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REVIEW 3 major objections 5 minor 41 references

Nuclear dimension, pure infiniteness and real rank for higher rank graph $C^*$-algebras

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Pith's one-line read When a higher-rank graph C*-algebra is purely infinite and its primitive ideal space is zero-dimensional, it is strongly purely infinite, absorbs O∞, and has nuclear dimension one, even when non-simple.

desk verdict The graph-level pure-infiniteness characterisation is a real contribution; the O∞-stability/nuclear-dimension-one theorem is plausible but rests on an uncited pure-to-strong-pure equivalence that a referee must force into the open. read the letter →

arxiv 2607.27691 v1 pith:6WKKOOB3 submitted 2026-07-30 math.OA

David Pask This is my paper · ORCID
classification math.OA MSC 46L0546L3546L5546L80
keywords higher-rankgraphk-graphC*-algebranucleardimensionpureinfinitenessO∞-stabilityrealrankzerostrongaperiodicitygeneralisedcycles
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

Higher-rank graph C*-algebras—operator algebras built from combinatorial k-graphs—are shown to be purely infinite precisely when each maximal tail in the graph has a generalised cycle with an entrance reaching every vertex. The paper's main result (Proposition 7.4) is that whenever such an algebra is purely infinite and has zero-dimensional primitive ideal space—in particular whenever its ideal lattice is finite—it is strongly purely infinite, tensorially absorbs the Cuntz algebra O∞, and has nuclear dimension one, without any simplicity assumption. This extends the nuclear-dimension-one computation from simple purely infinite algebras to a large class of non-simple algebras, using only graph combinatorics and standard absorption theorems.

What carries the argument

The central objects are generalised cycles (pairs of distinct paths with common source and range that admit minimal common extensions with every continuation) and maximal tails (sets of vertices that are downward closed and mutually reachable). The paper's machinery is the equivalence (Theorem 3.7) between strong aperiodicity of the k-graph, the entrance condition for all generalised cycles in each maximal tail, and gauge-invariance of every ideal of C*(Λ); this reduces the ideal lattice to the finite combinatorics of saturated hereditary subsets. The upgrade to strong pure infiniteness in Proposition 7.4 then invokes the standard O∞-absorption theorem for strongly purely infinite nuclear C*

What would settle it

Search for—or construct—a separable, nuclear, purely infinite C*-algebra of topological dimension zero that is not strongly purely infinite; the paper cites no proof for the equivalence it uses, so any such example would directly refute the upgrade step of Proposition 7.4 and, with it, the O∞-stability conclusion for non-simple k-graph algebras.

Watch

Extended reading notes

Core claim

The paper establishes a graph-level dictionary for pure infiniteness: C*(Λ) is purely infinite exactly when every generalised cycle in every maximal tail has an entrance and every vertex of every maximal tail is reached by a generalised cycle (Theorem 3.9, with the entrance condition alone equivalent to strong aperiodicity and to gauge-invariance of all ideals). The central application is Proposition 7.4: if C*(Λ) is purely infinite and has topological dimension zero—which is automatic when the saturated-hereditary-subset lattice H(Λ) is finite—then C*(Λ) is strongly purely infinite, C*(Λ) ≅ C*(Λ)⊗O∞, and C*(Λ) has nuclear dimension one, even when not simple. The paper also proves that under

Load-bearing premise

The proof of Proposition 7.4 assumes, without citation, that for a separable nuclear C*-algebra with topological dimension zero, pure infiniteness is equivalent to strong pure infiniteness; if this equivalence fails, the conclusion that C*(Λ) is O∞-stable and of nuclear dimension one does not follow.

Editorial extensions

If this is right

  • Non-simple purely infinite k-graph algebras with zero-dimensional primitive ideal space are O∞-stable and of nuclear dimension one, placing them within the scope of the classification program for nuclear C*-algebras.
  • Pure infiniteness of C*(Λ) can now be read directly from the graph: check every maximal tail for generalised cycles with entrances reaching all vertices.
  • For algebras with finite ideal lattice, the hypothesis of topological dimension zero is automatic, so every such purely infinite k-graph algebra is automatically O∞-stable and has nuclear dimension one.
  • Real rank zero for these algebras is equivalent to a K0-liftability condition that, for k=2, reduces to an elementary homological check on the connectivity matrices.
  • The chain conditions on ideals are forced by topological dimension zero, yielding a trichotomy: purely infinite, stably finite, or neither.

