Brown-Voiculescu entropy revisited

Bhishan Jacelon; Robert Neagu

arxiv: 2506.18352 · v2 · submitted 2025-06-23 · 🧮 math.OA · math.DS

Brown-Voiculescu entropy revisited

Bhishan Jacelon , Robert Neagu This is my paper

Pith reviewed 2026-05-19 08:26 UTC · model grok-4.3

classification 🧮 math.OA math.DS

keywords Brown-Voiculescu entropynoncommutative topological entropyC*-algebrasdecomposition ranknuclear dimensionvariational principlequasidiagonal approximationsclassifiable C*-algebras

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The pith

Coloured versions of Brown-Voiculescu entropy satisfy a variational principle for simple C*-algebras with finite decomposition rank and finitely many traces.

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

The paper reexamines Brown-Voiculescu entropy for endomorphisms of nuclear C*-algebras through the lens of modern classification theory. It defines coloured versions of noncommutative topological entropy adapted to algebras of finite nuclear dimension or finite decomposition rank. For simple separable unital algebras that satisfy the UCT and possess only finitely many extremal traces, the authors establish a variational principle that equates entropy to a supremum taken over quasidiagonal approximations relative to those traces. They further prove that infinite entropy occurs generically for endomorphisms and automorphisms of certain classifiable C*-algebras that model noncommutative versions of topological manifolds.

Core claim

Assuming A is simple, separable, unital, satisfies the UCT and has finitely many extremal traces, the Brown-Voiculescu entropy of an endomorphism satisfies a variational principle expressed through quasidiagonal approximations relative to this finite set of traces. Building on earlier results, infinite entropy is shown to occur generically among endomorphisms and automorphisms of certain classifiable C*-algebras that serve as noncommutative spaces of observables for topological manifolds.

What carries the argument

Coloured versions of noncommutative topological entropy, which are formulated for C*-algebras of finite nuclear dimension or finite decomposition rank and support a variational description in terms of quasidiagonal approximations relative to extremal traces.

If this is right

Entropy values for endomorphisms can be computed or estimated by optimizing over quasidiagonal approximations that respect the finite set of extremal traces.
Infinite entropy is the generic case for endomorphisms and automorphisms within the class of algebras considered.
The coloured entropy supplies a dynamical invariant that distinguishes conjugacy classes of maps on these classifiable C*-algebras.

Where Pith is reading between the lines

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

The variational principle may allow entropy to serve as a complete invariant for outer conjugacy of automorphisms in some of these algebras.
Explicit calculations of the coloured entropy on concrete examples such as crossed products by minimal homeomorphisms could be carried out to test the formula.
Analogous variational statements might hold for algebras of infinite decomposition rank once the finite-trace hypothesis is relaxed or replaced by a suitable substitute.

Load-bearing premise

The C*-algebra A must be simple, separable, unital, satisfy the UCT, possess finitely many extremal traces, and have finite decomposition rank.

What would settle it

A counterexample consisting of a simple separable unital C*-algebra satisfying the UCT with finitely many extremal traces and finite decomposition rank, together with an endomorphism whose Brown-Voiculescu entropy does not equal the variational supremum over quasidiagonal approximations relative to those traces.

read the original abstract

Aided by the tools and outlook provided by modern classification theory, we take a new look at the Brown-Voiculescu entropy of endomorphisms of nuclear C*-algebras. In particular, we introduce `coloured' versions of noncommutative topological entropy suitable for C*-algebras A of finite nuclear dimension or finite decomposition rank. In the latter case, assuming further that A is simple, separable, unital, satisfies the UCT and has finitely many extremal traces, we prove a variational type principle in terms of quasidiagonal approximations relative to this finite set of traces. Building on work of Kerr, we also show that infinite entropy occurs generically among endomorphisms and automorphisms of certain classifiable C*-algebras that function as noncommutative spaces of observables of topological manifolds.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note private letter to a colleague

Jacelon and Neagu introduce coloured versions of Brown-Voiculescu entropy and prove a variational principle plus genericity of infinite entropy for certain classifiable C*-algebras.

read the letter

Jacelon and Neagu introduce coloured versions of the Brown-Voiculescu entropy for C*-algebras with finite nuclear dimension or decomposition rank. They prove a variational principle for simple separable unital UCT algebras with finitely many extremal traces and finite decomposition rank, expressing the entropy via quasidiagonal approximations relative to those traces. They also establish that infinite entropy is generic for endomorphisms and automorphisms of certain classifiable C*-algebras corresponding to topological manifolds, extending Kerr's results.

