# Equivariant property (SI) revisited

## TL;DR

This paper extends the concept of property (SI) for C*-algebras to the equivariant setting, especially for actions of amenable groups on non-unital algebras, and proves that such actions are equivariantly Jiang-Su stable under certain conditions.

## Contribution

It generalizes property (SI) to unbounded trace C*-algebras in an equivariant context and establishes equivariant Jiang-Su stability for actions of amenable groups.

## Key findings

- All actions of countable amenable groups on certain simple nuclear C*-algebras have equivariant property (SI).
- Such actions are equivariantly Jiang-Su stable if the algebra has finitely many extremal traces.
- The results extend to relative property (SI) for inclusions into ultraproducts.

## Abstract

We revisit Matui-Sato's notion of property (SI) for C*-algebras and C*-dynamics. More specifically, we generalize the known framework to the case of C*-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 non-unital C*-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 non-elementary separable simple nuclear C*-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.