Title: Boundary actions, traces and C*-simplicity of quasi-regular representations
Abstract: In the last 6 years, there have been great advances in understanding when the reduced C*-algebra of a discrete group is simple and when it has the unique trace property. A central role in these discoveries has been played by Furstenberg's notion of boundary action (specially the so called Furstenberg boundary), and Hamana's notion of injective envelopes.
For quasi-regular representations in general, the situation is much different, with Haagerup and Olesen showing that there is such a representations π of Thompson's group V such that C*π(V) is the Cuntz algebra O2 (hence nuclear, simple and without traces).
In this talk, we will review the notions above, and present applications of boundary actions to the study of traces and C*-simplicity of quasi-regular and Koopman representations. This is joint work with Mehrdad Kalantar.