Flow under a function and discrete decomposition of properly infinite W∗-algebras

Abstract

The purpose of this paper is to generalize the classical under a construction to non-abelian W/*-algebras. That is, given an automorphism θ of a H /*-algebra N and a positive self-adjoint operator φ affiliated to the centre of N we show how to construct a continuous action a of the reals on a W*-algebra M. The resulting covariant system {M, a, R} is called the flow built on {N, θ, Z} under the function φ. 1. Introduction. We obtain existence and uniqueness theorems for the representation of a given covariant system over the reals as a flow build under a function. As an application we generalize Connes' discrete decomposition theorems ([3] Theoreme 5.3.1 and Theoreme 5.4.2) using Takesaki's continuous decomposition theorems ([8], Theorem 8.1, Lemma 8.2 and Corollary 8.4). In §2 we fix notation and state some results on covariant systems. In §3 we define flow built under a function and give necessary and sufficient conditions for a covariant system over the reals to be isomorphic to a flow built under a function. §4 deals with the uniqueness problem. That is, we show the relationship between {Nλ,θvφλ} and {N2,θ2,φ2} when the corresponding flows are isomorphic. In §5 we derive discrete decomposition theorems for properly infinite ^*-algebras using Takesaki's continuous decomposition theorem and our results on flow built under a function. The results in this paper constitute the author's doctoral thesis written under the supervision of D. Bures. The author wishes to express his gratitude to Professor Bures.