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The universal enveloping von Neumann algebra of a \(C^*\)-algebra, and \(W^*\)-algebras

Originally written by Claude Opus 5.5 (Anthropic), September 2026, with a separate historical AI spot-check. GPT-6.1 Sol (OpenAI), at the Ultra setting, read and replayed the entire lesson and all thirteen solutions, compared the complete programme proof providers, and completed the Jordan, bidual, normality and boundary arguments, October 2026. Public domain (CC0).

A \(C^*\)-algebra \(A\) is seldom the dual of a Banach space, but its bidual \(A^{**}\) always is, and \(A^{**}\) carries a natural product that makes it a von Neumann algebra. Every representation of \(A\) extends to a normal representation of \(A^{**}\). So \(A^{**}\) is the universal enveloping von Neumann algebra of \(A\): the von Neumann algebra generated by any representation of \(A\) is isomorphic to a corner \(A^{**}z\), cut out by a central projection \(z\). For example, the bidual of \(c_0\) is \(\ell^\infty\), and the bidual of the compact operators \(\mathcal K(H)\) is \(B(H)\) (Examples 5.5 and 5.6).

The first part of the lesson, Sections 1 to 7, builds this tool and uses it. Functionals on \(A\) become normal functionals on \(A^{**}\). Representations become central projections: two representations generate isomorphic von Neumann algebras, compatibly with \(A\), exactly when their central projections agree. The norm-closed subspaces of \(A^*\) that are invariant under multiplication by \(A\) correspond to the weak\(^*\) closed one-sided ideals of \(A^{**}\). And for a pure state, the quotient of \(A\) by the left kernel of the state is already a Hilbert space.

The second part, Sections 8 to 13, is about \(W^*\)-algebras: the \(C^*\)-algebras that are isomorphic to von Neumann algebras. The key tool is Tomiyama's theorem: a projection of norm one onto a \(C^*\)-subalgebra is positive and a bimodule map. With it we prove Sakai's theorem: a \(C^*\)-algebra is a \(W^*\)-algebra exactly when it is the dual of a Banach space, and then the predual is unique. So normal functionals, the \(\sigma\)-weak topology and the splitting of functionals into normal and singular parts belong to the algebra, not to a chosen Hilbert space. The lesson ends with Kadison's characterization of \(W^*\)-algebras by monotone closedness and normal states.

We use the full earlier proofs in C*-algebras, GNS, the double commutant theorem, operator preduals and topologies, and Kaplansky density and monotone approximation, together with the Banach-space lessons specified below. The bidual construction is given before Section 2, its representation extensions are proved in Lemma 2.1, and the positive normal-map criterion is proved after Corollary 11.5. The full internal proofs in The universal C*-bidual and the selected Hilbert-tensor and preadjoint proofs in Concrete preduals and Normal positive maps provide a parallel programme treatment; the present arguments require no weight theory. Exercise 6.6 uses the full Haar measure proof.

Nothing here assumes separability, a unit or a faithful normal state unless this is said. Edge cases are worked out where they arise: degenerate representations (Example 2.2 and Remark 5.4), a state whose quotient is not a Hilbert space (Remark 7.4), a nonunital subalgebra with no norming element (Example 8.3), an increasing net whose norms exceed the norm of its supremum (Remark 13.5), a least upper bound that is not a strong limit (Exercise 13.10), and weak operator topologies that depend on the representation (Example 11.7).

The universal representation turns every bounded functional into a normal functional on an enveloping von Neumann algebra. The lesson proves the bidual structure, the normal/singular decomposition and the abstract characterization, including the required norm-one projection theorem. Freely readable comparisons are identified at the end.

Conventions

Full proof prerequisites

The thirty inputs below have exact full programme proof locations. The Jordan decomposition, bidual construction and representation extensions are proved here before their use. Intrinsic normality and the positive map criterion are proved in Section 11; the indicated forward reference has an acyclic proof chain, explained there. No separability or countability assumption is implicit. The references identify freely readable comparison sources.

Banach spaces.

  1. Hahn–Banach and Mazur. Bounded functionals separate a point from a norm-closed subspace, and from a norm-closed convex set, of a normed space. Consequently a convex subset of a normed space has the same closure in the weak and in the norm topology (Mazur's theorem). Full proofs: Hahn–Banach, Section 2, separation, Section 6, and Mazur, Theorem 4.1.
  2. Banach–Alaoglu. The closed unit ball of a dual Banach space is weak\(^*\) compact. Full proof: weak topologies, Theorem 3.1.
  3. Bipolars. A weak\(^*\) closed subspace \(J\) of a dual space \(E^*\) consists of the elements of \(E^*\) that vanish on every \(e\in E\) annihilated by \(J\). In the polar notation of the conventions: \(J^{\circ\circ}=J\) for every \(\sigma\)-weakly closed subspace \(J\) of a von Neumann algebra. Full proof: weak topologies, Theorem 5.1.
  4. Second adjoints. For a bounded linear map \(T:E\to F\) between Banach spaces, the second adjoint \(T^{**}:E^{**}\to F^{**}\) is weak\(^*\) continuous, \(\|T^{**}\|=\|T\|\), \(T^{**}\circ j_E=j_F\circ T\), and \((ST)^{**}=S^{**}T^{**}\). Full proof: weak topologies, Proposition 5.3.

\(C^*\)-algebras.

