Preliminaries

N properly proximal relative toLΓ, we show thatN either has a properly proximal direct summand or is amenable relative toLΓinsideL(ZoΓ). In this step, the notion of normal bidual developed in [DKEP22, Section 2] is extensively used. Lastly, using a technique from [Ioa15], we conclude in Section 3.4 that if N⊂L(ZoΓ)is amenable relative toLΓ, thenN must be rigid with respect to the s-malleable deformation {α_t}associated withL(ZoΓ). Altogether, we obtain that ifN⊂L(ZoΓ)has no amenable or properly proximal direct summand, thenNmust beα_t-rigid.

3.2.4.1 Boundary pieces

GivenM a finite von Neumann algebra, anM-boundary pieceXis a hereditary C^∗-subalgebra ofB(L²M) such thatM(X)∩M⊂MandM(X)∩JMJ⊂JMJare weakly dense, whereM(X)is the multiplier ofX. To avoid pathological examples, we will always assume thatX6={0}and it follows thatK(L²M)⊂Xfor any M-boundary pieceX.

Let M be a finite von Neumann algebra andXan M-boundary piece. Denote byK^L_X(M)⊂B(L²M) thek · k_∞,2-closure of the norm closed left idealB(L²M)X, wherekTk_∞,2=sup_a∈(M)

1kTakˆ ₂ for anyT ∈ B(L²M), and setK_X(M) = (K^L_X(M))^∗∩K^L_X(M)to be the hereditary C^∗-subalgebra generated byK^L_X(M).

The multiplier algebra ofKX(M)contains bothM andJMJand we denote byK^∞,1_X (M)thek · k_∞,1-closure ofKX(M), wherekTk_∞,1=sup_a,b∈(M)

1hTa,ˆ biˆ forT ∈B(L²M), andK^∞,1_X (M)coincides withX^k·k∞,1. And denote bySX(M)the following operator system that containsM,

SX(M) ={T ∈B(L²M)|[T,x]∈K^∞,1_X (M),for anyx∈JMJ}.

WhenX=K(L²M), we omitXin the above notations for simplicity.

LetN⊂Mbe a von Neuman subalgebra. We sayN⊂Mis properly proximal relative toXif there does not exist anyN-central stateϕ onS_X(M)such thatϕ|M is normal. And we sayM is properly proximal if M⊂Mproperly proximal relative toK(L²M). By [DKEP22, Theorem 6.2], a groupΓis properly proximal in the sense of [BIP21] if and only ifLΓis properly proximal.

One particular type of boundary pieces arise from subalgebras. LetN⊂Mbe a von Neumann subalgebra and we may associate withNanM-boundary pieceXN, which is the hereditary C^∗-subalgebra ofB(L²M) generated byxJyJe_Nforx,y∈M, wheree_N∈B(L²M)is the orthogonal projection fromL²MontoL²N.

Remark 3.2.2. LetΓ be a group that is not properly proximal, thenLΓ has no properly proximal direct summand. Indeed, supposez∈Z(LΓ)in a nonzero central projection such thatzLΓis not properly proximal, i.e., there exists azLΓ-central stateϕ:S(zLΓ)→Cthat is normal onzLΓ. We may consider theΓ-equivariant embeddingi:S(Γ)→S(LΓ)andE:=Ad(z):S(LΓ)→S(zLΓ)[DKEP22, Section 6]. Thenϕ◦E◦i:S(Γ)→ Cis then aΓ-invariant state, showing Γis not properly proximal. A similar argument shows that if Γis nonamenable, thenLΓhas no amenable direct summand.

Recall that a groupΓis biexact relative to a subgroupΛ<Γif the left action ofΓonSΛ(Γ) ={f ∈`^∞Γ| f−R_tf ∈c₀(Γ,{Λ})}is topologically amenable, wherec₀(Γ,{Λ})is functions onΓthat converge to 0 when t∈Γescapes subsets ofΓthat are small relative toΛ(See [BO08, Chapter 15] for the precise definition).

We remark that this is equivalent toΓy SI(Γ)is amenable. Indeed, since we may embed`^∞Γ,→I^∗∗ in a

Γ-equivariant way, we haveΓyI^∗∗⊕(SI(Γ)/I)^∗∗=SI(Γ)^∗∗is amenable, and it follows thatΓy SI(Γ)is an amenable action [BEW19, Proposition 2.7].

