Proof companion: normality and the tracial representation

Original OA-MOD programme proof excerpts and classical proof integration by GPT-6 Astra (OpenAI), Ultra, October 2026. Independently written AI programme text is CC0 1.0 under the programme's current dedication. Human source works retain their own rights and licences.

This companion supplies the order-normality and projection arguments used by Bounded ultrastrong topology and the semifinite tracial representation, Sections 5–6. It retains arbitrary Hilbert dimensions, arbitrary increasing nets and arbitrary weights in the theorems that require them. A faithful state is used only in supported corners during the normal-weight proof, with an explicit return to the original algebra. It is an accompanying proof document, not an additional numbered lesson.

The earlier bounded-topology companion supplies the already constructed preduals, Hilbert geometry, continuous bounded calculus, compactness and bounded monotone convergence. Its formatted proofs remain unchanged. Complete additional proofs follow here, ending with TP1–TP4 at the exact tracial interface. The entry assumptions are complete real and complex scalars, inner-product and normed-space axioms, elementary topology and the maximal principle.

Source and change notices. The CV exposition is credited in its original source to OpenAI Codex, GPT-6 Astra / Ultra. CV3 develops the ordinary real/complex specialization of Javier Falcó and Daniel Isert, G-strong subdifferentiability and applications to norm attaining subspaces, Section 3.4, Lemmas 32–33, Proposition 34 and Theorem 35, printed pages 254–258. Their article is available under CC BY 4.0. The programme's recorded changes are the finite-block enumeration, a full c0 coefficient argument, real parts for complex separation, a fixed separating vector, verified affine slices and the full completeness step. Its historical notice explicitly describes independently written expression; the human article's licence is not a blanket licence on this AI exposition. Jacob Lurie's Lecture 12, Theorem 6, page 3 supplied a separate finite-cover comparison. Credit, source links and these changes are retained; no source-author endorsement is implied.

The normal-weight construction records both its genuine historical correspondence with Masamichi Takesaki, Theory of Operator Algebras II, VII.1, Theorem 1.11 and preparatory lemmas and its later comparison with Uffe Haagerup, Normal weights on W-algebras* (1975), Lemmas 1.4–1.7/Theorem 1.8 and Proposition 2.1/Theorem 2.2. The programme proves the bounded GNS graph step by summable energy increments and uses direct downward-set closure for positive separation. These are existing OA-MOD proofs, not a claim of new mathematical results. The older NW records do not identify an exact authoring model; this integration's model is not assigned retrospectively to them.

The passages below are programme exposition, with source locations and the original source/change records in the accompanying manifest. Earlier agent-selected GFDL metadata does not override the current CC0 dedication for independently written AI text. No new blanket clearance of human-source expression is inferred from this selection. General Borel/unbounded spectral theory is not treated here.

Concrete topology and corner preduals

Programme source: OA-MOD, public/src/concrete-preduals.md, original lines 250–374.

OA-MOD-CP-08 — The sigma-strong seminorms are vector seminorms

For \(\xi\in\ell^2(H)\), set

This is a seminorm and \(q_\xi(x)^2=\omega_\xi(x^*x)\). Thus it is one of the seminorms

which are seminorms by (CP.13).

Conversely, choose a mixed series (CP.11) for \(\omega\in M_*^+\). For \(a\ge0\), Cauchy–Schwarz for \(a^{1/2}\) and \(2uv\le u^2+v^2\) give

All sums converge absolutely and \(\theta\in M_*^+\). If \(\zeta\) concatenates \(\xi/\sqrt2\) and \(\eta/\sqrt2\), then \(p_\omega(x)\le q_\zeta(x)\). Therefore the sigma-strong topology generated by the \(p_\omega\) is exactly the topology generated by the \(q_\xi\). Adding the seminorms at \(x^*\) gives the sigma-strong* topology.

This proof does not assume that a positive predual functional itself extends as a positive vector series on \(B(H)\). Domination by \(\theta\) suffices.

On a norm-bounded set, strong operator convergence is equivalent to sigma-strong convergence. If \(x_\alpha\to x\) strongly and \(\|x_\alpha\|,\|x\|\le R\), the tail of \(\sum_n\|(x_\alpha-x)\xi_n\|^2\) is at most \(4R^2\sum_{n>N}\|\xi_n\|^2\); the finite initial sum tends to zero. This proves the forward implication, while single-vector tests prove the reverse. Applying the argument also to adjoints gives the analogous strong* and sigma-strong* statement.

OA-MOD-CP-09 — The two continuous complex duals

Theorem. The continuous complex-linear dual of each of the sigma-strong and sigma-strong* topologies is exactly \(M_*\).

Proof for sigma-strong. The estimate

makes every member of \(M_*\) continuous.

Conversely, continuity at zero and homogeneity bound a continuous linear \(f\) by a constant times the maximum of finitely many defining seminorms. Concatenating their sequences gives \(|f(x)|\le Cq_\xi(x)\) for some \(\xi\in\ell^2(H)\). When the controlling seminorm vanishes, scaling forces \(f(x)=0\). Hence

is well-defined and bounded on its linear image in \(\ell^2(H)\). Extend it continuously to the closure of that image. The Riesz representation proof in BK-01, applied to that closed Hilbert subspace, supplies \(\eta\in\ell^2(H)\) with \(f(x)=\sum_n\langle x\xi_n,\eta_n\rangle\). Thus \(f\in M_*\).

Proof for sigma-strong.* The forward inclusion remains true because this topology is finer. For the reverse, continuity and concatenation give

Use the complex-linear map

The conjugate space is essential: \(x\mapsto x^*\eta_n\) alone is conjugate-linear. Factor \(f\) through the image of \(J\), extend to its closure, and apply the Riesz proof in BK-01. There exist \(u,v\in\ell^2(H)\) with

Concatenation is a vector-series representation, so \(f\in M_*\). ∎

Equal continuous duals do not mean that the topologies themselves are equal.

OA-MOD-CP-10 — Real duals and the convex-closure consequence

On the underlying real space of \(M\), a continuous real-linear \(r\) has continuous complexification

Thus the common continuous real dual for these three topologies is \(\{\operatorname{Re}f:f\in M_*\}\).

On \(M_{\rm sa}\), sigma-strong and sigma-strong* coincide. Suppose \(r\) is continuous real linear there. After finite concatenation, \(|r(a)|\le Cq_\xi(a)\) for self-adjoint \(a\). Define

It is complex linear and agrees with \(r\) on self-adjoint elements. The bound

makes it sigma-strong* continuous, hence a member of \(M_*\) by CP-09. It is hermitian. Conversely every hermitian predual functional restricts to a continuous real functional on \(M_{\rm sa}\) for all three topologies. Ultraweak continuity already implies sigma-strong continuity, so these exhaust the ultraweak real dual as well.

The argument does not presume that \(x\mapsto(x+x^*)/2\) is sigma-strong continuous on the full algebra.

Convex-closure corollary. By the full real locally convex separation and half-space proof in NP2, a convex subset of \(M\) has the same closure for the ultraweak, sigma-strong and sigma-strong* topologies. The same holds for convex subsets of \(M_{\rm sa}\). The empty convex set is closed for all three topologies. For a nonempty convex set, each closure is the intersection of closed half-spaces defined by its continuous real functionals; the dual identifications make those half-spaces identical. NP2 proves precisely this separation conclusion at arbitrary locally convex generality. There is no boundedness or sequential-closure restriction. Banach–Alaoglu and Krein–Smulian are different parts of that broader contract and are not needed for this conclusion.

OA-MOD-CP-11 — Corners have exactly their inherited predual

Let \(p\in M\) be a projection and let \(N=pMp\) act on \(pH\). It is a concrete von Neumann algebra there. To verify closedness, extend a weak operator convergent net on \(pH\) by zero on \((1-p)H\). The extensions converge weakly on \(H\), so a net from \(pMp\) has its limit in \(M\) and still satisfies \(x=pxp\).

Let \(j:N\to M\) be this inclusion, and let \(C:M\to N\) be \(C(x)=pxp\). Both are contractions and \(Cj=\operatorname{id}_N\). A vector-series functional on \(N\) extends through \(C\) using the same sequences in \(H\). Restricting a series from \(M\) through \(j\) replaces both vector sequences by their images under \(p\). Hence the ultraweak topology on \(N\) is exactly the inherited topology.

The maps

are contractions, \(rs=1\), and \(s\) is isometric: restriction gives the reverse norm inequality. They preserve positivity. Every \(g\in N_*\) has an extension of exactly its norm, and therefore

isometrically, where \(P_*=sr\) is a contractive idempotent. The second identification uses \(s\); a quotient is not identified with a subspace without a specified map.

At tensor level, \(p\otimes\overline p\) is contractive by (CP.2), extends to the completions, and gives the compression formulas. The construction includes \(p=0\), requires no centrality, and uses no characterization of normal maps.

Inverse order, supports and polar decomposition

Programme source: OA-MOD, public/src/bounded-operator-kernel.md, original lines 161–232.

OA-MOD-BK-05 — Inversion reverses order

If \(a,b\in M_+\) and \(mI\leq a\leq b\), \(m>0\), the calculus in BK-01 gives their inverses and inverse square roots in \(M\). We prove

Let \(c=a^{-1/2}ba^{-1/2}\). Testing quadratic forms gives \(c\geq I\), so its one-operator calculus gives \(0\leq c^{-1}\leq I\). Direct multiplication verifies \(b^{-1}=a^{-1/2}c^{-1}a^{-1/2}\). Congruence of the last order inequality by \(a^{-1/2}\) proves the claim. The proof applies to noncommuting \(a,b\).

For \(0\leq a\leq b\) and \(\lambda>0\), add \(\lambda I\), apply (BK.6), and subtract the scaled inverses from \(I\):

Injectivity alone supplies no positive lower bound; BK-09 gives an injective positive operator with unbounded inverse.

