Equivariant $KK$-theory of $r$-discrete groupoids and inverse semigroups

Abstract

For an $r$-discrete Hausdorff groupoid $\mathcal{𝒢}$ and an inverse semigroup $S$ of slices of $\mathcal{𝒢}$ there is an isomorphism between $\mathcal{𝒢}$-equivariant $KK$-theory and compatible $S$-equivariant $KK$-theory. We use it to define descent homomorphisms for $S$, and indicate a Baum–Connes map for inverse semigroups. Also findings by Khoshkam and Skandalis for crossed products by inverse semigroups are reflected in $KK$-theory.