## Reviews of Jacques Dixmier's books

**1. Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann (1957), by Jacques Dixmier.**

**1.1. Review by: Lynn Harold Loomis.**

*Mathematical Reviews*, MR0094722

**(20 3 1234)**.

This book presents a thorough reworking and systematic development of the theory of weakly-closed *-algebras of bounded operators on a Hilbert space, which the author renames von Neumann algebras. Although only twenty years old, this has been one of the most active and fruitful areas of modern functional analysis, as the author's bibliography of nearly two hundred items attests. The present exposition divides into three large chapters: the first on the global theory, the second on the direct integral reduction theory, and a final one presenting those standard but more technical topics which do not fit into the pattern of development chosen by the author.

**2. Problèmes de Mesure (1962), by H Cartan, J Dixmier, P Dubreil, A Lichnerowicz and A Revuz.**

**2.1. Review by: Robert P Gillespie.**

*The Mathematical Gazette*

**48**(364) (1964), 243-244.

The main part of the book and the part properly described by the title consists of the lectures on 'Volumes of Polyhedra' by Henri Cartan, on 'Measure of Angles' by Jacques Dixmier and on the 'Theory of Integration' by André Revuz. ... The second lecture on 'Measure of Angles' ... would provide ideal material for discussion in a sixth form which had already met the simpler ideas about groups and equivalence classes and was familiar with the properties of complex numbers and of the exponential function.

**2.2. Review by: Paul Richard Halmos.**

*Mathematical Reviews*, MR0159917 **(28 #3133)**.

This is a set of five expository articles, two of which have nothing to do with measure. ... Dixmier approaches the concept of angle (in the plane) via linear algebra. He defines angle for an ordered pair of unit vectors as the unique proper rotation that carries the first onto the second. There is a natural isomorphism between proper rotations and complex numbers of modulus 1; the trigonometric functions of an angle are defined via this isomorphism.

**3. Les C*-algèbres et leurs représentation (1964), by Jacques Dixmier.**

**3.1. Review by: Richard Friederich Arens.**

*Mathematical Reviews*, MR0171173

**(30 #1404)**.

A C*-algebra is a uniformly closed *-algebra of bounded operators on some Hilbert space. The "von Neumann algebras", which form the subject of the author's definitive treatise *Les algèbres d'opérateurs dans l'espace hilbertien* (*Algèbres de von Neumann*) [1957], form an important special case. A C*-algebra can be studied abstractly inasmuch as its definition involves only the norm and the *-operation. The material included here is mainly due to James M G Fell, James G Glimm, Richard Kadison, Irving Kaplansky, Segal, and (as he modestly fails to emphasize) the author.

**4. L'intégrale de Lebesgue (1964), by Jacques Dixmier.**

**4.1. Review by: Editors.**

*Mathematical Reviews*, MR0179320

**(31 #3568)**.

Text for a first course in integration.

**5. Cours de mathématiques du premier cycle (1967), by Jacques Dixmier.**

**5.1. Review by: Jean Dieudonné.**

*Mathematical Reviews*, MR0389454

**(52 #10285)**.

This book contains the material taught to first year students of mathematics of French universities, according to new programmes enacted in 1966. One finds here the remarkable qualities of exposition of the author, and one can say without exaggeration that it is a model of what a course for beginners should be. ... Everywhere the concepts studied are presented with the utmost brevity and clarity, unless the author is forced to refer to matters which should in principle be treated in secondary education, such as the notion of angle (which, as we know, the "classic" presentation is only a tissue of nonsense). The examples are always relevant and well chosen ....

**6. Cours de mathématiques du premier cycle. Deuxième année (1967), by Jacques Dixmier.**

**6.1. Review by: Jean Dieudonné.**

*Mathematical Reviews*, MR0389455

**(52 #10286)**.

This volume follows the one the author recently published for freshmen and covers substances of the mathematics curriculum of the second year, apart from the calculation of probabilities and mechanics. ... the author has collected a little less basic issues of algebra than the first volume ... 240 pages deal with issues of analysis, and the last three chapters of the book are devoted to the basic theory of curves (Frenet formulas) and classification of quadrics.

