Rings with involution as partially ordered abelian groups

Abstract

Let (S, *) be a ring with involution *. The involution is positive definite if, for all finite subsets {r,-} of S, £ w * = 0 implies all the r{ are zero. Then the set of self-adjoint elements of S, denoted S, possesses a natural partial ordering, with positive cone consisting of elements of the form Z l w * ; with this ordering, S is a directed partially ordered abelian group. Let Sb denote the set of bounded elements, that is, the set of elements s such that ss* is less than an integral multiple of 1 in this ordering. Then Sb is a *-subalgebra of S whenever S is an algebra over the rationals. We will be studying the objects Sb9 S , and (Sb) , from the point of view of their ordered structures. For instance, suppose S is a field, and * is the identity. Then S is formally real, and Sb must be Prufer domain, all of whose residue fields are themselves formally real (and in fact, are embeddable in the reals). Viewing Sb as a partially ordered abelian group with order unit 1 (indeed, Sb is the convex subgroup of S generated by 1), Sb has the Riesz decomposition property, and its normalized extremal states are precisely the ring homomorphisms into the reals. There is a natural mapping from the collection of total orderings of S to the set of extremal states of Sb9 and this in turn maps to Spec Sb (the prime ideal space of Sb); when S is even a real algebra much more can be said. If either S is a field and * is not the identity, or S is a quaternionic division algebra with the natural involution, essentially the same properties hold, with the appropriate modifications. A useful tool here is an involutory version of the Artin Schreier Theorem, about the existence of sufficiently many total orderings finer than the natural ordering. Studies are made of several specific bounded subrings. For instance, if S is the rational function field in one variable over the reals, then Sb is a Dedekind domain with class group of order 2, with spectrum the unit circle (in the point-open topology), and all of its maximal ideals are not principal. Expanding the scope of S somewhat, we next allow S to be a division