General Introduction

The theory of multilinear maps on a von Neumann algebra is developed in these notes and applied to the continuous Hochschild cohomology of von Neumann algebras. The methods used are those of von Neumann algebras and complete boundedness rather than of homological algebra, and only elementary cohomlogical techniques are employed in the proofs. We have chosen to base our presentation on the problem of whether the continuous cohomology groups Hn(M, M) of a von Neumann algebra M over itself are zero for all n. This, and closely related questions, has stimulated much of the recent development of the theory of completely bounded maps, and so we have adopted an approach which has wider applications beyond cohomology theory. The results in these notes have been proved in full generality, provided that they do not stray too far from the central topic of dual normal modules over von Neumann algebras.

There are two main reasons for investigating the Hochschild cohomology groups of operator algebras. When they are non-zero they provide invariants which can distinguish classes of algebras; when they are zero they lead to positive results on the stability of algebraic structures and on the space of bounded derivations on an operator algebra. Elliott's classification of separable AF-C* -algebras by K-theory is an example of the use of a homological invariant [El1].

Introduction

The continuous cohomology from a von Neumann algebra into itself is the principal topic of this chapter. The technical tools required in the proofs have been developed here, as in the last chapter, with a view that they may be useful in other von Neumann cohomology calculations.

The main technical result (6.2.1) of the second section is reminiscent of the theorems of Chapter 3, in that it says that continuous and completely bounded cohomology are equal under suitable hypotheses. This enables us to reduce continuous cohomology from a von Neumann algebra into dual normal modules to the completely bounded case when the algebra is type II∞, III, or II1 and stable under tensoring with R, and the module is an operator space with multiplication. The possible difference between completely bounded and continuous cohomology thus lies in the algebras of type II1 that are not stable under tensoring by R.

By analogy with Lie groups, a Cartan subalgebra A of a von Neumann algebra M is a masa A whose unitary normalizer generates M as a von Neumann algebra. The automatic complete boundedness conditions of Section 1.6 and the averaged Haagerup–Pisier–Grothendieck inequality (5.4.5) show that, with the Cartan algebra hypothesis, suitable cocycles are completely bounded.

The authors wish to thank all their friends and colleagues who sent them reprints and preprints, and explained parts of the theory to them. Barry Johnson, Dick Kadison and John Ringrose laid the foundation of the subject and wrote about it clearly. We are indebted to them, and particularly to John Ringrose for the two survey articles [Ri3, Ri6] which made the subject accessible and are still essential reading. Many of the beautiful and deep ideas in Chapters 1, 4 and 6 are due to Erik Christensen on his own or in collaboration with one or both authors. Without his mathematical insight, perseverance and encouragement this book would not have been written. It is a great pleasure to acknowledge our deep debt to him.

The patience and understanding of our wives Patricia and Virginia have been invaluable, and we have received much support and advice from Roger Astley and David Tranah of Cambridge University Press. Our special thanks go to Robin Campbell who typed the entire manuscript with great expertise.

While writing this book, both authors were supported by a NATO collaborative research grant, and the second author by a grant from the National Science Foundation. We take this opportunity to record our sincere gratitude to both agencies.

Introduction

This chapter presents some applications of cohomology to the structure theory of von Neumann algebras. Much of this material applies to arbitrary Banach algebras without modification, and so we have stated many of the results in full generality.

Section 7.2 explores the relationship between the first cohomology group and the principal component in the automorphism group of a Banach algebra. The main result (7.2.5) shows that, under the hypothesis H1(A, A) = 0, any automorphism in the principal component of Aut(A) is not only inner but is implemented by an element of the principal component of the group A−1 of invertible elements in A.

Section 7.3 contains an implicit function theorem for Fréchet differentiable maps on Banach spaces, which we apply to questions of stability in Section 7.4. Here we present two main results. If λ: A → A is an invertible linear map then (x, y) → λ−1(λ(x)λ(y)) defines a new associative product on A. Under the hypothesis of vanishing second and third cohomology groups, we show that any associative product sufficiently close to the original one has such a form, and λ can be chosen to be close to the identity map. The second stability result is a consequence of this: any von Neumann algebra sufficiently close to an injective von Neumann algebra must itself be injective, and the two algebras are isomorphic.

Introduction

This chapter contains three different techniques, each associated with hyperfinite von Neumann algebras, which play an important role in the next chapter on continuous cohomology. The technique in the first section is due to Kadison and Ringrose [KR3] and is based on a generalization of the old idea of Murray and von Neumann that if x ∈ M and y ∈ M′ with xy = 0 then there is a z ∈ M ∩ M′, the centre of both M and M′, such that xz = 0 and zy = y. For module maps over the centre this enables us to extend multilinear maps algebraically by a subalgebra of the commutant, provided this subalgebra is abelian. The extension can be shown to be norm continuous and behaves well with respect to the coboundary operator. This theory is developed in Section 5.2 and leads to the continuous cocycles from Mn into M being coboundaries into larger algebras.

Section 5.3 is devoted to three remarkable results of Popa on the existence of hyperfinite subfactors in type II1 factors with trivial relative commutants, and a result of his on maximal abelian self-adjoint subalgebras (masas). The first is that there is a hyperfinite subfactor N of a type II1 factor M with trivial relative commutant (that is, N′ ∩ M = C1) (5.3.6). In the second N is constructed to contain a preassigned Cartan subalgebra A(5.3.9). In the third a masa in B(L2(M)) is constructed from a Cartan subalgebra (5.3.11).

