2021/10/27 by Gardella, Eusebio, Hirshberg, Ilan, Vaccaro, Andrea
#37A55 #46L05 (primary) #FOS: Mathematics #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2110.14387
Let A be a separable, unital, simple, Z-stable, nuclear C^*-algebra, and let α\colon 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 α is strongly outer if and only if α\otimesidZ has the weak tracial Rokhlin property. If G is moreover residually finite, these conditions are also equivalent to α\otimesidZ having 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\otimesidZ, even if α is not strongly outer. In particular, in this case the equivalences above hold for α in place of α\otimesidZ. 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.