| import OAI.Analysis.FactorGeneration.AmplificationEncoding |
| import OAI.Analysis.FactorGeneration.Correlation |
| import OAI.Analysis.FactorGeneration.MatrixGenerators |
| import OAI.Analysis.FactorGeneration.PerturbationBounds |
| import OAI.Analysis.FactorGeneration.RelativeGeneration |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Generator.Tracial |
|
|
| /-! Relative generation under represented-factor hypotheses: unitaries in a dense G-delta set satisfy W*(N,u)=M. -/ |
| open Filter |
| open scoped Topology |
| section |
| open UltrapowerBridge |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) |
| variable (hNM : N ≤ M) |
| variable (hfactor : ∀ c : N.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : N.toStarSubalgebra)) |
| variable (hdiff : ∀ r : N.toStarSubalgebra,IsStarProjection r → r ≠ 0 → |
| ∃ q : N.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ r ∧ r*q=q ∧ q*r=q) |
| variable (hirr : ∀ c : M.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,coordinateInclusion hNM a*c=c*coordinateInclusion hNM a) → |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| variable (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) |
| include hfactor hdiff hirr hsep |
|
|
| theorem relative_generation_at_continuity (u : TraceUnitary τ) |
| (hcont : ∀ x : M.toStarSubalgebra, |
| ContinuousAt (fun v => projectionNorm (relativeTraceSpace (N := N) τ v) (baseVector τ x)) u) : |
| relativeGenerated (N := N) τ u=M := by |
| classical |
| by_contra hneq |
| let Q := relativeGenerated (N := N) τ u |
| have hQM : Q ≤ M := relativeGenerated_le τ hNM u |
| obtain ⟨z,hz⟩ := exists_unitary_outside hQM hneq |
| let xi := subalgebraResidual τ hQM (z : M.toStarSubalgebra) |
| let s : ℝ := τ.norm₂ xi |
| have hs : 0 < s := subalgebraResidual_norm_pos τ hQM _ hz |
| let α : ℝ := min (1/4) (s/32)/2 |
| have hα : 0 < α := half_pos (lt_min (by norm_num) (by positivity)) |
| have hq : α < 1/4 := by |
| have h := min_le_left (1/4 : ℝ) (s/32) |
| dsimp [α] at hα ⊢ |
| linarith |
| have hsmall : α < s/32 := by |
| have h := min_le_right (1/4 : ℝ) (s/32) |
| dsimp [α] at hα ⊢ |
| linarith |
| have hhalf : α < 1/2 := by linarith |
| let ω : Ultrafilter ℕ := Ultrafilter.of atTop |
| have hω : (ω : Filter ℕ) ≤ atTop := Ultrafilter.of_le atTop |
| have he (n : ℕ) := exists_represented_perturbation τ ω hNM hfactor hdiff hirr hsep hω |
| u.toUnitary z xi (n+1) (Nat.succ_pos _) α hα hhalf |
| choose v t hv ht using he |
| let V : ℕ → TraceUnitary τ := fun n => ⟨v n⟩ |
| have hV : Tendsto V atTop (𝓝 u) := by |
| apply (TraceUnitary.tendsto_iff τ atTop V u).mpr |
| exact squeeze_zero (fun _ => norm_nonneg _) hv (perturbation_size_tendsto α) |
| let F : TraceUnitary τ → ℝ := fun v => |
| projectionNorm (relativeTraceSpace (N := N) τ v) (baseVector τ xi) |
| have hFzero : F u=0 := by |
| change ‖(vectorOrbitClosure Q.toStarSubalgebra τ.vector).starProjection |
| (baseVector τ (subalgebraResidual τ hQM (z : M.toStarSubalgebra)))‖=0 |
| rw [subalgebraResidual_projection,norm_zero] |
| let δ : ℕ → ℝ := fun n => 1/((n+1 : ℕ) : ℝ) |
| have hδ : Tendsto δ atTop (𝓝 0) := |
| tendsto_one_div_atTop_nhds_zero_nat.comp (tendsto_add_atTop_nat 1) |
| let c : ℕ → ℂ := fun n => τ.trace (star xi*relativeUnitaryWord τ hNM (V n) (t n) (n+1)) |
| have hc (n : ℕ) : ‖c n-((α*s^2/2 : ℝ) : ℂ)‖ ≤ |
