Applications of deformation rigidity theory in Von Neumann algebras

Abstract

This work contains some structural results for von Neumann algebras arising from measure preserving actions by direct products of groups on probability spaces. The technology and the methods we use are a continuation of those used by Chifan and Sinclair in [Ionut Chifan and Thomas Sinclair. On the structural theory of II1 factors of negatively curved groups, ArXiv e-prints, March 2011]. By employing these methods, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. We show for instance that every II1 factor associated with a weakly amenable group in the class S of Ozawa is strongly solid [Narutaka Ozawa. Solid von Neumann algebras, Acta Math., 192 (2004), 111–117]. We also obtain a product version of this result: any maximal abelian *-subalgebra of any II1 factor associated with a finite direct product of weakly amenable groups in the class S of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with Ioana’s cocycle superrigidity theorem [Adrian Ioana. Cocycle superrigidity for profinite actions of property (T) groups, Duke Math. J., to appear ], we prove that compact actions by finite products of lattices in Sp(n, 1), n ≥ 2, are virtually W ∗-superrigid. The results presented here are joint work with Ionut Chifan and Thomas Sinclair. They constitute the substance of an article which has already been submitted for publication [Ionut Chifan, Thomas Sinclair, and Bogdan Udrea. On the structural theory of II1 factors of negatively curved groups, ii. actions by product groups, ArXiv e-prints, August 2011].