Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type

Hiroshi Ando; Yasumichi Matsuzawa; Andreas Thom; Asger Törnquist

doi:10.1515/crelle-2017-0047

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Unitarizability, Maurey–Nikishin factorization, and Polish groups of finite type

Hiroshi Ando , Yasumichi Matsuzawa , Andreas Thom and Asger Törnquist

Published/Copyright: November 17, 2017

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From the journal Journal für die reine und angewandte Mathematik (Crelles Journal) Volume 2020 Issue 758

Abstract

Let Γ be a countable discrete group, and let π:Γ→GL⁢(H) be a representation of Γ by invertible operators on a separable Hilbert space H. We show that the semidirect product group G=H⋊πΓ is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group 𝒰⁢(ℓ2⁢(ℕ))) if and only if π is uniformly bounded, and that π is unitarizable if and only if G is of finite type, that is, G embeds into the unitary group of a II1-factor. Consequently, we show that a unitarily representable Polish SIN group need not be of finite type, answering a question of Sorin Popa. The key point in our argument is an equivariant version of the Maurey–Nikishin factorization theorem for continuous maps from a Hilbert space to the space L0⁢(X,m) of all measurable maps on a probability space.

Funding statement: This research was supported by JSPS KAKENHI 16K17608 (HA), 6800055 and 26350231 (YM), ERC Starting Grant No. 277728 and ERC Consolidator Grant No. 681207 (AT) and the Danish Council for Independent Research through grant no. 10-082689/FNU (AT), from which HA was also supported while he was working at University of Copenhagen.

A Appendix

The following two results are well known, but we include the proofs for the reader’s convenience.

Proposition A.1.

Let A be a commutative von Neumann algebra with separable predual and τ a normal faithful tracial state on A. Let G be a Polish group and β:G→Aut⁢(A) a continuous τ-preserving action. Then there exist a separable compact metrizable space X, a Borel probability measure m on X, an m-preserving continuous action G×X∋(g,x)↦g⁢x∈X and a *-isomorphism Φ:A→L∞⁢(X,m) such that

(A.1)τ⁢(a)=∫XΦ⁢(a)⁢(x)⁢dm⁢(x),a∈𝒜,

(A.2)Φ⁢(βg⁢(a))⁢(x)=Φ⁢(a)⁢(g-1⁢x),g∈G,μ⁢-a.e. ⁢a∈𝒜.

Moreover, the G-action is continuous as a map G→Aut⁢(X,m), where the group Aut⁢(X,m) of all m-preserving automorphism of X is equipped with the weak topology.

Lemma A.2.

Let M,N be von Neumann algebras, and let ψ,φ be normal faithful states on M,N, respectively. Let M0 (resp. N0) be a *-strongly dense subalgebra of M (resp. N). If α:M0→N0 is a *-isomorphism such that φ⁢(α⁢(x))=ψ⁢(x) for every x∈M0, then α can be extended to a *-isomorphism from M onto N such that φ(α(x))=ψ(x)(x∈M).

Proof.

By the GNS construction, we may assume that M acts on L2⁢(M,ψ) and N acts on L2⁢(N,φ) so that ψ=〈⋅ξψ,ξψ〉, φ=〈⋅ξφ,ξφ〉 for unit vectors ξψ∈L2⁢(M,ψ) and ξφ∈L2⁢(N,φ). Define U0:M0⁢ξψ→N0⁢ξφ by

U0⁢x⁢ξψ:=α⁢(x)⁢ξφ,x∈M0.

Since ψ is faithful, U0 is well defined. Also, since M0 is *-strongly dense in M and ξψ is cyclic for M, M0⁢ξψ is dense in L2⁢(M,ψ). Similarly, N0⁢ξφ is dense in L2⁢(N,φ). By φ∘α=ψ on M0, we have

∥U0⁢x⁢ξψ∥2=φ⁢(α⁢(x*⁢x))=ψ⁢(x*⁢x)=∥x⁢ξψ∥2,x∈M0.

