| import OAI.Analysis.FactorGeneration.LaurentMoments |
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| namespace OAI |
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| noncomputable section |
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| namespace Generator.Tracial |
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| /-! Complete each supported partial unitary by the identity on its orthogonal |
| complement. If the quotient support is one, the quotient value is unchanged. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} |
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| def PartialUnitary.completed (v : PartialUnitary M) : M.toStarSubalgebra := |
| v.val+(1-v.support.val) |
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| theorem PartialUnitary.completed_unitary (v : PartialUnitary M) : |
| v.completed ∈ _root_.unitary M.toStarSubalgebra := by |
| have hp := v.support.property.isIdempotentElem.eq |
| have hs := v.support.property.isSelfAdjoint.star_eq |
| have hi := v.unitary.initial |
| have hf := v.unitary.final |
| have hleft := v.unitary.support_mul |
| have hright := v.unitary.mul_support |
| have hleft' := v.unitary.adjoint.support_mul |
| have hright' := v.unitary.adjoint.mul_support |
| dsimp only [CornerUnitary.adjoint] at hleft' hright' |
| constructor <;> |
| simp only [PartialUnitary.completed,star_add,star_sub,star_one,hs,add_mul,mul_add, |
| sub_mul,mul_sub,one_mul,mul_one,hi,hf,hleft,hright,hleft',hright',hp] <;> abel |
| |
| theorem PartialUnitary.completed_mem (v : PartialUnitary M) |
| (hv : v.val.val ∈ N) : v.completed.val ∈ N := by |
| have hp : v.support.val.val ∈ N := by |
| rw [← v.unitary.initial] |
| have hstar : star v.val.val ∈ N := star_mem hv |
| exact N.mul_mem hstar hv |
| exact N.add_mem hv (N.sub_mem N.one_mem hp) |
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| def boundedCompletedUnitary (u : ℕ → PartialUnitary M) : Bounded M := |
| boundedPartialUnitary u+(1-boundedPartialSupport u) |
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| @[simp] theorem boundedCompletedUnitary_apply (u : ℕ → PartialUnitary M) (j : ℕ) : |
| (boundedCompletedUnitary u).val j=(u j).completed := rfl |
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| theorem boundedCompletedUnitary_unitary (u : ℕ → PartialUnitary M) (j : ℕ) : |
| (boundedCompletedUnitary u).val j ∈ _root_.unitary M.toStarSubalgebra := |
| (u j).completed_unitary |
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| theorem quotient_completed_eq (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
| (u : ℕ → PartialUnitary M) |
| (hi : star (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u))* |
| quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u)=1) : |
| quotientMap τ (ω : Filter ℕ) (boundedCompletedUnitary u)= |
| quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary u) := by |
| have hp : quotientMap τ (ω : Filter ℕ) (boundedPartialSupport u)=1 := by |
| rw [← boundedPartialUnitary_initial,map_mul,quotientMap_star] |
| exact hi |
| let q := quotientMap τ (ω : Filter ℕ) |
| have hs : q (1-boundedPartialSupport u)=0 := |
| (map_sub q 1 (boundedPartialSupport u)).trans |
| ((congrArg₂ (fun a b => a-b) (map_one q) hp).trans (sub_self _)) |
| exact (map_add q (boundedPartialUnitary u) (1-boundedPartialSupport u)).trans |
