Equivariant property (SI) revisited

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.

arXiv commentv4 36 pages; this version has been accepted at Analysis & PDE

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Cited by in corpus (5)

• Strongly outer actions of amenable groups on $\mathcal{Z}$-stable nuclear $C^{\ast }$-algebras
• On tracial $\mathcal{Z}$-stability of simple non-unital C*-algebras
• Equivariant property Gamma and the tracial local-to-global principle for C*-dynamics
• Equivariant Kirchberg-Phillips type absorption for the Razak-Jacelon algebra
• Rohlin actions of finite groups on the Razak-Jacelon algebra