Von Neumann Algebras and Ergodic Theory of Group Actions

Von Neumann Algebras and Ergodic Theory of Group Actions

Von Neumann algebras and ergodic theory of group actions CEMPI Inaugural Conference Lille, September 2012 Stefaan Vaes∗ ∗ Supported by ERC Starting Grant VNALG-200749 1/21 Functional analysis Group theory I Hilbert space operators, I free groups, I Von Neumann algebras. I hyperbolic groups. Today's talk Ergodic theory I of group actions, I of equivalence relations. 2/21 Reminders : operators on a Hilbert space I Examples of bounded operators on a Hilbert space : • Operators on Cn $ n × n matrices. • Given f 2 L1(X ), we have the multiplication operator onL 2(X ) given by ξ 7! f ξ. 2 • The one-sided shift on ` (N) given by δn 7! δn+1. 2 Notation :( δn)n2N is the canonical orthonormal basis of ` (N). ∗ I Every bounded operator T has an adjoint T determined by hT ξ; ηi = hξ; T ∗ηi. ∗ I Self-adjoint operators : T = T , unitary operators : U∗U = 1 = UU∗, normal operators : T ∗T = TT ∗. 3/21 Von Neumann algebras The weak topology on B(H). Let Tn; T 2 B(H). We say that Tn ! T weakly if hTnξ; ηi ! hT ξ; ηi for all ξ; η 2 H. 2 n Example. If u 2 B(` (N)) is the one-sided shift, then u ! 0 weakly. Definition A von Neumann algebra is a weakly closed, self-adjoint algebra of operators on a Hilbert space. Examples : B(H), in particularM n(C), L1(X ) as multiplication operators onL 2(X ). Von Neumann's bicommutant theorem (1929) Let M ⊂ B(H) be a self-adjoint algebra of operators with1 2 M. TFAE I Analytic property : M is weakly closed. 00 0 I Algebraic property : M = M , where A is the commutant of A. 4/21 Group von Neumann algebras LetΓ be a countable group. I A unitary representation ofΓ on a Hilbert space H is a map π :Γ ! B(H): g 7! πg satisfying • all πg are unitary operators, • πg πh = πgh for all g; h 2 Γ. I The regular representation ofΓ is the unitary representation λ on 2 ` (Γ) given by λg δh = δgh, 2 where( δh)h2Γ is the natural orthonormal basis of ` (Γ). Definition (Murray - von Neumann, 1943) The group von Neumann algebra LΓ is defined as the weakly closed linear span of fλg j g 2 Γg. As we shall see, the relation between the groupΓ and its von Neumann algebraLΓ is extremely subtle. 5/21 The free groups The free group F2 is defined as \the group generated by a and b subject to no relations". −1 −1 I Elements of F2 are reduced words in the letters a; a ; b; b , like aba−1a−1b, or like bbbbbba−1bbbb. −1 −1 I Reduced means : no aa , no b b, ... in the word, because they \simplify". So bbaa−1a is not reduced. It reduces to bba. I Group operation : concatenation followed by reduction. Similarly : the free group Fn with n = 2; 3; ::: free generators. Big open problem (Murray - von Neumann, 1943) Are the free group factorsL Fn isomorphic for different n = 2; 3; ::: ? (Voiculescu 1990, Radulescu 1993) They are either all isomorphic, or all non-isomorphic. 6/21 II1 factors I A factor is a von Neumann algebra M with trivial center. Equivalently : M is not the direct sum of two von Neumann algebras. I A II1 factor is a factor M that admits a trace, i.e. a linear functional τ : M ! C such that τ(xy) = τ(yx) for all x; y 2 M. In a way, all von Neumann algebras can be assembled from II1 factors. The group von Neumann algebraLΓ is a factor iffΓ has infinite conjugacy classes : for every g 6= e, the set fhgh−1 j h 2 Γg is infinite. LΓ always has a trace, namely τ(x) = hxδe ; δe i, satisfying τ(λg ) = 0 for all g 6= e and τ(1) = τ(λe ) = 1. All icc groupsΓ give us II 1 factors LΓ, but their structure is largely non-understood. 7/21 Amenable groups and Connes's theorem Definition (von Neumann, 1929) A groupΓ is called amenable if there exists a finitely additive probability m on all the subsets ofΓ that is translation invariant : m(gU) = m(U) for all g 2 Γ and U ⊂ Γ. Non-amenable groups $ Banach-Tarski paradox. Theorem (Connes, 1976) All amenable groupsΓ with infinite conjugacy classes have isomorphic group von Neumann algebras LΓ. Actually : there is a unique \amenable" II1 factor. The following groups are amenable. • Abelian groups, solvable groups. • Stable under subgroups, extensions, direct limits. 