ABSTRACT

The account of the Effros Borel structure and direct integral theory of von Neumann algebras is seriously incomplete in that it fails to consider the usual classification theory of von Neumann algebras. This chapter proves two theorems which will remedy this omission; roughly speaking, these theorems will say that both the Effros Borel structure and the direct integral theory are “as compatible as possible” with the classification theory. It simply contains the statements of these two theorems. The actual proofs of the theorems are interwoven with one another and depend on a number of technical lemmas; the theorems will be proven bit–by–bit. If the set in question is the set of factors of type III acting on some separable Hilbert space then this effectively means that one has to, be able to distinguish the factors of type III from amongst all of the factors by countable many “Borel” conditions, and it is precisely that the difficulty resides.