Bounded variation and tensor products of Banach lattices
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.
scientific article; zbMATH DE number 2002003