The C*-algebras of Compact Transformation Groups

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 dosed 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 R-n > SO( n) is a Fell algebra. This uses the classical branching theorem for the special orthogonal group SO (n) with respect to SO (n-1). 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.