June 28th 2017; 12:00 auditorium of the CRM
Amenability and actions on strongly self-absorbing C*-algebras
Abstract: Amenability for discrete groups admits a surprisingly long list of equivalent formulations. Some of them are purely group-theoretic (like the Følner con- dition), some are measure-theoretic (existence of a left invariant mean), and some are phrased in terms of operator algebras (hyperfiniteness of the group von Neumann algebra, or nuclearity of the reduced group C∗ -algebra). Here, we are interested in certain characterizations of amenability that are related to actions of the group. These characterizations usually come in the form of a dichotomy: roughly speaking, they assert that there is an object in the relevant category (like the standard probability space, the hyperfinite II 1 -factor, etc), on which every amenable group acts in an essentially unique way, while every nona- menable group admits a continuum of non-equivalent actions (and the equivalence relation of such actions is not even Borel). In this talk, we will discuss the possibility of characterizing of amenability in terms of actions on strongly self-absorbing C∗ -algebras, with focus on the nonamenable part of the dichotomy. Our main result asserts that any group with property (T), which is a strong form of nonamenability, admits uncountably many non-cocycle equivalent actions on any UHF-algebra (of infinite type), and that this relation is not Borel. This is joint work with Martino Lupini