| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.Nonunital.RelativeCoefficient | |
| import OAI.Analysis.JiangSu.Absorption.OneSidedIntertwiningChain2 | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace NonunitalRelative | |
| open Pureness NonunitalGamma NormUltra NormUltra.QuotientAlgebra Filter CStarAbsorption | |
| open scoped CStarAlgebra ENNReal TensorProduct | |
| universe u | |
| variable {D : Type u} [CStarAlgebra D] [Nontrivial D] | |
| variable {A B E T : Type*} [NonUnitalCStarAlgebra A] [NonUnitalCStarAlgebra B] | |
| variable [CStarAlgebra E] [CStarAlgebra T] | |
| variable [PartialOrder A] [StarOrderedRing A] [TopologicalSpace.SeparableSpace A] | |
| variable (U : Ultrafilter ℕ) [Fact ((U : Filter ℕ)≤atTop)] | |
| variable (ι : A →⋆ₙₐ[ℂ] B) (k : B →⋆ₙₐ[ℂ] E) (κ : D →⋆ₐ[ℂ] E) | |
| variable (hc : ∀ a d,Commute (k (ι a)) (κ d)) | |
| variable (hj : ∃ a : A,k (ι a)≠0) (φ : D →⋆ₐ[ℂ] CentralSequenceAlgebra (A := A) U) | |
| variable (hD : IsNuclear.{u,u} D) (hs : Pureness.TopologicallySimple D) | |
| variable (f : D ⊗[ℂ] D →⋆ₐ[ℂ] T) (hf : Function.Injective f) (hd : DenseRange f) | |
| variable (hh : ConnectedApproximateInnerHalfFlip f) | |
| variable (hcoef : ∀ b : B,∀ ε : ℝ,0<ε → ∃ l : List (A×D), | |
| ‖coefficientSum (k.comp ι) κ l-k b‖<ε) | |
| include hc hj φ hD hs hf hd hh hcoef in | |
| theorem ambient_quotient_conjugacy : NonunitalIntertwining.AmbientQuotientConjugacy k U ι := by | |
| classical | |
| intro F G ε hε | |
| have hlist (b : G) : ∃ l : List (A×D),‖coefficientSum (k.comp ι) κ l-k b‖<ε/2 := | |
| hcoef b (ε/2) (half_pos hε) | |
| choose l hl using hlist | |
| obtain ⟨u,hu⟩ := finite_coefficient_approximation U (k.comp ι) κ hc hj φ hD hs f hf hd hh l (half_pos hε) | |
| let v : unitary (QuotientAlgebra (fun _ : ℕ => E) U) := | |
| Unitary.map (CStarConnectedLift.starMonoidHom (Centralizer U (k.comp ι)).subtype) u | |
| refine ⟨v,?_,?_⟩ | |
| · intro a ha | |
| have he := unitary_conjugate_inclusion U (k.comp ι) u a | |
| change ‖((u:Centralizer U (k.comp ι)):QuotientAlgebra (fun _ : ℕ => E) U)* | |
| normInclusion U (k (ι a))*star ((u:Centralizer U (k.comp ι)):QuotientAlgebra (fun _ : ℕ => E) U)- | |
| normInclusion U (k (ι a))‖<ε | |
| rw [show normInclusion U (k (ι a))=normInclusion U ((k.comp ι) a) from rfl,he,sub_self,norm_zero] | |
| exact hε | |
| · intro b hb | |
| obtain ⟨x,hx⟩ := hu ⟨b,hb⟩ | |
| refine ⟨x,?_⟩ | |
| have hnorm : ‖normInclusion U (coefficientSum (k.comp ι) κ (l ⟨b,hb⟩))-normInclusion U (k b)‖<ε/2 := by | |
| rw [←map_sub,normInclusion_norm] | |
| exact hl ⟨b,hb⟩ | |
| have htriangle := norm_sub_le_norm_sub_add_norm_sub | |
| ((v : QuotientAlgebra (fun _ : ℕ => E) U)*NonunitalNormMaps.quotientMap U (k.comp ι) x* | |
| star (v : QuotientAlgebra (fun _ : ℕ => E) U)) | |
| (normInclusion U (coefficientSum (k.comp ι) κ (l ⟨b,hb⟩))) (normInclusion U (k b)) | |
| have hx' : ‖(v : QuotientAlgebra (fun _ : ℕ => E) U)*NonunitalNormMaps.quotientMap U (k.comp ι) x* | |
| star (v : QuotientAlgebra (fun _ : ℕ => E) U)- | |
| normInclusion U (coefficientSum (k.comp ι) κ (l ⟨b,hb⟩))‖<ε/2 := hx | |
| linarith | |
| include hc hj φ hD hs hf hd hh hcoef in | |
| theorem equiv_of_central_embedding [TopologicalSpace.SeparableSpace B] | |
| (hi : Isometry ι) (hk : Isometry k) | |
| (hleft : ∀ (e : E) (b : B),e*k b∈Set.range k) | |
| (hright : ∀ (b : B) (e : E),k b*e∈Set.range k) : Nonempty (A ≃⋆ₐ[ℂ] B) := | |
| NonunitalIntertwining.equiv_of_ambient_quotient_conjugacy k hk hleft hright U ι hi | |
| (ambient_quotient_conjugacy U ι k κ hc hj φ hD hs f hf hd hh hcoef) | |
| end NonunitalRelative | |
| end | |
| end | |
| end OAI | |