Category Archives: Functional analysis

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.

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

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.

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

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

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

Souvenirs from Amsterdam

Conference, Functional analysis, Operator algebras, Operator theory, Research, Talks,

(I am writing a post on hot trends in mathematics in the midst of war, completely ignoring it. This seems like the wrong thing to do, but my urge to write has overcome me. To any reader of this blog: I wish you a peaceful night, wherever you are).

Last week I returned from the yearly “International Workshop on Operator Theory and Applications”, IWOTA 2014 for short (see the previous post for the topic of my own talk, or this link for the slides).

This conference was very broad (and IWOTA always is). One nice thing about broad conferences is that you are able sometimes to identify a growing trend. In this talk I got particularly excited by a series of talks on “noncommutative function theory” or “free analysis”. There was a special session dedicated to this topic, but I was mostly inspired by a semi-plenary talk by Jim Agler, and also by two interesting talks by Joe Ball and Spela Spenko. I also attended nice talks related to this subject by Victor Vinnikov, Dmitry Kalyuhzni-Verbovetskyi, Baruch Solel, Igor Klep and Bill Helton. This topic has attracted the attention of many operator theorists, for its applications as well as for its inherent beauty, and seems to be accelerating in the last several years; I will only try to give a taste of some neat things that are going on, by telling you about Agler’s talk. What I will not be able to do is to convey Agler’s intense and unique mathematical charisma.

Here is the program of the conference, so you can check out other things that were going on there.

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Daniel Spielman talks at HUJI – thoughts

Functional analysis, Open problem, Thoughts on mathematics,

I got an announcement in the email about the “Erdos Lectures”, that will be given by Daniel Spielman in the Hebrew University of Jerusalem next week (here is the poster on Gil Kalai’s blog). The title of the first lecture is “The solution of the Kadison-Singer problem”. Recall that not long ago Markus, Spielman and Srivastava proved Weaver’s KS2 conjecture, which implies a positive solution to Kadison-Singer (the full story been worked out to expository perfection on Tao’s blog).

My immediate response to this invitation was to start planning a trip to Jerusalem on Monday – after all it is not that far, it’s about a solution of a decades old problem, and Daniel Spielman is sort of a Fields medalist. I highly recommend to everyone to go hear great scientists live whenever they have the opportunity. At worst, their lectures are “just” inspiring. It is not for the mathematics that one goes for in these talks, but for all the stuff that goes around mathematics (George Mostow’s unusual colloquium given at BGU on May 2013 comes to mind).

But then I remembered that I have some obligations on Monday, so I searched and found a lecture by Daniel Spielman with the same title online: here. Watching the slides with Spielman’s voice is not as inspiring as hearing and seeing a great mathematician live, but quite good. He makes it look so easy!

In fact, Spielman does not discuss KS at all. He says (about a minute into the talk) “Actually, I don’t understand, really, the Kadison-Singer problem”. A minute later he has a slide where the problem is written down, but he says “let me not explain what it is”, and sends the audience to read Nick Harvey’s survey paper (which is indeed very nice). These were off-hand remarks, and I should not catch someone at his spoken word, (and I am sure that even things that Spielman would humbly claim to “not understand, really”, he probably understands as well as I do, at least), but the naturality in which the KS problem was pushed aside in a talk about KS made we wonder.

In the post I put up soon after appearance of the paper I wrote (referring to the new proof of KS2) that “… this looks like a very nice celebration of the Unity of Mathematics”. I think that in a sense the opposite is also true. I will try to reformulate what I wrote.

“The solution of KS is a beautiful and intriguing manifestation of the chaotic, sticky, psychedelic, thickly interwoven, tangled, scattered, shattered and diffuse structure of today’s mathematics.”

I don’t mean that in a bad way. I mean that a bunch of deep conjectures, from different fields, most of which, I am guessing, MSS were not worried about, were shown over several decades to be equivalent to each other, and were ultimately reduced (by Weaver) to a problem on the arrangement of vectors in finite dimensional spaces (Discrepancy Theory), and eventually solved, following years of hard work, by three brilliant mathematicians using ingenious yet mostly elementary tools. The problem solved is indeed interesting in itself, and the proof is also very interesting, but it seems that the connection with “Kadison-Singer” is more a trophy than a true reward.

