On the Construction of a Groupoid from an Ample Hausdorff Groupoid with Twisted Steinberg Algebra not Isomorphic to its Non-twisted Steinberg Algebra
DOI:
https://doi.org/10.29020/nybg.ejpam.v17i1.5051Keywords:
Groupoids, Steinberg Algebra, Twisted Steinberg Algebra, Non-twisted Steinberg AlgebraAbstract
This study introduces an ample Hausdorff groupoid $\hat{A} \rtimes \mathcal{R}$ extracted from an ample Hausdorff groupoid $\mathcal{G}$ and a unital commutative ring $R$; a Hausdorff groupoid $D$ which is the discrete twist over $\hat{A} \rtimes \mathcal{R}$. In the groupoid C*-algebra perspective, when $R = \mathbb{C}$ there is an isomorphism between the non-twisted groupoid C*-algebra $(C^*(\mathcal{G}))$ and the twisted groupoid C*-algebra $(C^*(\hat{A} \rtimes \mathcal{R};D))$. However, in this paper, in a purely algebraic setting, the non-twisted Steinberg algebra $(A_R(\mathcal{G}))$ and the twisted Steinberg algebra $(A_R(D; \hat{A} \rtimes \mathcal{R}))$ are non-isomorphic.
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