  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 $*$-algebras. A version of Stengle's Positivstellensatz for $n
\times n$ matrices of real polynomials is proved.
