Pure inductive limit state and Kolmogorov's property

Let $(\text{clb},{\lambda }_t,\psi )$ be a $C^{\ast }$-dynamical system where $(\lambda_t: t \in \IT_+)$beasemigroupofinjectiveendomorphismand$\psi$bean$(\lambda_t)$invariantstateonthe$C^*$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 $\clb \raro^{\lambda_t} \clb$canonicallyassociatedwith$\psi$ 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.