Orientations and von Neumann algebras

Abstract

As was discussed in the introduction to Chapter 5, the Jordan product \(a^\circ b = \frac{1}{2}\left( {ab + ba} \right)\) does not determine the associative product ab. In the case of C*-algebras, we showed that a notion of orientation of the state space is what is needed to determine the possible associative products compatible with the given Jordan product (Theorem 5.73). That notion does not apply directly to normal state spaces of von Neumann algebras, since it is defined in terms of pure states, and the normal state space of a von Neumann algebra might have no pure states. We will therefore introduce the notion of orientation in a different way. The theme of this chapter is the interplay between Jordan compatible associative products in a von Neumann algebra and global orientations of the algebra and its normal state space.