| import OAI.Analysis.IrrationalRotation.Model.Hyperfinite | |
| import OAI.Analysis.IrrationalRotation.Analysis.GeneratedVonNeumann | |
| import OAI.Analysis.IrrationalRotation.Analysis.VonNeumannStrongClosure | |
| namespace OAI | |
| noncomputable section | |
| open Filter | |
| open scoped Topology | |
| namespace Invariant233.HyperfiniteCriterion | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| def asStrong (T : H →L[ℂ] H) : H →Lₚₜ[ℂ] H := | |
| UniformConvergenceCLM.ofFun (RingHom.id ℂ) H {s : Set H | s.Finite} T | |
| omit [CompleteSpace H] in | |
| @[simp] theorem underlying_asStrong (T : H →L[ℂ] H) : | |
| strongOperatorUnderlying (asStrong T) = T := rfl | |
| omit [CompleteSpace H] in | |
| @[simp] theorem asStrong_apply (T : H →L[ℂ] H) (x : H) : | |
| asStrong T x = T x := rfl | |
| theorem mem_iSup_iff (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H)) | |
| (hA : Monotone A) (T : H →L[ℂ] H) : | |
| T ∈ iSup A ↔ ∃ n, T ∈ A n := by | |
| change T ∈ (↑(iSup A) : Set (H →L[ℂ] H)) ↔ _ | |
| rw [StarSubalgebra.coe_iSup_of_directed hA.directed_le] | |
| exact Set.mem_iUnion | |
| theorem strong_union_eq (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H)) | |
| (hA : Monotone A) : | |
| {B : H →Lₚₜ[ℂ] H | ∃ n, strongOperatorUnderlying B ∈ A n} = | |
| {B : H →Lₚₜ[ℂ] H | strongOperatorUnderlying B ∈ iSup A} := by | |
| ext B | |
| exact (mem_iSup_iff A hA (strongOperatorUnderlying B)).symm | |
| theorem mem_strong_union_closure_of_tendsto | |
| (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H)) | |
| (T : ℕ → H →L[ℂ] H) (S : H →L[ℂ] H) | |
| (hT : ∀ n, T n ∈ A n) | |
| (hlim : ∀ x, Tendsto (fun n => T n x) atTop (𝓝 (S x))) : | |
| asStrong S ∈ closure {B : H →Lₚₜ[ℂ] H | | |
| ∃ n, strongOperatorUnderlying B ∈ A n} := by | |
| have hstrong : Tendsto (fun n => asStrong (T n)) atTop (𝓝 (asStrong S)) := | |
| PointwiseConvergenceCLM.tendsto_iff_forall_tendsto.mpr hlim | |
| exact mem_closure_of_tendsto hstrong (Eventually.of_forall fun n => ⟨n, hT n⟩) | |
| theorem strong_union_closure_subset | |
| (M : VonNeumannAlgebra H) (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H)) | |
| (hAM : ∀ n, A n ≤ M.toStarSubalgebra) : | |
| closure {B : H →Lₚₜ[ℂ] H | ∃ n, strongOperatorUnderlying B ∈ A n} ⊆ | |
| {B : H →Lₚₜ[ℂ] H | strongOperatorUnderlying B ∈ M} := by | |
| apply closure_minimal | |
| · rintro B ⟨n, hn⟩ | |
| exact hAM n hn | |
| · exact vonNeumannAlgebra_stronglyClosed M | |
| theorem generated_iSup_eq_of_generator_approximation | |
| (s : Set (H →L[ℂ] H)) (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H)) | |
| (hAM : ∀ n, A n ≤ (vonNeumannGenerated s).toStarSubalgebra) | |
| (hs : ∀ T ∈ s, asStrong T ∈ closure {B : H →Lₚₜ[ℂ] H | | |
| ∃ n, strongOperatorUnderlying B ∈ A n}) : | |
| vonNeumannGenerated (↑(iSup A) : Set (H →L[ℂ] H)) = vonNeumannGenerated s := by | |
| apply le_antisymm | |
| · apply vonNeumannGenerated_le | |
| exact iSup_le hAM | |
| · apply vonNeumannGenerated_le | |
| intro T hT | |
| have hinclude : ∀ n, A n ≤ | |
| (vonNeumannGenerated (↑(iSup A) : Set (H →L[ℂ] H))).toStarSubalgebra := by | |
| intro n S hS | |
| exact subset_vonNeumannGenerated _ ((le_iSup A n) hS) | |
| exact strong_union_closure_subset _ A hinclude (hs T hT) | |
| end Invariant233.HyperfiniteCriterion | |
| end | |
| end OAI | |