Braided quantum groups, actions and von Neumann algebras

The project is divided into three parts. In the first, we work on braided quantum groups. They are generalisations of quantum groups where the usual tensor product is replaced by a non-commutative version. The main task in this part of the project is to develop their theory in the language of von Neumann algebras and weights, generalizing the theory of locally compact quantum groups introduced by Kustermans and Vaes. Subsequently we study their modular and approximation properties. Furthermore, we introduce new classes of examples: q-deformations of compact semisimple Lie groups for complex q, and a class of non compact braided quantum groups related to nilpotent Lie groups. In the second part, we study approximation properties of quantum groups as well as approximation and structural properties of their group von Neumann algebras. If a discrete quantum group is unimodular, or more generally inner amenable, then there is a close relation between an approximation property of a discrete quantum group and its group von Neumann algebra. We investigate how inner amenability behaves under graph product construction. We also determine the structure of von Neumann algebras of automorphism groups of quantum graphs and certain property (T) quantum groups. In the last part, we study structure of von Neumann algebras given by homogenous spaces of quantum SU(2) group