| import OAI.Analysis.FreeGroupFactor.Nielsen |
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| namespace OAI |
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| noncomputable section |
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| universe uA uG uι |
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| open Set Metric Filter |
| open scoped NNReal Topology BigOperators ContDiff |
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| namespace FreeGroupFactorMain.Generation |
| variable {A : Type uA} [CStarAlgebra A] {ι : Type uι} |
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| def word (U : ι → unitary A) : FreeGroup ι →* A := |
| (unitary A).subtype.comp (FreeGroup.lift U) |
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| @[simp] theorem word_of (U : ι → unitary A) (i : ι) : word U (FreeGroup.of i) = (U i : A) := by |
| simp [word] |
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| @[simp] theorem word_inv (U : ι → unitary A) (g : FreeGroup ι) : |
| word U g⁻¹ = star (word U g) := by |
| change ((FreeGroup.lift U g⁻¹ : unitary A) : A) = _ |
| rw [map_inv] |
| rfl |
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| @[fun_prop] theorem word_continuous (g : FreeGroup ι) : Continuous (fun U : ι → unitary A => word U g) := by |
| induction g using FreeGroup.induction_on with |
| | one => simpa only [map_one] using (continuous_const : Continuous (fun _ : ι → unitary A => (1 : A))) |
| | of i => |
| simp only [word_of] |
| exact continuous_subtype_val.comp (continuous_apply i) |
| | inv_of i ih => simpa only [word_inv] using ih.star |
| | mul g h hg hh => |
| simp only [map_mul] |
| exact hg.mul hh |
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| def poly (U : ι → unitary A) (p : FreeGroup ι →₀ ℂ) : A := p.sum (fun g c => c • word U g) |
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| @[simp] theorem poly_zero (U : ι → unitary A) : poly U 0 = 0 := by simp [poly] |
| @[simp] theorem poly_add (U : ι → unitary A) (p q : FreeGroup ι →₀ ℂ) : |
| poly U (p+q) = poly U p + poly U q := |
| (Finsupp.linearCombination ℂ (word U)).map_add p q |
| @[simp] theorem poly_single (U : ι → unitary A) (g : FreeGroup ι) (c : ℂ) : |
| poly U (Finsupp.single g c) = c • word U g := by simp [poly] |
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| @[fun_prop] theorem poly_continuous (p : FreeGroup ι →₀ ℂ) : Continuous (fun U : ι → unitary A => poly U p) := by |
| classical |
| unfold poly Finsupp.sum |
| exact continuous_finsetSum _ (fun g _ => (word_continuous g).const_smul _) |
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| variable {G : Type uG} [Group G] |
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| @[simp] theorem twoNorm_zero : twoNorm (0 : groupVonNeumann G) = 0 := by |
| simp [twoNorm] |
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| theorem twoNorm_add_le (x y : groupVonNeumann G) : twoNorm (x+y) ≤ twoNorm x + twoNorm y := by |
| change ‖groupL2 (x+y)‖ ≤ _ |
| rw [map_add] |
| exact norm_add_le _ _ |
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| theorem twoNorm_smul (c : ℂ) (x : groupVonNeumann G) : twoNorm (c • x) = ‖c‖ * twoNorm x := by |
| change ‖groupL2 (c • x)‖ = _ |
| rw [map_smul, norm_smul] |
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| theorem twoNorm_mul_sub_le (U V U' V' : unitary (groupVonNeumann G)) : |
| twoNorm ((U : groupVonNeumann G)*(V : groupVonNeumann G)- |
| (U' : groupVonNeumann G)*(V' : groupVonNeumann G)) ≤ |
| twoNorm ((U : groupVonNeumann G)-(U' : groupVonNeumann G)) + |
| twoNorm ((V : groupVonNeumann G)-(V' : groupVonNeumann G)) := by |
| have he : (U : groupVonNeumann G)*(V : groupVonNeumann G)- |
