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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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 | import Mathlib.LinearAlgebra.Complex.Module
import OAI.Analysis.FactorGeneration.MatrixEncoding
import OAI.Analysis.FactorGeneration.MatrixSequence
import OAI.Analysis.FactorGeneration.ProjectionEncoding
namespace OAI
noncomputable section
namespace Generator
/-! One arbitrary operator and a pair of self-adjoint operators generate the same weakly closed star algebra. -/
open scoped ComplexStarModule
section
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
local notation "Op" => H →L[ℂ] H
theorem realPart_mem (S : StarSubalgebra ℂ Op) {x : Op} (hx : x ∈ S) :
(ℜ x : Op) ∈ S := by
rw [realPart_apply_coe, ← Complex.coe_smul]
exact S.smul_mem (add_mem hx (star_mem hx)) _
theorem imaginaryPart_mem (S : StarSubalgebra ℂ Op) {x : Op} (hx : x ∈ S) :
(ℑ x : Op) ∈ S := by
rw [imaginaryPart_apply_coe, ← Complex.coe_smul]
exact S.smul_mem (S.smul_mem (sub_mem hx (star_mem hx)) _) _
theorem mem_iff_parts_mem (S : StarSubalgebra ℂ Op) (x : Op) :
x ∈ S ↔ (ℜ x : Op) ∈ S ∧ (ℑ x : Op) ∈ S := by
refine ⟨fun hx => ⟨realPart_mem S hx, imaginaryPart_mem S hx⟩, ?_⟩
rintro ⟨hr, hi⟩
rw [← realPart_add_I_smul_imaginaryPart x]
exact add_mem hr (S.smul_mem hi _)
theorem wstar_singleton_eq_parts (x : Op) :
wstar {x} = wstar {(ℜ x : Op), (ℑ x : Op)} := by
apply le_antisymm
· apply wstar_le (wstar_closed _)
intro y hy
have hyx : y = x := Set.mem_singleton_iff.mp hy
subst y
apply (mem_iff_parts_mem _ x).mpr
exact ⟨subset_wstar _ (by simp), subset_wstar _ (by simp)⟩
· apply wstar_le (wstar_closed _)
intro y hy
have hx := subset_wstar {x} (Set.mem_singleton x)
rcases Set.mem_insert_iff.mp hy with rfl | hy
· exact realPart_mem _ hx
· rcases Set.mem_singleton_iff.mp hy with rfl
exact imaginaryPart_mem _ hx
theorem realPart_pack {a b : Op} (ha : IsSelfAdjoint a) (hb : IsSelfAdjoint b) :
(ℜ (a + Complex.I • b) : Op) = a := by
simp only [map_add, realPart_I_smul, hb.imaginaryPart, neg_zero, add_zero]
exact ha.coe_realPart
theorem imaginaryPart_pack {a b : Op} (ha : IsSelfAdjoint a) (hb : IsSelfAdjoint b) :
(ℑ (a + Complex.I • b) : Op) = b := by
simp only [map_add, ha.imaginaryPart, imaginaryPart_I_smul, zero_add]
exact hb.coe_realPart
theorem wstar_pack {a b : Op} (ha : IsSelfAdjoint a) (hb : IsSelfAdjoint b) :
wstar {a + Complex.I • b} = wstar {a, b} := by
rw [wstar_singleton_eq_parts, realPart_pack ha hb, imaginaryPart_pack ha hb]
/-- Exact equivalence in thm:main, without proving either side for II1 factors. -/
theorem singlyGenerated_iff_two_selfAdjoint (S : StarSubalgebra ℂ Op) :
SinglyGenerated S ↔ ∃ a ∈ S, ∃ b ∈ S,
IsSelfAdjoint a ∧ IsSelfAdjoint b ∧ wstar {a, b} = S := by
constructor
· rintro ⟨x, hx, hgen⟩
exact ⟨ℜ x, realPart_mem S hx, ℑ x, imaginaryPart_mem S hx,
(ℜ x).property, (ℑ x).property, (wstar_singleton_eq_parts x).symm.trans hgen⟩
· rintro ⟨a, ha, b, hb, hsa, hsb, hgen⟩
exact ⟨a + Complex.I • b, add_mem ha (S.smul_mem hb _),
(wstar_pack hsa hsb).trans hgen⟩
end
/-! Two self-adjoint generators for an actual two-by-two amplification
whose actual commutant has four self-adjoint generators. Spectral separation
is proved, and the final generation uses the surjective coefficient map. -/
open scoped BigOperators
namespace MatrixUnitSystem
open MatrixEncoding
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H}
variable (E : MatrixUnitSystem (Fin 2) M.toStarSubalgebra)
def concreteCoefficientHom : Matrix (Fin 2) (Fin 2) E.commutant →⋆ₐ[ℂ] (H →L[ℂ] H) :=
M.toStarSubalgebra.subtype.comp E.coefficientHom
theorem coefficient_corner (a : E.commutant) :
E.concreteCoefficientHom (corner a)=a.val.val*(E.entry 0 0).val := by
