| import OAI.Analysis.FactorGeneration.HaarSequences |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Generator.Tracial |
|
|
| /-! Every nonempty reduced word in successive Haar variables has zero trace. |
| The induction eliminates the largest label, retaining centered lower-label gaps. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
|
|
| def unitaryWord {A : Type*} (v : A → unitary (Q τ ω)) (w : List (A × Bool)) : Q τ ω := |
| (w.map (fun x => algebraSignedLetter (v x.1 : Q τ ω) (!x.2))).prod |
|
|
| theorem unitaryWord_nil {A : Type*} (v : A → unitary (Q τ ω)) : |
| unitaryWord τ ω v []=1 := rfl |
|
|
| theorem unitaryWord_cons {A : Type*} (v : A → unitary (Q τ ω)) |
| (x : A × Bool) (w : List (A × Bool)) : |
| unitaryWord τ ω v (x::w)=algebraSignedLetter (v x.1 : Q τ ω) (!x.2)*unitaryWord τ ω v w := rfl |
|
|
| theorem unitaryWord_append {A : Type*} (v : A → unitary (Q τ ω)) |
| (a b : List (A × Bool)) : unitaryWord τ ω v (a++b)=unitaryWord τ ω v a*unitaryWord τ ω v b := by |
| simp [unitaryWord,List.map_append,List.prod_append] |
|
|
| theorem unitaryWord_mem {A : Type*} (v : A → unitary (Q τ ω)) |
| (D : StarSubalgebra ℂ (Q τ ω)) (w : List (A × Bool)) |
| (hv : ∀ x ∈ w,(v x.1 : Q τ ω) ∈ D) : unitaryWord τ ω v w ∈ D := by |
| induction w with |
| | nil => exact D.one_mem |
| | cons x w ih => |
| apply D.mul_mem _ (ih (fun y hy => hv y (by simp [hy]))) |
| have hx := hv x (by simp) |
| change algebraSignedLetter (v x.1 : Q τ ω) (!x.2) ∈ D |
| unfold algebraSignedLetter |
| split |
| · exact star_mem hx |
| · exact hx |
|
|
| / |
| theorem maximal_word_gaps (v : ℕ → unitary (Q τ ω)) |
| (D : StarSubalgebra ℂ (Q τ ω)) (n : ℕ) |
| (hv : ∀ i < n,(v i : Q τ ω) ∈ D) |
| (hzero : ∀ a : List (ℕ × Bool),FreeGroup.IsReduced a → a ≠ [] → |
| (∀ x ∈ a,x.1 < n) → quotientTrace τ ω (unitaryWord τ ω v a)=0) |
| (w : List (ℕ × Bool)) (s : Bool) |
| (hr : FreeGroup.IsReduced ((n,s)::w)) (hn : ∀ x ∈ w,x.1 ≤ n) : |
| ∃ xs b,b ∈ D ∧ SignedReducedGaps τ ω D.toSubalgebra (!s) xs ∧ |
| unitaryWord τ ω v ((n,s)::w)=betweenWord (v n : Q τ ω) (!s) xs b := by |
| classical |
| by_cases hex : ∃ x ∈ w,x.1=n |
| · obtain ⟨a,t,c,he,ha⟩ := word_split_first n w hex |
| have hlt : ∀ x ∈ a,x.1 < n := by |
| intro x hx |
| have := hn x (by rw [he]; simp [hx]) |
| have := ha x hx |
| omega |
| have har : FreeGroup.IsReduced a := reduced_append_left (he ▸ reduced_tail hr) |
| have hcr : FreeGroup.IsReduced ((n,t)::c) := reduced_append_right (he ▸ reduced_tail hr) |
| have hcn : ∀ x ∈ c,x.1 ≤ n := by |
| intro x hx |
| exact hn x (by rw [he]; simp [hx]) |
| obtain ⟨xs,b,hb,hxs,hev⟩ := maximal_word_gaps v D n hv hzero c t hcr hcn |
