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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 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 | import Mathlib
import OAI.Analysis.Commutator.AmplifiedProjectionBlocks
import OAI.Analysis.Commutator.CentralAtoms
import OAI.Analysis.Commutator.CentralAverage
import OAI.Analysis.Commutator.CentralBlockEstimate
namespace OAI
noncomputable section
noncomputable section
open scoped ComplexOrder InnerProductSpace
namespace Kadison.CentralAtoms
open ProjectionBlocks
variable {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
[TopologicalSpace.SeparableSpace H]
omit [TopologicalSpace.SeparableSpace H] in
lemma block_commutes (M : VonNeumannAlgebra H) (Y : H →L[ℂ] H)
(hy : ∀ p : Atom M,Y*projection M p=projection M p*Y)
(h : ℕ) (X : Matrix (Fin h) (Fin h) (H →L[ℂ] H)) (hX : ∀ i j,X i j∈M)
(p : Atom M) :
(diagonalAmplification h Y*matrixOperator h X-matrixOperator h X*diagonalAmplification h Y)*
diagonalAmplification h (projection M p)=
diagonalAmplification h (projection M p)*
(diagonalAmplification h Y*matrixOperator h X-matrixOperator h X*diagonalAmplification h Y) := by
let A := diagonalAmplification h Y
let B := matrixOperator h X
let Q := diagonalAmplification h (projection M p)
have ha : A*Q=Q*A := by
change OperatorAmplification.diagonal (H:=H) (Fin h) Y*
OperatorAmplification.diagonal (H:=H) (Fin h) (projection M p)=_
rw [←map_mul,hy p,map_mul]
rfl
have hb : B*Q=Q*B := (diagonal_matrix_commute h (projection M p) X
(fun i j => (((CentralCorner.mem_center M _).mp p.property.1).2 (X i j) (hX i j)).symm)).symm
change (A*B-B*A)*Q=Q*(A*B-B*A)
calc
_=A*(B*Q)-B*(A*Q) := by simp only [sub_mul,mul_assoc]
_=A*(Q*B)-B*(Q*A) := by rw [ha,hb]
_=(A*Q)*B-(B*Q)*A := by simp only [mul_assoc]
_=(Q*A)*B-(Q*B)*A := by rw [ha,hb]
_=_ := by simp only [mul_sub,mul_assoc]
theorem centered_bound (M : VonNeumannAlgebra H) (hat : IsAtomicCenter M)
(Y : H →L[ℂ] H) (hy : ∀ p : Atom M,Y*projection M p=projection M p*Y)
(h : ℕ) (hh : 1≤h) (X : Matrix (Fin h) (Fin h) (H →L[ℂ] H)) (hX : ∀ i j,X i j∈M) :
‖diagonalAmplification h Y*matrixOperator h X-matrixOperator h X*diagonalAmplification h Y‖≤
3122*commutatorSeminorm M Y*‖matrixOperator h X‖ := by
apply norm_le_of_blocks (fun p : Atom M => diagonalAmplification h (projection M p))
(diagonal_projection (projection M) (isProjection M) h)
(diagonal_orthogonal (projection M) (orthogonal M) h) _
(mul_nonneg (mul_nonneg (by norm_num) (commutatorSeminorm_nonneg M Y)) (norm_nonneg _))
(block_commutes M Y hy h X hX)
· intro p ξ
let pM : M := ⟨projection M p,((CentralCorner.mem_center M _).mp p.property.1).1⟩
have hp : IsStarProjection pM :=
⟨Subtype.ext p.property.2.1.isIdempotentElem.eq,Subtype.ext p.property.2.1.isSelfAdjoint.star_eq⟩
have hz : (pM:H →L[ℂ] H)∈Set.centralizer (M : Set (H →L[ℂ] H)) :=
((CentralCorner.mem_center M _).mp p.property.1).2
exact CentralCorner.projected_vector_bound M hp hz p.property Y (hy p) h hh
(fun i j => ⟨X i j,hX i j⟩) ξ
· exact diagonal_full (projection M) (isProjection M) (orthogonal M) h (part_eq_one_of_atomic M hat)
theorem uniform_commutator_atomic_center (M : VonNeumannAlgebra H) (hat : IsAtomicCenter M)
(Y : H →L[ℂ] H) (h : ℕ) (hh : 1≤h)
(X : Matrix (Fin h) (Fin h) (H →L[ℂ] H)) (hX : ∀ i j,X i j∈M) :
‖diagonalAmplification h Y*matrixOperator h X-matrixOperator h X*diagonalAmplification h Y‖≤
9368*commutatorSeminorm M Y*‖matrixOperator h X‖ := by
obtain ⟨Z,hz,hd,_⟩ := CentralAverage.exists_center_average M Y
have hc (p : Atom M) : Z*projection M p=projection M p*Z :=
hz _ ((CentralCorner.mem_center M _).mp p.property.1).1
((CentralCorner.mem_center M _).mp p.property.1).2
have hb := centered_bound M hat Z hc h hh X hX
convert matrix_commutator_perturb M Y Z hd 3122 (by norm_num) h X hb using 1
norm_num
end Kadison.CentralAtoms
end
end
end OAI
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