Noncommutative Analysis

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A few words on the book “Functional Analysis” by Peter Lax

I recently bought Peter Lax‘s textbook on Functional Analysis, with a clear intention of having it become my textbook of choice. I heard nice opinions of it. Especially, I thought that I would find useful Lax’s point of view that gives the Lebesgue spaces L^p the primary role, and pushes the Lebesgue measure to play a secondary role (I wrote about this subject before).

In fact, the first time I heard of this book was in the following MathOverflow question, that is likely to have been triggered by Lax’s comment (p. 282) “[It is] an open question if there are irreducible operators in Hilbert space, and it is an open question whether this question is interesting” (to spell it out, Lax is making the remark that it is an open question whether the invariant subspace problem is interesting!). I have nothing against the invariant subspace problem (of course it is interesting!), but I was sure that I would love reading a book by a mathematician with a bit of self humor (it turns our Lax’s remark appears at the end of a chapter devoted to invariant subspaces).

In one sense the book was a disappointment, in that I realized that I could not, or would not like to, use it as a textbook for courses I teach. I really don’t like its organization, and I don’t love his style. And the pushing back of Lebesgue measure is a very minor topic (which makes good sense because this is a textbook that was used for second year graduate students). I will have to write my own lecture notes.

On the other hand, the book contains a lot of very neat applications of functional analysis (I won’t spoil it for you, but some are really fun!), and so much better to have it coming from someone like Lax. That’s enough to justify the purchase.

But mathematics aside, this book will now stay close to my heart and change the way I approach the subject of functional analysis. This is because of several historical notes dotted throughout the book. Here is an example, which caught me completely unprepared at the end of Chapter 16 (p. 172) (I read the book non-linearly):

During the Second World War, Banach was one of a group of people whose bodies were used by the Nazi occupiers of Poland to breed lice, in an attempt to extract an anti-typhoid serum. He died shortly after the conclusion of the war.”

And so, when reading through the book, we meet some of our familiar (and also not-so-familiar) heroes of functional analysis being deported to concentration camps, or committing suicide knowing what awaits them from the hand of the Nazis, or somehow making it safely to the west. Some other players are involved in the race to construct a nuclear bomb, or to crack the Enigma code (as I learned from the book, Beurling also broke this code – well, we all knew that he was very smart, but this was surprising. By the time we reach the chapter on Beurling’s theorem, we are not surprised that Lax cannot just leave the anecdote there without mentioning also Turing’s tragedy).

I realize that I rarely connected the history of mathematics and the history of Europe in the twentieth century. It is an unusual and disturbing but also a good thing that Lax – who was born in the nineteen twenties in Hungary and lived through the war in the US – makes this connection. I find it very strange that I always knew well, for instance, that Galois died in a duel, but I never heard that Juliusz Schauder was murdered by the Nazis; I use the open mapping theorem all the time.

Thank you, Peter Lax.

Summer projects in Mathematics at the Technion

Small advertisement:

This summer there will be a special one week program for advanced undergraduate students at Department of Math at the Technion. See this page for information on projects and on how to apply. There is a very nice variety of topics to choose from.

Note that students from any university can apply (also from other countries).

An old mistake and a new version (or: Hilbert, Poincare, and us)

[Update June 28, 2014: This post originally included stories about Poincare and Hilbert making some mistakes. At some point after posting this I realised how unfair it is to talk about somebody else’s mistake (even if it is Hilbert and Poincare) without giving precise references. Instead of deleting the stories, I’ll insert some comments where I think I am unfair. Sorry!]

I was recently forced to reflect on mistakes in mathematics. The reason was that my collaborators and I discovered a mistake in an old paper (16 years old), which forced us to make a significant revision to two of our papers.

A young student of mathematics may consider a paper which contains a mistake to be a complete disaster. (By “mistake” I don’t mean a gap – some step that is not sufficiently well justified (where “sufficiently well” can be a source of great controversy). By “mistake” I mean a false claim). But it turns out that mistakes are inevitable. A paper that contains a mistake is a terrible headache, indeed, but not a disaster.

