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

Abstract

Abstract Given an ample, Hausdorff groupoid 𝒢 {\mathcal{G}} , and a unital commutative ring R, we consider the Steinberg algebra A R ⁢ ( 𝒢 ) {A_{R}(\mathcal{G})} . First we prove a uniqueness theorem for this algebra and then, when 𝒢 {\mathcal{G}} is graded by a cocycle, we study graded ideals in A R ⁢ ( 𝒢 ) {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.