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

Abstract

Let $\Gamma$ be a countable discrete group, and let $\pi :\Gamma \to \mathrm{GL}(H)$ be a representation of $\Gamma$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H{⋊}_{\pi }\Gamma$ 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 $\pi$ is uniformly bounded, and that $\pi$ is unitarizable if and only if $G$ is of finite type: that is, $G$ embeds into the unitary group of a II${}_1$-factor. Consequently, we show that a unitarily representable Polish SIN groups 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.

arXiv comment27 pages v2 minor changes: corrected the hypothesis in Corollary 3.16 and first few lines of the proof of Lemma 3.7