Referee Report

0 major / 3 minor

Summary. The paper revisits the Brown-Voiculescu entropy of endomorphisms of nuclear C*-algebras using tools from modern classification theory. It introduces coloured versions of noncommutative topological entropy adapted to C*-algebras of finite nuclear dimension or finite decomposition rank. Under the further assumptions that A is simple, separable, unital, satisfies the UCT, has finitely many extremal traces and finite decomposition rank, the authors establish a variational principle expressing the entropy via a supremum over quasidiagonal approximations relative to the extremal traces. Building on Kerr's constructions, they also prove that infinite entropy occurs generically for endomorphisms and automorphisms of certain classifiable C*-algebras serving as noncommutative models for spaces of observables of topological manifolds.

Significance. If the central claims hold, the work meaningfully extends the study of entropy in noncommutative dynamical systems by incorporating classification-theoretic hypotheses. The coloured entropy variants provide invariants suited to finite decomposition rank, and the variational principle supplies a concrete computational tool in terms of trace approximations. The genericity result for infinite entropy clarifies typical behavior in these algebras. The manuscript explicitly builds on and credits Kerr's prior results, and the logical chain from the stated hypotheses to the variational and genericity statements appears internally consistent.

minor comments (3)

The introduction would benefit from a short paragraph explicitly comparing the new coloured entropy to the classical Brown-Voiculescu definition, including a reference to the relevant equation or definition number where the reduction is shown.
In the statement of the variational principle (around Theorem 4.1), the dependence of the coloured entropy on the finite set of extremal traces should be notated uniformly in all subsequent statements to avoid ambiguity for readers.
A minor typographical inconsistency appears in the abstract: 'variational type principle' should be 'variational principle' for consistency with the body of the paper.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, recognition of its significance in extending noncommutative entropy via classification theory, and recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper introduces coloured variants of Brown-Voiculescu entropy tailored to C*-algebras of finite nuclear dimension or decomposition rank. The variational principle is derived under the explicit hypotheses of simplicity, separability, unitality, UCT, finitely many extremal traces and finite decomposition rank, equating entropy to a supremum over quasidiagonal approximations relative to those traces. This construction builds directly on Kerr's independent prior results rather than any self-citation chain or author-overlapping uniqueness theorem. The genericity statement for infinite entropy among endomorphisms of classifiable algebras likewise follows from the new entropy definitions and external classification theory without reducing any central claim to a fitted parameter, self-definition, or renamed known result. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions from C*-algebra theory and prior results in classification and entropy; no free parameters or invented entities are evident from the abstract.

axioms (1)

domain assumption A is simple, separable, unital, satisfies the UCT, has finitely many extremal traces, and finite decomposition rank.
Explicitly required for the variational principle in the abstract.

pith-pipeline@v0.9.0 · 5656 in / 1214 out tokens · 20254 ms · 2026-05-19T08:26:28.891823+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost Jcost uniqueness (washburn_uniqueness_aczel) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Definition 2.2 introduces (d,F,ε)-decomposable rank rnuc and contractive rdr; htnuc(α,F,ε)=lim sup (1/n)log rnuc(d,F∪α(F)∪⋯∪α^{n-1}(F),ε)
IndisputableMonolith/Foundation/AlexanderDuality alexander_duality_circle_linking (D=3 forcing) unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem A: htdr(α)=sup_{τ∈∂eT(A)} hqdτ(α) for simple separable unital Z-stable UCT algebras with finitely many extremal traces
IndisputableMonolith/Foundation/ArithmeticFromLogic LogicNat ≃ Nat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem B: infinite entropy generic on dense Gδ of endomorphisms when ∂eT(A) homeomorphic to compact manifold of dim≥2

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Works this paper leans on

49 extracted references · 49 canonical work pages · cited by 1 Pith paper

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Kerr and H

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E. Kirchberg and W. Winter. Covering dimension and quasidiagonality. Internat. J. Math., 15(1):63–85, 2004

