Real Algebraic and Analytic Geometry
Submission: 2008, November 18.
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.
Mathematics Subject Classification (2000): 11E25, 13J30, 16W10, 16R50.
Keywords and Phrases: central simple algebra, involution, quadratic form, ordering, trace, noncommutative polynomial.
Full text, 12p.: dvi 62k, ps.gz 63k, pdf 103k.