This book is an elementary self-contained exposition, with complete proofs, of the fundamentals of the theory of von Neumann algebras. Although this theory has considerably evolved in the last 40 years since the first edition of this book was printed, the fundamentals of the theory remain the same. Besides many achievements in the operator algebras themselves, new connected powerful mathematical theories have appeared, such as the Noncommutative Geometry of Alain Connes and the Free Probability Theory of Dan-Virgil Voiculescu, and this makes it even more necessary to have an elementary presentation of the fundamentals of the theory. The first edition was intensively used in the main mathematical centres in the world and was very soon out of print, and we received many demands for the book that we could not fulfil.
In this new edition we have added, among other things, some material concerning factors associated to ICC groups, the BT-Theorem of J. von Neumann, the Murray–von Neumann proof of the existence of the trace on type II1 factors, the Averaging Theorem of J. Dixmier in case of factors, and a preliminary presentation of Tomita's Theorem first in the classical case of a cyclic trace vector and then in an introduction to Chapter X, in the case of a cyclic and separating vector. In the comments sections, we have quoted some new results, among which are the Jones index of subfactors and the Connes classification of hyperfinite factors. The extensive bibliography of the first edition has been considerably reduced, mainly to quoted books or articles.
The main consequences of Tomita's theory, presented in Chapter 10, are the subject of a direct continuation of the present book, namely Modular Theory in Operator Algebras by Ş. Strătilă (1981).
We are grateful to V. Sunder (Chennai Institute of Mathematical Sciences, India) and Gadadhar Misra (Indian Institute of Science, Bangalore, India) for proposing a revised edition of this book, especially to Gadadhar Misra for the help in the LaTeX conversion of the book, and to the Cambridge University Press for publishing this second edition of the book.
We are also grateful to Alexandru Negrescu who typed in LaTeX all the supplementary material for the Second Edition (c. 60 p).