Arveson once told me: “Everybody makes mistakes. And I mean EVERYBODY”. And he was right. There are two well known stories about Hilbert and Poincare which I’d like to repeat for the reader’s entertainment, and also to make myself feel better before telling you about the mistake my collaborators and I overlooked.

First story: [I think I first read the story about Hilbert in Rota’s “Ten Lesson’s I wish I had been Taught” (lesson 6)]: When a new set of Hilbert’s collected papers was prepared (for his birthday, the story tells), it was discovered that the papers were full of mistakes and could not be published as they were. A young and promising mathematician (Olga Taussky-Todd) worked for three years to correct (almost) all the mistakes. Finally, when the new volume of collected (and corrected) papers was presented to Hilbert, he did not notice any change. What is the moral here? One moral, I suppose, is that even Hilbert made mistakes (hence we are all allowed to). The second is that many mistakes — say, the type of mistakes Hilbert would make — are not fatal: if the mistakes are planted in healthy garden, they can be weeded out and replaced by true alternatives, often-times leaving the important corollaries intact.

[Update June 28: A reference to Rota’s “10 Lessons” is not good enough, and neither is reference to the Wikipedia article on Olga Taussky-Todd, which in turn references Rota’s “Indiscrete Thoughts”, where “10 Lessons” appear.]

Second story: actually two stories, about Poincare.  Poincare made two very important mistakes! First mistake: in 1888 Poincare submitted a manuscript to Acta Mathematica – as part of a competition in honour of the King of Norway and Sweden – in which, among other things (for example inventing the field of dynamical systems), he claimed that the solutions of the 3-body problem (restricted to the plane) are stable (meaning roughly that the inhabitants of a solar system with a sun and two planets can rest assured that the planets in their solar system will continue orbiting more or less as they do forever, without collapsing to the sun or diverging to infinity). After winning the competition, and after the paper was published (and probably in part due to the assistant editor of Acta, Edvard Phragmen, asking Poincare for numerous clarifications during the editorial process), Poincare discovered that his manuscript had a serious error in it. Poincare corrected his mistake, inventing Chaos while he was at it.

[Update June 28: This story is well documented. I learned it from Donal Oshea’s book “The Poincare Conjecture: In Search of the Shape of the Universe” , but it is easy to find online references, too]. 

Second mistake: In 1900 Poincare claimed that if the homology of a compact 3 manifold is trivial, then it is homeomorphic to a sphere. He himself found out his mistake, and provided a counterexample. In order to show that his example is indeed a counter example he had to invent a new topological invariant: the fundamental group. He computed the fundamental group of his example and saw that it is different from the one of the sphere. But this led him to ask: if a closed manifold has a trivial fundamental group, must it be homeomorphic to the 3-sphere? This is known as the Poincare conjecture, of course, and the rest is history.

[Update June 28: Here I should have given a reference of where exactly Poincare claimed that trivial homology implies a space is a sphere. I don’t know it (it probably also appears in Oshea’s book)]

The moral here? I don’t know. But it is nice to add that after making his first mistake, Poincare and Mittag-Leffler (the editor) set a good example by recalling all published editions and replacing them with a new and correct version.

So that’s what I’ll try to imitate now.

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Advanced analysis – this week’s lectures

The semester here at BGU began, and I am teaching Advanced Analysis again. For the students’ convenience, I am putting up links to lecture notes which are relevant to this week.

Introduction, parts one and two.

First lecture on Hilbert spaces.

Survey on the Drury-Arveson space: more-or-less ready for use

Several weeks ago I posted a link to the survey I wrote: “Operator theory and function theory on Drury-Arveson space and its quotients“. Now after several rounds of corrections and additions I think that it is more or less in final form.

This survey is written for Handbook in Operator Theory, ed. Daniel Alpay, to appear in the Springer References Works in Mathematics series.

I wish to thank Joav Orovitz, Guy Salomon, Matthew Kennedy and Joseph Ball for finding many mistakes, suggesting additional topics and references, and other improvements. Their help was truly invaluable.

Shana Tova to all.