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arXiv · 1510.06022

Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture

Wen Huang·Zhengxing Lian·Song Shao·Xiangdong Ye
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Abstract

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

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Wen Huang, Zhengxing Lian, Song Shao, Xiangdong Ye. 2015-10-19. Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture. https://arxiv.org/abs/1510.06022

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