mathematics

Jacques Dixmier

Jacques Dixmier is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Jacques Dixmier rather than just read about it. In short: Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping.

Key takeaways

Jacques Dixmier belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Jacques Dixmier to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Jacques Dixmier from memory before moving on to harder problems.

Reference excerpt

Jacques Dixmier (born 24 May 1924) is a French mathematician. He worked on operator algebras, especially C*-algebras, and wrote several of the standard reference books on them, and introduced the Dixmier trace and the Dixmier mapping.

Biography Dixmier received his Ph.D. in 1949 from the University of Paris, and his students include Alain Connes. In 1949 upon the initiative of Jean-Pierre Serre and Pierre Samuel, Dixmier became a member of Bourbaki, in which he made essential contributions to the Bourbaki volume on Lie algebras. After retiring as professor emeritus from the University of Paris VI, he spent five years at the Institut des Hautes Études Scientifiques. Often, there is made the erroneous claim that Dixmier originated the name von Neumann algebra for the operator algebras introduced by John von Neumann, but Dixmier said in an interview that the name originated from a proposal by Jean Dieudonné. Dixmier was an invited speaker at the International Congress of Mathematicians in 1966 in Moscow with the talk Espace dual d'une algèbre, ou d'un groupe localement compact and again in 1978 in Helsinki with the talk Algèbres enveloppantes.

Publications J. Dixmier, C*-algebras. Translated from the French by Francis Jellett. North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. xiii+492 pp. ISBN 0-7204-0762-1 A translation of Les C*-algèbres et leurs représentations, Gauthier-Villars, 1969. Dixmier, Jacques (1996) [1974], Enveloping algebras, Graduate Studies in Mathematics, vol. 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0560-2, MR 0498740 A translation of Algèbres enveloppantes, Cahiers Scientifiques, Fasc. XXXVII. Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974. ii+349 pp. J. Dixmier, von Neumann algebras, Translated from the second French edition by F. Jellett. North-Holland Mathematical Library, 27. North-Holland Publishing Co., Amsterdam-New York, 1981. xxxviii+437 pp. ISBN 0-444-86308-7 A translation of Les algèbres d'opérateurs dans l'espace hilbertien: algèbres de von Neumann, Gauthier-Villars (1957), the first book about von Neumann algebras.

Books Dixmier, Jacques (1984). General Topology. Undergraduate Texts in Mathematics. Translated by Berberian, S. K. New York: Springer-Verlag. ISBN 978-0-387-90972-1. OCLC 10277303.

Worked examples

Example 1 — a first encounter with Jacques Dixmier

Start with the simplest possible case. Write down what Jacques Dixmier claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Jacques Dixmier before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Jacques Dixmier ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Jacques Dixmier

In research: Jacques Dixmier appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Jacques Dixmier in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Jacques Dixmier is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1924 births, 20th-century French mathematicians, 21st-century French mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Jacques Dixmier outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Jacques Dixmier in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Jacques Dixmier means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Jacques Dixmier out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Jacques Dixmier in simple terms?

Why does Jacques Dixmier matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Jacques Dixmier?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Jacques Dixmier.