Fiberwise amenability of \'{e}tale groupoids

Jianchao Wu; Xin Ma

REVIEW 2 major objections 3 minor 199 references

Fiberwise amenability of \'{e}tale groupoids

T0 review · 2 major / 3 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Every σ-compact étale groupoid carries a canonical invariant fiberwise extended metric, and minimal ones split into a Følner-like regime and a paradoxical regime according to fiberwise amenability.

desk verdict Genuinely new coarse-geometric framework for etale groupoids; the main results hold up, but the Local Slice Lemma has a real, repairable domain gap that should be patched before acceptance. read the letter →

arxiv 2608.09796 v1 pith:A32VMD65 submitted 2026-08-10 math.DS math.MGmath.OA

Xin Ma , Jianchao Wu This is my paper · ORCID

classification math.DSmath.MGmath.OA MSC 22A2246L3551F3037A5537B05

keywords coarsegeometryfiberwiseamenabilityétalegroupoidsubiquitousFølnersetsmetricinvariantmeasuresalmostfiniteness

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 is trying to establish that amenability of an étale groupoid is a fiberwise coarse-geometric phenomenon, not just a global topological one. It proves that every σ-compact étale groupoid admits a canonical invariant fiberwise extended metric, unique up to coarse equivalence, and it defines fiberwise amenability as metric amenability of the source fibers under that metric. On compact unit spaces, fiberwise amenability forces the existence of a groupoid-invariant probability measure; for minimal groupoids it coincides with the stronger ubiquitous version, producing a genuine Følner-versus-paradoxical dichotomy. A reader should care because this gives a common geometric generalization of amenability of discrete groups and metric amenability of coarse spaces, and because the dichotomy is designed as a tool for distinguishing finite-like from infinite-like groupoids in the sequel the authors announce.

What carries the argument

The load-bearing object is the canonical invariant fiberwise extended metric $\rho_{\mathcal G}$ induced by a coarse continuous length function $\ell$ on the groupoid $\mathcal G$: $\rho(x,y)=\ell(xy^{-1})$ when $s(x)=s(y)$, and $\rho(x,y)=\infty$ otherwise. Theorem A constructs $\ell$ from an arbitrary proper continuous function on $\mathcal G\setminus\mathcal G^{(0)}$ and shows any two coarse length functions are coarsely equivalent, so the metric is intrinsic. Uniform local finiteness makes every metric ball in a source fiber finite, and the Local Slice Lemma clones a ball in one source fiber homeomorphically onto nearby fibers with arbitrarily small metric distortion; this cloning mechanism carries Følner sets between fibers and is what upgrades fiberwise amenability to ubiquitous fiberwise amenability for minimal groupoids.

What would settle it

Build a minimal σ-compact étale groupoid with compact unit space whose canonical fiber metric is amenable, yet for some $R,\varepsilon$ the nearest $(R,\varepsilon)$-Følner sets to some units lie at radii tending to infinity; Theorem 5.14 predicts this cannot happen. A direct place to probe is the cloning step: test Lemma 5.11 on a groupoid where the length function has no largest value below $R+\varepsilon$ in a source fiber, to see whether balls of radius $S$ just below $R+\varepsilon$ can still be cloned exactly onto all nearby fibers.

Watch

Extended reading notes

Core claim

The central claim is that amenability of an étale groupoid is encoded in the large-scale geometry of its source fibers. Theorem A constructs a proper continuous length function on every σ-compact étale groupoid and proves any two such length functions are coarsely equivalent, so the induced invariant fiberwise extended metric is canonical. Theorem C shows that for minimal groupoids, fiberwise amenability, meaning the existence of $(K,\varepsilon)$-Følner sets for every compact $K$ and $\varepsilon>0$, is equivalent to ubiquitous fiberwise amenability, where such Følner sets appear uniformly in a compact enlargement of every unit. Theorem D then turns this into a dichotomy: in the ubiquitous fiberwise amenable case every finite set can be enlarged into a Følner set inside a fixed compact enlargement, while in the non-fiberwise amenable case any compact set has arbitrarily many disjoint translated copies packed into a bounded enlargement.

Load-bearing premise

Everything rests on the local slice lemma, that a finite metric ball in one source fiber can be copied homeomorphically to every nearby fiber with arbitrarily small metric distortion, because without that cloning step fiberwise amenability would not imply ubiquitous fiberwise amenability even for minimal groupoids.

Editorial extensions

If this is right

For a transformation groupoid $X\rtimes\Gamma$, fiberwise amenability is equivalent to amenability of the acting group $\Gamma$, not to topological amenability of the action.
For the coarse groupoid of a uniformly locally finite extended metric space, fiberwise amenability recovers metric amenability, and ubiquitous fiberwise amenability recovers its ubiquitous version.
A fiberwise amenable σ-compact étale groupoid with compact unit space has at least one invariant probability measure on the unit space.
Almost finiteness for ample groupoids implies ubiquitous fiberwise amenability, and the dichotomy in Theorem 5.22 is intended as a tool for the announced sequel on almost-elementariness.
There are minimal principal almost finite ample groupoids that are ubiquitously fiberwise amenable yet not topologically amenable, showing the new notion does not imply topological amenability.

Reading between the lines

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

The paper leaves implicit that fiberwise amenability is a coarse invariant of the groupoid: because the canonical metric is unique up to coarse equivalence, any property defined through it does not depend on the auxiliary continuous function used to build the length function.
A testable extension is to push the same dichotomy beyond minimal groupoids by isolating the role of recurrence, since the proof of Theorem 5.14 uses minimality only to transport a Følner set from one unit to all units.
The entourage formulation sketched in Remark 4.16 suggests the definitions could extend verbatim to non-σ-compact or non-Hausdorff étale groupoids, where continuous length functions may fail to exist but the coarse structure is still present.
One consequence the paper points toward but does not prove is that, for transformation groupoids, the dichotomy reinstates group amenability as the dividing line between finite-like and infinite-like behaviour of the associated C*-algebras.

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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

Short version: this is a genuinely new framework, not a repackaging. The canonical coarse length function (Theorem A) is a real extension of Struble and of Oyono-Oyono-Yu, and the Følner–paradoxical dichotomy (Theorem D) is a useful structural tool. The paper deserves a serious referee and, after small patches, acceptance.

What is new: Definition 5.4 (fiberwise and ubiquitous fiberwise amenability) is not in prior literature; Theorem B (invariant measure from fiberwise amenability on compact unit space) and Theorem C (equivalence for minimal groupoids) are solid. The coarse geometry machinery in Sections 3–4 is mostly standard but carefully assembled, and the construction in Theorem 4.11 via the n-ary multiplication maps and Lemma 2.7 is convincing. I think the main claims hold.

