Closure properties of order continuous operators
type: Item
instance of: scholarly article
title: Closure properties of order continuous operators
zbMATH Open document ID: 0604.47023
author: David C. Carothers
published in: Real Analysis Exchange
publication date: 1986-01-01T00:00:00Z
review text: Let X be a compact Hausdorff space and let C(X) (or simply C) be the space of all real valued continuous functions of X. C'(X) and C''(X) (or C',C'' respectively) represent the first and second norm duals of C(X). C may be imbedded in C''. U is the set of limits in C'' of order convergent nets from C. If \(L^ r(E,F)\) and \(L^ c(E,F)\) represent respectively the spaces of regular and order continuous operators from Riesz space E to Riesz space F, then \(L^ r(C,C)\) and \(L^ r(C,U)\) may be imbedded in \(L^ c(C'',C'')\). We show that under this imbedding, both \(L^ r(C,C)\) and \(L^ r(C,U)\) are dense in the band which they generate in \(L^ c(C'',C'')\) in the topology defined (for fixed \(\mu \in C_+')\) by \[ \| T\|_{\mu}=<\mu,| T| 1>,\quad T\in L^ c(C'',C'') \] where 1 is the unit in C''. This is used to show that \(L^ r(C,C)\) and \(L^ r(C,U)\) are order dense in the band which they generate in \(L^ r(C'',C'')\). In addition, if X is a metric space, the band generated by \(L^ r(C,U)\) is all of \(L^ c(C'',C'')\).
Mathematics Subject Classification ID: 47B60
Mathematics Subject Classification ID: 46A40
Mathematics Subject Classification ID: 47L05
Mathematics Subject Classification ID: 46E05
zbMATH DE Number: 3975710
zbMATH Keywords: spaces of regular and order continuous operators
zbMATH Keywords: Riesz space
zbMATH Keywords: band
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 3975710