import OAI.Analysis.InterpolatedFactors.Subalgebras namespace OAI noncomputable section universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 u_10 u_11 u_12 universe u_13 u_14 u_15 u_16 u_17 u_18 u_19 u_20 u_21 u_22 u_23 u_24 universe u_25 u_26 u_27 u_28 u_29 u_30 u_31 namespace FreeGroupFactorMain namespace InfiniteICC theorem reduced_append_fresh {α : Type u_1} {L : List (α × Bool)} (hL : FreeGroup.IsReduced L) (a : α) (b : Bool) (ha : ∀ x ∈ L, x.1 ≠ a) : FreeGroup.IsReduced (L++[(a,b)]) := by induction L with | nil => exact FreeGroup.IsReduced.singleton | cons x L ih => cases L with | nil => rw [List.cons_append,List.nil_append,FreeGroup.isReduced_cons_cons] exact ⟨fun he => (ha x (by simp) he).elim,FreeGroup.IsReduced.singleton⟩ | cons y L => rw [FreeGroup.isReduced_cons_cons] at hL simp only [List.cons_append] rw [FreeGroup.isReduced_cons_cons] exact ⟨hL.1,ih hL.2 (fun z hz => ha z (List.mem_cons_of_mem _ hz))⟩ theorem reduced_conjugate_fresh {α : Type u_2} {L : List (α × Bool)} (hL : FreeGroup.IsReduced L) (hne : L ≠ []) (a : α) (ha : ∀ x ∈ L, x.1 ≠ a) : FreeGroup.IsReduced ((a,true)::(L++[(a,false)])) := by cases L with | nil => exact (hne rfl).elim | cons x L => rw [List.cons_append,FreeGroup.isReduced_cons_cons] refine ⟨fun he => (ha x (by simp) he.symm).elim,?_⟩ exact reduced_append_fresh hL a false ha theorem conjugate_word (g : FreeGroup ℕ) (hg : g ≠ 1) (a : ℕ) (ha : ∀ x ∈ g.toWord, x.1 ≠ a) : (FreeGroup.of a*g*(FreeGroup.of a)⁻¹).toWord= (a,true)::(g.toWord++[(a,false)]) := by have he : FreeGroup.of a*g*(FreeGroup.of a)⁻¹ = FreeGroup.mk ((a,true)::(g.toWord++[(a,false)])) := by rw [← List.singleton_append,← List.append_assoc,← FreeGroup.mul_mk,← FreeGroup.mul_mk, FreeGroup.mk_toWord] rfl rw [he,FreeGroup.toWord_mk] exact (reduced_conjugate_fresh FreeGroup.isReduced_toWord (FreeGroup.toWord_eq_nil_iff.not.mpr hg) a ha).reduce_eq theorem infinite_conjugates (g : FreeGroup ℕ) (hg : g ≠ 1) : ∃ c : ℕ → FreeGroup ℕ, Function.Injective c ∧ ∀ n, ∃ a, c n=a*g*a⁻¹ := by classical let S : Finset ℕ := (g.toWord.map Prod.fst).toFinset let M := S.sup id let a (n : ℕ) := M+1+n have hfresh (n : ℕ) (x) (hx : x ∈ g.toWord) : x.1 ≠ a n := by have hm : x.1 ≤ M := Finset.le_sup (f := id) (by simpa only [S,List.mem_toFinset,List.mem_map] using ⟨x,hx,rfl⟩) dsimp [a] omega refine ⟨fun n => FreeGroup.of (a n)*g*(FreeGroup.of (a n))⁻¹,?_,fun n => ⟨FreeGroup.of (a n),rfl⟩⟩ intro m n h have hw := congrArg FreeGroup.toWord h rw [conjugate_word g hg (a m) (hfresh m),conjugate_word g hg (a n) (hfresh n)] at hw have he : a m=a n := congrArg Prod.fst (List.cons.inj hw).1 dsimp [a] at he omega end InfiniteICC theorem central_coefficient_conjugate {G : Type u_3} [Group G] [DecidableEq G] (T : groupVonNeumann G) (hT : ∀ S : groupVonNeumann G, Commute T S) (a g : G) : ((T : GroupOperator G) deltaOne) (a*g*a⁻¹)=((T : GroupOperator G) deltaOne) g := by have he : (T : GroupOperator G)*leftRegular a=leftRegular a*(T : GroupOperator G) := congrArg (fun A : groupVonNeumann G => (A : GroupOperator G)) (hT (groupUnitary a)).eq have hd : leftRegular a deltaOne=lp.single 2 a 1 := by ext x simp [leftRegular_apply,deltaOne,lp.single_apply,Pi.single_apply,inv_mul_eq_one,eq_comm] have hx := congrArg (fun A : GroupOperator G => A deltaOne (a*g)) he simp only [mul_apply_eq_comp,hd,leftRegular_apply,inv_mul_cancel_left] at hx rw [groupVN_matrix] at hx exact hx theorem group_center_scalar_of_infinite_conjugates {G : Type u_4} [Group G] (hICC : ∀ g : G, g ≠ 1 → ∃ c : ℕ → G, Function.Injective c ∧ ∀ n, ∃ a, c n=a*g*a⁻¹) (T : groupVonNeumann G) (hT : ∀ S : groupVonNeumann G, Commute T S) : T=canonicalTrace T • (1 : groupVonNeumann G) := by classical have hz (g : G) (hg : g ≠ 1) : ((T : GroupOperator G) deltaOne) g=0 := by obtain ⟨c,hc,hconj⟩ := hICC g hg have hs : Summable (fun x : G => ‖((T : GroupOperator G) deltaOne) x‖^2) := by simpa only [ENNReal.toReal_ofNat,Real.rpow_two] using (lp.memℓp ((T : GroupOperator G) deltaOne)).summable (by norm_num) have hcst : Summable (fun _ : ℕ => ‖((T : GroupOperator G) deltaOne) g‖^2) := by have hh := hs.comp_injective hc change Summable (fun n : ℕ => ‖((T : GroupOperator G) deltaOne) (c n)‖^2) at hh have he : (fun n : ℕ => ‖((T : GroupOperator G) deltaOne) (c n)‖^2) = (fun _ : ℕ => ‖((T : GroupOperator G) deltaOne) g‖^2) := by funext n obtain ⟨a,ha⟩ := hconj n rw [ha,central_coefficient_conjugate T hT] rwa [he] at hh have hh := (summable_const_iff _).mp hcst exact norm_eq_zero.mp (by nlinarith [norm_nonneg (((T : GroupOperator G) deltaOne) g)]) apply sub_eq_zero.mp apply groupVN_separating rw [← deltaOne_eq] ext g change ((T : GroupOperator G) deltaOne) g-canonicalTrace T*deltaOne g=(0 : ℂ) by_cases hg : g=1 · subst g simp only [deltaOne,lp.single_apply,Pi.single_eq_same,mul_one,canonicalTrace] exact sub_self _ · rw [hz g hg] simp [deltaOne,lp.single_apply,hg] theorem infinite_free_group_center_scalar (T : groupVonNeumann (FreeGroup ℕ)) (hT : ∀ S : groupVonNeumann (FreeGroup ℕ), Commute T S) : T=canonicalTrace T • (1 : groupVonNeumann (FreeGroup ℕ)) := group_center_scalar_of_infinite_conjugates InfiniteICC.infinite_conjugates T hT end FreeGroupFactorMain namespace FreeGroupFactorMain.Amplification section variable {A : Type u_9} [CStarAlgebra A] def equivalentCorner (p q : A) (hp : IsStarProjection p) (hq : IsStarProjection q) (u : A) (hu : star u*u=p) (hu' : u*star u=q) : Corner p hp ≃⋆ₐ[ℂ] Corner q hq where toFun x := ⟨u*(x : A)*star u,by constructor · rw [← hu'] calc (u*star u)*(u*(x : A)*star u) = u*((star u*u)*(x : A))*star u := by noncomm_ring _ = u*(x : A)*star u := by rw [hu,x.property.1] · rw [← hu'] calc (u*(x : A)*star u)*(u*star u) = u*((x : A)*(star u*u))*star u := by noncomm_ring _ = u*(x : A)*star u := by rw [hu,x.property.2]⟩ invFun y := ⟨star u*(y : A)*u,by constructor · rw [← hu] calc (star u*u)*(star u*(y : A)*u) = star u*((u*star u)*(y : A))*u := by noncomm_ring _ = star u*(y : A)*u := by rw [hu',y.property.1] · rw [← hu] calc (star u*(y : A)*u)*(star u*u) = star u*((y : A)*(u*star u))*u := by noncomm_ring _ = star u*(y : A)*u := by rw [hu',y.property.2]⟩ left_inv x := by apply Subtype.ext change star u*(u*(x : A)*star u)*u = (x : A) rw [← mul_assoc,← mul_assoc,hu,x.property.1,mul_assoc,hu,x.property.2] right_inv y := by apply Subtype.ext change u*(star u*(y : A)*u)*star u = (y : A) rw [← mul_assoc,← mul_assoc,hu',y.property.1,mul_assoc,hu',y.property.2] map_mul' x y := by apply Subtype.ext change u*((x : A)*(y : A))*star u = (u*(x : A)*star u)*(u*(y : A)*star u) simp only [mul_assoc] rw [← mul_assoc (star u),hu,← mul_assoc p,y.property.1] map_add' x y := by apply Subtype.ext change u*((x : A)+(y : A))*star u = _ simp only [mul_add,add_mul] rfl map_smul' z x := by