Hostname: page-component-848d4c4894-wzw2p Total loading time: 0 Render date: 2024-05-25T11:01:44.176Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1. Baggett, L. and Kleppner, A., Multiplier representations of abelian groups, J. Funct. Anal. 74 (1973), 299–324.Google Scholar
2. Dixmier, J., C*-algebras (North Holland, 1977).Google Scholar
3. Eymard, P., L'algèbre de Fourier d'un group localement compact, Bull. Soc. Math. France 92 (1964), 181–236.Google Scholar
4. Holzherr, A. K., Discrete groups whose multiplier representations are type I, J. Austral. Math. Soc. (Series A) 30 (1981).Google Scholar
5. Hannabuss, K., Representations of nilpotent locally compact groups, J. Funct. Anal. 34 (1979), 146–165.Google Scholar
6. Kaniuth, E., Die Struktur der regulàren Darstellung lokalkompakter Gruppen mit invarianter Umgebungsbasis der Eins, Math. Ann. 194 (1971), 225–248.Google Scholar
7. Kleppner, A., The structure of some induced representations, Duke Math. J. 29 (1962), 555–572.Google Scholar
8. Kleppner, A., Continuity and measurability of mulitiplier and projective representations, J. of Functional Anal. 17 (1974), 214–226.Google Scholar
9. Mackey, G. W., Unitary representations of group extensions. I, Acta Math. 99 (1958), 265–311.Google Scholar
10. Moore, C. C., Groups with finite dimensional irreducible representations, Trans. Amer. Math. Soc. 166 (1972), 401–410.Google Scholar
11. Robertson, L. C., A note on the structure of Moore groups, Bull. Amer. Math. Soc. 75 (1969), 594–599.Google Scholar
12. Sakai, S., C*-algebras and W*-algebras (Springer, 1971).Google Scholar
13. Takesaki, M., A characterization of group algebras as a converse of Tannaka-Stinespring-Tatsuuma duality theorem, Amer. J. Math. 91 (1969), 529–564.Google Scholar
14. Taylor, K. F., The type structure of the regular representation of a locally compact group, Math. Ann. 222 (1976), 211–224.Google Scholar