Subquadratic growth and uniform property $\Gamma$

Andrew S. Toms; Ethan Kessinger

REVIEW 2 major objections 5 minor 20 references

Subquadratic growth and uniform property $\Gamma$

T0 review · 2 major / 5 minor · reviewed 2026-07-30 · grok-4.5

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Pith's one-line read Subquadratic dimension growth forces uniform property Γ in ASH C*-algebras, so quadratic growth is the exact threshold where that property can fail.

desk verdict Subquadratic growth forces uniform Γ, and with Toms’ quadratic counterexample this pins the exact geometric threshold; the only real soft spot is the nonsimple Vaccaro–Winter bridge in §2.3. read the letter →

arxiv 2607.23817 v1 pith:5S7LZ5IL submitted 2026-07-26 math.OA

Ethan Kessinger , Andrew S. Toms This is my paper · ORCID

classification math.OA MSC 46L3546L05

keywords uniformpropertyΓASHalgebrasdimensiongrowthToms–WinterconjecturerecursivesubhomogeneousrankavoidancetracialultrapowersC*-algebras

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that unital separable ASH algebras with subquadratic dimension growth always have uniform property Γ, provided they have no nonzero finite-dimensional representations. Uniform property Γ is a tracial divisibility condition that makes the Toms–Winter regularity conjecture hold automatically for simple nuclear C*-algebras. The authors show the same conclusion for the larger class of algebras that admit only local tracial approximation by recursive subhomogeneous building blocks at subquadratic scale. Paired with a recent quadratic-growth AH algebra that fails uniform property Γ, the result pins quadratic growth as the precise geometric cutoff: slower than quadratic, the property holds; at quadratic or faster, it may fail. The work matters because many natural nuclear C*-algebras are ASH, and any counterexample to Toms–Winter in that class would have to live at quadratic growth or above while still enjoying strict comparison—an unlikely pairing.

What carries the argument

Iterated rank-avoidance for self-adjoint sections of endomorphism bundles (and a relative form for RSH pullbacks). The eigenvalue-multiplicity locus of multiplicity at least k in Mr has real codimension k², so when base dimension d is less than k² one can perturb to keep spectral multiplicities below k, yielding the 2-norm estimates that force real rank zero of the uniform tracial ultrapower.

What would settle it

Exhibit a unital separable ASH algebra with no finite-dimensional representations, strictly subquadratic growth in every RSH decomposition, yet whose uniform tracial ultrapower fails real rank zero or fails to be uniformly McDuff; or produce a local tracial RSH approximation at subquadratic scale that still blocks the Vaccaro-style bridge.

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Extended reading notes

Core claim

Every unital separable ASH algebra with subquadratic growth has uniform property Γ whenever it has no nonzero finite-dimensional representations. Equivalently, separable unital C*-algebras with unital locally tracially subquadratic RSH approximation and no nonzero finite-dimensional representations have uniform property Γ. Together with the known quadratic counterexample, quadratic dimension growth is the precise geometric threshold governing potential failure of uniform property Γ.

Load-bearing premise

The step that turns real rank zero of the tracial ultrapower plus local finite nuclear dimension into uniform property Γ must still work for nonsimple algebras via small full projections inside the ultrapower.

Editorial extensions

If this is right

Simple non-elementary ASH algebras of subquadratic growth satisfy the Toms–Winter conjecture even when they fail finite nuclear dimension, Z-stability, and strict comparison.
Quadratic (equivalently 2-norm linear) dimension growth is the sharp geometric boundary for possible failure of uniform property Γ.
Locally tracially subquadratic homogeneous or RSH approximation alone is enough for uniform property Γ when there are no finite-dimensional representations.
Any ASH counterexample to Toms–Winter would need quadratic-or-faster growth paired with strict comparison.

Reading between the lines

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

If every unital simple separable nuclear C*-algebra with a trace is ASH, the result would cover the entire Toms–Winter landscape below the quadratic threshold.
The same codimension-k² geometry suggests that other multiplicity-controlled regularity properties may share the same quadratic cutoff.
Crossed-product applications that already use nonunital homogeneous local models can now run at the optimal subquadratic scale rather than the stricter flat-growth scale.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The punchline is clean: subquadratic ASH/RSH growth (or the corresponding local tracial approximation) implies uniform property Γ whenever there are no nonzero finite-dimensional representations. Paired with Toms’ quadratic AH counterexample, quadratic growth is now the precise scale where uniform Γ can fail. That is a genuine structural clarification for the regularity side of classification.

