An inner amenable group whose von Neumann algebra does not have property Gamma

c 2012 by Institut Mittag-Leffler. All rights reserved

An inner amenable group whose von Neumann algebra does not have property Gamma

by

Stefaan Vaes

KU Leuven Leuven, Belgium

1. Introduction

In order to exhibit two non-isomorphic II₁ factors, Murray and von Neumann defined in [MN] property Gamma as the existence of a non-trivial sequence of asymptotically central elements. They showed that the group von Neumann algebra LFn of the free groupFn, n2, does not have property Gamma, while the group von Neumann algebra LS_∞ of the group of finite permutations ofNhas property Gamma.

More precisely, a II₁ factor M with trace τ has property Gamma if there exists a sequence of unitary operatorsx_ninM satisfyingτ(x_n)=0 for allnandx_ny−yx_n2!0 for ally∈M. Here · ₂ denotes theL²-norm onM given byx₂=

τ(xx^∗).

In [E], Effros aims to express property Gamma for a group von Neumann algebraLG in terms of a group-theoretic property. In this respect, he introduced the notion ofinner amenability for a countable group G, by requiring the existence of a mean on G\{e} which is invariant under all inner automorphisms. More precisely,Gis inner amenable if there exists a finitely additive measuremon the subsets ofG\{e}, with total mass 1 and satisfying m(gXg⁻¹)=m(X) for all X⊂G\{e} and allg∈G. In [E], Effros proved that if LGis a II₁ factor with property Gamma, then Gis inner amenable. He posed the question whether the converse holds: doesLGhave property Gamma wheneverGis an inner amenable group with infinite conjugacy classes (icc)? This problem attracted a lot of attention over the years, see e.g. [H, Problem 2] and the survey [BH]. In attempts to answer Effros’ question, several groups were first shown to be inner amenable (e.g.

Thompson’s group [J1] and Baumslag–Solitar groups [St]), but later shown to satisfy property Gamma as well (e.g. [J2]).

Partially supported by ERC Starting Grant VNALG-200749, Research Programme G.0231.07 of the Research Foundation Flanders (FWO) and KU Leuven BOF research grant OT/08/032.

We solve Effros’ question in the negative by providing concrete examples of inner amenable icc groupsGsuch thatLGdoes not have property Gamma. Our construction is inspired by [Sc, Example 2.7], which provides examples of strongly ergodic group actions that do not have spectral gap.

2. Construction of the group G

Fix a sequence of distinct prime numbersp_n. We define as follows a countable groupG.

Define

H_n:=

Z pnZ

₃

and K:=

∞ n=0

H_n.

Put Λ=SL(3,Z), which acts onHnby automorphisms in the natural way. We denote this action by g·x wheneverg∈Λ and x∈Hn. We let Λ act onK diagonally: (g·x)n=g·xn

for allg∈Λ andn∈N. For everyN∈N, define the subgroupKN<K as

KN:=

∞ n=N

Hn.

We putG₀=KΛ and inductively defineG_N+1as the following amalgamated free prod- uct:

G_N  //G_N+1:=G_N∗K_N(K_N×Z).

Note here that we viewKN as a subgroup ofGN by consideringKN<K <G₀<GN. We finally defineGas the inductive limit of the increasing sequence of groupsG₀⊂G₁⊂.... Theorem 1. The group G is inner amenable and has infinite conjugacy classes, while the II₁ factor LG does not have property Gamma.

3. Proof of Theorem 1

We denote by LG the group von Neumann algebra of a countable group G, generated by the unitary operators {u_g}g∈G. We denote by {δ_g}g∈G the canonical orthonormal basis of²(G). Then,²(G) is anLG-LG-bimodule, given byu_gδ_ku_h=δ_gkh. OnLG, we consider the usual trace given byτ(x)=δ_e, xδ_e.

Lemma2. For every g∈G\K,the set {hgh⁻¹:h∈Λ}is infinite. Also,Ghas infinite conjugacy classes.

Proof. If g∈G\G₀, take N0 such that g∈GN+1\GN. From the description of GN+1 as the amalgamated free product GN+1=GN∗K_N(KN×Z), it follows that the elements hgh⁻¹, h∈GN, are all distinct. In particular, {hgh⁻¹:h∈Λ} is infinite. If g∈G₀\K, the set {hgh⁻¹:h∈Λ} is infinite because Λ has infinite conjugacy classes.

