Traces on non-commutative homogeneous spaces

Abstract

We study properties of C*-algebraic deformations of homogeneous spaces $G/\Gamma$ which are equivariant in the sense that they preserve the natural action of $G$ by left translation. The center is shown to be isomorphic to $C(G/G_{\rho }^0)$ for a certain subgroup $G_{\rho }^0$ of $G$, and there is a 1-1 correspondence between normalised traces and probability measures on $G/G_{\rho }^0$. This makes it possible to represent the deformed algebra as operators over $L^2(G/\Gamma )$. Applications to K-theory are also mentioned.

12 pages, an error in the first version has been corrected. To appear in J. Funct. Anal