8/21 Connes's rigidity conjecture I Open problem : are the group von Neumann algebras L(SL(n; Z)) for n = 3; 4; :::, non-isomorphic ? I Conjecturally, they are all non-isomorphic. I The groups SL(n; Z) with n = 3; 4; :::, have Kazhdan's property (T). Conjecture (Connes, 1980) LetΓ ; Λ be icc groups with Kazhdan's property (T). ThenLΓ =∼ LΛ if and only ifΓ =∼ Λ. 9/21 W∗-superrigidity for groups Theorem (Ioana-Popa-V, 2010) There are countable groups G such thatL G entirely remembers G : ifΛ is an arbitrary countable group withL G =∼ LΛ, then G =∼ Λ. (I ) These groups are of the form G = (Z=2Z) o Γ. Theorem (Berbec-V, 2012) (Γ) The same is true for G = (Z=2Z) o (Γ × Γ), for many groupsΓ, including the free groups and arbitrary free product groupsΓ=Γ 1 ∗ Γ2. 10/21 Group measure space version of the free group factor problem Recall : it is wide open whether theL Fn with n = 2; 3; :::, are isomorphic. Theorem (Popa-V, 2011) 1 The group measure space II1 factorsL (X ) o Fn, arising from free ergodic probability measure preserving actions Fn y X , are non-isomorphic for different values of n = 2; 3; :::, independently of the actions. GroupΓ: von Neumann algebraLΓ. 1 Group actionΓ y X : von Neumann algebraL (X ) o Γ. We will explain all these concepts, and some ideas behind the theorem. 11/21 Probability measure preserving group actions We study actions of countable groupsΓ on probability spaces( X ; µ) by • measurable transformations, • that preserve the measure µ. We callΓ y (X ; µ)a pmp action. Favorite examples : I Irrational rotation Z y T given by n · z = exp(2πiαn) z for a fixed α 2 R n Q. Γ I Bernoulli action Γ y (X0; µ0) given by( g · x)h = xhg . n n n I The action SL(n; Z) y T = R =Z . I The actionΓ y G=Λ for lattices Γ; Λ < G in a Lie group G. 12/21 Group measure space construction (MvN, 1943) Data : a pmp actionΓ y (X ; µ). 1 Output : von Neumann algebra M = L (X ) o Γ with trace τ : M ! C. I Notations : g 2 Γ acts on x 2 X as g · x. −1 Then, g acts on a function ξ : X ! C as( g · ξ)(x) = ξ(g · x). 1 1 I L (X ) o Γ is generated by unitaries( ug )g2Γ and a copy ofL (X ). I We have the algebraic relations (cf. semidirect products of groups) ∗ • ug uh = ugh and ug = ug −1 , ∗ • ug F ug = g · F . R I Trace : τ(F ) = F dµ and τ(F ug ) = 0 for g 6= e. 13/21 Essential freeness and ergodicity We do not want trivial actions. I Γ y (X ; µ) is called (essentially) free if for all g 6= e and almost every x 2 X , we have that g · x 6= x. 1 1 I Equivalently,L (X ) is a maximal abelian subalgebra ofL (X ) o Γ. We do not want the union of two actions. I Γ y (X ; µ) is called ergodic ifΓ-invariant subsets of X have measure 0 or 1. 1 IfΓ y (X ; µ) is free and ergodic, thenL (X ) o Γ is a II1 factor. 14/21 Actions of free groups Theorem (Popa-V, 2011) If Fn y X and Fm y Y are free ergodic pmp actions with n 6= m, then 1 ∼ 1 L (X ) o Fn =6 L (Y ) o Fm. Approach to this theorem : 1 1 I Special role ofL (X ) ⊂ L (X ) o Γ: a Cartan subalgebra. I Uniqueness problem of Cartan subalgebras. I Countable pmp equivalence relations. I Gaboriau's work on orbit equivalence relations for free groups. 15/21 Cartan subalgebras Definition A Cartan subalgebra A of a II1 factor M is 0 I a maximal abelian subalgebra : A \ M = A, ∗ I such that the normalizer NM (A) := fu 2 U(M) j uAu = Ag spans a weakly dense subalgebra of M. 1 1 Typical example : L (X ) is a Cartan subalgebra ofL (X ) o Γ whenever Γ y X is a free, ergodic, pmp action. Main questions, given a II1 factor M : I Does M have a Cartan subalgebra ? I If yes, is it unique ? • up to unitary conjugacy : A = uBu∗ for some u 2 U(M), • up to automorphic conjugacy : A = α(B) for some α 2 Aut(M). 16/21 Non-uniqueness and non-existene of Cartan subalgebras Connes-Jones, 1981 : II1 factors with at least two Cartan subalgebras that are non-conjugate by an automorphism. Uniqueness of Cartan subalgebras seemed hopeless. Voiculescu, 1995 : LFn,2 ≤ n ≤ 1, has no Cartan subalgebra. 1 2 2 Ozawa-Popa, 2007 : the II1 factor M = L (Zp) o (Z o SL(2; Z)) has 1 2 2 two non-conjugate Cartan subalgebras, namelyL (Zp) andL( Z ). Speelman-V, 2011 : II1 factors with many Cartan subalgebras (where \many" has a descriptive set theory meaning). 17/21 Uniqueness of Cartan subalgebras Theorem (Popa-V, 2011) 1 Let Fn y X be an arbitrary free ergodic pmp action.

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