The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution

Details The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution

2008-10-29

arxiv.org/abs/0810.5254v2

J. Algebra 324 (2010), no. 2, 256-268

Mathematics

Rings and Algebras

Final pre-publication version

Scientific paper

10.1016/j.jalgebra.2010.03.022

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.

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