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.