process

24 R. MeyerBy the universal property, K-theory descends to a functor on KK, that is, we getcanonical mapsKK 0 .A; B/ ! Hom K .A/; K .B/ for all separable C -algebras A; B, where the right hand side denotes grading-preservinggroup homomorphisms. For A D C, this yields a map KK 0 .C;B/ !Hom Z; K 0 .B/ Š K 0 .B/. Using suspensions, we also get a corresponding mapKK 1 .C;B/! K 1 .B/.Theorem 53. The maps KK .C;B/ ! K .B/ constructed above are isomorphismsfor all B 22 C sep.Thus Kasparov theory is a bivariant generalisation of K-theory. Roughly speaking,KK .A; B/ is the place where maps between K-theory groups live. Most constructionsof such maps, say, in index theory can, in fact, be improved to yield elements ofKK .A; B/. One reason why this has to be so is the Universal Coefficient Theorem(UCT), which computes KK .A; B/ from K .A/ and K .B/ for many C -algebrasA; B. If A satisfies the UCT, then any grading preserving group homomorphismK .A/ ! K .B/ lifts to an element of KK 0 .A; B/.4.1 Extending functors and identities to KK G . We can use the universal propertyto extend various functors G-C sep ! H -C sep to functors KK G ! KK H . Weexplain this by an example:Proposition 54. The full and reduced crossed product functorsG Ë r;GË W G-C alg ! C algextend to functors G Ë r ;GË W KK G ! KK called descent functors.Gennadi Kasparov [31] constructs these functors directly using the concrete descriptionof Kasparov cycles. This requires a certain amount of work; in particular, checkingfunctoriality involves knowing how to compute Kasparov products. The constructionvia the universal property is formal:Proof. We only write down the argument for reduced crossed products, the other caseis similar. It is well-known that G Ë r A ˝ K.H/ Š .G Ë r A/ ˝ K.H/ for anyG-Hilbert space H. Therefore, the composite functorG-C sep GË r! C sep KK ! KKis C -stable. Proposition 9 shows that this functor is split-exact as well (regardless ofwhether G is an exact group). Now the universal property provides an extension to afunctor KK G ! KK.Similarly, we get functorsA ˝min ;A˝max W KK G ! KK G