Amenable correspondences

III. STRONG SOLIDITY FOR GROUP FACTORS FROM LATTICES IN SO(N, 1) AND

III.3 Amenable correspondences

Definition III.2.6. A II₁ factor M is said to have the complete metric approximation property (CMAP) if there exists a net (ϕi) of finite rank, normal, completely bounded maps ϕi :M →M such that lim supkϕ_ik_cb≤1 and such that kϕ_i(x)−xk₂ →0, for allx∈M.

If Γ is an i.c.c. countable discrete group, thenLΓ has the CMAP if and only if the Cowling-Haagerup constant of Γ, Λ_cb(Γ), equals 1, and if Γ is a lattice in G, then Λ_cb(Γ) = Λ_cb(G), cf. §12.3 of [7]

and [32].

Theorem III.2.7(Ozawa–Popa, Theorem 3.5 of [60]). LetM be aII₁ factor which has the CMAP.

Then for any diffuse amenable∗-subalgebraA⊂M,N_M(A) acts weakly compactly onA by conju- gation.

3. there exists a P-central state Φ onN which restricts to τ on N.

We do so by constructing a normal, faithful, semi-finite (tracial) weight ¯τ onN which canonically realizesH⊗_MH¯asL²(N,τ¯). An identical construction to the one we propose has already appeared in the work of Sauvageot [85] for an arbitrary factor M. However, we present an elementary approach in the II1 case.

Before presenting the details, we pause here to illustrate how Theorem III.3.2 generalizes (rel- ative) amenability for II₁ factors. Let M be a II₁ factor and H = L²M ⊗L²M¯ be the coarse M–M correspondence. We then have explicitly that M = B(H)∩(M^o)⁰ = B(L²M) ¯⊗M; hence, M has an M-central state which restricts to the trace on M if and only if the trace on M ex- tends to a hypertrace on B(L²M), i.e., M is amenable. Similarly, if P, Q⊂ M are von Neumann subalgebras andH=L²hM, e_Qi, where hM, e_Qi denotes the basic construction of Jones [40], then hM, e_Qi ⊂B(H)∩(M^o)⁰. So, if P ⊂M satifies one of the conditions in the above theorem, thenP is relatively amenable to QinsideM (PlM Q) in the sense of Theorem 2.1 in [60]. Conversely, if PlMQ, then using condition (4) in Theorem 2.1 of [60], one can construct such a functional Φ on B(H)∩(M^o)⁰ as in condition (3) in the above theorem forP ⊂M.

We recall from Chapter II that the right-bounded vectors form a dense subspace ofHwhich we will denote by H_b. We also recall, regardingH as a right HilbertM-module, that we can define a natural M-valued inner product on H_b, which we will denote (ξ|η) ∈M for ξ, η ∈ H_b, by setting (ξ|η) to be the Radon-Nikodym derivative of the normal functionalx7→ hξx, ηi.

Let N = B(H) ∩πM^o(M^o)⁰. (For instance, if ˜M ⊃ M is a tracial inclusion of II1 factors and H= L²( ˜M), considered as a Hilbert M–M bimodule in the natural way, then we have that (ˆx|ˆy) =EM(y^∗x) for all x, y∈M˜ and B(H)∩(M^o)⁰ =hM , e˜ Mi, whereEM is the trace preserving conditional expectation from ˜M onto M.) For ξ, η ∈ H_b, let T_ξ,η : H_b → H_b be the “rank one operator” given by T_ξ,η(·) = ξ(· |η). Then T_ξ,η extends to a bounded operator with kT_ξ,ηk²_∞ ≤ k(ξ|ξ)k_∞k(η|η)k_∞ [81]. Notice that T_ξ,ξ ≥0 and that T_ξ,ξ is a projection if (ξ|ξ) ∈ P(M). Since T_ξ,ηπ_M^o(x) =π_M^o(x)T_ξ,η for allx∈M^o, we have thatT_ξ,η ∈ N. It is easy to see that the span of all such operatorsT_ξ,ηis a∗-subalgebra ofB(H) which we will denote byN_f. Noticing that for any S ∈ N,S(H_b)⊂ H_b, we have thatS T_ξ,η =T_Sξ,η and T_ξ,ηS =T_ξ,S^∗_η. It follows thatN_f is an ideal of N which can be considered as the analog of the finite rank operators in B(H). The following

lemma further cements this analogy.

