Noncommutative Analysis

The isomorphism problem for algebras of bounded noncommutative functions on subvarieties of nc operator balls

Last week, Jeet Sampat and I posted on the arxiv our new preprint, Weak-* and completely isometric structure of noncommutative function algebras. This is the second paper in which we study the isomorphism problem for algebras of bounded noncommutative (nc) functions on subvarieties of nc operator balls. The first paper on this theme, On the classification of function algebras on subvarieties of noncommutative operator balls, was posted on the arxiv last year and was recently published in the Journal of Functional Analysis.

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Souvenirs from John E. McCarthy’s 60 birthday conference in St. Louis

Last week I attended the Multivariable Operator Theory Conference in honor of John E. McCarthy on his 60th birthday, in Washington University in St. Louis. John McCarthy is one of the leaders of the field that is called multivariable operator theory, which is in essence the study of functional analysis and operator theory in conjunction with several complex variables. Besides being a remarkable researcher, John is also a great communicator (talks, papers, books, expository essays…), a wonderful person and has helped many mathematicians grow. I am grateful for being one of his many collaborators.

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June 13, 2024

Dilation distance and the stability of ergodic commutation relations

Malte Gerhold and I have recently uploaded to the ArXiv our preprint “Dilation distance the stability of ergodic commutation relations“. In this paper, we continue refining the dilation techniques that we have been developing in the past years to study when certain unitary tuples can be approximated, in a certain sense, by other unitary tuples. A baby version of the main result of the paper can be stated as follows.

Theorem: Fix a complex number $q$ on the unit circle which is not a root of unity. For every $\varepsilon > 0$ there is a $\delta>0$ such that if $u$ and $v$ are unitaries on a Hilbert space $H$ such that $\|uv - qvu\|< \delta$ , then there exist two $q$ -commuting unitaries $U$ and $V$ on $H \otimes \ell^2$ such that $\|u\otimes 1 - U\| + \|v \otimes 1 - V\| < \varepsilon$ .

Here as usual “ $q$ -commuting” means $UV = qVU$ . We also obtain higher dimensional versions, for tuples almost satisfying the relations of noncommutative tori. The “ergodic” in the title refers to the higher dimensional requirement corresponding to $q$ not being a root of unity.

The conclusion of the theorem is false if you remove the ampliation (tensoring with identity), although in certain cases we can bootstrap the result and obtain it with no ampliation.

It is interesting to mention, for the record, that we were attempting to find a new proof for a theorem of Lin, which is the above theorem for $q = 1$ . Our methods do not give this, unless the original tuple is gauge invariant. Thus, we discovered the above proof by mistake.

May 8, 2024

Noncommutative Function Theory and Free Probability at Oberwolfach

I spent the week 28.4 – 3.5 at the MFO at Oberwolfach in a workshop on noncommutative function theory and free probability (whatever the hell that means), where I gave (ahem, ahem) a three lecture mini-course “Noncommutative Function Theory ~~for Free Probabilists~~ for Everyone”. It is a curious exercise to give a mini-course to a crowd that consists of about 45 superb mathematicians, about a third of which know at least as much as I do about the subject, and another fraction know almost nothing about it. It was hard work, and so I did not bring back any souvenirs.

I spent the week sadly thinking to myself, how could such a place like MFO exist? How could it be that every week a group of 48 mathematicians get pampered and fed, rest, hike, drink, give some talks and hear some talks, and all this to foster research in (usually pure) mathematics. When one raises one’s head from the scribbles on one’s notepad, and looks at the state of the world, it is hard not to think: how can this be?

I told wise old Bill Helton, one of the kind godfathers of our field, that I can’t believe that this place exists, and asked him whether with time one gets used to the idea. Does he believe that Oberwolfach exists? He answered “Of course! The world would fall apart without it.”

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April 9, 2024

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 $A$ be a unital operator algebra generating a C*-algebra $B$ .