| 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.FiniteFactorPrimitives |
| import OAI.Analysis.BoundedHochschild.CornerCochains |
| import OAI.Analysis.BoundedHochschild.AtomicResolution |
|
|
| namespace OAI |
|
|
| /-! Bounded primitives over an atomic finite center. -/ |
|
|
| noncomputable section |
|
|
| open scoped ComplexOrder |
| namespace BoundedHochschild |
| open HochschildAtomic HochschildCorners |
| universe u |
| variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] |
| [NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X] |
| (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) (τ : M →ₚ[ℂ] ℂ) |
|
|
| theorem primitive_of_atomic_finite |
| (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) |
| {n : ℕ} (f : Cochain M (n+1)) |
| (hf : ∀ x : Fin (n+2) → M, differentialValue f x=0) : |
| ∃ g : Cochain M n, ‖g‖ ≤ (12+n)*‖f‖ ∧ ∀ x, differentialValue g x=f x := by |
| classical |
| let I := {p : M |
| have hex (p : I) : ∃ g : Cochain (Corner p.val p.property.1) n, |
| ‖g‖≤12*‖f‖ ∧ ∀ x, differentialValue g x=cornerRestrict p.val p.property.1 p.property.2.1 f x := by |
| let ec := cornerPredualEquiv p.val p.property.1 e |
| let tc := cornerTrace p.val p.property.1 τ |
| obtain ⟨g,hg,he⟩ := primitive_of_finite_factor ec tc |
| (cornerTrace_normal p.val p.property.1 e τ hτ) |
| (cornerTrace_one p.val p.property.1 τ hfaith p.property.2.2.1) |
| (cornerTrace_tracial p.val p.property.1 τ htr) |
| (cornerTrace_faithful p.val p.property.1 τ hfaith p.property.2.2.1) |
| (central_atom_corner_factor e p.property) |
| (cornerRestrict p.val p.property.1 p.property.2.1 f) |
| (fun x => by rw [differential_cornerRestrict,hf,map_zero]) |
| exact ⟨g,hg.trans (mul_le_mul_of_nonneg_left |
| (cornerRestrict_norm p.val p.property.1 p.property.2.1 f) (by norm_num)),he⟩ |
| choose g hg he using hex |
| let b : I → Cochain M n := fun p => cornerExtend p.val p.property.1 p.property.2.1 (g p) |
| obtain ⟨G,hG,hGF⟩ := primitive_of_central_cut_primitives e |
| (HochschildSakai.mul_hasPreadjoint e) (fun p : I => p.val) |
| (fun p => p.property.1) (fun p a => (p.property.2.1 a).eq) |
| (fun p q hne => central_atoms_orthogonal p.property q.property |
| (fun heq => hne (Subtype.ext heq))) |
| (fun x hx => central_atoms_total e τ hτ htr hfaith hatomic x (fun p hp => hx ⟨p,hp⟩)) |
| f hf b (fun p x => cornerExtend_support p.val p.property.1 p.property.2.1 (g p) x) |
| (fun p => cornerExtend_primitive p.val p.property.1 p.property.2.1 f (g p) (he p)) |
| (12*‖f‖) (by positivity) |
| (fun p => (cornerExtend_norm p.val p.property.1 p.property.2.1 (g p)).trans (hg p)) |
| exact ⟨G,by simpa only [add_mul] using hG,hGF⟩ |
|
|
| end BoundedHochschild |
|
|
| end |
|
|
| end OAI |
|
|
