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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 45 46 47 48 | 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
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