File size: 3,060 Bytes
67aee1d | 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 75 76 77 78 | import Mathlib
import OAI.Analysis.FreeWords.MatrixConditionalExpectation
import OAI.Analysis.FreeWords.RelativeConjugationDefinitions
namespace OAI
noncomputable section
namespace KKControlledConjugation
open KKRelativeMatrix KKMatrixExpectation KKTraceTransfer KKFreeLiftFrontier
noncomputable section
variable {A : Type*} [Ring A] [StarRing A] [Algebra ℂ A] [StarModule ℂ A]
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]
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
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
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
end KKControlledConjugation
end
end OAI
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