| import OAI.Analysis.InterpolatedFactors.CornerStabilization | |
| namespace OAI | |
| universe u_1 u_2 u_3 | |
| noncomputable section | |
| namespace FreeGroupFactorMain.Amplification | |
| variable {G : Type u_1} {H : Type u_2} [Group G] [Group H] | |
| theorem groupNormal_of_iso (h : HasNormalTracialIso G H) : | |
| Nonempty (NormalTracialEquiv (canonicalTrace (G := G)) (canonicalTrace (G := H)) | |
| (groupUltraweakTopology G) (groupUltraweakTopology H)) := by | |
| classical | |
| obtain ⟨Φ,hΦ,_⟩ := h | |
| obtain ⟨e,hδ,he⟩ := spatial_of_trace_iso Φ hΦ | |
| refine ⟨⟨spatialGroupIso e he,?_,spatialGroupIso_normal e he,spatialGroupIso_symm_normal e he⟩⟩ | |
| intro T | |
| rw [canonicalTrace_eq_inner,canonicalTrace_eq_inner] | |
| change inner ℂ deltaOne (e ((T : GroupOperator G) (e.symm deltaOne)))=_ | |
| rw [←hδ,e.symm_apply_apply,e.inner_map_map] | |
| theorem groupAllPositive_of_rankTwo (h : HasNormalTracialIso G (FreeGroupRank 2)) | |
| (t : ℝ) (ht : 0<t) : | |
| (∃p : stabilization G,IsStarProjection p ∧ stabilizedProjectionTrace p=ENNReal.ofReal t) ∧ | |
| ∀(p : stabilization G)(hp : IsStarProjection p),stabilizedProjectionTrace p=ENNReal.ofReal t → | |
| Nonempty (GroupCornerNormalIso t p hp) := by | |
| have hinv := h.symm | |
| obtain ⟨e,_,_⟩ := h | |
| apply allPositiveScaling_both (fun T hT => center_scalar_of_iso e rankTwo_center_scalar T hT) | |
| (fun s hs hs1 => smallScaling_transport_group hinv (rankTwo_smallScaling s hs hs1)) t ht | |
| end FreeGroupFactorMain.Amplification | |
| namespace FreeGroupFactorMain | |
| theorem groupIso_refl (G : Type u_3) [Group G] : HasNormalTracialIso G G := | |
| ⟨StarAlgEquiv.refl ℂ (groupVonNeumann G),fun _ => rfl,@continuous_id _ (groupUltraweakTopology G)⟩ | |
| theorem finiteRank_to_two (n : ℕ) (hn : 2≤n) : HasNormalTracialIso (FreeGroupRank n) (FreeGroupRank 2) := by | |
| induction n, hn using Nat.le_induction with | |
| | base => exact groupIso_refl _ | |
| | succ n hn ih => | |
| have he : HasNormalTracialIso (FreeGroupRank n) (FreeGroupRank (n+1)) := by | |
| rcases eq_or_lt_of_le hn with h | h | |
| · subst n | |
| exact rankTwo | |
| · exact rankStep n (by omega) | |
| exact he.symm.trans ih | |
| end FreeGroupFactorMain | |
| end | |
| end OAI | |