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

Proved

OAI.Kadison.FreeBlocks.free_stage_symmetrization

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 UniformSpace MeasureTheory, open scoped ComplexOrder InnerProductSpace, open FreeMoments FreeWordTensor FreeWordAction):

theorem free_stage_symmetrization (G : t.functional.GNS →L[ℂ] t.functional.GNS)
    (ε : ℝ) (hε : 0≤ε) (β : ℂ) (s : A) (hs : s∈S true)
    (hGD : ∀ a : D, G*t.functional.gnsStarAlgHom (f a)=t.functional.gnsStarAlgHom (f a)*G)
    (hC : ∀ x, CornerSymmetrization.CornerElement
      (factorOperators t f) (ambientOperators t S)
      (TracialGNS.vacuum t.functional) β x → ‖G*x-x*G‖≤ε*‖x‖) :
    ∃ V : t.functional.GNS →L[ℂ] t.functional.GNS,
      ‖V‖≤‖G‖ ∧
      (∀ n u, V*(innerRepresentation t htr f S false hS hfree hgen (TracialGNS.stageUnitary (φ n) u):
        t.functional.GNS →L[ℂ] t.functional.GNS)=
        (innerRepresentation t htr f S false hS hfree hgen (TracialGNS.stageUnitary (φ n) u):
          t.functional.GNS →L[ℂ] t.functional.GNS)*V) ∧
      (∀ a : D, V*t.functional.gnsStarAlgHom (f a)=t.functional.gnsStarAlgHom (f a)*V) ∧
      (∀ x, CornerSymmetrization.CornerElement
        (factorOperators t f) (ambientOperators t S)
        (TracialGNS.vacuum t.functional) β x → ‖V*x-x*V‖≤ε*‖x‖) ∧
      (ContinuousLinearMap.adjoint (𝕜 := ℂ) (E := multiplicityLevel t S true 0) (F := t.functional.GNS)
        (baseEmbedding t S hfree).toContinuousLinearMap).comp
        ((V*t.functional.gnsStarAlgHom s-t.functional.gnsStarAlgHom s*V).comp
          (baseEmbedding t S hfree).toContinuousLinearMap)=
      (ContinuousLinearMap.adjoint (𝕜 := ℂ) (E := multiplicityLevel t S true 0) (F := t.functional.GNS)
        (baseEmbedding t S hfree).toContinuousLinearMap).comp
        ((G*t.functional.gnsStarAlgHom s-t.functional.gnsStarAlgHom s*G).comp
          (baseEmbedding t S hfree).toContinuousLinearMap)

Definitions. The statement is written with OAI.Kadison.TracialGNS.vacuum (definitions bundle OAIQ039C003 of this split); OAI.Kadison.CornerSymmetrization.CornerElement (definitions bundle OAIQ039C004 of this split); OAI.Kadison.FreeMoments.Free, OAI.Kadison.FreeWordTensor.FaithfulState, OAI.Kadison.FreeWordTensor.preSpace and OAI.Kadison.FreeWordTensor.Space (definitions bundle OAIQ039C010 of this split); OAI.Kadison.TracialGNS.stageUnitary (definitions bundle OAIQ039C018 of this split); OAI.Kadison.FreeBlocks.factorSpace (definitions bundle OAIQ039C025 of this split); OAI.Kadison.FreeBlocks.multiplicityLevel, OAI.Kadison.FreeBlocks.multiplicityTensor and OAI.Kadison.FreeWordTensor.generatedOperators (definitions bundle OAIQ039C036 of this split); OAI.Kadison.FreeBlocks.ambientOperators, OAI.Kadison.FreeBlocks.baseEmbedding and OAI.Kadison.FreeBlocks.factorOperators (definitions bundle OAIQ039C055 of this split); OAI.Kadison.FreeBlocks.innerRepresentation (definitions bundle OAIQ039C059 of this split).

Used by. The proof of OAI.Kadison.FreeBlocks.normal_free_corner_bound of this split imports 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 UniformSpace MeasureTheory

open scoped ComplexOrder InnerProductSpace

namespace Kadison.FreeBlocks

open FreeMoments FreeWordTensor FreeWordAction

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

variable (t : FaithfulState A) (htr : ∀ a b : A, t.functional (a*b)=t.functional (b*a))
  (S : Bool → StarSubalgebra ℂ A) (f : D →⋆ₐ[ℂ] A) (hS : S false=f.range)
  (hfree : Free S t.functional.toLinearMap)
  (hgen : ∀ a : A, t.functional.gnsStarAlgHom a ∈
    Set.centralizer (Set.centralizer (generatedOperators t S : Set _)))

variable (I : ℕ → Type*) [∀ n, Fintype (I n)] [∀ n, DecidableEq (I n)] [∀ n, Nonempty (I n)]
  (φ : ∀ n, Matrix (I n) (I n) ℂ →⋆ₐ[ℂ] D)
  (inc : ∀ {n m : ℕ}, n ≤ m → Matrix (I n) (I n) ℂ →⋆ₐ[ℂ] Matrix (I m) (I m) ℂ)
  (hinc : ∀ n m (h : n ≤ m) a, φ m (inc h a)=φ n a)

omit [PartialOrder D] [StarOrderedRing D] [∀ n, Nonempty (I n)] in
include hinc in
theorem free_stage_symmetrization (G : t.functional.GNS →L[ℂ] t.functional.GNS)
    (ε : ℝ) (hε : 0≤ε) (β : ℂ) (s : A) (hs : s∈S true)
    (hGD : ∀ a : D, G*t.functional.gnsStarAlgHom (f a)=t.functional.gnsStarAlgHom (f a)*G)
    (hC : ∀ x, CornerSymmetrization.CornerElement
      (factorOperators t f) (ambientOperators t S)
      (TracialGNS.vacuum t.functional) β x → ‖G*x-x*G‖≤ε*‖x‖) :
    ∃ V : t.functional.GNS →L[ℂ] t.functional.GNS,
      ‖V‖≤‖G‖ ∧
      (∀ n u, V*(innerRepresentation t htr f S false hS hfree hgen (TracialGNS.stageUnitary (φ n) u):
        t.functional.GNS →L[ℂ] t.functional.GNS)=
        (innerRepresentation t htr f S false hS hfree hgen (TracialGNS.stageUnitary (φ n) u):
          t.functional.GNS →L[ℂ] t.functional.GNS)*V) ∧
      (∀ a : D, V*t.functional.gnsStarAlgHom (f a)=t.functional.gnsStarAlgHom (f a)*V) ∧
      (∀ x, CornerSymmetrization.CornerElement
        (factorOperators t f) (ambientOperators t S)
        (TracialGNS.vacuum t.functional) β x → ‖V*x-x*V‖≤ε*‖x‖) ∧
      (ContinuousLinearMap.adjoint (𝕜 := ℂ) (E := multiplicityLevel t S true 0) (F := t.functional.GNS)
        (baseEmbedding t S hfree).toContinuousLinearMap).comp
        ((V*t.functional.gnsStarAlgHom s-t.functional.gnsStarAlgHom s*V).comp
          (baseEmbedding t S hfree).toContinuousLinearMap)=
      (ContinuousLinearMap.adjoint (𝕜 := ℂ) (E := multiplicityLevel t S true 0) (F := t.functional.GNS)
        (baseEmbedding t S hfree).toContinuousLinearMap).comp
        ((G*t.functional.gnsStarAlgHom s-t.functional.gnsStarAlgHom s*G).comp
          (baseEmbedding t S hfree).toContinuousLinearMap) := 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)

View graph