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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
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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

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References

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