The following is an easy adaptation of [DKEP22, Theorem 7.1]. For completeness, we include the proof.

Proposition 3.2.3. Suppose a groupΓis biexact relative to a subgroupΛ<Γ. Then for every von Neumann subalgebra N⊂LΓ, either the inclusion N ⊂LΓ is properly proximal relative to XLΛ or else N has an amenable direct summand.

Proof. Suppose the inclusion N⊂LΓ is not properly proximal relative to XLΛ, and let φ :SXLΛ(LΓ)→ hp(LΓ)p,e_{N p}ibe ap(LΓ)p-bimodular u.c.p. map, where p∈Z(N)is a non-zero central projection.

If we consider theΓ-equivariant diagonal embedding`<sup>∞</sup>Γ⊂B(`²Γ), we see thatc₀(Γ,{Λ})is mapped toXLΛ. Restricting to SΛ(Γ)then gives a Γ-equivariant embedding into S_XLΛ(LΓ). We therefore obtain a ∗-homomorphism SΛ(Γ)of Γ→B(`²Γ) whose image is contained in SXLΛ(LΓ). Composing this ∗- homomorphism with the u.c.p. mapφ then gives a u.c.p. map ˜φ :SΛ(Γ)ofΓ→ hp(LΓ)p,e_{N p}isuch that φ(t) =˜ pu_tpfor allt∈Γ.

SinceΓis biexact relative toΛ, the actionΓy SΛ(Γ)is topologically amenable. Hence,SΛ(Γ)ofΓ= SΛ(Γ)orΓis a nuclear C^∗-algebra. We setϕ(·):=_τ(p)¹ hφ˜(·)p,ˆ piˆ and note that forx∈C_r^∗Γwe haveϕ(x) =

1

τ(p)τ(pxp). SinceC^∗_rΓis weakly dense inLΓan argument similar to Proposition 3.1 in [BC15] then gives a representationπϕ:SΛ(Γ)orΓ→B(Hϕ), a state ˜ϕ ∈B(Hϕ)∗ withϕ=ϕ˜◦πϕ, and a projectionq∈ πϕ(SΛ(Γ)orΓ)⁰⁰with ˜ϕ(q) =1 such that there is a normal unital∗-homomorphismi:LΓ,→qπ_ϕ(SΛ(Γ)or

Γ)⁰⁰q.

Since SΛ(Γ)orΓ is nuclear, we have thatπϕ(SΛ(Γ)orΓ)⁰⁰ is injective, and so there is a u.c.p. map i˜:B(`<sup>2</sup>Γ)→qπ<sub>ϕ</sub>(SΛ(Γ)orΓ)<sup>00</sup>qthat extendsi. Notice thatψ:=ϕ˜◦i˜is then anN p-central state onB(`²Γ) andψ(x) =_τ(p)¹ τ(pxp)forx∈LΓ. Therefore,N pis amenable.

3.2.4.2 A bidual characterization

Next we collect some basics of the normal bidual from [DKEP22, Section 2].

Given a finite von Neumann algebra M and a C^∗-subalgebraA⊂B(L²M) such thatM andJMJ are contained in the multiplier algebraM(A), we recall thatA^M]M(resp.A^JMJ]JMJ)denotes the space ofϕ∈A^∗ such that for each T ∈A the mapM×M 3(a,b)7→ϕ(aT b)(resp. JMJ×JMJ3(a,b)7→ϕ(aT b)) is separately normal in each variable. When there is no confusion about the von Neumann algebra that we are referring to, we will denoteA^M]MbyA^]andA^M]M∩A^JMJ]JMJbyA^]_J.

We may view (A^]_J)^∗as a von Neumann algebra as follows. Denote by pnor∈M(A)^∗∗ the supremum of support projections of states inM(A)^∗ that restrict to normal states onM andJMJ, so thatM andJMJ

may be viewed as unital von Neumann subalgebras ofp_norM(A)^∗∗p_nor, which is canonically identified with (M(A)^]_J)^∗. Letq_A∈P(M(A)^∗∗)be the central projection such thatq_A(M(A)^∗∗) =A^∗∗ and we may then identify(A^]_J)^∗withp_norq_AM(A)^∗∗p_nor=p_norA^∗∗p_nor. Furthermore, ifB⊂Ais another C^∗-subalgebra with M,JMJ⊂M(B), we may identify(B^]_J)^∗withq_Bp_norA^∗∗p_norq_B, which is a non-unital subalgebra of(A^]_J)^∗.

