| import Mathlib |
| import OAI.Analysis.FreeWords.MatrixConditionalExpectation |
| import OAI.Analysis.FreeWords.RelativeConjugationDefinitions |
|
|
| namespace OAI |
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| noncomputable section |
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| namespace KKControlledConjugation |
|
|
| open KKRelativeMatrix KKMatrixExpectation KKTraceTransfer KKFreeLiftFrontier |
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| noncomputable section |
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| variable {A : Type*} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A] |
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| def amplifyConj (n : ℕ) (u : unitary A) : |
| Matrix (Fin n) (Fin n) A →⋆ₐ[ℂ] Matrix (Fin n) (Fin n) A := |
| { (Unitary.conjStarAlgAut ℂ A u).toStarAlgHom.toAlgHom.mapMatrix with |
| map_star' := by |
| intro X |
| ext i j |
| exact map_star (Unitary.conjStarAlgAut ℂ A u) (X j i) } |
|
|
| omit [StarModule ℂ A] in |
| @[simp] theorem amplifyConj_apply (n : ℕ) (u : unitary A) |
| (X : Matrix (Fin n) (Fin n) A) (i j : Fin n) : |
| amplifyConj n u X i j=u.val*X i j*star u.val := rfl |
|
|
| omit [StarModule ℂ A] in |
| theorem conj_trace (τ : A →ₗ[ℂ] ℂ) (htr : ∀a b,τ (a*b)=τ (b*a)) |
| (u : unitary A) (a : A) : τ (u.val*a*star u.val)=τ a := by |
| rw [htr,← mul_assoc] |
| rw [show star u.val*u.val=1 from u.property.1,one_mul] |
|
|
| omit [StarModule ℂ A] in |
| theorem amplifyConj_trace (τ : A →ₗ[ℂ] ℂ) (htr : ∀a b,τ (a*b)=τ (b*a)) |
| (n : ℕ) (u : unitary A) (X : Matrix (Fin n) (Fin n) A) : |
| matrixTraceMap τ n (amplifyConj n u X)=matrixTraceMap τ n X := by |
| simp only [matrixTraceMap_apply,amplifyConj_apply,conj_trace τ htr] |
|
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| theorem amplifyConj_expectation (τ : A →ₗ[ℂ] ℂ) (htr : ∀a b,τ (a*b)=τ (b*a)) |
| (n : ℕ) (u : unitary A) (X : Matrix (Fin n) (Fin n) A) : |
| matrixExpectation τ n (amplifyConj n u X)=matrixExpectation τ n X := by |
| ext i j |
| simp only [matrixExpectation_apply,amplifyConj_apply,conj_trace τ htr] |
|
|
| theorem amplifyConj_numerical (n : ℕ) (u : unitary A) (b : Matrix (Fin n) (Fin n) ℂ) : |
| amplifyConj n u (numericalHom (A := A) n b)=numericalHom (A := A) n b := by |
| ext i j |
| change (Unitary.conjStarAlgAut ℂ A u) (algebraMap ℂ A (b i j))=algebraMap ℂ A (b i j) |
| exact (Unitary.conjStarAlgAut ℂ A u).toStarAlgHom.commutes (b i j) |
|
|
| theorem amplifyConj_entry_range {n d : ℕ} (u : unitary A) |
| (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] A) |
| (X : Matrix (Fin n) (Fin n) A) (hX : ∀i j,X i j∈ρ.range) (i j : Fin n) : |
| amplifyConj n u X i j∈((Unitary.conjStarAlgAut ℂ A u).toStarAlgHom.comp ρ).range := by |
| obtain ⟨a,ha⟩ := hX i j |
| refine ⟨a,?_⟩ |
| change (Unitary.conjStarAlgAut ℂ A u) (ρ a)= (Unitary.conjStarAlgAut ℂ A u) (X i j) |
| exact congrArg (Unitary.conjStarAlgAut ℂ A u) ha |
|
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| theorem amplifyConj_eq_scalarConjugate {M : VonNeumannAlgebra H} {n : ℕ} |
| (u : unitary (ConcreteAlgebra M)) (X : Matrix (Fin n) (Fin n) (ConcreteAlgebra M)) : |
| amplifyConj n u X=scalarConjugate u X := by |
| ext i j |
| simp only [amplifyConj_apply,scalarConjugate,Matrix.diagonal_mul,Matrix.mul_diagonal] |
| rfl |
|
|
| end |
|
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| end KKControlledConjugation |
| end |
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| end OAI |
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|
