Schubert Calculus and uniform property $\Gamma$

Andrew S. Toms

REVIEW 2 major objections 4 minor 1 cited by

Schubert Calculus and uniform property $\Gamma$

T0 review · 2 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper constructs a unital simple separable AH C*-algebra without uniform property Γ, answering a listed open problem and linking the failure to quadratic dimension growth.

desk verdict New mechanism (Thom–Porteous/Schubert) yielding a simple nuclear C*-algebra without uniform property Γ; construction looks sound, but the trace-comparison bridge has a hypothesis gap that must be checked. read the letter →

arxiv 2606.12188 v2 pith:Q6IY4HW4 submitted 2026-06-10 math.OA math.KT

Andrew S. Toms This is my paper · ORCID

classification math.OAmath.KT MSC 46L0546L3546L8014N1514C1755R25

keywords AHalgebrauniformpropertyΓThom–PorteousclassdegeneracylociSchubertcalculusquadraticdimensiongrowthtracialcompletionprojectioncomparison

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 proves that uniform property Γ can fail for simple, separable, unital, nuclear C*-algebras. The example is an AH algebra built from a new topological obstruction: Thom–Porteous classes force every bundle map between two equal-rank bundles to vanish at some point on the base space, and Schubert calculus shows this forced vanishing survives an inductive limit. The obstruction travels into the uniform tracial completion, where two projections have equal traces yet cannot be Murray–von Neumann equivalent. If accepted, the construction resolves an explicit open question and identifies quadratic dimension growth as the geometric threshold beyond which uniform property Γ may fail.

What carries the argument

The engine is the Thom–Porteous class of a virtual bundle F−E: for equal-rank bundles of rank d and integer s, Δ_s(E,F) = det(c_{s+i−j}(F−E)) lies in H^{2s²}(X), and nonvanishing of Δ_d forces every bundle map E→F to have a zero fiber. The propagation lemma uses the rectangular resultant identity for supersymmetric Schur functions — s_{(n^n)}(U−V)=∏_{ξ∈U,ν∈V}(ξ−ν) for equal-size alphabets — to show the square Thom–Porteous class at stage i+1 contains a distinguished, non-cancellable term built from the stage-i class and a power of the difference of two line-bundle Chern classes.

What would settle it

Carry out the induction for the first nontrivial rank (d=2): compute Δ_2(c(q_2−p_2)) on X_2 = Gr(2,4)×Gr(2,4)×CP^8 and check whether the predicted top-degree term involving the eighth power of the line-bundle class is nonzero; if it vanishes, the rectangular-resultant propagation step is wrong.

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Extended reading notes

Core claim

The central discovery is a mechanism for manufacturing C*-algebras whose uniform tracial completion contains equal-trace, non-equivalent projections. At every stage of the inductive system the author places two vector bundles, one tautological and one trivial, of equal rank d over the Grassmannian Gr(d,2d); the Thom–Porteous class of their difference lives in the top cohomological degree 2d² and is nonzero, so any bundle map between them must have a zero fiber. A rectangular-resultant identity for supersymmetric Schur functions propagates this 'total degeneracy-forcing' property from each stage to the next. In the limit the two projections have equal traces because ranks agree at finite stag

Load-bearing premise

The load-bearing premise is the external theorem that a unital simple separable nuclear C*-algebra with uniform property Γ compares projections in its uniform tracial completion by the values of all tracial states; if that theorem fails, the constructed algebra still has exotic projections but the conclusion 'no uniform property Γ' would not follow.

Editorial extensions

If this is right

If correct, it settles the open problem of whether every simple separable unital nuclear C*-algebra has uniform property Γ in the negative.
The constructed algebra demonstrates that traces need not compare projections in the uniform tracial completion even when the algebra is simple and nuclear, so the comparison theorem for uniform Γ is sharp.
Because recent results force stable rank one for simple AH algebras with uniform Γ, the example must have stable rank at least two, and the Thom–Porteous mechanism provides a new route to higher stable rank phenomena.
The cohomological degree of the obstruction scales with the square of the rank, matching quadratic dimension growth; the paper frames this as the threshold at which uniform Γ can first fail.
The same construction, with point-evaluation summands inserted slowly, gives simple examples, so the failure is not an artifact of non-simplicity.

