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OAI.Kadison.Decomposable.uniform_commutator_in_central_model: intermediate lemma from OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288)

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OAI.Kadison.Decomposable.uniform_commutator_in_central_model

by dbenbenn · Oct 28, 2026 · Mathlib 0df444a (Lean v4.33.1)

openai-math

An intermediate lemma of the split of OpenAI's proof of OAI.Kadison.uniform_commutator, carved from OpenAI's openai/math development for this platform. The statement and its proof are OpenAI's (OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, result family 288, github.com/openai/math, Apache License 2.0), ported to Lean 4.33.1 and Mathlib 0df444a.

The statement, as published (namespace OAI.Kadison.Decomposable, with open MeasureTheory Filter, open scoped Topology InnerProductSpace):

theorem uniform_commutator_in_central_model
    (P : VonNeumannAlgebra (Lp H 2 μ)) (Q R : Ω → VonNeumannAlgebra H)
    (hdiag : ∀ (s : Set Ω) (hs : MeasurableSet s),
      indicatorOperator (H:=H) μ s hs∈P ∧
      indicatorOperator (H:=H) μ s hs∈Set.centralizer (P : Set (Lp H 2 μ →L[ℂ] Lp H 2 μ)))
    (test cotest : ℕ → Ω → H →L[ℂ] H)
    (ht : ∀ n ξ,AEStronglyMeasurable (fun a => test n a ξ) μ)
    (hs : ∀ n ξ,AEStronglyMeasurable (fun a => cotest n a ξ) μ)
    (htb : ∀ n,∀ᵐ a ∂μ,‖test n a‖≤1)
    (hsb : ∀ n,∀ᵐ a ∂μ,‖cotest n a‖≤1)
    (htP : ∀ n,operator μ (test n) (ht n) (htb n)∈P)
    (hsP : ∀ n,operator μ (cotest n) (hs n) (hsb n)∈P.commutant)
    (htQ : ∀ n,∀ᵐ a ∂μ,test n a∈Q a)
    (hsR : ∀ n,∀ᵐ a ∂μ,cotest n a∈R a)
    (hfiberdense : ∀ᵐ a ∂μ,∀ A : H →L[ℂ] H,A∈Q a → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ ξ,Tendsto (fun n => test (k n) a ξ) atTop (𝓝 (A ξ)))
    (hcofiberdense : ∀ᵐ a ∂μ,∀ A : H →L[ℂ] H,A∈R a → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ ξ,Tendsto (fun n => cotest (k n) a ξ) atTop (𝓝 (A ξ)))
    (hglobaldense : ∀ A : Lp H 2 μ →L[ℂ] Lp H 2 μ,A∈P → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ f,Tendsto
        (fun n => operator μ (test (k n)) (ht (k n)) (htb (k n)) f) atTop (𝓝 (A f)))
    (hcoglobaldense : ∀ A : Lp H 2 μ →L[ℂ] Lp H 2 μ,A∈P.commutant → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ f,Tendsto
        (fun n => operator μ (cotest (k n)) (hs (k n)) (hsb (k n)) f) atTop (𝓝 (A f)))
    (hcenter : ∀ (Z : Lp H 2 μ →L[ℂ] Lp H 2 μ),Z∈P → Z∈P.commutant →
      ∀ B : H →L[ℂ] H,Commute
        (operator μ (fun _ => B) (fun _ => aestronglyMeasurable_const)
          (Eventually.of_forall (fun _ => le_rfl))) Z)
    (Y : Lp H 2 μ →L[ℂ] Lp H 2 μ) (h : ℕ) (hh : 1≤h)
    (X : Matrix (Fin h) (Fin h) (Lp H 2 μ →L[ℂ] Lp H 2 μ)) (hX : ∀ i j,X i j∈P) :
    ‖diagonalAmplification h Y*matrixOperator h X-
      matrixOperator h X*diagonalAmplification h Y‖≤
      9368*commutatorSeminorm P Y*‖matrixOperator h X‖

Definitions. The statement is written with OAI.Kadison.Decomposable.operator (definitions bundle OAIQ039C022 of this split); OAI.Kadison.Decomposable.indicatorOperator (definitions bundle OAIQ039C033 of this split); OAI.Kadison.commutatorSeminorm, OAI.Kadison.diagonalAmplification and OAI.Kadison.matrixOperator (the published bundle Def_UniformCommutator).

