On an intermediate bivariant $K$-theory for $C^{\ast }$-algebras

Abstract

We construct a new bivariant $K$-theory for $C^{\ast }$-algebras, that we call $KE$-theory. For each pair of separable graded $C^{\ast }$-algebras $A$ and $B$, acted upon by a locally compact $\sigma$-compact group $G$, we define an abelian group $K{E}_G(A,B)$. We show that there is an associative product $K{E}_G(A,D)\otimes K{E}_G(D,B)\to K{E}_G(A,B)$. Various functoriality properties of the $KE$-theory groups and of the product are presented. The new theory is intermediate between the $KK$-theory of G.G. Kasparov, and the $E$-theory of A. Connes and N. Higson, in the sense that there are natural transformations $K{K}_G\to K{E}_G$ and $K{E}_G\to E_G$ preserving the products. The motivations that led to the construction of $KE$-theory were: (1) to give a concrete description of the map from $KK$-theory to $E$-theory, abstractly known to exist because of the universal characterization of $KK$-theory, (2) to construct a bivariant theory well adapted to dealing with elliptic operators, and in which the product is simpler to compute with than in $KK$-theory, and (3) to provide a different proof to the Baum–Connes conjecture for a-T-menable groups. This paper deals with the first two problems mentioned above; the third one will be treated somewhere else.