Noncommutative Analysis

Introduction to von Neumann algebras (Topics in functional analysis 106433 – Spring 2017)

This coming spring semester, I will be giving a graduate course, “Introduction to von Neumann algebras”. This will be a rather basic course, since most of our graduate students haven’t had much operator algebras. (Unfortunately, most of our graduate students didn’t all take the topics course I gave the previous spring). In any sub-field of operator theory, operator algebras, and noncommutative analysis, von Neumann algebras appear and are needed. Thus, this course is meant first and foremost to give (prospective) students and postdocs in our group the opportunity to add this subject to the foundational part of their training. This course is also an opportunity for me to refurbish and reorganize the working knowledge that I acquired during several years of occasional encounters with this theory. Finally, I believe that this course could be really interesting to other serious students of mathematics, who will have many occasions to bump into von Neumann algebras, regardless of the specific research topic that they decide to devote themselves to (yes, you too!).

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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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Journal of Xenomathematics

I am happy to advertise the existence of a new electronic journal/forum/website: Journal of Xenomathematics. Don’t worry, it’s not another new research journal. The editor is John E. McCarthy. The purpose is to discuss mathematics that is out of this world. Aren’t you curious?

Multivariable Operator Theory workshop at the Technion (Haifa, June 2017)

I am happy to advertise the research workshop Multivariable Operator Theory, that will take place at the Technion, In June 18-22, 2017, on the occasion of Baruch Solel’s 65 birthday. Here is the workshop webpage, and here is a link to the poster. The website and poster contain a preliminary list of speakers, and some words of explanation of what the workshop is about, roughly.

The workshop proper (that is: lectures) will take place from Monday June 19 to Thursday June 22, morning to evening. Everyone is welcome to attend, and there is no registration fee, but if you are planning to come you better contact me so we make sure that there is enough room in the lecture room, enough fruit and cookies in the breaks, etc. The information on the website will be updated from time to time, and will probably converge as the time of the workshop comes near.

Please free to contact me if you have any questions.

New journal: Advances in Operator Theory

I am writing to let you know about a new journal: Advances in Operator Theory.

This is good news! There is certainly room for another very good journal in operator theory. Naturally, this journal will be open access, and, obviously, there will be no author fees (page charges, or whatever you want to call that). So this is just the kind of journal we need, granted that it will be able to maintain a high standard and slowly build its reputation.

The first step in establishing a reputation is achieved: AOT has a respectable editorial board, with several distinguished members.

The founding editor-in-chief is Mohammad Sal Moslehian, who has been making efforts on the open access front at least since he launched the Banach Journal of Mathematical Analysis, roughly ten years ago. The BJMA is a good example of an electronic journal that started from scratch, and slowly worked its way to recognition (e.g., is now indexed by MathSciNet, etc.). I hope AOT follows suit, and hopefully will do even better; I believe it should aim to be at the level of Journal of Operator Theory, so that it can relieve JOT of a part of the load.

(Too bad that the acronym AOT, when spelled out, sounds very much like JOT. This will certainly lead to some confusion…)

A proof of Holder’s inequality

One of the parts of this blog that I am most proud of is my series of “Souvenirs” post, where I report about my favorite new finds in conferences. In July I went to a big conference (IWOTA 2016 in St. Louis) that I was looking forward to going to for a long time, but I did not write anything after I returned. It’s not that there was nothing to report – there was a lot, it was great conference. I was just too busy with other things.

Why am I so busy? Besides being the father of seven people (most of them “kids”) and preparing for next year, I am in the last stages of writing a book, partly based on the lecture notes on “Advanced Analysis” that appeared in this blog, and on lecture notes that evolved from that. (When it will be ready I will tell you about it, you can be sure). I want to share here and now one small excerpt from it (thanks to Guy Salomon for helping me finesse it!)

Working on the final touches to the book, I decided to include a proof of Holder’s inequality in it, but I did not want to copy a proof from somewhere.  So I came up with the following proof, which I think is new (and out of curiosity I am asking you to please tell me if you have seen it before). The lazy idea of the proof is to use the fact that we already know – thanks to Cauchy-Schwarz – that the inequality holds in the p =2 case, and to try to show how the general case follows from that.