OA-MOD-BK-06 — Supports from bounded resolvent cutoffs

For \(a\in M_+\), let \(p\) project onto \((\ker a)^\perp=\overline{aH}\). Then \(p\in M\), \(pa=ap=a\), and, as \(\varepsilon\downarrow0\),

Indeed the scalar functions \(t/(t+\varepsilon)\) make \(f_\varepsilon\) positive contractions, increasing as \(\varepsilon\) decreases. They vanish on \(\ker a\). On \(aH\),

This proves convergence to the identity on a dense subset of \(pH\). The contraction bound extends it to that whole space: approximate \(\zeta\in pH\) by \(a\xi\) and bound the error by \(2\|\zeta-a\xi\|+\varepsilon\|\xi\|\). On its orthogonal complement all cutoffs are zero, proving the strong limit. BK-02 gives membership \(p\in M\); BK-03 and BK-04 give the ultraweak limit and the supremum assertion.

Any projection \(q\) with \(qa=a\) has closed range containing \(aH\), hence contains \(pH\). Thus \(p\) is the smallest such projection, called \(s(a)\). The equality \(\ker a=\ker a^{1/2}\) proves \(s(a)=s(a^{1/2})\). Reparameterizing gives \(ta(I+ta)^{-1}\uparrow s(a)\).

A useful domination consequence needs no commutation assumption: if \(p\leq e\leq I\) and \(p\) is a projection, then \(ep=pe=p\). Indeed \(0\leq p(I-e)p\leq p(I-p)p=0\), so \(\|(I-e)^{1/2}p\xi\|^2=0\) for every \(\xi\). Hence \((I-e)p=0\), and its adjoint gives the other equality.

OA-MOD-BK-07 — Polar decomposition and its regularized limit

For \(x\in M\), put \(a=|x|=(x^*x)^{1/2}\). There is a unique partial isometry \(u\in M\) with

A partial isometry here is zero on its kernel and isometric on its orthogonal complement. Its initial and final projections are \(u^*u=s(|x|)\) and \(uu^*=s(|x^*|)\). Its bounded regularizations \(u_\varepsilon=x(|x|+\varepsilon I)^{-1}\) are contractions and converge strongly* to \(u\).

To construct it, \(\|x\xi\|=\|a\xi\|\) shows that \(a\xi\mapsto x\xi\) is a well-defined isometry on \(aH\). Extend it to an isometry \(\overline{aH}\to\overline{xH}\), and let it vanish on \(\ker a\). This defines \(u\) on all of \(H\), with \(ua=x\). Its initial and final projections are the projections onto these two closed ranges. The range-kernel identity and \(\ker|x^*|=\ker x^*\) identify the second as \(s(|x^*|)\).

Although \(u\) was constructed as a bounded map on \(H\), it belongs to the algebra. The calculus places \(u_\varepsilon\) in \(M\), and \(u_\varepsilon=u f_\varepsilon\), where \(f_\varepsilon\) is the cutoff of \(a\). Since \(f_\varepsilon\to p=s(a)\) strongly and \(up=u\), these contractions converge strongly to \(u\); strong closedness gives \(u\in M\). For adjoints use the separate calculation \(u_\varepsilon^*=f_\varepsilon u^*\to pu^*=u^*\); taking adjoints of an arbitrary strong limit would not be valid. A second partial isometry satisfying (BK.10) agrees on \(aH\), hence its closure, and is zero on the same orthogonal complement. This proves uniqueness.

For \(C:H\to K\), the identical construction gives \(C=U|C|\) between two possibly different Hilbert spaces. Here \(U\) is zero on \(\ker C\) and unitary from \((\ker C)^\perp\) onto \(\overline{\operatorname{ran}C}\). The operators \(C(|C|+\varepsilon I_H)^{-1}\) and their adjoints converge strongly to \(U\) and \(U^*\) in the respective directions. There is also the left factorization

Indeed \(U|C|U^*\) is positive and has square \(U|C|^2U^*=CC^*\), since \(U^*U\) is the support of \(|C|\). Positive-square-root uniqueness from BK-01 gives \(|C^*|=U|C|U^*\), and multiplication by \(U\) proves (BK.11). If \(C\) is injective with dense range, the initial and final spaces are all of \(H,K\); thus \(U:H\to K\) is unitary.

Supports of positive normal functionals

NP3 — Supports of positive ultraweakly continuous functionals

Let \(M\subset B(H)\) be a weak operator closed unital *-algebra and let \(\omega:M\to\mathbb C\) be positive and linear. The bounded calculus in BK01 writes a self-adjoint element as a difference of positive elements, so \(\omega\) is real on self-adjoint elements and preserves adjoints. Positivity of \(\omega((x+\lambda y)^*(x+\lambda y))\), followed by minimizing in the complex scalar \(\lambda\), gives

If \(\omega(y^*y)=0\), varying the magnitude and argument of \(\lambda\) instead forces the mixed coefficient to be zero, proving the same inequality. Taking \(y=1\) and using \(x^*x\le\|x\|^2 1\) gives \(|\omega(x)|\le\omega(1)\|x\|\); testing the unit proves \(\|\omega\|=\omega(1)\). For the zero algebra all assertions are immediate.

Assume now that \(\omega\) is ultraweakly continuous. This is the hypothesis called normal here; no order-normal-to-ultraweak theorem is used. BK04 shows that a bounded increasing positive net converges strongly and ultraweakly to its supremum. Applying \(\omega\) proves preservation of such suprema.

For a positive \(a\in M\), BK06 constructs the range-support projection \(s(a)\) and proves

Consequently \(\omega(a)=0\) implies \(\omega(s(a))=0\). For two projections \(e,f\), the identity \(\langle(e+f)\xi,\xi\rangle=\|e\xi\|^2+\|f\xi\|^2\) gives \(\ker(e+f)=\ker e\cap\ker f\). Thus \(s(e+f)\) projects onto the closed span of their ranges: it is their least upper projection, denoted \(e\vee f\). If both projections are \(\omega\)-null, (NP.7) proves that their join is null as well. No commutation assumption is needed.

Let \(\mathcal N\) be the set of all \(\omega\)-null projections. For each finite subset \(F\subset\mathcal N\), let \(q_F\) be its join, with \(q_\varnothing=0\). The preceding argument proves that every \(q_F\) is null. These projections form an increasing net bounded by the identity. By BK04 its supremum \(q\in M\) is its strong and ultraweak limit. It is a projection: for every \(\xi\),

because \(\|q_F\|\le1\). Since \(q_F^2=q_F\to q\) strongly, we obtain \(q^2=q\); self-adjointness follows from the positive limit. Ultraweak continuity gives \(\omega(q)=0\). It contains every member of \(\mathcal N\), so it is the largest null projection.

Set \(p=1-q\). Using (NP.6) with the pair \(q,x^*\) gives \(\omega(xq)=0\); adjoint preservation gives \(\omega(qx)=0\). Expanding with \(1=p+q\) proves

If a projection \(e\) has this compression property, then \(\omega(1-e)=0\). Maximality of \(q\) implies \(1-e\le q\), hence \(p\le e\). Thus \(p\) is the least compression projection.

The restriction to \(pMp\) is faithful. If \(a\in pMp\) is positive and \(\omega(a)=0\), (NP.7) makes \(s(a)\) a null projection, so \(s(a)\le q\). Since the range of \(a\) lies in \(pH\), also \(s(a)\le p\). Hence \(s(a)=0\), and \(a=0\). The zero functional has \(q=1\) and \(p=0\). Otherwise \(\omega(p)=\omega(1)>0\), so division by this value gives a state on the support corner. None of these projections is required to be central.

The algebraic GNS construction for every weight

Programme source: OA-MOD, public/src/weight-gns.md, original lines 23–210.

OA-MOD-WG-002. The finite part of an extended-valued weight

A weight is a map \(\varphi:M_+\to[0,\infty]\) satisfying

We use \(0\cdot\infty=0\). It follows immediately that \(0\leq a\leq b\) implies \(\varphi(a)\leq\varphi(b)\), since \(b=a+(b-a)\).

Define three subsets with different purposes:

The first is a positive cone, the second will be the domain of the GNS map, and the third will be the domain of a finite-valued linear extension. In particular, the original weight is not a complex-linear function on all of \(M\).

We call \(\varphi\) faithful if \(a\in M_+\) and \(\varphi(a)=0\) imply \(a=0\). We call it normal if

We call it semifinite if \(\mathfrak m_\varphi\) is ultraweakly dense in \(M\). This definition is essential: for general weights it is not replaced by a requirement that every nonzero positive element dominate a nonzero element of finite weight. Item OA-MOD-WG-013 gives a counterexample to that replacement.

OA-MOD-WG-003. Domain algebra and its positive cone

Proposition. For every weight, \(F_\varphi\) is an additive hereditary cone, \(\mathfrak n_\varphi\) is a left ideal, and \(\mathfrak m_\varphi\) is a possibly nonunital *-subalgebra. More precisely,

Every element of \(\mathfrak m_\varphi\) has the form \((a-b)+i(c-d)\), where \(a,b,c,d\in F_\varphi\).

Proof. Additivity and homogeneity preserve finite values, and monotonicity makes \(F_\varphi\) hereditary. For \(x,y\in\mathfrak n_\varphi\),

this follows by expanding \((x-y)^*(x-y)\geq0\). Consequently \(x+y\in\mathfrak n_\varphi\). Scalar multiplication is immediate. For \(a\in M\),

so \(ax\in\mathfrak n_\varphi\). This proves the left ideal assertion.

Taking adjoints preserves the span defining \(\mathfrak m_\varphi\). Products of its spanning elements remain in that span because

and \(xv^*u\in\mathfrak n_\varphi\). Each \(y^*x\) itself belongs to \(\mathfrak n_\varphi\), by the left ideal property, and so does its adjoint. This proves the subalgebra and inclusion statements.

The polarization identity

puts every spanning element in \(\operatorname{span}_{\mathbb C}F_\varphi\). Conversely, for \(a\in F_\varphi\), the square root \(a^{1/2}\) lies in \(\mathfrak n_\varphi\), and \(a=(a^{1/2})^*a^{1/2}\). The two spans therefore coincide.

