  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 $S$ on given $C^*$-algebra
$\mathcal{A},$ we shall assign an inclusion system $F = (F_s)_{s\ge 0}$ of
Hilbert bimodules over $\mathcal{A}$ with a generating unit
$\xi^{\sigma}=(\xi^{\sigma}_s)_{s\ge 0}.$ Consider a von Neumann algebra
$\mathcal{B}$, and let $\mathfrak{T}=(\mathfrak{T}_s)_{s\ge 0}$ be a QDS over a
set $S$ on the algebra $M_2(\mathcal{B})$ with
$\mathfrak{T}_s=\begin{pmatrix}\mathfrak{K}_{s,1} &
\mathfrak{L}_s\\\mathfrak{L}_s^*& \mathfrak{K}_{s,2} \end{pmatrix}$ which acts
block-wise. Further, suppose that $(F^i_s )_{s\ge 0}$ is the inclusion system
affiliated to the diagonal QDS $(\mathfrak{K}_{s,i})_{s\ge 0}$ along with the
generating unit $(\xi^{\sigma}_{s,i} )_{s\ge 0},$ $\sigma\in S,i\in \{1,2\}$,
then we prove that there exists a unique contractive (weak) morphism $V =
(V_s)_{s\ge 0}:F^2_s \to F^1_s$ such that
$\mathfrak{L}_s^{\sigma,\sigma'}(b)=\langle \xi_{s,1}^{\sigma},V_s
b\xi_{s,2}^{\sigma'}\rangle$ for every $\sigma',\sigma\in S$ and $b\in
\mathcal{B}.$ We also study the semigroup version of a factorization theorem
for $\mathfrak{K}$-families.
