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The C*–algebras of Compact Transformation Groups

Published online by Cambridge University Press:  20 November 2018

Robert J. Archbold
Affiliation:
Institute of Mathematics, University of Aberdeen, Aberdeen AB24 3UE, Scotland, UK e-mail: r.archbold@abdn.ac.uk
Astrid an Huef
Affiliation:
Department of Mathematics and Statistics, University of Otago, Dunedin 9054, New Zealand e-mail: astrid@maths.otago.ac.nz
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Abstract

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We investigate the representation theory of the crossed-product ${{C}^{*}}$-algebra associated with a compact group $G$ acting on a locally compact space $X$ when the stability subgroups vary discontinuously. Our main result applies when $G$ has a principal stability subgroup or $X$ is locally of finite $G$-orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation $V$ of a stability subgroup is obtained by restricting $V$ to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of $V$. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup; the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the ${{C}^{*}}$-algebra of the motion group ${{\mathbb{R}}^{n}}\,\rtimes \,\text{SO}\left( n \right)$ is a Fell algebra. This uses the classical branching theorem for the special orthogonal group $\text{SO}\left( n \right)$ with respect to $\text{SO}\left( n-1 \right)$. Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2015

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