yn∈Nwe have
n→∞lim sup
x∈M,kxk≤1
|hxξyn,ξi|=0,
i.e., the operatorTξ is compact as an operator fromNintoL1M. By Lemma 2.3.9 we then have thatTξ is compact as an operator fromNintoL2M, which by [Oza10] agrees with thek · k∞,2-closure ofK(L2N,L2M) inB(L2N,L2M). Cutting down by projections as above then shows that a dense set of bounded vectorsξ∈H have the property thatTξ ∈K(L2N,L2M)showing that (1) holds.
Example 2.5.11. IfH is anM-N correspondence, then we letXH ⊂B(L2M)denote the hereditaryC∗- algebra generated by operators of the formT∗
ξTη whereξ,η ∈H are bounded vectors. We similarly let YH ⊂B(L2N)denote the hereditaryC∗-algebra generated by operators of the formTηT∗
ξ. ThenXH and YH give the smallest boundary pieces ofMandNrespectively so thatH is mixing relative toXH ×YH.
Note that ifB⊂Mis a von Neumann subalgebra and we considerL2Mas anM-Bcorrespondence, then the corresponding boundary pieceXL2Mis the one described in Example 2.3.4.
topology, and sinceVX−mixis a strongM-Mbimodule it then follows that[x,ξ]∈VX−mix. We similarly have [x∗,ξ]∈VX−mix, and sox∈M0.
We letSX(M)⊂B(L2M)be the set of operators that are properly proximal relative toXwhen we view B(L2M)as anM-bimodule under the actionsx·T·y=Jy∗JT Jx∗J, i.e.,
SX(M) ={T∈B(L2M)|[T,JxJ]∈K∞,1X for allx∈M}.
Note thatSX(M)is an operator system that containsM. In the case whenX=K(L2M)we writeS(M)instead ofSK(L2M)(M).
Theorem 2.6.2. Let M be a tracial von Neumann algebra,X⊂B(L2M)a boundary piece, and let B⊂M be a von Neumann subalgebra. The following are equivalent:
1. There exists a B-central stateϕonSX(M)such thatϕ|Mis normal.
2. There exists a non-zero projection p∈Z(B)and a B-central stateϕonSX(M)such thatϕ|pM p= 1
τ(p)τ.
3. There exists a non-zero projection p∈Z(B)and an M p-bimodular u.c.p. mapΦ:SX(M)→ hpM p,eBpi.
4. If E is any normal operator M-system such that there exists a stateϕ0∈(E\)X−proxwithϕ0|M=τ, then there exists an B-central stateϕon E such thatϕ|Mis normal.
Proof. The equivalences between (1), (2), and (3) are standard. To see that (4) =⇒ (1) simply observe that the stateϕ0(T) =hTˆ1,ˆ1iis in(SX(M)\)X−prox, which follows from the remark after Theorem 2.5.6.
Conversely, to see that (1) =⇒ (4) note that ifEis a normal operatorM-system andϕ0∈(E\)X−proxwith ϕ0|M=τ, thenϕ0corresponds to anM-bimodular u.c.p.Φ0:E→B(L2M)such thatϕ0(T) =hΦ0(T)ˆ1,ˆ1i, forT∈E. Sinceϕ0is a properly proximal point it follows that the range ofΦ0is contained inSX(M). Indeed, if we viewB(L2M)as anM-Mbimodule with the bimodule structure given byx·T·y=Jy∗JT Jx∗J, then for T ∈E,a,b,x∈M, and ifµi∈M⊗ConMopis uniformly bounded such thatµi→µin theX-topology, then since[x,ϕ0]∈(E\)X−mixwe see that
hµi(x·φ0(T)−φ0(T)·x)a,ˆ biˆ =µi([x,ϕ0])(b∗Ta)→ hµ(x·φ0(T)−φ0(T)·x)a,ˆ bi.ˆ
Sincea,b∈M are arbitrary it follows that[φ0(T),Jx∗J] =x·φ0(T)−φ0(T)·x∈B(L2M)X−mix. By Theo- rem 2.5.6 and Remark 2.5.8 it then follows thatφ0(T)∈SX(M)for eachT∈E.
