W -superrigidity for Bernoulli actions and wreath product group von - - PDF document

W∗-superrigidity for Bernoulli actions and wreath product group von Neumann algebras

Lecture notes VNG2011 – Institut Henri Poincar´ e – Paris, May 2011

by Stefaan Vaes(1)(2) These lecture notes are written in a very informal style. They give an account of the author’s lectures at the Institut Henri Poincar´ e in Paris, May 2011 during the program Von Neumann algebras and ergodic theory of group actions at the Centre ´ Emile Borel. It is not excluded that certain statements are not entirely accurate. Please send your ques- tions and comments to stefaan.vaes@wis.kuleuven.be.

Lecture 1 – May 10, 2011 1 Wreath product groups and Bernoulli actions

All results in these lectures deal with the II1 factors given by wreath product groups and/or Bernoulli actions. Before stating the main results, we introduce the following notations. If H, Γ are countable groups, we denote by H ≀Γ := H(Γ) ⋊Γ the wreath product group. Here

• H(Γ) is the direct sum of copies of H index by Γ. So,

H(Γ) = {ξ : Γ → H | ξg = e for all but finitely many g ∈ Γ }.

• Then Γ H(Γ) by automorphisms, shifting the indices: (g · ξ)h = ξg−1h.

More generally, given Γ I, an action of a countable group Γ on a countable set I, put H ≀I Γ := H(I) ⋊ Γ. This is called a generalized wreath product group. Recall the Bernoulli action Γ (X0, µ0)Γ. We call (X0, µ0) the base probability space. We similarly consider the generalized Bernoulli actions Γ (X0, µ0)I given Γ I.

(1)Partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0639.11 of the

Research Foundation – Flanders (FWO) and K.U.Leuven BOF research grant OT/08/032.

(2)Department of Mathematics; K.U.Leuven; Celestijnenlaan 200B; B–3001 Leuven (Belgium).

E-mail: stefaan.vaes@wis.kuleuven.be

Lecture 1 – page 1

2 Main results

The whole lecture series centers around the proof (or outline of the proof) of the following results. Theorem I (Ioana-Popa-V [IPV10]). For ‘quite some’ Γ I the groups G := Z nZ ≀I Γ , n square-free , are W∗-superrigid: whenever LG ∼ = LΛ, we must have G ∼ = Λ. In case where n = 2, 3, any isomorphism θ : LG → LΛ must be group-like: there must exist a group isomorphism δ : G → Λ, a character ω : G → T and a unitary V ∈ LΛ such that θ(ug) = ω(g) V vδ(g) V ∗ for all g ∈ Γ . In this expression we denote by (ug)g∈Γ the canonical unitaries that generate LG and we denote by (vs)s∈Λ the unitaries that generate LΛ. A large family of Γ I for which Theorem I holds arises as follows. Let S be an infinite amenable group and Γ0 a non-amenable group. Put Γ = Γ0 ≀ S. Put I = Γ/S and observe that one identifies I = Γ(S)

0 . Then,

Z nZ ≀I Γ is a W∗-superrigid group for every square-free integer n. We prove in parallel the following striking and strong result by Ioana. Theorem II (Ioana [Io10]). The Bernoulli action Γ (X, µ) = (X0, µ0)Γ of any icc property (T) group is W∗-superrigid: if L∞(X)⋊Γ ∼ = L∞(Y )⋊Λ for any other free action Λ (Y, η), then Γ X must be conjugate with Λ Y . For an overview of earlier (and later) W∗-superrigidity theorems, we refer to Ioana’s lectures and to [PV09]. Both conceptually and technically Theorems I and II rely heavily on the proof of the following result by Popa. Historically this was the first theorem where conjugacy of group actions was deduced from the mere isomorphism of group measure space II1 factors. Theorem III (Popa [Po03, Po04]). Let Γ (X, µ) be any free ergodic pmp action of any icc property (T) group. Let Λ (Y, η) = (Y0, η0)Λ be the Bernoulli action of an arbitrary icc group. If L∞(X) ⋊ Γ ∼ = L∞(Y ) ⋊ Λ, then Γ X must be conjugate with Λ Y . Lecture 1 – page 2

3 Wreath product groups versus Bernoulli actions

If H is an abelian group, LH = L∞ H, where the unitaries (ug)g∈H generating LH correspond to the functions (ω → ω(g))g∈H generating L∞ H. Then also L(H(I)) = L∞( HI), so that finally L(H ≀I Γ) = L∞( HI) ⋊ Γ . We work throughout with wreath product group von Neumann algebras rather than Bernoulli crossed products. Formally this means that we only prove results for Bernoulli actions where the base space has a uniform probability measure. But all results are of course true as well for arbitrary base spaces.

4 Why does Thm I look less elegant than Thm II

Compared to the very neat Theorem II by Ioana, the formulation of Theorem I looks a bit

• messy. One might hope to prove a W∗-superrigidity theorem for plain wreath product groups

H ≀ Γ of, say, property (T) groups. But things turn out to be more subtle. Theorem IV. Let Γ be an icc property (T) group and H a non-trivial abelian group. Assume that L(H ≀ Γ) ∼ = LΛ for some countable group Λ. Then, Λ ∼ = Σ⋊Γ for some abelian group Σ on which Γ acts by group automorphisms in such a way that the pmp actions Γ Σ and Γ HΓ are conjugate. The crucial point that we prove right away, is that there are typically a lot of non-isomorphic groups Σ ⋊ Γ such that Γ Σ is conjugate to the Bernoulli action Γ HΓ and hence L(Σ ⋊ Γ) ∼ = L(H ≀ Γ). Theorem (Ioana-Popa-V [IPV10]). If Γ is a torsion free group and H a non-trivial finite abelian group, there exists a torsion free group Λ such that L(H ≀ Γ) ∼ = LΛ . In particular, Λ ∼ = H ≀ Γ.

• Proof. Step 1: assume Γ = Z.

We construct a torsion free abelian group Σ0 and an action Z

α

Σ0 by group automorphisms such that Z Σ0 is conjugate with the Bernoulli action Z {1, . . . , |H|}Z. Take Σ0 = Z[|H|−1] and define Z

α

Σ0 generated by the automorphism of multiplication by |H|. We have LH = ℓ∞({1, . . . , |H|}) ⊂ L∞T = LZ ⊂ LΣ0 , where we view ℓ∞({1, . . . , |H|}) ⊂ L∞T as the functions that are constant on the intervals that arise by cutting the circle in |H| equal pieces. Lecture 1 – page 3

• Exercise. The subalgebras αk(LH) ⊂ LΣ0, k ∈ Z, are independent and generate LΣ0.

Hence Z Σ0 is the correct Bernoulli action. Step 2: co-induction given Z ≤ Γ. We first recall the general co-induction construction given a pmp action Γ0 (Y, η) and a larger group Γ0 ≤ Γ. Choose a section ρ : Γ/Γ0 → Γ with ρ(eΓ0) = e. Put ω : Γ × Γ/Γ0 → Γ0 : ω(g, t) = ρ(gt)−1gρ(t) . Then ω is a 1-cocycle in the sense that ω(gh, t) = ω(g, ht)ω(h, t) . The pmp action Γ Z = Y Γ/Γ0 given by (g−1 · ξ)t = ω(g, t)−1 · ξgt is called the co-induced action. Up to conjugacy it does not depend on the choice of section ρ : Γ/Γ0 → Γ. Exercises

• The action Γ Z is pmp.
• If Γ0 (Y, η) is a Bernoulli action, then the co-induced action is also Bernoulli with the

same base space.

• If Γ0 (Y, η) was the dual of an action Γ0 Σ0 by automorphisms of a countable abelian

group Σ0, then also the co-induced action is the dual of an action of Γ by automorphisms

• f Σ(Γ/Γ0)

. The previous list of exercises also ends the proof: choose an embedding Z ֒ → Γ and co-induce the Z Σ0 given by step 1.

• Remark. In concrete cases one can vary the embedding Z ֒

→ Γ and prove that the resulting groups Σ ⋊ Γ are mutually non-isomorphic. In particular, it is proven in [IPV10] that for all n ≥ 2 there are infinitely many non-isomorphic groups Λ satisfying L( Z 2Z ≀ PSL(n, Z)) ∼ = LΛ . The following set of exercises provides an abstract characterization of co-induced actions. Exercises

• The map π : Z → Y : ξ → ξeΓ0 is a Γ0-equivariant pmp factor map.
• The maps (ξ → π(g−1 · ξ))g∈Γ/Γ0 are independent and generate the σ-algebra on Z.
• The previous two properties provide an abstract characterization of the co-induced action

up to conjugacy. Lecture 1 – page 4

5 Softness results for group von Neumann algebras

• (Connes, 1976) All LΓ for Γ icc amenable, are isomorphic to the unique hyperfinite II1

factor.

• (Dykema, 1993) For all infinite amenable groups Γ1, . . . , Γn, n ≥ 2, one has

L(Γ1 ∗ · · · ∗ Γn) ∼ = LFn .

• (Ioana, 2006) All L(Fn ≀ Z), n ≥ 2, are isomorphic.
• (Bowen, 2009) Fix n ≥ 1. All L(H ≀ Fn), H non-trivial abelian, are isomorphic.

We explain Bowen’s result in slightly more detail. Bowen proved that for fixed n ≥ 1, the Bernoulli actions Fn XFn and Fn XFn

1

are orbit equivalent (and hence give rise to isomorphic group measure space II1 factors). In particular L(H0 ≀ Fn) ∼ = L(H1 ≀ Fn) for all non-trivial abelian groups H0, H1. For n = 1, orbit equivalence of Z XZ

0 and Z XZ 1 is provided by Dye’s theorem. Next

Bowen proves that orbit equivalence is preserved under certain co-inductions. More precisely if the free pmp actions Γ0 X0 and Γ1 X1 are orbit equivalent, then their co-inductions to Γ0 ∗ Λ, resp. Γ1 ∗ Λ are orbit equivalent. The result in the previous paragraph follows by considering Fn = Z ∗ Fn−1 and a combination of Dye’s and Bowen’s theorems. The proof of Bowen’s theorem can be sketched as follows. Let ∆ : X0 → X1 be an orbit equivalence between Γ0 X0 and Γ1 X1. Define the corresponding Zimmer cocycle ω : Γ0 × X0 → Γ1 given by ∆(g · x) = ω(g, x) · ∆(x) . Let Γ1 ∗ Λ Z be the co-induction of Γ1 X1 to Γ1 ∗ Λ, together with the canonical Γ1-equivariant quotient map π : Z → X1. Denote this action by ·. Define an action of Γ0 ∗ Λ on Z, denoted by ∗ and given by g ∗ z = ω(g, (∆−1 ◦ π)(z)) · z if g ∈ Γ0 , λ ∗ z = λ · z if λ ∈ Λ .

• Exercise. Prove that Γ0 ∗ Λ Z has the same orbits as Γ1 ∗ Λ Z and that it satisfies,

together with the Γ0-equivariant factor map Z → X0 : z → (∆−1 ◦ π)(z) the abstract characterization of the co-induction of Γ0 X0 to Γ0 ∗ Λ. This exercise ends the proof of Bowen’s theorem. Lecture 1 – page 5

6 Group algebras and reduced group C∗-algebras are rather rigid

In the passage G → CG → C∗

rG → LG the group more and more fades away. It is illustrative

to consider the case of torsion free abelian groups G. Then G can be recovered from C∗

rG =

C( G) as the group of connected components of the unitary group of C∗

• rG. Indeed, this holds

for G = Zn and is preserved under direct limits. On the other hand all LG are isomorphic to the unique diffuse abelian von Neumann algebra.

• Higman’s conjecture (1940): if G is torsion free, the only invertible elements in the algebra

CG are the non-zero multiples of the group elements. So, we can recover G from CG. Higman’s conjecture has been proved for a number of groups, including orderable groups and free groups. Note that Higman’s conjecture implies Kaplansky’s conjecture on the absence of non-trivial idempotents.

• As far as we know, there are no examples of torsion free G ∼

= Λ with C∗

rG ∼

= C∗

rΛ. It

sounds however very plausible that such examples should exist.

