Properly proximal actions and crossed products

LetM be a separable finite von Neumann with normal faithful trace τ and let Γbe a group that acts on M by trace-preserving automorphisms. Note that the action Γy^α(M,τ)is unitarily implemented by the unitaries α_t⁰(x) =ˆ α[t(x). Conjugation by these unitaries then implements an action ofΓ onB(L²(M,τ)), which preserves the spaceS(M).

IfX⊂B(L²M)is a boundary piece such that conjugation by unitaries in the Koopman representation of ΓpreservesX, then the action ofΓonB(L²M)will also preserveSX(M). We say that the actionΓyMis properly proximal relative toXif there does not exist a non-zeroΓ-invariant projectionp∈Z(M), such that ΓyM pis weakly compact, and such that there does not exist aΓ-invariantM-central state onSX(M)that vanishes onK^∞,1_X (M)and is normal when restricted toM.

Note that the same proof as in Corollary 2.6.9 shows that ifΓyMis properly proximal, then(MoΓ)⁰∩ M^ω cannot be diffuse. In particular, ifM is abelian and the action is ergodic, then the the actionΓyM is strongly ergodic. Also note that ifΓyZ(M)is ergodic, then the traceτis the uniqueΓ-invariantM-central normal state onM.

The definition of proper proximality for actions is slightly more technical than for von Neumann algebras.

This is done to accommodate the proof of Lemma 2.8.5 below. We will show in Proposition 2.8.2 that proper proximality for the trivial action is equivalent to proper proximality for the von Neumann algebra.

Note that ifMis abelian and ifϕ is a state onSX(M)that restricts to the trace onM, thenϕvanishes on K^∞,1_X (M)if and only ifϕvanishes onX. Indeed, ifT∈K^∞,1_X (M), then for eachε>0 there exists a projection

p∈P(M)withτ(p)>1−ε such thatp(J pJ)T(J pJ)p∈X. SinceMis abelian we haveJ pJ=pand by Cauchy-Schwarzϕ(T)≤2εkTk+ϕ(pT p). Sinceε>0 is arbitrary this then shows that ifϕvanishes onX, then it must also vanish onK^∞,1_X (M). Thus, in the case whenMis abelian, the following proposition can give an easier criterion to check whether or not the action is properly proximal.

Proposition 2.8.1. Let(M,τ)be a finite von Neumann algebra, supposeΓy(M,τ)is a trace-preserving action, andX⊂B(L²M)is aΓ-invariant boundary piece for M. If there exists aΓ-invariant, M-central state ϕ:S_X(M)→Csuch thatϕ|M is normal andϕ|X6=0, then there exists a non-zero Γ-invariant projection p∈Z(M)so thatΓyM p is weakly compact.

Proof. First notice that since SX(M) =SKX(M), we may assumeX=KX and thus M⊂M(X). Let ϕ : SX(M)→Cbe aΓ-invariant,M-central state that satisfiesϕ|M =τ and does not vanish onX. Take aΓ- approximate unit{A_i}_i⊂Xthat is quasi-central inM(X). SinceMis contained in the multiplier ofXand since{A_i}_iis quasi-central inM(X), we obtain a non-zeroΓ-invariant andM-central positive linear functional ϕ˜ onB(L²M)with ˜ϕ|Mnormal by setting

ϕ(·) =˜ lim

i→ωϕ(A^1/2_i ·A^1/2_i ),

for some free ultrafilterω. Since ˜ϕ_|M isΓ-invariant, if we letpbe the support of ˜ϕ_|M, thenpisΓ-invariant, and ˜ϕ_{|M p}is faithful, so thatΓyM pis weakly compact.

Proposition 2.8.2. Let M be a tracial von Neumann algebra, andX⊂B(L²M)a boundary piece. Then M is not properly proximal relative toXif and only if there either exists a central projection p∈Z(M), such that M p is amenable, or else such that there exists an M-central stateψ :SX(M)→Csuch thatψ_|Mis normal andψ_|

K^∞,1_X (M)=0.

Proof. SupposeM is not properly proximal relative toXand thatM has no amenable direct summand. Let ϕ:SX(M)→Cbe anM-central state such thatϕ|M is normal. A simple standard argument shows that we may then produce a newM-central state (which we still denote byϕ) and a central projection p∈Z(M) such thatϕ(x) = _τ(p)¹ τ(xp)forx∈M. SinceMhas no amenable direct summand, Proposition 2.8.1 shows thatϕ_|X6=0.