| ((congrArg (fun b => q (boundedPartialUnitary u)+b) hs).trans (add_zero _)) |
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| theorem bounded_unitary_of_coordinates (a : Bounded M) |
| (ha : ∀ j,a.val j ∈ _root_.unitary M.toStarSubalgebra) : a ∈ _root_.unitary (Bounded M) := by |
| constructor |
| · apply Subtype.ext; funext j; exact (ha j).1 |
| · apply Subtype.ext; funext j; exact (ha j).2 |
| |
| theorem quotient_unitary_of_coordinates (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
| (a : Bounded M) (ha : ∀ j,a.val j ∈ _root_.unitary M.toStarSubalgebra) : |
| quotientMap τ (ω : Filter ℕ) a ∈ _root_.unitary (Q τ ω) := by |
| have hu := bounded_unitary_of_coordinates a ha |
| constructor |
| · rw [← quotientMap_star,← map_mul,hu.1,map_one] |
| · rw [← quotientMap_star,← map_mul,hu.2,map_one] |
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| theorem bounded_coordinate_unitaries_inclusion (hNM : N ≤ M) (a : Bounded M) |
| (ha : ∀ j,a.val j ∈ _root_.unitary M.toStarSubalgebra ∧ (a.val j).val ∈ N) : |
| ∃ b : Bounded N,(∀ j,b.val j ∈ _root_.unitary N.toStarSubalgebra) ∧ boundedInclusion hNM b=a := by |
| let b : Bounded N := ⟨fun j => ⟨(a.val j).val,(ha j).2⟩,a.property⟩ |
| refine ⟨b,?_,rfl⟩ |
| intro j |
| constructor |
| · apply Subtype.ext |
| change star (a.val j).val*(a.val j).val=1 |
| exact congrArg (fun x : M.toStarSubalgebra => x.val) (ha j).1.1 |
| · apply Subtype.ext |
| change (a.val j).val*star (a.val j).val=1 |
| exact congrArg (fun x : M.toStarSubalgebra => x.val) (ha j).1.2 |
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| end |
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| /-! Ordered independence with actual coordinate unitaries in the included |
| algebra, obtained by completing the constructed partial-unitary coordinates. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) |
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| variable (hNM : N ≤ M) |
| variable (hfactor : ∀ c : N.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : N.toStarSubalgebra)) |
| variable (hdiff : ∀ r : N.toStarSubalgebra,IsStarProjection r → r ≠ 0 → |
| ∃ q : N.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ r ∧ r*q=q ∧ q*r=q) |
| variable (hirr : ∀ c : M.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,coordinateInclusion hNM a*c=c*coordinateInclusion hNM a) → |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| variable (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) |
| include hfactor hdiff hirr hsep |
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| theorem exists_coordinate_ordered_unitary |
| (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ) ≤ atTop) |
| (x : ℕ → Q τ ω) (hx : ∀ i,quotientTrace τ ω (x i)=0) : |
| ∃ b : Bounded M, |
| (∀ j,b.val j ∈ unitary M.toStarSubalgebra ∧ (b.val j).val ∈ N) ∧ |
| ExactOrderedMoments τ ω (quotientMap τ (ω : Filter ℕ) b) x := by |
| obtain ⟨u,hu,hi,hf,hm⟩ := exists_full_support_ordered_independence τ hNM |
| hfactor hdiff hirr hsep ω hω x hx |
| refine ⟨boundedCompletedUnitary u,?_,?_⟩ |
| · intro j |
| exact ⟨boundedCompletedUnitary_unitary u j,(u j).completed_mem (hu j)⟩ |
| · rwa [quotient_completed_eq τ ω u hi] |
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| end |
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| /-! Two coordinate-unitary witnesses produce the reduced-moment Haar precursor |
