| import OAI.Analysis.InterpolatedFactors.Haar |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| universe u_1 u_2 u_3 u_4 u_5 |
|
|
| namespace FreeGroupFactorMain.CyclicSpectral |
| open MeasureTheory ProbabilityTheory Set |
| variable {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] |
| variable (A : E →L[ℂ] E) (hA : IsSelfAdjoint A) (ξ : E) |
| variable [IsProbabilityMeasure (measure A hA ξ)] [NullSingletonClass (measure A hA ξ)] |
|
|
| abbrev UniformAlgebra := C(Icc (0:ℝ) 1,ℂ) |
|
|
| def cdfMap : C(Space A,Icc (0:ℝ) 1) := |
| ⟨fun x => ⟨cdf (realMeasure A hA ξ) x,cdf_nonneg _ _,cdf_le_one _ _⟩, |
| ((ProbabilityTransform.continuous_cdf (realMeasure A hA ξ)).comp |
| continuous_subtype_val).subtype_mk _⟩ |
|
|
| def uniformHom : UniformAlgebra →⋆ₐ[ℂ] (E →L[ℂ] E) := |
| (complexHom A hA).comp (ContinuousMap.compStarAlgHom' ℂ ℂ (cdfMap A hA ξ)) |
|
|
| def uniformState (f : UniformAlgebra) : ℂ := |
| ∫ x:ℝ, f (projIcc 0 1 (by norm_num) x) ∂volume.restrict (Icc (0:ℝ) 1) |
|
|
| theorem uniformHom_state (f : UniformAlgebra) : |
| inner ℂ ξ (uniformHom A hA ξ f ξ)=uniformState f := by |
| let g : ℝ → ℂ := fun x => f (projIcc 0 1 (by norm_num) x) |
| have hg : Continuous g := f.continuous.comp continuous_projIcc |
| have he (x : ℝ) : g (cdf (realMeasure A hA ξ) x)=f ⟨cdf (realMeasure A hA ξ) x, |
| cdf_nonneg _ _,cdf_le_one _ _⟩ := by |
| dsimp only [g] |
| congr 1 |
| exact projIcc_of_mem (by norm_num) ⟨cdf_nonneg _ _,cdf_le_one _ _⟩ |
| have hi := integral_map (μ := measure A hA ξ) measurable_subtype_coe.aemeasurable |
| (hg.comp (ProbabilityTransform.continuous_cdf (realMeasure A hA ξ))).aestronglyMeasurable |
| have hj := integral_map (μ := realMeasure A hA ξ) |
| (ProbabilityTransform.continuous_cdf (realMeasure A hA ξ)).aemeasurable hg.aestronglyMeasurable |
| rw [ProbabilityTransform.map_cdf] at hj |
| rw [uniformHom,StarAlgHom.comp_apply,← integral_complex] |
| calc |
| _ = ∫ x : Space A, g (cdf (realMeasure A hA ξ) x) ∂measure A hA ξ := by |
| apply integral_congr_ae |
| filter_upwards [] with x |
| exact (he x).symm |
| _ = ∫ x : ℝ, g (cdf (realMeasure A hA ξ) x) ∂realMeasure A hA ξ := hi.symm |
| _ = uniformState f := hj.symm |
|
|
| def uniformPhase : UniformAlgebra := |
| ⟨fun x => ProbabilityTransform.phase x,ProbabilityTransform.phase_continuous.comp continuous_subtype_val⟩ |
|
|
| theorem uniformHom_phase : uniformHom A hA ξ uniformPhase=(haarUnitary A hA ξ : E →L[ℂ] E) := rfl |
|
|
| theorem uniformHom_mem (M : StarSubalgebra ℂ (E →L[ℂ] E)) |
| (hM : IsClosed (M : Set (E →L[ℂ] E))) (hAM : A∈M) (f : UniformAlgebra) : |
| uniformHom A hA ξ f∈M := complexHom_mem A hA M hM hAM _ |
| end FreeGroupFactorMain.CyclicSpectral |
|
|
| section |
| open scoped ENNReal |
|
|
| namespace FreeGroupFactorMain.CyclicSpectral |
|
|
| section |
| open MeasureTheory |
