  Let M denote the space of Borel probability measures on the real line. For
every nonnegative t we consider the transformation $\mathbb B_t : M \to M$
defined for any given element in M by taking succesively the the (1+t) power
with respect to free additive convolution and then the 1/(1+t) power with
respect to Boolean convolution of the given element. We show that the family of
maps {\mathbb B_t|t\geq 0} is a semigroup with respect to the operation of
composition and that, quite surprisingly, every $\mathbb B_t$ is a homomorphism
for the operation of free multiplicative convolution.
  We prove that for t=1 the transformation $\mathbb B_1$ coincides with the
canonical bijection $\mathbb B : M \to M_{inf-div}$ 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 free additive
convolution. As a consequence, we have that $\mathbb B_t(\mu)$ is infinitely
divisible with respect to free additive convolution for any for every $\mu$ in
M and every t greater than or equal to one.
  On the other hand we put into evidence a relation between the transformations
$\mathbb B_t$ and the free Brownian motion; indeed, Theorem 4 of the paper
gives an interpretation of the transformations $\mathbb B_t$ as a way of
re-casting the free Brownian motion, where the resulting process becomes
multiplicative with respect to free multiplicative convolution, and always
reaches infinite divisibility with respect to free additive convolution by the
time t=1.
