File size: 4,528 Bytes
a97875e | 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 | import Mathlib
namespace OAI
noncomputable section
section
noncomputable section
open scoped Topology InnerProductSpace
open MeasureTheory Filter
namespace Kadison.WeakOperatorIntegral
variable {ι H : Type*} [MeasurableSpace ι]
[NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (μ : Measure ι) [IsProbabilityMeasure μ] (T : ι → H →L[ℂ] H)
(hI : ∀ x y, Integrable (fun i => inner ℂ (T i x) y) μ)
(V : H →L[ℂ] H)
(hV : ∀ x y, inner ℂ (V x) y = ∫ i, inner ℂ (T i x) y ∂μ)
omit [CompleteSpace H] in
include hV in
lemma norm_le_of_integral {C : ℝ} (hC : 0 ≤ C) (hT : ∀ i, ‖T i‖ ≤ C) : ‖V‖ ≤ C := by
apply ContinuousLinearMap.opNorm_le_bound V hC
intro x
have hi : ‖inner ℂ (V x) (V x)‖ ≤ C*‖x‖*‖V x‖ := by
rw [hV]
have h := norm_integral_le_of_norm_le_const (μ := μ)
(f := fun i => inner ℂ (T i x) (V x)) (C := C*‖x‖*‖V x‖)
(Eventually.of_forall (fun i => (norm_inner_le_norm _ _).trans
(mul_le_mul_of_nonneg_right (((T i).le_opNorm x).trans
(mul_le_mul_of_nonneg_right (hT i) (norm_nonneg x))) (norm_nonneg _))))
simpa only [probReal_univ,mul_one] using h
rw [← inner_self_re_eq_norm,inner_self_eq_norm_mul_norm] at hi
by_cases hv : ‖V x‖=0
· rw [hv]; positivity
· have hp : 0 < ‖V x‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hv)
exact (mul_le_mul_iff_left₀ hp).mp (by simpa [mul_comm,mul_left_comm,mul_assoc] using hi)
omit [IsProbabilityMeasure μ] in
include hI hV in
lemma commutator_coeff (a : H →L[ℂ] H) (x y : H) :
inner ℂ ((V*a-a*V) x) y = ∫ i, inner ℂ ((T i*a-a*T i) x) y ∂μ := by
simp only [sub_apply,mul_apply_eq_comp,inner_sub_left]
simp only [← ContinuousLinearMap.adjoint_inner_right a]
rw [hV,hV,integral_sub (hI (a x) y) (hI x (a.adjoint y))]
include hI hV in
lemma commutator_norm (a : H →L[ℂ] H) {C : ℝ} (hC : 0 ≤ C)
(hb : ∀ i, ‖T i*a-a*T i‖ ≤ C) : ‖V*a-a*V‖ ≤ C := by
exact norm_le_of_integral μ (fun i => T i*a-a*T i) (V*a-a*V)
(commutator_coeff μ T hI V hV a) hC hb
include hI hV in
lemma commutes (a : H →L[ℂ] H) (hb : ∀ i, T i*a=a*T i) : V*a=a*V := by
apply sub_eq_zero.mp
apply norm_eq_zero.mp
exact le_antisymm (commutator_norm μ T hI V hV a le_rfl (by intro i; rw [hb i,sub_self,norm_zero]))
(norm_nonneg _)
omit [IsProbabilityMeasure μ] in
include hV in
lemma sandwich_coeff (a b : H →L[ℂ] H) (x y : H) :
inner ℂ ((a*V*b) x) y = ∫ i, inner ℂ ((a*T i*b) x) y ∂μ := by
simp only [mul_apply_eq_comp]
simp only [← ContinuousLinearMap.adjoint_inner_right a]
exact hV _ _
include hV in
lemma preserves_sandwich (a b Z : H →L[ℂ] H) (hZ : ∀ i, a*T i*b=Z) : a*V*b=Z := by
ext x
apply ext_inner_right ℂ
intro y
rw [sandwich_coeff μ T V hV]
simp_rw [hZ]
simp only [integral_const,probReal_univ,one_smul]
end Kadison.WeakOperatorIntegral
namespace Kadison.WeakOperatorIntegral
open MeasureTheory Filter
variable {ι H : Type*} [MeasurableSpace ι]
[NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable (μ : Measure ι) [IsProbabilityMeasure μ] (T : ι → H →L[ℂ] H)
(hI : ∀ x y, Integrable (fun i => inner ℂ (T i x) y) μ)
(V : H →L[ℂ] H)
(hV : ∀ x y, inner ℂ (V x) y = ∫ i, inner ℂ (T i x) y ∂μ)
omit [CompleteSpace H] in
include hI hV in
lemma difference_coeff (W : H →L[ℂ] H) (x y : H) :
inner ℂ ((V-W) x) y = ∫ i, inner ℂ ((T i-W) x) y ∂μ := by
simp only [sub_apply,inner_sub_left]
rw [hV,integral_sub (hI x y) (integrable_const _)]
simp only [integral_const,MeasureTheory.probReal_univ,one_smul]
omit [CompleteSpace H] in
include hI hV in
lemma difference_norm (W : H →L[ℂ] H) {C : ℝ} (hC : 0≤C)
(hb : ∀ i, ‖T i-W‖≤C) : ‖V-W‖≤C :=
norm_le_of_integral μ (fun i => T i-W) (V-W) (difference_coeff μ T hI V hV W) hC hb
omit [IsProbabilityMeasure μ] in
include hI hV in
lemma compressed_commutator_coeff (a b s : H →L[ℂ] H) (x y : H) :
inner ℂ ((a*(V*s-s*V)*b) x) y = ∫ i, inner ℂ ((a*(T i*s-s*T i)*b) x) y ∂μ :=
sandwich_coeff μ _ (V*s-s*V) (commutator_coeff μ T hI V hV s) a b x y
include hI hV in
lemma preserves_compressed_commutator (a b s W : H →L[ℂ] H)
(hb : ∀ i, a*(T i*s-s*T i)*b=W) : a*(V*s-s*V)*b=W :=
preserves_sandwich μ _ _ (commutator_coeff μ T hI V hV s) a b W hb
end Kadison.WeakOperatorIntegral
end
end
end
end OAI
|