| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.Comparison.Corner |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace CurrentMain |
| open Pureness Filter |
| open scoped Topology ComplexOrder ENNReal |
|
|
| lemma positive_contraction_defect_nonneg |
| {D : Type*} [CStarAlgebra D] [PartialOrder D] [StarOrderedRing D] |
| {p : D} (hp : 0 ≤ p) (hpn : ‖p‖ ≤ 1) : 0 ≤ p-p*p := by |
| have hp1 : p ≤ 1 := hp.isSelfAdjoint.le_algebraMap_norm_self.trans (by |
| simpa only [Algebra.algebraMap_eq_smul_one, one_smul] using |
| (smul_le_smul_of_nonneg_right hpn (zero_le_one : (0 : D) ≤ 1))) |
| have hc : Commute p (1-p) := (Commute.one_right p).sub_right (Commute.refl p) |
| simpa only [mul_sub,mul_one] using Commute.mul_nonneg hp (sub_nonneg.mpr hp1) hc |
|
|
| variable {D I X : Type*} [CStarAlgebra D] [PartialOrder D] [StarOrderedRing D] |
| [Nontrivial D] (τ : I → TracialState D) [Nonempty I] (U : Ultrafilter X) |
|
|
| lemma limitTrace_detects_positive_zero (z : FamilyUltrapower τ U) (hz : 0 ≤ z) |
| (htr : ∀ s : X → I, ((familyLimitTrace τ U s).functional z).re = 0) : z = 0 := by |
| have hs : CFC.sqrt z*CFC.sqrt z = z := CFC.sqrt_mul_sqrt_self z hz |
| have hzero : CFC.sqrt z = 0 := by |
| apply limitTrace_detects_zero τ U |
| intro s |
| simp only [traceTwoNorm,(CFC.sqrt_nonneg z).isSelfAdjoint.star_eq,hs,htr,Real.sqrt_zero] |
| rw [←hs,hzero,zero_mul] |
|
|
| theorem projection_of_vanishing_tracial_defect |
| (p : lp (fun _ : X => D) ∞) (hp : 0 ≤ p) (hpn : ‖p‖ ≤ 1) |
| (δ : X → ℝ) (hδ : Tendsto δ (U : Filter X) (𝓝 0)) |
| (hdef : ∀ n i, ((τ i).functional (p n-p n*p n)).re ≤ δ n) : |
| IsStarProjection ((familyNullIdeal τ (U : Filter X)).quotientMap p) := by |
| let J := familyNullIdeal τ (U : Filter X) |
| let q := J.quotientMap p |
| have hpq : 0 ≤ q := map_nonneg J.quotientStarHom hp |
| have hnq : ‖q‖ ≤ 1 := (J.quotientMap_norm_le p).trans hpn |
| have hd : 0 ≤ q-q*q := positive_contraction_defect_nonneg hpq hnq |
| have hd0 : q-q*q = 0 := by |
| apply limitTrace_detects_positive_zero τ U _ hd |
| intro s |
| have hlim := Complex.continuous_re.continuousAt.tendsto.comp |
| (familyLimitTrace_tendsto τ U s (p-p*p)) |
| have hn (n : X) : 0 ≤ ((τ (s n)).functional (p n-p n*p n)).re := by |
| have hp' : 0 ≤ p n := map_nonneg (boundedFamilyEval n) hp |
| have hpn' : ‖p n‖ ≤ 1 := (lp.norm_apply_le_norm ENNReal.top_ne_zero p n).trans hpn |
| exact ((τ (s n)).functional.map_nonneg (positive_contraction_defect_nonneg hp' hpn')).1 |
| have ht : Tendsto (fun n => ((τ (s n)).functional (p n-p n*p n)).re) |
| (U : Filter X) (𝓝 0) := |
| squeeze_zero (fun n => hn n) (fun n => hdef n (s n)) hδ |
| have he := tendsto_nhds_unique hlim ht |
| simpa only [map_sub,map_mul] using he |
| refine ⟨?_,hpq.isSelfAdjoint⟩ |
| change q*q=q |
| exact (sub_eq_zero.mp hd0).symm |
|
|
| lemma limitTrace_eq_of_coordinate_errors |
| (p b : lp (fun _ : X => D) ∞) (δ : X → ℝ) |
| (hδ : Tendsto δ (U : Filter X) (𝓝 0)) |
| (herr : ∀ n i, ‖(τ i).functional (p n-b n)‖ ≤ δ n) (s : X → I) : |
