Relations between convolutions and transforms in operator-valued free probability

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