Therefore, U0 extends to an isometry U from L2⁢(M,ψ) into L2⁢(N,φ) and the range of U contains a dense subspace N0⁢ξφ. Therefore U is onto. If x∈M0, then for each y∈N0,

U⁢x⁢U*⁢y⁢ξφ=U⁢x⁢α-1⁢(y)⁢ξψ=α⁢(x)⁢y⁢ξφ,

so that by the density of N0⁢ξφ, we have

U⁢x⁢U*=α⁢(x),x∈M0.

Let now x∈M. Then there exists a net (xi)i∈I in M0 converging *-strongly to x. Then U⁢xi⁢U*=α⁢(xi)(∈N) converges *-strongly to U⁢x⁢U*. Therefore U⁢x⁢U*∈N. Then it is clear that M∋x↦U⁢x⁢U*∈N defines a *-isomorphism with inverse N∋y↦U*⁢y⁢U∈M. The uniqueness of the extension is clear by the *-strong density of M0 and the fact that isomorphisms between von Neumann algebras are normal. It is now clear that φ∘α=ψ on M. This finishes the proof. ∎

Proof of Proposition A.1.

For convenience, we restrict to the case that G is locally compact. The general case requires an additional argument using the assumption that the action is trace-preserving. Let

𝒜0:={x∈𝒜:limg→1⁡∥βg⁢(x)-x∥=0},

which is a *-strongly dense (𝒜0 is norm closed) C*-subalgebra of 𝒜. Let {an}n=1∞ be a *-strongly dense subset of 𝒜0, and let C=C*⁢({βg⁢(an):g∈G,n∈ℕ}∪{1}). Then since G is separable and g↦βg⁢(an) is norm-continuous for all n∈ℕ, C is a norm-separable *-strongly dense C*-subalgebra of 𝒜, and let X=Spec⁡(C) (the Gelfand spectrum), which is a separable and compact metrizable space, and let Φ:C→C⁢(X) be the Gelfand transform. Since τ∘Φ-1∈C⁢(X)+*, there exists a Borel probability measure m on X such that

τ⁢(c)=∫XΦ⁢(c)⁢(x)⁢dm⁢(x),c∈C.

We let C⁢(X) act on L2⁢(X,m) by multiplication, so C⁢(X)⊂L∞⁢(X,m) is a *-strongly dense *-subalgebra of L∞⁢(X,m). The map τm:L∞⁢(X,m)∋f↦∫Xf⁢dm∈ℂ is a normal faithful tracial state on L∞⁢(X,m). Moreover, τm⁢(Φ⁢(c))=τ⁢(c) for every c∈C. Therefore by Lemma A.2, Φ can be extended to a *-isomorphism Φ:C′′=𝒜→C⁢(X)′′=L∞⁢(X,m) such that τm⁢(Φ⁢(a))=τ⁢(a) for every a∈𝒜. This shows (A.1). Now let g∈G and x∈X. Let mx∈C⁢(X)* be the delta probability measure at x∈X. Let β* be the dual action of G on C⁢(X)* given by

〈βg*⁢(ν),f〉:=〈ν,Φ∘βg-1∘Φ-1⁢(f)〉,ν∈C⁢(X)*,f∈C⁢(X),g∈G.

Since β is an action, βg* maps pure states (on C≅C⁢(X)) to pure states, and pure states on C⁢(X) are precisely the delta probability measures, there exists a unique point g⋅x∈X such that

βg*⁢(mx)=mg⋅x.

We show that G×X∋(g,x)↦g⋅x∈X is a continuous m-preserving action. It is routine to check that this is indeed an action of G on X.