apply Subtype.ext change u*(z • (x : A))*star u = _ simp only [mul_smul_comm,smul_mul_assoc] rfl map_star' x := by apply Subtype.ext change u*star (x : A)*star u = star (u*(x : A)*star u) simp only [star_mul,star_star,mul_assoc] theorem equivalentCorner_trace (p q : A) (hp : IsStarProjection p) (hq : IsStarProjection q) (u : A) (hu : star u*u=p) (hu' : u*star u=q) (τ : A → ℂ) (hτ : ∀ x y, τ (x*y)=τ (y*x)) (t : ℝ) (x : Corner p hp) : normalizedCornerTrace τ t q hq (equivalentCorner p q hp hq u hu hu' x) = normalizedCornerTrace τ t p hp x := by change (t : ℂ)⁻¹ * τ (u*(x : A)*star u) = (t : ℂ)⁻¹ * τ (x : A) rw [hτ,← mul_assoc,hu,x.property.1] theorem equivalentCorner_normal (p q : A) (hp : IsStarProjection p) (hq : IsStarProjection q) (u : A) (hu : star u*u=p) (hu' : u*star u=q) (𝒯 : TopologicalSpace A) (hmul : @Continuous _ _ 𝒯 𝒯 (fun x : A => u*x*star u)) : @Continuous _ _ (cornerTopology 𝒯 p hp) (cornerTopology 𝒯 q hq) (equivalentCorner p q hp hq u hu hu') := by let : TopologicalSpace A := 𝒯 let : TopologicalSpace (Corner p hp) := cornerTopology 𝒯 p hp let : TopologicalSpace (Corner q hq) := cornerTopology 𝒯 q hq exact (hmul.comp continuous_subtype_val).subtype_mk _ end section open scoped ENNReal ComplexConjugate local instance (α : Type u_10) : DecidableEq α := Classical.decEq α variable {I : Type u_11} {J : Type u_12} end section open scoped ENNReal local instance (α : Type u_13) : DecidableEq α := Classical.decEq α variable {I : Type u_14} {J : Type u_15} def contractionFamilyValue (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) (ξ : NestedHilbert I J) : NestedHilbert I J := ⟨fun i => A i (ξ i),(lp.memℓp ξ).norm.mono (fun i => ((A i).le_opNorm (ξ i)).trans (by simpa using mul_le_mul_of_nonneg_right (hA i) (norm_nonneg (ξ i))))⟩ theorem contractionFamilyValue_norm_le (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) (ξ : NestedHilbert I J) : ‖contractionFamilyValue A hA ξ‖ ≤ ‖ξ‖ := by apply lp.norm_mono (by norm_num : (2 : ℝ≥0∞) ≠ 0) intro i exact ((A i).le_opNorm (ξ i)).trans (by simpa using mul_le_mul_of_nonneg_right (hA i) (norm_nonneg (ξ i))) def contractionFamily (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) : NestedHilbert I J →L[ℂ] NestedHilbert I J := LinearMap.mkContinuous { toFun := contractionFamilyValue A hA map_add' := fun ξ η => by ext i j; exact congrArg (fun v : GroupHilbert J => v j) (map_add (A i) (ξ i) (η i)) map_smul' := fun z ξ => by ext i j; exact congrArg (fun v : GroupHilbert J => v j) (map_smul (A i) z (ξ i)) } 1 (fun ξ => by simpa using contractionFamilyValue_norm_le A hA ξ) @[simp] theorem contractionFamily_apply (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) (ξ : NestedHilbert I J) (i : I) : contractionFamily A hA ξ i = A i (ξ i) := rfl theorem contractionFamily_star (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) : contractionFamily (fun i => star (A i)) (by intro i; exact (norm_star (A i)).le.trans (hA i)) = star (contractionFamily A hA) := by apply ContinuousLinearMap.ext intro ξ apply ext_inner_right ℂ intro η change inner ℂ (contractionFamily (fun i => star (A i)) _ ξ) η = inner ℂ ((ContinuousLinearMap.adjoint (contractionFamily A hA)) ξ) η rw [ContinuousLinearMap.adjoint_inner_left] simp only [lp.inner_eq_tsum,contractionFamily_apply] apply tsum_congr intro i simpa only [ContinuousLinearMap.star_eq_adjoint,lp.inner_eq_tsum] using