Introduction

Up to this point, our discussion of masas has focused on the separable case. However, all von Neumann algebras possess masas, and there are some interesting phenomena which appear only when we leave the separable setting. The ultrapower examples of Appendix A are important non-separable algebras which play a role even for separable algebras, so strict adherence to separability is not generally possible. In this chapter we investigate masas in non-separable factors, and the results that we present below are all due to Popa [136, 138]. Many of them concern the algebra Nω which is discussed in Appendix A.

The main results for Nω in this chapter are Theorems 15.2.3 and 15.2.8. They can be summarised as follows:

If ℕ is a separable II1 factor, then all masas in Nω are non-separable,

Nω has no Cartan subalgebras and Nω is prime.

This is proved in Section 15.2. The third section is devoted to showing that, for an uncountable set S, masas in are separable and that this factor has no Cartan masas. These algebras were the first examples of the absence of Cartan masas. The algebras, n ≥ 2, also have this property [199], but in the separable case the proof is substantially more dificult. We draw attention to the difierence between Nω and: the first has no separable masas and the second has no non-separable ones.