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.