Centrally Free Actions of Amenable C ^∗ -Tensor Categories on von Neumann Algebras

We will show a centrally free action of an amenable rigid C ^∗-tensor category on a properly infinite von Neumann algebra has the Rohlin property. Our main result is the classification of centrally free cocycle actions of an amenable rigid C ^∗-tensor category up to approximate inner conjugacy on properly infinite von Neumann algebras. This is regarded as a generalization of classification of amenable discrete groups due to A. Connes, V. Jones and A. Ocneanu. We have the following two applications: a classification of centrally free actions of amenable discrete quantum groups of Kac type on von Neumann algebras and another proof of S. Popa’s celebrated classification result of amenable subfactors. As another application of the Rohlin property, we will prove the fullness of the crossed product of a full factor by a minimal action of a compact group.