lean/OAI/Analysis/BackwardIntertwiners/ModelProperties.lean

import OAI.Analysis.BackwardIntertwiners.ProductSymbols

namespace OAI

namespace BackwardIntertwiners
open Filter
open scoped Topology
noncomputable section

open Topology MeasureTheory Set
open scoped ENNReal

open scoped Classical

universe u
variable {H : Type u} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]

attribute [local instance] realOperatorTower realOperatorCommute complexNormalCalculus complexClosedCalculus realSelfAdjointCalculus

open InnerProductSpace ContinuousLinearMap


structure OrdinaryStructuralProperties (D : DirectModelData) : Prop where
  tails : ∀ h : ℤ, ∃ L : Submodule ℂ GlobalHilbert,
    (∀ f : GlobalHilbert, f ∈ L ↔ ∀ᵐ x ∂baseMeasure, f x ∈ coordinateTail h) ∧
    IsClosed (L : Set GlobalHilbert) ∧ L ≠ ⊥ ∧ L ≠ ⊤ ∧
    OperatorInvariant L D.S ∧ ¬OperatorInvariant L D.V
  fiber_compact : ∀ x : Base, IsCompactOperator (D.Sx x)
  global_not_compact : ¬IsCompactOperator D.S
  multiplier_witness : ∃ Q : GlobalHilbert →L[ℂ] GlobalHilbert,
    IsStarProjection Q ∧
    (∀ f : GlobalHilbert, Q f =ᵐ[baseMeasure] fun x => lp.single 2 0 (f x 0)) ∧
    Q * D.S ≠ D.S * Q ∧
    (∀ g : Lp ℂ ∞ baseMeasure, Q * scalarMultiplier g = scalarMultiplier g * Q) ∧
    Q ∉ Set.range (scalarMultiplier (μ:=baseMeasure))




structure PolarFamilyProperties (D : DirectModelData)
    (U : (GlobalHilbert →L[ℂ] GlobalHilbert)ˣ)
    (A : ℤ → GlobalHilbert →L[ℂ] GlobalHilbert) : Prop where
  unitary : star (U : GlobalHilbert →L[ℂ] GlobalHilbert) =
    ((U⁻¹ : (GlobalHilbert →L[ℂ] GlobalHilbert)ˣ) : GlobalHilbert →L[ℂ] GlobalHilbert)
  unitary_coordinates : ∀ f : GlobalHilbert, ∀ᵐ x ∂baseMeasure, ∀ j : ℤ,
    (U : GlobalHilbert →L[ℂ] GlobalHilbert) f x j = f x (j-1)
  modulus : A 0 = CFC.sqrt (star D.S * D.S)
  polar : D.S = (U : GlobalHilbert →L[ℂ] GlobalHilbert) * A 0
  injective : Function.Injective (A 0)
  dense_range : DenseRange (A 0)
  conjugates : ∀ n : ℤ, A n =
    ((U^(-n) : (GlobalHilbert →L[ℂ] GlobalHilbert)ˣ) : GlobalHilbert →L[ℂ] GlobalHilbert) *
      A 0 * ((U^n : (GlobalHilbert →L[ℂ] GlobalHilbert)ˣ) : GlobalHilbert →L[ℂ] GlobalHilbert)
  diagonal : ∀ n : ℤ, ∀ f : GlobalHilbert, ∀ᵐ x ∂baseMeasure, ∀ j : ℤ,
    A n f x j = (explicitBeta x (j+n):ℂ) * f x j
  finite_atomic : ∀ F : Finset ℤ, AtomicAlgebra (strongGenerated (A '' (F : Set ℤ)))
  modulus_pure_point : PurePointResolution (A 0)
  full_multiplication : ∀ T : GlobalHilbert →L[ℂ] GlobalHilbert,
    T ∈ strongGenerated (Set.range A) ↔ IsProductMultiplier T
  abelian : ∀ T ∈ strongGenerated (Set.range A), ∀ R ∈ strongGenerated (Set.range A), T*R=R*T
  diffuse : DiffuseAlgebra (strongGenerated (Set.range A))
  not_contained_in_commutant : ∃ T ∈ strongGenerated (Set.range A), T * D.S ≠ D.S * T

