Disjointness preserving operators on complex Riesz spaces
type: Item
instance of: scholarly article
title: Disjointness preserving operators on complex Riesz spaces
zbMATH Open document ID: 0909.47029
DOI: 10.1023/A:1009746711470
published in: Positivity
publication date: 1999-04-08T00:00:00Z
review text: For complex Riesz spaces, an order bounded disjointness preserving operator \(T\) satisfies \(|Tz|=|T|z||\) for \(z\) in the domain. This result, due to M. Meyer, is proved here without using representation theory. The proof uses a functional calculus on uniformly complete Riesz spaces. In this setting, each complex Riesz space is viewed as the complexification of a real Riesz space. These techniques are then used to show that \(Tz= \text{sgn }T(|z|)\cdot|T|(z)\). This then leads to decomposition of \(T\) as the product of a Riesz homomorphism and an orthomorphism. This polar decomposition generalizes results of Meyer-Nieberg while relying on only elementary methods for the proof. Conditions implying the existence of the polar decomposition relate to the \(Z\)-extension property for principal ideals; a subspace \(L\) of a Riesz space \(E\) is said to have the \(Z\)-extension property if every orthomorphism in the center \(Z(L)\) of \(L\) has an extension to a central operator in \(Z(E)\). Further, \(\sigma\)-Dedekind completeness implies the \(Z\)-extension property for all principal ideals.
Mathematics Subject Classification ID: 47B65
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 1121763
zbMATH Keywords: complex Riesz spaces
zbMATH Keywords: order bounded disjointness preserving operator
zbMATH Keywords: functional calculus
zbMATH Keywords: product of a Riesz homomorphism and an orthomorphism
zbMATH Keywords: \(Z\)-extension property
zbMATH Keywords: \(\sigma\)-Dedekind completeness
author: Jacobus J. Grobler
author: Charles B. Huijsmans
reviewed by: William A. Feldman
MaRDI profile type: Publication
full work available at URL: a:1009746711470
OpenAlex ID: W2056914585
description: scientific article; zbMATH DE number 1121763