Lemma III.3.3. If M is a II₁ factor, we have that N_f⁰ ∩B(H) =π_M^o(M^o); hence, N_f +C1_B(H) is weakly dense in N.

Proof. The inclusionπM^o(M^o)⊂ N_f⁰ ∩B(H) is trivial. Conversely, let T ∈ N_f⁰ ∩B(H) and choose a non-zero ζ ∈ H_b such that (ζ|ζ) = p ∈ P(M). (One can always find such a ζ as H has an orthonormal basis of right bounded vectors as a right Hilbert M-module.) Choose a sequence η_i ∈ H_b such that kη_i −T ζk → 0 and let y_i = (η_i|ζ). Then for every ξ ∈ H_b we have that kT(ξp)−ξy_ik ≤ kT_ξ,ζk_∞kη_i−T ζk ≤ k(ξ|ξ)k^1/2_∞ kη_i−T ζk so, the sequence (π_M^o(y^o_i)) converges in the strong topology to T ◦π_M^o(p^o). Hence, T ◦π_M^o(p^o) ∈ π_M^o(M^o)⁰⁰ = π_M^o(M^o). Since M is a II1 factor, by repeating the argument with ζ⁰ = ζu for u ∈ U(M) and using standard averaging techniques, we conclude that there exists yT ∈ M such that T ξ = ξyT for all ξ ∈ H. Thus T =πM^o(y^o_T).

Now, consider an element ϕ∈M∗, and define a functional ¯ϕ∈(N_f)^∗ by ¯ϕ(T_ξ,η) =ϕ((ξ|η)). It is easy to see that ¯ϕis normal onN_f and so, by the preceding lemma, may be extended to a normal semi-finite weight on N. Hence, we may construct for each such ϕ a noncommutative L^p-space overN, L^p(N,ϕ) =¯ {T ∈ N :kTk_p = ¯ϕ(|T|^p)^1/p<∞}. If M is a II1 factor with trace τ, then ¯τ is a normal, faithful, semi-finite trace onN and we denoteL^p(N,¯τ) simply byL^p(N). In the case of L²(N), we compute that kT_ξ,ηk²₂ =τ((ξ|ξ)(η|η)) =hξ(η|η), ξi. This shows that the map which sends T_ξ,η to the elementary M-tensor ξ⊗_M η¯∈ H ⊗_M H¯ extends to an N-N bimodular Hilbert space isometry fromL²(N) toH ⊗_M H. We are now ready to prove the motivating result in this¯ section.

Proof of Theorem III.3.2. The proof of (1) ⇔ (3) follows the usual strategy. For (1) ⇒ (3), we have that there exists a net (ξ_n) of vectors in H ⊗_M H¯ such that hxξ_n, ξ_ni → τ(x) for all x ∈ N and k[u, ξ_n]k →0 for allu ∈ U(P). Viewing ξ_n as an element ofL²(N), let Φ_n∈ N_∗ be given by Φ_n(T) = ¯τ(ξ_nξ_n^∗T) for any T ∈ N. Then, by the generalized Powers-Størmer inequality (Theorem IX.1.2 in [95]), we have that |Φ_n(x)−τ(x)| → 0 for all x ∈N and kAd(u)Φ_n−Φnk₁ → 0 for all u∈ U(P). Taking a weak cluster point of (Φ_n) inN^∗gives the requiredN-tracialP-central state on N. Conversely, given such a state Φ, we can find a net (ηn) in L¹(N)+ such that Φn(T) = ¯τ(ηnT)

weakly converges to Φ. In fact, by passing to convex combinations we may assume k[u, η_n]k₁ →0 for allu∈ U(P). By another application of the generalized Powers-Størmer inequality, it is easy to check that ξn=ηn^1/2 ∈ L²(N)∼=H ⊗_MH¯ satisfies the requirements of (1).

