Central Limit Theorem for General Universal Products Philipp Var so - PowerPoint PPT Presentation

Central Limit Theorem for General Universal Products Philipp Varˇ so Institute of Mathematics and Computer Science, University Greifswald, Germany S´ eminaire Analyse Fonctionnelle, Laboratoire de Math´ ematiques de Besan¸ con, 12.12.2017

Motivation • investigate noncommutative notions of independence using an algebraic approach • for d, m ∈ ◆ define category algP d , m of ( d, m )-algebraic quantum probability spaces [MS17] 1 • model independence by so called universal products • Muraki has shown ∃ only 5 normal universal products in algP 1 , 1 , i.e. objects are algebras A equipped with ϕ ∈ A ∗ • What about a d -tuple of linear functionals? What about an m -fold free product of A ? • reasons why we should study such structures are e.g. bifreeness ( d = 1 , m = 2) [Voi14], c-freeness ( d = 2 , m = 1) [BS91] and bimonotonic independence of type II ( d = 1 , m = 2) [GHS17] [Ger17] 1 S. Manzel and M. Sch¨ urmann. “Non-commutative stochastic independence and cumulants”. In: Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20.2 (2017), pp. 1750010, 38. 1 / 27

Contents 1 Essential definitions 2 Definition of Lachs Functor 3 Mini digression: comonoids and cotensor functors 4 Lachs Functor is cotensor functor 5 Convolution products and exponential series 6 Central Limit Theorem

Notational conventions • for m ∈ ◆ set [ m ] := { 1 , . . . , m } • set of all linear functionals of vector space V denoted by V ∗ • S( V ) for symmetric tensor algebra of ❈ -vector space V • if A is unital algebra and f : V A is linear map with f ( x ) f ( y ) = f ( y ) f ( x ) f.a. x, y ∈ V , then S ( f ): S( V ) A is unique unital algebra homomorphism s.t. S ( f ) ◦ ι V = f . • let V, W be ❈ -vector spaces and g : V W linear map, we put S( g ) := S ( ι W ◦ g ): S( V ) S( W ) • all algebras of consideration are in particular ❈ -vector spaces, associative but not necessarilly unital • free product of algebras? 2 / 27

Digression: Free product of algebras • for arbitrary index set I define ❆ I := { ε = ( ε i ) i ∈ [ m ] ∈ I m | m ∈ ◆ , ε k � = ε k +1 , k = 1 , . . . , m − 1 } • given family of vector spaces ( V i ) i ∈ I , for ε ∈ ❆ I set V ε := V ε 1 ⊗ · · · ⊗ V ε m • given family of algebras ( A i ) i ∈ I we set ⊔ � A i := A ε i ∈ I ε ∈ ❆ I with multiplication given by � a 1 ⊗ · · · ⊗ a m ⊗ b 1 ⊗ · · · ⊗ b n if ε m � = δ 1 ( a 1 ⊗ · · · ⊗ a m ) ( b 1 ⊗ · · · ⊗ b n ) := a 1 ⊗ · · · ⊗ a m b 1 ⊗ · · · ⊗ b n if ε m = δ 1 � �� � � �� � ∈ A ε ∈ A δ • ⊔ is coproduct in category alg , i.e. for ( A i ) i ∈ I ⊆ Obj ( alg ), A ∈ Obj ( alg ) and family of morphisms ( f i : A i A ) i ∈ I exists unique morphism ⊔ i ∈ I f i : ⊔ i ∈ I A i A such that ⊔ i ∈ I f i ◦ ι i = f i . Moreover f i X i Y i ι ′ ι i i ⊔ i ∈ I X i ⊔ i ∈ I Y i ⊔ i ∈ I f i 3 / 27

Central Limit Theorem for General Universal Products 1 Essential definitions 2 Definition of Lachs Functor 3 Mini digression: comonoids and cotensor functors 4 Lachs Functor is cotensor functor 5 Convolution products and exponential series 6 Central Limit Theorem 4 / 27

Definition 1 (Category alg m ) • objects of category alg m are ordered pairs ( A , ( A ( i ) ) i ∈ [ d ] ) with following properties i.) A is an associative algebra ii.) ∀ i ∈ [ m ]: A ( i ) is a subalgebra of A iii.) finite family ( A ( i ) ) i ∈ [ m ] freely generates A , i.e. the algebra homomorphism from ⊔ i ∈ [ m ] A ( i ) A defined by A ε ∋ a 1 ⊗ · · · ⊗ a n Ñ a 1 · · · · · a n ∈ A is bijection � � ( B , ( B ( i ) ) i ∈ [ m ] ) , ( A , ( A ( i ) ) i ∈ [ m ] ) • morphisms j ∈ Morph alg m fullfill the following iv.) j ∈ Morph alg ( B , A ) v.) ∀ k ∈ [ m ]: j ( B ( k ) ) ⊆ A ( k ) 5 / 27