**7. Algèbres enveloppantes (1974), by Jacques Dixmier.**

**7.1. Review by: Anthony Joseph.**

*Mathematical Reviews*, MR0498740

**(58 #16803b)**.

The original French version is the fruitful outcome of many of the author's seminars in Paris-VI. It is a great pleasure to review a work of such excellence and which has done so much to promote the formation of this new branch of mathematics. For the graduate student it is a masterpiece of pedagogical writing, being succinct, wonderfully self-contained and of exceptional precision. Moreover the end of each chapter is generously supplied with research notes which together with the list of open problems at the end of the book are a rich contribution to further developments. However, it is not especially recommended for the casual reader, as only a short preface indicates the basic aims and techniques of the subject, whilst the text itself is not given to any superfluous discussion.

**8. Topologie générale (1981), by Jacques Dixmier.**

**8.1. Review by: Ivan L Reilly.**

*Mathematical Reviews*, MR0637202

**(83a:54002)**.

This book is designed to be an introduction to general topology, and is based on a one-semester course of lectures given in 1979-80 to third-year students at the University of Paris VI. The book contains the usual topics in a rudimentary course in this area. The range of material and the book's particular emphasis can probably best be indicated by listing the ten chapter headings: Topological spaces, Limits and continuity, Constructions of spaces (subspaces, products and quotients), Compact spaces, Metric spaces, Limits and convergence of functions, Real-valued functions, Normed spaces, Infinite series, and Connected spaces. Each chapter develops the appropriate theory interspersed with many relevant examples. An interesting feature is that each chapter is preceded by a short summary of the material to be discussed. Several times the author refers back to the book's introduction where he has attempted to set the development of topology in the context of nineteenth and twentieth century mathematics.

**8.2. Review by: Nick J Lord.**

*The Mathematical Gazette* **69** (449) (1985), 245-246.

This review relates to the English translation *General Topology* (1984).

This book offers an attractively succinct overview of undergraduate general (analytic) topology. The pace is brisk and the style of presentation recognisably Bourbaki so that a certain degree of mathematical maturity is required on the part of the reader: this is not an introductory text but would be a valuable adjunct to a second-level course. ... The book may not be to every student's taste - some will find it rather short on significant worked examples and motivational passages - but I would recommend it unreservedly to anyone who is fairly sure of the foundations and who wants a concise and elegant guide to the impressive edifice that can be erected on them.

**8.3. Review by: Robert F Brown.**

*The American Mathematical Monthly* **94** (5) (1987), 475-479.

This review relates to the English translation *General Topology* (1984).

Dixmier [is] written for the student's first course in topology. ... A quick look at Dixmier's table of contents brought immediately to my mind a sentence from the Preface of John Kelley's pioneering (1955) text for this very course: "I have, with difficulty, been prevented by my friends from labelling it: What Every Young Analyst Should Know." Putting the tables of contents of Dixmier and Kelley side by side, it is even possible to line up roughly equivalent chapters. This is not to suggest that Dixmier's book is a new incarnation of Kelley, but rather that any text that teaches topology with an eye towards more advanced analysis is bound to look superficially like any other one. ... When [a student] picks up Dixmier's text, she won't realize that Chapter VIII (" Normed Spaces") is intended to place her foot firmly on the first step of a staircase headed for the heights of Functional Analysis. She will however notice that the text looks a bit different from those she is familiar with. Each paragraph or two is headed with a boldface number such as 6.2.9, often followed by a familiar word; usually "Theorem" but sometimes " Corollary" and, less often, "Example," "Definition," or "Remark." What she won't see is the indication "Proof" after any of those theorems nor an end-of-proof symbol. The reason is that Dixmier doesn't need any of that apparatus because the paragraphs immediately after the statement of the theorem and up to the next boldface number invariably present the proof. But where, the experienced instructor may ask, are the motivating remarks, special cases, and other ruffles and flourishes with which so many authors like to decorate their arguments, especially the difficult ones? It seems there aren't any. The proofs are neat, formal, and boy are they condensed! They fit just right on the page, between those boldface numbers. ... An instructor who is willing to put an unreasonable amount of effort into the course could use Dixmier as what amounts to a very high-quality outline, but it would be an uphill battle to drag [a student] through it. There are other good texts intended for American students that would be a lot easier to use. However, a more mature mathematician who wanted to review point-set topology, especially as it relates to functional analysis, might find Dixmier right on target.

JOC/EFR November 2014

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