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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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.CuntzSemigroup.FiniteDimensionRegularization
namespace OAI
noncomputable section
namespace Pureness
open TraceCone MeasureTheory Set
open scoped ENNReal NNReal
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
def finiteDimensionDistribution (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞)
(n : ℕ) (M : Positive (FinMatrix A n)) (t : ℝ) : ℝ≥0∞ :=
d (finitePositiveClass n (Positive.cut t.toNNReal M))
lemma finiteDimensionDistribution_antitone
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(n : ℕ) (M : Positive (FinMatrix A n)) :
Antitone (finiteDimensionDistribution d n M) := by
intro s t hst
exact finiteCutValue_antitone d hd n M (Real.toNNReal_mono hst)
lemma finiteDimensionDistribution_measurable
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(n : ℕ) (M : Positive (FinMatrix A n)) :
Measurable (finiteDimensionDistribution d n M) :=
(finiteDimensionDistribution_antitone d hd n M).measurable
def finiteSpectralIntegral (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞)
(n : ℕ) (M : Positive (FinMatrix A n)) : ℝ≥0∞ :=
∫⁻ t in Ioi (0 : ℝ), finiteDimensionDistribution d n M t
lemma finiteSpectralIntegral_zero (d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (n : ℕ) :
finiteSpectralIntegral d n Positive.zero = 0 := by
have hcut (t : ℝ) : Positive.cut t.toNNReal (Positive.zero : Positive (FinMatrix A n)) = Positive.zero := by
apply Subtype.ext
simp [Positive.cut,Positive.zero]
simp [finiteSpectralIntegral,finiteDimensionDistribution,hcut,finitePositiveClass_zero]
theorem finiteSpectralIntegral_inclusion
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞)
{n m : ℕ} (h : n ≤ m) (M : Positive (FinMatrix A n)) :
finiteSpectralIntegral d m (Positive.map (matrixInclusion h) M) = finiteSpectralIntegral d n M := by
simp only [finiteSpectralIntegral,finiteDimensionDistribution,← Positive.map_cut,finitePositiveClass_inclusion]
theorem finiteSpectralIntegral_block_add
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(n : ℕ) (M N : Positive (FinMatrix A n)) :
finiteSpectralIntegral d (n*2) (Positive.map (matrixInterleave n) (positiveProduct M N)) =
finiteSpectralIntegral d n M+finiteSpectralIntegral d n N := by
simp only [finiteSpectralIntegral,finiteDimensionDistribution,finitePositiveClass_cut_add,map_add]
exact lintegral_add_left (finiteDimensionDistribution_measurable d hd n M) _
end Pureness
namespace Pureness
open TraceCone MeasureTheory Set
open scoped ENNReal NNReal
variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D]
theorem supported_cfc_polar_factor (x : D) (f : ℝ → ℝ)
(hf : Continuous f) (hfpos : ∀ t, 0 ≤ f t) (δ : ℝ≥0) (hδ : 0 < δ)
(hfzero : ∀ t : ℝ, t ≤ δ → f t = 0) :
∃ y : D, star y * y = cfcₙ f (star x * x) ∧
y * star y = cfcₙ f (x * star x) := by
obtain ⟨g, hg, hg0, hgf⟩ := supported_cfc_factor f hf hfpos δ hδ hfzero
let a := Positive.starMul x
let b := Positive.mulStar x
have ha : IsSelfAdjoint a.1 := IsSelfAdjoint.of_nonneg a.2
have hb : IsSelfAdjoint b.1 := IsSelfAdjoint.of_nonneg b.2
have hid (c : Positive D) : cfcₙ g c.1 * c.1 * cfcₙ g c.1 = cfcₙ f c.1 := by
calc
_ = cfcₙ (fun t : ℝ => g t * t * g t) c.1 := by
rw [cfcₙ_mul (fun t : ℝ => g t * t) g c.1,
cfcₙ_mul g (fun t : ℝ => t) c.1, cfcₙ_id' ℝ c.1]
_ = _ := cfcₙ_congr (fun t _ => hgf t)
have hcomm : x * cfcₙ g a.1 = cfcₙ g b.1 * x :=
cfc_intertwine a b x (by dsimp [a, b, Positive.starMul, Positive.mulStar]; exact (mul_assoc _ _ _).symm) g hg hg0
let y := x * cfcₙ g a.1
have hleft : star y * y = cfcₙ f a.1 := by
dsimp [y]
rw [star_mul, (cfcₙ_predicate g a.1).star_eq]
calc
_ = cfcₙ g a.1 * a.1 * cfcₙ g a.1 := by
dsimp [a, Positive.starMul]
simp only [mul_assoc]
_ = _ := hid a
have hright : y * star y = cfcₙ f b.1 := by
dsimp [y]
rw [hcomm, star_mul, (cfcₙ_predicate g b.1).star_eq]
calc
_ = cfcₙ g b.1 * b.1 * cfcₙ g b.1 := by
dsimp [b, Positive.mulStar]
simp only [mul_assoc]
_ = _ := hid b
exact ⟨y, hleft, hright⟩
theorem positive_cut_polar_cuntz (x : D) (δ : ℝ≥0) (hδ : 0 < δ) :
CuntzBelow (Positive.cut δ (Positive.starMul x)) (Positive.cut δ (Positive.mulStar x)) ∧
CuntzBelow (Positive.cut δ (Positive.mulStar x)) (Positive.cut δ (Positive.starMul x)) := by
obtain ⟨y,hy,hz⟩ := supported_cfc_polar_factor x (fun t => max (t-δ) 0)
((continuous_id.sub continuous_const).max continuous_const) (fun _ => le_max_right _ _)
δ hδ (fun t ht => max_eq_right (sub_nonpos.mpr ht))
have hl : Positive.starMul y = Positive.cut δ (Positive.starMul x) := Subtype.ext hy
have hr : Positive.mulStar y = Positive.cut δ (Positive.mulStar x) := Subtype.ext hz
rw [← hl, ← hr]
exact ⟨starMul_cuntzBelow_mulStar y, mulStar_cuntzBelow_starMul y⟩
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
lemma finitePositiveClass_cut_polar (n : ℕ) (x : FinMatrix A n) (δ : ℝ≥0) (hδ : 0 < δ) :
finitePositiveClass n (Positive.cut δ (Positive.starMul x)) =
finitePositiveClass n (Positive.cut δ (Positive.mulStar x)) := by
obtain ⟨h,h'⟩ := positive_cut_polar_cuntz x δ hδ
exact le_antisymm ((finitePositiveClass_le_iff n _ _).mpr ((cuntzSubequiv_iff_shared _ _).mpr h))
((finitePositiveClass_le_iff n _ _).mpr ((cuntzSubequiv_iff_shared _ _).mpr h'))
theorem finiteSpectralIntegral_polar
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (n : ℕ) (x : FinMatrix A n) :
finiteSpectralIntegral d n (Positive.starMul x) =
finiteSpectralIntegral d n (Positive.mulStar x) := by
apply setLIntegral_congr_fun measurableSet_Ioi
intro t ht
unfold finiteDimensionDistribution
rw [finitePositiveClass_cut_polar n x t.toNNReal (Real.toNNReal_pos.mpr ht)]
end Pureness
end
end OAI
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