Infinite groups, actions on the interval [0,1] and von Neumann - - PowerPoint PPT Presentation

Infinite groups, actions on the interval [0,1] and von Neumann algebras

Algemeen Wiskundecolloquium Radboud Universiteit Nijmegen, March 2009 Stefaan Vaes

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What are groups ?

◮ Abstract definition. ◮ Groups appear as symmetries of mathematical structures. ◮ Actions of groups on ...

Examples (biased towards the topic of this talk) :

◮ Action of Z on the circle, where n ∈ Z acts by rotation of the

circle over the angle nα. Tn : S1 → S1 : Tn(z) = exp(inα)z . From now on, we write n · z for the action of n ∈ Z on z ∈ S1.

◮ Action of SL(2, Z) on the torus S1 × S1 = R2/Z2.

• a

b c d

• ·
• y

z

• =
• yazb

yczd

• .

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A crash course in measure theory

What do the examples have in common.

◮ On S1 we have a notion of length and on S1 × S1 a notion

• f area. Our transformations preserve this length/area.

◮ Both S1 and S1 × S1 are probability spaces : a set with a family

• f measurable subsets and a measure thereon.

New notion : a measurable map is such that the inverse image of a measurable set is again measurable. Never mind, in practice every map that you can write down is measurable. Other notions : measure zero and ‘almost everywhere’.

◮ Both Z ↷ S1 and SL(2, Z) ↷ S1 × S1 are

actions of groups by measurable transformations preserving the probability measure.

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More examples of probability measure preserving group actions

• 1. Let K be a compact group (e.g. K = U(n), O(n), S1).

◮ K has a unique K-invariant probability measure,

called Haar measure.

◮ Choose Γ ⊂ K countable and dense. ◮ Then, Γ acts on K by translation, preserving the Haar measure.

• 2. Let Γ be any countable group.

◮ Build the probability space

(X, µ) = [0, 1]Γ =

• Γ

[0, 1] with its product measure.

◮ The measure of a ‘rectangle’ is what you expect. ◮ Then, Γ ↷ (X, µ) by (g · x)h = xg−1h .

This is called the Bernoulli action of Γ.

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The assumption of ergodicity

We impose a ‘minimality’ or ‘simplicity’ assumption : we look at Γ ↷ (X, µ) that cannot be decomposed ‘as the sum of two’, meaning X = X1⊔X2 where both X1, X2 are Γ-invariant. However : we do not bother if X1 or X2 has measure zero. Formal definition of ergodicity The probability measure preserving action Γ ↷ (X, µ) is called ergodic, if every globally Γ-invariant measurable subset Y ⊂ X has measure 0 or measure 1.

◮ Equivalent condition :

Every Γ-invariant function X → C is constant almost everywhere.

◮ We prove that Z ↷ S1 : n · z = exp(inα)z is ergodic

if and only if α/2π is irrational.

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Ergodicity of irrational rotation

Recall : Z ↷ S1 : n · z = exp(inα)z . Proof. Let F : S1 → C be a bounded measurable Z-invariant function. This means that F

• exp(inα)z
• = F(z) for all n ∈ Z, z ∈ S1.

◮ The Fourier coefficients satisfy

ˆ

F(n) = exp(inα)ˆ F(n) .

◮ For α irrational, we get ˆ

F(n) = 0 when n ≠ 0.

◮ So, ˆ

F(n) = ˆ G(n) for some constant function G.

◮ Hence, F(z) = G(z) = constant for almost all z ∈ S1.

By the way : construct yourself, for α rational, Z-invariant functions.

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Hilbert spaces

• Complex vector space H.
• Positive-definite scalar product H × H → C : (ξ, η) ֏ ξ, η .
• Completeness for the norm ξ =
• ξ, ξ.

Examples of Hilbert spaces

◮ Cn with ξ, η = n

• k=1

ξkηk and ℓ2(N) with ξ, η =

∞

• k=0

ξkηk .

◮ L2(S1) with ξ, η =

1 2π 2π ξ(exp(it))η(exp(it)) dt . New notions : Operator = linear map from Hilbert space to Hilbert space. Unitary operator = bijective operator preserving scalar product.

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Unitary representations of groups

A unitary representation π of a group Γ on a Hilbert space H is

◮ a map π : Γ → unitary operators on H, ◮ such that π(gh) = π(g)π(h) and π(e) = 1.

Again a natural appearance of groups : as symmetries of a Hilbert space. Regular representation of Γ Γ Γ

• Hilbert space is ℓ2(Γ) , with orthonormal basis (eg)g∈Γ ,
• Representation g ֏ λg where λgeh = egh .

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Amenability of groups

Let π : Γ → unitaries on H be a unitary representation of Γ. A sequence of almost invariant vectors is a sequence ξn of norm one vectors in H satisfying π(g)ξn − ξn → 0 for all g ∈ Γ .

• Example. The ξn =

1 √ 2n + 1

n

• k=−n

ek form a sequence of almost invariant vectors for the regular representation of Z. Definition The group Γ is called amenable if its regular representation admits a sequence of almost invariant vectors. Examples of amenable groups.

