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Exponent for commutators close to the identity (C85C_{85})

Exponent for commutators close to the identity: determine C85C_{85} or improve its bounds; the repository records 1 (lower) and 4 (upper).

MSC 47B47 · Commutators, derivations, elementary operators, etc.

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

Let HH be an infinite-dimensional complex Hilbert space and let B(H)B(H) be the Banach algebra of bounded operators on HH, equipped with the operator norm. For D,X∈B(H)D, X \in B(H) let [D,X]:=DX−XD[D,X] := DX - XD denote their commutator, and for 0<ε<10 < \varepsilon < 1 define

mH(ε):=inf⁡{∥D∥ ∥X∥:D,X∈B(H), ∥[D,X]−1B(H)∥≤ε}.m_H(\varepsilon) := \inf \{ \lVert D\rVert\ \lVert X\rVert : D, X \in B(H),\ \lVert [D,X] - 1_{B(H)}\rVert \le \varepsilon \}.

The exponent for commutators close to the identity is

C85:=lim sup⁡ε→0+log⁡mH(ε)log⁡log⁡(1/ε).C_{85} := \limsup_{\varepsilon \to 0^+} \frac{\log m_H(\varepsilon)}{\log \log(1/\varepsilon)}.

Equivalently, C85C_{85} is the infimum of c≥0c \ge 0 for which one has mH(ε)≪clog⁡c(1/ε)m_H(\varepsilon) \ll_c \log^c(1/\varepsilon) as ε→0+\varepsilon \to 0^+.

By a classical theorem of Wintner [Win47] and Wielandt [Wie49], the identity operator 1B(H)1_{B(H)} cannot be written exactly as [D,X][D,X] for bounded D,XD, X (indeed in any unital Banach algebra), but in infinite dimensions Brown and Pearcy [BP65] showed that it can be approximated arbitrarily well by bounded commutators, so mH(ε)<∞m_H(\varepsilon) < \infty for every ε>0\varepsilon > 0. Popa [Pop82] gave a quantitative Wintner–Wielandt lower bound mH(ε)≥12log⁡(1/ε)m_H(\varepsilon) \ge \tfrac{1}{2}\log(1/\varepsilon), so C85≥1C_{85} \ge 1. It is a conjecture (raised in [Pop82, Remark 2.9] and reiterated in [Tao19]) that Popa's exponent-11 bound is essentially sharp, i.e. C85=1C_{85} = 1.

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Background

  • The Brown–Pearcy construction [BP65], together with an elementary conjugation-by-diagonal step (Proposition 0.2 of [Tao19], attributed there to Popa), gives the earlier bound mH(ε)=O(ε−2)m_H(\varepsilon) = O(\varepsilon^{-2}). This is polynomial in 1/ε1/\varepsilon and hence does not give a finite upper bound on C85C_{85}; polylogarithmic control in 1/ε1/\varepsilon first appears in [Tao19].
  • The problem is intrinsically infinite-dimensional: if dim⁡H<∞\dim H < \infty then any commutator [D,X][D,X] has trace zero and therefore some eigenvalue outside the disk {z:∣z−1∣<1}\{z : \lvert z-1\rvert < 1\}, forcing ∥[D,X]−1∥≥1\lVert [D,X] - 1\rVert \ge 1. Thus mH(ε)=+∞m_H(\varepsilon) = +\infty for every ε<1\varepsilon < 1 in finite dimensions, and C85C_{85} is unambiguously defined by the infinite-dimensional case.
  • A Lasserre / Navascués–Pironio–Acín semidefinite-programming approach to bounding C85C_{85} from below was proposed in [Tao19, Remark 2.9] (an observation of Tobias Fritz); with computationally-feasible sets of noncommutative monomials it has so far not been observed to yield any nontrivial pairs (C,ε)(C,\varepsilon).
  • [Tao19, §2] also raises a related question in terms of the distance dist(A,C+K(H))\mathrm{dist}(A, \mathbb{C} + K(H)) from A∈B(H)A \in B(H) to the scalar-plus-compact operators: whether the exponent 2/32/3 in a result of [Pop82, Theorem 2.1] characterizing which AA are commutators of bounded operators can be replaced by a log⁡−C\log^{-C} factor.

References (from the repository page)

  • [Win47] Wintner, Aurel. The unboundedness of quantum-mechanical matrices. Physical Review 71 (1947), 738–739.
  • [Wie49] Wielandt, Helmut. Über die Unbeschränktheit der Operatoren der Quantenmechanik. Mathematische Annalen 121 (1949), 21.
  • [BP65] Brown, Arlen; Pearcy, Carl. Structure of commutators of operators. Annals of Mathematics 82 (1965), 112–127.
  • [Pop82] Popa, Sorin. On commutators in properly infinite W\*W^\*-algebras. In: Invariant Subspaces and Other Topics (Timişoara/Herculane, 1981), Operator Theory: Advances and Applications 6, Birkhäuser, Basel, 1982, pp. 195–207.
  • [Tao19] Tao, Terence. Commutators close to the identity. Journal of Operator Theory 82 (2019), no. 2, 369–382. arXiv:1805.11131.
  • [KJ22] Krishna, K. Mahesh; Johnson, P. Sam. Commutators close to the identity in unital C\*C^\*-algebras. Proceedings — Mathematical Sciences 132 (2022), Paper No. 11. DOI: 10.1007/s12044-022-00663-w. arXiv:2104.02035.
  • [Bil26] Bilich, Boris. An O(log⁡4(1/ε))O(\log^4(1/\varepsilon)) refinement of Tao's construction of commutators close to the identity. Note (2026); result and Lean 4 formalization produced autonomously by GPT-5.6 Sol High. Available at bilichboris.github.io/blog/2026/gpt-5-6-sol-improved-tao-bound (accessed 17 Jul 2026).

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1; best known upper bound 4. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
55 [Tao19] Almost-upper-triangular matrices in Mn(B(H))M_n(B(H)) with n≍log⁡(1/ε)n \asymp \log(1/\varepsilon), obtained by solving a nonlinear system in B(H)nB(H)^n via Cuntz isometries and a contraction-mapping argument. Tao's first-draft exponent was 1616; the anonymous referee reduced it to 55.
55 [KJ22] Krishna–Johnson extend Tao's construction from B(H)B(H) to unital C\*C^\*-algebras satisfying suitable structural hypotheses (in particular, algebras containing a unital copy of the Cuntz algebra O2\mathcal{O}_2), with the same exponent. Does not improve C85C_{85} but broadens the setting.
44 [Bil26] Same nonlinear system as [Tao19], with a sharper point-source Green function estimate on the right inverse (norm O(n)O(n) on scalar point sources, versus O(n2)O(n^2) in the worst case), proved by interpreting iterated Cuntz block coefficients as a killed walk and comparing them with binomial incidence matrices. This upgrades the fixed-point scale from δ≍n−5\delta \asymp n^{-5} to δ≍n−4\delta \asymp n^{-4} and improves the exponent from 55 to 44. Result and Lean 4 formalization produced autonomously by GPT-5.6 Sol High (13 min for the proof, <3{<}3 h for the formalization).

Known lower bounds

Bound Reference Comments
00 Trivial mH(ε)≥1m_H(\varepsilon) \ge 1 for all ε\varepsilon.
11 [Pop82] Popa's quantitative Wintner–Wielandt bound mH(ε)≥12log⁡(1/ε)m_H(\varepsilon) \ge \tfrac{1}{2}\log(1/\varepsilon); see [Tao19, Theorem 0.1] for a short reproduction. Conjectured to be sharp.
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Exponent for commutators close to the identity ($C_{85}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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