On a remarkable semigroup of homomorphisms with respect to free multiplicative convolution

Abstract

r. Let M denote the space of Borel probability measures on R. For every t > 0 we consider the transformation B t : M → W defined by B t (μ) =(μ□ (1+t) ) U(1/1+t) , μ∈Μ, where □ and U are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to □ and w are defined in the natural way. We show that B s o B t = B s+t, ∀ s, t ≥ 0 and that, quite surprisingly, every B t is a homomorphism for the operation of free multiplicative convolution □ (that is, B t (μ 14 ν) = B t (μ) □ B t (ν) for all p, ν ∈ M such that at least one of μ, ν is supported on [0, oo)). We prove that for t = 1 the transformation B 1 coincides with the canonical bijection B: M → M inf-div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here M inf-div stands for the set of probability distributions in M which are infinitely divisible with respect to the operation □. As a consequence, we have that B t (μ) is □-infinitely divisible for every μ ∈ M and every t ≥ 1. On the other hand we put into evidence a relation between the transformations B t and the free Brownian motion; indeed, Theorem 1.6 of the paper gives an interpretation of the transformations B t as a way of recasting the free Brownian motion, where the resulting process becomes multiplicative with respect to □, and always reaches □-infinite divisibility by the time t = 1.