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Amalgamated free product type III factors with at most one Cartan subalgebra. (English) Zbl 1308.46067
Summary: We investigate Cartan subalgebras in nontracial amalgamated free product von Neumann algebras \(M_1*_B M_2\) 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 \((M_1,\varphi_1)* (M_,\varphi_2)\) 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 \(\text{II}_1\) 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 \({\mathcal R}={\mathcal R}_1*{\mathcal R}_2\) of recurrent subequivalence relations gives rise to a nonamenable factor \(L({\mathcal R})\) with a unique Cartan subalgebra, up to unitary conjugacy.
Finally, we prove unique Cartan decomposition for a class of group measure space factors \(L^\infty(X)\rtimes\Gamma\) arising from nonsingular free ergodic actions \(\Gamma\to(X, \mu)\) on standard measure spaces of amalgamated groups \(\Gamma= \Gamma_1*_\Sigma\Gamma_2\) over a finite subgroup \(\Sigma\).

MSC:
46L10 General theory of von Neumann algebras
46L54 Free probability and free operator algebras
37A40 Nonsingular (and infinite-measure preserving) transformations
46L09 Free products of \(C^*\)-algebras
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