Flows of Weights of Crossed Products of Type III Factors by Discrete Groups

Abstract

In [17] C. Sutherland and M. Takesaki classified (up to cocycle conjugacy) actions of discrete amenable groups on injective factors of type III^, 0^/i<l. For the unique injective factor of type IIIi ([7]), similar classification was recently completed for discrete abelian groups ([14]). This classification is based on four invariants: a certain normal subgroup, the module, the characteristic invariant, and the modular invariant. Cocycle conjugate actions give rise to isomorphic crossed products, and the isomorphism class of an injective type III factor is known to be determined by its flow of weights. Therefore the flow of weights of the crossed product Mx aG should be uniquely determined by the four invariants of the action a. The purpose of this article is to compute the flow of weights of Mxi aG explicitly in the case where M is a type III factor and G is a discrete group. This means that we have to compute the center of the crossed product Mx«G by R relative to a modular automorphism group. Generally a continuous crossed product is difficult to handle because a general element does not admit a Fourier expansion. This difficulty will be avoided by using the canonical extension a