| s*(4*α^2/(1-2*α))+δ n := by |
| have hn := ht n |
| have hcorr : τ.trace (star xi*(z : M.toStarSubalgebra))=(s : ℂ)^2 := |
| subalgebraResidual_correlation τ hQM (z : M.toStarSubalgebra) |
| rw [hcorr] at hn |
| have heq : ((α*s^2/2 : ℝ) : ℂ)=(α : ℂ)/2*(s : ℂ)^2 := by |
| push_cast |
| ring |
| rw [heq] |
| exact hn |
| have hproj (n : ℕ) : ‖c n‖ ≤ F (V n) := |
| relativeUnitaryWord_correlation_le τ hNM (V n) (t n) (n+1) xi |
| exact no_continuous_correlated_sequence F (hcont xi) hFzero V hV δ hδ c |
| hs hα hq hsmall hc hproj |
|
|
| theorem represented_relative_generation : |
| ∃ C : Set (TraceUnitary τ), IsGδ C ∧ Dense C ∧ |
| ∀ u ∈ C, wstar ((N : Set (H →L[ℂ] H)) ∪ {(u.algebra τ).val})=M.toStarSubalgebra := by |
| obtain ⟨C,hC,hd,hcont⟩ := common_relative_projection_continuity_algebra τ hNM hsep |
| refine ⟨C,hC,hd,fun u hu => ?_⟩ |
| exact congrArg VonNeumannAlgebra.toStarSubalgebra |
| (relative_generation_at_continuity τ hNM hfactor hdiff hirr hsep u (hcont u hu)) |
|
|
| end |
|
|
| /-! Four generators under a faithful normalized vector trace and trace-vector separability. -/ |
| open scoped ComplexStarModule |
| section |
| open UltrapowerBridge |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {M : VonNeumannAlgebra H} |
|
|
| theorem represented_four_selfAdjoint_generators (τ : TracialRepresentation M) |
| (hfactor : ∀ c : M.toStarSubalgebra, |
| (∀ a : M.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| (hdiff : ∀ p : M.toStarSubalgebra,IsStarProjection p → p ≠ 0 → |
| ∃ q : M.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p*q=q ∧ q*p=q) |
| (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) : |
| ∃ a ∈ M,∃ b ∈ M,∃ c ∈ M,∃ d ∈ M, |
| IsSelfAdjoint a ∧ IsSelfAdjoint b ∧ IsSelfAdjoint c ∧ IsSelfAdjoint d ∧ |
| wstar {a,b,c,d}=M.toStarSubalgebra := by |
| obtain ⟨P,hPM,hP,hscalar,a,ha,b,hb,hsa,hsb,hPgen⟩ := |
| exists_irreducible_two_generated_subfactor τ le_rfl hfactor hdiff |
| (fun c hc => hfactor c hc) hsep |
| have hPfactor (c : P.toStarSubalgebra) (hc : ∀ a : P.toStarSubalgebra,a*c=c*a) : |
| ∃ z : ℂ,c=z • (1 : P.toStarSubalgebra) := by |
| obtain ⟨z,hz⟩ := hP.factor c hc |
| exact ⟨z,by simpa only [Algebra.algebraMap_eq_smul_one] using hz⟩ |
| have hirr (c : M.toStarSubalgebra) |
| (hc : ∀ a : P.toStarSubalgebra,coordinateInclusion hPM a*c=c*coordinateInclusion hPM a) : |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra) := |
| ⟨τ.trace c,hscalar c (fun a => congrArg Subtype.val (hc a))⟩ |
| obtain ⟨C,_,hC,hgen⟩ := represented_relative_generation τ hPM hPfactor hP.diffuse hirr hsep |
| let : Nonempty (TraceUnitary τ) := ⟨⟨1⟩⟩ |
| obtain ⟨u,hu⟩ := hC.nonempty |
| let v : H →L[ℂ] H := (u.algebra τ).val |
| have hv : v ∈ M.toStarSubalgebra := (u.algebra τ).property |
| have hvgen : wstar ((P : Set (H →L[ℂ] H)) ∪ {v})=M.toStarSubalgebra := hgen u hu |
| refine ⟨a,hPM ha,b,hPM hb,ℜ v,realPart_mem _ hv,ℑ v,imaginaryPart_mem _ hv, |
| hsa,hsb,(ℜ v).property,(ℑ v).property,le_antisymm ?_ ?_⟩ |
| · exact wstar_le (vonNeumann_wotClosed M) |
| (by rintro _ (rfl | rfl | rfl | rfl) |
| · exact hPM ha |
| · exact hPM hb |
| · exact realPart_mem _ hv |
| · exact imaginaryPart_mem _ hv) |
| · let Q := wstar {a,b,(ℜ v : H →L[ℂ] H),(ℑ v : H →L[ℂ] H)} |
| have hPQ : P.toStarSubalgebra ≤ Q := by |
| rw [← hPgen] |
