The unitary implementation of a locally compact quantum group action

Abstract

In this paper we study actions of locally compact quantum groups on von Neumann algebras and prove that every action has a canonical unitary implementation, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the study of quantum groups in action. We will use it in this paper to study subfactors and inclusions of von Neumann algebras. When alpha is an action of a locally compact quantum group on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion of the fixed point algebra in the algebra N in the crossed product, is a basic construction. When alpha is an outer and integrable action on a factor N we prove that the inclusion of the fixed point algebra in the algebra N is irreducible, of depth 2 and regular, giving a converse to the results of Enock and Nest. Finally we prove the equivalence of minimal and outer actions and we generalize a theorem of Yamanouchi: every integrable outer action with infinite fixed point algebra is a dual action.