openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /AtomicNonscalarProjection.lean
| import OAI.Analysis.IrrationalRotation.Model.ConcreteSpectral | |
| import OAI.Analysis.IrrationalRotation.Operators.JointEigenprojection | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233 | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| theorem exists_nontrivial_projection_of_atomic_nonscalar | |
| (M : VonNeumannAlgebra H) (B : H →L[ℂ] H) | |
| (hB : B ∈ M) (hatomic : HasCountablyAtomicSpectralType B) | |
| (hnonscalar : ¬ ∃ c : ℂ, B = c • (1 : H →L[ℂ] H)) : | |
| ∃ q : H →L[ℂ] H, q ∈ M ∧ IsStarProjection q ∧ q ≠ 0 ∧ q ≠ 1 := by | |
| obtain ⟨S, _, hdense⟩ := hatomic | |
| let D : Unit → H →L[ℂ] H := fun _ => B | |
| let ev : S → Unit → ℂ := fun z _ => z.val | |
| have hdense' : (⨆ z : S, jointEigenspace D (ev z)).topologicalClosure = ⊤ := by | |
| simpa only [jointEigenspace, D, ev, iInf_const] using hdense | |
| have hB0 : B ≠ 0 := by | |
| intro hz | |
| exact hnonscalar ⟨0, by simpa using hz⟩ | |
| obtain ⟨z, hz⟩ := exists_mul_jointEigenprojection_ne_zero D ev hdense' B hB0 | |
| let q := jointEigenprojection D (ev z) | |
| have hqmem : q ∈ M := by | |
| apply vonNeumannGenerated_le (Set.range D) M _ (jointEigenprojection_mem_generated D (ev z)) | |
| rintro A ⟨i, rfl⟩ | |
| exact hB | |
| refine ⟨q, hqmem, isStarProjection_jointEigenprojection D (ev z), ?_, ?_⟩ | |
| · intro hq0 | |
| apply hz | |
| change B * q = 0 | |
| rw [hq0, mul_zero] | |
| · intro hq1 | |
| apply hnonscalar | |
| refine ⟨z.val, ?_⟩ | |
| ext x | |
| have hx : x ∈ jointEigenspace D (ev z) := by | |
| apply Submodule.starProjection_eq_self_iff.mp | |
| change q x = x | |
| rw [hq1] | |
| rfl | |
| exact (mem_jointEigenspace.mp hx) () | |
| end Invariant233 | |
| end | |
| end OAI | |