| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.DimensionDrop.SequentialCStarChain | |
| import OAI.Analysis.JiangSu.DimensionDrop.PermutationUnitary | |
| import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropScalarFunctions | |
| import OAI.Analysis.JiangSu.CStarAlgebras.StarTermsNormLift | |
| namespace OAI | |
| noncomputable section | |
| open scoped ComplexOrder CStarAlgebra | |
| namespace DimensionDropRelations.Witness | |
| open DimensionDropModel StarTerms | |
| variable {I : Type*} [Fintype I] [DecidableEq I] | |
| local instance functionCStar : CStarAlgebra C(Interval,Mat I) := ContinuousMap.instCStarAlgebra | |
| variable {B : Type} [CStarAlgebra B] [PartialOrder B] [StarOrderedRing B] | |
| variable {p : I} (w : Witness (B := B) p) | |
| def generatedHom : DimensionDropModel.generated p →⋆ₐ[ℂ] B := | |
| closureLift (functionValues p) w.interpret w.polynomial_norm | |
| lemma generatedHom_gen (g : Gen I) : | |
| w.generatedHom ⟨functionValues p g,StarSubalgebra.le_topologicalClosure _ ⟨.gen g,rfl⟩⟩=w.interpret g := | |
| closureLift_gen (functionValues p) w.interpret w.polynomial_norm g | |
| end DimensionDropRelations.Witness | |
| end | |
| noncomputable section | |
| open scoped CStarAlgebra ComplexOrder Kronecker | |
| local instance (n : ℕ) : Semiring (JiangSuConstruction.DimensionDrop (n+1) n) := | |
| (inferInstance : CStarAlgebra (JiangSuConstruction.DimensionDrop (n+1) n)).toSemiring | |
| namespace DimensionDropModel | |
| open JiangSuConstruction | |
| def indexEquiv (n : ℕ) : (Fin (n+1)×Fin n)≃(Option (Fin n)×Fin n) := | |
| Equiv.prodCongr (finSuccEquiv n) (Equiv.refl _) | |
| def twistAmbient (n : ℕ) : C(Interval,TensorMat (n+1) n) →⋆ₐ[ℂ] C(Interval,Mat (Fin n)) := | |
| (continuousUnitaryConjugation (finiteMatrixUnitaryPath 1 | |
| (permutationUnitary (flipInner (I := Fin n)))).toContinuousMap).comp | |
| (continuousReindex (indexEquiv n)) | |
| lemma reindex_left (n : ℕ) (a : JiangSuConstruction.Mat (n+1)) : | |
| CStarMatrix.reindexₐ ℂ ℂ (indexEquiv n) (leftAmplification (n+1) n a)= | |
| amplify (J := Fin n) (CStarMatrix.reindexₐ ℂ ℂ (finSuccEquiv n) a) := rfl | |
| lemma reindex_right (n : ℕ) (a : JiangSuConstruction.Mat n) : | |
| CStarMatrix.reindexₐ ℂ ℂ (flipInner (I := Fin n)) | |
| (CStarMatrix.reindexₐ ℂ ℂ (indexEquiv n) (rightAmplification (n+1) n a))=upperEmbed a := by | |
| change CStarMatrix.reindexₐ ℂ ℂ (flipInner (I := Fin n)) _= | |
| CStarMatrix.reindexₐ ℂ ℂ (flipInner (I := Fin n)) | |
| (CStarMatrix.reindexₐ ℂ ℂ (Equiv.prodComm (Fin n) (Option (Fin n))) (amplify a)) | |
| apply congrArg (CStarMatrix.reindexₐ ℂ ℂ (flipInner (I := Fin n))) | |
| ext ⟨b,i⟩ ⟨c,j⟩ | |
| change (if (finSuccEquiv n).symm b=(finSuccEquiv n).symm c then (1:ℂ) else 0)*a i j= | |
| a i j*(if b=c then 1 else 0) | |
| simp only [(finSuccEquiv n).symm.injective.eq_iff, mul_comm] | |
| lemma twist_left (n : ℕ) (f : C(Interval,TensorMat (n+1) n)) : | |
| twistAmbient n f leftEndpoint=CStarMatrix.reindexₐ ℂ ℂ (indexEquiv n) (f leftEndpoint) := by | |
| change unitaryConjugation ((finiteMatrixUnitaryPath 1 | |
