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

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 Mn1⊕⋯⊕Mnr →A for some natural numbers r,n1,...,nr with nj≥n for all j if and only if A has no representation of dimension less than n. © 2004 Elsevier Inc. All rights reserved.