| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.Excision.ExcisionFullnessError |
| import OAI.Analysis.JiangSu.Excision.ExcisionCoefficientAssembly |
| import OAI.Analysis.JiangSu.Excision.ExcisionSoftCompression |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace PureStateExcision |
| open scoped ComplexOrder ComplexStarModule |
|
|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| variable {I : Type} [Fintype I] |
|
|
| theorem pure_state_finite_excision (f : A →ₚ[ℂ] ℂ) (hf1 : f 1=1) |
| (hpure : ofPositiveState f hf1∈(realStateCarrier (A := A)).extremePoints ℝ) |
| (a : I → A) {ε ρ : ℝ} (hε : 0<ε) (hρ : 0<ρ) : |
| ∃ g : A, 0 ≤ g ∧ ‖g‖ ≤ 1 ∧ 1-ρ<(f g).re ∧ |
| ∀ i, ‖g*a i*g-f (a i) • (g*g)‖<ε := by |
| let b : I×Bool → selfAdjoint A := fun p => if p.2 then ℑ (a p.1) else ℜ (a p.1) |
| obtain ⟨g,hg,hgn,hfg,herr⟩ := pure_state_finite_selfadjoint_excision f hf1 hpure b |
| (show 0<ε/2 by positivity) hρ |
| refine ⟨g,hg,hgn,hfg,fun i => ?_⟩ |
| have hR := herr (i,false) |
| have hI := herr (i,true) |
| change ‖g*(ℜ (a i) : A)*g-f (ℜ (a i)) • (g*g)‖<ε/2 at hR |
| change ‖g*(ℑ (a i) : A)*g-f (ℑ (a i)) • (g*g)‖<ε/2 at hI |
| have he : g*a i*g-f (a i) • (g*g)= |
| (g*(ℜ (a i) : A)*g-f (ℜ (a i)) • (g*g))+ |
| Complex.I • (g*(ℑ (a i) : A)*g-f (ℑ (a i)) • (g*g)) := by |
| conv_lhs => rw [←realPart_add_I_smul_imaginaryPart (a i)] |
| simp only [mul_add,add_mul,mul_smul_comm,smul_mul_assoc,map_add,map_smul, |
| smul_eq_mul,add_smul,smul_sub,smul_smul] |
| module |
| rw [he] |
| have hn := norm_add_le (g*(ℜ (a i) : A)*g-f (ℜ (a i)) • (g*g)) |
| (Complex.I • (g*(ℑ (a i) : A)*g-f (ℑ (a i)) • (g*g))) |
| rw [norm_smul,Complex.norm_I,one_mul] at hn |
| linarith |
|
|
| end PureStateExcision |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace PureStateExcision |
| open DirectComparison |
| open scoped ComplexOrder ComplexStarModule |
|
|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| private def inversePlateau (t : ℝ) : ℝ := (max (1/2) t)⁻¹ |
| private lemma inversePlateau_continuous : Continuous inversePlateau := by |
| apply Continuous.inv₀ (by fun_prop) |
| intro t |
| have h := le_max_left (1/2 : ℝ) t |
| dsimp |
| linarith |
| private lemma inversePlateau_bounds (t : ℝ) : |
| 0 ≤ inversePlateau t ∧ inversePlateau t ≤ 2 := by |
| have h := le_max_left (1/2 : ℝ) t |
| have hp : 0 < max (1/2 : ℝ) t := by linarith |
| constructor |
| · exact inv_nonneg.mpr hp.le |
| · dsimp [inversePlateau] |
| calc |
| _ ≤ (1/2 : ℝ)⁻¹ := inv_anti₀ (by norm_num) h |
| _=2 := by norm_num |
|
|
| lemma plateau_factorization (g : A) (hg : 0 ≤ g) : |
| directCutoff (1/2) g=cfc inversePlateau g*g ∧ |
| directCutoff (1/2) g=g*cfc inversePlateau g := by |
| have he : cfc inversePlateau g*g=directCutoff (1/2) g := by |
| have hid : cfc inversePlateau g*g=cfc (fun t : ℝ => inversePlateau t*t) g := by |
| rw [cfc_mul inversePlateau (fun t : ℝ => t) g inversePlateau_continuous.continuousOn continuousOn_id, |
| cfc_id' ℝ g hg.isSelfAdjoint] |
| rw [hid] |
| apply cfc_congr |
| intro t ht |
| change (max (1/2) t)⁻¹*t=min (t/(1/2)) 1 |
| have ht0 := spectrum_nonneg_of_nonneg hg ht |
| by_cases h : t ≤ 1/2 |
| · rw [max_eq_left h,min_eq_left ((div_le_one (by norm_num : (0:ℝ)<1/2)).mpr h)] |
| ring |
| · rw [max_eq_right (le_of_not_ge h),min_eq_right |
| ((le_div_iff₀ (by norm_num : (0:ℝ)<1/2)).mpr (by linarith))] |
| exact inv_mul_cancel₀ (by linarith) |
| refine ⟨he.symm,?_⟩ |
| have hc : cfc inversePlateau g*g=g*cfc inversePlateau g := by |
| have hz := cfc_commute_cfc inversePlateau (fun t : ℝ => t) g |
| simpa only [cfc_id' ℝ g hg.isSelfAdjoint] using hz.eq |
| exact he.symm.trans hc |
|
|
| lemma plateau_error_bound (g a : A) (z : ℂ) (hg : 0 ≤ g) : |
| ‖directCutoff (1/2) g*a*directCutoff (1/2) g- |
| z • (directCutoff (1/2) g*directCutoff (1/2) g)‖ ≤ |
| 4*‖g*a*g-z • (g*g)‖ := by |
| let k := cfc inversePlateau g |
| have hk : ‖k‖ ≤ 2 := norm_cfc_le (by norm_num) fun t _ => by |
| rw [Real.norm_eq_abs,abs_of_nonneg (inversePlateau_bounds t).1] |
| exact (inversePlateau_bounds t).2 |
| have hfact := plateau_factorization g hg |
| have he : directCutoff (1/2) g*a*directCutoff (1/2) g- |
| z • (directCutoff (1/2) g*directCutoff (1/2) g)= |
| k*(g*a*g-z • (g*g))*k := by |
| conv_lhs => lhs; lhs; rw [hfact.1] |
| conv_lhs => lhs; rhs; rw [hfact.2] |
| conv_lhs => rhs; rhs; lhs; rw [hfact.1] |
| conv_lhs => rhs; rhs; rhs; rw [hfact.2] |
| simp only [mul_sub,sub_mul,mul_smul_comm,smul_mul_assoc] |
| noncomm_ring |
| rw [he] |
| calc |
| _ ≤ (‖k‖*‖g*a*g-z • (g*g)‖)*‖k‖ := |
| (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _)) |
| _ ≤ 4*‖g*a*g-z • (g*g)‖ := by |
| have h0 := norm_nonneg (g*a*g-z • (g*g)) |
| have hk0 := norm_nonneg k |
| nlinarith [mul_nonneg (sub_nonneg.mpr hk) h0, |
| mul_nonneg (sub_nonneg.mpr hk) (mul_nonneg hk0 h0)] |
|
|
| theorem pure_state_plateau_excision {I : Type} [Fintype I] |
| (f : A →ₚ[ℂ] ℂ) (hf1 : f 1=1) |
| (hpure : ofPositiveState f hf1∈(realStateCarrier (A := A)).extremePoints ℝ) |
| (a : I → A) {ε : ℝ} (hε : 0<ε) : |
| ∃ h c : A, 0 ≤ h ∧ ‖h‖ ≤ 1 ∧ 0 ≤ c ∧ ‖c‖ ≤ 1 ∧ c≠0 ∧ h*c=c ∧ |
| ∀ i, ‖h*a i*h-f (a i) • (h*h)‖<ε := by |
| obtain ⟨g,hg,hgn,hfg,herr⟩ := pure_state_finite_excision f hf1 hpure a |
| (show 0<ε/4 by positivity) (show (0:ℝ)<1/4 by norm_num) |
| let h := directCutoff (1/2) g |
| let c := directPositiveCut (1/2) g |
| have hh : 0 ≤ h := cfc_nonneg fun t ht => |
| le_min (div_nonneg (spectrum_nonneg_of_nonneg hg ht) (by norm_num)) zero_le_one |
| have hh1 : h ≤ 1 := cfc_le_one _ _ fun t _ => min_le_right _ _ |
| have hc0 : c≠0 := by |
