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: ➺

R

f♣zq d♣µ1 ✝ µ2q♣zq ✏ ➺

R

➺

R

f♣x yq dµ1♣xq dµ2♣yq. Multiplicative: ➺

T

f♣zq d♣ν1 ❢ ν2q♣zq ✏ ➺

T

➺

T

f♣zwq dν1♣zq dν2♣wq. The wrapping map W : P♣Rq Ñ P♣Tq is d♣W♣µqq♣e✁ixq ✏ ➳

nPZ

dµ♣x 2πnq. Clearly W♣µ1 ✝ µ2q ✏ W♣µ1q ❢ W♣µ2q.

Michael Anshelevich Exponential homomorphism

Non-commutative independence.

Non-commutative convolutions: based on different independence rules. Tensor/classical E rxyxys ✏ E ✏ x2✘ E ✏ y2✘ . Free E rxyxys ✏ E ✏ x2✘ E rys2 E rxs2 E ✏ y2✘ ✁ E rxs2 E rys2. Boolean E rxyxys ✏ E rxs2 E rys2. Monotone E rxyxys ✏ E ✏ x2✘ E rys2.

Michael Anshelevich Exponential homomorphism

Convolutions.

Additive convolutions: for measures µ1, µ2 P P♣Rq. Classical µ1 ✝ µ2, free µ1 ❵ µ2, Boolean µ1 ❩ µ2, monotone µ1 ⊲ µ2. Multiplicative convolutions: for measures ν1, ν2 P P♣Tq. Classical ν1 ❢ ν2, free ν1 ❜ ν2, Boolean ν1 ✂ ❨ ν2, monotone ν1 ÷ ν2.

Michael Anshelevich Exponential homomorphism

Transforms.

The F-transform: µ P P♣Rq, Gµ♣zq ✏ ➺

R

1 z ✁ x dµ♣xq, Fµ♣zq ✏ 1 Gµ♣zq. Fµ : C Ñ C, limyÒ✽

ℑFµ♣iyq iy

✏ 1. The η-transform: ν P P♣Tq, ψν♣zq ✏ ➺

T

zζ 1 ✁ zζ dν♣ζq, ην♣zq ✏ ψν♣zq 1 ψν♣zq. ην : D Ñ D, ην♣0q ✏ 0.

Michael Anshelevich Exponential homomorphism

Convolutions in non-commutative probability.

Additive convolutions: Free µ ❵ ν : F ✁1

µ❵ν♣zq ✁ z ✏ ♣F ✁1 µ ♣zq ✁ zq ♣F ✁1 ν ♣zq ✁ zq,

Boolean µ ❩ ν : Fµ❩ν♣zq ✁ z ✏ ♣Fµ♣zq ✁ zq ♣Fν♣zq ✁ zq, Monotone µ ⊲ ν : Fµ⊲ν♣zq ✏ Fµ♣Fν♣zqq. Multiplicative convolutions: Free µ ❜ ν : η✁1

µ❜ν♣zq

z ✏ η✁1

µ ♣zq

z η✁1

ν ♣zq

z , Boolean µ ✂ ❨ ν : ηµx

❨ν♣zq

z ✏ ηµ♣zq z ην♣zq z , Monotone µ ÷ ν : ηµ÷ν♣zq ✏ ηµ ✆ ην♣zq.

Michael Anshelevich Exponential homomorphism

Homomorphisms.

W is certainly not a homomorphism between free additive and multiplicative convolutions. Example. Let µ ✏ 1

2♣δ✁2π δ2πq be a Bernoulli distribution.

Then W♣µq ✏ δ1. Also, it is well-known that µ ❵ µ is an arcsine dis- tribution, while δ1 ❜ δ1 ✏ δ1. Thus W♣µ ❵ µq ✘ W♣µq ❜ W♣µq. 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♣Rq and ν P P♣Tq by exp♣iFµ♣zqq ✏ ην♣eizq . Then “obviously” µ1 ❵ µ2 Ø ν1 ❜ ν2, µ1 ❩ µ2 Ø ν1 ✂ ❨ ν2, µ1 ⊲ µ2 Ø ν1 ÷ ν2. In fact also µ❩t Ø ν

x

❨t,

µ❵t Ø ν❜t, µ1 ✐ µ2 Ø ν1 ✪ ν2. (here Fµ1❵µ2 ✏ Fµ2 ✆ Fµ1✐µ2, ην1❜ν2 ✏ ην2 ✆ ην1 ✪ ν2). 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✁tq and Bt♣τ ✐ νq ✏ τ ✐ ♣ν ❵ τ ❵tq (particular case obtained in Zhong 2014). Mt ✏ multiplicative version of the Belinschi-Nica transformation

• Bt. 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♣µq.

