all \\(j\\).\n3. \\(Se_0=e_0S=e_0\\) for \\(S\\in\\operatorname{co}U(G)\\), and \\(e_0\\) commutes with \\(U(G)'\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1180,
        "through_line": 1187,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 23.2",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Theorem 23.2",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 23.2.** For \\(\\varphi\\in\\mathfrak S^G\\), with \\(\\pi=\\pi_\\varphi\\), the following are equivalent.\n\n- (i) \\(e_0\\pi(A)e_0\\) is commutative.\n- (ii\\(^*\\)) For all \\(x,y\\in A\\) and \\(\\zeta\\in H_0\\), \\[\n\\begin{gathered}\n\\inf\\{|\\langle\\pi(x'y-yx')\\zeta,\\zeta\\rangle|:\\\\\n\\ x'\\in K_G(x)\\}\\\\\n=0\n\\end{gathered}\n\\].\n\nConsider also the condition (ii), in which the infimum is taken only over the orbit, \\(x'=\\alpha_s(x)\\) with \\(s\\in G\\). It implies (ii\\(^*\\)), hence (i). But (i) does not imply (ii), as the end of the proof shows.\n\nThe theorem proves the equivalence with the convex-hull condition (ii\\(^*\\)). The end of the proof separates it from the orbit condition (ii), and Theorem 23.4(3) gives a generated face whose closure is not a simplex.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1188,
        "through_line": 1233,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 23.3",
      "kind": "proposition",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Proposition 23.3",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Proposition 23.3.** If (i) holds, then \\(\\mathfrak M_\\varphi=\\pi(A)'\\cap U(G)'\\) is abelian; in fact \\(\\mathfrak M_\\varphi=\\pi(A)'\\cap\\{e_0\\}'\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1234,
        "through_line": 1237,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 23.4",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Theorem 23.4",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 23.4.** Suppose (i) holds for \\(\\varphi\\in\\mathfrak S^G\\).\n\n1. The face \\(F^G_\\varphi\\) of \\(\\mathfrak S^G\\) generated by \\(\\varphi\\) has a lattice cone: \\(\\mathbb R_+F^G_\\varphi\\cong(\\mathfrak M_\\varphi)_+\\).\n2. \\(\\varphi\\) is the barycentre of exactly one maximal (boundary) probability measure on \\(\\mathfrak S^G\\). It is the orthogonal measure \\(\\mu\\) with \\(\\mathcal C_\\mu=\\mathfrak M_\\varphi\\), and \\(\\mu(B)=0\\) for every Baire subset \\(B\\) of \\(\\mathfrak S^G\\) that misses \\(\\partial_e\\mathfrak S^G\\); if \\(A\\) is separable, \\(\\mu\\) is concentrated on \\(\\partial_e\\mathfrak S^G\\).\n3. The closure \\(\\overline{F^G_\\varphi}\\) need not be a simplex.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1238,
        "through_line": 1245,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 23.5",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Theorem 23.5",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 23.5.** If (i) holds for every \\(\\varphi\\in\\mathfrak S^G\\) (the system is *\\(G\\)-abelian*), then \\(\\mathfrak S^G\\) is a simplex.",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1246,
        "through_line": 1249,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 23.6",
      "kind": "example",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Example 23.6",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Example 23.6** (Diagonal unitaries acting on \\(M_n\\)). Let \\(A=M_n(\\mathbb C)\\) with \\(n\\geq2\\), \\(G\\) the diagonal unitary matrices, \\(\\alpha_s=\\operatorname{Ad}s\\), and \\(\\varphi(x)=x_{11}\\).\n\nThis example separates conditions (i) and (ii): the compressed algebra is commutative, while the tested orbit commutator has constant modulus one.\n\n- *\\(\\mathfrak M_\\varphi\\) is one-dimensional.* \\(\\varphi\\) is the vector state of \\(e_1\\) in the identity representation on \\(\\mathbb C^n\\), which is irreducible, so \\(\\pi_\\varphi(A)'=\\mathbb C1\\).\n- *Condition (ii) fails.* \\(U_s=\\bar s_{11}s\\), and for \\(x=E_{12}\\), \\(y=E_{21}\\), \\(\\zeta=e_1\\): \\(\\langle[\\alpha_s(x),y]e_1,e_1\\rangle=s_{11}\\bar s_{22}\\), of modulus \\(1\\) for every \\(s\\).\n- *Condition (i) holds.* \\(H_0=\\{\\zeta:\\ \\bar s_{11}s_{jj}\\zeta_j=\\zeta_j\\ \\forall s\\}=\\mathbb Ce_1\\), so \\(e_0\\pi(A)e_0=\\mathbb Ce_0\\) is commutative.\n\nIn fact every invariant state satisfies (i). An invariant state has a diagonal density matrix \\(D\\); its GNS space is \\(\\{X\\in M_n:\\ X\\text{ vanishes on }\\ker D\\}\\) with \\(\\xi=D^{1/2}\\) and the Hilbert–Schmidt inner product, \\(U_sX=sXs^*\\), \\(H_0\\) is its diagonal part, and \\(e_0\\pi(x)e_0\\) is multiplication by \\(\\operatorname{diag}(x_{11},\\dots,x_{nn})\\). So the system is \\(G\\)-abelian, and \\(\\mathfrak S^G\\), the \\((n-1)\\)-simplex of diagonal states, is a simplex, as Theorem 23.5 predicts. The example shows that (ii) is strictly stronger than (i).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1250,
        "through_line": 1259,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 23.7",
      "kind": "exercise",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-25::Exercise 23.7",
      "anchor": "oa-fnd-ir-25",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Exercise 23.7** (medium; An invariant state under a finite group). Let \\(G=\\mathbb Z_2\\) act on \\(M_2(\\mathbb C)\\) by \\(\\operatorname{Ad}\\operatorname{diag}(1,-1)\\), and \\(\\varphi=\\tau\\). Find \\(\\mathfrak S^G\\), \\(\\partial_e\\mathfrak S^G\\), \\(\\mathfrak M_\\tau\\), and the unique maximal measure of \\(\\tau\\) on \\(\\mathfrak S^G\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1260,
        "through_line": 1263,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 24.1",
      "kind": "theorem",
      "unit": "integral-representations-of-states",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/integral-representations-of-states#oa-fnd-ir-26::Theorem 24.1",
      "anchor": "oa-fnd-ir-26",
      "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
      "statement_and_full_conditions": "**Theorem 24.1.**\n\n- (a) (L)\\(\\iff\\)(S), and (S)\\(\\Rightarrow\\)(L\\(_p\\)).\n- (b) If (S) holds, the system is \\(G\\)-abelian, so \\(\\mathfrak S^G\\) is a simplex.\n- (c) If (S) holds, then for \\(\\varphi\\in\\mathfrak S^G\\) the following are equivalent: (i) \\(\\varphi\\) is ergodic; (ii) \\(U_\\varphi(G)'\\cap\\mathcal Z_\\varphi=\\mathbb C1\\); (iii) \\(H_0=\\mathbb C\\xi_\\varphi\\).",
      "proof_locus": {
        "source": "src/integral-representations-of-states.md",
        "line": 1280,
        "through_line": 1305,
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 2.1",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-01::Definition 2.1",
      "anchor": "oa-fnd-af-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 2.1.** A *size vector* is \\(\\mathbf m=(m_1,\\dots,m_r)\\in\\mathbb N^r\\). The *multimatrix algebra* of \\(\\mathbf m\\) is\n\\[\nM_{\\mathbf m}=M_{m_1}\\oplus\\cdots\\oplus M_{m_r}.\n\\]\nWe write its elements as \\(x=(x_1,\\dots,x_r)\\) and call \\(M_{m_i}\\) its \\(i\\)-th *summand*. We allow \\(r=0\\), and then \\(M_{\\mathbf m}=0\\). Let \\(z_i\\) be the unit of the \\(i\\)-th summand and \\(e^{(i)}_{ab}\\) its matrix units. The minimal projections of \\(M_{\\mathbf m}\\) are the rank-one projections of the single summands. The *rank vector* of a projection \\(p=(p_1,\\dots,p_r)\\) is\n\\[\n\\begin{gathered}\n\\operatorname{rk}p\\\\\n=(\\operatorname{Tr}p_1,\\dots,\\operatorname{Tr}p_r)^T\\in\\mathbb Z^r_+,\\\\\n0\\\\\n\\leq\\operatorname{rk}p\\\\\n\\leq\\mathbf m .\n\\end{gathered}\n\\]\n\nThe first aim is to show that every finite-dimensional C\\*-algebra is isomorphic to some \\(M_{\\mathbf m}\\). We do not assume a unit.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 48,
        "through_line": 64,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.2",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-01::Lemma 2.2",
      "anchor": "oa-fnd-af-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 2.2.** Let \\(F\\) be a finite-dimensional C\\*-algebra.\n\n1. Every self-adjoint element of \\(F\\) is a real linear combination of mutually orthogonal projections of \\(F\\). In particular \\(F\\) is spanned by its projections.\n2. Mutually orthogonal nonzero projections of \\(F\\) are linearly independent, so there are at most \\(\\dim F\\) of them. Every nonzero projection of \\(F\\) dominates a minimal projection of \\(F\\).\n3. \\(F\\) has a unit.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 65,
        "through_line": 76,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.3",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-01::Lemma 2.3",
      "anchor": "oa-fnd-af-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 2.3** (Full matrix algebras). Let \\(E\\) be a nonzero finite-dimensional C\\*-algebra whose centre is \\(\\mathbb C1_E\\). Let \\(e\\) be a minimal projection of \\(E\\). Then there are \\(v_1=e,v_2,\\dots,v_k\\in E\\) with \\(v_a^*v_a=e\\) for all \\(a\\) and \\(\\sum_av_av_a^*=1_E\\). The elements \\(e_{ab}=v_av_b^*\\) satisfy \\(e_{ab}e_{cd}=\\delta_{bc}e_{ad}\\) and \\(e_{ab}^*=e_{ba}\\), and \\((\\xi_{ab})\\mapsto\\sum_{a,b}\\xi_{ab}e_{ab}\\) is an isomorphism of \\(M_k\\) onto \\(E\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 77,
        "through_line": 111,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 2.4",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-01::Theorem 2.4",
      "anchor": "oa-fnd-af-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 2.4** (Structure of finite-dimensional C\\*-algebras). Every finite-dimensional C\\*-algebra \\(F\\) is isomorphic to \\(M_{\\mathbf m}\\) for some size vector \\(\\mathbf m\\). The number \\(r\\) of summands is the dimension of the centre of \\(F\\), and the entries of \\(\\mathbf m\\) are determined by \\(F\\) up to their order.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 112,
        "through_line": 119,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 2.5",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-01::Lemma 2.5",
      "anchor": "oa-fnd-af-01",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 2.5** (Projections of a multimatrix algebra). For projections \\(p,q\\in M_{\\mathbf m}\\) the following are equivalent: (i) \\(\\operatorname{rk}p=\\operatorname{rk}q\\); (ii) \\(p\\sim q\\); (iii) \\(q=upu^*\\) for some \\(u\\in U(M_{\\mathbf m})\\). The rank vectors of the projections of \\(M_{\\mathbf m}\\) are exactly the \\(x\\in\\mathbb Z^r\\) with \\(0\\leq x\\leq\\mathbf m\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 120,
        "through_line": 123,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.1",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-02::Definition 3.1",
      "anchor": "oa-fnd-af-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 3.1.** Let \\(\\mathbf m\\in\\mathbb N^r\\), \\(\\mathbf n\\in\\mathbb N^s\\), and let \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) be a \\*-homomorphism. Write \\(\\varphi(x)=(\\varphi_1(x),\\dots,\\varphi_s(x))\\). The *multiplicity matrix* of \\(\\varphi\\) is the \\(s\\times r\\) matrix \\(\\alpha(\\varphi)\\) with entries\n\\[\n\\alpha(\\varphi)_{ji}=\\operatorname{Tr}\\varphi_j\\big(e^{(i)}_{11}\\big),\n\\]\nthe number of times the \\(i\\)-th summand of \\(M_{\\mathbf m}\\) enters the \\(j\\)-th summand of \\(M_{\\mathbf n}\\). Its *defect vector* is \\(d(\\varphi)=\\mathbf n-\\alpha(\\varphi)\\mathbf m\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 128,
        "through_line": 133,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 3.2",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-02::Theorem 3.2",
      "anchor": "oa-fnd-af-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 3.2** (Homomorphisms between multimatrix algebras). Let \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) be a \\*-homomorphism with multiplicity matrix \\(\\alpha=\\alpha(\\varphi)\\).\n\n1. For every projection \\(p\\in M_{\\mathbf m}\\),\n\\[\n\\operatorname{rk}\\varphi(p)=\\alpha\\operatorname{rk}p .\n\\tag{3.1}\n\\]\nIn particular \\(\\alpha_{ji}=\\operatorname{Tr}\\varphi_j(e)\\) for every minimal projection \\(e\\) of the \\(i\\)-th summand. If \\(\\psi:M_{\\mathbf n}\\to M_{\\mathbf k}\\) is another \\*-homomorphism, then \\(\\alpha(\\psi\\circ\\varphi)=\\alpha(\\psi)\\alpha(\\varphi)\\).\n2. \\(\\alpha\\mathbf m\\leq\\mathbf n\\), that is, \\(d(\\varphi)\\geq0\\), with equality if and only if \\(\\varphi\\) is unital. The map \\(\\varphi\\) is injective if and only if no column of \\(\\alpha\\) is zero.\n3. Conversely, let \\(\\alpha\\) be any \\(s\\times r\\) matrix with entries in \\(\\mathbb Z_+\\) and \\(\\alpha\\mathbf m\\leq\\mathbf n\\), and put \\(d=\\mathbf n-\\alpha\\mathbf m\\). The *standard homomorphism*\n\\[\n\\begin{gathered}\n\\varphi_\\alpha(x)_j\\\\\n=\\operatorname{diag}\\big(x_1^{(\\alpha_{j1})},x_2^{(\\alpha_{j2})},\\\\\n\\dots,x_r^{(\\alpha_{jr})},0_{d_j}\\big),\\\\\nj=1,\\dots,s,\n\\end{gathered}\n\\tag{3.2}\n\\]\nis a \\*-homomorphism with multiplicity matrix \\(\\alpha\\).\n4. There is \\(u\\in U(M_{\\mathbf n})\\) with \\(\\varphi=\\operatorname{Ad}u\\circ\\varphi_\\alpha\\). Consequently two \\*-homomorphisms \\(M_{\\mathbf m}\\to M_{\\mathbf n}\\) are unitarily equivalent if and only if they have the same multiplicity matrix.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 134,
        "through_line": 191,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
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            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
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              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
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              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 3.5",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-02::Definition 3.5",
      "anchor": "oa-fnd-af-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 3.5** (Bratteli diagram of a homomorphism). The *Bratteli diagram* of \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) is the graph with \\(r\\) vertices on the left, labelled \\(m_1,\\dots,m_r\\), and \\(s\\) vertices on the right, labelled \\(n_1,\\dots,n_s\\), in which the \\(i\\)-th left vertex and the \\(j\\)-th right vertex are joined by \\(\\alpha(\\varphi)_{ji}\\) edges. By Theorem 3.2(4) it determines \\(\\varphi\\) up to unitary equivalence.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 192,
        "through_line": 193,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
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          "local_scopes": [
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              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
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              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 3.6",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-02::Example 3.6",
      "anchor": "oa-fnd-af-02",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 3.6.** (a) Let \\(\\mathbf m=(1,2)\\), \\(\\mathbf n=(5,4)\\) and \\(\\alpha=\\begin{pmatrix}3&1\\\\0&2\\end{pmatrix}\\). Then \\(\\alpha\\mathbf m=(5,4)^T=\\mathbf n\\), so the standard homomorphism\n\\[\n\\begin{gathered}\n\\varphi_\\alpha(\\lambda,y)\\\\\n=\\big(\\operatorname{diag}(\\lambda,\\lambda,\\lambda,y),\\ \\operatorname{diag}(y,y)\\big),\\\\\n\\lambda\\in\\mathbb C,\\ y\\in M_2,\n\\end{gathered}\n\\]\nis unital and injective. In its diagram, the left vertex \\(1\\) is joined to the right vertex \\(5\\) by three edges, and the left vertex \\(2\\) is joined to \\(5\\) by one edge and to \\(4\\) by two edges.\n\n(b) The map \\(M_2\\to M_5\\), \\(x\\mapsto\\operatorname{diag}(x,x,0)\\), has multiplicity matrix \\((2)\\) and defect \\((1)\\): it is injective and not unital. The map \\(\\mathbb C^2\\to\\mathbb C\\), \\((\\lambda,\\mu)\\mapsto\\lambda\\), has multiplicity matrix \\((1\\ \\ 0)\\): its second column is zero, and it is not injective.\n\n(c) The two maps \\(M_2\\to M_4\\) given by \\(x\\mapsto\\operatorname{diag}(x,x)\\) and by \\(x\\mapsto\\begin{pmatrix}x_{11}1_2&x_{12}1_2\\\\x_{21}1_2&x_{22}1_2\\end{pmatrix}\\) both have multiplicity matrix \\((2)\\). By Theorem 3.2(4) they are unitarily equivalent; here the unitary is the permutation matrix that exchanges the second and third basis vectors.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 194,
        "through_line": 207,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.1",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Definition 4.1",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 4.1.** An *inductive sequence* \\((A_n,\\varphi_n)_{n\\geq1}\\) consists of C\\*-algebras \\(A_n\\) and \\*-homomorphisms \\(\\varphi_n:A_n\\to A_{n+1}\\). For \\(m>n\\) put \\(\\varphi_{m,n}=\\varphi_{m-1}\\circ\\cdots\\circ\\varphi_n:A_n\\to A_m\\), and \\(\\varphi_{n,n}=\\mathrm{id}\\). An *inductive limit* of the sequence is a C\\*-algebra \\(A\\) with \\*-homomorphisms \\(\\varphi_{\\infty,n}:A_n\\to A\\) such that\n\n1. \\(\\varphi_{\\infty,n+1}\\circ\\varphi_n=\\varphi_{\\infty,n}\\) for all \\(n\\);\n2. \\(\\bigcup_n\\varphi_{\\infty,n}(A_n)\\) is dense in \\(A\\);\n3. \\(\\|\\varphi_{\\infty,n}(a)\\|=\\lim_{m\\to\\infty}\\|\\varphi_{m,n}(a)\\|\\) for all \\(n\\) and \\(a\\in A_n\\).\n\nThe limit in (3) exists, because \\(\\|\\varphi_{m+1,n}(a)\\|=\\|\\varphi_m(\\varphi_{m,n}(a))\\|\\leq\\|\\varphi_{m,n}(a)\\|\\) by Section 1(a).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 212,
        "through_line": 219,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.2",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Proposition 4.2",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 4.2.** Let \\((A_n,\\varphi_n)\\) be an inductive sequence.\n\n1. It has an inductive limit.\n2. (*Universal property.*) Let \\((A,\\varphi_{\\infty,n})\\) be an inductive limit, \\(B\\) a C\\*-algebra and \\(\\sigma_n:A_n\\to B\\) \\*-homomorphisms with \\(\\sigma_{n+1}\\circ\\varphi_n=\\sigma_n\\). There is exactly one \\*-homomorphism \\(\\sigma:A\\to B\\) with \\(\\sigma\\circ\\varphi_{\\infty,n}=\\sigma_n\\) for all \\(n\\).\n3. If \\((A,\\varphi_{\\infty,n})\\) and \\((A',\\varphi'_{\\infty,n})\\) are inductive limits, there is exactly one isomorphism \\(\\theta:A\\to A'\\) with \\(\\theta\\circ\\varphi_{\\infty,n}=\\varphi'_{\\infty,n}\\) for all \\(n\\).\n4. If every \\(\\varphi_n\\) is injective, then every \\(\\varphi_{\\infty,n}\\) is isometric, and \\(A\\) is the closure of the increasing union of the subalgebras \\(\\varphi_{\\infty,n}(A_n)\\cong A_n\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 220,
        "through_line": 240,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
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          "unit": "af-algebras",
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              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
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              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
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              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 4.3",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Example 4.3",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 4.3** (Condition (3) matters). Let \\(A_n=C_0([n,\\infty))\\) and let \\(\\varphi_n\\) be restriction to \\([n+1,\\infty)\\). Every \\(\\varphi_n\\) is onto and every \\(A_n\\) is nonzero. For \\(a\\in A_n\\), \\(\\|\\varphi_{m,n}(a)\\|=\\sup_{t\\geq m}|a(t)|\\to0\\), because \\(a\\) vanishes at infinity. So \\(\\varphi_{\\infty,n}=0\\) for every \\(n\\), and the inductive limit is the zero algebra.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 241,
        "through_line": 242,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
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              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 4.4",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Lemma 4.4",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 4.4** (Ladders). Let \\((A_n,\\varphi_n)\\) and \\((B_n,\\psi_n)\\) be inductive sequences with limits \\(A\\) and \\(B\\), and let \\(\\theta_n:A_n\\to B_n\\) be \\*-homomorphisms with \\(\\theta_{n+1}\\circ\\varphi_n=\\psi_n\\circ\\theta_n\\). There is exactly one \\*-homomorphism \\(\\theta:A\\to B\\) with \\(\\theta\\circ\\varphi_{\\infty,n}=\\psi_{\\infty,n}\\circ\\theta_n\\) for all \\(n\\). If every \\(\\theta_n\\) is an isomorphism, so is \\(\\theta\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 243,
        "through_line": 248,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 4.5",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Theorem 4.5",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 4.5** (Unitary perturbation of the connecting maps). Let \\((A_n,\\varphi_n)\\) be an inductive sequence, and for each \\(n\\geq2\\) let \\(u_n\\) be a unitary of \\(A_n^+\\). Put \\(\\varphi_n'=\\operatorname{Ad}u_{n+1}\\circ\\varphi_n\\). Define unitaries \\(w_n\\in A_n^+\\) by\n\\[\n\\begin{gathered}\nw_1\\\\\n=1,\\\\\nw_{n+1}\\\\\n=u_{n+1}\\,\\varphi_n^+(w_n).\n\\end{gathered}\n\\tag{4.1}\n\\]\nThen \\(\\operatorname{Ad}w_{n+1}\\circ\\varphi_n=\\varphi_n'\\circ\\operatorname{Ad}w_n\\) for every \\(n\\). Consequently there is an isomorphism \\(\\theta:\\varinjlim(A_n,\\varphi_n)\\to\\varinjlim(A_n,\\varphi_n')\\) with \\(\\theta\\circ\\varphi_{\\infty,n}=\\varphi'_{\\infty,n}\\circ\\operatorname{Ad}w_n\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 249,
