# Journal of Operator Theory

Volume 72, Issue 2, Fall 2014 pp. 387-404.

Pure inductive limit state and Kolmogorov's
property. II

**Authors**:
Anilesh Mohari

**Author institution:** The Institute of Mathematical Sciences,
Tharamani, Chennai-600113, India

**Summary: ** A translation invariant state $\omega$ on
$C^*$-algebra $\clb=\bigotimes\limits_{k \in \IZ}\!M^{(k)}_d$, where
$\!M^{(k)}_d=\!M_d(\IC)$ is the $d$-dimensional matrices over field of
complex numbers,
give rises to a stationary quantum Markov chain and associates canonically a
unital completely positive normal map $\tau$ on a von Neumann algebra
$\clm_0$ with a faithful normal invariant
state $\phi_0$. We give an asymptotic criteria on the Markov map
$(\clm_0,\tau,\phi_0)$ for purity of $\omega$. Such a pure $\omega$ gives
only a type I or type III factor $\omega_\mathrm R$ once restricted to
one side of the chain $\clb_\mathrm R=\bigotimes\limits_{k \in
\IZ_+}\!M^{(k)}$. In case $\omega_\mathrm R$ is type I, $\omega$ admits
Kolmogorov's property.

**DOI: **http://dx.doi.org/10.7900/jot.2013apr11.1985

**Keywords: ** uniformly hyperfinite factors, Kolmogorov's property,
pure states

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