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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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.Comparison.ExactLocalSupply
namespace OAI
noncomputable section
namespace DirectPacking
open Pureness CurrentMain DirectComparison Filter
open scoped ComplexOrder
variable {D : Type*} [CStarAlgebra D] [PartialOrder D] [StarOrderedRing D] [TraceFamily D]
def CoefficientSupply : Prop :=
∀ (q : D), IsStarProjection q → ∀ s : ℝ, s ∈ Set.Icc 0 1 →
∀ F : Finset D, ∀ ε : ℝ, 0 < ε →
∃ e : D, IsStarProjection e ∧ e ≤ q ∧ ∀ b ∈ F, ∀ τ : RestrictedTracialState D,
‖τ.functional (e*b)-(s:ℂ)*τ.functional (q*b)‖ < ε
def InsertionSupply : Prop :=
∀ (z y : D), IsStarProjection z → IsStarProjection y →
∃ t : D, IsStarProjection t ∧ t ≤ 1-z ∧ ∀ τ : RestrictedTracialState D,
((τ.functional y).re-3*(τ.functional (y*z)).re ≤ (τ.functional t).re ∧
(τ.functional t).re ≤ (τ.functional y).re) ∧
traceTwoNorm τ (t-y)^2 ≤ 5*(τ.functional (y*z)).re ∧
∀ x : D, ‖x‖ ≤ 1 → ‖τ.functional ((t-y)*x)‖ ≤
2*Real.sqrt (τ.functional y).re*traceTwoNorm τ (t-y)
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
variable [Nontrivial A] [Nonempty (TracialState A)]
variable (U : Ultrafilter ℕ)
local notation "D₀" => FamilyUltrapower (fun τ : TracialState A => τ) U
@[instance_reducible] def limitTraceFamily : TraceFamily D₀ where
carrier := Set.range (familyLimitTrace (fun τ : TracialState A => τ) U)
lemma limitTraceFamily_nonempty :
let := limitTraceFamily (A := A) U
Nonempty (RestrictedTracialState D₀) := by
let := limitTraceFamily (A := A) U
let s : ℕ → TracialState A := fun _ => Classical.ofNonempty
exact ⟨⟨familyLimitTrace (fun τ : TracialState A => τ) U s,⟨s,rfl⟩⟩⟩
lemma source_tools (hfree : (U : Filter ℕ) ≤ atTop)
(hsimple : TopologicallySimple A) (hinf : ¬FiniteDimensional ℂ A) (hcmp : StrictComparison A) :
let := limitTraceFamily (A := A) U
SpectralProjectionSupply (fun τ : RestrictedTracialState D₀ => (τ : TracialState D₀)) ∧
CoefficientSupply (D := D₀) ∧ InsertionSupply (D := D₀) := by
let := limitTraceFamily (A := A) U
constructor
· intro h hh hhn α hα hα1
obtain ⟨p,hp,hl,hr,htr⟩ := projection_supply U hfree hsimple hinf hcmp h hh hhn α hα hα1
refine ⟨p,hp,hl,hr,?_⟩
rintro ⟨τ,⟨s,rfl⟩⟩
exact htr s
constructor
· intro q hq s hs F ε hε
obtain ⟨e,he,heq,ht⟩ := coefficient_split U hfree hsimple hinf hcmp hq hs F hε
refine ⟨e,he,heq,?_⟩
intro b hb
rintro ⟨τ,⟨t,rfl⟩⟩
exact ht b hb t
· intro z y hz hy
obtain ⟨t,ht,htz,htr⟩ := DirectComparison.orthogonal_insertion U hfree hsimple hinf hcmp hz hy
refine ⟨t,ht,htz,?_⟩
rintro ⟨τ,⟨s,rfl⟩⟩
exact htr s
end DirectPacking
namespace DirectPacking
open Pureness CurrentMain
open scoped ComplexOrder
variable {D I : Type*} [CStarAlgebra D] [PartialOrder D] [StarOrderedRing D]
[TraceFamily D] [Fintype I] {q : D} (U : ProjectionRow I q)
theorem sample_mass_overlap (hcoef : CoefficientSupply (D := D))
(z : D) {c ε : ℝ} (hc : c ∈ Set.Icc 0 1) (hε : 0 < ε) :
∃ e : D, IsStarProjection e ∧ e ≤ q ∧ ∀ τ : RestrictedTracialState D,
‖τ.functional (U.sample e) - (c : ℂ) * τ.functional (U.sample q)‖ < ε ∧
‖τ.functional (z * U.sample e) - (c : ℂ) * τ.functional (z * U.sample q)‖ < ε := by
classical
let k : ℝ := Fintype.card I
let δ := ε / (k + 1)
have hk : 0 ≤ k := Nat.cast_nonneg _
have hδ : 0 < δ := div_pos hε (by positivity)
have hδeq : δ * (k + 1) = ε := div_mul_cancel₀ _ (ne_of_gt (by positivity))
have hkδ : k * δ < ε := calc
k * δ < (k + 1) * δ := mul_lt_mul_of_pos_right (lt_add_one k) hδ
_ = ε := (mul_comm _ _).trans hδeq
have hδε : δ ≤ ε := div_le_self hε.le (le_add_of_nonneg_left hk)
obtain ⟨e, he, heq, htest⟩ := hcoef q U.projection c hc {1, U.overlapTest z} δ hδ
have heq' := (he.le_iff_mul_eq_left U.projection).mp heq
refine ⟨e, he, heq, ?_⟩
intro τ
constructor
· have hmass := htest 1 (Finset.mem_insert_self _ _) τ
simp only [mul_one] at hmass
rw [U.trace_sample τ e heq', U.trace_sample τ q U.projection.isIdempotentElem]
rw [show (Fintype.card I : ℂ) * τ.functional e -
(c : ℂ) * ((Fintype.card I : ℂ) * τ.functional q) =
(Fintype.card I : ℂ) * (τ.functional e - (c : ℂ) * τ.functional q) by ring,
norm_mul, Complex.norm_natCast]
exact (mul_le_mul_of_nonneg_left hmass.le hk).trans_lt hkδ
· rw [U.trace_overlap_sample, U.trace_overlap_sample]
exact (htest _ (Finset.mem_insert_of_mem (Finset.mem_singleton_self _)) τ).trans_le hδε
end DirectPacking
end
end OAI
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