Arveson’s hyperrigidity conjecture refuted by Bilich and Dor-On

Boom! This morning Boris Bilich and Adam Dor-On published a short preprint on the arXiv “Arveson’s hyperrigidity conjecture is false” in which they provide a counter example that refutes Arveson’s hyperrigidity conjecture. This is a fantastic achievement! It is one of the most interesting things that happened in my field lately and also somewhat of a surprise, a paper that is sure to make a significant impact on the subject.

(I should say that Adam was kind enough to let me read the manuscript a week ago, so that I had time already to check the details and as far as I can tell it looks correct.)

Let us recall quickly what the conjecture is (for more background see the series of posts that I wrote for the topics course I gave several years ago).

Let be a unital operator algebra generating a C*-algebra .

On dilations of CP-semigroups (my talk at the Midrasha at Weizmann Institute)

I had the privilege of being invited this year again to give a talk at the Group Theory Seminar at the Weizmann Institute (Here is a link to last year’s talk). I am pasting here a link to the recording of the talk; the link is to the third minute, the recording started three minutes before the lecture began.

Title: Dilations of CP-semigroups via subproduct systems and superproduct systems of C*-correspondences.

Abstract:

The title is a bit of a mouthful, so let us unpack it together:

1. A C*-correspondence is a certain kind of bimodule over a C*-algebra B that has a B-valued inner product.
2. Sub and super-product systems are families of C*-correspondences that enjoy certain semigroup-like properties under the tensor product. 
3. A CP-semigroup is a family of completely positive maps that form a semigroup under composition; the most important examples are when the semigroup is given by a family (T_t) parameterized by positive real t>0.
4. By dilation of CP-semigroup we mean a specific way of exhibiting one CP-semigroup as a part of another semigroup that belongs to a better understood category, typically a semigroup of *-homomorphisms.

The problem of constructing dilations for CP-semigroups or determining their existence/uniqueness has drawn some of the best minds in operator algebras. And still there are open problems. I will describe the elaborate framework of sub and super-product systems employed by Michael Skeide and myself to attack this problem, a framework which is built on top of decades of work by ourselves and others. Using this framework we resolve some problems in the multi-parameter case; more surprisingly, we also obtain new results in the one-parameter case.

“CP-semigroups and dilations… and beyond” is now in print

Michael Skeide’s and my big joint paper “CP-semigroups and dilations, subproduct systems and superproduct systems: the multi parameter case and beyond” is now in print. Actually, I received the copies a couple of months ago, but I didn’t feel like posting anything. I blogged a few years back about what a difficult project this was, and how long it took to complete it. It also took quite to get published and we had to overcome some very annoying technical issues – I wrote about that here. But in the end, finally, the printed version of the paper appeared as a standalone “book” in the excellent journal Dissertationes Mathematicae (it is the analogue of Memoirs of the AMS published by the Institute of Mathematics in the Polish Academy of Sciences). Now I know that the project is really over. It feels wonderful to hold a copy in my hand. It is much more fun to read a physical copy than an electronic one.

From idea to paper in just fourteen years

The paper “CP-semigroups and Dilations, Subproduct Systems and Superproduct Systems: The Multiparameter Case and Beyond” is finally published (electronically): see here for the journal page. As I wrote on this blog before, we started working on the project in 2009, we completed work and submitted to the arxiv im March 2020. It is a very long paper (now 233 pages) and there aren’t many venues that publish papers of this length. We then tried a couple of journals who rejected our work, and then in May 2020 we submitted to the journal Dissertationes Mathematicae, which is the Polish Academy of Sciences’s analogue of Memoirs of the AMS (every paper constitutes an entire volume). Our paper was accepted in May 2022, we received the first proofs in November 2022, and now, a year later, after a very painstaking and painful editorial process, our paper is published as Volume 585 of Dissertationes Math. I have given some talks on this work, and will probably give some more talks, and we already have a new (much shorter!) paper under preparation making use of the general framework. Right now I do not want to talk about anything except to record the fact that this project that we have been working on for fourteen years is finally complete. I leave it as an exercise for the reader to draw conclusions on publishing, the life of a research mathematician, and the world.

My talk at the Weizmann institute (“Dilation theory in action”)

Last week I was invited by Guy Salomon to give a talk at the Group Theory Midrasha at the Weizmann Institute (Midrasha is a fancy Hebrew word for seminar). Since the seminar there is two hours long, I took the opportunity to try something different, and for the first time in a long while I gave a whiteboard talk, going into the details of the proof of the main result in my paper with Gerhold on perturbation of the Heisenberg commutation relations. The group there at Weizmann is really fantastic with many young and curious (and bright!) students who bombarded me with questions, so the talk was quite alive and I think it was a successful experiment (yesterday I gave a similar talk at the Analysis Seminar at Bar-Ilan University; the crowd was full of strong analysts who also asked great questions, but since I aimed for an hour I got pressed for time, so I think in the end it wasn’t as good. That’s on me, because they actually let me choose whether I want to go for two hours or one, and again I wanted to try something a bit different).

Here is a video recording of the talk.

BTW: You can see that someone in the audience asked me a question that I, embarrassingly, blacked out on: do strongly commuting (unbounded) operators commute in the sense that there is some dense subspace on which the commutator is defined and equal to zero? The answer is yes and is actually not hard to show with basic semigroup theory techniques. A little trickier is to show that strongly commuting operators have commuting spectral projections – which is an equivalent and perhaps more natural definition of “strong commutation” than the one I gave.