Definition 2 (Category algP d , m [MS17, Sec. 2]) • objects of algP d , m are triples ( A , ( A ( i ) ) i ∈ [ m ] , ( ϕ ( i ) ) i ∈ [ d ] ), wherein ( A , ( A ( i ) ) i ∈ [ m ] ) ∈ Obj ( alg m ) and ( ϕ ( i ) ) i ∈ [ d ] ∈ ( A ∗ ) d • for morphisms � � ( B , ( B ( i ) ) i ∈ [ m ] , ( ψ ( i ) ) i ∈ [ d ] ) , ( A , ( A ( i ) ) i ∈ [ m ] , ( ϕ ( i ) ) i ∈ [ d ] ) j ∈ Morph algP d , m of algP d , m we demand i.) j ∈ Morph alg m ( B , A ) ii.) ∀ i ∈ [ d ]: ϕ ( i ) ◦ j = ψ ( i ) . Remark 1 there exist well known isomorphisms � d ∼ � Hom ❈ ( V d , ❈ ) ∼ = Hom ❈ ( V, ❈ d ) = Hom ❈ ( V, ❈ ) 6 / 27

Definition 3 (u.a.u.-product in algP d , m ) • universal product in the category algP d , m is bifunctor ⊙ of the form � � �  ( A 1 , ( A ( i ) , ( A 2 , ( A ( i ) Obj ( algP d , m × algP d , m ) ∋ ) i ∈ [ m ] , ϕ 1 ) i ∈ [ m ] , ϕ 2  1 2     � �  A 1 ⊔ A 2 , ( A ( i ) ⊔ A ( i )  ) i ∈ [ m ] , ϕ 1 ⊙ ϕ 2 ∈ Obj ( algP d , m )  Ñ 1 2 ⊙ : �� �� � �   Morph algPd , m × algPd , m ( B 1 , ψ 1 ) , ( B 2 , ψ 2 ) ) , ( A 1 , ϕ 1 ) , ( A 2 , ϕ 2 ) ∋ ( j 1 , j 2 )      � �  Ñ j 1 ⊔ j 2 ∈ Morph algPd , m ( B 1 ⊔ B 2 , ψ 1 ⊙ ψ 2 ) , ( A 1 ⊔ A 2 , ϕ 1 ⊙ ϕ 2 ) • universal product in algP d , m is called unital if ∀ i ∈ [ d ] , ∀ j ∈ [2] : ( ϕ 1 ⊙ ϕ 2 ) ( i ) ◦ ι j = ϕ ( i ) j • universal product in algP d , m is called associative if � � � � ( ϕ 1 ⊙ ϕ 2 ) ⊙ ϕ 3 = ϕ 1 ⊙ ( ϕ 2 ⊙ ϕ 3 ) . product having all these 3 properties is abbreviated by u.a.u.-product 7 / 27

• tensor category is category C with bifunctor ⊠ : C × C C , that is associative up to natural isomorphism, and an object E that is both a left and right identity for ⊠ . coherence conditions ensure that all relevant diagrams commute Remark 2 i.) if ⊙ is an u.a.u.-product in algP d , m ⇒ universality condition, i.e. for all B i ∈ Obj ( alg m ) and ( A i , ϕ i ) ∈ Obj ( algP d , m ) and j i ∈ Morph alg m ( B i , A i ) , i ∈ [2] holds � � ( ℓ ) = ( ϕ 1 ⊙ ϕ 2 ) ( ℓ ) ◦ ( j 1 ⊔ j 2 ) ( ϕ ( i ) ◦ j 1 ) i ∈ [ d ] ⊙ ( ϕ ( i ) ∀ ℓ ∈ [ d ]: ◦ j 2 ) i ∈ [ d ] 1 2 ii.) if ⊙ is an u.a.u.-product in algP d , m , then ( algP d , m , ⊙ , ( { 0 } , 0 Ñ 0)) is tensor category Definition 4 if ⊙ in algP d , m and ( A i , ( A ( k ) ) k ∈ [ d ] ) ∈ Obj ( alg m ) , i ∈ [2] are i given, then define for ϕ i ∈ (( A i ) ∗ ) d �� �� � � A 1 , ( A ( k ) A 2 , ( A ( k ) ϕ 1 ⊙ ϕ 2 := ) k ∈ [ d ] , ϕ 1 ⊙ ) k ∈ [ d ] , ϕ 2 1 2 where we identify ϕ 1 ⊙ ϕ 2 ∈ (( A 1 ⊔ A 2 ) ∗ ) d 8 / 27

