Functional Analysis: Operator Algebras

Job: develop C*-algebras and von Neumann algebras at frontier depth — the Gelfand–Naimark theorem (every C*-algebra embeds in \(B(H)\)), the GNS construction (states give representations), the Gelfand duality (commutative C* \(\leftrightarrow\) locally compact spaces), the von Neumann bicommutant theorem and the Murray–von Neumann type classification (I / II / III), and Connes' classification of injective / amenable factors with the free group factor isomorphism problem.

Core

C*-algebras

A C*-algebra \(A\) is a complex Banach *-algebra satisfying the C*-identity \(\|a^*a\|=\|a\|^2\) for all \(a\in A\). The C*-identity forces the involution to be isometric (\(\|a^*\|=\|a\|\)) and makes the algebra a closed *-subalgebra of \(B(H)\) once represented.

C*-algebra Description
\(B(H)\) all bounded operators on a Hilbert space \(H\) (the universal example)
\(\mathcal{K}(H)\) the compact operators (the unique separable simple C*-algebra, up to stable isomorphism)
\(C_0(X)\) continuous functions vanishing at infinity on a LCH space \(X\) (commutative)
\(C^*_r(G)\) the reduced group C*-algebra, closure of the left-regular representation of a locally compact group \(G\)
\(M_n(A)\) \(n\times n\) matrices over \(A\) (a C*-algebra with the unique C*-norm)

Theorem (Gelfand–Naimark). Every C*-algebra \(A\) is isometrically *-isomorphic to a norm-closed *-subalgebra of \(B(H)\) for some Hilbert space \(H\) (the universal representation \(A\hookrightarrow B(\bigoplus_\omega H_\omega)\) over all states \(\omega\)). Hence C*-algebras are exactly the norm-closed *-subalgebras of \(B(H)\).

Theorem (Gelfand duality, commutative case). The functor \(X\mapsto C_0(X)\) is a contravariant equivalence between the category of locally compact Hausdorff spaces (with proper maps) and the category of commutative C*-algebras (with *-homomorphisms). The inverse sends \(A\) to its spectrum \(\hat A\) (the characters, i.e. nonzero *-homomorphisms \(A\to\mathbb{C}\), with the weak-* topology); \(A\cong C_0(\hat A)\) via the Gelfand transform \(a\mapsto\hat a\) with \(\hat a(\chi)=\chi(a)\).

Commutative C*-algebra Locally compact Hausdorff space
\(C_0(X)\) \(X\)
\(C_b(X)\) (bounded) \(\beta X\) (Stone–Čech compactification)
\(C(K)\) (unital) compact \(K\)
\(L^\infty(X,\mu)\) (as a C*-algebra) the hyperstonean spectrum (a measure-theoretic space)
*-homomorphism \(A\to B\) continuous proper map \(\hat B\to\hat A\)
ideal \(I\subset A\) closed subset of \(\hat A\)

The GNS construction

For a state \(\omega\) on a C*-algebra \(A\) (a positive linear functional with \(\omega(1)=1\) in the unital case, or \(\|\omega\|=1\) in general), the GNS representation is

\[ \pi_\omega:A\to B(H_\omega),\qquad H_\omega=\overline{A/N_\omega}^{(\,\cdot|\cdot\,)},\quad N_\omega=\{a:\omega(a^*a)=0\},\quad \langle[a],[b]\rangle=\omega(a^*b), \]

with \(\pi_\omega(a)[b]=[ab]\) and the cyclic vector \(\xi_\omega=[1]\) satisfying \(\omega(a)=\langle\pi_\omega(a)\xi_\omega,\xi_\omega\rangle\). The universal representation \(\pi_u=\bigoplus_\omega\pi_\omega\) over all states is faithful (hence realizes Gelfand–Naimark). The GNS construction is the noncommutative analog of "a measure on the spectrum gives an \(L^2\) space": for \(A=C_0(X)\) and \(\omega(f)=\int f\,d\mu\), the GNS Hilbert space is \(L^2(X,\mu)\), the representation is multiplication, and \(\xi_\omega=1\).

Von Neumann algebras

A von Neumann algebra is a *-subalgebra \(M\subset B(H)\) with \(M=M''\) (the bicommutant \(M''=(M')'\) where \(M'=\{T\in B(H):[T,m]=0\,\forall m\in M\}\)).

