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

Abstract

We study the natural representation of the topological full group of an ample Hausdorff groupoid into the groupoid's complex Steinberg algebra. 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 extension of the representation of the topological full group into the full and reduced groupoid C*-algebras. We show that the extension into the full groupoid C*-algebra is not surjective unless the groupoid is a group, and we provide an example showing that the extension may still surject onto the reduced groupoid C*-algebra even when the groupoid is not a group.Comment: 12 page