Free products and continuous bundles of C*-algebras
The purpose of this thesis is to investigate the properties of free products of C*-algebras and continuous bundles of C*-algebras. We also consider how these two areas are connected. In the first chapter we present background material relevant to the thesis. We discuss nuclearity, exactness and Hilbert C*-modules. Then we review the definitions and properties of bundles and free products of C*-algebras. The second chapter considers reduced amalgamated free products of C*-algebras. We show that, if the initial conditional expectations involved are all faithful, then the resulting free product conditional expectation is also faithful. In the third chapter we are interested in the properties of reduced free product C*-algebras. We introduce the orthounitary basis concept for unital C*-algebras with faithful traces and show that reduced free products of C*-algebras with orthounitary bases are, except in a few special cases, not nuclear. Building on this, we then determine the ideals in a certain tensor product C v Cop of the reduced free product with its opposite C*-algebra. In the second half of the chapter, we use Cuntz-Pimsner C*-algebras to study reduced free products of nuclear C*-algebras with respect to pure states. We show that, if the G.N.S. representations of the C*-algebras involved contain the compact operators, then the reduced free product C* -algebra is also nuclear. Chapter four looks at the minimal tensor product operation on continuous bundles of C*-algebras. We construct, for any non-exact C*-algebra C, a continuous bundle A on the unit interval [0,1] such that A C is not continuous. This leads to a new characterisation of exactness for C*-algebras. These results are then extended to allow for any compact infinite metric space as the base space. Finally, we introduce free product operations on bundles of C*-algebras in chapter five. Both full and reduced free product bundles are constructed. We show that taking the free product (full or reduced) of two continuous bundles gives another continuous bundle, at least when the bundle C*-algebras are exact.