| import OAI.Analysis.InterpolatedFactors.Spatial |
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| namespace OAI |
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| noncomputable section |
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| universe u_1 u_2 u_3 |
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| namespace FreeGroupFactorMain.Amplification |
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| section |
| variable {G : Type u_1} {H : Type u_2} [Group G] [Group H] |
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| def stabilizationIso (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H) |
| (he : ∀ T : GroupOperator G, T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) : |
| stabilization G ≃⋆ₐ[ℂ] stabilization H where |
| toFun T := ⟨(productSpatial ℕ e).conjStarAlgEquiv T, (productSpatial_stabilization e he T).mp T.property⟩ |
| invFun T := ⟨(productSpatial ℕ e).conjStarAlgEquiv.symm T, |
| (productSpatial_stabilization e he _).mpr (by |
| rw [(productSpatial ℕ e).conjStarAlgEquiv.apply_symm_apply]; exact T.property)⟩ |
| left_inv T := Subtype.ext ((productSpatial ℕ e).conjStarAlgEquiv.symm_apply_apply T) |
| right_inv T := Subtype.ext ((productSpatial ℕ e).conjStarAlgEquiv.apply_symm_apply T) |
| map_mul' T U := Subtype.ext (map_mul (productSpatial ℕ e).conjStarAlgEquiv |
| (T : GroupOperator (ℕ × G)) (U : GroupOperator (ℕ × G))) |
| map_add' T U := Subtype.ext (map_add (productSpatial ℕ e).conjStarAlgEquiv |
| (T : GroupOperator (ℕ × G)) (U : GroupOperator (ℕ × G))) |
| map_star' T := Subtype.ext (map_star (productSpatial ℕ e).conjStarAlgEquiv (T : GroupOperator (ℕ × G))) |
| map_smul' c T := Subtype.ext (map_smul (productSpatial ℕ e).conjStarAlgEquiv c (T : GroupOperator (ℕ × G))) |
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| theorem stabilizationIso_normal (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H) |
| (he : ∀ T : GroupOperator G, T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) : |
| @Continuous _ _ (stabilizationTopology G) (stabilizationTopology H) (stabilizationIso e he) := by |
| let : TopologicalSpace (GroupOperator (ℕ × G)) := ultraweakOperatorTopology (ℕ × G) |
| let : TopologicalSpace (GroupOperator (ℕ × H)) := ultraweakOperatorTopology (ℕ × H) |
| let : TopologicalSpace (stabilization G) := stabilizationTopology G |
| let : TopologicalSpace (stabilization H) := stabilizationTopology H |
| exact ((spatial_ultraweak_continuous (productSpatial ℕ e)).comp continuous_subtype_val).subtype_mk _ |
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| theorem stabilizationDelta_eq_single [DecidableEq (ℕ × G)] (k : ℕ) : |
| stabilizationDelta G k = lp.single 2 (k,1) 1 := by |
| classical |
| ext x |
| by_cases hx : x=(k,1) <;> simp [stabilizationDelta,lp.single_apply,hx] |
| |
| theorem productSpatial_delta (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H) |
| (hδ : e deltaOne = deltaOne) (k : ℕ) : |
| productSpatial ℕ e (stabilizationDelta G k) = stabilizationDelta H k := by |
| classical |
| have hg : stabilizationDelta G k=productSingle ℕ G k deltaOne := (productSingle_single k (1:G)).symm |
| have hh : stabilizationDelta H k=productSingle ℕ H k deltaOne := (productSingle_single k (1:H)).symm |
| rw [hg,hh,productSpatial_productSingle,hδ] |
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| theorem stabilizationIso_projectionTrace (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H) |
| (he : ∀ T : GroupOperator G, T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) |
| (hδ : e deltaOne = deltaOne) (T : stabilization G) : |
| stabilizedProjectionTrace (stabilizationIso e he T) = stabilizedProjectionTrace T := by |