| (U' : groupVonNeumann G)*(V' : groupVonNeumann G) = |
| ((U : groupVonNeumann G)-(U' : groupVonNeumann G))*(V : groupVonNeumann G) + |
| (U' : groupVonNeumann G)*((V : groupVonNeumann G)-(V' : groupVonNeumann G)) := by noncomm_ring |
| rw [he] |
| exact (twoNorm_add_le _ _).trans_eq (by rw [twoNorm_mul_unitary_right, twoNorm_mul_unitary_left]) |
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| theorem word_twoNorm_bound (g : FreeGroup ι) : |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ (U V : ι → unitary (groupVonNeumann G)) (δ : ℝ), 0 ≤ δ → |
| (∀ j, twoNorm ((U j : groupVonNeumann G)-(V j : groupVonNeumann G)) ≤ δ) → |
| twoNorm (word U g-word V g) ≤ K*δ := by |
| induction g using FreeGroup.induction_on with |
| | one => |
| refine ⟨0,le_rfl,?_⟩ |
| intro U V δ _hδ _hUV |
| simp only [map_one, sub_self, twoNorm_zero, zero_mul, le_refl] |
| | of j => |
| refine ⟨1,by norm_num,?_⟩ |
| intro U V δ _hδ hUV |
| simpa only [word_of,one_mul] using hUV j |
| | inv_of j ih => |
| obtain ⟨K,hK,hb⟩ := ih |
| refine ⟨K,hK,?_⟩ |
| intro U V δ hδ hUV |
| rw [word_inv, word_inv, ← star_sub, twoNorm_star] |
| exact hb U V δ hδ hUV |
| | mul g h hg hh => |
| obtain ⟨K,hK,hb⟩ := hg |
| obtain ⟨L,hL,hl⟩ := hh |
| refine ⟨K+L,add_nonneg hK hL,?_⟩ |
| intro U V δ hδ hUV |
| rw [map_mul,map_mul] |
| exact (twoNorm_mul_sub_le (FreeGroup.lift U g) (FreeGroup.lift U h) |
| (FreeGroup.lift V g) (FreeGroup.lift V h)).trans |
| (by simpa only [add_mul, word, MonoidHom.comp_apply, Submonoid.subtype_apply] using add_le_add (hb U V δ hδ hUV) (hl U V δ hδ hUV)) |
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| theorem poly_twoNorm_bound (p : FreeGroup ι →₀ ℂ) : |
| ∃ K : ℝ, 0 ≤ K ∧ ∀ (U V : ι → unitary (groupVonNeumann G)) (δ : ℝ), 0 ≤ δ → |
| (∀ j, twoNorm ((U j : groupVonNeumann G)-(V j : groupVonNeumann G)) ≤ δ) → |
| twoNorm (poly U p-poly V p) ≤ K*δ := by |
| induction p using Finsupp.induction_linear with |
| | zero => |
| refine ⟨0,le_rfl,?_⟩ |
| intro U V δ _hδ _hUV |
| simp only [poly_zero,sub_self,twoNorm_zero,zero_mul,le_refl] |
| | add p q ihp ihq => |
| obtain ⟨K,hK,hb⟩ := ihp |
| obtain ⟨L,hL,hl⟩ := ihq |
| refine ⟨K+L,add_nonneg hK hL,?_⟩ |
| intro U V δ hδ hUV |
| simp only [poly_add] |
| have he : poly U p+poly U q-(poly V p+poly V q) = |
| (poly U p-poly V p)+(poly U q-poly V q) := by abel |
| rw [he] |
| exact (twoNorm_add_le _ _).trans (by |
| simpa only [add_mul] using add_le_add (hb U V δ hδ hUV) (hl U V δ hδ hUV)) |
| | single g c => |
| obtain ⟨K,hK,hb⟩ := word_twoNorm_bound (G := G) g |
| refine ⟨‖c‖*K,mul_nonneg (norm_nonneg _) hK,?_⟩ |
| intro U V δ hδ hUV |
| simp only [poly_single,← smul_sub,twoNorm_smul] |
| exact (mul_le_mul_of_nonneg_left (hb U V δ hδ hUV) (norm_nonneg c)).trans_eq (mul_assoc _ _ _).symm |
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| end FreeGroupFactorMain.Generation |
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| namespace FreeGroupFactorMain |
| open UnitaryEvaluation |
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| @[simp] theorem generator_haar {ι : Type uι} (i : ι) (m : ℤ) : |
| canonicalTrace (((generatorUnitary i)^m : unitary (groupVonNeumann (FreeGroup ι))) : groupVonNeumann (FreeGroup ι)) = |
| if m = 0 then 1 else 0 := by |
| classical |
| have hz : (((generatorUnitary i)^m : unitary (groupVonNeumann (FreeGroup ι))) : groupVonNeumann (FreeGroup ι)) = |
| groupUnitary ((FreeGroup.of i)^m) := |
| congrArg Subtype.val ((groupUnitaryUHom (FreeGroup ι)).map_zpow (FreeGroup.of i) m).symm |
| rw [hz,canonicalTrace_groupUnitary] |
| have hi : (FreeGroup.of i)^m = 1 ↔ m = 0 := by |
| constructor |
| · intro h |
| have he := congrArg (FreeGroup.lift (fun _ : ι => Multiplicative.ofAdd (1 : ℤ))) h |
