Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models

Gaute Schwartz

REVIEW 2 major objections 4 minor 4 references

Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read C*-regularity on the boundary is enough to give boundary quotients a groupoid model.

desk verdict 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. read the letter →

arxiv 2507.10168 v1 pith:DDXMYSZ6 submitted 2025-07-14 math.OA

Gaute Schwartz This is my paper · ORCID

classification math.OA MSC 46L0522A2220M1846L55

keywords boundaryquotientleftcancellativemonoidsemigroupC*-algebraétalegroupoidC*-regularityBorelamenabletightconstructiblerightideals

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

What 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.

Soft 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.

Who 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.

Referee Report

2 major / 4 minor

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.

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 (2)

[Section 3.3, final paragraph of proof of Theorem 3.1] 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.
[Proposition 3.14 and Theorem 3.19] 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.

minor comments (4)

[Abstract and Introduction] 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.
[Proposition 3.14, Step 3] 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 4] 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.
[Throughout] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are conditional statements with genuine proofs, and the example is verified by direct computation plus an external corollary.

full rationale

I found no circular step in the derivation chain. The central theorems (2.12 and 2.23) are explicitly conditional: they assume that an exotic norm from a short exact sequence coincides with the reduced norm and that a boundary regularity condition holds, and they then prove an isomorphism. These hypotheses are not derived from the conclusions, and no parameter is fitted to force the result. The proof of Corollary 2.7 is a genuine argument from the stated hypotheses (norm equality and weak containment) to the isomorphism, and it does not presuppose the conclusion. For the example monoid R, the paper supplies direct proofs: left cancellativity (Lemma 3.2), strong C*-regularity on the boundary (Proposition 3.8), Borel amenability of the boundary groupoid via an explicit approximate invariant mean (Theorem 3.19), and non-C*-regularity (Propositions 3.4 and 3.5). The equality of universal and reduced norms on the boundary groupoid is imported from the external result [ABGBHL25, Corollary 6.10], which is not a self-citation and is not fitted. Citations to the author's previous paper [NS23] provide published definitions and theorems with independent proofs; they are load-bearing in the ordinary sense of building on prior work, but they are not unverified assertions and are not used to forbid alternatives. Possible concerns about the applicability of [ABGBHL25, Corollary 6.10] to non-Hausdorff groupoids, or about the completeness of the boundary-character classification in Proposition 3.14, are correctness risks, not instances of circular reasoning, because no conclusion is shown to reduce to its own input by definition or by a self-citation chain.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new physical or algebraic entities beyond the new regularity conditions. The example monoid R is a newly defined object, but it is not an invented entity in the sense of a postulate introduced without evidence; it is constructed explicitly with generators and relations. The main unproved input is the norm equality condition for general monoids, which the author openly identifies. The classification of constructible ideals and boundary characters for R are heavy technical inputs, but they are proved inside the paper. The newly defined conditions are the core content, not hidden assumptions.

assumptions (5)

standard math The reduced groupoid C*-algebra and its ideals behave as in the short exact sequences (1.2) and (1.3).
The paper cites [CN24, Proposition 1.2] for the reduced sequence and states the full sequence as known. This is standard background in operator algebra.
domain assumption Borel amenability of a sigma-compact etale groupoid implies the universal and reduced norms coincide on the groupoid C*-algebra.
Invoked in Section 3 via [ABGBHL25, Corollary 6.10]. This is a recent theorem from the cited preprint, not proved in the paper.
domain assumption The classification of constructible right ideals of R in equation (5.1) is complete.
This classification is a long computation in the appendix (Section 5.1). It is essential for the boundary character analysis and the regularity arguments. The paper proves it in detail, but the computation is a self-contained technical result that the central example depends on.
ad hoc to paper The boundary characters of R are exactly the three families in Proposition 3.14.
This is a custom structural result for the example monoid R, proven in the paper with some steps deferred to the appendix. It is a load-bearing classification used to prove Borel amenability.
standard math Conditions such as C*-regularity on the boundary, when formulated for free products, reduce to the corresponding conditions on the factors as in Proposition 4.4.
This is a theorem proved in the paper, but it assumes a structural claim about the inverse hull of the free product that is established within the proof. It is used for the example monoids in Section 4.

