| 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 |
| |
