Noncommutative Analysis

Category: Research

Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball

Guy Salomon, Eli Shamovich and I recently uploaded to the arxiv our paper “Algebras of bounded noncommutative analytic functions on subvarieties of the noncommutative unit ball“. This paper blends in with the current growing interest in noncommutative functions theory, continues and unifies several strands of my past research.

A couple of years ago, after being inspired by lectures of Agler, Ball, McCarthy and  Vinnikov on the subject, and after years of being influenced by Paul Muhly and Baruch Solel’s work, I realized that many of my different research projects (subproduct systems, the isomorphism problem, space of Dirichlet series with the complete Pick property, operator algebras associated with monomial ideals) are connected by the unifying theme of bounded analytic nc functions on subvarieties of the nc ball. “Realized” is a strong word, because many of my original ideas on this turned out to be false, and others I still don’t know how to prove. Anyway, it took me a couple of years and a lot of help, and here is this paper.

In short, we study algebras of bounded analytic functions on subvarieties of the the noncommutative (nc) unit ball :

\mathfrak{B}_d = \{(X_1, \ldots, X_d) tuples of n \times n matrices,  \sum X_i X_i \leq I\}

as well as bounded analytic functions that extend continuously to the “boundary”. We show that these algebras are multiplier algebras of appropriate nc reproducing kernel Hilbert spaces, and are completely isometrically isomorphic to the quotient of H^\infty(\mathfrak{B}_d) (the bounded nc analytic functions in the ball) by the ideal of nc functions vanishing on the variety. We classify these algebras in terms of the varieties, similar to classification results in the commutative case. We also identify previously studied algebras (such as multiplier algebras of complete Pick spaces and tensor algebras of subproduct systems) as algebras of bounded analytic functions on nc varieties. See the introduction for more.

We certainly plan to continue this line of research in the near future – in particular, the passage to other domains (beyond the ball), and the study of algebraic/bounded isomorphisms.

Aleman, Hartz, McCarthy and Richter characterize interpolating sequences in complete Pick spaces

The purpose of this post is to discuss the recent important contribution by Aleman, Hartz, McCarthy and Richter to the characterization of interpolating sequences (for multiplier algebras of certain Hilbert function spaces). Their recent paper “Interpolating sequences in spaces with the complete Pick property” was uploaded to the arxiv about two weeks ago, and, as usual, writing this post is meant mostly as a diversion for me (somewhere between doing “real” work and getting frustrated about the news), just giving some background and stating the main result. (Even more recently these four authors released yet another paper that looks very interesting – this one.)

1. Background – interpolating sequences

We will be working with the notion of Hilbert function spaces – also called reproducing Hilbert spaces (see this post for an introduction). Suppose that H is a Hilbert function space on a set X, and k its reproducing kernel. The Pick interpolation problem is the following:

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Dilations, inclusions of matrix convex sets, and completely positive maps

In part to help myself to prepare for my talk in the upcoming IWOTA, and in part to help myself prepare for getting back to doing research on this subject now that the semester is over, I am going to write a little exposition on my joint paper with Davidson, Dor-On and Solel, Dilations, inclusions of matrix convex sets, and completely positive maps. Here are the slides of my talk.

The research on this paper began as part of a project on the interpolation problem for unital completely positive maps*, but while thinking on the problem we were led to other problems as well. Our work was heavily influenced by works of Helton, Klep, McCullough and Schweighofer (some which I wrote about the the second section of this previous post), but goes beyond. I will try to present our work by a narrative that is somewhat different from the way the story is told in our paper. In my upcoming talk I will concentrate on one aspect that I think is most suitable for a broad audience. One of my coauthors, Adam Dor-On, will give a complimentary talk dealing with some more “operator-algebraic” aspects of our work in the Multivariable Operator Theory special session.

[*The interpolation problem for unital completely positive maps is the problem of finding conditions for the existence of a unital completely positive (UCP) map that sends a given set of operators A_1, \ldots, A_d to another given set B_1, \ldots, B_d. See Section 3 below.]

