The characteristic square of a factor and the cocycle conjugacy of discrete group actions on factors

Abstract

This paper has three main purposes; ®rst, to associate to any separable factor M a canonical commuting exact square of groups called the characteristic square ofM; secondly, to formulate at least in a preliminary way, a methodology for approaching ``non-smooth'' classi®cation problems in analysis; and thirdly to use this approach in conjunction with the characteristic square to complete the cocycle conjugacy classi®cation of actions of discrete amenable groups on injective factors of type III1. This result, and previous works on the cocyle conjugacy classi®cation problem combine to produce a cocycle conjugacy classi®cation theorem which applies simultaneously to all countable amenable groups and all injective separable factors, and in which the characteristic square and our new methodology play a crucial role. The cocycle conjugacy classi®cation problem has by now a long and distinguished history, beginning with Connes' ground-breaking outer conjugacy classi®cation of single automorphisms of the approximately ®nite dimensional, or AFD, factor R0 of type II1, [C3]. In the intervening 20 years, enormous progress has been made. The ®rst step was taken by V. Jones, [J], who identi®ed the true nature of the cocycle conjugacy invariants, and completed the classi®cation of actions of ®nite groups on R0. Ocneanu, [O], re®ned this work and introduced new analytic techniques, enabling him to extend the classi®cation to countable amenable groups acting on the semi-®nite AFD factor R0;1, (and to some actions on AFD factors of type III also). Sutherland and Takesaki, using groupoid and duality techniques originating in [JT], extended the classi®cation results to Invent. math. 132, 331±380 (1998)