Structure and K-theory of crossed products by proper actions

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 $G\mathrm{\backslash }X$. We also give some results on the $\text{K}$-theory of such C*-algebras. These more or less compute the $\text{K}$-theory in the case of isolated orbits with non-trivial (finite) stabilizers. We also give a purely $\text{K}$-theoretic proof of a result due to Paul Baum and Alain Connes on (\K)-theory with complex coefficients of crossed products by finite groups.