Essential groupoid amenability and nuclearity of groupoid C*-algebras

Alcides Buss; Diego Mart\'inez

REVIEW 2 major objections 3 minor 1 cited by

Essential groupoid amenability and nuclearity of groupoid C*-algebras

T0 review · 2 major / 3 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Nuclearity of the essential C*-algebra of a non-Hausdorff étale groupoid forces a strictly weaker form of amenability.

desk verdict The essential half is the real contribution and it holds up; the reduced half overlaps with parallel independent work, and the authors name the main open point honestly. read the letter →

arxiv 2501.01775 v3 pith:GA3TE2K5 submitted 2025-01-03 math.OA

Alcides Buss , Diego Martínez This is my paper · ORCID

classification math.OA MSC 20L0546L5246L05

keywords étalegroupoidnon-HausdorffessentialC*-algebranuclearityamenabilitydangerousarrowsBorelHerz-Schurmultipliers

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

Groupoids encode partially defined symmetries, and an étale groupoid is a topologised version that can fail to be Hausdorff. The paper studies the C*-algebras built from such groupoids, especially the reduced algebra and the essential algebra, the latter being designed to ignore behaviour on the 'dangerous' arrows where non-Hausdorffness is witnessed. Its central claim is that for étale groupoids covered by countably many open bisections, the reduced C*-algebra is nuclear exactly when the groupoid is amenable, while nuclearity of the essential C*-algebra forces a new, strictly weaker property called essential amenability. The paper also constructs a maximal essential algebra and a faithful representation of the essential algebra on the complement of the dangerous arrows. This matters because nuclearity is the key finite-dimensional approximation property of C*-algebras, and non-Hausdorff groupoids arise naturally from inverse semigroup actions and algebraic actions of cancellative semigroups.

What carries the argument

The central object is the set $D$ of dangerous arrows: an arrow $g\in G$ is dangerous if a net can converge to $g$ and to a different arrow $h$, so $D$ is exactly the set of points where $G$ fails to be Hausdorff. Under the countable-bisection hypothesis, $D$ is meager, and the singular ideal of the essential algebra is characterized as functions supported on $D$. The second load-bearing tool is a family of Borel Herz-Schur multipliers $m_\phi(a)(g)=\phi(g)a(g)$ with $\phi=\xi^* * \xi$, which give completely positive maps on the maximal, reduced, and essential Borel algebras; these multipliers carry both directions of the nuclearity-amenability argument. The faithful representation of $C^*_{\mathrm{ess}}(G)$ on $\bigoplus_{x\in X\setminus D}\ell^2(G_x)$ completes the picture by tying the essential algebra to the non-dangerous part of the groupoid.

What would settle it

Find an étale groupoid satisfying the countable-bisection hypothesis whose essential C*-algebra is nuclear yet some non-dangerous isotropy group $xGx$ at a unit $x\in X\setminus D$ is non-amenable; Proposition 5.6(iii) would then rule out essential amenability, contradicting Theorem 5.12(i)$\Rightarrow$(ii).

Watch

Extended reading notes

Core claim

Let $G$ be an étale groupoid with locally compact Hausdorff unit space $X:=G^{(0)}$, and assume $G$ can be covered by countably many open bisections. Theorem A says that $C^*_{\mathrm{red}}(G)$ is nuclear if and only if $G$ is amenable, in which case the left regular representation $C^*_{\max}(G)\to C^*_{\mathrm{red}}(G)$ is an isomorphism, and that if $C^*_{\mathrm{ess}}(G)$ is nuclear, then $G$ is essentially amenable. Essential amenability asks for functions $\xi_i$ in the algebra $A_c(G)$ (the linear span of pointwise products of functions from $C_c(G)$) such that the convolution squares $\xi_i^**\xi_i$ converge to $1$ uniformly on compact subsets of $G\setminus D$, where $D$ is the set of dangerous arrows; this property is strictly weaker than amenability. The paper introduces the maximal essential algebra $C^*_{\mathrm{ess,max}}(G)$ and proves that the essential regular representation on $\bigoplus_{x\in X\setminus D}\ell^2(G_x)$ is faithful, exhibiting the essential algebra as the reduced algebra of the non-dangerous part of the groupoid. It also defines Borel versions of amenability and essential amenability and proves they are equivalent to the topological versions.

