The Exponential Map in Non-commutative Probability

Abstract

ABSTRACT. The wrapping transformation W is a homomorphism from the semigroup of probability measures on the real line, with the convolution operation, to the semigroup of probability measures on the circle, with the multiplicative convolution operation. We show that on a large class L of measures, W also transforms the three non-commutative convolutions-free, Boolean, and monotone-to their multiplicative counterparts. Moreover, the restriction of W to L preserves various qualitative properties of measures and triangular arrays. We use these facts to give short proofs of numerous known, and new, results about multiplicative convolutions.

Large time unimodality for classical and free Brownian motions with initial distributions

“…Actually this kind of similarity between additive and multiplicative free convolutions were observed in several situations, for example in [2]. Anshelevich and Arizmendi [1] succeeded in systematically explaining those similarities: they proved that various results on additive free convolution can be transferred to multiplicative free convolution on T using the exponential mapping. However, unimodality seems out of the applicability of their approach.…”

Limit theorems for free Lévy processes

“…The proof of (3) uses the (clockwise) exponential map x → e −ix , in order to reduce unitary MFLPs to AFLPs. In spite of the non-commutativity of the process, such a reduction is possible, thanks to the work of Anshelevich and Arizmendi [AA17]. This method of using the exponential map has been limited to multiplicative convolutions on T so far, and not available to positive multiplicative convolutions, and hence not available to (1).…”

“…Furthermore, according to [AA17], the map W restricted to a subclass of probability measures provides a homomorphism from additive free/Boolean convolutions to multiplicative ones on the unit circle. Define…”

Homomorphisms relative to additive convolutions and max-convolutions: Free, boolean and classical cases

“…Example 2.3. The Marchenko-Pastur law π λ with rate λ > 0 is defined by 2 . The special case π 1 is simply denoted by π and is called the standard Marchenko-Pastur law.…”

“…In probability theory, the asymptotic behavior of products of a large number of independent positive random variables reduces to the usual law of large numbers for addition of real-valued random variables, because the exponential mapping is a homomorphism from (R, +) onto ((0, ∞), •). However, for non-commutative random variables such as random matrices, law of large numbers for products of independent random variables is no longer obvious because the exponential mapping is not a homomorphism (an attempt to recover the homomorphism property is found in [2] in the unitary case).…”