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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 | import OAI.Analysis.FreeGroupFactor.RankTwoCoefficients
namespace OAI
namespace FreeGroupFactorMain
noncomputable section
open Set Metric Filter
open scoped NNReal Topology BigOperators ContDiff
attribute [local instance] Classical.propDecidable
open UnitaryEvaluation
theorem one_step_of_coefficient_estimate (n : ℕ)
(hc : ∀ η : ℝ, 0 < η → ∃ (w : FreeGroup (Fin n)) (h : Coeff (FreeGroup (Fin n))),
coefficientNorm h < η ∧
coefficientNorm (freeCocycle (firstCoefficients h) w-Finsupp.single 1 1) < η) {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⟩ := hc ((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
theorem one_step_two {ε : ℝ} (hε : 0 < ε) :
∃ α : groupVonNeumann (FreeGroup (Option (Fin 2))) ≃⋆ₐ[ℂ] groupVonNeumann (FreeGroup (Option (Fin 2))),
(∀ T, canonicalTrace (α T) = canonicalTrace T) ∧
@Continuous _ _ (groupUltraweakTopology (FreeGroup (Option (Fin 2))))
(groupUltraweakTopology (FreeGroup (Option (Fin 2)))) α ∧
∃ w : FreeGroup (Fin 2),
(∀ j : Fin 2, ‖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 :=
one_step_of_coefficient_estimate 2 smallCocycleTwo hε
theorem one_step_ge_two (n : ℕ) (hn : 2 ≤ 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
by_cases h : n = 2
· subst n
exact one_step_two hε
· exact one_step n (by omega) hε
namespace Generation
theorem orbit_step_ge_two (n : ℕ) (hn : 2 ≤ n)
(Φ : Ambient n ≃⋆ₐ[ℂ] Ambient n)
(ht : ∀ T, canonicalTrace (Φ T) = canonicalTrace T) {θ : ℝ} (hθ : 0 < θ) :
∃ V ∈ orbit n, dist V (mapTuple Φ (initial n)) < θ ∧
∃ w : FreeGroup (Fin n), ∀ j : Option (Fin n),
twoNorm ((wordTuple V (fun j => j.elim w FreeGroup.of) j : Ambient n) -
(mapTuple Φ generatorUnitary j : Ambient n)) < θ := by
obtain ⟨α,hαt,_hαn,w,hgen,hword⟩ := one_step_ge_two n hn hθ
let V : Tuple n := mapTuple (α.trans Φ) (initial n)
have hdist : dist V (mapTuple Φ (initial n)) < θ := by
apply (dist_pi_lt_iff hθ).mpr
intro j
change dist (Φ (α (groupUnitary (FreeGroup.of (some j))))) (Φ (groupUnitary (FreeGroup.of (some j)))) < θ
have hnorm (T : Ambient n) : ‖Φ T‖ = ‖T‖ :=
NonUnitalStarAlgHom.norm_map Φ.toStarAlgHom Φ.injective T
rw [dist_eq_norm, ← map_sub, hnorm]
exact hgen j
refine ⟨V,⟨α.trans Φ,?_,rfl⟩,hdist,w,?_⟩
· intro T
simp only [StarAlgEquiv.trans_apply,ht,hαt]
· intro j
cases j with
| none =>
change twoNorm (word V w-Φ (groupUnitary (FreeGroup.of none))) < θ
rw [show V = mapTuple (α.trans Φ) (initial n) from rfl,word_mapTuple,
word_initial,StarAlgEquiv.trans_apply,← map_sub,twoNorm_iso Φ ht]
exact hword
| some j =>
change twoNorm ((V j : Ambient n)-Φ (groupUnitary (FreeGroup.of (some j)))) < θ
have hj := (dist_pi_lt_iff hθ).mp hdist j
change dist (V j : Ambient n) (Φ (groupUnitary (FreeGroup.of (some j)))) < θ at hj
rw [dist_eq_norm] at hj
exact (twoNorm_le_norm _).trans_lt hj
