Continuous representations of infinite symmetric groups on reflexive Banach spaces

Abstract

Let $S$ be an arbitrary infinite set and let $G$ be the group of finitely supported permutations of $S$; give $G$ the topology of pointwise convergence on $S$. Let $B$ be a reflexive Banach space and let $\Gamma$ be a continuous representation of $G$ on $B$ such that $||\Gamma(g)||\leq M$ for all $g \epsilon G$ for some fixed positive number $M$. Through the use of a canonically defined dense subspace of cofinite vectors, it is shown that $\Gamma$ is strongly continuous and contains an irreducible subrepresentation. An equivalence relation of cofinite equivalence of representations is defined; if $\Gamma$ is irreducible, then $\Gamma$ is cofinitely equivalent to an irreducible weakly continuous unitary representation of $G$ on a Hilbert space.