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

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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.