  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.
