In this talk, I will present a noncommutative Nevo-Zimmer theorem for actions of lattices in higher rank simple algebraic groups on von Neumann algebras. This extends to the realm of algebraic groups defined over arbitrary local fields the noncommutative Nevo-Zimmer theorem we obtained with Rémi Boutonnet in 2019 for real Lie groups.
I will discuss various applications of the above theorem to topological dynamics, unitary representations and operator algebras. I will also present a noncommutative analogue of Margulis’ factor theorem and discuss its relevance regarding Connes’ rigidity conjecture for group von Neumann algebras of higher rank lattices.
This is based on joint work with Uri Bader and Rémi Boutonnet (arXiv:2112.01337)