| (familyLimitTrace τ U s).functional |
| ((familyNullIdeal τ (U : Filter X)).quotientMap p) = |
| (familyLimitTrace τ U s).functional |
| ((familyNullIdeal τ (U : Filter X)).quotientMap b) := by |
| have hnorm : Tendsto (fun n => ‖(τ (s n)).functional (p n-b n)‖) |
| (U : Filter X) (𝓝 0) := |
| squeeze_zero (fun n => norm_nonneg _) (fun n => herr n (s n)) hδ |
| have ht : Tendsto (fun n => (τ (s n)).functional (p n-b n)) |
| (U : Filter X) (𝓝 0) := tendsto_zero_iff_norm_tendsto_zero.mpr hnorm |
| have he := tendsto_nhds_unique (familyLimitTrace_tendsto τ U s (p-b)) ht |
| simpa only [map_sub,sub_eq_zero] using he |
|
|
| end CurrentMain |
|
|
| namespace DirectComparison |
| open Pureness CurrentMain Filter Topology |
| open scoped ComplexOrder ENNReal NNReal |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| lemma cfc_clamp_of_positive_contractive [Nontrivial A] |
| {h : A} (hh : 0 ≤ h) (hhn : ‖h‖ ≤ 1) : |
| cfc (fun t : ℝ => max 0 (min t 1)) h = h := by |
| calc |
| _ = cfc (fun t : ℝ => t) h := by |
| apply cfc_congr |
| intro t ht |
| have ht0 := spectrum_nonneg_of_nonneg hh ht |
| have ht1 : t ≤ 1 := (le_abs_self t).trans ((spectrum.norm_le_norm_of_mem ht).trans hhn) |
| change max 0 (min t 1) = t |
| rw [min_eq_left ht1, max_eq_right ht0] |
| _ = h := cfc_id' ℝ h hh.isSelfAdjoint |
| |
| lemma positive_contraction_lift (J : ClosedStarIdeal A) |
| {h : J.Quotient} (hh : 0 ≤ h) (hhn : ‖h‖ ≤ 1) : |
| ∃ a : A, 0 ≤ a ∧ ‖a‖ ≤ 1 ∧ J.quotientMap a = h := by |
| rcases subsingleton_or_nontrivial J.Quotient with hs | hn |
| · let := hs |
| exact ⟨0,le_rfl,by simp,Subsingleton.elim _ _⟩ |
| let := hn |
| let : ContinuousFunctionalCalculus ℝ J.Quotient IsSelfAdjoint := |
| IsSelfAdjoint.instContinuousFunctionalCalculus |
| obtain ⟨a,ha⟩ := J.quotientMap_surjective h |
| let b : A := (realPart a : A) |
| have hb : IsSelfAdjoint b := (realPart a).property |
| have he : J.quotientStarHom b = h := by |
| change J.quotientMap (realPart a : A) = h |
| rw [J.quotientMap_realPart,ha,hh.isSelfAdjoint.coe_realPart] |
| let f : ℝ → ℝ := fun t => max 0 (min t 1) |
| have hf : Continuous f := continuous_const.max (continuous_id.min continuous_const) |
| refine ⟨cfc f b,cfc_nonneg (fun _ _ => le_max_left _ _),?_,?_⟩ |
| · apply norm_cfc_le (by norm_num : (0 : ℝ) ≤ 1) |
| intro t _ |
| change |max 0 (min t 1)| ≤ 1 |
| rw [abs_of_nonneg (le_max_left _ _)] |
| exact max_le zero_le_one (min_le_right _ _) |
| · change J.quotientStarHom (cfc f b) = h |
| rw [J.quotientStarHom.map_cfc f b hf.continuousOn J.quotientStarHom_continuous hb (by simpa only [he] using hh.isSelfAdjoint),he] |
| exact cfc_clamp_of_positive_contractive hh hhn |
| |
| omit [StarOrderedRing A] in |
| lemma directPositiveCut_norm_le [Nontrivial A] (h : A) (_hh : 0 ≤ h) (hhn : ‖h‖ ≤ 1) |
| (η : ℝ) (hη : η ≤ 1) : ‖directPositiveCut η h‖ ≤ 1-η := by |
| apply norm_cfc_le (sub_nonneg.mpr hη) |
| intro t ht |
| have ht1 : t ≤ 1 := (le_abs_self t).trans ((spectrum.norm_le_norm_of_mem ht).trans hhn) |
