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On Popa’s factorial commutant embedding problem. (English) Zbl 07243365
Author’s abstract: An open question of Sorin Popa asks whether or not every \(\mathcal{R}^{\mathcal{U}}\)-embeddable factor admits an embedding into \(\mathcal{R}^{\mathcal{U}}\) with factorial relative commutant. We show that there is a locally universal McDuff \(\mathrm{II}_1\) factor \(M\) such that every property (T) factor admits an embedding into \(M^{\mathcal{U}}\) with factorial relative commutant. We also discuss how our strategy could be used to settle Popa’s question for property (T) factors if a certain open question in the model theory of operator algebras has a positive solution.
03C20 Ultraproducts and related constructions
03C66 Continuous model theory, model theory of metric structures
03C30 Other model constructions
46L10 General theory of von Neumann algebras
Full Text: DOI
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