Crossed Products by Endomorphisms, Vector Bundles and Group Duality

Abstract

We construct the crossed product of a C(X)-algebra by an endomorphism, in such a way that the endomorphism itself becomes induced by the bimodule of continuous sections of a vector bundle. Some motivating examples for such a construction are given. Furthermore, we study the C*-algebra of G-invariant elements of the Cuntz-Pimsner algebra associated with a G-vector bundle, where G is a (noncompact, in general) group. In particular, the C*-algebra of invariant elements w.r.t. the action of the group of special unitaries of the given vector bundle is a crossed product in the above sense. We also study the analogous construction on certain Hilbert bimodules, called 'noncommutative pullbacks'.

37 pages, uses xy. Revised version of the first part of the previous submission, to appear on Int. J. Math