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Amalgamated free product type III factors with at most one Cartan subalgebra

Published online by Cambridge University Press:  13 December 2013

Rémi Boutonnet*
Affiliation:
ENS Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France
Cyril Houdayer
Affiliation:
CNRS–ENS Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France email cyril.houdayer@ens-lyon.fr
Sven Raum
Affiliation:
ENS Lyon, UMPA UMR 5669, 69364 Lyon cedex 7, France email sven.raum@ens-lyon.fr
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Abstract

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We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras ${\mathop{M{}_{1} \ast }\nolimits}_{B} {M}_{2} $ over an amenable von Neumann subalgebra $B$. First, we settle the problem of the absence of Cartan subalgebra in arbitrary free product von Neumann algebras. Namely, we show that any nonamenable free product von Neumann algebra $({M}_{1} , {\varphi }_{1} )\ast ({M}_{2} , {\varphi }_{2} )$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established by A. Ioana [Cartan subalgebras of amalgamated free product ${\mathrm{II} }_{1} $factors, arXiv:1207.0054]. Next, we prove that any countable nonsingular ergodic equivalence relation $ \mathcal{R} $ defined on a standard measure space and which splits as the free product $ \mathcal{R} = { \mathcal{R} }_{1} \ast { \mathcal{R} }_{2} $ of recurrent subequivalence relations gives rise to a nonamenable factor $\mathrm{L} ( \mathcal{R} )$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors ${\mathrm{L} }^{\infty } (X)\rtimes \Gamma $ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X, \mu )$ on standard measure spaces of amalgamated groups $\Gamma = {\mathop{\Gamma {}_{1} \ast }\nolimits}_{\Sigma } {\Gamma }_{2} $ over a finite subgroup $\Sigma $.

Type
Research Article
Copyright
© The Author(s) 2013 

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