| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.Excision.PureStateExcisionNormUltrapower |
| import OAI.Analysis.JiangSu.Nonunital.ControlledComparison |
| import OAI.Analysis.JiangSu.Nonunital.EventualFullness |
|
|
| namespace OAI |
|
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|
|
| noncomputable section |
| namespace NonunitalSI |
| open Pureness CurrentMain DirectComparison NonunitalGamma PureStateExcision Filter Topology CStarAlgebra |
| open scoped ComplexOrder ENNReal |
|
|
| variable {A : Type} [NonUnitalCStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
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| theorem pure_state_small_excision [Nonempty (NormalizedTrace A)] [CompactSpace (NormalizedTrace A)] [Nontrivial A] |
| (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hprojectionless : ∀ p : A,IsStarProjection p → p=0) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) |
| (U : Ultrafilter ℕ) |
| (e f : lp (fun _ : ℕ => A) ∞) |
| (he : ∀ n,0 ≤ e n) (hen : ∀ n,‖e n‖ ≤ 1) |
| (hf : ∀ n,0 ≤ f n) (hfn : ∀ n,‖f n‖ ≤ 1) |
| (hcentral : ∀ a : A,Tendsto (fun n => ‖f n*a-a*f n‖) (U : Filter ℕ) (𝓝 0)) |
| (hsmall : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A, |
| (τ.functional (e n)).re<η) |
| {t : ℝ} (ht : 0<t) |
| (hlarge : ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A,t ≤ (τ.functional (f n)).re) |
| (hdefect : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A, |
| (τ.functional (f n-f n*f n)).re<η) |
| (φ : Unitization ℂ A →ₚ[ℂ] ℂ) (hφ1 : φ 1=1) |
| (hpure : ofPositiveState φ hφ1∈(realStateCarrier (A := Unitization ℂ A)).extremePoints ℝ) |
| {I : Type} [Fintype I] (a : I → Unitization ℂ A) {ε : ℝ} (hε : 0<ε) : |
| ∀ᶠ n in (U : Filter ℕ),∃ s : A, ‖s‖ ≤ 2 ∧ ‖star s*s-e n‖<ε ∧ |
| ‖f n*s-s‖<ε ∧ ∀ i,‖compression (a i) s-φ (a i) • e n‖<ε := by |
| obtain ⟨h,c,hh,hhn,hc,hcn,hc0,hhc,hex⟩ := pure_state_plateau_excision φ hφ1 hpure a |
| (show 0<ε/16 by positivity) |
| obtain ⟨β,hβ,hb⟩ := NonunitalSI.eventual_hereditary_upperCut_trace_lower hsimple hprojectionless U f hf hfn |
| hcentral ht hlarge hdefect c hc hcn hc0 |
| let M : ℝ := (∑ i,‖φ (a i)‖)+1 |
| have hM : 1 ≤ M := by |
| have hn := Finset.sum_nonneg (fun i (_ : i∈(Finset.univ : Finset I)) => norm_nonneg (φ (a i))) |
| dsimp [M]; linarith |
| have hM0 : 0<M := lt_of_lt_of_le zero_lt_one hM |
| have hiM (i : I) : ‖φ (a i)‖ ≤ M := by |
| have h := Finset.single_le_sum (fun j _ => norm_nonneg (φ (a j))) (Finset.mem_univ i) |
| dsimp [M]; linarith |
| let κ := min 1 (ε/(16*M)) |
| have hκ : 0<κ := lt_min zero_lt_one (by positivity) |
| have hκ1 : κ ≤ 1 := min_le_left _ _ |
| have hκb : 16*κ*M ≤ ε := by |
| have h := min_le_right (1 : ℝ) (ε/(16*M)) |
| rw [le_div_iff₀ (by positivity)] at h |
| dsimp [κ]; nlinarith |
| have hκe : κ+κ<ε := by |
| nlinarith [mul_nonneg hκ.le (sub_nonneg.mpr hM)] |
| let δ := min 1 (ε*β/16) |
| have hδ : 0<δ := lt_min zero_lt_one (by positivity) |
| have hδ1 : δ ≤ 1 := min_le_left _ _ |
| have hδb : δ ≤ ε*β/16 := min_le_right _ _ |
| have hsupport : 4*β⁻¹*δ ≤ ε/4 := by |
| have heq : 4*β⁻¹*δ=(4*δ)/β := by ring |
| rw [heq,div_le_iff₀ hβ] |
| linarith |
