An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity

Hossein Larki

REVIEW 3 major objections 3 minor 45 references

An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Every self-similar k-graph C*-algebra is the tight C*-algebra of an inverse semigroup.

desk verdict A genuinely useful inverse semigroup model with a convincing main isomorphism, but the simplicity theorems rest on a false ultra-filter claim and a missing converse; needs revision and refereeing. read the letter →

arxiv 2411.14027 v1 pith:LHQIR4YO submitted 2024-11-21 math.OA

Hossein Larki This is my paper · ORCID

classification math.OA MSC 46L05

keywords finitelyalignedk-graphself-similarinversesemigrouptightgroupoidofgermsC*-algebrasimplicityboundarypathspaceG-cofinality

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

This paper shows that every self-similar k-graph C*-algebra, defined for a group action on a finitely aligned higher-rank graph (sources allowed), is really the tight C*-algebra of a single inverse semigroup built from the graph and the action. The payoff is that the algebra's groupoid, its tight groupoid of germs, can be read off from graphical data: its unit space is the boundary path space of the graph, Hausdorffness is controlled by strongly fixed paths, minimality by G-cofinality, and effectiveness by an aperiodicity condition. From these, the paper derives concrete necessary and sufficient conditions for simplicity of the algebra, covering both Hausdorff and non-Hausdorff cases. A sympathetic reader should care because this unifies and extends earlier row-finite, source-free, pseudo-free results to a substantially larger class, and gives an inverse-semigroup machine for future questions.

What carries the argument

The central object is the inverse semigroup $S_{G,\Lambda}$: its elements are finite sets of pairwise orthogonal triples $(\mu,g,\nu)$ of paths and group elements with $s(\mu)=g\cdot s(\nu)$, the product is defined by splicing minimal common extensions, and the inverse reverses triples. It carries the argument because its tight C*-algebra is the universal algebra $\mathcal{O}_{G,\Lambda}$ (Theorem 4.7), its tight groupoid of germs is the groupoid whose C*-algebra is the same algebra, and its idempotent semilattice is exactly that of the underlying k-graph's inverse semigroup. That last fact, via a homeomorphism between ultrafilters in that semilattice and boundary paths, identifies the tight spectrum with $\partial\Lambda$, letting every later graphical criterion be read off from the k-graph itself.

What would settle it

Look for a self-similar k-graph over a finitely aligned k-graph for which two different boundary paths produce the same filter $\mathcal{F}_x$, or for which a tight filter contains no idempotent $\iota_{x(0,n)}$; either would disprove Proposition 6.3 and void the simplicity theorems. Concretely, testing a small example with a non-pseudo-free cocycle and comparing the tight-spectrum topology with the cylinder-set topology on $\partial\Lambda$ would settle it.

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Extended reading notes

Core claim

The paper's central claim is that the universal C*-algebra $\mathcal{O}_{G,\Lambda}$ of a self-similar k-graph $(G,\Lambda)$ over a finitely aligned k-graph $\Lambda$ admits a canonical inverse semigroup model: there is an inverse semigroup $S_{G,\Lambda}$ whose tight C*-algebra is canonically isomorphic to $\mathcal{O}_{G,\Lambda}$, and whose tight groupoid of germs $\mathcal{G}_{\mathrm{tight}}(S_{G,\Lambda})$ satisfies $C^*(\mathcal{G}_{\mathrm{tight}}(S_{G,\Lambda})) \cong C^*_{\mathrm{tight}}(S_{G,\Lambda}) \cong \mathcal{O}_{G,\Lambda}$. The elements of $S_{G,\Lambda}$ are finite sets of pairwise orthogonal triples $(\mu,g,\nu)$ with $s(\mu)=g\cdot s(\nu)$, with multiplication computed through minimal common extensions; its idempotents coincide with those of the inverse semigroup of the underlying k-graph, so the tight spectrum is homeomorphic to the boundary path space $\partial\Lambda$. On this model, Hausdorffness of the groupoid is equivalent to the set of strongly fixed paths by each nontrivial group element being locally exhausted; minimality is equivalent to G-cofinality; effectiveness is equivalent, under that Hausdorff condition, to a G-aperiodicity condition. These criteria combine into simplicity theorems for $\mathcal{O}_{G,\Lambda}$, including a non-Hausdorff case handled by a stronger condition on the semigroup action.

