Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory (Graduate Studies in Mathematics, Volume 15)

These volumes deal with a subject, introduced half a century ago, that has become increasingly important and popular in recent years. While they cover the fundamental aspects of this subject, they make no attempt to be encyclopaedic. Their primary goal is to teach the subject and lead the reader to the point where the vast recent research literature, both in the subject proper and in its many applications, becomes accessible. Although we have put major emphasis on making the material presented clear and understandable, the subject is not easy; no account, however lucid, can make it so. If it is possible to browse in this subject and acquire a significant amount of information, we hope that these volumes present that opportunity-but they have been written primarily for the reader, either starting at the beginning or with enough preparation to enter at some intermediate stage, who works through the text systematically. The study of this material is best approached with equal measures of patience and persistence.

This book, together with Fundamentals of the Theory of Operator Algebras. Volume II, Advanced Theory, Graduate Studies in Mathematics, vol. 16, present an introduction to functional analysis and the initial fundamentals of $C^*$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide a clear account of the introductory portions of this important and technically difficult subject. Major concepts are sometimes presented from several points of view; the account is leisurely when brevity would compromise clarity. An unusual feature in a text at this level is the extent to which it is self-contained; for example, it introduces all the elementary functional analysis needed. The emphasis is on teaching. Well supplied with exercises, the text assumes only basic measure theory and topology. The book presents the possibility for the design of numerous courses aimed at different audiences. The book is intended for graduate students and research mathematicians, and mathematical physicists interested in functional analysis, operator algebras, and applications. It can also be used as a text for a graduate course in any of these areas. Praise for both volumes ... ... these two volumes represent a magnificent achievement. They will be an essential item on every operator algebraist's bookshelves and will surely become the primary source of instruction for research students in von Neumann algebra theory. —Bulletin of the London Mathematical Society Volumes I and II were published in 1982 and 1983. Since then they have quickly established themselves as The Textbooks in operator algebra theory. —Bulletin of the American Mathematical Society One of the splendid features of the original two volumes is their large supply of exercises ... which illustrate the results of the text and expand its scope. —L'Enseignement mathematique

