Translation invariant state and its mean entropy-II

Abstract

Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ be the two sided infinite tensor product $C^{\ast }$-algebra of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$. Let $\omega$ be a translation invariant state of $\IM$. In a recent paper, we have proved that the mean entropy $s(\omega )$ is a complete invariant for certain classes of translation invariant state $\omega$ of $\IM$. In this paper, we have developed a general theory for dynamical entropy for an automorphism on an arbitrary $C^{\ast }$- or von-Neumann algebras based on repeated admissible measurement processes. In particular, we prove that dynamical entropy $h_{\omega }(\theta )$ for translation dynamics $(\IM,θ,ω)$ satisfies $s(\omega )\le h_{\omega }(\theta )\le 2s(\omega )$. In case $\omega$ is an infinite tensor product state of $\IM$ then $h_{\omega }(\theta )=s(\omega )$.

This is an enlarge version of an earlier preprint