Analyzing the Weyl construction for dynamical Cartan subalgebras

TL;DR

This paper explores the structure of Cartan subalgebras within reduced twisted groupoid $C^*$-algebras, revealing how the Weyl groupoid and twist relate to the original groupoid and subgroupoid, especially under certain conditions.

Contribution

It establishes a detailed relationship between the original groupoids, the Weyl groupoid, and the twist, including explicit descriptions when a continuous section exists.

Findings

The spectrum of the Cartan subalgebra is identified as a specific space $rak{B}$.

The quotient groupoid $rak{G}/rak{S}$ acts on $rak{B}$, forming the Weyl groupoid.

The Weyl twist can be explicitly described by a continuous 2-cocycle under certain conditions.

Abstract

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 $\mathcal{G}/\mathcal{S} \ltimes \mathfrak{B}$.