This book, together with Fundamentals of the Theory of Operator Algebras. Volume II, Advanced Theory, Graduate Studies in Mathematics, vol. 16, present an introduction to functional analysis and the initial fundamentals of $C^•$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide a clear account of the introductory portions of this important and technically difficult subject. Major concepts are sometimes presented from several points of view; the account is leisurely when brevity would compromise clarity. An unusual feature in a text at this level is the extent to which it is self-contained; for example, it introduces all the elementary functional analysis needed. The emphasis is on teaching. Well supplied with exercises, the text assumes only basic measure theory and topology. The book presents the possibility for the design of numerous courses aimed at different audiences. The book is intended for graduate students and research mathematicians, and mathematical physicists interested in functional analysis, operator algebras, and applications. It can also be used as a text for a graduate course in any of these areas. Praise for both volumes ... ... these two volumes represent a magnificent achievement. They will be an essential item on every operator algebraist's bookshelves and will surely become the primary source of instruction for research students in von Neumann algebra theory. —Bulletin of the London Mathematical Society Volumes I and II were published in 1982 and 1983. Since then they have quickly established themselves as The Textbooks in operator algebra theory. —Bulletin of the American Mathematical Society One of the splendid features of the original two volumes is their large supply of exercises ... which illustrate the results of the text and expand its scope. —L'Enseignement mathématique Fundamentals of the Theory of Operator Algebras......Page 4 Copyright Page......Page 5 Contents......Page 6 Preface......Page 8 Contents of Volume II......Page 14 1.1. Algebraic results......Page 18 1.2. Linear topological spaces......Page 29 1.3. Weak topologies......Page 45 1.4. Extreme points......Page 48 1.5. Normed spaces......Page 52 1.6. Linear functionals on normed spaces......Page 60 1.7. Some examples of Banach spaces......Page 65 1.8. Linear operators acting on Banach spaces......Page 76 1.9. Exercises......Page 82 2.1. Inner products on linear spaces......Page 92 2.2. Orthogonality......Page 102 2.3. The weak topology......Page 114 2.4. Linear operators......Page 116 2.5. The lattice of projections......Page 126 2.6. Constructions with Hilbert spaces......Page 137 2.7. Unbounded linear operators......Page 171 2.8. Exercises......Page 178 3.1. Basics......Page 190 3.2. The spectrum......Page 195 3.3. The holomorphic function calculus......Page 219 3.4. The Banach algebra C(X)......Page 227 3.5. Exercises......Page 240 4.1. Basics......Page 253 4.2. Order structure......Page 261 4.3. Positive linear functionals......Page 272 4.4. Abelian algebras......Page 286 4.5. States and representations......Page 292 4.6. Exercises......Page 302 5.1. The weak- and strong-operator topologies......Page 321 5.2. Spectral theory for bounded operators......Page 326 5.3. Two fundamental approximation theorems......Page 342 5.4. Irreducible algebras—an application......Page 347 5.5. Projection techniques and constructs......Page 349 5.6. Unbounded operators and abelian von Neumann algebras......Page 357 5.7. Exercises......Page 387 Bibliography......Page 401 Index of Notation......Page 404 Index......Page 408 Pure and Applied Mathematics......Page 416 These volumes are companions to the treatise; "Fundamentals of the Theory of Operator Algebras," which appeared as Volume 100 - I and II in the series, Pure and Applied Mathematics, published by Academic Press in 1983 and 1986, respectively. As stated in the preface to those volumes, "Their primary goal is to teach the sub ject and lead the reader to the point where the vast recent research literature, both in the subject proper and in its many applications, becomes accessible." No attempt was made to be encyclopcedic; the choice of material was made from among the fundamentals of what may be called the "classical" theory of operator algebras. By way of supplementing the topics selected for presentation in "Fundamentals," a substantial list of exercises comprises the last section of each chapter. An equally important purpose of those exer cises is to develop "hand-on" skills in use ofthe techniques appearing in the text. As a consequence, each exercise was carefully designed to depend only on the material that precedes it, and separated into segments each of which is realistically capable of solution by an at tentive, diligent, well-motivated reader." These volumes are companions to the treatise;'Fundamentals of the Theory of Operator Algebras,'which appeared as Volume 100 - I and II in the series, Pure and Applied Mathematics, published by Academic Press in 1983 and 1986, respectively. As stated in the preface to those volumes,'Their primary goal is to teach the sub­ ject and lead the reader to the point where the vast recent research literature, both in the subject proper and in its many applications, becomes accessible.'No attempt was made to be encyclopCEdic; the choice of material was made from among the fundamentals of what may be called the'classical'theory of operator algebras. By way of supplementing the topics selected for presentation in'Fundamentals,'a substantial list of exercises comprises the last section of each chapter. An equally important purpose of those exer­ cises is to develop'hand-on'skills in use of the techniques appearing in the text. As a consequence, each exercise was carefully designed to depend only on the material that precedes it, and separated into segments each of which is realistically capable of solution by an at­ tentive, diligent, well-motivated reader. This work and Fundamentals of the Theory of Operator Algebras. Volume II, Advanced Theory present an introduction to functional analysis and the initial fundamentals of $C^*$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide a clear account of the introductory portions of this important and technically difficult subject. Major concepts are sometimes presented from several points of view; the account is leisurely when brevity would compromise clarity. An unusual feature in a text at this level is the extent to which it is self-contained; for example, it introduces all the elementary functional analysis needed. The emphasis is on teaching. Well supplied with exercises, the text assumes only basic measure theory and topology. The book presents the possibility for the design of numerous courses aimed at different audiences. Presents an introduction to functional analysis and the initial fundamentals of $C^*$- and von Neumann algebra theory in a form suitable for both intermediate graduate courses and self-study. The authors provide an account of the introductory portions of this important and technically difficult subject. The goal of this text is to teach operator algebras and lead readers to where the literature - in the subject specifically and in its applications - becomes accessible. The choice of material was made from among the fundamentals of what may be called the "classical" theory of operator algebras. The primary purpose of this book is to teach, and to enable readers to study the recent literature on this subject and its many applications. It is suitable for graduate-level courses in functional analysis and operator algebras, and as a reference for self-study by graduates. This chapter contains an account of those basic aspects of linear functional analysis that are needed, later in the book, in the study of operator algebras. v. 1. Elementary theory v. 2. Advanced theory v. 3. Elementary theory, an exercise approach v. 4. Special topics advanced theory, an exercise approach. Discusses the fundamentals of classical theory of operator algebras. This book features written solutions to the exercises.