∑n≥12−nϕi(n)(pi(n)·pi(n))then is anM-central state oneS(M) such that ϕ0|M is normal and ∨i∈Ipi is the support ofϕ0. Suppose(ϕ,p)is a maximal element andq=p⊥>0. Denote byEq: Ad(q):B(L2M)→ B(L2(qM))and one checks that(Eq)∗mapsB(L2(qM))]JtoB(L2M)]J. Therefore dualizingEqyeilds a u.c.p.
map ˜Eq:(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=1viv∗i, wherevi∈Mare partial isometries such thatv∗ivi≤p. Sinceϕis normal and tracial onM, we haveϕ(z(p)) =∑∞i=1ϕ(viv∗i)≤∑∞i=1ϕ(p) =0, which shows thatp≤z(p) =0.
Recall from Section 3.2.4.2 thatι:KX(M)→KX(M)∗∗is the canonical embedding, pnor∈B(L2M)∗∗is the projection such that pnorKX(M)∗∗pnor= (KX(M)]J)∗and the embeddingιnor:KX(M)→(KX(M)]J)∗is given byιnor=Ad(pnor)◦ι.
Lemma 3.3.3. Let M be a finite von Neumann algebra andXan M-boundary piece. LetX0⊂KX(M)be a C∗-subalgebra and{en}n∈I an approximate unit ofX0. IfX0⊂K∞,1X (M)is dense in k · k∞,1 andι(en) commutes with pnorfor each n∈I, thenlimnιnor(en)∈(KX(M)]J)∗is the identity, where the limit is in the weak∗topology.
Proof. Since ιnor(KX(M))⊂(KX(M)]J)∗ is weak∗dense and functionals inKX(M)]J are continuous ink · k∞,1 topology by [DKEP22, Proposition 3.1], we haveιnor(X0)⊂(KX(M)]J)∗ is also weak∗ dense. Let e=limnιnor(en)∈(KX(M)]J)∗be a weak∗limit point and for anyT∈X0, we have
eιnor(T) =lim
n pnorι(en)ι(T)pnor=lim
n pnorι(enT)pnor=ιnor(T),
and similarlyιnor(T)e=ιnor(T). By density ofιnor(X0)⊂(KX(M)]J)∗, we conclude thateis the identity in (KX(M)]J)∗.
Lemma 3.3.4. Let M be a finite von Neumann algebra and B⊂M a von Neumann subalgebra. Let eB∈ B(L2M)be the orthogonal projection onto L2B. Thenι(eB)∈B(L2M)∗∗commutes with pnor.
Proof. SupposeB(L2M)∗∗⊂B(H)and notice thatξ ∈H is in the range ofpnorif and only ifM3x→ hι(x)ξ,ξiandJMJ3x→ hι(x)ξ,ξiare normal. Forξ ∈pnorH, we haveϕ(x):=hι(x)ι(eB)ξ,ι(eB)ξi= hι(EB(x))ξ,ξiis also normal forx∈MandJMJ, which implies thatι(eB)pnor=pnorι(eB)pnor. It follows thatι(eB)andpnorcommutes.
Lemma 3.3.5. LetΓbe a group andΛ<Γa subgroup. Let M=LΓ, B=LΛandX=XB. Denote by{tk}k∈K
a representative ofΓ/Λ, i.e.,Γ=tk∈KtkΛand uk:=λtk∈LΓthe canonical unitaries. For each finite subset F⊂K, let eF=Wk,`∈FukJu`JeBJu∗`Ju∗k. ThenlimFιnor(eF)∈(KX(M)]J)∗is the identity.
Proof. Denote by X0⊂B(L2M)the hereditary C∗-subalgebra generated byxJyJeN for x,y∈C∗r(Γ). It is clear thatX0is anM-boundary piece and by hereditariness we haveeF∈X0for eachF.
