Locally unitary groupoid crossed products (Q2898361)

In this paper, the connection between groupoid dynamical systems and the spectrum of the crossed product is studied. Let \(S\) be an abelian locally compact Hausdorff groupoid group bundle. The notion of the principal \(S\)-bundle is defined and it is shown that there is a one-to-one correspondence between principal \(S\)-bundles and elements of a sheaf cohomology group associated to \(S\). The notion of locally unitary action of \(S\) is defined and it is shown that the spectrum of the crossed product is a principal \(\hat{S}\)-bundle (where \(\hat{S}\) is the dual group bundle). It is also proved that the isomorphism class of the spectrum of the crossed product as \(\hat{S}\)-bundle determines the exterior equivalence class of the action and that every principal \(S\)-bundle can be realized as the spectrum of some locally unitary crossed product. Some of the results of this paper generalize similar results of \textit{J. Phillips} and \textit{I. Raeburn} [J. Oper. Theory 11, 215--241 (1984; Zbl 0538.46046)] for locally unitary group actions to groupoids.