Traces on non-commutative homogeneous spaces

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We study properties of C*-algebraic deformations of homogeneous spaces G/Γ 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/Gρ⁰) for a certain subgroup Gρ⁰ of G, and there is a 1-1 correspondence between normalised traces and probability measures on G/Gρ⁰. This makes it possible to represent the deformed algebra as operators over L²(G/Γ). Applications to K-theory are also mentioned.