Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture

In this paper we study $C^*$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_Theta$ be a noncommutative torus and $alpha_Theta$ be the noncommutative toral automorphism arising from a matrix $Sin GL(d,mathbb{Z})$. We show that if the Voiculescu-Brown entropy of $alpha_{Theta}$ is zero, then the sequence ${rho(alpha_{Theta}^nu)}_{nin mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $uin A_Theta$ and $rho$ is any state on $A_Theta$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Mobius function.