${}^{\ast }$-homomorphisms between groupoid $C^{\ast }$-algebras

Abstract

In this paper, we investigate ${}^{\ast }$-homomorphisms between $C^{\ast }$-algebras associated to étale groupoids. First, we prove that such a ${}^{\ast }$-homomorphism can be described by closed invariant subsets, groupoid homomorphisms and cocycles under some assumptions. Then we prove $C^{\ast }$-rigidity results for étale groupoids which are not necessarily effective. As another application, we investigate certain subgroups of the automorphism groups of groupoid $C^{\ast }$-algebras. More precisely, we show that the groups of automorphisms that globally preserve the function algebras on the unit spaces are isomorphic to certain semidirect product groups. As a corollary, we show that, if group actions on groupoid $C^{\ast }$-algebras fix the function algebras on the unit spaces, then the actions factors through the abelianizations of the acting groups.