| rw [Real.norm_eq_abs,abs_of_nonneg (le_max_right _ _)] |
| exact max_le (sub_le_sub_right ht1 η) (sub_nonneg.mpr hη) |
| |
| omit [PartialOrder A] [StarOrderedRing A] in |
| lemma boundedFamilyEval_continuous [Nontrivial A] (n : ℕ) : |
| Continuous (boundedFamilyEval (D := A) n) := |
| (continuous_apply n).comp lp.uniformContinuous_coe.continuous |
| |
| variable [Nontrivial A] [Nonempty (TracialState A)] |
| variable (U : Ultrafilter ℕ) (hfree : (U : Filter ℕ) ≤ atTop) |
| local notation "D" => FamilyUltrapower (fun τ : TracialState A => τ) U |
| local notation "Λ" => familyLimitTrace (fun τ : TracialState A => τ) U |
| |
| local instance : ContinuousFunctionalCalculus ℝ (lp (fun _ : ℕ => A) ∞) IsSelfAdjoint := |
| IsSelfAdjoint.instContinuousFunctionalCalculus |
| local instance (J : ClosedStarIdeal (lp (fun _ : ℕ => A) ∞)) : |
| ContinuousFunctionalCalculus ℝ J.Quotient IsSelfAdjoint := |
| IsSelfAdjoint.instContinuousFunctionalCalculus |
| |
| include hfree in |
| |
| theorem projection_supply (hsimple : TopologicallySimple A) (hinf : ¬FiniteDimensional ℂ A) |
| (hcmp : StrictComparison A) (h : D) (hh : 0 ≤ h) (hhn : ‖h‖ ≤ 1) |
| (η : ℝ) (hη : 0 < η) (hη1 : η < 1) : |
| ∃ p : D, IsStarProjection p ∧ |
| directCutoff η h*p=p ∧ p*directCutoff η h=p ∧ |
| ∀ s : ℕ → TracialState A, (Λ s).functional p = (Λ s).functional (directPositiveCut η h) := by |
| classical |
| let τ := fun t : TracialState A => t |
| let J := familyNullIdeal τ (U : Filter ℕ) |
| obtain ⟨a,ha0,han,ha⟩ := positive_contraction_lift J hh hhn |
| let b : lp (fun _ : ℕ => A) ∞ := directPositiveCut η a |
| have ha0' (n : ℕ) : 0 ≤ a n := map_nonneg (boundedFamilyEval n) ha0 |
| have hb (n : ℕ) : b n = directPositiveCut η (a n) := by |
| exact (boundedFamilyEval n).map_cfc (fun t : ℝ => max (t-η) 0) a |
| (by fun_prop) (boundedFamilyEval_continuous n) ha0.isSelfAdjoint (ha0' n).isSelfAdjoint |
| have han' (n : ℕ) : ‖a n‖ ≤ 1 := (lp.norm_apply_le_norm ENNReal.top_ne_zero a n).trans han |
| have hb0 (n : ℕ) : 0 ≤ b n := by rw [hb]; exact directPositiveCut_nonneg _ _ |
| have hbn (n : ℕ) : ‖b n‖ < 1 := by |
| rw [hb] |
| exact (directPositiveCut_norm_le (a n) (ha0' n) (han' n) η hη1.le).trans_lt (by linarith) |
| let δ : ℕ → ℝ := fun n => 1 / ((n : ℝ)+1) |
| have hδ (n : ℕ) : 0 < δ n := by dsimp [δ]; positivity |
| have hδlim : Tendsto (fun n => 2*δ n) (U : Filter ℕ) (𝓝 0) := by |
| have ht : Tendsto δ atTop (𝓝 0) := by |
| simpa only [δ] using (tendsto_one_div_add_atTop_nhds_zero_nat : |
| Tendsto (fun n : ℕ => 1/((n : ℝ)+1)) atTop (𝓝 0)) |
| simpa only [mul_zero] using |
| ((tendsto_const_nhds : Tendsto (fun _ : ℕ => (2 : ℝ)) atTop (𝓝 2)).mul ht).mono_left hfree |
| choose p hpmem hp0 hpn hpτ using |
| fun n => almost_projection hsimple hinf inferInstance hcmp (b n) (hb0 n) (hbn n) (δ n) (hδ n) |
| let P : lp (fun _ : ℕ => A) ∞ := |
| ⟨p,memℓp_infty_iff.mpr ⟨1,by rintro _ ⟨n,rfl⟩;exact hpn n⟩⟩ |
| have hP0 : 0 ≤ P := by |
| let Q : lp (fun _ : ℕ => A) ∞ := ⟨fun n => CFC.sqrt (p n),memℓp_infty_iff.mpr |
| ⟨1,by |
