On some group theoretic, ergodic theoretic and operator algebraic aspects of compact quantum groups[HBNI Th104]

Abstract

In this thesis, we make study compact quantum groups from the viewpoint of group theory, ergodic theory and operator algebras. We define and study the notion of inner automorphisms of compact quantum groups and topological properties of the group of inner automorphisms of a compact quantum group. We show the stability of any normal subgroup of a compact quantum group under the action of an inner automorphism but show that the converse is false. We then define and study the center of compact quantum groups and compute it for several examples. This is followed by a study of group actions on compact quantum groups, where the action is by quantum group automorphisms. We investigate spectral properties of such non-commutative dynamical systems, which we call CQG dynamical systems, like ergodicity, weak mixing, etc and obtain combinatorial characterizations of such spectral properties, which enables us to construct examples which possess these properties. Under mild conditions, we also show the existence and uniqueness of a maximal ergodic normal subgroup of given CQG dynamical systems. Finally, we make a comprehensive study of the bicrossed product and the crossed product quantum groups, their representation theory and approximation properties possessed by such quantum groups like Haagerup property, weak amenability, etc. We discuss several examples and in particular, construct an infinite family of non-isomorphic quantum groups possessing Property (T).