Reading between the lines

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

  • The combinatorial criteria suggest an algorithmic way to certify O∞-stability and nuclear dimension one for finite k-graphs: compute the maximal tails and verify the generalised-cycle entrance condition, which is finite data.
  • If the unproved pure-infiniteness-to-strong-pure-infiniteness equivalence for zero-dimensional algebras is ultimately shown to require an extra hypothesis, the class of k-graph algebras covered by Proposition 7.4 would shrink accordingly, but the combinatorial characterization of pure infiniteness in Theorem 3.9 would remain intact.
  • The same cycle-and-tail framework may extend to compute the nuclear dimension of mixed extensions (AF ideal with purely infinite quotient) by translating stability and fullness conditions into graph data, potentially closing the gap left open in Remark 7.6.
  • The paper's results suggest a dichotomy: on the stably finite branch, Z-stability; on the purely infinite branch, O∞-stability.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies row-finite, locally convex higher-rank graph C*-algebras. Its main results are: Theorem 3.9, a characterisation of pure infiniteness of C*(Λ) in terms of generalised cycles, maximal tails and strong aperiodicity; Theorem 4.1, a real-rank-zero criterion under a finiteness assumption on the lattice H(Λ) of saturated hereditary subsets; Theorem 6.2, claiming that topological dimension zero forces both ascending and descending chain conditions on ideals; and Proposition 7.4, the central application, which asserts that when C*(Λ) satisfies condition (5) of Theorem 3.9 and has topological dimension zero, it is strongly purely infinite, O∞-stable, and of nuclear dimension one, even when non-simple. The paper also discusses extremal richness, stable rank, a finite-ideal trichotomy, and Z-stability.

Significance. If correct, the results are significant: they give a purely combinatorial graph-level dictionary for pure infiniteness in the non-simple higher-rank setting and extend the nuclear-dimension-one computation beyond the simple case. The paper draws on deep external theorems (Kirchberg–Rørdam, Pasnicu–Rørdam, Szabó, Bosa–Gabe–Sims–White) and makes useful corrections to earlier graph-algebra literature. However, the advertised main theorem currently exceeds what is proved: Proposition 7.4 relies on an uncited equivalence between pure and strong pure infiniteness under topological dimension zero, and the abstract overstates the hypotheses under which the conclusions hold. The core graph-theoretic programme is promising, but the manuscript needs substantive revision before the central claims are supported.

major comments (3)
  1. [§7, Proposition 7.4] The proof contains the uncited assertion: 'A separable, nuclear C*-algebra of topological dimension zero is purely infinite if and only if it is strongly purely infinite: the two notions coincide once Prim has a basis of compact-open sets.' This is load-bearing: it is exactly the step that upgrades pure infiniteness to the strong version needed to apply the Kirchberg–Rørdam O∞-absorption theorem. Without it, the conclusions D ≅ D ⊗ O∞ and nuclear dimension one do not follow. The statement is not obviously in [KR] or [PR]; standard strong-pure-infiniteness criteria are usually phrased via real rank zero or the ideal property, not via zero-dimensionality of Prim alone. Please supply a proof or a precise reference. If the proof is graph-specific, it should be included, for instance using strong aperiodicity, gauge-invariant ideals, and the structure of H(Λ).
  2. [§6, Theorem 6.2] The theorem 'If C*(Λ) has topological dimension zero, then C*(Λ) satisfies both ACC and DCC on ideals' is false as stated. The commutative algebra C0(N) has zero-dimensional primitive ideal space (discrete topology, hence a basis of compact open sets), but it fails both chain conditions: the ascending chain of ideals corresponding to the open sets {0,1,...,n} does not stabilise, and the descending chain corresponding to {n,n+1,...} does not stabilise. The citation [Ped, Theorem 4.4.6] does not support the claim. The theorem needs an additional hypothesis (e.g. finiteness of the ideal lattice) or should be removed/restricted. Corollary 7.7 assumes chain stabilisation directly and is therefore not affected, but the theorem as stated cannot stand.
  3. [Abstract and Introduction vs. §7, Proposition 7.4] The abstract claims the conclusions hold 'whenever C*(Λ) is purely infinite of topological dimension zero—in particular whenever its ideal lattice is finite'. Proposition 7.4, however, assumes condition (5) of Theorem 3.9 (strong aperiodicity and reachability by generalised cycles in each maximal tail) in addition to topological dimension zero. Theorem 3.9 does not imply condition (5) from pure infiniteness when Λ0 is infinite; Remark 3.10 explicitly leaves the infinite-vertex case open. Moreover, finiteness of H(Λ) does not by itself imply condition (5). The manuscript should either restrict the abstract/introduction to the hypotheses actually used in Proposition 7.4, or prove that pure infiniteness plus topological dimension zero implies condition (5). The phrase 'In particular, the hypotheses hold whenever H(Λ) is finite' inside Proposition 7.4 is likewise too strong.
minor comments (5)
  1. [Throughout] The phrase 'connected to by a generalised cycle' appears repeatedly (Theorems 3.9, 4.1, 5.4, etc.) and should read 'connected to a generalised cycle' or similar.
  2. [§2.3, Theorem 2.11] The citation is garbled: 'the proof may be found in Theorem 5.5]S or [RSY1, Theorem 5.2]'. Please fix the bracket/reference.
  3. [§3.3, Definition 3.3] The notation 'α∈s(r(µ))Λ' in the definition of an entrance is confusing. Since r(µ) is a vertex, this should be written as 'α∈r(µ)Λ' or 'α∈s(r(µ))Λ' with the source/range conventions made explicit.
  4. [§5.1 and §6.3] Examples 5.6/5.7 and 6.7(2)/(3) describe the same graphs Λ_PI and Λ_mix with essentially identical computations. Consolidating them would improve readability.
  5. [§7, Remark 7.2] The example 'the minimal unitisation of O2 ⊗ K' is claimed to be both purely infinite and stably finite. Under Definition 3.1, pure infiniteness excludes characters, and the minimal unitisation has a character (the quotient map onto C). Please clarify the intended notion or replace the example.

Circularity Check

0 steps flagged · score 2.0 of 10

No derivation reduces to its own inputs; main concern is an uncited load-bearing equivalence in Proposition 7.4, which is a support gap rather than circularity.

full rationale

The claimed derivation chain is not circular. Theorem 3.9 proves pure infiniteness from generalized-cycle/maximal-tail conditions via [KaP], [RSY1], [ES], [S], [KR] and the cited [PSS2] theorem; these are published, parameter-free inputs whose assumptions do not include the target result. Real-rank, extremal-richness, stable-rank and Z-stability statements are assembled from external permanence theorems ([PR], [BP2], [BP3], [PRA1]), not from the paper's conclusions. The self-citations to [PSS2], [PSS3], [PSS4] are load-bearing but independent published results, so under the supplied rules they do not in themselves raise the circularity score. The one genuine concern is in Proposition 7.4: the step from pure infiniteness to strong pure infiniteness is made by the unproved, uncited assertion 'A separable, nuclear C*-algebra of topological dimension zero is purely infinite if and only if it is strongly purely infinite: the two notions coincide once Prim has a basis of compact-open sets.' This assertion is exactly the upgrade needed for O-infinity-stability and nuclear dimension one, and no proof or reference is supplied. That is an omitted-support/omitted-proof issue, not a case of the conclusion being built into the definitions or fitted parameters, so it does not make the derivation circular. Score 2 rather than 0 because the paper's headline regularity claim depends on this unverified step and on several self-citations; there is, however, no exhibited reduction of a prediction to its own inputs.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The paper's central claim rests on a network of external deep theorems (BGSW, KR, Szabó, Pasnicu–Rørdam, PSS2, RS1, ES, Ped). There are no fitted numerical parameters and no invented entities. The most significant unproved axiom introduced without citation is the equivalence between pure infiniteness and strong pure infiniteness under topological dimension zero, used in Proposition 7.4. The Pedersen citation in Theorem 6.2 is suspect.

