We show that any II${}_{1}$ factor that has the same 4-quantifier theory as the hyperfinite II${}_{1}$ factor $\mathcal{R}$ satisfies the conclusion of the Popa Factorial Commutant Embedding Problem (FCEP) and has the Brown property. These results improve recent results proving the same conclusions under the stronger assumption that the factor is actually elementarily equivalent to $\mathcal{R}$. In the same spirit, we improve a recent result of the first-named author, who showed that if (1) the amalgamated free product of embeddable factors over a property (T) base is once again embeddable, and (2) $\mathcal{R}$ is an infinitely generic embeddable factor, then the FCEP is true of all property (T) factors. In this paper, it is shown that item (2) can be weakened to assume that $\mathcal{R}$ has the same 3-quantifier theory as an infinitely generic embeddable factor.
Revised:
Accepted:
Published online:
Keywords: Continuous model theory, von Neumann algebras, II$_1$ factors, factorial commutant embedding problem
Isaac Goldbring ^{1}; Bradd Hart ^{2}
@article{CML_2020__12_2_37_0, author = {Isaac Goldbring and Bradd Hart}, title = {Properties expressible in small fragments of the theory of the hyperfinite {II}$_1$ factor}, journal = {Confluentes Mathematici}, pages = {37--47}, publisher = {Institut Camille Jordan}, volume = {12}, number = {2}, year = {2020}, doi = {10.5802/cml.67}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.67/} }
TY - JOUR AU - Isaac Goldbring AU - Bradd Hart TI - Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor JO - Confluentes Mathematici PY - 2020 SP - 37 EP - 47 VL - 12 IS - 2 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.67/ DO - 10.5802/cml.67 LA - en ID - CML_2020__12_2_37_0 ER -
%0 Journal Article %A Isaac Goldbring %A Bradd Hart %T Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor %J Confluentes Mathematici %D 2020 %P 37-47 %V 12 %N 2 %I Institut Camille Jordan %U https://cml.centre-mersenne.org/articles/10.5802/cml.67/ %R 10.5802/cml.67 %G en %F CML_2020__12_2_37_0
Isaac Goldbring; Bradd Hart. Properties expressible in small fragments of the theory of the hyperfinite II$_1$ factor. Confluentes Mathematici, Volume 12 (2020) no. 2, pp. 37-47. doi : 10.5802/cml.67. https://cml.centre-mersenne.org/articles/10.5802/cml.67/
[1] S. Atkinson, I. Goldbring, and S. Kunnawalkam Elayavalli. Factorial commutants and the generalized Jung property for II${}_{1}$ factors. Preprint. arXiv 2004.02293.
[2] I. Ben Yaacov, A. Berenstein, C. W. Henson, and A. Usvyatsov. Model theory for metric structures, in Model theory with applications to algebra and analysis. Vol. 2, London Math. Soc. Lecture Note Ser. 350, 315–427. Cambridge Univ. Press, Cambridge, 2008. | DOI | Zbl
[3] N. Brown. Topological dynamical systems associated to II${}_{1}$ factors. Adv. Math. 227 (2011), 1665–1699. With an appendix by N. Ozawa. | DOI | Zbl
[4] A. Connes and V. Jones. Property T for von Neumann algebras. Bull. London Math. Soc. 17 (1985), 57–62. | DOI | MR | Zbl
[5] I. Farah, I. Goldbring, B. Hart, and D. Sherman. Existentially closed II${}_{1}$ factors. Fund. Math. 233 (2016), 173–196. | DOI | MR | Zbl
[6] I. Farah, B. Hart, and D. Sherman. Model theory of operator algebras III: Elementary equivalence and II${}_{1}$ factors. Bull. London Math. Soc. 46 (2014), 1–20. | DOI | Zbl
[7] I. Goldbring. On Popa’s factorial commutant embedding problem. Proc. Amer. Math. Soc., to appear. | DOI | MR | Zbl
[8] I. Goldbring. Spectral gap and definability, Beyond First Order Model Theory Vol. 2, to appear.
[9] I. Goldbring and B. Hart. The universal theory of the hyperfinite II${}_{1}$ factor is not computable. Preprint. arXiv 2004.02299.
[10] I. Goldbring and B. Hart. On the theories of McDuff’s II${}_{1}$ factors. Int. Math. Res. Notices 27 (2017), 5609–5628. | DOI | Zbl
Cited by Sources: