{"id":"8e7ca96b-3806-41f2-b230-ec6620a084f1","arxiv_id":"1908.07786","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"c0 is not a projective Banach lattice, and c0 is complemented in FBL[c0], implying FBL[c0] is not projective.","lead":"This paper proves that c0, the space of sequences tending to zero, is not a projective Banach lattice, answering an open question of de Pagter and Wickstead. It also shows c0 embeds as a complemented sublattice of the free Banach lattice FBL[c0], which therefore is not projective.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3 defines functions g_nm that cannot exist as stated: degree-zero homogeneity plus continuity at 0 forces constant values, contradicting the required 0/1 boundary conditions.","rationale":"The proof that c0 itself is not projective in Section 2 is internally sound once Proposition 2.1 is granted, and Proposition 2.1 is an elementary categorical fact (an approximate retract of a projective object is projective), so the external reference cited by the reader is not the real risk. The genuinely load-bearing soft spot is in Section 3: the auxiliary functions g_nm, as defined, cannot exist if they are required to be continuous on all of ℓ1 while being homogeneous of degree 0 and taking different values on e_n and e_m. This directly affects the construction of the disjoint norm-one sequence with bounded partial sums, and hence the complementability of c0 in FBL[c0] and the non-projectivity of FBL[c0]. The issue is not fatal, because the intended construction can be repaired by requiring continuity only away from 0 and using the prefactor to control the limit at 0. Since a repair is straightforward and Section 2 is unaffected, the appropriate outcome remains conditional acceptance; the reader's specific concern about Proposition 2.1 is not the most load-bearing one, hence the disagreement.","tokens_in":9342,"tokens_out":35925,"duration_ms":333438,"concrete_test":"Check the stated existence claim in the simplest case N_m=2, n=1, m=2. If a continuous g:ℓ1→[0,1] satisfying the radial condition and the boundary conditions existed, then for every t>0 one would have g(t e_1)=1 and g(t e_2)=0, while t e_1→0 and t e_2→0 in norm; continuity at 0 is impossible. This confirms the inconsistency. To test the repair, define g on ℓ1∖{0} by continuous interpolation in the gap |x^*_2|/|x^*_1| ∈ (N_m−1, N_m), set g=0 where |x^*_1|=0<|x^*_2|, extend h_k by h_k(0)=0, and verify that the norm estimates in Lemma 3.3 still hold; if they do, the central theorem survives with a minor correction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"In §3, the construction of the disjoint separating sequence uses functions g_nm : c_0^* = ℓ1 → [0,1] required to be continuous, to satisfy g_nm(x^*) = 0 when N_m|x^*_n| ≤ |x^*_m|, to satisfy g_nm(x^*) = 1 when |x^*_m| ≤ (N_m−1)|x^*_n|, and to be radial: g_nm(x^*) = g_nm(x^*/‖x^*‖) for x^* ≠ 0. The radial condition makes each g_nm positively homogeneous of degree 0. On a normed space, a degree-0 homogeneous function that is continuous at 0 must be constant on all nonzero directions: along the rays t u and t v the limits as t→0 must both equal g(0), so g(u) = g(v) for all nonzero u,v. But the boundary conditions force conflicting values, e.g. for n=1, m=2, N_m=2, one has g(e_1) = 1 (since |e_1|_2 = 0 ≤ (N_m−1)·1) and g(e_2) = 0 (since N_m·0 ≤ 1). Hence no such continuous function on all of ℓ1 exists. Lemmas 3.3–3.5, and therefore Theorems 3.6–3.8, depend on these g_nm; as written, the construction of the complementation is not rigorous. The gap is repairable: define g_nm only on ℓ1∖{0} by interpolating between the two cones, and observe that the prefactor (|x^*_n| − N_n max_{k<n}|x^*_k|)_+ makes the approximants h_k extend continuously to 0. But the paper should state this explicitly; the current statement asserts existence of an impossible object.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9750,"tokens_out":24227,"duration_ms":740938,"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":[{"comment":"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.","section":"§3, definition of g_{nm} and Lemmas 3.3–3.5"},{"comment":"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].","section":"§2, Proposition 2.1 and its uses in Theorems 2.4 and 3.8"},{"comment":"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.","section":"§2, Lemma 2.2 and Theorem 2.4"}],"minor_comments":[{"comment":"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.","section":"Abstract and §3"},{"comment":"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.","section":"§3, Lemma 3.3"},{"comment":"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.","section":"§3, Theorem 3.6"}],"recommendation":"major_revision","confidential_remarks":"The mathematical program appears sound and likely correct, and the Section 3 construction is plausibly repairable with a modest rewrite. I recommend major revision rather than rejection. The main issues are the impossible definition of the g_{nm} functions, the unproved external Proposition 2.1, and the quotient-map identification in Section 2. None of these appears to require a new mathematical idea beyond what the authors already have, but each must be addressed before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nWorth your eyes: this short paper resolves a real open question—c0 is not a projective Banach lattice—and adds a slick complementability result for c0 inside FBL[c0]. The negative answer also settles Question 12.10. Section 2 is the stronger half: the quotient argument via FBL(L) and Lemma 2.3 is coherent, and the subsequence/norm estimate is a nice piece of work. The Claim in Lemma 3.3, controlling sums of coordinate evaluations, is also correct and useful.