Soft spots: - Lemma 5.11 (Local Slice Lemma) as printed has a real domain gap. The proof defines f(x,v)=f_x(v) for (x,v) in \bar B(u,S)×U, but (5.1) only gives \bar B(v,S)⊆∪_x U_x, so v need not lie in the source of the particular U_x used for coordinate x. The fix is immediate: replace U by U ∩ ⋂_x s(U_x), still an open neighborhood of u, and the rest of the argument goes through. Because the lemma is load-bearing for Lemma 5.12 and Theorem 5.14, the proof should be patched before publication. - Proposition 3.9 has a radius typo: to get disjoint r-balls from Lemma 3.7, the centers need distance > 2r, not > r. The surrounding choice of N_X(2r) shows the intended statement. - Minor: the proof of Proposition 3.8 writes F as M ∪ (⊔ F_{x_i}) without explaining overlaps. Since the F_{x_i} are pairwise disjoint and |F| ≥ Σ|F_{x_i}|, the estimate still works, but a clarifying sentence would help. - Self-citation is not an issue here; the cited sequel [MW20] is motivational rather than load-bearing.

Verdict: accept after minor revision. The framework is worth referee time and will likely be cited for Theorem A and Theorem D.

Referee Report

2 major / 3 minor

Summary. The paper introduces fiberwise amenability and ubiquitous fiberwise amenability for étale groupoids, viewed through a canonical coarse metric structure. The authors prove that every σ-compact étale groupoid admits a proper continuous length function unique up to coarse equivalence (Theorem A), that fiberwise amenability with compact unit space yields an invariant probability measure (Theorem B), that for minimal groupoids fiberwise amenability coincides with its ubiquitous variant (Theorem C), and that these notions yield a Følner-versus-paradoxical dichotomy (Theorem D). The framework connects metric amenability of coarse spaces with fiberwise amenability of coarse groupoids and distinguishes the new notion from topological amenability.

Significance. If the technical gaps are patched, this is a valuable contribution to the coarse geometry of étale groupoids. The canonical coarse metric construction is natural and extends the classical passage from countable groups to proper invariant metrics. The invariant-measure consequence and the Følner-paradoxical dichotomy are likely to be useful in the study of almost finiteness, pure infiniteness, and the authors' planned notion of almost elementariness. The paper is generally well organized, with explicit statements of dependencies between lemmas, and it correctly situates the new notions relative to known examples such as transformation groupoids and coarse groupoids.

major comments (2)

[Lemma 5.11] The proof of the Local Slice Lemma is incomplete because the map f is not defined on the stated domain. For v in U, equation (5.1) only ensures that v lies in the union of the source sets s(U_x) over x in the finite ball, not that v belongs to s(U_x) for every such x. Hence f(x,v) = f_x(v) is undefined when v is outside s(U_x). The gap is repairable: define U' = U ∩ ⋂_{x∈\bar B_ρ(u,S)} s(U_x). Since each s(U_x) is an open neighborhood of u, U' is an open neighborhood of u, and the covering condition (5.1) remains valid on U'. The rest of the proof, including injectivity and the metric estimates, then goes through. Because Lemma 5.11 is used in Lemma 5.12, Theorem 5.14, and Proposition 5.18, this patch is necessary before the main applications can be accepted.
[Theorem 5.22] The proof of Theorem D contains an incorrect reduction: it claims that by enlarging K we may assume K = ℓ^{-1}(r) for some r ≥ 0. An arbitrary compact set need not be contained in a level set of ℓ, and the metric propositions being invoked (Propositions 3.8 and 3.9) concern sublevel balls ℓ^{-1}([0,r]), not level sets. The reduction should presumably read K = ℓ^{-1}([0,r]) (or a sublevel set containing the original K). Without this correction, the conversion of Propositions 3.8 and 3.9 into the groupoid setting does not yield the stated conclusion about |KF| or about sets of the form Kx.

minor comments (3)

[Proposition 3.9] In the proof of Proposition 3.9, the inequality immediately after applying Lemma 3.7 says d(x_i,x_j) > r, but disjointness of the closed balls \bar B(x_i,r) requires d(x_i,x_j) > 2r. The use of N_X(2r) in the choice of k indicates that the intended application of Lemma 3.7 is with s = 2r, so the displayed inequality should be corrected to d(x_i,x_j) > 2r.
[Theorem 4.11] In the continuity argument after equation (4.3), the displayed equality (f^{-1}([1,N]))^j ∩ (˚δ^{(j)})^{-1}({y}) = {η_i^{(j)}(y) : i=1,...,m_j} is not literally true as stated, since some η_i^{(j)}(y) may lie outside f^{-1}([1,N])^j. The minimum formula for ℓ(y) remains valid because any tuple with ˚f^{(j)} < N automatically lies in f^{-1}([1,N])^j and tuples with larger ˚f^{(j)} do not affect the minimum, but the equality should be clarified.
[Section 5.2] In Remark 5.17, the statement that ℓ^{-1}([0,r)) is contained in the compact set E_r is slightly imprecise: the containment is of the intersection of ℓ^{-1}([0,r)) with the dense subset Y × Y, and one should say that the closure is contained in E_r to justify properness. The intended argument is clear, but the wording invites confusion.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: core theorems follow from independent constructions, with only minor non-load-bearing self-citations.

full rationale

The paper's derivation chain is self-contained. Theorem A's canonical length function is constructed from a proper continuous function via Lemma 4.10, and uniqueness is proved directly from the definitions of proper and controlled length functions in Lemma 4.8, not imported from prior work. Section 3 develops metric amenability and its ubiquitous variant using standard notions credited to [BW92] and [ALLW18b]; the key metric results, Propositions 3.8 and 3.9, are proved in the text. Fiberwise amenability is introduced independently in Definition 5.4, and Proposition 5.5 establishes equivalences with metric amenability under the canonical metric rather than defining the target theorem into existence. Theorem B follows from the Følner-measure argument in Proposition 5.9, which is a genuine estimate on invariant measures. The minimal-groupoid equivalence (Theorem 5.14) relies on the Local Slice Lemma 5.11 and the cloning Lemma 5.12; these are geometric transport arguments, not assumptions of the conclusion. Theorem 5.22 is a translation of Propositions 3.8 and 3.9 using Lemma 4.2, and no fitted parameter is renamed as a prediction. The self-citations present, namely [ALLW18b] for metric amenability background and [MW20] as an announced sequel, are motivational or background and are not load-bearing for the proofs. The only substantive concern is a correctness gap in the printed proof of Lemma 5.11: the map f(x,v)=f_x(v) requires v in s(U_x) for every x in the finite ball, while equation (5.1) only gives v in some U_x; this is repairable by intersecting with the finitely many open sets s(U_x), and it does not indicate circularity. Overall, no step of the derivation reduces to its own input.