Grouping the positive and negative real and imaginary coefficients of a finite linear combination of members of \(F_\varphi\) gives the four-term decomposition. If an element \(h\) of that span is self-adjoint, averaging a decomposition with its adjoint removes the imaginary part and expresses \(h=a-b\), with \(a,b\in F_\varphi\). If also \(h\geq0\), then \(0\leq h\leq a\), so heredity gives \(h\in F_\varphi\). The reverse inclusion was already established. ∎

Here “hereditary” for the subalgebra refers to its positive cone: if \(0\leq b\leq a\in(\mathfrak m_\varphi)_+\), then \(b\in\mathfrak m_\varphi\). No norm-closedness is asserted.

Corollary for arbitrary hereditary cones. Let \(P\subseteq M_+\) be a nonempty cone closed under addition and multiplication by nonnegative scalars, and suppose \(0\leq a\leq b\in P\) implies \(a\in P\). Set

Then \(\mathfrak n_P\) is a left ideal, \(\mathfrak m_P\) is a hereditary *-subalgebra,

and every element of \(\mathfrak m_P\) is a complex linear combination of four members of \(P\).

Proof. Define \(\varphi_P(a)=0\) for \(a\in P\) and \(\varphi_P(a)=+\infty\) otherwise. Nonemptiness and the cone property give \(0\in P\). For positive \(a,b\), heredity gives

It follows that \(\varphi_P(a+b)=\varphi_P(a)+\varphi_P(b)\), with the extended-value conventions of WG-002. For \(t>0\), \(ta\in P\) is equivalent to \(a\in P\), by applying the cone property also to \(t^{-1}\). This proves homogeneity for \(t>0\); for \(t=0\) it follows from \(0\cdot\infty=0\). Thus \(\varphi_P\) is a weight with \(F_{\varphi_P}=P\). The proposition applied to this weight gives all the stated conclusions. Neither normality nor semifiniteness of this auxiliary weight is claimed. ∎

OA-MOD-WG-004. Linear extension without infinite subtraction

Proposition. There is a unique positive complex-linear functional

whose restriction to \(F_\varphi\) equals \(\varphi\). It satisfies \(\widetilde\varphi(z^*)=\overline{\widetilde\varphi(z)}\).

Proof. For a self-adjoint element \(h=a-b\), with \(a,b\in F_\varphi\), set \(\widetilde\varphi(h)=\varphi(a)-\varphi(b)\). If also \(h=c-d\), then \(a+d=c+b\), and additivity gives

Every number here is finite. Subtracting proves independence of the decomposition. Addition of decompositions proves additivity on the self-adjoint part; positive scalar multiplication follows from homogeneity, and negative scalar multiplication follows by swapping the two terms. Thus this is a real-linear functional.

Every \(z\in\mathfrak m_\varphi\) has the unique self-adjoint decomposition

Define \(\widetilde\varphi(z)=\widetilde\varphi(h)+i\widetilde\varphi(k)\). Real linearity and the transformation \((h,k)\mapsto(-k,h)\) under multiplication by \(i\) prove complex linearity. Positivity follows from OA-MOD-WG-003, because the positive elements of the domain are exactly \(F_\varphi\). The formula for adjoints and uniqueness follow from the same decomposition. ∎

We retain the tilde when a complex argument is present, making the domain visible. A formula such as \(\widetilde\varphi(y^*a x)\) is legitimate for \(x,y\in\mathfrak n_\varphi\), \(a\in M\), because \(ax\in\mathfrak n_\varphi\). It is not legitimate for arbitrary \(x,y\in M\) merely because the product exists.

OA-MOD-WG-005. Cauchy-Schwarz and the null ideal

Lemma. On \(\mathfrak n_\varphi\), the formula

defines a positive semidefinite sesquilinear form, linear in \(x\), and

Its null space is the left ideal

Proof. Sesquilinearity and conjugate symmetry follow from OA-MOD-WG-004; positivity follows from the definition. Put \(A=B_\varphi(x,x)\), \(C=B_\varphi(y,y)\), and \(b=B_\varphi(x,y)\). For every \(t\in\mathbb C\),

If \(C>0\), choose \(t=-b/C\) to obtain \(|b|^2\leq AC\). If \(C=0\) and \(b\neq0\), choosing \(t=-r b\), with \(r>0\), would make the right side \(A-2r|b|^2\), which is eventually negative. Thus \(b=0\) when \(C=0\), proving the inequality in every case.

The inequality shows that a null vector is orthogonal to every element of \(\mathfrak n_\varphi\). Hence sums and scalar multiples of null vectors are null. The estimate from OA-MOD-WG-003 gives

so \(N_\varphi\) is a left ideal. ∎

OA-MOD-WG-006. The GNS construction for an arbitrary weight

Let \(H_\varphi\) be the Hilbert completion of \(\mathfrak n_\varphi/N_\varphi\). Write

The preceding lemma proves that the quotient inner product

is well-defined and positive definite. By construction, the range of \(\Lambda_\varphi\) is dense. For \(a\in M\), define initially

This is well-defined because both domain ideals are left ideals, and

It therefore extends uniquely to a bounded operator on \(H_\varphi\), with norm at most \(\|a\|\).

Theorem. This construction gives a unital *-representation \(\pi_\varphi:M\to B(H_\varphi)\), for every weight. Neither normality, faithfulness, nor semifiniteness is needed for this assertion. The triple \((H_\varphi,\pi_\varphi,\Lambda_\varphi)\) is the GNS semicyclic triple.

Proof. Linearity and multiplication hold on the dense range of \(\Lambda_\varphi\), because the left action of \(M\) has these properties. Boundedness extends these identities to all of \(H_\varphi\). For \(x,y\in\mathfrak n_\varphi\),

Density yields \(\pi_\varphi(a)^*=\pi_\varphi(a^*)\). Finally, \(\pi_\varphi(1)\Lambda_\varphi(x)=\Lambda_\varphi(x)\), so \(\pi_\varphi(1)=I\). In particular the representation is nondegenerate: \(\overline{\pi_\varphi(M)H_\varphi}=H_\varphi\). When \(H_\varphi=\{0\}\), its identity operator is the zero operator and these statements retain their usual meaning. ∎

Uniqueness. Suppose \((H',\pi',\Lambda')\) has a linear map \(\Lambda':\mathfrak n_\varphi\to H'\) with dense range, the same inner-product formula, and \(\pi'(a)\Lambda'(x)=\Lambda'(ax)\), with \(\pi'\) a bounded-operator representation. Then \(U\Lambda_\varphi(x)=\Lambda'(x)\) defines an isometry on a dense subspace: its well-definedness and norm preservation follow from the shared quadratic form. Its range is dense, so it extends to a unitary. The action identity gives \(U\pi_\varphi(a)=\pi'(a)U\) first on GNS vectors and then everywhere. Density also proves uniqueness of \(U\).

Here “semicyclic” means this representation together with its dense-range module map. Some treatments impose additional closedness conditions when defining an abstract semicyclic object. No such closedness is included silently in the present terminology.

Convex closure and Krein–Šmulian

Compactness and convex closure for normal-weight proofs

Historical provider status notice: Classical proof exposition assembled and corrected by OpenAI Codex, GPT-6 Astra / Ultra.

This is the original provider's recorded status.

Bounded slices are often easier to control than an entire convex set. This note proves the passage from those slices to weak-star closedness, together with the real/complex and product-topology facts needed by the normal-weight construction. No separability, boundedness of the whole convex set, balancedness or sequential weak-star test is assumed.

The full Hahn–Banach proof and point-separation argument are written in NP1–NP2. AB04–AB05 prove compact products and Banach–Alaoglu, including arbitrary normed preduals. The real and complex Hilbert representation arguments are in the real Hilbert proof and BK01. The norm topology, complete scalar fields and maximal principle are the common entry assumptions.

The Krein–Šmulian argument preserves the course's previously written finite-cover proof and its corrections. The human mathematical sources are Javier Falcó and Daniel Isert, G-strong subdifferentiability and applications to norm attaining subspaces, §3.4, an open-access article under CC BY 4.0, and Jacob Lurie, Math 261y, Lecture 12, Theorem 6, page 3. The latter gives a second finite-cover construction. The full coefficient and affine-slice arguments needed here are written below. No citation substitutes for a proof.

CV1 — Extension with a seminorm and convex separation

NP1 proves the real dominated-extension theorem for every finite sublinear function. In particular, let \(p\) be a seminorm and let \(h\) be linear on a subspace with \(|h|\leq p\). In the real case, extension dominated by \(p\), applied also to \(-x\), has the same absolute bound. In the complex case, extend \(\operatorname{Re}h\) as a real functional \(f\leq p\), then set

As in NP1, evaluation at \(ix\) proves complex linearity and agreement with \(h\). Choose \(|\lambda|=1\) with \(\lambda H(x)=|H(x)|\). Complex absolute homogeneity of the seminorm gives

Thus continuity of \(p\) implies continuity of \(H\).

If a functional is continuous on a subspace for the topology induced by a locally convex space, some finite maximum \(p_0\) of continuous seminorms has \(|h(x)|<1\) whenever \(p_0(x)<\varepsilon\) on that subspace, for an \(\varepsilon>0\). When \(p_0(x)>0\), apply the bound to \(tx\) for \(0<t<\varepsilon/p_0(x)\), then let \(t\) increase to that endpoint. This gives \(|h(x)|\leq p_0(x)/\varepsilon\). When \(p_0(x)=0\), arbitrary positive \(t\) gives \(h(x)=0\). The preceding extension with the continuous seminorm \(p_0/\varepsilon\) therefore proves continuous extension. No closedness or completeness of the subspace is required.

NP2 proves strict separation of a point from a nonempty closed convex set, by a continuous real functional with a positive margin. It also proves that equal continuous real duals give equal closures of convex sets. We will apply precisely this point-separation statement to the norm closure of a convex image in \(c_0\). This avoids requiring an additional open-set separation theorem.

CV2 — Weak compactness of Hilbert balls

Claim. Every closed norm ball of a real or complex Hilbert space \(H\) is compact for its weak topology. No dimension or separability restriction is imposed.

