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arxiv: 2605.21711 · v1 · pith:TKDPMQ55new · submitted 2026-05-20 · 🧮 math.FA

Primariness and the Primary Factorisation Property

Antonio Acuaviva , Tomasz Kania This is my paper

Pith reviewed 2026-05-22 07:45 UTC · model grok-4.3

classification 🧮 math.FA
keywords primarinessprimary factorisation propertyuniform primary factorisationBanach spacesℓ_p spacesnon-separable spacessymmetric sums
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The pith

The Banach space B(ℓ_p) satisfies the uniform primary factorisation property for 1 < p < ∞.

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

This paper connects primariness of Banach spaces to the stronger primary factorisation property and its uniform version. It revisits classical primariness arguments to isolate the extra conditions needed for factoring the identity through arbitrary operators. Support-reduction and free-selection tools are developed for uncountable direct sums to transfer these properties from countable building blocks. The main result establishes a uniform primary factorisation theorem for B(ℓ_p) using its finite-block representation.

Core claim

Using the finite-block representation of B(ℓ_p), the identity operator on B(ℓ_p) factors uniformly through an arbitrary bounded operator on B(ℓ_p) for 1 < p < ∞.

What carries the argument

Finite-block representation of B(ℓ_p), which reduces general operators to actions on finite-dimensional blocks to achieve uniform factorisation.

Load-bearing premise

The support-reduction and free-selection tools for uncountable direct sums transfer primariness and PFP from countable building blocks to non-separable cases.

What would settle it

An explicit bounded operator on B(ℓ_p) through which the identity cannot factor with any uniform constant would disprove the uniform primary factorisation theorem.

read the original abstract

We study the relation between primariness of Banach spaces and the stronger operator-theoretic notions of the primary factorisation property (PFP) and the uniform primary factorisation property (UPFP). We revisit several classical primariness arguments and isolate the additional information needed to factor the identity through arbitrary operators. In the separable setting, this recovers quantitative factorisation versions of the Casazza--Kottman--Lin method for spaces with symmetric bases and treats the exceptional cases of $\ell_1$ and $\ell_\infty$. We then develop support-reduction and free-selection tools for uncountable direct sums, allowing one to transfer primariness and the PFP/UPFP from countable building blocks to non-separable $\ell_p$-, $c_0$- and more general symmetric sums. As applications, we obtain, among others, the primariness of $C[0,1]^*$ under the negation of the Continuum Hypothesis and UPFP results for uncountable sums of ordinal $C(\alpha)$-spaces. Finally, using the finite-block representation of $\mathcal B(\ell_p)$, we prove a uniform primary factorisation theorem for the Banach space $\mathcal B(\ell_p)$, $1<p<\infty$, and end with open problems concerning the gap between primariness and factorisation.

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

1 major / 2 minor

Summary. The paper studies the relation between primariness of Banach spaces and the stronger operator-theoretic properties PFP and UPFP. It revisits classical primariness arguments (including Casazza-Kottman-Lin) to extract quantitative factorisation information, recovers versions for symmetric bases in the separable setting (including exceptional cases ℓ1 and ℓ∞), develops support-reduction and free-selection tools for uncountable direct sums, and transfers primariness/PFP/UPFP from countable blocks to non-separable ℓp, c0 and symmetric sums. Applications include primariness of C[0,1]* under ¬CH, UPFP for uncountable sums of ordinal C(α)-spaces, and a uniform primary factorisation theorem for B(ℓp) (1<p<∞) via the finite-block representation; the paper closes with open problems on the gap between primariness and factorisation.

Significance. If the central claims hold, the work supplies explicit, self-contained tools that upgrade classical primariness results to uniform factorisation statements and extends them to non-separable settings. The direct constructions for support-reduction and free-selection in both countable and uncountable cases, together with the finite-block representation argument for B(ℓp), constitute concrete technical advances that can be checked independently and applied to other symmetric sums.

major comments (1)
  1. [Uncountable direct sums] Uncountable direct sums section: the transfer of PFP/UPFP via support-reduction and free-selection is presented as a direct construction, but the argument for preserving uniformity when passing to uncountable sums of symmetric blocks should include an explicit verification that the resulting factorisation constants remain independent of the index set (this is load-bearing for the UPFP claims on uncountable ordinal sums).
minor comments (2)
  1. [Introduction] Introduction: the quantitative versions recovered from the Casazza-Kottman-Lin method are stated without recalling the precise constants or block sizes used in the original argument; a short reminder would clarify the improvement.
  2. [Applications] Applications to C[0,1]*: the primariness result under ¬CH is stated cleanly, but the dependence on the specific model of set theory could be flagged more explicitly in the statement of the theorem.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their thorough reading, positive evaluation, and constructive suggestion. We address the single major comment below and will incorporate the requested clarification in the revised manuscript.

read point-by-point responses

  1. Referee: [Uncountable direct sums] Uncountable direct sums section: the transfer of PFP/UPFP via support-reduction and free-selection is presented as a direct construction, but the argument for preserving uniformity when passing to uncountable sums of symmetric blocks should include an explicit verification that the resulting factorisation constants remain independent of the index set (this is load-bearing for the UPFP claims on uncountable ordinal sums).

    Authors: We agree that an explicit verification of the index-set independence of the factorisation constants would improve the exposition and strengthen the UPFP claims. In the revised manuscript we will add a short dedicated paragraph immediately after the statement of the main transfer result in the Uncountable direct sums section. This paragraph will track the constants through the support-reduction and free-selection steps, confirming that they depend only on the uniform constants of the countable symmetric blocks and are therefore independent of the cardinality or structure of the index set. The argument will be self-contained and will not alter any existing proofs. revision: yes

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper develops support-reduction and free-selection tools explicitly for countable and uncountable direct sums, then applies them via direct constructions to transfer primariness and PFP/UPFP. The uniform primary factorisation theorem for B(ℓ_p) is obtained from the finite-block representation through independent technical steps rather than any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. The central claims remain self-contained against external benchmarks and prior literature such as the Casazza-Kottman-Lin method.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard set theory to invoke negation of the Continuum Hypothesis and on classical Banach space results; no free parameters or new postulated entities are introduced.

axioms (1)
  • standard math ZFC set theory with the negation of the Continuum Hypothesis
    Invoked explicitly for the primariness result on C[0,1]*.

pith-pipeline@v0.9.0 · 5759 in / 1284 out tokens · 55718 ms · 2026-05-22T07:45:45.128383+00:00 · methodology

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Reference graph

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