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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 | import Mathlib
import OAI.Analysis.FreeWords.AmplifiedModelDefinitions
import OAI.Analysis.FreeWords.HaarEndpointsDefinitions
namespace OAI
noncomputable section
namespace KKFreeLiftFrontier
open KKTraceTransfer KKMatrixExpectation KKHaarEndpoints
noncomputable section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
def FiniteAmplifiedBlockModel : Prop :=
∀(M:VonNeumannAlgebra H) (τ:FaithfulNormalTrace M),IsFactor M → IsDiffuse M →
∀n:ℕ,0<n → ∀(K:Finset ℤ) (F:Finset (List (ℤ×MatrixAlgebra n M)))
(E:Finset (List (ℤ×MatrixAlgebra n M)×ℤ)),
(∀w∈F,w≠[] ∧ CenteredPowerWord (matrixTraceMap τ.map n) (numericalHom n).range.toSubalgebra w) →
(∀z∈E,CenteredPowerWord (matrixTraceMap τ.map n) (numericalHom n).range.toSubalgebra z.1 ∧ z.2≠0) →
∀ε:ℝ,0<ε → ∃d:ℕ,0<d ∧
∃ρ:Matrix (Fin d) (Fin d) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M,
Function.Injective ρ ∧ ∃U:unitary (MatrixAlgebra n M),
(∀i j,U.val i j∈ρ.range) ∧
(∀k∈K,k≠0 → ‖matrixTrace τ n (U^k).val‖<ε) ∧
(∀w∈F,‖matrixTrace τ n (powerWord U w)‖<ε) ∧
(∀z∈E,∀a b,‖τ.map ((powerWord U z.1*(U^z.2).val) a b)‖<ε)
end
end KKFreeLiftFrontier
end
end OAI
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