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Preprint update (Stable division and essential normality…)

Shibananda Biswas and I recently uploaded to the arxiv a new version of our paper “Stable division and essential normality: the non-homogeneous and quasi-homogeneous cases“. This is the paper I announced in this previous post, but we had to make some significant changes (thanks to a very good referee) so I think I have to re announce the paper.

I’ve sometimes been part of conversations where we mathematicians share with each other stories of how some paper we wrote was wrongfully (and in some cases, ridiculously) rejected; and then I’ve also been in conversations where we share stories of how we, as referees, recently had to reject some wrong (or ridiculous) paper. But I never had the occasion to take part in a conversation in which authors discuss papers they wrote that have been rightfully rejected. Well, thanks to the fact that I sometimes work on problems related to Arveson’s essential normality conjecture (which is notorious for having caused some embarrassment to betters-than-I), and also because I have become a little too arrogant and not sufficiently careful with my papers, I have recently become the author of a rightfully rejected paper. It is a good paper on a hard problem, I am not saying it is not, and it is (now (hopefully!)) correct, but it was rejected for a good reason. I think it is a story worth telling. Before I tell the story I have to say that both the referee and my collaborator were professional and great, and this whole blunder is probably my fault.

So Shibananda Biswas and I submitted this paper Stable division and essential normality: the non-homogeneous and quasi-homogeneous cases for publication. The referee sent back a report with several good comments, two of which turned out to be serious. The two serious comments concerned what appeared as Theorem 2.4 in the first version of the paper (and it appears as the corrected Theorem 2.4 in the current version, too). The first serious  issue was that in the proof of the main theorem we mixed up between t and t+1, and this, naturally, causes trouble (well, I am simplifying. Really we mixed between two Hilbert space norms, parametrised by t and t+1). The second issue (which did not seem to be a serious one at first) was that at some point of the proof we claimed that a particular linear operator is bounded since it is known to be bounded on a finite co-dimensional subspace; the referee asked for clarifications regarding this step.

The first issue was serious, but we managed to fix the original proof, roughly by changing t+1 back to t. There was a price to pay in that the result was slightly weaker, but not in a way that affected the rest of the paper. Happily, we also found a better proof of the result we wanted to prove in the first place, and this appears as Theorem 2.3 in the new and corrected version of the paper.

The second issue did not seem like a big deal. Actually, in the referee’s report this was just one comment among many, some of which were typos and minor things like that, so we did not really give it much attention. A linear operator is bounded on a finite co-dimensional subspace, so it is bounded on the whole space, I don’t have to explain that!

We sent the revision back, and after a while the referee replied that we took care of most things, but we still did not explain the part about the operator-being-bounded-because-it-is-bounded-on-a-finite-co-dimensional-space. The referee suggested that we either remove that part (since we already had the new proof), or we explain it. The referee added, that in either case he suggests to accept the paper.

Well, we could have just removed that part indeed and had the paper accepted, but we are not in the business of getting papers accepted for publication, we are in the business of proving theorems, and we believed that our original proof was interesting in itself since it used some interesting new techniques. We did not want to give up on that proof.

My collaborator wrote a revision with a very careful, detailed and rigorous explanation of how we get boundedness in our particular case, but I was getting angry and I made the big mistake of thinking that I am smarter than the referee. I thought to myself: this is general nonsense! It always holds. So I insisted on sending back a revision in which this step is explained by referring to a general principle that says that an operator which is bounded on a finite co-dimensional subspace is bounded.


That’s not quite exactly precisely true. Well, it depends what you mean by “bounded on a finite co-dimensional subspace”. If you mean that it is bounded on a closed subspace which has a finite dimensional algebraic complement then it is true, but one can think of interpretations of “finite co-dimensional” that make this is wrong: for example, consider an unbounded linear functional: it is bounded on its kernel, which is finite co-dimensional in some sense, but it is not bounded.

The referee, in their third letter, pointed this out, and at this point the editor decided that three strikes and we are out. I think that was a good call. A slap in the face and a lesson learned. I only feel bad for my collaborator, since the revision he prepared originally was OK.

