| import OAI.Analysis.InterpolatedFactors.Matrices |
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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 u_6 u_7 u_8 |
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| section |
| namespace FreeGroupFactorMain.HilbertUltraweak |
| variable {E : Type u_1} [NormedAddCommGroup E] [InnerProductSpace ℂ E] |
| variable {X : Type u_2} [TopologicalSpace X] |
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|
| theorem finset_sum_continuous {ι : Type u_3} (s : Finset ι) (f : ι → X → (E →L[ℂ] E)) |
| (hf : ∀ i∈s,@Continuous _ _ _ (topology E) (f i)) : |
| @Continuous _ _ _ (topology E) (fun x => ∑ i∈s,f i x) := by |
| classical |
| induction s using Finset.induction_on with |
| | empty => simpa only [Finset.sum_empty] using (@continuous_const X _ _ (topology E) (0 : E →L[ℂ] E)) |
| | @insert a s ha ih => |
| simp only [Finset.sum_insert ha] |
| exact add_continuous (hf a (Finset.mem_insert_self _ _)) |
| (ih (fun i hi => hf i (Finset.mem_insert_of_mem hi))) |
| end FreeGroupFactorMain.HilbertUltraweak |
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|
| namespace FreeGroupFactorMain.Amplification |
| variable {G : Type u_4} {H : Type u_5} |
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|
| theorem matrixEntry_normal (i j : ℕ) : |
| @Continuous _ _ (ultraweakOperatorTopology (ℕ × G)) (ultraweakOperatorTopology G) |
| (fun T : GroupOperator (ℕ × G) => matrixEntry T i j) := by |
| have hc := HilbertUltraweak.sandwich_continuous (productCoordinate ℕ G i) (productSingle ℕ G j) |
| convert hc using 1 |
| rfl |
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|
| theorem matrixBlock_normal (i j : ℕ) : |
| @Continuous _ _ (ultraweakOperatorTopology G) (ultraweakOperatorTopology (ℕ × G)) |
| (matrixBlock (G := G) i j) := |
| HilbertUltraweak.sandwich_continuous (productSingle ℕ G i) (productCoordinate ℕ G j) |
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| variable [Group G] [Group H] |
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| theorem finiteCornerEntry_normal (n : ℕ) (p : groupVonNeumann G) (hp : IsStarProjection p) (i j : Fin n) : |
| @Continuous _ _ (cornerTopology (stabilizationTopology G) (finiteProjection n p) (finiteProjection_projection n p hp)) |
| (cornerTopology (groupUltraweakTopology G) p hp) (fun x => finiteCornerEntry n p hp x i j) := by |
| let : TopologicalSpace (GroupOperator (ℕ × G)) := ultraweakOperatorTopology (ℕ × G) |
| let : TopologicalSpace (GroupOperator G) := ultraweakOperatorTopology G |
| let : TopologicalSpace (stabilization G) := stabilizationTopology G |
| let : TopologicalSpace (groupVonNeumann G) := groupUltraweakTopology G |
| let : TopologicalSpace (FiniteCorner n p hp) := cornerTopology (stabilizationTopology G) (finiteProjection n p) (finiteProjection_projection n p hp) |
| let : TopologicalSpace (Corner p hp) := cornerTopology (groupUltraweakTopology G) p hp |
| exact (((matrixEntry_normal i.val j.val).comp (continuous_subtype_val.comp continuous_subtype_val)).subtype_mk _).subtype_mk _ |
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| theorem finiteCornerOfMatrix_normal (n : ℕ) (p : groupVonNeumann G) (hp : IsStarProjection p) |
| {X : Type u_6} [TopologicalSpace X] (f : X → Fin n → Fin n → Corner p hp) |
| (hf : ∀ i j,@Continuous _ _ _ (cornerTopology (groupUltraweakTopology G) p hp) (fun x => f x i j)) : |
| @Continuous _ _ _ (cornerTopology (stabilizationTopology G) (finiteProjection n p) (finiteProjection_projection n p hp)) |
| (fun x => finiteCornerOfMatrix n p hp (f x)) := by |
| let : TopologicalSpace (GroupOperator (ℕ × G)) := ultraweakOperatorTopology (ℕ × G) |
