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On the structural theory of \(\mathrm{II}_1\) factors of negatively curved groups. II: Actions by product groups. (English) Zbl 1288.46037

Summary: This paper contains a series of structural results for von Neumann algebras arising from measure preserving actions by product groups on probability spaces. Expanding upon the methods used earlier by the first two authors, we obtain new examples of strongly solid factors as well as von Neumann algebras with unique or no Cartan subalgebra. For instance, we show that every \(\mathrm{II}_1\) factor associated with a weakly amenable group in the class \(S\) of Ozawa is strongly solid. There is also the following product version of this result: any maximal abelian \(\ast\)-subalgebra of any \(\mathrm{II}_1\) factor associated with a finite product of weakly amenable groups in the class \(S\) of Ozawa has an amenable normalizing algebra. Finally, pairing some of these results with a cocycle superrigidity result of A. Ioana [Duke Math. J. 157, No. 2, 337–367 (2011; Zbl 1235.37005)], it follows that compact actions by finite products of lattices in \(Sp(n,1)\), \(n\geq 2\), are virtually \(W^\ast\)-superrigid.
For Part I, see [I. Chifan and T. Sinclair, Ann. Sci. Éc. Norm. Supér. (4) 46, No. 1, 1–33 (2013; Zbl 1290.46053)].

MSC:

46L10 General theory of von Neumann algebras
43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc.
46L36 Classification of factors
46L55 Noncommutative dynamical systems
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