relative to LΓinside M. Then we haveαt→idN uniformly on the unit ball of N as t→0, whereαt is the s-malleable deformation described above.
Proof. Let ˜M= (AH⊗AH)oσπ⊗σπΓandαt,β∈Aut(M)˜ be as above. Suppose there exists some 0<δ≤1 such that case (1) of Lemma 3.4.1 does not hold. Then we have that there exists{ηk} ∈K⊥as in the second case of Lemma 3.4.1. Note that theM-MbimoduleL2(hM,˜ eLΓi) K is isomorphic toL2(M˜ M)⊗LΓL2M.˜ It is shown in [Bou12, Lemma 3.3] thatL2(M˜ M)is weakly contained in the coarseM-M bimodule as π≺λ, and hence we have
L2(hM,˜ eLΓi) K ≺L2M⊗(L2M⊗LΓL2M)˜ ≺L2M⊗L2M,
asM-Mbimodules. It follows that there exists a u.c.p. map
φ:B(L2M)→B(L2(hM,˜ eLΓi) K)∩(Mop)0,
such thatφ|M=idM. Therefore, we obtain a stateϕonB(L2M)given by
ϕ(·) =lim
k kpηkk−22 hφ(·)pηk,pηki,
which isN-central and restricts to a normal state on pM p. This contradicts the assumption thatN has no amenable direct summands.
Therefore, we have that limt→0infu∈U(N)kEM(αt(u))k2=kpk2. It follows that supu∈U(N)kαt(u)− EM(αt(u))k2→0 ast→0 and henceαt→id uniformly on(N)1by Popa’s transversality inequality [Pop08, Lemma 2.1].
Corollary 3.4.3. Let M, p∈P(M)and N⊂pM p be as in Proposition 3.4.2. Denote Q=NpM p(N)00. Ifπ is mixing, then Q≺MLΓ. Moreover, ifΓis an i.c.c. group, then there exists u∈U(M)such that u∗Qu⊂LΓ.
Proof. Since AH is abelian, N is diffuse and Qis type II1, the assertionQ≺M LΓfollows directly from [Bou12, Theroem 3.4] and Proposition 3.4.2. And the proof for the moreover part is contained in [Bou13, Proposition 2.3].
summand, where XLΓ is theM-boundary piece associated with LΓ. Moreover, sinceΓ<ZoΓ is almost malnormal andNhas no properly proximal direct summand, we haveNis amenable relative toLΓinsideM by Proposition 3.3.1. The rest follows from Proposition 3.4.2 by settingπ=λ.
Proof of Theorem 3.1.2. Letσ:ΓyXbe the Bernoulli action,M=L∞(X)oσΓ. Set ˜M= (L∞(X)⊗L∞(X))oσ˜
Γ, where ˜σ=σ⊗σ. If we denote byσt∈U(L2(X))for eacht∈Γthe unitary that implements the action σ, then we haveM⊂M˜ is generated by canonical unitaries{ut=σ˜t⊗λt|t∈Γ} andL∞(X)⊗C, where σ˜t=σt⊗σt.
LetΓ0<Γbe a nonamenable wq-normal subgroup that is not properly proximal. Ifω:Γ×X→Tis a 1- cocycle associated withσ, then for eacht∈Γ, we may considerωt∈U(L∞(X))given byωt(x) =ω(t,t−1x) and^L(Γ0):={u˜t:=ωtut|t∈Γ0}00⊂M, which is a von Neumann subalgebra isomorphic toL(Γ0).
Since ^L(Γ0)∼=L(Γ0)has no amenable and no properly proximal direct summand by Remark 3.2.2, it follows from Theorem 3.1.5 thatαt converges to identity uniformly on the unit ball of ^L(Γ0). The result follows from [Pop07a].
Proof of Theorem 3.1.1. This is an immediate result of Theorem 3.1.2 and [PS12, Theorem 1.1].
Proof of Theorem 3.1.3. SinceΛis exact, we haveL∞(Y)orΛis an exact C∗-algebra (e.g. [BO08, Theorem 10.2.9]) and it follows that(L∞(X)oΓ)t ∼=L∞(Y)oΛ is a weakly exact von Neumann algebra [Kir95].
Since weak exactness is stable under amplifications and passes to von Neumann subalgebras (with normal conditional expectations) [BO08, Corollary 14.1.5], we haveLΓis weakly exact, which impliesΓis exact [Oza07].
LetM=L∞(X)oΓ,N=L∞(Y)oΛandΛ0CΛbe the nonamenable normal subgroup that is not properly proximal. SinceN∼=Mt, we may denote byθ:N1/t→Ma∗-isomorphism, and identifyN1/twithpMn(N)p, wheren=d1/te,p=diag(1, . . . ,1,p0)∈Mn(N)andp0∈L(Λ0)withτN(p0) =1/t− b1/tc.
Note that by Remark 3.2.2,θ(pMn(L(Λ0))p)⊂Msatisfies the assumption of Theorem 3.1.5, and hence by Corollary 3.4.3 we may find someu∈U(M)such thatα(pMn(LΛ)p)⊂LΓ, whereα:=Ad(u)◦θ. Set e=diag(1,0, . . . ,0)∈Mn(LΛ) and we haveα(LΛ) =α(eMn(LΛ)e)⊂qLΓq, whereq=α(e)∈LΓand τM(q) =τN1/t(e) =t.
It then follows from Popa’s conjugacy criterion for Bernoulli actions [Pop06d, Theorem 0.7] (see also [Ioa11, Theorem 6.3]) thatt=1 and there exist a unitaryv∈M, a characterη∈Λand a group isomorphism δ :Λ→Γsuch thatα(L∞(Y)) =vL∞(X)v∗andα(λt) =η(t)vλδ(t)v∗for anyt∈Λ.
Proof of Theorem 3.1.4. A direct consequence of Theorem 3.1.3, [IPR19] and [Oza07].
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