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9eeb9c7 | 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 | 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
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