Cartan subalgebras for non-principal twisted groupoid C^⁎-algebras

2020 Elsevier Inc. Renault proved in 2008 [22, Theorem 5.2] that if G is a topologically principal groupoid, then C0(G(0)) is a Cartan subalgebra in Cr⁎(G,Σ) for any twist Σ over G. However, there are many groupoids which are not topologically principal, yet their (twisted) C⁎-algebras admit Cartan subalgebras. This paper gives a dynamical description of a class of such Cartan subalgebras, by identifying conditions on a 2-cocycle c on G and a subgroupoid S⊆G under which Cr⁎(S,c) is Cartan in Cr⁎(G,c). When G is a discrete group, we also describe the Weyl groupoid and twist associated to these Cartan pairs, under mild additional hypotheses.