| import OAI.Analysis.CStarAlgebra.PureInfiniteness.Signal |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open Filter |
| open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace |
|
|
| namespace MainGap |
| variable {A : Type*} [NonUnitalCStarAlgebra A] |
| local instance boundedDiagonalizationPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A |
| local instance boundedDiagonalizationStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A |
|
|
| theorem bounded_diagonalization (hP : PropertyP A) {a b : A} (ha : 0 ≤ a) (hb : 0 ≤ b) |
| (han : ‖a‖ ≤ 1) (hbn : ‖b‖ ≤ 1) (c : A) {δ ρ ζ : ℝ} |
| (hδ : 0 < δ) (hδhalf : δ < 1/2) (hρ : 0 < ρ) (hρδ : ρ ≤ 2*δ) (hζ : 0 < ζ) : |
| ∃ s t : A, ‖star s*a*s-a‖ < 2*δ+ρ ∧ ‖star t*b*t-b‖ < 2*δ+ρ ∧ |
| ‖star s*c*t‖ < ζ ∧ ‖s‖ ≤ Real.sqrt δ⁻¹ ∧ ‖t‖ ≤ Real.sqrt δ⁻¹ := by |
| let M := max 1 ‖c‖ |
| have hM1 : 1 ≤ M := le_max_left _ _ |
| have hM : 0 < M := lt_of_lt_of_le zero_lt_one hM1 |
| have hcn : ‖M⁻¹ • c‖ ≤ 1 := by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hM)] |
| calc M⁻¹ * ‖c‖ ≤ M⁻¹ * M := mul_le_mul_of_nonneg_left (le_max_right _ _) (inv_nonneg.mpr hM.le) |
| _ = 1 := inv_mul_cancel₀ hM.ne' |
| obtain ⟨p,q,κ,hp0,hpn,hpH,hq0,hqn,hqH,hκ0,hκζ,hcomp,hps,hqs⟩ := |
| iterated_separation hP (ramp_nonneg a (by linarith : δ ≤ 2*δ)) |
| (norm_ramp_le_one a (by linarith : δ ≤ 2*δ)) |
| (ramp_nonneg b (by linarith : δ ≤ 2*δ)) |
| (norm_ramp_le_one b (by linarith : δ ≤ 2*δ)) hcn |
| (by positivity : 0 < δ*ζ/M) |
| obtain ⟨w,hw,hsn⟩ := signal_coefficient hP ha han hδ hδhalf hρ hρδ hpH hps |
| obtain ⟨z,hz,htn⟩ := signal_coefficient hP hb hbn hδ hδhalf hρ hρδ hqH hqs |
| refine ⟨p*w,q*z,hw,hz,?_,hsn,htn⟩ |
| have hc : HerCompression p q c (M*κ) := by |
| intro x hx y hy |
| have hh := hcomp x hx y hy |
| rw [mul_smul_comm, smul_mul_assoc, norm_smul, Real.norm_eq_abs, |
| abs_of_pos (inv_pos.mpr hM)] at hh |
| have hh' := mul_le_mul_of_nonneg_left hh hM.le |
| calc ‖x*c*y‖ = M * (M⁻¹ * ‖x*c*y‖) := by rw [← mul_assoc, mul_inv_cancel₀ hM.ne', one_mul] |
| _ ≤ M*(κ*‖x‖*‖y‖) := hh' |
| _ = (M*κ)*‖x‖*‖y‖ := by ring |
| have hprod : ‖p*w‖*‖q*z‖ ≤ δ⁻¹ := by |
| calc ‖p*w‖*‖q*z‖ ≤ Real.sqrt δ⁻¹ * Real.sqrt δ⁻¹ := |
| mul_le_mul hsn htn (norm_nonneg _) (Real.sqrt_nonneg _) |
| _ = δ⁻¹ := Real.mul_self_sqrt (inv_nonneg.mpr hδ.le) |
| calc |
| ‖star (p*w)*c*(q*z)‖ ≤ (M*κ)*‖p*w‖*‖q*z‖ := |
| HerCompression.right_products (mul_nonneg hM.le hκ0) hp0 hq0 hc w z |
| _ = (M*κ)*(‖p*w‖*‖q*z‖) := by ring |