def PolarStructuralProperties (D : DirectModelData) : Prop :=
  ∃ (U : (GlobalHilbert →L[ℂ] GlobalHilbert)ˣ)
    (A : ℤ → GlobalHilbert →L[ℂ] GlobalHilbert), PolarFamilyProperties D U A


lemma coordinateTail_eq_closedSpan (h : ℤ) :
    coordinateTail h = (Submodule.span ℂ
      (Set.range (fun j : {j : ℤ // h ≤ j} => (lp.single 2 j.val (1:ℂ) : Fiber)))).topologicalClosure := by
  let P := Submodule.span ℂ (Set.range (fun j : {j : ℤ // h ≤ j} => (lp.single 2 j.val (1:ℂ) : Fiber)))
  apply le_antisymm
  · intro v hv
    apply (Submodule.isClosed_topologicalClosure P).mem_of_tendsto
      (lp.hasSum_single (by norm_num : (2:ENNReal) ≠ ∞) v)
    apply Eventually.of_forall
    intro s
    apply P.topologicalClosure.sum_mem
    intro j _
    by_cases hj : j < h
    · rw [hv j hj,lp.single_zero]
      exact P.topologicalClosure.zero_mem
    · apply Submodule.le_topologicalClosure
      have hg : (lp.single 2 j (1:ℂ) : Fiber) ∈ P :=
        Submodule.subset_span ⟨⟨j,by omega⟩,rfl⟩
      simpa only [←lp.single_smul,smul_eq_mul,mul_one] using P.smul_mem (v j) hg
  · apply Submodule.topologicalClosure_minimal _ ?_ (coordinateTail_isClosed h)
    apply Submodule.span_le.mpr
    rintro v ⟨j,rfl⟩
    exact single_mem_coordinateTail j.property 1

lemma explicitDirectModel_ordinary : OrdinaryStructuralProperties explicitDirectModel where
  tails h := ⟨globalTail h, fun f => mem_globalTail h f,
    globalTail_closed h,globalTail_nonzero h,globalTail_proper h,
    globalTail_forward h,globalTail_not_backward h⟩
  fiber_compact := forwardFiber_compact
  global_not_compact := forwardOperator_not_compact
  multiplier_witness := ⟨sliceProjection 0, sliceProjection_isStarProjection 0,
    sliceProjection_ae 0,sliceProjection_zero_not_commute,
    sliceProjection_scalar_commute 0,sliceProjection_zero_not_scalar⟩

lemma explicitDirectModel_polarFamily : PolarFamilyProperties explicitDirectModel shiftUnit polarDiagonal where
  unitary := coordinateShift_star 1
  unitary_coordinates := coordinateShift_ae 1
  modulus := polar_modulus.symm
  polar := forward_polar_factor
  injective := polarDiagonal_injective
  dense_range := polarDiagonal_denseRange
  conjugates n := by
    rw [shiftUnit_zpow,shiftUnit_zpow]
    exact (polarDiagonal_conjugate n).symm
  diagonal := polarDiagonal_ae
  finite_atomic := finitePolarAlgebra_atomic
  modulus_pure_point := polarDiagonal_purePoint
  full_multiplication := polarAlgebra_mem_iff_productMultiplier
  abelian := fun _ hT _ hR => polarAlgebra_commutative hT hR
  diffuse := polarAlgebra_diffuse
  not_contained_in_commutant := ⟨polarDiagonal 0,polarDiagonal_mem 0,polarDiagonal_not_commute⟩




theorem direct_model_complete : ∃ D : DirectModelData,
    DirectModelProperties D ∧ OrdinaryStructuralProperties D ∧ PolarStructuralProperties D := by
  exact ⟨explicitDirectModel,explicitDirectModel_properties,
    explicitDirectModel_ordinary,shiftUnit,polarDiagonal,explicitDirectModel_polarFamily⟩


end
end BackwardIntertwiners

end OAI