  In recent years, much work has been devoted to a systematic study of
polynomial identities certifying strict or non-strict positivity of a
polynomial on a basic closed semialgebraic set. The interest in such identities
originates not least from their importance in polynomial optimization. The
majority of the important results requires the archimedean condition, which
implies that the semialgebraic set has to be compact. This paper introduces the
technique of pure states into commutative algebra. We show that this technique
allows an approach to most of the recent archimedean Stellensaetze that is
considerably easier and more conceptual than the previous proofs. In
particular, we reprove and strengthen some of the most important results from
the last years. In addition, we establish several such results which are
entirely new. They are the first that allow the polynomial to have arbitrary,
not necessarily discrete, zeros on the semialgebraic set.
