Set Theory and von Neumann algebras

The aim of the lectures is to give a brief introduction to the area of von Neumann algebras to a typical set theorist. The ideal intended reader is a person in the field of (descriptive) set theory, who works with group actions and equivalence relations, and who is familiar with the rudiments of ergodic theory, and perhaps also orbit equivalence. This should not intimidate readers with a different background: Most notions we use in these notes will be defined. The reader is assumed to know a small amount of functional analysis. For those who feel a need to brush up on this, we recommend consulting [Ped89]. What is the motivation for giving these lectures, you ask. The answer is two-fold: On the one hand, there is a strong connection between (non-singular) group actions, countable Borel equivalence relations and von Neumann algebras, as we will see in Lecture 3 below. In the past decade, the knowledge about this connection has exploded, in large part due to the work of Sorin Popa and his many collaborators. Von Neumann algebraic techniques have lead to many discoveries that are also of significance for the actions and equivalence relations themselves, for instance, of new cocycle superrigidity theorems. On the other hand, the increased understanding of the connection between objects of ergodic theory, and their related von Neumann algebras, has also made it possible to construct large families of non-isomorphic von Neumann algebras, which in turn has made it possible to prove non-classification type results for the isomorphism relation for various types of von Neumann algebras (and in particular, factors). For these reasons, it seems profitable that (descriptive) set theorists should know more about von Neumann algebras, and we hope that these lectures can serve as a starting point for those who want to start learning about this wonderful area of mathematics. Before moving on to the mathematics, a warning or three: Theorems and Lemmas below that are not attributed are not due to the authors, and their origin can usually be deduced by perusing the surrounding text. Mistakes, however, are entirely due to the authors (specifically, the first author). On the other hand, it is often implicitly assumed below that Hilbert spaces are separable, and if a result is false without this assumption, then it is only a mistake if it also is false with this assumption.