Reading between the lines

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

If the paper's threshold principle is right, one should expect algebras with subquadratic dimension growth to always have uniform Γ; a direct proof of that converse would complete the taxonomy.
The equal-rank Thom–Porteous technique may apply to other regularity properties (Z-stability, strict comparison) whose standard obstructions are K-theoretic; forced degeneracy of bundle maps could yield new counterexamples or sharpen existing thresholds.
The role of the rectangular resultant identity suggests that other resultant formulas in Schubert calculus could translate into inductive limit constructions with prescribed dimension growth, giving a general machine for pathological C*-algebras.
The non-simple intermediate algebra is asserted (in later work) to lack tracial almost divisibility; if that holds, the method produces algebras with no nonzero homomorphism from M_2 despite abundant constant-trace projections, a stronger pathology than absence of uniform Γ.

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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

Colleague,

Here's my read on Toms's new preprint. If the main theorem holds, it resolves an explicit open question: there is a simple separable unital nuclear AH algebra without uniform property Γ. The genuinely new thing is the engine. Previous Villadsen-type constructions used Chern classes to obstruct large trivial subbundles; this paper uses Thom–Porteous degeneracy classes to force every bundle map between equal-rank bundles to vanish somewhere, then propagates the obstruction through an inductive system with a Schubert calculus computation. I went carefully through Lemma 1 and the simplicity argument in Section 7. The Schubert identity is standard, the degree-counting cancellation is handled cleanly, and the scheduled point-evaluation trick for simplicity is well executed. The construction does what it claims, and the quadratic dimension growth is really coming out of the cohomological degree of the obstruction.

The soft spot is precisely where the reader and the stress-test point: the final trace-theoretic bridge. Section 6 opens by citing an Evington–Tikuisis theorem for a 'type II_1 factorial tracially complete C*-algebra.' That is a strong hypothesis, and the paper never checks that A^u or B^u is factorial. §6.3 applies the comparison conclusion to the non-simple algebra A, and §7.5 applies it to the simple limit B without proving factoriality of B^u. If the comparison theorem in [CCE+23]/[CETW22] is known to hold without factoriality—say for all tracially complete algebras with uniform property Γ—then the gap is cosmetic and a referee can fix it with a precise citation. If factoriality is essential, the contradiction in §7.5 collapses and Theorem 1 is unsupported. This is not a minor typo; it is the load-bearing wall between the bundle obstruction and the failure of uniform property Γ. But it is also the kind of thing that can be settled by reading two preprints, not by redoing the geometry.

I would send this to a serious referee. The construction deserves scrutiny and the trace bridge needs a definite check. If the bridge holds, this is a major result in the area. If not, the author will need to add a lemma proving projection comparison for his specific tracial completions. Either way, it is not a desk-reject. The citation pattern is fine; the heavy self-citations are to his earlier Villadsen-type examples, which are the right background.

Referee Report

2 major / 4 minor

Summary. The paper constructs a unital, simple, separable, nuclear AH C*-algebra without uniform property Gamma, answering Question XIX of [STW25] in the negative. The construction builds an inductive system of homogeneous C*-algebras over products of Grassmannians and projective spaces. The new ingredient is an equal-rank Thom–Porteous obstruction: a nonvanishing rectangular Schur class forces every bundle map between two equal-rank bundles to vanish somewhere. A Schubert-calculus lemma (Lemma 1) propagates this nonvanishing across the inductive system using the rectangular resultant identity for supersymmetric Schur functions. The resulting compatible projections P and Q have equal values on all traces but are shown not to be Murray–von Neumann equivalent in the uniform tracial completion. A point-evaluation modification of the inductive system forces simplicity while preserving the obstruction, yielding Theorem 1.

Significance. If correct, this is a major result: it is the first example of a simple separable nuclear unital C*-algebra without uniform property Gamma, resolving a problem explicitly posed in [STW25]. The method is genuinely novel, moving beyond Villadsen-type Chern class obstructions by using Thom–Porteous classes and supersymmetric Schur functions. The paper also contains a careful and self-contained treatment of the Schubert-calculus core, and the simplicity argument via scheduled point evaluations is elegant. The geometric interpretation in terms of quadratic dimension growth is suggestive, though the paper correctly treats the threshold statement as a guiding principle rather than a fully proved theorem. The main fragility is the external trace-comparison theorem used to convert non-equivalence of equal-trace projections into failure of uniform property Gamma; this is the load-bearing point that needs attention.

major comments (2)