\n\nThe construction in Section 3, however, has a genuine flaw as written. The functions g_nm are required to be continuous on all of ℓ1, radial, and to take values 0 and 1 on certain cones. A radial function that is continuous at 0 must be constant on every nonzero ray: g(u)=g(0)=g(v). The stated boundary conditions force conflicting values (e.g. n=1, m=2, N_m=2 gives g(e1)=1 and g(e2)=0). So those functions don't exist on ℓ1. This isn't fatal—the repair is to define g_nm only on ℓ1∖{0} by interpolating between the cones and use the prefactor (|x*_n|−N_n max_{k<n}|x*_k|)_+ to extend the approximating h_k continuously to 0—but the preprint needs to say that. As it stands, Lemmas 3.3–3.5 depend on an impossible object. A referee should ask for a rewrite of this definition; the rest of Section 3 then goes through.\n\nThe other soft spot, already flagged by the reader, is Proposition 2.1. It is lifted from a to-appear paper by the same authors, and every main theorem—Theorems 2.4, 3.7, 3.8—relies on it. The proposition is independent of the new results, so this is not circular, but it is a black box. If it is correct, the paper's conclusions follow; if it needs extra hypotheses, the non-projectivity claim is unsupported. I'd want that proposition available or proved in a footnote.\n\nThe citation pattern is otherwise unremarkable: de Pagter–Wickstead, Avilés–Rodríguez–Tradacete, and Aliprantis–Burkinshaw are the right sources. No hidden fitting or invented objects beyond the g_nm issue.\n\nBottom line: this deserves a serious referee, not a desk reject. The Section 2 result alone is a substantial bite at an open question, and the Section 3 gap is localized and repairable. I'd send it out and ask for (i) a clean treatment of the g_nm functions and (ii) either a statement of Proposition 2.1 with proof sketch or a pointer to the published version.","headline":"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.","tokens_in":10223,"tokens_out":3933,"would_cite":true,"duration_ms":40153,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46B43","06BXX"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["c0","projective Banach lattice","free Banach lattice","FBL[c0]","Banach lattice homomorphism","complemented sublattice","quotient criterion","sequence space"],"falsifier":"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.","tokens_in":9184,"feed_emoji":"","tokens_out":10256,"duration_ms":95549,"temperature":0.7,"pith_summary":"This paper settles two open questions from the theory of free and projective Banach lattices in the negative. It proves that the sequence space $c_0$ — the Banach space of real sequences tending to zero, with the supremum norm — is not projective when regarded as a Banach lattice. It then proves that $c_0$ is complemented inside the free Banach lattice $FBL[c_0]$ that it generates as a Banach space, through an isometric lattice embedding whose inverse is a quotient map. Since a projective Banach lattice cannot have a non-projective quotient with an approximate lattice-homomorphism section, it follows that $FBL[c_0]$ is not projective. These results matter because projectivity is a structural lifting property: they show the property fails even for one of the most familiar separable Banach spaces.","feed_headline":"c0 is not projective, and neither is its free Banach lattice","feed_subtitle":"Negative answers to two open questions: c0 and the free lattice it generates both lack the lifting property.","key_machinery":"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 $FBL[c_0]$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies Proposition 2.1, the quotient criterion that converts projectivity of a quotient into the existence of approximate lattice-homomorphism sections; both main theorems use it as a black box.","marker":"[1]"},{"why":"Defines and explicitly describes the free Banach lattice generated by a Banach space, including the norm formula and the natural copy of the space inside it, which Section 3 relies on.","marker":"[2]"},{"why":"Introduces free and projective Banach lattices, proves basic projectivity examples, poses the questions answered negatively here, and supplies Proposition 5.3 used in Lemma 3.1.","marker":"[3]"},{"why":"Provides the classical lattice-embedding criterion for $c_0$ in terms of a disjoint positive sequence with norm-bounded partial sums, which the constructed sequence $(f_n)$ verifies.","marker":"[4]"}],"fun_headline_variants":["c0 not projective, free lattice also fails","c0 and its free Banach lattice: no lifting","Projectivity fails for c0 and its free lattice","c0 is not projective, nor is FBL[c0]","c0 and FBL[c0] both lack lifting property"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["c0 not projective, free lattice also fails","c0 and its free Banach lattice: no lifting","Projectivity fails for c0 and its free lattice","c0 is not projective, nor is FBL[c0]","c0 and FBL[c0] both lack lifting property"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000133,"raw_usage":{"total_tokens":1127,"prompt_tokens":928,"completion_tokens":199,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":544,"completion_tokens_details":{"reasoning_tokens":118}},"tokens_in":544,"tokens_out":199,"duration_ms":2965,"temperature":1.0,"reasoning_tokens":118,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:57:24.285084+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Avil\\'es, J","cited_arxiv_id":null,"evidence_quote":"Supplies Proposition 2.1, the quotient criterion that converts projectivity of a quotient into the existence of approximate lattice-homomorphism sections; both main theorems use it as a black box."},{"cited_title":"Avil\\'es, J","cited_arxiv_id":null,"evidence_quote":"Defines and explicitly describes the free Banach lattice generated by a Banach space, including the norm formula and the natural copy of the space inside it, which Section 3 relies on."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces free and projective Banach lattices, proves basic projectivity examples, poses the questions answered negatively here, and supplies Proposition 5.3 used in Lemma 3.1."},{"cited_title":"W.\\ Wickstead, Free and projective Banach lattices, Proc","cited_arxiv_id":null,"evidence_quote":"Provides the classical lattice-embedding criterion for $c_0$ in terms of a disjoint positive sequence with norm-bounded partial sums, which the constructed sequence $(f_n)$ verifies."}],"review_version":1}