NONCOMMUTATIVE PRINCIPAL BUNDLES: BEYOND THE COMPACT CASE

The notion of a compact noncommutative (or quantum) principal bundle, which generalizes the Cartan compact principal bundle from topology (local triviality not assumed), emerged in the literature almost 30 years ago. Recently, the difficulty of introducing the local-triviality condition to the noncommutative realm was overcome using the notion of the local-triviality dimension of an action of a compact quantum group on a unital C*-algebra. In this talk, I will propose a definition of a locally trivial noncommutative principal bundle in the setting of actions of locally compact Hausdorff groups on (possibly non-unital) C*-algebras. I will discuss various motivations and technical difficulties that appear in the non-compact case. I will also provide basic results and examples. The key difference is that, although the problem itself can be described in the language of C*-algebras, one is quickly led to the theory of pro-C*-algebras and Pedersen multiplier algebras. In particular, the Gelfand-Naimark duality has to be replaced by a duality between certain compactly generated spaces and unital commutative pro-C*-algebras introduced by N. C. Phillips.