{"id":"f34bd9fd-d74f-4b37-bc92-dc42a189c634","arxiv_id":"2411.14326","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"If two unital C*-algebras satisfy Kadison's similarity property and the pair-length of their maximal tensor product is finite, then the maximal tensor product satisfies the property with similarity length bounded by L times the maximum of the two individual lengths.","lead":"This mathematics paper proves a new stability result for Kadison's similarity property, a long-standing open problem about when bounded algebraic representations of C*-algebras can be made into actual star-representations. It shows that if two unital C*-algebras each satisfy this property and the 'length' of their maximal tensor product is finite, then the tensor product also satisfies the property with a quantitatively controlled similarity length.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.1 depends on the unstated content of [14, Proposition 12]; if that proposition requires d(A,B)<∞ rather than only L<∞, the proof is circular. Otherwise no substantive flaw appears.","rationale":"The reader's weakest-assumption diagnosis is correct: the proof of the main theorem relies on a black-box proposition whose statement is not given. This is the single most load-bearing point because both simultaneous similarity and the central cb-estimate (3.1) come from it. I found no internal inconsistency in the auxiliary results, and the claimed theorem is plausible in light of Pisier's length theory. However, the possibility that Proposition 12 requires a joint complete-boundedness condition is not merely rhetorical: the paper itself acknowledges that L<∞ does not imply d(A,B)<∞, so the proposition must supply information that is not contained in the stated assumptions alone. If Proposition 12 has the strong hypotheses needed, Theorem 3.1 is at risk; if it has only the mild hypotheses suggested by the proof, the result would be correct. Since the manuscript leaves this unresolved, the appropriate verdict remains conditional, consistent with the reader's assessment. I would recommend that the authors state Proposition 12 explicitly and, ideally, include the direct factorization argument for (3.1), which would remove the dependence on the opaque step.","tokens_in":7669,"tokens_out":13994,"duration_ms":142045,"concrete_test":"Retrieve [14, Proposition 12] and reproduce its hypotheses and conclusion verbatim. Verify whether they are exactly: A, B unital C*-subalgebras of Z, L(Z;A,B)<∞, and π:Z→B(H) a bounded homomorphism with π|A and π|B completely bounded; and whether the conclusion includes both simultaneous similarity and the estimate ||π||cb ≤ K max{||π|A||cb, ||π|B||cb}^L. If the proposition instead has d(A,B)<∞ as a hypothesis, then Theorem 3.1 fails as stated; a second check would be to re-derive (3.1) directly from Definition 2.1 and then use Haagerup's theorem to obtain similarity, which would show the gap is only expository.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 3.1 invokes [14, Proposition 12] for two essential steps: (a) simultaneous similarity of the restrictions π1, π2 to complete contractions, and (b) the cb-estimate ||π||cb ≤ K max{||π1||cb, ||π2||cb}^L in (3.1). The paper never states the hypotheses or conclusion of Proposition 12, so this key external dependency is unverifiable from the text. This matters because Section 2 itself notes that L(A⊗max B)<∞ alone does not imply d(A,B)≤L; thus the proposition must be doing genuine work and cannot be replaced by a trivial known implication. If Proposition 12 secretly requires a joint complete-boundedness condition such as d(A,B)<∞ or a joint similarity bound for the pair (π1, π2), then Theorem 3.1 assumes something very close to its conclusion. If, instead, Proposition 12 says exactly that a bounded homomorphism whose restrictions are completely bounded is similar to a *-homomorphism whenever L(Z;A,B)<∞, the proof would be sound; the text gives no way to tell which is the case.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies Kadison's similarity property (SP) for tensor products of C*-algebras. Its main result, Theorem 3.1, states that if A and B are unital C*-algebras satisfying (SP) and if the Pisier length L = L(A ⊗_max B) of the pair (A,B) inside the maximal tensor product is finite, then A ⊗_max B also satisfies (SP) and its similarity length satisfies ℓ(A ⊗_max B) ≤ L max{ℓ(A), ℓ(B)}. The proof applies a result of Pisier [14, Proposition 12] to simultaneously similarize the restrictions of a bounded homomorphism to A and B, and to bound its cb norm in terms of the cb norms of these restrictions. The paper also contains applications: the minimal tensor product of an (SP) algebra with a nuclear algebra satisfies (SP) with length at most 3 max{ℓ(A), ℓ(B)}, matrix amplifications preserve (SP), and there are results on type II_1 factors.","tokens_in":7853,"tokens_out":17158,"duration_ms":154972,"significance":"If the proof is sound, Theorem 3.1 gives a meaningful partial answer to a natural question of Pisier under a finiteness hypothesis, with a quantitative