| while retaining actual unitary coordinates in the included algebra. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) |
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| variable (hNM : N ≤ M) |
| variable (hfactor : ∀ c : N.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : N.toStarSubalgebra)) |
| variable (hdiff : ∀ r : N.toStarSubalgebra,IsStarProjection r → r ≠ 0 → |
| ∃ q : N.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ r ∧ r*q=q ∧ q*r=q) |
| variable (hirr : ∀ c : M.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,coordinateInclusion hNM a*c=c*coordinateInclusion hNM a) → |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| variable (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) |
| include hfactor hdiff hirr hsep |
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| theorem exists_coordinate_reduced_unitary |
| (ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ) ≤ atTop) |
| (x : ℕ → Q τ ω) (hx : ∀ i,quotientTrace τ ω (x i)=0) : |
| ∃ a : Bounded M, |
| (∀ j,a.val j ∈ unitary M.toStarSubalgebra ∧ (a.val j).val ∈ N) ∧ |
| ReducedOrderedMoments τ ω (quotientMap τ (ω : Filter ℕ) a) x := by |
| obtain ⟨a,ha,ham⟩ := exists_coordinate_ordered_unitary τ hNM hfactor hdiff hirr hsep ω hω x hx |
| let u := quotientMap τ (ω : Filter ℕ) a |
| have hu : u ∈ unitary (Q τ ω) := quotient_unitary_of_coordinates τ ω a (fun j => (ha j).1) |
| obtain ⟨b,hb,hbm⟩ := exists_coordinate_ordered_unitary τ hNM hfactor hdiff hirr hsep ω hω |
| (productCenteredFamily τ ω u x) (productCenteredFamily_centered τ ω u x hu.2 hx ham) |
| refine ⟨a*b,?_,?_⟩ |
| · intro j |
| exact ⟨(unitary M.toStarSubalgebra).mul_mem (ha j).1 (hb j).1,N.mul_mem (ha j).2 (hb j).2⟩ |
| · rw [map_mul] |
| intro c d sign xs hxs |
| exact product_reduced_moment τ ω u (quotientMap τ (ω : Filter ℕ) b) x hbm c d sign xs hxs |
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| end |
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| /-! Full integer endpoint moments and an actual included coordinate-unitary |
| Haar witness, before imposing an operator topology on generated algebras. -/ |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
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| variable (hNM : N ≤ M) |
| variable (hfactor : ∀ c : N.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,a*c=c*a) → ∃ z : ℂ,c=z • (1 : N.toStarSubalgebra)) |
| variable (hdiff : ∀ r : N.toStarSubalgebra,IsStarProjection r → r ≠ 0 → |
| ∃ q : N.toStarSubalgebra,IsStarProjection q ∧ q ≠ 0 ∧ q ≠ r ∧ r*q=q ∧ q*r=q) |
| variable (hirr : ∀ c : M.toStarSubalgebra, |
| (∀ a : N.toStarSubalgebra,coordinateInclusion hNM a*c=c*coordinateInclusion hNM a) → |
| ∃ z : ℂ,c=z • (1 : M.toStarSubalgebra)) |
| variable (hsep : TopologicalSpace.IsSeparable (Set.range (baseVector τ))) |
| include hfactor hdiff hirr hsep |
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| omit hfactor hdiff hirr hsep in |
| private theorem coordinate_haar_endpoints_of_coordinates |
| (D : Subalgebra ℂ (Q τ ω)) (x : ℕ → Q τ ω) |
| (hdense : ∀ d ∈ D,quotientTrace τ ω d=0 → d ∈ closure (Set.range x)) |
| (a : Bounded M) |
| (ha : ∀ j,a.val j ∈ unitary M.toStarSubalgebra ∧ (a.val j).val ∈ N) |
| (hm : ReducedOrderedMoments τ ω (quotientMap τ (ω : Filter ℕ) a) x) : |
| ∃ t : unitary (Q τ ω), |
