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Unique Cartan decomposition for \(\mathrm{II}_1\) factors arising from arbitrary actions of free groups. (English) Zbl 1307.46047
This paper makes substantial contributions towards the understanding of unique Cartan decompositions, classification, and rigidity phenomena related to group measure space algebras. The main results are concerned with free ergodic probability-measure-preserving (pmp) actions \(\Gamma \curvearrowright (X,\mu)\) of weakly amenable groups \(\Gamma\) on probability spaces \((X,\mu)\). One problem of major interest concerns the uniqueness of the Cartan subalgebra \(L^\infty(X)\) in the group measure space algebra \(L^\infty (X) \rtimes \Gamma\). One important result that applies to a broad class of infinite discrete groups states that if, in addition, \(\Gamma\) admits a proper \(1\)-cocycle into a non-amenable representation, or if (for instance) it has non-zero first \(\ell^2\)-Betti number, then \(L^\infty(X)\) is the unique Cartan subalgebra in \(L^\infty(X)\rtimes \Gamma\) up to unitary conjugacy.
To illustrate the strength of this unique Cartan decomposition result, suppose that \({\mathbb F}_m \curvearrowright (X,\mu)\) and \({\mathbb F}_n \curvearrowright (Y,\eta)\) are two free ergodic pmp actions of the free groups \({\mathbb F}_m\) and \({\mathbb F}_n\) on \(m\) and respectively \(n\) generators. Combining the above result with a result of D. Gaboriau [Publ. Math., Inst. Hautes Étud. Sci. 95, 93–150 (2002; Zbl 1022.37002)], one concludes that if \(L^\infty (X)\rtimes {\mathbb F}_m \cong L^\infty (Y) \rtimes {\mathbb F}_n\), then \(m=n\). Important applications concerning wreath product groups and treeable ergodic pmp equivalence relations are also given. For instance, it is shown that if the group von Neumann algebras defined by the groups \({\mathbb Z}^{({\mathbb F}_m)} \rtimes {\mathbb F}_m\) and \({\mathbb Z}^{({\mathbb F}_n)} \rtimes {\mathbb F}_n\) are isomorphic, then \(m=n\). It is also proved that if \(L{\mathcal R}_1 \cong L{\mathcal R}_2\) for two treeable countable ergodic pmp equivalence relations \({\mathcal R}_1\) and \({\mathcal R}_2\), then \({\mathcal R}_1 \cong {\mathcal R}_2\).
A relative strong solidity type dichotomy is established for a large class of weakly amenable groups \(\Gamma\): an amenable subalgebra \(A\) of an arbitrary crossed product \(B\rtimes \Gamma\), with \(B\) amenable, either embeds into \(B\) in Popa’s sense of “intertwining-by-bimodules” or it has amenable normalizer. This extends ground-breaking work of N. Ozawa [Acta Math. 192, No. 1, 111–117 (2004; Zbl 1072.46040)], N. Ozawa and S. Popa [Ann. Math. (2) 172, No. 1, 713–749 (2010; Zbl 1201.46054)], and I. Chifan and T. Sinclair [Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 1–33 (2013; Zbl 1290.46053)] on hyperbolic groups, and solidity and strong solidity in von Neumann algebras.
Another application of unique Cartan decomposition establishes \(W^*\)-superrigidity of quotients of the generalized Bernoulli action \(\Gamma_1 \times \Gamma_2 \curvearrowright [0,1]^{\Gamma_1 \times \Gamma_2}\), where \(\Gamma_1\) and \(\Gamma_2\) are weakly amenable icc groups that admit a proper \(1\)-cocycle into a non-amenable representation. Finally, it is proved that if \(\Gamma\) and \(\Lambda\) are icc groups that satisfy the requirements from the main unique Cartan decomposition theorem, and act outerly on the hyperfinite II\(_1\) factor \(R\) and such that \(R\rtimes \Gamma \cong R\rtimes \Lambda\), then \(\Gamma \cong \Lambda\) and the actions of \(\Gamma\) and \(\Lambda\) on \(R\) are cocycle conjugated.

46L36 Classification of factors
22D25 \(C^*\)-algebras and \(W^*\)-algebras in relation to group representations
28D15 General groups of measure-preserving transformations
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
46L10 General theory of von Neumann algebras
46L55 Noncommutative dynamical systems
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