Weak *-Orderings on *-Fields

Abstract

Analogous to the notion of natural valuations of ordered fields, we introduce the notion of order *-valuations for any Baer ordered *-fields. When the Bear ordered division rings are finite dimensional over their centers, we show that their order *-valuations are nontrivial. Using this, we study a new generalization of *-orderings, namely, weak *-orderings. Unlike *-orderings, weak *-orderings do exist in Bear ordered *-fields odd dimensional over their centers. Moreover, we prove that if the involution is of the first kind, these *-fields must be either commutative fields or standard quaternion algebras. Whereas in case the involution is of the second kind, the dimension of these *-fields over their centers must be odd. This strong result also implies that the restriction of weak *-ordering on any commutative subfield consisting of symmetric elements only is in fact an ordering (not just a semiordering) is these *-fields. © 1993 Academic Press. All rights reserved.