        "through_line": 281,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.7",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Definition 4.7",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 4.7.** A C\\*-algebra \\(A\\) is an *AF-algebra* if it contains an increasing sequence \\(A_1\\subseteq A_2\\subseteq\\cdots\\) of finite-dimensional C\\*-subalgebras whose union is dense. Such a sequence is a *generating sequence* of \\(A\\), and the \\*-subalgebra \\(A_\\infty=\\bigcup_kA_k\\) is its *local algebra*.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 282,
        "through_line": 284,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 4.8",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Proposition 4.8",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 4.8.**\n\n1. The inductive limit of any inductive sequence of finite-dimensional C\\*-algebras is an AF-algebra.\n2. An AF-algebra is separable.\n3. If \\(A\\) is a unital AF-algebra with generating sequence \\((A_k)\\), then \\(1_A\\in A_k\\) for all large \\(k\\). So every unital AF-algebra has a generating sequence of C\\*-subalgebras that contain \\(1_A\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 285,
        "through_line": 298,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 4.9",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Definition 4.9",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 4.9** (Bratteli diagram of a sequence). Let \\((A_k)\\) be a generating sequence of an AF-algebra, with \\(A_k\\cong M_{\\mathbf m(k)}\\), \\(\\mathbf m(k)\\in\\mathbb N^{r_k}\\), and let \\(\\alpha_k\\) be the multiplicity matrix of the inclusion \\(A_k\\subseteq A_{k+1}\\) (an \\(r_{k+1}\\times r_k\\) matrix). The *Bratteli diagram* of the sequence has, at level \\(k\\), one vertex for each summand of \\(A_k\\), labelled by its size, and \\((\\alpha_k)_{ji}\\) edges between the \\(i\\)-th vertex at level \\(k\\) and the \\(j\\)-th vertex at level \\(k+1\\). The same definition applies to any inductive sequence of multimatrix algebras. We write \\(\\alpha_{l,k}=\\alpha_{l-1}\\cdots\\alpha_k\\) for \\(l>k\\) and \\(\\alpha_{k,k}=1\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 299,
        "through_line": 300,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 4.10",
      "kind": "corollary",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-03::Corollary 4.10",
      "anchor": "oa-fnd-af-03",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Corollary 4.10** (The diagram determines the algebra). Let \\((M_{\\mathbf m(k)},\\varphi_k)\\) and \\((M_{\\mathbf m(k)},\\varphi'_k)\\) be inductive sequences with the same size vectors and with \\(\\alpha(\\varphi_k)=\\alpha(\\varphi'_k)\\) for all \\(k\\). Then their inductive limits are isomorphic. Conversely, for every sequence of size vectors \\(\\mathbf m(k)\\) and matrices \\(\\alpha_k\\) with entries in \\(\\mathbb Z_+\\) and \\(\\alpha_k\\mathbf m(k)\\leq\\mathbf m(k+1)\\), the standard homomorphisms \\(\\varphi_{\\alpha_k}\\) form an inductive sequence with these multiplicity matrices; its limit is an AF-algebra, and the connecting maps are injective exactly when no \\(\\alpha_k\\) has a zero column.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 301,
        "through_line": 304,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "2. Finite-dimensional C\\*-algebras",
              "line": 46,
              "through_line": 123,
              "anchors": [
                "OA-FND-AF-01"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "3. Homomorphisms between finite-dimensional C\\*-algebras",
              "line": 124,
              "through_line": 207,
              "anchors": [
                "OA-FND-AF-02"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "4. Inductive limits and AF-algebras",
              "line": 208,
              "through_line": 304,
              "anchors": [
                "OA-FND-AF-03"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
          "source_locus": "§8; PDF 44–48",
          "role": "proof comparison",
          "correspondence": "Introductory AF treatment, classification stated",
          "explanation": "Finite-dimensional decomposition and injective inductive limits compared; several proofs and classification are referenced or exercises. Course supplies full multiplicities, scaled K0, trace and classification arguments."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.1",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-04::Example 5.1",
      "anchor": "oa-fnd-af-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 5.1** (Compact operators). Let \\(H\\) be a Hilbert space with a countably infinite orthonormal basis \\((\\varepsilon_i)_{i\\geq1}\\), let \\(P_n\\) be the projection onto the span of \\(\\varepsilon_1,\\dots,\\varepsilon_n\\), and let \\(\\mathcal K\\) be the C\\*-algebra of compact operators on \\(H\\). The algebras \\(A_n=P_nB(H)P_n\\cong M_n\\) increase. Their union is dense in \\(\\mathcal K\\): for compact \\(T\\), Section 1(h) gives\n\\[\n\\begin{gathered}\n\\|T-P_nTP_n\\|\\\\\n\\leq\\|T-P_nT\\|+\\|P_n\\|\\,\\|(T^*-P_nT^*)^*\\|\\to0 .\n\\end{gathered}\n\\]\nSo \\(\\mathcal K\\) is an AF-algebra. The inclusion \\(A_n\\subseteq A_{n+1}\\) is \\(x\\mapsto\\operatorname{diag}(x,0)\\): its multiplicity matrix is \\((1)\\) and its defect is \\((1)\\). The Bratteli diagram has one vertex at each level, labelled \\(n\\) at level \\(n\\), with single edges between consecutive levels.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 307,
        "through_line": 315,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.2",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-04::Example 5.2",
      "anchor": "oa-fnd-af-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 5.2** (The unitization of the compact operators). The C\\*-algebra \\(\\mathcal K+\\mathbb C1\\subseteq B(H)\\) is isomorphic to \\(\\mathcal K^+\\): the map \\(k+\\lambda1\\mapsto k+\\lambda1_H\\) is an injective \\*-homomorphism (\\(1_H\\notin\\mathcal K\\)), hence an isomorphism onto its image (Section 1(a)). The algebras\n\\[\nA_n=P_nB(H)P_n+\\mathbb C(1-P_n)\\cong M_n\\oplus\\mathbb C\n\\]\nincrease and contain \\(1_H\\), and their union is dense in \\(\\mathcal K+\\mathbb C1\\), since \\(k+\\lambda1\\) is the limit of \\(P_nkP_n+\\lambda1\\in A_n\\). The inclusion \\(A_n\\subseteq A_{n+1}\\) is \\((x,\\lambda)\\mapsto(\\operatorname{diag}(x,\\lambda),\\lambda)\\). With the summands ordered as \\((M_n,\\mathbb C)\\), its multiplicity matrix is\n\\[\n\\alpha_n=\\begin{pmatrix}1&1\\\\0&1\\end{pmatrix},\\qquad \\alpha_n\\binom n1=\\binom{n+1}1 ,\n\\]\nso the inclusions are unital. In the Bratteli diagram, level \\(n\\) has two vertices, labelled \\(n\\) and \\(1\\); the vertex \\(n\\) is joined to the vertex \\(n+1\\) of the next level, and the vertex \\(1\\) is joined both to the vertex \\(1\\) and to the vertex \\(n+1\\) of the next level.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 316,
        "through_line": 325,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 5.3",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-04::Example 5.3",
      "anchor": "oa-fnd-af-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 5.3** (UHF algebras). Let \\(k_1,k_2,\\dots\\) be integers \\(\\geq2\\), put \\(d_1=1\\) and \\(d_{n+1}=k_1k_2\\cdots k_n\\), and let \\(A_n=M_{d_n}\\) with the unital connecting maps \\(x\\mapsto x^{(k_n)}\\) of multiplicity \\(k_n\\). The limit is the *UHF algebra of type \\((k_n)\\)*; its Bratteli diagram has one vertex at each level, labelled \\(d_n\\), and \\(k_n\\) edges between levels \\(n\\) and \\(n+1\\). When every \\(k_n=2\\) we get the *CAR algebra*. Identify \\(M_{d_n}=M_{2^{n-1}}\\) with the tensor product of \\(n-1\\) copies of \\(M_2\\) (for \\(n=1\\), with \\(\\mathbb C\\)). The map \\(x\\mapsto x\\otimes1\\) into the tensor product of \\(n\\) copies has multiplicity \\(2\\), so by Corollary 4.10 the CAR algebra is also the limit of the tensor powers of \\(M_2\\) under \\(x\\mapsto x\\otimes1\\): it is the infinite tensor product \\(M_2\\otimes M_2\\otimes\\cdots\\).\n\nFor a C\\*-algebra \\(A\\), let \\(M_\\infty(A)=\\bigcup_nM_n(A)\\), where \\(M_n(A)\\) sits in \\(M_{n+1}(A)\\) as the upper left corner, \\(x\\mapsto\\operatorname{diag}(x,0)\\). This inclusion is an injective \\*-homomorphism of C\\*-algebras (Section 1(f)), hence isometric, so \\(M_\\infty(A)\\) carries a norm. The completion lemma in Section 1(i) gives its Banach completion. Products and adjoints extend to it: for Cauchy sequences \\(a_n,b_n\\), their boundedness and\n\\[\n\\begin{gathered}\n\\|a_nb_n-a_mb_m\\|\\\\\n\\leq\\|a_n\\|\\,\\|b_n-b_m\\|+\\|a_n-a_m\\|\\,\\|b_m\\|\n\\end{gathered}\n\\]\nshow that \\(a_nb_n\\) is Cauchy and its class is independent of the approximants; adjoints are isometric. Associativity and the \\(C^*\\)-identity pass to the limit. Thus the completion is a \\(C^*\\)-algebra, called the *stable algebra* \\(A\\otimes\\mathcal K\\) [Blackadar 2006, II.6.6.11]; it is an inductive limit of the sequence \\((M_n(A))\\) under the corner inclusions.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 326,
        "through_line": 336,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 5.4",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-04::Proposition 5.4",
      "anchor": "oa-fnd-af-04",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 5.4** (Matrices and stabilization). Let \\(A\\) be an AF-algebra with generating sequence \\((A_k)\\), \\(A_k\\cong M_{\\mathbf m(k)}\\), and multiplicity matrices \\(\\alpha_k\\).\n\n1. For each \\(n\\), \\(M_n(A)\\) is an AF-algebra with generating sequence \\((M_n(A_k))_k\\), \\(M_n(A_k)\\cong M_{n\\mathbf m(k)}\\), and the multiplicity matrices are again the \\(\\alpha_k\\).\n2. \\(A\\otimes\\mathcal K\\) is an AF-algebra with generating sequence \\((M_k(A_k))_k\\), \\(M_k(A_k)\\cong M_{k\\mathbf m(k)}\\), and the multiplicity matrices are again the \\(\\alpha_k\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 337,
        "through_line": 345,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.1",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-05::Definition 6.1",
      "anchor": "oa-fnd-af-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 6.1.** An *ordered abelian group* \\((G,G^+)\\) is an abelian group \\(G\\) with a subset \\(G^+\\) such that \\(G^++G^+\\subseteq G^+\\), \\(G^+\\cap(-G^+)=\\{0\\}\\) and \\(G=G^+-G^+\\). We write \\(g\\leq h\\) if \\(h-g\\in G^+\\). A *scaled ordered group* \\((G,G^+,\\Sigma)\\) is an ordered abelian group with a subset \\(\\Sigma\\subseteq G^+\\), the *scale*. A *homomorphism* \\(h:(G,G^+,\\Sigma)\\to(H,H^+,\\Sigma')\\) is a group homomorphism with \\(h(G^+)\\subseteq H^+\\) and \\(h(\\Sigma)\\subseteq\\Sigma'\\); an *isomorphism* is a bijective homomorphism whose inverse is a homomorphism, that is, \\(h(G^+)=H^+\\) and \\(h(\\Sigma)=\\Sigma'\\).\n\nThe multimatrix algebra \\(M_{\\mathbf m}\\) gives the scaled ordered group \\((\\mathbb Z^r,\\mathbb Z^r_+,\\Sigma_{\\mathbf m})\\) with\n\\[\n\\Sigma_{\\mathbf m}=\\{x\\in\\mathbb Z^r:0\\leq x\\leq\\mathbf m\\},\n\\]\nthe set of rank vectors of projections of \\(M_{\\mathbf m}\\) (Lemma 2.5). A \\*-homomorphism \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) gives the homomorphism \\(x\\mapsto\\alpha(\\varphi)x\\) of scaled ordered groups: it maps \\(\\mathbb Z^r_+\\) into \\(\\mathbb Z^s_+\\), and \\(\\Sigma_{\\mathbf m}\\) into \\(\\Sigma_{\\mathbf n}\\) because \\(\\alpha(\\varphi)\\mathbf m\\leq\\mathbf n\\). By (3.1) it sends the rank vector of \\(p\\) to the rank vector of \\(\\varphi(p)\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 348,
        "through_line": 355,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 6.2",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-05::Definition 6.2",
      "anchor": "oa-fnd-af-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 6.2** (Scaled dimension group of a generating sequence). Let \\(A\\) be an AF-algebra with a generating sequence \\((A_k)\\), \\(A_k\\cong M_{\\mathbf m(k)}\\) with \\(\\mathbf m(k)\\in\\mathbb N^{r_k}\\), and multiplicity matrices \\(\\alpha_k\\). Let \\(G\\) be the direct limit of the groups \\(\\mathbb Z^{r_k}\\) under the maps \\(\\alpha_k\\): its elements are the classes \\(\\alpha_{\\infty,k}(x)\\) of pairs \\((k,x)\\), \\(x\\in\\mathbb Z^{r_k}\\), where \\((k,x)\\) and \\((l,y)\\) define the same class if \\(\\alpha_{N,k}x=\\alpha_{N,l}y\\) for some \\(N\\geq k,l\\); addition is computed at a common stage. Put\n\\[\nG^+=\\bigcup_k\\alpha_{\\infty,k}\\big(\\mathbb Z^{r_k}_+\\big),\\qquad \\Sigma=\\bigcup_k\\alpha_{\\infty,k}\\big(\\Sigma_{\\mathbf m(k)}\\big).\n\\]\nThe triple \\((G,G^+,\\Sigma)\\) is the *scaled dimension group* of the sequence. For a projection \\(p\\in A_k\\) we write \\([p]=\\alpha_{\\infty,k}(\\operatorname{rk}p)\\); by (3.1) this does not depend on \\(k\\).\n\nBy Remark 3.4, other identifications \\(A_k\\cong M_{\\mathbf m(k)}\\) change the triple only by an isomorphism. Section 7 shows that it does not depend on the generating sequence either.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 356,
        "through_line": 363,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
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          "unit": "af-algebras",
          "local_scopes": [
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              "heading": "6. The scaled dimension group",
              "line": 346,
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              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 6.3",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-05::Lemma 6.3",
      "anchor": "oa-fnd-af-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 6.3.** In the setting of Definition 6.2:\n\n1. \\((G,G^+)\\) is an ordered abelian group.\n2. If \\(0\\neq x\\in\\mathbb Z^{r_k}_+\\), then \\(\\alpha_{\\infty,k}(x)\\neq0\\).\n3. \\(\\alpha_{\\infty,k}(x)\\in G^+\\) if and only if \\(\\alpha_{l,k}x\\geq0\\) for some \\(l\\geq k\\); and \\(\\alpha_{\\infty,k}(x)\\in\\Sigma\\) if and only if \\(\\alpha_{l,k}x\\in\\Sigma_{\\mathbf m(l)}\\) for some \\(l\\geq k\\).\n4. The scale is *hereditary*: if \\(g\\in G\\), \\(h\\in\\Sigma\\) and \\(0\\leq g\\leq h\\), then \\(g\\in\\Sigma\\).\n5. If \\(A\\) is unital and \\(1_A\\in A_k\\) for all \\(k\\), then \\([1_A]=\\alpha_{\\infty,k}(\\mathbf m(k))\\) for every \\(k\\), and \\(\\Sigma=\\{g\\in G:0\\leq g\\leq[1_A]\\}\\).\n6. For the generating sequence \\((M_k(A_k))\\) of \\(A\\otimes\\mathcal K\\) (Proposition 5.4), the group and positive cone are those of \\((A_k)\\), and the scale is the whole positive cone.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 364,
        "through_line": 386,
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              "heading": "6. The scaled dimension group",
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              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
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              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
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          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 6.4",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-05::Example 6.4",
      "anchor": "oa-fnd-af-05",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 6.4** (Computations).\n\n(a) *Compact operators* (Example 5.1). All the matrices are \\((1)\\), so \\(G=\\mathbb Z\\) and \\(G^+=\\mathbb Z_+\\). The scale of \\(M_n\\) is \\(\\{0,1,\\dots,n\\}\\), so \\(\\Sigma=\\mathbb Z_+\\): the classes of the projections of rank \\(0,1,2,\\dots\\) of \\(\\mathcal K\\).\n\nThe finite-rank projections constructed here prove directly that the scale is all of \\(\\mathbb Z_+\\).\n\n(b) *Unitization of the compact operators* (Example 5.2). Define \\(h_n:\\mathbb Z^2\\to\\mathbb Z^2\\) by \\(h_n(x)=(x_1-nx_2,\\,x_2)\\). Then\n\\[\n\\begin{gathered}\nh_{n+1}(\\alpha_nx)\\\\\n=\\big(x_1+x_2-(n+1)x_2,\\ x_2\\big)\\\\\n=h_n(x),\n\\end{gathered}\n\\]\nso the \\(h_n\\) induce a homomorphism \\(G\\to\\mathbb Z^2\\), which is bijective because each \\(h_n\\) is. Under this identification,\n\\[\n\\begin{gathered}\nG^+\\\\\n=\\{(a,b):b\\geq1\\}\\cup\\{(a,0):a\\geq0\\},\\\\\n\\Sigma\\\\\n=\\{(a,0):a\\geq0\\}\\cup\\{(a,1):a\\leq0\\},\\\\\n[1]\\\\\n=(0,1).\n\\end{gathered}\n\\]\nIndeed, for \\(b\\geq1\\) the first coordinate \\(x_1-nb\\) of \\(h_n(x_1,b)\\) runs through all integers \\(\\geq-nb\\), and for \\(b=1\\) and \\(0\\leq x_1\\leq n\\) it runs through \\(-n,\\dots,0\\). In words: \\((a,0)\\) is the class of a projection of rank \\(a\\) in \\(\\mathcal K\\), and \\((-a,1)\\) is the class of \\(1-p\\) for a projection \\(p\\in\\mathcal K\\) of rank \\(a\\). The cone \\(G^+\\) is not finitely generated as a monoid. Suppose finitely many elements generated it. A sum of generators with second coordinate \\(1\\) contains exactly one generator \\((a,1)\\) and otherwise generators \\((c,0)\\) with \\(c\\geq0\\), so the first coordinates of the elements \\((a,1)\\) of \\(G^+\\) would be bounded below; but every \\((a,1)\\), \\(a\\in\\mathbb Z\\), lies in \\(G^+\\). Since \\(\\mathbb Z^2_+\\) is generated by two elements, \\((G,G^+)\\) is not isomorphic to \\((\\mathbb Z^2,\\mathbb Z^2_+)\\).\n\nThe explicit connecting maps and the non-finite-generation argument above distinguish this cone from \\(\\mathbb Z^2_+\\).\n\n(c) *UHF algebras* (Example 5.3). The maps \\(\\mathbb Z\\to\\mathbb Q\\), \\(x\\mapsto x/d_n\\), are compatible with multiplication by \\(k_n\\), because \\(k_nx/d_{n+1}=x/d_n\\). They identify \\(G\\) with the subgroup \\(\\bigcup_nd_n^{-1}\\mathbb Z\\) of \\(\\mathbb Q\\), with\n\\[\n\\begin{gathered}\nG^+\\\\\n=G\\cap[0,\\infty),\\\\\n\\Sigma\\\\\n=G\\cap[0,1],\\\\\n[1]\\\\\n=1 .\n\\end{gathered}\n\\]\nFor the CAR algebra, \\(G=\\mathbb Z[\\tfrac12]\\), the dyadic rationals.\n\n(d) *Stabilization.* By Lemma 6.3(6), \\(A\\otimes\\mathcal K\\) has the group and cone of \\(A\\) and the scale \\(G^+\\). For instance, \\(\\mathcal K\\otimes\\mathcal K\\) has the invariant \\((\\mathbb Z,\\mathbb Z_+,\\mathbb Z_+)\\), the same as \\(\\mathcal K\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 387,
        "through_line": 430,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
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          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
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              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
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              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
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          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.1",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Lemma 7.1",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 7.1** (Close projections are equivalent). Let \\(p,q\\) be projections of a C\\*-algebra \\(A\\) with \\(\\|p-q\\|<1\\). There is a unitary \\(u\\) in the C\\*-subalgebra of \\(A^+\\) generated by \\(1,p,q\\) with \\(upu^*=q\\). In particular \\(v=up\\in A\\) satisfies \\(v^*v=p\\) and \\(vv^*=q\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 435,
        "through_line": 460,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 7.2",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Lemma 7.2",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 7.2** (Projections near a subalgebra). Let \\(B\\) be a C\\*-subalgebra of a C\\*-algebra \\(A\\), and \\(p\\in A\\) a projection whose distance to \\(B\\) is less than \\(\\frac14\\). Then there is a projection \\(q\\in B\\) with \\(\\|p-q\\|<\\frac12\\); by Lemma 7.1, \\(p\\) and \\(q\\) are equivalent in \\(A\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 461,
        "through_line": 468,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.3",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Proposition 7.3",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 7.3** (Projections in the closure of a union). Let \\(A\\) be a C\\*-algebra and \\(A_1\\subseteq A_2\\subseteq\\cdots\\) C\\*-subalgebras with dense union.\n\n1. Every projection \\(p\\in A\\) is equivalent in \\(A\\) to a projection of some \\(A_k\\).\n2. If \\(p,q\\in A_k\\) are projections that are equivalent in \\(A\\), they are equivalent in \\(A_l\\) for some \\(l\\geq k\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 469,
