Introduction

In chapter 9, we developed the theory of perturbations of masas in type II1 factors making use of the special structure of such subalgebras. Here we turn to the general theory. The results are essentially the same since close subalgebras will be shown to have spatially isomorphic cutdowns by projections, and the two chapters could have been combined into this one. However, the techniques of the previous chapter do produce significantly better numerical estimates and also give Theorem 9.6.3 on normalising unitaries for which we know no general counterpart.

In Section 10.2, we give a very brief survey of the theory of subfactors, just those parts that we will use subsequently. Section 10.3 considers the situation of a containment M ⊆ N where these two algebras are close in an appropriate sense. The main result is Theorem 10.3.5, which shows that there is a large projection p in the relative commutant M′ ∩ N so that Mp = pNp. This is the crucial result for the perturbation theorems of Section 10.4, the most general one being Theorem 10.4.1.

Much of the material of this chapter is taken from [152], which was based on earlier results from [147].

The Jones index

In this section we will briey describe those parts of subfactor theory that we will use in this chapter. There are several good accounts of the theory in [95, 97, 144] and so we will only state the relevant results.