{"id":"d3da4ec2-ec16-4283-96b8-6b67bf5647c5","arxiv_id":"2506.04412","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Maps on B(X) or M_n preserving idempotency of the (scaled) Jordan product in both directions are signed similarity transformations (with field automorphisms and transposition in finite dimensions).","lead":"This paper classifies all maps on algebras of bounded operators that preserve, in both directions, whether a scaled Jordan product of two operators is an idempotent. It shows the only such maps are signed similarity transformations, with field automorphisms and transposition allowed in the matrix case, even when the map is not assumed linear or additive.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 21 is false as stated: A=diag(λ,0,0) in M_3 satisfies its hypothesis but not its conclusion, so the proof chain Lemma 21→22→25→Theorem 1 is currently unsound.","rationale":"The paper's central classification is plausible and the long proof contains many correct-looking reductions, but the proof as written relies on Lemma 21, which is false under its stated hypothesis. My counterexample A=diag(λ,0,0) is a concrete witness: it satisfies the hypotheses (only one nonzero spectral point, all required Jordan products with T_{k,R} are nonzero idempotents) yet violates the conclusion. The reader's weakest_assumption was the range condition, which is indeed a strong explicit assumption but not a flaw; the more serious issue is the invalid lemma inside the reduction. However, because Lemma 22's intended application may only need the stronger version spectrum(B_11)={λ}, the theorem may be repairable. I therefore keep the CONDITIONAL verdict, with the condition being that Lemma 21 be corrected and the applications re-verified. I did not find an ad hominem or theatrical issue; the concern is purely mathematical.","tokens_in":18281,"tokens_out":39433,"duration_ms":326646,"concrete_test":"Compute directly in M_3: set A=diag(1,0,0). For k=1 and arbitrary 1×2 R, and for k=2 and arbitrary 2×1 R, form T_{k,R} as in Lemma 21 and verify A∘T_{k,R} is a nonzero idempotent, while A e_2=0≠e_1+e_2. This settles that Lemma 21 is false as stated. To determine impact on the main theorem, re-check Lemma 22: does its proof establish that B_11 has only eigenvalue λ before invoking Lemma 21? If yes, restate Lemma 21 with spectrum(A)={λ}; if no, construct A,B meeting Lemma 22's hypotheses with B_11≠J_n(λ), and the central classification fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Lemma 21 claims: if A∈M_n has only one nonzero spectral point and A∘T_{k,R} is a nonzero idempotent for every k<n and every rank-one R, then A e_k = J_n(λ)e_k for k<n. This is false for n=3, λ=1, A=diag(1,0,0). Its spectrum is {1,0}, so it has only one nonzero spectral point. For k=1, T_{1,R}=[[1,R],[0,0]] and A∘T_{1,R}=[[1,R/2],[0,0]] (with R a 1×2 row) is idempotent. For k=2, T_{2,R}=[[J_2(1)^{-1},R],[0,0]] with J_2(1)^{-1}=[[1,-1],[0,1]] gives A∘T_{2,R}=[[1,-1/2,r_1/2],[0,0,0],[0,0,0]]-type (after halving), again a nonzero idempotent for every 2×1 R. But the conclusion requires A e_2=e_1+e_2, whereas A e_2=0. The proof breaks when it says A_11 is upper triangular with all diagonal entries λ; this uses the stronger 'single spectral point' assumption, not 'single nonzero spectral point'. Lemma 22 then cites Lemma 21 to conclude B_11=J_n(λ), so the main result depends on this lemma. The fix is to strengthen Lemma 21's hypothesis to spectrum(A)={λ} and confirm that in Lemma 22's application B_11 indeed has singleton spectrum; otherwise the chain of reductions is invalid.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper characterizes maps φ: B(X) → B(X), for a complex Banach space X of dimension at least three, that preserve in both directions the idempotency of the Jordan product A∘B = (1/2)(AB+BA), with no linearity or additivity assumption on φ. The main results are Theorem 1 (infinite-dimensional case): under the assumption that the range of φ contains all nonzero idempotents and anti-idempotents, φ has the form φ(X)=λT XT^{-1} with T bounded invertible linear or conjugate-linear and λ∈{−1,1}, or, only in the reflexive case, φ(X)=λT X' T^{-1}; and Theorem 2 (finite-dimensional case): the same conclusion up to a field automorphism and transposition. Theorems 4 and 5 are scaled versions obtained by a change of variables. The proof proceeds by first showing φ is injective, reducing the action