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namespace OAI
noncomputable section
universe u_1 u_2
namespace FreeGroupFactorMain.Amplification
section
open scoped ENNReal
variable {G : Type u_1} [Group G]
theorem cornerStabilization_weight_iff (a : ℝ) (ha : 0<a) (p : stabilization G) (hp : IsStarProjection p)
(t : ℝ) (_ht : 0≤t) (q : CornerStabilization p hp) :
cornerStabilizationProjectionTrace a p hp q=ENNReal.ofReal t ↔
stabilizedProjectionTrace q.val=ENNReal.ofReal (a*t) := by
have ha0 : ENNReal.ofReal a≠0 := ne_of_gt (ENNReal.ofReal_pos.mpr ha)
have hat : ENNReal.ofReal a≠⊤ := ENNReal.ofReal_ne_top
unfold cornerStabilizationProjectionTrace
rw [ENNReal.ofReal_mul ha.le]
constructor
· intro h
have hh := congrArg (fun z : ℝ≥0∞ => ENNReal.ofReal a*z) h
simpa only [←mul_assoc,ENNReal.mul_inv_cancel ha0 hat,one_mul] using hh
· intro h
rw [h,←mul_assoc,ENNReal.inv_mul_cancel ha0 hat,one_mul]
theorem doubleDiagonal_matrix (T : stabilization G) (i j : ℕ) :
matrixEntry (doubleDiagonal T : GroupOperator (ℕ × (ℕ × G))) i j=
if i=j then (T : GroupOperator (ℕ × G)) else 0 := by
change matrixEntry (productAmplification ℕ (ℕ × G) T) i j=_
ext ξ x
rw [matrixEntry_apply,productAmplification_productSingle,productSingle_apply]
by_cases h : i=j <;> simp [h]
theorem doubleFiniteDiagonal_supported (n : ℕ) (p : stabilization G) (hp : IsStarProjection p)
(q : Corner p hp) :
doubleDiagonal p * doubleFiniteDiagonal n q.val=doubleFiniteDiagonal n q.val ∧
doubleFiniteDiagonal n q.val * doubleDiagonal p=doubleFiniteDiagonal n q.val := by
constructor
· apply Subtype.ext
apply matrixEntry_ext
intro i j
change matrixEntry (productAmplification ℕ (ℕ × G) (p : GroupOperator (ℕ × G))*finiteDiagonal n (q.val : GroupOperator (ℕ × G))) i j=matrixEntry (finiteDiagonal n (q.val : GroupOperator (ℕ × G))) i j
rw [matrixEntry_diagonal_mul,finiteDiagonal_sum]
simp only [matrixEntry_sum,matrixEntry_matrixBlock,Finset.mul_sum]
apply Finset.sum_congr rfl
intro k _
split_ifs
· exact congrArg Subtype.val q.property.1
· exact mul_zero _
· apply Subtype.ext
apply matrixEntry_ext
intro i j
change matrixEntry (finiteDiagonal n (q.val : GroupOperator (ℕ × G))*productAmplification ℕ (ℕ × G) (p : GroupOperator (ℕ × G))) i j=matrixEntry (finiteDiagonal n (q.val : GroupOperator (ℕ × G))) i j
rw [matrixEntry_mul_diagonal,finiteDiagonal_sum]
simp only [matrixEntry_sum,matrixEntry_matrixBlock,Finset.sum_mul]
apply Finset.sum_congr rfl
intro k _
split_ifs
· exact congrArg Subtype.val q.property.2
· exact zero_mul _
theorem finiteFlatten_supported (n : ℕ) (p : stabilization G) (hp : IsStarProjection p)
(q : Corner p hp) : repeatFlatten p * finiteFlatten n q.val=finiteFlatten n q.val ∧
finiteFlatten n q.val * repeatFlatten p=finiteFlatten n q.val := by
