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

We construct a new bivariant theory, that we call $KE$-theory, which is intermediate between the $KK$-theory of G. G. Kasparov, and the $E$-theory of A. Connes and N. Higson. 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 $KE_G(A,D) \otimes KE_G(D,B) \to KE_G(A,B)$. Various functoriality properties of the $KE$-theory groups and of the product are presented. The new theory has a simpler product than $KK$-theory and there are natural transformations $KK_G \to KE_G$ and $KE_G \to E_G$. The complete description of these maps will form the substance of a second paper.