In the world of von Neumann algebras, factors can be classified into three types. The type III factors are those that do not have a trace. They are related to nonsingular ergodic actions, regular representations of non-unimodular groups and quantum field theory. Some of the key structural properties of this class of factors are still not well understood. In this mini-course, I will give a gentle introduction to the theory of type III factors and to the deepest open problem in the theory : Connes's Bicentralizer Problem (1979).[-]

I will present a recent result in the theory of unitary representations of lattices in semi-simple Lie groups, which can be viewed as simultaneous generalization of Margulis normal subgroup theorem and C*-simplicity and the unique trace property for such lattices. The strategy of proof gathers ideas of both of these results: we extend Margulis' dynamical approach to the non-commutative setting, and apply this to the conjugation dynamical system induced by a unitary representation. On the way, we obtain a new proof of Peterson's character rigidity result, and a new rigidity result for uniformly recurrent subgroups of such lattices. I will give some basics on non-commutative ergodic theory and explain-some steps to prove the main result and its applications. This is based on joint works with Uri Bader, Cyril Houdayer, and Jesse Peterson.[-]