Introduction

In this chapter we consider various useful technical results concerning projections, partial isometries and ∥·∥2-norm estimates arising from the polar decomposition. Many of these will be crucial when we come to discuss perturbations in Chapters 9 and 10. Everything in this chapter is well known and can be found in [36, 188], although the proofs are sometimes different from the originals.

We begin by considering two projections, and we show that the algebra they generate has only irreducible representations of dimensions 1 and 2. This reduces the study of such a pair to the 2×2 matrices, and we use this in obtaining Theorem 5.2.5. We also draw attention to Lemma 5.2.7 which contains estimates that we use repeatedly in the sequel. The third section is devoted to estimates concerning pairs of projections.

The concluding Sections 5.5 and 5.6 are devoted to some work of Kadison, [99, 100], which constructs abelian projections with desirable properties in type I von Neumann algebras. These will be relevant in Chapter 9 since masas in II1 factors can lead to problems in type I von Neumann algebras.

Comparison of two projections

We begin by proving that the von Neumann algebra generated by two projections is, in general, a direct sum of an abelian algebra and one of type I2. The first few lemmas lead in this direction.

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.

Introduction

This appendix contains sections on: ultrafilters and characters of l∞(ℕ); a discussion of maximal ideals in finite von Neumann algebras; the construction of the ultrapowers and ℕω; property Γ and relative commutants in ℕω.

In these notes ultrafilter is used for free ultrafilter in ℕ as these are the only ultrafilters discussed. See the article by Ge and Hadwin [78] for a detailed discussion of ultrafilters and ultraproducts directed at operator algebras, or the books [34, 90, 107, 204] for a discussion of filters and ultrafilters in set theory and general topology. It is convenient to think of ultrafilters as characters ω of l∞(ℕ) induced by points in βℕ\ℕ so this relationship is discussed briefly in the second section.

Section A.3 on maximal ideals in a finite von Neumann algebra contains a theorem due to Wright [211] that the quotient of a finite von Neumann algebra by a maximal two-sided ideal is a finite factor with trace arising from the original algebra and maximal ideal; Wright actually proved this for AW*-algebras. A theorem for AW*-algebras that yields this was rediscovered by Feldman [68], though he does not state this exact result or examine the norm closed ideals as Wright does. This result for von Neumann algebras appears in Sakai's Yale notes [165] with no reference, and there is an account by Takesaki [187, p. 357].

The authors wish to thank all the mathematicians who have contributed to the ideas that are presented in these notes. The informal discussions, seminars, papers and reprints of our friends and colleagues have shaped our approach to these notes. We are indebted to our co-authors, Erik Christensen, Ken Dykema, Florin Pop, Sorin Popa, Guyan Robertson, Stuart White and Alan Wiggins, who have directly, and indirectly, inuenced us.

We wish to thank Jan Cameron, Kunal Mukherjee, Gabriel Tucci and Alan Wiggins who, while research students, read parts of the notes and made many constructive comments on them. Particular thanks go to Stuart White who read the entire manuscript, made many suggestions for improvements, and saved us from numerous errors.

Our wives Pat and Ginny have been extremely patient and understanding during the long periods that we have been immersed in this project. We have received much support and advice from Roger Astley at Cambridge University Press. Our thanks go to Robin Campbell who with great expertise turned our manuscripts and rough LATEX files into this book.

We take this opportunity to record our sincere gratitude to the National Science Foundation. Grants to the second author have enabled us to meet on numerous occasions in the last few years, allowing us to bring this book to completion.

Introduction

Pukfianszky [154] defined an invariant for a masa A in a separable type II1 factor N based on the type I decomposition of A′ = (A ⋃ JAJ)′. The latter algebra is the commutant of an abelian algebra and so is type I. The standard theory then gives a decomposition as a direct sum of n-homogeneous algebras (where n = ∞ is allowed). The Pukánszky invariant is essentially the union of these numbers; the exact definition is given in Definition 7.1.2 whose wording takes account of a subtle point that we explain subsequently. The only previous invariants for masas were Cartan (= regularity), semi-regularity and singularity due to Dixmier [47]. Attention is restricted to separable type II1 factors to avoid pathologies and so there is only one infinite cardinal, which is denoted ∞. See the paper by Popa [139] for a discussion of some of the unusual problems that can arise in non-separable type II1 factors. In this chapter we define the Pukánszky invariant, prove the basic theorems concerning it and give some examples.

We briey review our standard notation from earlier chapters. Let A be a masa in a separable type II1 factor N with a faithful normal trace τ, let J be the conjugate linear isometry Jx = x* on L2(N) and let ξ be the vector in L2(N) corresponding to 1. We let eA be the projection of L2(N) onto L2(A).

Introduction

This chapter is devoted to the construction of irreducible hyperfinite subfactors R in a separable II1 factor N with suitable additional properties available for R in its embedding in N. All these results depend on inductive matrix methods that were developed by Popa [136]. The method has already been used extensively in Chapter 12 for the construction of singular and semiregular masas.

In Section 13.2, a basic method is presented to show that an irreducible hyperfinite subfactor exists in each separable II1 factor. Section 13.3 shows that if A is a Cartan masa in a separable II1 factor N, then there is an irreducible hyperfinite subfactor R in N with A ⊆ R and A Cartan in R (see [141]). Section 13.4 discusses the basic theory of property Γ factors, a topic which we will revisit in greater depth in Appendix A. This is applied in Section 13.5 to prove a useful result (Theorem 13.5.4) on the existence of a masa in a Γ factor that is Cartan in an irreducible hyperfinite subfactor and that contains unitaries that can be used in the Γ condition. This theorem combines methods from [140] and from [30, Theorem 5.3] that give the Γ condition.

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.

These notes are an introduction to some of the theory of finite von Neumann algebras and their von Neumann subalgebras, with the emphasis on maximal abelian self-adjoint subalgebras (usually abbreviated masas). Assuming basic von Neumann algebra theory, the notes are fairly detailed in covering the basic construction, perturbations of von Neumann subalgebras, general results on masas and detailed ones on singular masas in II1 factors. Due to the large volume of research on finite von Neumann algebras and their masas the authors have been forced to be selective of the topics included. Nevertheless, a substantial body of recent research has been covered.

Each chapter of the book has its own introduction, so the overview of the contents below will be quite brief. We have also included a discussion of a few important results which have been omitted from the body of the text. In each case, we felt that the amount of background required for a reasonably self-contained account was simply too much for a book of this kind.

We have tried to make the material accessible to graduate students who have some familiarity with von Neumann algebras at the level of a first course in the subject. The early chapters review some of this, but are best read by the beginner with one of the standard texts, [104, 105, 187], to hand to fill in any gaps.

Introduction

For the most part, these notes are concerned with the bounded operators which constitute the von Neumann algebras under consideration. However, results from the theory of unbounded operators have played a role in Chapter 9, and knowledge of this topic is essential for reading the literature in this area. Since we feel that this theory is less well known than its counterpart for bounded operators, we include here a brief exposition of the main theorems required in these notes. Most of what is needed may be found in [104, Section 5.6], and we follow their development to a considerable extent. However, we have specific goals for the theory and we do not offer a comprehensive treatment. The main objective is to understand the operators that arise as unbounded left multiplication operators on II1 factors.

Section B.2 contains the basic theory of closed and closable operators. In Section B.3, we develop as much of the functional calculus as we will need. We carry this out for positive operators, relating matters to the well known functional calculus for bounded positive operators. Along the way we establish the polar decomposition of a closed operator which appeared in Lemma 9.4.2. The important unbounded operators of II1 factor theory are those that arise from vectors in L2(N) and L1(N), and we lay out their theory in Sections B.4 and B.5 respectively.

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.