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namespace OAI
noncomputable section
open Filter
open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace
namespace MainGap
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance signalCutPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance signalCutStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
lemma Her_cut_lower {a : A} (ha : 0 ≤ a) {δ : ℝ} (hδ : 0 ≤ δ)
{p : A} (hp : p ∈ Her (cut a δ)) :
δ • (star p * p) ≤ star p * a * p := by
have hc : IsClosed {p : A | δ • (star p * p) ≤ star p * a * p} :=
isClosed_le (by fun_prop) (by fun_prop)
apply closure_minimal (s := Set.range (fun w : A => cut a δ * w * star (cut a δ))) _ hc hp
rintro _ ⟨w, rfl⟩
change δ • (star (cut a δ * w * star (cut a δ)) *
(cut a δ * w * star (cut a δ))) ≤
star (cut a δ * w * star (cut a δ)) * a * (cut a δ * w * star (cut a δ))
have hh := star_left_conjugate_le_conjugate (cut_sandwich_lower ha.isSelfAdjoint hδ)
(w * cut a δ)
simpa only [star_mul, (cut_nonneg a δ).isSelfAdjoint.star_eq,
mul_smul_comm, smul_mul_assoc, mul_assoc] using hh
lemma norm_cut_le_sub {a : A} (ha : 0 ≤ a) {M δ : ℝ} (han : ‖a‖ ≤ M)
(hδ : δ ≤ M) : ‖cut a δ‖ ≤ M - δ := by
apply norm_cfcₙ_le
intro t ht
have htM : t ≤ M := (le_abs_self t).trans
((NonUnitalIsometricContinuousFunctionalCalculus.norm_quasispectrum_le a ht ha.isSelfAdjoint).trans han)
rw [Real.norm_eq_abs, abs_of_nonneg (le_max_right _ _)]
exact max_le (by linarith) (sub_nonneg.mpr hδ)
lemma cut_eq_zero_of_norm_le_nonunital {a : A} (ha : 0 ≤ a) {δ : ℝ} (haδ : ‖a‖ ≤ δ) :
cut a δ = 0 := by
apply norm_eq_zero.mp
exact le_antisymm (by simpa using norm_cut_le_sub ha haδ le_rfl) (norm_nonneg _)
end MainGap
namespace MainGap
universe u
variable {A : Type u} [NonUnitalCStarAlgebra A]
local instance signalComparisonPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance signalComparisonStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
local instance signalOperatorPartialOrder (H : Type*) [NormedAddCommGroup H]
[InnerProductSpace ℂ H] [CompleteSpace H] :
PartialOrder (H →L[ℂ] H) := CStarAlgebra.spectralOrder _
local instance signalOperatorStarOrderedRing (H : Type*) [NormedAddCommGroup H]
[InnerProductSpace ℂ H] [CompleteSpace H] :
StarOrderedRing (H →L[ℂ] H) := CStarAlgebra.spectralOrderedRing _
lemma signal_cut_comparison (hP : PropertyP A) {a p : A} (ha : 0 ≤ a) {δ : ℝ}
(hδ : 0 < δ) (hpH : p ∈ Her (ramp a δ (2*δ)))
(hsignal : NormSignal (ramp a δ (2*δ)) p) :
ScalarCuntzLE (cut a (2*δ)) (star p * a * p) := by
let v := star p * a * p
have hv : 0 ≤ v := star_left_conjugate_nonneg ha p
let V : Unitization ℂ A := Unitization.inr v
have hV : 0 ≤ V := Unitization.inr_nonneg_iff.mpr hv
let J := closedIdeal V hV.isSelfAdjoint
have hcut : (Unitization.inr (cut a (2*δ)) : Unitization ℂ A) ∈ J := by
by_contra hh
obtain ⟨R,hRJ,hRcut⟩ := exists_rep_killing_ideal J
(closedIdeal_closed V hV.isSelfAdjoint)
(fun x z hz => closedIdeal_left hV.isSelfAdjoint hz x)
