It is a wide open problem whether the group von Neumann algebras of free groups with a different number of generators are isomorphic or not. However, Sorin Popa and I proved that their probability measure preserving actions never give rise to isomorphic von Neumann algebras. I will sketch the main methods to prove this result.

Stefaan Vaes is a full professor and the head of the Analysis section at KU Leuven (Belgium). His research focuses on operator algebras and their connections to group theory and ergodic theory. He was an invited speaker at the International Congress of Mathematicians in 2010 and at the European Congress of Mathematics in 2016. In 2015, he was awarded the Francqui Prize, which is the highest scientific distinction in Belgium. In the spring of 2017, he was a Rothschild Fellow at the Newton Institute in Cambridge. Stefaan Vaes is a member of the Royal Academy of Belgium (KVAB). He is one of the editors-in-chief of Journal of Functional Analysis.