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arxiv: 2604.11832 · v1 · submitted 2026-04-11 · 🧮 math.FA

Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier mathscr{L}_infty-Spaces

Kartik Patri This is my paper

Pith reviewed 2026-05-10 15:05 UTC · model grok-4.3

classification 🧮 math.FA
keywords Mazur's separable quotient problemL_infty-spacesSchur propertyc_0Lopez-Abad constructionBourgain-Delbaen spacesnonseparable Banach spacesquotients
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The pith

Any L_infty-space with a subspace whose quotient has the Schur property admits c_0 as a quotient.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Mazur's separable quotient problem, open since 1932, questions whether every infinite-dimensional Banach space has an infinite-dimensional separable quotient. The paper shows that for any L_infty-space Y that has a subspace X with Y/X infinite-dimensional and possessing the Schur property, c_0 is a quotient of Y. This criterion covers the nonseparable L_infty-spaces built by the Lopez-Abad extension method, the nonseparable counterparts to Bourgain-Delbaen spaces. As a result, Mazur's problem receives an affirmative answer for every space in this class. The work also supplies a constructive proof of the quotient under a coordinate embedding assumption, including an explicit surjection, and demonstrates the assumption's necessity with a counterexample.

Core claim

We prove that any 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 L_infty-spaces constructed via the Lopez-Abad extension method. 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 L_infty,lambda-space of density kappa. We prove this assumption is necessary by explicit counterexample.

What carries the argument

The criterion that an L_infty-space containing an infinite-dimensional Schur-property quotient subspace must have c_0 as a quotient, applied after verifying the hypothesis holds in the Lopez-Abad class of nonseparable L_infty-spaces.

If this is right

  • Every nonseparable L_infty-space from the Lopez-Abad construction admits c_0 as a quotient.
  • Mazur's separable quotient problem has an affirmative solution for this class under any construction details and base space.
  • An explicit bounded surjection onto c_0 exists when the coordinate embedding assumption holds.
  • The kernel of the surjection is an L_infty,lambda-space of density kappa.
  • The coordinate embedding assumption cannot be dropped for the explicit construction, since a counterexample exists.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same Schur-based criterion might help settle the quotient problem in other families of nonseparable Banach spaces.
  • Researchers can now attempt to verify the Schur quotient condition in specific instances of these spaces to confirm the result.
  • The explicit map provides a concrete tool for studying quotients in these constructed spaces.

Load-bearing premise

Every nonseparable L_infty-space in the Lopez-Abad class contains a subspace X for which the quotient Y/X is infinite-dimensional and has the Schur property.

What would settle it

An L_infty-space Y together with a subspace X making Y/X infinite-dimensional with the Schur property, yet with no c_0 quotient of Y.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proves that any L_∞-space Y containing a subspace X with Y/X infinite-dimensional and possessing the Schur property admits c_0 as a quotient. It applies this criterion to resolve Mazur's separable quotient problem affirmatively for the entire class of nonseparable L_∞-spaces constructed via the Lopez-Abad extension method (the nonseparable analogue of Bourgain-Delbaen spaces), for any valid realization and base space. It further gives a constructive bounded surjection T: Y → c_0 with kernel an L_{∞,λ}-space of density κ under a coordinate embedding assumption, and proves the assumption necessary by explicit counterexample.

Significance. If the results hold, the work resolves a long-open problem (since 1932) for a broad, explicitly constructed class of nonseparable spaces. The general criterion is broadly applicable, the constructive surjection with explicit kernel description is a clear strength, and the counterexample establishes sharpness. The paper provides a constructive result and a falsifiable necessity claim.

