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 .

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.

New preprint: Tensor algebras of subproduct systems and noncommutative function theory

For a several years I believed that every tensor algebra of a subproduct system with Hilbert space fibers is the algebra of uniformly continuous noncommutative functions on a homogeneous noncommutative variety. Now, since the isomorphism problem for tensor algebras has interested me for a long time, and was solved for the case of finite dimensional fibers, and since the isomorphism problem for algebras of uniformly continuous functions on homogeneous noncommutative varieties was, even for the case of infinitely many variables, solved by Salomon, Shamovich and myself, I returned several months ago to the problem of showing that every tensor algebra can indeed be represented as an algebra of nc functions.

Talk: CP-semigroups and dilations, subproduct systems and superproduct systems

Last Friday I gave a talk at the Athens Functional Analysis Seminar on my joint paper with Michael Skeide, which is to appear in Dissertationes Mathematicae. This time I tried to emphasize the appearance of Morita equivalence in our work.

Souvenirs from the children’s room, and the warmest recommendation for an online mini-course

Ilia Binder, Damir Kinzebulatov and Javad Mashreghi have organized a Focus Program on Analytic Function Spaces and their Applications at the Fields Institute, and this week, as part of this focus program, there was a Mini-course and Workshop on Drury-Arveson Space which I virtually attended (from the “children’s room” in our house, because that’s where we have the internet connection). The workshop is still not over, I have Ken Davidson’s talk to look forward to tonight.

I used to have a section in my blog Souvenirs from … where I would write about my favorite talks that I heard in recent conferences. This exercise helped concentrate during conferences (“hmmm, I wonder who’s going to be my souvenir?”) and also helped me get the most out of great talks after the conference (writing about stuff forces you to actually look up the paper or at least have another look at the notes you took during the talk). In fact, some of the souvenirs I brought home from conferences ended up becoming major parts of my own research program.

“Coming back” from the workshop on Drury-Arveson space, I can report that all the talks are recorded and can be found on the Fields Institute’s Youtube channel. To a certain extent that makes the task of reporting from conferences seem less needed.

Still, I will share my recommendations. And I want to give one very very warm recommendation for the Mini-course that Michael Hartz gave on the Drury-Arveson space. I have been to several minicourses in my life, and I also gave a couple, and I think that I have never seen a better prepared or more motivating mini-course. It was artful! Really, anybody going to work on Drury-Arveson space and the related operator theory should see it.

Here are the talks:

First lecture:

Mini-course on Drury-Arveson space, Lecture 1

Second Lecture:

Mini-course on Drury-Arveson space, Lecture 2

Third lecture:

Mini-course on Drury-Arveson space, Lecture 3

Since it was recorded, I can also put up here a link to my own talk at the workshop “Quotients of the Drury-Arveson space and their classification in terms of complex geometry”: