Translation invariant state and its mean entropy-II

Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ be the two sided infinite tensor product $C^*$-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 $C^*$- or von-Neumann algebras based on repeated admissible measurement processes. In particular, we prove that dynamical entropy $h_{\omega}(\theta)$ for translation dynamics $(\IM,\theta,\omega)$ satisfies $s(\omega) \le h_{\omega}(\theta) \le 2s(\omega)$. In case $\omega$ is an infinite tensor product state of $\IM$ then $h_{\omega}(\theta)=s(\omega)$.