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

Let M denote the space of Borel probability measures on R. For every t ≥ 0 we consider the transformation Bt : M → M defined by Bt(μ) = ( μ )⊎(1/(1+t)) , μ ∈ M, where ⊞ and ⊎ are the operations of free additive convolution and respectively of Boolean convolution on M, and where the convolution powers with respect to ⊞ and ⊎ are defined in the natural way. We show that Bs ◦ Bt = Bs+t, ∀ s, t ≥ 0 and that, quite surprisingly, every Bt is a homomorphism for the operation of free multiplicative convolution ⊠ (that is, Bt(μ ⊠ ν) = Bt(μ) ⊠ Bt(ν) for all μ, ν ∈ M such that at least one of μ, ν is supported on [0,∞)). We prove that for t = 1 the transformation B1 coincides with the canonical bijection B : M → Minf−div discovered by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. Here Minf−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 Bt(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 Bt and the free Brownian motion; indeed, Theorem 4 of the paper gives an interpretation of the transformations Bt as a way of re-casting the free Brownian motion, where the resulting process becomes multiplicative with respect to ⊠, and always reaches⊞-infinite divisibility by the time t = 1. ∗Research supported by a Discovery Grant of NSERC, Canada and by a PREA award from the province of Ontario.