Noncommutative Analysis

Tag: functional analysis

Course announcement: “Topics in Functional Analysis 106433 – Introduction to Operator Algebras”

My sabbatical is nearing its end and I starting to get used to the idea of getting back to teaching. Luckily (or is it really just luck?) I am going to have a very smooth return to teaching, because this coming fall I will be teaching a topics course of my choice, and it is going to be an introduction to operator algebras (the official course title and number are above). To be honest, the idea is to give the optimal course for students who will work with me, but I believe that other students will also enjoy it and find it useful. I will probably use this blog to post material and notes.

Here is the content of the info page that I will be distributing:

Topics in Functional Analysis 106433

Winter 2021

Introduction to Operator Algebras

Lecturer: Orr Shalit (, Amado 709)

Credit points: 3

Summary: The theory of operator algebras is one of the richest and broadest research areas within contemporary functional analysis, having deep connections to every subject in mathematics. In fact, this topic is so huge that the research splits into several distinct branches: C*-algebras, von Neumann algebras, non-selfadjoint operator algebras, and others. Our goal in this course is to master the basics of the subject matter, get a taste of the material in every branch, and develop a high-level understanding of operator algebras.

The plan is to study the following topics:

  1. Banach algebras and the basics of C*-algebras.
  2. Commutative C*-algebras. Function algebras.
  3. The basic theory of von Neumann algebras.
  4. Representations of C*-algebras. GNS representation. Algebras of compact operators.
  5. Introduction to operator spaces, non-selfadjoint operator algebras, and completely bounded maps.
  6. Time permitting, we will learn some additional advanced topics (to be decided according to the students’ and the instructor’s interests). Possible topics:
    1. C*-algebras and von Neumann algebras associated with discrete groups.
    1. Nuclearity, tensor products and approximation techniques.
    1. Arveson’s theory of the C*-envelope and hyperrigidity.
    1. Hilbert C*-modules.

Prerequisites: I will assume that the students have taken (or are taking concurrently) the graduate course in functional analysis. Exceptional students, who are interested in this course but did not take Functional Analysis, should talk to the instructor before enrolling.

The grade: The grade will be based on written assignments, that will be presented and defended by the students.


The following are good general references, though we shall not follow any of them very closely (at most a chapter here or there).   

  1. Orr Shalit’s lecture notes.
  2. K.R. Davidson, “C*-Algebras by Example”.
  3. R.V. Kadison and J. Ringrose, “Fundamentals of the Theory of Operator Algebras”.
  4. C. Anantharaman and S. Popa, “An Introduction to II_1 Factors”.
  5. N.P. Brown and N. Ozawa, “C*-Algebras and Finite Dimensional Approximations”
  6. V. Paulsen, “Completely Bounded Maps and Operator Algebras”.

A review of my book A First Course in Functional Analysis

A review for my book A First Course in Functional Analysis appeared in Zentralblatt Math – here is a link to the review. I am quite thankful that someone has read my book and bothered to write a review, and that zBMath publishes reviews. That’s all great. Now I have a few words to say about it. This is an opportunity for me to bring up the subject of my book and highlight some things worth highlighting.

I am not too happy about this review. It is not that it is a negative review – actually it has a rather kind air to it. However, I am somewhat disappointed in the information that the review contains, and I am not sure that it does the reader some service which the potential readers could not achieve by simply reading the table of contents and the preface to the book (it is easy to look inside the book in the Amazon page; of course, it is also easy to find a copy of the book online).

The reviewer correctly notices that one key feature of the book is the treatment of L^2[a,b] as a completion of C([a,b]), and that this is used for applications in analysis. However, I would love it if a reviewer would point out to the fact that, although the idea of thinking about L^2[a,b] as a completion space is not new, few (if any) have attempted to actually walk the extra mile and work with L^2 in this way (i.e., without requiring measure theory) all the way up to rigorous and significant applications in analysis. Moreover, it would be nice if my attempt was compared to other such attempts (if they exist), and I would like to hear opinions about whether my take is successful.

I am grateful that the reviewer reports on the extensive exercises (this is indeed, in my opinion, one of the pluses of new books in general and my book in particular), but there are a couple of other innovations that are certainly worth remarking on, and I hope that the next reviewer does not miss them. For example, is it a good idea to include a chapter on Hilbert function spaces in an introductory text to FA? (a colleague of mine told me that he would keep that out). Another example: I think that my chapter on applications of compact operators is quite special. This chapter has two halves: one on integral equations and one on functional equations. Now, the subject of integral equations is well trodden and takes a central place in some introductions to FA, and one might wonder whether anything new can be done here in terms of the organization and presentation of the material. So, I think it is worth remarking about whether or not my exposition has anything to add. The half on applications of compact operators to functional equations contains some beautiful and highly non-trivial material that has never appeared in a book before, not to mention that functional equations of any kind are rarely considered in introductions to FA; this may also be worth a comment.

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).

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.

Advanced Analysis, Notes 8: Banach spaces (application: weak solutions to PDEs)

Today I will show you an application of the Hahn-Banach Theorem to partial differential equations (PDEs). I learned this application in a seminar in functional analysis, run by Haim Brezis, that I was fortunate to attend in the spring of 2008 at the Technion.

As often happens with serious applications of functional analysis, there is some preparatory material to go over, namely, weak solutions to PDEs.

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