Dilations of non-Markovian dynamical systems on graphs

TL;DR

This paper develops dilation techniques for non-Markovian dynamical systems on graphs across various mathematical frameworks, enabling the transformation of indivisible families into divisible ones with desirable properties.

Contribution

It introduces a unified approach to dilate non-Markovian systems on graphs in Banach spaces, $C^{ ext{*}}$-algebras, and Hilbert spaces, extending existing dilation theorems.

Findings

Dilations to divisible families with isometries or automorphisms

Conditions for strong continuity of dilations on ordered graphs

Extension of existing dilation theorems to non-Markovian graph systems

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].