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9eeb9c7 | 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 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 | import OAI.Analysis.IrrationalRotation.Model.Hyperfinite
import OAI.Analysis.IrrationalRotation.Analysis.GeneratedVonNeumann
import OAI.Analysis.IrrationalRotation.Analysis.VonNeumannStrongClosure
namespace OAI
noncomputable section
open Filter
open scoped Topology
namespace Invariant233.HyperfiniteCriterion
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H]
def asStrong (T : H →L[ℂ] H) : H →Lₚₜ[ℂ] H :=
UniformConvergenceCLM.ofFun (RingHom.id ℂ) H {s : Set H | s.Finite} T
omit [CompleteSpace H] in
@[simp] theorem underlying_asStrong (T : H →L[ℂ] H) :
strongOperatorUnderlying (asStrong T) = T := rfl
omit [CompleteSpace H] in
@[simp] theorem asStrong_apply (T : H →L[ℂ] H) (x : H) :
asStrong T x = T x := rfl
theorem mem_iSup_iff (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H))
(hA : Monotone A) (T : H →L[ℂ] H) :
T ∈ iSup A ↔ ∃ n, T ∈ A n := by
change T ∈ (↑(iSup A) : Set (H →L[ℂ] H)) ↔ _
rw [StarSubalgebra.coe_iSup_of_directed hA.directed_le]
exact Set.mem_iUnion
theorem strong_union_eq (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H))
(hA : Monotone A) :
{B : H →Lₚₜ[ℂ] H | ∃ n, strongOperatorUnderlying B ∈ A n} =
{B : H →Lₚₜ[ℂ] H | strongOperatorUnderlying B ∈ iSup A} := by
ext B
exact (mem_iSup_iff A hA (strongOperatorUnderlying B)).symm
theorem mem_strong_union_closure_of_tendsto
(A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H))
(T : ℕ → H →L[ℂ] H) (S : H →L[ℂ] H)
(hT : ∀ n, T n ∈ A n)
(hlim : ∀ x, Tendsto (fun n => T n x) atTop (𝓝 (S x))) :
asStrong S ∈ closure {B : H →Lₚₜ[ℂ] H |
∃ n, strongOperatorUnderlying B ∈ A n} := by
have hstrong : Tendsto (fun n => asStrong (T n)) atTop (𝓝 (asStrong S)) :=
PointwiseConvergenceCLM.tendsto_iff_forall_tendsto.mpr hlim
exact mem_closure_of_tendsto hstrong (Eventually.of_forall fun n => ⟨n, hT n⟩)
theorem strong_union_closure_subset
(M : VonNeumannAlgebra H) (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H))
(hAM : ∀ n, A n ≤ M.toStarSubalgebra) :
closure {B : H →Lₚₜ[ℂ] H | ∃ n, strongOperatorUnderlying B ∈ A n} ⊆
{B : H →Lₚₜ[ℂ] H | strongOperatorUnderlying B ∈ M} := by
apply closure_minimal
· rintro B ⟨n, hn⟩
exact hAM n hn
· exact vonNeumannAlgebra_stronglyClosed M
theorem generated_iSup_eq_of_generator_approximation
(s : Set (H →L[ℂ] H)) (A : ℕ → StarSubalgebra ℂ (H →L[ℂ] H))
(hAM : ∀ n, A n ≤ (vonNeumannGenerated s).toStarSubalgebra)
(hs : ∀ T ∈ s, asStrong T ∈ closure {B : H →Lₚₜ[ℂ] H |
∃ n, strongOperatorUnderlying B ∈ A n}) :
vonNeumannGenerated (↑(iSup A) : Set (H →L[ℂ] H)) = vonNeumannGenerated s := by
apply le_antisymm
· apply vonNeumannGenerated_le
exact iSup_le hAM
· apply vonNeumannGenerated_le
intro T hT
have hinclude : ∀ n, A n ≤
(vonNeumannGenerated (↑(iSup A) : Set (H →L[ℂ] H))).toStarSubalgebra := by
intro n S hS
exact subset_vonNeumannGenerated _ ((le_iSup A n) hS)
exact strong_union_closure_subset _ A hinclude (hs T hT)
end Invariant233.HyperfiniteCriterion
end
end OAI
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