First we show that K∞,1X0(M) =K∞,1X (M), whereK∞,1X0(M)is obtained fromX0in the way described in Section 3.2.4.1. Notice that B(L2M)X0⊂KLX(M)is dense in k · k∞,2. Indeed, for any contractions T ∈ B(L2M)andx,y∈LΓ, we may find a net of contractionsTi∈B(L2M)X0such thatTi→TeNxJyJink · k∞,2, as it follows directly from [DKEP22, Proposition 3.1], the non-commutative Egorov theorem and the Kaplansky
density theorem. It then follows thatKX0(M)⊂K∞,1X (M)is dense ink · k∞,1and henceX0
∞,1=K∞,1X0(M) = K∞,1X (M)by [DKEP22, Proposition 3.6].
Next we show that{eF}F forms an approximate unit ofX0. Indeed, every element inX0can be written as a norm limit of linear spans consisting of elements of the from x1Jy1JT Jy2Jx2, where xi,yi∈Cr∗(Γ) andT ∈B(L2B). Write eachxi,yi as summations of ukλt, t ∈Λ, it suffices to check eF(ukJu`JeB)and (eBJu`Juk)eF agree withukJu`JeBandeBJu`JukwhenFis large enough, respectively, which follows easily from the construction ofeF.
By Lemma 3.3.4, it is easy to check thatι(eF)commutes with pnor for everyF. And it follows from Lemma 3.3.3 that limFιnor(eF)∈(KX(M)]J)∗is the identity.
Lemma 3.3.6. LetΓbe a group andΛ<Γa subgroup. Denote by qK∈(K(L2M)]J)∗the identity, M=LΓ and B=LΛ. Then pt,s=q⊥Kιnor(λtρseBρs∗λt∗)∈(B(L2M)]J)∗is a projection for t,s∈Γ. Moreover, ifΛ<Γ is almost malnormal, then pt,spt0,s0=0if t6=t0or s6=s0.
Proof. Since pnorcommutes withι(M)andι(JMJ)andqK∈(B(L2M)]J)∗is a central projection, together with Lemma 3.3.4, we see thatpt,sis a projection.
Note thatpt,spt0,s0=q⊥
Kpnorι(ProjtΛs∩t0Λs0)pnor, where ProjtΛs∩t0Λs0 denotes the orthogonal projection onto the sp{tΛs∩t0Λs0}, which is finite dimensional ift6=t0ors6=s0by the almost malnormality ofΛ<Γ. And it follows thatpt,spt0,s0 =0.
3.3.2 Proof of Proposition 3.3.1
Proof. SinceNhas no properly proximal direct summand, there exists anN-central stateµon ˜S(N)such that µ|N is normal and faithful by Lemma 3.2.5.
LetE:=Ad(eN):B(L2M)→B(L2N)and for the corresponding bidual ˜E: B(L2M)]J∗
→ B(L2N)]J∗
, we have a u.c.p. map ˜E|S(M)˜ : ˜S(M)→S˜(N)by Lemma 3.3.2. Thusϕ=µ◦E˜|S(M)˜ : ˜S(M)→C defines aN-central state that is faithful and normal on M. Let qK∈ K(L2M)]J∗
, and qX∈ KX(M)]J∗
be the corresponding identities in these von Neumann algebras. Note thatqK≤qXandqXcommutes withMand JMJ.
First we analyze the support ofϕ. Observe thatϕ(q⊥
K) =1. Indeed, ifϕ(qK)>0, i.e.,ϕdoes not vanish on(K(L2M)]J)∗, then we may restrictϕ toB(L2M), which embeds into(K(L2M)]J)∗as a normal operator M-system [DKEP22, Section 8], and this shows thatNwould have an amenable direct summand. We also haveϕ(qX) =1, since ifϕ(q⊥
X)>0, we would then have anN-central state 1
ϕ(q⊥
X)ϕ◦Ad(q⊥X): ˜SX(M)→C,
whose restriction toM is normal. This contradicts the assumption thatN⊂Mis properly proximal relative toX, sinceSX(M)embeds unitally intoeSX(M)throughιnor in Section 3.3.1. Therefore we conclude that ϕ(qXq⊥K) =1.
