Type III von Neumann algebras

Lectures byAmine Marrakchi, winner of the Claude-Antoine Peccot Prize for 2022

A type III von Neumann algebra is the quantum analog of a dynamical system without invariant measures. The aim of this lecture is to present some recent advances in the understanding of these von Neumann algebras. The lecture will begin by studying various examples of classical dynamical systems without invariant measure, taken from geometry or representation theory. As a common thread, we will focus on a fundamental object, the modular fibroid, which will allow us to introduce Tomita-Takesaki's modular theory in a natural way. Finally, we'll end the lecture with an open conjecture : the Connes bicentralizer problem.