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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropPartitionWeights
import OAI.Analysis.JiangSu.MatrixAlgebras.MatrixDiagonalComposition
import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropSquareHalfFlip
import OAI.Analysis.JiangSu.DimensionDrop.SequentialCStarInductiveLimit
import OAI.Analysis.JiangSu.TensorProducts.FaithfulTensorExtension
namespace OAI
noncomputable section
namespace JiangSuConstruction
open Pureness
open scoped CStarAlgebra ComplexOrder TensorProduct
lemma hom_conjugation_difference_bound {B C : Type*} [CStarAlgebra B] [CStarAlgebra C]
(θ : B →⋆ₐ[ℂ] C) (u : unitary B) (x y : B) :
‖(homUnitary θ u:C)*θ x*star (homUnitary θ u:C)-θ y‖≤
‖(u:B)*x*star (u:B)-y‖ := by
change ‖θ (u:B)*θ x*star (θ (u:B))-θ y‖≤_
rw [←map_star,←map_mul,←map_mul,←map_sub]
exact NonUnitalStarAlgHom.norm_apply_le θ _
variable {p q : ℕ} [NeZero p] [NeZero q]
variable {T : Type*} [CStarAlgebra T]
variable (g : DimensionDrop p q ⊗[ℂ] DimensionDrop p q →⋆ₐ[ℂ] T)
theorem tensor_connected_half_flip_of_matching (hg : Function.Injective g)
(F : Finset (DimensionDrop p q)) {ε : ℝ} (hε : 0<ε)
(W : C(Interval×Interval,unitary (TensorMat p q))) (hWd : ∀ s,W (s,s)=1)
(hW : ∀ f∈F,∀ s t,‖unitaryConjugation (W (s,t)) (f.val s)-f.val t‖<ε/42) :
∃ u : unitary T,Joined 1 u ∧ ∀ f∈F,
‖(u:T)*g (f ⊗ₜ[ℂ] 1)*star (u:T)-g (1 ⊗ₜ[ℂ] f)‖<ε := by
let θ := faithfulTensorExtension g dimensionDrop_nuclear (squareTensorMap p q p q)
(squareTensorMap_injective p q p q) (squareTensorMap_dense (p:=p) (q:=q) (r:=p) (s:=q)) hg
have hθ (z : DimensionDrop p q ⊗[ℂ] DimensionDrop p q) : θ (squareTensorMap p q p q z)=g z :=
faithfulTensorExtension_apply _ _ _ _ _ _ z
obtain ⟨u,huj,hu⟩ := square_connected_half_flip_of_matching F hε W hWd hW
refine ⟨homUnitary θ u,?_,?_⟩
· simpa only [homUnitary_one] using huj.map (homUnitary_continuous θ)
· intro f hf
have hn := hom_conjugation_difference_bound θ u
(squareTensorMap p q p q (f ⊗ₜ[ℂ] 1)) (squareTensorMap p q p q (1 ⊗ₜ[ℂ] f))
rw [hθ,hθ] at hn
exact hn.trans_lt (hu f hf)
end JiangSuConstruction
namespace SequentialCStar.System
open scoped CStarAlgebra
variable {C : ℕ → Type*} [∀ n,CStarAlgebra (C n)] [∀ n,Nontrivial (C n)]
variable (S : System C)
theorem finite_set_common_stage (F : Finset S.Limit) {ε : ℝ} (hε : 0<ε) :
∃ (n : ℕ) (b : F → C n),∀ x : F,‖(x:S.Limit)-S.inclusion n (b x)‖<ε := by
classical
choose n a ha using fun x : F => S.finite_stage_approximation (x:S.Limit) hε
let N := Finset.univ.sup n
have hN (x : F) : n x≤N := Finset.le_sup (f := n) (Finset.mem_univ x)
refine ⟨N,fun x => S.map (hN x) (a x),?_⟩
intro x
rw [S.inclusion_compatible]
exact ha x
variable {T : Type*} [CStarAlgebra T]
variable (ι κ : S.Limit →⋆ₐ[ℂ] T)
def ApproximateConjugacy : Prop :=
∀ (F : Finset S.Limit) (ε : ℝ),0<ε → ∃ u : unitary T,
∀ x∈F,‖(u:T)*ι x*star (u:T)-κ x‖<ε
def FiniteStageConjugacy : Prop :=
∀ (n : ℕ) (F : Finset (C n)) (ε : ℝ),0<ε → ∃ u : unitary T,
∀ x∈F,‖(u:T)*ι (S.inclusion n x)*star (u:T)-κ (S.inclusion n x)‖<ε
lemma conjugacy_difference_bound (u : unitary T) (x y : S.Limit) :
‖(u:T)*ι x*star (u:T)-(u:T)*ι y*star (u:T)‖≤‖x-y‖ := by
rw [← sub_mul,← mul_sub,←map_sub]
change ‖((u:T)*ι (x-y))*((star u : unitary T):T)‖≤‖x-y‖
rw [CStarRing.norm_mul_coe_unitary,CStarRing.norm_coe_unitary_mul]
exact NonUnitalStarAlgHom.norm_apply_le ι (x-y)
theorem approximateConjugacy_iff_finiteStages :
ApproximateConjugacy S ι κ ↔ FiniteStageConjugacy S ι κ := by
classical
constructor
· intro h n F ε hε
obtain ⟨u,hu⟩ := h (F.image (S.inclusion n)) ε hε
exact ⟨u,fun x hx => hu _ (Finset.mem_image.mpr ⟨x,hx,rfl⟩)⟩
· intro h F ε hε
obtain ⟨n,b,hb⟩ := S.finite_set_common_stage F (div_pos hε (by norm_num : (0:ℝ)<3))
obtain ⟨u,hu⟩ := h n (Finset.univ.image b) (ε/3) (by positivity)
refine ⟨u,?_⟩
intro x hx
let z : F := ⟨x,hx⟩
let y := S.inclusion n (b z)
have hxy : ‖x-y‖<ε/3 := hb z
have hmid : ‖(u:T)*ι y*star (u:T)-κ y‖<ε/3 :=
hu _ (Finset.mem_image.mpr ⟨z,Finset.mem_univ _,rfl⟩)
have hleft : ‖(u:T)*ι x*star (u:T)-(u:T)*ι y*star (u:T)‖<ε/3 :=
(S.conjugacy_difference_bound ι u x y).trans_lt hxy
have hright : ‖κ y-κ x‖<ε/3 := by
rw [←map_sub]
exact (NonUnitalStarAlgHom.norm_apply_le κ (y-x)).trans_lt (by simpa only [norm_sub_rev] using hxy)
have htri₁ := norm_sub_le_norm_sub_add_norm_sub
((u:T)*ι x*star (u:T)) ((u:T)*ι y*star (u:T)) (κ x)
have htri₂ := norm_sub_le_norm_sub_add_norm_sub
((u:T)*ι y*star (u:T)) (κ y) (κ x)
linarith
end SequentialCStar.System
end
end OAI
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