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H. Lin. Exponential rank of C ∗-algebras with real rank zero and the Brown–Pedersen conjectures. J. Funct. Anal., 114(1):1–11, 1993

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Matui and Y

H. Matui and Y. Sato. Strict comparison and Z-absorption of nuclear C ∗-algebras. Acta Math., 209(1):179–196, 2012

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Matui and Y

H. Matui and Y. Sato. Decomposition rank of UHF-absorbing C ∗-algebras. Duke Math. J., 163(14):2687–2708, 2014. BROWN–VOICULESCU ENTROPY REVISITED 25

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Neshveyev and E

S. Neshveyev and E. Størmer. The variational principle for a class of asymptotically abelian C ∗-algebras. Comm. Math. Phys. , 215(1):177–196, 2000

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Ornstein

D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Advances in Math., 4:337–352, 1970

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Petersen

K. Petersen. Ergodic theory, volume 2 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1983

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M. Rørdam. The stable and the real rank of Z-absorbing C ∗-algebras. Internat. J. Math., 15(10):1065–1084, 2004

work page 2004
[36]

Y. Sato, S. White, and W. Winter. Nuclear dimension and Z-stability. Invent. Math., 202(2):893–921, 2015

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Schafhauser

C. Schafhauser. A new proof of the Tikuisis-White-Winter theorem. J. Reine Angew. Math., 759:291–304, 2020

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Y. G. Sina˘ ı. On the concept of entropy for a dynamic system.Dokl. Akad. Nauk SSSR , 124:768–771, 1959

work page 1959
[39]

Skalski and J

A. Skalski and J. Zacharias. Noncommutative topological entropy of endomorphisms of Cuntz algebras. Lett. Math. Phys. , 86(2-3):115–134, 2008

work page 2008
[40]

G. Szab´ o. Rokhlin dimension: absorption of model actions.Anal. PDE, 12(5):1357– 1396, 2019

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Szab´ o, J

G. Szab´ o, J. Wu, and J. Zacharias. Rokhlin dimension for actions of residually finite groups. Ergodic Theory Dynam. Systems , 39(8):2248–2304, 2019

work page 2019
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Tikuisis

A. Tikuisis. Nuclear dimension, Z-stability, and algebraic simplicity for stably projec- tionless C∗-algebras. Math. Ann., 358(3-4):729–778, 2014

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Tikuisis, S

A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear C ∗-algebras. Ann. of Math. (2) , 185(1):229–284, 2017

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Voiculescu

D. Voiculescu. Dynamical approximation entropies and topological entropy in operator algebras. Comm. Math. Phys. , 170(2):249–281, 1995

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W. Winter. Covering dimension for nuclear C ∗-algebras. J. Funct. Anal., 199(2):535– 556, 2003

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W. Winter. Nuclear dimension and Z-stability of pure C ∗-algebras. Invent. Math., 187(2):259–342, 2012

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W. Winter and J. Zacharias. Completely positive maps of order zero. M¨ unster J. Math., 2:311–324, 2009

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K. Yano. A remark on the topological entropy of homeomorphisms. Invent. Math., 59(3):215–220, 1980. Bhishan Jacelon, Institute of Mathematics of the Czech Academy of Sciences, ˇZitn´a 25, 115 67 Prague 1, Czech Republic Email address: jacelon@math.cas.cz Robert Neagu, Department of mathematics, KU Leuven, Celestijnenlaan 200B, 3001, Leuven, Belgium. Email...

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[1] [1]

R. L. Adler, A. G. Konheim, and M. H. McAndrew. Topological entropy. Trans. Amer. Math. Soc., 114:309–319, 1965

work page 1965

[2] [2]

F. P. Boca and P. Goldstein. Topological entropy for the canonical endomorphism of Cuntz-Krieger algebras. Bull. London Math. Soc. , 32(3):345–352, 2000

work page 2000

[3] [3]

J. Bosa, N. P. Brown, Y. Sato, A. Tikuisis, S. White, and W. Winter. Covering dimension of C ∗-algebras and 2-coloured classification. Mem. Amer. Math. Soc. , 257(1233):vii+97, 2019. 1One can ask the same question if α has finite Rokhlin dimension (without commuting towers) if G is a group for which this is defined. 24 B. JACELON AND R. NEAGU

work page 2019

[4] [4]

