On the Construction of a Groupoid from an Ample Hausdorff Groupoid with Twisted Steinberg Algebra not Isomorphic to its Non-twisted Steinberg Algebra

Abstract

This study introduces an ample Hausdorff groupoid $\hat{A}⋊\mathcal{R}$ extracted from an ample Hausdorff groupoid $\mathcal{G}$ and a unital commutative ring $R$; a Hausdorff groupoid $D$ which is the discrete twist over $\hat{A}⋊\mathcal{R}$. In the groupoid C*-algebra perspective, when $R=\mathbb{C}$ there is an isomorphism between the non-twisted groupoid C*-algebra $(C^{\ast }(\mathcal{G}))$ and the twisted groupoid C*-algebra $(C^{\ast }(\hat{A}⋊\mathcal{R};D))$. However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra $(A_R(\mathcal{G}))$ and the twisted Steinberg algebra $(A_R(D;\hat{A}⋊\mathcal{R}))$ are non-isomorphic.