On the structure of separable \(\mathcal{L}_\infty\)-spaces (Q2827910)

In [Acta Math. 145, 155--176 (1980; Zbl 0466.46024)], \textit{J. Bourgain} and \textit{F. Delbaen} constructed examples of infinite-dimensional separable \(\mathcal{L}_\infty\)-spaces that do not contain copies of \(c_0\). Based on the Bourgain-Delbaen construction, the authors introduce and study here a general method to construct separable \(\mathcal{L}_\infty\)-spaces. They show that every infinite-dimensional separable \(\mathcal{L}_\infty\)-space can be obtained using this method, and that each of these spaces \(X\) contains an \(\mathcal{L}_\infty\)-subspace \(M\) with \(X/M\) isomorphic to \(c_0\). They also construct an isomorphic \(\ell_1\)-predual which is asymptotic \(c_0\) and does not contain copies of \(c_0\), answering a question of B.\ Sari. The authors think of this last example as a step towards answering the following problem formulated by \textit{G. Godefroy} and \textit{D. Li} [Math. Scand. 66, No. 2, 249--263 (1990; Zbl 0687.46010)]: Is \(c_0\) the only isomorphic predual of \(\ell_1\) that satisfies property (u)?