process

Categorical aspects of bivariant K-theory 29The Universal Coefficient Theorem and the universal property of KK imply thatvery few homology theories for (pointed compact metrisable) spaces can extend to thenon-commutative setting. More precisely, if we require the extension to be split-exact,C -stable, and additive for countable direct sums, then only K-theory with coefficientsis possible. Thus we rule out most of the difficult (and interesting) problems in stablehomotopy theory. But if we only want to study K-theory, anyway, then the operatoralgebraic framework usually provides very good analytical tools. This is most valuablefor equivariant generalisations of K-theory.Jonathan Rosenberg and Claude Schochet [50] have also constructed a spectralsequence that, in favourable cases, computes KK G .A; B/ from K G .A/ and KG .B/;they require G to be a compact Lie group with torsion-free fundamental group andA and B to belong to a suitable bootstrap class. This equivariant UCT is clarifiedin [39], [40].4.4 E-theory and asymptotic morphisms. Recall that Kasparov theory is only exactfor (equivariantly) cp-split extensions. E-theory is a similar theory that is exact for allextensions.Definition 58. We let E G W G-C sep ! E G be the universal C -stable, exact homotopyfunctor.Lemma 59. The functor E G is split-exact and factors through KK G W G-C sep !KK G . Hence it satisfies Bott periodicity.Proof. Proposition 48 shows that any exact homotopy functor is split-exact. The remainingassertions now follow from Corollary 51.The functor E (for trivial G) was first defined as above by Nigel Higson [25]. ThenAlain Connes and Nigel Higson [8] found a more concrete description using asymptoticmorphisms. This is what made the theory usable. The equivariant generalisation of thetheory is due to Erik Guentner, Nigel Higson, and Jody Trout [21].We write E G n .A; B/ for the space of morphisms A ! Susn .B/ in E G . Bott periodicityshows that there are only two different groups to consider.Definition 60. The asymptotic algebra ofaC -algebra B is the C -algebraAsymp.B/ WD C b .R C ;B/=C 0 .R C ;B/:An asymptotic morphism A ! B is a -homomorphism f W A ! Asymp.B/.Representing elements of Asymp.B/ by bounded functions Œ0; 1/ ! B, we canrepresent f by a family of maps f t W A ! B such that f t .a/ 2 C b .R C ;B/for each a 2A and the map a 7! f t .a/ satisfies the conditions for a -homomorphism asymptoticallyfor t !1. This provides a concrete description of asymptotic morphisms and explainsthe name.If a locally compact group G acts on B, then Asymp.B/ inherits an action of G bynaturality (which need not be strongly continuous).