Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture

Abstract

In this paper we study $C^{\ast }$-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 $S\in GL(d,\mathbb{Z})$. We show that if the Voiculescu-Brown entropy of ${\alpha }_{\Theta }$ is zero, then the sequence $\{\rho ({\alpha }_{\Theta }^{n}u){\}}_{n\in \mathbb{Z}}$ is a sum of a nilsequence and a zero-density-sequence, where $u\in A_{\Theta }$ and $\rho$ is any state on $A_{\Theta }$. Then by a result of Green and Tao, this sequence is linearly disjoint from the Möbius function.

29 pages

References in corpus (2)

• Nilsequences, null-sequences, and multiple correlation sequences
• M$\ddot{o}$bius Disjointness for Exact C${}^{\ast }$ Algebras

Cited by in corpus (2)

• Möbius disjointness for topological models of ergodic systems with discrete spectrum
• Möbius Disjointness for Skew Products on the Heisenberg Nilmanifold