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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.CentralSequences.NormCentralFunctionalCalculus
namespace OAI
noncomputable section
namespace DirectComparison
open Pureness CurrentMain Filter Topology
open scoped ComplexOrder ENNReal
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
omit [PartialOrder A] [StarOrderedRing A] in
lemma norm_two_sided_contractions (y x z : A) (hy : ‖y‖ ≤ 1) (hz : ‖z‖ ≤ 1) :
‖y*x*z‖ ≤ ‖x‖ := by
calc
_ ≤ (‖y‖*‖x‖)*‖z‖ := (norm_mul_le _ _).trans
(mul_le_mul_of_nonneg_right (norm_mul_le _ _) (norm_nonneg _))
_ ≤ (1*‖x‖)*1 := mul_le_mul
(mul_le_mul_of_nonneg_right hy (norm_nonneg _)) hz (norm_nonneg _)
(by positivity)
_=‖x‖ := by ring
lemma norm_sqrt_upperCut_le {δ : ℝ} (hδ : 0<δ) (f : A) (hfn : ‖f‖ ≤ 1) :
‖CFC.sqrt (upperCut δ f)‖ ≤ 1 := by
have h : ‖star (CFC.sqrt (upperCut δ f))*CFC.sqrt (upperCut δ f)‖=
‖CFC.sqrt (upperCut δ f)‖*‖CFC.sqrt (upperCut δ f)‖ := CStarRing.norm_star_mul_self
rw [(CFC.sqrt_nonneg _).isSelfAdjoint.star_eq,
CFC.sqrt_mul_sqrt_self _ (upperCut_nonneg hδ.le f)] at h
nlinarith [upperCut_norm_le hδ f hfn,norm_nonneg (CFC.sqrt (upperCut δ f))]
omit [PartialOrder A] [StarOrderedRing A] in
lemma norm_star_commutator (t x : A) (ht : IsSelfAdjoint t) :
‖t*star x-star x*t‖=‖x*t-t*x‖ := by
rw [←norm_star (t*star x-star x*t),star_sub,star_mul,star_mul,star_star,ht.star_eq]
omit [PartialOrder A] [StarOrderedRing A] in
lemma fullness_error_le {I : Type*} [Fintype I] (c t : A) (hc : ‖c‖ ≤ 1)
(ht : ‖t‖ ≤ 1) (hsa : IsSelfAdjoint t) (x : I → A) :
(∑ i,(‖t*star (x i)-star (x i)*t‖*‖(c*c)*x i*t‖+
‖star (x i)*t*(c*c)‖*‖x i*t-t*x i‖)) ≤
∑ i,2*‖x i‖*‖t*x i-x i*t‖ := by
have hc2 : ‖c*c‖ ≤ 1 := (norm_mul_le _ _).trans
(by nlinarith [norm_nonneg c])
apply Finset.sum_le_sum
intro i _
have h1 := norm_two_sided_contractions (c*c) (x i) t hc2 ht
have h2 : ‖star (x i)*t*(c*c)‖ ≤ ‖x i‖ := by
calc
_ ≤ ‖star (x i)‖*(‖t‖*‖c*c‖) := by
rw [mul_assoc]
exact (norm_mul_le _ _).trans (mul_le_mul_of_nonneg_left (norm_mul_le _ _) (norm_nonneg _))
_ ≤ ‖x i‖ := by
rw [norm_star]
have hp : ‖t‖*‖c*c‖ ≤ 1 := by
nlinarith [norm_nonneg t,norm_nonneg (c*c),
mul_nonneg (sub_nonneg.mpr ht) (norm_nonneg (c*c))]
simpa only [mul_one] using mul_le_mul_of_nonneg_left hp (norm_nonneg (x i))
rw [norm_star_commutator t (x i) hsa,norm_sub_rev (x i*t) (t*x i)]
nlinarith [norm_nonneg (t*x i-x i*t),
mul_le_mul_of_nonneg_left h1 (norm_nonneg (t*x i-x i*t)),
mul_le_mul_of_nonneg_right h2 (norm_nonneg (t*x i-x i*t))]
lemma hereditaryUpperCut_trace_lower_simple {I : Type*} [Fintype I]
(τ : TracialState A) {δ : ℝ} (hδ : 0<δ) (f c : A)
(hf : 0 ≤ f) (hfn : ‖f‖ ≤ 1) (hc : IsSelfAdjoint c) (hcn : ‖c‖ ≤ 1) (x : I → A)
(hframe : ∑ i,star (x i)*(c*c)*x i=1) :
(τ.functional f).re-(τ.functional (f-f*f)).re/δ ≤
(∑ i,‖x i‖^2)*(τ.functional (hereditaryUpperCut δ f c)).re+
∑ i,2*‖x i‖*‖CFC.sqrt (upperCut δ f)*x i-x i*CFC.sqrt (upperCut δ f)‖ := by
exact (hereditaryUpperCut_trace_lower τ hδ f c hf hfn hc x hframe).trans
(add_le_add le_rfl (fullness_error_le c _ hcn (norm_sqrt_upperCut_le hδ f hfn)
(CFC.sqrt_nonneg _).isSelfAdjoint x))
theorem central_fullness_error_tendsto {I : Type*} [Fintype I]
(U : Ultrafilter ℕ) (f : lp (fun _ : ℕ => A) ∞) (hf : ∀ n,0 ≤ f n)
(hcentral : ∀ a : A,Tendsto (fun n => ‖f n*a-a*f n‖) (U : Filter ℕ) (𝓝 0))
{δ : ℝ} (hδ : 0<δ) (hδ1 : δ ≤ 1) (x : I → A) :
Tendsto (fun n => ∑ i,2*‖x i‖*‖CFC.sqrt (upperCut δ (f n))*x i-x i*CFC.sqrt (upperCut δ (f n))‖)
(U : Filter ℕ) (𝓝 0) := by
have h := tendsto_finsetSum Finset.univ (fun i _ =>
(tendsto_const_nhds (x := 2*‖x i‖)).mul (norm_central_upperCut_sqrt U f hf (x i) (hcentral _) hδ hδ1))
simpa only [mul_zero,Finset.sum_const_zero] using h
theorem eventual_hereditary_upperCut_trace_lower
(hsimple : TopologicallySimple A) (U : Ultrafilter ℕ)
(f : lp (fun _ : ℕ => A) ∞) (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))
{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)
(c : A) (hc : 0 ≤ c) (hcn : ‖c‖ ≤ 1) (hc0 : c≠0) :
∃ β : ℝ,0<β ∧ ∀ δ : ℝ,0<δ → δ ≤ 1 →
∀ᶠ n in (U : Filter ℕ),∀ τ : TracialState A,
2*β ≤ (τ.functional (hereditaryUpperCut δ (f n) c)).re := by
have hc2 : 0 ≤ c*c := by simpa only [hc.isSelfAdjoint.star_eq] using star_mul_self_nonneg c
have hc20 : c*c≠0 := by
intro hz
apply hc0
apply norm_eq_zero.mp
have hn : ‖c‖*‖c‖=0 := by
rw [←CStarRing.norm_star_mul_self,hc.isSelfAdjoint.star_eq,hz,norm_zero]
nlinarith [norm_nonneg c]
obtain ⟨l,hl⟩ := exists_positive_fullness_frame hc2
(closure_span_singleton_eq_univ hsimple (c*c) hc20)
let x : Fin l.length → A := l.get
have hframe : ∑ i,star (x i)*(c*c)*x i=1 := by
rw [←List.sum_ofFn]
have hlist : List.ofFn (fun i => star (x i)*(c*c)*x i)=l.map (fun v => star v*(c*c)*v) := by
change List.ofFn ((fun v : A => star v*(c*c)*v) ∘ l.get)=_
rw [←List.map_ofFn,List.ofFn_get]
rw [hlist,hl]
let M : ℝ := (∑ i,‖x i‖^2)+1
have hM : 0<M := by dsimp [M]; positivity
let β := t/(8*M)
refine ⟨β,by dsimp [β]; positivity,?_⟩
intro δ hδ hδ1
have herr := central_fullness_error_tendsto U f hf hcentral hδ hδ1 x
have hevent := (tendsto_order.1 herr).2 (t/4) (by positivity)
have hdevent := hdefect (δ*(t/4)) (by positivity)
filter_upwards [hlarge,hevent,hdevent] with n hn he hd
intro τ
have hw := hereditaryUpperCut_nonneg hδ.le (f n) c hc.isSelfAdjoint
have hwp : 0 ≤ (τ.functional (hereditaryUpperCut δ (f n) c)).re :=
(τ.functional.map_nonneg hw).1
have hq := hereditaryUpperCut_trace_lower_simple τ hδ (f n) c (hf n) (hfn n)
hc.isSelfAdjoint hcn x hframe
have hdiv : (τ.functional (f n-f n*f n)).re/δ<t/4 :=
(div_lt_iff₀ hδ).mpr (by nlinarith [hd τ])
have hlow : t/2 ≤ M*(τ.functional (hereditaryUpperCut δ (f n) c)).re := by
dsimp [M]
nlinarith [hn τ]
have hβ : 2*β ≤ t/(2*M) := by
dsimp [β]
field_simp
nlinarith
exact hβ.trans ((div_le_iff₀ (by positivity : 0<2*M)).mpr (by nlinarith))
end DirectComparison
end
end OAI
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