Non-traditional C*-diagonals in twisted groupoid C*-algebras
Anna Duwenig
TL;DR
The paper develops a framework to understand non-traditional Cartan subalgebras arising from open normal subgroupoids $S$ inside étale groupoids with twists, and identifies when the corresponding reduced twisted C*-algebras form Cartan subalgebras or C*-diagonals in the ambient algebra. It gives a concrete description of the Weyl groupoid $W_{B\subset A}$ and Weyl twist $\\Sigma_{B\subset A}$ entirely in terms of the input data $\\uE \to G \\unrhd S$, showing that $W_{B\subset A}$ is the action groupoid $(G/S) \\ltimes \\\widehat{S}^{\\uE}$ and that $\\Sigma_{B\subset A}$ is a quotient twist of $(\\uE \\ltimes \\\widehat{S}^{\\uE})$ by a normal subgroupoid. The main results yield precise necessary and sufficient conditions for $C^{*}_{r}(S;\\uE_{S})$ to be a C*-diagonal, including abelianness and closure of the restricted twist and a commutation-type condition for elements outside $\\uE_{S}$. Consequences include characterizations of when Weyl twists are trivial, and specialized outcomes for untwisted data, with several open problems pointing to extensions beyond second countability and broader groupoid settings.
Abstract
We identify which conditions on an open normal subgroupoid of a LCH étale groupoid with twist are necessary and sufficient for the subgroupoid's reduced twisted C*-algebra to be a C*-diagonal in the ambient groupoid C*-algebra. We do so by first giving an explicit description of the Weyl groupoid and Weyl twist associated to any non-traditional Cartan subalgebra, that is, a Cartan subalgebra that is induced from a non-trivial open normal subgroupoid, as studied in [DWZ2025]. We then combine this description with Kumjian-Renault theory to establish the necessary and sufficient conditions to get a C*-diagonal.