Noncommutative Analysis

Month: July, 2014

Souvenirs from Amsterdam

(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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The isomorphism problem: update

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

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

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

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