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110f4f3 | 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 | 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
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