| import OAI.Analysis.BoundedHochschild.MultimodularNormalReduction |
|
|
| namespace OAI |
|
|
| /-! The properly infinite primitive endpoint needs module identities only |
| on the algebraic countable matrix units. Separate normality is supplied by |
| the proved normal reduction, retaining those identities. -/ |
|
|
| noncomputable section |
|
|
| namespace BoundedHochschild.KadisonRingrose |
|
|
| open HochschildComparison |
|
|
| universe u |
| variable {M X : Type u} [CStarAlgebra M] [PartialOrder M] [StarOrderedRing M] |
| [NormedAddCommGroup X] [NormedSpace ℂ X] [CompleteSpace X] |
|
|
| theorem primitive_of_algebraic_matrix_multimodular [Nontrivial M] |
| (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) (hno : NoFiniteCentralSummand M) |
| (c : ℕ → PartialIso M) (hi : ∀ i, (c i).initial = 1) |
| (ho : ∀ i j, i ≠ j → (c i).final * (c j).final = 0) |
| (ht : PredualTendsto e (fun s : Finset ℕ => ∑ i ∈ s, (c i).final) Filter.atTop 1) |
| {n : ℕ} (f : Cochain M (n + 2)) |
| (hM : MultimodularOver (countableMatrixAlgebra c) f) |
| (hf : ∀ x, differentialValue f x = 0) : |
| ∃ g : Cochain M (n + 1), ∀ x, differentialValue g x = f x := by |
| obtain ⟨F, b, hFN, hFM, hFc, hid⟩ := |
| normal_reduction_multimodular e (countableMatrixAlgebra c) f hf hM |
| obtain ⟨g, _, hg⟩ := primitive_of_normal_matrix_multimodular e hno c hi ho ht F hFN hFM hFc |
| refine ⟨g + b, fun x => ?_⟩ |
| rw [differentialValue_add, hg] |
| exact (hid x).symm |
|
|
| theorem primitive_of_matrix_units [Nontrivial M] |
| (e : StrongDual ℂ X ≃ₗᵢ⋆[ℂ] M) (hno : NoFiniteCentralSummand M) |
| (c : ℕ → PartialIso M) (hi : ∀ i, (c i).initial = 1) |
| (ho : ∀ i j, i ≠ j → (c i).final * (c j).final = 0) |
| (ht : PredualTendsto e (fun s : Finset ℕ => ∑ i ∈ s, (c i).final) Filter.atTop 1) |
| {n : ℕ} (f : Cochain M (n + 2)) |
| (hM : ∀ i j, MultimodularAt f (countableMatrixUnit c i j)) |
| (hf : ∀ x, differentialValue f x = 0) : |
| ∃ g : Cochain M (n + 1), ∀ x, differentialValue g x = f x := |
| primitive_of_algebraic_matrix_multimodular e hno c hi ho ht f |
| (multimodular_of_matrixUnits c hi ho f hM) hf |
|
|
| end BoundedHochschild.KadisonRingrose |
|
|
| end |
|
|
| end OAI |
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|
