As a preparation for the third and fourth lecture, I will introduce some of the basic concepts and results in the theory of von Neumann algebras. The main examples will come from actions of discrete groups on measure spaces. I will also discuss the connection to orbit equivalence of group actions and to measure equivalence of groups.

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.