The exponential homomorphism in non-commutative probability Michael - PowerPoint PPT Presentation

The exponential homomorphism in non-commutative probability Michael Anshelevich (joint work with Octavio Arizmendi) Texas A&M University March 25, 2016 Michael Anshelevich Exponential homomorphism

Classical convolutions. Additive: ➺ ➺ ➺ f ♣ z q d ♣ µ 1 ✝ µ 2 q♣ z q ✏ f ♣ x � y q dµ 1 ♣ x q dµ 2 ♣ y q . R R R Multiplicative: ➺ ➺ ➺ f ♣ z q d ♣ ν 1 ❢ ν 2 q♣ z q ✏ f ♣ zw q dν 1 ♣ z q dν 2 ♣ w q . T T T The wrapping map W : P ♣ R q Ñ P ♣ T q is ➳ d ♣ W ♣ µ qq♣ e ✁ ix q ✏ dµ ♣ x � 2 πn q . n P Z Clearly W ♣ µ 1 ✝ µ 2 q ✏ W ♣ µ 1 q ❢ W ♣ µ 2 q . Michael Anshelevich Exponential homomorphism

Non-commutative independence. Non-commutative convolutions: based on different independence rules. ✏ x 2 ✘ ✏ y 2 ✘ Tensor/classical E r xyxy s ✏ E E . E r y s 2 � E r x s 2 E ✁ E r x s 2 E r y s 2 . ✏ x 2 ✘ ✏ y 2 ✘ Free E r xyxy s ✏ E Boolean E r xyxy s ✏ E r x s 2 E r y s 2 . E r y s 2 . ✏ x 2 ✘ Monotone E r xyxy s ✏ E Michael Anshelevich Exponential homomorphism

Convolutions. Additive convolutions: for measures µ 1 , µ 2 P P ♣ R q . Classical µ 1 ✝ µ 2 , free µ 1 ❵ µ 2 , Boolean µ 1 ❩ µ 2 , monotone µ 1 ⊲ µ 2 . Multiplicative convolutions: for measures ν 1 , ν 2 P P ♣ T q . Classical ν 1 ❢ ν 2 , free ν 1 ❜ ν 2 , Boolean ν 1 ✂ ❨ ν 2 , monotone ν 1 ÷ ν 2 . Michael Anshelevich Exponential homomorphism

Transforms. The F -transform: ➺ 1 1 µ P P ♣ R q , G µ ♣ z q ✏ z ✁ x dµ ♣ x q , F µ ♣ z q ✏ G µ ♣ z q . R F µ : C � Ñ C � , lim y Ò✽ ℑ F µ ♣ iy q ✏ 1 . iy The η -transform: ψ ν ♣ z q ➺ zζ ν P P ♣ T q , ψ ν ♣ z q ✏ 1 ✁ zζ dν ♣ ζ q , η ν ♣ z q ✏ 1 � ψ ν ♣ z q . T η ν : D Ñ D , η ν ♣ 0 q ✏ 0 . Michael Anshelevich Exponential homomorphism

Convolutions in non-commutative probability. Additive convolutions: F ✁ 1 µ ❵ ν ♣ z q ✁ z ✏ ♣ F ✁ 1 µ ♣ z q ✁ z q � ♣ F ✁ 1 ν ♣ z q ✁ z q , Free µ ❵ ν : Boolean µ ❩ ν : F µ ❩ ν ♣ z q ✁ z ✏ ♣ F µ ♣ z q ✁ z q � ♣ F ν ♣ z q ✁ z q , Monotone µ ⊲ ν : F µ ⊲ ν ♣ z q ✏ F µ ♣ F ν ♣ z qq . Multiplicative convolutions: η ✁ 1 ✏ η ✁ 1 η ✁ 1 µ ❜ ν ♣ z q µ ♣ z q ν ♣ z q Free µ ❜ ν : , z z z η µ x ❨ ν ♣ z q ✏ η µ ♣ z q η ν ♣ z q Boolean µ ✂ ❨ ν : , z z z Monotone µ ÷ ν : η µ ÷ ν ♣ z q ✏ η µ ✆ η ν ♣ z q . Michael Anshelevich Exponential homomorphism