| let : TopologicalSpace (GroupOperator G) := ultraweakOperatorTopology G |
| let : TopologicalSpace (stabilization G) := stabilizationTopology G |
| let : TopologicalSpace (groupVonNeumann G) := groupUltraweakTopology G |
| let : TopologicalSpace (Corner p hp) := cornerTopology (groupUltraweakTopology G) p hp |
| have hc : Continuous (fun x => finiteMatrix n (fun i j => ((f x i j).val : GroupOperator G))) := by |
| apply HilbertUltraweak.finset_sum_continuous |
| intro i _ |
| apply HilbertUltraweak.finset_sum_continuous |
| intro j _ |
| exact (matrixBlock_normal i.val j.val).comp (continuous_subtype_val.comp (continuous_subtype_val.comp (hf i j))) |
| exact (hc.subtype_mk _).subtype_mk _ |
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| theorem finiteCornerIso_normal (n : ℕ) (p : groupVonNeumann G) (hp : IsStarProjection p) |
| (q : groupVonNeumann H) (hq : IsStarProjection q) (e : Corner p hp ≃⋆ₐ[ℂ] Corner q hq) |
| (he : @Continuous _ _ (cornerTopology (groupUltraweakTopology G) p hp) |
| (cornerTopology (groupUltraweakTopology H) q hq) e) : |
| @Continuous _ _ (cornerTopology (stabilizationTopology G) (finiteProjection n p) (finiteProjection_projection n p hp)) |
| (cornerTopology (stabilizationTopology H) (finiteProjection n q) (finiteProjection_projection n q hq)) |
| (finiteCornerIso n p hp q hq e) := by |
| let : TopologicalSpace (FiniteCorner n p hp) := cornerTopology (stabilizationTopology G) (finiteProjection n p) (finiteProjection_projection n p hp) |
| let : TopologicalSpace (Corner p hp) := cornerTopology (groupUltraweakTopology G) p hp |
| let : TopologicalSpace (Corner q hq) := cornerTopology (groupUltraweakTopology H) q hq |
| exact finiteCornerOfMatrix_normal n q hq _ (fun i j => he.comp (finiteCornerEntry_normal n p hp i j)) |
|
|
| end FreeGroupFactorMain.Amplification |
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| end |
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| namespace FreeGroupFactorMain.Amplification |
| open scoped ENNReal |
| variable {G : Type u_7} |
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| theorem productSingle_norm (i : ℕ) (ξ : GroupHilbert G) : ‖productSingle ℕ G i ξ‖=‖ξ‖ := by |
| classical |
| let : DecidableEq ℕ := Classical.decEq ℕ |
| simp only [productSingle,ContinuousLinearMap.comp_apply,lp.singleContinuousLinearMap_apply, |
| ContinuousLinearEquiv.coe_coe,LinearIsometryEquiv.coe_toContinuousLinearEquiv] |
| rw [LinearIsometryEquiv.norm_map,lp.norm_single (by norm_num : (0 : ℝ≥0∞)<2)] |
|
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| theorem matrixBlock_productSingle (i j k : ℕ) (T : GroupOperator G) (ξ : GroupHilbert G) : |
| matrixBlock i j T (productSingle ℕ G k ξ)=if j=k then productSingle ℕ G i (T ξ) else 0 := by |
| simp only [matrixBlock,ContinuousLinearMap.comp_apply,productCoordinate_productSingle] |
| split_ifs <;> simp |
|
|
| theorem finiteDiagonal_productSingle (n : ℕ) (T : GroupOperator G) (k : ℕ) (ξ : GroupHilbert G) : |
| finiteDiagonal n T (productSingle ℕ G k ξ)=if k<n then productSingle ℕ G k (T ξ) else 0 := by |
| classical |
| rw [finiteDiagonal_sum] |
| simp only [sum_apply,matrixBlock_productSingle] |
| by_cases hk : k<n |
| · rw [ite_eq_left hk] |
| let j : Fin n := ⟨k,hk⟩ |
| have he : ∀ i : Fin n,i.val=k ↔ i=j := fun i => (Fin.ext_iff (a := i) (b := j)).symm |
| simp only [he,Finset.sum_ite_eq',Finset.mem_univ,ite_true] |
| rfl |
| · rw [ite_eq_right hk] |
| apply Finset.sum_eq_zero |
| intro i _ |
| rw [ite_eq_right] |
| intro hi |
| exact hk (hi ▸ i.isLt) |
| |
| variable [Group G] |
| |
| theorem finiteProjection_trace (n : ℕ) (p : groupVonNeumann G) (hp : IsStarProjection p) |
