1.1. From the Preface.

During the last ten or fifteen years, much progress has been made in the theory of non-self-adjoint compact linear operators on Hilbert space. The present volume is intended to provide a brief (and far from exhaustive); introduction to the subject. For the greater part, it is a revised version of a graduate course given at the University of Pennsylvania during the academic year 1964-65; some further material has been added subsequently, notably in sections 4.4 and 4.5.

1.2. Review by: Takashi Yoshino.
zbMATH 0223.47012.

This book concentrates on recent works on compact non-self-adjoint operators acting on Hilbert spaces. Chapter 1 is an introductory chapter and it contains a number of notations that are used throughout the book, certain elementary topics in functional analysis and a number of results that are needed in later chapters. In Chapter 2, the author deals with certain classes $C_{p} (1 ≤ p < ∞)$ of compact operators on a Hilbert space which were introduced by von Neumann and Schatten. Chapter 3includes the Fredholm theory for trace class operators and this is used to obtain sharp estimates for the rate of growth on the resolvent of a quasi-nilpotent operator in general von Neumann-Schatten class. Chapter 4 contains some of the recent works of the Russian mathematicians on the representation of a compact operator on a Hilbert space as a "superdiagonal integral."

2.1. From the Publisher.

This valuable work provides the student with a careful treatment of elementary functional analysis and uses this foundation to explain clearly the fundamentals of operator algebras. Topics covered in this-the first of two volumes-include: - Linear spaces - Hilbert space and linear operators - Banach algebras - Elementary $C^{*}$-algebra theory - Elementary von Neumann algebra theory

This text will be suitable for both graduate-level courses in functional analysis and operator algebras, and for self-study by graduate students and research workers who have a basic knowledge of point set topology and measure theory.

2.2 From the Preface.

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.

Our starting point in Chapter I is finite-dimensional linear algebra. We assume that the reader is familiar with the results of that subject and begin by proving the infinite-dimensional algebraic results that we need from time to time. These volumes deal almost exclusively with infinite-dimensional phenomena. Much of the intuition that the reader may have developed from contact with finite-dimensional algebra and geometry must be abandoned in this study. It will mislead as often as it guides. In its place, a new intuition about infinite-dimensional constructs must be cultivated. Results that are apparent in finite dimensions may be false, or may be difficult and important principles whose application yields great rewards, in the infinite-dimensional case.

Almost as much as the subject matter of these volumes is infinite dimensional, it is non-commutative real analysis. Despite this description, the reader will find a very large number of references to the "abelian" or "commutative" case - an important part of this first volume is an analysis of the abelian case. This case, parallel to function theory and measure theory, provides us with a major tool and an important guide to our intuition. A good pan of what we know comes from extending to the noncommutative case results that are known in the commutative case. The "extension" process is usually difficult. The main techniques include elaborate interlacing of "abelian" segments. The reference to "real analysis" involves the fact that while we consider complex-valued functions and, non-commutatively, non-self-adjoint operators, the structures we study make simultaneously available to us the complex conjugates of those functions and, non-commutatively, the adjoints of those operators. In essence, we are studying the algebraic interrelations of systems of real functions and, non-commutatively, systems of self-adjoint operators. At its most primitive level, the non-commutativity makes itself visible in the fact that the product of a function and its conjugate is the same in either order while this is not in general true of the product of an operator and its adjoint.

In the sense that we consider an operator and its adjoint on the same footing, the subject matter we treat is referred to as the "self-adjoint theory." There is an emerging and important development of non-self-adjoint operator algebras that serves as a non-commutative analogue of complex function theory - algebras of holomorphic functions. This area is not treated in these volumes. Many important developments in the self-adjoint theory - both past and current-are not treated. The type I $C^{*}$-algebras and $C^{*}$-algebra $K$-theory are examples of important subjects not dealt with. The aim of teaching the basics and preparing the reader for individual work in research areas seems best served by a close adherence to the "classical" fundamentals of the subject. For this same reason, we have not included material on the important application of the subject to the mathematical foundation of theoretical quantum physics. With one exception, applications to the theory of representations of topological groups are omitted. Accounts of these vast research areas, within the scope of this treatise, would be necessarily superficial. We have preferred instead to devote space to clear and leisurely expositions of the fundamentals. For several important topics, two approaches are included.

