import OAI.Analysis.InterpolatedFactors.GroupBridge namespace OAI noncomputable section universe u_1 u_2 u_3 u_4 u_5 u_6 u_7 u_8 u_9 namespace FreeGroupFactorMain.HilbertUltraweak variable (E : Type u_1) [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] @[instance_reducible] def topology : TopologicalSpace (E →L[ℂ] E) := ⨅ (ξ : lp (fun _ : ℕ => E) 2) (η : lp (fun _ : ℕ => E) 2), TopologicalSpace.induced (fun T : E →L[ℂ] E => ∑' k : ℕ, inner ℂ (η k) (T (ξ k))) inferInstance variable {E} {F : Type u_2} [NormedAddCommGroup F] [InnerProductSpace ℂ F] [CompleteSpace F] def sequenceMap (L : E →L[ℂ] F) (ξ : lp (fun _ : ℕ => E) 2) : lp (fun _ : ℕ => F) 2 := ⟨fun k => L (ξ k),((lp.memℓp ξ).norm.const_mul ‖L‖).mono (fun k => L.le_opNorm (ξ k))⟩ omit [CompleteSpace E] in theorem functional_continuous (ξ η : lp (fun _ : ℕ => E) 2) : @Continuous _ _ (topology E) _ (fun T : E →L[ℂ] E => ∑' k : ℕ,inner ℂ (η k) (T (ξ k))) := continuous_iInf_dom (i := ξ) (continuous_iInf_dom (i := η) continuous_induced_dom) theorem sandwich_continuous (L : E →L[ℂ] F) (R : F →L[ℂ] E) : @Continuous _ _ (topology E) (topology F) (fun T : E →L[ℂ] E => L.comp (T.comp R)) := by apply continuous_iInf_rng.mpr intro ξ apply continuous_iInf_rng.mpr intro η apply continuous_induced_rng.mpr have he : (fun T : E →L[ℂ] E => ∑' k : ℕ,inner ℂ (η k) (L (T (R (ξ k)))))= (fun T : E →L[ℂ] E => ∑' k : ℕ,inner ℂ (sequenceMap (ContinuousLinearMap.adjoint L) η k) (T (sequenceMap R ξ k))) := by funext T apply tsum_congr intro k exact (ContinuousLinearMap.adjoint_inner_left L (T (R (ξ k))) (η k)).symm change @Continuous _ _ (topology E) _ (fun T : E →L[ℂ] E => ∑' k : ℕ,inner ℂ (η k) (L (T (R (ξ k))))) rw [he] exact functional_continuous _ _ theorem spatial_continuous (U : E ≃ₗᵢ[ℂ] F) : @Continuous _ _ (topology E) (topology F) U.conjStarAlgEquiv := sandwich_continuous U.toContinuousLinearEquiv.toContinuousLinearMap U.symm.toContinuousLinearEquiv.toContinuousLinearMap omit [CompleteSpace E] in theorem add_continuous {X : Type u_3} [TopologicalSpace X] {f g : X → (E →L[ℂ] E)} (hf : @Continuous _ _ _ (topology E) f) (hg : @Continuous _ _ _ (topology E) g) : @Continuous _ _ _ (topology E) (fun x => f x+g x) := by let : TopologicalSpace (E →L[ℂ] E) := topology E apply continuous_iInf_rng.mpr intro ξ apply continuous_iInf_rng.mpr intro η apply continuous_induced_rng.mpr have he (x : X) : (∑' k : ℕ,inner ℂ (η k) ((f x+g x) (ξ k)))= (∑' k : ℕ,inner ℂ (η k) (f x (ξ k)))+(∑' k : ℕ,inner ℂ (η k) (g x (ξ k))) := by simp only [add_apply,inner_add_right] exact (lp.summable_inner η (sequenceMap (f x) ξ)).tsum_add (lp.summable_inner η (sequenceMap (g x) ξ)) change Continuous (fun x : X => ∑' k : ℕ,inner ℂ (η k) ((f x+g x) (ξ k))) simp_rw [he] exact ((functional_continuous ξ η).comp hf).add ((functional_continuous ξ η).comp hg) end FreeGroupFactorMain.HilbertUltraweak section namespace FreeGroupFactorMain.CornerRerouting