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We exhibit examples of simple separable nuclear C*-algebras, along with actions of the circle group and outer actions of the integers, which are not equivariantly isomorphic to their opposite algebras. In fact, the fixed point subalgebras are not isomorphic to their opposites. The C*-algebras we exhibit are well behaved from the perspective of structure and classification of nuclear C*-algebras: they are unital C*-algebras in the UCT class, with finite nuclear dimension. One is an AH-algebra with unique tracial state and absorbs the CAR algebra tensorially. The other is a Kirchberg algebra.
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Marius Dadarlat, Ilan Hirshberg, N. Christopher Phillips, Simple nuclear C*-algebras not equivariantly isomorphic to their opposites. J. Noncommut. Geom. 12 (2018), no. 4, pp. 1227–1253