◮ Abelian groups, solvable groups. ◮ Closed under extensions, subgroups and direct limits. ◮ Open problem: is the group generated by a, b subject to the

relation that ab commutes with aba−1 and with a2ba−2, amenable (Thompson’s group F).

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Amenability of groups

Let π : Γ → unitaries on H be a unitary representation of Γ. A sequence of almost invariant vectors is a sequence ξn of norm one vectors in H satisfying π(g)ξn − ξn → 0 for all g ∈ Γ .

• Example. The ξn =

1 √ 2n + 1

n

• k=−n

ek form a sequence of almost invariant vectors for the regular representation of Z. Definition The group Γ is called amenable if its regular representation admits a sequence of almost invariant vectors. Examples of amenable groups.

◮ Abelian groups, solvable groups. ◮ Closed under extensions, subgroups and direct limits. ◮ Open problem: is the group generated by a, b subject to the

relation that ab commutes with aba−1 and with a2ba−2, amenable (Thompson’s group F). Theorem (Akhmedov, February 23, 2009) Thompson’s group F is non-amenable.

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Von Neumann algebras

The most uninteresting examples : Mn(C) and L∞(X). Notations.

◮ B(H) denotes the ∗-algebra of bounded operators on the

Hilbert space H. If H = Cn , then B(H) = Mn(C).

◮ The adjoint ∗ : ξ, Tη = T ∗ξ, η. ◮ One realizes L∞(X) ⊂ B(L2(X)) as multiplication operators. ◮ The maps T ֏ ξ, Tη induce the weak topology on B(H).

Von Neumann algebra : weakly closed ∗-subalgebra of B(H). Group von Neumann algebra Let Γ be a countable group and g ֏ λg its regular rep. on ℓ2(Γ).

◮ span{λg | g ∈ Γ} is the group algebra CΓ. ◮ Define L(Γ) ⊂ B(ℓ2(Γ)) as the weak closure of CΓ.

The L(Γ) are our first interesting von Neumann algebras.

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Classification of von Neumann algebras

Factor M : von Neumann algebra indecomposable ‘as a sum of two’. Equivalent condition : the center of M is trivial. Replaces the ergodicity assumption. Type classification for a factor M (Murray and von Neumann)

◮ Type I, if M ≅ B(H), ◮ Type II1, if M admits a finite trace :

τ : M → C , τ(xy) = τ(yx) , τ(1) = 1 . replaces ‘preserving a probability measure’.

◮ Type II∞, if M admits an infinite trace

replaces ‘preserving an infinite measure’.

◮ Type III, if M does not admit a trace.

• Example. The L(Γ) always admit a finite trace and are factorial if

and only if Γ has infinite conjugacy classes (ICC) : a lot of II1 factors.

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Amenability of von Neumann algebras

Connes : full classification of amenable von Neumann algebras. Definition A von Neumann algebra M ⊂ B(H) is called amenable if ...

◮ Example : L(Γ) is amenable iff Γ is amenable.

Theorem (Connes, 1975) All amenable II1 factors are isomorphic. In particular, all L(Γ) for Γ amenable and ICC are isomorphic ! There is also uniqueness of amenable factors of type II∞ and ... type IIIλ (0 ≤ λ ≤ 1). Finer classification into types IIIλ is the other seminal work of Connes from the 1970’s.

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Back to group actions

We assumed so far : ergodic, prob. measure preserving actions. We add one more condition : freeness. The probability measure preserving action Γ ↷ (X, µ) is called free, if almost every point x ∈ X has trivial stabilizer. If Γ ↷ (X, µ) = [0, 1]Γ is the Bernoulli action, certain x ∈ X have non-trivial stabilizer, but these x’s form a set of measure 0. Group measure space construction (Murray, von Neumann, 1943) Let Γ ↷ (X, µ) be free, ergodic, probability measure preserving. The II1 factor L∞(X) ⋊ Γ is generated by

◮ the subalgebra L∞(X), ◮ the subalgebra L(Γ) ∋ λg,

and, for F ∈ L∞(X) and g ∈ Γ, λ∗

g F λg = Fg with Fg(x) = F(g · x).

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Tons of II1 factors ... or maybe not

ICC group Γ Free ergodic p.m.p. Γ ↷ (X, µ) II1 factor L(Γ) L∞(X) ⋊ Γ Note : this was all known to Murray and von Neumann. But : they only knew two non-isomorphic II1 factors !

◮ L(S∞) where S∞ =

• n Sn and Sn = symmetric group,

This II1 factor has approximately central elements.

◮ L(F2) where F2 = free group on 2 generators.

This II1 factor has no approximately central elements. Connes’ theorem : all amenable data lead to the same II1 factor. Open problem. Is L(Fn) ≅ L(Fm) for n ≠ m ?

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Away from amenability, as far as possible

Recall : Γ is called amenable if the regular representation on ℓ2(Γ) admits a sequence of almost invariant vectors. Sequence of almost invariant vectors : π(g)ξn − ξn → 0, ∀g ∈ Γ. Definition The countable group Γ has Kazhdan’s property (T) if every unitary representation of Γ admitting a sequence of almost invariant vectors, actually has a non-zero invariant vector.

