%0 Journal Article
%T The Procesi-Schacher conjecture and Hilbert's 17th problem for algebras with involution
%A Igor Klep
%A Thomas Unger
%J Mathematics
%D 2008
%I arXiv
%R 10.1016/j.jalgebra.2010.03.022
%X 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.
%U http://arxiv.org/abs/0810.5254v2