  1. Functional calculus. If \(a\) is a normal element of a \(C^*\)-algebra \(A\) and \(g\) is continuous on the spectrum of \(a\), with \(g(0)=0\) when \(A\) has no unit, then \(g(a)\in A\), and \(\pi(g(a))=g(\pi(a))\) for every \(*\)-homomorphism \(\pi\). Full proofs: continuous calculus, Theorems 5.1 and 5.3 and Corollary 5.4.
  2. Homomorphisms. A \(*\)-homomorphism between \(C^*\)-algebras is contractive. An injective one is isometric, and it is an order isomorphism onto its image: \(\pi(a)\ge0\) only if \(a\ge0\). Full proofs: contractivity and isometry, Section 4, with spectral permanence, Theorem 3.2.
  3. Unitaries. Every element of a unital \(C^*\)-algebra is a linear combination of four unitaries. Full proof: continuous calculus, Proposition 7.3.
  4. Approximate units. Every \(C^*\)-algebra, and so every closed two-sided ideal of a \(C^*\)-algebra, has an increasing approximate unit: an approximate unit \((u_i)\) with \(u_i\le u_j\) for \(i\le j\). Full proof: continuous calculus, Theorem 11.4 and its increasing approximate identity.
  5. Unitization. A nonunital \(C^*\)-algebra \(A\) is a closed ideal of its unitization \(A^\sim=A+\mathbb C1\), and an element of \(A\) is positive in \(A\) exactly when it is positive in \(A^\sim\). Full proof: Banach algebras, Proposition 3.4 and Exercise 5, with positivity from continuous calculus, Section 8.
  6. States. The states (positive functionals of norm one) of a \(C^*\)-algebra norm its self-adjoint elements, \(\|h\|=\sup_\omega|\omega(h)|\), and they detect positivity: \(h\ge0\) exactly when \(\omega(h)\ge0\) for every state \(\omega\). The short proof immediately after this list supplies both assertions from the earlier faithful-representation and order proofs.
  7. Jordan decomposition. Every hermitian \(f\in A^*\) is a difference \(f=f_+-f_-\) of positive functionals with \(\|f\|=\|f_+\|+\|f_-\|\). It is restated as Proposition 1.1(1), and its uniqueness is proved in Proposition 3.4. Full existence proof: Proposition 1.1; no bidual is used. Uniqueness is proved in Proposition 3.4.
  8. Cyclic representations. For \(\omega\in A^*_+\) there are a Hilbert space \(H_\omega\), a nondegenerate representation \(\pi_\omega\) on it and a cyclic vector \(\xi_\omega\) with \(\omega(a)=\langle\pi_\omega(a)\xi_\omega,\xi_\omega\rangle\) and \(\|\xi_\omega\|^2=\|\omega\|\). The Cauchy–Schwarz inequality \(|\omega(b^*a)|^2\le\omega(b^*b)\,\omega(a^*a)\) holds, and \(|\omega(a)|^2\le\|\omega\|\,\omega(a^*a)\). Full proofs: GNS, Proposition 3.2, Theorem 4.7, and Construction 5.1–Theorem 5.4.
  9. Pure states. The positive functionals of norm at most one form the weak\(^*\) closed convex hull of \(0\) and the pure states; this follows from the Krein–Milman theorem. (Pure functionals are defined before Lemma 7.2.) Consequently \(\|x\|=\sup\{\rho(x):\rho\text{ a pure state}\}\) for every \(x\in A_+\). Full proofs: GNS, Proposition 8.4 and Theorem 8.5, with Krein–Milman, Theorem 6.1.

von Neumann algebras. Let \(M\subseteq B(H)\) be a von Neumann algebra.

