{"id":"7d7c21f2-ad44-4690-9b39-4b00572e6cd4","arxiv_id":"2607.17824","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every non-degenerate product system over a unital subsemigroup of a discrete group, the reduced Hao-Ng isomorphism holds for generalised gauge actions of any locally compact Hausdorff group.","lead":"This paper proves the reduced Hao-Ng isomorphism theorem for non-degenerate product systems over arbitrary subsemigroups of discrete groups, with actions by any locally compact Hausdorff group. It uses a Plancherel-weight averaging technique to bypass the missing conditional expectation in the non-discrete case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No internal flaw identified; the Hao–Ng conclusion is exactly as secure as the unverified preprint [15, Theorem 4.4] on which the final C*-envelope step rests.","rationale":"The reader's weakest-assumption analysis identified the same point: Theorem 4.7's final C*-envelope step depends on the Dor–On–Thompson preprint [15, Theorem 4.4]. My independent reading of the internal argument found no additional load-bearing flaw. The Plancherel-weight localisation in Lemmas 4.3–4.5 is used correctly in Theorem 4.6, and the application of [27, Corollary 4.1] is legitimate once injective Fock covariance and a normal coaction are established. The possible notational looseness in the proof of Lemma 4.3(iii) can be repaired by applying (3.2) directly to f=z u_i and h=w u_i, so it does not threaten the conclusion. Thus the only substantive exposure is external: the theorem is exactly as reliable as [15]. Since the reader already flagged this dependence and accepted with moderate confidence, my recommendation does not change the verdict.","tokens_in":23488,"tokens_out":42946,"duration_ms":414139,"concrete_test":"Obtain arXiv:2505.00587v3 and independently re-derive Theorem 4.4 in the special case A=Tλ(X)+: verify that the canonical map C*_env(A)⋊_{α,λ}H → C*_env(A⋊_{α,λ}H) is an isomorphism without invoking hyperrigidity, and that the proof remains valid for an arbitrary locally compact Hausdorff group H with no second-countability assumption. If the derivation succeeds, Theorem 4.7 follows; if extra hypotheses are needed, the central Hao–Ng claim must be weakened accordingly.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The new localisation machinery in Theorem 4.6 is internally coherent: the Plancherel-weight averaging argument, the approximate unit from non-degeneracy, and the verification of Fock covariance via Theorem 4.1 are all consistent, and together with [27, Corollary 4.1] they yield the completely isometric tensor-algebra isomorphism of Theorem 4.7. The only step that converts this into the asserted reduced Hao–Ng isomorphism is the identification C*_env(Tλ(X)+ ⋊_{α,λ}H) ≃ C*_env(Tλ(X)+) ⋊_{\\dot α,λ}H, imported from the preprint [15, Theorem 4.4]. That theorem is not proved or even sketched in this paper, and it is the unique bridge from non-selfadjoint Fock covariance to the selfadjoint strong-covariant C*-algebra isomorphism. If [15] contains an error or carries an unstated hypothesis not satisfied by Tλ(X)+ — for example hyperrigidity, separability, or second-countability of H — then the final ∗-isomorphism of Theorem 4.7 does not follow, even though Theorem 4.6 stands. The non-degeneracy of X is used precisely to supply the contractive approximate unit required by [15]; that hypothesis is satisfied, but the validity of [15] itself is not independently verified here.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves that for a non-degenerate concrete product system X over a unital subsemigroup P of a discrete group G and a generalised gauge action α of a locally compact Hausdorff group H on the Fock algebra T_λ(X), the identity representation ι⋊ : X⋊_{α,λ}H → T_λ(X)⋊_{α,λ}H is injective, Fock covariant, and admits a normal coaction by G (Theorem 4.6). It then derives a canonical completely isometric isomorphism T_λ(X⋊_{α,λ}H)+ ≃ T_λ(X)+ ⋊_{α,λ}H and, using the C*-envelope results of [15] and [53], the reduced Hao–Ng isomorphism (Theorem 4.7). The main technical novelty is a localisation argument using the averaging map associated with the Plancherel weight and the reduced dual coaction, which replaces the