On the socle of a class of Steinberg algebras

TL;DR

This paper investigates the structure of minimal left ideals in Steinberg algebras of Hausdorff groupoids, linking algebraic properties to topological features of the groupoid's unit space, and extends known results to more general graph algebras.

Contribution

It establishes a new relationship between minimal left ideals and open singleton sets in the unit space, broadening the understanding of the socle in Steinberg algebras, including higher-rank graph cases.

Findings

Characterizes minimal left ideals via open singleton sets

Provides new results on the socle of Steinberg algebras

Extends known algebraic results to higher-rank graphs

Abstract

We study minimal left ideals in Steinberg algebras of Hausdorff groupoids. We establish a relationship between minimal left ideals in the algebra and open singletons in the unit space of the groupoid. We apply this to obtain results about the socle of Steinberg algebras under certain hypotheses. This encompasses known results about Leavitt path algebras and improves on Kumjian-Pask algebra results to include higher-rank graphs that are not row-finite.