Noncommutative Analysis

Introduction to von Neumann algebras, addendum to Lecture 1 (solution of Exercise B: the norm of a selfadjoint operator)

One of the challenges I had in preparing this course, was to find a quick route to the modern theory that is different from the standard modern route, in order to save time and be able to reach significant results and examples in the limited time of a one semester course. A main issue was to avoid the (beautiful, beautiful, beautiful) Gelfand theory of commutative Banach and C*-algebras, and base everything on the spectral theorem for a single selfadjoint operator (which is significantly simpler than the one for normal operators). In the previous lecture, I stated Exercise B, which gave some important properties of the spectrum of a selfadjoint operator. Since my whole treatment is based on this, I felt that for completeness I should give the details.

Spoiler alert: If you are a student in the course and you plan to submit the solution of this exercise, then you shouldn’t read the rest of this post.

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Introduction to von Neumann algebras, Lecture 1 (Introduction to the course, and a crash course in operator algebras, the spectral theorem)

1. Micro prologue

Perhaps we cannot start a course on von Neumann algebras, without making a few historical notes about the beginning of the theory.

(To say it more honestly and openly, what I wanted to say is that perhaps I cannot teach a course on von Neumann algebras without finally reading the classical works by von Neumann and also learning a bit about the man. von Neumann was a true genius and has contributed all over mathematics, see the Wikipedia article).

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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.

A First Course in Functional Analysis (my book)

She’hechiyanu Ve’kiyemanu!

My book, A First Course in Functional Analysis, to be published with Chapman and Hall/CRC, will soon be out. There is already a cover, check it out on the CRC Press website.

This book is written to accompany an undergraduate course in functional analysis, where the course I had in mind is precisely the course that we give here at the Technion, with the same constraints. Constraint number 1: a course in measure theory is not mandatory in our undergraduate program. So how can one seriously teach functional analysis with significant applications? Well, one can, and I hope that this book proves that one can. I already wrote before, measure theory is not a must. Of course anyone going for a graduate degree in math should study measure theory (and get an A), but I’d like the students to be able to study functional analysis before that (so that they can do a masters degree in operator theory with me).

I believe that the readers will find many other original organizational contributions to the presentation of functional analysis in this book, but I leave them for you to discover. Instructors can request an e-copy for inspection (in the link to the publisher website above), friends and direct students can get a copy from me, and I hope that the rest of the world will recommend this book to their library (or wait for the libgen version).

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.