If we have aB-central stateϕ:SX(M)→Cwithϕ|M normal, thenϕ◦Φ0gives anB-central state onE withϕ◦Φ0|Mnormal.
We say that the inclusionB⊂M is properly proximal relative toX(or just properly proximal ifX= K(L2M)) if it fails to satisfy the conditions in the previous theorem. We say thatMis properly proximal if the inclusionM⊂Mis properly proximal. We remark that by condition (3) above proper proximality forM is independent of the given trace.
Remark 2.6.3. IfXandYare boundary pieces withX⊂Y, then we haveK∞,1X (M)⊂K∞,1Y (M)and hence SX(M)⊂SY(M). Thus, ifB⊂Mis properly proximal relative toX, then it is also properly proximal relative toY. In particular, ifB⊂Mis properly proximal, then the inclusion is properly proximal with respect to any boundary piece. Similarly, ifB⊂Q⊂M, andB⊂Mis properly proximal relative toX, thenQ⊂Mis also properly proximal relative toX.
Also note that ifB⊂Q⊂Mand we identifyL2Q=eQL2M, theneQKX(M)eQ⊂B(L2Q)gives a boundary piece forQ. Indeed, eQKX(M)eQ is aC∗-algebra, and if 0≤a≤b∈eQKX(M)eQ, then we may write a=b1/4cb1/4forc∈B(L2Q)[Bla06, II.3.2.5], and taking an approximate identity{pi}i⊂eQKX(M)eQwe then havea=limi→∞piapi∈eQKX(M)eQ, showing thateQKX(M)eQis hereditary. Moreover, sinceKX(M) contains bothM andJMJin its multiplier algebra we see thateQKX(M)eQcontains bothQandJQJin its multiplier algebra.
We then see thateQK∞,1X (M)eQ⊂K∞,1e
QKX(M)eQ(Q)and it follows thateQSX(M)eQ⊂SeQK
X(M)eQ(Q). We therefore see that ifB⊂M is properly proximal relative toX, thenB⊂Qis properly proximal relative to eQKX(M)eQ. In particular, ifB⊂M is properly proximal, thenB⊂Qis also properly proximal (andBis also properly proximal).
The same argument above shows that if p∈Z(M), then by identifyingL2(zM) =zL2M we have that p⊥Xp⊥is a boundary piece forzM.
Recall from [BIP21] that if Γis a group, then a boundary pieceX forΓis a non-empty closedΓ×Γ- invariant subspace ofβΓ\Γ. We have a bijective correspondence between boundary pieces forΓandΓ×Γ- invariant proper closed idealsc0Γ⊂I⊂`∞Γ, which is given byI={f ∈`∞Γ=C(βΓ)| f|X=0} ⊂`∞Γ. A groupΓis properly proximal relative toX (or relative toI) if there is no leftΓ-invariant state onC(X)R(Γ)= (`∞Γ/I)R(Γ).
We setSX(Γ) =SI(Γ) ={f ∈`∞Γ| f−Rtf ∈Ifor anyt∈Γ}, which is aC∗-subalgebra of`∞Γ. The groupΓacts on SX by left-translation, andΓis properly proximal relative to X if and only if there is no Γ-invariant state onSX(Γ).
Theorem 2.6.4. LetΓbe a group with aΓ×Γ-invariant proper closed ideal c0Γ⊂I⊂`∞Γ, and letXbe a boundary piece for LΓ. We let IX⊂`∞Γthe closed ideal generated by E`∞Γ(X), and we letXI denote the LΓ-boundary piece IB(`2Γ)Ik·kgenerated by I. The following statements are true:
1. If LΓis properly proximal relative toX, thenΓis properly proximal relative to IX. 2. Γis properly proximal relative to I if and only if LΓis properly proximal relative toXI.
