Journal
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 356, Issue 9, Pages 3721-3737Publisher
AMER MATHEMATICAL SOC
DOI: 10.1090/S0002-9947-04-03433-6
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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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