Noncommutative Analysis

Category: Topics in Operator Theory 106435

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.


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 »

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.

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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 »