Proof. Use the course convention that the inner product is linear in its first variable. Riesz representation gives the surjective isometry

Over the complex scalars \(J\) is conjugate linear. For every fixed \(x\), the function \(y\mapsto (Jy)(x)=\overline{\langle y,x\rangle}\) is weakly continuous. Since weak* topology is the initial topology of all evaluations, \(J\) is continuous from weak \(H\) to weak* \(H^*\). Conversely, every bounded linear functional on \(H\) is \(y\mapsto\langle y,x\rangle\) for some \(x\); its composite with \(J^{-1}\) is \(f\mapsto\overline{f(x)}\), which is weak* continuous. Thus \(J^{-1}\) is continuous into weak \(H\). This proves that \(J\) is a homeomorphism for these two topologies. Its isometry property carries a closed ball onto the corresponding closed ball in \(H^*\), which is weak* compact by AB05. Pull back this compact set by \(J^{-1}\). The real case is the same with conjugation removed. \(\square\)

CV3 — Krein–Šmulian at arbitrary Banach-space generality

Theorem. Let \(X\) be any real or complex Banach space, and let \(C\subseteq X^*\) be convex. Then \(C\) is weak* closed if and only if \(C\cap r\overline B_{X^*}\) is weak* closed for every \(r>0\). Equivalently one may test all closed norm balls, or just the zero-centered balls of positive integer radius. No boundedness, balancedness, separability or prior closedness assumption on \(C\) is imposed.

Proof of the separation mechanism. Suppose \(D\subseteq X^*\) is nonempty and convex, all its bounded ball slices are weak* closed, and \(D\cap\overline B_{X^*}=\varnothing\). We will construct \(x\in X\) such that

For a set \(A\subseteq X\), write \(P(A)=\{f:|f(a)|\leq1\text{ for all }a\in A\}\). Set \(F_0=\{0\}\). We construct finite sets \(F_n\subseteq n^{-1}\overline B_X\), each containing zero, such that

The case \(n=1\) is the assumed disjointness. Suppose \(KS1\) holds at \(n\), and put

This is weak* compact: the first ball slice is closed inside a weak* compact ball, and every polar constraint is weak* closed. Each \(f\in Q_n\) has \(\|f\|>n\), by \(KS1\). By the definition of operator norm there is \(a\in n^{-1}\overline B_X\) with \(|f(a)|>1\). Thus the weak* open sets \(\{f:|f(a)|>1\}\), indexed by such \(a\), cover \(Q_n\). Choose finitely many covering vectors, and add zero to obtain \(F_n\). If \(Q_n\) is empty, take \(F_n=\{0\}\). Intersecting \(Q_n\) with \(P(F_n)\) gives the empty set, which is \(KS1\) at \(n+1\).

List the finite blocks \(F_0,F_1,F_2,\ldots\) in order, keeping repetitions, to get a sequence \((u_j)_{j\geq1}\). Each block is nonempty; hence the sequence is infinite. It tends to zero in norm, because only finitely many vectors occur before any fixed block and every vector in \(F_n\) has norm at most \(1/n\). For any \(f\in D\), choose an integer \(n\geq\|f\|\). Formula \(KS1\) shows that some vector in the first \(n\) blocks satisfies \(|f(u_j)|>1\).

Define the bounded linear map

Indeed, \(|f(u_j)|\leq\|f\|\|u_j\|\to0\), and \(\|Tf\|_\infty\leq\|f\|\sup_j\|u_j\|\). Every element of \(T(D)\) has norm strictly greater than one. Continuity of the norm implies that its norm closure \(C_0\) is contained in \(\{z:\|z\|_\infty\geq1\}\); in particular \(0\notin C_0\). The closure is convex by NP2. Apply that note's point-separation theorem to \(0\) and \(C_0\), in the real normed space underlying \(c_0(\mathbb F)\). Reverse the sign and divide by the resulting positive separation margin. This gives a continuous real linear functional \(L\) with \(L(Tf)\geq1\) for every \(f\in D\). Its norm is finite and positive; it need not equal one.

Here is the coefficient representation, including its needed norm bound. In the real case put \(a_j=L(e_j)\); in the complex case put \(a_j=L(e_j)-iL(ie_j)\), where \(e_j\) is the \(j\)-th coordinate vector. For every finitely supported \(z\), real linearity gives

For every finite index set, choose the coordinates \(z_j\) of modulus one so that \(a_jz_j=|a_j|\), taking any modulus-one value if \(a_j=0\), and put the remaining coordinates equal to zero. Since \(\|z\|_\infty\leq1\), this yields \(\sum_{j\in F}|a_j|\leq\|L\|\). Thus \((a_j)\in\ell^1\). Truncations of any \(z\in c_0\) converge in the sup norm, so continuity gives

The last equality follows because the displayed representation also gives \(\|L\|\leq\sum_j|a_j|\). In the real case omit real parts and use signs instead of phases. The norm of any tail of \(\sum_j a_ju_j\) is at most \((\sup_j\|u_j\|)\sum_{j\text{ in the tail}}|a_j|\). The scalar tail tends to zero. The partial sums are therefore Cauchy, and completeness of \(X\) gives \(x=\sum_j a_ju_j\in X\). Each \(f\in X^*\) is continuous, hence

Proof of the full theorem. Assume the bounded-slice condition on \(C\). Then \(C\) is norm closed. To see this, a norm-convergent sequence in \(C\) is bounded and converges weak*, so its limit belongs to one of the weak* closed ball slices. For a point in the norm closure, choose a point of \(C\) within \(1/n\) for each \(n\); the resulting sequence converges in norm. The preceding observation puts its limit in \(C\), proving norm closedness. This step does not make any assertion that weak* closure can be detected by sequences.

If \(C\) is empty there is nothing to prove. Given \(f_0\notin C\), norm closedness supplies \(\delta>0\) with \((f_0+\delta\overline B_{X^*})\cap C=\varnothing\). Put \(D=\delta^{-1}(C-f_0)\). It is nonempty, convex, and disjoint from the closed unit ball. Its slices have the required weak* closedness: for \(r>0\), choose \(R\geq\|f_0\|+\delta r\). Then

The set inside the parentheses is weak* compact, using the hypothesis and Banach–Alaoglu. Translation and multiplication by a nonzero scalar are weak* homeomorphisms, so the displayed slice is compact and therefore closed in the Hausdorff weak* topology. Apply \(KS2\) to \(D\). There is a fixed \(x\in X\) with \(\operatorname{Re}(f-f_0)(x)\geq\delta\) for every \(f\in C\). The set

is a weak* open neighborhood of \(f_0\) disjoint from \(C\). Every point outside \(C\) therefore lies outside its weak* closure. This proves weak* closedness. The converse follows by intersecting two weak* closed sets; the alternate ball formulations follow from CV4. \(\square\)

CV4 — Products, real parts and arbitrary ball centers

Claim. If \(\tau_j\) and \(\rho_j\) are locally convex topologies on \(V_j\), for \(j=1,2\), with the same continuous scalar dual on each factor, then the product topologies \(\tau_1\times\tau_2\) and \(\rho_1\times\rho_2\) have the same convex closures.

Proof. A linear functional on \(V_1\times V_2\) has the unique expression \(L(x,y)=L_1(x)+L_2(y)\), with \(L_1(x)=L(x,0)\) and \(L_2(y)=L(0,y)\). If \(L\) is continuous, each restriction is continuous because the coordinate inclusions are continuous. If both restrictions are continuous, their sum after the coordinate projections is continuous. Equality of the duals on each factor therefore gives equality of the product duals. Apply NP2. \(\square\)

If \(V\) is a complex topological vector space and \(u:V\to\mathbb R\) is continuous and real linear, put \(F(x)=u(x)-i u(ix)\). Then \(F(ix)=iF(x)\), and additivity and real homogeneity show that \(F\) is complex linear. It is continuous because \(u\) and multiplication by \(i\) are continuous, and \(\operatorname{Re}F=u\). Conversely the real part of a continuous complex linear functional is continuous and real linear. This proves the real-part identification needed when comparing complex scalar duals for real-convex separation. It does not identify the continuous dual of \(M_{\mathrm{sa}}\) with a particular predual; that remains the separate predual contract.

For a subset \(C\subseteq X^*\), suppose \(C\cap n\overline B_{X^*}\) is weak* closed for every positive integer \(n\). Every closed ball \(\overline B(f,r)\), with \(r\geq0\), lies in \(n\overline B_{X^*}\) for an integer \(n\geq\|f\|+r\). Hence

is weak* closed, because the ball is weak* closed by AB05. The converse is immediate by choosing the centered integer balls. Thus the two formulations of the bounded-slice hypothesis are equivalent. CV3 supplies the further passage from these slices to the whole convex set.

The full normal-weight characterization

Programme source: OA-MOD, public/src/normal-weight-characterizations.md, original lines 1–447.

Detecting normal weights by finite observations

Historical provider status notice: OA-MOD-NW.

This is the original provider's recorded status.

An infinite weight cannot be tested by applying a bounded-functional continuity theorem to it. This unit proves the equivalence between preservation of increasing suprema, arbitrary positive summation, lower semicontinuity, and recovery from dominated normal positive functionals. The algebra and the weight are arbitrary: faithfulness, semifiniteness, separability, and the existence of a faithful normal state on the whole algebra are not hypotheses.

The main reference is Uffe Haagerup's Normal weights on \(W^*\)-algebras, Journal of Functional Analysis 19 (1975), 302–317. Lemmas 1.4–1.7 and Theorem 1.8 supply the localization and compactness mechanisms; Proposition 2.1 and Theorem 2.2 supply positive separation. Here the full proofs use the programme's written bounded-operator, predual and convex-analysis foundations. The graph-closing step is proved directly by summable GNS energy increments, so the predual-valued map in the paper's Lemmas 1.1–1.3 is not a prerequisite for this route.

OA-MOD-NW-01 — Objects, topologies, and written foundations

Let \(M\) be a unital von Neumann algebra and \(\varphi:M_+\to[0,\infty]\) a weight, with the conventions of OA-MOD-WG-002. Put

The linear map \(\Lambda_\varphi:\mathfrak n_\varphi\to H_\varphi\), its null quotient, and its Hilbert completion are those constructed in OA-MOD-WG-003–006. Inner products are linear in the first variable. Write \(M_*^+\) for the positive ultraweakly continuous linear functionals on \(M\).

