On the structural theory of II 1 factors of negatively curved groups - - PowerPoint PPT Presentation

On the structural theory of II1 factors of negatively curved groups

IONUT CHIFAN (joint with THOMAS SINCLAIR)

Vanderbilt University

Workshop “II1 factors: rigidity, symmetry and classification” IHP, Paris, May, 2011

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Summary

◮ Cocycles, quasi-cocycles, arrays associated with group

representations π : Γ → O(H) and the “small cancellation” property

◮ Structural results for von Neumann algebras arising from such

groups {LΓ, L∞(X) ⋊ Γ}; structure of normalizers for certain subalgebras, uniqueness of Cartan subalgebra and applications to W ∗-superrigidity; some structural results for the orbit equivalence class of such groups

◮ Brief outline of our approach

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Cocycles

Let π : Γ → O(H) be an orthogonal representation.

Definition

A cocycle is a map c : Γ → H satisfying the cocycle identity c(γ1γ2) = π(γ1)c(γ2) + c(γ1), for all γ1, γ2 ∈ Γ. A cocycle is proper if the map γ → c(γ) is proper, i.e., for all K > 0, {γ ∈ Γ : c(γ) ≤ K} < ∞. Examples: large classes of amalgamated free products, HNN extensions, etc

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Quasi-cocycles

Definition

Let (π, H) be a representation of Γ. A quasi-cocycle is a map q : Γ → H satisfying the cocycle identity up to bounded error, i.e., there exists D ≥ 0 such that q(γ1γ2) − πγ1(q(γ2)) − q(γ1) ≤ D, for all γ1, γ2 ∈ Γ. Examples: Gromov hyperbolic groups admit proper quasi-cocycles into left regular representation (Mineyev, Monod, Shalom ’04)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Arrays

Definition

Let (π, H) be a representation of Γ. An array is a map q : Γ → H satisfying the following properties :

• 1. πγq(γ−1) = −q(γ) (anti-symmetry);
• 2. supλ∈Γ πγq(λ) − q(γλ) < ∞ for all γ ∈ Γ (bounded

equivariance);

• 3. γ → q(γ) is proper.

proper cocycle ⇓ proper quasi-cocycle ⇓ array

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Examples

◮ If Γ is a lattice in Sp(n, 1), n ≥ 2 then Γ has no proper (even

unbounded) cocycle into any representation. (Delorme, Guichardet)

◮ Z2 ⋊ SL2(Z) has no proper quasi-cocycle into any

representation (Burger - Monod), but has an array into a representation weakly contained in the left regular representation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Arrays “prevent” commutation: A group Γ that admits an array into the left regular representation does not have non-amenable subgroups Λ < Γ with infinite centralizer CΓ(Λ).

Proof.

Λ is non-amenable there exist K > 0 and F ⊂ Λ finite s.t. for all ξ ∈ ℓ2(Γ) we have ξ ≤ K

s∈F πs(ξ) − ξ. For all λ ∈ CΓ(Λ):

q(λ) ≤ K1

• s∈F

πs(q(λ) − q(λ) ≤ K1

• s∈F

q(sλ) − q(λ) + K 2

1 |F|

= K1

• s∈F

q(λs) − q(λ) + K 2

1 |F|

= K1

• s∈F

− πλsq(s−1λ−1) + πλ(q(λ−1)) + K 2

1 |F|

= K1

• s∈F

− πs−1q(λ−1) − (q(s−1λ−1)) + K 2

1 |F| ≤ 2K 2 1 |F|

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Solidity and strong solidity

Definition (Ozawa)

A II1 factor M is called solid if for any diffuse A ⊂ M the relative commutant A′ ∩ M is amenable. Any solid, non-amenable factor M is prime, i.e., M ≇ M1 ¯ ⊗M2, for any M1, M2 diffuse factors.

