Lecture 1: We’ll briskly review basic properties of semi-finite von Neumann algebras. The standard representation, completely positive maps, group von Neumann algebras, the group-measure space construction, and some characterizations of the hyperfinite II1 factor.
Lecture 2: We discuss some approximation properties that are common in “rank 1” groups: Weak amenability and biexactness.
Lecture 3: We discuss properly proximal groups as defined by Boutonnet, Ioana, and myself, and give some applications to group von Neumann algebras associated to higher-rank groups.
Lecture 4: We’ll introduce measure equivalence (ME), W*-equivalence (W*E), and von Neumann equivalence (VNE). We’ll give examples and discuss invariants.
- Background on von Neumann algebras
- Some approximation properties
- Proper proximality
- Von Neumann equivalence
These videos were produced by the University of Münster, as part of the hybrid workshop YMC*A 2021.
