Sven Raum: Simplicity of essential groupoid C*-algebras

Abstract:

Groupoids merge the notion of space and group, yielding a well-adapted framework to study dynamical systems from an algebraic and analytic perspective. For decades, operator algebraists have studied various C*-algebras and von Neumann algebras associated to groupoids, because they provide a bridge to other areas like geometry and dynamics and describe important structural features of operator algebras themselves. In the focus of C*-algebraists, often are étale groupoids, which generalise the notion of discrete groups.

To every étale groupoid with locally compact Hausdorff space of units, one can associate an essential groupoid C*-algebra. For Hausdorff groupoids, it equals the better known reduced groupoid C*-algebra. Until recently, it had been an open problem to characterise simplicity of such essential groupoid C*-algebras. Even 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 simple. 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. 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*-simplicity of discrete groups, their topological dynamical systems and groupoids of germs.

I will keep this talk accessible for an audience of non-experts, starting with a motivation and explanation of groupoids themselves, before discussing operator algebraic aspects of our work.
 

Note: The passcode was sent to the AG and NT mailing lists. If you're not on these lists and would like to attend, or are having trouble accessing the meeting, please email Sven Raum at sven.raum@math.su.se . To be added to the AG mailing list, please email Jonas Bergström at jonasb@math.su.se .