REVIEW 3 major objections 3 minor 5 references
On the Banach lattice c_0
T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read The paper proves that $c_0$, viewed as a Banach lattice, is not projective, and that the free Banach lattice it generates is not projective either.
desk verdict Answers an open question of de Pagter–Wickstead with a clean Section 2 and a promising but under-specified construction in Section 3; the main results are probably true but the preprint needs a repair and an external check of Proposition 2.1. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The load-bearing machinery is the free Banach lattice construction together with a quotient criterion. $FBL(A)$ is the Banach lattice generated by evaluation functions $\delta_a$ on $[-1,1]^A$, and $FBL[E]$ is the analogous object generated by evaluations of functionals in $E^*$. Proposition 2.1, taken from a companion paper, says that for a projective Banach lattice $P$ and an ideal $I$, the quotient $P/I$ is projective exactly when, for every $\varepsilon>0$, the quotient map $\pi$ has a lattice-homomorphism section with norm at most $1+\varepsilon$. This reduces non-projectivity of $c_0$ to proving that no such section exists for a particular surjection $\Phi$ from a free lattice onto $c_0$. Lemma 2.3 is the engine of that proof, converting continuity in the product topology into unbounded partial sums for any candidate section. For the complementation half, the engine is an explicit disjoint sequence $(f_n)$ inside $FBL[c_0]$ whose partial sums are bounded by one and whose individual norms are one, which by a classical criterion embeds $c_0$ isometrically into $FBL[c_0]$.
What would settle it
Produce a bounded Banach lattice homomorphism $\varphi\colon c_0\to FBL(L)$ with $\Phi\circ\varphi=\mathrm{id}$, where $L$ is the set of nonempty finite subsets of $\mathbb{N}$ and $\Phi$ is the canonical surjection onto $c_0$; the partial sums $\sum_{k=1}^m \varphi(e_k)$ would have to stay bounded, while Lemma 2.3 forces their norm to be at least $m-\varepsilon$ for every $m$. Any such map would disprove the paper's central claim.
Extended reading notes
Core claim
The central discovery is that $c_0$ cannot satisfy the defining lifting property of a projective Banach lattice. The proof works by contradiction: if $c_0$ were projective, then using a quotient criterion from a companion paper, the canonical surjective lattice homomorphism $\Phi\colon FBL(L)\to c_0$ (where $L$ is the set of nonempty finite subsets of $\mathbb{N}$) would admit a bounded lattice-homomorphism splitting $\varphi\colon c_0\to FBL(L)$ with $\Phi\circ\varphi=\mathrm{id}$. A combinatorial lemma shows this is impossible: the images $f_n=\varphi(e_n)$ would be positive functions with $f_n(x_n^*)=1$ at carefully chosen evaluation points, and their partial sums would have norm at least $m-\varepsilon$ for every $m$, contradicting the boundedness of $\varphi$. In the second half, the authors construct a disjoint sequence of positive functions $f_n\in FBL[c_0]$ with $\|f_n\|=1$ and $\|\sum_{i=1}^n f_i\|\le 1$; by a standard characterization this gives an isometric lattice embedding $u\colon c_0\to FBL[c_0]$, and the evaluation map $T\colon FBL[c_0]\to c_0$ satisfies $T\circ u=\mathrm{id}$. Thus $c_0$ is complemented in $FBL[c_0]$, and if $FBL[c_0]$ were projective the quotient criterion would force $c_0$ to be projective as well, a contradiction.
Load-bearing premise
The argument leans on Proposition 2.1 from a companion paper — the characterization that a quotient of a projective Banach lattice is projective exactly when approximate lattice-homomorphism sections exist — and if that characterization has hidden hypotheses, both main theorems would lose their proof.
Editorial extensions
If this is right
- The open question asking whether $c_0$ is a projective Banach lattice is answered negatively.
- The related open question asking whether the free Banach lattice $FBL[c_0]$ is projective is also answered negatively.
- $c_0$ admits an isometric lattice embedding into $FBL[c_0]$, and the identity map on $c_0$ factors through $FBL[c_0]$ via a quotient map, so $c_0$ is complemented in $FBL[c_0]$.
- A free Banach lattice generated by an infinite-dimensional Banach space can fail projectivity even though free Banach lattices generated by plain sets are projective.
- Any attempt to split the natural quotient onto $c_0$ fails because of a norm-growth obstruction, not a topological one.
Reading between the lines
- The same disjoint-sequence construction may work for other Banach spaces with a normalized basis, yielding complemented copies inside their free Banach lattices and hence new non-projective free lattices.
- The proof strategy suggests that non-projectivity is inherited by any quotient of a projective lattice whose quotient map lacks approximate lattice-homomorphism sections; applying this to other classical sequence spaces could map the boundary of the projective class.