Finally, assume that g=e has a finite conjugacy class. By the first part of the proof, g∈K. TakingN large enough, g∈K\K_N. So, g∈G_N\K_N and we arrive at the contradiction thatg has a finite conjugacy class inG_N∗K_N(K_N×Z).

Denote by (A_n, τ) the tracial von Neumann algebra withA_n∼=C²and with minimal projectionse_n and 1−e_n such thatτ(e_n)=p⁻³_n .

Lemma 3. Define (A, τ):=∞

n=0(An, τ). There is a unique trace-preserving bijec- tive isomorphism

α:A−!LG∩(LΛ) satisfying

α(e_n) =p⁻³_n

h∈H_n

u_h for all n.

Moreover, α(en)lies in the center of LGn+1.

Proof. By Lemma 2,LG∩(LΛ)=LK∩(LΛ). PutBn=^∞(Hn) and define the trace τ onBn given by the normalized counting measure. ViewAn⊂Bn in a trace-preserving way and such thaten corresponds to the functionχ_{0}.

Define Λ^θB_n by (θ_g(F))(x)=F(g⁻¹·x) for all g∈Λ, x∈H_n and F∈B_n. Define Λ^σLKbyσ_g(u_x)=u_g·xfor allg∈Λ andx∈K. We haveH_n∼=H_n, and the Fourier trans- form yields a trace-preserving isomorphismα_n:B_n!LH_nsatisfyingα_n θ_g=σ_(g−1)^T α_n. Here,g^T denotes the transpose ofg∈Λ=SL(3,Z).

Put (B, τ)=∞

n=1(B_n, τ) and define Λ^θB diagonally. The isomorphismsα_n com- bine into a trace-preserving isomorphism α:B!LK satisfying α θ_g=σ_(g−1)^T α for all g∈Λ. We view A as a von Neumann subalgebra ofB. In order to prove thatα is an isomorphism ofA ontoLK∩(LΛ), we have to show thatB^Λ=A, where, by definition, B^Λ={b∈B:θ_g(b)=bfor allg∈Λ}.

The orbits of the diagonal action ΛH₀×...×HN are precisely the setsU0×...×UN, where everyUi is either{0} orHi\{0}. Hence,

N n=0

B_{n Λ}

=

N n=0

A_n.

LettingN!∞, we get thatB^Λ=A.

Denote byZ(M) the center of a von Neumann algebraM. Observe that Z(LGm)∩LKm⊂ Z(LGm+1).

Sinceα(e_n)∈Z(LG₀) andα(e_n)∈LK_m for allmn, we haveα(e_n)∈Z(LG_n+1).

Proof of Theorem 1. We saw in Lemma 2 thatGhas infinite conjugacy classes.

Embed LG !²(G) by x!xδ_e. Define ξ_n=p³n^/2α(e_n)δ_e. Then, ξ_n is a sequence of unit vectors in ²(G) satisfying δe, ξn=p⁻³n ^/2!0. Moreover, by Lemma 3, we have α(en)∈Z(LGn+1), so that ugξnu^∗_g=ξn wheneverg∈GN andNn+1. Hence, for every g∈G, the sequence ugξnu^∗_g−ξn2 is eventually zero. It follows that the adjoint repre- sentation of G on ²(G)Cδe weakly contains the trivial representation. Hence, G is inner amenable (see e.g. [BH, Th´eor`eme 1]).

Suppose thatxnis a sequence of unitary operators inLG, such thatxny−yxn2!0 for ally∈LG. We have to prove thatxn−τ(xn)12!0. Denote byπ:G!U(²(G)) the adjoint representation, defined byπ(g)ξ=ugξu^∗_g. Thenξn:=xnδeis a sequence of almost π-invariant unit vectors. Denote byP the orthogonal projection of²(G) onto the closed subspace of π(Λ)-invariant vectors. Since Λ has property (T), it follows that

ξn−P(ξn)2!0.

This means that x_n−y_n2!0, where y_n:=E_LG∩(LΛ)(x_n) and E_LG∩(LΛ) denotes the unique trace-preserving conditional expectation ofLGontoLG∩(LΛ).

As we have seen in the proof of Lemma 3, we have LG∩(LΛ)=LK∩(LΛ). In particular,y_n belongs to the unit ball ofLK. Recall that we inductively defined

G_N+1=G_N∗K_N(K_N×Z).

