| 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 |
|
|