assumptions (7)
  • standard math Separable, nuclear, O∞-stable C*-algebras have nuclear dimension one [BGSW, Theorem A]
    Invoked in Prop 7.4 to obtain nuclear dimension one.
  • standard math Separable, nuclear, strongly purely infinite C*-algebras are O∞-stable [KR]
    Invoked in Prop 7.4 for O∞-absorption.
  • standard math For a separable nuclear C*-algebra with topological dimension zero, pure infiniteness is equivalent to strong pure infiniteness
    Asserted without citation in the proof of Prop 7.4; load-bearing for the O∞-stability conclusion.
  • standard math [PSS2, Theorem 3.2, Corollary 3.9]: strong aperiodicity ⇔ every generalised cycle in each maximal tail has an entrance ⇔ every ideal is gauge-invariant
    Used as Theorem 3.7 and in the proof of Theorem 3.9.
  • standard math [Ped, Theorem 4.4.6]: separable C*-algebras with totally disconnected primitive ideal space satisfy both ACC and DCC on ideals
    Used in the proof of Theorem 6.2; appears to be a misquotation since c0 is a counterexample.
  • standard math [RS1]: cofinal and aperiodic k-graph implies simple C*-algebra
    Used in the proof of Theorem 3.9, (3)⇒(4).
  • standard math [ES, Prop 5.4]: a finite-vertex k-graph without generalised cycles has AF C*-algebra
    Used in the proof of Theorem 3.9, (3)⇒(4).

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Cite this review

Pith. "Pith review of Nuclear dimension, pure infiniteness and real rank for higher rank graph $C^*$-algebras." pith.science (2026). https://pith.science/paper/6WKKOOB3

@misc{pith2026260727691,
  author       = {Pith},
  title        = {Pith review of: Nuclear dimension, pure infiniteness and real rank for higher rank graph $C^*$-algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WKKOOB3}},
  note         = {Machine review of arXiv:2607.27691}
}
abstract

We study the structure and regularity of higher rank graph $C^*$-algebras, with particular emphasis on their nuclear dimension. For a row-finite, locally convex $k$-graph $\Lambda$ with no sources, we characterise pure infiniteness of $C^*(\Lambda)$ in terms of generalised cycles, maximal tails, and strong aperiodicity, and we relate these conditions to topological dimension zero of the primitive ideal space and to the structure of gauge-invariant ideals. Our main application is that whenever $C^*(\Lambda)$ is purely infinite of topological dimension zero---in particular whenever its ideal lattice is finite---it is strongly purely infinite, $\mathcal O_\infty$-stable, and of nuclear dimension one, \emph{even when $C^*(\Lambda)$ is not simple}. This extends to the non-simple, higher-rank setting the nuclear-dimension-one computation known for simple UCT-Kirchberg $2$-graph algebras. Along the way we refine and correct several results in the existing graph $C^*$-algebra literature.

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Reference graph

Works this paper leans on

41 extracted references · 1 linked inside Pith

  1. [1]

    A gauge invariant uniqueness theorem for corners of k -graph algebras, Rocky Mountain J.\ Math, 38 (2008), 1887--1907

    S.\ Allen. A gauge invariant uniqueness theorem for corners of k -graph algebras, Rocky Mountain J.\ Math, 38 (2008), 1887--1907

    2008

  2. [2]

    6 (2000), 307--324

    T.\ Bates, D.\ Pask, I.\ Raeburn and W.\ Szyma\'nski , The C^ -algebras of row-finite graphs , New York J.\ Math. 6 (2000), 307--324

    2000

  3. [3]

    The nuclear dimension of O_ -stable C^* -algebras, Adv.\ Math., 401 (2022), Paper No.\ 108250, 51 pp

