Topological invariants associated with the spectrum of crossed product \(C^*\)-algebras (Q1316402)

A separable \(C^*\)-dynamical system \((A,G, \alpha)\) in which \(A\) is a continuous-trace \(C^*\)-algebra and \(G\) is Abelian is called \(N\)- principal if \(N\) is a closed subgroup of \(G\) such that \(\alpha\) restricted to \(N\) is locally unitary and the action of \(G\) on \(\widehat {A}\) defines a principal bundle \(p(\alpha): \widehat {A}\to \widehat {A}/G\). In this event, it is known that the spectrum of \(A\rtimes_ \alpha G\) is a principal \(\widehat {N}\)-bundle \(q(\alpha)\) over \(\widehat {A}/G\). In this article the authors show that a pair \(([p ], [q ])\), where \(p: X\to Z\) is a principal \(G/N\)-bundle and \(q: Y\to Z\) is a principal \(\widehat {N}\)-bundle, determines a class in \(H^ 4 (Z)\) which vanishes if and only if there is a continuous-trace \(C^*\)-algebra \(A\) with spectrum \(X\) and a \(N\)-principal system \((A,G, \alpha)\) with \([p (\alpha)]= [p]\). More generally, given \(A\), \(G\), and \([p]\) as above, they consider the question of when systems \((A,G, \alpha)\) with \([p (\alpha)]= [p]\) exist.