The separable representations of $\textrm {U}(H)$

Abstract:

In this paper we show that the separable representation theory of ${\text {U(}}H)$ is completely analogous to that for ${\text {U(}}{{\mathbf {C}}^n})$, in that every separable representation is discretely decomposable and the irreducible representations all occur in the decomposition of the mixed tensor algebra of $H$. This was previously shown to be true (for all representations, separable and nonseparable) for the normal subgroup ${\text {U(}}H{)_\infty }$, consisting of operators which are compact perturbations of the identity, by Kirillov and Ol’shanskii. In particular we show that all nontrivial representations of the unitary Calkin group are nonseparable. The proof exploits the analogue of the following fact about the Calkin algebra: if $\pi$ is a nontrivial representation of the Calkin algebra and $T$ is a normal operator on $H$, then every point in the spectrum of $\pi (T)$ is an eigenvalue.