  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.

Operator Algebras;Quantum Algebra