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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 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.CbDerivation
import OAI.Analysis.Commutator.CyclicCbUnconditional
namespace OAI
noncomputable section
section
noncomputable section
namespace Kadison.CbFactor
open RectangularColumn CyclicColumns
variable {A K F : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
[NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K]
[NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F]
local instance : NormedAddCommGroup (K →L[ℂ] F) := ContinuousLinearMap.toNormedAddCommGroup
local instance : NormedSpace ℂ (K →L[ℂ] F) := ContinuousLinearMap.toNormedSpace
omit [CompleteSpace K] [CompleteSpace F] in
lemma matrixAction_smul {ι : Type*} [Fintype ι] (c : ℝ) (D : ι → ι → K →L[ℂ] F) :
matrixAction (fun i j => c • D i j)=c • matrixAction D := by
ext x i
simp [Finset.smul_sum]
theorem rectangular_implementation_quantitative (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K))
(lam : A →⋆ₐ[ℂ] (F →L[ℂ] F)) (D : A →L[ℂ] (K →L[ℂ] F))
(hD : ∀ a b, D (a*b)=(D a).comp (σ b)+(lam a).comp (D b))
{c : ℝ} (hc : 0≤c)
(hcb : ∀ (ι : Type) [Fintype ι] (X : CStarMatrix ι ι A),
‖matrixAction (fun i j => D (X i j))‖≤c*‖D‖*‖X‖) :
∃ V : K →L[ℂ] F,‖V‖≤((1+c)^2)*‖D‖ ∧ ∀ a : A,D a=V.comp (σ a)-(lam a).comp V := by
by_cases hz : D=0
· refine ⟨0,by simp [hz],?_⟩
intro a
simp [hz]
have hd : 0<‖D‖ := norm_pos_iff.mpr hz
let Δ : A →L[ℂ] (K →L[ℂ] F) := ‖D‖⁻¹ • D
have hΔ : ∀ a b,Δ (a*b)=(Δ a).comp (σ b)+(lam a).comp (Δ b) := by
intro a b
simp only [Δ,smul_apply,hD,smul_add,
ContinuousLinearMap.smul_comp,ContinuousLinearMap.comp_smul]
have hnorm : ∀ (ι : Type) [Fintype ι] (X : CStarMatrix ι ι A),
‖matrixAction (fun i j => Δ (X i j))‖≤c*‖X‖ := by
intro ι _ X
change ‖matrixAction (fun i j => ‖D‖⁻¹ • D (X i j))‖≤c*‖X‖
rw [matrixAction_smul]
calc
_ ≤ ‖‖D‖⁻¹‖*‖matrixAction (fun i j => D (X i j))‖ := ContinuousLinearMap.opNorm_smul_le _ _
_ ≤ ‖D‖⁻¹*(c*‖D‖*‖X‖) := by
rw [Real.norm_eq_abs,abs_of_pos (inv_pos.mpr hd)]
exact mul_le_mul_of_nonneg_left (hcb ι X) (inv_nonneg.mpr hd.le)
_ = c*‖X‖ := by field_simp
obtain ⟨V,hV,himpl⟩ := rectangular_implementation σ lam Δ hΔ hc hnorm
refine ⟨‖D‖ • V,?_,?_⟩
· exact (ContinuousLinearMap.opNorm_smul_le _ _).trans
((mul_le_mul_of_nonneg_left hV (norm_nonneg (‖D‖))).trans_eq (by
rw [Real.norm_eq_abs,abs_of_nonneg (norm_nonneg D)]; ring))
· intro a
have hh : ‖D‖ • Δ a=D a := by
simp only [Δ,smul_apply,smul_smul,mul_inv_cancel₀ hd.ne',one_smul]
rw [←hh,himpl,smul_sub,ContinuousLinearMap.smul_comp,ContinuousLinearMap.comp_smul]
theorem cyclic_implementation (σ : A →⋆ₐ[ℂ] (K →L[ℂ] K))
(lam : A →⋆ₐ[ℂ] (F →L[ℂ] F)) (D : A →L[ℂ] (K →L[ℂ] F))
(hD : ∀ a b, D (a*b)=(D a).comp (σ b)+(lam a).comp (D b))
(Ω : K) (hΩ : DenseRange (fun a : A => σ a Ω)) :
∃ V : K →L[ℂ] F,‖V‖≤289*‖D‖ ∧ ∀ a : A,D a=V.comp (σ a)-(lam a).comp V := by
have h := rectangular_implementation_quantitative σ lam D hD (c:=16) (by norm_num)
(fun _ _ X => cyclic_matrix_bound_unconditional σ lam D hD Ω hΩ X)
norm_num at h
exact h
end Kadison.CbFactor
end
end
end
end OAI
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