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110f4f3 | 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 | 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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