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ddc84c5 | 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 53 54 55 56 57 | 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
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