## 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 ⊗ idZ has the weak tracial Rokhlin property.

If G is moreover residually finite, the above conditions are also equivalent to

(3) α ⊗ idZ has finite Rokhlin dimension (in fact, at most 2).

If ∂eT(A) is furthermore compact, has finite covering dimension, and the orbit space ∂eT(A)/G is Hausdorff, we generalize results by Matui and Sato to show that α is cocycle conjugate to α ⊗ idZ , even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α⊗idZ . 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.

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 ⊗ idZ has the weak tracial Rokhlin property.

If G is moreover residually finite, the above conditions are also equivalent to

(3) α ⊗ idZ has finite Rokhlin dimension (in fact, at most 2).

If ∂eT(A) is furthermore compact, has finite covering dimension, and the orbit space ∂eT(A)/G is Hausdorff, we generalize results by Matui and Sato to show that α is cocycle conjugate to α ⊗ idZ , even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α⊗idZ . 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 GB |
---|---|

State | Published - 27 Oct 2021 |

## Keywords

- math.OA
- 46L05 (primary), 37A55

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