LetB:=LΛ⊂MandeB:L2M→L2Bthe orthogonal projection.
Claim.There exists a u.c.p. mapφ:hM,eBi →q⊥KqXS˜(M)qXsuch thatφ(x) =q⊥KqXxfor anyx∈M. This claim clearly implies thatN is amenable relative toBinsideM, asν=ϕ◦φ∈ hM,eBi∗ is anN- central state, which is a normal faithful state when restricted toM.
Proof of claim. Recall from Section 3.2.4.2 that we may embed B(L2M)into (B(L2M)]J)∗ through the u.c.p. mapιnor, which is given byιnor=Ad(pnor)◦ι, whereι :B(L2M)→B(L2M)∗∗ is the canonical∗- homomorphism into the universal envelope, andpnoris the projection inB(L2M)∗∗such thatpnorB(L2M)∗∗pnor= (B(L2M)]J)∗. We have that(ιnor)|M and(ιnor)|JMJare faithful normal representations ofMandJMJ, respec- tively, and to eliminate possible confusion, we will denote byιnor(M)andιnor(JMJ)the copies ofM and JMJ in(B(L2M)]J)∗. Restricting ιnorto C∗-subalgebraA⊂B(L2M) satisfyingM,JMJ⊂M(A)give rise to the embedding ofAinto(A]J)∗. Furthermore, althoughιnoris not a∗-homomorphism, by Lemma 3.3.4, spMeBMis in the multiplicative domain ofιnor.
Denote by{tk}k≥0⊂Γa representative of the cosetsΓ/Λwitht0being the identity ofΓ, i.e.,Γ=Fk≥0tkΛ, anduk:=λtk∈U(LΓ). We will construct the mapφin the following steps.
Step 1.For eachn≥0, consider the u.c.p. mapψn:hM,eBi → hM,eBigiven byψn(x) = (∑k≤nukeBu∗k)x(∑`≤nu`eBu∗`), and notice thatψnmapshM,eBiinto the∗-subalgebraA0:=sp{ukaeBu∗`|a∈B,k, `≥0}.
Step 2. By Lemma 3.3.6, we have{ιnor(JukJeBJu∗kJ)}k≥0⊂(B(L2M)]J)∗are pairwise orthogonal pro- jections. Sete=∑k≥0ιnor(JukJeBJu∗kJ)∈(B(L2M)]J)∗and notice thateis independent of the choice of the representativeΓ/Λ. Putφ0:A0→q⊥K(B(L2M)]J)∗to beφ0(uraeBu∗`) =q⊥Kιnor(ura)eιnor(u∗`)
It is easy to see thatφ0is well-defined. We then check thatφ0is a∗-homomorphism. For anyx∈M, we claim that
q⊥Keιnor(x)e=q⊥Kιnor(EB(x))e. (3.1) Indeed,
q⊥Keιnor(x)e
=q⊥K
∑
k,`≥0
ιnor (JukJeBJu∗kJ)x(Ju`JeBJu∗`J)
=q⊥Kιnor(EB(x))
∑
k≥0
ιnor(JukJeBJu∗kJ) +
∑
k6=`
ιnor (JukJeBJu∗kJ)x(Ju`JeBJu∗`J) .
SinceΛ<Γis almost malnormal which implies thatL2(M B)is a mixingB-bimodule, one may check that (JukJeBJu∗kJ)(x−EB(x))(Ju`JeBJu∗`J)∈B(L2M)is a compact operator fromMtoL2Mif`6=k. We also have
(JukJeBJu∗kJ)EB(x)(Ju`JeBJu∗`J) =0 if`6=k, and it follows that∑k6=`q⊥
Kιnor(JukJeBJu∗kJxJu`JeBJu∗`J) =0.
It then follows from (3.1) thatφ0is a∗-homomorphism.