Load-bearing premise

The whole argument leans on the assumption that the groupoid can be covered by countably many open bisections; if that fails, the dangerous arrows can be the entire groupoid and the essential machinery collapses, a limitation the paper explicitly notes.

Editorial extensions

If this is right

For a discrete group $\Gamma$ acting on a compact Hausdorff space $X$, nuclearity of $C(X)\rtimes_{\mathrm{red}} \Gamma$ is equivalent to amenability of the action, to nuclearity of $B_b(X)\rtimes_{\mathrm{red}} \Gamma$, and to Borel amenability of the action.
If $C^*_{\mathrm{ess}}(G)$ is nuclear, then every non-dangerous isotropy group $xGx$ with $x\in X\setminus D$ must be amenable, giving a concrete algebraic obstruction to nuclearity of the essential algebra.
Essential amenability is strictly weaker than amenability: for the groupoid $X\rtimes (\Gamma\sqcup\{0\})$ of Example 3.35, $G$ is essentially amenable for every group $\Gamma$, while $G$ is amenable only when $\Gamma$ is amenable.
For amenable $G$ with no nonzero meager-supported elements in $A^\infty_c(G)$, the canonical map $C^*_{\max}(G)\to C^*_{\mathrm{ess}}(G)$ is an isomorphism; in that case $C^*_{\max}(G)$ is simple exactly when $G$ is minimal and topologically free.
For exact algebraic actions of cancellative semigroups, the algebras $A_\sigma$ studied in the final section arise as quotients of $C^*_{\mathrm{ess,max}}(G_\sigma)$, and nuclearity of $A_\sigma$ implies essential amenability of $G_\sigma$.

Reading between the lines

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

The paper leaves open whether essential amenability alone implies nuclearity of $C^*_{\mathrm{ess}}(G)$; one concrete way to test this is to seek groupoids where the canonical map $\psi:C^*_{\mathrm{ess,max}}(G)\to B_{\mathrm{ess,max}}(G)$ fails to be max-injective, since the equivalence in Theorem 5.12 is proved precisely under that condition.
Because the singular ideal is exactly the set of functions supported on $D$, one can view $C^*_{\mathrm{ess}}(G)$ as a reduced algebra of the non-dangerous part of the groupoid; this suggests a broader principle that, under the countable-bisection hypothesis, non-Hausdorffness contaminates only a meager set and other ideal-structure questions could be studied by cutting out $D$.
The proved equivalence between Borel and topological (essential) amenability may make the property checkable by Borel Følner-type conditions, and the definition could plausibly be extended to groupoids without a countable bisection cover by quantifying over arbitrary meager sets, as the paper itself suggests.
For the algebras $A_\sigma$, nuclearity of $A_\sigma$ implies essential amenability of $G_\sigma$; deciding whether the converse holds for this class would amount to resolving the max-injectivity obstruction for these groupoids.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

Two things you should know. The genuinely new content is the essential half: a maximal essential C*-algebra, Borel essential algebras, and a notion of essential amenability strictly weaker than amenability, with the theorem that nuclearity of C*_ess(G) forces essential amenability. That direction is proved constructively, with explicit multipliers and norm estimates. The reduced-algebra equivalence—nuclearity of C*_red(G) iff G amenable—overlaps with the independent Brix–Gonzales–Hume–Li preprint [2], and the authors say so in their Note. Treat that half as parallel work, not the headline.

