About the properties (V) and (R. D. P.) in injective tensor products (Q1275742)

A bounded linear operator \(T\) from a Banach space \(X\) into a Banach space \(Y\) is said to be unconditionally converging if \(\sum_n Tx_n\) is unconditionally convergent whenever \(\sum_n x_n\) is weakly unconditionally Cauchy. \(T\) is a Dunford-Pettis operator if \(\lim_n\langle Tx_n,b_n\rangle= 0\) for each weak null sequence \((x_n)_n\subset X\) and each weak null sequence \((b_n)_n\subset Y^*\). A Banach space \(X\) has the property (V) of Pełczyński (resp. the reciprocal Dunford-Pettis property (RDP)) if every unconditionally converging operator (resp. every Dunford-Pettis operator) from \(X\) into any Banach space \(Y\) is weakly compact. The author proves that the injective tensor product \(X\widehat\otimes_\varepsilon Y\) of an \({\mathcal L}_\infty\)-space \(X\) with the property (V) (resp. with the RDP property) and a reflexive Banach space \(Y\) again has property (V) (resp. the RDP property). The main tool is a representation of the dual of the injective tensor product as a space of \(Y^*\)-valued measures on the unit ball of the triple dual of the \({\mathcal L}_\infty\)-space \(X\), endowed with its weak\(^*\)-topology. In the result on the RDP property, it is enough that the Banach space \(Y\) does not contain a complemented copy of \(\ell_1\) and that it has an unconditional Schauder decomposition \((Y_n)_n\) in which every \(Y_n\) is reflexive.