Pure inductive limit state and Kolmogorov's property

Abstract

Let $(\clb,λ_t,ψ)$ be a $C^*$-dynamical system where $(λ_t: t \in \IT_+)$ be a semigroup of injective endomorphism and $ψ$ be an $(λ_t)$ invariant state on the $C^*$ 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 $\clb \raro^{λ_t} \clb$ 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.