Topological Invariants Associated with the Spectrum of Crossed Product C*-Algebras

A separable C*-dynamical system (A, G, α) 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 α restricted to N is locally unitary and the action of G on  defines a principal bundle p(α):  → A/G. In this event, it is known that the spectrum of A ×|αG is a principal N̂-bundle q(α) over Â/G. In this article we show that a pair ([p], [q]), where p: X→ Z is a principal G/N-bundle and q: Y → Z is principal N̂bundle, determines a class in H4(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, α) with [p(α)] = [p]. More generally, given A, G, and [p] as above, we consider the question of when systems (A, G, α) with [p(α)] = [p] exist.