The ultraweak topology is \(\sigma(M,M_*)\). The sigma-strong topology is generated by

The sigma-strong* topology adds \(p_\omega(x^*)\). Throughout this unit, these are locally convex topologies on the entire algebra; when a continuity argument uses a norm bound, that bound is stated.

A family \((a_i)_{i\in I}\subseteq M_+\) is summable here if its finite partial sums have an upper bound in \(M_+\). OA-MOD-BK-04 then gives

The partial sums converge sigma-strongly and ultraweakly. Conversely, if the finite partial sums converge sigma-strongly to \(b\), then each partial sum is at most \(b\), by passing to the limit in its positive differences with all later partial sums. Thus this definition is equivalent to sigma-strong summability of a positive family. Indeed, if \(b_F\uparrow b\) and \(0\leq b-b_F\leq C1\), then \(p_\omega(b-b_F)^2\leq C\omega(b-b_F)\to0\). A scalar sum over an arbitrary set always means the supremum of its finite subsums.

The proofs use the following exact foundation contracts.

• OA-MOD-NW-DEP-DUAL: \(M=(M_*)^*\) isometrically; \(M_*^+\) separates positive elements and spans \(M_*\); the positive cone and norm balls are ultraweakly closed. Every normal positive functional has a support projection \(p\), satisfies \(\omega(x)=\omega(pxp)\), and is faithful on \(pMp\) after restriction. Corners have their inherited ultraweak topology and predual. Fixed multiplication and the adjoint operation are ultraweakly continuous.
• OA-MOD-NW-DEP-TOPO: the continuous linear duals of the sigma-strong and sigma-strong* topologies are \(M_*\). Thus a convex subset of \(M\) has the same closure for either of these topologies as for the ultraweak topology. The analogous real statement holds on \(M_{\mathrm{sa}}\).
• OA-MOD-NW-DEP-CONVEX: Hahn–Banach separation for locally convex spaces; weak* compactness of dual norm balls; weak compactness of Hilbert balls; and Krein–Smulian: a convex subset of a dual Banach space is weak* closed if and only if its intersection with every closed norm ball is weak* closed.

The exact written proofs for DUAL are CP06, CP07, CP11 and the bounded support proof in NP3. The topology proofs are CP08–CP10. Only the support part of CP12 is used; its later order-normal converse depends on the theorem proved here. The convex proofs are CV1–CV4, with the full Hahn–Banach and point-separation proofs in NP1–NP2. CV2 proves Hilbert-ball compactness, CV3 proves Krein–Šmulian at arbitrary real or complex Banach-space generality, and CV4 proves the product-topology and ball-test conversions.

For convex subsets of \(M_{\mathrm{sa}}\), no new predual theorem is needed to use Krein–Šmulian. The adjoint is ultraweakly continuous by CP07, so \(M_{\mathrm{sa}}\) is ultraweakly closed in \(M\). A bounded slice closed in its relative topology is therefore closed in \(M\). Apply CV3 in the complex dual \(M=(M_*)^*\), where convexity uses real coefficients. CP10 supplies the equality of convex closures on \(M_{\mathrm{sa}}\).

BK01 proves the bounded continuous calculus; BK03 proves the bounded strong-to-ultraweak limit statements; BK04 proves monotone-net convergence; BK05 proves inverse order; and BK06–BK07 prove support and polar decomposition. These inputs use no normality characterization for weights, unbounded spectral theorem, modular theorem or closed-form theorem.

For later use, bounded multiplication is sigma-strong* continuous. For example, if \(x_i\to x\), \(y_i\to y\) sigma-strong* and the two nets are norm bounded, then

The adjoints satisfy the analogous estimate. Uniform polynomial approximation therefore makes the square-root operation sigma-strongly continuous on every norm-bounded positive set. It also makes any fixed continuous real function on a common compact spectral interval continuous on bounded self-adjoint sets. The norm of each functional-calculus approximation error controls every \(p_\omega\). More explicitly, on a common spectral interval choose a polynomial \(p\) with \(\sup|f-p|<\varepsilon\). Each approximation error has \(p_\omega\)-seminorm at most \(\varepsilon\omega(1)^{1/2}\). The difference \(p(x_i)-p(x)\) tends to zero by repeated bounded multiplication. The triangle inequality, followed by \(\varepsilon\downarrow0\), proves the stated continuity for every \(\omega\).

OA-MOD-NW-02 — The implications requiring no localization

Consider these four assertions:

Proposition. \(\mathrm{P}\Rightarrow\mathrm{L}\Rightarrow\mathrm{N}\Rightarrow\mathrm{A}\).

Proof. Under P, a sublevel set is the intersection of \(M_+\) with the closed half-spaces \(\{\omega\leq c\}\), proving L. If \(a_\alpha\uparrow a\), the net converges ultraweakly by OA-MOD-BK-04. Write \(r=\sup_\alpha\varphi(a_\alpha)\). Monotonicity gives \(r\leq\varphi(a)\). If \(r=\infty\), equality already follows. Otherwise all \(a_\alpha\) lie in the closed sublevel set at \(r\), so \(a\) does too. This proves N. Finally, the finite partial sums of a summable family increase to its sum. N and finite additivity give A. \(\square\)

L is exactly lower semicontinuity of the extended-valued function on \(M_+\) with its relative ultraweak topology. Equivalently, \(\varphi(a)\leq\liminf_\alpha\varphi(a_\alpha)\) for every ultraweakly convergent net \(a_\alpha\to a\) in \(M_+\). The net formulation includes infinite values and does not require the original net to be norm bounded.

OA-MOD-NW-03 — Corners and the countability they actually provide

Call a projection \(p\) sigma-finite if \(pMp\) has a faithful normal state; include \(0\) in this class. Equivalently, a nonzero such \(p\) is the support of a normal state on \(M\): extend a state of \(pMp\) by \(x\mapsto\omega(pxp)\), or restrict a state with support \(p\).

Let \(\mathcal P\) be these projections and define

Projection facts. Subprojections and countable joins of members of \(\mathcal P\) again belong to \(\mathcal P\). Equivalent projections have this property simultaneously. Consequently \(J\) is a two-sided *-ideal, and the ultraweak limit of a sequence in \(J\) belongs to \(J\).

Joins used in the proof. For finitely many projections \(e_j\), the kernel of \(a=\sum_j e_j\) is \(\bigcap_j\ker e_j\), since \(\langle a\xi,\xi\rangle=\sum_j\|e_j\xi\|^2\). Hence \(s(a)\), constructed in BK06, is the projection onto the closed span of their ranges and is their join. For an arbitrary family take the increasing net of finite joins. BK04 gives its strong limit \(q\in M\). Bounded multiplication gives \(q^2=q=q^*\). It contains each original range, and every projection containing those ranges bounds every finite join and then \(q\); thus it is the required arbitrary join.

Proof. Restrict a faithful corner state to a nonzero subprojection and normalize it. For \(p_n\in\mathcal P\), choose normal states \(\omega_n\) supported on the nonzero \(p_n\). A sum with strictly positive summable coefficients has support \(\bigvee_n p_n\). To check this last assertion, put \(q=\bigvee_n p_n\). The sum vanishes on \(1-q\). If \(a\in(qMq)_+\) and every \(\omega_n(a)=0\), corner faithfulness gives \(p_nap_n=0\), hence \(a^{1/2}p_n=0\). Since the ranges of the \(p_n\) span a dense subspace of \(qH\), this forces \(a=0\). Thus the normalized sum is faithful on \(qMq\).

If \(u^*u=p\), \(uu^*=q\), a faithful state on \(pMp\) transports to \(qMq\) by \(a\mapsto\omega(u^*au)\). This proves invariance under equivalence.

An element belongs to \(J\) precisely when its left and right support projections belong to \(\mathcal P\): one direction follows by taking subprojections; the other uses their finite join. The right support of \(ax\) is at most the right support of \(x\). The two supports of \(ax\) are equivalent by polar decomposition, so \(ax\in J\) if \(x\in J\). Adjoints give the right ideal property, while finite joins handle sums. Finally, if \(x_n\in J\), one countable join \(q\in\mathcal P\) satisfies \(x_n=qx_nq\) for all \(n\). If \(x_n\to x\) ultraweakly, fixed multiplication gives \(x=qxq\). \(\square\)

There is an orthogonal family \((p_i)_{i\in I}\subseteq\mathcal P\) with sum \(1\). Indeed, every nonzero projection \(e\) contains a nonzero member of \(\mathcal P\): choose \(\omega\in M_*^+\) with \(\omega(e)>0\), compress to \(eMe\), and take its support. A maximal orthogonal family, obtained by Zorn's lemma, must therefore have supremum \(1\). No assertion that \(I\) is countable is made.

Metric lemma. If \(\omega\) has support \(p\), the metric

induces the sigma-strong topology on each norm ball of \(Mp\). On a norm ball of \(pMp\), the sum \(d_\omega(x,y)+d_\omega(x^*,y^*)\) induces the sigma-strong* topology.

Proof. For \(x= xp\), \(x^*x\in pMp\); faithfulness there shows that \(d_\omega\) separates points. Only one direction of the topology assertion needs proof. Suppose \(x_\alpha,x\in Mp\), all of norm at most \(R\), and \(d_\omega(x_\alpha,x)\to0\). Put

For any \(\psi\in M_*^+\), failure of \(\psi(t_\alpha)\to0\) gives a subnet on which \(\psi(t_\alpha)\geq\varepsilon>0\). The order interval \([0,4R^2p]\) is ultraweakly compact. A further subnet has an ultraweak limit \(t\) there. Then \(\omega(t)=0\), so \(t=0\) by faithfulness, whereas \(\psi(t)\geq\varepsilon\), a contradiction. Thus every sigma-strong seminorm tends to zero. Apply this argument also to adjoints inside \(pMp\) for the second assertion. \(\square\)

The existence of this metric on a corner does not assert separability of that metric space.

OA-MOD-NW-04 — Passing an energy bound through an unbounded majorant

For \(t>0\), define

whenever \(h=h^*\) and \(1+th\) is strictly positive. Inverse order gives operator monotonicity of \(r_t\) on this domain. For \(h\geq0\),

The last convergence is even in norm for a fixed bounded \(h\).

