Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences

Ralf Meyer

arxiv: 2506.19569 · v3 · pith:N6LOLOKInew · submitted 2025-06-24 · 🧮 math.OA

Groupoid models for relative Cuntz-Pimsner algebras of groupoid correspondences

Ralf Meyer This is my paper

Pith reviewed 2026-05-19 08:08 UTC · model grok-4.3

classification 🧮 math.OA

keywords groupoid correspondencerelative Cuntz-Pimsner algebragroupoid C*-algebraetale groupoidopen invariant subsettopological graphself-similar group

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The pith

Relative Cuntz-Pimsner algebras arising from groupoid correspondences are themselves groupoid C*-algebras.

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

This paper shows that starting from an etale locally compact groupoid and a groupoid correspondence on it, the relative Cuntz-Pimsner algebra with respect to the ideal coming from an open invariant subset is the C*-algebra of a new groupoid. The new groupoid is given by an explicit construction and is characterized by a universal property that describes its actions on topological spaces. This unifies the groupoid models used for the C*-algebras of topological graphs and of self-similar groups. A sympathetic reader cares because the result supplies a common groupoid-theoretic explanation for algebras arising in different contexts.

Core claim

A groupoid correspondence on an etale, locally compact groupoid induces a C*-correspondence on its groupoid C*-algebra. The Cuntz-Pimsner algebra for this C*-correspondence relative to an ideal associated to an open invariant subset of the groupoid is again a groupoid C*-algebra for a certain groupoid. We describe this groupoid explicitly and characterise it by a universal property that specifies its actions on topological spaces.

What carries the argument

The groupoid correspondence on an etale locally compact groupoid, which induces the C*-correspondence whose relative Cuntz-Pimsner algebra is modeled by the new groupoid constructed from the correspondence and the open invariant subset.

If this is right

The C*-algebras of topological graphs arise as relative Cuntz-Pimsner algebras from appropriate groupoid correspondences.
Self-similar group C*-algebras are realized as groupoid C*-algebras through the same construction.
The universal property classifies continuous actions of the new groupoid on topological spaces in terms of the original correspondence.
Separate earlier constructions for these algebras become special cases of one groupoid model.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The framework could extend to correspondences on non-etale groupoids, producing groupoid models for a wider range of C*-algebras.
Further connections to crossed-product constructions or other relative C*-algebra theories may emerge from the explicit groupoid description.
Testing the construction on additional examples of self-similar actions would check how broadly the unification applies.

Load-bearing premise

The input is an etale locally compact groupoid equipped with a groupoid correspondence whose induced C*-correspondence admits a relative Cuntz-Pimsner construction with respect to the ideal coming from an open invariant subset.

What would settle it

An explicit computation for a concrete etale groupoid and correspondence where the relative Cuntz-Pimsner algebra fails to be isomorphic to the C*-algebra of the constructed groupoid would disprove the claim.

read the original abstract

A groupoid correspondence on an etale, locally compact groupoid induces a C*-correspondence on its groupoid C*-algebra. We show that the Cuntz-Pimsner algebra for this C*-correspondence relative to an ideal associated to an open invariant subset of the groupoid is again a groupoid C*-algebra for a certain groupoid. We describe this groupoid explicitly and characterise it by a universal property that specifies its actions on topological spaces. Our construction unifies the construction of groupoids underlying the C*-algebras of topological graphs and self-similar groups.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Meyer builds an explicit groupoid model realizing relative Cuntz-Pimsner algebras from groupoid correspondences as groupoid C*-algebras, with a universal property that ties together topological graphs and self-similar groups.

