Dynamical comparison and Z-stability for crossed products of simple C∗-algebras

We establish Z-stability for crossed products of outer actions of amenable groups on Z-stable C*-algebras under a mild technical assumption which we call McDuff property with respect to invariant traces. We obtain such result using a weak form of dynamical comparison, which we verify in great generality. We complement our results by proving that McDuffness with respect to invariant traces is automatic in many cases of interest. This is the case, for instance, for every action of an amenable group G on a classifiable C*-algebra A whose trace space T(A) is a Bauer simplex with finite dimensional boundary partial differential eT(A), and such that the induced action G (-4. partial differential eT(A) is free. If G = Zd and the action G r partial differential eT(A) is free and minimal, then we obtain McDuffness with respect to invariant traces, and thus Z -stability of the corresponding crossed product, also in case partial differential eT(A) has infinite covering dimension. (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).