Noncommutative Analysis

Seminar talk by Salomon: Combinatorial and operator algebraic aspects of proximal actions

The Operator Algebras and Operator Theory Seminar is back (sort of). This semester we will have the seminar on Thursdays 15:30 (Israel time) about once in a while. Please send me an email if you want to join the mailing list and get the link for the zoom meetings. Here are the details for our first talk :

Title: Combinatorial and operator algebraic aspects of proximal actions

Speaker: Guy Salomon (Weizmann Institute)

Time: 15:30-16:30,Thursday Nov. 12, 2020

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

Zoom link: email me.

Abstract:

An action of a discrete group G on a compact Hausdorff space X is called proximal if for every two points x and y of X there is a net g_i \in G such that \lim(g_i x)=\lim(g_i y), and strongly proximal if the natural action of G on the space P(X) of probability measures on X is proximal. The group G is called strongly amenable if all of its proximal actions have a fixed point and amenable if all of its strongly proximal actions have a fixed point.

In this talk I will present relations between some fundamental operator theoretic concepts to proximal and strongly proximal actions, and hence to amenable and strongly amenable groups. In particular, I will focus on the C*-algebra of continuous functions over the universal minimal proximal G-flow and characterize it in the category of G-operator-systems.

I will then show that nontrivial proximal actions of G can arise from partitions of G into a certain kind of “large” subsets. If time allows, I will also present some relations to the Poisson boundaries of G. The talk is based on a joint work with Matthew Kennedy and Sven Raum.

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!

Michael Hartz awarded Zemanek prize in functional analysis

Idly skimming through the September issue of the Newsletter of the European Mathematical Society, I stumbled upon the very happy announcement that the 2020 Jaroslav and Barbara Zemanek prize in functional analysis with emphasis on operator theory was awarded to Michael Hartz.

The breakthrough result that every complete Nevanlinna-Pick space has the column-row property is one of his latest results and has appeared on the arxiv this May. Besides solving an interesting open problem, it is a really elegant and strong paper.

It is satisfying to see a young and very talented mathematician get recognition!

Full disclosure 😉 Michael is a sort of mathematical relative (he was a PhD student of my postdoc supervisor Ken Davidson), a collaborator (together with Ken Davidson we wrote the paper Multipliers of embedded discs) and a friend. I have to boast that from the moment that I heard about him I knew that he will do great things – in his first paper, which he wrote as a masters student, he ingeniously solved an open problem of Davidson, Ramsey and myself. Since then he has worked a lot on some problems that are close to my interests, and I have been following him with admiration.

Congratulations Michael!

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.

Read the rest of this entry »

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: