| import Mathlib |
| import OAI.Analysis.Commutator.AmplifiedTowerStage |
| import OAI.Analysis.Commutator.MatrixTensorRange |
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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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| attribute [-instance] NonUnitalStarSubalgebra.nonUnitalCStarAlgebra |
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| open scoped BigOperators InnerProductSpace Matrix.Norms.L2Operator |
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| namespace Kadison.OperatorAmplification |
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| variable {H : Type} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
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| variable (ι : Type) [Fintype ι] (i₀ : ι) (M : VonNeumannAlgebra H) |
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| local instance amplifiedTowerGeometryIndexDecidableEq : DecidableEq ι := Classical.decEq ι |
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| variable (E : ℕ → MatrixTower.Stage M) |
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| lemma amplifiedStage_monotone (hE : Monotone (fun n => (E n).hom.range)) : |
| Monotone (fun n => (amplifiedStage ι i₀ M (E n)).hom.range) := by |
| intro n m h |
| apply MatrixTensor.range_le (scalarMatrixHom ι i₀ M) ((diagonalHom ι i₀ M).comp (E n).hom) |
| (fun x y => scalar_diagonal_commute ι i₀ M x ((E n).hom y)) |
| ((amplifiedStage ι i₀ M (E m)).hom.range) |
| · rintro a ⟨b,rfl⟩ |
| exact amplifiedStage_scalar ι i₀ M (E m) b |
| · rintro a ⟨b,rfl⟩ |
| obtain ⟨c,hc⟩ := hE h (show (E n).hom b∈(E n).hom.range from ⟨b,rfl⟩) |
| change diagonalHom ι i₀ M ((E n).hom b)∈_ |
| rw [←hc] |
| exact amplifiedStage_diagonal ι i₀ M (E m) c |
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| lemma amplifiedStage_irreducible |
| (hirr : ∀ a : M,(∀ n b,(E n).hom b*a=a*(E n).hom b) → ∃ c : ℂ,a=c • (1:M)) |
| (Z : matrixVonNeumann ι i₀ M) |
| (hZ : ∀ n b,(amplifiedStage ι i₀ M (E n)).hom b*Z=Z*(amplifiedStage ι i₀ M (E n)).hom b) : |
| ∃ c : ℂ,Z=algebraMap ℂ (matrixVonNeumann ι i₀ M) c := by |
| have hs (i j : ι) : matrixUnit (H:=H) ι i j*(Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)= |
| (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)*matrixUnit ι i j := by |
| obtain ⟨b,hb⟩ := amplifiedStage_scalar ι i₀ M (E 0) (Matrix.single i j 1) |
| change (amplifiedStage ι i₀ M (E 0)).hom b=scalarMatrixHom ι i₀ M (Matrix.single i j 1) at hb |
| have hh := hZ 0 b |
| rw [hb] at hh |
| have hh' := congrArg (matrixVonNeumann ι i₀ M).toStarSubalgebra.subtype hh |
| change (scalarMatrixHom ι i₀ M (Matrix.single i j 1):SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)* |
| (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)= |
| (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)* |
| (scalarMatrixHom ι i₀ M (Matrix.single i j 1):SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι) at hh' |
| rw [scalarMatrixHom_single] at hh' |
| exact hh' |
| let a : M := ⟨entry ι (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι) i₀ i₀, |
| (mem_matrixVonNeumann ι i₀ M Z).mp Z.property i₀ i₀⟩ |
| have ha (n : ℕ) (b : Matrix (E n).I (E n).I ℂ) : (E n).hom b*a=a*(E n).hom b := by |
| obtain ⟨c,hc⟩ := amplifiedStage_diagonal ι i₀ M (E n) b |
| apply Subtype.ext |
| apply entry_commutes ι (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι) |
| change (amplifiedStage ι i₀ M (E n)).hom c=diagonalHom ι i₀ M ((E n).hom b) at hc |
| have hh := hZ n c |
| rw [hc] at hh |
| exact congrArg (matrixVonNeumann ι i₀ M).toStarSubalgebra.subtype hh |
| obtain ⟨c,hc⟩ := hirr a ha |
| refine ⟨c,?_⟩ |
| apply Subtype.ext |
| have hdiag := diagonal_of_matrixUnits ι i₀ (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι) hs |
| change (Z:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)=c • 1 |
| rw [hdiag] |
| change diagonal (H:=H) ι (a:H →L[ℂ] H)=_ |
| rw [hc] |
| change diagonal (H:=H) ι (c • 1)=c • 1 |
| rw [map_smul,map_one] |
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| lemma amplifiedStage_in_relative (R : VonNeumannAlgebra H) |
| (hE : ∀ n a,((E n).hom a:H →L[ℂ] H)∈R) (n : ℕ) |
| (a : Matrix (amplifiedStage ι i₀ M (E n)).I (amplifiedStage ι i₀ M (E n)).I ℂ) : |
| ((amplifiedStage ι i₀ M (E n)).hom a:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)∈matrixVonNeumann ι i₀ R := by |
| let S : StarSubalgebra ℂ (matrixVonNeumann ι i₀ M) := |
| (matrixVonNeumann ι i₀ R).toStarSubalgebra.comap (matrixVonNeumann ι i₀ M).toStarSubalgebra.subtype |
| apply MatrixTensor.range_le (scalarMatrixHom ι i₀ M) ((diagonalHom ι i₀ M).comp (E n).hom) |
| (fun x y => scalar_diagonal_commute ι i₀ M x ((E n).hom y)) S _ _ ⟨a,rfl⟩ |
| · rintro b ⟨x,rfl⟩ |
| change (scalarMatrixHom ι i₀ M x:SumSpace (H:=H) ι →L[ℂ] SumSpace (H:=H) ι)∈matrixVonNeumann ι i₀ R |
| rw [scalarMatrixHom_coe] |
| apply (matrixVonNeumann ι i₀ R).sum_mem |
| intro i _ |
| apply (matrixVonNeumann ι i₀ R).sum_mem |
| intro j _ |
| exact (matrixVonNeumann ι i₀ R).smul_mem (matrixUnit_mem ι i₀ R i j) _ |
| · rintro b ⟨x,rfl⟩ |
| exact diagonal_mem_matrixVonNeumann ι i₀ R (hE n x) |
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| lemma amplifiedStage_size (n : ℕ) : |
| Fintype.card (amplifiedStage ι i₀ M (E n)).I=Fintype.card ι*Fintype.card (E n).I := |
| Fintype.card_prod ι (E n).I |
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| end Kadison.OperatorAmplification |
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| end |
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| end |
| end |
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| end OAI |
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