Noncommutative Analysis

Category: Noncommutative function theory

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:

20180604_053844

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 »

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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 »

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).

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Guy Salomon, Eli Shamovich and I recently uploaded to the arxiv our paper “Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball“. This paper blends in with the current growing interest in noncommutative function theory, continues and unifies several strands of my past research.

A couple of years ago, after being inspired by lectures of Agler, Ball, McCarthy and  Vinnikov on the subject, and after years of being influenced by Paul Muhly and Baruch Solel’s work, I realized that many of my different research projects (subproduct systems, the isomorphism problem, space of Dirichlet series with the complete Pick property, operator algebras associated with monomial ideals) are connected by the unifying theme of bounded analytic nc functions on subvarieties of the nc ball. “Realized” is a strong word, because many of my original ideas on this turned out to be false, and others I still don’t know how to prove. Anyway, it took me a couple of years and a lot of help, and here is this paper.

In short, we study algebras of bounded analytic functions on subvarieties of the the noncommutative (nc) unit ball :

\mathfrak{B}_d = \{(X_1, \ldots, X_d) tuples of n \times n matrices,  \sum X_i X_i < I\}

as well as bounded analytic functions that extend continuously to the “boundary”. We show that these algebras are multiplier algebras of appropriate nc reproducing kernel Hilbert spaces, and are completely isometrically isomorphic to the quotient of H^\infty(\mathfrak{B}_d) (the bounded nc analytic functions in the ball) by the ideal of nc functions vanishing on the variety. We classify these algebras in terms of the varieties, similar to classification results in the commutative case. We also identify previously studied algebras (such as multiplier algebras of complete Pick spaces and tensor algebras of subproduct systems) as algebras of bounded analytic functions on nc varieties. See the introduction for more.

We certainly plan to continue this line of research in the near future – in particular, the passage to other domains (beyond the ball), and the study of algebraic/bounded isomorphisms.

Souvenirs from Amsterdam

(I am writing a post on hot trends in mathematics in the midst of war, completely ignoring it. This seems like the wrong thing to do, but my urge to write has overcome me. To any reader of this blog: I wish you a peaceful night, wherever you are).

Last week I returned from the yearly “International Workshop on Operator Theory and Applications”, IWOTA 2014 for short (see the previous post for the topic of my own talk, or this link for the slides).

This conference was very broad (and IWOTA always is). One nice thing about broad conferences is that you are able sometimes to identify a growing trend. In this talk I got particularly excited by a series of talks on “noncommutative function theory” or “free analysis”. There was a special session dedicated to this topic, but I was mostly inspired by a semi-plenary talk by Jim Agler, and also by two interesting talks by Joe Ball and Spela Spenko. I also attended nice talks related to this subject by Victor Vinnikov, Dmitry Kalyuhzni-Verbovetskyi, Baruch Solel, Igor Klep and Bill Helton. This topic has attracted the attention of many operator theorists, for its applications as well as for its inherent beauty, and seems to be accelerating in the last several years; I will only try to give a taste of some neat things that are going on, by telling you about Agler’s talk. What I will not be able to do is to convey Agler’s intense and unique mathematical charisma.

Here is the program of the conference, so you can check out other things that were going on there.

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Souvenirs from the Black Forest

Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.

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