Fine structure of the Mackey machine for actions of abelian groups with constant Mackey obstruction
If \(G\) is a locally compact Abelian group and \(\omega\in Z^2(G, \mathbb{T})\) is a (measurable) multiplier on \(G\), it is known that the primitive ideal space of \(C^*(G, \omega)\), the twisted group \(C^*\)-algebra of \(G\) defined by \(\omega\), is naturally homeomorphic to \(\widehat S_\omega\), where \(S_\omega\) is the ``symmetry group'' of \(\omega\). We show in fact that if \(\omega\) is type I, \(C^*(G, \omega)\) is Morita equivalent to \(C_0(\widehat S_\omega)\). In fact, if \(G\) is second-countable and we exclude the case where \(S_\omega\) is of finite index, \(C^*(G, \omega)\) is isomorphic to \(C_0(\widehat S_\omega)\otimes {\mathcal K}\), \(\mathcal K\) the compact operators on an infinite-dimensional separable Hilbert space. We also investigate crossed products \(A\rtimes_\alpha G\) such that \(A\) has continuous trace, the Abelian group \(G\) acts on \(\widehat A\) with constant isotropy group \(N\), and all Mackey obstructions are similar to a constant multiplier \(\omega\in Z^2(N, \mathbb{T})\). We study the question of such a crossed product \(A\rtimes_\alpha G\) has continuous trace, and in many cases, where this is so we compute the Dixmier-Douady class of the crossed product. The above results on the case \(A= {\mathcal K}\) may lead to the guess that if \(G= N\), \(A\rtimes_\alpha G\) is Morita equivalent to \(A\rtimes_{\alpha_S} S\), but we will see that this is not true in general. This work is the obvious starting point for the study of the ``fine structure of the Mackey machine'', for actions of Abelian groups on continuous-trace algebras with ``continuously varying'' stabilizers and Mackey obstructions.
scientific article; zbMATH DE number 910220