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Pith. "Pith review of Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models." pith.science (2026). https://pith.science/paper/DDXMYSZ6

@misc{pith2026250710168,
  author       = {Pith},
  title        = {Pith review of: Boundary quotients of C$^*$-algebras of left cancellative monoids and their groupoid models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DDXMYSZ6}},
  note         = {Machine review of arXiv:2507.10168}
}

abstract

For a left cancellative monoid $S$ we consider a quotient of the reduced semigroup C$^*$-algebra $C_r^*(S)$ known as the boundary quotient. We present two potential groupoid models for this boundary quotient, obtained as reductions of Paterson and Spielberg's groupoids associated to $S$, and formulate conditions on $S$ which guarantees that either is a groupoid model. We outline how these conditions are related to the notions (strong) C$^*$-regularity introduced in a previous paper, and construct an example of a left cancellative monoid which is not C$^*$-regular, but satisfies both of the new conditions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

4 extracted references · 2 canonical work pages

[1]

Aguyar Brix, J

[ABGBHL25] K. Aguyar Brix, J. Gonzales, J. B. Hume, and X. Li,On Hausdorff covers for non-Hausdorff groupoids, preprint (2025), available atarXiv:2503.23203v1 [math.OA] . [ADR00] C. Anantharaman-Delaroche and J. Renault,Amenable Groupoids, Monographie de l’Enseignement mathématique, L’Enseignement Mathématique,

arXiv 2025
[1999]

Renault, The Fourier Algebra of a Measured Groupoid and Its Multipliers, Journal of Functional Analysis 145 (1997), no

MR1724106 [Ren97] J. Renault, The Fourier Algebra of a Measured Groupoid and Its Multipliers, Journal of Functional Analysis 145 (1997), no. 2, 455-490,DOI:https://doi.org/10.1006/jfan.1996.3039. [Ren15] J. Renault,Topological Amenability is a Borel Property, Mathematica Scandinavica117 (2015), no. 1, 5–30, DOI:10.7146/math.scand.a-22235. [SY10] A. Sims a...

work page arXiv 1997
[2000]

Christensen and S

[CN24] J. Christensen and S. Neshveyev,Isotropy fibers of ideals in groupoid C∗-algebras, Advances in Math- ematics 447 (2024), 109696, DOI:https://doi.org/10.1016/j.aim.2024.109696. [CELY17] J. Cuntz, S. Echteroff, X. Li, and G. Yu,K-Theory for Group C∗-Algebras and Semigroup C∗-Algebras, Oberwolfach Seminars, vol. 47, Birkhäuser Cham,

arXiv 2024
[2017]

Left regular representations of Garside categories I. C*-algebras and groupoids

MR3618901 [Exe08] R. Exel, Inverse semigroups and combinatorial C ∗-algebras, Bulletin of the Brazilian Mathematical Society, New Series39 (2008), no. 2, 191–313,DOI:10.1007/s00574-008-0080-7. MR2419901 [KKLL22] E. Kakariadis, E. Katsoulis, M. Laca, and X. Li,Boundary quotient C*-algebras of semigroups, Journal of the London Mathematical Society105 (2022)...

work page Pith review arXiv 2008

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

Aguyar Brix, J

[ABGBHL25] K. Aguyar Brix, J. Gonzales, J. B. Hume, and X. Li,On Hausdorff covers for non-Hausdorff groupoids, preprint (2025), available atarXiv:2503.23203v1 [math.OA] . [ADR00] C. Anantharaman-Delaroche and J. Renault,Amenable Groupoids, Monographie de l’Enseignement mathématique, L’Enseignement Mathématique,

arXiv 2025

[2] [1999]

Renault, The Fourier Algebra of a Measured Groupoid and Its Multipliers, Journal of Functional Analysis 145 (1997), no

MR1724106 [Ren97] J. Renault, The Fourier Algebra of a Measured Groupoid and Its Multipliers, Journal of Functional Analysis 145 (1997), no. 2, 455-490,DOI:https://doi.org/10.1006/jfan.1996.3039. [Ren15] J. Renault,Topological Amenability is a Borel Property, Mathematica Scandinavica117 (2015), no. 1, 5–30, DOI:10.7146/math.scand.a-22235. [SY10] A. Sims a...

work page arXiv 1997

[3] [2000]

Christensen and S

[CN24] J. Christensen and S. Neshveyev,Isotropy fibers of ideals in groupoid C∗-algebras, Advances in Math- ematics 447 (2024), 109696, DOI:https://doi.org/10.1016/j.aim.2024.109696. [CELY17] J. Cuntz, S. Echteroff, X. Li, and G. Yu,K-Theory for Group C∗-Algebras and Semigroup C∗-Algebras, Oberwolfach Seminars, vol. 47, Birkhäuser Cham,

arXiv 2024

[4] [2017]

Left regular representations of Garside categories I. C*-algebras and groupoids

MR3618901 [Exe08] R. Exel, Inverse semigroups and combinatorial C ∗-algebras, Bulletin of the Brazilian Mathematical Society, New Series39 (2008), no. 2, 191–313,DOI:10.1007/s00574-008-0080-7. MR2419901 [KKLL22] E. Kakariadis, E. Katsoulis, M. Laca, and X. Li,Boundary quotient C*-algebras of semigroups, Journal of the London Mathematical Society105 (2022)...

work page Pith review arXiv 2008