  1. Bicommutant theorem. A \(*\)-algebra of operators acting nondegenerately on \(H\) is \(\sigma\)-weakly, weakly and strongly dense in its bicommutant. Full proofs: double commutant, Theorem 4.4 and Lemma 1.3.
  2. Topologies. On bounded sets the weak and the \(\sigma\)-weak topologies agree, and so do the strong and the \(\sigma\)-strong topologies; in particular a bounded net that converges strongly also converges \(\sigma\)-weakly. Multiplication is separately \(\sigma\)-weakly continuous and jointly strongly continuous on bounded sets, and the involution is \(\sigma\)-weakly continuous. Full proofs: operator topologies, Lemma 8.5 and Proposition 8.6, and double commutant, Lemma 1.2.
  3. Kaplansky's density theorem. If \(A\subseteq B(H)\) is a \(C^*\)-algebra acting nondegenerately, the unit ball of \(A\) is strongly dense in the unit ball of \(A''\) . Full proof, including positive contractions: density, Theorem 7.1.
  4. Preduals. \(M_*\) is a norm-closed subspace of \(M^*\), and evaluation is an isometric isomorphism of \(M\) onto \((M_*)^*\) that carries the \(\sigma\)-weak topology to the weak\(^*\) topology. Every \(\sigma\)-weakly continuous functional on \(B(H)\) is a norm-convergent sum \(\sum_n\omega_{\xi_n,\eta_n}\) of vector functionals \(\omega_{\xi,\eta}(x)=\langle x\xi,\eta\rangle\) with \(\sum_n\|\xi_n\|\|\eta_n\|<\infty\), and every weakly continuous functional is a finite such sum. The normal functionals on \(\ell^\infty\), acting on \(\ell^2\) by multiplication, are the maps \(x\mapsto\sum_nc_nx_n\) with \(c\in\ell^1\). Full proofs: operator preduals, Theorem 6.2 and Theorems 9.1 and 9.4. On diagonal operators the absolutely summable vector series gives the stated coefficient sequence; every such sequence is obtained by taking square-root absolute values and phases.
  5. Positive normal functionals. They span \(M_*\). A positive \(\omega\) satisfies \(\|\omega\|=\omega(1)\) and the Cauchy–Schwarz inequality. The positive cone and the norm balls of \(M\) are \(\sigma\)-weakly closed. An element \(x\in M\) is positive exactly when \(\omega(x)\ge0\) for every \(\omega\in M_*^+\). The \(\sigma\)-strong topology is defined by the seminorms \(x\mapsto\omega(x^*x)^{1/2}\), \(\omega\in M_*^+\). Full proofs: operator preduals, Proposition 6.3 and Lemmas 8.2 and 8.5. A normal functional on a subspace extends normally to \(B(H)\) by Theorem 9.1; restricting the four positive normal parts there proves the span assertion for \(M\). Positivity is tested by vector functionals. The mixed vector-series seminorms and the square-summable diagonal series prove the stated sigma-strong description.
  6. Monotone nets and normal maps. A norm-bounded increasing net of self-adjoint elements of \(M\) has a least upper bound in \(M\), and the net converges to it strongly and \(\sigma\)-weakly. A positive linear map between von Neumann algebras is normal exactly when it preserves these least upper bounds; in particular positive normal functionals preserve them. A bounded linear map \(T:M\to N\) is normal exactly when \(\psi\circ T\in M_*\) for every \(\psi\in N_*\). Every \(*\)-isomorphism between von Neumann algebras is normal, and so is its inverse; Corollary 11.4 proves this from predual uniqueness. Full bounded-monotone proof: density, Theorem 1.3. The elementary preadjoint test is proved below. The positive-map converse is proved after Corollary 11.5, and automatic normality of isomorphisms in Corollary 11.4. These are conclusions proved in this lesson, rather than assumed inputs to the bidual construction.
  7. Supports. For a positive \(h\in M\), the range projection \(s(h)\) lies in \(M\), and \(h(h+\varepsilon)^{-1}\) increases to \(s(h)\) strongly as \(\varepsilon\downarrow0\). If \(f\) is a projection and \(0\le v\le f\), then \(v=fvf\). For \(\omega\in M_*^+\) there is a least projection \(s(\omega)\) with \(\omega(1-s(\omega))=0\), and \(\{x\in M:\omega(x^*x)=0\}=M(1-s(\omega))\). Operator support and compression have full proofs in double commutant, Proposition 7.2 and Theorem 8.3. The short normal-functional support proof after this list supplies the remaining assertion before its uses.
  8. Polar decomposition. Every \(x\in M\) is \(x=u|x|\) with a partial isometry \(u\in M\) such that \(u^*u=s(x^*x)\) and \(uu^*\) is the range projection of \(x\). Full proof: double commutant, Proposition 7.2.
  9. Spectral projections. Every self-adjoint element of \(M\) is a norm limit of real linear combinations of its spectral projections, and these lie in \(M\). A positive element is a norm limit of nonnegative combinations of spectral projections for Borel sets bounded away from \(0\). Full proofs: spectral theorem, Theorem 3.1, Proposition 5.1, and density, Corollaries 4.3 and 4.6.
  10. Corners. For a central projection \(z\in M\), \(Mz\) acts on \(zH\) as a von Neumann algebra, and its predual is \(M_*z\), the restrictions of normal functionals. Full proof: double commutant, Theorem 5.8; the predual is obtained by restriction and extension \(\varphi(x)\mapsto\varphi(zxz)\), which preserve the norm. The same maps, restricted to the corner, prove the weak-star pairings.
  11. Weak\(^*\) closed ideals. A \(\sigma\)-weakly closed two-sided ideal \(J\) of \(M\) with \(J^*=J\) is \(Mz\) for a unique central projection \(z\). (Lemma 4.2 shows that \(J^*=J\) is automatic.) Full proof: double commutant, Theorem 8.3, including self-adjoint two-sided ideals and centrality.
  12. Monotone closure. A \(C^*\)-algebra \(A\subseteq B(H)\) acting nondegenerately, which contains the strong limit of every norm-bounded increasing net of its self-adjoint elements, equals its bicommutant. This is a consequence of the up-down-up theorem; Full proof: density, Lemmas 12.2, 12.4, 12.5, 12.7 and 12.8, Theorem 12.9 and Corollary 12.10. This is the general net theorem, with no sigma-finiteness assumption.

The bidual. Let \(A\) be a \(C^*\)-algebra.

  1. The von Neumann algebra \(A^{**}\). The bidual \(A^{**}\) carries a product and an involution that extend those of \(A\), are separately \(\sigma(A^{**},A^*)\)-continuous, and make \(A^{**}\) a von Neumann algebra with predual \(A^*\). Concretely, let \(\pi_S\) be the direct sum of the cyclic representations of all states of \(A\). Then \(A^{**}\) is isometrically \(*\)-isomorphic to \(\pi_S(A)''\) by a map that extends \(\pi_S\) and is a homeomorphism for \(\sigma(A^{**},A^*)\) and the \(\sigma\)-weak topology. The unit ball of \(j(A)\) is \(\sigma\)-weakly dense in the unit ball of \(A^{**}\). Full construction: the bidual construction before Section 2. Its proof uses Proposition 1.1 and the earlier concrete predual and positive Kaplansky density proofs.
  2. Normal extensions. Every possibly degenerate representation \(\pi:A\to B(H)\) has a normal extension \(\bar\pi:A^{**}\to B(H)\): the Banach adjoint of \(\rho\mapsto\rho\circ\pi\), from \(B(H)_*\) to \(A^*\), is a normal \(*\)-homomorphism with \(\bar\pi\circ j=\pi\). If \(\pi\) is nondegenerate, then \(\bar\pi(A^{**})=\pi(A)''\); the kernel of \(\bar\pi\) is \(A^{**}(1-z)\) for a central projection \(z\); and the restriction of \(\bar\pi\) to \(A^{**}z\) is an isometric \(*\)-isomorphism onto \(\pi(A)''\), with normal inverse, which maps the unit ball of \(A^{**}z\) onto the unit ball of \(\pi(A)''\). Full proof: Lemma 2.1, including the isometric preadjoint proof of the normal inverse.
  3. Second adjoints of homomorphisms. If \(\varphi:A\to B\) is a \(*\)-homomorphism, then \(\varphi^{**}:A^{**}\to B^{**}\) is a normal \(*\)-homomorphism. If \(B\) is a \(C^*\)-subalgebra of \(A\) with inclusion \(i\), then \(i^{**}:B^{**}\to A^{**}\) is an isometric normal \(*\)-homomorphism whose range is the \(\sigma\)-weak closure of \(j_A(B)\). Full proof: the second-adjoint paragraph after Lemma 2.1.

Other facts.

  1. Haar measure. A compact group \(G\) carries a Borel probability measure \(\mu\) that is invariant under left translation, \(\int F(vu)\,d\mu(u)=\int F\,d\mu\) for continuous \(F\) and \(v\in G\), and has \(\mu(U)>0\) for every nonempty open \(U\). See the existence of Haar measure and its uniqueness and positivity on open sets. These Haar proofs have been read and compared in full, including their Radon-measure construction and the support argument.
  2. Zorn's lemma, Tychonoff's theorem, and the existence of free ultrafilters on \(\mathbb N\). Full proofs: Hahn–Banach, Section 1 (with the axiom of choice), and weak topologies, Lemma 2.1 and Theorem 2.3. Extend the cofinite filter on \(\mathbb N\) to an ultrafilter by that lemma; it contains no finite set and is free.

State, support and continuity tools

States norm and detect the order. Choose a faithful nondegenerate representation \(\rho\) from GNS, Theorem 7.2. For a unit vector \(\xi\), \(a\mapsto\langle\rho(a)\xi,\xi\rangle\) is positive and has norm one: positive approximate identities tend strongly to \(1\), so their values tend to one. If \(h=h^*\), put \(c=\sup_{\|\xi\|=1}|\langle\rho(h)\xi,\xi\rangle|\). The defining quadratic-form test for positivity gives \(-c1\le\rho(h)\le c1\), whence \(\|\rho(h)\|\le c\) by the positive order and continuous calculus. The reverse inequality is immediate. Faithfulness is isometric, so these states norm \(h\); their values are nonnegative on \(h\) exactly when \(\rho(h)\ge0\), equivalently \(h\ge0\). The zero algebra has no states, and its only self-adjoint element is zero.

Support of a bounded normal positive functional. Cauchy–Schwarz gives \[ \begin{gathered} N_\omega\\ =\{x:\omega(x^*x)=0\} \\ =\bigcap_{y\in M}\ker(x\mapsto\omega(y^*x)). \end{gathered} \] This is a sigma-weakly closed left ideal: the displayed functionals are normal because fixed multiplication is sigma-weakly continuous. By the full one-sided-ideal theorem cited in (20), \(N_\omega=M(1-e)\) for a projection \(e\). Then \(\omega(1-e)=0\). If a projection \(f\) has \(\omega(1-f)=0\), \(1-f\in N_\omega\), so \((1-f)e=0\) and \(e\le f\). Thus \(e=s(\omega)\) is the least such projection, including \(e=0\) when \(\omega=0\). If \(x\in(eMe)_+\) and \(\omega(x)=0\), then \(x^{1/2}\in N_\omega\) and \(x^{1/2}e=x^{1/2}=0\). So \(\omega\) is faithful on \(eMe\).

The preadjoint test. For a bounded linear \(T:M\to N\), suppose \(\psi\circ T\in M_*\) for every \(\psi\in N_*\). Then \(T_*\psi=\psi\circ T\) is bounded with norm at most \(\|T\|\), and the defining pairings give \(T=T_*^*\). Equivalently each defining scalar test for the sigma-weak topology of \(N\) pulls back to a continuous scalar test on \(M\), so \(T\) is normal. The converse follows by composing continuous maps. This uses no order-normality converse.

1. Functionals as vector coefficients

Every 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.

For a possibly degenerate representation \(\pi\) of \(A\) on \(H\) and \(\xi,\eta\in H\) write \[ \begin{gathered} \omega_{\pi;\xi,\eta}(a)\\ =\langle\pi(a)\xi,\eta\rangle\\ (a\in A). \end{gathered} \tag{1.1} \] A functional \(f\) is hermitian when \(f(a^*)=\overline{f(a)}\).

Proposition 1.1. Let \(A\) be a \(C^*\)-algebra.

  1. Every hermitian \(f\in A^*\) is a difference \(f=f_+-f_-\) of positive functionals with \(\|f\|=\|f_+\|+\|f_-\|\).
  2. 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. (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 \(g(h)=|f(a)|\).

First 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 \[ \begin{gathered} K\\ =\{\alpha\rho-\beta\sigma: \\ \rho,\sigma\in S,\quad \alpha,\beta\ge0,\quad \\ \alpha+\beta\le1\}. \end{gathered} \] This 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. 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.

For 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.

(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 \[ \begin{gathered} |f_k(a)|^2\\ \le\|f_k\|\,f_k(a^*a)\\ \le\|f_k\|\,\psi(a^*a)\\ =\|f_k\|\,\|\pi(a)\xi\|^2 . \end{gathered} \] So \(\pi(a)\xi\mapsto f_k(a)\) 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 \[ \begin{gathered} \|\eta\|\\ \le\sum_k\|f_k\|^{1/2}\\ \le2\big(\sum_k\|f_k\|\big)^{1/2}\\ \le2(2\|f\|)^{1/2}, \end{gathered} \] so \(\|\xi\|\,\|\eta\|\le4\|f\|\). \(\square\)

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, and even of the direct sum of the cyclic representations of the states. The decomposition in (1) is unique; see Proposition 3.4.

Constructing the bidual before using it

For \(A\ne0\), 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 \[ \|\xi\|^2=\sup_{F\subseteq S,\ F\ {\rm finite}}\sum_{\rho\in F}\|\xi_\rho\|^2<\infty. \] Its 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 \[ R:M_*\longrightarrow A^*,\qquad R(\varphi)=\varphi\circ\pi_S. \] By 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 \[ \begin{gathered} \|R\varphi\|\\ =\sup_{\|a\|\le1}|\varphi(\pi_S(a))| \\ =\sup_{x\in M,\ \|x\|\le1}|\varphi(x)|\\ =\|\varphi\|. \end{gathered} \] The 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, \[ U=R^*:A^{**}\longrightarrow M \] is 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.

The 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.

2. Extending a representation to the bidual

A 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 \(\pi(A)\).

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 \(\pi(A)\).

  1. \(q\) 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 \[ \pi(A)''=N(\pi)+\mathbb C(1-q), \tag{2.1} \] and the sum is direct when \(q\neq1\). So \(N(\pi)=\pi(A)''\) exactly when \(\pi\) is nondegenerate.
  2. 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 \(\bar\pi(A^{**})=N(\pi)\).
  3. \(\bar\pi\) maps the closed unit ball of \(A^{**}\) onto the closed unit ball of \(N(\pi)\).
  4. 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.

For 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. (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 \(q\in N(\pi)\).

