process

2 R. Meyerassumptions must be closely related to K-theory. Thus special features of topologicalK-theory become more transparent when we work with C -algebras.On the other hand, analysis may create new difficulties, which appear to be very hardto study topologically. For instance, there exist C -algebras with vanishing K-theorywhich are nevertheless non-trivial in Kasparov theory; this means that the UniversalCoefficient Theorem fails for them. I know no non-trivial topological statement aboutthe subcategory of the Kasparov category consisting of C -algebras with vanishingK-theory; for instance, I know no compact objects.It may be necessary, therefore, to restrict attention to suitable “bootstrap” categoriesin order to exclude pathologies that have nothing to do with classical topology. Moreor less by design, the resulting categories will be localisations of purely topologicalcategories, which we can also construct without mentioning C -algebras. For instance,we know that the Rosenberg–Schochet bootstrap category is equivalent to a full subcategoryof the category of BU-module spectra. But we can hope for more interestingcategories when we work equivariantly with respect to, say, discrete groups.2 Additional structure in C -algebra categoriesWe assume that the reader is familiar with some basic properties of C -algebras, includingthe definition (see for instance [2], [13]). As usual, we allow non-unital C -algebras.We define some categories of C -algebras in §2.1 and consider group C -algebras andcrossed products in §2.2. Then we discuss C -tensor products and mention the notionsof nuclearity and exactness in §2.3. The upshot is that C alg and G-C alg carry twostructures of symmetric monoidal category, which coincide for nuclear C -algebras.We prove in §2.4 that C alg and G-C alg are bicomplete, that is, all diagrams in themhave both a limit and a colimit. We equip morphism spaces between C -algebras with acanonical base point and topology in §2.5; thus the category of C -algebras is enrichedover the category of pointed topological spaces. In §2.6, we define mapping cones andcylinders in categories of C -algebras; these rudimentary tools suffice to carry oversome basic homotopy theory.2.1 Categories of C -algebrasDefinition 1. The category of C -algebras is the category C alg whose objects arethe C -algebras and whose morphisms A ! B are the -homomorphisms A ! B;wedenote this set of morphisms by Hom.A; B/.AC -algebra is called separable if it has a countable dense subset. We often restrictattention to the full subcategory C sep C alg of separable C -algebras.Examples of C -algebras are group C -algebras and C -crossed products. Webriefly recall some relevant properties of these constructions. A more detailed discussioncan be found in many textbooks such as [44].