Proper proximality via malleable deformations

2.9 Proper proximality and deformation/rigidity theory

2.9.1 Proper proximality via malleable deformations

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.

Proof. We letV_n denote the isometry fromL²M to L²M˜ given byV_n(x) =ˆ α[_n(x)for x∈M. We also let φ_n:B(L²M)˜ →B(L²M)be theΓ-equivariant u.c.p. map given byφ_n(T) =V_n^∗TV_n, forT ∈B(L²M).˜

If x∈M, then we have kxV_n−V_nxk_∞,2=sup_a∈(M)

1kxα_n(a)−α_n(xa)k₂≤ kx−α_n(x)k₂. It therefore follows that for allx∈MandT∈B(L²M)˜ we have

kφ_n(T x)−φ_n(T)xk_∞,1,kφ_n(xT)−xφ_n(T)k_∞,1≤ kTkkx−αn(x)k₂. (2.17)

We similarly have

kφ_n(T JxJ)−φ_n(T)JxJk_∞,1,kφ_n(JxJT)−JxJφ_n(T)k_∞,1≤ kTkkx−α_n(x)k₂. (2.18)

We view each φ_n as a u.c.p. map into the von Neumann algebra (B(L²M)^]_J)^∗ and let φ :B(L²M)˜ → (B(L²M)^]_J)^∗denote a point ultraweak limit point of{φ_n}_n. From (2.17) and (2.18) we have thatφis bimodular with respect to bothMandJMJ. Moreover, sinceeM◦αn∈K^L_X(M,L²M)it follows thatφn(e_M)∈K^∞,1_X (M) for eachn≥1 and henceφ(e_M)∈(KX(M)^]_J)^∗.

IfΓyM were not properly proximal relative toX, then by Lemma 2.8.6, there exists aΓ-invariantM- central stateϕoneS_X(M)withϕ|M=τand such thatϕ vanishes on(K_X(M)^]_J)^∗. Sinceφ isJMJ-bimodular it follows that φ :JMJ⁰∩B(L²M)˜ →eS_X(M). Since φ is M-bimodular and Γ-equivariant it follows that ϕ◦φisM-central andΓ-invariant, and we also see thatϕ◦φ|M=τ. And sinceϕvanishes on(KX(M)^]_J)^∗and φ(e_M)∈(KX(M)^]_J)^∗it then follows that the stateϕ◦φ◦Ad(e^⊥_M)verifies the conclusion of the proposition.

As an application of the previous proposition, we give here examples of properly proximal actions stem- ming from Gaussian processes associated to an orthogonal representations. See, e.g., [KL16] for some details on Gaussian actions. LetH be a real Hilbert space, the Gaussian process gives a tracial abelian von Neumann algebraA_H, together with an isometryS:H →L²_R(A_H)so that orthogonal vectors are sent to independent Gaussian random variables, and so that the spectral projections of vectors in the range ofSgenerateA_H as a von Neumann algebra.

In this case, the complexification of the isometry S extends to a unitary operator from the symmetric Fock spaceS(H) =CΩ⊕^L∞_n=1(H ⊗C)ⁿinto L²(A_H). IfH =H1⊕H2, then conjugation by the unitary implementing the canonical isomorphismS(H1⊕H2)∼=S(H1)⊗S(H2)implements a canonical isomorphismA_H1⊕H2∼=A_H1⊗A_H2.

IfV :K →H is an isometry, then we obtain an isometryV^S:S(K)→S(H)on the level of the symmetric Fock spaces, and conjugation by this isometry gives an embedding of von Neumann algebras Ad(V^S):A_K →A_H. IfV were a co-isometry the conjugation byV^Simplements instead a conditional

expectation from A_H toA_K. In particular, ifU ∈O(H) is an orthogonal operator, then we obtain a trace-preserving∗-isomorphismαU =Ad(U^S)∈Aut(A_H). Ifπ:Γ→O(H)is an orthogonal represen- tation, then the Gaussian action associated to π is given by Γ3t 7→α_π(t)∈Aut(A_H). The correspond- ing Koopman representation is then canonically isomorphic to the symmetric Fock space representation π^S:Γ→U(S(H)).

Every contractionA:K →H can be written as the composition of an isometryV:K →L followed by a co-isometryW :L →H so that conjugation by A^S:S(K)→S(H)implements a u.c.p. map φA:B(L²A_H)→B(L²A_K), which mapsA_H intoA_K. Note that the associationA7→φAis continuous as a map from the space of contractions endowed with the strong operator topology into the space of u.c.p. maps endowed with the topology of point-ultraweak convergence.

