A simple AF algebra not isomorphic to its opposite

TL;DR

This paper demonstrates, within ZFC set theory, the existence of a simple nuclear non-separable C*-algebra that is not isomorphic to its opposite, constructed as an inductive limit of well-known C*-algebras.

Contribution

It provides the first known example of such an algebra, showing that simplicity and nuclearity do not guarantee isomorphism to the opposite algebra.

Findings

Existence of a simple nuclear non-separable C*-algebra not isomorphic to its opposite

Construction as an inductive limit of O_2 or CAR algebra

Consistency with ZFC set theory

Abstract

We show that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. We can furthermore guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O_2, or of the CAR algebra.