Denote bygN+1 the canonical generator of the copy ofZappearing in this definition of GN+1. Using the fact that ug_N+1 commutes withLKN, we get that

u_gN+1y_nu^∗_g

N+1−y_n=u_gN+1(y_n−E_LKN(y_n))u^∗_g

N+1+u_gN+1E_LKN(y_n)u^∗_g

N+1−y_n

=u_gN+1(y_n−E_LKN(y_n))u^∗_g

N+1+E_LKN(y_n)−y_n.

Because the setsgN+1(K\KN)g⁻¹_N+1 andK are disjoint, we have that the elements ug_N+1

yn−ELK_N(yn)

u^∗_gN+1 andELK_N(yn)−yn are orthogonal. Hence,

ug_N+1ynu^∗_gN+1−yn2ug_N+1(yn−ELK_N(yn))u^∗_gN+12=yn−ELK_N(yn)2.

So, for everyN, we get thaty_n−E_LKN(y_n)₂!0 asn!∞.

FixN. Asyncommutes withLΛ, alsoELK_N(yn) commutes withLΛ. By Lemma 3, take a sequencean in the unit ball of∞

k=N(Ak, τ) such that ELK_N(yn)=α(an). Since the sequencep⁻³_n is summable, the product of the projections 1−en,nN, converges to a minimal projectionfN in∞

k=N(Ak, τ), with τ(f_N) =

∞ n=N

(1−p⁻³_n ).

Putε_N=1−τ(f_N). An arbitraryain the unit ball of∞

k=N(A_k, τ) then satisfies a−τ(a)1₂4√ε_N.

Sinceτ(a_n)=τ(E_LKN(y_n))=τ(y_n), it follows that for allN andnwe have E_LKN(y_n)−τ(y_n)1₂4√ε_N.

AsεN!0 whenN!∞, and since for every fixedNwe haveyn−ELK_N(yn)2!∞when n!∞, we conclude thatyn−τ(yn)12!0. So, sincexn−yn2!0, also

xn−τ(xn)12!0.

4. Concluding remarks

The groupGconstructed above is not finitely generated. It seems impossible to modify our construction to provide finitely generated counterexamplesG, although we strongly believe that such examples exist.

The construction in this paper is inspired by the following similar phenomenology in ergodic theory of group actions, exhibited by Schmidt [Sc, Example 2.7]. LetG(X, μ) be a measure-preserving action of a countable groupGon a standard non-atomic prob- ability space (X, μ). The actionG(X, μ) is said to bestrongly ergodicif the following implication holds: wheneverUn⊂X is a sequence of almost invariant measurable subsets (i.e.μ(g·UnUn)!0 for allg∈G), thenμ(Un)(1−μ(Un))!0. The action G(X, μ) is said to have spectral gap if the Koopman representation G!U(L²(X)C1) does not weakly contain the trivial representation. It is easy to see (e.g. [BHV, Proposition 6.3.2]) that spectral gap implies strong ergodicity. In [Sc, Example 2.7], Schmidt shows that the converse can fail.

Finally, we illustrate the subtlety of the difference between inner amenability and property Gamma. Let G be an icc group and consider the Hilbert space ²(G) as an LG-LG-bimodule. We denote byC_r^∗Gthe reduced groupC^∗-algebra of G, viewed as a weakly denseC^∗-subalgebra ofLG.

The following are true:

• G is inner amenable if and only if there exists a sequence ξn of unit vectors in ²(G) such thatξn(e)=0 for allnandaξn−ξna2!0 for alla∈C_r^∗G.

This follows from the fact that Gis inner amenable if and only if the adjoint rep- resentation ofGon ²(G\{e}) weakly contains the trivial representation (see e.g. [BH, Th´eor`eme 1]).

• LGhas property Gamma if and only if there exists a sequenceξn of unit vectors in ²(G) such thatξ_n(e)=0 for all nand aξ_n−ξ_na₂!0 for alla∈LG.

This follows from the characterization of property Gamma in [C, Theorem 2.1 (c)].

Acknowledgment. I would like to thank Sorin Popa for pointing me towards [Sc].

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Stefaan Vaes

Department of Mathematics KU Leuven

Celestijnenlaan 200B BE-3001 Leuven Belgium

[email protected] Received March 18, 2010