Extendϕto a state, still denoted byϕ, onB(L²M). Sinceϕ_{|Z(M p)}=_τ(p)¹ τby the Dixmier property we may produce a net of u.c.p. mapsαi:B(L²M)→B(L²M)that are convex combinations of conjugation by unitaries inJMJ such that limi→∞|ϕ◦αi(x)−_τ(p)¹ τ(x)| →0 for eachx∈JMJ p. We may then let ˜ϕ be a weak^∗-limit point of{ϕ◦αi}_i. Since eachαi isM-bimodular and preservesS_X(M)(see the remark at the beginning of Section 2.6), it follows that ˜ϕ|S_X(M)is againM-central. Also, since eachαipreservesXit follows

that ˜ϕagain vanishes onX. Since ˜ϕrestricts to the canonical traces on bothM pandJMJ pit then follows from continuity that ˜ϕalso vanishes onK^∞,1_X (M).

It was shown in [IPR19] that proper proximality for a groupΓis an invariant for the orbit equivalence relation associated to any free probability measure-preserving action. Here we show a counterpart for proper proximality of an action.

Proposition 2.8.3. LetΓy^α(M,τ)be a trace-preserving action ofΓon a separable finite von Neumann algebra M with a normal faithful traceτ. SupposeX⊂B(L²(M,τ))is aΓ-boundary piece. ThenΓyM is properly proximal relative toXif and only ifNMoΓ(M)yM is properly proximal relative toX. In particular, proper proximality for a free probability measure-preserving action is an an invariant of the orbit equivalence relation.

Proof. SinceΓyMis weakly compact if and only ifNMoΓ(M)yMis weakly compact [OP10a, Proposition 3.4], we may assume that there is noΓ-invariant non-zero projectionp∈Z(M)such thatΓyMzis weakly compact.

SupposeΓyM is not properly proximal relative toXand letϕ be aΓ-equivariant andM-central state on SX(M)withϕ|M normal and ϕ_|

K^∞,1_X (M) =0. We claim that ϕ isNMoΓ(M)-equivariant. Indeed, take any u∈NMoΓ(M) and by [Dye59] there exists a partition of unity {p_t}t∈Γ⊂P(Z(M)) and unitaries {v_t}t∈Γ⊂U(M)so that{α_t−1(p_t)}t∈Γalso forms a partition of unity andu=∑t∈Γp_tv_tu_t. Letα_u⁰∈B(L²M) be a unitary given byα_u⁰(x) =ˆ uxud^∗. Notice thatα_u⁰=∑t∈Γp_tv_tJv_tJα_t⁰. ForF⊂Γfinite we also setα_u,F⁰ =

∑t∈Fp_tv_tJv_tJα_t⁰. We may extendϕ toB(L²M), still denoted byϕ, and compute that for anyT ∈B(L²M) andF⊂Γfinite

|ϕ(α_u⁰T(α_u⁰)^∗)−ϕ(α_u,F⁰ T(α_u,F⁰ )^∗)| ≤ |ϕ((α_u⁰−α_u,F⁰ )T(α_u⁰)^∗)|+|ϕ(α_u,F⁰ T(α_u⁰−α_u,F⁰ )^∗|

≤ |ϕ((α_u⁰−α_u,F⁰ )(α_u⁰−α_u,F⁰ )^∗)|^1/2|ϕ(α_u⁰T^∗T(α_u⁰)^∗)|^1/2 +|ϕ(α_u,F⁰ T T^∗(α_u,F⁰ )^∗)|^1/2|ϕ((α_u⁰−α_u,F⁰ )(α_u⁰−α_u,F⁰ )^∗)|^1/2

≤2kTkϕ(

∑

t6∈F

p_t)^1/2.

Notice that for S∈SX(M), we have α_u,F⁰ S(α_u,F⁰ )^∗= (∑t∈Fp_t)α_u⁰S(α_u⁰)^∗(∑t∈Fp_t)∈SX(M)and since ϕ|S_X(M)isM-central we have

ϕ(α_u,F⁰ S(α_u,F⁰ )^∗) =

∑

t∈F

ϕ(p_tv_tJv_tJα_t⁰S(α_t⁰)^∗Jv^∗_tJv_t) =

∑

t∈F

ϕ(p_tv_tα_t⁰S(α_t⁰)^∗v^∗_t) +ϕ(K_t)

=

∑

t∈F

ϕ(α_t⁰α_t⁻¹(p_t)S(α_t⁰)^∗) +ϕ(K_t) =ϕ(

∑

t∈F

α_t⁻¹(p_t)S) +ϕ(

∑

t∈F

K_t),

whereK_t=p_tv_t[Jv_tJ,α_t⁰S(α_t⁰)^∗]Jv^∗_tJv_t∈K^∞,1_X (M). Sinceϕvanishes onK^∞,1_X (M)we then have

|ϕ(S)−ϕ(α_uSα_u^∗)| ≤2kSkϕ(

∑

t6∈F

p_t)^1/2+kSkϕ(1−

∑

t∈F

α_t⁻¹(p_t))^1/2=3kSkϕ(

∑

t6∈F

p_t)^1/2.

