We revisit Matui and Sato’s notion of property (SI) for
-algebras and
-dynamics.
More specifically, we generalize the known framework to the case of
-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
-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
and any nonelementary separable simple nuclear
-algebra
with strict comparison,
every
-action
on
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
also has finitely many rays of extremal traces, then every
-action
on
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.
Keywords
$\mathrm{C}^*$-dynamics, cocycle conjugacy, Jiang–Su
algebra, Elliott program
