Strongly outer actions of amenable groups on Z-stable nuclear C*-algebras

Journal article, 2022

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.