Noncommutative ergodic theory of higher rank lattices

Abstract

We survey recent results regarding the study of dynamical properties of the space of positive definite functions and characters of higher rank lattices. These results have several applications to ergodic theory, topological dynamics, unitary representation theory and operator algebras. The key novelty in our work is a dynamical dichotomy theorem for equivariant faithful normal unital completely positive maps between noncommutative von Neumann algebras and the space of bounded measurable functions defined on the Poisson boundary of semisimple Lie groups.

arXiv comment22 pages; Prepared for the Proceedings of the ICM 2022

References in corpus (7)

• Operator-algebraic superrigidity for $S{L}_n(\mathbb{Z}),n\ge 3$
• Complete descriptions of intermediate operator algebras by intermediate extensions of dynamical systems
• Charmenability of arithmetic groups of product type
• Stationary characters on lattices of semisimple Lie groups
• Stationary C*-dynamical systems
• Charmenability of higher rank arithmetic groups
• Characters of the group ${\mathrm{EL}}_d(R)$ for a commutative Noetherian ring $R$

Cited by in corpus (4)

• The noncommutative factor theorem for lattices in product groups
• Wreath-like products of groups and their von Neumann algebras I: $W^{\ast }$-superrigidity
• Operator algebraic characterization of the noncommutative Poisson boundary
• Crossed product splitting of intermediate operator algebras via 2-cocycles