*-homomorphisms between groupoid C*-algebras

TL;DR

This paper characterizes *-homomorphisms between groupoid C*-algebras using invariant subsets, homomorphisms, and cocycles, and explores automorphism groups and rigidity properties of étale groupoid C*-algebras.

Contribution

It provides a new description of *-homomorphisms via geometric data and establishes C*-rigidity results for non-effective étale groupoids.

Findings

* -homomorphisms characterized by invariant subsets, homomorphisms, and cocycles

Automorphism groups preserving function algebras are semidirect products

Actions fixing function algebras factor through abelianizations

Abstract

In this paper, we investigate *-homomorphisms between C*-algebras associated to \'etale 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 \'etale 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.