| import Mathlib |
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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 |
| variable {A : Type*} [CStarAlgebra A] |
| variable [PartialOrder A] [StarOrderedRing A] |
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| 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 |
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| 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 |
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| 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) |
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| omit [PartialOrder A] [StarOrderedRing A] in |
| @[simp] lemma realPartCLM_apply (a : A) : realPartCLM a = (realPart a : A) := rfl |
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| 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 |
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| 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 |
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| end |
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| end OAI |
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