Translation invariant state and its mean entropy-I

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 $\omega$ be a translation invariant state of \IM. In this paper, we have proved that the mean entropy $s(\omega)$ and Connes-St\o rmer dynamical entropy h_{CS}(\IM,\theta,\omega) of $\omega$ are equal. Furthermore, the mean entropy $s(\omega)$ is equal to the Kolmogorov-Sinai dynamical entropy h_{KS}(\ID_{\omega},\theta,\omega) of $\omega$ when the state $\omega$ is restricted to a suitable translation invariant maximal abelian $C^*$ sub-algebra \ID_{\omega} of \IM. We have also proved that the mean entropy $s(\omega)$ is a complete invariant for certain classes of translation invariant states of \IM