File size: 3,984 Bytes
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 | import Mathlib
namespace OAI
noncomputable section
open Filter
open scoped Topology CStarAlgebra ComplexOrder InnerProductSpace
namespace MainGap
variable {A : Type*} [CStarAlgebra A]
variable [PartialOrder A] [StarOrderedRing A]
lemma norm_add_one_of_nonneg [Nontrivial A] {d : A} (hd : 0 ≤ d) :
‖d + 1‖ = ‖d‖ + 1 := by
apply le_antisymm
· simpa using norm_add_le d 1
have hh := norm_apply_le_norm_cfc (fun t : ℝ => t + 1) d
(CStarAlgebra.norm_mem_spectrum_of_nonneg d hd)
have he : cfc (fun t : ℝ => t + 1) d = d + 1 := by
rw [cfc_add_const (1 : ℝ) (fun t : ℝ => t) d, cfc_id' ℝ d, map_one]
simpa only [he, Real.norm_eq_abs, abs_of_nonneg (by positivity : 0 ≤ ‖d‖ + 1)] using hh
lemma real_norm_one_positive (F : A →L[ℝ] ℝ) (hF : ‖F‖ ≤ 1) (h1 : F 1 = 1)
{h : A} (hh : 0 ≤ h) : 0 ≤ F h := by
have hu : h ≤ algebraMap ℝ A ‖h‖ :=
IsSelfAdjoint.le_algebraMap_norm_self h (.of_nonneg hh)
have hp : 0 ≤ algebraMap ℝ A ‖h‖ - h := sub_nonneg.mpr hu
have hb : ‖algebraMap ℝ A ‖h‖ - h‖ ≤ ‖h‖ :=
(CStarAlgebra.norm_le_iff_le_algebraMap _ (norm_nonneg _) hp).2 (sub_le_self _ hh)
have he := F.le_opNorm (algebraMap ℝ A ‖h‖ - h)
have hx : F (algebraMap ℝ A ‖h‖ - h) = ‖h‖ - F h := by
rw [map_sub, Algebra.algebraMap_eq_smul_one, map_smul, h1, smul_eq_mul, mul_one]
have : F (algebraMap ℝ A ‖h‖ - h) ≤ ‖h‖ :=
(le_abs_self _).trans (he.trans ((mul_le_mul_of_nonneg_right hF (norm_nonneg _)).trans
(by simpa using hb)))
linarith
def realPartCLM : A →L[ℝ] A :=
((selfAdjoint.submodule ℝ A).subtype.comp (realPart : A →ₗ[ℝ] selfAdjoint A)).mkContinuous 1
(fun a => by
change ‖(realPart a : A)‖ ≤ 1 * ‖a‖
simpa using realPart.norm_le a)
omit [PartialOrder A] [StarOrderedRing A] in
@[simp] lemma realPartCLM_apply (a : A) : realPartCLM a = (realPart a : A) := rfl
omit [PartialOrder A] [StarOrderedRing A] in
lemma norm_realPartCLM : ‖(realPartCLM : A →L[ℝ] A)‖ ≤ 1 := by
apply ContinuousLinearMap.opNorm_le_bound _ zero_le_one
intro a
simpa using realPart.norm_le a
lemma exists_norming_positive [Nontrivial A] {d : A} (hd : 0 ≤ d) :
∃ f : A →ₚ[ℂ] ℂ, (∀ a, ‖f a‖ ≤ ‖a‖) ∧ f d = (‖d‖ : ℂ) := by
obtain ⟨F, hF, hFd⟩ := exists_dual_vector'' ℝ (d + 1)
have hFa (a : A) : F a ≤ ‖a‖ :=
(le_abs_self _).trans ((F.le_opNorm a).trans
(by simpa using mul_le_mul_of_nonneg_right hF (norm_nonneg a)))
have h1 : F 1 = 1 := by
rw [map_add, norm_add_one_of_nonneg hd] at hFd
simp only [RCLike.ofReal_real_eq_id, id_eq] at hFd
have h1' := hFa 1
have hd' := hFa d
simp only [norm_one] at h1'
linarith
have hFd' : F d = ‖d‖ := by
rw [map_add, norm_add_one_of_nonneg hd, h1] at hFd
simpa only [RCLike.ofReal_real_eq_id, id_eq, add_left_inj] using hFd
let G : A →L[ℝ] ℝ := F.comp realPartCLM
let H : A →L[ℂ] ℂ := StrongDual.extendRCLike G
have hG : ‖G‖ ≤ 1 := (ContinuousLinearMap.opNorm_comp_le _ _).trans
((mul_le_mul hF norm_realPartCLM (norm_nonneg _) zero_le_one).trans (by norm_num))
have hH (a : A) (ha : IsSelfAdjoint a) : H a = (F a : ℂ) := by
simp [H, StrongDual.extendRCLike_apply, G, ha.coe_realPart, realPart_I_smul,
ha.imaginaryPart]
let f : A →ₚ[ℂ] ℂ :=
{ H.toLinearMap with
monotone' := fun a b hab => by
have hn : 0 ≤ H (b - a) := by
rw [hH _ (.of_nonneg (sub_nonneg.mpr hab))]
exact_mod_cast real_norm_one_positive F hF h1 (sub_nonneg.mpr hab)
change H a ≤ H b
simpa only [map_sub, sub_nonneg] using hn }
refine ⟨f, fun a => ?_, ?_⟩
· change ‖H a‖ ≤ ‖a‖
exact (H.le_opNorm a).trans (by
simpa [H] using mul_le_mul_of_nonneg_right hG (norm_nonneg a))
· change H d = _
rw [hH d (.of_nonneg hd), hFd']
end MainGap
end
end OAI
|