IfAis self-adjoint this can be realized explicitly by considering the orthogonal matrix onH ⊕H given by ˜A=







A −√

1−A²

√

1−A² A





, so thatA=V^∗AV, where˜ Vξ =ξ⊕0, forξ∈H.

We recall that an orthogonal representationπ:Γ→O(H)is amenable in the sense of Bekka [Bek90]

if there exists a stateϕ onB(H ⊗C)that is invariant under the action given by conjugation byπ(t). If π is a nonamenable representation, then neither is π⊗ρ for any representation ρ [Bek90, Corollary 5.4].

In particular, ifπ is a nonamenable representation, then neither is the restriction ofπ^StoS(H) CΩ∼= H ⊗S(H)as it is isomorphic toπ⊗π^S.

Theorem 2.9.2. Letπ:Γ→O(H)be a nonamenable representation on a separable real Hilbert space, then the associated Gaussian actionΓy^απA_H is properly proximal.

Proof. Note that sinceπis a nonamenable representation we have thatΓyA_H is ergodic (see, e.g., [PS12]).

Denote byP_kthe orthogonal projection fromL²A_H ontoCΩ⊕^Lk_n=1(H ⊗C)ⁿfor eachk≥1 and setP₀ to be the projection ontoCΩ. Consider the hereditaryC^∗-algebraXF⊂B(L²A_H)generated by{P_k}k≥0in B(L²A_H), i.e.,T ∈XFif and only if limk→∞kT−T P_kk=limk→∞kT−P_kTk=0.

To see hatXFis a boundary piece it suffices to show that the multiplier algebra contains any von Neu- mann subalgebra of the formA_Rξ whereξ ∈H \ {0}. Considering the canonical isomorphismsS(H)∼= S(Rξ)⊗S(H Rξ)andA_H ∼=A_Rξ⊗A_{H Rξ} we may decomposeP_kasP_k=∑^k_j=0Q_j⊗Rk−j≤Q_k⊗1 whereQ_jdenotes the projection ontoCΩ⊕^L_n=1^j (Rξ⊗C)ⁿandRk−jdenotes the projection onto((H Rξ)⊗C)^k−j.

Since{Q_j}_j≥0gives an increasing family of finite rank projections such that∨_jQj=id_S(Rξ)we see that for eacha∈A_Rξ,j≥0, andε>0 there existsk>jso thatkQ_jaQ^⊥_k−jk<ε. Ifi≤jwe have(1⊗Rj−i)P_k^⊥=

Q^⊥_k−j+i⊗Rj−iand hence

kP_j(a⊗1)P_k^⊥k= max

0≤i≤jkQ_iaQ^⊥_k−j+ik ≤ kQ_jaQ^⊥_k−jk<ε.

From this fact it is then easy to see that(a⊗1)∈M(XF), and so we indeed haveA_Rξ ⊂M(XF)for each ξ ∈H \ {0}.

Fort≥0 we consider the orthogonal matrix







cos(πt/2) −sin(πt/2) sin(πt/2) cos(πt/2)





∈O(H ⊕H)and letαt ∈ Aut(A_H⊕H)be the associated automorphism.

Note thatαtcommutes with the action ofΓonA_H, and that limt→0kx−αt(x)k₂=0 for eachx∈A_H. We may compute E◦αt explicitly as E◦αt =P₀+∑^∞n=1cosⁿ(πt/2)(P_n−Pn−1), so that E◦αt∈XF for each 0<t<1. Using the isomorphismL²(A_H⊕H)∼=L²(A_H)⊗L²(A_H)∼=S(H)⊗S(H)we obtain an isomorphism ofA_H-modulesL²(A_H⊕H) L²(A_H⊕0)∼=L²(A_H⊕0)⊗(S(H) CΩ). So that we have an isomorphism

A⁰_H ∩B(L²(A_H⊕H) L²(A_H⊕0))∼=A_H⊗B(S(H) CΩ),

whereΓacts on the latter space byα_π(t)⊗Ad(π(t)^S). If we had aΓ-invariant state onA_H⊗B(S(H) CΩ), then restricting it to B(S(H) CΩ) would show that the restriction of the representation π^S to S(H) CΩis amenable, which would then imply thatπis an amenable representation, giving a contradic- tion. Thus, we conclude that there is noΓ-invariant state onA_H ⊗B(S(H) CΩ)and it then follows from Proposition 2.9.1 thatΓyA_H is properly proximal relative toXF.

