Hereditary properties for duals of Bochner \(L_p\)-function-spaces (Q1288332)

It is proved that the space \(Y=L_p^{w^*}(\Omega,X^*)\) \((1<p<\infty\), and \(\Omega\) a finite measure space) inherits certain geometric properties of \(X^*\). In particular, \(\ell_1\subset Y\) if and only if \(\ell_1\subset X^*\), and \(Y\) has the so-called complete continuity property (CCP) if and only if \(X^*\) has the CCP. Essentially the same method of proof is used to show also that \(L_p(\Omega,X)\) contains \(\ell_1\) resp.\ has the CCP if and only if \(X\) contains \(\ell_1\) resp.\ has the CCP. The latter has been obtained in [\textit{G. Pisier}, C. R. Acad. Sci., Paris, Sér. A 286, 747-749 (1978; Zbl 0373.46033) resp. \textit{N. Randrianantoanina} and \textit{E. Saab}, Proc. Am. Math. Soc 117, 1109-1114 (1993; Zbl 0812.46026)] by different means.