File size: 7,485 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegralAbelian
import OAI.Analysis.JiangSu.MatrixAlgebras.PositiveContinuousRepresentation
namespace OAI
noncomputable section
namespace Pureness
open TraceCone Set
open scoped NNReal ENNReal ComplexOrder IsMulCommutative
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
theorem finiteSpectralIntegral_add_commute
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(hlsc : ∀ n, LowerSemicontinuous (fun M : Positive (FinMatrix A n) => d (finitePositiveClass n M)))
(hfinite : ∀ x, d x ≠ ⊤)
(n : ℕ) (M N : Positive (FinMatrix A n)) (hMN : Commute M.1 N.1) :
finiteSpectralIntegral d n (Positive.add M N) =
finiteSpectralIntegral d n M + finiteSpectralIntegral d n N := by
let S : StarSubalgebra ℂ (FinMatrix A n) := (StarAlgebra.adjoin ℂ {M.1,N.1}).topologicalClosure
let : IsClosed (S : Set (FinMatrix A n)) := (StarAlgebra.adjoin ℂ {M.1,N.1}).isClosed_topologicalClosure
have hcomm : ∀ x ∈ ({M.1,N.1} : Set (FinMatrix A n)), ∀ y ∈ ({M.1,N.1} : Set (FinMatrix A n)), x*y = y*x := by
intro x hx y hy
simp only [mem_insert_iff,mem_singleton_iff] at hx hy
rcases hx with rfl | rfl <;> rcases hy with rfl | rfl
· rfl
· exact hMN.eq
· exact hMN.eq.symm
· rfl
let : IsMulCommutative (StarAlgebra.adjoin ℂ {M.1,N.1}) :=
StarAlgebra.isMulCommutative_adjoin ℂ (by
intro element helement
simp only [mem_insert_iff,mem_singleton_iff] at helement
rcases helement with rfl | rfl
· exact M.2.isSelfAdjoint.isStarNormal
· exact N.2.isSelfAdjoint.isStarNormal)
(fun left hleft right hright _ => hcomm left hleft right hright) (by
intro x hx y hy _
have hyStar : star y = y := by
simp only [mem_insert_iff,mem_singleton_iff] at hy
rcases hy with rfl | rfl
· exact M.2.star_eq
· exact N.2.star_eq
rw [hyStar]
exact hcomm x hx y hy)
let : IsMulCommutative S := (StarAlgebra.adjoin ℂ {M.1,N.1}).isMulCommutative_topologicalClosure
let : CommCStarAlgebra S := { }
have hMS : M.1 ∈ S := (StarAlgebra.adjoin ℂ {M.1,N.1}).le_topologicalClosure
(StarAlgebra.subset_adjoin ℂ _ (by simp))
have hNS : N.1 ∈ S := (StarAlgebra.adjoin ℂ {M.1,N.1}).le_topologicalClosure
(StarAlgebra.subset_adjoin ℂ _ (by simp))
let X := WeakDual.characterSpace ℂ S
let e := gelfandStarTransform S
let φ : C(X, ℂ) →⋆ₙₐ[ℂ] FinMatrix A n :=
(S.subtype.comp e.symm.toStarAlgHom).toNonUnitalStarAlgHom
have hφ : Function.Injective φ := Subtype.val_injective.comp e.symm.injective
let m : C(X, ℂ) := e ⟨M.1,hMS⟩
let q : C(X, ℂ) := e ⟨N.1,hNS⟩
have hm : φ m = M.1 := by change (e.symm (e ⟨M.1,hMS⟩)).1 = M.1; rw [e.symm_apply_apply]
have hq : φ q = N.1 := by change (e.symm (e ⟨N.1,hNS⟩)).1 = N.1; rw [e.symm_apply_apply]
have hmpos : 0 ≤ m := (nonUnitalStarAlgHom_nonneg_iff φ hφ m).mp (hm.symm ▸ M.2)
have hqpos : 0 ≤ q := (nonUnitalStarAlgHom_nonneg_iff φ hφ q).mp (hq.symm ▸ N.2)
obtain ⟨f,hf⟩ := continuousNNRealPositive_surjective (⟨m,hmpos⟩ : Positive C(X, ℂ))
obtain ⟨g,hg⟩ := continuousNNRealPositive_surjective (⟨q,hqpos⟩ : Positive C(X, ℂ))
have hfM : Positive.map φ (continuousNNRealPositive f) = M := by rw [hf]; exact Subtype.ext hm
have hgN : Positive.map φ (continuousNNRealPositive g) = N := by rw [hg]; exact Subtype.ext hq
simpa only [hfM,hgN] using finiteSpectralIntegral_image_add d hd hlsc hfinite n φ f g
end Pureness
namespace Pureness
open TraceCone MeasureTheory Set
open scoped ENNReal NNReal
variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D]
theorem positive_cut_smul (r : ℝ≥0) (hr : 0 < r) (ε : ℝ≥0) (M : Positive D) :
Positive.cut ε (Positive.smul r M) = Positive.smul r (Positive.cut (ε/r) M) := by
apply Subtype.ext
change cfcₙ (fun t : ℝ => max (t-ε) 0) ((r : ℝ) • M.1) =
(r : ℝ) • cfcₙ (fun t : ℝ => max (t-(ε/r : ℝ≥0)) 0) M.1
rw [← cfcₙ_comp_smul (r : ℝ) (fun t : ℝ => max (t-ε) 0) M.1,
← cfcₙ_smul (r : ℝ) (fun t : ℝ => max (t-(ε/r : ℝ≥0)) 0) M.1 (h0 := by simp only [zero_sub]; exact max_eq_right (neg_nonpos.mpr (NNReal.coe_nonneg _)))]
apply cfcₙ_congr
intro t ht
change max ((r : ℝ)*t-ε) 0 = (r : ℝ)*max (t-(ε/r : ℝ≥0)) 0
rw [NNReal.coe_div, mul_max_of_nonneg _ _ (show (0 : ℝ) ≤ r from r.2),mul_zero,mul_sub]
congr 1
field_simp
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
lemma finitePositiveClass_smul (n : ℕ) (M : Positive (FinMatrix A n))
(r : ℝ≥0) (hr : 0 < r) :
finitePositiveClass n (Positive.smul r M) = finitePositiveClass n M := by
have h := positive_smul_cuntz_equiv M.1 r hr
exact le_antisymm ((finitePositiveClass_le_iff n _ _).mpr h.1)
((finitePositiveClass_le_iff n _ _).mpr h.2)
lemma finiteDimensionDistribution_smul
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞)
(n : ℕ) (M : Positive (FinMatrix A n)) (r : ℝ≥0) (hr : 0 < r) (t : ℝ) :
finiteDimensionDistribution d n (Positive.smul r M) t =
finiteDimensionDistribution d n M (t/r) := by
unfold finiteDimensionDistribution
rw [positive_cut_smul r hr,finitePositiveClass_smul n _ r hr]
congr 3
exact by
by_cases ht : 0 ≤ t
· simp [Real.toNNReal_div ht, Real.toNNReal_coe]
· have htn : t ≤ 0 := (lt_of_not_ge ht).le
rw [Real.toNNReal_eq_zero.mpr htn, zero_div]
exact (Real.toNNReal_eq_zero.mpr (div_nonpos_of_nonpos_of_nonneg htn r.2)).symm
lemma positive_halfline_integral_div (f : ℝ → ℝ≥0∞) (hf : Measurable f)
(r : ℝ) (hr : 0 < r) :
(∫⁻ t in Ioi (0 : ℝ), f (t/r)) = ENNReal.ofReal r * ∫⁻ t in Ioi (0 : ℝ), f t := by
let g := (Ioi (0 : ℝ)).indicator f
have hg : Measurable g := hf.indicator measurableSet_Ioi
have heq : (Ioi (0 : ℝ)).indicator (fun t => f (t/r)) = fun t => g (r⁻¹*t) := by
funext t
by_cases ht : 0 < t
· simp [g,ht,inv_mul_eq_div,div_pos ht hr]
· have hn : ¬ 0 < t/r := by simpa only [div_pos_iff_of_pos_right hr] using ht
simp [g,ht,inv_mul_eq_div,hn]
rw [← lintegral_indicator measurableSet_Ioi, heq, ← lintegral_map hg (show Measurable (fun t : ℝ => r⁻¹*t) from measurable_const.mul measurable_id),
Real.map_volume_mul_left (inv_ne_zero hr.ne'), lintegral_smul_measure]
simp only [inv_inv, abs_of_pos hr, smul_eq_mul]
rw [lintegral_indicator measurableSet_Ioi]
theorem finiteSpectralIntegral_smul
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(n : ℕ) (M : Positive (FinMatrix A n)) (r : ℝ≥0) :
finiteSpectralIntegral d n (Positive.smul r M) = (r : ℝ≥0∞)*finiteSpectralIntegral d n M := by
by_cases hr : r = 0
· subst r
have hzero : Positive.smul 0 M = Positive.zero := by apply Subtype.ext; change (0 : ℝ) • M.1 = 0; exact zero_smul ℝ M.1
simp [hzero,finiteSpectralIntegral_zero]
· have hrpos : 0 < r := pos_iff_ne_zero.mpr hr
simp only [finiteSpectralIntegral,finiteDimensionDistribution_smul d n M r hrpos]
rw [positive_halfline_integral_div _ (finiteDimensionDistribution_measurable d hd n M) r hrpos,
ENNReal.ofReal_coe_nnreal]
end Pureness
end
end OAI
|