# ROOM TWO: Finite Rokhlin dimension of finite group actions on $\mathcal{Z}$-stable $C^\ast$-algebras

Finite Rokhlin dimension is one of the several ways in which the Rokhlin property, a concept originally generalized from ergodic theory to the framework of amenable actions on von Neumann algebras, has been adapted to C*-dynamics.

A nice feature of the notion of finite Rokhlin dimension is that, although it has weaker requirements compared to other adaptations of the Rokhlin property to actions on C*-algebras, it still induces useful regularity properties on the actions satisfying it. For instance, finite nuclear dimension and $\mathcal{Z}$-stability are preserved when taking the crossed product of a separable unital C*-algebra by a $\mathbb{Z}$-action which has finite Rokhlin dimension.

In this talk we show that for a finite group action $\alpha: G \to \text{Aut}(A)$ on a separable, simple, unital, $\mathcal{Z}$-stable, nuclear $C^\ast$-algebra $A$ with non-empty trace space, the action $\alpha$ is strongly outer if and only if $\alpha \otimes \text{id}_\mathcal{Z}$ has finite Rokhlin dimension.

The novelty of this result is that we make no topological assumption on the trace space $T(A)$ of $A$, in opposition to past works proving analogous statements, where $T(A)$ is always assumed to be a Bauer simplex.

This is a joint work with Ilan Hirshberg.