OAI.Kadison.FreeBlocks.finite_fejer_entry: intermediate lemma from OpenAI, Kadison's similarity theorem through uniform derivation estimates (result family 288)
ProvedOAI.Kadison.FreeBlocks.finite_fejer_entryAn 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, open scoped ComplexOrder InnerProductSpace, open FreeMoments FreeWordTensor DihedralCorner):
theorem finite_fejer_entry (e0 : D) (e : Corner hp)
(heq : f e0=(e:A)) (heproj : IsStarProjection e)
(he : e∈cornerFactor hp (S false) hpD)
(hr : 0<‖TracialGNS.algebraVector (factorTrace t f hf) e0‖)
(N : ℕ) (hN : 0 < N) (i j : Fin N) :
((ContinuousLinearMap.adjoint (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
(dihedralColumn t S hfree (e:A) (d*s)
‖TracialGNS.algebraVector (factorTrace t f hf) e0‖⁻¹ i.val)).comp
(t.functional.gnsStarAlgHom ((e:A)*((fejerCorner hp (d*s) (2*N):Corner hp):A)*(e:A)))).comp
(dihedralColumn t S hfree (e:A) (d*s)
‖TracialGNS.algebraVector (factorTrace t f hf) e0‖⁻¹ j.val)=
((halfCornerTrace hp t.functional.toLinearMap e).re:ℂ) •
((2*t.functional (wave (d*s) ((i.val:ℤ)-j.val)*
((fejerCorner hp (d*s) (2*N):Corner hp):A))) •
ContinuousLinearMap.id ℂ (multiplicityLevel t S true 0)+
(Complex.ofReal (((2*N:ℝ)-i.val-j.val-1)/(2*N:ℝ)^2)) • baseAction t S hfree (s:A))
Definitions. The statement is written with OAI.Kadison.TracialGNS.algebraVector (definitions bundle OAIQ039C003 of this split); OAI.Kadison.Corner, OAI.Kadison.cornerSubalgebra, OAI.Kadison.instAlgebraComplexCorner, OAI.Kadison.instModuleComplexCorner, OAI.Kadison.instRingCorner, OAI.Kadison.instStarModuleComplexCorner and OAI.Kadison.instStarRingCorner (definitions bundle OAIQ039C004 of this split); OAI.Kadison.DihedralCorner.wave (definitions bundle OAIQ039C005 of this split); OAI.Kadison.FreeMoments.Free, OAI.Kadison.FreeWordTensor.FaithfulState, OAI.Kadison.FreeWordTensor.faithfulNormed, OAI.Kadison.FreeWordTensor.preSpace and OAI.Kadison.FreeWordTensor.Space (definitions bundle OAIQ039C010 of this split); OAI.Kadison.FreeBlocks.factorSpace (definitions bundle OAIQ039C025 of this split); OAI.Kadison.FreeBlocks.multiplicityLevel, OAI.Kadison.FreeBlocks.multiplicityTensor and OAI.Kadison.FreeBlocks.pullbackState (definitions bundle OAIQ039C036 of this split); OAI.Kadison.FreeBlocks.factorTrace (definitions bundle OAIQ039C046 of this split); OAI.Kadison.halfCornerTrace (definitions bundle OAIQ039C054 of this split); OAI.Kadison.FreeBlocks.baseAction and OAI.Kadison.FreeBlocks.dihedralColumn (definitions bundle OAIQ039C055 of this split); OAI.Kadison.cornerFactor and OAI.Kadison.DihedralCorner.fejerCorner (definitions bundle OAIQ039C059 of this split).
Used by. The proof of OAI.Kadison.FreeBlocks.normalized_fejer_entry of this split imports it.
import Mathlib import Definitions.Def_OAIQ039C059
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
open scoped ComplexOrder InnerProductSpace
namespace Kadison.FreeBlocks
open FreeMoments FreeWordTensor DihedralCorner
variable {A D : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
[CStarAlgebra D] [PartialOrder D] [StarOrderedRing D]
variable (t : FaithfulState A) (S : Bool → StarSubalgebra ℂ A)
(hfree : Free S t.functional.toLinearMap)
variable (f : D →⋆ₐ[ℂ] A) (hf : Function.Injective f)
variable {p : A} (hp : IsStarProjection p)
variable (htr : ∀ a b : A, t.functional (a*b)=t.functional (b*a))
variable (d s : unitary A) (hd : (d:A)∈S false) (hs : (s:A)∈S true)
(hd0 : t.functional (d:A)=0) (hs0 : t.functional (s:A)=0)
(hrot : MulAut.conj d (d*s)=(d*s)⁻¹)
(hd2 : (d:A)*(d:A)=1) (hs2 : (s:A)*(s:A)=1)
variable (hpform : p=(1/2:ℂ) • (1+(d:A))) (hpd : p*(d:A)=p) (hdp : (d:A)*p=p)
(hpD : p∈S false)
include hfree htr hd hs hd0 hs0 hrot hd2 hs2 hpform hpd hdp in
theorem finite_fejer_entry (e0 : D) (e : Corner hp)
(heq : f e0=(e:A)) (heproj : IsStarProjection e)
(he : e∈cornerFactor hp (S false) hpD)
(hr : 0<‖TracialGNS.algebraVector (factorTrace t f hf) e0‖)
(N : ℕ) (hN : 0 < N) (i j : Fin N) :
((ContinuousLinearMap.adjoint (E:=multiplicityLevel t S true 0) (F:=t.functional.GNS)
(dihedralColumn t S hfree (e:A) (d*s)
‖TracialGNS.algebraVector (factorTrace t f hf) e0‖⁻¹ i.val)).comp
(t.functional.gnsStarAlgHom ((e:A)*((fejerCorner hp (d*s) (2*N):Corner hp):A)*(e:A)))).comp
(dihedralColumn t S hfree (e:A) (d*s)
‖TracialGNS.algebraVector (factorTrace t f hf) e0‖⁻¹ j.val)=
((halfCornerTrace hp t.functional.toLinearMap e).re:ℂ) •
((2*t.functional (wave (d*s) ((i.val:ℤ)-j.val)*
((fejerCorner hp (d*s) (2*N):Corner hp):A))) •
ContinuousLinearMap.id ℂ (multiplicityLevel t S true 0)+
(Complex.ofReal (((2*N:ℝ)-i.val-j.val-1)/(2*N:ℝ)^2)) • baseAction t S hfree (s:A)) := by
sorry
end Kadison.FreeBlocks
end
end
end
end OAI
end