| intro hz |
| have hn := norm_sub_directPositiveCut_le g hg (1/2) (by norm_num) |
| change ‖g-c‖ ≤ 1/2 at hn |
| rw [hz,sub_zero] at hn |
| have hs := f.norm_apply_le_of_nonneg g hg |
| rw [hf1,norm_one,one_mul] at hs |
| have hr := Complex.re_le_norm (f g) |
| linarith |
| refine ⟨h,c,hh,(CStarAlgebra.norm_le_one_iff_of_nonneg h hh).mpr hh1, |
| directPositiveCut_nonneg _ _,norm_directPositiveCut_le_one g hg hgn (1/2) (by norm_num),hc0,directCutoff_fixes_cut _ (by norm_num) g,fun i => ?_⟩ |
| exact (plateau_error_bound g (a i) (f (a i)) hg).trans_lt (by linarith [herr i]) |
|
|
| end PureStateExcision |
|
|
| end |
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace DirectComparison |
| open Pureness CurrentMain PureStateExcision Filter Topology |
| open scoped ComplexOrder ENNReal |
|
|
| variable {A : Type} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| theorem pure_state_small_excision [Nonempty (TracialState A)] |
| (hsimple : TopologicallySimple A) (hcmp : StrictComparison A) (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 : TracialState A, |
| (τ.functional (e n)).re<η) |
| {t : ℝ} (ht : 0<t) |
| (hlarge : ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A,t ≤ (τ.functional (f n)).re) |
| (hdefect : ∀ η : ℝ,0<η → ∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A, |
| (τ.functional (f n-f n*f n)).re<η) |
| (φ : A →ₚ[ℂ] ℂ) (hφ1 : φ 1=1) |
| (hpure : ofPositiveState φ hφ1∈(realStateCarrier (A := A)).extremePoints ℝ) |
| {I : Type} [Fintype I] (a : I → 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,‖star s*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⟩ := eventual_hereditary_upperCut_trace_lower hsimple 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 (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 := hereditaryUpperCut δ (f n) c |
| have hw : 0 ≤ w := hereditaryUpperCut_nonneg hδ.le (f n) c hc.isSelfAdjoint |
| have hwn : ‖w‖ ≤ 1 := hereditaryUpperCut_norm_le hδ (f n) c (hfn n) hcn |
| have hhw : h*w=w := hereditaryUpperCut_support δ (f n) c h hhc |
| obtain ⟨s,hsn,hse,hhs,hfs⟩ := small_trace_supported_isometry hcmp (e n) w h (f n) |
| (he n) (hen n) hw hwn hhw hκ hβ hκ hκ1 hne hn |
| refine ⟨s,hsn,hse.trans hκe,?_,?_⟩ |
| · have hwf := hereditaryUpperCut_error hδ (f n) c (hf n) (hfn n) hcn |
| have hcoef : 0 ≤ 2*β⁻¹ := by positivity |
| have hbound : ‖f n*s-s‖ ≤ 4*β⁻¹*δ := by |
| calc |
| _ ≤ 2*β⁻¹*(‖f n*c-c*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 hx := CStarExcision.compression_error_le h s (a i) (e n) (φ (a i)) |
| hh.isSelfAdjoint.star_eq hhs (hex i).le hse.le |
| have hsn2 : ‖s‖^2 ≤ 4 := by nlinarith [norm_nonneg s] |
| have hexc : ‖star s*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) |
|
|
| end DirectComparison |
|
|
| end |
| end |
|
|
| end OAI |
|
|
