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

TL;DR

This paper defines a noncommutative analogue of locally trivial principal G-bundles using group actions on C*-algebras, extending classical topology concepts to the noncommutative setting.

Contribution

It introduces the concept of locally trivial G-C*-algebras, broadening the understanding of noncommutative principal bundles beyond unital algebras.

Findings

New definition of locally trivial G-C*-algebra using Pedersen ideal multipliers

Illustrations with examples from C_0(Y)-algebras and graph C*-algebras

Freeness of actions on locally trivial G-C*-algebras in many cases

Abstract

We introduce a definition of the 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 $C_0(Y)$-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.