ContinuousLinearMap.adjoint_inner_left (A i) (η i) (ξ i) theorem contractionFamily_projection (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) (hp : ∀ i, IsStarProjection (A i)) : IsStarProjection (contractionFamily A hA) := by constructor · apply ContinuousLinearMap.ext intro ξ ext i j change (A i) (A i (ξ i)) j = A i (ξ i) j exact congrArg (fun B : GroupOperator J => B (ξ i) j) (hp i).isIdempotentElem.eq · change star (contractionFamily A hA) = contractionFamily A hA rw [← contractionFamily_star] congr 1 funext i exact (hp i).isSelfAdjoint.star_eq def productFamily (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) : GroupOperator (I × J) := (hilbertFlatten I J).conjStarAlgEquiv (contractionFamily A hA) @[simp] theorem productFamily_apply (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) (ξ : GroupHilbert (I × J)) (i : I) (j : J) : productFamily A hA ξ (i,j) = A i ((hilbertFlatten I J).symm ξ i) j := by change hilbertFlatten I J (contractionFamily A hA ((hilbertFlatten I J).symm ξ)) (i,j) = _ rw [hilbertFlatten_apply,contractionFamily_apply] theorem productFamily_projection (A : I → GroupOperator J) (hA : ∀ i, ‖A i‖ ≤ 1) (hp : ∀ i, IsStarProjection (A i)) : IsStarProjection (productFamily A hA) := (contractionFamily_projection A hA hp).map (hilbertFlatten I J).conjStarAlgEquiv variable {G : Type u_16} [Group G] theorem productFamily_mem_stabilization (A : ℕ → groupVonNeumann G) (hA : ∀ i, ‖(A i : GroupOperator G)‖ ≤ 1) : productFamily (fun i => (A i : GroupOperator G)) hA ∈ stabilization G := by rw [stabilization,StarSubalgebra.mem_centralizer_iff,stabilizedRightTranslations_eq_range] intro S hS obtain ⟨g,rfl⟩ := hS have hc (g : G) : productAmplification ℕ G (rightRegular g) * productFamily (fun i => (A i : GroupOperator G)) hA = productFamily (fun i => (A i : GroupOperator G)) hA * productAmplification ℕ G (rightRegular g) := by apply ContinuousLinearMap.ext intro ξ ext x rcases x with ⟨i,j⟩ simp only [mul_apply_eq_comp,productAmplification_apply,productFamily_apply, rightRegular_apply,hilbertFlatten_symm_apply] have he : (hilbertFlatten ℕ G).symm (productAmplification ℕ G (rightRegular g) ξ) i = rightRegular g ((hilbertFlatten ℕ G).symm ξ i) := by ext k simp only [productAmplification_apply,hilbertFlatten_symm_apply,rightRegular_apply] rw [he] simpa only [mul_apply_eq_comp,rightRegular_apply] using congrArg (fun B : GroupOperator G => B ((hilbertFlatten ℕ G).symm ξ i) j) (groupVN_commute_right (A i) g) constructor · exact hc g · rw [← map_star,rightRegular_star] exact hc g⁻¹ end section open scoped ENNReal variable {I : Type u_17} {J : Type u_18} def productSingle (I : Type u_19) (J : Type u_20) (j : I) : GroupHilbert J →L[ℂ] GroupHilbert (I × J) := by classical exact (hilbertFlatten I J).toContinuousLinearEquiv.toContinuousLinearMap.comp (lp.singleContinuousLinearMap ℂ (fun _ : I => GroupHilbert J) 2 j) @[simp] theorem productSingle_apply [DecidableEq I] (j i : I) (ξ : GroupHilbert J) (k : J) : productSingle I J j ξ (i,k) = if i = j then ξ k else 0 := by classical by_cases hi : i = j <;> simp [productSingle,hilbertFlatten_apply,lp.single_apply,hi] def matrixEntry (T : GroupOperator (I × J)) (i j : I) : GroupOperator J := (lp.evalCLM ℂ (fun _ : I => GroupHilbert J) 2 i).comp ((hilbertFlatten I J).symm.toContinuousLinearEquiv.toContinuousLinearMap.comp (T.comp (productSingle I J j))) @[simp] theorem matrixEntry_apply (T : GroupOperator (I × J)) (i j : I) (ξ : GroupHilbert J) (k : J) : matrixEntry T i j ξ k = T (productSingle I J j ξ) (i,k) := by exact hilbertFlatten_symm_apply _ _ _ theorem productSingle_single [DecidableEq I] [DecidableEq J] [DecidableEq (I × J)] (j : I) (g : J) : productSingle I J j (lp.single 2 g 1) = lp.single 2 (j,g) 1 := by classical ext x rcases x with ⟨i,k⟩ simp only [productSingle_apply,lp.single_apply,Pi.single_apply,Prod.mk.injEq] split_ifs <;> simp_all theorem matrixEntry_ext (T U : GroupOperator (I × J)) (h : ∀ i j, matrixEntry T i j = matrixEntry U i j) : T = U := by classical apply operator_matrix_ext intro x rcases x with ⟨j,g⟩ rw [← productSingle_single] ext y rcases y with ⟨i,k⟩ simpa only [matrixEntry_apply] using congrArg (fun A : GroupOperator J => A (lp.single 2 g 1) k) (h i j) theorem productAmplification_productSingle (S : GroupOperator J) (j : I) (ξ : GroupHilbert J) : productAmplification I J S (productSingle I J j ξ) = productSingle I J j (S ξ) := by classical ext x rcases x with ⟨i,k⟩ rw [productAmplification_apply,productSingle_apply] have he : (hilbertFlatten I J).symm (productSingle I J j ξ) i = if i=j then ξ else 0 := by ext l simp only [hilbertFlatten_symm_apply,productSingle_apply] split_ifs <;> rfl rw [he] split_ifs <;> simp theorem matrixEntry_diagonal_mul (S : GroupOperator J) (T : GroupOperator (I × J)) (i j : I) : matrixEntry (productAmplification I J S * T) i j = S * matrixEntry T i j := by apply ContinuousLinearMap.ext intro ξ ext k simp only [matrixEntry_apply,mul_apply_eq_comp,productAmplification_apply] rfl theorem matrixEntry_mul_diagonal (S : GroupOperator J) (T : GroupOperator (I × J)) (i j : I) : matrixEntry (T * productAmplification I J S) i j = matrixEntry T i j * S := by apply ContinuousLinearMap.ext intro ξ ext k simp only [matrixEntry_apply,mul_apply_eq_comp,productAmplification_productSingle] variable {G : Type u_21} [Group G] theorem mem_stabilization_iff_matrix (T : GroupOperator (ℕ × G)) : T ∈ stabilization G ↔ ∀ i j, matrixEntry T i j ∈ groupVonNeumann G := by rw [stabilization,StarSubalgebra.mem_centralizer_iff,stabilizedRightTranslations_eq_range] constructor · intro h i j rw [mem_groupVN_iff_commute_right] intro g have hh := (h _ ⟨g,rfl⟩).1 have he := congrArg (fun A => matrixEntry A i j) hh simpa only [matrixEntry_diagonal_mul,matrixEntry_mul_diagonal] using he · intro h S hS obtain ⟨g,rfl⟩ := hS have hc (g : G) : productAmplification ℕ G (rightRegular g) * T = T * productAmplification ℕ G (rightRegular g) := by apply matrixEntry_ext intro i j rw [matrixEntry_diagonal_mul,matrixEntry_mul_diagonal] exact (mem_groupVN_iff_commute_right _).mp (h i j) g exact ⟨hc g,by rw [← map_star,rightRegular_star]; exact hc g⁻¹⟩ end section variable {I : Type u_22} {J : Type u_23} def productCoordinate (I : Type u_24) (J : Type u_25) (i : I) : GroupHilbert (I × J) →L[ℂ] GroupHilbert J := (lp.evalCLM ℂ (fun _ : I => GroupHilbert J) 2 i).comp (hilbertFlatten I J).symm.toContinuousLinearEquiv.toContinuousLinearMap theorem productCoordinate_apply (i : I) (ξ : GroupHilbert (I × J)) (j : J) : productCoordinate I J i ξ j=ξ (i,j) := hilbertFlatten_symm_apply _ _ _ def matrixUnit (I : Type u_26) (J : Type u_27) (i j : I) : GroupOperator (I × J) := (productSingle I J i).comp (productCoordinate I J