It would be very interesting now to think of all the equivalent formulations with hindsight, and seek the unifying structure, and to try to glean a reward.

 

 

Souvenirs from the Black Forest

Conference, Functional analysis, Operator algebras, Operator theory, Research, Talks, , ,

Last week I attended a workshop titled “Hilbert modules and complex geometry” in MFO (Oberwolfach). In this post I wish to tell about some interesting things that I have learned. There were many great talks to choose from. Below is a sample, in short form, with links.

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K-spectral sets and the holomorphic functional calculus

Advanced Analysis, 20125401, Functional analysis, Operator algebras, Operator theory

In two previous posts I discussed the holomorphic functional calculus as part of a standard course in functional analysis (lectures notes 18 and 19). In this post I wish to discuss a slightly different approach, which relies also on the notion of K-spectral sets, and relies a little less on contour integration of Banach-space valued functions.

In my very personal opinion this approach is a little more natural then the standard one, and it would be even more natural if one was able to altogether remove the dependence on Banach-space valued integrals (unfortunately, right now I don’t know how to do this completely).

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Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties)

Advanced Analysis, 20125401, Complex variables, Functional analysis, Operator theory,

In this post we continue our discussion of the holomorphic functional calculus for elements of a Banach algebra (or operators). The beginning of this discussion can be found in Notes 18. Continue reading Advanced Analysis, Notes 19: The holomorphic functional calculus II (definition and basic properties) →

Major advances in the operator amenability problem

Open problem, Operator algebras,

Laurent Marcoux and Alexey Popov recently published a preprint, whose title speaks for itself :”Abelian, amenable operator algebras are similar to C*-algebras“. This complements another recent contribution, by Yemon Choi, Ilijas Farah and Narutaka Ozawa, “A nonseparable amenable operator algebra which is not isomorphic to a C*-algebra“.

The open problem that these two papers address is whether every amenable Banach algebra, which is a subalgebra of $latex B(H)$, is similar to a (nuclear) C*-algebra. As the titles clearly indicate (good titling!), we now know that an abelian amenable operator algebra is similar to a C*-algebra, and on the other hand, that a non-separable, non-abelian operator algebra is not necessarily similar to a C*-algebra.

I recommend reading the introduction to the Marcoux-Popov paper (which is very friendly to non-experts too) to get a picture of this problem, its history, and an outline of the solution.

Essential normality, essential norms and hyper rigidity

Arveson, d shift space, Open problem, Operator algebras, Operator theory, Research

Matt Kennedy and I recently posted on the arxiv a our paper “Essential normality, essential norms and hyper rigidity“. This paper treats Arveson’s conjecture on essential normality (see the first open problem in this previous post). From the abstract:

Let $latex S = (S_1, ldots, S_d)$ denote the compression of the $latex d$-shift to the complement of a homogeneous ideal $latex I$ of $latex mathbb{C}[z_1, ldots, z_d]$. Arveson conjectured that $latex S$ is essentially normal. In this paper, we establish new results supporting this conjecture, and connect the notion of essential normality to the theory of the C*-envelope and the noncommutative Choquet boundary.

Previous works on the conjecture verified it for certain classes of ideals, for example ideals generated by monomials, principal ideals, or ideals of “low dimension”. In this paper we find results that hold for all ideals, but – alas! – these are only partial results.

Denote by $latex Z = (Z_1, ldots, Z_d)$ the image of $latex S$ in the Calkin algebra (here as in the above paragraph, $latex S$ is the compression of the $latex d$-shift to the complement of an ideal $latex I$ in $latex H^2_d$). Another way of stating Arveson’s conjecture is that the C*-algebra generated by $latex Z$ is commutative. This would have implied that the norm closed (non-selfadjoint) algebra generated by $latex Z$ is equal to the sup-norm closure of polynomials on the zero variety of the ideal $latex I$. One of our main results is that we are able to show that the non-selfadjoint algebra is indeed as the conjecture predicts, and this gives some evidence for the conjecture. This is also enough to obtain a von Neumann inequality on subvarieties of the ball, what would have been a consequence of the conjecture being true.

Another main objective is to connect between essential normality and the noncommutative Choquet boundary (see this and this previous posts). A main result here is  we have is that the tuple $latex S$ is essentially normal if and only if it is hyperrigid  (meaning in particular that all irreducible representations of $latex C^*(S)$ are boundary representations).