Convolution powers in the operator-valued framework
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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 one can define convolution powers {squared plus} (with respect to free additive convolution) and {multiset union} (with respect to Boolean convolution), where the exponent is a suitably chosen linear map from B to B, instead of being a nonnegative real number. More precisely, {multiset union} is always defined when is completely positive, while {squared plus} is always defined when is completely positive (with "1" denoting the identity map on B). In connection to these convolution powers we define an evolution semigroup {B | : B B, completely positive}, 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 convolution powers {squared plus}. One of the results concerns the analytic subordination of the Cauchy-Stieltjes transform of {squared plus} with respect to the Cauchy-Stieltjes transform of . The other one 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. 2012 American Mathematical Society.