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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 | 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
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