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

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 KEG(A, B). We show that there is an associative product KEG(A, D) circle times KEG(D, B) -> KEG(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 KKG -> KEG and KEG -> 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.