AF-embeddings into C*-algebras of real rank zero

Abstract

It is proved that every separable $C^*$-algebra of real rank zero contains an AF-sub-$C^*$-algebra such that the inclusion mapping induces an isomorphism of the ideal lattices of the two $C^*$-algebras and such that every projection in a matrix algebra over the large $C^*$-algebra is equivalent to a projection in a matrix algebra over the AF-sub-$C^*$-algebra. This result is proved at the level of monoids, using that the monoid of Murray-von Neumann equivalence classes of projections in a $C^*$-algebra of real rank zero has the refinement property. As an application of our result, we show that given a unital $C^*$-algebra $A$ of real rank zero and a natural number $n$, then there is a unital $^*$-homomorphism $M_{n_1} \oplus ... \oplus M_{n_r} \to A$ for some natural numbers $r,n_1, ...,n_r$ with $n_j \ge n$ for all $j$ if and only if $A$ has no representation of dimension less than $n$.