        "through_line": 477,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.4",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Definition 7.4",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 7.4.** Let \\(A\\) be a C\\*-algebra. Two projections \\(p\\in M_n(A)\\) and \\(q\\in M_{n'}(A)\\) are *equivalent* if they are equivalent in \\(M_N(A)\\), \\(N=\\max(n,n')\\), where both sit as upper left corners; this does not depend on \\(N\\), because a partial isometry \\(v\\) with \\(v^*v=p\\) and \\(vv^*=q\\) satisfies \\(v=qvp\\). Let \\(V(A)\\) be the set of equivalence classes \\([p]\\) of projections in \\(M_\\infty(A)\\). For projections \\(p,q\\), view both in some \\(M_N(A)\\) and put\n\\[\n\\begin{gathered}\n{}[p]+[q]\\\\\n=[\\operatorname{diag}(p,q)],\\\\\n\\operatorname{diag}(p,q)\\in M_{2N}(A).\n\\end{gathered}\n\\]\nEnlarging \\(N\\) changes \\(\\operatorname{diag}(p,q)\\) by a permutation of the basis, which is implemented by a permutation matrix, so the class does not depend on \\(N\\). The addition is well defined and makes \\(V(A)\\) an abelian monoid with zero \\([0]\\): if \\(v,w\\) implement \\(p\\sim p'\\) and \\(q\\sim q'\\), then \\(\\operatorname{diag}(v,w)\\) implements \\(\\operatorname{diag}(p,q)\\sim\\operatorname{diag}(p',q')\\); and \\(\\begin{pmatrix}0&q\\\\p&0\\end{pmatrix}\\) implements \\(\\operatorname{diag}(p,q)\\sim\\operatorname{diag}(q,p)\\). A \\*-homomorphism \\(\\psi:A\\to B\\) induces the monoid homomorphism \\(\\psi_*[p]=[\\psi(p)]\\), where \\(\\psi\\) acts entrywise.\n\nFor orthogonal projections \\(p,q\\in A\\), \\(p+q\\sim\\operatorname{diag}(p,q)\\) through \\(\\begin{pmatrix}p&q\\\\0&0\\end{pmatrix}\\), so \\([p+q]=[p]+[q]\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 478,
        "through_line": 489,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 7.5",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Theorem 7.5",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 7.5** (The dimension group is an invariant). Let \\(A\\) be an AF-algebra with generating sequence \\((A_k)\\) and scaled dimension group \\((G,G^+,\\Sigma)\\). For a projection \\(p\\in M_N(A_k)\\cong M_{N\\mathbf m(k)}\\), let \\(\\operatorname{rk}p\\in\\mathbb Z^{r_k}_+\\) be its rank vector.\n\n1. The rule \\(\\alpha_{\\infty,k}(\\operatorname{rk}p)\\mapsto[p]\\) is a well-defined isomorphism of monoids \\(\\Phi:G^+\\to V(A)\\). In particular \\(V(A)\\) has cancellation: \\(a+c=b+c\\) implies \\(a=b\\).\n2. \\(\\Phi(\\Sigma)\\) is the set of classes of projections of \\(A\\) itself.\n3. \\(G\\) is the Grothendieck group of \\(V(A)\\): every monoid homomorphism from \\(V(A)\\) to an abelian group extends uniquely to a group homomorphism on \\(G\\), through \\(\\Phi^{-1}\\).\n4. Every \\*-homomorphism \\(\\psi:A\\to B\\) between AF-algebras induces a homomorphism of scaled dimension groups \\(\\psi_*\\) with \\(\\psi_*[p]=[\\psi(p)]\\) for projections \\(p\\in M_\\infty(A)\\); \\((\\psi\\circ\\psi')_*=\\psi_*\\psi'_*\\) and \\(\\mathrm{id}_*=\\mathrm{id}\\). Isomorphic AF-algebras have isomorphic scaled dimension groups, whatever generating sequences are used.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 490,
        "through_line": 506,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 7.6",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Definition 7.6",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 7.6.** For an AF-algebra \\(A\\) we write \\((K_0(A),K_0(A)^+,\\Sigma(A))\\) for its scaled dimension group, computed from any generating sequence: \\(K_0(A)\\) is the Grothendieck group of \\(V(A)\\), \\(K_0(A)^+\\) the image of \\(V(A)\\), and \\(\\Sigma(A)\\) the set of classes of projections of \\(A\\).\n\nFor unital C\\*-algebras, \\(K_0(A)\\) is defined in K-theory as the Grothendieck group of \\(V(A)\\), so the two notions agree. For a nonunital C\\*-algebra, K-theory defines \\(K_0(A)\\) as the kernel of the map \\(K_0(A^+)\\to K_0(\\mathbb C)=\\mathbb Z\\) induced by the quotient map \\(A^+\\to\\mathbb C\\) that kills \\(A\\), with positive cone the image of \\(V(A)\\) [Blackadar 2006, V.1.1.15–V.1.1.17]. The next proposition shows that this also agrees with Definition 7.6.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 507,
        "through_line": 510,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 7.7",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-06::Proposition 7.7",
      "anchor": "oa-fnd-af-06",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 7.7** (Nonunital AF-algebras). Let \\(A\\) be a nonunital AF-algebra. Then \\(A^+\\) is an AF-algebra, and the map \\(V(A)\\to V(A^+)\\) induced by the inclusion extends to an isomorphism of \\(K_0(A)\\) onto the kernel of \\(K_0(A^+)\\to\\mathbb Z\\), carrying \\(K_0(A)^+\\) onto the image of \\(V(A)\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 511,
        "through_line": 524,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.1",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Lemma 8.1",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 8.1** (Existence). Let \\(\\mathbf m\\in\\mathbb N^r\\), and let \\(h:\\mathbb Z^r\\to K_0(B)\\) be a group homomorphism with \\(h(\\mathbb Z^r_+)\\subseteq K_0(B)^+\\) and \\(h(\\mathbf m)\\in\\Sigma(B)\\). Then there are \\(l\\) and a \\*-homomorphism \\(\\varphi:M_{\\mathbf m}\\to B_l\\) with\n\\[\n\\beta_{\\infty,l}\\big(\\alpha(\\varphi)x\\big)=h(x)\\qquad(x\\in\\mathbb Z^r),\n\\]\nthat is, \\([\\varphi(p)]=h(\\operatorname{rk}p)\\) for every projection \\(p\\in M_{\\mathbf m}\\). If \\(h(e_i)\\neq0\\) for every \\(i\\), then \\(\\varphi\\) is injective.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 529,
        "through_line": 536,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 8.2",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Lemma 8.2",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 8.2** (Uniqueness). Let \\(\\varphi,\\psi:M_{\\mathbf m}\\to B_l\\) be \\*-homomorphisms with \\(\\beta_{\\infty,l}\\alpha(\\varphi)=\\beta_{\\infty,l}\\alpha(\\psi)\\). Then there are \\(l'\\geq l\\) and \\(u\\in U(B_{l'})\\) with \\(u\\varphi(x)u^*=\\psi(x)\\) for all \\(x\\in M_{\\mathbf m}\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 537,
        "through_line": 540,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.3",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Theorem 8.3",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 8.3** (Classification of AF-algebras). Let \\(A\\) and \\(B\\) be AF-algebras with generating sequences \\((A_k)\\) and \\((B_l)\\) and local algebras \\(A_\\infty\\) and \\(B_\\infty\\).\n\n1. For every homomorphism \\[\n\\begin{gathered}\nh:(K_0(A),K_0(A)^+,\\Sigma(A))\\\\\n\\to(K_0(B),K_0(B)^+,\\Sigma(B))\n\\end{gathered}\n\\] of scaled ordered groups there is a \\*-homomorphism \\(\\psi:A\\to B\\) with \\(\\psi(A_\\infty)\\subseteq B_\\infty\\) and \\(\\psi_*=h\\).\n2. For every isomorphism \\(\\theta\\) of the scaled dimension groups there is an isomorphism \\(\\Phi:A\\to B\\) with \\(\\Phi(A_\\infty)=B_\\infty\\) and \\(\\Phi_*=\\theta\\).\n3. The following are equivalent: (i) \\(A\\cong B\\); (ii) \\(A_\\infty\\) and \\(B_\\infty\\) are isomorphic \\*-algebras; (iii) the scaled dimension groups of \\(A\\) and \\(B\\) are isomorphic.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 541,
        "through_line": 568,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 8.4",
      "kind": "corollary",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Corollary 8.4",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Corollary 8.4.** Let \\(A\\) be an AF-algebra. (1) Any two generating sequences of \\(A\\) have isomorphic local algebras. (2) Every automorphism of the scaled dimension group of \\(A\\) is induced by an automorphism of \\(A\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 569,
        "through_line": 572,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 8.5",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Theorem 8.5",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 8.5** (Stable isomorphism). For AF-algebras \\(A\\) and \\(B\\), \\(A\\otimes\\mathcal K\\cong B\\otimes\\mathcal K\\) if and only if the ordered groups \\((K_0(A),K_0(A)^+)\\) and \\((K_0(B),K_0(B)^+)\\) are isomorphic.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 573,
        "through_line": 578,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Corollary 8.6",
      "kind": "corollary",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Corollary 8.6",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Corollary 8.6** (Classification of UHF algebras). The UHF algebra of type \\((k_n)\\) has the scaled dimension group \\((\\mathbb Z(q),\\mathbb Z(q)\\cap[0,\\infty),\\mathbb Z(q)\\cap[0,1])\\). Two UHF algebras are isomorphic if and only if their supernatural numbers are equal.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 579,
        "through_line": 585,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 8.7",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-07::Example 8.7",
      "anchor": "oa-fnd-af-07",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 8.7** (The scale cannot be dropped). Let \\(A\\) be the CAR algebra. By Proposition 5.4(1), \\(M_3(A)\\) is an AF-algebra with the same group and cone as \\(A\\), namely \\(\\mathbb Z[\\frac12]\\) with the usual order, but with unit class \\(3\\) and scale \\([0,3]\\cap\\mathbb Z[\\frac12]\\). A group homomorphism \\(\\theta\\) of \\(\\mathbb Z[\\frac12]\\) satisfies \\(2^n\\theta(2^{-n})=\\theta(1)\\), so \\(\\theta(x)=\\theta(1)x\\). If \\(\\theta\\) is bijective and positive, then \\(\\theta(1)\\) is a positive unit of the ring \\(\\mathbb Z[\\frac12]\\), that is, \\(\\theta(1)=2^j\\) for some \\(j\\in\\mathbb Z\\). Then \\(\\theta([0,1])=[0,2^j]\\neq[0,3]\\). So \\(M_3(A)\\not\\cong A\\), although \\(M_3(A)\\otimes\\mathcal K\\cong A\\otimes\\mathcal K\\) by Theorem 8.5. This agrees with Corollary 8.6: \\(M_3(A)\\) is the UHF algebra of type \\((3,2,2,\\dots)\\), with supernatural number \\(3\\cdot2^\\infty\\neq2^\\infty\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 586,
        "through_line": 587,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 9.2",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-08::Lemma 9.2",
      "anchor": "oa-fnd-af-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 9.2.** Let \\(H_{n,j}\\) be the span of the \\(\\varepsilon_\\xi\\) with \\(|\\xi|=j\\), of dimension \\(\\binom nj\\), and \\(P_{n,j}\\) the projection onto it. Then\n\\[\n\\begin{gathered}\nA_n\\\\\n=\\{x\\in B_n:xH_{n,j}\\subseteq H_{n,j}\\text{ for }j=0,\\dots,n\\}\\\\\n=\\bigoplus_{j=0}^nB(H_{n,j})\\cong\\bigoplus_{j=0}^nM_{\\binom nj}.\n\\end{gathered}\n\\]\nMoreover, for \\(x\\in B_n\\) and every integer \\(N>2n\\),\n\\[\n\\sum_{j=0}^nP_{n,j}\\,x\\,P_{n,j}=\\frac1N\\sum_{\\omega^N=1}\\sigma_\\omega(x).\n\\tag{9.1}\n\\]",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 600,
        "through_line": 624,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 9.3",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-08::Proposition 9.3",
      "anchor": "oa-fnd-af-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 9.3** (Pascal's triangle). The algebras \\(A_0\\subseteq A_1\\subseteq\\cdots\\) form a generating sequence of \\(A\\), so \\(A\\) is a unital AF-algebra. Index the summands of \\(A_n\\) by \\(j=0,\\dots,n\\), the \\(j\\)-th being \\(B(H_{n,j})\\) of size \\(\\binom nj\\). The inclusion \\(A_n\\subseteq A_{n+1}\\) is unital, and the \\(j\\)-th summand of \\(A_n\\) enters the summands \\(j\\) and \\(j+1\\) of \\(A_{n+1}\\), each with multiplicity one. So the Bratteli diagram is Pascal's triangle:\n\n```\nlevel 0                         1\nlevel 1                      1     1\nlevel 2                   1     2     1\nlevel 3                1     3     3     1\nlevel 4             1     4     6     4     1\nlevel 5          1     5    10    10     5     1\nlevel 6       1     6    15    20    15     6     1\n```\n\nHere each vertex is joined to the two vertices just below it, and the labels are the sizes \\(\\binom nj\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 625,
        "through_line": 650,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.4",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-08::Theorem 9.4",
      "anchor": "oa-fnd-af-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 9.4** (Pólya's theorem). Let \\(F(x,y)=\\sum_{j=0}^dc_jx^jy^{d-j}\\) be a homogeneous polynomial with real coefficients such that \\(F(x,y)>0\\) whenever \\(x,y\\geq0\\) and \\(x+y=1\\). Then for all sufficiently large \\(N\\), every coefficient of \\((x+y)^NF(x,y)\\) is strictly positive.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 651,
        "through_line": 681,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 9.5",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-08::Theorem 9.5",
      "anchor": "oa-fnd-af-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 9.5** (Dimension group of the gauge-invariant CAR algebra). For a minimal projection \\(e\\) of the \\(j\\)-th summand of \\(A_n\\), put \\(\\rho[e]=t^j(1-t)^{n-j}\\in\\mathbb Z[t]\\). This rule extends to an isomorphism of scaled ordered groups\n\\[\n\\rho:(K_0(A),K_0(A)^+,\\Sigma(A))\\longrightarrow(\\mathbb Z[t],\\mathcal P,\\mathcal S),\n\\]\nwhere\n\\[\n\\begin{gathered}\n\\mathcal P\\\\\n=\\{0\\}\\cup\\{f\\in\\mathbb Z[t]:f(t)>0\\\\\n\\text{ for all }0<t<1\\},\\\\\n\\mathcal S\\\\\n=\\{0,1\\}\\cup\\{f\\in\\mathbb Z[t]:0<f(t)<1\\\\\n\\text{ for all }0<t<1\\}.\n\\end{gathered}\n\\]\nThe class of the unit is the constant polynomial \\(1\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 682,
        "through_line": 724,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 9.6",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-08::Example 9.6",
      "anchor": "oa-fnd-af-08",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 9.6** (Edge cases). (a) The polynomial \\(t(1-t)\\) vanishes at both ends of \\([0,1]\\) and lies in \\(\\mathcal P\\): it is the class of a minimal projection of the middle summand of \\(A_2\\). (b) The polynomial \\((2t-1)^2\\) is \\(\\geq0\\) on \\([0,1]\\) but vanishes at \\(\\frac12\\), so it is not in \\(\\mathcal P\\); neither is its negative. So the order of \\(K_0(A)\\) is not the pointwise order of functions on \\([0,1]\\), and \\((2t-1)^2\\) is not the class of any projection in any matrix algebra over \\(A\\). (c) Neither \\(2t-1\\) nor \\(1-2t\\) lies in \\(\\mathcal P\\), so \\(K_0(A)\\) is not totally ordered, unlike the dimension groups of UHF algebras.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 725,
        "through_line": 726,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Definition 10.1",
      "kind": "definition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-09::Definition 10.1",
      "anchor": "oa-fnd-af-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Definition 10.1.** A *trace* on a C\\*-algebra \\(A\\) is a bounded linear functional \\(\\tau\\) with \\(\\tau(a)\\geq0\\) for \\(a\\geq0\\) and \\(\\tau(xy)=\\tau(yx)\\) for all \\(x,y\\in A\\). A *tracial state* is a trace of norm one. Let \\(T(A)\\) be the set of tracial states and \\(T_{\\leq1}(A)\\) the set of traces of norm at most one, both with the weak\\* topology (pointwise convergence on \\(A\\)).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 729,
        "through_line": 730,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.2",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-09::Lemma 10.2",
      "anchor": "oa-fnd-af-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 10.2** (Traces on \\(M_{\\mathbf m}\\)). For \\(t\\in\\mathbb R^r_+\\) put \\(\\tau_t(x)=\\sum_it_i\\operatorname{Tr}(x_i)\\). The traces of \\(M_{\\mathbf m}\\) are exactly the \\(\\tau_t\\), \\(t\\in\\mathbb R^r_+\\), and \\(t_i\\) is the value of \\(\\tau_t\\) on any minimal projection of the \\(i\\)-th summand. Moreover \\(\\|\\tau_t\\|=\\tau_t(1)=\\mathbf m^Tt\\). So \\(T(M_{\\mathbf m})\\) is identified with the simplex \\(\\Delta(\\mathbf m)=\\{t\\in\\mathbb R^r_+:\\mathbf m^Tt=1\\}\\), whose vertices are the \\(e_i/m_i\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 731,
        "through_line": 734,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 10.3",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-09::Lemma 10.3",
      "anchor": "oa-fnd-af-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 10.3** (Pulling back traces). If \\(\\varphi:M_{\\mathbf m}\\to M_{\\mathbf n}\\) has multiplicity matrix \\(\\alpha\\), then \\(\\tau_y\\circ\\varphi=\\tau_{\\alpha^Ty}\\) for \\(y\\in\\mathbb R^s_+\\), and \\(\\|\\tau_y\\circ\\varphi\\|\\leq\\|\\tau_y\\|\\), with equality when \\(\\varphi\\) is unital.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 735,
        "through_line": 738,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 10.4",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-09::Theorem 10.4",
      "anchor": "oa-fnd-af-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 10.4** (Traces as a projective limit). Let \\(A\\) be an AF-algebra with generating sequence \\((A_k)\\), \\(A_k\\cong M_{\\mathbf m(k)}\\), and multiplicity matrices \\(\\alpha_k\\). For a trace \\(\\tau\\) on \\(A\\), let \\(t^{(k)}(\\tau)\\in\\mathbb R^{r_k}_+\\) be the vector with \\(\\tau|_{A_k}=\\tau_{t^{(k)}(\\tau)}\\).\n\n1. The map \\(\\tau\\mapsto(t^{(k)}(\\tau))_k\\) is an affine bijection of \\(T_{\\leq1}(A)\\) onto the set of sequences \\((t^{(k)})\\) with \\(t^{(k)}\\in\\mathbb R^{r_k}_+\\), \\(\\alpha_k^Tt^{(k+1)}=t^{(k)}\\) and \\(\\mathbf m(k)^Tt^{(k)}\\leq1\\) for all \\(k\\). It is a homeomorphism for the weak\\* topology and the product topology. Moreover \\(\\|\\tau\\|=\\lim_k\\mathbf m(k)^Tt^{(k)}(\\tau)\\), and the sequence \\(\\mathbf m(k)^Tt^{(k)}(\\tau)\\) is nondecreasing.\n2. A trace \\(\\tau\\) has norm one exactly when \\(\\lim_k\\mathbf m(k)^Tt^{(k)}(\\tau)=1\\).\n3. If \\(A\\) is nonzero and unital and \\(1_A\\in A_k\\) for all \\(k\\), then \\(\\mathbf m(k)^Tt^{(k)}(\\tau)=\\tau(1)\\) for every \\(k\\). So \\(T(A)\\) is affinely homeomorphic to the projective limit of the simplices \\(\\Delta(\\mathbf m(k))\\) under the maps \\(t\\mapsto\\alpha_k^Tt\\), and \\(T(A)\\) is not empty.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 739,
        "through_line": 750,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 10.5",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-09::Example 10.5",
      "anchor": "oa-fnd-af-09",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 10.5.** (a) *Compact operators.* All matrices are \\((1)\\), so a compatible sequence is constant, \\(t^{(n)}=t\\), with \\(nt\\leq1\\) for all \\(n\\). So \\(t=0\\): the only trace of norm at most one is \\(0\\), and \\(T(\\mathcal K)=\\varnothing\\). So a nonunital AF-algebra may have no tracial state.\n\n(b) *Unitization of the compact operators.* With \\(t^{(n)}=(a_n,b_n)\\) and \\(\\alpha_n^T=\\begin{pmatrix}1&0\\\\1&1\\end{pmatrix}\\), compatibility says \\(a_n=a_{n+1}\\) and \\(b_n=a_{n+1}+b_{n+1}\\), and normalization says \\(na_n+b_n=1\\). So \\(a_n=a\\) is constant with \\(na\\leq1\\) for all \\(n\\), hence \\(a=0\\) and \\(b_n=1\\). So \\(\\mathcal K+\\mathbb C1\\) has exactly one tracial state, \\(\\tau(k+\\lambda1)=\\lambda\\).\n\nThe explicit quotient trace and Theorem 10.4(3) prove that every nonzero unital AF-algebra has a tracial state. The zero algebra has only the zero functional and no norm-one state.\n\n(c) *UHF algebras.* Compatibility forces \\(t^{(n)}=1/d_n\\): a UHF algebra has exactly one tracial state. By Exercise 3 below, the gauge-invariant CAR algebra has infinitely many.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 751,
        "through_line": 758,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [
        {
          "unit": "af-algebras",
          "local_scopes": [
            {
              "heading": "6. The scaled dimension group",
              "line": 346,
              "through_line": 430,
              "anchors": [
                "OA-FND-AF-05"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "7. Projections and the dimension group as an invariant",
              "line": 431,
              "through_line": 524,
              "anchors": [
                "OA-FND-AF-06"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "8. Elliott's classification theorem",
              "line": 525,
              "through_line": 587,
              "anchors": [
                "OA-FND-AF-07"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "9. The gauge-invariant CAR algebra and Pascal's triangle",
              "line": 588,
              "through_line": 726,
              "anchors": [
                "OA-FND-AF-08"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            },
            {
              "heading": "10. Traces",
              "line": 727,
              "through_line": 758,
              "anchors": [
                "OA-FND-AF-09"
              ],
              "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
            }
          ],
          "source_key": "human:blackadar-k@A20E676E9D400EBCD0CD07C13FBDBE1FAC759A30C739FAB23D6161ECFB1A23FB",
          "source_locus": "§§7.2–7.3; PDF 62–68, printed 48–54",
          "role": "proof comparison",
          "correspondence": "Ordered scaled AF classification including nonunital case",
          "explanation": "Finite positive homomorphism lift, unitary uniqueness and Elliott intertwining; general dimension-group realization and trace correspondence in these pages are only statements. Own K(H) scale, irrational cone, nonzero unital trace and Pascal/Pólya construction proofs remain."