| (a : ℝ) (_ha : 0≤a) (hpa : canonicalTrace p=(a:ℂ)) : |
| stabilizedProjectionTrace (finiteProjection n p)=ENNReal.ofReal (n*a) := by |
| classical |
| have hv : ‖(p : GroupOperator G) (lp.single 2 (1:G) (1:ℂ))‖^2=a := by |
| apply Complex.ofReal_injective |
| have hh := canonicalTrace_star_mul p |
| rw [hp.isSelfAdjoint.star_eq,hp.isIdempotentElem.eq,hpa] at hh |
| simpa only [Complex.ofReal_pow] using hh.symm |
| change (∑' k : ℕ,ENNReal.ofReal (‖finiteDiagonal n (p : GroupOperator G) (lp.single 2 (k,1) (1:ℂ))‖^2))=_ |
| have he (k : ℕ) : ENNReal.ofReal (‖finiteDiagonal n (p : GroupOperator G) (lp.single 2 (k,1) (1:ℂ))‖^2)= |
| if k<n then ENNReal.ofReal a else 0 := by |
| rw [←productSingle_single,finiteDiagonal_productSingle] |
| split_ifs |
| · rw [productSingle_norm,hv] |
| · simp |
| simp_rw [he] |
| rw [tsum_eq_sum (s := Finset.range n)] |
| · have hh : (∑ k∈Finset.range n,if k<n then ENNReal.ofReal a else 0)=∑ _k∈Finset.range n,ENNReal.ofReal a := |
| Finset.sum_congr rfl (fun k hk => ite_eq_left (Finset.mem_range.mp hk)) |
| rw [hh] |
| simp only [Finset.sum_const,Finset.card_range,nsmul_eq_mul] |
| rw [ENNReal.ofReal_mul (Nat.cast_nonneg n),ENNReal.ofReal_natCast] |
| · intro k hk |
| rw [ite_eq_right (fun h => hk (Finset.mem_range.mpr h))] |
|
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| theorem stabilizedTrace_matrix (T : stabilization G) : |
| stabilizedTrace T=∑' k : ℕ,canonicalTrace |
| (⟨matrixEntry (T : GroupOperator (ℕ × G)) k k,(mem_stabilization_iff_matrix _).mp T.property k k⟩ : groupVonNeumann G) := by |
| classical |
| apply tsum_congr |
| intro k |
| rw [←productSingle_single,productSingle_inner_left,canonicalTrace_eq_inner] |
| rfl |
| |
| theorem finiteCorner_trace (n : ℕ) (p : groupVonNeumann G) (hp : IsStarProjection p) (x : FiniteCorner n p hp) : |
| stabilizedTrace x.val=∑ i : Fin n,canonicalTrace (finiteCornerEntry n p hp x i i).val := by |
| classical |
| rw [stabilizedTrace_matrix] |
| have he (k : ℕ) : canonicalTrace |
| (⟨matrixEntry (x.val : GroupOperator (ℕ × G)) k k,(mem_stabilization_iff_matrix _).mp x.val.property k k⟩ : groupVonNeumann G)= |
| if hk : k<n then canonicalTrace (finiteCornerEntry n p hp x ⟨k,hk⟩ ⟨k,hk⟩).val else 0 := by |
| split_ifs with hk |
| · rfl |
| · have hz : matrixEntry (x.val : GroupOperator (ℕ × G)) k k=0 := by |
| rw [←finiteCorner_reconstruct n p hp x] |
| exact matrixEntry_finiteMatrix_out n _ k k (Or.inl (le_of_not_gt hk)) |
| have hs : (⟨matrixEntry (x.val : GroupOperator (ℕ × G)) k k,(mem_stabilization_iff_matrix _).mp x.val.property k k⟩ : groupVonNeumann G)=0 := Subtype.ext hz |
| rw [hs] |
| exact map_zero (canonicalTraceLinear (G := G)) |
| simp_rw [he] |
| rw [tsum_eq_sum (s := Finset.range n)] |
| · rw [←Fin.sum_univ_eq_sum_range] |
| apply Finset.sum_congr rfl |
| intro k _ |
| simp only [dite_eq_left k.isLt] |
| · intro k hk |
| simp only [dite_eq_right (fun h => hk (Finset.mem_range.mpr h))] |
| |
| variable {H : Type u_8} [Group H] |
| theorem finiteCornerIso_trace (n : ℕ) (p : groupVonNeumann G) (hp : IsStarProjection p) |
| (q : groupVonNeumann H) (hq : IsStarProjection q) (e : Corner p hp ≃⋆ₐ[ℂ] Corner q hq) |
| (c : ℂ) (hc : ∀ x,canonicalTrace (e x : groupVonNeumann H)=c*canonicalTrace (x : groupVonNeumann G)) |
| (x : FiniteCorner n p hp) : |
| stabilizedTrace (finiteCornerIso n p hp q hq e x).val=c*stabilizedTrace x.val := by |
| rw [finiteCorner_trace,finiteCorner_trace,Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro i _ |
| change canonicalTrace (finiteCornerEntry n q hq (finiteCornerOfMatrix n q hq _) i i).val=_ |
| rw [finiteCornerEntry_ofMatrix,hc] |
| |
| end FreeGroupFactorMain.Amplification |
| |
| end |
| |
| end OAI |
| |
