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Transactions of the American Mathematical Society

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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.

References [Enhancements On Off] (What's this?)

  • [1] J. Dixmier, Les algèbres d'opérateurs dans l'espace Hilbertien, 2nd ed., Gauthier-Villars, Paris, 1969.
  • [2] -, Les $ {C^ \ast }$-algèbres et leurs représentations, Cahiers Scientifique, Fasc. 29, Gauthier-Villars, Paris, 1964. MR 30 #1404. MR 0171173 (30:1404)
  • [3] K. Gödel, The consistency of the continuum hypothesis, Ann. of Math. Studies, no. 3, Princeton Univ. Press, Princeton, N. J., 1940. MR 2, 66. MR 0002514 (2:66c)
  • [4] M. A. Naĭmark, Normed rings, GITTL, Moscow, 1956; English transl., Noordhoff, Groningen, 1959. MR 19, 870; MR 22 #1824.
  • [5] G. B. Robinson, Representation theory of the symmetric group, Math. Expositions, no. 12, University of Toronto Press, Toronto, 1961. MR 23 #A3182. MR 0125885 (23:A3182)
  • [6] 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 18, 812. MR 0084122 (18:812f)
  • [7] E. Thoma, Die unzerlegbaren, positiv-definiten Klassenfunctionen der abzählbar unendlichen symmetrischen Gruppe, Math. Z. 85 (1964), 40-61. MR 30 #3382. MR 0173169 (30:3382)
  • [8] H. Weyl, The classical groups. Their invariants and representations, 2nd ed., Princeton Univ. Press, Princeton, N. J., 1946. MR 1, 42. MR 1488158 (98k:01049)

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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

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