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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | 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; omega) (by have := (OriginalJiangSu.pair n).q_gt; omega) ((OriginalJiangSu.pair n).coprime)
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
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