faithful conditional expectation available for discrete H.","tokens_in":1117,"tokens_out":1402,"duration_ms":240804,"significance":"If the proof is correct, this is a substantial result: it resolves the reduced Hao–Ng isomorphism problem for all non-degenerate product systems over arbitrary unital subsemigroups of discrete groups, for arbitrary locally compact Hausdorff group actions. The localisation technique via integrable elements of coactions is new in this context and is developed carefully: Lemmas 4.2–4.5 and the proof of Theorem 4.6 are internally coherent, and no fitted parameters or ad-hoc axioms are introduced. The paper is transparent about its main external dependency, which is the only substantive concern.","major_comments":[{"comment":"The final step of Theorem 4.7 relies on the C*-envelope isomorphism C*_env(T_λ(X)+ ⋊_{α,λ}H) ≃ C*_env(T_λ(X)+) ⋊_{α˙,λ}H, imported from [15, Theorem 4.4]. This preprint is neither proved nor stated in the manuscript, and it is the unique bridge from the Fock covariance of Theorem 4.6 to the asserted reduced Hao–Ng ∗-isomorphism. Since the entire selfadjoint conclusion of the paper rests on this external theorem, please include the exact statement of [15, Theorem 4.4], confirm explicitly that its hypotheses are satisfied (the contractive approximate unit supplied by non-degeneracy of X, and any further hypotheses such as separability or second-countability that [15] may require), and either prove it in an appendix or supply a peer-reviewed reference. If [15] is not yet published, Theorem 4.7 should at minimum be labelled as conditional on [15, Theorem 4.4].","section":"Section 4.3, Theorem 4.7"}],"minor_comments":[{"comment":"In the proofs of Lemma 4.3 and Lemma 4.4, the expression z^*u_i appears where the displayed formula actually computes the convolution of z with u_i; the notation should be changed to z*u_i (or the star explicitly explained) to avoid the impression that the adjoint of z is being used.","section":"Section 4.1, Lemmas 4.3–4.4"},{"comment":"The assertion that normality of the coaction by G on T_λ(X) 'yields in particular' normality of the coaction on ι⋊ is very compressed; one or two sentences spelling out the tensor-product argument via [55, Lemma 7.16] would help the reader.","section":"Theorem 4.6"},{"comment":"The statement that a contractive approximate unit for A gives a contractive approximate unit for T_λ(X)+ is used to enter [15, Theorem 4.4]; it would be useful to record this as a short lemma or to give a precise reference for the non-degenerate case.","section":"Section 4.3"}],"recommendation":"major_revision","confidential_remarks":"The paper is well-written and the technical core, Theorem 4.6, appears sound. My recommendation is driven solely by the unverified dependence on [15, Theorem 4.4] for the final Hao–Ng isomorphism. If the editor is satisfied that [15] is correct and will appear in a peer-reviewed venue, I would be willing to accept after the author confirms the reference and fixes the notation issues."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The main theorem is what it claims, but \"settles\" should be read as \"settles modulo Dor-On-Thompson [15].\" The genuinely new part is Theorem 4.6: the identity representation of X⋈H inside Tλ(X)⋈H is Fock covariant. The Plancherel-weight averaging machinery is a sensible substitute for the conditional expectation that made the discrete-H case work, and the localisation lemmas 4.3–4.5 are carefully proved. I checked the main line of Theorem 4.6 in detail: condition (ii) of [27, Thm 3.2] is verified correctly, the approximate unit coming from non-degeneracy is used legitimately, and the positivity argument turning Eδ⋈((c_ru_i)* (c_ru_i))j(λ_r(X_r))=0 into the required vanishing is sound. I did not find an internal error.