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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 57 58 59 60 61 62 63 | import Mathlib
import OAI.Analysis.FreeWords.FiniteMomentNet
import OAI.Analysis.FreeWords.HaarFiniteEndpointsDefinitions
namespace OAI
noncomputable section
namespace KKHaarEndpoints
noncomputable section
open Filter
open scoped Topology
variable {A : Type*} [Ring A] [StarRing A] [Algebra ℂ A]
abbrev CyclicTest (τ : A →ₗ[ℂ] ℂ) (B : Subalgebra ℂ A) :=
{k : ℤ // k≠0} ⊕ {w : List (ℤ×A) // w≠[] ∧ CenteredPowerWord τ B w}
abbrev CyclicIndex (τ : A →ₗ[ℂ] ℂ) (B : Subalgebra ℂ A) := Finset (CyclicTest τ B)×ℕ
theorem cyclic_approx_net (τ : A →ₗ[ℂ] ℂ) (B : Subalgebra ℂ A) (Good : unitary A → Prop)
(happ : FiniteCyclicApprox τ B Good) :
∃u:CyclicIndex τ B → unitary A, (∀j,Good (u j)) ∧
(∀k:ℤ,k≠0 → Tendsto (fun j => τ (u j^k).val) atTop (𝓝 0)) ∧
(∀w:List (ℤ×A),w≠[] → CenteredPowerWord τ B w →
Tendsto (fun j => τ (powerWord (u j) w)) atTop (𝓝 0)) := by
classical
let T := CyclicTest τ B
let f : T → unitary A → ℂ := fun t u => match t with
| .inl k => τ (u^k.val).val
| .inr w => τ (powerWord u w.val)
have hfinite : ∀L:Finset T, ∀η:ℝ,0<η → ∃u,Good u ∧ ∀t∈L,‖f t u‖<η := by
intro L η hη
let K₀ : Finset ℤ := L.image (fun t => match t with | .inl k => k.val | .inr _ => 0)
let F₀ : Finset (List (ℤ×A)) := L.biUnion (fun t => match t with | .inl _ => ∅ | .inr w => {w.val})
have hF₀ : ∀w∈F₀,w≠[] ∧ CenteredPowerWord τ B w := by
intro w hw
obtain ⟨t,ht,hw⟩ := Finset.mem_biUnion.mp hw
cases t with
| inl k => simp at hw
| inr v => have he := Finset.mem_singleton.mp hw; subst w; exact v.property
obtain ⟨u,hgood,hK₀,hF₀'⟩ := happ K₀ F₀ hF₀ η hη
refine ⟨u,hgood,?_⟩
intro t ht
cases t with
| inl k => exact hK₀ k.val (Finset.mem_image.mpr ⟨.inl k,ht,rfl⟩) k.property
| inr w => exact hF₀' w.val (Finset.mem_biUnion.mpr ⟨.inr w,ht,by simp⟩)
obtain ⟨u,hgood,hu⟩ := KKFiniteMomentNet.finite_moment_net Good f hfinite
have hH (k : ℤ) (hk : k≠0) : Tendsto (fun j => τ (u j^k).val) atTop (𝓝 0) := hu (.inl ⟨k,hk⟩)
have hC (w : List (ℤ×A)) (hne : w≠[]) (hc : CenteredPowerWord τ B w) :
Tendsto (fun j => τ (powerWord (u j) w)) atTop (𝓝 0) := hu (.inr ⟨w,hne,hc⟩)
exact ⟨u,hgood,hH,hC⟩
end
end KKHaarEndpoints
end
end OAI
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