Load-bearing premise

The whole bridge from algebras to graph conditions rests on the claim that the tight filters of $S_{G,\Lambda}$ correspond exactly to boundary paths of $\Lambda$ (Proposition 6.3); if that homeomorphism failed, the simplicity and minimality criteria would not transfer to the algebra.

Editorial extensions

If this is right

For every finitely aligned self-similar k-graph, $\mathcal{O}_{G,\Lambda}$ is canonically isomorphic to the tight C*-algebra of $S_{G,\Lambda}$ and to $C^*(\mathcal{G}_{\mathrm{tight}}(S_{G,\Lambda}))$, so inverse semigroup machinery applies even when $\Lambda$ has sources and the action is not pseudo-free.
The tight groupoid is Hausdorff exactly when strongly fixed paths are locally exhausted for every nontrivial group element; in particular every pseudo-free self-similar k-graph has a Hausdorff tight groupoid.
The tight spectrum is $\partial\Lambda$, so the groupoid is minimal exactly when $(G,\Lambda)$ is G-cofinal, and effective exactly under the stated G-aperiodicity conditions, with equivalence when Hausdorffness holds.
Under Hausdorffness and amenability of the groupoid, $\mathcal{O}_{G,\Lambda}$ is simple if and only if it is G-cofinal, G-aperiodic in the stated sense, and satisfies the local strong-fixed-point condition; a separate non-Hausdorff theorem gives simplicity of the reduced algebra from G-cofinality, aperiodicity, and eventual triviality of the cocycle along infinite paths.
These results extend earlier simplicity criteria that were restricted to row-finite, source-free k-graphs and pseudo-free actions.

Reading between the lines

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

Because the idempotent semilattice is independent of the group, the unit space of the tight groupoid is the same boundary-path space as for the underlying k-graph; the action's dynamics live entirely in the germs. A natural extension would be to classify ideals or K-theory of $\mathcal{O}_{G,\Lambda}$ through invariant subsets of $\partial\Lambda$, a route the paper does not take.
The non-Hausdorff simplicity result via Condition (S) suggests that the Zappa-Szep product picture for left-cancellative small categories should admit analogous simplicity criteria under weaker hypotheses than right cancelation, using the same strongly-fixed-path analysis.
One could test whether the G-cofinality criterion alone characterizes simplicity of the full algebra when the tight groupoid is amenable but non-Hausdorff; the paper only states this under additional Condition (S) hypotheses.
The finite-set form of $S_{G,\Lambda}$, rather than single triples as in the one-graph case, may be the right model for handling sources, and it invites explicit computation for examples where $\Lambda$ is not row-finite by replacing covers with finite exhaustive sets.

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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

The thing to know: the inverse semigroup S_{G,Λ} for finitely aligned self-similar k-graphs is a real contribution, and the main isomorphism theorem (4.7) is argued in enough detail that I find it convincing. The simplicity results, though, are not ready as written. Three specific gaps need work.

What is genuinely new: the S_{G,Λ} construction, the cover-to-join proof that its tight C*-algebra is O_{G,Λ}, the Hausdorffness criterion via SF_g locally exhausted, the G-cofinality minimality criterion, and the G-aperiodicity condition. These extend Li-Yang's row-finite pseudo-free theory to the non-pseudo-free case. The author also honestly connects to the Ortega-Pardo/Spielberg LCSC framework via Proposition 6.4, and that comparison is useful.

Now the soft spots.

1. Proposition 6.3 is false as stated. The claimed homeomorphism from ∂Λ onto the ultra-filter space fails. Take a vertex v with infinitely many edges e_n to distinct sinks. The empty path v is a boundary path (no finite exhaustive sets exist), and F_v = {E : ι_v ∈ E} is not an ultra-filter: the idempotent {ι_{e_n}} intersects every element of F_v (since {ι_{e_n}}ι_v = {ι_{e_n}}), yet {ι_{e_n}} is not in F_v. The proof's step where finiteness of E forces some x(0,n) into E breaks down at n=0. The tight spectrum may still equal ∂Λ via the known identification for S_Λ, since E(S_{G,Λ})=E(S_Λ), but the paper's argument—and the later equality E_tight = E_∞ used in Proposition 11.3—is unsupported.

2. Corollary 10.2 asserts an "if and only if" for simplicity, but only the forward direction (Proposition 10.1) is proven. The reverse implication is a missing theorem, not a minor detail.

