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On almost commuting unitary matrices

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Pith's one-line read This paper proves an explicit dimension-independent bound: two unitary matrices with zero winding number and commutator norm δ are δ^{1/30}-close to a commuting pair.

desk verdict Solid, significant paper with a real sign error in Appendix A that invalidates the printed proof of Theorem 1.2; probably repairable, but the lemma needs fixing first. read the letter →

arxiv 2510.03674 v3 pith:S46B6RNW submitted 2025-10-04 math.OA math.FAmath.SP

Adam Dor-On , Lucas Hall , Ilya Kachkovskiy This is my paper · ORCID
classification math.OAmath.FAmath.SP MSC 46L0547A60
keywords almostcommutingunitarieswindingnumberisospectralinvariantapproximantsoperatorLipschitzfunctionsspectralgapdimension-independentboundC*-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

The paper settles the quantitative version of a long-standing question: given two unitary matrices whose commutator norm is small and whose winding-number obstruction vanishes, how close are they to a pair of exactly commuting unitaries? The answer is an explicit, dimension-independent estimate: the distance is at most a universal constant times the 1/30 power of the commutator norm. The proof combines a quantitative isospectral homotopy lemma, which replaces the pair by a path with controlled commutators at the cost of a 2/5 exponent, with a commutator-controlled transfer of a gap-opening construction from C*-algebra theory, costing a further 1/12 exponent; balancing these two losses yields 1/30. A sympathetic reader should take away that the obstruction-vanishing case of the unitary almost-commuting problem now has effective bounds, where previously existence results gave no explicit dependence of ε on δ.

What carries the argument

The load-bearing object is the winding number of the curve det(t·uv+(1−t)·vu), identified with an isospectral invariant and shown to vanish precisely when a commuting approximation is possible. The proof then runs on two quantitative machines: a homotopy lemma (Lemma 1.3) that, under vanishing invariant, straightens v to the identity through a path whose commutator with u never exceeds C||[u,v]||^{2/5}; and a transfer (Theorem 1.4) of gap-opening, spectral-compression and dimension-reduction arguments from C*-algebras into commutator-controlled matrix statements, costing a 1/12 exponent. The estimate on the operator-Lipschitz norm of a branch of the argument on a punctured circle (Appendix A

What would settle it

One concrete falsifier: exhibit unitaries u_n,v_n with zero winding number, δ_n = ||[u_n,v_n]|| → 0, such that the distance from (u_n,v_n) to any commuting pair, divided by δ_n^{1/30}, tends to infinity. No such sequence is known, and the theorem asserts none can exist.

Watch

Extended reading notes

Core claim

The central claim is Theorem 1.2: there is an absolute constant C such that whenever u,v ∈ U(n) have winding number w(u,v)=0, there exist commuting u′,v′ ∈ U(n) with ||u−u′||+||v−v′|| ≤ C||[u,v]||^{1/30}. The vanishing of the winding number—shown here to be equivalent to the isospectral invariant—is exactly the condition that removes the known topological obstruction; under it, the paper proves that the obstruction is not only absent but quantitatively harmless.

Load-bearing premise

The whole estimate leans on the claim that the argument function on the unit circle minus a small arc of size ρ is operator-Lipschitz with norm proportional to 1/ρ; if the true rate were 1/ρ^2, the final 1/30 exponent would collapse.

Editorial extensions

If this is right

  • For any zero-winding pair, the distance to a commuting pair goes to zero as a fixed power of the commutator norm, independent of n; before this work only a qualitative existence was known.
  • The ε–δ relation in the unitary case is explicit: to be ε-close to commuting unitaries it suffices that the commutator norm be below (ε/C)^{30}.
  • The homotopy lemma gives path-length-independent commutator control, so the bound is stable under continuous deformations of v within the zero-winding class.
  • The equivalence of the winding number and the isospectral invariant is proved directly, making the two invariants interchangeable for quantitative estimates.
  • The dimension reductions show that after creating a microscopic spectral gap in an amplified space, one can descend back to the original space with only polynomial losses, so the technique applies to matrix algebras rather than to abstract C*-algebra quotient arguments.

