Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture

In this paper we study $C^{\ast }$-algebra version of Sarnak Conjecture for noncommutative toral automorphisms. Let $A_{\mathrm{\Theta }}$ be a noncommutative torus and ${\alpha }_{\mathrm{\Theta }}$ be the noncommutative toral automorphism arising from a matrix $S\in GL(d,\mathbb{Z})$. We show that if the Voiculescu-Brown entropy of ${\alpha }_{\mathrm{\Theta }}$ is zero, then the sequence $\{\rho ({\alpha }_{\mathrm{\Theta }}^{n}u){\}}_{n\in \mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $u\in A_{\mathrm{\Theta }}$ and $\rho$ is any state on $A_{\mathrm{\Theta }}$. Then by a result of Green and Tao, this sequence is linearly disjoint from the M"obius function.