This book is based on lectures delivered in July–August 1972 at the Suceava Summer School organized by the Institute of Mathematics of the Academy of the Socialist Republic of Romania in cooperation with the Society of Mathematical Sciences.
The study of the algebras of operators in Hilbert spaces was initiated by F. J. Murray and J. von Neumann in connection with some problems of theoretical physics. The wealth of the mathematical facts contained in their fundamental papers interested many mathematicians. This soon led to the crystallization of a new branch of mathematics: the theory of algebras of operators. The first systematic exposition of this theory appeared in the well-known monograph by J. Dixmier 1957/1969, which was subtitled ‘Algèbres de von Neumann’. It expounded almost all the significant results achieved until its appearance. Afterwards, the theory continued to develop, for it had important applications in the theory of group representations, in mathematical physics and in other branches of mathematics. Of great importance were the results obtained by M. Tomita, who exhibited canonical forms for arbitrary von Neumann algebras. In recent times, fine classifications and structure theorems have been obtained for von Neumann algebras especially by A. Connes.
The present book contains what we consider to be the fundamental part of the theory of von Neumann algebras. The book also contains the essential elements of the spectral theory in Hilbert spaces. The material is divided into 10 chapters; besides the basic text, each chapter has two complementary sections: exercises and comments (including bibliographical comments).
The reader is expected to know only some elementary facts from functional analysis. In writing this book, we made use of existing books and courses (J. Dixmier 1957/1969, I. Kaplansky 1955/1968, J. R. Ringrose 1966/1967, 1972a, b, S. Sakai 1962, 1971, M. Takesaki 1969/1970, 1970 and D. M. Topping 1971), as well as many articles, some of them available only as preprints. Some of the exercises are borrowed from J. Dixmier's book 1957/1969.
Thanks to Grigore Arsene and Dan Voiculescu for their help in the writing of this book, for the useful discussions and for the bibliographical information they gave us.