SinceF⊂Γwas an arbitrary finite set andϕ_|Mis normal we conclude thatϕ_|S

X(M)isNMoΓ(M)-invariant.

The following lemma is a simple application of the Hahn-Banach Theorem.

Lemma 2.8.4. Let E be an operator system. Suppose Γy^αE is an action ofΓon E by complete order isomorphisms. If E does not admit a Γ-invariant state, then there exist F ⊂Γ finite, {T_t}t∈F ⊂E, and T₀∈E₊such thatk1+T₀+∑t∈F(T_t−αt(T_t))k<1/4.

IfX⊂B(L²M)is a boundary piece, then we setK_X(M)^]_J=K_X(M)^M]M∩K_X(M)^JMJ]JMJ. Recall from Proposition 2.2.10 that(B(L²M)^]_J)^∗is a von Neumann algebra containingMandJMJas von Neumann sub- algebras, and(KX(M)^]_J)^∗is a corner of this von Neumann algebra with the support projection of(KX(M)^]_J)^∗ commuting with bothMandJMJ. The following technical lemma is adapted from Theorem 4.3 in [BIP21].

Lemma 2.8.5. Let M be a separable finite von Neumann algebra with normal faithful traceτ. Suppose that Γy^αM is trace-preserving action by a groupΓsuch thatΓyZ(M)is ergodic and letX⊂B(L²M)be a Γ-boundary piece

ThenΓyM is properly proximal relative toXif and only if there is no M-centralΓ-invariant stateϕon

eSX(M):=n T∈

B(L²M)^]_J∗

|[T,a]∈

KX(M)^]_J∗

for alla∈JMJo

such thatϕ|M=τ.

Proof. First note that asS_X(M) =S_KX(M)andeS_X(M) =eS_KX(M)we may assume thatX=K_X. In particular, we may assume that both M andJMJ are contained in the multiplier algebra of X. Also, note that by considering the natural map fromB(L²M)into(B(L²M)^]_J)^∗we getΓ-equivariantM-bimodular u.c.p. maps B(L²M)→

KX(M)^]_J∗

, andSX(M)→eSX(M). Thus, if there exists an M-centralΓ-invariant stateϕ on eSX(M)such thatϕ_|M=τ, then eitherϕ

| KX(M)^]_J

∗6=0, in which case restricting toB(L²M)shows thatΓyM is weakly compact [OP10a, Proposition 3.2] (and hence not properly proximal), or elseϕ

|

KX(M)^]_J∗=0, in which case restrictingϕtoSX(M)shows thatΓyMis not properly proximal.

We now show the converse. So suppose thateSX(M)has noM-centralΓ-invariant state that restricts to the trace onM. SetG =NMoΓ(M)and continue to denote byα the trace-preserving action ofG onMgiven by conjugation. SinceΓyZ(M)is ergodic it follows from [Dye63] that the trace is the uniqueG-invariant state onM, and hence there exists noG-invariant state oneS_X(M).

LetM₀⊂Mbe an ultraweakly dense∗-subalgebra that is generated, as an algebra, by a countable set of contractionsS₀={x₀=1,x₁, . . .}. By Lemma 2.8.4 there exists a finite setF⊂G,{T_g}g∈F⊂eSX(M), and T₀∈eSX(M)₊, so thatk1+T₀+∑g∈F(T_g−α_g(T_g))k<1/4. By enlargingF(possibly with repeated elements) we may assumekT_gk ≤1 for eachg∈F.

We letIdenote the set of all pairs(A,κ)whereA⊂B(L²M)^]_Jis a finite set andκ>0. We define a partial ordering onIby(A,κ)≺(A⁰,κ⁰)ifA⊂A⁰andκ⁰≤κ.

For eachTg, we may find a net{T_gⁱ}_i∈I⊂B(L²M)whose weak^∗limit isTg; moreover, by Goldstein’s theorem, we may assumekT_gⁱk ≤1 for alli∈I. Since an operatorT ∈

B(L²M)^]_J∗

is a positive contraction if and only ifϕ(T)∈[0,1]for all statesϕ∈B(L²M)^]_J, and since the set of positive contractions is convex, it then follows from the Hahn-Banach theorem that any positive contraction in

B(L²M)^]_J∗

is in the ultraweak closure of positive contractions inB(L²M). We may therefore also find a net of positive operators{T₀ⁱ}i∈I⊂ B(L²M)so thatkT₀ⁱk ≤ kT₀k, andT₀ⁱ→T₀weak^∗.

We also choose a net{S_i}i∈I∈B(L²M)such thatkS_ik<1/4, for eachi∈I, andS_i→1+T₀+∑g∈F(T_g− α_g(T_g))weak^∗.

