Inducing unitary representations

Chapter 4: Von Neumann equivalence and group approximation properties

4.2 Inducing unitary representations

As the set of suchAis dense inM_∗, the result follows.

Theorem 4.9. SupposeΓ×Λy(M,Tr)is a trace-preserving action such thatM^Γhas a normal Λ-invariant finite trace, the action of Λ on M has a finite-trace fundamental domain, and the Koopman representationΓyL²(M,Tr)is mixing. IfΛis properly proximal, then so isΓ.

Proof. This follows from Propositions 2.11, 4.3 and 4.8.

Proof of Theorem 1.4. From Proposition 3.3, the existence of a fundamental domain forΓimplies that the Koopman representation is a multiple of the left-regular representation, and hence is mixing for any infinite group. The result then follows from Theorem 4.9.

If we are also given a trace-preserving actionΓy^σ(M,Tr)that commutes with theΛ-action, then we see that (4.2) also holds for the Γ-action. Hence if Γ preserves the traceτ onM^Λ, we obtain a unitary representationΓy(M ⊗ H)^Λ_τ.

Definition 4.10. Suppose π : Λ → U(H)is a unitary representation andΓ×Λy(M,Tr)is a trace-preserving action on a semi-finite von Neumann algebra such that the action ofΛ admits a finite-trace fundamental domain. We let τ denote theΓ-invariant trace on M^Λ given by Propo- sition 3.3. We say that the representation Γy(M ⊗ H)^Λ_τ isinduced fromπ, and we denote this representation byπM.

As anM^Λ-correspondence, we say that(M ⊗ H)^Λ_τ is the correspondence induced fromπ.

Proposition 4.11. Suppose π : Λ → U(H) is a unitary representation and Λy^σ(M,Tr) is a trace-preserving action on a semi-finite von Neumann algebra that has a finite-trace fundamental domainp. There exists an isomorphism of dual HilbertM^Λ-modulesV_p :M^Λ⊗ H →(M ⊗ H)^Λ such that

V_p(x⊗ξ) = X

t∈Λ

σ_t(p)x⊗π(t)ξ for allx∈ M^Λandξ∈ H.

Proof. Note first that by Lemma 4.4, when restricted to the algebraic tensor product, the map V_p :M^Λ⊗ H →(M ⊗ H)^Λis well-defined. Moreover, forx, y ∈ Mandξ, η ∈ Hwe have

hV_p(x⊗ξ), V_p(y⊗η)iM = X

s,t∈Λ

hπ(s)η, π(t)ξix^∗σ_t(p)σ_s(p)y

=hη, ξix^∗y

=hx⊗ξ, y⊗ηi_MΛ.

As described in Section 2.6, it follows thatV_p has a weak^∗-continuous extensionV_p :M^Λ⊗ H → (M ⊗ H)^Λ that preserves the inner product; and to see thatV is surjective, it suffices to show that the range of V_p is dense when viewed as a map into (M ⊗ H)^Λ_τ, where τ is the trace given by Proposition 3.3, i.e.,τ(x) = Tr(pxp)forx∈ M^Λ.

Suppose therefore that we haveζ₀ ∈(M ⊗ H)^Λ_τ, orthogonal to the range ofV. Note that since

hξ, ηi_τ =τ(hξ, ηi) = Tr(phξ, ηip) = Tr(hξp, ηpi)

for allξ, η ∈ (M ⊗ H)^Λ, we may viewζ0 as an element in(Mp⊗ H)Tr, which is the completion ofMp⊗ Hwith respect to the inner product given byhξ, ηi_Tr = Tr(hξ, ηi)for allξ, η ∈ Mp⊗ H.

Fixings∈Λ,x∈ M^Λ,ζ ∈(M ⊗ H)^Λandξ∈ Hwe have

hσ_s(p)xp⊗ξ, ζpi_Tr= Tr(hσ_s(p)x⊗ξ, ζpi)

=X

t∈Λ

Tr(pσ_t(hσ_s(p)x⊗ξ, ζpi))

=τ(hX

t∈Λ

σ_t(p)x⊗π(ts⁻¹)ξ, ζi)

=hV_p(x⊗π(s⁻¹)ξ), ζi_τ.

Approximating ζ₀ by elements in (M ⊗ H)^Λ and viewing ζ₀ as an element in (Mp⊗ H)_Tr, it follows that

hσ_s(p)xp⊗ξ, ζ₀i_Tr = 0.

By part (b) of Proposition 3.3 we have that span{σ_s(p)x⊗ξ | s ∈ Λ, x ∈ M^Λ, ξ ∈ H}is weak^∗-dense inM ⊗ H, and hence it follows thatζ0 = 0.

