Group C*-algebras and Crossed Products

Abstract

The main focus of these notes is the structure of certain kinds of crossed products. The C*-algebra of a group is a special case of a crossed product—it comes from the trivial action of the group on ℂ—but not one of the ones we are mainly concerned with. We devote this section and Section 9.3 to group C*-algebras anyway, in order to provide an introduction to crossed products in a simpler case, and because understanding the group C*-algebra is helpful, at least at a heuristic level, for understanding more general crossed products. Section 9.4 treats crossed product C*-algebras and Section 9.5 treats reduced crossed product C*-algebras. In Section 9.6 we give a number of explicit computations of crossed product C*- algebras. The brief Section 9.2 contains a proof that the reduced C*-algebra of a finitely generated nonabelian free group is simple.