Noncommutative Analysis

Category: Functional analysis

New paper “Compressions of compact tuples”, and announcement of mistake (and correction) in old paper “Dilations, inclusions of matrix convex sets, and completely positive maps”

Ben Passer and I have recently uploaded our preprint “Compressions of compact tuples” to the arxiv. In this paper we continue to study matrix ranges, and in particular matrix ranges of compact tuples. Recall that the matrix range of a tuple A = (A_1, \ldots, A_d) \in B(H)^d is the the free set \mathcal{W}(A) = \sqcup_{n=1}^\infty \mathcal{W}_n(A), where

\mathcal{W}_n(A) = \{(\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n is UCP \}.

A tuple A is said to be minimal if there is no proper reducing subspace G \subset H such that \mathcal{W}(P_G A\big|_G) = \mathcal{W}(A). It is said to be fully compressed if there is no proper subspace whatsoever G \subset H such that \mathcal{W}(P_G A\big|_G) = \mathcal{W}(A).

In an earlier paper (“Dilations, inclusions of matrix convex sets, and completely positive maps”) I wrote with other co-authors, we claimed that if two compact tuples A and B are minimal and have the same matrix range, then A is unitarily equivalent to B; see Section 6 there (the printed version corresponds to version 2 of the paper on arxiv). This is false, as subsequent examples by Ben Passer showed (see this paper). A couple of other statements in that section are also incorrect, most obviously the claim that every compact tuple can be compressed to a minimal compact tuple with the same matrix range. All the problems with Section 6 of that earlier paper “Dilations,…” can be quickly  fixed by throwing in a “non-singularity” assumption, and we posted a corrected version on the arxiv. (The results of Section 6 there do not affect the rest of the results in the paper, and are somewhat not in the direction of the main parts of that paper).

In the current paper, Ben and I take a closer look at the non-singularity assumption that was introduced in the corrected version of “Dilations,…”, and we give a complete characterization of non-singular tuples of compacts. This characterization involves the various kinds of extreme points of the matrix range \mathcal{W}(A). We also make a serious invetigation into fully compressed tuples defined above. We find that a matrix tuple is fully compressed if and only if it is non-singular and minimal. Consequently, we get a clean statement of the classification theorem for compacts: if two tuples A and B of compacts are fully compressed, then they are unitarily equivalent if and only if \mathcal{W}(A) = \mathcal{W}(B).

 

The complex matrix cube problem (in “Summer Projects in Mathematics at the Technion”)

Next week I will participate as a mentor in the Technion’s Summer Projects in Mathematics. The project I offered is called “Numerical explorations of open problems from operator theory”, and it suggests three open problems in operator theory where theoretical progress seems to be stuck, and for which I believe that some computer experiments can help us get a feeling of what is going on. I also hope that thinking seriously about designing experiments can help us to understand some general facets of the theory.

I have been in contact with the students in the last few weeks and we decided to concentrate on “the matrix cube problem”. On Sunday, when the week begins, I will need to present the background to the project to all participants of this week, and I have seven minutes (!!) for this. As everybody knows, the shorter the presentation, the harder the task is, and the more preparation and thought it requires. So I will take use this blog to practice my little talk.

Introduction to the matrix cube problem

This project is in the theory of operator spaces. My purpose is to give you some kind of flavour of what the theory is about, and what we will do this week to contribute to our understanding of this theory.

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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:

20180604_053844

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:

Click here to download the journal version of the paper

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.

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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).

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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.

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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).

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