| (permutationUnitary (flipInner (I := Fin n)))) 0) _=_ | |
| rw [Path.source] | |
| change (1 : Mat (Fin n))*CStarMatrix.reindexₐ ℂ ℂ (indexEquiv n) (f leftEndpoint)* | |
| star (1 : Mat (Fin n))=CStarMatrix.reindexₐ ℂ ℂ (indexEquiv n) (f leftEndpoint) | |
| simp | |
| lemma twist_right (n : ℕ) (f : C(Interval,TensorMat (n+1) n)) : | |
| twistAmbient n f rightEndpoint=CStarMatrix.reindexₐ ℂ ℂ (flipInner (I := Fin n)) | |
| (CStarMatrix.reindexₐ ℂ ℂ (indexEquiv n) (f rightEndpoint)) := by | |
| change unitaryConjugation ((finiteMatrixUnitaryPath 1 | |
| (permutationUnitary (flipInner (I := Fin n)))) 1) _=_ | |
| rw [Path.target,permutationUnitary_conjugation] | |
| rfl | |
| def concreteToGenerated {n : ℕ} (p : Fin n) : DimensionDrop (n+1) n →⋆ₐ[ℂ] generated p := | |
| ((twistAmbient n).comp (dimensionDrop (n+1) n).subtype).codRestrict (generated p) (fun f => by | |
| apply endpoint_function_mem p | |
| · obtain ⟨a,ha⟩ := f.property.1 | |
| change leftAmplification (n+1) n a=(f : C(Interval,TensorMat (n+1) n)) leftEndpoint at ha | |
| refine ⟨CStarMatrix.reindexₐ ℂ ℂ (finSuccEquiv n) a,?_⟩ | |
| change _=twistAmbient n (f : C(Interval,TensorMat (n+1) n)) leftEndpoint | |
| rw [twist_left,←ha,reindex_left] | |
| · obtain ⟨a,ha⟩ := f.property.2 | |
| change rightAmplification (n+1) n a=(f : C(Interval,TensorMat (n+1) n)) rightEndpoint at ha | |
| refine ⟨a,?_⟩ | |
| change _=twistAmbient n (f : C(Interval,TensorMat (n+1) n)) rightEndpoint | |
| rw [twist_right,←ha,reindex_right]) | |
| end DimensionDropModel | |
| namespace DimensionDropRelations.Witness | |
| open JiangSuConstruction | |
| variable {B : Type} [CStarAlgebra B] [PartialOrder B] [StarOrderedRing B] | |
| variable {n : ℕ} {p : Fin n} (w : Witness (B := B) p) | |
| def dimensionDropHom : DimensionDrop (n+1) n →⋆ₐ[ℂ] B := | |
| w.generatedHom.comp (DimensionDropModel.concreteToGenerated p) | |
| end DimensionDropRelations.Witness | |
| end | |
| noncomputable section | |
| namespace DirectComparison | |
| open Pureness CurrentMain Filter Topology CompletelyPositiveOperations JiangSuConstruction | |
| open scoped ComplexOrder ENNReal CStarAlgebra | |
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| variable [Nontrivial A] [Nonempty (TracialState A)] [TopologicalSpace.SeparableSpace A] | |
| variable (U : Ultrafilter ℕ) | |
| local instance (n : ℕ) : Semiring (DimensionDrop (n+1) n) := | |
| (inferInstance : CStarAlgebra (DimensionDrop (n+1) n)).toSemiring | |
| theorem comparison_dimension_drop_hom (hfree : (U : Filter ℕ)≤atTop) | |
| (hnuc : IsNuclear.{0,0} A) (hsimple : TopologicallySimple A) (hinf : ¬FiniteDimensional ℂ A) | |
| (hcmp : StrictComparison A) (n : ℕ) (hn : 0<n) : | |
| Nonempty (DimensionDrop (n+1) n →⋆ₐ[ℂ] unitalNormCentralizer (A := A) U) := by | |
| obtain ⟨φ,s,hφn,hφrel,_hsn,hse,hsF⟩ := | |
| comparison_dimension_drop_relations U hfree hnuc hsimple hinf hcmp n hn | |
| let w : DimensionDropRelations.Witness (B := unitalNormCentralizer (A := A) U) (⟨0,hn⟩ : Fin n) := | |
| ⟨φ,hφn,hφrel,s,hse,hsF⟩ | |
| exact ⟨w.dimensionDropHom⟩ | |
