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

We revisit the theory of operator-valued free convolution powers given by a completely positive map η. We first give a general result, with a new analytic proof, that the η-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 η-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 η=nid.