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Groups with Finite Dimensional Irreducible Multiplier Representations

Published online by Cambridge University Press:  20 November 2018

A. K. Holzherr
A. K. Holzherr*
Affiliation:
G.P.O. Box 1086, Canberra, Australia
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Let G be a locally compact group and ω a normalized multiplier on G. Denote by V(G) (respectively by V(G, ω)) the von Neumann algebra generated by the regular representation (respectively co-regular representation) of G. Kaniuth [6] and Taylor [14] have characterized those G for which the maximal type I finite central projection in V(G) is non-zero (respectively the identity operator in V(G)).

In this paper we determine necessary and sufficient conditions on G and ω such that the maximal type / finite central projection in V(G, ω) is non-zero (respectively the identity operator in V(G, ω)) and construct this projection explicitly as a convolution operator on L2(G). As a consequence we prove the following statements are equivalent,

  • (i) V(G, ω) is type I finite,

  • (ii) all irreducible multiplier representations of G are finite dimensional,

  • (iii) Gω (the central extension of G) is a Moore group, that is all its irreducible (ordinary) representations are finite dimensional.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 4 , 01 August 1985 , pp. 635 - 643
Copyright
Copyright © Canadian Mathematical Society 1985

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