| _ ≤ (M*κ)*δ⁻¹ := mul_le_mul_of_nonneg_left hprod (mul_nonneg hM.le hκ0) |
| _ < ζ := by |
| rw [← div_eq_mul_inv] |
| apply (div_lt_iff₀ hδ).mpr |
| have hh := (lt_div_iff₀ hM).mp hκζ |
| nlinarith |
|
|
| end MainGap |
|
|
| namespace MainGap |
| variable {A : Type*} [NonUnitalCStarAlgebra A] |
| local instance diagonalizationPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A |
| local instance diagonalizationStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A |
|
|
| theorem individual_to_strong (hP : PropertyP A) : UnrestrictedDiagonalization A := by |
| intro a b ha hb c ε hε |
| let M := max 1 (max ‖a‖ ‖b‖) |
| have hM1 : 1 ≤ M := le_max_left _ _ |
| have hM : 0 < M := lt_of_lt_of_le zero_lt_one hM1 |
| have haM : ‖a‖ ≤ M := (le_max_left _ _).trans (le_max_right _ _) |
| have hbM : ‖b‖ ≤ M := (le_max_right _ _).trans (le_max_right _ _) |
| let δ := min (1/4 : ℝ) (ε/(8*M)) |
| have hδ : 0 < δ := lt_min (by norm_num) (div_pos hε (mul_pos (by norm_num) hM)) |
| have hδhalf : δ < 1/2 := (min_le_left _ _).trans_lt (by norm_num) |
| have hsmall : M*(2*δ+δ) < ε := by |
| have hh : δ ≤ ε/(8*M) := min_le_right _ _ |
| have hh' := (le_div_iff₀ (by positivity : 0 < 8*M)).mp hh |
| nlinarith |
| have hn {x : A} (hx : ‖x‖ ≤ M) : ‖M⁻¹ • x‖ ≤ 1 := by |
| rw [norm_smul, Real.norm_eq_abs, abs_of_pos (inv_pos.mpr hM)] |
| calc M⁻¹*‖x‖ ≤ M⁻¹*M := mul_le_mul_of_nonneg_left hx (inv_nonneg.mpr hM.le) |
| _ = 1 := inv_mul_cancel₀ hM.ne' |
| obtain ⟨s,t,hs,ht,hc,hsn,htn⟩ := bounded_diagonalization hP |
| (smul_nonneg (inv_nonneg.mpr hM.le) ha) (smul_nonneg (inv_nonneg.mpr hM.le) hb) |
| (hn haM) (hn hbM) c hδ hδhalf hδ (by linarith : δ ≤ 2*δ) hε |
| have hscale (x r : A) : M • (star r*(M⁻¹ • x)*r-(M⁻¹ • x)) = star r*x*r-x := by |
| simp only [mul_smul_comm, smul_mul_assoc, smul_sub, smul_smul, |
| mul_inv_cancel₀ hM.ne', one_smul] |
| refine ⟨s,t,?_,?_,hc⟩ |
| · rw [← hscale a s, norm_smul, Real.norm_eq_abs, abs_of_pos hM] |
| exact (mul_lt_mul_of_pos_left hs hM).trans hsmall |
| · rw [← hscale b t, norm_smul, Real.norm_eq_abs, abs_of_pos hM] |
| exact (mul_lt_mul_of_pos_left ht hM).trans hsmall |
|
|
| theorem individual_to_strong_spi (hP : PropertyP A) : StronglyPurelyInfinite A := by |
| intro x y hx hy ε hε |
| exact individual_to_strong hP (x*x) (y*y) |
| (by simpa only [hx.isSelfAdjoint.star_eq] using star_mul_self_nonneg x) |
| (by simpa only [hy.isSelfAdjoint.star_eq] using star_mul_self_nonneg y) (x*y) ε hε |
|
|
| end MainGap |
|
|
| end |
|
|
| end OAI |
|
|
