Complementably universal Banach spaces. II. (Q1039409)

A Banach space \(U\) is said to be \textit{complementably universal} for a class \({\mathcal B}\) of Banach spaces when every \(B \in {\mathcal B}\) is isomorphic to a complemented subspace of \(U\). The paper is a sequel to the authors' paper [``Complementably universal Banach spaces'', Stud. Math. 58, 91--97 (1976; Zbl 0341.46017)]. The authors prove two main results: (1) If \(U\) is a Banach space complementably universal for the family of all subspaces of \(c_0\) with the bounded approximation property, then \(U^*\) is non-separable. For that, they prove that no separable Banach space is complementably universal for the family \(\{X^*;\;X \subseteq c_0\}\), and use a result which is implicit in \textit{W. B. Johnson} and \textit{T. Oikhberg} [``Separable lifting property and extensions of local reflexivity'', Ill. J. Math. 45, No.~1, 123--137 (2001; Zbl 1004.46008)]: for every subspace \(X\) of \(c_0\), there is a subspace \(Y\) of \(c_0\) with an \(FDD\) such that \(X^*\) is isomorphic to a complemented subspace of \(Y^*\). (2) There is no separable Banach space \(X\) such that every compact operator factors through \(X\). This solves a problem open since the 1970s. The constructions use finite-dimensional blocks and a probabilistic lemma which allows to extract suitable submatrices of a given matrix, in order to get suitable orthogonal systems.