| 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 | |