For any given g∈ {0} ∪F andn∈N, we have [T_g,Jx_nJ]∈

(KX)^]_J∗

and so we may choose a net {K_g,nⁱ }_i∈I⊂KXsuch that limi→∞K_g,nⁱ −[T_gⁱ,JxnJ] =0 weak^∗. Furthermore, sinceB(L²M)^]_J is the space of linear functionals that are continuous for both the M-M andJMJ-JMJ-topologies we may pass to convex combinations and assume that there exists a net{e^n,g_i }i∈I,⊂P(M)such thate^n,g_i →1 strongly, and

i→∞limke^n,g_i Je^n,g_i J(K_g,nⁱ −[T_gⁱ,Jx_nJ])Je^n,g_i Je^n,g_i k=0. (2.9)

Since we also have that 1+T₀ⁱ+∑g∈F(T_gⁱ−αg(T_gⁱ))−S_i converges weak^∗ to 0, we may assume that, in addition to (2.9), we also have

e^n,g_i Je^n,g_i J 1+T₀ⁱ+

∑

g∈F

(T_gⁱ−α_g(T_gⁱ))−S_i

!

Je^n,g_i Je^n,g_i

→0. (2.10)

We fix ε>0 to be chosen later. For eachn∈N, we may then choosei(n)∈Isuch that setting f_n=

∧g∈F,m≤ne^m,g_i(n), we haveτ(f_n)>1−ε2⁻ⁿ⁻¹, and

f_nJ f_nJ

Kg,mⁱ⁽ⁿ⁾−[T_gⁱ⁽ⁿ⁾,Jx_mJ]

J f_nJ f_n

<2⁻ⁿ, (2.11)

for allm≤nandg∈F, and also

f_nJ f_nJ 1+T₀ⁱ⁽ⁿ⁾+

∑

g∈F

(T_gⁱ⁽ⁿ⁾−αg(T_gⁱ⁽ⁿ⁾))−S_i(n)

! J f_nJ f_n

<2⁻ⁿ/16. (2.12)

Settingp_n=∧k≥nf_kwe then have that{p_n}_nis an increasing sequence, andτ(p_n)>1−ε2⁻ⁿ, for alln≥1.

By Lemma 13.7 in [Dav88] there existsδn>0 so that whenK∈X+, andz∈M(X)are contractions satisfyingk[K,z]k<δn, then we havek[K^1/2,z]k<2⁻ⁿ/16(6|F|+4+kT₀k)<2⁻ⁿ.

If we consider the unitalC^∗-subalgebraBgenerated by

{Kg,mⁱ⁽ⁿ⁾,Jx_nJ,a_k,p_n,Jq_nJ|n≥1,g∈F,m≤n},

thenBis a separableC^∗-algebra andB∩Xis a closed ideal. It then follows from [Arv77] that there exists an increasing sequence{F_n}n≥1⊂X+∩Bsuch that that the following conditions are satisfied:

(a) kF_nK_g,mⁱ⁽ⁿ⁺¹⁾−K_g,mⁱ⁽ⁿ⁺¹⁾k<2⁻ⁿfor allg∈ {0} ∪Fandm≤n+1;

(b) k[F_n,Jx_mJ]k<δn/2 form≤n;

(c) kF_n−α_g(F_n)k<δ_n/2 for allg∈F;

(d) k[F_n,p_m]k,k[F_n,J p_mJ]k ≤δ_n/2 for allm≤n.

SetE1=F₁^1/2,E_n= (F_n−Fn−1)^1/2forn≥2, and defineT_g⁰=∑_nE_nTgⁱ⁽ⁿ⁾E_nforg∈ {0} ∪F, where the sum is SOT convergent askT_gⁱk ≤ kT_gkfor eachg. We claim thatT_g⁰∈SX(M). SinceK^∞,1_X (M)is a strong JMJ-JMJbimodule it suffices to show that[T_g⁰,Jx_mJ]∈K^∞,1_X (M)for any givenx_m∈S₀. We compute

[T_g⁰,Jx_mJ] =

∑

n

E_nTgⁱ⁽ⁿ⁾[E_n,Jx_mJ] +

∑

n

[E_n,Jx_mJ]T_gⁱ⁽ⁿ⁾E_n +

∑

n

E_n[T_gⁱ⁽ⁿ⁾,Jx_mJ]E_n.