In other words, instead of bringing you fancy souvenirs from St. Louis, I got you this little snack from the nearby mall (really, the proof crystallized in my head when my daughter, my dog and I were sitting and waiting on a bench in the mall until other members of our family finish shopping).

Definition. Two extended real numbers p,q \in [1, \infty] are said to be conjugate exponents if

\frac{1}{p} + \frac{1}{q} = 1.

If p=1 then we understand this to mean that q = \infty, and vice versa.

For any (finite or infinite) sequence x_1, x_2, x_3, \ldots, and and any p \in [1,\infty], we denote

\|x\|_p =\big(\sum |x_k|^p \big)^{1/p}.

Theorem (Holder’s inequality): Let p,q \in [1, \infty] be conjugate exponents.
Then for any two (finite or infinite) sequences x_1, x_2, \ldots and y_1, y_2, \ldots

\sum_k |x_k y_k| \leq \|x\|_p \|y\|_q.

Proof. The heart of the matter is to prove the inequality for finite sequences. Pushing the result to infinite sequences does not require any clever idea, and is left to the reader (no offense).
Therefore, we need to prove that for every x = (x_k)_{k=1}^n and y = (y_k)_{k=1}^n in \mathbb{C}^n,

(HI)   \sum |x_ky_k| \leq \big(\sum |x_k|^p \big)^{1/p} \big( \sum |y_k|^q \big)^{1/q}.

The case p=1 (or p=\infty) is immediate. The right hand side of (HI) is continuous in p when x and y are held fixed, so it enough to verify the inequality for a dense set of values of p in (1,\infty).

Define

S = \Big\{\frac{1}{p} \in (0,1) \Big| p satisfies  (HI)  for all x,y \in \mathbb{C}^n \Big\}.

Now our task reduces to that of showing that S is dense in (0,1). By the Cauchy-Schwarz inequality, we know that \frac{1}{2} \in S. Also, the roles of p and q are interchangeable, so \frac{1}{p} \in S if and only if 1 - \frac{1}{p} \in S.

Set a = \frac{q}{2p+q} (a is chosen to be the solution to 2ap = (1-a)q, we will use this soon). Now, if \frac{1}{p} \in S, we apply (HI) to the sequences (|x_k| |y_k|^{a})_k and (|y_k|^{1-a})_k, and then we use the Cauchy-Schwarz inequality, to obtain

\sum |x_k y_k| = \sum|x_k||y_k|^a |y_k|^{1-a}

\leq \Big(\sum |x_k|^p |y_k|^{ap} \Big)^{1/p}\Big(\sum |y_k|^{(1-a)q} \Big)^{1/q}

\leq \Big((\sum |x_k|^{2p})^{1/2} (\sum |y_k|^{2ap})^{1/2} \Big)^{1/p}\Big(\sum |y_k|^{(1-a)q} \Big)^{1/q}

= \Big(\sum |x_k|^{p'} \Big)^{1/p'} \Big(\sum|y_k|^{q'} \Big)^{1/q'}

where \frac{1}{p'} = \frac{1}{2p} and \frac{1}{q'} = \frac{1}{2p} + \frac{1}{q}. Therefore, if s = \frac{1}{p} \in S, then \frac{s}{2} = \frac{1}{2p} \in S; and if s = \frac{1}{q} \in S, then \frac{s+1}{2} = \frac{1}{2}\frac{1}{q}+\frac{1}{2} = \frac{1}{q} + \frac{1}{2}\frac{1}{p} is also in S.

Since \frac{1}{2} is known to be in S, it follows that \frac{1}{4} and \frac{3}{4} are also in S, and continuing by induction we see that for every n \in \mathbb{N} and m \in \{1,2, \ldots, 2^n-1\}, the fraction \frac{m}{2^n} is in S. Hence S is dense in (0,1), and the proof is complete.

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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Thirty one years later: a counterattack on Halmos’s critique of non-standard analysis

As if to celebrate in an original way the fifty year anniversary of Bernstein and Robinson’s solution to (a generalization of) the Smith-Halmos conjecture (briefly, that if T is an operator such that p(T) is compact for some polynomial p, then T has an invariant subspace), several notable mathematicians posted a interesting and very nonstandard (as they say) paper on the arxiv.