3. Proposition 11.3 invokes Theorem 9.6 to get effectiveness but does not verify condition 9.6(3)(b). Theorem 9.6(3)(b) requires a finite exhaustive set of strongly fixed paths when g fixes all boundary paths at a vertex; the paper's condition (3) is a pointwise statement about individual infinite paths. I don't see how the latter implies the former.

These are addressable. The central construction is sound, and the Hausdorffness and minimality criteria look right. A serious revision could fix the gaps. As it stands, I would not cite the simplicity claims, but I would send this to a referee. The referee should insist on a corrected Proposition 6.3 (or a direct appeal to the S_Λ tight spectrum result), a proof of the converse in 10.2, and a repair of 11.3.

Overall: deserves peer review, yes. I'd cite it for the inverse semigroup model once cleaned up.

Referee Report

3 major / 3 minor

Summary. The paper generalizes the Li-Yang notion of self-similar k-graphs and their C*-algebras to the setting of finitely aligned k-graphs, and introduces an inverse semigroup S_{G,Λ} whose tight C*-algebra is claimed to be canonically isomorphic to O_{G,Λ}. The main structural result, Theorem 4.7, asserts O_{G,Λ} ≅ C*_tight(S_{G,Λ}) ≅ C*(G_tight(S_{G,Λ})). The paper then characterizes Hausdorffness, minimality, effectiveness, and simplicity of the tight groupoid in terms of graphical conditions such as local exhaustion of strongly fixed paths, G-cofinality, and a G-aperiodicity condition, and applies these to obtain simplicity criteria for O_{G,Λ} in both Hausdorff and non-Hausdorff settings.

Significance. If the main structural theorem and the spectral identifications hold, the paper provides a genuinely useful framework: it extends self-similar k-graph C*-algebras to finitely aligned graphs with sources, constructs the first inverse semigroup model for this class, and connects the tight groupoid to both the earlier path-like groupoid of Li-Yang and to Spielberg's LCSC groupoids. The proof of Theorem 4.7 is detailed and has a clear universal-property strategy: an explicit representation π is shown to be cover-to-join, and universality is verified by deriving the Cuntz-Krieger and self-similarity relations. The paper also explicitly acknowledges the relationship with the more general LCSC framework, which is a useful contextualization. However, the later simplicity results rest on an identification of the tight spectrum with the boundary path space, and the proof of that identification contains a substantive error; consequently the non-Hausdorff simplicity theorem and the claimed iff in the Hausdorff simplicity corollary are not established as written.

major comments (3)

[Section 6, Proposition 6.3] The assertion that every boundary path x gives an ultrafilter F_x is false. Consider the finitely aligned 1-graph with one vertex v and countably many edges e_n to distinct sinks v_n. The empty path v is a boundary path vacuously, but F_v = {E ∈ E(S_G,Λ) : (v,e,v) ∈ E} is not an ultrafilter: for every n, ι_{e_n} intersects ι_v, yet ι_{e_n} ∉ F_v, contradicting Remark 6.2(1). Thus the map x ↦ F_x is not onto the ultrafilter space, and the conclusion after Proposition 6.3 that Ê_tight(S_G,Λ) = Ê_∞(S_G,Λ) = ∂Λ is unjustified. This equality is used in load-bearing ways: the explicit description of G_tight(S_G,Λ) in (6.1)-(6.2) and the proof of Proposition 11.3 both rely on it. The tight-spectrum homeomorphism might be recoverable from E(S_G,Λ) = E(S_Λ) and the known results of [14], but it is not proved by the argument given in the paper.
[Section 10, Corollary 10.2] Corollary 10.2 states an iff criterion for simplicity of O_{G,Λ}, but Proposition 10.1 only proves the forward direction. The converse direction—that simplicity of O_{G,Λ} forces G-cofinality, the G-aperiodicity condition (A), and the strongly-fixed exhaustive condition—is not argued. To make the claimed equivalence valid, the author needs to invoke the converse of the Brown-Clark-Farthing-Sims theorem and the converses of Theorems 8.3 and 9.6, and to spell out how those converses apply under the standing assumptions. As written, the corollary exceeds what is proved.
[Section 11, Proposition 11.3] The proof asserts that G_tight(S_G,Λ) is minimal and effective by Theorems 8.3 and 9.6, but Theorem 9.6(3) requires condition (3)(b): if g fixes every x ∈ v∂Λ, then there exists a finite exhaustive set X such that every τ ∈ X is strongly fixed by g. Proposition 11.3's hypothesis (3) only gives, for each individually fixed path x, a prefix with trivial cocycle; it does not, as stated, supply the uniform finite exhaustive set required by Theorem 9.6(3)(b). Even if row-finiteness and the source-free assumption can be used to extract such an X by compactness, the argument is not given. Additionally, the proof uses the equality Ê_tight(S_G,Λ) = Ê_∞(S_G,Λ), which is false by the counterexample in the first major comment.

