| import Mathlib |
| import OAI.Analysis.FreeWords.FiniteMomentNet |
| import OAI.Analysis.FreeWords.HaarFiniteEndpointsDefinitions |
|
|
| namespace OAI |
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| noncomputable section |
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|
| namespace KKHaarEndpoints |
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| noncomputable section |
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|
| open Filter |
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| open scoped Topology |
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| variable {A : Type*} [Ring A] [StarRing A] [Algebra ℂ A] |
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| abbrev CyclicTest (τ : A →ₗ[ℂ] ℂ) (B : Subalgebra ℂ A) := |
| {k : ℤ // k≠0} ⊕ {w : List (ℤ×A) // w≠[] ∧ CenteredPowerWord τ B w} |
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| abbrev CyclicIndex (τ : A →ₗ[ℂ] ℂ) (B : Subalgebra ℂ A) := Finset (CyclicTest τ B)×ℕ |
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| 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⟩ |
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| end |
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| end KKHaarEndpoints |
| end |
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| end OAI |
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