Equivariant Kasparov Theory and Generalized Homomorphisms

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 (, , F) for A, B 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 ; and more generally if the group action on A is proper in the sense of Exel and Rieffel.