Residually finite dimensional C*-algebras and subquotients

Abstract. It is proved that the cone of a separable nuclearly embeddable resid-ually finite-dimensional C*-algebra embeds in the CAR algebra (the UHF algebra of type 2∞). As a corollary we obtain a short new proof of Kirchberg’s theorem asserting that a separable unital C*-algebra A is nuclearly embeddable if and only there is a semisplit extension 0 → J → E → A → 0 with E a unital C*-subalgebra of the CAR algebra and the ideal J an AF-algebra. The new proof does not rely on the lifting theorem of Effros and Haagerup. 1