| import Mathlib |
| import OAI.Analysis.FreeWords.AmplifiedModelDefinitions |
| import OAI.Analysis.FreeWords.BlockMatrixEmbedding |
| import OAI.Analysis.FreeWords.NumericalModel |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKFreeLiftFrontier |
|
|
| open KKTraceTransfer KKVariableUltra KKMatrixExpectation KKBlockMatrix KKFactorSupport KKProjectionSupport |
|
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| noncomputable section |
|
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| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
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| local instance amplifiedModelOrder (M : VonNeumannAlgebra H) : PartialOrder (ConcreteAlgebra M) := concreteOrder M |
|
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| local instance amplifiedModelPreorder (M : VonNeumannAlgebra H) : Preorder (ConcreteAlgebra M) := (concreteOrder M).toPreorder |
|
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| local instance amplifiedModelLE (M : VonNeumannAlgebra H) : LE (ConcreteAlgebra M) := (concreteOrder M).toLE |
|
|
| local instance amplifiedModelSOR (M : VonNeumannAlgebra H) : StarOrderedRing (ConcreteAlgebra M) := concreteSOR M |
|
|
| theorem finite_amplified_numerical_model : FiniteAmplifiedNumericalModel (H := H) := by |
| classical |
| intro M τ hF hD n hn K F hne hc ε hε |
| obtain ⟨d,hd,V,hVK,hVF⟩ := finite_numerical_model M τ hF hD n hn K F hne hc ε hε |
| have : NeZero d := ⟨hd.ne'⟩ |
| obtain ⟨ρ,hρ,hτρ⟩ := exists_matrix_subfactor M τ hF hD d hd |
| let φ := blockHom (ι := Fin n) ρ |
| let U : unitary (MatrixAlgebra n M) := ⟨φ V.val,Unitary.map_mem φ V.property⟩ |
| have hp (k : ℤ) : φ (V^k).val=(U^k).val := unitary_preimage_zpow φ V U rfl k |
| have hτ (X) : matrixTrace τ n (φ X)=((n*d:ℕ):ℂ)⁻¹*Matrix.trace X := |
| blockHom_trace ρ τ.map τ.normalized τ.tracial X |
| refine ⟨d,hd,ρ,hρ,U,?_,?_,?_⟩ |
| · intro i j |
| exact blockHom_entry_mem ρ V.val i j |
| · intro k hk hk0 |
| rw [← hp k,hτ] |
| exact hVK k hk hk0 |
| · intro w hw |
| have he : φ ((w.map (fun z => (V^z.1).val*Matrix.kronecker z.2 1)).prod)= |
| ((w.map (fun z => (U^z.1).val*numericalHom (A := ConcreteAlgebra M) n z.2)).prod) := by |
| rw [map_list_prod,List.map_map] |
| apply congrArg List.prod |
| apply List.map_congr_left |
| intro z hz |
| exact (map_mul φ _ _).trans (congrArg₂ (·*·) (hp z.1) (blockHom_kronecker ρ z.2)) |
| rw [← he,hτ] |
| exact hVF w hw |
|
|
| end |
|
|
| end KKFreeLiftFrontier |
| end |
|
|
| end OAI |
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|
