Math 247A: von Neumann algebras, Winter 2019

Homework assignments: homework 1 (due January 30th) and homework 2 (optional).

Lecture notes (in progress).

Course description: This course will provide an introduction to von Neumann algebras while emphasizing the connections with ergodic theory and group theory. One goal is to highlight some of the exciting developments at the intersection of these fields. We will start by discussing the results needed from functional analysis, abelian C*-algebras, the spectral theorem, and the basic theory of von Neumann algebras. We will then cover the construction of von Neumann algebras from groups and actions, and explain how representation theoretic properties of groups (amenability, property (T)) and orbit equivalence of actions are connected to the study of these algebras. Time permitting, we will also cover a selection of topics from Popa's deformation/rigidity theory. We will assume no background except basic knowledge of real analysis. The pace of the class and the material covered will be adjusted depending on the audience's knowledge and interests.

Resources:

• The topics covered are mostly contained in the book in preparation An introduction to II_1 factors by Claire Anantharaman and Sorin Popa.
• The topics covered in the first part of the class are largely contained in A course in operator theory by John B. Conway.
• Here are online lecture notes on von Neumann algebras by Cyril Houdayer, Vaughan Jones, and Jesse Peterson .
• For an overview of some recent developments in von Neumann algebras, see the surveys by Popa (2007) , Stefaan Vaes (2010) and Ioana (2012 and 2017).

FA Seminar: You may want to check out UCSD's Functional Analysis Seminar which will feature several research talks on von Neumann algebras.