Operator models and analytic subordination for operator-valued free convolution powers
Abstract
We revisit the theory of operator-valued free convolution powers given by a completely positive map $\eta $. We first give a general result, with a new analytic proof, that the $\eta $-free convolution power of the law of $X$ is realized by $V^*XV$ for any operator $V$ satisfying certain conditions, which unifies Nica and Speicher’s construction in the scalar-valued setting and Shlyakhtenko’s construction in the operator-valued setting. Second, we provide an analog, for the setting of $\eta $-free convolution powers, of the analytic subordination for conditional expectations that holds for additive free convolution. Finally, we describe a Hilbert-space manipulation that explains the equivalence between the $n$-fold additive free convolution and the convolution power with respect to $\eta = n \,{\rm id}$.