Used by. The proof of OAI.Kadison.Decomposable.uniform_commutator_in_central_spectral_model_nonzero of this split imports it.

Preamble
import Mathlib
import Definitions.Def_UniformCommutator
import Definitions.Def_OAIQ039C033
Formal statement
section

set_option linter.all false
set_option linter.unusedVariables false
set_option linter.unusedTactic false
set_option linter.unreachableTactic false

namespace OAI

noncomputable section

section

noncomputable section

open MeasureTheory Filter

open scoped Topology InnerProductSpace

namespace Kadison.Decomposable

variable {Ω H : Type} [MeasurableSpace Ω] (μ : Measure Ω)
  [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
  [IsFiniteMeasure μ] [TopologicalSpace.SeparableSpace H]

theorem uniform_commutator_in_central_model
    (P : VonNeumannAlgebra (Lp H 2 μ)) (Q R : Ω → VonNeumannAlgebra H)
    (hdiag : ∀ (s : Set Ω) (hs : MeasurableSet s),
      indicatorOperator (H:=H) μ s hs∈P ∧
      indicatorOperator (H:=H) μ s hs∈Set.centralizer (P : Set (Lp H 2 μ →L[ℂ] Lp H 2 μ)))
    (test cotest : ℕ → Ω → H →L[ℂ] H)
    (ht : ∀ n ξ,AEStronglyMeasurable (fun a => test n a ξ) μ)
    (hs : ∀ n ξ,AEStronglyMeasurable (fun a => cotest n a ξ) μ)
    (htb : ∀ n,∀ᵐ a ∂μ,‖test n a‖≤1)
    (hsb : ∀ n,∀ᵐ a ∂μ,‖cotest n a‖≤1)
    (htP : ∀ n,operator μ (test n) (ht n) (htb n)∈P)
    (hsP : ∀ n,operator μ (cotest n) (hs n) (hsb n)∈P.commutant)
    (htQ : ∀ n,∀ᵐ a ∂μ,test n a∈Q a)
    (hsR : ∀ n,∀ᵐ a ∂μ,cotest n a∈R a)
    (hfiberdense : ∀ᵐ a ∂μ,∀ A : H →L[ℂ] H,A∈Q a → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ ξ,Tendsto (fun n => test (k n) a ξ) atTop (𝓝 (A ξ)))
    (hcofiberdense : ∀ᵐ a ∂μ,∀ A : H →L[ℂ] H,A∈R a → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ ξ,Tendsto (fun n => cotest (k n) a ξ) atTop (𝓝 (A ξ)))
    (hglobaldense : ∀ A : Lp H 2 μ →L[ℂ] Lp H 2 μ,A∈P → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ f,Tendsto
        (fun n => operator μ (test (k n)) (ht (k n)) (htb (k n)) f) atTop (𝓝 (A f)))
    (hcoglobaldense : ∀ A : Lp H 2 μ →L[ℂ] Lp H 2 μ,A∈P.commutant → ‖A‖≤1 →
      ∃ k : ℕ → ℕ,∀ f,Tendsto
        (fun n => operator μ (cotest (k n)) (hs (k n)) (hsb (k n)) f) atTop (𝓝 (A f)))
    (hcenter : ∀ (Z : Lp H 2 μ →L[ℂ] Lp H 2 μ),Z∈P → Z∈P.commutant →
      ∀ B : H →L[ℂ] H,Commute
        (operator μ (fun _ => B) (fun _ => aestronglyMeasurable_const)
          (Eventually.of_forall (fun _ => le_rfl))) Z)
    (Y : Lp H 2 μ →L[ℂ] Lp H 2 μ) (h : ℕ) (hh : 1≤h)
    (X : Matrix (Fin h) (Fin h) (Lp H 2 μ →L[ℂ] Lp H 2 μ)) (hX : ∀ i j,X i j∈P) :
    ‖diagonalAmplification h Y*matrixOperator h X-
      matrixOperator h X*diagonalAmplification h Y‖≤
      9368*commutatorSeminorm P Y*‖matrixOperator h X‖ := by
  sorry

end Kadison.Decomposable

end

end
end

end OAI
end
Source
OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288), OpenAI Math Release, September 23, 2026, https://github.com/openai/math (result family 288; paper: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/Kadisons-similarity-theorem-through-uniform-derivation-estimates-September-23-2026/paper.pdf; Apache License 2.0)

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