openAI-math-dataset / lean /OAI /Analysis /IrrationalRotation /Operators /NormalTraceProjectionRigidity.lean
| import OAI.Analysis.IrrationalRotation.Model.NormalAffiliation | |
| import Mathlib.Algebra.Star.StarProjection | |
| import Mathlib.Tactic | |
| namespace OAI | |
| noncomputable section | |
| namespace Invariant233.NormalTraceProjectionRigidity | |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] | |
| [CompleteSpace H] {M : VonNeumannAlgebra H} | |
| theorem trace_im_eq_zero (τ : TracialState M) {p : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) : (τ p).im = 0 := by | |
| have hstar : star p = p := Subtype.ext hp.isSelfAdjoint.star_eq | |
| have heq : star (τ p) = τ p := (τ.map_star p).symm.trans (congrArg τ hstar) | |
| have him := congrArg Complex.im heq | |
| simp only [Complex.star_def, Complex.conj_im] at him | |
| linarith | |
| theorem trace_re_nonneg (τ : TracialState M) {p : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) : 0 ≤ (τ p).re := by | |
| have hstar : star p = p := Subtype.ext hp.isSelfAdjoint.star_eq | |
| have hmul : p * p = p := Subtype.ext hp.isIdempotentElem.eq | |
| simpa only [hstar, hmul] using τ.positive p | |
| theorem trace_re_eq_zero_iff (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) {p : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) : (τ p).re = 0 ↔ p = 0 := by | |
| constructor | |
| · intro hz | |
| apply hτ.1 p (ContinuousLinearMap.IsPositive.of_isStarProjection hp) | |
| exact Complex.ext hz (trace_im_eq_zero τ hp) | |
| · rintro rfl | |
| simp only [map_zero, Complex.zero_re] | |
| theorem trace_mono (τ : TracialState M) {p q : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) | |
| (hq : IsStarProjection (q : H →L[ℂ] H)) | |
| (hpq : (p : H →L[ℂ] H) * (q : H →L[ℂ] H) = p) : | |
| (τ p).re ≤ (τ q).re := by | |
| have hdiff : IsStarProjection ((q - p : M) : H →L[ℂ] H) := | |
| hp.sub_of_mul_eq_left hq hpq | |
| have hnonneg := trace_re_nonneg τ hdiff | |
| change 0 ≤ (τ.toContinuousLinearMap (q - p)).re at hnonneg | |
| rw [map_sub, Complex.sub_re] at hnonneg | |
| exact sub_nonneg.mp hnonneg | |
| theorem trace_re_eq_iff_of_nested (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) {p q : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) | |
| (hq : IsStarProjection (q : H →L[ℂ] H)) | |
| (hpq : (p : H →L[ℂ] H) * (q : H →L[ℂ] H) = p) : | |
| (τ p).re = (τ q).re ↔ p = q := by | |
| constructor | |
| · intro heq | |
| have hdiff : IsStarProjection ((q - p : M) : H →L[ℂ] H) := | |
| hp.sub_of_mul_eq_left hq hpq | |
| have hz : (τ (q - p)).re = 0 := by | |
| rw [map_sub, Complex.sub_re, heq, sub_self] | |
| exact (sub_eq_zero.mp ((trace_re_eq_zero_iff τ hτ hdiff).mp hz)).symm | |
| · rintro rfl | |
| rfl | |
| theorem trace_re_eq_one_iff (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) {p : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) : (τ p).re = 1 ↔ p = 1 := by | |
| have hone : (τ (1 : M)).re = 1 := by rw [τ.map_one]; rfl | |
| rw [← hone] | |
| exact trace_re_eq_iff_of_nested τ hτ hp (IsStarProjection.one _) (mul_one _) | |
| theorem trace_mem_Ioo_iff (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) {p : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) : | |
| (τ p).re ∈ Set.Ioo (0 : ℝ) 1 ↔ p ≠ 0 ∧ p ≠ 1 := by | |
| have hlo := trace_re_nonneg τ hp | |
| have hhi : (τ p).re ≤ 1 := by | |
| simpa only [τ.map_one, Complex.one_re] using | |
| trace_mono τ (q := (1 : M)) hp (IsStarProjection.one _) (mul_one (p : H →L[ℂ] H)) | |
| constructor | |
| · intro h | |
| exact ⟨fun hz => (ne_of_gt h.1) ((trace_re_eq_zero_iff τ hτ hp).mpr hz), | |
| fun ho => (ne_of_lt h.2) ((trace_re_eq_one_iff τ hτ hp).mpr ho)⟩ | |
| · rintro ⟨hz, ho⟩ | |
| exact ⟨lt_of_le_of_ne hlo (Ne.symm (fun h => hz ((trace_re_eq_zero_iff τ hτ hp).mp h))), | |
| lt_of_le_of_ne hhi (fun h => ho ((trace_re_eq_one_iff τ hτ hp).mp h))⟩ | |
| theorem nontrivial_iff_of_trace_eq (τ : TracialState M) | |
| (hτ : IsNormalFaithfulTracialState M τ) {p q : M} | |
| (hp : IsStarProjection (p : H →L[ℂ] H)) | |
| (hq : IsStarProjection (q : H →L[ℂ] H)) (htrace : τ p = τ q) : | |
| (p ≠ 0 ∧ p ≠ 1) ↔ (q ≠ 0 ∧ q ≠ 1) := by | |
| rw [← trace_mem_Ioo_iff τ hτ hp, ← trace_mem_Ioo_iff τ hτ hq, htrace] | |
| end Invariant233.NormalTraceProjectionRigidity | |
| end | |
| end OAI | |