| import Mathlib | |
| import OAI.Analysis.Commutator.FactorCorner | |
| import OAI.Analysis.Commutator.MinimalProjection | |
| import OAI.Analysis.Commutator.OrthogonalLinkSums | |
| namespace OAI | |
| noncomputable section | |
| section | |
| noncomputable section | |
| open scoped ComplexOrder InnerProductSpace Topology BigOperators | |
| open Filter | |
| universe u | |
| namespace Kadison.AtomicFactor | |
| open PartialIsometryOrder | |
| variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] | |
| structure Decomposition (M : VonNeumannAlgebra H) (p : H →L[ℂ] H) where | |
| I : Type u | |
| v : I → H →L[ℂ] H | |
| mem : ∀ i,v i∈M | |
| initial : ∀ i,star (v i)*v i=p | |
| cross : ∀ i j,i≠j → star (v i)*v j=0 | |
| source : ∀ i,v i*p=v i | |
| strong : ∀ x : H,Tendsto (fun s : Finset I => (∑ i∈s,v i*star (v i)) x) atTop (𝓝 x) | |
| theorem exists_decomposition (M : VonNeumannAlgebra H) (hM : FactorCorner.IsFactor M) | |
| {p : H →L[ℂ] H} (hp : IsMinimalProjection M p) : Nonempty (Decomposition M p) := by | |
| classical | |
| obtain ⟨S,hS⟩ := exists_maximal_orthogonalLinks M p 1 | |
| let T : Set (Link M p 1) := {v | v∈S ∧ v.val≠0} | |
| have hTS : T⊆S := fun _ h => h.1 | |
| have hT : OrthogonalLinks M p 1 T := fun _ hv _ hw hne => hS.1 (hTS hv) (hTS hw) hne | |
| have hspaces : familySpace M p 1 T=familySpace M p 1 S := by | |
| apply le_antisymm | |
| · apply Submodule.topologicalClosure_mono | |
| apply Submodule.span_mono | |
| rintro x ⟨v,hv,ξ,rfl⟩ | |
| exact ⟨v,hTS hv,ξ,rfl⟩ | |
| · apply Submodule.topologicalClosure_minimal | |
| · apply Submodule.span_le.mpr | |
| rintro x ⟨v,hv,ξ,rfl⟩ | |
| change v.val ξ∈familySpace M p 1 T | |
| by_cases hz : v.val=0 | |
| · rw [hz] | |
| exact (familySpace M p 1 T).zero_mem | |
| · exact familySpace_generator M p 1 T ⟨hv,hz⟩ ξ | |
| · exact Submodule.isClosed_topologicalClosure _ | |
| have hspan : (familySpace M p 1 T).starProjection=1 := by | |
| have he : (familySpace M p 1 T).starProjection=(familySpace M p 1 S).starProjection := by | |
| congr 1 | |
| rw [he] | |
| exact maximal_family_spans M p 1 hM hp.1 M.one_mem hp.2.1 (IsStarProjection.one _) hp.2.2.1 S hS | |
| refine ⟨⟨T,fun i => i.val.val,fun i => i.val.mem,?_,?_,?_,?_⟩⟩ | |
| · intro i | |
| rcases hp.2.2.2 i.val.initial (M.mul_mem (M.star_mem' i.val.mem) i.val.mem) | |
| i.val.initial_projection i.val.source_le with hz|hz | |
| · have he := i.val.mul_initial | |
| rw [hz,mul_zero] at he | |
| exact (i.property.2 he.symm).elim | |
| · exact hz | |
| · intro i j hne | |
| exact family_cross M p 1 T hT hne | |
| · intro i | |
| exact i.val.mul_source hp.2.1 | |
| · intro x | |
| simpa only [finiteProjection,Link.final,one_apply_eq_self] using | |
| finiteProjection_tendsto M p 1 T hT (IsStarProjection.one _) hspan x | |
| end Kadison.AtomicFactor | |
| end | |
| end | |
| end | |
| end OAI | |