| import Mathlib |
| import OAI.Analysis.BoundedHochschild.CocycleNormalParts |
| import OAI.Analysis.BoundedHochschild.FactorFinitePaths |
| import OAI.Analysis.BoundedHochschild.FreeWords |
| import OAI.Analysis.BoundedHochschild.MatrixDisplacement |
| import OAI.Analysis.BoundedHochschild.UnitaryLifts |
| import OAI.Analysis.BoundedHochschild.FactorPrimitive |
| import OAI.Analysis.BoundedHochschild.TypeIICorners |
| import OAI.Analysis.BoundedHochschild.CenterPowers |
| import OAI.Analysis.BoundedHochschild.CentralAtoms |
|
|
| namespace OAI |
|
|
| /-! Orthogonal atomic central resolution. -/ |
|
|
| noncomputable section |
|
|
| open Filter Topology |
| open scoped ComplexOrder |
| namespace HochschildAtomic |
| open BoundedHochschild HochschildTypeII FinitePathMain |
| universe u |
| variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] |
| [NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X] |
|
|
| def AtomicCenter (M : Type u) [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] : Prop := |
| ∀ p : M, IsStarProjection p → (∀ a : M, Commute p a) → p ≠ 0 → |
| ∃ q : M, CentralAtom q ∧ q ≤ p |
|
|
| lemma central_atoms_orthogonal {p q : M} (hp : CentralAtom p) (hq : CentralAtom q) (hne : p ≠ q) : |
| p*q=0 := by |
| have hpq : IsStarProjection (p*q) := hp.1.mul hq.1 (hp.2.1 q) |
| have hc : ∀ b : M, Commute (p*q) b := fun b => (hp.2.1 b).mul_left (hq.2.1 b) |
| have hpqp : p*q ≤ p := (hpq.le_iff_mul_eq_right hp.1).mpr |
| (by rw [← mul_assoc,hp.1.isIdempotentElem.eq]) |
| have hpqq : p*q ≤ q := (hpq.le_iff_mul_eq_left hq.1).mpr |
| (by rw [mul_assoc,hq.1.isIdempotentElem.eq]) |
| rcases hp.2.2.2 (p*q) hpq hc hpqp with hz | hpp |
| · exact hz |
| rcases hq.2.2.2 (p*q) hpq hc hpqq with hz | hqq |
| · exact hz |
| exact (hne (hpp.symm.trans hqq)).elim |
|
|
| variable (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) (τ : M →ₚ[ℂ] ℂ) |
|
|
| include e in |
|
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| theorem central_atoms_total |
| (hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) (hatomic : AtomicCenter M) |
| (x : M) (hx : ∀ p : M, CentralAtom p → p*x=0) : x=0 := by |
| classical |
| let S : Set M := {p | CentralProjection p ∧ p*x=0} |
| have hzero : (0:M) ∈ S := ⟨⟨IsStarProjection.zero M,fun _ => Commute.zero_left _⟩,zero_mul _⟩ |
| obtain ⟨q,_hq0,hmax⟩ := zorn_le_nonempty₀ S (fun c hc hchain y hy => by |
| obtain ⟨q,hq,hqx,hupper⟩ := central_annihilator_chain e τ hτ htr hfaith |
| x c hchain ⟨y,hy⟩ (fun p hp => hc hp) |
| exact ⟨q,⟨hq,hqx⟩,hupper⟩) 0 hzero |
| have hq : CentralProjection q := hmax.1.1 |
| have hqx : q*x=0 := hmax.1.2 |
| have hone : q=1 := by |
| by_contra hne |
| have hr : CentralProjection (1-q) := |
| ⟨hq.1.one_sub,fun a => (Commute.one_left a).sub_left (hq.2 a)⟩ |
| obtain ⟨p,hp,hpq⟩ := hatomic (1-q) hr.1 hr.2 (sub_ne_zero.mpr (Ne.symm hne)) |
| have hqp : q*p=0 := projection_orthogonal_of_le_complement hq.1 hp.1 hpq |
| have hnew : q+p ∈ S := ⟨⟨hq.1.add hp.1 hqp,fun a => (hq.2 a).add_left (hp.2.1 a)⟩, |
| by rw [add_mul,hqx,hx p hp,zero_add]⟩ |
| have he : q+p=q := le_antisymm (hmax.2 hnew (le_add_of_nonneg_right hp.1.nonneg)) |
| (le_add_of_nonneg_right hp.1.nonneg) |
| exact hp.2.2.1 (add_eq_left.mp he) |
| simpa only [hone,one_mul] using hqx |
|
|
| include e in |
|
|
| theorem typeII_no_minimal |
| (hτ : TraceInPredual e τ) (htr : ∀ a b : M, τ (a*b)=τ (b*a)) |
| (hfaith : ∀ a : M, τ (star a*a)=0 → a=0) (hII : TypeII1 M) : |
| HochschildDiffuse.NoMinimalProjections M := by |
| intro p hp hp0 |
| by_contra hn |
| have hmin (q : M) (hq : IsStarProjection q) (hqp : q ≤ p) : q=0 ∨ q=p := by |
| by_cases hq0 : q=0 |
| · exact Or.inl hq0 |
| · right |
| by_contra hne |
| exact hn ⟨q,hq,hqp,hq0,hne⟩ |
| have hab : AbelianProjection p := by |
| refine ⟨hp,fun a b => ?_⟩ |
| apply HochschildCorners.commute_corner_of_projections e hp |
| (b:=p*b*p) _ (HochschildCorners.compress_mem_corner hp a) |
| intro q hq hqp |
| rcases hmin q hq hqp with hz | he |
| · rw [hz]; exact Commute.zero_left _ |
| · rw [he] |
| change p*(p*b*p)=(p*b*p)*p |
| rw [(HochschildCorners.compress_mem_corner hp b).1, |
| (HochschildCorners.compress_mem_corner hp b).2] |
| exact hp0 (no_abelian_projection e τ hτ htr hfaith hII hab) |
|
|
| end HochschildAtomic |
|
|
| end |
|
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| end OAI |
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|
