Conne’s bicentralizer problem and uniqueness of the injective factor of type III1

Abstract

In Connes' fundamental work Classification of injective [7], it is proved that injective factors of type III,t, 2 . 1 on a separable Hilbert space are completely classified by their smooth flow of weights. Since the flow of weights of factors of type III1 is trivial, one would expect that there is only one isomorphism class of injective factors of type IIIt. During the years 1976-78, Connes spent much effort to prove that there is only one injective factor of type III1, and found a number of conditions for an injective factor of type III1 to be isomorphic to the Araki-Woods' factor R~o. One of these conditions is the following: Let q0 be a normal faithful state on a yon Neumann algebra M, and let the bicentralizer of q0 be the set B. of operators a in M for which