Theorem (von Neumann bicommutant). For a unital *-subalgebra \(M\subset B(H)\), TFAE: (i) \(M=M''\); (ii) \(M\) is closed in the weak operator topology (WOT); (iii) \(M\) is closed in the strong operator topology (SOT). Hence von Neumann algebras are the WOT/SOT-closed *-subalgebras of \(B(H)\), equivalently the C*-algebras that are dual Banach spaces (\(M=(M_\ast)^*\) for the predual \(M_\ast\) of ultraweakly continuous functionals).

Examples: \(B(H)\) (type I\(_\infty\)); the group von Neumann algebra \(VN(G)\) (closure of the left-regular representation in the WOT); the hyperfinite \(\mathrm{II}_1\) factor \(R\) (the closure of the increasing union of matrix algebras \(M_{2^n}(\mathbb{C})\) with normalized trace); the free group factors \(L(\mathbb{F}_n)=VN(\mathbb{F}_n)\) for the free group \(\mathbb{F}_n\).

Factors and the Murray–von Neumann type classification

A factor is a von Neumann algebra with trivial center \(Z(M)=M\cap M'=\mathbb{C}\). The Murray–von Neumann classification is by the structure of the projection lattice:

Type Definition Trace Examples
I \(M\simeq B(H)\) (or \(M_n(\mathbb{C})\)) the usual trace (semi-finite, not finite on \(B(H)\) for \(\dim H=\infty\)) \(B(H)\) (I\(_\infty\)), \(M_n(\mathbb{C})\) (I\(_n\))
II\(_1\) finite (a finite trace \(\tau:M\to[0,1]\) with \(\tau(1)=1\)), diffuse projections (no minimal projections) a faithful normal finite trace \(\tau\) the hyperfinite \(R\), the free group factors \(L(\mathbb{F}_n)\)
II\(_\infty\) \(M=N\overline\otimes B(\ell^2)\) with \(N\) a II\(_1\) factor semi-finite trace (infinite) \(R\overline\otimes B(\ell^2)\)
III no semi-finite trace; all nonzero projections equivalent no trace (only the modular flow) the Powers factors \(R_\lambda\) (\(0<\lambda<1\)), the Araki–Woods factors

The Connes classification of type III: type III\(_0\), III\(_\lambda\) (\(0<\lambda<1\)), III\(_1\) by the modular flow (the Connes \(T\) invariant = the spectrum of the modular operator \(\Delta\)), with the Powers factors \(R_\lambda\) realizing III\(_\lambda\) and the Araki–Woods factor realizing III\(_1\).

Theorem (Connes 1976, injective \(\mathrm{II}_1\) = hyperfinite). A \(\mathrm{II}_1\) factor \(M\) is injective (the identity \(M\hookrightarrow B(H)\) has a conditional expectation \(B(H)\to M\)) iff \(M\) is hyperfinite (the WOT closure of an increasing union of finite-dimensional subalgebras), iff \(M\cong R\) (the unique hyperfinite \(\mathrm{II}_1\) factor). This is the classification of amenable / injective \(\mathrm{II}_1\) factors — the analog of the Connes injectivity / amenability in the type II setting.

Free group factors and the isomorphism problem

The free group factor \(L(\mathbb{F}_n)\) is the von Neumann algebra of the free group \(\mathbb{F}_n\) (the closure of the left-regular representation); it is a \(\mathrm{II}_1\) factor with the canonical trace \(\tau(\lambda(g))=\delta_{g,e}\).

Open problem (free group factor isomorphism). For \(m\ne n\) (\(m,n\ge 2\)), is \(L(\mathbb{F}_m)\cong L(\mathbb{F}_n)\) as von Neumann algebras? The free probability theory of Voiculescu gives partial invariants (the free entropy, the \(L^2\) invariants) but the full problem is open. The free absorption result (Voiculescu) \(L(\mathbb{F}_m)\ast L(\mathbb{F}_n)\cong L(\mathbb{F}_{m+n})\) (free product) suggests the factors may be isomorphic for \(m,n\ge 2\), but this is unproven.

See it

The GNS construction for \(C[0,1]\) with the Lebesgue state, and for \(M_n(\mathbb{C})\) with the normalized trace.

