Category: Functional analysis

Souvenirs from the Red River

Last week I attended the annual Canadian Operator Symposium, better known in its nickname: COSY. This conference happens every year and travels between Canadian universities, and this time it was held in the University of Manitoba, in Winnipeg. It was organized by Raphaël Clouâtre and Nina Zorboska, who altogether did a great job.

My first discovery: Winnipeg is not that bad! In fact I loved it. Example: here is the view from the window of my room in the university residence:

Not bad, right? A very beautiful sight to wake up to in the morning. (I got the impression, that Winnipeg is nothing to look forward to, from Canadians. People of the world: don’t listen to Canadians when they say something bad about any place that just doesn’t quite live up to the standard of Montreal, Vancouver, or Banff.) Here is what you see if you look from the other side of the building:  Read the rest of this entry »

Minimal and maximal matrix convex sets

The final version of the paper Minimal and maximal matrix convex sets, written by Ben Passer, Baruch Solel and myself, has recently appeared online. The publisher (Elsevier) sent us a link through which the official final version is downloadable, for anyone who clicks on the following link before May 26, 2018. Here is the link for the use of the public:

Of course, if you don’t click by May 26 – don’t panic! We always put our papers on the arXiv, and here is the link to that. Here is the abstract:

Abstract. For every convex body $K \subseteq \mathbb{R}^d$, there is a minimal matrix convex set $\mathcal{W}^{min}(K)$, and a maximal matrix convex set $\mathcal{W}^{max}(K)$, which have $K$ as their ground level. We aim to find the optimal constant $\theta(K)$ such that $\mathcal{W}^{max}(K) \subseteq \theta(K) \cdot \mathcal{W}^{min}(K)$. For example, if $\overline{\mathbb{B}}_{p,d}$ is the unit ball in $\mathbb{R}^d$ with the $p$-norm, then we find that

$\theta(\overline{\mathbb{B}}_{p,d}) = d^{1-|1/p-1/2|}$ .

This constant is sharp, and it is new for all $p \neq 2$. Moreover, for some sets $K$ we find a minimal set $L$ for which $\mathcal{W}^{max}(K) \subseteq \mathcal{W}^{min}(L)$. In particular, we obtain that a convex body $K$ satisfies $\mathcal{W}^{max}(K) = \mathcal{W}^{min}(K)$ only if $K$ is a simplex.

These problems relate to dilation theory, convex geometry, operator systems, and completely positive maps. For example, our results show that every $d$-tuple of self-adjoint contractions, can be dilated to a commuting family of self-adjoints, each of norm at most $\sqrt{d}$. We also introduce new explicit constructions of these (and other) dilations.

“A toolkit for constructing dilations on Banach spaces”, by Fackler and Gluck

About a week ago an interesting preprint appeared on the arxiv: “A toolkit for constructing dilations on Banach spaces“, by Stephan Fackler and Jochen Gluck. I have been studying various aspects of dilations for some years, but I haven’t really given much thought to dilation theory in general classes of Banach spaces. This paper – which is very clearly organized and written – was very refreshing for me, and in it a very general framework for proving existence of dilations in classes of Banach spaces is presented. The paper also contains a nice overview of the literature, and I was surprised by learning also about old results in, and application of, dilation theory, which I was not aware of and perhaps I should have been. The purpose of this post is to record my first impression of this paper and to put down some links to the references, which I would like to study better at some point.  Read the rest of this entry »

Tapioca on page 49

To my long camping vacation this year I took the book “Topological Vector Spaces” by Alex and Wendy Robertson. I “inherited” this book (together with a bunch of other classics) from an old friend after he officially decided to leave academic mathematics and go into high-tech. The book is a small and thin hard-cover, with pages of high quality that are starting to become a delicious cream color.

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 integral 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.

Introduction to von Neumann algebras, Lecture 7 (von Neumann algebras as dual spaces, various topologies)

Until this point in the course, we concentrated on constructions of von Neumann algebras, examples, and properties of von Neumann algebras as algebras. In this lecture we turn to study subtler topological and Banach-space theoretic aspects of von Neumann algebras. We begin by showing that every von Neumann algebra is the Banach-space dual of a Banach space. For this to have any hope of being true, it must be true for the von Neumann algebra $B(H)$; we therefore look there first.

(The reference for this lecture is mostly Takesaki, Vol. I, Chapters 2 and 3).

Introduction to von Neumann algebras, Lecture 6 (tensor products of Hilbert spaces and vN algebras; the GNS representation, the hyperfinite II_1 factor)

In this lecture we will introduce tensor products of Hilbert spaces. This construction is very useful for exhibiting various operators, and, in particular, it will enable us to introduce new von Neumann algebras. In particular, we will construct the so called hyperfinite $II_1$ factor.

Introduction to von Neumann algebras, Lecture 5 (comparison of projections and classification into types of von Neumann algebras)

In the previous lecture we discussed the group von Neumann algebras, and we saw that they can never be isomorphic to $B(H)$. There is something fundamentally different about these algebras, and this was manifested by the existence of a trace. von Neumann algebras with traces are special, and the existence or non-existence of a trace can be used to classify von Neumann algebras, into rather broad “types”. In this lecture we will study the theory of Murray and von Neumann on the comparison of projections and the use of this theory to classify von Neumann algebras into “types”. We will also see how traces (or generalized traces) fit in. (For preparing these notes, I used Takesaki (Vol I) and Kadison-Ringrose (Vol. II).)

Most of the time we will stick to the assumption that all Hilbert spaces appearing are separable. This will only be needed at one or two spots (can you spot them?).

In addition to “Exercises”, I will start suggesting “Projects”. These projects might require investing a significant amount of time (a student is not expected to choose more than one project).

The preface to “A First Course in Functional Analysis”

I am not yet done being excited about my new book, A First Course in Functional Analysis. I will use my blog to advertise my book, one last time. This post is for all the people who might wonder: “why did you think that anybody needs a new book on functional analysis?” Good question! The answer is contained in the preface to the book, which is pasted below the fold.

Introduction to von Neumann algebras, Lecture 4 (group von Neumann algebras)

As the main reference for this lecture we use (more-or-less) Section 1.3 in the notes by Anantharaman and Popa (here is a link to the notes on Popa’s homepage).

As for exercises:  Read the rest of this entry »