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

The results are pure mathematics and contain no fitted constants or data. The proofs rely on standard point-set and groupoid facts, including the Tietze extension theorem and bisection techniques, and on three imported constructions: metric amenability as developed by Block-Weinberger and Ara-Li-Lledo-Wu, the coarse groupoid of Skandalis-Tu-Yu, and Elek's geometric groupoids. The new definitions are introduced, not derived from fitted values; no new particles, dimensions, or entities are hypothesized.

assumptions (4)

standard math Tietze extension theorem and standard compactness-local homeomorphism arguments used to construct proper continuous functions in Lemma 4.10 and Lemma 2.7.
Used in Theorem 4.11 to seed the coarse continuous length function on a sigma-compact etale groupoid.
standard math The basis of open bisections for etale groupoids and the local homeomorphism property of n-ary multiplication (Proposition 2.4, Proposition 2.5, Corollary 2.6).
Used throughout the paper for local slice constructions, metric finiteness, and continuity arguments.
domain assumption The coarse groupoid construction of Skandalis-Tu-Yu is locally compact, Hausdorff, principal, and etale (Definition 5.16, cited to STY02 Proposition 3.2).
Used in Section 5.2 to connect metric amenability of coarse spaces to fiberwise amenability.
domain assumption Elek's geometric groupoids exist and are minimal, principal, almost finite, ample, and not topologically amenable (Example 5.23, cited to Ele18 Theorem 6).
Used to show that ubiquitous fiberwise amenability does not imply topological amenability in general.

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Cite this review

Pith. "Pith review of Fiberwise amenability of \'{e}tale groupoids." pith.science (2026). https://pith.science/paper/A32VMD65

@misc{pith2026260809796,
  author       = {Pith},
  title        = {Pith review of: Fiberwise amenability of \'etale groupoids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A32VMD65}},
  note         = {Machine review of arXiv:2608.09796}
}

read the original abstract

We introduce a new amenability property for \'{e}tale groupoids, termed \textit{fiberwise amenability}, along with a stronger variant termed \emph{ubiquitous fiberwise amenability}. (Ubiquitous) fiberwise amenability emerges naturally from a coarse-geometric perspective on \'{e}tale groupoids and, in the special case of transformation groupoids, it coincides precisely with the amenability of the acting group (rather than topological amenability of the action). It is also tightly linked to the existence of invariant measures on the unit space of the groupoid. The coarse-geometric framework for \'{e}tale groupoids that we develop systematically in this work allows us to establish several foundational properties of (ubiquitous) fiberwise amenability. As an application, we prove a F\o lner--paradoxical dichotomy for minimal \'{e}tale groupoids, which will serve as a key tool in a sequel on almost elementariness of \'{e}tale groupoids.

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Works this paper leans on

199 extracted references · 91 canonical work pages

[1]

The type semigroup, comparison, and almost finiteness for ample groupoids , journal =

Ara, Pere and B. The type semigroup, comparison, and almost finiteness for ample groupoids , journal =. 2023 , doi =

2023
[2]

Ara, Pere and Dykema, Kenneth J. and R. Correction of proofs in ``. Canad. J. Math. , Year =. doi:10.4153/CJM-2012-018-6 , Fjournal =

work page doi:10.4153/cjm-2012-018-6 2012
[3]

Amenability of coarse spaces and

Ara, Pere and Li, Kang and Lled\'. Amenability of coarse spaces and. Bull. Math. Sci. , Year =. doi:10.1007/s13373-017-0109-6 , Fjournal =

work page doi:10.1007/s13373-017-0109-6
[4]

Barlak, Sel. The. J. Noncommut. Geom. , Year =. doi:10.4171/JNCG/226 , Fjournal =

work page doi:10.4171/jncg/226
[5]

The real rank of inductive limit

Blackadar, Bruce and D. The real rank of inductive limit. Math. Scand. , Year =
[6]

K -Theory , Year =

Approximately central matrix units and the structure of noncommutative tori , Author =. K -Theory , Year =. doi:10.1007/BF00961466 , Fjournal =

work page doi:10.1007/bf00961466
[7]

Extending states on preordered semigroups and the existence of quasitraces on

Blackadar, Bruce and R. Extending states on preordered semigroups and the existence of quasitraces on. J. Algebra , Year =. doi:10.1016/0021-8693(92)90098-7 , Fjournal =

doi:10.1016/0021-8693(92)90098-7
[8]

An algebraic approach to the radius of comparison , Author =. Trans. Amer. Math. Soc. , Year =. doi:10.1090/S0002-9947-2012-05538-3 , Fjournal =

work page doi:10.1090/s0002-9947-2012-05538-3 2012

Show all 199 references

[9]

Properly infinite

Blanchard, Etienne and Rohde, Randi and R. Properly infinite. J. Noncommut. Geom. , Year =. doi:10.4171/JNCG/21 , Fjournal =

doi
[10]

Aperiodic tilings, positive scalar curvature and amenability of spaces , Author =. J. Amer. Math. Soc. , Year =. doi:10.2307/2152713 , Fjournal =

doi
[11]

and Sato, Yasuhiko and Tikuisis, Aaron and White, Stuart and Winter, Wilhelm , Journal =

Bosa, Joan and Brown, Nathanial P. and Sato, Yasuhiko and Tikuisis, Aaron and White, Stuart and Winter, Wilhelm , Journal =. Covering dimension of. 2019 , Number =

2019
[12]

The crossed product of a

Bratteli, Ola and St. The crossed product of a. Ergodic Theory Dynam. Systems , Year =. doi:10.1017/S0143385700007574 , Fjournal =

doi
[13]

Quasitraces are traces: a short proof of the finite-nuclear-dimension case , Author =. C. R. Math. Acad. Sci. Soc. R. Can. , Year =
[14]

Nuclear Dimension of Simple

Castillejos, Jorge and Evington, Samuel and Tikuisis, Aaron and White, Stuart and Winter, Wilhelm , Journal =. Nuclear Dimension of Simple. 2021 , Number =. doi:, Fjournal =

2021
[15]

and Karasev, A

Chigogidze, A. and Karasev, A. and R. Real rank and squaring mappings for unital. Proc. Amer. Math. Soc. , Year =. doi:10.1090/S0002-9939-03-07102-8 , Fjournal =

doi
[16]

and Smith, Roger R

Christensen, Erik and Sinclair, Allan M. and Smith, Roger R. and White, Stuart A. and Winter, Wilhelm , Journal =. Perturbations of nuclear. 2012 , Number =. doi:10.1007/s11511-012-0075-5 , Fjournal =

