On ideals generated by positive operators
type: Item
instance of: scholarly article
title: On ideals generated by positive operators
zbMATH Open document ID: 1059.47040
DOI: 10.1023/A:1025884519832
published in: Positivity
publication date: 2003-11-10T00:00:00Z
review text: Let \(E\) be a Riesz space with separating order dual and \(L_{b}(E)\) be the space of all order bounded operators. Let \(E^{\sim}\) denote the order dual of \(E\), let \(Z(E)\) denote the ideal center of \(E\) and \(\text{Orth}(E)\) denote the space of all orthomorphisms of \(E\). Let \(A_{T}\) and \(B_{T}\) denote the ideal generated by \(T\) in \(L_{b}(E)\) and the band generated by \(T\), respectively. A Riesz space \(E\) is said to have topologically full center if for each pair \(x,y\) in \(E\) with \(0 \leq y \leq x\), there exists a net \((\pi_{\alpha})\) in \(Z(E)\) with \(0 \leq {\pi}_{\alpha} \leq I\) for each \(\alpha\), such that \({\pi}_{\alpha}x \rightarrow y\) in \(\sigma(E,E^{\sim})\). In the paper under review, the authors study the algebraic structure of \(A_{T},B_{T}\) and their commutants \(\{A_{T}\}^{c}\) and \(\{B_{T}\}^{c}\). The bilinear map \(\phi: E \times E^{\sim} \rightarrow A_{T}^{\sim}\) is defined by \(\phi(x,f)= \mu_{x,f}\), where \(\mu_{x,f}(a)=f(a(x))\) for \(a \in A_{T}\). Let \(E\) have topologically full center and \(0 < T \in L_{b}(E)\) such that \(A_{T}\) has the Riesz Decomposition Property (R.D.P.). Among other things, using the bi-lattice homomorphism of the bilinear mapping \(\phi: E \times E^{\sim} \rightarrow A_{T}^{\sim}\), the authors show that \(\{A_{T} \}^{c} \subseteq Orth(E)\).
reviewed by: Ömer Gök
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 2002013
zbMATH Keywords: Riesz spaces
zbMATH Keywords: topologically full center
zbMATH Keywords: orthomorphism
zbMATH Keywords: f-algebra
zbMATH Keywords: order bounded operators
zbMATH Keywords: positive operators
zbMATH Keywords: ideal center
author: Şafak Alpay
author: Ayşe Uyar
MaRDI profile type: Publication
full work available at URL: a:1025884519832
OpenAlex ID: W10058679
description: scientific article; zbMATH DE number 2002013