Simplicity and the ideal intersection property for essential groupoid C*-algebras
To every étale groupoid with locally compact Hausdorff space of units, one can associate an essential groupoid C*-algebra, which is a suitable quotient of the reduced groupoid C*-algebra by an ideal of singular elements. For Hausdorff groupoids, it equals the reduced groupoid C*-algebra. Until recently, it had been an open question to characterise simplicity of such essential groupoid C*-algebras.
In this talk, I will report on joint work with Matthew Kenney, Se-Jin Kim, Xin Li and Dan Ursu, which characterises étale groupoids with locally compact Hausdorff space of units whose essential groupoid C*-algebra is has the ideal intersection property. Our characterisation is phrased in terms of what is called essentially confined amenable sections of isotropy groups, a notion that can be checked in concrete cases. This provides a complete solution of the open problem, combining the ideal intersection property with the dynamical requirement of minimality. In particular, it comes as a surprise that non-Hausdorff groupoids fit well into this general picture. Our work completes, extends and unifies previous results concerning C*-simplicty of topological dynamical systems (Kawabe), Hausdorff groupoids with compact space of units (Borys) and groupoids of germs (Kalantar-Scarparo). Even the notion of essential groupoid C*-algebras for non-Hausdorff groupoids was only developed recently (Kwaśniewski-Meyer).
An interesting consequence of our work is the fact that a relative Powers averaging property can be proven. To establish this result, the full extend of our work is necessary. As an application we prove relative Powers averaging property for suitable unitary representations of Thompson's group T into the Cuntz algebra O_2.