Types of Radon-Nikodým properties for the projective tensor product of Banach spaces
In this very interesting paper the authors show that the Radon-Nikodym property (RNP) and several of its variations, like the analytic Radon-Nikodým property and the near Radon-Nikodym property, are preserved under projective tensor products when one of the component spaces has an unconditional basis. Similar results also hold when one of the spaces has a boundedly complete basis (and hence has the RNP) and the other component has various of the Radon-Nikodym properties. Let \(U\) be a Banach space with a boundedly complete 1-unconditional normalized basis \(\{e_i\}_{i \geq 1}\). For a Banach space \(X\), let \(U(X) = \{\overline {x}= \{x_i\} : x_i \in X, \sum\| x_i\| e_i \text{ converges in }U\}\). This space equipped with the norm \(\| \sum\| x_i\| e_i\| \) is a Banach space. An important result proved in this paper is that the map \(\psi : U \widehat{\otimes} X \rightarrow U(X)\) defined by \(\psi(z) = ( \sum e_i^\ast(u_k)x_k)_i\) where \(\sum u_k\otimes x_k\) is a representation of \(z\), is a semi-embedding in the sense that \(\psi\) is injective and maps the closed unit ball to a closed set. To see a relation with the concepts considered earlier one notes that several of the Radon-Nikodým properties pass from the range space to the domain (separable) under a semi-embedding.
scientific article; zbMATH DE number 2028874