| variable {H : Type u_2} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| local instance : InnerProductSpace ℝ H := InnerProductSpace.rclikeToReal ℂ H |
| variable (A : H →L[ℂ] H) (hA : IsSelfAdjoint A) (ξ : H) |
|
|
| def evaluate : C(Space A,ℝ) →ₗ[ℝ] H := |
| (((ContinuousLinearMap.apply ℂ H ξ).restrictScalars ℝ).comp (cfcL hA)).toLinearMap |
|
|
| theorem evaluate_norm (f : C(Space A,ℝ)) : |
| ‖evaluate A hA ξ f‖=‖ContinuousMap.toLp 2 (measure A hA ξ) ℝ f‖ := by |
| have hs : ‖evaluate A hA ξ f‖^2=‖ContinuousMap.toLp 2 (measure A hA ξ) ℝ f‖^2 := by |
| calc |
| _ = ∫ x, (f x)^2 ∂measure A hA ξ := (integral_sq A hA ξ f).symm |
| _ = inner ℝ (ContinuousMap.toLp 2 (measure A hA ξ) ℝ f) |
| (ContinuousMap.toLp 2 (measure A hA ξ) ℝ f) := by |
| rw [MeasureTheory.ContinuousMap.inner_toLp] |
| congr 1 |
| funext x |
| simp only [pow_two,RCLike.conj_to_real] |
| _ = _ := real_inner_self_eq_norm_sq _ |
| nlinarith [norm_nonneg (evaluate A hA ξ f),norm_nonneg (ContinuousMap.toLp 2 (measure A hA ξ) ℝ f)] |
|
|
| def realize : Lp ℝ 2 (measure A hA ξ) →L[ℝ] H := |
| (evaluate A hA ξ).extendOfNorm (ContinuousMap.toLp 2 (measure A hA ξ) ℝ).toLinearMap |
|
|
| theorem realize_continuous (f : C(Space A,ℝ)) : |
| realize A hA ξ (ContinuousMap.toLp 2 (measure A hA ξ) ℝ f)=cfcHom hA f ξ := by |
| apply LinearMap.extendOfNorm_eq |
| · exact ContinuousMap.toLp_denseRange ℝ (measure A hA ξ) ℝ (p := 2) (by norm_num) |
| · refine ⟨1,fun g => ?_⟩ |
| change ‖evaluate A hA ξ g‖ ≤ 1 * ‖ContinuousMap.toLp 2 (measure A hA ξ) ℝ g‖ |
| rw [one_mul,evaluate_norm] |
|
|
| theorem realize_norm (f : Lp ℝ 2 (measure A hA ξ)) : ‖realize A hA ξ f‖=‖f‖ := by |
| have hd := ContinuousMap.toLp_denseRange ℝ (measure A hA ξ) ℝ (p := 2) (by norm_num : (2:ℝ≥0∞)≠∞) |
| refine hd.induction_on (p := fun f => ‖realize A hA ξ f‖=‖f‖) f |
| (isClosed_eq ((realize A hA ξ).continuous.norm) continuous_norm) ?_ |
| intro q |
| rw [realize_continuous] |
| exact evaluate_norm A hA ξ q |
|
|
| def isometry : Lp ℝ 2 (measure A hA ξ) →ₗᵢ[ℝ] H where |
| toLinearMap := (realize A hA ξ).toLinearMap |
| norm_map' := realize_norm A hA ξ |
|
|
| end |
| section |
| open MeasureTheory Filter |
| variable {H : Type u_3} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| local instance : InnerProductSpace ℝ H := InnerProductSpace.rclikeToReal ℂ H |
| variable (A : H →L[ℂ] H) (hA : IsSelfAdjoint A) (ξ : H) |
|
|
| theorem mul_memLp (g : C(Space A,ℝ)) (f : Lp ℝ 2 (measure A hA ξ)) : |
| MemLp (fun x => g x*f x) 2 (measure A hA ξ) := by |
| apply (Lp.memLp f).of_le_mul (c := ‖g‖) |
| · exact g.continuous.aestronglyMeasurable.mul (Lp.aestronglyMeasurable f) |
| · exact Eventually.of_forall (fun x => by rw [norm_mul];exact mul_le_mul_of_nonneg_right (g.norm_coe_le_norm x) (norm_nonneg _)) |
| |
| def multiply (g : C(Space A,ℝ)) : Lp ℝ 2 (measure A hA ξ) →L[ℝ] Lp ℝ 2 (measure A hA ξ) := |