(2.13)

The finite sums in the summations∑nE_nT_kⁱ⁽ⁿ⁾[E_n,Jx_mJ]and∑n[E_n,Jx_mJ]T_gⁱ⁽ⁿ⁾E_nbelong toXsinceXis hered- itary. The summations∑nE_nT_gⁱ⁽ⁿ⁾[E_n,Jx_mJ]and∑n[E_n,Jx_mJ]T_gⁱ⁽ⁿ⁾E_nare norm convergent sincek[E_n,Jx_mJ]k<

2⁻ⁿfor sufficiently largenand hence they also belong toX. Using a similar argument with (2.11) and (d) we deduce

p_{`</sub>J p<sub>`}J

∑

n

E_nKg,mⁱ⁽ⁿ⁾E_n−

∑

n

E_n[T_gⁱ⁽ⁿ⁾,Jx_mJ]E_n

J p_{`</sub>J p<sub>`}∈X,

for any fixed`≥1, and hence

∑

n

E_nKg,mⁱ⁽ⁿ⁾E_n−

∑

n

E_n[T_gⁱ⁽ⁿ⁾,Jx_mJ]E_n

∈K^∞,1_X . (2.14)

On the other hand, sinceE_n²=F_n−Fn−1, we have from (a) thatkE_nKg,mⁱ⁽ⁿ⁾E_nk<2⁻ⁿ⁺¹for large enough nand hence∑nEnKg,mⁱ⁽ⁿ⁾Enis also norm convergent, and thus contained inXas well. This then shows that [T_g⁰,Jx_mJ]∈K^∞,1_X (M), for allg∈ {0} ∪F, andm≥0, verifying the claim thatT_g⁰∈SX(M). Note that we also haveT₀⁰∈(S_X(M))₊sinceT₀⁰is the SOT-limit of positive operators.

We defineS=∑nE_nS_i(n)E_n. From (2.12), (c) and (d) we have

p₁J p₁J 1+T₀⁰+

∑

g∈F

(T_g⁰−α_g(T_g⁰))−S

! J p₁J p₁

≤2

∑

g∈F

∑

n

kE_n−αg(E_n)k+2

∑

n

k[p₁,E_n]k+

∑

n

k[J p₁J,E_n]k

(2+kT₀k+2|F|)

+k

∑

n

E_np₁J p₁J(1+T₀ⁱ⁽ⁿ⁾+

∑

g∈F

(T_gⁱ⁽ⁿ⁾−α_g(T_gⁱ⁽ⁿ⁾))−S_i(n))J p₁J p₁E_nk

≤1/16+k

∑

n

E_np₁J p₁J 1+T₀ⁱ⁽ⁿ⁾+

∑

g∈F

(T_gⁱ⁽ⁿ⁾−α_g(T_gⁱ⁽ⁿ⁾))−S_i(n)

!

J p₁J p₁E_nk ≤1/8.

Note that sinceE_n≥0 and∑nE_n²=1, the mapφ:`^∞(B(L²M))→B(L²M)given byφ((z_n)) =∑nE_nz_nE_n is unital completely positive. In particular, we have that φ is a contraction and it follows that kSk ≤ sup_nkS_i(n)k ≤1/4. Thus, from the previous inequalities we conclude that

p1J p1J 1+T₀⁰+

∑

g∈F

(T_g⁰−α_t(T_g⁰))

! J p1J p1

<3/8. (2.15)

Since τ(p₁)≥1−ε, it follows easily by considering the disintegration into factors that there exist a central projectionz≤z(p₁), withτ(z)>1−2ε, and two subprojectionsq₂,q₃≤zwithq₂+q₃=zsuch thatq₂andq₃are both Murray-von Neumann subequivalent top₁. Take partial isometriesv₂,v₃such that v^∗₂v₂,v^∗₃v₃≤p₁ andv₂v^∗₂=q₂, v₃v^∗₃=q₃. We define the u.c.p. map φ :B(L²M)→B(L²M)by φ(T) = Jv₂JT Jv^∗₂J+Jv₃JT Jv^∗₃J+z^⊥T z^⊥. Note thatφ mapsSX(M) into itself, and for allT ∈SX(M)we have φ(T)−T ∈K^∞,1_X (M).

It follows from (2.15) that

p₁z 1+φ(T₀⁰) +

∑

g∈F

(φ(T_g⁰)−φ(α_g(T_g⁰)))

! zp₁

<3/4. (2.16)

If we had a Γ-invariant stateϕ onS_X(M)such thatϕ|M =τ andϕ_|

K^∞,1_X (M)=0, then from (2.16), the Cauchy-Schwarz inequality, and the fact thatφ(T)−T∈K^∞,1_X (M)for eachT ∈SX(M), we would then have

1−3ε≤τ(p₁z) =ϕ(p₁z)

≤3/4+ϕ p₁z φ(T₀⁰) +

∑

g∈F

(φ(T_g⁰)−φ(α_g(T_g⁰)))

! zp₁

!

=3/4+ϕ p₁z T₀⁰+

∑

g∈F

(T_g⁰−α_g(T_g⁰))

! p₁z

!

≤3/4+2|F|(1−ϕ(p₁z))≤3/4+6|F|ε.

