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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 99 100 101 102 103 104 105 106 107 108 | import OAI.Analysis.IrrationalRotation.Operators.VectorTraceNormality
import OAI.Analysis.IrrationalRotation.Operators.VectorProjectionNormality
import OAI.Analysis.IrrationalRotation.Operators.AffiliatedTransportSupplement
namespace OAI
noncomputable section
open Filter
open scoped Topology
namespace Invariant233
namespace TracialState
variable {A : Type*} [CStarAlgebra A]
theorem apply_selfAdjoint_eq_re (τ : TracialState A) {a : A}
(ha : IsSelfAdjoint a) : τ a = ((τ a).re : ℂ) := by
apply (Complex.conj_eq_iff_re.mp ?_).symm
change star (τ a) = τ a
rw [← τ.map_star, ha.star_eq]
theorem projection_nonneg (τ : TracialState A) {p : A}
(hp : IsStarProjection p) : 0 ≤ (τ p).re := by
have h := τ.positive p
rwa [hp.isSelfAdjoint.star_eq, hp.isIdempotentElem.eq] at h
theorem projection_mono (τ : TracialState A) {p q : A}
(hp : IsStarProjection p) (hq : IsStarProjection q) (hpq : p * q = p) :
(τ p).re ≤ (τ q).re := by
have h := τ.projection_nonneg (hp.sub_of_mul_eq_left hq hpq)
change 0 ≤ (τ.toContinuousLinearMap (q - p)).re at h
rw [map_sub, Complex.sub_re] at h
linarith
theorem projection_re_mem_Icc (τ : TracialState A) {p : A}
(hp : IsStarProjection p) : (τ p).re ∈ Set.Icc (0 : ℝ) 1 := by
refine ⟨τ.projection_nonneg hp, ?_⟩
have h := τ.projection_mono hp (IsStarProjection.one A) (mul_one p)
simpa only [τ.map_one, Complex.one_re] using h
end TracialState
universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H]
namespace NormalTracialProjectionTransport
theorem tendsto_trace_projection_exhaustion
(M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
{ι : Type} [Preorder ι] [NeBot (atTop : Filter ι)]
(K : ι → Submodule ℂ H) [∀ i, (K i).HasOrthogonalProjection]
(hK : Monotone K) (hmem : ∀ i, (K i).starProjection ∈ M)
(hdense : (⨆ i, K i).topologicalClosure = ⊤) :
Tendsto (fun i => τ ⟨(K i).starProjection, hmem i⟩) atTop (𝓝 1) := by
have h := hτ.tendsto_of_strong_small atTop
(fun i => ⟨(K i).starProjection, hmem i⟩) 1
⟨1, fun i => (K i).starProjection_norm_le⟩ (fun x => ?_)
· simpa only [τ.map_one] using h
· simpa only [OneMemClass.coe_one, one_apply_eq_self] using
Submodule.starProjection_tendsto_self K hK x (by rw [hdense])
theorem tendsto_trace_projection_exhaustion_re
(M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
{ι : Type} [Preorder ι] [NeBot (atTop : Filter ι)]
(K : ι → Submodule ℂ H) [∀ i, (K i).HasOrthogonalProjection]
(hK : Monotone K) (hmem : ∀ i, (K i).starProjection ∈ M)
(hdense : (⨆ i, K i).topologicalClosure = ⊤) :
Tendsto (fun i => (τ ⟨(K i).starProjection, hmem i⟩).re) atTop (𝓝 1) := by
simpa only [Function.comp_def, Complex.one_re] using
Complex.continuous_re.continuousAt.tendsto.comp
(tendsto_trace_projection_exhaustion M τ hτ K hK hmem hdense)
theorem trace_eq_of_cutoff_estimates
(M : VonNeumannAlgebra H) (τ : TracialState M)
(hτ : IsNormalFaithfulTracialState M τ)
(p q : M) (hp : IsStarProjection p) (hq : IsStarProjection q)
(K : ℕ → Submodule ℂ H) [∀ n, (K n).HasOrthogonalProjection]
(hK : Monotone K) (hmem : ∀ n, (K n).starProjection ∈ M)
(hdense : (⨆ n, K n).topologicalClosure = ⊤)
(hforward : ∀ n, (τ p).re + (τ ⟨(K n).starProjection, hmem n⟩).re - 1 ≤
(τ q).re)
(hbackward : ∀ n, (τ (1 - p)).re +
(τ ⟨(K n).starProjection, hmem n⟩).re - 1 ≤ (τ (1 - q)).re) :
τ p = τ q := by
have he : Tendsto (fun n : ℕ => (τ ⟨(K n).starProjection, hmem n⟩).re)
atTop (𝓝 1) :=
tendsto_trace_projection_exhaustion_re M τ hτ (ι := ℕ) K hK hmem hdense
have hbackward' : ∀ n, (1 - (τ p).re) +
(τ ⟨(K n).starProjection, hmem n⟩).re - 1 ≤ 1 - (τ q).re := by
intro n
have h := hbackward n
change (τ.toContinuousLinearMap (1 - p)).re + _ - 1 ≤
(τ.toContinuousLinearMap (1 - q)).re at h
simpa only [map_sub, Complex.sub_re, τ.map_one, Complex.one_re] using h
have hre := Invariant233.AffiliatedTransport.trace_eq_of_cutoff_estimates
he hforward hbackward'
rw [τ.apply_selfAdjoint_eq_re hp.isSelfAdjoint,
τ.apply_selfAdjoint_eq_re hq.isSelfAdjoint, hre]
end NormalTracialProjectionTransport
end Invariant233
end
end OAI
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