2007/04/16 by Anilesh Mohari, Mohari, Anilesh
Economics, Econometrics and Finance · Mathematics · Physics and Astronomy · #FOS: Mathematics #Inductive limit #Kolmogorov's property #Markov shift #Mathematical Dynamics and Fractals #Operator Algebras (math.OA) #Probability (math.PR) #Pure state #Statistical Mechanics and Entropy #Stochastic processes and financial applications #math.OA #math.PR
paper · pdf · doi:10.48550/arxiv.0704.1987
arxiv created 2007/04/16 · openalex publication_date 2007/04/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (\clb,λt,ψ) be a C^*-dynamical system where (λt: t ∈ \IT+) be a semigroup of injective endomorphism and ψ be an (λt) invariant state on the C^* subalgebra \clb and \IT+ is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state \clb \raroλt \clb canonically associated with ψ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state ψ to prove that Kolmogorov's property [Mo2] of the Markov semigroup is a sufficient condition for the inductive state to be pure. As an application of this criteria we find a sufficient condition for a translation invariant factor state on a one dimensional quantum spin chain to be pure. This criteria in a sense complements criteria obtained in [BJKW,Mo2] as we could go beyond lattice symmetric states.