Equivariant Chern Character

Let G be a (countable) discrete group acting by a smooth action on a manifold M. There is no further hypothesis on the action. C*(G,M) denotes the reduced crossed-product C*-algebra arising from the action of G on M. If G is finite, then the K-theory of C*(G,M) is Atiyah-Segal equivariant K-theory. When G is not finite, the K-theory of C*(G,M) can be viewed as the natural generalization of Atiyah-Segal equivariant K-theory. What should be the target of the Chern character whose source is the K-theory of C*(G,M)? In this talk, the target is defined in terms of classical homological algebra. Two extreme cases are then examined: the case when the action of G on M is proper - and the case when the manifold M is a point.