7 Structure of the following lectures

We will outline in parallel the proofs of Theorems I, II and IV. This outline will be sufficiently detailed to provide a fairly complete proof of Theorem III. What do Theorems I and II have in common? We start with a wreath product (or Bernoulli) von Neumann algebra M = L(H ≀I Γ). In the case of Theorem I we are given another group von Neumann algebra decomposition M = LΛ. In the case of Theorem II we are given another group measure space decomposition M = B ⋊Λ. In both cases we associate to these ‘mysterious’ decompositions an embedding ∆ : M → M ⊗ M. ∆(vs) = vs ⊗ vs for all s ∈ Λ in the case M = LΛ, ∆(bvs) = bvs ⊗ vs for all b ∈ B, s ∈ Λ in the case M = B ⋊ Λ. We use Popa’s deformation/rigidity theory to give a rather complete classification of all possible embeddings ∆ : M → M ⊗ M. Lecture 2. Write ∆Λ(vs) = vs ⊗ vs for all s ∈ Λ. In the case M = LΛ and in the context

• f Theorem I, the classification theorem will allow us (in lecture 6) to conclude that the

embedding ∆Λ is unitarily conjugate with the embedding ∆Γ : M → M ⊗ M : ug → ug ⊗ ug for all g ∈ G = H ≀I Γ. So ∆Λ(x) = Ω∆Γ(x)Ω∗ for all x ∈ M and some unitary Ω ∈ M ⊗ M. This unitary Ω then (almost) satisfies the properties of a symmetric 2-cocycle on the ‘compact quantum group’ Γ. In lecture 2 we prove a cohomology vanishing result that then finishes the proof of Theorem I. We also deduce a general isomorphism criterion of group von Neumann algebras. Lecture 3. Note that M ⊗ M = L(H ≀I⊔I (Γ × Γ)) is itself a wreath product group von Neumann algebra. We prove that ‘rigid’ subalgebras of L(H ≀I Γ) can be virtually conjugated Lecture 1 – page 6

into LΓ. Applied to our embeddings ∆ and given a rigidity property on Γ we will be able to show that after a unitary conjugacy of ∆ we have ∆(LΓ) ⊂ L(Γ × Γ). Lectures 4 and 5. Put A = L(H(I)). By the previous lecture we now know that ∆(LΓ) ⊂ L(Γ × Γ).

• We prove a general principle of the following kind: if an abelian subalgebra B ⊂ L(H ≀I Γ)

is normalized by ‘enough’ unitaries in LΓ, then B can be virtually conjugated in either L(H(I)) or LΓ.

• In the group von Neumann algebra case M = LΛ we will deduce that after yet another

unitary conjugacy ∆(A) ⊂ A ⊗ A.

• In the case M = B ⋊ Λ we will deduce that either A and B are unitarily conjugate

(in which case Γ X and Λ Y are conjugate because of Popa’s orbit equivalence superrigidity theorem), or that ∆(A) ⊂ A ⊗ A after a unitary conjugacy. Lecture 6. We conclude the proofs of Theorems I, II and IV. Now ∆(A) can be conjugated into A⊗A and ∆(LΓ) can be conjugated into L(Γ×Γ). We deduce that it (virtually) follows that ∆ is an equivariant embedding of the action Γ A into the action of Γ × Γ on A ⊗ A. Lecture 1 – page 7

Lecture 2 – May 12, 2011 8 Isomorphism of group von Neumann algebras and dual 2-cocycles

Let M be a II1 factor. Whenever M is written as a group von Neumann algebra M = LΛ, we get an embedding ∆Λ : M → M ⊗M given by ∆Λ(vs) = vs ⊗vs for all s ∈ Λ. If the same II1 factor has two group von Neumann algebra decompositions LG = M = LΛ, we get two embeddings ∆G, ∆Λ : M → M ⊗ M. The very best deformation/rigidity theory can give us, is that ∆G and ∆Λ are unitarily conjugate, i.e. there exists a unitary Ω ∈ M ⊗ M such that ∆G(x) = Ω∆Λ(x)Ω∗ for all x ∈ M . (8.1) Denote by σ : M ⊗ M → M ⊗ M the flip automorphism: σ(a ⊗ b) = b ⊗ a. We claim that there exist µ, η ∈ T such that (∆G ⊗ id)(Ω)(Ω ⊗ 1) = η(id ⊗ ∆G)(Ω)(1 ⊗ Ω) , σ(Ω) = µΩ . (8.2) Indeed, using the fact that (∆G ⊗ id) ◦ ∆G = (id ⊗ ∆G) ◦ ∆G and that a similar property holds for ∆Λ, it follows that Z := (∆G ⊗ id)(Ω)(Ω ⊗ 1)(1 ⊗ Ω∗)(id ⊗ ∆G)(Ω∗) is a unitary in M ⊗ M ⊗ M that commutes with all ug ⊗ ug ⊗ ug, g ∈ G. Since M is a II1 factor, the group G is icc and it follows that Z must be a scalar: Z = η1. The formula for σ(Ω) follows similarly by using that σ ◦ ∆G = ∆G and σ ◦ ∆Λ = ∆Λ. We want to understand better what formulae (8.2) mean. Assume that G is an abelian group and that Ω ∈ LG ⊗ LG is a unitary satisfying (8.2). Identifying LG = L∞( G) we view Ω as a measurable map Ω : G × G → T satisfying Ω(xy, z)Ω(x, y) = ηΩ(x, yz)Ω(y, z) , Ω(y, x) = µΩ(x, y) for almost all x, y, z ∈

• G. This means that, up to the scalars η, µ, the map Ω is a measurable

symmetric 2-cocycle on the compact abelian group G. Symmetric 2-cocycles on abelian groups are coboundary. We prove that this is still the case for duals of discrete groups. Theorem 8.1 (Ioana-Popa-V [IPV10]). Let G be a countable group. Put M = LG generated by (ug)g∈G and ∆ : M → M ⊗ M : ∆(ug) = ug ⊗ ug. If Ω ∈ U(M ⊗ M) and η, µ ∈ T satisfy (∆ ⊗ id)(Ω)(Ω ⊗ 1) = η(id ⊗ ∆)(Ω)(1 ⊗ Ω) , σ(Ω) = µΩ , then η = µ = 1 and there exists a unitary w ∈ M such that Ω = ∆(w∗)(w ⊗ w) . Lecture 2 – page 1

Before proving Theorem 8.1 let me explain the impact on the W*-superrigidity problem for group von Neumann algebras. So assume that we are again in the situation of (8.1). It follows that ∆G(wvsw∗) = wvsw∗ ⊗ wvsw∗ for every s ∈ Λ. So, for every s ∈ Λ there is a unique δ(s) ∈ G and a unique ω(s) ∈ T such that wvsw∗ = ω(s)uδ(s). It follows that δ : Λ → G is a group isomorphism. We will also derive the following cute consequence of Theorem 8.1.

• Notations. Let (M, τ) be a tracial von Neumann algebra with corresponding · 2. Given

subsets U, V ⊂ (M)1, we consider the (asymmetric, upper) Hausdorff distance dist2(U, V) = sup

u∈U

dist2(u, V) = sup

u∈U

• inf

v∈V u − v2

• .

Note that if U, V are sets of unitaries, then dist2(TU, TV) ≤ √ 2. Theorem 8.2 (Ioana-Popa-V, 2010). Let G be an icc group and assume that M = LG has another group von Neumann algebra decomposition M = LΛ. Denote by (ug)g∈G and (vs)s∈Λ the corresponding unitaries. Then the following are equivalent.

• dist2(TG, TΛ) <

√ 2.

• There exists a unitary w ∈ M, a character ω : G → T and a group isomorphism δ : G → Λ

such that wugw∗ = ω(g)vδ(g) for all g ∈ G . Open problem. What can we conclude if G, Λ are countable groups, LG ⊂ LΛ and dist2(TG, TΛ) < √

• 2. Does the same conclusion hold with an injective group homomorphism

δ : G → Λ ?

Proof of Theorem 8.1

• Idea. Define the cocycle twisted group von Neumann algebra for
• G. Prove that it is a

discrete abelian von Neumann algebra (i.e. ℓ∞(G), the non twisted group von Neumann algebra of G). This comes down to proving that the 2-cocycle is coboundary. We have M = LG. We realize M explicitly by right translation operators. So:

• H = ℓ2G.
• λgδh = δgh and ρgδh = δhg−1.
• M := {ρg | g ∈ G}′′.
• ℓ∞(G) ⊂ B(ℓ2G) as multiplication operators: Fδg = F(g)δg for all F ∈ ℓ∞(G).

Define the unitary W ∈ ℓ∞(G) ⊗ M given by the function g → ρg. So, W(δg ⊗ δh) = δg ⊗ δhg−1 . Lecture 2 – page 2

One checks that ∆(x) = W ∗(x ⊗ 1)W and λg ⊗ ρg = W(λg ⊗ 1)W ∗ . (8.3) Note that (id ⊗ ∆)(W) = W13W12. Here W12 = W ⊗ 1 and W13 acts on the first and third tensor factor of H ⊗ H ⊗ H. One should think of W as the regular representation of

• G. We now define the Ω-regular
• representation. Define the unitary

X = WΩ ∈ B(H) ⊗ M . The 2-cocycle relation for Ω now becomes (id ⊗ ∆)(X)(1 ⊗ Ω) = ηX13X12 . (8.4)

• Notation. We denote by [V] the norm closed linear span of the set V.

We next define the twisted reduced C∗-algebra of

• G. Put

A := [(id ⊗ ω)(X) | ω ∈ M∗] = [(id ⊗ ω)(X) | ω ∈ B(H)∗] .

• Proposition. The closed subspace A ⊂ B(H) is a C∗-algebra that acts non-degenerately on

H (i.e. [AH] = H).

• Proof. The inclusion AA ⊂ A follows by applying id⊗ω1⊗ω2 to the equality (8.4). It indeed

follows that [AA] = [(id ⊗ ω1 ⊗ ω2)(ηX13X12) | ω1, ω2 ∈ M∗] = [(id ⊗ ω)((id ⊗ ∆)(X)(1 ⊗ Ω)) | ω ∈ (M ⊗ M)∗] = [(id ⊗ ω)(id ⊗ ∆)(X) | ω ∈ (M ⊗ M)∗] = A . To prove that A∗ = A we rewrite (8.4). Recall that ∆(x) = W ∗(x ⊗ 1)W. Hence (8.4) reads W ∗

23X12W23Ω23 = ηX13X12 .

(8.5) Note that the left hand side equals W ∗

• 23X12X23. So we get

X∗

13W ∗ 23X12 = ηX12X∗ 23 .

Denote by Pg ∈ ℓ∞(G) the projection onto Cδg. Note that A = [(id ⊗ ω1 ⊗ ω2)(ηX12X∗

23) | ω1, ω2 ∈ B(H)∗] .

So, A = [(id ⊗ ω1 ⊗ ω2)(X∗

13W ∗ 23X12) | ω1, ω2 ∈ B(H)∗]

= [(id ⊗ ω1Pg ⊗ ω2)(X∗

13W ∗ 23X12) | ω1, ω2 ∈ B(H)∗, g ∈ G] .

Lecture 2 – page 3

One computes that (1 ⊗ Pg ⊗ 1)X∗

13W ∗ 23X12 = X∗ 13(1 ⊗ Pg ⊗ ρ∗ g)X12 = (1 ⊗ Pg ⊗ 1)X∗ 13(X ⊗ ρ∗ g) .

We conclude that A = [(id ⊗ ω1Pg ⊗ ρ∗

gω2)(X∗ 13X12) | ω1, ω2 ∈ B(H)∗, g ∈ G]

= [(id ⊗ ω1Pg ⊗ ω2)(X∗

13X12) | ω1, ω2 ∈ B(H)∗, g ∈ G]

= [(id ⊗ ω1 ⊗ ω2)(X∗

13X12) | ω1, ω2 ∈ B(H)∗]

= [A∗A] . So A = [A∗A] and in particular A = A∗. The non-degeneracy [AH] = H follows easily: [AH] = [(1 ⊗ ξ∗

1)X(ξ2 ⊗ ξ3) | ξi ∈ H]

= [(1 ⊗ ξ∗

1)Xξ | ξ1 ∈ H, ξ ∈ H ⊗ H] = H

because X(H ⊗ H) = H ⊗ H.

