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

Abstract

We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras $M_1{\ast }_B{M}_2$ 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 $(M_1,{\phi }_1)\ast (M_2,{\phi }_2)$ with respect to faithful normal states has no Cartan subalgebra. This generalizes the tracial case that was established in \cite{Io12a}. 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 $\rL(\mathcal R)$ with a unique Cartan subalgebra, up to unitary conjugacy. Finally, we prove unique Cartan decomposition for a class of group measure space factors $\rL^\infty(X) \rtimes Γ$ arising from nonsingular free ergodic actions $\Gamma \curvearrowright (X,\mu )$ on standard measure spaces of amalgamated groups $\Gamma ={\Gamma }_1{\ast }_{\Sigma }{\Gamma }_2$ over a finite subgroup $\Sigma$.

arXiv comment30 pages

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Cited by in corpus (17)

• CCAP for universal discrete quantum groups
• Asymptotic structure of free product von Neumann algebras
• Asymptotic structure of free Araki-Woods factors
• Strong solidity of free Araki-Woods factors
• Rigidity of free product von Neumann algebras
• Primeness results for von Neumann algebras associated with surface braid groups
• Strongly ergodic equivalence relations: spectral gap and type III invariants
• Tensor product decompositions of II${}_1$ factors arising from extensions of amalgamated free product groups
• Structure of extensions of free Araki-Woods factors
• Factoriality, Connes' type III invariants and fullness of amalgamated free product von Neumann algebras
• Some prime factorization results for free quantum group factors
• Cartan subalgebras of tensor products of free quantum group factors with arbitrary factors
• Unique prime factorization for infinite tensor product factors
• On bi-exactness of discrete quantum groups
• Classification of Tensor Decompositions of II${}_1$ Factors Associated With Poly-Hyperbolic Groups
• Absence of Cartan subalgebras in continuous cores of free product von Neumann algebras
• Classification of tensor decompositions for II${}_1$ factors