2 00 7 Pure inductive limit state and Kolmogorov ’ s property

Let (B, λt, ψ) be a C-dynamical system where (λt : t ∈ IT+) be a semigroup of injective endomorphism and ψ be an (λt) invariant state on the C ∗ subalgebra B 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 B →t B canonically associated with ψ 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 ψ 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.