Anyway, in the situation studied in our paper, the linear subspace on which the operator is bounded is a finite co-dimensional ideal in the ring of polynomials. It’s closure has zero intersection with the finite dimensional complement (the proof of this is not very hard, but is indeed non-trivial and makes use of the nature of the spaces in question), and everything is all right.  Having learned our lessons, we explain everything in detail in the current version. I hope that carefully enough.

I think that what caused us most trouble was that I did not understand what the referee did not understand. I assumed (very incorrectly, and perhaps arrogantly) that they did not understand a basic principle of functional analysis; it turned out that the referee did not understand why we are in a situation where we can apply this principle, and with hindsight this was worth explaining in more detail.

One of the most outrageous open problems in operator/matrix theory is solved!

I want to report on a very exciting development in operator/matrix theory: the von Neumann inequality for 3 \times 3 matrices has been shown to hold true. I learned this from a recent paper (with the irresistible title) “The von Neumann inequality for 3 \times 3 matrices“, posted on the arxiv by Greg Knese. In this paper, Knese explains how the solution of this outstanding open problem follows from results in a paper by Lukasz Kosinski, “The three point Nevanlinna-Pick problem in the polydisc” that appeared on the arxiv about a half a year ago. Beautifully, and not surprisingly, the solution of this operator/matrix theoretic problem follows from deep new facts in complex function theory in several variables.

To recall the problem, let us denote \|A\| the operator norm of a matrix A, and for every polynomial p in d variables we denote by \|p\|_\infty the supremum norm

\|p\|_\infty = \sup_{|z_i|\leq 1} |p(z_1, \ldots, z_d)|.

A matrix A is said to be contractive if \|A\| \leq 1.

We say that d commuting contractions A_1, \ldots, A_d satisfy von Neumann’s inequality if 

(*)  \|p(A_1,\ldots, A_d)\| \leq \|p\|_\infty.

It was known since the 1960s that (*) holds when d \leq 2. Moreover, it was known that for d \geq 3, there are counter examples, consisting of d contractive 4 \times 4 matrices that do not satisfy von Neumann’s inequality. On the other hand, it was known that (*) holds for any d if the matrices A_1, \ldots, A_d are of size 2 \times 2. Thus, the only missing piece of information was whether or not von Neumann’s inequality holds or not for three or more contractive 3 \times 3 matrices. To stress the point: it was not known whether or not von Neumann’s inequality holds for three three-by-three matrices. The problem in this form has been open for 15 years  – but the problem is much older: in 1974 Kaiser and Varopoulos came up with a 5 \times 5 counter-example, and since then both the 3 \times 3  and the 4 \times 4 cases were open until Holbrook in 2001 found a 4 \times 4 counter example. You have to agree that this is outrageous, perhaps even ridiculous, I mean, three 3 \times 3 matrices, come on!

In Knese’s paper this story and the positive solution to the problem is explained very clearly and succinctly, and is recommended reading for any operator theorist. One has to take on faith the paper of Kosinski which, as Knese stresses, is where the major new technical advance has been made (though one should not over-stress this fact, because tying things together, the way Knese has done, requires a deep understanding of this problem and of the various ingredients). To understand Kosinki’s paper would require a greater investment of time, but it appears that the paper has already been accepted for publication, so I am quite confident and happy to see this problem go down.

Spaces of Dirichlet series with the complete Pick property (or: the Drury-Arveson space in a new disguise)

John McCarthy and I have recently uploaded a new version of our paper “Spaces of Dirichlet series with the complete Pick property” to the arxiv. I would like to advertise the central discovery of this paper here.

Recall that the Drury-Arveson space H^2_d is the reproducing kernel Hilbert space on the open unit ball of a d dimensional Hilbert space, with reproducing kernel

k(z,w) = \frac{1}{1 - \langle z, w \rangle}.

