Functors between Kasparov categories from étale groupoid correspondences

Abstract

For an étale correspondence $\Omega :G\to H$ of étale groupoids, we construct an induction functor ${\mathrm{Ind}}_{\Omega }:{\mathrm{KK}}^H\to {\mathrm{KK}}^G$ between equivariant Kasparov categories. We introduce the crossed product of an $H$-equivariant correspondence by $\Omega$, and use this to build a natural transformation ${\alpha }_{\Omega }:K_{\ast }(G⋉{\mathrm{Ind}}_{\Omega }-)\Rightarrow K_{\ast }(H⋉-)$. When $\Omega$ is proper these constructions naturally sit above an induced map in K-theory $K_{\ast }(C^{\ast }(G))\to K_{\ast }(C^{\ast }(H))$.

Version published in the Journal of Functional Analysis