Von Neumann algebras, countable groups and ergodic theory Workshop - PowerPoint PPT Presentation

Von Neumann algebras, countable groups and ergodic theory Workshop Young Researchers in Mathematics Madrid, 2011 Stefaan Vaes ∗ ∗ Supported by ERC Starting Grant VNALG-200749 1/20

Von Neumann algebras Group theory Today’s talk Group actions on probability spaces 2/20

Abstract von Neumann algebras The space B( H ) of bounded operators on a Hilbert space admits several topologies. ◮ Norm topology given by � T � = sup { � T ξ � | � ξ � ≤ 1 } . ◮ Weak topology where T i → T iff � T i ξ, η � → � T ξ, η � for all ξ, η ∈ H . The commutant M ′ of M ⊂ B( H ) is defined as M ′ := { T ∈ B( H ) | ∀ S ∈ M : ST = TS } Observation. We have M ⊂ M ′′ and M ′ , M ′′ are weakly closed. Von Neumann’s bicommutant theorem (1929) Let M ⊂ B( H ) be a ∗ -algebra of operators with 1 ∈ M . TFAE • M is weakly closed. • M = M ′′ . Von Neumann algebras 3/20

Concrete von Neumann algebras Group von Neumann algebras Let Γ be a countable group. Consider the Hilbert space ℓ 2 (Γ) and the unitary operators ( u g ) g ∈ Γ given by u g δ h = δ gh . These operators satisfy u g u h = u gh . Define LΓ as the weakly closed linear span of { u g | g ∈ Γ } . Crossed product von Neumann algebras Let Γ � ( X , µ ) be an action of a countable group by non-singular transformations of a measure space. The crossed product L ∞ ( X ) ⋊ Γ is generated by L ∞ ( X ) and unitary operators ( u g ) g ∈ Γ satisfying u g u h = u gh and u ∗ g F ( · ) u g = F ( g · ). Central problem: Classify LΓ and L ∞ ( X ) ⋊ Γ in terms of the group (action) data. Huge progress by Popa’s deformation/rigidity theory, 2001- · · · 4/20

II 1 factors Simple von Neumann algebras: those that cannot be written as a direct sum of two. We call them factors. Murray - von Neumann classification of factors: types I, II and III. II 1 factors M are those factors that admit a trace τ : M → C : τ ( xy ) = τ ( yx ). Arbitrary von Neumann algebras can be ‘assembled’ from II 1 factors (Connes, Connes & Takesaki). Von Neumann algebras coming from groups and group actions ◮ LΓ is a II 1 factor if and only if Γ has infinite conjugacy classes (icc). ◮ L ∞ ( X ) ⋊ Γ is a II 1 factor if Γ � ( X , µ ) is • free : for all g � = e and almost every x ∈ X we have g · x � = x , • ergodic : Γ-invariant subsets have measure 0 or 1, • probability measure preserving (pmp). 5/20

Favorite free ergodic pmp actions ◮ Irrational rotation Z � T given by n · z = exp(2 π i α n ) z for a fixed irrational number α ∈ R \ Q . ◮ Bernoulli action Γ � ( X 0 , µ 0 ) Γ given by ( g · x ) h = x hg . ◮ The action SL( n , Z ) � T n = R n / Z n . They give all rise to II 1 factors L ∞ ( X ) ⋊ Γ. 6/20

Amenability and softness of II 1 factors Definition (von Neumann) A countable group Γ is amenable if there exists a finitely additive probability measure m on the subsets of Γ such that m ( gA ) = m ( A ) for all g ∈ Γ , A ⊂ Γ . Equivalent condition. Existence of a sequence of finite subsets A n ⊂ Γ such that for all g ∈ Γ we have | g · A n △ A n | → 0 . | A n | First example. The group Z with A n = {− n , . . . , n } . Counterexample. The free group F 2 is non-amenable. This explains the Banach-Tarski paradox. Further examples ◮ Abelian groups, solvable groups. ◮ Closed under extensions, subgroups, direct limits. 7/20

Amenability and softness of II 1 factors Theorem (Connes, 1975) All the following II 1 factors are isomorphic. ◮ Group von Neumann algebras LΓ for Γ amenable and icc. ◮ All L ∞ ( X ) ⋊ Γ for Γ amenable and Γ � ( X , µ ) free ergodic pmp. Another characterization of amenability. ◮ Given a unitary representation π : Γ → U ( H ), we say that ξ n ∈ H , � ξ n � = 1, is a sequence of almost invariant vectors if � π ( g ) ξ n − ξ n � → 0 for all g ∈ Γ. ◮ A group Γ is amenable iff the regular representation on ℓ 2 Γ given by π ( g ) δ h = δ gh admits a sequence of almost invariant vectors. To retrieve Γ or the action Γ � X from LΓ or L ∞ ( X ) ⋊ Γ we have to move far away from amenability. 8/20

Rigidity : Kazhdan’s property (T) ◮ A group Γ has property (T) if every unitary representation with almost invariant vectors, has a non-zero invariant vector. Ex. SL( n , Z ) for n ≥ 3, lattices in higher rank simple Lie groups. ◮ A subgroup Λ < Γ has relative property (T) if every unitary rep. of Γ with almost invariant vectors, has a non-zero Λ-invariant vector. Example: Z 2 < SL(2 , Z ) ⋉ Z 2 . Illustration Connes’ rigidity conjecture (1980). If G and Λ are icc property (T) groups and L G ∼ = LΛ, then G ∼ = Λ. Theorem (Popa, 2006). The conjecture holds up to countable classes. If G is an icc property (T) group, there are at most countably many non isomorphic groups Λ with L G ∼ = LΛ. 9/20