| simp only [map_zpow,FreeGroup.lift_apply_of,map_one] at he |
| simpa using congrArg Multiplicative.toAdd he |
| · rintro rfl; simp |
| simp only [hi] |
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| theorem trace_symm {G : Type uG} [Group G] |
| (γ : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann G) |
| (ht : ∀ T, canonicalTrace (γ T) = canonicalTrace T) (T : groupVonNeumann G) : |
| canonicalTrace (γ.symm T) = canonicalTrace T := by |
| simpa only [γ.apply_symm_apply] using (ht (γ.symm T)).symm |
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| theorem norm_symm_sub {A : Type uA} [CStarAlgebra A] |
| (γ : A ≃⋆ₐ[ℂ] A) (X Y : A) : ‖γ.symm X - Y‖ = ‖X - γ Y‖ := by |
| have he : γ.symm (X - γ Y) = γ.symm X - Y := by simp only [map_sub,γ.symm_apply_apply] |
| rw [← he,StarAlgEquiv.norm_map] |
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| theorem twoNorm_symm_sub {G : Type uG} [Group G] |
| (γ : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann G) |
| (ht : ∀ T, canonicalTrace (γ T) = canonicalTrace T) (X Y : groupVonNeumann G) : |
| twoNorm (γ.symm X-Y) = twoNorm (X-γ Y) := by |
| have he : γ.symm (X-γ Y) = γ.symm X-Y := by simp only [map_sub,γ.symm_apply_apply] |
| rw [← he,twoNorm_iso γ.symm (trace_symm γ ht)] |
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| theorem laurentValue_eq_value {A : Type uA} [CStarAlgebra A] (C : unitary A) (l : ℤ →₀ ℂ) : |
| laurentValue C l = LaurentApprox.value C l := rfl |
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| theorem firstCoefficients_norm_le {n : ℕ} (h : Coeff (FreeGroup (Fin n))) (j : Fin n) : |
| coefficientNorm (firstCoefficients h j) ≤ coefficientNorm h := by |
| unfold firstCoefficients |
| split_ifs |
| · exact le_rfl |
| · simp only [coefficientNorm,Finsupp.sum_zero_index,Real.sqrt_zero] |
| exact Real.sqrt_nonneg _ |
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| theorem one_step_generator_estimate {G : Type uG} [Group G] |
| (β γ : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann G) |
| (A : groupVonNeumann G) (hγ : γ A = A) |
| {K ε c : ℝ} (hK : 0 < K) (hε : 0 < ε) |
| (hb : ‖β A-A‖ ≤ 3*K*c) (hc : c < (ε/2)/(3*K)) : |
| ‖γ.symm (β A)-A‖ < ε := by |
| rw [norm_symm_sub,hγ] |
| have hd := (lt_div_iff₀ (by positivity : 0 < 3*K)).mp hc |
| nlinarith |
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| theorem one_step_word_estimate {G : Type uG} [Group G] |
| (β γ : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann G) |
| (hγt : ∀ T, canonicalTrace (γ T) = canonicalTrace T) |
| (C W : unitary (groupVonNeumann G)) (E : groupVonNeumann G) (z : unitary ℂ) |
| (hγC : γ (C : groupVonNeumann G) = (z : ℂ) • ((C : groupVonNeumann G)*(W : groupVonNeumann G))) |
| {K ε c : ℝ} (hK : 0 < K) |
| (hb : ‖β (W : groupVonNeumann G)-E*(W : groupVonNeumann G)‖ ≤ 3*K*c) |
| (hc : c < (ε/2)/(3*K)) (hze : twoNorm (E-(z : ℂ) • (C : groupVonNeumann G)) < ε/2) : |
| twoNorm (γ.symm (β (W : groupVonNeumann G))-(C : groupVonNeumann G)) < ε := by |
| rw [twoNorm_symm_sub γ hγt,hγC] |
| have he : β (W : groupVonNeumann G) - (z : ℂ) • ((C : groupVonNeumann G)*(W : groupVonNeumann G)) = |
| (β (W : groupVonNeumann G)-E*(W : groupVonNeumann G)) + |
| (E-(z : ℂ) • (C : groupVonNeumann G))*(W : groupVonNeumann G) := by |
| simp only [sub_mul,smul_mul_assoc] |
| abel |
| rw [he] |
| have ht := Generation.twoNorm_add_le |
| (β (W : groupVonNeumann G)-E*(W : groupVonNeumann G)) |
| ((E-(z : ℂ) • (C : groupVonNeumann G))*(W : groupVonNeumann G)) |
| rw [twoNorm_mul_unitary_right] at ht |
| have hle := twoNorm_le_norm (β (W : groupVonNeumann G)-E*(W : groupVonNeumann G)) |
| have hd := (lt_div_iff₀ (by positivity : 0 < 3*K)).mp hc |
| nlinarith |
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| theorem one_step_from_laurent (n : ℕ) {ε : ℝ} (hε : 0 < ε) |
| (l : ℤ →₀ ℂ) (hl0 : l 0 = 0) |
| (hlS : IsSelfAdjoint (laurentValue (generatorUnitary (none : Option (Fin n))) l)) |
| (hln : ‖laurentValue (generatorUnitary (none : Option (Fin n))) l‖ ≤ 2*(Real.pi+1)) |
| (z : unitary ℂ) (hze : twoNorm (NormedSpace.exp (Complex.I • laurentValue |
| (generatorUnitary (none : Option (Fin n))) l) - (z : ℂ) • groupUnitary (FreeGroup.of none)) < ε/2) |
| (w : FreeGroup (Fin n)) (h : Coeff (FreeGroup (Fin n))) |
| (hh : coefficientNorm h < (ε/2)/(3*(2*(Real.pi+1)))) |
| (hw : coefficientNorm (freeCocycle (firstCoefficients h) w-Finsupp.single 1 1) < |
| (ε/2)/(3*(2*(Real.pi+1)))) : |
| ∃ α : groupVonNeumann (FreeGroup (Option (Fin n))) ≃⋆ₐ[ℂ] groupVonNeumann (FreeGroup (Option (Fin n))), |
| (∀ T, canonicalTrace (α T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (Fin n)))) |
| (groupUltraweakTopology (FreeGroup (Option (Fin n)))) α ∧ |
| ∃ w : FreeGroup (Fin n), |
| (∀ j : Fin n, ‖α (groupUnitary (FreeGroup.of (some j))) - groupUnitary (FreeGroup.of (some j))‖ < ε) ∧ |
| twoNorm (α (groupUnitary (FreeGroup.map Option.some w)) - groupUnitary (FreeGroup.of none)) < ε := by |
| let K : ℝ := 2*(Real.pi+1) |
| have hK : 0 < K := by dsimp [K]; positivity |
| obtain ⟨β,hβt,hβn,_hβC,hβgen,hβword⟩ := |
| laurent_flow_displacement (ι := Fin n) (firstCoefficients h) l hl0 hlS K hK.le hln |
| obtain ⟨γ,hγt,_hγn,hγi,hγgen,hγC⟩ := gauged_nielsen_iso (ι := Fin n) w z |
| have hnormal : @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (Fin n)))) |
| (groupUltraweakTopology (FreeGroup (Option (Fin n)))) (β.trans γ.symm) := by |
| let : TopologicalSpace (groupVonNeumann (FreeGroup (Option (Fin n)))) := |
| groupUltraweakTopology (FreeGroup (Option (Fin n))) |
| exact hγi.comp hβn |
| refine ⟨β.trans γ.symm,?_,hnormal,w,?_,?_⟩ |
| · intro T |
| change canonicalTrace (γ.symm (β T)) = canonicalTrace T |
| rw [trace_symm γ hγt,hβt] |
| · intro j |
| exact one_step_generator_estimate (G := FreeGroup (Option (Fin n))) β γ _ (hγgen j) hK hε (hβgen j) |
| ((firstCoefficients_norm_le h j).trans_lt hh) |
| · exact one_step_word_estimate (G := FreeGroup (Option (Fin n))) β γ hγt (generatorUnitary (none : Option (Fin n))) |
| (groupUnitaryUHom (FreeGroup (Option (Fin n))) (FreeGroup.map Option.some w)) |
| _ z hγC hK (hβword w) hw hze |
|
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| theorem one_step (n : ℕ) (hn : 3 ≤ n) {ε : ℝ} (hε : 0 < ε) : |
| ∃ α : groupVonNeumann (FreeGroup (Option (Fin n))) ≃⋆ₐ[ℂ] groupVonNeumann (FreeGroup (Option (Fin n))), |
| (∀ T, canonicalTrace (α T) = canonicalTrace T) ∧ |
| @Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (Fin n)))) |
| (groupUltraweakTopology (FreeGroup (Option (Fin n)))) α ∧ |
| ∃ w : FreeGroup (Fin n), |
| (∀ j : Fin n, ‖α (groupUnitary (FreeGroup.of (some j))) - groupUnitary (FreeGroup.of (some j))‖ < ε) ∧ |
| twoNorm (α (groupUnitary (FreeGroup.map Option.some w)) - groupUnitary (FreeGroup.of none)) < ε := by |
| obtain ⟨l,hl0,hlS,hln,z,hze⟩ := haar_centered_laurent (G := FreeGroup (Option (Fin n))) |
| (generatorUnitary (none : Option (Fin n))) (generator_haar (none : Option (Fin n))) (half_pos hε) |
| rw [← laurentValue_eq_value] at hlS hln hze |
| obtain ⟨w,h,hh,hw⟩ := smallCocycle n hn ((ε/2)/(3*(2*(Real.pi+1)))) (by positivity) |
| exact one_step_from_laurent n hε l hl0 hlS hln z hze w h hh hw |
|
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| end FreeGroupFactorMain |
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| end |
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| end OAI |
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