What is new is the matching positive theorem at the optimal scale, plus the relative rank-avoidance needed for arbitrary RSH pullbacks. The geometry is elementary and well-used: the exact-nullity locus has codimension k² (Lemma 3.1), so when dim X < k² you can iteratively avoid the strata via Jacob’s bundle transversality (Prop. 3.2 and the relative Prop. 7.4). That feeds property (FG), real rank zero of the uniform tracial ultrapower, and then the Vaccaro–Winter bridge. Parallel tracks for unital AH, non-unital homogeneous models, and full RSH are written carefully; the induction through pullbacks does not accumulate error with length. Citations look right and the argument is one-directional, not circular.

The soft spot is exactly where the stress-test points: Lemma 2.5 / Theorem 2.6. The authors claim that Winter’s order-zero almost-divisibility construction works without simplicity once tracial almost divisibility is assumed, and that the small full projections from ⊗(M_{2}⊕M_{3}) plus Vaccaro’s lemma give uniform tracial largeness on every closed face of T(A). I did not find a counter-example in the text, and for the simple non-elementary case that the main corollaries care about the issue evaporates. Still, “inspection shows simplicity is unused” is the least secure sentence in the paper; if the unit of the ultralimit map fails to be 1 uniformly, uniform McDuff and the final Γ computation break. Everything upstream of that bridge looks solid.

This is for people working on Toms–Winter, nuclear dimension, and dimension-growth thresholds. It deserves a serious referee. I would cite the threshold statement and the RSH local theorem. Send it out.

Referee Report

2 major / 5 minor

Summary. The paper proves that unital separable ASH algebras with subquadratic RSH dimension growth and no nonzero finite-dimensional representations have uniform property Γ (Theorem 1.2, via Theorem 7.11). The argument introduces property (FG), shows that it implies real rank zero of the uniform tracial ultrapower, and obtains (FG) from an iterated codimension-$k^2$ rank-avoidance theorem for self-adjoint sections. Relative and nonunital versions handle RSH pullbacks and locally tracial approximation. Real rank zero is then converted into uniform McDuffness and uniform property Γ through a Vaccaro–Winter-type bridge. Together with the quadratic-growth counterexample in [Tom26], this is presented as identifying quadratic growth as the sharp threshold for possible failure of uniform property Γ.

Significance. If the bridge in §2.3 is fully secured, this is a substantial and sharp threshold result. It gives a Toms–Winter corollary for a broad class of ASH algebras and extends Elliott–Niu/Vaccaro-type local approximation results to the optimal subquadratic scale. A particular strength is that the quadratic threshold is derived from an explicit geometric mechanism—the codimension-$k^2$ eigenvalue-multiplicity loci—rather than introduced as an unexplained technical hypothesis. The iterative Jacob transversality argument, its relative RSH form, and the parallel unital, nonunital, and RSH tracks are clearly organized and give quantitative constants that appear internally consistent. The optimality comparison with the quadratic counterexample makes the result especially useful.

major comments (2)

[§2.3, Lemma 2.5] §2.3, Lemma 2.5: the proof uses [Win12, Lemma 5.11] outside its stated simple-algebra setting, with only an “inspection” assertion. The required conclusion is strong: order-zero maps that almost commute with the approximants and satisfy τ(φ_j(1_n))>1−ε_j uniformly for every τ∈T(A). In a nonsimple algebra, trace estimates may degenerate on closed faces, so the precise reason that no faithfulness, fullness, or simplicity step is needed should be proved or cited from a result whose hypotheses explicitly cover this case. This lemma feeds Theorem 2.6 and hence Theorems 7.11 and 1.2. The later fullness and lifting steps in Theorem 2.6 appear standard once this unital uniform-trace estimate is in place.
[§7.3, Corollary 7.7] The passage to a uniform θ_0<θ is not justified by Theorem 7.6 as stated. That theorem gives dim ker(π(b)−λ1)<θ dim π for each fixed finite-dimensional representation, which by itself allows ratios approaching θ across representations. The proof then defines θ_0 as a maximum over the finitely many blocks and asserts the bound for every irreducible representation. Please strengthen Theorem 7.6 to record the blockwise bound for every factor representation—e.g. via the classification of such representations as amplifications of block evaluations—or add the missing induction/classification argument. This estimate is the input to Theorem 7.9.

minor comments (5)