(fun x z hz => closedIdeal_right hV.isSelfAdjoint hz x) hh
let π := R.hom.toNonUnitalStarAlgHom.comp (Unitization.inrNonUnitalStarAlgHom ℂ A)
have hvzero : π v = 0 := hRJ V (self_mem_closedIdeal V hV.isSelfAdjoint)
have hlower := Her_cut_lower ha hδ.le
(Her_mono_of_mem (ramp_mem_Her_cut ha hδ.le) hpH)
have hpos := map_nonneg π (sub_nonneg.mpr hlower)
have hsmul (r : ℝ) (z : A) : π (r • z)=r • π z :=
LinearMapClass.map_smul_of_tower π r z
change 0 ≤ π (v - δ • (star p * p)) at hpos
rw [map_sub, hvzero, hsmul, map_mul, map_star] at hpos
have hz : δ • (star (π p) * π p) = 0 :=
le_antisymm (by simpa only [zero_sub, neg_nonneg] using hpos)
(smul_nonneg hδ.le (star_mul_self_nonneg _))
have hpzero : π p = 0 := (CStarRing.star_mul_self_eq_zero_iff _).mp
((smul_eq_zero.mp hz).resolve_left hδ.ne')
have hnorm : 2*δ < ‖π a‖ := by
by_contra! hn
apply hRcut
change π (cut a (2*δ)) = 0
rw [map_cut π ha (by positivity), cut_eq_zero_of_norm_le (map_nonneg π ha) hn]
have hramp : ‖π (ramp a δ (2*δ))‖ = 1 := by
rw [map_ramp π ha hδ.le]
exact norm_ramp_eq_one (map_nonneg π ha) hδ.le (by linarith) hnorm.le
have hpn := hsignal R (by change (1/2:ℝ) < ‖π (ramp a δ (2*δ))‖; rw [hramp]; norm_num)
change ‖π p‖ = 1 at hpn
rw [hpzero, norm_zero] at hpn
norm_num at hpn
apply scalarCuntzLE_of_inr hv
exact closedIdeal_comparison (Unitization.inr_nonneg_iff.mpr (cut_nonneg _ _)) hV
((hP v hv).map (Unitization.inrNonUnitalStarAlgHom ℂ A)) hcut
end MainGap
namespace MainGap
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance signalCoefficientPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance signalCoefficientStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
lemma signal_coefficient (hP : PropertyP A) {a p : A} (ha : 0 ≤ a) (han : ‖a‖ ≤ 1)
{δ ρ : ℝ} (hδ : 0 < δ) (hδhalf : δ < 1/2) (hρ : 0 < ρ) (hρδ : ρ ≤ 2*δ)
(hpH : p ∈ Her (ramp a δ (2*δ))) (hsignal : NormSignal (ramp a δ (2*δ)) p) :
∃ w : A, ‖star (p*w)*a*(p*w)-a‖ < 2*δ+ρ ∧
‖p*w‖ ≤ Real.sqrt δ⁻¹ := by
have hcomp := signal_cut_comparison hP ha hδ hpH hsignal
obtain ⟨w,hw⟩ := hcomp.exists_approx hρ
have hw' : ‖star (p*w)*a*(p*w) - cut a (2*δ)‖ < ρ := by
simpa only [star_mul, mul_assoc] using hw
refine ⟨w, ?_, ?_⟩
· calc
‖star (p*w)*a*(p*w)-a‖ ≤
‖star (p*w)*a*(p*w)-cut a (2*δ)‖ + ‖cut a (2*δ)-a‖ := by simpa only [dist_eq_norm] using dist_triangle (star (p*w)*a*(p*w)) (cut a (2*δ)) a
_ < ρ + 2*δ := add_lt_add_of_lt_of_le hw'
(by simpa only [norm_sub_rev] using norm_sub_cut_le ha (by positivity : 0 ≤ 2*δ))
_ = 2*δ+ρ := by ring
· have hlower := star_left_conjugate_le_conjugate
(Her_cut_lower ha hδ.le (Her_mono_of_mem (ramp_mem_Her_cut ha hδ.le) hpH)) w
have hlower' : δ • (star (p*w)*(p*w)) ≤ star (p*w)*a*(p*w) := by
simpa only [star_mul, mul_smul_comm, smul_mul_assoc, mul_assoc] using hlower
have hn := CStarAlgebra.norm_le_norm_of_le_of_nonneg hlower'
(smul_nonneg hδ.le (star_mul_self_nonneg (p*w)))
simp only [norm_smul, Real.norm_eq_abs, abs_of_pos hδ, CStarRing.norm_star_mul_self] at hn