Lemma. Suppose \(\varphi\) satisfies N. Let \(y_n\in M_+\) be norm bounded and converge sigma-strongly to \(y\). Suppose \(b_n\in M_+\) increases, \(y_n\leq b_n\), and

The norms of the \(b_n\) need not be bounded. Then \(\varphi(y)\leq C\).

Proof. Fix \(t>0\). The bounded increasing sequence \(r_t(b_n)\) has a supremum \(c_t\in M_+\). By N,

Monotonicity gives \(r_t(y_n)\leq r_t(b_n)\leq c_t\). The bounded functional-calculus continuity from NW-01 gives \(r_t(y_n)\to r_t(y)\) sigma-strongly and ultraweakly. The positive cone is ultraweakly closed, so \(r_t(y)\leq c_t\). Hence \(\varphi(r_t(y))\leq C\). Apply N once more to \(r_t(y)\uparrow y\) to obtain the result. \(\square\)

The order supremum of the \(b_n\) themselves was never asserted to exist in \(M\). This is why the bounded functions \(r_t\) enter the argument.

OA-MOD-NW-05 — Closing a GNS graph by summable increments

Theorem. Suppose \(\varphi\) satisfies N. If \(x_n\in\mathfrak n_\varphi\) is norm bounded, \(x_n\to x\) sigma-strong*, and \(\Lambda_\varphi(x_n)\to\xi\) in Hilbert norm, then

This sequential statement holds on an arbitrary von Neumann algebra.

Proof. Choose a subsequence, relabeled \(u_k\), so fast that, for \(z_k=u_{k+1}-u_k\),

This is possible because the given GNS vectors are Cauchy. Let \(\lambda_k=2^{-k}\) and \(L_k=\sum_{j=k}^\infty\lambda_j=2^{1-k}\). For \(N\geq k\), weighted Cauchy–Schwarz on every vector in a faithful concrete representation gives

Explicitly, \(\|\sum z_jv\|^2 \leq(\sum\lambda_j)\sum\lambda_j^{-1}\|z_jv\|^2\); enlarging the first scalar sum to \(L_k\) proves the operator inequality.

For fixed \(k\), the left side is a norm-bounded sequence converging sigma-strongly to \((x-u_k)^*(x-u_k)\), by the strong* assumption and bounded multiplication. The right side increases and

Here \(C_k<\infty\) and \(C_k\to0\). NW-04 yields

Thus \(x-u_k\in\mathfrak n_\varphi\). Since \(u_k\) lies there and the domain is linear, \(x\in\mathfrak n_\varphi\). Moreover \(\|\Lambda_\varphi(x)-\Lambda_\varphi(u_k)\|^2\leq C_k\to0\). The same subsequence converges to \(\xi\), so \(\Lambda_\varphi(x)=\xi\). \(\square\)

This proof neither assumes \(|h|\in\mathfrak m_\varphi\) for self-adjoint \(h\in\mathfrak m_\varphi\), nor applies the weight to a norm limit using an unproved semicontinuity assertion.

OA-MOD-NW-06 — Lower semicontinuity on a sigma-finite corner

Proposition. On a sigma-finite von Neumann algebra, A implies N.

Proof. Choose a faithful normal state \(\omega\). Given \(a_\alpha\uparrow a\), inductively choose an increasing sequence of indices \(\alpha_n\) such that

Normality of the bounded functional \(\omega\), or simply ultraweak convergence of the increasing net, supplies each choice. The sequence \(a_{\alpha_n}\) has a supremum \(b\leq a\). Ultraweak continuity gives \(\omega(b)=\omega(a)\), so faithfulness implies \(b=a\). Set \(a_{\alpha_0}=0\) and \(d_n=a_{\alpha_n}-a_{\alpha_{n-1}}\geq0\). Then \(\sum_nd_n=a\), and A gives

No infinite subtraction occurs. \(\square\)

Proposition. On a sigma-finite von Neumann algebra, N implies L.

Proof. For finite \(r,s\geq0\), consider the bounded part of the graph

It is convex. The product of sigma-strong* topology in \(M\) and norm topology in \(H_\varphi\) is metrizable on the containing norm balls, by NW-03. A point of its closure is therefore the limit of a sequence in \(G_{r,s}\). NW-05 proves that the point belongs to the graph; the two norm bounds persist. Thus \(G_{r,s}\) is closed for that product topology.

By the compatible-dual contract and Hahn–Banach separation, the closure of this convex set is the same for the product of ultraweak and weak Hilbert topologies. It is consequently a closed subset of the product of two compact balls, hence compact.

Its projection to \(M\) is

This set is ultraweakly compact and therefore closed. The set \(E_s=\{x:\varphi(x^*x)\leq s^2\}\) is convex, because it is the inverse image of a Hilbert ball under the linear GNS map on its linear domain. Krein–Smulian now shows that \(E_s\) is ultraweakly closed.

Fix \(c\geq0\). In a norm ball, if \(a_\alpha\geq0\), \(\varphi(a_\alpha)\leq c\), and \(a_\alpha\to a\) sigma-strongly, then \(a_\alpha^{1/2}\to a^{1/2}\) sigma-strongly and ultraweakly. Every square root lies in \(E_{\sqrt c}\), hence so does the limit. Therefore \(\varphi(a)\leq c\). The bounded slice of this sublevel set is convex, so compatible duals make it ultraweakly closed. A second use of Krein–Smulian proves that the entire sublevel set is ultraweakly closed. This proves L, including \(c=0\). \(\square\)

The sequence in this proof exists because the algebra is sigma-finite. The next two items remove that local hypothesis.

OA-MOD-NW-07 — A bounded gluing lemma for hereditary sets

Let \(F\subseteq M_+\) be convex and hereditary, meaning

Assume \(F\cap pMp\) is ultraweakly closed for every \(p\in\mathcal P\).

Lemma. If \(a_\alpha\in F\cap J\) is norm bounded and converges sigma-strongly to \(a\in J\), then \(a\in F\).

Proof. The assertion is empty if \(F=\varnothing\), so assume \(F\neq\varnothing\). Put

This set is convex: for \(0\leq t\leq1\),

and then convexity and heredity of \(F\) apply. It is also invariant under left multiplication by contractions, since \((vx)^*(vx)\leq x^*x\).

For \(q\in\mathcal P\), each bounded slice of \(E\cap qMq\) is sigma-strong* closed: it is the inverse image of the closed set \(F\cap qMq\) under \(x\mapsto x^*x\), using bounded multiplication. Convexity and compatible duals make these slices ultraweakly closed; Krein–Smulian gives ultraweak closedness of \(E\cap qMq\).

Fix \(p\in\mathcal P\). We claim that \(E^*p\), and therefore \(pE\), is ultraweakly closed in \(M\). Consider a sigma-strong limit point \(z\) of a bounded slice of \(E^*p\). It belongs to the same norm ball of \(Mp\). The metric in NW-03 gives a sequence \(z_n\in E^*p\) converging sigma-strongly to \(z\). Choose a countable join \(q\in\mathcal P\) containing both supports of every \(z_n\). Since \(z_n\to z\) ultraweakly, \(z=qzq\) too. Write \(z_n=e_n^*p\) with \(e_n\in E\). Then \(z_n^*=pe_n\in E\), by contraction invariance. Hence \(z_n^*\in E\cap qMq\), whose ultraweak closedness gives \(z^*\in E\). Also \(zp=z\), so \(z=(z^*)^*p\in E^*p\).

Thus each bounded slice of the convex set \(E^*p\) is sigma-strongly closed. Compatible duals and Krein–Smulian prove the claim; the adjoint homeomorphism gives it for \(pE\).

Return to the given positive net. Its limit is positive. For \(p=s(a)\in\mathcal P\), bounded square-root continuity gives

sigma-strongly and ultraweakly. Since \(a_\alpha^{1/2}\in E\), the net lies in \(pE\). Closedness gives \(a^{1/2}\in pE\subseteq E\), so \(a\in F\). \(\square\)

The lemma asserts exactly the bounded convergence needed below. It does not infer a statement about unbounded nets from an argument using bounded square-root continuity.

OA-MOD-NW-08 — Arbitrary positive sums imply global lower semicontinuity

Theorem. A implies L on every von Neumann algebra.

Proof. The zero algebra is immediate. Assume otherwise and put \(F=\{a\in M_+:\varphi(a)\leq1\}\). It is convex and hereditary. On any sigma-finite corner, the restricted weight satisfies A: the sums computed in the corner are the same positive operator sums computed in \(M\). NW-06 proves that its sublevel set is ultraweakly closed. Thus \(F\) meets the hypothesis of NW-07.

Take a norm-bounded net \(a_\alpha\in F\) converging sigma-strongly to \(a\geq0\). Choose an orthogonal family \((p_i)_{i\in I}\subseteq\mathcal P\) with sum \(1\). For a finite subset \(K\subseteq I\), write \(q_K=\sum_{i\in K}p_i\in\mathcal P\). The positive net

is norm bounded and converges sigma-strongly. Every term and the limit lie in \(J\), since \(J\) is an ideal. Moreover

so the terms belong to \(F\). NW-07 gives \(\varphi(a^{1/2}q_Ka^{1/2})\leq1\). The family \((a^{1/2}p_ia^{1/2})_{i\in I}\) is summable, with sum \(a\). Apply A to obtain

Thus every bounded slice of \(F\) is sigma-strongly closed. Convexity, compatible duals, and Krein–Smulian imply that \(F\) is ultraweakly closed. For \(c>0\), its scalar multiple \(cF\) is the sublevel set at \(c\). The zero sublevel set is \(\bigcap_{n\geq1}n^{-1}F\). All are closed, proving L. \(\square\)

The finite subsets \(K\) form a directed set. Replacing them with a sequence would lose this argument when \(I\) is uncountable.

OA-MOD-NW-09 — Positive separation needs a downward closure

Theorem. If \(E\subseteq M_+\) is a nonempty ultraweakly closed hereditary convex set, then

Consequently, for each \(a\in M_+\setminus E\), there is \(\omega\in M_*^+\) such that

Proof of the closure identity. Work in the real space \(M_{\mathrm{sa}}\), and write \(D=E-M_+\). This is convex and downward closed: \(h\in D\), \(k\leq h\) imply \(k\in D\). Since \(0\in E\), it contains \(-M_+\).

