Pre-Riesz spaces
type: Item
instance of: scholarly article
title: Pre-Riesz spaces
zbMATH Open document ID: 1430.46002
DOI: 10.1515/9783110476293
author: Anke Kalauch
author: O. W. van Gaans
published in: De Gruyter Expositions in Mathematics
publication date: 2018-09-26T00:00:00Z
review text: 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 pres…
Mathematics Subject Classification ID: 46-02
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 06F20
zbMATH DE Number: 6944287
zbMATH Keywords: pre-Riesz spaces
reviewed by: Mohamed Ali Toumi
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1515/9783110476293
OpenAlex ID: W2903069912
description: scientific article; zbMATH DE number 6944287