A motivating example is when M = B(`<sup>2</sup>Λ)and the actionσ : Λ → Aut(B(`²Λ))is given byσ_t(T) = ρ_tT ρ^∗_t, whereρ : Λ → U(`²Λ)is the right-regular representation. Then M^Λ = LΛ and the above process describes a method of inducing representations ofΛto normal HilbertLΛ- bimodules.

There is another, extensively used, method of inducing representations to normal Hilbert bi- modules, which was originally discovered by Connes (see [Con82, Cho83, CJ85, Pop86]). Given a unitary representationπ : Λ→ U(H), setK=`²Λ⊗ H, and consider the representationsλ⊗π, and1⊗ρ ofΛinU(K). The Fell unitary U : K → Kgiven by U(δ_t⊗ξ) =δ_t⊗π(t)ξsatisfies

U(λ⊗π)U^∗ = λ⊗1, and thus both representationsλ⊗π andρ⊗1extend to give commuting normal representations ofLΛandLΛ^op inB(K).

The following proposition shows that, for this example, the induced bimodule described in Definition 4.10 is isomorphic to Connes’ induced bimodule.

Proposition 4.12. Let Λ be a discrete group, and π : Λ → U(H) a unitary representation.

Then there exists a unitaryV : `<sup>2</sup>Λ⊗ H → (B(`²Λ)⊗ H)^Λ_τ that induces an isomorphism ofLΛ- bimodules.

Proof. Forr ∈ Λwe letp_r be the rank-one projection onto Cδ_r ⊂ `<sup>2</sup>Λ. We letV<sub>pe</sub> : LΛ⊗ H → (B(`²Λ)⊗ H)^Λbe as in Proposition 4.11. Ifs, t∈Λandξ ∈ H, then

(λ_s⊗1)V_pe(δ_t⊗ξ) = (λ_s⊗1)X

r∈Λ

p_rλ_t⊗π(r⁻¹)ξ

=X

r∈Λ

p_srλ_st⊗π(r⁻¹)ξ

=V_pe(λ_s⊗π(s))(δ_t⊗ξ).

Viewing LΓ as a dense subspace of `<sup>2</sup>Γand taking completions shows that Vpe extends to a unitaryVpe :`²Γ⊗ H →(B(`²Λ)⊗ H)^Λ_τ that intertwines theLΓ-module structures defined above.

AsV_pe is also rightLΓ-modular, the result follows easily.

Lemma 4.13. Suppose π : Λ → U(H) and ρ : Λ → U(K) are unitary representations and Λy^σ(M,Tr) is a trace-preserving action on a semi-finite von Neumann algebra. For finite- trace fundamental domainsp, q ∈ M, letV_p andV_q respectively be the maps defined in Proposi- tion 4.11. SupposeGis a finite set and we have functionsξ : G → Kandξ_i :G → Hsuch that sup_i,k∈Gkξ_i^kk<∞, and for allt ∈Λandk, `∈Gwe have

hπ(t)ξ^k_i, ξ^{`</sup><sub>i</sub>i → hρ(t)ξ<sup>k</sup>, ξ<sup>`}i.

Then for allx, y ∈ M^Λand for allk, `∈G, we have

hV_p(x⊗ξ_i^k), V_q(y⊗ξ_i^{`</sup>)i<sub>τ</sub> → hV<sub>p</sub>(x⊗ξ<sup>k</sup>), V<sub>q</sub>(y⊗ξ<sup>`})i_τ.

Proof. We compute

hV_p(x⊗ξ_i^k), V_q(y⊗ξ_i^`)i= X

s,t∈Λ

hσ_s(p)x⊗π(s)ξ_i^k, σ_t(q)y⊗π(t)ξ_i^`i

= X

s,t∈Λ

x^∗σ_s(p)σ_t(q)yhπ(t)ξ^`_i, π(s)ξ_i^ki

=X

s∈Λ

x^∗ X

t∈Λ

σ_t(σ_s(p)q)

!

yhξ_i^`, π(s)ξ_i^ki. (4.3)

We have

X

s∈Λ

τ X

t∈Λ

σ_t(σ_s(p)q)

!

=X

s∈Λ

Tr(pσ_s⁻¹(q)) = Tr(p)<∞, and hence givenε >0there exists a finite setF ⊂Λsuch that setting

y_F =X

s6∈F

X

t∈Λ

σ_t(σ_s(p)q),

for allk, `∈Gwe have

|τ(x^∗y_Fy)| ≤τ(x^∗y_Fx)^1/2τ((y)^∗y_Fy)^1/2 ≤ kxkkykτ(y_F)< ε.