minor comments (3)

[Section 7, Proposition 7.2] The converse implication in the proof of (7.2) is left to the reader, but injectivity of the proposed groupoid isomorphism is central to the comparison with Li-Yang's groupoid; it should be written out.
[Section 4, Remark 4.5] The decomposition of a cover C for J_F into covers C_i for J_{ι_{μ_i}} is stated very tersely and would benefit from a sentence explaining how the outer-cover and cover conditions interact under the union.
[General] There are several typographical issues, e.g., 'beacuse' in the proof of Lemma 3.3 and inconsistent spacing in 'C ∗-algebras'; these do not affect the mathematics but should be cleaned up.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the inverse-semigroup model is proved directly, and the boundary-path identifications rely on external results.

full rationale

The central claim of the paper, Theorem 4.7, is proved by a direct universal-property argument rather than by importing its own conclusion. The paper constructs the inverse semigroup S_{G,Λ} from the self-similar k-graph data, then shows in Lemmas 4.3–4.6 that tight representations of S_{G,Λ} correspond exactly to (G,Λ)-families, and finally uses the universal property of O_{G,Λ} together with Exel's results [9, Theorem 13.3 and Corollary 10.16] to obtain C*_tight(S_{G,Λ}) ≅ C*(G_tight(S_{G,Λ})) ≅ O_{G,Λ}. No step in this chain assumes the isomorphism it is proving. The identification of the tight spectrum with the boundary path space in Section 6 rests on Corollary 3.5, which identifies the idempotent semilattice of S_{G,Λ} with that of S_Λ, and on the external result [14] identifying the tight spectrum of S_Λ with ∂Λ. Proposition 6.3 gives an additional ultrafilter description, but the later results do not depend on a purported circular reduction; they depend on a known external theorem. The only self-citation in the paper is the use of [22, Proposition 2.7 and Lemma 2.6] in Proposition 2.6 to justify a spanning-set identity. This is an auxiliary algebraic fact from a published paper, it does not presuppose Theorem 4.7, and the main structural theorem is proven independently of it. The paper also explicitly acknowledges possible overlap with the LCSC framework but proves an isomorphism with the Ortega–Pardo groupoid in Proposition 6.4 rather than treating that framework as an input. There are no fitted parameters, no data-fitting steps, and no uniqueness theorem imported from the author's own prior work to force a conclusion. Even if the proof of Proposition 6.3 were subject to a correctness challenge, that would be a mathematical-error concern, not an instance of the paper's derivation reducing to its own inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are present; the construction is purely combinatorial. The axioms are standard operator-algebraic tools and the paper's standing assumption of finite alignment. No new physical entities are introduced.

assumptions (4)

domain assumption Standing assumption: Λ is a finitely aligned k-graph (Definition 2.1).
The entire paper restricts to finitely aligned k-graphs, possibly with sources, and defines S_{G,Λ} using finite sets of triples and minimal common extensions.
domain assumption Standard Cuntz-Krieger relations for finitely aligned k-graphs (Def 2.2) and the universality of O_{G,Λ} (Def 2.4).
The inverse semigroup is modeled on the generators s_μ, u_{v,g} of O_{G,Λ}; if these relations were not the right universal relations the isomorphism in Theorem 4.7 would fail.
standard math Exel's inverse semigroup tight C*-algebra theory ([9], [12]).
The paper uses tight representations, cover-to-join, tight groupoids, and characterizations from Exel and Exel-Pardo as black boxes.
standard math Simplicity criteria for groupoid C*-algebras ([5, Theorem 5.1], [7, Lemma 5.6, Corollary 4.12]).
The simplicity results in Sections 10 and 11 are deduced from these external theorems.

