Operator models and analytic subordination for operator-valued free convolution powers

TL;DR

This paper develops a unified operator model for operator-valued free convolution powers, introduces an analytic subordination result, and explains the relation between multiple free convolutions and convolution powers via Hilbert-space techniques.

Contribution

It provides a new analytic proof for operator-valued free convolution powers and extends subordination results to this setting, unifying previous scalar and operator-valued frameworks.

Findings

Unified operator model for $ ext{η}$-convolution powers

Analytic subordination for operator-valued free convolution

Hilbert-space approach to multiple free convolutions

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 $\eta$-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 $\eta$-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 $\eta = n \operatorname{id}$.