Types of von Neumann Algebras and Traces

The material presented in this chapter is directly related to the dimension theory of Murray and von Neumann. The projections of a von Neumann algebra form a complete lattice. A partial ordering and an equivalence relation on the projection lattice of a von Neumann algebra are introduced by means of the partial isometries in the algebra in Section 1. According to the structure of the projection lattice with this ordering, the algebras are classified into those of type I, type II l, type II∞, and type III. It will be shown that every von Neumann algebra is decomposed into the direct sum of the algebras of these types. We will see that the von Neumann algebras of type I behave most naturally from the classical point of view. The structure of such an algebra will be completely determined in terms of the spectrum of the center and a set of cardinal numbers. At this point, it should be mentioned that the main task of the theory of von Neumann algebras is to establish methods for analyzing von Neumann algebras of type II or type III. Indeed, von Neumann algebras of type I do not introduce anything mysterious into the frame of the classical point of view, while there is no other theory which can handle mathematical phenomena related to von Neumann algebras of nontype I. Section 2 is devoted to the study of traces on semi-finite von Neumann algebras. It will be seen that the relative dimension of projections in a factor of type II takes continuous values.