| rintro _ ⟨n,rfl⟩ |
| change ‖CFC.sqrt (p n)‖ ≤ 1 |
| exact (CFC.norm_sqrt (p n) (hp0 n)).le.trans (Real.sqrt_le_one.mpr (hpn n))⟩⟩ |
| have he : P=star Q*Q := by |
| apply Subtype.ext; funext n |
| change p n = star (CFC.sqrt (p n))*CFC.sqrt (p n) |
| rw [(CFC.sqrt_nonneg _).isSelfAdjoint.star_eq,CFC.sqrt_mul_sqrt_self _ (hp0 n)] |
| rw [he]; exact star_mul_self_nonneg Q |
| have hPn : ‖P‖ ≤ 1 := lp.norm_le_of_forall_le zero_le_one hpn |
| have hPJ : IsStarProjection (J.quotientMap P) := |
| projection_of_vanishing_tracial_defect τ U P hP0 hPn _ hδlim |
| (fun n t => (hpτ n t).2.2.2.le) |
| have hfix : directCutoff η a*P=P ∧ P*directCutoff η a=P := by |
| constructor |
| · apply Subtype.ext; funext n |
| change (directCutoff η a) n*p n=p n |
| have hc : (directCutoff η a) n = directCutoff η (a n) := by |
| exact (boundedFamilyEval n).map_cfc (fun t : ℝ => min (t/η) 1) a |
| (by fun_prop) (boundedFamilyEval_continuous n) ha0.isSelfAdjoint (ha0' n).isSelfAdjoint |
| rw [hc] |
| exact (directCutoff_fixes_hereditary η hη (a n) (by simpa only [hb] using hpmem n)).1 |
| · apply Subtype.ext; funext n |
| change p n*(directCutoff η a) n=p n |
| have hc : (directCutoff η a) n = directCutoff η (a n) := by |
| exact (boundedFamilyEval n).map_cfc (fun t : ℝ => min (t/η) 1) a |
| (by fun_prop) (boundedFamilyEval_continuous n) ha0.isSelfAdjoint (ha0' n).isSelfAdjoint |
| rw [hc] |
| exact (directCutoff_fixes_hereditary η hη (a n) (by simpa only [hb] using hpmem n)).2 |
| have hc : J.quotientMap (directCutoff η a) = directCutoff η h := by |
| change J.quotientStarHom (cfc _ a) = cfc _ h |
| rw [J.quotientStarHom.map_cfc _ a (by fun_prop) J.quotientStarHom_continuous ha0.isSelfAdjoint (by simpa only [show J.quotientStarHom a = h from ha] using hh.isSelfAdjoint)] |
| rw [show J.quotientStarHom a = h from ha] |
| rfl |
| have hbJ : J.quotientMap b = directPositiveCut η h := by |
| change J.quotientStarHom (cfc _ a) = cfc _ h |
| rw [J.quotientStarHom.map_cfc _ a (by fun_prop) J.quotientStarHom_continuous ha0.isSelfAdjoint (by simpa only [show J.quotientStarHom a = h from ha] using hh.isSelfAdjoint)] |
| rw [show J.quotientStarHom a = h from ha] |
| rfl |
| refine ⟨J.quotientMap P,hPJ,?_,?_,?_⟩ |
| · rw [←hc,←map_mul,hfix.1] |
| · rw [←hc,←map_mul,hfix.2] |
| · intro s |
| rw [←hbJ] |
| apply limitTrace_eq_of_coordinate_errors τ U P b _ hδlim _ s |
| intro n t |
| change ‖t.functional (p n-b n)‖ ≤ 2*δ n |
| have hpR : (t.functional (p n)).im=0 := (t.functional.map_nonneg (hp0 n)).2.symm |
| have hbR : (t.functional (b n)).im=0 := (t.functional.map_nonneg (hb0 n)).2.symm |
| have he : t.functional (p n-b n) = |
| (((t.functional (p n)).re - (t.functional (b n)).re : ℝ) : ℂ) := by |
| apply Complex.ext <;> simp only [map_sub,Complex.sub_re,Complex.sub_im, Complex.ofReal_re,Complex.ofReal_im,hpR,hbR,sub_self] |
| rw [he,Complex.norm_real,Real.norm_eq_abs] |
| rw [abs_of_nonpos (by linarith [(hpτ n t).2.1])] |
| linarith [(hpτ n t).1] |
|
|
| end DirectComparison |
| end |
|
|
| end OAI |
|
|