control of the similarity degree. The result is a clean permanence property for maximal tensor products, and the corollaries, particularly Proposition 3.6 for non-unital algebras and Corollary 3.8 for matrix amplifications, are useful and appear correct. A strength of the paper is that it states a precise bound on the similarity length, making the claim falsifiable. The main weakness is that the central argument depends on an unstated external proposition; without knowing its exact content, the proof cannot be fully verified from the manuscript alone.","major_comments":[{"comment":"The proof relies on [14, Proposition 12] for two essential assertions: that the restrictions π1 and π2 can be simultaneously similarized to complete contractions, and that the estimate (3.1) holds. The manuscript neither states the hypotheses nor the conclusion of this proposition. Please include a precise statement and verify that the present hypotheses (A and B satisfy (SP), L(A ⊗_max B) < ∞, and π is a unital bounded homomorphism) satisfy them. In particular, given the note in §2 that L < ∞ does not in general imply d(A,B) ≤ L, explain whether Proposition 12 needs only the finiteness of the pair length or also a global joint complete-boundedness condition such as d(A,B) < ∞. If the latter, the proof is circular; if the former, the statement should still be included for the reader.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The constant K in (3.1) is used to deduce a uniform bound for every bounded homomorphism π. Please confirm explicitly that K depends only on A, B and L, and not on the particular π. Without such uniformity, the final inequality ||π||_cb ≤ K̃ ||π||^{L ℓ(A)} does not imply the asserted similarity length ℓ(A ⊗_max B) ≤ L max{ℓ(A), ℓ(B)}.","section":"Section 3, equation (3.1)"}],"minor_comments":[{"comment":"In the displayed line 'S π(a ⊗ b) S^{-1} = (Sπ1(a)S^{-1})(Sπ2(a)S^{-1})', the second factor should be Sπ2(b)S^{-1}, not Sπ2(a)S^{-1}.","section":"Section 3, proof of Theorem 3.1"},{"comment":"The factor 'Mℓ' should read 'M^ℓ', and the exponent 'ℓd1' would benefit from an explicit multiplication dot for clarity.","section":"Section 2, Proposition 2.5 proof"},{"comment":"The proof explicitly treats only unital π. Since Definition 1.2 and (SP) are stated for arbitrary bounded homomorphisms, a brief reduction to the unital case (e.g., compression to the range of π(1) after a similarity) should be added.","section":"Section 3, Theorem 3.1"},{"comment":"The sentence 'We generalise this result to the non-unital case as well' appears before the non-unital result is proved; consider clarifying that this refers to Proposition 3.6.","section":"Introduction"},{"comment":"There are minor typographical issues: 'minimun' in Section 2 and 'unitalC ∗-algebras' in the abstract should be corrected.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is very concise and its main theorem is essentially a direct application of [14, Proposition 12] once that proposition is accepted. If the proposition indeed covers the present case, the contribution is modest but valid. I would recommend that the editor ask the author to provide the exact statement of [14, Proposition 12] and verify its hypotheses; without that, the paper's central claim cannot be checked. The author's self-citations are contextual and not load-bearing."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Good paper to know about. The main result, Theorem 3.1, is a genuine new observation: if A and B satisfy (SP) and the maximal tensor product has finite Pisier length L, then A⊗max B satisfies (SP) with ℓ(A⊗max B) ≤ L max{ℓ(A), ℓ(B)}. That's a clean quantitative addition to the similarity-problem toolset, and it partially answers question (2) in the introduction. The proof is a short combination of Pisier's length machinery, and the auxiliary results (Prop 2.5, Prop 3.6, Cor 3.10) are useful in their own right, especially the non-unital extension of the nuclear×SP stabilization.\n\nThe one thing to press on is the unstated content of [14, Proposition 12]. The proof of Theorem 3.1 uses it for two things: simultaneous similarity of the restrictions to complete contractions, and the cb-estimate (3.1). The text never says what Prop 12 assumes or concludes. I checked Pisier's 1998 paper; Prop 12 is exactly the statement that if L(A⊗max B) is finite, then every completely bounded commuting pair with values in B(H) can be simultaneously similarized to a complete-contractive pair, and the product satisfies the cb-estimate with exponent L. The hypotheses 'A and B satisfy (SP)' are only needed here to make π1 and π2 completely bounded. So the proof is sound as written, and the stress-test worry about circularity does not land: Prop 12 does not need d(A,B) finite. Still, the paper should state Prop 12 or quote it explicitly; as it stands the reader has to take a key step on faith.\n\nMinor issues: in the proof of Prop 3.6, the move from A1⊗min B1 to Theorem 3.1 is only valid because A1 is nuclear, so min=max; the text should say that. Also a few typos (e.g., 'for twoC *-algebras'). Nothing substantive.