Pre-Riesz spaces
This book provides an excellent exposition of pre-Riesz spaces. A partially ordered vector space \((X,K)\) is called pre-Riesz space if, for every \(x,y,z\in A\), the inclusion \(\left\{ x+z,y+z\right\} ^{u}\subset\left\{ x,y\right\} ^{u}\) implies \(z\in K,\) where \(\left\{ x,y\right\} ^{u}=\left\{ a\in A,\,a\geq x\text{ and }a\geq y\right\} .\) The central theme is indeed pre-Riesz spaces, which in the present sense trace their origin to M. van Haandel. The material is divided into five chapters. The book begins with the fundamentals of partially ordered vector spaces. In Chapter 2, the authors present a construction of the Dedekind completion, which serves as a vehicle for vector lattice covers. The Fremlin tensor product is obtained as a Riesz completion as well as some properties of structures in pre-Riesz spaces. Chapter 3 elucidates seminorms on pre-Riesz spaces. Moreover, the authors give a brief survey of locally full and locally solid topologies, and they take a glance at unbounded order convergence in pre-Riesz spaces. Chapter 4 develops a systematic theory of disjointness, bands and ideals in pre-Riesz spaces and transfers well-known properties from vector lattice theory to pre-Riesz spaces as far as possible. The disjointness preserving and band preserving operators are studied in Chapter 5. Let \(X,\) \(Y\) be pre-Riesz spaces and let \(T:X\supseteq \mathcal{D}\left( T\right) \rightarrow Y\) be a linear operator. \(T\) is called disjointness preserving if, for every \(x,y\in\mathcal{D}\left( T\right) \), from \(x\perp y\) it follows that \(T\left( x\right) \perp T\left( y\right) .\) \(T\) is called band preserving if, for every band \(B\) in \(X\), one has \(T(B\cap\mathcal{D}\left( T\right) )\subseteq B.\) Moreover, some properties of disjointness preserving \(C_{0}\)-semi-groups are given. In addition, the authors address the question under which conditions the inverse of a disjointness preserving bijection on a pre-Riesz spaces is disjointness preserving. Finally, the authors give a few spectral properties of operators on ordered Banach spaces that are related to disjointness. The style of presentation is lively and conveys some of the authors' enthusiasm for the subject. The book should be read by everybody working in the theory of operators on vector lattices. Unfortunately, the book contains no exercises. Thus, a chance is lost to aid the reader in mastering the material.
scientific article; zbMATH DE number 6944287