# Jacques Dixmier

### Quick Info

Born
1 January 1924
Saint-Étienne, France

Summary
Jacques Dixmier is a French mathematician who worked on C*-algebras and other operator algebras.

### Biography

Jacques Dixmier's parents were both school teachers. His father taught languages, French, Latin and Greek. Although Jacques was born in Saint-Étienne, near Lyon, he wasn't brought up there since his father was employed in schools in different towns and, of course, the family moved. From Saint-Étienne they moved to Rouen and then to Lille. In 1933, when Jacques was nine years old, the family moved to Versailles. His father was a professor at the Lycée in Versailles while his mother taught at the elementary school there.

It was in Versailles that Jacques studied for his baccalaureate. The first year that he studied there he had a poor mathematics teacher but that was only for one year and, for every year after that, he had an outstanding mathematics teacher. In fact he said later in life that it was almost certainly due to the high quality of the mathematics teaching that he decided that mathematics was the subject for him. Now he wasn't in Versailles for all the years of World War II (1939-1945) for he spent one of them in Saint-Brieuc, a town in the Brittany region in northwest France. He said [6]:-
Even there I had a very good teacher in mathematics.
After his baccalaureate, Dixmier remained in Versailles to prepare for his university studies in mathematics. His father advised him to study engineering but Dixmier had his heart set on mathematics and his mother supported his plans so his father, slightly reluctantly, agreed that he could aim at being admitted to the École normale supérieure. He spent two preparatory years, the mathématiques supérieures year and then the mathématiques spéciales year, studying in Versailles.

In October 1942 Dixmier entered the École Normale Supérieure in Paris. His main teachers were Henri Cartan and Gaston Julia. In fact he was fortunate to have Henri Cartan as a lecturer since he was on the faculty at Strasbourg but he could not return there since the Germans controlled the university. He said [6]:-
I followed [Julia's] course in 1943 during my first year at École Normale. Julia gave a course every year in advanced mathematical analysis. This sort of course ... was too difficult for us students. But we were very anxious to have a diploma as soon as possible because of the war; you did not know what could happen in a few months! Julia was an extremely good teacher and we understood him. In fact the level of his course about Hilbert space was not very high but we understood almost everything. As a result, Hilbert space became for me as familiar as ordinary space. That was certainly the reason for my research orientation later on. It is very important to be completely familiar with the basics of your subject.
Henri Cartan taught a calculus course but he had modified the course to present it in the style of Bourbaki [3]:-
The main point was to introduce us to so-called modern algebra (groups, rings, vector spaces, ...). Therefore, Henri Cartan's lectures were combining a rather standard course on calculus but in the spirit of the not yet published volume of Bourbaki, entitled 'Functions of a real variable'. ... In the more "modern" or "advanced" part, we were told about integers mod p, Grassmann calculus, Fourier transforms, and fixed-point theorems. Henri Cartan knew how to involve us in the class ... We were seventeen in the class ...
In his second year at the École Normale, 1943-44, Dixmier was again taught by Henri Cartan. However, the class was [3]:-
... reduced to nine during the second year, because of the hardships of the war and also because in the second year he taught only the students who had chosen mathematics as a major. After consulting the class, he decided to teach a course on Lie groups.
Dixmier joined the resistance which was a dangerous thing to do. He attended a few meetings but was never part of any armed resistance. During the vacation between Dixmier's second and third years the war came to an end. However, in the last days of the German occupation of France tragedy struck the École Normale. On the night of 4 August 1944, the Gestapo arrested a student who was absent from Rue d'Ulm. They then carried out a systematic search of the École Normale. Dixmier was there at the time [6]:-
There is something which concerned me personally that I still do not understand. I was in the École Normale Supérieure doing something that was probably illegal. I heard some noise but I did not know what had happened. So, I went to the entrance of the École Normale Supérieure. Two officers of the Gestapo were there. They shouted out and I was taken completely by surprise. They fired and I don't understand how I was not wounded. They took me and I was sent to detention in an office.
The director of the École Normale, Georges Bruhat (the father of Yvonne Choquet-Bruhat and François Bruhat), was not present and neither was Jean Baillou, the general secretary of the École Normale Supérieure. Their wives, however, were present and the Gestapo took Mrs Bruhat and Mrs Baillou hostage. The next day they released the students, including Dixmier, and Mrs Bruhat and Mrs Baillou but took Georges Bruhat and Jean Baillou prisoner. They were deported, and Bruhat died at the Buchenwald concentration camp.