2012 doi
[17]

and Smith, Roger R

Christensen, Erik and Sinclair, Allan M. and Smith, Roger R. and White, Stuart A. and Winter, Wilhelm , Journal =. The spatial isomorphism problem for close separable nuclear. 2010 , Number =. doi:10.1073/pnas.0913281107 , Fjournal =

2010 doi
[18]

Endomorphisms of

Conti, Roberto and R. Endomorphisms of. J. Funct. Anal. , Year =. doi:10.1016/j.jfa.2010.03.027 , Fjournal =

2010 doi
[19]

Cuntz, Joachim , TITLE =. Math. Ann. , FJOURNAL =. 1978 , NUMBER =. doi:10.1007/BF01421922 , URL =

1978 doi
[20]

The homotopy groups of the automorphism group of

Dadarlat, Marius , Journal =. The homotopy groups of the automorphism group of. 2007 , Number =. doi:10.4171/JNCG/3 , Fjournal =

2007 doi
[21]

and Winter, Wilhelm , Journal =

Dadarlat, Marius and Hirshberg, Ilan and Toms, Andrew S. and Winter, Wilhelm , Journal =. The. 2009 , Number =. doi:10.4310/MRL.2009.v16.n1.a3 , Fjournal =

2009 doi
[22]

Strongly self-absorbing

Dadarlat, Marius and R. Strongly self-absorbing. M\"unster J. Math. , Year =
[23]

Dadarlat, Marius and Winter, Wilhelm , Journal =. On the. 2009 , Number =

2009
[24]

Trivialization of

Dadarlat, Marius and Winter, Wilhelm , Journal =. Trivialization of. 2008 , Number =

2008
[25]

Tilings of amenable groups , Author =. J. Reine Angew. Math. , Year =. doi:10.1515/crelle-2016-0025 , Fjournal =

2016 doi
[26]

Correction to: ``

Dykema, Ken and Haagerup, Uffe and R. Correction to: ``. Duke Math. J. , Year =. doi:10.1215/S0012-7094-98-09410-8 , Fjournal =

doi
[27]

The stable rank of some free product

Dykema, Ken and Haagerup, Uffe and R. The stable rank of some free product. Duke Math. J. , Year =. doi:10.1215/S0012-7094-97-09004-9 , Fjournal =

doi
[28]

Dykema, K. J. and R. Erratum to: ``. Geom. Funct. Anal. , Year =. doi:10.1007/PL00001644 , Fjournal =

doi
[29]

Dykema, Kenneth J. and R. Projections in free product. Math. Z. , Year =. doi:10.1007/s002090050505 , Fjournal =

doi
[30]

Dykema, K. J. and R. Projections in free product. Geom. Funct. Anal. , Year =. doi:10.1007/s000390050046 , Fjournal =

doi
[31]

Dykema, Kenneth J. and R. Purely infinite, simple. Canad. J. Math. , Year =. doi:10.4153/CJM-1998-017-x , Fjournal =

1998 doi
[32]

Qualitative graph limit theory

Elek, G\'. Qualitative graph limit theory. 2018 , Owner =

2018
[33]

and Gong, Guihua and Lin, Huaxin and Niu, Zhuang , title =

Elliott, George A. and Gong, Guihua and Lin, Huaxin and Niu, Zhuang , title =. J. Noncommut. Geom. , volume =. 2025 , mrnumber =

2025
[34]

The cyclic homology of algebras with adjoined unit , Author =. Proc. Amer. Math. Soc. , Year =. doi:10.2307/2048523 , Fjournal =

doi
[35]

and Niu, Zhuang , Journal =

Elliott, George A. and Niu, Zhuang , Journal =. The. 2017 , Number =. doi:10.1215/00127094-2017-0033 , Fjournal =

2017 doi
[36]

Elliott, George A. and R. Perturbation of. Operator. 2006 , Address =. doi:10.1007/978-3-540-34197-0_5 , Mrclass =

2006 doi
[37]

Elliott, George A. and R. Classification of certain infinite simple. Comment. Math. Helv. , Year =. doi:10.1007/BF02566025 , Fjournal =

doi
[38]

Elliott, George A. and R. The automorphism group of the irrational rotation. Comm. Math. Phys. , Year =
[39]

and Toms, Andrew S

Elliott, George A. and Toms, Andrew S. , Journal =. Regularity properties in the classification program for separable amenable. 2008 , Number =. doi:10.1090/S0273-0979-08-01199-3 , Fjournal =

2008 doi
[40]

Approximation with normal operators with finite spectrum, and an elementary proof of a

Friis, Peter and R. Approximation with normal operators with finite spectrum, and an elementary proof of a. Pacific J. Math. , Year =. doi:10.2140/pjm.2001.199.347 , Fjournal =

2001 doi
[41]

Almost commuting self-adjoint matrices---a short proof of

Friis, Peter and R. Almost commuting self-adjoint matrices---a short proof of. J. Reine Angew. Math. , Year =. doi:10.1515/crll.1996.479.121 , Fjournal =

1996 doi
[42]

Subshifts and perforation , Author =. J. Reine Angew. Math. , Year =. doi:10.1515/CRELLE.2010.012 , Fjournal =

2010 doi
[43]

Ergodic Theory Dynam

The absorption theorem for affable equivalence relations , Author =. Ergodic Theory Dynam. Systems , Year =. doi:10.1017/S0143385707000946 , Fjournal =

doi
[44]

Affable equivalence relations and orbit structure of

Giordano, Thierry and Putnam, Ian and Skau, Christian , Journal =. Affable equivalence relations and orbit structure of. 2004 , Number =. doi:10.1017/S014338570300066X , Fjournal =

2004 doi
[45]

Gong, Guihua and Lin, Huaxin and Niu, Zhuang , Note =
[46]

Gong, Guihua and Lin, Huaxin and Niu, Zhuang , title =. C. R. Math. Rep. Acad. Sci. Canada , volume =
[47]

Perturbations of the rotation

Haagerup, Uffe and R. Perturbations of the rotation. Duke Math. J. , Year =. doi:10.1215/S0012-7094-95-07720-5 , Fjournal =

doi
[48]

Haagerup, Uffe and R. J. Operator Theory , Year =
[49]

K -Theory , Year =

Higson, Nigel and Pedersen, Erik Kj. K -Theory , Year =. doi:10.1023/A:1007705726771 , Fjournal =

doi
[50]

Higson, Nigel and R. The. Canad. J. Math. , Year =. doi:10.4153/CJM-1991-018-9 , Fjournal =

1991 doi
[51]

