## Anilesh Mohari (Chennai, India)

### Translation invariant pure state in quantum spin chain and its split property

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 $\clb_R=\otimes_{\IZ_+}\!M^{( k)}$. We prove that a real lattice symmetric reflection positive translation invariant pure state of $\clb=\otimes_{\IZ}M_d(\IC)$ give rises type-I $\omega_R$ if its two points spatial correlation functions decay exponentially.