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

Proved

OAI.Kadison.FreeBlocks.innerRepresentation_commutes_other

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.FreeBlocks, with open scoped ComplexOrder InnerProductSpace Matrix.Norms.L2Operator, open MeasureTheory MeasureTheory.Measure, open FreeWordTensor FreeMoments FreeWordAction):

lemma innerRepresentation_commutes_other (hji : j≠i) (u : unitary D) (a : A) (ha : a∈S j) :
    (innerRepresentation t ht f S i hS hfree hgen u : t.functional.GNS →L[ℂ] t.functional.GNS)*
      t.functional.gnsStarAlgHom a = t.functional.gnsStarAlgHom a*
    (innerRepresentation t ht f S i hS hfree hgen u : t.functional.GNS →L[ℂ] t.functional.GNS)

Definitions. The statement is written with OAI.Kadison.FreeMoments.Free and OAI.Kadison.FreeWordTensor.FaithfulState (definitions bundle OAIQ039C010 of this split); OAI.Kadison.FreeWordTensor.generatedOperators (definitions bundle OAIQ039C036 of this split); OAI.Kadison.FreeBlocks.innerRepresentation (definitions bundle OAIQ039C059 of this split).

Used by. The proofs of OAI.Kadison.FreeBlocks.exists_free_symmetrization and OAI.Kadison.FreeBlocks.innerRepresentation_fixes_baseProjection of this split import it.

Preamble

import Mathlib
import Definitions.Def_OAIQ039C059

Formal statement

section

attribute [-instance] _root_.OAI.Kadison.ConcreteCorner.closedCarrier _root_.OAI.Kadison.CyclicColumns.instDecidableEq_solutions _root_.OAI.Kadison.UnitaryDualAction.action _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 ComplexOrder InnerProductSpace Matrix.Norms.L2Operator

open MeasureTheory MeasureTheory.Measure

namespace Kadison.FreeBlocks

open FreeWordTensor FreeMoments FreeWordAction

variable {A D : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
  [CStarAlgebra D]

variable (t : FaithfulState A) (ht : ∀ a b : A, t.functional (a*b)=t.functional (b*a))

variable (f : D →⋆ₐ[ℂ] A)

variable {ι : Type*} (S : ι → StarSubalgebra ℂ A) (i j : ι) (hS : S i=f.range)

variable (hfree : Free S t.functional.toLinearMap)
  (hgen : ∀ a : A, t.functional.gnsStarAlgHom a ∈
    Set.centralizer (Set.centralizer (generatedOperators t S : Set _)))

lemma innerRepresentation_commutes_other (hji : j≠i) (u : unitary D) (a : A) (ha : a∈S j) :
    (innerRepresentation t ht f S i hS hfree hgen u : t.functional.GNS →L[ℂ] t.functional.GNS)*
      t.functional.gnsStarAlgHom a = t.functional.gnsStarAlgHom a*
    (innerRepresentation t ht f S i hS hfree hgen u : t.functional.GNS →L[ℂ] t.functional.GNS) := by
  sorry

end Kadison.FreeBlocks

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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