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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 | 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
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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