  Let $(\clb,\lambda_t,\psi)$ be a $C^*$-dynamical system where $(\lambda_t: t
\in \IT_+)$ be a semigroup of injective endomorphism and $\psi$ be an
$(\lambda_t)$ invariant state on the $C^*$ subalgebra $\clb$ and $\IT_+$ is
either non-negative integers or real numbers. The central aim of this
exposition is to find a useful criteria for the inductive limit state $\clb
\raro^{\lambda_t} \clb$ canonically associated with $\psi$ to be pure. We
achieve this by exploring the minimal weak forward and backward Markov
processes associated with the Markov semigroup on the corner von-Neumann
algebra of the support projection of the state $\psi$ to prove that
Kolmogorov's property [Mo2] of the Markov semigroup is a sufficient condition
for the inductive state to be pure. As an application of this criteria we find
a sufficient condition for a translation invariant factor state on a one
dimensional quantum spin chain to be pure. This criteria in a sense complements
criteria obtained in [BJKW,Mo2] as we could go beyond lattice symmetric states.