Dixmier's third year of study was spent working towards the agrégation where the aim was to qualify students to teach. At this point the three year degree was changed into a four year degree but, although technically still at the École Normale, Dixmier worked at the Centre National de la Recherche Scientifique. It was during this year 1945-46 that he worked on his thesis, advised by Gaston Julia. Now Julia had close links with the Dixmier family, having been a student at the École normale at the same time as Dixmier's father. Julia also lived in Versailles and his children had been taught by Dixmier's parents. However, Dixmier did not find relations with Julia straightforward [6]:-
I think he was a very good mathematician. ... He was not an easy person. As you probably know, he was politically at the extreme right and hoped for a German victory during the war.
Although Julia did not directly suggest the problem that Dixmier worked on for his thesis, it was Julia's lectures on Hilbert spaces that inspired him [6]:-
I was certainly inspired by some of the things he had told us - generally speaking, on Hilbert space. In my thesis, I studied some particular subspaces, non-closed but not arbitrary - very special subspaces; my entire thesis is concerned with this class of subspaces.
He published his first paper Les idéaux dans l'ensemble des variétés d'un espace hilbertien in 1946. In 1946-47 he continued to work at the Centre National de la Recherche Scientifique and was helped in his research by Roger Godement. Although both had been students at the École Normale Supérieure, Godement was two years older than Dixmier and they had not known each other at that time. However, Godement now was a great help to Dixmier, telling him about unitary representations and showing him results on this topic which he had proved but had not published.

Towards the end of this year at the Centre National de la Recherche Scientifique he was offered an appointment as maître de conférences in Toulouse. By this time Dixmier was married and his wife continued to work as a secondary school teacher. They had a one room apartment in Paris and travelled a lot to their respective jobs. He said [6]:-
We travelled a lot and travelling by train from Paris to Toulouse was not easy at all in 1947. The bridge over the Loire at Orléans had not been completely repaired.
His next paper was L'adjoint du produit de deux opérateurs fermés (1947) which he published in the Annales de la Faculté des sciences de Toulouse. In the following year, he delivered the lecture Homologie et cohomologie singulières to Henri Cartan's seminar in Paris. He had completed the work for his thesis before taking up the position in Toulouse and his thesis Étude sur les variétés et les opérateurs de Julia, avec quelques applications was published in 1949. Also in 1949 he published the paper Les anneaux d'opérateurs de classe finie . This was the year that he was invited to become a member of the Bourbaki team [6]:-
One day in 1949, Serre and Samuel, who both knew me, approached me and asked whether I would accept becoming a member of Bourbaki. At the time, this was extremely flattering; I jumped at it!
In 1949 a new position was created in Dijon. This was just one of many new positions that were being created around France since there was a large increase in student numbers and the government tried to cope with this increase by expanding the universities. Living in Paris and working in Toulouse was certainly not an ideal thing for anyone to do, so Dixmier decided that, since Dijon was much closer to Paris, he would apply for the position there. Dixmier was on the faculty at Dijon from 1949 to 1955 when he was nominated for a position at the Institut Henri Poincaré in Paris. Speaking about the beginning of his time in this position, he said [6]:-
The number of students was enormous. I held my courses in a place called Conservatoire des Arts et Métiers; there was a big auditorium. During the first 2 or 3 years, I had 500 students.
He spent the rest of his career in Paris.

Dixmier wrote some excellent books. These include: Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann (1957); Les $C^{*}$-algèbres et leurs représentation (1964); L'intégrale de Lebesgue (1964); Cours de mathématiques du premier cycle (1967); Cours de mathématiques du premier cycle. Deuxième année (1967); Algèbres enveloppantes (1974); and Topologie générale (1981).

Extracts of reviews of these books may be seen at THIS LINK.

However, to indicate the quality of these texts, we should give here a very short extract from a review by Jean Dieudonné of Dixmier's undergraduate textbook Cours de mathématiques :-
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 ...
We also give a very short extract from a review by Anthony Joseph (1942-) of Dixmier's monograph Algèbres enveloppantes :-
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.
In fact these books give a good indication of the areas to which Dixmier made important contributions, namely operator algebras, von Neumann algebras, C*-algebras and unitary representation theory. Later he worked on enveloping algebras and finally on invariant theory and partition theory. MathSciNet lists 187 published items for Dixmier (up to 2014). There his main areas of research are given as 'Functional analysis', 'Nonassociative rings and algebras', and 'Topological groups and Lie groups'.

