Submodules of normalisers in groupoid C*-algebras and discrete group coactions

TL;DR

This paper explores the structure of submodules generated by normalizers in groupoid C*-algebras, linking coactions to cocycles and establishing a Galois correspondence for discrete group coactions.

Contribution

It demonstrates that submodules generated by normalizers are closures of functions on open sets and characterizes coactions via cocycles, providing a Galois correspondence in this context.

Findings

Submodules generated by normalizers are closures of compactly supported functions.

Discrete group coactions are induced by cocycles under certain conditions.

A Galois correspondence for discrete group coactions is established.

Abstract

In this paper, we investigate certain submodules in C*-algebras associated to effective \'etale groupoids. First, we show that a submodule generated by normalizers is a closure of the set of compactly supported continuous functions on some open set. As a corollary, we show that discrete group coactions on groupoid C*-algebras are induced by cocycles of \'etale groupoids if the fixed point algebras contain C*-subalgebras of continuous functions vanishing at infinity on the unit spaces. In the latter part, we prove the Galois correspondence result for discrete group coactions on groupoid C*-algebras.