1-bounded entropy and regularity problems in von Neumann algebras

Abstract

We investigate the singular subspace of an inclusion of tracial von Neumann algebras. The singular subspace is a canonical N-N subbimodule of L^{2}(M) and it contains the quasinormalizer introduced by Popa, one-sided quasinormalizer introduced by Fang-Gao-Smith, and wq-normalizer introduced in Galatan-Popa (following upon work in Ioana-Peterson-Popa and Popa). We then obtain a weak notion of regularity (called spectral regularity) by demanding that the singular subspace of N in M generates M. By abstracting Voiculescu's original proof of absence of Cartan subalgebras, we show that there can be no diffuse, hyperfinite subalgebra of L(\FF_{n}) which is spectrally regular. Our techniques are robust enough to repeat this process by transfinite induction and rule out chains of spectrally regular inclusions of algebras starting from a diffuse, hyperfinite algebra and ending in L(\FF_{n}). We use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II_{1}-factors. Our results may be regarded as a consistency check for the possibility of existence of a "good" cohomology theory of II_{1}-factors. Lastly, we deduce nonisomorphism results for crossed products of q-deformed free group factors by Bogoliubov actions, as well as for the continuous core of q-deformed Free Araki-Woods algebras. This extends work of Houdayer-Shlyakhtenko as well as Shlyakhtenko.