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ddc84c5 | 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 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 | 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
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