We now need only show (2)⇒(3) as (3)⇒(2) is trivial. But this is exactly the averaging trick found in the proof of (2) ⇒ (1) in Theorem 2.1 of [60]. We repeat the argument here for the sake of completeness. Since Φ is normal onN, we have that for someη∈ L¹(N)+, Φ(x) =τ(ηx) for all x ∈N. In fact, η ∈ L¹(P⁰∩N)₊ since Φ is P-central. Denoting by F the net of finite subsets of U(P⁰∩N) under inclusion, for anyε >0, we setη_F = _|F¹_|P

u∈Fuηu^∗ andξ_F,ε= (χ_[ε,∞)(η_F))η_F^−1/2. We now let Ψ_F,ε(T) = _|F¹_|P

u∈FΦ(uξ_F,εT ξ_F,εu^∗). Note that Ψ_F,ε is stillP-central. Now it is easy to see that limF,εχ_[ε,∞)(η_F) =z, where zis the central support of η. But by the faithfulness of Φ on Z(P⁰∩N), we see that z= 1. Hence, any weak cluster point of (Ψ_F,ε)F,ε inN^∗ is aP-central state which when restricted to N isτ.

Corollary III.3.4(generalized Haagerup’s criterion for amenability). Let N,M be II₁ factors and H anN-M correspondence. If P ⊂N is a von Neumann subalgebra, then His left amenable over P (in the sense of Theorem III.3.2) if and only if for every non-zero projection p∈ Z(P⁰∩N) and finite subset F ⊂ U(P), we have

kX

u∈F

up⊗upk_H⊗

MH,∞¯ =|F|, where k · k_H⊗

MH,∞¯ denotes the operator norm on B(H ⊗_M H).¯

Proof. Since we have obtained a “hypertrace” characterization of amenability for correspondences in Theorem III.3.2, the result follows by the same arguments as in Lemma 2.2 in [31].

Definition III.3.5 (cf. Definition 1.3 in [65]). Let M be a II₁ factor,H anM-M correspondence and P^c the orthogonal projection onto the subspace H^c = {ξ ∈ H : xξ = ξx, ∀x ∈ M}. The correspondence H has spectral gap if for every ε > 0 there exist δ > 0 and x₁, . . . , x_n ∈ M such that ifkx_iξ−ξx_ik< δ,i= 1, . . . , n, thenkξ−P^cξk< ε. The correspondenceHhasstable spectral gap ifH ⊗_M H¯ has spectral gap.

Note that if H has stable spectral gap, then H is amenable if and only if (H ⊗_M H)¯ ^c 6=

{0}. Hence, we say an M-M correspondence H is nonamenable if it has stable spectral gap and

(H ⊗_M H)¯ ^c = {0}. The following theorem is the analog of Lemma 3.2 in [77] for the category of correspondences. N.B. Stable spectral gap as defined in [77] corresponds to our definition of nonamenability.

Theorem III.3.6. LetM be aII₁ factor andHanM-M correspondence. ThenHis nonamenable if and only if H ⊗_M K¯ has spectral gap and for any M-M correspondenceK.

Proof. LetH_b,K_bdenote subspaces of right-bounded vectors inHandK, respectively. Givenξ ∈ H_b and η ∈ K_b, by the same arguments as above we can define a bounded operator T_ξ,η :K → H by T_ξ,η(·) =ξ(· |η). As above, one may check that k(T^∗T)^1/2k₂ =kξ⊗_M ηk¯ =k(T T^∗)^1/2k₂ so that H ⊗_MK is isometric to a Hilbert-normed subspace of the bounded right M-linear operators fromH toK, which we denote L²(H,K). Moreover, this identification is natural with respect to theM-M bimodular structure on L²(H,K) given by x Tξ,ηy=Txξ,y^∗η.

We need now only prove the forward implication, as the converse is trivial. Let us fix some arbitraryM-M correspondenceK. From Proposition 1.4 in [65], we have that if (H ⊗_MH)¯ ^c={0}, then (H ⊗_M K)¯ ^c = {0}. So, by way of contradiction, we may assume that for every ε > 0 and x₁, . . . , x_n ∈ M, there exists a unit vectorξ ∈ H ⊗_M K¯ such thatkx_iξ−ξx_ik₂ ≤ ε, i = 1, . . . , n.

Without loss of generality, we may assume x₁, . . . , x_n are unitaries. Viewing ξ as an element of L²(H,K), let η = (ξ^∗ξ)^1/2 ∈ L²(H). By the generalized Powers-Størmer inequality, we have kx_iηx^∗_i −ηk²₂ ≤ 2kx_iξx^∗_i −ξk₂ ≤ 2ε, i = 1, . . . , n. Hence, η ∈ L²(H) ∼= H ⊗_M H¯ is a unit vector such that kx_iη−ηxik₂ ≤√

2ε,i = 1, . . . , n. Thus, H ⊗_M H¯ does not have spectral gap, a contradiction.