read the letter

The main thing to know is that this paper gives a concrete groupoid whose C*-algebra recovers the relative Cuntz-Pimsner algebra of a correspondence on an etale locally compact groupoid, taken relative to the ideal coming from an open invariant subset, and it characterizes that groupoid by a universal property for actions on spaces. The construction unifies the groupoid pictures already used for topological graphs and for self-similar groups. Meyer starts from the given groupoid and correspondence, induces the C*-correspondence, and then defines a new groupoid via a pullback or equivalence relation that folds in the ideal; the claim is that the resulting C*-algebra matches the relative one and that the universal property holds by direct verification from the correspondence data. That explicitness and the unification are the useful parts. The setup is standard in this corner of operator algebras, and the paper organizes some previously separate examples under one description without introducing extra parameters or circular definitions. The soft spots are limited. The verification that the new groupoid operations are well-defined and that the isomorphism and universal property go through relies on checking compatibility with the original correspondence and the invariant subset; the outline suggests this is straightforward, but a referee would still want to see the lemmas that confirm the actions on spaces are captured exactly and that no hidden restrictions on the correspondence are needed. Nothing in the stated hypotheses looks inconsistent or under-justified. This is written for people already working with groupoid C*-algebras, Cuntz-Pimsner constructions, and dynamical systems in operator algebras. A reader who wants a common language for these models will find it helpful; someone outside the subfield will not. The work shows clear engagement with the literature and a direct construction rather than fitting or self-reference. I would send it to peer review. The unification and explicit model are worth having experts check the details.

Referee Report

1 major / 2 minor

Summary. The paper considers an étale locally compact groupoid equipped with a groupoid correspondence that induces a C*-correspondence on the groupoid C*-algebra. It claims to construct an explicit new groupoid whose C*-algebra is isomorphic to the relative Cuntz-Pimsner algebra of the original correspondence with respect to the ideal associated to an open invariant subset, and to characterize this new groupoid via a universal property for its actions on topological spaces. The construction is presented as unifying the groupoid models underlying C*-algebras of topological graphs and self-similar groups.

Significance. If the explicit construction and universal property hold, the result supplies a groupoid model for a class of relative Cuntz-Pimsner algebras, extending existing work on groupoid correspondences and permitting groupoid techniques (such as amenability or K-theory computations) to be applied in this relative setting. The unification of topological-graph and self-similar-group constructions is a concrete strength that could streamline future comparisons between these classes.

major comments (1)

The manuscript asserts an explicit description of the new groupoid and a direct verification of the isomorphism with the relative Cuntz-Pimsner algebra, yet the provided abstract and summary contain no lemmas, intermediate steps, or verification details for the pullback/equivalence-relation construction or the universal property. A load-bearing step appears to be the verification that the C*-algebra of the constructed groupoid coincides with the relative Cuntz-Pimsner algebra; without the concrete steps or a cited proposition, soundness cannot be assessed from the given information.

minor comments (2)

The abstract states the main result but does not reference a theorem number or section where the explicit groupoid is defined; adding such a pointer would improve readability.
Notation for the open invariant subset and the induced ideal is introduced without a preliminary subsection summarizing the standing assumptions on the original groupoid; a short notation table or paragraph would help.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for their positive evaluation of its potential significance in providing groupoid models that unify constructions for topological graphs and self-similar groups. We address the single major comment below.

read point-by-point responses

Referee: The manuscript asserts an explicit description of the new groupoid and a direct verification of the isomorphism with the relative Cuntz-Pimsner algebra, yet the provided abstract and summary contain no lemmas, intermediate steps, or verification details for the pullback/equivalence-relation construction or the universal property. A load-bearing step appears to be the verification that the C*-algebra of the constructed groupoid coincides with the relative Cuntz-Pimsner algebra; without the concrete steps or a cited proposition, soundness cannot be assessed from the given information.

Authors: The manuscript does contain an explicit construction of the new groupoid (via the indicated pullback and equivalence-relation approach) together with a verification that its C*-algebra realizes the relative Cuntz-Pimsner algebra. This verification proceeds by establishing the universal property for actions on topological spaces, which is the characterizing feature stated in the abstract. The intermediate steps supporting the construction and the isomorphism are developed in the body of the paper. We acknowledge, however, that the introduction could more clearly signpost these steps and cite the key propositions, thereby making the soundness of the argument easier to assess on a first reading. We will therefore add a short proof outline and improved cross-references in the revised introduction. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper constructs an explicit groupoid from the input etale locally compact groupoid and groupoid correspondence by incorporating the open invariant subset and associated ideal via a direct pullback or equivalence relation construction. It then verifies by direct computation that the C*-algebra of this new groupoid recovers the relative Cuntz-Pimsner algebra and satisfies the stated universal property for actions on spaces. All steps proceed from the given definitions and hypotheses without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations; the argument is self-contained and unifies prior constructions for topological graphs and self-similar groups through this explicit model.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions about etale locally compact groupoids and the correspondence between groupoid and C*-structures; no free parameters or new invented entities are introduced in the abstract.

