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namespace OAI
noncomputable section
namespace Generator.Tracial
/-! Extension of actual partial unitaries and ordered words from pH.
The empty word maps to p, so word transport retains a final p. -/
section
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (p : AlgebraProjection M)
def PartialUnitary.fromRepresentedCorner (v : PartialUnitary (representedCorner p)) :
PartialUnitary M where
support := ⟨cornerEmbedding p v.support.val,v.support.property.map (cornerEmbedding p)⟩
unitary := ⟨cornerEmbedding p v.val, by
exact (congrArg (fun x => x * cornerEmbedding p v.val)
(map_star (cornerEmbedding p) v.val).symm).trans
((map_mul (cornerEmbedding p) (star v.val) v.val).symm.trans
(congrArg (cornerEmbedding p) v.unitary.initial)), by
exact (congrArg (fun x => cornerEmbedding p v.val * x)
(map_star (cornerEmbedding p) v.val).symm).trans
((map_mul (cornerEmbedding p) v.val (star v.val)).symm.trans
(congrArg (cornerEmbedding p) v.unitary.final))⟩
theorem PartialUnitary.fromRepresentedCorner_support_le
(v : PartialUnitary (representedCorner p)) : (v.fromRepresentedCorner p).support ≤ p :=
cornerEmbedding_left p v.support.val
theorem cornerEmbedding_signedLetter (v : (representedCorner p).toStarSubalgebra) (s : Bool) :
cornerEmbedding p (signedLetter v s)=signedLetter (cornerEmbedding p v) s := by
cases s with
| false => rfl
| true => exact map_star (cornerEmbedding p) v
theorem cornerEmbedding_patchingWord (v : (representedCorner p).toStarSubalgebra)
(xs : List (Bool × (representedCorner p).toStarSubalgebra)) :
cornerEmbedding p (patchingWord v xs)=
patchingWord (cornerEmbedding p v) (xs.map (fun a => (a.1,cornerEmbedding p a.2)))*p.val := by
induction xs with
| nil => simp [patchingWord]
| cons a xs ih =>
simp only [patchingWord,List.map_cons,map_mul,cornerEmbedding_signedLetter,ih,mul_assoc]
theorem cornerEmbedding_patchingWord_nonempty (v : (representedCorner p).toStarSubalgebra)
(xs : List (Bool × (representedCorner p).toStarSubalgebra)) (hx : xs ≠ []) :
cornerEmbedding p (patchingWord v xs)=
patchingWord (cornerEmbedding p v) (xs.map (fun a => (a.1,cornerEmbedding p a.2))) := by
induction xs with
| nil => exact (hx rfl).elim
| cons a xs ih =>
cases xs with
| nil => simp [patchingWord,cornerEmbedding_signedLetter]
| cons b ys =>
change cornerEmbedding p (signedLetter v a.1*a.2*patchingWord v (b::ys)) =
signedLetter (cornerEmbedding p v) a.1*cornerEmbedding p a.2*
patchingWord (cornerEmbedding p v) ((b::ys).map (fun c => (c.1,cornerEmbedding p c.2)))
exact (map_mul (cornerEmbedding p) (signedLetter v a.1 * a.2)
(patchingWord v (b :: ys))).trans
(congrArg₂ (fun x y : M.toStarSubalgebra => x * y)
((map_mul (cornerEmbedding p) (signedLetter v a.1) a.2).trans
(congrArg (fun x => x * cornerEmbedding p a.2)
(cornerEmbedding_signedLetter p v a.1)))
(ih (by simp)))
theorem cornerEmbedding_trace (τ : TracialRepresentation M) (hp : p.val ≠ 0)
(a : (representedCorner p).toStarSubalgebra) :
τ.trace (cornerEmbedding p a)=τ.trace p.val*(representedCornerTrace τ p hp).trace a := by
rw [representedCornerTrace_trace]
have ht : τ.trace p.val ≠ 0 := by
intro h
