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

TL;DR

This paper characterizes when the primitive ideal space of a crossed product of a C*-algebra by an abelian group is a generalized trivial bundle, linking topological and algebraic conditions including the existence of a Green twisting map.

Contribution

It provides necessary and sufficient conditions for the primitive ideal space to be a σ-trivial G-hat bundle, generalizing previous results with a new characterization involving generalized Green twisting maps.

Findings

Primitive ideal space is σ-trivial iff the quasi-orbit space is Hausdorff.

Continuity of the subgroup assignment relates to the structure of the crossed product.

Existence of a generalized Green twisting map characterizes the crossed product as a generalized induced algebra.

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 $\sigma$-trivial $\ghat G$-space for the dual $\ghat G$-action. Specifically, we show that $\Prim(\acg)$ is $\sigma$-trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit $\w$ a certain subgroup $\ttg(\alpha^\w)$ of the Connes spectrum of the system $(A_\w,G,\alpha^\w)$ is continuous, and there is a generalized Green twisting map for $(A,G,\alpha)$. 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 $(A,G,\alpha)$ if and only if $\acg$ is isomorphic to a generalized induced algebra.