*-homomorphisms between groupoid C*-algebras

Abstract

In this paper, we investigate *-homomorphisms between C*-algebras associated to étale groupoids. First, we prove that such a *-homomorphism can be described by closed invariant subsets, groupoid homomorphisms and cocycles under some assumptions. Then we prove C*-rigidity results for étale groupoids which are not necessarily effective. As another application, we investigate certain subgroups of the automorphism groups of groupoid C*-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*-algebras fix the function algebras on the unit spaces, then the actions factors through the abelianizations of the acting groups.

24 pages, with minor changes(ver2). In ver3, assumptions about effectiveness of étale groupoids are relaxed. arXiv admin note: text overlap with arXiv:2206.11713