Block quantum dynamical semigroups of completely positive definite kernels
Santanu Dey, Dimple Saini, Harsh Trivedi
TL;DR
This work analyzes block quantum dynamical semigroups built from completely positive definite kernels (CPD-kernels) on a unital $C^*$-algebra by leveraging Kolmogorov representations. It develops a block-structure theory for CPD-kernels, introducing a contracting morphism $V$ that encodes off-diagonal blocks and yields a block CPD-kernel via $\mathfrak{L}^{\sigma,\sigma'}(b)=\langle\xi_{1}^{\sigma}, V b \xi_{2}^{\sigma'}\rangle$, while decomposing associated Kolmogorov representations into diagonal components. The paper then extends these ideas to the continuous-time setting, proving a semigroup version of a factorization theorem for $\mathfrak{K}$-families and establishing $E_0$-dilations and lifting results for block quantum Markov semigroups. These results provide a structural framework for block CPD-kernel semigroups, enabling dilation, factorization, and morphism-lifting techniques relevant to quantum stochastic processes in operator-algebraic contexts.
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 $ξ^σ=(ξ^σ_s)_{s\ge 0}.$ 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 $(ξ^σ_{s,i} )_{s\ge 0},$ $σ\in S,i\in \{1,2\}$, 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 $\mathfrak{L}_s^{σ,σ'}(b)=\langle ξ_{s,1}^σ,V_s bξ_{s,2}^{σ'}\rangle$ for every $σ',σ\in S$ and $b\in \mathcal{B}.$ We also study the semigroup version of a factorization theorem for $\mathfrak{K}$-families.