When the reduced twisted $C^*$-algebra $C^*_r({\mathcal{G}}, c)$ of a non-principal groupoid ${\mathcal{G}}$ admits a Cartan subalgebra, Renault’s work on Cartan subalgebras implies the existence of another groupoid description of $C^*_r({\mathcal{G}}, c)$. In an earlier paper, joint with Reznikoff and Wright, we identified situations where such a Cartan subalgebra arises from a subgroupoid ${\mathcal{S}}$ of ${\mathcal{G}}$. In this paper, we study the relationship between the original groupoids ${\mathcal{S}}, {\mathcal{G}}$ and the Weyl groupoid and twist associated to the Cartan pair. We first identify the spectrum ${\mathfrak{B}}$ of the Cartan subalgebra $C^*_r({\mathcal{S}}, c)$. We then show that the quotient groupoid ${\mathcal{G}}/{\mathcal{S}}$ acts on ${\mathfrak{B}}$, and that the corresponding action groupoid is exactly the Weyl groupoid of the Cartan pair. Lastly, we show that if the quotient map ${\mathcal{G}}\to{\mathcal{G}}/{\mathcal{S}}$ admits a continuous section, then the Weyl twist is also given by an explicit continuous $2$-cocycle on ${\mathcal{G}}/{\mathcal{S}} \ltimes{\mathfrak{B}}$.

中文翻译：

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分析动态Cartan子代数的Weyl构造

当非主群 ${\mathcal{G}}$ 的简化扭曲 $C^*$-代数 $C^*_r({\mathcal{G}}, c)$ 承认 Cartan 子代数时，雷诺的工作在 Cartan 子代数上意味着存在 $C^*_r({\mathcal{G}}, c)$ 的另一个群描述。在与 Reznikoff 和 Wright 合作的早期论文中，我们确定了这样的 Cartan 子代数来自 ${\mathcal{G}}$ 的子群 ${\mathcal{S}}$ 的情况。在本文中，我们研究了原始群群 ${\mathcal{S}}、{\mathcal{G}}$ 与与 Cartan 对相关的外尔群群和扭曲之间的关系。我们首先确定 Cartan 子代数 $C^*_r({\mathcal{S}}, c)$ 的谱 ${\mathfrak{B}}$。然后我们证明商群 ${\mathcal{G}}/{\mathcal{S}}$ 作用于 ${\mathfrak{B}}$，并且对应的动作群正是嘉当对的外尔群。最后，我们证明如果商映射 ${\mathcal{G}}\to{\mathcal{G}}/{\mathcal{S}}$ 承认连续部分，则外尔扭曲也由显式给出${\mathcal{G}}/{\mathcal{S}} \ltimes{\mathfrak{B}}$ 上的连续 $2$-cocycle。