• Proposition. The C∗-algebra A is abelian and µ = 1.
• Proof. Apply id ⊗ σ to (8.4). It follows that X12X13 = µX13X12. Applying id ⊗ ω1 ⊗ ω2 we

conclude that ab = µba for all a, b ∈ A. Since A is a C∗-algebra that acts non-degenerately

• n H, the same equality must hold for a and b in the weak closure of A and, in particular,

for a = b = 1. So, µ = 1 and A is abelian. Final step. The von Neumann algebra A′′ is discrete (i.e. 1 is the sum of minimal projec- tions). Once the final step is proven, Theorem 8.1 follows: take a minimal projection p ∈ A′′. Since A′′ is abelian and X is a unitary in A′′ ⊗ M, we find a unitary w ∈ M such that (p ⊗ 1)X = p ⊗ ηw . Putting p ⊗ 1 ⊗ 1 in front of (8.4), it follows that p ⊗ ∆(w)Ω = p ⊗ w ⊗ w yielding the required formula Ω = ∆(w∗)(w ⊗ w). Proof of the final step. Denote by τ the canonical trace on M = LG. We show that the formula E(x) := (id ⊗ τ)(X(x ⊗ 1)X∗) is a normal conditional expectation of B(H) onto A′′. Once this is proven, A′′ follows discrete and we are done. Since A′′ is abelian and X ∈ A′′ ⊗ M, it follows that E(a) = a for all a ∈ A′′. Remains to be proven: for all x ∈ B(H) we have E(x) ∈ A′′. One checks that for all x ∈ B(H) of the form x = aλg, a ∈ A, g ∈ G, we have E(x) = E(aλg) = aE(λg) = a(id ⊗ τ)(X(λg ⊗ 1)X∗) = a(id ⊗ τ)(λg ⊗ ρg) = a or 0 Lecture 2 – page 4

depending on g = e or g = e. Final statement to be proven: the linear span of {aλg | a ∈ A, g ∈ G} is weakly dense in B(H). Step 1. The closed linear span B := [aλg | a ∈ A, g ∈ G] is a C∗-algebra that acts non-degenerately on H. Since X(λg ⊗ 1)X∗ = λg ⊗ ρg, we also have (λ∗

g ⊗ 1)X(λg ⊗ 1) = (1 ⊗ ρg)X .

Applying id ⊗ ω yields λ∗

gAλg = A. Then B is a C∗-algebra and B acts non-degenerately

since already A does. Step 2. B′ = C1. Assume x ∈ B′. Since x commutes with all λg, g ∈ G, it follows that x ∈ M. So, x ∈ M ∩ A′. Since A′′ is abelian, we know that X13X12 = X12X13. Together with (8.5) this implies that W ∗

23X12W23Ω23 = ηX12X13 .

But then X(1 ⊗ x)X∗ ⊗ 1 = ηX12X13(1 ⊗ x ⊗ 1)(ηX12X13)∗ = W ∗

23X12X23(1 ⊗ x ⊗ 1)(W ∗ 23X12X23)∗

= W ∗

23X12(1 ⊗ x ⊗ 1)(W ∗ 23X12)∗

(because x ⊗ 1 and X commute) = (id ⊗ ∆)(X(1 ⊗ x)X∗) . Put y = (id ⊗ τ)(X(1 ⊗ x)X∗) ∈ A′′. Applying id ⊗ id ⊗ τ to the above computation yields X(1 ⊗ x)X∗ = y ⊗ 1 . Since A′′ is abelian, it follows that 1 ⊗ x = X∗(y ⊗ 1)X = y ⊗ 1. Hence x ∈ C1. This ends the proof of Theorem 8.1.

9 Proof of the intertwining theorem 8.2

• Terminology. A unitary representation π : Γ → U(H) is called weakly mixing if {0} is the
• nly finite dimensional π(Γ)-invariant subspace.
• Exercise. Let π : Γ → U(H) be a unitary representation. Prove that the following state-

ments are equivalent.

• π is weakly mixing.
• Whenever ρ is a unitary representation, π ⊗ ρ has no non-zero invariant vectors.
• There exists a sequence gn ∈ Γ such that π(gn) → 0 weakly.
• Exercise. Prove that the adjoint representation Adg δh = δghg−1 on ℓ2(G − {e}) is weakly

mixing iff G is icc. Lecture 2 – page 5

Reinterpretation of the condition dist2(TG, TΛ) < √

• 2. Every element x ∈ M has a

Fourier decomposition x =

• s∈Λ

(x)svs with (x)s ∈ C . Note that (x)s = τ(xv∗

s). For any x ∈ M we write

hΛ(x) = sup

s∈Λ

|τ(xv∗

s)| ,

i.e. the absolute value of the largest Fourier coefficient of x.

• Exercise. If w ∈ LΛ is unitary, then

dist2(w, TΛ) =

• 2(1 − hΛ(w)) .

Proof of Theorem 8.2. We only prove the non-trivial implication from the first to the second

• statement. By the previous exercise, assume that there exists a δ > 0 such that

hΛ(ug) ≥ δ for all g ∈ G . Denote ∆ : M → M ⊗ M : ∆(vs) = vs ⊗ vs. Writing ug in its Fourier decomposition, we have (τ ⊗ τ ⊗ τ)

• (∆(ug) ⊗ ug) (ug ⊗ ∆(ug))∗

=

• s∈Λ

|(ug)s|4 ≥ hΛ(ug)4 ≥ δ4 for all g ∈ G. As usual let X be the unique element of minimal · 2 in the weakly closed convex hull of {(∆(ug)⊗ug) (ug ⊗∆(ug))∗ | g ∈ G}. It follows that X is a non-zero element of M ⊗M ⊗M and that (∆(ug) ⊗ ug)X = X(ug ⊗ ∆(ug)) for all g ∈ G . So XX∗ commutes with all ∆(ug) ⊗ ug. Since G is icc, we get that XX∗ = T ⊗ 1 for some T ∈ M ⊗ M ∩ ∆(M)′. Since Λ is icc, T must be scalar. So X is a multiple of a unitary and we may assume right away that X is unitary. Put Y = (X ⊗ 1)(1 ⊗ X). It follows that (∆(ug) ⊗ ug ⊗ ug)Y = Y (ug ⊗ ug ⊗ ∆(ug)) for all g ∈ G . Hence the unitary representation ξ → (ug ⊗ ug)ξ∆(u∗

g) on L2(M ⊗ M) is not weakly mixing.

So it admits a non-zero finite dimensional irreducible subrepresentation. So we find an irreducible finite dimensional unitary representation π : G → U(K) and a non-zero vector ξ ∈ K ⊗ L2(M ⊗ M) such that (π(g) ⊗ ug ⊗ ug)ξ = ξ∆(ug) for all g ∈ G . Since G is icc and π is irreducible, ξξ∗ is scalar. Since (Tr ⊗τ ⊗ τ)(ξξ∗) ≤ 1, it follows that K is one-dimensional and that ξ is a multiple of a unitary Ω ∈ M ⊗ M. So π is a character and ∆G(ug) = ug ⊗ ug = π(g)Ω∆Λ(ug)Ω∗ for all g ∈ G . It follows that Ω satisfies (8.2). Theorem 8.1 yields a unitary w ∈ M such that ∆Λ(wugw∗) = wugw∗ ⊗ wugw∗ for all g ∈ G . This means that wugw∗ is of the required form. Lecture 2 – page 6

Lecture 3 – May 13, 2011

We prove that ‘rigid’ subalgebras of wreath product group von Neumann algebras L(H ≀ Γ), and even generalized wreath products L(H ≀I Γ), can be virtually conjugated into LΓ. This virtual conjugacy is actually Popa’s intertwining-by-bimodules that we will recall as well. These results go back to Popa’s [Po03] in the case of plain wreath products and were generalized over the years. Here the ‘rigidity’ can obviously come from relative property (T), but also from spectral gap rigidity, as we will see.

10 Popa’s intertwining-by-bimodules

Let (M, τ) be a tracial von Neumann algebra and P, Q ⊂ M von Neumann subalgebras. Then the following are equivalent.

• There exist projections p ∈ P and q ∈ Q, a non-zero partial isometry v ∈ pMq and a

normal ∗-homomorphism ϕ : pPp → qQq satisfying xv = vϕ(x) for all x ∈ pPp.

• There is no sequence of unitaries wn ∈ U(P) satisfying EQ(awnb)2 → 0 for all a, b ∈ M.

If these conditions hold, we write P ≺M Q, or simply P ≺ Q. In the situation where M = LG and Q = LΣ for some subgroup Σ ≤ G, the intertwining criterion can be made slightly more intuitive.

• Notation. Assume that M = LG. We identify L2(M) with ℓ2(G) where ug ∈ M corresponds

to the base vector δg ∈ ℓ2(G). Whenever S ⊂ G is a subset, we denote by PS the orthogonal projection of ℓ2(G) onto ℓ2(S). Hence, PS(ug) = ug if g ∈ S and PS(ug) = 0 if g ∈ S.

• Terminology. If Σ ≤ G is a subgroup and S ⊂ G a subset, we say that S is small relative

to Σ if S can be written as a finite union of subsets gΣh, g, h ∈ G.

• Exercise. We have P ≺ LΣ iff there exists a sequence of unitaries wn ∈ U(P) such that

PS(wn)2 → 0 for every subset S ⊂ G that is small relative to Σ. The result in the previous exercise can be improved as follows. This generalization is not entirely trivial, but requires a ‘classical’ argument with a diagonal inclusion into matrices

• ver LG. If S is a family of subgroups of G, we say that S ⊂ G is small relative to S if S

can be written as a finite union of subsets gΣh, g, h ∈ G, Σ ∈ S. Lemma 10.1. Let S be a countable family of subgroups of G and P ⊂ LG a von Neumann

• subalgebra. Then the following are equivalent.
• For every Σ ∈ S we have P ≺ LΣ.
• There exists a sequence of unitaries wn ∈ U(P) such that PS(wn)2 → 0 for every subset

S ⊂ G that is small relative to S. Lecture 3 – page 1

11 Control of relative commutants and normalizers

We prove the following theorem due to Popa. Theorem 11.1 (Popa [Po03, Po04]). Let Σ ≤ Λ ≤ G be subgroups and Q ⊂ LΣ a von Neumann subalgebra. Assume that for all g ∈ G − Λ we have that Q ≺ L(Σ ∩ gΣg−1). Then, Q′ ∩ LG ⊂ LΛ. Even more, whenever v ∈ LG and xv = vα(x) for all x ∈ Q and some homomorphism α : Q → Q, we must have v ∈ LΛ.

• Proof. It suffices to prove the second statement. Replace v by v − ELΛ(v). We may then

assume that v ∈ LG ⊖ LΛ and we have to prove that v = 0. Define S := {Σ ∩ gΣg−1 | g ∈ G − Λ}. By Lemma 10.1 take a sequence (wn) of unitaries in Q such that PS(wn)2 → 0 for every subset S ⊂ G that is small relative to S.

• Claim. PΣ(awnb)2 → 0 for all a, b ∈ LG ⊖ LΛ. It suffices to prove this claim when a = ug

and b = uh with g, h ∈ G − Λ. Observe however that for all x ∈ Q we have PΣ(ugxuh) = ugPΣ∩g−1Σh−1(x)uh . Since Σ ∩ g−1Σh−1 is small relative to S, the claim is proven. On the other hand we have PΣ(v∗wnv)2 = PΣ(v∗v)α(wn)2 = PΣ(v∗v)2 . The claim implies that PΣ(v∗v) = 0. A fortiori τ(v∗v) = 0 and hence v = 0. Example 11.2. We will apply Theorem 11.1 in the following two cases.

• Consider G = H ≀ Γ and let Q ⊂ LΓ be a diffuse subalgebra. Then the normalizer of Q

inside L(H ≀ Γ) lies inside LΓ. Indeed, for all g ∈ G − Γ, the group Γ ∩ gΓg−1 is finite and hence Q ≺ L(Γ ∩ gΓg−1).

• The previous example can be generalized as follows.

Let G = H1 ≀ Γ, H ≤ H1 and G = H ≀ Γ. If Q ⊂ L(G) and Q ≺ L(H(Γ)), then the normalizer of Q inside L G lies inside LG.

• Exercise. Prove the following generalization of Theorem 11.1. Let Σ and Λ be subgroups
• f G. Assume that Q ⊂ LΣ is a von Neumann subalgebra satisfying Q ≺ L(Σ ∩ gΛg−1) for

all g ∈ G. Prove if α : Q → LΛ is a homomorphism and w ∈ LG satisfies aw = wα(a) for all a ∈ Q, then w = 0.

