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

Abstract

AbstractGiven 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.