Credit where due. Proposition B (a function in C_c(G) vanishing on a dense set is supported only on the dangerous arrows D) is clean and does real work, as does the faithful representation of C*_ess(G) on the fibers over X\D in Proposition 3.32. The Herz-Schur multiplier machinery behind Proposition 5.15 is substantial, and the proof of (i)=> (ii) in Theorem 5.12 is spelled out in genuine detail. The Bruce-Li application in Section 7 is new and concrete. The paper is also honest about its own history: it names the earlier error found by Xin Li, flags a claimed mistake in Renault's [24, Prop 2.8] with a concrete counterexample, and repeatedly says where proofs are sketched.

The soft spots are mostly the ones the authors admit. Theorem 5.11 is sketchy in places, and (iii)=> (iv) in Theorem 5.12 is deferred as "similar to (i)=> (ii)"; a referee should ask for those details. The countable-bisection cover is genuinely load-bearing—D meager, the singular ideal identified with support in D, and faithfulness of the essential regular representation all depend on it—but it is a stated hypothesis, so a scope limitation rather than a hidden flaw. The open point, max-injectivity of the map from C*_ess,max to B_ess,max and whether essential amenability alone gives the essential containment property, is theirs to leave open; they flag it in Remarks 4.16 and 5.13, and the strong converse is explicitly conditional. The Renault claim deserves independent checking, but they supply an example. I agree with the stress-test read: no load-bearing flaw in the central claims as stated.

Who this is for: people working on non-Hausdorff étale groupoid C*-algebras, essential crossed products, and semigroup C*-algebras. It deserves a serious referee. My recommendation: send it to review; ask for the sketched implications to be completed or explicitly deferred, and for the Renault counterexample to be verified. Conditional, not reject.

Referee Report

2 major / 3 minor

Summary. The paper introduces a notion of essential amenability for étale groupoids with locally compact Hausdorff unit space, constructs a maximal essential C*-algebra C*_ess,max(G) and a family of Borel algebras, and proves a theorem relating nuclearity of the reduced and essential groupoid C*-algebras to amenability and essential amenability, respectively. The main results are Theorem A (Theorems 5.11 and 5.12): for étale groupoids covered by countably many open bisections, C*_red(G) is nuclear iff G is amenable, and if C*_ess(G) is nuclear then G is essentially amenable. Applications are given to crossed products and to the Bruce-Li algebras of algebraic actions of cancellative semigroups, which are realized as quotients of the maximal essential C*-algebra of a groupoid of germs.

Significance. If the results are correct, the paper is a substantive contribution to the C*-algebra theory of non-Hausdorff étale groupoids. The new notion of essential amenability is strictly weaker than amenability and is shown to be a necessary condition for nuclearity of the essential algebra; the proof of this direction is explicit and constructive, built on carefully justified Herz-Schur multipliers. The construction of the maximal essential algebra and of Borel groupoid algebras is versatile and likely to be reused. The paper also corrects an error in Renault's treatment of Borel amenability (Remark 5.3) and applies the machinery to Bruce-Li algebras. Importantly, the authors are transparent about the limitations: the converse implication (essential amenability ⇒ nuclearity of C*_ess) and the max-injectivity of ψ remain open, and the one-way direction is clearly stated. The potential significance is high for the non-Hausdorff groupoid literature.

major comments (2)

[Section 5.2, Theorem 5.11] The equivalence between nuclearity of C*_red(G) and amenability of G is one of the paper's main theorems, but its proof is only sketched. The text says 'we will sometimes be sketchy with the proofs of the former theorem' and the combined proof does not spell out the implications (i)⇒(ii) and (ii)⇒(i) for the reduced algebra, leaving the reader to adapt the essential-case arguments by replacing X\D with X. Because Theorem A(i) is a central claim, full details of these implications should be provided, or a precise reference to an existing proof in the non-Hausdorff setting should be given.
[Section 5.2, Theorem 5.12] The implication (iii)⇒(iv), asserting that nuclearity of Bess(G) implies Borel essential amenability, is only sketched. Although this implication is not needed for the one-way statement in Theorem A(ii), it forms part of the claimed equivalence (iii)⇔(iv) and of the conditional equivalence of all four conditions under max-injectivity. A rigorous proof should be supplied, or the theorem statement should be adjusted to reflect exactly which implications are fully proven.

