Crossed products whose primitive ideal spaces are generalized trivial?-bundles

Abstract

We characterize when the primitive ideal space of a crossed product A ⋊α G of a C * -algebra A by a locally compact abelian group G is a σtrivial G-space for the dual G-action. Specifically, we show that Prim(A ⋊α G) is σ-trivial if and only if the quasi-orbit space is Hausdorff, the map which assigns to each quasi-orbit x a certain subgroupΓ(α x ) of the Connes spectrum of the system (Ax, G, α x ) 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 A ⋊α G is isomorphic to a generalized induced algebra.1991 Mathematics Subject Classification. Primary 46L05, 46L55.

Structure of crossed products by strictly proper actions on continuous-trace algebras

“…In the case where one assumes away the difficulty -that is, where we assume all the Mackey obstructions vanish -there has been considerable progress if it is also assumed that the stabilizers are constant, see, for example [19,21,23,26], and when the stabilizer map is continuous [25]. When nontrivial Mackey obstructions are allowed, the progress has been more modest and is usually accompanied with robust hypotheses [5,6,9,10,14]. In this article, we take on the case where G acts properly on the spectrum X of a continuous-trace C * -algebra B.…”