Definition 5 (Catgegory alg ◆ 0 and calg ◆ 0 ✶ ) m • objects of alg ◆ 0 are triples ( A , ( A ( i ) ) i ∈ [ m ] , (( A ( i ) ) � α � ) ( i,α ) ∈ [ m ] × ◆ 0 ) i.) m where ( A , ( A ( i ) ) i ∈ [ m ] ) ∈ Obj ( alg m ) and α ∈ ◆ 0 , such that A ( i ) = ( A ( i ) ) � α � � � ∀ i ∈ [ m ] ∃ a family of subspaces A ( i ) � � α � and A ( i ) is a graded algebra � � α ∈ ◆ 0 • morphisms j ∈ Morph alg ◆ 0 m ( B , A ) are elements of Morph alg m ( B , A ) and j ↾ B ( i ) , i ∈ [ m ] is also homogeneous • objects of calg ◆ 0 ii.) are commutative, unital, ◆ 0 -graded algebras ✶ • morphisms are homogeneous, unital algebra homomorphisms Lemma 1 ( alg ◆ 0 m , ⊔ , { 0 } ) and ( calg ◆ 0 ✶ , ⊗ , ❈ ) are tensor categories 9 / 27

Central Limit Theorem for General Universal Products 1 Essential definitions 2 Definition of Lachs Functor 3 Mini digression: comonoids and cotensor functors 4 Lachs Functor is cotensor functor 5 Convolution products and exponential series 6 Central Limit Theorem 10 / 27

Proposition 1 ([MS17, Prop. 5.1]) if ⊙ is u.a.u.-product in algP d , m ⇒ ∀ A i ∈ Obj ( alg m ) , i ∈ [2] ∃ ! linear mapping S( A 1 ) ⊗ d ⊗ S( A 2 ) ⊗ d σ ⊙ A 1 . A 2 : ( A 1 ⊔ A 2 ) d � ( A i ) d � ∗ , i ∈ [ d ] the diagram is commutative such that for all ϕ i ∈ σ ⊙ A 1 , A 2 S( A 1 ) ⊗ d ⊗ S( A 2 ) ⊗ d ( A 1 ⊔ A 2 ) d . ϕ 1 ⊙ ϕ 2 S ( ϕ 1 ) ⊗ S ( ϕ 2 ) ❈ Lemma 2 stated linear mappings σ ⊙ A 1 , A 2 are even homogeneous 11 / 27

Definition 6 (Lachs Functor, [Lac15] 2 )  m ) ∋ ( A , ( A ( i ) ) i ∈ [ m ] , (( A ( i ) ) � α � ) ( i,α ) ∈ ([ m ] × ◆ 0 ) ) Obj ( alg ◆ 0    � �  � (S( A d )) � α � � S( A d ) , ∈ Obj ( calg ◆ 0 ✶ ) S : Ñ α ∈ ◆ 0    m ( B , A ) ∋ j Ñ S( j d ) ∈ Morph calg ◆ 0 ✶ (S( B d ) , S( A d ))  Morph alg ◆ 0 Theorem 1 ([Lac15, Thm. 5.2.4]) if ⊙ u.a.u.-product, g 0 : S( { 0 } d ) ❈ with g 0 ( ✶ S( { 0 } d ) ) = 1, then ( S , S ( σ ⊙ ) , g 0 ) is cotensor functor • What does this mean??? • What are the consequences??? 2 S. Lachs. “A New Family of Universal Products and Aspects of a Non-Positive Quantum Probability Theory”. PhD thesis. Ernst-Moritz-Arndt-Universit¨ at Greifswald, 2015. 12 / 27

Central Limit Theorem for General Universal Products 1 Essential definitions 2 Definition of Lachs Functor 3 Mini digression: comonoids and cotensor functors 4 Lachs Functor is cotensor functor 5 Convolution products and exponential series 6 Central Limit Theorem 13 / 27