{"id":"d633561e-776c-40f4-8514-6692dabcbbc4","arxiv_id":"2608.11894","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A non-zero symmetric sequence ideal E ⊆ c_0 is a continuous linear image of C_p(X) only if E = c_0, so proper ideals such as (ℓ_q)_p never appear.","lead":"The paper proves that the only symmetric sequence ideal inside c_0 that can be a continuous linear image of a pointwise-convergence function space C_p(X) is c_0 itself. This rules out surjections onto (ℓ_q)_p for every finite q and completes a characterization with the Josefson-Nissenzweig property.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified. The central proof is sound; the only point needing care is that 'span' in F and M_S must be read as the closed linear span.","rationale":"The paper proves Theorem 1.1 by reducing to Theorem 3.1, a Banach-space statement. I checked the reduction: Theorem 2.1 correctly gives functional boundedness of S and replacement by C_b(X); the quotient norm on E is complete and the inclusion into c0 is continuous; M_S is the closed span of evaluations, isometric to ell_1(S), hence has the Schur property; and T_b^* delta_n = mu_n in M_S. In Theorem 3.1, Lemmas 3.2 and 3.3 are sound closed-graph arguments; Lemma 3.4 correctly proves that (delta_n) is a seminormalized unconditional basis of the closed span. Lemma 3.5 is a standard and elementary dichotomy, not a source of risk. The first alternative leads to a contradiction via the Schur property and the bounded-below adjoint of a surjection; the second alternative yields the ell_1 lower bound, which with uniform diagonal projections gives norm comparison (6) and hence E = c0. The only imprecision is the word 'span' in F and M_S, which must be read as closed linear span; this is a notational patch and does not affect the mathematics. No load-bearing concern remains, so the reader's ACCEPT verdict is unchanged.","tokens_in":8535,"tokens_out":38958,"duration_ms":396356,"concrete_test":"Inspect the definitions of F in Theorem 3.1 (paragraph before Lemma 3.4) and of M_S in the proof of Theorem 1.1. If the manuscript uses 'span' to mean the algebraic linear span, replace it by the norm closure in E* and C_b(X)*, respectively, and re-verify that Lemma 3.4 still constructs an unconditional Schauder basis of the closure and that M_S is isometric to ell_1(S) and therefore has the Schur property. These are the only steps affected; if both checks pass, the theorem is correct as intended.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is supported by a correct argument. I re-checked the two load-bearing external inputs. Lemma 3.5 is the standard unconditional-sequence dichotomy and is elementary: a non-weakly-null seminormalized unconditional basic sequence has a functional and a subsequence with |f(u_n)| >= epsilon, giving the lower ell_1 estimate; it is valid. The Schur-property step in the first alternative is also valid because the relative weak topology on a closed subspace coincides with its own weak topology. The only genuine point of care is notational: 'span' in the definitions of F (before Lemma 3.4) and M_S (in the proof of Theorem 1.1) must denote the closed linear span. If taken as the algebraic linear span, F would not be a Banach space and Lemma 3.5 could not be applied, and M_S would not be complete, so its Schur property would not be available. The surrounding text, including calling M_S isometrically isomorphic to ell_1(S) and requiring M closed in Theorem 3.1, shows the intended reading is the closure, and the patch is immediate. With that reading, Theorem 3.1's case split is valid and Theorem 1.1 follows.