Minimal Inclusions of C*-algebras of Groupoids

A non-degenerate inclusion of C*-algebras B ⊂ A is considered minimal if the only fully normalised ideals are trivial. An ideal is fully normalised if the set of normalisers of B are contained in the set of normalisers of the ideal. We begin by considering when the inclusions C₀(G⁽⁰⁾) ⊂ Cᵣ*(G) and C₀(G⁽⁰⁾) ⊂ C*(G) are minimal inclusions for an étale groupoid G. Then we consider étale groupoids graded by a cocycle c: G → Γ for a discrete group Γ, writing Gₑ for the identity-graded subgroupoid. We investigate the inclusions Cᵣ*(Gₑ) ⊂ Cᵣ*(G) and C*(Gₑ) ⊂ C*(G) of groupoid C*-algebras. We give conditions on the groupoids G and Gₑ under which the inclusions are minimal.

We generalise these results by considering non-degenerate inclusions of twisted groupoid C*-algebras and graded twisted groupoid C*-algebras. We finds conditions on G (and Gₑ for the twisted graded inclusion), which make the inclusions minimal. We conclude by applying our results to higher rank-graphs. In their 2020 paper Crytser and Nagy found simplicity criteria for the ambient C*-algebras depending on the type of inclusion. We use this criteria to show how some of our results align literature results.