Noncommutative Fibrations via Positive Maps and Bimodules

We construct a Leray-Serre spectral sequence for fibrations for de Rham cohomology on noncommutative algebras. The fibrations are bimodules with zero-curvature extendable bimodule connections satisfying an additional condition. By the KSGNS construction, completely positive maps between C*-algebras correspond to Hilbert C*-bimodules. We give examples of fibrations on group algebras and matrix algebras.