        }
      ],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Lemma 11.1",
      "kind": "lemma",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-10::Lemma 11.1",
      "anchor": "oa-fnd-af-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Lemma 11.1.** Let \\(H=\\bigoplus_i(\\mathbb C^{k_i}\\otimes K_i)\\oplus L\\) with finite-dimensional \\(K_i\\) and \\(L\\), and let \\(Q\\subseteq B(H)\\) consist of the operators \\(\\bigoplus_i(y_i\\otimes1_{K_i})\\oplus0_L\\), \\(y_i\\in M_{k_i}\\). Then\n\\[\n\\begin{gathered}\nQ'\\\\\n=\\bigoplus_i\\big(1_{k_i}\\otimes B(K_i)\\big)\\oplus B(L)\\\\\n\\cong\\bigoplus_{i:K_i\\neq0}B(K_i)\\ \\oplus\\ B(L).\n\\end{gathered}\n\\]",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 763,
        "through_line": 773,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Theorem 11.2",
      "kind": "theorem",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-10::Theorem 11.2",
      "anchor": "oa-fnd-af-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Theorem 11.2** (Commutants reflect the diagram). Put \\(P=\\pi(M_{\\mathbf n})\\) and \\(Q=\\pi(\\varphi(M_{\\mathbf m}))\\subseteq P\\), so that \\(P'\\subseteq Q'\\). Let \\(\\mathbf m'=\\alpha^T\\mathbf n'\\) (so \\(m_i'=\\sum_j\\alpha_{ji}n_j'\\)) and \\(d'=\\sum_jd_jn_j'\\). Then:\n\n1. \\(P'\\cong\\bigoplus_{j:n_j'>0}M_{n_j'}\\);\n2. \\(Q'\\cong\\bigoplus_{i:m_i'>0}M_{m_i'}\\oplus M_{d'}\\), where the last summand is present only when \\(d'>0\\);\n3. the inclusion \\(P'\\subseteq Q'\\) is unital, the summand \\(M_{n_j'}\\) of \\(P'\\) enters the summand \\(M_{m_i'}\\) of \\(Q'\\) with multiplicity \\(\\alpha_{ji}\\), and it enters \\(M_{d'}\\) with multiplicity \\(d_j\\).\n\nIn particular, if \\(\\varphi\\) is unital and injective and \\(\\pi\\) is faithful, then \\(P'\\cong M_{\\mathbf n'}\\), \\(Q'\\cong M_{\\alpha^T\\mathbf n'}\\), and the multiplicity matrix of \\(P'\\subseteq Q'\\) is the transpose \\(\\alpha^T\\): the Bratteli diagram of \\(P'\\subseteq Q'\\) is the diagram of \\(\\varphi\\) read from right to left, with the new sizes \\(\\mathbf n'\\) and \\(\\alpha^T\\mathbf n'\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 774,
        "through_line": 794,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 11.3",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-10::Example 11.3",
      "anchor": "oa-fnd-af-10",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 11.3.** (a) Take \\(\\varphi_\\alpha:\\mathbb C\\oplus M_2\\to M_5\\oplus M_4\\) from Example 3.6(a), with \\(\\alpha=\\begin{pmatrix}3&1\\\\0&2\\end{pmatrix}\\), and the identity representation of \\(M_5\\oplus M_4\\) on \\(\\mathbb C^5\\oplus\\mathbb C^4\\), with \\(\\mathbf n'=(1,1)\\). Then \\(P'=\\mathbb C\\oplus\\mathbb C\\) is the centre of \\(M_5\\oplus M_4\\), and \\(\\mathbf m'=\\alpha^T\\mathbf n'=(3,3)\\), so \\(Q'\\cong M_3\\oplus M_3\\). Directly: \\(Q\\) acts on \\(\\mathbb C^9\\) as \\(\\lambda\\) on a three-dimensional subspace and as \\(y\\otimes1\\) on \\(\\mathbb C^2\\otimes\\mathbb C^3\\), so its commutant is \\(M_3\\oplus(1_2\\otimes M_3)\\). The pair \\((t_1,t_2)\\in P'\\) becomes \\((\\operatorname{diag}(t_1,t_1,t_1),\\operatorname{diag}(t_1,t_2,t_2))\\), with multiplicity matrix \\(\\begin{pmatrix}3&0\\\\1&2\\end{pmatrix}=\\alpha^T\\).\n\n(b) *A nonunital inclusion.* Take \\(\\varphi:M_2\\to M_5\\), \\(x\\mapsto\\operatorname{diag}(x,x,0)\\), with \\(\\alpha=(2)\\) and \\(d=(1)\\), and \\(\\pi\\) the identity on \\(\\mathbb C^5\\). Then \\(P'=\\mathbb C1\\), and \\(Q'=(1_2\\otimes M_2)\\oplus\\mathbb C\\cong M_2\\oplus\\mathbb C\\): the defect produces the extra summand \\(M_{d'}=\\mathbb C\\). The scalar \\(t\\in P'\\) becomes \\((t1_2,t)\\), with multiplicities \\(2=\\alpha\\) and \\(1=d\\). Without unitality of \\(\\varphi\\), the commutant \\(Q'\\) is not \\(M_{\\alpha^T\\mathbf n'}=M_2\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 795,
        "through_line": 798,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.1",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-11::Proposition 12.1",
      "anchor": "oa-fnd-af-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 12.1.** Let \\(\\Gamma\\) be a finite group.\n\n1. The centre of \\(\\mathbb C[\\Gamma]\\) consists of the functions that are constant on conjugacy classes, so the number of summands of \\(\\mathbb C[\\Gamma]\\) is the number of conjugacy classes.\n2. Unitary representations of \\(\\Gamma\\) on finite-dimensional Hilbert spaces correspond to unital representations of \\(\\mathbb C[\\Gamma]\\) by \\(\\rho(f)=\\sum_yf(y)\\rho(y)\\), with the same invariant subspaces. So the summands of \\(\\mathbb C[\\Gamma]\\) correspond to the unitary equivalence classes of irreducible representations of \\(\\Gamma\\), and the size of a summand is the dimension of the representation.\n3. If \\(\\Gamma_0\\subseteq\\Gamma\\) is a subgroup, the inclusion \\(\\mathbb C[\\Gamma_0]\\subseteq\\mathbb C[\\Gamma]\\) (extension by zero) is a unital embedding, and its multiplicity matrix has, in the row of an irreducible representation \\(\\rho\\) of \\(\\Gamma\\) and the column of an irreducible representation \\(\\sigma\\) of \\(\\Gamma_0\\), the multiplicity of \\(\\sigma\\) in the restriction of \\(\\rho\\) to \\(\\Gamma_0\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 803,
        "through_line": 810,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Proposition 12.2",
      "kind": "proposition",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-11::Proposition 12.2",
      "anchor": "oa-fnd-af-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Proposition 12.2** (Locally finite groups). Let \\(\\Gamma\\) be the union of an increasing sequence \\(\\Gamma_1\\subseteq\\Gamma_2\\subseteq\\cdots\\) of finite subgroups, and let \\(\\mathbb C[\\Gamma]\\) be the \\*-algebra of finitely supported functions on \\(\\Gamma\\), with convolution and involution as above. Then \\(\\mathbb C[\\Gamma]\\) has exactly one C\\*-norm, namely \\(\\|\\lambda(f)\\|\\) for the left regular representation on \\(\\ell^2(\\Gamma)\\). Its completion \\(C^*(\\Gamma)\\) is an AF-algebra with generating sequence \\((\\mathbb C[\\Gamma_n])\\), and the multiplicities in its Bratteli diagram are the restriction multiplicities of Proposition 12.1(3).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 811,
        "through_line": 814,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Example 12.3",
      "kind": "example",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-11::Example 12.3",
      "anchor": "oa-fnd-af-11",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Example 12.3** (The infinite symmetric group). Let \\(S_n\\) be the group of permutations of \\(\\{1,\\dots,n\\}\\), embedded in \\(S_{n+1}\\) as the permutations fixing \\(n+1\\). The union \\(S_\\infty\\) is the group of permutations of \\(\\mathbb N\\) that move only finitely many points, and \\(C^*(S_\\infty)\\) is an AF-algebra by Proposition 12.2. We compute the first levels of its diagram.\n\n\\(\\mathbb C[S_1]=\\mathbb C\\). The group \\(S_2\\) has two conjugacy classes and order \\(2\\), so \\(\\mathbb C[S_2]\\cong\\mathbb C\\oplus\\mathbb C\\); the two summands are the trivial and the sign representation. The group \\(S_3\\) has three conjugacy classes and order \\(6\\), and the only way to write \\(6\\) as a sum of three squares of positive integers is \\(1+1+4\\). So \\(\\mathbb C[S_3]\\cong\\mathbb C\\oplus\\mathbb C\\oplus M_2\\): the trivial representation, the sign representation and a two-dimensional irreducible representation \\(\\rho\\).\n\nWe identify \\(\\rho\\). Let \\(S_3\\) permute the coordinates of \\(V=\\{v\\in\\mathbb C^3:v_1+v_2+v_3=0\\}\\), a two-dimensional unitary representation. A one-dimensional invariant subspace would be spanned by a common eigenvector \\(v\\) of the transpositions \\((12)\\) and \\((23)\\), which generate \\(S_3\\); since they are involutions, each acts on \\(v\\) by \\(+1\\) or \\(-1\\). If both eigenvalues are \\(+1\\), the coordinates of \\(v\\) are equal and their sum zero forces \\(v=0\\). If \\((12)\\) has eigenvalue \\(-1\\), then \\(v=(a,-a,0)\\). For \\((23)\\) to have eigenvalue \\(+1\\) requires \\(-a=0\\), and eigenvalue \\(-1\\) requires \\(a=0\\). In the remaining case, \\((12)\\) has eigenvalue \\(+1\\) and \\((23)\\) has eigenvalue \\(-1\\); then \\(v=(0,a,-a)\\) and \\(v_1=v_2\\) again forces \\(a=0\\). So \\(V\\) is irreducible, and since \\(\\mathbb C[S_3]\\) has only one summand of size \\(2\\), \\(V\\) is \\(\\rho\\). On \\(V\\), the transposition \\((12)\\) fixes \\((1,1,-2)\\) and negates \\((1,-1,0)\\), so \\(\\rho\\) restricted to \\(S_2\\) is the sum of the trivial and the sign representations. The trivial and sign representations of \\(S_3\\) restrict to those of \\(S_2\\). With the summands ordered as (trivial, sign, \\(\\rho\\)) and (trivial, sign), the multiplicity matrices of \\(\\mathbb C[S_1]\\subseteq\\mathbb C[S_2]\\subseteq\\mathbb C[S_3]\\) are\n\\[\n\\begin{pmatrix}1\\\\1\\end{pmatrix},\\qquad\\begin{pmatrix}1&0\\\\0&1\\\\1&1\\end{pmatrix}.\n\\]\n*Unused extension, not proved here.* The general Young-diagram branching rule is not used in the finite-level computations, the locally finite group theorem, or any subsequent proof of this lesson. In general, the irreducible representations of \\(S_n\\) are indexed by the partitions of \\(n\\), drawn as Young diagrams, and the restriction of the representation of a diagram to \\(S_{n-1}\\) is the sum, with multiplicity one each, of the representations of the diagrams obtained by removing one box . So the Bratteli diagram of \\(C^*(S_\\infty)\\) is the graph of Young diagrams ordered by adding one box, and all its multiplicities are \\(0\\) or \\(1\\). The case \\(n\\leq3\\) above agrees: \\((2)\\) and \\((1,1)\\) come from \\((1)\\), and the diagram \\((2,1)\\) of \\(\\rho\\) contains both \\((2)\\) and \\((1,1)\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 815,
        "through_line": 824,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 1",
      "kind": "exercise",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-12::Exercise 1",
      "anchor": "oa-fnd-af-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Exercise 1** (medium; The Cantor set). Let \\(X=\\{0,1\\}^{\\mathbb N}\\) with the product topology. For a word \\(w\\in\\{0,1\\}^n\\) let \\([w]\\) be the set of sequences that begin with \\(w\\), and let \\(A_n\\subseteq C(X)\\) be the span of the indicator functions \\(\\chi_{[w]}\\), \\(w\\in\\{0,1\\}^n\\).\n(a) Show that \\(C(X)\\) is an AF-algebra with generating sequence \\((A_n)\\) and that its Bratteli diagram is the binary tree: all sizes are \\(1\\), and each vertex at level \\(n\\) is joined by one edge to each of two vertices at level \\(n+1\\).\n(b) Show that the scaled dimension group of \\(C(X)\\) is isomorphic to \\((C(X,\\mathbb Z),C(X,\\mathbb Z_+),\\{\\chi_U:U\\subseteq X\\text{ clopen}\\})\\).",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 827,
        "through_line": 833,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 2",
      "kind": "exercise",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-12::Exercise 2",
      "anchor": "oa-fnd-af-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Exercise 2** (medium; Two diagrams, one algebra). Let \\(A\\) be the limit of \\(A_1=\\mathbb C^2\\to A_2\\to\\cdots\\), where \\(A_k=M_{2^{k-1}}\\oplus M_{2^{k-1}}\\) and every connecting map has multiplicity matrix \\(\\begin{pmatrix}1&1\\\\1&1\\end{pmatrix}\\). Show that \\(A\\) is isomorphic to the CAR algebra, although no \\(A_k\\) is a full matrix algebra.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 834,
        "through_line": 837,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 3",
      "kind": "exercise",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-12::Exercise 3",
      "anchor": "oa-fnd-af-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Exercise 3** (easy; Traces of the gauge-invariant CAR algebra). For \\(s\\in[0,1]\\) define vectors \\(t^{(n)}\\in\\mathbb R^{n+1}_+\\) by \\(t^{(n)}_j=s^j(1-s)^{n-j}\\). Show that they define a tracial state \\(\\tau_s\\) of the gauge-invariant CAR algebra \\(A\\), that \\(\\tau_s\\neq\\tau_{s'}\\) for \\(s\\neq s'\\), and that \\(\\tau_s(p)=\\rho[p](s)\\) for every projection \\(p\\in A\\), with \\(\\rho\\) as in Theorem 9.5.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 838,
        "through_line": 856,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 4",
      "kind": "exercise",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-12::Exercise 4",
      "anchor": "oa-fnd-af-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Exercise 4** (medium; AF-algebras are finite). (a) Show that a projection \\(q\\) in an AF-algebra with \\([q]=0\\) in \\(K_0\\) is zero. (b) Deduce that in a unital AF-algebra every \\(v\\) with \\(v^*v=1\\) satisfies \\(vv^*=1\\). (c) Show that the C\\*-subalgebra of \\(B(\\ell^2)\\) generated by the unilateral shift is not an AF-algebra.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 857,
        "through_line": 860,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    },
    {
      "local_label": "Exercise 5",
      "kind": "exercise",
      "unit": "af-algebras",
      "corpus_qualified_locator": "foundations-of-von-neumann-algebras/af-algebras#oa-fnd-af-12::Exercise 5",
      "anchor": "oa-fnd-af-12",
      "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
      "statement_and_full_conditions": "**Exercise 5** (easy; Stable isomorphism of UHF algebras). Show that the CAR algebra and the UHF algebra of type \\((3,3,3,\\dots)\\) are not stably isomorphic, while the CAR algebra and the UHF algebra of type \\((4,4,4,\\dots)\\) are isomorphic.",
      "proof_locus": {
        "source": "src/af-algebras.md",
        "line": 861,
        "through_line": 864,
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
      },
      "source_uses": [],
      "proof_basis": "Complete argument at the identified source locus.",
      "global_tag_allocated": false
    }
  ],
  "required_provider_edges": [
    {
      "id": "OA-FND.CP.B1",
      "required_statement_and_full_conditions": "- **(B1) Positive elements**, [Theorem 8.2 and Proposition 8.5(10) of the C\\(^*\\)-algebra lesson](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-15). Let \\(x\\) be a self-adjoint element of a C\\(^*\\)-algebra \\(A\\). The following are equivalent: \\(x\\ge0\\), that is, the spectrum of \\(x\\) lies in \\([0,\\infty)\\); \\(x=y^*y\\) for some \\(y\\in A\\); \\(x=h^2\\) for some self-adjoint \\(h\\in A\\). The positive elements form a norm-closed convex cone \\(A_+\\) with \\(A_+\\cap(-A_+)=\\{0\\}\\). Every \\(a\\in A_+\\) has exactly one positive square root \\(a^{1/2}\\). It is a norm limit of polynomials in \\(a\\) without constant term, so it commutes with every element that commutes with \\(a\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "locus": "Theorem 8.2; Proposition 8.5(10)",
        "anchor": "oa-fnd-cf-15",
        "anchors": [],
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "reader": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html#oa-fnd-cf-15"
      },
      "correspondence": "exact",
      "dependencies": [
        "CF Theorems 5.1 and 5.3; Proposition 7.2",
        "Stone–Weierstrass compact form"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.B2",
      "required_statement_and_full_conditions": "- **(B2) Spectral permanence**, [Theorem 3.2 of the C\\(^*\\)-algebra lesson](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-06). If \\(B\\) is a C\\(^*\\)-subalgebra of \\(A\\), an element of \\(B\\) has the same spectrum in \\(B\\) as in \\(A\\), apart from the point \\(0\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "locus": "Theorem 3.2",
        "anchor": "oa-fnd-cf-06",
        "anchors": [],
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "reader": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html#oa-fnd-cf-06"
      },
      "correspondence": "exact",
      "dependencies": [
        "CF Proposition 1.5",
        "BN Neumann series and inversion continuity"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.B3",
      "required_statement_and_full_conditions": "- **(B3) Homomorphisms**, [Theorem 4.2 and Corollary 4.6 of the C\\(^*\\)-algebra lesson](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-12). A \\(*\\)-homomorphism between C\\(^*\\)-algebras is contractive. An injective one is isometric, so its range is closed.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "locus": "Theorem 4.2; Theorem 4.4(2); Corollary 4.6",
        "anchor": "oa-fnd-cf-12",
        "anchors": [],
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "reader": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html#oa-fnd-cf-12"
      },
      "correspondence": "stronger",
      "dependencies": [
        "CF normal spectral radius",
        "BN characters and spectrum",
        "Stone–Weierstrass and Urysohn"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.B4",
      "required_statement_and_full_conditions": "- **(B4) Functional calculus**, [Theorems 5.1 and 5.3](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-07) and [Proposition 7.2](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-09) of the C\\(^*\\)-algebra lesson. Let \\(h\\) be a self-adjoint element of a unital C\\(^*\\)-algebra, with spectrum \\(\\sigma(h)\\). For continuous \\(f\\) on \\(\\sigma(h)\\) the element \\(f(h)\\) is defined, and \\(f\\mapsto f(h)\\) is an isometric \\(*\\)-homomorphism of \\(C(\\sigma(h))\\) into the algebra that sends the identity function to \\(h\\). In particular, the norm of a self-adjoint element equals its spectral radius. Without a unit the same holds in the unitization, and \\(f(h)\\) lies in the algebra when \\(f(0)=0\\). With \\(f(t)=\\max(\\pm t,0)\\) this gives \\(h=h_+-h_-\\) with \\(h_\\pm\\ge0\\), \\(h_+h_-=0\\) and \\(\\|h_\\pm\\|\\le\\|h\\|\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "locus": "Theorems 5.1 and 5.3; Proposition 7.2",
        "anchor": "oa-fnd-cf-07",
        "anchors": [],
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "reader": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html#oa-fnd-cf-07"
      },
      "correspondence": "stronger",
      "dependencies": [
        "CF Theorems 2.1 and 3.2",
        "BN character theorem",
        "Stone–Weierstrass"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.B5",
      "required_statement_and_full_conditions": "- **(B5) Commutative C\\(^*\\)-algebras**, [Theorem 2.1 of the C\\(^*\\)-algebra lesson](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-04) (the commutative Gelfand–Naimark theorem). Let \\(B\\) be an abelian C\\(^*\\)-algebra, and \\(\\Omega\\) its space of characters, a locally compact Hausdorff space. The Gelfand transform \\(b\\mapsto\\hat b\\), \\(\\hat b(\\chi)=\\chi(b)\\), is an isometric \\(*\\)-isomorphism of \\(B\\) onto \\(C_0(\\Omega)\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "locus": "Theorem 2.1",
        "anchor": "oa-fnd-cf-04",
        "anchors": [],
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "reader": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html#oa-fnd-cf-04"
      },
      "correspondence": "exact",
      "dependencies": [
        "CF Theorem 1.3 and Proposition 1.5",
        "BN Gelfand representation",
        "Stone–Weierstrass"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.B6",
      "required_statement_and_full_conditions": "- **(B6) Approximate identities**, [Theorem 11.4, applied to the whole algebra](c-algebras-continuous-functional-calculus-automatic-continuity-positiv.md#oa-fnd-cf-19). Every C\\(^*\\)-algebra \\(A\\) has an increasing net \\((u_i)\\) in \\(A_+\\) with \\(\\|u_i\\|\\le1\\), \\(\\|u_ix-x\\|\\to0\\) and \\(\\|xu_i-x\\|\\to0\\) for all \\(x\\in A\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
        "locus": "Theorem 11.4; Corollary 11.5(1)",
        "anchor": "oa-fnd-cf-19",
        "anchors": [],
        "sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "source_key": "programme:foundations-of-von-neumann-algebras/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones@1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.md",
        "reader": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones.html#oa-fnd-cf-19"
      },
      "correspondence": "stronger",
      "dependencies": [
        "CF Lemma 11.2",
        "CF Proposition 8.5 inverse order"
      ],
      "complete_internal_proof": true,
      "new_source_uses": [
        {
          "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
          "local_labels": [
            "Lemma 11.2",
            "Theorem 11.4",
            "Corollary 11.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.4.1.1–4",
          "role": "proof comparison",
          "correspondence": "weaker",
          "evidence": "content-level comparison",
          "explanation": "Blackadar proves positive increasing units from the open positive ball for a dense two-sided ideal. The retained theorem treats arbitrary, possibly nonclosed one-sided ideals and their closures; its full proof stays internal."
