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

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”.

Introduction to von Neumann algebras (Topics in functional analysis 106433 – Spring 2017)

This coming spring semester, I will be giving a graduate course, “Introduction to von Neumann algebras”. This will be a rather basic course, since most of our graduate students haven’t had much operator algebras. (Unfortunately, most of our graduate students didn’t all take the topics course I gave the previous spring). In any sub-field of operator theory, operator algebras, and noncommutative analysis, von Neumann algebras appear and are needed. Thus, this course is meant first and foremost to give (prospective) students and postdocs in our group the opportunity to add this subject to the foundational part of their training. This course is also an opportunity for me to refurbish and reorganize the working knowledge that I acquired during several years of occasional encounters with this theory. Finally, I believe that this course could be really interesting to other serious students of mathematics, who will have many occasions to bump into von Neumann algebras, regardless of the specific research topic that they decide to devote themselves to (yes, you too!).

Topic in Operator Theory 106435 (Spring 2016)

I am happy to announce that in spring semester I will be teaching a “topics” course in operator theory here at the Technion. This course is a graduate course, and is suitable for anyone who took the graduate course in functional analysis and enjoyed it. This is not exactly a course in C*-algebras, and not exactly a course in operator theory, but rather very particular blend of operator theory and operator algebras that starts from very basic material but proceeds very quickly (covering “Everything one should know”), and by the end of the course I hope some of the students will be able to consider research in this area. The official information page is available here.

I might use this blog for posting information or even notes from time to time. Here is the first piece of important information:

I will assume that students taking the course know some basic Banach algebra theory, including commutative Banach and C*-algebras. If you are interested in taking the course, make sure you know this stuff. A good source from which you can refresh your memory is Arveson’s book “A Short Course on Spectral Theory”. Start reading at the start all the way up to Section 2.3.

 

 

Course announcement : Advanced Analysis, 20125401

In the first term of the 2012/2013, I will be giving the course “Advanced Analysis” here at BGU. This is the department’s core functional analysis course for graduate students, though ambitious undergraduate students are also encouraged to take this course, and some of them indeed do. The price to pay is that we do not assume that the students know any functional analysis, and the only formal requisites are a course  in complex variables and a course in (point set) topology, as well as a course in measure theory which can be taken concurrently. The price to pay for having no requisites in functional analysis, while still aiming at graduate level course, is that the course is huge: we have five hours of lectures a week. In practice we will actually have six hours of lectures a week, because I will go abroad in the middle of the semester to this conference and workshop in Bangalore. The official syllabus of the course is as follows:

Banach spaces and Hilbert spaces. Basic properties of Hilbert spaces. Topological vector spaces. Banach-Steinhaus theorem; open mapping theorem and closed graph theorem. Hahn-Banach theorem. Duality. Measures on locally compact spaces; the dual of C(X). Weak and weak-* topologies; Banach-Alaoglu theorem. Convexity and the Krein-Milman theorem. The Stone-Weierstrass theorem. Banach algebras. Spectrum of a Banach algebra element. Gelfand theory of commutative Banach algebras. The spectral theorem for normal operators (in the continuous functional calculus form).

I plan to cover all these topics (with all that is implicitly implied), but I will probably give the whole course a little bend towards my own area of expertise, especially in the exercises and examples. We do have to wait and see who the students are and what their background is before deciding precisely how to proceed. Some notes for the course will appear (in the English language) on this blog. The official course webpage (which is in Hebrew) is behind this link.