12 Support length deformation on wreath products

Let G = H ≀ Γ be a wreath product. Whenever ξ ∈ H(Γ) denote supp ξ := {g ∈ Γ | ξg = e} Lecture 3 – page 2

and denote by | supp ξ| the cardinality of supp ξ. We claim that for all 0 < ρ < 1 the formula ϕρ(ξg) = ρ| supp ξ| for ξ ∈ H(Γ), g ∈ Γ, defines a function of positive type on G. To prove this claim, first observe that the function H → C : a →

• 1

if a = e , ρ if a = e . is of positive type. One can view ξ → ρ| supp ξ| as an infinite product of such positive type

• functions. So this function is of positive type as well. Moreover it is Γ-invariant and we are

done.

• Notation. Put M = L(H ≀ Γ). We denote by θρ : M → M the unital completely positive

maps that go with ϕρ, namely θρ : M → M : θρ(uξg) = ρ| supp ξ|uξg for all ξ ∈ H(Γ), g ∈ Γ . Intuitively θρ is close to the identity on elements ‘with short support’. In the following theorem we characterize on which von Neumann subalgebras θρ is uniformly close to the

• identity. The crucial point is that in a subalgebra one can multiply two elements. If we

consider in H ≀ Γ two elements ξg and ηh with | supp ξ| and | supp η| small and if we then multiply these elements, two things can happen: if g does not move the support of η, then the support of the product of ξg and ηh will remain unchanged. If however g is ‘far away’, then it will move the support of η disjoint from the support of ξ and so, the supports add up in length.

• Exercise. If Σ ≤ H ≀ Γ is a subgroup such that supξg∈Σ | supp ξ| < ∞, then Σ can be

conjugated into either Γ or H(F) ⋊ Norm F for some finite subset F ⊂ Γ. Here we denote Norm F := {g ∈ Γ | gF = F}. The theorem that we prove for von Neumann algebras is not as precise as the previous exercise, but it could be made as precise, see [Io06]. Theorem 12.1 (Popa [Po03], Ioana [Io06]). Let Q ⊂ L(H ≀ Γ) be a diffuse von Neumann

• subalgebra. Assume that

θρ(b) − b2 → 0 uniformly on (Q)1 as ρ → 1. Then at least one of the following statements holds.

• Q ≺ LΓ.
• Q ≺ L(H(Γ)).

The proof of Theorem 12.1 consists of several steps. We assume that Q ≺ L(H(Γ)) and prove that Q ≺ LΓ. Step 1. The malleable deformation corresponding to the support length deformation. Lecture 3 – page 3

Put G = H ≀ Γ and G = (H ∗ Z) ≀ Γ. Denote by α0 the inner automorphism of H ∗ Z given by Ad 1 with 1 ∈ Z. Denote by α the infinite direct sum of such α0 which is an automorphism

• f (H ∗ Z)(Γ) which is no longer inner. Note that α is Γ-equivariant and hence extends to an

automorphism of G that is the identity on Γ and that we still denote by α. We even denote by α the corresponding automorphism of L G given by α(ux) = uα(x) for all x ∈ G. We shall construct a malleable deformation (αt) by automorphisms of L G such that α1 = α. Choose a = a∗ in LZ with spectrum [−π, π] such that u1 = exp(ia). The formula α0

t =

Ad exp(ita) defines an inner automorphism of L(H ∗ Z). Taking the infinite tensor product

• f these α0

t and taking the identity map on LΓ, we have found the required one-parameter

group of automorphisms of L G. Moreover the malleable deformation (αt) is symmetric: define the automorphism β0 of H ∗Z that is the identity on H and that multiplies by −1 on Z. We again take the infinite product and find an automorphism β of

• G. We still denote by β the corresponding automorphism of

L

• G. By construction β = id on LG, β ◦ β = id and β ◦ αt = α−t ◦ β.
• Exercise. Put ρt = | sin(πt)/(πt)|2. Prove that

ELG(αt(x)) = θρt(x) for all x ∈ LG . Step 2. Since θρ → id uniformly on the unit ball of Q, we can take t > 0 such that τ(aαt(a∗)) ≥ 1 2 for all a ∈ U(Q) . Define v as the element of minimal · 2 in the weakly closed convex hull of {aαt(a∗) | a ∈ U(Q)}. It follows that v ∈ L G is a non-zero element satisfying av = vαt(a) for all a ∈ U(Q) and hence, for all a ∈ Q as well. Replacing v by its polar part, we may assume that v is a non-zero partial isometry. We may also assume that t is of the form t = 2−k for some integer k. Step 3. We construct a non-zero partial isometry w ∈ L G such that aw = wα1(a) for all a ∈ Q. In the previous step we have found a partial isometry v ∈ L G such that av = vαt(a) for all a ∈ Q. So vv∗ is a projection in L G that commutes with Q. By the second case of Example 11.2 and our assumption that Q ≺ L(H(Γ)), it follows that vv∗ ∈ L(G). Hence β(vv∗) = vv∗. From this it follows that β(v∗)v is a partial isometry with the same right support as v. Putting v′ = αt(β(v∗)v) one checks that av′ = v′α2t(a) for all a ∈ Q and that v′ = 0. Since we can double t, we can as well continue up to t = 1. To conclude the proof of the theorem, you first make the following combinatorial exercise.

• Exercise. Whenever F ⊂ Γ is finite, we denote Norm F := {g ∈ Γ | gF = F}. We also

consider the subgroup H(F) of G. By convention Norm ∅ = Γ and H(∅) = {e}. Prove that for every x ∈ G the subgroup G∩xα(G)x−1 is of the form H(F) ⋊Norm F for some, possibly empty, finite subset F ⊂ Γ. If F is non-empty, then Norm F is a finite group so that Q ≺ L(H(F) ⋊ Norm F). Since aw = wα(a) for all a ∈ Q, the previous exercise together with the exercise at the end of Section 11 imply that Q ≺ LΓ. Lecture 3 – page 4

13 How to use (a generalization of) Theorem 12.1 to prove Theorems I–IV

Part 1. How to get automatically that θρ → id uniformly on the unit ball of Q ?

• We of course have θρ → id uniformly on the unit ball of Q if the inclusion Q ⊂ LG has

the relative property (T) or if Q is diffuse with property (T).

• But the uniformity of θρ on the unit ball of Q can also come from spectral gap rigidity. In
• rder to see this, first make the following exercise.
• Exercise. Assume that H is amenable. Prove that the adjoint action (Adg)g∈G on the

set G − G given by Adg(h) = ghg−1 has amenable stabilizers. Deduce that the adjoint representation (Adg)g∈G on ℓ2( G − G) given by Adg δx = δgxg−1 is weakly contained in the regular representation. Put M = L G and M = LG. Once the above exercise is made, it follows that ML2( M ⊖ M)M is weakly contained in the coarse M-bimodule. Assume now that Q ⊂ M is such that P := Q′ ∩ M has no amenable direct summand. Since PL2( M ⊖ M)P is weakly contained in the coarse P-bimodule, this bimodule does not weakly contain PL2(Pz)P whenever z ∈ Z(P) is a non-zero central projection. So, given ε > 0, κ > 0, there exist finitely many d1, . . . , dn ∈ P and δ > 0 with the following property: if x ∈ M ⊖ M, x∞ ≤ κ and dix − xdi2 ≤ δ for all i = 1, . . . , n, then x2 ≤ ε. (13.1) Choose ε > 0 and put κ = 2. Take the corresponding d1, . . . , dn ∈ P and δ > 0. For t small enough, αt(di) ≈ di for all i = 1, . . . , n. It then follows that (13.1) can be applied to all x = αt(y) − EM(αt(y)), y ∈ (Q)1. It follows that αt(y) − EM(αt(y))2 ≤ ε forall y ∈ (Q)1 and all t close enough to 0 . Recall that θρt(y) = EM(αt(y)). Observe that for all y ∈ (Q)1 and all t close enough to 0, y − EM(αt(y))2

2 = y2 2 − EM(αt(y))2 2 ≤ 2(y2 − EM(αt(y))2)

= 2(αt(y)2 − EM(αt(y))2) ≤ 2αt(y) − EM(αt(y))2 ≤ 2ε . So indeed, for all y ∈ (Q)1 and all ρ close enough to 1 we have y − θρ(y)2 ≤ 2ε. Part 2. How to deal with generalized wreath products ? You first have to redo Example 11.2 in the case of generalized wreath products. You will see that the stabilizer subgroups Stab i play an important rˆ

• le. As a result one gets the

following: if Q ⊂ L(H ≀I Γ) and θρ → id uniformly on the unit ball of Q, then

• either Q ≺ L(H(I) ⋊ Stab i) for some i ∈ I,
• or Q ≺ LΓ.

Lecture 3 – page 5

Also spectral gap rigidity becomes more tricky: for the exercise in Part 1 to hold, we need to assume that H and all Stab i are amenable. Under these assumptions: if Q ⊂ L(H ≀I Γ) has a relative commutant without amenable direct summand, then θρ → id uniformly on the unit ball of Q. Part 3. How to deal with the tensor square M ⊗ M given M = LG ? If G = H ≀I Γ, then G × G = H ≀I⊔I (Γ × Γ) so that M ⊗ M is itself a generalized wreath product group von Neumann algebra. So we still have a version of Theorem 12.1 as explained in Part 2 above. Assume that ∆ : M → M ⊗M is the comultiplication given by either a group von Neumann algebra or a group measure space decomposition of M. If Q ⊂ M has no amenable direct summand and if P ⊂ M is amenable, one can show that ∆(Q) ≺ M ⊗P and ∆(Q) ≺ P ⊗M. We will only consider Γ I with Stab i amenable. So if θρ → id uniformly on the unit ball

• f ∆(Q), it follows that ∆(Q) ≺ L(Γ × Γ).

A similar remark holds concerning spectral gap rigidity. If Q1 ⊂ M is non-amenable and P ⊂ M is amenable, then ∆(Q1) is non-amenable relative to M ⊗P and P ⊗M. As a result, if we assume that all Stab i are amenable and if Q′ ∩ M has no amenable direct summand, it still follows that θρ → id uniformly the unit ball of ∆(Q).

14 Conclusion of lecture 3

Under the circumstances of Theorems I, II or IV we are given M = L(H ≀I Γ) with H abelian and Γ I satisfying ‘certain conditions’. We have another group von Neumann algebra or group measure space decomposition for M yielding ∆ : M → M ⊗ M. We impose that all Stab i are amenable. If Γ has property (T) we can right away conclude that ∆(LΓ) ≺ LΓ ⊗ LΓ. By spectral gap rigidity we can reach the same conclusion if

• Γ0 ⊳ Γ1 ⊳ Γ,
• Γ0 is non-amenable and has non-amenable centralizer.

Indeed, by spectral gap rigidity we will get that ∆(LΓ0) ≺ L(Γ × Γ). A version of Theorem 11.1 will imply first that ∆(LΓ1) ≺ L(Γ × Γ) and finally, ∆(LΓ) ≺ L(Γ × Γ).

• Example. Take Γ = Γ0 ≀ S acting on Γ/S whenever Γ0 is non-amenable and S is amenable.

The stabilizer subgroups Stab i are all conjugate with S and hence amenable. Also, Γ0 sits as a normal subgroup with non-amenable centralizer inside Γ(S)

0 , which in turn sits as a normal

subgroup in Γ. To finally conclude that ∆(LΓ) can really be unitarily conjugated into LΓ⊗LΓ and not only virtually (i.e. ∆(LΓ) ≺ LΓ ⊗ LΓ, we need to impose that Γ is an icc group. Lecture 3 – page 6

Lecture 4 – May 17, 2011

Let M = L(H ≀I Γ) as in Theorems I - IV. Let ∆ : M → M ⊗ M be the comultiplication coming from another group von Neumann algebra or group measure space decomposition. At the end of the previous lecture we concluded that in certain cases, ∆(LΓ) can be unitarily conjugated into L(Γ × Γ). In a next step towards proving Theorems I - IV we deduce that in certain circumstances also ∆(L(H(I))) can be unitarily conjugated into L(H(I) × H(I)). As in the previous lecture we rather formulate and prove the following easier –but still pretty hard– version of this phenomenon. This whole lecture and the beginning of the next lecture are devoted to the proof of the following theorem by Ioana [Io10]. Theorem 14.1 (Ioana [Io10]). Let H be a non-trivial abelian group and G = H ≀Γ. Assume that B ⊂ LG is an abelian von Neumann subalgebra that is normalized by a sequence of unitaries vn ∈ LΓ. Assume that vn → 0 weakly. Then one of the following statements hold.

