| import Mathlib |
| import OAI.Analysis.FreeWords.MatrixConditionalExpectation |
| import OAI.Analysis.FreeWords.MatrixTraceUnique |
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| namespace OAI |
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| noncomputable section |
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| namespace KKBlockMatrix |
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| open KKMatrixExpectation |
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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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| variable {ι κ : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] |
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| def blockHom (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) : |
| Matrix (ι×κ) (ι×κ) ℂ →⋆ₐ[ℂ] Matrix ι ι A := |
| { (ρ.toAlgHom.mapMatrix).comp (Matrix.compAlgEquiv ι κ ℂ ℂ).symm.toAlgHom with |
| map_star' := by |
| intro X |
| ext i j |
| exact map_star ρ (fun a b => X (j,a) (i,b)) } |
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| omit [StarModule ℂ A] in |
| @[simp] theorem blockHom_apply (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) |
| (X : Matrix (ι×κ) (ι×κ) ℂ) (i j : ι) : |
| blockHom ρ X i j=ρ (fun a b => X (i,a) (j,b)) := rfl |
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| omit [StarModule ℂ A] in |
| theorem blockHom_injective (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) (hρ : Function.Injective ρ) : |
| Function.Injective (blockHom (ι := ι) ρ) := by |
| intro X Y he |
| ext ⟨i,a⟩ ⟨j,b⟩ |
| exact congrFun (congrFun (hρ (congrFun (congrFun he i) j)) a) b |
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| theorem blockHom_entry_mem (ρ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) |
| (X : Matrix (ι×κ) (ι×κ) ℂ) (i j : ι) : blockHom ρ X i j∈ρ.range := |
| ⟨fun a b => X (i,a) (j,b),rfl⟩ |
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| theorem blockHom_kronecker {n d : ℕ} (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] A) |
| (b : Matrix (Fin n) (Fin n) ℂ) : |
| blockHom ρ (Matrix.kronecker b 1)=numericalHom (A := A) n b := by |
| ext i j |
| change ρ (fun a c => b i j*(1 : Matrix (Fin d) (Fin d) ℂ) a c)=algebraMap ℂ A (b i j) |
| have he : (fun a c => b i j*(1 : Matrix (Fin d) (Fin d) ℂ) a c)= |
| algebraMap ℂ (Matrix (Fin d) (Fin d) ℂ) (b i j) := by |
| ext a c |
| simp [Matrix.algebraMap_eq_diagonal,Matrix.diagonal_apply,Matrix.one_apply] |
| rw [he] |
| exact ρ.commutes (b i j) |
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| omit [StarModule ℂ A] in |
| theorem blockHom_trace {n d : ℕ} (ρ : Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] A) |
| [NeZero d] (τ : A →ₗ[ℂ] ℂ) (hτ1 : τ 1=1) (htr : ∀x y,τ (x*y)=τ (y*x)) |
| (X : Matrix (Fin n×Fin d) (Fin n×Fin d) ℂ) : |
| matrixTraceMap τ n (blockHom ρ X)=((n*d:ℕ):ℂ)⁻¹*Matrix.trace X := by |
| change (n:ℂ)⁻¹*(∑i:Fin n,τ (ρ (fun a b => X (i,a) (i,b))))=_ |
| have he : (∑i:Fin n,τ (ρ (fun a b => X (i,a) (i,b))))= |
| ∑i:Fin n,(d:ℂ)⁻¹*(∑a:Fin d,X (i,a) (i,a)) := by |
| apply Finset.sum_congr rfl |
| intro i hi |
| simpa only [Fintype.card_fin,Matrix.trace,Matrix.diag] using KKVariableUltra.matrixHom_trace ρ τ hτ1 htr (fun a b => X (i,a) (i,b)) |
| rw [he,← Finset.mul_sum] |
| simp only [Matrix.trace,Matrix.diag,Fintype.sum_prod_type,Nat.cast_mul,mul_inv_rev] |
| ring |
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| end |
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| end KKBlockMatrix |
| end |
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| end OAI |
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