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

TL;DR

This paper proves that certain nonseparable $ ext{L}_ ext{infty}$-spaces constructed via Lopez-Abad methods have $c_0$ as a quotient, affirmatively resolving Mazur's separable quotient problem for these spaces.

Contribution

It establishes that nonseparable $ ext{L}_ ext{infty}$-spaces from Lopez-Abad construction admit $c_0$ as a quotient, providing a constructive resolution under specific assumptions.

Findings

Spaces in the Lopez-Abad class have $c_0$ as a quotient.

A constructive surjection $T: Y o c_0$ is provided under certain conditions.

Counterexamples show the assumptions are necessary.

Abstract

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 $\mathscr{L}_\infty$-space $Y$ containing a subspace $X$ such that $Y/X$ is infinite-dimensional with the Schur property admits $c_0$ as a quotient. The natural class to which this criterion applies is the nonseparable $\mathscr{L}_\infty$-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 $\mathscr{L}_{\infty,\lambda}$-space of density $\kappa$. We prove this assumption is necessary by explicit counterexample.