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Pith. "Pith review of An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity." pith.science (2026). https://pith.science/paper/LHQIR4YO

@misc{pith2026241114027,
  author       = {Pith},
  title        = {Pith review of: An inverse semigroup approach to self-similar k-graph $C^*$-algebras and simplicity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LHQIR4YO}},
  note         = {Machine review of arXiv:2411.14027}
}

abstract

We generalize the Li-Yang notion of self-similar $k$-graph $(G,\Lambda)$ and its $C^*$-algebra $\mathcal{O}_{G,\Lambda}$ to any finitely aligned $k$-graph $\Lambda$. We then introduce an inverse semigroup model for $\mathcal{O}_{G,\Lambda}$ and analyze its tight groupoid and $C^*$-algebra via inverse semigroup methods.

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Works this paper leans on

45 extracted references · 44 canonical work pages

[14]

Farthing, P.S

C. Farthing, P.S. Muhly, and T. Yeend, Higher-rank graph C ∗ -algebras: an inverse semigroup and groupoid approach , Semigroup Forum 71(2) (2005), 159- 187

work page 2005
[1]

Anantharaman-Delaroche and J

C. Anantharaman-Delaroche and J. Renault. Amenable groupoids , vol. 36 of Monographs of L’Enseignement Math´ ematique, Geneva, 2000

work page 2000
[2]

B´ edos, S

E. B´ edos, S. Kaliszewski, and J. Quigg, On Exel-Pardo algebras , J. Operator Theory 78(2) (2017), 309-345

work page 2017
[3]

B´ edos, S

E. B´ edos, S. Kaliszewski, and J. Quigg, Skew products of ﬁnitely aligned left cancellative small categories and Cuntz-Krieger algebras , M¨ unster J. Math. 14(1) (2021), 59-99

work page 2021
[4]

B´ edos, S

E. B´ edos, S. Kaliszewski, J. Quigg, and J. Spielberg, On ﬁnitely aligned left can- cellative small categories, Zappa-Sz ´ep products and Exel-Pardo algebras , The- ory Appl. Categ. 33 (2018), 1346-1406

work page 2018
[5]

Brown, L.O

J. Brown, L.O. Clark, C. Farthing, and A. Sims, Simplicity of algebras associ- ated to ´etale groupoids, Semigroup Forum 88 (2014), 433-452

work page 2014
[6]

Burris and H.P

S. Burris and H.P. Sankappanavar, A course in universal algebra , vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York- Berlin, 1981

work page 1981
[7]

Clark, R

L. Clark, R. Exel, E. Pardo, A. Sims, and C. Starling Simplicity of algebras associated to non-Hausdorﬀ groupoids , Trans. Amer. Math. Soc. 372(5) (2019), 3669-3712

work page 2019

Show all 45 references

[8]

Donsig and D

A.P. Donsig and D. Milan, Joins and covers in inverse semigroups and tight C ∗ -algebras, Bull. Aust. Math. Soc. 90(1) (2014), 121-133

2014
[9]

Ruy Exel, Inverse semigroups and combinatorial C ∗ -algebras, Bull. Braz. Math. Soc. 39(2) (2008), 191-313. 3A subset B ⊆ G tight(SG,Λ ) is called regular open if ( B)0 = B self-similar k-graph C∗-algebras 37

2008
[10]

Exel, Tight and cover-to-join representations of semilattices a nd in- verse semigroups , Operator Theory, Functional Analysis and Applications

R. Exel, Tight and cover-to-join representations of semilattices a nd in- verse semigroups , Operator Theory, Functional Analysis and Applications. Birkh¨ auser, Cham (2021), 183-192

2021
[11]

Exel and E

R. Exel and E. Pardo, Self-similar graphs, a uniﬁed treatment of Katsura and Nekrashevych C ∗ -algebras, Adv. Math. 306 (2017), 1046-1129

2017
[12]

Exel and E

R. Exel and E. Pardo, The tight groupoid of an inverse semigroup , Semigroup Forum 92 (2016), 274-303

2016
[13]

R. Exel, E. Pardo, and C. Starling, C ∗ -algebras of self-similar graphs over arbitrary graphs, arXiv:1807.01686 (2018)

2018 arXiv
[15]

Hazrat, D

R. Hazrat, D. Pask, A. Sierakowski, and A. Sims, An algebraic analogue of Exel-Pardo C ∗ -algebras, Algebra Represent. Theory 24(4) (2021), 877-909

2021
[16]

Katsura, A construction of actions on Kirchberg algebras which induc e given actions on their K-groups, J