| exact wstar_mono (by intro x hx; simp only [Set.mem_insert_iff,Set.mem_singleton_iff] at hx ⊢; tauto) |
| rw [← hvgen] |
| apply wstar_le (wstar_closed _) |
| rintro x (hx | rfl) |
| · exact hPQ hx |
| · apply (mem_iff_parts_mem Q v).mpr |
| exact ⟨subset_wstar _ (by simp),subset_wstar _ (by simp)⟩ |
|
|
| end |
|
|
| /-! Single generation for the explicit faithful normalized vector-trace |
| representation with separable trace-vector range. -/ |
| open UltrapowerBridge |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {M : VonNeumannAlgebra H} |
|
|
| theorem represented_two_selfAdjoint_generators (τ : TracialRepresentation M) |
| (hfactor : ∀ c : M.toStarSubalgebra, |
| (∀ a : M.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| (hdiff : ∀ p : M.toStarSubalgebra,IsStarProjection p → p ≠ 0 → |
| ∃ q : M.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p*q=q ∧ q*p=q) |
| (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) : |
| ∃ a ∈ M,∃ b ∈ M,IsSelfAdjoint a ∧ IsSelfAdjoint b ∧ wstar {a,b}=M.toStarSubalgebra := by |
| obtain ⟨E,_,_,_,_,_⟩ := exists_commuting_matrix_extension τ |
| (FiniteMatrixStage.scalarStage M).units ⟨0,(FiniteMatrixStage.scalarStage M).positive⟩ hfactor hdiff 2 (by norm_num) |
| let N := E.relativeVonNeumann |
| let τN := restrictedTrace τ E.relativeVonNeumann_le |
| have hsepN : TopologicalSpace.IsSeparable (Set.range (baseVector τN)) := by |
| apply hsep.mono |
| rintro y ⟨a,rfl⟩ |
| exact ⟨coordinateInclusion E.relativeVonNeumann_le a,rfl⟩ |
| obtain ⟨a,ha,b,hb,c,hc,d,hd,hsa,hsb,hsc,hsd,hgen⟩ := represented_four_selfAdjoint_generators |
| τN (E.relative_scalar_center 0 hfactor) (E.relative_diffuse 0 hdiff) hsepN |
| let a' : E.commutant := E.relativeEquiv.symm ⟨a,ha⟩ |
| let b' : E.commutant := E.relativeEquiv.symm ⟨b,hb⟩ |
| let c' : E.commutant := E.relativeEquiv.symm ⟨c,hc⟩ |
| let d' : E.commutant := E.relativeEquiv.symm ⟨d,hd⟩ |
| have hs (x : H →L[ℂ] H) (hx : x ∈ N) (h : IsSelfAdjoint x) : |
| IsSelfAdjoint (E.relativeEquiv.symm ⟨x,hx⟩) := |
| (show IsSelfAdjoint (⟨x,hx⟩ : N.toStarSubalgebra) from Subtype.ext h.star_eq).map E.relativeEquiv.symm |
| exact E.two_generators_of_four_commutant_generators a' b' c' d' |
| (hs a ha hsa) (hs b hb hsb) (hs c hc hsc) (hs d hd hsd) hgen |
|
|
| theorem represented_single_generator (τ : TracialRepresentation M) |
| (hfactor : ∀ c : M.toStarSubalgebra, |
| (∀ a : M.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| (hdiff : ∀ p : M.toStarSubalgebra,IsStarProjection p → p ≠ 0 → |
| ∃ q : M.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ p*q=q ∧ q*p=q) |
| (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) : |
| SinglyGenerated M.toStarSubalgebra := by |
| obtain ⟨a,ha,b,hb,hsa,hsb,hgen⟩ := represented_two_selfAdjoint_generators τ hfactor hdiff hsep |
| exact ⟨a+Complex.I • b,M.add_mem ha (M.smul_mem hb _),(wstar_pack hsa hsb).trans hgen⟩ |
|
|
| theorem singlyGenerated_of_tracialRepresentation (hM : IsII1Factor M.toStarSubalgebra) |
| (τ : TracialRepresentation M) |
| (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) : |
| SinglyGenerated M.toStarSubalgebra := by |
| apply represented_single_generator τ _ hM.diffuse hsep |
| intro c hc |
| obtain ⟨z,hz⟩ := hM.factor c hc |
| exact ⟨z,by simpa only [Algebra.algebraMap_eq_smul_one] using hz⟩ |
|
|
| end Generator.Tracial |
|
|
| end |
|
|
| end OAI |
|
|
