Noncommutative Analysis

Category: Functional analysis

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.

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 Dor-On: Quantum symmetries in the representation theory of operator algebras

NOTE: THE SEMINAR WAS POSTPONED TO DECEMBER 10.

On next Thursday the Operator Algebras and Operator Seminar will convene for a talk by Adam Dor-On.

Title: Quantum symmetries in the representation theory of operator algebras

Speaker: Adam Dor-On (University of Illinois, Urbana-Champaign)

Time: AFTERNOON Thursday Dec. 10, 2020 (NOTE: THE SEMINAR WAS POSTPONED BY ONE WEEK FROM ORIGINAL DATE).

(Zoom room will open about ten minutes earlier, and the talk will begin at 15:30)

Zoom link: email me.

Abstract:

We introduce a non-self-adjoint generalization of Quigg’s notion of coaction of a discrete group G on a C*-algebra. We call these coactions “quantum symmetries” because from the point of view of quantum groups, coactions on C*-algebras are just actions of a quantum dual group of G on the C*-algebra. We introduce and develop a compatible C*-envelope, which is the smallest C*-coaction system which contains a given operator algebra coaction system, and we call it the cosystem C*-envelope.

It turns out that the new point of view of quantum symmetries of non-self-adjoint algebras is useful for resolving problems in both C*-algebra theory and non-self-adjoint operator algebra theory. We use quantum symmetries to resolve some problems left open in work of Clouatre and Ramsey on finite dimensional approximations of representations, as well as a problem of Carlsen, Larsen, Sims and Vitadello on the existence of a co-universal C*-algebra for product systems over arbitrary right LCM semigroup embedded in groups. This latter problem was resolved for abelian lattice ordered semigroups by the speaker and Katsoulis, and we extend this to arbitrary right LCM semigroups. Consequently, we are also able to extend the Hao-Ng isomorphism theorems of the speaker with Katsoulis from abelian lattice ordered semigroups to arbitrary right LCM semigroups.

*This talk is based on two papers. One with Clouatre, and another with Kakariadis, Katsoulis, Laca and X. Li.