Noncommutative Analysis

My talk at the Weizmann institute (“Dilation theory in action”)

Last week I was invited by Guy Salomon to give a talk at the Group Theory Midrasha at the Weizmann Institute (Midrasha is a fancy Hebrew word for seminar). Since the seminar there is two hours long, I took the opportunity to try something different, and for the first time in a long while I gave a whiteboard talk, going into the details of the proof of the main result in my paper with Gerhold on perturbation of the Heisenberg commutation relations. The group there at Weizmann is really fantastic with many young and curious (and bright!) students who bombarded me with questions, so the talk was quite alive and I think it was a successful experiment (yesterday I gave a similar talk at the Analysis Seminar at Bar-Ilan University; the crowd was full of strong analysts who also asked great questions, but since I aimed for an hour I got pressed for time, so I think in the end it wasn’t as good. That’s on me, because they actually let me choose whether I want to go for two hours or one, and again I wanted to try something a bit different).

Here is a video recording of the talk.

BTW: You can see that someone in the audience asked me a question that I, embarrassingly, blacked out on: do strongly commuting (unbounded) operators commute in the sense that there is some dense subspace on which the commutator is defined and equal to zero? The answer is yes and is actually not hard to show with basic semigroup theory techniques. A little trickier is to show that strongly commuting operators have commuting spectral projections – which is an equivalent and perhaps more natural definition of “strong commutation” than the one I gave.

Proof of the ergodic theorem and the H-theorem in quantum mechanics – notes on von Neumann’s paper

In this post I will work through von Neumann’s paper “Besweis des Ergodensatzes und des H-Theorems in der neuen Mechanik” from 1929 (“Proof of the ergodic theorem and the H-theorem in quantum mechanics”). This paper was brought to my attention in the introduction to a very nice talk by Guy Salomon in IWOTA about “stability” (this is not one of von Neumann’s papers that one usually sees cited in my area). Guy mentioned that von Neumann starts by considering the canonical position and momentum operators that almost commute in some sense (they satisfy PQ-QP = \frac{\hbar}{2i} and \hbar is a tiny number) and contemplates that if one could approximate these operators with a pair of commuting operators then one could do some calculations and derive some results. That’s what got me interested in looking up the paper, since this is very closely related to my recent work with Malte Gerhold on bounded perturbations of the canonical commutation relations. However, when I tried to read the paper, I quickly saw that it doesn’t have much to say on the issue of stability, however it does seem to have some elaborate considerations with physical consequences.

What is it that von Neumann was trying to do in that paper? I would like to understand this paper – the math as well as the physical consequences. It turns out that the only way I can really understand something is to try to explain it – hence this blog post. Luckily, this paper, which was originally written in German, was translated to English by Roderich Tumulka and published in the European Physical Journal H (and one can also find it on the arxiv). So following Tumulka’s translation I will now try to produce an annotated summary of the main part (Sections 0-2) of von Neumann’s paper.

With hindsight, after reading through the paper, I found that it very little to do with the stability questions that interested me, but it was a nice exercise.

UPDATE: After publishing the post, I continued looking and somehow I reached the following commentary on von Neumann’s paper which was published by four mathematical/theoretical physicists including the translator (ironically, I wasn’t aware of it and did not think of looking for something like this until I finished wrestling with the paper by myself). I am sure that it will be more useful than this post for people who are interested in understanding von Neumann’s work:

Long time behavior of macroscopic quantum systems“, by Goldstein, Lebowitz, Tumulka dn Zanghi, The European Journal of Mathematics, 2010.

Below is my work-through the main parts of the paper. Sections (and subsections) as well as their numbering follow the sections in the paper.

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New preprint: Tensor algebras of subproduct systems and noncommutative function theory

For a several years I believed that every tensor algebra of a subproduct system with Hilbert space fibers is the algebra of uniformly continuous noncommutative functions on a homogeneous noncommutative variety. Now, since the isomorphism problem for tensor algebras has interested me for a long time, and was solved for the case of finite dimensional fibers, and since the isomorphism problem for algebras of uniformly continuous functions on homogeneous noncommutative varieties was, even for the case of infinitely many variables, solved by Salomon, Shamovich and myself, I returned several months ago to the problem of showing that every tensor algebra can indeed be represented as an algebra of nc functions.

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Notes and references for my lecture series on dilation theory (Copenhagen, October 2022)

As I recently advertised, I am giving a lecture series on dilation theory and applications in the workshop Dilation and Classification in Operator Algebra Theory , October 17-21. I am putting up this post to have a place in which I will summarize the content of the daily lecture, list key references for the talks, or add notes that with hindsight should be added. This post will be updated on a daily basis.

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Souvenirs from Polania

I recently returned from the 33rd International Workshop on Operator Theory and its Applications (IWOTA). IWOTA is a series of large conferences, that has become so central to the field that it attracts hundreds of participants and also some big names. Now IWOTA has its own wikipedia page and even a youtube channel (this year’s plenary talks are not yet uploaded). I like going to IWOTA because I get to hear talks in a very large range of topics, on the one hand, while also having the opportunity to go to very specialized small parallel sessions.

Every time IWOTA is held at a different location – this year it was held in Krakow, Poland. This was the first time I traveled abroad since January 2020 and it was also the first conference in a couple of years for many of the participants and that was maybe the main (unofficial) theme of the conference: so nice to get back together, so nice get back to normal (honestly, I feel somewhat corrupt to speak of flying to conferences in a different continent, sleeping in hotels and eating out every day for a week as “normal” … about this we shall need to talk another day). It was nice to meet and shake hands with people whose papers I have read and admired, and exchange a few words. There was also a scientific program – check it here (clicking the speaker name leads to a pdf file with title and abstract).

My talks

I was invited to speak in two special sessions: the first one on “Functional Calculus and Spectral Constraints” and in it I spoke about “A von Neumann type inequality for commuting row contractions”. I spoke about my joint work with Hartz and Richter, which has been recently published in Math Z. In the comments and questions part of the talk I learned about a very interesting conjecture or Matsaev’s, which has been apparently solved in the negative by Drury a little more than a decade ago. The conjecture is that for every p \in (1,\infty) every polynomial p and every contraction T \in B(\ell^p), it holds that

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