\n\nThe one real soft spot is the final step of Theorem 4.7. After Theorem 4.6 and [27, Cor 4.1], the paper obtains the completely isometric isomorphism of tensor algebras. To convert that into the C*-isomorphism of the reduced strong-covariant algebras, it invokes [15, Thm 4.4], a preprint, for C*_env(Tλ(X)+ ⋈H) ≃ C*_env(Tλ(X)+)⋈H. Everything after that is Sehnem [53], which is published. So the advertised Hao-Ng isomorphism is exactly as secure as [15]. That is not a flaw in the internal argument, but it is a load-bearing dependence, and a referee should be asked to check [15] against this application. The non-degeneracy of X is used precisely to supply the contractive approximate unit required by [15]; that hypothesis is satisfied here, but the preprint itself is neither proved nor sketched. I would also ask the author to flag this dependence in the abstract, since the current abstract says \"we prove\" without mentioning the external preprint.\n\nI agree with the reader's scores. The novelty is real: the localisation technique is not an incremental tweak, and the statement covers the previously missing full-generality case. No circularity, no fitted parameters. The citation pattern looks appropriate; the self-citation to [27] is for the established discrete case and the Fock-covariance characterisation, not for the new claim.\n\nWho is this for? Anyone working on Hao-Ng isomorphisms, tensor algebras of product systems, or coaction techniques. It deserves a serious referee. If I were the editor I would send it out, with the specific instruction to verify [15, Thm 4.4] and its hypotheses; if that preprint checks out, this paper really does complete the programme.","headline":"A clean, significant proof that the reduced Hao-Ng isomorphism holds for non-degenerate product systems over arbitrary unital subsemigroups of discrete groups with locally compact Hausdorff actions, conditional on one external preprint.","tokens_in":24284,"tokens_out":6280,"would_cite":true,"duration_ms":57599,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L08","47L55","46L05","46K50","46L55"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the reduced Hao–Ng isomorphism problem has an affirmative answer for every non-degenerate product system over a unital subsemigroup of a discrete group, for actions of any locally compact Hausdorff group.","keywords":["Product systems","Fock space","Fock covariance","Hao–Ng isomorphism","Reduced crossed products of operator algebras","Non-degenerate product systems","Generalised gauge action","Plancherel weight"],"falsifier":"Compute both sides of the asserted isomorphism for a non-degenerate product system over $P=\\mathbb{Z}_+$ (a single C*-correspondence) under an action of a non-discrete locally compact group $H$, and check whether the canonical map is a $*$-isomorphism; a mismatch would disprove Theorem 4.7. A more focused check is to test the cited outer-hull theorem [15] on the tensor algebra of any C*-correspondence, looking for an operator algebra with a contractive approximate unit whose C*-envelope does not commute with the reduced crossed product by $H$.","tokens_in":23176,"feed_emoji":"🔁","tokens_out":9948,"duration_ms":77528,"temperature":0.7,"pith_summary":"This paper proves a commutation theorem: for a non-degenerate product system $X$ over a unital subsemigroup $P$ of a discrete group $G$, forming the reduced crossed product by a generalised gauge action of any locally compact Hausdorff group $H$ can be interchanged with forming the tensor algebra of the induced product system. That is, $T_{\\lambda}(X\\rtimes_{\\alpha,\\lambda}H)_+$ is canonically completely isometrically isomorphic to $T_{\\lambda}(X)_+\\rtimes_{\\alpha,\\lambda}H$, and the reduced strong covariant algebra of the induced system is canonically $*$-isomorphic to the reduced crossed product of the reduced strong covariant algebra of the original system by the induced action. This resolves the reduced Hao–Ng isomorphism problem in full generality, removing earlier restrictions to discrete, abelian, or amenable groups and to right-LCM or compactly aligned semigroups. The proof shows the identity representation of the induced product system inside the reduced crossed product of the Fock algebra is Fock covariant, using an averaging map built from the Plancherel weight when no faithful conditional expectation is available.","feed_headline":"Reduced Hao–Ng isomorphism proved for non-degenerate product systems","feed_subtitle":"Fock covariance and a Plancherel-weight localisation make tensor algebras and reduced crossed products commute.","key_machinery":"The central object is the averaging map $E_{\\delta^{\\rtimes}}=(\\mathrm{id}\\otimes\\omega_H)\\circ\\delta^{\\rtimes}$, formed from the reduced dual coaction $\\delta^{\\rtimes}$ on the reduced crossed product and the Plancherel weight $\\omega_H$ on the reduced group C*-algebra $C^*_{\\lambda}(H)$. This map replaces the faithful conditional expectation that exists when $H$ is discrete but generally does not exist for locally compact Hausdorff groups. Non-degeneracy of $X$ yields a contractive approximate unit $u_i$ supported in the coefficient algebra, and multiplication by these $u_i$ localises arbitrary elements inside the square-integrable domain where the averaging map is well defined. The Fock-covariance criterion of Theorem 4.1 then verifies the two $K$-core conditions for the identity representation, and the cited results [53] and [15] carry the conclusion to the C*-envelope and the strong covariant algebra.","core_discovery":"Let $P$ be a unital subsemigroup of a discrete group $G$, $X$ a non-degenerate product system over $P$, and $\\alpha$ a generalised gauge action of a locally compact Hausdorff group $H$ on the Fock algebra $T_{\\lambda}(X)$. The identity representation $\\iota^{\\rtimes}$ of $X\\rtimes_{\\alpha,\\lambda}H$ inside $T_{\\lambda}(X)\\rtimes_{\\alpha,\\lambda}H$ is injective, Fock covariant, and admits a normal coaction by $G$. From this, the tensor algebra of the induced system is canonically completely isometrically isomorphic to the reduced crossed product of the tensor algebra by $\\alpha$, and consequently $(A\\rtimes_{\\alpha,\\lambda}H)\\times_{X\\rtimes_{\\alpha,\\lambda}H,\\lambda}P$ is canonically $*$-isomorphic to $(A\\times_{X,\\lambda}P)\\rtimes_{\\dot{\\alpha},\\lambda}H$, where $\\dot{\\alpha}$ is the induced action. In short, the reduced strong covariant functor commutes with the reduced crossed product functor under the non-degeneracy assumption.","pith_inferences":["The averaging-map localisation should extend beyond product systems to any reduced crossed product where a contractive approximate unit supported in the coefficient algebra exists, so similar Fock-covariance arguments may prove Hao–Ng-type isomorphisms for other families of Fell bundles.","If the cited outer-hull theorem [15] turns out to be false, the paper's Theorem 4.6 would remain a standalone Fock-covariance statement, while the Hao–Ng isomorphism would remain open; the contribution would then be conditional.","A natural stress test is to drop non-degeneracy: for a degenerate product system the tensor algebra may lack a contractive approximate unit, and constructing a counterexample to the tensor-algebra isomorphism there would show the assumption is genuinely load-bearing."],"forward_implications":["For every non-degenerate product system over a unital subsemigroup of a discrete group and every generalised gauge action of a locally compact Hausdorff group, the reduced Hao–Ng isomorphism holds by a canonical $*$-isomorphism.","The tensor algebra of the induced product system is completely isometrically isomorphic to the reduced crossed product of the original tensor algebra, so the non-selfadjoint algebra behaves predictably under reduced crossed products.","The identity representation of the induced system is injective, Fock covariant, and normally coacted, so the full Fell-bundle machinery applies to the induced system.","The result unifies and extends the previously known cases (single correspondences, abelian lattice-ordered semigroups, right-LCM semigroups, and discrete acting groups) to arbitrary unital subsemigroups and arbitrary locally compact Hausdorff $H$, at the cost of non-degeneracy.","The C*-envelope of the reduced crossed product of a tensor algebra under a generalised gauge action is the reduced crossed product of its C*-envelope by the induced action."],"supporting_citations":[{"why":"Supplies the characterisation of injective Fock covariant representations (Theorem 4.1) and the discrete-acting-group case whose strategy is extended here.","marker":"[27]"},{"why":"Supplies the final C*-envelope crossed-product isomorphism used to pass from the tensor-algebra isomorphism to the Hao–Ng isomorphism.","marker":"[15]"},{"why":"Identifies the C*-envelope of the tensor algebra with the reduced strong covariant algebra, carrying the conclusion to $A\\times_{X,\\lambda}P$.","marker":"[53]"},{"why":"Provides the Fock covariant bundle, the normal coaction on $T_\\lambda(X)$, and the strong covariant bundle framework.","marker":"[13]"},{"why":"Defines reduced crossed products of operator algebras and identifies $T_\\lambda(X)_+\\rtimes_{\\alpha,\\lambda}H$ inside $T_\\lambda(X)\\rtimes_{\\alpha,\\lambda}H$.","marker":"[33]"},{"why":"Constructs the strong covariant C*-algebra and gives the injectivity and co-universality properties used in the reduction.","marker":"[52]"},{"why":"Supplies the integrability of dual coactions and the averaging-map identities behind Lemmas 4.3–4.5.","marker":"[8]"},{"why":"Provides the slice-map and reduced crossed-product machinery for locally compact Hausdorff groups used in Sections 3–4.","marker":"[16]"}],"fun_headline_variants":["Hao–Ng holds for non-degenerate product systems","Plancherel weight proves reduced Hao–Ng for product systems","Fock covariance yields Hao–Ng isomorphism for product systems","Non-degenerate product systems satisfy reduced Hao–Ng"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The paper assumes the system's fibres are generated by the coefficient algebra (non-degeneracy), which supplies the approximate unit that makes the localisation argument work, and the final step assumes a still-unpublished result that taking the outer hull of an operator algebra commutes with taking reduced crossed products; if that outer-hull result fails, the Fock-covariance theorem survives but the Hao–Ng isomorphism does not follow.","fun_headline_variants_meta":{"raw":{"variants":["Hao–Ng holds for non-degenerate product systems","Plancherel weight proves reduced Hao–Ng for product systems","Fock covariance yields Hao–Ng isomorphism for product systems","Non-degenerate product systems satisfy reduced Hao–Ng"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000376,"raw_usage":{"total_tokens":1967,"prompt_tokens":869,"completion_tokens":1098,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":485,"completion_tokens_details":{"reasoning_tokens":1031}},"tokens_in":485,"tokens_out":1098,"duration_ms":8935,"temperature":1.0,"reasoning_tokens":1031,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T15:34:26.487653+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute both sides of the asserted isomorphism for a non-degenerate product system over $P=\\mathbb{Z}_+$ (a single C*-correspondence) under an action of a non-discrete locally compact group $H$, and check whether the canonical map is a $*$-isomorphism; a mismatch would disprove Theorem 4.7. A more focused check is to test the cited outer-hull theorem [15] on the tensor algebra of any C*-correspondence, looking for an operator algebra with a contractive approximate unit whose C*-envelope does not commute with the reduced crossed product by $H$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the characterisation of injective Fock covariant representations (Theorem 4.1) and the discrete-acting-group case whose strategy is extended here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Identifies the C*-envelope of the tensor algebra with the reduced strong covariant algebra, carrying the conclusion to $A\\times_{X,\\lambda}P$."},{"cited_title":"Dor-On, E","cited_arxiv_id":null,"evidence_quote":"Provides the Fock covariant bundle, the normal coaction on $T_\\lambda(X)$, and the strong covariant bundle framework."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines reduced crossed products of operator algebras and identifies $T_\\lambda(X)_+\\rtimes_{\\alpha,\\lambda}H$ inside $T_\\lambda(X)\\rtimes_{\\alpha,\\lambda}H$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Constructs the strong covariant C*-algebra and gives the injectivity and co-universality properties used in the reduction."},{"cited_title":"Buss,Integrability of dual coactions on Fell bundleC ∗-algebras, Bull","cited_arxiv_id":null,"evidence_quote":"Supplies the integrability of dual coactions and the averaging-map identities behind Lemmas 4.3–4.5."},{"cited_title":"Echterhoff, S","cited_arxiv_id":null,"evidence_quote":"Provides the slice-map and reduced crossed-product machinery for locally compact Hausdorff groups used in Sections 3–4."}],"review_version":2}