Analyzing the Weyl Construction for Dynamical Cartan Subalgebras

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