[§6 and §7.5] The trace-comparison bridge is not adequately justified. Section 6 opens by citing a theorem of Evington–Tikuisis about a "type II_1 factorial tracially complete C*-algebra" [ET26], and then immediately applies the conclusion to the non-simple algebra A of Section 4 and later to the simple algebra B. No factoriality of A^u or B^u is established, and in fact T(A) and T(B) have many extreme points, so the uniform tracial completions are far from factorial. If the cited theorem is restricted to factorial algebras, Theorems 4 and 6 do not follow. The paper must either state the precise comparison theorem from [CETW22] or [CCE+23] that applies to the present algebras, verify its hypotheses explicitly, or prove the needed comparison result directly. This is not a cosmetic issue: the non-equivalence of equal-trace projections is the only mechanism preventing uniform property Gamma.
[§6.3, proof of Theorem 4] The one-sentence inference in Theorem 4 is a non sequitur as written: "The algebra A is the unital separable nuclear stably finite inductive limit constructed in Section 4, and it has no finite-dimensional representations by construction. It follows that projections in matrix amplifications of A^u are compared by their values on the designated traces coming from T(A) [CETW22, CCE+23]." The listed properties do not by themselves imply trace comparison unless the cited references contain a theorem with exactly these hypotheses. Please quote the theorem being invoked and check every hypothesis. In particular, [ET26] as stated in the text does not apply to non-factorial completions.

minor comments (4)

[§1] The introduction says "The algebras we construct here will satisfy (3) for f(x)=x^2," which is immediate from the construction. The stronger threshold claim that every AH presentation of the limit has positive quadratic dimension growth is not proved and should be labeled as a conjecture or forthcoming work, not as an established property of the constructed algebra.
[§3.3 and §5] The notation for Schur classes is inconsistent: s_{(d^d)}(S) appears as s^{(dd)} or s_{(d_i^{d_i})} in different places. Please standardize the notation, e.g., always use s_{(d^d)}.
[References] Minor reference typos: [ES24] is listed as "arxiv:2407:16612" and [ET26] as "arxiv:2604:24206"; the colons should be dots (arXiv:2407.16612, arXiv:2604.24206).
[§7.4, Lemma 5] The "diagonal subnet argument" is a little terse; it would help the reader to specify the directed set and why the limits of restrictions are compatible, though the argument is standard and correct.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the construction is self-contained and the load-bearing external theorems are used as premises, not as conclusions.

full rationale

The paper's main derivation is an inductive topological construction culminating in non-equivalent projections with equal traces in the uniform tracial completion. The key external input is the theorem that simple separable nuclear unital C*-algebras with uniform property Γ compare projections in matrix amplifications of the uniform tracial completion by designated traces. This theorem is cited to [CETW22], [CCE+23], and [ET26]; it is used in the contrapositive direction (failure of trace comparison implies failure of uniform property Γ), not assumed as the conclusion. The paper never fits parameters to target data, and no 'prediction' is equivalent to an input by construction. The equal-trace property of P and Q follows from the explicit constant-rank equality rank(p_i)=rank(q_i), while their non-equivalence follows from the independently proven Thom–Porteous degeneracy-forcing result. Even if the trace-comparison theorem were inapplicable or false, that would be a correctness gap, not circularity. The author's self-citations appear only in motivational examples and surveys, not as load-bearing steps in the proof. The quadratic-dimension-growth narrative is a consequence of the chosen parameters (j_i=2d_i^2), not an input used to define the obstruction. Therefore the derivation chain is not circular.

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

No free parameters are fitted to data: the integers d and the sequences (ε_i) can be chosen arbitrarily within the stated inequalities, and no numerical constants are tuned to obtain the result. The argument relies on standard geometric facts (Thom-Porteous support, resultant identity, Künneth theorem, splitting principle) and on the cited comparison theorem for uniform tracial completions.

assumptions (7)