Let \(\pi_0\) 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 \(N(\pi)=\pi_0(A)''\oplus0\).

Every \(T\in\pi(A)'\) 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).

(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 \[ T(j(a)Y)=\pi(a)T(Y). \] Next 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.

(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 \[ V=T|_{A^{**}z} \] is 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.

For 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 \(H_0=0\). \(\square\)

Second adjoints of homomorphisms

For 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 \[ \begin{gathered} \theta^{**}(XY)\\ =\theta^{**}(X)\theta^{**}(Y), \\ \theta^{**}(X^*)\\ =\theta^{**}(X)^*. \end{gathered} \] For 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.

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 \(\mathcal M(\pi)\).

3. The universal enveloping von Neumann algebra

One 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.

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 \(A\).

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. Universality supplies normal maps \(\theta\) and \(\theta'\). The composite \(\theta'\theta\) is normal and fixes \(\pi_1(A)\) pointwise. By Lemma 2.1(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. \(\square\)

Theorem 3.3 (The universal representation). Put \[ \begin{gathered} \pi_u\\ =\bigoplus_{\omega\in A^*_+}\pi_\omega\\ \text{on}\\ H_u\\ =\bigoplus_{\omega\in A^*_+}H_\omega , \end{gathered} \tag{3.1} \] where 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. Each summand is nondegenerate, so \(\pi_u\) is. Let \(\bar\pi_u(X)=0\) and let \(f\in A^*\). By Proposition 1.1(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(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.

Universality: 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. \(\square\)

From 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.

A first use of \(\tilde A\): the decomposition of Proposition 1.1(1) is unique, and it can be read off from one norming element.

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 \[ \begin{gathered} f_+(a)\\ =\hat f(aE_+),\\ f_-(a)\\ =-\hat f(aE_-)\\ (a\in A). \end{gathered} \tag{3.2} \] In particular the decomposition of Proposition 1.1(1) is unique, and the supports of \(\hat f_+\) and \(\hat f_-\) are orthogonal, lying under \(E_+\) and \(E_-\).

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 \(\|X\|\le1\).

Take 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 \(-1\le X\le1\), \[ \begin{gathered} \|f\|\\ =\hat f_+(X)-\hat f_-(X)\\ \le\hat f_+(1)+\hat f_-(1)\\ =\|f\| . \end{gathered} \] So \(\hat f_+(1-X)=0\) 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 \[ \begin{gathered} |\hat f_+(Y(1-X))|^2\\ =|\hat f_+((Yd)d)|^2\\ \le\hat f_+(Y(1-X)Y^*)\,\hat f_+(1-X)\\ =0 . \end{gathered} \] Realize \(\tilde A\) 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 \(\hat f_-(Y)=\hat f_-(YE_-)=\hat f_-(E_-Y)\).

The kernels of \(1-X\) and \(1+X\) are eigenspaces of the self-adjoint \(X\) for different eigenvalues, so \(E_+E_-=0\). Therefore \[ \begin{gathered} \hat f(YE_+)\\ =\hat f_+(YE_+)-\hat f_-(E_-YE_+E_-)\\ =\hat f_+(Y), \end{gathered} \] and 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. \(\square\)

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 \[ \begin{gathered} |\hat f|(YW)\\ =\hat f_+(YWE_+)+\hat f_-(YWE_-)\\ =\hat f_+(Y)-\hat f_-(Y), \end{gathered} \] 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.

4. Invariant subspaces of the dual and weak\(^*\) closed ideals

Multiplication 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.

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.

Two facts organise what follows.

  1. 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.
  2. 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(1) with \(S=j(A)\). So it makes no difference whether "invariant" refers to \(A\) or to \(\tilde A\).

The correspondence rests on a description of the weak\(^*\) closed one-sided ideals of a von Neumann algebra.

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. 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 \(p\in J\).

If also \(J=p'M\), then \(p'\in pM\) gives \(pp'=p'\), and \(p\in p'M\) gives \(p'p=p\). Hence \(p=p^*=(p'p)^*=pp'=p'\).

A \(\sigma\)-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.) \(\square\)

Theorem 4.3 (Invariant subspaces and weak\(^*\) closed ideals). Let \(M\) be a von Neumann algebra.

  1. 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 \(M\).
  2. 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 \(J\mapsto J^\circ\).
  3. 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 \(M_*e=eM_*\).

For 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, 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. 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).

(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.

(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.

(3) Let \(V\) be norm closed and left invariant. By (2) and Lemma 4.2, \(V^\circ=pM\) for a unique projection \(p\). Put \(e=1-p\). Then \[ \begin{gathered} V\\ =(pM)^\circ\\ =\{\varphi:\varphi(px)=0\ \forall x\}\\ =\{\varphi:\varphi p=0\}\\ =\{\varphi:\varphi=\varphi e\}\\ =M_*e . \end{gathered} \tag{4.1} \] Uniqueness 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 \((\varphi e)^*=e\varphi^*\).

For 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. \(\square\)

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 \(\tilde A\).

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 \[ \begin{gathered} M_*p\\ =\bigcap_iM_*e_i,\\ pM_*\\ =\bigcap_ie_iM_*,\\ M_*q\\ =\overline{\operatorname{span}}\bigcup_iM_*e_i,\\ qM_*\\ =\overline{\operatorname{span}}\bigcup_ie_iM_* . \end{gathered} \tag{4.2} \]

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.

For 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 \(\bigcap_iM_*e_i=((1-p)M)^\circ=M_*p\).

For 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. \(\square\)

5. Central supports of representations and quasi-equivalence

Each 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.

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(4), so that \(\ker\bar\pi=\tilde A(1-z(\pi))\). The subspace associated with \(\pi\) is \[ V(\pi)=\{\psi\circ\pi:\psi\in\mathcal M(\pi)_*\}\subseteq A^* . \]

We 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.

Proposition 5.2.

  1. \(\psi\mapsto\psi\circ\pi\) is an isometry of \(\mathcal M(\pi)_*\) onto \(V(\pi)\).
  2. \(V(\pi)=A^*z(\pi)\): 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(3) is \(z(\pi)\).
  3. Every norm-closed invariant subspace \(V\subseteq A^*\) equals \(V(\rho)\) for some representation \(\rho\).

