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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.

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

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