Representing topological full groups in Steinberg algebras and C*-algebras

TL;DR

This paper investigates the representation of topological full groups within Steinberg algebras and C*-algebras, characterizing injectivity, surjectivity, and density properties, especially for discrete groupoids.

Contribution

It provides a precise characterization of when the natural representation is injective and explores its density properties in full and reduced C*-algebras for discrete groupoids.

Findings

Representation is injective under certain conditions.

Rarely surjective in general.

Image is not dense in full C*-algebra unless the groupoid is a group.

Abstract

We study the natural representation of the topological full group of an ample Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full and reduced C*-algebras. We characterise precisely when this representation is injective and show that it is rarely surjective. We then restrict our attention to discrete groupoids, which provide unexpected insight into the behaviour of the representation of the topological full group in the full and reduced groupoid C*-algebras. We show that the image of the representation is not dense in the full groupoid C*-algebra unless the groupoid is a group, and we provide an example showing that the image of the representation may still be dense in the reduced groupoid C*-algebra even when the groupoid is not a group.