Crossed products whose primitive ideal spaces are generalized trivial $\widehat G$-bundles

Abstract

We characterize when the primitive ideal space of a crossed product $\acg$ of a \cs-algebra $A$ by a locally compact abelian group $G$ is a $σ$-trivial $\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that $\Prim(\acg)$ is $σ$-trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup $\ttg(α^\w)$ of the Connes spectrum of the system $(A_\w,G,α^\w)$ is continuous, and there is a generalized Green twisting map for $(A,G,α)$. Our proof requires a substantial generalization of a theorem of Olesen and Pedersen in which we show that there is a generalized Green twisting map for $(A,G,α)$ if and only if $\acg$ is isomorphic to a generalized induced algebra.