- If the companion paper's quotient criterion later needs strengthening, the complementability theorem may survive independently, while the non-projectivity of $FBL[c_0]$ would need a different proof.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims two main results about the Banach lattice c0: (1) c0 is not a projective Banach lattice, answering a question of de Pagter and Wickstead; and (2) c0 is complemented in the free Banach lattice FBL[c0] generated by c0 as a Banach space, from which it follows that FBL[c0] is not projective. Section 2 proves non-projectivity by assuming a projective c0 and using a quotient map from a free Banach lattice FBL(L) onto c0 together with a combinatorial subsequence argument (Lemma 2.3) to contradict boundedness of the induced right inverse. Section 3 constructs a disjoint positive sequence in FBL[c0] with norm-one partial sums, yielding an isometric lattice embedding and a norm-one projection onto c0. Both arguments depend on Proposition 2.1 from a to-appear paper by two of the same authors.
Significance. If the proofs are made fully rigorous, the paper answers de Pagter and Wickstead's Questions 12.10 and 12.11 negatively and gives a clean example of a complemented non-projective sublattice inside a free Banach lattice. The combinatorial core of Section 2, especially Lemma 2.3, is elegant and the norm estimates in Section 3 are carefully designed. The main obstruction is not the mathematical strategy but the current presentation of the Section 3 construction, which asserts the existence of an impossible family of functions, and the reliance on an unproved external proposition for the central deduction.
major comments (3)
- [§3, definition of g_{nm} and Lemmas 3.3–3.5] The functions g_{nm}: c_0^* → [0,1] do not exist as stated. The condition g_{nm}(x^*) = g_{nm}(x^*/‖x^*‖) for x^*≠0 makes g_{nm} positively homogeneous of degree 0. If such a function were continuous at 0, then for any nonzero u and v one would have g_{nm}(u) = lim_{t→0+} g_{nm}(tu) = g_{nm}(0) = lim_{t→0+} g_{nm}(tv) = g_{nm}(v), so g_{nm} would be constant on ℓ1∖{0}. But the boundary conditions are incompatible with constancy: at x^*=0 both N_m|x^*_n|≤|x^*_m| and |x^*_m|≤(N_m−1)|x^*_n| hold, forcing 0=1; and for n=1, m=2 one has g_{12}(e_1)=1 and g_{12}(e_2)=0. Since Lemma 3.3 proves h_k∈FBL[c0] using these g_{nm}, and Lemmas 3.4–3.5 and Theorems 3.6–3.8 depend on that construction, the embedding and complementation results are not rigorously established as written. The gap appears repairable by defining g_{nm} only on ℓ1∖{0} and using the prefactor (|x^*_n|−N_n max_{m<n}|x^*_m|)_+ to ensure the approximants h_k extend continuously to 0, but this repair must be stated and proved explicitly.
- [§2, Proposition 2.1 and its uses in Theorems 2.4 and 3.8] The main non-projectivity deduction uses Proposition 2.1 as a black box taken verbatim from the to-appear paper [1] by two of the same authors, and no proof is included. This proposition is load-bearing: it is exactly what converts the nonexistence of a bounded lattice right inverse into non-projectivity, and it is also used in Theorem 3.8 to derive non-projectivity of FBL[c0] from non-projectivity of c0 and the complementation constructed in Section 3. The sufficiency direction is not an immediate consequence of the definition of projectivity. Please include a proof of Proposition 2.1 in this paper, or give a precise reference to a numbered statement in a published version of [1].
- [§2, Lemma 2.2 and Theorem 2.4] The assertion immediately after Lemma 2.2 that 'Φ is a quotient map' needs justification beyond surjectivity. To apply Proposition 2.1 with P=FBL(L) and P/I identified with c0, the quotient norm on FBL(L)/ker Φ must coincide, or be suitably comparable, with the c0 norm. A surjective contractive lattice homomorphism is not automatically a metric quotient map. Please prove that for every x∈c0 there exists f∈FBL(L) with Φ(f)=x and ‖f‖_{FBL(L)} ≤ ‖x‖∞, or otherwise explain why Proposition 2.1 is applicable in this situation. Without such a proof, the identification of the quotient FBL(L)/ker Φ with the projective Banach lattice c0 is not fully established.
minor comments (3)
- [Abstract and §3] There are a few typographical errors, e.g. 'spac e' in the abstract and 'F LB[c0]' in Theorem 3.7; these should be corrected.
- [§3, Lemma 3.3] In the proof of Lemma 3.3, after repairing the definition of g_{nm}, the continuity of the auxiliary function \tilde h_k on [-1,1]^{B_{c0}} at points where the finite-coordinate vector vanishes should be checked explicitly; the current text does not address this point.
