Noncommutative Analysis

A survey (another one!) on dilation theory

I recently uploaded to the arxiv my new survey “Dilation theory: a guided tour“. I am pretty happy and proud of the result! Right now I feel like it is the best survey ever written (honest, that’s how I feel, I know that its an illusion), but experience tells me that two months from now I might be a little embarrassed (like: how could I be so vain to think that I could pull of a survey on this gigantic topic?).

(Well, these are the usual highs and lows of being a mathematician, but since this is a survey paper and not a research paper, I feel comfortable enough to share these feelings).

This survey was submitted (and will hopefully appear in) to the Proceedings of the International Workshop on Operator Theory and its Applications (IWOTA) 2019, Portugal. It is an expanded version of the semi-plenary talk that I gave in that conference. I used a preliminary version of this survey as lecture notes for the mini-course that I gave at the recent workshop “Noncommutative Geometry and its Applications” at NISER Bhubaneswar.

I hope somebody finds it useful or entertaining 🙂

Janos Aczel (1924-2020)

I was saddened to find out that Janos Aczel passed away earlier this month. In my early days, after Boris Paneah got me hooked on functional equations, Aczel’s books caught my eye. Since then I am a fan of his. In particular, I was drawn by a small booklet by him whose title I am not able to reconstruct, and his two larger books “Lectures on Functional Equations and Their Applications” (1966) and “Functional Equations in Several Variables” (1987, co-authored with, Dhombres), which for a couple of years were to me among the most interesting and useful books I knew.

I have two stories two tell about Janos.

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My talks at NCGA Bhubaneswar

This week I am giving a series of five lectures on dilation theory in the workshop Noncommutative Geometry and its Applications , at NISER Bhubaneswar (India). I am putting my talks up here in case anybody would like to see them (and also as backup, in case my stick doesn’t work).

Lecture number 4.

Lecture number 5.

(Lectures 1,2,3 were board talks).

New paper: "On the matrix range of random matrices"

Malte Gerhold and I recently posted our new paper “On the matrix range of random matrices” on the arxiv, and I want to write a few words about it.

Recall that the matrix range of a d-tuple of operators A = (A_1, \ldots, A_d) \in B(H)^d is the noncommutative set W(A) = \cup_n W_n(A), where

W_n(A) = \{ (\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n is UCP \}.

The matrix range appeared in several recent papers of mine (for example this one), it is a complete invariant for the unital operator space generated by A_1 \ldots, A_d, and is within some classes is also a unitary invariant.

The idea for this paper came from my recent (last couple of years or so) flirt with numerical experiments. It has dawned on me that choosing matrices randomly from some ensembles, for example by setting

G = randn(N);

X = (G + G')/sqrt(2*N);

(this is the GOE ensemble) is a rather bad way for testing “classical” conjectures in mathematics, such as what is the best constant for some inequality. Rather, as N increases, random N \times N behave in a very “structured” way (as least in some sense). So we were driven to try to understand, roughly what kind of operator theoretic phenomena do we tend to observe when choosing random matrices.

The above paragraph is a confession of the origin of our motive, but at the end of the day we ask and answer honest mathematical questions with theorems and proofs. If X^N = (X^N_1, \ldots, X^N_d) is a d-tuple of N \times N matrices picked at random according to the Matlab code above, then experience with the law of large numbers, the central limit theorem, and Wigner’s semicircle law, suggests that W(X^N) will “converge” to something. And by experience with free probability theory, if it converges to something, then is should be the matrix range of the free semicircular tuple. We find that this is indeed what happens.

Theorem: Let X^N be as above, and let s = (s_1, \ldots, s_d) be a semicircular family. Then for all n,

\lim_{N \to \infty} d_H(W_n(X^N),W(s)) = 0 almost surely

in the Hausdorff metric.

The semicircular tuple s = (s_1, \ldots, s_d) is a certain d-tuple of operators that can be explicitly described (see our paper, for example).

We make heavy use of some fantastic results in free probability and random matrix theory, and our contribution boils down to finding the way to use existing results in order to understand what happens at the level of matrix ranges. This involves studying the continuity of matrix ranges for continuous fields of operators, in particular, we study the relationship between the convergence

(*) \lim_{N \to \infty} \|p(X^N)\| = \|p(X)\|

(which holds for X^N as above and X = s by a result of Haagerup and Torbjornsen) and

(**) \lim_{N \to \infty} d_H(W_n(X^N),W(X)) = 0.

To move from (*) to (**), we needed to devise a certain quantitative Effros-Winkler Hahn-banach type separation theorem for matrix convex sets.

Boris Paneah (1936-2019)

Last Thursday (10.10.2019) my beloved teacher and master’s thesis advisor Boris Paneah passed away. Boris was a great mathematician who worked most of his career on PDEs and harmonic analysis. In the last decade of his mathematical activity, he founded a new direction in the theory of functional equations, using dynamical methods. His publications can be found on his homepage. For me, he was most of all a very special teacher. In fact, he played an important role in my decision to continue to graduate studies. I want to share here an obituary that I wrote for him (it is in Hebrew).