Thus, we obtain a contradiction by choosingε>0 so that(3+6|F|)ε<1/4.

The following slight variant of of the previous lemma will be of more use in the sequel.

Lemma 2.8.6. Using the same notation as in the previous lemma, the actionΓyM is properly proximal if and only ifΓyM is not weakly compact and there is no M-centralΓ-invariant stateϕ oneS_X(M)such that ϕ|M=τandϕ

|

K_X(M)^]_J∗=0.

Proof. Using the previous lemma, supposeϕ:eSX(M)→Cis anM-centralΓ-invariant state such thatϕ_|M=τ.

By considering the naturalM andJMJ-bimodular u.c.p. map fromB(L²M)into

KX(M)^]_J∗

we see that if ϕdoes not vanish on

KX(M)^]_J∗

, then we get a state onB(L²M)that isΓ-invariant,M-central, and normal onM, which shows thatΓyMis weakly compact by Proposition 3.2 in [OP10a].

Lemma 2.8.7. Let M be a finite von Neumann algebra and B⊂M a von Neumann subalgera. Let e_B:L²M→ L²B denote the orthogonal projection. Then e_BK^∞,1(M)e_B⊂K^∞,1(B), and e_BS(M)e_B⊂S(B).

Proof. For anyT∈K^∞,1(M), denote by{T_n}n∈N⊂K(L²M)a sequence that converges toT ink · k_∞,1. Notice that

ke_B(T−T_n)e_Bk∞,1= sup

a,b∈(B)₁

he_B(T−T_n)e_Ba,bi ≤ kTˆ −T_nk∞,1,

and we conclude thate_BTe_B∈K^∞,1(B). SinceJBJcommutes withe_B, we also havee_BS(M)e_B⊂S(B).

Theorem 2.8.8. Let M₁ and M₂ be separable finite von Neumann algebras and suppose we have trace- preserving actionsΓyM_i. Then the actionΓyM₁⊗M₂is properly proximal if and only if the actionΓyM_i is properly proximal for each i=1,2.

Proof. If the actionΓyM₁⊗M₂is weakly compact, then so isΓyM_i, for eachi. We may therefore restrict to the case whenΓyM is not weakly compact. Also, ifZ(M_i)^Γis diffuse for somei, then the same is

true forZ(M₁⊗M₂)^Γ, and the result is easy. We may therefore also restrict to the case whenZ(M_i)^Γis completely atomic for eachi=1,2. It is easy to see that a direct sum of trace-preserving actions is properly proximal if and only if each summand is properly proximal, and hence by restricting to each atom we may further reduce to the case whenΓyZ(M_i)is ergodic for eachi=1,2.

SupposeΓyMis not properly proximal. By Lemma 2.8.6 there is anM-centralΓ-invariant stateϕ on eS(M)withϕ|M =τ andϕ_|(K∞,1

(M)^M]M_J )^∗ =0. LetA_i∈K(L²M1)⊂B(L²M1)be an increasing quasi-central approximate unit. We consider the von Neumann algebra(B(L²M)^M]M_J )^∗and letA∈(B(L²M)^M]M_J )^∗be the weak^∗-limit of the increasing netA_i⊗1. We then define anM₂-bimodularΓ-equivariant completely positive mapΦ:S(M₂)→(B(L²M)^M]M_J )^∗byΦ(T) =A^1/2(1⊗T)A^1/2.

Note that forx₁∈M₁we have

[Φ(T),Jx₁J] =lim

i [A_i,Jx₁J]⊗T=0, and forx₂∈M₂we have

[Φ(T),Jx2J] =lim

i A_i⊗[T,Jx2J]∈(K^∞,1(M)^M]M_J )^∗.

Since(K^∞,1(M)^M]M_J )^∗is a strongJMJ-bimodule, we have[Φ(T),JxJ]∈(K^∞,1(M)^M]M_J )^∗forx∈Min general.

Therefore we haveΦ:S(M₂)→eS(M). Ifϕ(A)6=0, then composingΦwithϕ gives anM₂-centralΓ- invariant positive linear functional onS(M₂)that restricts toϕ(A)τ onM₂. As we haveΦ:K^∞,1(M₂)→ (K^∞,1(M)^M]M_J )^∗this then shows thatΓyM₂is not properly proximal.

If we let B_j∈K(L²M₂)⊂B(L²M₂)be an increasing quasi-central approximate unit and if we letB∈ (B(L²M)^M]M_J )^∗be a weak^∗-limit of the increasing net 1⊗B_j, then just as above ifϕ(B)6=0 it would then follow thatΓyM₁is not properly proximal.

