Traces on non-commutative homogeneous spaces

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.