| import Mathlib |
| import OAI.Analysis.FreeWords.FiniteFreeDefinitions |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace KKFreeLiftFrontier |
|
|
| open KKTraceTransfer Filter Topology |
|
|
| noncomputable section |
|
|
| def DenseTestMatrixFreeSequence : Prop := |
| ∀ (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| [TopologicalSpace.SeparableSpace H] [Nontrivial H] (M : VonNeumannAlgebra H) |
| (τ : FaithfulNormalTrace M), IsFactor M → IsDiffuse M → |
| ∀ n : ℕ, 0 < n → ∃ a : ℕ → MatrixAlgebra n M, |
| (∀ j, matrixTrace τ n (a j) = 0) ∧ (∀ j i k, ‖a j i k‖ ≤ 1) ∧ |
| (∀ x : MatrixAlgebra n M, matrixTrace τ n x = 0 → (∀ i k, ‖x i k‖ ≤ 1) → |
| ∀ ε : ℝ, 0 < ε → ∃ j, ∀ i k, traceNorm τ (a j i k-x i k) < ε) ∧ |
| ∃ d : ℕ → ℕ, (∀ j, 0 < d j) ∧ |
| ∃ ρ : ∀ j, Matrix (Fin (d j)) (Fin (d j)) ℂ →⋆ₐ[ℂ] ConcreteAlgebra M, |
| (∀ j, Function.Injective (ρ j)) ∧ ∃ U : ℕ → unitary (MatrixAlgebra n M), |
| (∀ j i k, (U j).val i k ∈ (ρ j).range) ∧ |
| (∀ k : ℤ, k ≠ 0 → Tendsto (fun j => matrixTrace τ n (U j ^ k).val) atTop (𝓝 0)) ∧ |
| (∀ w : List (ℤ × ℕ), w ≠ [] → (∀ z ∈ w, z.1 ≠ 0) → |
| Tendsto (fun j => mixedMoment τ (U j) (w.map fun z => (z.1,a z.2))) atTop (𝓝 0)) |
|
|
| end |
|
|
| end KKFreeLiftFrontier |
| end |
|
|
| end OAI |
|
|
