Category Archives: Expository

Topological K-theory of C*-algebras for the Working Mathematician – Lecture 3 (Topological K-theory and three big theorems)

Expository, K-theory course, Operator algebras,

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.

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Topological K-theory of C*-algebras for the Working Mathematician – Lecture 2 (Definitions and core examples)

Expository, K-theory course, Operator algebras,

This is a write-up of the second lecture in the course given by Haim Schochet. For the first lecture and explanations, see the previous post.

I will very soon figure out how to put various references online and post links to that, too.

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Topological K-theory of C*-algebras for the Working Mathematician – Lecture 1

Expository, K-theory course, Operator algebras, ,

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):

  1. A crash course in C*-algebras.
  2. K-theory by axioms and core examples.
  3. K-theory strengths and limitations.
  4. Payoffs in functional analysis: elliptic operators on compact spaces, essentially normal and Toeplitz operators.
  5. Payoffs in algebraic topology: bivariant K-theory by axioms, core examples, and the UCT.
  6. Modelling of groups, groupoids, and foliations.
  7. Payoffs in geometry: Atiyah-Singer and Connes index theorems.

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

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The isomorphism problem: update

Complex variables, Expository, Research, , ,

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

$latex M_V = Mult(H^2_d)big|_V$

where $latex V$ is a subvariety of the unit ball and $latex Mult(H^2_d)$  denotes the multiplier algebra of Drury-Arveson space (see this survey), and therefore $latex M_V$ is the space of all restrictions of multipliers to $latex V$. The hope is to show that the geometry of the variety $latex V$ is a complete invariant for the algebras $latex M_V$, in various senses that will be made precise below.

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Where have all the functional equations gone (the end of the story and the lessons I’ve learned)

Expository, Fun stuff, Functional equations, Thoughts on mathematics

This will be the last of this series of posts on my love affair with functional equations (here are links to parts one, two and three).

1. A simple solution of the functional equation

In the previous posts, I told of how I came to know of the functional equations

(*)  $latex f(t) = fleft(frac{t+1}{2}right) + f left( frac{t-1}{2}right) ,, , ,, t in [-1,1]$

and more generally

(**) $latex f(t) = f(delta_1(t)) + f(delta_2(t)) ,, , ,, t in [-1,1]$

(where $latex delta_1$ and $latex delta_2$ satisfy some additional conditions) and my long journey to discover that these equations have, and now I will give it away… Continue reading Where have all the functional equations gone (the end of the story and the lessons I’ve learned) →

Where have all the functional equations gone (part III)

Expository, Functional equations, Paneah, Thoughts on mathematics

The last post ended with the following problem:

Problem: Find all continuous solutions to the following functional equation:

(FE) $latex f(t) = fleft(frac{t+1}{2} right) + f left(frac{t-1}{2} right) ,, , ,, t in [-1,1] .$

In the previous post I explained why all continuously differentiable solutions of the functional equation (FE) are linear, that is, of the form $latex f(x) = cx$, 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 $latex C^1[-1,1]$ to the much larger $latex C[-1,1]$, then new solutions will appear. On the other hand, the dynamical system affiliated with this problem (the dynamical space generated by the maps $latex delta_1(t) = frac{t+1}{2}$ and $latex delta_2(t) = frac{t-1}{2}$ on the space $latex [-1,1]$) 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.

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Where have all the functional equations gone (part II)

Dynamical systems, Expository, Fun stuff, Functional equations, Thoughts on mathematics,

I’ll start off exactly where I stopped in the previous post: I will tell you my solution to the problem my PDEs lecturer (and later master’s thesis advisor) Paneah gave us:

Problem: Find all continuously differentiable solutions to the following functional equation:

(FE) $latex f(t) = fleft(frac{t+1}{2} right) + f left(frac{t-1}{2} right) ,, , ,, t in [-1,1] .$

Before writing a solution, let me say that I think it is a fun exercise for undergraduate students, and only calculus is required for solving it, so if you want to try it now is your chance.

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Where have all the functional equations gone (part I)

Expository, Functional equations, Paneah, Thoughts on mathematics

My first encounter with research mathematics was in the last term of my undergraduate studies (spring 2003). My professor in the course “Introduction to Partial Differential Equations”, Prof. Boris Paneah, thought that it is pointless to give standard homework problems to students of pure mathematics, and instead he gave us several problems which were either extremely challenging, related to his research or related to advanced courses that he was going to give. This was a thrilling experience for me, and is one of the reasons why I decided not long after to do my master’s thesis under his supervision, since no other faculty member came even close to engaging us like Paneah (another reason was that the lectures themselves were fantastic). For example he suggested that we explore the ultrahyperbolic equation

$latex u_{tt} + u_{ss} – u_{xx} – u_{yy} = 0 , $    in     $latex mathbb{R}^4$,

or that we try to prove the existence of solutions to the two dimensional heat equation in a non-rectangular bounded region of the plane. I remember spending hours on the heat equation, unsuccessfully of course (if I was successful I would have probably become a PDE person). Especially memorable is the one time that he ended a lecture with the following three problems, which were, as you may guess, quite unrelated to the content of the lecture: Continue reading Where have all the functional equations gone (part I) →

Arveson memorial article

Arveson, Expository, memorial, , ,

Palle Jorgensen and Daniel Markiewicz have put together a beautiful tribute to the late Bill Arveson, with contributions from about a dozen mathematicians as well as a more personal piece by Lee Ann Kaskutas. This memorial article might appear later elsewhere in shorter form, but I think it would be interesting for many people to see the full tribute, with all the various points of view and pieces of life that it contains. I wrote a post dedicated to Arveson’s memory about half a year ago, where I put links to two recent surveys (1 by Davidson and and 2 by Izumi), and in about a month there will be a big conference in Berkeley dedicated to Arveson’s legacy; still I feel that this tribute really fills a hole, and conveys in broader, fuller way what a remarkable mathematician he was, and how impacted so many so strongly. Please share the link with people who might be interested.

Forty five years later, a major open problem in operator algebras is solved

Arveson, Expository, Operator algebras

A couple of days ago, Ken Davidson and Matt Kennedy posted a preprint on the arxiv, “The Choquet boundary of an operator system“. In this paper they solve a major open problem in operator algebras, showing that every operator system has sufficiently many boundary representations. 

In 1969, William Arveson published the seminal paper, [“Subalgebras of C*-algebras”, Acta Math. 123, 1969], which is one of the cornerstones, (if not the cornerstone) of the theory of operator spaces and nonself-adjoint operator algebras. In that paper, among other things, Arveson introduced and put to good use the notion of a boundary representation. I wrote on “Subalgebras of C*-algebras” in a previous post dedicated to Arveson, and for some background material the reader is invited to look into that old post. I did not, however, write much about boundary representations (because I was emphasizing his contributions rather what he has left open). Below I wish to explain what are boundary representations, what does it mean that there are sufficiently many of these, and where Davidson and Kennedy’s new results fits in the chain of results leading to the solution of the problem. The paper itself is accessible to anyone who understands the problem, and the main ideas are clearly presented in its introduction.

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