Reading between the lines

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

  • The exponent 1/30 is a by-product of balancing two losses; the paper's structure suggests 1/12 is the genuine analytic bottleneck, and a sharper estimate for the argument-branch operator-Lipschitz constant would likely improve the exponent.
  • The methods should extend to Schatten p-norms or to pairs of unitaries in finite von Neumann algebras with a trace, where the winding-number condition would need an appropriate analytic replacement.
  • One could test near-sharpness numerically: sample random zero-winding pairs with small commutators, compute their distance to the nearest commuting pair, and fit the exponent; this would show how far 1/30 is from the true rate.
  • The amplification-and-descend scheme is a template for making other nonconstructive C*-algebra existence proofs effective; the same 'gap opening in amplified space, then two-step dimension reduction' pattern may apply to tuples of almost-commuting unitaries.
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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

1 major / 5 minor

Summary. The paper proves a quantitative version of the Halmos-type question for pairs of unitary matrices whose winding-number obstruction vanishes: for such u,v ∈ U(n) there exist commuting unitaries u',v' with ||u−u'||+||v−v'|| ≤ C||[u,v]||^{1/30}. The proof has two main ingredients: a quantitative isospectral homotopy lemma (Lemma 1.3), which connects v to the identity while keeping the commutator with u of size O(||[u,v]||^{2/5}), and a quantitative Lin-type theorem (Theorem 1.4), which converts a small-commutator path of one unitary to the identity into a commuting pair at distance O(δ^{1/12}). Theorem 1.4 is proved by opening a spectral gap in an amplified pair, applying a spectral-gap almost-commutation lemma (Lemma 4.3), and then descending back to the original space in two dimension-reduction steps. The paper also gives a self-contained proof of the equivalence between the winding number and the isospectral invariant, and includes a survey of the needed operator-Lipschitz machinery.

Significance. If established, Theorem 1.2 would resolve the long-standing quantitative question of whether approximately commuting unitary matrices with vanishing winding number are nearly commuting, with an explicit power bound. The paper is well structured and the two-step strategy is appealing: the quantitative homotopy lemma and the amplification/dimension-reduction argument are nontrivial and potentially influential. The proof also provides a useful self-contained treatment of the relation between the winding-number and isospectral invariants. The main issue is that the proof of Lemma A.1, which is load-bearing for the final exponent, is flawed as written; however, the flaw appears local and repairable, and the overall approach is likely correct.

major comments (1)
  1. [Appendix A, Lemma A.1; Eq. (A.4)] Equation (A.4) contains a sign error invalidating Lemma A.1. The paper correctly states in (A.2) that arg_{-}(z)=arg_{+}(z)−π(1−sign y). But the φ-term in (A.4), φ(z)(arg_{+}(z)+π(1−s_ρ(z))), on T_ρ∩suppφ with y<0 gives arg_{+}(z)+2π, whereas arg_{-}(z)=arg_{+}(z)−2π: a difference of 4π. Hence arg_ρ≠arg_{-} on the lower arc; the subsequent equality claim is false. This matters because Lemma 4.3 uses the ρ^{-1} bound to get loss ρ^{-1/2}δ^{1/2}, which fixes ε=δ^{1/3}, γ=Cδ^{1/3}, and the final δ^{1/12} (hence δ^{1/30}). The likely fix is a sign change to φ(z)(arg_{+}(z)−π(1−s_ρ(z))), after which the equality and OL bound must be re-verified. If only a ρ^{-2} bound held, the final exponent would degrade.
minor comments (5)
  1. [Appendix A, (OL3)] The parenthetical 'for F replaced by C' appears to be a typo; it should read 'for C replaced by F'.
  2. [Appendix A, proof of Lemma A.1] In the norm estimate after (A.4), the symbol 's_z' is undefined; it should be 's_ρ'.
  3. [Lemma 4.3] The notation 'arg u = 1/2π arg_ρ u' is confusing. It should refer explicitly to the corrected extension arg_ρ from Lemma A.1, or to the branch arg_{-}.
  4. [Throughout] There are several typographical slips: 'acheive' should be 'achieve' in §1.1, and 'Propostion' should be 'Proposition' in the proof of Proposition 3.1.
  5. [Proof of Theorem 1.2] The application of Theorem 1.4 via Lemma 1.3 implicitly swaps the roles of u and v; this should be stated explicitly for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the main theorem is assembled from independent external results and in-paper lemmas; the Appendix A sign concern is a correctness issue, not circularity.

