| import Mathlib.Analysis.VonNeumannAlgebra.Basic |
| import Mathlib.Topology.Order |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace FiniteFactor |
| variable (H : Type*) [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| /-- The actual ultraweak topology, generated by absolutely summable vector |
| functionals. No continuity of a chosen trace is built into this definition. -/ |
| @[instance_reducible] def ultraweakTopology : TopologicalSpace (H →L[ℂ] H) := |
| ⨅ (ξ : ℕ → H) (η : ℕ → H) (_ : Summable (fun n => ‖ξ n‖ * ‖η n‖)), |
| TopologicalSpace.induced (fun T : H →L[ℂ] H => ∑' n, inner ℂ (ξ n) (T (η n))) |
| inferInstance |
|
|
| variable {H} |
| omit [CompleteSpace H] in |
| theorem vectorFunctional_ultraweak_continuous [CompleteSpace H] (ξ η : H) : |
| @Continuous _ _ (ultraweakTopology H) inferInstance |
| (fun T : H →L[ℂ] H => inner ℂ ξ (T η)) := by |
| let x : ℕ → H := fun n => if n = 0 then ξ else 0 |
| let y : ℕ → H := fun n => if n = 0 then η else 0 |
| have hs : Summable (fun n => ‖x n‖ * ‖y n‖) := by |
| exact (hasSum_ite_eq (0 : ℕ) (‖ξ‖ * ‖η‖)).summable.congr (by intro n; by_cases h : n = 0 <;> simp [x,y,h]) |
| have hc : @Continuous _ _ (ultraweakTopology H) inferInstance |
| (fun T : H →L[ℂ] H => ∑' n, inner ℂ (x n) (T (y n))) := |
| continuous_iInf_dom (i := x) (continuous_iInf_dom (i := y) |
| (continuous_iInf_dom (i := hs) continuous_induced_dom)) |
| have he (T : H →L[ℂ] H) : |
| (fun n => inner ℂ (x n) (T (y n))) = |
| (fun n => if n = 0 then inner ℂ ξ (T η) else 0) := by |
| funext n |
| by_cases h : n = 0 <;> simp [x,y,h] |
| have hf : (fun T : H →L[ℂ] H => ∑' n, inner ℂ (x n) (T (y n))) = |
| (fun T => inner ℂ ξ (T η)) := by |
| funext T |
| rw [he] |
| exact tsum_ite_eq 0 _ |
| rw [hf] at hc |
| exact hc |
|
|
| end FiniteFactor |
|
|
| end |
|
|
| end OAI |
|
|
