Noncommutative Analysis

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

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.

 

Souvenirs from the Rocky Mountains

I recently returned from the Workshop on Multivariate Operator Theory at Banff International Research Station (BIRS). BIRS is like the MFO (Oberwolfach): a mathematical resort located in the middle of a beautiful landscape, to where mathematicians are invited to attend/give talks, collaborate, interact, catch up with old friends, make new friends, have fun hike, etc.

As usual I am going over the conference material the week after looking for the most interesting things to write about. This time there were two talks that stood out from my perspective, the one by Richard Rochberg (which was interesting to me because it is on a problem that I have been thinking a lot about), and the one by Igor Klep (which was fascinating because it is about a subject I know little about but wish to learn). There were some other very nice talks, but part of the fun is choosing the best; and one can’t go home and start working on all the new ideas one sees.

A very cool feature of BIRS is that now they automatically shoot the talks and put the videos online (in fact the talks are streamed in real time! If you follow this link at the time of any talk you will see the talk; if you follow the link at any other time it is even better, because there is a webcam outside showing you the beautiful surroundings.

I did not give a talk in the workshop, but I prepared one – here are the slides on the workshop website (best to download and view with some viewer so that the talk unfolds as it should). I also wrote a nice “take home” that would be probably (hopefully) what most people would have taken home from my talk if they heard it, if I had given it. The talk would have been about my recent work with Evgenios Kakariadis on operator algebras associated with monomial ideals (some aspects of which I discussed in a previous post), and here is the succinct Summary (which concentrates on other aspects).  Read the rest of this entry »

A few words on the book “Functional Analysis” by Peter Lax

I recently bought Peter Lax‘s textbook on Functional Analysis, with a clear intention of having it become my textbook of choice. I heard nice opinions of it. Especially, I thought that I would find useful Lax’s point of view that gives the Lebesgue spaces L^p the primary role, and pushes the Lebesgue measure to play a secondary role (I wrote about this subject before).

In fact, the first time I heard of this book was in the following MathOverflow question, that is likely to have been triggered by Lax’s comment (p. 282) “[It is] an open question if there are irreducible operators in Hilbert space, and it is an open question whether this question is interesting” (to spell it out, Lax is making the remark that it is an open question whether the invariant subspace problem is interesting!). I have nothing against the invariant subspace problem (of course it is interesting!), but I was sure that I would love reading a book by a mathematician with a bit of self humor (it turns our Lax’s remark appears at the end of a chapter devoted to invariant subspaces).

In one sense the book was a disappointment, in that I realized that I could not, or would not like to, use it as a textbook for courses I teach. I really don’t like its organization, and I don’t love his style. And the pushing back of Lebesgue measure is a very minor topic (which makes good sense because this is a textbook that was used for second year graduate students). I will have to write my own lecture notes.

On the other hand, the book contains a lot of very neat applications of functional analysis (I won’t spoil it for you, but some are really fun!), and so much better to have it coming from someone like Lax. That’s enough to justify the purchase.

But mathematics aside, this book will now stay close to my heart and change the way I approach the subject of functional analysis. This is because of several historical notes dotted throughout the book. Here is an example, which caught me completely unprepared at the end of Chapter 16 (p. 172) (I read the book non-linearly):

During the Second World War, Banach was one of a group of people whose bodies were used by the Nazi occupiers of Poland to breed lice, in an attempt to extract an anti-typhoid serum. He died shortly after the conclusion of the war.”

And so, when reading through the book, we meet some of our familiar (and also not-so-familiar) heroes of functional analysis being deported to concentration camps, or committing suicide knowing what awaits them from the hand of the Nazis, or somehow making it safely to the west. Some other players are involved in the race to construct a nuclear bomb, or to crack the Enigma code (as I learned from the book, Beurling also broke this code – well, we all knew that he was very smart, but this was surprising. By the time we reach the chapter on Beurling’s theorem, we are not surprised that Lax cannot just leave the anecdote there without mentioning also Turing’s tragedy).

I realize that I rarely connected the history of mathematics and the history of Europe in the twentieth century. It is an unusual and disturbing but also a good thing that Lax – who was born in the nineteen twenties in Hungary and lived through the war in the US – makes this connection. I find it very strange that I always knew well, for instance, that Galois died in a duel, but I never heard that Juliusz Schauder was murdered by the Nazis; I use the open mapping theorem all the time.

Thank you, Peter Lax.

Just a link to another blog I would like to put up

One of my favorite blogs written by a mathematician is Izabella Laba’s “The accidental mathematician“. The title of her blog in itself is enough to make it one of my favourites. Some of her posts on political-academic issues, especially gender, were eye openers for me. Her recent post is another powerful piece.

There is one particularly troubling paragraph. (Brought here out of context. You have to read her post for context).

Here is the first half of the paragraph, which is a statement worth considering in the context of academia, even without the context of discrimination:

We gerrymander research areas so as to keep in the people we choose and exclude those we would rather keep out. Even those gerrymandered borders can fluctuate, expanding when more names are needed on a funding application and then shrinking back when the benefits are shared. We define “interesting and exciting” as that which interests and excites those colleagues whose opinions we respect, and we respect them the most when they agree with us. We cite the “enthusiasm” of colleagues, or lack thereof, as though it were an objective and quantifiable measure of worth.

For completeness, here is the last part of the paragraph – a statement that can be made also outside the context of math, and is too worthy of consideration:

We care deeply about those women and minorities who are absent, hypothetical, or nonexistent, devising elaborate strategies to attract them and treat them fairly, but ignore those who are already there, standing right in front of us and asking for the same resources that their colleagues have been enjoying all along.

Summer projects in Mathematics at the Technion

Small advertisement:

This summer there will be a special one week program for advanced undergraduate students at Department of Math at the Technion. See this page for information on projects and on how to apply. There is a very nice variety of topics to choose from.

Note that students from any university can apply (also from other countries).

“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.

Read the rest of this entry »

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.

Topological K-theory of C*-algebras for the Working Mathematician – closure (Lectures 5,6 and 7)

The mini course in K-theory given by Haim (Claude) Schochet here at the Technion continued as planned until its end, with lectures 5,6 and 7 following the first four lectures. The topics of these lectures were

Lecture 5 – Kasparov’s KK-theory

Lecture 6 – Foliated spaces and C*-algebras of foliated spaces

Lecture 7 – Applications.

As Haim told us, each of these topics could be a one semester course. The scope and speed were such that a detailed account was impossible for me to produce. However, I will still like to record here the fact that this course ended, since I wrote summaries of the first four lectures and someone may find these and look for the rest of the notes. I cannot write such notes because it takes a master of this field like Schochet to give a brief and colourful overview; an amateur like me will only make a mess.

In the last three lectures, we learned that there is something called KK-theory, which is at once both a generalisation of K-theory and of K-homology (see this survey article by Nigel Higson), we learned that there is a geometrical object called a foliated space (or foliated manifold, see wiki article), we learned that with a foliated space one may associated a groupoid C*-algebra (see this survey by Debord and Lescure), and finally, we were told that all of this can be used to prove an index theorem for foliated spaces (the whole story can be found in the book by Moore and Schochet).

I am somewhat of a mathematical frog (or maybe a mathematical chicken would be a better description of what I am), and I cannot take much from such speedy talks except motivation and inspiration. Motivation and inspiration are important, but you have to be there to get them. I have not much to pass on.

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