have he : corner a=Matrix.single 0 0 a := by ext i j; fin_cases i <;> fin_cases j <;> simp [corner,Matrix.single]
change (E.coefficientHom (corner a)).val=_
rw [he,E.coefficientHom_single]
rfl
theorem coefficient_diagonal (a : E.commutant) :
E.concreteCoefficientHom (corner a+E21*corner a*E12)=a.val.val := by
have he : corner a+E21*corner a*E12=Matrix.diagonal (fun _ : Fin 2 => a) := by
ext i j
fin_cases i <;> fin_cases j <;> simp [corner,E21,E12,Matrix.diagonal]
rw [he]
change (E.coefficientEval (Matrix.diagonal (fun _ : Fin 2 => a))).val=a.val.val
congr 1
simp [coefficientEval,Matrix.diagonal,apply_ite,ite_mul,← Finset.mul_sum,E.total]
theorem two_generators_of_four_commutant_generators (a₁ a₂ a₃ a₄ : E.commutant)
(h₁ : IsSelfAdjoint a₁) (h₂ : IsSelfAdjoint a₂)
(h₃ : IsSelfAdjoint a₃) (h₄ : IsSelfAdjoint a₄)
(hgen : wstar {a₁.val.val,a₂.val.val,a₃.val.val,a₄.val.val}=E.relativeVonNeumann.toStarSubalgebra) :
∃ A ∈ M,∃ B ∈ M,IsSelfAdjoint A ∧ IsSelfAdjoint B ∧ wstar {A,B}=M.toStarSubalgebra := by
let f := E.concreteCoefficientHom
let t := f (encodeA a₁ a₂ 0)
let q := f E22
let C : ℝ := 2*‖t‖+1
let A := f (encodeA a₁ a₂ (algebraMap ℂ E.commutant (C : ℂ)))
let B := f (encodeB a₃ a₄)
have hA : A ∈ M := (E.coefficientHom _).property
have hB : B ∈ M := (E.coefficientHom _).property
have hsa : IsSelfAdjoint A :=
(encodeA_selfAdjoint h₁ h₂ (scalar_shift_selfAdjoint C)).map f
have hsb : IsSelfAdjoint B := (encodeB_selfAdjoint h₃ h₄).map f
have hq : q*q=q :=
(map_mul f E22 E22).symm.trans (congrArg f E22_idempotent)
have hqt : Commute q t := (E22_commute_encodeA a₁ a₂).map f
have hsplit : A=C • q+t := by
change f _=C • f E22+f (encodeA a₁ a₂ 0)
exact (congrArg f (encodeA_scalar_shift a₁ a₂ C)).trans
((map_add f _ _).trans (congrArg (fun x => x + f (encodeA a₁ a₂ 0))
((map_smul f (C : ℂ) E22).trans (Complex.coe_smul C (f E22)))))
let Q := wstar {A,B}
let S := Q.comap f
have hAm : encodeA a₁ a₂ (algebraMap ℂ E.commutant (C : ℂ)) ∈ S := subset_wstar _ (by simp [A])
have hBm : encodeB a₃ a₄ ∈ S := subset_wstar _ (by simp [B])
have h22 : (E22 : Matrix (Fin 2) (Fin 2) E.commutant) ∈ S := by
change q ∈ wstar {A,B}
apply (wstar_mono (s := {A}) (t := {A,B}) (by simp))
rw [hsplit]
exact idempotent_mem_wstar_of_gap q t hq hqt C (by dsimp [C]; linarith)
have h11 : (E11 : Matrix (Fin 2) (Fin 2) E.commutant) ∈ S := by
have he : (1 : Matrix (Fin 2) (Fin 2) E.commutant)-E22=E11 := by rw [← complement]; abel
rw [← he]
exact S.sub_mem S.one_mem h22
obtain ⟨_,h12,h21⟩ := matrix_units_mem S h11 hBm
obtain ⟨hc₁,hc₂,hc₃,hc₄⟩ := encoded_corners_mem_scalar S (C : ℂ) h11 hAm hBm
have hcoeff (a : E.commutant) (ha : corner a ∈ S) : a.val.val ∈ Q := by
rw [← E.coefficient_diagonal a]
exact S.add_mem ha (S.mul_mem (S.mul_mem h21 ha) h12)
have hN : E.relativeVonNeumann.toStarSubalgebra ≤ Q := by
rw [← hgen]
exact wstar_le (wstar_closed _) (by
rintro _ (rfl | rfl | rfl | rfl)
· exact hcoeff a₁ hc₁
· exact hcoeff a₂ hc₂
· exact hcoeff a₃ hc₃
· exact hcoeff a₄ hc₄)
have hcorner (a : E.commutant) : corner a ∈ S := by
change f (corner a) ∈ Q
rw [E.coefficient_corner a]
apply Q.mul_mem (hN ⟨a,rfl⟩)
have hp := E.coefficient_corner (1 : E.commutant)
have he : corner (1 : E.commutant)=E11 := rfl
rw [he] at hp
have hp' : f E11=(E.entry 0 0).val := by simpa using hp
have hh : f E11 ∈ Q := h11
rwa [hp'] at hh
have htop : S=⊤ := eq_top_of_corner_and_units S hcorner h12 h21
refine ⟨A,hA,B,hB,hsa,hsb,le_antisymm ?_ ?_⟩
· exact wstar_le (Tracial.vonNeumann_wotClosed M) (by rintro _ (rfl | rfl) <;> assumption)
· intro x hx
obtain ⟨a,ha⟩ := E.coefficientHom_surjective 0 ⟨x,hx⟩
have hm : a ∈ S := by rw [htop]; trivial
change (E.coefficientHom a).val ∈ Q at hm
rwa [ha] at hm
end MatrixUnitSystem
end Generator
end
end OAI
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