• B ≺ LΓ.
• B′ ∩ LG ≺ L(H(Γ)). Note here that B ⊂ B′ ∩ LG since B is abelian.

The roots of this theorem lie in Popa’s [Po04] who proved that if LG = B ⋊Λ with B abelian and LΛ = LΓ, then B ≺ L(H(Γ)). Note that Ioana actually could prove a statement about abelian von Neumann subalgebras

• f L(G × G) that are normalized by ‘enough’ unitaries in L(Γ × Γ). This is even harder to

prove but an essential ingredient to establish W∗-superrigidity for Bernoulli actions. Finally in [IPV10] we also allow certain generalized wreath products. Things get again more com- plicated but essential to establish W∗-superrigidity for certain group von Neumann algebras. Open problem. Assume that Γ (X, µ) is a free pmp and mixing action, i.e. lim

g→∞

• X

ξ(g · x)η(x) dµ(x) =

• X

ξ(x) dµ(x)

• X

η(x) dµ(x) for all ξ, η ∈ L2(X) . Put A = L∞(X) and M = A ⋊ Γ. Assume that B ⊂ M is an abelian von Neumann subalgebra that is normalized by a sequence of unitaries vn ∈ LΓ tending to 0 weakly. Is it true that either B ≺ LΓ or B′ ∩ M ≺ A ? Note that Theorem 14.1 says that this is indeed true if Γ XΓ

0 is the Bernoulli action.

15 Proof of Theorem 14.1, part 1

A notation that we use all the time. We have G = H ≀ Γ. Whenever S ⊂ G is a subset, we denote by PS the orthogonal projection of ℓ2(G) onto ℓ2(S). We specifically use the following subsets of G = H ≀ Γ:

• H(F)Γ = {ξg | ξ ∈ H(F), g ∈ Γ} whenever F ⊂ Γ (finite or infinite subset),
• H(Γ)S

Lecture 4 – page 1

• the intersection of both, i.e. H(F)S.

Fix the notation and the assumptions of Theorem 14.1. Lemma 15.1. If B ≺ LΓ and ε > 0, there exists a unitary a ∈ U(B) s.t. ELΓ(a)2 < ε.

• Proof. This is just a special case of the definition of B ≺ LΓ.

Lemma 15.2. For all x ∈ LG and all finite subsets F ⊂ Γ we have vnxv∗

n − PH(Γ−F)Γ(vnxv∗ n)2 → 0 .

• Proof. Intuitively the lemma is not so surprising. The analogous statement on the group

level is the following: assume gn ∈ Γ, gn → ∞. Then for all ξg ∈ G we have that gnξgg−1

n

eventually lies in H(Γ−F)Γ. It suffices to prove the lemma for x = uξg, ξ ∈ H(Γ), g ∈ Γ. Put wn = ugv∗

• n. Then wn is a

sequence of unitaries in LΓ and we have to prove that vnuξwn − PH(Γ−F)Γ(vnuξwn)2 → 0 . Write vn in its Fourier expansion: vn =

g∈Γ(vn)gug with (vn)g ∈ C. Then,

PH(Γ−F)Γ(vnuξwn) =

• g∈Γ

(vn)gPH(Γ−F)Γ(uguξwn) =

• g∈Γ

(vn)gPH(Γ−F)Γ(ug·ξugwn) =

• g∈Γ,g·ξ∈H(Γ−F)

(vn)gug uξwn . We finally conclude that vnuξwn − PH(Γ−F)Γ(vnuξwn) =

• g∈Γ,g·ξ∈H(Γ−F)

(vn)gug

• uξwn .

Since the sum on the right hand side is finite and vn → 0 weakly, the conclusion of the lemma follows. We will specifically apply Lemma 15.2 to x ∈ B and consider sequences vnav∗

n for a ∈ B.

Since B is abelian we can say more about these sequences vnav∗

n in the following lemma.

Lemma 15.3. Assume that B ≺ LΓ. Let (an) be a sequence of unitaries in B satisfying an − PH(Γ−F)Γ(an)2 → 0 for all finite subsets F ⊂ Γ (e.g. an = vnav∗

n with a ∈ U(B), thanks to Lemma 15.2 and the

fact that the unitaries vn normalize B). Then for every ε > 0 there exists a finite subset S ⊂ Γ such that an − PH(Γ)S(an)2 ≤ ε for all n . Lecture 4 – page 2

• Proof. Assume by contradiction that we have found ε > 0 such that for every S ⊂ Γ finite

the following holds: lim inf

n

PH(Γ)S(an)2 < 1 − ε . We will deduce the lemma out of two claims. Claim 1. For every b ∈ M ⊖ LΓ and every δ > 0, there exists a finite subset F ⊂ Γ such that lim sup

n

PH(Γ−F)Γ(ban)2 < δ . Claim 2. For every b ∈ M, δ > 0 and F ⊂ Γ finite, there exists S ⊂ F finite such that lim inf

n

(1 − PH(Γ−F)Γ)(anb)2 ≤ δ + lim inf

n

b∞ PH(Γ)S(an)2 . Proof of claim 1. The group version of this statement is again obvious: b corresponds to a fixed element ξg with ξ ∈ H(Γ) − {e} and g ∈ Γ. The sequence an corresponds to a sequence ξngn such that ξn eventually lies in H(Γ−F) whenever F ⊂ Γ is finite. Take F to be the support of ξ. One easily checks that for n large the element ξgξngn never lies in H(Γ−F) since there is a support in F that cannot be canceled by g · ξn. The von Neumann algebra version of claim 1 is slightly more technical but intrinsically

• identical. We may assume that b is a finite linear combination of uξg with ξ ∈ H(Γ) − {e}

and g ∈ Γ. Define F as the union of the supports of these finitely many ξ. It now suffices to prove that PH(Γ−F)Γ(uξgan)2 → 0 whenever supp ξ ⊂ F and ξ = e. (For those who wonder where δ went: this is needed when approximating b by a finite linear combination of uξg.) To prove this statement consider PH(Γ−F)Γ(uξguηh) . If supp η ⊂ Γ − g−1F, the outcome is zero. Hence PH(Γ−F)Γ(uξgan) = PH(Γ−F)Γ

• uξg(1 − PH(Γ−g−1F)Γ)(an)
• .

The right hand side converges to 0 in · 2 by the assumptions on (an). This ends the proof

• f claim 1.

Proof of claim 2. First we do again the group theory analogy. As with claim 1, we have that an corresponds to ξngn and that b corresponds to ηh. Consider ξngnηh. Assume that F ⊂ Γ is a finite subset. For n large enough ξn belongs to H(Γ−F). Define S as the finite set of k ∈ Γ such that k · supp η ∩ F = ∅. Whenever gn lies outside S, also gn · η belongs to H(Γ−F)Γ and hence ξngnηh as well. For gn ∈ S this need no longer be true and this explains the term PH(Γ)S(an) at the right hand side of claim 2. Now we do the real proof. We may again assume that b is a finite linear combination of uηihi, i = 1, . . . , N. Define S ⊂ Γ as the finite subset such that for all k ∈ Γ − S we have that k · supp ηi ⊂ Γ − F for all i = 1, . . . , N. Observe that for all i = 1, . . . , N we have PH(Γ−F)Γ

• PH(Γ)(Γ−S)(an)uηihi
• = PH(Γ−F)(Γ−S)(an)uηihi .

Lecture 4 – page 3

Our assumption on (an) then implies that (1 − PH(Γ−F)Γ)

• PH(Γ)(Γ−S)(an)b
• 2 → 0

It follows that lim inf

n

(1 − PH(Γ−F)Γ)(anb)2 ≤ lim inf

n

(1 − PH(Γ−F)Γ)

• PH(Γ)S(an)b
• 2

≤ lim inf

n

b∞ PH(Γ)S(an)2 . (Again δ disappeared but was needed when approximating b by a finite linear combination

• f uξg.)

We have proven our two claims and now deduce the lemma. Choose ε > 0. Take δ ≪ ε (and we will specify δ later). By Lemma 15.1 take b ∈ U(B) satisfying ELΓ(b)2 < δ. Apply Claim 1 to b − ELΓ(b). This gives us a finite subset F ⊂ Γ such that lim sup

n

PH(Γ−F)Γ(ban)2 ≤ lim sup

n

PH(Γ−F)Γ

• (b − ELΓ(b))an
• 2 + ELΓ(b)2 < 2δ .

Apply Claim 2 to the element b and to the subset F that we have just found. We get a finite subset S ⊂ Γ such that lim inf

n

(1 − PH(Γ−F)Γ)(anb)2 ≤ δ + lim inf

n

PH(Γ)S(an)2 < δ + (1 − ε) . Since B is abelian, we know that anb = ban for all n. So, 1 = anb2

2 = PH(Γ−F)Γ(ban)2 2 + (1 − PH(Γ−F)Γ)(anb)2 2 .

Taking the lim inf we arrive at 1 < (2δ)2 + (δ + (1 − ε))2 which is absurd if we choose in the beginning δ sufficiently small w.r.t. ε. Recall from Lecture 2 the notion of height of an element in a group von Neumann algebra, i.e. the absolute value of the largest Fourier coefficient: whenever x ∈ LΓ, put h(x) = sup

g∈Γ

|τ(xu∗

g)| = sup g∈Γ

|(x)g| where we denote by (x)g ∈ C the g-th Fourier coefficient of x. x =

• g∈Γ

(x)gug . Lemma 15.4. Assume that B ≺ LΓ. Then there exists a δ > 0 such that h(vn) ≥ δ for all n ∈ N. Lecture 4 – page 4

• Proof. Claim. If vn, wn are bounded sequences in LΓ such that h(vn) → 0, then

PH(Γ)S(vnawn)2 → 0 for all a ∈ M ⊖ LΓ and all finite subsets S ⊂ Γ . It suffices to prove the claim for a singleton S = {h} and an element a = uξg with ξ ∈ H(Γ) − {e} and g ∈ Γ. Replacing wn by ugwnu∗

h, it even suffices to take ξ ∈ H(Γ) − {e} and

to prove that EL(H(Γ))(vnuξwn)2 → 0 . One computes that EL(H(Γ))(vnuξwn) =

• g∈Γ

(vn)g(wn)g−1 ug·ξ and therefore EL(H(Γ))(vnuξwn)2

2 =

• g,h∈Γ,g·ξ=h·ξ

(vn)g(wn)g−1(vn)h(wn)h−1 . Denote by F the (non-empty) support of ξ. Define S ⊂ Γ as the finite subset such that kF ∩ F = ∅ if k ∈ Γ − S. So, g · ξ = h · ξ implies that g−1h ∈ S. Hence, EL(H(Γ))(vnuξwn)2

2 ≤

• g∈Γ

|(vn)g| |(wn)g−1|

• h∈gS

|(vn)h| |(wn)h−1| ≤

• g∈Γ

|(vn)g| |(wn)g−1| |S| h(vn) ≤ |S|h(vn) → 0 . So the claim is proven. We now deduce the lemma. Assume by contradiction that, after passage to a subsequence, h(vn) → 0. By Lemma 15.1 take a unitary a ∈ U(B) such that ELΓ(b)2 < 1/3. Using Lemmas 15.2 and 15.3 take a finite subset S ⊂ Γ such that vnav∗

n − PH(Γ)S(vnav∗ n)2 < 1/3

for all n . By the claim above we know that PH(Γ)S(vn(a − ELΓ(a))v∗

n)2 → 0 .

It follows that lim sup

n

PH(Γ)S(vnav∗

n)2 ≤ lim sup n

ELΓ(a)2 < 1/3 . But then lim supn vnav∗

n2 < 2/3, which is absurd since vnav∗ n2 = 1.

Conclusion of Lecture 4 : Lemmas 15.2, 15.3 and 15.4 combined

We combine all lemmas into the following statement that we will use in the next lecture. Under the assumption that B ≺ LΓ, there exists a sequence an ∈ U(B) with the following properties.

• 1. For every finite subset F ⊂ Γ we have that (1 − PH(Γ−F)Γ)(an)2 → 0.

