Free entropy theory and rigid von Neumann algebras

Abstract

In this thesis, we study rigid von Neumann algebras from the framework of Voiculescu’s free probability. A rigid von Neumann algebra or a Property (T) von Neumann algebra generalizes the well known notion of Property (T) for groups. These are central objects that are studied in the program of classifying finite von Neumann algebras. For instance Popa’s program of deformation-rigidity crucially plays these rigid inclusions against deformations arising from various natural constructions to deduce classification results. Property (T) von Neumann algebras can be viewed as the ‘trivial’ objects from the perspective of Popa’s deformation-rigidity theory. A parallel approach to studying classification that predates Popa’s program is Voiculescu’s free entropy theory. This theory was the first successful attempt at showing primeness and absence of Cartan subalgebras for the von Neumann algebras associated to non abelian free groups. There has been a significant thread of research studying rigid von Neumann algebras from the perspective of Voiculescu’s theory. An important missing piece in this area was whether rigid von Neumann algebras are also ‘trivial’ objects in the sense of Voiculescu’s entropy. We settle this open problem in joint work with Ben Hayes and David Jekel. We show that all von Neumann algebras with a Kazhdan set (which includes factors with Property (T)) are strongly 1-bounded in the sense of Jung, which is the correct notion of ‘trivial’ in this setting. This result has applications to certain classification problems in von Neumann algebras. We also study groups with vanishing L2-cohomology, and establish this notion of strong 1-boundedness for the von Neumann algebras associated to these groups using Voiculescu’s theory of microstates free entropy. This result is not new. This has been obtained by Jung and by Shlyakhtenko. Our proof follows, refines and simplifies the ideas of Jung.