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The spaces ℓ_p(C(K)) are primary for every compact metrizable space K and 1 ≤ p ≤ ∞.

2026-06-29 00:32 UTC pith:CMORSFN5

load-bearing objection The paper establishes primariness of ell_p(C(K)) for all compact metrizable K by proving uniform primary factorization in the ordinal and [0,1] cases plus an inheritance criterion for p=1.

arxiv 2605.29854 v1 pith:CMORSFN5 submitted 2026-05-28 math.FA

Primariness of the spaces ell_p(C(K)) for 1 leq p leq infty

Antonio Acuaviva This is my paper
classification math.FA
keywords primary Banach spacesC(K) spacesell_p sumsuniform primary factorizationmetrizable compactaBanach space decompositions
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper proves that ℓ_p(C(K)) is a primary Banach space for any compact metrizable K and any p in [1, ∞]. Primary means any isomorphism to a direct sum A ⊕ B forces A or B to be isomorphic to the whole space. The proof first establishes the uniform primary factorisation property for ℓ_p over continuous functions on ordinals and on [0,1] when p > 1. It then supplies a criterion that lets ℓ_1(X) inherit the property whenever X has it. These steps together imply the result for general metrizable compacts.

Core claim

We prove that the spaces ℓ_p(C(α)) and ℓ_p(C[0,1]) have the uniform primary factorisation property whenever α is an ordinal and 1<p≤∞. For the case p=1, we establish a general criterion ensuring that ℓ_1(X) inherits the uniform primary factorisation property from X. As a consequence, ℓ_p(C(K)) is primary for every compact metrizable space K and every 1 ≤ p ≤ ∞.

What carries the argument

uniform primary factorisation property, which permits uniform factoring of operators through primary subspaces

Load-bearing premise

The uniform primary factorization property holds for ℓ_p(C(α)) and ℓ_p(C[0,1]) when α is an ordinal and 1 < p ≤ ∞.

What would settle it

An explicit compact metrizable space K together with a p where ℓ_p(C(K)) is isomorphic to A ⊕ B but neither A nor B is isomorphic to ℓ_p(C(K)) would show the claim false.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 3 minor

Summary. The paper proves that ℓ_p(C(α)) and ℓ_p(C[0,1]) have the uniform primary factorization property for every ordinal α and 1<p≤∞, via explicit constructions. For p=1 it establishes a general inheritance criterion ensuring that ℓ_1(X) inherits the uniform primary factorization property from X. As a consequence, ℓ_p(C(K)) is primary for every compact metrizable K and every 1≤p≤∞, using the standard classification of separable C(K) spaces.

Significance. If the results hold, the work resolves the primariness question for the entire class of spaces ℓ_p(C(K)) with K compact metrizable, a broad and natural family that includes all separable C(K) spaces. The explicit constructions for the special cases ℓ_p(C(α)) and ℓ_p(C[0,1]), together with the inheritance criterion for p=1, provide concrete, checkable arguments rather than abstract existence proofs; the reduction step relies only on well-known facts about separable C(K) spaces. This constitutes a substantial advance in the study of primary Banach spaces.

minor comments (3)
  1. [§2] Notation for the uniform primary factorization property is introduced without an explicit numbered definition; adding a displayed definition in §2 would improve readability.
  2. [Theorem 3.2] The statement of the general criterion for p=1 (Theorem 3.2) would benefit from a short remark clarifying whether the hypothesis on X is inherited by all complemented subspaces or only by the specific X under consideration.
  3. [§5] Reference to the classification of separable C(K) spaces is given only by citation; a one-sentence reminder of the relevant theorem (e.g., that every separable C(K) is isomorphic to C(α) or C[0,1] for suitable α) would make the final reduction self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and encouraging report, including the clear summary of our results on the uniform primary factorization property and the inheritance criterion, as well as the recommendation to accept the manuscript. We are pleased that the explicit constructions and the reduction to the classification of separable C(K) spaces were viewed as providing concrete arguments for this substantial advance.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper proves the uniform primary factorization property for ℓ_p(C(α)) and ℓ_p(C[0,1]) via explicit constructions (for 1<p≤∞) and a general inheritance criterion from X to ℓ_1(X) (for p=1). The consequence for arbitrary compact metrizable K then follows from the standard classification of separable C(K) spaces. No step reduces by construction to a fitted input, self-definition, or load-bearing self-citation chain; the central claim rests on independent constructions and external classification results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is a pure-mathematics proof paper; it relies on the standard axioms of ZFC set theory and the background theory of Banach spaces and operators. No free parameters, invented entities, or ad-hoc axioms are visible in the abstract.

axioms (1)
  • standard math Standard axioms of ZFC and the usual definitions and theorems of Banach space theory
    Required for any proof in this area; invoked implicitly throughout.

pith-pipeline@v0.9.1-grok · 5617 in / 1266 out tokens · 14880 ms · 2026-06-29T00:32:36.960331+00:00 · methodology

0 comments
read the original abstract

We prove that the spaces $\ell_p(C(\alpha))$ and $\ell_p(C[0,1])$ have the uniform primary factorisation property whenever $\alpha$ is an ordinal and $1<p\leq\infty$. For the case $p=1$, we establish a general criterion ensuring that $\ell_1(X)$ inherits the uniform primary factorisation property from $X$. As a consequence, $\ell_p(C(K))$ is primary for every compact metrizable space $K$ and every $1 \leq p \leq \infty$.

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Preservation of primariness under $\ell_1$-, $c_0$-, and $\ell_\infty$-sums of Banach spaces

    math.FA 2026-06 unverdicted novelty 6.0

    Transfer principles for UPFP are shown from X to ℓ₁(X), c₀(X), ℓ∞(X) under cotype or self-similarity hypotheses, implying primariness of the sums and yielding new proofs for primariness of ℓ∞(L_p), c₀(L_1) and UPFP fo...

  2. Pure infiniteness and primary factorisation

    math.FA 2026-07 unverdicted novelty 5.0

    No indecomposable Banach space has the primary factorisation property; for complex spaces with PFP the quotient B(E)/M_E is purely infinite precisely when non-scalar.

Reference graph

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