full rationale

The derivation chain of Theorem 1.2 is not circular. Theorem 1.2 is obtained by combining Corollary 3.4 (equivalence of the winding number and isospectral invariants), Lemma 1.3 (quantitative isospectral homotopy), and Theorem 1.4 (commuting approximants from a path with small commutators). Corollary 3.4 is proved in the paper from Lemma 1.3, additivity, and an explicit computation for Voiculescu's unitaries; it is not assumed as an input. Lemma 1.3 is proved from the in-paper Proposition 2.12 and operator-Lipschitz estimates; Theorem 1.4 is proved by gap-opening, amplification, and two dimension-reduction steps. The only author-overlap citation is [21], used in Proposition 4.1 as the quantitative Lin theorem. That is a published, parameter-free external result whose assumptions do not include the target theorem, so it counts as independent support rather than a smuggled conclusion. No fitted parameter is renamed as a prediction, no uniqueness theorem from the authors' prior work is used to forbid alternatives, and no definition of an invariant is chosen so that the main conclusion holds by construction. Internal verification note: Lemma A.1 is load-bearing for Lemma 4.3, and as printed Eq. (A.4) appears sign-inconsistent with Eq. (A.2) for y<0, giving arg_+ + 2π where arg_− = arg_+ − 2π. This is a real correctness/verification concern, but it is not circularity: the ρ^{-1} operator-Lipschitz bound is an analytical input to the theorem, not a restatement or fitted version of the theorem's conclusion.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central claim rests on standard operator-Lipschitz/spectral theory and three external theorems ([21] Prop 4.1, [26] Prop 4.5, [5,11,12] invariants). No free parameters are fitted; no new entities are introduced. The only author-overlap citation is [21] (Kachkovskiy), used as an independent black box whose statement does not include the unitary-with-obstruction result.

assumptions (5)
  • standard math Operator Lipschitz functions satisfy (OL1)–(OL8), including commutator-Lipschitz property (OL6).
    Used throughout Section 2 and Appendix A to control spectral projections and smooth functional calculus when perturbing unitaries.
  • domain assumption Quantitative Lin theorem: for self-adjoint (or normal) t,s with ||[t,s]||=ε, there are commuting t',s' with distance ≤ C ε^{1/2} (Proposition 4.1, from [21]).
    Invoked repeatedly in Section 4 as a black box to produce commuting approximants; the main theorem inherits this external theorem.
  • standard math Winding number w(u,v) is a homotopy invariant within {||[u,v]||<2} and equals the isospectral invariant (Corollary 3.4, using [11,12,5]).
    Bridges the hypothesis w(u,v)=0 to the isospectral setting used by Lemma 1.3.
  • domain assumption Gap-opening formula for diag(u,u*) via rotation (Proposition 4.5, from [26]) opens a spectral gap of size ε with commutator loss factor 2.
    Used in §4.2 to construct amplified almost-commuting pairs with a spectral gap.
  • standard math Spectral calculus and projection perturbation facts (Propositions 2.5, 2.8, 2.11, 2.12).
    Background used throughout, mostly proved in paper.

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Pith. "Pith review of On almost commuting unitary matrices." pith.science (2026). https://pith.science/paper/S46B6RNW

@misc{pith2026251003674,
  author       = {Pith},
  title        = {Pith review of: On almost commuting unitary matrices},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/S46B6RNW}},
  note         = {Machine review of arXiv:2510.03674}
}
read the original abstract

A question going back to Halmos asks when two approximately commuting matrices of a certain kind are close to exactly commuting matrices of the same kind. It has long been known that there is a winding number obstruction for approximately commuting unitary matrices to be close, in a dimension-independent way, to genuinely commuting unitary matrices. In this paper, under the vanishing of the said obstruction, we obtain effective bounds for the distance to commuting unitary matrices in terms of the commutator of the original matrices.