Lecture 4 – page 5

• 2. For every ε > 0, there exists a finite subset S ⊂ Γ such that (1 − PH(Γ)S)(an)2 < ε

for all n.

• 3. There exists δ > 0 and a sequence of finite subsets Fn ⊂ Γ satisfying

3.1. supn |Fn| < ∞, 3.2. Fn → ∞, 3.3. lim infn P(H(Fn)−{e})Γ(an)2 > δ. We find an of the form an = vnav∗

n for a well chosen element a ∈ U(B). By Lemma 15.4

put δ1 = lim infn h(vn) > 0. Take ε ≪ δ1. Choose a ∈ U(B) with ELΓ(a)2 < ε. Define an = vnav∗

n.

We claim that the sequence (an) satisfies all desired properties. Property 1 is given by Lemma 15.2 and property 2 by Lemma 15.3. Since δ1 = lim infn h(vn), take a sequence gn ∈ Γ such that lim infn |(vn)gn| = δ1. Put µn = (vn)gn and note that lim sup

n

vn − µnugn2 =

• 1 − δ2

1 .

Since vn → 0 weakly, we have that gn → ∞. Take a finite subset F ⊂ Γ such that we can approximate a by an element a0 ∈ CH(F)Γ satisfying a0∞ ≤ 1 and a − a02 < ε. Define Fn := gnF. Properties 3.1 and 3.2 are obvious. It remains to prove property 3.3. For n large enough vnav∗

n is at distance at most

• 1 − δ2

1 + ε of µnugnav∗ n.

Next µnugnav∗

n is at distance at most ε from µnugn(a − ELΓ(a))v∗ n.

This last element is at distance at most 2ε from µnugna1v∗

n where we defined a1 = a0−ELΓ(a0).

Note that by construction a1 belongs to C(H(F) − {e})Γ. Therefore µnugna1v∗

n lies in the

image of P(H(gnF)−{e})Γ. We conclude that lim sup

n

(1 − P(H(gnF)−{e})Γ)(an)2 ≤

• 1 − δ2

1 + 4ε .

Taking ε ≪ δ1 in the beginning we can make sure that

• 1 − δ2

1 + 4ε <

√ 1 − δ2 for some δ > 0. Then property 3.3 follows. Lecture 4 – page 6

Lecture 5 – May 18, 2011 16 Ioana’s cute intertwining criterion

Assume that Γ (N, τ) is any trace preserving action and put M = N ⋊ Γ. Every x ∈ M has a Fourier decomposition x =

• g∈Γ

(x)gug with (x)g ∈ N . This Fourier decomposition converges in · 2. We call the elements (x)g the Fourier coefficients of x ∈ N ⋊ Γ. Let P ⊂ N ⋊ Γ be a von Neumann subalgebra. Similarly to the first exercise in Section 10

• ne proves that

P ≺ N iff ∃wn ∈ U(P) such that (wn)g2 → 0 for all g ∈ Γ . In words: P ≺ N iff P contains a sequence of unitaries whose Fourier coefficients tend to zero pointwise in norm · 2. Very remarkably Ioana could prove in [Io10] that pointwise convergence can be replaced by uniform convergence. Theorem 16.1 (Ioana [Io10]). Assume that Γ (N, τ) is any trace preserving action and put M = N ⋊ Γ. Let P ⊂ N ⋊ Γ be a von Neumann subalgebra. Then P ≺ N iff there exists a sequence of unitaries (wn) in U(P) such that sup

g∈Γ

(wn)g2 → 0 as n → ∞.

• Proof. One implication being obvious, assume that δ > 0 and

sup

g∈Γ

(w)g2 ≥ δ for all w ∈ U(P) . We have to prove that P ≺ N. Consider the tensor square M2 = M ⊗ M. Define the subalgebra M1 ⊂ M2 generated by N ⊗ N and the unitaries (ug ⊗ ug)g∈Γ. Observe that EM1(aug ⊗ buh) =

• aug ⊗ buh

if g = h , if g = h . It follows that for all w ∈ M we have EM1(w ⊗ w) =

• g∈Γ
• (w)g ⊗ (w)g
• (ug ⊗ ug) .

So, for all w ∈ U(P) we have EM1(w ⊗ w)2

2 =

• g∈Γ

(w)g4

2 ≥ δ4 .

Lecture 5 – page 1

Denote by P1 the von Neumann algebra generated by the group of unitaries {w ⊗ w | w ∈ U(P)}. The previous line implies that P1 ≺ M1. Denote by σ the flip automorphism of M ⊗ M.

• Claim. P1 = (P ⊗ P)σ where (P ⊗ P)σ := {x ∈ P ⊗ P | σ(x) = x}.

Proof of the claim. The inclusion P1 ⊂ (P ⊗ P)σ is obvious. To prove the converse inclusion, observe that a ⊗ b + b ⊗ a = (a ⊗ 1 + 1 ⊗ a)(b ⊗ 1 + 1 ⊗ b) − (ab ⊗ 1 + 1 ⊗ ab) . Since the elements a ⊗ b + b ⊗ a generate (P ⊗ P)σ it suffices to prove that a ⊗ 1 + 1 ⊗ a belongs to P1 for all a ∈ P. It hence even suffices to prove that p ⊗ 1 + 1 ⊗ p belongs to P1 for every projection p ∈ P. Take a projection p ∈ P. Whenever α ∈ S1 − {1} define the unitary wα ∈ U(P) given by wα = p + α(1 − p). One checks that 1 1 − α(wα ⊗ wα − α21 ⊗ 1) = α(p ⊗ 1 + 1 ⊗ p) . Since the left hand side belongs to P1, the same is true for the right hand side. We let α → 1 to reach the desired conclusion. This ends the proof of the claim. Combining the claim with the statement P1 ≺ M1 proven above, we get that (P ⊗P)σ ≺ M1. If P is not diffuse, the statement P ≺ N always holds. So we may assume that L∞[0, 1] ⊂ P. One then easily finds a unitary w0 ∈ U(P ⊗ P) such that σ(w0) = −w0 and w2

0 = 1. It

follows that w0 normalizes (P ⊗ P)σ and that together they generate P ⊗ P. So, (P ⊗ P)σ is of index 2 in P ⊗ P. Hence, P ⊗ P ≺ M1 as well. A fortiori P ⊗ 1 ≺ M1. If we would have P ≺ N, we can take a sequence of unitaries wn ∈ U(P) such that (wn)g2 → 0 for every g ∈ Γ. It follows that EM1(a(wn ⊗ 1)b)2 → 0 for all a, b ∈ M ⊗ M . (Indeed, it suffices to check this for a = ug ⊗ uh and b = ug′ ⊗ uh′. Then the computation is an exercise.) We have reached a contradiction with P ⊗ 1 ≺ M1.

17 Proof of Theorem 14.1, Part 2

We continue our proof of Theorem 14.1.

• Setup. We have M = L(H ≀ Γ) and B ⊂ M is an abelian von Neumann subalgebra that

is normalized by a sequence of unitaries vn ∈ LΓ tending to zero weakly. We assume that B ≺ LΓ and we have to prove that B′ ∩ M ≺ L(H(Γ)) .

• Terminology. We call a bounded sequence (an) in M clustering if the following two prop-

erties hold.

• For every finite subset F ⊂ Γ we have that (1 − PH(Γ−F)Γ)(an)2 → 0.
• For every ε > 0, there exists a finite subset S ⊂ Γ such that (1 − PH(Γ)S)(an)2 < ε for

all n. Lecture 5 – page 2

End of the proof of Theorem 14.1

At the end of the last lecture we have found δ > 0, a clustering sequence an ∈ U(B) and a sequence of finite subsets Fn ⊂ Γ such that

• supn |Fn| < ∞,
• Fn → ∞,
• lim infn P(H(Fn)−{e})Γ(an)2 > δ.

Assume by contradiction that B′ ∩ M ≺ L(H(Γ)). We will prove that there exists a sequence

• f finite subsets Hn ⊂ Γ such that
• supn |Hn| < ∞,
• Hn → ∞,
• lim infn P(H(Hn)−{e})Γ(an)2 >

√ 2δ. Since this procedure can be iterated, we ultimately find the absurd statement that lim infn an2 can be arbitrarily large. To construct Hn we use Ioana’s intertwining Theorem 16.1. Since we assumed by contra- diction that B′ ∩ M ≺ L(H(Γ)), we can find for every ε > 0 and every κ ∈ N, a unitary d ∈ U(B′ ∩ M) such that PH(Γ)L(d)2 < ε whenever L ⊂ Γ with |L| ≤ κ . We shall exploit the relation an = dand∗ whenever d ∈ U(B′ ∩ M) and we will take a d with extremely spread out Fourier coefficients as given by Ioana’s theorem. Put δ1 := lim infn P(H(Fn)−{e})Γ(an)2. So, δ1 > δ. Take ε ≪ δ1 − δ. Choose S ⊂ Γ finite such that an − PH(Γ)S(an)2 ≤ ε for all n ∈ N . Put κ = supn |Fn|. Take d ∈ U(B′ ∩ M) such that PH(Γ)L(d)2 ≤ ε 2|S| whenever L ⊂ Γ with |L| ≤ κ2 . For n large enough an is at distance at most

• 1 − δ2

1 + ε of P(H(Fn)−{e})Γ(an). Also an is at

distance at most ε from PH(Γ)S(an). Since moreover d is unitary and an = dand∗ we conclude that an is at distance at most

• 1 − δ2

1 + 2ε from

d P(H(Fn)−{e})S(an) d∗ . We approximate d and d∗ by elements in CH(Γ)Γ. More precisely, take finite subsets G, T ⊂ Γ and elements d0, d1 ∈ CH(G)T satisfying d0∞ ≤ 1 , d1∞ ≤ 1 , d − d02 ≤ ε 2|S| , d∗ − d12 ≤ ε 2|S| . Lecture 5 – page 3

Observe moreover that the formula P(HFn−{e})S(x) =

• g∈S
• EL(H(Fn))(xu∗

g) − τ(xu∗ g)1

• ug

proves that P(HFn−{e})S(an)∞ ≤ 2|S|. It follows that d P(H(Fn)−{e})S(an) d∗ is at distance at most 2ε from d0 P(H(Fn)−{e})S(an) d1 . Define Ln := {g ∈ Γ | gFn ∩ Fn = ∅}. So Ln = FnF−1

n

and hence |Ln| ≤ κ2 for all n. Therefore PH(Γ)Ln(d)2 ≤ ε 2|S| . Since also d − d02 ≤ ε/(2|S|), it follows that PH(Γ)Ln(d0)2 ≤ ε |S| . Put Tn = T − Ln. Since d0 ∈ CH(G)T, we conclude that d0 is at distance at most ε/|S| from PH(G)Tn(d0) . Putting all our estimates together, we have found that for all n large enough an is at distance at most

• 1 − δ2

1 + 6ε from

PH(G)Tn(d0) P(H(Fn)−{e})S(an) d1 . Note that the right hand side belongs to ℓ2 H(G∪TSG)(H(TnFn) − {e})Γ

• .

Since moreover (1 − PH(Γ−(G∪T SG))Γ)(an)2 → 0 we conclude that for all n large enough an is at distance at most

• 1 − δ2

1 + 7ε from

ℓ2 (H(TnFn) − {e})Γ

• .

Taking ε ≪ δ1 − δ, we can make sure that

• 1 − δ2

1 + 7ε <

√ 1 − δ2. We conclude that lim inf

n

P(H(TnFn)−{e})Γ(an)2 > δ . By construction the sets Fn and TnFn are disjoint. Putting Hn = Fn ∪ TnFn, it follows that lim inf

n

P(H(Hn)−{e})Γ(an)2 > √ 2δ . Since Tn ⊂ T, we also get that Hn → ∞ and supn |Hn| < ∞. This contradiction concludes the proof of Theorem 14.1. Lecture 5 – page 4

18 How to deal with generalized wreath products and tensor squares

Theorem 18.1 ([IPV10]). Let Γ I be an action such that there exists κ ∈ N with the property that Stab F is finite whenever |F| > κ. Let H be an abelian group. Put G = H ≀I Γ and M = LG. Denote by S the family of subgroups S := {Γ × Stab i, Stab i × Γ | i ∈ I}. Assume that B ⊂ M ⊗ M is an abelian von Neumann subalgebra that is normalized by a sequence of unitaries vn ∈ L(Γ × Γ) such that PS(vn)2 → 0 whenever S ⊂ Γ × Γ is small relative to S. Denote by P the normalizer of B inside M ⊗ M. Write A := L(H(I)). Then at least one of the following statements hold.

