Hostname: page-component-54dcc4c588-b5cpw Total loading time: 0 Render date: 2025-10-03T20:17:54.946Z Has data issue: false hasContentIssue false
  • English
  • Français

THE UNIVERSAL THEORY OF THE HYPERFINITE II$_1$ FACTOR IS NOT COMPUTABLE

Published online by Cambridge University Press:  16 February 2024

ISAAC GOLDBRING  and
BRADD HART
ISAAC GOLDBRING
Affiliation:
DEPARTMENT OF MATHEMATICS UNIVERSITY OF CALIFORNIA IRVINE, 340 ROWLAND HALL (BLDG.# 400), IRVINE CA 92697-3875 USA E-mail: isaac@math.uci.edu URL: http://www.math.uci.edu/~isaac
BRADD HART
Affiliation:
DEPARTMENT OF MATHEMATICS AND STATISTICS MCMASTER UNIVERSITY 1280 MAIN STREET HAMILTON, ON L8S 4K1 CANADA E-mail: hartb@mcmaster.ca URL: http://ms.mcmaster.ca/~bradd/
Get access Rights & Permissions [Opens in a new window]

Abstract

We show that the universal theory of the hyperfinite II$_1$ factor is not computable. The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg’s QWEP Conjecture and Tsirelson’s Problem.

Keywords

Connes embedding problemhyperfinite II1 factorundecidable universal theory

Information

Type
Article
Information
Bulletin of Symbolic Logic , Volume 30 , Issue 2 , June 2024 , pp. 181 - 198
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Archbold, R., Robert, L., and Tikuisis, A., The Dixmier property and tracial states for ${C}^{\ast }$ -algebras . Journal of Functional Analysis , vol. 273 (2017), pp. 2655–2718.CrossRefGoogle Scholar
Ben Yaacov, I., Berenstein, A., Henson, C. W., and Usvyatsov, A., Model theory for metric structures , Model Theory with Applications to Algebra and Analysis , vol. 2 , London Mathematical Society Lecture Note Series, vol. 350, Cambridge University Press, Cambridge, 2008, pp. 315–427.CrossRefGoogle Scholar
Ben Yaacov, I. and Pederson, A. P., A proof of completeness for continuous first order logic . Journal of Symbolic Logic , vol. 75 (2010), no. 1, pp. 168–190.CrossRefGoogle Scholar
Brown, N. and Ozawa, N., C*-Algebras and Finite-Dimensional Approximations , AMS Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, 2008.Google Scholar
Camburd, C., Goldbring, I., and McNicholl, T., On the complexity of the theory of a computably presented metric structure . Archive for Mathematical Logic , vol. 62 (2023), pp. 1111–1129.Google Scholar
Farah, I., Hart, B., Lupini, M., Robert, L., Tikuisis, A., Vignati, A., and Winter, W., Model Theory of C*-Algebras , Memoirs of the American Mathematical Society, vol. 271, American Mathematical Society, Providence, 2018, 147 pp.Google Scholar
Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras III: Elementary equivalence and II ${}_1$ factors. Bulletin of the London Mathematical Society , vol. 46 (2014), pp. 609–628.CrossRefGoogle Scholar
Fritz, T., Tsirelson’s problem and Kirchberg’s conjecture . Reviews in Mathematical Physics , vol. 24 (2012), no. 5, p. 1250012.CrossRefGoogle Scholar
Fritz, T., Netzer, T., and Thom, A., Can you compute the operator norm? Proceedings of the American Mathematical Society , vol. 142 (2014), no. 12, pp. 4265–4276.CrossRefGoogle Scholar
Goldbring, I. and Hart, B., A computability-theoretic reformulation of the Connes embedding problem, this Journal, vol. 22 (2016), no. 2, pp. 238–248.Google Scholar
Goldbring, I. and Hart, B., Operator algebras with hyperarithmetic theories . Journal of Logic and Computation , vol. 31 (2021), pp. 612–629.Google Scholar
Haagerup, U. and Zsidó, L., Sur la propriété de Dixmier pour les ${C}^{\ast }$ -algèbres . Comptes Rendus de l’Académie des Sciences—Series I—Mathematics , vol. 298 (1984), pp. 173–176.Google Scholar
Ji, Z., Natarajan, A., Vidick, T., Wright, J., and Yuen, H., MIP* = RE, preprint, 2022, arXiv:2001.04383.Google Scholar
Junge, M., Navascues, M., Palazuelos, C., Perez-Garcia, D., Scholz, V., and Werner, R., Connes’ embedding problem and Tsirelson’s problem . Journal of Mathematical Physics , vol. 52 (2011), no. 1, p. 012102.CrossRefGoogle Scholar
Kim, S., Paulsen, V., and Schafhauser, C., A synchronous game for binary constraint systems . Journal of Mathematical Physics , vol. 59 (2018), p. 032201.CrossRefGoogle Scholar
Kirchberg, E., On nonsemisplit extensions, tensor products and exactness of group ${C}^{\ast }$ -algebras . Inventiones Mathematicae , vol. 112 (1993), no. 3, pp. 449–489.CrossRefGoogle Scholar
Ozawa, N., About the Connes embedding conjecture . Japanese Journal of Mathematics , vol. 8 (2013), no. 1, pp. 147–183.CrossRefGoogle Scholar
Paulsen, V., Severini, S., Stahlke, D., Todorov, I., and Winter, A., Estimating quantum chromatic numbers . Journal of Functional Analysis , vol. 270 (2016), pp. 2180–2222.CrossRefGoogle Scholar