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

Abstract

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

Graded Steinberg algebras and partial actions

“…Let G be a G-graded ample Hausdorff groupoid such that G ε is strongly effective. It was proved in [12,Theorem 5.3] that the correspondence U −→ A K (G| U ) is an isomorphism from the lattice of open invariant subsets of G (0) to the lattice of graded ideals in A K (G).…”

“…Recall that when G is an ample, Hausdorff groupoid with a continuous cocycle c : G − → G and c −1 (ε) is strongly effective, there is a one-to-one correspondence between the open invariant subsets of G (0) and the graded ideals of the Steinberg algebra A K (G) of G (see [12,Theorem 5.3]). We observe that the associated groupoid G X is an ample, Hausdorff groupoid if X is locally compact Hausdorff with a basis of compact open sets.…”

Simplicity of skew inverse semigroup rings with applications to Steinberg algebras and topological dynamics

“…Lately, Steinberg algebras have attracted a lot of attention, partly since they include all Kumjian-Pask algebras of higher-rank graphs introduced in [1] and therefore also all Leavitt path algebras. For some examples of the theory of Steinberg algebras, we refer the reader to [9,11,14,34].…”