We now supposeϕ(A) =ϕ(B) =0 and consider theM₁-bimodularΓ-equivariant u.c.p. mapΨ:S(M₁)→ (B(L²M)^M]M_J )^∗given byΨ(T) = (1−A)^1/2(T⊗1)(1−A)^1/2. Similar to the calculation above, we have that [Ψ(T),JxJ] =0 for allx∈M1⊗_algM2and hence[Ψ(T),JxJ] =0 for allx∈M. Therefore,Ψ(T)∈eS(M), and sinceϕ(A) =0 it then follows thatϕ◦Ψdefines anM₁-centralΓ-equivariant state onS(M₁)that restricts to the trace onM₁. Moreover, sinceϕ(B) =0 we have thatϕ◦Ψ_|K∞,1(M₁)=0 and henceΓyM₁is not properly proximal.

Conversely, supposeΓyM₁is not properly proximal. Letϕ:S(M₁)→Cbe anM₁-centralΓ-invariant state satisfyingϕ_|M1=τandϕ_|K∞,1(M₁)=0

We consider the u.c.p. map

Ad_e1:B(L²(M₁⊗M₂))→B(L²(M₁))

given by Ad_e1(T) =e₁Te₁wheree₁:L²(M₁⊗M₂)→L²(M₁)is the orthogonal projection. By Lemma 2.8.7 Ad_e1mapsS(M₁⊗M₂)intoS(M₁)and mapsK^∞,1(M₁⊗M₂)intoK^∞,1(M₁). We obtain a stateψ:S(M₁⊗M₂)→ Cby settingψ=ϕ◦Ade₁.

Note thatψ|M=τ◦EM₁=τ. Also note that since Ad_e1 isM1-bimodular andΓ-equivariant we see that ψ is M₁-central. If u∈U(M₂), then we have e₁uJuJ =e₁, and since ϕ_|K∞,1(M₁) =0 it follows that for T ∈S(M₁⊗M₂)we have

ψ(uT) =ϕ(e₁uTe₁) =ϕ(e₁(uJuJ)T(Ju^∗J)e₁) =ϕ(e₁T(Ju^∗J)e₁)

=ϕ(e₁T(Ju^∗J)(uJuJ)e₁) =ψ(Tu).

Hence,M₂ is also in the centralizer ofψ, so that the centralizer ofψ contains the entirety ofM₁⊗M₂= W^∗(M₁,M₂). Sinceψ_|K∞,1(M1⊗M₂)=0 this then shows thatΓyM₁⊗M₂is not properly proximal.

Remark 2.8.9. LetMbe a properly proximal separable II₁factor. Then the amplificationM^tis also properly proximal for anyt>0. Indeed, it follows from the previous theorem thatM⊗M∼=M^t⊗M^1/t is properly proximal and thusM^t is also properly proximal. Also, ifN⊂Mis a finite index subfactor in a separable II₁ factorM, and ifNis properly proximal, then so ishM,e_Nisince it is an amplification ofN. Combining with Proposition 2.6.6, we then see thatMis properly proximal as well.

Theorem 2.8.10. LetΓy(X,µ)be a probability measure preserving action, and letπ:(X,µ)→(Y,ν)be a factor map with(Y,ν)diffuse. IfΓy(X,µ)is properly proximal, then so isΓy(Y,ν).

Proof. SupposeΓy(X,µ)is properly proximal. We setM=L^∞(X,µ)andM₁=L^∞(Y,ν), and we view M₁⊂Mvia the embeddingπ^∗(f) = f◦π. If we consider the map Ad_e1 :B(L²M)→B(L²(M₁))as in the second half of the proof of Theorem 2.8.8, then we see from the proof of Theorem 2.8.8 that the only property used to show thatΓyM₁is properly proximal is thatM=W^∗(M₁,M₁⁰∩M), which obviously holds here since Mis abelian.

Proposition 2.8.11. SupposeΓy^α(M,τ)is a trace-preserving action on a finite von Neumann algebra. If MoΓis properly proximal, then the actionΓyM is properly proximal.

Proof. This is also similar to the second half of the proof of Theorem 2.8.8. SupposeΓyMis not properly proximal, and letϕ:S(M)→Cbe aΓ-invariantM-central state such thatϕ|Mis normal andϕ_|K∞,1(M)=0.

Consider the u.c.p. map Ad_eM :B(L²(MoΓ))→B(L²M) given by Ad_eM(T) =e_MTe_M, wheree_M : L²M⊗`²Γ→L²Mis the orthogonal projection. By Lemma 2.8.7 we have Ad_eM:S(MoΓ)→S(M)and and Ad_eM :K^∞,1(MoΓ)→K^∞,1(M). We then consider the stateψ=ϕ◦Ad_eM onS(MoΓ).

