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67aee1d | 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 | import Mathlib
import OAI.Analysis.FreeWords.RestrictedDerivationDefinitions
import OAI.Analysis.FreeWords.ScalarMatrixCommutator
namespace OAI
noncomputable section
namespace KKMatrixHilbert
noncomputable section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
theorem controlled_lift_commutator {n d : ℕ}
(S : StarSubalgebra ℂ (H →L[ℂ] H))
(ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] (H →L[ℂ] H))
(hρ : ∀ a, ρ a ∈ S) (h : H →L[ℂ] H) (_ : IsSelfAdjoint h)
(s : unitary (Matrix (Fin n) (Fin n) (H →L[ℂ] H)))
(hs : ∀ i j, s.val i j ∈ ρ.range) :
‖matrixOperator (Matrix.diagonal fun _ : Fin n => h) * matrixOperator s.val -
matrixOperator s.val * matrixOperator (Matrix.diagonal fun _ : Fin n => h)‖ ≤
2 * ‖restrictedDerivation S h‖ := by
apply scalar_matrix_commutator_bound ρ h _ _ s hs
intro u
let a : S := ⟨ρ u.val,hρ u.val⟩
have ha : a ∈ unitary S := by
constructor
· apply Subtype.ext
change star (ρ u.val) * ρ u.val = 1
rw [←map_star,←map_mul,u.property.1,map_one]
· apply Subtype.ext
change ρ u.val * star (ρ u.val) = 1
rw [←map_star,←map_mul,u.property.2,map_one]
have hnorm : ‖a‖ ≤ 1 := by
cases subsingleton_or_nontrivial S with
| inl h => have := h; simp [Subsingleton.elim a 0]
| inr h => have := h; exact (CStarRing.norm_of_mem_unitary ha).le
change ‖restrictedDerivation S h a‖ ≤ ‖restrictedDerivation S h‖
calc
_ ≤ ‖restrictedDerivation S h‖ * ‖a‖ := (restrictedDerivation S h).le_opNorm a
_ ≤ ‖restrictedDerivation S h‖ * 1 := mul_le_mul_of_nonneg_left hnorm (norm_nonneg _)
_ = _ := mul_one _
end
end KKMatrixHilbert
end
end OAI
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