Isomorphism theorem for Kolmogorov states of $\clb = \dsp{\otimes_{j \in \IZ}}\!M^{(j)}_d(\IC)$

Abstract: We consider the translation dynamics on the $C^*$-algebra $\IM =\otimes_{n \in \IZ}!M^{(n)}(\IC)$ of two sided infinite tensor product of $d$ dimensional matrices $!M^{(n)}(\IC)=!M_d(\IC)$ over the field of complex numbers $\IC$ and its restriction to the maximal abelian $C^*$ sub-algebra $\ID^e = \otimes_{n \in \IZ}!D_e^{(n)}(\IC)$ of $\IM$, where each $!D_e^{(n)}(\IC)=!D_d(\IC)$ is the algebra of $d$ dimensional diagonal matrices with respect to an orthonormal basis $e=(e_i)$ of $\IC^d$. We prove that any two translation invariant Kolmogorov pure states of $\IM$ give unitarily equivalent dynamics in their Gelfand-Naimark-Segal spaces. Furthermore, for a class of Kolmogorov pure states of $\IM$ satisfying some additional invariance, we prove Kolmogorov states give isomorphic translation dynamics if their restrictions to the maximal abelian $C^*$ sub-algebra $\ID^e$ of $\IM$ are isomorphic. On the other hand, we prove that the translation dynamics with two infinite tensor product translation invariant faithful states of $\IM$ are isomorphic if and only if their mean entropies are equal.