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Non-traditional Cartan subalgebras in twisted groupoid C*-algebras

Anna Duwenig, Dana P. Williams, Joel Zimmerman

TL;DR

The paper develops a general criterion for when the image of an open subgroupoid's twisted C*-algebra $i(C^{*}_{r}(S;\mathcal{E}_{S}))$ is a Cartan subalgebra of the larger reduced twisted groupoid C*-algebra $C^{*}_{r}(G;\mathcal{E})$, extending Renault's Cartan correspondence to non‑effective groupoids via twists. It introduces the invariants $\Omega_{S}$ and $\operatorname{Ad}_{S}$ to quantify conjugacy behavior and uses interior conditions $\operatorname{int}_{\mathcal{E}}(\Omega_{S}^{-1}({1}))=\mathcal{E}_{S}$ and $\operatorname{int}_{\mathcal{E}}(\Omega_{S}^{-1}(\mathbf{Z}_{>1}))=\emptyset$, together with maximality among open subgroupoids of $\operatorname{int}_{G}(G')$, to characterize when $i(C^{*}_{r}(S;\mathcal{E}_{S}))$ is Cartan. A conditional-expectation map is constructed to realize the inclusion, and the results recover known theorems in the untwisted and cocycle cases while providing a unified framework for twisted groupoid Cartan theory and connections to Weyl groupoids. The work broadens Renault’s classification program and yields concrete criteria for identifying Cartan subalgebras in a broad class of twisted groupoid C*-algebras.

Abstract

Well-known work of Renault shows that if $\mathcal{E}$ is a twist over a second countable, effective, étale groupoid $G$, then there is a naturally associated Cartan subalgebra of the reduced twisted groupoid C*-algebra $C^*_{r}(G; E)$, and that every Cartan subalgebra of a separable C*-algebra arises in this way. However twisted C*-algebras of non-effective groupoids $G$ can also possess Cartan subalgebras: In work by the first author together with Gillaspy, Norton, Reznikoff, and Wright, sufficient conditions on a subgroupoid $S$ of $G$ were found that ensure that $S$ gives rise to a Cartan subalgebra in the cocycle-twisted C*-algebra of $G$. In this paper, we extend these results to general twists $\mathcal{E}$, and we refine the conditions on the subgroupoid for $C^*_{r}(S;\mathcal{E}_S)$ to be a Cartan subalgebra of $C^*_{r}(G;\mathcal{E})$.

Non-traditional Cartan subalgebras in twisted groupoid C*-algebras

TL;DR

The paper develops a general criterion for when the image of an open subgroupoid's twisted C*-algebra is a Cartan subalgebra of the larger reduced twisted groupoid C*-algebra , extending Renault's Cartan correspondence to non‑effective groupoids via twists. It introduces the invariants and to quantify conjugacy behavior and uses interior conditions and , together with maximality among open subgroupoids of , to characterize when is Cartan. A conditional-expectation map is constructed to realize the inclusion, and the results recover known theorems in the untwisted and cocycle cases while providing a unified framework for twisted groupoid Cartan theory and connections to Weyl groupoids. The work broadens Renault’s classification program and yields concrete criteria for identifying Cartan subalgebras in a broad class of twisted groupoid C*-algebras.

Abstract

Well-known work of Renault shows that if is a twist over a second countable, effective, étale groupoid , then there is a naturally associated Cartan subalgebra of the reduced twisted groupoid C*-algebra , and that every Cartan subalgebra of a separable C*-algebra arises in this way. However twisted C*-algebras of non-effective groupoids can also possess Cartan subalgebras: In work by the first author together with Gillaspy, Norton, Reznikoff, and Wright, sufficient conditions on a subgroupoid of were found that ensure that gives rise to a Cartan subalgebra in the cocycle-twisted C*-algebra of . In this paper, we extend these results to general twists , and we refine the conditions on the subgroupoid for to be a Cartan subalgebra of .
Paper Structure (8 sections, 23 theorems, 111 equations, 2 figures)

This paper contains 8 sections, 23 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Twists and their $C^{*}$-Algebras
  3. The Main Theorem
  4. The conditional expectation
  5. Maximality of $B$
  6. The proof of Theorem \ref{['thm:May17']}
  7. Corollaries and Applications
  8. Open questions

Key Result

Theorem A

Suppose $\mathcal{E}$ is a twist over a locally compact, Hausdorff, étale groupoid $G$, and $S$ is an open subgroupoid of $G$. Then the following statements are equivalent.

Figures (2)

  • Figure 1: We shrink the open bisections ( drawn with solid lines in color on the left) around the $S$-conjugates $g_{0},\ldots, g_{n}$ to make them disjoint and homeomorphic ( circular neighborhoods drawn with dashed lines ).
  • Figure 2: Conjugating $g_0(u)$ by $s_{i}(u)$ yields $g_i(u)$.

Theorems & Definitions (70)

  • Theorem A
  • Example 2.1
  • Remark 2.2: Pedantry
  • Lemma 2.3
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • Proposition 2.6: BFPR:GammaCartan*Proposition 2.8
  • Remark 2.7
  • ...and 60 more