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

Antonio Acuaviva

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Finite-cotype or self-similarity assumptions on a Banach space X transfer the uniform primary factorisation property to the spaces ℓ₁(X), c₀(X) and ℓ∞(X), making them primary.

2026-06-26 00:44 UTC pith:XRDU3C7O

load-bearing objection Transfer principles move UPFP to l1/c0/l∞ sums under cotype or self-similarity, recovering l∞(Lp) primariness without Bourgain and adding c0(L1) plus C[0,1]*.

arxiv 2606.26417 v1 pith:XRDU3C7O submitted 2026-06-24 math.FA

Preservation of primariness under ell₁-, c₀-, and ell_infty-sums of Banach spaces

Antonio Acuaviva This is my paper

classification math.FA

keywords Banach spacesprimarinessuniform primary factorisation propertyvector-valued sequence spacesfinite cotypeself-similarityℓ1-sumsc0-sums

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 transfer principles that carry the uniform primary factorisation property from a Banach space X to its vector-valued sequence spaces ℓ₁(X), c₀(X) and ℓ∞(X). The required hypotheses are finite-cotype conditions on X or on X*, or natural self-similarity conditions on X. A sympathetic reader cares because these conditions then guarantee that the three sequence spaces are primary, a structural property that limits how they can decompose. The same principles recover the primariness of ℓ∞(L_p) for 1 ≤ p < ∞ without localisation methods and establish the primariness of c₀(L_1). They also show that ℓ₁(Γ, L_1[0,1]) has the uniform primary factorisation property for every index set Γ, which implies that C[0,1]* has the property and is primary.

Core claim

Transfer principles are proved for the uniform primary factorisation property from X to the spaces ℓ₁(X), c₀(X) and ℓ∞(X). The hypotheses used are finite-cotype assumptions on X or on X*, or natural self-similarity assumptions on X. Under these conditions the vector-valued sequence spaces are therefore primary. Applications include a new proof of the primariness of ℓ∞(L_p) spaces for 1 ≤ p < ∞, the primariness of c₀(L_1), and the fact that ℓ₁(Γ, L_1[0,1]) has the UPFP for every index set Γ, which implies that the dual of C[0,1] has the UPFP and is primary.

What carries the argument

The uniform primary factorisation property (UPFP), which supplies a uniform factorisation of operators through primary spaces and thereby transfers primariness to the three sequence spaces.

Load-bearing premise

Finite-cotype assumptions on X or X* or self-similarity assumptions on X are what enable the transfer of the uniform primary factorisation property.

What would settle it

A Banach space X satisfying finite cotype on X or X* for which ℓ∞(X) fails to have the uniform primary factorisation property.

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

If this is right

ℓ∞(L_p) is primary for every 1 ≤ p < ∞
c₀(L_1) is primary
ℓ₁(Γ, L_1[0,1]) has the UPFP for arbitrary index sets Γ
C[0,1]* has the UPFP and is primary

Where Pith is reading between the lines

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

The transfer may extend to other classical constructions such as direct integrals or iterated sums built from the same base spaces.
Similar inheritance could hold when the base space X is replaced by other classical Banach spaces that satisfy the same cotype or self-similarity hypotheses.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Transfer principles move UPFP to l1/c0/l∞ sums under cotype or self-similarity, recovering l∞(Lp) primariness without Bourgain and adding c0(L1) plus C[0,1]*.

read the letter

The main point is that the paper gives transfer results taking the uniform primary factorisation property from X to its l1(X), c0(X), and l∞(X) versions when X or X* has finite cotype or when X satisfies a self-similarity condition. Those sums then turn out to be primary. The applications include a proof of primariness for l∞(Lp) that skips Bourgain's localisation, a new claim that c0(L1) is primary, and the statement that l1(Γ, L1) has UPFP for any index set, which yields UPFP and primariness for C[0,1]*.

The work is clearest on the hypotheses and on how they feed the conclusions. Avoiding the localisation method for the l∞(Lp) case is a concrete improvement if the new route is shorter. The self-similarity condition looks tailored to the setting and covers the examples they need.