This paper briefly tells the story regarding the publication of this paper, in which Bernstein and Robinson use Robinson’s new theory of non-standard analysis (NSA) to prove the above mentioned conjecture in operator theory. This was one of the first major successes of NSA, and perhaps one would think that all of the operator theory community should have accepted the achievement with nothing but high praise. Instead, it was received somewhat coldly: Halmos went to work immediately to translate the NSA proof and published a paper proving the same result, with a proof in “standard” operator theoretic terms. (See the paper, I am leaving out the juicy parts). And then, since 1966 until 2000 (more or less), Halmos has been apparently at “war” with NSA (in the paper the word “battle” is used), and has also had criticism of logic; for example, it is implied in his book that he did not always consider logic to be a part of mathematics, worse, it seems that he has not always considered logicians to be mathematicians. (When I wrote about Halmos’s book a few months ago, I wrote that I do not agree with all the opinions expressed in the book, and I remember having the issue with logic and logicians in my mind when writing that).

In the paper that appeared on the arxiv today, the authors take revenge on Halmos. Besides a (convincing) rebuttal of Halmos’s criticisms, the seven authors hand Halmos at least seven blows, not all of them below the belt. The excellent and somewhat cruel title says it all: A non-standard analysis of a cultural icon: the case of Paul Halmos.

Besides some feeling of uneasiness in seeing a corpse being metaphorically stabbed (where have you been in the last thirty years?), the paper raises interesting issues (without wallowing too much on either one), and may serve as a lesson to all of us. There is nothing in this story special to operator theory versus model theory, or NSA, or logic. The real story here is the suspicion and snubbish-ness of mathematicians towards fields in which they do not work, and towards people working in these fields.

I see it all the time. Don’t kid me: you have also seen quite a lot of it. It is possible, I confess, that I have exercised myself a small measure of suspicion and contempt to things that I don’t understand. As the authors of the paper hint, these things are worse than wrong – they might actually hurt people.

Anyway, many times people who are ignorantly snobbish to other fields end up looking like idiots. Stop doing that, or thirty years from now a mob of experts will come and tear you to shreds.

P.S. – It seems that the question of who was the referee of the Bernstein-Robinson paper is not settled, though some suspect it was Halmos. Well, if someone could get their hands on the (anonymous!) referee report (maybe Bernstein or Robinson kept the letter?), I am quite sure that if it was Halmos, it would be clear. In other words, if Bernstein or Robinson suspected that it was him on account of the style, then I bet it was.

 

P.P.S. – regarding the theorem starting this discussion the quickest way to understand it is via Lomonosov’s theorem. The invariant subspace theorem proved by Bernstein and Robinson (polynomially compact operator has an invariant subspace) is now superseded by Lomonosov’s theorem (google it for a simple proof), which says that every bounded operator on a Banach space that commutes with a nonzero compact operator has a non-trivial invariant subspace.

Summer projects in math at the Technion 2016

This year, the Faculty of Math at the Technion is continuing with its recently founded tradition of summer projects. As in last year’s week of summer projects, the Faculty of Math at the Technion is inviting advanced undergraduates from Israel and from around the world to get a little taste of research in mathematics. This is a nice opportunity, especially for someone who is considering graduate studies in math.

For a list of topics with abstracts, and for other important details (like how to apply), see this page.

 

Revising and resubmitting my opinions on refereeing

With time, with age, having done already quite a few paper-refereeing jobs, I have come to change some of my opinions on refereeing.

Anonymous refereeing. I used to think that anonymous refereeing was not important. Why can’t I (as referee) just write back to the authors and discuss the weak points of the paper with them. Wouldn’t that be much better and faster? Besides, if I have a certain opinion about a paper, I should be willing to back it with my name, publicly.

Yes, I was innocent and was not yet aware of the endless ways in which some people will try to get back at you if your report includes anything but praise and/or typo corrections. But besides the usual reasons for or against anonymous refereeing, here is something I overlooked.