Rigidity for crossed product II 1 factors Free ergodic pmp action Γ � ( X , µ ) Orbit equivalence relation (Dye, 1958) : x ∼ y iff Γ · x = Γ · y . II 1 factor L ∞ ( X ) ⋊ Γ. Definition Two actions Γ � ( X , µ ) and Λ � ( Y , η ) are called • conjugate, if there exist ∆ : X → Y and δ : Γ → Λ s.t. ∆( g · x ) = δ ( g ) · ∆( x ). • orbit equivalent, if there exist ∆ : X → Y s.t. ∆(Γ · x ) = Λ · ∆( x ). • W*-equivalent, if L ∞ ( X ) ⋊ Γ ∼ = L ∞ ( Y ) ⋊ Λ. ◮ Obviously, conjugacy implies orbit equivalence. ◮ Singer (1955): an orbit equivalence amounts to a W ∗ -equivalence mapping L ∞ ( X ) onto L ∞ ( Y ). ◮ Rigidity: prove W*-equivalence ⇒ OE ⇒ conjugacy! 10/20

W*-superrigidity Popa’s seminal “strong rigidity” theorem (2004) Let Γ be a property (T) group and Γ � ( X , µ ) a free ergodic pmp action. Let Λ be an icc group and Λ � ( Y , η ) = ( Y 0 , η 0 ) Λ its Bernoulli action. If L ∞ ( X ) ⋊ Γ ∼ = L ∞ ( Y ) ⋊ Λ, then the groups Γ and Λ are isomorphic and their actions conjugate. First theorem ever deducing conjugacy out of isomorphism of II 1 factors. Definition A free ergodic pmp action Γ � ( X , µ ) is called W ∗ -superrigid if any W ∗ -equivalent action must be conjugate. In other words: L ∞ ( X ) ⋊ Γ remembers the group action. Compare: the assumptions in Popa’s theorem are asymmetric. 11/20

First W ∗ -superrigidity theorem Theorem (Popa-V, 2009) For a large family of amalgamated free product groups Γ = Γ 1 ∗ Σ Γ 2 the Bernoulli action Γ � ( X 0 , µ 0 ) Γ is W ∗ -superrigid. • Concrete examples: Γ = SL(3 , Z ) ∗ T 3 ( T 3 × Λ) with T 3 the upper triangular matrices and Λ � = { e } arbitrary. • Theorem covers more general families of group actions. Theorem (Houdayer - Popa - V, 2010) using Kida’s work. All free ergodic pmp actions of SL(3 , Z ) ∗ Σ SL(3 , Z ) are W ∗ -superrigid, with Σ < SL(3 , Z ) the subgroup of matrices x with x 31 = x 32 = 0. • Peterson (2009) proved existence of virtually W ∗ -superrigid actions. 12/20

How to establish W ∗ -superrigidity for Γ � (X , µ ) W ∗ -superrigidity of Γ � ( X , µ ) arises as the sum of the following two properties. Put M = L ∞ ( X ) ⋊ Γ. Uniqueness of the group measure space Cartan subalgebra Whenever M = L ∞ ( Y ) ⋊ Λ is another crossed product decomposition, the subalgebras L ∞ ( X ) and L ∞ ( Y ) should be unitarily conjugate : there should exist a unitary u ∈ M such that u L ∞ ( X ) u ∗ = L ∞ ( Y ). Orbit equivalence superrigidity Any action orbit equivalent with Γ � X should be conjugate with Γ � X . ◮ Zimmer, Furman, Monod-Shalom, Popa, Ioana, Kida, ... ◮ For several Γ the Bernoulli action is OE superrigid (Popa). Both parts are very hard. Even more difficult: both together for the same group action. 13/20

Uniqueness of Cartan subalgebras Definition A Cartan subalgebra A of a II 1 factor M is a maximal abelian subalgebra such that { u ∈ U ( M ) | uAu ∗ = A } generates M (i.e. its linear span is weakly dense in M ). Example : L ∞ ( X ) ⊂ L ∞ ( X ) ⋊ Γ, which we call a group measure space Cartan subalgebra. Not all Cartan subalgebras arise like this. Theorem (Popa-V, 2009) Let Γ = Γ 1 ∗ Γ 2 be the free product of an infinite property (T) group Γ 1 and a non-trivial group Γ 2 . Then L ∞ ( X ) ⋊ Γ has unique group measure space Cartan subalgebra for arbitrary free ergodic pmp Γ � ( X , µ ). ◮ Open problem: uniqueness of arbitrary Cartan subalgebras? ◮ Generalizations by Chifan-Peterson (2010) and V (2010) : groups Γ satisfying a cohomological condition. 14/20

Uniqueness of Cartan subalgebras Theorem (Ozawa-Popa, 2007) Let F n � ( X , µ ) be a free ergodic profinite action, n ≥ 2. Then, L ∞ ( X ) ⋊ F n has a unique Cartan subalgebra up to unitary conjugacy. ◮ Profinite action : Γ � lim − Γ / Γ n with [Γ : Γ n ] < ∞ . ← ◮ (Chifan-Sinclair, 2011) Theorem holds for profinite actions of hyperbolic groups. ◮ All known uniqueness theorems for Cartan subalgebras are restricted to profinite actions ! 15/20