| import OAI.Analysis.FiniteFactor.Diffuse.OperatorMeanErgodic |
| import Mathlib.Analysis.VonNeumannAlgebra.Basic |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology |
| namespace Nonhyperlinear.Diffuseness |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
|
|
| / |
| def rangeSupport (T : H →L[ℂ] H) : H →L[ℂ] H := |
| T.range.topologicalClosure.starProjection |
|
|
| lemma rangeSupport_projection (T : H →L[ℂ] H) : IsStarProjection (rangeSupport T) := |
| isStarProjection_starProjection |
|
|
| lemma rangeSupport_range (T : H →L[ℂ] H) : |
| (rangeSupport T).range=T.range.topologicalClosure := |
| Submodule.range_starProjection _ |
|
|
| lemma rangeSupport_mul_self (T : H →L[ℂ] H) : rangeSupport T*T=T := by |
| ext x |
| apply Submodule.starProjection_eq_self_iff.mpr |
| exact Submodule.le_topologicalClosure _ ⟨x,rfl⟩ |
|
|
| lemma rangeSupport_ne_zero {T : H →L[ℂ] H} (hT:T≠0) : rangeSupport T≠0 := by |
| intro h |
| have he:=rangeSupport_mul_self T |
| rw [h,zero_mul] at he |
| exact hT he.symm |
|
|
| / |
| theorem rangeSupport_mem (M : VonNeumannAlgebra H) {T:H→L[ℂ]H} (hT:T∈M) : |
| rangeSupport T∈M := by |
| apply (VonNeumannAlgebra.IsStarProjection.mem_iff (rangeSupport_projection T) M).mpr |
| intro y hy |
| rw [rangeSupport_range] |
| apply Submodule.topologicalClosure_mem_invtSubmodule |
| intro x hx |
| obtain ⟨z,rfl⟩:=hx |
| refine ⟨y z,?_⟩ |
| have hc := VonNeumannAlgebra.mem_commutant_iff.mp hy T hT |
| exact congrArg (fun A:H→L[ℂ]H=>A z) hc |
|
|
| lemma projection_mul_rangeSupport {p T:H→L[ℂ]H} (hp:IsStarProjection p) (h:p*T=T) : |
| p*rangeSupport T=rangeSupport T := by |
| have hs : Set.EqOn p id (T.range:Set H) := by |
| rintro _ ⟨x,rfl⟩ |
| have hx : T x ∈ p.range := ⟨T x, congrArg (fun A:H→L[ℂ]H=>A x) h⟩ |
| obtain ⟨y,hy⟩ := hx |
| change p (T x) = T x |
| rw [← hy] |
| exact congrArg (fun A:H→L[ℂ]H=>A y) hp.isIdempotentElem.eq |
| have hc : Set.EqOn p id (closure (T.range:Set H)) := |
| hs.closure p.continuous continuous_id |
| ext x |
| exact hc (T.range.topologicalClosure.starProjection_apply_mem x) |
|
|
| lemma rangeSupport_mul_projection {p T:H→L[ℂ]H} (hp:IsStarProjection p) (h:p*T=T) : |
| rangeSupport T*p=rangeSupport T := by |
| have he:=congrArg star (projection_mul_rangeSupport hp h) |
| simpa only [star_mul,hp.isSelfAdjoint.star_eq,(rangeSupport_projection T).isSelfAdjoint.star_eq] using he |
|
|
| end Nonhyperlinear.Diffuseness |
|
|
| end |
|
|
| end OAI |
|
|
