Analysis Seminar: "Simplicity and the ideal intersection property for essential groupoid C*-algebras"
Abstract: Groupoid operator algebras are of important objects, since on the one hand they link the field of C*-algebra and von Neumann algebra to other areas like geometry and dynamics and on the other hand describe important structural features of operator algebras themselves. 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. Even in for Hausdorff groupoids, only partial results were known.
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 extends and unifies previous results among others on C*-simplicty of discrete groups, their topological dynamical systems and groupoids of germs.
Host: Xiang Tang