variable {A : Type u_4} [Ring A] [StarRing A] theorem lift_commute_four {p v t z : A} (hp : IsStarProjection p) (hpv : p*v=v) (hvr : v*(1-p)=v) (hvi : star v*v=1-p) (ht : Supported p t) (htr : Commute t (v*star v)) (ha : Commute t (p*z*p)) (hb : Commute t (p*z*star v)) (hc : Commute t (v*z*p)) (hd : Commute t (v*z*star v)) : Commute (lift v t) z := by have hvp := right_zero hvr have hvsp : star v*p=star v := by simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hpv have hsupport (a b : A) (hal : p*a=a) (hbr : b*p=b) : Supported p (a*z*b) := by exact ⟨by rw [← mul_assoc,← mul_assoc,hal],by rw [mul_assoc,hbr]⟩ have ha' := lift_commute_supported hp.isSelfAdjoint hvp (hsupport p p hp.isIdempotentElem.eq hp.isIdempotentElem.eq) ha have hb' := lift_commute_supported hp.isSelfAdjoint hvp (hsupport p (star v) hp.isIdempotentElem.eq hvsp) hb have hc' := lift_commute_supported hp.isSelfAdjoint hvp (hsupport v p hpv hp.isIdempotentElem.eq) hc have hd' := lift_commute_supported hp.isSelfAdjoint hvp (hsupport v (star v) hpv hvsp) hd have hv' := lift_commute_edge hpv hvp hvi hvr ht htr have hvs' := lift_commute_edge_star hp.isSelfAdjoint hpv hvp hvi hvr ht htr have hz : z=p*z*p+(p*z*star v)*v+star v*(v*z*p)+star v*(v*z*star v)*v := by calc z = (p+star v*v)*z*(p+star v*v) := by rw [hvi]; noncomm_ring _ = _ := by noncomm_ring rw [hz] exact ((ha'.add_right (hb'.mul_right hv')).add_right (hvs'.mul_right hc')).add_right ((hvs'.mul_right hd').mul_right hv') end FreeGroupFactorMain.CornerRerouting namespace FreeGroupFactorMain.FullReducingCorner open Amplification CornerRerouting variable {A : Type u_5} [CStarAlgebra A] def bicomm (s : Set A) : StarSubalgebra ℂ A := StarSubalgebra.centralizer ℂ (StarSubalgebra.centralizer ℂ s : Set A) def compress (p : A) (hp : IsStarProjection p) (a : A) : Corner p hp := ⟨p*a*p,by rw [← mul_assoc,← mul_assoc,hp.isIdempotentElem.eq],by rw [mul_assoc,hp.isIdempotentElem.eq]⟩ theorem compress_star (p : A) (hp : IsStarProjection p) (a : A) : compress p hp (star a)=star (compress p hp a) := by apply Subtype.ext change p*star a*p=star (p*a*p) simp only [star_mul,hp.isSelfAdjoint.star_eq,mul_assoc] def compressedSet (p : A) (hp : IsStarProjection p) (s : Set A) : Set (Corner p hp) := compress p hp '' s variable (p : A) (hp : IsStarProjection p) (s : Set A) variable (hps : p∈StarSubalgebra.centralizer ℂ s) theorem supported_commutant (x : Corner p hp) (hx : (x:A)∈StarSubalgebra.centralizer ℂ s) : x∈StarSubalgebra.centralizer ℂ (compressedSet p hp s) := by rw [StarSubalgebra.mem_centralizer_iff] rintro _ ⟨y,hy,rfl⟩ have hpy : Commute p (x:A) := x.property.1.trans x.property.2.symm have hyx : Commute y (x:A) := ((StarSubalgebra.mem_centralizer_iff ℂ).mp hx y hy).1 have he : Commute (compress p hp y) x := by apply Subtype.ext exact ((hpy.mul_left hyx).mul_left hpy).eq have hyx' : Commute (star y) (x:A) := ((StarSubalgebra.mem_centralizer_iff ℂ).mp hx y hy).2 have he' : Commute (compress p hp (star y)) x := by apply Subtype.ext exact ((hpy.mul_left hyx').mul_left hpy).eq rw [compress_star] at he' exact ⟨he.eq,he'.eq⟩ include hps in theorem commutant_of_supported (x : Corner p hp) (hx : x∈StarSubalgebra.centralizer ℂ (compressedSet p hp s)) : (x:A)∈StarSubalgebra.centralizer ℂ s := by have htest (y : A) (hyP : Commute y p) (hh : Commute (compress p hp y) x) : Commute y (x:A) := by have hh' := congrArg Subtype.val hh.eq change (p*y*p)*(x:A)=(x:A)*(p*y*p) at hh' change y*(x:A)=(x:A)*y calc y*(x:A)= (p*y*p)*(x:A) := by calc y*(x:A) = (y*p)*(x:A) := by rw [mul_assoc,x.property.1] _ = (p*y)*(x:A) := by rw [hyP] _ = (p*y*p)*(x:A) := by rw [mul_assoc (p*y),x.property.1] _ = (x:A)*(p*y*p) := hh' _ = (x:A)*y := by calc (x:A)*(p*y*p) = (x:A)*(y*p) := by rw [← mul_assoc,← mul_assoc,x.property.2,mul_assoc] _ = (x:A)*(p*y) := by rw [hyP] _ = (x:A)*y := by rw [← mul_assoc,x.property.2] rw [StarSubalgebra.mem_centralizer_iff] intro y hy have hc := (StarSubalgebra.mem_centralizer_iff ℂ).mp hx _ ⟨y,hy,rfl⟩ have hpY := (StarSubalgebra.mem_centralizer_iff ℂ).mp hps y hy refine ⟨(htest y hpY.1 hc.1).eq,?_⟩ apply htest (star y) hpY.2 change compress p hp (star y)*x=x*compress p hp (star y) rw [compress_star] exact hc.2 include hps in theorem compress_mem (a : bicomm s) : compress p hp (a:A)∈bicomm (compressedSet p hp s) := by rw [bicomm,StarSubalgebra.mem_centralizer_iff] intro x hx have ht (z : Corner p hp) (hz : z∈StarSubalgebra.centralizer ℂ (compressedSet p hp s)) : Commute z (compress p hp (a:A)) := by have hzA := commutant_of_supported p hp s hps z hz have hza := ((StarSubalgebra.mem_centralizer_iff ℂ).mp a.property (z:A) hzA).1 have hzp : Commute (z:A) p := z.property.2.trans z.property.1.symm apply Subtype.ext exact ((hzp.mul_right hza).mul_right hzp).eq exact ⟨(ht x hx).eq,(ht (star x) (star_mem hx)).eq⟩ variable (v : A) (hvs : v∈StarSubalgebra.centralizer ℂ s) variable (hpv : p*v=v) (hvr : v*(1-p)=v) (hvi : star v*v=1-p) include hps hvs hpv hvr hvi in theorem lift_mem (x : bicomm (compressedSet p hp s)) : lift v (x.val:A)∈bicomm s := by have hsupport (a b : A) (hal : p*a=a) (hbr : b*p=b) : Supported p (a*b) := ⟨by rw [← mul_assoc,hal],by rw [mul_assoc,hbr]⟩ have hvsp : star v*p=star v := by simpa only [star_mul,hp.isSelfAdjoint.star_eq] using congrArg star hpv have hx (z : A) (hz : z∈StarSubalgebra.centralizer ℂ s) (hzp : Supported p z) : Commute (x.val:A) z := by let z' : Corner p hp := ⟨z,hzp⟩ have hm := supported_commutant p hp s z' hz have hh := ((StarSubalgebra.mem_centralizer_iff ℂ).mp x.property z' hm).1 exact (congrArg Subtype.val hh).symm have htest (z : A) (hz : z∈StarSubalgebra.centralizer ℂ s) : Commute (lift v (x.val:A)) z := by apply lift_commute_four hp hpv hvr hvi x.val.property · exact hx _ (mul_mem hvs (star_mem hvs)) (hsupport v (star v) hpv hvsp) · exact