National University of Singapore, Institute of Mathematical Sciences

Model theory of tracial von Neumann algebras

Lecturer: Isaac Goldbring

July 16-20, 2022

In this lecture series, we will introduce these mathematical objects and explain the logic appropriate for studying them. We will then move on to some applications of model-theoretic ideas to the study of tracial von Neumann algebras, including the model-theoretic take on the recent negative resolution of the famous Connes Embedding Problem as well as how model-theoretic techniques can be used to shed light on some problems concerning embeddings into ultrapowers. We will conclude the course by discussing a recent application of model-theoretic ideas by David Jekel to free probability.

The course will be fairly self-contained and only prior exposure to basic (classical) model theory will be assumed.

- Lecture 1
- Introduction to tracial von Neumann algebras
- We introduce the class of tracial von Neumann algebras, giving a variety of examples. We also discuss the important subclass of II_1 factors as well as introduce the tracial ultraproduct construction.

- Introduction to tracial von Neumann algebras
- Lecture 2
- Tracial von Neumann algebras as metric structures
- We introduce the appropriate continuous logic used for studying tracial von Neumann algebras and show that, in the right continuous language, they form an elementary class. We then move on to discussing the problem of identifying many different first-order theories of II_1 factors.

- Tracial von Neumann algebras as metric structures
- Lecture 3
- Model theory and the Connes Embedding Problem
- We discuss the famous Connes Embedding Problem (CEP) and explain how model-theoretic techniques can be used to simplify the recent negative resolution of the CEP from a landmark result in quantum complexity theory known as MIP*=RE.

- Model theory and the Connes Embedding Problem
- Lecture 4
- Applications to problems around embeddings with factorial commutants
- We discuss two recent applications of model theory to von Neumann algebra theory: progress on Popa's Factorial Relative Commutant Problem and a new characterization of the hyperfinite II_1 factor in terms of embeddings into ultrapowers.

- Applications to problems around embeddings with factorial commutants
- Lecture 5
- Model theory and free probability
- We discuss recent results of David Jekel applying model-theoretic ideas to prove theorems about free entropy with applications to embeddings in matrix ultraproducts.

- Model theory and free probability

- A gentle introduction to von Neumann algebras for model theorists (by I. Goldbring)
- Lecture notes on von Neumann algebras (by A. Ioana)
- Model theory of operator algebras II: Model theory (by I. Farah, B. Hart, and D. Sherman)
- The Connes Embedding Problem: a guided tour (by I. Goldbring)