Equivariant $${\mathcal {Z}}$$-stability for single automorphisms on simple $$C^*$$-algebras with tractable trace simplices

Chaotic tracial dynamics

“…An important problem in the study of group actions on simple, separable, Z-stable C * -algebras is to determine when a given action 𝛼 : 𝐺 → Aut( 𝐴) is cocycle conjugate to its tensor product 𝛼 ⊗ id Z with the trivial action on Z. (As discussed in the introduction of [94], these are the actions one can expect to classify.) In [46], the problem is solved for actions of discrete, countable, amenable groups G on unital C * -algebras A under the assumptions that the extreme boundary 𝜕 𝑒 (𝑇 ( 𝐴)) of 𝑇 ( 𝐴) is compact and of finite covering dimension and that the action of G on 𝜕 𝑒 (𝑇 ( 𝐴)) has finite orbits of bounded size, with Hausdorff orbit space 𝜕 𝑒 (𝑇 ( 𝐴))/𝐺.…”

“…In short, the following theorem indicates that, when there are infinitely many extremal traces, the typical situation can be very different to the one considered in [46]. (Note, however, that it does not represent an obstruction to generalising [46] for single automorphisms; in the breakthrough article [94], it is shown how to treat the portion of 𝜕 𝑒 (𝑇 ( 𝐴)) with unbounded orbits.) Theorem A.…”