Finally, after a long delay, a package arrived containing some hard copies of my book.
In this lecture we will describe some projection construction in von Neumann algebras, and we will classify commutative von Neumann algebras.
So far (the first two lectures and in this one), the references I used for preparing these notes are Conway (A Course in Operator Theory) Davidson (C*-algebras by Example), Kadison-Ringrose (Fundamentals of the Theory of Operator Algebras, Vol .I), and the notes on Sorin Popa’s homepage. But since I sometimes insist on putting the pieces together in a different order, the reader should be on the look out for mistakes.
In this second lecture we start a systematic study of von Neumann algebras.
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.
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).
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 :
tuples of matrices,
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 (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.
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).
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!).
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; here I will just give some background and state the main result. (Even more recently these four authors released yet another paper that looks very interesting – this one.)
We will be working with the notion of Hilbert function spaces – also called reproducing Hilbert spaces (see this post for an introduction). Suppose that is a Hilbert function space on a set , and its reproducing kernel. The Pick interpolation problem is the following: