| import Mathlib |
| import OAI.Analysis.Commutator.PaulsenDilation |
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| namespace OAI |
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| noncomputable section |
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| section |
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| noncomputable section |
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| open scoped BigOperators CStarAlgebra |
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| namespace Kadison.CbFactor |
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| variable {A H : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
| [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| def HasMatrixBound (f : A →ₗ[ℂ] (H →L[ℂ] H)) (c : ℝ) : Prop := |
| ∀ (ι : Type) [Fintype ι] (X : CStarMatrix ι ι A),‖CStarMatrix.mapₗ f X‖≤c*‖X‖ |
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| lemma normalized_contracts (f : A →ₗ[ℂ] (H →L[ℂ] H)) {c : ℝ} (hc : 0<c) |
| (hf : HasMatrixBound f c) : Paulsen.IsCompletelyContractive ((c⁻¹ : ℂ) • f) := by |
| intro ι _ X |
| have he : CStarMatrix.mapₗ ((c⁻¹ : ℂ) • f) X=(c⁻¹ : ℂ) • CStarMatrix.mapₗ f X := by |
| ext i j |
| rfl |
| rw [he,norm_smul,norm_inv,Complex.norm_real,Real.norm_eq_abs,abs_of_pos hc] |
| calc |
| c⁻¹*‖CStarMatrix.mapₗ f X‖≤c⁻¹*(c*‖X‖) := |
| mul_le_mul_of_nonneg_left (hf ι X) (inv_nonneg.mpr hc.le) |
| _ = ‖X‖ := by rw [←mul_assoc,inv_mul_cancel₀ hc.ne',one_mul] |
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| variable [Nontrivial A] (f : A →ₗ[ℂ] (H →L[ℂ] H)) (c : ℝ) (hc : 0<c) (hf : HasMatrixBound f c) |
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| abbrev Space := Paulsen.DilationSpace ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf) |
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| def representation : A →⋆ₐ[ℂ] (Space f c hc hf →L[ℂ] Space f c hc hf) := |
| Paulsen.dilationRepresentation ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf) |
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| def leftOperator : H →L[ℂ] Space f c hc hf := |
| Paulsen.leftOperator ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf) |
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| def rightOperator : H →L[ℂ] Space f c hc hf := |
| (c : ℂ) • Paulsen.rightOperator ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf) |
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| lemma factorization (a : A) : |
| f a=(leftOperator f c hc hf).adjoint.comp |
| ((representation f c hc hf a).comp (rightOperator f c hc hf)) := by |
| dsimp only [leftOperator,representation,rightOperator] |
| rw [ContinuousLinearMap.comp_smul,ContinuousLinearMap.comp_smul, |
| ←Paulsen.dilation_factorization] |
| simp only [LinearMap.smul_apply,smul_smul] |
| rw [mul_inv_cancel₀ (Complex.ofReal_ne_zero.mpr hc.ne'),one_smul] |
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| lemma leftOperator_norm_le : ‖leftOperator f c hc hf‖≤1 := |
| Paulsen.leftOperator_norm_le_one _ _ |
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| lemma rightOperator_norm_le : ‖rightOperator f c hc hf‖≤c := by |
| unfold rightOperator |
| have h := ContinuousLinearMap.opNorm_smul_le (c : ℂ) |
| (Paulsen.rightOperator ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf)) |
| rw [Complex.norm_real,Real.norm_eq_abs,abs_of_pos hc] at h |
| exact h.trans ((mul_le_mul_of_nonneg_left (Paulsen.rightOperator_norm_le_one _ _) hc.le).trans_eq (mul_one c)) |
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| end Kadison.CbFactor |
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| end |
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| end |
| end |
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| end OAI |
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