Block quantum dynamical semigroups of completely positive definite kernels

Kolmogorov decomposition for a given completely positive definite kernel is a generalization of Paschke’s GNS construction for the completely positive map. Using Kolmogorov decomposition, to every quantum dynamical semigroup (QDS) for completely positive definite kernels over a set [Formula: see text] on given [Formula: see text]-algebra [Formula: see text], we shall assign an inclusion system [Formula: see text] of Hilbert bimodules over [Formula: see text] with a generating unit [Formula: see text] Consider a von Neumann algebra [Formula: see text], and let [Formula: see text] be a QDS over a set [Formula: see text] on the algebra [Formula: see text] with [Formula: see text] which acts block-wise. Further, if [Formula: see text] is the inclusion system affiliated to the diagonal QDS [Formula: see text] along with the generating unit [Formula: see text] [Formula: see text], then we prove that there exists a unique contractive (weak) morphism [Formula: see text] such that [Formula: see text] for every [Formula: see text] and [Formula: see text] We also study the semigroup version of a factorization theorem for [Formula: see text]-families.