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

Proved

OAI.Kadison.MatrixGNS.flattenCopies_commutator_norm

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.MatrixGNS, with open scoped BigOperators InnerProductSpace, open ContinuousLinearMap, open OperatorAmplification CyclicColumns):

lemma flattenCopies_commutator_norm (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H))
    (Y : H →L[ℂ] H) (X : CStarMatrix κ κ (CStarMatrix ι ι A)) :
    ‖diagonal (H:=SumSpace (H:=H) ι) κ (diagonal (H:=H) ι Y)*
        matrixRepresentation (ι:=κ) (H:=SumSpace (H:=H) ι) (matrixRepresentation (ι:=ι) ρ) X-
      matrixRepresentation (ι:=κ) (H:=SumSpace (H:=H) ι) (matrixRepresentation (ι:=ι) ρ) X*
        diagonal (H:=SumSpace (H:=H) ι) κ (diagonal (H:=H) ι Y)‖=
    ‖diagonal (H:=H) (κ×ι) Y*matrixRepresentation ρ (flatten ι κ X)-
        matrixRepresentation ρ (flatten ι κ X)*diagonal (H:=H) (κ×ι) Y‖

Definitions. The statement is written with OAI.Kadison.OperatorAmplification.SumSpace and OAI.Kadison.OperatorAmplification.diagonal (definitions bundle OAIQ039C002 of this split); OAI.Kadison.CyclicColumns.instDecidableEq_solutions (definitions bundle OAIQ039C004 of this split); OAI.Kadison.MatrixGNS.flatten and OAI.Kadison.MatrixGNS.matrixFlattenRepresentationCopiesDecidableEq (definitions bundle OAIQ039C013 of this split); OAI.Kadison.CyclicColumns.matrixRepresentation (definitions bundle OAIQ039C022 of this split).

Used by. The proof of OAI.Kadison.MatrixGNS.flatBalanced_commutator_dominates of this split imports it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C013
import Definitions.Def_OAIQ039C022

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier _root_.OAI.Kadison.VonNeumannCStar.cstarAlgebra _root_.OAI.Kadison.VonNeumannCStar.isClosed _root_.OAI.Kadison.VonNeumannCStar.smulMemClass _root_.OAI.Kadison.VonNeumannCStar.starOrderedRing
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 BigOperators InnerProductSpace

open ContinuousLinearMap

namespace Kadison.MatrixGNS

open OperatorAmplification CyclicColumns

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

variable (ι κ : Type*) [Fintype ι] [Fintype κ]

attribute [local instance] OAI.Kadison.MatrixGNS.matrixFlattenRepresentationCopiesDecidableEq
lemma flattenCopies_commutator_norm (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H))
    (Y : H →L[ℂ] H) (X : CStarMatrix κ κ (CStarMatrix ι ι A)) :
    ‖diagonal (H:=SumSpace (H:=H) ι) κ (diagonal (H:=H) ι Y)*
        matrixRepresentation (ι:=κ) (H:=SumSpace (H:=H) ι) (matrixRepresentation (ι:=ι) ρ) X-
      matrixRepresentation (ι:=κ) (H:=SumSpace (H:=H) ι) (matrixRepresentation (ι:=ι) ρ) X*
        diagonal (H:=SumSpace (H:=H) ι) κ (diagonal (H:=H) ι Y)‖=
    ‖diagonal (H:=H) (κ×ι) Y*matrixRepresentation ρ (flatten ι κ X)-
        matrixRepresentation ρ (flatten ι κ X)*diagonal (H:=H) (κ×ι) Y‖ := by
  sorry

end Kadison.MatrixGNS

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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