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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 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 | import Mathlib
import OAI.Analysis.FreeWords.FiniteMatrixFreeApproximation
namespace OAI
noncomputable section
namespace KKFreeLiftFrontier
open KKTraceTransfer Filter Topology
noncomputable section
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⟩
end
end KKFreeLiftFrontier
end
end OAI
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