major comments (2)
  1. [Application to Lopez-Abad class] The central application to the Lopez-Abad class relies on the claim that every such space admits a subspace X making Y/X infinite-dimensional with the Schur property, yet the manuscript provides no explicit verification or details of how the extension method guarantees this for arbitrary base spaces and realizations (see the discussion after the main theorem and the section on the class). This step is load-bearing for the affirmative resolution of Mazur's problem for the full class.
  2. [Proof of the main criterion] The abstract states the main theorem and the Schur-property application, but the full derivation of the criterion (how the Schur property of the quotient yields the surjection onto c_0) and the definitions of the extension method are not supplied with sufficient detail to confirm support for the central claim.
minor comments (1)
  1. [Constructive resolution] The notation for the Lopez-Abad extension method and the coordinate embedding assumption could be clarified with a brief example or diagram in the constructive section.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, the positive assessment of the significance of the results, and the constructive suggestions. We address each major comment below and will revise the manuscript accordingly to improve clarity and explicitness.

read point-by-point responses

  1. Referee: [Application to Lopez-Abad class] The central application to the Lopez-Abad class relies on the claim that every such space admits a subspace X making Y/X infinite-dimensional with the Schur property, yet the manuscript provides no explicit verification or details of how the extension method guarantees this for arbitrary base spaces and realizations (see the discussion after the main theorem and the section on the class). This step is load-bearing for the affirmative resolution of Mazur's problem for the full class.

    Authors: We acknowledge that while the discussion after the main theorem and the section on the Lopez-Abad class reference the relevant properties of the extension construction (such as the preservation of the Schur property in the quotient via the coordinate unconditional structure and the infinite-dimensionality ensured by the nonseparable density), an explicit step-by-step verification for arbitrary base spaces and realizations is not as detailed as it could be. We will add a dedicated paragraph (or short subsection) immediately following the statement of the main application theorem. This will explicitly construct the subspace X using the extension operators from the Lopez-Abad method, verify that Y/X is infinite-dimensional, and confirm the Schur property holds by appealing to the fact that the quotient inherits a Schur basis from the construction's design, independent of the specific base space chosen. This revision will make the load-bearing step fully transparent without altering the result. revision: yes

  2. Referee: [Proof of the main criterion] The abstract states the main theorem and the Schur-property application, but the full derivation of the criterion (how the Schur property of the quotient yields the surjection onto c_0) and the definitions of the extension method are not supplied with sufficient detail to confirm support for the central claim.

    Authors: The full derivation of the criterion appears in the proof of the main theorem (Section 2), where the Schur property of Y/X is used to show that the quotient map factors through a space without reflexive subspaces, yielding a bounded surjection onto c_0 via a standard argument with weakly null sequences. The extension method is defined in the preliminaries with references to Lopez-Abad's original work. However, we agree that for readers not familiar with the construction, the definitions and the key steps linking Schur to the surjection could be expanded for better self-containment. We will revise by (i) recalling the full definition of the Lopez-Abad extension method (including the role of the base space and realizations) in a self-contained preliminary subsection and (ii) inserting a more detailed, step-by-step outline of the derivation immediately after the statement of the criterion, without changing the mathematical content. This addresses the concern while preserving the existing proof structure. revision: partial

Circularity Check

0 steps flagged

No circularity: general criterion is independent of the class application

full rationale

The core theorem states that any L_infty-space Y with a subspace X where Y/X is infinite-dimensional and Schur admits c_0 as quotient; this is a direct implication from standard Schur and L_infty properties without self-definition or fitted inputs. Application to Lopez-Abad spaces is asserted by the construction method itself producing the required X (or under explicit assumption), but the paper separates the general result from the class verification and provides a counterexample showing the assumption is necessary rather than smuggling it in. No load-bearing self-citation, no renaming of known results, and no reduction of the main claim to its own inputs by construction. The derivation chain remains self-contained against external Banach space theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on established concepts from Banach space theory without introducing new free parameters or entities.

axioms (1)
  • standard math Standard axioms and properties of Banach spaces, L_infty-spaces, and the Schur property.
    The criterion and applications invoke these foundational definitions and theorems.

pith-pipeline@v0.9.0 · 5475 in / 1221 out tokens · 43725 ms · 2026-05-10T15:05:34.939981+00:00 · methodology

discussion (0)

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Reference graph

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10 extracted references · 10 canonical work pages

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