Noncommutative Analysis

Category: operator spaces

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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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 1: Introduction

This is a summary of the first lecture, which was introductory in nature.

H will always denote a Hilbert space over \mathbb{C}. B(H) will always denote the algebra of bounded operators on H. I am interested in operators on Hilbert space; various subspaces and algebras of operators that come with various structures, as well as the relationship between these subspaces and structures; and connections and applications of the above to other areas, in particular complex function theory and matrix theory.

I expect students to know the spectral theorem for normal operators on Hilbert space (see here. A proof in the selfadjoint case that assumes very little from the reader can be found in my notes, see Section 3 and 4). I also will assume some familiarity with Banach algebras and commutative C*-algebras – the student should contact me for references.

We begin by surveying different kinds of structures of interest.  Read the rest of this entry »