We give examples of purely infinite simple separable nuclear C*-algebras
equipped with actions of the integers and the circle group which are
not equivariantly isomorphic to the corresponding actions on the
To put our result in context: Examples are known of type II and type III
factors not isomorphic to their opposite algebras, of separable unital
type I C*-algebras not isomorphic to their opposite algebras, and of
separable simple unital Z-stable exact C*-algebras not isomorphic to
their opposite algebras. No example is known of a simple nuclear
C*-algebra not isomorphic to its opposite algebra.
This is joint work with Marius Dadarlat and Ilan Hirshberg.