Noncommutative Analysis

Souvenirs from the Red River

Last week I attended the annual Canadian Operator Symposium, better known in its nickname: COSY. This conference happens every year and travels between Canadian universities, and this time it was held in the University of Manitoba, in Winnipeg. It was organized by Raphaël Clouâtre and Nina Zorboska, who altogether did a great job.

My first discovery: Winnipeg is not that bad! In fact I loved it. Example: here is the view from the window of my room in the university residence:


Not bad, right? A very beautiful sight to wake up to in the morning. (I got the impression, that Winnipeg is nothing to look forward to, from Canadians. People of the world: don’t listen to Canadians when they say something bad about any place that just doesn’t quite live up to the standard of Montreal, Vancouver, or Banff.) Here is what you see if you look from the other side of the building:  Read the rest of this entry »


The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field \mathbb{C} (e.g., there are many polynomials, such as  x^{2n} +1, that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write \mathbb{C}[z] for the algebra of polynomials in one complex variable with complex coefficients, and consider it as a function of the complex variable z \in \mathbb{C}. We will also write \mathbb{C}[z_1, \ldots, z_d] for the algebra of polynomials in d (commuting) variables, and think of it – at least initially – as a function of the variable z = (z_1, \ldots, z_d) \in \mathbb{C}^dRead the rest of this entry »

Minimal and maximal matrix convex sets

The final version of the paper Minimal and maximal matrix convex sets, written by Ben Passer, Baruch Solel and myself, has recently appeared online. The publisher (Elsevier) sent us a link through which the official final version is downloadable, for anyone who clicks on the following link before May 26, 2018. Here is the link for the use of the public:

Click here to download the journal version of the paper

Of course, if you don’t click by May 26 – don’t panic! We always put our papers on the arXiv, and here is the link to that. Here is the abstract:

Abstract. For every convex body K \subseteq \mathbb{R}^d, there is a minimal matrix convex set \mathcal{W}^{min}(K), and a maximal matrix convex set \mathcal{W}^{max}(K), which have K as their ground level. We aim to find the optimal constant \theta(K) such that \mathcal{W}^{max}(K) \subseteq \theta(K) \cdot \mathcal{W}^{min}(K). For example, if \overline{\mathbb{B}}_{p,d} is the unit ball in \mathbb{R}^d with the p-norm, then we find that 

\theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p-1/2|} .

This constant is sharp, and it is new for all p \neq 2. Moreover, for some sets K we find a minimal set L for which \mathcal{W}^{max}(K) \subseteq \mathcal{W}^{min}(L). In particular, we obtain that a convex body K satisfies \mathcal{W}^{max}(K) = \mathcal{W}^{min}(K) only if K is a simplex.

These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every d-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most \sqrt{d}. We also introduce new explicit constructions of these (and other) dilations.

Ronald G. Douglas (1938-2018)

A couple of weeks ago I learned from an American colleague that Ron Douglas passed away. This loss saddens me very much. Ron Douglas was a leader in the Operator Theory community, an inspiring mathematician, a person of the kind that they don’t make like any more.

The first time that I met him was in the summer of 2009, in a workshop on multivariable operator theory at the Fields Institute in Toronto. I walked up to him and asked him what he thought of some proposed proof of the invariant subspace problem (let’s say that I don’t remember exactly which one), and he didn’t even want to hear about it! At the time I was still rather fresh and didn’t understand why (I later learned that he has had his fair share of checking failed attempts). After this first encounter I thought for some time that he was a scary person, only to discover slowly through the years that he was actually a very very generous, gentle, and kind person. And he was very sharp, that was really scary.

The last time that I met him, it was the spring of 2014, and we were riding a train from Oberwolfach to Frankfurt. I think we were Douglas, Ken Davidson, Brett Wick and I. Davidson, Wick and I were going to Saarbrucken, and Douglas was supposed to be on another train, but he joined us because by that time he was half blind and thought that it was better to travel at least part of the way with friends. (The conductor found him out, but decided to let the old man be). At some point we had to switch trains and we left him and I was worried how can we leave a half blind man to travel alone (he made it home safely). On the train he talked about the corona theorem something, and I was sitting on the edge of my sit trying to keep up. I don’t remember what he said about the corona theorem, but I remember clearly that he told me that I shouldn’t have nausea because it is only psychological (you see, even very smart people occasionally say silly things). He also talked about black jack. That was the last time I saw him.

When I was a postdoc I became obsessed with the Arveson-Douglas conjecture, and I worked on this conjecture on and off for several years (see here, here, here and here for earlier posts of mine mentioning this conjecture). That’s one way I got to know some of Douglas’s later works. Douglas motivated many people to work on this problem, and was also responsible for some of the most recent breakthroughs. Just last semester, in our Operator Theory and Operator Algebras Seminar at the Technion, I gave a couple of lectures on two of his very last papers on this topic, which were written together with his PhD student Yi Wang: “Geometric Arveson-Douglas Conjecture and Holomorphic Extension” and “Geometric Arveson-Douglas Conjecture – Decomposition of Varieties“. These are very difficult papers, written with a rare combination of technical ability and vision.

