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

We characterize when the primitive ideal space of a crossed product $\text{acg}$ of a \cs-algebra $A$ by a locally compact abelian group $G$ is a $\sigma$-trivial $\text{ghat}G$-space for the dual $\text{ghat}G$-action. Specifically, we show that $\text{Prim}(\text{acg})$ is $\sigma$-trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit $\text{w}$ a certain subgroup $\text{ttg}({\alpha }^{\text{w}})$ of the Connes spectrum of the system $(A_{\text{w}},G,{\alpha }^{\text{w}})$ 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 $\text{acg}$ is isomorphic to a generalized induced algebra.