| import OAI.Analysis.InterpolatedFactors.NormalEquiv |
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| namespace OAI |
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| noncomputable section |
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| universe u_1 u_2 u_3 u_4 u_5 |
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| namespace FreeGroupFactorMain.Amplification |
| open scoped ENNReal |
| variable {G : Type u_1} [Group G] |
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| theorem finiteProjection_trace_div (n : ℕ) (hn : 0<(n:ℝ)) (t : ℝ) (ht : 0≤t) |
| (p : groupVonNeumann G) (hp : IsStarProjection p) (hpτ : canonicalTrace p=((t/(n:ℝ):ℝ):ℂ)) : |
| stabilizedProjectionTrace (finiteProjection n p)=ENNReal.ofReal t := by |
| rw [finiteProjection_trace n p hp _ (div_nonneg ht hn.le) hpτ] |
| congr 1 |
| field_simp |
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| theorem preferred_scaling_model |
| (hbase : ∀ T : groupVonNeumann G, (∀ S : groupVonNeumann G,Commute T S) → |
| ∃ c : ℂ,T=c • (1 : groupVonNeumann G)) |
| (hsmall : ∀t : ℝ,0<t → t≤1 → HasCornerScaling (canonicalTrace (G := G)) (groupUltraweakTopology G) t) |
| (t : ℝ) (ht : 0<t) : |
| ∃ (p : stabilization G) (hp : IsStarProjection p), |
| stabilizedProjectionTrace p=ENNReal.ofReal t ∧ Nonempty (GroupCornerNormalIso t p hp) := by |
| obtain ⟨n,hn⟩ := exists_nat_gt (max t 1) |
| have hnt : t<(n:ℝ) := lt_of_le_of_lt (le_max_left _ _) hn |
| have hn1 : 1<(n:ℝ) := lt_of_le_of_lt (le_max_right _ _) hn |
| have hn0 : 0<(n:ℝ) := by linarith |
| have ha : 0<t/(n:ℝ) := div_pos ht hn0 |
| have ha1 : t/(n:ℝ)≤1 := (div_le_one hn0).mpr hnt.le |
| have hb : 0<1/(n:ℝ) := div_pos zero_lt_one hn0 |
| have hb1 : 1/(n:ℝ)≤1 := (div_le_one hn0).mpr hn1.le |
| rcases hsmall (t/(n:ℝ)) ha ha1 with ⟨q,hq,f,hqτ,hfn,hfin,hfτ⟩ |
| rcases hsmall (1/(n:ℝ)) hb hb1 with ⟨r,hr,g,hrτ,hgn,hgin,hgτ⟩ |
| refine ⟨finiteProjection n q,finiteProjection_projection n q hq, |
| finiteProjection_trace_div n hn0 t ht.le q hq hqτ,?_⟩ |
| have hrtr := finiteProjection_trace_div n hn0 1 zero_le_one r hr hrτ |
| obtain ⟨D⟩ := preferred_unit_iso (G := G) hbase (finiteProjection n r) |
| (finiteProjection_projection n r hr) hrtr |
| let e : Corner q hq ≃⋆ₐ[ℂ] Corner r hr := f.symm.trans g |
| have heτ (x : Corner q hq) : canonicalTrace (e x).val=(t:ℂ)⁻¹*canonicalTrace x.val := |
| scaling_ratio_trace t ht n hn0 q r hq hr f g hfτ hgτ x |
| let : TopologicalSpace (groupVonNeumann G) := groupUltraweakTopology G |
| let : TopologicalSpace (Corner q hq) := cornerTopology (groupUltraweakTopology G) q hq |
| let : TopologicalSpace (Corner r hr) := cornerTopology (groupUltraweakTopology G) r hr |
| have hen : Continuous e := hgn.comp hfin |
| have hein : Continuous e.symm := hfn.comp hgin |
| exact ⟨(finiteScaledIso n t q r hq hr e heτ hen hein).trans D⟩ |
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| theorem allPositiveScaling_both |
| (hbase : ∀ T : groupVonNeumann G, (∀ S : groupVonNeumann G,Commute T S) → |
| ∃ c : ℂ,T=c • (1 : groupVonNeumann G)) |
| (hsmall : ∀t : ℝ,0<t → t≤1 → HasCornerScaling (canonicalTrace (G := G)) (groupUltraweakTopology G) t) |
| (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 |
| obtain ⟨q,hq,hqτ,⟨E⟩⟩ := preferred_scaling_model (G := G) hbase hsmall t ht |
| refine ⟨⟨q,hq,hqτ⟩,?_⟩ |
| intro p hp hpτ |
| obtain ⟨f,hfτ,hfn,hfin⟩ := finite_corner_choice_both (G := G) hbase t p q hp hq |
| (hpτ.trans hqτ.symm) (hpτ ▸ ENNReal.ofReal_ne_top) |
| let F : NormalTracialEquiv (normalizedCornerTrace stabilizedTrace t p hp) |
| (normalizedCornerTrace stabilizedTrace t q hq) |
| (cornerTopology (stabilizationTopology G) p hp) (cornerTopology (stabilizationTopology G) q hq) := |
| ⟨f,hfτ,hfn,hfin⟩ |
| exact ⟨F.trans E⟩ |
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| theorem infinite_allPositiveScaling (t : ℝ) (ht : 0<t) : |
