File size: 2,756 Bytes
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 | import Mathlib
import OAI.Analysis.Commutator.PaulsenDilation
namespace OAI
noncomputable section
section
noncomputable section
open scoped BigOperators CStarAlgebra
namespace Kadison.CbFactor
variable {A H : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A]
[NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
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‖
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]
variable [Nontrivial A] (f : A →ₗ[ℂ] (H →L[ℂ] H)) (c : ℝ) (hc : 0<c) (hf : HasMatrixBound f c)
abbrev Space := Paulsen.DilationSpace ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf)
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)
def leftOperator : H →L[ℂ] Space f c hc hf :=
Paulsen.leftOperator ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf)
def rightOperator : H →L[ℂ] Space f c hc hf :=
(c : ℂ) • Paulsen.rightOperator ((c⁻¹ : ℂ) • f) (normalized_contracts f hc hf)
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]
lemma leftOperator_norm_le : ‖leftOperator f c hc hf‖≤1 :=
Paulsen.leftOperator_norm_le_one _ _
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))
end Kadison.CbFactor
end
end
end
end OAI
|