DP1 and completely continuous operators (Q1415090)

Summary: \textit{W. Freedman} [Stud. Math. 125, 143--159 (1997; Zbl 0897.46007)] introduced an alternate to the Dunford-Pettis property, called the DP1 property. He showed that for \(1\leq p \leq \infty\), \((\bigoplus_{\alpha\in\mathcal{A}} X_{\alpha})_{p}\) has the DP1 property if and only if each \(X_\alpha\) does. This is not the case for \((\bigoplus_{\alpha\in\mathcal{A}} X_\alpha)_\infty\). In fact, we show that \((\bigoplus_{\alpha\in\mathcal{A}} X_\alpha)_\infty\) has the DP1 property if and only if it has the Dunford-Pettis property. A similar result also holds for vector-valued continuous function spaces.