Ideals and bands in principal modules
The main purpose of the paper is to present abstract versions in the framework of the theory of principal modules of some results of \textit{B. de Pagter} [Indag. Math. 45, 229-241 (1983; Zbl 0521.47018)] and \textit{C. D. Aliprantis} and \textit{O. Burkinshaw} [Math. Z. 184, 245-257 (1983; Zbl 0503.47035)]. For the reader's convenience, the paper begins with a self-contained introduction to the author's theory of principal modules; thus, the section reproduces some main results of the theory published by the author in some earlier papers. Recall that a principal module is a topological Riesz space E which is a module over an f-algebra A with a strong order unit such that the submodule generated by an arbitrary \(x\in E\) is dense in the principal order ideal generated by x. Among the results in the section we mention: a) The characterization of principal modules: an Archimedean Riesz space E endowed with a locally solid topology and with a structure of A-module is principal iff for every \(x_ 1,x_ 2\in E\) with \(x_ 1\wedge x_ 2=0\) and every neighborhood W of O and E there are \(a_ 1,a_ 2\in A\) with \(a_ 1\wedge a_ 2=0\) and \(x_ i-a_ ix_ i\in W\) \((i=1,2);\) b) The permanence theorem (proved in a preprint from 1981 and stated (without proof) in some papers from 1983-1984): let E be a principal A- module and let F be a principal B-module whose topology is order continuous. Then \(L'_ r(E,F)\) is a principal A\({\bar \otimes}B\)-module for the solid strong topology. The next section presents monotone approximation theorems in principal modules. For instance, it is proved that in a principal module E whose topology is separated and Fatou, we have \([-| x|,| x|]=([-e,e]x)^{0\omega \omega}\) for any \(x\in E\) (e denotes the unit of A and \(M^ 0\) (respectively \(M^{\omega})\) stands for the set of limits of order convergent sequences (respectively nets) in M). As a particular case, de Pagter's result on the monotone approximation of the components of a positive operator by its elementary components is reobtained; also, combining the above monotone approximation theorem with the permanence theorem, an extension of Aliprantis' and Burkinshaw's result of the relation between the order ideal and the closed algebraic ideal generated by a positive operator is proved (for instance, non positive regular operators are considered). The paper continues by giving a characterization of the elements in the band generated by an element in a principal module; as an application, a new proof of the Radon-Nikodym theorem is obtained. In the final section, a formula for computing the component of J(y) in the band generated by J(x) \((x,y\in E_+)\) is presented; here J denotes the canonical map from a principal module E into its second dual. The formula so-obtained provides a principal module-theoretic version for Aliprantis' and Burkinshaw's formula for the order projection on the band generated by a positive operator.
scientific article; zbMATH DE number 3929766
scientific article