    J.\ Bosa, J.\ Gabe, A.\ Sims and S.\ White. The nuclear dimension of O_ -stable C^* -algebras, Adv.\ Math., 401 (2022), Paper No.\ 108250, 51 pp

    2022

  4. [4]

    L.G.\ Brown and G.K.\ Pedersen, C^* -algebras of real rank zero, J.\ Funct.\ Anal., 99 (1991), 131--149

    1991

  5. [5]

    L.G.\ Brown and G.K.\ Pedersen, On the geometry of the unit ball of a C^* -algebra, J.\ reine angew.\ Math., 469 (1995), 113--147

    1995

  6. [6]

    L.G.\ Brown and G.K.\ Pedersen, Ideal structure and C^* -algebras of low rank, Math.\ Scand., 100 (2007), 5--33

    2007

  7. [7]

    AF-embeddability of 2 -graph algebras and quasidiagonality of k -graph algebras, J.\ Funct.\ Anal., 271 (2016), 958--991

    L.O.\ Clark, A.\ an Huef and A.\ Sims. AF-embeddability of 2 -graph algebras and quasidiagonality of k -graph algebras, J.\ Funct.\ Anal., 271 (2016), 958--991

    2016

  8. [8]

    Periodicity in rank 2 graph algebras, Canad.\ J.\ Math., 61 (2009), 1239--1261

    K.R.\ Davidson and D.\ Yang. Periodicity in rank 2 graph algebras, Canad.\ J.\ Math., 61 (2009), 1239--1261

    2009

Show all 41 references
  1. [9]

    Amplified graph C^* -algebras, M u nster J.\ Math., 5 (2012), 121--150

    S.\ Eilers, E.\ Ruiz and A.P.W.\ S rensen. Amplified graph C^* -algebras, M u nster J.\ Math., 5 (2012), 121--150

    2012

  2. [10]

    When is the C untz- K rieger algebra of a higher-rank graph approximately finite-dimensional? , J.\ Funct.\ Anal., 263 (2012), 183--215

    D.G.\ Evans and A.\ Sims. When is the C untz- K rieger algebra of a higher-rank graph approximately finite-dimensional? , J.\ Funct.\ Anal., 263 (2012), 183--215

    2012

  3. [11]

    Nuclear dimension of extensions of O_ -stable algebras, J.\ Operator Theory, 88 (2022), 171--187

    S.\ Evington. Nuclear dimension of extensions of O_ -stable algebras, J.\ Operator Theory, 88 (2022), 171--187

    2022

  4. [12]

    Z -stable graph algebras, arXiv:2511.02760

    G.\ Faurot. Z -stable graph algebras, arXiv:2511.02760

    arXiv

  5. [13]

    Nuclear dimension of graph C^* -algebras with condition (K), Proc.\ Amer.\ Math.\ Soc., 152 (2024), 4421--4435

    G.\ Faurot and C.\ Schafhauser. Nuclear dimension of graph C^* -algebras with condition (K), Proc.\ Amer.\ Math.\ Soc., 152 (2024), 4421--4435

    2024

  6. [14]

    Hazlewood, I.\ Raeburn, A.\ Sims, and S.B.G.\ Webster

    R. Hazlewood, I.\ Raeburn, A.\ Sims, and S.B.G.\ Webster. Remarks on some fundamental results about higher-rank graphs and their C^* -algebras. Proc. Edinb. Math. Soc. (2) , 56 (2013), 575--597

    2013

  7. [15]

    Purely Infinite Cuntz-Krieger Algebras of Directed Graphs, Bull.\ London Math.\ Soc.\ 35 (2003), 689--696

    J.H.\ Hong and W.\ Szyma\'nski. Purely Infinite Cuntz-Krieger Algebras of Directed Graphs, Bull.\ London Math.\ Soc.\ 35 (2003), 689--696

    2003

  8. [16]

    Jeong , G.H.\ Park and D.Y.\ Shin

    Ja.\ A. Jeong , G.H.\ Park and D.Y.\ Shin. Stable rank and real rank of graph C^* -algebras, Pacific J.\ Math., 200 (2001), 331--342

    2001

  9. [17]

