Narrow Results

In 1965, Tauer produced a countably infinite family of semi-regular masas in the hyperfinite II₁ factor, no pair of which are conjugate by an automorphism. This was achieved by iterating the process of passing to the algebra generated by the normalizers and, for each n ∈ ℕ, finding masas for which this procedure terminates at the nth stage. Such masas are said to have length n. In this paper, we consider a transfinite version of this idea, giving rise to a notion of ordinal valued length. We show that all countable ordinals arise as lengths of semi-regular masas in the hyperfinite II₁ factor. Furthermore, building on work of Jones and Popa, we obtain all possible combinations of regular inclusions of irreducible subfactors in the normalizing tower.

We construct a group measure space II₁ factor that has two nonconjugate Cartan subalgebras. We show that the fundamental group of the II₁ factor is trivial, while the fundamental group of the equivalence relation associated with the second Cartan subalgebra is nontrivial. This is not absurd as the second Cartan inclusion is twisted by a 2-cocycle.

We survey some of the progress made recently in the classification of von Neumann algebras arising from countable groups and their measure preserving actions on probability spaces. We emphasize results which provide classes of (W*-superrigid) actions that can be completely recovered from their von Neumann algebras and II₁ factors that have a unique Cartan subalgebra. We also present cocycle superrigidity theorems and some of their applications to orbit equivalence. Finally, we discuss several recent rigidity results for von Neumann algebras associated to groups.