The structure of certain unitary representations of infinite symmetric groups

Abstract:

Let S be an infinite set, $\beta$ an infinite cardinal number, and ${G_\beta }(S)$ the group of those permutations of S whose support has cardinal number less than $\beta$. If T is any nonempty set, ${S^T}$ is the set of functions from T to S. The canonical representation $\Lambda _\beta ^T$ of ${G_\beta }(S)$ on ${L^2}({S^T})$ is the direct sum of factor representations. Factor representations of types ${{\text {I}}_\infty },{\text {II}_1}$, and ${\text {II}_\infty }$ occur in this decomposition, depending upon S, $\beta$, and T; the type ${\text {II}_1}$ factor representations are quasi-equivalent to the left regular representation. Let ${G_\beta }(S)$ have the topology of pointwise convergence on S. ${G_\beta }(S)$ is a topological group but is not locally compact. Every continuous representation of ${G_\beta }(S)$ is the direct sum of irreducible representations. Let $\Gamma$ be a nontrivial continuous irreducible representation of ${G_\beta }(S)$. Then $\Gamma$ is continuous iff $\Gamma$ is equivalent to a subrepresentation of $\Lambda _\beta ^T$ for some nonempty finite set T iff there is a nonempty finite subset Z of S such that the restriction of $\Gamma$ to the subgroup of those permutations which leave Z pointwise fixed contains the trivial representation of this subgroup.