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 $\hat{I}\cdot$. We first give a general result, with a new analytic proof, that the $\hat{I}\cdot$-convolution power of the law of $X$ is realized by $V^{\ast }XV$ 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 $\hat{I}\cdot$-valued 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 $\hat{I}\cdot =n\mathrm{id}$.

17 pages, revised