Block quantum dynamical semigroups of completely positive definite kernels

Abstract

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 $\hat{I}{¾}^{\ddot{I}}ƒ=(\hat{I}{¾}^{\ddot{I}}{ƒ}_s{)}_{s\ge 0}.$ 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 ${\mathfrak{T}}_s=\left(\begin{array}{cc}{\mathfrak{K}}_{s,1}& {\mathfrak{L}}_s\\ {\mathfrak{L}}_s^{\ast }& {\mathfrak{K}}_{s,2}\end{array}\right)$ 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 $(\hat{I}{¾}^{\ddot{I}}{ƒ}_{s,i}{)}_{s\ge 0},$ $\ddot{I}ƒ\in S,i\in \{1,2\}$, 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 ${\mathfrak{L}}_s^{\ddot{I}ƒ,\ddot{I}{ƒ}^{\prime }}(b)=⟨\hat{I}{¾}_{s,1}^{\ddot{I}}ƒ,V_{s}b\hat{I}{¾}_{s,2}^{\ddot{I}{ƒ}^{\prime }}⟩$ for every $\ddot{I}{ƒ}^{\prime },\ddot{I}ƒ\in S$ and $b\in \mathcal{B}.$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families.