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On the characters of unitary representations

Part of: Locally compact groups and their algebras

Published online by Cambridge University Press:  09 April 2009

Michael Cowling
Michael Cowling
Affiliation:
School of MathematicsUniversity of New South WalesP. O. Box 1Kensington. N.S.W. 2033, Australia
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Abstract

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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.

MSC classification

Secondary: 22D10: Unitary representations of locally compact groups 22D25: $C^*$-algebras and $W^*$-algebras in relation to group representations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 45 , Issue 1 , August 1988 , pp. 62 - 65
Copyright
Copyright © Australian Mathematical Society 1988

References

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