Commutative example. Let \(A=C[0,1]\) and \(\omega(f)=\int_0^1 f(x)\,dx\) (the Lebesgue state). The GNS construction: \(N_\omega=\{f:\int|f|^2=0\}=\{0\}\) (Lebesgue measure has full support), so \(H_\omega=L^2[0,1]\) (the closure of \(C[0,1]\) in the \(L^2\) norm). The representation \(\pi_\omega(f)\) is multiplication by \(f\) on \(L^2[0,1]\): \(\pi_\omega(f)[g]=[fg]\), i.e. \(\pi_\omega(f)\psi=f\cdot\psi\) for \(\psi\in L^2\). The cyclic vector is \(\xi_\omega=[1]\) (the constant function \(1\), since \(\omega(f)=\langle\pi_\omega(f)1,1\rangle=\int_0^1 f\cdot 1\,dx=\int_0^1 f\,dx\)). The image \(\pi_\omega(C[0,1])\) is the algebra of multiplication operators by continuous functions on \(L^2[0,1]\), a maximal abelian von Neumann subalgebra of \(B(L^2[0,1])\) (its WOT closure is \(L^\infty[0,1]\), the multiplication operators by \(L^\infty\) functions). This is the Gelfand-duality side: the commutative C*-algebra \(C[0,1]\) is represented as multiplication on \(L^2\) of its spectrum \([0,1]\) with the Lebesgue measure.

Noncommutative example. Let \(A=M_n(\mathbb{C})\) and \(\omega=\tau=\frac1n\mathrm{tr}\) (the normalized trace, a state). The GNS construction: \(N_\tau=\{a:\mathrm{tr}(a^*a)=0\}=\{0\}\) (the trace is faithful), so \(H_\tau=M_n(\mathbb{C})\) with the Hilbert–Schmidt inner product \(\langle a,b\rangle=\tau(a^*b)=\frac1n\mathrm{tr}(a^*b)\) (the Frobenius inner product up to the \(1/n\) normalization). The representation \(\pi_\tau(a)\) is left multiplication: \(\pi_\tau(a)b=ab\) (an \(n^2\times n^2\) matrix acting on the vectorized \(M_n(\mathbb{C})\)). The cyclic vector is \(\xi_\tau=I_n\) (the identity matrix, since \(\tau(a)=\langle\pi_\tau(a)I,I\rangle=\frac1n\mathrm{tr}(a^*I)=\frac1n\mathrm{tr}(a)\)). The image \(\pi_\tau(M_n(\mathbb{C}))\) is the left-regular representation of \(M_n(\mathbb{C})\) on itself, isomorphic to \(M_n(\mathbb{C})\otimes I_n\subset M_{n^2}(\mathbb{C})\) — a type I\(_n\) factor. The two GNS examples illustrate the commutative-vs-noncommutative dichotomy: the commutative case gives a maximal abelian subalgebra of \(B(L^2)\) (the "multiplication operator" picture of the spectral theorem); the noncommutative case gives the Hilbert–Schmidt representation, the finite-dimensional prototype of the type I factor.

Check

Explain why a C*-algebra is norm-closed but a von Neumann algebra is WOT/SOT-closed, and identify the precise sense in which a von Neumann algebra is 'larger' than a C*-algebra on the same generators.

A C*-algebra \(A\subset B(H)\) is closed in the norm topology of \(B(H)\); a von Neumann algebra \(M\subset B(H)\) is closed in the weak operator topology (or SOT, equivalently \(M=M''\) by the bicommutant theorem). The WOT is strictly coarser than the norm topology on \(B(H)\) (e.g. the unilateral shift \(S\) and its adjoint generate the Toeplitz C*-algebra \(\mathcal{T}=C^\ast(S)\) in the norm topology, but the WOT closure of \(\mathcal{T}\) is \(B(\ell^2)\), the type I\(_\infty\) factor — much larger). Hence a von Neumann algebra is "larger" in the sense that the WOT closure of a C*-algebra is a von Neumann algebra containing it strictly in general. The C*-algebra retains the norm / topological structure (continuous functional calculus, K-theory); the von Neumann algebra retains the measure-theoretic structure (the predual, the trace, the type classification). The same set of generators (e.g. the left-regular representation of a group \(G\)) gives the reduced C*-algebra \(C^*_r(G)\) (norm closure) and the group von Neumann algebra \(VN(G)\) (WOT closure), with \(C^*_r(G)\subset VN(G)\) and typically \(VN(G)\) much larger (a factor for ICC groups, while \(C^*_r(G)\) is not).

State the Murray–von Neumann criterion distinguishing type I from type II\(_1\) via the projection lattice, and explain why the hyperfinite \(R\) is type II\(_1\) but \(B(H)\) is type I.

