Convolution powers in the operator-valued framework

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 μone 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 ηfrom B to B, instead of being a non-negative real number. More precisely, the Boolean convolution power is defined whenever ηis completely positive, while the free additive convolution power is defined whenever η- 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 η. 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.

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