minor comments (3)

[Abstract and Section 2.1] There are several typographical issues, e.g. 'W e' and 'th e' in the abstract and 'M orrally' in Section 2.1; these should be corrected in a final polish.
[Section 1, final paragraph] The phrase 'we characterize when C*_ess(G) is nuclear in terms of a certain essential amenability' is stronger than what is proven; only the implication nuclearity ⇒ essential amenability is established, with the converse left open in Remarks 4.16 and 5.13. Please rephrase to avoid overstatement.
[Corollary 3.42, Claim 3.43] The assertion that ‖λ_{γ_n}(a)‖ → ‖λ_{x_0}(a)‖ is justified by the statement that 'any finite behaviour at x0 can be witnessed at γ_n for all large n'; this is plausible but deserves a more formal argument, since the claim is used to factor the essential regular representation through πmax.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: Theorem A's essential-amenability implication is derived, not built into the definition; self-citations are not load-bearing.

full rationale

The paper's central claim (ii) of Theorem A is a one-way implication: nuclearity of C*_ess(G) implies essential amenability. Essential amenability (Definition 5.4) is an independent notion stated in terms of nets of Borel/pointwise-multiplier functions controlling ξ*ξ near 1 outside the dangerous set D; it is not defined as 'nuclear C*_ess' nor is it obtained by fitting parameters to the target claim. The proof of (i)=>(ii) in Theorem 5.12 genuinely constructs witnessing nets from finite-dimensional c.c.p. approximations of the identity on C*_ess(G), using the essential quotient only to identify functions on the meager dangerous set D. Conversely, (ii)=>(iii) follows from Herz-Schur multiplier technology, and the unresolved converse—whether essential amenability alone suffices for essential containment/nuclearity—is explicitly acknowledged in Remark 5.13, which would be impossible if the equivalence were built into the definitions. The countable-bisection hypothesis is stated transparently and used rather than hidden. Self-citations (e.g., [6]-[8]) support standard inverse-semigroup facts and are not load-bearing for the main theorems; key external anchors are Kwasniewski-Meyer [18], Neshveyev-Schwartz [20], and the independent work [2] noted in the text. The paper also candidly reports errors in an earlier version and in Renault's [24], further confirming that the implication is a substantive derivation rather than a restatement of inputs.

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

This is a purely proof-based paper: there are no fitted parameters, no physical constants, and no postulated new particles, forces, or dimensions. The new algebraic objects and notions, such as C*_ess,max(G) and essential amenability, are definitions rather than unexplained entities.

assumptions (5)

domain assumption G is an étale groupoid with locally compact Hausdorff unit space X, not necessarily Hausdorff itself.
This is the standing class of groupoids throughout the paper, stated in the conventions and used in every definition and theorem.
domain assumption G can be covered by countably many open bisections.
Used in Lemma 3.14, Corollary 3.15, Proposition 3.32, and the main Theorems 5.11 and 5.12. Without it the dangerous-arrow set D may not be meager and the essential representation need not be faithful.
standard math Standard C*-algebra facts: nuclearity, Stinespring dilation, max-injective inclusions, and generalized conditional expectations behave as stated.
Invoked throughout, especially in Proposition 5.15, Lemma 5.29, Corollary 4.15, and the proofs of the main theorems.
standard math The convolution algebra Bc(G) admits a largest C*-norm, so Bmax(G) and Bess,max(G) exist.
Used in Definition 4.1; the authors cite a standard argument from [28] rather than proving it.
standard math Baire Category Theorem for locally compact Hausdorff spaces.
Used in Corollary 3.13 to conclude that co-meager sets in the unit space are dense.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Essential groupoid amenability and nuclearity of groupoid C*-algebras." pith.science (2026). https://pith.science/paper/GA3TE2K5

