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

Proved

OAI.Kadison.IntertwinerCompression.pullback_matrix_commutator_norm_le

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.IntertwinerCompression, with open scoped InnerProductSpace ComplexOrder, open ContinuousLinearMap):

lemma pullback_matrix_commutator_norm_le (hcon : ‖T‖≤1) (h : ℕ)
    (X : Matrix (Fin h) (Fin h) A) (Z : G →L[ℂ] G) :
    ‖diagonalAmplification h (transfer T.adjoint Z)*matrixOperator h (fun i j => φ (X i j))-
      matrixOperator h (fun i j => φ (X i j))*diagonalAmplification h (transfer T.adjoint Z)‖≤
    ‖diagonalAmplification h Z*matrixOperator h (fun i j => ψ (X i j))-
      matrixOperator h (fun i j => ψ (X i j))*diagonalAmplification h Z‖

Definitions. The statement is written with OAI.Kadison.IntertwinerCompression.transfer (definitions bundle OAIQ039C012 of this split); OAI.Kadison.diagonalAmplification and OAI.Kadison.matrixOperator (the published bundle Def_UniformCommutator).

Used by. The proofs of OAI.Kadison.CentralSpectral.separable_uniform_bound and OAI.Kadison.NormalRepresentation.finite_standard_compression_bound of this split import it.

Preamble

import Mathlib
import Definitions.Def_UniformCommutator
import Definitions.Def_OAIQ039C012

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 scoped InnerProductSpace ComplexOrder

open ContinuousLinearMap

namespace Kadison.IntertwinerCompression

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

variable {A : Type*} [CStarAlgebra A]

variable (φ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (ψ : A →⋆ₐ[ℂ] (G →L[ℂ] G))

variable (T : H →L[ℂ] G) (hT : ∀ a,T.comp (φ a)=(ψ a).comp T)

include hT

lemma pullback_matrix_commutator_norm_le (hcon : ‖T‖≤1) (h : ℕ)
    (X : Matrix (Fin h) (Fin h) A) (Z : G →L[ℂ] G) :
    ‖diagonalAmplification h (transfer T.adjoint Z)*matrixOperator h (fun i j => φ (X i j))-
      matrixOperator h (fun i j => φ (X i j))*diagonalAmplification h (transfer T.adjoint Z)‖≤
    ‖diagonalAmplification h Z*matrixOperator h (fun i j => ψ (X i j))-
      matrixOperator h (fun i j => ψ (X i j))*diagonalAmplification h Z‖ := by
  sorry

end Kadison.IntertwinerCompression

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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