平移不变态及其平均熵- II
Translation invariant state and its mean entropy-II
摘要：
设$\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$是复数域$\IC$上$d$维矩阵$\!M^{(n)}(\IC)=\!M_d(\IC)$的双边无限张量积$C^*$ -代数.设$ω$是$\IM$的一个平移不变态。在最近的一篇论文中，我们已经证明了平均熵$s(ω)$是一个完整的不变量的某些类的平移不变状态$ω$的$\IM$。在本文中，我们发展了一个一般理论的自同构的$C^*$ -或冯-诺依曼代数的基础上的重复容许测量过程。特别地，我们证明了平移动力学$(\IM,θ,ω)$的动力学熵$h_ω(θ)$满足$s(ω) \le h_ω(θ) \le 2s(ω)$。如果$ω$是$\IM$的无限张量积状态，则$h_ω(θ)=s(ω)$。
英文摘要：
Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ 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 $\IC$. Let $ω$ be a translation invariant state of $\IM$. In a recent paper, we have proved that the mean entropy $s(ω)$ is a complete invariant for certain classes of translation invariant state $ω$ of $\IM$. In this paper, we have developed a general theory for dynamical entropy for an automorphism on an arbitrary $C^*$- or von-Neumann algebras based on repeated admissible measurement processes. In particular, we prove that dynamical entropy $h_ω(θ)$ for translation dynamics $(\IM,θ,ω)$ satisfies $s(ω) \le h_ω(θ) \le 2s(ω)$. In case $ω$ is an infinite tensor product state of $\IM$ then $h_ω(θ)=s(ω)$.
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