## Abstract

Let A be a separable, unital, simple, Z-stable, nuclear C^{⁎}-algebra, and let α:G→Aut(A) be an action of a discrete, countable, amenable group. Suppose that the orbits of the action of G on T(A) are finite and that their cardinality is bounded. We show that the following are equivalent: (1) α is strongly outer; (2) α⊗id_{Z} has the weak tracial Rokhlin property. If G is moreover residually finite, the above conditions are also equivalent to (3) α⊗id_{Z} has finite Rokhlin dimension (in fact, at most 2). If ∂_{e}T(A) is furthermore compact, has finite covering dimension, and the orbit space ∂_{e}T(A)/G is Hausdorff, we generalize results by Matui and Sato to show that α is cocycle conjugate to α⊗id_{Z}, even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α⊗id_{Z}. In the course of the proof, we develop equivariant versions of complemented partitions of unity and uniform property Γ as technical tools of independent interest.

Original language | English |
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Pages (from-to) | 76-123 |

Number of pages | 48 |

Journal | Journal des Mathematiques Pures et Appliquees |

Volume | 162 |

DOIs | |

State | Published - 1 Jun 2022 |

## Keywords

- Amenability
- Group action
- Jiang-Su algebra
- Strong outerness

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics