Noncommutative Analysis

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

\|p(T)\| \leq \|p(S)\|

where S is the unilateral shift on \ell^p. It seemed to me that some of the participants in the session who knew about this paper didn’t completely believe the counter example because it requires numerical verification. Drury’s paper has a link to code that one can download and check. This problem and its solution is definitely something I’d like to learn more about, and maybe this will be next undergraduate project that I will offer. In any case, a friend told me that it is still an open question whether \|p(T)\| \leq C\|p(S)\| with some constant C.

The second session I spoke in was “Operator Space Techniques in Operator Algebras”, and in it I spoke about “Dilations of unitary tuples and their surprising applications”, highlighting my recent work with Malte Gerhold. I also wanted to advertise my survey on dilation theory (because I recently recalled how good it is) but I ran out of time so I forgot to do the little advertisement in the end.

My favorite talks

I usually actively look for talks that would excite and inspire me, to note to myself directions that I should follow up on after the conference. Let me tell you about the three talks that did it for me this time.

The first one was also the first talk in the conference, Nikhil Srivastava’s talk on numerical and algorithmic aspects of diagonalizing a matrix. I find it quite comforting that supposedly completely solved problems like the diagonalization of a matrix are actually not completely settled once you look at them in practice. There are incredible software packages that one can use that actually usually work excellently; however, the speaker pointed out that that he is after algorithm algorithms that provably always work with provable bounds on running time and precision. Building on decades of remarkable work in numerical algebra, they obtain some striking improvements to what was known. If I understood correctly, the algorithms they use are have been around in one form or another for decades, and the main new ingredient that one can tell of without going into details, is the analysis of what happens to the matrix A + Gduring the algorithms, where A is the matrix you want to diagonalize, and G is a random perturbation. This is good because if you diagonalize A + G and G is small, then you have found a diagonal D and an invertible V such that A - U^{-1}DV is small, which is one reasonable way to state the problem of numerically diagonalizing a matrix A.

The second talk that I want to mention was the talk by Ion Nechita, which highlighted connections between stuff I work on (free spectrahedra and matrix convexity) and stuff I don’t work on: quantum information. I was already aware of Nechita’s work but this talk really inspired me to look more into the interesting connections and applications of the mathematics I do to quantum information.

The last talk I want to highlight is the talk by Mirte van der Eyden, whose talk title was “Halos and the undecidability of tensor stable positive maps”, where she presented this paper with the same title. The talk was captivating both because of the very interesting content as well as for that fact that it was also the best talk.

The problem treated is that of existence of essential tensor stable positive maps. A map P: M_n \to M_k is said to be tensor stable if the maps P, P \otimes P, P \otimes P \otimes P, etc. are all positive. If a map P is completely positive, then it is tensor stable. Also, if P is a completely positive map followed by a transpose, then it is also tensor stable. A map that is of the previous two kinds is said to be a trivial tensor stable map, a map that is tensor stable but not of the previous two kinds is said to be an essential tensor stable map. The question is: do there exist any essential tensor stable maps?

The authors treat this problem in two interesting and non-standard ways. The first attack is literally based on nonstandard analysis. They work over the hypercomplex numbers (warning: on wikipedia these are called surcomplex numbers) and show that in that setting there do exist essential tensor stable maps (they can actually construct such a map). The “halo” in the title refers to the image in the hypercomplex world, where the tensor stable maps are surrounded by a halo consisting of the hypercomplex tensor stable maps.

The second approach to the problem is the following: they tried to show that the problem of deciding whether a given map is tensor stable is undecidable. Why would this be helpful? Well, deciding whether a map is completely positive, or a composition of a completely positive map and a transposition, is decidable. Thus, if they prove that the problem of deciding whether a map is tensor stable is an undecidable problem then there has to be an essential one! That’s quite clever. Unfortunately they can only show that a related problem is undecidable, but this is still very interesting.

Auschwitz-Birkenau

Krakow is not far from the small town Staszow in Poland (which my grandmother would call Polania) which is where my father’s mother was born and raised. My grandmother immigrated to Israel before the war, but her family stayed behind and her parents and some of her siblings were murdered in the holocaust (two brothers fled the Nazis, and two of sisters survived the death camp, I think Treblinka). I have a friend who is a mathematician that works and lives in Krakow, and (knowing about my family’s origins) on the last day of the conference he very kindly suggested to drive me around the area, and since another friends of ours (a German) was already planning to go to Auschwitz, we decided to go there. And so we went to Auschwitz, a Pole, a German and a Jew. This was a very special experience that I will not forget.

What is there to say? By the end of the day I forgot that I was at a conference that ended that same morning. I don’t know what is the lesson to be learned and what I am supposed to do now. Just one thing is very clear: we should always remember and continue to remind.

Auschwitz-Birkenau

Bounded perturbations of the Heisenberg commutation relation via dilation theory (preprint)

Malte Gerhold and I recently uploaded our new preprint, “Bounded perturbations of the Heisenberg commutation relation via dilation theory“, to the arxiv. In the paper we modify the notion of dilation distance that was introduced in a previous paper of ours (joint with Satish Pandey and Baruch Solel) and we apply it to one-parameter unitary groups. Our main result is that a small dilation distance between groups implies that their generators are a bounded perturbation one of the other, and our main application is a new (and easy, if you ask me) proof of the theorem of Haagerup and Rordam that the infinite ampliation of the canonical position and momentum operators satisfying the Heisenberg commutation relation [P,Q]=iI is a bounded perturbation of a pair of strongly commuting selfadjoint operators. This result is striking, since it can be shown that the finite ampliations of the position and momentum operators are not a bounded perturbation of strongly commuting operators.

It was a nice surprise to discover that our dilation techniques have applications also in the realm of unbounded operators. It was also a nice surprise to find myself writing a 9 page paper, I forgot that this is even possible!

I will be speaking about this result in the upcoming IWOTA and then in the IMU meeting.

Dilation and classification in operator algebra theory (workshop)

I am really looking forward to the following workshop “Dilation and classification in operator algebra theory” to take place in the University of Copenhagen this October. I will be speaking about… dilation theory and applications, what else.

Noncommutative analysis at the Technion 2022

Group photo NCAT 2022

The workshop NCAT 2022 that was held in honour of Paul Muhly at the Technion two weeks ago ended, to my great relief as organizer, with no disasters, thank god 🙂 In fact it was a delightful event, with some excellent talks, some new collaborations starting up, a lot of words of kindness and friendship, old friendships and partnerships rekindled, I am very happy to have taken a part in the organization, together with my friends Baruch Solel and Adam Dor-On. Needless to say, all of the actual physical organization was done by the current and former CMS coordinators, the excellent Karla Konik and Yael Stern.

Check out the website to see the program, abstracts, and the slides of some of the talks. As we say in Hebrew: it was good, and it is good that it was.

Workshop in honour of Paul Muhly: Noncommutative Analysis at the Technion, June 2022

Registration is open until june 19th

https://cms-math.net.technion.ac.il/noncommutative-analysis-at-the-technion-2/