An operator algebra is said to have the Kadison property if all its bounded representations are completely bounded. It is a long-standing open problem to determine whether this is satisfied by every $C^*$-algebra. On the other hand, due to work of Haagerup and Gifford, it is known that the Kadison property for $C^*$-algebras is equivalent to a weaker version of amenability, called the total reduction property.

In this talk, we investigate whether non self-adjoint operator algebras with the total reduction property necessarily have the Kadison property. We obtain positive results in the case where either the domain or codomain of the representation is residually finite dimensional. We also explain why these facts are meaningful with regards to the general problem. Finally, we exhibit connections to the harder question of determining whether operator algebras with the total reduction property are necessarily similar to $C^*$-algebras.

This is joint work with Laurent Marcoux.