| import Mathlib |
| import OAI.Analysis.Commutator.CentralPinching |
| import OAI.Analysis.Commutator.CommutatorPerturbation |
| import OAI.Analysis.Commutator.ProjectionCommutant |
| import OAI.Analysis.Commutator.WeakOperatorAlgebra |
| import OAI.Analysis.Commutator.WeakOperatorLimit |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| section |
|
|
| noncomputable section |
|
|
| open Filter |
|
|
| open scoped InnerProductSpace Topology |
|
|
| namespace Kadison.CentralAverage |
|
|
| open UnitaryOrbit |
|
|
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| theorem exists_center_average (M : VonNeumannAlgebra H) (Y : H →L[ℂ] H) : |
| ∃ Y₁ : H →L[ℂ] H, |
| (∀ a∈M,a∈Set.centralizer (M : Set (H →L[ℂ] H)) → Y₁*a=a*Y₁) ∧ |
| ‖Y₁-Y‖≤commutatorSeminorm M Y ∧ |
| commutatorSeminorm M Y₁≤3*commutatorSeminorm M Y := by |
| classical |
| choose T hT hcomm using exists_finite_average M Y |
| let ω : Ultrafilter (Finset (CentralProjection M)) := Ultrafilter.of atTop |
| have hω : (ω : Filter (Finset (CentralProjection M)))≤atTop := Ultrafilter.of_le atTop |
| obtain ⟨Y₁,_,hlim⟩ := WeakOperatorLimit.exists_ultralimit ω T |
| (commutatorSeminorm M Y+‖Y‖) (add_nonneg (commutatorSeminorm_nonneg M Y) (norm_nonneg Y)) |
| (fun s => hull_norm M (hT s)) |
| have hd : ‖Y₁-Y‖≤commutatorSeminorm M Y := by |
| apply WeakOperatorLimit.norm_le_of_tendsto (l := (ω : Filter (Finset (CentralProjection M)))) |
| (T := fun s => T s-Y) |
| (fun x y => ?_) (commutatorSeminorm_nonneg M Y) |
| (Filter.Eventually.of_forall fun s => hull_distance M (hT s)) |
| simpa only [sub_apply,inner_sub_left] using |
| (hlim x y).sub_const (inner ℂ (Y x) y) |
| have hc (p : CentralProjection M) : Y₁*p.val=p.val*Y₁ := by |
| apply WeakOperatorLimit.intertwine hlim p.val p.val |
| have he := (Filter.eventually_ge_atTop ({p} : Finset (CentralProjection M))).filter_mono hω |
| filter_upwards [he] with s hs |
| exact hcomm s p (hs (Finset.mem_singleton_self p)) |
| refine ⟨Y₁,?_,hd,?_⟩ |
| · intro a ha haz |
| apply commutes_algebra_of_commutes_projections (vonNeumannIntersection M M.commutant) Y₁ |
| (fun p hp hpp => hc ⟨p,hp.1,hpp,by |
| simpa only [VonNeumannAlgebra.coe_toStarSubalgebra,VonNeumannAlgebra.coe_commutant] using hp.2⟩) a |
| exact ⟨ha,by |
| simpa only [VonNeumannAlgebra.coe_toStarSubalgebra,VonNeumannAlgebra.coe_commutant] using haz⟩ |
| · exact commutatorSeminorm_three M Y Y₁ hd |
|
|
| end Kadison.CentralAverage |
|
|
| end |
|
|
| end |
| end |
|
|
| end OAI |
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|
