Given a C*-algebra A, we can construct the opposite algebra as being the same Banach space with the same involution, but with the reverse multiplication. In the separable setting, there are known examples of simple non-nuclear C*-algebras which aren't isomorphic to their opposite algebras, and there are non-simple nuclear examples. Whether there exists a separable nuclear example is a difficult open problem, and is of interest from the perspective of structure and classification: the Elliott invariant and the Cuntz semigroup cannot distinguish a C*-algebra from its opposite, so if such an example exists, it would most likely reveal new phenomena.

In the non-separable setting, things become stranger. I'll discuss a recent preprint in which it is shown that it is consistent with ZFC that there is a simple nuclear non-separable C*-algebra which is not isomorphic to its opposite algebra. One can in fact guarantee that this example is an inductive limit of unital copies of the Cuntz algebra O_2, or of the CAR algebra.

This is joint work with Ilijas Farah.