Crossed products by $C_0(X)$-actions

Abstract

Suppose that $G$ has a representation group $H$, that $G_{ab}:= G/\bar{[G,G]}$ is compactly generated, and that $A$ is a \cs-algebra for which the complete regularization of $\Prim(A)$ is a locally compact Hausdorff space $X$. In a previous article, we showed that there is a bijection $α\mapsto (Z_α,f_α)$ between the collection of exterior equivalence classes of locally inner actions $α:G\to\Aut(A)$, and the collection of principal $\hgab$-bundles $Z_α$ together with continuous functions $f_α:X\to H^2(G,\T)$. In this paper, we compute the crossed products $A\rtimes_αG$ in terms of the data $Z_α$, $f_α$, and~$\cs(H)$.