Classifying actions of groups on von Neumann algebras

Abstract

The objective of this paper is to provide an overview of results obtained recently by the author and M. Takesaki on the classification, up to cocycle conjugacy, of actions of discrete amenable groups on injective factor von Neumann algebras of type $III_{\lambda}, \lambda \neq 1$. Details will appear elsewhere, [14]. We also describe the conclusions of preceding work of Oceanu, [10] extending previous work of Connes [1], [2] and Jones [7], and of Sutherland and Takesaki, [13], on actions of discrete amenable groups and groupoids on injective semi finite von Neumann algebras.