Noncommutative Analysis

Dilations of q-commuting unitaries

Malte Gerhold and I just have just uploaded a revision of our paper “Dilations of q-commuting unitaries” to the arxiv. This paper has been recently accepted to appear in IMRN, and was previously rejected by CMP, so we have four anonymous referees and two handling editors to be thankful to for various corrections and suggested improvements (though, as you may understand, one editor and two referees have reached quite a wrong conclusion regarding our beautiful paper :-).

This is a quite short paper (200 full pages shorter than the paper I recently announced), which tells a simple and interesting story: we find that optimal constant c_\theta, such that every pair of unitaries u,v satisfying the q-commutation relation

vu = e^{i\theta} uv

dilates to a pair of commuting normal operators with norm less than or equal to c_\theta (this problems is related to the “complex matrix cube problem” that we considered in the summer project half year ago and the one before). We provide a full solution. There are a few ramifications of this idea, as well as surprising connections and applications, so I invite you to check out the nice little introduction.

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 “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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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 »