| import Mathlib | |
| import OAI.Analysis.TracialSplitting.PreparedBlocks | |
| import OAI.Analysis.JiangSu.Comparison.LocalTools | |
| import OAI.Analysis.JiangSu.Comparison.FamilyExactLocalSupply | |
| namespace OAI | |
| section | |
| noncomputable section | |
| namespace FamilyComparison | |
| open Pureness CurrentMain DirectComparison DirectPacking Filter | |
| open scoped ComplexOrder | |
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] | |
| variable [Nontrivial A] {I : Type} [Nonempty I] (θ : I → TracialState A) | |
| variable (U : Ultrafilter ℕ) | |
| local notation "D" => FamilyUltrapower θ U | |
| local notation "Λ" => familyLimitTrace θ U | |
| @[instance_reducible] def limitTraceFamily : TraceFamily D where | |
| carrier := Set.range Λ | |
| lemma limitTraceFamily_nonempty : | |
| let := limitTraceFamily θ U | |
| Nonempty (RestrictedTracialState D) := by | |
| let := limitTraceFamily θ U | |
| let s : ℕ → I := fun _ => Classical.ofNonempty | |
| exact ⟨⟨Λ s,⟨s,rfl⟩⟩⟩ | |
| lemma source_tools (hfree : (U : Filter ℕ) ≤ atTop) | |
| (hsupply : SpectralProjectionSupply Λ) : | |
| let := limitTraceFamily θ U | |
| SpectralProjectionSupply (fun τ : RestrictedTracialState D => (τ : TracialState D)) ∧ | |
| CoefficientSupply («D» := D) ∧ InsertionSupply («D» := D) := by | |
| let := limitTraceFamily θ U | |
| constructor | |
| · intro h hh hhn α hα hα1 | |
| obtain ⟨p,hp,hl,hr,htr⟩ := hsupply h hh hhn α hα hα1 | |
| refine ⟨p,hp,hl,hr,?_⟩ | |
| rintro ⟨τ,⟨s,rfl⟩⟩ | |
| exact htr s | |
| constructor | |
| · intro q hq s hs F ε hε | |
| obtain ⟨e,he,heq,ht⟩ := finite_coefficient_split Λ hsupply | |
| (fun q hq t ht => proportional_subprojection θ U hsupply hq ht) hq hs F hε | |
| refine ⟨e,he,heq,?_⟩ | |
| intro b hb | |
| rintro ⟨τ,⟨t,rfl⟩⟩ | |
| exact ht b hb t | |
| · intro z y hz hy | |
| obtain ⟨t,ht,htz,htr⟩ := FamilyComparison.orthogonal_insertion θ U hfree hsupply hz hy | |
| refine ⟨t,ht,htz,?_⟩ | |
| rintro ⟨τ,⟨s,rfl⟩⟩ | |
| exact htr s | |
| end FamilyComparison | |
| end | |
| end | |
| end OAI | |