### 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).

Read the rest of this entry »

### 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. Read the rest of this entry »

### 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 »

### Topics in Operator Theory, Lecture 3: Dilations of commuting operators

We continue in this lecture to consider dilation theory of contractions. In the theory of Sz.-Nagy and Foias, the main route proceeded from the existence of the minimal isometric and unitary dilations, to the study of how such dilations look like, and to use them to extract information about an operator from its dilations. The only application we saw until now was von Neumann’s inequality, which is not a trivial fact, but let’s admit it: somewhat rinky dinky. But after a deeper look is taken into the structure of the minimal unitary dilation, the way for more significant applications opens. One of these applications is a functional calculus (for non-selfadjoints) that extends the holomorphic functional calculus. Another application is an affirmative solution to the invariant subspace problem for certain classes of operators. The main parts of this theory are laid down in the book Harmonic Analysis of Operators On Hilbert Space.

We will not follow that route. Rather, we will see what dilation theory can help us to understand regarding tuples of commuting operators (which is also treated to some extent in the book). Surprisingly, this will lead to a truly nifty application in function theory.  Read the rest of this entry »

### Topics in Operator Theory, Lecture 2: Dilations of contractions

In this lecture we will study the first chapter in the theory of dilations of contractions. To proceed in our study of operator spaces and operator algebras, the material we will cover is not strictly needed. However, this is where I want to begin, for several reasons:

1. The objects and theorems here motivate (and have motivated historically) the development of the general theory, and help understand it better and appreciate it more.
2. We will reach very quickly a nontrivial application of operator theory to function theory, which is quite different from what you all saw in your studies, probably.
3. I am stalling, so that the students who need to fill in some prerequisites (like commutative C*-algebras and the spectral theorem, will have time to do so).
4. I love this stuff!

Okay, enough explaining, let us begin.

### 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 »

### Course announcement: Operator Spaces, Operator Algebras and Related Topics (Topics in Operator Theory 106435)

Next week I will begin teaching a topics course “Topics in Operator Theory – 106435”. This is an advanced graduate course, where “advanced” means that I expect students to be familiar with graduate functional analysis.

The official name of the course is “Topics in Operator Theory” but the true title is “Operator Spaces, Operator Algebras and Related Topics”. There are two somewhat competing goals driving this course: the first goal is to give students a taste of the beautiful subjects of operator spaces and operator algebras, broadening their view of functional analysis, and giving those who wish enough tools to delve into the literature in this subject. The second goal is to train students to understand the problems in which I am interested and to get acquainted with the methods of the theory so that they will be able to carry out research in my group. The choice of topics will therefore be somewhat eclectic. In fact, I have several different plans for this course, and I am keeping things vague on purpose so that I am free to change course as the wind blows (and as I see who the students are, what their background is and where their interests lie).

What else? The course will be given in English. There is no official web page for the course – I might open exercises online on this blog. The grade will be based on some exercises that I will give throughout the semester, and a final “big homework” project.

### New paper “Compressions of compact tuples”, and announcement of mistake (and correction) in old paper “Dilations, inclusions of matrix convex sets, and completely positive maps”

Ben Passer and I have recently uploaded our preprint “Compressions of compact tuples” to the arxiv. In this paper we continue to study matrix ranges, and in particular matrix ranges of compact tuples. Recall that the matrix range of a tuple $A = (A_1, \ldots, A_d) \in B(H)^d$ is the the free set $\mathcal{W}(A) = \sqcup_{n=1}^\infty \mathcal{W}_n(A)$, where

$\mathcal{W}_n(A) = \{(\phi(A_1), \ldots, \phi(A_d)) : \phi : B(H) \to M_n$ is UCP $\}$.

A tuple $A$ is said to be minimal if there is no proper reducing subspace $G \subset H$ such that $\mathcal{W}(P_G A\big|_G) = \mathcal{W}(A)$. It is said to be fully compressed if there is no proper subspace whatsoever $G \subset H$ such that $\mathcal{W}(P_G A\big|_G) = \mathcal{W}(A)$.

In an earlier paper (“Dilations, inclusions of matrix convex sets, and completely positive maps”) I wrote with other co-authors, we claimed that if two compact tuples $A$ and $B$ are minimal and have the same matrix range, then $A$ is unitarily equivalent to $B$; see Section 6 there (the printed version corresponds to version 2 of the paper on arxiv). This is false, as subsequent examples by Ben Passer showed (see this paper). A couple of other statements in that section are also incorrect, most obviously the claim that every compact tuple can be compressed to a minimal compact tuple with the same matrix range. All the problems with Section 6 of that earlier paper “Dilations,…” can be quickly  fixed by throwing in a “non-singularity” assumption, and we posted a corrected version on the arxiv. (The results of Section 6 there do not affect the rest of the results in the paper, and are somewhat not in the direction of the main parts of that paper).

In the current paper, Ben and I take a closer look at the non-singularity assumption that was introduced in the corrected version of “Dilations,…”, and we give a complete characterization of non-singular tuples of compacts. This characterization involves the various kinds of extreme points of the matrix range $\mathcal{W}(A)$. We also make a serious invetigation into fully compressed tuples defined above. We find that a matrix tuple is fully compressed if and only if it is non-singular and minimal. Consequently, we get a clean statement of the classification theorem for compacts: if two tuples $A$ and $B$ of compacts are fully compressed, then they are unitarily equivalent if and only if $\mathcal{W}(A) = \mathcal{W}(B)$.

### The complex matrix cube problem summer project – summary of results

In the previous post I announced the project that I was going to supervise in the Summer Projects in Mathematics week at the Technion. In this post I wish to share what we did and what we found in that week.

I had the privilege to work with two very bright students who have recently finished their undergraduate studies: Mattya Ben-Efraim (from Bar-Ilan University) and Yuval Yifrach (from the Technion). It is remarkable the amount of stuff they learned for this one week project (the basics of C*-algebras and operator spaces), and that they actually helped settle the question that I raised to them.

I learned a lot of things in this project. First, I learned that my conjecture was false! I also learned and re-learned some programming abilities, and I learned something about the subtleties and limitations of numerical experimentation (I also learned something about how to supervise an undergraduate research project, but that’s besides the point right now).