Pełczyński's property (V) on positive tensor products of Banach lattices (Q6614515)

Recall that a Banach space \(X\) has Pełczyński's property (V) whenever a set \(A\subset B_{X^*}\) is relatively weakly compact if, for each wuC series $\sum_n x_n$, \(\sup\{|x^*(x_n)|:x^*\in A\}\rightarrow0\) as \(n\rightarrow0\). In the Banach lattice context, this property has been useful for instance in the study the Grothendieck property. In the present paper the authors show that if \(E\) is an atomic reflexive Banach lattice and \(X\) is a Banach lattice, then the \textit{positive injective} tensor product \(E\breve{\otimes}_{|\varepsilon|}X\) has Pełczyński's property (V) if and only if so does \(X\). A similar result is also proved for the \textit{positive projective} tensor product.