| import OAI.Analysis.InterpolatedFactors.Scaling | |
| namespace OAI | |
| noncomputable section | |
| universe u_1 u_2 u_3 u_4 u_5 | |
| namespace FreeGroupFactorMain.Amplification | |
| section | |
| variable {A : Type u_1} {B : Type u_2} {C : Type u_3} [CStarAlgebra A] [CStarAlgebra B] [CStarAlgebra C] | |
| namespace NormalTracialEquiv | |
| variable {τA : A → ℂ} {τB : B → ℂ} {τC : C → ℂ} | |
| variable {𝒯A : TopologicalSpace A} {𝒯B : TopologicalSpace B} {𝒯C : TopologicalSpace C} | |
| def symm (e : NormalTracialEquiv τA τB 𝒯A 𝒯B) : NormalTracialEquiv τB τA 𝒯B 𝒯A where | |
| equiv := e.equiv.symm | |
| trace_eq x := by simpa only [StarAlgEquiv.apply_symm_apply] using (e.trace_eq (e.equiv.symm x)).symm | |
| normal := e.inverse_normal | |
| inverse_normal := e.normal | |
| def trans (e : NormalTracialEquiv τA τB 𝒯A 𝒯B) (f : NormalTracialEquiv τB τC 𝒯B 𝒯C) : | |
| NormalTracialEquiv τA τC 𝒯A 𝒯C where | |
| equiv := e.equiv.trans f.equiv | |
| trace_eq x := (f.trace_eq (e.equiv x)).trans (e.trace_eq x) | |
| normal := by | |
| let : TopologicalSpace A := 𝒯A | |
| let : TopologicalSpace B := 𝒯B | |
| let : TopologicalSpace C := 𝒯C | |
| exact f.normal.comp e.normal | |
| inverse_normal := by | |
| let : TopologicalSpace A := 𝒯A | |
| let : TopologicalSpace B := 𝒯B | |
| let : TopologicalSpace C := 𝒯C | |
| exact e.inverse_normal.comp f.inverse_normal | |
| end NormalTracialEquiv | |
| end | |
| open scoped ENNReal | |
| variable {G : Type u_4} [Group G] | |
| abbrev GroupCornerNormalIso (t : ℝ) (p : stabilization G) (hp : IsStarProjection p) := | |
| NormalTracialEquiv (normalizedCornerTrace stabilizedTrace t p hp) canonicalTrace | |
| (cornerTopology (stabilizationTopology G) p hp) (groupUltraweakTopology G) | |
| def unitNormalIso {A : Type u_5} [CStarAlgebra A] (τ : A → ℂ) (𝒯 : TopologicalSpace A) : | |
| NormalTracialEquiv (fun x : Corner (1:A) (IsStarProjection.one A) => τ x.val) τ | |
| (cornerTopology 𝒯 1 (IsStarProjection.one A)) 𝒯 where | |
| equiv := unitCornerEquiv.symm | |
| trace_eq _ := rfl | |
| normal := by | |
| let : TopologicalSpace A := 𝒯 | |
| exact continuous_subtype_val | |
| inverse_normal := by | |
| let : TopologicalSpace A := 𝒯 | |
| exact continuous_id.subtype_mk _ | |
| def finiteSingletonNormalIso (p : groupVonNeumann G) (hp : IsStarProjection p) : | |
| NormalTracialEquiv (normalizedCornerTrace stabilizedTrace 1 (finiteProjection 1 p) | |
| (finiteProjection_projection 1 p hp)) (fun x : Corner p hp => canonicalTrace x.val) | |
| (cornerTopology (stabilizationTopology G) (finiteProjection 1 p) (finiteProjection_projection 1 p hp)) | |
| (cornerTopology (groupUltraweakTopology G) p hp) where | |
| equiv := finiteCornerSingletonIso p hp | |
| trace_eq x := by | |
| simp only [normalizedCornerTrace,Complex.ofReal_one,inv_one,one_mul] | |
| exact finiteCornerSingletonIso_trace p hp x | |
| normal := finiteCornerSingletonIso_normal p hp | |
| inverse_normal := finiteCornerSingletonIso_symm_normal p hp | |
| def finiteUnitCornerIso : GroupCornerNormalIso (G := G) 1 | |
| (finiteProjection 1 (1 : groupVonNeumann G)) (finiteProjection_projection 1 1 (IsStarProjection.one _)) := | |
| (finiteSingletonNormalIso (1 : groupVonNeumann G) (IsStarProjection.one _)).trans | |
| (unitNormalIso canonicalTrace (groupUltraweakTopology G)) | |
| theorem canonicalTrace_one : canonicalTrace (1 : groupVonNeumann G)=1 := by | |
| classical | |
| simp [canonicalTrace,lp.single_apply] | |
