process

Categorical aspects of bivariant K-theory 9The category of commutative C -algebras is equivalent to the opposite of the categoryof pointed compact spaces by the Gelfand–Naimark Theorem. It is frequentlyconvenient to replace a pointed compact space X with base point ? by the locallycompact space X nf?g. A continuous map X ! Y extends to a pointed continuousmap X C ! Y C if and only if it is proper. But there are more pointed continuousmaps f W X C ! Y C than proper continuous maps X ! Y because points in X maybe mapped to the point at infinity 12Y C . For instance, the zero homomorphismC 0 .Y / ! C 0 .X/ corresponds to the constant map x 7! 1.Example 20. If U X is an open subset of a locally compact space, then C 0 .U / isan ideal in C 0 .X/. No map X ! U corresponds to the embedding C 0 .U / ! C 0 .X/.Example 21. Products of commutative C -algebras are again commutative and correspondby the Gelfand–Naimark Theorem to coproducts in the category of pointedcompact spaces. The coproduct of a set of pointed compact spaces is the Stone–Čechcompactification of their wedge sum. Thus infinite products in C alg and G-C algdo not behave well for the purposes of homotopy theory.The coproduct of two non-zero C -algebras is never commutative and hence has noanalogue for (pointed) compact spaces. The smash product for pointed compact spacescorresponds to the tensor product of C -algebras becauseC 0 .X ^ Y/Š C 0 .X/ ˝min C 0 .Y /:2.5 Enrichment over pointed topological spaces. Let A and B be C -algebras. It iswell-known that a -homomorphism f W A ! B is automatically norm-contracting andinduces an isometric embedding A= ker f ! B with respect to the quotient norm onA= ker f . The reason for this is that the norm for self-adjoint elements in a C -algebraagrees with the spectral radius and hence is determined by the algebraic structure; bythe C -condition kak 2 Dka ak, this extends to all elements of a C -algebra.It follows that Hom.A; B/ is an equicontinuous set of linear maps A ! B. We alwaysequip Hom.A; B/ with the topology of pointwise norm-convergence. Its subbasicopen subsets are of the formff W A ! B jk.ff 0 /.a/k