Some properties of the vector-valued Banach ideal space \(E(X)\) derived from those of \(E\) and \(X\) (Q685933)

Let \((\Omega,\Sigma,\mu)\) be an arbitrary measure space, \(E\) a Banach ideal space (Köthe function space) on \(\Omega\), \(X\) a Banach space, and \(E(X)\) the ``vector-valued Banach ideal space'' composed of \(E\) and \(X\). By a general method based on semi-embeddings, it is proved for certain properties \(P\) of Banach spaces that if \(E\) and \(X\) have \(P\) then \(E(X)\) has \(P\) as well \((*)\). Examples for \(P\) are the analytic Radon-Nikodým property and the property not to contain \(c_ 0\), thus simplifying results of Bukhvalov. \((*)\) is also true for the type \(\Pi\)-\(\Lambda\)- Radon-Nikodým property and the separable complementation property, and somewhat weaker versions of \((*)\) hold for the type \(I\)-\(\Lambda\)-Radon- Nikodým property, the property \((P)\) of Costé and Lust-Piquard, and for the near Radon-Nikodým property.