The structure of certain unitary representations of infinite symmetric groups

Lieberman, Arthur

doi:10.1090/S0002-9947-1972-0286940-2

The structure of certain unitary representations of infinite symmetric groups
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by Arthur Lieberman PDF

Trans. Amer. Math. Soc. 164 (1972), 189-198 Request permission

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.

References

Les algèbres d’opérateurs dans l’espace Hilbertien

Jacques Dixmier, Les $C^{\ast }$-algèbres et leurs représentations, Cahiers Scientifiques, Fasc. XXIX, Gauthier-Villars & Cie, Éditeur-Imprimeur, Paris, 1964 (French). MR 0171173
Kurt Gödel, The Consistency of the Continuum Hypothesis, Annals of Mathematics Studies, No. 3, Princeton University Press, Princeton, N. J., 1940. MR 0002514, DOI 10.1515/9781400881635

Normed rings

19

22

G. de B. Robinson, Representation theory of the symmetric group, Mathematical Expositions, No. 12, University of Toronto Press, Toronto, 1961. MR 0125885
I. E. Segal, The structure of a class of representations of the unitary group on a Hilbert space, Proc. Amer. Math. Soc. 8 (1957), 197–203. MR 84122, DOI 10.1090/S0002-9939-1957-0084122-8
Elmar Thoma, Die unzerlegbaren, positiv-definiten Klassenfunktionen der abzählbar unendlichen, symmetrischen Gruppe, Math. Z. 85 (1964), 40–61 (German). MR 173169, DOI 10.1007/BF01114877
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158