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f1f7953 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 | 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
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