A note on the approximation of elements in the Riesz tensor product
type: Item
instance of: scholarly article
title: A note on the approximation of elements in the Riesz tensor product
zbMATH Open document ID: 0648.46062
DOI: 10.1007/BF01160682
published in: Mathematische Zeitschrift
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/174052
review text: Let E and F be Archimedean Riesz spaces (vector lattices). A bilinear map \(\psi\) from \(E\times F\) to a Riesz space G is called positive if \(\psi(x,y)\in G_ +\) for all \((x,y)\in E_+\times F_+\) and it is called a Riesz bimorphism whenever \(| \psi(x,y)| =\psi (| x|,| y|)\) for all \((x,y)\in E\times F\). A pair \((R_{E,F},\sigma)\), with \(R_{E,F}\) an Archimedean Riesz space and \(\sigma: E\times F\to R_{E,F}\) a Riesz bimorphism is called a Riesz tensor product of E and F if the linearization \(\sigma ^{\ell}: E\otimes F\to R_{E,F}\) is injective and if, for every Riesz space G and every Riesz bimorphism \(\phi: E\times F\to G\) there exists a unique Riesz homomorphism \(\tau: R_{E,F}\to G\) such that \(\tau\circ\sigma= \phi.\) We show that every positive element in the Riesz tensor product \(R_{E,F}\) can be approximated relatively uniformly by elements of the projective cone \[ E_+\otimes F_+:= \{\sum ^{n}_{i=1}x_ i\otimes y_ i:(x_ i,y_ i)\in E_+\times F_+,\quad n\in {\mathbb{N}}\}. \] This improves a theorem of D. H. Fremlin. As an application an easy proof is given of the Riesz tensor product's universal mapping property for positive bilinear mappings.
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46A40
zbMATH DE Number: 4058218
zbMATH Keywords: Archimedean Riesz spaces
zbMATH Keywords: vector lattices
zbMATH Keywords: bilinear map
zbMATH Keywords: Riesz bimorphism
zbMATH Keywords: Riesz tensor product
zbMATH Keywords: projective cone
zbMATH Keywords: universal mapping property for positive bilinear mappings
author: Jacobus J. Grobler
author: Coenraad C. A. Labuschagne
MaRDI profile type: Publication
cites work: Tensor Products of Archimedean Vector Lattices
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cites work: Q4049667
cites work: Q3957439
cites work: Q3669960
OpenAlex ID: W1976997710
description: scientific article; zbMATH DE number 4058218