Full k-simplicity of Steinberg algebras over Clifford semifields with application to Leavitt path algebras

Abstract

As a continuation of the study of the Steinberg algebra of a Hausdorff ample groupoid \({\mathcal {G}}\) over commutative semirings by Nam et al. (J. Pure Appl. Algebra 225, 2021), we consider here the Steinberg algebra \(A_S({\mathcal {G}})\) with coefficients in a Clifford semifield S. We obtain a complete characterization of the full k-ideal simplicity of \(A_S({\mathcal {G}})\). Using this result for the Steinberg algebra \(A_S({\mathcal {G}}_\Gamma )\) of the graph groupoid \({\mathcal {G}}_\Gamma \), where \(\Gamma \) is a row-finite digraph and S is a Clifford semifield, we characterize the full k-simplicity of the Leavitt path algebra \(L_S(\Gamma )\).