• 1. B ≺ M ⊗ 1 or B ≺ 1 ⊗ M.
• 2. P ≺ M ⊗ LΓ or P ≺ LΓ ⊗ M.
• 3. B′ ∩ M ⊗ M ≺ A ⊗ A.

It is illustrative to compare the assumptions and conclusion of Theorems 18.1 and 14.1. Theorem 18.1 PS(vn)2 → 0 for S small relative to S. B ≺ M ⊗ 1 or B ≺ 1 ⊗ M. P ≺ M ⊗ LΓ or P ≺ LΓ ⊗ M. B′ ∩ M ⊗ M ≺ A ⊗ A. Theorem 14.1 vn → 0 weakly. B is not diffuse. B ≺ LΓ (and hence P ≺ LΓ by Thm 11.1). B′ ∩ M ≺ A. We apply Theorem 18.1 when ∆ : M → M ⊗ M is the comultiplication given by another group von Neumann algebra or group measure space decomposition of M. By the conclusion

• f Lecture 3, if we impose enough rigidity on Γ, we may assume that after a unitary conjugacy

∆(LΓ) ⊂ L(Γ × Γ). We put B = ∆(A) where A = L(H(I)). Notice that the normalizer of B contains ∆(M). Several of the potential conclusions of Theorem 18.1 can now be ruled out. Group von Neumann algebra case Now ∆(vs) = vs ⊗ vs for some group von Neumann algebra decomposition M = LΛ.

• 1. B ≺ M ⊗ 1 or B ≺ 1 ⊗ M would imply that A is non-diffuse, which is absurd.
• 2. P ≺ M ⊗ LΓ or P ≺ LΓ ⊗ M would imply that ∆(M) ≺ M ⊗ LΓ or ∆(M) ≺ LΓ ⊗ M,

which in turn would imply that LΓ has finite index in M, which is absurd. So the conclusion ∆(A)′ ∩ M ⊗ M ≺ A ⊗ A holds. Lecture 5 – page 5

Group measure space case Now ∆(bvs) = bvs ⊗ vs for some group measure space decomposition M = L∞(Y ) ⋊ Λ.

• 1. B ≺ M ⊗ 1 would imply that A ≺ L∞(Y ). This means that the Cartan subalgebras

L∞(Y ), A ⊂ M are unitarily conjugate. By OE superrigidity for Bernoulli actions of property (T) groups, the conclusion of Theorem II would follow. On the other hand B ≺ 1 ⊗ M would imply that A is non-diffuse.

• 2. P ≺ LΓ ⊗ M would, as above, imply that LΓ has finite index in M, which is absurd.

This leaves us with two remaining conclusions.

• Either ∆(A)′ ∩ M ⊗ M ≺ A ⊗ A,
• or ∆(M) ≺ M ⊗ LΓ.

Note that the second option is the ‘canonical’ one: viewing M = A ⋊ Γ the canonical comultiplication is given by ∆(aug) = aug ⊗ ug for all a ∈ A and g ∈ Γ. Then ∆(M) ⊂ M ⊗ LΓ. If the second option holds, i.e. ∆(M) ≺ M ⊗ LΓ, it will follow that LΛ ≺ LΓ. It follows from (a variant of) Theorem 14.1 that either L∞(Y ) ≺ LΓ or L∞(Y ) ≺ A. The first

• ption is impossible since Theorem 11.1 would imply that M ≺ LΓ, which is absurd. So

L∞(Y ) ≺ A, meaning that the Cartan subalgebras L∞(Y ), A ⊂ M are unitarily conjugate. By OE superrigidity for Bernoulli actions of property (T) groups, the conclusion of Theorem II would follow. So in the last lecture we will concentrate on the first option: ∆(A)′ ∩ M ⊗ M ≺ A ⊗ A. Lecture 5 – page 6

Lecture 6 – May 20, 2011

We can finally finish the sketch of the proof of Theorems I, II and IV. Assume that H is a non-trivial abelian group and assume that Γ I satisfies the following conditions.

• Γ is icc.
• Γ0 ⊳ Γ1 ⊳ · · · ⊳ Γn = Γ with Γ0 non-amenable.
• Γ0 ⊳ Γ1 either has the relative property (T) or Γ0 has a non-amenable centralizer inside

Γ1.

• There exists κ > 0 such that Stab F is finite whenever F ⊂ I satisfies |F| > κ.
• All Stab i, i ∈ I are amenable.

The two main examples are Γ Γ when Γ is an icc property (T) group and Γ = Γ0 ≀ S acting on I = Γ/S whenever Γ0 is non-amenable and S is infinite amenable. Put G = H ≀I Γ and M = LG. Assume that ∆ : M → M ⊗ M comes from another group von Neumann algebra or group measure space decomposition. Write A = L(H(I)). At the end of the last lecture we arrived at a point where we may assume that after a unitary conjugacy of ∆

• ∆(LΓ) ⊂ L(Γ × Γ),
• ∆(A)′ ∩ M ⊗ M ≺ A ⊗ A.

We again cheat a little bit and assume for simplicity that D := ∆(A)′ ∩ M ⊗ M can be unitarily conjugated into A ⊗ A, i.e. WDW ∗ ⊂ A ⊗ A for some unitary W ∈ M ⊗ M. Since A ⊗ A is abelian, this actually means that WDW ∗ = A ⊗ A. View M ⊗ M as the group measure space construction M ⊗ M = (A ⊗ A) ⋊ (Γ × Γ) . In this picture (W∆(ug)W ∗)g∈Γ is a group of unitaries in (A ⊗ A) ⋊ (Γ × Γ)

• that normalize A ⊗ A,
• that can be unitarily conjugated into L(Γ × Γ) (namely, by Ad W ∗),
• whose action on A ⊗ A is weakly mixing (we de not discuss this in detail, but this is

essentially because ∆(LΓ) cannot be conjugated into L(Γ × Stab i) or L(Stab i × Γ)). So we can apply the following theorem. To keep notations simpler and since the result is entirely general, we call A ⊗ A simply A and Γ × Γ simply Γ. Lecture 6 – page 1

19 A conjugacy criterion for group actions

In the simplified form as we state it here, the following theorem is due to Popa [Po03, Po04]. But above we have cheated by assuming that ∆(A)′ ∩ M ⊗ M can be really unitarily conjugated into A ⊗ A. To keep track of all ‘finite index virtualities’ one has to apply the more general conjugacy criteria in [Io10, IPV10]. Theorem 19.1. Let Γ (X, µ) be a free ergodic pmp action. Denote A = L∞(X) and M = A ⋊ Γ. Let π : Λ → U(M) be a group homomorphism and W ∈ U(M) a unitary with the following properties.

• The unitaries (π(s))s∈Λ normalize A.
• The action (Ad π(s))s∈Λ of Λ on A is weakly mixing.
• We have W ∗π(s)W ∈ L(Γ) for all s ∈ Λ.

Then there exists a unitary U ∈ LΓ, a character ω : Λ → T and a group homomorphism δ : Λ → Γ such that writing V := WU, we have V ∗AV = A and V ∗π(s)V = ω(s)uδ(s) for all s ∈ Λ . Before proving the above conjugacy theorem we recall a few properties of the normalizer of A = L∞(X) inside M = A ⋊ Γ given a free ergodic pmp action Γ (X, µ). Consider the

• rbit equivalence relation R := R(Γ X) and its full group

[R] = {α ∈ Aut(X, µ) | α(x) ∈ Γ · x for a.e. x ∈ X} . There is a natural embedding of [R] into NM(A) : given α ∈ [R] there is a unique unitary Vα ∈ NM(A) satisfying Vαpα

g = ugpα g

where pα

g is the projection in A with support {x ∈ X | α(x) = g · x}.

Moreover V ∗

α a(·)Vα = a(α(·)) for all a ∈ A. Also, NM(A) is the product of the normal

subgroup U(A) and the above embedding of [R]. More concretely, given V ∈ NM(A) define α ∈ [R] such that V ∗a(·)V = a(α(·)). There is a unique unitary Wα ∈ U(A) such that V = VαWα.

• Notation. Given V ∈ NM(A) with corresponding α ∈ [R], define the map ωV : X → TΓ

given by ωV (x) = Wα(x)g if α(x) = g · x .

• Notation. Consider the embedding η : A ⋊ Γ → ℓ2(Γ) ⊗ L2(X) given by η(uga) = δg ⊗ a.

Identify ℓ2(Γ) ⊗ L2(X) with L2(X, ℓ2(Γ)). View ℓ2(Γ) as an LΓ-bimodule. Under all these identifications, if V ∈ NM(A) with corresponding α ∈ [R], then the following holds: η(dV )(x) = η(d)(α(x)) · ωV (x) for all d ∈ M . Lecture 6 – page 2

It suffices to check this equality for those x ∈ X satisfying α(x) = g · x. Then, η(dV )(x) = η(dV pα

g )(x) = η(d ugpα g Wα)(x) = η(dug)(x) Wα(x)

= η(d)(g · x) · ug Wα(x) = η(d)(α(x)) · ωV (x) . Finally, and this is straightforward to check, if V ∈ LΓ then η(V d)(x) = V · η(d)(x) for all d ∈ M and a.e. x ∈ X. A reminder on weak mixing. An ergodic pmp action Λ (X, µ) is called weakly mixing if the diagonal action Λ X ×X given by s·(x, y) = (s·x, s·y) is still ergodic. Assume that Λ (X, µ) is weakly mixing, that (Z, d) is a Polish space and that Λ Z by isometries. If F : X → Z is a measurable map satisfying F(s · x) = s · F(x) for all s ∈ Λ and a.e. x ∈ X, then F is essentially constant. Indeed, the map (x, y) → d(F(x), F(y)) is invariant under the diagonal action, hence constant. The separability of Z forces this constant to be 0 so that F is essentially constant. Proof of Theorem 19.1. Using the above notations, define the measurable map ω(s, ·) : X → TΓ : ω(s, x) := ωπ(s)(x) . Denote by ∗ the action of Λ on (X, µ) induced by Ad π(s), namely π(s)∗a(·)π(s) = a(s ∗ ·). Then η(dπ(s))(x) = η(d)(s ∗ x) · ω(s, x) for all d ∈ M . It follows that ω : Λ × X → TΓ is a 1-cocycle. Define the measurable map ξ : X → ℓ2(Γ) given by ξ = η(W ∗). Denote ρ(s) := W ∗π(s)W. Note that ρ(s) ∈ U(LΓ). By construction ρ(s)W ∗ = W ∗π(s) for all s ∈ Λ. Applying η it follows that ρ(s) · ξ(x) = ξ(s ∗ x) · ω(s, x) for all s ∈ Λ and a.e. x ∈ X . View ℓ2(Γ) as L2(LΓ) so that the formula P(x) := ξ(x)ξ(x)∗ yields a measurable map P : X → L1(LΓ). By construction P(s ∗ x) = ρ(s)P(x)ρ(s)∗ for all s ∈ Λ and a.e. x ∈ X. By weak mixing of Λ X, the map P is essentially constant. Also, for all d ∈ LΓ we have

• X

τ(dP(x)) dµ(x) =

• X

d · η(W ∗)(x), η(W ∗)(x) dµ(x) = η(dW ∗), η(W ∗) = τ(dW ∗W) = τ(d) . Since we already know that P is essentially constant, we must have that P(x) = 1 for a.e. x ∈ X. So ξ(x) ∈ U(LΓ) for a.e. x ∈ X. View TΓ as a closed subgroup of U(LΓ). The formula Q(x) := ξ(x)TΓ defines a measurable map from X to the Polish space U(LΓ)/(TΓ) satisfying Q(s ∗ x) = ρ(s) · Q(x). By weak mixing, Q is essentially constant. So we find a unitary U ∈ LΓ such that U ∗ξ(x) ∈ TΓ for a.e. x ∈ X. By construction U ∗ρ(s)U = ω(s) uδ(s) where ω : Λ → T is a character and δ : Λ → Γ is a group homomorphism. So, U ∗W ∗π(s)WU = ω(s) uδ(s) . It remains to prove that U ∗W ∗AWU = A. But this follows from the fact that U ∗W ∗ is a unitary in M with the property that η(U ∗W ∗) takes values a.e. in TΓ. Lecture 6 – page 3

