Explicit computations of all finite index bimodules for a family of - - PowerPoint PPT Presentation

Explicit computations of all finite index bimodules for a family of II1 factors

Fields Institute Workshop Von Neumann Algebras Stefaan Vaes

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II1 factor without finite index subfactors

Trivial subfactor : 1 ⊗ N ⊂ Mn(C) ⊗ N. Take Γ = SL(2, Q) ⋉ Q2, with Γ ↷ (X, µ) = (X0, µ0)Q2 . Theorem (V, 2007) The II1 factor M = L∞(X) ⋊Ωα Γ has no non-trivial finite index subfactors if

◮ (X0, µ0) atomic with atoms of different weights, ◮ α ∈ R \ {0} and Ωα ∈ Z2(Γ, S1) defined by

Ωα x1

x2

• ,

y1

y2

• = exp(2πiα(x1y2 − x2y1))
• n Q2 and extended to Γ by SL(2, Q)-invariance.

Moreover, the II1 factor M remembers (X0, µ0) and α.

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Other results and plan of talk

◮ Introduce finite index bimodules

and fusion algebra of II1 factors.

◮ Present the first explicit computations of fusion algebras

• f II1 factors :
• Identification with Hecke algebras.
• Crucial ingredients :

Popa’s deformation/rigidity and cocycle superrigidity.

◮ Theorem (V, 2007, generalizing Popa – V, 2006).

Every countable group arises as the outer automorphism group

• f a II1 factor.

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Connes’ correspondences

A representation theory of II1 factors

Let M be a type II1 factor with trace τ.

◮ A right M-module is a Hilbert space with a right action of M.

Example : L2(M, τ)M.

◮ Always, HM ≅

• i∈I piL2(M)

and one defines dim(HM) =

• i τ(pi) ∈ [0, +∞].

Complete invariant of right M-modules. Definition An M-M-bimodule of finite Jones index, is an M-M-bimodule MHM satisfying dim(HM) < ∞ and dim(MH) < ∞. Denote FAlg(M) the set of (equiv. classes) of finite index bimodules.

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The structure of the fusion algebra FAlg(M )

Example : let α ∈ Aut(M) and define the M-M-bimodule H(α)

• n L2(M) by

a · ξ · b = aξα(b) Precisely those MHM with dim(MH) = 1 = dim(HM). Example : let M ⊂ M1 be a finite index subfactor. Then, ML2(M1)M belongs to FAlg(M). In general, FAlg(M ) carries the following structure.

◮ Direct sum of bimodules. ◮ Identity element : ML2(M)M. ◮ Connes’ tensor product H ⊗M K.

Example : H(α) ⊗M H(β) ≅ H(α ◦ β).

◮ Contragredient MHM of MHM.

FAlg(M) consists of the generalized symmetries of M.

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Group-like elements in FAlg(M )

We call MHM group-like if H ⊗M H is the identity. Example of group-like element MHM :

◮ H = L2(M)p. ◮ α : M → pMp an isomorphism. ◮ a · ξ · b = aξα(b).

Set Out(M) = Aut(M) Inn(M) . Observation We have a short exact sequence e → Out(M) → {group-like bimodules} → F(M) → e where F(M) is the fundamental group of M.

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Abstract fusion algebras

Definition A fusion algebra A is a free N-module N[G], equipped with

◮ an associat. distribut. product :

x ∗ y =

• z∈G

mult(z, x ∗ y)z,

◮ a multiplicative neutral element e ∈ G, ◮ a contragredient map x ֏ x which is ...,

such that Frobenius reciprocity holds : for all x, y, z ∈ G, we have mult(z, x ∗ y) = mult(x, z ∗ y) = mult(y, x ∗ z) Examples

◮ N[Γ] for a group Γ. ◮ Rep(G), the finite dim. unitary rep. of a compact group G. ◮ FAlg(M) of a II1 factor M.

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Fusion algebra of a Hecke pair

Let Γ < G be a Hecke pair, i.e. [Γ : gΓg−1 ∩ Γ] < ∞ for all g ∈ G. Hecke fusion algebra H (Γ < G) = {ξ : Γ\G/Γ → N | ξ has finite support } (ξ ∗ η)(g) =

• h∈G/Γ

ξ(h) η(h−1g) Γ\G/Γ is the set of irreducibles with fusion rules ... Example Let T be k-valent tree and G < Aut(T) countable dense subgroup. Choose a vertex e and set Γ = Stab e. Identify Γ\G/Γ = Γ\T ≅ N via the distance to e. Then, mult(n, a ∗ b) = #{t ∈ T | |et| = a , |ts| = b} when |es| = n. A way to understand the Hecke fusion algebra of PSL(2, Z) < PSL(2, Q).