Michael Anshelevich Exponential homomorphism

Class L.

exp♣iFµ♣zqq ✏ ην♣eizq. Domain: tµ P P♣Rq : Fµ♣z 2πq ✏ Fµ♣zq 2π✉ ✏ L. Range: ✥ ν P P♣Tq : η✶

ν♣0q ✘ 0, and ην♣zq ✏ 0 ô z ✏ 0

✭ ✏ ID

x

❨ ✝.

ID

x

❨ ✝ ✏

✥ ν P P♣Tq : ν

x

❨t exists for t ➙ 0, ν ✘ Lebesgue

✭ .

Michael Anshelevich Exponential homomorphism

The wrapping homomorphism.

• Theorem. (A, Arizmendi 2015)

When restricted to L, the wrapping map W satisfies exp♣iFµ♣zqq ✏ ηW♣µq♣eizq. 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 δ2π, where any of the four convolu- tions with δ2π 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 Bt. 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. ID

x

❨ ✝ is closed under the three multiplicative convolution

• perations ✂

❨ , ❜, ÷, under the subordination operation ✪ , and under Boolean and free (whenever defined) multiplicative convolution powers. ID

x

❨ ✝ contains ID❜ ✝ and ID÷ ✝ .

If ν P ID❜

✝ , then every element of W ✁1♣νq is in ID❵ ❳ L.

If ν P ID÷

✝ , then there is µ P ID⊲ ✝ ❳ L such that W♣µq ✏ ν.

Michael Anshelevich Exponential homomorphism

Example of µ P L I.

Cauchy distribution µ ✏ 1 π a ♣x ✁ bq2 a2 dx. Wrapped Cauchy distribution W♣µq ✏ 1 2π 1 ✁ e✁2a 1 e✁2a ✁ 2e✁a cos♣θ ✁ bq dθ.

Michael Anshelevich Exponential homomorphism

Example of µ P L II.

Pre-image of the multiplicative Boolean Gaussian. µ ✏ ➳ αkδxk, where xk ✏ cot xk 2 , xi P ✁ ✁π 2 πk, π 2 πk ✠ and αk ✏ 1

3 2 1 2x2 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

• f 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 e❵♣µq ✏ lim

nÑ✽

✁ W♣µ❵ 1

n q

✠❜n . 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❵♣σq ✏ ν for the semicircular distribution σ, with canonical pair ♣0, δ0q. But σ ❘ L, and W♣σq ✘ ν. Instead, W♣µq ✏ ν for µ P L with canonical pair ✄ ➳

k✘0

1 2πk♣1 ♣2πkq2q, ➳

kPZ

1 1 ♣2πkq2 δ2πk ☛ . 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♣µacq ✏ R, then supp♣♣W♣µqqacq ✏ T. W sends infinitesimal triangular arrays tµni, 1 ↕ i ↕ kn✉nPN

• f measures in P♣Rq to infinitesimal triangular arrays of

measures in P♣Tq.

Michael Anshelevich Exponential homomorphism

Converses.

Theorem. For µ P L, W maps the atoms of µ bijectively onto the atoms of W♣µq, and preserves the weights. If supp♣♣W♣µqqacq ✏ T, then supp♣µacq ✏ R. If tνni, 1 ↕ i ↕ kn✉nPN is an infinitesimal triangular arrays of measures in ID

x

❨ ✝, then νni ✏ W♣µniq for some infinitesimal

triangular array of measures in L. For µ P L, Fµ is injective if and only if ηW♣µq is.

Michael Anshelevich Exponential homomorphism

Corollaries I.

Can re-prove many results, but only for measures in ID

x

❨ ✝.

Atoms, singular continuous part, components of the absolutely continuous part of µ ❜ ν. Some of these are new. Atoms, singular continuous part, components of the absolutely continuous part of ν❜. These only make sense for ν P ID

x

❨ ✝.

Corollary. Let ν P ID

x

❨ ✝ and t ➙ 1. Denote νt ✏ ν❜t. ζ is an atom of νt if and

• nly if for some α P R, e✁itα ✏ ζ and e✁iα is an atom of ν, with

ν♣ ✥ e✁iα✭ q → 1 ✁ 1④t, in which case νt♣tζ✉q ✏ tν♣ ✥ e✁iα✭ ✁ ♣t ✁ 1q.

Michael Anshelevich Exponential homomorphism

Corollaries II.

Limit theorems for ❢, ❜, ✂ ❨ , ÷ from those for ✝, ❵, ❩, ⊲. First examples of limit theorems for non-identically distributed monotone arrays beyond the finite variance case.

Michael Anshelevich Exponential homomorphism

Thank you!

Michael Anshelevich Exponential homomorphism