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f1f7953 | 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 58 59 60 | import Mathlib
import OAI.Analysis.TracialSplitting.PreparedBlocks
import OAI.Analysis.JiangSu.CStarAlgebras.ContinuousPositiveEmbedding
namespace OAI
noncomputable section
namespace Pureness
open TraceCone Set Filter
open scoped ComplexOrder NNReal Topology
variable {X : Type*} [TopologicalSpace X] [CompactSpace X]
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
end OAI
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