Introduction

The topic for this and the succeeding chapter is the theory of perturbations. Before giving a detailed description, we discuss the ideas in general terms without reference to norms or metrics. If we have a von Neumann subalgebra A of a II1 factor N, and u ∈ N is a unitary close to 1, then the algebras A and uAu* are close and we think of uAu* as a small perturbation of A. Conversely, if we have two algebras A and B which are close to one another, then we might expect to find a unitary u ∈ N close to 1 so that B = uAu*. This is too much to ask for in general. In these two chapters we explore whether suitable modifications can be made so that results of this type hold true. Although there are circumstances where unitary equivalence is possible, it is usually necessary to cut the algebras by projections and ask only for a partial isometry which implements a spatial isomorphism of the compressions. As we will see, the strength of the results will depend on the norms and metrics selected to define the notion of close operators and close algebras. Some theorems in this chapter are formulated for general subalgebras and thus apply in the next chapter. The main focus here is on masas, and some of these results are only valid in that case.