Fibrations with noncommutative fibers

Abstract

We study an analogue of fibrations of topological spaces with the homotopy lifting property in the setting of C � -algebra bundles. We then derive an analogue of the Leray-Serre spectral sequence to compute the K-theory of the fibration in terms of the cohomology of the base and the K-theory of the fibres. We present many examples which show that fibrations with noncommutative fibres appear in abundance in nature. In recent years the study of the topological properties of C*-algebra bundles plays a more and more prominent role in the field of Operator algebras. The main reason for this is two-fold: on one side there are many important examples of C*-algebras which do come with a canonical bundle structure. On the other side, the study of C*-algebra bundles over a locally compact Hausdorff base space X is the natural next step in classification theory, after the far reaching results which have been obtained in the classification of simple C*-algebras. To fix notation, by a C*-algebra bundle A(X) over X we shall simply mean a C0(X)-algebra in the sense of Kasparov (see (17)): it is a C*-algebra A together with a non-degenerate ∗-homomorphism