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a Tychonoff space X and a non-zero symmetric sequence ideal E⊆c0, the existence of a continuous linear surjection T:C_p(X)→E_p (where E_p is E with the pointwise topology inherited from R^N) forces E=c0. The proof has two main parts. Theorem 2.1 shows that a pointwise bounded sequence of finitely supported sign-measures on X has a functionally bounded union of supports and that every f∈C(X) can be replaced by a bounded continuous function with the same values under all the measures. The second part, Theorem 3.1, is an abstract Banach-space result: if E is a non-zero symmetric ideal with a Banach norm continuously embedded in c0, and a bounded surjection Q:Z→E sends the adjoints of the coordinate functionals into a closed Schur subspace M of Z*, then E=c0 as a set. The proof uses the standard dichotomy for unconditional basic sequences (Lemma 3.5) and a careful norm comparison. Theorem 1.1 follows by applying Theorem 3.1 to Z=C_b(X), Q=T_b induced by the original surjection, and M the closed span of point evaluations over the support set S. The paper also proves Theorem 1.3 (every continuous linear operator C_p(X)→(c00)_p has finite-dimensional range) and derives a complete characterization (Corollary 1.4) of when a surjection exists, using the known Josefson–Nissenzweig characterization for C_p(X).","tokens_in":8758,"tokens_out":35149,"duration_ms":343606,"significance":"If the result holds, it is a clean and definitive negative answer to a natural analogue of Rosenthal's quotient theorem for C_p-spaces in the class of symmetric sequence ideals: (c0)_p is the only non-zero symmetric ideal in c0 that can occur as a continuous linear image of a C_p-space. The abstract Banach-space Theorem 3.1 is independently useful and is proved in full detail. The paper is careful and rigorous: the arguments use standard tools (Baire category, Banach–Steinhaus, closed graph theorem, Schur property, unconditional basis dichotomy) and contain no free parameters or circular reasoning. The only substantive external input is Lemma 3.5, a standard result which is cited and valid. The paper also credits the earlier characterization of Banakh–Kąkol–Śliwa and uses it transparently to obtain the final equivalence. Overall, this is a valuable contribution to the Cp-theory and Banach-space literature.","major_comments":[],"minor_comments":[{"comment":"The symbol 'span' must be explicitly declared to mean the closed linear span. With the literal algebraic reading, F is not a Banach space, so Lemma 3.5 cannot be applied, and M_S is not complete, so the Schur property statement is not justified. The intended reading is clear from the context (M is required to be closed in Theorem 3.1, and M_S is said to be isometrically isomorphic to ℓ1(S)), but the notation should be made explicit in a revision.","section":"§3, definition of F before Lemma 3.4; §4, definition of M_S"},{"comment":"The Josefson–Nissenzweig characterization is cited as [2, Theorem 1] in the Introduction and in the proof of Corollary 1.4, but the preamble to Corollary 1.4 refers to [3, Theorem 1]. Reference [3] is the paper 'Josefson–Nissenzweig property for Cp-spaces' and appears to be the intended source; please harmonize the citations.","section":"Introduction and proof of Corollary 1.4"},{"comment":"Reference [5] (Cembranos) is not cited anywhere in the text; it should either be cited in an appropriate place or removed from the bibliography.","section":"References"},{"comment":"There are several small typos: 're-produced' in the abstract, 'Does i the space' in Problem 1.1, 'Kąkol, Saxon initiated' in the Introduction, and a duplicated entry '11' in the citation list '[11, 8, 2, 3, 11, 15, 12]'.","section":"Throughout"},{"comment":"The notation E0 = c00^{||·||_E} is slightly compressed; writing E0 = \\overline{c_{00}}^{||·||_E} would make the definition of E0 as the closure of c00 in E unambiguous.","section":"§3, notation for E0"}],"recommendation":"minor_revision","confidential_remarks":"The paper is sound and well within the journal's scope. The only issue that needs attention is the closed-span convention, which is a local fix; the citation mismatch and minor typos are also easy to correct. I see no reason why the paper should not be published after these small revisions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this paper proves the right theorem and the proof is solid. The main result — if a Tychonoff space X admits a continuous linear surjection C_p(X) → E_p for a non-zero symmetric sequence ideal E ⊆ c_0, then E = c_0 — is genuinely new. Prior work had embedding theorems and the JN-characterization for (c_0)_p, but nobody had excluded surjections onto proper ideals like (ℓ_q)_p. The authors do it by isolating an abstract Banach-space criterion (Theorem 3.1) and applying it via the quotient norm on the range. The dichotomy argument with unconditional bases is clean: either the coordinate functionals are weakly null, which contradicts the open mapping theorem plus Schur, or they contain an ℓ_1-subsequence, which forces the ideal norm to be equivalent to the sup norm on c_00 and hence E = c_0. I checked the two load-bearing inputs. The Baire-category theorem in Theorem 2.1 is correct, and Lemma 3.5 is the standard unconditional dichotomy; the Schur-property steps are valid. The paper also proves a nice standalone result (Theorem 1.3) that every operator C_p(X) → (c_00)_p has finite-dimensional range.\n\nThe only real soft spot is notational: 'span' in the definitions of F (Lemma 3.4) and M_S (proof of Theorem 1.1) must be the closed linear span. As written, the algebraic span is not complete and the Schur property would not apply. But the context makes the intended reading clear — M_S is explicitly said to be isometrically isomorphic to ℓ_1(S), which is the closed span — and the fix is immediate. That is a minor patch, not a gap.\n\nThe citation pattern is honest. The paper leans on [2, Theorem 1] for the full characterization in Corollary 1.4, which is a published external benchmark, and on standard basis theory. No fitted parameters, no circular reasoning. The scope is deliberately narrow: this settles the surjection problem for symmetric sequence ideals in c_0, not the general metrizable quotient problem, and the authors do not oversell it.\n\nWho is this for: people working in Cp-theory, topological function spaces, and maybe Banach-space theorists interested in the pointwise topology. It deserves a serious referee; I would send it out. The referee should ask the authors to clarify the closed-span convention and maybe add a remark to that effect, but the mathematics is sound.","headline":"A clean classification: among symmetric sequence ideals in c_0, only (c_0)_p can be a continuous linear image of a C_p-space; the proof is sound and the only real issue is a shorthand 'span' that should be read as closed span.","tokens_in":9304,"tokens_out":8741,"would_cite":true,"duration_ms":82728,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["54C35","46A03","46B15","46B45"],"pacs":[],"model":"deepseek-v4-flash","headline":"A continuous linear surjection from a C_p-space onto a symmetric sequence ideal E⊆c0 forces E=c0; hence only c0 can be such an image.","keywords":["C_p-space","symmetric sequence ideal","pointwise convergence topology","continuous linear surjection","Josefson–Nissenzweig property","unconditional basic sequence","Schur property","c_0"],"falsifier":"A direct counterexample would settle the question: find a Tychonoff space $X$ and a proper symmetric sequence ideal $E\\subset c_0$ — for instance $E=\\ell_1$ or $E=\\ell_2$ — together with a continuous linear surjection $T:C_p(X)\\to E_p$. The theorem asserts that no such pair exists, so exhibiting one would refute it.","tokens_in":8339,"feed_emoji":"📐","tokens_out":14120,"duration_ms":119222,"temperature":0.7,"pith_summary":"The paper proves that the pointwise topology severely restricts which classical sequence spaces can appear as continuous linear images of a function space $C_p(X)$. Its main theorem: if $E\\subseteq c_0$ is a non-zero symmetric sequence ideal carrying the topology inherited from $\\mathbb{R}^{\\mathbb{N}}$, then a continuous linear surjection $T:C_p(X)\\to E_p$ exists only when $E=c_0$. Hence no proper symmetric ideal — in particular no $\\ell_q$ with $0<q<\\infty$ — can occur as such an image. This settles the surjection problem for the whole class of symmetric sequence ideals in $c_0$ and, combined with the Josefson–Nissenzweig characterization, gives a complete description of all pairs $(X,E)$ for which such a surjection exists.","feed_headline":"Surjections from Cp(X) onto symmetric