The finite-cotype requirement is standard but narrows the range of spaces. Self-similarity may be even narrower, so the transfer applies mainly to spaces already known to behave well. The abstract states the proofs exist under the listed conditions, but the actual lemmas and estimates are not visible here; a referee would need to check whether the arguments stay clean or pick up hidden technical overhead.

This is for people already working on primariness, factorisation, and structural properties of classical Banach spaces and their sums. A reader who follows results on Lp, C[0,1], and their sequence-space versions will see direct value. The combination of alternative proofs and one or two new cases is enough to justify sending it to a serious referee rather than desk-rejecting it.

Referee Report

0 major / 3 minor

Summary. The manuscript proves transfer principles for the uniform primary factorisation property (UPFP) from a Banach space X to the vector-valued sequence spaces ℓ₁(X), c₀(X) and ℓ∞(X). The hypotheses are finite-cotype assumptions on X or X*, or natural self-similarity assumptions on X. Consequently the resulting spaces are primary. Applications recover primariness of ℓ∞(L_p) for 1≤p<∞ without Bourgain's localisation method, establish primariness of c₀(L₁), and show that ℓ₁(Γ,L₁[0,1]) has the UPFP for every Γ (hence C[0,1]* has the UPFP and is primary).

Significance. If the transfer principles hold, the work supplies a unified mechanism for establishing primariness of vector-valued sequence spaces under standard hypotheses from Banach space theory. It recovers classical results by new means and yields new conclusions for c₀(L₁) and C[0,1]*. The explicit hypotheses (cotype or self-similarity) make the statements falsifiable and applicable to many concrete spaces.

minor comments (3)

Clarify the precise definition of 'natural self-similarity assumptions' early in the introduction, with a forward reference to the section where they are used.
In the applications section, add a short paragraph verifying that the cited spaces (e.g., L_p, L₁) satisfy the finite-cotype or self-similarity hypotheses invoked in the transfer theorems.
Ensure that the statement of each transfer theorem explicitly lists the required hypothesis on X (cotype of X, cotype of X*, or self-similarity) rather than referring only to 'the listed hypotheses'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript. The recommendation for minor revision is noted. No specific major comments were listed in the report, so we address the overall assessment below and confirm that any minor editorial points will be incorporated in the revision.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper proves transfer of the uniform primary factorisation property (UPFP) to vector-valued sequence spaces under independent hypotheses (finite cotype on X or X*, or self-similarity on X), which are standard external Banach space notions not defined in terms of the target primariness result. The primariness conclusions and applications (e.g., to ℓ∞(Lp), c0(L1), ℓ1(Γ,L1)) are presented as direct consequences of these transfers without any reduction to fitted parameters, self-definitional equations, or load-bearing self-citations that collapse the argument. The derivation remains self-contained against external benchmarks such as known cotype and primariness definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard definition of Banach spaces, the notion of finite cotype, and the definition of self-similarity; no free parameters or invented entities appear in the abstract.

axioms (2)

domain assumption Finite cotype is a well-defined property of Banach spaces controlling Rademacher averages.
Invoked as one of the two main hypotheses enabling the transfer (abstract).
domain assumption Natural self-similarity assumptions on X are sufficient to trigger the same transfer.
Alternative hypothesis stated in the abstract.

pith-pipeline@v0.9.1-grok · 5697 in / 1460 out tokens · 25889 ms · 2026-06-26T00:44:26.343923+00:00 · methodology

0 comments

read the original abstract

We prove transfer principles for the uniform primary factorisation property (UPFP) from a Banach space $X$ to the vector-valued sequence spaces $\ell_1(X)$, $c_0(X)$ and $\ell_\infty(X)$. The hypotheses are either finite-cotype assumptions on $X$ or $X^*$, or natural self-similarity assumptions on $X$. Consequently, under these conditions, the resulting vector-valued sequence spaces are primary. As applications, we recover the primariness of $\ell_\infty(L_p)$ for $1\leq p<\infty$ without using Bourgain's localisation method, and obtain the primariness of $c_0(L_1)$. We also show that $\ell_1(\Gamma,L_1[0,1])$ has the UPFP for every set $\Gamma$, and consequently that $C[0,1]^*$ has the UPFP and is primary.

