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The C*–algebras of Compact Transformation Groups

  • Robert J. Archbold (a1) and Astrid an Huef (a2)

Abstract

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.

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References

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The C*–algebras of Compact Transformation Groups

  • Robert J. Archbold (a1) and Astrid an Huef (a2)

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