{"id":"cba89aee-29a6-4319-ac00-bbd1a44ab86e","arxiv_id":"2507.10168","paper_version":1,"verdict":"CONDITIONAL","confidence":"LOW","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Boundary quotients of reduced semigroup C*-algebras are modeled by reductions of Paterson and Spielberg groupoids under new boundary regularity conditions.","lead":"This paper proposes two groupoid models for the boundary quotient of the reduced C*-algebra of a general left cancellative monoid. It identifies conditions under which these models work, and constructs a monoid example to test the boundaries of the theory.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The example's norm equality rests on Borel amenability of ∂G(R), whose proof depends on the unverified boundary description in Proposition 3.14; if that decomposition misses any tight character, the approximate invariant mean is undefined and the main isomorphism collapses.","rationale":"The reader's weakest assumption was that the exotic norm e equals the reduced norm in the short exact sequences, and that this is hard to verify for general monoids. I agree this is the structurally most fragile input to the general theorems. However, for the paper's central example the author attempts to prove this equality via Borel amenability of ∂G(R). My concern sharpens the reader's: the Borel amenability proof is only as sound as the description of ∂Ω in Proposition 3.14, which is a long, computation-heavy classification of tight characters that the reader could not certify. A concrete failure there would mean the approximate invariant mean is not defined on some boundary character, so the norm equality and the isomorphism C*_r(∂G(R)) ≅ C*_r(R)/I would not follow. I found no confirmed error, but the verification burden is real. The reader's CONDITIONAL verdict remains appropriate; my read does not move the verdict, hence UNCHANGED. I partially agree with the reader because the concern is not exactly the abstract difficulty of the norm equality but its concrete implementation in the example through Proposition 3.14 and the cited non-Hausdorff amenability result.","tokens_in":50948,"tokens_out":36325,"duration_ms":413636,"concrete_test":"Verify Proposition 3.14 by computer-assisted enumeration for increasing word-length bounds L: enumerate all constructible ideals of R generated by words of length at most L, compute the tight characters on this finite sub-semilattice, and check that they are exactly the restrictions of the characters listed in Proposition 3.14. Explicitly test χ_{b x0}: either exhibit a finite foundation set for R that it fails to recognize (confirming it is not tight), or find that it is tight, which would refute the boundary description and invalidate the Borel amenability argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The general theorems 2.12 and 2.23 are explicitly conditional on the exotic norm e (resp. e2) in (2.3)/(2.13) coinciding with the reduced norm, and the author states that finding general conditions for this is difficult. In the central example, this equality is obtained by proving that ∂GP(R) = ∂G(R) is Borel amenable and then citing [ABGBHL25, Corollary 6.10] to conclude that universal and reduced norms coincide on this non-Hausdorff σ-compact groupoid. This is the load-bearing step: if the corollary does not apply to non-Hausdorff Borel amenable groupoids, or if the Borel mean constructed in Theorem 3.19 is not defined on all of ∂G(R), then the bottom maps in diagrams (2.4) and (2.15) are not known to be isomorphisms, and Theorem 3.1 fails. The mean is defined only on the three invariant pieces of ∂Ω listed in Proposition 3.14: {⟨wX⟩}, {⟨wY⟩}, and {χ_w | w infinite reduced}. That classification is proved using the lengthy and unverified technical claims in Lemmas 3.3, 3.6, 5.1–5.4 and the classification of constructible ideals (5.1). In particular, Proposition 3.14 excludes all principal characters such as χ_{b x0}; if such a character were actually tight, the mean would not be defined on it and the amenability proof would be incomplete. I found no internal contradiction, but this is the most fragile unverified premise supporting the paper's main example.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines a boundary quotient of the reduced semigroup C*-algebra of an arbitrary left cancellative monoid and proposes two groupoid models for it, obtained as reductions of Paterson's and Spielberg's groupoids. The main general results, Theorem 2.12 and Theorem 2.23, are explicitly conditional: they show that the relevant bottom map in diagram (2.4) or (2.15) is an isomorphism whenever the exotic norm e (resp. e2) from the associated short exact sequence equals the reduced norm and the monoid is strongly C*-regular (resp. C*-regular) on the boundary. The paper then constructs a monoid R, given by the presentation (3.1), and proves in Theorem 3.1 that R is left cancellative, strongly C*-regular on the boundary, not C*-regular, and that its boundary groupoid is Borel amenable; from this it concludes that the two boundary groupoid models coincide and give an isomorphism with the boundary quotient. Section 4 gives a systematic inventory of the logical relations among six regularity and groupoid-equality properties and provides nine monoid constructions realizing the possible combinations.","tokens_in":51214,"tokens_out":15801,"duration_ms":186585,"significance":"If the example is correct, the paper makes a genuine contribution by exhibiting a non-C*-regular left