Crossed Product Decompositions of a Purely Infinite Von Neumann Algebra with Faithful, Almost Periodic Weight

For M a separable, purely infinite von Neumann algebra with almost periodic weight φ, a decomposition of M as a crossed product of a semifinite von Neumann algebra by a trace–scaling action of a countable abelian group is given. Then Takasaki’s continuous decomposition of the same algebra is related to the above discrete decomposition via Takesaki’s notion of induced action, but here one induces up from a dense subgroup. The above results are used to give a model for the one–parameter trace–scaling action of R∗+ on the injective II∞ factor. Finally, another model of the same action, due to work of Aubert and explained by Jones, is described. Introduction. A crucial part of the present–day understanding of type III factors is their decomposition as crossed products of type II∞ von Neumann algebras by groups of trace–scaling (or trace–decreasing) automorphisms. This was accomplished by Connes in [4] and by Takesaki in [14]. Connes defined the classification of type III factors as type IIIλ, 0 ≤ λ ≤ 1 and showed that a type IIIλ factor where 0 ≤ λ < 1 is isomorphic to the crossed product of a type II∞ von Neumann algebra, N by a single automorphism (i.e. by the group Z). When λ > 0 N can be chosen to be a factor and the automorphism trace–scaling, and in the case λ = 0 the automorphism is ergodic on the center of N and a trace can be chosen such that the automorphism is strictly decreasing of the trace. Takesaki developed the theory of crossed products of a von Neumann algebra by actions of locally compact groups, including his duality theory and his theory of induced actions. He thereby proved the continuous decomposition for a factor M: if M is type III1 then M is the crossed product of a type II∞ factor N by a one–parameter group of trace–scaling automorphisms. Almost periodic weights (the definition is reviewed in §1.3) were defined by Connes in [4] and can be used to elucidate the structure of certain type III1 factors. Connes defined the invariant Sd for a full type III1 factor in terms of its almost periodic weights in [6], where he also showed that there is a type III1 factor having no almost periodic weights. However, many type III1 factors of interest have them. For example, the injective type III1 factor, which was shown to be unique by Haagerup [9], has many almost periodic weights, (cf §4). Also, the free products of certain finite dimensional algebras with respect to certain states that are not traces are known to be type III1 factors (see [3] and [7]) and the free product states on them are almost periodic. This work was supported by a National Science Foundation Postdoctoral Fellowship. Typeset by AMS-TEX 1