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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 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.DimensionFunctions.FiniteSpectralIntegralBounds
import OAI.Analysis.JiangSu.CStarAlgebras.FiniteDimensionPowers
namespace OAI
noncomputable section
namespace Pureness
open TraceCone MeasureTheory Set Filter
open scoped ENNReal NNReal Topology
variable {D : Type*} [NonUnitalCStarAlgebra D] [PartialOrder D] [StarOrderedRing D]
theorem positive_root_cut_monotone (M : Positive D) (hM : ‖M.1‖ ≤ 1) (ε : ℝ≥0) :
Monotone (fun k => Positive.cut ε (Positive.power (Positive.rootExponent k) M)) := by
intro j k hjk
change cfcₙ (fun t : ℝ => max (t-ε) 0) (M.1 ^ Positive.rootExponent j) ≤
cfcₙ (fun t : ℝ => max (t-ε) 0) (M.1 ^ Positive.rootExponent k)
rw [CFC.nnrpow_eq_cfcₙ_real M.1 _ M.2,CFC.nnrpow_eq_cfcₙ_real M.1 _ M.2,
← cfcₙ_comp' (fun t : ℝ => max (t-ε) 0) (fun t : ℝ => t ^ (Positive.rootExponent j : ℝ)) M.1
(hf0 := Real.zero_rpow (show (Positive.rootExponent j : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos j).ne')),
← cfcₙ_comp' (fun t : ℝ => max (t-ε) 0) (fun t : ℝ => t ^ (Positive.rootExponent k : ℝ)) M.1
(hf0 := Real.zero_rpow (show (Positive.rootExponent k : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos k).ne'))]
apply (cfcₙ_le_iff _ _ M.1
(hf0 := by simp [Real.zero_rpow (show (Positive.rootExponent j : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos j).ne')])
(hg0 := by simp [Real.zero_rpow (show (Positive.rootExponent k : ℝ) ≠ 0 by exact_mod_cast (Positive.rootExponent_pos k).ne')])
(ha := IsSelfAdjoint.of_nonneg M.2)).mpr
intro t ht
have ht0 := quasispectrum_nonneg_of_nonneg M.1 M.2 t ht
have ht1 : t ≤ 1 := (le_abs_self t).trans
((NonUnitalIsometricContinuousFunctionalCalculus.norm_quasispectrum_le M.1 ht
(IsSelfAdjoint.of_nonneg M.2)).trans hM)
exact max_le_max (sub_le_sub_right (Real.rpow_le_rpow_of_exponent_ge' ht0 ht1
(Positive.rootExponent k).2 (Positive.rootExponent_antitone hjk)) ε) le_rfl
theorem positive_cut_le_smul_root_cut (M : Positive D) (hM : ‖M.1‖ ≤ 1)
(δ r α : ℝ≥0) (hr : 0 < r) (hα : (α : ℝ) < (δ : ℝ) ^ (r : ℝ)) :
(Positive.cut δ M).1 ≤ (1/((δ : ℝ)^(r : ℝ)-α)) •
(Positive.cut α (Positive.power r M)).1 := by
change cfcₙ (fun t : ℝ => max (t-δ) 0) M.1 ≤
(1/((δ : ℝ)^(r : ℝ)-α)) • cfcₙ (fun t : ℝ => max (t-α) 0) (M.1 ^ r)
rw [CFC.nnrpow_eq_cfcₙ_real M.1 r M.2,
← cfcₙ_comp' (fun t : ℝ => max (t-α) 0) (fun t : ℝ => t^(r : ℝ)) M.1
(hf0 := Real.zero_rpow (show (r : ℝ) ≠ 0 by exact_mod_cast hr.ne')),
← cfcₙ_smul (1/((δ : ℝ)^(r : ℝ)-α)) (fun t : ℝ => max (t^(r : ℝ)-α) 0) M.1
(h0 := by simp [Real.zero_rpow (show (r : ℝ) ≠ 0 by exact_mod_cast hr.ne')])]
apply (cfcₙ_le_iff _ _ M.1 (hf0 := by simp)
(hg0 := by simp [Real.zero_rpow (show (r : ℝ) ≠ 0 by exact_mod_cast hr.ne')])
(ha := IsSelfAdjoint.of_nonneg M.2)).mpr
intro t ht
change max (t-δ) 0 ≤ (1/((δ : ℝ)^(r : ℝ)-α))*max (t^(r : ℝ)-α) 0
by_cases htδ : t ≤ δ
· rw [max_eq_right (sub_nonpos.mpr htδ)]
exact mul_nonneg (one_div_nonneg.mpr (sub_pos.mpr hα).le) (le_max_right _ _)
· have hpow : (δ : ℝ)^(r : ℝ) ≤ t^(r : ℝ) :=
Real.rpow_le_rpow δ.2 (le_of_not_ge htδ) r.2
have ht1 : t ≤ 1 := (le_abs_self t).trans
((NonUnitalIsometricContinuousFunctionalCalculus.norm_quasispectrum_le M.1 ht
(IsSelfAdjoint.of_nonneg M.2)).trans hM)
calc
max (t-δ) 0 ≤ 1 := max_le ((sub_le_self t δ.2).trans ht1) zero_le_one
_ = (1/((δ : ℝ)^(r : ℝ)-α))*((δ : ℝ)^(r : ℝ)-α) := by field_simp [(sub_pos.mpr hα).ne']
_ ≤ _ := mul_le_mul_of_nonneg_left
((sub_le_sub_right hpow α).trans (le_max_left _ _))
(one_div_nonneg.mpr (sub_pos.mpr hα).le)
variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
lemma finiteSpectralIntegral_roots_monotone
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) :
Monotone (fun k => finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M)) := by
intro j k hjk
apply lintegral_mono
intro t
apply hd
apply (finitePositiveClass_le_iff n _ _).mpr
exact (cuntzSubequiv_iff_shared _ _).mpr
(CuntzBelow.of_le _ _ (positive_root_cut_monotone M hM t.toNNReal hjk))
lemma finiteSpectralIntegral_root_le
(d : finiteCuntzSubmonoid (A := A) →+ ℝ≥0∞) (hd : Monotone d)
(n : ℕ) (M : Positive (FinMatrix A n)) (hM : ‖M.1‖ ≤ 1) (k : ℕ) :
finiteSpectralIntegral d n (Positive.power (Positive.rootExponent k) M) ≤
d (finitePositiveClass n M) := by
have h := finiteSpectralIntegral_le d hd n (Positive.power (Positive.rootExponent k) M)
rw [finitePositiveClass_power n M _ (Positive.rootExponent_pos k)] at h
exact h.trans (mul_le_of_le_one_right bot_le (by
simpa using ENNReal.ofReal_le_ofReal (Positive.power_norm_le_one _ (Positive.rootExponent_pos k) M hM)))
end Pureness
end
end OAI
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