| import OAI.Analysis.CStarAlgebra.PureInfiniteness.Diagonalization |
| import OAI.Analysis.CStarAlgebra.PureInfiniteness.MatrixEntries |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace |
|
|
| namespace MainGap.Manuscript |
| variable {A : Type*} [NonUnitalCStarAlgebra A] |
| local instance positivityOrder : PartialOrder A := CStarAlgebra.spectralOrder A |
| local instance positivityStarOrder : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A |
|
|
| def CuntzLE (d v : Stable.Stabilization A) : Prop := |
| ∃ r : ℕ → Stable.Stabilization A, |
| Tendsto (fun j => star (r j) * v * r j) atTop (𝓝 d) |
|
|
| def ProperlyInfinite (h : A) : Prop := |
| CuntzLE ((⟦⟨2, MainGap.doubleCorner h⟩⟧ : Stable.Algebraic A) : Stable.Stabilization A) |
| ((⟦⟨1, MainGap.scalarCorner h⟩⟧ : Stable.Algebraic A) : Stable.Stabilization A) |
|
|
| def PropertyP (A : Type*) [NonUnitalCStarAlgebra A] : Prop := |
| ∀ h : A, 0 ≤ h → ProperlyInfinite h |
|
|
| lemma finiteCuntzLE_of_stable {m n : ℕ} (d : Stable.Mat A n) (v : Stable.Mat A m) |
| (h : CuntzLE (Stable.corner n d) (Stable.corner m v)) : MainGap.FiniteCuntzLE d v := by |
| obtain ⟨r,hr⟩ := h |
| refine ⟨fun j => Stable.block m n (r j), ?_⟩ |
| have ht := (Stable.continuous_block (A := A) n n).continuousAt.tendsto.comp hr |
| simpa only [Function.comp_def, Stable.block_conjugate, Stable.block_corner_self] using ht |
|
|
| lemma PropertyP.finite (hP : PropertyP A) : MainGap.PropertyP A := by |
| intro h hh |
| exact finiteCuntzLE_of_stable _ _ (hP h hh) |
|
|
| theorem individual_to_strong (hP : PropertyP A) : MainGap.UnrestrictedDiagonalization A := |
| MainGap.individual_to_strong hP.finite |
|
|
| theorem individual_to_strong_spi (hP : PropertyP A) : MainGap.StronglyPurelyInfinite A := |
| MainGap.individual_to_strong_spi hP.finite |
|
|
| theorem individual_to_strong_main (hP : PropertyP A) : |
| (∀ a b : A, 0 ≤ a → 0 ≤ b → ∀ c : A, ∀ ε : ℝ, 0 < ε → |
| ∃ s t : A, |
| ‖star s * a * s - a‖ < ε ∧ |
| ‖star t * b * t - b‖ < ε ∧ |
| ‖star s * c * t‖ < ε) ∧ |
| MainGap.StronglyPurelyInfinite A := by |
| exact ⟨individual_to_strong hP, individual_to_strong_spi hP⟩ |
|
|
| end MainGap.Manuscript |
|
|
| end |
|
|
| end OAI |
|
|