have := projection_trace_pos τ p.val p.property hp
simp [h] at this
exact (mul_div_cancel₀ _ ht).symm
theorem cornerEmbedding_trace_re (τ : TracialRepresentation M) (hp : p.val ≠ 0)
(a : (representedCorner p).toStarSubalgebra) :
(τ.trace (cornerEmbedding p a)).re=
(τ.trace p.val).re*((representedCornerTrace τ p hp).trace a).re := by
rw [cornerEmbedding_trace p τ hp,projection_trace_real τ p,Complex.mul_re]
simp
end
/-! Finite partial independence in every nonzero represented subcorner,
with actual ambient support, membership and trace estimates. -/
section
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M)
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
theorem exists_represented_corner_partial_independence
(p : AlgebraProjection M) (hpN : p.val.val ∈ N) (hp : p.val ≠ 0)
(F : Finset (representedCorner p).toStarSubalgebra)
(hF : ∀ x ∈ F,‖x‖ ≤ 1) (hc : ∀ x ∈ F,τ.trace (cornerEmbedding p x)=0)
(n : ℕ) (ε : ℝ) (hε : 0 < ε) :
∃ v : PartialUnitary M,v.val.val ∈ N ∧ v.support ≤ p ∧
(τ.trace p.val).re/4 < (τ.trace v.support.val).re ∧
∀ x₀ ∈ (F : Set (representedCorner p).toStarSubalgebra) ∪ {1},
∀ xs : List (Bool × (representedCorner p).toStarSubalgebra),
1 ≤ xs.length → xs.length ≤ n →
patchingCoefficients (F : Set (representedCorner p).toStarSubalgebra) xs →
‖τ.trace (cornerEmbedding p x₀ *
patchingWord v.val (xs.map (fun a => (a.1,cornerEmbedding p a.2))))‖ ≤
(τ.trace p.val).re*ε := by
let σ := representedCornerTrace τ p hp
have hf := representedCorner_factor (restrictedTrace τ hNM)
(projectionInSubalgebra p hpN) hfactor
have hd := representedCorner_diffuse (projectionInSubalgebra p hpN) hdiff
have hi := representedCorner_irreducible τ p hpN hNM hirr
have hs := representedCorner_separable τ p hp hsep
have hcenter : ∀ x ∈ F,σ.trace x=0 := by
intro x hx
rw [representedCornerTrace_trace,hc x hx,zero_div]
obtain ⟨u,hu,ht,hw⟩ := exists_finite_partial_independence σ
(representedSubcorner_le p hpN hNM) hf hd hi hs F hF hcenter n ε hε
refine ⟨u.fromRepresentedCorner p,hu,u.fromRepresentedCorner_support_le p,?_,?_⟩
· change (τ.trace p.val).re/4 < (τ.trace (cornerEmbedding p u.support.val)).re
rw [cornerEmbedding_trace_re p τ hp]
have hpos := projection_trace_pos τ p.val p.property hp
have h := mul_lt_mul_of_pos_left ht hpos
dsimp [σ] at h
linarith
· intro x₀ hx₀ xs hlen hn hcoeff
have hne : xs ≠ [] := by intro h; simp [h] at hlen
have he : cornerEmbedding p x₀ *
patchingWord (u.fromRepresentedCorner p).val
(xs.map (fun a => (a.1,cornerEmbedding p a.2))) =
cornerEmbedding p (x₀*patchingWord u.val xs) := by
rw [map_mul,cornerEmbedding_patchingWord_nonempty p _ _ hne]
rfl
rw [he,cornerEmbedding_trace p τ hp,norm_mul,projection_trace_real τ p,
Complex.norm_real,Real.norm_of_nonneg ((projection_trace_pos τ p.val p.property hp).le)]
exact mul_le_mul_of_nonneg_left (hw x₀ hx₀ xs hlen hn hcoeff)
((projection_trace_pos τ p.val p.property hp).le)
end
/-! Ordered-word transport with ambient unit endpoints, and contractive
compression of supported centered coefficients. -/
section
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {M : VonNeumannAlgebra H} (p : AlgebraProjection M)