Our emphasis on instruction rather than comprehensive coverage has led us to settle on a very brief bibliography. We cite just three textbooks for background information on general topology and measure theory, and for this first volume, include only 25 items from the literature of our subject. Several extensive and excellent bibliographies are available, and there would be little purpose in reproducing a modified version of one of the existing lists. We have included in our references items specifically referred to in the text and others that might provide profitable additional reading. As a consequence, we have made no attempt, either in the text or in the exercises, to credit sources on which we have drawn or to trace the historical background of the ideas and results that have gone into the development of the subject.

Each of the chapters of this first volume has a final section devoted to a substantial list of exercises, arranged roughly in the order of the appearance of topics in the chapter. They were designed to serve two purposes: to illustrate and extend the results and examples of the earlier sections of the chapter, and to help the reader to develop working technique and facility with the subject matter of the chapter. For the reader interested in acquiring an ability to work with the subject, a certain amount of exercise solving is indispensable. We do not recommend a rigid adherence to order - each exercise being solved in sequence and no new material attempted until all the exercises of the preceding chapter are solved. Somewhere between that approach and total disregard of the exercises a line must be drawn congenial to the individual reader's needs and circumstances. In general, we do recommend that the greater proportion of the reader's time be spent on a thorough understanding of the main text than on the exercises. In any event, all the exercises have been designed to be solved. Most exercises are separated into several parts with each of the parts manageable and some of them provided with hints. Some are routine, requiring nothing more than a clear understanding of a definition or result for their solutions. Other exercises (and groups of exercises) constitute small (guided) research projects.

2.3. Review by: Joachim Cuntz.
zbMATH 0518.46046.

This is the long awaited first volume of the introduction to the theory of operator algebras (i.e. $C^{*}$- and von Neumann algebras) by R Kadison and J Ringrose. This first volume is largely preparation tor the second one which contains the bulk of the material. The first three chapters are devoted to a basic course in functional analysis with the intention, as the authors say, to cultivate intuition about infinite dimensional constructs. The contents here are (topological) linear spaces. Hilbert. spaces and operators on Hilbert spaces, (commutative) Banach algebras, Gelfand transform and Fourier analysis. Complete ease with these topics is indispensable for a serious study of operator algebras.

Chapters 3 and 5 treat the elementary theory or $C^{*}$- and von Neumann algebras, such as the order structure, functional calculus and GNS construction on the $C^{*}$-algebra side, strong and weak topologies, spectral theory for bounded and unbounded operators, the double commutant theorem, Kaplansky's density theorem on the von Neumann algebra side. These are the fundamentals on which the whole theory is based.

The emphasis in all this is on the pedagogical side. At the end of each chapter there is a large number of very carefully designed exercises.

Drawing on their rich experience the authors have succeeded in presenting a very attractive and well written book that conveys the flavour and the beauty of classical operator algebra theory and that should be ideally suited as a text tor a graduate course on the subject.

2.4. Review by: Robert S Doran.
Mathematical Reviews MR0719020 (85j:46099).

Selfadjoint algebras of bounded linear operators on a Hilbert space, closed in the weak operator topology, were introduced in 1929 by J von Neumann [Math. Ann. 102 (1929), 370–427; Jbuch 55, 825]. Such algebras were called "rings of operators'' by von Neumann and, a bit later, $W^{*}$-algebras by several other mathematicians. In 1957 J Dixmier called them "von Neumann algebras'' in his definitive monograph [Algebras of operators in Hilbert space (von Neumann algebras) (French), Gauthier-Villars, Paris, 1957]. The foundations for their study were laid in a fundamental series of papers by von Neumann and his collaborator F J Murray during the 1930s and the early 1940s. On the other hand, I M Gelfand's important work in 1939 on the representation of commutative Banach algebras as continuous functions on their maximal ideal space and his 1943 paper with M A Naimark [Mat. Sb. (N.S.) 12(54) (1943), 197–213] characterising uniformly closed selfadjoint algebras of bounded operators on a Hilbert space by a few simple axioms opened the way for a systematic study of what are known today as $C^{*}$-algebras.

