Noncommutative numerable principal bundles from group actions on C⁎-algebras

Abstract

We introduce the notion of a locally trivial G-C⁎-algebra, which is a noncommutative counterpart of the total space of a locally compact Hausdorff numerable principal G-bundle. To obtain this generalization, we have to go beyond the Gelfand–Naimark duality and use the multipliers of the Pedersen ideal. Our new concept enables us to investigate local triviality of noncommutative principal bundles coming from group actions on non-unital C⁎-algebras, which we illustrate through examples coming from C0(Y)-algebras and graph C⁎-algebras. In the case of an action of a compact Hausdorff group on a unital C⁎-algebra, local triviality in our sense is implied by the finiteness of the local-triviality dimension of the action. Furthermore, we prove that if A is a locally trivial G-C⁎-algebra, then the G-action on A is free in a certain sense, which in many cases coincides with the known notions of freeness due to Rieffel and Ellwood.