Noncommutative Analysis

Category: operator spaces

My talk at BIRS on “Noncommutative convexity, a la Davidson and Kennedy”

Update August 5: here is the link to the video recording of the talk: link.

I was invited to speak in the BIRS workshop Multivariable Operator Theory and Function Spaces in Several Variables. Surprise: the organizers asked each of the invited speakers (with the exception of some early career researchers, I think) to speak on somebody else’s work. I think that this is a very nice idea for two reasons.

First, it is very healthy to encourage researchers to open their eyes and look around, instead of concentrating always on their own work – either racing for another publication or “selling” it. At the very least being asked to speak about somebody else’s work, it is guaranteed that I will learn something new in the workshop!

The second reason why I think that this is a very welcome idea is maybe a bit deeper. Every mathematician works to solve their favorite problems or develop their theories, but every once in a while it is worthwhile to stop and think: what do we make out of all this? What are the results/theories/points of view that we would like to carry forward with us? The tree can’t grow in all directions with no checks – we need to prune it. We need to bridge the gap between the never stopping flow of papers and results, on one side, and the textbooks of the future, on the other side.

With these ambitious thoughts in mind, I chose to speak about Davidson and Kennedy’s paper “Noncommutative Choquet theory” in order to force myself to digest and internalize what looked to me to be an important paper from the moment it came out, and with this I hoped to stop a moment and rearrange my mental grip on noncommutative function theory and noncommutative convexity.

The theory developed by Davidson and Kennedy and its precursors were inspired to a very large extent by classical Choquet theory. It therefore seems that to understand it properly, as well as to understand the reasoning behind some of the definitions and approaches, one needs to be familiar with this theory. So one possible natural way to start to describe Davidson and Kennedy’s theory is by recalling the classical theory that it generalizes.

But I didn’t want to explain it in this way, because that is the way that Davidson and Kennedy’s exposition (both in the papers and in some talks that I saw) goes. I wanted to start with the noncommutative point of view from the outset. I did use the classical (i.e. commutative case) for a tiny bit of motivation but in a somewhat different way, which rests on stuff everybody knows. So, I did a little expository experiment, and if you think it blew up then everybody can simply go and read the original paper.

Here are my “slides”:

The conference webpage will have video recordings of all talks at some point.

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.

The complex matrix cube problem – new results from summer projects

In this post I will summarize the results obtained by my group in the “Summer Projects Week” that took place two weeks ago at the Technion. As in last time (see here for a summary of last year’s project) the title of the project I suggested was “Numerical Explorations of Open Problems from Operator Theory”. This time, I was lucky to have Malte Gerhold and Satish Pandey, my postdocs, volunteer to help me with the mentoring. The students who chose our project were Matan Gibson and Ofer Israelov, and they did fantastic work.

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Topics in Operator Theory, Lecture 10: hyperrigidity

In this lecture we discuss the notion of hyperrigidity, which was introduced by Arveson in his paper The noncommutative Choquet boundary II: Hyperrigidity, shortly after he proved the existence of boundary representations (and hence the C*-envelope) for separable operator systems. Most of the results and the examples that we will discuss in this lecture come from that paper, and we will certainly not be able to cover everything in that paper. In the last section of this post I will put some links concerning a result of Kennedy and myself which connects hyperrigidity to the Arveson’s essential normality conjecture.

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Topics in Operator Theory, Lecture 9: the boundary theorem

In this post, we come back to boundary representations and the C*-envelope, prove an important theorem, and see some examples. It is interesting to note that the theory has interesting consequences even for operators on finite dimensional spaces. Here is a link to a very interesting paper by Farenick giving an exposition of Arveson’s boundary theorem in the setting of operators on finite dimensional spaces.

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