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 $C^*$-algebra of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$. Let $ω$ be a translation invariant state of $\IM$. In a recent paper, we have proved that the mean entropy $s(ω)$ is a complete invariant for certain classes of translation invariant state $ω$ 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 $h_ω(θ)$ for translation dynamics $(\IM,θ,ω)$ satisfies $s(ω) \le h_ω(θ) \le 2s(ω)$. In case $ω$ is an infinite tensor product state of $\IM$ then $h_ω(θ)=s(ω)$.