  We consider the framework of an operator-valued noncommutative probability
space over a unital C*-algebra B. We show how for a B-valued distribution \mu
one can define convolution powers with respect to free additive convolution and
with respect to Boolean convolution, where the exponent considered in the power
is a suitably chosen linear map \eta from B to B, instead of being a
non-negative real number. More precisely, the Boolean convolution power is
defined whenever \eta is completely positive, while the free additive
convolution power is defined whenever \eta - 1 is completely positive (where 1
stands for the identity map on B).
  In connection to these convolution powers we define an evolution semigroup
related to the Boolean Bercovici-Pata bijection. We prove several properties of
this semigroup, including its connection to the B-valued free Brownian motion.
  We also obtain two results on the operator-valued analytic function theory
related to the free additive convolution powers with exponent \eta. One of the
results concerns analytic subordination for B-valued Cauchy-Stieltjes
transforms. The other gives a B-valued version of the inviscid Burgers
equation, which is satisfied by the Cauchy-Stieltjes transform of a B-valued
free Brownian motion.