Figures

Figures reproduced from arXiv: 2510.03674 by the authors.

Figure 1
Figure 1. Proof. Let r2k−1 := PkQk−1Pk, r2k := PkQkPk, s2k := QkPkQk, s2k+1 := QkPk+1Qk. We visualize these operators according to [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Noncommutative “Change of Variables,” see Remark 4.2. In this section we prove Theorem 1.4. By applying the quantitative version of Lin’s theorem we first reduce the problem to the question of opening a uniform gap in the spectrum of one of the unitaries. However, we do not apply this reduction as-is. In order to obtain sharper bounds we apply this reduction to sufficiently large ampliated pairs of matrices, and a p… view at source ↗
Figure 3
Figure 3. When the obstruction vanishes, we decouple red edges. Proof. Without loss of generality, one can assume that the spectral gap of u is around −1. Let x := 1 2π arg u = 1 2π argρ u as considered in Appendix A. From Lemma A.1 and (OL6), we have ∥[x, v]∥ ≤ Cρ−1 ∥[u, v]∥. Apply the last claim of Proposition 4.1 with t = x and s = v, thus arriving to a commuting pair of x ′ and v ′ . Since the function x 7→ e ix is operat… view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: A “high order” spectral decomposition for u ⊕ u ′ path(ε) into the unitary u ∈ A and two semicircular arcs dividing u ′ path(ε). After slight modifications (see equations (4.5) and (4.10) below), the spectral projections for the black, red, and blue arcs implement dime…

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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

  1. $\mathbb{Z}^2$ is flexibly stable in the operator norm

    math.OA 2026-07 conditional novelty 7.0 of 10

    Z² is flexibly stable in the operator norm: almost-commuting unitary pairs admit commuting corrections after an o(d)-dimensional enlargement, making flexible stability strictly weaker than stability for the first time.

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Works this paper leans on

28 extracted references · 2 linked inside Pith · cited by 1 Pith paper

  1. [21]

    Kachkovskiy I., Safarov Y.,Distance to normal elements in real rank zeroC ∗-algebras, J. Amer. Math. Soc. 29 (2016), no. 1, 61 – 80

    2016

  2. [1]

    Aleksandrov A., Peller V., Potapov D., Sukochev F.,Functions of normal operators under perturbations, Adv. Math. 226 (2011), No. 6, 5216 – 5251

    2011

  3. [2]

    Aleksandrov A., Peller V.,Estimates of operator moduli of continuity, J. Funct. Anal. 261 (2011), No. 10, p. 2741–2796; arXiv:1104:3553

    2011

  4. [3]

    4 (2012), p

    Aleksandrov A., Peller V.,Operator and commutator moduli of continuity for normal operators, Proceedings of the London Mathematical Society 105, No. 4 (2012), p. 821–851

    2012

  5. [4]

    Berg I.,On approximation of normal operators by weighted shifts, Michigan Math. J. 21 (1975), no. 4, 377 – 383

    1975

  6. [5]

    Bratteli O., Elliott G., Evans D., Kishimoto A.,Homotopy of a pair of approximately commuting unitaries in a simpleC ∗-algebra, J. Func. Anal. 160 (1998), no. 2, 466 – 523. ON ALMOST COMMUTING UNITARY MATRICES 23

    1998

  7. [6]

    Brown L., Douglas R., Fillmore P., Unitary equivalence modulo the compact operators and extensions ofC∗-algebras, Proc. Conf. Operator Theory (Dalhousie Univ., Halifax, N. S. 1973), Lecture Notes in Mathematics 345, Springer, 1973, p. 58 – 128

    1973

  8. [7]

    Davidson, K,Almost commuting Hermitian matrices, Math. Scand. 56 (1985), p. 222–240

    1985

Show all 28 references
  1. [8]

    I, 317–366

    Davidson K., Szarek S.,Local operator theory, random matrices, and Banach spaces, Handbook of the geometry of Banach spaces, Vol. I, 317–366. North-Holland Publishing Co., Amsterdam, 2001

