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

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 C_r ^⁎(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 C_r ^⁎(S,c) is Cartan in C_r ^⁎(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.