| import Mathlib.Analysis.VonNeumannAlgebra.Basic |
| import Mathlib.Analysis.SpecialFunctions.Pow.Real |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open Filter |
| open scoped Topology |
| universe u |
|
|
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| / |
| def HasScalarCenter (M : VonNeumannAlgebra H) : Prop := |
| ∀ a ∈ M, (∀ b ∈ M, a * b = b * a) → ∃ z : ℂ, a = z • (1 : H →L[ℂ] H) |
|
|
| / |
| def IsFiniteAlgebra (M : VonNeumannAlgebra H) : Prop := |
| ∀ v ∈ M, star v * v = 1 → v * star v = 1 |
|
|
| / |
| def HasNoMinimalProjection (M : VonNeumannAlgebra H) : Prop := |
| ∀ p ∈ M, IsStarProjection p → p ≠ 0 → |
| ∃ q ∈ M, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ q * p = q |
|
|
| / |
| def HasSeparablePredual (M : VonNeumannAlgebra H) : Prop := |
| ∃ (E : Type u) (_ : NormedAddCommGroup E) (_ : NormedSpace ℂ E) |
| (_ : CompleteSpace E) (_ : TopologicalSpace.SeparableSpace E), |
| Nonempty (StrongDual ℂ E ≃ₗᵢ⋆[ℂ] M.toStarSubalgebra) |
|
|
| def MainTarget : Prop := |
| ∃ (H : Type u) (_ : NormedAddCommGroup H) (_ : InnerProductSpace ℂ H) |
| (_ : CompleteSpace H) (M : VonNeumannAlgebra H), |
| HasScalarCenter M ∧ IsFiniteAlgebra M ∧ HasNoMinimalProjection M ∧ |
| HasSeparablePredual M ∧ |
| ∃ T : H →L[ℂ] H, T ∈ M ∧ T ≠ 0 ∧ |
| Tendsto (fun n : ℕ => ‖T ^ n‖ ^ (1 / (n : ℝ))) atTop (𝓝 0) ∧ |
| ∀ p ∈ M, IsStarProjection p → (1 - p) * T * p = 0 → p = 0 ∨ p = 1 |
|
|
| end FiniteFactor |
|
|
| end OAI |
|
|
