A note on the approximation of elements in the Riesz tensor product
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.
universal mapping property for positive bilinear mappings
scientific article; zbMATH DE number 4058218