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

TL;DR

This paper characterizes when certain groups formed from a discrete group and a Hilbert space representation are unitarily representable and of finite type, linking these properties to boundedness and unitarizability of the representation.

Contribution

It establishes a precise connection between unitarizability of group representations and the finite type property of associated groups, and provides a counterexample to a question about Polish SIN groups.

Findings

Group $G=H\rtimes_{\pi}\Gamma$ is SIN and unitarily representable iff $\pi$ is uniformly bounded.

$\pi$ is unitarizable iff $G$ embeds into a II$_1$-factor's unitary group.

Unitarily representable Polish SIN groups need not be of finite type.

Abstract

Let $\Gamma$ be a countable discrete group, and let $\pi\colon \Gamma\to {\rm{GL}}(H)$ be a representation of $\Gamma$ by invertible operators on a separable Hilbert space $H$. We show that the semidirect product group $G=H\rtimes_{\pi}\Gamma$ 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 $\pi$ 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.