Pure inductive limit state and Kolmogorov's property

Let be a C^*-dynamical system where be a semigroup of injective endomorphism and ψ be an (λ_t) invariant state on the C^* subalgebra and 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 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 A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003) 189–209] 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 O. Bratteli, P.E.T. Jorgensen, A. Kishimoto, R.F. Werner, Pure states on , J. Operator Theory 43 (1) (2000) 97–143; A. Mohari, Markov shift in non-commutative probability, J. Funct. Anal. 199 (2003) 189–209] as we could go beyond lattice symmetric states.