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

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