Translation invariant state and its mean entropy-I
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 $\IC$ and $\omega$ 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 $\IM$. Futhermore, a translation invariant factor state of $\IM$ is pure if and only if its mean entropy is zero. The last statement can be regarded as a non commutative extension of Rokhlin-Sinai positive entropy theorem for non-pure factor states.