# Pure inductive limit state and Kolmogorov&#x27;s property - Mathematics Operator Algebras

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Abstract: Let $\clb,\lambda t,\psi$ be a $C^*$-dynamical system where $\lambda t: t\in \IT +$ be a semigroup of injective endomorphism and $\psi$ be an$\lambda t$ invariant state on the $C^*$ subalgebra $\clb$ and $\IT +$ iseither non-negative integers or real numbers. The central aim of thisexposition is to find a useful criteria for the inductive limit state $\clb aro^{\lambda t} \clb$ canonically associated with $\psi$ to be pure. Weachieve this by exploring the minimal weak forward and backward Markovprocesses associated with the Markov semigroup on the corner von-Neumannalgebra of the support projection of the state $\psi$ to prove thatKolmogorovs property Mo2 of the Markov semigroup is a sufficient conditionfor the inductive state to be pure. As an application of this criteria we finda sufficient condition for a translation invariant factor state on a onedimensional quantum spin chain to be pure. This criteria in a sense complementscriteria obtained in BJKW,Mo2 as we could go beyond lattice symmetric states.

Autor: Anilesh Mohari

Fuente: https://arxiv.org/