Modular Automorphism Groups

Abstract

As we have seen in the previous chapters, the modular operator gives rise to a one parameter automorphism group on the von Neumann algebra in question. First, we are going to identify this group. The first section is devoted to this task. This group is characterized by the modular condition, Definition 1.1. Unlike the classical integration theory, the non-commutative theory comes with the intrinsic dynamics of the modular automorphism group. This chapter is devoted to the elementary theory of the modular automorphism groups. In fact, the modular automorphism group is the key to the entire theory of this volume. Thus, the theory does not end in this chapter. It will continue to the rest of this book. Section 1 is devoted to the characterization of the modular automorphism group of a weight by the modular condition, Theorem 1.2. In Section 2, the analysis and the characterization of the fixed point algebra under the modular automorphism group, called the centralizer, of a weight are given. We also discuss the perturbation of a weight by an operator affiliated to the centralizer of the weight, which is needed in the following section. The classical Radon-Nikodym theorem in the integration theory is extended to the non-commutative setting in a surprisingly simple manner, Theorem 3.3. In the course of proving the theorem, we will see that the semi-cyclic representation of a von Neumann algebra induced by a faithful semi-finite normal weight is unique up to unitary equivalence, Theorem 3.2. One of the consequences of Theorem 3.3 is that the modular automorphism group is unique up to unitary one cocycles, hence their images in the quotient group Out(M) = Aut(M)/Int(.M) is an intrinsic quantity of the algebra. Many results follow from this theorem. Only immediate consequences are discussed in this section, and many others are spread to the rest of this book. The semi-finiteness of the algebra is then equivalent to the innerness of the modular automorphism group, Theorem 3.14.