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 AΘ be a noncommutative torus and αΘ be the noncommutative toral automorphism arising from a matrix S∈GL(d,Z). We show that if the Voiculescu-Brown entropy of αΘ is zero, then the sequence {ρ(αΘnu)}n∈Z is a sum of a nilsequence and a zero-density-sequence, where u∈AΘ and ρ is any state on AΘ. Then by a result of Green and Tao [9], this sequence is linearly disjoint from the Möbius function.