Polish groups of unitaries

TL;DR

This paper investigates which Polish groups can be embedded into unitary groups of infinite-dimensional Hilbert spaces, exploring properties like (OB), (FH), and (T) in various contexts of C*-algebras and von Neumann algebras.

Contribution

It characterizes when the identity component of unitary groups of C*-algebras has property (OB) and shows that certain p-unitary groups lack property (FH), solving an open problem.

Findings

The identity component of unitary groups of C*-algebras has property (OB) iff the algebra has finite exponential length.

The full unitary group often does not have property (T).

p-unitary groups of properly infinite semifinite von Neumann algebras lack property (FH).

Abstract

We study the question of which Polish groups can be realized as subgroups of the unitary group of a separable infinite-dimensional Hilbert space. We also show that for a separable unital C$^*$-algebra $A$, the identity component $\mathcal{U}_0(A)$ of its unitary group has property (OB) of Rosendal (hence it also has property (FH)) if and only if the algebra has finite exponential length (e.g. if it has real rank zero), while in many cases the unitary group $\mathcal{U}(A)$ does not have property (T). On the other hand, the $p$-unitary group $\mathcal{U}_p(M,\tau)$ where $M$ is a properly infinite semifinite von Neumann algbera with separable predual, does not have property (FH) for any $1\le p<\infty$. This in particular solves a problem left unanswered in the work of Pestov \cite{Pestov18}.