A note on the groups of finite type and the Hartman-Mycielski construction
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Vladimir G. PestovInstituto de Matemática e Estatística,Universidade Federal da Bahia,Ondina, Salvador, BA, 40.170-115, Brasil and
Department of Mathematics and Statistics,University of Ottawa,Ottawa, ON, K1N 6N5, CanadaE-mail: [email protected] of June 14, 2020
Abstract
Ando, Matsuzawa, Thom, and Törnquist have resolved a prob-lem by Sorin Popa by constructing an example of a Polish group ofunitary operators with the strong operator topology, whose left andright uniform structures coincide, but which does not embed intothe unitary group of a finite von Neumann algebra. The question re-mained whether such a group can be connected. Here we observe thata connected (in fact, homeomorphic to the Hilbert space) example isobtained from the example of the above authors via the Hartman–Mycielski construction.
A Polish topological group is of finite type if it topologically embeds intothe unitary group, U ( A ) , of a finite von Neumann algebra, equipped with Mathematics Subject Classification : Primary 22A10; Secondary 46L10.
Key words and phrases : group of finite type, SIN group, unitarily representable group,Hartman–Mycielski construction. V.G. Pestov the strong topology. Any such group G is a SIN group , that is, a base ofneighbourhoods at identity is formed by open sets V invariant under conju-gations: g − V g = V for all g ∈ G . Also, G is unitarily representable , that is,it admits a unitary representation π : G → U ( H π ) that is an embedding oftopological groups with regard to the strong operator topology. For a whileit remained unclear if those two properties were enough to characterize thegroups of finite type. The question was asked in print by Popa [9] and re-solved in the negative in the article [1], to which we refer for more detaileddefinitions and references.The central result of the article states that the semidirect product Γ ⋉ π ℓ of a discrete group Γ and the additive topological group of the Hilbert space ℓ with regard to a representation π : Γ → GL ( ℓ ) is unitarily representableand SIN if and only if π is bounded, and is of finite type if and only if π is unitarizable. Since there are many known examples of bounded non-unitarizable representations, this construction provides a counter-exampleto Popa’s question. However, the Polish group Γ ⋉ π ℓ is disconnected, sothe authors have asked whether a connected Polish group with the samecombination of properties exists.Here we note that such an example is derived from the example of theabove authors through a classical construction going back to Hartman andMycielski [5]. In this paper, it was shown that if G is any topological group,then the group of all maps from the interval [0 , to G , constant on half-open intervals and taking finitely many values, equipped with the topologyof convergence in Lebesgue measure, is a path-connected and locally pathconnected topological group containing G as a topological subgroup formedby constant maps. It was further remarked in [3] that this group is con-tractible and locally contractible.One can form a larger group, consisting of all (equivalence classes of)strongly (or: Bourbaki) measurable maps f : X → G from a standardLebesgue space ( X, µ ) to G , meaning that for every ε > there is a compactsubset K ⊆ X of measure > − ε on which f is continuous. We denotethis group L ( X, µ, G ) . The topology is still the topology of convergence inmeasure, whose basis is formed by the sets [ V, ε ] = { f ∈ L ( X, µ, G ) : µ { x ∈ X : f ( x ) ∈ V } > − ε } , where V runs over a neighbourhood base at the identity of G and ε > .The group L ( X, µ, G ) contains the original group of Hartman–Mycielskias an everywhere dense subgroup. Even in this generality, the constructionyields interesting results, see e.g. [8]. roups of finite type and Hartman–Mycielski construction V is conjugation invariant, then so is [ V, ε ] ,so we have: Lemma 1.
If a topological group G is SIN, then the group L ( X, µ, G ) isSIN as well. If G is metrizable, so is the group L ( X, µ ; G ) . For example, if d is a right-invariant metric generating the topology of G , then the following metricgenerates the topology of convergence in measure and is right-invariant (wefollow Gromov’s notation [4], p. 115):me ( f, g ) = inf { ε > µ { x ∈ X : d ( f ( x ) , g ( x )) > ε } < ε } . For G separable metric, strongly measurable maps from ( X, µ ) to G are ofcourse just the µ -measurable maps. Here is one of the main results aboutthe Hartman–Mycielski construction. Theorem 2 (Bessaga and Pełczyński [2]) . Whenever the group G is sepa-rable metrizable, the topological group L ( X, µ ; G ) is homeomorphic to theseparable Hilbert space ℓ provided G is nontrivial. In fact, the above result was stated and proved for any separable metricspace instead of G , but the motivation was to answer, in the affirmative, aquestion of Michael [6]: does every separable metrizable topological groupembed into a topological group homeomorphic to ℓ ?The following observation is surely folklore, but we do not have a refer-ence. The argument is the same as in [7], Lemma 6.5, for G locally compact,and of course it could be generalized further if need be. Lemma 3.
Let G be a unitarily representable Polish group. Then L ( X, µ ; G ) is unitarily representable as well.Proof. Fix a topological group embedding ρ : G → U ( H ) . Define a unitaryrepresentation π of L ( X, µ ; G ) in L ( X, µ ; H ) as the direct integral of copiesof the representation ρ : for every f ∈ L ( X, µ ; G ) and φ ∈ L ( X, µ ; H ) , π f ( ψ )( x ) = ρ f ( x ) ( ψ ( x )) , for µ -a.e. x. For each simple function ψ ∈ L ( X, µ ; H ) , one can see that the correspond-ing orbit map L ( X, µ ; G ) ∋ f π f ( ψ ) ∈ L ( X, µ ; H ) is continuous at identity, and consequently π is strongly continuous. V.G. Pestov
Fix a right-invariant compatible pseudometric d on G , and let ε > bearbitrary. There are ξ , . . . , ξ n ∈ H and a δ > such that ∀ h ∈ G, if k ρ g ( ξ i ) − ξ i k < δ for all i, then d ( g, e ) < ε. Let ¯ ξ i be a constant function on X taking value ξ i . For any function f ∈ L ( X, µ ; G ) , if ∀ i = 1 , , . . . , n (cid:13)(cid:13) π f ( ¯ ξ i ) − ¯ ξ i (cid:13)(cid:13) < δ r εn , then one must have ∀ i, (cid:13)(cid:13) ρ f ( x ) ( ξ i ) − ξ i (cid:13)(cid:13) < δ on a set of x having measure at least − ε/n . This implies me ( f, e ) < ε ,concluding the argument.Now it is enough to apply the construction to the Polish group Γ ⋉ π H from [1]. The topological group L ( X, µ ; Γ ⋉ π H ) is homeomorphic tothe Hilbert space ℓ (theorem 2), in particular Polish and contractible, isSIN (lemma 1), unitarily representable (lemma 3), yet not of finite type,because it contains Γ ⋉ π H as a closed topological subgroup made of constantfunctions. References [1] H. Ando, Y. Matsuzawa, A. Thom, and A. Törnquist,
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