Residually finite dimensional C*-algebras and subquotients of the CAR algebra

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.

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Published 1 January 2001