If we denote byι:M(A)→M(A)^∗∗ the canonical embedding, we may then viewM(A)as an operator subsystem of(M(A)^]_J)^∗through the isometric u.c.p. mapιnor:M(A)3T →pnorι(T)p_nor∈(M(A)^]_J)^∗, and the its restriction toAgives a natural embedding ofA⊂(A^]_J)^∗as an operator system.

It is worth noting that ιnorandι are different in a few ways. On one hand, ι:M(A)→M(A)^∗∗ is a

∗-homomorphism whileιnor:M(A)→(M(A)^]_J)^∗is a u.c.p. map; on the other hand,ιnorgives rise to normal faithful representations when restricted toMandJMJ, butι|M andι|JMJare not normal in general.

The following is a bidual characterization of properly proximal.

Lemma 3.2.4. [DKEP22, Lemma 8.5] Let M be a separable tracial von Neumann algebra with an M- boundary pieceX. Then M is properly proximal relative to Xif and only if there is no M-central stateϕ on

eSX(M):=n T∈

B(L²M)^]_J∗

|[T,a]∈

KX(M)^]_J∗

for alla∈JMJo

such thatϕ|Mis normal.

WhenX=K(L²M), we will abbreviateXfor simplicity. It is worth noting thateS_X(M)is a von Neumann algebra which containsMas a von Neumann subalgebra, whileS_X(M)is only an operator system. Following the above discussion, we note that ˜S_X(M)may be identified with

eSX(M) ={T ∈p_norB(L²M)^∗∗p_nor|[T,a]∈q_X M(KX(M))∗∗

q_X,for anya∈JMJ},

whereq_X is the identity of (KX(M)^]_J)^∗⊂(M(KX(M))^]_J)^∗. If we setq_K=q_K(L2M) to be the identity of (K(L²M)^]_J)^∗⊂(B(L²M)^]_J)^∗, then using the above description of ˜SX(M), we haveq^⊥_XeSX(M)q^⊥

X⊂q^⊥_KeS(M), as q_Xcommutes withMandJMJ.

Lemma 3.2.5. Let M be a separable tracial von Neumann algebra. Suppose M has no properly proximal direct summand, then there exists an M-central stateϕoneS(M)such thatϕ|Mis faithful and normal.

Proof. First we show that there exists anM-central stateϕ oneS(M)such thatϕ|Z(M) is faithful. Consider a pair (ϕ,p), where ϕ ∈eS(M)^∗ is anM-central state such that ϕ|M is normal and p∈Z(M) is support projection ofϕ. And we may order such pairs by the order onZ(M), i.e.,(ϕ₁,p₁)≤(ϕ₂,p₂)ifp₁≤p₂. If{(ϕ_i,p_i)}i∈I is a chain, then we may find a subsequence p_i(n) such that lim_np_i(n)=∨i∈Ip_i, andϕ₀(·) =

∑n≥12⁻ⁿϕ_i(n)(p_i(n)·p_i(n))then is anM-central state oneS(M) such that ϕ0|M is normal and ∨i∈Ip_i is the support ofϕ0. Suppose(ϕ,p)is a maximal element andq=p^⊥>0. Denote byE_q: Ad(q):B(L²M)→ B(L²(qM))and one checks that(E_q)^∗mapsB(L²(qM))^]_JtoB(L²M)^]_J. Therefore dualizingE_qyeilds a u.c.p.

map ˜E_q:(B(L²M)^]_J)^∗→(B(L²(qM))^]_J)^∗, and(E˜_q)_|eS(M):eS(M)→eS(qM). SinceqMis not properly proximal, there exists a stateψ∈eS(qM)^∗that isqM-central andψ_|qMis normal. Setϕ⁰(T) =ϕ(pT p) +ψ(E˜_q(qT q)), which is anM-central state on eS(M) that is normal on M with support strictly larger than p, which is a contradiction.

Now supposeϕis such a state withϕ|Z(M)faithful, andϕ(p) =0 for somep∈P(M), then we may write the central supportz(p) =∑^∞_i=1v_iv^∗_i, wherev_i∈Mare partial isometries such thatv^∗_iv_i≤p. Sinceϕis normal and tracial onM, we haveϕ(z(p)) =∑^∞_i=1ϕ(v_iv^∗_i)≤∑^∞_i=1ϕ(p) =0, which shows thatp≤z(p) =0.