| have hcomm := (tendsto_order.1 (norm_central_boundedInr U f hcentral c)).2 δ hδ |
| filter_upwards [hb δ hδ hδ1,hcomm,hsmall (κ*β) (mul_pos hκ hβ)] with n hn hnc hne |
| let w : A := (hereditaryUpperCut δ (Unitization.inr (f n)) c).snd |
| have hwI : (Unitization.inr w : Unitization ℂ A)=hereditaryUpperCut δ (Unitization.inr (f n)) c := |
| inr_hereditaryUpperCut hδ1 (f n) c |
| have hw : 0≤w := Unitization.inr_nonneg_iff.mp (hwI.symm ▸ |
| hereditaryUpperCut_nonneg hδ.le (Unitization.inr (f n)) c hc.isSelfAdjoint) |
| have hwn : ‖w‖≤1 := by |
| rw [←Unitization.norm_inr (𝕜 := ℂ),hwI] |
| exact hereditaryUpperCut_norm_le hδ (Unitization.inr (f n)) c |
| (by simpa only [Unitization.norm_inr] using hfn n) hcn |
| have hhw : h*Unitization.inr w=Unitization.inr w := by |
| rw [hwI] |
| exact hereditaryUpperCut_support δ (Unitization.inr (f n)) c h hhc |
| have hn' (τ : NormalizedTrace A) : 2*β≤(τ.functional w).re := by |
| rw [←NormalizedTrace.unitized_inr,hwI] |
| exact hn τ |
| obtain ⟨s,hsn,hse,hhs,hfs⟩ := small_trace_supported_isometry_inr hcone hcmp (e n) w h |
| (Unitization.inr (f n)) (he n) (hen n) hw hwn hhw hκ hβ hκ hκ1 hne hn' |
| refine ⟨s,hsn,hse.trans hκe,?_,?_⟩ |
| · change ‖Unitization.inr (f n)*c-c*Unitization.inr (f n)‖<δ at hnc |
| have hwf := hereditaryUpperCut_error hδ (Unitization.inr (f n)) c |
| (Unitization.inr_nonneg_iff.mpr (hf n)) |
| (by simpa only [Unitization.norm_inr] using hfn n) hcn |
| rw [←hwI] at hwf |
| rw [←Unitization.inr_mul,←Unitization.inr_sub,Unitization.norm_inr] at hfs |
| have hcoef : 0 ≤ 2*β⁻¹ := by positivity |
| have hbound : ‖f n*s-s‖ ≤ 4*β⁻¹*δ := by |
| calc |
| _ ≤ 2*β⁻¹*(‖Unitization.inr (f n)*c-c*Unitization.inr (f n)‖+δ) := hfs.trans |
| (mul_le_mul_of_nonneg_left hwf hcoef) |
| _ ≤ 4*β⁻¹*δ := by nlinarith [mul_le_mul_of_nonneg_left hnc.le hcoef] |
| exact lt_of_le_of_lt (hbound.trans hsupport) (by linarith) |
| · intro i |
| have hseI : ‖star (Unitization.inr s : Unitization ℂ A)*Unitization.inr s-Unitization.inr (e n)‖≤κ+κ := by |
| rw [←Unitization.inr_star,←Unitization.inr_mul,←Unitization.inr_sub,Unitization.norm_inr] |
| exact hse.le |
| have hx := CStarExcision.compression_error_le h (Unitization.inr s) (a i) (Unitization.inr (e n)) |
| (φ (a i)) hh.isSelfAdjoint.star_eq hhs (hex i).le hseI |
| rw [Unitization.norm_inr,←norm_inr_compression_sub] at hx |
| have hsn2 : ‖s‖^2 ≤ 4 := by nlinarith [norm_nonneg s] |
| have hexc : ‖compression (a i) s-φ (a i) • e n‖ ≤ 4*(ε/16)+M*(κ+κ) := |
| hx.trans (add_le_add (mul_le_mul_of_nonneg_right hsn2 (by positivity)) |
| (mul_le_mul_of_nonneg_right (hiM i) (by positivity))) |
| exact lt_of_le_of_lt hexc (by nlinarith) |
|
|
| theorem pure_state_excision_norm_ultrapower [Nonempty (NormalizedTrace A)] [CompactSpace (NormalizedTrace A)] [Nontrivial A] [TopologicalSpace.SeparableSpace A] |
| (hsimple : NonunitalGamma.TopologicallySimple A) |
| (hprojectionless : ∀ p : A,IsStarProjection p → p=0) |
| (hcone : AllDenselyFiniteTracesBounded A) (hcmp : NonunitalGamma.StrictComparison hcone) |
| (U : Ultrafilter ℕ) |
| (hfree : (U : Filter ℕ) ≤ atTop) |
| (e f : lp (fun _ : ℕ => A) ∞) |
| (he : ∀ n,0 ≤ e n) (hen : ∀ n,‖e n‖ ≤ 1) |