axioms (1)

domain assumption The groupoid is etale and locally compact.
This is the explicit setting stated in the abstract for the groupoid correspondence.

pith-pipeline@v0.9.0 · 5615 in / 1170 out tokens · 48975 ms · 2026-05-19T08:08:22.887628+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages

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Thesis, Georg-August- Universität Göttingen, 2015, http://hdl.handle.net/11858/00-1735-0000-0028-87E8-C

Suliman Albandik, A colimit construction for groupoids, Ph.D. Thesis, Georg-August- Universität Göttingen, 2015, http://hdl.handle.net/11858/00-1735-0000-0028-87E8-C . available athttp://hdl.handle.net/11858/00-1735-0000-0028-87E8-C

work page 2015

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Thesis, Georg-August-Universität Göttingen, 2024

Celso Antunes, C*-algebras of relatively proper groupoid correspondences, Ph.D. Thesis, Georg-August-Universität Göttingen, 2024. doi: 10.53846/goediss-10930

work page doi:10.53846/goediss-10930 2024

[3] [3]

Math.28 (2022), 1329–1364, available athttps://nyjm.albany.edu/j/2022/28-56

Celso Antunes, Joanna Ko, and Ralf Meyer,The bicategory of groupoid correspondences, New York J. Math.28 (2022), 1329–1364, available athttps://nyjm.albany.edu/j/2022/28-56. html. MR 4489452

work page 2022

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Alcides Buss, Rohit Holkar, and Ralf Meyer,A universal property for groupoid C∗-algebras. I, Proc. Lond. Math. Soc. (3)117 (2018), no. 2, 345–375, doi: 10.1112/plms.12131.MR3851326 34 RALF MEYER

work page doi:10.1112/plms.12131.mr3851326 2018

[5] [5]

Alcides Buss and Ralf Meyer,Inverse semigroup actions on groupoids, Rocky Mountain J. Math. 47 (2017), no. 1, 53–159, doi: 10.1216/RMJ-2017-47-1-53. MR3619758

work page doi:10.1216/rmj-2017-47-1-53 2017

[6] [6]

Alcides Buss, Ralf Meyer, and Chenchang Zhu, A higher category approach to twisted actions on C∗-algebras, Proc. Edinb. Math. Soc. (2) 56 (2013), no. 2, 387–426, doi: 10.1017/S0013091512000259. MR 3056650

work page doi:10.1017/s0013091512000259 2013

[7] [7]

Ruy Exel,Inverse semigroups and combinatorialC∗-algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313, doi: 10.1007/s00574-008-0080-7. MR2419901

work page doi:10.1007/s00574-008-0080-7 2008

[8] [8]

Math.306 (2017), 1046–1129, doi: 10.1016/j.aim.2016.10.030

Ruy Exel and Enrique Pardo,Self-similar graphs, a unified treatment of Katsura and Nekra- shevych C∗-algebras, Adv. Math.306 (2017), 1046–1129, doi: 10.1016/j.aim.2016.10.030. MR 3581326

work page doi:10.1016/j.aim.2016.10.030 2017

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Higgins,Techniques of semigroup theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992

Peter M. Higgins,Techniques of semigroup theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1992. With a foreword by G. B. Preston. MR 1167445

work page 1992

[10] [10]

Takeshi Katsura,A class ofC∗-algebras generalizing both graph algebras and homeomorphism C∗-algebras. I. Fundamental results, Trans. Amer. Math. Soc.356 (2004), no. 11, 4287–4322, doi: 10.1090/S0002-9947-04-03636-0. MR2067120

work page doi:10.1090/s0002-9947-04-03636-0 2004

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Math.184 (1998), no

Alex Kumjian, David Pask, and Iain Raeburn,Cuntz–Krieger algebras of directed graphs, Pacific J. Math.184 (1998), no. 1, 161–174, doi: 10.2140/pjm.1998.184.161. MR1626528

work page doi:10.2140/pjm.1998.184.161 1998

[12] [12]