The really nice thing about anonymous refereeing is this: not only does it free the referee to say bad things, it also frees the referee to say good things. There was a paper I was reviewing for a good journal, and I really wanted the paper to get accepted. I thought it was very good, and that it should be accepted to this good journal. I wanted to be very clear that this paper should be accepted (sometimes, a lukewarm report is not enough to get a paper accepted), and being anonymous made it easier for me to use superlatives that I rarely feel comfortable using in front of someones face. The fact that I was anonymous, and the fact that the editor knows that I am anonymous, also makes it easier for the editors to take my praise seriously.

Therefore, the identity of referees should be kept secret, so we can all be kind to each other. (And please don’t ever ask me if it was me who refereed your paper).

Is the paper interesting? When I first heard that papers get rejected because they are “not interesting”, I was a little surprised. “Interesting” is not an objective criterion. It might be interesting to one person, and not interesting to another. Certainly the author thinks it is interesting!

Certainly? Well, I have seen some papers, unfortunately, about which I cannot say that I am certain that even the author thought that they are interesting. I have seen some papers that were written only because they could be written. Nobody ever wrote that particular proof to that particular proposition, with this set of assumptions, so this is a “new contribution”. But sometimes, a paper contains nothing which has appeared before, but does not really contain anything new. If there is nothing new, then it is boring, not interesting.

It is very hard to say what makes a good paper, and what makes a bad one. What makes good scientific research? I believe that judging the value of scientific research is not a scientific activity in itself. Deciding whether a mathematical paper is good is a job for mathematicians, but it is not a mathematical problem.

So when I evaluate a paper, I check if it is correct and new, of course, but I also cannot help but thinking whether or not it is interesting. What does interesting mean? It means interesting to me, of course! But that’s OK, because if the editor asked for my opinion, then it is my opinion that I am going to give.

Do I work for the journal. The editors of Journal A say that they want to publish only the best research articles. What does that mean? How can I compare? Let me tell you if the paper is new, correct, and interesting. What do I care that Journal A wants to remain prestigious? In fact, I never published in Journal A, and as far as I care its reputation can go to hell.

And really, to be honest, there are many factors that may affect my decision to recommend acceptance of  a paper to Journal A: 1) The authors are young researchers and this could help them in their career. 2) The paper is in my field, and I want to use the reputation of Journal A to increase the prestige of my field. 3) etc., etc., one can think of all kinds of impure reasons to be consciously biased for accepting a paper. In any case, if the paper is in my opinion a good, solid contribution, then why is it my business that the editor wants only to publish spectacular papers?

I now look at it differently. It is an honour to be approached by Journal A and be asked for their opinion. The editor is asking my professional opinion, based on my reputation. I should keep in mind that my answer, among other things, affects my reputation. I have to behave like a professional, and answer the question asked. Of course, I still don’t work for the journal, and I am free to be very enthusiastic about papers that are important in my opinion.

More on the pecking order. I have heard more than once of the following scenario: an editor of Journal B tells a referee that his journal (Journal B) is now only accepting papers that would be good enough for Journal A.

Excuse me!? If the authors thought their paper was good enough for Journal A, then they would submit it to Journal A, and not to B! And anyway, I don’t work for the journal! Clearly the journal has its goals, it wants to increase its prestige (or whatever), but I also have my own priorities, and in any case I don’t care about the prestige of Journal B. I’m already being very nice that I am willing to referee this paper for free, so don’t ask me to work for your prestige (if it is good enough for me to referee, then it’s good enough for you to publish).

Actually, the idea that Journal B aims to be at the “quality” of Journal A (whatever that means) is not so ridiculous. Journal A rejects most of the papers submitted to it, in fact it rejects some excellent papers. Where are all these papers supposed to go? So I don’t mind answering the question asked. (What I once wanted to write, but did not, is this: “Yes, I would recommend it for Journal A, and in fact this paper is too good for you, Journal B! I recommend rejecting the paper on the grounds that it is too good for this journal…”)

Submitting my review in a timely manner. I have not changed my mind about that. I always give an estimate of when I will submit my report, and I always submit on (or before) time. This means that I have to say “no” to a large fraction of referee requests (I try to referee at least about as many papers as I publish every year), otherwise I would not be able to do it in a timely manner. Naturally, I try to accept for review the papers that are more interesting.