Boundary Actions and Applications in Operator Algebras

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The theory of Operator Algebras was developed in the 1930s as the mathematical foundation of quantum mechanics. This theory also provides the natural mathematical framework in which physical systems as they evolve in time (called dynamical systems) can be represented and studied; Groups are the algebraic structures that represent the time in dynamical systems. Today, each of the theories of Operator Algebras, Groups, and Dynamical Systems are independently of each other among most important and well-established parts of modern mathematics and mathematical physics. But the framework of operator algebras still allows deep interactions between these mathematical concepts. The aim of this project is to exploit the existing, and develop new bridges between these different theories. This allows one to utilize the current advanced mathematical technology of each area to help overcome some open problems in the others.This project is concerned with various aspects of interaction between analytic group theory (and its quantum version) and operator algebras, particularly via the theory of boundaries. Measure-theoretical boundaries (e.g. the Poisson boundary), and topological boundaries (e.g. the Furstenberg boundary) of groups were introduced and developed in the 1960s and 1970s in the seminal work of Furstenberg. These concepts were used as a tool to prove certain rigidity results for lattices in Lie groups. The former type of boundaries have since been vastly investigated and used as the main tool in some of the most substantial results in the past few decades in both ergodic theory of groups and rigidity theory of von Neumann algebras. However, topological boundaries are much less understood, and surely have not been fully exploited as a similar powerful tool in continuous ergodic theory or C*-algebraic rigidity problems. These boundaries are defined abstractly in general for all discrete (and also non-discrete) groups, and they compromise all natural notions of boundaries such as Gromov's boundaries of hyperbolic groups or Furstenberg boundaries of semisimple Lie groups. The project aims to further develop the understanding of such boundaries. The goal is to apply them to generalize various results in operator algebra theory that have been proven by using special cases of such boundary actions, for example, the existing results concerning maximal injective von Neumann subalgebras and character-rigidity. In another direction, the project aims to develop and study the boundary theory of quantum groups and investigate their possible applications to problems such as C*-simplicity in the quantum setting.