Tracially

Hirshberg, Ilan and Orovitz, Joav , Journal =. Tracially. 2013 , Number =. doi:10.1016/j.jfa.2013.05.005 , Fjournal =

2013 doi
[52]

Hirshberg, Ilan and R. Math. Ann. , Year =. doi:10.1007/s00208-007-0129-8 , Fjournal =

doi
[53]

Rokhlin dimension for flows , Author =. Comm. Math. Phys. , Year =. doi:10.1007/s00220-016-2762-0 , Fjournal =

doi
[54]

Permutations of strongly self-absorbing

Hirshberg, Ilan and Winter, Wilhelm , Journal =. Permutations of strongly self-absorbing. 2008 , Number =. doi:10.1142/S0129167X08005011 , Fjournal =

2008 doi
[55]

Rokhlin actions and self-absorbing

Hirshberg, Ilan and Winter, Wilhelm , Journal =. Rokhlin actions and self-absorbing. 2007 , Number =. doi:10.2140/pjm.2007.233.125 , Fjournal =

2007 doi
[56]

Rokhlin dimension and

Hirshberg, Ilan and Winter, Wilhelm and Zacharias, Joachim , Journal =. Rokhlin dimension and. 2015 , Number =. doi:10.1007/s00220-014-2264-x , Fjournal =

2015 doi
[57]

2023 , Owner =

Hirshberg, Ilan and Jianchao, Wu , Note =. 2023 , Owner =

2023
[58]

Hjelmborg, Jacob v. B. and R. On stability of. J. Funct. Anal. , Year =. doi:10.1006/jfan.1997.3221 , Fjournal =

1997
[59]

Topology Appl

Free continuous actions on zero-dimensional spaces , Author =. Topology Appl. , Year =. doi:10.1016/j.topol.2005.03.003 , Fjournal =

2005 doi
[60]

Classification of tiling

Ito, Luke and Whittaker, Michael and Zacharias, Joachim , Note =. Classification of tiling. 2019 , Owner =

2019
[61]

2014 , Number =

Jacelon, Bhishan and Winter, Wilhelm , Journal =. 2014 , Number =. doi:10.4171/JNCG/176 , Fjournal =

2014 doi
[62]

On a simple unital projectionless

Jiang, Xinhui and Su, Hongbing , Journal =. On a simple unital projectionless. 1999 , Number =

1999
[63]

The similarity problem for

Johanesov. The similarity problem for. Bull. Lond. Math. Soc. , Year =. doi:10.1112/blms/bds048 , Fjournal =

doi
[64]

Non-supramenable groups acting on locally compact spaces , Author =. Doc. Math. , Year =
[65]

Dimension, comparison, and almost finiteness , Author =. J. Eur. Math. Soc. (JEMS) , Year =. doi:10.4171/jems/995 , Fjournal =

doi
[66]

Almost finiteness and the small boundary property , Author =. Comm. Math. Phys. , Year =. doi:10.1007/s00220-019-03519-z , Fjournal =

doi
[67]

Central sequence -algebras and tensorial absoption of the

Kirchberg, Eberhard and R. Central sequence -algebras and tensorial absoption of the. J. Reine Angew. Math. , Year =
[68]

2014 , Bdsk-url-1 =

When central sequence -algebras have characters , Author =. 2014 , Bdsk-url-1 =

2014
[69]

Kirchberg, E. and R. Purely infinite. Geom. Funct. Anal. , Year =. doi:10.1007/s00039-005-0510-2 , Fjournal =

doi
[70]

Infinite non-simple

Kirchberg, Eberhard and R. Infinite non-simple. Adv. Math. , Year =. doi:10.1006/aima.2001.2041 , Fjournal =

2001
[71]

Non-simple purely infinite

Kirchberg, Eberhard and R. Non-simple purely infinite. Amer. J. Math. , Year =
[72]

Internat

Covering dimension and quasidiagonality , Author =. Internat. J. Math. , Year =. doi:10.1142/S0129167X04002119 , Fjournal =

doi
[73]

On a dimension for a class of homeomorphism groups , Author =. Math. Ann. , Year =. doi:10.1007/BF01420115 , Fjournal =

doi
[74]

Algebraic

L. Algebraic. K -Theory , Year =. doi:10.1007/BF00961216 , Fjournal =

doi
[75]

Every classifiable simple C^* -algebra has a Cartan subalgebra , Author =. Invent. Math. , Year =. doi:10.1007/s00222-019-00914-0 , Fjournal =

doi
[76]

Pacific J

Full extensions and approximate unitary equivalence , Author =. Pacific J. Math. , Year =. doi:10.2140/pjm.2007.229.389 , Fjournal =

2007 doi
[77]

An introduction to the classification of amenable

Lin, Huaxin , Publisher =. An introduction to the classification of amenable. 2001 , Address =. doi:10.1142/9789812799883 , ISBN =

2001 doi
[78]

Tracially

Lin, Huaxin , Journal =. Tracially. 2001 , Number =. doi:10.1090/S0002-9947-00-02680-5 , Fjournal =

2001 doi
[79]

Extensions of inductive limits of circle algebras , Author =. J. London Math. Soc. (2) , Year =. doi:10.1112/jlms/51.3.603 , Fjournal =

doi
[80]

Israel J

Mean topological dimension , Author =. Israel J. Math. , Year =. doi:10.1007/BF02810577 , Fjournal =

doi
[81]

Ma, Xin , title =. Int. Math. Res. Not. IMRN , year =. doi:10.1093/imrn/rnaa360 , fjournal =

doi
[82]

Fiberwise amenability of ample \'

Ma, Xin , Note =. Fiberwise amenability of ample \'. 2021 , Owner =

2021
[83]

Invariant ergodic measures and the classification of crossed product

Ma, Xin , Journal =. Invariant ergodic measures and the classification of crossed product. 2019 , Number =. doi:10.1016/j.jfa.2018.06.013 , Fjournal =

2019 doi
[84]

Topological full groups of \'

Matui, Hiroki , Booktitle =. Topological full groups of \'. 2017 , Pages =

2017
[85]

Topological full groups of one-sided shifts of finite type , Author =. J. Reine Angew. Math. , Year =. doi:10.1515/crelle-2013-0041 , Fjournal =

2013 doi
[86]

Homology and topological full groups of \'

Matui, Hiroki , Journal =. Homology and topological full groups of \'. 2012 , Number =. doi:10.1112/plms/pdr029 , Fjournal =

2012 doi
[87]

Matui and Y

H. Matui and Y. Sato , Journal =. 2014 , Pages =

2014
[88]

Matui and Y

H. Matui and Y. Sato , Journal =. Decomposition rank of. 2014 , Number =

2014
[89]

