A Generalised uniqueness theorem and the graded ideal structure of Steinberg algebras

TL;DR

This paper establishes a general uniqueness theorem for Steinberg algebras of ample, Hausdorff groupoids and explores their graded ideal structure, with applications to groupoid constructions from graph actions and Boolean dynamical systems.

Contribution

It introduces a broad uniqueness theorem for Steinberg algebras and analyzes the structure of graded ideals in these algebras for specific classes of groupoids.

Findings

Proved a general uniqueness theorem for Steinberg algebras.

Characterized graded ideals in Steinberg algebras for graded groupoids.

Applied results to groupoids from graph actions and Boolean dynamical systems.

Abstract

Given an ample, Hausdorff groupoid $\mathcal{G}$, and a unital commutative ring $R$, we consider the Steinberg algebra $A_R(\mathcal {G})$. First we prove a uniqueness theorem for this algebra and then, when $\mathcal{G}$ is graded by a cocycle, we study graded ideals in $A_R(\mathcal {G})$. Applications are given for two classes of ample groupoids, namely those coming from actions of groups on graphs, and also to groupoids defined in terms of Boolean dynamical systems.