Whittaker,Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs, Adv

Marcelo Laca, Iain Raeburn, Jacqui Ramagge, and Michael F. Whittaker,Equilibrium states on operator algebras associated to self-similar actions of groupoids on graphs, Adv. Math.331 (2018), 268–325, doi: 10.1016/j.aim.2018.03.030. MR3192463

work page doi:10.1016/j.aim.2018.03.030 2018

[13] [13]

arXiv: 2203.12068

Ralf Meyer,Groupoid models for diagrams of groupoid correspondences(2022), eprint. arXiv: 2203.12068

work page arXiv 2022

[14] [14]

Ralf Meyer and Ryszard Nest,C∗-Algebras over topological spaces: the bootstrap class, Mün- ster J. Math. 2 (2009), 215–252, available athttp://nbn-resolving.de/urn:nbn:de:hbz: 6-10569452982. MR 2545613

work page 2009

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Scand.125 (2019), no

Ralf Meyer and Camila Fabre Sehnem,A bicategorical interpretation for relative Cuntz– Pimsner algebras, Math. Scand.125 (2019), no. 1, 84–112, doi: 10.7146/math.scand.a-112630. MR 4009326

work page doi:10.7146/math.scand.a-112630 2019

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Ralf Meyer and Chenchang Zhu,Groupoids in categories with pretopology, Theory Appl. Categ. 30 (2015), 1906–1998, available athttp://www.tac.mta.ca/tac/volumes/30/55/30-55abs. html. MR 3438234

work page 2015

[17] [17]

arXiv: 2409.02359

Alistair Miller and Benjamin Steinberg,Homology and K-theory for self-similar actions of groups and groupoids(2024), eprint. arXiv: 2409.02359

work page arXiv 2024

[18] [18]

Muhly, Jean N

Paul S. Muhly, Jean N. Renault, and Dana P. Williams,Equivalence and isomorphism for groupoid C∗-algebras, J. Operator Theory 17 (1987), no. 1, 3–22, available athttp: //www.theta.ro/jot/archive/1987-017-001/1987-017-001-001.html . MR 873460

work page 1987

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Reine Angew

Volodymyr Nekrashevych,C∗-algebras and self-similar groups, J. Reine Angew. Math.630 (2009), 59–123, doi: 10.1515/CRELLE.2009.035. MR2526786

work page doi:10.1515/crelle.2009.035 2009

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Alan L. T. Paterson,Graph inverse semigroups, groupoids and theirC∗-algebras, J. Operator Theory 48 (2002), no. 3, suppl., 645–662, available athttp://www.theta.ro/jot/archive/ 2002-048-003/2002-048-003-012.html . MR 1962477

work page 2002

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Pimsner, A class of C∗-algebras generalizing both Cuntz–Krieger algebras and crossed products byZ, Free probability theory (Waterloo, ON, 1995), Fields Inst

Mihai V. Pimsner, A class of C∗-algebras generalizing both Cuntz–Krieger algebras and crossed products byZ, Free probability theory (Waterloo, ON, 1995), Fields Inst. Commun., vol. 12, Amer. Math. Soc., Providence, RI, 1997, pp. 189–212. MR1426840

work page 1995

[22] [22]

103, Amer

Iain Raeburn,Graph algebras, CBMS Regional Conference Series in Mathematics, vol. 103, Amer. Math. Soc., Providence, RI, 2005. MR2135030

work page 2005

[23] [23]

793, Springer, Berlin, 1980

Jean Renault,A groupoid approach toC∗-algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. doi: 10.1007/BFb0091072 MR584266

work page doi:10.1007/bfb0091072 1980

[24] [24]

Math.9 (2004), 565–597, available athttp://www.math.uni-bielefeld.de/documenta/vol-09/26.html

Jean-Louis Tu,Non-Hausdorff groupoids, proper actions andK-theory, Doc. Math.9 (2004), 565–597, available athttp://www.math.uni-bielefeld.de/documenta/vol-09/26.html. MR 2117427

work page 2004

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G. T. Whyburn,A unified space for mappings, Trans. Amer. Math. Soc.74 (1953), 344–350, doi: 10.2307/1990886. MR0052762 Email address: rmeyer2@uni-goettingen.de Mathematisches Institut, Universität Göttingen, Bunsenstraße 3–5, 37073 Göttingen, Germany

work page doi:10.2307/1990886 1953