Rokhlin dimension: obstructions and permanence properties

TL;DR

This paper extends the concept of finite Rokhlin dimension to nonunital C*-algebras, explores its properties, and identifies K-theoretic obstructions to certain group actions with finite Rokhlin dimension.

Contribution

It introduces a nonunital version of finite Rokhlin dimension, proves its good behavior under extensions, and establishes K-theoretic obstructions to specific finite group actions.

Findings

Finite Rokhlin dimension is well-behaved under extensions.

No nontrivial finite group actions with finite Rokhlin dimension exist on Jiang-Su or Cuntz algebra O_infinity.

K-theoretic obstructions prevent certain group actions with finite Rokhlin dimension.

Abstract

This paper is a further study of finite Rokhlin dimension for actions of finite groups and the integers on C*-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 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 O_\infty with finite Rokhlin dimension in this sense.