        }
      ]
    },
    {
      "id": "OA-FND.CP.B7",
      "required_statement_and_full_conditions": "- **(B7) Positive functionals**, [Proposition 3.2](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-03) and [Theorem 4.7](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-04) of the GNS lesson. A positive linear functional \\(f\\) on \\(A\\) is bounded, and \\(f(x^*)=\\overline{f(x)}\\). The form \\((x,y)\\mapsto f(y^*x)\\) is positive semidefinite, so \\(|f(y^*x)|^2\\le f(x^*x)f(y^*y)\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Proposition 3.2; Theorem 4.7",
        "anchor": "oa-fnd-gn-03",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-03"
      },
      "correspondence": "stronger",
      "dependencies": [
        "CF positive cone",
        "CF increasing positive contractive approximate identity"
      ],
      "complete_internal_proof": true,
      "new_source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Proposition 3.2",
            "Theorem 4.7"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.2.1–5",
          "role": "proof comparison",
          "correspondence": "weaker for 3.2; exact C*-scope for 4.7",
          "evidence": "content-level comparison",
          "explanation": "The free source supplies the C*-positive functional norm and Schwarz arguments; the retained algebraic and involutive Banach generality is not replaced."
        }
      ]
    },
    {
      "id": "OA-FND.CP.B8",
      "required_statement_and_full_conditions": "- **(B8) The GNS construction**, [Construction 5.1 and Theorems 5.4–5.5 of the GNS lesson](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-05). For every positive linear functional \\(f\\) on \\(A\\) there are a Hilbert space \\(H_f\\), a representation \\(\\pi_f\\) of \\(A\\) on \\(H_f\\) and a vector \\(\\xi_f\\in H_f\\) with \\(f(a)=\\langle\\pi_f(a)\\xi_f,\\xi_f\\rangle\\) for all \\(a\\in A\\), such that \\(\\pi_f(A)\\xi_f\\) is dense in \\(H_f\\). Moreover \\(\\|\\xi_f\\|^2=\\|f\\|\\), and the triple is unique up to the unitary that preserves this vector and intertwines the representations.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Construction 5.1; Lemma 5.2; Theorems 5.3–5.5",
        "anchor": "oa-fnd-gn-05",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-05"
      },
      "correspondence": "stronger",
      "dependencies": [
        "GN Lemma 3.3 and Lemma 4.5",
        "GN Theorem 4.7 and Proposition 1.4",
        "HS Riesz–Frechet"
      ],
      "complete_internal_proof": true,
      "new_source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Construction 5.1",
            "Lemma 5.2",
            "Theorem 5.3",
            "Theorem 5.4",
            "Theorem 5.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
          "correspondence": "weaker",
          "evidence": "content-level comparison",
          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Construction 5.1",
            "Lemma 5.2",
            "Theorem 5.5"
          ],
          "source_key": "erdman-faoa-2015@F320B16AF7448FBB43582C21569840FE657FCCF6F31D97F176913FDD0E1EB823",
          "source_locus": "GNS_construction.tex, theorem label 0029",
          "role": "statement comparison",
          "correspondence": "weaker",
          "evidence": "content-level comparison",
          "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Worked checkpoint: a state’s seminorm remembers one column",
            "Construction 5.1",
            "Lemma 5.2",
            "Theorem 7.2"
          ],
          "source_key": "westerbaan-category-von-neumann-algebras@bff9e58239a125af7d77a7ebacd686a21f761e42",
          "source_locus": "cstar.tex, omega-norm-basic, gns, rho-omega-miu and proto-gelfand-naimark",
          "role": "permitted adaptation / proof comparison",
          "correspondence": "weaker outside the unital C*-case",
          "evidence": "content-level comparison",
          "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
        }
      ]
    },
    {
      "id": "OA-FND.CP.B9",
      "required_statement_and_full_conditions": "- **(B9) Enough representations**, [Theorem 7.2](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-07) and [Theorem 8.5(2)](representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md#oa-fnd-gn-10) of the GNS lesson. For every nonzero \\(b\\in A\\) there is a positive linear functional \\(f\\) whose GNS representation \\(\\pi_f\\) is irreducible and has \\(\\pi_f(b)\\ne0\\). So the irreducible representations of \\(A\\) separate its points, and the direct sum of the GNS representations of all positive functionals of \\(A\\) is faithful. In particular, every C\\(^*\\)-algebra has a faithful representation (the Gelfand–Naimark theorem).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Lemma 7.1; Theorem 7.2; Theorem 8.5(2)",
        "anchor": "oa-fnd-gn-10",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-10"
      },
      "correspondence": "stronger",
      "dependencies": [
        "GN Lemma 8.1 and Theorem 8.3",
        "GN Proposition 8.4",
        "GN Cohen–Hewitt and Varopoulos proof",
        "WT Banach–Alaoglu and Krein–Milman",
        "HB separation"
      ],
      "complete_internal_proof": true,
      "new_source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Worked checkpoint: a state’s seminorm remembers one column",
            "Construction 5.1",
            "Lemma 5.2",
            "Theorem 7.2"
          ],
          "source_key": "westerbaan-category-von-neumann-algebras@bff9e58239a125af7d77a7ebacd686a21f761e42",
          "source_locus": "cstar.tex, omega-norm-basic, gns, rho-omega-miu and proto-gelfand-naimark",
          "role": "permitted adaptation / proof comparison",
          "correspondence": "weaker outside the unital C*-case",
          "evidence": "content-level comparison",
          "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Proposition 7.1a",
            "Theorem 7.2"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.2.5, II.6.3.1–3, II.6.4.10",
          "role": "alternative proof and strengthening",
          "correspondence": "exact C*-scope",
          "evidence": "content-level comparison",
          "explanation": "Added norm-at-one positivity criterion, common-unit state extension and a norming state for any nonzero positive element, with the full independent proof. The old separating-functional argument remains."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
            "Lemma 8.1",
            "Theorem 8.3",
            "Theorem 8.5"
          ],
          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.4.6, II.6.4.8–10",
          "role": "proof comparison",
          "correspondence": "weaker outside C*-algebras",
          "evidence": "content-level comparison",
          "explanation": "Free commutant Radon–Nikodym and pure-state proofs match the C*-case. The retained Banach version uses its proved factorization and represents dominated functionals on products before extending equality to all elements."
        }
      ]
    },
    {
      "id": "OA-FND.CP.B10",
      "required_statement_and_full_conditions": "- **(B10) Operators on Hilbert space**, [Theorem 3.1 and Corollary 3.2 of Hilbert spaces and compact operators](hilbert-spaces-and-compact-operators.md#oa-fnd-hs-03), and [Corollary 5.2 of the Hahn–Banach and Baire lesson](hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md#oa-fnd-hb-05). Every bounded sesquilinear form \\(\\beta\\) on a Hilbert space \\(H\\) has the form \\(\\beta(\\xi,\\eta)=\\langle T\\xi,\\eta\\rangle\\) for a unique \\(T\\in B(H)\\). A self-adjoint \\(T\\in B(H)\\) has \\(\\|T\\|=\\sup_{\\|\\xi\\|=1}|\\langle T\\xi,\\xi\\rangle|\\), so \\(\\|T\\|=\\sup_{\\|\\xi\\|=1}\\langle T\\xi,\\xi\\rangle\\) when \\(T\\ge0\\). A bounded linear bijection between Banach spaces has a bounded inverse (the open mapping theorem).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "hilbert-spaces-and-compact-operators",
        "locus": "Theorem 3.1; Corollary 3.2; HB Corollary 5.2",
        "anchor": "oa-fnd-hs-03",
        "anchors": [],
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
        "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/hilbert-spaces-and-compact-operators.md",
        "reader": "hilbert-spaces-and-compact-operators.html#oa-fnd-hs-03"
      },
      "correspondence": "exact",
      "dependencies": [
        "HS Proposition 1.1; projection and Riesz–Frechet",
        "HB Baire and open mapping"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.B11",
      "required_statement_and_full_conditions": "- **(B11) Preduals**, [Theorem 9.4(a) of Compact and trace-class operators, the predual of B(H), and the operator topologies](compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md#oa-fnd-lt-10). For a von Neumann algebra \\(N\\), the space \\(N_*\\) of ultraweakly continuous linear functionals on \\(N\\) is norm-closed in the dual space \\(N^*\\). The earlier theorem proves this for every ultraweakly closed linear subspace of \\(B(H)\\), by identifying its space of normal functionals isometrically with a quotient of \\(B(H)_*\\).",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Theorem 9.4(a)",
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    {
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      "provider": {
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    {
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        "proof_evidence": "Read current complete Sections 1-4 (lines 9-146) and complete Lemma 6.1/proof (188-223). Compare full textual diff against actual local original a6d03...; this original does not equal older active EE22... pin.",
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      "dependencies": [],
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        "anchors": [],
        "sha256": "EE22F2F44DD2C089B2BFB10D25FD5B36EBD9D7C245EA3206A1ABB71E393E2E12",
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        "proof_evidence": "Complete proof at the identified source locus.",
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        "reader": "../harmonic-analysis-on-locally-compact-groups/reader/measure-and-hilbert-space-tools.html#2-integration-and-convergence-without-countability-assumptions"
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    },
    {
      "id": "OA-FND.IR.IntegrableRadonDensity",
      "required_statement_and_full_conditions": "An integrable density against a finite Radon measure has finite Radon variation.",
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        "course": "foundations-of-von-neumann-algebras",
        "unit": "abelian-operator-algebras",
        "locus": "Section 2 full small-set integral lemma and regularity consequence",
        "anchor": "oa-fnd-ao-23",
        "anchors": [],
        "sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5",
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        "proof_evidence": "Complete proof at the identified source locus.",
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    {
      "id": "OA-FND.IR.FiniteMeasureUniqueness",
      "required_statement_and_full_conditions": "Finite measures agreeing on a generating pi-system and on the whole space agree.",
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        "course": "foundations-of-von-neumann-algebras",
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        "locus": "Section 9 full finite-measure uniqueness lemma",
        "anchor": "oa-fnd-bi-18",
        "anchors": [],
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
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    {
      "id": "OA-FND.IR.HahnBanach",
      "required_statement_and_full_conditions": "Real/complex extension and strict compact/closed convex separation.",
      "provider": {
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        "locus": "Full Theorem 2.1, Corollary 2.3, Theorem 6.3 and Corollary 6.4",
        "anchor": "oa-fnd-hb-06",
        "anchors": [],
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
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        "proof_evidence": "Complete proof at the identified source locus.",
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        "reader": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html#oa-fnd-hb-06"
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      "correspondence": "exact",
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          "source_locus": "reader-work-selected.tex, CH03 RW001",
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          "explanation": "Selected Zorn/one-dimensional companion proof read; the course keeps the full complex norm-preserving proof."
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    {
      "id": "OA-FND.IR.WeakDuality",
      "required_statement_and_full_conditions": "The continuous dual of sigma(X,Xprime) is Xprime when it separates points.",
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        "locus": "Full Theorem 1.2, finite-coordinate proof",
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        "anchors": [],
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        "proof_evidence": "Complete proof at the identified source locus.",
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    {
      "id": "OA-FND.IR.Alaoglu",
      "required_statement_and_full_conditions": "Arbitrary normed-space dual unit balls are weak-star compact.",
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        "course": "foundations-of-von-neumann-algebras",
        "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
        "locus": "Full Tychonoff Theorem 2.1 and Banach–Alaoglu Theorem 3.1",
        "anchor": "oa-fnd-wt-03",
        "anchors": [],
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "reader": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html#oa-fnd-wt-03"
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    {
      "id": "OA-FND.IR.KreinMilman",
      "required_statement_and_full_conditions": "A nonempty compact convex set is the closed convex hull of its extreme points.",
      "provider": {
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        "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
        "locus": "Full Theorem 6.1, minimal faces and separation",
        "anchor": "oa-fnd-wt-06",
        "anchors": [],
        "sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
        "source_key": "programme:foundations-of-von-neumann-algebras/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian@0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.md",
        "reader": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html#oa-fnd-wt-06"
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      "correspondence": "exact",
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    {
      "id": "OA-FND.IR.PositiveGNS",
      "required_statement_and_full_conditions": "Positive Cauchy–Schwarz, bounded positive functionals, states, cyclic GNS and unitary uniqueness.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Full Proposition 3.2, Theorem 4.7, Construction 5.1, Lemma 5.2 and Theorems 5.3–5.5; Lemma 7.1/Theorem 7.2",
        "anchor": "oa-fnd-gn-05",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-05"
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      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": [
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
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            "Theorem 4.7"
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          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.2.1–5",
          "role": "proof comparison",
          "correspondence": "weaker for 3.2; exact C*-scope for 4.7",
          "evidence": "content-level comparison",
          "explanation": "The free source supplies the C*-positive functional norm and Schwarz arguments; the retained algebraic and involutive Banach generality is not replaced."
        },
        {
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          "local_labels": [
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            "Lemma 5.2",
            "Theorem 5.3",
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          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
          "correspondence": "weaker",
          "evidence": "content-level comparison",
          "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
        },
        {
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          "local_labels": [
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          "source_key": "erdman-faoa-2015@F320B16AF7448FBB43582C21569840FE657FCCF6F31D97F176913FDD0E1EB823",
          "source_locus": "GNS_construction.tex, theorem label 0029",
          "role": "statement comparison",
          "correspondence": "weaker",
          "evidence": "content-level comparison",
          "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
        },
        {
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          "local_labels": [
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            "Construction 5.1",
            "Lemma 5.2",
            "Theorem 7.2"
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          "source_key": "westerbaan-category-von-neumann-algebras@bff9e58239a125af7d77a7ebacd686a21f761e42",
          "source_locus": "cstar.tex, omega-norm-basic, gns, rho-omega-miu and proto-gelfand-naimark",
          "role": "permitted adaptation / proof comparison",
          "correspondence": "weaker outside the unital C*-case",
          "evidence": "content-level comparison",
          "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
        },
        {
          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
          "local_labels": [
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          "source_locus": "II.6.2.5, II.6.3.1–3, II.6.4.10",
          "role": "alternative proof and strengthening",
          "correspondence": "exact C*-scope",
          "evidence": "content-level comparison",
          "explanation": "Added norm-at-one positivity criterion, common-unit state extension and a norming state for any nonzero positive element, with the full independent proof. The old separating-functional argument remains."
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    {
      "id": "OA-FND.IR.Fuglede",
      "required_statement_and_full_conditions": "Commutation with a bounded normal operator implies commutation with its adjoint.",
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        "unit": "kaplansky-s-density-theorem-and-its-consequences",
        "locus": "Full Theorem 3.1 exponential-conjugation and Liouville proof",
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        "sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89",
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    {
      "id": "OA-FND.IR.Kaplansky",
      "required_statement_and_full_conditions": "Bicommutant self-adjoint contractions are strong limits of contractions from a unital star algebra.",
      "provider": {
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      "correspondence": "exact",
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    {
      "id": "OA-FND.IR.OperatorTopology",
      "required_statement_and_full_conditions": "Weak compactness of bounded weakly closed operator sets, convex weak/strong closure equality, bounded weak/sigma-weak equality and separate multiplication continuity.",
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        "unit": "integral-representations-of-states",
        "locus": "New full Lemma 9.0, three parts",
        "anchor": "oa-fnd-ir-29",
        "anchors": [],
        "sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
        "source_key": "programme:foundations-of-von-neumann-algebras/integral-representations-of-states@9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/integral-representations-of-states.md",
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      },
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    {
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        "locus": "Full Theorem 9.1, positive weighting and constant-vector proof",
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        "anchors": [],
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/hilbert-spaces-and-compact-operators.md",
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    {
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        "locus": "Full Theorem 4.4 and Proposition 5.1",
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        "sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD",
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    {
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      "required_statement_and_full_conditions": "Convergence and L2/simple-function tools on arbitrary measure spaces.",
      "provider": {
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        "locus": "Full Theorems 2.1–2.2 and 3.1–3.2",
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      "id": "OA-FND.LT.ProductFubini",
      "required_statement_and_full_conditions": "Tonelli and absolutely integrable Fubini for arbitrary sigma-finite products, including completed measurable classes.",
      "provider": {
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        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Full Lemma 0.1(1)",
        "anchor": "oa-fnd-lt-17",
        "anchors": [],
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        "proof_evidence": "Complete proof at the identified source locus.",
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        "reader": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html#oa-fnd-lt-17"
      },
      "correspondence": "exact",
      "dependencies": [],
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    },
    {
      "id": "OA-FND.LT.LebesgueNullMaps",
      "required_statement_and_full_conditions": "Normalized Lebesgue measure, interval null covers, Borel null supersets, reflection/scaling and locally Lipschitz null images.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Full Lemma 0.1(2)",
        "anchor": "oa-fnd-lt-17",
        "anchors": [],
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "reader": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html#oa-fnd-lt-17"
      },
      "correspondence": "exact",
      "dependencies": [],
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    },
    {
      "id": "OA-FND.LT.SmoothDensity",
      "required_statement_and_full_conditions": "Compact smooth density and Gaussian approximate identities on the real line in L2.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Full Lemma 0.1(3)",
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        "anchors": [],
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "reader": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html#oa-fnd-lt-17"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
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    {
      "id": "OA-FND.LT.GaussianTransform",
      "required_statement_and_full_conditions": "Gaussian normalization and Fourier transform with the exact variance and exponential constants.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Full Lemma 0.2",
        "anchor": "oa-fnd-lt-18",
        "anchors": [],
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
        "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "reader": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html#oa-fnd-lt-18"
      },
      "correspondence": "exact",
      "dependencies": [],
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    {
      "id": "OA-FND.LT.PlancherelReal",
      "required_statement_and_full_conditions": "The real-line Fourier transform extends to a unitary with the stated 2pi convention.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Full Theorem 0.3",
        "anchor": "oa-fnd-lt-19",
        "anchors": [],
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
        "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "reader": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html#oa-fnd-lt-19"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
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    {
      "id": "OA-FND.HB.CardinalSquare",
      "required_statement_and_full_conditions": "Every set can be well ordered; every infinite cardinal equals its square, and every uncountable set partitions into that many equicardinal subsets.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
        "locus": "Full Theorem 8.5",
        "anchor": "oa-fnd-hb-09",
        "anchors": [],
        "sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
        "source_key": "programme:foundations-of-von-neumann-algebras/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces@7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.md",
        "reader": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces.html#oa-fnd-hb-09"
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      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
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    },
    {
      "id": "OA-FND.LT.SchattenDiagonalization",
      "required_statement_and_full_conditions": "Every bounded self-adjoint operator is diagonal modulo an arbitrarily small Schatten p perturbation for finite p greater than one.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
        "locus": "Full Proposition 11.18 with Lemma 11.1 and Theorem 11.2",
        "anchor": "oa-fnd-lt-20",
        "anchors": [],
        "sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
        "source_key": "programme:foundations-of-von-neumann-algebras/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies@65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.md",
        "reader": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies.html#oa-fnd-lt-20"
      },
      "correspondence": "exact",
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    {
      "id": "OA-FND.GN.VectorIntegration",
      "required_statement_and_full_conditions": "Banach-valued L1 integration, compact continuous density, sigma-finite-support Radon-product Fubini and Hilbert-valued L2/tensor identification.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Full Lemma 0.1 and following point-norm continuity proof",
        "anchor": "oa-fnd-gn-16",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-16"
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      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
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    },
    {
      "id": "OA-FND.GN.CohenFactorization",
      "required_statement_and_full_conditions": "Cohen–Hewitt factorization and automatic continuity of positive functionals under the stated bounded approximate-identity hypotheses.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Full Theorem 6.1, Lemma 6.2 and Corollaries 6.3–6.4",
        "anchor": "oa-fnd-gn-06",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-06"
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      "correspondence": "exact",
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      "complete_internal_proof": true,
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    {
      "id": "OA-FND.GN.PositiveStates",
      "required_statement_and_full_conditions": "Positive Cauchy–Schwarz and full C-star positivity, norm and continuity estimates.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Full Sections 3–4, especially Proposition 3.2 and Theorem 4.7",
        "anchor": "oa-fnd-gn-03",
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        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-03"
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      "correspondence": "exact",
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      "complete_internal_proof": true,
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          "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
          "source_locus": "II.6.2.1–5",
          "role": "proof comparison",
          "correspondence": "weaker for 3.2; exact C*-scope for 4.7",
          "evidence": "content-level comparison",
          "explanation": "The free source supplies the C*-positive functional norm and Schwarz arguments; the retained algebraic and involutive Banach generality is not replaced."