| have ham : unitaryWord τ ω v a ∈ D := unitaryWord_mem τ ω v D a (fun x hx => hv _ (hlt x hx)) |
| have hac : (!s) ≠ (!t) → quotientTrace τ ω (unitaryWord τ ω v a)=0 := by |
| intro hst |
| apply hzero a har _ hlt |
| intro hempty |
| have hr' : FreeGroup.IsReduced ((n,s)::(n,t)::c) := by |
| simpa only [he,hempty,List.nil_append] using hr |
| have hst' : s=t := (FreeGroup.isReduced_cons_cons.mp hr').1 rfl |
| exact hst (congrArg Bool.not hst') |
| refine ⟨(unitaryWord τ ω v a,!t)::xs,b,hb,⟨ham,hac,hxs⟩,?_⟩ |
| rw [unitaryWord_cons,he,unitaryWord_append,hev] |
| simp only [betweenWord,mul_assoc] |
| · refine ⟨[],unitaryWord τ ω v w,?_,trivial,rfl⟩ |
| apply unitaryWord_mem τ ω v D w |
| intro x hx |
| apply hv |
| have := hn x hx |
| have : x.1 ≠ n := fun he => hex ⟨x,hx,he⟩ |
| omega |
| termination_by w.length |
| decreasing_by simp only [he,List.length_append,List.length_cons]; omega |
|
|
| theorem successive_reduced_word_zero (v : ℕ → unitary (Q τ ω)) |
| (D : ℕ → StarSubalgebra ℂ (Q τ ω)) |
| (hv : ∀ i j,i < j → (v i : Q τ ω) ∈ D j) |
| (hep : ∀ j,HaarEndpoints τ ω (D j).toSubalgebra (v j)) |
| (n : ℕ) (w : List (ℕ × Bool)) (hr : FreeGroup.IsReduced w) (hw : w ≠ []) |
| (hn : ∀ x ∈ w,x.1 < n) : quotientTrace τ ω (unitaryWord τ ω v w)=0 := by |
| classical |
| induction n generalizing w with |
| | zero => |
| cases w with |
| | nil => exact (hw rfl).elim |
| | cons x w => have := hn x (by simp); omega |
| | succ n ih => |
| by_cases hex : ∃ x ∈ w,x.1=n |
| · obtain ⟨a,s,b,he,ha⟩ := word_split_first n w hex |
| have halt : ∀ x ∈ a,x.1 < n := by |
| intro x hx |
| have := hn x (by rw [he]; simp [hx]) |
| have := ha x hx |
| omega |
| have hbr : FreeGroup.IsReduced ((n,s)::b) := reduced_append_right (he ▸ hr) |
| have hbn : ∀ x ∈ b,x.1 ≤ n := by |
| intro x hx |
| have := hn x (by rw [he]; simp [hx]) |
| omega |
| obtain ⟨xs,c,hc,hxs,hev⟩ := maximal_word_gaps τ ω v (D n) n |
| (fun i hi => hv i n hi) ih b s hbr hbn |
| rw [he,unitaryWord_append,hev] |
| exact (hep n).signed_gaps τ ω _ |
| (unitaryWord_mem τ ω v (D n) a (fun x hx => hv _ _ (halt x hx))) c hc _ xs hxs |
| · apply ih w hr hw |
| intro x hx |
| have := hn x hx |
| have : x.1 ≠ n := fun he => hex ⟨x,hx,he⟩ |
| omega |
|
|
| end |
|
|
| /-! The actual free-group representation furnished by successive Haar variables. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {N M : VonNeumannAlgebra H} (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
|
|
| def unitaryRepresentation {A : Type*} (v : A → unitary (Q τ ω)) : |
| FreeGroup A →* unitary (Q τ ω) := FreeGroup.lift v |
|
|
| theorem unitaryRepresentation_mk {A : Type*} (v : A → unitary (Q τ ω)) |
