Dilations of non-Markovian dynamical systems on graphs

Abstract

To generalise evolution families we consider systems of contractions $\{\varphi(u, v)\}_{(u, v) \in E}$ defined on the edges of a graph $\mathcal{G} = (\Omega, E)$. In this setup the Markov property, or \emph{divisibility}, can be modelled via $\varphi(u, v)\varphi(v, w) = \varphi(u, w)$ for edges $(u, v), (v, w), (u, w) \in E$. We obtain results in three settings: 1) contractive Banach space operators; 2) positive unital maps on $C^{\ast}$-algebras; and 3) CPTP-maps on trace class operators on a Hilbert space. In the discrete setting, we are able to dilate possibly indivisible families of contractions to divisible families of operators with 'nice' properties (viz. surjective isometries resp. $C^{\ast}$-algebraic automorphisms resp. unitary representations). In the special case of linearly ordered graphs equipped with the order topology, we establish sufficient conditions for strongly continuous dilations of possibly indivisible families in the Banach space and $C^{\ast}$-algebra contexts. To achieve these results we work with string-rewriting systems, and make use of and extend dilation theorems of Stroescu [44], Kraus [23, 24], and vom Ende--Dirr [50].