Embedding ∗-ordered domains into skew fields

We study the problem of embedding domains with ∗-orderings into skew fields. Assuming that the natural valuation associated to a ∗-ordered domain satisfies an Ore-type condition, we prove that the domain embeds in an order-preserving way into a ∗-ordered skew field. We call this the ∗-ordered version of the Dauns embedding theorem for domains with valuations. A number of concrete examples, where this result can be applied, is given. Moreover, a question of Murray Marshall regarding ∗-orderable groups is answered in the affirmative.