The book under review, written by two eminent mathematicians each of whom has made major contributions to the theory of operator algebras, is the first of two volumes devoted to a "textbook'' treatment of the theory of $C^{*}$-algebras and von Neumann algebras. The authors' purpose is clearly stated in the first paragraph of the preface: "[Our] primary goal is to teach the subject and lead the reader to the point where the vast recent literature, both in the subject proper and in its many applications, becomes accessible." This point of view is to be contrasted with the many (excellent) existing books on the subject, which are primarily directed toward the research mathematician, or exist to provide background on operator algebras for applications in theoretical physics.
...
 Although the material in this first volume is standard and well known to experts, the authors have presented it in a fresh and attractive way which conveys the spirit and beauty of the subject. They are to be commended for writing a beautiful book which, in the reviewer's opinion, fulfils all of the promises made in the preface.

3.1. From the Preface.

Most of the comments in the preface appearing at the beginning of Volume 1 are fully applicable to this second volume. This is particularly so for the statement of our primary goal: to teach the subject rather than be encyclopaedic.

3.2. Review by: Robert S Doran.
Mathematical Reviews MR0859186 (88d:46106).

This book is the second of two volumes devoted to a "textbook treatment'' of the theory of $C^{*}$-algebras and von Neumann algebras. As with the first volume [1983] the primary goal is to teach the subject rather than try to be encyclopaedic. For this reason and also because of the enormous present-day size, scope, and complexities of the theory, the authors have made no attempt to include more recent developments involving the "noncommutative aspects'' of topology, differential geometry, shape theory, $K$-theory, etc. Lest one feel that this is cause for alarm let me hasten to say that what is presented here is an exciting and careful account of some very substantial mathematics! For example, the reader will find, among many other things, a detailed treatment of the Tomita-Takesaki modular theory, the Powers construction of a continuum of pairwise non-isomorphic factors of type III, and the theory of discrete and continuous crossed products. In general, the authors have treated those topics which they consider to be fundamental for an understanding of current research in operator algebras.
...
As with the first volume, this book has an outstanding collection of exercises (and groupings of exercises). Many of these constitute small guided tours, with hints, to nontrivial and important results. (Two volumes containing solutions to many of the exercises are to appear in the future.) The bibliography consists of a selected list of 108 items, with no attempt at completeness. The book closes with an index of symbols and a nicely cross-referenced general index that contains results from both volumes.

Although the material in this second volume is substantially more advanced than that in the first volume, the authors have maintained the same spirit and freshness of the presentation given there. Once again, they are to be commended for placing in our hands a beautiful volume which lives up to the promises made in the preface.

3.3 Review by: Joachim Cuntz.
zbMATH 0601.46054.

This book, written by two of the main contributors to the subject, is the first introduction to operator algebras that aims at being pedagogical. It concentrates on the classical non-commutative measure theory aspect of operator algebras (as opposed to "non-commutative topology" and "differential geometry" which have been developed much more recently), thus on von Neumann algebras - and on representation theory as far as $C^{*}$-algebras are concerned. Even with this restriction the theory has by now grown so complicated that it seems to be impossible to give more than the fundamentals of the theory in a pedagogical text. The authors do a very fine job in exposing these fundamentals up to and including Tomita-Takesaki theory.

The second volume, under review, starts with chapter 6 on Murray-von Neumann comparison theory for projections which remains up to this day one of the most beautiful and fertile ideas in von Neumann-algebra theory, allowing to associate a real valued dimension to certain subspaces of a Hilbert space.

The following chapter 7 develops the theory of normal states and contains a nice proof of the result that algebraic isomorphisms between von Neumann algebras with cyclic and separating vectors are implemented by unitaries. Chapter 8 is devoted to the existence of the trace which is the "integral" associated to the "measure" given by the dimension function alluded to above. It also contains the construction of some further examples of factors (others had been constructed in chapter 6).

