Schauder decompositions and the Fremlin projective tensor product of Banach lattices
When are certain properties of the Banach spaces \(X\) and \(Y\) inherited by tensor products of \(X\) and \(Y\)? Some of these questions turn out to be extremely difficult. As an example, it is very hard to prove that RNP is not inherited by \(X\widehat\bigotimes_\pi Y\). In this paper, the Fremlin projective tensor product \(X\widehat\bigotimes_F Y\) of two Banach lattices (which is a Banach lattice itself) is considered and it is proven that, if \(X\) is atomic, then \(X\widehat\bigotimes_F Y\) enjoys the RNP [is a KB-space] if and only if both \(X\) and \(Y\) have the RNP [are KB-spaces]. Quite a lot of machinery is built up, giving results interesting in themselves. For example, it is shown that, if the Banach lattice \(X\) has a lattice decomposition \((P_i)_{i=1}^\infty\) such that each \(P_i[X]\) has finite dimension, then the following are equivalent statements: (i) \(X\) is a \(\sigma\)-Levi space, (ii) \(X\) is a Levi space, (iii) \(X\) is a KB-space and (iv) \(X\) has the RNP. In a remark at the end of the paper, it is commented that the theorem concerning when the RNP is inherited by \(X\widehat\bigotimes_F Y\) is claimed in [\textit{D. Puglisi}, Quaest. Math. 30, No. 1, 45--56 (2007; Zbl 1139.46047)], but with a wrong proof. The error is pointed out. It is also asked whether it is necessary that one of \(X\) and \(Y\) be atomic for \(X\hat\otimes_F Y\) to inherit the RNP [being a KB-space].
scientific article; zbMATH DE number 5555255
scientific article