We show that the action is continuous. Since G is a Polish group and X is a Polish space, it suffices to show that the action is separately continuous (see e.g. [16, Theorem 9.14]). Suppose that (xn)n=1∞⊂X converges to x∈X and let g∈G. Then by the definition of the Gelfand spectrum, mxn→n→∞mx (weak*). Since βg induces an action on C (C is β-invariant by construction), we have

βg*⁢(mxn)=mg⋅xn→n→∞βg*⁢(mx)=mg⋅x,

which implies g⋅xn→g⋅x as n→∞ (again by the definition of the topology on the Gelfand spectrum). Let now x∈X and let (gn)n=1∞⊂G be a sequence converging to g∈G. Then for each f∈C⁢(X),

〈βgn*⁢(mx),f〉=〈mx,Φ∘βgn-1∘Φ-1⁢(f)〉→n→∞〈mx,Φ∘βg-1∘Φ-1⁢(f)〉=〈βg*⁢(mx),f〉,

whence gn⋅x→g⋅x as n→∞. Therefore the G-action on X is continuous. Next, we show that for each g∈G, the map X∋x↦g⋅x∈X preserves m. It suffices to show that

∫Xf⁢(g-1⁢x)⁢dm⁢(x)=∫Xf⁢(x)⁢dm⁢(x),f∈C⁢(X).

Let ν=∑i=1nλimxi∈conv{mx:x∈X}=:𝒟. Then

βg*(ν)=∑i=1nλiδg⋅xi=ν(g-1⋅).

By the Krein–Milman Theorem, 𝒟 is weak*-dense in the state space of C⁢(X). This shows that

(A.3)βg*(ν)=ν(g-1⋅),ν∈C(X)+*.

In particular, by τ∘βg=τ,

∫Xf⁢(g-1⁢x)⁢dm⁢(x)=∫Xf⁢(x)⁢dm⁢(g⁢(x))⁢=(A⁢.3)⁢〈βg-1*⁢(m),f〉

=〈m,Φ∘βg∘Φ-1⁢(f)〉

=τm⁢(Φ∘βg∘Φ-1⁢(f))=τ⁢(βg∘Φ-1⁢(f))

=τ⁢(Φ-1⁢(f))=τm⁢(f)

=∫Xf⁢(x)⁢dm⁢(x).

Therefore the G-action on X is m-preserving. Finally, we show (A.2). Since the map

βg′:L∞(X,m)∋f↦f(g-1⋅)∈L∞(X,m)

defines a τm-preserving action and since C⁢(X) is *-strongly dense in L∞⁢(X,m), it suffices to show that (A.2) holds for every f=Φ⁢(a)∈C⁢(X) (a∈C), x∈X and g∈G. But

f⁢(g-1⁢x)=〈μg-1⋅x,f〉=〈βg-1*⁢(μx),f〉

=〈μx,Φ∘βg∘Φ-1⁢(f)〉

=Φ∘βg∘Φ-1⁢(f)⁢(x).

Finally, assume that gn→g as n→∞ in G. Then for every measurable set A⊂X, we have

m⁢(△⁡gn⁢Ag⁢A)=∫X|Φ∘βgn∘Φ-1⁢(𝟙A)⁢(x)-Φ∘βg∘Φ-1⁢(𝟙A)⁢(x)|⁢dm⁢(x)

=∥βgn∘Φ-1⁢(𝟙A)-βg∘Φ-1⁢(𝟙A)∥22→n→∞0,

because the u-topology on Aut⁢(𝒜) is the same as the topology of pointwise ∥⋅∥2-convergence. Thus the G-action defines a continuous homomorphism G→Aut⁢(X,m). This finishes the proof. ∎

Acknowledgements

We thank Professors Łukasz Grabowski, Magdalena Musat, Yuri Neretin, Izumi Ojima and Mikael Rørdam for helpful discussions on the 𝒰fin-problem at early stages of the project. The final part of the manuscript was completed while Hiroshi Ando was visiting the Mittag-Leffler Institute for the program “Classification of operator algebras: Complexity, rigidity, and dynamics”. He thanks the institute and the workshop organizers for the invitation and for their hospitality. Last but not least, we would like to thank the anonymous referee for suggestions which improved the presentation of the paper.

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Received: 2017-02-09

Revised: 2017-09-07

Published Online: 2017-11-17

Published in Print: 2020-01-01

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