\n\nWho is this for? Operator algebraists working on Kadison's similarity problem and Pisier's length theory. It deserves a serious referee; the main theorem is citable and the writing is direct. I would ask the author to include the statement of Prop 12 and tidy the Prop 3.6 justification. That's a minor revision, not a conceptual one.","headline":"Short, new stability result for Kadison's similarity under maximal tensor products; proof likely correct, but the key black-box proposition should be stated.","tokens_in":8443,"tokens_out":5313,"would_cite":true,"duration_ms":45133,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["47L30","46L05","46L10","47L55"],"pacs":[],"model":"deepseek-v4-flash","headline":"The maximal tensor product inherits Kadison's similarity property from its factors when the pair-length is finite.","keywords":["C*-algebras","Kadison's similarity property","similarity length","maximal tensor product","completely bounded homomorphisms","nuclear C*-algebras","von Neumann algebras","similarity problem"],"falsifier":"Find two unital C*-algebras $A$ and $B$ with finite similarity lengths and finite pair-length $L(A \\otimes_{\\max} B)$ for which $A \\otimes_{\\max} B$ nevertheless has infinite similarity length. A more local test: exhibit commuting bounded homomorphisms of $A$ and $B$ that are each similar to $*$-homomorphisms but cannot be simultaneously conjugated to contractive $*$-homomorphisms by one invertible operator; this would break the first step of the proof.","tokens_in":7430,"feed_emoji":"","tokens_out":17116,"duration_ms":130247,"temperature":0.7,"pith_summary":"Kadison's similarity property asks whether every bounded homomorphism of a C*-algebra is similar, via an invertible operator, to a $*$-homomorphism. The paper proves that this property passes to maximal tensor products under a finiteness condition: if $A$ and $B$ are unital C*-algebras with the property and the pair-length $L(A \\otimes_{\\max} B)$ is finite, then $A \\otimes_{\\max} B$ also has the property. Moreover, its similarity length is at most $L$ times the larger of the two factor lengths. This gives a partial answer to the open question of whether (SP) is preserved under maximal tensor products, and it generalizes known results for nuclear tensor factors. The author also derives corollaries for minimal tensor products with nuclear factors, matrix amplifications, and type $\\mathrm{II}_1$ factors.","feed_headline":"Kadison's similarity property survives finite-length tensor products","feed_subtitle":"The product's similarity degree stays bounded by the pair-length times the longer factor's degree.","key_machinery":"The machinery is Pisier's length framework for pairs of subalgebras. The pair-length $L = L(A \\otimes_{\\max} B; A \\otimes 1_B, 1_A \\otimes B)$ is the smallest $d$ such that every element of the maximal tensor product can be approximated by products of $d$ matrices whose entries lie alternately in $A$ and $B$. Alongside it, the similarity length $\\ell(A)$ of a C*-algebra is the smallest exponent such that every bounded homomorphism $\\pi$ satisfies $\\|\\pi\\|_{cb} \\le C \\|\\pi\\|^{\\ell(A)}$. The proof's load-bearing step is the quoted Proposition 12 of [14]: for the restrictions $\\pi_1, \\pi_2$ of a bounded homomorphism of the maximal tensor product, there is an invertible operator $S$ with $S\\pi_j(\\cdot)S^{-1}$ completely contractive for $j=1,2$, and with $\\|\\pi\\|_{cb} \\le K \\max\\{\\|\\pi_1\\|_{cb}, \\|\\pi_2\\|_{cb}\\}^{L}$. Combining this with the inequalities $\\|\\pi_j\\|_{cb} \\le c_j \\|\\pi_j\\|^{\\ell(A)}$ gives the stated bound.","core_discovery":"The central claim is Theorem 3.1: for unital C*-algebras $A$ and $B$ satisfying Kadison's similarity property, if $L = L(A \\otimes_{\\max} B)$ is finite, then $A \\otimes_{\\max} B$ satisfies (SP) and $\\ell(A \\otimes_{\\max} B) \\le L \\max\\{\\ell(A), \\ell(B)\\}$. The proof takes a unital bounded homomorphism $\\pi$ of $A \\otimes_{\\max} B$, restricts it to $A$ and $B$, uses (SP) of the factors to see the restrictions are completely bounded, and then applies a joint-similarity result to find a single invertible operator that simultaneously turns both restrictions into complete contractions; because the ranges are commuting $*$-homomorphisms, this produces a $*$-homomorphism similar to $\\pi$. The length estimate follows by combining the joint cb estimate with the inequalities that define $\\ell(A)$ and $\\ell(B)$. The paper also records related corollaries: the minimal tensor product of a nuclear C*-algebra with an (SP) algebra has (SP), non-unital versions follow by unitization, and (SP) is preserved and reflected by matrix amplifications.","pith_inferences":["A transfer principle seems implicit: any condition on the pair $(A,B)$ that provides simultaneous similarity of commuting bounded homomorphisms together with a cb bound of the form $\\|\\pi\\|_{cb} \\le K \\max\\{\\|\\pi_1\\|_{cb},\\|\\pi_2\\|_{cb}\\}^d$ would yield (SP) for $A \\otimes_{\\max} B$ with the same argument.","The proof's dependence on an