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 \[ \begin{gathered} \|\psi\circ\pi\|\\ =\sup_{\|X\|\le1}|\psi(\bar\pi(X))|\\ =\sup_{\|y\|\le1,\,y\in\mathcal M(\pi)}|\psi(y)|\\ =\|\psi\| . \end{gathered} \]

(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.

(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), 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. \(\square\)

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. (ii)\(\Leftrightarrow\)(iii): \(V(\pi_k)=A^*z(\pi_k)\), and the support of an invariant subspace is unique (Theorem 4.3(3)).

(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 \(\theta(\pi_1(a))=\pi_2(a)\).

(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, 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 \(\pi_1(A)\). \(\square\)

Remark 5.4 (Degenerate representations). With \(N(\pi)\) of Lemma 2.1 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.

Two examples show the bidual concretely.

For 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.

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 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 \(1_{\{1,\dots,n\}}\).

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 \(\mathcal K(H)^{**}\cong B(H)\).

A representation of an ideal extends to the whole algebra, and the bidual shows why.

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. 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 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). Define \[ \pi^A(a)=\bar\pi\big((i^{**})^{-1}(j_A(a)z)\big)\qquad(a\in A). \] The 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. \(\square\)

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 \(N(\pi)\).

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 holds, in agreement with (2.1): \(\pi(A)''=\{0\}+\mathbb C1\). \(\square\)

6. Minimal projections and finite-dimensional von Neumann algebras

This 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 \(f\ne0\).

Lemma 6.1 (Minimal projections). Let \(N\) be a \(C^*\)-algebra and \(f,g\) minimal projections.

  1. \(Nf\) is a Hilbert space for \(\langle x,y\rangle f=y^*x\), and its Hilbert norm is the norm of \(N\).
  2. \(\dim fNg\le1\).
  3. Left multiplication \(\lambda(a)x=ax\) is an irreducible representation of \(N\) on \(Nf\).
  4. If \(N\) is a von Neumann algebra, every bounded linear functional on \(Nf\) is the restriction of a normal functional on \(N\).

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 \(x\). \(\square\)

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. 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.

For 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 \(N\). \(\square\)

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. 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. \(\square\)

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. 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 \(UxU^*=\theta(x)\). \(\square\)

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. 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 \(\dim eNe\le m^2\). \(\square\)

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.

We 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), 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 \(2\).)

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.

(\(\Rightarrow\)) 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). 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).

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 \(M\).

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.

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 \(\sum_cx_c\in M\).

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. \(\square\)

7. Pure states and Hilbert space quotients

Let \(\omega\) 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.

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. 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 \(Tx=y\). \(\square\)

A positive functional \(\varphi\) is pure if every positive \(\psi\le\varphi\) is \(\lambda\varphi\) for some \(\lambda\in[0,1]\).

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. 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 \(\psi=\lambda\omega\). \(\square\)

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 \[ \begin{gathered} \|a+N_\omega\|\\ =\omega(a^*a)^{1/2}\\ (a\in A). \end{gathered} \tag{7.1} \] Consequently 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. 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).

For \(\le\), let \(R:A\to H\), \(R(a)=\pi(a)\xi\). By Lemma 7.2, \(\mathcal M(\pi)=B(H)\). By Lemma 2.1(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 \(\|a+N_\omega\|\le\omega(a^*a)^{1/2}\).

The 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 \(H\). \(\square\)

Remark 7.4.

  1. 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.
  2. 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.
  3. 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 determines exactly when \(A/N_\omega\) is reflexive.

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 \(A\).

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 \(x^*x\ne0\).

First 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.

Now 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 \(\dim B=1\).

(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.

(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. \(\square\)

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.

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.

(\(\Rightarrow\)) 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, \(\pi(B_A)\) is \(\sigma\)-weakly dense in the unit ball of \(M\) (Lemma 2.1(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.

The 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 \(\psi_l(a)q_l=q_l\pi(a)q_l\).

Each \(\psi_l\) 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.

(\(\Leftarrow\)) 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 \(\sigma(\tilde A,A^*)\).

Consider \(T:A\to\tilde Ae\), \(T(a)=j(a)e\). 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. \(\square\)

For a pure state this is again Proposition 7.3, since then \(e\) is minimal and \(\tilde Ae\) is a Hilbert space.

8. Projections of norm one

A 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.

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 \(\|a\|\le1\).

The 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. 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.

If \(A\) 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 \(\omega\). \(\square\)

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. 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. \(\square\)

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.

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.)

Theorem 8.5 (Tomiyama's theorem). Let \(E\) be a projection of norm one of \(A\) onto \(B\). Then

  1. \(E\) is positive, \(E(x^*x)\ge0\), and hence \(E(x^*)=E(x)^*\);
  2. \(E(bxc)=bE(x)c\) for \(b,c\in B\) and \(x\in A\);
  3. \(E(x)^*E(x)\le E(x^*x)\) for \(x\in A\);
  4. if \(A\) has a unit and \(B\ne0\), then \(B\) has a unit and \(E(1_A)=1_B\).

Proof. If \(B=0\) all claims are trivial, so let \(B\ne0\).

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.

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 \(a=0\).

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, \(\psi\circ P\) 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.

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.

Fix \(x\) with \(\|x\|\le1\) and put \(y=P(fx(1-f))\in R\).

(a) \(fyf=0\). If \(f=0\), this is immediate. For \(f\ne0\) and \(t>0\), \(P(fx(1-f)+tf)=y+tf\), and \[ \begin{gathered} \|fx(1-f)+tf\|^2\\ =\|(fx(1-f)+tf)(fx(1-f)+tf)^*\|\\ =\|fx(1-f)x^*f+t^2f\|\\ \le1+t^2 . \end{gathered} \] On 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 \(fyf=0\).

