Equivariant Kasparov theory and generalized homomorphisms

TL;DR

This paper extends Kasparov theory to the equivariant setting for locally compact groups, representing KK^G elements via equivariant homomorphisms and providing new proofs of fundamental properties.

Contribution

It introduces a method to describe KK^G elements through equivariant homomorphisms, generalizing Cuntz's approach and establishing the universal property in the equivariant context.

Findings

Representation of KK^G elements via equivariant homomorphisms

Proof of the universal property of KK^G as a split exact stable homotopy functor

Conditions for making the Fredholm operator equivariant in Kasparov triples

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.