Strict comparison and

Matui, Hiroki and Sato, Yasuhiko , Journal =. Strict comparison and. 2012 , Number =. doi:10.1007/s11511-012-0084-4 , Fjournal =

2012 doi
[90]

Matui and Y

H. Matui and Y. Sato , Journal =. Strict comparison and. 2012 , Number =

2012
[91]

2012 , Number =

Matui, Hiroki and Sato, Yasuhiko , Journal =. 2012 , Number =. doi:10.1007/s00220-011-1392-9 , Fjournal =

2012 doi
[92]

Ergodic Theory Dynam

Simple groups of dynamical origin , Author =. Ergodic Theory Dynam. Systems , Year =. doi:10.1017/etds.2017.47 , Fjournal =

2017 doi
[93]

Nuclear dimension and the corona factorization property , Author =. Int. Math. Res. Not. IMRN , Year =. doi:10.1093/imrn/rnp125 , Fjournal =

doi
[94]

Commutative

Ng, Ping Wong and Winter, Wilhelm , Journal =. Commutative. 2008 , Number =. doi:10.1512/iumj.2008.57.3415 , Fjournal =

2008 doi
[95]

A note on subhomogeneous

Ng, Ping Wong and Winter, Wilhelm , Journal =. A note on subhomogeneous. 2006 , Number =

2006
[96]

Niu, Zhuang , title =. Trans. Amer. Math. Soc. , volume =. 2021 , doi =

2021
[97]

Niu, Zhuang , title =. Can. J. Math. , volume =. 2024 , zbl =

2024
[98]

Niu, Zhuang , title =. J. Anal. Math. , volume =. 2022 , doi =

2022
[99]

2012 , Series =

Large scale geometry , Author =. 2012 , Series =. doi:10.4171/112 , ISBN =

2012 doi
[100]

The corona factorization property, stability, and the

Ortega, Eduard and Perera, Francesc and R. The corona factorization property, stability, and the. Int. Math. Res. Not. IMRN , Year =
[101]

The corona factorization property and refinement monoids , Author =. Trans. Amer. Math. Soc. , Year =. doi:10.1090/S0002-9947-2011-05480-2 , Fjournal =

2011 doi
[102]

Ortega, Eduard and R. The. J. Funct. Anal. , Year =. doi:10.1016/j.jfa.2011.02.017 , Fjournal =

2011 doi
[103]

Groups Geom

Almost finiteness and homology of certain non-free actions , author =. Groups Geom. Dyn. , volume =
[104]

Quantitative. J. Funct. Anal. , Year =. doi:10.1016/j.jfa.2019.01.009 , Fjournal =

2019 doi
[105]

Rank-two graphs whose

Pask, David and Raeburn, Iain and R. Rank-two graphs whose. J. Funct. Anal. , Year =. doi:10.1016/j.jfa.2006.04.003 , Fjournal =

2006 doi
[106]

Purely infinite

Pasnicu, Cornel and R. Purely infinite. J. Reine Angew. Math. , Year =. doi:10.1515/CRELLE.2007.091 , Fjournal =

2007 doi
[107]

Tensor products of

Pasnicu, Cornel and R. Tensor products of. J. Funct. Anal. , Year =. doi:10.1006/jfan.2000.3630 , Fjournal =

2000
[108]

Perera, Francesc and R. A. J. Funct. Anal. , Year =. doi:10.1016/j.jfa.2004.05.001 , Fjournal =

2004 doi
[109]

Perera, Francesc and Toms, Andrew and White, Stuart and Winter, Wilhelm , Journal =. The. 2014 , Number =. doi:10.2140/apde.2014.7.929 , Fjournal =

2014 doi
[110]

2014 , Owner =

Large subalgebras , Author =. 2014 , Owner =

2014
[111]

Christopher , Journal =

Phillips, N. Christopher , Journal =. Crossed products of the. 2005 , Number =. doi:10.1007/s00220-004-1171-y , Fjournal =

2005 doi
[112]

Christopher , Journal =

Phillips, N. Christopher , Journal =. A classification theorem for nuclear purely infinite simple. 2000 , Pages =

2000
[113]

Putnam, Ian F. and R. The maximum unitary rank of some. Math. Scand. , Year =
[114]

Classification of nuclear

R. Classification of nuclear. 2002 , Address =

2002
[115]

Classification of

R. Classification of. 2001 , Date-added =

2001
[116]

R. The. J. Reine Angew. Math. , Year =
[117]

Structure and classification of

R. Structure and classification of. International. 2006 , Pages =

2006
[118]

The real rank of certain simple

R. The real rank of certain simple. Advances in operator algebras and mathematical physics , Publisher =. 2005 , Pages =

2005
[119]

A purely infinite

R. A purely infinite. Israel J. Math. , Year =. doi:10.1007/BF02772211 , Fjournal =

doi
[121]

A simple

R. A simple. Acta Math. , Year =. doi:10.1007/BF02392697 , Fjournal =

doi
[122]

Classification of nuclear, simple

R. Classification of nuclear, simple. Classification of nuclear. 2002 , Address =

2002
[123]

Extensions of stable

R. Extensions of stable. Doc. Math. , Year =
[124]

Operator algebras and operator theory (

On sums of finite projections , Author =. Operator algebras and operator theory (. 1998 , Address =. doi:10.1090/conm/228/03295 , Mrclass =

1998 doi
[125]

Classification of extensions of certain

R. Classification of extensions of certain. Math. Ann. , Year =. doi:10.1007/s002080050067 , Fjournal =

doi
[126]

Classification of certain infinite simple

R. Classification of certain infinite simple. Operator algebras and their applications (. 1997 , Address =

1997
[127]

Stability of

R. Stability of. Doc. Math. , Year =
[128]

The stable rank of

R. The stable rank of. Operator algebras and quantum field theory (. 1997 , Pages =

1997
[129]

Classification of certain infinite simple

R. Classification of certain infinite simple. J. Funct. Anal. , Year =. doi:10.1006/jfan.1995.1095 , Fjournal =

1995
[130]

Classification of

R. Classification of. K -Theory , Year =. doi:10.1007/BF00965458 , Fjournal =

doi
[131]

A short proof of

R. A short proof of. C. R. Math. Rep. Acad. Sci. Canada , Year =
[132]

Classification of inductive limits of

R. Classification of inductive limits of. J. Reine Angew. Math. , Year =. doi:10.1515/crll.1993.440.175 , Fjournal =

1993 doi
[133]

On the structure of simple

R. On the structure of simple. J. Funct. Anal. , Year =. doi:10.1016/0022-1236(92)90106-S , Fjournal =

doi
[134]