@misc{pith2026250101775,
  author       = {Pith},
  title        = {Pith review of: Essential groupoid amenability and nuclearity of groupoid C*-algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GA3TE2K5}},
  note         = {Machine review of arXiv:2501.01775}
}

abstract

We give an alternative construction of the essential $C^*$-algebra of an \'etale groupoid, along with an ``amenability'' notion for such groupoids that is implied by the nuclearity of this essential $C^*$-algebra. In order to do this we first introduce a maximal version of the essential $C^*$-algebra, and prove that every function with dense co-support can only be supported on the set of ``dangerous'' arrows. We then introduce an essential amenability condition for a groupoid, which is (strictly) weaker than its (topological) amenability. As an application, we describe the Bruce-Li algebras arising from algebraic actions of cancellative semigroups as exotic essential $C^*$-algebras.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

The ideal structure of C*-algebras of etale groupoids with isotropy groups of local polynomial growth
math.OA 2024-12 accept novelty 8.0 of 10

For amenable etale groupoids with isotropy groups of local polynomial growth, the Jacobson topology on Prim C*(G) is characterized by closure in the Fell topology on subgroup-representation pairs.

Reference graph

Works this paper leans on

32 extracted references · 30 canonical work pages · cited by 1 Pith paper

[2]

Hume, and Xin Li, On Hausdorﬀ covers for non-Hausdorﬀ groupoids (2025)

Keving Aguyar Brix, Julian Gonzales, Jeremy B. Hume, and Xin Li, On Hausdorﬀ covers for non-Hausdorﬀ groupoids (2025). arXiv: 2503.23203

arXiv 2025
[1]

arXiv: 2406.05717

Krzysztof Bardadyn, Bartosz Kwaśniewski, and Andrew Mc Kee, Banach algebras associated to twisted étale groupoids: simplicity and pure inﬁnitenes s (2024). arXiv: 2406.05717

arXiv 2024
[3]

Brown and Narutaka Ozawa, C*-algebras and ﬁnite-dimensional approxima- tions, Graduate studies in mathematics, American Mathematical S ociety, Providence, R.I, 2008 (en)

Nathanial P. Brown and Narutaka Ozawa, C*-algebras and ﬁnite-dimensional approxima- tions, Graduate studies in mathematics, American Mathematical S ociety, Providence, R.I, 2008 (en). OCLC: ocn180190949

work page 2008
[4]

88, Amer

, C∗ -algebras and ﬁnite-dimensional approximations , Graduate Studies in Mathemat- ics, vol. 88, Amer. Math. Soc., 2008. MR2391387

work page 2008
[5]

C*-algebras and groupoids , Journal of Functional Analysis 286 (2024), no

Chris Bruce and Xin Li, Algebraic actions I. C*-algebras and groupoids , Journal of Functional Analysis 286 (2024), no. 4, 57 pp

work page 2024
[6]

Operator Theory 67 (2012), no

Alcides Buss and Ruy Exel, Fell bundles over inverse semigroups and twisted étale grou poids, J. Operator Theory 67 (2012), no. 1, 153–205. MR2881538

work page 2012
[7]

Alcides Buss, Ruy Exel, and Ralf Meyer, Reduced C∗ -algebras of Fell bundles over inverse semigroups, Israel J. Math. 220 (2017), no. 1, 225–274. MR3666825

work page 2017
[8]

Alcides Buss and Diego Martínez, Approximation properties of Fell bundles over inverse semigroups and non-Hausdorﬀ groupoids , Adv. Math. 431 (2023), pp. 54

work page 2023

Show all 32 references

[9]

