| 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 |
|
|
