On the structural theory of II_1 factors of negatively curved groups, II: Actions by product groups

Abstract

This paper includes a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors \cite{CS}, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance we show that every II${}_1$ factor associated with a weakly amenable group in the class $\mathcal{S}$ of Ozawa is strongly solid, \cite{OzSolid}. There is also the following product version of this result: any maximal abelian $\star$-subalgebra of any II${}_1$ factor associated with a finite product of weakly amenable groups in the class $\mathcal{S}$ of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with cocycle superrigidity results from \cite{IoaCSR}, it follows that compact actions by finite products of lattices in $Sp(n,1)$, $n\ge 2$, are virtually $W^{\ast }$-superrigid.

final version

References in corpus (9)

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Cited by in corpus (22)

• Unique prime factorization and bicentralizer problem for a class of type III factors
• Amalgamated free product type III factors with at most one Cartan subalgebra
• W*-superrigidity for arbitrary actions of central quotients of braid groups
• Structure of $\mathrm{I}{\mathrm{I}}_1$ factors arising from free Bogoljubov actions of arbitrary groups
• W${}^{\ast }$-Rigidity for the von Neumann Algebras of Products of Hyperbolic Groups
• OE and W* superrigidity results for actions by surface braid groups
• Unique Cartan decomposition for II_1 factors arising from arbitrary actions of hyperbolic groups
• Constructing MASAs with prescribed properties
• Inner amenability for groups and central sequences in factors
• Primeness results for von Neumann algebras associated with surface braid groups
• Absence of Cartan subalgebras for right-angled Hecke von Neumann algebras
• On fundamental groups of tensor product $\mathrm{I}{\mathrm{I}}_1$ factors
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups
• Acylindrical hyperbolicity of groups acting on trees
• Examples of factors which have no Cartan subalgebras
• Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors
• On bi-exactness of discrete quantum groups
• Cocycle superrigidity for coinduced actions
• Weak Exactness for C*-algebras and Application to Condition (AO)
• Classification of Tensor Decompositions of II${}_1$ Factors Associated With Poly-Hyperbolic Groups
• Riesz transforms on compact quantum groups and strong solidity
• Classification of tensor decompositions for II${}_1$ factors