Chapter 9 contains the study of the close connections that exist between a von Neumann algebra and its commutant and of modular (Tomita-Takesaki) theory. This theory is the basis for all of the more recent work in the classification of von Neumann algebras (Connes, Haagerup etc. ..) and is explained here with the care that it deserves. The necessary prerequisites on unbounded operators are contained in volume I. Chapter 10 contains some material on representations of $C^{*}$ algebras derived from von Neumann algebra theory. Chapter 11 treats (finite and infinite) tensor products of $C^{*}$- and von Neumann algebras in a standard way.

Chapter 12 then studies the very important examples of $C^{*}$- and von Neumann algebras obtained as infinite tensor products of matrix algebras and their classification. ...

Chapter 13 studies discrete and continuous crossed products of von Neumann algebras, and the particular case of the crossed product of a type III factor by its modular automorphism group. The last chapter 14 finally contains the theory of the decomposition of a general von Neumann algebra as a direct integral (a generalisation of the direct sum) of factors (a particular case is the disintegration of a unitary representation of a locally compact group). This theory permits to reduce most questions about von Neumann algebras to questions about factors.

Globally, this book is extremely clear and well written and ideally suited for an introductory course on the subject or for a student who wishes to learn the fundamentals of the classical theory of operator algebras. The authors plan two further volumes containing solutions to the many exercises in the text.

4.1. From the Publisher.

This volume is the companion volume to Fundamentals of the Theory of Operator Algebras. Volume I - Elementary Theory (Graduate Studies in Mathematics series, Volume 15). The goal of the text proper is to teach the subject and lead readers to where the vast literature - in the subject specifically and in its many applications - becomes accessible. The choice of material was made from among the fundamentals of what may be called the "classical" theory of operator algebras. This volume contains the written solutions to the exercises in the Fundamentals of the Theory of Operator Algebras. Volume I - Elementary Theory.

4.2. From the American Mathematical Society.

The many readers of the highly acclaimed treatise on which this volume is based will enthusiastically welcome the authors' decision to share with them their detailed solutions to the stimulating and penetrating exercises that appeared in the first volume. With clarity, thoroughness, and depth, these solutions supplement the basic theory and provide an invaluable tool for students and researchers in mathematics and theoretical physics and engineering. They provide models which test understanding and suggest alternate methods and styles for producing further solutions and extending knowledge in such areas as functional analysis and quantum physics.

4.3. Review by: Herbert Halpern.
zbMATH 0869.46028.

This volume 3 is reviewed together with volume 4 below.

4.4. Review by: Robert S Doran.
Mathematical Reviews MR1134132 (92m:46084).

This book contains complete detailed statements and solutions to the exercises in Volume I of the authors' two-volume work with the same title [Vol. I, Academic Press, New York, 1983; Vol. II, Academic Press, Orlando, FL, 1986]. It is a fitting companion to the existing volumes and a welcome addition to the literature on functional analysis. The exercises, 265 of them in all (which have been broken down into 573 manageable parts), were carefully designed by the authors to illustrate the results of the text and to expand its scope. Taken as a whole they constitute an elegant presentation of topics that are fundamental to the subject and provide the reader with the opportunity to develop "hands-on'' skills in the use of techniques appearing in the text. To facilitate this approach the authors designed the exercises so that they depend only on material that precedes them. Further, they separated them into smaller parts whose solutions are actually within reach of a diligent, well-motivated reader. Finally, it should be pointed out that the authors' solutions, which were developed from scratch specifically for this volume, are models of clarity and efficiency, reflecting their vast experience and insight into the subject.

4.5. Review by: Gert K Pedersen.
Bull. Amer. Math. Soc. 31 (2) (1994), 275-277.

When Johann von Neumann in 1929 proved the double commutant theorem (the weak closure of a unital *-invariant algebra of operators on a Hubert space is precisely the set of operators that commute with every operator commuting with the algebra), he set a train of thoughts in motion, whose destination even he could not have foreseen. The ensuing theory of operator algebras was originally motivated by the desire to create an axiomatic foundation for the emerging quantum theory, but during the long years of close collaboration between Johnny Neumann and Frank Murray things "got out of hand". The result was the monolith in twentieth century mathematics known as Rings of operators, I-IV. Respected by all, it was also daunting in its extreme technicality, and only a few dedicated mathematicians dared approach its summits.

