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Type III factors with unique Cartan decomposition. (English) Zbl 1291.46052
Summary: We prove that, for any free ergodic nonsingular nonamenable action $$\Gamma\curvearrowright(X,\mu)$$ of all $$\Gamma$$ in a large class of groups including all hyperbolic groups, the associated group measure space von Neumann algebra $$L^\infty(X)\rtimes\Gamma$$ has $$L^\infty(X)$$ as its unique Cartan subalgebra, up to unitary conjugacy. This generalizes the probability measure preserving case that was established in [S. Popa and S. Vaes, Acta Math. 212, No. 1, 141–198 (2014; Zbl 1307.46047), arXiv:1201.2824]. We also prove primeness and indecomposability results for such crossed products, for the corresponding orbit equivalence relations and for arbitrary amalgamated free products $$M_{1^\ast B}M_2$$ over a subalgebra $$B$$ of type I.

##### MSC:
 46L10 General theory of von Neumann algebras 46L54 Free probability and free operator algebras 37A40 Nonsingular (and infinite-measure preserving) transformations
Zbl 1307.46047
Full Text:
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