obtain ⟨hl,hr⟩ := doubleFiniteDiagonal_supported n p hp q
constructor
· have h := congrArg (flattenStabilizationIso (G := G)) hl
simpa only [map_mul,repeatFlatten,finiteFlatten] using h
· have h := congrArg (flattenStabilizationIso (G := G)) hr
simpa only [map_mul,repeatFlatten,finiteFlatten] using h
def cornerFiniteProjection (n : ℕ) (p : stabilization G) (hp : IsStarProjection p)
(q : Corner p hp) : CornerStabilization p hp :=
⟨finiteFlatten n q.val,finiteFlatten_supported n p hp q⟩
theorem cornerFiniteProjection_projection (n : ℕ) (p : stabilization G) (hp : IsStarProjection p)
(q : Corner p hp) (hq : IsStarProjection q) : IsStarProjection (cornerFiniteProjection n p hp q) := by
have hh := finiteFlatten_projection n q.val (nested_projection p hp q hq)
constructor
· exact Subtype.ext hh.isIdempotentElem.eq
· exact Subtype.ext hh.isSelfAdjoint.star_eq
end
open scoped ENNReal
variable {G : Type u_2} [Group G]
theorem projection_weight_of_trace (p : stabilization G) (hp : IsStarProjection p)
(hfin : stabilizedProjectionTrace p≠⊤) (a : ℝ) (hτ : stabilizedTrace p=(a:ℂ)) :
stabilizedProjectionTrace p=ENNReal.ofReal a := by
have h := stabilizedTrace_star_mul p hfin
rw [hp.isSelfAdjoint.star_eq,hp.isIdempotentElem.eq,hτ] at h
have he : (stabilizedProjectionTrace p).toReal=a := Complex.ofReal_injective h.symm
rw [←he,ENNReal.ofReal_toReal hfin]
theorem corner_projection_weight (a : ℝ) (ha : 0<a) (p : stabilization G) (hp : IsStarProjection p)
(hpτ : stabilizedProjectionTrace p=ENNReal.ofReal a) (s : ℝ)
(q : Corner p hp) (hq : IsStarProjection q)
(hqτ : normalizedCornerTrace stabilizedTrace a p hp q=(s:ℂ)) :
stabilizedProjectionTrace q.val=ENNReal.ofReal (a*s) := by
apply projection_weight_of_trace q.val (nested_projection p hp q hq)
(finite_weight_corner p hp (hpτ ▸ ENNReal.ofReal_ne_top) q) (a*s)
have ha0 : (a:ℂ)≠0 := Complex.ofReal_ne_zero.mpr ha.ne'
have h := congrArg (fun z : ℂ => (a:ℂ)*z) hqτ
change (a:ℂ)*((a:ℂ)⁻¹*stabilizedTrace q.val)=(a:ℂ)*(s:ℂ) at h
rw [←mul_assoc,mul_inv_cancel₀ ha0,one_mul] at h
simpa only [Complex.ofReal_mul] using h
theorem cornerStabilization_projection_exists
(hpos : ∀(t : ℝ),0<t → ∀(p : stabilization G)(hp : IsStarProjection p),
stabilizedProjectionTrace p=ENNReal.ofReal t → Nonempty (GroupCornerNormalIso t p hp))
(hsmall : ∀t : ℝ,0<t → t≤1 → HasCornerScaling (canonicalTrace (G := G)) (groupUltraweakTopology G) t)
(a : ℝ) (ha : 0<a) (p : stabilization G) (hp : IsStarProjection p)
(hpτ : stabilizedProjectionTrace p=ENNReal.ofReal a) (t : ℝ) (ht : 0<t) :
∃q : CornerStabilization p hp,IsStarProjection q ∧
cornerStabilizationProjectionTrace a p hp q=ENNReal.ofReal t := by
obtain ⟨n,hn⟩ := exists_nat_gt t
have hn0 : 0<(n:ℝ) := ht.trans hn
have hs : 0<t/(n:ℝ) := div_pos ht hn0
