### Introduction to von Neumann algebras, Lecture 2 (Definitions, the double commutant theorem, etc.)

In this second lecture we start a systematic study of von Neumann algebras.

In this second lecture we start a systematic study of von Neumann algebras.

One of the challenges I had in preparing this course, was to find a quick route to the modern theory that is different from the standard modern route, in order to save time and be able to reach significant results and examples in the limited time of a one semester course. A main issue was to avoid the (beautiful, beautiful, beautiful) Gelfand theory of commutative Banach and C*-algebras, and base everything on the spectral theorem for a single selfadjoint operator (which is significantly simpler than the one for normal operators). In the previous lecture, I stated Exercise B, which gave some important properties of the spectrum of a selfadjoint operator. Since my whole treatment is based on this, I felt that for completeness I should give the details.

Spoiler alert: If you are a student in the course and you plan to submit the solution of this exercise, then you shouldn’t read the rest of this post.

Perhaps we cannot start a course on von Neumann algebras, without making a few historical notes about the beginning of the theory.

(To say it more honestly and openly, what I wanted to say is that perhaps I cannot teach a course on von Neumann algebras without finally reading the classical works by von Neumann and also learning a bit about the man. von Neumann was a true genius and has contributed all over mathematics, see the Wikipedia article).