Journal of Operator Theory

Volume 62, Issue 1, Summer 2009 pp. 171-198.

Extension problems for representations of crossed-product $C^*$-algebras

Authors: Astrid an Huef (1), S. Kaliszewski (2), Iain Raeburn (3), and Dana P. Williams (4)
Author institution: (1) School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia
(2) Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, U.S.A.
(3) School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia
(4) Department of Mathematics, Dartmouth College, Hanover, NH 03755, U.S.A.

Summary: A classical problem in representation theory asks which unitary representations $U$ of a closed subgroup $H$ of a locally compact group $G$ are the restrictions $V|_H$ of unitary representations $V$ of $G$. We have recently shown that this extension problem has a dual formulation involving representations of crossed products of $C^*$-algebras by coactions, and this dual formulation raises many interesting test questions for the theory of non-abelian duality. In this paper, we consider the extension problem in the context of covariant representations for actions of $G$ on $C^*$-algebras, and the analogous problem for coactions. Each of our three main theorems has two main ingredients: a theorem describing some aspect of the duality between induction and restriction of representations, and an imprimitivity theorem. Some of these ingredients are available in the literature, but others are new and should be of independent interest. For example, we prove a version of Green's imprimitivity theorem for reduced crossed products, and this seems to be an interesting new application of non-abelian duality in itself.

Keywords: Unitary representation, crossed product, extension, restriction, induction, coaction, non-abelian duality.

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