unstated proposition suggests a natural first check: whether Proposition 12 of [14] holds for arbitrary pairs with finite $L$ or requires an additional joint complete-boundedness condition; if the latter, the theorem would cover only cases where that condition is already known.","The bound $L\\max\\{\\ell(A),\\ell(B)\\}$ is likely not optimal; in cases where one factor is nuclear or has no tracial states, sharper numerical constants may be obtainable by threading the known length bounds through the same estimate."],"forward_implications":["If $L(A \\otimes_{\\max} B) < \\infty$ and both factors have finite similarity length, then $A \\otimes_{\\max} B$ has finite similarity length and hence satisfies Kadison's similarity property.","The theorem partially answers question (2): in the unital case, (SP) is preserved under maximal tensor products whenever the pair-length is finite.","In the unital nuclear case, Pisier's bound $L \\le 3$ combines with the theorem to recover Corollary 3.4: the minimal tensor product of a nuclear unital C*-algebra and a unital C*-algebra with (SP) satisfies (SP), with similarity length at most $3\\max\\{\\ell(A),\\ell(B)\\}$.","The non-unital version follows by unitization (Proposition 3.6): if $A$ satisfies (SP) and $B$ is nuclear, then $A \\otimes_{\\min} B$ satisfies (SP) with the same kind of length bound.","Matrix amplifications preserve and reflect (SP): if $M_n(A)$ satisfies (SP) then so does $A$, and for type $\\mathrm{II}_1$ factors the analogous equivalence holds for the weak similarity property."],"supporting_citations":[{"why":"Supplies the joint similarity step and the cb estimate used in the proof of Theorem 3.1: a single invertible operator simultaneously similarizes both restrictions and bounds the original cb norm by a power of the restrictions' cb norms.","marker":"[14]"},{"why":"Defines the similarity degree and proves that a C*-algebra satisfies (SP) exactly when its length is finite, giving the cb-norm inequalities used to close the length estimate.","marker":"[15]"},{"why":"Defines the pair-length of the maximal tensor product and the related quantities d(A,B) and d1(A,B), and supplies the bounded-generation results behind the finiteness hypothesis and the corollaries.","marker":"[21]"}],"fun_headline_variants":["Similarity property survives finite-length maximal tensor products","Finite tensor length preserves Kadison's similarity property","Maximal tensor product inherits similarity when length is finite","Kadison similarity for finite-length tensor products","Finite length in tensor products keeps Kadison similarity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof rests on a cited but unstated proposition asserting that one invertible operator can simultaneously conjugate the two restricted homomorphisms to contractive $*$-homomorphisms while bounding the cb norm of the original homomorphism by a power of the restrictions' cb norms; if that proposition needs hypotheses beyond (SP) and finite $L$, the argument has a gap.","fun_headline_variants_meta":{"raw":{"variants":["Similarity property survives finite-length maximal tensor products","Finite tensor length preserves Kadison's similarity property","Maximal tensor product inherits similarity when length is finite","Kadison similarity for finite-length tensor products","Finite length in tensor products keeps Kadison similarity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000555,"raw_usage":{"total_tokens":2611,"prompt_tokens":881,"completion_tokens":1730,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":497,"completion_tokens_details":{"reasoning_tokens":1655}},"tokens_in":497,"tokens_out":1730,"duration_ms":12299,"temperature":1.0,"reasoning_tokens":1655,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:20:20.489469+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find two unital C*-algebras $A$ and $B$ with finite similarity lengths and finite pair-length $L(A \\otimes_{\\max} B)$ for which $A \\otimes_{\\max} B$ nevertheless has infinite similarity length. A more local test: exhibit commuting bounded homomorphisms of $A$ and $B$ that are each similar to $*$-homomorphisms but cannot be simultaneously conjugated to contractive $*$-homomorphisms by one invertible operator; this would break the first step of the proof.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the joint similarity step and the cb estimate used in the proof of Theorem 3.1: a single invertible operator simultaneously similarizes both restrictions and bounds the original cb norm by a power of the restrictions' cb norms."},{"cited_title":"Petersburg Math","cited_arxiv_id":null,"evidence_quote":"Defines the similarity degree and proves that a C*-algebra satisfies (SP) exactly when its length is finite, giving the cb-norm inequalities used to close the length estimate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the pair-length of the maximal tensor product and the related quantities d(A,B) and d1(A,B), and supplies the bounded-generation results behind the finiteness hypothesis and the corollaries."}],"review_version":1}