We now take aΓ-almost invariant approximate identityA_n∈K(H)withkA_nk<1 for eachn≥0, and letαA˜n be the automorphisms ofA_H⊕H as given above. As in the proof of Proposition 2.9.1 we letV_n: L²(A_H)→L²(A_H⊕H)be given byV_n(x) =ˆ α\A˜n(x), forx∈A_H. We also denote byφ_n:B(L²(A_H⊕H))→ B(L²(A_H))the u.c.p. mapφ_n(T) =V_n^∗TV_n, forT∈B(L²(A_H⊕H)).

We view eachφ_nas a u.c.p. map into the von Neumann algebra(XF]

J)^∗, which we view as a corner of (B(L²(A_H))^]_J)^∗. We then letφ:B(L²(A_H⊕H))→(XF]

J)^∗be a point ultraweak cluster point of these maps.

SinceA_n→1 in the strong operator topology it follows thatkα_A˜

n(x)−xk₂→0 for eachx∈A_H⊕H. Also, since eachA_nis compact, and sincekA_nk<1, it follows thatA^S_n ∈K(S(H))and the same computation as above then shows thatE_AH⊕0◦αA˜n ∈K(L²M). The proof of Proposition 2.9.1 then shows thatφ isA_H- bimodular and satisfiesφ(e_AH⊕0)∈(K(M)^]_J)^∗.

If we taket∈Γ, then askπ(t)A_nπ(t⁻¹)−A_nk →0 we have that limn→∞k(α_{π(t)⊕π(t)}V_n−V_nα_π(t))P_kk=0

for eachk≥1, and hence for eachT∈B(L²(A_H⊕H))we have

n→∞limkP_k(α_π(t)φn(T)α_π(t−1)−φ_n(α_{π(t)⊕π(t)}Tα_π(t⁻¹_)⊕π(t⁻¹₎))P_kk=0.

Since{P_k}_kgives an approximate identity forXFit then follows by passing to a limit thatφisΓ-equivariant.

Suppose now that ΓyA_H is not properly proximal, then by Lemma 2.8.5 there exists anA_H-central andΓ-invariant stateϕon ˜S(M). We may further assume it vanishes on((K^∞,1_AH)^])^∗. AsΓyA_H is properly proximal relative to XF we must then have thatϕ is supported on((XF)^]_J)^∗, but then by considering the composition ofϕwith the restriction ofφtoA⁰_H∩B(L²(A_H⊕H) L²(A_H⊕0))we obtain aΓ-invariant state on this space, which would give a contradiction, as noted above.

Proof of Theorem 2.1.2. From the proof of Theorems 2.9.1 and 2.9.2 we show, in fact, that ifπ:Γ→O(H) is an orthogonal representation, then there are A_H-bimodular Γ-equivariant maps ψ :A_H ⊗B(S(H) CΩ)→S˜XF(A_H) and φ :A_H⊗B(S(H) CΩ)→((XF)^]_J)^∗∩S˜(A_H). If R is a subequivalence re- lation of the orbit equivalence relation associated to the action ΓyA_H, then just as above, if the ac- tion NLR(A_H)yA_H is not properly proximal, then we would obtain aNLR(A_H)-invariant state ϕ on

A_H ⊗B(S(H) CΩ)such thatϕ|A_H is normal with supportp∈A^Γ_H. Under the isomorphismN_LR(A_H)/U(A_H)∼= [R], where[R]denotes the full group of the equivalence relation we then see thatϕis[R]-invariant for the

natural action of[R]onA_H ⊗B(S(H) CΩ).

By a standard argument using Day’s trick (e.g., as in [Bek90]) this would then give a net of vectors ξi∈L²(A_H)⊗HS(S(H) CΩ)such thathaξ_i,ξii=_τ(p)¹ τ(ap)fora∈A_H and such that{ξ_i}_iis asymp- totically[R]-invariant. Ifπ≺λ, then this, in turn, gives an asymptotically[R]-invariant net of vectors in L²(A_H)⊗`²Γ, which would show thatRpis an amenable equivalence relation [ADR00, Theorem 6.1.4]. A simple maximality argument then gives the result.

More generally, if some tensor powerπ^⊗ksatisfiesπ^⊗k≺λ, then by considering self-tensor powers of the net{ξ_i}_iit would again follow thatRpis an amenable equivalence relation.