It has the remarkable universal property that every Hilbert function space with the complete Pick property is naturally isomorphic to the restriction of H^2_\infty to a subset of the unit ball (see Theorem 6 and its corollary in this post), and consequently, every complete Pick algebra is a quotient of the multiplier algebra \mathcal{M}_\infty = Mult(H^2_\infty). To the best of my knowledge, no other Hilbert function spaces with such a universal property have been studied.

John and I discovered another reproducing kernel Hilbert space that turns out to be “the same” as the Drury-Arveson space H^2_\infty. Since the space H^2_\infty as been so well studied, it interesting to discover a new incarnation. The really interesting part is that the space we discovered is a space of analytic functions on a half plane (that is, a space of functions in one complex variable), rather than a space of analytic functions in infinitely many variables on the unit ball of a Hilbert space.

To be precise, the spaces we consider are spaces of Dirichlet series \mathcal{H}, of the form

\mathcal{H} = \{f(s) = \sum_{n=1}^\infty \gamma_n n^{-s} : \sum |\gamma_n|^2 a_n^{-1} < \infty \}.

(Here a_n is a sequence of positive numbers). These are Hilbert function spaces on some half plane that have a kernel of the form k(s,u) = \sum a_n n^{-s-\bar u}.

We first answer the question which of these spaces \mathcal{H} have the complete Pick property. This problem has a simple solution (which has been anticipated by similar results on spaces on the disc): if we denote by g(s) = \sum a_n n^{-s} the “generating function” of the space, and if we write

\frac{1}{g(s)} = \sum c_n n^{-s},

then \mathcal{H} is a complete Pick space if and only if c_n \leq 0 for all n \geq 2.

After we know to tell when these spaces are complete Pick, it is natural to ask which complete Pick spaces arise like this? We do not give a complete answer, but our surprising discovery is that things can easily be cooked up so to obtain the Drury-Arveson space H^2_d, where d can be any cardinal number in \{1,2,\ldots, \infty\}. For example, \mathcal{H} turns out to be “the same” as H^2_\infty if the kernel k is given by

k(s,u) = \frac{P(2)}{P(2) - P(2+s+\bar u)},

where P(s) = \sum_{p} p^{-s} is the prime zeta function (the sum is taken over all primes p).

 Now, I have been a little vague about what it means that \mathcal{H} is “the same” as H^2_\infty. In fact, this is a subtle question, and we devote a part of our paper what it means for two Hilbert function spaces to be the same — something that has puzzled us for a while.

What does this appearance of Drury-Arveson space as a space of Dirichlet series mean? Can we use this connection to learn something new on multivariable operator theory, or on Dirichlet series? How did the prime zeta function smuggle itself into this discussion? This requires further thought.

Stable division and essential normality: the non-homogeneous and quasi homogeneous cases

Update (January 29, 2016): paper revised, see this post


Several months ago Shibananda Biswas (henceforth: Shibu) and I posted to the arxiv our paper “Stable division and essential normality: the non-homogeneous and quasi homogeneous cases“. I was a little too busy to write about it at the time, but now that it is summer it seems like a good time to do it, since I am too busy, and I need a break from work. Nothing like going back and thinking about papers you have already written when you are overwhelmed by your current project.

Anyway, the main problem the paper I wrote with Shibu deals with, is the essential normality of submodules of various Hilbert modules (closely related to the Drury-Arveson module that I wrote about in the past: one, two, three, or if you are really asking for trouble, look at this survey). This paper is highly technical, and I want to try to explain it in a non-technical fashion. Read the rest of this entry »

A corrigendum

Matt Kennedy and I have recently written a corrigendum to our paper “Essential normality, essential norms and hyperrigidity“. Here is a link to the corrigendum. Below I briefly explain the gap that this corrigendum fills.

A corrigendum is correction to an already published paper. It is clear why such a mechanism exists: we want the papers we read to represent true facts, so false claims, as well as invalid proofs or subtle gaps should be pointed out to the community. Now, many many papers (I don’t want to say “most”) have some kind of mistake in them, but not every mistake deserves a corrigendum – for example there are mistakes that the reader will easily spot and fix, or some where the reader may not spot the mistake, but the fix is simple enough.