Proof. First assume thatΓis not properly proximal relative toIX, i.e., there exists aΓ-left invariant stateϕ onSIX(Γ). AsE`∞Γ:B(`2Γ)→`∞Γis continuous from thek·k∞,1-topology to the norm-topology it follows thatE`∞ΓmapsK∞,1X (LΓ)into IX, and consequently,E`∞ΓmapsSX(LΓ)into SIX(Γ), since ifT ∈SX(LΓ), thenE`∞Γ(T)−Rt(E`∞Γ(T)) =E`∞Γ(T−ρtTρt∗)∈IX, for eacht∈Γ.
We may therefore consider the stateψ:=ϕ◦E`∞Γ:SX(LΓ)→C. Observe that for anyx∈LΓ,E`∞Γ(x) = τ(x)and henceψ|LΓ=τ. Moreover, note thatE`∞Γis leftΓ-equivariant, i.e.,Ls(E`∞Γ(T)) =E`∞Γ(λsTλs∗)for anys∈ΓandT ∈SX(LΓ)and thusψ(λsT) =ψ(λsTλsλs∗) =ψ(Tλs). Finally, using the factψis normal on LΓand the Cauchy-Schwarz inequality, we conclude thatψ isLΓ-central.
Next, supposeLΓis not properly proximal relative toXIand letψbe anLΓ-central state onSXI(LΓ). We claim thatSI(Γ),→SXI(LΓ)by viewing f ∈SI(Γ)as a multiplierMf. Indeed, for any f∈SI(Γ)andt∈Γ, we have[Mf,ρt] =Mf−Rt(f)ρt∈XI, and henceMf ∈SXI(LΓ)by Lemma 2.6.1. SinceλtMfλt∗=MLt(f)for t∈Γit follows thatψ gives aΓ-invariant state onSI(Γ). The “only if” direction of the second statement follows from the first statement upon noticing thatE`∞(IB(`2Γ)I) =I`∞ΓI=I.
We recall that an von Neumann subalgebraN⊂Mis co-amenable if there exists a conditional expectation fromhM,eNitoM(see, e.g., [MP03]).
Lemma 2.6.5. Let (M,τ)be a finite von Neumann algebra with a faithful normal trace τ, N ⊂M a von Neumann subalgebra and E an M-system. Suppose there exists an N central stateϕon E such thatϕ|M=τ.
If N⊂M is co-amenable, then there exists an M-central stateψ on E withψ|M=τ.
Proof. As in Section 2.4.1, there exists an M-bimodular u.c.p. map Φ:E →B(L2M) such thatϕ(T) = hΦ(T)ˆ1,ˆ1ifor anyT ∈E. Sinceϕ isN-central, it is clear thatΦ:E→ hM,eNi. Denote byE a conditional expectationhM,eNi →Mgiven by co-amenability, thenτ◦E◦Φis anM-central state onEthat restricts to τonM.
Proposition 2.6.6. Let (M,τ)be a finite factor and N ⊂M a co-amenable subfactor. If M is properly proximal, then so is N.
Proof. SupposeN is not properly proximal, then there exists an N-central state ϕ onS(N)withϕ|N =τ.
Notice that Ad(eN):S(M)→S(N)and henceψ:=Ad(eN)◦ϕis anN-central state onS(M)withψ|N=τ.
Now it follows from Proposition 2.6.5 that there exists anM-central state onS(M)and thusMis not properly proximal.
Proposition 2.6.7. Let M1and M2be diffuse tracial von Neumann algebras. Then M=M1∗M2is properly proximal.
Proof. We will show that the usual paradoxical decomposition that proves nonamenability forM1∗M2also works to show proper proximality. Fori=1,2, letHi=L2(Mi)andHi0=Hi Cˆ1=Mi0ˆ1k·k2, whereMi0 denotes the kernel of the trace. Recall thatH =L2(M1∗M2)decomposes as
H =Cˆ1⊕M
n≥1
M
i16=i26=···6=in
Hi10⊗ · · · ⊗Hin0
! .
