On a class of ${\mathrm{II}}_1$ factors with at most one Cartan subalgebra

Abstract

We prove that the normalizer of any diffuse amenable subalgebra of a free group factor $L({\mathbb{F}}_r)$ generates an amenable von Neumann subalgebra. Moreover, any II${}_1$ factor of the form $Q \vt L(\Bbb F_r) $, with $Q$ an arbitrary subfactor of a tensor product of free group factors, has no Cartan subalgebras. We also prove that if a free ergodic measure preserving action of a free group ${\mathbb{F}}_r$, $2\le r\le \infty$, on a probability space $(X,\mu )$ is profinite then the group measure space factor $L^{\infty }(X)⋊{\mathbb{F}}_r$ has unique Cartan subalgebra, up to unitary conjugacy.

arXiv comment27 pages; minor modifications; 10/27/07: New version with improved statements, new applications, and simplifications in proofs

References in corpus (4)

• Weak amenability of CAT(0) cubical groups
• L^2-rigidity in von Neumann algebras
• Non-Orbit Equivalent Actions of ${\mathbb{F}}_n$
• Proper actions of lamplighter groups associated with free groups

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• A non-commutative Path Space approach to stationary free Stochastic Differential Equations
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• Noncommutative Riesz transforms -- a probabilistic approach
• A note on weak amenability for reduced free products of discrete quantum groups
• Weak amenability and simply connected Lie groups