Noncommutative Analysis

Category: Research

Three classification results in the theory of weighted hardy spaces in the ball – summary of summer project

Last month we had the Math Research Week here at the Technion, and I promised in a previous post to update if there would be any interesting results (see that post for background on the problems). Well, there are! I am writing this short post just to update as promised on the interesting results.

The two excellent students that worked with us – Danny Ofek and Gilad Sofer – got some nice results. They almost solved to a large extent the main problems mentioned in my earlier post. See this poster for a concise summary of the main results:

Danny and Gilad summarized their results in the following paper. Just take a look. They have some new results that I thought were true, they have some new results that I didn’t guess were true, and they also have some new and simplified proofs for a couple of known results. Their work fits in the long term research project to discover how the structure of Hilbert function spaces and their multiplier algebras encodes the underlying structures, and especially the geometry of sets in the unit disc or the unit ball. More on that soon!

Summer project 2020 – Hilbert function spaces of analytic functions in a complex variable

In the week of September 6-11 the Math Department at the Technion will again host the “Math Research Week“, or what we refer to as the the “summer projects week”. As in previous years, I will be offering a project, and this year, with the help of Ran Kiri and Satish Pandey, it will be a project on Hilbert function spaces. See here for the abstract. The purpose of this post is to collect my thoughts and my plans for this project.

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My talk at Fields – video available

Hi just got an email from the Fields Institute that the video recording of my talk “CP-semigroups and Dilations, Subproduct Systems and Superproduct systems,: the Multiparameter Case and Beyond” (on my paper with Michael Skeide I announced here) that I gave at COSY, is now available. The slides are available here. This is the first time I see (and hear!) a recording of myself giving a talk in English and, wow, it’s devastating. Thanks for bearing with me 🙂

Here is a link to the talk:

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.

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.