Set
H`(1) =Cˆ1⊕M
n≥1
M
i16=i2···6=in i16=1
Hi10⊗ · · · ⊗Hin0
,
and letP∈B(H)be the orthogonal projection ontoH`(1)⊥.
Ifz∈M20, then asJzJandJz∗Jpreserve the spaceH`(1)we have[P,JzJ] =0. Also, ifz∈M10, then asJzJ andJz∗Jpreserve the subspaceH`(1) Cˆ1 we have[P,JzJ] = [JzJ,ProjCˆ1]is finite-rank. By Lemma 2.6.1 we then haveP∈S(M).
Similarly, if we byQthe projection ontoH`(2)⊥, where
H`(2) =Cˆ1⊕M
n≥1
M
i16=i2···6=in i16=2
Hi10⊗ · · · ⊗Hin0
,
thenQ∈S(M). SinceM2is diffuse, we may choose orthogonal trace-zero unitariesu1,u2∈U(M2)so that we then haveu∗1Pu1+u∗2Pu2≤Q. Similarly we may choose a trace-zero unitaryv∈U(M1)and obtain that v∗Qv≤P.
If there were an M-central stateϕ on S(M), we would then have 2ϕ(P)≤ϕ(Q)≤ϕ(P) and hence ϕ(P+Q) =0. Since 1−P−Qis the projection ontoCˆ1 andϕ isM-central we haveϕ(1−P−Q) =0, which then gives a contradiction.
Theorem 2.6.8. Let M be a finite von Neumann algebra,X⊂B(L2M)an M-boundary piece and letG ⊂ U(M)be a subgroup. Suppose there exists a stateϕ∈B(L2M)∗such that
1. ϕrestricts to the canonical traces on M and JMJ.
2. ϕ◦Ad(u) =ϕ◦Ad(Ju∗J)for all u∈G. 3. ϕ|X=0,
thenϕ|SX(M)isG00-central, so that the inclusionG00⊂M is not properly proximal.
Proof. Note that sinceϕ|M=τis normal, the set of elementsx∈M such that[x,ϕ|SX(M)] =0 forms a von Neumann subalgebra ofM, thus it suffices to show thatG is contained in this set. Also, asϕ∈B(L2M)M]MJ andϕ|X=0 we haveϕ|
K∞,1X (M)=0.
IfT ∈SX(M)andu∈G, then(Ju∗J)T(JuJ)−T = [Ju∗J,T](JuJ)∈K∞,1X (M). Therefore,
ϕ◦Ad(u)(T) =ϕ◦Ad(Ju∗J)(T) =ϕ(T).
Corollary 2.6.9. Let M be a tracial von Neumann algebra with property (Gamma). Then M is not properly proximal.
Proof. SupposeMhas property (Gamma). Letun∈U(M)such thatun→0 ultraweakly andk[x,un]k2→0 for allx∈M. Letϕbe any weak∗-limit point of the statesB(L2M)3T 7→ hTuˆn,uˆni. Then it is easy to see thatϕsatisfies the hypotheses of Theorem 2.6.8, forG =U(M)andX=K(L2M).
Lemma 2.6.10. Let M be a finite von Neumann algebra and Q⊂M a regular von Neumann subalgebra. If un∈U(M)is such thatkEQ(aunb)k2→0for all a,b,∈M, then for all S,T∈B(L2M)of the form aJbJeQJcJd with a,b,c,d∈M we havekS∗unTk∞,1→0.
Proof. Suppose{un}n⊂U(M)is given as above. SincekT xJyJk∞,1≤ kTk∞,1kxkkyk, it is enough to check thatkS∗unTk∞,1→0 whenSandT are each of the formaJbJeQ, witha,b∈M.
