Representing topological full groups in Steinberg algebras and C*-algebras
Becky Armstrong, Lisa Orloff Clark, Mahya Ghandehari, Eun Ji Kang, Dilian Yang
TL;DR
This work investigates natural representations of the topological full group $F(\mathcal{G})$ of an ample Hausdorff groupoid $\mathcal{G}$ into the Steinberg algebra $A(\mathcal{G})$ and into the full and reduced groupoid C*-algebras. It provides precise necessary-and-sufficient conditions for when the associated map $\pi: \mathbb{C}F(\mathcal{G}) \to A(\mathcal{G})$ is injective or surjective, revealing that injectivity holds only in two narrow structural cases and surjectivity holds exactly when $\mathcal{G}$ is a group. In the discrete, finite-unit setting, the paper shows the full C*-algebra completion of the representation is an isomorphism iff $\mathcal{G}$ is a group, while in the reduced setting, density phenomena can occur even for non-group groupoids (e.g., $\mathcal{G}=\mathbb{F}_2\sqcup\mathbb{F}_2$). The results connect diagonal-preserving representations, Steinberg algebras, and groupoid C*-algebras, with implications for the structure of topological full groups and related operator-algebraic invariants.
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.