| 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 |
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|
|
| 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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