Noncommutative Analysis

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

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:

Read the rest of this entry »

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”.

Slides of my talk at the seminar “in” Bucharest

This Wednesday I gave a talk at the Institute of Mathematics in Bucharest, live on zoom. Here are the slides:

In this talk, I decided to put an emphasis on telling the story of how we found ourselves working on this problem, rather than giving a logical presentation of the results in the paper that I was trying to advertise (this paper). I am not sure how much of this story one can get from the slides, but here they are.

Seminar talk by Pandey – Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

In our next Operator Algebras/Operator Theory Seminar, Satish Pandey will present our recently published online paper (together with Danny Ofek and myself) “Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball” (arxiv version).

Time: 15:30-16:30

Date: May 6th, 2021

Title: Distance between reproducing kernel Hilbert spaces and geometry of finite sets in the unit ball

Abstract:

We study the relationships between a reproducing kernel Hilbert space, its multiplier algebra, and the geometry of the point set on which they live. We introduce a variant of the Banach-Mazur distance suited for measuring the distance between reproducing kernel Hilbert spaces, that quantifies how far two spaces are from being isometrically isomorphic as reproducing kernel Hilbert spaces. We introduce an analogous distance for multiplier algebras, that quantifies how far two algebras are from being completely isometrically isomorphic. We show that, in the setting of finite dimensional quotients of the Drury-Arveson space, two spaces are “close” to one another if and only if their multiplier algebras are “close”, and that this happens if and only if one of the underlying point sets is close to an image of the other under a biholomorphic automorphism of the unit ball. These equivalences are obtained as corollaries of quantitative estimates that we prove.

This is joint work with Danny Ofek and Orr Shalit.

If you are interested in the zoom link, let me know.