    Aperiodicity and primitive ideals of row-finite k -graphs, Int.\ J.\ Math., 25 (2014), 1450022 (25 pages)

    S.\ Kang and D.\ Pask. Aperiodicity and primitive ideals of row-finite k -graphs, Int.\ J.\ Math., 25 (2014), 1450022 (25 pages)

    2014

  10. [18]

    Non-simple purely infinite C^* -algebras, Amer.\ J.\ Math.\ 122 (2000), 637--666

    E.\ Kirchberg and M.\ R rdam. Non-simple purely infinite C^* -algebras, Amer.\ J.\ Math.\ 122 (2000), 637--666

    2000

  11. [19]

    Higher rank graph C^* -algebras, New York J.\ Math., 6 (2000), 1--20

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

    2000

  12. [20]

    Cuntz-Krieger algebras of directed graphs, Pacific J

    A.\ Kumjian, D.\ Pask and I.\ Raeburn. Cuntz-Krieger algebras of directed graphs, Pacific J. Math., 184 (1998) 161--174

    1998

  13. [21]

    Graphs, groupoids, and Cuntz--Krieger algebras, J.\ Funct.\ Anal., 144 (1997), 505--541

    A.\ Kumjian, D.\ Pask, I.\ Raeburn and J.\ Renault. Graphs, groupoids, and Cuntz--Krieger algebras, J.\ Funct.\ Anal., 144 (1997), 505--541

    1997

  14. [22]

    On the K -theory of twisted higher--rank graph C^* --algebras, J.\ Math.\ Anal.\ Appl., 401 (2013), 104---113

    A.\ Kumjian, D.\ Pask and A.\ Sims. On the K -theory of twisted higher--rank graph C^* --algebras, J.\ Math.\ Anal.\ Appl., 401 (2013), 104---113

    2013

  15. [23]

    Erratum to ``Higher-rank graph C^* -algebras'', New York J.\ Math., 30 (2024), 1029--1031

    A.\ Kumjian, D.\ Pask and A.\ Sims. Erratum to ``Higher-rank graph C^* -algebras'', New York J.\ Math., 30 (2024), 1029--1031

    2024

  16. [24]

    Simplicity of 2 -graph algebras associated to dynamical systems, Bull.\ Malaysian Math.\ Sci.\ Soc., 33 (2010), 177--196

    P.\ Lewin and D.\ Pask. Simplicity of 2 -graph algebras associated to dynamical systems, Bull.\ Malaysian Math.\ Sci.\ Soc., 33 (2010), 177--196

    2010

  17. [25]

    Rank-Two Graphs whose C^* -algebras are direct limits of Circle Algebras, J.\ Funct.\ Anal., 239 (2006), 137--178

    D.\ Pask, I.\ Raeburn, M.\ R rdam and A.\ Sims. Rank-Two Graphs whose C^* -algebras are direct limits of Circle Algebras, J.\ Funct.\ Anal., 239 (2006), 137--178

    2006

  18. [26]

    Real rank and topological dimension of higher rank graph algebras, Indiana Univ.\ Math.\ J., 66 (2017), 2137--2168

    D.\ Pask, A.\ Sierakowski and A.\ Sims. Real rank and topological dimension of higher rank graph algebras, Indiana Univ.\ Math.\ J., 66 (2017), 2137--2168

    2017

  19. [27]

    Unbounded quasitraces, stable finiteness and pure infiniteness, Houston J.\ Math., 45 (2019), 763--814

    D.\ Pask, A.\ Sierakowski and A.\ Sims. Unbounded quasitraces, stable finiteness and pure infiniteness, Houston J.\ Math., 45 (2019), 763--814

    2019

  20. [28]

    Structure theory and stable rank for C^* -algebras of finite higher-rank graphs, Proc.\ Edinb.\ Math.\ Soc.\ (2), 64 (2021), 822--847

    D.\ Pask, A.\ Sierakowski and A.\ Sims. Structure theory and stable rank for C^* -algebras of finite higher-rank graphs, Proc.\ Edinb.\ Math.\ Soc.\ (2), 64 (2021), 822--847

