| import Mathlib |
| import OAI.Analysis.FreeWords.FiniteMatrixFreeApproximation |
|
|
| namespace OAI |
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| noncomputable section |
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| namespace KKFreeLiftFrontier |
|
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| open KKTraceTransfer Filter Topology |
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| noncomputable section |
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| theorem countable_matrix_free_approximation |
| (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| [TopologicalSpace.SeparableSpace H] [Nontrivial H] (M : VonNeumannAlgebra H) |
| (τ : FaithfulNormalTrace M) (hfac : IsFactor M) (hdiff : IsDiffuse M) |
| (n : ℕ) (hn : 0 < n) (T : ℕ → List (ℤ × MatrixAlgebra n M)) |
| (hT : ∀ i, IsCenteredTest τ (T i)) : |
| ∃ 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)) ∧ |
| (∀ i, Tendsto (fun j => mixedMoment τ (U j) (T i)) atTop (𝓝 0)) := by |
| classical |
| have hε (j : ℕ) : 0 < (1 : ℝ) / (j + 1) := by positivity |
| have ex (j : ℕ) := finite_matrix_free_approximation H M τ hfac hdiff n hn |
| ((Finset.range (j+1)).image T) |
| (by intro w hw; obtain ⟨i, hi, rfl⟩ := Finset.mem_image.mp hw; exact hT i) |
| (Finset.Icc (-(j:ℤ)) (j:ℤ)) ((1:ℝ)/(j+1)) (hε j) |
| choose d hd ρ hρ U hU hH hM using ex |
| refine ⟨d,hd,ρ,hρ,U,hU,?_,?_⟩ |
| · intro k hk |
| apply squeeze_zero_norm' _ tendsto_one_div_add_atTop_nhds_zero_nat |
| filter_upwards [eventually_ge_atTop k.natAbs] with j hj |
| apply (hH j k _ hk).le |
| apply Finset.mem_Icc.mpr |
| have hc : (k.natAbs:ℤ) ≤ (j:ℤ) := by exact_mod_cast hj |
| have hpos : k ≤ (k.natAbs:ℤ) := Int.le_natAbs |
| have hneg : -k ≤ (k.natAbs:ℤ) := by simpa using (Int.le_natAbs (a := -k)) |
| constructor <;> omega |
| · intro i |
| apply squeeze_zero_norm' _ tendsto_one_div_add_atTop_nhds_zero_nat |
| filter_upwards [eventually_ge_atTop i] with j hj |
| apply (hM j (T i) _).le |
| apply Finset.mem_image.mpr |
| exact ⟨i,Finset.mem_range.mpr (by omega),rfl⟩ |
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| end |
|
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| end KKFreeLiftFrontier |
| end |
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| end OAI |
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|
