Equivariant Kasparov theory and generalized homomorphisms

Abstract

Let G be a locally compact group. We describe elements of KK^G (A,B) by equivariant homomorphisms, following Cuntz's treatment in the non-equivariant case. This yields another proof for the universal property of KK^G: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, phi, F) by an equivariant homomorphism, we have to arrange for the Fredholm operator F to be equivariant. This can be done if A is of the form K(L^2G) otimes A' and more generally if the group action on A is proper in the sense of Rieffel and Exel.