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Computations of FAlg(M )

Data : action of Γ on a countable set I, base probability space (X0, µ0). Action : Γ ↷ (X, µ) = (X0, µ0)I. II1 factor : M = L∞(X) ⋊ Γ. Theorem (V, 2007) Under the right conditions on Γ ↷ I and for (X0, µ0) atomic with distinct weights, we have Repfin(Γ) → FAlg(M) → H (Γ < G) where G = CommPerm I(Γ) = commensurator of Γ inside Perm I. For every Hecke pair Γ < G, one defines Hrep(Γ < G). Then, FAlg(M) ≅ Hrep(Γ < G) .

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Concrete examples

Recall : M = L∞(X) ⋊ Γ and Γ ↷ (X, µ) = (X0, µ0)I. Γ ↷ I FAlg(M) SLn(Z) ⋉ Qn ↷ Qn for n ≥ 2. Hrep(SLn(Z) < GLn(Q)) Λ < PSLn(Q) proper subgroup, relative ICC, Γ = Λ × PSLn(Q) ↷ PSLn(Q) by left-right action. Hrep(Λ < CommPGLn(Q)(Λ)). Z < R < Q strict inclusions of rings, i.e. R = Z[P−1] Write some Λ0 < Λ = SL2(Q)⋉Q2. Take Λ0 × Λ ↷ Λ. Add 2-cocycle. Hrep(R∗ ⋉ R < Q∗ ⋉ Q) Relation with Bost- Connes Hecke algebra. SL2(Q) ⋉ Q2 ↷ Q2 Add 2-cocycle. Trivial.

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We continue with M = L∞(X) ⋊ Γ and Γ ↷ (X, µ) = (X0, µ0)I. Our isomorphism FAlg(M) ≅ Hrep(Γ < CommPerm I(Γ)) works for

Good actions of good groups

A condition on the group Γ :

◮ Γ admits an infinite, almost normal subgroup with relative (T).

Some conditions on the action Γ ↷ I :

◮ Transitivity. ◮ Stab i0 acts with infinite orbits on I − {i0}. ◮ Minimal condition on stabilizers :

no infinite sequence (in) with Stab(i0, . . . , in) strictly decreasing.

◮ A faithfulness condition of Γ → Perm I.

About the minimal condition on stabilizers. Automatic if Γ ↷ I embeds in GL(V) ⋉ V ↷ V for fin.dim. V. For left-right action Λ × Λ ↷ Λ : equivalent with minimal condition on centralizers of Λ.

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How to find all finite index M - M - bimodules

Take Γ ↷ I a good action of a good group. Let M = L∞(X) ⋊ Γ and Γ ↷ (X, µ) = (X0, µ0)I. Take a finite index M-M-bimodule MHM.

1 The bimodule H contains a finite index L(Γ)-L(Γ)-subbimodule.

Main ingredients : Popa’s deformation/rigidity and the minimal condition on stabilizers.

2 The bimodule H contains a finite index L∞(X)-L∞(X)-subbimodule. 3 The bimodule H belongs to Hrep(Γ < CommAut(X,µ)(Γ))

Main ingredient : Popa’s cocycle superrigidity.

4 Identification of CommAut(X,µ)(Γ) and CommPerm I(Γ)

Main ingredient : Stab i0 acts with infinite orbits on I − {i0}. Attention : two difficult slides follow with steps 1 and 2.

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Every bimodule is L(Γ)-preserving (Step 1)

Let M = L∞(X) ⋊ Γ and Γ ↷ (X, µ) = (X0, µ0)I. Take a finite index M-M-bimodule MHM. Simplifying assumptions : H = H(ψ) for ψ ∈ Aut(M) and Γ has property (T). Aim : ψ(L(Γ)) and L(Γ) are unitarily conjugate.

◮ Popa’s malleability (roughly) :

• flow (αt)t∈R on L∞(X × X),
• commuting with diagonal Γ-action,
• connecting id = α0 with flip = α1.

◮ Extend αt to L∞(X × X) ⋊ Γ. Set P = ψ(L(Γ)) ⊂ (L∞(X) ⊗ 1) ⋊ Γ.

By property (T), α0(P) and α1(P) are unitarily conjugate. Then, ψ(L(Γ)) and L(Γ) are unitarily conjugate.

◮ Popa proves this for Γ ↷ (X, µ) mixing

(meaning here that Stab i is finite for all i ∈ I)

Mixing is replaced by minimal condition on stabilizers.

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Every L(Γ)-preserving bimodule is Cartan-preserving (Step 2) Let M = L∞(X) ⋊ Γ and Γ ↷ (X, µ) = (X0, µ0)I. Take a finite index M-M-bimodule MHM. Simplifying assumptions : H = H(ψ) for ψ ∈ Aut(M) and ψ(L(Γ)) = L(Γ) (because of Step 1). Aim : ψ(L∞(X)) and L∞(X) are unitarily conjugate.

◮ Fix i0 ∈ I and set Γ0 = Stab i0.

Recall : Stab i0 acts with infinite orbits on I − {i0}. First aim : ψ(L(Γ0)) and L(Γ0) are unitarily conjugate.

