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a97875e | 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 85 86 | 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
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