| 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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