Equivariant property (SI) revisited, II

TL;DR

This paper generalizes the concept of property (SI) for C*-dynamics to include actions of non-amenable groups, establishing that amenable group actions on certain C*-algebras possess this property.

Contribution

It extends the understanding of property (SI) to non-amenable group actions on simple nuclear C*-algebras, broadening the scope of previous results.

Findings

Amenable actions of any countable group on specified C*-algebras have property (SI).

The result applies to non-elementary separable simple nuclear C*-algebras with strict comparison.

The work involves a general statement about relative property (SI) in ultraproducts.

Abstract

We investigate Matui-Sato's notion of property (SI) for C*-dynamics, this time with a focus on actions of possibly non-amenable groups. The main result is a generalization of earlier work: For any countable group $\Gamma$ and any non-elementary separable simple nuclear C*-algebra $A$ with strict comparison, every amenable $\Gamma$-action on $A$ has equivariant property (SI). This is deduced from a more general statement involving relative property (SI) for certain inclusions into ultraproducts. The article concludes with a few consequences of this result.