- [§3, Theorem 3.6] The notation u(x)=∑_{i=1}^∞ x_i f_i is used before proving that the series converges in FBL[c0]; convergence follows from Lemma 3.4 and the fact that x∈c0, but this could be stated for clarity.
Circularity Check
No significant circularity: the main claims are derived from explicit constructions and independent general criteria; the only self-citation is a general quotient criterion that does not presuppose the target results.
full rationale
The paper's derivation chain does not reduce to its own inputs. The non-projectivity of c0 (Theorem 2.4) is proved by contradiction: assuming c0 projective, using the explicit quotient map Phi : FBL(L) -> c0 constructed in Lemma 2.2, and the norm-growth estimate of Lemma 2.3, the authors obtain a subsequence whose FBL(L)-norm is at least m - epsilon while boundedness of the supposed lifting forces the same norm to be at most C. This is a genuine argument not presupposing the conclusion. The complementation of c0 in FBL[c0] (Theorems 3.6 and 3.7) is built constructively by defining functions f_n and proving disjointness, membership in FBL[c0], norm-one behaviour, and bounded partial sums; this is independent of the non-projectivity result. The final theorem that FBL[c0] is not projective uses Proposition 2.1, cited from the authors' own to-appear paper [1]. This is a self-citation, but it is not circular: Proposition 2.1 is a general criterion about projective Banach lattices and quotient ideals, with no assumption that includes c0 or FBL[c0], and it is not a fitted parameter or a renamed version of the conclusion. The paper could have supplied a proof of Proposition 2.1, and a reader may want external verification of the to-appear reference, but that is a completeness or correctness-risk concern, not circularity. The skeptical issue about the existence of continuous functions g_nm satisfying homogeneity and boundary conditions is a mathematical correctness gap if valid, not a circularity: the construction would be flawed or incomplete, but it would not make the theorem equivalent to its assumptions. No circular step was identified.
Assumptions & free parameters
free parameters (1)
- N_n =
arbitrary strictly increasing sequence of natural numbers
assumptions (5)
- standard math Free Banach lattices FBL(A) and FBL[E] exist with the stated explicit representation and universal properties.
- domain assumption Proposition 2.1: projectivity of P/I is equivalent to existence of (1+epsilon)-bounded right inverses for every epsilon > 0.
- standard math Theorem 4.50 of [4]: c0 embeds lattice-isometrically into a Banach lattice iff there is a disjoint positive sequence with bounded partial sums and norm not tending to zero.
- domain assumption Functions in FBL(A) and FBL[E] are continuous with respect to the product topology on [-1,1]^A and weak*-continuous on bounded dual balls, and are positively homogeneous.
- standard math For any strictly increasing sequence N_n, continuous positively homogeneous functions g_nm satisfying the stated interpolation conditions exist.
Cite this review
Pith. "Pith review of On the Banach lattice c_0." pith.science (2026). https://pith.science/paper/VN3FUUXT
@misc{pith2026190807786,
author = {Pith},
title = {Pith review of: On the Banach lattice c_0},
year = {2026},
howpublished = {\url{https://pith.science/paper/VN3FUUXT}},
note = {Machine review of arXiv:1908.07786}
}
abstract
We show that $c_0$ is not a projective Banach lattice, answering a question of B. de Pagter and A. Wickstead. On the other hand, we show that $c_0$ is complemented in the free Banach lattice generated by itself (seen as a Banach space). As a consequence, the free Banach lattice generated by $c_0$ is not projective.
Reference graph
Works this paper leans on
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[1]
A. Avil\'es, J. D. Rodr\'iguez Abell\'an, The free Banach lattice generated by a lattice, Positivity. To appear
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[2]
A. Avil\'es, J. D. Rodr\'iguez Abell\'an, Projectivity of the free Banach lattice generated by a lattice, Archiv der Mathematik. To appear
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[3]
A.\ Avil\'es, J.\ Rodr\'iguez, P.\ Tradacete, The free Banach lattice generated by a Banach space, J. Funct. Anal. 274 (2018), 2955--2977
work page 2018
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[4]
W.\ Wickstead, Free and projective Banach lattices, Proc
B.\ de Pagter, A. W.\ Wickstead, Free and projective Banach lattices, Proc. Royal Soc. Edinburgh Sect. A, 145 (2015), 105--143
work page 2015
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[5]
C. D. Aliprantis, O. Burkinshaw, Positive Operators, Handbook of the Geometry of Banach Spaces, Springer 2006
work page 2006
Reviewed August 14, 2026 · model on record in the stance chip above.
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