◮ If not, ψ(L(Γ0)) will be far from L(Γ0) leading to

ψ(L(Γ0))′ ∩ M ⊂ L(Γ). Contradiction.

◮ So, we may assume ψ(L(Γ0)) = L(Γ0).

Take relative commutants : ψ

• L∞

(X0, µ0){i0} ⋊ Γ0

• = L∞

(X0, µ0){i0} ⋊ Γ0.

◮ Play a game to get ψ(L∞(X)) and L∞(X) conjugate.

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Cartan preserving bimodules and cocycle superrigidity

Let M = L∞(X) ⋊ Γ and Γ ↷ (X, µ) = (X0, µ0)I. Take a finite index M-M-bimodule MHM. Simplifying assumptions : H = H(ψ) for ψ ∈ Aut(M) and ψ(L∞(X)) = L∞(X) (because of Step 2). Restriction of ψ to L∞(X) yields an

• rbit equivalence ∆ : X → X : ∆(Γ · x) = Γ · ∆(x)

a.e. Associated Zimmer 1-cocycle : ∆(g · x) = ω(g, x) · ∆(x). Cocycle superrigidity theorem (Popa, 2005) Let Γ ↷ (X, µ) = (X0, µ0)I. Assume

◮ H < Γ almost normal and relative (T), ◮ H acts with infinite orbits on I, ◮ G either a countable or a compact group.

Every 1-cocycle for Γ ↷ (X, µ) with values in G, is cohomologous with a homomorphism Γ → G.

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Every Cartan preserving bimodule is elementary (Step 3)

Still M = L∞(X) ⋊ Γ and ψ ∈ Aut(M) satisfies ψ(L∞(X)) = L∞(X).

◮ Apply cocycle superrigidity with target group Γ :

• ψ(F) = F ◦ ∆−1

and ψ(ug) = ωg uδ(g),

• with ∆ ∈ NormAut(X,µ)(Γ),
• ωg ∈ L∞(X)

1-cocycle ω with values in S1.

◮ Apply cocycle superrigidity with target group S1 :

• ψ is given by

∆ ∈ NormAut(X,µ)(Γ) and ω ∈ Char(Γ).

What if the bimodule MHM is no longer of the form H(ψ) If Γ ↷ I is a good action of a good group, every finite index bimodule MHM is described by

◮ An element of

CommAut(X,µ)(Γ) instead of NormAut(X,µ)(Γ).

◮ A fin.dim. rep.

π ∈ Repfin(Γ) instead of ω ∈ Char(Γ).

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The commensurator of Γ Γ Γ inside Aut(X ,µ µ µ) (Step 4)

Consider the plain Bernoulli action Γ ↷ (X, µ) = (X0, µ0)Γ . Open problem : determine the centralizer (let alone the commensurator) of Γ inside Aut(X, µ). Consider next generalized Bernoulli action Γ ↷ (X, µ) = (X0, µ0)I. Observation If Γ ↷ I is transitive and Stab i0 acts with infinite orbits on I − {i0}, there is a natural isomorphism CommAut(X,µ)(Γ) ≅ CommPerm I(Γ) × Aut(X0, µ0) . We kill Aut(X0, µ0) by taking µ0 atomic with distinct atoms. Explanation of the observation : L∞ (X0, µ0){i0} =

• L∞(X, µ)

Stab i0

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About the outer automorphism group Out( M )

Theorem (V, 2007, generalizing Popa - V, 2006) Let Γ ↷ I be a good action of a good group. Set Γ0 = Stab i0 and M = L∞ (X0, µ0)I ⋊ Γ . Then, Out(M) ≅ Aut(X0, µ0) ×

• Char Γ ⋊ Aut(Γ0 ⊂ Γ)

Ad Γ0

• and these II1 factors remember (X0, µ0) and the inclusion Γ0 ⊂ Γ.

Example Taking (X0, µ0) atomic with distinct weights and taking PSL(n, Z) ↷ P(Qn) for n ≥ 3 and odd, we get a continuum of II1 factors M with Out(M) trivial.

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Every countable group arises as Out(M )

Let Λ be an arbitrary ICC group satisfying the minimal condition on centralizers and assume moreover that Λ is indecomposable as a direct product. With (PSL(n, Z) × Λ × Λ) ↷ P(Qn) × Λ , we get Out(M) ≅ Out Λ

× Z/2Z × Char Λ × Char Λ

(Bumagin, Wise, 2004) Every countable group arises as Out Λ for such kind of group Λ. Λ is given quite concretely as a subgroup of a C′(1/6)-small cancelation group (Rips’ construction). Being a little smarter, one can kill Z/2Z and assume Char Λ = {e}. Every countable group arises as the outer automorphism group of a II1 factor.

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Existence theorems about Out( M )

We just saw : Out(M) can be any countable group in a constructive manner. But, we also have the following existence theorems :

1 Ioana, Peterson, Popa, 2004

Out(M) can be any compact abelian, second countable group.

2 Falgui`

eres - V, 2007 Out(M) can be any compact, second countable group.

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