## Illinois Journal of Mathematics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 20 October 2009

**Permanent link to this document**

https://projecteuclid.org/euclid.ijm/1256051611

**Digital Object Identifier**

doi:10.1215/ijm/1256051611

**Mathematical Reviews number (MathSciNet)**

MR0320220

**Zentralblatt MATH identifier**

0261.20017

**Subjects**

Primary: 22A25: Representations of general topological groups and semigroups

Secondary: 22D10: Unitary representations of locally compact groups

#### Citation

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. https://projecteuclid.org/euclid.ijm/1256051611