on rank-one idempotents to a bijective orthogonality-preserving map (hence applying Semrl's classification), then extending the identification to nilpotents, tripotents, and Jordan blocks via a long series of auxiliary lemmas.","tokens_in":18618,"tokens_out":21025,"duration_ms":185569,"significance":"If the proof is repaired, this is a substantial result: it removes linearity and additivity assumptions in a preserver problem for the Jordan product and reduces the classification to Semrl's theorem on orthogonality-preserving maps on rank-one idempotents. The paper is self-contained modulo well-established external results, and the step-by-step reduction is clearly organized. The range condition (range containing I±(X)) is strong and is explicitly used in Steps 5 and 7; the theorems are conditional on it, which is a legitimate and clearly stated assumption. The main obstacle to accepting the paper is that Lemma 21 is false as stated, and the proof chain Lemma 21 → Lemma 22 → Lemma 25 → Theorems 1/2 therefore contains a load-bearing error. The error appears to be locally repairable by strengthening the lemma's hypothesis, but repair requires also filling a gap in Lemma 22.","major_comments":[{"comment":"Lemma 21 is false as stated. The hypothesis allows A to have 0 as an additional spectral point, but the proof uses the stronger assumption that A has a single spectral point. A concrete counterexample is A=diag(1,0,0) ∈ M_3 with λ=1. For every k=1,2 and every rank-one R, A∘T_{k,R} is a nonzero idempotent, as can be checked directly from the formula for T_{k,R}; however the conclusion would require Ae_2=e_1+e_2, whereas Ae_2=0. The failure occurs at the line \"Since A is assumed to have a single spectral point, we obtain that A11 is an upper-triangular matrix with all diagonal entries equal to λ\": when the additional spectral point is 0, the (k+1,k+1) entry of A11 can be 0, and A11∘J_{k+1}(λ)^{-1} can be a nonzero idempotent without forcing A11=J_{k+1}(λ). This lemma is load-bearing because Lemma 22 cites it to conclude B11=J_n(λ), and Lemma 22 is used in Step 13 of the proof of Theorems 1/2 and in Lemma 25. The statement should be corrected to require spectrum(A)={λ}, and the proof should be aligned with that hypothesis.","section":"Section 2, Lemma 21"},{"comment":"In the proof of Lemma 22, the sentence \"Since the only eigenvalue of B11 is λ, otherwise we are in a contradiction with Corollary 18 or Lemma 19, it follows that B11 = Jn(λ) from Lemma 21\" is not a complete argument. To apply a corrected Lemma 21 one must prove that spectrum(B11)={λ}; the current text merely asserts this, and the referenced contradictions with Corollary 18 or Lemma 19 are not spelled out. This is not a cosmetic gap: if B11 had an additional spectral point 0, the (false) form of Lemma 21 would be the only available tool, and it cannot yield the conclusion. The authors need to provide a detailed proof that B11 has no eigenvalues other than λ, using the equivalences for F1(X) and T(n−1;λ^{−1}), before invoking Lemma 21. This is load-bearing for Step 13 and hence for Theorems 1 and 2.","section":"Section 2, Lemma 22"},{"comment":"The final reduction in Lemma 25 — \"apply Lemma 22 if λ_k≠0 and Lemma 24 otherwise\" to a direct sum of distinct Jordan blocks — is only sketched. Lemmas 22 and 24 are stated for a single Jordan block J_n(λ) in the top-left corner of a 2×2 block decomposition, not for a direct sum of several blocks with distinct eigenvalues. The intended argument is plausible: one reorders the decomposition so that each Jordan block in turn occupies the distinguished subspace Y and applies the corresponding lemma, using the block-diagonal structure of A11. However, this needs to be written out, because Lemma 25 is used in Step 2 to prove injectivity of φ and again in Step 14 to complete the proof of Theorems 1/2. As it stands, the proof of Lemma 25 relies on an unstated generalization of Lemmas 22 and 24.","section":"Section 2, Lemma 25"}],"minor_comments":[{"comment":"The phrase \"and cα, c2 α = σ(α)/α\" is malformed; it should assert the existence of a scalar cα satisfying cα² = σ(α)/α.","section":"Theorem 5 statement"},{"comment":"In the finite-dimensional part of the proof, \"by Theorem 5\" should read \"by Theorem 2\", since Theorem 5 is the statement being proved.","section":"Proof of Theorems 4 and 5"},{"comment":"The sentence \"Lemma 16 provides that φ(T)=B11⊕0\" is a mis-citation: the conclusion that φ(T) has the block-diagonal form B11⊕0 follows from Lemma 14, not Lemma 16.","section":"Step 13 of the proof of Theorems 1 and 2"},{"comment":"References [16] and [17] are identical (both list Semrl, \"Non-linear commutativity preserving maps\", Acta Sci. Math. (Szeged) 71 (2005), 781–819); the duplicate entry should be removed.","section":"References"},{"comment":"In reference [9], \"2th ed.