Simple nuclear C*-algebras not equivariantly isomorphic to their opposites

TL;DR

This paper constructs examples of simple nuclear C*-algebras with group actions that are not equivariantly isomorphic to their opposites, highlighting new phenomena in the classification of such algebras.

Contribution

It provides the first known examples of simple nuclear C*-algebras with group actions that are not equivariantly isomorphic to their opposites, even at the fixed point level.

Findings

Examples include an AH-algebra with unique trace that absorbs the CAR algebra.

A Kirchberg algebra example is also constructed.

Fixed point subalgebras are not isomorphic to their opposites.

Abstract

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.