Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-Spaces
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866917404155576320 |
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| author | Patri, Kartik |
| author_facet | Patri, Kartik |
| contents | 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 $\mathscr{L}_{\infty,λ}$-space of density $κ$. We prove this assumption is necessary by explicit counterexample. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_11832 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-Spaces Patri, Kartik Functional Analysis 46B26 (Primary), 46B03, 46B25 (Secondary) 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 $\mathscr{L}_{\infty,λ}$-space of density $κ$. We prove this assumption is necessary by explicit counterexample. |
| title | Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-Spaces |
| topic | Functional Analysis 46B26 (Primary), 46B03, 46B25 (Secondary) |
| url | https://arxiv.org/abs/2604.11832 |