Non-traditional C*-diagonals in twisted groupoid C*-algebras

TL;DR

This paper characterizes conditions under which a reduced twisted C*-algebra from a non-traditional Cartan subalgebra in a twisted groupoid setting forms a C*-diagonal, expanding understanding of subgroupoid structures.

Contribution

It provides explicit criteria linking open normal subgroupoids to C*-diagonals in twisted groupoid C*-algebras, extending Kumjian-Renault theory to non-traditional Cartan subalgebras.

Findings

Identifies necessary and sufficient conditions for C*-diagonals

Describes Weyl groupoid and twist for non-traditional Cartan subalgebras

Extends Kumjian-Renault theory to new subgroupoid contexts

Abstract

We identify which conditions on an open normal subgroupoid of a LCH \'etale 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.