| import Mathlib |
| import OAI.Analysis.Commutator.Definitions |
| import OAI.Analysis.Commutator.StrongContractionLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| noncomputable section |
|
|
| open scoped InnerProductSpace Topology |
|
|
| open Filter |
|
|
| namespace Kadison.StrongContractionLimit |
|
|
| variable {ι H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| variable {l : Filter ι} |
|
|
| omit [CompleteSpace H] in |
| lemma piLp_tendsto {κ : Type*} [Fintype κ] (u : ι → PiLp 2 (fun _ : κ => H)) |
| (x : PiLp 2 (fun _ : κ => H)) (hu : ∀ k,Tendsto (fun i => u i k) l (𝓝 (x k))) : |
| Tendsto u l (𝓝 x) := by |
| have h := (PiLp.continuousLinearEquiv 2 ℂ (fun _ : κ => H)).symm.continuous.tendsto |
| (fun k => x k) |
| exact h.comp (tendsto_pi_nhds.mpr hu) |
|
|
| omit [CompleteSpace H] in |
| lemma diagonal_tendsto (P : ι → H →L[ℂ] H) (Y : H →L[ℂ] H) |
| (hstrong : ∀ x,Tendsto (fun i => P i x) l (𝓝 (Y x))) |
| (h : ℕ) (x : PiLp 2 (fun _ : Fin h => H)) : |
| Tendsto (fun i => diagonalAmplification h (P i) x) l (𝓝 (diagonalAmplification h Y x)) := by |
| apply piLp_tendsto |
| intro k |
| exact hstrong (x k) |
|
|
| omit [CompleteSpace H] in |
| lemma commutator_tendsto (P : ι → H →L[ℂ] H) (Y : H →L[ℂ] H) |
| (hstrong : ∀ x,Tendsto (fun i => P i x) l (𝓝 (Y x))) |
| (h : ℕ) (X : Matrix (Fin h) (Fin h) (H →L[ℂ] H)) |
| (x : PiLp 2 (fun _ : Fin h => H)) : |
| Tendsto (fun i => (diagonalAmplification h (P i)*matrixOperator h X- |
| matrixOperator h X*diagonalAmplification h (P i)) x) l |
| (𝓝 ((diagonalAmplification h Y*matrixOperator h X- |
| matrixOperator h X*diagonalAmplification h Y) x)) := by |
| exact (diagonal_tendsto P Y hstrong h (matrixOperator h X x)).sub |
| (((matrixOperator h X).continuous.tendsto (diagonalAmplification h Y x)).comp |
| (diagonal_tendsto P Y hstrong h x)) |
|
|
| omit [CompleteSpace H] in |
| lemma commutator_bound_limit [l.NeBot] (P : ι → H →L[ℂ] H) (Y : H →L[ℂ] H) |
| (hstrong : ∀ x,Tendsto (fun i => P i x) l (𝓝 (Y x))) |
| (h : ℕ) (X : Matrix (Fin h) (Fin h) (H →L[ℂ] H)) (C : ℝ) (hC : 0≤C) |
| (hbound : ∀ᶠ i in l,‖diagonalAmplification h (P i)*matrixOperator h X- |
| matrixOperator h X*diagonalAmplification h (P i)‖≤C) : |
| ‖diagonalAmplification h Y*matrixOperator h X- |
| matrixOperator h X*diagonalAmplification h Y‖≤C := |
| norm_le_of_strong _ _ (commutator_tendsto P Y hstrong h X) hC hbound |
|
|
| end Kadison.StrongContractionLimit |
|
|
| end |
|
|
| end |
| end |
|
|
| end OAI |
|
|
