  In this paper we prove a strong Hahn-Banach theorem: separation of disjoint
convex sets by linear forms is possible without any further conditions, if the
target field $\R$ is replaced by a more general real closed extension field.
From this we deduce a general Positivstellensatz for *-algebras, involving
representations over real closed fields. We investigate the class of group
algebras in more detail. We show that the cone of sums of squares in the
augmentation ideal has an interior point if and only if the first cohomology
vanishes. For groups with Kazhdan's property (T) the result can be strengthened
to interior points in the $\ell^1$-metric. We finally reprove some strong
Positivstellens\"atze by Helton and Schm\"udgen, using our separation method.