There are no rules as to what kind of errors require a corrigendum. This depends, among other things, on the authors. Some mistakes are corrected by other papers. I believe that very quickly some sort of mechanism – say google scholar, or mathscinet – will be able to tell if the paper you are looking up is referenced by another paper pointing out a gap, so such a correction-in-another-paper may sometimes serve as legitimate replacement for a corrigendum, when the issue is a gap or minor mistake.

There is also a question of why publish a corrigendum at all, instead of updating the version of the paper on the arxiv (and this is exactly what the moderators of the arxiv told us at first when we tried to upload our corrigendum there. In the end we convinced them that the corrigendum can stand by itself). I think that once a paper is published, it could be confusing to have a version more advanced than the published version; it becomes very clumsy to cite papers like that.

The paper I am writing about (see this post to see what its about) had a very annoying gap: we justified a certain step by citing a particular proposition from a monograph. The annoying part is that the proposition we cite does not exactly deal with the situation we deal with in the paper, but our idea was that the same proof works in our situation. We did not want to spell out the details because we considered that to be very easy, and in any case it was not a new argument. Unfortunately, the same proof does work when working with homogeneous ideals (which was what first versions of the paper treated) but in fact it is not clear if they work for non-homogeneous ideals. The reason why this gap is so annoying, is that it leads the reader to waste time in a wild goose chase: first the reader goes and finds the monograph we cite, looks up the result (has to read also a few extra pages to see he understands the setting and notation in the monograph), realises this is is not the same situation, then tries to adapt the method but fails. A waste of time!

Another problem that we had in our paper is that one requires our ideals to be “sufficiently non-trivial”. If this were the only problem we would perhaps not bother writing a corrigendum just to introduce a non-triviality assumption, since any serious reader will see that we require this.

If I try to take a lesson from this, besides a general “be careful”, it is that it is dangerous to change the scope of the paper (for us – moving form homogeneous to non-homogeous ideals) in late stages of the preparation of the paper. Indeed we checked that all the arguments work for the non-homogneous case, but we missed the fact that an omitted argument did not work.

Our new corrigendum is detailed and explains the mathematical problem and its solutions well, anyone seriously interested in our paper should look at it. The bottom line is this as follows.

Our paper has two main results regarding quotients of the Drury-Arveson module by a polynomial ideal. The first is that the essential norm in the non selfadjoint algebra associated to a the quotient module, as well as the C*-envelope, are as the Arveson conjecture predicts (Section 3 in the paper) . The second is that essential normality is equivalent to hyperrigidity (Section 4 in the paper).

Under the assumption that all our ideals are sufficiently non-trivial (and some other standing assumptions stated in the paper), the situation is as follows.

The first result holds true as stated.

For the second result, we have that hyperrigidity implies essential normality (as we stated), but the implication “essential normality implies hyperrigidity” is obtained for homogeneous ideals only.


“Guided” and “quantised” dynamical systems

Evegenios Kakariadis and I have recently posted our paper “On operator algebras associated with monomial ideals in noncommuting variables” on the arxiv. The subject of the paper is several operator algebras (at the outset, there are seven algebras, but later we prove that some are isomorphic to others) that one can associate with each monomial ideal, in such a way that these algebras encode various aspects of the relations defining the ideal.

I refer you to the abstract and intro of that paper for more information about we do there. In this post I would like to discuss at some length an issue that came up writing the paper, and the paper itself was not an appropriate place to have this discussion.

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The isomorphism problem for complete Pick algebras: a survey

Guy Salomon and I recently finished preparing a survey article for the Proceedings of IWOTA 2014. The talk I gave at the conference was an overview of the current state of this problem, so it made sense to prepare a contribution that did the same. I discussed this at length in a previous post. I think that the current survey is currently the best overview of the subject, and also contains some modest improvements and corrections to what appears in the literature. Here is the link: The isomorphism problem for complete Pick algebras: a survey.