cancellative monoid whose boundary quotient nevertheless admits a groupoid model, thereby going beyond the right-LCM and group-embeddable settings. The conditional structure of Theorems 2.12 and 2.23 is honest and clearly stated, and the author explicitly identifies the norm-equality hypothesis as the difficult input. The proof of Proposition 2.4 and the related diagram chase are clearly written, and the paper contains substantial original constructions, including the negative answer to Li's question about GP(S) versus G(S). The appendix provides detailed proofs of the technical lemmas used in the example, and Section 4's classification of the six properties is useful. The main weaknesses are that the central example rests on a long classification of boundary characters (Proposition 3.14) and on an external, partly preprint-based result to pass from Borel amenability to equality of universal and reduced norms on a non-Hausdorff groupoid; these points deserve a more self-contained and auditable presentation.","major_comments":[{"comment":"The step from Borel amenability of ∂G(R) to equality of the exotic norms e and e2 with the reduced norm is load-bearing for Theorem 3.1, but it is justified only by citing [ABGBHL25, Corollary 6.10] and by a parenthetical remark that the same conclusion also follows from [ADR00, Proposition 6.1.8] and [Ren97, Proposition 3.4]. Because ∂G(R) is non-Hausdorff, the exact hypotheses of the cited results must be stated explicitly and checked. Please include a precise theorem statement (with all hypotheses, including σ-compactness and non-Hausdorff étale) and either prove it or give a complete reference with theorem numbers; as written, a failure of this external result would invalidate the isomorphism statement in Theorem 3.1.","section":"Section 3.3, final paragraph of proof of Theorem 3.1"},{"comment":"The Borel approximate invariant mean in Theorem 3.19 is defined separately on the three invariant families listed in (3.21), so the classification of ∂Ω(R) in Proposition 3.14 is not an auxiliary detail: if any tight character were missing from the classification, the mean would not be defined on it and Definition 3.18 would cease to hold. The proof of Proposition 3.14 is a long case analysis that depends on Lemmas 3.3, 3.6, 5.1–5.4 and on the classification of constructible ideals in §5.1; I did not find an internal contradiction, but the logical path is hard to audit. Please add an explicit verification that principal characters such as ⟨b x0 R⟩ are not tight, and make the dependencies between Proposition 3.14, the decomposition (3.21), and the definition of the mean in (3.22)–(3.24) transparent.","section":"Proposition 3.14 and Theorem 3.19"}],"minor_comments":[{"comment":"The abstract says the paper formulates conditions on S that guarantee that either reduction is a groupoid model, but it does not mention that the main theorems also require the exotic norm e (resp. e2) to coincide with the reduced norm; since this hypothesis is explicitly acknowledged as difficult to verify, it should appear in the abstract.","section":"Abstract and Introduction"},{"comment":"In the argument that B := w1···wn x0 R does not intersect an A_i ideal, the word 'subword' is used where the intended meaning appears to be 'prefix' (or a prefix relation after applying the orthogonality property). Rephrasing this step would remove a source of ambiguity in an already intricate proof.","section":"Proposition 3.14, Step 3"},{"comment":"The constructions of S4, S5, and S7 and the claim that they realize the required combinations are sketched rather than proved; for S5 the text explicitly says 'We omit the proof here.' Since Section 4 is not the central claim of the paper this is acceptable, but a note stating that full verifications are available from the author or in a companion document would help the reader.","section":"Section 4"},{"comment":"There are several places where notation is introduced without being defined at first use, for example the use of ⟨wX⟩ in Section 3 before Definition 3.12; adding a short table of notation near the beginning of Section 3 would improve readability.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is mathematically ambitious and the conditional main theorems are clearly stated, but the central example depends on an extremely long verification of the boundary character classification and on a norm-equality step that is currently justified by citation to a recent arXiv preprint. I would recommend sending the manuscript back with a request to make those two points fully self-contained or precisely referenced, and I would encourage the editor to seek an additional expert opinion on the Appendix, since the correctness of Proposition 3.14 and Lemma 3.6 is not something I could certify with confidence from a single reading."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Honest take first: this is a worthwhile paper. Schwartz isolates two new boundary regularity conditions, proves conditional isomorphism theorems identifying the boundary quotient with the reduced C*-algebras of reductions of Paterson and Spielberg groupoids, and constructs a monoid R that is strongly regular on the boundary but not C*-regular. The structural half is clean and useful. The author states up front that the missing hypothesis — equality of an exotic norm with the reduced norm — is hard to check in general, and the example is specifically designed to verify it by proving Borel amenability of the boundary groupoid. That is a lot of machinery, and it is handled with unusual candor.