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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
Citations:
Zbl 1307.46047
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