| theorem finiteUnitProjection_trace : | |
| stabilizedProjectionTrace (finiteProjection 1 (1 : groupVonNeumann G))=ENNReal.ofReal 1 := by | |
| rw [finiteProjection_trace 1 1 (IsStarProjection.one _) 1 zero_le_one canonicalTrace_one] | |
| norm_num | |
| theorem preferred_unit_iso | |
| (hbase : ∀ T : groupVonNeumann G, (∀ S : groupVonNeumann G,Commute T S) → | |
| ∃ c : ℂ,T=c • (1 : groupVonNeumann G)) | |
| (p : stabilization G) (hp : IsStarProjection p) (hptr : stabilizedProjectionTrace p=ENNReal.ofReal 1) : | |
| Nonempty (GroupCornerNormalIso 1 p hp) := by | |
| obtain ⟨e,heτ,hen,hein⟩ := finite_corner_choice_both (G := G) hbase 1 p | |
| (finiteProjection 1 (1 : groupVonNeumann G)) hp (finiteProjection_projection 1 1 (IsStarProjection.one _)) | |
| (hptr.trans finiteUnitProjection_trace.symm) (hptr ▸ ENNReal.ofReal_ne_top) | |
| let E : NormalTracialEquiv (normalizedCornerTrace stabilizedTrace 1 p hp) | |
| (normalizedCornerTrace stabilizedTrace 1 (finiteProjection 1 (1 : groupVonNeumann G)) | |
| (finiteProjection_projection 1 1 (IsStarProjection.one _))) | |
| (cornerTopology (stabilizationTopology G) p hp) | |
| (cornerTopology (stabilizationTopology G) (finiteProjection 1 (1 : groupVonNeumann G)) | |
| (finiteProjection_projection 1 1 (IsStarProjection.one _))) := ⟨e,heτ,hen,hein⟩ | |
| exact ⟨E.trans finiteUnitCornerIso⟩ | |
| theorem scaling_ratio_trace (t : ℝ) (ht : 0<t) (n : ℕ) (_hn : 0<(n:ℝ)) | |
| (q r : groupVonNeumann G) (hq : IsStarProjection q) (hr : IsStarProjection r) | |
| (f : groupVonNeumann G ≃⋆ₐ[ℂ] Corner q hq) (g : groupVonNeumann G ≃⋆ₐ[ℂ] Corner r hr) | |
| (hf : ∀ x,canonicalTrace (f x).val=((t/(n:ℝ):ℝ):ℂ)*canonicalTrace x) | |
| (hg : ∀ x,canonicalTrace (g x).val=((1/(n:ℝ):ℝ):ℂ)*canonicalTrace x) | |
| (x : Corner q hq) : canonicalTrace (g (f.symm x)).val=(t:ℂ)⁻¹*canonicalTrace x.val := by | |
| have hh := hf (f.symm x) | |
| rw [StarAlgEquiv.apply_symm_apply] at hh | |
| rw [hg,hh] | |
| push_cast | |
| field_simp [Complex.ofReal_ne_zero.mpr ht.ne'] | |
| def finiteScaledIso (n : ℕ) (t : ℝ) | |
| (q r : groupVonNeumann G) (hq : IsStarProjection q) (hr : IsStarProjection r) | |
| (e : Corner q hq ≃⋆ₐ[ℂ] Corner r hr) | |
| (heτ : ∀x,canonicalTrace (e x).val=(t:ℂ)⁻¹*canonicalTrace x.val) | |
| (hen : @Continuous _ _ (cornerTopology (groupUltraweakTopology G) q hq) | |
| (cornerTopology (groupUltraweakTopology G) r hr) e) | |
| (hein : @Continuous _ _ (cornerTopology (groupUltraweakTopology G) r hr) | |
| (cornerTopology (groupUltraweakTopology G) q hq) e.symm) : | |
| NormalTracialEquiv (normalizedCornerTrace stabilizedTrace t (finiteProjection n q) (finiteProjection_projection n q hq)) | |
| (normalizedCornerTrace stabilizedTrace 1 (finiteProjection n r) (finiteProjection_projection n r hr)) | |
| (cornerTopology (stabilizationTopology G) (finiteProjection n q) (finiteProjection_projection n q hq)) | |
| (cornerTopology (stabilizationTopology G) (finiteProjection n r) (finiteProjection_projection n r hr)) where | |
| equiv := finiteCornerIso n q hq r hr e | |
| trace_eq x := by | |
| simp only [normalizedCornerTrace,Complex.ofReal_one,inv_one,one_mul] | |
| exact finiteCornerIso_trace n q hq r hr e (t:ℂ)⁻¹ heτ x | |
| normal := finiteCornerIso_normal n q hq r hr e hen | |
| inverse_normal := by | |
| rw [finiteCornerIso_symm] | |
| exact finiteCornerIso_normal n r hr q hq e.symm hein | |
| end FreeGroupFactorMain.Amplification | |
| end | |
| end OAI | |