The unitary implementation of a locally compact quantum group action

In this paper we study actions of locally compact quantum groups on von Neu mann algebras and prove that ever action has a canonical unitary implementa tion, paralleling Haagerup's classical result on the unitary implementation of a locally compact group action. This result is an important tool in the stud of quantum groups in action. We will use it in this paper to study su bfactors and inclusions of von Neumann algebras. When alpha is an action of the locally compact quantum group (M, Delta) on the von Neumann algebra N we can give necessary and sufficient conditions under which the inclusion N (alpha)subset ofN(subset of)-->M(alpha)xN is a basic construction. Here N-a lpha denotes the fixed pont algebra and M(alpha)xN is the crossed product. When alpha is an outer and integrable action on a factor N we prove that th e inclusion N(alpha)subset ofN is irreducible, of depth 2 and regular, givi ng a converse to the results of M. Enock and R. Nest (1996, J Funct. Anal. 137, 466 543; 1998, J. Funct. Anal. 154. 67-109). Finally we prove the equi valence of minimal and outer actions and we generalize the main theorem of Yamanouchi (1999, Math. Scand. 84, 297-319); every integrable outer action with infinite fixed point algebra is a dual action. (C) 2001 Academic Press .