Hermitian Forms over Ordered * -Fields

LetDbe a division ring with an involution. Assuming thatDadmits Baer orderings, we can study the Witt group of Hermitian forms overDby observing its image in the ring of continuous functions on the space of orderings. We are led to define a new class of rings which, when viewed in an abstract setting, provide a natural generalization of the spaces of orderings and real spectra studied in real algebraic geometry.