Also, note that ifa,b,c,d∈M, then forx∈Mwe have
keQaJbJxJcJdeQk∞,1= sup
y,z∈(Q)1
|τ(axdyc∗b∗z)| ≤ kbck2kaxdk2.
It therefore follows that by taking spans and using density ink · k2, to prove the lemma it suffices to show that kS∗unTk∞,1→0 whenSandT are each of the formeQaJbJ, wherea∈Mandb∈NM(Q). Finally, note that forb∈NM(Q)we haveeQJbJ=JbJbeQb∗, and from this we then see that it suffices to consider the case whenSandT are each of the formeQafora∈M. This is easily verified, for ifa1,a2∈M, then
keQa∗1una2eQk∞,1=kEQ(a∗1unca)k1→0.
For the next theorem, recall from Example 2.3.4 that if Q⊂M is a von Neumann subalgebra, then we denote by XQ the M-boundary piece consisting of the norm closed span of all operators of the from x1Jy1JT Jy2Jx2, withx1,x2,y1,y2∈MandT∈eQB(L2M)eQ.
Theorem 2.6.11. Let M be a finite von Neumann algebra, and Q⊂M a regular von Neumann subalgebra such that M is properly proximal relative to the M-boundary pieceXQ. If P⊂M is a weakly compact regular von Neumann subalgebra, then P≺MQ. In particular, if M is properly proximal, then M has no diffuse weakly compact regular von Neumann subalgebras.
Proof. By the weak compactness ofP⊂M, there exists a stateϕ:B(L2M)→Csatisfying the following properties:
(i)ϕis the canonical normal trace onMandJMJ;
(ii)ϕ(xT) =ϕ(T x)for allT ∈B(L2M),x∈P;
(iii)ϕ◦Ad(u)(T) =ϕ◦Ad(Ju∗J)(T)for allT ∈B(L2M)andu∈NM(P).
IfP6≺MQ, then by Lemma 2.6.10 for eachT ∈B(L2M)of the forma∗Jb∗JeQJbJawitha,b∈Mthere exists a sequence{un}n⊂U(P)such thatkT∗u∗munTk∞,1,kT∗u∗numTk∞,1<2−nwhenevern>m, and such thatkTub∗nk2<2−nfor eachn≥1. We then have
k1 N
N n=1
∑
u∗nTunk2∞,2≤ 1 N2
N n,m=1
∑
ku∗mTumu∗nTunk∞,1
≤ 2 N2
∑
1≤m<n≤N
2−n+ 1 N2
N
∑
n=1
ku∗nT2unk∞,1
≤ 2
N+kTk2 N →0.
Sinceϕis continuous in thek · k∞,2-norm on bounded sets it then follows from (ii) that
ϕ(T) = lim
N→∞
1 N
N
∑
n=1
ϕ(u∗nTun) =0.
If we now havea,b∈MandT∈eQB(L2M)eQwithT≥0, then
ϕ(a∗Jb∗JT JbJa)≤ kTkϕ(a∗Jb∗JeAJbJa) =0
and henceϕ(a∗Jb∗JT JbJa) =0. By polarization it then follows thatϕ(aJbJT JcJd) =0 for alla,b,c,d∈M andT ∈eQB(L2M)eQ. Since the span of such elements is norm dense inXQit follows thatϕ|XQ =0, and henceMis not properly proximal relative toXQby Theorem 2.6.8.
2.6.1 Proper proximality relative to the amenable boundary piece
We see from Theorem 2.6.4 that a groupΓis properly proximal if and only if the group von Neumann algebra LΓis properly proximal. In this section give an application of the development of boundary pieces for von Neumann algebras by showing that this also holds for proper proximality relative to a canonical “amenable”
boundary piece.
