kk-Theory for Banach algebras I: The non-equivariant case

Abstract

kkban is a bivariant K-theory for Banach algebras that has reasonable homological properties, a product and is Morita invariant in a very general sense. We define it here by a universal property and ensure its existence in a rather abstract manner using triangulated categories. The definition ensures that there is a natural transformation from Lafforgue's theory KK ban into it so that one can take products of elements in KK ban that lie in kk ban . A first attempt to define such a theory would be to adapt the definitions of [Cun97] or [CMR07] directly. This means to substitute the algebras of "compact operators" that are used as stabilisations in the several definitions of kk with a limit over all possible Morita equivalences of Banach algebras. But the relation of the non-commutative suspension functor J used by Cuntz and this stabilisation is somewhat unclear and this ansatz seems to lead to a dead end. KeywordsIn principle, it is still possible -though somewhat forced -to adapt Cuntz's general theory by using its triangulated structure. But it seems cleaner and more conceptual to use the so-called SpanierWhitehead construction -without furthur ado -as laid out in a slightly different framework in the appendix of [Del08]. It does not make use of Cuntz's non-commutative suspension functor J but just of the ordinary suspension functor which we call Σ. Several paragraphs of the present article are transferred rather directly from [Del08] into our context. The functor J is analysed a posteriori in Paragraph 5.2.In this article, we define several theories with more and more desirable properties: A theory ΣHo ban which comes right out of the Spanier-Whitehead construction and can be thought of as a stable homotopy category of Banach algebras. As a "quotient" of this theory we obtain a theory EHo ban which is comparable to the theory ΣHo of [CMR07] and has long exact sequences in both 1