Equivariant Kasparov Theory and Generalized Homomorphisms

Let G be a locally compact group. We describe elements of KK(A,B) by equivariant homomorphisms, following Cuntz’s treatment in the non-equivariant case. This yields another proof for the universal property of KK: It is the universal split exact stable homotopy functor. To describe a Kasparov triple (E, φ, 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 K(LG) ⊗ A and more generally if the group action on A is proper in the sense of Exel and Rieffel.