C*-algebras are algebras which are isomorphic to subalgebras of the algebra of bounded operators on a complex Hilbert space which are closed under conjugation and are closed in the norm topology. Von Neumann algebras are C*-algebras which are furthermore closed in the weak operator topology. They are called approximately finite dimensional if any finite subset is approximately contained in a finite dimensional *-subalgebra (where "approximately" means in the suitable topology). We shall focus on ones which are simple (that is, have no nontrivial ideals), and have a trace (that is, a positive linear functional which is invariant under unitary conjugation).

In the separable setting, Connes proved that there is a unique such von Neumann algebra, and Bratteli and Elliott classified AF C*-algebras via the K_0 functor, which associates to any C*-algebra a partially ordered abelian group.

To any C*-algebra or von-Neumann algebra A, one can associate the opposite C*-algebra A^op, obtained by reversing the order of multiplication. It follows from the classification results above that any AF algebra is isomorphic to its opposite. There are examples of C*-algebras and von-Neumann algebras which are not isomorphic to their opposites, however it is a hard and long-standing open problem whether there exists a simple separable amenable C*-algebra which is not isomorphic to its opposite (amenability being a weaker condition than AF in the C*-setting, whereas the von Neumann analogue turns out to be the same as AF, due to a deep theorem of Connes).

It turns out that without separability, one can construct AF algebras in both of those categories with density character \aleph_1 which are not isomorphic to their opposites, using Jensen's Diamond, and which furthermore exhibit additional rigidity phenomena which cannot happen in the separable setting.

I will explain the definitions in the above paragraph, outline the proof in one of those cases, and discuss some open problems. No background in operator algebras will be assumed.

This is joint work with Ilijas Farah.