Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-23T13:55:37.306Z Has data issue: false hasContentIssue false

On the characters of unitary representations

Published online by Cambridge University Press:  09 April 2009

Michael Cowling
Affiliation:
School of MathematicsUniversity of New South WalesP. O. Box 1Kensington. N.S.W. 2033, Australia
Rights & Permissions [Opens in a new window]

Abstract

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 let D(G) be a dense subalgebra of the convolution algebra L1(G). Suppose that π is a unitary representation of G and that, for each u in D(G), π(u)) is a trace-class operator. Then the linear functional u → tr(π(u)) (the trace of π(u)) is called the D-character of π. We give a simple proof that the D-character of such a representation determines the representation up to unitary equivalence. As an application, we give an easy proof of the result of Harish-Chandra that the K-finite characters of unitary representations of semisimple Lie groups determine the representations.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Cowling, M., ‘Some applications of Grothendieck's theory of topological tensor products in harmonic analysis’, Math. Ann. 232 (1978), 273285.Google Scholar
[2]Dixmier, J., C*-algebras (North-Holland Publishing Co., Amsterdam, New York, Oxford, 1977).Google Scholar
[3]Eymard, P., ‘L'algèbre de Fourier d'un groupe localement compact’, Bull. Soc. Math. France 92 (1964), 181236.CrossRefGoogle Scholar
[4]Warner, G., Harmonic analysis on semi-simple Lie groups. I (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[5]Wolf, J. A., Cahen, M. and DeWilde, M. (eds), Harmonic analysis and representations of semisimple Lie groups (D. Reidel Publishing Company, Dordrecht, Boston, London, 1980).CrossRefGoogle Scholar