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

Abstract

AbstractLet Γ be a countable discrete group, and let {\pi\colon\Gamma\to{\rm{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\rtimes_{\pi}\Gamma} is SIN (G admits a two-sided invariant metric compatible with its topology) and unitarily representable (G embeds into the unitary group {\mathcal{U}(\ell^{2}(\mathbb{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 {\mathrm{II}_{1}}-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 {L^{0}(X,m)} of all measurable maps on a probability space.