| (w : List (A × Bool)) : |
| (unitaryRepresentation τ ω v (FreeGroup.mk w) : Q τ ω)=unitaryWord τ ω v w := by |
| rw [unitaryRepresentation,FreeGroup.lift_mk] |
| induction w with |
| | nil => rfl |
| | cons x w ih => |
| simp only [List.map_cons,List.prod_cons,Submonoid.coe_mul,unitaryWord_cons,ih] |
| cases x.2 <;> rfl |
|
|
| theorem successive_freeGroup_trace (v : ℕ → unitary (Q τ ω)) |
| (D : ℕ → StarSubalgebra ℂ (Q τ ω)) |
| (hv : ∀ i j,i < j → (v i : Q τ ω) ∈ D j) |
| (hep : ∀ j,HaarEndpoints τ ω (D j).toSubalgebra (v j)) |
| (g : FreeGroup ℕ) (hg : g ≠ 1) : |
| quotientTrace τ ω (unitaryRepresentation τ ω v g : Q τ ω)=0 := by |
| rw [← FreeGroup.mk_toWord (x := g),unitaryRepresentation_mk] |
| apply successive_reduced_word_zero τ ω v D hv hep |
| (g.toWord.toFinset.sup Prod.fst+1) g.toWord FreeGroup.isReduced_toWord |
| (fun h => hg (FreeGroup.toWord_eq_nil_iff.mp h)) |
| intro x hx |
| have := Finset.le_sup (f := Prod.fst) (List.mem_toFinset.mpr hx) |
| omega |
|
|
| / |
| theorem unitaryRepresentation_injective {A : Type*} (v : A → unitary (Q τ ω)) |
| (hzero : ∀ g : FreeGroup A,g ≠ 1 → |
| quotientTrace τ ω (unitaryRepresentation τ ω v g : Q τ ω)=0) : |
| Function.Injective (unitaryRepresentation τ ω v) := by |
| apply (injective_iff_map_eq_one (unitaryRepresentation τ ω v)).mpr |
| intro g hg |
| by_contra hne |
| have ht := hzero g hne |
| rw [hg] at ht |
| have : (1 : ℂ)=0 := (quotientTrace_one τ ω).symm.trans ht |
| exact one_ne_zero this |
|
|
| theorem unitaryRepresentation_gram {A : Type*} [DecidableEq A] |
| (v : A → unitary (Q τ ω)) |
| (hzero : ∀ g : FreeGroup A,g ≠ 1 → |
| quotientTrace τ ω (unitaryRepresentation τ ω v g : Q τ ω)=0) |
| (g h : FreeGroup A) : |
| quotientTrace τ ω (star (unitaryRepresentation τ ω v g : Q τ ω)* |
| (unitaryRepresentation τ ω v h : Q τ ω))=if g=h then 1 else 0 := by |
| have he : star (unitaryRepresentation τ ω v g : Q τ ω)* |
| (unitaryRepresentation τ ω v h : Q τ ω)= |
| (unitaryRepresentation τ ω v (g⁻¹*h) : Q τ ω) := by |
| exact (congrArg (fun a : unitary (Q τ ω) => (a : Q τ ω)) |
| (((unitaryRepresentation τ ω v).map_mul (g⁻¹) h).trans |
| (congrArg (fun a => a*unitaryRepresentation τ ω v h) |
| ((unitaryRepresentation τ ω v).map_inv g)))).symm |
| apply (congrArg (quotientTrace τ ω) he).trans |
| by_cases hgh : g=h |
| · subst h |
| rw [_root_.inv_mul_cancel,map_one,ite_eq_left rfl] |
| exact quotientTrace_one τ ω |
| · rw [ite_eq_right hgh] |
| exact hzero _ (fun heq => hgh (_root_.inv_mul_eq_one.mp heq)) |
|
|
| variable {hNM : N ≤ M} {seed : Set (Q τ ω)} |
|
|
| theorem SuccessiveCoordinateHaar.freeGroup_trace |
| (F : SuccessiveCoordinateHaar τ ω hNM seed) |