| LinearMap.mkContinuous |
| { toFun := fun f => (mul_memLp A hA ξ g f).toLp (fun x => g x*f x) |
| map_add' := by |
| intro f k |
| apply Lp.ext |
| filter_upwards [(mul_memLp A hA ξ g (f+k)).coeFn_toLp, |
| (mul_memLp A hA ξ g f).coeFn_toLp,(mul_memLp A hA ξ g k).coeFn_toLp, |
| Lp.coeFn_add f k,Lp.coeFn_add ((mul_memLp A hA ξ g f).toLp _) ((mul_memLp A hA ξ g k).toLp _)] with x h₁ h₂ h₃ h₄ h₅ |
| simp only [h₁,h₅,Pi.add_apply,h₂,h₃,h₄] |
| exact mul_add _ _ _ |
| map_smul' := by |
| intro c f |
| apply Lp.ext |
| filter_upwards [(mul_memLp A hA ξ g (c • f)).coeFn_toLp, |
| (mul_memLp A hA ξ g f).coeFn_toLp,Lp.coeFn_smul c f, |
| Lp.coeFn_smul c ((mul_memLp A hA ξ g f).toLp _)] with x h₁ h₂ h₃ h₄ |
| simp only [RingHom.id_apply,h₁,h₄,Pi.smul_apply,smul_eq_mul,h₂,h₃] |
| ring } |
| ‖g‖ (by |
| intro f |
| change ‖(mul_memLp A hA ξ g f).toLp (fun x => g x*f x)‖ ≤ ‖g‖ * ‖f‖ |
| apply Lp.norm_le_mul_norm_of_ae_le_mul |
| filter_upwards [(mul_memLp A hA ξ g f).coeFn_toLp] with x hx |
| rw [hx,norm_mul] |
| exact mul_le_mul_of_nonneg_right (g.norm_coe_le_norm x) (norm_nonneg _)) |
|
|
| theorem multiply_ae (g : C(Space A,ℝ)) (f : Lp ℝ 2 (measure A hA ξ)) : |
| multiply A hA ξ g f =ᵐ[measure A hA ξ] (fun x => g x*f x) := |
| (mul_memLp A hA ξ g f).coeFn_toLp |
|
|
| theorem multiply_continuous (g f : C(Space A,ℝ)) : |
| multiply A hA ξ g (ContinuousMap.toLp 2 (measure A hA ξ) ℝ f)= |
| ContinuousMap.toLp 2 (measure A hA ξ) ℝ (g*f) := by |
| apply Lp.ext |
| filter_upwards [multiply_ae A hA ξ g (ContinuousMap.toLp 2 (measure A hA ξ) ℝ f), |
| f.coeFn_toLp (𝕜 := ℝ) (p := 2) (measure A hA ξ), |
| (g*f).coeFn_toLp (𝕜 := ℝ) (p := 2) (measure A hA ξ)] with x h₁ h₂ h₃ |
| rw [h₁,h₂,h₃] |
| rfl |
|
|
| theorem realize_multiply (g : C(Space A,ℝ)) (f : Lp ℝ 2 (measure A hA ξ)) : |
| realize A hA ξ (multiply A hA ξ g f)=cfcHom hA g (realize A hA ξ f) := by |
| have hd := ContinuousMap.toLp_denseRange ℝ (measure A hA ξ) ℝ (p := 2) (by norm_num : (2:ℝ≥0∞)≠∞) |
| refine hd.induction_on (p := fun f => realize A hA ξ (multiply A hA ξ g f)=cfcHom hA g (realize A hA ξ f)) f (isClosed_eq ((realize A hA ξ).continuous.comp (multiply A hA ξ g).continuous) |
| ((cfcHom hA g).continuous.comp (realize A hA ξ).continuous)) ?_ |
| intro q |
| rw [multiply_continuous,realize_continuous,realize_continuous,map_mul,mul_apply_eq_comp] |
|
|
| theorem realize_coordinate (f : Lp ℝ 2 (measure A hA ξ)) : |
| realize A hA ξ (multiply A hA ξ ((ContinuousMap.id ℝ).restrict (spectrum ℝ A)) f)= |
| A (realize A hA ξ f) := by |
| rw [realize_multiply,cfcHom_id] |
|
|
| end |
| open MeasureTheory Filter |
| variable {H : Type u_4} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| local instance : InnerProductSpace ℝ H := InnerProductSpace.rclikeToReal ℂ H |
| variable (A : H →L[ℂ] H) (hA : IsSelfAdjoint A) (ξ : H) |