    2001

  2. [9]

    46, Conference Board of Mathematical Sciences, Washington, DC, 1981

    Effros E.,Dimensions andC ∗-algebras, CBMS Regional Conference Series in Mathematics, vol. 46, Conference Board of Mathematical Sciences, Washington, DC, 1981

    1981

  3. [10]

    Eilers S., Loring T., Pedersen G.,Morphisms of extensions ofC ∗-algebras: pushing forward the Busby invariant, Adv. Math. 147 (1999), no. 1, 74 – 109

    1999

  4. [11]

    Exel R., Loring T.,Invariants of almost commuting unitaries, J. Funct. Anal. 95 (1991), no. 2, 364 – 376

    1991

  5. [12]

    Exel R.,The soft torus and applications to almost commuting matrices, Pacific J. Math. 160 (1993), no. 2, 207 – 217

    1993

  6. [13]

    Reine Angew

    Friis P., Rørdam M.,Almost commuting self-adjoint matrices — a shortproof of Huaxin Lin ’s theorem, J. Reine Angew. Math. 479 (1996), 121–131

    1996

  7. [14]

    Filonov N., Safarov Y.,On the relation between the operator and its self-commutator, J. Funct. Analysis 260 (2011), 2902 – 2932

    2011

  8. [15]

    Operator Theory 40 (1998), no

    Gong G., Lin H.,Almost multiplicative morphisms and almost commuting matrices, J. Operator Theory 40 (1998), no. 2, 217 – 275

    1998

  9. [16]

    R.,Some unsolved problems of unknown depth about operators in Hilbert space, Proc

    Halmos P. R.,Some unsolved problems of unknown depth about operators in Hilbert space, Proc. Roy. Soc. Edinburgh Sect., A 76 (1976), 67–76

    1976

  10. [17]

    Hastings M.,Making almost commuting matrices commute, Comm. Math. Phys. 291 (2009), No. 2, p. 321–345

    2009

  11. [18]

    Hastings M., Loring T.,Almost commuting matrices, localized Wannier functions, and the quantum Hall effect, J. Math. Phys. 51 (2010), no. 1, 015214

    2010

  12. [19]

    Herrera D.,On Hastings’ approach to Lin ’s Theorem for Almost Commuting Matrices, preprint (2020),https: //arxiv.org/abs/2011.11800

    2020 arXiv

  13. [20]

    32, Springer, 1995

    Kato T.,Perturbation Theory for Linear Operators, 2nd edition, Classics in Mathematics, vol. 32, Springer, 1995

    1995

  14. [22]

    Operator Algebras and Their Applications

    Lin H.,Almost commuting selfadjoint matrices and applications, in “Operator Algebras and Their Applications”, Fields Inst. Commun. 13 (1997), p. 193 – 233

    1997

  15. [23]

    Lin H.,Almost commuting unitaries and classification of purely infinite simpleC ∗-algebras, J. Funct. Anal. 155 (1998), 1 – 24

    1998

  16. [24]

    Lin H.,Exponential rank ofC ∗-algebras with real rank zero and Brown–Pedersen ’s conjecture, J. Funct. Anal. 114 (1993), p. 1–11

    1993

  17. [25]

    Osborne T.,Almost commuting unitaries with a spectral gap are near commuting unitaries, Proc. Amer. Math. Soc. 137 (2009), no. 12, 4043 – 4048

    2009

  18. [26]

    Phillips, N.C.,Approximation by unitaries with finite spectrum in purely infiniteC ∗-algebras, J. Funct. Anal. 120 (1994), no. 1, 98 – 106

    1994

  19. [27]

    Thiel H.,The real rank of some multiplier algebras, preprint (2024),https://arxiv.org/abs/2402.01022

    2024 arXiv

  20. [28]

    Voiculescu D.,Asymptotically commuting finite rank unitary operators without commuting approximants, Acta Sci. Math. 45 (1983), 429 – 431 Department of Mathematics, University of Haifa, 199 Aba Khoushi A ve, Mount Carmel, Haifa, 3103301, Israel Email address:adoron.math@gmail....

    1983

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