On operator-valued free convolution powers

Abstract

We give an explicit realization of the $\eta$-convolution power of an $A$-valued distribution, as defined earlier by Anshelevich, Belinschi, Fevrier and Nica. If $\eta :A\to A$ is completely positive and $\eta \ge \mathrm{id}$, we give a short proof of positivity of the $\eta$-convolution power of a positive distribution. Conversely, if $\eta \ngeqq \mathrm{id}$, and $s$ is large enough, we construct an $s$-tuple whose $A$-valued distribution is positive, but has non-positive $\eta$-convolution power.