2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/138555We propose and discuss how basic notions (quadratic modules, positive elements, semialgebraic sets, Archimedean orderings) and results (Positivstellensaetze) from real algebraic geometry can be generalized to noncommutative $*$-algebras. A version of Stengle's Positivstellensatz for $n \times n$ matrices of real polynomials is proved.Operator Algebras13J30, 47L60, 15A48 (Primary); 11E25 (Secondary)text