| import Mathlib |
| import OAI.Analysis.TracialSplitting.PreparedBlocks |
| import OAI.Analysis.JiangSu.CStarAlgebras.ContinuousPositiveEmbedding |
|
|
| namespace OAI |
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|
| noncomputable section |
| namespace Pureness |
| open TraceCone Set Filter |
| open scoped ComplexOrder NNReal Topology |
|
|
| variable {X : Type*} [TopologicalSpace X] [CompactSpace X] |
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|
| theorem continuousNNRealCut_le_mul_of_support (f g : C(X, ℝ≥0)) |
| (hfg : ∀ x, 0 < f x → 0 < g x) (ε : ℝ≥0) (hε : 0 < ε) : |
| ∃ C : ℝ≥0, ∀ x, f x-ε ≤ C*g x := by |
| let K : Set X := {x | ε ≤ f x} |
| have hK : IsCompact K := (isClosed_le continuous_const f.continuous).isCompact |
| by_cases hne : K.Nonempty |
| · obtain ⟨x,hx,hmin⟩ := hK.exists_isMinOn hne g.continuous.continuousOn |
| have hgx : 0 < g x := hfg x (hε.trans_le hx) |
| obtain ⟨C,hC⟩ := (isCompact_range f.continuous).bddAbove |
| refine ⟨C/g x, ?_⟩ |
| intro y |
| by_cases hy : y ∈ K |
| · have hgy : g x ≤ g y := hmin hy |
| calc |
| f y-ε ≤ f y := tsub_le_self |
| _ ≤ C := hC (mem_range_self y) |
| _ = (C/g x)*g x := (div_mul_cancel₀ C hgx.ne').symm |
| _ ≤ (C/g x)*g y := mul_le_mul_of_nonneg_left hgy bot_le |
| · rw [tsub_eq_zero_of_le (le_of_not_ge hy)] |
| exact bot_le |
| · refine ⟨0,?_⟩ |
| intro x |
| have hx : f x ≤ ε := le_of_not_ge (fun h => hne ⟨x,h⟩) |
| simp only [tsub_eq_zero_of_le hx,zero_mul,le_refl] |
|
|
| theorem continuousNNRealPositive_cuntzBelow_of_support (f g : C(X, ℝ≥0)) |
| (hfg : ∀ x, 0 < f x → 0 < g x) : |
| CuntzBelow (continuousNNRealPositive f) (continuousNNRealPositive g) := by |
| apply CuntzBelow.of_tendsto (fun k => Positive.cut (Positive.rootExponent k) (continuousNNRealPositive f)) |
| · apply Positive.tendsto_cut |
| simpa [Positive.rootExponent] using (tendsto_one_div_add_atTop_nhds_zero_nat : |
| Tendsto (fun k : ℕ => 1/((k : ℝ)+1)) atTop (𝓝 0)) |
| · intro k |
| obtain ⟨C,hC⟩ := continuousNNRealCut_le_mul_of_support f g hfg (Positive.rootExponent k) (Positive.rootExponent_pos k) |
| apply CuntzBelow.of_le_smul _ _ C C.2 |
| rw [← continuousNNRealPositive_cut] |
| intro x |
| change ((((f x-Positive.rootExponent k : ℝ≥0) : ℝ) : ℂ)) ≤ (C : ℝ) • (((g x : ℝ) : ℂ)) |
| rw [Complex.real_smul] |
| exact_mod_cast hC x |
|
|
| end Pureness |
| end |
|
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| end OAI |
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