Homomorphisms. W is certainly not a homomorphism between free additive and multiplicative convolutions. Example. Let µ ✏ 1 2 ♣ δ ✁ 2 π � δ 2 π q be a Bernoulli distribution. Then W ♣ µ q ✏ δ 1 . Also, it is well-known that µ ❵ µ is an arcsine dis- tribution, while δ 1 ❜ δ 1 ✏ δ 1 . Thus W ♣ µ ❵ µ q ✘ W ♣ µ q ❜ W ♣ µ q . Successful homomorphisms between ❵ and ❜ on the level of power series: (Mastnak, Nica 2010), (Friedrich, McKay 2012, 2013). A homomorphism between ❵ and ❜ infinitely divisible distributions: (Cebron 2014). Michael Anshelevich Exponential homomorphism

Homomorphisms II. Define an implicit relation between µ P P ♣ R q and ν P P ♣ T q by exp ♣ iF µ ♣ z qq ✏ η ν ♣ e iz q . Then “obviously” µ 1 ❵ µ 2 Ø ν 1 ❜ ν 2 , µ 1 ❩ µ 2 Ø ν 1 ✂ ❨ ν 2 , µ 1 ⊲ µ 2 Ø ν 1 ÷ ν 2 . In fact also µ ❩ t Ø ν µ ❵ t Ø ν ❜ t , ❨ t , x µ 1 ✐ µ 2 Ø ν 1 ✪ ν 2 . (here F µ 1 ❵ µ 2 ✏ F µ 2 ✆ F µ 1 ✐ µ 2 , η ν 1 ❜ ν 2 ✏ η ν 2 ✆ η ν 1 ✪ ν 2 ). µ ⊲ t Ø ν ÷ t ? Michael Anshelevich Exponential homomorphism

Identities. Easily obtain multiplicative identities from additive ones, for example of µ ✏ µ ❵ t ⊲ µ ❩♣ 1 ✁ t q and B t ♣ τ ✐ ν q ✏ τ ✐ ♣ ν ❵ τ ❵ t q (particular case obtained in Zhong 2014). M t ✏ multiplicative version of the Belinschi-Nica transformation B t . Use these to define multiplicative and additive free divisibility indicators. Proposition. For µ P L , the additive divisibility indicator of µ is equal to the multiplicative divisibility indicator of W ♣ µ q . Michael Anshelevich Exponential homomorphism

Class L . exp ♣ iF µ ♣ z qq ✏ η ν ♣ e iz q . Domain: t µ P P ♣ R q : F µ ♣ z � 2 π q ✏ F µ ♣ z q � 2 π ✉ ✏ L . Range: ν P P ♣ T q : η ✶ ❨ ✥ ✭ x ν ♣ 0 q ✘ 0 , and η ν ♣ z q ✏ 0 ô z ✏ 0 ✏ ID ✝ . ❨ t exists for t ➙ 0 , ν ✘ Lebesgue ❨ ✥ ✭ x x ID ✝ ✏ ν P P ♣ T q : ν . Michael Anshelevich Exponential homomorphism

The wrapping homomorphism. Theorem. (A, Arizmendi 2015) When restricted to L , the wrapping map W satisfies exp ♣ iF µ ♣ z qq ✏ η W ♣ µ q ♣ e iz q . Therefore this restriction is a homomorphism for all four addi- tive convolutions, and has the additional properties mentioned ❨ above. The pre-image of each ν P ID x ✝ is an equivalence class modulo the relation mod δ 2 π , where any of the four convolu- tions with δ 2 π is used. Proof of the Theorem. Poisson summation. Michael Anshelevich Exponential homomorphism

Domain. Proposition. L is closed under the three additive convolution operations ❩ , ❵ , ⊲ , under the subordination operation ✐ , under Boolean and free (whenever defined) additive convolution powers, and under the Belinschi-Nica transformation B t . If µ P L ❳ ID ⊲ , then µ ⊲ t P L for all t → 0 . All the elements in L which are not point masses are in the classical, Boolean, free, and monotone (strict) domains of attraction of the Cauchy law. The Bercovici-Pata bijections between P ✏ ID ❩ , ID ⊲ , and ID ❵ restrict to bijections between L , L ❳ ID ⊲ and L ❳ ID ❵ . Michael Anshelevich Exponential homomorphism