|
|
| def atom (x : Space A) : Lp ℝ 2 (measure A hA ξ) := |
| indicatorConstLp 2 (measurableSet_singleton x) (measure_ne_top _ _) (1:ℝ) |
|
|
| theorem atom_ae (x : Space A) : |
| atom A hA ξ x =ᵐ[measure A hA ξ] ({x} : Set (Space A)).indicator (fun _ => (1:ℝ)) := |
| indicatorConstLp_coeFn |
|
|
| theorem multiply_atom (x : Space A) : |
| multiply A hA ξ ((ContinuousMap.id ℝ).restrict (spectrum ℝ A)) (atom A hA ξ x)= |
| (x:ℝ) • atom A hA ξ x := by |
| classical |
| apply Lp.ext |
| filter_upwards [multiply_ae A hA ξ ((ContinuousMap.id ℝ).restrict (spectrum ℝ A)) (atom A hA ξ x), |
| atom_ae A hA ξ x,Lp.coeFn_smul (x:ℝ) (atom A hA ξ x)] with y h₁ h₂ h₃ |
| rw [h₁,h₃] |
| change (y:ℝ)*atom A hA ξ x y=(x:ℝ)*atom A hA ξ x y |
| rw [h₂] |
| by_cases hy : y=x |
| · subst y; rfl |
| · simp [Set.indicator_of_notMem,hy] |
| |
| theorem realize_atom_eigen (x : Space A) : |
| A (realize A hA ξ (atom A hA ξ x))=(x:ℝ) • realize A hA ξ (atom A hA ξ x) := by |
| rw [← realize_coordinate,multiply_atom,map_smul] |
|
|
| theorem realize_one : realize A hA ξ (ContinuousMap.toLp 2 (measure A hA ξ) ℝ (1:C(Space A,ℝ)))=ξ := by |
| rw [realize_continuous,map_one,one_apply_eq_self] |
|
|
| theorem realize_atom_inner (x : Space A) : |
| inner ℝ ξ (realize A hA ξ (atom A hA ξ x))=(measure A hA ξ).real {x} := by |
| have hm := (isometry A hA ξ).inner_map_map |
| (ContinuousMap.toLp 2 (measure A hA ξ) ℝ (1:C(Space A,ℝ))) (atom A hA ξ x) |
| change inner ℝ (realize A hA ξ (ContinuousMap.toLp 2 (measure A hA ξ) ℝ (1:C(Space A,ℝ)))) |
| (realize A hA ξ (atom A hA ξ x))=_ at hm |
| rw [realize_one] at hm |
| rw [hm,real_inner_comm] |
| change inner ℝ (indicatorConstLp 2 (measurableSet_singleton x) (measure_ne_top _ _) (1:ℝ)) |
| (ContinuousMap.toLp 2 (measure A hA ξ) ℝ (1:C(Space A,ℝ)))=_ |
| rw [L2.inner_indicatorConstLp_one] |
| calc |
| _ = ∫ _ in ({x}:Set (Space A)), (1:ℝ) ∂measure A hA ξ := by |
| apply setIntegral_congr_ae (measurableSet_singleton x) |
| exact ((1:C(Space A,ℝ)).coeFn_toLp (𝕜 := ℝ) (p := 2) (measure A hA ξ)).mono (fun y hy _ => hy) |
| _ = _ := by simp [Measure.real] |
|
|
| theorem singleton_eq_zero (x : Space A) |
| (he : ∀ η : H, A η=(x:ℝ) • η → inner ℝ ξ η=0) : measure A hA ξ {x}=0 := by |
| have hm : (measure A hA ξ).real {x}=0 := |
| (realize_atom_inner A hA ξ x).symm.trans (he _ (realize_atom_eigen A hA ξ x)) |
| exact (ENNReal.toReal_eq_zero_iff _).mp hm |>.resolve_right (measure_ne_top _ _) |
|
|
| theorem noAtoms (he : ∀ (r:ℝ) (η:H), A η=r • η → inner ℝ ξ η=0) : |
| NullSingletonClass (measure A hA ξ) := by |
| constructor |
| intro x |
| exact singleton_eq_zero A hA ξ x (he x) |
|
|
| end FreeGroupFactorMain.CyclicSpectral |
|
|
| end |
|
|
| namespace FreeGroupFactorMain.WeightedFock |
|
|
| section |
| open FreeFock Amplification MeasureTheory CyclicSpectral |
