Pure inductive limit state and Kolmogorov's property-II

Abstract

A translation invariant state $\omega$ on $C^*$-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 $\phi$. We give an asymptotic criteria on the Markov map $(\clm,\tau,\phi)$ 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.