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      ]
    },
    {
      "id": "OA-FND.GN.CyclicGNS",
      "required_statement_and_full_conditions": "Nonunital cyclic GNS with vector norm squared equal to functional norm and unitary uniqueness.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
        "locus": "Full Construction 5.1, Lemma 5.2 and Theorems 5.3–5.5",
        "anchor": "oa-fnd-gn-05",
        "anchors": [],
        "sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "source_key": "programme:foundations-of-von-neumann-algebras/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark@DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.md",
        "reader": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark.html#oa-fnd-gn-05"
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      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
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          "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
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          "source_locus": "II.6.4.1–3",
          "role": "proof construction comparison",
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          "source_key": "erdman-faoa-2015@F320B16AF7448FBB43582C21569840FE657FCCF6F31D97F176913FDD0E1EB823",
          "source_locus": "GNS_construction.tex, theorem label 0029",
          "role": "statement comparison",
          "correspondence": "weaker",
          "evidence": "content-level comparison",
          "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
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          "source_key": "westerbaan-category-von-neumann-algebras@bff9e58239a125af7d77a7ebacd686a21f761e42",
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          "role": "permitted adaptation / proof comparison",
          "correspondence": "weaker outside the unital C*-case",
          "evidence": "content-level comparison",
          "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
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    {
      "id": "OA-FND.WT.EberleinSmulian",
      "required_statement_and_full_conditions": "Relative weak compactness, subsequential weak convergence and weak cluster-point criteria in Banach spaces.",
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        "locus": "Full Lemma 7.1 and Theorem 7.2",
        "anchor": "oa-fnd-wt-07",
        "anchors": [],
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        "reader": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian.html#oa-fnd-wt-07"
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      "correspondence": "exact",
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      "id": "OA-FND.WT.ComplexSequenceModels",
      "required_statement_and_full_conditions": "Complex c0 duality, ell1 extreme points and the obstruction to c0 being linearly isometric to a dual.",
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        "locus": "Full Exercises 1–4 and solutions",
        "anchor": "exercises",
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    {
      "id": "OA-FND.GN.OA-FLOW-L24",
      "required_statement_and_full_conditions": "Exact group integration/recovery/regular and covariance statements enumerated in GNS Examples 7.6–7.7.",
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        "unit": "support-group-representations",
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          },
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          },
          {
            "anchor": "OA-FLOW.GRP.ALGEBRA",
            "line": 79,
            "through_line": 135
          },
          {
            "anchor": "OA-FLOW.GRP.CONTRACTIVITY",
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            "through_line": 158
          },
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            "anchor": "OA-FLOW.GRP.INTEGRATION",
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            "through_line": 184
          },
          {
            "anchor": "OA-FLOW.GRP.RECOVERY",
            "line": 185,
            "through_line": 236
          },
          {
            "anchor": "OA-FLOW.GRP.INTERTWINERS",
            "line": 237,
            "through_line": 269
          },
          {
            "anchor": "OA-FLOW.GRP.REGULAR",
            "line": 270,
            "through_line": 288
          },
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            "anchor": "OA-FLOW.GRP.MODELS",
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        "sha256": "445B3BBC54045434ED3AA70805766EB9B4B9C164ED2436ECE10E732C43F33010",
        "source_key": "programme:foundations-of-von-neumann-algebras/support-group-representations@445B3BBC54045434ED3AA70805766EB9B4B9C164ED2436ECE10E732C43F33010",
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      "id": "OA-FND.AB.CompletedRegularity",
      "required_statement_and_full_conditions": "Finite Radon inner and outer regularity extend to completed measurable sets.",
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        "locus": "Full completed-regularity argument in Section 2",
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      "required_statement_and_full_conditions": "Scalar L-infinity acts faithfully, isometrically and as a maximal abelian von Neumann algebra for arbitrary sigma-finite or Radon measures under local conventions.",
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        "proof_evidence": "Complete proof at the identified source locus.",
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      "id": "OA-FND.AB.StoneanOrder",
      "required_statement_and_full_conditions": "Stonean compact spaces are precisely those with conditionally complete continuous real functions; bounded functions extend from dense or open subspaces.",
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      "required_statement_and_full_conditions": "Normal measures vanish on meager sets; each bounded measurable class has a continuous representative on the support, with unique normal-singular decomposition.",
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        "proof_evidence": "Complete proof at the identified source locus.",
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    {
      "id": "OA-FND.AB.HyperstoneanSpatial",
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        "proof_evidence": "Complete proof at the identified source locus.",
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      "id": "OA-FND.AB.DiffuseClassification",
      "required_statement_and_full_conditions": "Countably generated abelian algebras have one self-adjoint generator; nonzero sigma-finite diffuse examples are isomorphic to Lebesgue L-infinity.",
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        "locus": "Full Lemmas 8.1 and 8.3, Theorems 8.2 and 8.4",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/abelian-operator-algebras.md",
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      "correspondence": "exact",
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    {
      "id": "OA-FND.AB.BorelMeager",
      "required_statement_and_full_conditions": "Borel functions modulo meager functions give stonean spectra, with explicit examples having no nonzero normal measure.",
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        "locus": "Full Section 9, including base-condition counterexample",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/abelian-operator-algebras.md",
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      "id": "OA-FND.AB.PhillipsSchur",
      "required_statement_and_full_conditions": "Phillips lemma and Schur norm convergence hold for arbitrary-index summable families.",
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        "locus": "Full Section 10 and local ell-one completeness/duality",
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        "proof_evidence": "Complete proof at the identified source locus.",
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    {
      "id": "OA-FND.AB.Injectivity",
      "required_statement_and_full_conditions": "For stonean compact spaces, a positive contraction onto C(Omega) has the module property and gives norm-preserving operator extension.",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/abelian-operator-algebras.md",
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      "id": "OA-FND.AB.RadonDecompositionBackground",
      "required_statement_and_full_conditions": "Scalar local measurability, disjoint compact Radon decomposition with gluing and locally null complement, and compact-supremum integral/Hilbert-sum identity.",
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        "locus": "Full Definition 1.1, Lemma 1.2, Lemma 2.1, Definitions 2.2/2.4 and Lemma 2.3",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "support/src/local-radon-integration.md",
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      "upstream_provider": {
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        "locus": "Full Definition 1.1, Lemma 1.2, Lemma 2.1, Definitions 2.2/2.4 and Lemma 2.3",
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      "id": "OA-FND.BI.OperatorClosures",
      "required_statement_and_full_conditions": "The weak/strong/strong-star closures of a linear subspace coincide, as do its three sigma closures; all six coincide for nondegenerate star algebras.",
      "provider": {
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        "locus": "Full Lemma 1.3 and Theorem 4.4",
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      "id": "OA-FND.BI.Amplification",
      "required_statement_and_full_conditions": "Set-indexed amplification preserves six operator topologies and computes the commutant by operator-matrix entries.",
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        "unit": "the-double-commutant-theorem",
        "locus": "Full Section 3",
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        "proof_evidence": "Complete proof at the identified source locus.",
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      "id": "OA-FND.BI.DoubleCommutant",
      "required_statement_and_full_conditions": "For any star subalgebra M of B(H), its bicommutant is the six-topology closure plus scalar identity; the closure is the supporting-projection summand.",
      "provider": {
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
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        "anchors": [],
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
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      "required_statement_and_full_conditions": "A nondegenerate norm-closed star subalgebra of K(H) is its bicommutant intersected with K(H); a simple C*-algebra with a nonzero minimal projection is a compact-operator algebra.",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
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      "required_statement_and_full_conditions": "Polar partial isometries belong to a von Neumann algebra, and all two-sided ideals are self-adjoint; the asymmetric Riesz decomposition holds for arbitrary-index positive sums.",
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        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
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      "required_statement_and_full_conditions": "Cyclic and separating sets are exchanged by commutants; sigma-finiteness is equivalent to faithful positive normal functionals and countable separating families.",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
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      "required_statement_and_full_conditions": "A positive sigma-strongly continuous functional on a unital operator star algebra is a countable square-summable sum of vector functionals, finite for strong continuity.",
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        "anchor": "oa-fnd-bi-14",
        "anchors": [],
        "sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-double-commutant-theorem@879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-double-commutant-theorem.md",
        "reader": "the-double-commutant-theorem.html#oa-fnd-bi-14"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PB.MetricCompletion",
      "required_statement_and_full_conditions": "Every metric space embeds densely and isometrically in a complete metric space; separability passes to the completion.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polish-spaces-and-standard-borel-spaces",
        "locus": "Full metric-completion lemma before Lemma 8.6",
        "anchor": "completing-a-metric-space",
        "anchors": [],
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
        "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polish-spaces-and-standard-borel-spaces.md",
        "reader": "polish-spaces-and-standard-borel-spaces.html#completing-a-metric-space"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed EF/HB/Haar/measure-tools proofs; scope limits retained in self-check"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PB.BinaryInverse",
      "required_statement_and_full_conditions": "The binary-expansion map on the Cantor space with noncanonical eventually-one expansions removed has a Borel inverse given by floor digits, with all-one expansion for 1.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polish-spaces-and-standard-borel-spaces",
        "locus": "Full Exercise 5 solution",
        "anchor": "exercises",
        "anchors": [],
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
        "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polish-spaces-and-standard-borel-spaces.md",
        "reader": "polish-spaces-and-standard-borel-spaces.html#exercises"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed EF/HB/Haar/measure-tools proofs; scope limits retained in self-check"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PB.FullSelectors",
      "required_statement_and_full_conditions": "The complete-class Borel transversal, universally measurable section and Polish quotient/open-mapping conclusions have the full constructions and hypotheses stated in Sections 7–8.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polish-spaces-and-standard-borel-spaces",
        "locus": "Full Sections 7–8 and all eight solutions",
        "anchor": "oa-fnd-pb-10",
        "anchors": [],
        "sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
        "source_key": "programme:foundations-of-von-neumann-algebras/polish-spaces-and-standard-borel-spaces@81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polish-spaces-and-standard-borel-spaces.md",
        "reader": "polish-spaces-and-standard-borel-spaces.html#oa-fnd-pb-10"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed EF/HB/Haar/measure-tools proofs; scope limits retained in self-check"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.HS.Completion",
      "required_statement_and_full_conditions": "Every normed space embeds densely and isometrically in a Banach completion; an inner product extends to a Hilbert completion, and bounded maps into complete spaces extend uniquely with the same norm.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "hilbert-spaces-and-compact-operators",
        "locus": "Full completion lemma in Section 1",
        "anchor": "completing-normed-and-inner-product-spaces",
        "anchors": [],
        "sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
        "source_key": "programme:foundations-of-von-neumann-algebras/hilbert-spaces-and-compact-operators@833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/hilbert-spaces-and-compact-operators.md",
        "reader": "hilbert-spaces-and-compact-operators.html#completing-normed-and-inner-product-spaces"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed CF/GN/HS/HB/LT/BI/SW and measure-tools proofs; scope limits retained"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.StinespringFull",
      "required_statement_and_full_conditions": "Every completely positive map has the stated minimal Stinespring dilation, with the commuting action, norm equality, normality assertion and unitary uniqueness under the stated hypotheses.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "completely-positive-maps",
        "locus": "Full Theorem 6.1, Theorem 6.2 and Examples 6.3–6.4",
        "anchor": "oa-fnd-cm-05",
        "anchors": [],
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/completely-positive-maps.md",
        "reader": "completely-positive-maps.html#oa-fnd-cm-05"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed CF/GN/HS/HB/LT/BI/SW and measure-tools proofs; scope limits retained"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.GroupDilation",
      "required_statement_and_full_conditions": "A positive-definite operator-valued group function continuous at the identity has a minimal unitary dilation with global strong continuity; the minimal realization is unique.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "completely-positive-maps",
        "locus": "Entire Exercise 6.7 and solution",
        "anchor": "oa-fnd-cm-12",
        "anchors": [],
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/completely-positive-maps.md",
        "reader": "completely-positive-maps.html#oa-fnd-cm-12"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed CF/GN/HS/HB/LT/BI/SW and measure-tools proofs; scope limits retained"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.MatrixDuality",
      "required_statement_and_full_conditions": "The entrywise dual pairing, positivity tests and transpose preserve the stated n-positivity and complete positivity properties.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "completely-positive-maps",
        "locus": "Full Proposition 7.1 and matrix computations of Proposition 7.5",
        "anchor": "oa-fnd-cm-09",
        "anchors": [],
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/completely-positive-maps.md",
        "reader": "completely-positive-maps.html#oa-fnd-cm-09"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed CF/GN/HS/HB/LT/BI/SW and measure-tools proofs; scope limits retained"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CP.GNSCommutant",
      "required_statement_and_full_conditions": "Positive functionals dominated by multiples of a fixed positive functional correspond to positive commutant operators, with the stated complete order inverse and bounded-inverse criterion.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "completely-positive-maps",
        "locus": "Full Lemma 7.2, Proposition 7.3 and Example 7.4",
        "anchor": "oa-fnd-cm-10",
        "anchors": [],
        "sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "source_key": "programme:foundations-of-von-neumann-algebras/completely-positive-maps@DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/completely-positive-maps.md",
        "reader": "completely-positive-maps.html#oa-fnd-cm-10"
      },
      "correspondence": "exact",
      "dependencies": [
        "Exact consumed CF/GN/HS/HB/LT/BI/SW and measure-tools proofs; scope limits retained"
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.AF.FiniteStructure",
      "required_statement_and_full_conditions": "Finite-dimensional C*-algebras are finite sums of full matrix algebras; homomorphisms are classified by multiplicity and defect.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "af-algebras",
        "locus": "Full Sections 2–3",
        "anchor": "oa-fnd-af-01",
        "anchors": [],
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/af-algebras.md",
        "reader": "af-algebras.html#oa-fnd-af-01"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.AF.InductiveLimits",
      "required_statement_and_full_conditions": "Sequential injective C*-inductive limits, their stabilization and dimension group/cone/scale are constructed explicitly.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "af-algebras",
        "locus": "Full Sections 4–7",
        "anchor": "oa-fnd-af-03",
        "anchors": [],
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/af-algebras.md",
        "reader": "af-algebras.html#oa-fnd-af-03"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.AF.Classification",
      "required_statement_and_full_conditions": "Scaled ordered dimension groups classify AF-algebras, via finite-stage existence/uniqueness and exact intertwining.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "af-algebras",
        "locus": "Full Section 8 and Exercise 5",
        "anchor": "oa-fnd-af-07",
        "anchors": [],
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/af-algebras.md",
        "reader": "af-algebras.html#oa-fnd-af-07"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.AF.PascalInvariant",
      "required_statement_and_full_conditions": "The Pascal fixed algebra has polynomial dimension group, strict interior positive cone and the stated scale.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "af-algebras",
        "locus": "Full Section 9, including Polya coefficient proof",
        "anchor": "oa-fnd-af-08",
        "anchors": [],
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/af-algebras.md",
        "reader": "af-algebras.html#oa-fnd-af-08"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.AF.Traces",
      "required_statement_and_full_conditions": "AF traces are the finite-stage projective-limit traces; every nonzero unital AF-algebra has a tracial state.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "af-algebras",
        "locus": "Full Section 10 and Exercise 3",
        "anchor": "oa-fnd-af-09",
        "anchors": [],
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/af-algebras.md",
        "reader": "af-algebras.html#oa-fnd-af-09"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.AF.GroupsCommutants",
      "required_statement_and_full_conditions": "Finite commutants have transposed multiplicities and defect blocks; locally finite group algebras are AF with the unique finite-level norm.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "af-algebras",
        "locus": "Full Sections 11–12",
        "anchor": "oa-fnd-af-10",
        "anchors": [],
        "sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "source_key": "programme:foundations-of-von-neumann-algebras/af-algebras@3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/af-algebras.md",
        "reader": "af-algebras.html#oa-fnd-af-10"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.VectorJordan",
      "required_statement_and_full_conditions": "Every bounded functional is a vector coefficient; hermitian functionals admit norm-additive positive Jordan decompositions.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full Section 1 and state/support tools",
        "anchor": "oa-fnd-wa-01",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-01"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.UniversalBidual",
      "required_statement_and_full_conditions": "The state direct sum and concrete predual construct the universal enveloping von Neumann algebra, isometrically and weak-star homeomorphically identified with the bidual.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full construction before Section 2 and Section 3",
        "anchor": "oa-fnd-wa-03",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-03"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.NormalExtension",
      "required_statement_and_full_conditions": "Representations extend normally to the bidual; the central-kernel quotient is a normal isomorphism, with a normal inverse proved by preadjoints.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full Section 2, including second adjoints",
        "anchor": "oa-fnd-wa-02",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-02"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.CentralSupports",
      "required_statement_and_full_conditions": "Invariant dual subspaces correspond to weak-star closed ideals; representations are compared through central supports and quasi-equivalence.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full Sections 4–5",
        "anchor": "oa-fnd-wa-09",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-09"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.DualCstar",
      "required_statement_and_full_conditions": "Norm-one projections give conditional expectations; a C-star algebra which is a dual Banach space has its von Neumann algebra realization.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full Sections 8–9",
        "anchor": "oa-fnd-wa-22",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-22"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.IntrinsicNormality",
      "required_statement_and_full_conditions": "Normal/singular splitting and complete additivity characterize normal functionals; the predual and operator topologies are intrinsic, and positive maps preserve increasing suprema exactly when normal.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full Sections 10–12 and positive-map completion",
        "anchor": "oa-fnd-wa-24",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-24"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.UE.MonotoneClosure",
      "required_statement_and_full_conditions": "The monotone closure criterion yields the stated weak-operator closure and normal-image consequences, with explicit counterexamples and thirteen worked solutions.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
        "locus": "Full Section 13 and exact own KD Section 12 proof",
        "anchor": "oa-fnd-wa-27",
        "anchors": [],
        "sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "source_key": "programme:foundations-of-von-neumann-algebras/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras@E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.md",
        "reader": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras.html#oa-fnd-wa-27"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.LatticeSupports",
      "required_statement_and_full_conditions": "Projections form a complete lattice; central supports and corner centres have their stated commutant descriptions.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Sections 3–4",
        "anchor": "oa-fnd-ty-01",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-01"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.Comparison",
      "required_statement_and_full_conditions": "Schroeder–Bernstein and central projection comparison imply total subequivalence comparison in factors.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Section 5 and Exercise 5.6",
        "anchor": "oa-fnd-ty-03",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-03"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.FourTypes",
      "required_statement_and_full_conditions": "Finite and properly infinite projections yield the unique central decomposition into types I, II1, II-infinity and III.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Sections 6–7",
        "anchor": "oa-fnd-ty-10",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-10"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.TypeIClassification",
      "required_statement_and_full_conditions": "Matrix entries, abelian comparison and bounded cardinal index sets give homogeneous type-I classification and its spatial commutant form.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Sections 8–11",
        "anchor": "oa-fnd-ty-14",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-14"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.TwoProjections",
      "required_statement_and_full_conditions": "The generic two-projection corner has the cosine/sine matrix form, with the four intersection corners separated.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Section 12",
        "anchor": "oa-fnd-ty-21",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-21"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.HalvingFiniteness",
      "required_statement_and_full_conditions": "Properly infinite projections halve and absorb countable sums; finite projection lattices obey the modular law.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Sections 13–14 and worked solutions",
        "anchor": "oa-fnd-ty-18",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-18"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.AmplificationIdeals",
      "required_statement_and_full_conditions": "Local sigma-finiteness gives amplification forms; projection lattices describe ideals and simplicity, with a nonempty input lattice.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Section 15 and Exercises 15.4–15.5",
        "anchor": "oa-fnd-ty-19",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-19"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PT.TransferSpatial",
      "required_statement_and_full_conditions": "The transfer theorem compares cyclic projections of the algebra and commutant; positive normal functionals become vector functionals and yield spatial isomorphisms.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "projections-and-types-of-von-neumann-algebras",
        "locus": "Full Sections 17–18",
        "anchor": "oa-fnd-ty-08",
        "anchors": [],
        "sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "source_key": "programme:foundations-of-von-neumann-algebras/projections-and-types-of-von-neumann-algebras@470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/projections-and-types-of-von-neumann-algebras.md",
        "reader": "projections-and-types-of-von-neumann-algebras.html#oa-fnd-ty-08"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.Supports",
      "required_statement_and_full_conditions": "Two-cut norm inequalities determine the least left and right supports of a functional.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 1",
        "anchor": "oa-fnd-pd-01",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-01"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.Polar",
      "required_statement_and_full_conditions": "Every normal functional has the unique normalized polar decomposition, with the stated module convention.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 2",
        "anchor": "oa-fnd-pd-02",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-02"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.AbsoluteValue",
      "required_statement_and_full_conditions": "Absolute values are characterized by the sharp quadratic bound and positive Jordan decomposition.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 3",
        "anchor": "oa-fnd-pd-04",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-04"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.SpectralDomination",
      "required_statement_and_full_conditions": "Repeated squaring establishes the spectral-radius domination bound.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 4",
        "anchor": "oa-fnd-pd-06",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-06"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.Continuity",
      "required_statement_and_full_conditions": "Absolute value is norm continuous and has the stated weak-star-plus-norm continuity.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 5",
        "anchor": "oa-fnd-pd-07",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-07"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.HereditaryCones",
      "required_statement_and_full_conditions": "Closed hereditary cones and pure-state annihilator ideals have the proved projection descriptions.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 6",
        "anchor": "oa-fnd-pd-09",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-09"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.PhillipsSchur",
      "required_statement_and_full_conditions": "Phillips lemma and Schur convergence hold with the full humps and countable-support arguments.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 7",
        "anchor": "oa-fnd-pd-12",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-12"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.SequenceSplitting",
      "required_statement_and_full_conditions": "Normal and singular sequence components separate; countable singular spans are weak-star closed.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 8",
        "anchor": "oa-fnd-pd-13",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-13"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.CompleteMetrics",
      "required_statement_and_full_conditions": "Faithful-state metrics give the stated complete weak-compact operator-ball metrics.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 9",
        "anchor": "oa-fnd-pd-15",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-15"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.WeakCompactness",
      "required_statement_and_full_conditions": "All seven weak-compactness conditions are equivalent, with uniform absolute-value control.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 10",
        "anchor": "oa-fnd-pd-16",
        "anchors": [],
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-16"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.MackeyTopologies",
      "required_statement_and_full_conditions": "The direct weak-compact-seminorm topology agrees with sigma-strong-star on bounded sets.",
      "provider": {
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        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 11",
        "anchor": "oa-fnd-pd-19",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-19"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.AtomicNorm",
      "required_statement_and_full_conditions": "Positive weak convergence in an atomic von Neumann predual implies norm convergence.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 12",
        "anchor": "oa-fnd-pd-20",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-20"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.NormalExtension",
      "required_statement_and_full_conditions": "Normal functionals on a weakly closed nonunital subalgebra extend normally.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 13",
        "anchor": "oa-fnd-pd-21",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-21"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.PD.DualClassification",
      "required_statement_and_full_conditions": "The weakly compact and dual C-star characterizations follow from the full finite-rank and central-factor proofs.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 14",
        "anchor": "oa-fnd-pd-22",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "source_key": "programme:foundations-of-von-neumann-algebras/polar-decomposition-of-functionals-and-weak-compactness-in-preduals@16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-22"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
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      "required_statement_and_full_conditions": "All five exercise statements and worked solutions are checked, including normalized supports and Schur compactness.",
      "provider": {
        "course": "foundations-of-von-neumann-algebras",
        "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
        "locus": "Full Section 15",
        "anchor": "oa-fnd-pd-25",
        "anchors": [],
        "sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412",
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        "proof_evidence": "Complete proof at the identified source locus.",
        "editable_source": "src/polar-decomposition-of-functionals-and-weak-compactness-in-preduals.md",
        "reader": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals.html#oa-fnd-pd-25"
      },
      "correspondence": "exact",
      "dependencies": [],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CT.complex-exponential-and-the-circle",
      "required_statement_and_full_conditions": "Convergent exponential series for all complex inputs; exact real period 2π, parametrization of the unit circle, and continuous angles on open semicircles and compact intervals.",
      "provider": {
        "course": "elementary-analysis",
        "unit": "complex-exponential-and-the-circle",
        "locus": "Lemma 1, Theorems 2–4, Corollary 5, Proposition 6 and Exercise 4 solution",
        "anchor": "",
        "anchors": [],
        "sha256": "018FF214884260328BCB4477D76EC49696777D009E58C781E99B6798513C6ED9",
        "source_key": "programme:elementary-analysis/complex-exponential-and-the-circle@018FF214884260328BCB4477D76EC49696777D009E58C781E99B6798513C6ED9",
        "proof_evidence": "Complete earlier programme proofs replayed at exact stated conditions; acyclic ordered-field → interval-analysis → complex-exponential → CT chain.",
        "editable_source": "../../human/elementary-analysis/complex-exponential-and-the-circle.md",
        "reader": "../../human/elementary-analysis/complex-exponential-and-the-circle.html"
      },
      "correspondence": "exact",
      "dependencies": [
        "Earlier ordered-field construction; rational arithmetic, induction and elementary sets as explicitly stated starting inputs."
