# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.