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 $S$ on given $C^{\ast }$-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 ${\xi }^{\sigma }=({\xi }_s^{\sigma }{)}_{s\ge 0}\mathrm{.}$ 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 ${\mathfrak{T}}_s=\left(\begin{array}{cc}{\mathfrak{K}}_{s,1}& {\mathfrak{L}}_s\\ {\mathfrak{L}}_s^{\ast }& {\mathfrak{K}}_{s,2}\end{array}\right)$ which acts block-wise. Further, suppose that $(F_s^i{)}_{s\ge 0}$ is the inclusion system affiliated to the diagonal QDS $({\mathfrak{K}}_{s,i}{)}_{s\ge 0}$ along with the generating unit $({\xi }_{s,i}^{\sigma }{)}_{s\ge 0},$ $\sigma \in S,i\in \{1,2\}$, then we prove that there exists a unique contractive (weak) morphism $V=(V_s{)}_{s\ge 0}:F_s^2\to F_s^1$ such that ${\mathfrak{L}}_s^{\sigma ,{\sigma }^{\mathrm{\prime }}}(b)=⟨{\xi }_{s,1}^{\sigma },V_{s}b{\xi }_{s,2}^{{\sigma }^{\mathrm{\prime }}}⟩$ for every ${\sigma }^{\mathrm{\prime }},\sigma \in S$ and $b\in \mathcal{B}\mathrm{.}$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families.