ideals: only c0 works","feed_subtitle":"No proper symmetric sequence ideal in c0, such as any ℓ_q, can be hit by such a map.","key_machinery":"The central objects are symmetric sequence ideals $E\\subseteq c_0$ — linear subspaces closed under domination of decreasing rearrangements, hence solid, permutation-invariant, and containing $c_{00}$ — and the coordinate functionals $\\delta_n$ on $E$. The proof isolates an abstract Banach-space theorem: if a Banach space $Z$ surjects boundedly onto such an $E$ and the functionals $Q^*\\delta_n$ all lie in a closed subspace $M$ of $Z^*$ with the Schur property, then $E=c_0$. The mechanism is to show that $(\\delta_n)$ is a seminormalized unconditional basis of its span and then apply the classical dichotomy for such bases: either it is weakly null, contradicting the Schur property and the boundedness-from-below of $Q^*$, or it has a subsequence equivalent to the $\\ell_1$ basis, which yields a norm comparison forcing the $c_0$-norm and the $E$-norm to be equivalent on $c_{00}$, and hence $E=c_0$. In the $C_p(X)$ setting, a preliminary theorem constructs the Schur subspace: the finitely supported sign-measures carried by a functionally bounded set $S$ form a subspace of $C_b(X)^*$ isometrically isomorphic to $\\ell_1(S)$, and every coordinate functional $\\pi_n\\circ T$ is one of these sign-measures.","core_discovery":"Let $X$ be a Tychonoff space and let $E\\subseteq c_0$ be a non-zero symmetric sequence ideal, endowed with the pointwise topology $E_p$ inherited from $\\mathbb{R}^{\\mathbb{N}}$. The paper establishes that the existence of a continuous linear surjection $T:C_p(X)\\to E_p$ forces $E=c_0$; equivalently, $(c_0)_p$ is the only non-zero symmetric sequence ideal in $c_0$ that can be a continuous linear image of a $C_p$-space. The proof also yields the stronger statement that every continuous linear operator $T:C_p(X)\\to (c_{00})_p$ has finite-dimensional range, so even non-surjective maps into finite-support sequences are trivial. Combining Theorem 1.1 with the known Josefson–Nissenzweig characterization, the paper concludes that such a surjection exists exactly when $E=c_0$ and $C_p(X)$ has the Josefson–Nissenzweig property, a condition equivalent to $C_p(X)$ containing a complemented copy of $(c_0)_p$ or admitting a quotient isomorphic to $(c_0)_p$.","pith_inferences":["The abstract Banach-space theorem suggests a broader principle: whenever coordinate functionals of a symmetric sequence ideal fall into a Schur subspace of the dual of a surjecting Banach space, the ideal must be $c_0$; one could test this on other spaces of continuous functions, such as $C_b(X)$ with different topologies or spaces of measures.","Because every infinite $C_p(X)$ contains subspaces isomorphic to $(\\ell_q)_p$ for every $0<q\\le\\infty$ (a result the paper cites), the contrast drawn here is clear: such spaces are abundant as subspaces but essentially forbidden as surjective images, pointing to the quotient or surjection structure rather than containment as the restrictive feature.","The proof works for any Banach norm on $E$ compatible with the inclusion into $c_0$; an extension to non-normable locally convex topologies on $E$ beyond the pointwise one might reveal whether the rigidity is purely a Banach-space phenomenon or a feature of the pointwise topology.","Since the only unresolved compact case for metrizable quotients is Efimov compacta, this result narrows what a counterexample would have to look like: if an Efimov compactum $X$ admitted an infinite-dimensional metrizable quotient, that quotient could not be a proper symmetric sequence ideal in $c_0$."],"forward_implications":["For every $0<q<\\infty$, there is no continuous linear surjection $C_p(X)\\to(\\ell_q)_p$; in particular, no $\\ell_q$ with its pointwise topology is a quotient of any $C_p$-space.","A continuous linear surjection $C_p(X)\\to E_p$ exists if and only if $E=c_0$ and $C_p(X)$ has the Josefson–Nissenzweig property; equivalently, $C_p(X)$ contains a complemented copy of $(c_0)_p$ or has a quotient isomorphic to $(c_0)_p$.","Every continuous