Noncommutative Analysis

Black Flag

Following the dishonorable exploitation of the coronavirus outbreak for stalling the activity of the newly elected parliament in Israel and for silencing protests, I am putting up a black flag on this blog.

Solutions to the situation can be found. Right now everywhere in the world meetings are made, decisions are taken, and classes are taught using video conferencing solutions. Many essential services go on as usual (construction, super-markets, the army, the police not to mention hospitals and so forth); they continue to operate, taking care to follow precautions.

There is absolutely no excuse for the newly elected Knesset not to begin operating immediately in full capacity, and first things first, replace the current chairman, who is behaving in a shameful way. Later, to approve new laws and a new government, according to the decisions made by the majority of the elected parliament members, and in compliance with the law.

Hopefully, the prime minister and the government will not abuse their power to stall this process or to gain leverage beyond the power they hold in the Knesset.

Here is an article in English (note that it is an opinion piece), that contains a lot of the facts:

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

Michael Skeide and I have recently uploaded our new paper to the arxiv: CP-Semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multi-Parameter Case and Beyond. In this gigantic (219 pages) paper, we propose a framework for studying the dilation theory of CP-semigroups parametrized by rather general monoids (i.e., semigroups with unit), and we use this framework for obtaining new results regarding the possibility or impossibility of constructing or having a dilation, we use it also for obtaining new structural results on the “mechanics” of dilations, and we analyze many examples using our tools. We present results that we have announced long ago, as well as some surprising discoveries.

This is an exciting moment for me, since we have been working on this project for more than a decade.

“Excuse me, did you really say decade?”

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A survey (another one!) on dilation theory

I recently uploaded to the arxiv my new survey “Dilation theory: a guided tour“. I am pretty happy and proud of the result! Right now I feel like it is the best survey ever written (honest, that’s how I feel, I know that its an illusion), but experience tells me that two months from now I might be a little embarrassed (like: how could I be so vain to think that I could pull of a survey on this gigantic topic?).

(Well, these are the usual highs and lows of being a mathematician, but since this is a survey paper and not a research paper, I feel comfortable enough to share these feelings).

This survey was submitted (and will hopefully appear in) to the Proceedings of the International Workshop on Operator Theory and its Applications (IWOTA) 2019, Portugal. It is an expanded version of the semi-plenary talk that I gave in that conference. I used a preliminary version of this survey as lecture notes for the mini-course that I gave at the recent workshop “Noncommutative Geometry and its Applications” at NISER Bhubaneswar.

I hope somebody finds it useful or entertaining 🙂

Janos Aczel (1924-2020)

I was saddened to find out that Janos Aczel passed away earlier this month. In my early days, after Boris Paneah got me hooked on functional equations, Aczel’s books caught my eye. Since then I am a fan of his. In particular, I was drawn by a small booklet by him whose title I am not able to reconstruct, and his two larger books “Lectures on Functional Equations and Their Applications” (1966) and “Functional Equations in Several Variables” (1987, co-authored with, Dhombres), which for a couple of years were to me among the most interesting and useful books I knew.

I have two stories two tell about Janos.

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My talks at NCGA Bhubaneswar

This week I am giving a series of five lectures on dilation theory in the workshop Noncommutative Geometry and its Applications , at NISER Bhubaneswar (India). I am putting my talks up here in case anybody would like to see them (and also as backup, in case my stick doesn’t work).

Lecture number 4.

Lecture number 5.

(Lectures 1,2,3 were board talks).