Introduction

This chapter is concerned with the basic theory of finite von Neumann algebras, with an emphasis on those which arise from discrete groups. This is described in Section 3.2, with the objective of reaching quickly the examples of masas which we present in Section 3.3. These are based on the work of Dixmier, [47], who found algebraic conditions on an abelian subgroup H of a discrete group G which are sufficient to imply that L(H) is a masa in the group von Neumann algebra L(G), and to determine the type of this masa. We present various matrix groups which satisfy these conditions and thus obtain examples of singular, Cartan and semi-regular masas.

Section 3.4 discusses various other ways in which naturally occurring masas can arise. These are based on tensor products and crossed products, and we give a detailed construction of a crossed product with two natural Cartan masas. We also give a brief discussion of free products in which examples of semi-regular masas are easy to exhibit. In Section 3.5 we prove that all difiuse separable abelian von Neumann algebras with a faithful normal trace are ✻-isomorphic to L∞[0, 1]. In particular this applies to masas.

Section 3.6 of the chapter returns to the basic theory of general finite von Neumann algebras, and we construct the unique trace preserving conditional expectation onto a subalgebra, which is fundamental throughout these notes.

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.