Noncommutative Fibre Bundles via Bimodules

Abstract

We construct a Leray-Serre spectral sequence for fibre bundles for de Rham sheaf cohomology on noncommutative algebras. The morphisms are bimodules with zero-curvature extendable bimodule connections. This generalises definitions involving differentiable algebra maps to differentiable completely positive maps by using the KSGNS construction and Hilbert $C^{\ast }$-bimodules with bimodule connections. We give examples of noncommutative fibre bundles, involving group algebras, matrix algebras, and the quantum torus.