| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.Nuclearity.NuclearUnitization |
| import OAI.Analysis.JiangSu.Excision.OrthogonalExcision |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace NonunitalSI |
| open Pureness CurrentMain NonunitalGamma PureStateExcision Filter Topology CStarAlgebra NuclearNorm |
| open scoped ComplexOrder ENNReal |
|
|
| 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) |
| local notation "N" => NormUltra.QuotientAlgebra (fun _ : ℕ => A) U |
| local notation "D" => Unitization ℂ N |
| local notation "B" => Unitization ℂ A |
| local notation "q" => NonunitalGamma.normToTracial (A := A) U |
| local notation "ρ" => unitizedNormInclusion (A := A) U |
| local notation "ι" => NonunitalGamma.normInclusion (A := A) U |
| local notation "L" => familyLimitTrace (fun τ : NormalizedTrace A => NormalizedTrace.unitized τ) U |
|
|
| theorem finite_nuclear_identity_excision |
| (hnuc : NuclearUnitization.IsNuclear.{0,0} A) (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hprojectionless : ∀ p : A,IsStarProjection p → p=0) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) |
| (hfree : (U : Filter ℕ) ≤ atTop) |
| (E F : N) (he : 0≤E) (hen : ‖E‖≤1) (hsmall : q E=0) |
| (hec : ∀ a : A,Commute E (ι a)) |
| (hf : 0≤F) (hfn : ‖F‖≤1) (hfc : ∀ a : A,Commute F (ι a)) |
| (hprojection : IsStarProjection (q F)) |
| {t : ℝ} (ht : 0<t) (hlarge : ∀ s : ℕ → NormalizedTrace A,2*t ≤ ((L s).functional (q F)).re) |
| {I : Type} [Fintype I] (a : I → selfAdjoint A) {ε : ℝ} (hε : 0<ε) : |
| ∃ v : N,‖v‖≤2 ∧ F*v=v ∧ ‖star v*v-E‖<ε ∧ |
| ∀ i,‖star v*ι (a i : A)*v-ι (a i : A)*E‖<ε := by |
| classical |
| let : TopologicalSpace.SeparableSpace B := |
| (Unitization.uniformEquivProd (𝕜 := ℂ) (A := A)).symm.surjective.denseRange.separableSpace |
| (Unitization.uniformEquivProd (𝕜 := ℂ) (A := A)).symm.continuous |
| obtain ⟨φ,hφ1,hpure,hfaith⟩ := exists_faithful_pure_unitized_state hsimple hprojectionless |
| let b : Option I → selfAdjoint B |
| | none => 1 |
| | some i => ⟨Unitization.inr (a i : A),(a i).property.map (Unitization.inrNonUnitalStarAlgHom ℂ A)⟩ |
| let δ := min ε 1 |
| have hδ : 0<δ := lt_min hε zero_lt_one |
| obtain ⟨R,hR⟩ := nuclear_state_coefficient_approximation |
| (NuclearUnitization.nuclear_unitization hnuc) φ hfaith b hδ |
| obtain ⟨v,_hvq,hvF,hv⟩ := nuclear_list_excision U hsimple hprojectionless hcone hcmp hfree |
| E F he hen hsmall hec hf hfn hfc hprojection ht hlarge φ hφ1 hpure R |
| have herr (i : Option I) : |
| ‖star (Unitization.inr v)*ρ (b i : B)*Unitization.inr v-ρ (b i : B)*Unitization.inr E‖<δ := by |
| rw [hv,←sub_mul,←map_sub] |
| calc |
| _≤‖ρ ((stateRowSum φ R (b i) : B)-(b i :B))‖*‖(Unitization.inr E : D)‖ := norm_mul_le _ _ |
| _≤‖(stateRowSum φ R (b i) : B)-(b i :B)‖*1 := |
| mul_le_mul (NonUnitalStarAlgHom.norm_apply_le (ρ).toNonUnitalStarAlgHom _) (by simpa only [Unitization.norm_inr] using hen) |
| (norm_nonneg _) (norm_nonneg _) |
| _=‖b i-stateRowSum φ R (b i)‖ := by rw [mul_one,norm_sub_rev]; rfl |
| _≤‖b-fun i => stateRowSum φ R (b i)‖ := norm_le_pi_norm (b-fun i => stateRowSum φ R (b i)) i |
| _<δ := hR |
| have hsq : ‖star v*v-E‖<δ := by |
| have hh := herr none |
| simpa only [b,selfAdjoint.val_one,map_one,mul_one,one_mul,←Unitization.inr_star, ←Unitization.inr_mul,←Unitization.inr_sub,Unitization.norm_inr] using hh |
| have hnorm : ‖v‖≤2 := by |
| have ht := norm_add_le (star v*v-E) E |
| rw [sub_add_cancel] at ht |
| have hvsq : ‖star v*v‖=‖v‖*‖v‖ := CStarRing.norm_star_mul_self |
| have hδ1 : δ≤1 := min_le_right _ _ |
| nlinarith [norm_nonneg v] |
| refine ⟨v,hnorm,hvF,hsq.trans_le (min_le_left _ _),fun i => ?_⟩ |
| have hh := (herr (some i)).trans_le (min_le_left _ _) |
| simpa only [b,unitizedNormInclusion_inr,←Unitization.inr_star,←Unitization.inr_mul, ←Unitization.inr_sub,Unitization.norm_inr] using hh |
|
|
| end NonunitalSI |
|
|
| end |
| end |
|
|
| end OAI |
|
|
