Block quantum dynamical semigroups of completely positive definite kernels (2401.16846v2)

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 $C^*$-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.