linear operator $T:C_p(X)\\to(c_{00})_p$ has finite-dimensional range, so even non-surjective maps into finite-support sequences are trivial.","Any metrizable quotient of $C_p(X)$ that is a symmetric sequence ideal inside $c_0$ must be $(c_0)_p$; proper ideals such as $(\\ell_q)_p$ are excluded as quotients, not merely as surjective images."],"supporting_citations":[{"why":"It identifies the dual of $C_p(X)$ with finitely supported sign-measures, so the coordinate functionals $\\pi_n\\circ T$ are legitimate measures to which Theorem 2.1 applies.","marker":"[1]"},{"why":"It supplies the Josefson–Nissenzweig characterization used in Corollary 1.4 to turn Theorem 1.1 into a complete equivalence for the existence of surjections onto $E_p$.","marker":"[3]"},{"why":"It provides the basis criterion (Proposition 4.36) used in Lemma 3.4 to show that the coordinate functionals $(\\delta_n)$ form an unconditional Schauder basis.","marker":"[6]"},{"why":"It supplies the closed graph theorem invoked repeatedly to prove boundedness of diagonal multipliers, coordinate permutations, and the operator $T_b$.","marker":"[7]"},{"why":"It provides the dichotomy lemma (Lemma 4.1) for seminormalized unconditional basic sequences that structures the entire proof of Theorem 3.1.","marker":"[14]"}],"fun_headline_variants":["Only c0 is a symmetric ideal image of C_p(X)","Surjections from C_p(X) to ideals force c0","No proper symmetric ideal is a C_p(X) image","C_p(X) never hits a proper symmetric ideal"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof's two-case analysis rests on the classical dichotomy that a seminormalized unconditional basic sequence is either weakly null or contains a subsequence equivalent to the $\\ell_1$ basis; the norm comparison that forces $E=c_0$ is unavailable if that dichotomy fails.","fun_headline_variants_meta":{"raw":{"variants":["Only c0 is a symmetric ideal image of C_p(X)","Surjections from C_p(X) to ideals force c0","No proper symmetric ideal is a C_p(X) image","C_p(X) never hits a proper symmetric ideal"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001107,"raw_usage":{"total_tokens":4701,"prompt_tokens":1118,"completion_tokens":3583,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":734,"completion_tokens_details":{"reasoning_tokens":3514}},"tokens_in":734,"tokens_out":3583,"duration_ms":29265,"temperature":1.0,"reasoning_tokens":3514,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:24:04.644628+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct counterexample would settle the question: find a Tychonoff space $X$ and a proper symmetric sequence ideal $E\\subset c_0$ — for instance $E=\\ell_1$ or $E=\\ell_2$ — together with a continuous linear surjection $T:C_p(X)\\to E_p$. The theorem asserts that no such pair exists, so exhibiting one would refute it.","supporting_citations":[{"cited_title":"Arkhangel’skii,Topological Function Spaces, Mathematics and its Applications, vol","cited_arxiv_id":null,"evidence_quote":"It identifies the dual of $C_p(X)$ with finitely supported sign-measures, so the coordinate functionals $\\pi_n\\circ T$ are legitimate measures to which Theorem 2.1 applies."},{"cited_title":"Banakh, J","cited_arxiv_id":null,"evidence_quote":"It supplies the Josefson–Nissenzweig characterization used in Corollary 1.4 to turn Theorem 1.1 into a complete equivalence for the existence of surjections onto $E_p$."},{"cited_title":"Fabian, P","cited_arxiv_id":null,"evidence_quote":"It provides the basis criterion (Proposition 4.36) used in Lemma 3.4 to show that the coordinate functionals $(\\delta_n)$ form an unconditional Schauder basis."},{"cited_title":"Jarchow,Locally Convex Spaces, B.G","cited_arxiv_id":null,"evidence_quote":"It supplies the closed graph theorem invoked repeatedly to prove boundedness of diagonal multipliers, coordinate permutations, and the operator $T_b$."},{"cited_title":"Laustsen, J","cited_arxiv_id":null,"evidence_quote":"It provides the dichotomy lemma (Lemma 4.1) for seminormalized unconditional basic sequences that structures the entire proof of Theorem 3.1."}],"review_version":1}