discussion (0)

Reference graph

Works this paper leans on

16 extracted references · 3 canonical work pages · 2 internal anchors

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work page arXiv 2025
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Primariness of the spaces $\ell_p(C(K))$ for $1 \leq p \leq \infty$

A. Acuaviva,Primariness of the spacesℓp(C(K))for1≤p≤∞, arXiv:2605.29854, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Primariness and the Primary Factorisation Property

A. Acuaviva and T. Kania,Primariness and the primary factorisation property, arXiv:2605.21711, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Acuaviva,C(K)-spaces with few operators relative to posets, arXiv:2511.22339 (2025)

A. Acuaviva,C(K)-spaces with few operators relative to posets, arXiv:2511.22339 (2025)

work page arXiv 2025

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

A. Acuaviva,Primariness of the spacesℓp(C(K))for1≤p≤∞, arXiv:2605.29854, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

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Primariness and the Primary Factorisation Property

A. Acuaviva and T. Kania,Primariness and the primary factorisation property, arXiv:2605.21711, 2026

work page internal anchor Pith review Pith/arXiv arXiv 2026

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D. E. Alspach and Y. Benyamini,Primariness of spaces of continuous functions on ordinals, Israel J. Math.27(1977), 64–92

1977

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Alspach, P

D. Alspach, P. Enflo, and E. Odell,On the structure of separableLp spaces(1< p < ∞), Studia Math.60(1977), no. 1, 79–90

1977

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Capon,Primarité deℓp(L1), Math

M. Capon,Primarité deℓp(L1), Math. Ann.250(1980), no. 1, 55–63

1980

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Dosev and W

D. Dosev and W. B. Johnson,Commutators onℓ ∞, Bull. London Math. Soc.42 (2010), 155–169

2010

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Hajnal,Proof of a conjecture of S

A. Hajnal,Proof of a conjecture of S. Ruziewicz, Fund. Math.50(1961), 123–128

1961

[9] [9]

N. J. Kalton,The endomorphisms ofLp, Indiana Univ. Math. J.27(1978), 353–381. PRESER V ATION OF PRIMARINESS UNDER VECTOR-V ALUED SUMS 39

1978

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N. J. Laustsen,Matrix multiplication and composition of operators on the direct sum of an infinite sequence of Banach spaces, Math. Proc. Cambridge Philos. Soc.131 (2001), no. 1, 165–183

2001

[11] [11]

Lechner and T

R. Lechner and T. Speckhofer,Factorization in Haar system Hardy spaces, Ark. Mat. 63(2025), no. 1, 149–205

2025

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Lindenstrauss and L

J. Lindenstrauss and L. Tzafriri,Classical Banach Spaces II. Function Spaces, Springer, 1979

1979

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Maurey,Sous-espaces complémentés deLp, d’après P

B. Maurey,Sous-espaces complémentés deLp, d’après P. Enflo, Séminaire Maurey– Schwartz (1974–1975), Exp. No. III, 1–14

1974

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P. F. X. Müller,Two remarks on primary spaces, Math. Proc. Cambridge Philos. Soc. 153(2012), no. 3, 505–523

2012

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Pełczyński,On strictly singular and strictly cosingular operators

A. Pełczyński,On strictly singular and strictly cosingular operators. I. Strictly singu- lar and strictly cosingular operators inC(S)-spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys.13(1965), 31–36

1965

[16] [16]

H. M. Wark,Theℓ∞direct sum ofLp (1< p <∞)is primary, J. Lond. Math. Soc. (2)75(2007), no. 1, 176–186. School of Mathematical Sciences, Fylde College, Lancaster University, LA1 4YF, United Kingdom Email address:ahacua@gmail.com

2007