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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.
MSC:
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
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[1] Atkinson, Scott, Convex sets associated to \(C^*\)-algebras, J. Funct. Anal., 271, 6, 1604-1651 (2016) · Zbl 1358.46049
[2] AGK S. Atkinson, I. Goldbring, and S. Kunnawalkam Elayavalli, On \(II_1\) factors with the generalized Jung property, manuscript in preparation.
[3] Ben Yaacov, Ita\"\i; Berenstein, Alexander; Henson, C. Ward; Usvyatsov, Alexander, Model theory for metric structures. Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser. 350, 315-427 (2008), Cambridge Univ. Press, Cambridge · Zbl 1233.03045
[4] Brown, Nathanial P., Topological dynamical systems associated to \(\text{II}_1\)-factors, Adv. Math., 227, 4, 1665-1699 (2011) · Zbl 1229.46041
[5] Dixmier, J.; Lance, E. C., Deux nouveaux facteurs de type \(\text{II}_1 \), Invent. Math., 7, 226-234 (1969) · Zbl 0174.18701
[6] Farah, Ilijas; Goldbring, Isaac; Hart, Bradd; Sherman, David, Existentially closed \(\text{II}_1\) factors, Fund. Math., 233, 2, 173-196 (2016) · Zbl 1436.03213
[7] Farah, Ilijas; Hart, Bradd; Sherman, David, Model theory of operator algebras III: elementary equivalence and \(\text{II}_1\) factors, Bull. Lond. Math. Soc., 46, 3, 609-628 (2014) · Zbl 1303.46049
[8] Gold1 I. Goldbring, Enforceable operator algebras, to appear in the Journal of the Institute of Mathematics of Jussieu.
[9] Gold2 I. Goldbring, Spectral gap and definability, to appear in Beyond First-order Model Theory Volume 2. \arXiv 1805.02752.
[10] Goldbring, Isaac; Hart, Bradd; Sinclair, Thomas, The theory of tracial von Neumann algebras does not have a model companion, J. Symbolic Logic, 78, 3, 1000-1004 (2013) · Zbl 1316.03019
[11] Jung, Kenley, Amenability, tubularity, and embeddings into \(\mathcal{R}^\omega \), Math. Ann., 338, 1, 241-248 (2007) · Zbl 1121.46052
[12] Macintyre, Angus, On algebraically closed groups, Ann. of Math. (2), 96, 53-97 (1972) · Zbl 0254.20021
[13] quantum Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen, \(MIP^*=\) RE, \arXiv 2001.04383.
[14] Popa, Sorin, On the classification of inductive limits of \(II_1\) factors with spectral gap, Trans. Amer. Math. Soc., 364, 6, 2987-3000 (2012) · Zbl 1252.46063
[15] Popa, Sorin, Independence properties in subalgebras of ultraproduct \(\text{II}_1\) factors, J. Funct. Anal., 266, 9, 5818-5846 (2014) · Zbl 1305.46052
[16] Usvyatsov, Alexander, Generic separable metric structures, Topology Appl., 155, 14, 1607-1617 (2008) · Zbl 1148.54019
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