| classical |
| unfold stabilizedProjectionTrace |
| apply tsum_congr |
| intro k |
| rw [← stabilizationDelta_eq_single,← stabilizationDelta_eq_single] |
| change ENNReal.ofReal (‖(productSpatial ℕ e) ((T : GroupOperator (ℕ × G)) |
| ((productSpatial ℕ e).symm (stabilizationDelta H k)))‖^2) = _ |
| rw [← productSpatial_delta e hδ,(productSpatial ℕ e).symm_apply_apply,(productSpatial ℕ e).norm_map] |
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| theorem stabilizationIso_trace (e : GroupHilbert G ≃ₗᵢ[ℂ] GroupHilbert H) |
| (he : ∀ T : GroupOperator G, T ∈ groupVonNeumann G ↔ e.conjStarAlgEquiv T ∈ groupVonNeumann H) |
| (hδ : e deltaOne = deltaOne) (T : stabilization G) : |
| stabilizedTrace (stabilizationIso e he T) = stabilizedTrace T := by |
| classical |
| unfold stabilizedTrace |
| apply tsum_congr |
| intro k |
| rw [← stabilizationDelta_eq_single,← stabilizationDelta_eq_single] |
| change inner ℂ (stabilizationDelta H k) ((productSpatial ℕ e) ((T : GroupOperator (ℕ × G)) |
| ((productSpatial ℕ e).symm (stabilizationDelta H k)))) = _ |
| rw [← productSpatial_delta e hδ,(productSpatial ℕ e).symm_apply_apply,(productSpatial ℕ e).inner_map_map] |
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| theorem amplification_functoriality (h : HasNormalTracialIso G H) : |
| ∃ Φ : stabilization G ≃⋆ₐ[ℂ] stabilization H, |
| (∀ p, stabilizedProjectionTrace (Φ p) = stabilizedProjectionTrace p) ∧ |
| (∀ T, stabilizedTrace (Φ T) = stabilizedTrace T) ∧ |
| @Continuous _ _ (stabilizationTopology G) (stabilizationTopology H) Φ := by |
| obtain ⟨Ψ,ht,_⟩ := h |
| obtain ⟨e,hδ,he⟩ := spatial_of_trace_iso Ψ ht |
| exact ⟨stabilizationIso e he,stabilizationIso_projectionTrace e he hδ, |
| stabilizationIso_trace e he hδ,stabilizationIso_normal e he⟩ |
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| end |
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| open scoped ENNReal |
| variable {G : Type u_3} |
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| @[simp] theorem pairedIndex_apply (i j : ℕ) (g : G) : pairedIndex G (i,j,g)=(Nat.pair i j,g) := rfl |
| variable [Group G] |
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| instance doubleStabilization_isClosed : IsClosed (doubleStabilization G : Set (GroupOperator (ℕ × (ℕ × G)))) := by |
| rw [doubleStabilization,StarSubalgebra.coe_centralizer] |
| exact Set.isClosed_centralizer _ |
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| instance doubleStabilization_CStarAlgebra : CStarAlgebra (doubleStabilization G) := inferInstance |
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| theorem mem_doubleStabilization_iff_matrix (T : GroupOperator (ℕ × (ℕ × G))) : |
| T ∈ doubleStabilization G ↔ ∀i j,matrixEntry T i j ∈ stabilization G := by |
| rw [mem_doubleStabilization_iff_right] |
| constructor |
| · intro h i j |
| rw [mem_stabilization_iff_right] |
| intro g |
| have hh := congrArg (fun S => matrixEntry S i j) (h g) |
| simpa only [doubleRight,matrixEntry_diagonal_mul,matrixEntry_mul_diagonal] using hh |
| · intro h g |
| apply matrixEntry_ext |
| intro i j |
| change matrixEntry (productAmplification ℕ (ℕ × G) (productAmplification ℕ G (rightRegular g))*T) i j=_ |
| rw [matrixEntry_diagonal_mul] |
| change _=matrixEntry (T*productAmplification ℕ (ℕ × G) (productAmplification ℕ G (rightRegular g))) i j |
| rw [matrixEntry_mul_diagonal] |
| exact (mem_stabilization_iff_right _).mp (h i j) g |
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| end FreeGroupFactorMain.Amplification |
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| end |
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| end OAI |
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