Relations between convolutions and transforms in operator-valued free probability

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 additive convolutions, we will focus on two other important additive 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.