[§3.2 and §4.1] Corollary 3.4 assumes compactness of T, but the displayed proof appears to choose δ uniformly from compactness of X and finiteness of Λ, with no use of compactness of T. In Theorem 4.2, the assertion that T_{n,i} is closed/compact is not justified by the displayed argument. Either prove that assertion or state Corollary 3.4 for all traces, if intended.
[§1 and §5.2] Theorem 1.1 is stated for a unital simple non-elementary AH algebra, while Theorem 5.4 explicitly adds separability and nuclearity. Please harmonize the hypotheses or state that separability and nuclearity are part of the paper’s AH convention.
[§6.2–§6.3] In the nonunital arguments, η_δ(b−λ) is interpreted variously in a multiplier algebra, a corner of A**, and then in A. The intended componentwise functional calculus is understandable, but consistently distinguishing 1_{A**}, s_i, and 1_{M(B_i)} would make the estimates easier to audit.
[Figure 1] The phrase “quadratic or faster / 2-norm linear or faster: uniform Γ may fail” should be identified as an existence/threshold statement, not as a claim that every algebra in those regimes fails uniform Γ. A reference or definition for the asserted equivalence with the 2-norm slow-growth scale would also help.
[References] Several central citations are to recent arXiv preprints, especially [Vac26, Proposition 1.6 and Lemma 2.1], [EN25], and [Tom26]. Please pin the arXiv versions used, since lemma and theorem numbers may change.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: subquadratic geometry independently forces rank-avoidance and property (FG), with self-citation used only as an optimality benchmark.

full rationale

The derivation runs one way from a geometric hypothesis (lim inf dim(X)/r^{2} = 0, or the local tracial RSH analogue) through an independent codimension count (Lemma 3.1: exact-nullity strata have real codimension j^{2}) and Jacob bundle transversality (Prop. 3.2, relative Prop. 7.4) to spectral multiplicity bounds, property (FG), real rank zero of A_U, and finally uniform property Γ via the Vaccaro–Winter bridge (Thm. 2.6). None of these steps defines the conclusion into the hypothesis: the growth scale is an external inductive-limit datum, not fitted from Γ data. The second author’s quadratic counterexample [Tom26] is invoked solely to mark optimality of the threshold and does not enter the positive proofs (Thms. 1.2, 4.4, 6.7, 7.11). Background citations (Jacob, Winter–Zacharias, Phillips RSH, Vaccaro, Elliott–Niu) supply external machinery rather than smuggled ansätze or uniqueness theorems that force the result. No equation reduces to its input by construction; the paper is self-contained against its stated geometric benchmark.

Assumptions & free parameters 0 free parameters · 8 assumptions · 2 invented entities

Load-bearing inputs are standard C*-algebra and bundle geometry facts plus several external theorems (Jacob transversality, Vaccaro’s ultrapower criteria, Winter nuclear-dimension permanence, Elliott–Rørdam embeddings, Phillips RSH structure, Ng–Winter ASH≈RSH). No numerical free parameters are fitted. The paper invents named technical criteria (property (FG), subquadratic RSH growth, locally tracially subquadratic approximation) as definitions, not as new physical entities.

assumptions (8)

standard math Jacob’s bundle-valued avoidance / transversality theorems for Lipschitz-codimension obstacles in paracompact spaces ([Jac13, Thms 3.4 and 5.1]), including the relative form over a closed subset.
Used as the black-box engine for Propositions 3.2 and 7.4; without them the spectral multiplicity perturbation fails.
standard math Self-adjoint matrices with exact nullity j form a smooth submanifold of real codimension j² (Lemma 3.1).
Elementary dimension count that supplies the quadratic threshold; standard linear algebra / stratified geometry.
domain assumption Vaccaro’s criteria relating tracially locally finite nuclear dimension, tracial almost divisibility, real rank zero of the uniform tracial ultrapower, uniform McDuff, and uniform property Γ ([Vac26]), extended by the authors to the nonsimple setting under absence of finite-dimensional representation
Theorem 2.6 is the bridge from rr(A_U)=0 to uniform Γ; the paper inspects simplicity-free steps but still rests on Vaccaro’s framework.
domain assumption Winter–Zacharias permanence of finite nuclear dimension under matrix amplification, hereditary subalgebras, minimal unitization, and finite direct sums ([WZ10]); Winter’s decomposition-rank result for subhomogeneous algebras ([Win04]).
Used to obtain tracially locally finite nuclear dimension from homogeneous or RSH local models (Lemmas 4.3, 6.6, 7.10).
domain assumption Every unital simple separable ASH algebra is an inductive limit of recursive subhomogeneous algebras (Ng–Winter [NW06]); Phillips’ structure theory of RSH algebras ([Phi07]).
Justifies treating ASH via RSH decompositions and the ideal-quotient description of irreducible representations of pullbacks.
domain assumption Elliott–Rørdam embedding of the infinite tensor product of (M₂⊕M₃) into real-rank-zero ultrapowers without finite-dimensional representations ([ER06]).
Supplies the small full projections q_N used to prove tracial almost divisibility in Theorem 2.6.
domain assumption Absence of nonzero finite-dimensional representations (equivalently non-elementarity when simple) is enough to run the ultrapower fullness and almost-divisibility arguments.
Explicit hypothesis of the main theorems; without it the bridge to uniform Γ is not claimed.
ad hoc to paper Property (FG) is a sufficient finite-grid criterion for real rank zero of the uniform tracial ultrapower (Proposition 2.3).
Technical intermediate introduced here (variant of Elliott–Niu property (S)); proved in the paper but definitional to the route chosen.

