  Given a quadratic module, we construct its universal C*-algebra, and then use
methods and notions from the theory of C*-algebras to study the quadratic
module. We define residually finite-dimensional quadratic modules, and
characterize them in various ways, in particular via a Positivstellensatz. We
give unified proofs for several existing strong Positivstellens\"atze, and
prove some new ones. Our approach also leads naturally to interesting new
examples in free convexity. We show that the usual notion of a free convex hull
is not able to detect residual finite-dimensionality. We thus propose a new
notion of free convexity, which is coordinate-free. We characterize
semialgebraicity of free convex hulls of semialgebraic sets, and show that they
are not always semialgebraic, even at scalar level. This also shows that the
membership problem for quadratic modules has a negative answer in the
non-commutative setup.
