| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropGeneratedHom |
| import OAI.Analysis.JiangSu.DimensionDrop.DimensionDropRelations2 |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace NonunitalSI |
| open Pureness CurrentMain NonunitalGamma Filter Topology CompletelyPositiveOperations 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 (p q : ℕ) : Semiring (DimensionDrop p q) := |
| (inferInstance : CStarAlgebra (DimensionDrop p q)).toSemiring |
|
|
| theorem comparison_dimension_drop_hom |
| (hnuc : NuclearUnitization.IsNuclear.{0,0} A) (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hstable : StablyProjectionless A) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) |
| (n : ℕ) (hn : 0<n) : |
| Nonempty (DimensionDrop (n+1) n →⋆ₐ[ℂ] Fω) := by |
| obtain ⟨φ,s,hφn,hφrel,_hsn,hse,hsF,_hφ1⟩ := comparison_dimension_drop_relations U |
| hnuc hsimple hstable hcone hcmp n hn |
| let w : DimensionDropRelations.Witness (B := Fω) (⟨0,hn⟩ : Fin n) := |
| { φ := φ |
| contractive := hφn |
| orderZero := hφrel |
| s := s |
| small := hse |
| support := hsF } |
| exact ⟨w.dimensionDropHom⟩ |
|
|
| end NonunitalSI |
|
|
| 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 instance (p q : ℕ) : Semiring (DimensionDrop p q) := |
| (inferInstance : CStarAlgebra (DimensionDrop p q)).toSemiring |
|
|
| theorem comparison_prime_drop_hom |
| (hnuc : NuclearUnitization.IsNuclear.{0,0} A) (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hstable : StablyProjectionless A) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) |
| (p q : ℕ) (hp : 0<p) (hq : 0<q) (hco : p.Coprime q) : |
| Nonempty (DimensionDrop p q →⋆ₐ[ℂ] CentralSequenceAlgebra (A := A) U) := by |
| obtain ⟨n,hn,⟨ψ⟩⟩ := prime_drop_into_consecutive hp hq hco |
| obtain ⟨φ⟩ := comparison_dimension_drop_hom U hnuc hsimple hstable hcone hcmp n hn |
| exact ⟨φ.comp ψ⟩ |
|
|
| end NonunitalSI |
|
|
| end |
| end |
|
|
| end OAI |
|
|
