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

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Von Neumann algebras Group theory Group actions

• n probability spaces

Today’s talk

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Abstract von Neumann algebras

The space B(H) of bounded operators on a Hilbert space admits several topologies.

◮ Norm topology given by T = sup{ Tξ | ξ ≤ 1 }. ◮ Weak topology where Ti → T iff Tiξ, η → Tξ, 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

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Concrete von Neumann algebras

Group von Neumann algebras Let Γ be a countable group. Consider the Hilbert space ℓ2(Γ) and the unitary operators (ug)g∈Γ given by ugδh = δgh. These operators satisfy uguh = ugh. Define LΓ as the weakly closed linear span of { ug | 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

• perators (ug)g∈Γ satisfying uguh = ugh and u∗

gF( · )ug = 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-· · ·

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II1 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. II1 factors M are those factors that admit a trace τ : M → C : τ(xy) = τ(yx). Arbitrary von Neumann algebras can be ‘assembled’ from II1 factors (Connes, Connes & Takesaki). Von Neumann algebras coming from groups and group actions

◮ LΓ is a II1 factor if and only if Γ has infinite conjugacy classes (icc). ◮ L∞(X) ⋊ Γ is a II1 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).

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Favorite free ergodic pmp actions

◮ Irrational rotation Z T given by n · z = exp(2πiαn) z for a fixed

irrational number α ∈ R \ Q.

◮ Bernoulli action Γ (X0, µ0)Γ given by (g · x)h = xhg. ◮ The action SL(n, Z) Tn = Rn/Zn.

They give all rise to II1 factors L∞(X) ⋊ Γ.

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Amenability and softness of II1 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 An ⊂ Γ such that for all g ∈ Γ we have |g · An △ An| |An| → 0 . First example. The group Z with An = {−n, . . . , n}.

• Counterexample. The free group F2 is non-amenable. This explains the

Banach-Tarski paradox. Further examples

◮ Abelian groups, solvable groups. ◮ Closed under extensions, subgroups, direct limits.

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Amenability and softness of II1 factors

Theorem (Connes, 1975) All the following II1 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 ξn ∈ H,

ξn = 1, is a sequence of almost invariant vectors if π(g)ξn − ξn → 0 for all g ∈ Γ.

◮ A group Γ is amenable iff the regular representation on ℓ2Γ given by

π(g)δh = δgh 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.

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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: Z2 < SL(2, Z) ⋉ Z2. Illustration Connes’ rigidity conjecture (1980). If G and Λ are icc property (T) groups and LG ∼ = 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 LG ∼ = LΛ.

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Rigidity for crossed product II1 factors

Free ergodic pmp action Γ (X, µ) Orbit equivalence relation (Dye, 1958) : x ∼ y iff Γ · x = Γ · y . II1 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!

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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) = (Y0, η0)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 II1 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.

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First W∗-superrigidity theorem

Theorem (Popa-V, 2009) For a large family of amalgamated free product groups Γ = Γ1 ∗Σ Γ2 the Bernoulli action Γ (X0, µ0)Γ is W∗-superrigid.

• Concrete examples: Γ = SL(3, Z) ∗T3 (T3 × Λ) with T3 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 x31 = x32 = 0.

• Peterson (2009) proved existence of virtually W∗-superrigid actions.

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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 uL∞(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.

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Uniqueness of Cartan subalgebras

Definition A Cartan subalgebra A of a II1 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 = Γ1 ∗ Γ2 be the free product of an infinite property (T) group Γ1 and a non-trivial group Γ2. 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.

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Uniqueness of Cartan subalgebras

Theorem (Ozawa-Popa, 2007) Let Fn (X, µ) be a free ergodic profinite action, n ≥ 2. Then, L∞(X) ⋊ Fn has a unique Cartan subalgebra up to unitary conjugacy.

◮ Profinite action : Γ lim

← − Γ/Γn with [Γ : Γn] < ∞.

◮ (Chifan-Sinclair, 2011) Theorem holds for profinite actions of

hyperbolic groups.

◮ All known uniqueness theorems for Cartan subalgebras are restricted

to profinite actions !

