Structure and K-theory of crossed products by proper actions

We study the C*-algebra crossed product C0(X)⋊GC0(X)⋊G of a locally compact group GG acting properly on a locally compact Hausdorff space XX. Under some mild extra conditions, which are automatic if GG is discrete or a Lie group, we describe in detail, and in terms of the action, the primitive ideal space of such crossed products as a topological space, in particular, with respect to its fibring over the quotient space G∖XG∖X. We also give some results on the K-theory of such C*-algebras. These more or less compute the K-theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely K-theoretic proof of a result due to Paul Baum and Alain Connes on K-theory with complex coefficients of crossed products by finite groups.