Mathematical Research Letters
Volume 8 (2001)
Residually finite dimensional C*-algebras and subquotients of the CAR algebra
Pages: 545 – 555
It is proved that the cone of a separable nuclearly embeddable residually finite-dimensional C*-algebra embeds in the CAR algebra (the UHF algebra of type $2^\infty$). 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 $0 \to J \to E \to A \to 0$ 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.