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Uniqueness of Cartan subalgebras: open problems

Open problem Does L∞(X) ⋊ Fn have a unique Cartan subalgebra for arbitrary free ergodic pmp actions Fn (X, µ) ?

• Answer is conjecturally yes.
• Maybe even for all groups having non-zero first L2-Betti number.
• (Ioana, 2011) Uniqueness of group measure space Cartan for rigid

actions of groups with non-zero first L2-Betti number. Open problem Let Γ (X, µ) = (X0, µ0)Γ be the Bernoulli action of a non-amenable group Γ. Does L∞(X) ⋊ Γ have a unique Cartan subalgebra ? Answer is again conjecturally yes. There is even no counterexample when Γ (X, µ) is a mixing action of any non-amenable group Γ. Ioana (2010) Bernoulli actions Γ X Γ

0 of icc property (T)

groups are W∗-superrigid.

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Groups and algebras

Countable group G a variety of algebras, like CG, C∗

r G, LG,

how does the isomorphism class depend on G ? The group algebra CG acts on ℓ2G by left convolution operators.

◮ The C∗-algebra C∗ r G is the norm closure of CG. ◮ The von Neumann algebra LG is the weak closure of CG.

General principle In the passage from CG to LG the memory of G tends to fade away. Illustration for torsion free abelian groups

◮ C∗ r G remembers G as the group of connected components of U(C∗ r G).

Indeed, G = Zn1 ֒ → Zn2 ֒ → · · · .

◮ All LG are the same diffuse abelian von Neumann algebra.

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C∗ versus von Neumann

Both CG and C∗

r G tend to remember G: ◮ Higman’s conjecture: if G is torsion free, the only invertible

elements in CG are the multiples of elements of G. (Proven for orderable groups. Implies Kaplansky’s conjecture.)

◮ There are no examples of torsion free G ∼

= Λ with C∗

r G ∼

= C∗

r Λ.

Group von Neumann algebras LG are very flexible:

◮ (Connes, 1976) All LG for G icc and amenable, are isomorphic. ◮ (Dykema, 1993) If n ≥ 2 and Γ1, . . . , Γn infinite amenable, then

L(Γ1 ∗ · · · ∗ Γn) ∼ = LFn.

◮ (Ioana, 2006) The L(Fn ≀ Z), n ≥ 2, are isomorphic.

Recall: H ≀ Γ = H(Γ) ⋊ Γ.

◮ (Bowen, 2009) The L(H ≀ F2), H non-trivial abelian, are isomorphic.

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The big open problems

◮ Are the free group factors LFn, n ≥ 2, isomorphic? ◮ Connes’ rigidity conjecture: if G and Λ are icc property (T) groups

and LG ∼ = LΛ, then G ∼ = Λ.

◮ Are the L(SL(n, Z)), n ≥ 3, isomorphic?

Note: Connes’ rigidity conjecture would imply that the LG for G icc property (T), remember the group G. Indeed, whenever LG ∼ = LΛ, the group Λ must be icc property (T). W∗-superrigidity of group von Neumann algebras (Ioana-Popa-V, ’10) We prove the first W∗-superrigidity theorem for certain group von Neumann algebras LG : whenever Λ is a group and LG ∼ = LΛ, one must have G ∼ = Λ.

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W∗-superrigidity theorem

Ioana-Popa-V, 2010 Let Γ0 be any non-amenable group. Consider the wreath product group G0 := Γ0 ≀ Z := Γ(Z) ⋊ Z and the generalized wreath product G = Z 2Z (I) ⋊ G0 where I = G0/Z = Γ(Z)

0 .

If Λ is any group such that LΛ ∼ = LG, then Λ ∼ = G.

◮ We can actually treat a wider class of generalized wreath product

groups (Z/nZ)(I) ⋊ Γ. Plain wreath products never work though, because... (IPV 2010) Let Γ be any torsion-free group and H0 any non-trivial finite abelian group. There exists a torsion-free group Λ such that LΛ ∼ = L(H0 ≀ Γ). In particular, Λ ∼ = H0 ≀ Γ.

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