Abstract
Let C be an ordered algebra with a unit e. The class of orthomorphism elements Orthe C of C was introduced and studied by Alekhno in ”The order continuity in ordered algebras”. If C = L G , where G is a Dedekind complete Riesz space, this class coincides with the band Orth G of all orthomorphism operators on G. In this study, the properties of orthomorphism elements similar to properties of orthomorphism operators are obtained. Firstly, it is shown that if C is an ordered algebra such that Cr , the set of all regular elements of C , is a Riesz space with the principal projection property and Orthe C is topologically full with respect to Ie , then Be = Orthe C holds, where Be is the band generated by e in Cr . Then, under the same hypotheses, it is obtained that Orthe C is an f -algebra with a unit e.
Highlights
All vector spaces are considered over the reals only
A Riesz algebra C is called an f -algebra if C has the additional property that a ∧ b = 0 implies ac ∧ b = ca ∧b = 0 for each c ∈ C+
An element a ∈ C is called a regular element if a = b − c with b and c positive, the space of all regular elements of C will be denoted by Cr
Summary
All vector spaces are considered over the reals only. An ordered vector space (Riesz space) C under an associative multiplication is said to be an ordered algebra (Riesz algebra) whenever the multiplication makes C an algebra, and in addition it satisfies the following property: a, b ∈ C+ implies ab ∈ C+. If C = L(G) is taken, where G is a Dedekind complete Riesz space, the set of all order idempotents OI(C) of C is the set of all order projections on G [3, Theorem 3.10] and the band Be generated by e in Cr is equal to Orth(G) = Orthe(C) [3, Theorem 8.11] . Proposition 2.1 Let C be an ordered algebra such that Cr is a Riesz space.
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