| (u : unitary (Q τ ω)) (hu : (u : Q τ ω) ∈ seed) |
| (g : FreeGroup ℕ) (hg : g ≠ 1) : |
| quotientTrace τ ω |
| (unitaryRepresentation τ ω (prefixVariable τ ω u (fun k => (F.step k).t)) g : Q τ ω)=0 := |
| successive_freeGroup_trace τ ω _ F.algebra |
| (fun _ _ hij => F.variable_mem τ ω u hu hij) (F.variable_endpoints τ ω u hu) g hg |
|
|
| theorem SuccessiveCoordinateHaar.freeGroup_injective |
| (F : SuccessiveCoordinateHaar τ ω hNM seed) |
| (u : unitary (Q τ ω)) (hu : (u : Q τ ω) ∈ seed) : |
| Function.Injective (unitaryRepresentation τ ω (prefixVariable τ ω u (fun k => (F.step k).t))) := |
| unitaryRepresentation_injective τ ω _ (F.freeGroup_trace τ ω u hu) |
|
|
| theorem SuccessiveCoordinateHaar.freeGroup_gram |
| (F : SuccessiveCoordinateHaar τ ω hNM seed) |
| (u : unitary (Q τ ω)) (hu : (u : Q τ ω) ∈ seed) (g h : FreeGroup ℕ) : |
| quotientTrace τ ω |
| (star (unitaryRepresentation τ ω (prefixVariable τ ω u (fun k => (F.step k).t)) g : Q τ ω)* |
| (unitaryRepresentation τ ω (prefixVariable τ ω u (fun k => (F.step k).t)) h : Q τ ω))= |
| if g=h then 1 else 0 := |
| unitaryRepresentation_gram τ ω _ (F.freeGroup_trace τ ω u hu) g h |
|
|
| end |
|
|
| /-! Reduced-word moments for p*y*(star p). Distinct labels have centered ratios; |
| reducedness requires distinct labels precisely at opposite-sign gaps. -/ |
| section |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
| variable {A : Type*} |
|
|
| def conjugateVariables (p : A → unitary (Q τ ω)) (y : unitary (Q τ ω)) |
| (a : A) : unitary (Q τ ω) := p a*y*star (p a) |
|
|
| def conjugateGaps (p : A → unitary (Q τ ω)) (a : A × Bool) : |
| List (A × Bool) → List (Q τ ω × Bool) |
| | [] => [] |
| | b::w => ((↑(star (p a.1)*p b.1) : Q τ ω),!b.2)::conjugateGaps p b w |
|
|
| def conjugateEnd (p : A → unitary (Q τ ω)) (a : A × Bool) : List (A × Bool) → Q τ ω |
| | [] => ↑(star (p a.1)) |
| | b::w => conjugateEnd p b w |
|
|
| theorem signed_conjugate (p y : unitary (Q τ ω)) (s : Bool) : |
| algebraSignedLetter (↑(p*y*(star p)) : Q τ ω) s= |
| (p : Q τ ω)*algebraSignedLetter (y : Q τ ω) s*↑(star p) := by |
| cases s with |
| | false => rfl |
| | true => |
| change star ((p : Q τ ω)*(y : Q τ ω)*star (p : Q τ ω))= |
| (p : Q τ ω)*star (y : Q τ ω)*star (p : Q τ ω) |
| rw [star_mul,star_mul,star_star,mul_assoc] |
|
|
| theorem conjugateWord_formula (p : A → unitary (Q τ ω)) (y : unitary (Q τ ω)) |
| (a : A × Bool) (w : List (A × Bool)) : |
| unitaryWord τ ω (conjugateVariables τ ω p y) (a::w)= |
| (p a.1 : Q τ ω)*betweenWord (y : Q τ ω) (!a.2) |
| (conjugateGaps τ ω p a w) (conjugateEnd τ ω p a w) := by |
| induction w generalizing a with |