Let \(G\) consist of the self-adjoint \(h\) such that

Scalar functional calculus gives \(r_t(h)\leq h\). Hence \(D\subseteq G\). Also \(r_t(h)\to h\) in norm as \(t\downarrow0\), so \(G\subseteq\overline D^{\,\mathrm{uw}}\).

We first show that every bounded slice of \(G\) is sigma-strongly closed. Suppose \(h_\alpha\in G\), \(\|h_\alpha\|\leq R\), and \(h_\alpha\to h\) sigma-strongly, where \(R>0\). Fix \(0<t<1/(2R)\). Choose \(e_\alpha\in E\) with \(r_t(h_\alpha)\leq e_\alpha\). The domain bounds allow a second application of \(r_t\), giving

Here \(0\leq r_t(e_\alpha)\leq e_\alpha\), so these elements belong to \(E\), and they are bounded above by \(t^{-1}1\). Choose an ultraweakly convergent subnet, with limit \(e_t\in E\). Bounded functional-calculus continuity and closedness of the positive cone give \(r_{2t}(h)\leq e_t\). Thus \(r_{2t}(h)\in D\).

Given any \(\beta>0\) for which \(1+\beta h\) is strictly positive, choose the preceding \(t\) small enough that \(2t<\beta\). The scalar function \(r_s(\lambda)\) decreases with \(s\) throughout its domain, so \(r_\beta(h)\leq r_{2t}(h)\in D\). Downward closedness gives \(r_\beta(h)\in D\). Thus \(h\in G\). A ball of radius zero is trivial.

Next, if \(0\leq r<s\) and \(B\) denotes the self-adjoint unit ball, then

For the forward inclusion, approximate \(h\in G\cap rB\) in norm by \(r_t(h)\in D\); for small \(t\), these approximants belong to \(sB\). Conversely, \(D\cap sB\subseteq G\cap sB\), and the latter is sigma-strongly closed by the preceding paragraph. The right side is convex. Hence each bounded slice of \(G\) is convex, and \(G\) itself is convex.

Compatible duals now make each bounded slice of \(G\) ultraweakly closed. Krein–Smulian makes \(G\) ultraweakly closed. Since \(D\subseteq G\subseteq\overline D^{\,\mathrm{uw}}\), this proves \(G=\overline D^{\,\mathrm{uw}}\).

If \(a\in G\cap M_+\), then \(r_t(a)\in D\cap M_+\). It is positive and dominated by an element of \(E\), so heredity gives \(r_t(a)\in E\). Their norm limit \(a\) belongs to \(E\). The reverse inclusion is immediate, proving the identity.

Proof of positive separation. If \(a\geq0\) is outside \(E\), the identity puts it outside the closed convex set \(\overline D^{\,\mathrm{uw}}\). Real Hahn–Banach separation supplies a continuous real linear functional \(f\) with

The number \(c\) is finite and nonnegative, since \(0\in D\). Because \(-tb\in D\) for all \(b\geq0\) and \(t\geq0\), boundedness above forces \(f(b)\geq0\). Complexifying \(f\) gives a positive normal linear functional. If \(c>0\), divide by \(c\). If \(c=0\), multiply by a positive scalar making its value at \(a\) exceed one. In either case the resulting \(\omega\) has the asserted properties. \(\square\)

Subtracting the positive cone before separating is what forces the separating functional to be positive. An arbitrary real functional separating \(a\) from \(E\) need not have that property.

OA-MOD-NW-10 — Recovering every value from dominated normal functionals

Theorem. L implies P.

Proof. Put \(E=\{b\geq0:\varphi(b)\leq1\}\). This is nonempty, ultraweakly closed, hereditary, and convex. If \(\omega\in M_*^+\) satisfies \(\omega(e)\leq1\) for every \(e\in E\), then \(\omega\leq\varphi\) on all of \(M_+\). Indeed, when \(0<\varphi(b)<\infty\), apply the bound to \(b/\varphi(b)\). When \(\varphi(b)=0\), every \(tb\), \(t>0\), lies in \(E\), forcing \(\omega(b)=0\). At an infinite value the domination inequality is automatic.

Fix \(a\geq0\). Every member of \(\mathcal F_\varphi\) has value at most \(\varphi(a)\). Conversely, if \(0<r<\varphi(a)\), then \(a/r\notin E\). NW-09 gives a positive normal \(\omega\) with \(\omega|_E\leq1\) and \(\omega(a/r)>1\). By the preceding paragraph \(\omega\in\mathcal F_\varphi\), and \(\omega(a)>r\). Let \(r\) increase to \(\varphi(a)\) when that value is finite, or let \(r\to\infty\) when it is infinite. If \(\varphi(a)=0\), equality follows from positivity and the zero functional. This proves P in all cases. \(\square\)

The set \(\mathcal F_\varphi\) need not be asserted to be upward directed for this theorem. The supremum is taken separately at each positive element.

OA-MOD-NW-11 — The full characterization

Theorem. For an arbitrary weight on an arbitrary von Neumann algebra, N, A, L, and P of NW-02 are equivalent.

Proof. NW-02 gives \(\mathrm{P}\Rightarrow\mathrm{L}\Rightarrow\mathrm{N}\Rightarrow\mathrm{A}\). NW-08 gives \(\mathrm{A}\Rightarrow\mathrm{L}\), and NW-10 gives \(\mathrm{L}\Rightarrow\mathrm{P}\). These implications cover all four assertions. \(\square\)

Thus normality as defined by monotone nets permits a finite-observation test: whenever \(a\geq0\) and \(r<\varphi(a)\) with \(r\) finite and nonnegative, some positive normal functional \(\omega\leq\varphi\) satisfies \(\omega(a)>r\). This remains meaningful when \(\varphi(a)=\infty\).

A faithful normal state appeared only inside sigma-finite corners. Neither the original weight nor the auxiliary restrictions were assumed faithful or semifinite.

The scalar normality criterion

Programme source: OA-MOD, public/src/normal-positive-maps.md, original lines 1–71.

Normal positive maps and their preadjoints

Historical provider status notice: OA-MOD-NP.

This is the original provider's recorded status.

For a positive map, preserving increasing suprema is an order property. Ultraweak continuity is a topology property. This unit proves their equivalence and identifies the bounded map on preduals which records both. The proof applies to arbitrary von Neumann algebras and arbitrary bounded increasing nets, without separability, sigma-finiteness, faithfulness or identity preservation.

OA-MOD-NP-01 — Objects and the direction of the proof

Let \(M\subseteq B(H)\) and \(N\subseteq B(K)\) be concrete von Neumann algebras on arbitrary complex Hilbert spaces. Use the preduals constructed in OA-MOD-CP-06, so

isometrically, with the concrete ultraweak topologies \(\sigma(M,M_*)\) and \(\sigma(N,N_*)\). Statements transfer along already established concrete von Neumann algebra isomorphisms; no abstract representation theorem is assumed here. Inner products are linear in the first variable. Zero algebras and zero maps are included.

A bounded complex-linear map \(T:M\to N\) is positive when \(T(M_+)\subseteq N_+\). Say it is order normal when

The domain net is indexed by an arbitrary nonempty directed set. Positivity makes the image increasing, and boundedness makes it norm bounded; BK-04 supplies its supremum.

Why boundedness is automatic here. Suppose only that \(T\) is positive and complex linear. Positive/negative parts from BK01 show that \(T(h)\) is self-adjoint for self-adjoint \(h\). Apply \(T\) to \(-\|h\|1_M\le h\le\|h\|1_M\). Since \(T(1_M)\ge0\), this gives

For \(x=h+ik\), with \(h=(x+x^*)/2\) and \(k=(x-x^*)/(2i)\), both self-adjoint parts have norm at most \(\|x\|\). Therefore

This bound suffices to apply the theorems below to every positive complex-linear map between these algebras. It is not asserted to be the optimal norm formula. The zero-domain case gives the zero map directly.

The exact inputs are:

• OA-MOD-CP-06/07: predual duality, identification with the ultraweakly continuous functionals, norm closure of the predual, positive spanning and positivity detection; for a positive functional \(\omega\), \(\|\omega\|=\omega(1)\).
• OA-MOD-BK-03/04: concrete ultraweak topology and convergence of bounded increasing positive nets to their suprema.
• OA-MOD-NW-11: an arbitrary order-normal weight is recovered pointwise on its positive cone as the supremum of its dominated positive ultraweakly continuous functionals.
• OA-MOD-CP-08: the sigma-strong seminorms \(p_\omega(x)=\omega(x^*x)^{1/2}\), and their comparison with concrete strong topology on explicitly bounded sets.

This ordering is acyclic. The actual proof of NW-11 uses the finite-domain construction WG-003–006, not the representation-normality theorem WG-007. Its route from increasing-net normality to domination recovery is NW-02, NW-08 and NW-10. The predual and topology facts used there are constructed in CP without the scalar converse below. The written convex inputs are now in CV1–CV4, and NW01 identifies their exact uses. NW11 proves the weight theorem; CP06–CP07 and CP08 and BK03–BK04 supply the bounded predual and topology facts. These are written programme proofs, not external citations in place of arguments.

OA-MOD-NP-02 — The scalar normality criterion

Theorem. A bounded positive linear functional \(\omega:M\to\mathbb C\) preserves bounded increasing positive suprema if and only if \(\omega\in M_*^+\), equivalently if and only if it is ultraweakly continuous.

Proof. If \(\omega\in M_*^+\), BK-04 gives \(a_\alpha\to a\) ultraweakly whenever \(0\le a_\alpha\uparrow a\) is bounded. Continuity and positivity give \(\omega(a_\alpha)\uparrow\omega(a)\).

