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 to another given set . See Section 3 below.]
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.
My notes of Haim’s Schochet’s fourth lecture in this series is here below.
It is impossible to start without mention that Alexander Grothendieck passed away last week. Grothendieck is considered by many as one of the greatest mathematicians of 20th century, and his contributions affect the material in this lecture series in at least two significant ways. As we mentioned, a first version of K-theory was developed by Grothendieck opening the door for topological K-theory (which, in turn, opened the door for K-theory of C*-algebras). Grothendieck also developed the theory of nuclear topological vector spaces and tensor products of topological vector spaces, a theory that has influenced the development of the concepts of nuclearity and tensor product which are central to contemporary C*-algebra theory. Read the rest of this entry »
Here is a write up of the third lecture. (Here are links to the first and second ones.) I want to stress that although Haim is giving me a lot of support in preparing these notes (thanks!), any mistakes you find here are my own.
In this lecture we briefly heard about the origin of K-theory (topological K-theory) and then we learned about three theorems (of Connes, Pimsner-Voiculescu and Schochet) describing how to compute the K-theory of various C*-algebras constructed from given C*-algebras in a given way.
Claude (Haim) Schochet is spending this semester at the Technion, and he kindly agreed to give a series of lectures on K-theory. This mini-course is called “Topological K-theory of C*-algebras for the Working Mathematician”.
There will be seven lectures (they take place in Amado 814, Mondays 11:00-12:30):
Since the pace will be really fast and the scope very broad, I plan to write up some of the notes I take, to help myself keep track of these lectures. When I write I will probably introduce some mistakes, and this is completely my fault. I will also probably not be able to hold myself from making some silly remarks, for which only I am responsible.
I also hope that these notes I post may help someone who has missed one or several of the talks make up and come to the next one.
The first talk took place last Monday. To be honest I wasn’t 100% on my guard since I heard such crash courses so many times, I was sure that I’ve heard it all before but very soon I was in territory which is not so familiar to me (The title “crash course” was justified!). Maybe I will make up some of the things I write, or imagine that I heard them.
(The next lectures will be on stuff that is more advances and I will take better notes, and hopefully provide a more faithful representation of the actual lecture).
I will refer in short to the following references:
1. Pedersen – C*-algebras and their automorphism groups.
2. Brown and Ozawa – C*-algebras and finite dimensional approximation.
3. Davidson – C*-algebras by example.
4. Dixmier – C*algebras
5. Blackadar – K-theory for operator algebras
Ken Davidson, Chris Ramsey and I recently uploaded a new version of our paper “Operator algebras for analytic varieties” to the arxiv. This is the second paper that was affected by a discovery of a mistake in the literature, which I told about in the previous post. Luckily, we were able to save all the results in that paper, but had to work a a little harder than what we thought was needed in our earlier version. The isomorphism problem for complete Pick algebras (which I like to call simply “the isomorphism problem”) has been one of my favorite problems during the last five years. I wrote four papers on this problem, with five co-authors. I want to give a short road-map to my work on this problem. Before I do so, here is link to the talk that I will give in IWOTA 2014 about this stuff. I think (hope) this talk is a good introduction to the subject. The problem is about the classification of a large class of non-selfadjoint operator algebras – multiplier algebras of complete Pick spaces – which can also be realized as certain algebras of functions on analytic varieties. These algebras all have the form
where is a subvariety of the unit ball and denotes the multiplier algebra of Drury-Arveson space (see this survey), and therefore is the space of all restrictions of multipliers to . The hope is to show that the geometry of the variety is a complete invariant for the algebras , in various senses that will be made precise below.
In the previous posts, I told of how I came to know of the functional equations
and more generally
(where and satisfy some additional conditions) and my long journey to discover that these equations have, and now I will give it away… Read the rest of this entry »
The last post ended with the following problem:
Problem: Find all continuous solutions to the following functional equation:
In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form , but now we remove the assumption that the solution be continuously differentiable and ask whether the same conclusion holds. I found this problem to be extremely interesting, and at this point I will only give away that I eventually solved it, but after five (!) years.
In principle, it is plausible that, when one enlarges the space of functions in which one is searching for a solution from to the much larger , then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps and on the space ) is minimal, and therefore one expects the functional equation to be rigid enough to allow only for the trivial solutions (at least under some mild regularity assumptions). In short, a good case can be made in favor of either a conjecture that all the continuous solutions are linear or a conjecture that there might be new, nonlinear solutions.