On \(\mathcal L_{\infty}\)-envelopes of Banach spaces
Given a reasonable class of Banach spaces \({\mathcal A}\), and an arbitrary Banach space \(X\), can one always find an \({\mathcal A}\)-space \(Y\) containing \(X\), so that every operator from \(X\) into an \({\mathcal A}\)-space admits a norm preserving extension to \(Y\)? This is well known for the class of \(L_\infty(\mu)\) spaces. Here it is shown that the conclusion holds for the most interesting classes of \({\mathcal L}_\infty\) spaces. This is established by devising a flexible construction, which not only unifies several known cases (\(C(K)\) spaces, Lindenstrauss spaces, \({\mathcal L}_{\infty,\lambda}\) spaces), but also yields some new cases (\(\lambda\)-separably-injective spaces, universally \(\lambda\)-separably-injective spaces, \(\lambda\)-Lindenstrauss-Pełczyński spaces).
scientific article; zbMATH DE number 6062809
scientific article