| import OAI.Analysis.CStarAlgebra.PureInfiniteness.Products |
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| namespace OAI |
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| noncomputable section |
| open Filter |
| open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace |
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| namespace MainGap |
| universe u |
| variable {A : Type u} [NonUnitalCStarAlgebra A] |
| local instance separationPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A |
| local instance separationStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A |
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| lemma ramp_mem_Her {a : A} (ha : 0 ≤ a) {l u : ℝ} (hl : 0 ≤ l) : |
| ramp a l u ∈ Her a := |
| Her_mono_of_mem (cut_mem_Her ha l) (ramp_mem_Her_cut ha hl) |
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| lemma NormSignal.cutoff_ramp {a p : A} (h : NormSignal a p) (hp : 0 ≤ p) : |
| NormSignal a (ramp p (1/2) 1) := by |
| intro R hR |
| let π := R.hom.toNonUnitalStarAlgHom.comp (Unitization.inrNonUnitalStarAlgHom ℂ A) |
| change ‖π (ramp p (1/2) 1)‖=1 |
| rw [map_ramp π hp (by norm_num : (0:ℝ)≤1/2)] |
| apply norm_ramp_eq_one (map_nonneg π hp) (by norm_num) (by norm_num) |
| exact (h R hR).symm.le |
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| lemma HerCompression.mono {p q c : A} {κ τ : ℝ} (h : HerCompression p q c κ) |
| (hκτ : κ ≤ τ) : HerCompression p q c τ := by |
| intro x hx k hk |
| exact (h x hx k hk).trans (by gcongr) |
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| lemma iteration_step {p q c : A} {r θ : ℝ} (hp : 0 ≤ p) (hq : 0 ≤ q) |
| (hr : 0 < r) (_h : HerCompression p q c r) |
| {p' q' : A} (hp'H : p' ∈ Her (ramp p (1/2) 1)) |
| (hq'H : q' ∈ Her (ramp q (1/2) 1)) |
| (h' : HerCompression p' q' |
| (r⁻¹ • (ramp p (1/4) (1/2)*c*ramp q (1/4) (1/2))) θ) : |
| HerCompression p' q' c (r*θ) := by |
| have hpg : ramp p (1/4) (1/2)*cut p (1/2)=cut p (1/2) := |
| ramp_mul_cut hp (by norm_num) (by norm_num) le_rfl |
| have hqg : ramp q (1/4) (1/2)*cut q (1/2)=cut q (1/2) := |
| ramp_mul_cut hq (by norm_num) (by norm_num) le_rfl |
| have hpg' : cut p (1/2)*ramp p (1/4) (1/2)=cut p (1/2) := by |
| simpa only [star_mul, (cut_nonneg p (1/2)).isSelfAdjoint.star_eq, |
| (ramp_nonneg p (by norm_num : (1/4:ℝ)≤1/2)).isSelfAdjoint.star_eq] using congrArg star hpg |
| intro x hx k hk |
| have hxcut : x ∈ Her (cut p (1/2)) := |
| Her_mono_of_mem (ramp_mem_Her_cut hp (by norm_num : (0:ℝ)≤1/2)) |
| (Her_mono_of_mem hp'H hx) |
| have hkcut : k ∈ Her (cut q (1/2)) := |
| Her_mono_of_mem (ramp_mem_Her_cut hq (by norm_num : (0:ℝ)≤1/2)) |
| (Her_mono_of_mem hq'H hk) |
| have hxg := Her_fixed_right (cut_nonneg p (1/2)).isSelfAdjoint hpg' hxcut |
| have hgk := Her_fixed_left hqg hkcut |
| have he : x*(r⁻¹ • (ramp p (1/4) (1/2)*c*ramp q (1/4) (1/2)))*k = |
| r⁻¹ • (x*c*k) := by |
| rw [mul_smul_comm, smul_mul_assoc] |
| congr 1 |
| simp only [mul_assoc, ← mul_assoc x (ramp p (1/4) (1/2)), hxg, hgk] |