Note thatψ_|M=ϕ◦E_M is normal andψ isM-central since Ad_eM isM-bimodular. For anyt∈Γ, denote byu_t∈U(MoΓ)andα_t⁰∈U(B(L²M))the corresponding unitaries. Notice thate_Mu_tJu_tJ=u_tJu_tJe_Mand u_tJu_tJ_|L2M=α_t⁰. Sinceϕ_|K∞,1(M)=0, it follows that for anyT∈S(MoΓ)we have

ψ(u_tT) =ϕ(e_MutTeM) =ϕ(e_M(u_tJutJ)T(Ju_tJ)^∗eM) =ϕ(α_t⁰eMT(Ju_tJ)^∗eM)

=ϕ(e_MT(Ju_tJ)^∗e_Mα_t⁰) =ϕ(e_MT(Ju_tJ)^∗u_tJu_tJe_M) =ψ(Tu_t).

Therefore,MoΓis contained in the centralizer ofψ. Sinceψ_|K∞,1(MoΓ)=0 this then shows thatMoΓis not properly proximal.

We note here that non-strong ergodicity and the existence of weakly compact factors are not the only obstructions for an action proper proximality. Ford≥3 the groupSL_d(Z)n(Z[¹_p])^dis not properly prox- imal by [IPR19], and sinceSL_d(Z)is properly proximal it then follows from Theorem 2.8.12 that the dual actionSL_d(Z)y(\Z[¹_p])^dis not properly proximal. However, this action has stable spectral gap (and hence is strongly ergodic and has no weakly compact diffuse factor) since it is a weak mixing action of a property (T) group.

We also mention that proper proximality ofMoΓdoes not imply thatΓis properly proximal, asΛoΓis always properly proximal for nontrivialΛand nonamenableΓ[DE21].

Theorem 2.8.12. LetΓy^α(M,τ)be a trace-preserving action of a countable groupΓon a separable finite von Neumann algebra M with a normal faithful traceτ. IfΓy(M,τ)andΓare both properly proximal, then MoΓis properly proximal.

Proof. This is similar to the first half of the proof of Theorem 2.8.8. We first note that ifZ(M)^Γis diffuse, then the result is easy. As above we may then restrict to the case whenZ(M)^Γis completely atomic, and then by considering atoms we may restrict to the case whenΓyZ(M)is ergodic.

Suppose then thatΓyZ(M)is ergodic andMoΓis not properly proximal, then by Lemma 2.8.6 there exists anM-centralΓ-invariant stateϕon ˜S(M)such thatϕ|M=τandϕ_|(K∞,1

(MoΓ)^]_J)^∗=0.

LetAi∈c0(Γ)⊂B(`²Γ)be an increasing approximate unit that is almostΓ-invariant. LetAdenote the weak^∗-limit of the increasing net 1⊗A_iwhen viewed inside of

B(L²M⊗`²Γ)^]_J∗

, where we identifyL²(Mo Γ)withL²M⊗`²Γby mappingxuc_tto ˆx⊗δ_t for eachx∈Mandt∈Γ. Note that under this identification we have thatMis represented asM⊗1, and ifx∈M, thenJxJis represented by the operator∑t∈ΓJ_Mαt(x)J_M⊗P_t

whereP_tdenotes the rank-one projection ontoCδt⊂`<sup>2</sup>Γ. In particular, we see thatMandJMJboth commute withC⊗`^∞Γ.

We then consider the u.c.p. mapΦ:B(L²M)→

B(L²M⊗`²Γ)^]_J∗

given byΦ(T) =A^1/2(T⊗1)A^1/2. SinceA_i∈c₀(Γ)it then follows thatΦis bothM andJMJ-bimodular. Also sinceA_i is almostΓ-invariant it follows thatΦisΓ-equivariant, where the action on

B(L²M⊗`²Γ)^]_J∗

is given by Ad(u_t). We also have that the range ofΦmaps into the subspace of invariant operators under the action of Ad(1⊗ρ_t). Hence, it follows that the restriction ofΦtoS(M)gives aM-bimodularΓ-equivariant u.c.p. map intoS(MoΓ).

Ifϕ(A)6=0, then consideringϕ◦Φwe obtain anM-centralΓ-invariant positive linear functional onS(M) that restricts toϕ(A)τonM. Moreover, sinceϕ_|(

K^∞,1(MoΓ)^]_J)^∗=0 we then have thatϕ◦Φ_|K∞,1(M)=0, which shows thatΓyMis not properly proximal.

Otherwise, ifϕ(A) =0, then we may consider theΓ-equivariant u.c.p. mapΨ:`^∞Γ→

B(L²M⊗`²Γ)^]_J∗

given byΨ(f) =1⊗(1−A_n)^1/2M_f(1−A_n)^1/2. As above, we see that the restriction ofΨtoS(Γ)maps into S(MoΓ), and then the stateϕ◦Ψdefines aΓ-invariant state onS(Γ), which shows thatΓis not properly proximal.