Convolution powers in the operator-valued framework

We consider the framework of an operator-valued noncommutative probability space over a unital $C^*$-algebra $\mathcal {B}$. We show how for a $\mathcal {B}$-valued distribution $\mu$ one can define convolution powers $\mu ^{\boxplus \eta }$ (with respect to free additive convolution) and $\mu ^{\uplus \eta }$ (with respect to Boolean convolution), where the exponent $\eta$ is a suitably chosen linear map from $\mathcal {B}$ to $\mathcal {B}$, instead of being a nonnegative real number. More precisely, $\mu ^{\uplus \eta }$ is always defined when $\eta$ is completely positive, while $\mu ^{\boxplus \eta }$ is always defined when $\eta - 1$ is completely positive (with ``$1$'' denoting the identity map on $\mathcal {B}$).

In connection to these convolution powers we define an evolution semigroup $\{ \mathbb{B}_{\eta } \mid \eta : \mathcal {B} \to \mathcal {B}$, completely positive$\}$, related to the Boolean Bercovici-Pata bijection. We prove several properties of this semigroup, including its connection to the $\mathcal {B}$-valued free Brownian motion.

We also obtain two results on the operator-valued analytic function theory related to convolution powers $\mu ^{\boxplus \eta }$. One of the results concerns the analytic subordination of the Cauchy-Stieltjes transform of $\mu ^{\boxplus \eta }$ with respect to the Cauchy-Stieltjes transform of $\mu$. The other one gives a $\mathcal {B}$-valued version of the inviscid Burgers equation, which is satisfied by the Cauchy-Stieltjes transform of a $\mathcal {B}$-valued free Brownian motion.