| import Mathlib |
| import OAI.Analysis.Commutator.CbDilation |
| import OAI.Analysis.Commutator.FactorMetric |
| import OAI.Analysis.Commutator.GramBounds |
| import OAI.Analysis.Commutator.TriangularCb |
| import OAI.Analysis.Commutator.TriangularMetric |
|
|
| 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 |
|
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| local instance : NormedSpace ℂ (K →L[ℂ] F) := ContinuousLinearMap.toNormedSpace |
|
|
| theorem rectangular_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)) |
| {c : ℝ} (hc : 0≤c) |
| (hcb : ∀ (ι : Type) [Fintype ι] (X : CStarMatrix ι ι A), |
| ‖matrixAction (fun i j => D (X i j))‖≤c*‖X‖) : |
| ∃ V : K →L[ℂ] F,‖V‖≤(1+c)^2 ∧ ∀ a : A,D a=V.comp (σ a)-(lam a).comp V := by |
| cases subsingleton_or_nontrivial A with |
| | inl hA => |
| let := hA |
| refine ⟨0,by simp; positivity,?_⟩ |
| intro a |
| simp [Subsingleton.elim a 0] |
| | inr hA => |
| let := hA |
| let π := triangularHom σ lam D hD |
| have hp : 0<1+c := by linarith |
| have hπ : HasMatrixBound π.toLinearMap (1+c) := by |
| intro ι _ X |
| exact triangular_matrix_norm_le σ lam D hD hc (hcb ι) X |
| let ρ := representation π.toLinearMap (1+c) hp hπ |
| let V := leftOperator π.toLinearMap (1+c) hp hπ |
| let W := rightOperator π.toLinearMap (1+c) hp hπ |
| have hfactor : ∀ a : A,π a=V.adjoint.comp ((ρ a).comp W) := by |
| intro a |
| exact factorization π.toLinearMap (1+c) hp hπ a |
| let L := leftCompression ρ V W |
| have hLW : L.comp W=1 := leftCompression_right π ρ V W hfactor |
| have hint : ∀ a : A,(π a).comp L=L.comp (ρ a) := |
| leftCompression_intertwines π ρ V W hfactor |
| exact implementer_of_factor_metric σ lam D L W hLW |
| (gram_intertwines π ρ L hint) hp.le |
| (leftCompression_norm_le ρ V W (leftOperator_norm_le _ _ _ _)) |
| (rightOperator_norm_le _ _ _ _) |
|
|
| end Kadison.CbFactor |
|
|
| end |
|
|
| end |
| end |
|
|
| end OAI |
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|
