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On almost commuting unitary matrices

Adam Dor-On, Lucas Hall, Ilya Kachkovskiy

TL;DR

The paper resolves a dimension-free version of Halmos’ question for unitary matrices under the vanishing winding-number obstruction. It introduces a quantitative framework based on the isospectral invariant and a quantitative isospectral homotopy lemma, paired with a gap-opening technique via amplification and two successive dimension reductions, to produce exactly commuting unitary approximants with explicit bounds in terms of the original commutator. The core contributions are the isospectral–winding equivalence, a constructive path-connectivity result with controlled commutator growth, and a robust amplification-and-reduction strategy that yields a final bound of the form ||u−u'||+||v−v'|| ≤ C||[u,v]||^{1/30}. These methods advance the quantitative understanding of proximity to commuting unitaries and connect topological obstructions to explicit operator-norm estimates, with potential implications for finite-dimensional approximations of noncommutative topological problems.

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.

On almost commuting unitary matrices

TL;DR

The paper resolves a dimension-free version of Halmos’ question for unitary matrices under the vanishing winding-number obstruction. It introduces a quantitative framework based on the isospectral invariant and a quantitative isospectral homotopy lemma, paired with a gap-opening technique via amplification and two successive dimension reductions, to produce exactly commuting unitary approximants with explicit bounds in terms of the original commutator. The core contributions are the isospectral–winding equivalence, a constructive path-connectivity result with controlled commutator growth, and a robust amplification-and-reduction strategy that yields a final bound of the form ||u−u'||+||v−v'|| ≤ C||[u,v]||^{1/30}. These methods advance the quantitative understanding of proximity to commuting unitaries and connect topological obstructions to explicit operator-norm estimates, with potential implications for finite-dimensional approximations of noncommutative topological problems.

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.

Paper Structure

This paper contains 19 sections, 20 theorems, 140 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Almost commuting unitary matrices: overview and the main result
  3. Structure of the paper
  4. Preliminaries
  5. Behavior of spectral projections under small perturbations
  6. Perturbations of orthogonal projections and unitary operators
  7. Unitary equivalence between intertwined families of projections
  8. Isospectral invariant, homotopy, and the winding number
  9. Definition of the isospectral invariant
  10. Proof of Lemma \ref{['lemma_homotopy_intro']}
  11. Equivalence between the invariants
  12. Related results: a trace formula and infinite-dimensional amplifications
  13. Commuting approximants and topological triviality
  14. Almost commuting matrices with spectral gaps
  15. Proof of Theorem \ref{['th_gap_opening_intro']}: the amplified almost commuting pair with a spectral gap
  16. ...and 4 more sections

Key Result

Theorem 1.2

There exists an absolute constant $C>0$ such that for any $u,v\in {\mathrm U}(n)$ with $w(u,v)=0$ one can find a pair $u',v'\in {\mathrm U}(n)$ such that

Figures (4)

  • Figure 1:
  • Figure 2: Noncommutative "Change of Variables," see Remark \ref{['rmk:joint_spectra']}.
  • Figure 3: When the obstruction vanishes, we decouple red edges.
  • Figure 4: A "high order" spectral decomposition for $u\oplus u'_{\mathrm{path}}(\varepsilon)$ into the unitary $u \in A$ and two semicircular arcs dividing $u'_{\mathrm{path}}(\varepsilon)$. After slight modifications (see equations \ref{['eq_sigma_properties']} and \ref{['eq_tau_close_1']} below), the spectral projections for the black, red, and blue arcs implement dimension reductions to $u'_{\mathrm{amp}}({\varepsilon})$ (with a controlled penalty to commutators).

Theorems & Definitions (42)

  • Theorem 1.2
  • Lemma 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 32 more