openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Spectral /FullPolarDiffuseCriterion.lean
| import OAI.Analysis.IrrationalRotation.Spectral.PolarOrbitInvariance | |
| import OAI.Analysis.IrrationalRotation.Operators.AtomicNonscalarProjection | |
| import OAI.Analysis.IrrationalRotation.Operators.ProjectionAtomOrbit | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.PolarOrbit | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] | |
| theorem fullPolarAlgebra_isDiffuse_of_periodic_projections | |
| (U : unitary (H →L[ℂ] H)) (B : H →L[ℂ] H) | |
| (N : VonNeumannAlgebra H) (ξ : H) | |
| (hU : (U : H →L[ℂ] H) ∈ N) (hB : B ∈ N) | |
| (hξ : IsFaithfulTracialVector N ξ) | |
| (hab : IsAbelianVonNeumannAlgebra (fullPolarAlgebra (U : H →L[ℂ] H) B)) | |
| (hatomic : HasCountablyAtomicSpectralType B) | |
| (hnonscalar : ¬ ∃ c : ℂ, B = c • (1 : H →L[ℂ] H)) | |
| (htotal : ∀ n : ℕ, 0 < n → ∀ p : H →L[ℂ] H, | |
| p ∈ fullPolarAlgebra (U : H →L[ℂ] H) B → IsStarProjection p → | |
| (U : H →L[ℂ] H) ^ n * p * star (U : H →L[ℂ] H) ^ n = p → | |
| p = 0 ∨ p = 1) : | |
| IsDiffuseVonNeumannAlgebra (fullPolarAlgebra (U : H →L[ℂ] H) B) := by | |
| have hMN := fullPolarAlgebra_le (U : H →L[ℂ] H) B N hU hB | |
| apply ProjectionAtomOrbit.isDiffuse_of_total_ergodicity ξ | |
| (fun A hA C hC => hab A C hA hC) | |
| (ProjectionAtomOrbit.faithfulTracialVector_restrict hMN ξ hξ) | |
| (Unitary.conjStarAlgAut ℂ (H →L[ℂ] H) U) | |
| · exact fun A => fullPolarAlgebra_mem_iff U B A | |
| · intro A hA | |
| exact ProjectionAtomOrbit.vectorTrace_unitary_conjugation ξ hξ U hU (hMN hA) | |
| · intro n hn p hp hproj hperiod | |
| apply htotal n hn p hp hproj | |
| rw [iterate_conjStarAlgAut] at hperiod | |
| simpa only [Unitary.conjStarAlgAut_apply, SubmonoidClass.coe_pow, star_pow] using hperiod | |
| · exact exists_nontrivial_projection_of_atomic_nonscalar _ B | |
| (mem_fullPolarAlgebra (U : H →L[ℂ] H) B) hatomic hnonscalar | |
| end Invariant233.PolarOrbit | |
| end | |
| end OAI | |