Noncommutative Analysis

Month: December, 2018

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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Topics in Operator Theory, Lecture 7: boundary representations

In this lecture we will present a proof that boundary representations exist in abundance, following Davidson and Kennedy’s breakthrough paper. Davidson and Kennedy’s paper was in the spirit of Arveson’s paper from 1969, and followed Arveson’s solution in the separable case from 2007. (BTW, I wrote about Davidson and Kennedy’s solution in a an old blog post). 

1. The unique extension property and maximal representations

Recall the definition of a boundary representation. 

Our setting will be of an operator system S contained in a C*-algebra B = C^*(S). Recall that earlier we discussed the situation of a unital operator algebra A \subseteq B = C^*(A), and later we extended our attention to unital operator spaces. In this post we will consider only operator systems, but there will be no loss of generality (because every unital completely contractive map A \mapsto B(H) extends to a unique unital completely positive map S: A + A^* \to B(H), and vice versa). 

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Topics in Operator Theory, Lecture 6: an overview of noncommutative boundary theory

The purpose of this lecture is to introduce some classical notions in uniform algebras that motivated Arveson’s two seminal papers, “Subalgebras of C*-algebras I + II”, and then to introduce the basic ideas on how to generalize to the noncommutative setting, which were introduced in those papers.

Note: If you are following the notes of this course, please note that the previous lecture has been updated with quite a lot of material.  Read the rest of this entry »