| (∃ b : Bounded N,(∀ j,b.val j ∈ unitary N.toStarSubalgebra) ∧ |
| quotientInclusion τ hNM (ω : Filter ℕ) |
| (quotientMap (restrictedTrace τ hNM) (ω : Filter ℕ) b)=(t : Q τ ω)) ∧ |
| (∀ n : ℤ,n ≠ 0 → quotientTrace τ ω (↑(t^n) : Q τ ω)=0) ∧ |
| ∀ c ∈ D,∀ d ∈ D,∀ n : ℤ,n ≠ 0 → ∀ xs : List (Q τ ω × ℤ), |
| (∀ z ∈ xs,z.1 ∈ D ∧ quotientTrace τ ω z.1=0 ∧ z.2 ≠ 0) → |
| quotientTrace τ ω (c*integerValueWord τ ω t n xs d)=0 := by |
| let t : unitary (Q τ ω) := ⟨quotientMap τ (ω : Filter ℕ) a, |
| quotient_unitary_of_coordinates τ ω a (fun j => (ha j).1)⟩ |
| obtain ⟨b,hb⟩ := bounded_coordinate_unitaries_inclusion hNM a ha |
| have hhaar := reducedOrderedMoments_haar τ ω t x hm |
| refine ⟨t,⟨b,hb.1,?_⟩,hhaar,?_⟩ |
| · change quotientMap τ (ω : Filter ℕ) (boundedInclusion hNM b)=quotientMap τ (ω : Filter ℕ) a |
| exact congrArg (quotientMap τ (ω : Filter ℕ)) hb.2 |
| · intro c hc d hd n hn xs hxs |
| apply reduced_integer_moment_closedSpan τ ω t x hm c d |
| (subalgebra_mem_centered_closedSpan τ ω D x hdense c hc) |
| (subalgebra_mem_centered_closedSpan τ ω D x hdense d hd) n hn xs |
| intro z hz |
| exact ⟨Submodule.closure_subset_topologicalClosure_span _ |
| (hdense z.1 (hxs z hz).1 (hxs z hz).2.1),(hxs z hz).2.2⟩ |
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| private theorem coordinate_haar_endpoints_of_dense_family (hω : (ω : Filter ℕ) ≤ atTop) |
| (D : Subalgebra ℂ (Q τ ω)) (x : ℕ → Q τ ω) |
| (hx : ∀ i,x i ∈ D ∧ quotientTrace τ ω (x i)=0) |
| (hdense : ∀ d ∈ D,quotientTrace τ ω d=0 → d ∈ closure (Set.range x)) : |
| ∃ t : unitary (Q τ ω), |
| (∃ b : Bounded N,(∀ j,b.val j ∈ unitary N.toStarSubalgebra) ∧ |
| quotientInclusion τ hNM (ω : Filter ℕ) |
| (quotientMap (restrictedTrace τ hNM) (ω : Filter ℕ) b)=(t : Q τ ω)) ∧ |
| (∀ n : ℤ,n ≠ 0 → quotientTrace τ ω (↑(t^n) : Q τ ω)=0) ∧ |
| ∀ c ∈ D,∀ d ∈ D,∀ n : ℤ,n ≠ 0 → ∀ xs : List (Q τ ω × ℤ), |
| (∀ z ∈ xs,z.1 ∈ D ∧ quotientTrace τ ω z.1=0 ∧ z.2 ≠ 0) → |
| quotientTrace τ ω (c*integerValueWord τ ω t n xs d)=0 := by |
| obtain ⟨a,ha⟩ := exists_coordinate_reduced_unitary (H := H) (N := N) (M := M) τ hNM hfactor hdiff hirr hsep |
| ω hω x (fun i => (hx i).2) |
| exact coordinate_haar_endpoints_of_coordinates τ ω hNM D x hdense a ha.1 ha.2 |
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| theorem exists_coordinate_haar_endpoints (hω : (ω : Filter ℕ) ≤ atTop) |
| (D : Subalgebra ℂ (Q τ ω)) (hD : TopologicalSpace.IsSeparable (D : Set (Q τ ω))) : |
| ∃ t : unitary (Q τ ω), |
| (∃ b : Bounded N,(∀ j,b.val j ∈ unitary N.toStarSubalgebra) ∧ |
| quotientInclusion τ hNM (ω : Filter ℕ) |
| (quotientMap (restrictedTrace τ hNM) (ω : Filter ℕ) b)=(t : Q τ ω)) ∧ |
| (∀ n : ℤ,n ≠ 0 → quotientTrace τ ω (↑(t^n) : Q τ ω)=0) ∧ |
| ∀ c ∈ D,∀ d ∈ D,∀ n : ℤ,n ≠ 0 → ∀ xs : List (Q τ ω × ℤ), |
| (∀ z ∈ xs,z.1 ∈ D ∧ quotientTrace τ ω z.1=0 ∧ z.2 ≠ 0) → |
| quotientTrace τ ω (c*integerValueWord τ ω t n xs d)=0 := by |
| obtain ⟨x,hx⟩ := exists_centered_dense_quotient_family (H := H) (M := M) τ ω D hD |
| exact coordinate_haar_endpoints_of_dense_family τ ω hNM hfactor hdiff hirr hsep |
| hω D x hx.1 hx.2 |
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| end Generator.Tracial |
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| end |
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| end OAI |
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