      ],
      "complete_internal_proof": true,
      "new_source_uses": []
    },
    {
      "id": "OA-FND.CT.real-analysis-on-closed-intervals",
      "required_statement_and_full_conditions": "Completeness and compactness of closed bounded real intervals; real EVT, uniform continuity, IVT and MVT; Riemann integral and both FTC directions at the stated integrability/continuity conditions, with componentwise complex extension.",
      "provider": {
        "course": "elementary-analysis",
        "unit": "real-analysis-on-closed-intervals",
        "locus": "Proposition 1, Lemma 2, Theorems 3–6, 8, 12–13 and full intervening derivative/integration providers",
        "anchor": "",
        "anchors": [],
        "sha256": "D5FBA78BE82C1EB252E6993E2534C61E3416C1AD08F81193BCE24C685379255C",
        "source_key": "programme:elementary-analysis/real-analysis-on-closed-intervals@D5FBA78BE82C1EB252E6993E2534C61E3416C1AD08F81193BCE24C685379255C",
        "proof_evidence": "Complete earlier programme proofs replayed at exact stated conditions; acyclic ordered-field → interval-analysis → complex-exponential → CT chain.",
        "editable_source": "../../human/elementary-analysis/real-analysis-on-closed-intervals.md",
        "reader": "../../human/elementary-analysis/real-analysis-on-closed-intervals.html"
      },
      "correspondence": "exact",
      "dependencies": [
        "Earlier ordered-field construction; rational arithmetic, induction and elementary sets as explicitly stated starting inputs."
      ],
      "complete_internal_proof": true,
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  ],
  "comparisons": [
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Lemma 11.2",
        "Theorem 11.4",
        "Corollary 11.5"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.4.1.1–4",
      "role": "proof comparison",
      "correspondence": "weaker",
      "explanation": "Blackadar proves positive increasing units from the open positive ball for a dense two-sided ideal. The retained theorem treats arbitrary, possibly nonclosed one-sided ideals and their closures; its full proof stays internal."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Proposition 13.3"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.4.2.1–5",
      "role": "proof comparison",
      "correspondence": "partial",
      "explanation": "Strict positivity and sequential units are compared. The free text/errata give complementary constructions; the lesson retains its complete functional-calculus/compactness proof."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Proposition 13.3"
      ],
      "source_key": "blackadar-operator-algebras-errata@7FA0252E0A61D00400737947869400637088B25399457FCAB41E0618AB4CE472",
      "source_locus": "PDF 6–7, correction to II.4.2.3",
      "role": "correction comparison",
      "correspondence": "partial",
      "explanation": "The missing original converse proof is supplied in the errata. The current internal proof already includes the relevant compactness argument; the unrelated nonunital state-space claim is not imported."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Theorem 15.1",
        "Corollary 15.4"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.5.1.1–3",
      "role": "proof construction comparison",
      "correspondence": "exact",
      "explanation": "Closed two-sided ideals are self-adjoint; cutoff units compute the quotient norm and prove the quotient C*-identity. Complete independent internal proof retained."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Proposition 17.1"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.5.1.5",
      "role": "alternative proof comparison",
      "correspondence": "exact",
      "explanation": "Norm-attaining lifts and positive/self-adjoint lifts compared at the stated scope; internal proof remains the provider."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Proposition 17.2"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.5.1.6",
      "role": "statement comparison",
      "correspondence": "exact statement only",
      "explanation": "The free source states constrained lifting without a proof. It cannot replace the complete internal argument."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_labels": [
        "Example 11.7"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.5.2.1(i)",
      "role": "example",
      "correspondence": "same phenomenon, independent calculation",
      "explanation": "The new profile f(t)=t+i|t| and its exact closure are independently calculated; no PDF wording is imported."
    },
    {
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "local_labels": [
        "Proposition 3.2",
        "Theorem 4.7"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.6.2.1–5",
      "role": "proof comparison",
      "correspondence": "weaker for 3.2; exact C*-scope for 4.7",
      "explanation": "The free source supplies the C*-positive functional norm and Schwarz arguments; the retained algebraic and involutive Banach generality is not replaced."
    },
    {
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "local_labels": [
        "Construction 5.1",
        "Lemma 5.2",
        "Theorem 5.3",
        "Theorem 5.4",
        "Theorem 5.5"
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      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.6.4.1–3",
      "role": "proof construction comparison",
      "correspondence": "weaker",
      "explanation": "Full nonunital C*-GNS and uniqueness proof read. Internal 4.5/5.2 handle arbitrary involutive Banach algebras; 5.3 gives an exact representability criterion and 5.4 retains the bounded-approximate-identity constants."
    },
    {
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "local_labels": [
        "Construction 5.1",
        "Lemma 5.2",
        "Theorem 5.5"
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      "source_key": "erdman-faoa-2015@F320B16AF7448FBB43582C21569840FE657FCCF6F31D97F176913FDD0E1EB823",
      "source_locus": "GNS_construction.tex, theorem label 0029",
      "role": "statement comparison",
      "correspondence": "weaker",
      "explanation": "Erdman assumes unital states and conventions for unital representations; GNS proof is explicitly reader work, not an external complete proof provider."
    },
    {
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "local_labels": [
        "Worked checkpoint: a state’s seminorm remembers one column",
        "Construction 5.1",
        "Lemma 5.2",
        "Theorem 7.2"
      ],
      "source_key": "westerbaan-category-von-neumann-algebras@bff9e58239a125af7d77a7ebacd686a21f761e42",
      "source_locus": "cstar.tex, omega-norm-basic, gns, rho-omega-miu and proto-gelfand-naimark",
      "role": "permitted adaptation / proof comparison",
      "correspondence": "weaker outside the unital C*-case",
      "explanation": "The matrix checkpoint adaptation already has a CC BY 4.0 boundary; new source reading verifies the unital GNS construction. Inner-product conventions are translated explicitly, not conflated."
    },
    {
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "local_labels": [
        "Proposition 7.1a",
        "Theorem 7.2"
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.6.2.5, II.6.3.1–3, II.6.4.10",
      "role": "alternative proof and strengthening",
      "correspondence": "exact C*-scope",
      "explanation": "Added norm-at-one positivity criterion, common-unit state extension and a norming state for any nonzero positive element, with the full independent proof. The old separating-functional argument remains."
    },
    {
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      "local_labels": [
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        "Theorem 8.3",
        "Theorem 8.5"
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      "source_locus": "II.6.4.6, II.6.4.8–10",
      "role": "proof comparison",
      "correspondence": "weaker outside C*-algebras",
      "explanation": "Free commutant Radon–Nikodym and pure-state proofs match the C*-case. The retained Banach version uses its proved factorization and represents dominated functionals on products before extending equality to all elements."
    },
    {
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      "source_key": "dixmier-stone-1951@931EBC486E9A20F65E7C1A976F64F4F2D6809D5621298BE27B959A81D65E7A2A",
      "source_locus": "§4 Proposition 9, printed 165–166",
      "role": "proof comparison",
      "correspondence": "exact after relabelling",
      "explanation": "Dixmier E1 is the dense-meager part and E2 hyperstonean; the lesson uses Ω2 and Ω1 respectively. Three clopen parts, uniqueness and rare-supported measures are retained with full internal proof."
    },
    {
      "unit": "abelian-operator-algebras",
      "local_labels": [
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      ],
      "source_key": "dixmier-stone-1951@931EBC486E9A20F65E7C1A976F64F4F2D6809D5621298BE27B959A81D65E7A2A",
      "source_locus": "§5 Theorem 1 and §6 Theorem 2, printed 169–174",
      "role": "proof construction comparison",
      "correspondence": "exact at stated abstract/spatial scope",
      "explanation": "Arbitrary Radon model uses local null sets, not a sigma-finite restriction. Full own normal-measure, gluing and operator proof/dependencies remain internal; external preceding lemmas are not claimed freshly audited."
    },
    {
      "unit": "abelian-operator-algebras",
      "local_labels": [
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      ],
      "source_key": "bouafia-de-pauw-2105.11331v1",
      "source_locus": "§4.7 and §6.1–4; measure-space definition in §6.11 footnote 1",
      "role": "generalization / proof mechanism",
      "correspondence": "partial source, complete independent operator proof",
      "explanation": "Finite-measure pieces glue over arbitrary index sets. The paper studies more general negligible-set categories; no claim equating all its hypotheses with the lesson is made."
    },
    {
      "unit": "abelian-operator-algebras",
      "local_labels": [
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        "Proposition 3.2b"
      ],
      "source_key": "fremlin-measure-theory-vol4-2013-source",
      "source_locus": "415A, 416A–B",
      "role": "context / statement",
      "correspondence": "different measure conventions",
      "explanation": "Fremlin uses complete, locally determined Radon measures. The lesson’s outer-regular Radon local-integral convention is instead supplied by the exact programme compact-piece proof. No unproved convention interchange is used."
    },
    {
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      "local_labels": [
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      "source_key": "programme:D20-companion@059bda086dfd6e6aa80f2077b2338c5d15039057",
      "source_locus": "reader-work-selected.tex, CH03 RW001",
      "role": "proof comparison",
      "correspondence": "real dominated extension exact; complex extension internal",
      "explanation": "Selected Zorn/one-dimensional companion proof read; the course keeps the full complex norm-preserving proof."
    },
    {
      "unit": "hilbert-spaces-and-compact-operators",
      "local_labels": [
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      ],
      "source_key": "programme:D20-companion@059bda086dfd6e6aa80f2077b2338c5d15039057",
      "source_locus": "reader-work-selected.tex, CH04 RW001",
      "role": "proof comparison",
      "correspondence": "exact Hilbert duality mechanism",
      "explanation": "Riesz–Fréchet proof read with first-variable linear convention. Actual own statement is bound through its section source, not inferred from the core title."
    },
    {
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      "local_labels": [
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      "source_key": "programme:D20-companion@059bda086dfd6e6aa80f2077b2338c5d15039057",
      "source_locus": "compact-spectral-svd.tex, O008 bridge LEM001 and THM002",
      "role": "proof comparison",
      "correspondence": "same compact Hilbert scope",
      "explanation": "Selected full proof passages read; the core Riesz–Schauder step consumes its earlier Fredholm alternative. Own source keeps its complete prerequisite chain. The subsequently read core SVD is related context, not asserted as a new HS theorem."
    },
    {
      "unit": "hahn-banach-baire-and-the-basic-theorems-on-banach-spaces",
      "local_scopes": [
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          "heading": "2. The Hahn–Banach theorem",
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          "heading": "3. The Baire category theorem",
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          "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
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          "heading": "4. Uniform boundedness",
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        {
          "heading": "5. The open mapping and closed graph theorems",
          "line": 152,
          "through_line": 183,
          "anchors": [
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          "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
        },
        {
          "heading": "6. Topological vector spaces and separation of convex sets",
          "line": 184,
          "through_line": 257,
          "anchors": [
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          "source_sha256": "7B79A7F7E1F1CDD6B9132FB4A684056329D588D6041F706A993D62E83932A5CD"
        }
      ],
      "source_key": "human:van-neerven@827AAD144C608177EF28A6883473FD16A50899CF7C12852AE584AF4C9F837CD4",
      "source_locus": "§§4.2, 5.1–5.3; PDF 140–144 and 183–190",
      "role": "proof comparison",
      "correspondence": "Classical Banach-space scope",
      "explanation": "Dominated extension, separation, Baire, uniform boundedness, successive open-mapping approximations and the closed graph argument; own maximality/cardinal proofs remain complete."
    },
    {
      "unit": "weak-topologies-tychonoff-banach-alaoglu-mazur-bipolars-krein-milman-and-eberlein-smulian",
      "local_scopes": [
        {
          "heading": "7. The Eberlein–Šmulian theorem",
          "line": 195,
          "through_line": 240,
          "anchors": [
            "OA-FND-WT-07"
          ],
          "source_sha256": "0B07A69CED011F1B54A1A6E657F752466EFF4202ED1CC43A85A209E645D4A8DA"
        }
      ],
      "source_key": "human:vogt@9534B2B9478680458400CDC966BE6B17E0260E4B8916EEF7A74191910706B0A7",
      "source_locus": "Full four-page paper, Theorems 3–4",
      "role": "proof comparison",
      "correspondence": "Related weak-star theorem with additional hypothesis",
      "explanation": "Tail closed convex hulls require the extra intersection condition; not an unrestricted weak-star version of Eberlein–Šmulian."
    },
    {
      "unit": "hilbert-spaces-and-compact-operators",
      "local_scopes": [
        {
          "heading": "5. Compact operators",
          "line": 172,
          "through_line": 225,
          "anchors": [
            "OA-FND-HS-05",
            "OA-FND-HS-10"
          ],
          "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
        },
        {
          "heading": "6. The spectral theorem for compact self-adjoint operators",
          "line": 226,
          "through_line": 255,
          "anchors": [
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          ],
          "source_sha256": "833A2D60BB99E63875CE10BF363953C0BB265F92996ECC8648FC8C6A105CB1FE"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§3.3; PDF 47–49",
      "role": "proof comparison",
      "correspondence": "Compact-operator proof mechanisms",
      "explanation": "Finite-rank approximation on the unit ball, compact approximate eigenvector argument, finite eigenspaces and norm-convergent spectral sum; the lesson supplies its own complete operators-between-two-spaces arguments."
    },
    {
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "local_scopes": [
        {
          "heading": "4. Cauchy's theorem for cycles",
          "line": 291,
          "through_line": 378,
          "anchors": [
            "OA-FND-CT-04"
          ],
          "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
        }
      ],
      "source_key": "human:mit-dixon@28860604809CA6E48B2EF1A1F946DB9DD92B8B8F0CD801FA3D2917EF8CF7A71C",
      "source_locus": "PDF 2–5, printed 1–4",
      "role": "proof comparison",
      "correspondence": "Closed-curve treatment; finite cycles supplied internally",
      "explanation": "Divided-difference continuity, Morera, holomorphic gluing and Liouville. Original Dixon attribution retained; actual reading is the MIT exposition."
    },
    {
      "unit": "cauchy-s-theorem-for-cycles-and-its-consequences",
      "local_scopes": [
        {
          "heading": "4. Cauchy's theorem for cycles",
          "line": 291,
          "through_line": 378,
          "anchors": [
            "OA-FND-CT-04"
          ],
          "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
        },
        {
          "heading": "5. Cycles that surround a compact set",
          "line": 379,
          "through_line": 404,
          "anchors": [
            "OA-FND-CT-05"
          ],
          "source_sha256": "B4B3DA6A8DC08A301C31700DAB99DF62DE210494E16ED863AD333DC18981318A"
        }
      ],
      "source_key": "human:cerny@1940CC2DCFFBFEA58C26CDDA70F2B38613960A4EA039FED5875726199B587174",
      "source_locus": "Main proof pp.1–5; comments pp.6–7 also read",
      "role": "proof comparison",
      "correspondence": "Alternative finite-cycle proof",
      "explanation": "Grid side cancellation and contour interchange; source assumes rectangular Cauchy formula, supplied by the lesson’s full local proof."
    },
    {
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "local_scopes": [
        {
          "heading": "6. The holomorphic functional calculus",
          "line": 420,
          "through_line": 608,
          "anchors": [
            "OA-FND-BN-10",
            "OA-FND-BN-11",
            "OA-FND-BN-12",
            "OA-FND-BN-22"
          ],
          "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
        }
      ],
      "source_key": "human:shirbisheh@067CBC91AA7578A2E6A45AD54DDDB9DE6035F847AC037F0231C37EF70963544D",
      "source_locus": "§2.5, Theorems 2.5.2 and 2.5.5",
      "role": "proof comparison",
      "correspondence": "Source contour formulation narrower",
      "explanation": "Multiplicativity, spectral mapping and composition compared. The lesson retains exact arbitrary-cycle index conditions for disconnected spectra and supplies full scalar Cauchy inputs."
    },
    {
      "unit": "banach-algebras-spectrum-holomorphic-functional-calculus-and-gelfand-theory",
      "local_scopes": [
        {
          "heading": "4. The spectrum",
          "line": 212,
          "through_line": 304,
          "anchors": [
            "OA-FND-BN-06",
            "OA-FND-BN-07"
          ],
          "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
        },
        {
          "heading": "5. Nonempty spectrum and the spectral radius formula",
          "line": 305,
          "through_line": 419,
          "anchors": [
            "OA-FND-BN-08",
            "OA-FND-BN-09"
          ],
          "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
        },
        {
          "heading": "10. Characters",
          "line": 700,
          "through_line": 738,
          "anchors": [
            "OA-FND-BN-16",
            "OA-FND-BN-17"
          ],
          "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
        },
        {
          "heading": "11. The Gelfand representation",
          "line": 739,
          "through_line": 782,
          "anchors": [
            "OA-FND-BN-18",
            "OA-FND-BN-19"
          ],
          "source_sha256": "C1CDDCF9028D7F51F891AE87F4EE53F270EEAAB5C49665BDF5542B43A86BC3C4"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.1.4–II.1.5 and II.2.1–II.2.2; PDF 64–70",
      "role": "proof comparison",
      "correspondence": "Partial textbook treatment",
      "explanation": "Spectral facts and character/maximal-ideal arguments; several spectral/calculus steps are summarized. Full spectral-radius and contour proofs remain internal."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_scopes": [
        {
          "heading": "4. Homomorphisms: contractivity, isometry and automatic continuity",
          "line": 159,
          "through_line": 258,
          "anchors": [
            "OA-FND-CF-12",
            "OA-FND-CF-13",
            "OA-FND-CF-14"
          ],
          "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
        },
        {
          "heading": "5. The continuous functional calculus",
          "line": 259,
          "through_line": 346,
          "anchors": [
            "OA-FND-CF-07",
            "OA-FND-CF-08",
            "OA-FND-CF-31"
          ],
          "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.1.6 and II.2.2–II.2.3; PDF 67–73",
      "role": "proof comparison",
      "correspondence": "Classical C*-algebra mechanisms",
      "explanation": "Spectral-radius automatic contractivity, injective isometry and normal continuous calculus, including functions vanishing at zero in the nonunital case."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_scopes": [
        {
          "heading": "8. The positive cone and the order",
          "line": 416,
          "through_line": 483,
          "anchors": [
            "OA-FND-CF-15",
            "OA-FND-CF-16"
          ],
          "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
        },
        {
          "heading": "9. Operator monotone powers and a Hölder estimate",
          "line": 484,
          "through_line": 615,
          "anchors": [
            "OA-FND-CF-17"
          ],
          "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.3.1; PDF 73–76",
      "role": "proof comparison",
      "correspondence": "Classical positive-cone and power-order mechanisms",
      "explanation": "Positive cone, x*x positivity and square-root order; source typographical slips are not imported."
    },
    {
      "unit": "c-star-algebras-continuous-functional-calculus-automatic-continuity-positive-cones",
      "local_scopes": [
        {
          "heading": "14. The asymmetric Riesz decomposition",
          "line": 1187,
          "through_line": 1261,
          "anchors": [
            "OA-FND-CF-21"
          ],
          "source_sha256": "1CAF1BA3FDEE1409E497FBAA4884317CDE491805A8CEBA6F7293AE88F56793B1"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.3.2.1–II.3.2.7; PDF 77–79",
      "role": "proof comparison",
      "correspondence": "Partial external proof; asymmetric conclusion preserved",
      "explanation": "Soft polar approximation is outlined; asymmetric Riesz decomposition II.3.2.7 refers to Pedersen for the proof. The lesson’s full proof and exact distinction between u*u and uu* remain the provider."
    },
    {
      "unit": "the-spectral-theorem-for-bounded-self-adjoint-operators",
      "local_scopes": [
        {
          "heading": "2. The spectral measure of a vector",
          "line": 81,
          "through_line": 98,
          "anchors": [
            "OA-FND-ST-02"
          ],
          "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
        },
        {
          "heading": "3. The Borel functional calculus",
          "line": 99,
          "through_line": 202,
          "anchors": [
            "OA-FND-ST-03"
          ],
          "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
        },
        {
          "heading": "4. Projection-valued measures and the spectral theorem",
          "line": 203,
          "through_line": 297,
          "anchors": [
            "OA-FND-ST-04"
          ],
          "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
        },
        {
          "heading": "5. Spectral projections and polar decomposition",
          "line": 298,
          "through_line": 350,
          "anchors": [
            "OA-FND-ST-05"
          ],
          "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
        },
        {
          "heading": "6. Monotone convergence of operators",
          "line": 351,
          "through_line": 381,
          "anchors": [
            "OA-FND-ST-06"
          ],
          "source_sha256": "5652A228DCD49EAB4E79F30010CEF368778FFC92A743C4DC10E517E8D87F58DD"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§3.7; PDF 58–63",
      "role": "proof comparison",
      "correspondence": "Spectral-measure construction compared",
      "explanation": "Monotone operator limits, scalar Radon measures, bounded forms, indicator multiplication and simple-function limits; own exact null-set and essential-range conclusions retained."
    },
    {
      "unit": "representations-and-positive-functionals-the-gns-construction-and-the-gelfand-naimark",
      "local_scopes": [
        {
          "heading": "3. Positive functionals",
          "line": 77,
          "through_line": 116,
          "anchors": [
            "OA-FND-GN-03"
          ],
          "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
        },
        {
          "heading": "4. Continuity and norms of positive functionals",
          "line": 117,
          "through_line": 356,
          "anchors": [
            "OA-FND-GN-04"
          ],
          "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
        },
        {
          "heading": "5. The Gelfand–Naimark–Segal construction",
          "line": 377,
          "through_line": 452,
          "anchors": [
            "OA-FND-GN-05"
          ],
          "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
        },
        {
          "heading": "7. The Gelfand–Naimark theorem and enveloping C\\*-algebras",
          "line": 453,
          "through_line": 488,
          "anchors": [
            "OA-FND-GN-07"
          ],
          "source_sha256": "DF95CB29F15166B9BA3B189FF601707228EAE822B968F3A80D61EA975D670220"
        }
      ],
      "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
      "source_locus": "§7.8–§7.12; PDF 39–41",
      "role": "proof comparison",
      "correspondence": "Unital C*-case; general Banach-* and nonunital cases broader",
      "explanation": "Quotient by the null left ideal, representation norm bound and norming-state extension. The source leaves *-homomorphism verification and nonunital construction as exercises."
