Noncommutative Analysis

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

Reblogging the announcement of my recent giant paper with Michael Skeide. I want this to be at the head of my blog for a while.

Noncommutative Analysis

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 โ€œmechanicsโ€ 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.

โ€œExcuse me, did you really say decade?โ€


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White flag

After the previous post, the political system in Israel spiraled into a farce. Things are not done yet, but the not-very-big chance of changing things for the better was thrown out the window for the purpose of a so-called “emergency government to battle the corona crisis”. Nevermind, the point is that I couldn’t leave the previous post as the top one, as if there is some battle between good and bad forces. Apparently the political system got tired of itself, gave up on law and order, democracy, and basic principles of decency, and there is no hope that I will have a government that I am proud of in the coming years. I will try to follow politics as little as I can, and comment on politics even less.

Because the previous post was obsolete and because I really did not feel like updating on the sorry situation, I made this blog private for a short while. I am now bringing it back, with this explanation. I am leaving the black banner though, until there is reason to remove it.

I hope to continue to post on mathematics soon.

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:

https://www.washingtonpost.com/opinions/2020/03/19/with-pandemic-cover-netanyahu-is-carrying-out-coup-israel/

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?”

Read the rest of this entry »

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 ๐Ÿ™‚