Gelfand's work, which reached the West after World War II, resulted in the theory of abstract C*-algebras and gave operator algebras a new direction, aimed at unitary representations of topological groups, mainly Lie groups. Although this multiplied the number of first-rank mathematicians working with operator algebras, the 1945-70 era may be labelled "the silent growth". Largely unnoticed by most of the mathematical community and with only marginal interaction with other branches of mathematics, the theory gathered a number of powerful techniques and strong results.

The integration in mainstream mathematics and the recognition came in the seventies and culminated in the Fields medals awarded to Alain Connes (1982) and Vaughan Jones (1990). Now we see operator algebras used in geometry (foliations), PDE (index theorems), algebraic topology (AT-theory and cyclic cohomology), and as the link between knot theory and models in statistical quantum mechanics. The theory has certainly come of age and has become fashionable. At the same time, the demand for concise textbooks that cover the subject has become noticeable, and this brings us to the books under review. Fundamentals in the theory of operator algebras, Volumes I & II were published in 1982 and 1986. Since then they have quickly established themselves as The Textbooks in Operator Algebra Theory. To be sure, they do not match Dixmier's books in elegance, nor do they have the relentless drive in Sakai's slender volume; but, as the authors claim, they teach the subject. They do so in a sober, timeless manner, with respect for the student's intelligence and anticipation of his shortcomings.

One unavoidable consequence of the detailed, painstakingly accurate presentation in the Kadison-Ringrose volumes is the limited area they cover. The authors do not (by a long shot) tell all they know but give only the fundamentals. This is partially compensated by the generous number of exercises accompanying the text. The 715 exercises serve to illustrate and extend the results and examples in the text, but also help the reader to develop working techniques and facility with the subject matter. Some are routine, requiring nothing more than a clear understanding of a definition or a result for their solution. Other exercises (and groups of exercises) constitute small (guided) research projects. Anyone who claims to have solved them all is either boasting or is truly a master of our subject.

The present volumes, III and IV, contain the solutions of the exercises in the first two volumes, in what the authors feel to be optimal form (and few would quibble with that claim). This gives the student a model with which his own solutions can be compared and an indicator of the method and style for producing further solutions on an individual basis. For the more experienced and, maybe, impatient researcher it gives a speedy route through one or another of the many special topics that supplement those of the text proper of "Fundamentals". These topics include:

Algebras of affiliated operators
Approximate identities in $C^{*}$-algebras
β-compactification of $\mathbb{N}$
Canonical anticommutation relations
Characterisations of von Neumann algebras among $C^{*}$-algebras
Compact operators
Completely positive maps
Conditional expectations
The Connes invariant
Coupling constant and operator
Derivations and automorphisms
Diagonalisation of operator-valued matrices
Dixmier approximation theorem
Examples of extreme points
Extremely disconnected spaces
Flip automorphisms
Friedrichs extension
The fundamental group of a $II_{1}$-factor
Generalised Schwarz inequality
Elements of harmonic analysis
Ideals in operator algebras
Isometries and Jordan homomorphisms
Modular theory
Non-normal traces
Relative commutants
Representations
Stone-Weierstrass theorems for $C^{*}$-algebras
Strong continuity of operator functions
Tensor products of operator algebras
Unitary implementation of automorphisms
Unitary elements of $C^{*}$-algebras
Vectors and vector states

Presented in alphabetical order, as above, the list could give the reader a dizzying feeling of embarras de richesse, but, of course, each topic is presented in the book where it connects with the fundamental part of the theory. It is instructive, therefore, also to give the headings of the fourteen chapters that constitute volumes I and II:

Linear spaces
Basics of Hubert space and linear operators
Banach algebras
Elementary $C^{*}$-algebra theory
Elementary von Neumann algebra theory
Comparison theory of projections
Normal states and unitary equivalence of von Neumann algebras
The trace
Algebra and commutant
Special representations of $C^{*}$-algebras
Tensor products
Approximation by matrix algebras
Crossed products
Direct integrals and decompositions

Many years ago Paul Halmos published a delightful book called A Hilbert space problem book that presented the elementary theory of operators in a series of problems (with hints and solutions). Appealing though this approach may be, it will probably not work in a highly technical field like operator algebras, where the teacher must step in from time to time to tell the student about heavy machinery that has to be developed before further progress can be made. Yet Halmos's dictum stands: The only way to learn mathematics is to do mathematics.

The completed four-volume treatise by Kadison and Ringrose seems to me to utilise the best of both methods: The fundamentals are explained as text to be read. The numerous exercises are inserted to challenge the curiosity, to develop "hands-on" skills, and to give a glimpse of wider spaces. Now the solutions, as in Halmos's book, appear at the end as the logical conclusion. The authors have erected a monument in mathematics in the tradition of Courant-Hilbert, Dunford-Schwartz, Hewitt-Ross, and Reed-Simon.

5.1. From the Publisher.

This volume is the companion volume to Fundamentals of the Theory of Operator Algebras. Volume II - Advanced Theory (Graduate Studies in Mathematics series, Volume 16). The goal of the text proper is to teach the subject and lead readers to where the vast literature - in the subject specifically and in its many applications - becomes accessible. The choice of material was made from among the fundamentals of what may be called the "classical" theory of operator algebras. This volume contains the written solutions to the exercises in the Fundamentals of the Theory of Operator Algebras. Volume II - Advanced Theory.

5.2. Review by: Gert K Pedersen.
Bull. Amer. Math. Soc. 31 (2) (1994), 275-277.

Volume IV is reviewed together with Volume III. See above.

Review by: Nick Lord.
The Mathematical Gazette 82 (493) (1998), 156-157.

Kadison and Ringrose's Fundamentals of the theory of operator algebra was originally published by Academic Press in the 1980s. This two-volume work (split into 'Elementary theory' and 'Advanced theory') met with immediate acclaim from functional analysts as a clear, careful, self-contained introduction to C*- and von Neumann algebra theory - an area in which it is notoriously easy to intimidate rather than initiate graduate students! Experts too relished the fresh gloss that the immensely experienced duo of authors brought to the development of the theory, and lecturers appreciated the large and eclectic set of exercises provided at the end of each chapter which reflected the authors' total familiarity with the literature. The volume under review is a companion book to Volume II and contains very full solutions to the 450 exercises therein: there is also a similar companion book (Volume III) to Volume I which was not sent for review. The exercises are presented in the order that they occur in Volume II, but there is also a useful cross-grouping by themes, index and bibliography. Of particular note in this technically demanding area is that exercises and solutions are precisely matched to the stage of development reached in the text: no rabbits are pulled out of hats! Moreover, and this is a real tribute to the author's skill, many of the solutions have an effortless inevitably about them which makes the reader feel that he or she could have devised them themselves: this is enormously encouraging to the novice.

I am happy to report that Volumes I and II have just been reissued by the
American Mathematical Society in their Graduate Studies series: this new lease of life is much to be welcomed since these books are 'must-haves' for operator algebraists: the companion solution books, Volumes III and IV, which are about to be republished by the AMS, significantly extend their usefulness to graduate students and lecturers

5.3. Review by: Robert S Doran.
Mathematical Reviews MR1170351 (93g:46052).

This book is the final volume of a four-volume set on operator algebras. It contains complete detailed statements and solutions to the exercises in Volume II [1986]. As with Volume III [1991], which contains statements and solutions to the exercises in Volume I, this volume contains carefully designed solutions that have been developed from scratch and which depend on techniques and methods in Volume II. A tremendous amount of work has obviously gone into the writing of these solutions, and the authors are to be highly commended for their efforts.

5.4. Review by: Herbert Halpern.
zbMATH 0869.46029.

These two volumes, which appeared several years ago, are the complete solutions of problems found in the first two volumes of R V Kadison's and J R Ringrose's treatise "Fundamentals of the theory of operator algebras", volumes I and II, Academic Press (1983) and (1985). The problems themselves range from observations of detailed constructions taking several pages and are grouped in the same 14 chapters as the treatise: in volume III are Linear Spaces, Basics of Hilbert space and Linear Operators, Banach Algebras, Elementary C*-algebra Theory, Elementary von Neumann Algebra Theory; and in Volume IV are Comparison of Projections, Normal States and Unitary Equivalence, the Trace, Commutant, Special Representations of C*-algebras, Tensor Products, Approximation by Matrix Algebras, Crossed Products, Direct Integrals and Decompositions.

