Noncommutative Analysis

Category: Functional analysis

Around and under my talk at Fields

This week I am attending a Workshop on Developments and Technical Aspects of Free Noncommutative Functions at the Fields Institute in Toronto. Since I plan to give a chalk-talk, I cannot post my slides online (and I cannot prepare for my talk by preparing slides), so I will write here what some ideas around what I want to say in my talk, and also some ramblings I won’t have time to say in my talk.

[Several years ago I went to a conference in China and came back with the insight that in international conferences I should give a computer presentation and not a blackboard talk, because then people who cannot understand my accent can at least read the slides. It’s been almost six years since then and indeed I gave only beamer-talks since. My English has not improved over this period, I think, but I have several reasons for allowing myself to give an old fashioned lecture – the main ones are the nature of the workshop, the nature of the audience and the kind of things I have to say]. 

In the workshop Guy Salomon, Eli Shamovich and I will give a series of talks on our two papers (one and two). These two papers have a lot of small auxiliary results, which in usual conference talk we don’t get the chance to speak about. This workshop is a wonderful opportunity for us to highlight some of these results and the ideas behind them, which we feel might be somewhat buried in our paper and have gone unnoticed. 

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Topics in Operator Theory, Lecture 10: hyperrigidity

In this lecture we discuss the notion of hyperrigidity, which was introduced by Arveson in his paper The noncommutative Choquet boundary II: Hyperrigidity, shortly after he proved the existence of boundary representations (and hence the C*-envelope) for separable operator systems. Most of the results and the examples that we will discuss in this lecture come from that paper, and we will certainly not be able to cover everything in that paper. In the last section of this post I will put some links concerning a result of Kennedy and myself which connects hyperrigidity to the Arveson’s essential normality conjecture.

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Topics in Operator Theory, Lecture 9: the boundary theorem

In this post, we come back to boundary representations and the C*-envelope, prove an important theorem, and see some examples. It is interesting to note that the theory has interesting consequences even for operators on finite dimensional spaces. Here is a link to a very interesting paper by Farenick giving an exposition of Arveson’s boundary theorem in the setting of operators on finite dimensional spaces.

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Topics in Operator Theory, Lecture 8: matrix convexity

In this lecture we will encounter the notion of matrix convexity. Matrix convexity is an active area of research today, and an important tool in noncommutative analysis. We will define matrix convex sets, and we will see that closed matrix convex sets have matrix extreme points which play a role similar to extreme points in analysis. As an example of a matrix convex set, we will study the set of all matrix states. We will use these notions to outline the proof that there are sufficiently many pure UCP maps, something that was left open from the previous lecture. 

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The never ending paper

My paper On operator algebras associated with monomial ideals, written jointly with Evgenios Kakariadis, has recently appeared in Journal of Mathematical Analysis and Applications. They gave me a link to share (the link will work for the next several weeks): click here for an official version of the paper.

The paper is a very long paper, so it has a very long introduction too. To help to get into the heart of editors and referees, we wrote, at some point, a shorter cover letter which attempts to briefly explain what the main achievements are. See below the fold for that.

But first, a rant!

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