Ideals of Steinberg algebras of strongly effective groupoids, with applications to Leavitt path algebras

Abstract

We consider the ideal structure of Steinberg algebras over a commutative ring with identity. We focus on Hausdorff groupoids that are strongly effective in the sense that their reductions to closed subspaces of their unit spaces are all effective. For such a groupoid, we completely describe the ideal lattice of the associated Steinberg algebra over any commutative ring with identity. Our results are new even for the special case of Leavitt path algebras; so we describe explicitly what they say in this context, and give two concrete examples.

Condition (K) for inverse semigroups and the ideal structure of their C⁎-algebras

“…Our first result in this direction is a direct translation of [5,Theorem 3.1] to the context of the present paper.…”

“…First we show that Condition (K) is related to the property that G tight (S) is strongly effective, as in the case of the path groupoid of a directed graph. Our goal now is to describe the ideals in C * (G tight (S)) and the basic ideals in the Steinberg algebras A R (G tight (S)), where R is any commutative ring with identity, in terms of saturated ideals in S. Our analysis involves a direct application of the main results of [2] and [5]. Since we will not work directly with Steinberg algebras in this paper, we refer the reader to those papers for definitions.…”

“…The second, Condition (K), is that every vertex that is the base of a cycle has at least two distinct return paths. This condition is equivalent to the property that the reduction of the path groupoid to any closed invariant subset of its unit space is effective [5,Corollary 6.5]. That is, Condition (K) is equivalent to the property that the path groupoid is strongly effective.…”

Prime étale groupoid algebras with applications to inverse semigroup and Leavitt path algebras

“…It is well known that G E is effective if and only if each cycle has an exit, cf. [16]. Since many of the references assume that E is countable or rowfinite, we shall prove it here.…”

“…A basis of compact open subsets for the topology on G There is an isomorphism L R (E) −→ RG E sending v ∈ E (0) to the characteristic function of Z(ε v , ε v ) and, for e ∈ E (1) , sending e to the characteristic function of Z(e, ε r(e) ) and e * to the characteristic function of Z(ε r(e) , e), cf. [13,16,18,39] or [19,Example 3.2].…”