A translation invariant state in quantum spin chain is determined uniquely
upto isomorphism by a Markov map on the support projection of an associated
Cuntz's state. We prove that Kolmogorov's property of the Markov map is a
necessary and sufficient condition for such a state to be pure. Kolmogorov's
property naturally give rise to a Mackey's system of imprimitivity for the
group of integers. A duality argument originated from non-commutative
probability theory is employed to prove an elegant alternative necessary and
sufficient condition for pureness.