Definition (Ozawa-Popa)

A II1 factor M is called strongly solid if for any amenable subalgebra A ⊂ M its normalizing algebra NM(A)′′ is amenable. Any strongly solid, non-amenable factor M does not have Cartan subalgebra. In particular, it cannot be decomposed as group measure space construction M ∼ = L∞(X) ⋊ Γ.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

History

◮ Popa (’81) - LFS is prime and has no Cartan subalgebras for

any S uncountable

◮ Voiculescu (’96) - LFn, n ≥ 2 have no Cartan subalgebras ◮ Ge (’98) - LFn, n ≥ 2 is prime ◮ Ozawa (’03) - LΓ solid, Γ Gromov hyperbolic ◮ Popa (’06) M solid, M admits“free malleable” deformation ◮ Peterson (’06) LΓ solid, b : Γ → ℓ2Γ, proper cocycle ◮ Ozawa - Popa (’07) LFn, n ≥ 2 strongly solid ◮ Ozawa - Popa (’08) LΓ strongly solid, Γ a lattice in SO(2, 1),

SO(3, 1), or SU(1, 1)

◮ Ozawa (’08) L(Z2 ⋊ SL2(Z)) is solid ◮ Sinclair (’10) LΓ strongly solid, Γ a lattice in SO(n, 1) or

SU(n, 1)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

History

◮ Popa (’81) - LFS is prime and has no Cartan subalgebras for

any S uncountable

◮ Voiculescu (’96) - LFn, n ≥ 2 have no Cartan subalgebras ◮ Ge (’98) - LFn, n ≥ 2 is prime ◮ Ozawa (’03) - LΓ solid, Γ Gromov hyperbolic ◮ Popa (’06) - M solid, M admits“free malleable” deformation ◮ Peterson (’06) - LΓ solid, b : Γ → ℓ2Γ, proper cocycle ◮ Ozawa - Popa (’07) - LFn, n ≥ 2 strongly solid ◮ Ozawa - Popa (’08) - LΓ strongly solid, Γ a lattice in

SO(2, 1), SO(3, 1), or SU(1, 1)

◮ Ozawa (’08) - L(Z2 ⋊ SL2(Z)) is solid ◮ Sinclair (’10) - LΓ strongly solid, Γ a lattice in SO(n, 1) or

SU(n, 1)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: solidity and strong solidity

Theorem (C - Sinclair, ’11)

Let Γ be an icc, exact group that admits an array into the left regular representation. Then LΓ is solid. It recovers some of the earlier results of Ozawa ’03 and Peterson ’06.

Theorem (C - Sinclair, ’11)

Let Γ be an icc, weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. Then LΓ is strongly solid. Examples: all hyperbolic groups (by De Canniere-Haagerup, Cowling-Haagerup, Ozawa)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique prime decomposition

Theorem (C - Sinclair ’11)

Let {Γi}n

i=1 be exact, icc groups that admit an array into the left

regular representation. If LΓ1 ¯ ⊗LΓ2 ¯ ⊗ · · · ¯ ⊗LΓn ∼ = N1 ¯ ⊗N2 ¯ ⊗ · · · ¯ ⊗Nm then n=m and there exist t1 · · · tn = 1 such that after a permutation of indices (LΓi)ti ∼ = Ni for all 1 ≤ i ≤ n. This result was proven by Ozawa - Popa ’03 for Γi hyperbolic or a lattice is a rank one, connected, simple, Lie group, and by Peterson ’06, for Γi admitting a proper 1-cocycle into the left regular representation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Theorem (C - Sinclair - Udrea ’11)

Let {Γi}n

i=1 be an icc, weakly amenable, exact group that admits a

proper quasi-cocycle into the left regular representation. If A ⊂ LΓ1 ¯ ⊗LΓ2 ¯ ⊗ · · · ¯ ⊗LΓn = M is an amenable subalgebra such that A′ ∩ M is amenable (e.g. A is either a MASA or an irreducible subfactor of M) then NM(A)′′ is amenable.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique Cartan subalgebra

Theorem (Ozawa - Popa ’07)

For any Fn X free, ergodic, p.m.p. weakly compact action L∞(X) is the unique Cartan subalgebra of L∞(X) ⋊ Fn, up to unitary conjugation.

