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

by Orr Shalit

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!).

The goals are to introduce the students first to the basic notions in operator algebras and operator theory (this part will be more of a crash course: commutative C*-algebras, the spectral theorem), and then to study the basic theory of von Neumann algebras: density theorems, topologies, the predual (and hopefully, its uniqueness), commutative von Neumann algebras, Murray von Neumann equivalence of projections and classification into types, factors, examples of all types, crossed products. Perhaps a realistic goal is to reach the level where we can discuss some von Neumann algebras arising from group actions, to say something about the relationship between the dynamical structure of the action, on the one hand, and the algebraic structure of the von Neumann algebra, on the other. I also wish that by the end of the course students will be able to read papers on von Neumann modules, W*-correspondences, and noncommutative dynamics. I suppose that Sakai’s theorem will be left out, so will decomposition theory, and maybe I will have time to say something about standard form and survey Tomita-Takesaki theory (maybe some of these topics will be chosen by students for a project).

The Prerequisites: This is tricky. In the winter semester, Baruch Solel taught a graduate course in functional analysis, and I would be very happy to assume that everyone attending my course already aced Solel’s course (or an equivalent). I know that this will not be the case. However, I have to make some assumptions about what students know, so we can get somewhere. Thus, I will take it for granted that everybody knows about Hilbert and Banach spaces, their operators, their dual spaces, the fundamental theorems (Hah-Banach, open mapping, etc.). I will very quickly review what is a Banach algebra and the theory of commutative Banach algebras, but I will do it as a review, knowing that this is usually covered in the graduate functional analysis course. Measure theory is also a key tool. Students who think they can still enjoy the course without all this background will have to work very hard between lectures to fill in their gaps. On the other hand, I won’t kick anyone out of class, and I don’t object that people learn the subject in the order that von Neumann did (more on this later). The key requirement is mathematical maturity (definition: a mathematically mature student is a student that already truly understands what mathematically mature means).

I wrote a syllabus: here, but since this is my first time teaching such a course, I am not really sure what I will manage to cover in class. In a sense, the main merit of an old fashioned lecture based course, is that it creates a pivot around which we can all self-study, and I hope to be able at least to open a door.

Incidentally, it turns out that Daniel Markiewicz will be giving an introductory course in von Neumann algebras in Ben-Gurion University in the spring. For students from Israeli universities (except those situated in Haifa) that want to attend a course in von Neumann algebras, I highly recommend to go to the course at Ben-Gurion.