\" should be \"2nd ed.\".","section":"References"},{"comment":"The word \"sclalars\" should be \"scalars\".","section":"Page 13, Proposition 23"}],"recommendation":"major_revision","confidential_remarks":"The paper has a genuine and significant contribution to make, and the overall strategy is convincing. The false Lemma 21 is a load-bearing error, but it is localized and appears repairable by strengthening the hypothesis to spectrum(A)={λ} and supplying the missing argument in Lemma 22. I therefore favor major revision rather than rejection. The range condition in Theorems 1 and 2 is very strong; the authors should perhaps state more explicitly that it is essential and not merely a technical convenience. The paper fits the scope of math.FA and this journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Let me get straight to it: the stress-test note is right, and it matters. Lemma 21, as stated, is false. Take n=3, λ=1, A=diag(1,0,0). A has exactly one nonzero spectral point. For k=1, with T_{1,R}=[[1,R],[0,0]], the Jordan product A∘T_{1,R}=2[[1,R/2],[0,0]], which is a nonzero idempotent for every row R. For k=2, T_{2,R}=[[J_2(1)^{-1},R],[0,0]] gives A∘T_{2,R}=[[2,-1,r],[0,0,0],[0,0,0]] (up to halving), again a nonzero idempotent. But A e_2=0, while the conclusion demands A e_2=e_1+e_2. So the lemma's hypothesis does not force the claimed conclusion. The proof step 'single spectral point' is doing work that the stated 'single nonzero spectral point' does not license; zero eigenvalues are allowed.\n\nThis is not a cosmetic issue. Lemma 22 cites Lemma 21 to get B11=J_n(λ); Lemma 25 uses Lemma 22 (and 24) to get injectivity of φ at Step 2 and the final identity at Step 14, and Step 13 uses Lemma 22 directly. The main classification therefore currently rests on a false statement. The likely fix is to strengthen Lemma 21 to spectrum(A)={λ}, and to show separately in Lemma 22 that B11 indeed has no zero eigenvalue—the present one-line 'otherwise contradiction' is not enough. The direct-sum situation in Lemma 25 is also only sketched; that is minor compared with Lemma 21 but should be spelled out.\n\nAll that said, this is not a crank paper. The classification result is genuinely new, the no-linearity/no-additivity setting is the right one, and the overall strategy—reduce to Semrl's theorem on rank-one idempotents, then extend by technical matrix lemmas—is sensible and mostly executed carefully. Lemmas 7-20 and the Step 1-14 architecture are readable and largely correct. The range condition is strong but explicit and load-bearing. The duplicated reference [16]/[17] is a typo.\n\nMy bottom line: this paper deserves a serious referee, but it is not ready as-is. I would send it to review with a specific request to fix Lemma 21 or replace the chain through Lemma 22. If the fix is routine (and I suspect it is), the paper becomes a solid contribution to preserver theory. If the spectral gap cannot be closed, the main theorems are unsupported.","headline":"Solid classification result, but Lemma 21 is false as stated, and the main proof currently depends on it.","tokens_in":19163,"tokens_out":6008,"would_cite":false,"duration_ms":54403,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47B49"],"pacs":[],"model":"deepseek-v4-flash","headline":"Any map on bounded operators that preserves, in both directions, whether a Jordan product is idempotent must be a similarity by an invertible linear or conjugate-linear operator, up to sign; in finite dimensions only field automorphisms…","keywords":["Jordan product","idempotency preserver","bounded linear operators","Banach space","rank-one idempotents","tripotents","field automorphisms","non-additive maps"],"falsifier":"Search, for instance in $M_3(\\mathbb C)$, for a map $\\phi$ whose range contains all nonzero idempotent and anti-idempotent matrices and such that $A\\circ B$ is idempotent exactly when $\\phi(A)\\circ\\phi(B)$ is idempotent, yet $\\phi(P)$ is not a rank-one idempotent for some rank-one idempotent $P$. Step 5 of the