Functors between Kasparov categories from étale groupoid correspondences (Q6611761)

This article studies the functoriality of Kasparov's equivariant bivariant KK-theory and of the \(K\)-theory of crossed products with respect to correspondences between étale groupoids. Many interesting \( C^\ast \)-algebras may be described as \( C^\ast \)-algebras of étale groupoids. In particular, this applies to many \( C^\ast \)-algebras associated to semigroups. This article develops tools to study their \(K\)-theory, constructing maps on \(K\)-theory and homomorphisms between equivariant Kasparov categories from correspondences between the underlying groupoids. Such a correspondence is a space with commuting actions of the two groupoids, subject to some conditions. In particular, any Morita equivalence is also a groupoid correspondence. A groupoid correspondence induces both a homomorphism between the equivariant Kasparov categories and a \( C^\ast \)-correspondence between the groupoid \( C^\ast \)-algebras of the two groupoids. More generally, this article also builds \( C^\ast \)-correspondences between crossed products, using an extra \( C^\ast \)-correspondence between the coefficient \( C^\ast \)-algebras on which the action takes place. The resulting construction looks like an induction process for \( C^\ast \)-correspondences. A \( C^\ast \)-correspondence induces a map on \(K\)-theory if it is proper. This property may be read off an underlying groupoid correspondence. The article also establishes functorial properties of these constructions, such as compatibility with the composition of correspondences.