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9eeb9c7 | 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 | 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
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