Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T18:16:29.905Z Has data issue: false hasContentIssue false
  • English
  • Français

The Theory of Tracial Von Neumann Algebras Does Not Have A Model Companion

Published online by Cambridge University Press:  12 March 2014

Isaac Goldbring ,
Bradd Hart  and
Thomas Sinclair
Isaac Goldbring
Affiliation:
Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Science and Engineering Offices M/C 249, 851 S. Morgan St., Chicago, IL, 60607-7045, USA, E-mail: isaac@math.uic.edu, URL: http://www.math.uic.edu/~isaac
Bradd Hart
Affiliation:
Department of Mathematics and Statistics, McMaster University, 1280 Main Street W., Hamilton, Ontario, L8S 4K1, Canada, E-mail: hartb@mcmaster.ca, URL: http://www.math.mcmaster.ca/~bradd
Thomas Sinclair
Affiliation:
Department of Mathematics, University of California, Los Angeles, 520 Portola Plaza, Box 951555, Los Angeles, CA, 90095-1555, USA, E-mail: thomas.sinclair@math.ucla.edu, URL: http://www.math.ucla.edu/~thomas.sinclair
Get access Rights & Permissions [Opens in a new window]

Abstract

In this note, we show that the theory of tracial von Neumann algebras does not have a model companion. This will follow from the fact that the theory of any locally universal, McDuff II1 factor does not have quantifier elimination. We also show how a positive solution to the Connes Embedding Problem implies that there can be no model-complete theory of II1 factors.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 78 , Issue 3 , September 2013 , pp. 1000 - 1004
Copyright
Copyright © Association for Symbolic Logic 2013

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.)

References

REFERENCES

[1] Adler, H., A geometrie introduction to forking and thorn-forking, Journal of Mathematical Logic, vol. 9 (2009), pp. 1–20.Google Scholar
[2] Anantharaman-Delaroche, C., Amenable correspondences and approximation properties for von Neumann algebras, Pacific Journal of Mathematics, vol. 171 (1995), no. 2, pp. 309–341.Google Scholar
[3] Ben Yaacov, I., Henson, C. W., Junge, M., and Raynaud, Y., Report on non-commutative probabilities and von neumann algebras, http://www.aimath.org/WWN/continuouslogic/ncpreport.pdf. Google Scholar
[4] Brown, N., Topological dynamical systems associated to II1 factors, Advances in Mathematics, vol. 227 (2011), pp. 1665–1699.Google Scholar
[5] Brown, N., Dykema, K., and Jung, K., Free entropy in amalgamated free products, Proceedings of the London Mathematical Society, vol. 97 (2008), pp. 339–367.Google Scholar
[6] Brown, N. and Ozawa, N., C*-algebras and finite-dimensional approximations, Graduate Studies in Mathematics, vol. 88, American Mathematical Society, Providence, RI, 2008.Google Scholar
[7] Farah, I., Hart, B., and Sherman, D., Model theory of operator algebras I: Stability, The Bulletin of the London Mathematical Society, to appear.Google Scholar
[8] Farah, I., Model theory of operator algebras II: Model theory, preprint.Google Scholar
[9] Farah, I., Model theory of operator algebras III. Elementary equivalence and II1 factors, preprint.Google Scholar
[10] Jung, K., Amenability, tubularity, and embeddings into R ω , Mathematische Annalen, vol. 338 (2007), pp. 241–248.Google Scholar