| import Mathlib | |
| import OAI.Analysis.Commutator.AtomicDecomposition | |
| import OAI.Analysis.Commutator.AtomicFinite | |
| import OAI.Analysis.Commutator.AtomicInfinite | |
| import OAI.Analysis.Commutator.FiniteDimensionalMain | |
| import OAI.Analysis.Commutator.ProperlyInfiniteMain | |
| namespace OAI | |
| noncomputable section | |
| section | |
| noncomputable section | |
| open scoped InnerProductSpace | |
| namespace Kadison | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| theorem uniform_commutator_atomic_factor (P : VonNeumannAlgebra H) | |
| (hfac : FactorCorner.IsFactor P) | |
| (hatom : ¬DiffuseProjection.HasNoMinimalProjections P) | |
| (Y : H →L[ℂ] H) (h : ℕ) | |
| (X : Matrix (Fin h) (Fin h) (H →L[ℂ] H)) (hX : ∀ i j,X i j∈P) : | |
| ‖diagonalAmplification h Y*matrixOperator h X- | |
| matrixOperator h X*diagonalAmplification h Y‖ ≤ | |
| 9*commutatorSeminorm P Y*‖matrixOperator h X‖ := by | |
| obtain ⟨p,hp⟩ := AtomicFactor.exists_minimal_of_not_diffuse P hatom | |
| obtain ⟨D⟩ := AtomicFactor.exists_decomposition P hfac hp | |
| rcases finite_or_infinite D.I with hD|hD | |
| · let := hD | |
| let := D.finite_dimensional hp | |
| exact (uniform_commutator_finiteDimensional P Y h X hX).trans | |
| (mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (by norm_num : (2:ℝ)≤9) | |
| (commutatorSeminorm_nonneg P Y)) (norm_nonneg _)) | |
| · let := hD | |
| obtain ⟨s,t,hs,ht,hst⟩ := D.two_isometries | |
| exact uniform_commutator_of_two_isometries P s t hs ht hst Y h X hX | |
| end Kadison | |
| end | |
| end | |
| end | |
| end OAI | |