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August 28, 2015

1-bounded entropy and regularity problems in von Neumann algebras

We introduce and investigate the singular subspace of an inclusion of a tracial von Neumann algebra N into another tracial von Neumann algebra M. The singular subspace is a canonical N-N subbimodule of L^{2}(M) containing the normalizer, the quasi-normalizer (introduced by Izumi-Longo-Popa), the one-sided quasi-normalizer (introudced by Fang-Gao-Smith), and the wq-normalizer (introduced by Galatan-Popa). By abstracting Voiculescu’s original proof of absence of Cartan subalgebras, we show that the von Neumann algebra generated by the singular subspace of a diffuse, hyperfinite subalgebra is not L(F_{2}). We rely on the notion of being strongly 1-bounded, due to Jung, and the 1-bounded entropy, a quantity which measures “how” strongly 1-bounded an algebra is. Our methods are robust enough to repeat this process by transfinite induction and we use this to prove some conjectures made by Galatan-Popa in their study of smooth cohomology of II_{1}-factors. We also present applications to nonisomoprhism problems for Free-Araki woods factors, as well as crossed products by Free Bogoliubov automorphisms in the spirit of Houdayer-Shlyakhtenko.