| have hb := h' x hx k hk |
| rw [he, norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hr)] at hb |
| have hbr := mul_le_mul_of_nonneg_left hb hr.le |
| simpa only [← mul_assoc, mul_inv_cancel₀ hr.ne', one_mul] using hbr |
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| theorem iterated_separation (hp : PropertyP A) {a b c : A} (ha : 0 ≤ a) (han : ‖a‖ ≤ 1) |
| (hb : 0 ≤ b) (hbn : ‖b‖ ≤ 1) (hcn : ‖c‖ ≤ 1) {ζ : ℝ} (hζ : 0 < ζ) : |
| ∃ (p q : A) (κ : ℝ), 0 ≤ p ∧ ‖p‖ ≤ 1 ∧ p ∈ Her a ∧ |
| 0 ≤ q ∧ ‖q‖ ≤ 1 ∧ q ∈ Her b ∧ 0 ≤ κ ∧ κ < ζ ∧ |
| HerCompression p q c κ ∧ NormSignal a p ∧ NormSignal b q := by |
| obtain ⟨θ,hθ0,hθ1,huniform⟩ := uniform_shrinking hp |
| have hi : ∀ n : ℕ, ∃ p q : A, 0 ≤ p ∧ ‖p‖ ≤ 1 ∧ p ∈ Her a ∧ |
| 0 ≤ q ∧ ‖q‖ ≤ 1 ∧ q ∈ Her b ∧ |
| HerCompression p q c (θ^(n+1)) ∧ NormSignal a p ∧ NormSignal b q := by |
| intro n |
| induction n with |
| | zero => simpa only [Nat.zero_add, pow_one] using huniform a b c ha han hb hbn hcn |
| | succ n ih => |
| obtain ⟨p,q,hp0,hpn,hpH,hq0,hqn,hqH,hcomp,hps,hqs⟩ := ih |
| let f : A → A := fun p => ramp p (1/2) 1 |
| let g : A → A := fun p => ramp p (1/4) (1/2) |
| let r := θ^(n+1) |
| have hr : 0 < r := pow_pos hθ0 _ |
| have hgH (p : A) (hp : 0 ≤ p) : g p ∈ Her p := ramp_mem_Her hp (by norm_num) |
| have hgn (p : A) : ‖g p‖ ≤ 1 := norm_ramp_le_one p (by norm_num) |
| have hf0 (p : A) : 0 ≤ f p := ramp_nonneg p (by norm_num) |
| have hfn (p : A) : ‖f p‖ ≤ 1 := norm_ramp_le_one p (by norm_num) |
| have hc' : ‖r⁻¹ • (g p*c*g q)‖ ≤ 1 := by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hr)] |
| have hh : ‖g p*c*g q‖ ≤ r := (hcomp _ (hgH p hp0) _ (hgH q hq0)).trans (by |
| calc r*‖g p‖*‖g q‖ ≤ r*1*1 := by gcongr <;> apply hgn |
| _ = r := by ring) |
| calc r⁻¹*‖g p*c*g q‖ ≤ r⁻¹*r := mul_le_mul_of_nonneg_left hh (inv_nonneg.mpr hr.le) |
| _ = 1 := inv_mul_cancel₀ hr.ne' |
| obtain ⟨p',q',hp'0,hp'n,hp'H,hq'0,hq'n,hq'H,hcomp',hps',hqs'⟩ := |
| huniform (f p) (f q) (r⁻¹ • (g p*c*g q)) (hf0 p) (hfn p) (hf0 q) (hfn q) hc' |
| refine ⟨p',q',hp'0,hp'n,?_,hq'0,hq'n,?_,?_,?_,?_⟩ |
| · exact Her_mono_of_mem hpH (Her_mono_of_mem (ramp_mem_Her hp0 (by norm_num)) hp'H) |
| · exact Her_mono_of_mem hqH (Her_mono_of_mem (ramp_mem_Her hq0 (by norm_num)) hq'H) |
| · have hh := iteration_step hp0 hq0 hr hcomp hp'H hq'H hcomp' |
| simpa only [r, pow_succ] using hh |
| · intro R hR |
| exact hps' R (by rw [hps.cutoff_ramp hp0 R hR]; norm_num) |
| · intro R hR |
| exact hqs' R (by rw [hqs.cutoff_ramp hq0 R hR]; norm_num) |
| obtain ⟨n,hn⟩ := exists_pow_lt_of_lt_one hζ hθ1 |
| obtain ⟨p,q,hp0,hpn,hpH,hq0,hqn,hqH,hcomp,hps,hqs⟩ := hi n |
| refine ⟨p,q,θ^(n+1),hp0,hpn,hpH,hq0,hqn,hqH,(pow_pos hθ0 _).le,?_,hcomp,hps,hqs⟩ |
| exact (by rw [pow_succ]; nlinarith [pow_pos hθ0 n] : θ^(n+1)≤θ^n).trans_lt hn |
| end MainGap |
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| end |
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| end OAI |
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