U-algebras and the Hahn–Banach extension theorem

Abstract

SynopsisAn Archimedean unital f-algebra A is called a U-algebra if, for every a∊A, there exists an invertible element u∊A such that a = u |a|. Characterisations of a U-algebra are established. As an application, an extension theorem of Hahn–Banach type on modules over a U-algebra and over the complexification of a Dedekind complete unital f-algebra is given.

Feebly projectable ℓ-groups

“…Notice that the property of being a Bézout ring (that is, every finitely generated ideal is principal) is stronger than being normal, and for some classes of rings (e.g., uniformly complete frings), the two notions coincide (see [14]). Our next result generalizes parts of Proposition 3.1 of [20]. Condition (b) below is new.…”

Order-coherent Archimedean f -algebras

“…[3,Definition 5.1] Lavrič defined U-algebras ([10, Section 3]): An Archimedean unital f -algebra A is called a U-algebra if for every a ∈ A there exists an invertible element u ∈ A with a = u|a|. Proposition 3.1 and Theorem 3.9 in [10] contain a list of seven equivalent conditions. We shall only need the following three:…”