Rokhlin dimension: obstructions and permanence properties

Abstract

This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on $C^{\ast }$-algebras, introduced by the first author, Winter, and Zacharias. We extend the definition of finite Rokhlin dimension to the nonunital case. This definition behaves well with respect to extensions, and is sufficient to establish permanence of finite nuclear dimension and $\mathcal{Z}$-absorption. We establish $K$-theoretic obstructions to the existence of actions of finite groups with finite Rokhlin dimension (in the commuting tower version). In particular, we show that there are no actions of any nontrivial finite group on the Jiang-Su algebra or on the Cuntz algebra ${\mathcal{O}}_{\infty }$ with finite Rokhlin dimension in this sense.