BOUTONNET, Rémi; HOUDAYER, Cyril; RAUM, Sven

doi:10.1112/S0010437X13007537

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BOUTONNET, Rémi
Institut de Mathématiques de Bordeaux [IMB]

HOUDAYER, Cyril
Université Paris Descartes - Paris 5 [UPD5]

RAUM, Sven
Ecole Polytechnique Fédérale de Lausanne [EPFL]

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Compositio Mathematica. 2014, vol. 150, p. 143 - 174

Foundation Compositio Mathematica

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We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras M1 * B M2 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 (M1, ϕ1) * (M2, ϕ2) with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in [Io12a]. Next, we prove that any countable nonsingular ergodic equivalence relation R defined on a standard measure space and which splits as the free product R = R1 * R2 of recurrent subequivalence relations gives rise to a nonamenable factor L(R) with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors L ∞ (X) ⋊ Γ arising from nonsingular free ergodic actions Γ (X, µ) on standard measure spaces of amalgamated groups Γ = Γ1 * Σ Γ2 over a finite subgroup Σ.< Leer menos

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

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