Pure inductive limit state and Kolmogorov's property-II

Abstract

A translation invariant state $\omega$ on $C^{\ast }$-algebra $\clb=\otimes_{k \in \IZ}M^{(k)}$, where $M^{(k)}=M_d(\IC)$ is the $d-$dimensional matrices over field of complex numbers, give rises a stationary quantum Markov chain and associates canonically a unital completely positive normal map $\tau$ on a von-Neumann algebra $\clm$ with a faithful normal invariant state $\varphi$. We give an asymptotic criteria on the Markov map $(\clm,τ,ϕ)$ for purity of $\omega$. Such a pure $\omega$ gives only type-I or type-III factor ${\omega }_R$ once restricted to one side of the chain $\clb_R=\otimes_{\IZ_+}M^{(k)}$. In case ${\omega }_R$ is type-I, $\omega$ admits Kolmogorov's property.

Paper is now divided into two part. This is part one which is accepted for publication by Journal of Operator Theory. Some typo corrections and citations