Non-Commutative Ergodic Theory

Abstract

The structure of a factor M is best understood through the study of symmetry of the factor, i.e. the study of the group Aut(M.) of automorphisms of M. We have been experiencing this through the structure analysis of factors of type III for instance. Apart from the modular automorphism groups, we do not have a systematic way of constructing an automorphism of a given factor M. It is still unknown if every separable factor of type II₁ admits an outer automorphism. Thus we restrict ourselves to AFD factors in most cases, where we have many different ways of constructing automorphisms. The counter part in analysis of the theory of automorphisms is ergodic theory or the theory of non-singular transformations on a σ-finite standard measure space. As we have seen in Chapter XIII, the Rokhlin’s tower theorem played a fundamental role in the theory of AF measured groupoids. We will present first the non-commutative analogue of this basic result in ergodic theory in §1. Unlike other parts, we need this theory for non-separable von Neumann algebras. It is interesting to note that whilst our primary interests are rest upon separable factors some results valid for non-separable von Neumann algebras are badly needed to advance our separable theory. One might be tempted to have a philosophical discussion about this irony. The results there will be applied to the analysis of outer conjugacy of single automorphisms in subsequent sections.