| 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 | |