Ideals in the multiplier algebra of a stable

R. Ideals in the multiplier algebra of a stable. J. Operator Theory , Year =
[135]

Advances in the theory of unitary rank and regular approximation , Author =. Ann. of Math. (2) , Year =. doi:10.2307/1971465 , Fjournal =

doi
[136]

An introduction to

R. An introduction to. 2000 , Address =

2000
[137]

Purely infinite

R. Purely infinite. Ergodic Theory Dynam. Systems , Year =. doi:10.1017/S0143385710000829 , Fjournal =

doi
[138]

On the ordered

R. On the ordered. K -Theory , Year =. doi:10.1023/A:1007731001836 , Fjournal =

doi
[139]

R. The. J. Reine Angew. Math. , Year =. doi:10.1515/CRELLE.2010.039 , Fjournal =

2010 doi
[140]

The stable and the real rank of

R rdam, Mikael , Journal =. The stable and the real rank of. 2004 , Number =. doi:10.1142/S0129167X04002661 , Fjournal =

2004 doi
[141]

A groupoid approach to

Renault, Jean , Publisher =. A groupoid approach to. 1980 , Series =

1980
[142]

Divisibility properties for

Robert, Leonel and R. Divisibility properties for. Proc. Lond. Math. Soc. (3) , Year =. doi:10.1112/plms/pds082 , Fjournal =

doi
[143]

2003 , Address =

Lectures on coarse geometry , Author =. 2003 , Address =

2003
[144]

Nuclear dimension and

Sato, Yasuhiko and White, Stuart and Winter, Wilhelm , Journal =. Nuclear dimension and. 2015 , Number =. doi:10.1007/s00222-015-0580-1 , Fjournal =

2015 doi
[145]

2017 , Owner =

Sims, Aidan , Note =. 2017 , Owner =

2017
[146]

The coarse

Skandalis, Georges and Tu, Jean-Louis and Yu, Guoliang , Journal =. The coarse. 2002 , Number =. doi:10.1016/S0040-9383(01)00004-0 , Fjournal =

2002 doi
[147]

and Winter, Wilhelm , Journal =

Strung, Karen R. and Winter, Wilhelm , Journal =. U. 2014 , Number =. doi:10.1142/S1793525314500198 , Fjournal =

2014 doi
[148]

and Winter, Wilhelm , Journal =

Strung, Karen R. and Winter, Wilhelm , Journal =. Minimal dynamics and. 2011 , Number =. doi:10.1142/S0129167X10006665 , Fjournal =

2011 doi
[149]

Almost finiteness for general \'

Suzuki, Yuhei , Journal =. Almost finiteness for general \'. 2020 , Number =. doi:10.1093/imrn/rny187 , Fjournal =

2020 doi
[150]

Tang, Xiang and Willett, Rufus and Yao, Yi-Jun , Journal =. Roe. 2018 , Number =. doi:10.1007/s00209-018-2064-7 , Fjournal =

2018 doi
[151]

The generator problem for

Thiel, Hannes and Winter, Wilhelm , Journal =. The generator problem for. 2014 , Number =. doi:10.1090/S0002-9947-2014-06013-3 , Fjournal =

2014 doi
[152]

Quasidiagonality of nuclear

Tikuisis, Aaron and White, Stuart and Winter, Wilhelm , Journal =. Quasidiagonality of nuclear. 2017 , Number =. doi:10.4007/annals.2017.185.1.4 , Fjournal =

2017 doi
[153]

Decomposition rank of

Tikuisis, Aaron and Winter, Wilhelm , Journal =. Decomposition rank of. 2014 , Number =. doi:10.2140/apde.2014.7.673 , Fjournal =

2014 doi
[154]

and White, Stuart and Winter, Wilhelm , Journal =

Toms, Andrew S. and White, Stuart and Winter, Wilhelm , Journal =. 2015 , Number =. doi:10.1093/imrn/rnu001 , Fjournal =

2015 doi
[155]

and Winter, Wilhelm , Journal =

Toms, Andrew S. and Winter, Wilhelm , Journal =. Minimal dynamics and. 2013 , Number =. doi:10.1007/s00039-012-0208-1 , Fjournal =

2013 doi
[156]

and Winter, Wilhelm , Journal =

Toms, Andrew S. and Winter, Wilhelm , Journal =. The. 2009 , Number =. doi:10.1016/j.jfa.2008.12.015 , Fjournal =

2009 doi
[157]

and Winter, Wilhelm , Journal =

Toms, Andrew S. and Winter, Wilhelm , Journal =. Minimal dynamics and the classification of. 2009 , Number =. doi:10.1073/pnas.0903629106 , Fjournal =

2009 doi
[158]

and Winter, Wilhelm , Journal =

Toms, Andrew S. and Winter, Wilhelm , Journal =. 2008 , Number =. doi:10.4153/CJM-2008-031-6 , Fjournal =

2008 doi
[159]

and Winter, Wilhelm , Journal =

Toms, Andrew S. and Winter, Wilhelm , Journal =. Strongly self-absorbing. 2007 , Number =. doi:10.1090/S0002-9947-07-04173-6 , Fjournal =

2007 doi
[160]

Winter, Wilhelm , Booktitle =. Q. 2017 , Pages =

2017
[161]

Classifying crossed product

Winter, Wilhelm , Journal =. Classifying crossed product. 2016 , Number =. doi:10.1353/ajm.2016.0029 , Fjournal =

2016
[162]

Localizing the

Winter, Wilhelm , Journal =. Localizing the. 2014 , Pages =

2014
[163]

Nuclear dimension and

Winter, Wilhelm , Journal =. Nuclear dimension and. 2012 , Number =. doi:10.1007/s00222-011-0334-7 , Fjournal =

2012 doi
[164]

Strongly self-absorbing

Winter, Wilhelm , Journal =. Strongly self-absorbing. 2011 , Number =. doi:10.4171/JNCG/74 , Fjournal =

2011 doi
[165]

Decomposition rank and

Winter, Wilhelm , Journal =. Decomposition rank and. 2010 , Number =. doi:10.1007/s00222-009-0216-4 , Fjournal =

2010 doi
[166]

Covering dimension for nuclear

Winter, Wilhelm , Journal =. Covering dimension for nuclear. 2009 , Number =. doi:10.1090/S0002-9947-09-04602-9 , Fjournal =

2009 doi
[167]

Winter, Wilhelm , Journal =. Simple. 2007 , Number =. doi:10.1016/j.jfa.2006.11.001 , Fjournal =

2007 doi
[168]

On the classification of simple

Winter, Wilhelm , Journal =. On the classification of simple. 2006 , Number =. doi:10.1112/S0024610706022903 , Fjournal =

