On an intermediate bivariant $K$-theory for $C^*$-algebras

Abstract

We construct a new bivariant $K$-theory for $C^$-algebras, that we call $KE$-theory. For each pair of separable graded $C^$-algebras $A$ and $B$, acted upon by a locally compact $\sigma$-compact group $G$, we define an abelian group $KE\_G(A,B)$. We show that there is an associative product $KE\_G(A,D) \otimes KE\_G(D,B) \rightarrow KE\_G(A,B)$. 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 $KK\_G \rightarrow KE\_G$ and $KE\_G \rightarrow E\_G$ 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.