\n\nWhat is genuinely new: the two definitions, the diagram reduction of the boundary quotient to those groupoids, the criterion for ∂G_P = ∂G, and the taxonomy of nine property combinations in Section 4. The proofs of Propositions 2.3, 2.4, 2.17, 2.18, and 2.25 that I checked are correct. The discussion of the exotic norm short exact sequence and its role is helpful. For someone working in semigroup C*-algebras, this is a real advance.\n\nSoft spots: the Borel amenability proof for ∂G(R) depends on the classification of tight characters in Proposition 3.14, and that classification in turn depends on long computations in the appendix (Lemmas 3.3, 3.6, 5.1–5.4 plus the constructible ideal classification). The calculations are dense and I did not certify every line. I found no concrete error, but a referee will need to sit down with the appendix. The Section 4 examples are asserted with varying levels of proof; that is acceptable for a research paper but makes the nine-way classification more of a map than a fully verified atlas. The reliance on the recent non-Hausdorff groupoid result [ABGBHL25] is fine, though the author's aside that classical arguments also work softens the dependency.\n\nWho should read this: people in the subfield will want it. It is not a paper for a general audience. It deserves a serious referee: the conditional framework is solid and the example, if confirmed, is a genuine separation result.","headline":"A solid structural paper whose conditional groupoid-model theorems are worth refereeing; the intricate example monoid is ambitious and needs a careful referee rather than blind trust.","tokens_in":51792,"tokens_out":2902,"would_cite":true,"duration_ms":34238,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["46L05","22A22","20M18","46L55"],"pacs":[],"model":"deepseek-v4-flash","headline":"C*-regularity on the boundary is enough to give boundary quotients a groupoid model.","keywords":["boundary quotient","left cancellative monoid","semigroup C*-algebra","étale groupoid","C*-regularity","Borel amenable groupoid","tight groupoid","constructible right ideals"],"falsifier":"Take a left cancellative monoid satisfying strong C*-regularity on the boundary and compute the reduced and universal norms of a characteristic function supported on an open bisection of $\\partial G_P(S)$; if the two norms differ, then the exotic norm in the short exact sequence (2.3) is not reduced and the hypothesis of Theorem 2.12 fails.","tokens_in":50649,"feed_emoji":"🧮","tokens_out":7202,"duration_ms":76618,"temperature":0.7,"pith_summary":"The paper asks when the boundary quotient of a reduced semigroup C*-algebra can be described by an étale groupoid. For any left cancellative monoid it proposes two boundary groupoids, reductions of the Paterson and Spielberg groupoids to the boundary characters. It proves that either one is a genuine model under a new, weaker condition — strong C*-regularity on the boundary for one, C*-regularity on the boundary for the other — together with equality of an exotic norm with the reduced norm. The point is that these conditions are weaker than the full regularity notions used earlier, so they enlarge the class of monoids whose boundary quotients are groupoid algebras. The paper also constructs a monoid satisfying the new conditions while failing full C*-regularity, showing the weakening is real.","feed_headline":"Boundary quotients get groupoid models under weaker regularity","feed_subtitle":"Two boundary groupoids model the boundary quotient when boundary C*-regularity and a norm condition hold.","key_machinery":"The central objects are the boundary groupoids $\\partial G_P(S)$ and $\\partial G(S)$: reductions of Paterson's universal groupoid and Spielberg's quotient groupoid to $\\partial\\Omega(S)$, the closure of the set of maximal characters (tight characters) on the constructible right ideals of $S$. The argument runs through two commutative diagrams built from short exact sequences $0 \\to C_r^*(G\\setminus\\partial G) \\to C_r^*(G) \\to C^*_e(\\partial G) \\to 0$, where $e$ is an exotic C*-norm; the bottom map is an isomorphism exactly when the kernel of the left regular representation is contained in the ideal coming from the complement of the boundary. In the example, the load-bearing mechanism is Borel amenability of $\\partial G(R)$, which forces the exotic norm to coincide with the reduced norm, and an explicit classification of boundary characters as infinite reduced words or principal filters on two special constructible ideals.","core_discovery":"The central claim is that the boundary quotient $C_r^*(S)/I$ of a left cancellative monoid $S$ is isomorphic to the reduced C*-algebra of a boundary groupoid whenever the relevant exotic norm from the short exact sequence separating the boundary from its complement equals the reduced norm, and $S$ is strongly C*-regular on the boundary (for the Paterson model) or C*-regular on the boundary (for the Spielberg model). These boundary regularity conditions are defined by replacing the covering requirement in ordinary (strong) C*-regularity with the weaker requirement that a family be a foundation set for the ideal. The isomorphism is established through two commutative diagrams in which the bottom map is induced by the left regular representation and the ideal generated by foundation sets. The constructed monoid $R$ shows the hypotheses can hold without full C*-regularity: $R$ is left