# Operator algebras seminar

Speaker: Asger Törnquist (University of Copenhagen)

Title: A new proof of Thoma's theorem on type I groups.

Abstract: A classical theorem due to Elmar Thoma (1964) states that a countable discrete group is type I (meaning that all factor representations are of type I) if and only if the group is Abelian-by-finite. Together with Glimm's famous Type I if and only if smooth dual'', this theorem shows that a countable discrete group which is not abelian by finite has many infinite-dimensional irreducible unitary representation. This fact, in turn, has given rise to many applications of Thoma's theorem (e.g. in ergodic theory through the Gaussian construction).

Unfortunately, Thoma's proof is rather difficult and long, and relies on setting up a direct integral representation theory for a certain class of positive definite functions. In this talk, I will give a rather simple and direct proof of Thoma's theorem which relies only on elementary von Neumann algebra techniques (mainly, the factor decomposition of a unitary representation).

This is joint work with my former PhD student Fabio Tonti.