20 Proof of W*-superrigidity for group von Neumann algebras

Let H be a non-trivial abelian group and assume that Γ I satisfies the above conditions. Put G = H ≀I Γ and M = LG. Theorem IV : If M = LΛ for some countable group Λ, then Λ ∼ = Σ ⋊ Γ for some action Γ Σ by group automorphisms of the abelian group Σ such that Γ Σ is conjugate with Γ HI as pmp actions. Denote by ∆ : M → M ⊗ M : ∆(vs) = vs ⊗ vs for all s ∈ Λ, the comultiplication that comes from the group von Neumann algebra decomposition M = LΛ. The conclusion of Lecture 3 was that ∆ can be unitarily conjugated such that ∆(LΓ) ⊂ L(Γ × Γ). Put A = L(H(I)). Next, the conclusion of Lecture 4 was that ∆(A)′∩M ⊗M ≺ A⊗A. We cheated by assuming the existence of a unitary conjugacy of ∆(A)′ ∩ M ⊗ M into A ⊗ A. From Theorem 19.1 we finally find a unitary Ω ∈ M ⊗ M such that Ω∗∆(A)Ω ⊂ A ⊗ A and Ω∗∆(ug)Ω = ω(g) uδ1(g) ⊗ uδ2(g) for all g ∈ Γ , where δ1, δ2 : Γ → Γ are group homomorphisms and ω : Γ → T is a character. Write Γi := δi(Γ). So, Ω∗∆(M)Ω ⊂ (A ⋊ Γ1) ⊗ (A ⋊ Γ2). Since ∆ is the comultiplication associated with a group von Neumann algebra decomposition, this implies that Γi ≤ Γ are subgroups of finite index. Since (∆ ⊗ id) ◦ ∆ = (id ⊗ ∆) ◦ ∆ it also follows that the unitary representations ω(g) ω(δ2(g)) uδ1(g) ⊗ uδ1(δ2(g)) ⊗ uδ2(δ2(g)) and ω(g) ω(δ1(g)) uδ1(δ1(g)) ⊗ uδ2(δ1(g)) ⊗ uδ2(g) are conjugate by a unitary in M ⊗ M ⊗ M. In particular, the unitary representation on L2(M) given by ξ → uδ1(g)ξu∗

δ1(δ1(g)) cannot by

weakly mixing. This means that there exists an h ∈ Γ such that {ghδ1(g)−1 | g ∈ Γ1 = δ1(Γ)} is finite. So there exists a finite index subgroup Γ0 ≤ Γ1 such that δ1(g) = h−1gh for all g ∈ Γ0. Then Γ0 also has finite index in Γ and since Γ is icc it follows that δ1(g) = h−1gh for all g ∈ Γ. Indeed, since δ1(g) = h−1gh for all g ∈ Γ0 it follows that {ghδ1(g−1k)h−1k−1 | g, k ∈ Γ} is a finite subset of Γ that is stable under conjugacy. So this finite subset is reduced to {e} and δ1 = Ad h−1. We similarly prove that δ2 is an inner automorphism of Γ. So after changing Ω we may assume that Ω∗∆(A)Ω ⊂ A ⊗ A and Ω∗∆(ug)Ω = ω(g) ug ⊗ ug for all g ∈ Γ . As we have seen in Lecture 2 this implies that Ω ∈ LΛ ⊗ LΛ is almost a 2-cocycle, as in the assumptions of Theorem 8.1. By Theorem 8.1 we find a unitary W ∈ M such that Ω = ∆(W ∗)(W ⊗ W). We conclude that ∆(WAW ∗) ⊂ WAW ∗⊗WAW ∗ and ∆(WugW ∗) = ω(g) WugW ∗⊗WugW ∗ for all g ∈ Γ . Theorem IV then follows from the following lemma. Lecture 6 – page 4

Lemma 20.1. Let Λ be a countable group and M = LΛ generated by the unitaries (vs)s∈Λ. Write ∆ : M → M ⊗ M : ∆(vs) = vs ⊗ vs.

• If u ∈ M is a unitary satisfying ∆(u) = u ⊗ u, there exists a unique s ∈ Λ such that

u = vs.

• If A ⊂ M is a von Neumann subalgebra satisfying ∆(A) ⊂ A ⊗ A, there exists a unique

subgroup Σ ≤ Λ such that A = LΣ.

• Proof. Assume that u ∈ M is a unitary satisfying ∆(u) = u ⊗ u. Write u =

s∈Λ(u)svs in

its Fourier decomposition and take s ∈ Λ with (u)s = 0. Note that (u)s = τ(uv∗

s). Hence

(u)s u = (id ⊗ τ)((u ⊗ u)(1 ⊗ v∗

s)) = (id ⊗ τ)(∆(u)(1 ⊗ v∗ s)) = (u)s vs .

Dividing by (u)s gives the desired conclusion. Next let A ⊂ M be a von Neumann subalgebra satisfying ∆(A) ⊂ A⊗A. Define the subsets Σ0 := {s ∈ Λ | vs ∈ A} and Σ1 := {s ∈ Λ | ∃a ∈ A s.t. (a)s = 0} . It is clear that Σ0 is a subgroup of Λ, that Σ0 ⊂ Σ1 and that A ⊂ ℓ2(Σ1). To conclude the proof it suffices to show that Σ1 ⊂ Σ0. Take s ∈ Σ1 and take a ∈ A with (a)s = 0. Note that (id ⊗ τ)(∆(a)(1 ⊗ v∗

s)) = (a)s vs .

By our assumption on A, the left hand side belongs to A. Since (a)s = 0, also vs ∈ A and hence s ∈ Σ0. To also conclude the proof of Theorem I we finally prove the following. Proposition 20.2. Let Γ I be a transitive action with the property that (Stab i) · j is infinite for all i = j. Let H, Σ be non-trivial abelian groups and Γ Σ an action by group

• automorphisms. Assume that

∆ : HI → Σ is a pmp Borel automorphism satisfying ∆(g · x) = g · ∆(x) for all g ∈ Γ and almost all x ∈

• HI. Then there exists an abelian group H1 with |H1| = |H| and a group isomorphism

δ : Σ → H(I)

1

such that δ(g · s) = g · δ(s) for all g ∈ Γ, s ∈ Σ . In the assumptions of Theorem I we also put that |H| is a square-free integer. There is then

• nly one abelian group of order |H| and hence in the above proposition we also get that

H1 = H and hence G = H ≀I Γ = H1 ≀I Γ = Σ⋊Γ = Λ, proving the W∗-superrigidity theorem for group von Neumann algebras.

• Proof. Denote by θ : L(H(I)) → LΣ the isomorphism induced by ∆. Denote by (σg)g∈Γ the

action of Γ by automorphisms of L(H(I)) and LΣ. Fix i0 ∈ I and put Γ0 := Stab i0. View H as a subgroup of H(I) sitting in position i0. Since Γ0 · j is infinite for all j = i0, the following Lecture 6 – page 5

property holds: if X ⊂ L(H(I)) is a finite dimensional vector subspace such that σg(X) = X for all g ∈ Γ0, then X ⊂ LH and σg(x) = x for all x ∈ X and g ∈ Γ0. Define the subgroups H1 ≤ H2 ≤ Σ as H1 := {s ∈ Σ | g · s = s for all g ∈ Γ0} , H2 := {s ∈ Σ | Γ0 · s is a finite set } .

• Claim. We have H1 = H2 and θ(LH) = L(H1). If x ∈ LH, then σg(θ(x)) = θ(x) for all

g ∈ Γ0 and hence θ(x) ∈ LH2. If s ∈ H2, we have that the linear span of all θ−1(ug·s) is a finite dimensional Γ0-invariant subspace of L(H(I)). By the remark above, θ−1(us) ∈ L(H). So we have proven that θ−1(LH2) ⊂ LH and hence LH2 ⊂ θ(LH). We already know the converse inclusion and conclude that θ(LH) = LH2. So all elements in θ(LH) are invariant under (σg)g∈Γ0. This is in particular true for the elements us, s ∈ H2. Hence, H2 ⊂ H1 and the claim is proven. Since θ is a ∗-isomorphism of LH onto LH1 we have |H| = |H1|. Identify I = Γ/Γ0. Since the subalgebras (σg(LH))g∈Γ/Γ0 are independent and generate L(H(I)), it follows that the subgroups (g · H1)g∈Γ/Γ0 of Σ are in a direct sum position and generate Σ. Hence Σ ∼ = H(I)

1

and the proof is done.

21 Proof of Ioana’s W*-superrigidity for Bernoulli actions of property (T) groups

Let Γ be an icc property (T) group. Put A = L∞(XΓ

0 ) and M = A ⋊ Γ.

Theorem II. If M = L∞(Y ) ⋊ Λ is another group measure space decomposition, then the actions Γ XΓ

0 and Λ Y are conjugate.

Put B = L∞(Y ). By Popa’s OE superrigidity for Bernoulli actions of property (T) groups we only have to prove that A and B are unitarily conjugate. It even suffices to prove that A ≺ B or that B ≺ A. Denote by ∆ : M → M ⊗ M the comultiplication given by the group measure space decom- position, i.e. ∆(bvs) = bvs ⊗ vs for all b ∈ B, s ∈ Λ. The conclusion of Lecture 3 was that we can unitarily conjugate ∆(LΓ) into L(Γ × Γ). Next the conclusion of Lecture 5 was that

• either ∆(M) ≺ M ⊗ LΓ,
• or ∆(A)′ ∩ M ⊗ M ≺ A ⊗ A.

At the end of Lecture 5 we also discussed that the first possibility ∆(M) ≺ M⊗LΓ ultimately leads to B ≺ A and hence to the desired conclusion. If the second possibility ∆(A)′ ∩ M ⊗ M ≺ A ⊗ A holds, we cheat a bit and assume that ∆(A)′ ∩ M ⊗ M can be unitarily conjugated into A ⊗ A. By Theorem 19.1 we can unitarily conjugate ∆ and assume that Ω∗∆(A)Ω ⊂ A ⊗ A and Ω∗∆(ug)Ω = ω(g) uδ1(g) ⊗ uδ2(g) for all g ∈ Γ , (21.1) where δ1, δ2 : Γ → Γ are group homomorphisms. Lecture 6 – page 6

• Claim. The kernel K := Ker δ2 is finite. Indeed, we have ∆(LK) ≺ M ⊗ 1. Taking into

account the special form of ∆ given as ∆(bvs) = bvs ⊗ vs, it follows that LK ≺ B. Taking relative commutants we get that B ≺ (LK)′ ∩ M. If K would be infinite, Theorem 11.1 implies that (LK)′ ∩ M ⊂ LΓ and hence B ≺ LΓ. Applying once more Theorem 11.1 leads to the contradiction M = B ⋊ Λ ≺ LΓ. This proves the claim. Assume now that B ≺ A. So we find a sequence of unitaries bn ∈ U(B) such that all Fourier coefficients (bn)g w.r.t. M = A ⋊ Γ tend to zero in · 2. Since Ker δ2 is finite, the special form of ∆ given by (21.1) then implies that EM⊗1(∆(bn))2 → 0 . But ∆(bn) = bn ⊗ 1 by the definition of ∆, which yields a contradiction. This ends the proof

• f Theorem II.

Lecture 6 – page 7

References

[Io06]

• A. Ioana, Rigidity results for wreath product II1 factors. J. Funct. Anal. 252 (2007),

763-791. [Io10]

• A. Ioana, W∗-superrigidity for Bernoulli actions of property (T) groups. J. Amer. Math.
• Soc. 24 (2011), 1175-1226.

[IPV10]

• A. Ioana, S. Popa and S. Vaes, A class of superrigid group von Neumann algebras. Annals
• f Math., to appear. arXiv:1007.1412

[Po03]

• S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I.
• Invent. Math. 165 (2006), 369-408.

[Po04]

• S. Popa, Strong rigidity of II1 factors arising from malleable actions of w-rigid groups,
• II. Invent. Math. 165 (2006), 409-452.

[PV09]

• S. Popa and S. Vaes, Group measure space decomposition of II1 factors and W∗-
• superrigidity. Invent. Math. 182 (2010), 371-417.

References