theorem cornerCompress_norm_le (a : M.toStarSubalgebra) : ‖cornerCompress p a‖ ≤ ‖a‖ := by
apply ContinuousLinearMap.opNorm_le_bound _ (norm_nonneg _)
intro y
change ‖p.val.val (a.val y.val)‖ ≤ ‖a‖*‖y‖
exact ((p.val.val.le_opNorm _).trans
(mul_le_of_le_one_left (norm_nonneg _) p.property.norm_le)).trans (a.val.le_opNorm y.val)
theorem cornerCompress_supported (a : M.toStarSubalgebra)
(hl : p.val*a=a) (hr : a*p.val=a) : cornerEmbedding p (cornerCompress p a)=a := by
rw [cornerEmbedding_compress,hl,hr]
theorem cornerEmbedding_ordered_tail (v : (representedCorner p).toStarSubalgebra)
(x : ℕ → (representedCorner p).toStarSubalgebra) (xs : List (Bool × ℕ))
(s : Bool) (last : Option ℕ) :
cornerEmbedding p (patchingWord v
(xs.map (fun a => (a.1,x a.2)) ++ [(s,optionalCoefficient x last)]))=
patchingWord (cornerEmbedding p v)
(xs.map (fun a => (a.1,cornerEmbedding p (x a.2))) ++
[(s,optionalCoefficient (fun i => cornerEmbedding p (x i)) last)]) := by
induction xs with
| nil =>
cases last <;>
simp only [List.map_nil,List.nil_append,patchingWord,mul_one,optionalCoefficient,
map_mul,← cornerEmbedding_signedLetter]
| cons a xs ih =>
change cornerEmbedding p (signedLetter v a.1*x a.2*patchingWord v
(xs.map (fun b => (b.1,x b.2)) ++ [(s,optionalCoefficient x last)])) = _
exact (map_mul (cornerEmbedding p) (signedLetter v a.1 * x a.2) _).trans
(congrArg₂ (fun y z : M.toStarSubalgebra => y * z)
((map_mul (cornerEmbedding p) (signedLetter v a.1) (x a.2)).trans
(congrArg (fun y => y * cornerEmbedding p (x a.2))
(cornerEmbedding_signedLetter p v a.1))) ih)
theorem cornerEmbedding_ordered_moment (v : (representedCorner p).toStarSubalgebra)
(x : ℕ → (representedCorner p).toStarSubalgebra) (first last : Option ℕ)
(xs : List (Bool × ℕ)) (s : Bool) :
cornerEmbedding p (optionalCoefficient x first * patchingWord v
(xs.map (fun a => (a.1,x a.2)) ++ [(s,optionalCoefficient x last)]))=
optionalCoefficient (fun i => cornerEmbedding p (x i)) first *
patchingWord (cornerEmbedding p v)
(xs.map (fun a => (a.1,cornerEmbedding p (x a.2))) ++
[(s,optionalCoefficient (fun i => cornerEmbedding p (x i)) last)]) := by
cases first with
| none => simpa only [optionalCoefficient,one_mul] using
cornerEmbedding_ordered_tail p v x xs s last
| some i =>
rw [map_mul,cornerEmbedding_ordered_tail]
rfl
end
/-! Diagonal partial independence in genuinely varying Hilbert-space corners.
The result retains an actual ambient partial-unitary sequence. -/
open Filter
open scoped Topology
section
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M)
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
theorem exists_varying_corner_partial_independence
(p : ℕ → AlgebraProjection M) (hpN : ∀ j,(p j).val.val ∈ N)
(hp : ∀ j,(p j).val ≠ 0)
(x : (j : ℕ) → ℕ → (representedCorner (p j)).toStarSubalgebra)
(hx : ∀ j i,‖x j i‖ ≤ 1)
(hc : ∀ j i,τ.trace (cornerEmbedding (p j) (x j i))=0) :
∃ v : ℕ → PartialUnitary M,
(∀ j,(v j).val.val ∈ N) ∧ (∀ j,(v j).support ≤ p j) ∧
(∀ j,(τ.trace (p j).val).re/4 < (τ.trace (v j).support.val).re) ∧
∀ first last : Option ℕ,∀ xs : List (Bool × ℕ),∀ s : Bool,
Tendsto (fun j => τ.trace (
optionalCoefficient (fun i => cornerEmbedding (p j) (x j i)) first *
patchingWord (v j).val
