Graded quasi-Baer $\ast$-ring characterization of Steinberg algebras

TL;DR

This paper characterizes when Steinberg algebras associated with graded ample Hausdorff groupoids are graded quasi-Baer *, linking algebraic properties to groupoid conditions.

Contribution

It provides necessary and sufficient conditions on groupoids for Steinberg algebras to be graded quasi-Baer *, a property previously not fully understood.

Findings

Characterization of graded quasi-Baer * Steinberg algebras

Conditions on groupoids for algebraic properties

Application to graded Leavitt path algebras

Abstract

Given a graded ample, Hausdorff groupoid $G$, and an involutive field $K$, we consider the Steinberg algebra $A_K(G)$. We obtain necessary and sufficient conditions on $G$ under which the annihilator of any graded ideal of $A_K(G)$ is generated by a homogeneous projection. This property is called graded quasi-Baer $\ast$. We use the Steinberg algebra model to characterize graded quasi-Baer $\ast$ Leavitt path algebras.