Continuity of linear maps on \(\mathcal L_1\)-spaces
Let \(L(X,Y)\) be the space of all linear (but not necessarily continuous) maps between two Banach spaces \(X\) and \(Y\). Let \(L_D(X,Y)\) be those \(T \in L(X,Y)\) whose restriction to some dense linear subspace of \(X\) is continuous. Inspired by a question of Bogachev, Kirchheim and Schachermayer the authors prove that a separable Banach space \(X\) is an \(\mathcal{L}_1\)-space if and only if \(L(X,c_0) = L_D(X,c_0)\). The proof is based on their characterization of \(\mathcal{L}_1\)-spaces: A separable Banach space \(X\) is an \(\mathcal{L}_1\)-space if and only if \(\mathrm{Ext}(X,H) = 0\) for every subspace \(H\) of \(c_0\). The condition \(\mathrm{Ext}(X,H) = 0\) means that whenever \(H\) is a subspace of a Banach space \(E\) in such a way that \(E/H\) is isomorphic to \(X\) then \(H\) must be complemented in \(E\).
scientific article; zbMATH DE number 5962582
scientific article