(xs.map (fun a => (a.1,cornerEmbedding (p j) (x j a.2))) ++
[(s,optionalCoefficient (fun i => cornerEmbedding (p j) (x j i)) last)])))
atTop (𝓝 0) := by
classical
let σ := fun j => representedCornerTrace τ (p j) (hp j)
let F := fun j => (Finset.range (j+1)).image (x j)
have hstage (j : ℕ) := exists_finite_partial_independence (σ j)
(representedSubcorner_le (p j) (hpN j) hNM)
(representedCorner_factor (restrictedTrace τ hNM)
(projectionInSubalgebra (p j) (hpN j)) hfactor)
(representedCorner_diffuse (projectionInSubalgebra (p j) (hpN j)) hdiff)
(representedCorner_irreducible τ (p j) (hpN j) hNM hirr)
(representedCorner_separable τ (p j) (hp j) hsep)
(F j) (by
intro a ha
obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp ha
exact hx j i) (by
intro a ha
obtain ⟨i,_,rfl⟩ := Finset.mem_image.mp ha
rw [representedCornerTrace_trace,hc j i,zero_div])
(j+1) (1/((j : ℝ)+1)) (by positivity)
choose u hu ht hw using hstage
let v := fun j => (u j).fromRepresentedCorner (p j)
refine ⟨v,hu,fun j => (u j).fromRepresentedCorner_support_le (p j),?_,?_⟩
· intro j
change (τ.trace (p j).val).re/4 < (τ.trace (cornerEmbedding (p j) (u j).support.val)).re
rw [cornerEmbedding_trace_re (p j) τ (hp j)]
have h := mul_lt_mul_of_pos_left (ht j)
(projection_trace_pos τ (p j).val (p j).property (hp j))
dsimp [σ] at h
linarith
· intro first last xs s
apply tendsto_iff_norm_sub_tendsto_zero.mpr
simp only [sub_zero]
apply squeeze_zero' (Eventually.of_forall (fun _ => norm_nonneg _)) ?_
tendsto_one_div_add_atTop_nhds_zero_nat
have hopt (i : Option ℕ) : ∀ᶠ j : ℕ in atTop,
optionalCoefficient (x j) i ∈ (F j : Set _) ∪ {1} := by
cases i with
| none => exact Eventually.of_forall (fun _ => Or.inr rfl)
| some i =>
filter_upwards [eventually_ge_atTop i] with j hj
exact Or.inl (Finset.mem_image.mpr ⟨i,Finset.mem_range.mpr (by omega),rfl⟩)
filter_upwards [eventually_list_indices (xs.map Prod.snd),eventually_ge_atTop xs.length,
hopt first,hopt last] with j hj hlen hfirst hlast
have he := cornerEmbedding_ordered_moment (p j) (u j).val (x j) first last xs s
change ‖τ.trace (optionalCoefficient (fun i => cornerEmbedding (p j) (x j i)) first *
patchingWord (cornerEmbedding (p j) (u j).val) _)‖ ≤ _
rw [← he,cornerEmbedding_trace (p j) τ (hp j),norm_mul]
have hb : ‖(σ j).trace (optionalCoefficient (x j) first *
patchingWord (u j).val (xs.map (fun a => (a.1,x j a.2)) ++
[(s,optionalCoefficient (x j) last)]))‖ ≤ 1/((j : ℝ)+1) := by
apply hw j _ hfirst _ (by simp) (by simpa using Nat.add_le_add_right hlen 1)
apply patchingCoefficients_append_endpoint _ _ ?_ (s,optionalCoefficient (x j) last) hlast
intro a ha
obtain ⟨b,hb,rfl⟩ := List.mem_map.mp ha
exact Finset.mem_image.mpr ⟨b.2,Finset.mem_range.mpr (by
have hh := hj b.2 (List.mem_map.mpr ⟨b,hb,rfl⟩)
omega),rfl⟩
have htrace : ‖τ.trace (p j).val‖ ≤ 1 :=
((τ.norm_trace_le_norm₂ _).trans (τ.norm₂_le_norm _)).trans (p j).property.norm_le
exact (mul_le_of_le_one_left (norm_nonneg _) htrace).trans hb
end
/-! Actual coordinate partial-unitary witnesses in supported corners, followed
by exact quotient moments and the ambient quarter-support bound. -/
open Filter
open scoped Topology
open UltrapowerBridge
variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H]
variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M)
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
theorem exists_supported_corner_partial_sequence
(p : ℕ → AlgebraProjection M) (hpN : ∀ j,(p j).val.val ∈ N)
(hp : ∀ j,(p j).val ≠ 0)
(x : ℕ → ℕ → M.toStarSubalgebra) (hx : ∀ i j,‖x i j‖ ≤ 1)
(hc : ∀ i j,τ.trace (x i j)=0)
(hl : ∀ i j,(p j).val*x i j=x i j) (hr : ∀ i j,x i j*(p j).val=x i j) :
∃ v : ℕ → PartialUnitary M,
(∀ j,(v j).val.val ∈ N) ∧ (∀ j,(v j).support ≤ p j) ∧
(∀ j,(τ.trace (p j).val).re/4 < (τ.trace (v j).support.val).re) ∧
∀ first last : Option ℕ,∀ xs : List (Bool × ℕ),∀ s : Bool,
Tendsto (fun j => τ.trace (indexedCoefficient x j first *
patchingWord (v j).val (xs.map (fun a => (a.1,x a.2 j)) ++
[(s,indexedCoefficient x j last)]))) atTop (𝓝 0) := by
let y := fun j i => cornerCompress (p j) (x i j)
have he (j i : ℕ) : cornerEmbedding (p j) (y j i)=x i j :=
cornerCompress_supported (p j) (x i j) (hl i j) (hr i j)
obtain ⟨v,hv,hs,ht,hw⟩ := exists_varying_corner_partial_independence τ hNM hfactor hdiff hirr hsep
p hpN hp y (fun j i => (cornerCompress_norm_le (p j) (x i j)).trans (hx i j))
(fun j i => by rw [he]; exact hc i j)
refine ⟨v,hv,hs,ht,?_⟩
intro first last xs s
have hh := hw first last xs s
cases first <;> cases last <;>
simpa only [he,optionalCoefficient,indexedCoefficient] using hh
theorem exists_quotient_corner_partial_sequence
(ω : Ultrafilter ℕ) (hω : (ω : Filter ℕ) ≤ atTop)
(g : Bounded M) (hg : ∀ j,IsStarProjection (g.val j))
(hgN : ∀ j,(g.val j).val ∈ N) (hg0 : ∀ j,g.val j ≠ 0)
(x : ℕ → Bounded M) (hx : ∀ i j,‖(x i).val j‖ ≤ 1)
(hc : ∀ i j,τ.trace ((x i).val j)=0)
(hl : ∀ i,g*x i=x i) (hr : ∀ i,x i*g=x i) :
∃ v : ℕ → PartialUnitary M,
(∀ j,(v j).val.val ∈ N) ∧ (∀ j,g.val j*(v j).support.val=(v j).support.val) ∧
(quotientTrace τ ω (quotientMap τ (ω : Filter ℕ) g)).re/4 ≤
(quotientTrace τ ω (quotientMap τ (ω : Filter ℕ) (boundedPartialSupport v))).re ∧
∀ first last : Option ℕ,∀ xs : List (Bool × ℕ),∀ s : Bool,
quotientTrace τ ω (optionalCoefficient (fun i => quotientMap τ (ω : Filter ℕ) (x i)) first *
algebraPatchingWord (quotientMap τ (ω : Filter ℕ) (boundedPartialUnitary v))
(xs.map (fun a => (a.1,quotientMap τ (ω : Filter ℕ) (x a.2))) ++
[(s,optionalCoefficient (fun i => quotientMap τ (ω : Filter ℕ) (x i)) last)]))=0 := by
let p := fun j => (⟨g.val j,hg j⟩ : AlgebraProjection M)
obtain ⟨v,hv,hs,ht,hw⟩ := exists_supported_corner_partial_sequence τ hNM hfactor hdiff hirr hsep
p hgN hg0 (fun i => (x i).val) hx hc
(fun i j => congrArg (fun a : Bounded M => a.val j) (hl i))
(fun i j => congrArg (fun a : Bounded M => a.val j) (hr i))
refine ⟨v,hv,hs,?_,?_⟩
· apply le_of_tendsto_of_tendsto
((Complex.continuous_re.continuousAt.tendsto.comp (traceRep_tendsto τ ω g)).div_const 4)
(Complex.continuous_re.continuousAt.tendsto.comp (traceRep_tendsto τ ω (boundedPartialSupport v)))
exact Eventually.of_forall (fun j => (ht j).le)
· intro first last xs s
exact quotient_ordered_moment_zero τ ω hω v x first last xs s (hw first last xs s)
end Generator.Tracial
end
end OAI
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