Cₒ(X)-structure in C*-algebras, multiplier algebras and tensor products

Abstract

We begin in Chapter 2 with an introduction to the various notions of a bundle of C*-algebras that have appeared throughout the literature, and clarify the definitions of upper- and lower-semicontinuous C*-bundles not explicitly defined in a formal way elsewhere. The definition of C0(X)-algebra, introduced by Kasparov [38], and its relation to C*-bundles is discussed in this chapter also. The purpose of this chapter is to bring together concepts that we will refer to in subsequent sections and which are described using various notations by different authors. Most of this is implicitly understood elsewhere, though Theorem 2.3.12, relating sub-modules of C0(X)-modules and subbundles of C*-bundles, is a new result.