Actions on classifiable C*-algebras without equivariant property (SI)

Abstract: We exhibit examples of actions of countable discrete groups on both simple and non-simple nuclear stably finite C*-algebras that are tracially amenable but not amenable. We furthermore obtain that, under the additional assumption of strict comparison, amenability is equivalent to tracial amenability plus the equivariant analogue of Matui--Sato's property (SI). By virtue of this equivalence, our construction yields the first known examples of actions on classifiable C*-algebras that do not have equivariant property (SI). We moreover show that such actions can be chosen to absorb the trivial action on the universal UHF algebra, thus showing that equivariant $\mathcal{Z}$-stability does not in general imply equivariant property (SI).