Non-commutative Real Algebraic Geometry - Some Basic Concepts and First Ideas

We 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 $\ast$-algebras. A version of Stengle’s Positivstellensatz for $n \times n$ matrices of real polynomials is proved.