Introduction

This chapter contains the basic averaging techniques and the method for lifting a continuous multilinear map to the weak closure of the algebra. The averaging first leads to the cocycles being replaced by cocycles that are module maps over hyperfinite (= injective) subalgebras, and then to completely bounded maps in suitable situations (see Section 6.2). The weak lifting results are used to prove that continuous and normal cohomology are equal for dual normal modules over von Neumann algebras. Both the above methods will be used on continuous and completely bounded multilinear maps, with the proofs for the latter being just a minor modification of the continuous case developed by Johnson, Kadison and Ringrose.

This introduction will contain a discussion of the averaging techniques and the normal operator lifting method, and the rest of the chapter will be devoted to the details. The averaging techniques over C*-algebras are in Section 3.2 and the normal operator lifting methods are in Section 3.3, as these require the averaging technique at one critical point. The two sections come together in Section 3.4 in averaging over a hyperfinite von Neumann subalgebra.

The averaging technique used is basically just to integrate over the compact unitary group of a finite dimensional C*-algebra. Taking suitable weak limits as the finite dimensional algebras increase in size leads to averages that are essentially over infinite dimensional algebras. A similar route is to use an amenable group of unitary operators and the (right) invariant mean to average.

The theory of differential equations is one of the outstanding creations of the human mind. Its influence upon the development of physical science would be hard to exaggerate. The long history and many applications of the theory, however, make it almost impossible to write a balanced account of the subject. Thus authors of student texts are confronted with the choice between writing rather superficially on a range of topics or in more depth on some narrow field, in which they have a particular interest.

In this book I have given a simple introduction to the spectral theory of linear differential operators. This spectral theory is an outgrowth of fundamental work of David Hilbert between 1900 and 1910 on the analysis of integral operators on infinite-dimensional spaces – now called Hilbert spaces. However, like almost every important new development in mathematics, it was preceded by much related work, for example Poincare's analysis of the Dirichlet problem and associated eigenvalues (1890–6). One could maintain that the subject started with the seminal work of Fourier on the solution of the heat equation using series expansions in sines and cosines, which was published by the Académie Française in 1822. Fourier submitted this work in 1807, during the Napoleonic era, and an account of his misfortunes during the fifteen year period before publication is given by Korner (1988). I have included the names and dates associated with a few of the key ideas in the text; a much more comprehensive account may be found in Dieudonné (1981).

Classification of the spectrum

In this chapter we investigate the spectral properties of abstract selfadjoint operators in more detail, making use of the spectral theorems (Theorems 2.3.1 and 2.5.1). In contrast with Chapter 3, we focus mainly on abstract situations in which the spectrum of a self-adjoint operator consists simply of isolated eigenvalues of finite multiplicity. Several classes of operators of this type are studied in later chapters. We use the notation of Theorem 2.5.1 freely within this section, but frequently suppress explicit reference to the unitary operator U of that theorem. Many of the considerations of this chapter become trivial or do not make sense if the Hilbert space ℋ is finite-dimensional, so we assume throughout that it is infinite-dimensional. We assume as always that ℋ is separable, or equivalently that any complete orthonormal set is countable.

The following lemma from measure theory will be important.

Lemma 4.1.1Let μ be a finite measure on RN, and let E be a Borel subset of RN. Then the space L := L2(E,dμ) has finite dimension m if and only if there are m distinct points x1, …,xm in E of positive measure such that E\Umr=1 {xr} has zero measure.

Proof If such points exist then it is clear that L has a basis consisting of the characteristic functions of these points. Conversely suppose that L has finite dimension m. For any integer r ≥ 1 we can partition RN into a countable number of cubes C(r,s), 1 ≤ S < ∞, with edge length 2−r; we take these cubes to be in standard position, that is every coordinate of every vertex of C(r,s) is of the form t/2r where t is an integer;…

Introduction

The general spectral theorem for self-adjoint operators was proved independently by Stone and von Neumann during the period 1929–1932. There have been several other proofs since that time, the most popular being that based upon Gelfand's theory of commutative Banach algebras. There are also several different ways of stating the spectral theorem, one in terms of a functional calculus, one using a family of spectral projections and one in terms of a measure-theoretic representation formula; all of these have advantages.

In this chapter we describe an approach to the spectral theorem which originates from a paper written in 1989 by Helffer and Sjöstrand. The approach is very explicit and has proved of great value in doing computations in n-body scattering theory. It has been rewritten in an axiomatic form applicable to suitable operators on Banach spaces in Davies (1994), but here we shall present only the simpler theory for self-adjoint operators. Another advantage of this approach is that it uses techniques of a type which are useful in other parts of the theory of partial differential equations, rather than abstract functional analysis. The material in this chapter is of fundamental importance to later work, but the reader may choose to defer reading the proofs to a later stage. The key results are Theorems 2.3.1, 2.5.1 and 2.5.3.

Our proof of the spectral theorem is based upon presenting an explicit formula for f(H) as an integral over resolvents, for a fairly large class of functions f. The work then consists of showing that this formula has all of the properties required of a functional calculus.