Relations between convolutions and transforms in operator-valued free probability

TL;DR

This paper explores the relationships between various convolutions and transforms in operator-valued free probability, introducing a class of independence relations and extending results to multi-variable cases using matricial extension properties.

Contribution

It introduces a new class of independence relations with a matricial extension property, enabling generalized analysis of convolutions in operator-valued free probability.

Findings

Operator-valued subordination functions are reciprocal Cauchy transforms.

Many convolution results extend to multi-variable cases.

Relations between convolutions and transforms in $C^*$-operator valued probability are established.

Abstract

We introduce a class of independence relations, which include free, Boolean and monotone independence, in operator valued probability. We show that this class of independence relations have a matricial extension property so that we can easily study their associated convolutions via Voiculescu's fully matricial function theory. Based the matricial extension property, we show that many results can be generalized to multi-variable cases. Besides free, Boolean and monotone independence convolutions, we will focus on two important convolutions, which are orthogonal and subordination additive convolutions. We show that the operator-valued subordination functions, which come from the free additive convolutions or the operator-valued free convolution powers, are reciprocal Cauchy transforms of operator-valued random variables which are uniquely determined up to Voiculescu's fully matricial function theory. In the end, we study relations between certain convolutions and transforms in $C^*$-operator valued probability.