Bounded variation and tensor products of Banach lattices
type: Item
instance of: scholarly article
title: Bounded variation and tensor products of Banach lattices
zbMATH Open document ID: 1037.46062
DOI: 10.1023/A:1025898718431
published in: Positivity
publication date: 2003-11-10T00:00:00Z
review text: The main result of this paper shows that for each bilinear map \(T\) from the product of normed Riesz spaces \(E\) and \(F\) into a Banach lattice \(G\) which is of bounded variation, i.e., \[ \begin{aligned} \text{Var\,}T:=&\sup\Biggl\{ \biggl\| \sum_{n,m}| T(x_n,y_m)| \biggl\|\;:\;x_1,\ldots x_N\in E^+,\;\Bigl\| \sum x_n \Bigr\| \leq 1,\\ &y_1,\ldots y_M\in F^+, \;\Bigl\| \sum y_m\Bigr\| \leq 1\Biggr\}<\infty, \end{aligned} \] there is a unique continuous linear map \(T^\otimes\) from \(E\overline{\otimes} F\), the tensor product of \(E\) and \(F\) in the sense of \textit{D. H. Fremlin} [Am. J. Math. 94, 778--798 (1972; Zbl 0272.46050)], into \(G\) such that \(T(x,y)=T^\otimes(x\otimes y)\) for all \(x\in E\), \(y\in F\). Moreover, \(T^\otimes\geq 0\) if and only if \(T\) is positive and the norm variation of \(T^\otimes\), that is \(\text{Var} (T^\otimes): \sup\{\| \sum_{n,m}| T^\otimes (x_n)|\,\| :x_1,\ldots x_N\in E^+\), \(\| \sum x_n\| \leq 1\}\) is equal to \(\text{Var\,}T\). An example of a bilinear map on \({\mathbb R}^2\times {\mathbb R}^2\) shows that the operator norm of \(T\) may be different from that of \(T^\otimes\). In the case where \(G\) is only a Banach space, it is shown that for every bilinear map on \(E\times F\) which is of finite semivariation, i.e., \[ \begin{aligned} ||| T||| := &\sup \Biggl\{ \biggl\| \sum_{n,m} \varepsilon_{nm} T(x_n,y_m)\biggr\| :x_1,\ldots x_N\in E^+,\Bigl\| \sum x_n\Bigr\| \leq 1,\\ & y_1,\ldots y_M\in F^+, \Bigl\| \sum y_m\Bigr\| \leq 1,\;\varepsilon_{n,m}= \pm 1\Biggr\}<\infty, \end{aligned} \] there is a unique continuous \(T^\otimes\) from \(E\overline{\otimes} F\) into \(G\) such that \(T(x,y)=T^\otimes(x\otimes y)\) with the operator norm of \(T^\otimes\) equal to \(||| T||| \). Results are also obtained for bilinear maps on Archimedean Riesz spaces with values in a Dedekind complete Archmedean Riesz space.
reviewed by: Christopher Boyd
Mathematics Subject Classification ID: 46M05
Mathematics Subject Classification ID: 46B40
Mathematics Subject Classification ID: 46B42
zbMATH DE Number: 2002003
zbMATH Keywords: tensor product
zbMATH Keywords: Banach lattice
author: Gerard Buskes
author: A. C. M. van Rooij
MaRDI profile type: Publication
full work available at URL: a:1025898718431
OpenAlex ID: W124242031
description: scientific article; zbMATH DE number 2002003