Let Γ be a group, and let π :Γ→U(H)be a universal representation that is weakly contained in the left regular representation, i.e.,π is the restriction of the universal representation ofC∗
λΓ. Associated to this representation is a boundary piece Xamen as described in [BIP21], where a net (ti)i has a limit in Xamen⊂βΓif and only ifπ(ti)→0 ultraweakly, i.e.,ti→0 in the weak-topology inC∗
λΓ. Alternatively, we can view the corresponding idealIamen⊂`∞Γas the ideal generated by the setBr(Γ)of all matrix coefficients ϕξ,η(t) =hπ(t)ξ,ηiforξ,η∈H. Note that since weak containment is preserved under tensor products it follows from Fell’s absorption principle that Br(Γ)is an ideal in the Fourier-Stieltjes AlgebraB(Γ). In particular,Br(Γ)is a self-adjoint subalgebra of`∞Γ, and sof ∈Iamenif and only if|f| ≤gfor someg∈Br(Γ).
IfΓis a nonamenable group, then the following lemma shows that we often havec0Γ(Iamen(`∞Γ.
Lemma 2.6.12. SupposeΣ<Γis a subgroup. ThenΣis amenable if and only if1Σ∈Iamen.
Proof. IfΣis amenable, then the quasi-regular representation`2(Γ/Σ)is weakly contained in the left regular representation and 1Σis a matrix coefficient. Conversely, if 1Σ∈Iamen, then there exists ϕ∈Br(Γ), say ϕ(t) =hπ(t)ξ,ηiso that 1Σ≤g, wheregis some element inBr(Γ)withkg−ϕk∞<1/2. Hence for allt∈Σ we haveℜ(hπ(t)ξ,ηi)≥1/2. If we letξ0denote the minimal norm element in the closed convex hull of {π(t)ξ}t∈Σ, it then follows thatξ0is aΣ-invariant vector and is non-zero sinceℜ(hξ0,ηi)≥1/2. It therefore follows that the trivial representation forΣis weakly contained inπ≺λ and henceΣis amenable.
We now fix a tracial von Neumann algebraM, and by analogy with above we consider a universalM-M correspondenceH that is weakly contained in the coarse correspondence L2M⊗L2M. Note that we may assume that as anM-Mcorrespondence we haveH ∼=H and we then have a boundary pieceXamen=XH
as defined in Example 2.5.11
Similar to Lemma 2.6.12, we have the following lemma.
Lemma 2.6.13. Let M be a tracial von Neumann algebra and A⊂M a von Neumann subalgebra such that A does not have an amenable direct summand, andG ⊂U(A)a subgroup that generates A as a von Neumann algebra. Then there exists a net{ui}i⊂G such that ui⊗uopi →0in theXamen-topology.
Proof. Fix a universalM-McorrespondenceH that is weakly contained in the coarse correspondence. Note that just as in the case for groups the spanB0of operators of the formTξ for some bounded vectorξ∈H
forms a∗-subalgebra ofB(L2M). Also,MandJMJare contained in the multiplier algebra ofB0inB(L2M).
Thus if we denote by B=B0, then we haveXamen=BB(L2M)B. In particular, a net {ui}i⊂G satisfies ui⊗uopi →0 in the Xamen-topology if and only ifuiTξ1STξ2u∗i →0 ultraweakly for anyS∈B(L2M)and ξ1,ξ2∈H bounded vectors. Moreover, by the polarization identity we see that this is if and only if for each bounded vectorξ∈H,S∈B(L2M), anda∈Mwe havekSTξJaJub∗ik22→0, and for this it suffices to consider only the case whenS=1.