• Observe. Property (T) and amenability = finiteness.

Examples.

◮ SL(n, Z) for n ≥ 3, lattices in higher rank simple Lie groups, ◮ certain random groups `

a la Gromov.

◮ Open problem : does Aut(Fn) have property (T) ?

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Applications of property (T)

• A countable property (T) group is finitely generated and has

finite abelianization. So, this holds in particular for lattices in higher rank simple Lie groups : Kazhdan’s original motivation.

• Margulis’ explicit construction of a family of expanding graphs.
• Uniqueness of finitely additive rotation invariant measure on

the n-sphere (Rosenblatt, Margulis, Sullivan).

• Von Neumann algebra theory : see the rest of this lecture.

Connes’ rigidity conjecture If Γ is an ICC property (T) group and L(Γ) ≅ L(Λ), then Γ ≅ Λ (at least virtually). Theorem (Popa). Connes’ rigidity conjecture holds up to countable

• classes. (and this is a non-empty statement since Gromov constructed

uncountably many property (T) groups)

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Popa’s von Neumann rigidity theorem

Recall : Γ ↷ (X, µ) free ergodic p.m.p. II1 factor L∞(X) ⋊ Γ. Definition : Γ ↷ (X, µ) and Λ ↷ (Y, η) are called conjugate if

◮ there exists a measure preserving bijection ∆ : X → Y, ◮ and there exists a group isomorphism δ : Γ → Λ

such that ∆(g · x) = δ(g) · ∆(x) almost everywhere. Recall the Bernoulli action Γ ↷ [0, 1]Γ . Theorem (Popa, 2005) Let Γ be a property (T) group and Γ ↷ (X, µ) free ergodic. Let Λ be an ICC group and Λ ↷ (Y, η) = [0, 1]Λ Bernoulli action. If L∞(X) ⋊ Γ ≅ L∞(Y) ⋉ Λ, then the groups Γ and Λ are isomorphic and their actions conjugate.

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Invariants of II1 factors

Since it is extremely hard to distinguish between II1 factors, why not introduce a few invariants that do the job. Murray and von Neumann (1943).

◮ A qualitative invariant called property (Γ) : they can distinguish

between L(S∞) and L(F2).

◮ A quantitative invariant called fundamental group : subgroup of

R+ that we define now. Projection p in a von Neumann algebra M : p2 = p and p = p∗. pMp is again a II1 factor. Definition of the fundamental group Let M be a II1 factor. Set F(M) = { τ(p)/τ(q) | pMp ≅ qMq } . Then, F(M) ⊂ R+ is a subgroup, called fundamental group of M.

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Examples of fundamental groups

1 We have F(L(S∞)) = R+. (Murray, von Neumann, 1943) 2 If Γ is ICC property (T), F(L(Γ)) is countable. (Connes, 1980) 3 We have F(L(F∞)) = R+. (R˘

adulescu, 1992)

Theorem (Voiculescu, 1989). If p ∈ L(Fn) has trace 1/k, then p L(Fn)p ≅ L(F(n−1)k2+1). Note the resemblance : If [Fn : G] = k, then G ≅ F(n−1)k+1.

4 Either L(Fn) ≅ L(F∞) for all n and the fundamental group is R+,

• r all the L(Fn), 2 ≤ n < ∞, are non-isomorphic and the

fundamental group is {1}. (Dykema, R˘

adulescu, 1992)

5 We have F(L∞(T2) ⋊ SL(2, Z)) = {1}. (Popa, 2001) 6 All countable subgroups of R+ arise as F(M). (Popa, 2003)

Open question : Can F(M) be uncountable without being R+ ? (Of course, remaining in the ‘real world’ of separability.)

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Uncountable fundamental groups

Theorem (Popa - V, 2008) For all subgroups H ⊂ R belonging to a large class S, there exist uncountably many free ergodic probability measure preserving actions σi : F∞ ↷ (Xi, µi) such that

• L∞(Xi) ⋊ F∞ has fundamental group exp(H),
• the II1 factors L∞(Xi) ⋊σi F∞ are non-isomorphic for different i.

The large class S includes

◮ all countable subgroups of R, ◮ subgroups of R with prescribed Hausdorff dimension in (0, 1), ◮ Certain

• x ∈ R
• ∞
• n=1

γn αnx < ∞

• , where x = d(x, Z).

Source of non-pathological (Borel, even σ-compact) uncountable subgroups of R.

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Ergodic measures and definition of S S S

Definition (Aaronson, Nadkarni, 1987) An ergodic measure ν on R is a σ-finite measure on the Borel sets

• f R such that

◮ for every x ∈ R, either ν ◦ λx = ν or ν ◦ λx ⊥ ν, ◮ ∃ countable Q ⊂ R, preserving ν, such that a Q-invariant Borel

function is ν-almost everywhere constant. We set Hν = {x ∈ R | ν ◦ λx = ν}. S = {Hν | ν is ergodic measure on R } We get in S, certain groups of the form

• x ∈ R
• ∞
• n=1

γn αnx < ∞

• by using a kind of Cantor measure construction.

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