We also show φ0 is norm continuous. Set ∑di=1ukiaieBu∗`
i ∈A0, and note that we may assume ki6=
kj and`i6=`j for i6= j. Consider Pk=q⊥K∑di=1ιnor(Projt
`iΛtk−1)andQk =q⊥K∑di=1ιnor(Projt
kiΛtk−1), where Projt
`iΛtk−1∈B(`2Γ)is the orthogonal projection onto the subspace sp{δt|t∈t`iΛtk−1}k·k, i.e., Projt
`iΛtk−1= JukJu`ieBu∗`
iJu∗kJ. By Lemma 3.3.6, we havePkandQkare a projections andPkPr=QkQr=0 ifk6=r. More- over, note that for eachi,ιnor(eBu∗`
iJu∗kJ)Pk=q⊥
Kιnor(eBu∗`
iJu∗kJ)andιnor(eBu∗k
iJu∗kJ)Qk=q⊥
Kιnor(eBu∗k
iJu∗kJ).
LetH be the Hilbert space where(B(L2M)]J)∗is represented on. Forξ,η∈(H)1, we compute
|hφ0(
d i=1
∑
ukiaieBu∗`
i)ξ,ηi| ≤
∑
k≥0
|
d i=1
∑
hq⊥Kιnor(eBu∗`
iJu∗kJ)ξ,ιnor(JukJukieBai)∗ηi|
=
∑
k≥0
|
d
∑
i=1
hιnor(eBu∗`
iJu∗kJ)Pkξ,ιnor(JukJukieBai)∗Qkηi|
≤
∑
k≥0
kιnor(JukJ(
d i=1
∑
ukiaieBu∗`
i)Ju∗kJ)kkPkξkkQkηk
≤ k
d
∑
i=1
ukiaieBu∗`
ik(
∑
k≥0
kPkξk2)1/2(
∑
k≥0
kQkηk2)1/2
≤ k
d i=1
∑
ukiaieBu∗`
ik,
where the last inequality follows from the orthogonality of{Pk}and{Qk}.
Lastly, notice thatφ0mapsA0intoq⊥
K
S˜(M). In fact, for anys∈Γ, we have
ιnor(ρs)eιnor(ρs∗) =
∑
k≥0
ιnor(J(λsuk)JeBJ(λsuk)∗J) =e,
asFk≥0stkΛj=Γ, and it follows thatφ0(A0)commutes withιnor(JMJ).
Therefore, we conclude thatφ0is a norm continuous∗-homomorphism from A0toq⊥KS˜(M)and hence extends to the C∗-algebraA:=A0k·k.
Step 3.For eachn≥0, setφn:=φ0◦ψn:hM,eBi →q⊥
KS˜(M), which is c.p. and subunital by construction.
We may then pickφ∈CB(hM,eBi,q⊥
KS˜(M))a weak∗limit point of{φn}n, which exists asq⊥
KS˜(M)is a von Neumann algebra.
We claim that
Ad(qX)◦φ:hM,eBi →q⊥KqXeS(M)qX
is anM-bimodular u.c.p. map, which amounts to showingφ(x) =q⊥KqXιnor(x)for anyx∈M.
In fact, for anyx∈M, we have
φ(x) =lim
n→∞φ0
0≤k,`≤n
∑
(ukEB(u∗kxu`)eBu∗`)
=q⊥Klim
n→∞
∑
0≤k,`≤n
ιnor(ukEB(u∗kxu`))eιnor(u∗`)
=q⊥Klim
n→∞
∑
0≤k,`≤n
ιnor(uk)eιnor(u∗k)
ιnor(x) ιnor(u`)eιnor(u∗`) ,
where the last equation follows from (3.1). Finally, note that by Lemma 3.3.6{pk}k≥0is a family of pairwise orthogonal projections, where
pk:=q⊥Kιnor(uk)eιnor(u∗k) =q⊥K
∑
r≥0
ιnor(JurJukeBu∗kJu∗rJ),
and∑k≥0pk=∑k,r≥0q⊥
Kιnor(JurJukeBu∗kJu∗rJ) =q⊥
KqXby Lemma 3.3.5. Therefore, we conclude thatφ(x) = q⊥
KqXιnor(x), as desired.