2025/01/16 by Charlesworth, Ian, Jekel, David
#46L54 #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2501.09690
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-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-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 η= n id.