have hs1 : t/(n:ℝ)≤1 := (div_le_one hn0).mpr hn.le
obtain ⟨E⟩ := hpos a ha p hp hpτ
have hx := HasCornerScaling.transport (canonicalTrace (G := G))
(normalizedCornerTrace stabilizedTrace a p hp) (groupUltraweakTopology G)
(cornerTopology (stabilizationTopology G) p hp) E.equiv.symm E.inverse_normal E.normal
E.symm.trace_eq (hsmall _ hs hs1)
obtain ⟨q,hq,_,hqτ,_,_,_⟩ := hx
refine ⟨cornerFiniteProjection n p hp q,cornerFiniteProjection_projection n p hp q hq,?_⟩
apply (cornerStabilization_weight_iff a ha p hp t ht.le _).mpr
change stabilizedProjectionTrace (finiteFlatten n q.val)=_
rw [finiteFlatten_trace,corner_projection_weight a ha p hp hpτ (t/(n:ℝ)) q hq hqτ]
rw [←ENNReal.ofReal_natCast,←ENNReal.ofReal_mul (Nat.cast_nonneg n)]
congr 1
field_simp
def nestedNormalizedIso (a t : ℝ) (p : stabilization G) (hp : IsStarProjection p)
(q : CornerStabilization p hp) (hq : IsStarProjection q) :
NormalTracialEquiv
(normalizedCornerTrace (cornerStabilizationTrace a p hp) t q hq)
(normalizedCornerTrace stabilizedTrace (a*t) q.val
(nested_projection (repeatFlatten p) (repeatFlatten_projection p hp) q hq))
(cornerTopology (cornerStabilizationTopology p hp) q hq)
(cornerTopology (stabilizationTopology G) q.val
(nested_projection (repeatFlatten p) (repeatFlatten_projection p hp) q hq)) where
equiv := nestedCornerEquiv (repeatFlatten p) (repeatFlatten_projection p hp) q hq
trace_eq x := by
change ((a*t:ℝ):ℂ)⁻¹ * stabilizedTrace x.val.val=(t:ℂ)⁻¹*((a:ℂ)⁻¹*stabilizedTrace x.val.val)
rw [Complex.ofReal_mul,mul_inv_rev, mul_assoc]
normal := nestedCornerEquiv_normal _ _ _ _ _
inverse_normal := nestedCornerEquiv_symm_normal _ _ _ _ _
theorem cornerStabilization_allPositive
(hpos : ∀(t : ℝ),0<t → ∀(p : stabilization G)(hp : IsStarProjection p),
stabilizedProjectionTrace p=ENNReal.ofReal t → Nonempty (GroupCornerNormalIso t p hp))
(a : ℝ) (ha : 0<a) (p : stabilization G) (hp : IsStarProjection p)
(hpτ : stabilizedProjectionTrace p=ENNReal.ofReal a) (t : ℝ) (ht : 0<t)
(q : CornerStabilization p hp) (hq : IsStarProjection q)
(hqτ : cornerStabilizationProjectionTrace a p hp q=ENNReal.ofReal t) :
Nonempty (NormalTracialEquiv
(normalizedCornerTrace (cornerStabilizationTrace a p hp) t q hq)
(normalizedCornerTrace stabilizedTrace a p hp)
(cornerTopology (cornerStabilizationTopology p hp) q hq)
(cornerTopology (stabilizationTopology G) p hp)) := by
have hqa := (cornerStabilization_weight_iff a ha p hp t ht.le q).mp hqτ
obtain ⟨E⟩ := hpos (a*t) (mul_pos ha ht) q.val
(nested_projection (repeatFlatten p) (repeatFlatten_projection p hp) q hq) hqa
obtain ⟨F⟩ := hpos a ha p hp hpτ
exact ⟨((nestedNormalizedIso a t p hp q hq).trans E).trans F.symm⟩
end FreeGroupFactorMain.Amplification
end
end OAI
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