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

Proved

OAI.Kadison.GeneratedMatrixBicommutant.in_range

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.GeneratedMatrixBicommutant, with open scoped ComplexOrder Matrix.Norms.L2Operator):

lemma in_range (hφ : Set.centralizer (Set.centralizer (Set.range φ))=Set.range φ)
    (a : algebra size stage φ) : (a:H →L[ℂ] H)∈Set.range φ

Definitions. The statement is written with OAI.Kadison.GeneratedMatrixBicommutant.algebra (definitions bundle OAIQ039C011 of this split).

Used by. The proofs of OAI.Kadison.GeneratedMatrixBicommutant.factor and OAI.Kadison.TracialSequence.freeCornerPosition_of_unitary_abstract of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C011

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

attribute [-instance] NonUnitalStarSubalgebra.nonUnitalCStarAlgebra

open scoped ComplexOrder Matrix.Norms.L2Operator

namespace Kadison.GeneratedMatrixBicommutant

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

variable (size : ℕ → ℕ)

variable (stage : ∀ n,Matrix (Fin (size n+1)) (Fin (size n+1)) ℂ →⋆ₐ[ℂ] A)

variable (φ : A →⋆ₐ[ℂ] (H →L[ℂ] H))

lemma in_range (hφ : Set.centralizer (Set.centralizer (Set.range φ))=Set.range φ)
    (a : algebra size stage φ) : (a:H →L[ℂ] H)∈Set.range φ := by
  sorry

end Kadison.GeneratedMatrixBicommutant

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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