A projection \(p\in M\) is minimal if \(pMp=\mathbb{C}p\). Two projections \(p,q\) are Murray–von Neumann equivalent (\(p\sim q\)) if there exists \(v\in M\) with \(p=v^*v\), \(q=vv^*\) (a partial isometry). A factor \(M\) is type I if it has a nonzero minimal projection (e.g. \(B(H)\): the rank-one projections are minimal); type II\(_1\) if it has no minimal projections but has a finite faithful normal trace \(\tau\) with \(\tau(1)=1\) (the projections are "diffuse" — for any \(p\) there is \(q<p\) with \(q\ne 0,p\) — and \(\tau\) takes all values in \([0,1]\) by the intermediate-value property of diffuse projections). The hyperfinite \(R\) is the WOT closure of \(\bigcup_n M_{2^n}(\mathbb{C})\) (with the embeddings \(a\mapsto a\otimes I_2\)); it has no minimal projections (the diagonal matrix algebras have rank-one projections in \(M_{2^n}\), but in the limit every projection can be further halved via \(p\mapsto p\otimes e_{11}\)), and the normalized trace \(\tau=\lim 2^{-n}\mathrm{tr}\) is a finite faithful normal trace. Hence \(R\) is type II\(_1\). \(B(H)\) for \(\dim H=\infty\) has the rank-one projections (minimal), so it is type I\(_\infty\); for \(\dim H<\infty\) it is type I\(_n\) (the finite-dimensional case). The distinction is the existence of minimal projections (type I) vs diffuse + finite trace (type II\(_1\)).

Pitfalls

  • C* is norm-closed, von Neumann is WOT/SOT-closed (different closures): the WOT closure of a C*-algebra is a von Neumann algebra strictly containing it in general (Toeplitz \(\to B(\ell^2)\)). Conflating the closures is the standard error.
  • The bicommutant theorem requires a faithful representation: the equivalence \(M=M''\) \(\Leftrightarrow\) WOT-closed holds for a unital *-subalgebra \(M\subset B(H)\) represented faithfully on \(H\); for an abstract C*-algebra, the WOT closure depends on the representation (different faithful representations give different von Neumann closures — the universal representation gives the universal von Neumann envelope).
  • A factor is a von Neumann algebra with trivial center; not every von Neumann algebra is a factor: general von Neumann algebras decompose as a direct integral of factors (the central decomposition); the type classification applies to the factors.
  • The hyperfinite \(R\) is the unique injective \(\mathrm{II}_1\) factor (Connes), but there are many non-injective \(\mathrm{II}_1\) factors: the free group factors \(L(\mathbb{F}_n)\) are non-injective (not hyperfinite) by Connes' theorem (since they contain free subgroups that obstruct injectivity), and the isomorphism problem among them is open.
  • The free group factor isomorphism problem is open: whether \(L(\mathbb{F}_m)\cong L(\mathbb{F}_n)\) for \(m\ne n\ge 2\) is open; Voiculescu's free probability gives partial invariants but no full resolution.
  • The type III classification (Connes) is by the modular flow / \(S\) and \(T\) invariants, not by the trace (there is no trace in type III): the modular theory (Tomita–Takesaki) gives the modular automorphism group, and the Connes invariants (\(S(M),T(M)\)) subclassify type III into III\(_0\), III\(_\lambda\), III\(_1\).

Refs

  • Pedersen, C*-algebras and their Automorphism Groups; Murphy, C*-algebras and Operator Theory
  • Takesaki, Theory of Operator Algebras I–III (von Neumann algebras, type classification, modular theory)
  • Connes, Classification of injective factors (Annals 1976); Brown–Ozawa, C*-algebras and Finite-Dimensional Approximations
  • Voiculescu–Dykema–Nica, Free Random Variables (free group factors); Murray–von Neumann, On rings of operators

Next

Spectral Theory · Function Spaces · Harmonic Analysis · FA index · Foundations · Noncommutative geometry · NCG spectral triples — Connes spectral triples are built on C*/von Neumann algebras · AQFT / algebraic — Haag–Kastler nets are von Neumann algebras; Tomita–Takesaki modular theory in QFT · Free probability · FP / operator-algebraic freeness — the free group factors \(L(\mathbb{F}_n)\), the free product, the isomorphism problem · Random matrices · RMT / determinantal — the DPP kernels are trace-class / projection operators