Structure and K-theory of crossed products by proper actions

Abstract

We study the C*-algebra crossed product C 0 ( X ) ⋊ G of a locally compact group G acting properly on a locally compact Hausdorff space X . Under some mild extra conditions, which are automatic if G 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 ∖ 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.