| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.CuntzSemigroup.RationalConeStates |
| import OAI.Analysis.JiangSu.CuntzSemigroup.FiniteCuntzFullness |
|
|
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| namespace Pureness |
| open TraceCone OrderedSemigroup |
| open scoped ENNReal |
|
|
| variable {A : Type*} [CStarAlgebra A] [PartialOrder A] [StarOrderedRing A] |
|
|
| theorem finiteCuntz_le_of_states (hsimple : TopologicallySimple A) |
| (hAU : ∀ (u v : CuntzClass (MatrixLimit A)) (k : ℕ), 1 ≤ k → |
| (k+1) • u ≤ k • v → u ≤ v) |
| (y : finiteCuntzSubmonoid (A := A)) (hy : y ≠ 0) |
| (x : finiteCuntzSubmonoid (A := A)) |
| (hgap : ∀ f : StateSpace y, f.1 x < 1) : x ≤ y := |
| le_of_states finiteCuntz_zero_le (finiteCuntz_almostUnperforated hAU) y x |
| (finiteCuntz_nonzero_orderUnit hsimple y hy) hgap |
|
|
| theorem finiteComparison_of_states (hsimple : TopologicallySimple A) |
| (hAU : ∀ (u v : CuntzClass (MatrixLimit A)) (k : ℕ), 1 ≤ k → |
| (k+1) • u ≤ k • v → u ≤ v) |
| (L : ℕ) (M N : Positive (FinMatrix A L)) (hN : N.1 ≠ 0) |
| (hgap : ∀ f : StateSpace (finitePositiveClass L N), f.1 (finitePositiveClass L M) < 1) : |
| CuntzSubequiv M.1 N.1 := by |
| apply (finitePositiveClass_le_iff L M N).mp |
| exact finiteCuntz_le_of_states hsimple hAU _ |
| (fun h => hN ((finitePositiveClass_eq_zero_iff L N).mp h)) _ hgap |
|
|
| theorem finiteCuntz_le_of_no_states (hsimple : TopologicallySimple A) |
| (hAU : ∀ (u v : CuntzClass (MatrixLimit A)) (k : ℕ), 1 ≤ k → |
| (k+1) • u ≤ k • v → u ≤ v) |
| (y : finiteCuntzSubmonoid (A := A)) (hy : y ≠ 0) |
| (hstates : IsEmpty (StateSpace y)) (x : finiteCuntzSubmonoid (A := A)) : x ≤ y := by |
| exact finiteCuntz_le_of_states hsimple hAU y hy x (fun f => (hstates.false f).elim) |
|
|
| theorem finiteCuntz_no_states_iff_properlyInfinite (hsimple : TopologicallySimple A) |
| (hAU : ∀ (u v : CuntzClass (MatrixLimit A)) (k : ℕ), 1 ≤ k → |
| (k+1) • u ≤ k • v → u ≤ v) |
| (y : finiteCuntzSubmonoid (A := A)) (hy : y ≠ 0) : |
| IsEmpty (StateSpace y) ↔ 2 • y ≤ y := |
| ⟨fun h => finiteCuntz_le_of_no_states hsimple hAU y hy h _, |
| no_state_of_properlyInfinite y⟩ |
|
|
| end Pureness |
| end |
| end |
|
|
| end OAI |
|
|
