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

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 $\mathscr{L}<em>\infty$-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 $\mathscr{L}</em>\infty$-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.

• The paper establishes a criterion showing that any L∞-space with an infinite-dimensional Schur quotient must admit a c₀ quotient.
• It employs the Lopez-Abad construction to provide an explicit bounded surjection from nonseparable L∞-spaces to c₀ under coordinate embedding assumptions.
• The work clarifies the technical necessities and structural differences in quotient constructions, advancing our understanding of nonseparable Banach space geometry.

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

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Introduction and Problem Significance

Mazur’s separable quotient problem, dating to 1932, asks whether every infinite-dimensional Banach space admits an infinite-dimensional separable quotient. While positive resolutions are known for classes such as reflexive, weakly compactly generated, and dual spaces, the general status of Mazur’s problem remains open, particularly in the context of nonseparable spaces without the structural properties that facilitate direct quotient constructions.

Nonseparable Bourgain-Pisier $\mathscr{L}_\infty$-spaces, especially those generated by the Lopez-Abad extension of the original Bourgain-Pisier method, present a paradigmatic case for the investigation of Mazur’s problem. These spaces—constructed for arbitrary uncountable densities—feature nonseparable quotients with the Schur and Radon-Nikodym properties, placing them outside previously settled classes and presenting substantial technical difficulty for separable quotient construction.

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Results: General Criterion and Explicit Realization

General Existence Theorem

The paper establishes a general criterion for a large class of $\mathscr{L}_\infty$-spaces:

Theorem: Let $Y$ be an $\mathscr{L}_\infty$-space containing a subspace $X$ such that $Y/X$ is infinite-dimensional with the Schur property. Then $Y$ admits $c_0$ as a quotient.

This result is direct in its scope: any $\mathscr{L}_\infty$-space possessing an infinite-dimensional Schur quotient must admit $\mathscr{L}_\infty$0 as a quotient. This argument is notably non-constructive and circumvents the lack of explicit FDD, standard coordinate functionals, and other compactness-based tools that are available in the separable regime.

The key analytic point is a contradiction arising from the fact that an infinite-dimensional Banach space cannot simultaneously possess the Schur and Grothendieck properties. Since every Grothendieck $\mathscr{L}_\infty$1-space lacks $\mathscr{L}_\infty$2 quotients, a Schur quotient blocks the possibility of $\mathscr{L}_\infty$3 being Grothendieck and compels the existence of a $\mathscr{L}_\infty$4 quotient.

Application to Lopez-Abad Bourgain-Pisier Spaces

Lopez-Abad’s generalized construction produces, over any infinite-dimensional base $\mathscr{L}_\infty$5 and arbitrary uncountable density $\mathscr{L}_\infty$6, an $\mathscr{L}_\infty$7-space $\mathscr{L}_\infty$8 such that $\mathscr{L}_\infty$9 is infinite-dimensional with both the Radon-Nikodym and Schur properties. Therefore, the general existence theorem applies immediately and affirmatively resolves Mazur’s separable quotient problem in this setting.

Explicit Construction Under Coordinate Embedding

The paper goes beyond existence, constructing, under a coordinate embedding assumption, an explicit bounded surjection $\mathscr{L}_\infty$0. For every sequence of singleton stages indexed by an uncountable set, coordinate functionals are defined, leveraged via an orthogonal extension process to yield uniformly bounded, disjointly supported functionals on $\mathscr{L}_\infty$1. The quotient map $\mathscr{L}_\infty$2 given by $\mathscr{L}_\infty$3 is surjective and admits $\mathscr{L}_\infty$4.

The kernel of $\mathscr{L}_\infty$5 is shown to be an $\mathscr{L}_\infty$6-space of density $\mathscr{L}_\infty$7—matching the ambient space—and the short exact sequence

$\mathscr{L}_\infty$8

is realized entirely within the $\mathscr{L}_\infty$9-framework.

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Necessity of Coordinate Embedding

The explicit construction of $Y$0 depends crucially on the coordinate embedding property at each stage of the Lopez-Abad construction. The coordinate embedding ensures (1) the well-definedness of coordinate functionals on the quotient, and (2) the $Y$1-equivalence required for iterative approximation and surjectivity.

It is proved that, for general selections of the gluing map $Y$2, the coordinate functional $Y$3 can fail to be well defined on the pushout quotient, even when choosing free directions outside the image of $Y$4. Additionally, the crucial upper bound required for the norm control of sums corresponding to $Y$5 basis elements breaks down without the coordinate embedding. Constructed counterexamples are furnished to demonstrate these failures.

Thus, the coordinate embedding is shown to be an essential technical substitute in the nonseparable regime for the FDD (finite-dimensional decomposition) structure inherent to separable Bourgain-Delbaen spaces.

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Kernel Structure and Related Properties

The kernel $Y$6 is analyzed systematically. Each finite-dimensional approximation $Y$7 is identified, up to constant $Y$8, with finite-dimensional $Y$9 spaces minus the coordinates corresponding to the quotient, yielding $\mathscr{L}_\infty$0, where $\mathscr{L}_\infty$1 counts quotient coordinates within the finite stage indexed by $\mathscr{L}_\infty$2.

The union of all $\mathscr{L}_\infty$3 is dense in $\mathscr{L}_\infty$4, and $\mathscr{L}_\infty$5 is itself an $\mathscr{L}_\infty$6-space of density $\mathscr{L}_\infty$7. The structure of the kernel is thus shown to be directly compatible with the operational structure of $\mathscr{L}_\infty$8, and the constant bounds are preserved.

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Implications and Future Directions

The results of the paper formally settle Mazur’s separable quotient problem within the rich and subtle class of nonseparable Bourgain-Pisier $\mathscr{L}_\infty$9-spaces constructed by the Lopez-Abad method. This both enlarges the known validity of Mazur’s conjecture and ties the resolution to precise quotient and embedding structure, anchored in detailed constructive and nonconstructive methodologies.

Implications include:

• Demonstration that explicit surjective mappings to separable quotients in $X$0-spaces of arbitrary uncountable density can be achieved by exploiting coordinate subspace embeddings within the inductive construction;
• Identification of intrinsic limitations and necessary assumptions for explicit quotient constructions, illuminating structural differences from the separable, Bourgain-Delbaen regime;
• Extension of techniques for decomposing nonseparable Banach spaces into well-controlled quotients and kernels within the $X$1-category.

Directions for further research include:

• Investigating whether all valid realizations of the Lopez-Abad construction are isomorphic as Banach spaces;
• Developing explicit quotient projections for more general (non-coordinate) realizations;
• Determining if the kernel structure found here is canonical among all quotient maps to $X$2 from these nonseparable $X$3-spaces.

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Conclusion

The paper delivers a comprehensive resolution to Mazur’s separable quotient problem for a broad new class of nonseparable Banach spaces using both general and explicit arguments. The analysis clarifies the precise technical conditions required for explicit separable quotient mappings and locates the structural features essential for such constructions, deepening understanding of the interplay between quotient properties, embedding techniques, and the geometry of nonseparable $X$4-spaces.

Reference: Mazur's Separable Quotient Problem for Nonseparable Bourgain-Pisier $X$5-Spaces (2604.11832)