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ddc84c5 | 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 | 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]
/-- A concrete factor has scalar center. -/
def HasScalarCenter (M : VonNeumannAlgebra H) : Prop :=
∀ a ∈ M, (∀ b ∈ M, a * b = b * a) → ∃ z : ℂ, a = z • (1 : H →L[ℂ] H)
/-- Finiteness: every isometry in the algebra is unitary. -/
def IsFiniteAlgebra (M : VonNeumannAlgebra H) : Prop :=
∀ v ∈ M, star v * v = 1 → v * star v = 1
/-- No nonzero projection is minimal, excluding matrix factors. -/
def HasNoMinimalProjection (M : VonNeumannAlgebra H) : Prop :=
∀ p ∈ M, IsStarProjection p → p ≠ 0 →
∃ q ∈ M, IsStarProjection q ∧ q ≠ 0 ∧ q ≠ p ∧ q * p = q
/-- A separable Banach space whose dual is conjugate-linearly isometric to M. -/
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
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