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

Proved

OAI.Kadison.CentralAverage.exists_center_average

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.CentralAverage, with open Filter, open scoped InnerProductSpace Topology):

theorem exists_center_average (M : VonNeumannAlgebra H) (Y : H →L[ℂ] H) :
    ∃ Y₁ : H →L[ℂ] H,
      (∀ a∈M,a∈Set.centralizer (M : Set (H →L[ℂ] H)) → Y₁*a=a*Y₁) ∧
      ‖Y₁-Y‖ ≤ commutatorSeminorm M Y ∧
      commutatorSeminorm M Y₁≤3*commutatorSeminorm M Y

Definitions. The statement is written with OAI.Kadison.commutatorSeminorm (the published bundle Def_UniformCommutator).

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

Preamble

import Mathlib
import Definitions.Def_UniformCommutator

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 Filter

open scoped InnerProductSpace Topology

namespace Kadison.CentralAverage

variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

theorem exists_center_average (M : VonNeumannAlgebra H) (Y : H →L[ℂ] H) :
    ∃ Y₁ : H →L[ℂ] H,
      (∀ a∈M,a∈Set.centralizer (M : Set (H →L[ℂ] H)) → Y₁*a=a*Y₁) ∧
      ‖Y₁-Y‖ ≤ commutatorSeminorm M Y ∧
      commutatorSeminorm M Y₁≤3*commutatorSeminorm M Y := by
  sorry

end Kadison.CentralAverage

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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