By the way, I have heard wonderful things about Douglas as a mentor and PhD supervisor. In July 2013 I attended a conference in Shanghai in honour of Douglas’s 75th birthday. At the banquet many of his students and collaborators got up to say some words of thanks and to tell about nice memories. After several have already spoken, the master of ceremony walked up to me with his wireless microphone and announced: “and now, to close this evening, the last student, Piotr Nowak!” Perhaps this is a good place to point out that I was not Douglas’s student, nor is my name Piotr Nowak (I think Piotr Nowak also was not a student, but he was a postdoc or at least spent some time at Texas A&M). I took the mic in my hand, but didn’t have the guts to play along, and handed it over to Piotr.

(I wrote above that I was not a student of Douglas, but in some sense I am his mathematical step-grandchild. Douglas’s first PhD student was Paul Muhly, who is mathematically married to Baruch Solel, my PhD supervisor, hence is my mathematical step-father.)

Another completely different work of his that I had the pleasure of studying is his beautiful little textbook “Banach Algebra Techniques in Operator Theory“, which I read cover-to-cover with no particular purpose in mind, just for the joy of it.

I think that perhaps Douglas’s greatest contribution to mathematics is the Brown-Douglas-Fillmore (BDF) theory. The magic ingredient of using sophisticated algebraic and topological intuition and machinery appears in much of Douglas’s work, but in BDF it had wonderful consequences as well as incredible impact. If one wants to get an idea of what this theory is about (and what kind of problems in operator theory motivated it), perhaps the best person to explain is Douglas himself. To this end, I recommend reading the introduction to Douglas’s small book on the subject, “C*-Algebra Extensions and K-Homology” (Annals of Mathematics Studies Number 95).

[Update, March 17th: I later checked my records and realised that the way I remembered things is not the way they were! I am leaving the memory as I wrote it, but for the record, that train ride was not the last time that I saw Douglas, I suppose that it was simply the most memorable and symbolic goodbye. The last time I met Douglas was in Banff, in 2015. (In my memory, I mixed Oberwolfach 2014 with Banff 2015). If I am not mistaken, he was there with his wife Bunny, and we did not interact much. I met him four other times: in June 2010 at the University of Waterloo, when he received a honorary doctorate, later that summer in Banff, at IWOTA 2012 which took place at Sydney, and at IWOTA 2014, in Amsterdam (which was also after our goodbye on the train). ]


Souvenirs from San Diego

Every time that I fly to a conference, I think about the airport puzzle that I once read in Terry Tao’s blog. Suppose that you are trying to get quickly from point A to point B in an airport, and that part of the way has moving walkways, and part of it doesn’t. Suppose that you can either walk or run, but you can only run for a certain small amount of the time. Where is it better to spend that amount of time running: on the moving walkways or in between the moving walkways? Does it matter?

Another question that continues to puzzle me (and to which I still don’t have a complete answer to) is: why do I continue to inflict upon myself the tortures of international travel, such as ten hour jet lag or trans-atlantic flights? More generally, I spent a lot of time wondering: why do I continue going to conferences? Is it worth it for me? Is it worth the university’s money? Is it worth it for mankind? 

Last week I attended the Joint Mathematics Meeting in San-Diego. It was my first time in such a big conference. I will probably not return to such a conference for a while, since it is not so “cost effective”. I guess that I am a small workshop kind of person.

I spoke in and attended all the talks in the Free Convexity and Free Analysis special session, which was excellent. Here is the abstract and here are the slides of my talk (the slides).  I also attended some of the talks in the special sessions on Advances in Operator AlgebrasOperators on Function Spaces in One and Several Variables, and another one on Advances in Operator Theory, Operator Algebras, and Operator SemigroupsI also attended several plenary talks, which were all quite entertaining.

I am happy to report that the field of free analysis and free convexity is in really good shape! There was a sequence of talks in the first day (Hartz, Passer, Evert and Kriel) by three very young researchers on free convexity that really put me into high spirits! The field is blossoming and the competition is healthy and friendly. But the talk that got me most excited was the talk by Jim Agler, who gave a preliminary report on joint work with John McCarthy and Nicholas Young regarding noncommutative complex manifolds. Now, at first it might seem that nc manifolds will be hard to make sense of, because how can you take direct sums of points in a manifold, etc. Moreover, the only take on the free manifolds that I met before was Voiculescu’s construction of the free projective plane, which I found hard to swallow and kind of ruined my appetite for the subject.