cancellative, strongly C*-regular on the boundary, its two boundary groupoids coincide and are Borel amenable, so the exotic norm is reduced and the boundary quotient is the reduced groupoid algebra. The paper also shows that the non-boundary groupoids $G_P(S)$ and $G(S)$ need not coincide, answering a question in the literature.","pith_inferences":["A natural next step would be to search for general sufficient conditions for the exotic norm on the boundary groupoid to equal the reduced norm; Borel amenability is one such route, and the techniques used for $R$ may extend to monoids with similar rewriting structure.","Because the boundary regularity conditions are formulated entirely in terms of constructible right ideals and partial bijections, they give a combinatorial criterion that could be tested on other monoids presented by generators and relations.","The classification of boundary characters in the example suggests that for monoids with strong cancellation properties, boundary characters may often decompose into infinite reduced words and principal filters on a few distinguished constructible ideals, which would make Borel amenability and norm computations more accessible."],"forward_implications":["For any left cancellative monoid satisfying strong C*-regularity on the boundary together with the norm condition, the boundary quotient is isomorphic to $C_r^*(\\partial G_P(S))$, with the analogous statement for Spielberg's groupoid under C*-regularity on the boundary.","The monoid $R$ is not C*-regular yet has a good groupoid model for its boundary quotient, so the boundary regularity conditions cover examples outside the scope of the earlier full regularity notions.","The example also shows that the left regular representation $C_r^*(G(R)) \\to C_r^*(R)$ can fail to be an isomorphism even when the boundary quotient is well modelled, so boundary behaviour is genuinely independent of the non-boundary regular representation.","The two boundary groupoids $\\partial G_P(S)$ and $\\partial G(S)$ can differ, so the two candidate models are not interchangeable in general; the paper gives a condition equivalent to their equality.","Combining C*-regularity on the boundary with equality of the two boundary groupoids forces strong C*-regularity on the boundary, and the analogous implication holds for the non-boundary notions."],"supporting_citations":[{"why":"Introduces the groupoids $G_P(S)$ and $G(S)$ and the notions of strong C*-regularity and C*-regularity that this paper extends to the boundary.","marker":"[NS23]"},{"why":"Supplies Spielberg's groupoid and its boundary reduction in the setting of left cancellative small categories.","marker":"[Spi20]"},{"why":"Provides Paterson's universal groupoid for inverse semigroups, from which $G_P(S)$ is obtained by reduction.","marker":"[Pat99]"},{"why":"Identifies the reduction to tight characters with the tight groupoid and gives the C*-algebra of tight representations.","marker":"[Exe08]"},{"why":"Gives the result that a sigma-compact Borel amenable groupoid has coinciding full and reduced norms, which verifies the norm hypothesis for the example monoid.","marker":"[ABGBHL25]"},{"why":"Provides the condition on nets in the unit space that guarantees weak containment of left regular representations, which the boundary regularity definitions negate.","marker":"[KS02]"},{"why":"Supplies the short exact sequence with an exotic norm for reduced groupoid C*-algebras of non-Hausdorff étale groupoids.","marker":"[CN24]"},{"why":"Shows that maximal characters form an invariant subset and contributes arguments used for the boundary space and the diagonal map.","marker":"[Li23]"}],"fun_headline_variants":["Boundary quotients: groupoid models from weaker regularity","Weaker regularity provides groupoid models for boundary quotients","Boundary quotients modeled by groupoids without full C*-regularity","Groupoid models for boundary quotients with relaxed conditions","Relaxed hypotheses yield groupoids for boundary quotient algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The weakest premise is that the exotic norm attached to the boundary groupoid equals the ordinary reduced norm; the paper says no general condition for this is known, and the example verifies it only by proving Borel amenability.","fun_headline_variants_meta":{"raw":{"variants":["Boundary quotients: groupoid models from weaker regularity","Weaker regularity provides groupoid models for boundary quotients","Boundary quotients modeled by groupoids without full C*-regularity","Groupoid models for boundary quotients with relaxed conditions","Relaxed hypotheses yield groupoids for boundary quotient algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000827,"raw_usage":{"total_tokens":3596,"prompt_tokens":906,"completion_tokens":2690,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":522,"completion_tokens_details":{"reasoning_tokens":2621}},"tokens_in":522,"tokens_out":2690,"duration_ms":21326,"temperature":1.0,"reasoning_tokens":2621,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:36:38.072274+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a left cancellative monoid satisfying strong C*-regularity on the boundary and compute the reduced and universal norms of a characteristic function supported on an open bisection of $\\partial G_P(S)$; if the two norms differ, then the exotic norm in the short exact sequence (2.3) is not reduced and the hypothesis of Theorem 2.12 fails.","supporting_citations":[],"review_version":1}