R. Bowen. Entropy for group endomorphisms and homogeneous spaces. Trans. Amer. Math. Soc., 153:401–414, 1971

work page 1971

[5] [5]

L. G. Brown and G. K. Pedersen. C ∗-algebras of real rank zero. J. Funct. Anal. , 99(1):131–149, 1991

work page 1991

[6] [6]

N. P. Brown. Topological entropy in exact C ∗-algebras. Math. Ann., 314(2):347–367, 1999

work page 1999

[7] [7]

N. P. Brown. Characterizing type I C ∗-algebras via entropy. C. R. Math. Acad. Sci. Paris, 339(12):827–829, 2004

work page 2004

[8] [8]

N. P. Brown. Invariant means and finite representation theory of C ∗-algebras. Mem. Amer. Math. Soc., 184(865):viii+105, 2006

work page 2006

[9] [9]

N. P. Brown and N. Ozawa. C ∗-algebras and finite-dimensional approximations , vol- ume 88 of Graduate Studies in Mathematics . American Mathematical Society, Provi- dence, RI, 2008

work page 2008

[10] [10]

J. R. Carri´ on, J. Gabe, C. Schafhauser, A. Tikuisis, and S. White. Classification of ∗-homomorphisms I: Simple nuclear C ∗-algebras. arXiv:2307.06480 [math.OA], 2023

work page arXiv 2023

[11] [11]

Castillejos and S

J. Castillejos and S. Evington. Nuclear dimension of simple stably projectionless C∗-algebras. Anal. PDE, 13(7):2205–2240, 2020

work page 2020

[12] [12]

Castillejos, S

J. Castillejos, S. Evington, A. Tikuisis, and S. White. Classifying maps into uniform tracial sequence algebras. M¨ unster J. Math., 14(2):265–281, 2021

work page 2021

[13] [13]

Castillejos, S

J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter. Nuclear dimension of simple C ∗-algebras. Invent. Math., 224(1):245–290, 2021

work page 2021

[14] [14]

Castillejos Lopez

J. Castillejos Lopez. Decomposable approximations and coloured isomorphisms for C∗-algebras. PhD thesis, University of Glasgow, 2016

work page 2016

[15] [15]

M. Choda. Entropy of Cuntz’s canonical endomorphism. Pacific J. Math. , 190(2):235– 245, 1999

work page 1999

[16] [16]

Connes, H

A. Connes, H. Narnhofer, and W. Thirring. Dynamical entropy of C ∗ algebras and von Neumann algebras. Comm. Math. Phys. , 112(4):691–719, 1987

work page 1987

[17] [17]

Connes and E

A. Connes and E. Størmer. Entropy for automorphisms of II1 von Neumann algebras. Acta Math., 134(3-4):289–306, 1975

work page 1975

[18] [18]

G. A. Elliott, G. Gong, H. Lin, and Z. Niu. On the classification of simple amenable C∗-algebras with finite decomposition rank, ii. arXiv:1507.03437 [math.OA], 2016

work page arXiv 2016

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Glasner and B

E. Glasner and B. Weiss. The topological Rohlin property and topological entropy. Amer. J. Math. , 123(6):1055–1070, 2001

work page 2001

[20] [20]

Hirshberg, W

I. Hirshberg, W. Winter, and J. Zacharias. Rokhlin dimension and C ∗-dynamics. Comm. Math. Phys. , 335(2):637–670, 2015

work page 2015

[21] [21]

Hurewicz and H

W. Hurewicz and H. Wallman.Dimension Theory, volume 4 of Princeton Mathematical Series. Princeton University Press, 1948

work page 1948

[22] [22]

B. Jacelon. Chaotic tracial dynamics. Forum Math. Sigma, 11:Paper No. e39, 21, 2023

work page 2023

[23] [23]

Jiang and H

X. Jiang and H. Su. On a simple unital projectionless C ∗-algebra. Amer. J. Math. , 121(2):359–413, 1999

work page 1999

[24] [24]

D. Kerr. Entropy and induced dynamics on state spaces.Geom. Funct. Anal., 14(3):575– 594, 2004

work page 2004

[25] [25]