However, it turns out that one can define a noncommutative complex manifold as topological space X that carries an atlas of charts (U,f) where U is an open subset of X and f : \Omega \to U is a homeomorphism from an nc domain \Omega onto U, such that given two intersecting charts (U,f), (U',f'), the map f^{-1} \circ f' going from f'(U \cap U') to f(U \cap U') is an nc biholomorphism. This definition is so natural and clear that I want to shout! Agler went on and showed us how one can construct a noncommutative Riemann surface, for example the Riemann surface corresponding to the noncommutative square root function. How can one not want to hear more of this? I am looking forward very enthusiastically to see what Agler, McCarthy and Young are up to this time; it looks like a very promising direction to study.

Among the plenary talks that I attended (see here for description), the one given by Avi Wigderson struck me the most. I went to the talk simply for mathematical entertainment (a.k.a. to broaden my horizons), but I was very pleasantly surprised to find completely positive maps and free functions in a talk that was supposed to be about computational complexity. I went to the first two talks but missed the third one because I had an opportunity to have lunch with a friend and collaborator, which in any respect was more important to me than the lecture. The above link (here it is again) contains links to a tutorial and papers related to Wigderson’s talks, and I hope to find time to study that, and at least catch up on what I missed in the third talk.

One more thing: there was one quite eminent operator theorist who is long retired, and came to several of the sessions that I attended. At some point I noticed that after every talk a came up to the speaker and said several words of encouragement or advice. Seeing such a pure expression of kindness and love of humanity was touching and inspiring. Upon later reflection, I noticed that such expressions were happening around me all the time, for example when another “celebrity” in our field arrived and a hugging (!) session began. This memory brings a smile to my face. Well, maybe going to San-Diego was worth it, after all.

Additional thoughts January 26: 

  1. The tutorial that you can find in “the above link” seems to cover all of Wigderson’s talk.
  2. I have had some more thoughts on “big conferences”. The good thing about them is that it gives an opportunity to interact with people people outside one’s own academic bubble, and attend high level talks by prominent mathematicians. The bad thing is that you fly far away, waste tons of grant money, and in the end have only a small time to discuss your research topic with experts. So: to go or not to go? I’ve found a solution! Attend local big conferences. Fly across the world only to meet with special colleagues or participate in focused and effective workshops or conferences on your subject of main interest. (And if they invite you to give a plenary talk at the ICM, then, OK, you should probably go).

The nightmare

In September 30 the mathematician Vladimir Voevodsky passed away. Voevodsky, a Fields medalist, is a mathematician of whom I barely heard earlier, but after bumping into an obituary I was drawn to read about him and about his career. His story is remarkable in many ways. Voevodsky comes out as brilliant, intellectually honest giant, who bravely and honestly confronted the crisis that he observed “higher dimensional mathematics” was in.

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I quit (from MathSciNet and ZbMath)

(This is the post that I wanted to write this weekend.)

Several months ago I informed both MathSciNet as well as Zentralblatt that I would like to stop reviewing papers for these repositories. If you don’t know what I am talking about (your PhD thesis advisor should be fired!), then MathSciNet and Zentralblatt are databases that index published papers in mathematics, contains some bibliographic information (such as a reference list for every paper, as well as a list of papers that reference it), and, significantly, has a review for every indexed paper. The reviews are written by mathematicians who do so voluntarily (they get AMS points or something). If the editors find nobody willing to review, then the abstract appears instead of a review. This used to a very valuable tool, and is still quite valuable.

I quit because:

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A comment on the sowa versus Gowers affair

I wanted to write about something else this weekend, but I got distracted and ended up writing this post. O well…

This is post is reply to (part of) a post by Scott Aaronson. I got kind of heated up by his unfair portrayal of the blog “Stop Timothy Gowers!!!“, and started writing a reply which got to be ridiculously long, so I moved it here.

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“A toolkit for constructing dilations on Banach spaces”, by Fackler and Gluck

About a week ago an interesting preprint appeared on the arxiv: “A toolkit for constructing dilations on Banach spaces“, by Stephan Fackler and Jochen Gluck. I have been studying various aspects of dilations for some years, but I haven’t really given much thought to dilation theory in general classes of Banach spaces. This paper – which is very clearly organized and written – was very refreshing for me, and in it a very general framework for proving existence of dilations in classes of Banach spaces is presented. The paper also contains a nice overview of the literature, and I was surprised by learning also about old results in, and application of, dilation theory, which I was not aware of and perhaps I should have been. The purpose of this post is to record my first impression of this paper and to put down some links to the references, which I would like to study better at some point.  Read the rest of this entry »

Tapioca on page 49

To my long camping vacation this year I took the book “Topological Vector Spaces” by Alex and Wendy Robertson. I “inherited” this book (together with a bunch of other classics) from an old friend after he officially decided to leave academic mathematics and go into high-tech. The book is a small and thin hard-cover, with pages of high quality that are starting to become a delicious cream color.

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