T. Katsura, A construction of actions on Kirchberg algebras which induc e given actions on their K-groups, J. Reine Angew. Math. 617 (2008), 27-65

2008
[17]

Kumjian and D

A. Kumjian and D. Pask, Higher rank graph C ∗ -algebras, New York J. Math. 6 (2000), 1-20

2000
[18]

M. Laca, I. Raeburn, J. Ramagge, and M.F. Whittaker, Equilibrium states on operator algebras associated to self-similar actions of gr oupoids on graphs, Adv. Math. 331 (2018), 268-325

2018
[19]

LaLonde and D

S.M. LaLonde and D. Milan, Amenability and uniqueness for groupoids asso- ciated with inverse semigroups , Semigroup Forum 95 (2016), 321-344

2016
[20]

LaLonde, D

S.M. LaLonde, D. Milan, and J. Scott, Condition (K) for inverse semigroups and the ideal structure of their C ∗ -algebras, J. Algebra 523 (2019), 119-153

2019
[21]

Larki, A dichotomy for simple self-similar graph C ∗ -algebras, J

H. Larki, A dichotomy for simple self-similar graph C ∗ -algebras, J. Math. Anal. Appl. 494(2) (2021), 124622

2021
[22]

Larki, Exel-Pardo algebras of self-similar k-graphs, Bull

H. Larki, Exel-Pardo algebras of self-similar k-graphs, Bull. Malays. Math. Soc. 46(2):82 (2023)

2023
[23]

Lawson, Inverse semigroups: The theory of partial symetries , World Sci- entiﬁc Publishing Co., Inc., River Edge, NJ, 1998

M.V. Lawson, Inverse semigroups: The theory of partial symetries , World Sci- entiﬁc Publishing Co., Inc., River Edge, NJ, 1998

1998
[24]

Li, Left regular representations of Garside categories I

X. Li, Left regular representations of Garside categories I. C ∗ -algebras and groupoids, Glasg. Math. J. 65 (2023), S53-S86

2023
[25]

Li and D

H. Li and D. Yang, Boundary quotient C ∗ -algebras of products of odometers , Canad. J. Math. 71(1) (2019), 183-212

2019
[26]

Li and D

H. Li and D. Yang, KMS states of self-similar k-graph C ∗ -algebras, J. Funct. Anal. 276(12) (2019), 3795-3831

2019
[27]

Li and D

H. Li and D. Yang, Self-similar k-graph C ∗ -algebras, Int. Math. Res. Not. 15 (2021), 11270-11305

2021
[28]

Li and D

H. Li and D. Yang, The ideal structures of self-similar k-graph C ∗ -algebras, Ergodic Theory Dynam. Systems 41(8) (2021), 2480-2507

2021
[29]

Mesyan and J.D

Z. Mesyan and J.D. Mitchell, The structure of a graph inverse semigroup , Semi- group Forum 93(1) (2016), 111-130

2016
[30]

Milan and B

D. Milan and B. Steinberg, On inverse semigroup C ∗ -algebras and crossed products, Groups Geom. Dyn. 2 (2014), 485-512. 38 Hossein Larki

2014
[31]

Nekrashevych, Cuntz-Pimsner algebras of group actions , J

V. Nekrashevych, Cuntz-Pimsner algebras of group actions , J. Operator Theory 52 (2004), 223-249

2004
[32]

Nekrashevych, Growth of ´etale groupoids and simple algebras , Int

V. Nekrashevych, Growth of ´etale groupoids and simple algebras , Int. J. Alg. Comp. 26 (2016), 375-397

2016
[33]

Nekrashevych, Self-Similar Groups , Mathematical Surveys and Mono- graphs, vol

V. Nekrashevych, Self-Similar Groups , Mathematical Surveys and Mono- graphs, vol. 117, Amer. Math. Soc., Providence RI 2005

2005
[34]

Ortega and E

E. Ortega and E. Pardo, The tight groupoid of the inverse semigroups of left cancellative small categories , Trans. Amer. Math. Soc. 373(7) (2020), 5199- 5234

2020
[35]

Ortega and E

E. Ortega and E. Pardo, Zappa-Sz´ep products for partial actions of groupoids on left cancellative small categories , J. Noncommut. Geom. 17(4) (2023), 1335- 1366

2023
[36]

Raeburn, Graph Algebras, CBMS Regional Conference Series in Mathemat- ics, vol

I. Raeburn, Graph Algebras, CBMS Regional Conference Series in Mathemat- ics, vol. 103, 2005

2005
[37]

