LetM be a finite von Neumann algebra acting on the standard Hilbert space L2(M)

NARUTAKA OZAWA

Abstract. LetM be a finite von Neumann algebra acting on the standard Hilbert space L²(M). We look at the space of those bounded operators on L²(M) that are compact as operators fromM intoL²(M). The case whereM is the free group factor is particularly interesting.

1. Introduction

In the paper [Oz1], it is proved that the free group factorLFr is solid, i.e., the relative commutantB⁰∩ LFr of any diffuse subalgebraB is amenable. The proof relies on C^∗-algebra techniques. (See [Pe, Po2] for purely von Neumann algebraic proofs of this fact.) In particular, the crucial ingredient in [Oz1] is Akemann and Ostrand’s theorem ([AO]) stating that the ∗-homomorphism

µ: C_λ^∗Fr⊗_algC_ρ^∗Fr 3X

a_k⊗x_k 7→X

a_kx_k+K(`<sup>2</sup>Fr)∈B(`²Fr)/K(`²Fr) is continuous w.r.t. the minimal tensor norm. It would be interesting to know how much of the proof in [Oz1] can be carried out at the level of von Neumann algebras.

In this paper, we will prove a version of Akemann and Ostrand’s theorem in the von Neumann setting. For this purpose, we consider the set of those operators in B(L²(M)) that are compact as operators from M into L²(M), where M = LF^r or any finite von Neumann algebra.

2. Compact operators

Let H be a Hilbert space. We denote by B(H) (resp. K(H)) the C^∗-algebra of all bounded (resp. compact) linear operators on H. Let Ω ⊂ H be a closed balanced bounded convex subset. (Recall that Ω is said to be balanced ifαΩ⊂Ω for α∈C with |α| ≤1.) We define the closed left ideal K^LΩ of B(H) by

K^LΩ ={x∈B(H) :xΩ is norm compact in H}.

We define a seminormk · k_Ω onB(H) by

kxk_Ω = sup{kxξk:ξ∈Ω}.

2000Mathematics Subject Classification. Primary 46L10; Secondary 46L07.

Key words and phrases. von Neumann algebras, free group factors.

The author was supported by JSPS.

1

We will use the following trivial proposition without quoting it.

Proposition. Let Ω ⊂ H be as above. Then, for x ∈ B(H), the following are equivalent.

(1) x∈K^LΩ.

(2) x is weak-norm continuous on Ω.

(3) For every weakly null sequence (ξ_n) in Ω, one has kxξ_nk →0.

(4) For every sequence of (finite-rank) projections (Q_n)strongly converging to 1 on H, one has kx−Q_nxk_Ω →0.

(5) There exists a sequence (x_n) in K(H) such that kx−x_nk_Ω →0.

Definition. We denote by KΩ the hereditary C^∗-subalgebra of B(H) associated with the left ideal K^LΩ:

K^Ω = (K^LΩ)^∗∩K^LΩ = (K^LΩ)^∗·K^LΩ.

Let x ∈ KΩ. For finite-rank projections Q_n % 1 on H, we define x_n =aQ_nb, where b=|x|^1/2 and a=xb⁻¹ are in KΩ. Then, x_n ∈K(H) satisfies kx_nk ≤ kxk and kx−x_nk_Ω+kx^∗−x^∗_nk_Ω →0.

3. Finite von Neumann algebras

Let M be a finite von Neumann algebra with a distinguished faithful normal trace τ, and L²(M) be the GNS-Hilbert space associated with (M, τ). We will write ˆafora∈M when viewed as a vector inL²(M), andkak₂ =kˆak=τ(a^∗a)^1/2. From now on, we set

Ω ={ˆa:a∈M, kak ≤1} ⊂L²(M)

and writeK^M instead ofK^Ω. It is clear that bothM andM⁰ are in the multiplier of K^M. The C^∗-algebra K^M is much larger than K(L²(M)). Indeed, if pn are mutually orthogonal projections inM (or inM⁰) andx_n are compact contractive operators such that x_n = p_nx_np_n, then P

x_n ∈ K^M. The following is useful in understanding the nature of the norm k · k_Ω.

Lemma. For every x∈B(H), one has

kxk_Ω ≤inf{kykkbk₂+kzkkc⁰k₂} ≤4kxk_Ω,

where the infimum is taken over all possible decomposition x = yb+zc⁰ with y, z ∈B(H), b ∈M and c⁰ ∈M⁰.