j) theorem matrixUnit_apply [DecidableEq I] (i j k : I) (ξ : GroupHilbert (I × J)) (g : J) : matrixUnit I J i j ξ (k,g)=if k=i then ξ (j,g) else 0 := by simp only [matrixUnit,ContinuousLinearMap.comp_apply,productSingle_apply,productCoordinate_apply] theorem matrixUnit_productSingle [DecidableEq I] (i j l : I) (ξ : GroupHilbert J) : matrixUnit I J i j (productSingle I J l ξ)=if j=l then productSingle I J i ξ else 0 := by by_cases h : j=l · subst l rw [ite_eq_left rfl] ext x rcases x with ⟨k,g⟩ simp only [matrixUnit_apply,productSingle_apply,ite_true] · rw [ite_eq_right h] ext x rcases x with ⟨k,g⟩ simp only [matrixUnit_apply,productSingle_apply,ite_eq_right h,ite_self,lp.coeFn_zero,Pi.zero_apply] theorem matrixEntry_matrixUnit_mul [DecidableEq I] (i j k l : I) (T : GroupOperator (I × J)) : matrixEntry (matrixUnit I J i j*T) k l=if k=i then matrixEntry T j l else 0 := by apply ContinuousLinearMap.ext intro ξ ext g simp only [matrixEntry_apply,mul_apply_eq_comp,matrixUnit_apply] split_ifs <;> simp only [matrixEntry_apply,zero_apply,lp.coeFn_zero,Pi.zero_apply] theorem matrixEntry_mul_matrixUnit [DecidableEq I] (i j k l : I) (T : GroupOperator (I × J)) : matrixEntry (T*matrixUnit I J i j) k l=if j=l then matrixEntry T k i else 0 := by apply ContinuousLinearMap.ext intro ξ ext g simp only [matrixEntry_apply,mul_apply_eq_comp,matrixUnit_productSingle] split_ifs <;> simp theorem matrixEntry_matrixUnit [DecidableEq I] (i j k l : I) : matrixEntry (matrixUnit I J i j) k l=if k=i ∧ j=l then 1 else 0 := by apply ContinuousLinearMap.ext intro ξ ext g simp only [matrixEntry_apply,matrixUnit_apply,productSingle_apply] split_ifs <;> simp_all theorem matrixEntry_productAmplification [DecidableEq I] (S : GroupOperator J) (i j : I) : matrixEntry (productAmplification I J S) i j=if i=j then S else 0 := by apply ContinuousLinearMap.ext intro ξ ext g rw [matrixEntry_apply,productAmplification_productSingle,productSingle_apply] split_ifs <;> rfl variable {G : Type u_28} [Group G] theorem matrixUnit_mem_stabilization (i j : ℕ) : matrixUnit ℕ G i j ∈ stabilization G := by rw [mem_stabilization_iff_matrix] intro k l rw [matrixEntry_matrixUnit] split_ifs · exact one_mem _ · exact zero_mem _ theorem diagonal_mem_stabilization (S : groupVonNeumann G) : productAmplification ℕ G (S : GroupOperator G) ∈ stabilization G := by rw [mem_stabilization_iff_matrix] intro k l rw [matrixEntry_productAmplification] split_ifs · exact S.property · exact zero_mem _ theorem stabilization_center_diagonal (T : stabilization G) (hc : ∀ S : stabilization G, Commute T S) : (T : GroupOperator (ℕ × G))=productAmplification ℕ G (matrixEntry (T : GroupOperator (ℕ × G)) 0 0) := by have hm (i j : ℕ) : (T : GroupOperator (ℕ × G))*matrixUnit ℕ G i j= matrixUnit ℕ G i j*(T : GroupOperator (ℕ × G)) := congrArg (fun A : stabilization G => (A : GroupOperator (ℕ × G))) (hc ⟨_,matrixUnit_mem_stabilization i j⟩).eq have hoff (i j : ℕ) (hij : i ≠ j) : matrixEntry (T : GroupOperator (ℕ × G)) i j=0 := by have he := congrArg (fun A => matrixEntry A i j) (hm j j) simpa only [matrixEntry_mul_matrixUnit,matrixEntry_matrixUnit_mul,ite_true,ite_eq_right hij] using he have hdiag (i : ℕ) : matrixEntry (T : GroupOperator (ℕ × G)) i i=matrixEntry (T : GroupOperator (ℕ × G)) 0 0 := by have he := congrArg (fun A => matrixEntry A i 0) (hm i 0) simpa only [matrixEntry_mul_matrixUnit,matrixEntry_matrixUnit_mul,ite_true] using he apply matrixEntry_ext intro i j rw [matrixEntry_productAmplification] split_ifs with hij · subst j exact hdiag i · exact hoff i j hij theorem stabilization_center_scalar (hbase : ∀ T : groupVonNeumann G, (∀ S : groupVonNeumann G, Commute T S) → ∃ c : ℂ, T=c • (1 : groupVonNeumann G)) (T : stabilization G) (hc : ∀ S : stabilization G, Commute T S) : ∃ c : ℂ, (T : GroupOperator (ℕ × G))=c • (1 : GroupOperator (ℕ × G)) := by let A : groupVonNeumann G := ⟨matrixEntry (T : GroupOperator (ℕ × G)) 0 0, (mem_stabilization_iff_matrix _).mp T.property 0 0⟩ have hA : ∀ S : groupVonNeumann G, Commute A S := by intro S apply Subtype.ext have he := congrArg (fun U : stabilization G => matrixEntry (U : GroupOperator (ℕ × G)) 0 0) (hc ⟨_,diagonal_mem_stabilization S⟩).eq change matrixEntry ((T : GroupOperator (ℕ × G))*productAmplification ℕ G (S : GroupOperator G)) 0 0= matrixEntry (productAmplification ℕ G (S : GroupOperator G)*(T : GroupOperator (ℕ × G))) 0 0 at he change matrixEntry (T : GroupOperator (ℕ × G)) 0 0*(S : GroupOperator G)= (S : GroupOperator G)*matrixEntry (T : GroupOperator (ℕ × G)) 0 0 simpa only [matrixEntry_mul_diagonal,matrixEntry_diagonal_mul] using he obtain ⟨c,hcA⟩ := hbase A hA refine ⟨c,?_⟩ rw [stabilization_center_diagonal T hc] have he : matrixEntry (T : GroupOperator (ℕ × G)) 0 0=c • (1 : GroupOperator G) := congrArg (fun U : groupVonNeumann G => (U : GroupOperator G)) hcA rw [he,map_smul,map_one] theorem stabilization_center_projection (hbase : ∀ T : groupVonNeumann G, (∀ S : groupVonNeumann G, Commute T S) → ∃ c : ℂ, T=c • (1 : groupVonNeumann G)) (P : GroupOperator (ℕ × G)) (hP : P ∈ stabilization G) (hp : IsStarProjection P) (hc : ∀ S ∈ stabilization G, Commute P S) : P=0 ∨ P=1 := by classical have hc' : ∀ S : stabilization G, Commute (⟨P,hP⟩ : stabilization G) S := fun S => Subtype.ext (hc S S.property).eq obtain ⟨c,he⟩ := stabilization_center_scalar hbase ⟨P,hP⟩ hc' change P=c • (1 : GroupOperator (ℕ × G)) at he have hh := congrArg (fun A : GroupOperator (ℕ × G) => A (lp.single 2 (0,1) (1 : ℂ)) (0,1)) hp.isIdempotentElem rw [he] at hh have hcc : c*c=c := by change c*(c*(lp.single 2 (0,1) (1 : ℂ) : GroupHilbert (ℕ × G)) (0,1))= c*(lp.single 2 (0,1) (1 : ℂ) : GroupHilbert (ℕ × G)) (0,1) at hh simpa only [lp.single_apply,Pi.single_eq_same,mul_one] using hh have hzero : c*(c-1)=0 := by linear_combination hcc rcases mul_eq_zero.mp hzero with hz|ho · left simpa only [hz,zero_smul ℂ (A := GroupOperator (ℕ × G))] using he · right have ho : c=1 := sub_eq_zero.mp ho simpa only [ho,one_smul] using he theorem infinite_stabilization_center_projection (P : GroupOperator (ℕ × FreeGroup ℕ)) (hP : P ∈ stabilization (FreeGroup ℕ)) (hp : IsStarProjection P) (hc : ∀ S ∈ stabilization (FreeGroup ℕ), Commute P S) : P=0 ∨ P=1 := stabilization_center_projection (fun T hT => ⟨canonicalTrace T,infinite_free_group_center_scalar T hT⟩) P hP hp hc end open scoped ENNReal variable {I : Type u_29} {J : Type u_30} theorem hilbert_ofReal_norm_sq (ξ : lp (fun _ : I => GroupHilbert J) 2) : ENNReal.ofReal (‖ξ‖^2)=∑' i, ENNReal.ofReal (‖ξ i‖^2) := by have hs : HasSum (fun i => ‖ξ i‖^2) (‖ξ‖^2) := by simpa only [ENNReal.toReal_ofNat,Real.rpow_two] using lp.hasSum_norm (by norm_num : (0:ℝ)<(2:ℝ≥0∞).toReal) ξ rw [← hs.tsum_eq,ENNReal.ofReal_tsum_of_nonneg (fun i => sq_nonneg _) hs.summable] theorem productSingle_inner_right (i : I) (ξ : GroupHilbert (I × J)) (η : GroupHilbert J) : inner ℂ ξ (productSingle I J i η)=inner ℂ (productCoordinate I J i ξ) η := by classical change inner ℂ ξ ((hilbertFlatten I J) (lp.single 2 i η))= inner ℂ (((hilbertFlatten I J).symm ξ) i) η have he := (hilbertFlatten I J).inner_map_map ((hilbertFlatten I J).symm ξ) (lp.single 2 i η) simpa only [LinearIsometryEquiv.apply_symm_apply,lp.inner_single_right] using he theorem productSingle_inner_left (i : I) (ξ : GroupHilbert (I × J)) (η : GroupHilbert J) : inner ℂ (productSingle I J i η) ξ=inner ℂ η (productCoordinate I J i ξ) := by have he := congrArg (star : ℂ → ℂ) (productSingle_inner_right i ξ η) change (starRingEnd ℂ) (inner ℂ ξ (productSingle I J i η))= (starRingEnd ℂ) (inner ℂ (productCoordinate I J i ξ) η) at he simpa only [inner_conj_symm] using he theorem matrixEntry_star (T : GroupOperator (I × J)) (i j : I) : matrixEntry (star T) i j=star (matrixEntry T j i) := by apply ContinuousLinearMap.ext intro ξ apply ext_inner_left ℂ intro η have he (T : GroupOperator (I × J)) (i j : I) (ξ : GroupHilbert J) : matrixEntry T i j ξ=productCoordinate I J i (T (productSingle I J j ξ)) := rfl rw [he,← productSingle_inner_left,ContinuousLinearMap.star_eq_adjoint, ContinuousLinearMap.adjoint_inner_right,productSingle_inner_right] change inner ℂ (matrixEntry T j i η) ξ=inner ℂ η (star (matrixEntry T j i) ξ) rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right] variable {G : Type u_31} [Group G] [DecidableEq G] theorem stabilizedProjectionTrace_eq_matrix (T : stabilization G) : stabilizedProjectionTrace T=∑' j : ℕ, ∑' i : ℕ, ENNReal.ofReal (‖matrixEntry (T : GroupOperator (ℕ × G)) i j (lp.single 2 (1 : G) (1 : ℂ))‖^2) := by classical apply tsum_congr intro j rw [← (hilbertFlatten ℕ G).symm.norm_map,hilbert_ofReal_norm_sq] apply tsum_congr intro i congr 2 congr 1 ext g rw [matrixEntry_apply,productSingle_single,hilbertFlatten_symm_apply] apply congrArg (fun ξ : GroupHilbert (ℕ × G) => (T : GroupOperator (ℕ × G)) ξ (i,g)) ext x simp only [lp.single_apply,Pi.single_apply] split_ifs <;> rfl theorem groupVN_adjoint_delta_norm (T : groupVonNeumann G) : ‖(star T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))‖^2= ‖(T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))‖^2 := by classical have ht : canonicalTrace (T*star T)=(‖(star T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))‖^2 : ℂ) := by simpa only [star_star,StarMemClass.coe_star] using canonicalTrace_star_mul (star T) have he' : (‖(T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))‖^2 : ℂ)= (‖(star T : GroupOperator G) (lp.single 2 (1 : G) (1 : ℂ))‖^2 : ℂ) := by rw [← canonicalTrace_star_mul,← ht] exact canonicalTrace_mul (star T) T have hh := congrArg Complex.re he' simpa only [← Complex.ofReal_pow,Complex.ofReal_re] using hh.symm theorem stabilizedProjectionTrace_star (T : stabilization G) : stabilizedProjectionTrace (star T)=stabilizedProjectionTrace T := by classical rw [stabilizedProjectionTrace_eq_matrix,stabilizedProjectionTrace_eq_matrix,ENNReal.tsum_comm] apply tsum_congr intro i apply tsum_congr intro j congr 1 simp only [StarMemClass.coe_star] rw [matrixEntry_star] exact groupVN_adjoint_delta_norm ⟨_,(mem_stabilization_iff_matrix _).mp T.property j i⟩ end FreeGroupFactorMain.Amplification end end OAI