Translation invariant state and its mean entropy-I

2017/05/31 by Anilesh Mohari, Mohari, Anilesh
Mathematics · #Advanced Operator Algebra Research #Spectral Theory in Mathematical Physics #Advanced Topics in Algebra

paper · pdf · doi:10.48550/arxiv.1705.11038

Abstract

Let \IM =⊗_{n ∈ \IZ} M⁽ⁿ⁾(\IC) be the two sided infinite tensor product C^*-algebra of d dimensional matrices M⁽ⁿ⁾(\IC)= M_d(\IC) over the field of complex numbers \IC and ω be a translation invariant state of \IM. In this paper, we have proved that the mean entropy s(ω) and Connes-Størmer dynamical entropy h_CS(\IM,θ,ω) of ω are equal. Furthermore, the mean entropy s(ω) is equal to the Kolmogorov-Sinai dynamical entropy h_KS(\ID_ω,θ,ω) of ω when the state ω is restricted to a suitable translation invariant maximal abelian C^* sub-algebra \ID_ω of \IM. Futhermore, a translation invariant factor state of \IM is pure if and only if its mean entropy is zero. The last statement can be regarded as a non commutative extension of Rokhlin-Sinai positive entropy theorem for non-pure factor states.

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