Theorem (C - Sinclair ’11)

Let Γ be an weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. For any Γ X p.m.p. weakly compact action L∞(X) is the unique Cartan subalgebra of L∞(X) ⋊ Γ, up to unitary conjugation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique Cartan subalgebra, cont.

Theorem (C - Sinclair - Udrea ’11)

Let Γi be an icc, weakly amenable, exact group that admits a proper quasi-cocycle into the left regular representation. For any Γ1 × Γ2 × · · · × Γn X p.m.p. weakly compact action L∞(X) is the unique Cartan subalgebra of L∞(X) ⋊ (Γ1 × Γ2 × · · · × Γn), up to unitary conjugation.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: W ∗-superrigidity results

Theorem (Ioana, ’08)

Let Γ be a property (T) group and Γ X be a profinite, free, ergodic, p.m.p. action. If Λ Y is any p.m.p. action that is orbit equivalent to Γ X then the two actions are virtually conjugate.

Theorem (C - Sinclair ’11)

Let Γ be an icc, property (T), hyperbolic group (e.g. Γ a lattice in Sp(n, 1) with n ≥ 2). Then any p.m.p. compact action Γ X is virtually W ∗-superrigid. Applying the previous theorems we obtain the same result for actions by products of such groups.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: Structural results for orbit equivalence class

• f hyperbolic groups

Theorem (C - Sinclair ’11)

Let Γ be a countable discrete group which is orbit equivalent to a hyperbolic group. If Σ < Γ is an infinite, amenable subgroup, then its normalizer NΓ(Σ) is amenable. The method we employ is purely von Neumann algebraic. This theorem recovers and extends upon early results of Adams ’94 and Monod-Shalom ’04.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Main results: unique measure-equivalence decomposition results for product groups

Theorem (C - Sinclair ’11)

Let Γ = Γ1 × · · · × Γn be a product of countable discrete groups which admit arrays in the left regular representation and let Λ = Λ1 × · · · × Λm be a product of arbitrary countable discrete

• groups. If Γ ∼

=ME Λ then n = m and after permuting indices then Γi ∼ =ME Λi for all 1 ≤ i ≤ n. This should be compared with earlier results Monod - Shalom ’04 and Sako ’09.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Ideas behind the proofs of main results

General strategy: combines the full strength from the following methods of approach:

◮ topological (Ozawa) - C ∗-algebraic techniques (exactness,

local reflexivity)

◮ cohomological (Peterson, Popa, Ozawa - Popa, Sinclair) -

proper cocycles into good representations; von Neumann algebraic techniques (Popa’s deformation/rigidity theory, closable derivations, etc)

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: “small action” by normalizers

Theorem (Ozawa - Popa ’07, Ozawa ’10)

Let Γ be an i.c.c. hyperbolic group and let A ⊂ LΓ = M be a diffuse amenable subalgebra. Then the natural action NM(A)

ad(u)

A is weakly compact: There exists a net of vectors ηn ∈ L2(A¯ ⊗A)+ such that

• 1. (v ⊗ ¯

v)ηn − ηn → 0 for all v ∈ U(A);

• 2. [u ⊗ ¯

u, ηn] → 0, for all u ∈ NM(A);

• 3. (x ⊗ 1)ηn, ηn = τ(x) = (1 ⊗ ¯

x)ηn, ηn, for all x ∈ M.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: a family of deformations

• I. Exponentiate quasi-cocycle q : Γ → ℓ2(Γ) to obtain a

“deformation” on LΓ: Γ X gaussian action; H = L2(X) ⊗ ℓ2(Γ) For t ∈ R consider υt : Γ → U(L∞(X)) defined by υt(γ)(x) = exp(itq(γ)(x)) Vt ∈ B(H) unitaries by letting Vt(ξ ⊗ δγ) = (υt(γ)ξ) ⊗ δγ αt(x) = VtxV ∗

t ∈ Aut(B(H))

Let P : H → ℓ2(Γ) is the orthogonal projection. Note: in general αt moves elements from LΓ ⊂ B(H) outside LΓ; however it moves elements from C ∗

r (Γ) into the extended uniform

Roe algebra C ∗

u (Γ X).