hx _ (mul_mem (mul_mem hps hz) hps) (hsupport (p*z) p (by rw [← mul_assoc,hp.isIdempotentElem.eq]) hp.isIdempotentElem.eq) · exact hx _ (mul_mem (mul_mem hps hz) (star_mem hvs)) (hsupport (p*z) (star v) (by rw [← mul_assoc,hp.isIdempotentElem.eq]) hvsp) · exact hx _ (mul_mem (mul_mem hvs hz) hps) (hsupport (v*z) p (by rw [← mul_assoc,hpv]) hp.isIdempotentElem.eq) · exact hx _ (mul_mem (mul_mem hvs hz) (star_mem hvs)) (hsupport (v*z) (star v) (by rw [← mul_assoc,hpv]) hvsp) rw [bicomm,StarSubalgebra.mem_centralizer_iff] intro z hz exact ⟨(htest z hz).symm.eq,(htest (star z) (star_mem hz)).symm.eq⟩ include hps hvs hpv hvi in theorem lift_compress (a : bicomm s) : lift v (compress p hp (a:A):A)=(a:A) := by have hap := ((StarSubalgebra.mem_centralizer_iff ℂ).mp a.property p hps).1 have havs := ((StarSubalgebra.mem_centralizer_iff ℂ).mp a.property v hvs).2 change p*(a:A)*p+star v*(p*(a:A)*p)*v=(a:A) rw [hap,mul_assoc,hp.isIdempotentElem.eq,← mul_assoc (star v),havs,mul_assoc,hpv,mul_assoc,hvi] noncomm_ring include hvr in theorem compress_lift (x : Corner p hp) : compress p hp (lift v (x:A))=x := by have hvp := right_zero hvr apply Subtype.ext change p*lift v (x:A)*p=(x:A) rw [mul_assoc,lift_mul_supported hvp ⟨hp.isIdempotentElem.eq,hp.isIdempotentElem.eq⟩,x.property.2,x.property.1] def iso : bicomm s ≃⋆ₐ[ℂ] bicomm (compressedSet p hp s) where toFun a := ⟨compress p hp a,compress_mem p hp s hps a⟩ invFun x := ⟨lift v (x.val:A),lift_mem p hp s hps v hvs hpv hvr hvi x⟩ left_inv a := Subtype.ext (lift_compress p hp s hps v hvs hpv hvi a) right_inv x := Subtype.ext (compress_lift p hp v hvr x.val) map_mul' a b := by apply Subtype.ext apply Subtype.ext have hpA := ((StarSubalgebra.mem_centralizer_iff ℂ).mp a.property p hps).1 change p*((a:A)*(b:A))*p=(p*(a:A)*p)*(p*(b:A)*p) calc p*((a:A)*(b:A))*p = (p*(a:A))*((b:A)*p) := by simp only [mul_assoc] _ = ((a:A)*p)*((b:A)*p) := by rw [hpA] _ = (p*(a:A)*p)*(p*(b:A)*p) := by rw [hpA] simp only [mul_assoc,← mul_assoc p p,hp.isIdempotentElem.eq] map_add' a b := by apply Subtype.ext; apply Subtype.ext change p*((a:A)+(b:A))*p=p*(a:A)*p+p*(b:A)*p simp only [mul_add,add_mul] map_smul' c a := by apply Subtype.ext; apply Subtype.ext change p*(c • (a:A))*p=c • (p*(a:A)*p) simp only [mul_smul_comm,smul_mul_assoc] map_star' a := Subtype.ext (compress_star p hp (a:A)) end FreeGroupFactorMain.FullReducingCorner end namespace FreeGroupFactorMain.ReducingSubspace variable {E : Type u_6} [NormedAddCommGroup E] [InnerProductSpace ℂ E] variable (K : Submodule ℂ E) [CompleteSpace K] def reduce (T : E →L[ℂ] E) : K →L[ℂ] K := K.orthogonalProjectionOnto.comp (T.comp K.subtypeL) theorem reduce_apply (T : E →L[ℂ] E) (hT : ∀ ξ∈K,T ξ∈K) (ξ : K) : (reduce K T ξ:E)=T ξ := K.starProjection_eq_self_iff.mpr (hT ξ ξ.property) theorem reduce_one : reduce K (1:E →L[ℂ] E)=1 := by apply ContinuousLinearMap.ext intro ξ exact