| import OAI.Analysis.IrrationalRotation.Model.JointEigenspace | |
| import OAI.Analysis.IrrationalRotation.Model.OperatorProperties | |
| import OAI.Analysis.IrrationalRotation.Analysis.GeneratedVonNeumann | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| variable {ι H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| omit [CompleteSpace H] in | |
| @[simp] | |
| theorem mem_jointEigenspace {D : ι → H →L[ℂ] H} {ev : ι → ℂ} {x : H} : | |
| x ∈ jointEigenspace D ev ↔ ∀ i, D i x = ev i • x := by | |
| simp [jointEigenspace] | |
| omit [CompleteSpace H] in | |
| theorem map_mem_jointEigenspace {D : ι → H →L[ℂ] H} {ev : ι → ℂ} | |
| {T : H →L[ℂ] H} (hT : ∀ i, Commute (D i) T) | |
| {x : H} (hx : x ∈ jointEigenspace D ev) : T x ∈ jointEigenspace D ev := by | |
| rw [mem_jointEigenspace] at hx ⊢ | |
| intro i | |
| calc | |
| D i (T x) = T (D i x) := DFunLike.congr_fun (hT i).eq x | |
| _ = ev i • T x := by rw [hx i, map_smul] | |
| theorem commute_starProjection_of_reducing (K : Submodule ℂ H) | |
| [K.HasOrthogonalProjection] (T : H →L[ℂ] H) | |
| (hT : ∀ x ∈ K, T x ∈ K) (hTstar : ∀ x ∈ K, star T x ∈ K) : | |
| Commute K.starProjection T := by | |
| apply ContinuousLinearMap.ext | |
| intro x | |
| change K.starProjection (T x) = T (K.starProjection x) | |
| apply Submodule.eq_starProjection_of_mem_orthogonal | |
| · apply hT | |
| exact Submodule.coe_mem (K.orthogonalProjectionOnto x) | |
| · have horth : Kᗮ ∈ Module.End.invtSubmodule T.toLinearMap := | |
| ContinuousLinearMap.orthogonal_mem_invtSubmodule hTstar | |
| rw [Module.End.mem_invtSubmodule_iff_forall_mem_of_mem] at horth | |
| rw [← map_sub] | |
| exact horth _ (K.sub_starProjection_mem_orthogonal x) | |
| @[simp] | |
| theorem isStarProjection_jointEigenprojection (D : ι → H →L[ℂ] H) (ev : ι → ℂ) : | |
| IsStarProjection (jointEigenprojection D ev) := | |
| isStarProjection_starProjection | |
| theorem jointEigenprojection_mem_generated (D : ι → H →L[ℂ] H) (ev : ι → ℂ) : | |
| jointEigenprojection D ev ∈ vonNeumannGenerated (Set.range D) := by | |
| rw [VonNeumannAlgebra.IsStarProjection.mem_iff | |
| (isStarProjection_jointEigenprojection D ev)] | |
| intro T hT | |
| change (jointEigenspace D ev).starProjection.range ∈ Module.End.invtSubmodule T | |
| rw [Submodule.range_starProjection] | |
| rw [Module.End.mem_invtSubmodule_iff_forall_mem_of_mem] | |
| apply map_mem_jointEigenspace | |
| intro i | |
| exact ((mem_commutant_vonNeumannGenerated_iff _ T).mp hT (D i) ⟨i, rfl⟩).1 | |
| theorem jointEigenprojection_mem_commutant (D : ι → H →L[ℂ] H) (ev : ι → ℂ) | |
| (hD : ∀ i, IsSelfAdjoint (D i)) : | |
| jointEigenprojection D ev ∈ (vonNeumannGenerated (Set.range D)).commutant := by | |
| rw [mem_commutant_vonNeumannGenerated_iff] | |
| rintro _ ⟨i, rfl⟩ | |
| have h : Commute (jointEigenprojection D ev) (D i) := by | |
| apply commute_starProjection_of_reducing | |
| · intro x hx | |
| rw [(mem_jointEigenspace.mp hx) i] | |
| exact (jointEigenspace D ev).smul_mem _ hx | |
| · intro x hx | |
| rw [(hD i).star_eq, (mem_jointEigenspace.mp hx) i] | |
| exact (jointEigenspace D ev).smul_mem _ hx | |
| exact ⟨h.eq.symm, by simpa only [(hD i).star_eq] using h.eq.symm⟩ | |
| theorem exists_mul_jointEigenprojection_ne_zero {A : Type*} | |
| (D : ι → H →L[ℂ] H) (ev : A → ι → ℂ) | |
| (hdense : (⨆ a, jointEigenspace D (ev a)).topologicalClosure = ⊤) | |
| (p : H →L[ℂ] H) (hp : p ≠ 0) : | |
| ∃ a, p * jointEigenprojection D (ev a) ≠ 0 := by | |
| by_contra! h | |
| apply hp | |
| have hker : (⨆ a, jointEigenspace D (ev a)) ≤ p.ker := by | |
| apply iSup_le | |
| intro a x hx | |
| change p x = 0 | |
| have hxq : jointEigenprojection D (ev a) x = x := | |
| Submodule.starProjection_eq_self_iff.mpr hx | |
| have heq := DFunLike.congr_fun (h a) x | |
| simpa only [mul_apply_eq_comp, hxq, zero_apply] using heq | |
| have htop : (⊤ : Submodule ℂ H) ≤ p.ker := by | |
| rw [← hdense] | |
| exact Submodule.topologicalClosure_minimal _ hker p.isClosed_ker | |
| ext x | |
| exact htop (Submodule.mem_top : x ∈ (⊤ : Submodule ℂ H)) | |
| theorem atomic_generated_of_jointEigenprojection_atoms {A : Type*} | |
| (D : ι → H →L[ℂ] H) (ev : A → ι → ℂ) | |
| (hD : ∀ i, IsSelfAdjoint (D i)) | |
| (hdense : (⨆ a, jointEigenspace D (ev a)).topologicalClosure = ⊤) | |
| (hatom : ∀ a, jointEigenprojection D (ev a) ≠ 0 → | |
| IsProjectionAtom (vonNeumannGenerated (Set.range D)) (jointEigenprojection D (ev a))) : | |
| IsAtomicVonNeumannAlgebra (vonNeumannGenerated (Set.range D)) := by | |
| intro p hp hpp hp0 | |
| obtain ⟨a, ha⟩ := exists_mul_jointEigenprojection_ne_zero D ev hdense p hp0 | |
| let q := jointEigenprojection D (ev a) | |
| have hqm : q ∈ vonNeumannGenerated (Set.range D) := jointEigenprojection_mem_generated D _ | |
| have hqp : IsStarProjection q := isStarProjection_jointEigenprojection D _ | |
| have hq0 : q ≠ 0 := by | |
| intro hq | |
| apply ha | |
| change p * q = 0 | |
| rw [hq, mul_zero] | |
| have hpq : Commute p q := | |
| VonNeumannAlgebra.mem_commutant_iff.mp (jointEigenprojection_mem_commutant D _ hD) p hp | |
| have hqa := hatom a hq0 | |
| have hpqq : (p * q) * q = p * q := by | |
| rw [mul_assoc, hqp.isIdempotentElem.eq] | |
| have hpeq : p * q = q := | |
| (hqa.2.2.2 (p * q) (mul_mem hp hqm) (hpp.mul hqp hpq) hpqq).resolve_left ha | |
| exact ⟨q, hqa, hpq.eq.symm.trans hpeq⟩ | |
| end Invariant233 | |
| end | |
| end OAI | |