| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.DimensionDrop.SequentialLimitSimplicity |
| import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropHom |
| import OAI.Analysis.JiangSu.CentralSequences.CentralSaturation |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace SequentialCStar.System |
| open NormUltra NormUltra.QuotientAlgebra RationalTerms Filter Topology NonunitalGamma |
| open scoped ENNReal |
|
|
| local instance nonunitalSequentialRationalAlgebra {E : Type*} [CStarAlgebra E] : NormedAlgebra ℚ E := .restrictScalars ℚ ℂ E |
| local instance nonunitalSequentialRationalTower {E : Type*} [CStarAlgebra E] : IsScalarTower ℚ ℂ E := |
| IsScalarTower.restrictScalars ℚ ℂ E |
| variable {B : ℕ→Type} [∀ n,CStarAlgebra (B n)] [∀ n,Nontrivial (B n)] |
| variable (S : System B) |
| variable {A : Type} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable [TopologicalSpace.SeparableSpace A] (U : Ultrafilter ℕ) [Fact ((U : Filter ℕ)≤atTop)] |
| local notation "Fω" => CentralSequenceAlgebra (A := A) U |
|
|
| theorem nonunital_finite_stage_models (d : ℕ→S.Limit) (hdI : d 0=algebraMap ℂ S.Limit Complex.I) |
| (hstage : ∀ j,∃ (n : ℕ) (a : B n),S.inclusion n a=d j) |
| (hsupply : ∀ n,Nonempty (B n →⋆ₐ[ℂ] Fω)) : |
| NonunitalCentralSaturation.FiniteModels (A := A) U d := by |
| classical |
| choose n b hb using hstage |
| intro E |
| let F : Finset ℕ := insert 0 (E.biUnion Term.support) |
| let M : ℕ := F.sup n |
| have hn (j : ℕ) (hj : j∈F) : n j≤M := Finset.le_sup hj |
| let y : ℕ→B M := fun j=>if hj:j∈F then S.map (hn j hj) (b j) else 0 |
| have hy (j : ℕ) (hj : j∈F) : S.inclusion M (y j)=d j := by |
| dsimp [y] |
| rw [dite_eq_left hj,S.inclusion_compatible,hb] |
| obtain ⟨φ⟩ := hsupply M |
| let w : ℕ→Fω := fun j=>φ (y j) |
| refine ⟨w,?_,?_,?_⟩ |
| · intro j |
| by_cases hj:j∈F |
| · have he := S.inclusion_norm M (y j) |
| rw [hy j hj] at he |
| exact (NonUnitalStarAlgHom.norm_apply_le φ (y j)).trans he.symm.le |
| · simp only [w,y,dite_eq_right hj,map_zero,norm_zero] |
| exact norm_nonneg _ |
| · have he : y 0=algebraMap ℂ (B M) Complex.I := by |
| apply (S.inclusion_isometry M).injective |
| rw [hy 0 (Finset.mem_insert_self 0 _),hdI] |
| exact ((S.inclusion M).commutes Complex.I).symm |
| change φ (y 0)=_ |
| rw [he] |
| exact φ.commutes Complex.I |
| · intro e he |
| have hinc : (S.inclusion M) (e.eval y)=e.eval d := by |
| have hm := e.map_eval y ((S.inclusion M).restrictScalars ℚ) |
| exact hm.trans (e.eval_agree _ d (fun j hj=>hy j |
| (Finset.mem_insert_of_mem (Finset.mem_biUnion.mpr ⟨e,he,hj⟩)))) |
| have hφe : φ (e.eval y)=e.eval w := e.map_eval y (φ.restrictScalars ℚ) |
| rw [←hφe] |
| have hi := S.inclusion_norm M (e.eval y) |
| rw [hinc] at hi |
| exact (NonUnitalStarAlgHom.norm_apply_le φ (e.eval y)).trans hi.symm.le |
|
|
| theorem nonunital_embedding_of_stage_maps [∀ n,TopologicalSpace.SeparableSpace (B n)] |
| (hsupply : ∀ n,Nonempty (B n →⋆ₐ[ℂ] Fω)) : Nonempty (S.Limit →⋆ₐ[ℂ] Fω) := by |
| obtain ⟨d,hd,hdI,hs⟩ := S.exists_dense_stage_sequence |
| exact NonunitalCentralSaturation.rational_saturation U d hd hdI |
| (S.nonunital_finite_stage_models U d hdI hs hsupply) |
| end SequentialCStar.System |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace NonunitalSI |
| open Pureness CurrentMain NonunitalGamma Filter Topology JiangSuConstruction |
| open scoped ComplexOrder ENNReal CStarAlgebra |
|
|
| variable {A : Type} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable [Nontrivial A] [Nonempty (NormalizedTrace A)] [CompactSpace (NormalizedTrace A)] [TopologicalSpace.SeparableSpace A] |
| variable (U : Ultrafilter ℕ) [Fact ((U : Filter ℕ)≤atTop)] |
| local notation "Fω" => CentralSequenceAlgebra (A := A) U |
| local instance originalNonunitalStageSeparable (n : ℕ) : TopologicalSpace.SeparableSpace (OriginalJiangSu.Stage n) := by |
| infer_instance |
| local instance (n : ℕ) : Semiring (OriginalJiangSu.Stage n) := |
| (inferInstance : CStarAlgebra (OriginalJiangSu.Stage n)).toSemiring |
|
|
| theorem comparison_original_jiang_su_embedding |
| (hnuc : NuclearUnitization.IsNuclear.{0,0} A) (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hstable : StablyProjectionless A) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) : |
| ∃ φ : OriginalJiangSu.Algebra →⋆ₐ[ℂ] Fω,Function.Injective φ := by |
| have hs (n : ℕ) : Nonempty (OriginalJiangSu.Stage n →⋆ₐ[ℂ] Fω) := |
| comparison_prime_drop_hom U hnuc hsimple hstable hcone hcmp _ _ |
| (by have := (OriginalJiangSu.pair n).p_gt |
| obtain ⟨φ⟩ := OriginalJiangSu.system.nonunital_embedding_of_stage_maps U hs |
| let : Nontrivial Fω := ⟨⟨0,1,fun h=>by |
| have he := congrArg (centralSequenceToTracial (A := A) U) h |
| exact zero_ne_one (by simpa only [map_zero,map_one] using he)⟩⟩ |
| exact ⟨φ,CurrentMain.simple_starAlgHom_injective OriginalJiangSu.simple φ⟩ |
|
|
| end NonunitalSI |
|
|
| end |
| end |
|
|
| end OAI |
|
|
