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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 | 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
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