| (∃p : stabilization (FreeGroup ℕ),IsStarProjection p ∧ stabilizedProjectionTrace p=ENNReal.ofReal t) ∧ |
| ∀(p : stabilization (FreeGroup ℕ))(hp : IsStarProjection p),stabilizedProjectionTrace p=ENNReal.ofReal t → |
| Nonempty (GroupCornerNormalIso t p hp) := |
| allPositiveScaling_both (fun T hT => ⟨canonicalTrace T,infinite_free_group_center_scalar T hT⟩) |
| infinite_hasCornerScaling t ht |
| end FreeGroupFactorMain.Amplification |
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| namespace FreeGroupFactorMain |
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| theorem HasNormalTracialIso.symm {G : Type u_2} {H : Type u_3} [Group G] [Group H] |
| (h : HasNormalTracialIso G H) : HasNormalTracialIso H G := by |
| classical |
| obtain ⟨Φ,hΦ,_⟩ := h |
| obtain ⟨e,hδ,he⟩ := Amplification.spatial_of_trace_iso Φ hΦ |
| refine ⟨(spatialGroupIso e he).symm,?_,spatialGroupIso_symm_normal e he⟩ |
| intro T |
| rw [canonicalTrace_eq_inner,canonicalTrace_eq_inner] |
| change inner ℂ deltaOne (e.symm ((T : GroupOperator H) (e deltaOne)))=_ |
| rw [hδ,←e.inner_map_map,e.apply_symm_apply,hδ] |
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| theorem center_scalar_of_iso {G : Type u_4} {H : Type u_5} [Group G] [Group H] |
| (e : groupVonNeumann G ≃⋆ₐ[ℂ] groupVonNeumann H) |
| (hH : ∀ T : groupVonNeumann H,(∀ S : groupVonNeumann H,Commute T S) → ∃ c : ℂ,T=c • 1) |
| (T : groupVonNeumann G) (hT : ∀ S : groupVonNeumann G,Commute T S) : |
| ∃ c : ℂ,T=c • 1 := by |
| have he : ∀ S : groupVonNeumann H,Commute (e T) S := by |
| intro S |
| have hh := (hT (e.symm S)).map e |
| simpa only [StarAlgEquiv.apply_symm_apply] using hh |
| obtain ⟨c,hc⟩ := hH (e T) he |
| have hec : e (c • (1 : groupVonNeumann G))=c • (1 : groupVonNeumann H) := by |
| change e.toFun (c • (1 : groupVonNeumann G))=_ |
| rw [e.map_smul'] |
| change c • e (1 : groupVonNeumann G)=_ |
| rw [map_one] |
| exact ⟨c,e.injective (hc.trans hec.symm)⟩ |
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| end FreeGroupFactorMain |
| namespace FreeGroupFactorMain.Amplification |
| open scoped ENNReal |
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| theorem rankTwo_smallScaling (t : ℝ) (ht : 0<t) (ht1 : t≤1) : |
| HasCornerScaling (canonicalTrace (G := FreeGroupRank 2)) (groupUltraweakTopology (FreeGroupRank 2)) t := |
| smallScaling_transport_group ((rankTwo.trans rankThreeInfinite).symm) (infinite_hasCornerScaling t ht ht1) |
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| theorem rankTwo_center_scalar (T : groupVonNeumann (FreeGroupRank 2)) |
| (hT : ∀ S : groupVonNeumann (FreeGroupRank 2),Commute T S) : ∃c : ℂ,T=c • 1 := by |
| obtain ⟨e,_,_⟩ := rankTwo.trans rankThreeInfinite |
| exact center_scalar_of_iso e (fun S hS => ⟨canonicalTrace S,infinite_free_group_center_scalar S hS⟩) T hT |
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| theorem rankTwo_allPositiveScaling (t : ℝ) (ht : 0<t) : |
| (∃p : stabilization (FreeGroupRank 2),IsStarProjection p ∧ stabilizedProjectionTrace p=ENNReal.ofReal t) ∧ |
| ∀(p : stabilization (FreeGroupRank 2))(hp : IsStarProjection p),stabilizedProjectionTrace p=ENNReal.ofReal t → |
| Nonempty (GroupCornerNormalIso t p hp) := |
| allPositiveScaling_both rankTwo_center_scalar rankTwo_smallScaling t ht |
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| theorem positiveRealSelfSimilarity (t : ℝ) (ht : 0 < t) |
| (p : stabilization (FreeGroupRank 2)) (hp : IsStarProjection p) |
| (htrace : stabilizedProjectionTrace p = ENNReal.ofReal t) : |
| HasAmplificationIso (FreeGroupRank 2) (FreeGroupRank 2) t p hp := by |
| obtain ⟨E⟩ := (rankTwo_allPositiveScaling t ht).2 p hp htrace |
| exact ⟨E.equiv,E.trace_eq,E.normal⟩ |
| end FreeGroupFactorMain.Amplification |
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| end |
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| end OAI |
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