Archimedean Quadratic Modules and Positivstellensätze

Let Q be a quadratic module of a unital \(*\)-algebra A. The set of bounded elements of A with respect to Q forms a \(*\)-subalgebra which carries a natural C\(^{*}\)-seminorm. If this \(*\)-algebra of bounded elements coincides with A, then Q is called Archimedean. In this case each Q-positive \(*\)- representation of A acts by bounded operators and the corresponding C\(^{*}\)-seminorm can be characterized in terms of the Q-positive representations. Two abstract Stellensatze for Archimedean quadratic modules and their applications give a glimpse into noncommutative real algebraic geometry. One application is a strict Positivstellensatz for the Weyl algebra. Finally, a theorem about the closedness of the cone of finite sums of hermitian squares in certain \(*\)-algebras is proved. For these \(*\)-algebras sums of hermitian squares are precisely those elements that are mapped into positive operators under all \(*\)-representations.