J Austral Math Soc Ser A 45 pp62--65, 1988.

On the Characters of Unitary Representations

Michael Cowling

Abstract

Let G be a locally compact group, and let D(G) be a dense subalgebra of the convolution algebra L¹(G). Suppose that p is a unitary representation of G and that, for each u in D(G), p(u) is a trace-class operator. Then the linear functorial u ® tr(p(u)) (the trace of p(u)) is called the D-character of p. 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.

1980 AMS Subject Classification: 22D10, 22D25

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Authors

Michael Cowling

School of Mathematics, University of New South Wales, P. O. Box 1, Kensington, N.S.W. 2033, Australia.