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ddc84c5 | 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 | import OAI.Analysis.FactorGeneration.Generation
import OAI.Analysis.FactorGeneration.PredualContinuity
import OAI.Analysis.FactorGeneration.TraceTransport
namespace OAI
namespace Generator
/-! Single generation of a finite diffuse factor with a separable Banach predual,
using its faithful normal trace and given-predual continuity. -/
noncomputable section
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
theorem single_generation_of_II1_separable_predual
(S : StarSubalgebra ℂ (H →L[ℂ] H)) (hS : IsII1Factor S)
(hsep : HasSeparablePredual S) : SinglyGenerated S := by
let M := vonNeumannOfWOTClosed S hS.weaklyClosed
have hf : ∀ c : M.toStarSubalgebra,
(∀ a : M.toStarSubalgebra, a*c=c*a) → ∃ z : ℂ, c=z • (1 : M.toStarSubalgebra) := by
intro c hc
obtain ⟨z, hz⟩ := hS.factor c hc
exact ⟨z, hz.trans (Algebra.algebraMap_eq_smul_one z)⟩
have hunit : (1 : M.toStarSubalgebra) ≠ 0 := by
intro h
exact hS.nonzero (congrArg Subtype.val h).symm
let E := Tracial.TraceFreePreHilbert hf hS.diffuse hS.finite hunit
let : NormedAddCommGroup E :=
Tracial.traceFreePreHilbertNormedAddCommGroup hf hS.diffuse hS.finite hunit
let : InnerProductSpace ℂ E :=
Tracial.traceFreePreHilbertInnerProductSpace hf hS.diffuse hS.finite hunit
let N := Tracial.traceFreeHilbertClosure hf hS.diffuse hS.finite hunit
let τ := Tracial.traceFreeHilbertClosureTrace hf hS.diffuse hS.finite hunit
have hN := Tracial.traceFreeHilbertClosure_isII1Factor hf hS.diffuse hS.finite hunit
have hpre := Tracial.traceFreeHilbertClosure_hasSeparablePredual hf hS.diffuse hS.finite hunit hsep
have htrace := Tracial.traceOrbit_isSeparable_of_separablePredual τ hpre
exact Tracial.traceFreeHilbert_singlyGenerated_pullback hf hS.diffuse hS.finite hunit
(Tracial.singlyGenerated_of_tracialRepresentation hN τ htrace)
theorem mainTarget : @MainTarget H _ _ _ :=
single_generation_of_II1_separable_predual
end
end Generator
end OAI
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