| (hf : ∀ n,0 ≤ f n) (hfn : ∀ n,‖f n‖ ≤ 1) |
| (hcentral : ∀ a : A,Tendsto (fun n => ‖f n*a-a*f n‖) (U : Filter ℕ) (𝓝 0)) |
| (hsmall : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A, |
| (τ.functional (e n)).re<η) |
| {t : ℝ} (ht : 0<t) |
| (hlarge : ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A,t ≤ (τ.functional (f n)).re) |
| (hdefect : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : NormalizedTrace A, |
| (τ.functional (f n-f n*f n)).re<η) |
| (φ : Unitization ℂ A →ₚ[ℂ] ℂ) (hφ1 : φ 1=1) |
| (hpure : ofPositiveState φ hφ1∈(realStateCarrier (A := Unitization ℂ A)).extremePoints ℝ) |
| : ∃ s : NormUltra.QuotientAlgebra (fun _ : ℕ => A) U, ‖s‖ ≤ 1 ∧ |
| star s*s=NormUltra.QuotientAlgebra.mk U e ∧ |
| NormUltra.QuotientAlgebra.mk U f*s=s ∧ |
| ∀ a : A,star s*NonunitalGamma.normInclusion U a*s= |
| φ (Unitization.inr a) • NormUltra.QuotientAlgebra.mk U e := by |
| classical |
| obtain ⟨a,ha⟩ := TopologicalSpace.exists_dense_seq A |
| let g : ℕ → ℕ → A → ℝ |
| | 0,k,s => ‖star s*s-e k‖ |
| | 1,k,s => ‖f k*s-s‖ |
| | j+2,k,s => ‖star s*a j*s-φ (Unitization.inr (a j)) • e k‖ |
| have hg (j k : ℕ) (s : A) : 0 ≤ g j k s := by |
| rcases j with _|(_|j) <;> exact norm_nonneg _ |
| have hsupply (m : ℕ) (ε : ℝ) (hε : 0<ε) : |
| ∀ᶠ k in (U : Filter ℕ),∃ s : A,‖s‖ ≤ 2 ∧ ∀ j ≤ m,g j k s<ε := by |
| have hs := NonunitalSI.pure_state_small_excision hsimple hprojectionless hcone hcmp U e f he hen hf hfn hcentral |
| hsmall ht hlarge hdefect φ hφ1 hpure (fun i : Fin (m+1) => Unitization.inr (a i)) hε |
| filter_upwards [hs] with k hk |
| obtain ⟨s,hsn,hse,hsf,hsa⟩ := hk |
| refine ⟨s,hsn,?_⟩ |
| intro j hj |
| rcases j with _|(_|j) |
| · exact hse |
| · exact hsf |
| · simpa only [compression_inr] using hsa ⟨j,by omega⟩ |
| obtain ⟨z,hzn,hzg⟩ := NormUltra.bounded_epsilon_test U hfree 2 (by norm_num) g hg hsupply |
| let q := NormUltra.QuotientAlgebra.mkHom (A := fun _ : ℕ => A) (U := U) |
| have hse : star (q z)*q z=q e := |
| (NormUltra.QuotientAlgebra.mk_eq_mk_iff (star z*z) e).mpr (hzg 0) |
| have hsf : q f*q z=q z := |
| (NormUltra.QuotientAlgebra.mk_eq_mk_iff (f*z) z).mpr (hzg 1) |
| have hcoeff (i : ℕ) : star (q z)*NonunitalGamma.normInclusion U (a i)*q z= |
| φ (Unitization.inr (a i)) • q e := |
| (NormUltra.QuotientAlgebra.mk_eq_mk_iff (star z*boundedConstant (A := A) (a i)*z) |
| (φ (Unitization.inr (a i)) • e)).mpr (hzg (i+2)) |
| have hnorme : ‖q e‖ ≤ 1 := (NormUltra.QuotientAlgebra.norm_mk_le e).trans |
| (lp.norm_le_of_forall_le zero_le_one hen) |
| have hnorm : ‖q z‖ ≤ 1 := by |
| have hh : ‖star (q z)*q z‖=‖q z‖*‖q z‖ := CStarRing.norm_star_mul_self |
| rw [hse] at hh |
| nlinarith [norm_nonneg (q z)] |
| refine ⟨q z,hnorm,hse,hsf,?_⟩ |
| have hc1 : Continuous (fun b : A => star (q z)*NonunitalGamma.normInclusion U b*q z) := by fun_prop |
| have hc2 : Continuous (fun b : A => φ (Unitization.inr b) • q e) := by fun_prop |
| have hge := ha.equalizer hc1 hc2 (funext hcoeff) |
| exact fun b => congrFun hge b |
|
|
| end NonunitalSI |
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| end |
|
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| end OAI |
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