Therefore, if no such net of unitaries existed, then there would exist bounded vectorsξ1, . . . ,ξn∈H, a1, . . . ,an∈Mandc>0 such that for allu∈G we would have∑nk=1kTξ
kJakJukˆ 22≥c. We letξ=⊕nk=1ξk∈
⊕nk=1H ⊗MH and we let ˜ξ=⊕nk=1a∗kξkak. We also letC denote the closed convex set generated by vectors of the formu∗ξu, foru∈G. Then for eachu∈G we havehu∗ξu,ξ˜i=∑nk=1kTξ
kJakJukˆ 22≥c, and hence for anyη∈C we havehη,ξ˜i ≥c>0. Hence, if we take η∈C to be the vector of minimal norm, then η is a non-zero vector that is invariant under conjugation byG. SinceG generatesAit then follows thatη is a non-zeroA-central vector. Since⊕nk=1H ⊗MH is weakly contained in the coarse correspondence this would then show thatApis amenable wherep∈P(Z(A))is the non-zero support projection forξ. Theorem 2.6.14. LetΓbe a discrete group, then Iamen⊂`∞Γis the closed ideal generated by E`∞Γ(Xamen) andXamen⊂B(L2(LΓ))contains the boundary piece generated by Iamen.
Proof. LetH be a universalM-Mcorrespondence that is weakly contained in the coarse correspondence, and letξ,η∈H be bounded vector. ThenH ⊗MH is also weakly contained in the coarse correspondence.
In particular if we consider the representationπ:Γ→U(H ⊗MH)given by conjugation, thenπis weakly contained in the conjugation action associated to the coarse correspondenceL2M⊗L2M, which is easily seen to be a multiple of the left regular representation. Fort∈Γwe compute
E`∞Γ(T∗
ξTη)(t) =E`∞Γ(Tη⊗ξ)(t) =hTη⊗ξδt,δti=hut∗(η⊗ξ)ut,η⊗ξi.
Sinceπ≺λwe then haveE`∞Γ(T∗
ξTη)∈Iamen, and it then follows thatE`∞Γ(Xamen)⊂Iamen.
On the other hand, if π:Γ→U(H)is a representation that is weakly contained in the left regular representation, then we may consider theLΓ-LΓcorrespondence`2Γ⊗H where the first copy ofLΓacts asλt⊗π(t)(which is conjugate toλt⊗1 by Fell’s absorption principle), and the second copy ofLΓas as ρt⊗1. As is well known, ifπ≺λ, then`2Γ⊗H is weakly contained in the coarse correspondence. Also, ifξ ∈H, then it is easy to check thatδe⊗ξ is a bounded vector andTδe⊗ξ is the diagonal multiplication operator corresponding toΓ3t7→ hπ(t)ξ,ξi. It then follows thatIamen⊂Xamen.
Moreover, ifϕ(t) =hπ(t)ξ,ξiandMϕdenotes the diagonal multiplication operator, then sinceE`∞(Mϕ) =
ϕwe see thatIamenis, in fact, equal toE`∞Γ(Xamen).
Corollary 2.6.15. LetΓbe a discrete group, then LΓis properly proximal relative toXamenif and only ifΓ is properly proximal relative to Iamen.
Proof. This follows from the previous theorem, together with Theorem 2.6.4.
IfΓis properly proximal, then clearlyΓis properly proximal relative toIamen, but also ifΓhas a normal amenable subgroupΣCΓsuch thatΓ/Σis properly proximal, then it follows from [BIP21] thatΓis also properly proximal relative toIamen. Examples of groups and von Neumann algebras that are not properly proximal relative to the amenable boundary piece are infinite direct products of nonamenable groups or infinite tensor products of nonamenable II1factors, which follows from the following proposition.
Proposition 2.6.16. Suppose M is a tracial von Neumann algebra and Bn⊂M is a decreasing sequence of von Neumann subalgebras such that each Bn has no amenable summand and such that∪n(B0n∩M)is ultraweakly dense in M. Then M is not properly proximal relative toXamen.
Proof. This is similar to the proof of Corollary 2.6.9. Given any asymptotically central net{ui}i⊂U(M) we may consider a stateϕ onSXamen(M)which is a weak∗-limit point of the vector statesSXamen(M)3T 7→
hTubi,ubii. Since each Bn has no amenable summand, it follows from Lemma 2.6.13 that we may find an asymptotically central net{ui}i⊂U(M)so thatϕvanishes onXamen, and it then follows thatϕsatisfies the hypotheses of Theorem 2.6.8.