| import OAI.Analysis.InterpolatedFactors.Positive |
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| namespace OAI |
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| noncomputable section |
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| universe u_1 u_2 |
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| namespace FreeGroupFactorMain.Amplification |
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| section |
| open scoped ENNReal |
| variable {G : Type u_1} [Group G] |
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| 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] |
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| 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] |
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| 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 _ |
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| 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 |
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| 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⟩ |
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| 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 |
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| open scoped ENNReal |
| variable {G : Type u_2} [Group G] |
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| 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] |
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| 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 |
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| 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 |
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| 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 _ _ _ _ _ |
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| 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 |
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| end |
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| end OAI |
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