Unique Cartan decomposition for II1 factors arising from arbitrary actions of free groups

By Sorin Popa, S. Vaes. 2011

174 citations recorded by the source corpus.

We prove that for any free ergodic probability measure-preserving action $${\mathbb{F}_n \curvearrowright (X, \mu)}$$Fn↷(X,μ) of a free group on n generators $${\mathbb{F}_n, 2\leq n \leq \infty}$$Fn,2≤n≤∞, the associated group measure space II1 factor $${L^\infty (X)\rtimes \mathbb{F}_n}$$L∞(X)⋊Fn has L∞(X) as its unique Cartan subalgebra, up to unitary conjugacy. We deduce that group measure space II1 factors arising from actions of free groups with different number of generators are never isomorphic. We actually prove unique Cartan decomposition results for II1 factors arising from arbitrary actions of a much larger family of groups, including all free products of amenable groups and their direct products.

Subfields: Endocrinology, Diabetes and Metabolism, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fbbc54c989ab12ff60c7cbf5e146545c9dcac9f2

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In Endocrinology, Diabetes and Metabolism

S. Vaes

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Type III factors with unique Cartan decomposition (2012, background): “se of [Oz07], this covers all nonelementary hyperbolic groups. This result was then extended in [CSU11] to cover as well direct products of nonamenable, weakly amenable group Γ in the class QHreg. In [PV11], the above conjecture was entirely solved and it was proven that for arbitrary free ergodic p”