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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 88 89 90 91 92 93 94 95 96 97 | import Mathlib
import OAI.Analysis.Commutator.HilbertPadding
namespace OAI
noncomputable section
section
noncomputable section
open scoped BigOperators InnerProductSpace
open ContinuousLinearMap
namespace Kadison.HilbertPadding
open OperatorAmplification CyclicColumns
variable {A H K : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
[NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
[NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]
variable (ι : Type*) [Fintype ι]
local instance : DecidableEq ι := Classical.decEq ι
def amplifyIsometry (E : H →ₗᵢ[ℂ] K) : SumSpace (H:=H) ι →ₗᵢ[ℂ] SumSpace (H:=K) ι where
toFun v := WithLp.toLp 2 (fun i => E (v i))
map_add' v w := by ext i; simp
map_smul' z v := by ext i; simp
norm_map' v := by
apply (sq_eq_sq₀ (norm_nonneg _) (norm_nonneg _)).mp
simp [PiLp.norm_sq_eq_of_L2]
omit [CompleteSpace H] [CompleteSpace K] in
@[simp] lemma amplifyIsometry_apply (E : H →ₗᵢ[ℂ] K) (v : SumSpace (H:=H) ι) (i : ι) :
amplifyIsometry ι E v i=E (v i) := rfl
lemma amplifyIsometry_adjoint (E : H →ₗᵢ[ℂ] K) (v : SumSpace (H:=K) ι) :
(amplifyIsometry ι E).toContinuousLinearMap.adjoint v=
WithLp.toLp 2 (fun i => E.toContinuousLinearMap.adjoint (v i)) := by
apply ext_inner_left ℂ
intro w
rw [adjoint_inner_right]
simp only [PiLp.inner_apply,amplifyIsometry_apply,adjoint_inner_right,
LinearIsometry.coe_toContinuousLinearMap]
lemma diagonal_extend (E : H →ₗᵢ[ℂ] K) (Y : H →L[ℂ] H) :
diagonal (H:=K) ι (extend E Y)=extend (amplifyIsometry ι E) (diagonal (H:=H) ι Y) := by
ext v i
simp [extend,amplifyIsometry_adjoint]
lemma amplifyIsometry_matrix_intertwines (E : H →ₗᵢ[ℂ] K)
(ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K))
(hE : ∀ a,(σ a).comp E.toContinuousLinearMap=E.toContinuousLinearMap.comp (ρ a))
(X : CStarMatrix ι ι A) :
(matrixRepresentation σ X).comp (amplifyIsometry ι E).toContinuousLinearMap=
(amplifyIsometry ι E).toContinuousLinearMap.comp (matrixRepresentation ρ X) := by
ext v i
change matrixAction (fun i j => σ (X i j)) (amplifyIsometry ι E v) i=
E (matrixAction (fun i j => ρ (X i j)) v i)
simp only [matrixAction_apply,amplifyIsometry_apply,map_sum]
apply Finset.sum_congr rfl
intro j _
exact congrArg (fun T : H →L[ℂ] K => T (v j)) (hE (X i j))
lemma amplifyIsometry_adjoint_matrix_intertwines (E : H →ₗᵢ[ℂ] K)
(ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K))
(hE : ∀ a,(σ a).comp E.toContinuousLinearMap=E.toContinuousLinearMap.comp (ρ a))
(X : CStarMatrix ι ι A) :
(amplifyIsometry ι E).toContinuousLinearMap.adjoint.comp (matrixRepresentation σ X)=
(matrixRepresentation ρ X).comp (amplifyIsometry ι E).toContinuousLinearMap.adjoint := by
have h := congrArg (fun T => T.adjoint) (amplifyIsometry_matrix_intertwines ι E ρ σ hE (star X))
simpa only [adjoint_comp,←star_eq_adjoint,←map_star,star_star] using h
theorem amplified_extension_commutator_norm (E : H →ₗᵢ[ℂ] K)
(ρ : A →⋆ₐ[ℂ] (H →L[ℂ] H)) (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K))
(hE : ∀ a,(σ a).comp E.toContinuousLinearMap=E.toContinuousLinearMap.comp (ρ a))
(Y : H →L[ℂ] H) (X : CStarMatrix ι ι A) :
‖diagonal (H:=K) ι (extend E Y)*matrixRepresentation σ X-
matrixRepresentation σ X*diagonal (H:=K) ι (extend E Y)‖=
‖diagonal (H:=H) ι Y*matrixRepresentation ρ X-
matrixRepresentation ρ X*diagonal (H:=H) ι Y‖ := by
rw [diagonal_extend]
exact norm_extend_commutator _ _ _ _ (amplifyIsometry_matrix_intertwines ι E ρ σ hE X)
(amplifyIsometry_adjoint_matrix_intertwines ι E ρ σ hE X)
end Kadison.HilbertPadding
end
end
end
end OAI
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