openAI-math-dataset / lean /OAI /RepresentationTheory /VertexAlgebra /GeneratedNormalUniqueness.lean
| import OAI.RepresentationTheory.VertexAlgebra.NormalUniqueness | |
| import OAI.RepresentationTheory.VertexAlgebra.SelfAdjointHomExt | |
| namespace OAI | |
| noncomputable section | |
| namespace StronglyRationalVOA.LocalOperatorAlgebra | |
| open BicommutantDensity | |
| universe u v | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| {K : Type v} [NormedAddCommGroup K] [InnerProductSpace ℂ K] [CompleteSpace K] | |
| theorem normal_ext_generated (s : Set (H →L[ℂ] H)) | |
| (π ρ : (generated s).toStarSubalgebra →⋆ₐ[ℂ] (K →L[ℂ] K)) | |
| (hπ : IsNormal (generated s) π) (hρ : IsNormal (generated s) ρ) | |
| (he : ∀ (x : H →L[ℂ] H) (hx : x ∈ s), | |
| π ⟨x, subset_generated s hx⟩ = ρ ⟨x, subset_generated s hx⟩) : π = ρ := by | |
| let N := generated s | |
| let B := (StarAlgebra.adjoin ℂ s).topologicalClosure | |
| let : IsClosed (N.toStarSubalgebra : Set (H →L[ℂ] H)) := vonNeumann_isClosed N | |
| let : IsClosed (B : Set (H →L[ℂ] H)) := StarSubalgebra.isClosed_topologicalClosure _ | |
| have hB : B ≤ N.toStarSubalgebra := | |
| StarSubalgebra.topologicalClosure_adjoin_le_centralizer_centralizer ℂ s | |
| have hgen : N.toStarSubalgebra ≤ Bicommutant B := by | |
| apply generated_mono | |
| intro x hx | |
| exact StarSubalgebra.le_topologicalClosure _ (StarAlgebra.subset_adjoin ℂ s hx) | |
| have hcπ : Continuous π := | |
| AddMonoidHomClass.continuous_of_bound π 1 (fun a => by | |
| simpa only [one_mul, coe_nnnorm] using (NNReal.coe_le_coe.mpr (NonUnitalStarAlgHom.nnnorm_apply_le π a))) | |
| have hcρ : Continuous ρ := | |
| AddMonoidHomClass.continuous_of_bound ρ 1 (fun a => by | |
| simpa only [one_mul, coe_nnnorm] using (NNReal.coe_le_coe.mpr (NonUnitalStarAlgHom.nnnorm_apply_le ρ a))) | |
| let E : StarSubalgebra ℂ (H →L[ℂ] H) := (StarAlgHom.equalizer π ρ).map N.toStarSubalgebra.subtype | |
| have hE : IsClosed (E : Set (H →L[ℂ] H)) := by | |
| change IsClosed (Subtype.val '' {a : N.toStarSubalgebra | π a = ρ a}) | |
| exact ((vonNeumann_isClosed N).isClosedMap_subtype_val) _ (isClosed_eq hcπ hcρ) | |
| have hBE : B ≤ E := by | |
| apply StarSubalgebra.topologicalClosure_minimal _ hE | |
| apply StarAlgebra.adjoin_le | |
| intro x hx | |
| refine ⟨⟨x,subset_generated s hx⟩, ?_, rfl⟩ | |
| exact he x hx | |
| have heB (b : B) : π ⟨b.val,hB b.property⟩ = ρ ⟨b.val,hB b.property⟩ := by | |
| obtain ⟨d,hd,heq⟩ := hBE b.property | |
| have hd' : π d = ρ d := hd | |
| have hde : d = ⟨b.val,hB b.property⟩ := Subtype.ext heq | |
| simpa only [hde] using hd' | |
| apply starAlgHom_ext_selfAdjoint | |
| intro D hD | |
| exact normal_eq_selfAdjoint N B hB hgen π ρ hπ hρ heB D | |
| (hD.map N.toStarSubalgebra.subtype) | |
| end StronglyRationalVOA.LocalOperatorAlgebra | |
| end | |
| end OAI | |