have hupper : ‖star (p*w)*a*(p*w)‖ < 1 := by
calc
‖star (p*w)*a*(p*w)‖ ≤
‖star (p*w)*a*(p*w)-cut a (2*δ)‖ + ‖cut a (2*δ)‖ := norm_le_norm_sub_add _ _
_ < ρ + (1-2*δ) := add_lt_add_of_lt_of_le hw'
(norm_cut_le_sub ha han (by linarith))
_ ≤ 1 := by linarith
have hsq : ‖p*w‖ ^ 2 ≤ δ⁻¹ := by
have hh : δ * (‖p*w‖ ^ 2) ≤ 1 := by nlinarith
have hh' := (le_inv_mul_iff₀ hδ).mpr hh
simpa only [mul_one] using hh'
have hsqrt := Real.sq_sqrt (inv_nonneg.mpr hδ.le)
nlinarith [Real.sqrt_nonneg δ⁻¹, norm_nonneg (p*w)]
end MainGap
namespace MainGap
variable {A : Type*} [NonUnitalCStarAlgebra A]
local instance signalProductsPartialOrder : PartialOrder A := CStarAlgebra.spectralOrder A
local instance signalProductsStarOrderedRing : StarOrderedRing A := CStarAlgebra.spectralOrderedRing A
lemma HerCompression.right_products {p q c : A} {κ : ℝ} (hκ : 0 ≤ κ)
(hp : 0 ≤ p) (hq : 0 ≤ q) (h : HerCompression p q c κ) (w z : A) :
‖star (p*w) * c * (q*z)‖ ≤ κ * ‖p*w‖ * ‖q*z‖ := by
let : PartialOrder (Her p) := CStarAlgebra.spectralOrder _
let : StarOrderedRing (Her p) := CStarAlgebra.spectralOrderedRing _
let : PartialOrder (Her q) := CStarAlgebra.spectralOrder _
let : StarOrderedRing (Her q) := CStarAlgebra.spectralOrderedRing _
let ap := CStarAlgebra.increasingApproximateUnit (Her p)
let aq := CStarAlgebra.increasingApproximateUnit (Her q)
let pp : Her p := ⟨p, self_mem_Her hp⟩
let qq : Her q := ⟨q, self_mem_Her hq⟩
have htp : Tendsto (fun e : Her p => (e : A) * (p*w))
(CStarAlgebra.approximateUnit (Her p)) (𝓝 (p*w)) := by
have hh := ((continuous_subtype_val.tendsto pp).comp (ap.tendsto_mul_right pp)).mul
(tendsto_const_nhds (x := w))
change Tendsto (fun e : Her p => ((e : A) * p) * w) _ (𝓝 (p*w)) at hh
simpa only [mul_assoc] using hh
have htq : Tendsto (fun e : Her q => (e : A) * (q*z))
(CStarAlgebra.approximateUnit (Her q)) (𝓝 (q*z)) := by
have hh := ((continuous_subtype_val.tendsto qq).comp (aq.tendsto_mul_right qq)).mul
(tendsto_const_nhds (x := z))
change Tendsto (fun e : Her q => ((e : A) * q) * z) _ (𝓝 (q*z)) at hh
simpa only [mul_assoc] using hh
have ht := ((htp.comp tendsto_fst).star.mul (tendsto_const_nhds (x := c))).mul
(htq.comp tendsto_snd)
apply le_of_tendsto ht.norm
filter_upwards [ap.eventually_norm.prod_mk aq.eventually_norm] with ef hef
rcases ef with ⟨e,f⟩
change ‖star ((e : A)*(p*w))*c*((f : A)*(q*z))‖ ≤ _
have he : ‖(e : A)‖ ≤ 1 := hef.1
have hf : ‖(f : A)‖ ≤ 1 := hef.2
calc
‖star ((e : A)*(p*w))*c*((f : A)*(q*z))‖ =
‖star (p*w) * (star (e : A)*c*(f : A)) * (q*z)‖ := by
simp only [star_mul, mul_assoc]
_ ≤ ‖p*w‖ * ‖star (e : A)*c*(f : A)‖ * ‖q*z‖ := by
grw [norm_mul_le, norm_mul_le, norm_star]
_ ≤ ‖p*w‖ * (κ*‖(e : A)‖*‖(f : A)‖) * ‖q*z‖ := by
gcongr
simpa only [norm_star] using h (star (e : A)) (star_mem e.property) (f : A) f.property
_ ≤ ‖p*w‖ * (κ*1*1) * ‖q*z‖ := by gcongr
_ = κ * ‖p*w‖ * ‖q*z‖ := by ring
end MainGap
end
end OAI
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