| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.Comparison.FiniteCoefficients |
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| namespace OAI |
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| section |
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| noncomputable section |
| namespace DirectComparison |
| open Pureness CurrentMain Filter |
| open scoped Topology ComplexOrder |
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|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable [Nontrivial A] [Nonempty (TracialState A)] |
| variable (U : Ultrafilter ℕ) (hfree : (U : Filter ℕ) ≤ atTop) |
| variable (hsimple : TopologicallySimple A) (hinf : ¬FiniteDimensional ℂ A) |
| variable (hcmp : StrictComparison A) (hnuc : IsNuclear A) |
| local notation "τ" => (fun t : TracialState A => t) |
| local notation "D" => FamilyUltrapower τ U |
| local notation "Λ" => familyLimitTrace τ U |
| include hfree hsimple hinf hcmp hnuc in |
| theorem comparison_dyadic_approximation (S : Set D) (hS : S.Countable) (n : ℕ) |
| {r : D} (hr : IsStarProjection r) (hrc : ∀ a ∈ S, Commute r a) |
| {t : ℝ} (ht : t ∈ Set.Icc 0 1) : |
| ∃ p : D, ∃ α : ℝ, IsStarProjection p ∧ p ≤ r ∧ α ∈ Set.Icc 0 1 ∧ |
| |α-t| ≤ (1/2:ℝ)^n ∧ ∀ a ∈ S, Commute p a ∧ ∀ s : ℕ → TracialState A, |
| (Λ s).functional (p*a)=(α:ℂ)*(Λ s).functional (r*a) := by |
| induction n generalizing r t with |
| | zero => |
| refine ⟨0,0,IsStarProjection.zero _,hr.nonneg,⟨le_rfl,zero_le_one⟩,?_,?_⟩ |
| · simpa only [zero_sub,abs_neg,abs_of_nonneg ht.1,pow_zero] using ht.2 |
| · intro a ha |
| exact ⟨Commute.zero_left a,by intro s; simp⟩ |
| | succ n ih => |
| obtain ⟨h,hh,hhr,hht⟩ := comparison_corner_halving U hfree hsimple hinf hcmp hnuc S hS hr hrc |
| have hhc : ∀ a ∈ S, Commute h a := fun a ha => (hht a ha).1 |
| by_cases ht2 : t ≤ 1/2 |
| · have h2t : 2*t ∈ Set.Icc 0 1 := ⟨by linarith [ht.1],by linarith⟩ |
| obtain ⟨p,α,hp,hph,hα,herr,hpt⟩ := ih hh hhc h2t |
| refine ⟨p,α/2,hp,hph.trans hhr,⟨by linarith [hα.1],by linarith [hα.2]⟩,?_,?_⟩ |
| · rw [abs_sub_le_iff] at herr ⊢ |
| rw [pow_succ] |
| constructor <;> nlinarith [herr.1,herr.2] |
| · intro a ha |
| refine ⟨(hpt a ha).1,?_⟩ |
| intro s |
| rw [(hpt a ha).2 s,(hht a ha).2 s] |
| push_cast |
| ring |
| · have ht2' : 1/2 ≤ t := (lt_of_not_ge ht2).le |
| have h2t : 2*t-1 ∈ Set.Icc 0 1 := ⟨by linarith,by linarith [ht.2]⟩ |
| have hhr' : h*r=h := (hh.le_iff_mul_eq_left hr).mp hhr |
| have hrem : IsStarProjection (r-h) := hh.sub_of_mul_eq_left hr hhr' |
| have hremc : ∀ a ∈ S, Commute (r-h) a := fun a ha => (hrc a ha).sub_left (hhc a ha) |
| obtain ⟨p,α,hp,hprem,hα,herr,hpt⟩ := ih hrem hremc h2t |
| have hremp : (r-h)*p=p := (hp.le_iff_mul_eq_right hrem).mp hprem |
| have horth : h*p=0 := by |
| calc |
| h*p=h*((r-h)*p) := by rw [hremp] |
| _=(h*(r-h))*p := (mul_assoc _ _ _).symm |
| _=0 := by rw [mul_sub,hhr',hh.isIdempotentElem.eq,sub_self,zero_mul] |
| have hsum : h+p ≤ r := by |
| have hhadd := add_le_add_left hprem h |
| simpa only [sub_add_cancel,add_comm p h] using hhadd |
| refine ⟨h+p,(1+α)/2,hh.add hp horth,hsum, |
| ⟨by linarith [hα.1],by linarith [hα.2]⟩,?_,?_⟩ |
| · rw [abs_sub_le_iff] at herr ⊢ |
| rw [pow_succ] |
| constructor <;> nlinarith [herr.1,herr.2] |
| · intro a ha |
| refine ⟨(hhc a ha).add_left (hpt a ha).1,?_⟩ |
| intro s |
| rw [add_mul,map_add,(hpt a ha).2 s,sub_mul,map_sub,(hht a ha).2 s] |
| push_cast |
| ring |
|
|
| end DirectComparison |
| end |
| end |
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| end OAI |
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