Von Neumann Algebras C*-Algebras Dimensions and C*-Algebras Nigel Higson Department of Mathematics Pennsylvania State University British Mathematics Colloquium, 2006 Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras C*-Algebras Outline 1 Von Neumann Algebras Classification into Types Group Measure Space Construction 2 C*-Algebras Dimensions and de Rham Theory The Baum-Connes Conjecture Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Rings of Operators Definition π :Γ → U(H) a unitary representation of a group. Commutant: M = { T ∈ B(H): T π(g) = π(g)T }. Commutant = algebra of operators—a von Neumann algebra: T ∈ M ⇒ T ∗ ∈ M. John von Neumann Tn ∈ M & Tnv → Tv, ∀v ⇒ T ∈ M. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Projections and Subrepresentations M = T ∈ B(H): T π(g) = π(g)T , ∀g ∈ Γ Subrepresentations of π correspond to projections in M. Equivalence of subrepresentations corresponds to Murray-von Neumann equivalence of projections: P ∼ Q ⇔ P = UV , Q = VU. Definition Denote by Dim(M) the set of equivalence classes of projections—it is a partially ordered set Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Factors and Isotypical Representations Definition A unitary representation π :Γ → U(H) is isotypical if any two subrepresentations have equivalent subsubrepresentations. Theorem Every representation decomposes canonically into isotypical representations. Theorem The commutants of isotypical representations are factors—von Neumann algebras with trivial centers. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Factors and Isotypical Representations Type I Isotypical Representations In simple situations, an isotypical representation is necessarily a multiple of a single irreducible representation. The corresponding factor is isomorphic to the algebra of all bounded operators on a Hilbert space K , where the dimension of K is equal to the multiplicity of the irreducible representation. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Factors and Isotypical Representations In general, isotypical representations can be far more complicated! Example Regular representation of a free group F on `2(F). It can be decomposed into irreducible representations in two (indeed many) totally disjoint ways. The complete structure of this representation is unknown. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Murray-von Neumann Classification Theorem If M is a factor, then the partially ordered set Dim(M) of equivalence classes of projections in M is isomorphic one of: {0, 1, 2,..., N} Type I [0, K ] Type II {0, ∞} Type III Remark Possibly N = ∞ and K = ∞. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Examples? (X, µ): measure space. G: discrete group of transformations of X (preserving the class of µ) Definition M(G, X) is the von Neumann algebra of operators on L2(G × X) generated by: ∞ Multiplication operators Mf , f ∈ L (X) Unitary translation operators Ug, g ∈ G such that ∗ UgMf Ug = Mg(f ). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Group Measure Space Construction Key Idea: The unitary operators Ug ∈ M(G, X) implement the action of G on the algebra of measurable functions on X. Theorem Assume the action of G on X is free. Then M(G, X) is a factor if and only if the action of G on X is ergodic. Theorem Assume that the action of G on X is ergodic. Then M(G, X) is of type I or II if and only if there is an invariant measure ν in the class of µ, in which case DimM(G, X) =∼ ν(E): E ⊆ X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Classification into Types C*-Algebras Group Measure Space Construction Examples X = Z, G = Z, action by translations. Type I. X = S1, G = Z, action by irrational rotations. Type II. X = S1, G = SL(2, Z), action by projective transformations of 1 ∼ 1 S = RP . Type III. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture C*-Algebras Definition A C∗-algebra is a norm-closed ∗-algebra of operators on a Hilbert space. ∗ C -algebra: Tn ∈ A & kTn − T k → 0 ⇒ T ∈ A. Von Neumann algebra: Tn ∈ M & Tnv → Tv, ⇒ T ∈ M. Abelian C∗-algebra: C(X). Abelian von Neumann algebra: L∞(X). Abelian group C∗-algebra: C∗(G) = C(Gb). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Crossed Product Algebras X: smooth manifold. G: discrete group of diffeomorphisms of X. Definition ∗ ∗ The crossed product algebra Cλ(G, X) is the C -algebra of operators on L2(G × X) generated by: Multiplication operators Mf , where f is smooth and compactly supported. Unitary translation operators Ug, g ∈ G such that ∗ UgMf Ug = Mg(f ). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Traces and Dimensions Lemma Assume that there is a G-invariant measure ν on X. The formula X Z τ : Mfg Ug 7→ fe(x) dν(x) g∈G X ∗ defines a trace on Cλ(G, X): ∗ τ(xy) = τ(yx), ∀x, y ∈ Cλ(G, X). If p and q are equivalent projections, then τ(p) = τ(q). We can ask: what is the set of numerical dimensions ∗ τ(p): p ∈ Cλ(G, X) ? Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Traces and Dimensions Improved Question Trace τ : A → C What is abelian group generated by Dimτ A = ? τ(p), where p ∈ Mn(A) Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture The Irrational Rotation C*-Algebra Example Let X = S1, and let G = Z act on the circle by rotations through multiples of the angle 2πθ, where θ is irrational. The C∗-algebra ∗ Cλ(G, X) is the irrational rotation algebra Aθ. Theorem Dimτ Aθ = Z + θZ. Theorem (Rieffel) τ(p): p ∈ A = Z + θZ ∩ [0, 1]. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Twisted Product M: Connected manifold with fundamental group G. X: Manifold with smooth action of G. Definition M#X = (Me × X)/G. Me × X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Twisted Product Me × X M#X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Ruelle-Sullivan Current M: Connected manifold with fundamental group G. X: Manifold with smooth action of G. ν: Invariant measure on X Definition n Let n = dim(M). Define Cν :Ωc (M#X) → R as follows: If ω is supported in a small set W ⊆ M#X diffeomorphic to U × V ⊆ Me × X, Z Z ! Cν(ω) = ω dν(x). V U×{x} For general ω, use a partition of unity. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Ruelle-Sullivan Current Me × X U V Local Definition of Current Projection Cν(ω) = Z Z ! W ω dν(x). V U×{x} M#X Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Conjecture on Dimensions Theorem n−1 Cν is closed: Cν(dη) = 0, for all η ∈ Ωc (M#X). n As a result, Cν defines a map Cν : Hc (M#X) → R. We are interested in values on integral cohomology classes . Conjecture (Approximate Form) ∗ Dimτ Cλ(G, X) = Cν(ω): ω closed and integral }. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Irrational Rotations Example — Irrational Rotation Cν(ω): ω closed, integral = Z + θZ Alternate Definition of Current L3 1 Z Cν(ω) = lim ω N→∞ N LN Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Some Consequences Corollary (of the Conjecture) ∗ ∗ Take X = pt, so that Cλ(G, X) = Cλ(G). Then ∗ ∗ Dimτ Cλ(G) = Z. As a result, Cλ(G) has no nontrivial projections. Corollary (of the Conjecture, due to Connes) If the group G = Z acts on X = S3 via a minimal ∗ diffeomorphism, then Dimτ Cλ(G, X) = Z. As a result, ∗ Cλ(G, X) has no nontrivial projections. Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Von Neumann Algebras, Revisited Assume G acts on (X, µ). Define an action on (X × R, µ × Lebesgue) by dg µ g :(x, t) 7→ g · x, log ∗ (x) + t. dµ Definition The module of the given action of G on X is the pair Mod(G, X) = (A, α), where ∞ A = abelian von Neumann algebra = L (X × R)G α = action of R = translation action. Theorem The module is an invariant of M(G, X). Nigel Higson Dimensions and C*-Algebras Von Neumann Algebras Dimensions and de Rham Theory C*-Algebras The Baum-Connes Conjecture Type III Factors Assuming the action of G on X is ergodic, the module, Mod(G, X), is an ergodic flow.