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

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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.
Original languageEnglish
JournalJournal fuer die Reine und Angewandte Mathematik
Pages (from-to)223-251
Number of pages29
Publication statusPublished - 2020

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