This book is designed as a “second course in C*-algebras”. It presupposes a familiarity with the elementary theory of C*-algebras (the GNS construction, the functional calculus) such as may be found in any of the several excellent texts now available—[Dix 2], [KadRin], [Mur], [Ped], [Tak], for example. The aim, as indicated by the subtitle, is to provide the student with a collection of techniques that have shown themselves to be useful in a variety of contexts in modern C*-algebra theory.

These techniques centre round the quite elementary and natural concept of a Hilbert C*-module. As explained in Chapter 1, this is an object like a Hilbert space except that the inner product is not scalar-valued, but takes its values in a C*-algebra. The first three chapters present the elementary theory of Hilbert C*-modules and their bounded adjointable operators. From Chapter 4 onwards, tensor products figure prominently, and some knowledge of tensor products of C*-algebras (summarised at the beginning of Chapter 4) is needed.

Hilbert C*-modules have had three main areas of applications:

• the work of Rieffel and others on induced representations and Morita equivalence ([BroGreRie], [Rie 1], [Rie 2]);

• the work of Kasparov and others on KK-theory ([BaaJul], [Kas 1], [Kas 2]);

• the work of Woronowicz and others on C*-algebraic quantum group theory ([BaaSka], [Wor 5]).

There is not very much information on any of these topics in this book, since the aim is to develop a toolkit rather than to demonstrate the tools in use.

When first learning potential theory, as a new graduate student, I experienced some difficulty with the literature then available. The choice lay between several excellent but encyclopaedic treatises on the subject, from which it was hard work to extract what was needed, and several equally excellent books on complex variable, each containing a useful chapter on potential theory, but which did not go nearly far enough. This book is an attempt to bridge that gap—indeed it was consciously written as the book that I should have liked to read all those years ago.

Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, harmonic measure, Green's functions, potentials and capacity. It can be developed in many contexts, ranging from classical potential theory in ℝn and pluripotential theory in ℂn to axiomatic theories in very general spaces. In between there are versions relating to Riemann surfaces and other manifolds, uniform algebras and analytic multifunctions, to say nothing of the connections with Brownian motion and other stochastic processes. However, there is one case which is common to them all: potential theory in the plane. As it contains all the essential ingredients of the subject, yet is relatively easy and quick to treat, it seems to me to be well worth mastering first. This is the subject of the book.

There is also a further goal, hinted at by the use of the word ‘complex’ in the title. It is to emphasize the very close connection between potential theory and complex analysis.

Introduction

This appendix contains a section on bounded group cohomology and one on its relation with the ℓ1-group algebra cohomology. Though there is currently no link between bounded group cohomology and that of the reduced group or von Neumann group algebras it is an obvious question to ask if the subjects are related. Given that bounded group cohomology is a topic unknown to most operator algebraists, it seemed worth introducing it in Section 8.2 and linking it with Hochschild cohomology in Section 8.3. There is a list of problems in Section 8.4.

Bounded Group Cohomology

Remarks. Bounded group cohomology was related to corresponding geometrical and topological ideas for manifolds by Gromov in (1982) [Grom] following work of Hirsch and Thurston (1975) [HiT]. Earlier Johnson (1972) [J3] had used bounded cohomology of groups to show that H2(ℓ1(G), ℓ1(G)) ≠ 0 for G the free group on two generators. Bounded group cohomology is defined and a few of its properties are given in these notes. The theory is only developed as far as its current relevance to the Hochschild cohomology of Banach algebras warrants. For further details of the theory see the paper by Gromov [Grom] (beware there are errors), the survey by Ivanov [Iv1] and the paper by Grigorchuk [Gri2]. The authors are indebted to Professor Grigorchuk for the preprint [Gri2], which is recommended reading.

An elementary concrete approach is taken to the bounded cohomology of groups analogous to the Hochschild cohomology discussion.

Introduction

This chapter contains much of the background material which is necessary for the study of the cohomology theory of von Neumann algebras. In Sections 1.2 and 1.3 we introduce the basic concepts: operator systems, operator spaces, completely positive maps and completely bounded maps. The two fundamental results in the subject are the Stinespring representation theorem (Theorem 1.2.1) and Arveson's Hahn–Banach theorem for completely positive maps (Theorem 1.2.3). We then discuss matrix ordered spaces, and obtain an important abstract characterization of operator systems (Theorem 1.2.7). With these results established, the representation of a completely bounded map as V*πW (Theorem 1.3.1) is easily obtained.

The fourth section is devoted to the Haagerup tensor product of operator spaces, in preparation for the succeeding section where complete boundedness is introduced for multilinear maps. The point is that multilinear maps can be viewed as linear maps on tensor products. The Haagerup tensor product norm is the correct one for compatibility with the completely bounded norm, and this allows us to prove multilinear results by appealing to the linear theorems of the second and third sections. The most important theorems here are 1.5.6 and 1.5.8. The first gives a general representation theorem for multilinear completely bounded maps on operator spaces, while the second describes an improved version for completely bounded maps on von Neumann algebras which are separately normal in each variable.

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.