Nuclear Dimension of Twisted $C^{\ast }$-Algebras of Virtually Abelian Groups

Abstract

Let $G$ be a finitely generated virtually abelian group and $[\sigma ]\in H^2(G;\mathbb{T})$ such that $\sigma (x,y)$ is always a root of unity. We show that the nuclear dimension of the twisted group $C^{\ast }$-algebra $C^{\ast }(G,\sigma )$ is equal to the rank of a finite index abelian subgroup of $G$. We also show that $\mbox{dim}_{\text{nuc}}(C^*(\mathbb{Z}^r,σ))=r$ if and only if $\sigma$ is type I.

Updated with referee comments. To appear in Canadian Mathematical Bulletin