Noncommutative Fibre Bundles via Bimodules
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 $C^*$-bimodules with bimodule connections. We give examples of noncommutative fibre bundles, involving group algebras, matrix algebras, and the quantum torus.
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