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.

我们重新审视 Matui 和 Sato 的财产概念（SI） $C^{＊}$-代数和 $C^{＊}$-动力学。更具体地说，我们将已知框架推广到以下情况$C^{＊}$-具有可能无界迹的代数。这种方法的新颖之处在于等变上下文，以前的工作都不允许（直接）将这种方法应用于高度非统一的服从群体的行动。$C^{＊}$-代数，特别是建立等变江-苏稳定性。我们的主要结果是佐藤观察的扩展：对于任何可数的服从组$\Gamma$ 和任何非基本可分离的简单核 $C^{＊}$-代数 $\mathrm{一种}$ 严格比较，每 $\Gamma$- 行动 $\mathrm{一种}$具有等变特性 (SI)。还证明了涉及超产品中夹杂物的相对属性 (SI) 的更一般性陈述。因此我们证明如果$\mathrm{一种}$ 也有有限多条极值迹线，那么每个 $\Gamma$- 行动 $\mathrm{一种}$等变江-苏稳定。此外，我们还提供了主要结果在强外部动作的上下文中的应用，例如 Nawata 在（稳定的）Razak-Jacelon 代数上对强外部自同构的分类的推广。