D. Kerr. Generically infinite entropy in a simple AF algebra. Bull. Lond. Math. Soc. , 39(2):265–271, 2007

work page 2007

[26] [26]

Kerr and H

D. Kerr and H. Li. Dynamical entropy in Banach spaces.Invent. Math., 162(3):649–686, 2005

work page 2005

[27] [27]

Kirchberg and W

E. Kirchberg and W. Winter. Covering dimension and quasidiagonality. Internat. J. Math., 15(1):63–85, 2004

work page 2004

[28] [28]

A. N. Kolmogorov. A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces. Dokl. Akad. Nauk SSSR (N.S.) , 119:861–864, 1958

work page 1958

[29] [29]

H. Lin. Exponential rank of C ∗-algebras with real rank zero and the Brown–Pedersen conjectures. J. Funct. Anal., 114(1):1–11, 1993

work page 1993

[30] [30]

Matui and Y

H. Matui and Y. Sato. Strict comparison and Z-absorption of nuclear C ∗-algebras. Acta Math., 209(1):179–196, 2012

work page 2012

[31] [31]

Matui and Y

H. Matui and Y. Sato. Decomposition rank of UHF-absorbing C ∗-algebras. Duke Math. J., 163(14):2687–2708, 2014. BROWN–VOICULESCU ENTROPY REVISITED 25

work page 2014

[32] [32]

Neshveyev and E

S. Neshveyev and E. Størmer. The variational principle for a class of asymptotically abelian C ∗-algebras. Comm. Math. Phys. , 215(1):177–196, 2000

work page 2000

[33] [33]

Ornstein

D. Ornstein. Bernoulli shifts with the same entropy are isomorphic. Advances in Math., 4:337–352, 1970

work page 1970

[34] [34]

Petersen

K. Petersen. Ergodic theory, volume 2 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1983

work page 1983

[35] [35]

M. Rørdam. The stable and the real rank of Z-absorbing C ∗-algebras. Internat. J. Math., 15(10):1065–1084, 2004

work page 2004

[36] [36]

Y. Sato, S. White, and W. Winter. Nuclear dimension and Z-stability. Invent. Math., 202(2):893–921, 2015

work page 2015

[37] [37]

Schafhauser

C. Schafhauser. A new proof of the Tikuisis-White-Winter theorem. J. Reine Angew. Math., 759:291–304, 2020

work page 2020

[38] [38]

Y. G. Sina˘ ı. On the concept of entropy for a dynamic system.Dokl. Akad. Nauk SSSR , 124:768–771, 1959

work page 1959

[39] [39]

Skalski and J

A. Skalski and J. Zacharias. Noncommutative topological entropy of endomorphisms of Cuntz algebras. Lett. Math. Phys. , 86(2-3):115–134, 2008

work page 2008

[40] [40]

G. Szab´ o. Rokhlin dimension: absorption of model actions.Anal. PDE, 12(5):1357– 1396, 2019

work page 2019

[41] [41]

Szab´ o, J

G. Szab´ o, J. Wu, and J. Zacharias. Rokhlin dimension for actions of residually finite groups. Ergodic Theory Dynam. Systems , 39(8):2248–2304, 2019

work page 2019

[42] [42]

Tikuisis

A. Tikuisis. Nuclear dimension, Z-stability, and algebraic simplicity for stably projec- tionless C∗-algebras. Math. Ann., 358(3-4):729–778, 2014

work page 2014

[43] [43]

Tikuisis, S

A. Tikuisis, S. White, and W. Winter. Quasidiagonality of nuclear C ∗-algebras. Ann. of Math. (2) , 185(1):229–284, 2017

work page 2017

[44] [44]

Voiculescu

D. Voiculescu. Dynamical approximation entropies and topological entropy in operator algebras. Comm. Math. Phys. , 170(2):249–281, 1995

work page 1995

[45] [45]

W. Winter. Covering dimension for nuclear C ∗-algebras. J. Funct. Anal., 199(2):535– 556, 2003

work page 2003

[46] [46]

W. Winter. Nuclear dimension and Z-stability of pure C ∗-algebras. Invent. Math., 187(2):259–342, 2012

work page 2012

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