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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 | 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
noncomputable section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
local instance amplifiedModelOrder (M : VonNeumannAlgebra H) : PartialOrder (ConcreteAlgebra M) := concreteOrder M
local instance amplifiedModelPreorder (M : VonNeumannAlgebra H) : Preorder (ConcreteAlgebra M) := (concreteOrder M).toPreorder
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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