Traicial ultraproducts of von Neumann algebras and free probability

Explanation

A research talk at the SNU Operator Seminar on June 20 and June 24, 2025.

Abstract

Two von Neumann algebras are said to be elementary equivalent if they have a pair of isomorphic ultrapowers. In this series of talks, we review how the authors of [2] proved that the II_1 factor, which was constructed in [1] and proven to be without the Gamma property, is not elementary equivalent to any free product of diffuse tracial von Neumann algebras. On the first day, we review basic terminologies of free probability theory and how they are related to (tracial) von Neumann algebras. I will also introduce the problem and main results of [2]. On the second day, we continue to follow the main steps of the proof for the results, particularly focusing on how the language of free probability was used.

[1] I. Chifan, A. Ioana, S. Kunnawalkam Elayavalli, An exotic II_1 factor without property Gamma. To appear in Geom. Funct. Anal.
[2] C. Houdayer, A. Ioana, Asymptotic freeness in tracial ultraproducts, To appear in Forum of Mathematics, Sigma.