proof derives that implication directly, so finding such an example would contradict the theorem; a small matrix search is a concrete way to test the classification.","tokens_in":18054,"feed_emoji":"🔁","tokens_out":13720,"duration_ms":113360,"temperature":0.7,"pith_summary":"This paper determines the structure of arbitrary maps on the algebra of bounded linear operators over a complex Banach space that preserve, in both directions, the statement \"the Jordan product of two operators is idempotent\". No linearity or additivity is assumed. If the map's range contains all nonzero idempotent and anti-idempotent operators, then in infinite dimensions the map must be a similarity by a bounded invertible linear or conjugate-linear operator, up to a sign, or a similarity through the adjoint when the space is reflexive; in finite dimensions the same rigidity holds up to a field automorphism and transposition. The point is that a purely relational condition, idempotency of $A\\circ B$, is rigid enough to recover the full algebraic form of the map.","feed_headline":"Jordan-idempotency preserving maps reduce to similarities","feed_subtitle":"A purely relational condition on operator pairs is rigid enough to recover the full algebraic form of the map.","key_machinery":"The proof is carried by probe sets inside $\\mathcal B(X)$: rank-one idempotents, nilpotents of rank one, tripotents (operators with $A^3=A$), and Jordan-block-like operators $T(k;\\lambda)$. The core identity is Lemma 16, which says that for nontrivial idempotents $P,Q$ the orthogonality $P\\perp Q$ is equivalent to $P\\circ Q=0$ and to $-P\\circ Q$ being idempotent. This converts the global idempotency condition into orthogonality preservation on rank-one idempotents, where a known classification of bijective maps preserving orthogonality in both directions produces the underlying operator $T$. Lemmas 7 through 10 transfer the relation from rank-one idempotents to nilpotents and tripotents, Lemma 21 identifies Jordan blocks through a Sylvester equation, and Lemma 25 shows that two operators coincide once their Jordan products with every probe in a similarity-invariant test set have the same idempotency and zero pattern.","core_discovery":"For $X$ of dimension at least three, let $\\phi:\\mathcal B(X)\\to\\mathcal B(X)$ have range containing $I^{\\pm}(X)$, that is, every nonzero idempotent $P$ with $P^2=P$ and every anti-idempotent $Q\\neq0$ with $Q^2=-Q$, and assume that $A\\circ B$ is idempotent exactly when $\\phi(A)\\circ\\phi(B)$ is idempotent, where $A\\circ B=\\frac12(AB+BA)$. Theorem 1 says that in infinite dimensions $\\phi$ must be $\\phi(X)=\\lambda TXT^{-1}$ for a bounded invertible linear or conjugate-linear operator $T$ and $\\lambda\\in\\{1,-1\\}$, or $\\phi(X)=\\lambda TX'T^{-1}$ implemented through the adjoint, the second possibility only when $X$ is reflexive. Theorem 2 says that on $M_n$, $n\\ge3$, the same assumption yields $\\phi([x_{ij}])=\\lambda T[\\sigma(x_{ij})]^{\\diamond}T^{-1}$, where $\\sigma$ is a field automorphism of $\\mathbb C$, $\\diamond$ is the identity or transposition, and $\\lambda=\\pm1$. This is the authors' sense of determination: the idempotency equivalence, with no additivity assumed, has only these algebraic solutions.","pith_inferences":["One testable extension is whether the range condition can be weakened to containing only nonzero idempotents; the proof uses anti-idempotents essentially in Step 6 to establish $\\phi(-P)=-\\phi(P)$, so the symmetric condition $I^{\\pm}$ appears necessary for the stated conclusion.","The probe-set strategy seems ready-made for longer Jordan products $A_1\\circ\\cdots\\circ A_k$ or for the Jordan triple product $ABA$; the paper's lemmas already isolate the local information such variants would need.","The theorem implies a practical rigidity statement for any setting that models operations by idempotent transformations: an exact preserver of the idempotency relation is, up to the listed alterations, a change of basis.","Dimensions one and two are excluded; checking whether the conclusion holds there would show whether the dimensional restriction is technical or intrinsic."],"forward_implications":["Idempotency preservation for the Jordan product is a complete invariant: the equivalence alone, together with the range condition, forces the map to be a global similarity or its adjoint analogue.","The scaled variants for a fixed nonzero $\\alpha$, preserving when $\\alpha(AB+BA)$ is idempotent, follow by the change of variables $\\psi(X)=\\sqrt{2\\alpha}\\,\\phi(X/\\sqrt{2\\alpha})$, so the same rigidity holds for every rescaling of the Jordan product.","In finite dimensions the only extra freedom is a field automorphism and transposition; for continuous automorphisms of $\\mathbb C$, which are only the identity and complex conjugation, the classification reduces to linear or conjugate-linear similarities.","Any such map must send rank-one idempotents bijectively onto rank-one idempotents and preserve orthogonality in both directions, so the whole classification is controlled by the action on one-dimensional idempotents."],"supporting_citations":[{"why":"supplies the classification of bijective maps on rank-one idempotents preserving orthogonality in both directions, invoked in Step 7 to produce the operator $T$.","marker":"[16]"},{"why":"provides Lemma 7's criterion for $A\\circ(x\\otimes f)$ to be a nonzero idempotent in terms of $Ax$ or $A'f$; the probe arguments use this throughout.","marker":"[6]"},{"why":"gives the representation of tripotents as $P-Q$ with unique orthogonal idempotents $P,Q$, used in Step 9 and Corollary 20.","marker":"[2]"},{"why":"supplies the uniqueness theorem for the Sylvester equation that identifies Jordan-block matrices in Lemma 21.","marker":"[9]"},{"why":"shows the existence of many non-continuous automorphisms of $\\mathbb C$, which motivates the field-automorphism term in the finite-dimensional classification.","marker":"[12]"}],"fun_headline_variants":["No additivity needed: Jordan-idempotency forces similarity","Jordan-idempotency equivalence forces operator similarities","Purely relational condition recovers similarity form","Jordan idempotency preservation: maps are similarities","No linearity assumed: Jordan idempotency implies similarity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing assumption is that the map's range contains every operator $P$ with $P^2=P$ and every $Q\\neq0$ with $Q^2=-Q$; without that supply of test operators the proof cannot establish bijectivity and orthogonality preservation on rank-one idempotents, and the classification argument stops.","fun_headline_variants_meta":{"raw":{"variants":["No additivity needed: Jordan-idempotency forces similarity","Jordan-idempotency equivalence forces operator similarities","Purely relational condition recovers similarity form","Jordan idempotency preservation: maps are similarities","No linearity assumed: Jordan idempotency implies similarity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000743,"raw_usage":{"total_tokens":3293,"prompt_tokens":904,"completion_tokens":2389,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":520,"completion_tokens_details":{"reasoning_tokens":2312}},"tokens_in":520,"tokens_out":2389,"duration_ms":18012,"temperature":1.0,"reasoning_tokens":2312,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T10:44:12.564955+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Search, for instance in $M_3(\\mathbb C)$, for a map $\\phi$ whose range contains all nonzero idempotent and anti-idempotent matrices and such that $A\\circ B$ is idempotent exactly when $\\phi(A)\\circ\\phi(B)$ is idempotent, yet $\\phi(P)$ is not a rank-one idempotent for some rank-one idempotent $P$. Step 5 of the proof derives that implication directly, so finding such an example would contradict the theorem; a small matrix search is a concrete way to test the classification.","supporting_citations":[{"cited_title":"Fang, Linear maps preserving the idempotency of Jordan products of operators, Electron","cited_arxiv_id":null,"evidence_quote":"provides Lemma 7's criterion for $A\\circ(x\\otimes f)$ to be a nonzero idempotent in terms of $Ax$ or $A'f$; the probe arguments use this throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"gives the representation of tripotents as $P-Q$ with unique orthogonal idempotents $P,Q$, used in Step 9 and Corollary 20."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the uniqueness theorem for the Sylvester equation that identifies Jordan-block matrices in Lemma 21."},{"cited_title":"Kestelman, Automorphisms in the field of complex numbers , Proc","cited_arxiv_id":null,"evidence_quote":"shows the existence of many non-continuous automorphisms of $\\mathbb C$, which motivates the field-automorphism term in the finite-dimensional classification."}],"review_version":1}