Raeburn, A

I. Raeburn, A. Sims, and T. Yeend, The C ∗ -algebras of ﬁnitely aligned higher- rank graphs, J. Funct. Anal. 213 (2004), 206-240

2004
[38]

Renault, A groupoid approach to C ∗ -algebras, Lecture Notes in Mathematics, vol

J. Renault, A groupoid approach to C ∗ -algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980

1980
[39]

Renault, Cartan subalgebras in C ∗ -algebras, Irish Math

J. Renault, Cartan subalgebras in C ∗ -algebras, Irish Math. Soc. Bull. 61 (2008), 29-63

2008
[40]

Spielberg, C ∗ -algebras for categories of paths associated to the Baumsla g- Solitar groups , J

J. Spielberg, C ∗ -algebras for categories of paths associated to the Baumsla g- Solitar groups , J. Lond. Math. Soc. 86(3) (2012), 728-754

2012
[41]

Spielberg, Groupoids and C ∗ -algebras for categories of paths , Trans

J. Spielberg, Groupoids and C ∗ -algebras for categories of paths , Trans. Amer. Math. Soc. 366(11) (2014), 5771-5819

2014
[42]

Spielberg, Groupoids and C ∗ -algebras for left cancellative small categories , Indiana Univ

J. Spielberg, Groupoids and C ∗ -algebras for left cancellative small categories , Indiana Univ. Math. J. 69(5) (2020), 1579-1626

2020
[43]

Steinberg, A groupoid approach to discrete inverse semigroup algebras , Adv

B. Steinberg, A groupoid approach to discrete inverse semigroup algebras , Adv. Math. 223(2) (2010), 689-727

2010
[44]

Steinberg and N

B. Steinberg and N. Szak´ acs,Simplicity of inverse semigroup and ´etale groupoid algebras, Adv. Math. 380 (2021): 107611

2021
[45]

Webster, The path space of a higher-rank graph , Stud

S.B.G. Webster, The path space of a higher-rank graph , Stud. Math. 204 (2011), 155-185. Hossein Larki Department of Mathematics Faculty of Mathematical Sciences and Computer Shahid Chamran University of Ahvaz P.O. Box: 83151-61357 Ahvaz Iran e-mail: h.larki@scu.ac.ir

2011

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [14]

Farthing, P.S

C. Farthing, P.S. Muhly, and T. Yeend, Higher-rank graph C ∗ -algebras: an inverse semigroup and groupoid approach , Semigroup Forum 71(2) (2005), 159- 187

work page 2005

[2] [1]

Anantharaman-Delaroche and J

C. Anantharaman-Delaroche and J. Renault. Amenable groupoids , vol. 36 of Monographs of L’Enseignement Math´ ematique, Geneva, 2000

work page 2000

[3] [2]

B´ edos, S

E. B´ edos, S. Kaliszewski, and J. Quigg, On Exel-Pardo algebras , J. Operator Theory 78(2) (2017), 309-345

work page 2017

[4] [3]

B´ edos, S

E. B´ edos, S. Kaliszewski, and J. Quigg, Skew products of ﬁnitely aligned left cancellative small categories and Cuntz-Krieger algebras , M¨ unster J. Math. 14(1) (2021), 59-99

work page 2021

[5] [4]

B´ edos, S

E. B´ edos, S. Kaliszewski, J. Quigg, and J. Spielberg, On ﬁnitely aligned left can- cellative small categories, Zappa-Sz ´ep products and Exel-Pardo algebras , The- ory Appl. Categ. 33 (2018), 1346-1406

work page 2018

[6] [5]

Brown, L.O

J. Brown, L.O. Clark, C. Farthing, and A. Sims, Simplicity of algebras associ- ated to ´etale groupoids, Semigroup Forum 88 (2014), 433-452

work page 2014

[7] [6]

Burris and H.P

S. Burris and H.P. Sankappanavar, A course in universal algebra , vol. 78 of Graduate Texts in Mathematics, Springer-Verlag, New York- Berlin, 1981

work page 1981

[8] [7]

Clark, R

L. Clark, R. Exel, E. Pardo, A. Sims, and C. Starling Simplicity of algebras associated to non-Hausdorﬀ groupoids , Trans. Amer. Math. Soc. 372(5) (2019), 3669-3712

work page 2019