| | nil => |
| rw [unitaryWord_cons,unitaryWord_nil,mul_one] |
| change algebraSignedLetter (↑(p a.1*y*star (p a.1)) : Q τ ω) (!a.2)=_ |
| rw [signed_conjugate] |
| exact mul_assoc _ _ _ |
| | cons b w ih => |
| rw [unitaryWord_cons,ih] |
| change algebraSignedLetter (↑(p a.1*y*star (p a.1)) : Q τ ω) (!a.2)*_= _ |
| rw [signed_conjugate] |
| simp only [conjugateGaps,conjugateEnd,betweenWord,Submonoid.coe_mul,mul_assoc] |
|
|
| theorem conjugateEnd_mem (p : A → unitary (Q τ ω)) |
| (D : StarSubalgebra ℂ (Q τ ω)) (hp : ∀ a,(p a : Q τ ω) ∈ D) |
| (a : A × Bool) (w : List (A × Bool)) : conjugateEnd τ ω p a w ∈ D := by |
| induction w generalizing a with |
| | nil => exact star_mem (hp a.1) |
| | cons b w ih => exact ih b |
|
|
| theorem conjugateGaps_reduced (p : A → unitary (Q τ ω)) |
| (D : StarSubalgebra ℂ (Q τ ω)) (hp : ∀ a,(p a : Q τ ω) ∈ D) |
| (hzero : ∀ a b,a ≠ b → quotientTrace τ ω (↑(star (p a)*p b) : Q τ ω)=0) |
| (a : A × Bool) (w : List (A × Bool)) (hr : FreeGroup.IsReduced (a::w)) : |
| SignedReducedGaps τ ω D.toSubalgebra (!a.2) (conjugateGaps τ ω p a w) := by |
| induction w generalizing a with |
| | nil => trivial |
| | cons b w ih => |
| have ha : star (p a.1 : Q τ ω) ∈ D := star_mem (hp a.1) |
| refine ⟨D.mul_mem ha (hp b.1),?_,ih b (reduced_tail hr)⟩ |
| intro hsign |
| apply hzero |
| intro he |
| exact hsign (congrArg Bool.not ((FreeGroup.isReduced_cons_cons.mp hr).1 he)) |
|
|
| theorem conjugate_reduced_word_zero (p : A → unitary (Q τ ω)) |
| (D : StarSubalgebra ℂ (Q τ ω)) (hp : ∀ a,(p a : Q τ ω) ∈ D) |
| (hzero : ∀ a b,a ≠ b → quotientTrace τ ω (↑(star (p a)*p b) : Q τ ω)=0) |
| (y : unitary (Q τ ω)) (hy : HaarEndpoints τ ω D.toSubalgebra y) |
| (w : List (A × Bool)) (hr : FreeGroup.IsReduced w) (hw : w ≠ []) : |
| quotientTrace τ ω (unitaryWord τ ω (conjugateVariables τ ω p y) w)=0 := by |
| cases w with |
| | nil => exact (hw rfl).elim |
| | cons a w => |
| rw [conjugateWord_formula] |
| exact hy.signed_gaps τ ω _ (hp a.1) _ (conjugateEnd_mem τ ω p D hp a w) |
| _ _ (conjugateGaps_reduced τ ω p D hp hzero a w hr) |
|
|
| theorem conjugate_freeGroup_trace [DecidableEq A] (p : A → unitary (Q τ ω)) |
| (D : StarSubalgebra ℂ (Q τ ω)) (hp : ∀ a,(p a : Q τ ω) ∈ D) |
| (hzero : ∀ a b,a ≠ b → quotientTrace τ ω (↑(star (p a)*p b) : Q τ ω)=0) |
| (y : unitary (Q τ ω)) (hy : HaarEndpoints τ ω D.toSubalgebra y) |
| (g : FreeGroup A) (hg : g ≠ 1) : |
| quotientTrace τ ω (unitaryRepresentation τ ω (conjugateVariables τ ω p y) g : Q τ ω)=0 := by |
| rw [← FreeGroup.mk_toWord (x := g),unitaryRepresentation_mk] |
| exact conjugate_reduced_word_zero τ ω p D hp hzero y hy g.toWord |