2006 doi
[169]

On topologically finite-dimensional simple

Winter, Wilhelm , Journal =. On topologically finite-dimensional simple. 2005 , Number =. doi:10.1007/s00208-005-0657-z , Fjournal =

2005 doi
[170]

Decomposition rank of subhomogeneous

Winter, Wilhelm , Journal =. Decomposition rank of subhomogeneous. 2004 , Number =. doi:10.1112/S0024611504014716 , Fjournal =

2004 doi
[171]

Covering dimension for nuclear

Winter, Wilhelm , Journal =. Covering dimension for nuclear. 2003 , Number =. doi:10.1016/S0022-1236(02)00109-X , Fjournal =

2003 doi
[172]

Covering dimension for nuclear

Winter, Wilhelm , Booktitle =. Covering dimension for nuclear. 2000 , Address =

2000
[173]

The nuclear dimension of

Winter, Wilhelm and Zacharias, Joachim , Journal =. The nuclear dimension of. 2010 , Number =. doi:10.1016/j.aim.2009.12.005 , Fjournal =

2010 doi
[174]

M\"unster J

Completely positive maps of order zero , Author =. M\"unster J. Math. , Year =
[175]

Fiberwise amenability for \'

Ma, Xin and Wu, Jianchao , Note =. Fiberwise amenability for \'. 2026 , Owner =

2026
[176]

Classifying * -homomorphisms

Carri\'. Classifying * -homomorphisms. 2023 , Owner =

2023
[177]

Almost elementary \'

Ma, Xin and Wu, Jianchao , Note =. Almost elementary \'. 2020 , Owner =

2020
[178]

Matui, Hiroki , year =. \'. Adv. Math. , publisher =
[179]

Amenability and uniform. J. Math. Anal. Appl. , author =. 2018 , pages =. doi:10.1016/j.jmaa.2017.10.063 , language =

2018 doi
[180]

doi:10.48550/arXiv.2410.01757 , urldate =

Naryshkin, Petr , year =. doi:10.48550/arXiv.2410.01757 , urldate =

doi arXiv
[181]

Elementary amenability and almost finiteness , volume =. Compos. Math. , author =. 2025 , pages =. doi:10.1017/S0010437X26102899 , abstract =

2025 doi
[182]

Almost finiteness and groups of dynamical origin , volume =. Int. Math. Res. Not. IMRN , author =. doi:10.1093/imrn/rnaf016 , abstract =

doi
[183]

Using the

Matou. Using the. 2008 , keywords =. doi:10.1007/978-3-540-76649-0 , language =

2008 doi
[184]

Banach, Stefan and Tarski, Alfred , title =. Fund. Math. , volume =. 1924 , url =

1924
[185]

von Neumann, John , title =. Fund. Math. , volume =. 1929 , url =

1929
[186]

On groups with full

F. On groups with full. Math. Scand. , volume =. 1955 , doi =

1955
[187]

, title =

Zimmer, Robert J. , title =. J. Funct. Anal. , volume =. 1978 , doi =

1978
[188]

Ergodic Theory and Dynam

Connes, Alain and Feldman, Jacob and Weiss, Benjamin , title =. Ergodic Theory and Dynam. Systems , volume =. 1981 , doi =

1981
[189]

1980 , isbn =

Renault, Jean , title =. 1980 , isbn =

1980
[190]

2000 , isbn =

Anantharaman-Delaroche, Claire and Renault, Jean , title =. 2000 , isbn =

2000
[191]

Block, Jonathan and Weinberger, Shmuel , title =. J. Amer. Math. Soc. , volume =. 1992 , doi =

1992
[192]

Yu, Guoliang , title =. Invent. Math. , volume =. 2000 , doi =

2000
[193]

2003 , isbn =

Roe, John , title =. 2003 , isbn =

2003
[194]

Sako, Hiroki , title =. J. Lond. Math. Soc. , volume =. 2020 , doi =

2020
[195]

Topology , volume =

Adams, Scot , title =. Topology , volume =. 1994 , doi =

1994
[196]

Topology , volume =

Skandalis, Georges and Tu, Jean-Louis and Yu, Guoliang , title =. Topology , volume =. 2002 , doi =

2002
[197]

, title =

Struble, Raimond A. , title =. Compos. Math. , volume =. 1974 , url =

1974
[198]

Proceedings of the London Mathematical Society , volume =

Matui, Hiroki , title =. Proceedings of the London Mathematical Society , volume =. 2012 , doi =

2012
[199]

Non-simple purely infinite

Kirchberg, Eberhard and R. Non-simple purely infinite. American Journal of Mathematics , volume =. 2000 , doi =

2000
[200]

2026 , Owner =

Strict comparison holds in the uniform Roe algebra of a discrete amenable group , Author =. 2026 , Owner =

2026

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [1]

The type semigroup, comparison, and almost finiteness for ample groupoids , journal =

Ara, Pere and B. The type semigroup, comparison, and almost finiteness for ample groupoids , journal =. 2023 , doi =

2023

[2] [2]

Ara, Pere and Dykema, Kenneth J. and R. Correction of proofs in ``. Canad. J. Math. , Year =. doi:10.4153/CJM-2012-018-6 , Fjournal =

work page doi:10.4153/cjm-2012-018-6 2012

[3] [3]

Amenability of coarse spaces and

Ara, Pere and Li, Kang and Lled\'. Amenability of coarse spaces and. Bull. Math. Sci. , Year =. doi:10.1007/s13373-017-0109-6 , Fjournal =

work page doi:10.1007/s13373-017-0109-6

[4] [4]

Barlak, Sel. The. J. Noncommut. Geom. , Year =. doi:10.4171/JNCG/226 , Fjournal =

work page doi:10.4171/jncg/226

[5] [5]

The real rank of inductive limit

Blackadar, Bruce and D. The real rank of inductive limit. Math. Scand. , Year =

[6] [6]

K -Theory , Year =

Approximately central matrix units and the structure of noncommutative tori , Author =. K -Theory , Year =. doi:10.1007/BF00961466 , Fjournal =

work page doi:10.1007/bf00961466

[7] [7]

Extending states on preordered semigroups and the existence of quasitraces on

Blackadar, Bruce and R. Extending states on preordered semigroups and the existence of quasitraces on. J. Algebra , Year =. doi:10.1016/0021-8693(92)90098-7 , Fjournal =

doi:10.1016/0021-8693(92)90098-7

[8] [8]

An algebraic approach to the radius of comparison , Author =. Trans. Amer. Math. Soc. , Year =. doi:10.1090/S0002-9947-2012-05538-3 , Fjournal =

work page doi:10.1090/s0002-9947-2012-05538-3 2012