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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A positivstellensatz for non-commutative polynomials


Authors: J. William Helton and Scott A. McCullough
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3721-3737
MSC (2000): Primary 47A13
DOI: https://doi.org/10.1090/S0002-9947-04-03433-6
Published electronically: March 23, 2004
MathSciNet review: 2055751
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Abstract: A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the commutative case.

A broader issue is, to what extent does real semi-algebraic geometry extend to non-commutative polynomials? Our ``strict" Positivstellensatz is positive news, on the opposite extreme from strict positivity would be a Real Nullstellensatz. We give an example which shows that there is no non-commutative Real Nullstellensatz along certain lines. However, we include a successful type of non-commutative Nullstellensatz proved by George Bergman.


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Additional Information

J. William Helton
Affiliation: Department of Mathematics, University of California, San Diego, California 92093
Email: helton@osiris.ucsd.edu

Scott A. McCullough
Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611-8105
Email: sam@math.ufl.edu

DOI: https://doi.org/10.1090/S0002-9947-04-03433-6
Received by editor(s): January 6, 2003
Received by editor(s) in revised form: June 5, 2003
Published electronically: March 23, 2004
Additional Notes: The first author was partially supported by the the NSF, DARPA and Ford Motor Co.
The second author was partially supported by NSF grant DMS-0140112
Article copyright: © Copyright 2004 American Mathematical Society