Thus,

lim sup

i→∞

|hV_p(x⊗ξ_i^k), V_q(y⊗ξ_i^{`</sup>)i<sub>τ</sub>− hV<sub>p</sub>(x⊗ξ<sup>k</sup>), V<sub>q</sub>(y⊗ξ<sup>`})i_τ|

≤εsup

i

kξ^k_ik^1/2kξ^`_ik^1/2+ lim sup

i→∞

X

s∈F

τ x^∗ X

t∈Λ

σ_t(σ_s(p)q)

! y

!

· |hξ_i^{`</sup>, π(s)ξ<sub>i</sub><sup>k</sup>i − hξ<sup>`}, ρ(s)ξ^ki|

=εsup

i

kξ_i^kk^1/2kξ_i^`k^1/2.

Asε >0was arbitrary, the result follows.

Lemma 4.14. Supposeπ : Λ → U(H)is a mixing representation andΛy^σ(M,Tr)is a trace- preserving action on a semi-finite von Neumann algebra. Suppose we have finite-traceΛ-fundamental domainsp_i ∈ M such thatp_i → 0 in the weak operator topology. Then for anyΛ-fundamental domainpandξ, η ∈ H, we have

i→∞lim sup

x,y∈(M^Λ)1

|hV_p(x⊗ξ), V_pi(y⊗η)i_τ|= 0.

Proof. Fixp∈ Ma finite-trace fundamental domain andξ, η ∈ H. Then forx, y ∈ M^Λwe may compute as in (4.3)

hV_p(x⊗ξ), V_pi(y⊗η)i_τ =X

s∈Λ

x^∗ X

t∈Λ

σ_t(σ_s(p)p_i)

!

yhη, π(s)ξi.

Fixε >0. Sinceπis a mixing representation, there existsF ⊂Λfinite so that|hη, π(s)ξi|< ε for alls6∈F. Asp_i →0weakly, we have

i→∞lim X

s∈F

τ X

t∈Λ

σ_t(σ_s(p)p_i)

!

= 0.

Hence

lim sup

i→∞

sup

x,y∈(M^Λ)1

|hV_p(x⊗ξ), V_pi(y⊗η)i_τ|

≤lim sup

i→∞

sup

x,y∈(M^Λ)1

X

s6∈F

τ x^∗ X

t∈Λ

σ_t(σ_s(p)p_i)

! y

!

hη, π(s)ξi

< ε.

Since ε > 0 was arbitrary, and since vectors of the form x ⊗ξ span a weak^∗-dense subset of M^Λ⊗ H, the result follows.

The following proposition generalizes results in Section 8 from [Fur99b]. In the case when Mis associated to aW^∗-equivalence as in Proposition 4.12, this follows from results in [Cho83, CJ85].

Proposition 4.15. Supposeπ : Λ →U(H)andρ: Λ →U(K)are two unitary representations of Λ, andΓ×Λy(M,Tr)is a von Neumann coupling. The following hold:

(i) Ifπ≺ρ, thenπM ≺ρM.

(ii) Ifπis mixing, thenπ_M is mixing.

(iii) λMis a multiple of the left-regular representation ofΓ.

(iv) Ifπis weak mixing, thenπMhas no non-zero invariant vectors.

Proof. Suppose first that π ≺ ρ. Replacing ρwithρ^⊕∞, we may assume thatρ has infinite mul- tiplicity. Fix Ga finite set, and suppose ξ : G → K is a map. Since π ≺ ρ, there exists a net ξi :G → Hsuch that for allt ∈ Λ, we havehπ(t)ξ_i^k, ξ_i^{`</sup>i → hρ(t)ξ<sup>k</sup>, ξ<sup>`}i. By Lemma 4.13, for all x, y ∈ M^Λandγ ∈Γ, we then have

h(σ_γ⊗1)V_p(x⊗ξ_i^k), V_p(y⊗ξ_i^{`</sup>)i<sub>τ</sub> =hV<sub>σγ(p)</sub>(σ<sub>γ</sub>(x)⊗ξ<sub>i</sub><sup>k</sup>), V<sub>p</sub>(y⊗ξ<sub>i</sub><sup>`})i_τ

→ hV_σγ(p)(σ_γ(x)⊗ξ^k), V_p(y⊗ξ^`)i_τ

=h(σ_γ⊗1)V_p(x⊗ξ^k), V_p(y⊗ξ^`)i_τ.

As elements of the formx⊗ξspan a dense subset of(M^Λ⊗ H)_τ this then shows (i).

If π is mixing and γ → ∞, then for a fixed Λ-fundamental domain p ∈ M, we have that σ_γ(p)→0weakly. Hence Lemma 4.14 shows that for allξ, η∈ Handx, y ∈ M^Λ, we have

γ→∞limh(σγ⊗1)Vp(x⊗ξ), Vp(y⊗η)iτ = lim

γ→∞hVσγ(p)σγ(x)⊗ξ, Vp(y⊗ηiτ = 0.

ThusπM is also mixing, which then shows (ii).

We define the mapF : (M ⊗`²Λ)^Λ → MbyF(ξ) = h1⊗δ_e, ξi_M. Forx ∈ M^Λ andt ∈Λ, we then haveF(V_p(x⊗δ_t)) = h1⊗δ_e, V_p(x⊗δ_t)i = σ_t⁻¹(p)x. Hence if we also havey ∈ M^Λ ands ∈Λ, then

hF(V_p(x⊗δ_t)),F(V_p(y⊗δ_s))i_Tr =δ_s,tTr(x^∗σ_t⁻¹(p)y)

=δ_s,tτ(σ_t(yx^∗))

=δ_s,tτ(x^∗y)

=hx⊗δ_t, y⊗δ_si_τ

=hV_p(x⊗δ_t), V_p(y⊗δ_s)i_τ.

Thus,F extends to an isometryF : (M ⊗`²Λ)^Λ_τ → L²(M,Tr). Moreover, by part (b) of Propo- sition 3.3 we see thatspan{F(Vp(x⊗δt)) | x ∈ M^Λ, t ∈ Λ}is dense inL²(M,Tr), henceF is unitary.

AsFcommutes with the action ofΓ, we then see thatFimplements an intertwiner between the representationλM and the Koopman representation on L²(M,Tr). Since Γhas a finite-measure fundamental domain, the latter representation is isomorphic to an amplification of the left regular representation by part (ii) of Proposition 3.3. This then establishes (iii).

We now suppose that πM has a non-zero invariant vector in (M ⊗ H)^Λ_τ. First, note that this then implies that there is a non-zeroΓ-invariant vector in(M ⊗ H)^Λ. Indeed, ifξ∈(M ⊗ H)^Λ_τ is a Γ-invariant vector, then we may approximateξby someη∈(M ⊗ H)^Λso thatkξ−ηk_τ < ¹₂kξk.

If we let ξ₀ be the unique element of minimal k · k_τ in the k · k_τ-closed convex closure hull of {πM(γ)η| γ ∈ Γ}, then ξ₀ is alsoΓ-invariant, and we havekξ₀−ξk ≤ kη−ξk< ¹₂kξk, so that ξ₀ is non-zero. Closed balls in M ⊗ Hare weak^∗-compact by the Banach-Alaoglu theorem and hence we see thatξ₀ ∈(M ⊗ H)^Λ⊂(M ⊗ H)^Λ_τ.

We therefore have a non-zero vector in (M ⊗ H)^Γ×Λ ∼= (M^Γ⊗ H)^Λ. Recall that we endow H with its column operator space structure coming from the isomorphism H ∼= B(C,H). We therefore considerξ₀ ∈(M^Γ⊗ B(C,H))^Λ, and we then obtain a non-zero positive operator|ξ₀| ∈

(M^Γ⊗HS(H))^Λ, whereHS(H)denotes the space of Hilbert-Schmidt operators onH. Asτ_Γ⊗Tr gives a faithful trace onM^Γ⊗ B(H), we then obtain a non-zeroΛ-invariant vector(τ_Γ⊗id)(|ξ₀|)∈ HS(H). This then shows thatπis not weak mixing, establishing (iv).

Proof of Theorem 1.2. Amenability is characterized by having the left regular representation weakly contain the trivial representation, thus (i) and (iii) in Proposition 4.15 show that amenability is pre- served under von Neumann equivalence.

Similarly, the Haagerup property is characterized by having a mixing representation that weakly contains the trivial representation. Thus, (i) and (ii) in Proposition 4.15 show that the Haagerup property is preserved under von Neumann equivalence.

Finally, if Γ has property (T) and π is a representation of Λ that weakly contains the trivial representation, then since 1M contains the trivial representation for Γ, it follows that πM also weakly contains the trivial representation. Property (T) then implies that π_M contains non-zero Γ-invariant vectors, and by (iv) in Proposition 4.15 it follows that π is not weak mixing. It then follows from [BV93, Theorem 1] thatΛalso has property (T).