Course announcement: “Topics in Functional Analysis 106433 – Introduction to Operator Algebras”

by Orr Shalit

My sabbatical is nearing its end and I starting to get used to the idea of getting back to teaching. Luckily (or is it really just luck?) I am going to have a very smooth return to teaching, because this coming fall I will be teaching a topics course of my choice, and it is going to be an introduction to operator algebras (the official course title and number are above). To be honest, the idea is to give the optimal course for students who will work with me, but I believe that other students will also enjoy it and find it useful. I will probably use this blog to post material and notes.

Here is the content of the info page that I will be distributing:

Topics in Functional Analysis 106433

Winter 2021

Introduction to Operator Algebras

Lecturer: Orr Shalit (oshalit@technion.ac.il, Amado 709)

Credit points: 3

Summary: The theory of operator algebras is one of the richest and broadest research areas within contemporary functional analysis, having deep connections to every subject in mathematics. In fact, this topic is so huge that the research splits into several distinct branches: C*-algebras, von Neumann algebras, non-selfadjoint operator algebras, and others. Our goal in this course is to master the basics of the subject matter, get a taste of the material in every branch, and develop a high-level understanding of operator algebras.

The plan is to study the following topics:

  1. Banach algebras and the basics of C*-algebras.
  2. Commutative C*-algebras. Function algebras.
  3. The basic theory of von Neumann algebras.
  4. Representations of C*-algebras. GNS representation. Algebras of compact operators.
  5. Introduction to operator spaces, non-selfadjoint operator algebras, and completely bounded maps.
  6. Time permitting, we will learn some additional advanced topics (to be decided according to the students’ and the instructor’s interests). Possible topics:
    1. C*-algebras and von Neumann algebras associated with discrete groups.
    1. Nuclearity, tensor products and approximation techniques.
    1. Arveson’s theory of the C*-envelope and hyperrigidity.
    1. Hilbert C*-modules.

Prerequisites: I will assume that the students have taken (or are taking concurrently) the graduate course in functional analysis. Exceptional students, who are interested in this course but did not take Functional Analysis, should talk to the instructor before enrolling.

The grade: The grade will be based on written assignments, that will be presented and defended by the students.

References:

The following are good general references, though we shall not follow any of them very closely (at most a chapter here or there).   

  1. Orr Shalit’s lecture notes.
  2. K.R. Davidson, “C*-Algebras by Example”.
  3. R.V. Kadison and J. Ringrose, “Fundamentals of the Theory of Operator Algebras”.
  4. C. Anantharaman and S. Popa, “An Introduction to II_1 Factors”.
  5. N.P. Brown and N. Ozawa, “C*-Algebras and Finite Dimensional Approximations”
  6. V. Paulsen, “Completely Bounded Maps and Operator Algebras”.