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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 | import OAI.Analysis.IrrationalRotation.Operators.JointEigenprojection
import OAI.Analysis.IrrationalRotation.Operators.JointEigenspaceAlgebra
import Mathlib.Analysis.InnerProductSpace.Spectrum
namespace OAI
noncomputable section
open scoped Pointwise
namespace Invariant233
variable {ι H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H]
[CompleteSpace H]
theorem coe_generated_range_selfAdjoint (D : ι → H →L[ℂ] H)
(hD : ∀ i, IsSelfAdjoint (D i)) :
(vonNeumannGenerated (Set.range D) : Set (H →L[ℂ] H)) =
Set.centralizer (Set.centralizer (Set.range D)) := by
rw [coe_vonNeumannGenerated]
have hstar : star (Set.range D) ⊆ Set.range D := by
intro T hT
change star T ∈ Set.range D at hT
obtain ⟨i, hi⟩ := hT
refine ⟨i, ?_⟩
calc
D i = star (D i) := (hD i).star_eq.symm
_ = T := by rw [hi, star_star]
rw [Set.union_eq_left.mpr hstar]
theorem jointEigenprojection_isProjectionAtom (D : ι → H →L[ℂ] H)
(hD : ∀ i, IsSelfAdjoint (D i)) (ev : ι → ℂ)
(hp0 : jointEigenprojection D ev ≠ 0) :
IsProjectionAtom (vonNeumannGenerated (Set.range D)) (jointEigenprojection D ev) := by
have hK0 : jointEigenspace D ev ≠ ⊥ := by
intro hK
apply hp0
ext x
have hx := (jointEigenspace D ev).starProjection_apply_mem x
change (jointEigenspace D ev).starProjection x = 0
exact (le_of_eq hK) hx
obtain ⟨x, hxK, hx0⟩ := Submodule.exists_mem_ne_zero_of_ne_bot hK0
have hreal : ∀ i, ((ev i).re : ℂ) = ev i := by
intro i
apply Complex.conj_eq_iff_re.mp
apply (hD i).isSymmetric.conj_eigenvalue_eq_self
apply Module.End.hasEigenvalue_of_hasEigenvector
exact ⟨Module.End.mem_eigenspace_iff.mpr ((mem_jointEigenspace.mp hxK) i), hx0⟩
refine ⟨jointEigenprojection_mem_generated D ev,
isStarProjection_jointEigenprojection D ev, hp0, ?_⟩
intro q hqm hqp hle
apply JointEigenspaceAlgebra.eq_zero_or_eq_joint_projection_of_le
D hD (fun i => (ev i).re) (jointEigenspace D ev)
· intro i y hy
rw [hreal i]
exact (mem_jointEigenspace.mp hy) i
· exact hqp
· change q ∈ (vonNeumannGenerated (Set.range D) : Set (H →L[ℂ] H)) at hqm
rwa [coe_generated_range_selfAdjoint D hD] at hqm
· exact (hqp.le_iff_mul_eq_left (isStarProjection_jointEigenprojection D ev)).mpr hle
theorem atomic_generated_of_dense_jointEigenspaces {A : Type*}
(D : ι → H →L[ℂ] H) (ev : A → ι → ℂ)
(hD : ∀ i, IsSelfAdjoint (D i))
(hdense : (⨆ a, jointEigenspace D (ev a)).topologicalClosure = ⊤) :
IsAtomicVonNeumannAlgebra (vonNeumannGenerated (Set.range D)) := by
apply atomic_generated_of_jointEigenprojection_atoms D ev hD hdense
intro a
exact jointEigenprojection_isProjectionAtom D hD (ev a)
end Invariant233
end
end OAI
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