theorem orbit_local_approximation_ge_two (n : ℕ) (hn : 2 ≤ n)
(U : Tuple n) (hU : U ∈ orbit n) (δ : ℝ) (hδ : 0 < δ)
(g : FreeGroup (Option (Fin n))) (ε : ℝ) (hε : 0 < ε) :
∃ V ∈ orbit n, dist V U < δ ∧ ∃ p : FreeGroup (Fin n) →₀ ℂ,
dist (groupL2 (poly V p)) (lp.single 2 g (1 : ℂ)) < ε := by
classical
obtain ⟨Φ,ht,rfl⟩ := hU
obtain ⟨p,hp⟩ := automorphism_poly_dense Φ ht (groupUnitary g) (half_pos hε)
obtain ⟨K,hK,hbound⟩ := poly_twoNorm_bound (G := FreeGroup (Option (Fin n))) p
let θ := min δ (ε/(2*(K+1)))
have hθ : 0 < θ := lt_min hδ (div_pos hε (by positivity))
obtain ⟨V,hV,hd,w,hw⟩ := orbit_step_ge_two n hn Φ ht hθ
refine ⟨V,hV,hd.trans_le (min_le_left _ _),substitute (fun j => j.elim w FreeGroup.of) p,?_⟩
have hb := hbound (wordTuple V (fun j => j.elim w FreeGroup.of)) (mapTuple Φ generatorUnitary)
θ hθ.le (fun j => (hw j).le)
have hθe : θ*(2*(K+1)) ≤ ε :=
(le_div_iff₀ (by positivity : 0 < 2*(K+1))).mp (min_le_right _ _)
have hKe : K*θ < ε/2 := by nlinarith
rw [dist_eq_norm,← groupL2_groupUnitary g,← map_sub]
change twoNorm (poly V (substitute (fun j => j.elim w FreeGroup.of) p)-groupUnitary g) < ε
rw [poly_substitute]
have he : poly (wordTuple V (fun j => j.elim w FreeGroup.of)) p-groupUnitary g =
(poly (wordTuple V (fun j => j.elim w FreeGroup.of)) p-poly (mapTuple Φ generatorUnitary) p)+
(poly (mapTuple Φ generatorUnitary) p-groupUnitary g) := by abel
rw [he]
exact (twoNorm_add_le _ _).trans_lt (by linarith)
end Generation
open Generation
theorem generating_haar_tuple_ge_two (n : ℕ) (hn : 2 ≤ n) :
∃ U : Tuple n,
(∀ g : FreeGroup (Fin n), canonicalTrace (word U g) = if g = 1 then 1 else 0) ∧
⊤ ≤ (Submodule.span ℂ (Set.range (fun g => groupL2 (word U g)))).topologicalClosure := by
classical
obtain ⟨U,hU,happrox⟩ := baire_local_approximation (orbit n) (orbit_nonempty n)
(fun p U => groupL2 (poly U p)) (poly_groupL2_continuous n)
(fun g : FreeGroup (Option (Fin n)) => lp.single 2 g (1 : ℂ))
(orbit_local_approximation_ge_two n hn)
exact ⟨U,closure_orbit_trace n U hU,total_of_poly_approx U happrox⟩
theorem rank_step_option_ge_two (n : ℕ) (hn : 2 ≤ n) :
HasNormalTracialIso (FreeGroupRank n) (FreeGroup (Option (Fin n))) := by
classical
obtain ⟨U,htrace,htotal⟩ := generating_haar_tuple_ge_two n hn
apply normalTracialIso_of_haar_total (word U)
· intro g
exact (word_inv U g).symm
· exact htrace
· exact htotal
theorem rank_step_ge_two (n : ℕ) (hn : 2 ≤ n) :
HasNormalTracialIso (FreeGroupRank n) (FreeGroupRank (n+1)) :=
(rank_step_option_ge_two n hn).trans (normalTracialIso_of_groupEquiv (optionRankEquiv n))
theorem rankTwo : HasNormalTracialIso (FreeGroupRank 2) (FreeGroupRank 3) :=
rank_step_ge_two 2 le_rfl
theorem rankStep (n : ℕ) (hn : 3 ≤ n) :
HasNormalTracialIso (FreeGroupRank n) (FreeGroupRank (n + 1)) := rank_step n hn
end
end FreeGroupFactorMain
end OAI
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