The structure of certain unitary representations of infinite symmetric groups

Abstract

Let S be an infinite set, β \beta an infinite cardinal number, and G β ( S ) {G_\beta }(S) the group of those permutations of S whose support has cardinal number less than β \beta . If T is any nonempty set, S T {S^T} is the set of functions from T to S. The canonical representation Λ β T \Lambda _\beta ^T of G β ( S ) {G_\beta }(S) on L 2 ( S T ) {L^2}({S^T}) is the direct sum of factor representations. Factor representations of types I ∞ , II 1 {{\text {I}}_\infty },{\text {II}_1} , and II ∞ {\text {II}_\infty } occur in this decomposition, depending upon S, β \beta , and T; the type II 1 {\text {II}_1} factor representations are quasi-equivalent to the left regular representation. Let G β ( S ) {G_\beta }(S) have the topology of pointwise convergence on S. G β ( S ) {G_\beta }(S) is a topological group but is not locally compact. Every continuous representation of G β ( S ) {G_\beta }(S) is the direct sum of irreducible representations. Let Γ \Gamma be a nontrivial continuous irreducible representation of G β ( S ) {G_\beta }(S) . Then Γ \Gamma is continuous iff Γ \Gamma is equivalent to a subrepresentation of Λ β T \Lambda _\beta ^T for some nonempty finite set T iff there is a nonempty finite subset Z of S such that the restriction of Γ \Gamma to the subgroup of those permutations which leave Z pointwise fixed contains the trivial representation of this subgroup.