(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: \[ \begin{gathered} \|fx(1-f)+tf'\|^2\\ =\|(fx(1-f)+tf')^*(fx(1-f)+tf')\|\\ =\|(1-f)x^*fx(1-f)+t^2f'\|\\ \le1+t^2 , \end{gathered} \] because \(ff'=0\) kills the cross terms. The argument of (a), with \(f'\) in place of \(f\), gives \(f'yf'=0\).

(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 \(t>0\), \[ P\big(fx(1-f)+tf'yf\big)=fyf'+(1+t)f'yf . \] The 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 \(y=fyf'\).

(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 \[ \begin{gathered} x\\ =fxf+fx(1-f)\\ +(1-f)xf+(1-f)x(1-f) \end{gathered} \] and multiply \(P(x)\) by \(f\) on the left: \[ fP(x)=P(fxf)+y=P(fx). \] By scaling, \(P(fx)=fP(x)\) for all \(x\), and taking adjoints, \(P(xf)=P(x)f\).

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 \(b,c\in R\).

Step 5: the Schwarz inequality. With \(r=X-P(X)\), bimodularity and \(P(P(X))=P(X)\) give \[ 0\le P(r^*r)=P(X^*X)-P(X)^*P(X). \]

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 \(b\in B\). \(\square\)

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.

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.

Solution. \(\ell^\infty\) has a unit, so by Tomiyama's theorem (Theorem 8.5(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 \[ \begin{gathered} E(x)_ne_n\\ =e_nE(x)\\ =E(e_nx)\\ =E(x_ne_n)\\ =x_ne_n. \end{gathered} \] Hence \(E(x)=x\) for every \(x\in\ell^\infty\), which is impossible for a non-convergent \(x\). \(\square\)

9. \(W^*\)-algebras and dual \(C^*\)-algebras

A 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.

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.

A 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 \(\iota_F(F)\).

Theorem 9.2 (Sakai's theorem). For a \(C^*\)-algebra \(A\) the following are equivalent:

  1. \(A\) is a \(W^*\)-algebra;
  2. \(A\) has a predual \(F\).

When (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 \(\iota_F(F)\).

Reference: [Sakai 1956] gives the original characterization using normal-state representations. The complete proof below uses the norm-one projection and bidual route.

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).

(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) \(E\) 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.

\(E\) 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 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.

Topologies. 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. \(\square\)

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 shows that \(\iota_F(F)\) does not depend on \(F\).

10. Normal and singular parts

Every 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.

Let \(M\subseteq B(H)\) 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(2), \[ M_*=V(\pi_0)=M^*z_0 . \tag{10.1} \]

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.)

For 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. Since \(\mathcal M(\pi)=\pi(A)\), Proposition 5.2(2) gives the normal ones as \(\iota_F(F)=V(\pi)=A^*z(\pi)\), and the singular ones are \(A^*(1-z(\pi))\).

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. 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 \[ \begin{gathered} \varphi(w)\\ =\varphi(zx)+\varphi((1-z)y)\\ \ge\|\varphi z\|+\|\varphi(1-z)\|-2\varepsilon. \end{gathered} \] \(\square\)

Theorem 10.3 (The normal–singular splitting).

  1. Every norm-closed left or right invariant subspace \(V\subseteq M^*\) splits as \[ \begin{gathered} V\\ =(V\cap M_*)\\ \oplus_1(V\cap M_*^\perp),\\ V\cap M_*\\ =Vz_0,\\ V\cap M_*^\perp\\ =V(1-z_0), \end{gathered} \tag{10.2} \] where \(\oplus_1\) means \(\|\varphi+\psi\|=\|\varphi\|+\|\psi\|\) for \(\varphi\in V\cap M_*\) and \(\psi\in V\cap M_*^\perp\).
  2. 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. (1) Let \(V\) be left invariant; the right case is the same. By Theorem 4.3(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 \(\tilde M\).

(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 \(M^*(1-z_0)\). \(\square\)

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(2)), so is \(M_*^\perp\). By (2), every representation splits as \(\pi=\pi_n\oplus\pi_s\), a normal part and a singular part.

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 \(\psi\in N_*\).

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. 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). \(\square\)

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.

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(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 \(z_\varphi=z_\psi\).

Suppose \(z_\varphi z_\psi=0\). Since \(\varphi\in V(\pi_\varphi)\) and \(\psi\in V(\pi_\psi)\), Proposition 5.2(2) gives \(\hat\varphi=\hat\varphi z_\varphi\) and \(\hat\psi=\hat\psi z_\psi\), so \(\hat\psi z_\varphi=0\). Lemma 10.2, with the central projection \(z_\varphi\), gives \[ \begin{gathered} \|\varphi-\psi\|\\ =\|(\hat\varphi-\hat\psi)z_\varphi\|+\|(\hat\varphi-\hat\psi)(1-z_\varphi)\|\\ =\|\hat\varphi\|+\|\hat\psi\|\\ =2 . \end{gathered} \] So \(\|\varphi-\psi\|<2\) forces \(z_\varphi=z_\psi\). By Theorem 5.3 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 \(U\pi_\varphi(a)U^*=\pi_\psi(a)\). \(\square\)

The 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.

11. Singular functionals and the uniqueness of the predual

A 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.

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. Realise \(A\) as a von Neumann algebra with \(\sigma\)-weak topology \(\sigma(A,A_*)\) (Theorem 9.2). 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(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 \(\omega=e\omega e\).

Let \(x\in A\) 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 \(\omega\ne0\).

Uniqueness. 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 \(q\le1-e\). \(\square\)

A 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).

Throughout the rest of this section, \(A\) is a \(W^*\)-algebra, realised as a von Neumann algebra by Theorem 9.2, 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.

Theorem 11.2 (Singular functionals). For a positive \(\omega\in A^*\), the following are equivalent:

  1. \(\omega\) is singular;
  2. every nonzero projection \(e\in A\) majorises a nonzero projection \(e_0\) with \(\omega(e_0)=0\).

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). 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 \(\omega_n=0\).

(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 \(\omega(e_0)=(e_0\omega e_0)(1)=0\). \(\square\)

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. By Theorem 9.2, \(F_k\) 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). 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 \(\iota_{F_1}(F_1)=\iota_{F_2}(F_2)\). \(\square\)

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. 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. \(\square\)

The 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.

