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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The structure of certain unitary representations of infinite symmetric groups

Author: Arthur Lieberman
Journal: Trans. Amer. Math. Soc. 164 (1972), 189-198
MSC: Primary 22.60
MathSciNet review: 0286940
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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.

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Keywords: Infinite symmetric groups, representations of not locally-compact groups, representations of discrete groups
Article copyright: © Copyright 1972 American Mathematical Society