| import Mathlib |
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| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| open scoped Topology InnerProductSpace |
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| open MeasureTheory Filter |
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| namespace Kadison.WeakOperatorIntegral |
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| variable {ι H : Type*} [MeasurableSpace ι] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| 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) |
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| 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) |
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| 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))] |
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| 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 |
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| 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 _) |
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| 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 _ _ |
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| 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] |
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| end Kadison.WeakOperatorIntegral |
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| namespace Kadison.WeakOperatorIntegral |
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| open MeasureTheory Filter |
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| variable {ι H : Type*} [MeasurableSpace ι] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| 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) |
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| 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] |
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| 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 |
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| 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 |
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| 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 |
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| end Kadison.WeakOperatorIntegral |
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| end |
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| end |
| end |
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| end OAI |
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