Noncommutative Analysis

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 »

Topics in Operator Theory, Lecture 5 and on

Last week (which was the fourth week, not really the fourth lecture) we finished the proof of Pick’s interpolation theorem, and then I gave a one hour crash course in C*-algebras. The main topics we covered were:

  1. Positive functionals and states on C*-algebras and the GNS construction.
  2. For a linear functional f on a C*-algebra, f\geq 0 \Leftrightarrow f(1) = \|f\|.
  3. The Gelfand-Naimark theorem .
  4. A Hahn-Banach extension theorem: If A is a unital C*-algebra and B is a unital C*-subalgebra, then every state on B extends to a state on A.

From now on we will begin a systematic study of operator spaces, operator systems, and completely positive maps. I will be following my old notes, which for this part are based on Chapters 2 and 3 from Vern Paulsen’s book , and I will make no attempt at writing better notes.

As I start with some basic things this week, the students should brush up on tensor products of vector spaces and of Hilbert spaces.

UPDATE DECEMBER 4th: 

I decided to record here in some more details the material that I covered following Paulsen’s book, since my presentation was not 1-1 according to the book. In what follows, M will denote a unital operator space, S an operator system, and A and B are C*-algebras. Elements in these spaces will be denoted as a,b etc.

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Topics in Operator Theory, Lecture 4: Pick interpolation via commutant lifting

Finally we reached the point where we can apply the general theory that we developed in the last two weeks to obtain an interesting application to function theory, namely, the Pick interpolation theorem. Read the rest of this entry »