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