invented entities (2)

Property (FG) (finite-grid property (S))
purpose: Finite spectral-grid criterion that implies rr(A_U)=0 and is checkable via rank-avoidance estimates.
Named definition internal to the paper’s proof architecture; not an external physical or algebraic object beyond a convenient property.
Unital / nonunital locally tracially subquadratic homogeneous and RSH approximation
purpose: Local approximation hypotheses that organize the AH, Vaccaro-style, and full RSH theorems at the subquadratic scale.
Definitional extensions of prior locally flat / LTH conditions to the optimal growth scale; the content is the inequality dim/r²→0, not a new substance.

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Pith. "Pith review of Subquadratic growth and uniform property $\Gamma$." pith.science (2026). https://pith.science/paper/5S7LZ5IL

@misc{pith2026260723817,
  author       = {Pith},
  title        = {Pith review of: Subquadratic growth and uniform property \(\Gamma\)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5S7LZ5IL}},
  note         = {Machine review of arXiv:2607.23817}
}

abstract

We prove that every unital separable ASH algebra with subquadratic growth has uniform property $\Gamma$ whenever it has no nonzero finite-dimensional representations. When simple and non-elementary, these algebras therefore satisfy the Toms--Winter regularity conjecture despite the fact that they generally fail its three conjecturally equivalent properties. In light of the second author's recent construction of a unital simple separable AH algebra of quadratic growth which fails uniform property $\Gamma$, we conclude that the quadratic dimension growth scale (equivalently, the 2-norm slow dimension growth scale) is the precise geometric threshold governing the potential failure of uniform property $\Gamma$. We also extend recent work of Elliott--Niu and Vaccaro to the optimal subquadratic scale by proving that separable unital $C^*$-algebras with locally tracially subquadratic RSH approximation have uniform property $\Gamma$, provided that they have no nonzero finite-dimensional representations.

Figures

Figures reproduced from arXiv: 2607.23817 by the authors.

**Figure 1.** A coarse taxonomy of geometric dimension-growth regimes. The sequel is devoted essentially to proving the statement under left-hand brace in [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

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Works this paper leans on

20 extracted references · 2 canonical work pages

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Pith tools

Reviewed July 30, 2026 · model on record in the stance chip above.

[1] [1]

L. G. Brown and G. K. Pedersen , title =. J. Funct. Anal. , volume =

[2] [2]

Castillejos and S

J. Castillejos and S. Evington and A. Tikuisis and S. White , title =. Int. Math. Res. Not. IMRN , volume =. 2022 , doi =

2022

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Castillejos and S

J. Castillejos and S. Evington and A. Tikuisis and S. White and W. Winter , title =. Invent. Math. , volume =

[4] [4]

G. A. Elliott and Z. Niu , title =. 2025 , eprint =

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G. A. Elliott and M. R. Perturbation of. Operator

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Jacob, Beno\^it , TITLE =. Canad. Math. Bull. , FJOURNAL =. 2013 , NUMBER =. doi:10.4153/CMB-2011-158-8 , URL =

work page doi:10.4153/cmb-2011-158-8 2013

[7] [7]

F. J. Murray and J. von Neumann , title =. Ann. of Math. (2) , volume =

[8] [8]

Ng, Ping Wong and Winter, Wilhelm , TITLE =. C. R. Math. Acad. Sci. Soc. R. Can. , FJOURNAL =. 2006 , NUMBER =

2006