Noncommutative Analysis

von Neumann’s inequality for row contractive matrix tuples

Update: we showed that the constants C_{d,n} are uniformly bounded in d for a fixed n.
Here is a link to the second version:
https://arxiv.org/abs/2109.08550

Noncommutative Analysis

Michael Hartz, Stefan Richter and I recently uploaded our paper von Neumann’s inequality for row contractive matrix tuples to the arxiv.

The main result is the following.

We prove that for all $latex d,nin mathbb{N}$, there exists a constant $latex C_{d,n}$ such that for every row contraction $latex T$ consisting of $latex d$ commuting $latex n times n$ matrices and every polynomial $latex p$, the following inequality holds:

$latex  |p(T)| le C_{d,n} sup_{z in mathbb{B}_d} |p(z)|$ .

We then apply this result and the considerations involved in the proof to several open problems from the literature. I won’t go into that because I think that the abstract and introduction do a good job of explaining what we do in the paper. In this post I will write about how this collaboration with Michael and Stefan started, and give some heuristic explanation why our result is not trivial.


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von Neumann’s inequality for row contractive matrix tuples

Michael Hartz, Stefan Richter and I recently uploaded our paper von Neumann’s inequality for row contractive matrix tuples to the arxiv.

The main result is the following.

We prove that for all d,n\in \mathbb{N}, there exists a constant C_{d,n} such that for every row contraction T consisting of d commuting n \times n matrices and every polynomial p, the following inequality holds:

 \|p(T)\| \le C_{d,n} \sup_{z \in \mathbb{B}_d} |p(z)| .

We then apply this result and the considerations involved in the proof to several open problems from the literature. I won’t go into that because I think that the abstract and introduction do a good job of explaining what we do in the paper. In this post I will write about how this collaboration with Michael and Stefan started, and give some heuristic explanation why our result is not trivial.

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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.

Why a “scientific approach” to science education is something I reject

Our Department has a new Teaching Seminar (concerned with teaching mathematics at the university level) which is led by legendary math professor Aviv Censor. The first lecture that I attended this semester was given by Professor Emeritus Avinoam Kolodny (Hebrew abstract here. A link to the talk – works only for Technion accounts – here). In the compelling lecture Kolodny started by mentioning the assumptions that we make when teaching (students come to class, they listen, they understand what we say, they then go home and solve homework problems) and contrasts this with empiric reality (a huge portion of students don’t come to class, the ones that do don’t listen, the ones that do don’t understand, and then they go and copy homework or solve routine problems like robots). Prof. Kolodny – an esteemed and decorated lecturer – said that he was troubled and puzzled by his students’ lack of success, and that at some point he became aware of the paper “Why not try a scientific approach to science education?” by eminent physicist, educationist and Nobel Prize laureate Carl Wieman. Kolodny explained various ideas of how to improve science (or engineering) education at the university level, to a large extent in line with ideas presented in Wieman’s paper.

The bottom line of Kolodny’s talk and Wieman’s paper is that the university lecture as we know it doesn’t work and is a waste of time. They have some ideas how to fix it, an approach that – as a first approximation – we can call “technology driven flipped classroom”. To me, the most disturbing parts of their approach are (1) that they believe that their opinions are “science based”, and therefore (2) they believe in promoting institutional change. These two aspects worry more than any technical discussion whether we should flip the classroom sideways or upside-down.

Kolodny remarked during his talk (I am paraphrasing): “I am not here to bury the concept of a lecture. Lectures are good and important. In fact, I am giving a lecture at this very moment. But you should remember that lectures are no good at passing information. In a lecture you motivate, you stimulate, you do propaganda. I’m here to do propaganda”.

Certainly I was stimulated by the talk, I was motivated to look up and then read Wieman’s paper, but most of all I was angry, I felt that someone was trying to brainwash me to believe in a certain ideology, rather than sharing some insights on teaching. Part of what made me feel this way was the “scientific approach” rhetoric. Another thing that bothered me was the jump from facts (some problems that almost everybody will agree on) to conclusions (a particular pedagogical methodology is the only way that works), disregarding tradition as not much more than momentum. Indeed, it felt like propaganda.

In this post I want to record my thoughts on some arguments raised by flipped classroom enthusiasts, and in particular on two aspects: the “scientific approach” approach, and with it the claim that lectures don’t work and we have to revolutionize the whole structure of courses to make them work.

I wish to recommend reading Wieman’s paper. Not only so that you can appreciate my criticism, but because it is a well reasoned piece of work by someone who has not only thought deeply about, but also researched the subject. I have a lot of respect for his efforts.

I am focusing my criticism on his paper, because it is written and available and interesting. But I am really arguing with talks, lectures, discussions, blog posts etc. that I have seen through the years, and have got me thinking for a long time. Now is just an opportunity to pour all of this out.

So, why not try a scientific approach to science education? Here’s why not:

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Course announcement: “Topics in Functional Analysis 106433 – Introduction to Operator Algebras”

My sabbatical is nearing its end and I starting to get used to the idea of getting back to teaching. Luckily (or is it really just luck?) I am going to have a very smooth return to teaching, because this coming fall I will be teaching a topics course of my choice, and it is going to be an introduction to operator algebras (the official course title and number are above). To be honest, the idea is to give the optimal course for students who will work with me, but I believe that other students will also enjoy it and find it useful. I will probably use this blog to post material and notes.

Here is the content of the info page that I will be distributing:

Topics in Functional Analysis 106433

Winter 2021

Introduction to Operator Algebras

Lecturer: Orr Shalit (oshalit@technion.ac.il, Amado 709)

Credit points: 3

Summary: The theory of operator algebras is one of the richest and broadest research areas within contemporary functional analysis, having deep connections to every subject in mathematics. In fact, this topic is so huge that the research splits into several distinct branches: C*-algebras, von Neumann algebras, non-selfadjoint operator algebras, and others. Our goal in this course is to master the basics of the subject matter, get a taste of the material in every branch, and develop a high-level understanding of operator algebras.

The plan is to study the following topics:

  1. Banach algebras and the basics of C*-algebras.
  2. Commutative C*-algebras. Function algebras.
  3. The basic theory of von Neumann algebras.
  4. Representations of C*-algebras. GNS representation. Algebras of compact operators.
  5. Introduction to operator spaces, non-selfadjoint operator algebras, and completely bounded maps.
  6. Time permitting, we will learn some additional advanced topics (to be decided according to the students’ and the instructor’s interests). Possible topics:
    1. C*-algebras and von Neumann algebras associated with discrete groups.
    1. Nuclearity, tensor products and approximation techniques.
    1. Arveson’s theory of the C*-envelope and hyperrigidity.
    1. Hilbert C*-modules.

Prerequisites: I will assume that the students have taken (or are taking concurrently) the graduate course in functional analysis. Exceptional students, who are interested in this course but did not take Functional Analysis, should talk to the instructor before enrolling.

The grade: The grade will be based on written assignments, that will be presented and defended by the students.

References:

The following are good general references, though we shall not follow any of them very closely (at most a chapter here or there).   

  1. Orr Shalit’s lecture notes.
  2. K.R. Davidson, “C*-Algebras by Example”.
  3. R.V. Kadison and J. Ringrose, “Fundamentals of the Theory of Operator Algebras”.
  4. C. Anantharaman and S. Popa, “An Introduction to II_1 Factors”.
  5. N.P. Brown and N. Ozawa, “C*-Algebras and Finite Dimensional Approximations”
  6. V. Paulsen, “Completely Bounded Maps and Operator Algebras”.