Operator Algebras

Preface (P. 11)

This volume attempts to give a comprehensive discussion of the theory of operator algebras (C*-algebras and von Neumann algebras.) The volume is intended to serve two purposes: to record the standard theory in the Encyclopedia of Mathematics, and to serve as an introduction and standard reference for the specialized volumes in the series on current research topics in the subject.

Since there are already numerous excellent treatises on various aspects of the subject, how does this volume make a signi.cant addition to the literature, and how does it differ from the other books in the subject? In short, why another book on operator algebras?

The answer lies partly in the first paragraph above. More importantly, no other single reference covers all or even almost all of the material in this volume. I have tried to cover all of the main aspects of "standard" or "classical" operator algebra theory, the goal has been to be, well, encyclopedic. Of course, in a subject as vast as this one, authors must make highly subjective judgments as to what to include and what to omit, as well as what level of detail to include, and I have been guided as much by my own interests and prejudices as by the needs of the authors of the more specialized volumes.

A treatment of such a large body of material cannot be done at the detail level of a textbook in a reasonably-sized work, and this volume would not be suitable as a text and certainly does not replace the more detailed treatments of the subject. But neither is this volume simply a survey of the subject (a .ne survey-level book is already available [Fil96].)

My philosophy has been to not only state what is true, but explain why: while many proofs are merely outlined or even omitted, I have attempted to include enough detail and explanation to at least make all results plausible and to give the reader a sense of what material and level of diffculty is involved in each result. Where an argument can be given or summarized in just a few lines, it is usually included, longer arguments are usually omitted or only outlined.

More detail has been included where results are particularly important or frequently used in the sequel, where the results or proofs are not found in standard references, and in the few cases where new arguments have been found. Nonetheless, throughout the volume the reader should expect to have to fill out compactly written arguments, or consult references giving expanded expositions.

I have concentrated on trying to give a clean and efficient exposition of the details of the theory, and have for the most part avoided general discussions of the nature of the subject, its importance, and its connections and applications in other parts of mathematics (and physics), these matters have been amply treated in the introductory article to this series. See the introduction to [Con94] for another excellent overview of the subject of operator algebras. There is very little in this volume that is truly new, mainly some simplified proofs.