    2021

  21. [29]

    The -function in operator algebras, J.\ Operator Theory, 26 (1991), 345--381

    G.\ Pedersen. The -function in operator algebras, J.\ Operator Theory, 26 (1991), 345--381

    1991

  22. [30]

    Purely infinite C^* -algebras of real rank zero, J.\ Reine Angew.\ Math., 613, (2007), 51--73

    C.\ Pasnicu, M.\ R rdam. Purely infinite C^* -algebras of real rank zero, J.\ Reine Angew.\ Math., 613, (2007), 51--73

    2007

  23. [31]

    C^* -algebras and their automorphism groups, London Math.\ Soc.\ Monogr., 14, Academic Press, London, 1979

    G.K.\ Pedersen. C^* -algebras and their automorphism groups, London Math.\ Soc.\ Monogr., 14, Academic Press, London, 1979

    1979

  24. [32]

    Chain conditions for graph C^* -algebras, Forum Math., 32 (2020), 491--500

    M.\ Pourgholamhossein, M.\ Rouzbehani and M.\ Amini. Chain conditions for graph C^* -algebras, Forum Math., 32 (2020), 491--500

    2020

  25. [33]

    On finiteness properties of Noetherian (Artinian) C^* -algebras, Linear and Multilinear Algebra, 70 (2022), 419--430

    M.\ Pourgholamhossein, M.\ Rouzbehani and M.\ Amini. On finiteness properties of Noetherian (Artinian) C^* -algebras, Linear and Multilinear Algebra, 70 (2022), 419--430

    2022

  26. [34]

    Higher-rank graphs and their C^* -algebras , Proc.\ Edinb.\ Math.\ Soc.\ (2), 46 (2003), 99--115

    I.\ Raeburn, A.\ Sims and T.\ Yeend. Higher-rank graphs and their C^* -algebras , Proc.\ Edinb.\ Math.\ Soc.\ (2), 46 (2003), 99--115

    2003

  27. [35]

    Simplicity of C^ -algebras associated to higher-rank graphs , Bull.\ Lond.\ Math.\ Soc., 39 (2007), 337--344

    D.I.\ Robertson and A.\ Sims. Simplicity of C^ -algebras associated to higher-rank graphs , Bull.\ Lond.\ Math.\ Soc., 39 (2007), 337--344

    2007

  28. [36]

    UCT-Kirchberg algebras have nuclear dimension one, Adv.\ Math., 279 (2015), 1--28

    E.\ Ruiz, A.\ Sims and A.P.W.\ S rensen. UCT-Kirchberg algebras have nuclear dimension one, Adv.\ Math., 279 (2015), 1--28

    2015

  29. [37]

    The nuclear dimension of graph C^* -algebras, Adv.\ Math., 272 (2015), 96--123

    E.\ Ruiz, A.\ Sims and M.\ Tomforde. The nuclear dimension of graph C^* -algebras, Adv.\ Math., 272 (2015), 96--123

    2015

  30. [38]

    Gauge-invariant ideals in the C^* -algebras of finitely aligned higher-rank graphs , Canad.\ J.\ Math., 58(2006), 1268--1290

    A.\ Sims. Gauge-invariant ideals in the C^* -algebras of finitely aligned higher-rank graphs , Canad.\ J.\ Math., 58(2006), 1268--1290

    2006

  31. [39]

    On the nuclear dimension of strongly purely infinite C^* -algebras, Adv.\ Math., 306 (2017), 1262--1268

    G.\ Szab\'o. On the nuclear dimension of strongly purely infinite C^* -algebras, Adv.\ Math., 306 (2017), 1262--1268

    2017

  32. [40]

    J.\ Tyler, Every AF algebra is Morita equivalent to a graph algebra, Bull.\ Austral.\ Math.\ Soc., 69 (2004), 237--240

    2004

  33. [41]

    The nuclear dimension of C^* -algebras, Adv.\ Math., 224 (2010), 461--498

    W.\ Winter and J.\ Zacharias. The nuclear dimension of C^* -algebras, Adv.\ Math., 224 (2010), 461--498

    2010

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