Yeong Chyuan Chung, Diego Martínez, and Nóra Szakács, Quasi-countable inverse semi- groups as metric spaces, and the uniform roe algebras of loca lly ﬁnite inverse semigroups (2022)

2022
[10]

5, 3669–3712

Lisa Orloﬀ Clark, Ruy Exel, Enrique Pardo, Aidan Sims, a nd Charles Starling, Simplicity of algebras associated to non-hausdorﬀ groupoids , Transactions of the American Mathematical Society 372 (2019), no. 5, 3669–3712

2019
[11]

Elliott and Zhuang Niu, On the classiﬁcation of simple amenable C*-algebras with ﬁnite decomposition rank , arXiv: Operator Algebras (2015)

George A. Elliott and Zhuang Niu, On the classiﬁcation of simple amenable C*-algebras with ﬁnite decomposition rank , arXiv: Operator Algebras (2015)

2015
[12]

Ruy Exel, Inverse semigroups and combinatorial C∗ -algebras, Bull. Braz. Math. Soc. (N.S.) 39 (2008), no. 2, 191–313. MR2419901

2008
[13]

Pitts, Characterizing groupoid C∗ -algebras of non-Hausdorﬀ étale groupoids , Lecture Notes in Mathematics, vol

Ruy Exel and David R. Pitts, Characterizing groupoid C∗ -algebras of non-Hausdorﬀ étale groupoids , Lecture Notes in Mathematics, vol. 2306, Springer, Cham, [ 2022] ©2022. MR4510931

2022
[14]

Systems 37 (2017), no

Ruy Exel and Charles Starling, Amenable actions of inverse semigroups , Ergodic Theory Dynam. Systems 37 (2017), no. 2, 481–489. MR3614034

2017
[15]

Reine Angew

Mahmood Khoshkam and Georges Skandalis, Regular representation of groupoid C∗ - algebras and applications to inverse semigroups , J. Reine Angew. Math. 546 (2002), 47–72. MR1900993

2002
[16]

Operator Theory 89 (2023), no

Julian Kranz, The weak containment problem for étale groupoids which are s trongly amenable at inﬁnity , J. Operator Theory 89 (2023), no. 2, 349–360. MR4591645

2023
[17]

Alexander Kumjian, On C∗ -diagonals, Canad. J. Math. 38 (1986), no. 4, 969–1008. MR854149

1986
[18]

Bartosz Kosma Kwaśniewski and Ralf Meyer, Essential crossed products for inverse semi- group actions: simplicity and pure inﬁniteness , Doc. Math. 26 (2021), 271–335

2021
[19]

Xin Li, Every classiﬁable simple C*-algebras has a Cartan subalgeb ra, Inventiones Mathe- maticae 219 (2020), 653–699

2020
[20]

Sergey Neshveyev and Gaute Schwartz, Non-Hausdorﬀ étale groupoids and C*-algebras of left cancellative monoids , Münster J. Math. 16 (2023), no. 1, 147–175

2023
[21]

96, Cambr idge University Press, Cambridge, 2020

Gilles Pisier, Tensor products of C∗ -algebras and operator spaces—the Connes-Kirchberg problem, London Mathematical Society Student Texts, vol. 96, Cambr idge University Press, Cambridge, 2020. MR4283471

2020
[22]

793, Springer, Berlin, 1980

Jean Renault, A groupoid approach to C∗ -algebras, Lecture Notes in Mathematics, vol. 793, Springer, Berlin, 1980. MR584266

1980
[23]

, Cartan subalgebras in C∗ -algebras, Irish Math. Soc. Bull. 61 (2008), 29–63. MR2460017

2008
[24]

, Topological amenability is a borel property , Math. Scand. 117 (2015), 5–30. ESSENTIAL GROUPOID AMENABILITY AND NUCLEARITY 47

2015
[25]

Aidan Sims, Hausdorﬀ étale groupoids and their C*-algebras (Francesc Perera, ed.), Advanced Courses in Mathematics, Birkhäuser/Springer, CRM Barcelo na, 2020

