  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.