When writing books, Dixmier liked to follow the style of Bourbaki but when he lectured his style was very different from that of his colleagues. He spoke about this when interviewed [6]:-
I have certainly a philosophy [about lecturing] that differs from most of my colleagues. In order to make good talks, I prepared them word for word. Most of my colleagues think that this is a far too formal attitude, that you have to rely on inspiration. I do not believe that at all. I have often heard the opinion that it is good to get stuck; otherwise students do not understand that there is a real difficulty. I must say, I do not agree at all. I was usually very careful to write in big letters, to avoid talking into the blackboard and so on. I really wanted to be different from Arnaud Denjoy; it is said that, during a lecture, he thought a, he said b, he wrote c, and d would have been correct! In fact, when I was a student at the École normale supérieure, during my second year, I tried to follow a course given by Denjoy. After the second lecture, I left; it was hopeless for me. Well, not for everyone; Gustave Choquet was a pupil of Denjoy. I know that the written work of Denjoy is very good. But as a teacher, he was terrible. For written text, I must admit that I like the Bourbaki style!
During his career, he supervised 20 Ph.D. students, the most famous being Alain Connes. In 1984 Dixmier reached the age of 60 and, since a new law had been passed making it possible to retire at that age, Dixmier chose to retire but continued to undertake research at the Institut des Hautes Études Scientifiques at Bures-sur-Yvette. He spent five years at IHES. However, after that he did other things in addition to mathematics, for example, writing two science fiction books L'Aurore des dieux (1993) and Le Septième arrhe (1995). In 2013, in collaboration with Alain Connes and Danye Chéreau, he published the book Le Théâtre quantique . Here is the publisher's description:-
For years, you've been hearing about quantum mechanics, a major 20th -century discovery, so certain terms have become familiar and they stir your imagination: wave functions, particle interaction, uncertainty principle, Feynman diagrams. At times, you find yourself dreaming about whimsical photons, disturbing experiments and their unexpected results, but you don't have the time to spend hundreds of hours on difficult mathematical and scientific studies that, at best, would make you understand the relevance of Richard Feynman's quip: "If you think you understand quantum mechanics, you don't understand quantum mechanics." But this book offers a quick and entertaining look into this magical universe. It gives you a front-row seat in the quantic theatre, and you'll be amazed at what you see.
Dixmier received several prestigious prizes and medals for his outstanding contributions. He was awarded the Ampère prize by the Académie des Sciences in 1976, and the Leroy P Steele prize from the American Mathematical Society in 1993:-
.. for his books "von Neumann Algebras (Algèbres de von Neumann)" (1957), "$C^{*}$-Algebras (Les $C^{*}$-Algèbres et leurs Representations)" (1964), and "Enveloping Algebras (Algèbres Enveloppantes)" (1974).
He was also awarded the Émile Picard medal by the Académie des Sciences in Paris in 2001. The citation states:-
The Émile Picard medal is awarded to Jacques Dixmier for the whole of his work collecting into a major synthesis the analysis of operator algebras of von Neumann with the algebraic theory of enveloping algebras, allowing the current progress in the theory of quantum groups. Author of books used for decades in France as well as Russia and the United States, Jacques Dixmier has been for thirty years at the University Pierre and Marie Curie a generous head of the school, who has promoted theses that are landmarks.

### References (show)

1. A Borel, Twenty-five years with Nikolas Bourbaki (1949-1973), Notices Amer. Math. Soc. 45 (3) (1998), 373-380
2. R F Brown, Review: General topology, by Jacques Dixmier, The American Mathematical Monthly 94 (5) (1987), 475-479.
3. P Cartier, J Dixmier, A Douady, C Houzel, J-P Kahane and M Karoubi, Cartan as a Teacher, Notices Amer. Math. Soc. 57 (8) (2010), 961-971.
4. R P Gillespie, Review: Problèmes de Mesure (1962), by H Cartan, J Dixmier, P Dubreil, A Lichnerowicz and A Revuz, The Mathematical Gazette 48 (364) (1964), 243-244.
5. N J Lord, Review: General topology, by Jacques Dixmier, The Mathematical Gazette 69 (449) (1985), 245-246.
6. M Raussen, Interview with Jacques Dixmier, European Mathematical Society Newsletter 72 (2009), 34-41.