Noncommutative Analysis

Category: Talks

Seminar talk at the BGU OA Seminar

This coming Thursday (July 2nd, 14:10 Israel Time) I will be giving a talk at the Ben-Gurion University Math Department’s Operator Algebras Seminar. If you are interested in a link to the Zoom please send me an email.

I will be talking mostly about these two papers of mine with co-authors: older one, newer one. Here is the title and abstract:

Title: Matrix ranges, fields, dilations and representations

Abstract: In my talk I will present several results whose unifying theme is a matrix-valued analogue of the numerical range, called the matrix range of an operator tuple. After explaining what is the matrix range and what it is good for, I will report on recent work in which we prove that there is a certain “universal” matrix range, to which the matrix ranges of a sequence of large random matrices tends to, almost surely. The key novel technical aspects of this work are the (levelwise) continuity of the matrix range of a continuous field of operators, and a certain quantitative matrix valued Hahn-Banach type separation theorem. In the last part of the talk I will explain how the (uniform) distance between matrix ranges can be interpreted equivalently as a “dilation distance”, which can be interpreted as a kind of “representation distance”. These vague ideas will be illustrated with an application: the construction of a norm continuous family of representations of the noncommutative tori (recovering a result of Haagerup-Rordam in the d=2 case and of Li Gao in the d>2 case).

Based on joint works with Malte Gerhold, Satish Pandey and Baruch Solel.

My slides for the COSY talk and the seminar talk

Here is a link to the slides for the short talk that I am giving in COSY.

This talk is a short version of the talk I gave at the Besancon Functional Analysis Seminar last week; here are the slides for that talk.

Seminar talk

Next Tuesday, May 19th, at 14:30 (Israeli time), I will give a video talk at the Séminaire d’Analyse Fonctionnelle “in” Laboratoire de mathématiques de Besançon. It will be about my recent paper with Michael Skeide, the one that I announced here.

Title: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: the Multi-Parameter Case and Beyond.

Abstract: We introduce a framework for studying dilations of semigroups of completely positive maps on von Neumann algebras. The heart of our method is the systematic use of families of Hilbert C*-correspondences that behave nicely with respect to tensor products: these are product systems, subproduct systems and superproduct systems. Although we developed our tools with the goal of understanding the multi-parameter case, they also lead to new results even in the well studied one parameter case. In my talk I will give a broad outline and a taste of the dividends our work.

The talk is based on a recent joint work with Michael Skeide.

Assumed knowledge: Completely positive maps and C*-algebras.

Feel free to write to me if you are interested in a link to the video talk.

Around and under my talk at Fields

This week I am attending a Workshop on Developments and Technical Aspects of Free Noncommutative Functions at the Fields Institute in Toronto. Since I plan to give a chalk-talk, I cannot post my slides online (and I cannot prepare for my talk by preparing slides), so I will write here what some ideas around what I want to say in my talk, and also some ramblings I won’t have time to say in my talk.

[Several years ago I went to a conference in China and came back with the insight that in international conferences I should give a computer presentation and not a blackboard talk, because then people who cannot understand my accent can at least read the slides. It’s been almost six years since then and indeed I gave only beamer-talks since. My English has not improved over this period, I think, but I have several reasons for allowing myself to give an old fashioned lecture – the main ones are the nature of the workshop, the nature of the audience and the kind of things I have to say]. 

In the workshop Guy Salomon, Eli Shamovich and I will give a series of talks on our two papers (one and two). These two papers have a lot of small auxiliary results, which in usual conference talk we don’t get the chance to speak about. This workshop is a wonderful opportunity for us to highlight some of these results and the ideas behind them, which we feel might be somewhat buried in our paper and have gone unnoticed. 

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The complex matrix cube problem (in “Summer Projects in Mathematics at the Technion”)

Next week I will participate as a mentor in the Technion’s Summer Projects in Mathematics. The project I offered is called “Numerical explorations of open problems from operator theory”, and it suggests three open problems in operator theory where theoretical progress seems to be stuck, and for which I believe that some computer experiments can help us get a feeling of what is going on. I also hope that thinking seriously about designing experiments can help us to understand some general facets of the theory.

I have been in contact with the students in the last few weeks and we decided to concentrate on “the matrix cube problem”. On Sunday, when the week begins, I will need to present the background to the project to all participants of this week, and I have seven minutes (!!) for this. As everybody knows, the shorter the presentation, the harder the task is, and the more preparation and thought it requires. So I will take use this blog to practice my little talk.

Introduction to the matrix cube problem

This project is in the theory of operator spaces. My purpose is to give you some kind of flavour of what the theory is about, and what we will do this week to contribute to our understanding of this theory.

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