| end DirectComparison | |
| end | |
| noncomputable section | |
| namespace JiangSuConstruction | |
| local instance (priority := 2000) dropStarRing (p q : ℕ) : StarRing (DimensionDrop p q) := | |
| CStarAlgebra.toStarRing | |
| local instance (p q : ℕ) : Semiring (DimensionDrop p q) := | |
| (inferInstance : CStarAlgebra (DimensionDrop p q)).toSemiring | |
| local instance (priority := 2000) dropStar (p q : ℕ) : Star (DimensionDrop p q) := | |
| (CStarAlgebra.toStarRing (A := DimensionDrop p q)).toStarMul.toInvolutiveStar.toStar | |
| theorem consecutive_multiplicities {p q : ℕ} (hp : 0<p) (hq : 0<q) | |
| (hco : p.Coprime q) : ∃ (n : ℕ),0<n ∧ ∃ m : Multiplicities p q, | |
| p*m.k=n+1 ∧ q*m.l=n := by | |
| let r := Nat.chineseRemainder hco 0 1 | |
| let N := (r:ℕ)+16*p*q | |
| have hN : 16*p*q≤N := by dsimp [N]; omega | |
| have hNp : p∣N := by | |
| apply Nat.modEq_zero_iff_dvd.mp | |
| exact r.2.1.add ((show p∣16*p*q by use 16*q; ring).modEq_zero_nat) | |
| have hNq : N ≡ 1 [MOD q] := by | |
| have hh := r.2.2.add ((show q∣16*p*q by use 16*p; ring).modEq_zero_nat) | |
| simpa only [Nat.add_zero] using hh | |
| have hNpos : 1≤N := by have hh := Nat.mul_pos hp hq; nlinarith | |
| have hd : q∣N-1 := by | |
| apply Nat.modEq_zero_iff_dvd.mp | |
| simpa only [Nat.sub_self] using hNq.sub_right hNpos (by omega : 1≤1) | |
| obtain ⟨k,hkN⟩ := hNp | |
| obtain ⟨l,hlN⟩ := hd | |
| have he : p*k=q*l+1 := by omega | |
| have hk : 8*q≤k := by nlinarith | |
| have hl : 8*p≤l := by nlinarith | |
| let m := multiplicitiesOfLarge hp hq k l hk hl | |
| refine ⟨q*l,by have h := Nat.mul_pos hq (show 0<l by omega); exact h,m,he,rfl⟩ | |
| theorem prime_drop_into_consecutive {p q : ℕ} (hp : 0<p) (hq : 0<q) | |
| (hco : p.Coprime q) : ∃ n : ℕ,0<n ∧ | |
| Nonempty (DimensionDrop p q →⋆ₐ[ℂ] DimensionDrop (n+1) n) := by | |
| obtain ⟨n,hn,m,hle,hri⟩ := consecutive_multiplicities hp hq hco | |
| refine ⟨n,hn,?_⟩ | |
| have h := concreteStep m | |
| rw [hle,hri] at h | |
| exact ⟨h⟩ | |
| end JiangSuConstruction | |
| namespace DirectComparison | |
| open Pureness CurrentMain Filter Topology CompletelyPositiveOperations JiangSuConstruction | |
| open scoped ComplexOrder ENNReal CStarAlgebra | |
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| variable [Nontrivial A] [Nonempty (TracialState A)] [TopologicalSpace.SeparableSpace A] | |
| variable (U : Ultrafilter ℕ) | |
| local instance (p q : ℕ) : Semiring (DimensionDrop p q) := | |
| (inferInstance : CStarAlgebra (DimensionDrop p q)).toSemiring | |
| theorem comparison_prime_drop_hom (hfree : (U : Filter ℕ)≤atTop) | |
| (hnuc : IsNuclear.{0,0} A) (hsimple : TopologicallySimple A) (hinf : ¬FiniteDimensional ℂ A) | |
| (hcmp : StrictComparison A) (p q : ℕ) (hp : 0<p) (hq : 0<q) (hco : p.Coprime q) : | |
| Nonempty (DimensionDrop p q →⋆ₐ[ℂ] unitalNormCentralizer (A := A) U) := by | |
| obtain ⟨n,hn,⟨ψ⟩⟩ := prime_drop_into_consecutive hp hq hco | |
| obtain ⟨φ⟩ := comparison_dimension_drop_hom U hfree hnuc hsimple hinf hcmp n hn | |
| exact ⟨φ.comp ψ⟩ | |
| end DirectComparison | |
| end | |
| end OAI | |