K.orthogonalProjectionOnto_mem_subspace_eq_self ξ theorem reduce_add (T U : E →L[ℂ] E) : reduce K (T+U)=reduce K T+reduce K U := by apply ContinuousLinearMap.ext intro ξ exact K.orthogonalProjectionOnto.map_add _ _ theorem reduce_smul (c : ℂ) (T : E →L[ℂ] E) : reduce K (c • T)=c • reduce K T := by apply ContinuousLinearMap.ext intro ξ exact K.orthogonalProjectionOnto.map_smul c _ theorem inner_reduce (T : E →L[ℂ] E) (ξ η : K) : inner ℂ ξ (reduce K T η)=inner ℂ (ξ:E) (T η) := K.inner_orthogonalProjectionOnto_eq_of_mem_left ξ (T η) variable [CompleteSpace E] theorem reduce_star (T : E →L[ℂ] E) : reduce K (star T)=star (reduce K T) := by apply ContinuousLinearMap.ext intro ξ apply ext_inner_left ℂ intro η rw [inner_reduce,ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right] rw [ContinuousLinearMap.star_eq_adjoint,ContinuousLinearMap.adjoint_inner_right] exact (K.inner_orthogonalProjectionOnto_eq_of_mem_right ξ (T η)).symm omit [CompleteSpace E] in theorem reduce_mul (T U : E →L[ℂ] E) (hU : ∀ ξ∈K,U ξ∈K) : reduce K (T*U)=reduce K T*reduce K U := by apply ContinuousLinearMap.ext intro ξ change K.orthogonalProjectionOnto (T (U ξ))=K.orthogonalProjectionOnto (T (reduce K U ξ:E)) rw [reduce_apply K U hU] end FreeGroupFactorMain.ReducingSubspace namespace FreeGroupFactorMain.GaussianRecognition.System open ReducingSubspace variable {A : Type u_7} {E : Type u_8} {I : Type u_9} [Ring A] [StarRing A] [Algebra ℂ A] variable [NormedAddCommGroup E] [InnerProductSpace ℂ E] [CompleteSpace E] variable {covariance : I → I → A → A} (X : System A E I covariance) def reducedCoeff : A →⋆ₐ[ℂ] (X.normalHull →L[ℂ] X.normalHull) where toFun a := reduce X.normalHull (X.coeff a) map_zero' := by simp [reduce] map_one' := by rw [map_one,reduce_one] map_add' a b := by rw [map_add,reduce_add] map_mul' a b := by rw [map_mul,reduce_mul X.normalHull _ _ (fun _ h => X.normalHull_coeff b h)] commutes' c := by rw [Algebra.algebraMap_eq_smul_one,map_smul,map_one,reduce_smul,reduce_one,Algebra.algebraMap_eq_smul_one] map_star' a := by rw [map_star,reduce_star] def reducedSystem : System A X.normalHull I covariance where coeff := X.reducedCoeff root := ⟨X.root,X.normalHull_root⟩ create i := reduce X.normalHull (X.create i) contract i j a := by change star (reduce X.normalHull (X.create i))*reduce X.normalHull (X.coeff a)* reduce X.normalHull (X.create j)=reduce X.normalHull (X.coeff (covariance i j a)) rw [← reduce_star,← reduce_mul X.normalHull _ _ (fun _ h => X.normalHull_coeff a h), ← reduce_mul X.normalHull _ _ (fun _ h => X.normalHull_create j h),X.contract] annihilate i a := by apply Subtype.ext rw [← reduce_star,reduce_apply X.normalHull _ (fun _ h => X.normalHull_annihilate i h)] change star (X.create i) ((reduce X.normalHull (X.coeff a) ⟨X.root,X.normalHull_root⟩):E)=0 rw [reduce_apply X.normalHull _ (fun _ h => X.normalHull_coeff a h),X.annihilate] theorem reducedSystem_normal (a : A) (w : List (I × A)) : (X.reducedSystem.normal a w:E)=X.normal a w := by induction w generalizing a with | nil => exact reduce_apply X.normalHull _ (fun _ h => X.normalHull_coeff a h) _ | cons q w ih => rcases q with ⟨i,b⟩ change (reduce X.normalHull (X.coeff a) (reduce X.normalHull (X.create i) (X.reducedSystem.normal b w)):E)=_ rw [reduce_apply X.normalHull _ (fun _ h => X.normalHull_coeff a h), reduce_apply X.normalHull _ (fun _ h => X.normalHull_create i h),ih] rfl theorem reducedSystem_family (w : Word A I) : (X.reducedSystem.family w:E)=X.family w := X.reducedSystem_normal _ _ theorem reducedSystem_state (a : A) : inner ℂ X.reducedSystem.root (X.reducedSystem.coeff a X.reducedSystem.root)= inner ℂ X.root (X.coeff a X.root) := inner_reduce X.normalHull _ _ _ theorem reducedSystem_total : (Submodule.span ℂ (Set.range X.reducedSystem.family)).topologicalClosure=⊤ := by have hi : (Subtype.val '' (Submodule.span ℂ (Set.range X.reducedSystem.family) : Set X.normalHull))= (X.normalSpan:Set E) := by change ((Submodule.span ℂ (Set.range X.reducedSystem.family)).map X.normalHull.subtype : Set E)=_ rw [Submodule.map_span,← Set.range_comp] change (Submodule.span ℂ (Set.range (fun w => (X.reducedSystem.family w:E))) : Set E)=_ simp only [X.reducedSystem_family,normalSpan] apply top_unique intro ξ _ change ξ∈closure (Submodule.span ℂ (Set.range X.reducedSystem.family) : Set X.normalHull) rw [Topology.IsEmbedding.subtypeVal.closure_eq_preimage_closure_image,hi] exact ξ.property end FreeGroupFactorMain.GaussianRecognition.System namespace FreeGroupFactorMain.WeightedFock open FreeFock Amplification CyclicSpectral GaussianRecognition variable (u v : ℝ) (hv : 0 (reduced_weighted_scalar_state u v hv hvu a).symm) (weightedSystem u v hv hvu).reducedSystem_total scalarSystem_total scalarFlip theorem reducedGaussianIso_normal : @Continuous _ _ ((weightedSystem u v hv hvu).reducedSystem.generatedTopology scalarFlip) (scalarSystem.generatedTopology scalarFlip) (reducedGaussianIso u v hv hvu) := (weightedSystem u v hv hvu).reducedSystem.generatedIso_normal scalarSystem (fun a => (reduced_weighted_scalar_state u v hv hvu a).symm) (weightedSystem u v hv hvu).reducedSystem_total scalarSystem_total scalarFlip theorem reducedGaussianIso_trace (T : (weightedSystem u v hv hvu).reducedSystem.generated scalarFlip) : inner ℂ scalarSystem.root ((reducedGaussianIso u v hv hvu T : FreeFock.Operator ScalarIndex) scalarSystem.root)= inner ℂ (weightedSystem u v hv hvu).reducedSystem.root ((T : (weightedSystem u v hv hvu).normalHull →L[ℂ] (weightedSystem u v hv hvu).normalHull) (weightedSystem u v hv hvu).reducedSystem.root) := (weightedSystem u v hv hvu).reducedSystem.generatedIso_trace scalarSystem (fun a => (reduced_weighted_scalar_state u v hv hvu a).symm) (weightedSystem u v hv hvu).reducedSystem_total scalarSystem_total scalarFlip T end FreeGroupFactorMain.WeightedFock end end OAI