Range. Proposition. ❨ x ✝ is closed under the three multiplicative convolution ID ✪ , operations ✂ ❨ , ❜ , ÷ , under the subordination operation and under Boolean and free (whenever defined) multiplicative convolution powers. ❨ ✝ contains ID ❜ ✝ and ID ÷ x ✝ . ID ✝ , then every element of W ✁ 1 ♣ ν q is in ID ❵ ❳ L . If ν P ID ❜ If ν P ID ÷ ✝ , then there is µ P ID ⊲ ✝ ❳ L such that W ♣ µ q ✏ ν . Michael Anshelevich Exponential homomorphism

Example of µ P L I. Cauchy distribution µ ✏ 1 a ♣ x ✁ b q 2 � a 2 dx. π Wrapped Cauchy distribution 1 ✁ e ✁ 2 a W ♣ µ q ✏ 1 1 � e ✁ 2 a ✁ 2 e ✁ a cos ♣ θ ✁ b q dθ. 2 π Michael Anshelevich Exponential homomorphism

Example of µ P L II. Pre-image of the multiplicative Boolean Gaussian. ➳ µ ✏ α k δ x k , where x k ✏ cot x k ✁ π 2 � πk, π ✁ ✠ 2 , x i P 2 � πk and 1 α k ✏ . 3 2 � 1 2 x 2 k A similar formula for the pre-image of multiplicative Boolean compound Poisson. Michael Anshelevich Exponential homomorphism

Unimodality. Proposition. The only unimodal measures in L are delta measures and Cauchy distributions. This provides many examples of measures µ with connected support such that µ ❵ t is never unimodal, answering a question of Hasebe and Sakuma. Michael Anshelevich Exponential homomorphism

Relation to C´ ebron’s map I. In (C´ ebron 2014), he defined a homomorphism e ❵ : ID ❵ Ñ ID ❜ which satisfies W ✏ BP ❜ Ñ❢ ✆ e ❵ ✆ BP ✝Ñ ❵ . He also proved that ✠ ❜ n ✁ W ♣ µ ❵ 1 n q e ❵ ♣ µ q ✏ lim . n Ñ✽ Thus on ID ❵ ❳ L , e ❵ ✏ W . He also observed that W (roughly speaking) wraps the L´ evy measure of µ . Therefore on L , it does the same with its free, Boolean, monotone L´ evy measures. Michael Anshelevich Exponential homomorphism

Relation to C´ ebron’s map II. Example. Let ν be the multiplicative free Gaussian measure. Of course e ❵ ♣ σ q ✏ ν for the semicircular distribution σ , with canonical pair ♣ 0 , δ 0 q . But σ ❘ L , and W ♣ σ q ✘ ν . Instead, W ♣ µ q ✏ ν for µ P L with canonical pair ✄ ➳ ☛ 1 1 ➳ 2 πk ♣ 1 � ♣ 2 πk q 2 q , 1 � ♣ 2 πk q 2 δ 2 πk . k ✘ 0 k P Z Corollary. W intertwines the restrictions of the Bercovici-Pata maps to L with their multiplicative counterparts. Michael Anshelevich Exponential homomorphism

Obvious properties of W . W sends atoms to atoms. If supp ♣ µ ac q ✏ R , then supp ♣♣ W ♣ µ qq ac q ✏ T . W sends infinitesimal triangular arrays t µ ni , 1 ↕ i ↕ k n ✉ n P N of measures in P ♣ R q to infinitesimal triangular arrays of measures in P ♣ T q . Michael Anshelevich Exponential homomorphism

Converses. Theorem. For µ P L , W maps the atoms of µ bijectively onto the atoms of W ♣ µ q , and preserves the weights. If supp ♣♣ W ♣ µ qq ac q ✏ T , then supp ♣ µ ac q ✏ R . If t ν ni , 1 ↕ i ↕ k n ✉ n P N is an infinitesimal triangular arrays of ❨ measures in ID x ✝ , then ν ni ✏ W ♣ µ ni q for some infinitesimal triangular array of measures in L . For µ P L , F µ is injective if and only if η W ♣ µ q is. Michael Anshelevich Exponential homomorphism