*-orderings on a Ring with Involution

The object of the paper is to extend part of the theory of-orderings on a skewweld with involution to a general ring with involution. The valuation associated to a-ordering is examined. Every-ordering is shown to extend.-orderings are shown to form a space of signs as deened by Brr ocker and Marshall. In case the involution is the identity, the ring under consideration is commutative and the-orderings are just the usual orderings making up the usual real spectrum of a commutative ring as deened by Coste and Roy. On a skewweld with involution, various sorts of orderings have been considered, e.g., Baer orderings 2]. More recently c-orderings and-orderings (also called Jordan orderings) are studied by Chacron, Craven, Holland, Idris, Leung, Wadsworth and others. The survey article by Craven 7] is a useful reference in this regard. In this work, there is a nice connection with valuations and a nice application to Hermitian forms. Of course, much of this work is a generalization of well-known results for commutative elds, e.g., see 14] for a survey. In 11], Handelman looks at the partial ordering determined by the sums of norms and the corresponding subring of bounded elements in a ring with involution. Handelman's work seems to be more or less disjoint from what is done here (although, of course, there are connections). Some of the prettiest results concerning the real spectrum of a commutative ring are the results (by Brr ocker and Scheiderer initially) on minimal generation of constructible sets. Later, abstract spaces of signs were introduced to axiomatize parts of the theory (including the results on minimal generation) 1] 4] 18]. Still later, orderings on a noncommutative ring were also shown to form a spaces of signs 16] 19]. In the present paper, we generalize the notions of Baer ordering and-ordering to a ring A with involution. We show that the natural-valuation associated to a-ordering is well-behaved and use this to show that-orderings extend (as in the skewweld case). We establish an appropriate version of the Positivstellensatz and use this to prove that the-orderings on A form a space of signs (as in the commutative case). Of course, this means that the results on minimal generation of constructible sets carry over as well. The proofs are non-trivial. We conclude by considering some examples, including C-algebras.