We construct a family of operator systems and k-AOU spaces generated by a finite number of projections satisfying a set of linear relations. This family is universal in the sense that the map sending the generating projections to any other set of projections that satisfy the same relations is completely positive. These operator systems are constructed as inductive limits of explicitly defined operator systems. By choosing the linear relations to be the non-signaling relations from quantum correlation theory, we obtain a hierarchy of ordered vector spaces dual to the hierarchy of quantum correlation sets. By considering another set of relations, we also find a new necessary condition for the existence of a SIC-POVM.

I will survey recent results regarding the dynamics of positive definite functions and character rigidity of irreducible lattices in higher-rank semisimple algebraic groups. These results have several applications to ergodic theory, topological dynamics, unitary representation theory, and operator algebras. In the case of lattices in higher-rank simple algebraic groups, I will explain the key operator algebraic novelty, which is a noncommutative Nevo-Zimmer theorem for actions on von Neumann algebras. I will also present a noncommutative Margulis' factor theorem and discuss its relevance regarding Connes' rigidity conjecture for group von Neumann algebras of higher-rank lattices.