| import Mathlib |
| import OAI.Analysis.FreeWords.KronExpansion |
| import OAI.Analysis.FreeWords.MatrixCommutantTrace |
| import OAI.Analysis.FreeWords.MatrixProduct |
| import OAI.Analysis.FreeWords.MatrixReindex |
| import OAI.Analysis.FreeWords.MatrixSubfactor |
| import OAI.Analysis.FreeWords.MatrixTraceUnique |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKVariableUltra |
|
|
| open KKTraceTransfer KKFreeLiftFrontier KKProjectionSupport KKMatrixQuantization |
|
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| noncomputable section |
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|
| variable {ι κ A : Type*} [Fintype ι] [DecidableEq ι] [Fintype κ] [DecidableEq κ] |
| [Ring A] [Algebra ℂ A] [StarRing A] [StarModule ℂ A] |
|
|
| theorem matrixProduct_kronecker_one |
| (ρ : Matrix ι ι ℂ →⋆ₐ[ℂ] A) (σ : Matrix κ κ ℂ →⋆ₐ[ℂ] A) |
| (hc : ∀ i j k l, Commute (matrixUnit ρ i j) (matrixUnit σ k l)) (a : Matrix ι ι ℂ) : |
| matrixProduct ρ σ hc (Matrix.kronecker a 1)=ρ a := by |
| rw [kronecker_one_expansion] |
| simp only [map_sum,map_smul] |
| change (∑ i,∑ j,∑ k, a i j • matrixUnit (matrixProduct ρ σ hc) (i,k) (j,k))=ρ a |
| simp only [matrixProduct_unit,productUnits,← Finset.smul_sum,← Finset.mul_sum, |
| matrixUnit_sum_diag,mul_one] |
| exact (matrix_representation ρ a).symm |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| theorem matrix_extension_canonical (M : VonNeumannAlgebra H) (τ : FaithfulNormalTrace M) |
| (hF : IsFactor M) (hD : IsDiffuse M) (n k : ℕ) (hn : 0<n) (hk : 0<k) |
| (β : Matrix (Fin n) (Fin n) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M) : |
| ∃ ρ : Matrix (Fin (n*k)) (Fin (n*k)) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M, |
| Function.Injective ρ ∧ |
| (∀ a, ρ (matrixReindex finProdFinEquiv (Matrix.kronecker a (1 : Matrix (Fin k) (Fin k) ℂ)))=β a) ∧ |
| (∀ x, τ.map (ρ x)=((n*k:ℕ):ℂ)⁻¹*Matrix.trace x) := by |
| classical |
| have : NeZero n := ⟨Nat.ne_of_gt hn⟩ |
| have : NeZero k := ⟨Nat.ne_of_gt hk⟩ |
| obtain ⟨σ,hσ,_⟩ := exists_matrix_subfactor (matrixCommutant β) (commutantTrace β τ) |
| (matrixCommutant_factor β hF) (matrixCommutant_diffuse β hD) k hk |
| let σ' := (commutantInclusion β).comp σ |
| have hc (i j : Fin n) (a b : Fin k) : Commute (matrixUnit β i j) (matrixUnit σ' a b) := |
| (inclusion_commute β (matrixUnit σ a b) i j).symm |
| let P := matrixProduct β σ' hc |
| let e : Fin n × Fin k ≃ Fin (n*k) := finProdFinEquiv |
| let Q := P.comp (matrixReindex e.symm) |
| have : Nontrivial (ConcreteAlgebra M) := ⟨⟨0,1,fun h => by |
| have ht := congrArg τ.map h |
| rw [map_zero,τ.normalized] at ht |
| exact zero_ne_one ht⟩⟩ |
| have : NeZero (n*k) := ⟨Nat.ne_of_gt (Nat.mul_pos hn hk)⟩ |
| refine ⟨Q,Q.toAlgHom.toRingHom.injective,?_,?_⟩ |
| · intro a |
| change P (matrixReindex e.symm (matrixReindex e (Matrix.kronecker a 1)))=β a |
| have he (b : Matrix (Fin n × Fin k) (Fin n × Fin k) ℂ) : |
| matrixReindex e.symm (matrixReindex e b)=b := by |
| ext i j |
| change b (e.symm (e i)) (e.symm (e j))=b i j |
| simp only [Equiv.symm_apply_apply] |
| rw [he] |
| exact matrixProduct_kronecker_one β σ' hc a |
| · intro x |
| simpa only [Fintype.card_fin] using matrixHom_trace Q τ.map τ.normalized τ.tracial x |
|
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| end |
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| end KKVariableUltra |
| end |
|
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| end OAI |
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|