Conversely, suppose \(\omega\) preserves these suprema. Its restriction to \(M_+\) is a finite-valued normal weight. By NW-11,

If \(\omega(1)=0\), the positive norm formula gives \(\omega=0\). Otherwise, use (NP.2) only at \(a=1\) to choose \(\psi_n\in M_*^+\) with \(\psi_n\le\omega\) and

The difference is positive, so CP-07 gives

By norm closure from CP-06, \(\omega\in M_*\), and it is positive by hypothesis. ∎

Neither the family of dominated functionals nor the chosen sequence is asserted to be increasing. A sequence suffices here because it approximates one finite scalar supremum. This does not replace the arbitrary nets in the definition of normality.

TP1. Positive-functional order normality and ultraweak continuity

Theorem. Let \(\omega:M\to\mathbb C\) be a bounded positive complex-linear functional. The following assertions are equivalent:

1. For every bounded increasing positive net \(a_\alpha\uparrow a\) in \(M\), \(\omega(a)=\sup_\alpha\omega(a_\alpha)\).
2. \(\omega\in M_*^+\), for the concrete predual constructed in CP06.
3. \(\omega\) is continuous for the full ultraweak topology \(\sigma(M,M_*)\) on \(M\).

Proof. CP06 identifies the ultraweak continuous complex-linear dual with \(M_*\), so the second and third assertions are equivalent. If \(\omega\in M_*^+\), BK04 proves that every net in the first assertion converges ultraweakly to its supremum; positivity and continuity give that assertion.

Conversely, restrict \(\omega\) to \(M_+\). This is a finite-valued weight with property N of NW02. The fully proved theorem NW11, with no faithfulness, semifiniteness, separability or cardinality restriction, gives

For clarity, that provider has the following acyclic proof: NW02 proves N implies arbitrary positive summation A; NW03–NW08 prove A implies ultraweakly closed positive sublevel sets L by localizing to supported corners and returning through arbitrary positive sums; NW09–NW10 prove L implies the domination recovery P in (TP.1). Its GNS input is only the algebraic construction WG002–WG006. It does not use WG007, NP02, or the normality of any newly constructed representation. NW03's faithful states exist only on the supported corners used there, and NW08 retains the full possibly uncountable family of corners.

If \(\omega(1)=0\), the positive norm formula \(\|\omega\|=\omega(1)\), proved in CP07, gives \(\omega=0\). Otherwise apply (TP.1) at \(a=1\). For each positive integer \(n\), choose \(\psi_n\in M_*^+\) dominated by \(\omega\) with

The bounded functional \(\omega-\psi_n\) is positive. Its norm is therefore

CP06 proves that the predual is a norm-closed subspace of \(M^*\). Thus \(\omega\in M_*\), and it is positive by hypothesis. This proves the theorem. The auxiliary sequence approximates a single finite real supremum; it does not replace any of the arbitrary increasing nets in the theorem. No norm boundedness is assumed for a general ultraweakly convergent net. \(\square\)

This is the proof of OA-MOD-NP02 with its full transitive providers made explicit. It proves unrestricted ultraweak continuity by membership in the predual, not by a bounded-ball extension argument.

TP2. Arbitrary joins and orthogonal sums of projections

Lemma. Every family \((p_j)_{j\in J}\) of projections in \(M\), with arbitrary index set \(J\), has a least upper projection \(p\in M\). Its range is \(\overline{\operatorname{span}}\{p_jH:j\in J\}\). The increasing net of finite joins converges strongly and ultraweakly to \(p\). If the family is pairwise orthogonal, those finite joins are the sums \(\sum_{j\in F}p_j\). The empty join is zero.

Proof. For finite \(F\subseteq J\), let \(a_F=\sum_{j\in F}p_j\). Since

its kernel is \(\bigcap_{j\in F}\ker p_j\). BK06 constructs the support \(q_F=s(a_F)\in M\) as the projection onto \((\ker a_F)^\perp\). It is therefore the projection onto the closed span of the finitely many ranges and is their least upper projection. Set \(q_\varnothing=0\). The net \((q_F)\), directed by inclusion of finite subsets, is increasing and bounded by \(1\). BK04 gives its positive strong and ultraweak limit \(p\in M\). It is a projection, since for every \(\xi\in H\),

where \(\|q_F\|\leq1\). Thus \(p^2=p\); it is self-adjoint because it is positive. Every original range lies in \(pH\). Conversely, a projection containing all those ranges bounds every \(q_F\), and hence bounds \(p\) by passage to the strong limit. The projection onto their closed total span has exactly this leastness property, so it is \(p\). For pairwise orthogonal projections, their finite sum is a projection with that finite span as range, proving the last assertion. \(\square\)

This expands the complete arbitrary-join argument in NW03, original line 105, with BK04 and BK06. It applies independently of TP1 and of any trace.

TP3. The exact half-closed spectral cuts used by finite traces

Lemma. Let \(b\in M_+\) and \(\varepsilon>0\). The projection

commutes with \(b\) and is the half-closed threshold projection customarily denoted \(1_{[\varepsilon,\infty)}(b)\). It satisfies

If \(0<\varepsilon<\|b\|\), then \(q_\varepsilon\ne0\). If \(h\in M\) is a projection and \(b=hbh\), then \(q_\varepsilon\leq h\). Consequently, for any weight \(\tau\) with \(\tau(b)<\infty\),

Proof. The continuous calculus BK01 constructs \(c=(\varepsilon1-b)_+\) and \(a=(b-\varepsilon1)_+\), both positive, with \(b-\varepsilon1=a-c\) and \(ac=ca=0\). BK06 constructs \(s(c)\) in \(M\) by the strong limit of \(c(c+\delta1)^{-1}\). These operators commute with \(b\), so their limit and \(q_\varepsilon=1-s(c)\) do too. The range of \(q_\varepsilon\) is \(\ker c\); hence \(cq_\varepsilon=0\). All of \(a,c,b,q_\varepsilon\) commute. It follows that

Here each summand is positive, for instance \(b(1-q_\varepsilon)=(1-q_\varepsilon)b(1-q_\varepsilon)\). This proves (TP.4).

The same construction gives the exact endpoint convention. The continuous scalar functions

decrease pointwise to \(1_{[\varepsilon,\infty)}(t)\). At the operator level, \(f_n(b)=(1+nc)^{-1}\) decreases strongly and ultraweakly to \(1-s(c)=q_\varepsilon\), by the resolvent support formula in BK06. Thus (TP.3) constructs this particular spectral projection directly, including its value at the endpoint. No full Borel-calculus theorem is needed to construct or use this cut.

If \(0<\varepsilon<\|b\|\), then \(a\ne0\): otherwise \(b\leq\varepsilon1\), which implies \(\|b\|\leq\varepsilon\) by the positive norm formula. Since \(ca=0\), the range of \(a\) lies in \(\ker c\), so \(q_\varepsilon a=a\ne0\), proving nonzeroness. At \(\varepsilon=\|b\|\), nonzeroness is not asserted.

If \(b=hbh\), then \(h\) commutes with \(b\) and all the operators constructed above. On \((1-h)H\), \(b=0\), so continuous calculus gives \(c(1-h)=\varepsilon(1-h)\). Consequently \(q_\varepsilon(1-h)=\varepsilon^{-1}q_\varepsilon c(1-h)=0\). This proves \(q_\varepsilon\leq h\). Monotonicity and homogeneity of the weight, applied to (TP.4), prove (TP.5). No normality, faithfulness or semifiniteness of this last weight was needed. \(\square\)

The spectral statement proved here is exactly the threshold-projection statement consumed in Section 5. It does not claim a complete construction of arbitrary bounded Borel calculus or of unbounded spectral integrals; those are distinct theorems.

TP4. Exact application to the tracial representation

Let \(R\) carry the faithful normal semifinite trace \(\tau\) of Sections 4–6, with normality of the trace defined by arbitrary bounded increasing positive nets. For \(a\in\mathfrak n_\tau\), define \(\varphi_a(t)=\tau(a^*ta)\) for \(t\in R_+\). This function is additive and positively homogeneous and obeys

Every self-adjoint element is a difference of positive ones by BK01. Define \(\varphi_a(u-v)=\varphi_a(u)-\varphi_a(v)\). If \(u-v=u'-v'\), additivity applied to \(u+v'=u'+v\) proves independence; all four values are finite. Taking the real and imaginary self-adjoint parts gives a positive complex-linear functional on \(R\). CP07 proves its boundedness and \(\|\varphi_a\|=\varphi_a(1)=\tau(a^*a)\).

For \(0\leq t_\alpha\uparrow t\), the final congruence assertion of BK04 proves \(a^*t_\alpha a\uparrow a^*ta\). Normality of the trace gives order normality of \(\varphi_a\). TP1 therefore gives \(\varphi_a\in R_*^+\), at the full scope used in (6.1).

The existing estimate (6.2) proves (6.3) for the representation \(\lambda_\tau:R\to B(H_\tau)\), using its actual bounded increasing net and its dense square-integrable vectors. For every \(\omega\in B(H_\tau)_*^+\), the bounded positive functional \(\omega\circ\lambda_\tau\) preserves those suprema: (6.3) and BK04 give ultraweak convergence in \(B(H_\tau)\), and \(\omega\) is ultraweakly continuous. TP1 gives \(\omega\circ\lambda_\tau\in R_*^+\). CP07 shows that positive predual functionals span the entire predual, so this conclusion holds for each member of \(B(H_\tau)_*\). The defining evaluations of the full ultraweak topology therefore all pull back to continuous functionals. This proves global ultraweak continuity of \(\lambda_\tau\), without a boundedness claim about an arbitrary convergent net.

For Section 5, TP3 supplies its exact nonzero finite-trace cut from any nonzero positive finite-trace \(b=hbh\). If the initial convention supplies \(0\leq b\leq h\), then \(b=hbh\): positivity gives \((1-h)b(1-h)=0\), hence \(b^{1/2}(1-h)=0\) and its adjoint counterpart. TP2 supplies the arbitrary orthogonal sum used by Zorn's argument and the full net \(e_F\uparrow1\). The alternate convention in Section 5 supplies \(b=haa^*h\) directly. These arguments retain the possibly uncountable index set; no faithful state on \(R\) is introduced.

TP1–TP4, with all the exact preceding providers attached, discharge these specified order-normality and projection inputs. Standard-form, center-valued-trace and other free-action prerequisites remain separate.