The statements of the exercises are repeated with reference numbers of the Volumes I and II and are completely solved so the solutions in Volumes III and IV can be read for the most part independently of Volumes I and II and even independently of outside references. For example, a summation theorem of positive real numbers is worked out in detail in 13.4.11. However, reference back to theorems in Volume I and II appear and statements like "With the notation of Theorem 13.1.15 ..." which is in Volume II occasionally appear. But the 4 volume set is almost completely self-contained.

The problems come from the literature but, as the authors note, no attempt is made to identify the source since the proofs have been entirely re-worked with rigorous attention to precision, clarity and completeness of detail. The book might be said to be the culmination of the classical theory as outlined in the two books of J Dixmier ["Les algèbres d'opérateurs dans l'espace hilbertien", Gauthier-Villars, Paris (1957)] and ["Les $C^{*}$-algèbres et leur représentations", Gauthier-Villars, Paris (1964)]. The current volumes have some interesting things perhaps not found or not found in detail in the Dixmier's treatises, for example, infinite tensor products and ultraproducts, Jordan homomorphisms, the fundamental group of a type $II_{1}$ matricial (i.e., hyperfinite) factor, completely positive maps (and Stinespring's Theorem), the modular operator, monotone closure, strong operator continuous functions, nuclear $C^{*}$-algebras, the geometry of unitary operators, the Dauns-Hofmann theorem, CAR algebras and some detailed work on matricial $C^{*}$-algebras, the operator algebra generated by two projections, crossed products and the crossed product by the modular action.

6.1. From the Publisher.

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.

6.2. Review by: Editors.
Mathematical Reviews MR1468229 (98f:46001a).

A good many textbooks on operator algebras have appeared during the last 20 years, each with its own focus and features, and all of them valuable. Kadison and Ringrose's masterly treatise has maintained its popularity
...

6.3. Review by: Petr P Zabreiko.
zbMATH 0888.46039

This is the second edition of the well-known advanced textbook. The book consists of 5 chapters ("Linear spaces", "Basics of Hilbert spaces and linear operators", "Banach algebras", "Elementary $C^{*}$-algebra theory", "Elementary von Neumann algebra theory"), Bibliography, Index of notations, and Index.

In the fifties L V Kantorovi
wrote that functional analysis can be divided into five directions: "Analysis in Hilbert spaces", "Analysis in Banach spaces", "Analysis in topological linear spaces", "Analysis in ordered linear spaces", and "Theory of Banach algebras and representations". The treatise by R V Kadison and J R Ringrose is one of the best among books devoted to the first half of the fifth direction. In the first volume one can find basics of Banach and Hilbert spaces and the theory of linear operators between them, tensor products and Hilbert-Schmidt operators, elements of general theory of Banach algebras including spectral theory and holomorphic functional calculus, theory of positive linear functionals in $C^{*}$-algebras, projection technique and construction in von Neumann algebras. Although the main aim of this book is to give elements of non-commutative real analysis, the theory of commutative (abelian) algebras is presented with exhaustive completeness. In general, the description of the mathematical theory presented in this book is clear and accurate; exercises given in each chapter make the essential supplement to the basic text. The book can serve as a textbook for a one semester course of elementary functional analysis (Ch. 1-3), of $C^{*}$-Banach algebras (Ch. 3-5), or a two-semester course of Banach algebras; however, the book can also be used for individual study as well as courses or seminars. Undoubtedly, the acquaintance with this book is useful for all teachers in functional analysis and researchers in the corresponding field.

7.1. From the Publisher.

This work and Fundamentals of the Theory of Operator Algebras. Volume I, Elementary 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.

7.2. Review by: Editors.
Mathematical Reviews MR1468230 (98f:46001b).

A good many textbooks on operator algebras have appeared during the last 20 years, each with its own focus and features, and all of them valuable. Kadison and Ringrose's masterly treatise has maintained its popularity
...