Convolution powers in the operator-valued framework

TL;DR

This paper develops a framework for defining convolution powers in operator-valued noncommutative probability, extending classical concepts to B-valued distributions with new semigroup and analytic properties.

Contribution

It introduces convolution powers with respect to free and Boolean convolutions using linear maps, and explores their properties, including an evolution semigroup and connections to free Brownian motion.

Findings

Defined Boolean convolution power for completely positive ta.

Established free additive convolution power when ta - 1 is completely positive.

Connected the semigroup to B-valued free Brownian motion.

Abstract

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.