Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier ${\mathcal{L}}_{\mathrm{\infty }}$-Spaces

Mazur’s separable quotient problem, open since 1932, asks whether every infinite-dimensional Banach space admits an infinite-dimensional separable quotient. We prove that any ${\mathcal{L}}_{\mathrm{\infty }}$-space $Y$ containing a subspace $X$ such that $Y\mathrm{/}X$ is infinite-dimensional with the Schur property admits $c_0$ as a quotient. The natural class to which this criterion applies is the nonseparable ${\mathcal{L}}_{\mathrm{\infty }}$-spaces constructed via the Lopez-Abad extension method, the nonseparable analogue of the Bourgain–Delbaen spaces. For every space in this class, Mazur’s problem is thereby resolved affirmatively, for any valid realization of the construction and any base space. We further provide a constructive resolution under a coordinate embedding assumption via an explicit bounded surjection $T:Y\to c_0$ whose kernel is an ${\mathcal{L}}_{\mathrm{\infty },\lambda }$-space of density $\kappa$. We prove this assumption is necessary by explicit counterexample.