A structure theorem for asymptotically abelian W*-dynamical systems

Abstract

We prove a partial non-commutative analogue of the Furstenberg-Zimmerman Structure Theorem, originally proved by Tim Austin, Tanya Eisner and Terence Tao. In Chapter 1, we review the GNS construction for states on von Neumann algebras and the related semicyclic representation for tracial weights. We look at Tomita- Takasaki theory in the special case of traces. This will allow us to introduce the Jones projection and conditional expectations of von Neumann algebras. We then de ne the basic construction and its associated nite lifted trace. We also introduce the notion of projections of nite lifted trace and how they relate to right submodules. Chapter 2 introduces dynamics in the form of automorphisms on von Neumamnn algebras. We will see how the dynamics is represented on the GNS Hilbert space using a cyclic and separating vector. It is then shown how the dynamics is extended to the basic construction and the semicyclic representation. The last three chapters form the \core". At the beginning of each aforementioned chapter, we present a summary of the required theory, before providing detailed proofs. In Chapter 3, we prove one of two \fundamental lemmas" where we introduce some non-commutative integration theory. We use a version of the spectral theorem expressed in terms of a spectral measure to produce a certain projection of nite lifted trace. In Chapter 4, we prove our next fundamental lemma. We use direct integral theory in order to obtain a representation of the dynamics, in terms of a module basis, on the image of the projection of nite lifted trace. In Chapter 5, we apply our previous results to asymptotically abelian W*-dynamical systems, culminating in the proof of the titular theorem.