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: a family of deformations

Proposition

LΓ ⊂ B(H); for every t > 0 the operator mt = P · αt · P is “compact”; it transforms WOT-convergent sequences of LΓ into SOT-convergent sequences. properness, quasi-cocycle

Proposition

The family αt well behaved on C ∗

r (Γ) ⊂ LΓ as t → 0

(αt(x) − x) · P∞ → 0 as t → 0, for all x ∈ C ∗

r (Γ).

bounded equivariance is essentially used!

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: deformation/rigidity arguments a la Ozawa-Popa

Let A ⊂ LΓ = M and assume that N = NM(A)′′ is irreducible. From Ozawa -Popa result let ˜ ηn,t = (Vt ⊗ 1)ηn ζn,t = (P ⊗ 1)˜ ηn,t ξn,t = (P⊥ ⊗ 1)˜ ηn,t ∈ L2

0(X) ⊗ ℓ2(Γ)

Lemma

lim

n ξn,t ≥ 5

12. Compactness of mt = P · αt · P together with the fact that ηn “converges to the diagonal” is essentially used! traciality of ηn’s is not necessary in this case.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Outline: deformation/rigidity arguments a la Ozawa-Popa

Define a state Φt on P = B(H) ∩ ρ(Mop)′ by letting Φt(x) =

1 Limnξn,t2 (x ⊗ 1)ξn,t, ξn,t for every x ∈ P.

Lemma

For every ǫ > 0 and every finite set K ⊂ C ∗

r (Γ) with

dist·2(y, (N)1) ≤ ǫ for all y ∈ K one can find tǫ > 0 and a finite set LK,ǫ ⊂ NM(A) such that ((yx − xy) ⊗ 1)ξn,t, ξn,t ≤ 4ǫ + 2

• v∈LK,ǫ

[v ⊗ ¯ v, ηn], for all y ∈ K, x∞ ≤ 1,tǫ > t > 0, and n. Use successively triangle inequality, deformation property and traciality.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Lemma

For every ǫ > 0 and any finite set F0 ⊂ U(N) there exist a finite set F0 ⊂ F ⊂ M, a c.c.p. map ΨF,ǫ : span(F) → C ∗

r (Γ), and

tǫ > 0 such that Φtǫ(ΨF,ǫ(u)∗xΨF,ǫ(u)) − Φtǫ(x) ≤ 47ǫ, for all u ∈ F0 and x∞ ≤ 1. local reflexivity (implied by exactness) to locally approximate LΓ through C ∗

r (Γ) completely contractively: ∀ finite subset F ⊂ LΓ

and ǫ > 0, there exists φ : span{F ∪ F∗} → C ∗

r (Γ) c. c.p. , such

that φ(x) − x2 < ǫ for all x ∈ F. use Haagerup criterion to show that N is amenable.

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Open Problems

Problem

Describe the class of countable discrete groups which are orbit equivalent to some hyperbolic group.

Problem

If Γ admits an array into the left regular representation, then does Z ≀ Γ admit an array into a representation weakly contained in the left regular?

Conjecture

If Γ is a hyperbolic group, then any free, ergodic p.m.p. action Γ X gives rise to a von Neumann algebra with unique Cartan

• subalgebra. Open even for Γ = Fn with n ≥ 2!

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups

Thank you for listening!

Ionut Chifan(Vanderbilt University) II1 factors of negatively curved groups