When the Range of Every Orthomorphism is an Order Ideal
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When the Range of Every Orthomorphism is an Order Ideal Mohamed Ali Toumi1 Received: 19 February 2019 / Revised: 20 February 2020 © Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020
Abstract In this paper, it is proven that a uniformly complete vector lattice A is normal if and only if the range of every extended orthomorphism in A is an order ideal of A. As an application, it is shown that a vector lattice A is hyper-Archimedean if and only if the range of every extended orthomorphism in A is a uniformly closed order ideal of A. Moreover, a complete description of Archimedean f -algebras with unit elements such that the range of every orthomorphism is an order ideal is given in terms of order, algebraic and topological properties. Keywords Hyper-Archimedean vector lattice · Uniformly complete · Normal · Order ideal · Orthomorphism · f -algebra Mathematics Subject Classification 06A65 · 06A70; Secondary 46A40
1 Introduction A vector lattice A is called normal if A = {u}d + {v}d whenever u ∧ v = 0 in A. Over the course of time, much attention has been paid to the study of normal vector lattices of the form C(X ). A highlight in this connection is the paper by Gillman and Henriksen [5], which includes, among others, the following remarkable result: (G–H) For X a topological Hausdorff space, the following statements are equivalent:
This paper is dedicated to the memory of my father and of professor Abdelmajid Triki. Communicated by See Keong Lee.
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Mohamed Ali Toumi [email protected] Laboratoire d’Ingénierie Mathématique LIM, Ecole Polytechnique de Tunisie, Département de Mathématiques, Faculté des Sciences de Bizerte, Université de Carthage, Carthage, Tunisia
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M. A. Toumi
(i) Every finitely generated algebra ideal is a principal algebra ideal. (ii) Every algebra ideal is an order ideal. (iii) C(X ) is a normal vector lattice. Gillman and Henriksen’s paper aroused interest of a number of authors; see [6–9]. Huijsmans and de Pagter [8] (respect. de Pagter [4]) generalized the result (G– H)) to the more general case of uniformly complete f -algebras with unit elements. Particularly, they proved that for a uniformly complete f -algebra A with unit element every algebra ideal is an order ideal if and only if A is normal. Furthermore, they proved another result, as interesting as the previous one, which states what follows (see [8, Theorem 5.2]): The range of every orthomorphism in a uniformly complete normal vector lattice is an order ideal. Moreover, they give an example indicating that the condition of uniform completeness cannot be dropped. Motivated by Huijsmans and de Pagter’s paper [8] and by the fact that exist Archimedean vector lattices that are neither unital f -algebras nor uniformly complete, we will consider the following problems: • Is it possible to give a new characterization of a uniformly complete normal vector lattice A (which would comprise the converse of the above-cited [8, Theorem 5.2]) by studying the range of every orthomorphism in A? • Fin
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