The generators and relations picture of $KK$-theory

Abstract

This is half an overview article since what we describe here is essentially known. We describe $KK$-theory by generators and relations in a formal sum of formal products of $\ast$-homomorphisms and some synthetical morphisms. What comes out is a category. The Kasparov product is then just the composition of morphisms. This description may be interesting to anyone who wants a quick and elementary definition of $KK$-theory. The description could also be used for other categories of algebras than $C^{\ast }$-algebras endowed with group actions, for example, $C^{\ast }$-algebras equipped with an action by a semigroup, a category et cetera.

Minor typos corrections. Proof of Section 5 simplified. Section 5 newly written

References in corpus (1)

• Equivariant Kasparov theory and generalized homomorphisms

Cited by in corpus (1)

• A kind of $KK$-theory for rings