Noncommutative Analysis

Category: Expository

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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The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field \mathbb{C} (e.g., there are many polynomials, such as x^{2n} + 1, that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write \mathbb{C}[z] for the algebra of polynomials in one complex variable with complex coefficients, and consider a polynomial as a function of the complex variable z \in \mathbb{C}. We will also write \mathbb{C}[z_1, \ldots, z_d] for the algebra of polynomials in d (commuting) variables, and think of polynomials in \mathbb{C}[z_1, \ldots, z_d] – at least initially – as a functions of the variable z = (z_1, \ldots, z_d) \in \mathbb{C}^d

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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?

Dilations, inclusions of matrix convex sets, and completely positive maps

In part to help myself to prepare for my talk in the upcoming IWOTA, and in part to help myself prepare for getting back to doing research on this subject now that the semester is over, I am going to write a little exposition on my joint paper with Davidson, Dor-On and Solel, Dilations, inclusions of matrix convex sets, and completely positive maps. Here are the slides of my talk.

The research on this paper began as part of a project on the interpolation problem for unital completely positive maps*, but while thinking on the problem we were led to other problems as well. Our work was heavily influenced by works of Helton, Klep, McCullough and Schweighofer (some which I wrote about the the second section of this previous post), but goes beyond. I will try to present our work by a narrative that is somewhat different from the way the story is told in our paper. In my upcoming talk I will concentrate on one aspect that I think is most suitable for a broad audience. One of my coauthors, Adam Dor-On, will give a complimentary talk dealing with some more “operator-algebraic” aspects of our work in the Multivariable Operator Theory special session.

[*The interpolation problem for unital completely positive maps is the problem of finding conditions for the existence of a unital completely positive (UCP) map that sends a given set of operators A_1, \ldots, A_d to another given set B_1, \ldots, B_d. See Section 3 below.]

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Topological K-theory of C*-algebras for the Working Mathematician – closure (Lectures 5,6 and 7)

The mini course in K-theory given by Haim (Claude) Schochet here at the Technion continued as planned until its end, with lectures 5,6 and 7 following the first four lectures. The topics of these lectures were

Lecture 5 – Kasparov’s KK-theory

Lecture 6 – Foliated spaces and C*-algebras of foliated spaces

Lecture 7 – Applications.

As Haim told us, each of these topics could be a one semester course. The scope and speed were such that a detailed account was impossible for me to produce. However, I will still like to record here the fact that this course ended, since I wrote summaries of the first four lectures and someone may find these and look for the rest of the notes. I cannot write such notes because it takes a master of this field like Schochet to give a brief and colourful overview; an amateur like me will only make a mess.

In the last three lectures, we learned that there is something called KK-theory, which is at once both a generalisation of K-theory and of K-homology (see this survey article by Nigel Higson), we learned that there is a geometrical object called a foliated space (or foliated manifold, see wiki article), we learned that with a foliated space one may associated a groupoid C*-algebra (see this survey by Debord and Lescure), and finally, we were told that all of this can be used to prove an index theorem for foliated spaces (the whole story can be found in the book by Moore and Schochet).

I am somewhat of a mathematical frog (or maybe a mathematical chicken would be a better description of what I am), and I cannot take much from such speedy talks except motivation and inspiration. Motivation and inspiration are important, but you have to be there to get them. I have not much to pass on.