Proof. Since

kybk_Ω = sup

a∈(M)1

kybˆak ≤ kyk sup

a∈(M)1

kbak ≤ kykb sup

a∈(M)1

kˆbkkak=kykkbk₂ and kzc⁰k_Ω≤ kzkkc⁰k₂ similarly, one has kyb+zc⁰k_Ω≤ kykkbk₂+kzkkc⁰k₂.

To prove the other inequality, let x∈ B(H) be given such that kxkΩ = 1. We observe thatkxk_Ω is nothing but the norm as an operator fromM intoL²(M). It follows from the noncommutative little Grothendieck inequality (Theorem 9.4 in [Pi]), that there are unit vectors ζ, η ∈L²(M) such that kxˆak² ≤ kaζk²+kηak² for all a ∈ M. We view ζ and η as square integrable operators affiliated with M (see Appendix F in [BO] or Chapter IX in [Ta]), and let q = χ_(kxk²_,∞)(ζζ^∗), p=χ_(kxk²_,∞)(η^∗η). It follows that

kxˆak² ≤2 kx(p^⊥ˆaq^⊥)k²+kx(pˆaq^⊥)k²+kx(ˆaq)k²

≤2 kaq^⊥ζk²₂+kηp^⊥ak²₂+kxk²kpak²₂+kxk²kaqk²₂

= 2 kbˆak²₂+kc⁰ˆak²₂ ,

where b is the left multiplication operator by (p^⊥η^∗ηp^⊥+kxk²p)^1/2 and c⁰ is the right multiplication operator by (q^⊥ζζ^∗q^⊥+kxk²q)^1/2. Note that one has b ∈M, kbk ≤ kxk and kbk₂ ≤ kζk₂ = 1; and likewise for c⁰ ∈ M⁰. It follows that there are operators y, z ∈B(H) withyy^∗+zz^∗ ≤2 such that x=yb+zc⁰.

The “cb-version” of the normk · kΩ is defined to be kxk_τ = sup{(X

kxˆa_nk²)^1/2 : (a_n)^∞_n=1 ∈M such that X

a_na^∗_n≤1}

=kx: M⁰ 3a⁰ 7→xa⁰ˆ1∈L²(M)_colk_cb

= inf{kykkbk₂ :y∈B(H) and b ∈M with x=yb}.

We do not elaborate on this norm here. See [Ma] for more information about the topology associated with this norm.

4. Free group factors

We write λ and ρ respectively for the left and the right regular representation of a countable discrete group Γ on `²Γ. Recall that the group Γ is in the classS if it is exact and the ∗-homomorphism

µ: C_λ^∗Γ⊗_algC_ρ^∗Γ3X

a_k⊗x_k7→X

a_kx_k+K(`<sup>2</sup>Γ)∈B(`²Γ)/K(`<sup>2</sup>Γ) is continuous w.r.t. the minimal tensor norm. Free groups as well as hyperbolic groups are in the classS. (See [Oz2].) Let Γ be an ICC group so that the group von Neumann algebra LΓ = λ(Γ)<sup>00</sup> ⊂ B(`²Γ) is a factor. We note that L²(LΓ) is canonically isomorphic to `²Γ. We denote RΓ = (ρ(Γ))⁰⁰ = (LΓ)⁰ and consider the ∗-homomorphism

π: C^∗(LΓ,RΓ)3X

a_kx_k 7→X

a_k⊗x_k ∈ LΓ⊗_minRΓ⊂B(`<sup>2</sup>Γ⊗`²Γ), which is well-defined by Takesaki’s theorem on the minimal tensor norm. The following theorem extends Akemann and Ostrand’s theorem ([AO]).

Theorem. Let Γ be an ICC group which is in the class S. Then, one has kerπ =K^LΓ∩C^∗(LΓ,RΓ).

Proof. Take any sequence (ξ_n) of unit vectors in Ω = {ˆa : a ∈ LΓ, kak ≤ 1}, which weakly converges to 0. We define a stateω onC^∗(LΓ,RΓ) by the Banach limit ω(x) = Limhxξ_n, ξ_ni. Let y ∈ alg(LΓ,RΓ) be arbitrary. Since yΩ ⊂ K_yΩ for some constant K_y >0, one hasω(y^∗ · y)≤K_y²τ(·) both on LΓ and on RΓ.

Therefore, the GNS representation π_ω of ω is binormal on C^∗(LΓ,RΓ). More- over, sinceK^LΓ∩C^∗(LΓ,RΓ)⊂kerπ_ω, the ∗-homomorphism fromC_λ^∗Γ⊗_algC_ρ^∗Γ into π_ω(C^∗(LΓ,RΓ)) is continuous w.r.t. the minimal tensor norm. It follows from Lemma 9.2.9 in [BO] that the ∗-homomorphism from LΓ ⊗_alg RΓ into π_ω(C^∗(LΓ,RΓ)) is continuous w.r.t. the minimal tensor norm, too. Because of simplicity ofLΓ, this means thatπ_ω(C^∗(LΓ,RΓ)) = LΓ⊗_minRΓ, or equivalently that kerπ_ω = kerπ. Therefore, K^LΓ∩C^∗(LΓ,RΓ)⊂ kerπ. On the other hand, if x≥0 and x /∈ K^LΓ, then there is a normalized weakly null sequence (ξ_n) in Ω such thatω(x²) = Limkxξ_nk² >0 anda fortiori x /∈kerπ_ω. It follows thatK(`²Γ)⊂kerπ ⊂K^LΓ. The first inclusion is strict. Indeed, it is not hard to show that kerπ is non-separable. It is likely that the second is strict as well.

Recall that a finite von Neumann algebra N has the property (Γ) if there is a sequence (u_n) of unitary elements in N such that u_n → 0 ultraweakly and [u_n, a] → 0 ultrastrongly for every a ∈ N. We observe the following: Let M ⊂ B(L²(M)) be a finite von Neumann algebra and N ⊂ M be a von Neumann subalgebra with the property (Γ). Then, one hasK^M∩C^∗(N, M⁰) ={0}. Indeed, if (un) is as above, then on the one hand u^∗_nxun → x for every x ∈ C^∗(N, M⁰), but on the other handu^∗_nxu_n→0 for everyx∈K^M. This observation, combined with the above theorem, implies the main theorem of [Oz1]: A von Neumann subalgebra of LΓ which has the property (Γ) is necessarily amenable.

5. Boundary of free group factors

LetFr be the free group of rank r ∈ N. For each t ∈Fr, we define χ_t ∈ `^∞Fr

to be the characteristic function of the set of those elements in Fr whose last segments in the reduced forms aret. Let

A =C^∗({χ_t:t ∈Fr})⊂`^∞Fr

and observe that [A, C_λ^∗Fr]⊂K(`²Fr). Indeed, λ(s)χ_tλ(s)^∗δ_x =χ_tδ_x if|x| ≥ |s|+

|t|, and hence [χ_t, λ(s)] has finite rank. It is well-known thatB :=C^∗(A, ρ(Fr))∼= Ao Fr is nuclear. Akemann and Ostrand’s theorem stating thatFr is in the class S follows from this and

[C_λ^∗Fr, B]⊂norm-cl C_λ^∗Fr·[C_λ^∗Fr, A]

⊂K(`²Fr).

It would be interesting to know whether a similar fact holds true at the level of von Neumann algebras. Namely,

Problem. Is it true that [A,LFr]⊂K^LFr?

We recall Popa’s theorem ([Po1]) stating that every derivation from a von Neumann algebra M ⊂ B(H) into K(H) is inner. In particular, [x, M] ⊂ K(H) only ifx∈K(H) +M⁰. Nevertheless, the above problem has a positive answer if r= 1, i.e., if Fr =Z. Indeed, letχ=χn≥0 for simplicity. Then, the projection χ is the Riesz projection which is bounded onL⁴(bZ) (or on anyL^q with 1< q <∞, see [Ga]). It follows from the H¨older inequality that for everya ∈ L(Z)∼=L^∞(bZ), one has

k[χ, a]k_Ω ≤ kχk_2,4kak4,∞+kak_2,4kχk4,∞≤Ckak_L4(Z)b ,

wherek · k_p,q stands for the operator norm fromL^q(bZ) into L^p(bZ). Since C_λ^∗Zis dense inLZw.r.t. theL⁴-norm and [χ, C_λ^∗Z]∈K(`²Z), one obtains that [χ,LZ]∈ K^LZ. The author is unable to extend this argument to Fr with r≥2, because he does not know whether the ‘Riesz projection’ on Fr is a bounded operator from LFr intoL⁴(LFr).

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Department of Mathematical Sciences, University of Tokyo, 153-8914 E-mail address: [email protected]