Corollary 11.5 (Complete additivity). For \(\omega\in A^*\) on a \(W^*\)-algebra \(A\), the following are equivalent:

  1. \(\omega\) is normal;
  2. \(\omega\) 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. (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.

(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), \(\omega=0\). \(\square\)

Positive maps and increasing suprema

For 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.

Let \(T:M\to N\) 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, so this also applies without an added boundedness hypothesis.

The 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.

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.

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 \(\varphi(x)=\sum_nn^{-2}|\langle\zeta,e_n\rangle|^2>0\).

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.

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.

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(2)) is the first coordinate, \(x\mapsto x_1\), and its singular part is the second, \(x\mapsto\chi(x)\). \(\square\)

Exercise 11.10 (medium) (When every state is normal). Prove that all states of a von Neumann algebra \(M\) are normal exactly when \(\dim M<\infty\).

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). \(\square\)

12. Normal representations

The 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.

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. 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.

Let \(\omega\) be normal and positive with cyclic representation \((\pi_\omega,H_\omega,\xi_\omega)\). For \(a,b,x\in A\), \[ \begin{gathered} \langle\pi_\omega(x)\pi_\omega(a)\xi_\omega,\pi_\omega(b)\xi_\omega\rangle\\ =\omega(b^*xa)\\ =(a\omega b^*)(x), \end{gathered} \] and \(a\omega b^*\in A_*\) 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. \(\square\)

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) \(e_0\in\mathcal K\).

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).

(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 \(Me_0\subseteq\mathbb Ce_0\).

(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). For \(u\in G\), \(u\xi_0\in\mathfrak L\) has the same norm, so \(u\xi_0=\xi_0\).

(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 \(\pi(G)\).

We 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 \(\zeta=\pi(e_0)\zeta\).

Hence \(\eta=\pi(e_0)\eta=\pi(e_0k)\xi=\pi(e_0)\xi\), and \(\pi(e_0)\xi\in\pi(\mathcal K)\xi\).

(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. \(\square\)

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: (a) \(M\) has a faithful normal state \(\varphi\); (b) every nonzero projection \(e\in M\) majorises a nonzero positive element of \(A\), that is \(eMe\cap A\ne\{0\}\); (c) no singular state of \(M\) is faithful on \(A\); (d) if a state \(\omega\) of \(M\) is faithful on \(A\), its normal part \(\omega_n\) is faithful on \(M\); (e) \(A\) has a faithful state \(\omega=\sum_n\lambda_n\omega_n\) with pure states \(\omega_n\), \(\lambda_n\ge0\), \(\sum_n\lambda_n=1\); (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; (g) \(M\) is generated by its minimal projections, and \(A\) contains all of them.

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.

(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 \(\psi(e)=0\).

(c) Let \(\omega\) be a singular state. By Theorem 11.2, \(1\) 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 \(\omega(a)=0\).

(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 \(s(\omega_n)=1\).

(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.

(f) By Corollary 8.2, \(\bar\omega\) 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. For atomicity, write \(\rho=\bigoplus_n\pi_{\omega_n}\) on \(K=\bigoplus_nK_n\), a direct sum of irreducible representations (Lemma 7.2), and \(N=\rho(A)''\).

\(N\) 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.

\(\pi_\varphi(M)\) 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 \(M\cong\pi_\varphi(M)\).

(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 \(e\in A\). \(\square\)

13. Monotone closed \(C^*\)-algebras

In 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.

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 \(\omega(x)>0\).

Proposition 13.2.

  1. A \(W^*\)-algebra is monotone closed, and for it the two meanings of normal positive functional agree.
  2. A monotone closed \(C^*\)-algebra has a unit.

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. (The background fact on positive normal maps gives a second proof.)

(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 \(e=1\).) \(\square\)

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).

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. 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 \(i\), \[ \begin{gathered} \omega(x_i)\\ \ge\omega'(x_i)-\varepsilon C\\ \ge\omega'(x)-\varepsilon-\varepsilon C\\ \ge\omega(x)-\varepsilon(1+2C). \end{gathered} \] So \(\sup_i\omega(x_i)\ge\omega(x)\). \(\square\)

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.

Proposition 13.6. Let \(A\) be monotone closed and \(\omega\) a normal positive functional with cyclic representation \((\pi,H,\xi)\). Then \(\pi(A)=\pi(A)''\).

The 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. 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 \[ \|d_i^{1/2}\pi(u)\xi\|^2=\omega(u^*xu)-\omega(u^*x_iu)\to0 . \] \(A\) 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.

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 \(z=1-p\).

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 \[ \begin{gathered} (1-z)c\\ =(1-z)c(1-z)\\ \ge(1-z)c_i(1-z)\\ =0. \end{gathered} \] 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.

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, now gives \(\pi(A)=\pi(A)''\). \(\square\)

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. 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 gives \(\pi(A)=\pi(A)''\), so \(A\) is a \(W^*\)-algebra. \(\square\)

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.

Exercise 13.9. (easy) Show that a von Neumann algebra that is separable in the norm topology is finite-dimensional.

Solution. By Lemma 6.2, 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. \(\square\)

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.

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) \(\varepsilon\) 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 \(A\).

It 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 \[ \pi(a)\delta_i=\chi_i(a)\delta_i,\qquad M=\ell^\infty(I). \] Principal 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 \[ \varepsilon(g)=\pi((g(i_n))_{n\ge1})\qquad(g\in M). \] It 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.

(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.

Suppose \(\dim A=\infty\). 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 \(c\in C\).

\(C\) 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 \(p'\ne0\), \(p\notin\{0,1_A\}\).

The 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. \(\square\)

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.

Solution. By Sakai's theorem (Theorem 9.2) such an algebra would be a \(W^*\)-algebra, hence unital and monotone closed (Proposition 13.2(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 \([0,\tfrac12]\). \(\square\)

Where this leads

The 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.

References

Freely accessible reading: Shoichiro Sakai, A characterization of W*-algebras, printed pp. 763–773 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.

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