On subspaces and quotients of Banach spaces \(C(K,X)\)
Let \(C(K,X)\) denote the continuous vector-valued functions on a dispersed compact set \(K\). It is proved that every infinite-dimensional closed subspace of \(C(K,X)\) that is totally incomparable to \(X\) contains a complemented subspace isomorphic to \(c\). It is also shown that complemented subspaces that fail to contain \(c\) are isomorphic to a complemented subspace of a direct sum of finitely many copies of \(X\), and that the same is true with ``complemented'' replaced by ``quotient''. Other geometric properties of these spaces \(C(K,X)\) are also given.
scientific article; zbMATH DE number 1798909