Submission: 2008, Nov 17
In 1976 Procesi and Schacher developed an Artin-Schreier type theory for central simple algebras with involution and conjectured that in such an algebra a totally positive element is always a sum of hermitian squares. In this paper elementary counterexamples to this conjecture are constructed and cases are studied where the conjecture does hold. Also, a Positivstellensatz is established for noncommutative polynomials, positive semidefinite on all tuples of matrices of a fixed size.
2000 Mathematics Subject Classification: Primary 11E25, 13J30; Secondary 16W10, 16R50
Keywords and Phrases: central simple algebra, involution, quadratic form, ordering, trace, noncommutative polynomial
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