    },
    {
      "unit": "compact-and-trace-class-operators-the-predual-of-b-h-and-the-operator-topologies",
      "local_scopes": [
        {
          "heading": "4. Trace-class operators and the trace",
          "line": 438,
          "through_line": 611,
          "anchors": [
            "OA-FND-LT-05"
          ],
          "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
        },
        {
          "heading": "5. Duality between compact, trace-class and bounded operators",
          "line": 612,
          "through_line": 692,
          "anchors": [
            "OA-FND-LT-06"
          ],
          "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
        },
        {
          "heading": "6. The predual of B(H)",
          "line": 693,
          "through_line": 726,
          "anchors": [
            "OA-FND-LT-07"
          ],
          "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
        },
        {
          "heading": "8. Seven topologies on B(H)",
          "line": 822,
          "through_line": 916,
          "anchors": [
            "OA-FND-LT-09"
          ],
          "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
        },
        {
          "heading": "9. Continuous functionals and closed convex sets",
          "line": 917,
          "through_line": 1046,
          "anchors": [
            "OA-FND-LT-10"
          ],
          "source_sha256": "65E04118C1D72459208F78F956003075CF7D453C6DFB79B3A4E35B94512BA974"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§§3.1–3.4; PDF 40–52",
      "role": "proof comparison",
      "correspondence": "Trace duality and topological mechanisms",
      "explanation": "Rank-one sesquilinear forms prove B(H)=trace-class dual; finite-rank/compact duality is left as an exercise in the source. The lesson supplies that duality, Schatten-p estimates, Krein–Šmulian and scattering proofs itself."
    },
    {
      "unit": "the-double-commutant-theorem",
      "local_scopes": [
        {
          "heading": "4. The double commutant theorem",
          "line": 234,
          "through_line": 319,
          "anchors": [
            "OA-FND-BI-06",
            "OA-FND-BI-07"
          ],
          "source_sha256": "879E1B018A9E98436C3E209447F183B3D6B5517ED17E062F6EFFA720C2D60883"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§3.5; PDF 53–55",
      "role": "proof comparison",
      "correspondence": "Bicommutant proof at arbitrary Hilbert dimension",
      "explanation": "One-vector invariant subspace followed by finite diagonal amplification; own degenerate essential-subspace and all-six-closure statements retained."
    },
    {
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "local_scopes": [
        {
          "heading": "7. Kaplansky's density theorem",
          "line": 407,
          "through_line": 465,
          "anchors": [
            "OA-FND-KD-07"
          ],
          "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§3.6; PDF 56–57",
      "role": "proof comparison",
      "correspondence": "Standard bounded-density proof",
      "explanation": "Self-adjoint clamping and matrix amplification compared; own strong* conclusion and nonclosed *-algebra treatment retained."
    },
    {
      "unit": "kaplansky-s-density-theorem-and-its-consequences",
      "local_scopes": [
        {
          "heading": "7. Kaplansky's density theorem",
          "line": 407,
          "through_line": 465,
          "anchors": [
            "OA-FND-KD-07"
          ],
          "source_sha256": "7C11A93F8858656CD095E10B0AFF54BD3019B93AE9AEECDF2DC3776D94889C89"
        }
      ],
      "source_key": "human:elliott-griffin@DA2CCC95ADD8212265B23ABDABC18DE6CE5B8E9DCE1BBFC629286C51BC8C66ED",
      "source_locus": "Entire main argument and concluding remark",
      "role": "proof comparison",
      "correspondence": "New full proof of bounded-net method",
      "explanation": "Adjoint closure via convex balls, bounded continuous calculus with f(0)=0, M2 strong* control, trace duality and Krein–Šmulian. BI Theorem 4.4 justifies the final ultraweak-to-strong algebra closure step.",
      "local_labels": [
        "Theorem 7.2"
      ]
    },
    {
      "unit": "abelian-operator-algebras",
      "local_scopes": [
        {
          "heading": "3. Measures and cyclic representations",
          "line": 172,
          "through_line": 277,
          "anchors": [
            "OA-FND-AO-03",
            "OA-FND-AO-04"
          ],
          "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
        },
        {
          "heading": "8. Countably generated abelian von Neumann algebras",
          "line": 638,
          "through_line": 726,
          "anchors": [
            "OA-FND-AO-14",
            "OA-FND-AO-15"
          ],
          "source_sha256": "46AA4EAD63111B14A2D11BFDC0143261787FC0D4F3584FBD6C19A2140C9C0EC5"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§3.8; PDF 63–66",
      "role": "proof comparison",
      "correspondence": "Cyclic model; arbitrary decomposition sketched",
      "explanation": "Cyclic L2 Radon model and ternary coding of countably generated abelian algebras. The course retains full arbitrary finite-piece gluing, arbitrary Hilbert dimension, local-null conventions and non-semifinite counterexample."
    },
    {
      "unit": "the-universal-enveloping-von-neumann-algebra-of-a-c-star-algebra-and-w-star-algebras",
      "local_scopes": [
        {
          "heading": "1. Functionals as vector coefficients",
          "line": 106,
          "through_line": 189,
          "anchors": [
            "OA-FND-WA-01"
          ],
          "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
        },
        {
          "heading": "2. Extending a representation to the bidual",
          "line": 190,
          "through_line": 243,
          "anchors": [
            "OA-FND-WA-02"
          ],
          "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
        },
        {
          "heading": "3. The universal enveloping von Neumann algebra",
          "line": 244,
          "through_line": 327,
          "anchors": [
            "OA-FND-WA-03",
            "OA-FND-WA-04"
          ],
          "source_sha256": "E07EA555C07D14FA449CF2A415A417D4157E4527F3BC6436609B2E305C035F1B"
        }
      ],
      "source_key": "human:peterson-vn@5A30C5DB90BD240415C6C095AC426C6CD307EC8E681768AA0BEA777E9191FB7C",
      "source_locus": "§4.6, Theorem 4.6.1; PDF 68–69",
      "role": "proof comparison",
      "correspondence": "Universal bidual mechanism; own dependency order retained",
      "explanation": "Weak-star dense functionals, unit-ball surjectivity via density, and separation/faithful universal representation. Positive decomposition is a later source result; the lesson’s prerequisites provide it without that citation ordering."
    },
    {
      "unit": "projections-and-types-of-von-neumann-algebras",
      "local_scopes": [
        {
          "heading": "3. The projection lattice and equivalence of projections",
          "line": 49,
          "through_line": 96,
          "anchors": [
            "OA-FND-TY-01",
            "OA-FND-TY-02"
          ],
          "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
        },
        {
          "heading": "4. Supports, cyclic projections and the parallelogram law",
          "line": 97,
          "through_line": 131,
          "anchors": [
            "OA-FND-TY-04",
            "OA-FND-TY-07"
          ],
          "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
        },
        {
          "heading": "5. The comparison theorem",
          "line": 132,
          "through_line": 237,
          "anchors": [
            "OA-FND-TY-03",
            "OA-FND-TY-05",
            "OA-FND-TY-06"
          ],
          "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
        },
        {
          "heading": "6. Finite, infinite and abelian projections",
          "line": 238,
          "through_line": 271,
          "anchors": [
            "OA-FND-TY-09"
          ],
          "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
        },
        {
          "heading": "7. The type decomposition",
          "line": 272,
          "through_line": 312,
          "anchors": [
            "OA-FND-TY-10"
          ],
          "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "III.1.1–III.1.4; PDF 244–254",
      "role": "proof comparison",
      "correspondence": "Projection comparison and type mechanisms",
      "explanation": "Orthogonal sums, central support, maximal matching, Schröder–Bernstein and finite/properly-infinite/type splitting. Some implications are only sketched externally; own full statements and all cardinal cases remain."
    },
    {
      "unit": "projections-and-types-of-von-neumann-algebras",
      "local_scopes": [
        {
          "heading": "15. Countable sums and properly infinite algebras",
          "line": 784,
          "through_line": 879,
          "anchors": [
            "OA-FND-TY-19",
            "OA-FND-TY-20"
          ],
          "source_sha256": "470558882F5CEC1BBA862B5D5B65D4E196A3E2B7B07916B92E9932F5753AF11D"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "III.1.3.6; PDF 251–252",
      "role": "proof comparison",
      "correspondence": "Locally countably decomposable hypothesis is essential",
      "explanation": "Countable decomposition on central pieces compared; course retains its explicit nonseparable rank obstruction and does not extend the conclusion to arbitrary algebras."
    },
    {
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "local_scopes": [
        {
          "heading": "2. The polar decomposition",
          "line": 168,
          "through_line": 369,
          "anchors": [
            "OA-FND-PD-02",
            "OA-FND-PD-03"
          ],
          "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
        },
        {
          "heading": "3. How the absolute value is determined",
          "line": 370,
          "through_line": 447,
          "anchors": [
            "OA-FND-PD-04",
            "OA-FND-PD-05"
          ],
          "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
        }
      ],
      "source_key": "human:peterson-oa@309A2DB968DB2173C431BDE8A42379EE66E58ED7C6B3A4FB41A59A66934F90AD",
      "source_locus": "§3.11; PDF 78–81",
      "role": "proof comparison",
      "correspondence": "Polar and Jordan mechanisms",
      "explanation": "Support ideals, norm-attaining partial isometry and Jordan positivity; module conventions reconciled to the lesson’s (aφ)(x)=φ(xa). Own continuity modulus and invariant-subspace proofs remain."
    },
    {
      "unit": "polar-decomposition-of-functionals-and-weak-compactness-in-preduals",
      "local_scopes": [
        {
          "heading": "10. Weak compactness in the predual",
          "line": 801,
          "through_line": 969,
          "anchors": [
            "OA-FND-PD-16",
            "OA-FND-PD-17",
            "OA-FND-PD-18"
          ],
          "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
        },
        {
          "heading": "11. The Mackey topology on bounded sets",
          "line": 970,
          "through_line": 1021,
          "anchors": [
            "OA-FND-PD-19"
          ],
          "source_sha256": "16875E65B1EA176C7E1AC95329892B348E6AC399C099FA104661880DE9007412"
        }
      ],
      "source_key": "human:hamhalter2019@69D30F1F9F0503096906EB37AC2F652E857737D66FF1B7968B0A050BFCCA3563",
      "source_locus": "§11, Theorems 11.1 and 11.3; PDF 52–53",
      "role": "proof comparison",
      "correspondence": "Further reading, not complete foundation",
      "explanation": "Modern weak-noncompactness formulation extends to JBW*-triples. The criterion is quoted there from earlier sources, so it is not counted as a full proof replacement for the course’s predual or Mackey theorem."
    },
    {
      "unit": "completely-positive-maps",
      "local_scopes": [
        {
          "heading": "3. Completely positive maps",
          "line": 131,
          "through_line": 184,
          "anchors": [
            "OA-FND-CM-03"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "4. The Kadison–Schwarz inequality",
          "line": 185,
          "through_line": 249,
          "anchors": [
            "OA-FND-CM-07"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "5. When positivity implies complete positivity",
          "line": 250,
          "through_line": 336,
          "anchors": [
            "OA-FND-CM-04",
            "OA-FND-CM-08"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "6. Stinespring's dilation theorem",
          "line": 337,
          "through_line": 565,
          "anchors": [
            "OA-FND-CM-05",
            "OA-FND-CM-06",
            "OA-FND-CM-12"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "7. Completely positive maps and dual spaces",
          "line": 566,
          "through_line": 757,
          "anchors": [
            "OA-FND-CM-09",
            "OA-FND-CM-10",
            "OA-FND-CM-11"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        }
      ],
      "source_key": "human:goals2024@82BAA561ADE7E58AADDFDED386894E5FED8D8BDF01887AB9F3D2B9283E11FC04",
      "source_locus": "§10; PDF 54–61",
      "role": "proof comparison",
      "correspondence": "Some complete matrix steps, Stinespring/Arveson sketches",
      "explanation": "Scalar positivity implies CP, block positivity and multiplicative domain compared; unitization and finite-dimensional CP/positive-functional correspondence have omitted proofs. All nonunital and transpose-convention details remain proved internally."
    },
    {
      "unit": "completely-positive-maps",
      "local_scopes": [
        {
          "heading": "3. Completely positive maps",
          "line": 131,
          "through_line": 184,
          "anchors": [
            "OA-FND-CM-03"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "4. The Kadison–Schwarz inequality",
          "line": 185,
          "through_line": 249,
          "anchors": [
            "OA-FND-CM-07"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "5. When positivity implies complete positivity",
          "line": 250,
          "through_line": 336,
          "anchors": [
            "OA-FND-CM-04",
            "OA-FND-CM-08"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "6. Stinespring's dilation theorem",
          "line": 337,
          "through_line": 565,
          "anchors": [
            "OA-FND-CM-05",
            "OA-FND-CM-06",
            "OA-FND-CM-12"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        },
        {
          "heading": "7. Completely positive maps and dual spaces",
          "line": 566,
          "through_line": 757,
          "anchors": [
            "OA-FND-CM-09",
            "OA-FND-CM-10",
            "OA-FND-CM-11"
          ],
          "source_sha256": "DB4F81967D780A922DC4390F378123CC2B4FB3D077147EBE0BB732D03C8A8550"
        }
      ],
      "source_key": "human:blackadar-oa@8CB61A8348EFE6E4B36DBEFED4EAC2C35498D28C03E160638AD7AF952457F88A",
      "source_locus": "II.6.9; PDF 141–145",
      "role": "proof comparison",
      "correspondence": "Classical CP treatment with outlines",
      "explanation": "Stinespring and Arveson are brief checks/outlines; abelian positivity to CP uses finite representations. Own full construction, minimality/faithfulness, θω factorization and CP pairing proof remain the providers."
    },
    {
      "unit": "integral-representations-of-states",
      "local_scopes": [
        {
          "heading": "3. Barycentres",
          "line": 160,
          "through_line": 197,
          "anchors": [
            "OA-FND-IR-04"
          ],
          "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
        },
        {
          "heading": "4. The Choquet order",
          "line": 198,
          "through_line": 282,
          "anchors": [
            "OA-FND-IR-03",
            "OA-FND-IR-05",
            "OA-FND-IR-15"
          ],
          "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
        },
        {
          "heading": "5. Boundary measures",
          "line": 283,
          "through_line": 332,
          "anchors": [
            "OA-FND-IR-06"
          ],
          "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
        },
        {
          "heading": "6. The metrizable case and Baire sets",
          "line": 333,
          "through_line": 416,
          "anchors": [
            "OA-FND-IR-07",
            "OA-FND-IR-08"
          ],
          "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
        },
        {
          "heading": "7. Simplices",
          "line": 417,
          "through_line": 451,
          "anchors": [
            "OA-FND-IR-23",
            "OA-FND-IR-27"
          ],
          "source_sha256": "9158B7C473EA7357C8D742CC8883822E0B7EAFF4372BDF8E9552F86082AD81EB"
        }
      ],
      "source_key": "human:fremlin46@FA399BA84CDB22A1173699F85330972DE225743A71113FAFC28F650B3A2A323B",
      "source_locus": "§461A–§461P; PDF 1–9",
      "role": "proof comparison",
      "correspondence": "General compact convex scope, not only metrizable",
      "explanation": "Barycenters, Jensen, maximal measure and midpoint spreading. Nonmetrizable conclusion is Baire pseudo-support; actual Radon mass on extreme points is distinguished from it. Operator-algebra orthogonal/central measures and G-abelian proofs remain internal."
    },
    {
      "unit": "polish-spaces-and-standard-borel-spaces",
      "local_scopes": [
        {
          "heading": "1. Trees in Polish spaces",
          "line": 67,
          "through_line": 95,
          "anchors": [
            "OA-FND-PB-01"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        },
        {
          "heading": "2. Souslin sets",
          "line": 96,
          "through_line": 197,
          "anchors": [
            "OA-FND-PB-02",
            "OA-FND-PB-03",
            "OA-FND-PB-12"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        },
        {
          "heading": "3. Lusin spaces and one-to-one images",
          "line": 198,
          "through_line": 304,
          "anchors": [
            "OA-FND-PB-04",
            "OA-FND-PB-05"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        },
        {
          "heading": "4. Borel maps between Souslin spaces",
          "line": 305,
          "through_line": 353,
          "anchors": [
            "OA-FND-PB-06"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        },
        {
          "heading": "5. Standard Borel spaces",
          "line": 354,
          "through_line": 427,
          "anchors": [
            "OA-FND-PB-07",
            "OA-FND-PB-08"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        },
        {
          "heading": "8. Topological quotients and open homomorphisms",
          "line": 685,
          "through_line": 841,
          "anchors": [
            "OA-FND-PB-13",
            "OA-FND-PB-14",
            "OA-FND-PB-15",
            "OA-FND-PB-16"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        }
      ],
      "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
      "source_locus": "§§1–6, 12A–12C and 13A–13C; PDF 4–21, 46–49, 51–54",
      "role": "proof comparison",
      "correspondence": "Polish and standard Borel proofs; exercises identified",
      "explanation": "Complete metric/product, closed-tree coding, analytic separation and Borel isomorphism compared; some elementary statements are exercises. Own non-Hausdorff boundary examples and complete group-image proof retained."
    },
    {
      "unit": "polish-spaces-and-standard-borel-spaces",
      "local_scopes": [
        {
          "heading": "5. Standard Borel spaces",
          "line": 354,
          "through_line": 427,
          "anchors": [
            "OA-FND-PB-07",
            "OA-FND-PB-08"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        },
        {
          "heading": "7. Choosing points in a Borel way",
          "line": 537,
          "through_line": 680,
          "anchors": [
            "OA-FND-PB-10",
            "OA-FND-PB-11"
          ],
          "source_sha256": "81EBB53C47695558CD8D1F6A1DA54BBE421384EF7B15F2AC94DC02322EDCB286"
        }
      ],
      "source_key": "human:fremlin42@45E89089200CD6FDEE5E04F43CEF94CB44E6832386CE4B50E7322CE8D91E15F3",
      "source_locus": "423J, 423N–423P and 424B–424G; PDF 23–26, 30–31",
      "role": "proof comparison",
      "correspondence": "Exact countable-generator and prefix-selector scope",
      "explanation": "Countable separating algebra equals Borel algebra; least-prefix selector has sigma/Souslin, not necessarily Borel, measurability. External Polish-group action theorem 424H is not counted as fully read."
    },
    {
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "local_scopes": [
        {
          "heading": "6. Trees on a countable set",
          "line": 572,
          "through_line": 652,
          "anchors": [
            "OA-FND-DT-06"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        },
        {
          "heading": "7. Countable orders and well-orders",
          "line": 653,
          "through_line": 693,
          "anchors": [
            "OA-FND-DT-07"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        },
        {
          "heading": "8. Every co-Souslin set comes from the well-orders",
          "line": 694,
          "through_line": 731,
          "anchors": [
            "OA-FND-DT-08"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        },
        {
          "heading": "9. Reduction and the second separation theorem",
          "line": 732,
          "through_line": 757,
          "anchors": [
            "OA-FND-DT-09"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        }
      ],
      "source_key": "human:marker@B67CCDA708A8E985C37E1B6EFCB0DBCE1927BD766F0F1F714C515C40534507F4",
      "source_locus": "§5, Theorems 5.16, 5.20 and 5.22; PDF 43–48",
      "role": "proof comparison",
      "correspondence": "Tree ranks and two-set reduction; own countable theorem stronger",
      "explanation": "Borel rank levels, analytic boundedness and comparison ties are compared. The lesson’s countable rank-minimum construction and second separation proof remain complete."
    },
    {
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "local_scopes": [
        {
          "heading": "5. Souslin and co-Souslin sets",
          "line": 555,
          "through_line": 571,
          "anchors": [
            "OA-FND-DT-05"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        }
      ],
      "source_key": "human:tserunyan@BC9F7F621D679199AE0B7CA0300EC9F9BFE490D632DCA530DA9392395E0FF0DA",
      "source_locus": "§12A–§12C; PDF 46–49",
      "role": "proof comparison",
      "correspondence": "Analytic/coanalytic notation and first separation",
      "explanation": "Closed projections, countable closure and analytic separation are fully compared; this is not a proof of the second separation theorem."
    },
    {
      "unit": "numerical-ranges-positive-matrices-and-co-souslin-sets",
      "local_scopes": [
        {
          "heading": "1. Unital functionals and the numerical range",
          "line": 66,
          "through_line": 126,
          "anchors": [
            "OA-FND-DT-01"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        },
        {
          "heading": "2. The numerical range from the norm",
          "line": 127,
          "through_line": 257,
          "anchors": [
            "OA-FND-DT-02"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        },
        {
          "heading": "3. Hermitian elements",
          "line": 258,
          "through_line": 416,
          "anchors": [
            "OA-FND-DT-03"
          ],
          "source_sha256": "78D5A58911EF11F927EB3A66DF16B5F98A308723E57DF363D9B15903C5997682"
        }
      ],
      "source_key": "human:numericalrange2025@6A3F3BEF49CEE91C525D8C699FD0D587852E840D08E22E240BEA5EFC52EE5A13",
      "source_locus": "Complete §2, especially Lemma 2.1",
      "role": "proof comparison",
      "correspondence": "General Banach-algebra scope; classical inputs largely referenced",
      "explanation": "Actual full Lemma 2.1 proof supplies the new perturbation complement. Algebraic numerical range is not silently replaced by Hilbert-operator numerical range; all classical norm/exponential arguments remain proved internally.",
      "local_labels": [
        "Proposition 1.4a"
      ]
    },
    {
      "unit": "af-algebras",
      "local_scopes": [
        {
          "heading": "6. The scaled dimension group",
          "line": 346,
          "through_line": 430,
          "anchors": [
            "OA-FND-AF-05"
          ],
          "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
        },
        {
          "heading": "7. Projections and the dimension group as an invariant",
          "line": 431,
          "through_line": 524,
          "anchors": [
            "OA-FND-AF-06"
          ],
          "source_sha256": "3F2B4B0C933C93B9EC11024F0F7A78D55089F6815CACE9F061B19161103EAEE1"
        },
        {
          "heading": "8. Elliott's classification theorem",
          "line": 525,
          "through_line": 587,
          "anchors": [
            "OA-FND-AF-07"
          ],
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