|
|
| instance oldBaseOp_probability (u v : ℝ) : |
| IsProbabilityMeasure (measure (oldBaseOp u v (0:ℕ) 1) (oldBaseOp_selfAdjoint u v 0 1) pRoot) := |
| probability _ _ _ pRoot_norm |
|
|
| theorem oldBaseOp_noAtoms (u v : ℝ) (hv : 0<v) (hvu : v<u) : |
| NullSingletonClass (measure (oldBaseOp u v (0:ℕ) 1) (oldBaseOp_selfAdjoint u v 0 1) pRoot) := by |
| let : InnerProductSpace ℝ (PRange ℕ) := InnerProductSpace.rclikeToReal ℂ (PRange ℕ) |
| apply noAtoms |
| intro r η hη |
| have he : oldBaseOp u v (0:ℕ) 1 η=(r:ℂ) • η := by |
| exact hη.trans (by |
| apply Subtype.ext |
| apply Subtype.ext |
| funext w |
| change r • (η:H ℕ) w=(r:ℂ)*(η:H ℕ) w |
| exact Complex.real_smul) |
| have hh := oldBaseOp_eigen_root_zero u v hv hvu (0:ℕ) 1 (r:ℂ) η he |
| change (inner ℂ (pRoot:PRange ℕ) η).re=0 |
| rw [hh,Complex.zero_re] |
|
|
| variable (u v : ℝ) |
|
|
| def weightedUniform (hv : 0<v) (hvu : v<u) : UniformAlgebra →⋆ₐ[ℂ] (PRange ℕ →L[ℂ] PRange ℕ) := by |
| letI := oldBaseOp_noAtoms u v hv hvu |
| exact uniformHom (oldBaseOp u v (0:ℕ) 1) (oldBaseOp_selfAdjoint u v 0 1) pRoot |
|
|
| def representedBase : StarSubalgebra ℂ (PRange ℕ →L[ℂ] PRange ℕ) := |
| (Base u v).comap (pRepresentation (ι := ℕ)).symm.toStarAlgHom |
|
|
| theorem pRepresentation_symm_val (T : PRange ℕ →L[ℂ] PRange ℕ) : |
| ((pRepresentation (ι := ℕ)).symm T : Operator ℕ)= |
| (PRange ℕ).subtypeL.comp (T.comp pProject) := by |
| apply ContinuousLinearMap.ext |
| intro ξ |
| have he := pRepresentation_apply ((pRepresentation (ι := ℕ)).symm T) (pProject ξ) |
| rw [StarAlgEquiv.apply_symm_apply,pProject_val] at he |
| have hr := DFunLike.congr_fun ((pRepresentation (ι := ℕ)).symm T).property.2 ξ |
| exact hr.symm.trans he.symm |
|
|
| theorem pRepresentation_symm_coe_continuous : |
| Continuous (fun T : PRange ℕ →L[ℂ] PRange ℕ => ((pRepresentation (ι := ℕ)).symm T : Operator ℕ)) := by |
| simp_rw [pRepresentation_symm_val] |
| exact continuous_const.clm_comp (continuous_id.clm_comp continuous_const) |
|
|
| theorem representedBase_closed : IsClosed (representedBase u v : Set (PRange ℕ →L[ℂ] PRange ℕ)) := by |
| have hF : IsClosed (filledEdgeAlgebra u v (0:ℕ) 1 : Set (Operator ℕ)) := Set.isClosed_centralizer _ |
| exact hF.preimage pRepresentation_symm_coe_continuous |
|
|
| theorem oldBaseOp_mem_representedBase (hv : 0<v) (hvu : v<u) : oldBaseOp u v (0:ℕ) 1∈representedBase u v := by |
| change ((pRepresentation (ι := ℕ)).symm (pRepresentation (oldBaseCorner u v 0 1)))∈Base u v |
| rw [StarAlgEquiv.symm_apply_apply] |
| exact diffuseBase_le_filled u v (hv.trans hvu).ne' (0:ℕ) 1 (oldBaseGenerator_mem_diffuseBase u v 0 1) |
|
|
| theorem weightedUniform_mem (hv : 0<v) (hvu : v<u) (a : UniformAlgebra) : weightedUniform u v hv hvu a∈representedBase u v := by |
| let := oldBaseOp_noAtoms u v hv hvu |
