Characterizing AF-embeddable \(C^*\)-algebras by representations

A major open problem of AF-embedding is whether every separable exact quasidiagonal \(C^*\) $C^{\ast }$ -algebra can be embedded into an AF-algebra. In this paper we characterize AF-embeddable \(C^*\) $C^{\ast }$ -algebras by representations to observe their similarity to the separable exact quasidiagonal \(C^*\) $C^{\ast }$ -algebras. As an application, we show that every separable exact quasidiagonal \(C^*\) $C^{\ast }$ -algebra is AF-embeddable if and only if every faithful essential representation of a separable exact quasidiagonal \(C^*\) $C^{\ast }$ -algebra is a certain kind of \(*\) $\ast$ -representation. We also show that a separable \(C^*\) $C^{\ast }$ -algebra is AF-embeddable if and only if it can be embedded into a particular \(C^*\) $C^{\ast }$ -algebra.