| FreeGroup.isReduced_toWord (fun h => hg (FreeGroup.toWord_eq_nil_iff.mp h)) |
|
|
| end |
|
|
| /-! Positive and negative conjugate terms in first-order correlation. Same-sign gaps require no centering hypothesis. -/ |
| open UltrapowerBridge Filter |
| variable {H : Type*} [NormedAddCommGroup H] [InnerProductSpace ℂ H] [CompleteSpace H] |
| variable {M : VonNeumannAlgebra H} (τ : TracialRepresentation M) (ω : Ultrafilter ℕ) |
| variable {D : StarSubalgebra ℂ (Q τ ω)} {W : unitary (Q τ ω)} |
|
|
| theorem HaarEndpoints.base_correlation (hW : HaarEndpoints τ ω D.toSubalgebra W) |
| (xi : Q τ ω) (hxi : xi ∈ D) : quotientTrace τ ω (star xi*(W : Q τ ω))=0 := by |
| have hstar : star xi ∈ D := star_mem hxi |
| have h := hW.signed_gaps τ ω (star xi) hstar 1 D.one_mem false [] trivial |
| simpa only [betweenWord,algebraSignedLetter,Bool.false_eq_true,ite_false,mul_one] using h |
|
|
| theorem HaarEndpoints.positive_conjugate_correlation |
| (hW : HaarEndpoints τ ω D.toSubalgebra W) |
| (xi : Q τ ω) (hxi : xi ∈ D) (p z : unitary (Q τ ω)) |
| (hp : (p : Q τ ω) ∈ D) (hz : (z : Q τ ω) ∈ D) |
| (hcenter : quotientTrace τ ω (star (p : Q τ ω))=0) : |
| quotientTrace τ ω (star xi*(↑(p*(z*star W)*star p) : Q τ ω)*(W : Q τ ω))=0 := by |
| have hstar : star xi ∈ D := star_mem hxi |
| have hpstar : star (p : Q τ ω) ∈ D := star_mem hp |
| have h := hW.signed_gaps τ ω (star xi*(p : Q τ ω)*(z : Q τ ω)) |
| (D.mul_mem (D.mul_mem hstar hp) hz) 1 D.one_mem true [(star (p : Q τ ω),false)] |
| ⟨hpstar,fun _ => hcenter,trivial⟩ |
| change quotientTrace τ ω ((star xi*(p : Q τ ω)*(z : Q τ ω))* |
| (star (W : Q τ ω)*star (p : Q τ ω)*((W : Q τ ω)*1)))=0 at h |
| simpa only [Submonoid.coe_mul,Unitary.coe_star,mul_one,mul_assoc] using h |
|
|
| theorem HaarEndpoints.negative_conjugate_correlation |
| (hW : HaarEndpoints τ ω D.toSubalgebra W) |
| (xi : Q τ ω) (hxi : xi ∈ D) (p z : unitary (Q τ ω)) |
| (hp : (p : Q τ ω) ∈ D) (hz : (z : Q τ ω) ∈ D) : |
| quotientTrace τ ω (star xi*star (↑(p*(z*star W)*star p) : Q τ ω)*(W : Q τ ω))=0 := by |
| have hstar : star xi ∈ D := star_mem hxi |
| have hpstar : star (p : Q τ ω) ∈ D := star_mem hp |
| have hzstar : star (z : Q τ ω) ∈ D := star_mem hz |
| have h := hW.signed_gaps τ ω (star xi*(p : Q τ ω)) (D.mul_mem hstar hp) |
| 1 D.one_mem false [(star (z : Q τ ω)*star (p : Q τ ω),false)] |
| ⟨D.mul_mem hzstar hpstar,fun hh => (hh rfl).elim,trivial⟩ |
| change quotientTrace τ ω ((star xi*(p : Q τ ω))* |
| ((W : Q τ ω)*(star (z : Q τ ω)*star (p : Q τ ω))*((W : Q τ ω)*1)))=0 at h |
| simpa only [Submonoid.coe_mul,Unitary.coe_star,star_mul,star_star,mul_one,mul_assoc] using h |
|
|
| end Generator.Tracial |
|
|
| end |
|
|
| end OAI |
|
|