| exact uniformHom_mem _ _ _ _ (representedBase_closed u v) (oldBaseOp_mem_representedBase u v hv hvu) a |
|
|
| def weightedUniformBase (hv : 0<v) (hvu : v<u) : UniformAlgebra →⋆ₐ[ℂ] Base u v := |
| (((pRepresentation (ι := ℕ)).symm.toStarAlgHom).comp (weightedUniform u v hv hvu)).codRestrict |
| (Base u v) (fun a => by |
| simpa only [representedBase, StarSubalgebra.mem_comap, StarAlgHom.comp_apply] using |
| (weightedUniform_mem u v hv hvu a)) |
|
|
| theorem weightedUniform_state (hv : 0<v) (hvu : v<u) (a : UniformAlgebra) : |
| inner ℂ (pRoot:PRange ℕ) (weightedUniform u v hv hvu a pRoot)=uniformState a := by |
| let := oldBaseOp_noAtoms u v hv hvu |
| exact uniformHom_state _ _ _ a |
|
|
| end |
|
|
| open FreeFock Amplification |
| variable {ι : Type u_5} [DecidableEq ι] |
|
|
| theorem pRepresentation_vacuumTrace (u v : ℝ) (hu : u≠0) (x : PCorner ι) : |
| pVacuumTrace u v (x:Operator ι)=inner ℂ (pRoot:PRange ι) (pRepresentation x pRoot) := by |
| have he : inner ℂ ((P:Operator ι) (vacuum u v)) ((x:Operator ι) (P (vacuum u v)))= |
| vacuumTrace u v (x:Operator ι) := by |
| rw [← ContinuousLinearMap.adjoint_inner_right] |
| change inner ℂ (vacuum u v) ((star (P:Operator ι)*(x:Operator ι)*P : Operator ι) (vacuum u v))=_ |
| rw [projection_P.isSelfAdjoint.star_eq,x.property.1,x.property.2] |
| rfl |
| rw [pRoot_vacuum u v, map_smul,inner_smul_left,inner_smul_right] at he |
| rw [pVacuumTrace,← he] |
| change (u:ℂ)⁻¹*(u:ℂ)⁻¹*(star (u:ℂ)*((u:ℂ)*inner ℂ ((pRoot:PRange ι):H ι) |
| ((x:Operator ι) ((pRoot:PRange ι):H ι))))=_ |
| rw [Complex.star_def,Complex.conj_ofReal] |
| have hu' := Complex.ofReal_ne_zero.mpr hu |
| calc |
| _ = inner ℂ ((pRoot:PRange ι):H ι) ((x:Operator ι) ((pRoot:PRange ι):H ι)) := by field_simp |
| _ = _ := by rw [← pRepresentation_apply]; rfl |
| |
| theorem pAnnihilator_root (u v : ℝ) (hu : u≠0) (X : PCorner ℕ) |
| (hX : X∈oldAnnihilators u v) (a : Base u v) : |
| star (pRepresentation X) (pRepresentation a.val pRoot)=0 := by |
| have hr (x : PCorner ℕ) : (x:Operator ℕ) (vacuum u v)=(u:ℂ) • (x:Operator ℕ) ((pRoot:PRange ℕ):H ℕ) := by |
| have he := DFunLike.congr_fun x.property.2 (vacuum u v) |
| change (x:Operator ℕ) (P (vacuum u v))=(x:Operator ℕ) (vacuum u v) at he |
| rw [pRoot_vacuum,map_smul] at he |
| exact he.symm |
| have hh := hX a |
| rw [hr,map_smul] at hh |
| have hz : star (X:Operator ℕ) ((a.val:Operator ℕ) ((pRoot:PRange ℕ):H ℕ))=0 := |
| (smul_eq_zero.mp hh).resolve_left (Complex.ofReal_ne_zero.mpr hu) |
| have he := (pRepresentation (ι := ℕ)).map_star' X |
| change pRepresentation (star X)=star (pRepresentation X) at he |
| rw [← he] |
| apply Subtype.ext |
| change (pRepresentation (star X) (pRepresentation a.val pRoot) : H ℕ)=0 |
| rw [pRepresentation_apply,pRepresentation_apply] |
| exact hz |
| end FreeGroupFactorMain.WeightedFock |
|
|
| end |
|
|
| end OAI |
|
|
