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e3e0a0d | 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 | 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
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