#### Vol. 14, No. 4, 2021

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Equivariant property (SI) revisited

### Gábor Szabó

Vol. 14 (2021), No. 4, 1199–1232
##### Abstract

We revisit Matui and Sato’s notion of property (SI) for ${C}^{\ast }$-algebras and ${C}^{\ast }$-dynamics. More specifically, we generalize the known framework to the case of ${C}^{\ast }$-algebras with possibly unbounded traces. The novelty of this approach lies in the equivariant context, where none of the previous work allows one to (directly) apply such methods to actions of amenable groups on highly nonunital ${C}^{\ast }$-algebras, in particular to establish equivariant Jiang–Su stability. Our main result is an extension of an observation by Sato: for any countable amenable group $\Gamma$ and any nonelementary separable simple nuclear ${C}^{\ast }$-algebra $A$ with strict comparison, every $\Gamma$-action on $A$ has equivariant property (SI). A more general statement involving relative property (SI) for inclusions into ultraproducts is proved as well. As a consequence we show that if $A$ also has finitely many rays of extremal traces, then every $\Gamma$-action on $A$ is equivariantly Jiang–Su stable. We moreover provide applications of the main result to the context of strongly outer actions, such as a generalization of Nawata’s classification of strongly outer automorphisms on the (stabilized) Razak–Jacelon algebra.

##### Keywords
$\mathrm{C}^*$-dynamics, cocycle conjugacy, Jiang–Su algebra, Elliott program
##### Mathematical Subject Classification 2010
Primary: 46L35, 46L55, 46L40