Illinois Journal of Mathematics

Continuous representations of infinite symmetric groups on reflexive Banach spaces

Arthur Lieberman

Full-text: Open access


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.

Article information

Illinois J. Math., Volume 17, Issue 3 (1973), 450-457.

First available in Project Euclid: 20 October 2009

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22A25: Representations of general topological groups and semigroups
Secondary: 22D10: Unitary representations of locally compact groups


Lieberman, Arthur. Continuous representations of infinite symmetric groups on reflexive Banach spaces. Illinois J. Math. 17 (1973), no. 3, 450--457. doi:10.1215/ijm/1256051611.

Export citation