standard math The Thom-Porteous support property: if Δ_s(E,F) ≠ 0 then D_s(T) ≠ ∅ for every bundle map T.
Used in Proposition 2 and throughout; cited to Fulton, Intersection Theory, Ch. 14.
standard math Rectangular resultant identity for supersymmetric Schur functions: s_{(n^n)}(U-V) = ∏_{ξ∈U,ν∈V}(ξ-ν) when |U|=|V|=n.
Used in Lemma 1, Step 2; cited to Macdonald I.3 Example 9.
domain assumption Comparison by traces in the uniform tracial completion: a simple separable nuclear unital C*-algebra with uniform property Γ has projections in matrix amplifications of A_u compared by values of all traces.
Bridges topological obstruction to failure of uniform Γ; cited to [CETW22, CCE+23, ET26].
standard math Künneth formula for integral cohomology of products of spaces with torsion-free cohomology.
Used in Lemma 1, Step 3 to show the distinguished product is nonzero.
standard math Splitting principle for complex vector bundles: there is an injective pullback ρ* to a flag space where bundles split into line bundles.
Used in Lemma 1 Step 1.
domain assumption Kaplansky density theorem for the uniform tracial completion: the unit ball of A is dense in the unit ball of A_u.
Used in Theorem 3 and Theorem 6 to approximate partial isometries by finite-stage contractions; cited to [CCE+23].
standard math For k sufficiently large, the trivial rank-d bundle embeds in S^⊕k over Gr(d,2d).
Used in Section 4 to regard p_1 as a projection in M_k(A_1); stated without proof but standard from vector-bundle obstruction theory.

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Pith. "Pith review of Schubert Calculus and uniform property $\Gamma$." pith.science (2026). https://pith.science/paper/Q6IY4HW4

@misc{pith2026260612188,
  author       = {Pith},
  title        = {Pith review of: Schubert Calculus and uniform property $\Gamma$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q6IY4HW4}},
  note         = {Machine review of arXiv:2606.12188}
}

abstract

We construct a simple, separable, unital, nuclear C$^*$-algebra without uniform property $\Gamma$. The construction is based on a new topological obstruction arising from the Thom-Porteous theory of degeneracy loci. Constructions of pathological nuclear C$^*$-algebras over the past 30 years have used Chern class calculations introduced by Villadsen to obstruct the existence of large trivial subbundles. Here, by contrast, we use determinantal Schur classes to force every bundle map between certain equal-rank vector bundles to vanish somewhere on the base space. A quadratic Schubert calculus computation shows that this obstruction can persist across an inductive system and ultimately obstructs the comparison of projections by traces in the uniform tracial completion. The relevant Thom-Porteous classes live in degree proportional to the square of the forced rank loss, which in turn forces dimension growth of the same order in the constituent homogeneous C$^*$-algebras of our example. This identifies a new geometric threshold in the structure theory of nuclear C$^*$-algebras, linking the presence or absence of uniform property $\Gamma$ to quadratic dimension growth.

Figures

Figures reproduced from arXiv: 2606.12188 by the authors.

**Figure 1.** A coarse taxonomy of geometric dimension-growth regimes. Every region of the diagram above is populated by simple separable unital nuclear C∗ -algebras: • Slow: These are precisely the AH algebras satisfying the Toms-Winter regularity conjecture, and are classified by the Elliott invariant (this is the work of many, but a good modern reference is [CGS+23]). • Linear: This region contains Villadsen’s algebras of the… view at source ↗

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Cited by 1 Pith paper

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

Subquadratic growth and uniform property $\Gamma$
math.OA 2026-07 accept novelty 7.0 of 10

Unital separable ASH algebras with subquadratic growth and no nonzero finite-dimensional representations have uniform property Γ, making quadratic growth the precise threshold for its failure.

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Works this paper leans on

3 extracted references · 3 linked inside Pith · cited by 1 Pith paper

[2]

[Con76] A

arXiv:2307.06480. [Con76] A. Connes. Classification of injective factors. Cases II 1, II ∞, III λ,λ̸= 1.Ann. of Math. (2), 104(1):73–115,

arXiv
[2023]

[CET+21] J

arXiv:2310.20594. [CET+21] J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter. Nuclear dimension of simple C∗-algebras.Invent. Math., 224(1):245–290,

arXiv
[2025]

[Tom06] A

arXiv:2506.10902. [Tom06] A. S. Toms. Flat dimension growth for C ∗-algebras.J. Funct. Anal., 238(2):678–708,

arXiv

Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [2]

[Con76] A

arXiv:2307.06480. [Con76] A. Connes. Classification of injective factors. Cases II 1, II ∞, III λ,λ̸= 1.Ann. of Math. (2), 104(1):73–115,

arXiv

[2] [2023]

[CET+21] J

arXiv:2310.20594. [CET+21] J. Castillejos, S. Evington, A. Tikuisis, S. White, and W. Winter. Nuclear dimension of simple C∗-algebras.Invent. Math., 224(1):245–290,

arXiv

[3] [2025]

[Tom06] A

arXiv:2506.10902. [Tom06] A. S. Toms. Flat dimension growth for C ∗-algebras.J. Funct. Anal., 238(2):678–708,

arXiv