File size: 2,405 Bytes
a458723 | 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 | import Mathlib
import OAI.Analysis.Commutator.BalancedSymmetry
import OAI.Analysis.Commutator.MatrixEntryWOT
namespace OAI
noncomputable section
section
noncomputable section
open scoped BigOperators InnerProductSpace ComplexOrder
namespace Kadison.OperatorAmplification
variable {A H ι : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
[NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] [Fintype ι]
local instance blockDilationIndexDecidableEq : DecidableEq ι := Classical.decEq ι
local instance blockDilationFinTwoDecidableEq : DecidableEq (Fin 2) := Classical.decEq (Fin 2)
lemma entry_starHom_matrixRepresentation (ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H))
(X : CStarMatrix ι ι A) (i j : ι) :
entry ι (CyclicColumns.matrixRepresentation ρ X) i j=ρ (X i j) := by
ext x
change CyclicColumns.matrixAction (fun i j => ρ (X i j)) (single ι j x) i=ρ (X i j) x
simp only [CyclicColumns.matrixAction_apply, single_apply]
simp only [apply_ite, map_zero, Finset.sum_ite_eq',Finset.mem_univ,ite_true]
lemma entry_diagonal_commutator (T : H →L[ℂ] H)
(X : SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι) (i j : ι) :
entry ι (diagonal (H:=H) ι T*X-X*diagonal (H:=H) ι T) i j=
T*entry ι X i j-entry ι X i j*T := by
ext v
simp [entry_apply,mul_apply_eq_comp,diagonal_single]
lemma commutator_entry_norm_le (T : H →L[ℂ] H)
(X : SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι) (i j : ι) :
‖T*entry ι X i j-entry ι X i j*T‖≤
‖diagonal (H:=H) ι T*X-X*diagonal (H:=H) ι T‖ := by
rw [←entry_diagonal_commutator]
exact norm_entry_le ι _ i j
theorem balanced_symmetry_commutator_dominates
(ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (Y : H →L[ℂ] H) (X : A) :
‖Y*ρ X-ρ X*Y‖≤
‖diagonal (H:=H) (Fin 2) Y*
CyclicColumns.matrixRepresentation ρ (BalancedDilation.symmetry X)-
CyclicColumns.matrixRepresentation ρ (BalancedDilation.symmetry X)*
diagonal (H:=H) (Fin 2) Y‖ := by
have he : entry (Fin 2) (CyclicColumns.matrixRepresentation ρ (BalancedDilation.symmetry X)) 0 1=ρ X := by
rw [entry_starHom_matrixRepresentation]
rfl
simpa only [he] using commutator_entry_norm_le Y
(CyclicColumns.matrixRepresentation ρ (BalancedDilation.symmetry X)) (0:Fin 2) 1
end Kadison.OperatorAmplification
end
end
end
end OAI
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