2020
[26]

Benjamin Steinberg and Nóra Szakács, Simplicity of inverse semigroup and étale groupoid algebras, Advances in Mathematics 386 (2021)

2021
[27]

, On the simplicity of Nekrashevych algebras of contracting s elf-similar groups, Math- ematische Annalen 386 (2023), 1391–1428

2023
[28]

Klaus Thomsen, Semi-étale groupoids and applications , Ann. Inst. Fourier (Grenoble) 60 (2010), no. 3, 759–800. MR2680816

2010
[29]

1, 229–284

Aaron Tikuisis, Stuart White, and Wilhelm Winter, Quasidiagonality of nuclear C*-algebras , Annals of Mathematics 185 (2017), no. 1, 229–284

2017
[30]

Stuart White and Rufus Willett, Cartan subalgebras in uniform Roe algebras , Groups Geom. Dyn. 14 (2020), no. 3, 949–989

2020
[31]

Rufus Willett, A non-amenable groupoid whose maximal and reduced C∗ -algebras are the same, Münster J. Math. 8 (2015), no. 1, 241–252. MR3549528

2015
[32]

2, 461–498

Wilhelm Winter and Joachim Zacharias, The nuclear dimension of C*-algebras , Advances in Mathematics 224 (2010), no. 2, 461–498. Departamento de Matemática, Universidade Federal de Santa Ca tarina, 88.040-900 Florianópolis-SC, Brazil Email address : alcides.buss@ufsc.br Depart...

2010

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

[1] [2]

Hume, and Xin Li, On Hausdorﬀ covers for non-Hausdorﬀ groupoids (2025)

Keving Aguyar Brix, Julian Gonzales, Jeremy B. Hume, and Xin Li, On Hausdorﬀ covers for non-Hausdorﬀ groupoids (2025). arXiv: 2503.23203

arXiv 2025

[2] [1]

arXiv: 2406.05717

Krzysztof Bardadyn, Bartosz Kwaśniewski, and Andrew Mc Kee, Banach algebras associated to twisted étale groupoids: simplicity and pure inﬁnitenes s (2024). arXiv: 2406.05717

arXiv 2024

[3] [3]

Brown and Narutaka Ozawa, C*-algebras and ﬁnite-dimensional approxima- tions, Graduate studies in mathematics, American Mathematical S ociety, Providence, R.I, 2008 (en)

Nathanial P. Brown and Narutaka Ozawa, C*-algebras and ﬁnite-dimensional approxima- tions, Graduate studies in mathematics, American Mathematical S ociety, Providence, R.I, 2008 (en). OCLC: ocn180190949

work page 2008

[4] [4]

88, Amer

, C∗ -algebras and ﬁnite-dimensional approximations , Graduate Studies in Mathemat- ics, vol. 88, Amer. Math. Soc., 2008. MR2391387

work page 2008

[5] [5]

C*-algebras and groupoids , Journal of Functional Analysis 286 (2024), no

Chris Bruce and Xin Li, Algebraic actions I. C*-algebras and groupoids , Journal of Functional Analysis 286 (2024), no. 4, 57 pp

work page 2024

[6] [6]

Operator Theory 67 (2012), no

Alcides Buss and Ruy Exel, Fell bundles over inverse semigroups and twisted étale grou poids, J. Operator Theory 67 (2012), no. 1, 153–205. MR2881538

work page 2012

[7] [7]

Alcides Buss, Ruy Exel, and Ralf Meyer, Reduced C∗ -algebras of Fell bundles over inverse semigroups, Israel J. Math. 220 (2017), no. 1, 225–274. MR3666825

work page 2017

[8] [8]

Alcides Buss and Diego Martínez, Approximation properties of Fell bundles over inverse semigroups and non-Hausdorﬀ groupoids , Adv. Math. 431 (2023), pp. 54

work page 2023